Verification of FPA and PSO algorithms for rule curve extraction and optimization of single- and multi-reservoir systems' operations considering their specific purposes

Population growth, extension of arid and semi-arid regions' agricultural lands, and unequal spatiotemporal distribution of fresh water in both quantitative and qualitative terms have made required water demand an important challenge in the present century (Datta & Houck. 1984; Chang & Chang 2009). Nowadays, the construction of large reservoirs is no longer considered the best solution for water supply due to many of their limitations and problems (Bozorg-Haddad et al. 2006; Geem 2007; Ghimire & Reddy 2013). Limited locations for reservoir construction, increasing evaporation losses, accumulation of sediments behind the reservoirs, high costs of discharging sediments, and the problems with increasing the height of existing reservoirs are some of these challenges (Wu & Chen 2013; Ming et al. 2017). Optimal operation of existing reservoirs is one of the available and effective ways to deal with water shortages. By taking into account the existing limitations, the purposes of reservoirs, such as water supply for different demands, hydropower generation, and flood control should be accomplished (Kuo et al. 1990; Mesbah et al. 2009; Liu et al. 2011). Optimization of reservoir operations can be achieved both by classical methods (Rosenthal 1981; Datta & Houck 1984; Barros et al. 2003; Davidsen et al. 2015; Freire-González et al. 2018) and also by meta-heuristic models.

Classical methods provide standard reliable solutions by using point-to-point approaches. Thus, they start with an initial random guess and then, based on a predetermined rule, seek to find the best path to the optimal point. This process continues until it reaches a point where there is no better point, and eventually an optimal answer is obtained. These methods have disadvantages such as dependency of the convergence rate to the optimal solution on the initial random guess, and trapping at local optima and so inapplicability in dealing with issues that have a discrete search space (Venter 2010; Tsai & Chen 2014; Bozorg-Haddad 2018). Due to the limitations and disadvantages of classical optimization methods, researchers in the field of water resources management have used evolutionary and meta-heuristic algorithms. These algorithms generally converge to a near-global optimum for any well-defined optimization problems with the trial- and-error process. The main disadvantage of these algorithms is their long processing time needed to converge to a solution. For this reason, many researchers started to develop newer, computationally more efficient algorithms (e.g., Orouji et al. 2014; Jahandideh-Tehrani et al. 2015; Asgari et al. 2016). Many evolutionary and meta-heuristic techniques, such as Genetic Algorithm (GA), PSO and Honey-Bees Mating Optimization (HBMO) (East & Hall 1994; Carlisle & Dozier 2001; Bozorg-Haddad et al. 2008, 2009; Soltanjalili et al. 2011; Sabbaghpour et al. 2012; Fallah-Mehdipour et al. 2013, 2014; Ghimire & Reddy 2013; Abdelaziz et al. 2016; Asgari et al. 2019), have been used for water resources management.

The PSO algorithm was introduced by Kennedy & Eberhart (1995). It is simple to implement, requires few parameters to adjust, and has fast convergence and high efficiency in finding the global optima. It also has some limitations such as difficulty of defining initial design parameters, premature convergence, and tendency to be trapped into a local minimum especially for complex problems (Abdmouleh et al. 2017). Ghimire & Reddy (2013) used the PSO algorithm for a single reservoir system to extract optimal operation policies for dry and normal years, which increased the production of hydropower power by 3%. SaberChenari et al. (2016) optimized Mahabad reservoir by using the PSO algorithm to minimize the difference between monthly downstream water demand and release, showing that the PSO model had good performance in minimizing the water loss of the reservoir and the operation strategy was appropriate in a drought condition. Asgari et al. (2019) developed the combined Weed Optimization Algorithm (WOA) and Particle Swarm Optimization (WOAPSO) algorithm for evaluating river basin management. Their algorithm minimized the objective function with superior efficiency compared with the WOA and PSO, in terms of the convergence rate. Akbarifard et al. (2020) developed a model based on the Moth Swarm Algorithm (MSA) for optimization of a hydropower reservoir system. Their results revealed that the MSA was better than the GA and PSO in the optimization of reservoir operation. Jahandideh-Tehrani et al. (2020) evaluated the PSO algorithm and its applications in water resources management and showed its proper convergence to optimal Pareto fronts and fast convergence rate. Due to its applicability to various reservoir optimization problems, the PSO algorithm was selected in this study for the comparison purpose.

The Flower Pollination Algorithm (FPA), first introduced by Yang (2012), is a population-based algorithm initiated with a set of random solutions (Nabil 2016). The FPA has its limitations such as incapability of making a proper balance between exploration and exploitation and low exploration capability (Singh et al. 2018). Sakib et al. (2014) used the FPA and the Bat Algorithm (BA) to solve continuous optimization problems. They tested and compared the two algorithms on some benchmark functions. Nabil (2016) developed a Modified Flower Pollination Algorithm (MFPA) from the hybridization of FPA with the Clonal Selection Algorithm (CSA) and tested it for 23 optimization benchmark problems to investigate the efficiency of the new algorithm. The results of MFPA were then compared with those of Simulated Annealing (SA), GA, FPA, BA, and the Firefly Algorithm (FA), showing that the proposed MFPA was able to find more accurate solutions than FPA and the four other algorithms. Abdelaziz et al. (2016) applied FPA to derive the optimal sizing and allocations of the capacitors in different water distribution systems.

Optimal operation of a multi-reservoir system is a complex problem with a large number of decision variables and constraints, which is normally hard to solve by classical methods. Thus, evolutionary algorithms are good candidates to effectively and efficiently search the decision space and determine the optimal/near-optimal solution for reservoir operation. In this study, FPA is employed to optimize: (1) Aidoghmoush single-reservoir system operation for agricultural water supply, (2) Bazoft single-reservoir system operation for hydropower generation, (3) multi-reservoir system operation of Karun 5, Karun 4, and Bazoft, and (4) Bazoft single-reservoir system for rule curve extraction. The results are also compared with those from the PSO algorithm. It is aimed to demonstrate the capabilities of FPA and evaluate its performance in optimizing single- and multi-reservoir system operation problems. The flowchart of the methodology is shown in Figure 1.

Increase in water demands, along with the growth and development of human societies, has led to disruption in water use sustainability and also the problems in water resources management. In this regard, paying attention to optimal reservoir operation is crucial for better water resources management. Therefore, instead of building new reservoirs, it is possible to make optimal use of the existing reservoirs to solve or mitigate the water scarcity problems. Generally, optimization models consist of objective function(s), constraints, and simulation models. In reservoir operation problems, the objective function can be of minimization or maximization type, depending on the type of problems being defined.

It should be noted that the calculation of involves a lot of complexities, which adds to the complexity of the operation model. In addition, it is assumed that is fixed for all operation periods.

In this research, the FPA and PSO algorithms are used to solve reservoir operation optimization problems. These two algorithms are introduced in the following subsections.

The PSO algorithm was first introduced by Kennedy & Eberhart (1995). PSO is a population-based method inspired by bird behavior (information exchange) in a swarm. In this algorithm, the population is called a swarm and the individuals are called particles. In the search space, each particle moves with a certain velocity (Ali & Tawhid 2017). The algorithm starts with a random population, in which each member is a particle, creating the set. This set moves towards the optimal point in the decision space according to the velocities of each member and the whole set:

• 1.

  The best position that the particle has ever reached ⁠.

• 2.

  The best position that the best member in the neighborhood of this particle has reached so far ⁠.

In Equation (21), c₁ and c₂ are introduced as cognitive and social parameters, respectively. It should be noted that proper selection of these two coefficients affects the speed of convergence. The flowchart of the PSO algorithm is shown in Figure 3.

In order to evaluate the efficiency of the FPA and PSO algorithms, several test functions were chosen, all the data, relationships, and constraints were defined, and the final solutions were identified. The test functions include 1 – sphere, which is the simplest form of De Jong's functions (De Jong 1975), 2 – Ackley test function (Ackley 1987), 3 – Styblinski–Tang (Styblinski & Tang 1990), 4- Rosenbrock (Rosenbrock 1960), and 5 – the Holder table function. The results of these test functions and the specifications of the algorithms are shown in Table 1 (Fallah-Mehdipour et al. 2013; Orouji et al. 2014). It should be noted that, in order to solve the problem by PSO and FPA, a preliminary sensitivity analysis was performed for parameters such as populations and iterations for each method.

In this study, both FPA and PSO algorithms were also tested by solving typical multi-reservoir optimization problems in discrete and continuous domains and also in real case studies.

The applicability of the FPA and PSO algorithms was tested with three benchmark multi-reservoir operation problems in both discrete and continuous domains. They include: (1) four-reservoir operation optimization problem in a continuous domain, (2) four-reservoir problem in a discrete domain, and (3) ten-reservoir problem in a continuous domain (more information can be found in Bozorg-Haddad et al. (2011)).

Chow & Cortes-Rivera (1974) designed and solved the four-reservoir operation optimization problem in a continuous domain for the first time and obtained the optimal value of the objective function (308.26 units) by using the linear programming (LP) method. Bozorg-Haddad et al. (2011) solved this optimization problem using the HBMO algorithm and showed that the objective function was 308.24 units, a 0.02% difference from the value of Chow & Cortes-Rivera (1974). The optimization results of the FPA and PSO from the best run are shown in Table 2. The FPA solution was 0.3% different from the LP solution. However, the best PSO solution was 1.63 times worse than the FPA solution.

Larson (1968) designed and solved the four-reservoir operation optimization problem in a discrete domain for the first time and obtained the optimal value of the objective function (401.3 units) by using the State Increment Dynamics Programming (SIDP) method. Bozorg-Haddad et al. (2011) solved this optimization problem using the HBMO algorithm and showed that the objective function was also 401.3 units. The best run results from the FPA and PSO for this problem are shown in Table 2. The FPA solution was approximately 0.1% different from the SIDP solution, while the PSO solution was about 0.66% different from the SIDP solution.

Murray & Yakowitz (1979) designed and solved the ten-reservoir operation optimization problem in a continuous domain for the first time and obtained the optimal value of the objective function (1,190.625 units) by using the Constrained Differential Dynamic Programming (CDDP) method. Bozorg-Haddad et al. (2011) solved this optimization problem using the HBMO algorithm for both short and long runs. The optimal results from long and short runs were about 99.84% and 96.85% of the global optimal solution obtained from CDDP (1,194.44), respectively. The best run results from FPA and PSO for this problem are shown in Table 2. The FPA and PSO solutions were about 0.2% and 0.8% different from the CDDP solution, respectively.

As aforementioned, the FPA and PSO algorithms were tested for the following reservoir systems: (1) Aidoghmoush single-reservoir system for agricultural water supply, (2) Bazoft single-reservoir system for hydropower generation, (3) multi-reservoir system of Karun 5, Karun 4, and Bazoft, and (4) Bazoft single-reservoir system for rule curve extraction.

Aidoghmoush reservoir is located 23 km southwest of Mianeh City in the province of East Azarbaijan (northwest of Iran). This reservoir is on the Aidoghmoush River in the Caspian Sea basin, one of the important and main branches of the Ghezel Ozan. The specifications of Aidoghmoush reservoir are shown in Table 3.

The monthly downstream agricultural water demand and the monthly inflow of the reservoir during the ten-year operation period (1991–2000) are shown in Figure 4. The minimum, average, and maximum water demands during this period were 0.00, 12.12, and 39.57 (10⁶ m³), respectively. As shown in Figure 4, in the majority of the period, the monthly reservoir inflow and downstream agricultural demand varied differently. In the high inflow periods, the downstream agricultural demand was lower. In the low inflow periods, however, the downstream agricultural demand was considerably higher than the inflow. This timing disparity was one of the reasons for the reservoir construction. The data related to the Aidoghmoush River and reservoir and the downstream agricultural network were mainly derived from the studies by Ashofteh et al. (2013).

Optimizing the Aidoghmoush reservoir operation for downstream agricultural water supply is a simple-reservoir system problem. However, due to some non-linear constraints, the non-linear programming (NLP) method was used. The problem was solved by using LINGO 14 software and the optimal value of the objective function of 3.37 was obtained.

In order to solve the problem by PSO and FPA, a preliminary sensitivity analysis was performed for each method. The best results from PSO and FPA are shown in Table 4. According to the number of decision variables and the number of iterations in Table 4, which came from the sensitivity analysis, the results of the PSO and FPA are from 100,010 rounds of calculations of the objective function. The results of the objective functions from five independent PSO and FPA executions are shown in Table 4, indicating that the optimal solution from FPA was the same as that of the NLP method, while the best PSO solution was about 0.2% different from the NLP answer.

The PSO and FPA convergence diagrams for the three modes of minimum, average, and maximum runs among the five independent runs are shown in Figure 5(a) and 5(b), respectively. It can be observed in Figure 5 that the initial intervals of the generated solutions in the PSO are appropriate with a large range. As shown in Figure 5(b), the difference in the final results of the curves of the minimum, average, and maximum values of the objective function in FPA is small, and hence the algorithm is more reliable or in other words, the FPA is able to achieve a solution that is closer to the optimum one with fewer iterations. Figure 6 shows a comparison of the mean values of the objective functions from five-time runs for the Aidoghmoush reservoir problem from PSO and FPA, indicating that the FPA curve is more convex than the PSO curve.

The storage volumes of the best run determined by PSO, FPA, and NLP for the Aidoghmoush single-reservoir problem are shown in Figure 8. The FPA storage volumes are more compatible with the NLP solution than those from PSO.

The Bazoft River is the second largest tributary of the Karun River and the reservoir is on this tributary. The average annual inflow of the reservoir is 2,012.6 × 10⁹ m³. The minimum and maximum storage volumes of this reservoir are 142.15 and 450.3 × 10⁹ m³, respectively (Table 6). Five-year data (1966–1970) of Bazoft reservoir were used in this case study (Figure 9).

Since the LINGO software could not find the global optimum solution of this problem, it cannot be solved by using the NLP method. In order to solve the problem using PSO and FPA, a preliminary sensitivity analysis was performed for each method and the best values selected for PSO and FPA are shown in Table 4. Given the population values of the decision variables and the number of iterations in Table 4, the results of PSO and FPA are from 2,500,050 rounds of calculations of the objective function.

The results of the objective functions obtained from the five independent PSO and FPA runs in Table 4 indicate that the best values of the objective function for PSO and FPA are 4.93 and 3.18, respectively. Thus, the best value of the objective function of the PSO method is approximately 3.5 times worse than that of the FPA method.

The PSO and FPA convergence diagrams are shown in Figure 10(a) and 10(b), respectively, for the three modes of minimum, average, and maximum of the five independent runs. From Table 4 and Figure 10, it can be seen that the FPA performed better than the PSO in achieving the best value of the objective function. The curves of the minimum, average, and the maximum values of the objective function are more convergent in the FPA method. The FPA method converged after approximately 800,000 iterations, while the PSO method was not convergent until 2,000,000 iterations.

Figure 11(a) shows a comparison of the mean values of the objective function from the five-time runs for the PSO and FPA. It can be observed from Figure 11(b) that the FPA is much more convex than PSO and its solution is closer to the optimal one. So, the FPA outperformed the PSO. Figure 12(a) and 12(b) show the PSO and FPA reservoir release volumes, and Figure 13(a) and 13(b) show the PSO and FPA reservoir storage volumes, respectively.

For a hydropower system operation problem, the main goal is to produce the maximum electricity in each period. Therefore, in addition to reservoir release and storage volumes, the production capacities in each period also need to be determined. Figure 14(a) and 14(b) show the production capacities during the operation period of the Bazoft reservoir based on the best runs of the PSO and FPA methods.

The Karun River is considered to be the most important and abundant source of surface water in the southwest of Iran. The catchment area of this river ranges from 49° 35′ to 50° 35′ E (longitude) and from 31° 40′ to 32° 40′ N (latitude). This case study includes three reservoirs: Karun 5, Bazoft, and Karun 4 (Figure 15) and the required data are shown in Table 7.

The optimization problem of this three-reservoir system operation for agricultural water supply cannot be solved by using the NLP method due to its complexity, including the non-linear objective functions and constraints. For solving the problem using PSO and FPA, a preliminary sensitivity analysis is performed for each method and the best selected values for PSO and FPA are presented in Table 4.

The results of the objective functions obtained from the five independent PSO and FPA runs are shown in Table 4. It can be observed that the best values of the objective function for PSO and FPA are 2.03 and 1.58, respectively. Thus, the best value of the objective function from the PSO method is approximately 28% worse than that of the FPA method.

The PSO and FPA convergence diagrams are respectively shown in Figure 16(a) and 16(b) for the three minimum, average, and maximum runs among the five independent runs. According to Table 4 and Figure 16, the FPA performed better than the PSO in achieving the best value of the objective function. The curves of the minimum, average, and maximum objective values from the FPA method exhibit faster convergence. The FPA converged after 20,000 iterations, while the PSO converged after 40,000 iterations. Figure 17 shows a comparison of the mean objective function values for the five-time runs of FPA and PSO for the three-reservoir system. As shown in Figure 17, FPA is more convex than PSO, which indicates the faster convergence of the FPA than the PSO. Figure 18(a) and 18(b) respectively show the release and storage volumes of the best PSO and FPA runs for the Bazoft, Karun 5, and Karun 4 reservoirs.

The 43-year inflow data (1955–1997) of the Bazoft reservoir were used in a case study (Figure 19). Since the LINGO software is unable to find the global optimum solution of this problem, it cannot be solved by using the NLP method. Thus, PSO and FPA were used to extract the operation rule curve. A preliminary sensitivity analysis was performed for each method. The best values selected for PSO and FPA are presented in Table 4. Given the population values of the decision variables and the number of iterations in Table 4, the results from PSO and FPA are from 500,010 rounds of calculations of the objective function.

The results of the objective functions obtained from the five independent PSO and FPA runs are displayed in Table 4. It can be seen that the best values of the objective function for the PSO and FPA methods are 117.29 and 81.66, respectively, indicating that the best value of the objective function of the PSO method is approximately 43% worse than that of the FPA method. Figure 20(a) shows a comparison of the mean values of the objective functions of the five-time runs of the PSO and FPA for extracting the operation rule curve of the Bazoft single-reservoir system. It can be observed that the FPA is much more convex than the PSO and its solution is closer to the optimal one. So, the FPA outperformed the PSO in the optimization.

The reservoir release volumes, reservoir storage capacities, and power production capacities from the best PSO and FPA runs are shown in Figure 20(b)–20(d), respectively. From Figure 20(b), it can be observed that in most periods, the release volumes from the PSO do not match those from the FPA. According to Figure 20(c), in most periods the reservoir storage volumes from the PSO also do not match those from the FPA. As shown in Figure 20(d), in most periods the FPA was able to reach the maximum production capacity or a very close value. In contrast, the PSO more frequently failed to reach its maximum production capacity.

Generally, water resources systems are complex and optimization techniques are needed to determine the best solution for their design and operation with the maximum efficiency. To achieve the best planning in the operation of reservoirs, a variety of software packages, such as the Water Evaluation And Planning (WEAP) system, can be used (Fallah-Mehdipour 2008). In the related modeling, the operating conditions of each period are directly dependent on the conditions of the previous period. Also, the personal preference of the operator is involved in the type of result and planning of each period (Bozorg-Haddad 2018). When optimization tools are used, all periods of the planning horizon are considered simultaneously, which provides the best solution(s) for design and operation of the water systems. The optimization techniques reduce the trial- and-error process, overcome system complexity, and provide suitable decision-making options. In addition to the aforementioned properties of most optimization methods, FPA, which is one of the new evolutionary algorithms, has more capabilities such as determining better (closer to the ideal solution) objective functions, decreasing the calculation time, simplifying the problem, and providing better solutions for decision makers.

The results for the Aidoghmoush single-reservoir system showed that the best FPA solution was similar to the NLP solution, while the best PSO solution was about 0.2% different from the NLP solution. The best values of the objective function of the PSO were approximately 3.5 times, 28%, and 43% worse than those of the FPA for the Bazoft single-reservoir system for hydropower generation, the multi-reservoir system, and the Bazoft single-reservoir system for rule curve extraction, respectively. This study demonstrates that FPA is able to yield better results in the reservoir operation problems than PSO.

Optimal operation of a reservoir system (especially the reservoir system for hydropower generation) is a complex and non-linear optimization problem. In this regard, evolutionary and meta-heuristic algorithms have been widely used due to their ease of use and the ability to achieve a near-optimal solution in less time than the classical optimization methods. In the present study, the applicability of the FPA for optimization of the operation of reservoir systems was first evaluated, and then its performance was compared with that of the PSO algorithm.

In order to evaluate and validate FPA applicability, five single-objective functional tests (including sphere, Ackley, Styblinski-Tang, Rosenbrock, and Holder table) were selected. The results showed that FPA was more applicable than PSO in finding optimal solutions in these functional tests. Then, the FPA was applied to solve three benchmark multi-reservoir operation problems in both discrete and continuous domains, including (1) four-reservoir operation optimization problem in a continuous domain, (2) four-reservoir problem in a discrete domain, and (3) ten-reservoir problem in a continuous domain. The results demonstrated that the FPA outperformed the PSO in finding the optimal solutions in both continuous and discrete domains.

In addition, the FPA and PSO were tested in three case studies to optimize: (1) Aidoghmoush single-reservoir system operation for agricultural water supply, (2) Bazoft single-reservoir system operation for hydropower generation, (3) multi-reservoir system operation of Karun 5, Karun 4. and Bazoft, and (4) Bazoft for operation rule curve extraction. The results for the Aidoghmoush single-reservoir system showed that the FPA achieved the same best optimal solution as the one from the NLP method. However, the best solution from the PSO was about 0.2% different from the NLP solution. Also, the PSO had 2.5 times more errors than the FPA. For the Bazoft single-reservoir system for hydropower generation, the best value of the objective function of the PSO method was approximately 3.5 times worse than that of the FPA method. For multi-reservoir system of Karun 5, Bazoft, and Karun 4 for agricultural water supply, the best value of the objective function of the PSO method was approximately 28% worse than that of the FPA method. For the reservoir operation rule curve extraction, the best value of the objective function of the PSO method was approximately 43% worse than that of the FPA method. Overall, the FPA outperformed the PSO in the optimization of reservoir systems. Also, it should be noted that there were certain uncertainties in the inflow and demands, which can be considered in future studies in order to improve the FPA optimizations.

The authors thank Iran's National Science Foundation (INSF) for its financial support of this research.

There is no conflict of interest.

All relevant data are included in the paper or its Supplementary Information.