Shuffled complex evolution coupled with stochastic ranking for reservoir scheduling problems

Jing-qio Mo , Ming-ming Tin , Teng-fei Hu , Kng Ji , Ling-qun Di , Hui-ho Di

a College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, China

b Ningbo Hongtai Water Conservancy Information Technology Co., Ltd., Ningbo 315040, China

c College of Hydraulic and Environmental Engineering, Three Gorges University, Yichang 443002, China

Abstract This paper introduces an optimization method (SCE-SR)that combines shuffled complex evolution (SCE)and stochastic ranking (SR)to solve constrained reservoir scheduling problems, ranking individuals with both objectives and constrains considered.A specialized strategy is used in the evolution process to ensure that the optimal results are feasible individuals.This method is suitable for handling multiple conflicting constraints,and is easy to implement,requiring little parameter tuning.The search properties of the method are ensured through the combination of deterministic and probabilistic approaches.The proposed SCE-SR was tested against hydropower scheduling problems of a single reservoir and a multi-reservoir system, and its performance is compared with that of two classical methods (the dynamic programming and genetic algorithm).The results show that the SCE-SR method is an effective and efficient method for optimizing hydropower generation and locating feasible regions quickly, with sufficient global convergence properties and robustness.The operation schedules obtained satisfy the basic scheduling requirements of reservoirs.

Keywords:Reservoir scheduling; Optimization method; Constraint handling; Shuffled complex evolution; Stochastic ranking

1.Introduction

Reservoir scheduling refers to the timing and amounts of water released to fulfill management objectives, e.g., to improve the operational efficiency of reservoirs for optimal economic benefits (Simonovic, 1992).However, reservoir scheduling problems are challenging to solve, as they are dynamic, high-dimensional, nonlinear, and nonconvex(Labadie, 2004; Mao et al., 2016), as well as being subject to multiple constraints, such as water balance, release bounds,reservoir level,and power output limitations(Castelletti et al.,2008).The importance of complex hydraulic connections(i.e.,continuity of hydraulic factors of upstream and downstream discharges)should be emphasized for the joint operation of reservoirs, which affects the inflow of downstream reservoirs as well as operation rules and corresponding hydropower generation (Teegavarapu and Simonovic, 2000).

Various optimization techniques have been applied to reservoir scheduling problems, among which the dynamic programming (DP)and genetic algorithm (GA)are most commonly used.Although DP is suited to the optimization of water resources systems (Buras, 1966), it requires customized programs and is often faced with the “curse of dimension”when applied to multi-reservoir systems (Yeh, 1985).Similarly,GA is effective in searching for optimal solutions(Chang and Chang, 2001; Liu et al., 2017; Li et al., 2018), but some essential difficulties still exist, such as the “premature convergence” caused by high-dimensional decision variables(Li et al., 2012), especially when solving constrained multireservoir optimization problems.The shuffled complex evolution(SCE)is a global optimization method introduced in the 1990s, which combines the strengths of the simplex method and the competitive complex evolution (CCE)algorithm(Duan, 1991).This method has been increasingly applied to calibration of conceptual watershed models since it can converge to the global optima in a consistent, efficient, and fast way (Tang and Luan, 2007).However, it encounters difficulty in solving constrained optimization problems due to the lack of inherent constraint-handling techniques (CHTs).

CHTs are used to guide the search towards global optima by reasonably assigning fitness to the individuals with consideration of both objective functions and constraint violations.The commonly used CHTs are divided into three categories (Hu et al., 2018):(1)penalty function techniques,(2)techniques preferring feasible individuals, and (3)stochastic ranking (SR).The penalty function techniques are effective in handling constraints.However, striking the right balance between objectives and penalty functions is challenging (Runarsson and Yao, 2000).Techniques preferring feasible individuals are free of tedious parameter tuning, but tend to be trapped in local optima since much information carried by nonfeasible individuals is ignored (Deb, 2000;Mezura-Montes and Coello, 2005).SR is capable of balancing objectives and penalty functions (Runarsson and Yao, 2000), easy to implement, and highly competitive with other methods, without any complicated and specialized operators (Coello and Montes, 2002; Cai and Wang, 2006).

In order to make the SCE suitable for constrained reservoir scheduling problems and to achieve global convergence properties, a new optimization method referred to as the shuffled complex evolution-stochastic ranking (SCE-SR)was developed by combining SCE with SR.It is easy to implement and requires little parameter tuning.In addition, both the objective functions and constraint violations are taken into consideration, which can guarantee the competitiveness and diversity of the sample.The proposed SCE-SR was tested in various test cases and quantitatively compared with two commonly used methods to evaluate its performance.

2.Problem formulation

The objective of reservoir scheduling is often to maximize the benefits of flood control, hydropower generation, water supply,navigation,and ecosystem services(Mao et al.,2015).For example, the objective of maximizing the long-term hydroelectricityEgenerated byMreservoirs overTperiods throughout the scheduling horizon can be described as follows:

whereNi,tis the power output of reservoiriin periodt; Δtis the time step;Kiis the overall conversion coefficient of reservoiri;Qei,tis the discharge of reservoirifor hydropower generation in periodt;hi,tis the average water head of reservoiriin periodt;Qi,tis the average outflow of reservoiriin periodt; andQsi,tis the surplus discharge of reservoiriin periodt, which means the part of outflow that exceeds the maximum discharge through the reservoir turbines.

Reservoir scheduling is subjected to multiple constraints,as follows:

Water balance equation:

Forebay elevation constraint:

Outflow discharge constraint:

Power output constraint:

whereVi,tis the storage of reservoiriat the beginning of periodt;qi,tis the average inflow into reservoiriin periodt;Q1i,tis the rate of storage loss (e.g., evaporation and seepage)of reservoiriin periodt;Qi,tminandQi,tmaxare the lower and upper outflows of reservoiriin periodt,respectively;Zi,tis the forebay elevation of reservoiriat the beginning of periodt;Zi,tminandZi,tmaxare the lower and upper forebay elevations of reservoiriat the beginning of periodt, respectively; andNi,tminandNi,tmaxare the firm output and installed capacity of reservoiriin periodt, respectively.

When considering a multi-reservoir scheduling problem(M≥2), hydraulic connection between reservoirs (shown in Fig.1, considering two reservoirs), i.e., upstream control(Khatibi,2003),should be taken into account.At this point,the average inflow of a reservoir is calculated according to Eq.(8).

Fig.1.Schematic diagram of hydraulic connection between two reservoirs.

whereqIi,tis the lateral inflow into reservoiriin periodt,andQqi,tis the other water consumption, such as the water transport loss, of reservoiriin periodt.

3.Methodology

3.1.Shuffled complex evolution

Duan et al.(1992)presented SCE by introducing a new concept of complex shuffling to solve nonlinear optimization problems.The method involves the following terminologies:points (candidate solutions), population (the community containing all points), complex (the community containing several points partitioned from the sample), and complex shuffling(points in complexes reassigned and mixed to generate a new community).The main idea of this algorithm is that the points evolve independently in each complex and shuffle in the sample population to guide the searching process towards the optimal solutionof the problem.A flow chart ofSCEisshown inFig.2(a).

One main component of SCE is the CCE algorithm,which can be described briefly as follows (Duan et al., 1994):

(1)Construction of a sub-complex (containingqpoints)according to the trapezoidal probability distribution.

(2)Ranking:identification of the worst pointuof the subcomplex and computation of the centroidgof theq-1 points without including the worst one.

(3)Reflection:reflection of pointuthrough the centroid to generate a new pointrand calculation of its objective function valuefr.If the newly generated pointris within the feasible space andfr＞fu, wherefuis the objective function value of pointu,uis replaced withr,and the process moves to step(6).Otherwise, it goes to step (4).

(4)Contraction:determination of a pointchalfway between the centroid and the worst point,and then calculation offc.If pointcis within the feasible space andfc＞fu,uis replaced with the contraction pointcand the process goes to step (6).Otherwise, it goes to step (5).

(5)Mutation:random generatetion of a pointzwithin the feasible space and replacement of the worst point withz.

(6)Steps(2)through(5)are repeated α times,where α ≥1 is the number of consecutive offspring generated by each subcomplex.

(7)Steps(1)through(6)are repeated β times,where β ≥1 is the number of evolution steps taken by each complex before complexes are shuffled.

The CCE algorithm can be illustrated in Fig.3 by the mathematical problem of finding the global optima using a two-dimensional case, with the region ofX∈［0,1］ andY∈［0,1］.In this case,the complex contains five sample points and each sub-complex (the triangle in Fig.3)contains three points.The contour lines represent a function surface,.The competitive mechanism is implemented by introducing the trapezoidal probability distribution,which favors better points over worse ones.That is, each point in a complex has a potential ability to reproduce offspring,and the stronger one has more chance to survive and generate better offspring.Such a mechanism guides the search towards improvement regions at a high speed (Duan et al., 1993).In addition, the offspring is generated randomly in the feasible space sometimes to ensure that the evolution process is not trapped in nonfeasible regions(Duan et al., 1994).

Fig.2.Flow chart of SCE-SR.

Fig.3.Illustration of path from newly generated point to global optima by CCE algorithm.

3.2.Stochastic ranking

The SR, proposed by Runarsson and Yao (2000), is capable of balancing objective and penalty functions and improving the search performance.The main idea is to compare two adjacent individuals according to the objective function values or the degree of constraint violations by introducing a predetermined parameterPf.An increase in the number of ranking sweeps (N)is effectively equivalent to changing parameterPf.Thus, the number of ranking sweeps is fixed toN=s(number of points in sample population generated by SCE),andPfis adjusted within［0,1］to achieve the best performance.The comparison mechanism of two adjacent individuals can be briefly described as follows:if both individuals are feasible, or a randomly generated numberw∈［0,1］ is less thanPf, they are compared according to the objective function values; otherwise, they are compared based on the degree of constraint violations.Ranking of the whole sample population is then achieved through a bubble-like procedure.A flow chart of SR is given in Fig.2(b).

3.3.SCE-SR

Although SCE is a competitive global optimization algorithm, it encounters difficulties in solving constrained reservoir scheduling problems due to the lack of CHTs.The ranking of SCE based only on the objective function values will output a nonfeasible optimal solution with no practical significance.On the other hand, SR is an independent competitive CHT that is easy to implement and incorporate with other algorithms.Therefore, it is necessary to replace the ranking mechanism of SCE with SR to balance the objective and constraints and enable it to be applicable for the constrained reservoir scheduling problems.Fig.2 illustrates how these two methods are coupled and shows that only the parameterPfneeds to be tuned.

There are two main characteristics of SCE-SR:(1)the combination of the deterministic approach and competitive evolution, and (2)the combination of the probabilistic approach and complex shuffling.The former is conducive to directing the search in an improving direction and improving global convergence efficiency by making use of information carried by both feasible and nonfeasible individuals.The latter guarantees the survivability of individuals and the flexibility and robustness of the algorithm (Duan et al., 1992).These characteristics ensure the global convergence properties of SCE-SR over a variety of problems.

According to Duan(1991),four vital parameters of SCE are assigned default values:(1)the number of points in a complex,m= 2n+ 1, wherenis the dimension of the decision vector;(2)the number of points in a sub-complex,q=n+1;(3)the number of consecutive offspring generated by each subcomplex, α = 1; and (4)the number of iterations taken by each complex, β =m.For SR, the suggested range of the parameter is 0.4 ＜Pf＜0.5(Runarsson and Yao,2000);it was set to 0.45 in this study.

The SCE-SR method is terminated whenever one of the following convergence criteria is satisfied:

(1)The objective function value is not significantly improved afterjtimes of iterations.Its expression is as follows:

wherefnandfn-j+1are the objective function values of an optimized individual afternandn-j+1 times of iterations,respectively;fis the average absolute value of the objective function of the optimized individual after each iteration; andTOLis the predetermined acceptable degree.

(2)The interval of variables is small enough.Its expression is as follows:

whereximaxandximinare the maximum and minimum values of theith variable in the population after each iteration,respectively;ciis the size of the feasible interval of theith variable;andTOLλis the allowable concentrated degree of the variables.

(3)The cumulative number of objective function calls(NOFC)reaches the predetermined value.

4.Case study

SCE-SR was applied to long-term reservoir scheduling of both a single reservoir and a multi-reservoir system in this study.The scheduling horizon for each test was set to one year and the time step was one month.The forebay elevation of different reservoirs at the beginning of each month was set as the decision vector.The convergence criteria were based on the predetermined NOFC.

To evaluate the performance of SCE-SR, its results were compared with those of two classical methods, DP and GA.The specialized strategy was used for GA and SCE-SR as follows:if a feasible individual was not found, the nonfeasible individual with the lowest constraint violations was selected as the best; otherwise, the best feasible individual was archived.GA includes three important operations:selection, crossover, and mutation.In this study, normalized geometric selection was used for selection operation, with the probability of selecting the best individual being set at 0.05; heuristic crossover was used for crossover operation,with the number of retries being set at 10; and for mutation operation,both multi non-uniform mutation and non-uniform mutation were used,with the shape parameters being set at 3.For fair comparison, GA was coupled with SR (denoted as GA+SR).

For SCE-SR and GA, the input data were the monthly inflow in different scenarios under multiple constraints (forebay elevation, outflow discharge, and power output)for each reservoir.The outputs were the optimized forebay elevation of each month and corresponding annual hydropower generation.The input and output for DP were the monthly forebay elevation and corresponding annual hydropower generation,respectively.

4.1.Single reservoir

The Wanjiazhai Reservoir(WJZR)is a major component of the Water Diversion Project from the Yellow River to Shanxi Province in China (Fig.4), which serves multiple purposes,including water supply, hydropower generation, and flood control.WJZR has a gross storage of 8.96 × 108m3at the normal water level of 977 m,and its total installed capacity is 1.08 × 106kW.In this study, three representative years were selected for WJZR, by considering the guaranteed rates of their inflow scenarios:the wet year withP=35%,the normal year withP=50%, and the dry year withP=75%,wherePis the guaranteed rate.Fig.5(a)shows the average monthly discharges of WJZR from May to the next April in different inflow scenarios.According to Yan et al.(2018), in general,the overall conversion coefficientKof hydropower plants for Chinese reservoirs is 8.5 for large power stations, 8.0-8.5 for medium-sized power stations, and 6.0-7.5 for small power stations.For WJZR, theKvalue was 8.3, and the guaranteed rate of a dry year was set at 80%, as the output constraint is often difficult to satisfy in dry years.

The SCE-SR parameters were set as follows:the dimension of the decision vector wasn=12 and the number of points in each complex wasm= 25.Since the results generated by SCE-SR are sensitive to the number of complexes (p), three different values ofp,i.e.,8,12,and 20,were considered in this study.Therefore,the total points in the sample population(s=pm)were 200,300,and 500,respectively.To ensure the same sample size, the population sizes of GA were set at 200, 300,and 500, while the maximum generation was 500.Then, to ensure the same NOFC for different methods, the monthly water levels of DP were divided into 112,137,and 177 states,respectively.Table 1 lists nine designed test cases with different inflow scenarios and NOFC values,for each of which 30 independent runs were performed with GA and SCE-SR.For DP, only one run was performed in each test case since the optimal result is only subjected to the inflow condition and the limitation of the water level (Labadie, 2004).

Fig.4.Location of reservoirs.

Fig.5.Average monthly discharges in different inflow scenarios.

4.2.Multi-reservoir system

SCE-SR was further tested against a multi-reservoir system on the Qingjiang River, located in the Yangtze River watershed of China.The total length of the main stream is 423 km, with a drainage area of 17000 km2and a hydraulic drop of 1430 m.The multi-reservoir system consists of the Shuibuya Reservoir (SBYR), Geheyan Reservoir (GHYR),and Gaobazhou Reservoir (GBZR)from upstream to downstream (Liu et al., 2011).Both SBYR and GHYR are large water resources projects for hydropower generation and flood control, and GBZR functions as a daily regulation reservoir(Fig.4(b)).

Table 1 Designed test cases of scheduling of WJZR.

This study mainly considered the joint scheduling of SBYR and GHYR.The overall conversion coefficientKwas 8.5 for both reservoirs.Ten continuous hydrological years(1977-1986)were selected as the inflow scenarios to observe the performance of the present method,since the optimization results vary significantly for different inflows.The average monthly discharges of SBYR from May to the next April are shown in Fig.5(b).Numbers 1 to 10 represent different years,with a declining trend in the average monthly discharge(1 represents the wettest year of 10 years, and 10 represents the driest year of 10 years).

The SCE-SR parameters were set as follows:the dimension of the decision vector wasn= 24, the number of points in each complex wasm= 49, and the numbers of complexespwere 4,6,and 10,so that the total points in sample population were 196, 294, and 490, respectively.The population sizes of GA were set at 196, 294, and 490, while the maximum generation was set at 500 to ensure the same sample size as SCESR.The number of monthly water level states when DP was applied was set at 10,12,and 14,respectively.Table 2 lists the 30 designed test cases.For each case, 30 independent runs were performed with GA and SCE-SR and only one with DP.

4.3.Performance metrics

Several statistical indices were used to assess the performance of different methods,including the Max,Min,Average,Std, FR, and NOFCFFI.The first three indices indicate the maximum, minimum, and average optimized annual hydropower of 30 independent runs for each test case, respectively.Std represents the standard deviation of the optimized hydropower of 30 independent runs in different test cases.FR exhibits the percentage of feasible runs,which generated at least one feasible individual among 30 runs,and its range was[0,1].NOFCFFI denotes the number of objective function calls when finding the first feasible individual.Note that only the results of feasible runs could be incorporated into the statistics,and DP only involved the concepts of Max and FR.

5.Results and discussion

5.1.Optimized hydropower generation

Figs.6 and 7 illustrate the comparison of the abovementioned six indices between SCE-SR, GA+SR, and DP (a single run in each test case)in test cases of WJZR (S-1 through S-9)and SBYR-GHYR (M-1 through M-30),respectively.Note that there was only one feasible run in case M-29 and zero feasible runs in case M-30 when GA was applied to the multi-reservoir system.

Since the differences in the Max,Min,and Average of these three methods were much smaller than the optimized results of these indices, the best results of these methods in each test case were recorded.Then,the relative values of the Max,Min,and Average to the best results, presented as the distances of those results to the best ones in the searching regions, were calculated,and are shown in Fig.6(a)through(c)and Fig.7(a)through (c).

As can be seen in Figs.6(a)and 7(a), the optimal hydropower generation of DP was 0.07%-2.61% less than that of SCE-SR in all test cases, and 0.01%-1.49% less than that of GA in more than one-half of all cases.In addition, when taking the multi-reservoir system (case M-1 through M-30)into consideration, the differences between the results of DP and other methods were larger.The main reason is that DP provides the optimal solution in a discrete sense, and the discrete precision limits the performance of DP.With a greater number of monthly water level states, the dispersion degree decreases, and the optimized results improve.

Relatively speaking, SCE-SR showed a better performance than GA in more than four-fifths of the 39 test cases in terms of the Max, especially in single reservoir cases and dry yearsof multi-reservoir system cases.However, nearly one-third of its Min and Average values were worse than those of GA,mainly occurring in wet years in multi-reservoir system cases.It is worth noting that the relative values of the Max,Min,and Average of SCE-SR to the best had a tendency to decrease in test cases M-1 through M-30.This means that SCE-SR continued to generate better offspring, rather than converging to the local optima.There are two main reasons for this phenomenon:(1)the constrained scheduling problem of the multi-reservoir system was much more complex than that of the single reservoir so that the NOFC of each iteration was much larger,and(2)the predetermined convergence condition for NOFC was not large enough.

Table 2 Designed test cases of scheduling of SBYR-GHYR system.

Fig.6.Performance comparison of methods in terms of six indices in test cases of WJZR (* means the relative value to the best).

Fig.7.Performance comparison of methods in terms of six indices in test cases of SBYR-GHYR (* means the relative value to the best).

For further comparison, some additional numerical experiments were carried out by increasing the number of complexes in four scenarios (i.e., the hydrological years 1-4 for the SBYR-GHYR system).The adjusted parameters in these cases were set as follows:the number of complexes of SCESR was 20; the population size and the maximum generation of GA were 980 and 5000,respectively;and the NWLS of DP was 27.In each test case,30 independent runs were performed with SCE-SR and GA.Detailed results are shown in Table 3,in which bold lettering indicates that SCE-SR obtained the best results among these methods.

According to Table 3, SCE-SR showed better performance than both GA and DP in terms of the Max,Min,and Average.Compared with test cases M-3,M-6,M-9,and M-12(cases in wet years of the multi-reservoir system), the maximum hydropower generation of SCE-SR increased by 0.163%-0.474%.These findings suggest that SCE-SR may produce better solutions with the increase of the number of complexes.

In summary, SCE-SR can improve the annual hydropower generation for both the single reservoir and multi-reservoir system.This is mainly due to the combination of the CCE algorithm and complex shuffling.As shown in Fig.3,the CCE algorithm can guide the search process in a complex towards improvement.Subsequently, the evolved complexes are shuffled and sorted using SR, ensuring that the results are not trapped in the local optima through the information sharing in all complexes.

5.2.Ability to locate feasible region

Sometimes, it is impossible to find a feasible solution due to various constraints.Therefore, FR was introduced to assess the reliability of different methods.Figs.6(e)and 7(e)show that FR of GA varies greatly (from 0 to 1).It was close to 1.0 when the inflow was abundant, but it decreased in dry years since the outflow discharge constraint and the power output constraint were difficult to satisfy.In contrast,the FR of SCESR and DP reached 1.0 in all test cases, exhibiting greater reliability.Such an advantage is clearer when taking NOFCFFI in Figs.6(f)and 7(f)into account.SCE-SR could find the first feasible individual quickly even when the constraints were hard to satisfy.In addition, the differences in NOFCFFI between GA and SCE-SR in multi-reservoir system cases were much more significant, sometimes even 360 times those in single reservoir cases.

These results imply that SCE-SR has a high degree of reliability and a high capacity to locate feasible regions,which is mainly attributed to the implementation of the CCE algorithm, SR, and specialized strategy.The CCE algorithm can make full use of information contained in all complexes to guide the search process towards the global optima instead of being interrupted in the local optima (Duan et al., 1994).Simultaneously, the influences of the objective function and constraints are balanced by SR so that information about nonfeasible individuals can be used to help find feasible ones in the initial stages.In addition, the specialized strategy can preserve the best feasible individual during each iteration.The combination of these methods enables SCE-SR to locate feasible regions quickly.

5.3.Convergence performance

The convergence performance of SCE-SR and GA is examined in this section.When taking the metric Std into consideration (Figs.6(d)and 7(d)), it can be seen that GA results have a certain degree of random fluctuation in almost all cases of the multi-reservoir system.As for SCE-SR, its results fluctuate over a very small range near the optima in WJZR cases, and fluctuate over a relatively small range in test cases M-20 through M-30.Although the Std of SCE-SR is sometimes larger than that of GA (cases M-1 through M-19),it is noteworthy that its values dwindle with the increase of the number of complexes (Fig.7(d)and Table 3).That is,SCE-SR has the potential to converge to the global optima when the number of complexes is large enough.In contrast,GA would quickly converge to the local optima and be trapped in it.

Table 3 Optimized results from numerical experiments for SBYR-GHYR system.

Fig.8.Convergence trend in cases S-7 and M-19.

Test cases S-7 and M-19 were selected to exhibit the evolutionary trajectories of the objective(Fig.8),in which the Std of SCE-SR reached a maximum value in WJZR cases and SBYR-GHYR cases, respectively.For GA, only 22 runs in case S-7 and 19 runs in case M-19 produced feasible solutions.Note that the horizontal axis in Fig.8 is logarithmic since the hydropower generation increased markedly with the number of objective function calls in the initial stages.

As seen in Fig.8, SCE-SR has a narrower convergence range of feasible individuals (the gray region in Fig.8)compared with GA, especially in the initial stages of the search process.As mentioned in section 5.2, SCE-SR has a strong ability to locate feasible regions.Once a feasible individual is found,the objective function value evolves efficiently and uniformly when SCE-SR is applied.These findings can be partly attributed to the reflection, contraction, and mutation operations in the CCE algorithm, which guide the search towards the space with promising individuals at a competitive speed.In addition, SR can promote convergence as it allows some nonfeasible individuals to have a chance to participate in the process of generating better offspring,especially when the global optima are near the boundary of feasible regions.That is to say,SCE-SR can converge to the global optima uniformly and effectively.

5.4.Reservoir operation schedules

Fig.9 presents the optimized reservoir operation schedules obtained with different methods in case S-9, in which constraints were difficult to satisfy.The basic scheduling requirements of WJZR are visualized as dashed lines representing the upper and lower forebay elevation boundaries in this figure.Obviously, the optimized WJZR forebay elevation of SCE-SR varies within the predetermined bounds throughout the scheduling horizon and generally lies on the upper bound, which helps to maximize the hydropower generation.During the flood season,the forebay elevation starts to decrease from the normal water level (977 m)in June to the dead water level (948 m)in September and remains at 948 m until October in order to guarantee the flood reserve capacity.The forebay elevation rises to the normal water level as quickly as possible in October and November and then remains essentially unchanged.This is of great significance to hydropower generation in the following periods as the difference between forebay elevation and tailwater elevation is large.In general,the optimized results obtained with SCE-SR satisfy the basic scheduling requirements of WJZR and increase the efficiency of hydropower generation.

Fig.10 shows the optimized reservoir operation schedules in different inflow scenarios of the SBYR-GHYR system obtained with SCE-SR.For each scenario,the forebay elevations of SBYR and GHYR vary within their respective bounds, and the optimized schedules meet the scheduling requirements of these two reservoirs.It is worth noting that the forebay elevations of SBYR tend to decline during the dry season in response to the decrease of inflows from November to March.The GHYR inflows (identical to the SBYR outflows)are not sufficient enough to generate the firm output in the dry season so that the GHYR forebay elevations drop further to compensate for the inflows.In addition,both reservoirs have a certain degree of forebay elevation fluctuation in each year,especially in wet years.This is mainly due to the limitation of the power output constraint.Since the power output obtained with SCE-SR has already reached its maximum, i.e., the installed capacity of reservoirs, the forebay elevation would decrease to release excessive inflows.

Fig.9.Optimized forebay elevations of WJZR in case S-9.

Fig.10.Optimized forebay elevations with SCE-SR.

6.Conclusions

An optimization method referred to as SCE-SR is proposed to solve the complex constrained reservoir scheduling problems.This method is easy to implement, requires little parameter tuning, and is characteristic of the combination of deterministic and probabilistic approaches.SCE-SR was tested and compared with DP and GA against hydropower scheduling problems in both a single reservoir (WJZR)and a multi-reservoir system (SBYR-GHYR)with different inflow scenarios and population sizes.The results show that the performance of SCE-SR is strong, with better stability and higher computational efficiency than other methods,and SCESR can converge to the global optima in a consistent,efficient,and fast way with both objectives and constraints considered.In summary, the proposed SCE-SR is an effective, efficient,and reliable optimization method for solving reservoir scheduling problems and paves the way for the application of unconstrained algorithms in this field.