Aplication Example With Pso Algorithm Book Tutorial Pdf
Graphical abstract
Introduction
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 [4]. 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 [5]. 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 [12], particle mutation [13, 14], and the hybrid algorithm [15, 16], have been proposed. Reference [6], in which different strategies for inertia weight adjustment were compared, formulated a sigmoidlike 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 [7] used the sigmoidbased acceleration coefficients to balance the global searching and local optimization. Both adaptive inertia weight and acceleration coefficients were expressed by nonlinear functions in [8], and the improved PSO algorithm was applied to the modified substitution production function komplet. In [9, 10, 11], the interval type2 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 [12] to avoid premature convergence. The mutation process based on the genetic algorithm was introduced in [13] to maintain the diversity of the particles. In [14], a chaoticCatmappingbased mutation method, together with a cloud model, was introduced into the PSO. In [15], 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 gradientbased 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 [16]. The heuristic algorithm based on the penalty function offers an effective avenue to handle the constraints. Reference [4] and [17] 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 biobjective 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 [18]. In [19], the filter approach based on the concept of domination was proposed. A chaotic PSO algorithm with dual fitness was proposed in [20], and it was applied in the optimal operation of an energy storage system. Besides, a roulette wheel redistribution mechanism for equality constraint handling was proposed by [5] 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 [21] proposed a parametricequationbased 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 (EDFCPSO) 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 chaoticPencelupmappingbased mutation is implemented to ensure the diversity of particles. The main contributions of this paper are as follows:
 1)

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 chaoticCatmappingbased 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.
 2)

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 EDFCPSO 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 EDFCPSO. Conclusions are drawn in Section 6.
Section snippets
Related works
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 sigmoidlike function was utilized to obtain the adaptive inertia weight in [6], 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 [7], the acceleration coefficients adaptively changed according to the sigmoid function
Duaja PSO
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.
where
X
_{
s
}
is the position vector of the sth particle, and
V
_{
s
}
is the velocity vector of the sth particle;
x
_{
s,k
}
and
v
_{
s,k
}
are correspondingly the kth components;
N_{x}
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 [23] 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 EDFCPSO proposed in this paper is compared with the standard PSO, the dual fitness PSO (DFPSO) [22], the dual fitness chaotic PSO based on the Logistic mapping (DFCPSO) [20], and the equality constraint handling PSO based on the parametric equation method (EPSO) [21]. The numerical tests are
Application of EDFCPSO algorithm in combined bidding of power system
To verify the applicability of the proposed EDFCPSO algorithm in engineering fields, we present a combined bidding model in the dayahead market for a wind farm and a retired electric vehicle battery storage system (RESS).
Conclusion
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 EDFCPSO algorithm performs better in the feasible region searching than the PSO, the DFPSO, and the DFCPSO. When the objective function is unimodal, the EDFCPSO and the EPSO have similar performance in both the
Author bios
Feixiang PENG
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.
Shubo HU
received the B.S. degree and Ph.D. degree in electrical engineering from Dalian University of Technology,
CRediT authorship contribution statement
Feixiang Peng:
Conceptualization, Writing – original draft, Methodology.
Shubo Hu:
Visualization, Writing – review & editing.
Zhengnan Gao:
Data curation, Software, Investigation.
Wei Zhou:
Funding acquisition, Resmi analysis.
Hui Sun:
Supervision, Project administration.
Peng Yu:
Resources, Validation.
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 work is supported by the
National Natural Sciences Foundation of China
(61873048) and
China Scholarship Council
(201906065025). The authors want to thank Mr. Oodo Stephen for his help in editing the language of this paper.
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Source: https://www.sciencedirect.com/science/article/abs/pii/S0045790621003724