Optimal operation of reservoir systems with the symbiotic organisms search (SOS) algorithm

The unwise operation of reservoirs is the main driving-force of various water resources crises such as degrading native aquatic ecosystems (Steinschneider et al. 2014), water pollution (Yuan et al. 2015), shrinkage of lakes (Azarnivand & Banihabib 2016), and other calamities in many regions of the world. The countries within the arid regions of the world are grappling with anthropogenic and climatic driving forces which pose a burden to water, energy, and food security. For this reason, it is vital to improve water resources planning and management, which includes reservoir operation as a key component.

The techniques used for obtaining optimal operation of reservoir systems can be divided into classical methods and evolutionary algorithms (EAs). Schardong & Simonovic (2015) warned about the curse of dimensionality that plagues classical methods such as dynamic programming (DP) and stochastic dynamic programming (SDP). Linear programming (LP) requires objective functions and constraints that are linear on the decision variables, this being a common limitation for practical modeling of real reservoir systems. Nonlinear programming (NLP) is heavily influenced by the choice of initial conditions, and most searches may lead to local optimal solutions instead of the global optima. On the other hand, EAs have amply demonstrated their ability of finding near-global solutions of complex optimization problems (Farhangi et al. 2012; Zufferey 2012; Ashofteh et al. 2013a, 2013b, 2015a, 2015b, 2015c; Bozorg-Haddad et al. 2013, 2014, 2015a, 2015b; Fallah-Mehdipour 2013a, 2013b, 2013c, 2014; Orouji et al. 2013, 2014a, 2014b; Shokri et al. 2013; 2014; Soltanjalili et al. 2013; Ahmadi et al. 2014, 2015; Beygi et al. 2014; Bolouri-Yazdeli et al. 2014; Jahandideh-Tehrani et al. 2015). There is a large volume of published papers dealing with EAs applied to reservoir operation problems (Ching et al. 2003; Kumphon 2013; Afshar et al. 2015; Bashiri-Atrabi et al. 2015; Li et al. 2015; Rampazzo et al. 2015; Amirkhani et al. 2016; Bozorg-Haddad et al. 2016a, 2016b).

An EA such as the standard genetic algorithm (GA) can be applied to solving many types of optimization problems. The GA mimics natural selection mechanisms and works on the basis of populations of solutions that are improved iteratively (i.e., population-by-population (Fogel 2000)). In the first computational step of the GA, an initial population is created composed of randomly generated solutions. The next population is produced to improve the objective function through an iterative process. The chromosomes (or solutions) of the current population at each step are selected to generate the next generation. The selection probability of chromosomes with superior fitness is larger than those of less fit chromosomes. Selected chromosomes generate the next population with crossover and mutation operators. Crossover generates two new chromosomes by exchanging genes between them. The mutation alters the chromosomes' genes to create diversity in their population. The process of improving chromosomal populations continues iteratively until fulfilling a termination criteria.

Based on the ‘no free-lunch’ theorem, it is impossible for an EA to optimally solve all the optimizing problems (Wolpert & Macready 1997). Thus, there are new EAs being introduced continually. Yazdi et al. (2016) compared three EAs for the optimal design of buildings. Hosseini-Moghari et al. (2015) compared a recently developed EA called imperialist competitive algorithm (ICA) against the cuckoo optimization algorithm (COA) in two separated optimization problems, in which ICA outperformed COA in solving single reservoir and multi-reservoir operation problems. The ICA also outperformed ant colony optimization (ACO) in deriving optimal operational policies of the Dez reservoir in Iran (Afshar et al. 2015). Li et al. (2015) compared seven typical heuristic algorithms with an application to a real-life multi-reservoir system in China. In that study, particle swarm optimization proved superior to other algorithms. Bashiri-Atrabi et al. (2015) developed an optimization model based on the comparison of the harmony search algorithm vs. the honey-bee mating optimization for reservoir operation optimization with respect to flood control in northern Iran. Akbari-Alashti et al. (2014) applied NLP, GA, and fixed-length gene genetic programming (FLGGP) to derive multi-reservoir real-time operation rules for a multi-reservoir system. The results indicated the superiority of FLGGP in reaching the global optimum value of NLP.

The selection of an EA to solve a specific problem remains non-trivial (Maier et al. 2014). EAs are targets of criticism because of the need for specifying algorithmic parameters. Parameter tuning (either implicitly or explicitly) of EAs can be complex and time-consuming (Lobo et al. 2007; Eiben & Smit 2011; Joan-Arinyo et al. 2011; Yeguas et al. 2014; Veček et al. 2016). Parameter tuning using full factorial design is computationally burdensome. Hence, practitioners often apply already-available tuning approaches or develop parameter specification methods (Joan-Arinyo et al. 2011; Lee et al. 2013; Montero et al. 2014; Veček et al. 2016). It appears practical to test an EA which does not involve parameter tuning in water resources management. Cheng & Prayogo (2014) introduced the symbiotic organisms search (SOS) algorithm to overcome the parameter-specification disadvantage. The SOS algorithm requires only the specification of the ‘maximum number of evaluations’ and the ‘population size’. The SOS is a nature-inspired optimization algorithm which simulates three different symbiosis interactions between organisms that dwell in an ecosystem.

This work focuses on the optimal hydropower production optimization of the Karun4 reservoir system in southwestern Iran. The novelty of this work consists in the introduction of the SOS algorithm, a recently developed EA which does not involve parameter tuning to water resources systems analysis. This work assesses the performance of the SOS algorithm in optimizing a single reservoir operation against the GA, the water cycle algorithm (WCA), and NLP. Moreover, the optimal operation of a hypothetical benchmark four-reservoir system is also tackled with SOS, GA, WCA, and NLP to verify the used EAs. The remainder of this paper first describes the SOS algorithm and its theoretical underpinnings. The SOS, GA, WCA, and NLP are then verified with a hypothetical four-reservoir system. Lastly, the EAs and NLP are applied to solving for the optimal optimization of the Karun4 reservoir of Iran. The performances of the SOS, GA, WCA, and NLP are compared using the solutions to the single-reservoir problems.

In the third phase, which entails the mutation operator of the SOS and is called parasitism, x_i and x_j are the artificial parasite and host, respectively. In this type of symbiosis relationship, one organism benefits while another one is harmed. The trademark of the parasite vector (PV) is that it competes against other randomly selected dimensions rather than its parent/creator with a range between given lower and upper bounds. In this phase, an initial PV is generated by duplicating organism x_j. Some of the decision variables from the PV are modified randomly to distinguish the PV from x_j. A random number must be generated in the range of [1, the number of decision variables] to represent the total number of modified variables. A uniform random number is generated for each dimension to obtain the location of the modified variables. Lastly, a uniform distribution within the search space is required to modify the variables and provide a PV for the parasitism phase. In synthesis the PV attempts to replace x_j which is selected randomly from the ecosystem. If the PV outperforms x_j it becomes part of the ecosystem, whereas if the PV does not outperform x_j it vanishes from the ecosystem. The PV is created by modifying x_j in random dimensions with random numbers instead of making small changes in x_j. If the current PV and x_j are not the last member of the ecosystem the algorithm returns to the step that selected X_best until reaching a specified termination criterion.

The EAs' computations for the benchmark functions, the operation of the Karun4 reservoir system, and the operation of the four-reservoir system were programmed with MATLAB (MATLAB 7.11.0 software). The NLP solution was obtained with Lingo based on generalized reduced gradient algorithm (Lingo 11.0 software). Multiple solutions (runs) are required to assess statistically the performance of the EAs due to their random nature. Hence, the results were obtained based on 10 runs for each applied EA.

The SOS algorithm's results for the Karun4 reservoir operation were compared with those obtained with the GA, WCA, and NLP. The NLP method was applied to evaluate the global optimal solution whereas the EAs determined the near-optimal solution. The crossover rate, mutation rate, and number of populations for GA were determined by trial and error identical to the approach by Bozorg-Haddad et al. (2011) and set equal to 0.60 (via two-point crossover function), 0.05 (via uniform function), and 70,000 (including population size [ecosystem size in the SOS] = 70 and number of generations =1,000), respectively. Moreover, the selection process applied the roulette wheel. Due to critique associated with the application of the roulette wheel by Bozorg-Haddad et al. (2011), the authors also tested other operators. The crossover rate and mutation rate were set equal to 0.70 and 0.06, respectively. The parameter values of the WCA were identical to those used by Bozorg-Haddad et al. (2015c). The number of objective function evaluations (NFE) for the four-reservoir problem equaled 500,000 with the EAs (the SOS algorithm, the WCA, and the GA).

The NLP solution was equal to 1.213 (Bozorg-Haddad et al. 2015c). The superiority of an EA over other ones can be obtained based on the similarity of its near-optimal solution to the global optimal solution. Each run of the SOS algorithm lasted approximately 30 seconds. Table 2 demonstrates the performance of the SOS algorithm vs. GA and WCA based on 10 runs. The main results to emerge from Table 2 are as follows. (1) The GA in its best performance converged to 1.535, while the SOS algorithm reached the value 1.240 in its best performance. The SOS also outdid the WCA, which converged to 1.260 in its best run. (2) It is noteworthy that even the worst performance of the SOS algorithm was better than the best performance of the GA. (3) The coefficient of variation of the GA's solutions was almost five times greater than that of the SOS algorithm. Yet, on the basis of standard deviation and coefficient of variation, the WCAs' runs exhibited smaller variability than the SOS algorithm. The coefficients of variation of the SOS algorithm and WCA were close to zero.

This work introduced the SOS algorithm to the optimization of reservoir system operation and applied it to the Karun4 reservoir system with hydropower purpose, and to the optimal operation of a four-reservoir system. Our results from these two optimization problems indicate that the SOS algorithm outperformed the GA and a relatively new EA called the WCA. In the single reservoir problem with global optimal value equal to 1.213, the SOS algorithm, the WCA, and the GA determined 1.240, 1.260, and 1.535 as the optimal solutions, respectively. The superiority of the SOS algorithm over the other EAs was repeated in the four-reservoir optimizing problem with global optimal value equal to 308.29. In this regard, the GA, the WCA, and the SOS algorithm in their best performance among 10 runs converged to 97.46%, 99.56%, and 99.86% of the global optimal solution. Besides closeness of the best performance result to the global optimal solution calculated with the SOS algorithm, other practical merits are germane to the SOS algorithm. In comparison with the GA the SOS algorithm did not exhibit premature convergence, it had lower standard deviation, and had lower and more stable coefficient of variation. Moreover, unlike the WCA and the GA, the SOS does not require calibration of algorithmic parameters. The results showed that despite the simple principles of the SOS algorithm, it is capable of dealing with complexities and constraints associated with multi-reservoir operation optimization problems. This renders the SOS algorithm an attractive solution method of optimization problems in water resources management.

The superiority of the SOS algorithm over other EAs established in this study should not be considered applicable to all optimization problems. This paper's results are in line with the aforementioned ‘no free-lunch’ theorem, which emphasizes that a particular EA cannot optimally solve all well-posed optimizing problems. However, the SOS algorithm has unquestionable advantages, such as the simplicity of parameter specification, adding a perturbation to the search domain, and substituting a solution by evaluating the difference between other solutions that render it a very attractive EA. Future applications of the SOS algorithm could include solutions to conflicting objectives that appear in water resources management. For this purpose, multi-objective algorithm benchmarking can be accomplished with the NSGA-II, Borg, AMALGAM, and other algorithms which have shown good performance in solving multi-objective optimization problems.