Evaluation of Multi-Objective Firefly Algorithm and its Application in Optimal Multi-Objective Operation of Reservoirs

Background and Objectives
Previous researches have often used the Firefly Algorithm (FA) as a single-purpose operation of the reservoir. For example, Garousi-Nejad and Bozorg-haddad (2014) used FA for the first time in water science and reservoir operation, using ten-year statistical data of the Aidoghmoush reservoir located in East Azerbaijan province of Iran. They used it to meet the needs of downstream irrigation. In order to show the efficiency of FA, the results of this algorithm were compared with the global optimum results obtained from the nonlinear programming (NLP) method. The results showed that FA responses differed by less than one percent from NLP. Garousi-Nejad et al. (2016) used the FA to operate the reservoir and used the superiority of this algorithm over GA using five mathematical tests, the problem of reservoir operation for agricultural water supply, and the problem of reservoir operation to generate electricity. The results showed the superiority of FA performance in terms of global optimum convergence rate compared to GA results. Therefore, further investigating the capability and application of the multi-objective of this algorithm in the multi-purpose reservoir operation is necessary.
In this study, first, the performance of the MOFA algorithm is compared with the well-known and common NSGA-II algorithm from three perspectives of execution time, generation distance, and distance metric in two mathematical test functions. After ensuring they perform correctly in the test functions, an evaluation is performed for a reservoir operation problem. Then, three reservoir operation policies are selected from the policy set and reviewed.

Methodology
Aidoghmoush dam is one of the dams built in East Azerbaijan Province, located at 19 km from Mianeh city. Aydoghmoush dam is located in the Sefidrood catchment area and Aidoghmoush sub-basin on the Aidoghmoush river. The structure of the dam is a clay-core rockfill. In this study, the Aidoghmoush reservoir's operation is examined from 1990 to 1996. The FA algorithm was developed by Yang (2008). This algorithm is based on the idealization of the behavior and flashing patterns of fireflies.The test functions used in the present study are (1) Schaffer test function (SCH) and (2) Fonesca test function (FON) (Coello et al. 2007). The MOFA algorithm was used to solve two multi-objective test functions, and its performance was compared with NSGA-II. The Generational Distance (GD) metric, introduced by Van Veldhuizen and Lamont (1998), was used to measure the proximity of the non-dominated solutions obtained to the true Pareto front (PFtrue) (Coello et al. 2007). The spacing (SP) metric introduced by Schott (1995) was also used to measure the distribution of the obtained non-dominated solutions.

Findings
In both test functions, the MOFA with good solution quality has a significant time difference in execution time compared to the NSGA-II algorithm; the MOFA is 30% faster in the SCH test function and 15% more quickly in the FON test function than the NSGA-II. Therefore, the MOFA can be a good algorithm for optimizing complex and time-consuming problems such as reservoir operation. The MOFA algorithm in two groups of parameters compared with the NSGA-II algorithm was evaluated in the problem of reservoir operation to maximize reservoir storage and minimize deficit. Extracted solutions from the MOFA algorithm improved by 11% compared to the NSGA-II algorithm by considering better parameter values in terms of proximity to the PFtrue. Also, the execution time of the MOFA algorithm was improved by 13% compared to the NSGA-II algorithm's execution time.

Conclusion
Nature-inspired meta-heuristic algorithms have been considered in recent decades due to their suitable ability to derive optimal reservoir operation policies compared to classical methods. In this research, the Multi-Objective Firefly Algorithm (MOFA) is compared with the widely-used Non-Dominated Genetic Algorithm (NSGA-II) in two test functions [Schaffer (SCH) and Fonseca (FON)]. These algorithms are then used to solve the optimizing the reservoir operation problem. Aidoghmoush reservoir, located in East Azerbaijan province, is used as a case study. Two conflicting objective functions are considered, including the first objective function of maximizing reservoir storage and the second objective function of minimizing deficiency. The algorithm runtime, generation distance, and Spacing matric are used to evaluate and compare the studied algorithms. The results showed that the execution time of the MOFA algorithm in the two test problems examined was, on average, 22% less than the execution time of the NSGA-II algorithm, and the quality of the answers could be better or worse than NSGA-II. The results obtained in the multi-objective operation of the reservoir showed that the set of solutions obtained from the MOFA algorithm became closer to the PFtrue averagely of 13% by considering the parameters α, β, and γ equal to 10, 1, and 0.1, instead of considering the parameters α, β, and γ equivalent to 0.25, 1 and 1, and the average execution time of the algorithm was improved by 3%. This improvement of the solution's quality and runtime was 13 and 11, respectively, compared to the NSGA-II algorithm.
Since the MOFA algorithm is fast, the quality of the resulting set of answers can be improved by adjusting parameters, modifying, or integrating with other methods, such as different optimization algorithms or machine learning algorithms, and using it to Optimize real-time problems that are important in terms of computational time and burden. It is further suggested that the MOFA algorithm be developed for more complex models such as multi-purpose reservoir operation with more than two objectives of multi-reservoir systems, and its performance in reservoir operation management problems be compared with other meta-heuristic algorithms.

Keywords: MOFA, Multi-Objective Optimization, NSGA-II, Optimal Reservoir Operation, Test Function