Capacitor placement of distribution systems using particle swarm optimization approaches
Introduction
The problem of capacitors placement in distribution systems of electrical energy involves the determination of the number, location, type and size of the capacitors to be placed on the distribution feeders such that the total cost of installation and operation of the system is minimum with respect to the load on the system. This problem is combinatorial in nature and the application of discrete capacitors will be considered to solve it in this paper. That is, there are a total of (L + 1)J possible solutions, where J is the bus number and L is the number of available capacitor sizes. It is worth mentioning that (L + 1)J possibilities will become a large number to evaluate exhaustively even for a medium-size distribution system.
In this context, considerable efforts have been devoted to solve the capacitor placement problem. A variety of solution approaches based on mathematical programming techniques and gradient-based algorithms [1], [2], [3], [4], [5] have been developed to solve the capacitor placement problem over the years. In contrast to these analytical optimization techniques, meta-heuristics of natural computing field have been recently proposed in literature [6], [7], [8], [9], [10], [11].
The purpose of developing such meta-heuristics is to decrease the exhaustive search space, while providing as the final result (objective function) an optimal or near-optimal value. Moreover, classical gradient-based algorithms have disadvantages, such as (i) requirement of continuous and differentiable objective functions, (ii) difficulty in escaping local minima, and (iii) difficulty to handle discrete control variables.
To overcome these disadvantages, flexible meta-heuristics, such as particle swarm optimization (PSO) approaches, have been studied. The PSO algorithm is a stochastic algorithm. It does not need gradient information about the objective function, as the gradient-based algorithm does. PSO uses the analogy of swarming and collaboration principles and is one powerful method for solving unconstrained and constrained nonlinear global optimization of optimal capacitor placement.
The framework of PSO is simple. PSO is easy to be implemented within a personal computer and is inexpensive in terms of memory requirements and computational time. Moreover, the PSO technique can generate a high quality solution within shorter calculation time and stable convergence characteristics than other stochastic methods.
This paper presents an optimal capacitor placement method that applies PSO approaches. The PSO originally developed by Kennedy and Eberhart in 1995 [12], [13] is a population-based algorithm of the swarm intelligence. Similarly to genetic algorithms [14], an evolutionary algorithm approach, PSO is an optimization tool where each member is seen as a particle, and each particle is a potential solution to the problem under analysis. Each particle in PSO has a randomized velocity associated to it, which moves through the space of the problem. However, unlike genetic algorithms, PSO does not have operators such as crossover and mutation. PSO does not implement the survival of the fittest individuals; rather, it implements the simulation of social behavior. PSO approaches have been utilized in optimization problems in power systems area, such as power flow [15], loss power minimization [16], reactive power dispatch [17], economic dispatch [18], [19], [20], stabilizers tuning [21], expansion planning [22], unit commitment problem [23], reactive power and voltage control [24], and so on. A survey of the PSO applications in electrical power systems is presented in [25].
In PSO, a uniform probability distribution to generate random numbers is used. However, the use of other probability distributions may improve the ability to fine-tune or even to escape from local optima. In the meantime, the use of the Gaussian and Cauchy probability distributions has been proposed to generate random numbers to update the velocity equation [26], [27] inspired by studies of mutation operators in fast evolutionary programming [28], [29]. All these approaches attempted to improve the performance of the standard PSO, but the amount of parameters of the algorithm to tune remained the same.
This paper employs the Gaussian and the Cauchy probability distributions and also chaotic sequences in PSO approaches to search the size and location of capacitors to be installed on a radial distribution feeder. In other words, the main contribution of the paper can be summarized as using different distribution to generate random numbers required by the PSO. During each iterative procedure referred to as a generation, a new set of particles is produced using rules of evolution with improved performance. The fitness value for each particle of PSO is composed of the total power losses and the cost of capacitors added for that corresponding configuration. To demonstrate the effectiveness, the proposed approach is applied to two application example systems.
The remaining sections of this paper are organized as follows. Section ‘Problem description and formulations’ describes the problem and its formulations. Section ‘Particle swarm optimization’ then describes the Gaussian, Cauchy and chaotic sequences for PSO approaches adopted here, while Sections ‘Computational procedures and The application examples’ discuss the computational procedures and analyze the results applied to two examples, respectively. Lastly, Section ‘Conclusions and future research’ presents our conclusions and future research work.
Section snippets
Problem description and formulations
This work discusses the capacitor placement problem of distribution systems. The objective is to minimize the annual cost of the system, subject to operating constraints under a certain load pattern. Mathematically, the objective function of the problem can be described as min F = min (COST), where COST includes the cost of power loss and capacitor placement, and will be further discussed later. The voltage magnitude at each bus must be maintained within its limits expressed as follows
Fundamentals
The proposal of PSO algorithm was put forward by several scientists who developed computational simulations of the movement of organisms such as flocks of birds and fish schools. Such simulations were heavily based on manipulating the distances between individuals, i.e., the synchrony of the behavior of the swarm was seen as an effort to keep an optimal distance between them. Sociobiologist Edward Osbourne Wilson outlined a link of these simulations for optimization problems [30].
Each particle
Computational procedures
The computational procedures of the proposed method are mainly composed of power loss computation, bus voltage determination and PSO algorithmic calculations. The overall computational procedure finds configurations with the addition of different capacitors such that the objective function value is successively reduced.
The solution procedures start off with performing a feeder-line flow study to calculate the bus voltages, line losses, and annual cost. Then, to determine the location and sizes
The application examples
The proposed method has been programmed using MATLAB and run on an IBM compatible personal computer. To illustrate the application and demonstrate the effectiveness of the proposed method, two application examples are presented. The real solution values of PSO approaches are obtained by rounding to the nearest integer, in order to calculate the practical size.
For the optimization problems studied in this paper, the population size was set to 10 particles for PSO approaches. The number of
Conclusions and future research
A new capacitor placement method that employs PSO as well as feeder-line flow formulations to reduce power losses and enhance voltage profile for primary distribution systems is presented. The method seeks the most effective buses to install compensation capacitors so that a maximum annual cost saving is attained.
From the first application example, it is observed that the computational results obtained by the proposed method are exactly the same as that obtained using an exhaustive search
Acknowledgments
This study is supported by National Council of Scientific and Technologic Development of Brazil (CNPq) under Grants 479764/2013-1 and 307150/2012-7/PQ. Furthermore, the authors wish to thank the editor and anonymous referees for their constructive comments and recommendations, which have significantly improved the presentation of this paper.
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