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

Authors: Xu Chen; Bin Xu; Wenli Du
Journal: Complexity
 (2018)

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## 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.


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## Body

## 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: Section2 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(1)minF=∑j=1NgFjPj=∑j=1Ngaj+bjPj+cjPj2,where Ng is the total number of generating units or generators, FjPj is the cost function of the jth generating unit ($/hr), Pj is the real output of the jth generating units (in MW), and aj, bj, and cj are fuel cost coefficients of the jth 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
(2)minF=∑j=1NgFjPj=∑j=1Ngaj+bjPj+cjPj2+ejsinfjPjmin−Pj,where ej and fj are nonsmooth fuel cost coefficients of the jth generator with valve-point effects and Pjmin is the minimum power generation limit of the jth 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 (PD) and the total transmission network loss (PL):
(3)∑j=1NgPj=PD+PL.TheB coefficient method is widely utilized to calculate the total transmission network loss PL. In such a way, PL can be calculated as follows:
(4)PL=∑j=1Ng∑i=1NgPjBjiPi+∑j=1NgB0jPi+B00,where Bji, Bj0, and B00 are the loss coefficients or B coefficients. It can be seen that Bji is an Ng×Ng matrix.


#### 2.2.2. Power Generation Limits

The power generation of each generator should be within its minimum and maximum limits:(5)Pjmin≤Pj≤Pjmax,where Pjmin and Pjmax are the minimum and maximum power generation limits of the jth 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(6)Pj−Pj0≤URj.Similarly, the decrease is limited by(7)Pj0−Pj≤DRj,where Pj0 is the previous output power and URj and DRj are the up-ramp limit and the down-ramp limit of the jth generator, respectively.Combining6 and 7 with 5 results in the change of the effective operating or generation limits to
(8)maxPjmin,Pj0−DRj≤Pj≤minPjmax,Pj0+URj.


#### 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 thejth generator can be described as follows:
(9)Pj∈Pjmin≤Pj≤Pj,1l,Pj,k−1u≤Pj≤Pj,kl,Pj,nj−1u≤Pj≤Pjmax,k=2,3,…,nj,j=1,2,…,Ng,where nj is the number of prohibited zones of the jth generator. Pj,kl and Pj,ku are the lower and upper power output of the kth prohibited zone of the jth generator, respectively.Combining the equations from 2 to 9, the ED problem can be formulated as(10)minF=∑j=1NgFjPj=∑j=1Ngaj+bjPj+cjPj2+ejsinfjPjmin−Pj,s.t.∑j=1NgPj=PD+PL,maxPjmin,Pj0−DRj≤Pj≤Pj,1l,Pj,k−1u≤Pj≤Pj,kl,Pj,k−1u≤Pj≤minPjmax,Pj0+URj,k=2,3,…,nj,j=1,2,…,Ng.


## 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(1)minF=∑j=1NgFjPj=∑j=1Ngaj+bjPj+cjPj2,where Ng is the total number of generating units or generators, FjPj is the cost function of the jth generating unit ($/hr), Pj is the real output of the jth generating units (in MW), and aj, bj, and cj are fuel cost coefficients of the jth 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
(2)minF=∑j=1NgFjPj=∑j=1Ngaj+bjPj+cjPj2+ejsinfjPjmin−Pj,where ej and fj are nonsmooth fuel cost coefficients of the jth generator with valve-point effects and Pjmin is the minimum power generation limit of the jth 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 (PD) and the total transmission network loss (PL):
(3)∑j=1NgPj=PD+PL.TheB coefficient method is widely utilized to calculate the total transmission network loss PL. In such a way, PL can be calculated as follows:
(4)PL=∑j=1Ng∑i=1NgPjBjiPi+∑j=1NgB0jPi+B00,where Bji, Bj0, and B00 are the loss coefficients or B coefficients. It can be seen that Bji is an Ng×Ng matrix.


### 2.2.2. Power Generation Limits

The power generation of each generator should be within its minimum and maximum limits:(5)Pjmin≤Pj≤Pjmax,where Pjmin and Pjmax are the minimum and maximum power generation limits of the jth 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(6)Pj−Pj0≤URj.Similarly, the decrease is limited by(7)Pj0−Pj≤DRj,where Pj0 is the previous output power and URj and DRj are the up-ramp limit and the down-ramp limit of the jth generator, respectively.Combining6 and 7 with 5 results in the change of the effective operating or generation limits to
(8)maxPjmin,Pj0−DRj≤Pj≤minPjmax,Pj0+URj.


### 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 thejth generator can be described as follows:
(9)Pj∈Pjmin≤Pj≤Pj,1l,Pj,k−1u≤Pj≤Pj,kl,Pj,nj−1u≤Pj≤Pjmax,k=2,3,…,nj,j=1,2,…,Ng,where nj is the number of prohibited zones of the jth generator. Pj,kl and Pj,ku are the lower and upper power output of the kth prohibited zone of the jth generator, respectively.Combining the equations from 2 to 9, the ED problem can be formulated as(10)minF=∑j=1NgFjPj=∑j=1Ngaj+bjPj+cjPj2+ejsinfjPjmin−Pj,s.t.∑j=1NgPj=PD+PL,maxPjmin,Pj0−DRj≤Pj≤Pj,1l,Pj,k−1u≤Pj≤Pj,kl,Pj,k−1u≤Pj≤minPjmax,Pj0+URj,k=2,3,…,nj,j=1,2,…,Ng.


## 2.2.1. Power Balance Constraint

The total generated power should be equal to the sum of the total system demand (PD) and the total transmission network loss (PL):
(3)∑j=1NgPj=PD+PL.TheB coefficient method is widely utilized to calculate the total transmission network loss PL. In such a way, PL can be calculated as follows:
(4)PL=∑j=1Ng∑i=1NgPjBjiPi+∑j=1NgB0jPi+B00,where Bji, Bj0, and B00 are the loss coefficients or B coefficients. It can be seen that Bji is an Ng×Ng matrix.


## 2.2.2. Power Generation Limits

The power generation of each generator should be within its minimum and maximum limits:(5)Pjmin≤Pj≤Pjmax,where Pjmin and Pjmax are the minimum and maximum power generation limits of the jth 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(6)Pj−Pj0≤URj.Similarly, the decrease is limited by(7)Pj0−Pj≤DRj,where Pj0 is the previous output power and URj and DRj are the up-ramp limit and the down-ramp limit of the jth generator, respectively.Combining6 and 7 with 5 results in the change of the effective operating or generation limits to
(8)maxPjmin,Pj0−DRj≤Pj≤minPjmax,Pj0+URj.


## 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 thejth generator can be described as follows:
(9)Pj∈Pjmin≤Pj≤Pj,1l,Pj,k−1u≤Pj≤Pj,kl,Pj,nj−1u≤Pj≤Pjmax,k=2,3,…,nj,j=1,2,…,Ng,where nj is the number of prohibited zones of the jth generator. Pj,kl and Pj,ku are the lower and upper power output of the kth prohibited zone of the jth generator, respectively.Combining the equations from 2 to 9, the ED problem can be formulated as(10)minF=∑j=1NgFjPj=∑j=1Ngaj+bjPj+cjPj2+ejsinfjPjmin−Pj,s.t.∑j=1NgPj=PD+PL,maxPjmin,Pj0−DRj≤Pj≤Pj,1l,Pj,k−1u≤Pj≤Pj,kl,Pj,k−1u≤Pj≤minPjmax,Pj0+URj,k=2,3,…,nj,j=1,2,…,Ng.


## 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 theD-dimensional space, with a position vector xit=xi1t,xi2t,…,xiDt and a velocity vector vit=vi1t,vi2t,…,viDt. The particle updates its velocity and position according to the following equations:
(11)vijt+1=wvijt+c1r1pbestijt−xijt+c2r2gbestjt−xijt,(12)xijt+1=xijt+vijt,where pbestit=pbesti1t,pbesti2t,…,pbestiDt is the personal best position of particle i; gbestt=gbest1t,gbest2t,…,gbestDt is the position of the best particle in the population; w is the inertia weight; c1 and c2 are acceleration coefficients; and r1 and r2 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:
(13)vijt+1=wvijt+cr1pbestτij,jt−xijt,where pbestτit=pbestτi1,1t,pbestτi2,2t,…,pbestτiD,Dt is the learning exemplar for particle i and τi=τi1,τi2,…,τiD is the learning exemplar indices for particle i, 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:
(14)xijt+1=xijt+Δxijtifpit≤PiL,xijt,otherwise,(15)Δxijt+1=r1Δxijt+r2xkjt−xijt+r3εx¯jt−xijt,where PiL is the learning probability for particle i, pit is a randomly generated probability that satisfies 0≤pit≤PiL≤1, xkjt is the demonstrator of particle i in the jth dimension, x¯jt=∑i=1Nxij/N is the mean position of the all particles in the current swarm, and ε 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 N, PiL the learning probability, and the social influence factor ε.


### 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:(16)vijt+1=wvijt+cr1pbestτij,jt−xijt,(17)xijt+1=xijt+vijt,where pbestτit=pbestτi1,1t,pbestτi2,2t,…,pbestτiD,Dt is the learning exemplar for particle i; τi=τi1,τi2,…,τiD is the learning exemplar indices for particle i, which is generated by the biogeographic migration.In the biogeographic migration, all particles are firstly sorted based on the value of theirpbest from best to worst and assigned with ranking values. For a minimization problem, assume
(18)fpbests1≤fpbests2≤⋯≤fpbestsN,where s1 is the subscript of the particle with the best pbest, s2 is the subscript of the particle with the second best pbest, and sN is the subscript of the particle with the worst pbest; N is the population size. Then, the rankings of particles are assigned as below:
(19)rankxsk=N−k,k=1,2,…,N.Second, immigration and emigration rates are assigned for all particles. The immigration and emigration rates for all particles can be calculated as follows:(20)λxsk=1−N−kN2,μxsk=N−kN2,k=1,2,…,N.According to20, the solution xs1 with the best pbests1 will have the lowest immigration rate and highest emigration rate; and the solution xsN with the worst pbestsN will have the highest immigration rate and lowest emigration rate.Third, the biogeography-based exemplar indicesτi=τi1,τi2,…,τiD for particle i can be generated based on the biogeography-based exemplar generation method, see Algorithm 1.Algorithm 1: (biogeography-based exemplar generation method).
1 Input: particle index i immigration rate λk and emigration rates μk
2Output: learning exemplar indices τi=τi1,…,τid,…,τiD
3For k=1 to D Do  // for each dimension
4If rand<λranki Then
5  Utilize a roulette wheel to select a particle indexj with probability ∝μrankj;
6τik=j; // learn from other particle
7Else
8τik=i; // learn from itself
9End If
10End For
11If τik=i in all dimension Then
12 Randomly select a particle indexll≠i;
13 Randomly select a dimensiond;
14τid=l;
15End If
16Return τi=τi1,…,τid,…,τiD


#### 3.4.2. Procedures of BLPSO

Using the BLS, the procedures of BLPSO can be outlined in Algorithm2. It can be seen from Algorithm 2 that the structure of BLPSO is as simple as the classic PSO.Algorithm 2: (BLPSO).
1Stage 1: Initialization
2For each particle i=1,2,…,N Do
3  Initialize positionxi and velocity vi;
4  Evaluate the fitnessfxi and store the personal best position pbesti;
5  Generate the learning exemplar indicesτi using Algorithm 1;
6  Set the refreshing gapG;
7End For
8Stage 2: Main loop
9For each particle i=1,2,…,N Do
10If the stagnation number stagnatedi>G Then
11   Generate the exemplar vector indexτi using Algorithm 1;
12   Setstagnatedi=0;
13End If
14  Update velocityvi using Eq.16;
15  Update positionxi using Eq.17;
16  Evaluate the new positionfxi;
17If xi is better than pbesti Then
18   Setpbesti=xi, stagnatedi=0;
19Else
20stagnatedi=stagnatedi+1;
21End If
22End For
23 Store swarm’sgbest with the best pbesti
24Stage 3: If the stop criterion is satisfied, the process is terminated. Otherwise, return to Stage 2.In addition, based on lines 10–13 in Algorithm2, it can be seen that Algorithm 1 is executed to generate new learning exemplar indices τi only when there is a stagnation for G generations, which is used to save computational cost of the BLPSO. In other words, if new learning exemplar indices τi are generated for all particles in each generation, Algorithm 2 will be executed too frequently, and this may cost a large computational time, which is inappropriate for real-world ED problems.


#### 3.4.3. Remarks

(1) Complexity Analysis. The computational costs of the original BLPSO algorithm involve the initialization (Tini), biogeography-based exemplar generation method (Tbio), velocity and position update (Tupd), and evaluation (Teva) for each particle. Assume D is the dimensionality of the optimization problem, N is the population size, and maxFES is maximum number of functional evaluations allowed for the algorithm. The complexity of initialization, velocity and position update, and evaluation are OD, O2D, and OD, respectively. The computational costs of biogeography-based exemplar generation method Tbio include population sorting ON·logN, ranking assignment ON, immigration and emigration rate assignment O2N, and migration operator ON·D. Therefore, Tbio=ON·logN+ON+O2N+ON·D=O3N+N·logN+N·D.The total computational complexity of BLPSO isTBLPSO=Tini+Teva+Tupd+1/GTbio·maxFE=OD+D+2D+3N+N·logN+N·D/G·maxFES. In general, the population size N is often set to be proportional to the problem dimension D (i.e., N=kD) [47]. Thus, the complexity of BLPSO is TBLPSO=OD+D+2D+3kD+kD·logkD+kD2/G·maxFES=OD2·maxFES.(2) Compared with Previous Hybrid PSO/BBO Algorithms. Several hybrid PSO/BBO algorithms have been proposed in the literature. For example, Guo et al. [51] presented a biogeography-based particle swarm optimization with fuzzy elitism (BPSO-FE) for constrained engineering problems. In this BPSO-FE algorithm, the whole population is split into several subgroups, and BBO is employed to search within each subgroup while PSO for the global search. Mo and Xu [52] applied the position updating strategy of PSO to increase the diversity of population in BBO and develop a biogeography particle swarm optimization algorithm (BPSO) to optimize the paths in path network. However, there are some differences between BLPSO and them. First, the hybrid strategies of BLPSO, BPSO-FE, and BPSO are different. In BLPSO, the biogeography-based migration is used to generate the learning exemplar for each particle; while in BPSO-FE and BPSO, the biogeography-based migration is used as search operator. Second, the application areas of BLPSO, BPSO-FE, and BPSO are different. BLPSO is presented for ED problems, while BPSO-FE and BPSO are proposed for classical engineering optimization problems and robot path planning, respectively.


## 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 theD-dimensional space, with a position vector xit=xi1t,xi2t,…,xiDt and a velocity vector vit=vi1t,vi2t,…,viDt. The particle updates its velocity and position according to the following equations:
(11)vijt+1=wvijt+c1r1pbestijt−xijt+c2r2gbestjt−xijt,(12)xijt+1=xijt+vijt,where pbestit=pbesti1t,pbesti2t,…,pbestiDt is the personal best position of particle i; gbestt=gbest1t,gbest2t,…,gbestDt is the position of the best particle in the population; w is the inertia weight; c1 and c2 are acceleration coefficients; and r1 and r2 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:
(13)vijt+1=wvijt+cr1pbestτij,jt−xijt,where pbestτit=pbestτi1,1t,pbestτi2,2t,…,pbestτiD,Dt is the learning exemplar for particle i and τi=τi1,τi2,…,τiD is the learning exemplar indices for particle i, 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:
(14)xijt+1=xijt+Δxijtifpit≤PiL,xijt,otherwise,(15)Δxijt+1=r1Δxijt+r2xkjt−xijt+r3εx¯jt−xijt,where PiL is the learning probability for particle i, pit is a randomly generated probability that satisfies 0≤pit≤PiL≤1, xkjt is the demonstrator of particle i in the jth dimension, x¯jt=∑i=1Nxij/N is the mean position of the all particles in the current swarm, and ε 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 N, PiL the learning probability, and the social influence factor ε.


## 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:(16)vijt+1=wvijt+cr1pbestτij,jt−xijt,(17)xijt+1=xijt+vijt,where pbestτit=pbestτi1,1t,pbestτi2,2t,…,pbestτiD,Dt is the learning exemplar for particle i; τi=τi1,τi2,…,τiD is the learning exemplar indices for particle i, which is generated by the biogeographic migration.In the biogeographic migration, all particles are firstly sorted based on the value of theirpbest from best to worst and assigned with ranking values. For a minimization problem, assume
(18)fpbests1≤fpbests2≤⋯≤fpbestsN,where s1 is the subscript of the particle with the best pbest, s2 is the subscript of the particle with the second best pbest, and sN is the subscript of the particle with the worst pbest; N is the population size. Then, the rankings of particles are assigned as below:
(19)rankxsk=N−k,k=1,2,…,N.Second, immigration and emigration rates are assigned for all particles. The immigration and emigration rates for all particles can be calculated as follows:(20)λxsk=1−N−kN2,μxsk=N−kN2,k=1,2,…,N.According to20, the solution xs1 with the best pbests1 will have the lowest immigration rate and highest emigration rate; and the solution xsN with the worst pbestsN will have the highest immigration rate and lowest emigration rate.Third, the biogeography-based exemplar indicesτi=τi1,τi2,…,τiD for particle i can be generated based on the biogeography-based exemplar generation method, see Algorithm 1.Algorithm 1: (biogeography-based exemplar generation method).
1 Input: particle index i immigration rate λk and emigration rates μk
2Output: learning exemplar indices τi=τi1,…,τid,…,τiD
3For k=1 to D Do  // for each dimension
4If rand<λranki Then
5  Utilize a roulette wheel to select a particle indexj with probability ∝μrankj;
6τik=j; // learn from other particle
7Else
8τik=i; // learn from itself
9End If
10End For
11If τik=i in all dimension Then
12 Randomly select a particle indexll≠i;
13 Randomly select a dimensiond;
14τid=l;
15End If
16Return τi=τi1,…,τid,…,τiD


### 3.4.2. Procedures of BLPSO

Using the BLS, the procedures of BLPSO can be outlined in Algorithm2. It can be seen from Algorithm 2 that the structure of BLPSO is as simple as the classic PSO.Algorithm 2: (BLPSO).
1Stage 1: Initialization
2For each particle i=1,2,…,N Do
3  Initialize positionxi and velocity vi;
4  Evaluate the fitnessfxi and store the personal best position pbesti;
5  Generate the learning exemplar indicesτi using Algorithm 1;
6  Set the refreshing gapG;
7End For
8Stage 2: Main loop
9For each particle i=1,2,…,N Do
10If the stagnation number stagnatedi>G Then
11   Generate the exemplar vector indexτi using Algorithm 1;
12   Setstagnatedi=0;
13End If
14  Update velocityvi using Eq.16;
15  Update positionxi using Eq.17;
16  Evaluate the new positionfxi;
17If xi is better than pbesti Then
18   Setpbesti=xi, stagnatedi=0;
19Else
20stagnatedi=stagnatedi+1;
21End If
22End For
23 Store swarm’sgbest with the best pbesti
24Stage 3: If the stop criterion is satisfied, the process is terminated. Otherwise, return to Stage 2.In addition, based on lines 10–13 in Algorithm2, it can be seen that Algorithm 1 is executed to generate new learning exemplar indices τi only when there is a stagnation for G generations, which is used to save computational cost of the BLPSO. In other words, if new learning exemplar indices τi are generated for all particles in each generation, Algorithm 2 will be executed too frequently, and this may cost a large computational time, which is inappropriate for real-world ED problems.


### 3.4.3. Remarks

(1) Complexity Analysis. The computational costs of the original BLPSO algorithm involve the initialization (Tini), biogeography-based exemplar generation method (Tbio), velocity and position update (Tupd), and evaluation (Teva) for each particle. Assume D is the dimensionality of the optimization problem, N is the population size, and maxFES is maximum number of functional evaluations allowed for the algorithm. The complexity of initialization, velocity and position update, and evaluation are OD, O2D, and OD, respectively. The computational costs of biogeography-based exemplar generation method Tbio include population sorting ON·logN, ranking assignment ON, immigration and emigration rate assignment O2N, and migration operator ON·D. Therefore, Tbio=ON·logN+ON+O2N+ON·D=O3N+N·logN+N·D.The total computational complexity of BLPSO isTBLPSO=Tini+Teva+Tupd+1/GTbio·maxFE=OD+D+2D+3N+N·logN+N·D/G·maxFES. In general, the population size N is often set to be proportional to the problem dimension D (i.e., N=kD) [47]. Thus, the complexity of BLPSO is TBLPSO=OD+D+2D+3kD+kD·logkD+kD2/G·maxFES=OD2·maxFES.(2) Compared with Previous Hybrid PSO/BBO Algorithms. Several hybrid PSO/BBO algorithms have been proposed in the literature. For example, Guo et al. [51] presented a biogeography-based particle swarm optimization with fuzzy elitism (BPSO-FE) for constrained engineering problems. In this BPSO-FE algorithm, the whole population is split into several subgroups, and BBO is employed to search within each subgroup while PSO for the global search. Mo and Xu [52] applied the position updating strategy of PSO to increase the diversity of population in BBO and develop a biogeography particle swarm optimization algorithm (BPSO) to optimize the paths in path network. However, there are some differences between BLPSO and them. First, the hybrid strategies of BLPSO, BPSO-FE, and BPSO are different. In BLPSO, the biogeography-based migration is used to generate the learning exemplar for each particle; while in BPSO-FE and BPSO, the biogeography-based migration is used as search operator. Second, the application areas of BLPSO, BPSO-FE, and BPSO are different. BLPSO is presented for ED problems, while BPSO-FE and BPSO are proposed for classical engineering optimization problems and robot path planning, respectively.


## 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:(16)vijt+1=wvijt+cr1pbestτij,jt−xijt,(17)xijt+1=xijt+vijt,where pbestτit=pbestτi1,1t,pbestτi2,2t,…,pbestτiD,Dt is the learning exemplar for particle i; τi=τi1,τi2,…,τiD is the learning exemplar indices for particle i, which is generated by the biogeographic migration.In the biogeographic migration, all particles are firstly sorted based on the value of theirpbest from best to worst and assigned with ranking values. For a minimization problem, assume
(18)fpbests1≤fpbests2≤⋯≤fpbestsN,where s1 is the subscript of the particle with the best pbest, s2 is the subscript of the particle with the second best pbest, and sN is the subscript of the particle with the worst pbest; N is the population size. Then, the rankings of particles are assigned as below:
(19)rankxsk=N−k,k=1,2,…,N.Second, immigration and emigration rates are assigned for all particles. The immigration and emigration rates for all particles can be calculated as follows:(20)λxsk=1−N−kN2,μxsk=N−kN2,k=1,2,…,N.According to20, the solution xs1 with the best pbests1 will have the lowest immigration rate and highest emigration rate; and the solution xsN with the worst pbestsN will have the highest immigration rate and lowest emigration rate.Third, the biogeography-based exemplar indicesτi=τi1,τi2,…,τiD for particle i can be generated based on the biogeography-based exemplar generation method, see Algorithm 1.Algorithm 1: (biogeography-based exemplar generation method).
1 Input: particle index i immigration rate λk and emigration rates μk
2Output: learning exemplar indices τi=τi1,…,τid,…,τiD
3For k=1 to D Do  // for each dimension
4If rand<λranki Then
5  Utilize a roulette wheel to select a particle indexj with probability ∝μrankj;
6τik=j; // learn from other particle
7Else
8τik=i; // learn from itself
9End If
10End For
11If τik=i in all dimension Then
12 Randomly select a particle indexll≠i;
13 Randomly select a dimensiond;
14τid=l;
15End If
16Return τi=τi1,…,τid,…,τiD


## 3.4.2. Procedures of BLPSO

Using the BLS, the procedures of BLPSO can be outlined in Algorithm2. It can be seen from Algorithm 2 that the structure of BLPSO is as simple as the classic PSO.Algorithm 2: (BLPSO).
1Stage 1: Initialization
2For each particle i=1,2,…,N Do
3  Initialize positionxi and velocity vi;
4  Evaluate the fitnessfxi and store the personal best position pbesti;
5  Generate the learning exemplar indicesτi using Algorithm 1;
6  Set the refreshing gapG;
7End For
8Stage 2: Main loop
9For each particle i=1,2,…,N Do
10If the stagnation number stagnatedi>G Then
11   Generate the exemplar vector indexτi using Algorithm 1;
12   Setstagnatedi=0;
13End If
14  Update velocityvi using Eq.16;
15  Update positionxi using Eq.17;
16  Evaluate the new positionfxi;
17If xi is better than pbesti Then
18   Setpbesti=xi, stagnatedi=0;
19Else
20stagnatedi=stagnatedi+1;
21End If
22End For
23 Store swarm’sgbest with the best pbesti
24Stage 3: If the stop criterion is satisfied, the process is terminated. Otherwise, return to Stage 2.In addition, based on lines 10–13 in Algorithm2, it can be seen that Algorithm 1 is executed to generate new learning exemplar indices τi only when there is a stagnation for G generations, which is used to save computational cost of the BLPSO. In other words, if new learning exemplar indices τi are generated for all particles in each generation, Algorithm 2 will be executed too frequently, and this may cost a large computational time, which is inappropriate for real-world ED problems.


## 3.4.3. Remarks

(1) Complexity Analysis. The computational costs of the original BLPSO algorithm involve the initialization (Tini), biogeography-based exemplar generation method (Tbio), velocity and position update (Tupd), and evaluation (Teva) for each particle. Assume D is the dimensionality of the optimization problem, N is the population size, and maxFES is maximum number of functional evaluations allowed for the algorithm. The complexity of initialization, velocity and position update, and evaluation are OD, O2D, and OD, respectively. The computational costs of biogeography-based exemplar generation method Tbio include population sorting ON·logN, ranking assignment ON, immigration and emigration rate assignment O2N, and migration operator ON·D. Therefore, Tbio=ON·logN+ON+O2N+ON·D=O3N+N·logN+N·D.The total computational complexity of BLPSO isTBLPSO=Tini+Teva+Tupd+1/GTbio·maxFE=OD+D+2D+3N+N·logN+N·D/G·maxFES. In general, the population size N is often set to be proportional to the problem dimension D (i.e., N=kD) [47]. Thus, the complexity of BLPSO is TBLPSO=OD+D+2D+3kD+kD·logkD+kD2/G·maxFES=OD2·maxFES.(2) Compared with Previous Hybrid PSO/BBO Algorithms. Several hybrid PSO/BBO algorithms have been proposed in the literature. For example, Guo et al. [51] presented a biogeography-based particle swarm optimization with fuzzy elitism (BPSO-FE) for constrained engineering problems. In this BPSO-FE algorithm, the whole population is split into several subgroups, and BBO is employed to search within each subgroup while PSO for the global search. Mo and Xu [52] applied the position updating strategy of PSO to increase the diversity of population in BBO and develop a biogeography particle swarm optimization algorithm (BPSO) to optimize the paths in path network. However, there are some differences between BLPSO and them. First, the hybrid strategies of BLPSO, BPSO-FE, and BPSO are different. In BLPSO, the biogeography-based migration is used to generate the learning exemplar for each particle; while in BPSO-FE and BPSO, the biogeography-based migration is used as search operator. Second, the application areas of BLPSO, BPSO-FE, and BPSO are different. BLPSO is presented for ED problems, while BPSO-FE and BPSO are proposed for classical engineering optimization problems and robot path planning, respectively.


## 4. Implementation of BLPSO for ED Problems

When solving the ED problems using BLPSO, the following three important issues should be considered: initialization of population, constraint handling, and stopping criterion.


### 4.1. Initialization of Population

In BLPSO, each individual of the population is a solution of an ED problem. If there areNg units that must be operated to provide power to load, then the current position of the ith particle can be given by
(21)Xi=Pi1,Pi2,…,Pij,…,PiNg,k=1,2,…,N,where N is the population size, j is index of the generating unit, and Pij is the generation power output of the jth generating unit in the ith particle.


### 4.2. Constraint Handling

One of the most important issues in solving ED problems is how to handle the quality and inequality constraints. There are four types of constraints in the ELD problems: power generation limits, ramp-rate limits, prohibited operating zone, and power balance constraint.For power generation limit and ramp-rate limit constraints, the following strategy is employed:(22)Pij=maxPjmin,Pj0−DRjifPij≤maxPjmin,Pj0−DRj,minPjmax,Pj0+URjifPij≥minPjmax,Pj0+URj,Pij,otherwise.For prohibited operating zone constraints, ifPij is located in the kth prohibited operating zone, that is, Pj,kl≤Pij≤Pj,ku, it is truncated to the closest boundary of the kth prohibited operating zone as follows:
(23)Pij=Pj,klifPj,kl<Pij≤Pj,kl+Pj,ku2,Pj,kuifPj,kl+Pj,ku2<Pij<Pj,ku,where Pj,kl and Pj,ku denote the lower and the upper bounds of prohibited operation zone k of generator j, respectively.For the power balance constraint, a repaired operator together with a common penalty is employed [28]. The repaired operator is shown in Algorithm 3, and the objective function becomes
(24)minF=∑j=1NgFjPj+K∑j=1NgPj−PD−PL,where K is the penalty coefficient and the penalty term ∑j=1NgPj−PD−PL is the measure of violation of the equality constraint.Algorithm 3: (the repair operator for power balance constraint).
1 Initialize a setI=1,2,…,D
2If ∑j=1NgPij≤PD+PL Then
3 Randomly select a componentk from the set I;
4While Pik=Pimax
5  Excludek from I, and let the new set be I′;
6  Randomly select a componentk′ from I′;
7k=k′,I=I′;
8End While
9 Add an amountw=∑j=1NgPij≤PD−PL to Pik, such as Pik=minPik+w,Pimax;
10Else If ∑j=1NgPij≤PD+PL
11 Randomly select a componentk from the set I;
12While Pik=Pimin
13  Excludek from I, and let the new set be I′;
14  Randomly select a componentk′ from I′;
15k=k′,I=I′;
16End While
17 Add an amountw=∑j=1NgPij≤PD−PL to Pik, such as Pik=maxPik+w,Pimin;
18End If


### 4.3. Stopping Criterion

The BLPSO algorithm will be terminated if the maximum number of functional evaluationsmaxFES is reached.


## 4.1. Initialization of Population

In BLPSO, each individual of the population is a solution of an ED problem. If there areNg units that must be operated to provide power to load, then the current position of the ith particle can be given by
(21)Xi=Pi1,Pi2,…,Pij,…,PiNg,k=1,2,…,N,where N is the population size, j is index of the generating unit, and Pij is the generation power output of the jth generating unit in the ith particle.


## 4.2. Constraint Handling

One of the most important issues in solving ED problems is how to handle the quality and inequality constraints. There are four types of constraints in the ELD problems: power generation limits, ramp-rate limits, prohibited operating zone, and power balance constraint.For power generation limit and ramp-rate limit constraints, the following strategy is employed:(22)Pij=maxPjmin,Pj0−DRjifPij≤maxPjmin,Pj0−DRj,minPjmax,Pj0+URjifPij≥minPjmax,Pj0+URj,Pij,otherwise.For prohibited operating zone constraints, ifPij is located in the kth prohibited operating zone, that is, Pj,kl≤Pij≤Pj,ku, it is truncated to the closest boundary of the kth prohibited operating zone as follows:
(23)Pij=Pj,klifPj,kl<Pij≤Pj,kl+Pj,ku2,Pj,kuifPj,kl+Pj,ku2<Pij<Pj,ku,where Pj,kl and Pj,ku denote the lower and the upper bounds of prohibited operation zone k of generator j, respectively.For the power balance constraint, a repaired operator together with a common penalty is employed [28]. The repaired operator is shown in Algorithm 3, and the objective function becomes
(24)minF=∑j=1NgFjPj+K∑j=1NgPj−PD−PL,where K is the penalty coefficient and the penalty term ∑j=1NgPj−PD−PL is the measure of violation of the equality constraint.Algorithm 3: (the repair operator for power balance constraint).
1 Initialize a setI=1,2,…,D
2If ∑j=1NgPij≤PD+PL Then
3 Randomly select a componentk from the set I;
4While Pik=Pimax
5  Excludek from I, and let the new set be I′;
6  Randomly select a componentk′ from I′;
7k=k′,I=I′;
8End While
9 Add an amountw=∑j=1NgPij≤PD−PL to Pik, such as Pik=minPik+w,Pimax;
10Else If ∑j=1NgPij≤PD+PL
11 Randomly select a componentk from the set I;
12While Pik=Pimin
13  Excludek from I, and let the new set be I′;
14  Randomly select a componentk′ from I′;
15k=k′,I=I′;
16End While
17 Add an amountw=∑j=1NgPij≤PD−PL to Pik, such as Pik=maxPik+w,Pimin;
18End If


## 4.3. Stopping Criterion

The BLPSO algorithm will be terminated if the maximum number of functional evaluationsmaxFES is reached.


## 5. Results and Discussion

To test the effectiveness of the proposed BLPSO algorithm, five different test systems of varying computational difficulty levels have been solved using BLPSO. The results obtained by BLPSO are compared with two PSO algorithms, comprehensive learning PSO (CLPSO) [46] and social leaning PSO (SLPSO) [47]. The results are also compared with several techniques reported in the literature whose abbreviations are listed in Table 1.Table 1
Table of abbreviations.
Optimization algorithms
Abbreviation
Backtracking search algorithm
BSA
Biogeography-based optimization
BBO
Chaotic bat algorithm
CBA
Chaotic improved honey bee mating optimization
CIHBMO
Continuous quick group search optimizer
CQGSO
Differential evolution
DE
Enhanced Hopfield neural network
EHNN
Firefly algorithm
FA
Genetic algorithm-ant colony optimization
GAAPI
Group search optimizer
GSO
Honey bee mating optimization
HBMO
Hybrid chemical reaction optimization with differential evolution
HCRO-DE
Hybrid differential evolution with biogeography-based optimization
DE/BBO
Harmony search
HS
Hopfield modeling framework
HM
Hybrid harmony search
HHS
Hybrid differential evolution with biogeography-based optimization
DE/BBO
Hybrid differential evolution with particle swarm optimization
DEPSO
Immune algorithm
IA
Improved differential evolution
IDE
Improved orthogonal design particle swarm optimization-global version
IODPSO-L
Improved orthogonal design particle swarm optimization-local version
IODPSO-G
Improved random drift particle swarm optimization
IRDPSO
Modified artificial bee colony
MABC
Multiple tabu search
MTS
Multistrategy ensemble biogeography-based optimization
MsEBBO
New particle swarm optimization
New-PSO
New particle swarm optimization with local random search
NPSO-LRS
Oppositional invasive weed optimization
OIWO
Oppositional real-coded chemical reaction optimization
ORCCRO
Particle swarm optimization
PSO
Particle swarm optimization with chaotic sequences and crossover operator
CCPSO
Particle swarm optimization with time-varying acceleration coefficients
PSO-TVAC
Random drift particle swarm optimization
RDPSO
Self-tuning improved random drift particle swarm optimization
ST-IRDPSO
Simulated annealing
SA
Stochastic weight trade-off particle swarm optimization
SWT-PSO
Tabu search
TSTo compare the performance of the BLPSO, 50 independent trial runs are made, and the statistical results including the minimum, mean, maximum fuel cost, and standard deviation, as well as average run time, are tabulated for each test system. The parameters of BLPSO are set as follows: population sizeN=40, inertia weight w linearly decreases from 0.9 to 0.2, acceleration coefficient c=1.496, and refreshing gap G=5. The parameters of CLPSO and SLPSO are set as those recommended in their original papers. The maximum number of functional evaluations maxFES is set as 10,000; 50,000; 50,000; 50,000; and 200,000 for the five test systems, respectively. The programs are implemented in MATLAB language on a personal computer with a 3.2 GHz processor and 8 GB RAM.


### 5.1. Test System 1

This is a small system comprising 6 generators and meeting a load demand of 1263 MW and includes transmission loss, POZ, and ramp-rate limits. The system data are taken from [8, 53] and listed in Table S1. Table 2 presents the optimal generation values and fuel cost obtained by BLPSO. The obtained optimal cost is 15447.34 $/hr. It can be seen that the generation values satisfy the generation limit constraints and do not fall in the POZs.Table 2
Optimal generations and cost obtained by BLPSO for test system 1 (6-unit system,PD=1263 MW).
Unit
Pjmin
Pjmax
POZ
Generation
1
100
500
(210, 240); (350, 380)
447.0682
2
50
200
(90, 110); (140, 160)
173.5899
3
80
300
(150, 170); (210, 240)
263.2341
4
50
150
(80, 90); (110, 120)
142.6879
5
50
200
(90, 110); (140, 150)
162.5776
6
50
120
(75, 85); (100, 105)
86.5033
Cost ($/hr)
15447.34
Transmission loss (MW)
12.6619Table3 shows the comparison of the statistical results of different algorithms. In the table, the results obtained by BLPSO are compared with CLPSO, SLPSO, NPSO-LRS [54], MTS [55], TS [55], SA [55], GAAPI [56], HCRO-DE [42], DE [57], MABC [31], CBA [58], RDPSO [26], IRDPSO [26], and ST-IRDPSO [26]. It can be seen that the minimum and mean fuel costs obtained by BLPSO are similar to SLPSO and less than all the other methods with the exceptions of HCRO-DE [42]. In addition, the smaller value of standard deviation indicates that BLPSO is consistent. It is also important to note that the BLPSO is very efficient according to the average computational time (0.50 s), which is less than most of other methods. Figure 1 presents the convergence characteristics obtained by CLPSO, SLPSO, and BLPSO. From Figure 1, SLPSO has the fastest convergence speed, and BLPSO has the second. Both BLPSO and SLPSO can converge to the optimal cost after about 6000 functional evaluations.Table 3
Comparison of fuel costs and statistical results for test system 1 (6-unit system,PD=1263 MW).
Algorithm
Minimum cost ($/h)
Mean cost ($/h)
Maximum cost ($/h)
Standard deviation
Time (s)
NPSO-LRS [54]
15,450
15,454
15,452
NA
NA
MTS [55]
15450.06
15451.17
15450.06
0.9287
1.29
TS [55]
5454.89
15472.56
15454.89
13.7195
20.55
SA [55]
15461.1
15488.98
15461.1
28.3678
50.36
GAAPI [56]
15449.78
15449.81
15449.85
NA
NA
HCRO-DE [42]
15443.075
15443.327
15443.916
0.067
4.17
DE [57]
15449.5826
15449.62
15449.6508
NA
3.634
MABC [31]
15449.8995
15449.8995
15449.8995
6.04E-08
0.62
CBA [58]
15450.2381
15454.76
15518.6588
2.965
0.704
RDPSO [26]
15449.89
15458.01
NA
13.647
0.707
IRDPSO [26]
15449.89
15456.55
NA
10.9865
0.676
ST-IRDPSO [26]
15449.89
15450.7
NA
1.416
0.727
CLPSO
15447.72
15449.83
15452.88
1.28
0.48
SLPSO
15447.34
15447.46
15447.62
0.08
0.84
BLPSO
15447.34
15447.45
15447.67
0.09
0.50
NA means the data are not available in the literature.Figure 1
Convergence characteristics for test system 1.


### 5.2. Test System 2

This test system consists of 15 generators meeting a load demand of 2630 MW and includes transmission loss, POZ, and ramp-rate limits. The system data are taken from [8, 59] and listed in Table S2. Table 4 presents the optimal generations and the costs obtained. The optimal cost obtained by BLPSO is 32587.33 $/hr, and the generations satisfy the generation limit constraints.Table 4
Optimal generations and cost obtained by BLPSO for test system 2 (15-unit system,PD=2630 MW).
Unit
Pjmin
Pjmax
Generation
1
150
455
455.0000
2
150
455
450.0000
3
20
130
130.0000
4
20
130
130.0000
5
150
470
200.0000
6
135
460
460.0000
7
135
465
430.0000
8
60
300
60.0000
9
25
162
35.1998
10
25
160
91.2383
11
20
80
80.0000
12
20
80
80.0000
13
25
85
25.0000
14
15
55
15.0000
15
15
55
15.0000
Cost ($/hr)
32587.33
Transmission loss (MW)
12.6619Table5 shows the comparison of the statistical results of the BLPSO and other algorithms, including CLPSO, SLPSO, CCPSO [60], HBMO [59], CIHBMO [59], FA [15], MsEBBO [35], DEPSO [40], SWT-PSO [61], IA [62], IODPSO-G [63], and IODPSO-L [63]. The minimum and mean fuel costs obtained by BLPSO are the least of all methods with the exceptions of HBMO [59]. The average computation time of BLPSO (2.85 s) is also very small. The convergence characteristics obtained by CLPSO, SLPSO, and BLPSO are plotted in Figure 2. SLPSO has the fastest convergence speed in the beginning, but it is surpassed by BLPSO and CLPSO in the end. Only BLPSO can get the optimal cost in this case.Table 5
Comparison of fuel costs and statistical results for test system 2 (15-unit system,PD=2630 MW).
Algorithm
Minimum cost ($/h)
Mean cost ($/h)
Maximum cost ($/h)
Standard deviation
Time (s)
CCPSO [60]
32704.4514
32704.4514
32704.4514
0
16.2
HBMO [59]
32637.6219
32663.19
32676.07
NA
2.8
CIHBMO [59]
32548.58588
32548.58588
32548.58588
NA
3.1
FA [15]
32,704.50
32,856.10
33,175.00
147.17
NA
MsEBBO [35]
32,692.40
32,692.40
32,692.40
0
NA
DEPSO [40]
32588.81
32588.99
32591.49
4.02
1.88 s
SWT-PSO [61]
32704.45
NA
NA
NA
NA
IA [62]
32,698.20
32,750.22
32,823.78
9.3
NA
IODPSO-G [63]
32,692.39
32,692.39
32,692.39
NA
NA
IODPSO-L [63]
32,692.39
32,692.39
32,692.39
NA
NA
CLPSO
32608.83
32649.9
32705.5
23.13
2.77
SLPSO
32674.25
32707.46
32758.69
14.67
4.93
BLPSO
32587.33
32607.17
32667.2
17.06
2.85
NA means the data are not available in the literature.Figure 2
Convergence characteristics for test system 2.


### 5.3. Test System 3

This test system consists of 20 generators supplying a demand of 2500 MW. Transmission losses are included in this system. The cost coefficient andB coefficient data are taken from [22, 64] and listed in Table S3. Table 6 presents the optimal generation values and fuel cost obtained by the BLPSO. The optimal obtained fuel cost is 62456.58 $/hr. It is seen that the all the generation limit constraints are satisfied.Table 6
Optimal generations and cost obtained by the BLPSO for test system 3 (20-unit system,PD=2500 MW).
Unit
Pjmin
Pjmax
Generation
1
150
600
512.2358
2
50
200
169.7731
3
50
200
126.4272
4
50
200
102.6131
5
50
160
113.9049
6
20
100
73.6208
7
25
125
115.506
8
50
150
116.631
9
50
200
100.2842
10
30
150
105.5532
11
100
300
150.2329
12
150
500
293.3209
13
40
160
118.8701
14
20
130
30.6567
15
25
185
115.3421
16
20
80
36.2973
17
30
85
67.0567
18
30
120
87.9775
19
40
120
101.2398
20
30
100
54.4584
Cost ($/hr)
62456.58
Transmission loss (MW)
92.0046Table7 shows the comparison of the statistical results of different algorithms. In the table, the results obtained by BLPSO are compared with CLPSO, SLPSO, EHNN [65], λ-iteration [64], HM [64], GSO [66], CQGSO [66], BBO [22], BSA [22], and CBA [58]. It can be seen that the best fuel cost obtained by BLPSO is the least of all methods, and the mean fuel cost is the least of all methods with the sole exception of CBA [58]. In addition, the standard deviation and average computation time of BLPSO are both very small. Again, the BLPSO is efficient for this case. The convergence characteristics obtained by CLPSO, SLPSO, and BLPSO are plotted in Figure 3.Table 7
Comparison of fuel costs and statistical results for test system 3 (20-unit system,PD=2500 MW).
Algorithm
Minimum cost ($/h)
Mean cost ($/h)
Maximum cost ($/h)
Standard deviation
Time (s)
EHNN [65]
62,610
NA
NA
NA
0.11
λ-iteration [64]
62456.6391
NA
NA
NA
33.757
HM [64]
62456.6341
NA
NA
NA
6.355
GSO [66]
62456.6332
62456.6336
62456.6353
NA
30.45
CQGSO [66]
62456.633
62456.6331
62456.63337
NA
11.13
BBO [22]
62456.7793
62456.7928
62456.7928
NA
NA
BSA [22]
62456.6925
62457.1517
62458.1272
NA
14.477
CBA [58]
62456.6328
62456.6348
62501.6714
0.3879
1.16
CLPSO
62456.44
62456.84
62457.10
0.17
2.68
SLPSO
62456.92
62457.38
62458.06
0.28
3.80
BLPSO
62456.58
62456.64
62456.65
0.01
2.75
NA means the data are not available in the literature.Figure 3
Convergence characteristics for test system 3.


### 5.4. Test System 4

This test system consists of 38 generators, and the demand of this system is 6000 MW. The system data are taken from [28, 41] and listed in Table S4. Table 8 presents the optimal generation values and cost obtained by BLPSO. The optimal cost is 9417208.19 $/hr. It is seen that the generations satisfy the generation limit constraints.Table 8
Optimal generations and cost obtained by the BLPSO for test system 4 (38-unit system,PD=6000 MW).
Unit
Pjmin
Pjmax
Generation
1
220
550
426.5025
2
220
550
426.7865
3
200
500
429.5064
4
200
500
429.6401
5
200
500
429.7661
6
200
500
429.5265
7
200
500
429.7083
8
200
500
429.5542
9
114
500
114
10
114
500
114
11
114
500
119.7279
12
114
500
127.1661
13
110
500
110
14
90
365
90
15
82
365
82
16
120
325
120
17
65
315
159.6378
18
65
315
65
19
65
315
65
20
120
272
272
21
120
272
272
22
110
260
260
23
80
190
130.682
24
10
150
10
25
60
125
113.3391
26
55
110
88.0312
27
35
75
37.5497
28
20
70
20
29
20
70
20
30
20
70
20
31
20
70
20
32
20
60
20
33
25
60
25
34
18
60
18
35
8
60
8
36
25
60
25
37
20
38
21.7594
38
20
38
21.0902
Cost ($/hr)
9417208.19The results obtained by BLPSO are compared with those obtained by CLPSO, SLPSO, New-PSO [67], PSO-TVAC [67], HS [68], HHS [68], BBO [41], DE/BBO [41], MsEBBO [35], and IDE [28], as shown in Table 9. It can be seen that the minimum and mean fuel costs obtained by BLPSO are the least of all the methods. The average computation time of BLPSO is 2.89 s, smaller than all methods, with the exception of CLPSO. The convergence characteristics obtained by CLPSO, SLPSO, and BLPSO are plotted in Figure 4.Table 9
Comparison of fuel costs and statistical results for test system 4 (38-unit system,PD=6000 MW).
Algorithm
Minimum cost ($/h)
Mean cost ($/h)
Maximum cost ($/h)
Standard deviation
Time (s)
New-PSO [67]
9516448.312
NA
NA
NA
NA
PSO-TVAC [67]
9500448.307
NA
NA
NA
NA
HS [68]
9,419,960
9,421,056
9,427,466
NA
10.02
HHSÂ [68]
9,417,325
9,417,336
9,417,466
NA
5.06
BBOÂ [41]
9417633.638
NA
NA
NA
NA
DE/BBO [41]
9417235.786
NA
NA
NA
NA
MsEBBO [35]
9417235.776
9417235.779
9417235.778
0.0032
NA
IDE [28]
9417235.786
9417235.786
9417235.786
6.00E-09
9.149
CLPSO
9418283.79
9419255.18
9420192.8
526.51
2.86
SLPSO
9418407.11
9419560.62
9425492.54
1312.2
3.53
BLPSO
9417208.19
9417234.16
9417235.9
5.23
2.89
NA means the data are not available in the literature.Figure 4
Convergence characteristics for test system 4.


### 5.5. Test System 5

In order to study the performance of the BLPSO on high-dimensional ED problems, a large system with 110 generators is considered. The demand of this system is 15,000 MW, and the system data are taken from [69, 70] and listed in Table S5. Table 10 presents the optimal generation values and cost obtained by BLPSO. The optimal cost is 197988.16 $/hr.Table 10
Optimal generations and cost obtained by the CBA for test system 5 (110-unit system,PD=5000 MW).
Unit
Generation
Unit
Generation
Unit
Generation
Unit
Generation
Unit
Generation
1
2.4
23
68.9
45
660
67
70
89
82.7308
2
2.4
24
350
46
616.4766
68
70
90
89.7172
3
2.4
25
400
47
5.4
69
70
91
57.9161
4
2.4
26
400
48
5.4
70
360
92
100
5
2.4
27
500
49
8.4
71
400
93
440
6
4
28
500
50
8.4
72
400
94
500
7
4
29
200
51
8.4
73
105.0721
95
600
8
4
30
100
52
12
74
190.995
96
471.2608
9
4
31
10
53
12
75
90
97
3.6
10
64.6059
32
20
54
12
76
50
98
3.6
11
62.3474
33
80
55
12
77
160
99
4.4
12
36.3769
34
250
56
25.2
78
295.3172
100
4.4
13
56.6463
35
360
57
25.2
79
174.949
101
10
14
25
36
400
58
35
80
98.2904
102
10
15
25
37
40
59
35
81
10
103
20
16
25
38
70
60
45
82
12
104
20
17
155
39
100
61
45
83
20
105
40
18
155
40
120
62
45
84
200
106
40
19
155
41
156.791
63
185
85
325
107
50
20
155
42
220
64
185
86
440
108
30
21
68.9
43
440
65
185
87
13.9066
109
40
22
68.9
44
560
66
185
88
24.4992
110
20
Cost ($/hr)
197988.16Table11 shows the comparison of the statistical results of BLPSO and other algorithms, including CLPSO, SLPSO, SAB [71], SAF [71], SA [71], ORCCRO [72], BBO [72], DE/BBO [72], and OIWO [69]. The minimum, mean, and maximum fuel costs obtained by BLPSO are the least of all the methods. Meanwhile, the smaller value of standard deviation indicates that BLPSO is consistent. The average computation time of BLPSO is also very small compared with other methods. The convergence characteristics obtained by CLPSO, SLPSO, and BLPSO are presented in Figure 5.Table 11
Comparison of fuel costs and statistical results for test system 5 (110-unit system,PD=15,000 MW).
Algorithm
Minimum cost ($/h)
Mean cost ($/h)
Maximum cost ($/h)
Standard deviation
Time (s)
SAB [71]
206912.9057
207764.73
NA
NA
NA
SAF [71]
207380.5164
207813.37
NA
NA
NA
SA [71]
198352.6413
201595.19
NA
NA
NA
ORCCRO [72]
198016.29
198016.32
198016.89
NA
0.15
BBO [72]
198241.166
198413.45
199,102.59
NA
115
DE/BBO [72]
198231.06
198326.66
198,828.57
NA
132
OIWO [69]
197989.14
197989.41
197989.93
NA
31
CLPSO
198137.7
198200.37
198257.56
27.88
23.43
SLPSO
198240.4
198351.62
198602.16
96
22.59
BLPSO
197988.16
197988.18
197988.19
0
22.81
NA means the data are not available in the literature.Figure 5
Convergence characteristics for test system 5.


## 5.1. Test System 1

This is a small system comprising 6 generators and meeting a load demand of 1263 MW and includes transmission loss, POZ, and ramp-rate limits. The system data are taken from [8, 53] and listed in Table S1. Table 2 presents the optimal generation values and fuel cost obtained by BLPSO. The obtained optimal cost is 15447.34 $/hr. It can be seen that the generation values satisfy the generation limit constraints and do not fall in the POZs.Table 2
Optimal generations and cost obtained by BLPSO for test system 1 (6-unit system,PD=1263 MW).
Unit
Pjmin
Pjmax
POZ
Generation
1
100
500
(210, 240); (350, 380)
447.0682
2
50
200
(90, 110); (140, 160)
173.5899
3
80
300
(150, 170); (210, 240)
263.2341
4
50
150
(80, 90); (110, 120)
142.6879
5
50
200
(90, 110); (140, 150)
162.5776
6
50
120
(75, 85); (100, 105)
86.5033
Cost ($/hr)
15447.34
Transmission loss (MW)
12.6619Table3 shows the comparison of the statistical results of different algorithms. In the table, the results obtained by BLPSO are compared with CLPSO, SLPSO, NPSO-LRS [54], MTS [55], TS [55], SA [55], GAAPI [56], HCRO-DE [42], DE [57], MABC [31], CBA [58], RDPSO [26], IRDPSO [26], and ST-IRDPSO [26]. It can be seen that the minimum and mean fuel costs obtained by BLPSO are similar to SLPSO and less than all the other methods with the exceptions of HCRO-DE [42]. In addition, the smaller value of standard deviation indicates that BLPSO is consistent. It is also important to note that the BLPSO is very efficient according to the average computational time (0.50 s), which is less than most of other methods. Figure 1 presents the convergence characteristics obtained by CLPSO, SLPSO, and BLPSO. From Figure 1, SLPSO has the fastest convergence speed, and BLPSO has the second. Both BLPSO and SLPSO can converge to the optimal cost after about 6000 functional evaluations.Table 3
Comparison of fuel costs and statistical results for test system 1 (6-unit system,PD=1263 MW).
Algorithm
Minimum cost ($/h)
Mean cost ($/h)
Maximum cost ($/h)
Standard deviation
Time (s)
NPSO-LRS [54]
15,450
15,454
15,452
NA
NA
MTS [55]
15450.06
15451.17
15450.06
0.9287
1.29
TS [55]
5454.89
15472.56
15454.89
13.7195
20.55
SA [55]
15461.1
15488.98
15461.1
28.3678
50.36
GAAPI [56]
15449.78
15449.81
15449.85
NA
NA
HCRO-DE [42]
15443.075
15443.327
15443.916
0.067
4.17
DE [57]
15449.5826
15449.62
15449.6508
NA
3.634
MABC [31]
15449.8995
15449.8995
15449.8995
6.04E-08
0.62
CBA [58]
15450.2381
15454.76
15518.6588
2.965
0.704
RDPSO [26]
15449.89
15458.01
NA
13.647
0.707
IRDPSO [26]
15449.89
15456.55
NA
10.9865
0.676
ST-IRDPSO [26]
15449.89
15450.7
NA
1.416
0.727
CLPSO
15447.72
15449.83
15452.88
1.28
0.48
SLPSO
15447.34
15447.46
15447.62
0.08
0.84
BLPSO
15447.34
15447.45
15447.67
0.09
0.50
NA means the data are not available in the literature.Figure 1
Convergence characteristics for test system 1.


## 5.2. Test System 2

This test system consists of 15 generators meeting a load demand of 2630 MW and includes transmission loss, POZ, and ramp-rate limits. The system data are taken from [8, 59] and listed in Table S2. Table 4 presents the optimal generations and the costs obtained. The optimal cost obtained by BLPSO is 32587.33 $/hr, and the generations satisfy the generation limit constraints.Table 4
Optimal generations and cost obtained by BLPSO for test system 2 (15-unit system,PD=2630 MW).
Unit
Pjmin
Pjmax
Generation
1
150
455
455.0000
2
150
455
450.0000
3
20
130
130.0000
4
20
130
130.0000
5
150
470
200.0000
6
135
460
460.0000
7
135
465
430.0000
8
60
300
60.0000
9
25
162
35.1998
10
25
160
91.2383
11
20
80
80.0000
12
20
80
80.0000
13
25
85
25.0000
14
15
55
15.0000
15
15
55
15.0000
Cost ($/hr)
32587.33
Transmission loss (MW)
12.6619Table5 shows the comparison of the statistical results of the BLPSO and other algorithms, including CLPSO, SLPSO, CCPSO [60], HBMO [59], CIHBMO [59], FA [15], MsEBBO [35], DEPSO [40], SWT-PSO [61], IA [62], IODPSO-G [63], and IODPSO-L [63]. The minimum and mean fuel costs obtained by BLPSO are the least of all methods with the exceptions of HBMO [59]. The average computation time of BLPSO (2.85 s) is also very small. The convergence characteristics obtained by CLPSO, SLPSO, and BLPSO are plotted in Figure 2. SLPSO has the fastest convergence speed in the beginning, but it is surpassed by BLPSO and CLPSO in the end. Only BLPSO can get the optimal cost in this case.Table 5
Comparison of fuel costs and statistical results for test system 2 (15-unit system,PD=2630 MW).
Algorithm
Minimum cost ($/h)
Mean cost ($/h)
Maximum cost ($/h)
Standard deviation
Time (s)
CCPSO [60]
32704.4514
32704.4514
32704.4514
0
16.2
HBMO [59]
32637.6219
32663.19
32676.07
NA
2.8
CIHBMO [59]
32548.58588
32548.58588
32548.58588
NA
3.1
FA [15]
32,704.50
32,856.10
33,175.00
147.17
NA
MsEBBO [35]
32,692.40
32,692.40
32,692.40
0
NA
DEPSO [40]
32588.81
32588.99
32591.49
4.02
1.88 s
SWT-PSO [61]
32704.45
NA
NA
NA
NA
IA [62]
32,698.20
32,750.22
32,823.78
9.3
NA
IODPSO-G [63]
32,692.39
32,692.39
32,692.39
NA
NA
IODPSO-L [63]
32,692.39
32,692.39
32,692.39
NA
NA
CLPSO
32608.83
32649.9
32705.5
23.13
2.77
SLPSO
32674.25
32707.46
32758.69
14.67
4.93
BLPSO
32587.33
32607.17
32667.2
17.06
2.85
NA means the data are not available in the literature.Figure 2
Convergence characteristics for test system 2.


## 5.3. Test System 3

This test system consists of 20 generators supplying a demand of 2500 MW. Transmission losses are included in this system. The cost coefficient andB coefficient data are taken from [22, 64] and listed in Table S3. Table 6 presents the optimal generation values and fuel cost obtained by the BLPSO. The optimal obtained fuel cost is 62456.58 $/hr. It is seen that the all the generation limit constraints are satisfied.Table 6
Optimal generations and cost obtained by the BLPSO for test system 3 (20-unit system,PD=2500 MW).
Unit
Pjmin
Pjmax
Generation
1
150
600
512.2358
2
50
200
169.7731
3
50
200
126.4272
4
50
200
102.6131
5
50
160
113.9049
6
20
100
73.6208
7
25
125
115.506
8
50
150
116.631
9
50
200
100.2842
10
30
150
105.5532
11
100
300
150.2329
12
150
500
293.3209
13
40
160
118.8701
14
20
130
30.6567
15
25
185
115.3421
16
20
80
36.2973
17
30
85
67.0567
18
30
120
87.9775
19
40
120
101.2398
20
30
100
54.4584
Cost ($/hr)
62456.58
Transmission loss (MW)
92.0046Table7 shows the comparison of the statistical results of different algorithms. In the table, the results obtained by BLPSO are compared with CLPSO, SLPSO, EHNN [65], λ-iteration [64], HM [64], GSO [66], CQGSO [66], BBO [22], BSA [22], and CBA [58]. It can be seen that the best fuel cost obtained by BLPSO is the least of all methods, and the mean fuel cost is the least of all methods with the sole exception of CBA [58]. In addition, the standard deviation and average computation time of BLPSO are both very small. Again, the BLPSO is efficient for this case. The convergence characteristics obtained by CLPSO, SLPSO, and BLPSO are plotted in Figure 3.Table 7
Comparison of fuel costs and statistical results for test system 3 (20-unit system,PD=2500 MW).
Algorithm
Minimum cost ($/h)
Mean cost ($/h)
Maximum cost ($/h)
Standard deviation
Time (s)
EHNN [65]
62,610
NA
NA
NA
0.11
λ-iteration [64]
62456.6391
NA
NA
NA
33.757
HM [64]
62456.6341
NA
NA
NA
6.355
GSO [66]
62456.6332
62456.6336
62456.6353
NA
30.45
CQGSO [66]
62456.633
62456.6331
62456.63337
NA
11.13
BBO [22]
62456.7793
62456.7928
62456.7928
NA
NA
BSA [22]
62456.6925
62457.1517
62458.1272
NA
14.477
CBA [58]
62456.6328
62456.6348
62501.6714
0.3879
1.16
CLPSO
62456.44
62456.84
62457.10
0.17
2.68
SLPSO
62456.92
62457.38
62458.06
0.28
3.80
BLPSO
62456.58
62456.64
62456.65
0.01
2.75
NA means the data are not available in the literature.Figure 3
Convergence characteristics for test system 3.


## 5.4. Test System 4

This test system consists of 38 generators, and the demand of this system is 6000 MW. The system data are taken from [28, 41] and listed in Table S4. Table 8 presents the optimal generation values and cost obtained by BLPSO. The optimal cost is 9417208.19 $/hr. It is seen that the generations satisfy the generation limit constraints.Table 8
Optimal generations and cost obtained by the BLPSO for test system 4 (38-unit system,PD=6000 MW).
Unit
Pjmin
Pjmax
Generation
1
220
550
426.5025
2
220
550
426.7865
3
200
500
429.5064
4
200
500
429.6401
5
200
500
429.7661
6
200
500
429.5265
7
200
500
429.7083
8
200
500
429.5542
9
114
500
114
10
114
500
114
11
114
500
119.7279
12
114
500
127.1661
13
110
500
110
14
90
365
90
15
82
365
82
16
120
325
120
17
65
315
159.6378
18
65
315
65
19
65
315
65
20
120
272
272
21
120
272
272
22
110
260
260
23
80
190
130.682
24
10
150
10
25
60
125
113.3391
26
55
110
88.0312
27
35
75
37.5497
28
20
70
20
29
20
70
20
30
20
70
20
31
20
70
20
32
20
60
20
33
25
60
25
34
18
60
18
35
8
60
8
36
25
60
25
37
20
38
21.7594
38
20
38
21.0902
Cost ($/hr)
9417208.19The results obtained by BLPSO are compared with those obtained by CLPSO, SLPSO, New-PSO [67], PSO-TVAC [67], HS [68], HHS [68], BBO [41], DE/BBO [41], MsEBBO [35], and IDE [28], as shown in Table 9. It can be seen that the minimum and mean fuel costs obtained by BLPSO are the least of all the methods. The average computation time of BLPSO is 2.89 s, smaller than all methods, with the exception of CLPSO. The convergence characteristics obtained by CLPSO, SLPSO, and BLPSO are plotted in Figure 4.Table 9
Comparison of fuel costs and statistical results for test system 4 (38-unit system,PD=6000 MW).
Algorithm
Minimum cost ($/h)
Mean cost ($/h)
Maximum cost ($/h)
Standard deviation
Time (s)
New-PSO [67]
9516448.312
NA
NA
NA
NA
PSO-TVAC [67]
9500448.307
NA
NA
NA
NA
HS [68]
9,419,960
9,421,056
9,427,466
NA
10.02
HHSÂ [68]
9,417,325
9,417,336
9,417,466
NA
5.06
BBOÂ [41]
9417633.638
NA
NA
NA
NA
DE/BBO [41]
9417235.786
NA
NA
NA
NA
MsEBBO [35]
9417235.776
9417235.779
9417235.778
0.0032
NA
IDE [28]
9417235.786
9417235.786
9417235.786
6.00E-09
9.149
CLPSO
9418283.79
9419255.18
9420192.8
526.51
2.86
SLPSO
9418407.11
9419560.62
9425492.54
1312.2
3.53
BLPSO
9417208.19
9417234.16
9417235.9
5.23
2.89
NA means the data are not available in the literature.Figure 4
Convergence characteristics for test system 4.


## 5.5. Test System 5

In order to study the performance of the BLPSO on high-dimensional ED problems, a large system with 110 generators is considered. The demand of this system is 15,000 MW, and the system data are taken from [69, 70] and listed in Table S5. Table 10 presents the optimal generation values and cost obtained by BLPSO. The optimal cost is 197988.16 $/hr.Table 10
Optimal generations and cost obtained by the CBA for test system 5 (110-unit system,PD=5000 MW).
Unit
Generation
Unit
Generation
Unit
Generation
Unit
Generation
Unit
Generation
1
2.4
23
68.9
45
660
67
70
89
82.7308
2
2.4
24
350
46
616.4766
68
70
90
89.7172
3
2.4
25
400
47
5.4
69
70
91
57.9161
4
2.4
26
400
48
5.4
70
360
92
100
5
2.4
27
500
49
8.4
71
400
93
440
6
4
28
500
50
8.4
72
400
94
500
7
4
29
200
51
8.4
73
105.0721
95
600
8
4
30
100
52
12
74
190.995
96
471.2608
9
4
31
10
53
12
75
90
97
3.6
10
64.6059
32
20
54
12
76
50
98
3.6
11
62.3474
33
80
55
12
77
160
99
4.4
12
36.3769
34
250
56
25.2
78
295.3172
100
4.4
13
56.6463
35
360
57
25.2
79
174.949
101
10
14
25
36
400
58
35
80
98.2904
102
10
15
25
37
40
59
35
81
10
103
20
16
25
38
70
60
45
82
12
104
20
17
155
39
100
61
45
83
20
105
40
18
155
40
120
62
45
84
200
106
40
19
155
41
156.791
63
185
85
325
107
50
20
155
42
220
64
185
86
440
108
30
21
68.9
43
440
65
185
87
13.9066
109
40
22
68.9
44
560
66
185
88
24.4992
110
20
Cost ($/hr)
197988.16Table11 shows the comparison of the statistical results of BLPSO and other algorithms, including CLPSO, SLPSO, SAB [71], SAF [71], SA [71], ORCCRO [72], BBO [72], DE/BBO [72], and OIWO [69]. The minimum, mean, and maximum fuel costs obtained by BLPSO are the least of all the methods. Meanwhile, the smaller value of standard deviation indicates that BLPSO is consistent. The average computation time of BLPSO is also very small compared with other methods. The convergence characteristics obtained by CLPSO, SLPSO, and BLPSO are presented in Figure 5.Table 11
Comparison of fuel costs and statistical results for test system 5 (110-unit system,PD=15,000 MW).
Algorithm
Minimum cost ($/h)
Mean cost ($/h)
Maximum cost ($/h)
Standard deviation
Time (s)
SAB [71]
206912.9057
207764.73
NA
NA
NA
SAF [71]
207380.5164
207813.37
NA
NA
NA
SA [71]
198352.6413
201595.19
NA
NA
NA
ORCCRO [72]
198016.29
198016.32
198016.89
NA
0.15
BBO [72]
198241.166
198413.45
199,102.59
NA
115
DE/BBO [72]
198231.06
198326.66
198,828.57
NA
132
OIWO [69]
197989.14
197989.41
197989.93
NA
31
CLPSO
198137.7
198200.37
198257.56
27.88
23.43
SLPSO
198240.4
198351.62
198602.16
96
22.59
BLPSO
197988.16
197988.18
197988.19
0
22.81
NA means the data are not available in the literature.Figure 5
Convergence characteristics for test system 5.


## 6. Conclusion

This paper has presented a biogeography-based learning particle swarm optimization (BLPSO) for solving the economic dispatch (ED) problems, which is nonlinear, nonconvex, and discontinuous in nature, with numerous equality and inequality constraints. In the BLPSO, a biogeography-based learning strategy is used to generate the learning exemplar for each particle, in which particles learn more from high-quality particles. The biogeography-based learning strategy can provide a more effective balance between exploration and exploitation for the BLPSO.The BLPSO was applied to five test systems with various constraints such as power balance, POZs, and ramp-rate limits. Transmission losses have also been included in some systems. The experimental results show that the fuel costs obtained by BLPSO are either comparable or lower than those reported by other methods. The application to a 110-unit system shows that the BLPSO is also capable of handling high-dimensional ED problems.In the future, we are planning to extend the BLPSO to solve other more complicated ED problems, such as dynamical ED problems and environmental ED problems. We are also interested in applying the BLPSO to other optimization problems in energy field such as solar photovoltaic modeling [73].



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*Source: 7289674-2018-07-12.xml*