Forecasting Residential Water Demand: Case Study

Forecasting Residential Water Demand: Case Study

Downloaded from ascelibrary.org by RMIT UNIVERSITY LIBRARY on 04/23/15. Copyright ASCE. For personal use only; all rights reserved.

Shirley Gato1; Niranjali Jayasuriya2; and Peter Roberts3 Abstract: A new daily time series model for East Doncaster, Melbourne, Australia, is being evaluated. The model depends on the postulate that total water use is made up of base use and seasonal use, where base use is characterized by the water use during winter months and seasonal use on seasonal, climatic, and persistence components. Using the daily data collected by Yarra Valley Water for East Doncaster water supply distribution zone and the corresponding rainfall and temperature data from the Bureau of Meteorology from 1990 to 2000, the base values were calculated based on the lowest months of water usage in a year and were correlated with the day of the week and temperature and rainfall. Results revealed these three factors to be statistically significant and therefore, base use to be climate dependent. The seasonal water use is modeled by a series of three equations. The separation of the random component from the climatic variable resulted to a better R2 of 86%. The model is further validated using different set of data from 2000 to 2001 yielding a R2 of 86%. DOI: 10.1061/共ASCE兲0733-9496共2007兲133:4共309兲 CE Database subject headings: Water demand; Water supply; Forecasting; Regression analysis; Residential location; Case reports; Australia.

Introduction Residential demand modeling is important in the current climate of water restrictions and water conservation in Melbourne, Australia. For the past nine years, the city has been experiencing low rainfall conditions which forced water authorities to impose water restrictions after 20 years of unrestricted supply. A 12% per capita consumption reduction target has been set to be achieved by the year 2010 in the Water Resources Strategy developed for Greater Melbourne. The need to evaluate the potential benefits of incorporating information from seasonal climatic forecasts and short-term weather forecasts in operational water management decisions is also recommended 共Water Resources Strategy Committee for the Melbourne Area 2001兲. In line with this, water conservation efforts and initiatives are being encouraged and implemented especially in household usage. The interest in quantifying the impact of water conservation programs on reducing water demand is increasing. The existing demand models are based on studies of bigger cities and on a proposition that total daily water use is made up of base use, which is weather insensitive, and seasonal use. A previous model by Gato et al. 共2003兲 overcomes these difficulties by 1 Ph.D. Student, School of Civil and Chemical Engineering, RMIT Univ., GPO Box 2476V, Melbourne, Victoria, 3001 Australia. E-mail: [email protected] 2 Senior Lecturer, School of Civil and Chemical Engineering, RMIT Univ., GPO Box 2476V, Melbourne, Victoria, 3001 Australia. E-mail: [email protected] 3 Demand Forecasting Manager, Yarra Valley Water, Private Bag 1, Mitcham, Victoria, 3132 Australia. E-mail: [email protected] Note. Discussion open until December 1, 2007. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on December 9, 2004; approved on May 3, 2006. This paper is part of the Journal of Water Resources Planning and Management, Vol. 133, No. 4, July 1, 2007. ©ASCE, ISSN 0733-9496/2007/4309–319/$25.00.

focusing on modeling consumption patterns observed at the water supply distribution zone level and allows both base and seasonal uses to be weather dependent. The model adopted 共Maidment et al. 1985兲 a heat function approach in calculating the potential component of seasonal water use. The model is being refined and other methods and approaches are being evaluated to improve the model. The aim of this paper is to describe a model of consumption patterns observed at the water supply distribution zone level based on the Zhou et al. 共2000兲 approach. The results are compared with a previous model by Gato et al. 共2003兲. The improved model has a potential to be used as a tool for better determining impacts of water conservation initiatives and restrictions on water demand.

Literature Review Previous papers on forecasting urban water demand consider annual or monthly data and only few address daily water use. Maidment et al. 共1985兲 formulated a short-term forecasting model that rests on the following three propositions: 1. Total urban water use consists of base use and seasonal use, with base use considered as weather insensitive and observed as average use in the winter months; and seasonal use as weather dependent and is the difference between base use and total use during the other months of the year; 2. In the absence of rainfall, seasonal use follows a characteristic pattern over the year that is dependent on temperature conditions; and 3. A sudden drop in seasonal use is observed due to rainfall but gradually diminishes over time. Maidment and Miaou 共1986兲 applied the model to daily water use data from nine cities in the United States, three each from Florida, Pennsylvania, and Texas. The overall coefficient of determination, R2, for the nine cities averaged 96% in Texas, 73% in Florida, and 61% in Pennsylvania. They concluded that, as a proportion of the mean annual use, the average seasonal use for the

JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT © ASCE / JULY/AUGUST 2007 / 309

J. Water Resour. Plann. Manage. 2007.133:309-319.


three cities in each state were 23% in Texas, 15% in Florida, and 5% in Pennsylvania. The response of water use to rainfall depends first on the occurrence of rainfall and second on its magnitude. In Austin, Texas, a spatially averaged rainfall series showed a clearer relationship with water use than rainfall series from a single gauge. There was a nonlinear response of water use to air temperature changes. They also concluded that for small cities, such as College Station, Tex., there was a relatively higher inherent randomness in the daily water use data than in larger cities, so smaller cities would be harder to model than larger cities. Zhou et al. 共2000兲 also adopted the methodology based on time series analysis in which daily water consumption is considered to be the sum of base consumption and seasonal consumption, the latter comprising of seasonal, climatic, and persistence components. They formulated a time series model as a set of equations to forecast daily urban water demand using the daily water use from Greater Melbourne, Australia. These equations represent the effects of four factors, namely, trend, seasonality, climatic correlation, and autocorrelation on water use. The annual trend in base consumption was represented by a polynomial as a function of time. Seasonal water use was modeled by seasonal, climatic, and persistence components in consideration of the summer and winter six months separately. The recommended model from the study has a R2 value of 90% and has performed satisfactorily when tested using a cross-validation procedure and an independent data set during the summer period from December 1, 1996 to January 31, 1997. Both of these models considered base use as weather independent and represented it by a polynomial as a function of time. However, in modeling seasonal use, Maidment and Miaou 共1986兲 employed a nonlinear heat function relating water use to air temperature during rainless periods to deseasonalize the series while Zhou et al. 共2000兲 used the Fourier series to estimate the seasonal cycle in water use. While Maidment and Miaou 共1986兲 relied on just two climatic variables, rainfall and temperature, Zhou et al. 共2000兲 in addition to these two variables included Class A pan evaporation, Antecedent Precipitation Index 共API兲, and the number of days since previous rainfall. Maidment et al. 共1985兲 reported that the reason for using the heat function approach in estimating seasonal use rather than more conventional methods such as fitting Fourier series is that changes in air temperature can either increase or decrease water consumption, but a rainfall occurrence can only reduce consumption. A Fourier series model of historical seasonal water use may not represent these effects adequately as it contains rainfall as well as temperature effects. Maidment et al. 共1985兲 also analyzed the short memory series which is composed of the rainfall effects and the random component using Box–Jenkins time series analysis techniques, whereas Zhou et al. 共2000兲 determined climatic consumption and persistence components by individual regression. Zhou et al. 共2000兲 reported that there was no consideration given to component interaction in their model although it is accepted that they are present. This study applied the Zhou et al. 共2000兲 deseasonalizing approach but did not include Class A pan evaporation in modeling daily water use. This study focuses on East Doncaster water supply distribution zone, which is mostly residential, whereas Zhou et al. 共2000兲 covered the Greater Melbourne area, which includes residential, industrial, and commercial areas. The same water supply distribution zone adopted by Gato et al. 共2003兲 is used in this study in order to compare the results of using a heat function and the Fourier analysis in deseasonalizing the seasonal series. A

Fig. 1. Locality map for East Doncaster, Victoria, Australia

simple linear regression analysis to forecast short-term water use is also adopted.

Study Area As reported in Gato et al. 共2003兲, East Doncaster is located 18 km east of Melbourne Central Business District, Victoria, Australia 共Fig. 1兲. East Doncaster is a large residential suburb in Melbourne’s outer east. It is a fully developed “residential area” that has little to no commercial and industrial areas. It has been one of the areas in the eastern suburbs with high residential developments over the last 20 years. It is for this reason that East Doncaster was chosen for this study. Commercial and industrial areas have different water needs, thus exhibiting different consumption trends. This study assumes that East Doncaster is purely a residential zone. East Doncaster’s population has increased by 5,301 or 25.5% over the 1981–1996 period. In 1996, the population reached 26,063 persons. Between 1991 and 1996, the population decreased by 256 persons or 1%. The largest changes in age structure have been in the 50–59 age group 共+1,714兲 and the 5–17 age groups 共−1,348兲. The proportion of homeowners has grown from 29.4% in 1981 to 56.5% in 1996 共www.doi.vic.gov.au兲. East Doncaster has a total area of 1,161 ha. This is made up primarily of residential use, covering 66% 共770 ha兲 of the total land area. Other major land uses include parkland 共12%兲, open areas 共11%兲, commercial use 共1%兲, sporting 共7%兲, and schools 共3%兲共Yarra Valley Water, private communication, 2004兲.

Methodology Water consumption varies over time due to the effects of population, socioeconomic factors, climate and as a result of restriction

310 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT © ASCE / JULY/AUGUST 2007


in water use because of supply limitations. There are several approaches to water demand forecasting and due to the complexity of other approaches; a simple time series approach is employed in this study. Maidment et al.’s 共1985兲 proposition that daily water use is made up of base and seasonal use, both exhibiting trends through time, was also adopted in this study.


Base Use, Wb Base use, Wb, is defined as the portion of water use 共mainly indoor use兲 that is insensitive to climatic conditions and is typically characterized by the winter use. However, studies indicate that winter use can be quite weather sensitive in some areas, for example, Florida 共Gibbs 1978兲, southern California 共Miaou 1987兲, and East Doncaster, Australia 共Gato et al. 2003兲. Base use exhibits trends due to population and socioeconomic factors such as household income and water price. There are various methods available for estimating base use including regression against population and socioeconomic variables such as household income and water price or by fitting a polynomial function of time to the lowest month of water consumption in each year. In this study, a more general model framework by Miaou 共1990兲, which allows base use to be weather dependent, is adopted. Base use, Wb, in megaliters per day 共ML/day兲 can be represented by a function of socioeconomic and climatic variables: Wb = hd共T,R,X;␤兲 + vd

共1兲

where d⫽daily time index; hd共兲⫽function h at day d ; T⫽daily maximum air temperature; R⫽daily total rainfall; X⫽set of socioeconomic variables; ␤⫽parameter vector to be estimated; and v⫽model residuals. Seasonal Use, Ws Seasonal use, Ws, is the remaining water use 共mainly outdoor use兲 after the expected base use, Wb, is subtracted from the total daily water use, Wd Ws共d兲 = Wd共d兲 − Wb共d兲

共2兲

The variation in seasonal use is expected to be influenced by prevailing weather conditions such as air temperature, evaporation, and rainfall. Water consumption tends to exhibit a similar pattern to that of the air temperature and evaporation that vary periodically over a year. Rainfall varies in a more random fashion, occurring in bursts, but zero most of the time. Maidment et al. 共1985兲 postulated that seasonal use, Ws, has three components, namely, potential seasonal water use that is dependent on temperature in the absence of rainfall; a water use reduction due to rainfall; and the zero mean random component.

which is modeling two aspects of seasonal water use, periodic variation through time and the response of water use to changes in air temperature during rainless periods, is adopted. Thus, the data are selected such that there is no rain in the day under consideration. The method adopted by Zhou et al. 共2000兲 in determining potential seasonal use using Fourier series is used in this study. This allows the results of this study to be compared to the results obtained in Gato et al. 共2003兲 where potential seasonal use was modeled as a heat function. According to Zhou et al. 共2000兲 the Fourier series can adequately represent seasonal cycles within a year and its coefficients can be easily obtained from the measured data. The seasonal cycle as a Fourier series adopted in their study k1

Wsp共d兲 = a0 +

兺 k=1

冋冉冊 ak cos

冉冊册

2␲k 2␲k d + bk sin d 365 365

共3兲

where d⫽daily time index; a0⫽mean of the values being evaluated; coefficients ak and bk are computed by using the discrete Fourier transform of the arithmetic daily means; and k⫽number of harmonics which is found to be 8 for this study; and d = 1 , 2 , . . . , 365 where d = 1 corresponds to the first day of each year, and the seasonal pattern repeats on a yearly basis. Weather Component, Wsc The effects of rainfall and air temperature are then isolated as Wsc共d兲 = Ws共d兲 − Wsp共d兲

共4兲

where d⫽daily time index; Wsc⫽daily water use due to weather component such as rainfall and air temperature; Ws⫽daily seasonal use as explained in Eq. 共2兲; and Wsp⫽daily potential seasonal use as explained in Eq. 共3兲. The weather component is regressed against time, daily rainfall, or a substitute variable for rainfall effects and daily average air temperature. Persistence Component, Wsr The persistence component as employed in Zhou et al. 共2000兲 represents a short memory process. An autoregressive procedure is fitted to the residual time series to account for the dependencies of water use on its past values r r r Wsr共d兲 = ␾0 + ␾1Ws共d−1兲 + ␾2Ws共d−2兲 + ¯ + ␾tWs共d−t兲

共5兲

Wsr⫽daily

where water use component that accounts to its past use values; d⫽daily time index; ␾1 , ␾2 , . . . , ␾t⫽coefficients; and t⫽order of the autoregressive procedure.

Results and Discussions Potential Seasonal Water Use,

Wsp

There are various methods available in modeling seasonal variation in water consumption. This includes 共1兲 the formation of 12 differences of data 共Box and Jenkins 1976兲; 共2兲 the development of an autoregressive model of Wsp whose coefficients reflect seasonality; 共3兲 the use of the arithmetic or Fourier-smoothed daily means of Wsp; and 共4兲 the regression of Wsp against a seasonal variable such as maximum temperature. Gato et al. 共2003兲 adopted the fourth method, which assumed that there is a functional relationship between water use and temperature that is valid in the absence of rainfall. A heat function

As this study does not include variables representing population and other socioeconomic variables that may affect trends, time is employed as a substitute in explaining trend variations. The evaluation begins with an estimation of base use. Model comparisons are then performed for the components of the seasonal water use, potential water use, and short memory series. The Microsoft EXCEL package is used to carry out the stepwise regression analysis. Model performances are evaluated according to two criteria: standard error 共SE兲 and the coefficient of determination, R2. A favorable model is the one with high R2, but low residual standard error.




Fig. 2. Daily urban water demand, East Doncaster, Victoria, Australia 共April 1, 1991 to April 1, 2004兲

All parameters considered in this study are tested of its statistical level of significance by its P value. The P value is the smallest level of significance at which the parameter is significant 共Devore 1990兲. Conventionally 共and arbitrarily兲 a p value of 0.05 共5%兲 is generally regarded as sufficiently small. The 5% value is called the significance level of the test 共Campbell and Machin 1999兲. In this study, a P value lower than 0.05 is considered statistically significant. Data Used The same set of data used in Gato et al. 共2003兲 is employed for this study. This includes the daily water use 共ML/day兲 for East Doncaster water supply distribution zone 共Fig. 2兲, total daily rainfall 共mm兲, and daily maximum temperature 共°C兲 recorded at Mel-

bourne Regional Office 共Station No. 86071兲 from April 1, 1991 to April 1, 2001. Missing data and invalid values as noted in the record are not included in the analysis. Base Use As found in Gato et al. 共2003兲, the lowest level of monthly water use in East Doncaster is from May to September. The month with the lowest total monthly water use is selected for each year to estimate the base use in Eq. 共1兲. Fig. 3 shows all the days of the month with the lowest water usage in a year. In 1991, September has the lowest total monthly water use, thus the first day number 共X-axis兲 in Fig. 3 corresponds to September 1, 1991. The base use exhibits a step change in May 1994 and a different growth rate before May 1994 and after May 1997 共Fig. 3兲. From 1994 to

Fig. 3. Daily base water use, East Doncaster Zone, Victoria, Australia 共1991–1999兲 312 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT © ASCE / JULY/AUGUST 2007



• Calculating base use from Eq. 共6兲 during the month of lowest consumption in a year from 1994 to 1997; and • Estimating ␩ = 2.77 ML/ day, as the average of the difference between the calculated and recorded base values from 1994 to 1997.

Fig. 4. Average 7-day water use pattern, East Doncaster, Victoria, Australia 共1991–2001兲

1997, maintenance problems were encountered in the collection of water use data according to Yarra Valley Water staff, which could explain the lower consumption during these periods. While Gato et al. 共2003兲 included three dummy variables to represent these water usage changes during these periods, in this study, water usage is adjusted as shown in Fig. 3. Date Adjustment As the data during the period 1994–1997 encountered some maintenance problems and were lower compared to other water usage in other periods, it was assumed that the values recorded by the logger is short by a value which is constant throughout the period in question. Therefore, the daily water usage is adjusted by adding a constant value, ␩, which is determined by • Regressing the water usage during the month of lowest usage in a year with time excluding years 1994–1997. The regression equation Wb共calc兲 = 4.379 + 共0.0004d兲

共6兲

where Wb共calc兲⫽adjusted daily water usage during the lowest month 共ML/day兲 and d⫽daily time index;

Base Use Model’s Parameters The following variables were included in the regression model: • Weekends or weekdays as preliminary analysis of East Doncaster daily water demand data revealed that water demand was higher during weekends than on weekdays 共Fig. 4兲; • Total daily rainfall, to determine its significance into the daily base water consumption; and • Maximum temperature to determine its effect on daily base use. Table 1 shows the coefficients and associated statistics obtained from the regression analysis when different combinations of variables are considered. Based on the regression analysis, the following equation is obtained and considered as the final model to estimate the base use: ˆ 共d兲 = 3.656 + 0.0004X + 0.0368X − 0.035X + 0.042X W b 1 2 3 4

共7兲

ˆ 共d兲⫽estimated daily base water use 共ML/day兲; d⫽daily where W b time index from beginning of series 共1 on April 1, 1991兲; X1 = d; X2 represents weekends or weekdays 共weekends⫽1 and weekdays ⫽ 0兲; X3⫽daily total rainfall 共mm兲; and X4⫽maximum daily temperature 共°C兲. The addition of variables such as temperature and rainfall into the model increased the R2 and reduced the SE values 共Table 1兲. It is evident from the model obtained that the base use is dependent on temperature and rainfall and hence weather dependent. This could be attributed to people watering their gardens in winter 共Gato et al. 2004兲 or special school activities and parks watering. The result of base use being weather dependent is similar to results obtained by Gato et al. 共2003兲. The average effect of weather 共rainfall and temperature兲 on the base demand over the nine years 共3,285 days兲 can be evaluated by using the average recorded daily rainfall of 1.8 mm and the average recorded maximum temperature of 20° C. Substituting these values in Eq. 共7兲 resulted in the following:

Table 1. Coefficients and Associated Statistics of Alternative Base Water Use Models

Model B1

Time 共X1兲

Intercept

4.415 共5.2E − 119兲兵41.227其 4.527 B2 共6E − 120兲兵41.710其 3.702 B3 共1.35E − 20兲兵10.113其 3.656 B4 共1.07E − 20兲兵10.147其 Note: Values inside the fences such as statistically insignificant.

Total rainfall 关mm 共X2兲兴

Maximum temperature 关°C 共X3兲兴

Weekends or weekdays 共X4兲

SE 共%兲

R2 共%兲

0.0004 90.56 12.39 共2.06E − 9兲兵6.202其 0.0004 −0.043 88.38 16.86 共3.16E − 9兲共1.65E − 4兲兵6.126其兵−3.820其 0.0004 −0.038 0.046 87.64 18.55 共2.07E − 10兲共0.001兲共0.019兲兵6.608其兵−3.299其兵2.359其 0.0004 −0.035 0.042 0.368 86.22 21.47 共9.66E − 11兲共0.002兲共0.030兲共0.002兲兵6.739其兵−3.087其兵2.175其兵3.166其 parentheses and curly braces are p and t-statistic values of the corresponding coefficients. A p value ⬎0.05 is



Table 2. Fourier Coefficients for Seasonal Cycle of Daily Water Consumption, April 1, 1991 to December 31, 1999 Coefficients


Number of harmonics, K 1 2 3 4 5 6 7 8 Note: Coefficients a0 共mean兲⫽3.52.

ak

bk

3.91 0.91 −0.19 −0.24 −0.20 −0.28 −0.31 −0.39

0.42 −0.11 −0.62 −0.53 −0.25 −0.22 −0.21 0.04

Wb共weekday兲 = 3.656 + 0.0004共3,285/2兲 + 0.368共0兲 − 0.0385共1.80兲 + 0.0420共20兲 = 5.08 ML/day Wb共weekend兲 = 3.656 + 0.0004共3,285/2兲 + 0.368共1兲 − 0.0385共1.80兲 + 0.0420共20兲 = 5.45 ML/day The average daily weather effect is =−0.0385共1.80兲 + 0.042共20兲 = 0.77 ML/ day • For weekdays ⫽ 0.77/5.08 ⫽ 15.1%; and • For weekends ⫽ 0.77/5.45 ⫽ 14.1%. Rainfall and temperature are negatively correlated; the temperature drops due to the occurrence of rainfall. The recorded temperature during days with rainfall over the nine-year period averaged 17.66° C and rainfall averaged 4.45 mm, whereas during days without rainfall the temperature averaged 21.67° C. Using Eq. 共7兲 the effect of rainfall and temperature can be evaluated during rainless days and days without rainfall. The average daily weather effect is During rainless days = − 0.0385共0兲 + 0.042共21.67兲 = 0.91 ML/day During days with rain = − 0.0385共4.45兲 + 0.042共17.66兲 = 0.57 ML/day The base-use equation 关Eq. 共7兲兴 suggests that the average daily weather effect during rainless days is 1.6% higher than days with rainfall. Seasonal Use, Ws The seasonal daily use is estimated by subtracting the estimated base use from the observed daily water use. Regression models are developed to predict the potential seasonal water use and the short memory residual. The potential seasonal water use varies over the year with normal temperature and short memory residual representing the quick responses of people to weather changes. Potential Seasonal Water Use, Wsp The potential seasonal use is first analyzed by identifying the seasonal cycles as expressed in Eq. 共3兲. Table 2 shows the Fourier coefficients of the seasonal cycle of daily water consumption. Wsc

Weather Component The effects of rainfall and maximum air temperature are examined through a forward selection method of stepwise regression using the following variables:

• Daily maximum temperature and lagged maximum temperature, T j−1 is the maximum temperature on day j − 1 • Daily rainfall 艋25 mm as adopted in Zhou et al. 共2000兲. • Number of days after rainfall to prove the postulate that the occurrence of rainfall causes a temporary reduction in water use that diminishes over time. This “dynamic effect” of the occurrence of rainfall in a typical summer period in East Doncaster is reported in Gato et al. 共2003兲. • Antecedent precipitation index 共API兲 is also included since past techniques tended to rely on this variable for climatic explanation of water use. The API is calculated by 共8兲

API = kAPI j−1 + R j−1

where R j−1⫽rainfall on day j − 1. The value of k is dependent on the potential loss of moisture and varies seasonally. In this study a value of 0.85 for k is used as adopted by Bruce and Clark 共1966兲 and Zhou et al. 共2000兲. • Number of previous days with temperature over 30° C to ascertain the effects of hot consecutive days on short-term response to water use. Graeser 共1958兲 revealed that maximum daily demands in Dallas were significantly related to the number of previous days with maximum temperature over 38° C. Analysis of East Doncaster data also revealed that at temperature over 30° C daily water use is at its maximum. All the variables considered are significant having p values lower than 0.05. The R2 and SE values of the weather component model are 32% and 2.84, respectively. The coefficients and associated statistics of alternative weather component models are shown in Table 3. Based on these analyses, the following equation is obtained for the climatic component: ˆ 共d兲 = − 0.898 + 0.0002X + 0.196X − 0.078X + 0.214X W c 1 4 5 6 共9兲

− 0.065X7 + 2.698X8 − 0.09X9 − 0.052X10

ˆ 共d兲⫽estimated effects of rainfall and temperature 共ML/ where W c day兲; X1 and X4 are as explained in Eq. 共7兲; X5⫽daily rainfall of less than or equal to 25 mm 共mm兲; X6⫽number of days after rainfall; X7⫽API; X8⫽number of previous days with temperature over 30° C; X9⫽temperature lagged for one day 共°C兲; and X10 ⫽temperature lagged for two days 共°C兲. Persistence Component, Wsr The remaining seasonal use, which is called as the random or error or as Zhou et al. 共2000兲 called, the persistence component is modeled using the autoregressive procedure described in Eq. 共5兲. The resulting equation for the persistence component, Wsr, has R2 and SE values of 74% and 1.87, respectively r r r Wsr共d兲 = 0.015 + 0.527Ws共d−1兲 + 0.088Ws共d−2兲 + 0.066Ws共d−3兲 r r + 0.052Ws共d−6兲 + 0.13Ws共d−7兲

共10兲

Wsr共d兲

where is in megaliters/day; d⫽daily time index, d = 1 at the start of the data series, April 1, 1991. Only the lag days with P values lower than 0.05 are considered statistically significant and are included in Eq. 共14兲. The coefficients and associated statistics of alternative models are presented in Table 4. Total Daily Water Use Model, Wd The daily total water use is then calculated as the sum of the estimated daily base use and daily seasonal use of water. The components of the total water use models and the corresponding equations are as follows:



Table 3. Coefficients and Associated Statistics of Alternative Weather Component Models

Model C1

C2

Time 共X1兲

Intercept 0.521

0.0004

共7.25E − 5兲兵−3.974其

共1.04E − 5兲兵6.125其

−4.995

0.0003


共5.16E − 94兲共1.03E − 6兲兵4.898其兵−21.403其 C3

−4.493

C5

C6

C7

C8

C9

Rainfall 艋25 mm 共X5兲

DAR 共X6兲

API 共X7兲

Temp ⬎30° C 共X8兲

T-1 关day 共X9兲兴

T-2 关day 共X10兲兴

R2 共%兲

SE

1.37 3.41

0.237

16.78 3.13

共6.4E − 102兲兵22.380其

0.0003

共5.84E − 75兲共4.67E − 6兲兵4.588其兵−18.904其 C4

Temperature Rainfall 关°C 共X2兲兴关mm 共X3兲兴

0.223

−0.106

共1.23E − 91兲兵21.103其

共1.57E − 17兲

18.99 3.09

兵−8.58其

−4.087

0.0002

0.176

−0.072

0.266

7.49E − 65

共0.0004兲

共9.68E − 9兲

兵−17.472其

兵3.537其

共8.38E − 55兲兵15.950其

兵−5.754其

共6.91E − 33兲兵12.106其

−3.142

0.0001

0.177

−0.067

0.206

共6.94E − 37兲

共0.025兲兵2.242其

共2.85E − 58兲兵16.488其

共3.82E − 8兲

兵−12.88其 −3.056

0.0001

0.177

共1.65E − 34兲

共0.030兲

兵−12.426其

兵2.172其

共7.32E − 58兲兵16.426其

−1.564

0.0001

0.087

共2.31E − 8兲

共0.015兲

兵−5.604其

兵2.436其

共1.23E − 10兲兵6.461其

−1.083 共0.0001兲

0.0002

0.189

共0.011兲

兵−3.852其 −0.898 共0.0002兲兵−3.149其

兵2.658其

兵−5.515其

23.16 3.01

共2.03E − 20兲共2.49E − 28兲兵9.336其兵−11.165其 0.198 −0.105 共3.16E − 9兲兵−5.943其 −0.112

26.55 2.94

−0.061

26.68 2.94

−0.062

共1.85E − 18兲共5.21E − 29兲兵8.830其兵−11.310其 0.202 −0.058

3.140

共1.39E − 10兲共5.92E − 20兲共5.77E − 27兲兵9.218其兵−6.442其兵−10.870其 0.214 −0.088 −0.064

共8.38E − 26兲兵10.612其 2.803

−0.125

共3.75E − 7兲

共3.90E − 21兲兵9.514其

0.0002

0.196

−0.078

共1.08E − 22兲共4.70E − 32兲兵9.893其兵−0.065其 0.214 −0.065

共2.37E − 17兲

兵2.545其

共1.33E − 25兲兵10.567其

2.693

−0.09

共0.008兲

共2.73E − 27兲兵10.941其

共8.86E − 6兲

共9.76E − 23兲共2.74E − 33兲兵9.904其兵−12.185其

共1.55E − 19兲兵9.111其

共3.62E − 7兲

−0.05 共0.0003兲

兵−5.101其

兵−3.592其

兵−5.094其

兵−4.452其

兵−8.532其

Note: Values inside the fences, such as parentheses and curly braces, are p and t-statistic values of the corresponding coefficients. A p value ⬎0.05 is statistically insignificant.

• Base use, Wb Wb = 3.656 + 0.0004X1 + 0.368X2 − 0.035X3 + 0.042X4 共11兲 • Seasonal component, k1

Wsp = a0 +

兺 k=1

Wsp

冋冉冊 ak cos

冉冊册

2␲k 2␲k d + bk sin d 365 365

共12兲

coefficients are defined in Table 2. • Weather component, Wsc Wsc = − 0.898 + 0.0002X1 + 0.196X4 − 0.078X5 + 0.214X6 − 0.065X7 + 2.698X8 − 0.09X9 − 0.052X10

共13兲

• Persistence component, Wsr r r r Wsr共d兲 = 0.015 + 0.527Ws共d−1兲 + 0.088Ws共d−2兲 + 0.066Ws共d−3兲 r r + 0.052Ws共d−6兲 + 0.13Ws共d−7兲

共14兲

The relative contribution of each component to the estimated daily water use is presented in Fig. 5. The estimated daily total water use for the East Doncaster water supply distribution zone is compared with the recorded daily water use 共Fig. 6兲 yielding a

correlation coefficient R2 of 86%. The R2 value per component of the total water use model is shown in Table 5. The strong correlation between the estimated and the recorded values is an improvement on the previous model by Gato et al. 共2003兲. Splitting the years based on seasons, temperature, and rainfall depth as applied in previous daily demand models has not been considered in this study as further refinement to the model is still being done. An independent check is undertaken using the daily water use for East Doncaster zone from January 1, 2000 to April 1, 2001. A linear regression between the forecast daily water use and the recorded 共Fig. 7兲 yields R2 of 86%. Validity Check of the Model To check the validity of the model, a statistical test known as relative error as employed by Zhou et al. 共2000兲 was adopted. The relative error determines the ratio of absolute error to the measured consumption in percent. As reported by Zhou et al. 共2000兲 this statistic eliminates the bias of high consumption errors that mask those of low consumption by sheer magnitude and is a standard measure of error in data analysis. Relative errors of estimates based on the total water use model developed indicate that 66% of model estimates fall within the error band of ±15% and over 95% are within the range of ±40%



Table 4. Coefficients and Associated Statistics of Alternative Persistence Component Models Model

Intercept

r Ws共d−1兲

r Ws共d−2兲

r Ws共d−3兲

r Ws共d−4兲

r Ws共d−5兲

r Ws共d−6兲


P1

r Ws共d−7兲

−0.018 0.718 共0.641兲共0兲兵−0.467其兵53.071其 0.001 0.599 0.172 P2 共0.987兲共3E − 183兲共5.14E − 19兲兵0.016其兵31.309其兵8.978其 0.005 0.571 0.116 0.112 P3 共0.886兲共2.7E − 164兲共1.72E − 7兲共7.37E − 9兲兵0.143其兵29.394其兵5.241其兵5.801其 0.012 0.576 0.094 0.073 0.071 P4 共0.746兲共1.2E − 165兲共2.39E − 5兲共0.001兲共0.002兲兵0.324其兵29.558其兵4.233其兵3.279其兵3.675其 0.015 0.566 0.092 0.061 0.021 0.090 P5 共0.681兲共4.6E − 157兲共4.57E − 5兲共0.007兲共0.347兲共3.62E − 6兲兵0.412其兵28.678其兵4.084其兵2.712其兵0.941其兵4.642其 0.013 0.541 0.094 0.063 0.038 0.114 P6 共0.730兲共4E − 144兲共3.36E − 5兲共0.003兲共0.068兲共7.56E − 9兲兵0.345其兵27.297其兵4.155其兵3.012其兵1.826其兵5.798其 0.015 0.528 0.088 0.066 0.052 0.130 P7 共0.694兲共5.7E − 140兲共8.29E − 140兲共0.001兲共0.009兲共3.7E − 11兲兵0.393其兵26.850其兵3.942其兵3.268其兵2.608其兵6.645其 Note: Values inside the fences, such as parentheses and curly braces, are p and t-statistic values of the corresponding coefficients. A statistically insignificant.

共Fig. 8兲. During summer period, 62% of model estimates fall within the error band of ±15% and over 93% are within the range of ±40% compared with 71 and 98%, respectively, during the winter period 共Fig. 9兲. In the calibration period from January 1, 2000 to April 1, 2001, 83% of the daily estimates are within ±15% error band 共Fig. 8兲. Comparison with Previous Model A comparison between this model using the Fourier series function and the previous model by Gato et al. 共2003兲 using the heat function as described by Maidment et al. 共1985兲 was undertaken by comparing the relative errors in the modeled demand values

Fig. 5. Relative contribution of each component to the estimated daily water use

R2 共%兲

SE

51.35

1.98

53.31

1.93

54.34

1.91

55.12

1.89

55.28

1.88

55.04

1.89

55.54

1.87

p value ⬎0.05 is

from April 1, 1991 to December 31, 1999 and also during the calibration period from January 1, 2000 to April 1, 2001. Although relative errors of estimates based on the total water use model developed using the Fourier series indicate that 66% of model estimates fall within the error band of ±15% and over 95% are within the range of ±40%, the previous model developed by Gato et al. 共2003兲 using the heat function yield 55 and 89%, respectively 共Fig. 10兲. In the calibration period, 84% of model estimates using the heat function fall within the error band of ±15% compared to 83% using the Fourier series 共Fig. 11兲.

Conclusions Using East Doncaster daily water demand data from 1991 to 1999, the result confirms Gato et al.’s 共2003兲 findings that base use is weather sensitive 共dependent on temperature and rainfall兲 and is affected if the day is on a weekend or a weekday. The base use is based on the lowest month of the year which could have some outdoor component and therefore it is believed that identifying threshold level in temperature and rainfall would appropriately identify a base model that is weather insensitive and would represent purely indoor water use. The model if refined and validated using data from other water supply distribution zones has the potential to be used to forecast the amount of water to be supplied to water supply distribution zones taking into account the weather forecasted by the Bureau of Meteorology. It can also be used to determine the trend in water usage and the effectiveness of water conservation programs implemented with the water restriction in place. Based on the result of the study, the recommended total daily water use model has a coefficient of determination, R2, of 86%, which is considered a strong correlation. The model also performed strongly when tested in a forecast mode using indepen-




Fig. 6. Comparison between recorded and estimated daily water demand, East Doncaster zone, Victoria, Australia 共1991–1999兲

Table 5. Summary of the R2 Value of Each Component of the Total Water Use Model Component Base and seasonal use Weather component Auto regression

R2 共%兲

Sum of R2

46 23 17

46 69 86

dent daily data during the period January 1, 2000 to April 1, 2001, yielding an R2 of 86%. The result also revealed that the separation of the persistence component from the climatic effects and modeling it using an autoregressive approach greatly improved the model from a R2 of 69–86%.

Acknowledgments The daily water consumption data were provided by Yarra Valley Water through Stephen Sonnenberg, the then Growth Planning Team Leader of the Service Enhancement Water Section. The

Fig. 7. Comparison between recorded and estimated daily water demand, East Doncaster zone, Victoria, Australia 共2000–2001兲 JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT © ASCE / JULY/AUGUST 2007 / 317



Fig. 8. Relative errors of modeled 共April 1, 1990 to December 31, 1999兲 and of Calibrated 共January 1, 2000 to April 1, 2001兲 daily consumption, East Doncaster zone, Victoria, Australia

Fig. 9. Comparison of relative errors between summer and winter modeled values, East Doncaster zone, Victoria, Australia 共April 1, 1990 to December 31, 1999兲

Fig. 10. Comparison of relative errors in modeled demand values by using Fourier series and heat function, East Doncaster zone, Victoria, Australia 共April 1, 1991 to December 31, 1999兲 318 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT © ASCE / JULY/AUGUST 2007



Fig. 11. Comparison of relative errors in modeled demand between using the Fourier series and heat function, East Doncaster zone, Victoria, Australia 共January 1, 2000 to April 1, 2001兲

daily maximum temperature and rainfall data were obtained from Australian Bureau of Meteorology.

References Box, G. E. P., and Jenkins, G. M. 共1976兲. Time series analysis: Forecasting and control, Holden-Day, San Francisco. Bruce, J. P., and Clark, R. H. 共1966兲. Introduction to hydrometeorology, Pergamon, Oxford, U.K. Campbell, M. J. and Machin, D. 共1999兲. “Medical statistics: A commonsense approach.” The p-value (level of significance), 具http:// hsc.uwe.ac.uk/dataanalysis/quant_iss_pval.htm典共Jul. 25, 2005兲. Devore, J. L. 共1990兲. “Tests of hypotheses based on a single sample.” Probability and statistics for engineering and the sciences, 3rd Ed., Duxbury Press, Wadsworth, Inc., Belmont, Calif., 313–317. Gato, S., Jayasuriya, N., and Hadgraft, R. 共2003兲. “A simple time series approach to modelling urban water demand.” 28th Int. Hydrology and Water Resources Symp., Institution of Engineers, Australia. Gato, S., Jayasuriya, N., Roberts, P., and Hadgraft, R. 共2004兲. “Understanding residential water use.” ENVIRO 2004, Sydney, Australia.

Gibbs, K. C. 共1978兲. “Price variable in residential water demand models.” Water Resour. Res., 14共1兲, 15–18. Graeser, H. J., Jr. 共1958兲. “Meter records in system planning.” J. Am. Water Works Assoc., 50共11兲, 1395–1402. Maidment, D. R., and Miaou, S. P. 共1986兲. “Daily water use in nine cities.” Water Resour. Res., 22共6兲, 845–851. Maidment, D. R., Miaou, S. P., and Crawford, M. M. 共1985兲. “Transfer function models of daily urban water use.” Water Resour. Res., 21共4兲, 425–432. Miaou, S. P. 共1987兲. “Metropolitan’s daily water use—Analysis and forecasting.” Technical Rep., Metropolitan Water District of Southern California, Los Angeles. Miaou, S. P. 共1990兲. “A class of time series urban water demand models with nonlinear climatic effects.” Water Resour. Res., 26共2兲, 169–178. Water Resources Strategy Committee for the Melbourne Area. 共2001兲. “Discussion starter: Stage 1 in developing a water resources strategy for the greater Melbourne area.” Melbourne, Australia. Zhou, S. L., McMahon, T. A., Walton, A., and Lewis, J. 共2000兲. “Forecasting daily urban water demand: A case study of Melbourne.” J. Hydrol., 236共3兲, 153–164.

