Put forward by Kennedy and Eberhart since 1995, particle swarm optimization (PSO) has been widely studied and used to solve complex problems [1, 2, 3]. The principle of the PSO algorithm is simple, and there are no requirements for the function form of optimization models, rendering this algorithm easy to implement. There remain, however, some limitations in the PSO algorithm. For one thing, premature convergence may occur in this algorithm, resulting in the local optimal solutions for complex multimodal problems, especially for those with high dimensions . For another, lack of constraint handling mechanism leads to poor applicability of this algorithm to constrained optimization models; thus, a reasonable constraint handling mechanism is necessary . To overcome the limitations mentioned above, researchers have proposed different methods.
For avoiding premature convergence, several approaches, such as adaptive parameters [6, 7, 8, 9, 10, 11], improvement of the particle update rule , particle mutation [13, 14], and the hybrid algorithm [15, 16], have been proposed. Reference , in which different strategies for inertia weight adjustment were compared, formulated a sigmoid-like inertia weight to balance the mondial and local searching, ensuring good features of the algorithm in both the test functions and the application in standard image segmentation. Reference  used the sigmoid-based acceleration coefficients to balance the global searching and local optimization. Both adaptive inertia weight and acceleration coefficients were expressed by nonlinear functions in , and the improved PSO algorithm was applied to the modified substitution production function komplet. In [9, 10, 11], the interval type-2 fuzzy logic was utilized to adjust the parameters so that the performance of the optimization algorithm can be improved. The modified algorithm has good property in optimization and result stability. A modified position update formula was proposed in  to avoid premature convergence. The mutation process based on the genetic algorithm was introduced in  to maintain the diversity of the particles. In , a chaotic-Cat-mapping-based mutation method, together with a cloud model, was introduced into the PSO. In , the hybrid algorithm of PSO, combined with a fuzzy logic method, was proposed. As premature convergence is the main komplikasi to be solved in the above literature, the constraints are not mentioned in most of these works. Therefore, unconstrained test functions are utilized in these works to verify the performance of the modified PSO algorithm.
For implementing constraint handling, the conventional Lagrangian algorithm and the gradient-based method can be utilized in optimization models. However, the constrained optimization models may lack an explicit mathematical formulation or have discrete definition domains. Hence, they cannot be solved via conventional methods . The heuristic algorithm based on the penalty function offers an effective avenue to handle the constraints. Reference  and  utilized the penalty function to transform constrained optimization problems into unconstrained ones by introducing the allowance tolerance factor. Because of the significant influence on the optimization results, a suitable penalty factor is hard to choose. The constraints and the objective function are divided into bi-objective criteria in some works [18, 19, 20], avoiding the influence of the penalty factor on the results. The total constraints violation was processed as an objective function in . In , the filter approach based on the concept of domination was proposed. A chaotic PSO algorithm with dual fitness was proposed in , and it was applied in the optimal operation of an energy storage system. Besides, a roulette wheel re-distribution mechanism for equality constraint handling was proposed by  in power dispatch. No matter what kind of constraints handling is adopted, the equality constraints are still hard to satisfy, affecting the feasible solution searching. Reference  proposed a parametric-equation-based equality constraint handling method, but the premature convergence komplikasi is not considered.
In this paper, a dual fitness value chaotic PSO algorithm with equality constraint handling (EDFC-PSO) is proposed to improve the performance of the PSO algorithm. In the proposed algorithm, both the limitations of premature convergence and constraint handling are considered. The equality constraints are solved by parametric equations, and the inequality constraints are handled by dual fitness evaluation criteria. The chaotic-Pencelup-mapping-based mutation is implemented to ensure the diversity of particles. The main contributions of this paper are as follows:
A modified PSO algorithm, aiming at improving the global searching ability and the application performance of the PSO, is proposed on the basis of parametric equations, the dual fitness evaluation, and the chaotic-Cat-mapping-based mutation. The equality constraints are satisfied by parametric equation solutions. The diversity of particles is ensured by the mutation process based on the chaotic Cat mapping.
Several mathematical test functions are utilized to illustrate the effectiveness of the proposed algorithm. An application in combined bidding of a power system is implemented to prove the application value of the algorithm.
The rest of this paper is organized as follows. More details of the related works are given in Section 2, and the differences between this work and others are clarified. In Section 3, the proposed EDFC-PSO algorithm is introduced. In Section 4, the numerical results of several test functions are given. In Section 5, the strategy of combined bidding is presented, and a case study of a benaran system in northeast China is carried out to verify the applicability of the EDFC-PSO. Conclusions are drawn in Section 6.
To clarify the differences between the related works and the modified PSO algorithm, we make comparisons, and the results are shown in Table 1.
A sigmoid-like function was utilized to obtain the adaptive inertia weight in , in which the inertia weight nonlinearly changed along with the increase in iteration times. Besides, the wavelet mutation was used for the diversity of particle swarm positions. In , the acceleration coefficients adaptively changed according to the sigmoid function
For a particle in the standard PSO, the vectors of the position and the velocity, shown in Eq. (1) and Eq. (2), are defined to describe the current searching state.
is the position vector of the s-th particle, and
is the velocity vector of the s-th particle;
are correspondingly the k-th components;
is the dimension of the particle, which is also the number of variables to be optimized.
In the iteration process,
Test functions and numerical results
Test functions  are utilized to test the algorithm performance in this section, which are shown in Eq. (17), Eq. (18), Eq. (19), Eq. (20), Eq. (21), Eq. (22), Eq. (23), Eq. (24), Eq. (25). The EDFC-PSO proposed in this paper is compared with the standard PSO, the dual fitness PSO (DF-PSO) , the dual fitness chaotic PSO based on the Logistic mapping (DFC-PSO) , and the equality constraint handling PSO based on the parametric equation method (E-PSO) . The numerical tests are
Application of EDFC-PSO algorithm in combined bidding of power system
To verify the applicability of the proposed EDFC-PSO algorithm in engineering fields, we present a combined bidding model in the day-ahead market for a wind farm and a retired electric vehicle battery storage system (RESS).
In this paper, a dual fitness value chaotic particle swarm optimization algorithm with equality constraint handling is proposed to solve the nonlinear optimization teladan with equality constraints. Tested by two simple unimodal functions and seven multimodal functions, the proposed EDFC-PSO algorithm performs better in the feasible region searching than the PSO, the DF-PSO, and the DFC-PSO. When the objective function is unimodal, the EDFC-PSO and the E-PSO have similar performance in both the
received the B.S. degree in electrical engineering from Dalian University of Technology, Dalian, China, in 2022. Currently, he is a doctoral candidate in the Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology. His research interests include optimal operation of power systems with renewable energy and electricity market.
received the B.S. degree and Ph.D. degree in electrical engineering from Dalian University of Technology,
CRediT authorship contribution statement
Conceptualization, Writing – original draft, Methodology.
Visualization, Writing – review & editing.
Data curation, Software, Investigation.
Funding acquisition, Resmi analysis.
Supervision, Project administration.
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.
This work is supported by the
National Natural Sciences Foundation of China
China Scholarship Council
(201906065025). The authors want to thank Mr. Oodo Stephen for his help in editing the language of this paper.
© 2022 Elsevier Ltd. All rights reserved.