Abstract

The aim of this study is to improve the performance of the shuffled complex evolution (SCE) algorithm used in the optimization of hydropower generation in reservoirs as a complex issue in water resources. First, the SCE algorithm is merged with the differential evolution (DE) algorithm to form the SCE-DE algorithm. Then, a complex mathematical function is used as a benchmark to evaluate the performance and validate the SCE-DE algorithm and the outcomes are compared with the original SCE algorithm to show the superiority of the proposed SCE-DE algorithm. In addition, the two-reservoir system of Dez-Gotvand is considered as a real optimization problem to evaluate the performance of the SCE-DE algorithm. It is revealed that optimization by SCE-DE is much better than SCE. In conclusion, the results show that the proposed SCE-DE algorithm is a reasonable alternative to optimizing resource systems and can be used to solve complex issues of water resources.

INTRODUCTION

The paramount issue facing water resources globally is the exploitation of water reservoirs. The use of an optimization method is essential when designing such an exploitation program. Optimization management of a reservoir is of considerable complexity and a systemic approach to reservoir management can reveal the inefficiency of classic methods. The most complex issue regarding optimization of water reservoirs is optimization of the hydropower system.

The complexity of hydropower reservoir optimization stems from its nonlinear objective function and constraints; however, several conventional techniques have been developed to optimize hydropower systems. Researchers have listed the following challenges to the optimization of multi-reservoir hydropower systems: (1) imprecise and uncertain input (Simonovic 1987), (2) the dynamic process of the stages of decision-making (Simonovic 1987), (3) the intricate problem caused by coupling reservoirs hydraulically (Orero & Irving 1998), (4) the issue of optimization because of its various variables and constraints (Labadie 2004), (5) nonlinear (El-Hawary & Christensen 1979), non-convex (Tauxe et al. 1980), discontinuous, indistinguishable (Lyra & Ferreira 1995) and evenly mixed hydropower performance models (Wang & Zhang 2012), (6) the ordeal of finding a solution to the constraints because of their physical and operational characteristics (Simonovic 1987), and (7) multiple conflicting objectives (Tauxe et al. 1980; Lyra & Ferreira 1995).

The use of meta-heuristic techniques is recommended to meet these challenges. Many attempts have been made to optimize reservoirs using meta-heuristic algorithms such as the genetic algorithm (GA) (Chang et al. 2005; Hınçal et al. 2011; Louati et al. 2011). Karamouz et al. (2003) applied a GA to the optimization of hydropower generation and established its efficient utilization in the development of operation policies for a multipurpose hydropower reservoir system. A GA was developed by Jian-Xia et al. (2005) and Li et al. (2013) to maximize hydropower generation. Ant colony optimization algorithms were used by Jalali et al. (2006, 2007) and Kumar & Reddy (2006). Hydropower scheduling in multi-reservoir systems was optimized using an improved particle swarm optimization algorithm (Moeini etal. 2011).

The shuffled complex evolution (SCE) algorithm introduced by Duan et al. (1993) has been shown to be effective for calibration of hydrological models. The efficacy of this algorithm, a form of differential evolution (DE), comes from its use of geometric operations to find possible optimal solutions to space parameters. Kuczera (1997) believed that SCE is more robust than a traditional GA, although it may not be most effective.

This optimization method has been employed much less in reservoirs and has not been used to optimize hydropower reservoir systems. In the current study, this optimization method has been examined and developed to optimize a hydropower reservoir system. The DE algorithm, rarely used in the optimization of reservoirs, has been employed to improve the SCE algorithm.

Price & Storn (1995) provided a DE algorithm as a global optimization method. Yin & Liu (2010) modified this algorithm to optimize hydropower production. Optimization is the result of an increase in the convergence ratio caused by a corrected mutation operator when choosing four individuals. They found that the computation speed of the DE is greater than for dynamic programming and presented their new technique to optimize reservoir operation.

As it is necessary to obviate the management problems in practical water resources by providing a new algorithm, in the present study, attempts have been made to present a detailed comprehensive SCE algorithm. To this end, a local search was performed to improve the original SCE algorithm by employing the DE algorithm. To show the ability of the proposed algorithm, the general SCE algorithm was analyzed while in use to solve a benchmark function. The efficiency of the improved SCE-DE algorithm was compared with the general SCE algorithm. The proposed algorithm then was used to solve a well-known hydropower operation of a multi-reservoir system, the Dez-Gotvand reservoir problem, in water resources management.

METHODS

Introduction of SCE algorithm

Simplex optimization, competitive evolution and complex shuffling have been employed to make the SCE algorithm effective and robust and improve its yielding and flexibility. A deterministic search strategy was used to effectively explore the global optimum. From the available literature, one can understand that the SCE algorithm is well-suited for solving nonlinear and non-convex high dimensional problems because of its convergence and robustness.

The parameters used in SCE are individual numbers in each complex (m), individual numbers in each sub-complex (q), each sub-complex iteration (α), each complex iteration (β), the number of complexes (p), the minimum number of individuals in each complex (pmin) and the dimension of the problem (n) formulated as m = 2n + 1, q = n + 1, α = 1, β = m, p = pmin. In the present paper, the parameters in these formulas are applied as presented under experimental calibration by Duan etal. (1994).

SCE algorithm

  • 1.

    Launch the parameters of SCE (p and m) and calculate the initial population size .

  • 2.

    Randomly create s individuals x{x1, x2, … , xs} in the feasible space.

  • 3.

    Investigate the objective function.

  • 4.

    Categorize and arrange the s individuals in array D in ascending fashion according to the function D = {(xi, fi), i = 1, 2, … , s}, where i = 1 and has the smallest function value in the population.

  • 5.

    Sort the s individuals so that the first individual represents the individual with the smallest objective function.

  • 6.

    Divide the population into q complexes, A1, A2, … , Ap with each complex containing m individuals where .

  • 7.

    Modify each complex Ak using the Downhill Simplex method (reflection and contraction).

  • 8.

    Substitute A1, A2, … , Ap into array D according to ascending function value, then sort them such that D = {Ak, k = 1, … , p}.

  • 9.

    If the outcome satisfies the criteria, the operation stops; otherwise return to step 5.

Differential evolution

DE has been very recently developed as a global heuristic optimization method. The algorithm can adapt continual, discrete, nonlinear and complicated objective functions, regardless of the many types of constraints. They are classified by the notation DE/x/y/z, in which x denotes the mutated vector, y denotes different vectors and z denotes the binomial or exponential crossover scheme (Price & Storn 1995). Individuals are mutated using the mutation operator as follows:  
formula
(1)
where r1 ≠ r2 ≠ r3 are vector indices of the current population chosen by chance, g is the generation number, η∈[0.2, 0.8] is the vector drawn from a uniform distribution chosen by chance and is the mutant vector. The DE algorithm performs a crossover operation on with the likelihood of Cr of forming a new solution, . The polynomial mutation creates from as follows:  
formula
(2)
and  
formula
(3)
where the ith individual of the jth real-valued vector is represented by , is the crossover constant ratio between [0,1], a uniform random number from [0,1] is represented by rand, the distribution index is a control parameter, Lb is the lower and Ub is the upper bound of the decision variables (Deb 2001), and is a randomly chosen index from 1 to n which guarantees that can at least reach one parameter of .

Improved SCE algorithm

Multi-reservoir systems are the most complex problems in water resources management systems (Labadie 2004). SCE was developed to solve such problems, but requires improvements to allow computation of the best solution. In the following section, the procedure of improving the performance of the SCE algorithm is described.

SCE-DE algorithm

The proposed SCE-DE algorithm presents the DE algorithm to be used together with an adaptation of the downhill simplex. In SCE-DE, steps 4 to 6 of the SCE algorithm are modified and implemented by a variant of the DE algorithm, DE/best/1/bin as follows:

  • 1.

    The initial parameters of SCE-DE (q, α, β, Cr and F) are applied to the program.

  • 2.

    Each individual in Ak is assigned a triangular probability distribution pi = {2(m + 1 − i)/m(m + 1)}, where i = 1, … , m, has the greatest probability p1 = 2/(m + 1) and has the least likelihood pm = 2/{m(m + 1)}.

  • 3.

    Next q individuals from Ak are randomly selected with regard to their probability distribution and are stored in their position in array B = {(ui, vi), i = 1, … , q} and L, where vi is the function value of ui.

  • 4.

    The detailed creation of the offspring is as follows:

    • 4.1.
      Array B and L are sorted such that q individuals are ascending in terms of function value and then a mutated individual is calculated through DE as:  
      formula
      (4)
      where the population of an individual index is denoted as i = 1, 2, … , s. The position of the ith individual of the population of q real-valued n-dimensional vectors is denoted as Bi = [b1, b2, … , bn]. The position of the ith individual of a mutant vector is denoted as Vi = [v1, v2, … , vn]. A real parameter, termed as the mutation factor, prevents search stagnation by controlling the expansion difference between two individuals and is denoted as F > 0. The target vector is randomly selected with ii1. Then, individuals and are randomly chosen such that ab and difference vector is computed. and are the best and worst individuals of the population of q, respectively; sigma is a positive constant that is typically recommended as being between 2 and 3.
    • 4.2.
      Crossover is applied after the mutation operation. To this end, the intended vector is integrated into the mutated vector to reach the trial vector as:  
      formula
      (5)
  • 5.

    If is located in a feasibility environment, then calculate its function value fz and go to (6); otherwise go to (7).

  • 6.

    If fz<fq, then replace individual by and go to (10); otherwise, go to (7).

  • 7.

    Calculate a new individual by applying , where r is the reflection of and is the lowest individual among q individuals.

  • 8.

    If r is located in a feasibility environment, then calculate its function value fr and go to (9); otherwise create an individual y by chance then compute its value fy and set .

  • 9.

    If fr < fq, then substitute individual r with y and go to (10). Otherwise, generate an individual y by chance, compute its value fy and set .

  • 10.

    Repeat (1)–(9) α times.

  • 11.

    Substitute the offspring in array B with the parents according to their initial location set in array L, and then sort these individuals according to their ascending function value.

  • 12.

    Repeat (1)–(11) β times.

To reinforce and upgrade the performance of the SCE algorithm, the main objective of this research, and to attain better quality in optimization of water resources, especially optimization of the use of hydropower, this algorithm should be integrated into the DE algorithm. The flowchart of the SCE upgrade program made through integration with the DE algorithm produces the SCE-DE algorithm and is shown in detail in Figure 1.

Figure 1

The flowchart of the SCE-DE algorithm.

Figure 1

The flowchart of the SCE-DE algorithm.

Test function

Ackley ̛is a continuous and multimodal benchmark function obtained by modulating an exponential function with a cosine wave of moderate amplitude. Ackley causes moderate complications in the search because a strictly local optimization algorithm performing hill-climbing would surely get trapped in a local optimum. A search strategy that scans a slightly bigger neighborhood can cross intervening valleys toward increasingly better optima. Therefore, the Ackley function provides one reasonable test case for the SCE-DE algorithm. The Ackley function is as follows:  
formula
(6)

Despite being restricted to a smaller domain, the evaluation of the function can be computed by hypercube xi ∈ [−32.768, 32.768] for all i = 1, … , d where d represents the dimension of the functions and d = 30. The global minimum for this function is , at .

Optimization of the Ackley function was checked using the SCE and SCE-DE methods to verify and determine the functionality of the codes that have been provided in MATLAB. The results of ten runs of the SCE and SCE-DE methods, including 10,000 iterations for each run, are given in Table 1 and Figure 2.

Table 1

Values of the objective function calculated with SCE and SCE-DE in ten different runs for the benchmark function

Number of runSCESCE-DE
3.39 7.99 × 10−15 
3.69 7.99 × 10−15 
2.98 7.99 × 10−15 
5.30 7.99 × 10−15 
5.26 7.99 × 10−15 
4.49 7.99 × 10−15 
5.28 7.99 × 10−15 
4.53 7.99 × 10−15 
4.42 7.99 × 10−15 
10 3.87 7.99 × 10−15 
Best 2.98 7.99 × 10−15 
Worst 5.30 7.99 × 10−15 
Average 4.42 7.99 × 10−15 
Standard deviation 0.80 
Number of runSCESCE-DE
3.39 7.99 × 10−15 
3.69 7.99 × 10−15 
2.98 7.99 × 10−15 
5.30 7.99 × 10−15 
5.26 7.99 × 10−15 
4.49 7.99 × 10−15 
5.28 7.99 × 10−15 
4.53 7.99 × 10−15 
4.42 7.99 × 10−15 
10 3.87 7.99 × 10−15 
Best 2.98 7.99 × 10−15 
Worst 5.30 7.99 × 10−15 
Average 4.42 7.99 × 10−15 
Standard deviation 0.80 
Figure 2

Convergence curves of the best objective function of the SCE and SCE-DE algorithms for the Ackley function.

Figure 2

Convergence curves of the best objective function of the SCE and SCE-DE algorithms for the Ackley function.

The values (minimum, maximum, mean and standard deviation) of the solutions acquired over ten non-associated runs are tabulated in Table 1. The strongest algorithm is the one with the smallest standard deviation for ten runs. As expected, the mechanisms suggested upgrading the quality of the end solution saliently.

The results seen in Table 1 indicate high accuracy and greater functionality for the SCE-DE method when compared with the SCE method. The results of the SCE-DE method are close to the main results of the mathematical function (0) and the standard deviation of the results is zero. The convergence curves in Figure 2 indicate that the operation of SCE-DE approaches the result more quickly and effectively than the SCE method. In conclusion, the efficacy of the SCE-DE method in this mathematical function shows its superiority to SCE.

Reservoir operation model

The reservoirs operate according to the continuity equation, which stabilizes water equilibrium during every period of reservoir operation. The continuity equation for a multi-reservoir system is:  
formula
(7)
where t denotes the number of periods assumed, i denotes the number of reservoirs, and denote the storage in the ith reservoir at the beginning and end period of t, respectively, denotes the inflow volume into the ith reservoir in period t, denotes the release volume from the ith reservoir in period t, denotes the overflow volume from the ith reservoir in period t, denotes the evaporation loss from the ith reservoir surface in period t, n denotes the number of reservoirs and T denotes the total period of operation. The water loss from a reservoir can be described as follows:  
formula
(8)
 
formula
(9)
where denotes the net evaporation from the ith reservoir surface in period t, denotes the average ith reservoir area in period t and and denote the ith reservoir areas in period t. The overflow volume or spill from the reservoir is calculated as:  
formula
(10)
where is the maximum volume of the ith reservoir over period t. The constraints on reservoir release and storage are:  
formula
(11)
 
formula
(12)
 
formula
(13)
in which and denote the minimum and maximum permissible release from the ith reservoir during period t, respectively, and denote the minimum and maximum storage of the ith reservoir at the beginning of period t, respectively, denotes the storage of the ith reservoir at the beginning of the operation period and denotes the storage in the ith reservoir at the end of the operation period. Equations (7)–(13) are applied to simulate single-reservoir and multiple-reservoir systems.
The goal of hydropower reservoir operation is to maximize hydropower performance or minimize hydropower loss. Equation (14) describes the purposed function that minimizes hydropower production loss as:  
formula
(14)
where denotes the production power (MW) in period t, PPC denotes the installed capacity of the power plants in terms (MW), M denotes the number of reservoirs and T denotes the period.
In order to calculate the production power, both the volume of water released from the reservoir and the effect of water volume on the turbines should be defined. As has been shown, the issue of optimum hydropower utilization of the dam reservoir is a nonlinear constraint; hence, its quarry space is non-convex:  
formula
(15)
 
formula
(16)
 
formula
(17)
where g denotes gravitational acceleration (m/s2), denotes the efficiency of the power plant, PF denotes the power plant function coefficient, denotes the effective head of the hydroelectric plant (calculated in Equation (16)), denotes the reservoir level in period t (m), denotes reservoir storage at the beginning of period t and TWL denotes the tail water elevation (m). To calculate the volume-elevation of the reservoirs, a third-degree polynomial function for each reservoir is individually fitted in Equation (17).

Case study

Multi-reservoir system

The application of the algorithm was checked monthly to develop operation-based policies for the Dez-Gotvand multi-reservoir system over a 20-year period. These are the most significant reservoirs in southwestern Iran (Figure 3) and were built for hydropower generation. Dez dam is on the Dez River in Khuzestan province and located 25 km north of the city of Dezful. Dez River is the second largest in Iran in terms of discharge. The river joins the Karun River. Gotvand Dam is one of the largest dams in Iran and was built on the Karun River. The dam is located 25 km north of the city of Shushtar in Khuzestan province. Table 2 lists the details for the Dez and Gotvand dams.

Table 2

Characters of hydropower plants

Power plantMin reservoir storage (106 m3)Max reservoir storage (106 m3)Power plant capacity (106 W)Plant factorEfficiency
Dez 468 2,492 520 0.48 0.90 
Gotvand 2,040 4,327 2,000 0.60 0.89 
Power plantMin reservoir storage (106 m3)Max reservoir storage (106 m3)Power plant capacity (106 W)Plant factorEfficiency
Dez 468 2,492 520 0.48 0.90 
Gotvand 2,040 4,327 2,000 0.60 0.89 
Figure 3

(a) Location and (b) schematic of the Gotvand-Dez multi-reservoir system.

Figure 3

(a) Location and (b) schematic of the Gotvand-Dez multi-reservoir system.

RESULTS AND DISCUSSION

To compare the functions of the SCE and SCE-DE algorithms for increasing the energy efficiency of hydropower reservoirs, the Dez-Gotvand two-reservoir system was selected for study over a period of 20 years. The plant factor of Gotvand Dam is 0.48 and that of Dez Dam is 0.6. The decision variables for the 20-year statistical study period including the release reservoir volume were 240 for each dam, totaling 480 decision variables.

The codes were written in MATLAB to improve the SCE algorithm when merging it with the DE algorithm to optimize water reservoirs. Optimization operates according to the objective function (Equation (14)) to minimize the differences between production power and the installed capacity of the power plant. Using the volume-elevation data for each reservoir, a third-degree polynomial function was fitted in Equation (17) for each reservoir and the following coefficients for the Dez and Gotvand dams were obtained: (a = 225.61, b = 0.117, c = −6.25 × 10−5, d = 12.6 × 10−9) and (a = 115.74, b = 0.074, c = −2 × 10−5, d = 2.1 × 10−9), respectively. The correlation coefficients obtained for the third-degree polynomial function for the Dez and Gotvand dams were 0.984 and 0.985, respectively. The downstream level as a function of reservoir release was obtained at the beginning period as a second-degree function according to the stage-discharge rating curves of the power plants downstream.

The objective function values were obtained for SCE and SCE-DE after performing ten runs and 10,000 iterations for each run as shown in Table 3. As seen, the best result for the objective function in SCE-DE was 103.97 and for SCE was 130.32. The worst result obtained by SCE-DE was 105.90 and for SCE was 158.52. The mean of all results obtained by SCE-DE in comparison with the best answer was 0.8 (less than 1) with a standard deviation of 0.61. The standard deviation for SCE was 8.59. The results clearly show that SCE-DE was more successful in obtaining an optimal solution than SCE. In fact, the proposed algorithm enhanced the exploration and exploitation abilities of the original SCE algorithm by the use of the efficient operators in the DE algorithm to reach a global optimum. The mean values and standard deviations are given in Table 3.

Table 3

Objective function values of SCE and SCE-DE for two-reservoir system

Number of runSCESCE-DE
135.78 103.97 
144.74 105.32 
136.11 104.75 
158.52 104.86 
130.32 104.88 
135.78 105.90 
133.88 105.00 
135.39 105.33 
148.44 104.38 
10 134.65 103.98 
Best 130.32 103.97 
Worst 158.52 105.90 
Average 139.36 104.84 
Standard deviation 8.59 0.61 
Number of runSCESCE-DE
135.78 103.97 
144.74 105.32 
136.11 104.75 
158.52 104.86 
130.32 104.88 
135.78 105.90 
133.88 105.00 
135.39 105.33 
148.44 104.38 
10 134.65 103.98 
Best 130.32 103.97 
Worst 158.52 105.90 
Average 139.36 104.84 
Standard deviation 8.59 0.61 

The convergence of SCE and SCE-DE after 10,000 iterations of the objective function is depicted in Figure 4. The objective function values obtained by SCE-DE were more accurate than those of SCE. The results depicted in Figure 4 show that SCE-DE convergence was superior to that of SCE after 2,500 iterations of the objective function.

Figure 4

Convergence curves of the best objective function of SCE and SCE-DE for the two-reservoir problem.

Figure 4

Convergence curves of the best objective function of SCE and SCE-DE for the two-reservoir problem.

Figure 5 shows the optimum hydropower generation for a period of 240 months for the Dez and Gotvand dams using SCE-DE and SCE. These two methods generated 307 and 273 GW of total power, respectively, which demonstrates the efficacy of SCE-DE. Figure 6 shows the optimum volumes of water released from the reservoirs of the Dez and Gotvand dams that were obtained monthly using SCE-DE and SCE. Figure 7 shows the optimum storage in the Dez and Gotvand reservoirs obtained monthly using SCE-DE and SCE.

Figure 5

(a) Monthly optimal power generated by the Dez reservoir. (b) Monthly optimal power generated by the Gotvand reservoir.

Figure 5

(a) Monthly optimal power generated by the Dez reservoir. (b) Monthly optimal power generated by the Gotvand reservoir.

Figure 6

(a) Monthly optimal release of the Dez reservoir. (b) Monthly optimal release of the Gotvand reservoir.

Figure 6

(a) Monthly optimal release of the Dez reservoir. (b) Monthly optimal release of the Gotvand reservoir.

Figure 7

(a) Monthly optimal storage of the Dez reservoir. (b) Monthly optimal storage of the Gotvand reservoir.

Figure 7

(a) Monthly optimal storage of the Dez reservoir. (b) Monthly optimal storage of the Gotvand reservoir.

CONCLUSIONS

The aim of this paper was to evaluate the performance of the proposed SCE-DE algorithm for improving optimization of water reservoir systems. The SCE algorithm was introduced and enhanced by integration with the DE algorithm to produce the proposed SCE-DE algorithm. To test and evaluate the performance of SCE-DE, it was compared with that of SCE when solving the Ackley benchmark mathematical function and for optimization of hydropower in a real two-reservoir system.

The Ackley benchmark was solved and the results obtained by SCE-DE were compared with those of SCE. The results show that SCE-DE provided a better result than SCE such that its worst answer was much better than that of SCE. In addition, SCE-DE provided a better mean value and standard deviation than the SCE method. The convergence curves of the two methods for solving the mathematical function show that the SCE-DE convergence speed to reach the correct answer was better than that of SCE.

The second problem that the real two-reservoir system must address is the complex issue of the high number of variables used to optimize hydropower for the optimization of water resources. The results show that SCE-DE was implemented in ten runs, each in 10,000 iterations. The SCE-DE provided a much better answer than the SCE algorithm. In addition, the worst answer of the proposed method was much better than that obtained by SCE. The mean value of all results obtained by SCE-DE was only 0.8 (less than 1). The standard deviation obtained for this method by SCE-DE was 0.61; however, this value for the SCE method was 8.59. Moreover, the total power generated by SCE-DE was 34 GW, which was more than that for the SCE method.

It can be concluded that the SCE-DE algorithm was more successful in obtaining an optimal solution than the SCE alone. In sum, the proposed SCE-DE algorithm is a reasonable choice for optimizing reservoir systems and can be applied to the complex issue of water resources.

ACKNOWLEDGEMENTS

We would like to thank the editor and anonymous reviewers for their useful comments that improved this study.

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