Complexity

Volume 2018, Article ID 7289674, 15 pages

https://doi.org/10.1155/2018/7289674

## An Improved Particle Swarm Optimization with Biogeography-Based Learning Strategy for Economic Dispatch Problems

^{1}School of Electrical and Information Engineering, Jiangsu University, Zhenjiang, 212013 Jiangsu, China^{2}Key Laboratory of Advanced Control and Optimization for Chemical Processes, Ministry of Education, East China University of Science and Technology, Shanghai 200237, China^{3}School of Mechanical Engineering, Shanghai University of Engineering Science, Shanghai 201620, China

Correspondence should be addressed to Xu Chen; nc.ude.sju@nehcux

Received 3 April 2018; Revised 2 May 2018; Accepted 14 May 2018; Published 12 July 2018

Academic Editor: Zhile Yang

Copyright © 2018 Xu Chen 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

Economic dispatch (ED) plays an important role in power system operation, since it can decrease the operating cost, save energy resources, and reduce environmental load. This paper presents an improved particle swarm optimization called biogeography-based learning particle swarm optimization (BLPSO) for solving the ED problems involving different equality and inequality constraints, such as power balance, prohibited operating zones, and ramp-rate limits. In the proposed BLPSO, a biogeography-based learning strategy is employed in which particles learn from each other based on the quality of their personal best positions, and thus it can provide a more efficient balance between exploration and exploitation. The proposed BLPSO is applied to solve five ED problems and compared with other optimization techniques in the literature. Experimental results demonstrate that the BLPSO is a promising approach for solving the ED problems.

#### 1. Introduction

Economic dispatch (ED) is an important optimization task in power system operation and planning. The main objective of ED problems is to allocate generation among the committed generating units so as to meet the required load demand at minimum operating cost, with various physical constraints [1]. The cost of power generation is high, and economic dispatch can help in saving a significant amount of revenue [2].

In the original ED problem, the cost function for each generation unit is approximately represented by a single quadratic function, and traditional approaches based on mathematical programming techniques have been utilized to solve the ED problem, including the lambda-iteration method, gradient method, Newton’s method, linear programming, interior point method, and dynamic programming [3–5]. Usually, these methods are highly sensitive to starting points and rely on the assumption that the cost function needs to be continuous and convex. However, the practical ED problems exhibit nonconvex and nonsmooth characteristics because of valve-point effects, ramp-rate limit, multifuel cost, prohibited operating zones, and so on [6]. The traditional methods are not capable of efficiently solving the ED problems with these characteristics.

In the past decades, more and more researchers are turning to metaheuristic search (MS) algorithms for solving the ED problems. These methods have the ability to identify higher-quality solutions and can be grouped into three categories, as original, improved, and hybrid MS algorithms.

The first category consists of methods applied in their original version, such as genetic algorithm (GA) [7], particle swarm optimization (PSO) [8], differential evolution (DE) [9], ant colony optimization (ACO) [10], harmony search (HS) [11], artificial bee colony (ABC) [12], teaching-learning-based optimization (TLBO) [13], gravitational search algorithm (GSA) [14], firefly algorithm (FA) [15], biogeography-based optimization (BBO) [16, 17], bacterial foraging optimization (BFO) [18], imperialist competitive algorithm (ICA) [19], seeker optimization algorithm (SOA) [20], grey wolf optimization (GWO) [21], backtracking search algorithm (BSA) [22], and root tree optimization (RTO) [23].

The second refers to improved or modified methods derived from the original version, and the following are included: self-adaptive real-coded genetic algorithm (SARGA) [24], random drift PSO (RDPSO) [25, 26], fuzzy adaptive modified PSO (FAMPSO) [27], improved differential evolution (IDE) [28], shuffled differential evolution (SDE) [29], improved harmony search (IHS) [30], modified artificial bee colony (MABC) [31], incremental artificial bee colony (IABC) [32], ramp-rate biogeography-based optimization (RRBBO) [33], dynamic nondominated sorting biogeography-based optimization (Dy-NSBBO) [34], multistrategy ensemble biogeography-based optimization (MsEBBO) [35], and modified group search optimizer (MGSO) [36].

The third is the hybrid method in which two or more optimization techniques are combined, including hybrid genetic algorithm (HGA) [37], chaotic PSO with sequential quadratic programming (CPSO-SQP) [38], hybrid PSO and gravitational search algorithm (HPSO-GSA) [39], hybrid differential evolution algorithm based on PSO (DEPSO) [40], hybrid differential evolution with biogeography-based optimization (DE/BBO) [41], hybrid chemical reaction optimization with differential evolution (HCRO-DE) [42], and hybrid imperialist competitive-sequential quadratic programming (HIC-SQP) [43].

In this paper, an improved PSO algorithm with biogeography-based learning strategy is proposed to solve the ED problems. The main contributions of this paper are listed as follows: (1)A biogeography-based learning particle swarm optimization (BLPSO) algorithm which employs a biogeography-based learning strategy (BLS) is presented. The computational complexity of BLPSO is also analyzed.(2)By combining the feature of ED problems, the BLPSO-based economic dispatch method is developed.(3)BLPSO is applied to solve five ED problems with various practical constraints, and the experimental results demonstrate that the proposed method can obtain promising results for ED problems.

This paper is organized as follows: Section 2 briefly introduces the formulation of ED problems. Section 3 introduces the original PSO and its three variants. In addition, a biogeography-based learning particle swarm optimization algorithm is presented in this section. Section 4 addresses the implementation of BLPSO for solving ED problems. Section 5 provides the experimental results on five test systems. Finally, the paper is concluded in Section 6.

#### 2. Formulation of ED Problems

The objective of the ED problem is to minimize the fuel cost of thermal power plants for a given load demand subject to various physical constraints.

##### 2.1. Objective Function

The traditional fuel cost or objective function of the ED problem is the quadratic fuel cost equation of the thermal generating units and is given by where is the total number of generating units or generators, is the cost function of the th generating unit ($/hr), is the real output of the th generating units (in MW), and , , and are fuel cost coefficients of the th generator.

In some ED problems, the admission valves control the steam entering the turbine through separate nozzle groups. When the valve opens, the fuel cost will increase dramatically because of the wire drawing effect, and this makes the practical objective function have many nondifferentiable points [44]. Therefore, the fuel cost function often contains many nonsmooth ripple curves due to the presence of valve-point effects. The objective function when the valve-point effect is taken into account is represented as where and are nonsmooth fuel cost coefficients of the th generator with valve-point effects and is the minimum power generation limit of the th generator (in MW).

##### 2.2. Optimization Constraints

###### 2.2.1. Power Balance Constraint

The total generated power should be equal to the sum of the total system demand () and the total transmission network loss ():

The coefficient method is widely utilized to calculate the total transmission network loss . In such a way, can be calculated as follows: where , , and are the loss coefficients or coefficients. It can be seen that is an matrix.

###### 2.2.2. Power Generation Limits

The power generation of each generator should be within its minimum and maximum limits: where and are the minimum and maximum power generation limits of the th generator.

###### 2.2.3. Ramp-Rate Limits

The physical limitations of starting up and shutting down of generators impose ramp-rate limits, which are modeled as follows. The increase in generation is limited by

Similarly, the decrease is limited by where is the previous output power and and are the up-ramp limit and the down-ramp limit of the th generator, respectively.

Combining 6 and 7 with 5 results in the change of the effective operating or generation limits to

###### 2.2.4. Prohibited Operating Zones

The prohibited operating zones (POZ) are due to steam valve operation or vibration in shaft bearing. The feasible operating zones of the th generator can be described as follows: where is the number of prohibited zones of the th generator. and are the lower and upper power output of the th prohibited zone of the th generator, respectively.

Combining the equations from 2 to 9, the ED problem can be formulated as

#### 3. Particle Swarm Optimization and Its Three Variants

##### 3.1. Particle Swarm Optimization

The PSO algorithm is a population-based metaheuristic algorithm which was firstly proposed by Eberhart and Kennedy [45]. It is based on the swarm intelligence theory, and the fundamental idea is that the optimal solution can be found through cooperation and information sharing among individuals in the swarm. In the past decade, PSO has gained increasing popularity due to its effectiveness in performing difficult optimization tasks.

In PSO, each individual is treated as a particle in the -dimensional space, with a position vector and a velocity vector . The particle updates its velocity and position according to the following equations: where is the personal best position of particle ; is the position of the best particle in the population; is the inertia weight; and are acceleration coefficients; and and are two random real numbers distributed uniformly within [0,1].

##### 3.2. Comprehensive Learning Particle Swarm Optimization

Liang et al. [46] proposed a comprehensive learning PSO (CLPSO) which uses a novel comprehensive learning strategy whereby all other particle personal best positions are used to update a particle velocity. This strategy can preserve the diversity of the swarm to discourage premature convergence. CLPSO uses the following velocity updating equation: where is the learning exemplar for particle and is the learning exemplar indices for particle , which is generated based on tournament selection procedure. The CLPSO does not introduce any complex operations to the original simple PSO framework, and the main difference from the original PSO is the velocity update equation.

##### 3.3. Social Leaning Particle Swarm Optimization

Cheng and Jin [47] proposed a social learning PSO (SLPSO) inspired by learning mechanisms in social learning of animals. The SLPSO is performed on a sorted swarm, and particles learn from any better particles in the current swarm. The particles learn from different particles based on the following equations: where is the learning probability for particle , is a randomly generated probability that satisfies , is the demonstrator of particle in the th dimension, is the mean position of the all particles in the current swarm, and is the social influence factor. In addition, the SLPSO adopts dimension-dependent parameter control methods to determine the three parameters, that is, the swarm size , the learning probability, and the social influence factor .

##### 3.4. Biogeography-Based Learning Particle Swarm Optimization

In this paper, a biogeography-based learning particle swarm optimization (BLPSO) which employs a new biogeography-based learning strategy (BLS) [48] is proposed for the ED problems.

###### 3.4.1. Biogeography-Based Learning Strategy

BLS is inspired from both from the comprehensive learning strategy of CLPSO [46] and biogeography-based optimization [49, 50]. It has two characteristics: (1)Each particle updates itself by using the combination of its own personal best position and personal best positions of all other particles, which is similar to the comprehensive learning strategy of CLPSO. This updating method enables the diversity of the swarm to be preserved to discourage premature convergence.(2)The migration operator of biogeography-based optimization is used to generate the learning exemplar for each particle, in which a ranking technique is employed to make particles learn more from particles with high-quality personal best positions. This can provide a more effecient balance between exploration and exploitation for the new PSO algorithm.

In BLS, each particle updates its velocity and position according to the following equations: where is the learning exemplar for particle ; is the learning exemplar indices for particle , which is generated by the biogeographic migration.

In the biogeographic migration, all particles are firstly sorted based on the value of their **pbest** from best to worst and assigned with ranking values. For a minimization problem, assume
where is the subscript of the particle with the best **pbest**, is the subscript of the particle with the second best **pbest**, and is the subscript of the particle with the worst **pbest**; is the population size. Then, the rankings of particles are assigned as below:

Second, immigration and emigration rates are assigned for all particles. The immigration and emigration rates for all particles can be calculated as follows:

According to 20, the solution with the best will have the lowest immigration rate and highest emigration rate; and the solution with the worst will have the highest immigration rate and lowest emigration rate.

Third, the biogeography-based exemplar indices for particle can be generated based on the biogeography-based exemplar generation method, see Algorithm 1.