Mathematical Problems in Engineering

Volume 2016, Article ID 6519678, 19 pages

http://dx.doi.org/10.1155/2016/6519678

## A Chaos-Enhanced Particle Swarm Optimization with Adaptive Parameters and Its Application in Maximum Power Point Tracking

^{1}Department of Electrical Engineering, Chung Yuan Christian University, Chung Li District, Taoyuan City 320, Taiwan^{2}Center for Research & Development and Department of Electronics Engineering, Adamson University, 1000 Manila, Philippines^{3}School of Graduate Studies, Mapua Institute of Technology, 1002 Manila, Philippines^{4}School of Electrical Electronics Computer Engineering, Mapua Institute of Technology, 1002 Manila, Philippines

Received 11 April 2016; Accepted 5 July 2016

Academic Editor: Zhen-Lai Han

Copyright © 2016 Ying-Yi Hong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This work proposes an enhanced particle swarm optimization scheme that improves upon the performance of the standard particle swarm optimization algorithm. The proposed algorithm is based on chaos search to solve the problems of stagnation, which is the problem of being trapped in a local optimum and with the risk of premature convergence. Type constriction is incorporated to help strengthen the stability and quality of convergence, and adaptive learning coefficients are utilized to intensify the exploitation and exploration search characteristics of the algorithm. Several well known benchmark functions are operated to verify the effectiveness of the proposed method. The test performance of the proposed method is compared with those of other popular population-based algorithms in the literature. Simulation results clearly demonstrate that the proposed method exhibits faster convergence, escapes local minima, and avoids premature convergence and stagnation in a high-dimensional problem space. The validity of the proposed PSO algorithm is demonstrated using a fuzzy logic-based maximum power point tracking control model for a standalone solar photovoltaic system.

#### 1. Introduction

Swarm intelligence is becoming one of the hottest areas of research in the field of computational intelligence especially with regard to self-organizing and decentralized systems. Swarm intelligence simulates the behavior of human and animal populations. Several swarm intelligence optimization algorithms can be found in the literature, such as ant colony optimization, artificial bee colony optimization, the firefly algorithm, differential evolution, and others. These are biologically inspired optimization and computational techniques that are based on the social behaviors of fish, birds, and humans. Particle swarm optimization (PSO) is a nature-inspired algorithm that draws on the behavior of flocking birds, social interactions among humans, and the schooling of fish. In fish schooling, bird flocking, and human social interactions, the population is called a swarm and candidate solutions, corresponding to the individuals or members in the swarm, are called particles. Birds and fishes generally travel in a group without collision. Accordingly, using the group information for finding the shelter and food, each particle adjusts its corresponding position and velocity, representing a candidate solution. The position of a particle is influenced by neighbors and the best found solution by any particle. PSO is a population-based search technique that involves stochastic evolutionary optimization. It is originally developed in 1995 by Eberhart and Kennedy [1, 2] to optimize constrained and unconstrained, continuous nonlinear, and nondifferentiable multimodal functions [1, 3]. PSO is a metaheuristic algorithm that was inspired by the collaborative or swarming behavior of biological populations [4]. Since then, it has been applied to solve a wide range of optimization problems, such as constrained and unconstrained problems, multiobjective problems, problems with multiple solutions, and optimization in dynamic environments [5–8]. Some of the advantages of particle swarm optimization are the following: (a) computational efficiency [6], (b) effective convergence and parameter selection [7], (c) simplicity, flexibility, robustness, and ease of implementations [9], (d) ability to hybridize with other algorithms [10], and many others. PSO has few parameters to adjust and a small memory requirement and uses few CPU resources, making it computationally efficient. Unlike simulated annealing which can work only with discrete variables, PSO can work for both discrete and analog variables without ADC or DAC conversion. Also, genetic algorithm optimization requires crossover, selection, and mutation operators, whereas PSO utilizes only the exchange of information among individuals searching the problem space repeatedly [11]. In recent years, the use of particle swarm optimization has been investigated with focus on its use to solve a wide range of scientific and engineering problems such as fault detection [12], parameter identification [13, 14], power systems [15–17], transportation [18], electronic circuit design [19], and plant control design [20]. Most relevant research focuses on either constrained or unconstrained optimization problems.

The particle swarm optimization was developed to optimally search for the local best and the global best; these searches are frequently known as the exploitation and exploration of the problem space, respectively. Hong et al. [21] stated that exploitation involves an intense search of particles in a local region while exploration is a long term search, whose main objective is to find the global optimum of the fitness function. Although particle swarm optimization rapidly searches the solution of many complex optimization problems, it suffers from premature convergence, trapping at a local minimum, the slowing down of convergence near the global optimum, and stagnation in a particular region of the problem space especially in a multimodal functions and high-dimensional problem space. If a particle is located at the position of the global best and the preceding velocity and weight inertia are non-zero, then the particle is moving away from that particular point [16, 22]. Premature convergence happens if no particle moves and the previous velocities are near to zero. Stagnation thus occurs if the majority of particles are concentrated at the best position that is disclosed by the neighbors or the swarm. This fact has in recent years motivated various investigations by several researchers on variants of the particle swarm optimization, in an attempt to improve the performance of exploitation and exploration and to eliminate the aforementioned problems. The various methods of particle swarm optimization have been used for several purposes, including scheduling, classification, feature selection, and optimization.

Mendes et al. [23] presented fully informed particle swarm optimization, in which, during the optimization search, particles are influenced by the best particles in their neighborhood and information is evenly distributed during the generations of the algorithm. Liang et al. [24] proposed a comprehensive learning PSO in which each particle learns from the other neighborhood personal best at different dimensions. Accordingly, particles update their velocity based on the history of the their own personal bests. Wang et al. [25, 26] developed the opposition-based PSO with Cauchy mutation. Their technique uses an opposition learning scheme in which the Cauchy mutation operator helps the particles move to the best positions. Pant et al. [27] modified the inertia weight to exhibit a Gaussian distribution. Xiang et al. [28] applied the time delay concept PSO to enable the processing of information by particles to find the global best. Cui et al. [29] presented the fitness uniform selection strategy (FUSS) and the random walk strategy (RWS) to enhance the exploitation and exploration capabilities of PSO. Montes de Oca et al. [30] developed Frankenstein’s PSO, which incorporates several variants of PSO in the literature such as constriction [31], the time-varying inertia weight optimizer [32, 33], the fully informed particle swarm optimizer [23], and the adaptive hierarchical PSO [34]. The adaptive PSO that was proposed by Zhan et al. [35] utilized the information that was obtained from the population distribution and fitness of particles to determine the status of the swarm and an elitist learning strategy to search for the global optimum. Juang et al. [36] presented the use of fuzzy set theory to tune automatically the acceleration coefficients in the standard PSO. A quadratic interpolation and crossover operator is also incorporated to improve the global search ability. The literature includes hybridization of particle swarm optimization with other stochastic or evolutionary techniques [10, 37–39] to realize all of their strengths.

Every modification of the particle swarm optimization uses a different method to solve optimization problems. The investigations cited above therefore elucidate some improvements of the standard particle swarm optimization. However, variants of particle swarm optimization generally exhibit the following limitations. (a) The particles may be positioned in a region that has a lower quality index than previously, leading to a risk of premature convergence, trapping in local optima, and the impossibility of further improvement of the best positions of the particles because the inertia weight, cognitive factors, and social learning factors in the algorithm are not adaptive or self-organizing. (b) The inclusion of the mutation operator may improve the speed of convergence. Nevertheless, global convergence is not guaranteed because the method is likely to become trapped in the local optimum during local searches of several functions. (c) The probability in the algorithm may improve the updated positions of particles. However, the changes in the new positions of particles, consistent with the probabilistic calculations, can move the particles into the worst positions. (d) Improving information sharing and the particle learning process capability of the algorithm can provide several benefits, but doing so often increases CPU times for computing the global optimum. (e) Integrating particle swarm optimization with other evolutionary or stochastic algorithms may increase the number of required generations, the complexity of the algorithm, and the number of required calculations.

This paper proposes a novel particle swarm optimization framework. The primary merits of the proposed variant of particle swarm optimization are as follows. (a) A modified sine chaos inertia weight operator is introduced, overcoming the drawback of trapping in a local minimum which is commonly associated with an inertia weight operator. Chaos search improves the best positions of the particles, favors rapid finding of solutions in the problem space, and avoids the risk of premature convergence. (b) Type 1′′ constriction coefficient [40] is incorporated to increase the convergence rate and stability of the particle swarm optimization. (c) Self-organizing, adaptive cognitive, and social learning coefficients [41] are integrated to improve the exploitation and exploration search of the particle swarm optimization algorithm. (d) The proposed optimization algorithm has simple structure reducing the required memory demands and the computational burden on the CPU. It can therefore easily be realized using a few, low-cost test modules.

The remainder of this paper is organized as follows. Section 2 presents the standard particle swarm optimization algorithm. Section 3 describes the proposed variant of particle swarm optimization algorithm. Section 4 discusses the performance of the proposed variant of the particle swarm optimization and compares results obtained when well known optimization methods are applied to benchmark functions. The proposed variant of particle swarm optimization is further utilized in maximum power point tracking control using fuzzy logic for a standalone photovoltaic system. Finally, a brief conclusion is drawn.

#### 2. Standard Particle Swarm Optimization

The particle swarm optimization is a simulating algorithm, evolutionary, and a population-based stochastic optimization method that originates in animal behaviors such as the schooling of fish and the flocking of bird, as well as human behaviors. It has best position memory of all optimization methods and a few adjustable parameters and is easy to implement. The standard PSO does not use the gradient of an objective function and mutation [11]. Each particle randomly moves throughout the problem space, updating its position and velocity with the best values. Each particle represents a candidate solution to the problem and searches for the local or global optimum. Every particle retains a memory of the best position achieved so far, and it travels through the problem space adaptively. The personal best () is the best solution so far achieved by an individual in the swarm within the problem space while the global best () refers to globally obtained best solution by any particle within the swarm in the problem space dimension . The position and velocity of particle in the problem space dimension are thus given by and , respectively. The velocity and position of a particle are adjusted as follows [1, 2]:where the superscript is the generation index, whereas and are cognitive and social parameters, which are frequently known as acceleration constants and which are mainly responsible for attracting the particles toward and . The terms , , and denote uniform random numbers and inertia weight , respectively. These factors are mainly responsible for balancing the local and global optima search capabilities of the particles in problem space. Every generation, the velocity of individuals in the swarm is computed and which adjusted velocity is used to compute the next position of the particle. To determine whether the best solution is achieved and to evaluate the performance of each particle, the fitness function is included. The best position of each particle is relayed to all particles in the neighborhood. The velocity and the position of each particle are repeatedly adjusted until the halting criteria are satisfied or convergence is obtained.

#### 3. Chaos-Enhanced Particle Swarm Optimization with Adaptive Parameters

This section demonstrates that the proposed variant of particle swarm optimization improves upon the performance of the standard particle swarm optimization consistent with (1). The novel scheme improves upon the performance of other population-based algorithms in solving high-dimensional or multimodal problems. Chaos operates in a nonlinear fashion and is associated with complex behavior, unpredictability, determinism, and high sensitivity to initial conditions. In chaos, a small perturbation in the initial conditions can produce dramatically different results [42, 43]. In 1963, Lorenz [44] presented an autonomous nonlinear differential equation that generated the first chaotic system. In recent years, the scientific community has paid increasing attention to the chaotic systems and their applications in various areas of science and engineering. Such systems have been investigated in such fields as parameter identifications [14], optimizations [45], electronic circuits [46], electric motor drives [47, 48], power electronics [49], communications [50], robotics [51], and many others.

Feng et al. [52] introduced two means of modifying the inertia weight of a PSO using chaos. The first type is the chaotic decreasing inertia weight and the second type is the chaotic random inertia weight. In this paper, the latter is considered intensifying the inertia weight parameter of the PSO. The dynamic chaos random inertia weight is used to ensure a balance between exploitation and exploration. A low inertia weight favors exploitation while a high inertia weight favors exploration. A static inertia weight influences the convergence rate of the algorithm and often leads to premature convergence. Chaotic search optimization in all instances was used herein because of its highly dynamic property, which ensures the diversity of the particles and escape from local optimum in the process of searching for the global optimum.

The logistic map [53, 54], where is a very common chaotic map, which is found in much of the literature on chaotic inertia weight; it does not guarantee chaos on initial values of that may arise during the initial generation process. In this paper, the sine chaotic map [54] given by (2) was utilized to avoid this shortcoming. Its simplicity eliminates complex calculations, reducing the CPU time: where , and is the generation number. Figure 1 presents the bifurcation diagram of the sine chaotic map. In some instances of generations, has relatively very small values. Hence, to improve the effectiveness of the chaos random inertia weight of particle swarm optimization, the original sine chaotic map is lightly modified as follows: where and ; the absolute sign ensures that the next-generation process in chaos space has . Therefore, the chaotic random inertia weight is given by