A modified sine cosine algorithm for accurate global optimization of numerical functions and multiple hydropower reservoirs operation
Introduction
Compared with the fossil fuels, hydropower has the advantages of low environmental pollution, and high operational flexibility, making it become one of the most important renewable energies [1], [2], [3], [4]. With the increasingly severe environmental problems, hydropower is gaining more and more attention and a large number of hydropower reservoirs are built around the world [5], [6], [7], [8]. In China, about 20% power generation and installed capacity of the power industry are provided by hydropower [9], [10], [11]. Meanwhile, to achieve the goal of energy savings and pollutions reduction, hydropower is given a high priority in the mid-to-long-term electric power development plan of China and then a growing number of reservoirs will be put into production in the coming future. Due to the hydraulic-electrical connections, spatiotemporal correlated constraints and intrinsic nonlinearity, the rapid expansions of system scale will sharply increase the modeling difficulty and pose unprecedented challenges for managers in water resource system and electrical power systems [12], [13], [14]. Motivated by this practical necessity, multiple hydropower reservoirs operation optimization is chosen as the focus of this paper.
Generally, the goal of hydropower operation is often set to find the best scheduling scheme that can maximize the total generation benefit of all the reservoirs while satisfying a group of physical constraints (like water balance equation, turbine discharge or power output limits), which can be mathematically categorized as a typical constrained optimization problem. To address this kind of problem, many attempts have been made to find effective methods over the past decades [15], [16], [17], [18]. Based on the employed search principles, the existing approaches can be roughly divided into two different kinds of categories: mathematical programming methods and evolutionary algorithms [19], [20], [21]. For the former, the representative includes linear programming, nonlinear programming, dynamic programming, and network optimization [22], [23], [24], [25]; for the latter, the representatives are genetic algorithm [26], differential evolution [27], particle swarm optimization [28], grey wolf optimizer [29], gravitational search algorithm [30], artificial ecosystem optimization [31], cuckoo search [32], quantum-behaved particle swarm optimization [33], and Harris Hawks optimizer [34]. Although varying degree of success has been achieved in practice, the applications of the above methods are still limited by some shortcomings [35], like dimensionality problem [36], [37], [38], high computation burden [39], parameter tuning [40], [41], [42], [43] and premature convergence [44], [45], [46]. Hence, it is of great importance to develop some effective optimization methods suitable for the complex engineering optimization problem [47], [48].
Sine cosine algorithm is a novel meta-heuristic method for global optimization problem [49], [50], [51]. During the evolutionary process, the sine and cosine functions are employed to guide the search direction in the entire decision space [52], [53], [54], which can increase the probability of obtaining global optima. Based on the simulation results in previous literature, SCA can yield better results than the classical particle swarm optimization and genetic algorithm in numerical optimization problems [55]. However, owing to the loss of swarm diversity, the standard SCA method still suffers from some defects as used to solve large and complex real-world engineering problems, like slow convergence and local minimum. To enhance the SCA performance, this paper aims at developing a novel quasi-opposition SCA (QSCA for short) using the elite-guide evolutionary strategy to enhance the convergence speed, the quasi-opposition learning strategy to balance exploration and exploitation, and the adaptive mutation strategy to improve the swarm diversity. To verify its reliability, the QSCA method was firstly tested on 12 famous benchmark functions, 24 large-scale CEC2005 problems and 30 CEC2017 benchmark functions, and then used to address the optimal operation of multiple hydropower reservoirs in China’s Wu River. The results in both numerical tests and engineering problems indicate that the QSCA method can improve the search efficiency and solution accuracy of the standard SCA method.
To clearly understand this paper, the key contributions are given as below: (1) a novel QSCA method is proposed for the complex global optimization problem, where the elite-guide evolution strategy, quasi-opposition learning strategy and adaptive mutation strategy are embedded into the standard SCA method; (2) a practical operation optimization model is developed to maximize the generation benefit of multiple hydropower reservoirs; (3) the QSCA method outperforms several famous evolutionary algorithms in the numerical experiments and multiple hydropower reservoirs operation, providing an effective optimizer tool for the complex global optimization problems.
The remainder of this paper is organized as below: the detailed of the QSCA method is given in Section 2; the QSCA performances in benchmark functions are compared with several famous methods in Section 3; the technical details of the QSCA method in multiple hydropower reservoirs operation are given in Section 4; Next, the Wu hydropower system in China is chosen to test the practicability of the QSCA method; finally, the conclusions are given at the end.
Section snippets
Sine cosine algorithm (SCA)
Sine cosine algorithm (SCA) is a new meta-heuristic method inspired by the sine and cosine mathematical functions [56], [57], [58]. After randomly generating the positions of the initial swarm, each solution dynamically updates the positions under the guidance of the global best-known solution. For the minimization problem, the evolutionary procedure of the SCA method can be expressed as
Benchmark functions
Here, 12 famous benchmark functions are chosen to test the performance of the QSCA method, which can be broadly divided into unimodal functions () and multimodal functions (). Unimodal functions with one extreme value in the problem space can investigate the convergence speed of the method; multimodal functions with one or more global optima in the search area can evaluate the ability of escaping from local minimum. Table 1 and Fig. 1 show the detailed information and 2D shapes of
Objective function
Under the market environment, the operational goal of hydropower enterprise is often chosen to maximize the total generation benefit of all the hydropower reservoirs in the scheduling horizon [77], [78], [79], which can be expressed as below: where E is the total generation benefit of all the reservoirs. N is the number of reservoirs. T is the number of periods. is the number of hours at the tth period. and denote the power output and electricity price of
Engineering background
In this study, five huge reservoirs on the mainstream of Wu River in China is chosen to test the feasibility of the proposed method. As one of the largest hydropower bases, the Wu River plays an essential role in the promoting the sustainable economic-social development of the western region in China. In the following sections, the scheduling horizon is set as 1 year split into 12 months, while several methods are introduced for comparison under the parameter settings: the number of solutions
Conclusions
For the past few years, a growing number of meta-heuristic methods based on natural biological behaviors have been successfully developed to solve the complicated multiple reservoir operation optimization problem. In this paper, the emerging SCA method is introduced and then improved in three ways, including the quasi-opposition learning strategy for achieving a comprise between local exploration and global exploitation, the elite-guide evolutionary strategy for enhancing the convergence rate,
CRediT authorship contribution statement
Zhong-kai Feng: Modeling, Finalization of the manuscript. Shuai Liu: Programming implementation, Data curation. Wen-jing Niu: Modeling, Finalization of the manuscript. Bao-jian Li: Data analysis. Wen-chuan Wang: Data analysis. Bin Luo: Literature review. Shu-min Miao: Literature review.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This paper is supported by the National Key R&D Program of China (2017YFC0405900), National Natural Science Foundation of China (52009012 and 51709119), Natural Science Foundation of Hubei Province, China (2020CFB340 and 2018CFB573) and Fundamental Research Funds for the Central Universities, China (HUST: 2017KFYXJJ193). The writers would like to thank editors and reviewers for their valuable comments and suggestions.
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All authors contributed extensively to the work presented in this paper.