Complexity
Volume 2018 (2018), Article ID 7289674, 15 pages
https://doi.org/10.1155/2018/7289674

Research Article

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

Xu Chen,1,2 Bin Xu,³ and Wenli Du²

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

Correspondence should be addressed to Xu Chen; xuchen@ujs.edu.cn

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.

1 Input: particle index immigration rate and emigration rates

2 Output: learning exemplar indices

3 For toDDo        // for each dimension

4    If     Then

5        Utilize a roulette wheel to select a particle index with probability ;

6        ;    // learn from other particle

7    Else

8        ;    // learn from itself

9    End If

10 End For

11 If in all dimension Then

12    Randomly select a particle index ;

13    Randomly select a dimension ;

14    ;

15 End If

16 Return

Algorithm 1: (biogeography-based exemplar generation method).

3.4.2. Procedures of BLPSO

Using the BLS, the procedures of BLPSO can be outlined in Algorithm 2. It can be seen from Algorithm 2 that the structure of BLPSO is as simple as the classic PSO.

1 Stage 1: Initialization

2    For each particle Do

3        Initialize position and velocity ;

4        Evaluate the fitness and store the personal best position ;

5        Generate the learning exemplar indices using Algorithm 1;

6        Set the refreshing gap ;

7    End For

8 Stage 2: Main loop

9    For each particle Do

10        If the stagnation number Then

11            Generate the exemplar vector index using Algorithm 1;

12            Set ;

13        End If

14        Update velocity using Eq.¹⁶;

15        Update position using Eq.¹⁷;

16        Evaluate the new position ;

17        If is better than     Then

18            Set , ;

19        Else

20            ;

21        End If

22    End For

23 Store swarm’s with the best

24 Stage 3: If the stop criterion is satisfied, the process is terminated. Otherwise, return to Stage 2.

Algorithm 2: (BLPSO).

In addition, based on lines 10–13 in Algorithm 2, it can be seen that Algorithm 1 is executed to generate new learning exemplar indices only when there is a stagnation for generations, which is used to save computational cost of the BLPSO. In other words, if new learning exemplar indices 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 (), biogeography-based exemplar generation method (), velocity and position update (), and evaluation () for each particle. Assume is the dimensionality of the optimization problem, is the population size, and is maximum number of functional evaluations allowed for the algorithm. The complexity of initialization, velocity and position update, and evaluation are , , and , respectively. The computational costs of biogeography-based exemplar generation method include population sorting , ranking assignment , immigration and emigration rate assignment , and migration operator . Therefore, .

The total computational complexity of BLPSO is . In general, the population size is often set to be proportional to the problem dimension (i.e., ) [47]. Thus, the complexity of BLPSO is .

(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 are units that must be operated to provide power to load, then the current position of the th particle can be given by

where is the population size, is index of the generating unit, and is the generation power output of the th generating unit in the th 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:

For prohibited operating zone constraints, if is located in the th prohibited operating zone, that is, , it is truncated to the closest boundary of the th prohibited operating zone as follows:

where and denote the lower and the upper bounds of prohibited operation zone of generator , 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

where is the penalty coefficient and the penalty term is the measure of violation of the equality constraint.

1 Initialize a set

2 If     Then

3    Randomly select a component from the set ;

4    While

5        Exclude from , and let the new set be ;

6        Randomly select a component from ;

7        ;

8    End While

9    Add an amount to , such as ;

10 Else If

11    Randomly select a component from the set ;

12    While

13        Exclude from , and let the new set be ;

14        Randomly select a component from ;

15        ;

16    End While

17    Add an amount to , such as ;

18 End If

Algorithm 3: (the repair operator for power balance constraint).

4.3. Stopping Criterion

The BLPSO algorithm will be terminated if the maximum number of functional evaluations 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
TS

To 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 size , inertia weight linearly decreases from 0.9 to 0.2, acceleration coefficient , and refreshing gap . The parameters of CLPSO and SLPSO are set as those recommended in their original papers. The maximum number of functional evaluations 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,  MW).

Unit

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)
12.6619

Table 3 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,  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,  MW).

Unit

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)
12.6619

Table 5 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,  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 and 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,  MW).

Unit

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)
92.0046

Table 7 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], -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,  MW).

Algorithm
Minimum cost ($/h)
Mean cost ($/h)
Maximum cost ($/h)
Standard deviation
Time (s)

EHNN [65]
62,610
NA
NA
NA
0.11

-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,  MW).

Unit

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)
9417208.19

The 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,  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,  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)
197988.16

Table 11 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,  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].

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported in part by the Natural Science Foundation of Jiangsu Province (Grant no. BK 20160540), China Postdoctoral Science Foundation (Grant no. 2016M591783), and Fundamental Research Funds for the Central Universities (Grant no. 222201717006).

Supplementary Materials

Table S1∼Table S5 provides the system data of the five test systems studied in this paper. (1) Table S1: the system data of test system 1 (6-unit system). (2) Table S2: the system data of test system 2 (15-unit system). (3) Table S3: the system data of test system 3 (20-unit system). (4) Table S4: the system data of test system 4 (38-unit system). (5) Table S5: the system data of test system 5 (110-unit system). (Supplementary Materials)

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