Optimal operation of multi-reservoir systems: comparative study of three robust metaheuristic algorithms

In this study, the capability of the recently introduced moth swarm algorithm (MSA) was compared with two robust metaheuristic algorithms: the harmony search (HS) algorithm and the imperialist competitive algorithm (ICA). First, the performance of these algorithms was assessed by seven benchmark functions having 2 – 30 dimensions. Next, they were compared for optimization of the complex problem of four-reservoir and 10-reservoir systems operation. Furthermore, the results of these algorithms were compared with nine other metaheuristic algorithms. Sensitivity analysis was performed to determine the appropriate values of the algorithms ’ parameters. The statistical indices coef ﬁ cient of determination (R 2 ), root mean square error (RMSE), mean absolute error (MAE), mean square error (MSE), normalized MSE (NMSE), mean absolute percentage error (MAPE), and Willmott ’ s index of agreement (d) were used to compare the algorithms ’ performance. The results showed that MSA was the superior algorithm for solving all benchmark functions in terms of obtaining the optimal value and saving CPU usage. ICA and HS were ranked next. When the dimensions of the problem were increased, the performance of ICA and HS dropped but MSA has still performed extremely well. In addition, the minimum CPU usage and the best solutions for the optimal operation of the four-reservoir system were obtained by MSA, with values of 269.7 seconds and 308.83, which are very close to the global optimum solution. Corresponding values for ICA were 486.73 seconds and 306.47 and for HS were 638.61 seconds and 264.61, which ranked them next. Similar results were observed for the 10-reservoir system; the CPU time and optimal value obtained by MSA were 722.5 seconds and 1,195.58 while for ICA they were 1,421.62 seconds and 1,136.22 and for HS they were 1,963.41 seconds and 1,060.76. The R 2 and RMSE values achieved by MSA were 0.951 and 0.528 for the four-reservoir system and 0.985 and 0.521 for the 10-reservoir system, which demonstrated the outstanding performance of this algorithm in the optimal operation of multi-reservoir systems. In a general comparison, it was concluded that among the 12 algorithms investigated, MSA was the best, and it is recommended as a robust and promising tool in the optimal operation of multi-reservoir systems


INTRODUCTION
The optimal operation of reservoirs is one the most important issues in water resources management, especially for multi-reservoir systems. Many control variables determine the operation strategies for scheduling a sequence of releases to meet a large number of demands for different users. Optimal reservoir operation defines the optimum policies to retain or release water from a reservoir at different periods of the year according to the inflows and demands.
To develop a comprehensive and optimal operation policy for a reservoir, the sources of uncertainties should be considered. The uncertainties may be due to different factors such as inaccurate predictive models, data scarcity, measurement and observation errors, the measurement site being unrepresentative, or problems in aggregating or disaggregating data. For a multi-reservoir system, the uncertainties may be caused by the stochastic nature of the system inputs (rainfall, reservoir inflow, evaporation, leakage, release, etc.), the nonlinearity of functions, the multiple constraints, the large number of decision variables, and other spatial and temporal variations of system components. Consideration of these uncertainties can help in the design of efficient operational policies and to develop robust predictive models.
Over the past few years, several evolutionary algorithms have been developed and applied to solving reservoir optimization problems (Table 1). They are considered to be very effective alternatives for solving complex optimization problems with either single or multiple objectives. These algorithms offer an expanded capability to systematically select the optimal solutions given the objectives and con-  • Extraction of optimal operation rules in an aquifer-dam system with developed version of GP in comparison with GA ✓ Developed GP is more flexible and effective in determining optimal rule curves for a conjunctive aquifer-dam system. • Using constrained PSOs (CPSO) for optimization of large reservoir operation compared to GA and conventional PSO ✓ CPSOs were superior to conventional PSO and GA in locating near-optimal solutions and convergence characteristics. ✓ CPSOs were more insensitive to the swarm size and initial swarm.
(continued) • HBMO was compared with linear programming (LP), DP, differential DP, discrete differential DP and GA in the optimal operation of multi-reservoir systems ✓ The high efficiency and rapid convergence rate of HBMO compared to other algorithms make it a robust tool for the optimal operation of reservoirs.

Soghrati & Moeini ()
• Performance of HBMO for optimization of Dez hydropower reservoir operation was compared with artificial bee colony (ABC) algorithm, GA, improved particle swarm optimization (IPSO) algorithm, ACO and GSA ✓ Using ABC gave the best results with low computational costs. The main question that this paper seeks to answer is that, among the several evolutionary algorithms that have been recently introduced by different researchers in different engineering fields, which one is the most appropriate for the optimal operation of multi-reservoir systems? Accordingly, this paper investigates the capability of the recently intro-

MATERIALS AND METHODS
As previously stated, this study investigates the capability of three robust algorithms, MSA, HS and ICA, for the optimal operation of multi-reservoir systems. The overall methodology of this study is shown in Figure 1. More details of methodology are presented below.

Moth swarm algorithm (MSA)
MSA is a robust metaheuristic algorithm which originates from moth behaviour in nature. Moths hide from predators during the day, and at night they use celestial navigation to orient themselves in the dark and exploit food sources. In     All the algorithms were coded in MATLAB 2016a using a PC with i7 CPU 1.8 GHz/16GB RAM/2TB HDD.

Verification of the algorithms
In order to evaluate the efficiency and validation of the developed models for the optimal operation of multi-reservoir systems, a set of standard benchmark functions was selected, as presented in Table 3.
The performance of MSA in solving these functions was compared with HS and ICA. The population size and the number of evaluations of benchmark functions in all the algorithms was identical and proportional to the dimensions of each function. As seen in Table 3, for benchmark functions with lower dimensions, the performance of all the algorithms was approximately similar, but for large dimension problems (Rosenbrock function with dimension of 10 and 30), MSA was the only algorithm which was capable of solving the problem. The performance of the other algorithms was so weak, the results of them dramatically diverged from the optimal value. Figure 2 shows the convergence rate of the algorithms in reaching the optimal value for standard benchmark functions. It indicates that for all benchmark functions, MSA reached its optimal value by the smallest number of iterations. The difference was more obvious in high dimension problems.
After successful verification of utilized algorithms by benchmark functions, their performance was investigated in the optimization of two multi-reservoir systems.

Sensitivity analysis of model parameters
Before using the algorithms to optimize the multi-reservoir system operation, a sensitivity analysis was performed to determine the best values for the algorithm parameters.

Test cases: optimization of multi-reservoir systems
The four-reservoir problem was first formulated and solved by Larson () as a linear problem with a known global  The 10-reservoir problem is more complicated, not only in terms of size, but also because of the many time-dependent constraints on storage. The layout of the 10-reservoir problem is shown in Figure 3(b). The system comprises reservoirs in series and in parallel, and a reservoir may receive supplies from one or more upstream reservoirs.
Inflows are defined for each of the upstream reservoirs, and initial storage and target storage at the end of the operating period are specified for each reservoir. In addition, there are minimum operating storage amounts in each reservoir that must be satisfied, as well as constraints on minimum and maximum reservoir releases. Operation of the system is optimized over 12 operating periods to maximize hydropower production. More details of these problems can be found in Murray & Yakowitz ().
The objective function is the maximization of benefits from the systems over 12 two-hour operating periods, defined as follows: where K is the number of reservoir; NT ¼ total number of periods; R K (t) ¼ releases in time period t from reservoir k (k ¼ 1, …, K); and b k (t) is the benefit functions for the k th reservoir. The function F, which should be maximized, is the sum of the returns due to power generated by the power plants and the return from the diversion of the irrigation project.
The fundamental constraints include the continuity constraints for each reservoir over each operating period t, defined as: Constraints on reservoir storage: And constraints on releases from the reservoirs: Here S k (t) ¼ storage at time t in reservoir k (k ¼ 1, …., K); I k (t) ¼ inflows in time period t to reservoir k; S min k (t) ¼ minimum storage in reservoir k; S max k (t) ¼ maximum storage in reservoir k; R min k (t) ¼ minimum release from reservoir k; R max k (t) ¼ maximum release from reservoir k.

Evaluation criteria
In Re opt t À Re t Re opt t (10) In the above equations, Re t is the releases in time period t from the investigated algorithms, Re opt t is the optimum release in time period t, Re opt is the mean of the optimum release, Re is the mean of the release, and n is the number of total time periods. The variation domain of Willmott's index of agreement ranges from À∞ to 1, so that 1 indicates perfect agreement between the optimum release and releases from the investigated algorithms.
A low value of RMSE and a high value of R 2 indicate acceptable accuracy of the algorithm and correlation between the data, and also imply its superiority over the other algorithms. Each of the MAE, MSE, NMSE, and MAPE parameters shows the difference between the optimum release and releases from the investigated algorithms; the lower the values of these parameters, the more efficienct the algorithm. MSE highlights the difference between the data, and its normalized form (NMSE) can be compared with other algorithms.

RESULTS AND DISCUSSION
Results of sensitivity analysis As previously described, to ensure the reliability and validity of the models' outputs, MSA, HS and ICA were put through 10 runs with different iterations. To achieve the appropriate number of iterations, a sensitivity analysis on the number of iterations was performed. Accordingly, each algorithm was run with 500, 1,000, 2,000 and 5,000 iterations. Table 4 shows the values of objective function for different numbers of iterations in the four-reservoir system.
Only MSA could reach the values very close to the optimal value after 1,000 iterations. Although the two other algorithms converged after 1,000 iterations, they have a large difference from the optimal value. For iterations over 1,000, while the execution time of the algorithms dramatically increased, the variations of the objective function were negligible. Therefore, the number of iterations in each algorithm was considered to be 1,000. The results of sensitivity analysis for the four-reservoir system are shown in Figures S1-S4 in the Supplementary Information. In addition, Figure 4 shows the variations of objective function by increasing the number of iterations. Similar results were obtained for the 10-reservoir system. Table 5 shows the final values of algorithm parameters for multi-reservoir systems, all of which were derived by sensitivity analysis.
Convergence rate and models performance    value of the objective function for the 10-reservoir system achieved by MSA was 12.5% and 6.6% better than that of HS and ICA, respectively. The calculation time of the 10reservoir system optimization by MSA, HS and ICA were 722 seconds, 1,963 seconds, and 1,421 seconds, respectively, indicating the superior performance of MSA.
As previously described, for a better comparison of the three utilized algorithms in the optimal operation of multireservoir systems, seven statistical indices were employed.
According to Table 8  be employed by water policymakers as a guide (rule curve) to schedule water releases from multi-reservoir systems in a way that gives the most benefits. It can be seen from Figure 6 that HS and ICA failed to produce reasonable results for multi-reservoir systems. | Convergence rate of applied algorithms in (a) the four-reservoir system and (b) the 10-reservoir system.    Table 9 compares the results of this study with those of previous studies on the optimization of four-and 10-reservoir system operation. It can be seen that MSA has the highest performance among all evolutionary algorithms. For the four-reservoir system, MSA gave the optimal value of 308.83, the closest to the global optimum. After that, GSA with the optimum values of 308. 21, 308.17, 308.05, 307.50, 307.26, 306.47, 306.39, 305.51 and 264.61, respect-ively. This confirms the superiority of MSA to the 11 other metaheuristic algorithms in the optimal operation of the four-reservoir system.

Comparison with previous research
For the case of the 10-reservoir system, it was also findings of this research and the application of the same methodology for the optimal operation of multi-reservoir systems will enable decision makers to make informed choices on water development, conservation, allocation, and use in the context of growing demands for all uses and increased scarcity. In addition to the low cost, easy implementation and simple procedure of the methodology presented, it has many capabilities that make it attractive to be used by water policymakers, water resources planners, and reservoir managers.