Application of Harmony Search Algorithm to Reservoir ...

Water Resour Manage DOI 10.1007/s11269-015-1143-3

Application of Harmony Search Algorithm to Reservoir Operation Optimization Hamid Bashiri-Atrabi 1 & Kourosh Qaderi 1 & David E. Rheinheimer 2 & Erfaneh Sharifi 1

Received: 28 October 2014 / Accepted: 15 September 2015 # Springer Science+Business Media Dordrecht 2015

Abstract In this study, a meta-heuristic technique called harmony search (HS) algorithm is developed for reservoir operation optimization with respect to flood control. The HS algorithm is used to minimize the water supply deficit and flood damages downstream of a reservoir. The GIS database is used to determine the flood damage functions. The efficacy of HS algorithm is evaluated in comparison with other techniques by using a benchmark problem for a single reservoir operation optimization problem. HS showed promising results in terms of speed of convergence to an optimal objective function value compared with other techniques such as honey-bee mating optimization (HBMO) and a global optimization model (LINGO 8.0 NLP solver). The HS algorithm is then applied to the Narmab reservoir, north of Iran, as a case study. Narmab reservoir serves multiple purposes including irrigation, flood control, and drinking water requirements. The developed model is applied for monthly operation. The results show that the HS algorithm can be effectively used for operation of reservoir for flood management. Keywords Flooddamages . Reservoir operation optimization . Harmonysearch algorithm . GIS . HEC-RAS

1 Introduction Flood management is complex, requiring the simultaneous consideration of hydrologic, hydraulic, environmental, and economic aspects (Ahmad and Simonovic 2006). Reservoir operation can play an important role in pre-flood management as a non-structural method to reduce flood damages.

* Hamid Bashiri-Atrabi [email protected] 1

Department of Water Engineering, Shahid Bahonar University of Kerman, Kerman 76169133, Iran

2

School of Engineering, University of California, Merced, CA 95343, USA

H. Bashiri-Atrabi et al.

During the flood period, reservoir operations for flood moderation, one of several flood management options, plays an important role in protecting people and their socio-economic activities in floodplains. Most reservoir operations optimization methods and strategies that have been proposed, including short-term and long-term problems and single-purpose and multi-purpose reservoirs, are collectively described by Loucks et al. (1981), Labadie (2004), Yeh (1985), and Ko et al. (1992). Several studies have incorporated reservoir operation optimization into flood control models. For example, Wurbs et al. (1985) focused on the particular management strategy of reallocating storage capacity between flood control and conservation purposes in response to changing conditions. They proposed reallocations of storage capacity between flood control and municipal and industrial water supply by a hydrologic and economic evaluation procedure. Marien (1984) presented some necessary controllability conditions for a multi-reservoir flood control system under which no flooding occurs at a flood damage center situated downstream. Kelman et al. (1989) used Marien’s results to determine the optimal flood control volumes. Marien et al. (1994) presented a methodology to build flood control rule curves for multi-purpose multi-reservoir systems. Wang et al. (2003) proposes a reliability analysis system, which can be widely applied to the cases in which a reservoir is operated to meet several purposes such as flood control, energy generation, irrigation, domestic and industrial water supply, etc. Ahmad and Simonovic (2006) simulated a single multi-purpose reservoir for flood management using the system dynamics model (SD). In recent years, various meta-heuristic techniques have also been used to solve reservoir operation optimization problems. Applications of metaheuristic techniques to flood problems are described in several studies (Chang and Chen 1998; Karaboga et al. 2004; Ngo et al. 2007; Kumar et al. 2010; Malekmohammadi et al. 2010). However, one particularly promising metaheuristic technique, the harmony search (HS) algorithm, described below, has not yet been tested or applied to flood models. Previous research has shown that the HS algorithm has advantages compared with other meta-heuristics such as genetic algorithm (GA), which is particularly popular. Specifically, the HS algorithm needs fewer mathematical requirements and does not need an initial value to be set for the decision variables (Lee and Geem 2005; Geem 2007). There is a need to better understand whether or not HS might be useful in reservoir operations studies. The aim of the present work is to apply the HS algorithm to single, multi-purpose reservoir operation problem and to assess its efficacy. The objective function of the optimization model comprises real economic values of downstream flood damages and irrigation demands. This study combines the HS algorithm with the U.S. Army Corps of Engineers Hydrological Engineering Centers River Analysis System (HEC-RAS 2010) river hydraulics simulation model and a geographic information system (GIS) to minimize flood damages. In the following sections, the HS algorithm is described and has used for a benchmark study and details of the case study and the model formulation are explained, followed by a discussion of results.

2 Harmony Search (HS) Algorithm Harmony Search (HS) algorithm is an optimization algorithm inspired by the musical improvisation process which occurs in musical bands. It was first proposed by Geem

Application of Harmony Search Algorithm to Reservoir Operation

et al. (2001) and has been used as an optimization technique in various water engineering problems. The harmony in music is analogous to the optimization solution vector, and the musician’s improvisations whereby the musicians continually search for harmony are analogous to local and global search schemes in optimization techniques. The steps in the HS algorithm are as follows and depicted in Fig. 1: (1) Initialize the problem and algorithm parameters. The objective function is defined as follows: Minimize f ðxÞ subject to xi ∈X i ; xiL ≤X i ≤xiU ði ¼ 1; 2; …; N Þ

ð1Þ

where f(x) is the objective function, xi is the set of each design variable, Xi is the set of the possible range of values for each decision variable, xiL and xiU are the lower and upper bounds for decision variables, and N is the number of design variables. The HS algorithm parameters consist of the harmony memory size (HMS), the harmony memory considering rate (HMCR), the distance bandwidth (bw), the pitch adjusting rate (PAR), and the stopping criterion (K). Each of these parameters is specified in the first step.

Fig. 1 The steps in the HS algorithm (adapted from Lee and Geem 2005)

Start

Step 1

Define objective function and specify HS parameters

Step 2

Initialize HM and sort resulting objective function values

Step 3

Generate new harmony from HM

Is new harmony better?

Yes Step 4

Update HM

Step 5

Is stopping criterion satisfied?

Yes Stop

No

No


(2) Initialize the harmony memory (HM). The HM matrix, given by Eq. (2), is filled with randomly generated solution vectors and sorted by their respective objective function values. The initial HM is randomly generated by Eq. (3). The domain of initial HM is [xiL, xiU]. 2

3 x1 6 x2 7 7 HM ¼ 6 4 ⋮ 5 xHMS

ð2Þ

xij ¼ xiL þ randðÞ ðxiU −xiL Þ j ¼ 1; 2; …; HMS

ð3Þ

where HMS is number of solution vectors in harmony search memory, rand() is a random number from a uniform distribution of [0, 1]. (3) Improvise a new harmony from the HM. A new harmony vector is generated from the HM based on pitch adjustment rate (PAR), random selection, and harmony memory consideration rate (HMCR). The new value of x is chosen probabilistically with HMCR parameter, which varies between 0 and 1 (inclusive), as follows: ( 0 0 xi ∈ x1i ; x1i ; …; xHMS with probability HMCR i ð4Þ xi ← 0 with probability ð1−HMCRÞ xi ∈X i where HMCR is the probability of choosing one value from the values which are stored in the HM, and (1-HMCR) is the probability of randomly choosing a value from the possible range of values. Then, each new variable is either pitch-adjusted or not, depending on the PAR parameter, as follows: 0 Yes with probability of PAR ð5Þ decision of pitch adjusting f or xi ← No with probability of ð1−PARÞ The pitch adjustment process, similar to the mutation in GA, modifies the variable probabilistically. The pitch adjustment is given as (Ayvaz 2009): 0

0

xi ¼ xi rand ð0; 1Þ bw

ð6Þ

where bw is an arbitrary distance bandwidth. (4) Update the HM. in this step, if the new harmony vector, x′=(x′1, x′2, …, x′N), is better than the worst harmony in the HM in terms of the objective function magnitude, the new harmony is replaced in the HM as a new member. (5) Check the stopping criteria. Repeat steps 3 and 4 until the stopping criteria is satisfied. HS applications in water engineering include: optimal design of water distribution networks (Geem 2006; Geem and Cho 2011), optimal operation of a


multi-reservoir system for hydropower and irrigation (Geem 2007), simulation of irrigation systems (Cisty 2008), optimization model for groundwater management objectives (Ayvaz 2009), and identification of unknown groundwater pollution sources (Ayvaz 2010). The advantages of HS algorithm are: HS imposes fewer mathematical requirements and doesn’t require initial value setting of the decision variables; uses stochastic random searches; in contrast to the GA, which considers only the two parent vectors, it considers all of the existing vectors to generate a new vector (Mahdavi et al. 2007).

3 Methodology A complete flow chart of the process used to model reservoir operations is depicted in Fig. 2. The main components of the process are: collect the required data (land use, river cross section, inflows, reservoir characteristics, and water demands), run a hydraulic flood routing model, estimate the flood damage function, and run a reservoir operation optimization model using HS algorithm. In flood damage estimation model, peak flow frequency analysis was conducted. Flood damages were estimated by linking a hydraulic flood routing model with GIS tools for considering land-use information and flood damage functions in the downstream floodplain of the reservoir (Malekmohammadi et al. 2010). Different flood return periods were used for developing the flood damage function, which was used in one of the objective functions in the monthly operation optimization model. The optimization model optimized the monthly release and storage values to reduce flood damages and water deficits.

Start

Data of land use in floodplain, river cross secons, and hydraulic model inputs

Historical records of inflow flood hydrographs

Hydraulic flood roung model

Analyzing historical flood events and generang flood hydrographs with various return periods for each month

Geographical informaon systems (GIS) Flood damage funcons based on floods

Input data of physical characteriscs and reservoir

Input data of agricultural land informaon and water demands

Reservoir operaon opmizaon model using HS algorithm

Fig. 2 Flowchart of the proposed methodology for developing optimization of reservoir operation


4 Application 4.1 Case Study: Narmab Reservoir The case study considered in this paper is Narmab reservoir in the province of Golestan, north of Iran (Fig. 3). Although, the main climate of Iran is arid and semiarid, but the northern part of the country enjoys the Mediterranean precipitation regime with 400–1900 mm (Sharifi et al. 2012). The Narmab reservoir is located on the eastern part of the Caspian Sea coast and lies at latitude 37°6 N and longitude 55°16 E. The study area is about 207 km2. The flash floods of 2001, 2002, and 2005 show that extreme flash floods are severe disasters in the eastern part of the province (Sharifi et al. 2012), necessitating the incorporation of a flood damage function into the model. Figure 4 shows monthly maximum inflow observed at Jangaldeh station for 1996–2010. The Narmab reservoir serves multiple purposes in the following order of decreasing priority: flood control, drinking water, and irrigation. The study area extends to 7.3 km

Gonbad Minoodasht

(a)

Chehelchay dam Narmab dam

Chehelchay river Narmab river Study area

Narmab dam

Chehelchay dam

Bagherabad

Azadshahr Khormaloo dam

(b)

(c) Fig. 3 (a) Location of the Golestan province; (b) the Gorganrood Basin; and (c) downstream of the Narmab reservoir


Fig. 4 Maximum monthly precipitation data observed at Jangaldeh

downstream of the Narmab reservoir. Table 1 shows the main characteristics of the Narmab reservoir.

4.2 Flooding Model and Damage Estimation Flood damage functions were developed to estimate damage costs associated with monthly floods downstream of Narmab reservoir. The flood damages were assessed using statistical modeling technique, hydraulic modeling with HEC-RAS and GIS. Estimation of the flood damage function involved the following steps: (1) collect the flow data, topographic data, and land used data; (2) digitalize data in ArcView; (3) estimate floods using Storm-water Management and Design Aid (SMADA) with 15 years (1996–2010) of peak gauge level data from the Jangaldeh hydrometric station; (4) calculate flood inundation of the Narmab river with HEC-RAS; (5) process HEC-RAS results using GIS tools to estimate flood damages; and (6) estimate flood damage functions. Flood depth is usually calculated for a flood event with a specific return period (de Moel et al. 2009). Flood maps for return periods of 5, 10 and 25 years were generated based on predicted floods using the computer application SMADA. Historical monthly peck discharge data from Jangaldeh station were collected from the Golestan regional water company, Iran. Monthly flow predictions and distributions fitting were conducted using subprogram DISTRUB 2.0 (Eaglin et al. 1996). Log Pearson type ІІІ was found to best describe the peak Table 1 The main characteristics of the Narmab reservoir

Characteristics

Value

Height (m)

60

Crest length (m) Maximum storage (mcm)

807 115

Active storage (mcm)

107

Dead storage (mcm)

8


flow data. The following probability density represents the Log Pearson type ІІІ distribution (Singh and Yadava 2003): y β =δ −y=δ Px ðyÞ ¼ Py0 1 þ e a

ð7Þ

where δ is the difference between the mean and mode (d = m – Xm), Y is the mode of population y, β is a scaling parameter, and Py0 is value of Px(y) at the mode. To process the geographic data, we used ArcView GIS. HEC-GeoRAS was used to combine the HEC-RAS hydraulic modeling output and the GIS data. The unsteady HEC-RAS model, version 4.1 (HEC-RAS 2010) was used for hydraulic modeling of the floodplain in the river channel network, downstream of the Narmab reservoir. A total of 98 cross-sections were created. Based on field assessments and channel characteristics, different Manning’s n coefficients were used for overbank and in-channel river sections. By running the HEC-RAS model with the discharge estimated using SMADA, flood levels for different flood frequencies were simulated. These flood levels were then exported to the GIS geodatabase and overlain on a Digital Elevation Model (DEM) to estimate total inundation depth and area. Narmab river cross sections and benchmark elevations from 1: 1000 scale topographic maps of the region were used to create the DEM. The floodplain downstream of Narmab reservoir consists mostly of agricultural land. Therefore, estimation of flood damage functions was based primarily on agricultural use, with some residential use. Table 2 shows flood damage values based on the flood level and land use downstream of Narmab reservoir (Iranian ministry of energy 2006). The data Table 2 was used to estimate nonlinear flood damage curves (nonlinear regression) for each of Apr-Sep and Oct-Mar with 15 years (1996–2010), shown in Fig. 5. The flood damage curves for Apr-Sep and Oct-Mar (Fig. 5) are represented, respectively, as: DA ¼ −0:074F 2 þ 44:3F þ 3664

ð8Þ

DO ¼ −0:111F 2 þ 25:86 F þ 1111

ð9Þ

Table 2 Estimated peak discharges of Narmab River in Jangaldeh station using Log Pearson Type ІІІ distribution (SMADA) and their flood damages (1996–2010) Month

F for T=5 years Flood damages F for T=10 (m3/s) (Thousand $) years (m3/s)

Flood damages F for T=25 (Thousand $) years (m3/s)

Flood damages (Thousand $)

January

5.07

136.2

7.92

142.2

12.71

151.1

February

16.42

157.3

22.32

177.6

31.51

203.3

March April

27.22 20.55

194.0 460.8

40.32 31.89

222.6 565.4

61.75 52.04

253.0 674.4 1120.1

May

40.89

622.4

96.29

779.0

254.06

June

8.25

395.8

13.15

420.9

20.31

457.1

July

3.27

363.1

5.35

379.0

8.47

396.9

August

26.07

779.5

531.6

52.64

676.1

96.64

September 5.66

381.0

25.07

526.3

187.05

1047.1

October

3.96

110.2

7.5

141.3

14.91

154.8

November 5.44 December 6.43

137.0 139.1

9.68 8.42

145.6 143.1

17.57 10.75

159.4 147.6

T is the return period (year)

Application of Harmony Search Algorithm to Reservoir Operation 1.1 1

Damages(million $)

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

(a)

0

0

50

100

150

200

250

Flow (m3/s)

Damages(million $)

0.3

(b)

0.2

0.1

0

10

20

30

40

50

60

70

Flow (m3/s)

Fig. 5 Flood damage function for downstream of Narmab reservoir for (1996–2010): (a) Apr–Sep; (b) Oct–Mar

where DA and DO are the flood damages ($) and F is the flood discharge (m3/s). These functions were used in the optimization model to estimate downstream flood damages.

4.3 Reservoir Optimization Model This study aims to investigate the applicability of HS algorithm to optimize multi-purpose reservoir operations over 12 months. Herein, the reservoir operation problem is timedependent and nonlinear. In this optimization model, water supply and flood control purposes are considered, with one general objective function for each purpose. The first scenario is to minimize the water supply deficit by minimizing the sum of squared deviations of water supply deficit (Wu and Chen 2013): Minimize

f 1 ðxÞ ¼

N X t¼1

ðRt −Dt Þ2

ð10Þ


where f1(x) is the water supply objective function, Rt is monthly release from the reservoir, Dt is monthly agricultural demand (Iranian Ministry of Energy 2001), and t is the time step number, and N is the last time step. The second scenario is to minimize flood damages, represented by the flood damage curves: Minimize

f 2 ðxÞ ¼

N X

ðf

FD Þ

ð11Þ

t¼1

where f2(x) is the flood damage objective function to be minimized, and fFD is the monthly flood damage function from Eqs. (8) and (9). The optimization problem is subject to the following constraints. Operations are subject to a mass balance constraint, expressed as: S tþ1 ¼ S t þ I t −Rt −Spt ;

t ¼ 1; 2; …; 12

ð12Þ

where for each month t, St is the reservoir storage at the end of month, It is the inflow, and Spt is spill volume. Local gains and losses such as net evaporation are assumed negligible. Storage and release variables are subject to lower and upper bounds, imposed by physical limitations and expressed as: S min ≤S t ≤S max ;

∀t

ð13Þ

Rmin ≤Rt ≤Rmax ;

∀t

ð14Þ

where Smax and Smin are the maximum and minimum allowable storage volumes in month t, respectively and Rmax and Rmin are the maximum and minimum outflow capacity in month t, respectively. In addition, spill is calculated by (Buras 1972) S t þ I t −Rt −S max ; S t þ I t −Rt ≥ S max ∀t ð15Þ Spt ¼ 0 ; S t þ I t −Rt ≤ S max Therefore, spill will occur if I t ≥S max −ðS t −Rt Þ

ð16Þ

In this study, Smin is considered as dead storage and Rmax =53 m. For further information please refer to Table 1. The model is implemented in MATLAB and solved by the HS algorithm.

4.4 Benchmark Model The purpose of the benchmark model is to evaluate the ability of the HS algorithm to obtain a global optimum solution for a single reservoir operation problem. A Honey-bee mating optimization (HBMO) algorithm was used for the benchmark model, studied by Afshar et al. (2007) for a single reservoir problem. The nonlinear objective function of their study was to minimize the total squared deviation (TSD) of releases (Rt) from the target demands (Dt), normalized by maximum demand, as below: Minimize T SD ¼

N X

ððRt −Dt Þ=Dmax Þ2

t¼1

where Dmax is maximum demand in all months of operation (=831.1 mcm).

ð17Þ


The Dez reservoir in Iran was selected as a case study to test the performance of HBMO algorithm. Afshar et al. (2007) provides information about inflows and demands. Releases from the reservoir were considered as decision variables, resulting in reservoir storage as a continuous state variable (Afshar et al. 2007). Comparison of the HS model with the HBMO and the global optimization models (LINGO 8.0 NLP solver) for monthly storage volume at the end of each period and releases during each period are shown in Figs. 6 and 7, respectively. Table 3 presents the results of the HS, HBMO, and global optimum for the objective functions. The best and worst objective function values are 0.8062 and 0.8436, respectively, in the HS model and 0.8104 and 0.8493, respectively, in the HBMO model. As results show, the HS algorithm performance is better than HBMO in terms of objective function after 65,000 iterations.

4.5 Harmony Search Algorithm Results The procedure of HS algorithm has been carried out using a computer program that was developed in MATLAB. The HS model was tested with various algorithm parameter values: harmony memory size values ranged between 5 and 20; harmony memory considering rate values ranged between 0.8 and 0.95; and pitch adjusting rate values ranged between 0.05 and 0.5 (Step 1). The HS algorithm randomly generated solution vectors as harmony memory size within the specified bounds, and then stored them according to the values of the objective functions (Eqs. (10) and (12)) in harmony memory (Step 2). As described in step 3, a new harmony vector was generated based on memory considerations with 63 % (0.9×0.7) and 46.75 % probability, pitch adjustment with 27 % (0.9×0.3) and 38.25 % (0.85×0.45) probability, and randomization with 10 % (1–0.9) and 15 % (1–0.85) probability for the first and the second objective function, respectively. The objective function value of the new harmony was included in the HM and the worst harmony was removed from the HM (step 4). Finally, after 5000 and 50,000 searches, the HS algorithm converged to optimal values of the first and the second objective functions, respectively (Fig. 8a and b). For this study, the best (lowest) objective function values were obtained with HMS = 5 and 10; HMCR = 0.90 and

Fig. 6 Monthly optimum storage resulting from HS algorithm, HBMO algorithm and global optimum in a single reservoir problem


Fig. 7 Monthly optimum release resulting from HS algorithm, HBMO algorithm and global optimum in a single reservoir problem

0.85; and PAR = 0.3 and 0.45, for the first and the second scenarios, respectively. These parameters were obtained from the sensitivity analysis.

4.6 Operating Rules The development of reservoir operating rules is one of the crucial problems in water resources engineering. The operating rules are as indicators for decision makers to keep water in storage or release it. The observed flood hydrographs between the years of 1996–2010 for the Narmab river (measured in Jangaldeh hydrometric station) were used to determine the monthly reservoir inflow. The critical hydrographs (historical hydrographs in different months) are selected based on the magnitude of peak and volume of the floods. Monthly reservoir inflow data, downstream demands, and main characteristics of Narmab reservoir were used for testing the model. Figure 9a–c shows rule curves determined with the HS algorithm for the first objective. The reservoir release and storage values are affected by the water supply objective function. Figure 10a–c shows the rule curves for the second scenario. As this figure shows, if the dam capacity is insufficient for flood control, the system may increase the dam release to create enough capacity for flood control. Comparison of Figs. 9 and 10 show that in contrast to the first scenario, the second one is better for flood control. Releases in Fig. 9a, in contrast to

Table 3 Comparison of the HS model with the HBMO model and the global optimum Values of objective function

TSD HS

HBMO

Global optimum 0.7961

Best

0.8062

0.8104

Worst Objective function after 65,000 iterations

0.8436 0.7910

0.8493 0.8031

f1 (mcm)


f2 ($)

(a)

(b) Fig. 8 Progress of HS algorithm in: (a) the first objective function; (b) the second objective function

Fig. 10a, show that the first objective function is applicable to satisfy the agricultural demands in all months. In addition, Fig. 10b shows that due to the second scenario, reservoir storages are reduced to keep the capacity for flood management. It should be pointed out that the variations of rule curves decrease with increasing the operation period.

4.7 Reliability, Resiliency, and Vulnerability Criteria Reliability, resiliency, and vulnerability criteria are often used to evaluate the performance of reservoir operations. These three criteria are proposed by Hashimoto et al. (1982) as follows:


(a)

(b)

(c)

.

Fig. 9 Control curves for the first scenario: (a) Reservoir optimal release and demand (mcm); (b) storage (volume)-time continuum; (c) Reservoir spillway (mcm)


(a)

(b)

(c) Fig. 10 Control curves for the second scenario: (a) Reservoir optimal release and demand (mcm); (b) storage (volume)-time continuum; (c) Reservoir spillway (mcm)


4.7.1 Reliability This criterion indicates how likely a system is to not fail, which in this study is the probability that the reservoir meets demands. At any time t, the reliability can be calculated in several ways. Two ways are considered here (McMohan et al. 2006; Ahmadi et al. 2014): Time-based Reliability is defined as the number of periods that reservoir release satisfies the desired demands in all operation periods according to Eq. (18). N

RT ¼

N ðRt ≥α:Dt Þ

t¼1

N

;

0 < RT ≤1

ð18Þ N

where RT is the temporal reservoir reliability, α is the demand efficiency threshold, and N

t¼1

ðRt ≥α:Dt Þ is the number of periods that release is equal to or greater than α% of Dt. Table 4 contains several summary statistics of the two scenarios. Column 3 of the table shows the number of months the demand was not met divided by the number of months simulated (12). According to definition of Loucks et al. (1981), reliability can be expressed by Reliability ¼ 1−ðsystem’s failureÞ

ð19Þ

This definition is same as Eq. (18). Volumetric Reliability is defined as the volume of water supplied to a demand center divided by the total target demand in all operation periods: N X

RV ¼ 1−

ðDt −Rt Þ

t¼1 N X

;

0 < RV ≤ 1

ð20Þ

Dt

t¼1

where RV is the volumetric reservoir reliability and other terms are as previously defined. RV= 1 if Dt is totally satisfied.

4.7.2 Resiliency Resiliency describes how quickly a system can recover from failure, expressed as: N −1

R¼

N ðRt < α:Dt jRtþ1 ≥α:Dt Þ

t¼1

N

ð21Þ

N ðRt < α:Dt Þ

t¼1

where R is the resiliency of the reservoir, The numerator is the number of periods in which the system recovers from failure, and the denominator is the total number of failures in all periods of operation. Rapid system recovery (higher values of R) is better.

2

0.42

0.08

10

25

0

0.42

25

5

0

0

5

1

Frequency of failure (α=100 %)

10

T (years)

Scenario

0.08

0.33

0.42

0

0

0


Table 4 Interpretation of reservoir outputs

0.08

0.33

0.42

0

0

0


0.08

0.33

0.42

0

0

0


22.87

58.55

123.52

0

0

0

Total shortage (TS) (mcm)

0.92

0.58

0.58

1

1

1

RT (α=100 %)

0.89

0.72

0.40

1

1

1

RV (α=100 %)

1

0.8

0.8

0

0

0

R (α=100 %)

1.00

0.28

0.99

0

0

0

V (α=100 %)



4.7.3 Vulnerability Vulnerability indicates how severe the consequences of failure may be in a system by describing the average number of failures or deficits during operation period, expressed as: N

V ¼

N ðα:Dt −Rt jRt < α:Dt ; 0jRt ≥α:Dt Þ

t¼1

N

ð22Þ

N

where V is the vulnerability of the reservoir and N ðα:Dt −Rt jRt < α:Dt ; 0jRt ≥α:Dt Þ is the t¼1

total demand deficits. Vulnerability is dimensionless and is in the range of [0,1]. The time-based reliability for the first and the second scenario with T=5, 10 and 25 years are 1, 1, 1, 0.58, 0.58, and 0.92, respectively. These results indicate that for the first scenario, in which the system reliability is 1 for all return periods, the HS algorithm can optimize the first objective function well. In the second scenario, although the reliability is less than the first scenario, the reliability increases with the return period by increasing the release values for more months. The vulnerability and resiliency of the two scenarios are listed in Table 4.

5 Conclusions One of the main problems in reservoir operation optimization during floods is management of reservoir levels to provide sufficient flood space. The information provided by an optimization model can provide technical advice to decision-makers. In addition to measure flood damages, HEC-RAS and GIS are important because they can help decision-makers to model scenarios to minimize damages during a real disaster by providing geospatial information about flood extents. In this study, a simulation-optimization approach for flood management in a reservoirriver system is presented. The lack of significant application of HS in reservoir operation optimization persuaded us to do this research. This paper presents a HS algorithm for minimizing the flood damages and water shortages downstream of a reservoir, applied to a single, multi-purpose reservoir operation. The efficacy of HS algorithm is validated in the comparison with HBMO, and results shows better agreement between HS and global optimum for the objective function. Results of the case study in north of Iran shows that the developed model can help decisionmakers to estimate the flood damages in a larger floodplain and using them to control the floods with respect to the other contrasts such as agricultural demands. The formulation of the reservoir operation optimization problem and its application using the HS algorithm, as presented in this study, is a general formulation which can be extended to other cases and scenarios. The developed monthly model and estimated damages can be used also in a shortterm reservoir operation optimization model. In addition, the optimization model developed in this study can be extended for multi-reservoir systems. Further, models can concentrate on using a more accurate estimation model of flood damages and compare the results of HS algorithm with other reservoir operation optimization models, as was done here with the benchmark HBMO model. The first and second scenarios indicated the ecological and economical objectives, respectively. It’s an interesting issue to compare and balance between the optimization results derived under these two objectives in the further studies.

Application of Harmony Search Algorithm to Reservoir Operation Acknowledgments The authors would like to thank Golestan Regional Water Company, Iran, for providing data used in this study. In addition, the authors thank the reviewers and editors for their valuable comments and suggestions.

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