Optimal operation of single and multi-reservoir systems via hybrid shuffled grey wolf optimization algorithm (SGWO) Open Access

The scale of most water systems, the reciprocal impacts of various social, economic, and political issues and water resources management on the livelihood of humans, increasing demand for water and the scarcity of water resources, and its asymmetric temporal and spatial distribution enlighten the importance of water resources management. In this regard, optimal operation of the water resource system and especially water reservoirs is an imperative part of water resources management. Surface water allocation regulations occur within the reservoir operation rules to decide how water should be released in different system conditions to satisfy the system's goals. Most of the time, the reservoirs are built on a watershed system on different tributaries and have multiple operational objectives such as irrigation water supply, hydropower generation, flood control, and recreational aquacultural activities. This situation results in the emergence of multi-objective and complex reservoir operation optimization problems.

Although the literature consists of research done using classic optimization methods such as linear, non-linear, and dynamic programming and their variations (Barros et al. 2003; Mariano et al. 2008; Nagesh Kumar et al. 2010; Kamodkar & Regulwar 2014), Having multiple objectives and numerous complex constraints with high dimensions, reservoir operation optimization problems are hardly possible to solve or require high computational cost (Afshar et al. 2015a, 2015c).

From the beginning of the development of meta-heuristic optimization algorithms, natural inspiration has been considered to develop these algorithms properly. Hence, nature-inspired evolutionary and meta-heuristic optimization algorithms have aided researchers in finding various efficient optimization methods. Although these algorithms do not guarantee the optimum solution, they have been widely used because of their flexibility and ease of use. Some significant disadvantages of these algorithms include unmatured convergence, getting trapped in local optima, and multiple parameter configuration and calibration, which requires highly accurate sensitivity analysis (Chang et al. 2010; Afshar et al. 2015c; Choong et al. 2017). These algorithms tend to behave differently in the solution of different problems. Thus, it is necessary to investigate their performance with real-world problems alongside purely mathematical benchmark functions.

Meta-heuristic optimization can be traced in multiple fields of water resources engineering and management such as reservoir operation (Fallah-Mehdipour et al. 2013; Afshar et al. 2015b; Asgari et al. 2016), hydrologic models (Zhang et al. 2010), water resources management (Wu et al. 2015), and water distribution networks (Maier et al. 2003; Ostfeld 2015).

Multiple studies have been conducted concerning reservoir operation optimization problems to find a better algorithm to solve multiple aspects of these problems. Some of the notable research in this category includes Afshar (2012), who developed two adaptive constrained PSO algorithms to keep the decision variables in the feasible search space and solve two problems of water supply and hydropower generation of Dez reservoir in Iran. In another research, Afshar et al. (2015a) used the imperialist competitive algorithm (ICA) on reservoir operation of the Dez reservoir in Iran, at two states of simple and hydropower usage. Then, to show the superiority of the proposed algorithm, they compared the results with the ACO algorithm. In research conducted by Bozorg-Haddad et al. (2015), the bat algorithm was implemented to optimize the operation of the Karoun-e 4 reservoir to maximize the produced energy. The researchers claimed that the annual electrical energy produced using this algorithm was higher than other meta-heuristic algorithms. In another research, Bozorg-Haddad et al. (2016) applied the gravity search algorithm (GSA) to a single and a four-reservoir problem and compared the results with GA, which indicated the superiority of GSA. Zhang et al. (2014) proposed an algorithm called improved adaptive particle swarm optimization (IAPSO), which is an evolved type of particle swarm optimization (PSO). The researchers claimed that the newly developed algorithm has a better computational capability compared to PSO. Azizipour et al. (2016) utilized the invasive weed optimization (IWO) algorithm in single and multi-reservoir problems over short, medium, and long-term operation periods and compared the results with the findings obtained by GA and PSO algorithms. They found that the IWO is more efficient and effective than PSO and GA for single reservoir and multi-reservoir hydropower operation problems. Moeini et al. (2017) used GSA to solve an optimum reservoir operation problem from the Dez reservoir on a large scale. The results indicated that this algorithm is more efficient on larger scales, reducing the search space size considerably. Ehteram et al. (2017) applied the shark algorithm approach to optimize reservoir operation. The researchers compared the results with GA and PSO algorithms. Asgari et al. (2019) utilized the hybrid WOAPSO algorithm to solve a ten and a three-reservoir operation optimization problem. They claimed that the WOAPSO algorithm performance was superior compared to WOA and PSO algorithms. Despite the popularity of the grey wolf optimizer (GWO), there is no record of its application in water resources management-related optimization problems.

GWO is a population-based stochastic meta-heuristic algorithm that was developed by Mirjalili et al. (2014). Inspired by the grey wolves hunting and prey searching mechanism, this algorithm uses the social dominance hierarchy among wolf packs to simulate their hunting. GWO is proven to be competitive with other popular meta-heuristic algorithms such as DE, GA, and PSO (Mirjalili et al. 2014). The main advantage of GWO compared to other algorithms is its few required parameters. Application of this algorithm has been reflected in multiple studies in various fields such as optimizing controller gains (Sharma & Saikia 2015), power flow optimization (El-Fergany & Hasanien 2015), the optimal configuration of PID fuzzy logic controller system (Noshadi et al. 2016) and optimal reactive power dispatch (Sulaiman et al. 2015).

As previously mentioned, despite the advantages of metaheuristic algorithms, they are prone to entrapment in local optima. Researchers have attempted to modify the algorithms or apply hybridization (Zhang et al. 2016; Rezaei et al. 2017; Moeini & Afshar 2018; Pan et al. 2019). The GWO algorithm has also been modified by some researchers using various methods. Dudani & Chudasama (2016) introduced an adaptive grey wolf optimizer to detect partial discharge in transformers. They used the adaptive technique to change the updating mechanism of the wolves. Malik et al. (2016) utilized a weighted average instead of the simple average defined in the GWO algorithm's updating equation. Kohli & Arora (2018) implemented chaos functions for the parameter ‘a’ in the initial GWO algorithm.

Despite the popularity of the GWO algorithm, this algorithm has yet to be used in the water resources management field. The pre-analysis of this algorithm on reservoir operation optimization problems showed that early convergence and getting trapped in local optima were an issue. Monitoring the process of optimization by GWO for reservoir optimization problems revealed that some compatible answers were neglected during the population updating procedure of the GWO since this algorithm only pays attention to the three best answers. This seemed to hinder the information exchange flow between the members. Although as mentioned earlier, multiple studies have modified the GWO, the modifications were mainly to the position updating section of the algorithm. Thus, to see how altering population handling mechanism of the GWO can help, in this research, inspired by the Shuffled Complex Evolution (SCE-UA), a hybrid ‘Shuffled Grey Wolf Optimizer’ (SGWO) was developed and utilized to solve reservoir operation optimization problems with different conditions. This hybridized algorithm both altered the population handling and evolution of results to reach a robust solution.

The flowchart of SGWO is presented below and illustrated in Figure 1.

The Steps of the optimization are as follows:

Step 1: Initialization. Defining the number of updates in each iteration ⁠, the dimension of the problem D, the number of subpacks m, and the population of each subpack ⁠. Therefore, the total size of the sample will be ⁠.

Step 2: Using a uniform sampling distribution in the feasible space of Ω and computing the objective function value at each point of S, sample points are generated.

Step 3: Sample points sorting for the objective function value and storing the results in the vector in which corresponds to the point with the minimum value for the objective function.

Step 5: Evolve subpack considering the GWO algorithm (more details are presented in Appendix B).

• (a)

  Initialization. Select ⁠, and ⁠, a, where ⁠.

• (b)

  Update parameters A and C.

• (c)

  Calculate and for each member of the subpack.

• (d)

  Update each member's position to considering ⁠, ⁠, and (Equation (8)).

Step 6: Update ⁠, and (if needed).

Step 7: Iterate Updating. Repeat Steps (5) and (6) for λ times for subpack ⁠.

Step 8: Replace into P and resorting in an increasing value for the objective function.

Step 9: If convergence criteria are satisfied, stop the process; otherwise, return to step 4.

However, the nature of the gray wolves directly inspired the GWO, but the way it is altered to SGWO is not presumed to mimic the exact hunting mechanism of the pack. Hence, it can be considered that the wolves are approaching the prey not as one pack but as several smaller packs. This modification creates a more sophisticated network in which the information is exchanged at two levels: firstly between the global superiors and the pack, secondly between the local superior and the subpack.

Due to the simplicity, the benchmark functions are often used in the literature to compare the performance of algorithms. Thus, the performance of SGWO has been tested using 23 benchmark functions presented in Appendices C and D with different attributes. In these tables, Dim shows the number of decision variables, range indicates the boundary of functions' search space, and denotes the minimum value of the corresponding function. To conduct a fair comparison, for all four algorithms of SGWO, GWO, SCE, and PSO, the configuration of each algorithm was defined such that it resulted in the best outcome for the given 15,000 number of function evaluations (NFEs). Each algorithm was run 20 times for each benchmark function. The mean and standard deviation for each of the five algorithms were also computed. Results are tabulated in Appendix I.

The best results for each function have been highlighted in Appendix I. Note that the first criterion chosen to determine the superiority of an algorithm over others was the arithmetic mean of the result of the 20 consecutive solutions. If there was little difference in the mean result, the algorithm with the lower standard deviation was chosen as, the better one since it had proven to be more reliable than others. As can be concluded from Appendix I, the SGWO has won the competition among other algorithms in 12 of the total 23 benchmark problems. The amount of which the SGWO had improved the result considering the GWO varied a lot because of the nature of each problem. But it can be concluded that the hybrid algorithm performed better than its parent algorithms.

Two hypothetical systems comprising four and ten reservoirs (Figure 2) were analyzed to assess the performance of the SGWO algorithm on multi-reservoir systems. The discrete four-reservoir operation problem (DFRO) was first developed by Larson (1968) using discrete-time formulation. Implementing differential dynamic programming, Murray & Yakowitz (1979) further studied the problem. This discrete-time four reservoir operation problem has been studied ever since to evaluate multiple optimization tools and turned this problem into a benchmark problem in the reservoir operation optimization field.

In these equations, and are the minimum and maximum permitted reservoir releases for storage m, respectively, while and denote the minimum and maximum storage volumes for storage m. Also, is the release of reservoir m within time i, and is the storage volume of reservoir m within the time of i.

Concerning the ten-reservoir system, the problem presented by Murray & Yakowitz (1979) can be considered as a developed kind of the previously discussed four-reservoir operation problem. The system consists of reservoirs in series and parallel connections, which may receive supplies from one or more upstream reservoirs.

The operation of the system was performed over 12 months to maximize hydropower generation. The decision variables of this problem include reservoir releases each month. For each upstream reservoir, inflows were defined. Initial storage volumes were set to be (6,6,3,8,8,7,15,6,5,15), and the terminal volumes were constrained to be equal to the initial storage volumes. As with previous problems, this one also has minimum and maximum required storage volumes and reservoir releases, constraining the reservoir operation optimization problem (Murray & Yakowitz 1979).

As mentioned earlier, for the second approach, the monthly releases, and storage volumes in the next month are computed using the continuity equation (Equation (17)).

The following settings were applied to the SGWO algorithm to solve the DFRO problem. Number of subpacks = 5, population of each subpack = 49, maximum iteration = 205. λ was set to 2, signifying that at each iteration of the algorithm, the subpacks run twice through the GWO algorithm to update their position. For other algorithms, after a thorough sensitivity analysis, the parameters were set such that the algorithm functioned the best. The problem was solved 20 times with each algorithm. According to Table 1, which summarizes the 20 consecutive solutions by each algorithm, concerning the best answer over 20 solutions, and since in this problem the best answer is the largest one, the GWO algorithm has a final optimum value of 402.24, which is 0.05% more than the best answer achieved by the SGWO. However, concerning the average of the 20 results, the SGWO algorithm with a mean of 400.90 has exhibited the best result being 0.26% greater than the second-best algorithm (PSO). Another point derived from the results was the low standard deviation of the SGWO compared to other algorithms. Having the lowest standard deviation, accompanying the highest average optimum objective function value, asserts the fact that the SGWO has exhibited a reliable performance.

An equally principal factor in comparing meta-heuristic optimization algorithms is the NFE. The algorithm requires that the lower the NFE the less computational effort to reach the optimum point. In this regard, the SGWO using only 100,695 NFE has shown much better efficiency than other algorithms.

Depicting the optimization path, using the best value obtained for the objective function value, the convergence curve for the best, worst, and average results over 20 solutions are displayed in Figure 3. The behavior of the algorithm in the final iterations is notable here. As shown, at some point in the process, the best answer obtained for the objective function is constant, but in the final iterations, the answer tends to improve considerably. This is since the parameter ‘a’, which controls the exploration and exploitation, is altered at each iteration with a step size dependent on maximum iterations. The dependency of this parameter on the maximum number of iterations creates a situation in which increasing or decreasing the maximum number of iterations does not necessarily improve the results. Although this can result in a faster convergence at lower NFEs, a precise sensitivity analysis is required as the maximum iteration number directly affects the result. The results of reservoir releases and storages are shown in Appendix E and Appendix F, respectively, at each monthly period for SGWO and GWO algorithms against the maximum and minimum releases and storage volumes corresponding to the problem's constraints. These results indicate that the obtained answers are well within the boundaries of the constraints.

Regarding the ten-reservoir system, Table 2 lists the results according to 20 solution attempts for each algorithm. The proposed SGWO algorithm shows promising results in all aspects of the average answer, best answer (being the maximum of the 20 solutions), and required NFE.

The noteworthy point is that, in this problem, which has a considerably higher number of decision variables, the worst answer obtained by the SGWO algorithm, 1,189.12 is still better than the best answer resulted from the GWO and PSO algorithms. Having the highest average value of 1,200.68, being 0.99% higher than what was achieved by the second-best algorithm (GA), and the lowest standard deviation compared to other algorithms, yet again we have observed the high reliability of the SGWO algorithm. Efficiency-wise, the SGWO reached the top place while using 120,000 NFE which is 5 times less than GA, which was its closest rival in this problem.

The convergence curve for the best, worst, and average answers for the ten-reservoir problem has been illustrated in Figure 4. The same behavior observed in Figure 3 arising from the iteration number-based updating of the parameter ‘a’ is noticeable in the final iterations.

Appendices G and H display the computed storage volumes and monthly releases for each reservoir for SGWO and GWO algorithms corresponding to the problem's constraints. As illustrated, the obtained values are well within the boundaries of the constraints.

Finally, considering the single reservoir problem, which has essentially turned into 5 problems with different modeling approaches, each algorithm was tuned to perform at its best using preliminary sensitivity analysis.

For the first approach, the parameters of the SGWO algorithm were set as follows: Number of subpacks = 3, the population of each subpack = 37, maximum iteration = 300, where the λ was set to 2, meaning GWO will run twice on each subpack in each iteration. For the second approach, the number of subpacks has been set to 5, where the population of each subpack is set 1+number of decision variables; that is, 61, 121, and 241, respectively. Table 3 summarizes the results of 20 consecutive runs for each algorithm and each problem. In this table, 1st and 2nd approaches correspond to the rule curve and simple time-series approach. The former comprises 36 decision variables using a linear rule curve, and the latter has a decision variable for each operational month since the released water volume is the decision variable.

According to the derived results, the SGWO has arguably seized the top place among other algorithms. Concerning the best answer derived (the minimum column). The SGWO has the lowest value among all combinations. Reliability-wise, the SGWO has reached better results to average since the mean value of the optimum objective function is lower than other algorithms. The same thing is observable in the standard deviation of the results.

Furthermore, the low NFE required to the optimum point is again considerably lower than other algorithms. This number is especially important in large-scale problems. For instance, in the 2nd approach and during a 240-month operational period, the NFE required by the SGWO to reach the best answer is one-sixth of what PSO requires, which holds the second-best algorithm position for this problem.

In this paper, a novel hybrid meta-heuristic optimization algorithm was proposed to alleviate the local optima entrapment of the GWO algorithm and inspire by the population dividing and shuffling procedure in the SCE-UA algorithm. The proposed SGWO algorithm's performance was first assessed using 23 benchmark purely mathematical optimization problems. After the promising performance of this algorithm, the performance was tested using single and multi-reservoir operation optimization problems.

The results indicated that the modification imposed on the population handling mechanism of the GWO positively affected the performance of this algorithm. The first thing that was noticed was the reliability of the newly developed algorithm. The SGWO algorithm showed a low standard deviation parallel with the best average optimum value in all problems. And this happened while the SGWO required lower NFE. The optimization problems in different fields can be very complex, and the objective function they require to optimize might require a high computational effort to process. The SGWO algorithm can reach a reliable neighboring of the optimum value with few NFE, which means less computation time and economic feasibility. Concerning the best answer derived by the algorithms, aside from the four-reservoir problem in which the GWO offered an optimum point 0.05% higher than what SGWO resulted into, in other problems, the SGWO improved the result of the GWO from as low as about 4% to even as high as 37%. Especially in the large-scale problems, where the GWO seemed to face difficulty getting to a near-optimum solution, the SGWO showed a significantly better performance.

Although the SGWO showed promising performance, it must be mentioned that there was a trade-off for this better performance, and that is the complexity added to the algorithm. This algorithm has 3 more parameters to tune, which requires a delicate trial and error procedure. It is worth mentioning that the iteration number can immensely influence the result in this algorithm as the parameter ‘a’, which plays a crucial role in balancing exploring and exploiting procedure, increases in step size depending on the number of iterations.

The authors declare no conflict of interest.

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