1 Introduction

In order to increase the efficiency of water resources usage for various purposes, utilizing compromise policies for reservoir dams in a form of modeling the multi-reservoir and multi-objective systems in the watershed scale is necessary. A review of the previous studies showed that various systems including Standard Operating Policy (SOP), the yield model and the hedging policies were used as the reservoir operation rules. The Standard Operating Policy (SOP) was first developed and established by Loucks et al. (1981) as the simplest and easiest way for reservoir operating, and many researchers also employ the optimized and simulated models based on this method (Hassanzadeh et al. 2014; Sulis and Sechi 2013). WEAP model has been used for better planning and management of water resources to reduce drought risks and help decision makers in this field (Levite et al. 2003; Assaf and Saadeh 2008; Li et al. 2015). According to standard operating policy (SOP), simulation-optimization models are been used to determining the rule curve or the reservoir outflow strategies based on the reservoir storage volume and the reservoir inflow during the period of operation.

In the last few years, with the development of optimization methods and the introduction of the meta-heuristic methods in general, and genetic algorithm (GA) in particular, optimum operation of the reservoir systems has been entered into a new stage. Given the various issues related to water resources, different researches have been done on the application of genetic algorithm in this field, e.g. Guven and Gunal (2008) use GA as a new tool for prediction of local scour downstream of grade-control structures. Rabunal et al. (2007) determinate the unit hydrograph of a typical urban basin using GA-ANN program. Orouji et al. (2013) use ANFIS-GA programming to modeling of water quality parameters in Sefidrood River, Iran. Khu et al. (2001) use GA for calibration of MIKE-NAM model in real-time forecasting. Also Guven and Kisi (2010) apply LGA and ANN for estimation of suspended sediment yield in natural rivers in two stations on the Tongue River in Montana, USA.

Asiabar et al. (2010) found that although the genetic algorithm is able to solve complex nonlinear, non-convex and multi-objective problems, it has some drawbacks such as inaccuracy of the intensification process near the optimal set and a long convergence time. Dariane and Momtahen (2009) used Modified Direct Search Genetic Algorithm (DSGA) in comparison with the dynamic stochastic programming and the dynamic programming covering linear regression for optimizing multi-reservoir systems in Greater Karoon System. They found that DSGA is superior to the other two methods in computing speed and convergence time.

Non-dominated Sorting Genetic Algorithm-II (NSGA-II), which in fact is a multi-objective genetic algorithm, uses a group of possible solutions at each step. These algorithms have been used in various fields of water resources (Fu et al. 2008; Ercan and Goodall 2016). Due to the appropriate structure of the model and the selection of non-dominated solutions at each iteration of the equations, the algorithm has a better convergence speed in comparison to the conventional genetic algorithm (Deb et al. 2002; Chang and Chang 2009). Chang and Chang (2009) applied NSGA-II for identifying optimal operating strategies in a multi-reservoir system in Taiwan. The results indicated that by using the alternative management strategies presented in the Pareto-front, the water shortage during the drought years can be reduced.

Zhang et al. (2017) showed that Large-scale reservoir operating rules are derived by sensitive analysis and an adaptive surrogate model. In this case, the weighted crowding distance of NSGAII contributes to better performance than the classical NSGAII. In another study, the TRWD’s network that simulated by the river system modeling tool (RiverWare) is coupled with the MOEA to solve a seven-objective problem formulation (Smith et al. 2016). Nicklow et al. (2010) reviewed recent Evolutionary Computation applications and the state-of-the-art, with particular focus on GAs given their dominant use in the historical literature of the water resources planning and management field. They demonstrated that an immediate need to formally characterize operational successes and failures of EAs used in water resources engineering practice.

Studies show that the superiority of a specific procedure for optimizing cannot be conclusively proved in all optimization problems and empirical studies are required to evaluate the performance of NSGA-II compared with the other optimizing methods using the same problem formulation (Wolpert and Macready 1997; Mansoor et al. 2015). Because of the mentioned problems regarding the conventional genetic algorithm as well as the complexity of the problem and the number of decision variables, NSGA-II method was used in the optimization process of this study. In the reservoir operation based on SOP method, water withdrawal is set to the amount of demand. When the reservoir fails to fully meet the demand, a percentage of it is met. In this policy, the general water deficit is minimized, but deficits are more intense. The method requires some modification during drought. Therefore, various methods such as the hedging rule are recommended. During drought or semi-drought periods, the hedging rule is a common practice in the management of water resources. Therefore, although covering the whole or a big part of the demand may sometimes be possible, only a part of the desired output will be released which saves water for more intense possible deficits in the future.

In another word, demand reduction is suggested during pre-drought periods, even if the current storage and reservoir input can cover the demand during current period. Such a reduction will prevent larger deficits in subsequent periods. Therefore, accepting a small deficit in the current period to moderate a more sever deficit in the next periods was investigated considering important economic and social aspects (Draper and Lund 2004). In practice, the application of the hedging rule will distribute water deficits over a longer time horizon and improve reservoir operation efficiency (Neelakantan and Pundarikanthan 1999). In this regard, Taghian et al. (2014) conducted a case study on a multi-purpose reservoir in the south of Iran and used the conventional genetic algorithm along with a simulation model to determine the optimum water allocation and the water storage level in the reservoir. The results showed that the model performed well in deriving the optimum policy for reservoir operation in both normal and drought conditions (Taghian et al. 2014). In addition, Felfelani et al. (2013) investigated the application of the hedging policy in the Dez dam by using system dynamics approach (Dariane and Momtahen 2009).

The importance of the present research is to develop a suitable hybrid model (coupled simulation-optimization) based on hedging policy for optimize the reservoir operation in areas with water scarcity. Whit using this method, the failure intensity in supplying the demands of the area can be significantly reduced during the operation period. In order to achieve this objective, the Non-dominated Sorting Genetic Algorithm-II (NSGA-II, Deb et al. 2002), Pareto Envelope-based Selection Algorithm II (PESA-II, Corne et al. 2001) and Strength Pareto Evolutionary Algorithm II (SPEA-II, Zitzler et al. 2001) were combined separately with the simulating model called Water Evaluation And Planning System (WEAP) and their results were compared to produce optimal solution. Another purpose of this study is to employ a new formulations for imposing reservoir hedging rules by defining two parameters, including the hedging level (the active volume of the reservoir at this level) and the coefficient of reservoir hedging (percentage), as the decision variables. In this context, a modern structure was developed and evaluated in order to optimize the multi-reservoir system operation. With this multi-objective structure the periods of drought or water shortage can be detected and the water released from reservoir will be optimized based on the hedging policy approach in WEAP-NSGA-II coupling model. Using this structure, in the real operating of the reservoir, questions such as starting time, the hedging distribution and the water allocation limitations during droughts can be answered based on the hydrologic process of the basin. So the system would yield a maximum demand coverage reliability and a minimum violation fine due to failure in meeting demands and violation of the reservoir operation capacity.

2 Material and Methods

2.1 Study Area

The study area is located in the west of Iran and includes two reservoir dams called Shohada dam and Gavoshan dam (Fig. 1). The Shohada dam is located in the upstream of the Gavoshan dam. In the Gavoshan project, after stored the flow of the Gaveh River by the Gavoshan reservoir dam, the water is transferred to the Razavar river basin through the Gavoshan tunnel. While supplying drinking water of Kermanshah and Kamyaran, the water flow is combined with the surface water resources of the Razavar River and covers the irrigation water demand of Bilevar and MianDarband plains. Shohada reservoir dam is located in the north west of Songhor on Gavehrood River in Kermanshah province. The purpose of constructing this dam was to regulate the Gaveh Rood River runoff to supply drinking water of Songhor as well as meeting the irrigation water demand of Songhor and Koliaie plains. A schematic view and the designed framework for the WEAP model are shown in Fig. 1.

Fig. 1
figure 1

Location of the study area and schematic view in the study area

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2.2 Model Configuration and Simulation

As shown in Fig. 1, in WEAP model, river paths, hydrometric stations, location of the dams (Gavoshan and Shohada dams), water withdrawal channels, city and demand site nodes and etc. were digitized using the available GIS based maps and tools. The simulation time steps and the computational units were considered on monthly and metric units respectively. Hydrological and meteorological time series of the recorded data and the information about the monthly demands (agricultural and drinking), reservoir and withdrawal location data, required coefficients and parameters, etc. were introduced to the model using text files with CSV extension.

Firstly the model was calibrated and verified for a 6 years statistical period (Oct. 2008 to Sep. 2013) given the starting time of the operation of these dams and considering the observed and simulated discharges at the hydrometric stations, as well as the observed and simulated reservoirs’ volume in each dam. For this purpose the first 4 years were used for calibration and the last 2 years were considered for validation. The RSR (Legates and McCabe 1999), NSE (Nash and Sutcliffe 1970), and PBIAS (Gupta et al. 1999) evaluation tests were employed to check the accuracy of the calibration and validation steps. The evaluation tests of RSR, NSE and PBIAS are defined in Eqs. 1 to 3.

$$ \mathrm{RSR}=\frac{RMSE}{STDEV_{obs}}=\left[\sqrt{\sum \limits_{i=1}^n{\left({X}_i^{obs}-{X}_i^{sim}\right)}^2}/\sqrt{\sum \limits_{i=1}^n{\left({X}_i^{obs}-{X}^{mean}\right)}^2}\right] $$
(1)
$$ NSE=1-\sum \limits_{i=1}^n{\left({X}_i^{obs}-{X}_i^{sim}\right)}^2/\sum \limits_{i=1}^n{\left({X}_i^{obs}-{X}^{mean}\right)}^2 $$
(2)
$$ PBIAS=\left(\sum \limits_{i=1}^n\left({X}_i^{obs}-{X}_i^{sim}\right)\times 100\right)/\sum \limits_{i=1}^n\left({X}_i^{obs}\right) $$
(3)

Where \( {X}_i^{obs} \) is the ith observation for the constituent being evaluated, \( {X}_i^{sim} \) is the ith simulated value for the constituent being evaluated, Xmean is the mean of observed data for the constituent being evaluated, and n is the total number of observations.

After ensuring the well performance of the model, periods of water shortage were simulated for the next 20 years by defining a reference scenario and applying the operation policy based on the current situation. So a 20 years period of discharge recorded at Khalife Bapir station located in the upstream of the studied dams, which includes wet and dry periods, was considered as the inflow into the reservoirs of the dams. Afterwards, the operation data of Gavoshan and Shohada dams were interred into the model, accordance with Table 1. The Gavehrood and Razavar river discharges were defined in the model at the stations as well as the discharge of Alak Minor River in Alak gauging station for the future 20 years period (2013–2033) in the form of monthly time series. The average discharges at the stations during the 20 years period are listed in Fig. 2.

Table 1 Operational specifications of the Gavoshan and Shohada dams during simulation period
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Fig. 2
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The average monthly discharges at the gauging stations (m3/s) and the average net evaporation from the reservoirs of Gavoshan and Shohada dams (mm)

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The net evaporation from the reservoirs was calculated according to the surface evaporation data of the reservoirs of Shohada and Gavoshan dams and the amount of rainfall. Average of net evaporation during the 20 years period was defined in the WEAP model which is shown in Fig. 2.

The water demand in the Bilevar and Miandarband plains was supplied through the regulated water from Gavoshan dam and the water demand of the Songhor plain was supplied through the Shohada dam. A part of the water demand of the Miandarband plain is supplied through the regulated water from Gavoshan dam and another part is supplied through Razavar diversion dam. In all the mentioned plains, the water demand was calculated based on the crop patterns of the plains and the other related parameters. The average water demand (in cubic meters) of the located lands in the Miandarband, Bilevar and Songhor plains are listed in Fig. 3. These average water demands were defined in the WEAP model.

Fig. 3
figure 3

Water demand of different study plains in different months (MCM)

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Songhor drinking water demand is supplied through the Shohada dam and a part of drinking water demand of Kermanshah is supplied by the Gavoshan dam and the rest is supplied through underground water resources. The water of Gavoshan dam is also used to supply the drinking water demand of Kamyaran. In order to calculate the drinking water demand of Songhor, Kermanshah and Kamyaran in the simulation period, demographic census data of 2007 and 2012 as well as per capita water consumption data of these areas were used. Per capita water consumption data of these cities are 223, 225 and 220 (lit-person/day), respectively. Finally, considering the population of each area in each period and per capita water consumption data, monthly drinking demand was calculated during the simulation period.

In order to determine the environmental flow, the nodes related to the minimum environmental flow were established in the model for the downstream of the dams of study area and Razavar River. Environmental flow was estimated based on the natural river flow in the desired nodes (Table 2). Tennant (or Montana) method, which is a hydrologic water allocating method, was employed to estimate the minimum downstream environmental flow (Tennant 1976).

Table 2 Minimum environmental flow of Gavehrood and Razavar rivers (m3/s)
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2.3 The Structure of the Proposed Multi-Objective Operation Model

In this study the NSGA-II algorithm which has been formulated as a multi-objective model was used to apply the heading policy. To ensure the superiority of the NSGA-II, this method was compared with the results of the PESA-II and SPEA-II multi-objective algorithms. In order to optimize the system, these algorithms were coupled with the WEAP model by employing a VBScript developed by the authors of this study. By running this script, decision variables generated by optimization algorithm were set in the WEAP active area path. Afterwards WEAP active scenario was defined and the program run automatically. For evaluation, the WEAP results were exported to the MATLAB path. Finally, WEAP will be closed automatically when the script get done, and the changes will be saved to the WEAP active area. In the structure of the optimizer algorithm two parameters, including the hedging level (the active volume of the reservoir at this level) and the coefficient of reservoir hedging (percentage), were utilized as the decision variables.

First in this structure, Reservoir storage was divided into four zones. These include, from top to bottom, the flood-control zone, conservation zone, buffer zone and inactive zone (Fig. 4). System allows the reservoir to freely release water from the conservation pool to fully meet withdrawal and other downstream requirements. Once the storage level drops into the buffer pool, the release will be restricted according to the buffer coefficient, to conserve the reservoir’s dwindling supplies.

Fig. 4
figure 4

Dividing the reservoir volume into four zones during the operation of Gavshan and Shohada dams

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  • Hedging level (storage) or top of buffer: when the storage level falls into the buffer zone (Hedging level (storage)), system uses the hedging fraction (buffer coefficient) to slow releases. This parameter was employed as the first decision variable series in the body of the model. So that two variables a year (one variable for each six-month period) were considered as the hedging level (storage) for Gavshan dam. Two variables a year were also defined for Shohada dam. Hence, a total of 80 variables of this type were considered for these two dams in the entire period of the operation (20 years).

  • Hedging fraction (buffer coefficient): the buffer coefficient is the fraction of the water in the buffer zone available each month for release. Thus, a coefficient close to 1.0 will cause demands to be met more fully while rapidly emptying the buffer zone, while a coefficient close to 0 will leave demands unmet while preserving the storage in the buffer zone. This parameter was also utilized as the second decision variable series in the body of the model. So that four variables a year (one variable for each three-month period) were considered as the hedging fraction (the buffer coefficient) for Gavshan dam. Four variables a year were defined for Shohada dam as well. Hence, a total of 160 variables of this type were considered for these two dams in the entire period of the operation (20 years). Therefore, the total number of the variations in the optimization process equaled 240.

The first objective of the NSGA-II algorithm, was to maximize the reliability of the demand sites coverage during the planning period and the second objective was to minimize the violation due to the failure in meeting water demands and reservoir capacity violation (damage function) during operation. The objective functions and constraints of the study were defined as following:

  1. 1.

    To maximize the reliability of the demand site coverage Eq. 4 was employed:

$$ {F}_1=\mathit{\operatorname{Maximize}}(Reliability)=\mathit{\operatorname{Maximize}}\left(\sum \limits_{t=1}^m\sum \limits_{z=1}^{nz}\sum \limits_{s=1}^{ns}\frac{TAW_{tzs}}{DM_{tzs}}\right) $$
(4)

Eq. 4 re-defined as a minimization objective, as shown in Eq. 5. Because the NSGA-II multi-objective optimization algorithm finds the minimum objective functions, hence:

$$ {F}_1=\mathit{\operatorname{Minimize}}\left(\sum \limits_{t=1}^m\sum \limits_{z=1}^{nz}\sum \limits_{s=1}^{ns}\left(1-\frac{TAW_{tzs}}{DM_{tzs}}\right)\right) $$
(5)

Where DM tzs is the water demand of the section s in the period t of the zone z, and TAW tzs is the total amount of water allocated to the section s in the period t of the zone z.

  1. 2.

    To minimize the violation due to the failure in meeting water demands and reservoir capacity violation the Eq. 6 was employed:

$$ {F}_2=\mathit{\operatorname{Minimize}}\left(\alpha D+\beta A+\boldsymbol{\upgamma} R\right) $$
(6)

Where α, β and γ are constants that are used to coordinate the dimensions of the penalty functions. In this study, α, β and γ were considered as 10, 1 and 1 respectively. It should be noted that the coefficients were determined by trial and error. The penalty functions of R, D and A are defined as Eqs. (7) to (9):

$$ R=\left(\sum \limits_{t=1}^m\sum \limits_{r=1}^n\left(\mathit{\operatorname{Max}}\left(\left(1-\frac{S_{tr}}{S\min_r}\right),0\right)\right)\right) $$
(7)
$$ D=\left(\sum \limits_{t=1}^m\sum \limits_{z=1}^{nz}\sum \limits_{d=1}^{nd}{\left({DM}_{tzd}-{TAW}_{tzd}\right)}^2\right) $$
(8)
$$ A=\left(\sum \limits_{t=1}^m\sum \limits_{z=1}^{nz}\sum \limits_{a=1}^{na}{\left({DM}_{tza}-{TAW}_{tza}\right)}^2\right) $$
(9)

Where DM tzd is the drinking water demand in the consumption point d in the period t of the zone z, TAW tzd is the total drinking water allocated to the consumption point d in the period t of the zone z, DM tza is the agricultural water demand in the consumption point a in the period t of the zone z, TAW tza is the total agricultural water allocated to the consumption point a in the period t of the zone z, S tr is the volume of the reservoir of the dam r at the normal level of period t, and Smin r is the volume of the reservoir of dam r at the minimum level.

Constraints:

$$ {Smin}_r<{S}_{tr}<{Smax}_r $$
(10)

Smax r is the Storage Capacity in Max Level of dam r.

$$ {TSR}_t={R}_{t1}+{R}_{t2}+{R}_{t3}+{R}_{t4} $$
(11)

TSR t is the total volume of the allocated surface water in the period t, Rt1 is the volume of water withdrawal from Gavoshan dam in the period t (which includes the total volume of the water taken from the drinking water conveyance canals of Gavoshan dam, the volume of the water taken from Razavar diversion dam and the volume of the allocated water to Bilevar (b1b2, b3b4) and Miandarband lands). In addition, Rt2 is the volume of the water taken from Shohada dam in the period t (which includes the volume of the water allocated to Songhor lands and the volume of the water taken from drinking water conveyance canals of Shohada dam), Rt3 is the volume of the water taken from Gavehrood river in the period t, and Rt4 is the volume of the water taken from Razavar river in the period t.

$$ {\displaystyle \begin{array}{c}{ARS}_{tzs}=\left\{\begin{array}{c}{DM}_{tzs}\kern0.5em if\left({TSR}_t-\sum \limits_{z=1}^{z-1}\sum \limits_{s=1}^s{DM}_{tzs}-\sum \limits_{z=1}^z\sum \limits_{s=1}^{s-1}{DM}_{tzs}\ge {DM}_{tzs}\right)\\ {}\left({TSR}_t-\sum \limits_{z=1}^{z-1}\sum \limits_{s=1}^s{DM}_{tzs}-\sum \limits_{z=1}^z\sum \limits_{s=1}^{s-1}{DM}_{tzs}\right)\kern0.5em otherwise\end{array}\right\}\\ {}\begin{array}{cc}Z= IZ(1),\dots, IZ(nz)& S= IS(1),\dots, IS(ns)\end{array}\end{array}} $$
(12)

ARS tzs is the total volume of the surface water allocated to the section s in the period t of the zone z (considering the demand priority).

$$ Q\le 30{m}^3/s $$
(13)
$$ M1<\mathrm{Tb}1<\mathrm{N}1 $$
(14)
$$ M2<\mathrm{Tb}2<\mathrm{N}2 $$
(15)

Where Q is the discharge of Gavoshan water conveyance tunnel (m3/s), Tb1 is the hedging level of Shohada dam(m), Tb2 is the hedging level of Gavoshan dam(m), M1 is the inactive level of Shohada dam(m), M2 is the inactive level of Gavoshan dam(m), N1 is the maximum level of Shohada dam(m), and N2 is the maximum level of Gavoshan dam (m).

The seasonal hedging coefficients and the six-months hedging level coefficients were used in the body of the NSGA-II optimization algorithm and the coefficients were finally introduced to the model as the decision-making variables (240 variables).

Two scenarios including the reference and the optimum scenarios were applied in the model and the results were evaluated. The reference scenario was run as following:

  1. 1.

    The simulation year started in October 2013 and ended in September 2033.

  2. 2.

    By taking the population growth rates of cities into account, the drinking water demand of Kermanshah, Kamyaran and Songhor to be increased in the next periods in successive 5 years period by the end of a 20 years period based on per capita consumption of the drinking water and the future population.

  3. 3.

    Given the conditions of regional development and the tendency to increase the area under cultivation, it was assumed that the agricultural demand of Bilevar plain (b1b2 and b3b4) and Songhor and Miandarband plains would increase about 10% and 20%, respectively in the next 20-year period.

  4. 4.

    The system was required to comply with the downstream minimum environmental flows of Gavoshan and Shohada dams as well as Razavar River. Finally, the drinking water supply, the environmental flows, the traditional water right, and the agricultural demands of the plains were considered as the 1st, 2nd, 3rd and 4th priorities, respectively.

The optimum scenario was run based on the reference scenario with the following characteristics:

  1. 1.

    The same allocation priorities of the reference scenario were considered.

  2. 2.

    The drinking and agricultural demands conformed to the reference scenario. In addition, input discharges into the system were considered to be similar to the reference scenario.

  3. 3.

    In this scenario, the optimized hedging coefficients on a monthly basis were used.

  4. 4.

    The allocated volumes to each demand in different months conformed to the optimum values based on the mentioned objective functions in the previous sections.

In the optimum scenario, NSGA-II was coupled with the WEAP model to optimize the system operation. WEAP-NSGA-II coupling process was shown in Fig. 5.

Fig. 5
figure 5

Flowchart for WEAP-NSGA-II coupling process

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3 Results and Discussion

3.1 Calibration and Validation Results of Model

Results of the calibration and validation of the WEAP model for the operation period of 4 and 2 years respectively, is given in Table 3.

Table 3 The values of different criteria in the calibration and validation steps
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The values of the evaluation tests in the Table 3 indicates the correct simulation of the operation model in two calibration and validation steps.

3.2 Reference Scenario Results

The results of demand site coverage and the reliability (%) of the reference scenario are summarized in Fig. 6.

Fig. 6
figure 6

Average amount of demand site coverage and reliability of reference scenario

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According to Fig. 6, it can be concluded that for some demands, including the demands of Miandarband and Bilevar plains, the reliability percentage decreased significantly (about 43%) in compared with the historical period registered (1993–2013). The reason of this fact is came back to the land development due to the regional conditions and consequently water demand increase. In these lands, the percentages of demand coverage decreased significantly in all months excluding February and March. Regarding the drinking and agricultural demands of the lands of Songhor plain, demand site coverage reliability decreased about 9% and 4%, respectively. In addition, because of the numerous consumers known as the traditional water rights (WR), the distribution of water withdrawal sites along the river and not totally controlled traditional water rights, it can be indicated that these demands inevitably have a higher priority than the other agriculture demands of the study area. This is one of the reasons for the high percentages of coverage and reliability. The highest reliability percentages of the environmental demands were observed 95.8, 90.8 and 84.1 (%) for downstream of Gavoshan dam, Razavar River and Shohada dam, respectively in the reference scenario. Figures 7 and 8 show the storage changes in the reservoirs of Shohada and Gavoshan dams in the reference scenario.

Fig. 7
figure 7

The storage changes of the reservoir of Shohada dam in the reference scenario (MCM)

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Fig. 8
figure 8

The storage changes of the reservoir of Gavoshan dam in the reference scenario

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It is worth mentioning that in Figs. 7 and 8, the buffer level is the minimum level. In order to prevent reservoir storage levels being lower than the buffer level in model simulation, the reservoir outlet was set at the zero buffer level. Figures 7 and 8 also show that in the reference scenario the reservoir storage volume was defined at the levels above the minimum level (the buffer level) in all periods. It can attribute to the low percentages of the demand coverage and the reliability, especially for the agricultural demand.

3.3 Optimization Results

The system optimization results using the combination of the NSGA-II, PESA-II and SPEA-II algorithms with WEAP model are given in Fig. 9. Based on this figure the NSGA-II algorithm is able to generate a better Pareto front in terms of minimizing the objective functions. But SPEA2 and PESA-II has a better diversity and wider collection of optimal solutions. So finally, due to the quality of the optimal solutions, the NSGA-II algorithm was selected to be used in this study.

Fig. 9
figure 9

Nondominated solutions of NSGA-II, PESA-II and SPEA-II in the 500th iteration

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In the optimum scenario, runs the repeated model showed that in order to achieve better results, the initial population of chromosomes should be at least twice the number of the decision variables. In this study, the initial population of model was selected to be about 480. Due to the complexity of the problem and the large number of variables, the number of iterations for achieving the convergence was considered about 500. When running the algorithm for 500 iterations, Pareto curve will be updated in each iteration. Continuous review of the Pareto curve for 500 times run of algorithm showed that in a less iteration, both the reliability and the penalty functions changed significantly. However, fixed changes of the reliability function were observed in higher repetitions and the model focused on decreasing the penalty. Finally, considering the population size of 480 and running the NSGA-II algorithm for 500 iterations, close to optimum solutions were obtained within 12 days, 3 h and 20 min and the optimum exchange curve (Pareto-optimal Front) between optimization objectives (the maximum reliability function and the minimum function of penalty due to the system violations from demands coverage) was obtained. Accordance to the Non-dominated Sorting Genetic Algorithm (NSGA-II), the best solutions are selected based on the objective functions and elitism process in each iteration and are stored as F1 optimum set to be transferred into the next generation. Points on the Pareto graph are the optimum solutions of the model and the axis of the graph are the desired objective functions. This graph which is based on ten optimum solutions considering F1 and F2 objective functions in the last iteration is shown in Fig. 10. By choosing the answer 10 the average reliability of the entire system is 84.5% and by choosing the answer 6 it is about 88.5%. Therefore, in this respect, the answer 6 is perform better. As this figure illustrates, there is no big difference between answers 1 and 6 in terms of reliability. However, answer 6 has less penalty function, so finally answer 6 was selected as the best answer in this study.

Fig. 10
figure 10

The optimum exchange graph between the optimization goals (Pareto-optimal Front) in the 500th iteration

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After selecting the answer 6 as the best answer, the optimum variables were introduced to the WEAP surface water model and the results were evaluated. The coverage and reliability percentages of the optimum scenario are shown in Fig. 11.

Fig. 11
figure 11

The average amounts of the demand coverage and reliability (Re) in optimum scenario

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Figure 11 shows that coverage and the reliability percentages of most demands increased compared to the reference scenario by taking into account the hedging policy in the optimum scenario, indicating the efficiency of the multi-objective optimization based on the hedging policy in the study area. In the optimum scenario, it can be concluded that the demand coverage of all lands of the studied plains was absolutely desirable in December, January, February, March and April and there has been a significant increase in the demand coverage percentages in most of the other months. In addition, the reliability percent of Bilevar lands increased about 20% in the optimum scenario where the reliability percentage was the lowest in the reference scenario. The reliability percentage of the drinking water demand of Kermanshah decreased in comparison to the reference scenario, but the application of the hedging policy in the optimum scenario for this demand has reduced the intensity of the failures in the months with intense water deficits. The highest reliability percentages of environmental demands were observed 100, 95.41 and 84.6 (%) for downstream of Gavoshan dam, Razavar River and Shohada dam, respectively in the optimum scenario. Results shows that the reliability percentage of the environmental demand in Shohada dam in the optimum scenario does not change significantly in comparison to the reference scenario. The increase in reliability percentage of the other environmental demands is about 4 to 5% compared to the reference scenario, however in general it can be said that the coverage of the environmental demands in the optimum scenario is acceptable. Figures 12 and 13 show the storage changes of the reservoir of Shohada and Gavoshan Dams in the optimum scenario.

Fig. 12
figure 12

The storage changes of the reservoir of Shohada dam in the optimum scenario (MCM)

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Fig. 13
figure 13

The storage changes of the reservoir of Gavoshan dam in the optimum scenario (MCM)

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According to Figs. 12 and 13, it can be noted that due to the application of the hedging coefficient during the planning period in the optimum scenario, the reservoir storage did not become less than the reservoir storage corresponding to the minimum operation level of dam except for a few cases. The water hedging helps to manage the reservoir storage in intense droughts so that in addition to achieving the demand coverage reliability, the intensity of most failures in the months with intense water deficits reduces. However, due to the defined constraints in the reservoir in case of the reference scenario, although the water level was not lower than the minimum operation level in the reservoir, the demand coverage reliability was low. Therefore the intensity of failure in some dry months is high that can cause irreparable damages. Accordingly, it can be admitted that the implementation of hedging policy in real reservoir, by using two hedging level and hedging fraction parameters, will produce satisfactory results. So that during the operation period, the number of failures months and the failure intensity in the needs coverage will be reduced. This was proved by applying these parameters in reservoirs A and B using the combination of NSGAII optimization algorithm and WEAP simulation model. Originally, using this method, by reservoir parameterization based on these parameters, the release of reservoir can be optimized in low flow conditions.

4 Conclusion

From the results of this pare it can be concluded that the developed model in this study which is based on a combination of the NSGA-II multi-objective algorithm and the WEAP model is appropriately able to solve complex and nonlinear problems and present optimum solutions in order to apply the hedging policy in multi-reservoir systems. Also the NSGA-II algorithm has a superiority to other algorithms in this regard. Results showed that with considering the hedging policy in the optimum scenario, the coverage and the reliability percentages of most demands will increased in compared with the reference scenario and the demands were covered appropriately. In addition to offering optimal solution with maximum reliability and minimum penalty, this coupled model has the capability to significantly decrease the failure rate in dry months. Upon the completion of the algorithm and after deriving the optimized variables, it can be concluded there is a significant association between the monthly inflows to the reservoir, the volume of water in the reservoir, the downstream demands (as the independent variables) and the optimum release variable (as the dependent variable). Therefore, this model can be converted into the real time by employing intelligent methods such as the support vector machine. This means that in any given simulation for future period, if the three first parameters are determined at the beginning of each month, the optimum release will be known in the real time. With apply of this model and changing the inflow, there will be no need to repeat the optimization process to find out the optimal coefficients, but by using the relationship derived from the support vector machine, it will be possible to determine the optimum release or the hedging level as well as the hedging coefficient in the real time based on the inflow to the reservoir (at the beginning of each month), the volume of water stored in the reservoir (at the beginning of the month) and the downstream demands in the current months.