The basic function of an artificial reservoir is to change the rate of flow in the ...... TAX. The duration of the respective AX interval is expressed by T ; e.g., for the ...... mass demand curve & and the mass pumping curves 1: from 10p.m. till 6a.m.
WATER MANAGEMENT IN RESERVOIRS LADISLAV VOTRUBA
and
VOJTECH B R O ~ A Faculty of Civil Engineering of the Technical University Prague, Czechoslovakia
ELSEVlER
-
Amsterdam - Oxford New York
- Tokyo 1989
Published in co-edition with SNTL - Publishers of Technical Literature, Prague Distribution of this book is being handled by the following publishers for the U.S.A. and Canada ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52 Vanderbilt Avenue, New York, N.Y.10017
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Library of Congress Cataloging-in-Pnbllcation Data Votruba, Ladislav. [Hospodafeni s vodou v nadrtich. English] Water management in reservoirs/Ladislav Votruba and Vojttch BroZa; [translation by Eva Turkovii]. p. cm.--(Developments in water science; 33) Translation of: Hospodaieni s vodou v nadrZich. Bibliography: p. Includes index. 1. Reservoirs. 2. Water-supply engineering. 3. Water, Underground. 4. Hydrology. I. BroZa, Vojttch, 193511. Title. Ill. Series. TD395, V613 1988 628.1'324~19
87-27551 CIP
ISBN 0-444-98933-1 (Vol. 33) ISBN 0-444-41669-2 (Series) Copyright @ by Prof. lng. Dr. Ladislav Votruba, DrSc..and Prof. Ing. Vojttch Broia, DrSc.. 1989 Translation @ by Ing. Eva Turkova, 1989 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the copyright owner. Printed in Czechoslovakia
PREFACE Reservoirs help to maintain the balance between water resources and water demand. The water management of reservoirs is an important part of any flow regulation scheme. In economically and socially advanced countries with high demands on water relative to their water resources, multi-purpose flow-regulation systems ensure a rational exploitation of what is available. Reservoirs play an important part in these systems and a system approach should therefore be applied in their design. One of the more recent aspects of technical/scientific development is that more consideration must be shown for the environment, for nature. Water is one of its basic elements. Reservoirs have a fundamental impact on the environment, a fact which cannot be overlooked when designing and operating a reservoir. Reservoirs are designed not only for immediate needs, but also for the future, and any concept concerning them must therefore be based on the dynamics of development and on prognoses. In both parts of the water management balance-resource and demand-stochastic elements can be observed; a probability approach and the methods of mathematical statistics must therefore be applied to arrive at any solutions. Reservoir operation influences both the economic and non-economic spheres of the life of any society. In evaluating the efficiency of reservoirs, both these influences must therefore be borne in mind and the optimization criteria must be adjusted accordingly. To what extent the various aspects are weighted depends on the natural, economic and social conditions in the country concerned. All possible approaches have to be explored in these reservoir projects to arrive at a technical/economic/socialoptimum. The 1966 edition of this book could not at that time provide approaches to all these new progressive goals; these only appeared, and the means to their solution have only been developed, in the past ten years. The subject therefore has to be treated in a new way. Additional chapters deal with water-management systems and the environment. However, the technical approach to the subject has also been completely changed. The aim of the first edition was to build a bridge between the classical and probability methods, while, here, attention is focussed on the stochastic character of the problems.
The first edition included only an outline of modern computer calculations, while they are now of course, an essential tool in tackling the problems. We have, within the space available, been unable to deal with all aspects equally. A choice has had to be made concerning the problems to be discussed and the methods of their resolution. Full reference has however been made to the literature and other sources. The authors are only too well aware that this book does not close a chapter on reservoirs. Questions relating to water management in reservoirs will continue to occupy the attention of experts for many years to come. L. Votruba, V. Broia
LIST OF SYMBOLS
ki
gross capacity of reservoir, gross storage, storage capacity, total storage capacity, reservoir storage live storage capacity flood surcharge storage capacity flood control storage capacity inactive storage, dead storage active storage capacity coefficient of skewness coefficient of variation coefficient of excess, excess coefficient drainage basin area, backwater (inundated) area [m2 ;ha ; km2] pole distance stage, water head [m] depth of evaporation [mm] precipitation depth [mm] altitude [metres above sea level] depth, spill height [m] module discharge coefficient Qr,i
1
module discharge coefficient with a probability p central moment of ith order reservoir bottom level maximum water level (as a rule during the so-called design flood) level of flood surcharge storage capacity level of flood control storage capacity level of inactive storage, dead storage level of active storage capacity moment (general) of the ith order initial moment of the ith order probable return (exceedance)period [years] reservoir release or withdrawal [m3 s-']
16 Ond OP
Co co 0
P
non-damaging discharge [m3 s - '1 yield (withdrawal) with probability p [m3 s-'I amount of water released from a reservoir within a given period [m3] mass curve of reservoir discharge pole probability of exceedance design reliability of water supply output [kW;MW] i d o w to a reservoir [m3 s design reliability of water supply according to occurrence (o), time (duration) (t), or volume of supplied water (d) theoretical flood characterized by a hydrograph (time behaviour), its maximum peak discharge being equal to the N-year flood discharge amount of water flowing into a reservoir within a given period [m3] mass curve of inflow discharge and inflow to a reservoir [m3 s- '1 mean discharge in an over-year period [m3 s- '1 mean daily discharge [m3 s-'1 mean daily discharge with the probability of M-day exceeding [m3 s-'I mean monthly discharge [m3 s- '1 maximum discharge with probable N-year return (exceedance) period [m3 s-'1 minimum discharge with probable N-year return (exceedance) period [m3 s-'1 non-damaging discharge [m3 s- '1 mean daily discharge with a probability of exceedancep per cent [m3 s- '1 mean annual discharge [m3 smean annual discharge with a probability of exceedance p [m3 s- '1 mean annual runoff from a catchment (mean discharge volume) per year [m3] -also W, discharge mass curve specific runoff from a catchment [I s-' km-2] correlation coefficient correlation function discharge area [m2] sample standard deviation sample variance chronological time aggregate discharge curve storage capacity, reservoir capacity [m3; lo6 m3] total storage capacity [m3; lo6 m3]
'1
cp C P
Q
Qll
Qd QMMd
Qm QN
(Irnin, Qnci QP Qr Qr,p
CQ.
CQ 4
r
44
S S
S2
t
UQ V
v,
'1
17
flood control storage capacity [m3; lo6 m3] active storage capacity [m3; lo6 m3] seasonal component of active storage capacity [m3; 106m3] over-year component of active storage capacity [m3; lo6 m3] flood volume beyond the chosen discharge value Q with a probable N-year period of return (exceedance) [m3] volume of flood wave [m3] volume of theoretical flood wave with N-year maximum discharge [m3] arithmetic mean of sample median mode coefficient of the (safe)yield, relative yield
relative storage capacity relative active storage capacity relative seasonal component of active storage capacity relative over-year component of active storage capacity mean value of the population standard deviation of the population dispersion variance of the population duration discharge-travel time Less frequently used symbols are explained in the text.
A RESERVOIR FUNCTION AND BASIC TOOLS FOR ITS ANALYSIS
1 BASIC FUNCTION OF WATER RESERVOIRS A water reservoir is an enclosed area for the storage of water to be used at a later date; it can also serve to catch floods to protect valleys downstream of it; to establish an aquatic environment; or to change the properties of the water. A reservoir can be created by building a dam across a valley, or by using natural or man-made depressions.The main parameters of the reservoir are the volume, the area inundated and the range that the water level can fluctuate.
1.1 FUNCTION AND SIGNIFICANCE OF RESERVOIRS IN WATER MANAGEMENT
The basic function of an artificial reservoir is to change the rate of flow in the stream, or to store water for more expedient use. Reservoirs are among the more useful means of controlling the natural character of water flows, instead of depending on nature. Active storage (uccumulation,conservation) reservoirs help to overcome dependence on the natural hydrological regime of a territory during droughts. If the water consumption in the given area exceeds the water yield, and if the building of reservoirs does not suffice to meet the demand for water, then water has to be diverted to the area of shortage from another catchment, or better use must be made of the existing water. These measures can be economically effective even before the demand for water reaches the point of safe yield. The territory can be protected against floods by increasing the bankfull discharge, or by increasing the reservoir volume to hold the water surplus. These reservoirs are known as flood control (retention) reservoirs. The water can also be diverted to another catchment area. Impounding reservoirs on streams usually serve a dual purpose: water supply and flood protection. A particular part of each reservoir serves to carry out one or other of these tasks which can be partly complementary. The water accumulated in conseruurion reservoirs can be withdrawn for various purposes: irrigation, industrial, agricultural and public water supply, power generation, maintenance of navigation depth, dilution for sanitary and other purposes. A proper aquatic environment, once established in the reservoir, can serve for
20
recreation purposes and water sports; fish and duck farming; cultivation of aquatic plants; treatment of the water; improvement of the natural environment. If a reservoir serves several purposes, it is called a multi-purpose reservoir. Large impounding reservoirs are usually of this type. Even if the reservoir is a single-purpose one, it usually also serves for flood control. The integral task of a reservoir is to change the flow regime of the stream. Therefore a water volume created by a weir, the only purpose of which is to raise the water level, is not considered to be a reservoir. Such a volume is called a backwater reuch (i.e. a volume created by weirs to increase the navigational depth or to facilitate the direct withdrawal of water from the river). However, backwater reaches created by weirs, which change the flow regime, are also considered to be reservoirs. There is significant interaction between reservoirs and the environment : the reservoirs affect the environment and vice-versa. This is true for both the quantitative and the qualitative aspects of the water. The effect of a reservoir on the quality of the water downstream of the dam (pollution, temperature, etc.) and the impact of human activities on the quality of the inflowing water and on the mechanical, chemical and biological processes in the reservoir make the qualitative parameters of the water just as important as the quantitative parameters. A reservoir can accomplish its function as an isolated unit, if it is the only reservoir in the system “water source - user’’, or it can be one of the elements in a system of reservoirs, which cooperate mutually in the water management of a given region. These days, with increasing frequency reservoirs form systems, including older ones which were originally built as single reservoirs. 1.2 THE DEVELOPMENT OF WATER RESERVOIRS
The construction of reservoirs goes back to the most ancient cultures. In the days of slavery when, compared with other crafts, construction was of a very high standard, noteworthy structures were erected, including dams and water mains. Remnants of the first rock-fill dam, built in 3000 B.C., still stand, 30 km south of Cairo. It can be presumed that earth dams were built even earlier. Another reservoir in Egypt, Moeris, was built in about 2900 B.C. as a lateral reservoir of the Nile with a volume of 12. lo9 m3 (Tolke, 1938). Reservoirs can also be traced a long way.back in India; on the outskirts of Madras alone, there were about 50 OOO reservoirs. King Solomon built a system of reservoirs near Jerusalem as far back as the 10th century B.C. Ancient Persia (Iran) could not have reached its high cultural standard without irrigation. In the 6th century B.C.,the most important reservoir was Bend-e-Ramdjerd, near Persepolis, on the river Kor. It was reconstructed many times and to a limited degree is still in operation. About 500 B.C., Dareios I built “bridge weirs” on this river with many outlets and probably with wooden stoplogs, to supply water to
the irrigation system (Kuros, 1943). The reservoirs Saveh (about 40m deep) and Bend-e-Emir with gravity dams and the reservoir Kabar with an arch dam are about lo00 years old; the last two are still in operation. Over the years, their role changed, due to silting, elevation, damage or catastrophe. This also applies to the reservoirs of the turn of the 15th and 16th centuries. At that time, reservoirs for irrigation were also built in Spain (Almanza, Alicante and Tibi) with masonry gravity dams which were sometimes up to 40 m high. The Proserpina reservoir in Spain goes back to the time of the Romans. Lake Fucino was used as the source of the great Roman water main built in the first half of the 1st century A.D. by the Emperor Claudius. During feudalism, Europe was divided into numerous small states. These rather poor countries did not have the means to build any large constructions, and any reservoirs used were only of the pond type. 1.2.1 Construction of reservoirs on the territory of present-day Czechoslovakia
a) T h e ponds of Bohemia (up to the end of the 16th century) On the territory of present-day Czechoslovakia, the first constructions built to hold water were ponds for fish farming. The building of ponds was greatly developed in the 13th-16th centuries, thus establishing the tradition of water structures and water management. The largest system of ponds was located at Tieboil, on the estate of the Roimberks, founded by Josef Netolicky Stepanek (who died in 1538); the builder of the largest ponds was Jakub KrEin of JelEany and Sedlfdny (1535- 1604). However, the pond-building tradition in that area was already 200 years old by then. Stepanek built the 45 km long Golden Channel (Zlata stoka) and KrEin included a new element in the system - he re-routed the flood discharges of the LuZnice through the New River (Nova ieka) to the Neiarka to protect the Roimberk pond on the Ldnice (surface area 8.6 km2, depth 11.5 m and volume 6 . lo6 m3; however, during the flood of 1890 it held a volume of 50. lo6 m3) from the flood waters. Pond building and fish farming at that time were described by Vilem of PernStejn; Jan Skala of Doubrava (1480- 1553) wrote “Five Books about Ponds” ... in Latin which were translated into English, Polish and German and published by the Czechoslovak Academy of Sciences (Dubravius, 1953) in 1953.
Tuble 1.f Small reservoirs and ponds in Czechoslovakia in 1970
IIldC\
State
number (thousand) volume (mil m3) area (km’)
23 720 516.7 548.9
in
1070
Projects
936 119.3 50.6
22 Besides their use for fish farming, ponds help to balance the flow, but they only play a significant role if part of a larger system. In recent years, some of these ponds have been renewed and new ones built to cover the increasing needs of agriculture and to protect the environment. (b) Reservoirs for mining purposes (up to the 19th century) The need for water and hydro-power for the mining and processing industries gave rise to the second stage in reservoir construction in Czechoslovakia. Here, too, ingenious systems were constructed, some of which still exist, for example, in the KruSne hory and the Slavkov forest (which date back to the 16th century) in the neighbourhood of Banska Stiavnica and Pfibram (dating back to the 18th century), as well as others elsewhere. Mine timber was floated on some of the canals connecting the reservoirs, for example on the Slavkov canal (built in the second half of the 15th century) which was later extended to a length of 24 km (Majer, 1975). The reservoirs in the neighbourhood of Banska Stiavnica have a total volume of 7 . lo6 m3 and are notable for the height of their earth dams, the Rozgrund reservoir (1744) being the highest, at 30 m.
(c) Reservoirs for floating timber (up to the end of 19th century)
A flood wave, caused by discharging the water stored in the reservoir, was used for floating timber. These smnll reservoirs with earth dams were mainly constructed in the Sumava region (on the rivers Vydra and Kfemelna). The reservoir in the Korytnickl valley, a unique structure built i n 1882, demonstrates the first use of rubble masonry (I6 in high).
(d) Flood control reservoirs (end of 19th and early 20th century) After the great floods of 1890 and 1897, special flood control reservoirs started to be built. Their sole task was to contain the water during floods. The first reservoir to be built was on the JeviSovice rivulet, near JeviSovice, Moravia. Completed in 1897; it has a curved gravity dam, 25 m high, of masonry and cement mortar. In the Nisa catchment in north Bohemia there are five flood-control reservoirs built between 1902 and 1909, they are all rubble masonry gravity dams. Their total volume amounts to 6 . lo6 m3; the dams are 15.8-23.5 m high. By 19 18, another 11 reservoirs had been built, mainly to control floods; of the 17 reservoirs built between 1919 and 1945, over half are flood-control reservoirs; but none was built after 1945.
Flood-control reservoirs have been in operation for 50 to 90 years; however, many were later turned into conservation reservoirs. The new demands on water, better hydrological data and the advances in science make us regard the older reservoirs from a new angle, such as the safe yield, etc. Thus a better economic effect can be reached without further capital investments.
23 (e) H y d r o - p o w e r reservoirs The first hydro-power reservoir in Czechoslovakia was built on the Zelivka near Sedlice, with a volume of 2 . lo6 m' and a power plant of 2.36 MW. Of the reservoirs built in the period up to 1945, only the Vranov reservoir has a volume (122.5. lo6 m') which can effectively influence the runoff from a larger catchment area. Large reservoirs capable of affecting runoff were only built after 1948, mainly for hydro-power. These reservoirs include the ones at Ustie on the river Orava and at Lipno, Slapy and Orlik on the river Vltava with storage capacities of between 270 and 704. lo6 m3. These capacities not only ensure a large output from the power plants, but for Lipno and Ustie, an over year discharge control the favourable effect of which is also reflected in all the power plants down river.
Although these reservoirs were built mainly to produce electricity, they are in fact multi-purpose reservoirs which have a beneficial effect on the flow regime of the respective catchment areas. (f) C o n s e r v a t i o n reservoirs for public water s u p p l y a n d i n d u s t r y After the three reservoirs built at the beginning of the century: Jezefi, Kamenifka and Janov, only one more conservation reservoir was added in the period between 1918 and 1945, i.e., on the FryStak rivulet near Gottwaldov. After 1945, reservoirs were mainly built for waterworks. Of 35 reservoirs built between 1945 and 1960,20 were for public water supply, two for agriculture and 13 for power generation. Of the 57 reservoirs built between 1961 and 1967,42 serve for public water supply. Their total storage capacity comes to 1317.2. 10" m3.
According to the Czechoslovak Water Management Plan, the number of reservoirs used for waterworks is due to increase by a factor of three by the year 2000 as compared to 1970. (g) Agricultural reservoirs One of the primary aims of the oldest conservation reservoirs was to accumulate water for irrigation. In Czechoslovakia, irrigation water was at first withdrawn from ponds. The first irrigation reservoir was built in 1939 near Husinec on Blanice. Tuble 1.2 Water demand for agriculture in Czechoslovakia
Index
Specific Unit 1970
Year 1985
2000
1133
2257
145
161
annual withdrawal for irrigation in a design dry year
10' m3 yr-'
100.1
annual water demand for livestock breeding
lo6 m3 yr-I
116
Table 1.3 Water resources and reservoirs in Czechoslovak catchments
Long-term average annual precipitations
Territory Catchment
Czechoslovakia total
I. 11. 111. IV. V. VI.
Upper and middle Labe Vltava Berounka Lower Labe Odra Morava
Bohemia and Moravia total
VII. VIII. IX. X.
Danube Vah Hron Bodrog
Slovakia total
*
h)
P
Annual long-term average outflow
[mm]
[mil. m3]
[mil. m']
690
89 OOO
27 740
706 657 594 657 825 641
10 145 12 015 5 509 6 275 5 158 13 528
3 340 3 353 1257 2 105 1953 3 140
668
52 630
15 148
582 812 747 718
3 138 13 627 9 199 10 406
2918 3 838
743
36 370
12 592
') excluding reservoirs on the Danube
in 1970 number capacity [mil. m3] 185
159
3569
2504
355 5 481
26
1065
Reservoirs by the year 2000 number storage accumulation capacity rate in catchment [mil. m3] [XI 310
6127.7
22.1
37 57 22 33 19 67
678.8 1223.0 339.4 473.2 310.3 1016.0
20.3 36.5 27.0 22.5 15.9 32.4
235
4040.7
26.7
6 30 16 23
23.0') 1041.0 280.0 743.0
6.5 19.0 9.6 19.4
75
2087.0
16.6
25 It was not until 1960 that irrigation was considered in the design of larger reservoirs. By 1970, ten such reservoirs had been built. Those at Vihorlat in east Slovakia (336. lo6 m3), RozkoS near Ceska Skalice (76.15 10" m3)and Nove Mlyny on the river Dyje (145.3. lo6 m3) serve mainly for irrigation.
The construction of reservoirs for agriculture is closely linked with the development of agriculture and the available water resources. Table 1.2. shows the increased demand for water for irrigation and cattle breeding. It is important to remember that irrigation water is irretrievable. Agriculture will constantly continue to need more water and it is presumed that by the year 2000, seventy-one large, and several small, reservoirs will be used for irrigation. (h) R e s e r v o i r s f o r r e c r e a t i o n p u r p o s e s Many Czech reservoirs are also used for recreation (Liberec, BystWka, Vranov, Set, Kninitky, Orava, Slapy, Lipno, etc.); however, this is not the purpose for which they were originally built. The first impounding reservoir built exclusively for recreation was that on the Botit at Hostivaf, built in 1962 and large enough for 36 000 visitors. A recreation reservoir was built in 1973 for the Plzeii region, in ceske Udoli (Litice) on the river Radbuza. The reservoirs at Slapy and Orlik give many people the chance to spend their leisure tim; in a pleasant environment; however, the water in the river Vltava in Prague is now much colder due to the Vltava cascade of reservoirs. Small recreation reservoirs are therefore being built in Prague on small rivulets (Motol, Sarka, etc.).
The main problem with these reservoirs is to ensure the required volume of water, especially during a hot summer and its quality. Reservoirs which supply drinking water must not as a rule be used for recreation. Table 1.4 Data on reservoir utilization in Czechoslovakia up to the year 2000
Territory
Reservoirs
Purpose of reservoir
volume [mil. m']
Czechoslovakia Bohemia and Moravia Slovakia
surface water supply for
310
6128 8466
101
91
71
26
161
75
235
4041 5352 2087 3114
82
62
39
21
103
46
19
29
32
5
58
29
75
135 85 50
100
53 47
26 (i) New reservoirs t o be built i n Czechoslovakia
With an average annual precipitation of 690mm, about 89.109m3 of water falls on an area of 127 800 km2 per year. Of this, the runoff amounts to 31%, i.e., 27.7. lo9 m3 in an average year. In a low-flow year the runoff is only 16.3 - lo9 m3. Table 1.3 shows the distribution of the total average annual runoff in the respective watersheds, and the volumes of the 185 reservoirs built up to 1970 and the volumes of the 310 reservoirs planned up to the year 2000. The total accumulation of water is to increase in Czechoslovakia from 12.9% in 1970 to 22.1% in the year 2000. On a nationwide average this is a large volume, so that it is possible to make good use of the available water. Table 1.4 gives the data on the reservoirs and their exploitation in Czechoslovakia as given by the Water Management Plan up to the year 2000.
Fig. I . 1 Reservoir construction in Czechoslovakia (World, 1976).
Figure 1.1 shows the number of reservoirs and their volumes included in the World Register of Dams of the International Commission on Large Dams. This list includes only those reservoirs with dams at least 15 m high: the Czechoslovak Water Management Plan therefore has a greater number of reservoirs. 1.2.2 Worldwide construction of reservoirs
Many reservoirs are being built both in developed and developing countries. The greatest pre-war reservoir volume (Hoover 36.7 * lo9 m3)has already been exceeded by 15 others built during the last 20 years. Table 1.5 lists the 25 largest reservoirs in the world (Mermel, 1975).
27 Tohle 1.5 World's largest capacity reservoirs in 1988 (Mermel, 1988) No. 1
2 3 4 5 6 7 8 9 10 II I2
13 14 15 16 I7 18 19 20 21 22 23 24 25
Name
Country
Year
Volume [ I O ' ~ ' ]
Owen Falls') Kariba Bratsk Aswan (High) Akosombo Daniel Johnson Guri (Raul Leoni) Krasnoyarsk Bennett W. A. C. (Portage Mt.) Zeya Cabora Bassa La Grande 2 Chapeton La Grande 3 Ust-llini Boguchany Volga-V. 1. Lenin (Kuibyshev) Serra da Mesa Caniapiscau (KA-3, KA-4. KA-5) Upper Wainganga Bukhtarma Atatiirk lrkutsk Tucurui Turukhansk (Lower Tunguska)
Uganda ZimbabweiZambia USSR Egypt Ghana Canada Venezuela USSR Canada USSR Mozambique Canada Argentina Canada USSR USSR USSR Brazil Canada India USSR Turkey USSR Brazil USSR
1954 1959 1964 I970 1965 1968 1986 I967 1967 1978 1974 1978 U.C. 1981 1977 U.C. 1955 U.C. 1981 I987 1960
2700.000 180.600 169.270 168.900 148.000 141.852 138.000 73.300 70.309 68.400 63.000 61.715 60.600 60.020 59.300 58.200 58.000 54.000 53.800 50.700 49.800 48.700 46.000 45.800 45.000
U.C.
1956 I984 U.C.
denotes under construction ') The major part of this reservoir is the natural capacity of a lake
U.C.
To forecast the future development of water management, not only must the future requirements for water be known, but also the future regime of the water resources, which will continue to be increasingly influenced by human intervention. To make such a forecast is not easy, as - there is no strict dividing line between the water consumers (people, industry, agriculture) and between the water users (hydro-power, navigation, fish farming, recreation), as, e.g., hydro-power, by its large reservoirs, greatly changes the hydrological regime and the properties of water, and evaporation causes water losses; - the hydrological regime and the properties of the water are greatly changed by agrotechnical and forestry measures, urbanization, etc., which are neither consumers nor users of water.
28
Withdrawals of water decrease the flow in rivers and reservoirs influence the water sources. The need for water, as well as its irretrievable consumption, is rapidly growing in all continents. The value of the annual water need ( p ) on Earth came to totals of 400,1100 and 2600 km3 in 1900,1950 and 1970, and the annual irretrievable consumption (s) to 270, 650 and 1550 km3, respectively. According to forecasts p = 6000 and s = 3000 km3 in the year 2000. In 1970, the withdrawal was divided as follows:public water supply 120,industry 500and irrigation 1900km3. Irretrievable annual consumption for irrigation came to 1300 and evaporation from reservoirs to 70 km3. The total reservoir volume is 5000 km3 and the surface area 300 000 km2 (Mirovoj, 1974). Large reservoirs are usually multi-purpose reservoirs. The Aswan High Dam has a capacity of about 30. lo9 mSof dead storage, 70. 10’ m3 of active storage for electrical power and irrigation and 30- lo9 m3 for flood prevention. The complex system in the Snow Mountains of Australia has 2.25. lo9 m3 water for irrigation and a total output of 2 770 MW from 15 hydro-power plants. The Bulgarian reservoirs (the largest is Isker-670. lo6 m3) store water for irrigation and hydropower. Typical for Bulgaria are systems where large reservoirs upstream regulate the flow for the hydro-power plants, and the large reservoirs downstream of the last power plant change the flow according to the irrigation needs.
Plans for the construction of reservoirs cannot be made without determining their future tasks in the cascade of reservoirs and in the water management systems (see Chapter B.l). Due to the greater exploitation of water an ever-increasing nuiiiber of reservoirs are becoming a part of extensive systems. The designers of thesc 5 1 stcms are faced with qualitatively new and much more difficult tasks than designers of individual reservoirs. Probability and system approaches will become the rule and the application of system sciences and modern computer technology will become a necessity. 1.3 MODERN TRENDS IN RESERVOIR-DESIGN
The problem of water storage in reservoirs was solved according to the function of the reservoir and according to the construction means of the given period. In the course of time new solutions were found and the development of the respective sciences made it possible to perfect the methods. Czechoslovakia stood at the forefront of reservoir development worldwide. In Czechoslovakia the development of the analysis of water management of reservoirs can be divided into three stages: (a) the solution of the function of an isolated reservoir with a short-term, or withinyear control cycle, by direct methods in real series (up to 1960), (b) the solution of the function of isolated reservoirs with a within-year and over-year control cycle, using probability methods (since the 1950s),
29 (c) the solution of the function of reservoirs in single purpose and multi-purpose systems (since 1970). The design of the reservoirs must take into consideration the relationships between the water resources (hydrological conditions), the need for water (economico-social conditions) and the size of the reservoirs (accumulation conditions). Hydrological d a t a Long-term measurement of precipitations and runoffs, processing the results of the measurements, up-to-date hydrological and hydrometeorological information ensure a good knowledge of the hydrological conditions. On the initiative of J. Stepling, meteorological data began to be recorded in the Prague Klementinum as early as 1752, which makes it the oldest meterological station of the former Austro-Hungarian empire. In Slovakia, the first station was established in 1789 in Ketmarok. Bohemia was also among the first countries to organize hydrographic services. After the floods of 1872 and after the exceptionally low-water period in 1874, a hydrographical service was established for the Kingdom of Bohemia in 1875. Thanks to this initiative long series of measurements exist. However, exceptionally high- or low-water periods were recorded even earlier: the great flood in Prague, Usti on the Labe and DEin have been measured since about the year 1O00, water stages have been measured regularly by the water gauge near the StaromBstsky dam in Prague since 1825. After 1851, further water gauges were installed on the Labe in MBlnik, LitomBfice, Usti and DEfin. The oldest water gauges in Slovakia can be found in Bratislava (1823) and Komarno (1830) on the Danube. Hydrological services were introduced in 1886. Since 1919, hydrological services for the whole country have been supervised by the State Hydrological Institute in Prague, which in 1954 merged with the Meteorological Institute to form the Hydrometeorological Institute.
Nine hundred and seventy four water gauges have been installed in Czechoslovakia, thus there is one gauge for every 130 km’. The long-term hydrological measurements were processed. After 1960, data from the most important water gauges were processed for the thirty-year period from 1931 to 1960 to serve as a basis for future studies. The result of this work was the study Hydrological Conditions in the CSSR ( I 965-1 970), the contents of which are unique. The fifty-yearseries: 193 1 to 1980, has been processed to serve as the hydrological data for future periods. In view of the constantly more progressive non-stationary character of hydrological series, it is rather doubtful whether more recent, and shorter series, can be considered as representative in view of theexistence of the longer series which date back a long time.
The Hydrometeorological Institute processes the basic hydrological data in three cycles, with monthly, annual and long-term (10, 20 years, etc.) intervals. Figure 1.2 shows a diagram of a monthly balance (Daiikovil.et al., 1975).It can be seen that this is not only a calculation of the average daily flow, but the optimization of the number of hydrometric measurements, in order to check the accuracy of the stage-discharge
. .
W
--------operutor
I hydrometric
I
0
1
I I
Fig. 1.2 Automation of monthly flow balance.
31
curve. The diagram also shows the links between the hydrological data bank, the Hydrofund and the calculation process. Data in the bank are processed in annual or long-term cycles. Hydrology is one of the basic factors of the information system of water management (ISVH), which is part of the national information system on the territory (ISU), established in 1970. The basic system of hydrological information consists of information files and sub-files according to the hydrological division of water occurrence in nature; a survey is shown in Table 1.6. The library of the hydrological sub-files programmes is constantly being supplemented. Table 1.6 Survey of registers and sub-registers on surface water in the hydrological data-bank
Register
A.I. basic register of discharges
B.I. basic register of temperatures
Sub-register
11. sub-register of derived water
discharge indices Ill. sub-register of extreme discharges phases 11. subregister of derived water temperatures
C.I. basic register of water stages at selected stations
The Hydrofund records, registers, documents and balances all activities relating to water management in an attempt to obtain knowledge of, and to assess the natural water fund. The Hydrofund therefore is able to give information about the natural water circulation regime and about the capacity of resources in the respective regions, and it systematically supplements and defines the hydrological balance. The Hydrofund thus helps to determine the water balance, and its control, over the whole country. To evaluate the natural amount of water, the Hydrofund registers the results of observations and measurements at the gauging stations of surface water and subsurface water, of the precipitation gauges and atmometers of the Hydrometeorological Institute’s network, as well as of the stations of other organizations, data on the withdrawal of water and on mineral and mine water. This information is processed by the Hydrofund from various angles, in order to compile balances, hydrological studies, studies of the thermal regime, hydrological manuals and other documents which are indispensable in designing reservoirs.
32 Hydrological forecasts play a large part in water management in reservoirs. However, in our physico-geographic conditions the hydrometeoric forecasts only have a limited prediction period. It can be increased, and this is especially important during floods, by hydrosynoptic forecasts which use synoptic maps with actual data on the state or the presumed development of the atmosphere (Kakos, 1976). Even though the meteorological forecasts are not yet sufficiently reliable and accurate, the hydrosynoptic forecasts of the Central Forecasting and Water Management Information Service (UPVIS) have already provided timely notifications or warnings several times since they started being issued in 1971. T h e first water management calculations Hydrotechnical projects have been designed on the basis of water management calculations since the end of the 19th century. At that time these calculations were only simple empirical calculations, resulting in inaccurate basic parameters, frequently leading to damage or even catastrophes. The first hydrological study was undertaken in 1893for the dam on the lower Morava. The first books which reflect the standard of the water management calculations of that time are by Jilek (1904, 1907, 1909) and Stupecky (1909, 1911). Graeff (1873) gives the basic equation of the performance of a reservoir in a differential form Fdh = P d t
-
Odt
-
0 At
as well as
F Ah
=
PAt
where F is the surface area of the reservoir at the level h ; h - water level in the reservoir. The analytical solution of the differential equation is derived and an empirical calculation of the flood storage effect of the reservoir is carried out under simplified presumptions. The graphical solution of the basic equation of reservoir function was introduced in the 1880s. In 1897, Kresnik published a graphical solution of the flood-control effect of a reservoir using a mass curve of the inflow and outflow. Stupeckf (1909) published a graphical and numerical (in the form of a table) solution for conservation and flood-control reservoirs. However, it was JeZdik (1936) who systematically elaborated graphical methods in water management. Development of the calculations of the flood-control effect of reservoirs Most of the older reservoirs served only for flood protection. The obvious relationship between the level of the reservoir and the outflow from the reservoir was
33 the reason why the flood-control effect of reservoirs was solved earlier than the designs of conservation reservoirs. The first solution, from which others were derived, was presented by 0.Z . Ekdahl in Sweden (1888). This solution does not require the drawing of a mass curve and chooses a time interval At, for which the change of the level Ah in the reservoir is found (other authors used the change of the outflow 0); the result is the time behaviour of the water level h = f(t). Thanks to T. Jeidik the most frequently used Ekdahl type of solution in Czechoslovakia is that by Visentini (1932).This calculation requires, besides the given flood P = f,(r), the construction of two auxiliary curves
0 -
2
= f,(h)
and
o v
-
2
+ Ar = f3(h) -
where P is the inflow to the reservoir, 0 - the outflow, t - time, h - water level, V - reservoir volume. The result is the outflow curve
0 = fk(4 A step forward was Potapov’s (1933) calculation, which was the first to introduce the relation 0 = f(V ) ,by which he replaced the two relations 0 = f ’ ( h )and I/ = f ( h ) , which means that besides the given flood P = fi(t) only one auxiliary curve 0 = = f2( V )was necessary; the result of this solution is the time behaviour of the outflow 0 = f3(t). Potapov’s calculation also used the variable interval At for the first time. In the 1950s, several other new calculations needing only one auxiliary curve were used. These can be found in Mallet-Pacquant’s book on earth dams (1951), in the works Qf Sorensen (1952), Ho-Ci-Huan (1954), Urban (1956) and Zaruba (1961). Zaruba’s calculation is the simplest. The only auxiliary line is the stage-discharge curve of the outflow device 0 = f ( h ) . The depth-storage relation is approximated (with sufficient accuracy) by a straight line or by a broken line. An accurate calculation using the mass curve of the inflow was presented by Koieny (1914), who worked with subsidiary curves I/ = f,(b),0 = f3(h), X P = f$); the result of the calculation is the outflow curve 0 = f4(t). This calculation differs from the previous ones; here the depth interval of the change of the water level in the reservoir Ah is chosen and the respective time interval At is sought. Work with a mass curve takes more time and makes the solution less accurate. A solution for the regulation of a flood during its passage through the reservoir by operating the gates was presented by A. Bratranek (1939) both for the lowering of the gates up to the spillway crest and for raising them above the crest. This solution
34
is both numerical and graphical and is based on the presumption that the gates move at a constant rate and that the inflow is constant. J. Urban modified the graphical solution for a general rate of inflow and a general rate of the lowering and raising of the gates. Urban’s work (1956) with about twenty-five solutions reflects the great contribution made by Czech engineers in solving the retention effect of reservoirs. To solve the retention problem successfully,not only the maximum flood discharge must be known, but also its duration, i.e., its volume. For studies of probability and statistics the flood volume is just as important as the maximum peak discharges. Large over-year reservoirs play an important role in flood control even without limiting the conservation function. An economically justified solution of the flood control effect of a reservoir is no longer based on the design of a control volume for the given normal runoff, but on a probability technico-economical analysis, which takes into account. - the flood control effect of the active storage capacity of the reservoir, - the transformation effect of the surcharge capacity on the lowering of the maximum discharges, - it is only for such a modified set of transformed floods that the actual flood control volume can be designed, taking both economic and other factors into consideration. As the result of more effective exploitation of water resources, no single-purpose reservoirs have been built in Czechoslovakia for the last 40 years and even the multipurpose reservoirs have a relatively small flood-control volume. The development of solutions for conservation reservoirs Jilek (1909) suggested a solution of the reservoir volume with the help of a direct method expressed in basic curves and mass curves of real hydrological series, both for constant and variable release. In pre-revolutionary Russia, also, the first solutions to the problem of controlled release were based on the balance between the given laws of inflow and release. The first work, published by Masticky (1912), dealt with water used for irrigation. Jeidik (1936) published a general solution for a conservation reservoir and an analysis of the solution for a balancing reservoir. JeBdik was the first person to publish a complete water management plan for a hydro-power plant with a reservoir. Later a graphical solution was added, which took into consideration the losses caused by evaporation. The advantages of this solution were reflected in the optimum alternative solution and later in the solution with discharge hydrographs, introducing various reliability water supply. These solutions partly lost their importance with the introduction of the probability theory and modern computer methods. However, this solution has retained its significance up to the present for its general character
35 and clear approach. This is proved, e.g., by the fact that random flow series in which flow regime calculations use the same methods as real series are used with increasing frequency. Water management calculations, however, depend not only on technology, but mainly on the principles of solving the problem of the best utilization of water. Up to the 1930s the problems were simple: to solve, for real time series of hydrologic data and the presumed use of water, the necessary reservoir volume for a guaranteed release, i.e., one hundred per cent security with regard to the given observed hydrological series. This practice was in use up to the 1960s. These simple problems of release control were overcome in practice by Soviet experts, first of all in hydroenergetics: Morozov (1948), Gubin (1949), Zolotarev (1950) - and not much later also for release control and water management: volumes of the edition AN USSR Problemy regulirouanija rechnogo stoka, Kritsky and Menkel (1950, 1952) and others. Hazen (1914) was the first person to use mathematical statistics for reservoirs calculations. In his work, he derived graphs for the calculation of over-year reservoirs in relation to the mean annual runoff, the variation coefficient and the dependable yield. Hazen used the runoff from fourteen rivers to compile a 300-year series. However, his method did not take root. Kritsky and Menkel(l932) found a method of determining the over-year reservoir volume for a uniform withdrawal and a given component of reliability. At the same time, they also elaborated a method to determine the seasonal component of the reservoir volume. In 1935 the same authors presented a more perfect method based on an accurate statistical calculation of the outflow. On the basis of this method, Pleshkov (1939) compiled a diagram which greatly facilitated the calculation of the reservoir volumes for C, = 2Cv for the reliabilites p = 75, 80, 85, 90, 95 and 97% and for the coefficient of the yield a up to 0.90. Ivanov’s (1946; Morozov’s 1948) calculation method and graphs are based on * hydrological series of eighteen rivers (sixteen Soviet rivers plus the Rhine, and the Croton in the USA) with a total duration of 1000 years. This method helps to determine the over-year component of active storage capacity. A computed skewness coefficient C, (i.e., not only C, = 2Cv)can be used and the a-values reach up to the value ct = 1.0. Savarensky’s method (1940) makes it possible to determine not only the reliability of the water supply, but also e.g., the distribution of the probability of the filling of the reservoir, the probability and depth of failures, the probability of overflow and the conditions of the functioning of the reservoir during the years of the first filling, etc. These methods were all based on the white noise random character of the annual discharge values. However, studies of the hydrological series showed that the years were apt to form either low or high flow groups (Yefimovich, 1936). This tendency can be quantitatively expressed by the correlation function of the flow series. Kritsky
and Menkel’s (1959) method introduced into the calculation of the over-year component of the active reservoir storage capacity the correlation co&icient of the discharge in the successive years; Gugli used this method to construct graphs for the value r1 = 0.30. Methods were also developed to calculate the seasonal volume of a reservoir. The problems of reservoir volumes for variable withdrawals during the year, e.g., for irrigation, electrical power, etc., were studied to a lesser extent. L. Votruba solved the problem of the reliability of supply of water, according to the occurrence, the duration and the volume supplied, both for balanced and variable withdrawal (Votruba, 1962; Votruba and Broia, 1963). The methods described above for the calculation of reservoir volumes are inadequate for some of the more complicated cases of flow regulation. Soviet authors have worked with exact analytical methods with general statistical characteristics of hydrological data*) which made it impossible to use them for more complex problems. However, even this exact concept of a stochastic approach had to be proved, against the “classical” direct methods. In Czechoslovakia, a decisive stand on the use of direct and indirect methods for the release control by reservoirs was taken at the meeting of the Czechoslovak Scientifico-technical Society in April 1946 (Manual, 1965). It was confirmed ihat the indirect methods were correct, determining the room for the use of direct methods. A new qualitative element in the calculation of reservoirs was the nontraditional processing of hydrological data with the help of a mathematical model of a random process, based on a synthetic hydrological series and not on composing real series from several rivers (Hazen, 1914; Ivanov, 1946), nor on any statistical “treatment” (Lyapichov, 1955), but on modelling according to the principles of the theory of stationary stochastic processes (including their correlation functions). To apply these methods it was necessary to determine whether they suited this purpose, and it was also necessary to use modern computer methods without which the calculations could not be made. An approach, referred to as the Monte Carlo method (random sample, random experiments),made it possible to reach mathematical solutions by multiple random experiments. This method is especially expedient where the algorithm is so complicated that an analytical calculation is practically impossible. In hydrology, it is thus possible to obtain a synthetic series of an arbitrary length, which corresponds to the logic of the modelled process, i.e., the process of the fluctuation of the river runoff. The first primitive experiment to model hydrological series in this way was made *) The analytical method means that general probability characteristics of release, filling of the reservoir and other variables determiningthe regime of the given reservoir are constructed according to the general probability characteristics of the flow series and in accordance with the given rules of controlled outflow. This method does not study the respective realization of random processes.
37 by Sudler (1927).Kritsky, Menkel and Rybkin published their work on the modelling of hydrological series in 1946. From 1954 to 1959 Moran and Gani published studies on the application of the Monte Carlo method in the calculation of reservoirs; management of water in reservoirs was seen by them as one group of a class of probability problems known as “accumulation”. The most complete study of the use of the Monte Carlo method for the calculation of over-year control of the release from reservoirs was presented by Svanidze (1961, 1964) and with application to hydro-energy b;y Reznikovsky (1969, 1974). Just as in the previous studies, here too, the modelling of the series was based on the principle of the simple Markov chain. The results of these studies were graphs similar to those by Pleshkov and Gugli, however, for a greater number of values of the coefficient of skewness and coefficient of correlation of the release for the previous and following years. An analysis of the hydrological series, however, showed that their correlation functions have, in most cases, a harmonious character; hydrological series can therefore be realizations of sequences with a periodic component or also of a higher order Markov chain. From an analysis of the hydrological series, Nachazel(l965,1975) derived a solution of the over-year component of the active reservoir storage capacity using a composite Markov chain with a damped harmonic correlation function, which is closer to the actual hydrological regime and gives more probable results than the calculation based on the simple Markov chain. In the USSR, Chomeriki (1963, 1964) studied the cyclic fluctuation of river runoff and its influence on the over-year release control. A drawback of all mentioned probability calculations of the over-year release control was that the reservoir volume was divided into the over-year and the within-year components. The total volume could only be calculated by the balance method with an adequately short time interval. The interval of one month is sufficient for an over-year control. Votruba (Votruba and Broh, 1966) performed such a graphical-numerical solution for a 1 10-year series, 1851 - 1960, for the Labe in D&Ein. Even a series of such length, however, is too short for reliable over-year control. Therefore, another step forward in the calculation of the over-year storage function of the reservoir was the modelling of synthetic, arbitrarily long series which divided the flow within the year. Two methods of modelling were developed, i.e., the so-called fragment method of Svanidze (1962) and the regression model of flow series (Kos, 1969). The current balancing method is then used for the calculations. Its advantage is that even the most complicated water management problems can be solved in this way. The disadvantage of the large number of calculations have been overcome by the use of computers. A further development stage in the calculation of the conservation function of reservoirs was the introduction of the non-stationary approach to the processing of
38
hydrological data by stochastic methods. Nachazel and Patera (1975,1976) did much pioneer work in this field in Czechoslovakia. As a result of human activities, the non-stationarity of the hydrological regime is going to increase in the future. It is therefore important for the calculation of the conservation function of a reservoir to predict the main characteristics of the future hydrological regime some decades hence, i.e., at a time when the reservoir designed today actually starts to serve its purpose. This forecast should be reflected in the reservoirs being designed today. The trend of research was outlined in the resolution adopted at the Symposium on the Methods of Flow Control through Reservoirs, held at the Technical University in Prague in 1974 (BroZa, 1975). Solution of the storoqe function o f ~ s e r u n b s I nwmgement of s wir W m
1 determinotion of the Efive stomge mpocrty
I
onoij4iml
methods
I
!
reser-
I
diqitol- gmphim methods
in synthetic methods
-mnogement system
Fig. 1.3 Methods for analysis of the reservoir storage function.
Research into the conservation function of a reservoir should also consider the role of the respective reservoirs in the water-management systems (see Chapter 15). This aspect must be reflected in the design as well as in the control of operations. Figure 1.3 shows a survey of various solutions of the reservoir conservation function. Direct methods can be considered as closed while the others are still open and will be the subject of further research. 1.4 PERFORMANCE OF THE RESERVOIR
A reservoir, in the broadest sense of the word, is an enclosed volume (vessel)which
can be tilled and emptied repeatedly over a period of time, with solid, liquid or gaseous substances. Thus, for example, the term reservoir may be applied to a cement bin which is used to make up the difference in the amount of cement supplied from the cement works and the amount used for construction. A bin of gravel aggregates for concrete has a similar regulatory function in the links between production, transport and
39 utilization of the aggregates. Silos for grain and storehouses for building and other materials have a similar function. All the above-mentioned reservoirs have one and the same task : to accommodate fluctuations in the supply of substances (inflow)to the reservoir with fluctuations in the withdrawal of substances (outflow)from it; in other words, to permit the proper management of the flow of materials. If the rate of supply were identical to the rate of withdrawal, no reservoir would be necessary. A reservoir becomes necessary only where inflow and withdrawal are temporarily out of balance. The solution principle is therefore essentially the same for all reservoirs; the problem becomes more difficult if the rate of inflow cannot be regulated, as is the case with impounding reservoirs. Among reservoirs in general may also be included those which do not have a clear release control, because the release is not subject of our interest. The purpose of such reservoirs is to create a particular type of environment: a recreation reservoir, a pond for poultry farming, etc. The water is not accumulated to be used for a specific purpose, but rather to maintain a particular environment. 1.4.1 Basic performance of the water storage reservoir
The basic data for the solution of the behaviour of reservoirs are those concerning the inflow into the reservoir and the demands on the withdrawal, or release from it. Solution of the function of the reservoir is unambiguous if the law of inflow and the law of outflow are determined by the relations
p
=
fdt)
(1.1)
0 = .f$)
(1.2)
The two relations may be mutually dependent or independent. As a rule, the input and discharge behaviour are so complicated that they cannot be expressed analytically, and therefore tables or graphs (time curves) are used. These may be continuous or discrete, and for the solution of some of the problems their statistical characteristics can be used. The relationship inflow, outflow and retention of a reservoir is given by the equation (1.3)
P-O=R
i.e.: inflow - outflow = retention (all in m3 s- '). IfP > 0, R is positive, the amount of water in the reservoir increases (the reservoir is being filled); if P 0, R is negative, the amount of water in the reservoir decreases (the reservoir is being emptied). Equation (1.3) is the basic equation of reservoir performance; it is very simple, but because of the complicated behaviour in time of the quantities therein, it cannot, as a rule, be solved analytically.
-=
40 Figure 1.4 shows the general pattern of inflow, P, and outflow, 0, and the resulting retention behaviour, R. The value R,, is obtained by moving the line 0 in the direction of the ordinates until it becomes a tangent to the line P. The tangents to the lines P and 0 at time t, must therefore be parallel.
Fig. 1.4 Schematic representation of the basic reservoir function
-?
At times t , and t,, P = 0, i.e., R = 0 and is transitional between positive and negative values. However, points t , and t , are not the same, since at time c, the sign of R changes from negative to positive, and vice-versa at f2. At time t , the reservoir starts to fill and continues to fill until time t,, when the amount contained reaches its maximum and the reservoir starts to empty. The basic equation of reservoir performance (1.3) is derived with respect to t :
dP dO dR _-- =dt dt dt When R = R,,
then dRldt = 0 and therefore
dP dO - --dt
dt
Since the terms dR/dt, dP/dt, dO/dt express the gradients of the tangents to the respective curves, we find that in time ,r when R = Rmax: (a) the tangent to line R and the axis t are parallel; (b) the tangents to the lines P and 0 are parallel. We usually solve the basic equation of reservoir performance numerically or graphically, by dividing the time, t, into finite intervals, At, for which we determine
41
the mean values of P and 0, being taken as constant for the period At. The precision of the procedure using these discrete values depends on the length of the intervals and is usually not less than the accuracy of the initial data. 1.4.2 Classification of reservoirs according to origin and location
Generally speaking, any enclosed volume with a variable water content has the function of a reservoir. Such a system can, for example, be a lake, a storage reservoir, a pond. a special-purpose reservoir. a water-works reservoir, etc. However, other formations not generally thought of as reservoirs have a similar effect on the runoff from within the watershed: inundations arising from accumulated flood water, depressions in the terrain, the soil, vegetation, snow etc. Reservoirs are either natural or artificial : 1. A natural reservoir is a hollow (a basin) or cavity filled with water which arrives without human intervention. It can be of tectonic, volcanic, glacial or karst origin, etc. To increase its regulatory effect, it may be equipped with a regulating structure at the outlet. mox. level ------
A I
i -0
om,
0
Fig. 1.5 Schematic representation for use of a lake for release control
Figure 1.5 shows a longitudinal section of a lake outlet (lakes being the most common natural reservoirs), and the relationship between outflow, 0, and the water level in the lake, H. The height of the dead storage level, M,,and the active storage between M , and the maximum level are determined by the condition that the maximum backwater must not be exceeded, and that the minimum regulated flow should be Omin. The active storage can be increased in two ways: (a) by increasing the maximum backwater above the maximum water level, (b) by dredging the bottom and the outlet from the lake so as to enable, for M,, a discharge of Ominto occur. Figure 1.5 shows the required dredging depth and specific curve, 0, at which M , = = min. level. The depth of the active storage, h,, of lakes serving to control outflow, is usually small since the large area of the lake in itself ensures a large storage volume. The cascade of reservoirs on the River Ankara is able to control runoff by virtue of a rise in the level of Lake Baikal to 1.46 m above the average level, equivalent to a storage
42
volume of 46 - lo9 m3. As can be seen, a lake makes it possible to gain, at relatively low cost, a large man-made reservoir volume. 2. A man-made reservoir is one made specially the purpose of managing natural processes. It may be a reservoir without through-flow (dioersion reservoir), i.e., not forming part of the course of a stream and having its own artificial supply of water, or it may be a through-flow reservoir. Reservoirs withour through-flow can be made (a) by excavation of the ground, by erecting a dam enclosing a basin, or most frequently by the combination of both; or (b) as lateral reservoirs. Through-flow reserooirs are usually built as impounding reservoirs. If it is not the water of the main stream that is impounded, but rather that of a tributary (with the inflow of water from the main stream), then the reservoir is called a lateral reservoir. An excavated reservoir, which is usually bordered by dams made from the excavated material, is built where it is most expedient from a functional point of view, and where a cheaper and simpler reservoir cannot be built (e.g., by building a dam across a valley). Today, excavated reservoirs are mainly built as upper reservoirs for pumped-storage power plants, e.g., Stikhovice (1945), Fig. 1.6; Amalienhohe in the GDR (1963), Cierny Vah (1981) and others.
Fig. 1.6 Upper reservoir of a pumped-storage hydro-power plant, Sttchovice (1945).
43 A lateral reservoir is built by dividing and closing off a part of the valley beside the stream by means of a side dam. Such a dam is not a barrage, as it does not dam the valley transversely and does not create an impounding reservoir. For a fairly large lateral reservoir, the average cost of the dam (per unit volume of the reservoir) is usually higher than the cost of that for an impounding reservoir. However, it can still be more advantageous than an impounding reservoir if it helps to prevent the flooding of important installations, housing estates, factories, roads, fertile fields, etc.
Fig. 1.7 Layout of a lateral reservoir.
Another advantage of lateral reservoirs is that the inflow of water can be stopped at any time, so that it can be protected against floods, polluted water, silting, ice, etc. There are also smaller demands on the outlet and safety devices, and maintenance of the reservoir is reduced. However, compared with the impounding reservoir, the lateral reservoir has a smaller regulatory effect as it can only control that part of the flow in the river which can be brought to the reservoir by diverting structures. Impounding and lateral reservoirs are compared in Fig. 1.7. If the impounding dam were built in the profile I - I’ the reservoir would be larger and would have a relatively short dam, but it would flood a railroad and the buildings in the valley. Expediency suggests shifting the side dam of the lateral reservoir towards the river into the hatched part, a small increase in height and lengthening of the dam greatly increasing the volume of the reservoir. Lateral reservoirs are ponds without throughflow; here the advantages of lateral reservoirs can be exploited fully as the object of the pond is not to control the outflow, but to create an environment for fish farming where perfect control of the inflow to the pond helps to create the best possible conditions for fish farming. Recreation reservoirs are intended mainly to create a suitable environment for recreation purposes, with clean water, a suitable temperature, an absence of deposits, and no flood problems, etc.; here too, lateral reservoirs prove the most suitable. A special case of the lateral type of reservoir is the dry reservoir (polder),i.e., a nat-
44 urally- or artificially-bounded basin near a stream, which is filled with water only during floods. After the flood recedes it is again completely empty (dry) and can therefore be used for agricultural purposes; it has a retention function, as it reduces the flood discharge in the stream.
shore line _._._.-. -.-.-.-.--
a)
--.-.__.-. *
P
Fig. 1.8 Schematic representation of an impounding reservoir (a)layout; (b) longitudinal section of main stream
An impounding reservoir is made by building a dam across the valley of a watercourse, such that the shape of the dam is dictated by the bottom and slopes of the valley, with the upstream face rising up to the backwater elevation (Fig. 1.8). It receives the total discharge of both the main stream, P , and of the tributary, P', and the water leaves it through outlets or spillways discharging on to the riverbed downstream of the dam, or as withdrawal that is taken from the reservoir for any specific purpose. The inflow into the reservoir is given by the total discharge from the watershed to the dam site. For control purposes, the volume contained between the dead storage level and the active-storage level, possibly also the volume of the cavities in the valley rocks between those two levels, is used. The reservoir is filled above the level of active storage only during floods, and thus decreases the flood discharges. A tributary lateral reservoir is an impounding reservoir on a tributary (Fig. 1.9); it is therefore lateral with reference to the main stream. It can control the total discharge of a tributary leading to the dam, and that part of the discharge of the main stream which is diverted to the reservoir. Figure 1.9 shows three ways of diverting water from the main stream to the lateral reservoir. (a) A water diversion device is built on the main stream and from there the water
45
is led to the lateral reservoir on the tributary. To obtain a natural flow, the diversion facility must be situated above the highest operational backwater in the lateral reservoir. Part of the supply conduit is usually a tunnel, as it is necessary for the flow to traverse the watershed divide between the main stream and the tributary.
C)
reservoir
station
Fig. 1.9 Layout of a lateral reservoir (a) with diversion canal; (b)with a pumping station downstream of the dam; (c) with a pumping station on the main stream
(b) Downstream of the confluence of the main stream and the tributary, an impounding structure is built so that the backwater reaches up to the dam of the lateral reservoir. Here a pumping station is built which pumps the water from the main stream to the lateral reservoir. To cover part of the power needed, the power of the water flowing back from the reservoir can be used. (c) The pumping station is built on the main stream in such a way as to make the length of the supply conduit to the lateral reservoir as short as possible. The water from the reservoir can be returned in the same way (0,- arrangement of the pumping station as in case (b)),or it can be discharged into the tributary (0,)if it is used down-
46
lakes
excuwted
tributary
/ ' Fig. 1.I 1 Schematic representation of a cascade (series) of impounding reservoirs
Figure 1.10 gives a diagrammatic classification of reservoirs according to their origin and location. A cascade of impounding reservoirs is a series of functionally integrated impounding reservoirs on the same stream. Figure I .11 is a diagram of an as yet incomplete cascade of eight reservoirs. A connected cascade of reservoirs is considered mainly where the river water power is to be used in a cascade of reservoirs with hydropower plants. The segmentation of the stream by dams, the determination of the backwater level and the degrees of utilization of the respective reservoirs, are functions not only of the volumes of water and withdrawals, but also of the head. The analysis of a cascade of reservoirs and its operation is one of the most complex problems of water management and hydro-power engineering.
47
Examples of cascades of reservoirs are those on the rivers Vltava, Volga, Dnepr, Ankara and others. A sysfem of’ impounding reservoirs is a group of reservoirs on different streams comprising a main stream and its tributaries, the reservoirs cooperating mutually (Fig. 1.12). Operation of the reservoirs must be coordinated if they are power reservoirs and the power plants are supplying power to a combined power network. All or some of the reservoirs can also function co-operatively to protect the downstream part of the valley from floods, and control the flow regime in the downstream river reaches (e.g., the cooperative action of the Vltava reservoirs and the proposed Kfivoklat reservoir on the river Berounka, and that of the Vah reservoirs with the Orava reservoir). The cooperation of reservoirs in generating power is not limited to those within one and the same watershed, as the high-voltage network links power plants in different watersheds. The cooperation of reservoirs with respect to water management used to be limited by the boundaries of the watersheds in which they were situated; today, significant diversions from one watershed to another are no longer rare.
Fig. 1.12 Schematic representation of a systern of impounding reservoirs
Fig. 1.13 Schematic representation of water accumulation in flood plains
Underground reseruoirs are created by damming the permeable soil in a valley with a watertight barrier (underground dam). To let the water infiltrate more easily and more rapidly, it is usually diverted from the watercourse during high discharges by means of deep infiltrations ditches to the sides. The water is then collected from the underground reserves for irrigation purposes, etc. Underground reservoirs’require particularly favourable conditions, with respect to the width, depth and properties of the permeable sediment deposits, economic utilization of the valley, etc., if they are to be of any importance for water management. Accumulation in flood plains (Fig. 1.13) is similar in effect to that of the flood surcharge storage of reservoirs, in decreasing flood discharges. As the area of inundation is usually large, the retention effect of the inundation is also quite significant. If inundation is prevented (by enlarging and cleaning the channel, etc.), the retention eiTect is also removed and discharge conditions in the lower reaches deteriorate. It can, therefore, be dangerous to proceed with river training which
48 ensures the full discharge of floods from upstream. The mathematical analysis of the retention effect of inundation is, with the exception of the gradient of the water level, similar to that for reservoirs, Water storuge within the watershed greatly influences the discharge regime of the stream. Water is stored - in cavities and on uneven terrain, - in the vegetation cover (these two effects influence the runoff mainly during periods of precipitation), - in the soil as soil and ground water (stabilizing the rate of runoff), - in the snow cover; a supply of water is created which can dangerously increase spring floods, or can itself cause floods during rapid thawing; on the other hand, the retardation of the infiltration of water into the soil up to the spring is, from an agricultural point of view, a favourable phenomenon; also favourable is the thawing of glaciers and high-mountain snows in the warmer weather, as this helps to raise the low summer flows. - in ice cover (this decreases the flow in the winter months). 1.4.3 Classification of reservoirs according to purpose
The purpose of a reservoir is to regulate (control) discharges, i.e., to adjust the natural behaviour of the flow in the stream according to the requirements of various water-management schemes. According to the basic purpose, reservoirs are divided into : - those which ensure withdrawals of water, or yield (conservation reservoirs), - those which lower flood discharges (flood control reservoirs), - those which create an aquatic environment, - those which are used to adjust the quality of the water, - those which trap bedload and wastes. Frequently, one reservoir fulfils several functions (e.g., water supply and flood protection), and is then referred to as multi-purpose. A multi-purpose reservoir may also be one which is used in various ways (e.g., a multi-purpose conservation reservoir may have withdrawal for water supplies, irrigation and hydro-power). Smaller reservoirs serving local requirements (fire protection, small-scale withdrawal, irrigation, watering of cattle, etc.) are also usually of the multi-purpose type. Figure 1.14 shows a detailed diagram of the classification of reservoirs according to purpose. A conservation reservoir serves to store water, obtained during periods of excess, for use during periods of shortage. If the inflow exceeds the water demand, the reservoir begins to fill; if the demand exceeds the inflow, it empties. The result is that’the required water supply is ensured or that low flows are augmented, i.e., 0 > Pmin
49 sinq/e-purpose]
b ,b
Ic(I""'"tion
for:
households
icultum
cycle:
overyear
,
,h
multi-purpose
h
fir: mtim
~
&(hen
retention
,
weekly
WmPWe
iption
Fig. 1.14 Classification of reservoirs as to their purpose
(Fig. 1.15a).As far as possible, the reservoir is kept full so as to provide a sufficient supply of water in the low-flow period. The flood-control reservoir catches the peaks of flood discharges and thus protects the land downstream from flooding. If the flow exceeds the non-damaging value, On& the reservoir fills and the result is a decrease in the maximum flow, i.e. 0 < P,,, (Fig. 1.15b). After the flood wave, it is emptied as quickly as possible (0 = Ond) to be ready to catch the next flood peak. A reservoir can fulfil both a storage and a flood protection function; it then has an active storage and a flood-control storage. However, the active storage can also fulfil a flood-control function, provided that this volume is at least partially emptied before a flood wave arrives. In particular, the larger over-year storage reservoirs, also lower the flood discharges on account of their supply function, i.e., the intake of water to replenish the storage volume. They also decrease the frequency of floods.
Fig. 1.15 Schematic representation of water supply and flood control functions of a reservoir (11) water supply; (b)flood control
50
According to the duration of the discharge control cycle, basic control methods may be divided into : - ouer-year -the control is longer than one year, - within-year (seasonal) - the control cycle is completed within a single watermanagement year, - weekly - the control cycle is completed in one week, - daily - the control cycle lasts 24 hours. Figure 1.16 shows the function of reservoirs with various cycle durations in terms of inflow, outflow and water level fluctuation curves. a) over -yeor
b) within-year
e ) occasional w i t h d r a w l
T
0
t
i
f
c ) weekly
d ) doily
Fig. I .J 6 Schematic representation of reservoir function with different cycles
The cycle of' the reservoirs is a time interval in a recurring sequence of operations during which the storage volume is filled and the accumulated water used, the space again being completely (or partly) emptied. If we classify reservoirs according to the duration of the cycle, we have to consider the conditions which will determine the size of the storage volume; reservoirs with long cycles (e.g. over-year) always also have shorter superimposed cycles (seasonal, weekly, daily, etc.). Some reservoirs do not have a regular cycle of filling and emptying, but rather provide for: - occasional withdrawals for emergency use (during breakdowns, etc.) - sudden withdrawals - e.g., for the finer control or river flow regulation (buffer reservoir).
51
Besides the above-mentioned basic methods of withdrawal control there are some more complicated methods, amongst which we may consider: - river flow regulation - with reaches between the reservoir and the place of withdrawal, - in cascades of reservoirs, - in systems of reservoirs. Over-year control of the release from a reservoir (Fig. 1.16b) involves a filling and emptying cycle that lasts longer than one year; the annual water demand may be greater than some of the annual discharge amounts, and therefore the excess water from wet years must be used. A reservoir with an over-year control cycle therefore tends to equalize the discharges of consecutive years, and approximates the total annual release, XO,,from the reservoir to the long-term mean annual discharge, ZQ,, in the river. The within-year control of release from a reservoir (Fig. 1.16b) involves an explicit annual cycle of filling and emptying; the water demand does not exceed the discharge of the current water-management year. Annual control is therefore less demanding with respect to the size of the reservoir volume than over-year control. The coefficient of the yield, a, is defined by the relationship between the required withdrawal, O,, and the long-term mean flow, Q,, i.e. 0 Q a
It follows that with control aimed at uniform withdrawal, the value of ct can be a 5 1, and therefore 0, S Q,.The value of a will increase with greater evenness of the flow in the river during the year, and with increasing volume of the reservoir. Seasonal, or incomplete annual control of the flow is a particular type of within-year control in which the reservoir functions only for a limited period of the year which ' may be related to vegetation growth, navigation requirements, the winter season, etc. The control is incompletewhen the volume is not large enough to ensure complete control. Weekly control of the flow (Fig. 1.16~) presupposes smaller withdrawals on some days of the week, e.g., non-working days; there are also usually quite considerable variations during the day. The reservoir is filled, for example, on Saturday and Sunday and is gradually emptied from Monday to Friday, so that one cycle of filling and emptying the reservoir takes a week. The specific characteristics of the weekly cycle depend on the composition of the withdrawal diagram; continuous operations wipe out the differences between working days and rest days. Daily release control ensures that the difference between inflow and withdrawal is accommodated during the day; one cycle of filling and emptying takes one day (24 hours). Figure 1.16d gives a general example of the variation of inflow, P , and outflow, 0,during the day; the inflow to the reservoir is given by the constant pumping
52
rate, P , for a period of 16 hours, and the withdrawal, 0, by the hourly averages of water consumption. A common case of a daily control is the matching of a varying inflow with a constant outflow (Fig. 1.17a), or the distribution of a constant inflow into fluctuating withdrawal (Fig. 1.17b). b)
0
Cl-
-
t Chl
t
-
fChl
Fig. 1.17 Reservoir functioning with a daily cycle (a) balancing; (b) distributing
Storage or conservation reservoirs providingfor particular types of discharge control are named according to the type of control afforded. We therefore have reservoirs with over-year release control, those with within-year release control, etc., or more briefly, over-year, within-year, etc., reservoirs. We also use the terms balancing reservoir and distributing reservoir to refer to two kinds of storage reservoir with daily or weekly discharge control. The equalizing reservoir (Fig. 1.17a) accommodates the differences between a non-uniform inflow and a uniform outflow during the day; equalizing reservoirs are frequently located down-stream of peak hydro-power plants. The distribution reservoir (Fig. 1.17b) has a function opposite to that of the equalizing reservoir; it converts a uniform flow into the reservoir into a non-uniform outflow from it. In plotting the performance of the two types, it therefore sufficesto transpose the inflow and outflow lines (Fig. 17b). Discharge control with a cycle of up to few days (daily, weekly) is termed short-term release control. A control facility allowing for occasional withdrawal keeps a certain amount of water in the reservoir for any need that might arise at any time. Such a reservoir might be a fire-protection tank, a locally managed reservoir with a water supply for fire protection, or an emergency store of water for use in the event of failure of a power system. An occasional outflow wave from the reservoir to the river may be allowed to occur for the purpose of transporting wood, for water sports, for flushing the stream, manipulating ice, etc. When the distance of the intake from the stream is so far from the main reservoir that the outflow from the main reservoir ( I ) cannot be accurately adjusted to provide the required withdrawal, especially if the outflow varies, then the control of sudden
53 discharges is carried out in a smaller buffer reservoir (2, Fig. 1.16f)at the intake point, as a supplement to the main river regulation system. Reservoir I takes care of general discharge control, and reservoir 2 provides for finer discharge adjustment to the withdrawal requirements; in other words, the reservoir accommodates the differences between the outflow, caused by withdrawal variations (shocks),and the manipulation of reservoir 1. The river flow regulation increases the reliability of the yield in the river profile at a greater distance downstream of the reservoir, compared with natural (uncontrolled) discharge from the interbasin. According to the diagram in Fig. 1.16f, reservoir I performs the function of river flow regulation, as only that amount of water is released which supplements (compensates) the insufficient inflow from the interbasin to bring it up to the needed withdrawal, 0. If there is approximate concurrence of high flow or low flow conditions both within the watershed of the reservoir and in the interbasin, the reservoir will experience difficult operational conditions: in the high flow period the outflow from the reservoir will need to be small or zero, and conversely, in the low flow period the outflow from the reservoir will need to be great. Such conditions place greater demands on the reservoir volume, compared with discharge control for withdrawal just downstream of the dam. However, for withdrawal at some distance downstream of the reservoir (i.e., with an interbasin area), river flow regulation offers, under some conditions, a higher safe yield than that obtainable without river flow regulation; i.e., if the problem is solved by locating the withdrawal point just downstream of the dam, and to this withdrawal the natural discharges from the interbasin are added. Discharge control in a cascade of reservoirs is more complicated than that provided by a single reservoir, as the upstream reservoirs affect the pattern of inflow to all the downstream reservoirs. A reverse influence of the downstream reservoirs on the upstream reservoirs can also be observed, if their optimal operation is solved as an integrated operational entity. Such a solution is essential for the hydro-power exploitation of reservoirs, and is relevant also to the use of multi-purpose reservoirs for supply and flood protection. The control of release of water in a system of reservoirs is the most complex problem of all in the control of discharge. The task required cannot be solved without optimizing the structure and behaviour of the system. Technico-economico-social optimization is made the more difficult by the fact that the system is usually a multi-purpose one with conflicting demands on the water, with stochastic inputs, with a dynamic pattern of behaviour, and with important intangible functions (see Chapter 15). Different types of release control place different demands on the volume of the reservoir. For a stream of a given size and discharge variability, it must generally be larger for longer cycle of release control.
54
Where long-term release control is practised on a reservoir, there is usually also some short-term control, i.e., in an over-year reservoir, annual, weekly and daily control also takes place. If‘ a solution is sought for over-year control, it does not suffice to consider the mean annual values; the pattern of release and withdrawal within the year must also be taken into account. The survey in Fig. 1.14 also includes those types of reservoirs in which water is not managed in the proper sense of the word. Sludge-settling ponds (tailing dams) are also mentioned as areas that are naturally or artificially bounded, serving as permanent or temporary storage places for sludge and waste that is transported mainly hydraulically. Although these are not reservoirs in the sense of being within the context of water management, sludge-settling ponds have to be approved by the water management authorities. Sludge-settlingponds by their number, size, and prospective further growth, present a serious technico-economic problem and have a great impact on the environment, necessarily involving water management authorities in the design and operation of sludge-settling systems. 1.4.4 Influences affecting the performance of the reservoir (flow regime, human intervention, complex utilization)
(a) Influence of t h e flow regime on the reservoir performance The performance of a reservoir is mainly influenced by the changes of the flow in the river, which depends on weather conditions and on how the river receives its water. The flow in the river Q is derived from a genetic equation (Ogievski, 1952):
Q = Q, + Q, + Qg +
Q V
+ Qd
where Q, is the surface runoff from snow water, Q, - surface runoff from rainfall, Q, - surface runoff from glaciers, Q, - groundwater aMuent from valley alluvium, Q, - groundwater affluent from the deep aquifer of the catchment. The terms of the equation have different meanings for rivers of different types and change in time; some can equal zero. The water supply from groundwater can amount to more than half of the annual runoff (rivers in the plains with much alluvia),however, it can also equal zero (in small streams with an impermeable drainage basin). The valley alluvium helps to balance the seasonal runoff in the river and the value Q , can even be a negative value if the river water level is higher than the surrounding groundwater level. The inflow of groundwater from deep layers changes more slowly and is actually an over-year natural regulator. Depending on the dominant member in the genetic equation, Lvovich (Dub, 1957)
55
defined four basic types of river regimes: rainfall (R), snow (S),glacier (G) and groundwater ( U ) .He then further divided every type into three groups according to the share of the basic type of supply in the total runoff. He also took into account whether the greatest runoffs occur in spring, summer, autumn or winter. The rivers of Czechoslovakia, with the exception of the Danube, belong to the so-called Oder type, where the rivers receive most of their water from rainfall and an increased rate of streamflow occurs in the spring. According to their hydrographs, the various types of rivers need various reservoirs volumes to ensure .the same safe withdrawal. LFt us presume, e.g., an annual control of the outflow for a constant release. Then the hydrograph with one very short high-flow period will require largest reservoir volume.
100
h
.$ 80
1;
0
60 0
tig 1.18 Annual rclalrkr discharge c u r w
the Danube, Volga and Mekong rivers
011
Fig. 1.19 Relationship between reservoir storage capacity and type of river ( a ) storage capacity for the 100% relative yield for two types of rivers; (b) influence of the shift in hydrograph on the reservoir storage capacity
Figure 1.18 gives the annual hydrographs for the Danube at Bratislava, the Volga at Gorky and the Mekong at Phnom-Penh. Each of those rivers belongs to a different type: the Danube reflects the glacier character with the greatest runoffs in June, the Volga has the greatest runoff in April and May after the snow thaw and the Mekong belongs to the monsoon region with the highest flow in September and October. What all three rivers have in common is one main wave of increased discharge during the year, which requires the reservoirs to have a large volume. However, the Danube reaches the same yield coefficient ct = O,/Qa with a smaller relative volume p, as it has a more favourable ratio Qmin: Qmax and a longer period ,,, is roughly of increased discharge. For the Volga and the Mekong the ratio Qmin: Q the same; however, the Volga will require a larger reservoir relative volume fi, as the period of increased discharge is much shorter than that of the Mekong. Rivers with two or more periods of increased discharges during one year generally require smaller reservoir volumes than rivers with a single high-flow period. The diagram in Fig. 1.19a shows the complete balancing of the discharge of the two types
56
of discharge curves. The curve QI with one wave requires a volume bigger by A V than the curve Qll with two waves, for which a volume t;, is sufficient. Figure 1.19b shows the required withdrawal 0 and two discharge curves Q which have the same shape and differ only in time At. The full curve Q = f,(t) must have a reservoir volume V,, the dotted curve Q = f,(t) a volume bigger by AV. However, rivers with one high-flow period have, as a rule, a more regular discharge during the year and the reservoir volume can be designed with a greater accuracy. Rivers fed by rain which can occur at any time during the year, and rivers where snow can start to thaw any time during the winter, have hydrographs with a very accidental character. This fact makes it more difficult to design a reservoir and also to operate it. The rivers of Czechoslovakia are among those with difficult discharge conditions. If the river regime is stochastic, forecasts for at least the medium-term are of great importance. Attempts at long-term forecasts have so far been imperfect. The design of reservoirs and the directives for their operation are therefore based on past observations. The information gained from actually observed data is increased by methods of mathematical statistics. However, not even the law of release can be predicted with a sufficient certainty. Water demand will depend on the development of the economy and forecasts about the future demand of water cannot be any more accurate than forecasts of economic and social development. Useful for the conditions in Czechoslovakia are the medium-term (seasonal)forecasts for at least several weeks or months which - forecast the discharge according to the recession curve in the dry period, - forecast of the spring runoff from snow thaw. Even though the forecast of the spring runoff does not concern such a large part of the whole annual runoff as it does for the rivers in the plains of the countries of the north, it is of importance for the management of water in reservoirs at the end of the winter period, which tends to be critical especially for the hydro-power reservoirs. In the slightly subnormal year, 1935, the approximate runoff from snow was 82 lo6 m3 in the catchment of the Lipno reservoir on the Vltava, which is more than 40% of the active storage capacity. The methods and results of these forecasts can be found in the works by Votruba, 1949; Martinec, 1961, 1963; Dub and N5mec, 1969. Although the general equation for the volume of the spring runoff from snow V,, has about 10 variables, a linear equation proved to be suitable for conditions in Czechoslovakia
V,,= a H , , +
c
where H,, is the amount of snow storage, a - a constant, c - a coefficient.
(1.8)
57 For the decreased flow from groundwater sources in a rainless period, the following equation is applied for the flow Q ’
Q=Qoe
- bl”
(1.9)
which for some of the Czechoslovak rivers takes the form of Q = Q o e- b u t
(1.10)
where Qo is the initial flow, b - depletion coefficient(natural logarithm of the recession constant), t - time (in days) from the initial flow to flow Q. The method of calculation is given in the literature (Balco, 1958, and others). Rivers in which the flow is regulated by lakes, ponds, inundation, etc., usually need only a small reservoir volume. (b) Influence of h u m a n i n t e r v e n t i o n o n t h e water c o u r s e regime Today there is no longer any large catchment that has not been changed by human intervention. Agricultural measures, changes in the composition of plants, decay of forests, building of housing estates, mining, river training, etc., cause a change in the surface runoff which is reflected in the design and operation of reservoirs In processing hydrological data the development in time of the catchment and the runoff from it must be taken into consideration. If two sections of the hydrological series have different conditions, then the series cannot be considered to be homogeneous and cannot be considered as a set of data of the same importance. If the conditions in the catchment changed continuously during the observation period, it would be possible to use mathematical processing of the runoff characteristics on the principle of observations of unequal weight as in geodesy, etc. (Votruba and Broia, 1966).These calculations, however, would not reflect the changes in nature. It is therefore necessary to determine, by an analysis of the hydrological series, whether it is non-stationary and then to assess the influence of the non-stationarity on the water-management solution. It is also necessary to bear in mind any essential intervention in the flow, e.g., the construction of a reservoir or the diversion of part of the flow to another catchment. If we want to prolong the previous “natural” series after the intervention, we must change the measured flow by the influence of intervention. The corrections are introduced in the hydrological data and the corrected data are then used in the same way as the observed data of the same weight. The transfer of the influenced flow series to the natural series is a demanding problem as the size of the influence can usually be quantified only inaccurately and sometimes there are no observations.
58
However, in designing the water-management measures, the future hydrological and discharge conditions, as they will exist during the operation of the designed structures, must be considered. Long-term forecasts should therefore be elaborated as they give a more pertinent hydrological basis than any statistical processing of data measured in the past or present. These forecasts are especially important where human intervention has greatly changed the flow. (c) Influence of the comprehensive utilization of a reservoir o n its performance The respective tasks of a reservoir can be competitive. Table 1.7 shows the matrix of competition of five tasks of a reservoir with four types of competition. The greatest competition is in the required reservoir volume (1) and the operation (4). In the demand for the amount of water (2) the competition is only between the public supply of water and irrigation; in the demand on the quality of water (3) only between the supply of water (drinking water) and recreation. Completely without competition are recreation and flood control. The aims of the design and the operation of a reservoir are to solve all these contradictions to ensure a technico-economico-social optimum. The construction of reservoirs may also have unfavourable consequences: (a) fertile plots, housing estates and buildings become inundated, (b) communication between the two banks is hampered, (c) during fluctuations of the water level in a reservoir mud deposits and banks are eroded with aesthetic and environmental impacts, (d) waves and ice can cause erosion of the banks, Table 1.7 Competition matrix of reservoir purposes
Purpose
1.
(PWS)
2. (HP)
3. (1)
4. (R)
5. (FC)
-
L4
1, 2
334
1
2. hydro-power (HP)
1, 4
-
194
4
1
3. irrigation (I)
1, 2
1.4
394
4
4 -
1
4. recreation (R)
4
1
1
1
1. public water supply
5. flood control (FC)
(PWS)
1 - competition as to the volume demand 2 - competition as to the demand on the amount ofwater 3 - competition as to the demand on the water quality 4 - competition as to the manipulation demands
-
59
(e) the water quality can deteriorate: the impact of great drops in the temperature of the water on recreation and irrigation, and the anaerobic processes in the reservoir, etc. Consequences (a) and (b) are included in the assessment of the capital investment, (c) to (e) can be eliminated by suitable design. The functions of a reservoir should and sometimes even must be combined with other hydraulic structures. The comprehensive utilization of water sources usually gives the most satisfactory results as to the need of water and also as to the best economic effect. A set of various purposes of water-management is called a water-management complex and the respective purposes are its components. Schemes or structures with several purposes are called multi-purpose structures. Water reservoirs are, by their very nature, the most complex structures. The complexity of the structure is justified when it brings an economic effect, i.e., when comprehensive structure is more economical than any single-purpose alternation. In designing a new reservoir it is necessary to see to it (a) that the future aim should not be difficult or impossible to attain; (b) that the present utilization of the river should not be harmed; however, benefits can be eliminated if they are replaced by better ones (VySSi Brod - Lipno, Dixence Grand Dixence, Aswan - Sad el Aali, etc.). The competition of the components of a water-management complex does not make it impossible to reach a comprehensive solution, it only makes it more difficult. According to the character of their need of water, the users can be divided into two groups : (a) users who only use the water without consuming it or deteriorating its quality (hydro-power, navigation, fish farming, recreation, etc.); (b) consumers who either completely or partly consume the water or deteriorate its quality (water supply, irrigation, etc.). The water consumer greatly limits utilization of the source by others. .Ihc user limits the other components by changing the time pattern of the natural discharges. It is therefore more difficult to control the release which requires a larger reservoir volume. If the water source cannot fully satisfy all those who need water or if its use is too expensive it is possible to - bring water from an other source, e.g., the neighbouring catchment; - exclude the least important or otherwise replaceable component of the complex, or limit the respective components according to their significance to attain the total optimum effect of water resource utilization. In the multi-purpose reservoir both the economic effectiveness of the reservoir as a whole and the respective components of the water-management complex must be taken into consideration.
60
To solve this complicated economic problem, the effectiveness has to be proved when this comprehensive alternative is compared with an alternative solution which can be another multi-purpose scheme, a set of single-purpose reservoirs or any other form aimed at the same target. To determine the economic effect of the respective purposes, the investment and operation costs must be divided into individual components of the whole complex. For the analysis of this economic problem many methodological approaches have been used. Often the analysis of the main component of the complex with a prevailing effect can be preferred symplifying thus the analysis. For the further development of the economic evaluation of the multipurpose reservoirs it is necessary : - to use the more comprehensive methodology, - to acquire the necessary technological and economic parameters, - to seek the best possibilities, how to incorporate the intangible aspects in the analysis.
2 BASIC METHODS AND TOOLS FOR THE CALCULATION OF RESERVOIR FUNCTION The methods for the analysis of reservoirs in technological parameters can be divided according to the type of hydrological data available, into those based on - real hydrological data - statistically processed hydrological data; according to the procedure used into - progressive balancing (simulation) in chronological series - general statistical flow characteristics. Solutions expressed in real hydrological data are based on direct measurements during a given observation period. However, it is presumed that the groups of flow observations compiled in the past, also sufficiently characterize the time pattern of the future exploitation of the reservoir. These groups include calculations with natural chronological flow series, as well as those with chronological series including partial corrections meant to eliminate atypical properties of the series, e.g., concerning the beginning of a wet period, etc. - although statistical methods are used for such modifications. Calculations based on statistically processed hydrological data utilize hydrological data processed by probability theory methods and mathematical statistics, to obtain general statistical characteristics or synthetic hydrological pseudochronological series. Progressive balancing (simulation)can be performed in real as well as in synthetic hydrological series. The method is chosen according to the nature of the problem. Most generally valid and applicable for very complicated cases offlow regimes are methods based on modelling time chronological series (synthetic flows). Real hydrological data are used only where they can evidently assure a sufficient reliability of the results. 2.1 RESERVOIRS DESIGNED ACCORDING TO NATURAL CHRONOLOGICAL FLOW SERIES
Analysis of reservoirs according to natural chronological series is the oldest (conventional) method applied to solve any water-management problem. It was highly perfected and is still applied because of its certain inherent advantages.
62 In Fig. 2.la we see the line y = ,f(.x) in a rectangular coordinate system. In Fig. way. The main disadvantage consists in the results being random according to the length and representative character of the hydrological series. It can be eliminated by a suitable choice of the design period, by evaluating how representative the series is, by modifications of the series to eliminate the anomalies and finally by statistical processing of the parameters of the series or of results. The design principle of all water-management reservoirs, based on chronological series, consists in the successive construction of balances between the given inflow into the reservoir and the release (or withdrawal and losses) from it, dependent on or independent of the water level. The balance may be constructed numerically, resulting in a chronological series of water volumes or water levels in the reservoir with the sought characteristic and extreme values, or graphically where the result is given by a chronological curve with the same parameters. 2.1.1 Gruphical methods
The advantage of graphical methods is that - they provide a clear description of the flow control time pattern in the reservoir and thereby form an enormous aid in understanding the phenomena taking place in it, - they are easy to survey - serious or basic mistakes become apparent at first sight, - they permit a solution of problems which can be hardly calculated without computers, - they permit a very prompt solution of many problems. The accuracy of these graphical methods (with a deviation less than 2%) is quite satisfactory. Basic tools of graphical methods Graphical presentation use lines in plane to describe the calculated relationships. They must be arranged in such a way as to provide a quick and clear orientation. Complicated problems are graphically transformed into simpler ones needing only lines in the plane; it is also possible to combine a graphical method with a numerical one. For a graphical presentation we have to choose - a coordinate system, - a scale for the most important quantities. The coordinate system is often rectangular. Such a system is used as long as there is no special reason for abandoning it. If the coordinates reach values rendering their scale-which is limited by the size of the drawing board-inadmissably small, an oblique-angled system is chosen: the coordinates are plotted from the oblique X”)in the direction of the Y-axis which remains vertical (see Fig. 2.1). axis (X’,
63 In Fig. 2.la we see the line y = f ( x ) in a rectangular coordinate system. In Fig. 2.lb we see the same line y’ = f ( x ) plotted on the same scale from the real X’-axis and line y;’ = f ( x )from the real X”-axis. Finally the line y’; is plotted from the same X”-axis, but at twice the scale. It is sufficient to draw only the segments X” shifted vertically by the rounded-off values (the segments) plotted on the y-axis.
3
0 -X
Fig. 2.1 Relationship of graphs in the rectangular and oblique coordinate systems
Figure 2.1 shows how bevelling of the X-axis results in a larger scale for the y quantities. On a rectangular raster paper (mm) the coordinate scale is uniform and not, for instance, logarithmic. Oblique coordinates are used mainly to draw mass curves of long-term-phenomena. For graphic methods the choice of independent scales and derivations of dependent scales is of essential significance because of their great influence on accuracy and clarity. The derived scale follows from the explicit analytical expression for the respective quantity, if on the right-hand side we substitute for all quantities units in the chosen linear scale. If, e.g., the required quantity D is expressed analytically as D[m3] =
A [m3 s-’]. B [s]
C
64 we choose the scales for the given quantities: A - 1 m3 s - l = a [cm] B-ls=b[cm] C (dimensionlessnumber) - 1 = c [cm]
and derive the scale for the requested quantity D: 1 m 3 = d[cm]
=
a [cm] b [cm]
c [cml If the requested quantity is, e.g., the dimensionless number C, the method will be the same :
C=
A [m3 s-']. B [s]
D [m31 When using the same chosen scales for A, B, D, the scale resulting for the required C will be: I = c [cm] =
a [cm] b [cm]
d [cml
Curves used in graphical presentations Graphical methods in water management use curves which can or cannot be expressed analytically (straight lines, curves); all of which can be sub-divided into four groups: - relation curves, - time curves (chronological curves), - exceedance curves, - auxiliary curves. All types of curves can either be given or derived. The most important derived curves are the mass curves. The relation curues express the relationship between two quantities, one of which is expressed by means of the second; the relation is permanent, independent of time.
-
Q
Im'i'l
a)
-
F
b)
[MI
-V
/mil m'] C)
Fig. 2.2 Relation curves ((0 elevation-discharge; (h) elrvation-area; (c) elevation-storage
65
Relation curves are usually applied whenever a parameter which is difficult to measure directly is to be calculated with the aid of a parameter which can be measured more easily (e.g., the discharge by measuring the water stage, Fig. 2.2a). Time curves express the change of the quantity with time. These curves are frequently used in water management; they help to determine the variability of the flow with time (day, year, over-year period), changes in the free water level with time, etc. Parameter X therefore depends on time t, i.e., X = F(t).
Fig. 2.3 Time pattern curves (u) water stage curve and discharge curve; (b)fluctuation
c)
0 t,
72
t2
24
of water level curve in the compensating basin; (c) diagram of power system load
Figure 2.3a shows two time curves which are dependent of one another, i.e., the time course of water states in river H = f(t) and the resulting time course of flow Q = F(t). The two phenomena are joined by the rating curve Q = Y(h).The construction of the points of curve Q from the points of curve H is given for time t, in two ways and is shown by arrows. Figure 2.3b shows the time course of the changes in the water level in the surge chamber on the conduit to the hydro-power plant for the case of a sudden interruption of release. Figure 2 . 3 shows ~ the daily course of the load of the power system P = f(t). Figure 2.3a shows also the time curve = F(t) which is the mass curve to the flow time curve Q = f(t). In a similar way, a time curve may also express the amount of power used, E, in the electric power system by the mass curve to the daily load curve in Fig. 2.3~.
cp
66
The energy consumed in the time interval t , - t , is given by the difference in the coordinates of the mass curve at times t , and I , . For some solutions it is sufficient to know the time interval during which a value of the phenomenon is exceeded; the phenomenon is then characterized by the so-called exceedance curve.
interval duration
0
Fig. 2.4 Derivation of the discharge exceedance curve from the hydrograph (Vltava - hchovice, 1973)
The initial curve is the time curve. In Fig. 2.4 is the chronological curve of flow Q = f(t) for the river Vltava at Stkhovice for the hydrological year 1937. From this curve we derived the exceedance curve Q = f ( ~in)three ways: (a) the flow curve is divided into vertical parts with duration At, these are ranked in order of magnitude, regardless of their chronological order; (b) the flow curve is divided into horizontal parts which have heights carresponding to a flow interval AQ. The lengths of the corresponding time intervals are added, and their total value is horizontally plotted from the coordinate axis; this again gives us the exceedance curve which, however, should rather be denoted according to its structure T = f(Q); (c) the exceedancecurve can also be obtained as a mass curve corresponding to the interval duration curve At = q(Q) derived from the chronological curve: it is divided into intervals AQ as in (b).The values At, i.e., the time of flow within the given interval, are plotted horizontally from the vertical axis as segments with the value AT. The mass curve is drawn from pole 0, which in our case was chosen at the pole distance f = AQ. In this way we reach equal scales for t and T. Thereafter, it is no longer necessary to work with pole 0. A parallel to the diagonal of the respective rectangle (AT, AQ) within the respective interval AQ can then be plotted. If the hydrological data are given numerically, the most convenient way to draw the exceedance curve is to rank the numerical values of the parameter in order of magnitude and plot them directly as ordinates of the exceedance curve to the suitably chosen duration scale T.
67
When the character of the task permits it (e.g. in the analysis of the run-of-the-river power plant), we prefer to use in the analysis the exceedance curves rather than the chronological curves, as the pattern of the former is often simpler and the analysis is less complicated. Auxiliary curves are those which cannot be included in any of the three abovementioned groups (e.g., the turbine efficiency curve, secants, etc.) and all auxiliary structures, helping us to reach a final graphical solution.
Fig. 2.5 Derivation of mass curves P, and P, vs. base curve y = f ( x ) -X
Xl
The niass curve. In water-management calculations we often have to determine the area between a curve y = f ( x ) , the x-axis and the ordinates of two points M I and M, on the curve with abscissas x, and x2 (or lying inside the curve y = f(x), the y-axis and the abscissa of points M , and M, with the ordinates y, and y 2 ) Fig. 2.5. The area base (x2 - x , ) is divided into n equal intervals Ax; the areas of the individual parts modified to rectangles APi = yiAx are calculated and then added. The volume of the entire area lying between the vertical ordinates will therefore be given by P, = y , AX +
n
AX + ... + AX =- C Y , A X
(2.5)
i= 1
Similarly, the area between the horizontal lines will be
Ps = X , Ay
n
+ x2Ay + ... + xnAy = iC= x i A y
The limit of the sum for Ax + 0 (or Ay P, = r X*I y d x = jx>(x)dx
1
+ 0)
and n
+
co is the definite integral (2.7)
68
The integration can be performed if the function y = f ( x ) can be expressed analytically. If that is not possible, graphical methods can be applied. To avoid working with two-dimensional planar figures, mass curves are used to reduce the problem to a single-dimensionalone. The mass curve P, = F ( x ) (Fig. 2.5) is obtained by successively plotting the elementary areas y Ax as coordinates in the chosen scale; the finite coordinate P, is the expression of the area x l , x2, M , , M I in the base curve 4’ = f’(x).Analogously we obtain the mass curve Py = F(y).
Fig. 2.6 Construction of mass discharge curve
xu
The dimensions of the elementary areas APx = y Ax or AP,, = x Ay can be determined graphically.The method is shown in Fig. 2.6- using the example of hydrograph Q =.j’(t). The mass hydrograph is the mass curve corresponding to the chronological hydrograph Q = f ( t ) and indicates by its coordinates in m3 the water volume discharged during the chosen period t. Construction: The area of the basic curve Q is divided into At wide vertical strips. Their centre lines Q iare projected to the vertical. The pole o is chosen at a certain distance f’ from the vertical and from there lines are drawn through the finite points of the respective projections Qi. In the strips Ati parallels to the lines are drawn and by doing so the mass curve is obtained.
xu
La
69 The unit value of the pole distance f’ = “1” follows from the scale chosen for Q, t and
c,:
Q : 1 m3s-l = u [cm] t :Is = b [cm] 1 m3 = c [cm]
CQ:
therefore ub f=c
=d[cm]
Figure 2.7 shows the basic curve Q in wet and dry periods. A mass curve is plotted to it from pole o and from pole o’ which is higher, but at the same pole is the axis t, for the curve the real basis is the distance f.The basis for curve axis t‘.
CQ
xQ
zh
Fig. 2.7 Relationship of mass curves drawn from a raised and not raised pole with the same pole distance f
As both mass curves have a common basic curve and as the pole distances are identical, the scale of these two mass curves is also the same. That means that also the finite coordinates r, - r = t i - r’ must be the same. That can be proved with the aid of similar triangles A(o 21) and A(354) as well as A(o‘21) and A(678), or by projecting the same Qkfrom the pole o l r0 2 , o3 at various heights but at the same distance f ;in all these cases the segments of the mass curve A CQkare equal. In order not to have to draw the real time axis of time t‘ to its full length, further
70
Lk
+
auxiliary axes t’ + c, t’ 2c, etc., are drawn plus the same constant value + A = = +c. The connecting line j7 between the beginning and end points of the mass curve is the mass curve for the average flow Q,. If a parallel to is drawn through pole 0,the value Q, is plotted on the vertical axis Q. in relation to the higher pole. As pole 0’ The same is true for the mass curve was placed exactly at the height Q,, the connecting line of the beginning and end points of the mass curve jF is horizontal. The identity of the end point of the mass curve I’ and point t , on the apparent time axis proves the accuracy of the drawing.
cQ
zQo
zQo
Some characteristics of mass curves: 1. The absolute value of the basic quantity determines the slope of the tangent to the mass curve. 2. A break in the mass curve can only appear where there is a discontinuous basic function. 3. A point of inflexion on the mass curve corresponds to a relative maximum or minimum on the basic curve. 4. If the basic quantity is constant then the mass curve is a straight line. 5. At the intersection point of two base curves, the respective mass curves have parallel tangents. 6. If the pole moves on a parallel to the axis of the coordinates then the scale of the mass curves does not change. Only the direction of the real axis of the abscissae, given by the connection of the pole with the origin changes. 7. If the ordinates of the basic curve remain non-negative, the mass curve cannot decrease if the pole lies on the horizontal axis of abscissae. 8. If the value of the basic quantity is zero, the tangent of the mass curve is parallel to the connecting line between pole and origin. This rule is valid for any position of the pole and helps to define the real axis of the abscissae as a connecting line between the pole and the origin of the coordinates (or their parallel lines). 9. The parallel to the connecting line between the beginning and end points of the mass curve, drawn through the pole, gives the average value for the basic quantity on the axis of the ordinates.
Characteristics of a reservoir The characteristics of the reservoir (Fig. 2.8) consist of two lines depicting the shape and size of the topographical configuration created by the bed and slopes of the reservoir, i.e., the depth-area curve and the curve of the reservoir volumes. The depth-area curue F = q ( h ) expresses the areas of water surface corresponding to any water depth in the reservoir or to any elevation. It can be plotted
71 from the contour line plan by determining planimetrically the areas restricted by the chosen contour line in the terrain and on the upstream face of the dam. In shallow reservoirs with intensive discharge and at the end of the backwater, hydrodynamic accumulation is more significant. The depth-area curve is not sufficient in such a case and we have to “characterize” the reservoir by cross-sections.
‘m3
Fig. 2.8 Construction of the elevation-storage curve from the reservoir elevation-area curve
Horizontal sections at the level of the contour lines 6 divide the volume of the reservoir into A H i layers. Their thicknesses correspond to the vertical distance of the contour lines. The volume of one layer is equal to A K = Fi
+ Fi+, A H i = Fi AHi 2
where Fi ... is the mean size of a flooded area between two neighbouring contour lines. The reservoir volume from the bottom to the selected level H , + l is obtained as the sum of the volumes of the individual layers, i.e. V = rFi AHi i
(2.10)
or, if A H i = AH = constant
v = AH ‘pi 1
(2.11)
By adding on the coordinate axes the respective reservoir volumes to the level, we obtain the depth-volume curve V = @(If). Both curves, F = q ( H ) and V = q H ) , are relation curves. The depth-volume curve V = @ ( H )can be plotted as a mass curve to the depth-area curve of flooded areas F = q ( H ) (Fig. 2.8). Pole o is chosen on the prolonged axis
12
of the ordinates at a distance f. This point is then connected with the end points of the mean coordinates Fi.Parallels to these connecting lines are drawn within the boundaries of the respective layers in the reservoir H i + Hi+l. The distance f is that obtained by substituting units for all quantities: f=--Fi AHi - 1 [m’] . 1 [m] = 1 AY 1 [m31
(2.12)
The pole distance does not depend on the AH size, but only on its scale. If F = constant, the volume curve is a straight line; if F changes according to the straight line, i.e., F = tan uH, the volume line, becomes a second-degreeparabola, because for infinitely small distances it follows from relation (2.9) =
(2.13)
d V = F d H = tanuHdH V
=
r F d H = tana
1
H d H = -tanaH’ 2
(2.14)
In impounding reservoirs the inundated area increases with increasing depth, which is why the deviation of the volume curve from the perpendicular increases with increasing depth. The practical result is that in a higher position in the reservoir, a larger volume AV increase corresponds to the same height interval AH. Summation curve The summation curve is the mass curve corresponding to the chronological line of parameter X, on which the added elementary values are represented by the product TAX. The duration of the respective AX interval is expressed by T ; e.g., for the summation hydrograph (2.15)
73 Addition in vertical direction was applied by Jeidik (1936) in his solution of compensation and distribution reservoirs. The same method was used in our derivation of the summation curve corresponding to the daily load curve when seeking the part of the load diagram which can be covered by a pumped-storage power plant. Figure 2.9 shows the construction of a summation curve using an example from hydro-power engineering. The question is, which part of the output can be covered by a pumped storage hydro-power plant, with a power equivalent of E , = 1 mil. kWh of its accumulation reservoir, in the daily load diagram. As the output of the pumped storage hydro-power plant is placed at the peak of the diagram, we have to separate the highest part of the load diagram, whose area represents 1 mil. kWh. To avoid handling the areas in the basic curve, a mass curve permitting the addition of the strips in vertical direction-i.e. the summation curve-is plotted. Construction of the summation curve. The load diagram P = f ( t ) is divided into horizontal strips of a constant width AP, the exceedance curve T = f(P)is plotted and divided into similar strips, then the summation curve is drawn from pole o at the distance f.From similar triangles A(oOT,) and A(123) it follows that
f=,,
T
AP
(2.16)
According to the chosen scale: f= 1 =
1 [h]. 1 [kW] - a.b 1 [kWh] C
Parallels to the rays from the pole trace on the strip boundaries the points through which the continuous curve E = f(P)is drawn. Its coordinates E indicate the energy for the load diagram from zero to any optional output value P. The maximum coordinate Etota,is at the level P,,. If we subtract from the terminal point 4 of the curve E = f(P),the power equivalent of the reservoir E , = 1 mil. kWh, and if we draw a vertical, this line intersects the summation curve at point 5, indicating the lower limit of the peak output which can be covered by the pumped-storage hydro-power plant. The diagram shows that the pumped-storage power plant would have to have a capacity of 160 MW, while 340 MW would be left for the other power plants in the system. The energy delivered by the pumped-storage power plant is indicated by the hatched section in this load diagram. Summation curves can be said to have the same properties as other mass curves. Duration T = constant means that a straight line represents the summation curve. The output P from 0 to Pminlasts a full 24 hours and therefore its summation curve is a straight line E , parallel to the respective pole ray. It is easy to prove that the full load curve E , intersects the vertical leading through the final point 4 of curve E = f(P) at the mean output Pmedium.
74
The horizontal distance between curves E and E , determines the energy deficit in the given load diagram P needed for a full-load capacity of 24 hours. It is obvious this deficit equals the energy in the diagram above Pmedium, that at the height Pmedium which also defines the mean capacity. One more question: “What should the minimum capacity of the other power plants in the system be if they are also to supply the energy needed for pumping the water into the accumulation reservoir of the pumped-storage power plant for a total efficiency of qc = 70%”’ The energy demand for the pumping would amount to approximately 1.43 mil. kWh; the minimum capacity of the other power plants will be on the level at which the horizontal distance between the curves E and E, corresponds to 1.43 mil. kWh, i.e., about 300 MW. The energy needed for pumping E,, is the hatched part of the load diagram. The properties of the summation curve in the load diagram are: 1. It indicates the respective energy for an optional output P or for the capacity scope P, to P2. 2. The summation curve for full-load E, indicates on the finite vertical line the mean output Pmedi,m. A prolongation of the E, curve to P,,, indicates the energy to which a maximum load during 24 hours would amount, e.g., Eotot,,= 24P,,,. Analytical presentation of t h e summation curve The continuous shape of the summation curve induced several authors to present it analytically. Comparison of the load diagram of the Czechoslovak Power System on the day of the annual maximum in 1950, with the calculations performed according
to Mostkov, showed a very good coincidence. The maximum deviations amounted to 0.5-1% of the total daily energy consumption and to 2.5-6% of the peak energy (Votruba and Tvarfiiek, 1960). 2.1.2 Mathematical methods
Even before computers were discovered, predetermined simple problems were solved by mathematical methods. After the introduction of computers, mathematical methods started to be applied more than graphical methods, even for complicated analytical problems for which the graphical methods had been particularly suitable. The calculation of the function of a reservoir can be subdivided into analytical and successively balancing methods. Analytical methods cannot be applied frequently since the functional relations are rarely able to express the basic laws of the reservoir function and its characteristics. An exact description is usually only possible in case of a simple, fully controlled inflow and withdrawal and only for reservoirs with a simple geometric shape. The successively balancing solution (simulation) is usually presented in the form of
75 Table 2.1 Tabular calculations of reservoir storage function Month (Year 1932)
I
XI XI1 I I1 Ill IV V VI VII VIll IX X 1 month
Inflow
Q
Withdrawal 0,
Storage Filling and emptying (Q - 0,). of reservoir (Q - Op) 1 month C(Q - Op).1 month
[m-l s-'1
[m-l s-'1
[m3 s-'1
[mil. m']
2
3
4
5
0.90 1.13 0.80 1.08 2.00 2.48 1.69 0.79 0.51 0.40 0.58 3.02
1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10
-0.20 +0.03 -0.30 -0.02 +0.90 +1.38 +0.59 -0.31 -0.59
-0.52 +0.08 -0.78 -0.05 +2.33 +3.58 1.53 -0.80 -1.53 - 1.82
- number
-0.70 -0.52
+ 1.92
of seconds per month.
+
[mil. m3] 7
6
Water volume in reservoir [mil. m'] at end in middle of month of month 8
9
1.27,0.75 0.83 0.05 0
1.01 0.79 0.44 0.02 1.17 4.10 5.50 5.10 3.93 2.26 0.68 2.49
2.33i ;;:
0
5.91 7.44
4.70
-1.35
3.17 1.35 0
+4.97
4.97
(average 2.63)
a table. For simple problems, calculators are quite satisfying, while complicated, extensive questions require computers. For an identical task, a graphical and a mathematical solution require a similar mental process. Let us take the hydrological data entered in Table 2.1. and plotted
2
n
Fig. 2.10 Graphical solution of reservoir storage capacity V, for a given yield 0,and vice-versa.
76 in Fig. 2.10. Let us also presume that the discharge is controlled for a constant dependable withdrawal (safe yield) 0, in a given year. and The graphical solution is then performed with the mass curves Laand for better orientation, base curves Q and 0, are also introduced. The example can be plotted in two ways: (a) For agiven 0, we have to find the corresponding volume V , for the active storage capacity of the reservoir. From the end of the 5th month the mass curve of the safe discharge is drawn, starting on the curve At the end of each month, the vertical distance between the two mass curves indicates the partial withdrawal from the active storage capacity, which we presume to be full at the beginning of the low-flow period. At the end of the 9th month the distance between the two curves is the longest; this vertical distance indicates, on the scale of the mass curves, the active storage capacity necessary to assure the constant withdrawal 0, under the given hydrological conditions. The real point of intersection C of the mass curves and (going through point b parallel to proves that the spring period yields enough water for the reservoir to fill up. Similar considerations are shown in the mathematical solution in Table 2.1. We presume that the reservoir is full at the end of the 5th month and empty at the end of the 9th month. The calculation of the necessary V, is best performed by summarizing, backwards, the multiples (Q - o,,)* number of seconds per month (for brevity “1 month”, etc., is used) in the 5th column from the end of the low-flow period. These sums are then entered in column 6 ; they indicate the content of the reservoir necessary at the turn of the month to assure that it is empty at the end of the low-flow period at a given 0,. The highest value in column 6 shows the necessary volume of the active storage V,. A control filling of the reservoir in the spring period is performed by adding the values in column 5 from the beginning of the high-water period; the results are entered in column 7. The reservoir is full if at the moment at which it starts to discharge the value in column 7 is higher than, or at least equal to, the value in column 6 . The graphical and mathematical solutions for an over-year control are similar. In the mathematical determination of V, we again have to calculate successively backwards from the end of the most unfavourable grouping of low-flow years. (b) For a given V, we have to find the achievable, dependable constant withdrawal 0,; at the end of the low-flow period (Q is no longer smaller than the 0,sought) we plot V, vertically from point b and from the finite point b‘ we draw the tangent back to the curve. In this way we get the mass curve of the dependable withdrawal Copand the value 0, on the parallel pole ray. The control filling of the reservoir in the previous high-water period is performed as in (a). The mathematical solution of a task formulated in this manner is more difficult than the graphical one, especially in a complicated hydrological series in which the
co,,
cop
cu.
cu zop
&
cop)
77 periods with excess and shortage of water in the stream are not clearly separated. Here we have to use a trial and error method in which we mark on the values taken from Table 2.1 for a given V, = 5.5 * lo6 m3. To get an idea of the yield that such an active storage capacity could offer, we transferred it to the value of a single month of yield 0,'= 5.5 * lo6 m3 : 1 month = = 2.12 m3 s - '. The values in column 2 indicate that a yield is achieved between the 6th to 9th months. Under more complicated hydrological conditions we would not be able to determine this so definitely; we therefore have to estimate and verify the estimation afterwards. For a dependable withdrawal we can use the natural flow in those months and the water in the storage reservoir, which we presume to be full at the beginning of the 6th month - i.e., a total volume of V = (0.79
+ 0.51 + 0.40 + 0.58 + 2.12) - 1 month
The dependable withdrawal 0, results from dividing this water volume evenly into 4 months (6th to 9th):
0,=
V
4months
-
4.40 4
-
1.10~3~-~
The proof of having estimated the shortage period correctly is the statement that before and after this period there are months (pentads, decades, years) with a greater flow than the calculated 0,. If that were not the case, then we would also include the month with the flow Q < 0, in the shortage period. This is the answer to the main part of the question. If we wish to observe the changes in the reservoir water-stage, we have to fill in column 4,5,6 and 7 as in case a), but with the difference that in column 6 we need not work backwards from the end of the low-flow period, but start directly from its beginning when the reservoir is full. Since estimation of the low-flow period is difficult, the task is often solved in the following way: several values of 0, are chosen, then we find the respective V, and interpolation determines which 0, belongs to the given V,. All these mathematical and graphical methods have a serious disadvantage - they do not include the water losses. These are usually a function of the level in the reservoir. We therefore add more columns to our table. 9 - Water volume in the reservoir in the middle of the month ( lo6 m3). 10 - The respective. elevations of the mid-month level in the reservoir (m above sea-level). 11 - The respective inundated areas - mid-month (hectares). 12 - Water losses related to these parameters in the middle of the respective months which are applied to the entire months (m3s-'). 13 - Safe net withdrawals (m3s - ').
78
With the method described above we do not arrive at the required constant safe withdrawal 0, = 1.1 m3 s - l , but rather the withdrawal minus the losses and, since the losses are variable, the withdrawal is not constant. If we wish to keep to the required constant net withdrawal O,,, we have to determine the course of the gross regulated withdrawal resulting after the substraction of the water losses in the required Opn. evoporwfion [mm per month J
H-
Fig. 2.1 1 Analysis of a storage reservoir including evaporation losscs
Jeidik (1946) introduced the consideration of water losses by evaporation into the calculation of storage reservoirs. Figure 2.1 1 presents the graphical solution of the storage volume for reservoir V, in mass curves with raised poles; the loss per time interval is derived from the level at the beginning of the interval, giving a slightly safer result than the reality. The basis for plotting this diagram is the reservoir volume curve V and the time behaviour of the inflow P , and of the release 0, and are drawn from the pole at the height 0,. evaporation H,. The mass curves and Without considering the losses by evaporation, the size of the storage volume is V,. The loss by evaporation is introduced into the diagram in the following way: On the gepth-volume curve we subtract the evaporation depth HE = 140 mm from the water stage on June 1st to find directly on the Vcurve the evaporation loss Z:, which we subtract on the mass curve from the total inflow at the end of June. In this way we obtain the amount of water which can be used for release from the June inflow. From the water stage on July 1st we again subtract the evaporation depth H i = = 160 mm, and so on. If we wish to maintain the net release 0, = 0,, the reservoir is emptied mo!e than if the evaporation were not considered. The difference is exactly the volume of the evaporated water. We have to refill not only the release 0, from the storage volume, but also the evaporation loss Z,. The reservoir volume must then necessarily amount to V,, > V,. Evaporation can be introduced into the calculation in a simpler manner if it is not an important factor in the balance.
cp co
cp
79 The above-described table and graphical solution were based on the principle that a certain V , has to be matched with the largest possible safe withdrawal 0, that is able to guarantee it during the given period, or to match a given 0, with the smallest possible active storage capacity V, resulting in a 100% reliability for that period. As the hydrological series is limited, the real reliability is p < 100%. Using the tabular form it is possible to solve the tasks with reliability of p < 100)w We achieve this by eliminating one or several years (periods) with the lowest flow. In practice we often analyse in the given case the function
fP,, o,, P ) = 0
(2.17)
or in relative values (2.18)
f@,, a, P) = 0
Suitable for such analyses are the “relative” storage-yield curves p, = f(a) or a statistical processing of the results of calculation in hydrological series (see Chapter 6).
P
I
Fig. 2.12 Storage-yield curves = /(O,,)
= / ( x ) or
/{,
=
Fig. 2.13 The principle of the determination of the storage-yield curves for various probabilities p (a) diagram of values; (6) decisive low-flow period
The curves p, = f ( a ) for a given hydrological series can, e.g., be calculated or plotted using graphical analysis for chosen values a l , a2,... by determining the respective values p,,, fizz,... in the individual low-flow periods (Fig. 2.12). With their aid we determine which period is decisive for the chosen a and which p, value belongs to it (according to Fig. 2.12 the 2nd period is decisive for a c a, and the first period for a > ax). In the given hydrological series the reliability p = 100% corresponds to the upper envelope of all storage-yield relation curves. The lower envelope, plotted from a larger number of relation curves, belongs to the reliability p < 100%. To arrive at reliable value for the reliability p , especially for a high p -, 0.99, we need a statistical processing of the results or computation using synthetic hydrological series.
80 The program can be compiled according to the instructions given in Fig. 2.12. For a chosen aiwe get n fi!, fit,..., fiy values (Fig. 2.13a). For the requisite p = 0.99 we have to have n 5 100, or even better a multiple of n. With n = 100, the storage-yield relation curve for p = 0.99 consists of the points /31- ', fi;- ..., K- belonging to /3;-', ...,fi;-;, the values ai ( i = 1,2, ..., m);for p = 0.98 these will be the points /3;-,' etc. When determing the values f i j ( j = 1,2, ..., n) care must be taken to determine correctly the low-flow period in which the respective /3j appears (Fig. 2.13b).
',
2.2 PROBABILITY AND STATISTICAL METHODS
The application of probability theory and of mathematical statistics to the solution of problems relating to flow control by reservoirs is justified by two conditions: (a) that the problems to be solved deal with processes having random (probability, stochastic) character, (b) that the historical observed series is only one of the infinite number of possible realizations of the random process which will never be repeated in an identical form in the future. A random phenomenon is the basic concept of the probability theory, its occurrence cannot be forecast exactly in each single case, even if a certain complex of conditions is preserved. Random can be a quantity (variable) or an entire process. A random variable is characterized by the set of all its possible values and by its probability distribution; a random process is characterized by the set of all its possible time functions and the laws determining the statistical properties of this set. In water management there are many quantities and processes with a random character and for that reason the probability theory and methods of mathematical statistics can be applied to them. They can be used for calculating the release control of reservoirs, if at least some of the parameters and phenomena have a stochastic character. The final function of a reservoir follows from the relationship between the time behaviour of water inflow and outflow. The release is determined by the water need which may either be planned or may show a random course. The natural inflow into the reservoir includes some random elements, such as the average annual flow (runoff from the watershed), the synchronous runoff from the watershed for certain periods in the respective years (months, etc.),the maximum or minimum annual flows, the beginning of the higher spring flow, the duration of the ice-cover on the river or reservoir, etc. All these elements can appear in any year to a certain extent. Other elements of the natural runoff from the watershed have an essentially cyclic character, with only some random elements. This concerns mainly the annual course of the flow, with the characteristic cycle repeated every year (again within certain limits and with varying regularity according to the geographical region).
81 It is therefore obvious that the methods of mathematical statistics and probability theory can be applied more freely to elements with an annual interval, or for flood waves, than for changes in the flow and other hydrological parameters within the year where the sequence of the members and their arbitrary values cannot be freely interchanged. Long-term fluctuations in hydrological phenomena were first of all considered to be completely random, i.e., without any statistical relationships between the individual years. It was only in 1936 that Yefimovich published a thesis that the correlation between runoff of neighbouring years should be taken into consideration and in 1959 Kritsky and Menkel proposed a method, and Gugli constructed nomograms, for the mathematical solution of over-year flow control including these correlations. The designs of impounding reservoirs indicated that even with flow fluctuations lasting many years a trend towards grouping of years with high- or low-flow periods could be observed. The genetic concept of the hydrological phenomenon leads to the study of the reason for its cyclic nature based on the reasons for the cyclic character of its genetic elements (i.e., runoff in terms of precipitation, air temperature, etc.). Empirically we seek a coincidence between hydrological phenomena and a regularly repeated solar activity pattern. Such studies are not yet satisfactorilycompleted, but even the present results are encouraging (see Section 3.2). The runoff from a watershed is obviously influenced by so far unknown factors of a random nature, which are the reason why at present its long-term variability is considered to be more or less pseudocyclic. For their mathematical expression of the relationships appearing as a result of random influenceson the runoff and influences of inertia in the state of the watershed Kritsky and Menkel (1946) used processes (series, chains) developed by Markov (1951), which represent the generalization of a set of independent observations. The authors were of the opinion that the application of the simple Markov series would agree sufficiently well with the present knowledge about the correlation between the fluctuations in the solar activity pattern and those of the runoff. 2.2.1 Markov processes
Let us consider a series of observations. In each of them there might occur one, and only one, of k non-interchangeablecases A',"),A:', ...,A:) (the superscript marking the observation number). The series of observations creates a simple Markov series, if the probability of the appearance of case A:+') ( i = 1,2, ...,k) in the (s + 1)th observation (s = 1,2,...) depends only on which case occurred in the sth observation and is not changed by the knowledge as to which data on cases occurred in the earlier observations or which appeared later on. We could also speak about the physical system S which at each moment can be in one of the states A,, A,, ..., A, and changes its state only in the intervals t,, t,, ..., t,, ..., while the probability of
82
transition into an arbitrary state Ai (i = 1,2, ...,k) at the moment 7 (t, < 7 < t,, 1) depends only on the state the system was in at the moment t (ts- < f < t,) and does not change by reason of the fact that its states at earlier moments are known. The Markov process, therefore, is a special case of a random process. If discrete time is used (as is usual in water-management reservoir design), it is called a Markou chain. By only taking into consideration the relationships between two neighbouring states, we arrive at the Markou process of the first-order, also called the simple Markou process. If we consider n relationships in the process we obtain the Markov process of the nth order or the composite Markou process (or chain in the case of discrete time). The Markov series permit a quantitative comparison of the studied relationships and may be applied to water management problems. Most frequently we use only the simple Markov series, presuming a correlation between the neighbouring members of a series. The transition from the series of independent random quantities xl, xq, ..., xi, ...
to the Markov standardized series (with the same indices) y,, y2, *.*, Y ,
a * *
follows from the presumption that the distribution of the probability of an arbitrary member in the ith place in the series is correlated to the value of the previous (i - 1)th member. If the correlation coefficient is r, the mean value of the ith member of the Markov series is
Li = ryi-,
(2.19)
and its standard deviation
(2.20) In the case of a normal distribution ( X i = 0, cxi= 1) we have yi - yi - xi - xi --= xi ‘Yi
=Xi
and therefore yi = yi
+ xpYi = ryi- + xi J i - 7
(2.21)
According to the given series of random quantities, we can calculate from equation (2.21) the individual members in the simple Markov series connected by the correlation coefficient of the neighbouring members r. Let us consider the series of observed hydrological parameters as a simple Markov series with the correlation coefficient of the neighbouring members r = r l r the cor-
83 relation coefficient of members with distance two is r2 = r: and that of members with distance three is r 3 = r;, etc. This correlation (auto-correlation)function therefore has the power form
r
=
(2.22)
ri
where T is the distance (time lag) in the sequence of correlated elements. It has a rapidly decreasing course, but with a positive r l it cannot pass to negative values (Fig. 2.14).
Fig. 2.14 Correlation power function rr = r;
-r
We can then determine the parameters of the series by means of the expressions mentioned later (derivation: Kritsky and Menkel, 1946). S t a n d a r d d e v i a t i o n of t h e series We investigated a series consisting of a large number ( N )of samples from a Markov process, the size of each being equal to n. 02 3
o=
1-
(2.23)
2r n(n - 1)(1 - r)
in which S2 is the mean value of the sample variances of all N samples. M e a n s q u a r e e r r o r of t h e s a m p l e m e a n
an
A section chosen at random from a simple Markov series including n members is
examined
o"=J;lJ+----y--
I-r)
where for a we can substitute the expression from equation (2.23).
(2.24)
84
Correlation coefficient r between neighbouring members We investigate the relationship between the sample correlation coefficient rb and the population value of the correlation coeficient r. r2 ri =
+ n(n - 1)2r(1 - r ) ( n -
z)
2 1-r" I + n(n - 1 ) ( 1 - r ) ( n - 5)
(2.25)
Arbitrarily chosen sections can be studied as a part of the simple Markov series, by calculating first, on the basis of the equation (2.25), the correlation coefficient r and then determining the standard deviation 0 according to equation (2.23) and the mean square error of the average sample mean 6, according to equation (2.24).
I
I
tk
I
I
Fig. 2.15 Scheme of transition probability
fk+,
The Markov processes are also studied with the aid of a transition probability which is the conditional probability p i j that from a certain state Xi the process passes at the following moment into' another state Xj.If at any moment the process has a finite number of n states X , , X , , ..., X,,then n is the probability p i j that the state X i ( i = 1,2, ...,n) passes to states X,,X,,...,X,,or j = 1,2, ...,n. The number of transition probabilities from each state at the moment t k into each state at the moment tk+ is n2 (Fig. 2.15). The transition probabilities are recorded clearly in the transition matrix which naturally in this case is of the square type (n * n):
,
(2.26)
85
The sum of the values in a row of the matrix is
n
p i j = 1, since it expresses the
j= 1
probability of the transition from state X i to all n states. Each column of the matrix shows the probability of the transition from all n states into the state Xj;for this reason, in general their sums vary. 2.2.2 Application of probubility and statistical methods in water management
The basic hydrological material for the solution of water-management problems in reservoirs is the result of measurements performed in situ; but even long hydrological series do not usually include extreme values, which is why the observed hydrological material is corrected by the probability theory and mathematical statistics. Their first application in water-management calculations is in the processing of basic data (see Chapter 3). Elements with one or more common symbols are collected into populations and various characteristic values are determined for them, such as arithmetic mean X, extreme values x, and xmin,variation ranges x, - xmin, the mean and the standard deviation, the frequency of occurrence, the variation coefficient C,, the skewness coefficient C,, the theoretical exceedance probabilities curve, the correlation between two or more statistical sets, etc. The difference between the empirical measured values and their statistical processing is quite obvious if we compare the real and the theoretical exceedance probability curves for the flows. The theoretical curves not only give a more reliable guarantee, especially for extreme flows, but the statistical methods also permit a generalization of the results and their application to other hydrological conditions on other rivers. Thus we have a tool with which to solve the water-management problems of rivers with only few measured data. Statistical populations include members with one or more common symbols; but their values are random. Such an indicator of the annual runoff is, e.g., the total annual runoff (without any relationship to the runoff in the neighbouring years), the maximum annual flow, floods of various magnitude, etc. Mathematical statistics so far used in hydrology have mostly been applied to annual runoff and flood waves. In order to be able to solve questions of irrigation and drainage, hydrological parameters must be found (precipitations,runoff, temperatures, etc.) in the vegetation period. For hydro-power problems we need to know the flow in the winter months, for pollution prevention of the rivers we need to know the flow at the time of the maximum load of waste water, etc. These seasonal discharges of the respective years also form statistical populations. With these we can work in the same way as with annual flow. The common indicator for all set members (i.e., of the runoff, precipitation, etc.) is the value at a certain period of the year. The second field in which probability and statistical methods are used in water management, is for calculation of the flow control of reservoirs. The rules for inflow
86
and withdrawal from the reservoir are generally independent of one an other; only in some cases can a certain relationship exist between them. Otherwise, withdrawals are determined by the need for water, while inflow into the reservoir is mainly determined by natural laws and they can be processed by statistical methods. As the character of random quantities is especially evident in annual hydrological parameters, statistical methods were used for the first time to determine the over-year components in reservoir volume. The statistical set was used to compile yearly flows (Hazen, 1914). It is logical that the main factor for the determination of reservoir volumes on the basis of statistically processed hydrological data is the over-year flow control. An annual flow control each year furnishes a complete closed reservoir cycle filling and withdrawal. This means that for each of the n years in a chronological flow series we have to calculate the dependable withdrawal 0, for a given storage volume V,, or vice-versa, to get n results. If the series has several dozen members, these create the same number of examined and mutually independent dependable withdrawals (or necessary storage volumes), a large enough set to be processed and to supply results which can be generalized. If the outflow is controlled over several years, the number of groups of years decisive for the evaluation of the results is much lower than the total number of years and therefore the number of independent results is also much smaller. To deduce general conclusions from these results is therefore also more difficult; even more so, if the coefficient of the safe yield a = O,/Q, approaches unity or the longer the period of the flow control is. It is obvious that a prolongation of the control cycle means a reduction in the reliability of the hydrological series regarding their ability to reflect the regime to be expected in the future operation of the reservoir. Therefore, it becomes necessary to compile a synthetically larger number of flow combinations than that offered by the observed hydrological series and to solve with them the problems arising from over-year cycles. The tools for this are supplied by probability theory and by mathematical statistics. The reservoirs completed during the last twenty years have often volumes for a flow control over several years. Probability methods were necessary in their design. Since one chronological series is more specific than the character of the parameters derived from the same number of values, regardless of their chronological sequence, the method based on the probability distribution of the basic hydrological data may offer a more reliable result than the method using one real chronological arrangement of these data. Judging the results of the solution from the point of view of the extfnt to which the water supply is guaranteed, it remains impossible to say in advance (if there is no analogy with other longer series) that the shorter period has a lower dependable yield (in %) than the longer one. If the observations happen to be performed in unfavourable years, an identical dependable yield would mean lower withdrawals
87
than those based on a longer series which would include years with more favourable flows than in the unfavourable period considered above. If, however, the observations happen to take place in a period with a favourable flow, the same dependable yield results in withdrawals that are higher than in the longer series. Short hydrological series might therefore furnish results that deviate considerably from the results of the longer series. The deviation may be in either direction. This is the reason why we have to evaluate the representative character of the series; for instance, its analogy with another longer series. The results gained from a chronological series will be more susceptible to the incorrect selection of hydrological data than results obtained by probability methods. Naturally, not even probability methods can furnish correct results, if the data are incorrect or unverified. The relationships between direct methods in observed chronological hydrological series and indirect methods based on general hydrological statistical characteristics, can be defined in the following way: 1. On the basis of the present knowledge of the respective methods, it can be recommended that mainly indirect methods should be used. Direct methods can only be applied for solving the function of reservoirs with a low coefficient of safe yield and a lower reliability required, e.g., mainly with a short-term or annual flow control, for the following reasons: (a) For the usual length of hydrological series (30 to 40 years) control with a short cycle furnishes such a large number of mutually independent results that (after probability evaluation) the reliability of the resulting parameters can be presumed. (b) For an annual or low over-year flow control, the entire storage volume of the reservoir, or its decisive part, is created by the seasonal part of the reservoir volume resulting from the annual variability cyclic flow. 2. Solutions in chronological series are clear and easy to survey, which is of advantage for the explanation of the release control by reservoirs and for the solution of complicated water-management problems. However, the reliability of such a solution always has to be assessed. 3. The flow characteristics as measured by hydrometry without processing should not always be accepted as a basis for water-management solutions. Time series characteristics should be assessed more accurately for values in a hydrological series which are not sufficiently realiable - especially the extreme values. For short available periods, especially, they should be processed in order to eliminate random irregularities; such assessments use statistical methods. The measured time series characteristics make it possible to use genetic methods of deterministic hydrological research for analyses of the outflow conditions (important, e.g., for outflow forecasts). 4. The question of safe yields in flow control by reservoirs cannot be solved without probability methods. 5. When using general statistical characteristics, the entire storage volume is
divided into the over-year component and the seasonal component; the seasonal component is determined by dividing up the outflow during the year. 6. General solutions based on probability methods using statistical Characteristics have the advantage that the relationship between the parameter values of the reservoir and its regime can also be applied to other rivers. They are therefore useful in regions where there is a lack of hydrological data. 7. General methods make it simpler and easier to solve flow regime problems, as the calculation of statistical characteristics is easier than finding solutions by simulation. 8. Synthetic hydrological series combine the advantages of the balancing method in chronological series (simulation) and of the probability theory (see Chapter 3). 9. Probability methods are indispensable for the solution of problems of watermanagement systems with reservoirs.
B STORAGE FUNCTION OF RESERVOIRS
The aim of the storage function of reservoirs is to assure a better supply of water for users and consumers than withdrawal from natural river flow can provide. The design of a reservoir therefore depends mainly on the needs to be me1 and on the water sources available. A comparison of the water demand with water resources available (e.g., the quantity and quality of the water) forms part of the Czechoslovak Water Management Plan. The individual steps in the water supply balance are shown in Fig. 3.1.
s2 qwlitutive
m t i o n ond liquidofion o f pOrlutibn
(waste- wokr, . nn-point pdlutron
Fig. 3. I Diagram of sub-division and contents of water-management balance
Descriptions of reservoirs construction of which started in 1975, and those which are planned up to the year 2000 and even later, were prepared for the Water Management Plan of the Czech Socialist Republic (1975) with a layout for each at a scale of 1 : 50 OOO to outline the purpose, and to present the technical and economic parameters of the reservoirs. This plan serves as the basis for any future design. From the very beginning it must be clear whether the reservoir will be (Fig. 3.2) (a) - a single reservoir (b)- a single lateral reservoir (c) - a single compensation reservoir (d)- part of a cascade (e) - part of a system
90
The next step is to determine the water yield and the demand, i.e., the gross balance of the water supply: whether the required annual withdrawal EO, will be covered by the water hource with the required reliability of EO,,,or at least the mean long-term A minimum permissible maintained discharge downstream of the dam, flow due to water losses, etc., must also be taken into consideration.
CQa.
Fig. 3.2 Schematic representation of reservoir positions
c Baloncinq plan
I
suitoble Fig. 3.3 Flow chart of basic considerations for the design of a storage reservoir
Finally, the realistic localization of the necessary reservoir volume I/ has also to be evaluated with a view to the morphological and geological possibilities, Vmorf,of the dam site and the valley. The scheme of such a diagram of considerations and their results is shown in Fig. 3.3. If Vmorf> V, we have to evaluate whether the suitable locality could not be utilized to satisfy further needs of water management, apart from the main purpose to be satisfied by the investment task. The solution of the reservoir consists in the preparation of the basic data and plans and the actual technico-economic plan.
3 BASIC DATA FOR CALCULATING THE STORAGE FUNCTION OF RESERVOIRS The basic data for the calculation of the storage function of reservoirs consist of
- hydrological records - the needs of users and consumers.
3.1 HYDROLOGICAL RECORDS AND THEIR PROCESSING
The basic hydrological records for the mathematical solution of the reservoir function are the flow series of the stream cross-sections. To what extent and how these hydrological records are processed depends on - the aim and significance of the reservoir, - the stage of the design, - the character and range of the hydrological data. 3.1.1 Selection and evaluation of hydrological series Longer hydrological series are usually a more reliable basis than shorter ones. However, the so-called representative character of the series is important. A series represents the hydrological conditions in the watershed well if it includes data leading to reliable results in the solution of flow-regime problems. Other observations (outflow from adjacent watersheds, precipitation, etc.) can increase the representative value of a given shorter series within the framework of a long series, if - they cover a long period - they include the years of the studied short-term hydrological series - their relationship to the data in the hydrological series is close enough. If these conditions are fulfilled, the given hydrological series can be prolonged. Sufficiently reliable parameters can be expected in the prolonged series, if the series in question is not shorter than 15 years and if the correlation coefficient for the synchronous values of the series and their analogues does not amount to less than 0.90. A period of about 20 years is considered to be long enough for the calculation of seasonal control. For over-year flow periods we need at least 30 years of data which naturally have to be assessed as to their representativeness and suitability.
92 In Czechoslovakia better results arr usua' \ achieved by hydrological analogies of the discharges from two analogous watersheds, since Czechoslovakia hw . dense network of watergauges with comparatively long series: the relationship betxeen precipitation snd discharge will usually be less close.
Reservoirs on streams usually have more complete and reliable hydrological records than those on small rivers. The disadvantage of real years is that they introduce a random element into the results of the calculations. Statistical methods are used to adjust the measured data. The adjusted measured data, especially for the decisive years or periods, result in hypothetical hydrological records which are more reliable than the measured ones.
Fig. 3.4 Hydrograph of mean monthly discharges of the river Labe at Dtfin
In the past, a hypothetical year was considered to be favourable basis for calculating reservoirs with annual flow control. It was worked out by selecting in the hydrological series of the flow, in individual decades or months, those with the same required probability of exceedance. This is how a hypothetical year is created, and it usually has a smoother pattern than any real year. Its demands regarding the reservoir volume are lower than for a real year with the same exceedance probability; it cannot therefore be recommended as a basis for the calculation of reservoir volumes.
93 More promising is the method of hydrological phase characteristics (Potapov, 1951) which is based on the assumption that a number of characteristic phases (periods) of the annual discharge regime, repeated every year and differing only in intensity or in the date of their beginning or end, can be found for each type of river. The flow conditions of Middle European rivers are much more complicated and heterogeneous (as shown in Fig. 3.4, depicting the average monthly Q , discharges of the river Labe in DtfEin for the decade 1931-1940). The sum of the monthly discharges for the identical months of the entire decade (Q, curve for 1931-1940) has a higher discharge in March and April and a minimum discharge in July and August, as well as higher flows in autumn and winter. In winter the flow is not very low, as even in winter there is rainfall and ice melts in the catchment area of the Labe. Comparing the flow in the respective years to this cumulative curve for the whole decade, a good similarity can only be seen for the years 1931, 1935, 1937 and 1940; in the other years the flow is distributed differently over the year. The spring flow from snow water is not always very high (in 1936, 1938); the flow is also strongly influenced by summer rainfall (in 1932, 1936). A good example is the comparison of the two years with the highest flow in the entire 100-year period, i.e., 1941 (Fig. 3.4b) and 1926 (Fig. 3.4~).In 1941, the highest flow period occurred in March and April, while in 1926 it occurred in the summer months of June and July. This unsteady nature of the flow has to be taken into account when designing reservoirs in such hydrological conditions, and only after careful deliberation can methods which proved successful on rivers with a simpler flow regime be used. For the design-but mainly for the operation of these reservoirs-it is important to know that one cannot rely on regular filling with the spring floods; these often have values of a long-term average Q, or even less (1932,1933,1934,1936). Often, not 'only the annual discharge regime, but also the water demand (withdrawal from the reservoirs)changes considerably during the year. Water is withdrawn in the vegetation period for irrigation purposes and in the winter mainly for power production. The relationship between the flow in the rivers and the withdrawal from the reservoirs-their size and their timing (see Fig. 1.19b)-always has to be taken into consideration. The variability of the beginning and end of the individual flow periods (phases) and the flow in the individual periods must be analysed. A hydrograph adjusted according to such analyses does not lead to an unrealistic equalization of the flow, but preserves the character of the annual discharge regime. A design period of several years is also selected according to the analysis of the entire series. The chosen period should: 1. have a mean discharge close to the longterm average, 2. include years with various stream flows in characteristic groups, 3. have a variation coefficient of the annual discharges close to that for the entire series.
94
The fulfillment of these conditions is obvious from the mass curve of the deviation from the mean of the module coefficients (ki - 1) and the mass curve of the values (ki - 1)2 where Qr,i k. = -
’
Qa
Qr,iis the mean annual flow in an optional ith year, Q , is the long-term mean annual flow. The sum of all n values (ki - 1) in an n-year long hydrological series is zero. The mass curve c(ki- 1) = f(t) plotted in right angle coordinates therefore terminates at the end of the series in the horizontal axis of the abscissae. Any other horizontal straight line intersecting the mass curve of the module coefficients is always traced between two points of intersection of the period in which z ( k - 1) = 0, or the period with the same average flow as the whole of the series. A period selected in this way therefore fulfills the first condition for a correct determination of the design period. = f ( t ) can serve the same Obviously the mass curve of the mean annual flow purpose. If drawn from the pole at the height Q,, it terminates at the end of the series in the horizontal straight line drawn through the beginning of the mass curve. Any other horizontal straight line intersecting this mass curve also traces, always between two points of intersection, periods with a flow identical with the entire series. If, generally speaking, the pole is not at height Q,, all data given so far about the horizontal straight line are true for the line connecting the beginning and the end of the mass curve, and for all lines parallel to it, intersecting the mass curve. The mass curve c ( k i - 1)2 = f ( t ) (Andreyanov, 1957) also helps to determine the period in which the variation coefficient is close to the value C, of the entire n-year series. It is not difficult to draw this curve, so that at the end of the n-year series, it ends in a horizontal line drawn from the starting point of the mass curve. n
We either first calculate the sum x ( k i - 1)’ and draw the base of the oblique i=1
coordinates of the mass curve to intersect its starting point at the end of the n-year series, at a perpendicular distance of
n
-
1(ki - 1)’ from the horizontal line, or we
i= 1
draw the mass curve l ( k i - 1)2 from the pole at the point of the mean value
Thus each horizontal line intersecting the mass curve x ( k
-
l)’, always traces
95
an m-year period between two points of intersection. Its mean value for (k - 1)2 corresponds to the quantity for the entire period, i.e. n
r+m
i= 1
(ki - 1)’ n
. c ( k i - 1)’
- i=x -
(3.3)
m
If we multiply both sides of the equation by the fraction 1
1--
1 n
and extract the root, we obtain I n
\i
n-l
\i
(3.4)
m m-n
(3.5)
Fig. 3.5 Selection of a design period froin
ii
40-year series (Slapy on ilic Vltaba)
96
Jm
as the value of does not differ greatly from the value of Jm-1 if rn has a high value. Instead of the mass curve of the (ki- 1)2 values, the mass curve for (Q,,i - Q a ) 2 can be used, since these are values proportional to the proportion coefficient 1/Qf. This method is advantageous if the module coefficients (k - 1) need not be calculated for other purposes. A 40-year long hydrological series of the river Vltava in Slapy, with data for the mean annual flow recorded from 191 1 to 1950 (Fig. 3.5), was chosen as an example. From the entire series, a common period of several years was sought in which the connecting lines between the beginning and the end of the respective sections of the two mass curves would be approximately horizontal. The period 1931 to 1940 fulfilled both conditions. The selected decade contains elements with various water yields in suitable groupings; years with a water yield very similar to the average years (1931 and 1938), a one very high-flow year (1940) the water yield of which was only surpassed in 1941, a year with an extraordinarily high-flow. The selected decade suits all the requirements of the period to solve the problem of within-year control. However, the discharge distribution in the given years will have to be checked. This favourable conclusion was reached by comparison of a 10-year series with a 40-year series (1911 to 1950). The question remains: to what extent are the above-mentioned forty years representative. Thus, comparisons for other cross-sections with longer hydrological series were carried out; the results are recorded in Table 3.1.
Table 3.1 Comparison of statistical characteristics for various periods
Profile
193 1- 1940
1931-1960
Q.
Long-term period
Q.
C,
Labe-DEEin
305.5
0.35
305.0
0.36
1851-1960 1851-1900
Berounka-Kfivoklht
32.7
0.47
31.8
0.46
1887-1960
[m3 s-'1
Vltava-St&chovice Mean annual precipitations (in Bohemia)
[m' s-
water year XII-XI 86
682
[mm]
0.41
0.152
C,
'1
Qm
C, Note
[m3 s-'1 300 293
30.1
0.29 year 0.27 0.36 water
years water year XI-X discharge influenced by reservoirs 669
[mml
0.157
water year XII-XI 1891- 1 9 4
84
0.34 water
years XU-XI 1876-1960
679
[mml
0.133 calendar
years
97 The table shows that the ten years 1931 to 1940 and the thirty years 1931 to 1960 have similar average flows as well as a similar variation coeficient; but both these values are much higher than the respective values for long-term periods reaching further back into the past. The table also indicates that the river Berounka has a much higher C, than the rivers Labe and Vltava, i.e., their flow variability is lower. The last line of the table lists the average annual precipitations in Bohemia for these periods, as well as their variation coefficients. We can see that the precipitation variability is much smaller than the flow variability, but even the precipitation values indicate that their variability was much higher during the last thirty years than in a longer period going far back to the past. That the variability of the annual flow is larger than that of the annual precipitations is also evident from the relationship between the maximum and minimum values. As indicator for the degree of representativeness of the selected period is also the distance between the exceedance curves: the empirical and the theoretical curves for the period.
2W 780
760
.!!140 'v)
120
OF
h
o ' 7m
1:
40 2y7
'0
5
10
20
30
40 - p
50
60
[%I
70
80
G9 95 1 W
Fig. 3.6 Fitting of theoretical exceedance curve of mean monthly flows with empirical curves (Slapy on the river Vltava)
In Fig. 3.6, the theoretical exceedance curves are plotted according to Rybkin's table. The empirical points of the 40-year series are denoted by circles and the 10-year series (1931-1940) by crosses. The exceedance probability of the individual members of the empirical series was calculated from Chegodayev's formula
P[%]
=
m - 0.3 -'
n
+ 0.4
100
where m is the ordinal number of the member in the degressive sequence, and n the number of members in the whole sequence (here 40 and 10). The empirical points of the 40-year series can easily be plotted along the theoretical exceedance curve. The deviations of the 10-year series from the theoretical curve are the same as the respective deviations of the 40-year series. This means that according to this indicator it can be stated that the 10-year series (1931 to 1940) has a good representative character within the 40-year series (1911-1950).
98 As it is usually not possible to find a given period with an average flow Qb equalling the long-term average Q , this drawback has either to be eliminated by multiplying the flow Q, by Q./Qb or by correcting the
results of the solution.
The compilation of synthetic hydrological series by modelling is a further stage in hydrological data processing (see Section 3.2). 3.1.2 Solutions where there is a lack of direct hydrological observations
For reservoirs on streams, the flow of which has not been measured or has only short hydrological series, hydrological data have to be procured in an indirect manner (Dub, 1963; Ogievski, 1952). If there are no direct observations, we can work in the following way: (a) The long-term mean flow, as well as the statistical parameters (Cv,C,) determining the flow distribution, can be ascertained from the empirical dependence of the mean annual flow on the climatic conditions. The results of such calculations are often unsatisfactory and cannot be recommended generally. An analogy with the nearest observed catchment area usually supplies better data. If the flow parameters are to be studied by the most reliable analogy, a careful analysis of all circumstances which might influence it is necessary, both in the catchment area where there are no direct measurements, as well as in the watershed of the similar stream which is furnishing the information. First of all, we have to compare the annual and seasonal precipitations, but also the slope conditions in the catchment area, the geologicalconditions and soil characteristics,the vegetation, air temperature, saturation deficit, etc. In the case of streams in mountainous aceas, anyone using the analogy must be aware of the fact that discharges depend greatly on the height of the waterhed above sea-level. (b) The flow distribution during the year is also determined according to a suitable, analogous stream; the lack of directly measured data is balanced by the selection of a most unfavourable flow distribution. In this way a fictive hydrograph is compiled but, of course, based on hydrological analogy. According to this analogy we also determine the discharge of the required exceedance probability. If the direct observations supply a short hydrological series (4 to 10 years), their representative nature has to be tested by comparison with a related profile and if it is not representative, it has to be prolonged according to hydrological analogy with that profile. The size of the catchment area also has to be taken into consideration, as the correlation between small and much larger watersheds is often not very close. Correlation methods serve to express approximate relationships between two quantities x and y , as compared to a functional relationship in which a certain exact value y corresponds to a certain x, i.e., y is a “function” of x. In a graphic presentation of such relationships the points found by measurement do not form a continuous line, they are scattered around the curve. Correlation
99
analyses are used for a quantitative expression of a close relationship between two quantities. In hydrology, a linear correlation relarionship is most frequently used, with the regression equation
and with the correlation coefficient
where ax,uy are the standard deviations of x and y from their mean values R and 7:
Equation (3.7) is the equation of a regression of y according to x; for the same values we can write the equation of regression of x according to y: x - R = r -OX( y OY
7)
(3.10)
Both equations express straight lines with the slopes (3.11)
and the relationship between the two slopes must be bxb, = r2
(3.12)
The probable error E of the correlation coefficient r is calculated from E
1 - r2
= k0.674-
4
(3.13)
and the extreme error is usually assumed to be four times the probable error. Therefore, the complete expression for the correlation coefficient is r k 4c
In the case of a definite functional dependence, both regression lines are identical and r = 1. If r < 1 the two lines intersect in a point, with the coordinates K and j j at an angle-the larger the angle, the smaller the r-i.e., the less close is the relationship. If we search for the respective y values with respect to the selected x values according to the regression lines, we use the line (3.7); if we search for the respective x for a selected y, we use line (3.10).The line going through the point of intersection (X,7)
100 and halving the angle of the regression lines, expresses the approximate relationship between x and y, but does not express how close that relationship is. The correlation method can also be applied for the dependence of one variable y on two variables x and z , i.e., for three variables (Ogievski, 1952, p. 273). There are also curvilinear correlations in hydrology and mathematical statistics can also solve these. Such calculations are complicated and laborious, thus Curvilinear relationships are usually defined graphically. A curved relationship can frequently be expressed satisfactorily by the exponential equation y = ax". The relationship between the logarithms of the coordinates of such curves, the so-called logarithmic transformations, are then expressed in a straight line. This is then no longer a correlation between x and y, but between their logarithms. The results are equations of straight lines of the regression of the logarithmic transformations, from which we pass on to the curves themselves. 200 180 160 140 it
$--
120
2 2-J ,
$100
40
20 0
o w 20304050 a m m m
-
-
0,-Oh& - Louny
0 Qp
50 100 150 200 -[m3s'II - V lfovo Modr'any
250
-
Fig. 3.7 Linear and non-linear correlation between the flows of the Vltava, Labe and Ohie rivers (1925-1946)
When applying correlation methods, we first have to predetermine graphically the general character of the correlation and then afterwards to perform the calculation. The correlation methods give an accurate description of the hydrological relations and offer a more objective criterion for judging how close the relationship between the variable quantities is; these methods are rather laborious and that is why often only the graphical presentation of these relationships, which is fully satisfactory, is used.
101
If the variation coefficient is known in both profiles, the correlation between the annual flows can be determined by the Pearson curves or Rybkin tables for the theoretical exceedance curves. Figure 3.7 shows the correlation between certain profiles on Czech rivers. A linear correlation was used
Q, = aQx + b
(3.14)
and a curvilinear correlation with equation
Q,
=
kQ:
(3.15)
in which Q,, is the value of the required flow, Q, is the known flow of the site used as a basis and a, b, k, n are the constant parameters for a certain correlation. Results of the calculation of some correlations for some rivers in the catchment area of the Labe can be found in the book by Votruba and Broia (1966). 3.1.3 Statistical and probability processing of hydrological series
The aim of the processing of measured hydrological quantities is to gain more useful information for the calculation of a reservoir than the original series could supply. This is why the probability theory, mathematical statistics and the theory of random processes, are used. The results are - general statistical characteristics of the series, - the laws of distribution - synthetic (modelled)pseudo-chronological series. In water management the above-mentioned methods are used to study random phenomena, i.e., phenomena which cannot be forecast for each individual case generally, although a certain complex of conditions is adhered to. For instance, we cannot forecast the spring flow from the snow cover of the same watershed, although we know the water content of the snow cover in the watershed and other genetic elements at a certain date; we cannot determine the flow process exactly, although we know the precipitation rate. One or more variables or even entire processes might be random. We shall first of all analyse the properties of the random variable and afterwards those of the random process. In both cases there might be one or more random phenomena involved. A frequently applied characteristic of the random variable is the probability of exceedance (reliability)expressed for the flow Q by P(Q 2 Q,), in other words: the probability that the flow Q, in the series Q1 2 Qz 2 ... 2 Q , 2 ... >= Q, has been exceeded. This probability can be expressed by a formula derived by Chegodayev (equation (3.6)).
102 In the probability theory, it is usual to characterize a random variable by its cumulative distribution function. Expressed in PA), the relationship between the exceedance probabilities curve P(x) and the cumulative distribution function F ( x ) is P ( x ) = 100 - F ( x )
(3.16)
or for a general random variable t (3.17)
2 x) F(x) = P ( t 5 x)
P(x) = P(t
(3.18)
A random variable can be discrete, if all its possible, but countable values can be expressed in integer numbers, such as 0, 1, 2, ... - continuous, if its set of possible values is uncountable and continuously fills -
at least a part of the axis of real numbers. The random variable, therefore, here represents the value of the random time function, i.e., of a random process (in the case of a continuous variable) or a random sequence (in the case of a discrete variable). The probability of each of the possible values of a continuous variable equals zero, because the number of possible values is infinite. We therefore divide the entire range of possible values of continuous random variable into a finite number of intervals and introduce the concept of the probability that the value of the continuous random variable will lie within the range of some interval, analogous to the work with the discrete random variable. In hydrology and water management most phenomena take a continuous character: water levels, flows, reservoir fillings, precipitations, etc. For calculations or graphical presentations we transform them into “discrete” variables in the sense of the previous paragraph. Even when measuring in intervals (e.g.,once in 24 hours) and not practising continuous measurings (e.g., on the recording gauge) we have to bear in mind that these are “discrete” expressions of the continuous variable within the range of the respective interval. In this respect, mean monthly flows are, e.g., discrete variables. From the nature of the distribution function, according to equation (3.18), it follows that for an increasing x it has to rise continuously and when reaching x =, , ,< it must be ~ ( 3x & ),, = 1 (or 100%). More generally: from the fact that the values of the distribution function express a probability and, therefore, are in the range from 0 to 1, it follows that for an impossible phqomenon the value F(x) is: F(-co) = P ( t 4 - 0 0 ) = 0
(3.19)
for a certain phenomenon F(co) = P(< 4
00)
=1
(3.20)
103 The derivative of the distribution function is the probability density f ( x ) expressing the probability that the random variable t will lie in a certain interval. Between the distribution function and the probability density the relation exists
P(( 5 x)
= F(x) =
j:J(x)dx
(3.21)
or
(3.22) As the distribution function does not decrease, its derivative (probability density) cannot be negative.
Fig. 3.8 Relationship between probability density f ( x ) , distribution function F ( x ) and exceedance probability curve P ( x )
The relationship between the probability density, the distribution function and the exceedance probabilities curve of the continuous random variable is graphically presented in Fig. 3.8. The diagram of the distribution shows the mass curve corresponding to the probability density curve; all the relationships valid between mass curve and their basic curves (see Section 2.1.1) are valid for them. From these relationships and equation (3.21), it follows that F(xk)is determined by the ordinate in the diagram of the distribution function vs. the abscissa x k , or in the basic curve (e.g., the probability density diagram) by an area limited by this curve above the horizontal axis to the ordinate of the respective abscissa xk.
104
The relation in Fig. 3.8 can be written as (3.23)
or u4ng equation (2.7) (3.24)
This means that the probability that the value of the continuous random variable will lie in the interval (x1,x2) equals the area below the probability density curve above the respective interval, or the difference between the ordinates of the distribution function belonging to the abscissas x t and x2. The inflect point I on the F(x) curve belongs to the coordinate x, with a maximum f’(x). Figure 3.8 shows-from the coordinate x p to &the division of the probability density of the continuous random variable into intervals which are supposed to have the mean value of the random variable within the range of the respective interval. The probability is marked as a step-shaped line. Compared to this is the distribution curve (broken line) with peaks on the distribution function curve to the continuous random variable. A similar case is that in which a very large set of discrete random variables (e.g., mean daily flows) is divided in equal intervals into groups (e.g., for every 5 days-pentads, 10 days-decades) denoted classes. The interval centre represents all the values of the symbol of this class interval. The number of values of the symbols in each interval is the frequency, and the graphical presentation of such a group division is called a histogram (frequency distribution). The relative frequency of the ith class is the ratio ni/n (usually in %), where n, is the number of values of the symbol (frequency) in the ith class interval and n number of all values of the symbol. The cumulatiue frequency in relation to the ith class is defined by the sum The cumulative relative frequency is denoted by
i
j= 1
nj.
The diagram of the exceedance curve P(x) is bound to the diagram of the distribution function F(x) according to the relation in equation (3.16), as shown in Fig. 3.8 for the coordinate xk. From equation (3.24), it follows that for x1 = --a0 and x2 = co (or according to Fig. 3.8 for xl = A and x2 = B), with regard to equations (3.19) and (3.20) (3.25)
105
The part of this area between A and xk (Fig. 3.8) then corresponds to probability p that the density of the random variable is smaller than, or equal to, the selected number xk. In such a case, the number xk is called a quantile corresponding to a certain probability p. A sample quantile is the characteristic which divides the set of sample values, ranked in ascending order of magnitude, into two appropriate parts. For example, the quintile divides the sample into and 4 of all its values ranked in order of magnitude, the quartile into and 2. The lower quartile x1 separates of the smallest sample values, the upper quartile x2 separates of the greatest values. A special case of a quantile is the median I (the sample centre), dividing the set of sample values ranked in order of magnitude into two parts containing the same number of values. Median is denoted 2 (in English literature also Me), the notation being read ‘‘x tilda”. If the number of sample values is odd and equals 2m + 1, the median is
a
j2 =
x,+,
(3.26)
If the number is even and equals 2m, the median is (3.27)
The median of the distribution of a continuous random variable is determined, according to (3.25),from the equation (3.28)
0, as for x is divergent. Mean value of the Weibull distribution p(x) = xo
variance
+)
=
;ti')
+ -r
-[+-) fl 1
-
a+2
-
4 0 the integral
(3.85)
rZ(-J] a+l
(3.86)
117
skewness
(3.87)
The skewness C, depends on the constant a and the relationship between the two parameters is shown in Fig. 3.11. +I
u“
to -1
0
2
6 8
4
-a
10 7 2 1 4
1618
Fig. 3.1 1 The relation C, = j ( a ) for Weibull’s distribution
For the standardized variable t = [x - dx)]/o(x) the Weibull distribution depends on one single constant a. In the analysis of minimum flows we can write the basic formulae in the form of: for the frequency function a
f(x) =
s:.
x -xo
(G)exp [-
for the distribution function F(x) =
f(x) dx = 1
-
[
exp -
(=)”I
(=TI
(3.88)
(3.89)
for the exceedance probabilities (3.90)
Using the form as above, the distribution depends on three constants: Q’ - the so-called characteristic of “minimal” flow corresponding to the flow with the exceedance probability P(Q’)= 0.368 = e-’; xo - the minimum possible value of x; a - the slope of the exceedance probabilities curve in straight-line transformation, depending on the variance.
118
Gumbel probability distribution E. J. Gumbel introduced his probability distribution law while investigating maximum age and called it the “law of maximum value”. In the U.S.A., the Gumbel distribution is commonly used for statistical analysis of flood flows. Sometimes, it is, according to its mathematical description,called the double exponential distribution. The double exponential distribution is unlimited from either side and its probability density is given by f ( x ) = e-2 e-e-z
(3.91)
where z denotes the standardized deviation from the mode and depends on the random variable x according to the linear relation 1 (X - AX) 0.7800
z=-
+ 0.4500)
(3.92)
or with the aid of general parameters it may be written as z = a(x -
fl)
(3.93)
Using sample characteristics the parameters can be estimated by the terms (3.94)
where
3.14 (Ludolph’s number) y = 0.577 (Euler’s constant). The preceeding formulae remain valid only in the asymptotic sense. For any finite sample they must be considered as approximations (Dub and Nemec, 1969). The distribution function of the Gumbel distribution is 7t =
~ ( x= ) e-e-z
(3.95)
hence the exceedance probability is P ( X )= 1 - e-‘-‘
(3.96)
A disadvantage of the Gumbel distribution is that there is only one constant value of the coefficient of skewness, namely C,= 1.139, however, the value is close to the coefficients of skewness for a large class of rivers. The exceedance probability for mode is P = 63.2% and that for the mean is P = 42.97%. In spite of having been recommended for analysing maximum peak discharges, the Gumbel distribution was found to be less suitable in some cases. Nevertheless, it is often used with probabilistic analysis of maximum precipitations (of n days, of n hours).
119
Pearson probability distribution
K. Pearson considered a very general differential equation x+a dY _ dx bo + b,x + b,x 2 y
(3.97)
where a, b,, b,, b, are real (constant) numbers. According to the explicit values of these parameters and the domain of the variable x, the Pearson curves are of twelve different types. The basic relation, definihg the probability density of Pearson distribution, assumes the form (3.98)
The type of distribution is determined by the values of the following quantities
(3.99)
where M,, M , and M4 are the second, the third and the fourth central moments (around the mean). For B1 = 0, B2 = 3 and k = 0 the Pearson distribution is identical with the normal distribution. The first and the third type curves are often used in the probabilistic analysis of hydrological data. Pearson distribution of the first type The first type is determined by the condition k < 0. The distribution is asymmetric, unbounded at either side, and usually bell-shaped. The first-type Pearson distribution density curve, also called beta distribution, may be expressed by the following analytic formula (3.100)
Symbol B is used to denote the beta function (first kind Euler's integral) defined by
Beta function, values of which are tabulated, is related to the function r(x) by the formula (3.102)
120
The mean value Ax), the standard deviation 4.) and two independent constants a, /? can be found in eqn. (3.100) while the auxiliary constant y simplifying the notation is connected with the constants a and /? by the relation
(3.103) With the aid of a and /? we can also express both the skewness C, and the excess E:
c, = 2y E = 3y2
/?-a a+/?+2 2(a (a
(3.104)
+ a)' + a/?(a+ /? - 6) - 3 + /? + 2)(a + /? + 3)
(3.105)
The domain of the variable x is bounded on both sides, the lower boundary , the upper boundary being [p(x) ay o(x)]. For a > 1, being [p(x) - ay ~ ( x ) ]and /? > 1 which is the most frequent case, the probability density curve f ( x ) is bellshaped. With a = /? the curve is symmetric with respect to the ordinate of the corresponding mean value p(x). For /? + 00 the first type curve turns into the third-type Pearson curve. The first-type Pearson curves may successfully be applied whenever the domain of variable is bounded on both sides. The advantage consists in the fact that we can choose the beginning of the curve, often choosing zero, and yet there are three independent parameters p(x), 4.) and S, left.
+
Pearson distribution of the third type The third type is recognizable from k = 00 or 2/?, = 3/?, + 6. The distribution is asymmetric, bounded on one side, and usually bell-shaped. The general analytical formula for the third-type Pearson curve is
[m
am2 exp a P ( X ) - a'] f ( x )=
r ( a 2 )aa2(x)
[ a o(x) - p(x)
+ x ] ~ ' - ' exp [-
-XI 44 a
(3.106)
where r denotes the gamma function. The curve depends on three parameters: the mean value p(x) the variance a(.) a constant a defining the skewness by the relation
c
'
to
2
=-
a
(3.107)
Under positive skewness the domain of the variable x is from [ d x ) - a o ( x ) ] + 00, under negative skewness from - co to [p(x) - a ~ ( x ) ]For . a tending to
121 infinity it holds that C, -,0, and the curve becomes symmetric and turns to normal distribution with parameters p(x) and a(x), hence normal distribution is a special case of the Pearson distribution of the third type. The curve is bell-shaped whenever a' > 1, i.e., C, c 2. If C, > 0 the curve is decreasing, if C, = 2Cvthe curve is increasing. If it holds that C , > 0 under C, = 2Cv,the curve has its origin in zero. Hence the variable assumes positive values providing C, 2 2Cv.
-f Fig. 3.12 Pearson curve of the third type (a) for standardized variable r ; (b) for xo > 0
Excess E depends on C, according to the relation (3.108) The third-type curve for standardized variable (Fig. 3.12a) is (3.109) The domain of the standardized variable is from - a to -co to - a under a < 0. Mode of the curve is
+
00
under a > 0, from
1
t = -a
The inflexion oints exist only under a' > 1, their distance from the mode being f , / w r i n both directions. The third-type curve is very general because it may be used to describe both asymmetric distributions with either positive or negative skewness and symmetric distributions. When IC,(is close to zero, the third-type curve is close to the normal distribution. Both the frequency and distribution functions are tabulated in detail, hence they are easy to deal with.
122
The probability density of the Pearson distribution of the third type may, for convenience, be expressed by (3.110)
where a is the distance between the mode and the origin of the curve, d is the distance between the ordinate of the centre of gravity and the mode, and x, is the minimum value of x (Fig. 3.12b). Moments of the distribution may be expressed by means of the parameters from equation (3.1 10)
4.) = x, + a + d 1 C , = -[d(a + d)]’’’ 44
(3.111) (3.1 12) (3.113)
From these relations it follows that x, = 4.) (1 -
3) cs
In almost all hydrological events it must hold that x, 2 0 and therefore C , 2 2Cv. Hence, the whole curve is situated in the domain of positive x’s; with C , = 2Cv the random variable could assume negative values as well, which is an impossible rule in hydrological events. Kritsky-Menkel’s three-parameter probability distribution In order to describe annual river discharge in the U.S.S.R., a theoretical distribution is often used which was recommended by Kritsky and Menkel under the following assumptions: (a) the distribution function differs from zero on the interval (0, a); (b) the distribution function depends on three parameters, i.e., an arbitrary choice of the first three moments must be possible. The analytic formula for the Kritsky-Menkel three-parameter distribution function is (for x 2 0) in the form
where X, y, b are the parameters.
123 For b = 1 the distribution function (3.114) is reduced to the third-type Pearson distribution with equality C, = 2C,, i.e., a two-parameter distribution. Reznikovski (1969) concluded, on the basis of investigations performed in the U.S.S.R., that for rivers with C, 5 0.5 and r 7 0.3 the relation C, = C,"+ 3C, according to equation (3.62) is closer to reality than the relation C, = 2C,, i.e., the log-normal distribution is more suitable than the third-type Pearson distribution in such cases. Exponential probability distribution The exceedance probability curve of exponential probability distribution is given by the exponential function P(z) = e-'
(3.115)
hence the distribution function is ~ ( x= ) 1 - e-'
(3.116)
where z = f ( x ) . Various expressions of the variable z yield various types of exponential probability laws. The Goodrich exponential distribution was found suitable for analysing some flood-flow samples. Goodrich distribution is usually given by (3.117) (3.118) (3.119) The best way to estimate the constants k, x' and a, is to express the exceedance probability curve in the Goodrich probability paper. The constant x' is chosen so that the empirical distribution in the Goodrich probability paper creates approximatelly a straight line (for P -,1 the deviation may be a bit larger). By double-logarithming the equation (3.1 17), rewritten in the form
we obtain 1
- (log k
1
+ log loge)
(3.120)
124
The constant a determining the slope of the approximating line is equal to a=
A log (X - x') A log (log
i)
(3.121)
For the Goodrich probability chart, the scale of 1/P is double logarithmic (log log l/P), and the scale of (x - x') is logarithmic. The ordinate of the exceedance probability curve corresponding to P = 0.10, i.e., with log (log 1/P) = 0, determines the value log (x - x') = -a(log k + log loge) whence k can be calculated. For an arbitrary P, the corresponding x, is calculated from equation (3.120). In Czechoslovakia, V. Brofa succeeded in applying the Goodrich probability distribution to analyses of samples of all maximum peak discharges (not the annual maximum discharges for which some other distributions are more suitable). Binomial probability distribution An asymmetric binomial distribution is frequently used in hydrological data processing. Explaining the nature of the binomial distribution, we must keep in mind that it is a distribution of a discrete random variable. In a sequence of independent trials performed under stable conditions (Bernoulli trials), we observe whether or not an even A occurs, the probability of which being p in every trial. Thus the probability p(n,x) = (;)P.4..
= (;)P'(l
-P
YX
(3.122)
here has the meaning of probability density, hence (3.123)
where x denotes the number of the occurrences of event A. The constants n, p are called parameters of the binomial distribution. The set of probabilities p(n, x) for x = 0, 1,2, ...,n, i.e., p(n,O), p(n, l), p(n, 2), ... ...,p(n, n), is called binomial probability distribution. As these probabilities correspond to disjointed events, forming the complete system of events, it follows that n
(3.124)
which may easily be checked because the values p(n, x) defined by eqn. (3.122) are the terms of the binomial expression (q + p)" from which their name was derived.
125
The mode of the binomial distribution is the value 2 corresponding to the greatest probability. As a rule, the binomial distribution has only one mode. The maximum value may, of course, correspond to either 0 or n, the probabilities p(n,x) either decrease or increase, respectively. Such cases may occur with small n and p close to either zero or one. With a large n, the mode is always somewhere in the central part of the distribution and the probabilities decrease in both directions from the mode. In order to facilitate the calculation of the moments of the binomial distribution, the so-called moment-generating function (P(E) is introduced. This function is used to calculate the moments especially of a discrete random variable, and is defied as follows: (3.125)
Summation concerns all possible values x = 0, 1,2, ..., n. Let us suppose the sum to converge at least for suficiently small E. Then it holds that (3.126)
The formula for calculating the kth order general moment is n
mk(’
p(n,O) = (1 - p)” = 1 - -
whence
2n
...
(3.134)
I27 If
I2
or p 4 -
-41
1
J;;
n
holds, we may reduce eqn. (3.134)to the first term and write
(3.135)
e-A
p(n,O)
Similarly it follows that d n , 1) = np(1 - p)”-’
nP
=-(I
1-P
and generally p(n, x) =
n(n - 1) ... (n - x
I
= Ie-A
- p)” = -p(n,O)
+ 1) p”(1
1--
I
(3.136)
n
I” - p)”-” = -e-l
X!
X!
(3.137)
The result may be formulated as follows: Providing that number n is very large, the occurrence probability p of an event is very small so that the mean number of occurrences remains equal to a constant positive number 1, the probability p(n, x) for binomial distribution tends for every x = 0, 1,..., to the limit value
p(I, x)
1”
=f
= -e-l X!
(4
(3.138)
This asymptotic expression is sometimes called Poisson’s formula. It holds that
f P(A
x=o
co Axe-).
x) =
--1
(3.139)
x!
*=o
The parameters are calculated again with the aid of the moment-generating function. Combining eqn. (3.138)with eqn. (3.125)we obtain
44 =
c OD
ecxp
e- 1.
x!
Ze-1
- e-Ae.lec for
C* -1”= I + + + -I2!2+
x=OX!
... = e l
whence according to eqns. (3.127)and (3.128),the general moments are given by q ( x ) = p(x) =
~‘(E)I,=~
= e - A el e C ~ e c ( e= , oI
m2(x) = ( P ” ( E ) J , = ~ = Ie-*eAecee(Aee +
I)I&=~
=
I
+ l2
(3.140)
128
hence
+)
=
M,
=
m, - p 2 = 1 + A2 - 1, = 1
(3.141)
From equation (3.140) and (3.141) we get the obvious identity between the mean value Ax) and the variance b2(x), which is characteristic for the Poisson probability distribution. The coefficient of skewness M3 1 1 c, = = -= -
(3.142)
G f l $
is always positive because 1 is always positive. The coefficient of excess y = - - M4
c4(4
3=--
3A2 + 1 A2
3=-
1
1
(3.143)
of the Poisson distribution is always positive, too. For calculating the probabilities for the Poisson distribution, as well as its distribution function (i.e., the probability of a maximum of n occurrences) (3.144)
tables (Reisenauer, 1970) have been elaborated. 3.1.6 Estimates of exceedance probability curve parameters by means of quantiles
Up to now, the method of moments has been used to determine the values of parameters A x ) , ~’(x),C,, C,, y in the appropriate types of distributions. Lately, using many random samples, the fit has been checked between the sample characteristics C, and C,and the respective values of parameters given by eqns. (3.49) and (3.50). The optimal estimates based on the random samples are greater than the corresponding parameters. This is why it is sometimes recommended that the estimates of C, and C, should be calculated with the aid of quantiles. A quantile denotes the point chosen so that the distribution function assumes a prescribed value, e.g., 5%, 25%, etc. The method is based on several values of quantiles obtained from the empirical curve (approximately constructed), and on the standardized deviations of exceedance probability curves @(Pi;C,) for the corresponding distribution, found in tables. The index of @ denotes the type of distribution (@* - binomial, ,@, - log-normal, aGGumbel). The quantiles for fixed probabilities Piare used taking into account the
129 relationship between the cumulative distribution function and the exceedance probability curve - see Fig. 3.8. An approach essentially simplifying the calculation as compared with the complicated calculation based on equations (3.49)and (3.50),was elaborated by Alexeyev (1960).Only three points of the empirical curve, corresponding to PI = 574, Pz = 50% and P3 = 9574, are considered. Binomial distribution The calculation is described here in detail: 1. The set of sample values (e.g., the average annual flows) are ranked in descending order of magnitude and the probabilities P are calculated according to Chegodayev. 2. The empirical curve in the probability paper is approxinlated by a straight line or a continuous curve (curve fitting). From this curve we obtain three values for fixed characteristic probabilities, namely P = 5%, 50% and 95%, the values being 0 xg5. denoted x5, ~ 5 and 3. We calculate the auxiliary quantity S defined by S=
x5
+ x95 - 2x50
x5 - x95 which is, according to Alexeyev, the function of sample skewness (Fig. 3.13).
(3.145)
7.0 08
1
' I ,
06 04 02
0.0 0
7
2
3
4
5
Fig. 3.13 The relation S = f(C,)for binomial distribution
cs
With binomial distribution it also holds that S=
@5
+ @95 @5
-
-
2@50
(3.146)
@95
where Q5,@ 5 0 and @95 are values of QB(P,C,) obtained from Foster-Rybkin tables for P = 5%, 50% and 95%.
130 Table 3.2 Auxiliary values to determine parametersof binomial distribution according
to G . A. Alexeyev
~
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.o 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6
3.28 3.28 3.28 3.27 3.27 3.26 3.25 3.24 3.22 3.21 3.20 3.17 3.16 3.14 3.12 3.09 3.07 3.04 3.01 2.98 2.95 2.92 2.89 2.86 2.82 2.79 2.76
0.00 -0.02 -0.03 -0.05 -0.07 -0.08 -0.10 -0.12 -0.13 -0.15 -0.16 -0.18
0.00 0.03 0.06 0.08 0.11 0.14 0.17 0.20 0.22 0.25 0.28 0.31
2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8
-0.21 -0.22 -0.24 -0.25 -0.27 -0.28 -0.29 -0.31 -0.32 -0.33 -0.34 -0.35 -0.36 -0.37
0.37 0.39 0.42 0.45 0.48 0.51 0.54 0.57 0.59 0.63 0.64 0.67 0.69 0.72
4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 I
2.74 2.71 2.68 2.64 2.62 2.59 2.56 2.53 2.50 2.48 2.45 2.43 2.41 2.40 2.38 2.36 2.34 2.32 2.30 2.28 2.26 2.23 2.21 2.18 2.15 2.15
-0.38 -0.39 -0.39 -0.40 -0.40 -0.41 -0.41 -0.41 -0.41 -0.42 -0.42 -0.42 -0.41 -0.41 -0.41 -0.41 -0.40 -0.40 -0.40 -0.40 -a40 -0.39 -0.39 -0.38 -0.38 -0.37
0.74 0.76 0.78 0.80 0.81 0.83 0.85 0.86 0.87 0.89 0.90 0.91 0.92 0.92 0.93 0.94 0.94 0.95 0.96 0.97 0.97 0.98 0.98 0.98 0.98 0.98 - -
S is obtained from Table 3.2 [according to equation (3.145)], along with the value
C,and other auxiliary values (9, - GgS), and
characteristics of the binomial distribution. 4. We calculate the following characteristics: The standard deviation s, =
-9 s -xc
xoz 7 -
which enable us to calculate other
(3.147)
995
the mean value
x = xso - s,Gso
(3.148)
10
-
probability
~
131 P
[%I
-
probability P C%l Fig. 3.14 Brovkovich's probability paper (Pearson C, = 2C,)
the coefficient of variation
c
(3.149)
=sx
, x
We compare C, and C,; for theoretical correctness of the following stepwise relation, C, 2 2C, must be valid. 5. From the general equation
xp = x
+ s,
@(P,c,)= $1
+ c, @(P,c,)]
(3.150)
we calculate values of the random variable x for any exceedance probability P. The function @,(P, C,) is tabulated in detail. The problem may be solved also with the aid of the graphico-numerical method using the Brovkovich probability paper (Fig. 3.14). Log-normal distribution The approach is again based on quantiles corresponding to the probabilities P, = 574, Pz = 50% and P3 = 95%. Steps 1 and 2 are the same as for the binomial distribution.
132 3. We calculate an auxiliary quantity xo given by xo =
-k
'5
(3.151)
- 2x50
'95
and the standard deviation of a new variable y according to equation (3.61) sY = 0.305 log
x5 - xo
(3.152)
x9s - xo
4. We calculate values x, corresponding to various probability P's from the
equation log (xp - xo) = log (XSO - XO)
+
sy
@B(P,cs = 0)
(3.153)
Values of the function djB(P,0) can be found in the tables mentioned above. The fit of the log-normal distribution to the investigated sample is checked by representing the exceedance probabilities curve in the logarithmic probability paper. If the sample points with coordinates (P,,x, - xo) are near a line, the log-normal distribution is well-suited. Gumbel distribution The Gumbel distribution depends on two parameters a, /3 according to equation (3.93), and therefore only two quantiles are used to estimate the parameters, x5 and xg5,corresponding to probabilities PI = 5% and P2 = 95%, respectively. Following steps 1 and 2, we determine the values x 5 and xgS.The parameters a and p are calculated from the relations a=
4.067 x5
(3.154)
+ x9s
p = 0 . 2 7 ~+~0
. 7 3 ~ ~ ~
(3.155)
The ordinate xp corresponding to various exceedance probabilities P may be calculated either from the relation xp = p
+ -a1z p
(3.156)
where z p is the standardized deviation from the mode according to the table, or from a general equation of type (3.150), substituting the values @@, C,) also according to the table (e.g., Votruba and Nachhzel, 1971). The other characteristics of equation (3.150) are given by
X
+
= 0 . 4 1 2 ~ ~0 . 5 8 8 ~ ~ ~
s, = 0.315(x5 - x
~ ~ )
(3.157) (3.158)
133 Exponential distribution Following steps 1 and 2, the values x,, x , ~ and x,, are obtained. Step 3. We calculate the auxiliary quantity s, =
2logx,, - logx, - logx,, logx, - logx,,
(3.159)
and the sample coefficient of skewness (3.160)
C,, = 7 . 1 5 ~ ~
Th'e calculation continues according to whether C,, > 0 or C,, < 0 (see Votruba and Nachazel, 1971,p. 64). 3.1.7 Tables and probability papers for constructing the exceedance probability curves
An analytical formula for a probability distribution is usually very difficult to obtain, therefore numerical and graphical aids, such as tables and probability charts, are used for constructing exceedance probability curves. For the third type of Pearson distribution tables were constructed by E. E. Foster and S . J. Rybkin. Rybkin's table, completed by B. I. Serpik, is very comprehensive (Alexeyev, 1960, pp. 136, 137). The values are calculated for p = 0.01% up to 100% and C, = 0.0 up to 5.2 under C , = 1 (Table 3.3). The table serves to determine the ordinates of the theoretical exceedance probabilities curve with given values K, C,, C,. In the row corresponding to a given value C,, the values @(p,C,) for various exceedance probabilities are found. The modulus factors k, = xp/X or (directly) the ordinates x, of the exceeding probability curve are calculated from the relations (3.161) (3.162) Example: The value of average annual flow corresponding t o p = 95% is to be calculated, providing that the Pearson probability law holds, with Q. = 300 m 3s- ', C, = 0.29, C, = 0.8. For C , = 0.8 and p = 95%, the value @(p,C,) = - 1.38 is found in the table; k,, = - 1.38.0.29 + 1 = = 0.60or Q,,95 = [ - 1.38 '0.29 + 11. 300 = 180 m3 s - ' is calculated.
Probability papers Exceedance probabilities curves represented in a coordinate system with a linear scale (especially for the horizontal axis P ) are not suitable because of inaccuracies which may occur for the extreme values of probability ( P 0, P + 1). Therefore, for a graphical description of both empirical and theoretical exceedance probability
134 xi - E Table 3.3 Deviations of ordinates for the binomial curve of probability exceedance -- @(P, C,) (according to B. L. Serpik) Reliability
c,
0.01
0.1
0.5
1
2
3
5
10
20
25
30
0.0 0.1 0.2 0.3 0.4 0.5
3.72 3.94 4.16 4.38 4.61 4.83
3.09 3.23 3.38 3.52 3.66 3.81
2.58 2.67 2.76 2.86 2.95 3.04
2.33 2.40 2.47 2.54 2.61 2.68
2.02 2.11 2.16 2.21 2.26 2.31
1.88 1.92 1.96 2.00 2.04 2.08
1.64 1.67 1.70 1.72 1.75 1.77
1.28 1.29 1.30 1.31 1.32 1.32
0.84 0.84 0.83 0.82 0.82 0.81
0.67 0.66 0.65 0.64 0.63 0.62
0.52 0.51 0.50 0.48 0.47 0.46
0.6 0.7 0.8 0.9 1.o
5.05 5.28 5.50 5.73 5.96
3.96 4.10 4.24 4.38 4.53
3.13 3.22 3.31 3.40 3.49
2.75 2.82 2.89 2.96 3.02
2.35 2.40 2.45 2.50 2.54
2.12 2.15 2.18 2.22 2.25
1.80 1.82 1.84 1.86 1.88
1.33 1.33 1.34 1.34 1.34
0.80 0.79 0.78 0.77 0.76
0.61 0.59 0.58 0.57 0.55
0.44 0.43 0.41 0.38
1.1 1.2 1.3 1.4 1.5
6.18 6.41 6.64 6.87 7.09
4.67 4.81 4.95 5.09 5.28
3.58 3.66 3.74 3.83 3.91
3.09 3.15 3.21 3.27 3.33
2.58 2.62 2.67 2.71 2.74
2.28 2.31 2.34 2.37 2.39
1.89 1.92 1.94 1.95 1.96
1.34 1.34 1.34 1.34 1.33
0.74 0.73 0.72 0.71 0.69
0.54 0.52 0.51 0.49 0.47
0.36 0.35 0.33 0.31 0.30
1.6 1.7 1.8 1.9 2.0
7.31 7.54 7.76 7.98 8.21
5.37 5.50 5.64 5.77 5.91
3.99 4.07 4.15 4.23 4.30
3.39 3.44 3.50 3.55 3.60
2.78 2.82 2.85 2.88 2.91
2.42 2.44 2.46 2.49 2.51
1.97 1.98 1.99 2.00 2.00
1.33 1.32 1.32 1.31 1.30
0.68 0.66 0.63 0.61
0.46 0.44 0.42 0.40 0.39
0.28 0.26 0.24 0.22 0.20
-
2.2 2.3 2.4 2.5
6.04 6.14 6.26 6.37 6.50
4.38 4.46 4.52 4.59 4.66
3.65 3.68 3.73 3.78 3.82
2.94 2.95 2.98 3.02 3.05
2.53 2.54 2.57 2.60 2.62
2.01 2.02 2.01 2.00 2.00
1.29 1.27 1.26 1.25 1.23
0.59 0.57 0.55 0.52 0.50
0.37 0.35 0.32 0.29 0.27
0.18 0.16 0.14 , 0.12 0.10
2.6 2.7 2.8 2.9 3.0
6.54 6.75 6.86 7.00 7.10
4.71 4.80 4.86 4.91 4.95
3.86 3.92 3.96 4.01 4.05
3.08 3.10 3.12 3.12 3.14
2.63 2.64 2.65 2.66 2.66
2.00 2.00 2.00 1.99 1.97
1.21 1.19 1.18 1.15 1.13
0.48 0.46 0.44 0.41 0.39
0.25 0.24 0.22 0.20 0.19
3.1 3.2 3.3 3.4 3.5
7.23 7.35 7.44 7.54 7.64
5.01 5.08 5.14 5.19 5.25
4.09 4.11 4.15 4.18 4.21
3.14 3.14 3.14 3.15 3.16
2.66 2.66 2.66 2.66 2.66
1.97 1.96 1.95 1.94 1.93
1.11 1.09 1.08 1.06 1.04
0.37 0.35 0.33 0.31 0.29
0.17 0.15 0.13 0.11 0.085
. 2.1
0.64
0.40
0.085 0.070 0.057 0.041 0.027 0.010 -0.006 -0.022 -0.036 -0.049
135
40
50
60
70
75
80
90
95
97
99
99.9
100
-1.27 -1.26 -1.24 -1.23 -1.22
-1.64 -1.61 -1.58 -1.55 -1.52 -1.49
-1.88 -1.84 -1.79 -1.75 -1.70 -1.66
-2.33 -2.25 -2.18 -2.10 -2.03 -1.96
-3.09 -2.95 -2.81 -2.67 -2.54 -2.40
-20.0 -10.0 -6.67 -5.00 -4.00
0.25 0.24 0.22 0.20 0.19 0.17
0.00-0.25-0.52-0.67-0.84-1.28 -0.02 -0.27 -0.53 -0.68 -0.85 -0.03 -0.28 -0.55 -0.69 -0.85 -0.05 -0.30 -0.56 -0.70 -0.85 -0.07 -0.31 -0.57 -0.71 -0.85 -0.08 -0.33 -0.58 -0.71 -0.85
0.16 0.14 0.12 0.11 0.09
-0.10 -0.12 -0.13 -0.15 -0.16
-0.34 -0.36 -0.37 -0.38 -0.39
-0.59 -0.60 -0.60 -0.61 -0.62
-0.72 -0.72 -0.73 -0.73 -0.73
-0.85 -0.85 -0.86 -0.85 -0.85
-1.20 -1.18 -1.17 -1.15 -1.13
-1.45 -1.42 -1.38 -1.35 -1.32
-1.61 -1.57 -1.52 -1.47 -1.42
-1.88 -1.81 -1.74 -1.66 -1.59
-2.27 -2.14 -2.02 -1.90 -1.79
-3.33 -2.86 -2.50 -2.22 -2.00
0.07 0.05 0.04 0.02 0.00
-0.18 -0.19 -0.21 -0.22 -0.24
-0.41 -0.42 -0.43 -0.44 -0.45
-0.62 -0.63 -0.63 -0.64 -0.64
-0.74 -0.74 -0.74 -0.73 -0.73
-0.85 -0.84 -0.84 -0.83 -0.82
-1.10 -1.08 -1.06 -1.04 -1.02
-1.28 -1.24 -1.20 -1.17 -1.13
-1.38 -1.33 -1.28 -1.23 -1.19
-1.52 -1.45 -1.38 -1.32 -1.26
-1.68 -1.58 -1.48 -1.39 -1.31
-1.82 -1.67 -1.54 -1.43 -1.33
-0.02 -0.03 -0.05 -0.07 -0.08
-0.25 -0.27 -0.28 -0.29 -0.31
-0.46 -0.47 -0.48 -0.48 -0.49
-0.64 -0.64 -0.64 -0.64 -0.64
-0.73 -0.72 -0.72 -0.72 -0.71
-0.81 -0.81 -0.80 -0.79 -0.78
-0.99 -0.97 -0.94 -0.92 -0.90
-1.10 -1.06 -1.02 -0.98 -0.95
-1.14 -1.10 -1.06 -1.01 -0.97
-1.20 -1.14 -1.09 -1.04 -0.99
-1.24 -1.17 -1.11 - 1.05 -1.00
-1.25 -1.18 -1.11 -1.05 -1.00
-0.10 -0.12 -0.13 -0.14 -0.16
-0.32 -0.33 -0.34 -0.35 -0.36
-0.50 -0.50 -0.50 -0.51 -0.51
-0.64 -0.64 -0.63 -0.62 -0.62
-0.70 -0.69 -0.68 -0.67 -0.66
-0.76 -0.75 -0.74 -0.72 -0.71
-0.886 -0.842 -0.815 -0.792 -0.768
-0.914 -0.882 -0.850 -0.820 -0.790
-0.930 -0.895 -0.860 -0.826 -0.795
-0.945 -0.905 -0.867 -0.830 -0.800
-0.952 -0.910 -0.870 -0.833 -0.800
-0.952 -0.910 -0.870 -0.833 -0.800
-0.17 -0.18 -0.20 -0.21 -0.22
-0.37 -0.38 -0.39 -0.39 -0.40
-0.51 -0.51 -0.51 -0.51 -0.51
-0.61 -0.61 -0.60 -0.60 -0.59
-0.66 -0.65 -0.64 -0.63 -0.62
-0.70 -0.68 -0.67 -0.65 -0.64
-0.746 -0.724 -0.703 -0.681 -0.661
-0.764 -0.736 -0.711 -0.689 -0.665
-0.766 -0.739 -0.714 -0.690 -0.666
-0.770 -0.740 -0.715 -0.690 -0.666
-0.770 -0.740 -0.715 -0.690 -0.667
-0.770 -0.740 -0.715 -0.690 -0.667
-0.23 -0.25 -0.26 -0.27 -0.28
-0.40 -0.41 -0.41 -0.41 -0.41
-0.51 -0.51 -0.50 -0.50 -0.50
-0.58 -0.57 -0.56 -0.55 -0.54
-0.60 -0.59 -0.58 -0.57 -0.55
-0.62 -0.61 -0.59 -0.58 -0.56
-0.641 -0.621 -0.605 -0.586 -0.570
-0.645 -0.625 -0.606 -0.588 -0.571
-0.646 -0.625
-0.646 -0.625 -0.606 -0.606 -0.588 -0.588 -0.571 -0.571
-0,646 -0.625 -0.607 -0.588 -0.572
-0.646 -0.625 -0.607 -0.588 -0.572
--CO
136 Table 3.3 (continued) Reliability
c,
0.1
0.5
1
2
3
5
10
20
3.6 3.7 3.8 3.9 4.0
7.72 7.86 7.97 8.08 8.17
5.30 5.35 5.40 5.45 5.50
4.24 4.26 4.29 4.32 4.34
3.17 3.18 3.18 3.20 3.20
2.66 2.66 2.65 2.65 2.65
1.93 1.91 1.90 1.90 1.90
1.03 1.01 1.00 0.98 0.96
0.28 0.26 0.24 0.23 0.21
4.1 4.2 4.3 4.4 4.5
8.29 8.38 8.49 8.60 8.69
5.55 5.60 5.65 5.69 5.74
4.36 4.39 4.40 4.42 4.44
3.22 3.24 3.24 3.25 3.26
2.65 2.64 2.64 2.63 2.62
1.89 1.88 1.87 1.86 1.85
0.95 0.93 0.92 0.91 0.89
0.20 0.09 0.17 0.15 0.14
4.6 4.7 4.8 4.9 5.0 5.1 5.2
8.79 8.89 8.96. 9.04 9.12 9.20 9.27
5.79 5.84 5.89 5.90 5.94 5.98 6.02
4.46 4.49 4.50 4.51 4.54 4.57 4.59
3.21 3.28 3.29 3.30 3.32 3.32 3.33
2.62 2.61 2.60 2.60 2.60 2.60 2.60
1.84 1.83 1.81 1.80 1.78 1.76 1.74
0.87 0.85 0.82 . 0.80 0.78 0.76 0.73
0.13 0.11 0.10 0.084 0.068 0.051 0.035
0.01
25
30
0.064 0.048 0.032 0.020 0.010
-0.072 -0.084 -0.095 -0.11 -0.12
O.Oo0
-0.010 -0.021 -0.032 -0.042
-0.13 -0.13 -0.14 -0.15 -0.16
-0.052 -0.064 -0.075 -0.087 -0.099 -0.11 -0.12
-0.17 -0.18 -0.19 -0.19 -0.20 -0.21 -0.21
curves (or distribution functions), and for graphical statistical analysis (extrapolation, graphico-numerical estimations of distribution parameters, etc.), we use papers with a special scale typical to the increasing scale for the probabilities P tending to the extreme values. Thus the exceedance probabilities curve is represented by a straight line, or by a line with low curvature. In the probability paper, corresponding to a probability distribution, the theoretical exceedance probabilities curve of the distribution is represented by a straight line. Therefore the probability paper is chosen according to which distribution we presume that the random sample is from. The probability paper of the normal distribution, having a linear vertical scale, is also called normal probability paper. The probability paper of the log-normal distribution differs from the preceding one by a logarithmic vertical scale. The log-normal curves are represented as straight lines providing the relation C, = 3Cv + C: holds. The same vertical scale, but a different horizontal scale for the relative values of the variable ki = xi/X, correspond to Brovkovich’s probability paper. Here, straight lines represent all binomial distribution curves, providing the relation C , = 2Cv holds.
137
40
50
60
70
75
80
-0.28 -0.29 -0.30 -0.30 -0.31
-0.42 -0.42 -0.42 -0.41 -0.41
-0.49 -0.48 -0.48 -0.47 -0.46
-0.54 -0.52 -0.51 -0.50 -0.49
-0.54 -0.53 -0.52 -0.51 -0.49
-0.55 -0.54 -0.52 -0.51 -0.50
-0.31 -0.31 -0.32 -0.32 -0.32
-0.41 -0.41 -0.40 -0.40 -0.40
-0.46 -0.45 -0.44 -0.44 -0.43
-0.48 -0.47 -0.46 -0.451 -0.441
-0.484 -0.473 -0.462 -0.454
-0.32 -0.32 -0.32 -0.33 -0.33 -0.33 -0.33
-0.40 -0.40 -0.39 -0.386 -0.380 -0.376 -0.370
-0.42 -0.42 -0.41 -0.401 -0.395 -0.388 -0.382
-0.432 -0.424 -0.416 -0.407 -0.399 -0.391 -0.384
97
99
99.9
100
-0.555 .-0.556 -0.541 -0.541 -0.526 -0.526 -0.513 -0.513 -0.500 -0.500
-0.556 -0.541 -0.526 -0.513 -0.500
-0.556 -0.541 -0.526 -0.513 -0.500
-0.556 -0.541 -0.527 -0.513 -0.500
-0.556 -0.541 -0.527 -0.513 -0.500
-0.486 -0.475 -0.465 -0.455 -0.444 -0.445
-0.487 -0.476 -0.465 -0.455 -0.445
-0.487 -0.476 -0.465 -0.455 -0.445
-0.487 -0.476 -0.465 -0.455 -0.445
-0.488 -0.477 -0.465 -0.455 -0.445
-0.488 -0.477 -0.465 -0.455 -0.445
-0.488 -0.477 -0.465 -0.455 -0.445
-0.434 -0.425 -0.416 -0.407 -0.400 -0.392 -0.385
-0.435 -0.426 -0.416 -0.408 -0.400 -0.392 -0.385
-0.435 -0.426 -0.416 -0.408 -0.400 -0.392 -0.385
-0.435 -0.426 -0.416 -0.408 -0.400 -0.392 -0.385
-0.435 -0.426 -0.417 -0.408 -0.400 -0.392 -0.385
-0.435 -0.426 -0.417 -0.408 -0.400 -0.393 -0.385
-0.435 -0.426 -0.417 -0.408 -0.400 -0.393 -0.385
-0.435 -0.426 -0.416 -0.407 -0.400 -0.392 -0.385
90
95
Using the horizontal scale linearizing the distribution of the standardized z, we obtain the Gumbel probability paper, the vertical scale of which is linear. The Frtchet probability paper differs from the preceding one by a logarithmic vertical scale. The horizontal scale of the Goodrich probability paper is log log 1/P' and the vertical is log x. The probability papers of the normal Gauss-Laplace distribution (and hence that of the log-normal one which has the same horizontal scale), the Gumbel distribution (hence the Frkchet distribution), and the Goodrich distribution may be constructed for probabilities in the range from 0.01% to 99.99%. Brovkovichs paper serves to illustrate the application of probability papers. The empirical distribution is approximated by a straight line tending to the same value C , in the appropriate scales both on the left and right sides of the chart. Brovkovich's paper may be used to check the correct calculation of C,. If the straight line, corresponding to the estimate of C,, approximates the empirical distribution well, then the calculation of C, is correct, if not, it is incorrect.
Table 3.4 Probability distributions used in water management
Type of distribution
Density f ( x ) Distribution function F(x)
normal (Gauss-Laplace) 1
standardized normal f ( t ) = -= e-''12 J2n 3parameters lognormal (Gauss-Gibrat)
F(x) =
A r Jn
c-
W
Domain of variable
Parameters
-wto+w
&a
--mto+w
-
Mean value p
0
e-yz dy
-m
0 to +cc
4parameters log-normal (Johnson)
rz) 1
f ( x )=
4 y ) exp
b-x
a to b
JG
3parameters exponential (Weibull)
f(x) = ab(x - xoP-' exp [-S(x
2parameters double exponential (GUUibel)
f ( x ) = e-'exp(-e-3
- xoP1
~ ( x= ) I - exp [ -b(x - xop]
Y
= a(x -
8)
~ ( x= ) exp ( -e-y)
xo to +-m
a.b.x,
-wto+w
4b
xo
+
1
Vs 0.577
8+-
r
(T)
a+l
Zparameters beta-distribution (Pearson I)
=
gamma-distribution Pearson I11
f(x)
=
binomial
f(x)
= (:)P14.-x
4
= 1
f(x)
= -e-A
Poisson
@ t o1
f - y l - .)8-’
f(z)
B(a9 B) 1
d(i
( x - xoy/dexp
a
a>0;B>0
a+B
a, d, xo
xo+a+d
0 5 x < n
n, P
nP
0sxsco
1.
1
m (number of degrees of freedom)
m
m (number ofdegrees of freedom)
0
m,,m,(degrees of freedom)
3 for m, > 2
(- x7)a. to + co -
a. B
xo
+ i)r(i + 1)
- p
P
X!
0
- w to + a
Student
Fisher (2)
to +co
logarithmic transformation of Fdistribution z
= +lgF
-co to +co
rn,
Table 3.4 (continued)
Type of distribution
Variance
Skewness
cT2
Cs
normal (Gauss-Laplace)
cT2
0
1
8
standardized normal 3parameters log-normal (Gauss-Gibrat) Zparameters log-normal (Ven-Te-Chow)
xo - close to min. observed value 3C,(x) + cg(x)
p2(y)(ea2(y) - 1)
4parameters log-normal (Johnson) 3parameters exponential (Weibull) Zparameters double exponential (Gumbel) 2parameters beta-distribution (Pearson I)
Note
-
x2
-
6a2
aB ( a + BIZ (a + B
+
1)
1.14 (const.)
y
=lnx
y
=In-
x - a 6-x
gamma distribution (Pearson 111)
J z ' d
d(a + d )
1
-
xo - min. value of x
a
- distance from
the mode to the origin
d -distance from the mode to the centre of gravity
2p
__
binomial
"PY
Poisson
A
xz
2m
2 J Im
Student
m m-2
0
"PY 1
for m
= p n , I > O
--t
rn close to normal
asymmetric
Snedecor ( F )
independent variables u, u are distributed'1 Fisher (z)
rn: (mz
-
2(m1
2)2
for rn, >.4
+ rn,
- 2)
m , ( m , - 4)
asymmetric
more symmetric than Fdistribution
142
3.1.8 Applicability of various probability distributions
The suitability of applying certain probability distributions (summary in Table 3.4) to a given sample is most effectively checked by marking several points on a probability paper; the paper is chosen so that the points are close to a straight line.
I
a)
-X
- x
Fig. 3.15 Typical shape of probability density of (a) Pearson, three parameter, log-normal, Gumbel and Frtchet distribution; (b) exponential distribution
Fig. 3.16 Frequency distribution (histograms) of flood maxima on the Otava river at Pisek (1931-1960) (a) annual; (b)thirty largest floods
The exponential distribution differs essentially from the other mentioned distributions by the course of the probability density y = f (x). This curve assumes its maximum (mode) at the point x’ to which the exceedance probability P = 1 corresponds. (Fig. 3.15b); the probability density of the Pearson, log-normal, Gumbel arld other distributions at first increases up to the mode then decreases (Fig. 3.15a). For example, the sample of observations of maximum flood discharges represented by annual maxima on the Otava river in Pisek from 1931 to 1960 yields a frequency distribution (histogram) (Fig. 3.16a) corresponding to the theoretical distribution in Fig. 3.15a. The samples of the thirty greatest observed flow maxima in the same period, however, have a distribution (Fig. 3.16b) corresponding to exponential law (Fig. 3.15). In spite of being similar in form, the probability density functions y = f ( x ) of various distribution laws differ essentially, namely in values of the radius of asymmetry b = Z - 2, the frequency of the mode ymax= f(a), and the asymptotic behaviour for P ( x ) + 0. The latter has a great influence on the extrapolation of the exceedance probability curve towards the extreme values. The convergence is very
143 slow in case of the Frichet distribution, as well as in the case of the Galton distribution. In water management the average annual flows and maximum flood discharges are most frequently analysed by means of statistics. Alexeyev (1960)stated that in the case of average annual flows, the use of the binomial, the three-parameter (Kritsky and Menkel), and the log-normal distributions yield approximately the same results. For the maximum annual flood discharges the three parameter and the lognormal distributions are recommended. In Czechoslovakia, particularly, distributions of all the maxima correspond, due to their nature, to the exponential distribution laws; the more general Goodrich exponential law is the most suitable.
The theoretical distribution law may be chosen from among those mentioned above, to fit the sample values. Extrapolation of the curves out of the range of the observed values towards the extreme values of probability P = 0 and P = 1, presumes an agreement between the theoretical and the empirical laws. The uncertainty decreases with the growing number of observations, but cannot be avoided completely because it is impossible to obtain sfliciently long series of observations to determine the probabilities of extreme values reliably. Therefore, the extrapolated values have only a conventional meaning; the advantage is that they are unambiguous for the chosen conditions. 3.1.9 Accuracy of statistical characteristics in water management
Let us evaluate the accuracy of the most frequently used statistical characteristics in hydrology and water management which are (a) long-term average (modulus),equation (3.47)
(b) sample standard deviation, equation (3.48)
S =
li=1
(c) sample coefficient of variation, equation (3.49)
J
2 (ki- 1)2
I n
c, =
i= 1
n-1
144 (d) sample coefficient of skewness, equation (3.51) ”
(e) sample coefficient of excess, equation (3.46)
The form of the exceedance probabilities curve is influenced the most by C,, less by C, and even less by the excess E (Fig. 3.17). The sample coefficient of variation C, = 0 corresponds to a sample consisting of values mutually identical; increasing C, means that the sample values differ essentially from the average. Figure 3.17d 2 75 250 225 200
2.00
I 80 I60
1.40
d)
1.75
z 120
.L:
-x
t
0.40 a20
m
250
I25
050 0.25 QDO
- Q25
-am cs to
-P
[%I
- 075O W 2 0
50 -PC%7
100
Fig. 3.17 Influence of C,, C, and E on the shape of the Pearson type Ill exceedance probability curve (a) influence of C , ( q = 0); (b)C, (C, = 0.5);(c) graphic illustration of positive and negative skewness; (d) influence of E (C, = 0.5, C, = 0)
shows both positive skewness (C, > 0, the mode to the left of the mean) and negative skewness (C, < 0,the mode to the right of the mean). On the diagram (Fig. 3.17b), a graphical descriptions of the curves with C, = 0.5 is given for three different values of C,. The course of the curves is less dissimilar than in Fig. 3.17a, especially in the middle part. The greatest difference can be observed between the respective ordinates corresponding to the extreme values of P . Greater values of C,yield greater values of the end ordinates. The curves corresponding to the smaller values of C, are more suitable from the low flow period viewpoint. Therefore, as a rule, the minimum admissible value of C, is chosen if the series of observations is not long enough to provide a reliable calculation of C,.
145 From Foster’s expression C, = 2b/C, (Morozov, vol. I, 1954) it follows that for the flows the lower boundary of C, is
c, 2 2 c v
(3.163)
The upper boundary follows from the equation of the Pearson type I11 curve
(3.164) The relation C, = 2C, is often used; it results from the theory of binomial distribution curves. The values of C, might, however, be smaller, close to zero or even negative, or on the other hand much greater. It is recommended not to insist strictly on the Foster relation C, 2 2C, and to choose according to circums!ances, even c, c 2c,. With C, < 2Cv the binomial distribution curve may, for a greater dependability of P, in case the of flows, have unrealistic negative values of k. The accuracy of statistical characteristicsis influenced not only by the observation time, but also by the variability of flows. The coefficient of variation of annual runoff for most rivers is in the range of 0.10 to 1.20. The influence of the number of observations n and the coefficient of variation C, on the accuracy of the characteristicsof runoff(X,s, C,, C, and E) is given by estimating their standard errors (Votruba - BroZa, 1966, p. 82). In order to illustrate the above, Table 3.5 shows the standard errors of all the statistical characteristics under C, = 0.1 and 1.0, providing that n = 50, C, = 2C, and E = $: = 6Ct. Table 3.5 Standard errors of statistical characteristicsi, a, C, C,, E for number of observations n = 50
C“
0;
[“XI
a,*%“
[“A1
[“A1
%.
aE
PA1
0.1
1.4
10
149
1400
1.o
14.0
20
60
210
The real hydrological series of many Czechoslovak rivers are sufficiently accurate to determine the characteristicsx,s and C,; the errors ac, and especially aEare, however, too great. The calculationsabove are based on the assumption that the observed random variables are mutually independent. If the assumption is not fulfilled the errors are even greater.
146
In dry countries there are rivers that in some years have zero runoff, hence corresponding to an exceedance probability of less than 100%. It therefore holds that C, < 2C,;according to the requirement of agreement between the theoretical and the empirical distributions, we choose as a rule C, = 1.5CVor C, = C, (Voskresenski, 1956).
The value C, may be influenced, even for very long series, by one highly extreme year, the theoretical probability of its occurrence being too small in a series of a given length (e.g., W, of the extraordinarily high water year 1941 in Czech and Moravian rivers). Svoboda (1964) used the 110 year runoff series obtained in DeEin on the Labe to calculate the long-term values of the coefficient of variation of annual runoff of Czech and Moravian rivers on the basis of shorter series of observations. For C, and C, empirical formulae were derived, applicable for preliminary calculations or for very short series. For Czechoslovak rivers the derivations were made by A. Bratranek, Dub (1953), Svoboda (1963); see also Dub and Nemec (1969). 3.1.10 Evaluation of'goodness-of-fitbetween empirical and theoretical distributions
When investigating the same quantity, but under different conditions (e.g., average annual flows, maximum peak discharges, flood volumes, etc.), by means of the probability theory, we cannot estimate the type of theoretical probability distribution unambiguously. Thus, the uncertainty contained in the estimate gives rise to the question as to what extent the chosen theoretical distribution agrees with the empirically obtained quantities. Using the same coordinatesystem for a graphical description of both the histogram of the sample and the theoretical frequency function, we obtain the most illustrative idea, at least qualitatively, about the agreement between the theoretical and the empirical distributions ; less illustrative is the conception of the table of numerical empirical values of frequency ni of a respective class interval and theoretical values nfi where n is the number of elements of the sample, fi is a part of the area limited by the theoretical distribution curve within the limits of the respective interval. We now proceed to solve the problem of testing the hypothesis that data form a sample of a random variable 5 with the distribution P(x) of a given type. Such tests are called tests of goodness offit, and are based on a choice of a certain measure of deviation between the theoretical (hypothetical) and the empirical distributions. If in the case investigated the deviation exceeds a fixed level, the hypothesis is rejected, and vice-versa. The tests most frequently used are the x2-test and the KolmogorovSmirnov test. Applying the x2 test introduced by Pearson, we calculate the quantity (3.165)
147
and compare it with the value x,” for m degrees of freedom and small level p (usually 5% or 1%).The number of degrees of freedom is (3.166)
m=I-c-1
where I is the number of classes into which the space of the variable is divided, and c is the number of sample characteristics used to estimate the unknown parameters of the distribution j(x). With x2 c xf the agreement between the distributions is considered to be sufficient, with x2 > x j it is not. Using this test we must bear in mind that it should hold if nA 2 5. As this condition might not be fulfilled, especially for the outer classes, we pool two or three of them if necessary. The agreement between the distributions is tested very simply with the aid of the Kolmogorou-Smirnov test, based on the probability P{max IF,(x,)
- F(XJ
> D,(n)}
(3.167)
=a
Tuhle 3.6 Critical values D,(n) of the Kolmogorov-Smirnov test n
a = 0.05
a = 0.01
1
0.975 0.842 0.708 0.624 0.563 0.519 0.483 0.454 0.430 0.409 0.39 I 0.375 0.36 1 0.349 0.338 0.327 0.318 0.309 0.301 0.294
0.995 0.929 0.829 0.734 0.669 0.617 0.576 0.542 0.513 0.489 0.468 0.449 0.432 0.418
?
. I
4 5 6 7 8 9 10 11
I2 13
14 15 16 17 18 19 20
0.404 0.392 0.381 0.371 0.361 0.352
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 50 60 over 60
a = 0.05
u = 0.01
0.287 0.281 0.275 0.269 0.264 0.259 0.254 0.250 0.246 0.242 0.238 0.234 0.231 0.227 0.224 0.221 0.2 18 0.2 15 0.213 0.210 0.190 0.175 1.358 ‘ -
0.344 0.337 0.330 0.323 0.317 0.311 0.305 0.300 0.295 0.290 0.285 0.281 0.277 0.273 0.269 0.265 0.262 0.258 0.255 0.252 0.228 0.210 1.628 -
Ji
Jrr
148
where the expression max IF&) - F(xi)( means the maximum value of the difference between the theoretical and the empirical distribution functions ; the test criterion Du(n)depends on the level a (usually a = 5% or 1%)and the number of observations. With n > 60 it holds that Dub) = D,(n) =
1.358 J;r 1.628 J;r
for a = 5%
(3.168)
for a = 1%
(3.169)
If the calculated value D = max IF,(xi) - F(xi)l exceeds the tabulated value D,(n) (Table 3.6), the hypothesis about the agreement between the theoretical and the empirical distributions is rejected on the given level of significance (the deviation between the theoretical and the empirical distributions is statistically significant). Statistical estimation of probability distribution parameters is one of the most important parts of mathematical statistics. So-called sampling distributions of the sample characteristics are used for the estimation, namely
x2 distribution - Student distribution - F-distribution (Snedecor) - Fisher distribution. -
Either point or interval estimates are used to estimate the mean and the variance. The interval estimate is more suitable as it also provides the accuracy of the result. The accuracy depends on the number n of the sample values. A frequent problem is what the number n should be for the unknown parameter to be estimated with a certain (chosen in advance) maximum permissible error or accuracy. The estimated values of parameters can be checked (tested) if the distribution law is considered as known. 3.1.11 Random processes and sequences in hydrology and water management
In a random (stochastic) process, the random variable X depends on a parameter which, as a rule, is the time t. If the variable X is continuous in time we use the term “random process”, if the parameter assumes only integers “random sequence” is used. The following considerations for random processes are also valid for random sequences. Just as a random variable is characterized by its distribution function F(x) and frequency function f ( x ) , a random process can similarly be characterized just by adding the variable parameter as another argument, hence, e.g., F(x, t).
149
Figure 3.18 gives a graphical description of several realizations x(’)(t),x(’)(t),... ..., x(k)(t)of a random process X(t). For a fixed value of parameter t = t , these realizations can be considered as a sample from a random variable which will be denoted by x 1 in agreement with the index oft,. The sample values x(’)(t,),x(’)(t,),... ...,x(k)(ti)are denoted in the figure by 1,2, ..., k. We calculate the number m of values less than or equal to, a given value x,. The relative frequency P = m/k is a function
\,,,---p
Fig. 3.18 Graph of several realizations of a random process
of x 1 and estimates the probability of the random variable X ,being less than or equal to x,. Hence, the distribution function of the random variable X, may be written in a form similar,to equation (3.18): F(x,)= P(Xl 4 xl)
(3.170)
This distribution function characterizing the random process is written as q x , , t , ) = p ( x , 5 x1)
(3.171)
and is called the first (one-dimensional) distribution function of a random process; it depends on the given parameter (time) t , and level x,. The first probability density of a random process is (3.172)
The random process is characterized at a fixed time (in static way) by (3.171) and (3.172). We may, however, simultaneously fix another value of parameter t,. The second (two-dimensional)distribution function of a random process is given by &(XI,
r , ; x,, t z ) = p ( x , 5 x , ; x2 4 x,)
(3.173)
and denotes the probability of the random process not exceeding the value x , at time t = t , and simultaneously not exceeding the value x2 at time t = t,. The second probability density of the random process is (3.174)
150
Analogously, the n-dimensional distribution function may be derived, expressing the probability of the random process X(t) not exceeding the levels x,, x2, ..., x, in the n values of parameter t ,t,, ..., in,respectively. In problems of water management the first and second-order distributions are usually sufficient. Similarly to the case of random variables, we use certain numerical characteristics of’ random process, namely (a) the first-order general moment determining the mean value of the random process
,
1-;
m,[x(t)l=
m
fl(x9r) dx = p(t);
(3.175)
(b) the second-order central moment determining the variance of the random process
j-? m
M,[X(r)] =
-
p)’j,(x, t)dx = m,
-
mf = a,(?)
(3.176)
(c) the first-order product general moment determining the correlation function of the random process m,[X(r,) X(t,)]
=
jm jm
X I X 2 f,(X,, f , ;
-m
-m
x2.tz) dx, dx, = R(t,, f 2 )
(3.177)
The investigation of the random processes is much simpler if they are stationary and ergodic. A random process is stationary if all the finite-dimensional densities remain the same whenever an arbitrary T is added to each parameter, hence (3.178)
In our investigations it will be sufficient to check the first- and the second-order densities. Similarly, only the invariance of the first and the second distribution functions is usually checked, and consequently that of their numerical characteristics - the mean value eqn. (3.175) and the variance eqn. (3.176) which are constant in the case of a stationary process, and the correlation function eqn. (3.177) which turns to a function of a unique time parameter. All random processes shown in Fig. 3.18 are stationary. Non-stationary random processes show a certain trend, e.g., either decreasing or increasing the mean value or variance. As an example of a non-stationary random process let us mention the time series of flows in a river site where water is withdrawn for irrigation, with a successive growth of the irrigated area. Since about 1975 we have been studying theoretically the non-stationary phenomena in hydrology and their consequences for the design of reservoirs (Nachrizel and
151 Patera, 1975; Nachazel, 1976). Since 1976 the non-stationarity has been examined in the Ohie basin. The non-stationarity may either grow continuously due to man’s interference in nature, or it may arise suddenly, e.g., as a consequence of building a large new reservoir on a river. The most significant non-stationarity lies in water demands which are quickly and progressively growing; in water management plans and projects this must not be overlooked. In the future, the non-stationarity of the hydrological quantities will also have to be taken into account. A random process is ergodic if the average of sufficiently many values in the realization of the process is close to the population mean value, with a probability close to one. In Fig. 3.18, the processes represented by the full line are stationary ergodic, those represented by hatched lines are stationary non-ergodic. In the case of an ergodic process, a large number of realizations need not be investigated, a single long one is sufficient. The assumptions of stationarity and ergodicity enable us to consider one realization as representative for many others. The fulfilment of the assumptions may be checked only with a very long series which may be divided into several parts and their characteristics compared. The first-order probability densities are mutually the same for a stationary random process, hence we may write f ( X P tl)
=fdx)
(3.179)
Thus, the calculation of the first-order density of a random process is transformed to the calculation of the density of the random variable X,the realizations of which are given by the values of the realizations of the random process in any fixed time. The mean value m , ( X ) of the random variable X is, therefore, the mean value m , [ X ( t ) ] of the random process X(t). 3.1.12 Correlation (autocorrelation) functions of random processes
Figure 3.19 shows two random processes X , ( t ) and X&) which can have the same mean value and variance, and yet differ essentially: in Fig. 3.19a the value of each realization in the process X , ( t ) is decreasing, while in Fig. 3.19b the realizations show
2 L
0 4
f
_ I
b
0 1,
-f
z
Fig. 3.19 Inner structure ofseveral realizations of a random process
152
considerablepulsations. This difference is described by the correlation function which is another characteristic of a random process, expressing its inner structure, i.e., the correlation dependence between the values of the process in various moments of time. For a pair with a time lag the value of the correlation function is equal to the coefficient of correlation of the corresponding random variables. According to equation (3.177), the correlation function R(t,, t 2 ) is given by the mean value of the product [X(t,) - X(t2)], hence R(t,, t 2 ) = m,[X(t,) X(t,)]
(equation 3.177’)
Sometimes it is useful to express the deviations of the random process from the mean u(t). Such a random process is called centred and is given by X(t) =
x(t)- a(t)
(3.180)
Its correlation function is R(tl, t 2 ) = m,[~(t,)X(t2)1= m,{[x(t,)
-4
1 ) l
[X(t2)-
4lll
(3.181)
and is sometimes called the correlation function of the pulsations of a random process. Between the correlation function of a random process [equation (3.17811 and the correlation function of the centred process [equation (3.181)] is the relation R(t1, t z ) = R(t1,
4 -4 1 )4 2 )
(3.182)
Dividing k(tl,t 2 ) by the product of the standard deviations o[X(t,)] o [ X ( t , ) ] of the corresponding random variables X(t,) and X(t2), we obtain the standardized correlation function of the random process (3.183)
It follows from equation (3.183)that the standardized correlation function is equal to the coefficient of correlation of the random variables X(t,) and X(t2). Thus for the normalized correlation function it holds that
Irl 5 1
(3.184)
If the random process is stationary, the respective correlation functions of the process and the centred process depend on the difference z = t , - t,, only, hence R(t,, t 2 ) = R(z) and
k(t,,t 2 ) = k(z)
(3.185)
The relationship between them is
k(t)= R(t) - a2
(3.186)
153 The standardized correlation function of a stationary random process is
r(z) =
R(t)- uz -A(t)
4
(3.187)
4
For example of correlation function see Fig. 3.20.
-
w l;;
$
’O
b L
2
:
t
~ +
07 2
01 2
--r
-Y
0 1 2 3 4 -Y
0 1
-
l
:
+
-Y
Fig. 3.20 Examples of random-processcorrelation functions (a) absolutely random process (“white noise”); (b) exponential correlation function; (c) power correlation function; (d) correlation function in the form of a damped harmonic motion
The correlation function in Fig. 3.20a may be described by the relation 47)=
/
1 for
‘ 0
for
T =0 t
+o
(3.188)
i.e., the values X ( t ) are uncorrelated for any two moments in time. Such processes are called absolutely random (white noise) and differ from each other only by either mean values and distribution functions. In spite of not being really possible in hydrology, they are used when the problem of storage reservoirs is solved with the aid of general statistical characteristics. The correlation function in Fig. 3.20b may be expressed by the exponential function r ( t ) = e-‘(’)
(a > 0)
(3.189)
The curve rapidly tends to zero. The constant tl determines the rate of disappearance of the correlation. The correlation function in Fig. 3.20~is given by the power function r ( t ) = r( 1)’
(3.190)
where r(1) = r(z)l= This curve also decreases rapidly; e.g., with 41) = 0.3 it holds that 43) = 0.027 for t = 3. The power correlation function is typical for the simple Markov process which is used to design the storage reservoir volume.
~
154
The correlation function in Fig. 3.2Od may be expressed as the damped harmonic motion r(T) =
e-a(r)cos PT
(3.19 1)
For small z the curve decreases, as in Fig. 3.20b, c; then it turns to negative values, and oscillates around zero, the amplitudes becoming smaller and smaller. It was proved that the empirical correlation functions of rivers in Czechoslovakia are of a harmonic nature; therefore they may be better approximated by the function (3.191), in spite of not exhibiting any perpetual damping. The possibility of negative correlation of more distant quantities substantially influences the reservoir volume under higher values of the coefficient of safe yield. Figure 3.21 shows three empirical correlation functions of a 30-year sample moving-average series (Votruba-Nachkel, 1971): (a) that of annual relative Wolfs numbers, (b) annual precipitations, (c) average annual flows.
-m+
Fig. 3.21 Empirical correlation function of (a)average annual relative Wolfs numbers (1831-1894); (b)annual total of precipitations in Prague-Klementinum (1851-1914); (c) average annual flowsof the river Berounka at Ktivoklat (191 1-1960)
155
The correlation function of the average annual relative Wolfs numbers (Fig. 3.21a) has a very regular harmonic behaviour with the values of the positive maxima being mutually similar. The 11-year period of solar activity is very significant. The correlation function of precipitations (Fig. 3.21b) is also periodic, although not as regular as in Fig. 3.21a. The negative maxima are even higher than the positive ones. The course of the correlation function of the annual flows (Fig. 3.21~)is similar to that in Fig. 3.21b which proves the similar behaviour of the long-time fluctuation of precipitation totals and runoffs. Calculation of sample correlation function In the case of an ergodic random sequence, two approaches may be applied (Fig. 3.22).
Fig. 3.22 Diagram of calculations of the correlation function of an ergodic random sequence (u)for every 7 the same number of sample value pairs; ( b )for every r the complete sample range
(a) From all the n values of the observation series a shorter sequence containing m members is taken and correlated to the sequence shifted by 1,2,. ..,7 (Fig. 3.22a). Hence, according to equation (3.8),it follows that (3.192) where xi Xi+r
t
are the sample values from x 1 to x,,, are the sample values from x1+ r to x,,,+*, is the time parameter of the correlation function (time lag), varying from T = 0 to T , , , ~ ~= n - tn (the complete range T ( 0 , ~ ~ ~ ~ ) need not be exhausted),
156
is the sample mean of the values xi, (xi+,), ~ ~ , ( c r ~ + ~is) sample standard deviation of the values xi, (xi+,). The advantage of this calculation method consists in assigning the same weight to all the r( l), r(2),..., r(z). All the empirical correlation functions in Fig. 3.21 were calculated in this way. (b) The complete range of the observation series (Fig. 3.22b) is used to calculate the correlation function r(z) for every T = 0,1,2, ... . Using the same notation as above we can write Xi, (Xi+J
n-r
r(T)
=
C
(xi i= 1
- 'i)(Xi+r - 'i+J
biui+r(n- T
-
1)
(3.193)
The number of correlated pairs is greater than in case (a), but with a growing T it decreases to (n - 2 ) pairs at the end. For a growing T,the reliability of the calculation ) therefore, reduced. of r ( ~ is, For method (a) the confidence limits are (according to R. L. Anderson): -1
-2
t,Jm
(3.194) m-1 where t, is a standardized random variable quantile corresponding to the significance level (1 - a). For the usual significance levels of 95% and 99%, the quantile values are t,,95 = = 1.645 and t,,99 = 2.326, respectively. For the method (b) the confidence limits are rr, =
=
-1
t,Jm - z - 2 m-2-1
(3.195)
where t, is the same as in equation (3.194). 3.1.13 Spectral densities of random processes
The spectral density S(W), being the Fourier transformation of the correlation function, is a very important characteristic used to investigate the periodicity properties of a random process. In the case of a centred random process S,(O) = Tc
s,"A(T)
cos WT dT
where o is the circular frequency 2n o=T providing the period is denoted by T (e.g., number of years).
(3.196)
(3.197)
157
According to N. Wiener and A. J. Chinchin the correlation function of a stationary random process may be expressed by means of the spectral density as follows (3.198) Hence, with the aid of equations (3.197) and (3.198), it is possible to transform S(w) into r(~),and vice-versa. If the complex representation eior
+
-ior
(3.199) 2 is used, we can derive (substituting S,(w) = 2S:(w) and integrating from - 00 to 00) the expressions coswz =
rm
I
(3.200) W
(3.201) The spectral densities Sx(w) and S,*(o)differ mutually by the scale and the range of frequency o (only non-negative w for S,(w)). Similarly to equation (3.200), we may write for a non-stationary random process (3.202) With hydrological series mostly stationary spectral densities have been investigated. Here, because of short observation series the spectral density is calculated on the basis of a single realization. Nachazel and Patera (1975) investigated the correlation functions and spectral densities of 17 river points, 13 in the catchment of the river Labe, 4 in the catchment of the rivers Morava and the Danube. They used the approximation expression of A. S. Monin (3.203) where T is the length of the period in years, m is the length of the interval of the correlation function argument T values, r ( ~ is) the value of the correlation function according to equation (3.193) where the values x, are substituted by annual discharges Q,. Figure 3.23 shows the correlation function m (Fig. 3.23a) calculated by method (b) (see Section 3.1.12), and the spectral density (Fig. 3.23b) of the flow series of the river Berounka at Kfivoklat (1911-1960). The course of the correlation function differs slightly from that in Fig. 3.21~which was calculated by means of method (a)in Section 3.1.12. The confidence limits corresponding to the levels of significance 95% and 99%, respectively, are the dashed lines. The curve of the spectral density assumes two strict maxima for T = 3 and 6 years, respectively, and another maximum for T = 13 years. The river Vltava at Kam);k and the river Sazava at Pofiti show a similar course in their series.
158 Reznikovski (1969) introduced analyses of 83 rivers in the U.S.S.R. In 25 of them periodic components appear, while for the others simple Markov chains without periodic components are better suited.
1:
i
-
99% 95%
a2
30
-02 2-04 5, - 06
,
95% 99%
-.?
1;
T
Q5
Lo4
0
0
5
70 -T
15
20
25
30
Fig. 3.23 Correlation function ( u ) and spectral density ( b ) of annual flows series of the river Berounka at Kiivoklat (191 1-1960)
The methods of filtration are supplementary in statistical analysis of hydrological series. Their essence consists in separating the random influences from the random sequence under investigation, e.g., the average annual flows; after filtration the properties (correlation, periodicity, etc.) briefly intervene and a more reliable basis for the mathematical model of the random sequence is obtained. An analysis of correlation functions and spectral densities of filtered series was performed by Nachlzel and Patera ( 1974) for annual flow series, precipitation series, temperatures, and relative Wolfs numbers.
Thus, a filter used in the filtration methods transforms the given random process. This leads to another process with the desired probability properties. The theory of transformation of random processes was first elaborated in radio technology (see e.g., Levin, 1965, p. 252). The function of the filter is expressed by the transfer response function h(z).A linear system input random process (t) is transformed to another random process c,m-kzc.m-l C
C
~c.1n-k~c.m-2 C
C
+
* * *
-t
'k.m
>c.m-kzc,m-k C
?+) +
2 (in years).
After dividing each equation of the system [eqn. (3.217)] by the number of correlated pairs reduced by one, i.e., by c - C, it follows that ml(zm-lzm)
= bl,mml(zm-
+ b2,mml(zm-
l'm-1)
+
1'911-2)
'**
+ bk,mml(zm-
lZm-k)
(3.218) ml(zm-kzm)
= bl.mml(Zm-kZm-
1)
+ b2,mml(zm-kzm-Z)
+
**'
+ bk,mml(zm-kZm-k)
The expressions m, mean the elements of the correlation matrix which are calculated from the given real flow series. The unknown regression coefficients bi,, are determined by solving the system in eqn. (3.218).Thus, r(1,m)=bl,;1 r(k,m) = b,,,r(k
+b2,,r(1,m)+...+bk,,r(k-
1, m)
+ b2,,r(k
- 2, m)
1,m)
+ ... +
(3.219) bk,,,
*
1
The meaning of the determinants D, and Di,,, (i = 1,2, ...,k ) is the same as in the model of the annual flows; however, they depend on every particular month m. The residual variance in every month m follows from the equation si=
) bl,mml(zmZm-l) = m ~ ( ~ ;-
- bZ,mml(zmzm-2)
According to the limited length of the series, it holds that
- . * * - bk,rnml(zmzm-k) (3.220) (3.221)
The previously calculated values of coefficients bi,,, and m,(z,z,- i) may be used advantageously; thus only m ( z i ) remains to be determined.
165
When testing for the proper length of the Markov chain, we follow the same approach as in the case of the model of annual flows, i.e., the residual variance is determined in dependence on the length of the Markov chain until the minimum is reached. The particular steps in the calculation with the aid of a computer are analogous to those of the annual flow modelling, the only difference is that the appropriate procedures are repeated for every month. The last step of the calculation is again the inverse transformation of the artificial flows; if the log-normal distribution is used for the direct transformation, then for the inverse transformation it holds that xc,m
=
exp [zc.m%l
+ LI + x,,o
(3.222)
Method of fragments The method of fragments (Svanidze, 1963) is based on double random sampling. With the aid of the method described above, the first random sampling of the average annual flows is performed (Fig. 3.24a).Then the second random sampling of fragments
",/ n 700
600
3cx)
200 100
0
Fig. 3.24 Modelling of monthly flow series by means of the fragment method (a) first random sample Qr.,; (b) second random sample (of fragments &)); (c) section of modelled series of average monthly flows
follows, corresponding to the real relative flows Q/Qr,i= q = q(t) in individual years of observation; thus in every year it holds that 4 = 1 (Fig. 3.24b). The fragments are numbered and with the aid of the random-numbers table are assigned to the individual years of the synthetic series. The ordinates of the fragments are multiplied by the
1 66
corresponding Qr,i of the synthetic series and an arbitrarily long, pseudo-chronological synthetic series with distribution of the flow within the years is obtained (Fig. 3.24~). Applying the method in this form, when no relation between Q, and the distribution of the flow within the year is supposed, may lead to unrealistic results*). By investigating relations between Q , and variability of flow during a year, the unrealistic randomness may be excluded. The method might be improved by increasing the number of fragments to be comparable with the number of years of observation by means of modelling. Thereby, however, the method loses its main advantage, i.e., simplicity, whereby it does not need modern computers. Since the introduction of numerical computers, the method of fragments has lost its importance as compared with the method of the linear regression model. There is, however, one advantage: it maintains the correlations between the annual flows Q,, whereas with the regression method the correlations between monthly flows Q, are retained, but the correlations between values of Q, might not be. For a long-term annual cycle, the method of fragments should give more exact results, because of the importance of a proper correlation between Q , values, whereas for a cycle of a few years, more exact results should be obtained from a series modelled by the regression method, because the correctness of the correlation between Q , values is ensured. 3.2.3 Modelling sequences of monthly flows in a system of stations
Between the empirical flow series at points on a river a significant correlation often exists (Fig. 3.7, Table 3.7). The table shows that the coefftcient of correlation r = 0.69 is high, even between distant catchments (the Labe to Josefov and the Vltava to Kamik). In two cases the coefficientis as high as, r = 0.94. If reservoirs work in a system, their collective efficiency is generally higher than the total of their isolated effects. The rate of the increased effect of the system, as compared with the isolated effects of individual reservoirs in the system, depends on the synchronization of their hydrological regimes and their relative volume. The more asynchronous the regimes, the higher the effect of the system. If flow series of a system of river profiles are modelled, a double correlation relationship has to be considered: - correlation (auto-correlation) within each series, - cross-correlation between synchronized flows of various stations of the system. *) If from the 30-year series from 1931 to 1960 for the river Berounka at Kfivoklat a fragment from 1954 is assigned to the highest-flow year, 1941, then in July we obtain an average flow equal to 2.5 times the greatest observed monthly maximum and greater than the maximum peak flow of a two-year flood. On the contrary by assigning a fragment of 1947 to the year 1934, the minimum monthly mean value is smaller by one third than the minimum daily flow of the whole thirty years (Votruba and Brora, 1966, p. 145).
167 Table 3.7 Correlation between the annual flows of some gauging stations in Bohemia (19261946)
Stream gauging site
Catchment area
the Labe-Josefov the Labpardubice the Vltava-Kam);k the Berounka-Beroun the Vltava-Modfany
12 213.0 8 287.9 26 706.1
the Ohfe-Louny the Labe-MElnik
4981.4 41 798.3
1840.2 5 987.3
Correlation coefficient r
-
0.92 -
0.69 0.76 -
0.71 0.78 0.89
-
0.71 0.79
0.94 0.93 -
0.75 0.80 0.85
0.79 0.86 0.92
0.89 0.87
0.92 0.94 0.89 -
-
-
Two basic methods are developed for modelling flow series of a system of stations: - method of central and satellite stations, - method of orthogonal transformation. (a) Method of central a n d satellite stations By this method we model first the flows in the central station (e.g., by means of the method of the linear regression stochastic model according to Sections 3.2.1 or 3.2.2)and then the temporally corresponding (coinciding) flows in the other satellite stations, according to the linear regression : (3.223)
where yi,j,k is the flow with the ordinal number i in monthj, in station k, L j , k - average monthly flow in month j, in station k, bj,k - regression coefficient in monthj for the relationship between the flows of the central station and of the satellite stations, x-i , j - flow of the central station, with the ordinal number i, in month j, x j - average monthly flow from the central station in monthj, ui - standardized random variable,
168 sj,k - standard deviation of flows in month j , in station k, - coefficient of correlation between flows in the central and the satellite station k, in month j . For any pair of the central and satellite stations we prepare twelve equations of type (3.223)and (3.224)and the calculation is carried out for the individual calendar months. The method is suitable mainly for relatively synchronic regimes; otherwise the auto-correlation of the series of the satellite stations may be considerably disrupted. In the case of a looser bond between the two stations, equation (3.223)can contain not only the contemporary flows in the central station xi, but also the preceding flows xj- and also, if need be, the other flows yj- in the satellite station.
,,
(b) Method of orthogonal transformation This method is more general and its principle consists in the transformation of the linearly dependent flows in any calendar month at various stations into independent (orthogonal) flows. The given real flows xi,j,kin the same month j , but at various stations k, form correlated random vectors which are transformed to independent vectors. First the real flow sequence in one calendar month and at all stations is transformed to a normally distributed sequence which, moreover, is standardized. Then the orthogonal transformation is performed, and a random series at hypothetical stations is modelled. By the inverse transformation we obtain the correlated random sequences in the real stations and by another inverse transformation we finally obtain the desired synthetic sequences with a given mean value and standard deviation. In the case of a large system of stations (reservoirs), we may be limited by the computer capacity. Then we cannot model the whole system at once, but parts (sub-systems) with closer connections have to be modelled separately. Hence, the same problem arises as with the independent modelling of isolated series: the possibility of deranging the real correlations between the individual sub-systems. The most important things are to analyse the problem, to prepare a program for the computer, and finally to interpret the results properly. 3.3 INFLOW TO A RESERVOIR
In designing a reservoir, it is absolutely imperative to know the law of inflow. Inflow can be - natural, influenced by man, or completely controlled by human intervention, - deterministic or stochastic, - constant or variable.
169
The characteristics of inflow to a reservoir depend on its function (flood control or direct supply) and on the duration of the control cycle (Table 3.8). For short-term control, a deterministic inflow can be considered; for over-year control, a stochastic one in necessary. Table 3.8 Inflow to a reservoir and withdrawal with various release control cycles
Control cycle
daily
weekly
annual
over-year
Inflow
deterministic
deterministic
deterministic or stochastic variable natural or changed
stochastic variable natural or changed
stochastic
stochastic
variable controlled
variable controlled
steady or variable steady as a rule natural or frequently controlled by man controlled Withdrawal and controlled release
deterministicor stochastic variable or steady controlled
deterministicor stochastic variable as a rule controlled
A constant inflow to a reservoir is introduced only for short-term release control by distribution reservoirs. Controlled inflow to a reservoir is frequently introduced for short-term discharge control (pumping of water to a water tank, to the reservoir of pumped storage hydropower plants, etc.). Lateral reservoirs have a more or less controlled inflow as the water is brought from the main stream, or generally speaking, reservoirs that receive their water from other catchments. The flood inflow to a reservoir must also be taken into consideration in the determination of its flood-control effect. This i d o w is short, but very variable. To solve the problem, the real pattern of a flood wave is sometimes used. However, more frequently, models of floods are constructed with various probabilities of occurrence, both as to their maximum peak discharge and to the form this takes, and thus the flood volume. This is usually a natural or influenced inflow, and is included in the design as deterministic, with a certain probability of being exceeded. For the inflow to a reservoir it is not sufficient to consider the statistical data of the past and the present states, but also a forecast of the development of inflow in the future has to be elaborated with regard to the long physical life span of a reservoir and of the more progressive facilities that control the inflow. More and more frequently, water for reservoirs will be diverted from one catchment area to another. Economic analysis optimizes the size of the transported flow and
170
determines the dimensions and costs of the conduits. For preliminary calculations it is possible to determine the amount of water diverted with a certain chosen reliability; this can be done either numerically or graphically by a summation curve, which is especially suited for an analysis of alternative conduit capacities. The diagram in Fig. 3.25 illustrates that - with a minimum maintained discharge in the river Qmin= 0.25Qr,the amount of water diverted in one year decreases by 25% of the annual discharge W,, - if the sum of the conduit capacity and the maintained minimum flow Q,,,,, + Qmin= l.SQ,, the amount of water diverted per year decreases by the excess discharge q,which is 18% W,, - if the conduit is Q,,,,, = 1.25Q,, the annual diversion has the value of Wdiy= = 0.57 W, of the design year, - the total water diversion Wdivis decreased to a greater extent by maintaining a minimum flow in the stream than by a small capacity of the conduit, as peak flows last only a short time so that the volume of the excess discharge W', is relatively small.
+
Fig. 3.25 Analysis of water diversion by means of the summation curve (u)time curve (hydrograph) and exceedance curve of discharges in the river from where the water is diverted; (b)summation curve corresponding to curves (a)
When diverting water directly from a river (without storage) it is usually not sufficient to work with monthly or ten-day discharges, but the continuous curve of discharges or at least the average daily discharges have to be considered. The calculations could be incorrect if the average discharges within the month are used. This error need not be significant if the conduit has a large capacity and if the flood peaks exceeding this capacity are only very short. It is more difficult to construct a chronological or pseudo-chronological series for the balance solution of a reservoir, if we do not want to limit ourselves to a direct solution on the basis of observed series. Essentially there are two such cases:
171 - the natural inflow to a reservoir is negligibly small (as for a lateral reservoir) so that the inflow is only the diverted water, - the design has to consider the natural inflow together with the diverted water. For the over-year reservoir, a long series of discharges in the river site from where the water is withdrawn and diverted to the reservoir will have to be modelled. Let us presume that mean daily discharges will be considered. No reliable method for direct modelling of the chronology of mean daily discharges with the help of a linear regression model has yet been elaborated. These are actually not natural daily discharges, but discharges from which the maintained minimum discharge
Qmi"
I
I
XI
I
I
I
I
I
xu
11
Ill
-t Fig. 3.26 Reservoir with water diversion ( u) layout; (h) adjustment of hydrological input data at site I; (c) construction of inflow to reservoir (QII + QCJ
172 and the excess discharge, exceeding the conduit capacity, will have to be deducted. Naturally, such a reduced discharge will have different characteristics from a natural discharge. For the fragment method, the method explained in Section 3.2.2 is applied. Only the pattern of the fragments would change : instead of monthly (or ten-day) discharges, the fragments consist of daily discharges. A model of the inflow to a reservoir, i.e., a series reduced by the minimum maintained and excess discharges, can be constructed as follows (Fig. 3.26), e.g. : 1. observed series of mean daily discharges Q, in the withdrawal profile I are reduced by the minimum maintained discharge Qmin and the excess discharge Qi (Fig. 3.26b),whereby a series of daily discharges through the conduit Q,,, is obtained; 2. the synchronous daily dischargesin profile I1 and the reduced discharges through the conduit Q,,, are added; from these the series of the mean monthly discharges to a reservoir (Q,, + Q,,,) are calculated; this is the input set for the modelling of the series; 3. from this initial series (Q,, + Q,,,),e.g., a linear regression model of a series of any length of average monthly flows to a reservoir is modelled; 4. the modelled series serves as a basis for the design. It is also possible to model the series of the mean monthly discharges in profiles I and I1 separately, bearing in mind the correlation relationship between the two profiles according to Section 3.2.3; but as the series in profile I was deformed by the reduction, it does not seem that this more complicated method can have more reliable and accurate results. If the discharge Q,,is negligible, modelling is limited to the model of the reduced series Q,,, using the above method, but substituting Q,,= 0.
4 RELEASE (WITHDRAWAL) FROM A RESERVOIR Besides inflow, the law of outflow from a reservoir is the most important factor in its design. Outflow is the water volume that leaves a reservoir per second by outlets, spillways or withdrawal; withdrawal is the water volume that leaves a reservoir per second only when it is specifically withdrawn. Outflow from a reservoir is determined by a reservoir release schedule, i.e. a set of principles, rules and directives, approved by water-management authorities. Surcharge over a spillway depends on the water level; the other components of outflow are regulated by various devices and can be adjusted to meet the given demands. The reservoir release schedule includes data about the maintained minimum outflow, i.e., the smallest outflow from a reservoir to a stream must be preserved to meet the demands of other users downstream of it, for sanitary reasons, etc. Withdrawal of water is given by the need for water, i.e., the sum of water demands from a reservoir with their time pattern and reliability of water supply. Outflow from a reservoir not used for the purpose for which the reservoir was built is called excess outflow. According to the function and control cycle (Table 3.8) withdrawal and controlled outflow can be - deterministic or stochastic, - constant or variable. In designing the storage volume of a reservoir, water losses must be taken into account, i.e., the amount of water escaping by seepage, evaporation, etc. 4.1 USERS’ WATER DEMAND
To determine the design water demands for the solution of the storage capacity of a reservoir is one of the most complicated problems. It must respect the following
principles: - bear in mind the needs of the region and determine all the various demands, - consider the order of importance of the respective demands, - consider not only the quantity of water, but also its quality, - select economically justified time levels for meeting the water demand, - determine a justified measure of the reliability of the water supply, - optimize the design by alternative solutions,
174
select a justified maintained minimum flow downstream of the reservoir, consider cooperation with other reservoirs in the system, - consider the influence of the environment on the reservoir and its influence on the environment. Every user has demands as to the time pattern of water supply: water supply for households and industry is relatively balanced throughout the year; irrigation needs are concentrated in the vegetation period; power resources needs are usually concentrated in the winter months. The designer of a reservoir cannot decide by himself the amount and quality of water for the various users. This is a problem in which many various branches of industry are involved; however, the water-management experts always hold a control and coordinating position. It is also necessary, although difficult, to determine the future withdrawal. In Czechoslovakia, the following forecasting methods are used : - analytical smoothing and extrapolation of past time series, - direct investigations in the most important enterprises, factories, etc., - determination of the total demands of the respective production branches, - application of prospective standards of water needs and demand, - estimation of the total water demand with respect to the number of people. All these methods serve extensive more than intensive uses of water resources. They do not incorporate the water price adjustment, which is reflected in an increasing demand for new water resources. In the nineteen-seventies the term demand has been introduced in forecasts of water needs. The future rate of water demand gives more reliable data for the planning of future capital investments for the exploitation of water resources. How these aspects will be applied in future constructions depends on the level of the exploitation of water resources and on the economic system of the country. -
-
The International Institute for Applied System Analysis (IIASA) is preparing, together with the member ‘countries, an extensive Survey on Methods f i r Eastimating Water Demands and Waste Water Discharges. In January 1977 a discussion was held on Modelling of Water Demands; further discussions will be organized. The IIASA considers two approaches to the problem of lack of water: - gaining new resources (extensive method), - more eflicient utilization of the existing resources (intensive method). Stress is being put on the second method, which so far has been applied only to a limited extent, as water was always considered a free or reasonably princed “commodity”. Fees for water supply and lines for water wastage are not effective enough. If they are low, their influence is very small, if they are high, they are unfavourably reflected in the final product, which in turn affects the consumer. Thompson and Young (1973) stressed the significance of the forecasting of the “law of demand” for water as: - enabling alternative proposals for the utilization of water as the factors influencing water demand changes,
175 - supplyingdata for the evaluation of whether capital investmentsfor flow regulation or water diversion between catchments are justified. For economic solutions of the questions of utilization of water and treatment of waste water, analytical methods (Thompson and Young, 1973) or programming methods (Calloway er a/., 1974) can be used.
4.1.1 Requirements of public water supply
Supplying households with drinking water is an important task in Czechoslovakia, as it greatly effects the living standards of the people. The greater part of the population receives its water from public water mains (in 1985 it was 76.2%, in the year 2000 it is presumed that it will be 90%).To rationalize the use of water, regional water mains and water mains systems are being built and are supplied from reservoirs. h
(Y
-€
Fig. 4.1 Exploitation of a new water resource in time
The need for a new water source must be determined well in advance, leaving sufficient time for the design and construction. For this, a forecast of the water need in a given area is indispensable (Fig. 4.1). To take a hypothetical case, the yield of an existing resource will be depleted in 1986 and a new resource 1 should take its place. This is to be a large over-year reservoir which will take ten years to be designed and built. Work on the reservoir should, therefore, have been started in 1976. The problem of the first filling of a reservoir has to be analysed carefully, as it is only when the reservoir is full that it can carry out its complete function. The following must be taken into consideration: - whether the reservoir could start to be filled already’ during its construction (which depends, among other things, on the type of dam), - the need for a full output right from the beginning. It is usually not required that a waterworks reservoir starts full operations right
176 from the first years, and it will only be used completely after several years. Such reservoirs can also be built in stages. According to Fig. 4.1 the yield of resource 1 is depleted in time t , (1993),when the water need will rise to 0,.By building the reservoir in stages, stage Z could supply a yield of 0, - 0,, which will be depleted in time t , (1986);then stage ZZ could be build. However, building in stages requires greater capital investment and must therefore be considered carefully. In our hypothetical case, construction would be stopped from 1976 to 1983. However, the new resource might be put into operation in time ti instead of t,. At the time between t , and t; the lack of water is balanced by measures designed to economize on water. These limiting measures can be mitigated by a more extensive use of other resources, which can be replenished when resource 1 has been completed. This means a depletion of storage volume in other reservoirs, mainly in over-year reservoirs. Studies of water needs in the region at various time levels help to determine the water demand increase. The storage capacity of new reservoirs is determined by methods corresponding to the respective outflow control (Chaps. 5 to 10).It is usually designed for a constant withdrawal according to the given reliability, which for drinking water is rather high (Section 4.4).Drinking water is largely returned to the stream (about 70%)after treatment in a sewage-treatment plant. 4.1.2 Demands on water for industry
Every branch of industry has its own specific demands as to the amount and quality of water. Only very few technological products need less technological water than their own weight; some need many thousand times more than the weight of the products. In 1985,the water demand in industry was five times higher than for households; of this amount, 12% was for consumption. The greatest amount of water is needed for power production. Hydro-power plants only make use of the mechanical energy of water; they do not consume water and do not change its quality, but only re-distribute the discharges. Steam power plants need water mainly for cooling purposes and for the transport of slag and ash. The demand for water is very high, especially if no recycling of water is used. In Czechoslovakia, withdrawal of water for the production of heat and electrical power came to 1700.lo6m3 in 1985. Other large groups of water users in industry are fuel production, products from coal and oil, the chemical industry and the paper and cellulose industry, which in 1970 needed more than lo00 lo6 m3 of water. Metallurgy, machinery and the metalworking industries required more than 380 * lo6 m3 of water. Even though about 90% of the water used by industry is returned to the streams, industry is still a very demanding consumer as to amount and negative impact on the quality of the returned water.
177
Industry is not always seriously effected if the supply of water is temporarily decreased. However, in designing new water resources for industry, a large enough reliability of the yield must be ensured. It is possible to use two values of reliabilities: a high value for the supply of water for technological purposes and a lower one for the total demand. 4.1.3 Demands on water for agriculture
Agriculture needs water for irrigation and for livestock. Water demand for agriculture in Czechoslovakia amounted to 225 * lo6 m3 in 1985, though only one ffth came from public water mains. Water demand for irrigation is increasing and the water is almost 100%consumed. The amount in 1970 was 79. lo6 m3,and about 176 * lo6m3 in 1985. Water demand changes greatly according to weather conditions and the amount and distribution of precipitation in time during the vegetation period. This is why the water demands also differ greatly. When designing the storage capacity of a reservoir for irrigation, the size of the area irrigated must be determined as well as the amount of irrigation water and its distribution in time. If no reliable data are available, the water demand in a so-called design dry year is used as a basis. 4.1.4 Demands on water for hydro-power production
Hydro-power systems must consider two aspects in their water demand : - power production, i.e., the needs of the electric power system, - water management, i.e., the needs of the other water users. Different types of hydro-power plants affect the interests of water management in different ways: (a) Run-of-the-river hydro-power plants do not change the flow regime, nor the characteristics of the water so that they affect other users only by a raised water level and any investments that have to be made to ensure navigation, etc. (b) Peak-load hydro-power plants incorporated in or adjacent to dams greatly influence the flow regime of the river without, however, changing the amount or quality (except for changes caused by the very existence of the reservoir). Any unfavourable changes of the discharge downstream of the power plant can be regulated by a daily or weekly storage reservoir. (c) Peak-load hydro-power plants on diversion canal have a similar effect on the flow regime as in (b). If the water is withdrawn from one stream and let out into another, the flow regime of both rivers changes essentially. (d) Pumped-storage hydro-power plants do not as a rule limit other users, except for losses from the upper and lower reservoirs. Run-of-the-river hydro-power plants do not work with accumulated storage water
178
and can therefore best be designed with the help offlow-duration curves. The volumes of the upper and lower reservoirs of pumped-storage power plants are determined by the demands of the power system, making the solution of water management issues rather simple. The design of a reservoir with a large share of peak-load power production is rather complicated. The parameters of the reservoir depend on the local conditions and must take into account other water users and also bear in mind the place it occupies in the power system. The reservoirs are usually of the over-year control type. The released water volume is larger in winter than in summer. Besides covering the peak-load parts of the daily diagram, these power plants ensure further services and needs of the power system. 4.1.5 Water demands of other users
Naoigation may need a controlled increase of discharge from a reservoir to make a river navigable or to supply water to a canal. For the controlled increase of river stages to be effective, the storage capacity of the reservoir must be large. The economic efficiency of navigation must therefore be considered from the point of view of the total optimal utilization of the water resource. Navigation on rivers can also make use of the controlled discharge from reservoirs for other purposes, e.g., power production or flood control. Navigation uses water, but does not consume it. It can, however, in certain cases increase the pollution of the river. A reservoir or a river downstream of a reservoir can be used for recreation purposes. Recreation has its demands as to the amount and quality of water. The water level in a reservoir should not change greatly and the banks should be firm. At the time that a river is used for recreation the flow should be large enough to ensure good water quality. This concerns the discharge during dry periods, when it is almost minimum. Summer recreation downstream of a reservoir is unfavourably affected by cold water released from a reservoir, and this influence can be observed up to some dozen kilometres away. The quality of water in a stream can be improved by an occasional flushing of the stream channel by an increased discharge. Water sports can also be performed on reservoirs and streams. Controlled increase of discharge in the dry summer months is of help. Competitions on “wild” water in the upper sections of rivers usually have to wait for a long discharge wave from a reservoir. Fish and poultry farming has its specific demands on water management. These can best be met by reservoirs (ponds) constructed for this purpose. Large reservoirs offer suitable conditions for fishing, if the water is clean enough. If, under certain conditions, zones with a low oxygen content appear in the lower levels of a reservoir, incorrect manipulation might cause large numbers of fish to perish.
179 4.2 HOW TO MEET THE WATER DEMAND
When deciding about how to meet the water demands of a certain region, the demands must be compared with the water resources both as to amount and quality. The results of this comparison supply the most important data for decisions concerning the measures that must be introduced for using the existing water resources and for introducing new ones. The solution is based on the water-management balance of the past period, the present balance and the forecast balance, for 5 years or up to the years 2000 and 2015. The design of a new reservoir must be based on the study of its function in the whole system and an analysis of its relationship with the environment. Optimization of the design is usually a complex technico-economic and social problem. Method for determining new resources f o r drinking-water demands within the framework of regional waterworks 1. Definition of the system and its environment
The final scope of the system (e.g.,up to the year 2015)must be determined according to the water plan and natural conditions (water resources,morphology of the territory) and according to the location of villages, industry and agricultural production. The environment of the system is defined and its interactions with the system too. 2. Meeting the needs of industry and agriculture
The system of water supply for a region is divided into sub-systems according to their purpose, e.g., waterworks sub-system, industrial water, water for agriculture, flood control, pollution control, etc. The water demands of these sub-systems must be assessed as well as their links with the waterworks system. 3. Assessment of the present state of drinking-water supply to the region The present state is described and analysed, including ground and surface water, water users and facilities for the treatment and transport of water. The present operations must be judged as well as all possibilities of a rational utilization of the present resources, e.g., by increasing the capacity of the water mains, a more systematic exploitation of the advantages of systems, etc. 4. Demographic prognosis
Wit! the help of existing data, the future number of people living in the respective parts and communities of a region must be determined to serve as a basis for the calculation of the drinking-water demand in the whole region and its respective parts.
180
5 . Future drinking-water demand The drinking-water demand is determined according to various spans of time, e.g., up to the years 2000 and 2015. The specific water demands according to the size of the community and the coefficients of unequal demands are determined according to water-management plans. The growing number of people supplied by public water mains must be taken into consideration. Drinking water for industry and agriculture must be determined separately. 6. Balance of demands and resources
For every balance unit the demands and local resources must be compared. From these the demands on the central resources can be determined. The capacity of the existing central resources is assessed with the help of the most recent hydrological data, progressive methods, etc. 7. Drinking-water deficits
Drinking-water deficits are calculated to the given time levels in mean and maximum values and the amount to be covered by central resources is determined. This gives us the year in which a new central resource must be operated.
8. Analyses of potential drinking-water resources Both ground and surface water is taken into consideration. The most promising resources are considered together with any possible further impacts. Water-management development plans in the given area and its surroundings are taken into consideration. 9. Basic demands on a new water resource (reservoir)
The design of a new reservoir must meet all the aims it is to serve (or the priorities should be stated), e.g., withdrawal of as large as possible an amount of drinking water, improved water quality in a stream for other water withdrawals, improved flood control, ensured recreation facilities, etc. 10. Forecast of the quality of withdrawn water and consequences of reservoir operat ions
The present and future quality of the water flowing into a reservoir and changes in the quality as a result of water storage in a reservoir must be judged. Protective zones are designed as well as other measures to ensure a good quality in the stream and tributaries. A forecast of the effect of waves on the reservoir banks, the effects of periodical emptying and filling of a reservoir, any possible eutrophication of
181
a reservoir, the winter regime, etc., must be determined. Figure 4.2 shows the time-related lowering of the water level in a reservoir which, if the banks are not under water for any length of time can lead to overgrowing of a reservoir banks by vegetation and the deterioration of water quality (Kubin, 1976).
Fig. 4.2 Continuous depression of the level below the spillway elevation in relation to the storage volume A ,
11. Determination of reservoir parameters
The parameters of a reservoir which form the basis of the conception and dimensions of the project are determined by a technico-economic evaluation of alternatives. 4.3 WATER LOSSES FROM RESERVOIRS
A reservoir changes the runoff from an inundated area as a result of evaporation, infiltration, formation of ice and changes in the water storage (ground water, snow, etc.). Changes in the runoff as a result of evaporation are caused by changes from the land surface to water surface. The difference in the water balance E caused by evap oration and changes in the water storage can be expressed by the relationship E = E,
-
E,
AW
(44
where E, is evaporation from the water level, E , - the total evaporation from the territory prior to inundation (from the soil and transpiration), A W - change in the water storage.
182
A W can influence the water balance of short seasons. Mean annual water storage does not change essentially, so that one can write
E
= E, - E ,
(44
Evaporation E , can be expressed as the difference between precipitations runoff 9 from the inundated area, therefore
E,=X-P
X
and (4.3)
so that
E
= E, - (X -
9)
(4.4)
If, in the time interval At [ s ] , E is given in metres, and if the increase in the water level caused by a reservoir is denoted by A F [m’], the mean change per second of the water balance (by evaporation losses) q [m3 s - l ] in interval At can be written as q
EAF
=
7
(4.5)
whereby AF changes according to the state of the water level in a reservoir. J. VaSa (VaSa and Wurm, 1975) calculated the evaporation from a pan 5 m in diameter at Hlasivo (547 m above sea level) with an average temperature slightly higher than the long-term average for the period from 1957 to 1974. Table 4.1 gives the values supplemented by measurements with a Wild atmometer in the winter months (November till March).. Ttrhle 4.1 Evaporation distribution during the year (J. VaSa)
1
2
3
4
5
6
Mean daily [mm d - ‘3 evaporation [“A]
(0.2) 1.19
(0.3) 1.54
(0.7) 4.35
1.5 8.70
2.2 12.70
3.0 16.97
Max. daily [mmd-’1 evaporation
(1.9)
(3.1)
(3.8)
5.7
7.0
7.1
7
8
9
10
II
12
Mean daily [mm d - ‘3 evaporation [“A]
2.9 17.16
2.8 16.39
1.9 10.72
1.0 5.80
(0.4) 2.41
(0.4) 2.09
Max. daily [mm d-’1 evaporation
7.8
6.5
5.3
5.1
(3.0)
(2.6)
Month
Month
183 The mean annual evaporation was 528 mm (1.45 mm day- '), the mean evaporation during the vegetation period (April - September) was 436 mm, mean evaporation during summer (June - August) was 267 mm (50% of the annual evaporation). P. Petrovic elaborated a nomograph to determine the mean daily evaporation from the water level according to the mean (monthly) pressure of water vapour in the air and according to the mean air temperature (Fig. 4.3).
Fig. 4.3 Determination of the mean daily evaporation from a free water Ic\cl h! nlc:1114 of
-
"
.I Il0mogr;lpIl .-
mwn monthly air fempemfum L O C I
If the influence of evaporation on a reservoir is small, it is suficient to include in the design the mean annual evaporation; if the influence is greater, a more detailed analysis must be worked out (Section 2.1.2). If the evaporation in the low-period is large, evaporation losses are included in the design with a certain probability where p is the design reliability of water supply. Seepage can be just as important for the design of a reservoir as evaporation. Water can escape through the banks or bottom of a reservoir, the dam or other functional structures (bad sealing, etc.). When including seepage losses in the design it must be decided whether the water escapes to another valley or whether it returns to the river downstream of the dam, and whether water is withdrawn from a reservoir or from the river downstream of a reservoir. In the first years seepage is rather extensive, later it usually decreases. The extent of seepage through the dam and subsoil can in simple cases be calculated or assessed by suitable analogues.
184 Table 4.2 Leakage in Czechoslovak reservoirs (M. Simek)
Type of dam
earth rockfill with earth sealing earth and rockfill with concrete, asphalt-concretesealing, PVC foil concrete and masonry gravity
Number of reservoirs wjth leakage (1 s - ' ) less than 5 5 a 20 more than 20
17
6
2
4
1
1
2
2
1
with some exceptions less than 0.15 1 s - ' per lo00 m2 of wetted area, only exceptionally more than 2 1 s-
'
Note: With half the depth of backwater, leakage drops below 20% of the values of the full depth of backwater.
Losses caused by bad sealing, can be determined by using the type of equipment described in the literature or from data supplied by manufacturers. Table 4.2 shows seepage through the dams of Czechoslovakia. Water losses caused by ice cover have a greater impact only in reservoirs with flat banks, which have a great drop in their water level in winter, when the ice cover rests on the banks and temporarily (until it melts or until the water rises again) decreases the available water volume in a reservoir. If the mean width of the ice resting on the banks is h and the water level at the beginning of the winter is Fb and at the end of the winter F,, the water losses caused by ice are AV = 0.9h(Fb- F,)
(4.7)
4.4 RELIABILITY OF WATER SUPPLY FROM RESERVOIRS
From the economic point of view it is indispensable to introduce the rate of reliability into the solution of technical and economic problems. It is a very important question, as only a small change in the value of the rate of reliability leads to an essential change in the basic parameters of a reservoir. Although the determination of an optimal rate of reliability of water supply for any purpose is mainly an economic problem, it also has its non-economic impact. Definition of t h e basic terms of reliability Reliability of water supply from a reservoir (or energy from a hydro-power plant) is the probability that the ensured parameter of water supply (or energy) will not
185
drop below the given value. The parameter can be the amount per year, the amount per second, output, etc. More generally, the design reliability of a reservoir (hydro-power plant) can be defined as the probability that the users will receive the full amount of water (energy), needed for their most expedient operations. Quantitatively the design reliability can be expressed (a) as the ratio (percentage) of the number of years in which the ensured supply of water (energy) is not limited, to the whole period in questiomccurrence-based reliability Po; (b) as the ratio of the duration of water (energy) supply without any breakdowns to the total duration of the period in question-time-based reliability Pt ; (c) as the ratio of the actually supplied water (or energy)to the value of the demand on water (energy)supply during the period in question-quantity-based reliability Pa In all three cases the rate of reliability of water or energy supply from a given reservoir is expressed as a percentage P of the value of the full amount meeting the needs during the period in question. The total deficit of water (energy) ACO, during the given period is the total amount of water (energy) in m3 (kWh) that is lacking, during low-flow periods, to ensure the volume of water (energy) that is to be supplied. The relatiue deficit of water (energy) is the ratio of the total deficit to the demanded quantity of supply during the whole period in question, expressed as a percentage (1 - Pd). Mean annual deficit of water (energy) is the share of the total defcit of water during the given period and the number of years. For flood control, the term reliability reflects the average time (number of years) in which the design value of non-damaging flow (the so-called recurrence interval of floods) is not exceeded. However, it is no longer only the quantity, but mainly the quality that has to be taken into consideration. Regulated flow in streams helps to dilute waste waters and to increase the water quality, relationships between the discharge and the water quality are not simple and are being studied further. Water also changes its quality in reservoirs, sometimes to its detriment. In determining the optimal reliability of water supply, we must always bear in mind not only the quantity, but also the quality, which can have direct economic and non-economic consequences. Economic aspects must be considered from the economic and not only from the technical point of view. Methods using economic parameters are less reliable than those using technical parameters, as the availability of statistical and economic data is limited. Difficulties arise as it is almost impossible to assess exactly the economic consequencesof a limited supply of water and it is also most difficult to evaluate the impacts caused by deficits of drinking water objectively, by maintaining the minimum flow in the river etc.
186 Water balances incorporate the meeting of water demand with various degrees of reliability for various users, depending on the consequences caused by incomplete water supplies, The rate of reliability of a full supply of water is given by a standard of design reliability, determined on the basis of the occurrence of unfavourable hydrological conditions in long-term perspectives. In hydrological series of several decades, the period of a minimum discharge corresponds to a reliability of 95 to 99%. This reliability has become the standard for important consumers. Where a decreased supply does not cause any great damage, failures are admitted to occur once every five to ten years, i.e., 80 to 90% probability of years without any failure. Sometimes two standards of reliability for two withdrawal values are given: a lower one for the full standard of needs, a higher one for a reduced minimum. Supplies (P = 97 to 99%) must be ensured for industries and housing estates. Supplies can be decreased for navigation, irrigation and hydro-power plants in large power systems with base-load thermal power-plants. The higher the standard of the design reliability, the smaller the losses caused by deficits of water, but the higher the capital investments that have to be put into the water resource. By comparing the two consequences, an economic optimum for the meeting of the demands of the respective users is found. As it is very difficult to solve these problems, the design standard is determined on the basis of experience from design and operation practice. In compiling a balance, the standards of the design reliability Podcs,are usually given as a share [%I of years, in which water is supplied without any failure. The values usually are: (a) supplies for households and industry (b) irrigation (for the growing season) (c) livestock farming (d) hydro-power production (e) navigation (f) maintained minimum discharge in streams
[“A1
95 - 99 90 - 95 95 - 97 80-95 90 - 95 95 -99
Over-year release control must have a higher design reliability than seasonal control. If water is withdrawn from a stream without a reservoir, the design reliability P,dcs.can be given as the share of the duration of time in which the supply is fully guaranteed. More accurate standards than Podes,should be used in those cases where they convincingly express the economic consequences of deficits of water.
187 4.5 LOCATION OF THE ACTIVE STORAGE CAPACITY IN A RESERVOIR
The total storage capacity o f a reservoir A , with a volume V,, confined by the bottom and sides of a reservoir, by the dam (or weir) and the maximum water level, is usually divided into smaller areas, according to the purpose of the reservoir (Fig. 4.4).
-v Fig. 4.4 General scheme of the division of a reservoir storage capacity
(a) dead storage capacity A,, which under normal conditions is not used for flow control, but for the settling of sediments, to ensure the required water quality, to create a head, to preserve a sflicient amount of water for fish when the higher regions of a reservoir are emptied in low-flow periods, etc.; a part of the permanent storage is the technological dead capacity below the lower outlets, which cannot be emptied without pumping, (b) active storage capacity A,, which helps to regulate the flow and withdrawals in low-flow periods, (c) flood control capacity A,, which catches the water from floods and transforms flood waves; it can be regulated (Aro), (d) or non-regulated (A,"),called also surcharge capacity (Aov),which is above the level of the overflow crest of an uncontrolled spillway or the overflow crest of a controlled spillway. The sum of the volume V, of the active storage capacity and the volume V, of the flood control capacity gives us the volume of the regulation capacity. The size of dead storage is usually given indirectly by the minimum permissible operation level of the active storage capacity; decisive therefore is its depth h, and not its volume V,, which we try to make as small as possible so as not to increase the total volume of a reservoir and thus its costs. This is also why we place the bottom outlets as low as possible to avoid any technological dead storage or making it only large enough for sediments to settle. To ensure a good quality of water (especially in reservoirs for drinking water) the
188
minimum level of the active storage capacity M,should be relatively high above the bottom of a reservoir; usually 12 to 15 m. The reason is that water to be withdrawn should be out of the reach of any chemical or biological factors near the bottom and that an adequate depth of water is preserved above the place of withdrawal even at the time of the reservoir’s lowest level (Fig. 4.5).
ho ‘d
Fig. 4.5 Depth of dead storage level M, corresponding to the lowest withdrawal level from a withdrawal device built in steps
From the point of view of the quality of water (especially its temperature), a great depth of the active storage capacity is important, as it enables withdrawals from various levels (or from several levels simultaneously), while preserving the quality of the withdrawn water. A suitable head is most important for hydro-power plants. The various types of turbines used in hydro-power plants need a relatively steady head, which is reflected in a given minimum level of the storage capacity. Reservoirs with hydro-power plants therefore usually have a relatively large dead storage capacity (e.g., Orlik reservoir A, = 280 lo6m3 and A, = 364.lo6m3). The minimum head needed for water withdrawal by diversion canal (for water treatment plants, for irrigation, etc.) can also be one of the deciding factors in the selection of the water level. In the interest of preserving the natural environment, the storage capacity should ensure that the flat muddy parts of the valley are under water level, even when a reservoir is being emptied; the water level should not change to any great extent when a reservoir is also used for recreation purposes. The storage capacity on gravel-bearing streams has its specific characteristics. The forecast of the gradual silting of a reservoir with gravel influences not only the position
Fig. 4.6 Diagram of the silting of a reservoir and the corresponding elevation-storage curve (a) change in the longitudinal profile due to silting; (b)corresponding change of elevation-storage curve
189
of the active storage capacity (dead storage serves to collect the sediments), but also its size (silting of a reservoir decreases its storage capacity). In Fig. 4.6, a cross-section of a reservoir shows the gradual filling with gross sediments: presuming that the flood control capacity A,, is empty, it is the active storage capacity A, that is the first to be filled. As sediments reach a reservoir mainly during flood flows, it is at that time that the flood-control capacity is filled (in Fig. 4.8 dotted, stage 1 ) ; in view of the large inundated area, the decrease in size is relatively small. In stage 2 the original active storage capacity decreases from the original V,,, by AV,,* to the value V,,z; the other capacities remain unchanged. In stage 3 the sediments reach the foot of the dam, partly silting the dead storage capacity A, and increasing the silting of the active storage capacity. In stage 4 the dead technological capacity is completely silted, the volume of dead storage decreases from the original V,,, to K,4 and the active storage capacity decreases to the value V,,& The whole silting process is, however, more complicated. Sediments have different grain size, from gravel to fine suspended particles. Gross sediments settle mainly at the end of the backwater due to the decreased carrying force, fine sediments usually go as far as the dam. Gross sediments shift constantly deeper into a reservoir as the result of the smaller depth of the stream channel due to silting, which again increases the flow rate and also as the result of the changes in the water level of a reservoir. Reservoirs are silted not only by sediments from the inflow, but also by abrasion of their banks caused by waves. If the active storage capacity is used for flood control in connection with the forecasting service, a rapid decrease of the reservoir volume must be possible. The same applies to reservoirs which are built to solve difficult sanitary conditions or removal of ice downstream of a reservoir with the help of an artificial discharge wave. If a reservoir does not have any flood control function, the level of the active storage capacity can be its maximum level. This solution allows for a sufficientlylarge gated spillway. If the spillway is not gated, there must always remain an uncontrolled capacity above the level of the active storage capacity, the depth of which is determined by the need to transfer the design flood over the spillway. If a reservoir also has a flood-controlfunction, a flood-control capacity can usually be found above the active storage capacity (see Chap. 12). Another way to solve the problem of special demands on the storage capacity is a reserve capacity, by which the design value V, has to be increased. The boundary between the dead storage and the active storage capacity and between the active storage and flood control capacity need not be fixed. The size of the storage capacity is designed with a water supply reliability P 100%. In a specially low-flow year, the storage capacity can be completelyemptied and a deficit in the water supply can occur. At that time water from the dead storage capacity can be used.
-=
The boundary between A, and A,, is less defined, if the active storage capacity is also used for flood control, either according to forecasts or after the partial or complete catching of a flood wave in the over-year storage capacity that has previously been slightly emptied.
5 OVER-YEAR RELEASE CONTROL Over-year reservoirs are currently used to solve the conflict between increasing water demand and limited resources. They form one of the most important links in systems which supply various users with water from surface resources. Growing demands and limited resources must be balanced by the regulation of natural discharges, which can only be done with the help of over-year discharge control. Let us consider a reservoir with a storage capacity (in this chapter storage capacity means active storage capacity) which has to ensure a relatively high yield. The coefficient of the relative yield a (= Op/Qn)is e.g. equal to three quarters, the required reliability Po = 99%. In the hydrological conditions pertaining in Czechoslovakia, low-flow periods occur during which, from time of the beginning up to the stage of
Fig. 5.1 Stages of the work regime of a reservoir with an over-year release control inflow and release; (b)illustration of the reservoir function by means of discharge and release mass curves; (c) time pattern of the emptying of the storage capacity
(u) time pattern of
192
complete emptying of the storage capacity takes longer than one year. The subsequent filling of the latter can take several years. The greatest emptying of the storage capacity of a reservoir does not take place at the end of the low-flow season of the first year of a longer period of low flows, but in one of the following years; the excess discharge of the high-flow seasons in between is not sufficient to refill the storage capacity (Fig. 5.1). Low-flow periods lasting for several years include an annual cycle which must be taken into account in the design. At the same time, the discharge is balanced for a short period, e.g., a month, a week, a day. In high-flow years, the storage capacity of a reservoir is emptied to some degree during low-flow periods of the year, but it is refilled in the following high-flow period. In such years only a seasonal or annual discharge regulation typical for within-year (seasonal) release control is applied. In high-flow years, the low-flow periods last only a few days and the inflow to a reservoir does not drop below the value of the outflow (withdrawal). The storage function of the reservoir is then made use of only in the few days of a low-flow period so that the storage capacity is mostly full. If several high-flow years follow one another, the storage capacity of a reservoir is only used to a limited extent, even in over-year release control. The storage capacity of a reservoir is designed on the basis of data obtained from a sequence of several low-flow years. Therefore the design need be based on the long-term flow-regime characteristics. An over-year reservoir cycle can occur for reservoirs with diversion (withdrawal), with river flow regulation, with reservoir in a cascade, and for those working in a system. This chapter deals with reservoirs with direct withdrawal. The storage capacity with an over-year release must be designed in such a way as to accumulate water in high-flow years and to raise the release (withdrawal) in low-flow years (when Q, < (Ip). The theoretical exceedance curves indicate that most of the cases where a reservoir has a storage function with direct withdrawal can theoretically be included in the over-year outflow control. Where the probability of exceedance P (abscissa) is very close to the value lOO%, the ordinates of the theoretical curve of exceedance (mean
Fig. 5.2 Considerations about release control in the exceedance curve of mean annual discharges (inflows to a reservoir)
193 annual discharges) are close to zero (presuming the validity of the Pearson type I11 distribution where C, = 2C, corresponds to the value P = 100% zero value of the mean annual discharge).Theoretically there is always a non-zero interval (Pa;loo%), Fig. 5.2. In this interval the values Q, of the mean annual discharges are smaller than the required mean annual withdrawal from a reservoir 0, [for Pa it holds that Q,(P,) = O,];it is therefore necessary to have a certain volume from the high-flow years in the storage capacity of a reservoir, which means that the design working cycle of water supply is an over-year cycle. If discharge regulation is not extensive, the probability that the mean annual discharge will be smaller than the withdrawal is very small; the range of the interval (Pa;100%) is, in this case, close to zero. To disregard this influence does not mean that it will cause an unacceptable error and we are therefore justified in using the method which is typical for within-year release control. With the design reliability Po, the following cases can occur (Fig. 5.2): (a) Pa 2 Po, which might be considered to be the sign of a within-year (seasonal) release control. To ensure the required withdrawal with the given reliability, it would suffice to stabilize the uneven discharge distribution within all the years of the interval (0; Po) in such a way as not to cause a failure in the water supply. It can, however, be presumed that in some years which, by the value of their mean annual discharge are in the interval (Po;Pa), withdrawal from the reservoir without any failure would be required; in an over-year relase control (Pa;loOo/,). over-year low-flow periods can occur in which the reservoir volume designed for 'discharge regulation in very low-flow seasons within the framework of within-year release control, will suffice to stabilize the discharge. It is therefore correct to consider this case, too, as an over-year release control; however for Po + 100% the method of within-year release control can also be used. (b) P, c Po, which is sufficient condition fo,r over-year release control. In practice the design reliability is usually very high (Po = 95 t 99%, rarely less). Then even if Pa < Po the interval (Pa;100%) has so small a range that the influence of over-year release control on the characteristics of the storage function cannot be expressed quantitatively and a simpler process must be used. Division into within-year (seasonal) and over-year outflow control is of great practical importance. In over-year outflow control the design size' of the storage capacity is given by low-flow periods lasting several years. These low-flow years can vary as to their number, the size of the mean discharge and the variability of flow during the year. The longer the reservoir cycle (i.e., the more extensive the regulation of the discharge), the more thoroughly the changes in long-term characteristics of inflow to a reservoir will have to be studied. At the present state of knowledge, the most expedient method is the study of synthetic sequences of mean annual discharges from which a mathematical model can be elaborated. In hydrological conditions with
194
irregular regimes of high-flow and low-flow periods attention must be paid to seasonal changes in the inflow. In within-year release control we study the variability of discharges within the respective years; the relationship between high-flow and low-flow years in their time sequence need not be studied. The complexities of changes in discharge during the course of a year, make it impossible to create a typical design annual hydrograph; here too probability methods should be used to determine the characteristics of release control. Hydrological data should therefore be prepared and other factors considered (e.g., water losses from the reservoir), according to the type of release control (see Chap. 3.4). Methods used for over-year release control can be divided into two groups: (a) methods based on the separation of the over-year and seasonal changes in the discharges, i.e., on the division of the over-year and within-year components of a reservoir storage capacity, (b) methods based on the chara6teristics of release control, including the laws of the discharge regimes as a whole; for this, synthetic discharge time series are used which reflect, besides the long-term variability, also the variable discharges in the respective years. To resolve the consequences of over-year and seasonal discharge fluctuations separately, two methods can be used: - analytical methods, which determine the influence of over-year discharge fluctuations on the release control parameters, on the basis of the general statistical C,,Cs)and the correlation function; characteristics of mean annual discharges (Qa, - methods using synthetic pseudo-chronologicalseries of mean annual discharges. 5.1 METHOD BASED ON ANALYTICAL SOLUTIONS OF THE STORAGE FUNCTION
As in complicated hydrological conditions only the influence of long-term discharge changes can be determined analytically, it is necessary to divide the storage capacity of a reservoir V, into two parts, i.e. the' over-year component V;, and the within-year (seasonal) component v,S so that
The over-year component of the storage capacity is determined from the mean annual inflows to a reservoir and the withdrawals, the seasonal component should express the demands on the storage derived from the variability of inflow and withdrawal during the year. In expressing mean annual discharges in water years, the seasonal component in low-flow periods lasting several years is usually divided into two parts (Fig. 5.3).
195 Figure 5.3 illustrates a case in which the two parts of the seasonal component are given by the low-flow seasons of the years which either precede (year 1) or follow (year 4) a low-flow period of several years. The size of the seasonal component can also be influenced by fluctuations of discharges in the years within the low-flow period.
2.
7.
3.
4.
Fig. 5.3 Over-year and seasonal Component of the active storage capacity
Equation (5.1) or equation (5.1‘) in relative values of storage capacity
&=E+E
(5.1‘)
theoretically has an infinite number of solutions, as various combinations of the sums y(Opp l ) + y(O,,p2), or /3:(a, pl) + s”,(a,p z ) can be applied to the unique solution of the function V, = f(O,, p ) or fl, = f ( a , p). Generally p 9 p1 9 p 2 . Therefore, another condition has be introduced for the unique solution of the over-year and seasonal component. Usually the reliability of the over-year component is chosen as equal to the design reliability of water supply P; to this is added the seasonal component so as to meet the condition
((O,,p ) =
yzy(op9 P) + y(0, P Z )
(5-2)
regardless of the reliability size p 2 of the seasonal component itself. This definition of the over-year and seasonal component of the storage capacity for over-year release control makes it possible to solve separately the relationships between the characteristics of release control 0,, P or a, E, P,resulting from the variability of the mean annual inflows to a reservoir.
v,
5.1 .I Over-year component of’ the storage capacity
The law of the sequences of mean annual discharges, presuming a stationary hydrological regime, is defined by basic statistical characteristics:
- long-term mean discharge Q,, - variation coefficient of mean annual discharges C,, - skewness coefficient of mean annual discharges C,, - correlation function r(z).
196 The first three characteristics are sufficient to define the theoretical curves of exceedance; the correlation function gives the internal sequence structure. If we neglect the influence of the correlation function (Kritsky and Menkel, 1932), in determining the over-year component, it is sufficient to know the duration of the design low-flow period n (number of years) and the mean discharge in this period Q,(p) (inflow into a reservoir), with the probability of exceeding p being equal to the design reliability P, according to the number of years without failure. Then or in relative values
R=
- kn(0p)I
(5.3‘)
where k,(p) = Q,(p)/Q, and T, is the number of seconds per year (T, = 31.6 - lo6 [s]). As the number of years of the design low-flow period is not known in advance and as it changes with 0, (or a) and with p, a trial-and-error method has to be applied. Let us presume that the design low-flow period is only a one-year period (n = 1). Then it is suf€icient to read the value Q,(p) for p = P from the theoretical curve of exceeding mean annual discharges Qr = f ( p ) and to calculate Y(1) = T[Op - Qi(p)]
(5.4)
For a two-year period we calculate first, from the sequence of mean annual discharges, Qr,l,Qr,2,Qr,3,..., Q,,n the mean values for two years:
From the values Q42) we compile an empirical exceedance curve to which we fit a theoretical curve and for p = P we read the mean discharge in the design two-year period Q2(p).Then
KY2) = 2T,[op - QhII The procedure is the same for three-year, four-year, etc., periods. For k-year design low--owperiod V;‘(k)= k q [ O p - Qk(P)I
v(k)
(5.6)
(5.7)
From a certain value k the values start to decrease, however, if 0, < Q, (a < 1.0). When 0, 5 Q., then the right-hand side of equatlon (5.7) exceeds all limits with the increase of k. Thedesign size of the over-year component of the storage capacity will then be = max { k q [ O ,
- Qk(P)]}
(5.8)
197
If the terms of the synthetic sequence of mean annual discharges [correlation function F(T) = 0 for T > 0; ( T = 0) = 11 are mutually independent, it holds that
If r(z) =+ 0 (also for T > 0) the variation coefficient Cv(k)must be determined from the relationship k-2
+ r3-k -k 3 + .*.)
(5.10)
and C,(k) : C,(k) = C, : C,
If we neglect r2, r3, etc., then k-1
(5.10)
Figure 5.4 illustrates the results of the calculations of the over-year component of the storage capacity for various relative yields a with a reliability of P = 97%, C, = 0.427, C, = 0.8, presuming that equation (5.9) is valid. h e dashed lines show
Fig. 5.4 Calculations of the over-year component of the storage volume by means of the exceedance curves of mean discharges during 11 years (Kritsky and Menkel, 1930)
the relationship of fi: = f(k) for a = 0.8, determined directly from the curves of exceeding mean discharges in the respective k-years. The results differ greatly (fi: = 0.86 as compared to the value 0.66).Simple and logical principles were used to prove that the curves of exceeding mean annual discharges are suitable to determine the over-year component of the storage capacity. As the problem can be solved
198
regardless of the observed chronological discharge series, we obtain much more general results, which include any discharge conditions. The real series Q, = f(t) only helps to determine the statistical characteristics and the correlation functions, which are considered to be the estimate of the parameters of the basic population. As the changes in the values of the statistical characteristics (especially C,) greatly influence the results of the calculations of the over-year release control (see Section 5.3), all possible means (Chap. 3) must be used to determine the most representative values Q,, C,, C, and r(o). Mean discharges with a given probability of exceedance in the respective k-years, which are the basis of the calculations, make it possible to get to know more comprehensively the laws of the river regime and can also be used for other purposes, such as a “safe” forecast of the first filling of a reservoir, for a statistical evaluation of low-flow periods lasting several years during operations, etc. One of the advantages of this method is that it can determine the relationship pz = f(k), which helps to evaluate some of the other influences on the size of a reservoir in relation to the duration of a reservoir refill cycle, e.g., water losses, etc. Other methods of assessment of the over-year component of the reservoir storage capacity are based on general statistical characteristics of the law of inflow to a reservoir and use the theory of stochastic processes.
Fig. 5.5 Calculations of the over-year component of the storage volume hy means of the composition of C\CCL.L~.IIILL’ L I I I \c\ of mean discharges during ir-jc‘ars (Kritsky and Menkel, 1935)
The principle of Kritsky’s and Menkel’s “second” method (derived in 1935 and generalized in 1956 to 1957) is based on the separation of the intervals of the years with failures and without failures in the curve of exceeding mean annual discharges Q, = f(p), or annual modulus coefficients k , = f(p) and on a detailed analysis of interval in which a failure to supply water can or need not occur (interval of conditional failure years). Stated is the curve of exceeding the annual modulus coefficients k, = f(p) (Q,, C,, C, and the type of a suitable theoretical function), the relative yield M (given 0,)
199
and the relative over-year component of the storage capacity p:. The unknown quantity is, therefore, reliability P. In the exceedance curve (Fig. 5.5) we can separate the years that must be without failure, in which k, 2 a in the interval (0;p a ) , and the years in which failures occur in which k, c (a - p:) in the interval 100%).In the remaining interval (pa;p(.-#)) there are the years for which a > k, 2 (a - /I) which can be without failure, if the water in a reservoir suplements the deficit (a - k,), or with failures, if the storage volume is so low that it is unable to cover the deficit. To determine the difference between the years with failures and those without failures in this interval, a two-year , ) ) from the previous years composition of the years in the interval ( ~ ~ ; p ( ~ : , . and must be calculated, then a three-year, four-year, etc., period until all years are divided into years with and without failures to supply water, whereby we pbtain the reliability P sought. Presuming that the sequence of the mean annual dischargesis independent [r(z) = 0 for z > 01,we obtain the exceedance curve of two-year modulus coefficients k,, = = f ( p ) by combining the exceedance curves k, = f ( p ) for p = 0 to 100% (curve I) and k, = f ( p ) for p in the interval (pa;p(,-@)) (curve 11),which must be extended to interval (0;100%)in such a way that value pa is ascribed p = 0 and value p(a-s) the limit p = 100%; between those boundary values the scale p is linear. Curves of exceedance (or reliability) can be compiled by various, mostly graphical or graphicalnumerical methods (Votruba and BroZa, 1974). In exceedance curve k,, = f ( p ) we again determine three intervals of probability of exceeding (Fig. 5.5): - interval (0;p,,) of two-year periods without failure, - interval (P(,~-@);lW/,) of a two-year period necessarily with failures, - interval ( p Z a ;p(?.-@)) of two-year periods with conditional failures. As the exceedance curve k,, = f ( p ) helps to decide in which years failures will or will not occur in interval (pa;pC-@,), the width of the respective intervals of the exceedance probability curve k,, = f(p) must be multiplied by the width of interval (pa;peps,), i.e., the value 0.01 .(p(,-@) - pa), ifp is given in percent. Then in the interval of years of inevitable failure 1WL)a supplement will be added, resulting from the years of inevitable failure, equal to the value of
and in the interval of a conditional failure two-year periods of the width ( ~ ( 2 a - p )-
Pza) .O.Ol(~(a-fi) - P a )
another year must be added to compile a curve of exceeding a three-year modulus coefficient k,, = f ( p ) , in which we again decide which years will be years of inevitable failure and which will be without failure.
200 The increments of years of inevitable failure rapidly converge towards zero so that in practical applications the number of steps is small. The sum of the width of intervals of inevitable failure years in the respective steps ZS,, gives us the unreliability of the water supply (in percent); by bringing this value up to one hundred we obtain the reliability of the required yield a (or 0,) with the given value /I: (or If we apply the method described above for a series of values /?: with a given relative yield a, we can construct the relationship /I: = f ( p ) and determine the resultant size @') for the design reliability of water supply P (occurrence-based). By repeated calculations for several values of a, data are obtained for the construction of the relationship a = f(p) with a constant and = f ( a ) with a given P. Pleshkov (1939) constructed graphs for the mutual independence of mean annual discharges [r(z) = 0 for z > 0; r(z) = 1 for z = 01 which express the relationship /I: = f ( C v )for values of a (from 0.15 to 0.90 with a step 0.05) and p (75, 80, 85, 90, 95,97%), presuming that C, = 2Cv(see, e.g., Votruba - Brob, 1966, p. 198). If the correlation function r(z) is not zero, the probability of encountering an arbitrary value k, (generally speaking, a k-year modulus coefficient)from the interval of conditional failure years (or conditional failure k-year periods) with an arbitrary value k, on the curve of exceeding annual modulus coefficients(from 0 to 100%)is not the same. This must be taken into consideration when constructing exceedance curves for k-year modulus coefficients, which will be conditional. Conditional parameters of curves of exceedance and of the conditional correlation coefficient for k-years must be estimated from unconditional parameters valid for (k - 1)-years, for (k 1)-years from parameters for k-years, etc. Presuming the stationarity of the random sequence of mean annual discharges and the same type of distribution of conditional and unconditional curves. Kritsky and Menkel(l959) derived the necessary relations valid for the general correlation function. In the conditional curves of exceeding the intervals without failures, necessarily with failures and conditionally with failures k-years are then determined;calculations then continue as for a zero correlation function. Kritsky and Menkel(l959) used the following correlation function
v).
+
r(z) = r;
, resp. r,
=
6
(5.11)
For this function, with rl = 0.30 and C, = 2C,, Gugli (1959) constructed graphs = f ( C v )for a given a and p, which are formally identical with Pleshkov's graphs. By analysing empirical correlation functions of mean annual discharges in the Labe catchment, N a c h h l (1965) reached a relationship expressing a damped harmonious motion
/I:
5 2n r, = - rI e-'." cos - (z 3 15
+ 1)
(5.12)
20 1
Figure 5.6 compares the course of the two correlation functions with rl = 0.3. The power correlation function always leads to positive values of r,, even though, roughly from value r4, one can speak of a practical independence. It reflects the greater probability that low-flow and high-flow years appear in groups and puts
Fig. 5.6 Comparison of the behaviour of the correlation function expressed by the equation of the damped harmonious motion and the power correlation of the function
- 031 2.0
15
T
1
lo
05
'04
05
06 0.7 08 09 -a
1
Fig. 5.7 Influence of the correlation function of mean annual dischargeson the size of the over-year component of the storage capacity (relationshipB: = f ( a ) )
greater demands on the over-year component of the storage capacity than in the case of independent mean annual discharges. The correlation function in the shape of a damped harmonious motion (Fig. 5.6) permits an indirect correlation dependence (with a negative extreme of r6). This fact is reflected mainly in a clearly over-year cycle of several years, i.e., one that requires a high degree of regulation a, where the demands on the over-year component will be smaller than for the power correlation function. A comparison of the relationship = f (a) throughout the whole course, introducing various correlation functions, can be found in Fig. 5.7 (Nachbel, 1965). Several sets of diagrams help to determine the over-year component of the storage
202
capacity j?;, with given values of ct and P, based purely on statistical characteristics and the correlation functions of mean annual discharges (e.g., Svanidze, 1964; Reznikovski et al., 1969; N a c h h l , 1973). For their close relationship to our hydrological conditions we chose the graphs constructed by K. Nachbzel with the correlation function(5.12) for rl = 0.30(Fig. 5.8); Pleshkov's and Gugli's graphs can be found in, e.g., Votruba and Brota (1966). These graphs make it possible to determine very easily one of the three quantities a, j?:, P , if the other two are given and have the known characteristics C,, C,and r(t). G-c"
-G
-
C"
Fig. 5.8 Graphs to determine the over-year component of the storage capacity (Nachazel, 1973), C(T) - damped harmonious ( r , = 0.30) (a) C, = C,; (b)C, = ZC,; (c) C, = 3C,
203
If the estimated quantities a, P and also C,, C,, rl are not identical with the quantities in the graphs, then the result can be obtained by interpolation (the best way is graphically after plotting several points of the studied relationship). Savarenski (1935- 1951), and P.A. P. Moran and others, chose a different method for release control. This method is based on the probability with which a certain chosen water volume will be accumulated in the storage capacity of a reservoir at the turn of two years. If this probability is determined for every size of the filling of the storage capacity
-
C"
-G
-c"
Fig. 5.8 Graphs to dctcrminc thc owr-year component of the storage cupacitj ( NachiLel. 1973). I . ( T ) - damped harmonious (rl = 0.30) ( d ) c. = c,: ( h ) c, = Zc,: (c) c, = 3c,
204
(from zero to maximum value), the rule for the distribution of the water volume in a reservoir is obtained, Savarenski(1950),Moran (1959),Gould (1961),White (1963), Lloyd (1963)and others introduced a simplified pattern of the discharge within a year into the calculations, but due to the complex conditions of Middle European rivers this grossly distorts the results. We therefore use Moran's formulation for the over-year component of the storage capacity of a reservoir, only slightly adjusted for practical applications. Let us presume that the mean annual discharges create a random sequence which
c, = 3C"
Fig. 5.8 Graphs to determine the over-year component of the storage capacity (Nachizel, 1973). r(7) - damped harinonious ( r , = 0.30) (a) C, = C,; (b) C, = ZC,;(c) C , = 3C,
205
is stationary. The calculations of the probability of the given filling of the storage volume at the end of a year must take into consideration the state at the beginning of a year, the size of the inflow and withdrawal. The required distribution of the probability of storage capacity filling V = f(P) that is continuous is approximated by a discrete one by dividing the given size of the storage volume into n - 2 equal parts and use their mean values; we add another two values: V, = 0, expressing complete emptying, and V,, which is the complete filling of the storage capacity (Fig. 5.9). Each interval AV, has a probability of occurrence AP,,,and the value V, has the probability of exceedance P,.
Fig. 5.9 Solution of the probability of the storage volumes (Moran's method)
The probability of exceeding a certain volume 5 is a conditional probability including the probability of occurrence of initial filling and also the probability of occurrence of inflow Q i ; the withdrawal is presumed to be constant. For all values V,, V,, ..., V,, ..., V ,the probability of exceedingP(5)can be expressed by the equations P(Vl) = a,, AP, P(V2)= a,, Ap1
+ a,, AP2 + ... 4- alkAPk+ ... +- a,,,&',, + + ... + U,,APk + ... -I-a 2 , , U , ,
a,,,
'12,
a219
a,,, .*-,
a2,,
an,,
an21
ann
coefficient matrix A = .
***,
Qln
is denoted as the matrix of transient probabilities, i.e., a transition matrix.
206
Figure 5.10 shows how to determine alk. We construct an exceedancc curve k , = f(p); on the axis of ordinates we put fl: and
(Pi + 4.
We divide the ordinate fl: into (n - 2) equal intervals A/? (usually 15 to 20 is sufficient)and in each interval we determine, in the centre of the interval, the ordinate flk’
Fig. 5.10 Procedure for determining the matrix of transitional probability coeficients (Mown’s method)
If we add to any of the values flk the ordinates of the curves of exceeding k, = f(p) and if we bound the thus gained curve by ordinates (fl: a) and fll = 0, we obtain a conditional curve of exceeding water supplies prior to withdrawal. After subtracting ordinate a from this curve, we obtain the conditional curve of exceedance after withdrawal fl (& k,, a) = f ( p ) ; the points of intersection with the boundaries of intervals Afl then determine intervals on the axis of the abscissas which are the conditional probabilities a,,, 02k, ...,ank sought. The following relationship is valid between the values of APk
+
n
C APk = 1
(5.15)
k= 1
The relationship between the prohahilities of occurrence and the probabilities of exceedance (illustrated in Fig. 5.9) is P(&)
+ AP3 + ... + + ... + + AP, = iAPz + AP3 + ... + m k + ... + AF,,-ll + AP,,
P(&)
=
P(Vl) = AP,
@k
iAPk+ ... + AP(,,-,, + APn
P ( K _ , ) = +AP(,,-,)+ AP,, P ( K ) = APn
1)
(5.16)
207 From equation (5.15) it follows that (5.17)
P(Vl) = 1 - APl
which brings us to the final equations
+
+
(all 1 ) M l a12 A P z a21 AP1 (U22 - 0.5)APz -k
+
-..+ U,,,AP, = 1 ... -k ( a 2 k - 1)APk + ... + (a2, - 1)AP, = 0 a l k M k
The matrix of coefficients of the unknown variables, which we denote by b,, can easily be calculated from the transition matrix A by adding matrix C:
(5.19)
where 1 0 0 o...o 0 -0.5 -1 - 1 ... -1 C = 0 0 -0.5 - I . . . -1 0
0
0
o...
(5.20)
-1
The results obtained from equations (5.18) are the probabilities of occurrence APk of filling of the storage capacity V,, i.e., a discrete distribution curve. By introducing this into equation (5.16), we determine the exceedance curve V = j(p). The value of the probability of exceedance corresponding to V, = 0 is the equivalent of the required reliability of withdrawal according to the number of years without failure (occurrence-basedreliability). The calculations can also include the influence of the non-zero correlation function of mean annual discharges(instead of the exceedance curves k, = f(p), the conditional exceedance curves would have to be included in the calculations), as well as the variable withdrawal from year to year (independent or correlationally dependent on the inflow to a reservoir).
208
5.1.2 Within-year(seasonal)component of the storage capacity
To determine the within-year component Esof the storage capacity the definition given by equation (5.2) should be used as the starting point. According to this, the seasonal component is determined as a certain supplement of the over-year component needed to ensure the reliability which equals the design water supply reliability P; we do not study the reliability p z of the seasonal component itself. is therefore the difference between the total size of The seasonal component the storage capacity and the over-year component
y(op, P z ) = K(Op,p ) - Y(OPYP )
(5.21)
or in relative values R(a, PZ) = /%(a,P )
- &(a,
4
(5.21')
In the hydrological conditions of many rivers, discharge time pattern in the course of a year is very complicated and it is, therefore, impossible according to some authors, e.g., Kritsky and Menkel (1952) or Pleshkov (1961), to apply a simplified schematic division of discharges. To include the variability of discharges within one year is also rather complicated. It is therefore only logical to avoid any complicated methods, such as replacing an analytical (indirect) solution by a direct solution in a synthetic chronological series of mean monthly discharges (see Section 5.2). Modelling of synthetic series gives rise to certain problems, mainly in relation to the
Fig. 5.1 1 Definition of initial, end and middle part of the seasonal component of the storage capacity (in years with Q, 2 0,)
209 over-year release control, which have not yet been solved satisfactorily. It can be expected that in the future methods based on the division of the storage capacity into over-year and seasonal components will no longer be used. First we analyse the influence of dischargefluctuations in all the years of the studied discharge series on the size of the storage capacity that is needed to ensure the required withdrawal; however, we must bear in mind that hydrological services usually base their observations on water years, which means that one low-flow period can be divided into two parts. In the years when Q, 2 O,, the over-year component is zero so that V, 3 T, or b, = pi. The seasonal component can be represented by three parts in one hydro(from logical year (Fig. 5.11). In Fig. 5.11a we can find the mass curve of inflow a raised pole) in one hydrological year. The parallels to the mass curve of withdrawal Copdetermine: (a) at the beginning of a year, the so-called initial part of the seasonal Component init AV,; its size is given (under the condition that the discharges in a year are expressed, e.g., by mean monthly discharges Qm,i)by the largest negative value K, of the sums (5.22)
where k has the values 1,2, ..., n (n is the number of intervals At in a year). initAV,. = K p ( = m u ( & } )
(5.23)
When all these sums are negative, init AV,. = 0 (Fig. 5.11a); (b) at the end of year, the so-called end part of the seasonal component end AT. For this equation (5.22) is used as for (a),but with the negative extreme, the absolute value of which is denoted by K,. Then
end AV: = K , -
i=n i= 1
(Q, - 0,)Ati = K, - 31.6. 106(Q, - Op)
[m3]
(5.24)
Besides those that occur at the beginning or the end of a year, low-flow seasons can occur within a year and can also influence the seasonal component. They can be included in the calculations by determining the size of the so-called middle part of the seasonal component, mid A v,S (Fig. 5.11b) in every year and by adding the values (0, - QmVi) At? Then the following conditions have to be fulfilled: - the mid AV,. can only be calculated after at least one negative value (0,- Qm,i) since the beginning of a year has been determined, which eliminates the possibility of confusing the initial and middle parts of the seasonal component; - when the added values c(0,- Qm,i)Ati reach a negative value, zero is considered;
2 10
- the maximum value C(Q,- Q,,d Ati may not occur at the end of a year. Should this be the case, then mid A c is considered as the second maximum (local) of the sum c(0,- Qm,i)Ati; if it does not exist, then mid A T = 0 is used; - for every year only one value of m i d A T is chosen, given by the maximum c(0,- Qm,i)Ati, although several low-flow periods might occur in one year. After determining init A T , end A C and mid AV,. in every year in which Q, 2 0,, we can add the end component of one year and the initial component of the following year [end AV,.(t) init A T ( t l)], disregarding in the chronological series the years with Q, < 0,.Thus, we obtain a set of values, from which, after ranking in an ascending order and adding the probability, we gain an empirical probability (reliability) curue c ( 1 ) = V'l) = f(p) reflecting the size of the seasonal component of the storage capacity in the years with Q, 2 0,.
+
+
-
pCXl
Fig. 5.12 Probability (reliability) curve of the seasonal component V,. (K + P), constructed in the probability paper of normal distribution
From the mid AT values we can also construct a probability (reliability)curve of mid A c = f(p) which we use to correct, if necessary, the results in the final stage of the calculations. In constructing probability curves T(1) = f(p) or mid AV,. = f(p), the special character of the initial values T(1) or mid AT must be considered, as usually several terms are zero. These values cannot be fitted by a theoretical distribution function according to some of the methods usually used (moments or quantiles). Therefore, we must draw the empirical probability curve through the observation points plotted on the probability paper (usually of a normal distribution-Fig. 5.12).
21 1
Certain experience is needed, especially in those cases where the "empirical" points are not grouped around a curve of a simple type, or when the extreme values deviate from the tendency of increased demand with increased reliability p. Sometimes it is sufficientto eliminate the doubts concerning the pattern of the probability curve 1) = f(p) to change the scale of ordinates. All these circumstances render a practical application very difficult and make it almost impossible to solve the problem with the help of a computer. Even more complicated is the question of including the influence of the seasonal component on the size of the storage capacity in over-year low-flow periods. Let us presume that before the first and after the last year of a low-flow period (in which Q, < 0, for all the years) the same probability applies to any of the years with Q, 2 0,. Let us therefore consider the sequence of the mean annual discharges as a completely random sequence. Figure 5.13 illustrates the various possible influences of the fluctuation of discharges within a year on that part of the seasonal component relevant to the limit of the over-year low-flow period. In Fig. 5.13a the value end AV,S of year i , which precedes the low-flow period, is the significant value for the seasonal component. However, a case might occur when the size of the seasonal component is determined by the Fig. 5.13b, or ( i - 2), etc.; the probability changes ofdischarges in the year ( i - 1)-see of such a state decreases with increasing distance from the low-flow period. The discharge fluctuations in the years of the low-flow periods (Q, < 0,)can also ~ a case where the determine the size of the seasonal component. Figure 5 . 1 3 illustrates
c(
boundary of over-ymri low- flow period
Fig. 5. I3 Influence of discharge fluctuations in years near boundary of an over-year low-flow period on the size of the seasonal component of the storage capacity
Fig. 5.14 Definition of initial and end part of seasonal component in lowflow years (Q, < 0,)
212
size of part of the seasonal component at the beginning of the low-flow period is given by the value end AV,. of year j, which is the first year of the over-year low-flow period; however, the year (1 1) can also be decisive-see Fig. 5.13d, ( j + 2), etc. Similar conditions can occur at the terminal boundary of the low-flow period. As we have proved, the influence of the years Q, < 0, on the size of the seasonal component, the initial and end parts of the seasonal component in those years (Fig. 5.14), must be defined. Similarly, as in the years with Q, 2 0, the mass discharge curve (with a raised pole) is illustrated and parallels with the mass curve of withare drawn in the terminal points at the turn of the year. The vertical drawal distance between those parallels illustrates the deficit between the annual volumes of outflow and of inflow, which means that it is a part of the over-year component of the storage capacity. The initial part of the seasonal component must ensure an augmented discharge in the low-flow periods at the beginning of a water year; however, only the part that exceeds the deficit of the annual volume of inflow (Fig. 5.14).The same applies to the end of a water year. ifwe use discrete values of Q,,i for the discharge distribution within a year, we add, similarly to the case for years with Q, 2 0,
+
%
where k has the value 1,2, ..., n (n is the number of discrete values of Qm,i,or intervals Ati in a year). We take the largest value of the positive sum (denoted by K,) and the negative extreme (if it exists), the absolute value of which we denote by K,. Then init AV,. = K,
+
c (Q, - O,)Ati
i=n
i= 1
=K,
+ 31.6. 106(Q,- 0,)
endAV,'= K ,
If all values
(5.25) (5.26)
i=n
1(0,- Q,,,i) Ati are positive, end A T = 0.
i= 1
That these algebraicterms are true can be seen from Fig. 5.14, which also illustrates the discharge distribution within a year, where the initAV,' or e n d A v , or both &., parts of the seasonal component are zero (co., First we include the influence of the years with Q, 2 0,, preceding or following an over-year low-flow period in the seasonal component. For the directly preceding to following years the values are end AV,S (of the preceding year) and init A T (of the following year) determined by equations (5.23)and (5.24).As there is always at least one low-flowyear between the correspondingvalues end A y and init A y,they can be considered independent.Therefore (from empirical values), probability curves init A V, = = f'(p) and end AV, = f ( p ) are constructed and by their composition the probability
L-).
213 curve y(i)= f(p) is obtained, which expresses the influence of the years directly preceding or following the low-flow period on the size of the seasonal component. The way to construct the probability curves by composition can be seen in Fig. 5.15. In suitably selected intervals one of the probability curves is replaced by mean values and to these a second probability curve is “added”, with the condition that the width of every interval is 0400%. By adding the horizontal partial coordinates of the thus-constructed probability curves for the selected values of the ordinates, the resulting probability curve is obtained.
Fig. 5.15 Diagram of the composition of probability curves (presuming mutual independence of the members of the two samples)
In considering the inluence of the years more distant from the low-flow year, it should not be overlooked that between the year under observation and the first low-flow year there are x years with Q, 20,.Then for the seasonal component of the ( i - x)th year
y(i - x) = y(i)-
x(Q,- 0,)T, i
(5.27)
i-x
where T, is the number of seconds per year (31.6. lo6). From the probability curves v,“(i), y(i - l), etc., those values must be chosen which, in every probability sphere, have a larger seasonal component v,S and are therefore signifcant values for the design of the storage capacity. The significant values are determined by dividing one of the probability curves (I) into intervals Ap, in which it is replaced by mean values. In each interval from O-lOO% a second probability curve (11) is drawn, which by its point of intersection with the mean value of the first curve (Fig. 5.16) divides every interval Ap into a part Ap’ in which the values v,S of the first curve are larger than those of the second curve and therefore are significant values, and into a part Ap”, with the opposite conditions. The segments of the first probability curve in the respective intervals are drawn by a linear reduction of the
214
abscissa into intervals Ap', where they are significant, while in the intervals Ap" the segments of the second probability curve are valid (as they are larger). By joining the horizontal coordinates of the probability curves in all partial intervals for the chosen ordinates v,S, we obtain the required probability curve.
I
1
0
- C%l . . 50
p
I.
L
.... LL
100 aA'
Fig. 5.16 Diagram of the choice of the significant values of two prob-
This method brings us to a single probability curve v,S (i, i - 1, i - 2, ...) = f(p) reflecting the influenceof the years with Q, 2 0,on the size of the seasonal component in over-year low-flow periods. We proceed similarly for years with Q, < 0,. From the probability curves init AT and end AI/,", curve v,S(j) = f(p) is obtained, expressing the influence of the years directly preceding or following the low-flow period. For years more distant from the low-flow period it follows that (5.28)
r(j), c(j
By selecting significant values from the probability curves + I), ... (see Fig. 5.16) the resulting probability curve I/,"(j,j 1, ...) = f ( p ) is obtained, expressing the influence of the years with Q, < 0, on the size of the seasonal component. As on the boundary of an over-year low-flow period, one group of years with Q, 2 0,meets a group of years with Q, < 0,,the significant value (i.e., larger) must be chosen, regardless of the group of years to which it belongs. We therefore choose significant values in the probability curves T(i, i - 1, ...) and v,s(j,j + 1, ...)(see Fig 5.17) and consider the width of the probability space to be the same (one value of the Mst group of years always meets one value of the second
+
215 group, even when the number of years in each group varies, i.e., the width of the intervals (0;p a ) and ( p m ;100%) is different). The result is a probability curve c(2) = f ( p ) which is an expression of the seasonal component of a reservoir storage capacity in over-year low-flow periods.
0
li
b)
C)
Fig. 5. I7 Determination of probability curves reflecting the influence of the years more distant from the boundary of the over-year low-flow period on the size of the seasonal component
Fig. 5.18 Construction of the resultant storage capitcity probability curve V, = f(p)
It is also necessary to consider the over-year component y. It can be presumed that the two quantities ( y ,Ks)are independent and we therefore obtain the resultant probability curve t ( 2 ) = f ( p ) by the composition of the curves V,.and v,S (Fig. 5.18a). Curve = f'(p) can best be constructed with the help of a nomograph (see Section 5.1) for pa, V:, = 0. As one value of V,. is put together with one value 6f Y, the width of the probability of both curves is identical. Curve t ( 2 ) determines the demand of the over-year low-fow periods on the size of the storage volume;from the whole probability of exceeding mean annual discharges it refers to the interval (pa; 100%) in which Q, < 0, (Figs. 5.17, 5.18). Curve K(1) = = f(p) is valid in the interval ( O ; p a ) where Q , 2 0, (Fig. 5.18b). As the storage
216
capacity can either provide the yield in a relatively deep discharge depression of a low-flow season in one single year, or cover over-year low-flow seasons (with a smaller discharge decrease), we put together the two probability curves simply by 5.18~. adding the horizontal coordinates and then obtain curve V, = f(p)-Fig. Finally, it is possible to take into consideration the influence of low-flow seasons within the respective years, expressed by the probability curve mid AV,. = f(p), and by selecting significant values to correct the curve V, = f ( p ) . This step, however, is not frequently taken as the curve mid AKs usually does not Munce the shape of curve V, in the sphere of higher probability values. The theoretical size of the storage capacity VJp) sought is thus determined for the given design reliability P (accordingto the number of years without failure). This method is so far the most perfect for the calculation of the seasonal component of the storage volume and is an original Czech contribution (Klemd, 1963). As it is methodologically as well as technically very demanding, it is not yet widely used in practice. It can be found in greater detail in other studies (e.g., Votruba and BroBa, 1974). This method can also be used in those cases where the value 0, changes during the year, but where the withdrawal diagram is the same in all the years. The over-year component is then determined for a mean withdrawal 6,; the variability of withdrawal has to be included in the calculations of the respective parts of the seasonal component. However, the method can be simplified. It is, e.g., possible to disregard the influence of the years (i - 1) and (j+ 1) on the size of the seasonal component. With a high relative yield a = O,/Qa (a > 0.7),the probability curve K(1) in the interval of the seasonal release (0; p a ) need not be constructed as it does not influence the shape of curve V, = f(p) in the sphere of high valuds of p (Fig. 5.18);the resultant value V,(P)can then be read directly from the probability curve V,(2). Another simplification which, however, can make the solution less accurate, is to disregard the influence of the years with Q, < 0, on the seasonal component.
0
-p C%7
Fig. 5.19 Diagram of the simplified method of determining the storage capacity probability curve (Andreianov, 1960)
217 Then the probability curve y(2) is identical to curve v ( i ) . This simplification is very similar to Andreianou’s method (1960), which is based on the so-called water management years, which make it possible to determine the seasonal components in the respective years as a whole; considering, however, only years with Q, 2 0,.It = f ( p ) , which is very close to curve v,S( 1) is presumed that the probability curve mentioned in the original method, is valid both in the interval ( O ; p , ) and in the interval of low-flow years (pa; 100%) where it meets the over-year component V;, (Fig. 5.19). Despite its methodological shortcomings, Andreianov’s method is satisfactory from the point of view of accuracy, especially with a high relative yield (a). 5.2 METHOD BASED ON SYNTHETIC DISCHARGE SERIES
Reservoirs designs based on simulation in time discharge series (Chap. 3) are simple, and make it possible to solve cases of complicated withdrawal (release) diagrams. “Classical” method determined the parameters of discharge control a and /3, (or 0, and V,) in real time series. The size of the storage capacity was therefore given by one single series of discharge conditions in the past which will not be repeated during the operation of a reservoir (i.e., in the next few decades). Even during relatively long observations (40to 50 years), in real discharge series only a very few low-flow periods lasting longer than one year occur which could greatly influence the size of the storage capacity (on the average one low-flow period in 10 to 12 years). Therefore, the shortest permissible length of time series in relation to the type of release control was recommended. In the conditions pertaining, e.g., in Middle Europe, such a recommendation is rather risky (Votruba and Broia, 1966); for over-year release control it is therefore necessary to apply, for the determination of the parameters of release control (a,3/, P), synthetic chronological discharge series (also called random, synthetic) of a sufficient length (as a rule 500-1000 years; Broia, 1973). In preliminary studies for the comparison of various alternatives (mainly from the qualitative point of view), solutions in real series cannot be excluded. A condition is a given withdrawal (release) from a reservoir in every step of the solution. Let us consider a long random discharge series, in which every member Q(Ati)is an expression of the mean discharge in the time interval Ati; the required release (withdrawal)from the reservoir is given at each interval Ati by the mean value Op(Ati)* The balance deficit AV(Ati)in the interval Ati is given by AV(Ati) = [O,(Ati) - Q(Ati)] Ati
(5.29)
In the problem, the withdrawal 0, (i.e., a) and the size of the storage capacity (i.e., 3/): are given; the reliability of water supply p is to be calculated.
V:
218
At the beginning of the series, the first filling of the storage volume has to be chosen. With a sufficiently long series, the choice of the first filling has practically no influence on the result. Usually the storage capacity is considered to be full. Additions are made i=k
i=k
i=1
i= I
v, = x - A v ( A t i )= - ~ ~ [ o , ( A t i-) Q(Ati)] Ati
(5.30)
with the following limiting conditions (Fig. 5.20): (a) if V, < 0, V, = 0 is considered as the result in this step, and this value is used to calculate K + I , & + 2 , etc.; (b) if the value of V, > the result of the calculations in the given step is V, = with the respective consequences in the further steps (k + 1,...).
c,
c,
Fig. 5.20 Simulation of the work regime of a reservoir by means of inflow and release mass curves
The first condition reflects the state with a full storage capacity, when the surplus inflow to a reservoir (above the value 0,, or any other real release value) cannot be accumulated, causing an excess outflow. The second condition reflects the emptying of the active storage capacity. If the inflow to a reservoir continues .to be less than the release (withdrawal),then a failure to supply water occurs as 0 = Q c 0,. To express the reliability of water supply in numbers according to various aspects, the following must be recorded: 1. number of years (water years), in which failures to supply water occur, 2. duration of failures to supply water AT, when V, = t$' and Q < 0,, 3. volume of water deficits with respect to the planned withdrawal, given by AK = ~ [ O , ( A t , )- Q(Ati)] Ati, starting with the moment when V, 2 v,l up to the moment when Q 2 0, (Fig. 5.20). It is also expedient to know the maximum deficit in every case; this is given by the difference A 0 = (0, - Q),,,ax while at the same time fulfilling V, = v,l (the active storage capacity is empty).
219
From these data it is possible to express the reliability of water supply according to the number of years without failure (occurrence-based reliability). (5.31)
where n is the number of years from the whole series in which failure to supply water occurs, and N is the total number of years of a synthetic series. With a suficiently long series (500-1000 years) the differences as compared with the usual relationship p,, = 100[(N - n) - 0.3]/(N + 0.4) [“A] are negligible. The time-based reliability (of water supply according to duration) p t is determined with regard to the length of the series with a duration T(in units of time) (5.32)
where C A T is the sum of the duration of all failures that occurred in the series.
g
Y
Fig. 5.21 Linear interpolation with time-based reliability
The failure usually starts somewhere in the middle of the interval Ar, at the beginning of which the value of the emptying of the storage value is V,- < while at the end it is V, > 1/1; therefore part of interval At, is usually considered as the failure part and is denoted by AT^, resulting from linear interpolation (Fig. 5.21)
c,
(5.33)
In calculating the quantity-based reliability (or withdrawal according to the volume of the supplied water) pd, the theoretical volume of supplied water without any failure for the whole period, i.e., for the whole random series must first of all be determined n
% = C Op(Ati)Ati i= I
(5.34)
220
The value of pd is calculated from the equation (5.35)
In turn, further values of v,l (or &) are chosen while preserving the rates of 0, (or a) and the calculations are repeated. Thus, the relationship V,, = f(p) or 3/: = f ( p ) is constructed and for the design value of reliability P, the size of the storage capacity is subtracted (Fig. 5.22).
-pC%l Fig. 5.22 Determination of the size of the storage capacity by means of the storage capacity probability curve (relationshipconstructed by repeated simulation)
.
Fig. 5.23 Relationship between the basic characteristics of the reservoir storage function (P.&, a) in terms of the results of the simulation in synthetic series (by computer)
The general algorithm of a direct solution could be perfected by applying an iteration method to a computer and thus obtalning the design value of the storage capacity V, directly; however, a more rational method is that which leads to the relationship V, = f ( ~ ) . A rough idea about the size of the storage capacity is obtained by calculating the which in the synthetic series ensures the required water supply with volume tm, a 1ooO/, reliability. One proceeds again according to equation (5.30), however, without the limiting condition’ V, 6 VL. The size of volume Crnequals the absolute maximum from among the values V,. If the size of the storage capacity V,(or 3/), and the design reliability P are given and calculations are to determine the size 0,, reliability p must be determined for several selected values of 0;(with a given b),which makes it possible to construct the re, for the design reliability P the sought size lationship 0;= f ( p ) (or a’ = f ( ~ ) )where of the yield (withdrawal)0, must be found. The relationships between the characteristics of release control can be expressed with the help of three-dimensional graphs (P, V,, 0,)or (P,3/, a) (Fig. 5.23). For
22 1 various suitably selected values of 0;(or a’), the relationship = f(p) [or fi: = f ( p ) ] is constructed. The vertical sections of this graph (P = const) are the storage-yield curves p, = f ( a ) or V, = f(0,). 5.2.1 Over-year component of the storage capacity
Synthetic series of mean annual discharges are usually not used for practical calculations of the over-year component of the storage capacity as there are simpler methods. However, as they were frequently used for calculations of a research nature we shall mention this method briefly as a special case of a general algorithm of simulation calculations. fbilure years
-
Fig. 5.24 Failure years in synthetic series of mean annual discharges using the simulation method
In a random series of mean annual discharges (At, = 1 year), only the reliability from the point of view of years without failure (occurrence-based reliability) is significant,as the duration and extent of the shortages are greatly distorted. Therefore, only the number of for direct calculations with the given values 0, and Vl or a and Fz, years in which failures to supply water occur are studied. Considered as failure years are those years in the course of which the reservoir volume V: is emptied (Fig. 5.24). The sought reliability p , is determined from equation (5.31). It is possible to determine the size V,(P,) with a given value 0,, which ensures a water supply with the selected (required) reliability Po, by approximation (selecting various V:). Some authors used this method instead of the analytical method to construct graphs for a quick determination of the over-year component of a storage capacity (see Section 5.1). The reason is that modelling of synthetic series (including indispensible testing), and the actual solution as well as the smoothing of the results can be carried out completely by a digital computer, whereas some of the operations included in the analytical calculations cannot be easily programmed and carried out by computer. If we consider that, with regard to the general character of this method, various values of C,, C,, r(r) or their combinations have to be chosen for modelling the series and that for the testing a large number of identical realizations of the random series (e.g., ten) are needed, it is obvious that many calculations have to be made to construct the nomograms p: = f(Cv) for various values of a and for Po, C, and r(z). Computers are able to handle this problem as a whole and therefore graphs constructed in this way are much more universal than Pleshkov’s or Gugli’s graphs.
222
Among the first persons to use synthetic series was Svanidze (1964),who worked with series of mean annual discharges of loo0 years; similar graphs were elaborated by Reznikovski (1969) and Nachhzel(l973). Direct calculations in synthetic series of mean annual discharges were further used, e.g., in the study of non-stationary discharge series and various types of their models (Nachiizel and Patera, 1975). 5.2.2 Total size o j the storage capacity The lack of analytical methods, i.e., the necessity to divide the total size of the storage capacity into the over-year and the seasonal components, led to research on methods which would include both the over-year and the seasonal cycle of a reservoir. The introduction of matrix methods (Moran et al., and especially Lloyd (1963)), in which the inflow to a reservoir is considered to be a simple Markov chain, represented a great progress in this respect. However, it was only by the elaboration of mathematical models of long discharge series, including the fluctuations of discharges within the respectioe years, that a qualitatively higher level was reached in the calculations of reservoir regulation. Synthetic series of mean monthly discharges are usually sufficientlydetailed for overyear release control. The error introduced by the gross character of these data is permissibleand can be compensated by a certain percentage increase of the theoretical size of the storage capacity (Section 5.3), or it can be neglected, In synthetic series of mean monthly discharges, algorithms described in the introduction to Section 5.2 are used, i.e., according to equations (5.29) to (5.35). If we study only the reliability, based on occurrence, the results of the calculations are equivalent to the results of the analytical method (using suitable graphs to determine the over-year component and a detailed solution of the seasonal component). However, from a long synthetic series further information can be gained making it possible, e.g., to analyse the failure periods, to determine the rules for reservoir operations, etc. The design reliability of water supply reflects the probability with which the consumer’s demands will be met over a long period; these are characteristics which cannot reflect the real conditions in the next several decades for which we are able to predict the demands on a reservoir. A method should therefore be used in which a long synthetic series is replaced by a set of short realizations of synthetic series of mean monthly discharges. Each such series can be considered to be one of the possible discharge series in the future reservoir operations (with the same probability of occurrence). In each series (e.g., of 50 years) direct calculations are carried out which lead to a set of results (e.g., the size of the storage capacity). Further statistical processing leads to the design size of the studied characteristics of release control. In Fig. 5.25 there are plotted the points of the “empirical” curve of exceeding the
223
relative sizes of storage volumes &, necessary to ensure withdrawal a = 0.7 in 50-year realizations of series of mean monthly discharges. The variability of the necessary volumes (with a realibility of 100%in each series) is relatively great, C , 0.21. When comparing the results gained from shorter random series with the results of 2000-year synthetic series (whichwere made up of 40fifty-year series),is was found that the value fl, = 0.358, which in the long series has a reliability of p , = 95%, has a reliability of 12%; similarly with p, = 0.424 with a reliability of p , = 97% in the long series, the reliability of the storage volume in 50-year series is 34% (Fig. 5.25). 07
06
.q0.5
I
0.4 03 02,
3 5
20 30 605060 iV 80
91) 9 5 9 7 99 -p [%I Fig. 5.25 Probability (reliability)curve of storage capacity gained from a sample of 50-year realizations of mean monthly discharge series x)
The idea to use a set of shorter series instead of one sufficiently long synthetic flow series originated from the need to simulate the possible changes of the hydrological regime in the future reservoir operation. As the analysis of the reservoir function can be done in every short series, it is possible to obtain by this method a set of results reflecting the possible variation of the necessary storage capacities of the reservoir for the given yield and, on the other hand, for the constructed reservoir, the probable variation of the yield in the relation to the hydrological conditions. From the view of the recent application of the theory of estimation on water resources analysis this method has some drawbacks. These drawbacks are given by the relationship between the sample of statistical characteristics of shorter series (i.e. the realizations of the population) and the parameters of the long series that simulate the population. The results of the research show that this relationship is very complicated as it depends on many factors (e.g. the type of probability distribution of the long series, its autocorrelation, length and number of the shorter series etc.). These relations can exercise negative impacts on the analysis of the reservoir function in which it is necessary to determine, for the given yield, the design active storage capacity based on the sample of values obtained by the analysis using the
224
shorter series. It was proved that this sample characteristics can be biased under the capacity value derived from the long series and that the results of such analysis can result in underestimating the reservoir storage capacities. The risk of such a fault is the greater, the smaller the set of synthetic series is (i.e. it includes a small number of series) and the synthetic series are shorter. In the analysis of the inverse task, when the active storage capacity is given and the yield is calculated, a similar risk occurs-there is a tendency to find higher effects (yield) as compared to the analysis based on the long series. The research activities (Nachhel, 1987) included further complicated issueM.g. the relation of the reliability of the yield, based on the set of shorter series and that based on the long series. It this case it was proved that the variation of the reliability values in case of the set of shorter series is extensive and may result in the unjustified increase of the reservoir yield. The research, therefore, resulted in the statement that the design of reservoir parameters should be based on the hydrological data formed by one sufficiently long synthetic flow series, derived from the representative input parameters. The method using a set of shorter series can be used as a supplementary aid for the estimation of the probable variation of the yield from the designed (constructed) reservoirs for different hydrological conditions. 5.3 EVALUATION OF THE RESULTS OF THEORETICAL CALCULATIONS AND FURTHER INFLUENCES ON THE DESIGN OF THE STORAGE CAPACITY
Theoretical calculations, which determine the basic characteristics of reservoir functions 0,,, V,, P (or a, b,, P) are the first stage in designing the size of the storage capacity. An important part of these calculations is the evaluation of the results attained and the inclusion of previously not considered influences (e.g., water losses from a reservoir). An evaluation of theoretical results must consider the following: (a) reliability of input data, mainly hydrological data and the method used to express the quantity of the withdrawal (release) from a reservoir, (b) theoretical and practical accuracy of the method used, (c) possible variability of results caused by random deviations of statistical characteristics of other elements of a real discharge series. The resultant (design) size of the storage capacity of a reservoir must further include - demands on the storage capacity to cover water losses, usually caused by evaporation from the water level, in some cases also caused by uncontrollable seepage of water from a reservoir, - application of “inaccurate” hydrological data resulting from the choice of mean
225 monthly discharges, such as the characteristics of discharge fluctuations during a year, - reserves which serve to meet various demands not included in the planned storage function (e.g., to flush a river downstream of a reservoir, t o influence the temperature of a stream in winter as well as in summer, to alleviate failures of water supply, etc.).
1910
1920
1950
19%
Fig. 5.26 Variability of statistical characteristics and sues of the over-year component of storage capacity determined from “moving” 50-year discharge series in DiSEin on the Labe (samples from the period 1851-1969)
226 Of great importance for the results of theoretical calculations of the characteristics of an over-year release control is the input real discharge series, especially its length and position in time. Not even probability methods can exclude the direct relationship of the statistical characteristics, correlation relations (or characteristics of the changes in the dischargesin the initial real series)and input data of reservoir regulation analysis. Already in 1963 L. Votruba and V. Brohpointed out the great variability of the statistical characteristics of mean annual discharges of the rivers in the Labe catchment in relation to time and the length of the period of observation. Since then several authors have studied this problem (e.g., Nachhzel and Patera, 1975). Figure 5.26 illustrates the changes in the statistical characteristics Qo,C,, C,/C, and selected values of the correlation function rl, r2, r6 in "moving" 50-year series from a 110-year series 1851-1960 at the river Labe- DEEin site. Also plotted, are the over-year components of the storage capacity with a yield (release) 0, = 210 m3 s- (a = 0.7) derived from the statistical characteristics and a correlation function with a reliability of 99%, and given by the direct solution for the respective fifty-year periods (classical method). The variability of the over-year component of the storage capacity caused by random deviations of statistical characteristics is very great. With regard to the value Vz derived from the whole 1 10-year series, deviations currently exceed f20% and in some cases even 50%. Figure 5.26 also shows that there is no direct relationship between the statistical characteristics of discharge in the given period and the demand on the size of the storage capacity in the same period.
'
7 3
"F"m
6 5
0
24 3
Lq
I 21 0
250
-3co4[m3s"I
350 02
-C"
a25
03
035
I
-2
WC"
3
-
p1
Fig. 5.27 Variability of over-year component of the storage capacity with the change of the respective statistical characteristics and correlation coefficient r l In Fig. 5.27 it can be seen how sensitive the over-year conlponent is to any change of one of the characteristics Q., C,, C, and r , , while the others remain constant. The statistical characteristics of the 110-year series of the river Labe - D W n were considered average, deviations were considered in the range of the changes of the "moving" 50-year series. None of the influences caused by changes of the considered characteristics on the size of Vzv can be considered to be insignificant. As the result of a change in the mean discharge Q. (Fig. 5.27a) and with a constant withdrawal 0,.the relative yield a changes (with an increasing Q, the value of a drops); in this is not equivalent to the relation V;, = f(Q,). case the relation 8: = f(Q,,)
221 A change in C , whilst preserving the value C , (Fig. 5.27b)automatically leads to a change in the CJC, ratio, which further increases the size Vzv (or &) with the increase of C,. In our case, the influence of the change in C , / C , ratio was practically negligible as it moved within the range of 2.7 to 4.7 where the change of AVzv and the change of A(CJCJ are very small. The influence of the change is C,/C, (Fig. 5.27~)is very significant if CJC, i2. This naturally cannot be generalized for higher C, (above OS), see Reznikovski er ul. (1969).
Tuble 5.1 Influence of the beginning of observations (length of series) on the characteristics Qo,C,, C,. r , and the size of the over-year component of the storage capacity (the river Berounka-Kfivoklat)
Period
1951-1970 1941-1970 1931-1970 1921-1970 1911-1970 1901-1970 1891-1970
Duration of series years
Q, [m’ s-’1
20 30 40 50 60 70 80
30.9 31.4 31.6 31.3 31.2 30.7 30.6
C,
Cs
C,/C,
rl
r.
0.393 0.35 0.89 0.464 1.35 2.91 0.455 1.25 2.74 0.532 0.440 1.19 2.69 0.475 0.414 1.20 2.90 0.427 0.396 1.34 3.40 0.425 0.379 1.37 3.60 0.411
V,. [lo6 m3] 0, = 0,= 0,= = 18 m3 s - ’ = 21 m3 s - ’ = 24m3 s - ’ 114 179 166 158 126 115 108
338
444 417 403 361 352 328
666 823 777 769 707 696 657
Usually discharge series are available from the time of the installation of a discharge-gauging site up to the present, and it should therefore be realized what consequences for the estimation of statistical parameters and for the over-year component the beginning of observations can have. Table 5.1 gives the statistical characteristics determined from 20 years, 30 years, ..., 80 years, always ending in 1970, and the size of the over-year component for various values of 0, (Po = 97%, presuming a correlation function in terms of K. Nachazel) on the river Berounka - Kfivoklat site. The last 30- as well as 40-year periods have very high C , values as compared to the 80-year series; coefficients of skewness are always very high. Differences in the values of are large. If, for example, the series 1941 to 1970 were used, the values of V,. would belarger by 65%with 0, = 10 m3 s-’, by 35% withO, = 21 m3 SKIand by 25%withO, = 24 m3s - ’ as compared with the values of the eighty-year series 1891-1970. For the discharge series Labe - DtEin and Berounka - Kfivoklht statistical characteristics were determined by the method of moments, which is suitable for computer processing, but does not enable a deeper analysis of any anomalies occurring in the observation period. In both series an exceptional influence of the mean discharge in 1940 and in 1941 on the moment characteristics can be observed. As this phenomenon can also be observed in other discharge series of the Labe catchment rivers and other rivers, the corrections of the statistical characteristics in shorter series were determined analogously to the experience from the IlO-year series on the Labe in DMin (Svoboda, 1964). It should not be forgotten, however, that in spite of several common traits the respective hydrological series also have specific characteristics (e.g., water-yield of 1941 need not be so expressive). Therefore, points of the “empirical” curve of exceeding mean annual discharges should be plotted in the probability paper and statistical parameters should be estimated by the quantile method (Chap. 3). Smaller random deviations of statistical characteristics can be expected that when using the method of moments. In the probability paper in Fig.
r
228 5.28 there are theoretical curves of exceedance of mean annual discharges for the period 1891-1970 and for 1931-1970. In both cases the statistical characteristics were determined by the method of quantiles. The noticeable difference in the two curves shows that the random deviations can occur throughout the whole curves of exceedance, which means that the variability of statistical characteristics is not only the result of random occurrence of extreme values of mean discharges. Much less attention has been paid to the influence of the observation period on the size of the seasonal component of the storage capacity. Votruba and Brota (1974) observed differences in the probability curves at the beginning and the end of the seasonal component in a 40-year series (1931-1970) and 80-year series (1891-1970) on the Berounka - Kfivoklat which were (0,= 25 m3 s- ') (Fig. 5.29).
610
m
; q
60
%o
a'40
Fig. 5.28 Difference in the theoretical exceedance curves of mean annual discharges (the river Berounka - Kfivoklat) with different input discharge observation periods (estimation of statistical characteristics by means of the quantile method)
-PC%I
Fig. 5.29 Differences in the probability curves of the respective parts of the seasonal component with various durations of the input discharge series (site Berounka river-Kiivoklat)
If we design a reservoir with over-year release control, we must be aware of possible random deviations in the results of the calculations. Even when applying the best available methods to determine the seasonal component, we reach a result which must be considered as an estimation with an error, which on the basis of a 30-year
229 series (e.g., 1931-1960) or a 40-year series (1931-1970) can be &20%, f30% or even more. Besides this error resulting purely from a limited length of discharge series, there can be others, caused, for example, by derived hydrological data (i.e. estimated, based on the measured data), etc. The influence of the initial real discharge series can also be observed when synthetic series are used for water management calculations. The observed series helps to estimate the statistical characteristics and correlation bonds, which are the input data of linear regression model of discharge series, and determined the discharge fluctuations within the respective years when using the fragment method, etc. When modelling the sequence of mean monthly discharges, a large number of numerical constants must be determined from the real series: 3 . 12 basic statistical characteristics, 12. 12 elements of the correlation matrix or further relationships when using the Markov chain of a higher order. Statistical samples of mean monthly discharges Q m , i have, as compared to mean annual discharges, a much greater variability (e.g., on the site at Kfivoklat on the river Berounka the values of C,, for the period 1931-1970 are in the interval (0.620; 0.964)) and also show much greater skewness. The estimation of statistical characteristics Qma,i.Cvm,i and C,,,, (i = November, December, ..., October) is very sensitive to any extreme discharges Q,., in the initial real series; their influence is also unfavourably reflected in the elements of the correlation matrix. This logically leads to the demand that the longest possible discharge series be used as a basis for the construction of a regression model. The variability of the values of C,,,, in relation to the length of the discharge series that was used (80 f 30 years) is shown in Table 5.2. A graphical-numerical analysis of the respective samples of mean monthly discharges can be found very useful. Tuhle 5.2 Variation coefficient of mean discharges in corresponding months with various lengths
of discharge series (the river Berounka-Kfivoklat)
Period
1891-1970 1901- 1970 19 I 1 - 1970 192I - 1970 193I - I970 1941- 1970
XI
XI1
I
I1
III
IV
v
VI
VII
VIIl
IX
x
0.70 0.71 0.72 0.75 0.79 0.83
0.70 0.70 0.73 0.77 0.78 0.71
0.71 0.71 0.74 0.60 0.64 0.69
0.65 0.67 0.69 0.74 0.73 0.77
0.64 0.66 0.68 0.72 0.73 0.67
0.68 0.66 0.68 0.68 0.72 0.74
0.58 0.56 0.59 0.60 0.63 0.71
0.91 0.95 0.96 0.97 0.96 1.06
0.95 0.97 1.00 1.03 0.96 1.05
0.75 0.74 0.76 0.77 0.77 0.84
0.70 0.68 0.69 0.71 0.73 0.59
0.62 0.60 0.63 0.60 0.62 0.59
Other sources of inaccuracies can be the numerical stability of the program, the transformation used, very small or randomly high elements in the correlation matrix, etc. It was found, for example, that synthetic series of mean monthly discharges do not automatically keep the statistical rates of mean annual discharges of the initial real series (Broia, 1973).This simple test of a modelled series is always useful. As a rule it will be necessary to construct several random series and to choose from them those which proved the most suitable in the testing. The length of a random series is important (for calculations corresponding to a theoretically infinite series. Votruba et ul. (1971) studied the deviations in the values of the relative yield a and the relative sizes of the storage volume fi, with various lengths of random series (50,100,200,500 years) from the values obtained from a 200-years series. A graphical illustration of the marginal deviations of storage-yield curves
230 a) B e m n h - Kfiivdtl6f
- cesk6 Skolice
c)
-a
-a
-a
Fig. 5.30 Limit deviations of relation fi, = f ( a ) for Po = 99% determined from 500-year and 200-year synthetic series of mean monthly discharges
fi, = f ( a )with a reliability Po = 99% for 200-year and 500-year series can be found in Fig. 5.30.If we do not study the deviations when a > 0.9, it is obvious that the relative deviations practically do not change with $change of a or fi,. Extreme relative deviations Aa/a and Afi,/fiz (in % of values from a 2000-year series) can be found in Table 5.3 and 5.4. For an accuracy of release (withdrawal) of +_5% it will suffice if the random series is 500 years long, or for greater reliability lo00 years long. However, this means that the size of the storage capacity will only be f 25% to f35% accurate (Table 5.4). If we allow a certain relative error in demands on the storage function of a reservoir (required withdrawal) we can accept a five- to seven-times larger relative error in the size of the storage capacity. Tuhle 5.3 Extreme deviations of relative yield from the value a2,,, with various lengths of synthetic series (Berounka r.-Kfivoklht, Labe r.-Brandys, Upa r.-C. Skalice)
Extreme deviation Aa (in % of value a2,,,)-length
Po [“A1 K
90
- 9 12
95
-12 13
91
-13 14
100 years B -12 7 -12 10 -15 11
99
K Kiivoklht-Berounka river
S
K
200 years B
- 7 8
- 7
- 5
6 -12 8 -12 10 -11 14
3 - 8 5 -11 6 -10
-10 8 -14 7
B BrandysLabe river
10
of series
S
K
-4 3 -6 2
-3 3
-7 5
-8 9
500 years B
-5 4 -4 4 -5
-3 2 -4 3 -5 3 -6
11
6
S &ski Skalice-Upa river
S
-2 1
-3 2 -3 4 -5 4
23 1 Table 5.4 Extreme deviations of the relative active storage capacity from the value /?z.2000
with various length of synthetic series (Berounka-r.-Ktivoklat, Labe r.-Brand?s, Upa r.-C. Skalice)
. Po
[XI K
Extreme deviation A& (in % of value / ? z , 2 0 0 0 ~ l e n gof t hseries 100 years 200 years 500 years B S K B S K B
90
-35 45
-25 50
-33 41
-29 38
-15 30
-13 19
-22 12
-10 55
95
-52 65
-31 53
-30 48
-35 70
-19 80
97
-56 58
-38 65
-31 52
-40 73
-22 81
-12 40 -24 41
-16 20 -22 24
-10 31 -18 43
-45 66
-37 66
-31 45
-39 26
-34 32
99
S - 7 12 - 8 18 -21 26 -24 40
For real accuracy of the theoretical size of the storage capacity, there is no need to apply a detailed balancing method to calculate the water losses from a reservoir by evaporation (Chap. 2). In this case we can use more general calculations, such as two time intervals in each year (period of emptying and period of partial filling, or winter and summer period) in which besides the evaporation depth the water pool level of a reservoir must also be considered. A simpler estimate with the help of the equation
z,
=
R,Pt
(5.36)
can also be used; here Z , is the water volume lost by evaporation from the water level during a certain “design” period of reservoir empying, RE- mean daily evaporation depth during the emptying period [mday-’I, F - estimated mean water
5
-a
-a
-a
Fig. 5.31 Probable duration of emptying period with over-year release control
232 level during the emptying period [m’] (from the characteristics of a reservoir), t - length of “design” period of reservoir emptying. However, it is always necessary to know the duration of the reservoir emptying in the period, which from the point of view of the characteristics of release control can be considered as the design duration. Calculations based on analytical or matrix methods determine the design duration of the “over-year” emptying, either directly during the calculations or by estimation with the help of Kritsky and Menkel’s (1952) graphs (Fig. 5.31). The direct method in random discharge series makes it possible to select one or several over-year low-flow periods, according to which the time of emptying typical for the given parameters of release control (cr, p,, P ) can be reliably estimated. It might also suffice to use only the most critical period of the real discharge series. In Czechoslovakia the supplementary volume needed to cover evaporation losses in over-year release control can come to more than 20% of the theoretical size V, (with relatively flat reservoirs), however, usually it is less (no more than 15%). When the water losses from a reservoir are an important part of the total balance of a reservoir regime, e.g., with flat reservoirs in dry, warm regions, these losses should be considered as part of the release (withdrawal) from a reservoir, which makes the calculations more complicated (the approximation method must be used). The error in the theoretical size of the storage capacity arising from the application of series of mean monthly discharges instead of more detailed, e.g., mean daily discharges, is not very great in over-year release control. It can be presumed (see Chap. 6) that the increase in the ordinates of the probability curves of the respective parts of the seasonal component, when using mean daily discharges, will not be greater than 10% as compared with the results gained from mean monthly discharges (the over-year component does not change). As the seasonal component is only a certain part of the theoretical size of the storage capacity (its share can be estimated), the total error will only be a small percentage. After the theoretical size of the storage capacity has been determined quantitatively (with a relatively large probable error) and after adding volumes for water losses, the resultant size of the storage capacity A , must be determined. Here not only the water-management aspects, but other aspects connected with reservoir design must be considered, such as the rate of exploitation of a given locality from the morphological, hydrological, etc., points of view, the importance of a reservoir in the complex of water resources in a given region, real conditions of the future reservoir operations, expected development of water demand in the future, development of the complete water-management system in a region and the construction of a reservoir in this system as well as the construction of further water resources. Such an analysis gives a comprehensive view of the design of a reservoir based so far on isolated calculations, as well as a qualitative evaluation of any reserves in the whole system of water resources; this facilitates decisions on the measure of reliability of the design. ’
233 The simulation method for the over-year release control using real discharge series cannot give satisfactory results. Even though in recent years great attention has been paid to discharge series of the period 193 1 to 1960, and in several important discharge-gauging sites comparative calculations of the classical method (in real series) and statistical methods were carried out, the conclusions cannot be generalized. In real discharge series, for example, the storage function of a reservoir should be compared with the characteristics of release control determined by probability methods. It can be used for a quick comparison of alternative designs in preliminary studies, for the solution of problems connected with the reservoir operations, etc.
Fig. 5.32 Ideal diagram of overyear release control
The literature (e.g., Morozov, 1954) describes over-year release control in which the storage volume is used not only for its basic function of augmenting the discharge to the required withdrawal O,, but also for the best possible discharge regulation during high-flow periods. This method is illustrated in Fig. 5.32, in which the mass curve of inflow to a reservoir, is drawn for several years and at a distance equal to the size of the storage capacity V, an equidistant is drawn. If we draw the mass curve of release we obtain a very balanced release. and from a reservoir as a “tight thread between the curve As for such a release control, discharges have to be known for several years in advance, it cannot be applied the term to real reservoir operations. From the method of constructing the mass curve release “tight-thread-method has been derived.
&,
LG
L.,
L,,
6 WITHIN-YEAR (SEASONAL) RELEASE CONTROL A reservoir works with a within-year (seasonal)cycle if it is able to distribute the discharges to cover the required withdrawal within a year, i.e., if the mean withdrawal is less than the mean discharge in the driest years. This definition explains the essence of within-year release control, it is, however, not completely accurate. In the introduction to Chapter 5, within-year release control is explained as a special case of over-year control, when the probability of the occuris very small. It is then possible to rence of years with a mean discharge Q, c
op
Fig. 6.1 Work regime of a reservoir with within-year release control for a constant reliable yield Lor) (withdrawal) 0,(in mass curves
L.
neglect the influence of these years on the reservoir regime without making the results less accurate. We actually have to neglect this influnce as there are no available data for its quantitative expression. This approach is significant from the technical point of view as the method for calculating the within-year release control can be derived from the methods of over-year release control. In Figure 6.1, a time curve of the filling (emptying)of the storage capacity derived, and withdrawal e.g., from the graphical solution in mass curves of inflow
(cQ)
(cop),
235 illustrates the regime of a reservoir with within-year release control in a ten-year period. Inflow to a reservoir is identical with the case in Fig. 5.1 ;presuming a constant safe yield 0, = 12 m3 s - ' (a = 0.4). The reservoir's working cycle is in no case longer than one year. The beginning, termination and duration of low-flow periods are very variable, which is typical for many rivers. The same applies to a quick refilling of the storage capacity. In view of the not very high relative yield (even if it is close to the upper limit for within-year release control), the probability of the occurrence of years in which the inflow to a reservoir does not drop below the value of the required withdrawal and a reservoir does not exert its influnce, is relatively high. In the ten-year period in Fig. 6.1 the probability of a full reservoir (in terms of duration) is -64% (for the over-year release control illustrated in Fig. 5.1 it was only 31%). The annual cycle of a reservoir is not identical with the water year (November October) nor with the calendar year (January - December). For rivers with a simple hydrological regime, the so-called wuter-management year can easily be defined; its beginning coincides, e.g., with the beginning of the wet period. For.irregular regimes this term is less strictly defined: usually the only rule is that the reservoir cycle should be completed within one water management year. The term wirhin-year release control refers to the annual discharge cycle which, in spite of the differences in the respective years, is a general phenomenon with a genetic justification. The equivalent seasonal release control is derived from the seasonal character of water withdrawal (e.g., for irrigation, hydro-power, food industry, etc.). In the framework of an annual cycle much shorter release control can also be applied, e.g., daily or weekly (if the reservoir also serves as a storage or balancing reservoir for peak load or pumped-storage hydro-power plants), which might affect the size of the storage volume; however, these calculations can be carried out separately. In view of the random discharge distribution in the respective years, probability methods are also used for within-year release control. One possibility is statistically to process significant factors of the annual discharge cycle, e.g., beginning of low-flow periods, duration and depth of discharge depressions, analogous signs of wet periods, etc., and to create an unreal hypothetical design year. This process, which is suitable for rivers with a simple discharge pattern, cannot be applied to complicated hydrological conditions (Votruba and BroZa, 1966). In this case, statistical methods should be used only after gaining further results in real series (necessarystorage capacities in the respective years or regulated discharges). Stochastic discharge series can be used similarly as for over-year release control.
236 6.1 STORAGE CAPACITY DETERMINED WITH THE HELP OF EXCEEDANCE PROBABILITY CURVES OF NECESSARY VOLUMES
Here we apply the method described in Section 5.1.2 in its general formulation for over-year release control, using the basis of water years. With given withdrawal values 0, and with a design reliability Po (occurrence-based),the theoretical storage capacity V, (Po)is sought. In within-year release control, the relationship Q, 2 0, holds for all the years of the real series. (If one or two years with Q, < 0, occur in the series, they can be neglected or the method for over-year release control can be applied.) In every year the respective parts of the storage capacity init AK, end A K and mid AVz are then determined by the same method as the respective parts of the seasonal componentSection 5.1.2, equation (5.22) to (5.24Fand after adding the value of end A K of one year ( t ) and init A K of the following year (t + l), an “empirical” probability curve 1/2( 1) = f’(p) is constructed and then a probability curve mid A< = f’(p), which is the component of the size of the storage capacity (Fig. 6.2).
Fig. 6.2 Construction of probability curve of necessary storage volumes V, = j ’ ( p )
-PC%l
Po
Over-year low-flow periods do not have to be taken into consideration (the probability curve V;. = f ( p ) is zero, therefore the method is greatly simplified. It only remains to select the significant values V,(1) and mid AV, (see Fig. 5.17), the result of which is the probability curue of the needed storage capacity V, = f’(p), from which we obtain, for the design realibility Po, the resultant size of the storage capacity 1/2(Po)(Fig. 6.2).
231 This method can also be applied if we look for the yield (withdrawal) with a given size of storage capacity. We select several 0, values (all within the seasonal release valid for the control), repeat the calculations, construct the relationship V, = f(0,) design reliability Po, and from that we determine the withdrawal 0, for the given size V,. Similarly as for over-year release control, calculations can be simplified by V. G. Andreianov’s method, however, in this case without any detriment to accuracy. The size of the volume necessary to ensure a withdrawal 0, is determined for every water management year by direct calculations. After arranging the values V, in order of increasing magnitude, whereby a certain number of necQsary volumes equal zero (in relation to the relative yield a), the probability curve V, = f(p) is constructed, giving the resultant storage capacity V,(P,) for the given reliability Po. 6.2 DETERMINATION OF THE YIELD (RELEASE, WITHDRAWAL) FROM A RESERVOIR USING FllTED THEORETICAL CURVES
The method presumes a given size of storage volume and the quantity to be determined is the augmented release from a reservoir. In every water management year of the real hydrological series, the theoretically possible withdrawal is determined; this is possible by distributing the discharge to be ensured by the given storage capacity. The basic relationship is simple: 0p.r
=
Qlf
+
v,
7
Tlf
(6.1)
where Q,f is the mean discharge in the low-flow period [m3 s-’1, V, - the size of the storage capacity [m3], T~~ - duration of low-flow period [s]. The term low-flow (dry) period in a year is a relative term, referring to the mean discharges in the year. In more complicated cases, where the duration of a low-flow period cannot be clearly estimated, it is indispensible to make the calculations for several estimated periods and to select the one with the smallest Op,rvalue. A mass discharge curve greatly facilitates the selection of the most critical low-flow period. The Op,rvalue can be determined with the help of a computer, but as all possible alternatives must be systematically tested, this is time consuming. A statistical population of theoretically possible O,,, withdrawals in which, as compared to Andreianov’s method, all members are non-zero, is arranged in order of decreasing magnitude; then the basic statistical characteristics and the curve of exceeding Op,r= f’(p) are ascertained, from which we determine the safe 0, (Po) yield corresponding to the required reliability Po. The author of this method Liapichov (1955) states that the types currently used in statistical evaluations of discharges can be used for curves of exceedance and
238 statistical characteristics of possible OPllwithdrawals. The skewness coefficient can be the same as for mean discharges in low-flow periods (this refers to rivers with a simple hydrological regime), or mean annual discharges. More suitable is a graphical-numerical method using quantiles (Fig. 6.3).
2
-p
I %I
Fig. 6.3 Exceedancecurveof theoretical withdrawals with a given size of storage capacity (Liapichov’s method)
In Fig. 6.3 one can also see the shortcomings of Liapichov’s method. The dotted curve of exceedance fits the sample of empirical points as well as the full curve; the most significant differences are in the interval of high P values (due to the different skewhess coefficients),which are usually design values. The dotted curve of exceedance leads with p = 95% to an 0, value which is 10% lower and with p = 99% more than 30% lower as compared with the original curve. 6.3 CALCULATIONS OF WITHIN-YEAR RELEASE CONTROL IN SYNTHETIC DISCHARGE SERIES
Synthetic discharge series is justified for within-year release control mainly in the boundary regions between seasonal and over-year release control and also for complicated release control pattern from a reservoir, especially in real-time reservoir operation (see Chap. 4). The same applies to reservoirs with an annual cycle, which
239 are important parts of a more complicated water resources system (e.g., to raise the yield by transbasin water diversion). Mostly synthetic series of mean monthly discharges are used, even though they supply only approximate data for seasonal control. The modelling of series of, e.g., mean ten-day discharges must have many input constants (3 .36 statistical characteristics, 36 .36 members of correlation matrix, etc.), including the risks they reflect (e.g., the errors in estimation of sample characteristics will increase). An experiment with synthetic series of mean daily discharges (Water Resources Development and Construction Institute, Prague, 1970), showed that such series should be used only for some special calculations. Solutions in synthetic series for within-year release control are identical with the algorithm in Section 5.2, equations (5.29) to (5.35). By repeated calculations for several suitably selected sizes of storage capacity with a given value of withdrawal O,,the relationship V: = f ( p ) is obtained (Fig. 5.22), where the desired size of storage capacity V,(P)for the design reliability P can also be found. For within-year release control it is also possible to apply in random series some of the statistical methods originally meant for real series (e.g., Andreianov’s method) as a suitable means to verify the results.
c,
6.4 ACCURACY OF RESULTS IN WITHIN-YEAR RELEASE CONTROL
6.4.1 Error in the size of the storage capacity resulting from the application of series of mean monthly discharges
In annual reservoir cycles mean monthly discharges are a rough schematization of the continuous time function of the discharge Q = f(t),especially with a low level of yield a, so that they can cause great errors in the determination of the release control parameters (a, B,, P). Bratrhek (1939) compared the results of real series of mean daily and mean monthly discharges. Presuming that at the beginning or at the end of the “design” low-flow period a month occurs in which the mean discharge is influenced by a flood, Bratranek derived a relationship for the estimation of the error in the size of the storage capacity when using mean monthly discharges E(&)
=
O.O8(a - 0.1)
(6.2) The probability method for release control was studied at the Technical University in Prague (1976) to supply information about the influence of errors caused by the use of mean monthly discharges to determine the storage capacity with the help of the probability curve of necessary volumes (Andreianov’s method). In several gauging sites in the Labe catchment, calculations were made for the period 1931 to 1970 in mean daily discharge series and in mean monthly discharge series for a = 0.2; 0.3; 0.4, and over. Probability curves of the necessary volumes were con-
240
structed and deviations A% were evaluated in per cent of volumes resulting from mean monthly discharge series (Fig. 6.4). However, no general conclusions can be drawn from these calculations. Clearly as the result of random influences on the pattern of the probability curves of necessary volumes, it is often difficult, to construct the relationship A K [“A]= f’(cc) as a continuous declining curve, as had been expected.
-U
-Jizem- Vil6mov Saiow- P0r;‘E;
---- Ohm- P k k
._..........Jizpm- T&ce
Fig. 6.4 Errors in determining the size of the storage volume using mean monthly discharge series
Of note is the relationship between the errors AV, PA] and reliability Po [“A]. In most cases the values A \ [“A]decrease with increasing reliability. The reason most probably is that the increments of AV, (absolute) are independent of the values of V, (month). If the probability of the occurrence of certain absolute increments A% is approximately the same for any V, value (month) a decrease of the relative error A t with increasing reliability (and therefore with increasing V,) is a logical consequence. As the probability curves of necessary volumes are usually constructed from a small number of “empirical” points, another random factor is introduced which distorts the results. A great number of hydrological series will have to be processed in order to generalize the results. It also follows from the analysis that the size of the errors resulting from the application of mean monthly discharges must be evaluated from the point of view of the method that was used. For example, when using Liapichov’s method the values of the theoretically possible withdrawals in the respective years will be lower when using mean daily discharges than when using mean monthly discharges. In this case, too it can be expected that the deviations AOp,, (absolute) are independent of the values 0,. The method using synthetic series of mean monthly discharges is essentially the same as when using probability curves of necessary volumes for the design of the storage capacity. A correction can therefore be made of the results gained from synthetic series according to the relationship between the “daily” and “monthly” quantities resulting from the real series. A theoretically more correct procedure would be to determine in the real series the deviation AVz (absolute) from calculations in
24 1
daily and monthly discharges; a probability curve AV, = f (p) is constructed which is then combined with the probability curve V, = f ( p ) gained from repeated calculations in synthetic series of mean monthly discharges (for various V, values with same Op). 6.4.2 Influence of the input real series on the characteristics of release control in an annual cycle
When using statistical methods for within-year release control, the influence of the input real discharge series on the results of the solution can also be observed.
50 €0
70
80
-
95
pt%l
97
99
Fig. 6.5.Variability of the probability curves of necessary storage volumes (Andreianov'smethod) in relation to the input observation period (the river Berounka - Kfivoklat)
Figure 6.5 shows a probability paper with plotted probability curves of necessary storage capacity (from mean monthly discharges) in the Kfivoklat site on the Berounka, presuming a withdrawal of 0, = 8 m3 s-' (a i 0.25) for an 80-year period from 1891to 1970and for several other selected periods. The differences are relatively large; e.g., the storage capacity V, with Po = 95% for the period 1931 to 1970 is 34% greater than the respective value from the 80-year series. The possibility of such great differences is in agreement with the results gained from the study of synthetic series, where approximately the same differences of the a values regardless of the size p, or of the b, values, regardless of the size of a could be observed (Fig. 5.30, Section 5.3).
242
In seasonal release control the risk of deviations (errors) in design values V,(Po)or O,(Po) has been generally considered to be much smaller than in over-year control. We have proved that in spite of using statistical methods and after correcting the results of calculations with series of mean monthly discharges, the reliability in determining the characteristics of within-year release control is approximately the same as for over-year control. 6.4.3 Corrections of the storage capacity with regard to evaporation losses
Evaporation losses can be calculated approximately without affecting the accuracy. The duration of a low-flow period typical for the given degree of discharge regulation, its time in the year, total evaporation depth x H E [m] and the average water level in a reservoir F [m’] for this period must be determined. Then the water volume lost by evaporation from the water level Z,[m3] is given by the relationship 2, = F C H ,
The extent of evaporation should be determined for several years which have had the highest claim on the size of the storage capacity; of particular significance are the summer and autumn low-flow periods. An inundated area related to threequarters of the depth of the storage capacity can be considered as a reliable estimate of F. The volume that should be added to the theoretical V, size to cover evaporation losses is, in annual control, usually only a few per cent of the V, value. 6.4.4 Water-management plan for reservoirs with power plants
If reservoirs which regulate discharges for peak-load hydro-power plants work in an annual cycle, a low-flow year (approximately the “design” year, or even a failure year), an average year, or a high-flow year can be selected in the real discharge series. Direct calculations are then carried out, including the time pattern of the filling of a reservoir, tailwater and headwater, the head and mean power-plant output. Figure 6.6 shows a low-flow year (the storage volume empties without causing any failure) and an average year. In order to illustrate this, a graphical method was used. In winter (November - February) release is 50% higher than in the rest of the year. the time pattern From direct graphical calculations on mass curves of & and of filling V = f ( t ) can be obtained by plotting the vertical distances between them from the horizontal base. With the help of volume-depth curves V = @(h)and the secant under a 45% angle, we obtain the time behaviour of the fluctuations of the level of the headwater h = f (t). Presuming that the tailwater level is constant (e.g., the mean level of balancing
243 reservoir), we obtain the head curve H = f(t) simply by shifting the scale of the ordinates. With a known release 0 (Fig. 6.6a) and gross head H (Fig. 6.6d), the time pattern of mean outputs can be calculated from
P
= 9.81qOH
which is plotted in Fig. 6.6e.
Fig. 6.6 Water-management plan of a reservoir for the generation of hydro-power in a low-flow and an average year
244
EF
On the curve of mean outputs P = f (t) a mass curve E = is constructed, the ordinates of which give us the annual production of electrical power. Calculations for three suitably selected years provide basic information about the electrical power parameters of a reservoir. More extensive information is obtained from calculations of the whole observation period (real hydrological series) and . statistical evaluations of the reliability of outputs and power production in the respective years. This applies all the more to over-year release control of reservoirs for hydro-power stations. 6.4.5 Within-year release control using curves of exceeding mean daily discharges
For small reservoirs with a direct supply function for various needs. which are mostly found on small streams for which there are no discharge observations available, the curves of exceeding mean daily discharges are sufficient as a basis for the estimation of the size of the storage capacity.
Fig. 6.7 Within-year release control calculated in the exceedance curve of mean daily discharges
From a comparison of calculations in time series and on the curve of exceedance (Fig. 6.7) it is possible to find out how successful was the estimation of the size of the storage capacityfrom the curve of exceeding mean daily discharges, with the following conditions: - release control must be within-year (seasonal), - water withdrawal is constant, - the period of augmented discharge is continuous, or interrupted only by insignificant discharge peaks. The size of the volume Y(T)determined by the curve of exceedance equals, or is greater than, the volume Y(t)ascertained on the time curve. Differences can vary from negligible values up to tens of per cent, depending on the discharge fluctuations (Votruba and Brofa, 1966). It can be presumed that in low-flow years with continuous discharge depressions, the storage capacity will not be over-estimated by more than 20 to 30% when using curves of exceedance.
245
An important problem is the definition of the “design” curve of exceeding mean daily discharges and its characteristics. So-called mean exceedance curves naturally cannot be used to estimate the size of the storage capacity. Much better from the point of view of low-flow periods is the so-called minimum exceedance curve (Hydrometeorological Institute, Prague, 1970), even though even this curve has no direct relationship to the specific low-flow years. Due to the above-mentioned unclarity, which continues to increase when introducing the rate of reliability of water supply in the solutions, it is better to use suitable estimated flows based on measured flows making it possible to work with time discharge series; shortcomings in hydrological data must be balanced by a greater “safety margin” in the calculations.
7 SHORT-TERM RELEASE CONTROL In short-term release control the cycle of filling and emptying a reservoir is of the order of days; in periodical release control the cycle is regularly repeated (usually with a one-day or weekly periodicity), in non-periodical release control a reservoir is filled and emptied irregularly, according to hydrological conditions or according to needs. Periodical control serves to meet the periodical variability of water demand during 24 hours or during the seven days of a week, if the water demand is not the same for all the days of the week. An example of a non-periodical short-term control is the accumulation of water from a small resource for short intensive irrigation, for additional re-regulation of the controlled discharge from reservoirs upstream, etc. 7.1 DAILY RELEASE CONTROL
Daily release control is applied when the periodically repeated reservoir cycle lasts for one day (24 hours). An example could be a water tank, a balancing reservoir for a peak-load hydro-power plant, reservoirs for pumped-storage hydro-power plants, etc. 7.1 .I Capacity of a water tank
'
The water demands of public water supply vary greatly during the day. This must be taken into consideration when assessing the yield of the water resource and in determining the parameters of the waterworks facilities, such as the pipelines, water tanks, etc. Fluctuations of water demand depend on the character of the community. If no more detailed data are available, then Table 7.1 can be used as a basis. The table also shows the distribution of water demand in Prague on June 12, 1964. From the values in Table 7.1 we can determine the necessary storage capacity (water tank) to meet the variable withdrawal, if we know the behaviour of the inflow to the water tank as a percentage of the water demand throughout one day. Thus we ascertain the size of the necessary storage as a percentage of the water demand in one day. Figure 7.1 gives a graphical solution for a water tank for a housing estate. The from 10p.m. till 6a.m. mass demand curve & and the mass pumping curves
1:
247 Table 7.1 Hourly water demand as a percentage of daily demand Hour
24- 1 1- 2 2- 3 3- 4 4- 5 5- 6 6- 7 7- 8 8- 9 9-10 10-11 11-12
For coefficient k,
Prague 12th June
1.8
2.1
1964
1.o 0.7
1.6 1.5 1.5 1.5 3.0 4.2 5.0 5.0 5.0 4.6 4.2 4.6
2.4 2.9 2.4 2.3 2.7 3.0
g
0.7 2.0 3.0 5.0 6.4 4.5 5.5 5.5 5.5
4.9 4.8 6.0 4.7
Hour
1 12-13
For coefficient k, 1.8
2.1
1964
5.0 5.0 4.0 5.0 5.0 6.0
63
4.6 4.8 4.6 4.6 4.6 5.0 6.5
72
@
5.0 3.9 6.0 4.0 5.7 5.3 3.9 6.1 4.9 4.5 4.0
13-14 14-15 15-16 16-17 17-18 18-19 19-20 20-21 21-22 22-23 23-24
5.0 4.0 1.5
5.0 4.6 3.2 2.0
told
100.0
100.0
ratio max: min
Prague 12th June
55
10.7
5.87
3.0 100.0 4.85
Fig. 7.1 Calculation of the volume of a water supply tank for a housing estate with different times of pumping
1:
x:
and from 12 a.m. till 6 p.m., for pumping only from 10 p.m. till 6 a.m. and for pumping throughout the day are plotted. It can be seen that in the first case the necessary storage capacity is 38.6%, in the second case 81.5% and in the third case 14.8% of the demand in one day. The way water is pumped to the water tank greatly
248
influences its size. If we want to limit pumping to night time, the volumes of the water tanks and the output of the pumps and engines must be much greater. When pumping proceeds throughout the day or when the inflow to a water tanks is uniform and if the all-day inflow and withdrawal are equal, 4.16% of the all-day demand flows into a water tank per hour and the necessary storage, in terms of Fig. 7.1, is 14.8%. A uniform inflow to a water tank is used mainly for filling without pumping, in which case the minimum yield of the resource must be verified; very frequently it is slightly more than the water demand. Under these circumstances the water tank volume can be much smaller (Volejnik, 1959). If the minimum yield of the resource exceeds the maximum daily water demand by 40 to Ox,there is no need for a water tank. 7.1.2 Daily release control for hydro-power generation
Figure 7.2 shows the load distribution of the electric power system of Czechoslovakia in the four seasons of 1975, and the changes of position, depth and duration of the peeks during a year can be seen. The load variability during a day is reflected in the daily release control in several ways : (a) In reservoirs with within-year or over-year release control a daily cycle is reflected only by small fluctuations of the water level so it can be disregarded when judging the work of a reservoir and power plant. The influence of great and sudden withdrawals causing currents of water in the reservoir which affect the quality characteristics of the water in reservoir and the water flowing out of it has not yet been determined.
2
in the month
I
1
-Chl
Fig. 7.2 Daily load of the power system in Czechoslovakia in 1975
249 (b) In the balancing reservoir of a peak-load hydro-power plant the inflow is given indirectly by the load of the plant. When the fluctuations of the head and the efficiency during the day are negligible, then the shape of the output diagram is identical to the shape of the inflow curve to the balancing reservoir, if the scale of ordinates is changed according to the relationship P = 9.81qQH = KQ
(7.1)
where P is the hydro-power plant output at the turbine shaft [kW], q - effectiveness of the turbine, Q - discharge through the turbines [m3 s- '1, H - hydro-power plant head [m]. An analysis of the work of the balancing reservoir of a peak-load hydro-power plant can be found in Fig. 7.3. A diagram of the load of a power system receiving its power from thermal-power plants (TPP) and hydro-power plants (HPP) is also plotted in Fig. 7.3. The insufficientoutput and production is to be augmented from the newly designed HPP,, which can store the necessary amount of water and can therefore be used to cover peak loads up to the 30% defkits of the total maximum load of the system. A summation curve E = f(P)can be used for an analysis of how to cover the load diagram-Fig. 7.3b. In point I the vertical (at the level of 70% of the maximum load) marks the value Pp = 11.6% on the line of mean peak outputs. The same value can be found between points 3 and 4 ; point 3 is the point of intersection of parallels I, 3 to the straight line of mean peak (or basic) outputs, with the vertical drawn through the end point of the summation curve. The summation curve defines that HPP, is to supply 14.5% of the total production in the system, which must be met by the power resource with the required rate of reliability. In Fig. 7 . 3 ~ the load diagram of HPP, takes the shape of two trapezoids (dot-anddash curve);attached to it, on the left, is a scale of P in kW. Presuming that in equation (7.1) the value 9 . 8 1 ~= 8.5 and the head H = 50m (i,e., constant with a negligible fluctuation &lo%),then the load diagram can be considered to be the discharge curve through the turbines and therefore the inflow curve to the balancing reservoir of a peak-load hydro-power plant, with the scale P = 9.81qHQ = 425Q
(7.2)
i.e., 425 kW corresponds to 1 m3 s-', or 100 MW ... 235 m3 s-'. In the same way, it is possible also to attach to the summation curve in Fig. 7.3d the power scale E [GWh] and the scale of the water volume, from the relationship 1 cm = 1 GWh = 100 OOO kW 10 h = 8.46 - lo6 m3, i.e., 10 * lo6 m3 = 1.18 cm. The peak of 300 MW in the summation curve corresponds to the production of E = = 3.06 GWh d-' and the necessary amount of water 25.9 - lo6 m3.
250
According to the summation curve E, about 1.1 GWh (i.e., 9.3 * lo6m3) are needed to distribute the flow to reach the mean value. However, as the diagram has two peaks, divided by a trough that is greater than Pp, these values are slightly higher than the accurate values that are absolutely necessary. These accurate values can be found from the summation curve & = E in Fig: 7.3e, drawn with a pole distance of f = 2.5 cm, for the scale of the mass curve to be the same as the scale of the summation curve. The size of the storage capacity needed to balance the inflow to meet a constant release is V, = 8 * lo6m3, which is the volume of the storage capacity of the balancing reservoir.
4
h
l
Fig. 7.3 Design and operation of balancing reservoir downstream of a peak-load hydro-power plant H = const.
25 1
cQ
co
From the relationship between the curves for mass intlow and release from the balancing reservoir (Fig. 7.3e),the curve of the tilling and releasing of the balancing reservoir V = f(t) can be transferred to Fig. 7.3f.
Fig. 7.4. Graphical-numerical solution of a balancing reservoir downstream of a peakload hydro-power plant, taking into account the variability of its head
If the depth-volume curve of the balancing reservoir V = @@)-Fig. 7.3g-is suitably joined to the curve of the filling and releasing, it is possible to draw over this curve, at a secant of a 45" angle, the curve of the fluctuation of the level h = j ( r ) ; the method is determined by points 2,3 and 4. The level fluctuates from evaluation 15 to 24. In view of the head of the peak-load H P P ( H = 50 m), fluctuations are rather extensive and can influence the volume of the balancing reservoir and the shape of curves V = f(t) and h = f(r). However, it can be expected that the influence is not of greater order than that of the schematization of the base curves Q and P (Fig. 7.3~). For a precise solution it would not be possible to have the curves Q and Pin Fig. 7 . 3 ~ identical, but we would have to start with the given curve P = f(t) and look for the curve Q = f ( t ) , e.g., using the graphical-numerical method shown in Fig. 7.4 and in Table 7.2. From Fig. 7.3c, e we first of all determine the value Fp,Q and A,. In Fig. 7.4a we find the moment for which we know the position of the water level in a reservoir; the best moment is when the reservoir is empty and is beginning to be filled; at that moment (7 a.m. o'clock) we start calculations. We solve the problem in intervals (1 or 2 hours) and the initial as well as the calculated, or graphically determined, values are then written down in Table 7.2 and plotted in Fig. 7.4.
252 Table 7.2 Balancing reservoir considering the variability of the head of a peak-load hydro-power plant Hour
Elevation main balancing
7.00a.m. 8.00a.m. 10.00a.m. 12.00a.m. 2.00 p.m. 4.00 p.m. 6.00 p.m.
70.00 70.00 70.00 70.00 70.00 70.00 70.00.
-
-
15.00 16.00 18.00 20.00 19.00 19.00 22.00 -
Head H
Output P
(4
(MW)
55.0 54.0 52.0 50.0 51.0 51.0 48.0
200 250 250 125 150 300
P Note n =8.5H [m's-'] 427.5
545 565 294 346 691
balancing reservoir empty
-
We consider the state of the level (70.0) in the main reservoir to be constant throughout the day. At 7 a.m. the balancing reservoir is empty, the level is at the elevation of dead storage M, = 15.00. The head is therefore 55m which we apply to the first interval from 7 to 8 a.m.; a certain inaccuracy arises from the fact that the initial state of the head is used for all the following intervals. From Fig. 7.4a we read the mean output of the interval (200 MW) and calculate the corresponding discharge Q = P/8.5H = 427.5 m3 s-'. To obtain the changes in the filling of the balancing reservoir, we draw the mass and the mass release curve from pole o at the height Q. Then inflow curve is a horizontal straight line, which at the same time is the axis for the mass curve the curve of filling V = f ( t ) , which is given by the vertical distance between the two mass curves. From the curve of filling and releasing and the depth-volume curve of the balancing reservoir V = @(H)and a secant under a 45" angle, we obtain the curve of the state of the levels in the balancing reservoir h = f(t), from which we start to solve the next interval. As we must know the state of the level in the reservoir for the further calculations, we write it down in the table-at 8 a.m. it is 16.0 m-and we calculate the head H = = 54 m for the next interval 8 till 10 a.m. with an output of 250 MW and a calculated discharge of Q = 545 m3 s-'. We draw the next section of the mass curve & within the limits of this interval and we transfer the filling at the end of this interval (about 1.7 - lo6 m3) to the state of the level (about 18.0m), whereby we obtain, for the next
c0
co
253
interval 10 till 12 a.m., a head of H = 52 m, etc. For a real project, larger scales of all curves should be used to make the values more accurate. Water-resources engineering also makes use of the opposite function of a reservoir with daily release control, i.e., to divide the uniform inflow to serve a non-uniform release. The reservoirs are called distribution reservoirs. A water tank with a constant inflow has a similar “distribution” function.
Fig. 7.5 Diagram of utilizing hydro-power from the distribution reservoir (a)power plant by the impounding structure; (b) on ‘the derivation canal
This can be the case of a peak-load hydro-power plant with an impounding structure forming a small reservoir, the volume of which is only large enough for daily or weekly release control (Fig. 7.5a). Inflow to the reservoir is considered to be constant and release according to the character of the structure and the needs of the power system. It can also be a case of utilization of hydro-power with a long derivation canal with a free water level and a peak-load hydro-power plant. Here the conduit is usually designed for peak discharges. The conduit is cheaper if designed only for a mean daily discharges which can be the case when there are suitable conditions at the end of the conduit for a distribution reservoir (Fig. 7.5b). In both cases the pressure pipeline (penstock)to the power plant must be able to meet a peak discharge. It must be considered from the economic point of view which of the alternatives is more suitable. A general solution for the capacity of a distribution reservoir is given in Fig. 7.6. Calculations were made on summation curves E and U,although it was a case of dnily dischnrges with two peaks. Figure 7.3 shows how to determine accurately the necessary storage capacity V , of a balancing reservoir on mass curves of and We shall explain how we can find the correct V, value using only summation curves, without having to draw mass curves. From the chronological curve P E 0 = f(r) we draw the duration curve P, = 0, and from it the summation curve E = U,, from pole 0’. The horizontal distance of the summation curve from the straight line of mean basic withdrawal determines, for any value 0 , how much water is needed to reach the amount AUi = Oi . 2 4 .3600 [m3]. On the level of Om,, this shortage is expressed in the scale of the sum-
za L.
254
-
Ch1
Fig. 7.6 Analysis of a distribution reservoir in summation curves
mation curve by the abscissa AUmaX. On the level of 6,this shortage equals the excess water during withdrawal peaks above this level. This value in the case of one or two equals the storage peaks with a slight trough (where Obi, between the two peaks 2 0) capacity necessary for the distribution of a constant inflow P = f ( t ) = const = 0 for the non-uniform withdrawal of 0 = f ( t ) ; in Fig. 7.6d there are three cases in which this simple solution was used to determine the correct volume of a distribution reservoir. An analysis of the distribution function of a reservoir can be shown graphically if we draw a summation curve UL of the above-mentioned water shortage into AUi. We draw this curve from pole 0’’ to the duration curve 0,, starting from point 0,. It follows that its distance from the vertical must be, at the same level, of the same size as the horizontal distance of the summation curve U , from the straight line of mean basic withdrawals; it can therefore be constructed also in this simple way. The two curves U , and UL intersect at the level 0 and the distance of this point of intersection from the vertical drawn through the end ordinate otthe summation curve U, gives us the volume of the distribution reservoir, with only a slight inaccuracy caused by the two peaks. The way to determine the precise V, value of a distribution reservoir for withdrawal with two pronounced peaks, using summation curves, is shown in Fig. 7.6a, b, c. It is clear that the necessary V, must be smaller than the V: value in Fig. 7.6b by the volume shown in Fig. 7.6a by the respective area of the trough of the curve 0 below the value P between both peaks. In Fig. 7 . 6 ~it is therefore sufficient to draw the summation curve U, to the trough (2, 3, 4) transferred into the duration curve 0;. On the level of P = 0,the horizontal distance U, from the vertical drawn through the initial point of the trough gives us the volume AV,, by which V: must be decreased to obtain the correct value V,. If all the summation curves are drawn at the same scale (with the same pole distance), it is sufficient to subtract the abscissae, i.e., V, = V: - A K to determine V,. To determine which withdrawn amount has to be replaced (hatched area at the
255
top 2 of the curve 0 = f ( t ) ) , we draw the summation curve U ,to the peak (I, 2,3) which we previously transferred to the duration curve 0:. The same method that has been explained for a distribution reservoir can also be used for a balancing reservoir; however, denotations P and 0 have to be exchanged. A summation curve makes it possible to analyse mainly those cases where the reservoir capacity is not large enough for complete balancing (Votruba and Broia, 1966). With daily release control for power plants, the problem is more complicated. In peak-load power plants great outputs start or stop within a few minutes. There is unsteady flow and the water level fluctuates in waves, which mostly affect low-pressure plants with small reservoirs and diversion canals of power plants, where the conduit also serves as a capacity able to baiance sudden changes in withdrawals. This changes the head of the power plant, which in turn changes the discharges. It is therefore not sufficient to simply determine only the reservoir volume, but the hydraulic regime in the upper and lower reservoir basin and its effect on the work of the hydro-power plants and on the function of reservoirs. A balancing reservoir downstream of a peak-load hydro-power plant sometimes also serves to pump water to a main reservoir or to a special storage reservoir. A case of this kind requiring increase of capacity is outlined in Fig. 7.7. The main reservoir regulates the discharge in the river to the yield Q, = 20 m3 s- = 1 728 OOO m3 dayIn the pumped-storage power plant there are turbines with design flow 400 m3 s through which 5 040 OOO m3 day-l flow during a 3.5 hour rectangular peak.
'
-
Ih1
Fig. 7.7 Analysis of a balancing reservoir with the function of a reservoir downstream of a pumped-storage hydro-power plant
'.
256
5 04OOOO - 1 728 OOO = 3 312 OOO m3 day-'; i.e., during 7 hours of pumping about
130 m3 s-' have to be pumped back to the main reservoir. These values are plotted in base curves in Fig. 7.7a; the amount pumped should be considered as a withdrawal from the balancing reservoir, so that we add it to the yield Q,. Mass curves in Fig. 7.7b solve the size of the balancing reservoir, V, = = 4.2 * lo6 m3. To give an idea of the influence of re-pumping on the balancing reservoir capacity, the volume of a balancing reservoir without re-pumping is shown in the same figure. The all-day augmented release of 1 728 0oO m3 is processed into two peaks of the same duration, 1.5 2.0 = 3.5 h. In this case, the volume of the balancing reservoir is V, = 1.02 * lo6 m3. It can be seen that re-pumping increased the volume of the balancing reservoir about four-fold. Power plants using tidal energy also work on a daily cycle; a description can be found in the book by Votruba and Brofa (1966, p. 159).
+
7.1.3 Daily release control for irrigation
For the irrigation, daily release control need not by typical. Conditions for a daily cycle are given, e.g., where land is irrigated only during the day. If, for such irrigation, water is withdrawn continuously and without storage, it is not used at night, and can even cause harm to land on lowest irrigated levels. This shortcoming can be avoided when reservoirs with daily release control are introduced. When placed at the end of a long canal, these reservoirs work as distribution reservoirs. The canal can then be designed only for mean daily and not for peak water demand and it helps to regulate the amount of water used. With a uniform inflow P to a reservoir throughout a day T and with a balanced withdrawal 0 for part of a day t, the necessary distributing reservoir volume V, can be determined in the following way:
For example,for an area of 10oOhectares with P = 0.4 1 s- per 1hectare and t = 10 h, the reservoir storage capacity would be V, = 20 OOO m3. The balance should also include evaporation and seepage losses from the reservoir. Daily discharge can also be controlled by a reservoir placed at the inflow to a canal. This enables better use of the water resource; however, the canal must have the correct dimensionsfor peak water demand. Certain difficulties of withdrawal control are caused by the time the water needs to flow from the inflow to the conduit to the place of withdrawal, because discharge in the network of canals is unsteady and complicated which can be investigated in operation. Short-term release control without a precise cycle duration can also be used for irrigation. If, for example, the yield of the water resource is small, a certain water
257
volume has to be stored in a reservoir to start with. If the power resource is unsteady (e.g., by a wind-driven engine),a certain water volume must also be stored. However, the most important factor for irrigation is seasonal and over-year release control. Daily release control is also used to increase the discharge for a few hours a day for floating timber. The reservoir volume is relatively small, about 85 to 90% of the daily release. A disadvantage of this method is that the wave reaches more distant places only at night, which causes difficulties for floating timber; much water is wasted by filling the empty parts of the stream channel, and the wave flattens quickly, which means an unfavourable discharge increase and a tise in the water level just downstream of the dam. Buffer-storage reservoirs with a daily periodical function can also have a daily cycle in river flow regulation. The difference in their function is that they must coordinate the inflow from an interbasin and the release of water from a distant reservoir upstream and regulate this inflow to meet the demands of non-uniform withdrawal. The simulation of the inflow regime to a reservoir is more difficult than the reservoir design. 7.2 WEEKLY RELEASE CONTROL
A weekly cycle is introduced if the water demand in the region (or power from the
hydro-power plants) differs in the respective days of the week. In this case, the reservoirs are usually distribution reservoirs. Figure 7.8 shows a general weekly load or
water demand diagram with mean daily values. The whole weekly demand calculated from the sum of demands on the respective days: m
= CQ, 1
+ (7 - M)Q"
cois (7.4)
'1
where 52 is the demand of one whole day [m3 d- from among the working days (index n) and on the days with smaller demands (index i = 1, ...,m).
258
The mean daily demand will be
si=
fai + (7 - m)a, 1
(7.5)
7
If the mean daily demand during the week a is greater than the greatest of the decreased daily demands a, to a,,,,then the reservoir volume at the inflow will be P = 0 = const
V, = (a, - bi)(7 - m)
(7.6) However, neither the water nor the energy demand is the same during the whole day. Figure 7.9 gives a diagram of a reservoir volume with regard to weekly and daily release control. Numerically the curve 0 = f ( t ) reads 7 1
=6.5Rn;
6.5 7
a=-R,;
V,=
These values were also determined graphically. A graphical method was also used to determine the storage capacity V,' for a weekly discharge distribution according to the detailed curve 0 = f ( t ) , which also reflects the daily variability of water
Fig. 7.9 Diagram showing weekly release control
withdrawal, i.e., V,' = 0.5952, which is larger than V, by the value 27.5%. It can be seen that calculations to determine the storage capacity for weekly release control cannot generally be limited to uniform daily discharges, but that discharge fluctuations within one day must also be taken into account. N o t e : The graphical solution in Fig. 7.9 uses a mass curve drawn from pole 0, and mass curve drawn from the raised pole 02. It can be seen that in the given case only the raising of the pole, by which the scale of the mass curve could be enlarged by five times, made it possible to read the value V, in the mass curves accurately (with an accuracy of 0.OlQn).
259 A balancing reservoir can also work on a weekly cycle. At weekends and holidays
the discharge downstream of the balancing reservoir does not then drop to a harmfully low value. It is, however, possible that the volume of a balancing reservoir with a daily cycle calculated for one specific day will also be satisfactory for a weekly cycle of a low-flow period, when the danger of a harmfully low release is the greatest. Of constantly greater importance in Czechoslovakiawill be peak-load hydro-power plants, the output of which will be used 8 to 11 h day-’ and 2000 to 2500 h yr-’ (Laudat, 1976). Pumped-storage hydro-power plants to be constructed in the future will no longer serve only 4 to 6 h day- and loo0 to 1500 h yr- as was previously the case. Weekly control cycles for pumped-storage hydro-power plants are one way a) PSHPP daily cycle-winter
“‘I’ 9‘L
d) PSHPP weekly cyclk-summer
LILA DL-’ ,
I
r.-
Fig. 7.10 Operations of a pumped-storage hydro-power plant (PSHPP)with a daily and weekly cycle from’ Monday to Fridny and at weekendsin December and June(f;i.audBt, 1976).P, -output with full turbine operations
of covering the growing needs of the power system. The volumes of the upper and lower reservoir of such power plants with a weekly cycle are about 4 to 5 times greater than for daily cycles. The difference between the function of pumped-storage hydropower plants (PSHPP) with a daily and weekly cycle can be seen from Fig. 7.10. With a weekly cycle the upper and lower reservoir must be large enough for three-days of pumping, i.e., from Friday night to Monday morning. If we use the maximum efficiency obtained from pumped-storage hydro-power
260 plants in Vianden, Luxembourg for 1964 (H = 292 m, P = 900 MW), i.e., pumping operations 85.7%and total effectiveness of re-pumping 7774, we can write P,
11.4QPH
PI
8.8Q1H
The volume of the upper reservoir. with a rcctnnplar peak and daily re-pumping cycle, is V,,d
= Qptp = QA
(7.9)
where - input
during pumping operations P, - output during turbine operations Q, (Q,)- discharge during pumping (turbine) operations t, (t,) - time of the day of pumping (turbine) operations H - mean head of pumped-storage hydro-power plant. From equations (7.7) to (7.9) it follows that P,
t Q 8.8PP A=S=-
t,
Q,
11.4P,
(7.10)
+
For P, = P, and for (t, t,) = 24 h, t , = 10.5 h and t, = 13.5 h. Under the above conditions a reservoir volume K,d can ensure an output of turbine operations PI for 10.5 hours. In Fig. 7.11 the function of the upper reservoir of a pumped-storage hydro-power plant with a weekly cycle of 12hours pumping on all days and of a 12-hour rectangular peak on work days is plotted in mass curves (for other pumping conditions the method is the same). Reservoir volume K,, = 100%. Under these conditions, the pumped daily volume is W = Qpt, = it,,and the daily water volume for turbine operations Ssd is W , d = Qttt = EK,~.
20. 10
n. " mn.
tue. wed. thu. fri.
sat.
sun. mon.
I'ip. 7.11 Function of the upper reservoir of a pumped-storage hydro-power plant with a weekly cycle (t, = t, = 12 h)
26 1
With these relationships any parameter can be calculated. For example, for the required turbine output P, we can calculate from equation (7.8) the discharge Q,for turbine operations and from this the necessary volume of the upper reservoir K., = %,t,, etc. A detailed comparison of a daily and weekly cycle of a pumpedstorage hydro-power plant was described by Laudit (1976). 7.3 SHORT-TERM NON-PERIODICAL RELEASE CONTROL
This type of short-term release control is applied when the cycle of the reservoir function is not repeated in regular intervals and when the duration of the cycle is never more than one week. A typical case is the floating of timber, when the discharge in the river is not sufficient to store water in a reservoir in one day to create a wave that would last at least three hours. In calculating the volume of a reservoir, we consider the inflow to be constant; however, release must be regulated according to the conditions (depth, duration of augmented water stages) which we wish to create in the river up to a distance of about 20 km downstream of the dam. We must therefore also calculate the transformation of the discharge wave in the stream channel. A discharge waue is typical for short-term release control as such, however, it also has other aspects (flood routing in the stream channel, daily release control in some hydro-power plants, a wave caused by the bursting of a dam, etc.). Several methods have been elaborated, but it is difficult to obtain accurate results due to the complicated conditions in nature, and therefore simplified presumptions with an acceptable reliability are frequently used. Short-term non-periodical release control is used wherever water management is connected with the exploitation of the power of the wind, which has short-term non-periodical characteristics (pumping water to a tank by wind engine). The best use of wind is made when wind-powered plants and hydro-power plants with storage reservoirs cooperate. As the output of wind-powered plants is never constant, the cooperating power plants must have a short-term, non-periodical control. This example can be applied to any power resource which does not have a constant and reliable output. Non-periodical, short-term changes can also occur with so-called bufSer release control. If the river-flow is regulated by a reservoir far upstream from the place of withdrawal, reservoir operations cannot meet the withdrawal changes, as the change of release from the reservoir is reflected in the place of withdrawal only after a certain period; tributaries on the way cause the discrepancies between the discharges in both the sites, etc. These discrepanciesmust be balanced by a reservoir with a smaller volume in the place of withdrawal; the main reservoir upstream carries out only rough release control and the “buffer” reservoir more sensitive regulation.
262 In a cascade of power plants, which essentially have a regular work cycle, an in-between reservoir can help to regulate the non-periodical changes of inflow from the upstream power plant. Non-periodical short-term release control for the irrigation was dealt with in Section 7.1.3. With small irrigated areas and resources with a small yield, water must be accumulated for several days (according to the present yield of the resource), to create a supply for 1 to 2 days of irrigation. Agricultural reservoirs placed in a network of canals also have short-term nonperiodical characteristics; they mainly balance the inflow (e.g., by pumping) and withdrawal, but can also serve as settling tanks, to warm cold water from wells, etc. (Cablik, 1960). Supply, fire control, etc., reservoirs, as well as cisterns, frequently have non-periodical operations. The volume of these reservoirs is mostly determined by withdrawal demands and less frequently by inflow conditions. Flood-control of release (Chap. 11 and 12), for which flood-control reservoirs, drainage ponds, pondage and flood control storage capacities of multi-purpose reservoirs are built, also has short-term, non-periodical characteristics.
8 RIVER FLOW REGULATION Let us consider a reservoir that is to ensure a given withdrawal at a given site downstream of the reservoir (Fig. 8.1); the natural discharge from the interbasin between the reservoir (dam site) and the withdrawal site is an important part of the water balance. Then the optimal reservoir operations will mean that they only supplement (compensate) the natural discharge from the interbasin, if it is unable to cover the required withdrawal at the withdrawal point. Release control from a reservoir by which the required withdrawal is ensured at the withdrawal point, together with the non-controlled discharge from the interbasin, is called river flow regulation (compensation control).
' 0
I
IV
by
v VI w YI IX x
XI x~iI
n
iii
0; -required release
P O*
t
IV v Y/ vn MI tx x xi A I
ii III
Fig. 8. I Diagram of a reservoir with river flow regulation Compensation in the broader sense of the word is, e.g., the case of drinking water supply from a reservoir, where withdrawal depends on the fluctuations of the groundwater yield, together with which it supplies water to the water mains; a reservoir compensates the differences between the total water demands and the capacity of the groundwater resource. However, in the more narrow sense, the essential trait is that a reservoir controls its own catchment, which is part of the catchment of the respective withdrawal point.
264 River flow regulation has often been used in reservoirsfor irrigation, thermal-power plants and public water supply demands. In multi-purpose works it is frequently combined with direct withdrawal. To explain the idea of river flow regulation let us consider the release control by the storage capacity placed either: (a) at the withdrawal point with direct withdrawal from a reservoir, (b) upstream of the withdrawal point with compensation operations, (c) as in (b), however, with release control ensuring a constant safe yield (regardless of discharges from the interbasin). Most efficient is (a), as the reservoir is in control of the whole catchment. If it is not possible to place a reservoir directly next to the withdrawal point, (b)is more expedient than (c), as it able to ensure a higher safe yield (withdrawal) at the withdrawal point. In case (c) the reliable discharge at the withdrawal point would only equal the sum of the yield at the reservoir site and the minimum natural discharge from the interbasin. A reservoir with a compensation regime can “save” water during higher discharges from the interbasin and can therefore ensure a more balanced discharge at the withdrawal point than constant release would be able to achieve. River flow regulation can be of an over-year or a seasonal type. It is not simple to estimate the reservoir cycle in advance, as in certain cases only insufficient discharges from the interbasin in short, closed low-flow seasons must be compensated, while over-year control must be used when the emptied volume is not refilled even in the high-flow seasons. The decision is only easy when over-year, low-flow periods have been observed at the withdrawal point. The basic rule for a reservoir with river flow regulation to ensure the required withdrawal O,(t) follows from the relationship
is the uncontrolled inflow from the interbasin (part of the discharge at the withdrawal point at the moment t, coming from the interbasin between a reservoir and the withdrawal point), OK(t - z) - release from a compensation reservoir, which must be let out into the stream at the moment (t - z), so that in the discharge-travel time z the discharge from the interbasin should be supplemented to the value O,(t). Another condition is
where QM(t)
OK 2 Q,
(8.2)
where Q, is the minimum yield in the stream downstream of the dam; at the most Q, = 0.
265
An indispensable condition for compensation operations is the forecasr oj’ discharges from the interbasin in advance of a minimum time 2. If there is no storage capacity near the withdrawal point, the forecast must be reliable, so as not to cause failures in water supply. Inflow from the interbasin is, however, frequently underestimated and a greater amount of water is released from the reservoir than necessary; water losses are then reflected in the capacity of the compensation reservoir. A buffer storage reservoir at the withdrawal point is able to balance the fluctuations of discharges. To construct an exact model of river flow regulation would be very complicated, as, e.g., the discharge-travel time t is not constant ;complicated hydrodynamic factors play their part in the stream channel, etc. Simplifications must therefore be introduced which must be taken into account when transferring the results to parameters of a project (Section 8.3). Calculations are usually made in daily to monthly time intervals; it is therefore dificult to include the relatively short discharge-travel time t (with the exception of daily discharge series) and the consequences of discharge forecasts. Then all the quantities in equation (8.1) are in the same time interval and we can simply write
+ 0,
QJ
(OK>= (8.3) The discharge from the interbasin is given by the difference in discharges at the withdrawal site Qo and the location of the reservoir QK,and therefore
0,= QM
Simulation of reservoirs with river flow regulation can be derived out in a chronological discharge series at the withdrawal site Qo = f(t); a discharge series at the reservoir site is indispensable for testing the conditions (8.2) during every step of the solution. If
0, = (0,- Qo)
+ QK < Q,
(8.6)
a supplementary release is added in the respective interval to the regulated discharge 0, A 0 = Q, - [(O, - Qo) + QK]
(8.7)
and a balance step is made for 0;= 0, + A 0 in the discharge series corresponding to the withdrawal site. This ensures that any surplusses (QM- 0,)from high discharges from the interbasin are not included in the calculation of the filling of the reservoir, as the discharge from the interbasin cannot be diverted to a reservoir without special measures; it is actually an idle outflow. The balance increments (0,- Qo) At, or (0;- Qo) At
266
and the balance sums c(0,- Qo) At can be determined successively in the chronological discharge series of the respective withdrawal site; this greatly simplifies the solution (0, < Q, occurs only rarely in low-flow periods).
Fig. 8.2 Calculations of river flow regulation in mass curves
LK
Lo
The graphical solution in mass discharge curves = f(t) and =f(t) (Fig. 8.2) concerns the same algorithm. The size of the storage capacity which is to ensure an augmented discharge 0, at the withdrawal point is to be determined; downstream of a reservoir there should be a minimum discharge Q,.
261
Lo;
The solution starts with a mass curve a tangent is drawn parallel to the mass and Co to curve Cop.By transferring the vertical distances between curves the as yet unverified mass release curve from the reservoir the mass curve is determined. If 0, Q, in any of the time intervals, we must correct the mass curve in that section in such a way as to make 0, = Q, (curve must be parallel to curve and project the consequences of this correction to the further shape of curve In Fig. 8.2 the condition that 0, 2 Q, was not met between two low-flow periods (1934), where the correction was reflected in the resultant size of the storage capacity V, and during the filling of a reservoir (in the high-flow periods of 1935 and 1936).
LK
xaK,-=
cQ,
roK
L,
L,)
co,.
3
A-
13
AQ -release fmreservoir required ot withdmkol site
V‘portial emptying dstcunpe capacity
Fig. 8.3 Over-year regime of a reservoir with river flow regulation due to insufficient inflow in wet periods (with greatly different areas of the reservoir catchment and the withdrawal site)
If the area of the reservoir catchment is much smaller than the area of the interbasin, over-year compensation release control must be introduced, as the inflow volume to a reservoir during a high-flow season is not large enough to refill the storage capacity emptied during the previous low-flow period (Fig. 8.3). If the mean release from a reservoir (given by the demands at the withdrawal site) were greater than the inflow, the required size of the storage capacity would increase with time. 8.1 RIVER FLOW REGULATION USING SYNTHETIC DISCHARGE SERIES
As the cycle of a reservoir with a compensation regime cannot always be defined in advance, and as the “compensation” release is usually combined with direct withdrawal, the method used must be a general and adaptable one. Most general, but also most time-consuming method is the analysis with synthetic discharge series, modelled simultaneously at the withdrawal site and the site of a re-
268
servoir, taking into account the cross-correlation relationships (See Chap. 3). Instead of the series Qo = f ( t ) , it is possible to model the inflow series from the interbasin QM= f ( t ) . Sometimes discharges at another site between the withdrawal point and the reservoir have to be used which means that another synthetic series has to be constructed at this control site. To create synthetic discharge series in a system ofriver points, the number of numerical characteristics needed to construct a mathematical model, which must be taken from real series, increases substantially; this, however, increases the risk that atypical influences might occur and makes any control very difficult. We shall therefore use monthly discharge series, though we are aware of the effect of this simplification on the results of the solution. The algorithm of a direct solution in discharge series is described by equations (8.3) to (8.7), from which a computer program can be compiled. We have proved that by introducing supplementary rcleases A 0 (equation 8.7). if the condition that 0, 2 Q, is not kept, the procedure ih practicallq the same as for the solution with direct withdrawal; a hypothetical reservoir is considered to be at the withdrawal site. With a selected size of storage capacity v,l and with a given augmented discharge 0, at the withdrawal site, the reliability of water supply p is determined by calculations in synthetic series. The equation i=k
V,
C (0,- Qo)Ati
= i=
(8.8)
1
-=
where the value 0, is replaced in the intervals Ati, in which 0, Q,, by the value 0; = 0, AO, determines the number of failure years and the duration of Pd l‘I LIfC!, to supply water as well as the volume of the deficit water. If V, < 0, the result of the balance step is V, = 0; the inequality V, > v,l signals the emptying of the storage capacity of the selected size and the beginning of water supply deficit. After evaluating the rate of water supply reliability p (most frequently occurrence based) one “point” of the relationship = f ( p ) is determined. This values and after constructing the relationship procedure is repeated for further v,l = f(p), the theoretical size of the storage capacity E ( P ) is found for the design reliability P. If the theoretical size of the storage capacity is estimated at first, then direct calculations are made with a given volume V,for various suitably chosen values of augmented discharge 0, (the minimum discharge Q, downstream of a reservoir does not change). The results are the “points” ( 0 , ; p ) of the relationship 0, = f(p). Using this relationship, the required flow O,(P) for the design reliability is determined. Similarly as for direct withdrawal, short synthetic series with a statistical evaluation of the results can be used for river flow regulation.
+
v:
v:
269 8.2 DETERMINATION OF THE SIZE OF THE STORAGE CAPACITY USING EXCEEDANCE PROBABILITY CURVES OF NECESSARY VOLUMES
This method, essentially the same as Andreianov’s method (Section 6.1) for seasonal release control, can only be used for compensation reservoirs with an annual cycle. In every water management year the value necessary to ensure the required augmented discharge 0, at the withdrawal site is determined by direct calculations, equations (8.3) to (8.7). At the same time, a test of the filling of this volume in the following high-flow period (i.e. the test of a reservoir’s seasonal cycle) is carried out. The set of necessary storage capacites determined for all the years of observations which are available (in real chronological discharge series) are arranged in ascending order of magnitude (whereby several members equal zero); the “empirical” points of probability paper are fitted by a smooth curve, i.e., the probability curue of necessary volumes. The size of the storage capacity K(P) for the given reliability P is thus found. 8.3 OVER-YEAR RIVER FLOW REGULATION
In probability methods of over-year river flow regulation, the reservoir volume is divided into over-year and seasonal components (e.g., Ivanov, 1964; Orlova, 1956; Gildenblat and Korenistov, 1960; KlemeS, 1965). The approach is similar to the analytical solution of over-year release control (Section 5.1). We shall describe how the matrix method, based on Moran’s methodological approach (Section 5. l), can be applied to river flow regulation, i.e., to determine the ouer-year component of the reservoir storage capacity. Real chronological discharge series are used as a basis. The volume of release from a reservoir, that is indispcnsable to ensure the required augmented discharge 0, (and also the minimum discharge downstream of a reservoir QJ, is determined by direct calculations at the withdrawal site for every year, using equations (8.3) to (8.7). From the set of release volumes from a reservoir an exceedance curve of release volumes is constructed, then the correlation coefficient between the inflow volumes to the reservoir in one year and the release volumes are determined. As the replenishment of the inflows from the interbasin is the greater the smaller the water yield in the low-flow period is, the correlation coefficient will be negative. In view of the dependence of the annual inflow volumes and outflow volumes conditional exceedance curves of the release volumes (w) must be constructed for \the selected inflow volumes (n).The exceedance curve of inflow volumes n = f ( p ) must be divided into a sufficiently large number of intervals, in which the mean values 2, are determined. The statistical characteristics of conditional exceedance
270
curves of release volumes oi = f ( p ) of the corresponding values Ei are determined, presuming a linear correlation, from the relationships for the conditional mean
ai = a + r n , o ) y 7 i i E) 6,
and for the conditional standard deviation UO,(Z)
= 6,
m
(8.10)
The conditional coefficient of skewness is usually the same as C, of the original exceedance curve of o = f ( p ) . However, it is also possible to transform the skew distribution o = f(p) into a normal distribution (with C, = 0) and after constructing the transformed conditional exceedance curves it is possible, by reversing the transformation, to obtain the resultant exceedance curves of mi = f ( p ) . We divide the selected size of the over-year component of the storage capacity into (TI - 2) parts (they should be of the same size) and determine the mean V, values in the respective intervals; the total number n is made up by V, = 0 and V, = V,.'. Then for all possible combinations of inflow volumes xi, initial filling of a reservoir h,inil. and final filling the probability of exceeding the release volume from can be read from the conditional exceedance a reservoir w = xi &,inil. curves of oi = f(p). After multiplying these probabilities by the corresponding width of intervals Api of the respective mean values xi,and after adding the obtained values for all pairs V,,init., &,end, the conditional probabilities of exceeding the volumes of water stored in the storage capacity at the turn of two years are obtained, i.e., the coefficients of the matrix of transition A, see equation (5.14). The following steps are the same as for direct withdrawal from a reservoir. The result is the exceedance curve of the water volumes in a reservoir, which for V, = 0 give the reliability of water supply. values, the relationship V,.' = f ( p ) By repeating the calculations for several is reached, from which it is possible to determine, for the design reliability, the resultant value V,.. In calculating the seasonal component, the principles are the same as for direct withdrawal from a reservoir (Section 5.1.2); however, the non-uniform release from a reservoir in the course of a year and the variability of annual release volumes must be taken into account. The method described above is rather time-consuming and requires a computer (with the exception of some stages). For preliminary studies it is not always possible to use these complicated methods. It is more expedient to make use of the similarity of river flow regulation and release control for an augmented withdrawal 0, from a hypothetical reservoir with an over-year cycle, placed at the withdrawal point. With a given 0,, the storage capacity of the compensation reservoir (river flow regulation) as well as the storage capacity
r'
+
r'
27 1
of the hypothetical reservoir (direct withdrawal 0, from a reservoir-usually for complete reliability in the given observation period), are determined in real time series. Then the over-year component of the storage capacity of a hypothetical reservoir is determined for the design reliability P from graphs, using the known statistical characteristics and correlation function as well as the over-year component V;. in real time series. The size of the storage capacity of a reservoir with a compensating regime is then corrected according to the ratio of both over-year components of the hypothetical reservoir. This approximate procedure is more reliable than calculations in real discharge series. 8.4 REAL OPERATION FOR RIVER FLOW REGULATION AND ITS CONSEQUENCES
For river flow regulation operations, short-term discharge forecasts at the withdrawal site, i.e., inflow from the interbasin (introductory part of Chap. 8), must be applied. The significance of real manipulations with river flow regulation increases with the distance of the withdrawal site from a reservoir and thus with the increasing size of the interbasin and the ratio of the mean discharges from the interbasin and the reservoir catchment. The discharge-travel time z from a reservoir is given by the distance of the withdrawal site. If the distance is great, water losses caused by incorrect flow regulation can play an important role in the total balance. As one case dlffers from another, it is rather difficult to give any general instructions as to how to include operation losses in the design of the storage capacity. However, the general method based on the analysis of the most important facts described below can also be used in other cases, especially if the withdrawal site is at a great distance from the reservoir. A basic condition of short-term discharge forecasts from an interbasin, in terms of which the release from a reservoir 0, is determined, is that the forecast values must be reliable. When water is withdrawn from a stream, an augmented discharge must be ensured at the withdrawal site Qb 2 0, (within the range of the design water supply reliability). It is therefore essential for discharge forecasts from an interbasin that forecast values Q, should be smaller than (or equal to) real values. The forecast of relatively low discharges from an interbasin is important; for higher discharges (Q, > OP)it will be sufficient to forecast their occurrence, regardless of their size. An absolute forecast reliability would lead to disproportionate increase of unused release and therefore to a large storage capacity. It can be presumed that users will be able to handle short-term fluctuations in water supply. Another reserve can be a minimum guaranteed discharge downstream of the withdrawal site. For buffer-storage reservoirs at the withdrawal site, the forecast rules can be less
.
272
strict, as an overestimation of the inflow from an interbasin does not increase the danger of water supply failure. Time factor of forecast is given by the discharge-travel time z and depends on the initial forecast information, on the time necessary to evaluate this information and on the adaptability of operation. The main source of information will be data on the discharge at the discharge-gauging sites in the catchment. The i d o w to a reservoir and the discharge at the withdrawal site will be studied (or the inflow to a bufferstorage reservoir). Information about precipitations, temperature, etc., can supplement the basic information, especially when assessing the development of the discharges. One of the simplest methods is the forecast of the discharge from an interbasin Q, in time t, in terms of the inflow to a reservoir Q K in time (t - T). This method is suitable for not too great a distance of the withdrawal site and a close relationship of &(t) = f [Q& - z)]. When the withdrawal site is at a great distance, this type of forecast is unreliable (as the conditions differ in the reservoir catchment and the interbasin). In this case forecastsbased on the inflow from the interbasin should be used. Value 0, is obtained as the difference between the flow at the withdrawal site and the release from a reservoir at time preceded by the discharge-travel time z. On the basis of information from the two main gauging sites, a combined forecast method can be used. Deficits of water and losses caused by operation are further given by the time during which release from the reservoir stays unchanged. This could require forecasts and operation changes within the shortest possible time intervals. If the distance between a reservoir and a withdrawal site is very great this kind of operation is unsuitable, not only because it causes practical difficulties, but because it can violate the “regulation stability” due to hydrodynamic factors in the stream channel. The risk of failure can be made smaller by changing the forecast technology to the benefit of reliability, however, at the cost of greater water losses. The greater the relative yield, the smaller the relative size of operation losses. The influence of the higher relative yield is reflected in the duration of the emptying period and in the days when the forecast value 0, is greater than the difference (0, - QJ, which is rather rare. It is rather difficult to evaluate short-tem jailures o j wafer supply caused by Operation. From the balance point of view they are not actually failures, as the unavailable water volume remains in the reservoir and, after determining a water supply shortage, can be utilized at any time. As compared to these failures, operation losses mean a real loss of water from a reservoir, for which the size of the storage capacity must be increased. As these losses can be extensive,they should, for economic reasons, be estimated most reliably. If the river flow regulation is to be calculated in mean monthly discharge series, the relationship between the values of mean
273
monthly discharges and the respective mean monthly operation losses must be taken into account. In real series of mean daily discharges, direct calculations are made, bearing in mind the proposed forecast method, which make it possible to determine the mean operation losses in each month. It is then possible, presuming an indirect linear correlation relationship between the mean monthly inflow to a reservoir QK and mean monthly operation losses, to construct conditional exceedance curves of these losses (valid for any month of a year or for selected periods) and, simultaneously with the modelling of synthetic series of mean monthly discharges, to model a parallel time series of mean operation losses. In the final solution, the thus determined water losses from a reservoir, are considered as part of the release. It can be expected that during reservoir operations these losses will be smaller than presumed by the original project, mainly as a result of experience, better forecasting methods, etc. The increased augmented discharge 0,will cover water demands for a longer period of time than presumed by the project. A very effective means of limiting water losses or failures to supply water due to operation is a bufier-storage reservoir at the withdrawal site (Chap. 7). Let us consider a buffer-storage reservoir with a working volume V,, in which part AVp is to cover operation failures and part A K to limit operation water losses (V, = A 5 + AK). The basic working stage of such a reservoir is the filled volume AVp and the empty volume AVz. During operations the discharge forecasts from the interbasin are taken into consideration, as well as the volume in the buffer-storage reservoir. The supplementary inflow AQ, determined from the filling of volume AK( +) at the forecast moment (presuming an equal distribution throughout the period between the respective operation intervals), or from the emptying of volume AVp(-), is added algebraically to the forecast value QM. The corrected values (QM+ AQ) are then used to determine the release from a reservoir 0, (2Q,). If we were to design the working volume of a buffer-storage reservoir V, = AVp A t as equal to the sum of the greatest determined plus and minus deviations of the forecast discharges QM from reality (in the sphere where the forecast of values Q , influences the release from a reservoir OK),we would eliminate water supply failures and operation losses. Such demands on the buffer-storage reservoir as far as water losses are concerned would, however, be too strict. It would be better to choose several values of V,, to carry out calculations in discharge series and derive from them the relationships between the size of volume V,, or its parts AVp and AK, and the frequency and depth of water supply deficits or the volume of operation water losses. The size of the buffer-storage reservoir can then be determined by an economic analysis. This method was used in the Water Resources Development and Construction Institute in Prague (Votruba Jr., 1975). Data on various sizes of main and bufferstorage reservoirs were fed into a computer. The basis was a two-day advance
+
274
discharge forecast from an interbasin. After assessing the real operations of the system, a simulation solution in real series of mean daily dischargesfor 1931to 1970was used. An analysis of river flow regulation showed that water-management operations can have a great impact on the reservoir design parameters, because of operation water losses, which are extensive especially if the distance between the reservoir and the withdrawal site is great. For great distances, a buffer-storage reservoir at the withdrawal site (or in the system of the user) is indispensible, as such a reservoir can remedy the consequences of imperfect operation (caused by inaccurate discharge forecasts) much more effectively than supplementary capacity in the main reservoir. There are, however, further reasons for a buffer-storage reservoir such as variability of discharge-travel time, hydrodynamic phenomena in the stream channel, random concentration of increased small withdrawals on the route between a reservoir and the withdrawal site, operation of weirs, etc. The theoretical size of the storage capacity of the main reservoir must be increased by the volume necessary to cover further water losses from it (mainly by evaporation from the water level) or to eliminate any errors resulting from mean monthly discharge series (if they are not already included in the estimate of operation losses). The procedure is analogous to that for direct withdrawal from a reservoir with an over-year or within year release control (Chaps. 5 and 6). In river-flow regulation, operation losses are only an additional factor which can make the solution less accurate. The danger of any distortions due to the initial real discharge series or any other factor is the same as that which can arise in over-year or within-year discharge control with direct withdrawal (Sections 5.3, 6.4).
9 RELEASE CONTROL IN A CASCADE OF RESERVOIRS (SEVERAL RESERVOIRS ON ONE STREAM) The construction of a cascade of reservoirs, i.e., a series of impounding reservoirs on one stream with a functional continuity, was originally connected with hydropower, production of which was a logical consequence of the efforts to make the best use of the natural power potential of a stream. However, during the water resources development the newly designed as well as the already constructed reservoirs have become a part of a system or cascade, even though they are no longer only used for power generation. A cascade can be an important part of a system of reservoirs. In designing a reservoir in a cascade, the water-management function of the other reservoirs must be respected; the work of one reservoir is usually designed so as to meet the interests of the cascade as a whole. Usually each reservoir in a cascade is partly used for water management and power production purposes while at the same time all reservoirs work together to ensure the total function. When a reservoir is to produce hydro-power and to supply water, the interests of the users are competitive: power production needs the greatest possible head for the peak-load power plants of the cascade to have a great output, while for w t e r supply, full use is made of the reservoir storage capacities. This means that reservoirs in a cascade must be designed individually, case by case. Using a cascade for water supply as an example, and excluding any influence of the downstream reservoirs on the function of the upstream reservoirs, we can formulate some basic principles for release control from reservoirs in a cascade that are generally valid. As compared with river flow regulation, in this case one has to start with the reservoir highest upstream, which influences the work of the downstream reservoir, mainly by essentially changing its law of inflow, even if the original law of streamflow is not changed by withdrawals or losses. The work of the downstream reservoir is usually made easier by the upstream reservoir as, with the same storage capacity due to a better distributed inflow, a higher yield (withdrawal) can be obtained. The upstream reservoir is all the more efficient, the greater the relative yield (a) and the greater its catchment.
276 9.1 ANALYSIS OF CASCADES OF RESERVOIRS USING SYNTHETIC DISCHARGE SERIES
As with any water management problem, solutions of reservoirs in cascades in which, besides the balance of inflow, release and volume, other quantities have to be studied (e.g., the head), direct solutions in synthetic discharge series are most frequently used as they best enable a detailed study and evaluation of the respective quantity in every time step. Just as for river flow regulation, the series have to be modelled simultaneously at several sites; between the corresponding discharge values a relatively close correlation can be expected (as the discharge series concern the same stream). For practical reasons (Chap. 8) we shall consider synthetic mean monthly discharges. In release control in a cascade of reservoirs the corresponding balance steps do not take place at the same time, but are shifted by discharge-travel time 71,2, z ~ ,...~ , between the respective reservoirs. As we are trying to analyse the storage function of reservoirs in which the shortest cycle is a seasonal cycle and where the natural discharges from all parts of catchments and interbasins flow into one of the reservoirs, the time lag can usually be neglected without affecting the accuracy of the results. In considering the time lags 7i,p it would be necessary to prepare the initial hydrological data, i.e., the discharge series, in such a way that the respective time intervals (in our case months) would be shifted at the respective sites by the value 71,2, 22,3,... Then the synchronous processes in reservoirs can be considered in the analysis. The upstream reserooir in a cascade can be solved in the same way as an individual reservoir with direct withdrawal with an over-year or within-year cycle (Section 5.2). Besides determining the main parameters of release control (the size of the storage capacity V,,,, yield O,,, and reliability P,) the time pattern of the release from a reservoir 0, = f(t) must be calculated in the whole series. As long as the emptying of a reservoir in any balance step V, meets the condition that 0 c V, c V,, release from the reservoir 0, equals the yield 0,. If V, = 0 (full storage capacity) with Ql > 0, in the next step, or V, = V, with Q1 < 0, in the next step (complete empying of the storage capacity water supply deficit), the release from reservoir 0, equals the inflow Q , (0, = Q , ) . Inflow P2 to the donvstrearn reserooir is given by the relationship
+
where X,
is withdrawal from a reservoir or stream downstream of the upstream reservoir, which is not returned to the catchment of the lower reservoir; if at least part of the withdrawn water is returned to the stream, only the consumption is considered,
277
z,
- water losses from the upstream reservoir or from the stream downstream of the reservoir [m3 s-'], Q,( 1,2) - inflow from the interbasin between the upstream, and downstream reservoir, which is equal to the difference (Q, - Q1).
N o t e : Inflow to a reservoir influenced by an upstream reservoir is denoted P, the natural inflow from the catchment is denoted Q (e.g.,Q 2 is the natural discharge at the site of the downstream reservoir).
After rearranging, P2
= (01 -
Qi)
+ Q2
-
(94
Xi - Z1
If X , = 0 (no withdrawal with consumption at the upstream reservoir of the cascade) and if the water losses are neglected (Z, = 0), the relationship can be simplified to P2
=
(01- Q i ) + Q2
If we are to study the inflow volume W, to a reservoir in individual time steps, we can write W2,k =
i=k
i=k
i = k.. .
i= 1
i= 1
i= 1
1 P2 Ati = C . Q , Ati + C (0,- Q1)Ati
(9.3)
Equation (9.3) is an algebraic expression of the graphical solution of a cascade of reservoirs in mass curves, shown in Fig. 9.1. The mass curve of the inflow to the downstream reservoir is constructed by adding to the mass curve of natural discharges the volumes augmented by the upstream reservoir, represented by the vertical distances between the mass curves and which stand €or the work of the upstream reservoir. The time series (inflow curve) P2 = f(t) is the basis for the direct solution of the downstream reservoir, the method being the same as for the upstream reservoir. The
cQ2
co, cQ,
8
\
1st year (design)
2nd yeor
Fig. 9.1 Analysis of a cascade (series) of reservoirs in mass discharge curves
278
parameters of release control I/Z,?, Op,z,Pz and the release time series 0, = f(t) are to be determined. In Fig. 9.1 we can also find the yield (withdrawal) OP,$in the mass curve CQ2which would make the storage capacity K2 possible without the balancing effect of the upstream reservoir. Then one can proceed to the third stage of the cascade, and so on, until the last downstream reservoir is reached. In simple cases where cascades consist of two reservoirs (or under special conditions of more reservoirs) analytical methods can be used as for independent reservoirs. One condition, however, is a synchronou\ discharge regime from the upstream reservoir catchment as well as from the interbasin between thc two reservoirs and the same reliability of water supply. Let us presume that the relative volume of the upstream reservoir is greater than the relative volume of the doinstream reservoir (for a greater number of reservoirs it holds that >&2 &, ...,etc.). Then the downstream reservoir is always filled sooner than the upstream and cannot use the idle outflow from the upstream reservoir. The downstream reservoir can only control the discharges from the interbasin, regardless of the upstream reservoir function. This cascade can therefore be solved as two independent reservoirs, the inflow to the downstream reservoir being given by the discharge from the interbasin. The total reliable withdrawal from the cascade equals the sum of yields (withdrawals) of the two reservoirs: it is upstream However, if the relative storage capacity of the downstream reservoir is larger (& > jz,l), reservoir that is filled first in a wet period (following after a low-flow season, during which the reservoir empties). In this case, the surplus outflow is used by the downstream reservoir until its storage capacity is filled. The cycle duration is decided by the time needed to fill the downstream reservoir. The problem can therefore be simplified by considering that the storage capacities of the two reservoirs are at the site of the downstream reservoir ( y = y.l + VZ.J.The characteristics of release control from a cascade (Op,P) are then the same as the corresponding values determined for independent reservoirs, with a storage capacity of V, = K,, + K,2 situated at the downstream reservoir site (hydrological data are discharge series Q2 = f(r)). This simple method does not give a detailed picture of the work regime of the respective reservoirs in a cascade; however, it helps to give a ready estimate of the regulating effect of release control of several reservoirs on the stream.
9.2 PRINCIPLES OF RELEASE CONTROL OF RESERVOIRS IN THE CASCADE FOR HYDROELECTRIC POWER PRODUCTION
In designing a cascade of reservoirs for hydroelectric power production, the role of the hydro-power plants in the power system, as well as the mutual relations between the electric power and water management demands, must be considered. In Czechoslovakia, where most of the electrical power is supplied by thermal-power plants, hydro-power plants supply only about 25% of the total output. They supply electric power only to cover the peaks in the daily load diagram, to help regulate the frequency, etc. Reservoirs of hydro-power plants are usually a part of more extensive water schemes, including the supply of water for various users.
The basic requirement of hydroelectric power production is the optimization of the design and operations of a cascade from'the point of view of the power output, with
279
increased demands in the winter months. With a given design flow capacity of the turbines, optimization from the point of view of power output is transferred to a search for the choices which would ensure a maximum head of the hydro-power plants in a cascade. This state is stressed in those cases where the primary energy potential of a stream is enforced by pumping power plant. Many books have been published on cascades of hydro-power plants, especially by Soviet authors (Kartvelishvili, 1961, 1967; Cvetkov, 1961; Reznikovski, 1964, 1969, 1974). Before computer technology was introduced, all the input quantities were usually given as deterministic quantities, contrary to the real conditions in the electric power systems, in which many data have probability characteristics (river runoff, including hydrological forecasts, energy demands for certain purposes, rated outputs of thermal and hydro-power plants, demands of other users from multipurpose reservoirs, etc.). .Deterministic quantities can be included in the calculations only when disregard of the probability characteristics will not influence the accuracy of the solution. These, however, certainly do not include discharges in the rivers, load of the power system, rated outputs of power plants or water withdrawals for irrigation. In the calculations of the regime of hydroelectric power cascades, the changes in the input quantities with time must be considered as a probability process, which for the sake of simplicity is usually introduced with discrete time as a Markov chain, although it is actually a continuous process.
--____
p-peak load
b-boslood
Fig. 9.2 Basic diagram of hydro-power reservoirs in a cascade
The probability concept is also reflected in the optimization, whatever its criteria1 function might be. The influence of water management operations of a cascade of reservoirs on the head of hydro-power plants depends on the arrangement of the reservoirs. Figure 9.2
280 gives a basic scheme of the relationship of reservoirs in a cascade from the point of view of the power-plant head. In Fig. 9.2a, the fluctuation of the head is given only by changes of the water level in the upstream reservoir. The diagram in Fig. 9.2b shows a state in which the size of the head is influnced by the position of the water level in the upstream and downstream reservoirs. Finally, Fig. 9 . 2 ~shows an arrangement with a balancing reservoir by a peak-load power plant, which is usually used if distance between the end oPTlii5backwater of the downstream reservoir and the peak-load power plant of the upstream reservoir is great. In low-flow periods, the demands of power production are at competition with the demands of water supply; this requires a planned emptying of the storage capacities of the cascade reservoirs. In the period that corresponds to the design conditions, all storage capacities of all reservoirs are emptied (if we do not limit the withdrawal) so that the minimum guaranteed output of the power plants in the cascade (with a given rate of reliability) is a design constant. However, at the time of emptying and filling of the reservoirs, the order of usage of the respective storage capacities, as well as the respective layers of the storage capacities of the reservoirs, can be changed in such a way that, while meeting the demands of the users, the greatest probability of exceeding the rated outputs of the cascade should be reached.
10 RELEASE CONTROL IN A SYSTEM OF RESERVOIRS A system of reservoirs is a set of reservoirs on various streams, which cooperate in release control. These reservoirs can be on a main stream and its tributaries in one catchment, where the upstream reservoirs can influnce the inflow to the downstream reservoirs (Fig. 10.la), or they can be in different catchments and can cooperate to cover the withdrawal demands (Fig. lO.1b). Most important is the relationship of their functions.
Fig. 10.1 Schematic representation of a system of reservoirs
The design of release control in a system of reservoirs is one of the most complicated problems in water management. A system of reservoirs serves many purposes, the priority of which can only be estimated objectively from the economic point of view. However, usually there are no available data that geflect the actual impact of the respective functions on the national economy, but there are also intangible aspects, e.g., the relationship of the reservoirs to the environment, social factors, etc. Another reason for this problem being complicated is that it takes a long time to build such a system. Even if a proper concept were used from the beginning of the construction of the system, demands on the respective functions would certainly change during the long period of construction and the development would deviate from the original prognosis. More frequently,however, a system consists of individual reservoirs that were built according to the demands of the given period. When water demand increases or flood control measures must be introduced, which require the construction offurther reservoirs, does a system that works as afunctional entirety become more expedient. The necessary technological measures must be taken that
282
enable the cooperation of reservoirs in the system. Further necessary assumption is the development of the methodology of the water resources systems with reservoirs analysis. The parameters of the existing reservoirs will be far from optimum with regard to their qualitatively new function in a system. Several books on comprehensiveanalysis of water resources systems with reservoirs have been published, e.g., Buras (1972) Votruba, Nachhzel and Patera (1974),Votruba et al. (1979, 1988). Characteristics of this relatively new branch can be found in Chap. 15. Even though the methods of operation research as applied to the solution of water resources systems are not currently used in operation (with the exception of technical simulation), it is possible to determine quantitatively the contribution brought about by the cooperation of reservoirs in systems by much simpler methods. If a system of reservoirs is able to meet all demands completely and there is no need to determine the order of importance of the respective demands, it is sufficient to use only technical parameters. This concerns mainly single-purpose systems (e.g., supply of drinking water), or systems with one dominant purpose. 10.1 TECHNICAL PARAMETERS OF RELEASE CONTROL IN A SYSTEM OF RESERVOIRS FOR PUBLIC WATER SUPPLY
Reservoirs that are to cover water demands effectively-which means that with a given storage capacity they should ensure a maximum yield or with a given withdrawal should have the smallest possible storage capacity-should cooperate in a system; this is especially advantageous if (a) the discharge regime of the streams with reservoirs working in a system is asynchronous, (b) the discharge regulation through the respective reservoirs differs greatly. In spite of the differences in the discharge time series in the streams of Czechoslovakia, the hydrological regime can be considered to be essentially synchronous (with the exception of the Danube). High-flow and low-flow periods occur in the same periods of the year, with differences in time, resulting only from the different altitudes (beginningof the winter period, spring thawing); over-year low-flow periods are also mostly identical. From the point of view of the water yield of these periods, using a five-point classification (very high-flow, high-flow, average, low-flow, exceptionally low-flow) the difference is never greater than one point. However, even these small differences in time can be put to good use, especially if the area of the system is large, and on the rivers with a different yield where even an insignificant asynchronous pattern on a stream with a large mean discharge can represent a great contribution to the whole system (as compared with other reservoirs on smaller streams).
283
A great difference in the regulation of the discharges (relative yield) in a system makes it possible to use the excess releases from reservoirs with a lower relative yield (the probability of occurrence is relatively high) and thus save the water in the storage capacities with an over-year cycle. The water volume gained by increased withdrawals from seasonal reservoirs at the time of a lack of water makes it possible either to decrease the storage capacity of the newly designed reservoir in the system, or to increase the overall reliability of withdrawal. An important condition, however, is that corresponding technical measures should be introduced, namely that withdrawal structures and canals of a sufficiently large capacity be built. A system of reservoirs should be analysed, similarly to other complicated methods of release control, in synthetic discharge series modelled simultaneously in a system of river sites. As it is presumed that all the reservoirs of a system have at least a seasonal cycle (with the exception of reservoirs with special functions, buffer-storage reservoirs, balancing reservoirs, etc.) monthly discharge series usually give a correct solution (analysis can then help to correct any errors). The method we have derived (BeEva!, Votruba and Broia, 1968; Nachhzel, 1970) is similar to river flow regulation. This is given by the regime of the reservoirs with high relative yield that augment the discharge to the required withdrawal at low-flow periods (withdrawal from seasonal reservoirs is identical to withdrawal from individual reservoirs with isolated function); however, in wet periods they only supplement the withdrawals from the seasonal reservoirs to meet the withdrawal demand in the whole system. The withdrawals from the seasonal reservoirs vary from the safe yield corresponding to their isolated function to the capacity of the withdrawal facilities which can be determined by a technical-economicanalysis; if the augmented release (withdrawal)flows through the stream channel downstream of a reservoir, the upper withdrawal limit need not be considered in the calculations. When increased withdrawals are made from reservoirs with a shorter cycle, discharge forecasts are used (as for compensation regimes), from which the size of withdrawals from reservoirs with a longer cycle are determined. Water losses can occur due to incorrect forecasts and operation, which theoretically lessen the effect of the cooperation of reservoirs. It is therefore necessary, at the end of the calculations, to correct the theoretical values of the release control characteristics or propose measures to eliminate operation losses in the system. A simple case of two reservoirs, where the first ( N , ) has an over-year cycle and the second ( N 2 )a within-year cycle, can serve as an example (Fig. 10.2). Reservoir N 2 , when working independently, has a release (withdrawal) 0,,2with the reliability P. Operation is based on a simple central rule curve (see Section 13.1), which is given by a so-called “anti-failure curve” of the storage capacity targets V, = = j ’ ( t ) during the year (Fig. l(X2b). If the water volume in a reservoir is, on a given date, smaller than or equal to the value given by the rule curve, withdrawal from the
284
reservoir equals O,,,; if the storage volume is larger, the surplus volume is used to increase the withdrawal. With a full storage capacity, withdrawal equals the inflow (up to the value given by the capacity of the withdrawal facility and the water supply to the consumer).
b)
t i
confinue wr‘th solution of reservoir N, with O,=
-4
Fig. 10.2 Schematic representation of the cooperation of an over-year and seasonal reservoir, based on the utilization of excess outflow from the seasonal reservoir
If the time needed to transport withdrawal 0, from reservoir N, to the user is shorter than’from reservoir N , , discharge forecasts should be used for the operation. This forecast need not be as accurate as for river flow regulation without a bufferstorage reservoir; deviations should, however, be balanced or be slightly more reliable as to excess discharges (i.e., over-estimations in the forecasts). However, operation need not be based on forecasts, when the volume of reservoir N , is used. The basic relationship reflecting the cooperation of reservoirs to cover the withdrawal demand Op,cin a system is
o , ,= 0 1 + 0 2
(10.1)
Release 0, from the reservoir with a shorter cycle (Fig. 10.2c, d) is, 0, = OPp2 when the reservoir storage volume is V V, 0, = Q , when the storage capacity is full, if Q , 5 C (C - capacity of withdrawal facility) or, on the contrary, when the storage capacity is empty and with inflow going to reservoir N,, for which it is
285
Q , < OP,, (failure in water supply, predicted within the framework of the given reliability rate P), 0,= C with a full storage capacity and Q , > C, 0,= OP,, + AO, when V > V,; for radical use of the water surplus in the storage volume 0, = C can be withdrawals as long as the volume stored in the storage capacity is not identical with the rule curve volume V, which ensures the greatest utilization of water surplusses. With a given theoretical size of storage capacity K,2, which when functioning in isolation ensures a withdrawal of O,,, with a reliability P, direct calculations are carried out in a synthetic series of mean montly discharges, bearing in mind the principles for a better utilization of reservoir N , . The result is a time release (withdrawal) series 0, (Fig. 10.2d),from which the time series of required withdrawals 0, from reservoir N , (with over-year release control) can be determined. With a known course of withdrawal 0, throughout the synthetic discharge series, direct calculations (simulation) bring us to the design size of the storage capacity of reservoir N , , with the corresponding required reliability P (see Section 5.2). If an already constructed reservoir N , were to be supplemented by a newly designed reservoir N , , the problem could be solved by successive approximations for several values of K,,. When already constructed reservoirs are to cooperate, a new reliable withdrawal from the system 0, (with a design reliability P ) must be determined. The procedure can be the same as for the cooperating reservoir N , . The result gives the size of the storage capacity &, which is smaller than the constructed volume K,, (theoretical size). The difference AV, = K+l- y,, is used for a uniform (or proportional) increase of withdrawal 0;throughout the time of emptying of the reservoir, corresponding to the design conditions. For computer processing several values of the withdrawal increment AO, are selected and for these the respective reliabilities P are determined. The resultant effect of the cooperation, AO,corresponding to the design reliability P, is then determined by interpolation. In 1969 the department of hydrotechnics of the Technical University in Prague studied the Eossibilities of cooperation between two reservoirs with an over-year
i 1i
\
’ >
,,
6 - Jonov
7 - Fl6je
-water tunnelsmains
Fig. 10.3 Public water-supply reservoirs in the North-Bohemian industrial region
286
cycle and several seasonal reservoirs in the north Bohemian industrial region (Fig. 10.3).The total theoretical effectof the cooperation, 0.33 m3 s- ',is 16% of the sum of isolated withdrawals and as compared to the already constructed reservoirs. This greatly exceeds the reliable withdrawal from any of the seasonal reservoirs in the system (Nachhzel et al., 1969; NachBzel, 1970).It follows from an analysis of the real conditions of the cooperation of reservoirs in the framework of a system supplying drinking water that the present concept, in which the treatment plants are joined to the public water supply reservoirs (the treatment plant capacity corresponds to the safe yield of a reservoir), will not be able to meet future requirements. To distribute water-treatment plants according to the needs of a region supplied by a system of public water supply reservoirs (in the optimal way from the aspect of raw water supply as well as treated water distribution) is an important precondition for the full exploitation of the effect of the cooperation of reservoirs without having to double their capacity. 10.2 SYSTEM WITH WATER DIVERSION
Diversion of water is frequently used in systems as it helps to intensify the natural hydrological potential of a reservoir catchment. As systems with water diversion are always greatly influnced by local conditions, the variability of designs is extensive. Figure 10.4 shows schematic representation of some water diversion systems designed in Czechoslovakia. The most usual case (Fig. 10.4a) is to increase the inflow to a reservoir on stream 1 by diversion from the neighbouring catchment 2 (by canal or pumping); this is especially suitable if a relatively large storage capacity can be constructed on stream I. Frequently, the water is brounght to a reservoir from the upstream catchment joining the recipient downstream of the dam site (Fig. 10.4b). Other, more complicated systems have lateral reservoirs. The diagram in Fig. 10.4~ illustrates a case where conditions in the upstream catchment of stream I are not suitable for building the main reservoir; therefore, a suitable site for a dam was chosen on tributary 2, to which water is diverted from the main stream. In Fig. 10.4d the arrangement is similar: but here water is also diverted from neighbouring catchment 3. In the case shown in Fig. 10.4e the diversion to a lateral reservoir holds only part of the upstream catchment of the main stream I and, therefore, besides direct withdrawal from the reservoir, water is also withdrawn from the stream (if hydrological conditions are suitable). In Fig. 10.4f water is diverted from stream I and the surplus water is diverted to catchment 4, which is a passive catchment from the water-management point of view. The lateral reservoir, which is filled by pumping, smoothes the pattern of diverted discharges. In the diagram in Fig. 10.4g, stream 2 flows in the opposite direction from the natural flow (the water from recipient I is pumped to a reservoir from a cascade of low stages). The diagram in Fig. 10.4h reflects specific
287
local conditions: Dam PI,built on stream 1 just downstream of the confluence with stream 2, creates a reservoir for the supply of drinking water. This reservoir may not be used as a recreation facility,which is a pity, as it is close to a town. In the catchment of stream 2 farming is going to become more intensive, which will necessarily igfluence the quality of the water. It was therefore decided to divide part of the reservoir in the valley of stream 2 by a dam P z ; the water is taken to the stream downstream of dam P, by a tunnel 0,. Reservoir N , ensures the required discharge in stream 1 downstream of the dam and serves a recreational function, while reservoir N supplies mainly drinking water.
Fig. 10.4 Arrangement of systems with water diversion
288 A universal method for the solution of such systems is the method based on simulation with synthetic discharge series. Ideal input data, i.e., discharge series observed simultaneously in a system of river sites, will be available only in very rare cases as the water is frequently diverted to the reservoir catchment from small streams without any hydrological observation. Generally, it is presumed that the discharges at the river sites are synchronous and that the corresponding values are proportionate to the long-term mean discharges. In the first stage, the total discharge Q, to a reservoir at each time step of simulation has to be determined :
Q, = Q, + ( R - Q, - U )
(1 0.2)
where Q, is the inflow from the catchment of the reservoir, R is the discharge in the adjacent stream from which water is diverted to the reservoir, Q, is the minimum guaranteed discharge in the adjacent stream downstream of the withdrawal site, U is the wasted discharge of the diverted stream due to the limited capacity of the water diversion facility. In using series of mean monthly discharges, a method must be elaborated to determine indirectly the wasted discharges U (mean monthly values U,,,). Let us presume that the minimum diversion capacity equals the value K . This is given, e.g., by the capacity of the inflow structure of the diversion (tunnel, canal, pipeline) or by the capacity of the pumping station. If there is no reservoir capacity at the withdrawal site or if the reservoir has only a daily storage cycle (e.g., when
Fig. 10.5 Relationship between the mean annual discharges, water diversion and wasted discharges and accurate values gained in exceedance curves of discharges in one month
289 water is pumped only at night), an analysis of discharge exceedance curves in the respective months in real series should be used to calculate the mean discharge losses U,. Figure 10.5 shows the exceedance curves of mean daily discharges R = f ( ~in) months with different mean discharges R, ( M - duration of the month), showing the various relationships between the respective discharge components R when water is being diverted. In the cases shown in Fig. 10.5a, b the wasted discharge U equals zero, which can be expected in low-flow months. The diagram in Fig. 10.5a shows a case in which the discharge fluctuations within a month do not influence the mean monthly values R and Q,, so that the difference (R, - Q,,,) is the accurate balance value. From Fig. 10.5b it follows that if discharge-R drops below the value Q, within a month, the mean monthly value would be Q,., < Q,. If this phenomenon is neglected, the solution is safer. If the discharge in a month is R > (Q, + K ) , discharge losses will occur. Figure 10.5~gives the duration of this state M I ; the hatched area is the unused water volume; if this is transferred to a rectangle, where the base equals the “width M of the time step (1 month), then its height determines the value U,. If discharge R throughout the month is greater than (Q, + K ) , then the diverted part of the discharge (mean monthly value) equals the maximum diversion capacity (Fig. 10.5d). An analysis of the above relationships in real series makes it possible to coqstruct the dependence between the diverted discharge components ( R - Q, - U), and the mean monthly discharges R,. In certain cases this can be functional (e.g., with large values of R,, when ( R - Q, - U ) , = K ; or, on the contrary, with small values of R,, when (R - Q, - U), = R, - Q,, otherwise correlational. Then the corresponding series of the components diverted to the reservoir (R - Q, - U ) , can be related to the random synthetic series R, and the diverted volume W,,,,,, can be determined in each step. This method is very time-consuming, but it can be simpliiied by using summation discharge curves in the respective months, which also gives sufficiently reliable results (Pilna and Votruba, 1971). Often the complicated scheme of diversions and withdrawals can be essentially simplified. Figure 10.6 shows the approximate conformity of a solution of a system
Fig. 10.6 An example of how a water-management solution ofasystem with water diversion can be simplified
290 with water diversion and a solution for an independent reservoir. One condition, however, is that the diversion capacity from stream 2 to the catchment of reservoir N , is large enough: for example, when water is diverted from a small stream (Q. 7 7 0.50 m3 s- ') by a tunnel with a free water level, then the smallest diameter of the tunnel (about 2 m) can divert all the water with the sole exception of flood peaks. The diversion capacity must always be considered in relation to the reservoir regime. For seasonal cycles results need not be distorted even if the diversion capacity is relatively limited. In low-flow periods, which are important for release control parameters, even a small diversion capacity can ensure that the discharge from the neighbouring stream (2)will be used without any losses, so that the alternative scheme, which considers that stream 2 flows into reservoir N , and the required minimum discharge Qz,, as part of the yield (withdrawal) from the reservoir, practically corresponds to reality. As it is not considered that discharges in stream 2 would drop below the value Qz,Z, which is not the case with a real operation, the results of the alternative solution are slightly biased so as to be on the safe side. In the following wet period, when the reservoir capacity is full, the alternative scheme is rather too optimistic compared with reality, because due to the limited diversion capacity, excess releases occur at the withdrawal site on stream 2 and the reservoir is therefore filled more slowly. However, if the reservoir cycle is no longer than one year, this inaccuracy does not affect the design size of the storage capacity, the augmented QZP1 ;Q z , 2 ) , nor the reliability. withdrawal (0"; With over-year release control, wet periods of over-year low-flow seasons can, if the whole problem is simplified, lead to an over-estimation of the release-control characteristics and the results must therefore be adjusted. The above simple methods show how the problem of the cooperation of reservoirs can be solved at a time when the more general methods are being studied. Including storage reservoirs in systems offers further advantages; e.g., respective resources can replace each other, breakdowns can be overcome more easily, etc. 10.3 SPECIAL CASES OF RELEASE CONTROL
Special cases of release control can concern reservoirs which work independently, in cascades or in systems. 10.3.1 Release control with various water supply reliabilities Individual withdrawals from multi-purpose reservoirs can supply water at different rates of reliability. This problem has not yet been solved and the following should therefore only be considered as an attempt to find a solution to this problem. Let us consider a reservoir that is to ensure a withdrawal O,,, with a reliability of P, and a withdrawal 0,,,with a reliability of P2, whereby P2 > PI.The size of the storage
29 1 capacity V , is to be determined. At the same time the minimum storage volume, which ensures only withdrawal O,,, must be determined so that withdrawals requiring a different water supply reliability can be implemented. Let us first consider a solution in synthetic series of mean monthly discharges. In the first step the size of volume for withdrawal 0, = 0,, + 0,, and the lower withdrawal reliability (i.e.,P,) must be determined. By a gradual increase of the volume above the value volume V, can be determined; this will ensure a greater reliability ( P J of withdrawal O,,. If the storage volume is V z K, both withdrawals are unlimited. When the water volume in the storage capacity decreases further, withdrawal 0,, is eliminated (which might cause problems). This solution can be transferred to an equivalent scheme of two reservoirs on one stream with zero inflow from the interbasin between the dam sites (Fig. 10.7). Applying the method described in Section 9.1, the size K(1)for the given withdrawal 0,, and reliability P, is determined in synthetic discharge series. As the same time, the time series 0, = f ( t ) of the release from reservoir N , must be calculated and considered equal to the inflow P2 = f ( t ) to reservoir N , . In series P, = f ( t )the size 1.;(2)necessary to ensure withdrawal 0,, with a reliability P, must be determined. The design size of the storage capacity is V, = K( 1) + K(2). In this case, too, volume must be determined to make real operations possible.
Q1
Fig. 10.7 Substitutional solution with various withdrawal reliabilities in a cascade (series) of reservoirs (calculations in synthetic series)
Q' l
II
Fig. 10.8 Substitutional solution with various withdrawal reliabilities in a system of reservoirs with an optimal distribution of inflow
Discussions are frequently held to decide what the probability of exceeding the minimum maintained discharge in the stream downstream of a reservoir should be; it is believed that it could be smaller than the probability of exceeding withdrawals needed for households, industry, etc. This is then also a case where the storage capacity must be determined with a different reliability, with the only difference that should a failure of withdrawal 0,, (P, < P,) occur, it may not drop to zero. This might also apply to other types of withdrawal.
292
In that case a certain minimum release Olminfrom reservoir N , (e.g., Q3,,d) can be chosen, which together with withdrawal 0,, flows into reservoir N , . This makes certain that even if reservoir N , fails, the chosen minimum outflow can be maintained in the stream or a limited withdrawal 0, can be ensured. Under these new conditions t ( 1 ) must be larger and t ( 2 ) smaller. For statistical solutions based on real discharge series (regardless of whether release control has an over-year or a within-year cycle), an alternative scheme consisting of two reservoirs (Fig. 10.8) can be used. In this case reservoir N , ensures withdrawal 0,, with a reliability P, and reservoir N , a withdrawal Op2 with a reliability P,, while discharge Q is divided into inflow Q , = k , Q to reservoir N , and into inflow Q , = k2Q to reservoir N , ( k , + k , = 1, or Q , + Q2 = Q). By trial and error, a division of discharges Q , and Q , must be found so as to lead to a minimum value of the sum Vz(1) + V,(2).This condition is derived from the logical presumption that one single volume V , = t ( 1 ) + t ( 2 ) enables an optimal “cooperation” between the storage volumes of reservoirs N , and N , and the volume can therefore be the smallest possible. For the design of the size of volumes V,(l), t ( 2 ) and also K’ (for 0, = 0,, + Op2 and PI)the same method as for the design of the storage capacity for direct withdrawal is used (see Chaps. 5 and 6). If the design reliability of the individual withdrawals does not differ greatly, the method can be simplified. For example, it is possible to determine the size of the storage capacity for a design reliability P‘, which can be calculated as a weighted mean : (10.3) When compared with the real reliability of the results (random deviations in the size of the storage volume) inaccuracies due to the simplified introduction of different withdrawal reliabilities can be disregarded. 10.3.2 Release control in the period of the first filling of a reservoir
Reservoirs cannot meet all water demands as soon as their construction has been completed. Water-management calculations, however, presume that the storage capacity is full before a low-flow period starts; this corresponds to established operational conditions, but not to the state existing at the end of construction and the beginning of operation. The first filling of a reservoir takes place under so-called test operations, during which operation is tested from the technical and safety points of view (limited changes in the water level of a reservoir, planned measurements, “flushing” of the reservoir with regard to the water characteristics, etc.). When there is an urgent need, reservoirs can be used, with certain limitations, during construction and test operations (supply
293 of drinking water when the reservoir is only partly filled, checking of the installations, etc.). Sometimes the flow regulation operations should also be checked (e.g., work of a reservoir in a system, compensation operations, etc.). For this period a special program should be prepared. Let us presume that when a reservoir isput into operation the storage capacity is empty. For within-year release control the development of the filling of the reservoir can be estimated even under complicated hydrological conditions, presuming the occurrence of a low-flow year in which the demands on the storage volume are close to the size of the design. At first, the consumers who receive their water from a new source are exposed to the “water management” risk that the supply might fail; however, the following year operations already correspond to the conditions as they were foreseen by the project. If at the time that a reservoir with ouer-year release control is being put into operation a low-flow period lasting several years should occur, water supply might not be ensured without any failures for several years. To avoid this, special operation rules have to be introduced. A smaller reliable withdrawal is not as a rule contradictory to the demands on a reservoir, as at the beginning of operation the design capacity is not completely being exploited. For quantitative calculations of the work of an over-year reservoir during initial filling, mean discharges for k-years with the probability of exceedance equal to the design water supply reliability P (Section 5.1) can be used. The most unfavourable case of first-year operations is the occurrence of a low-flow year with a mean discharge of a probability of exceedance P . The maximum withdrawal in that year is equal to the mean discharge Q1, = QI with the probability of exc‘eedance P; whereby the amount of water in the storage capacity at the end of the year again equals zero. The mean discharge in the first two years is considered to be equal to the theoretical discharge for the two year period Q,,(P) corresponding to the probability of exceedance P. As the first year was ascribed a mean discharge Qlr, the respective value QZris determined from the relationship Qzr
= 2Q1, -
Q1r
(10.4)
Reliable release in the second year can at best equal the value QZr;with 0 = Q2r the storage capacity is again empty at the end of the year. The same result is reached with any other couple of mean annual discharges, the mean value of which is Qll (in the first year it must be Q, > QJ. From the theoretical mean discharge for a three-year period Qlll(P),we can calculate (10.5) 3QI11 - Q2r - Q l r which equals the release from a reservoir in the third year, etc. As soon as the value Qirexceeds the design value of withdrawal from a reservoir 0,, Q3r
=
294
flow regulation .operations envisaged by the project can start, even though it is only then that the storage capacity begins to be filled (in the sense of the long-term trend). If the actual demands on withdrawal are smaller than the calculated limit values, the reservoir is filled more quickly. High-flow years during the initial stages of permanent operations also speed up the filling of a reservoir.
I.
9.
10.
Fig. 10.9 Approximate solution of the first filling of a reservoir with an over-year cycle
Figure 10.9 gives the results of calculations for a yield by an over-year reservoir during the initial filling with release control parameters of a = 0.75; B: = 0.6; Po = 97%;C, = 0.40; C , = 2C,; r(z) = 0. Withdrawal must be reduced only during the first two years of operations; however, the storage capacity is only filled after 9 to 10 years with discharge conditions corresponding to a 99”,,reliability. if a n over-year lo~-flowperiod of a 99% exceedance probability OCCLITS. The solution of the storage function of a reservoir during the period of initial filling is essentially simple. Even though we have not discussed the consequences of various discharge situations in detail and have dealt only with the “over-year component” of release control, the results can be considered reliable for the determination of operating rules for this period. Decision-makers must not consider a planned contribution of the newly built reservoirs immediately after completion, but after test operations and after the period of the initial filling. The logical consequence of these considerations should be, especially for reservoirs with over-year release control, that construction should start well in advance of time. In the working schedule of construction and the programs of test operations the first operational filling and beginning of water supply should be chosen so as to meet real withdrawals that might be needed, i.e., before the beginning of the spring wet period.
10 RELEASE CONTROL IN A SYSTEM OF RESERVOIRS A system of reservoirs is a set of reservoirs on various streams, which cooperate in release control. These reservoirs can be on a main stream and its tributaries in one catchment, where the upstream reservoirs can influnce the inflow to the downstream reservoirs (Fig. 10.la), or they can be in different catchments and can cooperate to cover the withdrawal demands (Fig. lO.1b). Most important is the relationship of their functions.
Fig. 10.1 Schematic representation of a system of reservoirs
The design of release control in a system of reservoirs is one of the most complicated problems in water management. A system of reservoirs serves many purposes, the priority of which can only be estimated objectively from the economic point of view. However, usually there are no available data that geflect the actual impact of the respective functions on the national economy, but there are also intangible aspects, e.g., the relationship of the reservoirs to the environment, social factors, etc. Another reason for this problem being complicated is that it takes a long time to build such a system. Even if a proper concept were used from the beginning of the construction of the system, demands on the respective functions would certainly change during the long period of construction and the development would deviate from the original prognosis. More frequently,however, a system consists of individual reservoirs that were built according to the demands of the given period. When water demand increases or flood control measures must be introduced, which require the construction offurther reservoirs, does a system that works as afunctional entirety become more expedient. The necessary technological measures must be taken that
282
enable the cooperation of reservoirs in the system. Further necessary assumption is the development of the methodology of the water resources systems with reservoirs analysis. The parameters of the existing reservoirs will be far from optimum with regard to their qualitatively new function in a system. Several books on comprehensiveanalysis of water resources systems with reservoirs have been published, e.g., Buras (1972) Votruba, Nachhzel and Patera (1974),Votruba et al. (1979, 1988). Characteristics of this relatively new branch can be found in Chap. 15. Even though the methods of operation research as applied to the solution of water resources systems are not currently used in operation (with the exception of technical simulation), it is possible to determine quantitatively the contribution brought about by the cooperation of reservoirs in systems by much simpler methods. If a system of reservoirs is able to meet all demands completely and there is no need to determine the order of importance of the respective demands, it is sufficient to use only technical parameters. This concerns mainly single-purpose systems (e.g., supply of drinking water), or systems with one dominant purpose. 10.1 TECHNICAL PARAMETERS OF RELEASE CONTROL IN A SYSTEM OF RESERVOIRS FOR PUBLIC WATER SUPPLY
Reservoirs that are to cover water demands effectively-which means that with a given storage capacity they should ensure a maximum yield or with a given withdrawal should have the smallest possible storage capacity-should cooperate in a system; this is especially advantageous if (a) the discharge regime of the streams with reservoirs working in a system is asynchronous, (b) the discharge regulation through the respective reservoirs differs greatly. In spite of the differences in the discharge time series in the streams of Czechoslovakia, the hydrological regime can be considered to be essentially synchronous (with the exception of the Danube). High-flow and low-flow periods occur in the same periods of the year, with differences in time, resulting only from the different altitudes (beginningof the winter period, spring thawing); over-year low-flow periods are also mostly identical. From the point of view of the water yield of these periods, using a five-point classification (very high-flow, high-flow, average, low-flow, exceptionally low-flow) the difference is never greater than one point. However, even these small differences in time can be put to good use, especially if the area of the system is large, and on the rivers with a different yield where even an insignificant asynchronous pattern on a stream with a large mean discharge can represent a great contribution to the whole system (as compared with other reservoirs on smaller streams).
283
A great difference in the regulation of the discharges (relative yield) in a system makes it possible to use the excess releases from reservoirs with a lower relative yield (the probability of occurrence is relatively high) and thus save the water in the storage capacities with an over-year cycle. The water volume gained by increased withdrawals from seasonal reservoirs at the time of a lack of water makes it possible either to decrease the storage capacity of the newly designed reservoir in the system, or to increase the overall reliability of withdrawal. An important condition, however, is that corresponding technical measures should be introduced, namely that withdrawal structures and canals of a sufficiently large capacity be built. A system of reservoirs should be analysed, similarly to other complicated methods of release control, in synthetic discharge series modelled simultaneously in a system of river sites. As it is presumed that all the reservoirs of a system have at least a seasonal cycle (with the exception of reservoirs with special functions, buffer-storage reservoirs, balancing reservoirs, etc.) monthly discharge series usually give a correct solution (analysis can then help to correct any errors). The method we have derived (BeEva!, Votruba and Broia, 1968; Nachhzel, 1970) is similar to river flow regulation. This is given by the regime of the reservoirs with high relative yield that augment the discharge to the required withdrawal at low-flow periods (withdrawal from seasonal reservoirs is identical to withdrawal from individual reservoirs with isolated function); however, in wet periods they only supplement the withdrawals from the seasonal reservoirs to meet the withdrawal demand in the whole system. The withdrawals from the seasonal reservoirs vary from the safe yield corresponding to their isolated function to the capacity of the withdrawal facilities which can be determined by a technical-economicanalysis; if the augmented release (withdrawal)flows through the stream channel downstream of a reservoir, the upper withdrawal limit need not be considered in the calculations. When increased withdrawals are made from reservoirs with a shorter cycle, discharge forecasts are used (as for compensation regimes), from which the size of withdrawals from reservoirs with a longer cycle are determined. Water losses can occur due to incorrect forecasts and operation, which theoretically lessen the effect of the cooperation of reservoirs. It is therefore necessary, at the end of the calculations, to correct the theoretical values of the release control characteristics or propose measures to eliminate operation losses in the system. A simple case of two reservoirs, where the first ( N , ) has an over-year cycle and the second ( N 2 )a within-year cycle, can serve as an example (Fig. 10.2). Reservoir N 2 , when working independently, has a release (withdrawal) 0,,2with the reliability P. Operation is based on a simple central rule curve (see Section 13.1), which is given by a so-called “anti-failure curve” of the storage capacity targets V, = = j ’ ( t ) during the year (Fig. l(X2b). If the water volume in a reservoir is, on a given date, smaller than or equal to the value given by the rule curve, withdrawal from the
284
reservoir equals O,,,; if the storage volume is larger, the surplus volume is used to increase the withdrawal. With a full storage capacity, withdrawal equals the inflow (up to the value given by the capacity of the withdrawal facility and the water supply to the consumer).
b)
t i
confinue wr‘th solution of reservoir N, with O,=
-4
Fig. 10.2 Schematic representation of the cooperation of an over-year and seasonal reservoir, based on the utilization of excess outflow from the seasonal reservoir
If the time needed to transport withdrawal 0, from reservoir N, to the user is shorter than’from reservoir N , , discharge forecasts should be used for the operation. This forecast need not be as accurate as for river flow regulation without a bufferstorage reservoir; deviations should, however, be balanced or be slightly more reliable as to excess discharges (i.e., over-estimations in the forecasts). However, operation need not be based on forecasts, when the volume of reservoir N , is used. The basic relationship reflecting the cooperation of reservoirs to cover the withdrawal demand Op,cin a system is
o , ,= 0 1 + 0 2
(10.1)
Release 0, from the reservoir with a shorter cycle (Fig. 10.2c, d) is, 0, = OPp2 when the reservoir storage volume is V V, 0, = Q , when the storage capacity is full, if Q , 5 C (C - capacity of withdrawal facility) or, on the contrary, when the storage capacity is empty and with inflow going to reservoir N,, for which it is
285
Q , < OP,, (failure in water supply, predicted within the framework of the given reliability rate P), 0,= C with a full storage capacity and Q , > C, 0,= OP,, + AO, when V > V,; for radical use of the water surplus in the storage volume 0, = C can be withdrawals as long as the volume stored in the storage capacity is not identical with the rule curve volume V, which ensures the greatest utilization of water surplusses. With a given theoretical size of storage capacity K,2, which when functioning in isolation ensures a withdrawal of O,,, with a reliability P, direct calculations are carried out in a synthetic series of mean montly discharges, bearing in mind the principles for a better utilization of reservoir N , . The result is a time release (withdrawal) series 0, (Fig. 10.2d),from which the time series of required withdrawals 0, from reservoir N , (with over-year release control) can be determined. With a known course of withdrawal 0, throughout the synthetic discharge series, direct calculations (simulation) bring us to the design size of the storage capacity of reservoir N , , with the corresponding required reliability P (see Section 5.2). If an already constructed reservoir N , were to be supplemented by a newly designed reservoir N , , the problem could be solved by successive approximations for several values of K,,. When already constructed reservoirs are to cooperate, a new reliable withdrawal from the system 0, (with a design reliability P ) must be determined. The procedure can be the same as for the cooperating reservoir N , . The result gives the size of the storage capacity &, which is smaller than the constructed volume K,, (theoretical size). The difference AV, = K+l- y,, is used for a uniform (or proportional) increase of withdrawal 0;throughout the time of emptying of the reservoir, corresponding to the design conditions. For computer processing several values of the withdrawal increment AO, are selected and for these the respective reliabilities P are determined. The resultant effect of the cooperation, AO,corresponding to the design reliability P, is then determined by interpolation. In 1969 the department of hydrotechnics of the Technical University in Prague studied the Eossibilities of cooperation between two reservoirs with an over-year
i 1i
\
’ >
,,
6 - Jonov
7 - Fl6je
-water tunnelsmains
Fig. 10.3 Public water-supply reservoirs in the North-Bohemian industrial region
286
cycle and several seasonal reservoirs in the north Bohemian industrial region (Fig. 10.3).The total theoretical effectof the cooperation, 0.33 m3 s- ',is 16% of the sum of isolated withdrawals and as compared to the already constructed reservoirs. This greatly exceeds the reliable withdrawal from any of the seasonal reservoirs in the system (Nachhzel et al., 1969; NachBzel, 1970).It follows from an analysis of the real conditions of the cooperation of reservoirs in the framework of a system supplying drinking water that the present concept, in which the treatment plants are joined to the public water supply reservoirs (the treatment plant capacity corresponds to the safe yield of a reservoir), will not be able to meet future requirements. To distribute water-treatment plants according to the needs of a region supplied by a system of public water supply reservoirs (in the optimal way from the aspect of raw water supply as well as treated water distribution) is an important precondition for the full exploitation of the effect of the cooperation of reservoirs without having to double their capacity. 10.2 SYSTEM WITH WATER DIVERSION
Diversion of water is frequently used in systems as it helps to intensify the natural hydrological potential of a reservoir catchment. As systems with water diversion are always greatly influnced by local conditions, the variability of designs is extensive. Figure 10.4 shows schematic representation of some water diversion systems designed in Czechoslovakia. The most usual case (Fig. 10.4a) is to increase the inflow to a reservoir on stream 1 by diversion from the neighbouring catchment 2 (by canal or pumping); this is especially suitable if a relatively large storage capacity can be constructed on stream I. Frequently, the water is brounght to a reservoir from the upstream catchment joining the recipient downstream of the dam site (Fig. 10.4b). Other, more complicated systems have lateral reservoirs. The diagram in Fig. 10.4~ illustrates a case where conditions in the upstream catchment of stream I are not suitable for building the main reservoir; therefore, a suitable site for a dam was chosen on tributary 2, to which water is diverted from the main stream. In Fig. 10.4d the arrangement is similar: but here water is also diverted from neighbouring catchment 3. In the case shown in Fig. 10.4e the diversion to a lateral reservoir holds only part of the upstream catchment of the main stream I and, therefore, besides direct withdrawal from the reservoir, water is also withdrawn from the stream (if hydrological conditions are suitable). In Fig. 10.4f water is diverted from stream I and the surplus water is diverted to catchment 4, which is a passive catchment from the water-management point of view. The lateral reservoir, which is filled by pumping, smoothes the pattern of diverted discharges. In the diagram in Fig. 10.4g, stream 2 flows in the opposite direction from the natural flow (the water from recipient I is pumped to a reservoir from a cascade of low stages). The diagram in Fig. 10.4h reflects specific
287
local conditions: Dam PI,built on stream 1 just downstream of the confluence with stream 2, creates a reservoir for the supply of drinking water. This reservoir may not be used as a recreation facility,which is a pity, as it is close to a town. In the catchment of stream 2 farming is going to become more intensive, which will necessarily igfluence the quality of the water. It was therefore decided to divide part of the reservoir in the valley of stream 2 by a dam P z ; the water is taken to the stream downstream of dam P, by a tunnel 0,. Reservoir N , ensures the required discharge in stream 1 downstream of the dam and serves a recreational function, while reservoir N supplies mainly drinking water.
Fig. 10.4 Arrangement of systems with water diversion
288 A universal method for the solution of such systems is the method based on simulation with synthetic discharge series. Ideal input data, i.e., discharge series observed simultaneously in a system of river sites, will be available only in very rare cases as the water is frequently diverted to the reservoir catchment from small streams without any hydrological observation. Generally, it is presumed that the discharges at the river sites are synchronous and that the corresponding values are proportionate to the long-term mean discharges. In the first stage, the total discharge Q, to a reservoir at each time step of simulation has to be determined :
Q, = Q, + ( R - Q, - U )
(1 0.2)
where Q, is the inflow from the catchment of the reservoir, R is the discharge in the adjacent stream from which water is diverted to the reservoir, Q, is the minimum guaranteed discharge in the adjacent stream downstream of the withdrawal site, U is the wasted discharge of the diverted stream due to the limited capacity of the water diversion facility. In using series of mean monthly discharges, a method must be elaborated to determine indirectly the wasted discharges U (mean monthly values U,,,). Let us presume that the minimum diversion capacity equals the value K . This is given, e.g., by the capacity of the inflow structure of the diversion (tunnel, canal, pipeline) or by the capacity of the pumping station. If there is no reservoir capacity at the withdrawal site or if the reservoir has only a daily storage cycle (e.g., when
Fig. 10.5 Relationship between the mean annual discharges, water diversion and wasted discharges and accurate values gained in exceedance curves of discharges in one month
289 water is pumped only at night), an analysis of discharge exceedance curves in the respective months in real series should be used to calculate the mean discharge losses U,. Figure 10.5 shows the exceedance curves of mean daily discharges R = f ( ~in) months with different mean discharges R, ( M - duration of the month), showing the various relationships between the respective discharge components R when water is being diverted. In the cases shown in Fig. 10.5a, b the wasted discharge U equals zero, which can be expected in low-flow months. The diagram in Fig. 10.5a shows a case in which the discharge fluctuations within a month do not influence the mean monthly values R and Q,, so that the difference (R, - Q,,,) is the accurate balance value. From Fig. 10.5b it follows that if discharge-R drops below the value Q, within a month, the mean monthly value would be Q,., < Q,. If this phenomenon is neglected, the solution is safer. If the discharge in a month is R > (Q, + K ) , discharge losses will occur. Figure 10.5~gives the duration of this state M I ; the hatched area is the unused water volume; if this is transferred to a rectangle, where the base equals the “width M of the time step (1 month), then its height determines the value U,. If discharge R throughout the month is greater than (Q, + K ) , then the diverted part of the discharge (mean monthly value) equals the maximum diversion capacity (Fig. 10.5d). An analysis of the above relationships in real series makes it possible to coqstruct the dependence between the diverted discharge components ( R - Q, - U), and the mean monthly discharges R,. In certain cases this can be functional (e.g., with large values of R,, when ( R - Q, - U ) , = K ; or, on the contrary, with small values of R,, when (R - Q, - U), = R, - Q,, otherwise correlational. Then the corresponding series of the components diverted to the reservoir (R - Q, - U ) , can be related to the random synthetic series R, and the diverted volume W,,,,,, can be determined in each step. This method is very time-consuming, but it can be simpliiied by using summation discharge curves in the respective months, which also gives sufficiently reliable results (Pilna and Votruba, 1971). Often the complicated scheme of diversions and withdrawals can be essentially simplified. Figure 10.6 shows the approximate conformity of a solution of a system
Fig. 10.6 An example of how a water-management solution ofasystem with water diversion can be simplified
290 with water diversion and a solution for an independent reservoir. One condition, however, is that the diversion capacity from stream 2 to the catchment of reservoir N , is large enough: for example, when water is diverted from a small stream (Q. 7 7 0.50 m3 s- ') by a tunnel with a free water level, then the smallest diameter of the tunnel (about 2 m) can divert all the water with the sole exception of flood peaks. The diversion capacity must always be considered in relation to the reservoir regime. For seasonal cycles results need not be distorted even if the diversion capacity is relatively limited. In low-flow periods, which are important for release control parameters, even a small diversion capacity can ensure that the discharge from the neighbouring stream (2)will be used without any losses, so that the alternative scheme, which considers that stream 2 flows into reservoir N , and the required minimum discharge Qz,, as part of the yield (withdrawal) from the reservoir, practically corresponds to reality. As it is not considered that discharges in stream 2 would drop below the value Qz,Z, which is not the case with a real operation, the results of the alternative solution are slightly biased so as to be on the safe side. In the following wet period, when the reservoir capacity is full, the alternative scheme is rather too optimistic compared with reality, because due to the limited diversion capacity, excess releases occur at the withdrawal site on stream 2 and the reservoir is therefore filled more slowly. However, if the reservoir cycle is no longer than one year, this inaccuracy does not affect the design size of the storage capacity, the augmented QZP1 ;Q z , 2 ) , nor the reliability. withdrawal (0"; With over-year release control, wet periods of over-year low-flow seasons can, if the whole problem is simplified, lead to an over-estimation of the release-control characteristics and the results must therefore be adjusted. The above simple methods show how the problem of the cooperation of reservoirs can be solved at a time when the more general methods are being studied. Including storage reservoirs in systems offers further advantages; e.g., respective resources can replace each other, breakdowns can be overcome more easily, etc. 10.3 SPECIAL CASES OF RELEASE CONTROL
Special cases of release control can concern reservoirs which work independently, in cascades or in systems. 10.3.1 Release control with various water supply reliabilities Individual withdrawals from multi-purpose reservoirs can supply water at different rates of reliability. This problem has not yet been solved and the following should therefore only be considered as an attempt to find a solution to this problem. Let us consider a reservoir that is to ensure a withdrawal O,,, with a reliability of P, and a withdrawal 0,,,with a reliability of P2, whereby P2 > PI.The size of the storage
29 1 capacity V , is to be determined. At the same time the minimum storage volume, which ensures only withdrawal O,,, must be determined so that withdrawals requiring a different water supply reliability can be implemented. Let us first consider a solution in synthetic series of mean monthly discharges. In the first step the size of volume for withdrawal 0, = 0,, + 0,, and the lower withdrawal reliability (i.e.,P,) must be determined. By a gradual increase of the volume above the value volume V, can be determined; this will ensure a greater reliability ( P J of withdrawal O,,. If the storage volume is V z K, both withdrawals are unlimited. When the water volume in the storage capacity decreases further, withdrawal 0,, is eliminated (which might cause problems). This solution can be transferred to an equivalent scheme of two reservoirs on one stream with zero inflow from the interbasin between the dam sites (Fig. 10.7). Applying the method described in Section 9.1, the size K(1)for the given withdrawal 0,, and reliability P, is determined in synthetic discharge series. As the same time, the time series 0, = f ( t ) of the release from reservoir N , must be calculated and considered equal to the inflow P2 = f ( t ) to reservoir N , . In series P, = f ( t )the size 1.;(2)necessary to ensure withdrawal 0,, with a reliability P, must be determined. The design size of the storage capacity is V, = K( 1) + K(2). In this case, too, volume must be determined to make real operations possible.
Q1
Fig. 10.7 Substitutional solution with various withdrawal reliabilities in a cascade (series) of reservoirs (calculations in synthetic series)
Q' l
II
Fig. 10.8 Substitutional solution with various withdrawal reliabilities in a system of reservoirs with an optimal distribution of inflow
Discussions are frequently held to decide what the probability of exceeding the minimum maintained discharge in the stream downstream of a reservoir should be; it is believed that it could be smaller than the probability of exceeding withdrawals needed for households, industry, etc. This is then also a case where the storage capacity must be determined with a different reliability, with the only difference that should a failure of withdrawal 0,, (P, < P,) occur, it may not drop to zero. This might also apply to other types of withdrawal.
292
In that case a certain minimum release Olminfrom reservoir N , (e.g., Q3,,d) can be chosen, which together with withdrawal 0,, flows into reservoir N , . This makes certain that even if reservoir N , fails, the chosen minimum outflow can be maintained in the stream or a limited withdrawal 0, can be ensured. Under these new conditions t ( 1 ) must be larger and t ( 2 ) smaller. For statistical solutions based on real discharge series (regardless of whether release control has an over-year or a within-year cycle), an alternative scheme consisting of two reservoirs (Fig. 10.8) can be used. In this case reservoir N , ensures withdrawal 0,, with a reliability P, and reservoir N , a withdrawal Op2 with a reliability P,, while discharge Q is divided into inflow Q , = k , Q to reservoir N , and into inflow Q , = k2Q to reservoir N , ( k , + k , = 1, or Q , + Q2 = Q). By trial and error, a division of discharges Q , and Q , must be found so as to lead to a minimum value of the sum Vz(1) + V,(2).This condition is derived from the logical presumption that one single volume V , = t ( 1 ) + t ( 2 ) enables an optimal “cooperation” between the storage volumes of reservoirs N , and N , and the volume can therefore be the smallest possible. For the design of the size of volumes V,(l), t ( 2 ) and also K’ (for 0, = 0,, + Op2 and PI)the same method as for the design of the storage capacity for direct withdrawal is used (see Chaps. 5 and 6). If the design reliability of the individual withdrawals does not differ greatly, the method can be simplified. For example, it is possible to determine the size of the storage capacity for a design reliability P‘, which can be calculated as a weighted mean : (10.3) When compared with the real reliability of the results (random deviations in the size of the storage volume) inaccuracies due to the simplified introduction of different withdrawal reliabilities can be disregarded. 10.3.2 Release control in the period of the first filling of a reservoir
Reservoirs cannot meet all water demands as soon as their construction has been completed. Water-management calculations, however, presume that the storage capacity is full before a low-flow period starts; this corresponds to established operational conditions, but not to the state existing at the end of construction and the beginning of operation. The first filling of a reservoir takes place under so-called test operations, during which operation is tested from the technical and safety points of view (limited changes in the water level of a reservoir, planned measurements, “flushing” of the reservoir with regard to the water characteristics, etc.). When there is an urgent need, reservoirs can be used, with certain limitations, during construction and test operations (supply
293 of drinking water when the reservoir is only partly filled, checking of the installations, etc.). Sometimes the flow regulation operations should also be checked (e.g., work of a reservoir in a system, compensation operations, etc.). For this period a special program should be prepared. Let us presume that when a reservoir isput into operation the storage capacity is empty. For within-year release control the development of the filling of the reservoir can be estimated even under complicated hydrological conditions, presuming the occurrence of a low-flow year in which the demands on the storage volume are close to the size of the design. At first, the consumers who receive their water from a new source are exposed to the “water management” risk that the supply might fail; however, the following year operations already correspond to the conditions as they were foreseen by the project. If at the time that a reservoir with ouer-year release control is being put into operation a low-flow period lasting several years should occur, water supply might not be ensured without any failures for several years. To avoid this, special operation rules have to be introduced. A smaller reliable withdrawal is not as a rule contradictory to the demands on a reservoir, as at the beginning of operation the design capacity is not completely being exploited. For quantitative calculations of the work of an over-year reservoir during initial filling, mean discharges for k-years with the probability of exceedance equal to the design water supply reliability P (Section 5.1) can be used. The most unfavourable case of first-year operations is the occurrence of a low-flow year with a mean discharge of a probability of exceedance P . The maximum withdrawal in that year is equal to the mean discharge Q1, = QI with the probability of exc‘eedance P; whereby the amount of water in the storage capacity at the end of the year again equals zero. The mean discharge in the first two years is considered to be equal to the theoretical discharge for the two year period Q,,(P) corresponding to the probability of exceedance P. As the first year was ascribed a mean discharge Qlr, the respective value QZris determined from the relationship Qzr
= 2Q1, -
Q1r
(10.4)
Reliable release in the second year can at best equal the value QZr;with 0 = Q2r the storage capacity is again empty at the end of the year. The same result is reached with any other couple of mean annual discharges, the mean value of which is Qll (in the first year it must be Q, > QJ. From the theoretical mean discharge for a three-year period Qlll(P),we can calculate (10.5) 3QI11 - Q2r - Q l r which equals the release from a reservoir in the third year, etc. As soon as the value Qirexceeds the design value of withdrawal from a reservoir 0,, Q3r
=
294
flow regulation .operations envisaged by the project can start, even though it is only then that the storage capacity begins to be filled (in the sense of the long-term trend). If the actual demands on withdrawal are smaller than the calculated limit values, the reservoir is filled more quickly. High-flow years during the initial stages of permanent operations also speed up the filling of a reservoir.
I.
9.
10.
Fig. 10.9 Approximate solution of the first filling of a reservoir with an over-year cycle
Figure 10.9 gives the results of calculations for a yield by an over-year reservoir during the initial filling with release control parameters of a = 0.75; B: = 0.6; Po = 97%;C, = 0.40; C , = 2C,; r(z) = 0. Withdrawal must be reduced only during the first two years of operations; however, the storage capacity is only filled after 9 to 10 years with discharge conditions corresponding to a 99”,,reliability. if a n over-year lo~-flowperiod of a 99% exceedance probability OCCLITS. The solution of the storage function of a reservoir during the period of initial filling is essentially simple. Even though we have not discussed the consequences of various discharge situations in detail and have dealt only with the “over-year component” of release control, the results can be considered reliable for the determination of operating rules for this period. Decision-makers must not consider a planned contribution of the newly built reservoirs immediately after completion, but after test operations and after the period of the initial filling. The logical consequence of these considerations should be, especially for reservoirs with over-year release control, that construction should start well in advance of time. In the working schedule of construction and the programs of test operations the first operational filling and beginning of water supply should be chosen so as to meet real withdrawals that might be needed, i.e., before the beginning of the spring wet period.
C
The flood control function of reservoirs is along with their storage function the basic component of the comprehensive water-management effect of the reservoirs. The regime of a reservoir includes both these basic functions, which means that it increases low discharges and decreases high discharges. There is no sharp boundary between these two tasks; a reservoir which theoretically establishes a completely uniform discharge (a = 1.0)has one single capacity which is perfectly able to ensure both the storage and flood control functions. Only a reservoir can ensure the storage function, flood control can, however, also be ensured by, e.g., river training or by technical and organizational measures during a flood. Flood-control reservoirs are therefore less frequently needed than storage reservoirs. Other reasons arise from the time development of streams and their environment. River training has been built so that the non-damaging flow (the flow without damaging inundations) reaches relatively high values, e.g., the flow with one year return period and even more. Economic losses are only caused by floods with a longer return period, i.e., with a smaller probability of exceedance. It need not always be economical to build reservoirs to reduce these damages. The basic characteristic of a reservoir with flood-control release is an empty volume which can hold part of the flood volume. The flood then causes less harm downstream of the reservoir.This empty volume can be a special volume, the so-called flood-control (retention) capacity (controlled or uncontrolled), or it can be gained by partly emptying the storage capacity before a flood. Frequently both of these methods are combined. Every reservoir with a storage function also has a certain flood-control effect. If a storage reservoir is at least slightly emptied, e.g., at the end of a low-flow period, at least part of a flood wave can be intercepted by the storage capacity. When the relative yield is high (continuous decreasing of the water volume even for several years) the control effect of the storage capacity can be very extensive. This effect is automatic, regardless of the flood-control function of the reservoir. The flood control effect of the storage capacity can be greatly increased without causing any harm to water supply, e.g., if water is released according to a rule curve (Chap. 13) which ensures that the storage capacity is partly emptied in a certain period of the year or by short-term discharge forecasts; this makes it possible to lower the level of the storage capacity just before a flood occurs. Flood-control release has its effect on the stream downstream of a reservoir and
296
on the environment of the stream. With the increasing area of the interbasin downstream the river the effect diminishes. The most usual task of a flood-control reservoir is to lower flood discharges to a value, which does not cause any damage to the stream downstream of the reservoir. It is usually not very economical to ensure flood control with a very small probability of exceedance; the control function is therefore usually designed with a certain, economically justified, rate of reliability. Flood control can also help to delay the damaging effects of a flood downstream of a reservoir, as time is gained to introduce indispensable technical measures to alleviate flood damage.
11 DATA FOR THE ANALYSIS OF FLOOD-CONTROL RESERVOIRS The solution of the function of a reservoir that is to protect the territory along a stream downstream of it concerns both technical and economic aspects. For this reason not only technical characteristics of flood control are needed, but also data on the damage caused in the respective territories; this means that the relationship between the flood characteristics and damage must be known. Only very rarely the design reliability of flood control is used as a basis. Flood-control release is not only reflected in the part of the stream that is to be protected, but also further downstream. Therefore, hydrological data characterizing the flood regime of the stream in the whole affected reach downstream of the reservoir must be available. If no such data are available, the calculations must be simplified or estimations must be used. 11.1 HYDROLOGICAL DATA
The time behaviour of floods in the hydrological conditions in Czechoslovakia usually has short-term characteristics-floods only last for a few days and on small streams only a few hours. To obtain a sufficiently correct pattern of a flood, the daily flows are not sufficient and a flood hydrograph must be constructed. Discharge measurements during floods are less accurate because - the stage-discharge curve at the gauging site in the area of high water stages is less “sensitive”; - possibilities of verifying the stage-discharge curve by direct measurements during floods with a small probability of exceedance are limited; - the relationship between the water stage and discharge during a flood due to changes in the water-level slope is not unique; - in winter the water level may be raised due to ice-pack. The best available hydrological data for flood regimes are chronological discharge series, e.g., of mean monthly discharges with, however, a more detailed description of the time function of the discharge Q = f(t) at the time of the flood. If a flood occurs in a certain month, then the discharge should by characterized by mean daily values up to the time that the flood occurs,during the flood by continuous “point” recordings and then until the end of the month, again by mean daily discharges. If a flood occurs
298
at the junction of two months, a detailed description of the discharge behaviour in both months must be determined. These data make it possible statistically to estimate the characteristics of the flood regime of a stream, e.g., the maximum flood discharges, flood volumes above a certain discharge value, etc.; at the same time it is possible to determine the exact state of a reservoir before a flood and therefore also the overal effect of flood-control release from it on the flood. For flood-control release the input observation period has a much greater influnce on the results of the solution (even when statistical methods are used) than for the storage function. In view of the length of Czechoslovak hydrological series, the solution of the flood-control function will probably be less accurate. The danger of underestimating maximum peak discharges in small basins was pointed out by Broia et al. (1978). On the basis of the Czechoslovak Standard 73 6805, the Hydrometeorological Institute supplied the following characteristic of flood regimes: (a) N-year maximum flood discharges for N = I , 2, 5, 10,50 and 100 years (for N > 100 years only if required) as one of the basic hydrological data. They are denoted Q1, Qz, ..., etc., generally Q N ; (b) pattern of flood waves at the given site on the stream, expressed by the discharge curve Q = f ( r ) , i.e., by a discharge hydrograph, supplemented by digital data about the maximum peak discharge (QmJ and volume ( W,); the date of occurrence is added for actually observed flood waves. Frequently, theoretical N-year flood waves are given, the maximum peak discharge of which equals the N-year maximum discharge ( Q N ) and the respective volume (denoted WF,N); (c) N-year flood volume above a certain selected discharge Q,.Q , can be the mean discharge Q,,, the mean daily discharge exceeded for 30 days in one year Q30d or, e.g., Q I , or any other suitable value. The N-year flood volume is denoted WN,axand it must be distinguished from the volume of the N-year flood wave W,, with which it need not be in a close correlation relationship and naturally it also does not have a direct connection with the N-year maximum discharge Q N .
11.1.1 Maximum flood discharges
Probability methods for determining the maximum flood discharges were applied in Czechoslovakia as early as 1933 (Vorel, 1933). However, only in recent years have maximum flood discharges been mapped for the whole country. The basic data used were the annual maximum discharges during the observation period. Calculations use graphical-digital methods estimating the statistical characteristics with the help of quantiles (Chap. 3). After ascribing the probabilities of exceedance p to the respective members of a sample of observations arranged in a decreasing order of magnitude of n members, the relationship p = m/(n + 1) (m = 1,2, ..., n), is used, which is more reliable for small probabilities of exceedance than the formula p = (m - 0.3)/(n + 0.4). From among the theoretical probability distributions it is possible to choose the Pearson type 111, log-normal or “exponential” (power transformation of the Pearson distribution type 111, according to Kritsky and Menkel); the goodness-of-fit with the points of the empirical exceedance curve is decisive.
299 Figure 11.1 gives the annual maximum discharges at Bechyng on the river Luinice (in the probability paper of normal distribution).
Fig. I I.I Statktical evnluation of annual discharge mmima (BechynE - the river LuZnice)
The probable time of exceedance N of the maximum flood discharge, currently denoted as the return period, can be calculated from the exceedance probability p , of this discharge by the relationship p
'=
1 - e-l/N
(11.1)
or after rearrangement (11.2) Values of p for values N = 1,2,,., lo00 years are given in Table 11.1 This method of determining the maximum flood discharges has practical advantages, especially for the processing of observation samples. As compared to
Tuhlr I I . / Relationship between various forms of probabilities of exceedance (occurrence periods) of floods
N [years]
1
2
5
10
20
50
100
lo00
[“A]
63.2
39.3
18.1
9.52
4.88
1.98
0.995
0.0999 5
P
N’ [years]
1.58
2.54
5.52
10.5
20.5
50.5
100.5
1000.5
selection,which included the maximum discharges of all floods that occurred (usually a certain value was selected as the lowest limit), the sample of annual maxima is clearly defined, and no problems are caused by floods with several maximum peaks, etc. However, it must be quite clear what exceedance probability the sample of annual maxima determines. The value p indicates the probability that in any year a certain maximum flood discharge will be exceeded, regardless of the number of times that this will occur in any one year. The time N‘ = l/p indicates that a certain maximum flood discharge will be exceeded on the average in the year out of ”years (or several times in that one year). In view of the fact that the occurrence of maximum flood discharges (as well as other characteristics of floods) is usually defined as an exceedance of the maximum ~ discharges on the average of once every N years. i.e.. different from the L I S L ~ : Iprobability estimation of suiiiples of annual maximum discharps. the results must be calculated according to equations (1 1.1) or (11.2). That this is necessary was proved by comparing the results of the statistical estimations of the two types of observations, i.e., annual maxima and maxima of “all” floods (Dzubak, 1950 and others). The same method can be used when estimating the maximum discharges for a certain calendar or functional period of a year, e.g., vegetation period, period of the occurrence of floods originating exclusively from rain, etc. 11.1.2 Flood volumes
The statistical evaluation of the volumes of flood waves is more complicated than the evaluation of maximum discharges. The compilation of observation samples frequently depends on the purpose the hydrological data are to serve. However, flood volumes are of foremost importance for flood-control release from reservoirs. In the Hydrological Conditions of Czechoslovakia - part I11 (Hydrometeorological Institute. 1970), annual maxima of discharge volumes for various periods of time Ar (2, 5, 10 and 30 days) have been determined. The method is similar to that for maximum discharges. The best fit was attained with the lognormal probability distribution. Besides annual maximum volumes the maxima for winter and summer periods were also studied; however, these calculations of flood volumes cannot be used for the flood-control function of a reservoir.
301
For flood control release there is no point in working with flood volumes in the hydrological sense, but with volumes which are connected with the exceedance of non-damaging discharges. As long as these volumes exceeding the value of nondamaging discharges are held by a reservoir, the nondamaging discharge will not be exceeded regardless of the maximum flood discharge. Statistical calculations of flood volumes exceeding the given discharge value are the data most frequently used to determine the flood-control function of a reservoir.
Fig. 11.2 Choice of a sample for statistical processing of annual flood maximum volumes above the discharge value Q,
Fig. 11.3 Method for the selection of a complete range of samples of flood volumes exceeding 'the value of non-damaging discharge Qnd
The method can be similar to that for flood discharges. In every year the largest (Fig. 11.2), is chosen and a sample Pood volume above discharge Q,, i.e., of annual maximum volumes compiled. With a graphical-digital procedure, using the quantile method, a suitable theoretical distribution is fitted to determine the N-year flood volumes (bearing in mind the relationships between the probability of exceedance p and the return period-Table 11.1). With a high non-damaging discharge Q, = Qndin some years the flood volume can equal zero. Then the sample of observations is not homogeneous and statistical estimations are difficult (e.g., the truncated probability distribution has to be considered). Another possibility is to go back to the sample of observations and include
vmax,Qx
302 in it all floods exceeding the value of the non-damaging discharge. Uncertainties in the selection of observation samples do not concern the volumes, as floods with several maximum peaks can be estimated with good results (Fig. 11.3). In the case in Fig. 11.3a the flood volume is unambiguous. In the case in Fig. 11.3b the significant volume is given by the algebraic sum of volumes W,, W, and W,; if the absolute value of volume W, is bigger than W,, the signlficant value of the volume is W,. In Fig. 11.3~a case is shown in which the floods are included in the observation sample independently of one another, as the discharge depression between the waves allows for a complete emptying of volume W,,
Goodrich's exponential probability distribution (Chap. 3) is best suited for this type of sample. In Fig. 11.4 the curve of exceedance of flood volumes exceeding the discharge Q, = 350 m3 s-' is drawn in the probability paper of Goodrich's distribution for the river Berounka at Kfivoklat for the period 1887 to 1940. When stating the probable time of exceeding (return period) N flood volumes it must be considered that in this case the sample of observations reflects the real frequency of flood occurrence and a generally different number of members of the sample of observations (n')than the number of years of observations (n). Then the probable time of exceedance N is given by the relationship 100
N=-v
P'
(11.3)
303 n/n‘ is the mean period of flood occurrence, is the probability of exceeding the flood volume (occurrence-based). P’ The volume of flood waves influnced by the more complicated operation of reservoirs can be calculated similarly. However, the same operating rules have to be applied to all floods of the observation samples. where v
=
11.1.3 Model oJ’ a flood regime on a stream
At present there is no perfect model of the discharge regime which would include the related creation of floods in a large catchment. Not even a simplified model, which would include discharge series in a system of water gauging stations on the river, has so far been elaborated for flood-control purposes in many basins. Hanzl(l971)elaborated an approximate procedure to determine the decomposition of observed floods at two neighbouring sites, which makes it possible to express the time behaviour of floods from the interbasins. This method considers the discharge-travel time between the two sites and what the effects of the inundation are between them. In compiling the theoretical flood waves from the partial interbasins, the method considers the shape and volume of the flood waves in the respective interbasins, the moment of the beginning of a flood in each partial catchment on a common time axis, the flood-travel time between the respective sites, and the changes of floods under the influnce of the inundation areas (flood routing). Random samples are made and the composite flood is checked to see whether it is realistic. The whole samples of composite flood waves must be compiled in such a way that the maximum discharges correspond to the recurrence curve of N-year discharges. For comprehensive solution of the control function of a reservoir (system of reservoirs), synthetic discharge series in a system of river sites should be used, reflecting the time behaviour of the discharge in such detail as to characterize the flood phenomena accurately. Regression models encounter numerous theoretical as well as practical problems. A way of avoiding these problems is to use a combination of genetic and statistical methods. Kos (1975)consideredgenetic methods as more suitable for flood discharges; he derived the time pattern of the floods from precipitations based on the relation of surface runoff and precipitation. Synthetic precipitation series can then serve as basic data. Other methods can also be used where for the modelling of the discharge behaviour the transformed white Gaussian noise can be used (random process with normal distribution with a spectral density independent of the frequency); the transformation can be carried out by the application of various ‘%filters”.Another possibility is to express the detailed time behaviour of discharges during one month by a sample of special “fragments” and by joining them to the mean monthly discharges of the synthetic series. However, the problems of synthetic series with
a detailed discharge time pattern have not yet been satisfactorily solved, not even for isolated sites. Generating in a system of stations is a very complicated process, as the model must also take into account the internal hydrodynamic relationships of the system, especially the discharge-travel time of the respective discharge waves. 11.1.4 Accuracy of the characteristics of flood regimes The Czechoslovak Standard 73 6805 gives the general probable errors of maximum discharges classified according to reliability (Table 11.2). The extreme errors can be much larger. Today we are not yet able to supply general characteristics of the accuracy of flood volumes. Tubk 11.2 Probable errors of N-year maximum discharges (“lo)
Reliability class
QI
+Qio
Q m + Qioo
1
I1
111
IV
k15 k25
+20 +30
+30
k40
+50 +60
Of importance is the change of the conditions of the development of flood phenomena caused by the construction of reservoirs or other water schemes. The progress of a flood wave in a reservoir is faster than in the natural stream channel, which becomes apparent especially with a large backwater in a cascade or system of reservoirs or when several floods from larger streams flow directly into a reservoir. The original “natural” characteristics of the flood regime can then differ greatly from those that should be considered in calculating the flood-control function of a reservoir and in designing the safety devices of dams. Full reservoirs in cascades greatly accelerate the travel time of flood waves; in the river Vltava the mean travel time in the original stream channel was, on the average, 10 km h - ; after the construction of reservoirs it increased to 30 km h (Water Research Institute, Prague). During the 20 years of operation of the Ustie reservoir on the river Orava, the discharge originally estimated as Qloo was exceeded several times: the probable reasons are the new conditions for the discharge-travel time and the presence of flood waves in the reservoir, especially from the rivers Bila and Cerna Orava. Changes of the inundation areas along the streams in the reservoir catchment can also play an important role. Flood characteristics can also be greatly influenced by the activities of man (changes in forestation, irrigation, etc.).
305 The new conditions must be estimated at least approximately and if quantitative estimations are not available the safety margin of the design must be raised. 11.2 RELEASE FROM A RESERVOIR
If a reservoir is designed as an independent measure to decrease flood damages, then the non-damaging discharge (safe canal capacity) in the given reach of the stream must be determined. Its size Qndis not the same for the whole reach. Non-damaging release from a reservoir is determined as the minimum of the difference of the nondamaging discharge Qnd and the increment of the discharge from the interbasin AQp:
Ond = min (Qnd
-
AQP)
(1 1.4)
where AQP is related to the considered reliability of flood control. The size of the non-damaging discharge can change during the year, e.g., in summer it will be lower than in winter, etc. Flood control is usually ensured by a combination of river training, channel improvement and by the flood-control function of a reservoir (or several reservoirs). We select several values of non-damaging release from a reservoir Ond,from which we determine the design safe channel capacity in the respective sections of the stream. Another variable in the optimization process is the rate of the flood control reliability. The result is given by optimizing the costs (in annual values) and benefits (reduction of flood damages). If the river training must also be built for reasons other than flood control (e.g., irrigation, navigation, etc.), then only the costs for flood control are included in the technical and economic considerations. A specific task of flood-control release from reservoirs is to hold part of the flood-wave volume at the beginning of the flood and to delay the increased discharge long enough to carry out all the necessary technical and organizational measures that help decrease flood damages. These measures can be, e.g., warning the people and even moving them temporarily from some houses, removal of ice blocks, opening of tilting gate weirs, and lessening of the effect of the encounter of flood waves from the main stream and tributaries flowing into the stream downstream of the reservoir. Release from a reservoir during floods can depend on the position of the water level in the reservoir which determines the pressure head of the bottom outlets or the overfall depth of an ungated spillway, etc. Then the differential equation of the reservoir flood-control function Pdt
-
Odt = dV
where P is the inflow 0 - release V - volume
(11.5)
306 must be integrated, taking into account the relation 0 = @), V = @(h), where h is the characteristic of the position of the water level. As equation (1 1.5) cannot be solved analytically, a graphical or digital integration is applied according to its differential form PAt
-
OAt
=
AV
(11.6)
which is adjusted in terms of the selected method of approximate integration; P and 0 are usually considered to be mean values in the interval At :
From among the many solutions of this “transformation of a flood wave in a reservoir” (flood routing), we shall describe the method derived by Urban (1956) and Zaruba (1961), which is very simple. Thirriot (1970) described a general method of a graphical integration of differential equations of this type. Urban’s m e t h o d (1965) As a starting point the author used equation (11.6) which he rearranged as follows
2K At
-
zy-,+ (P,-, At
- (Ik-,)+ (P, - 0;) - (0, - 0;)
(1 1.7)
where O:, is an auxiliary value which, on the right-hand side of the equation, is both added and subtracted. Curve P = ll(r), the depth-release curve 0 = t,(h) and the reservoir depth-volume curve V = @(h)are given. The release time pattern 0 = f2(r) should be determined. First of all, an auxiliary curve 0 = qz(2V/Ar) is determined from relationships 0 = b;. In thc case in Fig. 12.la the flood-control volume V, was filled only after the maximum-peak discharge was reached, so that the flood-control measured not
only decreased the volume of the flood wave, but also the maximum discharge. During the flood shown in Fig. 12.1b, the flood-control volume was filled already before the peak of the flood, so that the maximum release from the reservoir is identical with the maximum peak-flood discharge. However, even in this case the effect of the flood downstream of the reservoir is mitigated as at least part of the volume of the flood wave is held by the reservoir. A sudden transition from the value On, to release equal to the inflow (Fig. 12.1) is possible only in reservoirs with a gated spillway. With an ungated spillway, the transformation effect of the uncontrolled volume (Section 11.2) appears after the filling of the flood-control volume, so that the release from the reservoir will be as shown by the dot-and-dash section.
12.2 HOW TO USE THE ACTIVE STORAGE CAPACITY FOR FLOOD CONTROL
A partly emptied storage capacity can either completely or partly catch the flood wave, thus taking part in the flood-control regime. The storage capacity can be emptied partly, either automatically by the required withdrawals, or according to a plan with the aim of contributing to flood control as much as possible. However, the original function of the storage capacity must never be overlooked.
316 The cycle of emptying and filling of the storage capacity as well as the volume of emptying depends on the relative yield a = O,/Q, and therefore the automatic flood-control effect will also depend on this parameter. When floods occur the following cases can arise: (a) the flood wave flows into a full storage capacity (here and in the following text active storage capacity is often shortened to storage capacity); the maximum flood discharge is not decreased nor is any part of the flood volume held; (b) the flood wave flows into a partly emptied storage capacity, which holds the wave : - partly, however, before the maximum-peak discharge is reached, the storage capacity is again full so that the maximum discharge is not decreased, - partly, however, the storage capacity is filled only after the maximum flood discharge is reached, which decreases the maximum of the flood, - completely, so that the non-damaging discharge is not exceeded. Quantitative calculations of the automatic flood-control effect of the active storage capacity were made by BroZa (1964). In a chronological discharge series (mean monthly discharges) for 1887-1940 at the Kfivoklat site on the river Berounka, made more accurate at the time of floods by mean daily discharges (before and after the flood), and by detailed time recordings of the floods, the storage function was determined. At the time of the occurrence of floods it was taken into consideration how full the storage capacity was and the effect of the storage capacity on the maximum flood discharge and on the flood volume. Figure 12.2 shows a case in which the storage capacity holds an essential part of the volume AVpn of the flood wave and the decreased maximum discharge QLax,the volume WN,Ond that is not held exceeding the non-damaging release (hatched part). At the moment 1 , the partly emptied storage capacity was filled.
Fig. 12.2 Control effect of the storage capacity if a flood occurs immediately after a low-flow period (storage capacity not full)
Thejlood-control ej’ect ofthe active srorage capacity is the result of the combination of two phenomena of a random character: the occurrence of floods and the partial emptying of the storage capacity. The relationship of these two phenomena is in Central Europe a very loose one; a flood can occur, however full the storage capacity is. Therefore, the occurrence of floods and the rate at which these are held by the storage capacity can be considered as approximately independent. The solution is then simplified, as the probability of exceeding the flood characteristics (maximumpeak discharge, volume) influenced by the storage capacity P, is given by the product p, = p(1 - P,)
(12.2)
317
and the probable time of exceedance by the relationship N,= N -
1
1 - P,
(12.3)
where P,N is the probability of exceedance, or the probable time of exceeding the characteristics of the original (uncontrolled) sample of floods, p , - probability of holding floods, i.e., the ratio of the number of floods held by the storage capacity and the total number of floods. We can determine the number of floods which the storage capacity held completely during the period of observations. From the original curve of exceedance, e.g., volumes of floods, the curve of flood volume exceedance reduced by the control effect of the storage capacity is determined with the help of equations (12.2) or (12.3). Figure 12.3 shows the exceedance curves of flood volumes above the value On, influenced by the active storage capacity determined at the KfivoklAt site on the river Berounka for a 54-year period (1887- 1940). The value of the non-damaging discharge was considered to be On,= 350 mi s - I , a constant yield was presumed; calculations were made for a = 0.3 - 0.95.
Fig. 12.3 Influence of the flood-control effect of the active storage capacity on the probable time with a constant yield (a)(the river Berounka - Kiivoklrit) of exceeding flood volumes WN,on, With a 0.4 the storage capacity did not show any flood-control effect. Only with very high relative yield (a 2 0.9) did an automatic flood control with a roughly five-year reliability occur. With a = 1 all floods should (theoretically) be held by the storage capacity. Figure 12.3b shows the dependence of the probability of holding a flood in the active storage capacity P, with a relative yield a. The automatic control effect of the active storage capacity is relatively small. In reservoirs with an annual cycle it is practically nil; with over-year release control it is significant only for high values of a. However, it can still help the measures introduced to eliminate flood damages. For example, for Fig. 12.3 it is clear that with a = 0.8 and the rate offlood control N = 10 years, bearing in mind the control effect of the storage capacity, the necessary size of the flood-control capacity (given by the value WN.o,d - Fig. 12.2, 12.3) is smaller by 50% then the value V, if the control function is solved independently.
318
The flood-control effect of the active storage capacity can be further increased by adequate operations of the reservoir. Rules for flood operation have to be determined in advance as part of the original design. To increase the flood-control effect of the storage capacity, operating rules (Chap. 13) and forecasts on the inflow to a reservoir should be available. P a r t i a l d r a w d o w n of t h e a c t i v e s t o r a g e c a p a c i t y a c c o r d i n g t o o p e r a t i n g schedules The rule curve for the storage function of a reservoir usually determines the water volume in the storage capacity in relation to time; this is needed to ensure the planned withdrawal 0, with the required reliability. If the boundaries and durations of the low-flow periods differ greatly from year to year, then the rule curve is given by the upper envelope of the time behaviour of the necessary volumes in the storage
Fig. 12.4 Rule curve with within-year release control and how it can be applied to flood control
capacity. If a rule curve is thus reliably constructed, the volume in a reservoir can drop below the prescribed value without causing a failure in water supply. However, if the rule curve allows it, the storage capacity can be partly emptied in certain periods, regardless of discharge conditions. The rule curve therefore determines the minimum values to which the storage capacity can be emptied during the year (differencebetween V, - V ) which can be used for flood control (Fig. 12.4).The size ( V , - V )changes during the year and therefore the flood-control effect of the active storage capacity also changes. To what extent a rule curve can help flood control (concan be determined approximately by comparing the flood volumes Wr,Ond sidering the time of occurrence in a year), with the degree of the emptying of the storage capacity according to the rule curve (for the whole period of observation) and by determining the number offloods held by the storage capacity. For partly held floods, the volume that is not held and the maximum discharge must be determined. The effect of control according to a rule curve determined in this way is less favourable than in reality. A more accurate solution can be obtained by the simulation of release control in terms of the prescribed rule curve in a given discharge series (for the observation period).
319 Reliability of flood control is determined as the mean time of exceeding On, by the relationship ND
n’n s n’
n s
( 12.4)
where n’ is the number of floods exceeding the harmless discharge during the observation period, n - the duration of observation period (in years), s - the number of floods, which are not held by the active storage capacity or only partly held. The control effect of the storage capacity, when using a rule curve, can be expressed by exceedance curves of the parts of the flood volumes not held (above Ond)and by the maximum flood discharges influenced by the storage capacity. As according to the rule curve the storage capacity is emptied to different degrees during the year, and for several months the storage capacity is full (Fig. 12.4), the mean flood-control reliability for the whole year does not reflect the flood-control effect to a sufficient extent. The rule curve, for example, requires that the storage capacity be full during the growing period so that floods are held by a reservoir mainly at a time when they cause the least damage to agricultural products. The probability of the occurrence of floods differs at different times of the year. It can be presumed that there exists a certain relationship between the frequency of floods in the respective periods and the rule curve (concentrated occurrence of floods in a certain period ensures, e.g., that the inflow to a reservoir is sufficiently large to refill the partly emptied storage capacity, etc.). It can therefore be recommended to estimate the contribution of release according to a rule curve separately for certain periods of a year (one or two periods are sufficient). The reliability of flood control will be generally different in the respective periods of a year. Other measures can be introduced to increase the flood-control effect of a reservoir, which can be effective throughout the year or in certain periods of a year. No rule curve for the storage function of an over-year reservoir has as yet been elaborated. However, from the point of view of the flood-control function a rule curve can be constructed for an annual cycle as the upper envelope of the required seasonal components of the volume in the storage capacity; this makes it possible to partly empty the reservoir during the year without endangering the reliability of water supply. The advantages of this release control for the flood-control function can be observed in wet and average periods. The estimation of the flood-control effect of a reservoir can again be made by a simulation of the release control in a discharge series for the observation period (with statistical estimations), where the safe yield requirements (decisive mainly in the over-year low-flow periods) and the rule curve are applied. The advantages of the operating schedules need not be applied to flood-control
320
release only (Chap. 13). The demands of the storage and flood-control functions are usually antagonistic. While for the augmentation of discharges the surplus water is divided evenly throughout the low-flow period, for flood control the surplus volume is quickly emptied from the storage capacity to make the whole volume determined by the rule curve available for any flood that might occur. Power-plant reservoirs require the biggest possible head, especially in winter, which is again contradictory to the flood-control requirements. Priority is given to the purpose that brings the greatest benefits or is divided between the storage and flood-control functions. On the basis of the operating schedules, the flood-control effect of the storage capacity can be increased without any discharge forecasts, as compared with the automatic effect due to flow augmentation to the required withdrawal 0,. For example using the partly emptied storage, determined by the rule curve (Fig. 12.4), for the holding of floods, with a seasonal release control for a safe yield 0, ( a = 0.38), a two-year flood control was ensured (Votruba and Broia, 1974). It is expedient to combine the operating schedules and other means that help to increase the flood-control function of the storage capacity, including discharge forecasts. Emptying of the active storage capacity o n the basis of discharge forecasts In this case, the storage capacity is partly emptied just before the flood occurs. This method of flood-control release must be based on forecasts of the inflow to a reservoir (runoff from a catchment); their accuracy and how well in advance the information is received, determines the effect of the active storage capacity for flood-control purposes. Let us consider a work regime of the storage capacity of a reservoir during a flood, based on data obtained from forecasting services (Fig. 12.5). It is presumed that prior to the flood the storage capacity is full, i.e., that release equals inflow (whereby Q > Op).In time tl information is received that in time At (i.e., in time r, = t , + Ar) inflow to a reservoir will increase to the value Qz which exceeds the non-damaging release On,from the reservoir. The forecast can also include the expected tendencies of the flood. Release from a reservoir can be increased to value On, (Fig. 12.5 shows a marginal case, with a maximum emptying of V,; however, release from a reservoir can be increased gradually), which empties the storage capacity by the volume AVpn. This volume can then be used to hold the flood wave volume above the discharge Q, = On,.Figure 12.5 illustrates a case in which the storage capacity holds only part of the flood wave volume above On,, the rest can be held in the flood-control capacity with the given flood control reliability. Any hesitation that the supply function of a reservoir might be threatened can be
321
eliminated by forecasts on the further development of a flood or by further data supplied by the forecasting service (in Fig. 12.5 in time t , , for the moment = = tn,l + At). However, an experienced operator can estimate the risk caused by excessive emptying of the storage capacity without any forecasts. In Fig. 12.5 the
t
Fig. 12.5 Diagram of flood-control release using forecast data
hatched section shows the development of the flood, where the forecast discharge Q2 turned into a maximum peak discharge. From the moment t2 the water volume V, = x [ Q ’ ( t )- O,,]At can be used to supplement the storage volume; in terms of the mutual relationship of volumes AVPnand A&, suitable rules to be applied duringfloods can be proposed. To what extent the storage capacity can be emptied depends on how well in advance forecasts are received, on the increase of the discharge in time, on the size of the non-damaging release and other factors (e.g., the reliability of the forecast). Forecasts should be received as far in advance as possible so that emptying can start at a time when the inflow to a reservoir has not yet greatly increased, making the emptying of the storage capacity more effective. The best available forecast is the forecast of discharges from precipitations in the catchment. As compared to discharge forecasts based on data from gauging sites upstream, where the time of the forecast is given by the discharge-travel time, precipitation forecasts include the time needed for the water particles of the catchment to reach the stream. The operation rules often do not make it possible to introduce the complete value On, immediately. Time is needed to send the necessary information downstream. This time loss should, however, be as short as possible. Another possibility is to
322
gradually increase the discharge; here too, however, the respective authorities have to be informed. Operation rules must be chosen in accordance with the planned function of a multipurpose reservoir. The demand must be met that at the end of a flood the storage capacity be again filled to the level needed for its supply function. In the case in was therefore not able to Fig. 12.6 the flood was overestimated and the volume Wona supplement the emptied part of the storage volume I$,. However, this is still safe from the point of view of the storage function, as volume AVd can be used to supI plement the storage volume to the required level.
Fig. 12.6 Possibilities of filling up the storage volume when emptying of V, is very high due to inaccurate forecast data
Fig. 12.7 Application of the forecast of flood inflow volume for flood-control release
Short-term forecasts of discharges can be based on data from water gauging stations in the catchment area if information is to be received a few days in advance. Forecasts of the time behaviour of discharges based on a precipitation-runoff relationships have similar characteristics and are used mainly for reservoirs in smaller catchments. If in the time interval ( t i , t 2 ) a flood volume ~ r l + r z , is expected to flow into a reservoir, then the largest volume to be held by the reservoir is given approximately by the difference ytl +12) - Ond(fZ - tl), see Fig. 12.7. At the same time the greatest possible emptying of the storage capacity in interval (tl, f 2 ) can be determined, given by the difference yt,+12) - Op(tt - t i ) - wb,where w b is the adequate safe reserve. Seasonal forecasts can also be of use for flood-control release. In spring they are mainly forecasts of the runoff volume from snow supplies in the catchment, which ~ make it possible to empty the storage capacity of a reservoir by the volume ~
~
z
323
- Op(tZ- t l ) where is the smallest expected inflow volume during the spring months. These forecasts are useful for catchments with large amounts of snow. In recent years, further methods of seasonal discharge forecasting, based on the analysis of historical discharge series considered as a stochastic process, have been developed (e.g.,Andtl et a/., 1971; Buchtele, 1975). These forecasts could also be used to increase the flood-control effect of the storage capacity, especially in high-flow periods, when the discharge forecasts allow for the emptying of the storage capacity without any risk to its supply function. To estimate the benefits of forecasts correctly a project should be elaborated, including all forecast relationships and other conditions. The effect of flood-control release is assessed for all catchments during the observation period in relation to the forecasts (simulation method). Statistical processing of the flood characteristics (volumes not held and a maximum release) gives a general idea about the contribution of forecasts to the increase of the flood-control effect of the storage capacity. Soinetimes it is possible to determine only the number of floods, that are held by the storage capacity thanks to forecasts (On,, is not exceeded); then the contribution of forecasts is characterized only by the probable time of exceeding the non-damaging discharge. If various forecasting methods are used in one year, they should be estimated separately. To make the flood-control function of a reservoir as effective as possible we shall, as a rule, combine various measures enabling the use of the active storage capacity for this purpose: the automatic emptying of the storage capacity, operating ,?(: :?
--&.--
c
-t
b
Fig. 12.8 Influence of active storage capacity on flood characteristics: determination of maximum release (Omax)and uncontrolled parts of the flood volume (AV,)
324
schedules and forecasts of all types. Any solution of such a complex case should include: (a) the determination of the reliability of flood control, (b) estimation of the probability of holding floods in the respective parts of a year, (c) evaluation of the effect of the proposed measures on the floods which are not completely held by the storage capacity (Ondis exceeded) (Fig. 12.8). If the effect of the storage capacity is not sufficient according to the required reliability, another auxiliary capacity should be included in the reservoir. However, this does not necessarily mean that the total capacity has to be increased, causing further costs, If a reservoir includes a surcharge capacity (which is often the case in those on small streams with ungated spillways)it can at least partly be used for a more reliable flood control. 12.3 UTILIZATION OF THE SURCHARGE CAPACITY
The size of the surcharge capacity (throughout the book surcharge capacity is used instead of the longer term flood surcharge storage capacity) depends on the arrangement of the spillway devices and on the conditions for transferring the socalled design flood, which ensure the dam against overflow, regardless of the operations of the reservoir. A surcharge capacity can also be found in a reservoir with a controlled spillway, if the height of the spillway gate is smaller than the ovefflow height corresponding to the design flood (the capacity between the upper crest of the gate and the maximum water level of the reservoir).
-t v-f(f)
Fig. 12.9 Effect ofsurcharge reservoir capacity (with suitable operation) on the time pattern of the flood (release from the reservoir)
Part of the surcharge capacity can serve as flood-control capacity, as it can ensure a release of 0 Ond. It is given by the discharge-rating curve that determines the spillway height h,, which allows for the overflow of non-damaging release On,(Fig. 11.10). Up to that level it can be ensured that On,is not exceeded, by regulating the release through the outlets. To solve this problem the number of floods held by the
325 capacity AKn during the observation period must be determined and the effect of the remaining part of the surcharge capacity on the floods that were not completely held must be estimated (Fig. 12.9). Time pattern of the floods influenced by the effect of the storage capacity is used as basic data. The iuluines of inflow and release with a given Q and 0 must be “balanced’ with the reservoir storage volume and the transformation of the time pattern of release, that is dependent on the water level of the reservoir, must be calculated (Section 11.2) (flood routing). If the storage capacity and the respective part of the surcharge capacity A & are not able to ensure the required rate of reliability, another capacity has to be designed (Fig. 12.10). If the reservoir has a gated spillway, it is possible to make use of a part A K of the surcharge capacity to decrease the height of the gate (Fig. 12.11). This, however, does not increase the total volume of a reservoir.
d
Fig. 12.10 Flood-control capacity and the surcharge reservoir capacity (design with a free spillway)
-V
Fig. 12.11 Location on the floodcontrol capacity in the reservoir with a gated spillway and the possibility of lowering the height of the gate .
fKTp /
/
/’
I)= f ( t )
I
/’
-f
I I
,
h
.
,
AW=AV
Fig. 12.12 An estimate of the transformation effect of the surcharge reservoir capacity
326 The surcharge capacity automatically decreases the maximum peak discharges of all floods. Figure 12.12 gives a comparison of the volume AW that has to be held in the surcharge capacity with a given release and of the volume AV, corresponding to the overflow height for a maximum release (Omax).The two volumes should be approximately even (AH‘ AV). I
-I
-t F i g 12.13 Possibility of decreasing the maxiinurn release by gradual closing of the bottom outlets (with ungated spillway)
Fig. 12.14 Pre-enip! 111; I u \ d on forecast data (with gated spillna)) I U catch thc flood peak
With the help of discharge forecasts it is possible to increase the effect of the surcharge capacity on the maximum peak discharges. The simplest measure is step-bystep closing the lower outlets at the time when the maximum peak release from a reservoir is expected (Fig. 12.13). Forecasts can be made better use of in reservoirs with a gated spillway, where a partial emptying of the storage capacity makes it possible to catch the “peak” of the flood wave and thus greatly decrease the maximum peak discharge (Fig. 12.14). 12.4 FLOOD ROUTING IN THE STREAM CHANNEL
For flood control the unsteady flow downstream from a reservoir must also be taken into consideration. Flood waves with a small probability of exceedance that are held only partly by a reservoir come into a stream as transformed discharge waves combined with the flood from interbasin. The conditions for the flood routing are dlfferentfrom those in the original (natural) streams. Unsteady flow is one of the most complicated hydraulic phenomena. This problem was studied by mathematical models with the help of modern computer technology and verifications “in situ” e.g., by Gabriel (1975) and others. 12.5 FLOOD CONTROL AT THE TIME OF RESERVOIR CONSTRUCTION
When building a reservoir on a stream, the water (discharge) has to be diverted from the building site. For this purpose it is best to divide the construction period into stages and design the water-diversion facilities according to these stages.
321
As the dam grows, a storage capacity is created that can hold flood waves; this, however, represents a certain risk to the environment of the stream downstream of the dam. The project for the diversion of water from the construction site must include, besides the protection of the construction itself, also the consequences caused by backwater during floods. Damage can also be caused in the inundation areas, but the greatest damages occur downstream of the dam. Measures and operation schedules therefore have to be implemented to control release from a reservoir under construction, bearing in mind that discharges can be greater than the design capacity of water-diversion conduits.
For the design of a reservoir, the input assumptions of release control and the relationships between the regulation function variables have to be determined. The results determine the storage capacities of a reservoir or the regulated flow (release), safe yield with a given reliability, the non-damaging flow, etc. During the operation of the reservoir the operating rules included in the input assumptions of release control must be adhered to. Other factors that have to be taken into consideration are the principles of operation during the failures in water supply and flood waves surpassing the design reliability. As the design of a reservoir always includes a safety margin, the conditions for its various functions will, in most years, prove to be more favourable during the operations than was presumed by the design. During those years further profit can be gained from the reservoir by using extra water for, e.g., “flushing” of the river, regulating the winter regime, etc. As soon as a reservoir starts to operate, it should be observed how it fulfils the planned functions; experiences gained should help to define more accurately the operation rules; all economic factors should be assessed. The tasks of the water-management operations of a reservoir can be summarized as follows: - to secure proper manipulation from the point of view of the storage and protection, which ensures the regulated flow as determined by the design calculations; - to contribute to a better efficiency as compared to the design values at times when the hydrological conditions permit it; - systematically to study and assess the water-management functions of the reservoir. The demands on the operation of a reservoir are qualitatively higher if it forms part of a water-management system. Then it must fit in with the operation of the whole system with a higher level of operations, i.e., water-management dispatching.
13 RULES FOR RELEASE CONTROL FROM RESERVOIRS
IN DAY-TO-DAY OPERATION
The control of the outflow from a given reservoir at any water stage, at different flow rates, etc., is given by the operation rules, schedules and guides. These documents must be discussed and approved by the respective water-management bodies. From the point of view of reservoir function, the most important part of the operation rules is that dealing with the effective use of water and the operation during floods. It includes instructions for the use of water under normal conditions (within the given reliability range of the storage function), with steady storage, for emptying and filling, and for maintaining the proper water level; instructions for the operation during floods; and other principles, including provisions for cooperation with other reservoirs. Besides these rules, which mainly concern the quantitative aspects of the proper use of water, the operation rules also include provisions which deal, e.g., with the water quality, with sediments, the flow in winter, cooperation with flood warning service and other exceptional events. The design documentation consisting of the results of the calculations for a reservoir, of the respective economic aspect and the draft operation rules, form the input data for the operation rules and schedules. All these data have to be assessed and “brought up to date”, as several years will have passed between the design stage and the elaboration of these rules. Any operations of a reservoir should be based on a rule curve (table) which would clearly and explicitly determine the release. There can be several curves, each with a specific validity. One of the basic obstacles for the optimum utilization of a reservoir is the lack of advance information about the future inflow (according to a reservoir cycle). In this respect the day-to-day operation of a reservoir essentially dlffers from the technical simulation. Flow forecasts can be used only to a limited extent (usually only for a few days ahead). Therefore, information about the flow time-behaviour is analysed and used to determine the rules for the discharge control which, regardless of forecasts makes it possible to make the best use of a reservoir. These rules can be presented in the form of curves or tables. N o rule curve is needed for very simple reservoir operation when, e.g., the storage
33 1 capacity is not completely full and release from the reservoir = 0, (constant or variable in the course of the year); also with a full storage capacity 0 = Q (as well as with a completely empty storage capacity, i.e., during a failure). Such operations coincide with the reservoir parameters; however, with the exception of very low-flow periods they do not make full use of the reservoir’s possibilities to regulate the flow. Dispatching control of the discharge is therefore expedient for all reservoirs, in which it is useful from the point of view of the conservation and flood-control function or operational conditions (head, maintaning a higher or lower reservoir water level at certain times of the year etc.). 13.1 RULE CURVES AND THEIR CONSTRUCTION
Most frequently, a rule curve is constructed, based on the dependence of the required volumes of water (or water levels) on time during the year (Fig. 13.la). In the simplest case, from the point of view of the storage function of the reservoir, this dependence limits the area in which only 0, must be released (or less during failures in water supply). Only if a reservoir has the storage volume I/ > Vd it is possible to withdraw 0 > 0,. The rule curve does not determine to what extent the release can be increased. More advantageous therefore is the rule curve in Fig. 13.lb which, depending on the amount of water in a reservoir and on time, determines the amount that can be released.
A rule curve can be constructed if the inflows to the reservoir in years with very low-flow periods (close to the design interval) have an at least approximately identical time pattern. Of importance is the time of the high-flow and low-flow periods which should be identical in all the years. The existence of an annual flow cycle is a basic condition for the construction of a rule curve and therefore a dispatching control of the outflow is effective mainly for reservoirs where the filling and emptying cycle does not exceed the limit of one year (water-management year), regardless of whether it is a direct release, regulated flow of the river or a reservoir in a system. The possibilities are significantly smaller for over-year flow control; this is the result of the random character of the chronological sequencing of the mean annual flows.
332
The more regular the annual flow cycle, the more abcurate the rule curve. For a simple hydrological regime the rule curve defines, besides the basic relation V, = f ( t ) which is often defined for the safe yield (Fig. 13.la), different zones divided by further lines. From the point of view of the storage function, curves of reduced releases should be plotted and in the area of values higher than Qd,an “anti-overflow” curve should be plotted (Fig. 13.2). Further details and principles concerning the construction of these curves are given, e.g., by Votruba and Brota (1966,p. 260) where further references can be found.
-f
In the complicated hydrological conditions pertaining e.g. in Czechoslovakia, the construction of a safe rule curve is a very demanding task. It is especially difficult to determine the boundaries of the high-flow and low-flow periods. Studies of the latter showed, e.g., that their end can coincide roughly with the end of September until the middle of March, which is an almost six-month interval. Great differences can also be observed in the “spring” high-flow periods. However, in spite of this, the flow regime of the Czechoslovak rivers is cyclic to such an extent that it is still effective to use the dispatching control. The complex time pattern required for storage volumes is the reason why the chronological flow series should be used for the construction of the rule curve. If we bear in mind that the real series are not even long enough to assess a reservoir’s basic parameters (a, p,, P ) satisfactorily, then they will certainly not sflice to determine the filling and emptying of the storage volume. It is therefore essential to use sufficientlylong synthetic flow series reflecting the flow variability during the course of the year. Only if absolutely unavoidable the real flow series can be used; however, the specific traits of the respective low-flow periods must be excluded, which can be done, e.g., by transferring the flows of signifcant low-flow periods to design conditions and including a certain margin up to the values of the required storage volumes V,. Method of constructing a rule curve from sufficiently long synthetic series: those years in which the demanded theoretical size of the storage volume is greater than the design value should be excluded. These years are failure-years as far as the design
333 measure of release is concerned; these failures are planned and are therefore not taken into consideration in the construction of a rule curve. Another point that must be considered is that at the end of every low-flow period the storage capacity should just have been depleted. The “known” rate of inflow to a reservoir during the given simulated series of mean monthly flows and the demanded yield (release 0,)are used as a basis.
One should proceed from the end of the low-flow period (in time t,,), when Q c 0, (Fig. 13.3). In the last time interval Atk, the water volume which has to be supplemented from a reservoir to ensure a release 0, has to be calculated :
A b= (Op,k - Q k )
Atk
(13.1)
The storage volume must therefore be equal to A& in time (elf - Atk).Similarly, in time (tlf - Atk - Atj) the storage volume must be (A% + A&), where A% = = (O,,j - Qj)Atj, etc. Calculations continue until the moment when AV 5 0. The method is shown in the form of a diagraph in Fig. 13.3a, where the storage volumes A V needed to ensure the demanded 0, are plotted for one water-management year. Figure 13.3b shows numerical calculations (indispensable if synthetic series are used) with the help of inflow and release mass curves. Thedifferences between ordinates C Q and C O Pgive the A V values. The upper envelope of all C A V = f ( t ) curves corresponding to the respective water-management years of the synthetic series is the limit, above which it is possible to pass on to greater releases from a reservoir with Qd > 0, without the risk of an unplanned failure. On the other hand, if the water volume in the storage capacity is
1
334
smaller on the given date than is determined by the envelope, only 0,can be released. In view of the complicated hydrological conditions, no failure in water supply usually occurs even in this case. The upper envelope of the A V curves can be defined as the basic rule curve V, = f ( t ) which divides the zone of increased supply; this zone must be defined for increased releases from a reservoir during high-flow periods.
This method was used for the construction of the basic rule curve in Fig. 13.4b. The basis for this curve was a flow series of 80 years (Berounka- Kfivoklat, 1891-1970), which can be conceived as a certain sectio? of a random series. Figure 13.4a shows the time pattern of the necessary storage volume in the respective low-flow periods, which differ greatly in time as to the commencement and termination of the low-flow and high-flow periods. The upper envelope of the determined necessary storage volumes (Fig. I 3 I h ) is the rule curve sought bearing in mind, however, the limited length of the flow seric\. t rani I- 18. 13.4b it is possible to notice the danger of constructing the rule curve directly Iroiii r c d flow series that have only a limited duration of observations. If a curve were constructed from the period 1931-1970, or 1931-1960, the upper envelope (dash or dot-and-dash) would allow for an extensive emptying of the storage capacity at the end of February. In a very low-flow period (1921-1922) the consequences of regulating releases according to such a rule curve would be very unfavourable. The low flow in the spring months would not be sufficient even partly to supplement the water supply in a reservoir (on April lst, the volume would be only 4.5. lo6 m3)and a failure to supply water would last from the beginning of May until the end of the year; the extent of the failure would also be significant. Even though the flow distribution of that year is not typical, it must be taken into consideration for the impact of its consequences.
335 The above example proves the significance of the variability in time of the occurrence of high-flow periods in the construction of rule curve. It has to be taken into account particularly if the yield approaches the fuzzy boundary between the withinand over-year release control. This example also explains the reason why synthetic series should be used that give a broader choice of the flow behaviour in time than the observed series. For the construction of a rule curve the theoretical size of the storage capacity is usually considered, which generally differs from the design value. The theoretical ordinates of the curve should therefore be corrected in such a way that the maximum storage volume would be identical to the real volume of the storage capacity. The simplest way to do this is by a proportional increase of the theoretical values by the coefficient V,(des.) 5= I/,(theor.) Operation governed by this rule curve, compiled in this way, depends on the purpose or the method of utilization.
1-
I
- t
.
+i
Fig. 13.5 Operation based on the rule curve when the storage volumes given by this curve are exceeded
From the point of view of the storage function, the water surplus in the reservoir (above curve V, = j ( t )of the rule curve) is used evenly up to the end of the low-flow - b,l,which period (Fig. 13.5). In time t , the surplus volume was AV, = was divided for the whole period t l = t,, - t , so that the release from the reservoir increased by
ve,l
(13.2)
and reached the value O(7,) = 0, + AOl. At moment t , the surplus was AV, = - ye,,- V& and after calculations for the rest of the low-flow period T, = t,, - t , , the release could be changed to O(r,) = 0, + AO,, etc. If there is no special need for these more detailed operations, it is possible to release the largest useful (usable)amount 0, from the real storage volume (V,, > V,);
as soon as the surplus value is used up (Ke= b),the release is changed to 0, (two-stage release control). In complicated hydrological conditions the storage volume frequently drops below the values prescribed by the rule curve. This drawdown, however, need not cause a failure in water supply and it is possible to continue to release 0,with only a small risk. As the design reliability is always less than 100% (for a within-year flow control usually up to 97%; occurrence-based), a failure to supply water might occur. It is most important to become aware of such a danger in time, but this is rather dSicult for reasons of the complicated flow regime in some rivers. Of certain help can be an analysis of the time pattern of emptying of the reservoir at the end of the low-flow period and the determining of the storage volume which signals the danger of a failure in the water supply. The example in Fig. 13.4 can serve as an illustration. Let us presume that with an irregular flow rate there is only a short period in which preventive measures can be introduced to control any failures in water supply; this period might be, e.g., two months. From the pattern of outflow from the reservoir at the end of the low-flow period (Fig. 13.4a), it is clear that the greatest demand on the storage volume of the reservoir two months prior to complete emptying is 37 . lo6m3 (this happens to be the same as the minimum given by rule curve). This value can be accepted regardless of whether the low-flow period comes early or late (it need, however, not be the same). As soon as, roughly from the middle of July to the beginning of January, the storage volume drops below this value, operation prescribed for this case must be introduced. The further depletion of the reservoir must be systematically assessed with the help of the control curve-which is the dotted curve in Fig. 13.4b-reflecting the most intensive possible emptying of the reservoir at the end of the low-flow period. This curve is independent of time within the interval in which it is real. The basic measure to abate the consequences of a failure to supply water is to reduce the releases from a reservoir in time; this means that although the period of failure is prolonged (if as a failure we consider any withdrawal below the value Op), a complete breakdown of supply, i.e., when the storage area is completely empty and release equals inflow, can be avoided. In the case studied, water-saving measures can be started two months in advance. If the water in the storage volume decreases below 37. lo6 m3 with a withdrawal 0 = 0,, the whole deficit AVl is divided over the whole two-month period (T,) so that release from the reservoir Or, will equal (13.3)
At a later stage (e.g., after 2 weeks) the deficit AV2 (Fig. 13.6)is distributed over the remaining time z2 of the presumed low-flow period and therefore ( 1 3.4)
337 In applying this method, the storage volume can exceed the value of the control curve (in Fig. 13.6 by the value AV, in time c,). For reservoir operations this is a sign of a certain improvement of the critical situation; however, care should continue to be taken. In this case it is better to permit a safe yield of 0, and to presume that further emptying of the reservoir will continue according to the control curve transferred in time by the interval AC which is given by the storage volume in time t,. Short-term flow forecasts, e.g., according to the recession curves (Buchtele and Hladkj, 1976) can be of help in decision making in such cases and improve the operation.
j # T [ -t
Fig. 13.6 Operation based on the control curve of the rule curve when there is a threat of a failure in the water supply Fig. 13.7 Effect of operation rules with twostage manipulations (........ control for 0,; ---- rule curve) b
When evaluating the contribution of the rule curve for the supply function according to the rule curve in complicated hydrological conditions it is necessary to bear in mind that the water supply benefit, as compared with the control of the safe yield, will not be very expressive. In high-flow years, or periods, an increased release from a reservoir can be ensured without a rule curve; on the other hand, the fact that low-flow and high-flow periods can occur at different times greatly reduces the advantages of control according to the rule curve. Figure 13.7a compares the release from a reservoir in an average year, presuming that the release can be 50% greater than 0, both for normal operations of release (constant throughout the year) and for two-stage operations according to the rule curve (Fig. 13.7b). As a result of the rule curve an increased release can be ensured for roughly two months longer than when applying a constant 0,. This is ensured by
338
the better utilization of the reservoir storage capacity (compare the time behaviour of the storage volume shown by the dashed and the doted curves). If increased release due to the rule curve cannot be utilized effectively, there is no point in constructing this curve. However, as this method of outflow control ensures that every year the reservoir storage volume is decreased slightly, it also helps to increase the reservoir flood control function (Chap. 12), which is always useful. 13.2 FAILURES IN WATER SUPPLY
If the designed water supply reliability is less than loO%, certain failures are admissible, i.e., the release can be less than planned. However, this reliability only applies to a reservoir (i.e., the water source), not to the whole system up to the place where the water is used. It does not include failures in the devices for water release and transport (pumping stations, pipelines, etc.), failures in the reserves, etc., which are of importance to the operator. As the design reliability is a technico-economical issue (Section 4.4), standards frequently have to be used for lack of economic data. However, even with this lack of data any significant failures in water supply should be assessed and remedies introduced. The principle of a timely reduction of releases, thus controlling the extent of the failure in water supply mentioned in the previous section, can be applied in different ways by different users; in certain cases other measures can be more expedient. Using synthetic flow series of sufTicient length, a certain number of failure periods are also obtained and these can be applied to operation principles in those periods in which a water supply failure might occur. These principles should be elaborated with the participation of the water users (consumers),as this is the only way to minimize the consequences of failures (economic, political, sanitary, etc.). As it is difficult to assess the duration and extent of the failures in advance, the principles should be of a general type and measures should be introduced successively (reduction of water supply, use of reserves, etc.). Only recently has research begun to tackle these problems. 13.3 RESERVOIR OPERATION DURING FLOODS
Operation methods to control floods must include floods with any probability of exceedance and all real operational stages of a reservoir, including any breakdowns of water-diverting devices. The most important task is to ensure the planned flood-protection function of a reservoir. The complex concept of controlled outflow during floods (Chaps. 11 and
339 12) should include calculations for the protective function of a reservoir, including floods which exceed the design reliability. The draft of operation determines when and where the respective reservoir devices should be used to control the flood flows. It must also be determined to whom the flood operation, the increasingflood danger, and other data should be reported. This is absolutely vital as every reservoir forms part of flood-control plans. 13.4 DAY-TO-DAY OPERATION AND OVER-YEAR RESERVOIR CYCLES
The essentially random character of the time sequence of the mean annual flows makes it impossible to use the whole volume of the storage capacity for the efficient use of water. The operation rules can be applied only when the storage area is almost full, or at times when water supply failures might occur; but in spite of that, a rule curve is useful as it can be used for roughly one-third to one-half of a reservoir's operation time (depending on relative yield a (see Chap. 5)). We outline here the method that makes it possible to exploit the advantage of operating rules for over-year reservoirs. In constructing the basic curve of the rule curve, using a verified synthetic flow series of sufficient length, the years with average flow Q, 2 0, apply in isolation, and the years with Q, < 0, as the initial years of the over-year low-flow period. For safety reasons, the maximum emptying according to the rule curve should not exceed the value given by the difference between the total storage capacity K ( P ) and the over-year storage component r ( P )with the design reliability P, even though in real operations the two parts (that can be used only for theoretical calculations) are mutually exchangeable. In practice, it is possible to determine the value given by calculations in the years with Q, 2 0, and then, using the method described in Section 13.1, the upper envelope of necessary storage volume can be established. It then must be decided whether or not the years with Q, < 0,, considered as the initial years in the framework of the over-year design period, require a greater storage volume. If this is the case, then the upper envelope of the required storage volume must be corrected and the basic rule curve is obtained. At times when the storage capacity is almost empty, control curves can be used, which-as is the case for the within-year flow control-can contribute to a timely discovery of the approaching water supply failures. As the reservoir has an over-year cycle, analyses of the emptying of the reservoirs should cover a longer period than for within-year flow control (e.g.,4 months or more). The difficulties connected with a better exploitation of reservoirs with over-year flow control initiated studies to help to solve this problem, at least for special cases when operation is constrained by further conditions. An original answer to the problem of increased release from an over-year reservoir at the 'time of a balance
340 deficit, before a further reservoir with an over-year cycle is put into operation, was presented by Kubat (1976), as shown in Fig. 13.8. The continuous development of the water demand is covered by the construction of resources in stages. Figure 13.8a shows that the need in time ( t l - t 2 ) was not fully covered because construction of reservoir B was started too late. During this period, release from reservoir A would be increased (Fig. 13.8b). After reservoir B starts operation (in time rz), release from reservoir A would be decreased, thus “compensating” the increased release in the interval (tr - t z ) without any harm to the planned storage function, including the sufficient supply of water (theoretically release can be decreased from moment r 2 till moment r4).
Fig. 13.8 Cooperation of over-year reservoirs operations to cover increasing demands
Under complicated hydrological conditions, the possibilities of the operating scheduling are rather difficult to ensure and attention is therefore also paid to flow forecasts to ensure a better utilization of a reservoir. As the water-management usually covers large areas, the technical facilities for forecasting also gradually improve. The actual flow control according to data supplied by the forecasting service must be based on instruction included in the operating rules and schedules.
-
14 THE FUNCTION OF RESERVOIRS THEIR MONITORING AND WALUATION Water acts in many countries state that the most economic use should be made of water resources according to the needs and interests of society. Capital must be invested rationally and therefore optimum water-management operation must be ensured. An indispensable condition for the perfection of projects and the control of all operations is the systematic processing and assessment of technical and economic experience gained from existing water schemes and the verification of the project (design) parameters. To verify and assess quantitatively the planned water-management function of a reservoir, a large sample of various technical and economic data must be studied. There are still certain shortcomings in the studies of the technical characteristics of reservoir operations; however, the greatest difficulty is in the assessment of the economic consequences of the water-management function of a reservoir, or other water resources. The aim of the study and assessment of reservoir operations is - to gain data for the comparison of the real and designed (planned) water-management functions of a reservoir, or a system of reservoirs; - to obtain data for the improvement of operating rules and to improve the efficiency of a reservoir by effective operating schedules and guides; - to facilitate decisions concerning changes in a reservoirs function or operations of already constructed multipurpose reservoirs or systems; - to help improve the planning and design of new reservoirs; - to gain information about the impact of a reservoir on its environment; - to gain experience of operations during planned maintenance and repairs of hydrotechnical structures (e.g., the technological devices for the outlets and spillways); - to obtain data to determine the compensation for the water released from a reservoir or river, where a reservoir makes this release possible; - to obtain data for the economic assessment of the benefits from reservoirs, mainly to substantiate the designed reliability of storage and flood-control functions. These sets of characteristics that should be studied can be divided into the following groups:
342 1. The basic sample which makes it possible to study, in chronological order, the quantitative relationships (basic reservoir tasks) and the water-quality indices related to the purpose of the reservoir. Indispensable for quantitative assessments are data on the inflow to a reservoir (including diversion), outflow to the stream channel downstream of the reservoir, and on the water volume in a reservoir, as well as on other components of the total balance such as the effect of precipitations on the water level, evaporation from the water level or losses by seepage. The water-quality indices must be watched carefully when deciding the way of release, the amount to be released and the time of release. 2. The supplementary sample. concerned with operative measures and the raising of the efficiency of a reservoir. These are mainly hydrological or meteorological characteristics which help to forecast the inflow to a reservoir, e.g., precipitations in a catchment area, temperature of the atmoshpere, the change of the hydrological regime of the river upstream or downstream of a reservoir, discharge-travel time of different flows, etc. 3. Sample of economic effects resulting from reservoir functions. Most attention should be paid to exceptional situations, such as period of limited supply, profound water supply failures, occurrence of floods causing damage, etc. Of great importance is - the relation between the degree of water supply reduction and the economic damage resulting from this limitation (bearing in mind the duration and extent of the limited supply, occurrence in time, etc.), - the determination of economic damage caused by extensive water supply limitations, - the relation between the exceedance of a non-damaging discharge and the respective flood damages. Besides the above-mentioned samples of characteristics to be studied, other specific investigations can also be important, e.g., the non-planned or intangible effects of reservoirs. The processed and verified results of the investigation of reservoir operation then help to judge the reservoir function correctly from technical, economic and social points of view.
E EFFECTIVENESS OF WATER RESERVOIRS AND THEIR FUNCTION IN SYSTEMS AND IN THE ENVIRONMENT
15 RESERVOIR FUNCTION
IN WATER-MANAGEMENT SYSTEMS Reservoirs are the basic elements of water-management systems. Their function is determined in cooperation with and in relation to other elements of the system (by diverting the flow from one catchment area to another, by river training, etc.). In a system, all basic functions of a reservoir can be made use of: storage, water conservation, flood control, water treatment, aquatic environment. Reservoir systems can be either single- or multi-purpose systems. 15.1 CHARACTERISTICS OF WATER-MANAGEMENT SYSTEMS WITH RESERVOIRS
A water-management system can be described as a set of water-management elements linked by mutual relations into one wholeness with the aim of utilizing and protecting the water resources. Water-management systems (the terms water-management systems and water resources systems are synonymously used in this chapter), of which storage and flood-control reservoirs form an important part, greatly change the natural flow regime of streams as well as the properties of the water. The extent of these changes is determined by the relative size and function of a reservoir, but also by the hydrological regime of the inflows, by the release conditions, by the geomorphological conditions of a reservoir, the quality of the inflow water, etc. Water-management systems with reservoirs can be relatively simple if they have, e.g., one reservoir and several users, or compound if they have several reservoirs and many users, or if they have several purposes. Figure 15.1 shows some schematic representations of water-management systems with conservation reservoirs, floodcontrol reservoirs and reservoirs creating an aquatic environment. Figure 15.la illustrates a simple system with one conservation reservoir, two releases and a regulated flow downstream of the reservoir. The principles of the analysis of such a system optimize use of the water resource (reservoir) for three functions, 0, - 0,. Figure 15.1b illustrates a simple system with two conservation reservoirs and one water user. The analysis is based on the optimization of cooperation of two reservoirs to create the water supply for the single user. Figure 1 5 . 1 illustrates ~ a compound water management system with several (in
344 this case two) conservation reservoirs and several users. The optimization as well as the behaviour (operation) of the system is complicated, a': both sides of the water balance in the system are to be optimized mutually, i.e., utilization of the water resources and meeting the demand for water. The system becomes even more complicated if the water yield P, and Pz does not cover the demands 0, - 0,, and diverted water P3 from the neighbouring catchment has to be included in the system (the dashed line in the figure).
Fig. 15.1 Schematic representation of water-management systems with reservoirs
Figure 15.Id illustrates a simple water-management system with one flood-control reservoir, where the valley downstream of the confluence A with another nonregulated discharge Q is to be protected. In this case, the analysis is based on the optimization of the protection function of the reservoir with regard to the discharge Q. This is a certain type of river-flow regulation with regard to flood control. Figure 15.le illustrates a water-management system with several (in this case two) flood-control reservoirs, which are to protect the valley downstream of profile A, as well as the valley downstream of one of the reservoirs, from floods. The optimization of the control function of the system includes the calculation of the variants of co-ordinated operations of all reservoirs in any realistically possible flood situation in the system.
345
Figure 15.lf illustrates a system of several ponds through which water flows or does not flow and which has one common conduit. The aim of this system is mainly to establish an aquatic environment suitable for fish farming. As both the areas of the ponds and the discharge Q are rather small, the changes in the hydrological regime are insignificant. However, in the opposite case low discharges are significantly affected by evaporation from the water level and flood discharges by the storage capacity of the uncontrollable areas of the ponds. Figure 15.1g illustrates a compound multi-purpose water-management system with several reservoirs on different rivers, with both conservation and flood-control functions. Besides optimizing the supply and flood control function of the reservoirs, the hydraulic link between the respective water users is also optimized so that they can obtain their water supply from various water resources (reservoirs). This cooperation is the more effective the more the hydrological regimes of the respective dam sites differ and the greater the difference between the relative reservoir volumes. Further information about water-management systems can be found in books on this subject (Buras, 1972; Votruba, Nachazel and Patera, 1974; Vitha and Doleial, 1975; Kos and Zeman, 1976; IIASA publications; Votruba et al., 1979, 1988, with extensive bibliography, including explanations of general system terms). Here, the water-management systems are discussed only with regard to the reservoir function in these systems. 15.2 DEFINITION AND ANALYSIS OF WATER-MANAGEMENT SYSTEMS WITH RESERVOIRS
A systematic analysis of any system must include the optimization of its structure and behaviour. To facilitate the calculations, simplified systems are defined. For a correct simplification no element or link which is important for the correct result related to the investigated aim may be overlooked, but on the other hand every element or link which is not significant for the given aim should be eliminated. The defining of a system is one of the most responsible tasks of any water-management engineer. Mathematical modelling and computer technology are indispensable aids in solving optimization problems. The construction of a model is a creative work requiring the ability to simplify and abstract. For compound systems a suitable method to use is dccomposition into sub-sjistems, which are - spatial, to decrease the scope of the system, - purpose, to reduce the multipurpose character of the system, - time, to decrease the dynamics of the system. However, decomposition is only a methodological aid which should not completely change the concept of the system which is established by coordination. Therefore,
346
the interpretation of the model’s results to objective reality is just as significantly creative work as the construction of the model itself. The “interpreter” must therefore constantly compare the results gained from the model with practical experience, to verify the analogy of the behaviour of the model with reality, which is not general, but specifically typical for every given case. A skilled “interpreter” avoids incorrect conclusions, while an unskilled one can even depreciate the results of a correct abstraction. The interpreted result is implemented by a certain decision. Decisions concerning the use of water affect all people; these decisions have political consequences,because they influence people’s behaviour and opinions. Proper consideration must be given as to whether the recommended solution can be implemented. The responsibility is very great as these measures usually have both economic and intangible impacts with long-term consequences. Some examples of systems and an outline of how to analyse them are set out below. 15.2.1 Power and irrigation water-management systems
Water-management systems often combine power production and irrigation, although this usually requires a change in the regulation of the flow to serve one purpose or the other. For this reservoirs are needed. Figure 15.2 shows a section of a very complicated system, which will suffice to explain the principles of the method (the whole schematic representation of the system can be found in the AIRH Manual,
-
river B
Fig. 15.2 Schematic representation of power supply and irrigation systems
347 1973, Dl). The system consists of a cascade of hydro-power plants on the upper and middle reaches of river A and of numerous irrigation canals which divert water from the conduits of the hydro-power plant. Besides the main river A, the system includes other rivers (B, C and D),which, after meeting the demands of their “own” systems, supply the surplus of their discharges to the main system. The main system has three reservoirs, N , , N , and N,, of which reservoir N , is the first one on the stream, with a large volume and an enormous ability to regulate the flow. The main task of an optimal operation schedule for a given water-management system is to make maximum use of the local discharge to cover part of the irrigation withdrawalsfrom reservoir N It is therefore necessary to determine how to cover the demands for irrigation water from the water resources of the system, from the discharges of the local rivers from the upper to the lower reaches, from the discharge of the main river in the same order, from the reservoirs, and finally from reservoir N , . If the mathematical model is constructed according to the sequence of the releases for river flow regulation, release from N , can be minimized. A compensation balance is used for the calculations in each control point, taking into consideration the excess flow Q (not used in the upper reaches), lateral tributaries P, minimum maintained flow Q , in the main river and the water needed for irrigation 0,: (15.1)
The absolute value Qi, if negative, equals the demand on withdrawals. The balance calculations must take into consideration the limits of storage volumes ( Vmin,V,,,), the accumulation of the water inflow during the given period A,, the amount of water in the reservoir at the end of the previous period the outflow volume 0, and c \ ~ i p o ~ ~ i; i~i i di o\ccpaye ~i losses E , :
v-l,
Vmi,
s K - l + A , - 0,- E, = r/; 5 v,,,
(15.2)
From these balance equations flow charts were elaborated for all the river control points and reservoirs, bearing in mind their mutual relationship. The flow charts consist of the following groups: 1. Input data 2. River flow regulation for the demand on water along the upper part of the system from river A 3. Design of reservoirs on river A 4. Design of reservoirs on river D 5. River flow regulation for the demand on water along the middle reaches of the system from river A and in the subsystem along river D 6. River flow regulation for the demand on water in the lower reaches of the system of river A (downstream of the cascade of WPP)
348 7. Calculation of outflow from river C and river A in their downstream reaches 8. Determination of total releases from N , . From the flow charts, which included the algorithms of solution, a computer program was prepared. The model makes it possible to determine the reservoirs’ operation schedules and release from N , with the aid of computer, discharge hydrograph (for a series of years) and a water-demand time pattern. The regular annual values of the watermanagement indices are determined from statistical assessments. The results of the calculations can be used to plan the future operations of the system, making more effective use of the flow of local rivers and the water in reservoir N , . This solution is also used to construct a mathematical model for the operation schedules of the whole multi-purpose system. The above mathematical model includes, despite its individual character, given by the aim and aspects of the problem to be solved, also general elements and procedures. that can be used for the construction of other, similar models. It proves that mathematical modelling of water-management systems is a suitable method for solving the problems of multi-purpose water-management systems. 15.2.2 Irrigation wuter-management system
It is proposed to include in the power and irrigation system in the future (Fig. 15.2) new irrigation sub-systems to which water is pumped from rivers B and C. The function of this irrigation is illustrated in Fig. 15.3. The calculation scheme of the system (model) includes, besides rivers B and C, three direct-supply reservoirs ( N , , N , , N , ) with volumes V , , V,, V, (determined by the optimization method), three canals K , , K , , K , and six pumping stations, P I ,
-
+
for irrigation d
for irrigation
-
K3
river B
;pq
Fig. 15.3 Schematic representation of irrigation system
349 to PI,. According to the scheme, reservoirs N , and N , are filled by pumps PI, and PI,, reservoir N , by P I , and PI,. Withdrawals from the reservoirs are used for the upper reaches using PI, and PI,. The aim of this economico-mathematical model of the system is to determine optimal relationships between the annual releases 0, and 0,, supplied to the upper reaches from rivers B and C, and to determine the optimal reservoir volumes for flow regulation, with the possibility of flows from one reservoir to another (0,-,, 0,0, - ,) for some given total value 0, of pumped water volume. The following balance relationships were used as a basis fbr a mathematical model: (a) relationships reflecting the flow regime conditions:
,,
0 , = 0 , >= 0, V, = aOO, 01-, - 03-1 V, = uoO, - 0 , - 2- 0 3 - 2
+
V, = 0 3 - ,+ 0 3 - ,
where a,is the accumulation pumping coefficient, (1 - a,)-the used in transit. (b) relationships reflecting the reservoir volume values
(15.3)
part of the flow
( 15.4) An economic objective is the minimization of the total costs (capital investments Ii and annual costs Ci) of all n reservoirs during the standard life span T, therefore n
n
i= 1
i= 1
C E~ =
(Ii
+ TC,) = min
(15.5)
The balancing conditions (15.3)and (15.4) led to a system of linear equations
(15.6)
1
where xj ( j = 1 + 5 ) are basic variables; by comparison with equation (15.3) it follows that they correspond to the parameters O1 -,xl, 0, -+ x2, 0 , --t x3, 0 3 - -+ x4, 0 3 - ,-,x5, that are to be determined.
,
-,
3 50
tk (k
= 1
+ 7)
are complementary variables (supplementing further: 5, = x j
( j = 6 i 12)).
The economic condition (15.5) takes a linear form (objective function) (15.7)
The calculation matrix of the problem model (matrix coefficients in Fig. 15.4) gives a clear picture of the system of equations (15.6) and the objective function (15.7). This economico-mathematical problem can be solved by the simplex method of linear programming. The only difficult part is choosing the correct calculation procedure, to introduce the correct flow regime conditions and the corresponding economic characteristics and to consider non-linearity for the future.
Fig. 15.4 Calculation matrix of an irrigation system model
Flow regime conditions include: the regime of the resources (rivers B and C ) ; dates and durations of withdrawals, storage and flows between the reservoirs; the relationship of the flows through the canals and pumping stations and the outflow volumes (i.e., the basic parameters of the model of the system); any limiting conditions of water management, etc. The technical conditions that were considered are - simplification of the scheme for modelling purposes, - technical characteristics of reservoirs, canals and pumping stations, - schemes of their mutual links, etc. The economic characteristics were determined from the technical and watermanagement characteristics. Cost indices were used for canals, pumping stations and their pipelines. The economic characteristics of direct-supply reservoirs were derived from the hypothetical design of each one separately. The calculated economic characteristics give the ratio of the total costs for a reservoir during T years vs. the elementary parameters of the model of the system. Generally these characteristics are not linear. However, from them the coefficients l j
35 1 were determined as linear from the alternatives, with regard to the intervals and the coordination between the parameters of the reservoirs and the system parameters that are to be optimized. Then the problem was solved by computer, using the simplex method of linear programming. Irrigation systems are of great importance for the future development of water management and of agricultural production. Holy et a / .(1976)published a survey of 46 important irrigation systems in Czechoslovakia, put into operation between 1961 and 1975. Their total area takes up 143 OOO hectares, and the respective systems from 400 to 9131 hectares (Horn9 Zitny ostrov 11). The question of optimization of complex irrigation systems was tackled by Korsuh (1972).
15.2.3 Water-munugement system for the supply ojdrinking water
An example is given below as to how the public water supply is being solved in Czechoslovakia: The present resource is no longer sufficient (or will cease to be so in the near future), a new resource must be found and the possibilities investigated for extending the existing water mains to further users. This changes the former, essentially isolated, supply of water into a water-management system, usually a multi-purpose system, however, with a main aim being the supply of drinking water. The problem is to design a multi-purpose system, with the main purpose of water supply, and making the optimum use of the existing facilities. The problem has to be solved in several stages, by given dates. The example was taken from the public water supply for a region in BohemiaLiberec, Jablonec and Frjdlant (KubiEek, Technical University, 1972). Description of the system (Fig. 15.5): (a) local resources with a total yield of 170 I s - ' ; to increase to 300 1 s -' in 1985, (b) SOUS,impounding reservoir on the river Cerna Desna, the purpose and operation rules of which will change; the new resource is the Josefh Dbl reservoir on the Kamenice. In the future, Frydlant region will not have a sufficient supply of water. Regional water mains will be constructed in Jablonec-Liberec with a system of reservoirs and pumping stations. With the main consumers and the respective water mains, we obtain the general description of the physical structure of the designed system. Some of its parameters are: The SOUSreservoir on the cerna Desna has a total storage capacity of 4.888. lo6 m3 and a mean flow of 0.460 m3 s- To improve this condition it was suggested that water from the Bila Desna be diverted through the old canal, which was originally built to divert flood waves. With the flow from the Bila Desna the yield reached is 0, = 0.290 m3 s - I , while releasing Qme = 0.050 m3 s-' from the reservoir. The Josefbv Dbl reservoir on the Kamenice has a storage capacity of 24.63. lo6 m3 and a mean flow of 0.720 m3 s- I . After diverting water from the Jeleni brook, the reservoir will have an 0, of 0.700 m3 s including the released Qm,"= 0.125 m3 SKI.Without the water diverted from the Jeleni brook, the augmentation would be 107; less. When comparing the effect of the two reservoirs in isolation with the demand for drinking water in the region (Table 15. I), the shortage of water in the year 2015 is calculated at 8.2 I s- ' and at 120 Is- ' in the year 2030.
'.
'
352 When using the two reservoirs independently, the demand for drinking water in this region will be covered until roughly the year 2015, presuming that the conditions for the use of the water resources do not change.
5iI Tonvold, i B m d \' m
\i
woter treatment plant
I
Fig. 15.5 Schematic representation of water-management system for public water supply to Liberec and Jablonec in the year 2015
Tubk 15.1 Comparison of the effect of the Joseffiv DBI and SOUSreservoirs
Time level - year
Total demand o n drinking water in ihc Liberec - Jablonec - Frydlant (Is- I )
2000
2015
2030
931.6
1196.0
12Y8.7
1400.5
475.5
475.5
475.5
475.5
230.0 575.0
230.0 575.0
8.2
120.0
.~icii
Total yield of present resources (local) (I s- ') Effect of the considered reservoirs when used separately ( I s - ' ) Deficit of drinking water (1 SKI)
1985
SouS Joseffiv DfiI
353
Effects of system solution This multi-purpose system was identified by highlighting its main purpose, i.e., the public supply of water; the system will not only be extended, but new relationships will be established between all the elements. The two reservoirs become more efficient when they cooperate with other facilities, such as those for the diversion and treatment of water, than when the respective resources function independently. As the two reservoirs are relatively near each other, it was presumed that the hydrological regimes would be synchronous, which is unfavourable for their mutual cooperation. On the other hand, a favourable fact is the dfierent flow regulation rate in the two reservoirs. The first one at Sou5 has, together with the diverted water from the river Bila Desna, a relative yield a = 0.48 and the reservoir at Josefbv Dbl has an a = 0.87, including the water diverted from the Jeleni brook. It is therefore a combination of a seasonal and an over-year reservoir. Another condition for efficient cooperation of reservoirs is the capacity of the water-divertingfacilities, the capacity of the water-treatment plants and the pipelines; this condition is frequently a constraint. Cooperation was ensured by a rule curve, which eliminates the disadvantages of other methods. A rule curve was therefore prepared for the seasonal reservoir at Sou:, whereby the volume of water in the reservoir must correspond to the planned safe yield. If at any time the storage volume of the reservoir at Sou5 is greater than that given by the rule curve, more water will be released, which is the principle of cooperation with the over-year reservoir at Joseffiv DP11.
-
Fig. 15.6 Rule curve of a seasonal reservoir SOUS
month
Calculations according to the rule curve of the reservoir at Sou5 (Fig. 15.6) were based on the chronological series for 1931 to 1960. Even though the cooperation was solved according to synchronous series, the effect was not negligible.
',
With the total withdrawal of the system of reservoirs at Josefiiv DiiI and SOUSof 935 I s- the result of the cooperation came to 120 I s - ' . If the water volume in the reservoir at SOUSreaches above the rule curve, 460 I s - ' is released; if it is below that curve the minimum release is 240 1 s-'. In the same periods
354 354 475 I s - ' and 695 Is-' are released from the reservoir at JoseMv DOI. The improved performance of the Sou5 treatment plant of 460 1 s-' IS made full use of in 76.5% of the given period. The maximum emptying of the Joseffiv Dbl reservoir comes to 16.1 13 ' lo6 m3, which is less than when it was not
functioning in cooperation. The emptying and release of the two reservoirs, functioning separately and in cooperation, in 1936 and 1937, is shown in Fig. 15.7.
P -$2 Bc4 --
k
--.s8 O6 0
P
: 1:
QI
.-
L -. 16
I8
-
.5 Q2-c $4
P -6.-
.-
0
f38.s $W--
$12:7
114-
b-
15.2.4 Optimal cooperation of a system of reservoirs for public water supply The given problem is to make optimum use of water resources with the alreadyexisting reservoirs and one designed reservoir. Even though the formulation of the problem is similar to that in paragraph 15.2.3, a method is used here that con-
355 sistently respects the stochastic elements of the system. The system approach was adequate to the task as the problem concerned the cooperation of seven reservoirs in northern Bohemia of great economic and political importance (Fig. 10.3). The problem was solved at the Technical University in Prague (1969). The results help in decision-making about constructions and operation of the water resources in northern Bohemia. The problem was divided as follows: (a) water-management authorities formulated the problem, (b) the designer constructed the model, including the analysis, solution and implementation, (c) the water-management authorities, which need not be identical with the ones mentioned in (a), decide on the method to be used for the application of the results. As the problem could not be solved by previous methods of discharge regulation, the Monte Carlo method was used to simulate the operations of all the reservoirs. Natural flow series of 1931 to 1960 served as basic data. Synthetic monthly series were modelled for 500 years by computer, using the orthogonal transformation (principal components) method. In view of the capacity of the computer, the synthetic series were modelled in fours selected according to the resources that fed the respective water mains; the groups PfiseEnice, KPimov, KameniEka, Jirkov and the groups Flaje, Jirkov, KameniEka. The reliability of the modelled series was tested by statistical and water-management tests, which were in good agreement with the real series. The simulation model of the optimal cooperation of reservoirs was controlled step by step from the simplest couple of reservoirs to the final four reservoirs and to the whole system. The aim of the simulation model was to ensure effective operation with reservoirs: if one reservoir has a surplus supply of water or if there is an overflow, release from this reservoir can be increased, while, on the other hand, the release decreased from the cooperating reservoir and better use can be made of the water supply in the next low-flow period. This method can be applied to a whole system of reservoirs; however, this requires basic changes in the operation of the respective reservoirs. If, for example, in specific conditions reservoirs are exploited for a uniform yield, then cooperation requires a transition to nonuniform control of release, which further-in the case of a demand to uniform release in the area of water consumption-requires river flow regulation of all reservoirs. Releasefrom the respective reservoirs corresponds to the hydrological situation; however, as all reservoirs cooperate, the final release is a uniform one. The claim for a final uniform release in the area of the demand for water is not a limiting condition, however; generally, the cooperation between reservoirs can be based on any arbitrary release curve; the objective function that must be adhered to is the best possible exploitation of the whole system. While adhering to the required reliability (Po = 99%), rule curves were prepared
356 from synthetic series for all the seasonal reservoirs involved in the cooperation. No rule curves were prepared for over-year reservoirs and their regimes in the cooperation with other reservoirs were controlled only as far as their reliability were concerned. An example of the cooperation of the two reservoirs at PFiseEnice and Kiimov is shown in Fig. 15.8. Cooperation can ensure an increased release of up to 64 I s-'. In the whole region it is possible to gain an extra 330 1 s - ' , which is 16% of the sum of the isolated releases, 2.006 m3 s - ' . Analyses of the mutual relationships in the storage volumes in synthetic series showed that any compensation of deficits of water would be hydrologically risky. Cooperation in breakdown situations should be considered as a kind of reserve in the distribution plan, depending on the actual situation.
1086 1076 1066 1056
1046
1036
2l
C
1026
1016
'
-
m x . usefur yield K r h o v
C1 $-'I
Fig. 15.8 Results of cooperation of Kiimov and PfiseEnice reservoirs
Research has proved that simulation of reservoir water flow regulations is a method that, theoretically as well as practically, leads to the required rebults. Modelling and the use of computers make better analysis of water-management problems possible, and also make it possible to come to scientifically correct decisions. It has also been proved that methods of operations research can successfully resolve the question of cooperation between reservoirs in various watersheds with given capacities of release facilities. The described method can be applied to any case which has similar technical and hydrological conditions. 15.2.5 Optimization oj'multi-purpose systems with river jlow
rr(jirltitioti
The problem concerns a system with three reservoirs N , , N , , N , and with three water withdrawals S , , S,, S , for industry, irrigation and drinking water-all with full consumption (Fig. 15.9).The operations of the whole system are to be optimized. The
357 method of simulation modelling was used both for the design and operation of the system. The model was a simplified statistical and deterministic one. The state of the system was assessed in constant time intervals, given by the state at the beginning of the interval and the changes during the interval. Optimization is ensured by the economicassessment of the alternatives.A criterion, for example, can be the ratio between the benefits and the costs during the presumed life span.
s,
5--
-4
-----
p~
S- consumption
PiYerC
Fig. 15.9 Multi-purposewater-management systems with river flow regulation
Data and resolution Hydrological data consist of chronological series of mean monthly natural or modelled flows in the reservoir sites and at the point of withdrawals; this also determines the inflows from the inter-catchments Q M . 1 and Q M . 2 . Storage capacities are important parameters if the solution is only to be a quantitative one, regardless of the properties of the water. The capacities can be given in fixed or variant values. Water losses can be introduced in the first considerations by a decrease of the storage capacities (e.g., to 90%). The water demands S, and S , for industry and households are given by mean values in m3 s-’. The water demand S, for irrigation is given by a curve for the vegetation periods; it must be considered for the largest possible area to be irrigated and in the case of a lack of water for alternatives of smaller irrigated areas. The maintained minimum flow Qmindownstream of a reservoir and in the control points is determined by special calculations or, e.g., Q3s5d can be introduced for preliminary solutions. As this is a case of river flow regulation, the water demand will be met mainly by the outflow from the inter-catchmentsand if the need arises it will be supplemented by water from a reservoir. For a point 2 equation: (15.8)
358 where Q 2 is the flow in point 2, X S - the sum of withdrawals, QnliIl- the maintained minimum flow downstream of point 2, QJ,2 - the surplus (+) or lack (-) of flow, stored in a reservoir or released from it. Various orders of emptying and filling of a reservoir are studied. A comparison of the variants helps determine the optimum operation. If, in this case, the result is an impermissible shortage of water supply, the amount of the irrigation water is decreased to the amount of economically optimal utilization of the water resource.
%F %G OQJ
-tl
-
-
0 W M304050607118093XIM(7 mfer shortog e 0 , [ % 7
Fig. 15.10 Relationship of economic losses .ind
w t e r deficits
Important for the economic assessment of the supply of water for industry and irrigation is the relationship between the deficits in supply of water 0, and the respective economic losses 2 (Fig. 15.10) which, as a rule, is non-linear. Presuming that the release for drinking water will not be reduced and that maintained minimum outflow will continue, the objective function can be written as
Z
=
+
I
Oll,irCir O,,.illCill= min
( 15.9)
wlicuc O,l,ir(Oll,ill) arc the deficits in supply of water for irrigation (industry), - economic losses caused by the deficits in supply of water for Cir(Gin) irrigation (industry). 15.2.6 Water-management systems for flood control
Flood-control measures in a catchment (region) constitute a single-purpose complex system as these have all the signs of complexity: large scope, stochastic phenomena, a complex hydraulic regime, complex relations. The individual measures (floodcontrol reservoirs, flood-control volumes of multi-purpose reservoirs, flood control effect of active storage, training of rivers, influence of gated weirs, influence of the winter regime, etc.) cause changes which mutually interact. A flood-control system can also be abstracted from a multi-purpose system.
359 The system of reservoirs in a relatively small catchment in southern Bohemia serves as an example. Description of the system The valley of the Stropnice and the Svinensky brook can not be used intensively for farming, as the channel can only hold about 30 to 40% of a one-year flood. The problem is resolved by controlling the flood regime of the two streams with the help of a system of flood control reservoirs. If the plots in this area are to be used for agricultural purposes, then a five-year flood control must be ensured. This control can be ensured by river training or a combination of river training with small flood-control reservoirs; one of the systems to include small flood-control reservoirs can be found in Fig. 15.11. For flood control in the Stropnice catchment (a) Humenice reservoir and ZevlDv pond (b) Humenice reservoir, Zevlfiv pond with water diverted from Humenice to the near-by Zarsky pond. with an unused flood control capacity of 1 . 10" km3, by a 3.5 km-long canal. For flood control in the Suinenskjl catchmeni (a) Kamenna and h n b e r k reservoirs (b) Kamenna reservoir with a 2.75 km long canal to Zbrskl pond and Zumberk reservoir (c) the same as in (b) but with further reservoirs on the Keblansky and DluhoSt brooks (tributaries of the Svinensky brook).
Fig. 15. I I Map of flood control measures in the catchment of the Stropnice and Svinensky Brook
Hydrological data consisted of data on the size of the floods and their form in some of the sites; however, there were no data on the routing of the flood waves in the streams in this region. The problem was resolved in stages: stage 1 : assessment of the isolated flood control of the respective reservoirs; stage 2: assessment of the control of the respective reservoirs when functioning in cooperation (system approach). F l o o d c o n t r o l of individual reservoirs The aim was to ensure flood control without any manpower and with the simplest possible devices, so as to decrease the value .Qsas much as possible whilst making the maximum use of the reservoir’s flood-control capacity, and not to impair the conditions downstream of the reservoir even when Q, > Q5.The reservoirs were designed as dry reservoirs with bottom outlets without gates. They filled as the capacity of the outlets was smaller than the inflow during floods. The profile of the outlets was designed so that with a flow Q5 a reservoir would fill up to the crest of the spillway. Table 15.2 explains the transformation of three designed reservoirs with individual functions; these reservoirs are empty before the floods. Tuhle 15.2 Transformation effect of reservoirs considered independently
[m3 s - l ]
transform. [m’s-’1
10.3
0.67
Pond Zevldv
1.4
Kamenna
8.6
Reservoir
Humenice
QI
transform. [m3 s-’1
Qloo [m3 s - ’ 1
transform. [m’s-’]
19.0
0.9 1
79.3
75.4
0.32
13.5
0.36
56.5
49.2
1.66
15.8
2.14
66.0
66.0
Q5
[m’s-’1
The influence of the reservoirs on a decrease of the maximum peak discharges of selected n-year flood in a series of sites downstream of the reservoirs was also studied. Under the most favourable conditions, Q1decreases on eight profiles on the Stropnice river came to 18.7 - 49.4%of the original value and Qsto 16.0- 49.0%of the original value; on Svinenskg brook Q1 decreased to 31.1 -65.7% and Qs to 24.2 to 64.6%.On the other hand, under the most unfavourable conditions, Qr decreased to 40.3- 88.5% and Q, to 45.6 -86.3% of the original value.
36 1 F l o o d c o n t r o l of reservoirs c o o p e r a t i n g in a system After considering the various alternatives, the optimal flood-control design was selected. It was decided to construct the Humenice reservoir and the Zevlfiv pond without using the % - s k ypond in the Stropnice river catchment. Together the two reservoirs control a catchment of 54.44 km2 and this is practically sufficient to catch Qs,which downstream of the Humenice reservoir decreases to 4.8%, and downstream of the ZevlSv pond to 2.7%of the original value; 17.92 km of the stream is designed to be trained. Kamenna reservoir on the Suinenskjl brook should cooperate with reservoirs on Keblansky and DluhoSt'sky brooks and 13.5 km of the stream should be trained. h m b e r k reservoir, with its small storage volume and its special role in the protection of the environment, was not recommended in the final design. It proved to be inexpedient to include the Zar pond in the system. Canals from the pond to the Humenice and Kamenna reservoirs would be very expensive. The inclusion of the Zar pond in the system would also endanger the very profitable fish farming in this pond.
reservoir proJiscts with river tminr'ng Fig. 15.12 Schematic representation of flood control system in the Stropnice and Svinensky Brook
catchment
A schematic representation of the final design is in Fig. 15.12. The most eficient cooperation of the two sub-systems was recommended on the basis of a detailed analysis. Other examples of the analysis of water-management systems can be found in Votruba et al. (1974, 1988).
362 Besides water-management systems which are concerned with the best utilization of water, there are also systems which are solved mainly in technical (not in economic) parameters. These are, for example: - short-term flow regulation in a system of backwater of weirs (Gabriel, 1975), - temperature regimes in a system of reservoirs, - a complex watermain network (Serek, 1968, 1972). 15.3 FUNCTION OF SMALL RESERVOIRS
The aim of small reservoirs is to ensure release and flood control on small streams, to create a supply of water for common use, to change the properties of the water or establish an aquatic environment for fish of duck farming and recreation. Small reservoirs are usually shallow, with a mean depth up to 4 m (fish farming reservoirs are usually up to 1 m), with an inundated area of up to 100 hectares and a capacity of up to 3 * lo6 m3. The dams are usually up to 10 to 15 m in height and the catchment area is no more than 20 km2 (Pavlica, 1964). As there are many such small reservoirs, they are also of economic importance. They can be divided into ponds and small reservoirs (Cablik, 1960). Ponds are mainly for fish farming and are therefore emptied every year when the fish are being fished out. They can greatly affect the discharge, if they are part of a system, in the upper part of the catchment or in regions with small streams. Small reservoirs serve local needs and can again be classified into : - industrial reservoirs: to supply industry with water; - recirculation reservoirs: to balance the discrepancies in the demand for circulated water in industrial plants; - tire-protection reservoirs, to ensure a sufficient supply of water for fire-fighting; - irrigation reservoirs; - drainage reservoirs: to collect water from the drained plots; - retaining reservoirs: to catch gross sediments and impurities; - recreational reservoirs. Ponds and small reservoirs play an important role in the protection of the natural environment. They are also more suitable than streams for recreational purposes. .
15.3.1 Conservation junction of’ small reservoirs Even though the principle of the conservation function of all reservoirs is essentially the same (to store surplus water for periods where there is a lack of water), small reservoirs have some specific characteristics. They can be filled by either surjuce, ground or waste water. A pond fed by rain or snow water running down from the nearby ground is called
363
a “heaoenly pond”. The inflow is not continuous and occurs only during the spring [m3] can be calculated from snow melt and strong precipitations. The inflow W, = cpFH,
where cp is the surface runoff coefficient, F - the “Catchment” area in [m2], H , - precipitation depth or water value of the snow cover at the start of melting in [m]. Value cp is very variable in the same catchment (from 0 to about 90%)and depends on the soil (permeability, humidity, vegetation) and on the size and duration of the precipitations. Area F is in the order of several km’. Drainage water consisting of a part of rain and snow water has a similar inflow regime. Annual inflow to the “heavenly” pond is very variable, as it depends not only on the annual precipitations, but on its distribution throughout the year and on the intensity and duration of the respective rainfalls. Significant water losses are caused by evaporation and seepage or infiltration. Data on the annual inflow and total expected water losses are used to determine the volume of the pond. Water-management and economic calculations determine the probability of the filling of the pond in a water year. The lower limit of the volume is that which can be filled on average once every two years ( p = 50%). However, the value can be much larger, e.g., with a probability of filling p = 10’4, and even less (Cablik, 1960). Ponds are more frequently filled by surfhce water fi.om streams. Ponds can be either on, or near the streams in which case water is diverted to them by canals. Ponds without through-flow (lateral reservoirs) are more suitable as they have optimal conditions for fish farming and are not affected by floods and sediments. threshold
7 springs
Fig. 15.13 Layout of village pond fed by a pipeline from a brook and from springs
364
The same hydrological data are used to design these small reservoirs as for the large impounding, or lateral, reservoirs. Finding the correct resolution is more difficult, however, as it concerns the hydrology of small catchments which frequently do not have direct water gauge measurements, therefore indirect methods, most frequently methods of analogy, must be used to determine the law of inflow. Sometimes the pond is fed by ground (spring) water. Such an inflow is relatively constant, but rather small; this is why the spring is usually not the only source, but is supplemented by surface water (Fig. 15.13). Wusre water is rarely used to fill a pond, and then only to help treat it. Only those small reservoirs which supply water to industry and for irrigation have any significant storage function. The storage capacity of such a reservoir is usually calculated from the given inflow and release laws by the simulation method. Reservoirs which establish an aquatic environment must preserve the optimal characteristics of the water and replenish evaporation and seepage losses. Such handling of water can be contradictory to the demands of water management. The compensation of water losses in low-flow summer months can further decrease the low flow in the rivers. O n the other hand, emptying of the ponds during autumn fishing can increase the autumn flows dangerously, especially if it takes place at the time of high autumn flows in the streams. O n the other hand the refilling of the ponds can further decrease the winter flow. Therefore even fish-farming reservoirs should have operation schedules that adhere to the water management requirements. Fire-control reservoirs are of small importance as far as storage is concerned. They are usually in villages or near buildings that are to be protected against fire danger. They can be filled by surface or ground water or even from the water mains. The only rule is that the water level should not drop below the necessary limit and that the water should be easy to pump. Fish-farming and other small reservoirs can also serve for fire protection purposes.
15.3.2 Flood-control function of’ small reservoirs Small reservoirs and ponds usually do not have a special flood control capacity under the crest of the spillway. The floods are controlled simply by transforming the flood wave in the overflow space. The measure of decrease of the maximum peak discharge of a flood wave depends on its volume and shape, on the surface area of the ponds and on the length of the overflow crest. The spillway is usually ungated and the width of the overflow jet is small so that Zaruba’s method (Section 11.2) for the transformation of a flood wave can be used. The design flow for the calculation of the spillway is the flood flow with a selected probability of exceedance, QN.The transformation function of a reservoir is reflected in the decrease of the maximum peak discharge only in fairly large reservoirs on relatively small streams.
365 A greater decrease of flood flows can be attained by the cooperation of small reservoirs in a cascade or system. The difference in the function of the cascades of large reservoirs (Chap. 9) and the control function in the cascade of small reservoirs is that in the latter the asynchrony of the phenomena in the reservoirs, i.e., a shift in time in the downstream direction, can be observed more clearly. The reason is the short-term character of the phenomena during the transformation of the flood wave in a small catchment.
a)
Pf 7
0-
CJ-
t
i
-
t Chl
-
0 "
&
t
-€
[hl
Fig 15.14 Transformation of flood waves In in a cascade (series) of small reservoirs with a synchronous and asynchronous function ( u ) discharges in pf I : ( h ) discharges in pf 2 ; (c) layout of the reservoirs
Figure 15.14 illustrates the transformation of a flood wave in a cascade of two small reservoirs N , and N , in points pf 1 and pf 2 on a small stream. Non-transformed floods are denoted by simple triangles ( Q , = &(t) and Q2 = J;(t));it is assumed that a reservoir is full up to the crest and water only runs over the ungated spillway. Figure 15.14a illustrates the course in time of a non-transformed flood wave Q , (inflow to the reservoir) and a transformed flood wave 0, (outflow from a reservoir) in point pf 1.
366
Figure 15.14b illustrates the course of a non-transformed flood wave Q , prior to the construction of reservoirs, two inflow curves Q ; and QY"' into reservoir N , , with the considered flood-control effect of reservoir N and the respective outflow curves 0; and O\+A'. Curves Q\ and 0; are constructed on the premise that the function of the two reservoirs N , and N , is synchronous, i.e., presuming that the difference ( Q , - 0 , ) at moment t manifests itself at the same moment by the same difference (Q, - Qi) = = (Q, - 0,). Thus, a modified inflow Q\ to reservoir N , is obtained from which the transformation wave in reservoir N , can be calculated by the known method; the result is the course of outflow 0; from reservoir N , . Curves Q;'"' and O;+A' are constructed,, presuming that the functions of the two reservoirs N and N , are asynchronous, with a shift in time; i.e., that the difference (Q, - 0,)at moment t manifests itself by the same difference(Q, - Q;) = (Q, - 0 , ) in reservoir N , at the moment t + At, i.e., with a time lag of At. Thus, the modified inflow Q\+" to reservoir N , is obtained and from this the transformation of the flood wave in N , is calculated f the result is the course of the outflow O>+A'from reservoir N , . From Fig. 15.14b it is possible to see that the difference between the curves Q; and Q;'"' and between curves 0: and O;+A' is significant. This is obvious from the and O .,:,:; In our course of the curves and their peak values: Q,: and Q\;$,;O:,, case, the time lag of the reservoir function, and its inclusion in the calculations, was reflected in a decrease of the maximum peak discharges. However, there are innumerable combinations of the final effects of the two reservoirs N , and N , and the results can differ greatly, depending on - the volume and shape of the flood waves in the two points and on their mutual times, - on the inundation area of the two reservoirs and the dimensions of the spillways, - on the distance (time lag) between the two reservoirs. The flood-control function of a cascade of such reservoirs therefore cannot quantitatively be assessed sufficiently accurately, and it must be calculated in great detail. If the final results are to be optimized, the optimization method, which is essentially very simple, must be applied to many alternative solutions. The flood-control function of a system of small reservoirs can have a great impact on the river during the flood wave. The effect of the system of reservoirs must be determined by a systems approach (Section 15.2.6). Calculations with technical parameters are rather simple, but they become more complicated by the transition from a deterministic problem to a stochastic problem. Optimization of the structure and the behaviour of the system can be rather demanding if the system is a large one, as it requires many alternative solutions. Economic optimization is even more dlfficult as it is not easy to obtain all the necessary economic data for the calculations of costs and flood damages. Uncertainty increases in dynamic systems with different time horizons.
,
,
367 15.3.3 Function of' the aquatic environment of' small reservoirs
Small reservoirs are mainly used for fish or water fowl farming, recreation and aesthetic improvement of the environment. They are usually not important as far as water management is concerned. Fish-furminy reservoirs are demanding as to the physical and chemical characteristics of the water. Warm-wczter pond cultivation for carp breeding must have water with sufficient plant nutriments, with a summer temperature of 20 to 30 "Cand with a neutral or slightly alkaline pH. The most suitable ponds are those rich in organic matter, but without mud. Cold-water pond cultivation which mainly produce trout, grayling, huck must have water with a low organic-matter content, a high oxygen content and a low temperature (in summer a maximum of 16 to 20 "C). Recreation reservoirs must have clean water and a constant water level. Inflow must be suficient, especially in the summer low-flow period. Outlets should allow discharge at the top and the bottom so as to enable the regulation of the released water quality. The banks of the reservoir should be adjusted for sunbathing and games. Sanitary facilities and parking lots ensure the good quality of the water. Recreation reservoirs play an important part in the human environment. Lateral reservoirs are best suited to ensure the good quality of the water. However, many large impounding reservoirs are also used for recreation, and some impounding reservoirs are even built purely for such purposes. The use of ponds for recreation might seem contradictory. Thirty years ago, recreation could be included as one point in the general uses of water, but today it is a strong sociological phenomenon and recreation on the banks of ponds has become a special type of water use. The contradiction between fish farming and recreation lies in the demands on the quality of the water, but even large ponds can be fertilized in such a way as to meet the demands of holiday makers, who want clean water and a pleasant environment (Lavickg, 1969). Reservoirs can also form a part of architectonic layouts, parks, etc.; these are usually shallow, or formed by low dams on brooks.
16 ECONOMIC EFFECTIVENESS OF RESERVOIRS Reservoirs are a part of the economic potential of any country, and as they are capital investment constructions they must be evaluated as such. An objective measure is to determine how they help to raise industrial productivity, which is done with the help of quantitative economic indices. Some capital investment consequences can only be estimated qualitatively and not quantitatively. However, the qualitative evaluation is of great importance for reservoirs. Intangible effects can lead to a choice of an alternative, which is less effective economically, but more advantageous socially. 16.1 EVALUATION OF THE EFFECTIVENESS OF RESERVOIRS
The same needs can often be met by different (alternative) measures. The most effective option is determined by a comparison of technical and economic indices of all implementuble mid interchangeable options. The needs can be met by various resources or by a different exploitation of the same resource. All losses and benefits must be taken into account. How the water demand from a certain source is met depends on the natural conditions and on the parameters of a reservoir, i.e., its dimensions and methods of operation. These are derived from a synthesis of technical and economic calculations, including quantitative indices and qualitative characteristics. The effectiveness of capital investments is determined by a complex comparative analysis of all decisive factors which influence the demands and effect of the capital investments. Characteristic for reservoirs and dams is - long service life, - relationship with other branches of the economy and with the human environment, - functions in large and complicated systems, - multi-purpose use, - possibility of construction in stages. These characteristics influence the evaluation of their effectiveness; the methods used are:
369
- evaluation of alternatives by the method of comparative effectiveness, - evaluation by the method of total effectiveness, - evaluation of the alternatives by the method of decision analysis. Basic t e r m s The ejfectiveness of a capital investment is the sum of the effects by which the investment contributes to the optimal structure of the material-technical foundation of a society and the meeting of that society’s needs. Economic effectiveness is the relationship between the summary economic demands and the economic effects created by them. Intangible effectiveness reflects the inlluence of capital investments on cultural and social needs, and the improvement of the environment. The effects of the investment are the economic and intangible results arising from its construction. The demand of the capital investment is the sum of the demands on all sources and services in the non-economic sphere which are connected with the acquisition and operations of the basic tools. Investment costs are the sum of the costs covered by capital investments and operation resources made in connection with the construction. Direct investment costs are costs included in the capital investment. Derived investment costs are the costs connected with the acquisition of the capital investment, which will be administered by other investors and must be made in the construction of the given capital investment. Indirect investment costs are the costs that must be made in connection with the construction under consideration in other branches, to ensure the operation of the construction.
16.1.1 Evaluation of the effectiveness of reservoir construction by the method of comparative (relative)effectiveness
The method of comparative effectiveness should be used for the comparison of alternatives, which qualitatively and quantitatively fulfil the same aims at the same time, and are therefore interchangeable. Economically the most effective option is the one that has the smallest sum of annuity or transferred costs, which serve as evaluation criteria. The index of transferred costs P transfers OMR (operation, maintenance and repair) and investment costs to a form that can be added up by one of the following methods :
P = k,J + P, = min where P is the transferred costs of the ith alternative J - investment costs of the ith alternative (a)
!
(16.1)
Pp - OMR costs of the ith alternative k, - standard coefficient of economic efficiency of investments (e.g., k, = 0.1). This method should be used if the construction time is short (about 1 year) and if the OMR costs are constant throughout the service life. (b)
PT = k, P V ( J ) + P,
min
( 16.2)
370
where PT is the modified index of transferred costs P V ( J ) - the present value of investment costs and whereby rc PI/(J)= ~ ~ ~ r ( r c + p - f i J,r-r
5
T= -Tcp+ 1
j= 1
( 16.3)
'&
- construction time prior to the start of operations, - time of the service life of the capital investment in years, T, - construction time in years, T, - investment (j-th year of construction) costs in the T-th year of J, operation, T - years of operation, - years of construction, j r - standard time factor (e.g., r = 1.1) r = 1 + d; d is the discount factor. This modified method should be used for options with different construction times T, or with different distributions J in the respective construction years, expressed on the time axis of years of operations. It is presumed that the OMR costs of the options are constant during the servicelife, or at least that they change in the same way. '
+
Pa = k,[PV(J) PV(P,)] = D M ( J ) + DM(Po) I min (c) are the annuity costs during the service life, where Pa PO - OMR costs, - discount mean in terms of the relationships, DM
DM(J) = k, PV(J) DM(Po) = k, P V(Po)
k,
( 16.4)
(16.5) (16.6)
- coefficient of economic efficiency of investments (capital recovery factor), which is a function of r and T, in terms of (1 6.7)
Values k, for r = 1.06 and 1.1 for various T, values can be found in Table 16.1. This modified method should be applied if the options (alternatives) differ not only in their investment and OMR costs, which change in different ways with time, but also as to their service life. Note I . Provisions of the Czech Ministry of Investments prescribe uniform values of the standard time factor and efficiency coefficient: r = 1 . 1 ; k , = 0.1; k , = capital recovery factor for r = 1.1 and the respective time K.
37 1 Tubk 16.1 Values k, for various values 7 with r = 1.06 and 1.1
7, [years]
1
2 3 4 5 10
k, for r 1.06 1.1
1.06000 0.54544 0.374 1 1 0.288 59 0.23740 0.13587
1.10000 0.576 19 0.402 I2 0.31547 0.263 80 0.16275
T. [years]
15 20 25 30 35 40
k, for r 1.06 1.1
0.10296 0.087 19 0.07823 0.072 65 0.06897 0.06646
0.131 47 0.11746 0.110 17 0.10608 0.103 69 0.10226
T, [years]
50 60 70 80
90 100
k, for r 1.06 1.1 0.06344 0.061 88 0.061 03 0.06057 0.06032 0.06018
0.10086 0.10033 0.100 13 0.10005 0.10002 0.10001
As reservoirs have an exceptionally long service life and are of great importance for the whole sociciy, the Ministry of Water Management issued orders that for the comparison of the options the values should be: r = 1.06;k, = 0.06: k, = capital recovery factor for r = 1.06 and the respective time T,. Norr 2. In the USSR methods for determining the economic efficiency of investments were a d o p i d in 1969. According to these, the standard coeficient of economic efficiency of investments k, is in the range of 0.10 to 0.33, i.e., the time lo recover the costs is between 10 and 3 years. Zarubayev (1976)determined the following values:
Complex water management projects Navigation Irrigation Drainage Fish farming
0.10 0.10-0.15 0.17-0.33 0.1 1-0.25 0.17
It is recommended that a time factor be introduced. which for most water management projects is
5 k,.
Method: 1. Investment and OMR costs are distributed over the respective years of a time series. If Po values are constant, they can be calculated for one year only; if they change in all options in the same way (only in relation to the value of the capital investment), they can be compared only for the fina! year; if Po increases annually by the same amount, the marginal flow factor Z , should be introduced:
( 16.8) 2. The characteristics of the differences between the options is estimated. and a criterion selected; the values are calculated from equation (16.1) to (16.7). 3. The costs in the respective years are multiplied by the respective investment-rate factor; they are added for every option and the difference between those sums is determined.
312
Evaluation of t h e results Economically, the most expedient is that option which has the minimum sum of transferred or annuity costs. However, when choosing the optimal option secondary economic as well as intangible factors must also be taken into consideration. It must also be borne in mind that investment and OMR costs can be flexible. If the effect of interchangeable options differs only slightly (up to 5%), the specific costs (transferred or annuity) are calculated per unit of production (amount of water or energy supplied annually).
Table 16.2 Economic comparison of two alternatives of a reservoir with dam
Index
Total investment cost J (mil. KEs), where J is divided into the years of construction 1978 1979 1980 1981 1982 Number of workers Average annual wage ( K b yr- ') Life span (years) Discounted investment costs (mil. KEs) Factor r' : Year of construction: 1978 1.262 48 1979 1.191 02 1980 1.13460 1981 1.06000 1982 1.OOO 00
OMR costs (mil. Kfs yr-I): wages electricity maintenance, etc. Annuity with capital recovery factor: k,,,, = 0.061 03 k1.80 = 0.060 57 Total costs (mil. KEs yr-')
Number of workers greater smaller
400
600
50
100 150 I50 150 50
100
100 100
50 400
100
30 OOO 80
32 OOO 70
63.12 119.20 113.46 106.00 50.00 45 1.68
126.25 178.65 170.19 159.00 50.00 __ 684.09
12.00 4.00 3.00
3.20 5.00 4.00 41.75
27.36 46.36
__
53.95
373 Problem 16.1 The more advantageous of two alternative reservoirs with dams which differ greatly in their need for manpower during the construction is to be selected. The saving in manpower is balanced by higher costs for machinery and the resulting higher wages for skilled workers. It is presumed that both options have the same utility and therefore only their respective costs are compared differently. Calculations are given in Table 16.2. The results of the comparison show that the machinery was very expensive so that the option with a greater need for manpower would be the cheaper of the two. Even if the service life of both the options were T, = 80 years, the result would change slightly more in favour of the “machinery” option. With a total of capital investment for the second option of J = 500 mil. Kfs, equally distributed over the years of construction and with the same OMR costs, the costs come to a total of 46.60mil. Kts yr-’, so that the first and the second option are practically equal. When using a higher standard value, r = 1.1, the first option without the introduction of new technology, would be more expedient.
16.1.2 Evaluation of the effectiveness of reservoir construction by the method of total (absolute)effectiveness
The method of total effectiveness is used to evaluate the economic effectiveness of the final choice derived by the method of comparative effectiveness (Section 16.1.1) or in estimating options with greatly differing effects which can be expressed in the form of costs and prices. This method is suitable to judge projects of - single-purpose reservoirs for the supply of surface water or the production of electrical power, - multi-purpose reservoirs, - irrigation or flood control, - supply of drinking water or utility water, including the resource, treatment, transport and distribution of the water. The following conditions must be fulfilled: - the capital investments must form an independent operational unit (otherwise the evaluation has to be extended to a set of functionally related investments), - it must be possible to determine clearly all demands necessary to attain the required effect and to determine the entire effectivity of the investment, - it must be possible to quantify the demands and effects in technical and monetary units in valid prices or their equivalents.Examples of the calculations for four alternatives are given in Table 16.3. The most effective of the options I to IV is, according to the criterion of the maximum of the summary effect,option 11. The optimum will be between options 1-111. Figure 16.1 plots the relationships SE = f(0,) and S E = /’(. which I), are similar. It is clear that the calculations of the four options do not suffice to determine the optimum, as four points still leave much licence as to their connection (see the dashed branches u, 6 and a’, b’). It follows from line 24 of Table 16.3 that the locality of the reservoir is not very suitable, as with the chosen standards ( r = 1.06) and with the price of water at 0.46 Kfs m-3, S E , , , , hardly reaches positive values. There is no doubt that with r = 1.1, S E , , , would be negative in all cases.
374 Table 16.3 Optimization of reservoir storage capacity for public water supply Line Index
Alternative I1 111
Unit
I Yield 0, m's-' 1.2 mil. m3 yr - ' 37.84 Water supply capacity Amount of water supplied 2872.48 mil. m' during T. = 80 34.01 mil. m' yr-' 3a Average during one year Benefits W for water supplied with price of water 0.46 Kfs . m-' 4.60 mil. KEs yr-' - 1st year 9.20 mil. KEs yr- 6th year mil. KEs yr-' - 11th year 17.41 mil. Kfs yr- ' final year (final year) (year of operation) (9) I'resent value of bcncl'its P V( W ) 229.67 9 When r = 0.6 mil. KEs 13.91 10 Discounted average of benefits mil. KEs yr- ' II Investment costs J mil. KEs 250 12 - direct costs mil. KEs 210 13 - derived costs mil. Kfs 30 14 Construction period years 4 15 Present value J [ P Y ( J ) ] mil. K f s 274.05 16 Depreciation ofdirect costs(A) mil. Kfs yr-' 2.63 17 O M R costs (Po) mil. Kfs yr- ' I .47 18 Total production costs ( N P ) mil. KCs yr-l 4.10 I9 Present value of Po [PV(P,)] mil. Kfs 24.27 20 Annuity of P V ( J ) mil. K f s yr 16.60 21 Annuity costs (NA) mil. Kfs yr- I 18.07 22 P V ( J ) . k , (for k, = 0.1) mil. Kfs yr-' 27.41 23 Transferred costs PTV = k, . P Y ( J ) + P,, mil. KEs yr- ' 31.51 24 Total investment effect when r = 1.06 S E = P V ( W ) - P V ( J ) - PV(P,,) mil. KEs -68.65
3.4 94.6 I 4628.76 57.86
I
'
Explanation to Table 16.3
'
4.0 126.15
IV
5.5 173.45
8731.30 1 I 546.65 109.14 144.33
4.60 9.20 23.00 43.52 (16)
4.60 9.20 23.00 58.03 (19)
442.3 1 26.79 350 300 36
533.61 32.32 500 440
5
407.27 3.75 2.10 5.85 34.67 24.67 26.77 40.73
5 583.99 5.50 3.08 8.58 50.85 35.37 38.45 58.40
639.36 38.73 700 600 72 6 834.13 7.50 4.20 11.70 69.34 50.52 54.72 83.41
46.58
66.98
95.1 I
+ 0.37
- 101.23
-264.11
40
4.60 9.20 23.00 79.79 (24)
is calculated from line 1 by multiplying by 31 536 OOO s yrthe actual amount supplied is calculated, i.e., up to the complete depletion of the reservoir's capacity (linear extrapolation between the years 1-6,6-11 and 11-16) and then up to T, = 80; full capacity line 3a: from line 3 by dividing by T. = 80 lines 4-7: the actual amount supplied is multiplied by the price of water 0.46 KEs m- ' line 2: line 3:
315 Tuhk 16.3 (continued)
ordinal number of the year from the start of operations in which the reservoir capacity is fully exploited for the first time the actual annual incomes are converted to the present values by multiplying by r-' line 9: line 10: from line 9 by dividing by z, = 16.509 13 for T. = 80 line 15: direct costs (line 12) are distributed over the respective years of construction, updated to the start of operations und added to the derived costs line 16: from line 12 by multiplying by 0.0125 line 17: from line 12 by multiplying by 0.007 line 19: from line 17 by multiplying by z, = 16.509 13 for T, = 80 line 20: from line 15 by multiplying by k,,,, = 0.060 57 line 21 : from the sum of lines 15 and 19 by multiplying by k,, 8o = 0.06057 line 23: sum of lines 18 and 22 line 8:
\%
Fig. 16.1 Selection of optimal parameters of a reservoir from the graph of SE,
As the capacity of the reservoirs differs greatly, the values of the specific indices of the respective options were calculated in Table 16.4. According to these, the best option is either I1 or 111. According to all indices of specific costs the least suitable is option I, even though according to SE, .06 it follows just after option 11. In the given case all specific costs in the final year are lower for option 11, while for the whole service life of a reservoir they are lower for option 111. The differences are not as great as for SE. If the reservoir is in an area with little water yield and therefore the price of the water is higher, optimization can be repeated for this higher price. If, for example, the price were double, the summary effect of option I1 would be 442.68 and of option 111 432.38. It is clear that the optimum is between option I!
376 and Ill. After adding the secondary elfects, the difference between the summary effects would further decrease to 455.1 1 and 449.29. If other reservoirs are to be built, the average discounted productivity of investment costs (using valid prices) is even lower and does not reach the ARDJ > 5% as in option Ill (seeTable 16.5), it would be justified to recommend option III., especially if also used for power production. Tuble 16.4 Specific indices for various reservoirs for water supply
Index
Unit
I Specific J for 0, Specific N V for m3 of water supplied in final year Specific N V for m3 of average water supplied Specific annuity costs - for m3 of water in final year - for m3 of average water supplied Specific transfered costs - for m3 of water in final year - for m3 of average water supplied
mil. KEs m-3 s + l
208.3
Alternative I1 111
116.7
125.0
IV
127.3
ha1 m-’
10.84
6.18
6.80
6.75
ha1 m-’
12.04
10.11
7.86
8.11
ha1 m-3 ha1 m-3
45.32 50.35
28.30
30.48
46.27
?5.23
31.55 37.91
ha1 m-3 ha1 rn-’
83.27 92.51
49.23 80.50
61.37
53.10
54.83 65.90
1 ha1 = 0.01 KEs
Table 16.5 Comparison of gross productivity and mean annual discounted productivity
Index
Unit
I
Alternative I1 111
1v
Gross productivity
Yn
6.38
11.83
10.99
10.80
Mean annual discounted productivity
%
4.54
6.06
5.01
4.14
Economic effectiveness of t h e o p t i m al o p t i o n If we want to compare the advantages of two localities for reservoirs which are to supply water to the same region, we must find an optimal solution for every locality (Table 16.3 and 16.4) and for these we calculate the indices of the effectiveness of the investment
377
(a) The summary discounted productivity of investment costs SRDJ is calculated from the equation PV(W) - PV(P,)
SRDJ =
(16.9)
PV(J)
where PVis the present value: W - benefits; Po - OMR costs; J - investment costs. (b) The mean annual discounted productivity of investment costs ARDJ is given by the relationship ARDJ =
100[DM(W) - DM(P,)]
WJ)
= 100k,-SRDJ
(16.10)
where D M is the discouned mean: equation (16.5) and (16.6). (c) The mean discounted (reproduction)productivity of direct costs 8 is given by VRDP, =
100[DM( W ) - DM(NP)]
(16.11)
d‘
(d) The rate of reproduction recovery TJR is given by TJ TJR = 100-
(16.12)
T,
where TJ is the reproduction recovery in years. Note: if the net benefit from the investment ( V = W - Po)is constant throughout the service life, then TJ
J
=
(16.13)
w, - Po,
If the net benefit changes, the calculations have to be done in steps for the respective years (see Table 16.6). (e) The demand on the indirect investments ZZ is given by the relationship 11 = ’d. ‘d,
indirecl
(16.14)
direct
P r o b l e m 16.2 We use the values of option I1 in problem 16. I (a) according to equation (16.9) SRDJ
=
442.31 - 34.67 402.27
= 1.0009>
I
(b) According to equation (16.10) ARDJ = 100.0.06057. 1.OOO9 = 6.063% > 6%
378 Table 16.6 Calculations of reproduction recovery of reservoir TJ (problem 16.I, alternative 11)
Year Benefits
(W)
Annual values (mil. KEs yr- ') OMR costs (NP)
Net benefits (W - N P )
Balance J (mil. KEs)
350.00
4.60 5.52 6.44 7.36 8.28 9.20 I 1.96 14.72 17.48 20.24 23.00 27.60 32.20 36.80 4 1.40 43.52 43.52 43.52
1
2 3 4 5 6 7 8 9 10 11
12 13 14 15 16 17 18
TJ
=
17
2.10 2.10 2.10 2.10 2.10 2.10 2.10 2.10 2.10 2.10 2.10 2.10 2.10 2.10 2.10 2.10 2.10 2.10
2.50 3.42 4.34 5.26 6.18 7.10
9.86 12.62 15.38 18.14 20.90 25.50 30.10 34.70 39.30 4 I .42 41.42 41.42
347.50 344.08 339.74 334.48 328.30 321.20 31 1.34 298.72 283.34 265.20 244.30 218.811 188.70 154.00 114.70 73.28 31.86 0
3 1.86 + 41.42 __ = 17.77 year
Let us compare ARDJ of all four options of problem 16.1 with the benefit-costs ratio which is given by the relationship VJ =
-
loo(%
J
(16.15)
where index c denotes the values of the final year. The results are given in Table 16.5. They clearly show that the benefit-costs ratio, calculated from the values of the final year, over-estimates the effectiveness of the investment, in our case quite substantially. (c) According LO equation (16.1 I):
V R D Pd
-
-
(442.31 - 34.67).0.060 57 _____-___ loo = 8.23?, 300
379 (d) The reproduction recovery TJ (Table 16.6) is calculated and it comes to TJ = 17.77 years. Then according to equation (16.12)
TJR
=
17.77
100-
80
=
22.217';
(e) According to equation (16.14):
36 I I = - = 0.12
300
The role of secondary effects on the optimization of the capacity of a reservoir Even a single-purpose reservoir can have favourable secondary effects which help to improve its total effect. Favourable effects can be, for example, recreational facilities, influence on the surrounding environment, new buildings which replace those in the inundated area, etc. Effects, however, can also be unfavourable, e.g., cold water downstream of a reservoir in summer, limitation of new constructions in the catchment of the reservoir. Any secondary effects are estimated in valid prices or their equivalents from other branches. However, each case requires an individual analysis. Table 16.7 Inclusion of secondary effects in the optimization of the water supply capacity
Line Data
Unit
I
Presumed number of holiday-makers Benefits from recreation facilities:
2 3
-
in final year (already 1st year) - present value of benefits
thousand visitors
___
-~
days per year
30
Atternative I1 111
40
IV
50
70
mil. KEs y r - ' mil. KEs
0.45 7.43
0.60 9.91
0.75 12.38
1.05 17.33
mil. KEs
5.0 0.30
7.0 0.42
10.0 0.61
15.0 0.91
mil. KEs
12.43
16.91
22.38
32.33
Improvement of the state of buildings:
4 5
6 7
- present value of benefits - annual equivalent of benefit Total present value of secondary effects Total effect of iinrstment including~secondaryelTects
mil. KEs
Note: Line 7 is the sum of lines 6 and 24 of Table 16.3
-56.22
17.28
--78.85 -.231.78
380 Let us look at how the indices of the four options (l-1V) of our problem change, if we include in the calculations of the water supply and the income for water, secondary effects such as recreation facilities and the improvement of the present state of some buildings. (Tab. 16.7). From line 7 in Table 16.7 one can see that the summary effect of the investment improved by including secondary effects; however, this did not change the order of the options. To cause any change, the secondary effects would have to be much more extensive and there would have to be a greater difference in the respective options.
16.1.3 Evaluation of alternatives of reservoirs by the method of’decision analysis
Decision analysis is a method of selecting the optimal alternative; the decision must consider all economic as well as intangible effects. The process is a formal objective one; however, the evaluation of the respective aspects includes subjective judgements. The decision process consists of - the determination of the criteria to estimate the extent to which the required aim was reached, - a simple evaluation of the options, - a mutual comparison of the options, - a weighted evaluation of the options. This process results in the determination the hierarchic sequence of functions. The criteria are to reflect the essential functions of a reservoir and should be independent of one another, i.e., they should not overlap. The criteria for reservoirs must express their costs, economic impacts, relationships with the environment and cultural monuments, conditions of construction, operational reliability, etc. The selection of the criteria is the most responsible stage of the decision-making process ; it frequently has heuristic characteristics; it requires a creative approach and is specific for every reservoir. Criteria t h a t should be used in selecting a n o p t i o n 1. Cost criteria: (a) investment (b) OMR (operation, maintenance and repair) Investment costs can be compared with the given financial limit as a whole or for a certain period. 2. Perjormance criteria: (a) according to the quantity (amount of water supplied, hydro-power, navigation, flood control, etc.), (b) according to the quality (reliability of water supply, water quality, etc.), 3. Criteria for the injluence on the human environment: (a) protection and creation of a natural environment (good and bad influence on the environment and respective measures, elimination of the fear of floods, flood
38 1 damages or fear of lack of water, improvement or deterioration of sanitary conditions, etc.), (b) exploitation for leisure time and water sports, (c) confiscation of farm and forest land, (d) moving of people living in the inundated area, (e) protection of cultural and technical monuments. 4. Criteria of economic efjiciency: (a) the present value of investment costs and OMR costs or transferred costs (Sections 16.1.1 and 16.1.2); (b) mean discounted reproduction productivity (Sections 16.1.1 and 16.1.2), if can be expressed in terms of money. 5. Criteria of construction which favourably or unfavourably affect the reservoir construction and are not reflected in the investment costs: (a) construction time, (b) conditions of foundations not determined by geological and geotechnical research, (c) dependence of the construction on deliveries from abroad, (d) requirement on the accuracy of work and technological discipline, (e) difficulties of construction due to weather conditions, transport, lack of local manpower, topography of the building site, etc., (f) use of prefabs, (g) utilization of the suppliers' resources. 6. Criteria of operations: (a) reliability of operations and ability to overcome any breakdowns, (b) maintenance and repairs, (c) scope of limitations during reconstructions and repairs, (d) influence of weather conditions on the reservoir's operations. 7. Criteria of development: (a) harmony with other water-management projects, (b) adaptability to future changes in the demands on a reservoir, (c) conceptual reliability, i.e., ability to overcome breakdown situations in a system, (d) extent and time consistency in the ulization of a reservoir. 8. Other criteria:
(a) conflicts of interests with other branches (use of fertilizers in farming, cattle breeding, etc.), (b) state of readiness (territorial, planning, project, from the point of view of the supplier), (c) international character of a reservoir, with the participation of other countries, (d) defence, (e) agreement with the political aims for the development of the region.
382
The criteria that are chosen, are those that best characterize the efficiency of the evaluated alternative. Those criteria which do not differentiate the alternatives can be eliminated from the evaluation. Tucrblc 16.11 Simple and weighted estimation of two reservoirs
Weight
Criterion
I. investment costs 2. amount of water supplied 3. protection of the environment 4. recreation facilities 5. flood control 6. confiscation of land 7. people to be moved 8. economic effectiveness 9. time of construction 10. construction difficulties I I . agreement with water-management plans 12. competitive goals
10 7
n
3 9 5 4 I1 2 1
12 6
score
Estimation of alternative I simple weighted
Estimation of alternative I1 simple weighted
1 4 4 5 5 3 2 4 2 2 5 2
10 28 32 15 45 15 8 44 4 2 60 12
4 2 5 2 2 2 5 3 4 3 2 4
40 14
39
275
38
240
40
6 in 10 20 33
3 24 24
Ttrhlc, 16.9 Comparison of the weight of the criteria in the Fuller triangle
2
I
3
4
5
I
I 2 3
I I I ~ 5 2 2 8 5 3 3 8 5 6 7 8 4 5 5 8 6 8
3
6
7
8
n
9
10
II
12
I I I I 2 2 II 2 3 3 II 3 4 1 1 1 2 5 5 II 5 I1 12 6 6 7 7 II 12 8 8 1 1 8 9 II 12 II I2 I1 -
Criterion
1
2 3 4 5 6
I 8 9 10 I1 12
Number of advantage5
9 6 7 2
n
4 3 10 1
0 I1 5
383 Problem The water resources of an area are to be supplemented by a further resource of surface water. Two localities are considered and their utilizations optimized independently of each other. The most suitable locality should be determined. As the two reservoirs also have an extensive intangible impact, the decision analysis method is used. Srep f : criteria are chosen (a total of 12). which best characterize the differences in the evaluated options and do not overlap (Table 16.8). Srep 2: a simple evaluation of the options is carried out, by giving each criterion a certain number of points on a scale, e.g., from 0 to 5, whereby 0 is the lowest evaluation and 5 the highest. By adding up these points a simple score is obtained for each option (Table 16.8). Evaluation by points is a matter of qualified estimation with a certain element of subjectivity. It is less serious to estimate one of the criteria incorrectly than to leave out an important criterion. According to this simple score, option I appeared to be the more suitable; however, the difference of only one point is insignificant. From this estimation it is clear that two reservoirs are bcing compared, of which one is very large and the other much smaller. Even though the result of the comparison is uncertain, it draws attention to the most important criteria of the problem, mainly 1, I 1 and some others. Srep 3: A mutual comparison of all the criteria is carried out with the help of the Fuller triangle (Table 16.9). The result of the estimation of two criteria is then written at the point of intersection of the line and column with the number of the compared pair of criteria, by writing down the number that is given preference. The preferences gained for each criterion are then added and their number written in the last column of the table with the respective number of the criterion. The number of preferences gained characterizes the weight of the criterion. The greatest weight ( I 1) was ascribed to broader water-management aims (criterion I I), followed by economic eficiency (10 for criterion 8). investment costs, etc. This estimation in pairs is also subject to personal opinions. Srep 4 : Weighted estimation of the options is gained by the multiplication of their simple estimation (Table 16.8) by the weight of the respective criterion (Table 16.9) increased by one (to eliminate the possibility ofzero weight). The sum of all the weighted values of the criteria ofeach option gives its weighted score; the bigger one determines the more favourable of the two options from the point of view of the criteria that were used (Tab. 16.8). Option I proved to be the most advantageous by 35 points, i.e.. by about 15n4. The final decision will depend on the attitudes taken by various authorities concerning the decisive criteria 11, 8 and I. Watermanagcment authorities will prefer option I, with respect to criterion 1 I , while the authorities deciding the financial means will be inclined to support option II with respect to criterion 1.
Evaluation of the method of decision analysis The method has an objective part (formalized process) and a subjective part (selection and evaluation of criteria, simple evaluation of the options), which is an advantage, as well as a weakness, of this method. The advantage is that the formal process is the same for everyone and that subjective estimation requires a more profound information about the respective options. The weakness is that the objective rules need not necessarily reflect the proportions of the advantages correctly and that the results are greatly influenced by the subjective part. This influence can be limited by the proper choice of experts that can
384
ensure a comprehensive view of all the elements evaluated. Where the opinions of experts differ, open discussion can clarify the reasons for the different opinions. The main advantage of the decision analysis, however, is that it exceeds the limits of purely technical and economic estimations a.nd that intangible factors can be introduced. In spite of these advantages, decision analysis should not be expected to produce a final decision. However, those that are to make the decision will have valuable material on which to base it. 16.2 COST DISTRIBUTION OF MULTI-PURPOSE RESERVOIRS
Reservoirs usually serve several purposes and therefore investment and operation costs are divided among the respective uses (components of the water management complex). This is necessary to estimate the economic efficiency of every use of the reservoir correctly, to determine the technical and economic indices of the respective uses and to evaluate the whole system (water management, power, etc.). There are many methods by which this complicated problem can be solved (Matlin, 1961; Shchavelev, 1961, 1966). It is not simple to distribute the investment and operation costs according to the respective uses according to their proper contribution to the national economy. Interesting methods have been elaborated by the Leningrad Polytechnic Institute (LPI) and by the Ministry of Forestry and Water Management of CzechoslovakiaMLVH (1976). 16.2.1 The LPI method
The LPI method is based on the following conditions (Zarubayev, 1976): 1. The economic efficiency of a reservoir for the respective uses is determined by comparing the costs for two possible cases: (a) for shared costs, (b) for the realization of the most suitable interchangeable options, which would have the same effect. 2. The options compared must render the same production or services. 3. The economic efficiency of a reservoir for the respective users is determined on the basis of the time needed to repay the additional costs invested for this use, as compared to its alternative solution. 4. The costs of a multi-purpose reservoir are divided among the respective users in proportion to the economic efficiency of each use. All investment costs for the basic parts of a reservoir are divided between joint costs and special costs. Joint costs serve several users: these include dams, reservoirs,
385 large canals, etc. Special costs serve one user: lock chamber, building of the power plant, fish-passing facility, etc. The efficiency of a reservoir is determined for every user by comparing the costs of two alternatives. For example, the efficiency of navigation compared with rail and road transport. The comparison must be based on an extensive analysis of all natural, political, social, economic and technical factors.,Other costs connected with a reservoir must also be taken into consideration, e.g., transfer of electrical power, transport of water, etc. The repayment period for the ith use is determined from the equation (16.16) where J i
are the investment costs of the ith use of the whole scheme, Jis are the investment costs needed for the operation of the ith use, Jiz are the investment costs of the interchangeable alternative for the ith use, Po,,; Po,izare the OMR (operation, maintenance and repair) costs of the ith use of the whole scheme (other facilities needed for its operation; its interchangeable alternatives). The repayment period of a whole scheme with n uses is n
n
C
(Ji i=1
T=
+ J i s ) - iC Jiz = 1
(16.17)
The index of transferred costs is frequently used as the minimum criterion
P
=
k,,J
+ Pp = min I
[equation (16.1)]
If the investment costs are distributed unevenly over the respective years of construction and if the OMR costs change, equation 16.1 can take a more general form: r
Pr = k,
j= 1
Jjr7-J
+
7
Pp,jr7-J
(16.18)
j=bp
is the year to which the costs are brought up to date (e.g., 7 = q), the year in which operations start; Other symbols are as given for equations (16.1) to (16.7). If we select for t the year of the beginning of operations z = zop = T,, equation (16.18)can be written as equation (16.2).
where
t
topis
386
For a multi-purpose reservoir with n uses, equation (16.18) will take the form of n
r
(16.19) The productivity of the reservoir can be derived from the relationship SRDJ =
PV(W) - PV(P0)
WJ)
[equation (16.9)]
The costs can be distributed proportionally between the respective components of the whole complex: 1. according to technical indices (reservoir volume, amount of water supplied ;ind used for individual purposes); 2. according to the scope of the contribution of each use; 3. according to the economic eficiency of the measures introduced for a certain use. The third condition led to practical recommendations: Every ith use of the whole complex has the following investment costs J, and OMR costs Po,,: Ji
= Jjoint
- Pspec,i + Jspec,i cpz - Cpspec '2.i
pz,i
Po,i= Po.Jo'n' . . CP,
- pspec,i - CPspec + Po.spec.i
where Jjoint Po,joint (or Pp,joint) Jspec.i
(16.20) (16.21)
are the investment costs of joint structures and from among the special structures those parts that are under the joint costs, ' - OMR costs, - investment costs of special structures for the ith use,
- OM' costs, - the transferred costs of interchangeable structures of the ith use and of all the uses of the whole complex, - the transferred costs of special structures of the whole Pspec,i, C p s p e c , i complex. Transferred costs can be calculated from equation P0,spec.i (0' Pp.spec,i) Py.i,
P = kn
where T
Cpz
T
I
j= 1
a= 1
C Jj + C po.0
(1 6.22)
is the number of years, j - the ordinal number of the year for the period T, Jj - the increasing investment costs including the jth year, Po,a - the variable OMR costs for the period from a = 1 to a = t years.
387 As the respective components do not make full use of the whole complex at the same time, the indices of their efficiency also change with time, so that the value of the fraction in equations (16.20)and (16.21)is variable.
16.2.2 Directives issued by the Ministry of Forestry and Water Management of Czechoslovakia concerning the principles of evaluating the eflciency of’ investment for water management constructions and the distribution of costs of multi-purpose reservoirs
If the efficiency of the respective uses of a multi-purpose reservoir (MPR) has clearly been proved and its capacity has been optimized, then the overall efficiency does not have to be proved by distributing the costs over the respective uses. The components of a multi-purpose reservoir can be divided into special and joint component%.Investment and OMR costs for the special components are called special costs and concern only specific use. Investment and OMR costs of the joint components are called joint costs and these must be divided between the respective uses. If a special component also has other functions (for instance if a lock chamber or the construction of a multi-purpose hydro-power plant replaces part of the impounding structure), then its special costs are decreased by the amount also serving other purposes; these costs are then included in the joint costs.
Cost distribution To simplify this method, the term “summary costs, or SC” can be introduced. This is the sum of the present value of the investment and the respective OMR costs.
sc = P V ( J ) + PV(P,)
(16.23)
Summary costs (SC) concern the whole reservoir, special components (SCspec) and joint components (SCjoin,). 1. Special investment and OMR costs of the respective uses and their present value, i.e., SCspecare determined. 2. Joint investment and OMR costs and their present value, i.e., SCjoin,are determined (the difference of the total costs and the special costs). 3. The efficiency of the respective uses is checked according to the special costs. If the SCspecare higher than the present value of its performance or the present value of an interchangeable solution, the use of MPR is not effective and it is omitted. 4. The share of the summary costs of the whole MPR for the ith use, presuming the same discounted contribution WSD of the ith use and the whole MPR is det ermined, therefore P V ( W )- -P V ( Y . )- const WSD = --
sc
SC,
(16.24)
388 hence ( 1 6.25)
where PV(W) or P V ( 4 ) is the present value (PV) of the output of the whole MPR, or the output of the ith use, SCor SC, - the summary costs of the whole MPR, or of the ith use. 5. The share of the ith use in the summary joint costs is the difference of the summary costs for this use out of the total costs of the MPR and the summary costs of its special components are determined: SCjoin1.i
=
SCi
(16.26)
- 'Cspec.i
6. The coeffrcient of the proportional distribution of the summary costs for the joint components among the respective uses is determined by k.
'
=
SCjoint,i
(16.27)
~
''joint
which must fulfil the condition that
1k, = 1.0
( 1 6.28)
i= 1
7. The share of any part of the costs of the joint components for the ith use can be calculated with the coefficient k,: Jj0int.i
=
kiJjoint
Po,joint,i
=
kiPo,joint
etc*
8. The resulting costs of the ith use are calculated as the sum of the respective special costs and the share for these uses from among the joint costs: = Jspec.i Po,, = '0,spec.i Ji
+ Jjoin1.i
+ P0.joint.i
etc.
This is valid for the costs in the respective years as well as for their present values and discounted means. Note: The accuracy and objectiveness of the distribution of the costs of a multipurpose reservoir among the respective uses is given mainly by the accuracy of economic measurements of its output and secondary effects, which must be expressed in monetary units as costs. If there are no value equivalents for significant secondary effect, they can be expressed with the help of investment and OMR costs of an interchangeable alternative; from these the annuity or transferred costs are calculated, or they can be determined by an agreement between the users of a multi-purpose reservoir.
389 16.3 ECONOMIC SIGNIFICANCE OF THE RELIABILITY OF WATER SUPPLY AND FLOOD CONTROL
Output and services in water management have only a limited realiability. This reliability therefore determines the probability of satisfying the needs of the users completely, as far as water and the protection against the harmful consequences of water are concerned. This factor is important for any economic estimation of a reservoir. Even a small change in this rate can extensively change the basic parameters of a reservoir, as well as its costs and efficiency. To determine its optimal value is an economic problem that can be solved only by a very complex method and with the help of economic indices. In water management the term “gauranteed output has two values: the size of the output and its reliability, which is usually less than 100%. According to the type of reservoir, the economically justified values of the size and respective reliability concern, e.g. - release from a reservoir to the river to increase the discharge, - withdrawal of water from a reservoir for various purposes, - output and production of hydro-power plants, - all of these in a multi-purpose reservoir. These parameters determine the size of a reservoir and are indispensable in any economic considerations. 16.3.1 Significance and variability of the reliability of water supply
Water-management balances presume that the need for water will be met in various degrees of reliability according to the damage that might be caused if the water is not supplied to the full amount. The rate of reliability of a full water supply is sometimes mostly as a share (%) of years in which given by a standard of design reliability Podes,; the supply of water is ensured without any breakdowns. Data can be found in Section 4.4. The higher the standard of the design reliability, the smaller the losses caused by the deficits of water, but the higher the costs for the utilization of the water resource (reservoir, conduits, etc.). The economic optimum for meeting the needs of the respective users is found by comparing these two consequences. However, even an extensive economic analysis on the basis of which the standards for the supply of water are determined cannot accurately define all economic consequences. One of the reasons is that the economic losses caused by the deficits of water are non-linear and differ greatly. A certain decrease in the supply of water or a short period in which no water is supplied at all need not lead to economic losses. Therefore the standard of reliability Podes, sometimes has two values of withdrawal: for the full need the standard is lower and for reduced withdrawal it is higher.
390 The time factor in the system water resource - user has an even greater effect on the reliability of water supply. The demand-supply relationships in the system vary with time. A new water resource (reservoir) is usually being built at a time when the reliability of water supply is lower than the standard or the economic optimum. During the development of the system and also during the construction and even the filling of a reservoir the reliability is decreased. When a reservoir is full, the reliability of water supply is usually higher than the standard and then gradually decreases up to the moment when the relationship presumed by the design between the yield of the water resources and the need for water is established. Sudden changes in the system concern not only the water resources, but also the users if, e.g., a new factory is opened, or irrigation systems, housing estates, etc. are built. Cooperation in a system helps to raise the output, or if the output remains the same the reliability is increased. A new resource which is not fully exploited can serve as a reserve in the system. The reliability of developing dynamic systems should be investigated. Conditions in the second year and in the final target twelfth year after a new reservoir has been
-
f Crl Fig. 16.2 Changes of reliability in time ( a )increase of water demand up in time and opening of a new resource u = 0.8; ( b )time pattern of volumes p, = f ( t ) needed to cover the demand a,, with various probabilities p ; ( c )time pattern of probability p = f(r) for various 8. values
39 1
constructed to supply water to a certain region are plotted in Fig. 16.2. The time function of the demand on water is shown by the straight line a, = f(t). In the second year a, = 0.5, but by constructing a reservoir with a relative storage capacity /?, = 1.47 a relative water supply of a = 0.8 was ensured*); therefore when the new reservoir starts to operate the system has a surplus output of a - a, = 0.3. To ensure the required a, = 0.5 in the second year, a reservoir with a storage capacity of fi, = 0.22, instead of /?, = 1.47, would sflice. It follows that in constructing an over-year storage reservoir with full utilization of target parameters in the future it is not necessary to consider the moment that the reservoir is completely full as the beginning of operations, but rather the time when it is full enough to cover a,; in our case a filling of /? = 0.22 in the second year is enough to cover a, = 0.5 with a reliability of 99%. Curve &./, = f ( t ) in Fig. 16.2b determines the storage volume of the reservoir in relation to time t which is needed to cover ap = f(t) from Fig. 16.2 with a 99% reliability. The differenceup to the full volume /?, = 1.47 can be used to supplement the storage function in the system (if filled with water) or for better flood control (if left empty). Let us consider how great the need is to introduce another source in the system in the final target twelfth year. The required a, increases above the reservoir yield a = 0.8 and can therefore be covered only with a smaller reliability. In the fourteenth year the reliability drops to 97%, in the fifteenth year to 95% and in the seventeenth year to 90%. With a linear increase of water demand (0.03~per year) the reliability of water supply drops from 99 to 90% if the construction of a new resource is postponed by 5 years. It is a matter of economy to find the optimum moment for building a new resource. Here it can be presumed that it is economically expedient temporarily to decrease the design reliability and to postpone operations of a new resource after the twelfth year. Let us further consider whether it would not suffce to build a reservoir in the second year that would, in the final twelfth year ensure ap = 0.8, with a reliability of p = 97%, as already long before that, a reliability of p = 99% is obtained. The relative volume of the storage capacity would be only fl, = 0.96, i.e., 65% /?99./,. Figure 16.2~shows the relationships of p = f ( t ) for various volumes /?,. From the = f(r) one can see that a reservoir with a volume of /? = 0.96 would curve ensure-from the beginning of Operations in the second year until the tenth year-a reliable water supply of po.96 2 99%, by the twelfth year it would decrease to 97%and with this it could ensure a = 0.8 for the whole remaining life span. There is no doubt that eight years of fully covered operations with the required p 2 99% cannot balance the incomplete operations with p = 97% for the remaining life span. *) For simplicity’s sake the problem was solved with Svanidze’s graphs (1964, p. 145) for C , = 0.4; C , = = 2C,; r = 0.2; p = 99::.
392
However, if another resource is constructed in the twelfth year with a capacity of > a , - 1 2 ,the reliability of the former reservoir could again temporarily be raised to the required value of 99%. However, another factor that should be taken into consideration is the time-based reliability. In this case, the function p = f(t) must be found, which does not encounter any difficulties. These difficulties increase if conclusions as to the optimization of the changes of the structure of the system in time are to be drawn, especially for extensive multi-purpose systems. a
The most frequently used index of the reliability is the probability of the number (Section 4.4); however, this index does not of years with unlimited supply p , completely reflect losses or other difficulties caused by the deficits of water (energy). The number of failure years does not reflect the duration and the depth of the failure, nor the amount of deficit water or energy. The data p , PA] do not reflect the percentage of economic losses out of the total production. A more suitable index of the reliability of water or energy supply is an index based on failures in percentage in terms of their duration, as this is closer to the percentage of economic losses. However, there is no direct relationship between the duration of the inability to supply water or energy and economic losses. For the supply of the same volume a less severe, but longer failure is generally economically more favourable than a shorter, but more profound failure. The best way to express the design reliability is to use its probability characteristics in terms of the supplied water volume or energy. It is easier to find a relationship between the deficits of water or energy and social losses, than a relationship between the number or duration of failures. However, not even the volume of deficit water can clearly define the economic losses. If each unit of deficit water or energy leads to the same economic loss, then the value pd is a parameter which accurately reflects the economic consequences. Mostly, however, a unit of defkit water or energy causes different economic losses, depending on whether it is a long, light failure or a short, heavy failure. The failure must then be characterized not only by the deficit volume, but also, e.g., by the depth of the failure. The economic consequences of a unit of deficit water or energy also depend on the time of deficits. A deficit kilowatt-hour can cause different economic losses in winter or in summer; deficit cubic metres of water for irrigation have a different impact in various parts of the growing season, etc. The relationship between the data on the rate of reliability can only be determined by a more profound analysis; to express the rate of reliability as a percentage of water supply is justified only if the economic impact can clearly be determined. If there are no suitable statistical-economic data on the basis of which the deficits
[%I
393
’,
expressed in technical units (in m3, m3 s- kWh, kW) can be transferred to economic consequences, the simplest method to determine the rate of reliability in water management; e.g., p, [“A]can be used. However, here too, at least the approximate relationship between p,, prand Pd should be known. 16.3.3 Relationship between the respective reliability indices
The relationship between the time-based reliability (pl) and the occurrence-based reliability (p,) is important. These values differ greatly, especially in reservoirs that hardly regulate their discharges; in that case pI > p,. In reservoirs with long-term regulation values, pl and p, will be close. For the relationship between the occurrence-based reliability of a real design low-flow year p, and the design duration-based reliability of normal power supply prdes. Aivazian (1947) recommended the expression
- Prdes.
po=l-
(1 6.29)
P
where p is the ratio of the total duration of the shortage period vs. the duration of these years. Table 16.10 shows calculations of values p, for p = 0.20 and for various ptdes, values; it can be seen how greatly they change due to even small changes of Ptdes.. Tuble 16.10 Relationship between po and pIdcr,for p = 0.2 (according to V. G. Aivazian)
P1dss.
Po
(XI
(%I
95
96
91
98
99
100
75
80
85
90
95
100
According to the selected value of p, the change of occurrence-based reliability by ApIdes,derived from p , by Apo corresponds to the change of the reliability equation (16.29). ‘Po
=
1 -
P
‘PI
des.
(1 6.30)
and therefore the change Aprdes.= 1% corresponds to the change Apo = l/p [“A]. From equation (16.30), but also from simple observations, it is clear that the values of Aprdes,and Ap, differ more, the smaller the share of time falling within the range
394
of the shortage period in low-water years, i.e., the smaller the value of p. If p is close to zero, Apl is also close to zero, but if p = 1 (i.e., if the shortage period includes the whole duration of the low-water years) Ap, = Apldes.. The relationships between occurrence-based reliability p,, time-based reliability pl and volume-based reliability pd had been studied for 23 river sites in the Czech regions (Votruba and BroZa, 1966, p. 286). Mean monthly discharges were used. Relationships were derived for current operations (without a storage capacity) as well as for various sizes of storage capacities up to j?, = 1.0. Relationships were calculated for the various degrees of release control for these capacities: - dependence of time-based reliability on occurrence-based reliability p, = f(p,); - dependence of yield 0, on occurrence-based reliability 0, = f(p,); - dependence of total water deficits A C O P[mil. m3] during the observation period on occurrence-based reliability A COP= f(p,); - dependence of relative water deficits A C O P[“A] on occurrence-based reliability A Cop[“A]= f(P,). In all cases, uniform yield was considered. (a) Relationship pl = f(p,) Relationships pl = f(p,) for various sizes of storage capacities A, were drawn up for the respective sites. In Fig. 16.3 these relationships are shown for the DtEin site on the Labe. It can be seen that the values of pI are much higher than the values of
I
p , (especially for smaller reservoir volumes). With larger volumes, i.e., with over-year release control, pl with po decline more rapidly. Also at the other sites the relationship between pl and p , is not very close and is greatly influenced by the reservoir volume. Figure 16.4 plots the differences in relationships between 0,, p,, pt, pd for current operations (/3, = @),derived from mean monthly discharges and from mean daily
395 discharges in a 50-year series from the river Berounka at Kfivoklat (1891-1940). Relationships 0, = f(p,) differ greatly, the relationship pd = f(p,) is essentially identical within the studied range.
0)
b)
4
99
-pe,
945
[%I
XM
Fig. 16.4 Relationships between 0,, po, p,. p,, at Kfivoklat (1891-1940) for /I,= 0, derived from mean monthly and daily discharges
(b) Relationship C O P = f'(p,)
Figure 16.5 plots the relationships 1 0 , = f ( p , ) for the storage capacity size fl, = sevcn studied sites using a complete observation series. The closeness of
= 0.30 at
-p, [ X I
t ig, I ( I 5 K ~ . l i a b i l i t y o f w a i r ~ s u p p ] y ~ ~ ,=( " , , )Fig. 16.6 Relationship of LO,(: 96% corresponds to the reliability p , = 50%. For power plants the index of time-based reliability is more suitable. Reliability of augmented discharges f o r navigation Inland waterways are part of a transport system, but also of a water-management complex; these bounds with two branches should not be overlooked in any economic estimations. All the various aspects of navigation have been dealt with in specialized books (cabelka, 1963. 1976 and others). Navigation can be replaced by other means of transport and its economic effect should be compared with the effect of those other means.
403
Reservoirs can supply water for canals or augment water stages on natural or trained rivers. Any effective augmentation of water stages requires a great augmentation of discharges and therefore large reservoir volumes. Navigation can benefit from release control for other purposes (e.g., flood control, power plants); it uses water, but does not consume it. To determine the optimal reliability of water release from reservoirs to augment water stages for navigation, the following method can be used: (a) We determine the economic losses on a water-way caused by low water stages under natural conditions making it impossible to use the stream for navigation throughout the year. This loss equals zero if the streams are navigable for fully-loaded vessels throughout the year. Losses are caused if navigation has to be interrupted due to low water stages or floods, etc. Calculations are based on loaded cargo and number of trips. Operation costs are only reduced slightly if navigation has to be interrupted (saving of fuel). If streams became unnavigable other means of transport have to be used, e.g., railways. From the difference between the transport costs of boats and railways the losses can be calculated for every ton that is transported and from these the total losses. The values of the total annual losses make up a statistical sample from which the necessary statistical characteristics can be derived mainly mean annual economic losses. (b) For an augmented discharge that provides for uninterrupted navigability, and for some other selected values of augmented discharges Q, which ensure partial navigability, we determine the necessary volumes of storage capacities. (c) We select several smaller values of storage capacities V, and ascertain for the same augmented discharges as in (b),the rate of reliability of the augmented discharge and the mean annual losses 2. It is best to use the time-based reliability pr.For every selected value of augmented discharge, we ascertain the relationships pl = f ( y), 2 = f(V,) and Z = f ( p , ) . (d) We calculate the annual operation costs Po for the construction of the respective storage volumes V , and construct the relationship Po = f(V,). By comparing this with the relationship Z = f ( V,) and pl = f(K), we can determine the economically optimal values of V, and pl. Influence of the limited length of hydrological series on the reliability values In a long hydrological series less favourable low-flow periods can be expected than in a short series. However, it is not possible to ascribe lower values to the reliability values from shorter series than from longer series. To find out which periods are decisive for the water supply from different volumes of storage capacities, we constructed, for all the observed sites, graphs of the necessary
404 sizes of storage capacities for any required augmented discharges 0,for the 10 to 15 driest periods out of the whole hydrological series and similar graphs for the 110-year hydrological series 1851 to 1960 at Di5Ein on the Labe. If we presume that there is an analogy between the river Labe at DEEn and other rivers as far as low-flow periods are concerned, it is possible to consider the driest period of the last forty years as the decisive period of the last hundred years. Therefore it can be said that the last forty-year period has a greater weight than would correspond to the number of years. 16.3.5 Relationship between flood characteristics and economic losses
Flood-control measures are introduced to eliminate losses caused by floods. Besides direct damage to land and houses, floods cause indirect losses, e.g., interruption of transport or power supply, etc. The flood-control effect of a reservoir is designed with an economically justified measure of reliability. Economic losses caused by floods are determined by a detailed study of the flood plain and from older recordings of extensive floods. The best data for the relationship between flood characteristics and economic losses are a time series of floods during the observation period, in which each flood is characterized by technical parameters and by the extent of the damage it caused. For economic estimation of the flood-control effect of a reservoir it is advantageous to use the relationship between economic losses and one of the flood characteristics (maximum discharge, flood volume, etc.). Most frequently, economic losses depend on the maximum flood discharge Z = f(Q,,,). The basic economic index of the extent of flood damage is the mean annual economic loss Z , given by the relationship (16.32) where N is the probable time of exceedingflood discharges. With regard to the general pattern of the curve Z = f ( N ) the value Z can be suitably determined graphically. The exceedance curve of maximum flood discharges is drawn by plotting probabilities of exceeding p = 1/N on the abscissa and the corresponding values of maximum discharges on the ordinate. Using the relationship 2 = f(Qmax), the exceedance curve of economic losses is determined (Fig. 16.15). The area defined by the axes of the coordinates and the curve 2 = f(p) is transferred to a rectangle with a base equal to 1, i.e., from p = 1/N = 0 top = 1, the height of which on the ordinate is the mean annual economic loss. Flood-control reservoirs help to diminish economic losses. The mean annual benefit is given by the difference between the mean annual economic loss before and after the construction of a reservoir. Floods that are held completely by a reservoir
405
and do not exceed a non-damaging discharge do not cause any losses; losses are caused by floods not held by reservoirs or only partly held. We construct their exceedance curve and ascertain the mean annual economic loss 2, after the con; struction of a reservoir in the same way as above (Fig. 16.16). The economic loss Fig. 16.15 Determination of the exceedance curve of economic losses and mean annual losses caused by floods
N
-P
I
-P Fig. 16.16 Determination of the reduction of losses due to the flood control effect of a reservoir
equals zero at point Pnd = l/jvnd,where N,, is the probable time of exceeding a non-damaging discharge. Also plotted in the diagram are the original exceedance curve of economic losses 2 = f(p) and the mean annual loss 2.The differencebetween the losses Z and Z , is the mean annual benefit of the flood-control effect of a reservoir. For further economic estimations the utility of the flood-control effect on the active storage capacity, the flood-control capacity and surcharge capacity must be determined. The solution is similar, but should be carried out in stages. First we determine the exceedance curve of economic losses for the active storage capacity and the respective mean annual losses. Then we consider the effect of the surcharge capacity and finally the effect of the flood-control capacity. Special attention should be paid to the surcharge. If it is expedient to build an ungated spillway in a storage reservoir, the reservoir has a surcharge capacity which is given by the need to make the whole construction safe. The costs for this capacity are shared by the users of the active storage capacity. If the surcharge capacity is also used for flood control the cost remains the same, as the size of this capacity is given by the standards that ensure the safety of the reservoir which are always more demanding than flood control standards; it is, however, possible to include an adequate share of these costs in the flood-control costs. The flood-control effect of the active storage capacity is estimated in a similar way. Here, however, extra costs have to be considered that are introduced to raise the effect of the active storage capacity, e.g., costs for forecasting services, etc.
17 RESERVOIRS AND THE ENVIRONMENT Every individual reservoir has an impact on its environment. Proportionally to its size and function, it influences the natural environment of the water course and the surrounding regions, the life of the inhabitants and the economic activities in the region. A reservoir changes the physical and biological regime of a river, inundates territories, threatens its embankment by abrasion and landslides, affects the weather conditions. It helps to raise the living standard (Fig. 17.1). In designing, constructing and operating reservoirs, the water-management engineer must maximize their favourable effects and minimize the negative con-
-
d
e t.8
q
BP
‘1
4 0
,
I
t
of the countryside
H I
Fig. 17.1 Survey of natural and socio-economic aspects of reservoir construction
407 sequences of their construction on the environment. It is his social mission to control the change of the environment. In collaboration with other experts (biologists, etc.), he has to maintain the quality of the environment even though it changes due to economic activities and scientific-technological progress, and to intensify the utilization of water resources-an irreplaceable part of the other natural resources. The technical qualification of water-management engineers must help to implement the scientifically controlled changes that lead to such technical advances that are able to satisfy the main necessities of man. Water does not know any frontiers and is therefore a subject of international interest. The European Charter on Water (1948)states that water is common property, the value of which muSt be recognized by everyone; it must be kept, protected and preserved; water quality must meet the standards of public health and water must be used effectively and reasonably. Environmental problems are dealt with within the framework of the Council of Mutual Economic Assistance that established the solution of concrete tasks in the water-managementbranch at a Conference of Leading Authorities for Water Protection in the CMEA Countries. The Water Act of 1973 declared surface and ground waters to be one of the principle raw-material resources, representing an important component of the human environment which ensures economic and other social requirements.
Every water reservoir is principally the reflection of the respective catchment area. If in the catchment area, soil management, agricultural activities and the settlement of the inhabitants are properly regulated, then the conditions of the reservoir, water quality and required service life of the reservoir are also favourable. Scientific methods and technological solutions for the rational exploitation of water resources have already been elaborated. Less advanced is the unierstanding of the complex problem of the natural processes caused by the activities of man, undoubtedly because these processes are very complicated and dynamic, requiring long-term research. Scientific and technical experts in the field of water management should help to overcome this problem. 17.1 CONSEQUENCES FOR THE HUMAN ENVIRONMENT OF THE CONSTRUCTION OF RESERVOIRS
Reservoirs affect the environment during the years of their construction, and for the many decades of their service life a survey of the unfavourable effects and the measures taken to prevent them, is carried out (see Fig. 17.2). The classification of the effects reservoirs have on the environment and the respective measures taken to counteract them lead to the conclusion that - every research task concerning the construction and operation of a reservoir should be undertaken with a view to the relationship between the reservoir and the environment;
408
- every reservoir design should be undertaken with a view to the environment, with a separate section dealing with the utilization of natural resources and environmen t a1 protect ion. The environment is an ever present factor of life. It is a feature of scientific studies, from theory in basic research up to the technological implementation in practice, and is the subject of studies in sociological,natural and technological sciences.Taking into account these fields of science, the position of the water-management branch should be determined in relation to the environment (Table 17.1).
Mwsures : The pmjwt ond construetian technology must consider the envii-onment
c
s
.F
cu ing own o rees ftftem&ory L:ilities
8
t
borrow pits, quorries
Construction work with minimizotion o f new five enviranmental impmt Servi'ces on the building site
Systematic implementotion of these measui-es
~
's
i;
operution and the necessary measures
.
f
-m u d n g o f reservoir
- Measures in the catch-
W t i v e chemimlodholopimfp m s s e s in resewi'; eutrophication adder wr?e,r downsfreom of m r v o i r in summer
-
-
-I
chanqe of roundmter r#%e
ment upstream ond in the close vifinity of the ~ ~ ~ v o ~ r
M f M ! f of mrmer woter ond offemtive m s u r e s Proqnosis ond neces-
-
Fig. 17.2 Negative impacts of reservoirs on the environment during construction and operation
409 Table 17.1 Subjects studied by scientists relating to the human environment
Social sciences
Natural sciences
Technological sciences
Relationship between society and nature
Influence of production and technology on nature
The control of the relationship between production and environment
Impacts of the differences of regions
Stability and transformations of natural systems
Social and individual needs Economical tools Legal measures
Biological productivity of ecosystems Biosphere and its natural components
Intensity of technological activities in the environment Technological improvements for the utilization of natural resources
Control of the interaction between man and environment
Adaptability of organisms to changes in natural conditions
Procedures to reduce waste
Effects on soil and occurrence of erosion
Production of equipment for waste treatment
Technology for pollution prevention
Effects on water resources, etc.
From the table it is evident that the branch of water management and of water engineering takes its duties concerning environmental problems in the field of sociological and natural as well as technologicalsciences seriously. In the cycle: research + + development + design + production + operation, water engineering also plays a part in solving the problems of the environment as well as in all the other steps of the cycle. Thus, it can be concluded that the field of water engineering deals with all aspects of environmental problems. The effects of reservoirs on the environment can be divided into physical and biological and effects on man (Cheret, 1973; Votruba, 1973). 17.1.1 Physicul impacts of reservoirs on the environment
Physical impacts of reservoirs have to be considered from the beginning of the construction as they have a marked effect on the hydrological and sedimentary regime. The hydrological regime is influenced mainly by the reservoir volume. Relatively small reservoirs have a smaller effect on the discharge conditions, whereas large ones, with long-term release control have a substantial effect on the discharge regime in the range,of low, as well as flood discharges. The balancing effect of regulated discharges faiourably influences the environment, since it reduces the flood
410
damages and improves the general use and quality of the water by augmenting the dischargss in perids of low water stages. How the discharges are changed depends on the purpose for which the reservoir is used. A flood-control reservoir reduces floods, but does not regulate the low discharges. A conservation reservoir, in addition to its main function, improves the minimum maintained discharge which is higher than the natural discharge minima. Peak operations of hydro-power plants have a negative effect on the discharge conditions. The rapid changes in the discharge have an adverse effect on the riverbed as well as on the biological function of the river. For this reason, if they do not operate in series, peak-power plants are usually accompanied by a balancing reservoir. A similar effect is exhibited by river flow regulation during sudden changes in water withdrawal. This shortcoming can be removed by a buffer-storage reservoir whose volume is designed with a view to the permissible harmless gradient of discharge change. A deterioration of discharge conditions can occur with relatively high consumptive withdrawal from a reservoir. For this reason a relatively high yield has to be included in the design, as well as a certain water volume for occasional flushing of the riverbed. Reservoirs also alter the groundwater regime. If the water level reaches above the surrounding land, agricultural lands are threatened by water. On the other hand, excavations of the riverbed downstream of the dam can lower the groundwater table to a detrimental extent. Seepage from the reservoir can deteriorate groundwater used for drinking purposes. The surplus groundwater on the agricultural land can sometimes occur in the valley below the dam. A significant seepage problem through the protective dykes arose due to the impounding of the water for 270 km, by damming the straights of Dierdap on the Danube (Mostarlic, 1973). The building of the reservoir led to the following adverse consequences: flooding of the valley due to the water impoundment, raising of the groundwater table, deposition of bedload, and changes in the ice mass. The impounding endangered 200 km of old protective dams that protected over 90 OOO ha of arable land and a large number of villages and towns with industrial plants against floods. These dams had to be reconstructed, i.e., they had to be raised and reinforced. A drainage system was constructed in the protective section. The dams were elevated by 0.70 m above the height of the end of the wind wave, counting on a high wind velocity with an occurrence probability of 1%. The groundwater table was determined by agropedological criteria, separately for unmineralized groundwater that cannot cause salt contamination of agricultural land (depth during the growing season 0.90 m in very light soils and up to 1.30 m in heavy clayey soils) and for mineralized water that can cause salt contamination of the soil (depth in the growing season 1.00 to 2.00 m and outside the growing season 2.20 to 2.70 m). All measures taken in connection with the construction of the DZerpad water scheme, whose main function is power generation and navigation, were judged not only from the aspect of these main objectives, but also from broader social-economic aspects.
Present experience shows that the unfavourable effects of impoundments on groundwaters have to be overcome in time - by geological and hydrogeological surveys,
41 1 - by predicting the changes in the groundwater regime after the filling of a reservoir, - by proposing and executing the respective control measures, - by careful maintenance, and investigation of the effectiveness of the measures carried out. Siltating is one of the most imporant physical factors in reservoirs, since it reduces their volume and shortens their service life, which should be at least 250 to 300 years. Bulgarian engineers assume that the volume for siltating should be calculated for 100 to 200 years. The speed and manner of siltating varies greatly in different watersheds and reservoirs. Bristol University has been collecting data on 130 reservoirs of various sizes in the USA, India and Cyprus, where the siltating amounted to 0.17 to 92.9% and the annual volume decrease was 0.02 to 14.3%.The Mangla reservoir in Pakistan is expected to lose 1.233* lo9 m3, i.e., one-ninth of its total volume in 20 years.
In Lake Nasser, created by the Aswan Dam, a volume of 30. lo9 m’ of the total volume of 164. lo9m3 is set aside for siltating; it is assumed that it will be silted up in 500 years, although only 12’4 of suspended solids are retained in the reservoir. Sediments contain mainly fine mineral material composed of: fine sand silt clay
(0.2 t 0.02 mm) : 30% (0.02 t 0.002 mm) : 40% ( < 0.002 mm) : 30%.
The loss of these suspended solids in the Nile water used for irrigation purposes had no effect on the crop yields. On the contrary, the yields are higher due to controlled irrigation and the addition of cheap fertilizers. However, downstream of the dam, the riverbed became deeper due to the passage of clean water, which disturbed the former equilibrium. The water level sank by 40 to 80cm. This fact is being investigated by geological, topographical and hydrological surveys and control measures will be introduced (Kinawy et a/.. 1973). The volume loss of Czechoslovak reservoirs due to siltating is relatively small. The volume of the KniniEky reservoir on the river Svratka decreased by 3% during 30 years’ operation and its service life under present conditions in the watershed is approximately 800 years (Kratochvil, 1970). The conditions in many other Czechoslovak reservoirs are similar. However, greater difficulties due to siltating can arise at the inflows of rivers into reservoirs. KaliS et a/. (1972) resolved the siltating in the inflow reaches of the rivers Dyje, Jihlava and Svratka into the NovC Mlfny reservoir on an air model and predicted the bottom deformation by bedload.
To retain coarse bedload before it reaches a reservoir, reruining darns are sometimes built on the tributaries. Their function is, as a rule, of short duration as they soon become clogged and can cause deterioration of the water quality. When the retaining dam is placed in the inundation of the reservoir, in the form of a “submerged” dad, it fulfils yet another function by controlling the fluctuation of the water level in shallow parts of the reservoir at the end of the backwater. Some of the disadvantages of tipped rockfill retaining dams can be prevented by using coarse screens that are installed into the riverbed of the tributary at the end of the reservoir backwater (Priehradne dni - Dam Conference 1972, Loucky, p. 187).
412
They affect the discharge conditions to a lesser degree, retain mainly coarse bedload of plant origin and can be treated easily, thus prolonging their service life. A removable screen was installed on the main tributary to the reservoir e.g. near Ludkovice. Although many methods for the solution of siltation problems have been developed, there remains much uncertainty in predicting the bedload movement and siltation due to the raising of the water level by a weir or dam. Experience gained all over the world was concentrated in questions No. 40 and 47 discussed at the XIth and XIIth International Congress on Large Dams (XIth Congress, 1973, XIIth Congress, 1976) and, together with Czechoslovak experience, they were also the subject of discussions at Pi’ehradni dny (Dam Conference) 1972,1973 and 1976 (see references). The environmental impacts connected with the changes in reservoir topography are bank abrasion and slides. Bank abrasion is the result of wind waves, waves produced by vessesls, ice, water level fluctuations and man-made interventions. The extent to which the banks are changed depends on the nature of the banks along the reservoir shore line and on the magnitude of the influence. Even small changes are unfavourable from the environmental point of view. However, sometimes they reach significant or even catastrophic dimensions when great land slides into the reservoir occur. Due to wave action and water-level fluctuations, the banks of the Orava reservoir (surface area 35 km2) receded in some places by more than 45m and abrasion palisades up to 20m high were formed (PriehradnC dni - Dam Conference 1972, Horski, p. 121).Abrasion and bank-slide material reduced the reservoirs volume. The original banks, with the slope of now-steadied bank-abrasion platforms, are practically unchanged. Steep slopes in Neocene and Quaternary sediments were very quickly transformed by abrasion. Experience gained is a valuable basis for the prediction of reservoir bank transformation. This prediction should be part of any reservoir design (Kachugin, 1955, Pyshkin, 1963). Peter and LukaE (Priehradne dny - Dam Conference 1972, p. 131) determined the extent of wave abrasion for the Orava and Velka DomaSa reservoirs; as a relative criterion they introduced the so-called abrasion degrees I to V: I: most intensive abrasion with great abrasion volume and velocity; 11: intensive abrasion in steep, easily decomposable banks; 111: slight abrasion with abrasion palisade heights of 0.5 to 1.0 m; IV: very slight abrasion with abrasion palisade heights up to 0.5 m; V : banks without abrasion. The extent of the abrasion degrees is given in Table 17.2. The greatest abrasions in Slovakia are encountered in flysch regions. Economic impacts are significant as the bank lengths of six reservoirs (Orava, Velka DomaSa, Vihorlat, Nosice, Ruiin and Lipt. Mara) are about 300 km and according to Table 17.2 about one-half of the bank lengths is affected by abrasion. Bank slides due to impoundments can sometimes reach catastrophic dimensions. On the left bank of the Rumanian Izvorul Muntelui - Bistrita reservoir, several bank
413 Table 17.2 Abrasion in the Orava and Velkh Domda reservoirs
Abrasion degree Orava [kml I most intense I1 intense 111 slight IV very slight V without abrasion
1-V total
6.60 6.30
Length of reservoir banks Velka DomaSa [“A1 [kml [“A1
28.80
9.56 9.13 22.32 17.25 41.74
2.70 4.87 3.83 6.91 21.69
6.74 12.17 9.58 17.28 54.23
69.00
100.00
40.00
100.00
15.40 11.90
slides occurred, the greatest of which was 250 m long and 350 m wide. On the right bank of the Pingarati - Bistrita reservoir, a bank slide on the interface of diluvium and rock occurred, having a width of about 300 m, a length of 150 m and a thickness of 3 to 6 m (XIth Congress, 1973, Q. 40,R. 31, Diacon et al., p. 445). The landslide of Toc Mountain into the Vajont reservoir in northern Italy in 1963, where about 250 * lo6 m3 of rock material slid into the reservoir from the left side, forcing a water wave over the crest of the 261 m high arch dam, was catastrophic. The discharge wave demolished the village of Longarone in the Piave river valley near the mouth of the Vajont mountain stream and killed 2000 people. In large reservoirs the water quality improves, undoubtedly due to physical sedimentation. However, as a rule, more complex physical, chemical and biological processes take place in reservoirs. The result of all these processes are changes in water properties that can vary greatly and to a large extent are dependent on the properties of the inflowing water. Special problems arise during the building of reservoirs for fresh water in the vicinity of salty groundwater. Problems connected with the leakage of salt and fresh water require special research. In the seaside zone in Yugoslavia, this problem is further complicated by the presence of karst waters. The effects of large reservoirs or reservoir systems on the weather conditions have been studied for a long time and have led to conflicting opinions. Posekang (1969) reports the effect of the South-Bohemian ponds on the environment, especially on the microclimate. With a daily evaporation of 5 mm, 1500 m3 of water passes from 1 ha of pond area into the atmosphere in one month, thus increasing the air humidity near the ground surface which can be carried by the wind to more distant regions. The Swedish Meteorological and Hydrological Institute ascertained during 20 years of investigations that discharge control can alter the water temperature,
414
whereas the air temperature is mostly unaffected. In spite of this, small changes in extreme daily temperatures were encountered near the banks of a new reservoir. In autumn and at the beginning of winter, fogs increased due to the higher water temperature. Volta Lake in Ghana, with a volume of 165 lo9 m3 and a surface area of 8730 km2, exhibited a change in its precipitation regime, After the completion of the Lake in 1964, it seems that October was no longer the month for rains, but that July and August are now the rain months (XIth Congress, 1973, Kumi, p. 907). One of the most marked effects of reservoirs on the surroundings is their impact on the temperature and winter regime. The temperature regime of a river changes in relation to the relative reservoir volume. Shallow or deep reservoirs, as well as relatively small or large reservoirs, create different temperature conditions. The most regular temperature conditions are in large and deep reservoirs with relatively small inflow and stready outflow. In such a reservoir a regular vertical stratification of water temperature is produced, similar to natural deep lakes. The temperature of the outflowing water is determined, not only by the temperature of the water in the reservoir, but also by the position of the outflow devices and by the magnitude and time pattern of the outflow. Since water has its greatest density at 4 “C, in summer the water is coldest near the bottom, while in winter it is at its warmest near the bottom. The water leaving the reservoir through the bottom outlets or low-lying hydro-power plant intakes is therefore cooler in the summer than the water in the natural river and warmer in the winter. This affects the water course for tens of hundreds of km downstream of a reservoir. The fact that the water is cold in the summer is unfavourable for the natural environment. Proposed measures therefore plan for water withdrawals from the upper layers. For the dam on the river Blanice near Husinec F. Cech designed a shutter intake with its lower edge supported by a float at a constant shallow depth below the water surface. For the Slapy and Orlik reservoirs it was suggested that an “apron”should be built at a certain distance from the dam that should close the total cross-section of the valley except for a few metres below the backwater level. L. Zaruba prepared B design using plastic foil and L. Liskovec tested its effect on the temperature rise of the water flowing through turbines from a reservoir in the laboratory. The object was to improve the recreation conditions on the river Vltava by the outflow of warmer water in summer. The impact of the apron on the chemical and biological conditions of the water still has to be determined.
S.ummerwater temperature stratification in a reservoir is called direct stratification, in winter indirect stratijication. When direct stratification changes to indirect stratification (in autumn) and from indirect back to direct stratification (in spring), the reservoir exhibits homogeneity in temperature, i.e., the temperature in a vertical section from the bottom to the surface is the same. Especially in summer, when the temperature gradient in the vertical is usually not constant, there is a certain layer (about 5 to 15m below the water level) where the temperature gradient is higher.
415
This thermocline is called the metalimnion, the area underneath it is the hypolimnion and above it the epilimnion. The temperature conditions affect the biochemical processes and water characteristics. The hypolimnion is usually poor in dissolved oxygen and richer in iron and manganese. In shallow reservoirs, isotherms are, as a rule, not horizontal, but inclined to the vertical and reflect, e.g., the gradual warming of the water in the direction towards the dam. Only in the case of small inflows can a vertical stratification also occur in a small reservoir. Ponds are typical shallow reservoirs. For fish-farming, which requires warm water, the warming of water is also supported by leading the water to the spill crest from the bottom at the deepest point of the pond. Ice conditions are changed by reservoirs in the backwater reach to a certain distance upstream from the backwater end and in the reach downstream of the dam. As a rule, ice sheets form earlier on the backwater than on the river, are thicker and melt later. The ice sheet in a reservoir melts without the ice falling over the spillway as long as the velocity at the cross-section during ice motion is less than 0.4 to 0.5 m s-'. At the end of the backwater and above it, the ice conditions deteriorate, especially when ice floes and frazil ice approach it (hummocks, ice clogging). Downstream of the dam, ice conditions improve again: no ice mass arrives from the upstream range and warm water released from the reservoir keeps part of the river free of ice (Kratochvil et al., 1965; Votruba, Patera, 1983). The water from the Slapy reservoir affects the temperature conditions in the river Vltava in Prague, i.e., at a distance of 4Okm, although it passes the Stkhovice and VranC reservoirs and has two large tributariesthe rivers Shzava and Berounka. Freeze-up in winter is reduced and recreation suffers due to the cold water in summer. At low discharges, the Nechranice reservoir affects the water temperature in the river Ohfe to a distance of 100 km. By releasing water from the reservoir, the ice conditions on the river can be controlled. Ice sheets act by their dynamic and static effects on the banks and structures and, together with frost, threaten the operability of tlic gates. Ice pressure endangers thin structures, concrete bank reinforcements, tower structures in reservoirs, etc. To counter this threat a number of effective measures have been introduced (heating of all kinds, disturbing the ice sheets near structures using compressed air, water from the deeper layers, etc.). Seismic efects due to the filling of large reservoirs with water have been reported from several sites (Caribbean, Marckison, Volta). An increasing number of seismic vibrations in the vicinity of large reservoirs in the Sudan has been registered, but the question still remains of whether all these reported vibrations were really caused by the water load in reservoirs or whether they would have occurred anyway.
416 17.1.2 Biological and chemical impacts oj’ reservoirs on the environment
Biological and chemical changes brought about by reservoirs are even more complicated than physical changes. To predict them and to design control measures, engineers have to cooperate with experts--chemists, biologists, and others. All these changes are caused by a multitude of factors and are extraordinarily complex: physico-chemical changes lead to changes in the microbiological composition which in turn affect plants and higher animals. Aquatic ecosystems that existed before a reservoir was built change gradually until, in the course of time, a new equilibrium is established and only then can the biological impact of the reservoir be judged with final validity. The chemical composition of the reservoir waters changes, in consequence of the increased evaporation from the free water surface (increase of the relative mineral content), and due to biochemical processes taking place in the reservoirs. In a nutrient-rich reservoir (eutrophic) the oxygen balance changes to a great extent : the dissolved oxygen content decreases with the depth, to such an extent that mass fish deaths can occur when this water reaches the riverbed dowwtream of the reservoir or the following reservoirs of the reservoir cascades (Stkhovice, 1956, Slapy, winter 196211963, etc.). During peak operation of the hydro-power plant, the Orava reservoir causes considerable biological disturbances in the river due to sudden changes in discharge, water velocity, temperatures, etc. Degradation processes of organic matter in the hypolimnion of eutrophic reservoirs can exhaust the total dissolved oxygen in the summer stagnation period, increase the free carbon dioxide concentration, and result in the release of hydrogen sulphide. From his measurements on the KliEava and Slapy reservoirs, Fiala (1961) revealed that the dissolved oxygen stratification is adversely affected by small dams built inside a reservoir; e.g., a temporary dam serving to by-pass the water from the dam building site during the construction period. Behind such a dam, the water stagnates, having a very low dissolved oxygen content. Similar conditions can be encountered when the water intake from the reservoirs is at a greater distance from the dam and the bottom outlets are closed. The outlets in the small dam are not sufficient to move the stagnating water. The water quality in a reservoir is adversely affected by water bloom (algae) which negatively influences water treatment, recreation, etc. In addition, it also leads to secondary eutrophication of the hypolimnion, which in turn leads to a deterioration of the water quality in the reservoir and in the downstream river reaches. This secondary eutrophication can be prevented by water aeration (Fiala, 1972). A very effective aeration of reservoir water can be achieved by introducing a waterair mixture below the water surface (Haindl, 1975). The water is pumped from the bottom layers of the reservoir and passes through an earation device operating on
417
the principle of a ring jump. The ring jump produces a water-air mixture; due to the high turbulence behind the jump, an intensive oxygenation is attained that is in addition supported by the secondary flow and mixing in the reservoir. Radical changes in water quality occur in tropical countries by the action of bacteria (XIth Congress, 1973, p. 523). Water can become undrinkable or corrosive to metal structures, as was shown by investi-
gations on the Ayame I reservoir, on the Ivory Coast, situated in wooded landscape. The investigations require long-term extensive measurements that must follow the vertical stratillcation and the effect of the annual discharge variations and water-level changes in the reservoir. Investigations in Cameroon and Gabon confirmed that the principal and permanent causes of microbial processes and the increased aggressivity of water in reservoirs result from the properties of the water that flows into reservoirs from the tropical woodland region.
The floodingof the biomass by water polluted during the first filling of the reservoir and with every following rise of the water level can lead to increased nitrogen and phosphorus concentrations and thus to the development of bacteria and then algae (phytoplankton and biotecton). The water has an unpleasant taste and appearance and is odorous, the oxygen concentration varies up to values inadmissible for fish and anaerobic conditions are encountered. Sanitary measures combined with the construction of water-supply reservoirs are expensive. Often villages, churchyards, manure heaps, etc., must be demolished. However, the extent and manner of removal of vegetation from the inundated area has to be judged separately in each case. According to experience from the operation of Czechoslovak reservoirs, it is possible to leave tree stumps in the region of permanent inundation, since even in the course of decades they do not yield to fouling (experience with spruce stumps in the reservoirs of Sedlice and SOUSafter 40 to 60 years of operation).The mere ploughing of the natural vegetation cover is detrimental. Also the disturbance and incomplete exploitation of peat deposits in the inundation leads to a deterioration of the water quality in the reservoir due to increased eluation of humic acids (experience from the Lipno reservoir on the upper Vltava). The algal concentration can,be determined quantitatively in milligrams of chlorophyll per litre and its limit must be ascertained for each use. It is important that the daily, annual and long-term changes are predicted. In one Dutch reservoir, very strong eutrophication phenomena due to the inflow of polluted water from the rivers Maas and Rhine have been observed. However, 40% of the product of algal decay settles on the bottom and thus flocculation and sedimentation maintain aerobic conditions. In tropical regions, mass development of vegetation can obstruct navigation and fishing, can change the chemical composition of the water and produce conditions supporting the propagation of diseasecarrying insects. In the zone of water level fluctuations in the reservoir, special vegetation can be found according to the extent and duration of the water level fluctuations; in the moderate and cool zone it is usually bare. However, in reservoirs with long-term control, where the emptying of the reservoir can last ten years or over, vegetation can develop in the moderate zone which can be very harmful for the water quality during the refilling of the reservoir.
The water quality in reservoirs is also affected by the activities of man in the entire watershed. Control measures should be introduced, mainly into the watershed of
418 public water-supply reservoirs. These usually cause a certain limitation on agricultural production, due to the reduction of the use of industrial fertilizers and chemicals against weeds and insects, control measures introduced in the breeding of livestock, and the disposal of sludge. Crop spraying from aircraft in the vicinity of these reservoirs and water courses increases the danger of diffused pollution. Those industrial processes that work with petroleum are especially dangerous. The strict adherence to sanitary measures is essential; otherwise the difficulties connected with the pollution of the inflowing water and with the low quality of the water withdrawn from the reservoir would surpass the acceptable limits. Great care was taken with the water-supply reservoir near Svihov on the river Zelivka, with a watershed covering an area of 1178 km’, 27% of which is covered by forests, and is inhabited by 60 OOO people. From the inundated area and the hygienic protection zone I. 2166 inhabitants were evacuated and 710 buildings were demolished. The sources of pollution on thc ri\cr ?clivk;i wcrc (Prichradne dni 1972, Dajbych-Lepka, p. 275):
[“A1
- sheet erosion and rainwash 49.6 - wastewaters from farms 8.7 - inhabitants 4.4 - industry 6.5 - other sources (fall-out, groundwater) 30.8 A study on the Biotechnical Protection of the Suihou Reservoir, prepared by the Comission for Water Management at the Czechoslovak Academy of Sciences, assumes the best protection against sheet erosion and rainwash from fields to be a zone of mixed deciduous and coniferous forest, 20 to 30 m broad, along the inflow reaches of significant tributaries. This was achieved by amelioration, using natural elements without grossly interfering with the natural environment. This necessitates an early analysis of the sources of pollution and the execution of the respective measures long before the reservoir is put into operation.
Ofeconomical and ecological significance is the effect of a reservoir in fishing, which can be both favourable and unfavourable. The change of the flow velocity, water depth, temperature, and water-level viriations considerably affect the fish-breeding conditions so that the composition of fish species can also vary. PivniEka and HolEik (Priehradnedni 1972,p. 219 and 225) reported on two stages of the reservoir operation from the ichthyological point of view: - the first (initial) stage lasts until the maximum capacity of the ichthyomass is reached, - the second (stabilization stage) following the termination of the first stage. Table 17.3 shows the development of the ichthyomass and production (weight increase) in the reservoir on the river KliEava near ZbeEno since 1957, i.e., two years after the filling of the reservoir. At the beginning, the ichthyomass and production increases; however, in the stabilization period of the hydrological and biological regime in the reservoir these values decrease, as the fish find less nourishment. From Table 17.3 it is also evident that the change in the species composition of ichthyocoenoses is unsatisfactory; it is possible t i derive that per lo00 fish about 50 kg of fish perish annually. It is therefore
419 Ttrhle 17.3 lchthyomass values and fish production in the Klitava reservoir ~~
Indicator [kg ha-
'1
1957
1964
1967
1968
1970
ichthyomass production (weight increase) biomass: pike tench roach
59.1 154.I 10.5 4.1 7.4
97.4 971.5 3.3 1.4 106.0
194.6 512.6 165.0
161.3 402.0
122.9 285.4 1.3 0.6 92.0
Most important species [number of fish per ha] roach porch chufs rudd tench pike
75 890 15 15 6 8
7471 650 398 I26 3 3
1720 408 84 58 2 L
-
126.0
3450 377 174 80 2 1
918 149 56 60 2 2
advisable to place predator fish (pike, pike-perch, trout, etc.) into these reservoirs and to forbid fishing for such species. The use of impounding reservoirs for fish farming requires collaboration between water-management engineers and biologists. According to present experience, some measures could be taken that would contribute to the improvement of fisheries in reservoirs (adjustment of the bottom in sites of net fishing, damming up the tributaries entering the reservoir, preventation of the escape of fish from the reservoir, regulation of the water level, flunctuations, etc.). In countries situated in the north (Sweden, Canada, the USA) where salmon-like fish are of great commercial and sporting significance, dams act unfavourably, since they hinder migration of fish. On the Saint John River in Canada three dams obstruct the migration path of salmon. The problem of migration was solved radically by transporting the fish over these obstacles in trucks. It is a system of sophisticated devices that serves to transport 10 to 20 thousand fish upstream yearly. About lo00 breeding fish produce approximately 500000 young salmon that are again transported below the downstream dam (XIth Congress, 1973, O'Connor et al., Q 40, R 47, p. 755). Positive effects on fish farming, are caused by impounding the water by dams on the African continent (Cariba between Rhodesia and Zambia, Kainja in Nigeria, Ayame and Kossou on the Ivory Coast, Volta in Ghana, etc.). In the Volta reservoir the quantity of fish catch increased from 10 O00 t to 50 0oO t annually. In the Cariba reservoir the annual production is about 17 OOO t, i.e., 186 kg ha-', and might further
420
increase. In the Nasser reservoir in Egypt, 3930 t of fish were caught prior to 1971, when the biological equilibrium was not yet established. From these examples it can be seen that reservoirs can contribute to an increased fish production; it requires suitable exploitation of their possibilities and measures to control some adverse effects. The effect of tropical reservoirs on the development of diseases of man and domesticated animal is important. This is manifested either directly by the change in the water regime upstream of the impounding structure, or indirectly by the construction of irrigation canals downstream of the dam. The most important diseases are malaria, bilharzia and onchocerciasis. To fight against them it is necessary to know exactly how they are transmitted. There are measures that reduce the risk, if they are introduced in time. The use of insecticides is not recommended, as they can be harmful to other organisms. Often it is considered more effective and cheaper to cure sick people than to destroy the carrier organism. It is essential to instruct people about the necessary sanitary measures and how to fight the diseases. Even though certain biological effects of reservoirs are negative, they are not as great as the positive effects. It is the task of water-management engineers, in collaboration with the respective biologists, to overcome the adverse biological impacts. A reservoir affects the water quality in a river to a great distance downstream of a dam. The problem is to control the water quality by controlling the release from the reservoir. By augmenting the discharge, only those water-quality indicators that have more favourable values in the water flowing out of the reservoir than in the water in the river downstream of the pollution source improve. It is not only a question of diluting wastewaters, but of changing the flow conditions, i.e., vertical and horizontal velocity components, their distribution in the cross-section and the fluctuations in the direction of the river. Nejedly (1975) demonstrated both in the laboratory and by measurements in nature that longitudinal dispersion affects the rate of the biochemical oxygen demand and that its velocity coefficient in the river is a relatively simple function of the coefficient of longitudinal dispersion; deoxygenation increases with increasing longitudinal dispersion. This law is valid in the discharge range in which the deoxygenation process is evident and hence measurable. Figure 17.3 shows a diagram of a water-quality model in a polluted river reach expressed by the BOD, value. It consists of two parts. The dimensionless part B expresses the share of the pollution source considered, part A the share of the watershed-both in dependence on the exceedance of the discharge. For polluted river reaches (upstream of pollution sources or far downstream), part A is a complete model and can be constructed for any water-quality indicator. The relation B has two marked minima, the left in the region of mean discharge Q and the right in the range of low discharges; both demarcate the range of the maximum effect of the deoxygenation process between the site considered and the site
42 1
of the wastewater outfall. The magnitude and mutual position of the two minima often vary; the left is usually deeper, more extensive and simple, compared with the right one. A great discharge creates favourable conditions for a rapid course of the deoxygenation process; it has, however, only a short mean water detention time. With 10.* T
9
uo L
?
VL
B -
7a0
I
Fig. 17.3 Graphic illustration of a mathematical model of water quality expressed in BOD, values (Nejedlf, 1976): C - concentration of substances expressed as BOD,
low discharges the opposite holds true. With a certain ratio of the mean water detention time and flow conditions, the effect of the deoxygenation process is at maximum and the share of the pollution source in the considered river reaches the minimum. Of considerable significance is the relatively narrow range of the right minimum, where the coefficient of longitudinal dispersion reaches a transitional maximum, and hence there are favourable flow conditions for the deoxygenation process. If the discharges are low, and hence the waste load in the river especially high, and if the water temperature is also high, the risk that will perish increases. A small increase of the discharge is sufficient to moderate the intensity of the deoxygenation process considerably, even at the expense of a deterioration of the water quality, which is slightly compensated by the greater dilution of the wastewaters. The duration of the augmentation is limited by the water volume in the reservoir that can be released for the given purpose. Naturally, also another operation can be applied, i.e., to lower the discharge to shift it to the right of the right minimum of the pollution-source share and thus to attain again a reduction of the intensity of the deoxygenation process, even though it means a slight deterioration of the water quality in the river. On the contrary, if the water temperature is lower and there is no risk of high dissolved oxygen deficits with mass fish deaths, it is advisable to control the flow so as
422
to intensify the deoxygenation process as much as possible, i.e., to keep the discharge within the range of one of the two minima of the share of the pollution source and so to maximize the natural self-purification process of the river. Knowledge of the correlation between the water properties and the discharge permits an effective management of the water in the reservoir, also from the aspect of the water quality in rivers. The relationships seem more complicated than was hitherto assumed, however, their significance for respecting the natural environment in the management of surface waters justified the continuation of this demanding research work.
f7.1.3 Impact of reservoirs construction on man That reservoirs can change the life of man in many respects can be seen from Figs. 17.1 and 17.2. The aim of this book is to show how reservoirs can be used rationally for the purposes for which they were built, i.e., to supply water, control floods and establish an aquatic environment. This is certainly advantageous to man; man must have sufficient water in order to live and to work and should not have to rely on the random character of the natural hydrological regime. However, reservoirs also have adverse effects on man, some of which can be eliminated and others limited if suitable measures are introduced. A building site is a source of noise, dust, smoke, and water turbidity. All this is to the detriment of man and the human environment, both in rural areas and densely populated regions. The surroundings are affected by the construction of new roads. Traffic on public roads is hindered by heavy vehicles. The workmen’s barracks do not please the local people. The noise of a building site disturbs the surroundings and is bad for the health of the people working on the dam’s construction site. Blasting operations and production of aggregates is accompanied by much noise and dust. When the arch dam at Las Portas, in Spain, was built, over 5000 people were affected by the noise. This initiated studies to reduce this noise, in which engineers (measurements and control), physicians (traumas) and sociologists (changes observed in the people) participated. The noise intensity was determined [dB] : e.g., crushers (95), sorting plant (102), and concrete plant (89 dB). It is very difficult to reduce this noise; one way is to use the ground configuration as a natural screen. Other measures can be: - reduction of the intensity of the noise source, - isolation of the noise source, if its intensity cannot be lowered, - individual protection, if the first two measures are impracticable. In the US.,standards were issued by the Bureau of Reclamation (Environment Act, 1969) concerning exhaust fumes on building sites, dust from stone aggregates,cement and puccolane and waste water flowing from the building site.
423 Another environmental factor connected with a reservoir is the danger to the people living downstream of the dam. To rid the people living within the reach of a break wave of their fear, safety measures and a reliable alarm service must be introduced. Actually, the risk of dam failures is very small; out of loo00 reservoirs only 3 or 4 accidents occur annually and those are mostly caused by floods in regions with little hydrological information. Most dangerous is the first filling of a reservoir. At that time detailed observations and measurings and their immediate estimation are indispensable. To move people and industries from the inundation area of the future reservoir requires a very sensitive approach. To keep the people in the region, new houses have to be built and jobs found. Very important, but very costly, is the construction of new roads. One advantage is that these new facilities have a higher technical level, thus actually raising the living standard. In tropical regions, reservoirs and irrigation system even stop the migration of the population, which benefits their living conditions and health. For the construction of the Keban Dam in East Anatolia, it was necessary to move 30000 people; compensation for the expropriation exceeded the costs for the construction of the dam and power plant. Besides that, 300 km of roads, 48 km of railroads and a large factory had to be built. The building of the housing estates due to the construction of the Bigge dam in the river Ruhr basin took up two thirds of the o\erall costs. In Czechoslovakia, the greatest number of people had to be moved from the area of the Liptovska Mara reservoir with an inundation area of 21.6 km’. In all, 3400 people had to be evacuated from 30 villages; 17.5 km of railroadsand more than 30 km of roads had to be moved (Priehradnk dni 1972; Svetlik, p. 249; Mack0.p. 257).
The construction of a large reservoir is accompanied by the construction of new villages, industrial plants, farming and tourism. There is the danger of an uncontrolled development of these projects with all of its negative effects on the environment, including water pollution. It is therefore essential to predict the development of the reservoir surroundings and to control it by a well-prepared plan. New reservoirs often interfere with the scenic beauty of the countryside. In building them, often only their function and utility are considered and not how they fit into their surroundings. This is quite contradictory to ancient structures, such as aquaducts, bridges and dams, the creative beauty of which we still admire today. The simplicity of concrete dams, the simple arrangement of earthfill dams, and the harmony of the curves of arch dams can make a very pleasing scene. A dam, as an essential part of a reservoir, changes the landscape, but this does not mean that it has to be ugly. However, the untouched nature has to be replaced sensitively by a combination of the landscape and the man-made element. Dams are usually built in beautiful and attractive areas and those who build them must see to it that the scenery does not deteriorate. There is no reason why a well-built dam should not even improve the landscape.
424
The water surface of a reservoir is always pleasing. A sensitive place is the shore line with possible abrasion pallisades and surfaces that are bared-during fluctuations of the water level. Suitable measures help to minimize these negative consequences. When the construction is completed, good care should be taken to remove all the equipment no longer needed, and to leave the place tidy and pleasant. The impression of a reservoir differs when viewed from a distance or from close to. The total view of the dam should exhibit aesthetic lines, surfaces and materials; the railings, entries, etc., which the public pass must be executed perfectly in suitable colours, materials and shapes. These details reflect the cultural standard of the whole project. 17.2 RESERVOIRS AS AN IMPORTANT ELEMENT OF ENVIRONMENTAL POLICY
A construction as a work by humans, and thus an artificial element, should form a harmonious entity with the surrounding countryside. Large reservoirs have a rather complicated relationship vis a vis nature. This complexity arises from the functional structural relationship of an impounding reservoir and its environment (Fig. 17.4). It is given by - the large dimensions (of the order of tens of km),
Fig. 17.4 Functional structural relationships of an impounding reservoir and its environment
425
- complicated relationships between the variable, often random quantities, when the change of one causes the change in others: hydrological regime, recreation and tourism. With the exception of, e.g., recreation reservoirs, large reservoirs are as a rule built for economic reasons and not to improve the environment and therefore the whole project corresponds to these reasons. Of special importance are small reservoirs, which can greatly contribute to the environment and are attractive recreational facilities as they can often be found in regions without any large rivers. In Czechoslovakia the density of dams exceeding 15 m is one of the highest in the world, as one reservoir can be found for every lo00 km2. A high dam changes the landscape to such an extent that it deserves a careful architectural approach. Masonry and concrete dams cannot be hidden in the landscape, but they should therefore become an integral part of it. Dams of local material can be fitted to the slopes of the valley in such a way that only the horizontal line of the dam crest can be seen; the dam equipment (intake, outlets etc.) is usually submerged in the water and only the inflow from the lateral or side spillways sometimes outs obliquely into the side of the valley. Not only is the architecture important for the appearance of a dam, but also the quality of the work, a smooth and even surface of the concrete, straight curbs, railings and breakwaters, even pavements, good-quality coatings, etc. A high standard of the finishing operations is of the utmost importance. Even if a very little is left undone, this can easily spoil the total impression. The beauty of the water surface is spoilt by devastated banks and the uncontrolled erection of recreational facilities. From an analysis of the exceedance curves of the water level in the various seasons, a suitable vegetation cover of the banks and maintenance of the flat parts of the bottom under constant bakwater can be derived. Reservoirs in Marianske L k d , Bedfichov and oiliers prove how beautiful a reservoir can be if surrounded by forests.
Often the stream channel is affected just below the stilling basin. A proper balance between the absorption of kinetic energy of water in the stilling basin and the reinforcement of the channel below it must be found. The beauty of a natural river can be spoilt by diverting part of the discharge through canals to power plants or by withdrawals for water supply. A maintained minimum discharge downstream of a dam and an occasional flushing of the stream channel is not only an economic problem, but also has its impact 00 sanitary and aesthetic conditions. The construction of a reservoir can change the picture of the surrounding countryside either for the better or for the worse. An unfavourable impact is usually not caused by the construction of a reservoir itself, but is the result of badly carried out
426 or incomplete work. Projects should include scientific prognoses that can eliminate any unfavourable consequences. A price has to be paid to preserve the beauty of the environment, but even environmental protection has its limits; we cannot stop the construction or limit the main functions of a reservoir. Nature conservationists can ask for certain economic sacrifices and it is also their duty and their right to participate in the elaboration of projects, such as reservoirs that affect the environment extensively. Water management engineers should be able, with high professional erudition and social responsibility, to understand the present and future needs of a society and to choose such solutions that would meet all demands in the proper proportions.
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Bratranek A,, Influence of the Operation with Spillway Gates on the RunoB Downstream OF the Reservoir (in Czech). V U V - Works and Studies, 1956,92. Bratranek A., Research of Manipulation with Water in Reservoirs during a Floodwaves (in Czech). Annual Report, VUV, Prague, 1953. BroZa V., Lncreased Flood Control Effectiveness of Reservoirs (in Czech). CVUT, Prague, 1964. Brota V., The Use of Mathematical Statistics and Probability Calculations For Flood Control of Reservoir Outflows. Manual on Outflow Control from Reservoirs (in Czech). VUV, Prague, 1965. Brota V., Protective Effect of Multi-Purpose Reservoirs (in Czech). Dam Conference 1965, I. CSVTS. Prague, 1965, pp. 170-179.
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Abrasion, 412 Accumulation in flood plains, 47 Accuracy of statistical characteristics, 143 ff - - the characteristics of flood regimes, 304 - - the storage capacity, 231 - - within-year release control, 239 ff Analysis, decision, 380 ff - of cascades of reservoirs, 276 ff - - flood-control reservoirs, 297 ff Andreianov’s method, 217, 237, 239-240.269 Aspects, socioeconomic, 406 Auto-correlation, 83 Automation of monthly flow balance, 30 Balance deficit, 217 Balancing, progressive, 61 Bank abrasion, 412 - slides. 412 Biochemical oxygen demand (BOD,), 420-421 Bottom outlets, 326 Brovkovich’s probability paper, 136-137 Buffer release control, 261 - storage reservoir, 257,265 Capacity, controlled, 31 1
-, flood control, 187,311,325
187.324 -.-, surcharge, uncontrolled, 3 11
Cascade of power plants, 262, 278 ff - - reservoirs, 46,275 ff, 3 12 Character, representative, 91 Characteristics of a reservoir, 70 -, statistical, 96, 105, 143, 145, 225 ff, 269 Chegodayev’s formula, 97, 101
Classification of reservoirs, 41 ff, 46,48 ff Coefficient of economic efficiency, 369 - the yield, 51 Compensation, 263 -, over-year, 267 - regime, 264,267 Component, over-year, 194 ff, 225,269,294 -, within-year (seasonal), 194 & 208 ff, 215 Composition of probability curves, 213 Conservation reservoirs, 23,52 Control, compensation, 263 -, daily, 5 I, 246 ff - for occasional withdrawal, 52 - in a cascade of reservoirs, 53 - in a system of reservoirs, 53 -, long-term release, 54 - of sudden discharges, 52 -, over-year, 51. 191 ff, 233, 339 -, short-term release, 52 -, weekly, 51,257 ff -, within-year (seasonal), 51,234 ff, 339 Cooperation ofa system of reservoirs, 354 - - reservoirs, 284 Correlation, 83, 167 - coefficient, 85 Cost distribution, 384 IT, 387 Costs, investment, 369, 385 -,joint, 384 -, OMR, 369 ff, 385 -,special, 385 -,transferred, 369, 385 Criteria for decision analysis, 380 Curve, anti-failure, 283 -, rule, 283, 318 ff, 331 ff, 353 Curves, auxiliary, 64.67 -, exceedance, 64,66,97,103,128, 144,170,192, 198,244.288
-
440 Curves, fitted theoretical, 237 ff -, mass, 67 ff, 94, 191,234,258, 266, 277 -, probability, 213 ff, 236, 269
-, relation, 64 -, summation, 72 ff, 170,249,250,254,289 -, time (chronological), 64-65, 170 Cycle, annual, 241 - of the reservoirs, 50 - - _-, daily, 50, 259 - - - -, over-year, 50,283, 339 - - - -,seasonal (within-year), 50, 283, 290 - - - -, weekly, 50,259 Data-bank, 31 -,hydrological, 29 ff, 31,61, 171, 297 ff -, input, 171 Decomposition in the sub-systems 345 Deficit of water, 185,268,272,358 Demands on drinking water, 175, 179 ff - - water for agriculture, 177 - - - - fish and poultry farming, 178 - - - _hydro-power production, 177 - - - - industry, 176 - - - - navigation, 178 Density of random process, 149 - - - -,spectral, 156 ff, 303 Depth-area curve, 70 Depth-release curve, 308 Depth-volume curve, 71,251,308 Design dry year, I77 - low-flow period, 196,239 - period, 95 - reliability of a reservoir, 185 ff, 193,285,338 - waterdemands, 173 Determining new water resources, 179 Development of water reservoirs, 20 ff - - _ management calculations, 32 ff Deviations of relative active storage capacity, 231 - - _yield, 230 Discharge conditions, complicated, 334,336 - forecast, 320 - from the interbasin, 265 -, maintained minimum, 171 - maxima,annual, 299 -, m;iuimum flood, 298 ff -, non-damaging, 301-302,305,3 14
Discharge regime, 282 - regulation, 282 - series, synthetic (chronological), 21 7, 238,267,276 -, wasted, 288 Distribution, double exponential, 1 18 -, sampling, 148 Drainage water, 363 Duration of the reservoir emptying, 232 Effectiveness, economic, 368 ff -, intangible, 369 - of re-pumping, 260 - of reservoirs, 343 ff, 368 ff - of the optimal option, economic, 376 Effects, economic, 342 - of operation rules, 337 - of system solution, 353 -, secondary, 379 Environment 343,407 ff, 424 -, human, 407,409 - of small reservoirs, 367 Errors of maximum discharges, 304 - - the storage volume, 240 Estimation of exceedance probability curve parameters 128 ff - - probability distribution parameters, 148 -, statistical, 319 European Charter on Water, 407 Evaporation, 181,242 Exceedance probability curves, 269 ff Excess, 108, 144 - outflow, 173 Factors, economic, 392 Failures in water supply, 272, 338 Failure year, 221 Filling of the reservoir, first, 294 Fish farming, 419 Floating of timber, 261 Flood control, 185. 262.295 ff. 3 I5 - - capacity, 187,295.311, 325 - - effect, 295,314 ff, 405 - - - of the active storage capacity. 3 15 ff - - function, 295 IT,305, 364 - - - of small reservoirs, 364 - - release, 295, 311 ff
441 Flood-control reservoirs, 22, 295 ff - - systems, 312, 358 Flood discharges, 298 ff - regime, 303 - routing, 326 - volumes, 300 ff, 315 - wave, 304 Forecast data, inaccurate, 322 - of discharges, 265,271,320 - - the quality of withdrawn water, 180 - - the spring runoff, 56 Fragment method, 37, 165 ff, 172 Frequency, cummulative, 104 - - relative, 104 -* relative, 104 Fuller triangle, 382 Function, autocorrelation, 151 ff -, correlation, I5 1 ff, 155,200 ff -, distribution, 102-103, 149 - of reservoirs, evaluation, 341 - - -, monitoring, 341 -, recreational, 287 -, transfer response, 158 Goodness-of-fit (between probability distributions), 146 Graphs to determine the over-year component (Gugli’s, Nachazel’s, Pleshkov’s), 200 ff Ground water, 362, 364 Henry’s line, 112 Hydrofund, 31 Hydrology in Czechoslovakia, 29 ff Hydro-power plants, base-load, 279 - - -, peak-load, 177,250-251, 255,259,279 - - -, pumped-storage, 177, 255,259 - - -, run-of-the-river, 177 - - reservoirs, 23 Ice conditions, 415 184 IIASA, 174 Impacts of reservoirs, biological and chemical, 416 ff - - - construction on man, 422 ff - - -, negative, 408 - - -, physical, 409 ff
- cover,
Inflow, constant,
169
-, controlled, 169 -, deterministic, 169
-,flood, 169 - from the interbasin, 277 -, natural, 171 -, stochastic, 169 - to a reservoir, 168 N, 276 Interbasin. 265,277 Intervention, human, 57 Investment costs (derived, direct, indirect), 369 Investments, capital, 368,369 Kritsky’s and Menkel’s method,
- - - -,second, 198
196 ff
Law of maximum value, 118 - - proportional eNect, 113 Leakage, 184 Liapichov’s method, 238,240 Load diagram, 249 Losses, economic, 358,404 -, evaporation, 242 LPI method, 384 N Markov chain, 82,160,279 - processes, 81 N Matrix of reservoir purposes, 58 Mean square error, 83 Method Monte Carlo, 36,355 Method of central and satellite stations, 167 - - comparative (relative) effectiveness, 369 ff - - continuous functions. 159 - - decision analysis, 380 R - - fragments, 37, 165 ff, 172 - - linear regression model, 160, 163 - - non-continuous functions, 159 - - orthogonal transformation, 168 - - total (absolute) effectiveness, 373 ff Methods, analytical, 74 -, graphical, 62, 308 -, mathematical, 74 - offiltration, 158 -, probability, 80, 239 -, statistical, 80
442 Model, economico-mathematical, 349 - of a flood regime, 303 ff regression, 303 Modelling annual flow sequences, 159 ff - flow series, 159 ff - in a system of stations, 166 ff - monthly flow sequences, 162 ff Module coefiicient, 94, 109 Modulus, 143 Moment, central, 106 -, general, 106 -, initial, 106 Moran's method, 205 ff
-.
Network of canals, 262 Noise, white, 153, 303 Occurrence period, 300 OM R (operation, maintenance and repair) costs, 369 ff, 385 Operating schedules, 318 ff, 347 Operation, day-to-day, 330 ff, 339 - for river flow regulation, 271 ff - rules, 322 -, two-stage, 337 Optimization of a cascade, 278 ff - - multi-purpose systems, 356 ff _ - reservoir storage capacity, 374 Outflow, excess, 173 - from a reservoir, 173 Over-year release control, ideal, 233 Parameters, statistical, 105 Peak-load, 249 ff Performance of the reservoir, 38 ff, 54 _ _ water storage reservoir, 39 ff Period, high-flow, 332, 337 -, I O W - ~ ~ O W(dry), 196,21 I , 237,266, 280,316,332 -, wet, 267, 294 Polder, 43 Policy, environmental, 424 Ponds, 43,362 - of Bohemia, 21 Power of the wind, 261 - system, 248 Precipitation series, 303 Probability curves, 213 ff, 236,241 - density, 103, 149
Probability density, standardized, 11 1, 138 - distribution, binomial, 124, 129, 134, 136, 138 - -, empirical, 146 - -, exponential, 123, 133,299 - - -,Goodrich, 123, 137,302 - -,extreme, 116 - -, Fisher, 138, 148 - -, Galton log-normal, 114 - -, Gauss-Gibrat log-normal, 113, 138 - -, Gumbel, 118, 132,137, 138 - -,Johnson log-normal, 114, 138 - -, Kritsky-Menkel's three parameter, 122 - -, log-normal, 112, 131, 136,299 - -, normal (Gauss-Laplace), 110, 136, 138 - -, Pearson, 119 - - - of the first type, 119, 138 - - - of the third type, 120, 138, 144, 159,299 - -,Poisson, 126, 138 - -,Snedecor, 138, 148 - -, Student, 138, 148 - theoretical. 109 ff, 146 Ven-Te-Chow log-normal, 114, 138 - -, Weibull, 116, 138 - -, x2, 138, 148 Probability exceedance, 134, 300 Probability paper, 133 ff, 210 Prognosis, demographic, 179
-.
-.
Random phenomena, 80, 101 - process, 80, 101-102, 148 ff - -, ergodic, I5 I - -, non-stationary, 150 ,stationary, 150 - sequence, 102.148 ff, 155 - variable, 80, 101-102, 105 Records, hydrological, 91 Recreation reservoirs, 43, 367 Regime, groundwater, 410 -, hydrological, 409 -, stationary hydrological, 195 Relationship, curvilinear, 100 -, linear correlation, 99 Release, compensation, 267 Release control, bufTer, 261 - -, daily, 246 ff - - for irrigation, 256, 262 - - in a cascade of reservoirs, 275 ff
--
443 Release control in a system of reservoirs, 281 N - - in the period of first filling, 292 N - -, non-periodical, 246,261-262 - -, over-year, 191 N, 248, 290,293 - periodical, 246 - -, short-term, 246 N - - with various water supply reliabilities, ' 290N - -, within-year (seasonal), 234 N, 248,293,3 I8 Release from a reservoir, 173, 305 N Reliability, duration (time)-based, 219, 394 IT - indices, 392 N -, occurrence-based, 219. 394 N - of augmented discharges for navigation, 402 - - flood C < > l l I ~ O l . 309 N. 319. 389 water tor hydro-power plants, 402 - - water supply, 183 R 217 N, 285, 389,395 - - - _ for households and industry, 401 - - - _for irrigation, 399 -, quantity-based, 219,394 ff -, rate of, 296 Reproduction recovery of reservoir, 378 Requirements of public water supply, 175 Reservoir, balancing, 52, 249, 25 1-252, 279,283 -, buNer-storage, 257, 261. 265, 271, 273,283 - capacity, 31 1 -, conservation, 48, 52 - design, 28 -, distributing (distribution), 52, 253 N -, downstream, 250, 276 -,dry, 43 -, excavated, 42 -, flood-control, 49, 262. 297 N, 361 -, highest upstream, 275 -, hilltop, 46 -, impounding, 43-44,275,367 -, bateral, 42, 43, 286, 367 -, man-made, 42 -, natural, 41 - of a pumped-storage power plant, 42,260 - operation, 329 N, 341
-.
~
Reservoir parameters, 18I - release schedule, 173 -, seasonal, 284,353 -, storage, 52 -, through-flow, 42 -, tributary lateral, 44-45 -, underground, 47 -, upper, 260 -, upstream, 276 - with river flow regulation, 263 Reservoirs, agricultural, 23, 262 -, cascade of, 275 N -, cooperation of. 284,361 -, fish farming, 367 - for floating timber, 22, 261 - for minig purposes, 22 - for recreation purposes, 25, 367 -, hydro-power, 279 - in Czechoslovakia, 21, 24 - in the world, 26 -, multi-purpose, 384 -.shallow, 415 -, small, 362 ff -, system of, 281 N, 361 -, tropical, 420 -, water supply, 285 - with over-year release control, 228 -' with power plants, 242 Retaining dams, 41 1 River flow regulation (compensation control), 5 1,263 IT, 283.356 - - -, over-year, 269 N River training, 305 Rule curves, 331 N Rules for release control. 330 N Sampling distributions. 148 Sanitary measures, 41 7 Seepage, 183 Seismic effects, 415 Selection of a design period, 95 Sequence structure, internal, 196 Series, chronological, 61, 170 -, hydrological real, 145,239,292 natural, 61,74 -, pseudo-chronological, 101, 170 -, synthetic (modelled), 61, 101, 267,333 Siltating of a reservoir, 188,411
-.
444 Simulation, 61, 218 ff, 265,288 Skewness, 108, 144 Slides. 412 Sludge-settling ponds, 54 Solution, digital, 309 -, graphical, 75, 258,266, 308 -, graphical-numerical, 25 I, 309 - of conservation reservoirs, 34 ff - of flood control effect, 32 -, successively balancing, 74 Spillway, gated, 3 I I , 325-326 -, ungated (free), 31 I , 325-326 Standard deviation of a series, 83, 143 - errors, 145 Standards of design reliability, 186,389 Storage capacity, 173,222 ff _ - , active, 187, 315 - -,dead, 187 - -, total, 187 - function of reservoirs, 89 ff Storage-yield curve, 79 Stratification in reservoirs, 414 Surcharge capacity, 187,324 System, flood-control, 312 -, multi-purpose, 356 - of impounding reservoirs, 47,354 _ _ reservoirs, 47, 281 ff, 354 - - _ for public water supply, 282 ff - - river points, 268 - solution, 353 - with water diversion, 286 ff Systems, water-management, 343 Tailing dams, 54 Temperature regime, 414 - stratification, direct, 414 - -, indirect, 414 Test, Kolmogorov-Smitnov, 147 - of goodness of fit, 146 - operations, 292 -, x2. 146 Time factor of forecast, 272 - -, standard, 370 Transformation effect of the surcharge capacity, 325 - - _ reservoirs, 360 -, logarithmic, 100 - of flood waves in small reservoirs, 365
Urban's method, 306 ff Utilization of a reservoir, comprehensive, 58 Variability of over-year component, 226 Variable, random, 80, 101-102, 105 -, standardized, I I 1 Variation, 107, 143, 229 Water demand, 173 ff, 247,340 - diversion, I70 ff, 286 ff, 312 - losses, 77, 173, 181 ff, 272, 277 - - byevaporation, 78 Water management, 7, 19, 85 - - balance, 89 - - calculations, 32 Water Management Plan, 89, 309 Water management plan for reservoirs, 242-243 Water-management systems, 343 ff - - -, analysis, 345 - _ _ , definition, 345 - - - for flood control, 358 _ - _ , irrigation, 348 _ - _ , multi-purpose, 348, 357 _ _ _ , power and irrigation, 346 _ - - ,water supply, 351 _ - year, 235,333 Water quality, 413, 417 - recreation, 178 - sports, 178 - storage within the watershed, 48 - supply, public, I75 ff, 179 ff _ _ tank, 246 ff - year, 96 Weather conditions, 413 Winter regime, 414 Withdrawal diagram, weekly, 257 - from a reservoir, I73 ff - reliabilities, various, 291 - site, 273 Wolfs numbers, relative, 154 Year, calendar, 96 -,water, 96 -, water-management, 235 Zaruba's method,
308
