An improved particle swarm optimization with

An improved particle swarm optimization with biogeography-based learning strategy for economic dispatch problems Xu Chena,b,∗, Bin Xuc , Wenli Dub a School

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

b Key

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 effecient 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. Keywords: Economic dispatch, particle swarm optimization, biogeography-based learning strategy, metaheuristic algorithm

1

1. Introduction

2

Economic dispatch (ED) is an important optimization task in power system operation and

3

planning. The main objective of ED problems is to allocate generation among the committed

4

generating units so as to meet the required load demand at minimum operating cost, while

5

various physical constraints [1]. The cost of power generation is high, and economic dispatch can

6

help in saving a significant amount of revenue [2]. ∗ Corresponding

author: Xu Chen Email address: [email protected] (Xu Chen)

Preprint submitted to Journal Name

May 15, 2018

7

In the original ED problem, the cost function for each generation unit is approximately rep-

8

resented by a single quadratic function, and traditional approaches based on mathematical pro-

9

gramming techniques have been utilized to solve the ED problem, including the lambda-iteration

10

method, gradient method, Newtons method, linear programming, interior point method and dy-

11

namic programming [3, 4, 5]. Usually, these methods are highly sensitive to starting points and

12

rely on the assumption that the cost function needs to be continuous and convex. However, the

13

practical ED problems exhibit non-convex and non-smooth characteristics because of valve-point

14

effects, ramp rate limit, multi-fuel cost and prohibited operating zones, etc [6]. The traditional

15

methods are not capable of efficiently solving the ED problems with these characteristics.

16

In the past decades, more and more researchers are turning to meta-heuristic search (MS)

17

algorithms for solving the ED problems. These methods have the ability to identify higher

18

quality solutions and can be grouped into three categories, as original, improved and hybrid MS

19

algorithms.

20

The first category consists of methods applied in their original version, such as genetic al-

21

gorithm (GA) [7], particle swarm optimization (PSO) [8], differential evolution (DE) [9], ant

22

colony optimization (ACO) [10], harmony search (HS) [11], artificial bee colony (ABC) [12],

23

teaching-learning-based optimization(TLBO) [13], gravitational search algorithm (GSA) [14],

24

firefly algorithm (FA) [15], biogeography-based optimization (BBO) [16, 17], bacterial foraging

25

optimization (BFO) [18], imperialist competitive algorithm (ICA) [19], seeker optimization algo-

26

rithm (SOA) [20], grey wolf optimization (GWO) [21], backtracking search algorithm(BSA) [22],

27

and root tree optimization (RTO) [23].

28

The second refers to improved or modified methods derived from the original version, and

29

the following are included: self-adaptive real coded genetic algorithm (SARGA) [24], random

30

drift PSO (RDPSO) [25, 26] , fuzzy adaptive modified PSO (FAMPSO) [27], improved differ-

31

ential evolution (IDE) [28], Shuffled differential evolution (SDE)[29], improved harmony search

32

(IHS) [30], modified artificial bee colony (MABC) [31], incremental artificial bee colony (IABC)

33

[32], ramp rate biogeography based optimization (RRBBO) [33], dynamic non-dominated sorting

34

biogeography-based optimization (Dy-NSBBO) [34], multi-strategy ensemble biogeography-based

35

optimization (MsEBBO) [35], and modified group search optimizer (MGSO) [36].

36

The third are the hybrid methods in which two or more optimization techniques are com-

37

bined, including hybrid genetic algorithm (HGA) [37], chaotic PSO with sequential quadratic

38

programming (CPSO-SQP) [38], hybrid PSO and gravitational search algorithm (HPSO-GSA)

39

[39], hybrid differential evolution algorithm based on PSO (DEPSO) [40], hybrid differential evo-

2

40

lution with biogeography-based optimization (DE/BBO) [41], hybrid chemical reaction optimiza-

41

tion with differential evolution (HCRO-DE) [42], and hybrid imperialist competitive-sequential

42

quadratic programming (HIC-SQP) [43].

43

In this paper, an improved PSO algorithm with biogeography-based learning strategy is

44

proposed to solve the ED problems. The main contributions of this paper are listed as follows:

45

(1) A biogeography-based learning particle swarm optimization (BLPSO) algorithm which em-

46

ploys a biogeography-based learning strategy (BLS) is presented. The computational com-

47

plexity of BLPSO is also analyzed.

48

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(2) Combining the feature of ED problems, the BLPSO-based economic dispatch method is developed.

50

(3) BLPSO is applied to solve five ED problems with various practical constraints, and the

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experimental results demonstrate that the proposed method can obtain promising results for

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ED problems.

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This paper is organized as follows: Section 2 briefly introduces the formulation of ED problem-

54

s. Section 3 introduces the original PSO and its three variants. In addition, a biogeography-based

55

learning particle swarm optimization algorithm is presented in this section. Section 4 addresses

56

the implementation of BLPSO for solving ED problems. Section 5 provides the experimental

57

results on five test systems. Finally, the paper is concluded in Section 6.

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2. Formulation of ED problems

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The objective of the ED problem is to minimize the fuel cost of thermal power plants for a

60

given load demand subject to various physical constraints.

61

2.1. Objective function

62

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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:

min F =

Ng X j=1

Fj (Pj ) =

Ng X

(aj + bj Pj + cj Pj2 )

(1)

j=1

64

where Ng is the total number of generating units or generators, Fj (Pj ) is the cost function of

65

the j-th generating unit ($/hr), Pj is the real output of the j-th generating units (in MW), and

66

aj , bj and cj are fuel cost coefficients of j-th generator.

3

67

In some ED problems, the admission valves control the steam entering the turbine through

68

separate nozzle groups. When the valve opens, the fuel cost will increase dramatically because

69

of the wire drawing effect, and this makes the practical objective function have many non-

70

differentiable points [44]. Therefore, the fuel cost function often contains many non-smooth

71

ripple curves due to the presence of valve point effects. The objective function when the valve

72

point effect is taken into account is represented as:

min F =

Ng X

Fj (Pj ) =

j=1

Ng X

(aj + bj Pj + cj Pj2 ) + ej sin(fj (Pjmin − Pj ))

(2)

j=1

73

where ej and fj are non-smooth fuel cost coefficients of the j-th generator with valve-point

74

effects; Pjmin is the minimum power generation limit of the j-th generator (in MW).

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2.2. Optimization constraints

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2.2.1. Power balance constraint

77

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

P j = PD + PL

(3)

j=1 79

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The B coefficients method is widely utilized to calculate the total transmission network loss PL . In such a way, PL can be calculated as follows:

PL =

Ng Ng X X

Pj Bji Pi +

j=1 i=1

Ng X

B0j Pi + B00

(4)

j=1

81

where Bji , Bj0 and B00 are the loss coefficients or B-coefficients. It can be seen that [Bji ] is an

82

Ng × Ng matrix.

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2.2.2. Power generation limits

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The power generation of each generator should be within its minimum and maximum limits:

Pjmin ≤ Pj ≤ Pjmax

(5)

85

where Pjmin and Pjmax are the minimum and maximum power generation limits of the j-th

86

generator.

4

87

88

89

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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: Pj − Pj0 ≤ U Rj

(6)

Pj0 − Pj ≤ DRj

(7)

Similarly, the decrease is limited by:

91

where Pj0 is the previous output power, U Rj and DRj are the up-ramp limit and the down-ramp

92

limit of the j-th generator respectively.

93

94

Combining Eq.6 and 7 with Eq.5 results in the change of the effective operating or generation limits to: max(Pjmin , Pj0 − DRj ) ≤ Pj ≤ min(Pjmax , Pj0 + U Rj )

95

96

97

(8)

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 j-th generator can be described as follows:  l   P min ≤ Pj ≤ Pj,1   j u l , k = 2, 3, · · · , nj , j = 1, 2, · · · , Ng Pj ∈ Pj,k−1 ≤ Pj ≤ Pj,k     Pu ≤ P ≤ P max j

j,nj −1

(9)

j

98

l u where nj is the number of prohibited zones of j-th generator. Pj,k and Pj,k are the lower and

99

upper power output of the k-th prohibited zone of the j-th generator, respectively.

100

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

min F =

Ng P j=1

s.t.

Ng P

Ng P

Fj (Pj ) =

j=1

(aj + bj Pj + cj Pj2 ) + ej sin(fj (Pjmin − Pj ))

Pj = PD + PL

j=1 l max(Pjmin , Pj0 − DRj ) ≤ Pj ≤ Pj,1 u l Pj,k−1 ≤ Pj ≤ Pj,k u Pj,n ≤ Pj ≤ min(Pjmax , Pj0 + U Rj ) j −1

k = 2, 3, · · · , nj , j = 1, 2, · · · , Ng

5

(10)

101

3. Particle swarm optimization and its three variants

102

3.1. Particle swarm optimization

103

The PSO algorithm is a population-based metaheuristic algorithm which was firstly proposed

104

by Kennedy and Eberhart [45]. It is based on the swarm intelligence theory and the fundamental

105

idea is that the optimal solution can be found through cooperation and information sharing

106

among individuals in the swarm. In the past decade, PSO has gained increasing popularity due

107

to its effectiveness in performing difficult optimization tasks.

108

In PSO, each individual is treated as a particle in the D-dimensional space, with a position

109

vector xi (t) = [xi1 (t), xi2 (t), · · · , xiD (t)] and a velocity vector vi (t) = [vi1 (t), vi2 (t), · · · , viD (t)].

110

The particle updates its velocity and position according to the following equations: vij (t + 1) = wvij (t) + c1 r1 (pbestij (t) − xij (t)) + c2 r2 (gbestj (t) − xij (t))

(11)

xij (t + 1) = xij (t) + vij (t)

(12)

111

112

where pbesti (t) = [pbesti1 (t), pbesti2 (t), · · · , pbestiD (t)] is the personal best position of particle

113

i; gbest(t) = [gbest1 (t), gbest2 (t), · · · , gbestD (t)] is the position of the best particle in the popu-

114

lation; w is the inertia weight; c1 and c2 are acceleration coefficients; r1 and r2 are two random

115

real numbers distributed uniformly within [0,1].

116

3.2. Comprehensive learning particle swarm optimization

117

Liang et al.[46] proposed a comprehensive learning PSO (CLPSO) which uses a novel compre-

118

hensive learning strategy whereby all other particles personal best positions is used to update a

119

particles velocity. This strategy can preserve the diversity of the swarm to discourage premature

120

convergence. CLPSO uses the following velocity updating equation: vij (t + 1) = wvij (t) + cr1 pbestτi (j),j (t) − xij (t)

(13)

121

where pbestτ i (t) = [pbestτi (1),1 (t), pbestτi (2),2 (t), · · · , pbestτi (D),D (t)] is the learning exemplar

122

for particle i, τ i = [τi (1), τi (2), · · · , τi (D)] is the learning exemplar indices for particle i, which

123

is generated based on tournament selection procedure. The CLPSO does not introduce any

124

complex operations to the original simple PSO framework, and the main difference from the

125

original PSO is the velocity update equation.

6

126

3.3. Social leaning particle swarm optimization

127

Cheng and Jin [47] proposed a social learning PSO (SLPSO) inspired by learning mechanisms

128

in social learning of animals. The SLPSO is performed on a sorted swarm, and particles learn

129

from any better particles in the current swarm. The particles learn from different particles based

130

on the following equations:  x (t) + ∆x (t) ij ij xij (t + 1) = x (t) ij

if pi (t) ≤ PiL

(14)

otherwise

131

∆xij (t + 1) = r1 ∆xij (t) + r2 (xkj (t) − xij (t)) + r3 ε(¯ xj (t) − xij (t)) 132

133

134

(15)

where PiL is the learning probability for particle i; pi (t) is a randomly generated probability satisfies 0 ≤ pi (t) ≤ PiL ≤ 1; xkj (t) is demonstrator of particle i in the j-th dimension; x ¯j (t) = N P xij /N is the mean position of the all particles in the current swarm; and ε is the social i=1

135

influence factor. In addition, the SLPSO adopts dimension-dependent parameter control methods

136

to determine the three parameters, i.e., the swarm size N , PiL the learning probability , and the

137

social influence factor ε.

138

3.4. Biogeography-based learning particle swarm optimization

139

In this paper, a biogeography-based learning particle swarm optimization (BLPSO) which

140

employs a new biogeography-based learning strategy (BLS) [48] is proposed for the ED problems.

141

3.4.1. Biogeography-based learning strategy

142

BLS is inspired from both from the comprehensive learning strategy of CLPSO [46] and

143

biogeography-based optimization [49, 50]. It has two characteristics:

144

(1) Each particle updates itself by using the combination of its own personal best position

145

and personal best positions of all other particles, which is similar to comprehensive learning

146

strategy of CLPSO. This updating method enables the diversity of the swarm to be preserved

147

to discourage premature convergence.

148

(2) The migration operator of biogeography-based optimization is used to generate the learning

149

exemplar for each particle, in which a ranking technique is employed to make particles learn

150

more from particles with high quality personal best positions. This can provide a more

151

effecient balance between exploration and exploitation for the new PSO algorithm.

7

152

In BLS, each particle updates its velocity and position according to the following equations: vij (t + 1) = wvij (t) + cr1 pbestτi (j),j (t) − xij (t)

(16)

xij (t + 1) = xij (t) + vij (t)

(17)

153

154

where pbestτi (t) = [pbestτi (1),1 (t), pbestτi (2),2 (t), · · · , pbestτi (D),D (t)] is the learning exemplar for

155

particle i; τ i = [τi (1), τi (2), · · · , τi (D)] is the learning exemplar indices for particle i, which is

156

generated by the biogeographic migration.

157

158

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 f (pbests1 ) ≤ f (pbests2 ) ≤ · · · ≤ f (pbestsN )

(18)

159

where s1 is the subscript of the particle with the best pbest, s2 is the subscript of the particle

160

with the second best pbest, and sN is the subscript of the particle with the worst pbest; N is

161

the population size. Then, the rankings of particles are assigned as below:

rank(xsk ) = N − k, k = 1, 2, · · · , N 162

163

(19)

Second, immigration and emigration rates are assigned for all particles. The immigration and emigration rates for all particles can be calculated as follows:   λ(x ) = 1 − N −k 2 sk N  µ(x ) = N −k 2 sk

, k = 1, 2, · · · , N

(20)

N

164

According to Eq.20, the solution xs1 with the best pbests1 will have the lowest immigration

165

rate and highest emigration rate; and the solution xsN with the worst pbestsN will have the

166

highest immigration rate and lowest emigration rate.

167

Third, the biogeography-based exemplar indices τ i = [τi (1), τi (2), · · · , τi (D)] for particle i

168

can be generated based on the biogeography-based exemplar generation method, see Fig. 1.

169

3.4.2. Procedures of BLPSO

170

171

172

Using the BLS, the procedures of BLPSO can be outlines in Fig. 2. It can be seen from Fig. 2 that the structure of BLPSO is as simple as the classic PSO. In addition, based on lines 10-13 in Algorithm 2, it can be seen that Algorithm 1 is

8

Algorithm 1. Biogeography-based exemplar generation method

k , and emigration rates  k

1

Input: particle index i , immigration rate

2

Output: learning exemplar indices  i  [ i (1),

3 4 5 6 7 8 9 10 11

For k = 1 to D Do // for each dimension If rand  rank ( i ) Then Utilize a roulette wheel to select a particle index j with probability   rank ( j ) ;

 i (k )  j ; // learn from other particle Else

 i (k )  i ; // learn from itself End If End For If  i (k )  i in all dimension Then

12

Randomly select a particle index l (l  i ) ;

13

Randomly select a dimension d ;

14 15 16

, i (d ), , i ( D)]

 i (d )  l ; End If Return  i  [ i (1),

, i (d ), , i ( D)]

Figure 1: Biogeography-based exemplar generation method

173

executed to generate new learning exemplar indices τ i only when there is a stagnation for G

174

generations, which is used to save computational cost of the BLPSO. In other words, if new

175

learning exemplar indices τ i are generated for all particles in each generation, Algorithm 1 will

176

be executed too frequently, and this may cost a large computational times, which is inappropriate

177

for real-world ED problems.

178

3.4.3. Remarks

179

(1) Complexity analysis: The computational costs of the original BLPSO algorithm involve

180

the initialization (Tini ), biogeography-based exemplar generation method(Tbio ), velocity and po-

181

sition update (Tupd ), and evaluation (Teva ) for each particle. Assume D is the dimensionali-

182

ty of the optimization problem, N is the population size, and maxF ES is maximum number

183

of functional evaluations allowed for the algorithm. The complexity of initialization, velocity

184

and position update, and evaluation are O(D), O(2D), and O(D) respectively. The computa-

185

tional costs of biogeography-based exemplar generation method Tbio include population sorting

186

O(N · log(N )), rankings assignment O(N ), immigration and emigration rates assignment O(2N ),

187

and migration operator O(N ·D). Therefore, Tbio = O(N ·log(N ))+O(N )+O(2N )+O(N ·D) =

188

O(3N + N · log(N ) + N · D).

9

Algorithm 2 BLPSO 1 2 3

Stage 1: Initialization For each particle i  1, 2,

, N Do

Initialize position xi and velocity v i ;

4

Evaluate the fitness f ( xi ) and store the personal best position pbesti ;

5

Generate the learning exemplar indices  i using Algorithm 1;

6

Set the refreshing gap G ;

7 8 9 10

End For Stage 2: Main loop For each particle i  1, 2,

, N Do

If the stagnation number stagnated (i )  G Then

11

Generate the exemplar vector index  i using Algorithm 1;

12

Set stagnated (i )  0;

13

End If

14

Update velocity v i using Eq.16;

15

Update position xi using Eq.17;

16

Evaluate the new position f ( xi ) ;

17

If xi is better than pbesti Then

18 19 20 21

Set pbesti  xi , stagnated (i )  0; Else stagnated (i)  stagnated (i)  1;

End If

22

End For

23

Store swarm’s gbest with the best pbesti .

24

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

Figure 2: BLPSO

189

The total computational complexity of BLPSO is: TBLP SO = Tini + (Teva + Tupd + 3N +N ·log(N )+N ·D ) G

1 G Tbio )

·

· maxF ES). In general, the population size

190

maxF E = O(D + (D + 2D +

191

N is often set to be proportional to the problem dimension D ( i.e., N = kD) [47]. Thus,

192

the complexity of BLPSO is: TBLP SO = O(D + (D + 2D +

193

O(D2 · maxF ES).

3kD+kD·log(kD)+kD 2 ) G

· maxF ES =

194

(2) Compared with previous hybrid PSO/BBO algorithms: Several hybrid PSO/BBO al-

195

gorithms have been proposed in the literature. For example, Guo et al. [51] presented a

196

biogeography-based particle swarm optimization with fuzzy elitism (BPSO-FE) for constrained

197

engineering problems. In this BPSO-FE algorithm, the whole population is split into several sub-

198

groups, and BBO is employed to search within each subgroup while PSO for the global search.

199

Mo and Xu [52] applied the position updating strategy of PSO to increase the diversity of pop-

10

200

ulation in BBO and developed a biogeography particle swarm optimization algorithm (BPSO)

201

to optimize the paths in path network. However, there are some difference between BLPSO and

202

them. First, the hybrid strategies of BLPSO, BPSO-FE and BPSO are different. In BLPSO, the

203

biogeography-based migration is used to generate the learning exemplar for each particle; while

204

in BPSO-FE and BPSO, the biogeography-based migration is used as search operator. Second,

205

the application areas of BLPSO, BPSO-FE and BPSO are different. BLPSO is presented for

206

ED problems, while BPSO-FE and BPSO are proposed for classical engineering optimization

207

problems and robot path planning respectively.

208

4. Implementation of BLPSO for ED problems

209

When solving the ED problems using BLPSO, the following three important issues should be

210

considered: initialization of population, constraints handling, and stopping criterion.

211

4.1. Initialization of population

212

In BLPSO, each individual of the population is a solution of an ED problem. If there are

213

Ng units that must be operated to provide power to load, then the current position of the i-th

214

particle can be given by:

Xi = [Pi1 , Pi2 , · · · , Pij , · · · , PiNg ], k = 1, 2, · · · , N

(21)

215

where N is the population size, j is index of the generating unit, and Pij is the generation power

216

output of the j-th generating unit in the i-th particle.

217

4.2. Constraints handling

218

One of the most important issues in solving ED problems is how to handle the quality and

219

inequality constraints. There are four types of constraints in the ELD problems: power generation

220

limits, ramp rate limits, prohibited operating zone and power balance constraint.

221

222

For power generation limits and ramp rate limits constraints, the following strategy is employed.    max(Pjmin , Pj0 − DRj ), if Pij ≤ max(Pjmin , Pj0 − DRj )   Pij = min(Pjmax , Pj0 + U Rj ), if Pij ≥ min(Pjmax , Pj0 + U Rj )     P , otherwise ij

11

(22)

223

For prohibited operating zones constraints, if Pij locates in the k-th prohibited operating

224

l u zone, i.e. Pj,k ≤ Pij ≤ Pj,k , it is truncated to the closest boundary of the k-th prohibited

225

operating zone as follows:

Pij =

  Pl , j,k  Pu , j,k

. l l u if Pj,k < Pij ≤ (Pj,k + Pj,k ) 2 . l u u if (Pj,k + Pj,k ) 2 < Pij < Pj,k

(23)

226

l u where Pj,k and Pj,k denotes the lower and the upper bounds of prohibited operation zone k of

227

generator j, respectively.

228

229

For the power balance constraint, a repaired operator together with a common penalty is employed [28]. The repaired operator is shown in Fig.3, and the objective function becomes: Ng X Pj − PD − PL min F = Fj (Pj ) + K j=1 j=1 Ng X

(24)

231

P N g where K is the penalty coefficient; the penalty term Pj − PD − PL is the measure of violation j=1 of the equality constraint .

232

4.3. Stopping criterion

230

233

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

234

maxF ES is reached.

235

5. Results and discussion

236

To test the effectiveness of the proposed BLPSO algorithm, five different test systems of

237

varying computational difficulty levels have been solved using BLPSO. The results obtained by

238

BLPSO are compared with two PSO algorithms, comprehensive learning PSO (CLPSO) [46]

239

and social leaning PSO (SLPSO) [47]. The results are also compared with several techniques

240

reported in the literature whose abbreviations are listed in Table 1.

241

To compare the performance of the BLPSO, 50 independent trial runs are made, and the

242

statistical results including the minimum, mean, maximum fuel cost and standard deviation, as

243

well as average run time are tabulated for each test system. The parameters of BLPSO are set

244

as: population size N = 40, inertia weight w linearly decreases from 0.9 to 0.2, acceleration

245

coefficient c = 1.496, and refreshing gap G = 5. The parameters of CLPSO and SLPSO are set

246

as those recommended in their original papers. The maximum number of functional evaluations 12

Algorithm 3 The repair operator for power balance constraint 1

Initialize a set I  {1, 2, Ng

2

If

P j 1

ij

, D}

 PD  PL Then

3

Randomly select a component k from the set I ;

4

While Pik  Pi max

5

Exclude k from I , and let the new set be I ' ;

6

Randomly select a component k ' from I ' ;

7

k  k ', I  I ' ;

8

End While

9

Add an amount w |  Pij  PD  PL | to Pik , such as Pik  min ( Pik  w, Pi max ) ;

Ng

j 1

Ng

10

Else If

P j 1

ij

 PD  PL

11

Randomly select a component k from the set I ;

12

While Pik  Pi min

13

Exclude k from I , and let the new set be I ' ;

14

Randomly select a component k ' from I ' ;

15

k  k ', I  I ' ;

16

End While

17

Add an amount w |  Pij  PD  PL | to Pik , such as Pik  max ( Pik  w, Pi min ) ;

Ng

j 1

18

End If

Figure 3: Repair operator for power balance constraint

247

maxF ES is set as 10000, 50000, 50000, 50000, and 200000 for the five test systems, respectively.

248

The programs are implemented in MATLAB language on a personal computer with a 3.2 GHz

249

processor and 8 GB RAM.

250

5.1. Test system 1

251

This is a small system comprising 6 generators and meeting a load demand of 1263 MW, and

252

includes transmission loss, POZ and ramp rate limits. The system data are taken from [8, 53]

253

and listed in Table S1. Table 2 presents the optimal generation values and fuel cost obtained by

254

BLPSO. The obtained optimal cost is 15447.34 $/hr. It can be seen that the generation values

255

satisfy the generation limit constraints and do not fall in the POZs.

256

Table 3 shows the comparison of the statistical results of different algorithms. In the table,

257

the results obtained by BLPSO are compared with CLPSO, SLPSO, NPSO-LRS [54], MTS

13

Table 1: Table of abbreviations Optimization algorithms Backtracking search algorithm Biogeography-based optimization Chaotic bat algorithm Chaotic improved honey bee mating optimization Continuous quick group search optimizer Differential evolution Enhanced Hopfield neural network Firefly algorithm Genetic algorithm-ant colony optimization Group search optimizer Honey bee mating optimization Hybrid chemical reaction optimization with differential Evolution Hybrid differential evolution with biogeography-based optimization Harmony search Hopfield modeling framework Hybrid harmony search Hybrid differential evolution with biogeography-based optimization Hybrid differential evolution with particle swarm optimization. Immune algorithm Improved differential evolution Improved orthogonal design particle swarm optimization-global version Improved orthogonal design particle swarm optimization-local version Improved random drift particle swarm optimization Modified artificial bee colony Multiple tabu search Multi-strategy ensemble biogeography-based optimization New particle swarm optimization New particle swarm optimization with local random search Oppositional invasive weed optimization Oppositional real coded chemical reaction optimization Particle swarm optimization Particle swarm optimization with chaotic sequences and crossover operator Particle swarm optimization with time varying acceleration coefficients Random drift particle swarm optimization Self-tuning improved random drift particle swarm optimization Simulated annealing Stochastic weight trade-off particle swarm optimization Tabu search

Abbreviation BSA BBO CBA CIHBMO CQGSO DE EHNN FA GAAPI GSO HBMO HCRO-DE DE/BBO HS HM HHS DE/BBO DEPSO IA IDE IODPSO-L IODPSO-G IRDPSO MABC MTS MsEBBO New-PSO NPSO-LRS OIWO ORCCRO PSO CCPSO PSO-TVAC RDPSO ST-IRDPSO SA SWT-PSO TS

258

[55], TS [55], SA [55], GAAPI [56], HCRO-DE [42], DE [57], MABC [31], CBA [58], RDPSO

259

[26], IRDPSO [26], and ST-IRDPSO [26]. It can be seen that, the minimum and mean fuel

260

cost obtained by BLPSO are similar to SLPSO, and less than all the other methods with the

261

exceptions of HCRO-DE [42]. In addition, the smaller value of standard deviation indicates that

262

BLPSO is consistent. It is also important to note that the BLPSO is very efficient according to

263

average computational time (0.50 s), which is less than most of other methods. Fig.4 presents the

264

convergence characteristics obtained by CLPSO, SLPSO and BLPSO. From Fig.4, SLPSO has

265

the fastest convergence speed, and BLPSO the second. Both BLPSO and SLPSO can converge

266

to the optimal cost after about 6000 functional evaluations.

14

Table 2: Optimal generations and cost obtained by BLPSO for Test System 1 (6-unit system, PD =1263MW) Unit 1 2 3 4 5 6 Cost ($/hr) Transmission loss (MW)

Pjmin Pjmax 100 500 50 200 80 300 50 150 50 200 50 120 15447.34 12.6619

POZ [210, 240]; [350, 380] [90, 110]; [140, 160] [150, 170]; [210, 240] [80, 90]; [110, 120] [90, 110]; [140, 150] [75, 85]; [100, 105]

Generation 447.0682 173.5899 263.2341 142.6879 162.5776 86.5033

Table 3: Comparison of fuel costs and statistical results for Test System 1 (6-unit system, PD =1263MW) Algorithm NPSO-LRS [54] MTS [55] TS [55] SA [55] GAAPI [56] HCRO-DE [42] DE [57] MABC [31] CBA [58] RDPSO [26] IRDPSO [26] ST-IRDPSO [26] CLPSO SLPSO BLPSO

Minimum cost($/h) 15450 15450.06 5454.89 15461.1 15449.78 15443.075 15449.5826 15449.8995 15450.2381 15449.89 15449.89 15449.89 15447.72 15447.34 15447.34

Mean cost ($/h) 15454 15451.17 15472.56 15488.98 15449.81 15443.327 15449.62 15449.8995 15454.76 15458.01 15456.55 15450.7 15449.83 15447.46 15447.45

Maximum cost ($/h) 15452 15450.06 15454.89 15461.1 15449.85 15443.916 15449.6508 15449.8995 15518.6588 NA NA NA 15452.88 15447.62 15447.67

Standard deviation NA 0.9287 13.7195 28.3678 NA 0.067 NA 6.04E-08 2.965 13.647 10.9865 1.416 1.28 0.08 0.09

Time (s) NA 1.29 20.55 50.36 NA 4.17 3.634 0.62 0.704 0.707 0.676 0.727 0.48 0.84 0.50

NA means the data are not available in the literature.

267

5.2. Test system 2

268

This test system consists of 15 generators meeting a load demand of 2630 MW, and includes

269

transmission loss, POZ and ramp rate limits. The system data are taken from [8, 59] and listed

270

in Table S2. Table 4 presents the optimal generations and the costs obtained. The optimal cost

271

obtained by BLPSO is 32587.33 $/hr, and the generations satisfy the generation limit constraints.

272

Table 5 shows the comparison of the statistical results of the BLPSO and other algorithms,

273

including CLPSO, SLPSO, CCPSO [60], HBMO [59], CIHBMO [59], FA [15], MsEBBO [35],

274

DEPSO [40], SWT-PSO [61], IA [62], IODPSO-G [63], and IODPSO-L [63]. The minimum and

275

mean fuel costs obtained by BLPSO are the least of all methods with the exceptions of HBMO

276

[59]. The average computation time of BLPSO (2.85s) is also very small. The convergence

277

characteristics obtained by CLPSO, SLPSO and BLPSO are plotted in Fig.5. SLPSO has the

278

fastest convergence speed in the beginning, but it is surpassed by BLPSO and CLPSO in the

279

end. Only BLPSO can get the optimal cost in this case.

15

Table 4: Optimal generations and cost obtained by BLPSO for Test System 2 (15-unit system, PD =2630MW) Unit 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Cost ($/hr) Transmission loss (MW)

Pjmin Pjmax 150 455 150 455 20 130 20 130 150 470 135 460 135 465 60 300 25 162 25 160 20 80 20 80 25 85 15 55 15 55 32587.33 12.6619

Generation 455.0000 450.0000 130.0000 130.0000 200.0000 460.0000 430.0000 60.0000 35.1998 91.2383 80.0000 80.0000 25.0000 15.0000 15.0000

Table 5: Comparison of fuel costs and statistical results for Test System 2 (15-unit system, PD =2630MW) Algorithm CCPSO [60] HBMO [59] CIHBMO [59] FA [15] MsEBBO [35] DEPSO [40] SWT-PSO [61] IA [62] IODPSO-G [63] IODPSO-L [63] CLPSO SLPSO BLPSO

Minimum cost($/h) 32704.4514 32637.6219 32548.58588 32,704.50 32,692.40 32588.81 32704.45 32,698.20 32,692.39 32,692.39 32608.83 32674.25 32587.33

Mean cost ($/h) 32704.4514 32663.19 32548.58588 32,856.10 32,692.40 32588.99 NA 32,750.22 32,692.39 32,692.39 32649.9 32707.46 32607.17


16

Maximum cost ($/h) 32704.4514 32676.07 32548.58588 33,175.00 32,692.40 32591.49 NA 32,823.78 32,692.39 32,692.39 32705.5 32758.69 32667.2

Standard deviation 0 NA NA 147.17 0 4.02 NA 9.3 NA NA 23.13 14.67 17.06

Time (s) 16.2 2.8 3.1 NA NA 1.88 s NA NA NA NA 2.77 4.93 2.85

1.55

x 10

4

CLPSO SLPSO BLPSO

Average fuel Cost

1.549 1.548 1.547 1.546 1.545 1.544

0

2000

4000

6000

8000

10000

FES

Figure 4: Convergence characteristics for test system 1

280

5.3. Test system 3

281

This test system consists of 20 generators supplying a demand of 2500 MW. Transmission

282

losses are included in this system. The cost coefficients and B-coefficients data are taken from

283

[22, 64] and listed in Table S3. Table 6 presents the optimal generation values and fuel cost

284

obtained by the BLPSO. The optimal obtained fuel cost are 62456.58$/hr. It is seen that the all

285

the generation limit constraints are satisfied.

286

Table 7 shows the comparison of the statistical results of different algorithms. In the table,

287

the results obtained by BLPSO are compared with CLPSO, SLPSO, EHNN [65], λ-iteration

288

[64], HM [64], GSO [66], CQGSO [66], BBO [22], BSA [22], and CBA [58]. It can be seen that,

289

the best fuel cost obtained by BLPSO is the least of all methods, and the mean fuel cost is the

290

least of all methods with the sole exception of CBA [58].In addition, the standard deviation and

291

average computation time of BLPSO are both very small. Again, the BLPSO is efficient for this

292

case. The convergence characteristics obtained by CLPSO, SLPSO and BLPSO are plotted in

293

Fig.6.

294

5.4. Test system 4

295

This test system consists of 38 generators, and the demand of this system is 6000 MW.

296

The system data are taken from [28, 41] and listed in Table S4. Table 8 presents the optimal

297

generation values and cost obtained by BLPSO. The optimal cost is 9417208.19 $/hr. It is seen

298

that the generations satisfy the generation limit constraints.

17

Table 6: Optimal generations and cost obtained by the BLPSO for Test System 3 (20-unit system, PD =2500MW) Unit 1 2 3 4 5 6 7 8 9 10 Cost ($/hr) Transmission loss (MW)

Pjmin Pjmax 150 600 50 200 50 200 50 200 50 160 20 100 25 125 50 150 50 200 30 150 62456.58 92.0046

Generation 512.2358 169.7731 126.4272 102.6131 113.9049 73.6208 115.506 116.631 100.2842 105.5532

Unit 11 12 13 14 15 16 17 18 19 20

Pjmin 100 150 40 20 25 20 30 30 40 30

Pjmax 300 500 160 130 185 80 85 120 120 100

Generation 150.2329 293.3209 118.8701 30.6567 115.3421 36.2973 67.0567 87.9775 101.2398 54.4584


Table 7: Comparison of fuel costs and statistical results Test System 3 (20-unit system, PD =2500MW) Algorithm EHNN [65] λ-iteration [64] HM [64] GSO [66] CQGSO [66] BBO [22] BSA [22] CBA [58] CLPSO SLPSO BLPSO

Minimum cost($/h) 62610 62456.6391 62456.6341 62456.6332 62456.633 62456.7793 62456.6925 62456.6328 62456.44 62456.92 62456.58

Mean cost ($/h) NA NA NA 62456.6336 62456.6331 62456.7928 62457.1517 62456.6348 62456.84 62457.38 62456.64


18

Maximum cost ($/h) NA NA NA 62456.6353 62456.63337 62456.7928 62458.1272 62501.6714 62457.10 62458.06 62456.65

Standard deviation NA NA NA NA NA NA NA 0.3879 0.17 0.28 0.01

Time (s) 0.11 33.757 6.355 30.45 11.13 NA 14.477 1.16 2.68 3.80 2.75

3.32

x 10

4

CLPSO SLPSO BLPSO

Average fuel Cost

3.31 3.3 3.29 3.28 3.27 3.26 3.25

0

1

2

3 FES

4

5 x 10

4


299

The results obtained by BLPSO are compared with those obtained by CLPSO, SLPSO,

300

New-PSO [67], PSO-TVAC [67], HS [68], HHS [68], BBO [41], DE/BBO [41], MsEBBO [35], and

301

IDE[28], as shown in Table 9. It can be seen that, the minimum and mean fuel cost obtained

302

by BLPSO are the least of all the methods. The average computation time of BLPSO is 2.89s,

303

smaller than all methods, with the exception of CLPSO. The convergence characteristics obtained

304

by CLPSO, SLPSO and BLPSO are plotted in Fig.7.

305

5.5. Test system 5

306

In order to study the performance of the BLPSO on high dimensional ED problems, a large

307

system with 110 generators is considered. The demand of this system is 15000MW, and the

308

system data are taken from [69, 70] and listed in Table S5. Table 10 presents the optimal

309

generation values and cost obtained by BLPSO. The optimal cost is 197988.16 $/hr.

310

Table 11 shows the comparison of the statistical results of BLPSO and other algorithms,

311

including CLPSO, SLPSO, SAB [71], SAF [71], SA [71], ORCCRO [72], BBO [72], DE/BBO

312

[72], and OIWO [69]. The minimum, mean, and maximum fuel cost obtained by BLPSO are

313

the least of all the methods. Meanwhile, the smaller value of standard deviation indicates that

314

BLPSO is consistent. The average computation time of BLPSO is also very small compared with

315

other methods. The convergence characteristics obtained by CLPSO, SLPSO and BLPSO are

316

presented in Fig.8.

19

Table 8: Optimal generations and cost obtained by the BLPSO for Test System 4 (38-unit system, PD =6000MW) Unit 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Cost ($/hr)

Pjmin Pjmax 220 550 220 550 200 500 200 500 200 500 200 500 200 500 200 500 114 500 114 500 114 500 114 500 110 500 90 365 82 365 120 325 65 315 65 315 65 315 9417208.19

Generation 426.5025 426.7865 429.5064 429.6401 429.7661 429.5265 429.7083 429.5542 114 114 119.7279 127.1661 110 90 82 120 159.6378 65 65

Unit 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

Pjmin 120 120 110 80 10 60 55 35 20 20 20 20 20 25 18 8 25 20 20

Pjmax 272 272 260 190 150 125 110 75 70 70 70 70 60 60 60 60 60 38 38

Generation 272 272 260 130.682 10 113.3391 88.0312 37.5497 20 20 20 20 20 25 18 8 25 21.7594 21.0902

Table 9: Comparison of fuel costs and statistical results for Test System 4 (38-unit system, PD =6000MW) Algorithm New-PSO [67] PSO-TVAC [67] HS [68] HHS[68] BBO[41] DE/BBO [41] MsEBBO [35] IDE [28] CLPSO SLPSO BLPSO

Minimum cost($/h) 9516448.312 9500448.307 9419960 9417325 9417633.638 9417235.786 9417235.776 9417235.786 9418283.79 9418407.11 9417208.19

Mean cost ($/h) NA NA 9421056 9417336 NA NA 9417235.779 9417235.786 9419255.18 9419560.62 9417234.16


20

Maximum cost ($/h) NA NA 9427466 9417466 NA NA 9417235.778 9417235.786 9420192.8 9425492.54 9417235.9

Standard deviation NA NA NA NA NA NA 0.0032 6.00E-09 526.51 1312.2 5.23

Time (s) NA NA 10.02 5.06 NA NA NA 9.149 2.86 3.53 2.89

Table 10: Optimal generations and cost obtained by the CBA for Test System 5 (110-unit system, PD =15000MW) Unit 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Cost ($/hr)

Generation 2.4 2.4 2.4 2.4 2.4 4 4 4 4 64.6059 62.3474 36.3769 56.6463 25 25 25 155 155 155 155 68.9 68.9 197988.16

Unit 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

Generation 68.9 350 400 400 500 500 200 100 10 20 80 250 360 400 40 70 100 120 156.791 220 440 560

Unit 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

Generation 660 616.4766 5.4 5.4 8.4 8.4 8.4 12 12 12 12 25.2 25.2 35 35 45 45 45 185 185 185 185

Unit 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88

Generation 70 70 70 360 400 400 105.0721 190.995 90 50 160 295.3172 174.949 98.2904 10 12 20 200 325 440 13.9066 24.4992

Unit 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110

Generation 82.7308 89.7172 57.9161 100 440 500 600 471.2608 3.6 3.6 4.4 4.4 10 10 20 20 40 40 50 30 40 20

Table 11: Comparison of fuel costs and statistical results for Test System 5 (110-unit system, PD =15000MW) Algorithm SAB [71] SAF [71] SA [71] ORCCRO [72] BBO [72] DE/BBO [72] OIWO [69] CLPSO SLPSO BLPSO

Minimum cost($/h) 206912.9057 207380.5164 198352.6413 198016.29 198241.166 198231.06 197989.14 198137.7 198240.4 197988.16

Mean cost ($/h) 207764.73 207813.37 201595.19 198016.32 198413.45 198326.66 197989.41 198200.37 198351.62 197988.18


21

Maximum cost ($/h) NA NA NA 198016.89 199,102.59 198,828.57 197989.93 198257.56 198602.16 197988.19

Standard deviation NA NA NA NA NA NA NA 27.88 96 0

Time (s) NA NA NA 0.15 115 132 31 23.43 22.59 22.81

6.25

x 10

4

CLPSO SLPSO BLPSO

Average fuel Cost

6.249

6.248

6.247

6.246

6.245

0

1

2

3

4

FES

5 x 10

4


317

6. Conclusion

318

This paper has presented a biogeography-based learning particle swarm optimization (BLP-

319

SO) for solving the economic dispatch (ED) problems, which is nonlinear, nonconvex and dis-

320

continuous in nature, with numerous equality and inequality constraints. In the BLPSO, a

321

biogeography-based learning strategy is used to generate the learning exemplar for each particle,

322

in which particles learn more from high quality particles. The biogeography-based learning strat-

323

egy can provide a more effective balance between exploration and exploitation for the BLPSO.

324

The BLPSO was applied to five test systems with various constraints such as power balance,

325

POZs, and ramp rate limits. Transmission losses have also been included in some systems. The

326

experimental results show that the fuel costs obtained by BLPSO are either comparable or lower

327

than those reported by other methods. The application to a 110-unit system shows that the

328

BLPSO is also capable of handling high dimensional ED problems.

329

In the future, we are planning to extend the BLPSO to solve other more complicated ED

330

problems, such as dynamical ED problems and environmental ED problems. We are also in-

331

terested in applying the BLPSO to other optimization problems in energy field such as solar

332

photovoltaic modeling [73].

22

9.6

x 10

6

CLPSO SLPSO BLPSO

Average fuel Cost

9.55

9.5

9.45

9.4

0.5

1

1.5

2

2.5 FES

3

3.5

4

4.5 x 10

5 4


333

7. Acknowledgements

334

This work was supported in part by Natural Science Foundation of Jiangsu Province (Grant

335

No. BK 20160540), China Postdoctoral Science Foundation (Grant No. 2016M591783), Research

336

Talents Startup Foundation of Jiangsu University (Grant No. 15JDG139), National Natural

337

Science Foundation of China (Grant NO. 61703268), and Fundamental Research Funds for the

338

Central Universities under (Grant 222201717006).

339

8. Supplementary Materials

340

Table S1 ∼ Table S5 provides the system data of the five test systems studied in this paper.

341

(1) Table S1: The system data of test system 1 (6-unit system)

342


343


344


345


346

References

347

348

[1] N. Ghorbani, E. Babaei, Exchange market algorithm for economic load dispatch, International Journal of Electrical Power & Energy Systems 75 (2016) 19–27. 23

Table S1: The system data of test system 1 (6-unit system) Unit 1 2 3 4 5 6

a 240 200 220 200 220 190

b 7 10 8.5 11 10.5 12

c 0.007 0.0095 0.009 0.009 0.008 0.0075

P min 100 50 80 50 50 50

P max 500 200 300 150 200 120

P0 440 170 200 150 190 150

UR 80 50 65 50 50 50

DR 120 90 100 90 90 90

POZ [210, 240];[350, 380] [90, 110]; [140, 160] [150, 170]; [210, 240] [80, 90]; [110, 120] [90, 110]; [140, 150] [75, 85]; [100, 105]

Table S2: The system data of test system 2 (15-unit system) Unit 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

a 10.1 10.2 8.8 8.8 10.4 10.1 9.8 11.2 11.2 10.7 10.2 9.9 13.1 12.1 12.4

b 671 574 374 374 461 630 548 227 173 175 186 230 225 309 323

c 0.000299 0.000183 0.001126 0.001126 0.000205 0.000301 0.000364 0.000338 0.000807 0.001203 0.003586 0.005513 0.000371 0.001929 0.004447

P min 150 150 20 20 150 135 135 60 25 25 20 20 25 15 15

P max 455 455 130 130 470 460 465 300 162 160 80 80 85 55 55

P0 400 300 105 100 90 400 350 95 105 110 60 40 30 20 20

UR 80 80 130 130 80 80 80 65 60 60 80 80 80 55 55

DR 120 120 130 130 120 120 120 100 100 100 80 80 80 55 55

POZ [185 225];[305 335];[420 450]

[180 200];[305 335];[390 420] [230 255];[365 395];[430 455]

[30 40];[55 65]

Table S3: The system data of test system 3 (20-unit system) Unit 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

a 1000 970 600 700 420 360 490 660 765 770 800 970 900 700 450 370 480 680 700 850

b 18.19 19.26 19.8 19.1 18.1 19.26 17.14 18.92 18.27 18.92 16.69 16.76 17.36 18.7 18.7 14.26 19.14 18.92 18.47 19.79

c 0.00068 0.00071 0.0065 0.005 0.00738 0.00612 0.0079 0.00813 0.00522 0.00573 0.0048 0.0031 0.0085 0.00511 0.00398 0.0712 0.0089 0.00713 0.00622 0.00773

24

P min 150 50 50 50 50 20 25 50 50 30 100 150 40 20 25 20 30 30 40 30

P max 600 200 200 200 160 100 125 150 200 150 300 500 160 130 185 80 85 120 120 100

Table S4: The system data of test system 4 (38-unit system) Unit 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

a 64782 64782 64670 64670 64670 64670 64670 64670 172832 172832 176003 173028 91340 63440 65468 77282 190928 285372 271676 39197 45576 28770 36902 105510 22233 30953 17044 81079 124767 121915 120780 104441 83224 111281 64142 103519 13547 13518

b 796.9 796.9 795.5 795.5 795.5 795.5 795.5 795.5 915.7 915.7 884.2 884.2 1250.1 1298.6 1298.6 1290.8 238.1 1149.5 1269.1 696.1 660.2 803.2 818.2 33.5 805.4 707.1 833.6 2188.7 1024.4 837.1 1305.2 716.6 1633.9 969.6 2625.8 1633.9 694.7 655.9

c 0.3133 0.3133 0.3127 0.3127 0.3127 0.3127 0.3127 0.3127 0.7075 0.7075 0.7515 0.7083 0.4211 0.5145 0.5691 0.5691 2.5881 3.8734 3.6842 0.4921 0.5728 0.3572 0.9415 52.123 1.1421 2.0275 3.0744 16.765 26.355 30.575 25.098 33.722 23.915 32.562 18.362 23.915 8.482 9.693

25

P min 220 220 200 200 200 200 200 200 114 114 114 114 110 90 82 120 65 65 65 120 120 110 80 10 60 55 35 20 20 20 20 20 25 18 8 25 20 20

P max 550 550 500 500 500 500 500 500 500 500 500 500 500 365 365 325 315 315 315 272 272 260 190 150 125 110 75 70 70 70 70 60 60 60 60 60 38 38

Table S5: The system data of test system 5 (110-unit system) Unit 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

a 24.389 24.411 24.638 24.76 24.888 117.755 118.108 118.458 118.821 81.136 81.298 81.464 81.626 217.895 218.335 218.775 142.735 143.029 143.318 143.597 259.131 259.649 260.176 177.057 210.002 211.91 210 180 240 220 60 50 200 140 120 90 80 70 115 150 40 300 250 100 160 130 34.389 34.411 34.638 34.761 34.888 127.755 128.108 128.458 128.821

b 25.547 25.675 25.803 25.932 26.061 37.551 37.664 37.777 37.89 13.327 13.354 13.8 13.407 18 18.1 18.2 10.694 10.715 10.737 10.758 23 23.1 23.2 10.862 7.492 7.503 12 12.1 12.2 12.5 23 13.5 13.2 12.4 10.3 9.9 13.4 13.3 12.9 12.8 12.7 12.6 7.4 6.6 6.5 6.2 26.547 26.675 26.803 26.932 17.061 38.551 36.664 38.777 38.89

c 0.0253 0.0265 0.028 0.0284 0.0286 0.012 0.0126 0.0136 0.0143 0.0088 0.0089 0.0091 0.0093 0.0062 0.0061 0.006 0.0046 0.0047 0.0048 0.0049 0.0026 0.0026 0.0026 0.0015 0.0019 0.0019 0.0014 0.0013 0.0026 0.0039 0.0051 0.005 0.0078 0.0012 0.0038 0.0043 0.0011 0.0023 0.0034 0.0067 0.0056 0.0023 0.0012 0.0045 0.0022 0.0067 0.0353 0.0365 0.038 0.0384 0.0386 0.032 0.0326 0.0236 0.0243

P min 12 12 12 12 12 20 20 20 20 76 76 76 76 100 100 100 155 155 155 155 197 197 197 350 400 400 500 500 200 100 50 20 80 250 360 400 40 70 100 120 180 220 440 560 660 700 32 32 52 52 52 60 60 60 60

P max 2.4 2.4 2.4 2.4 2.4 4 4 4 4 15.2 15.2 15.2 15.2 25 25 25 54.3 54.3 54.3 54.3 68.9 68.9 68.9 140 100 100 140 140 50 25 10 5 20 75 110 130 10 20 25 20 40 50 120 160 150 200 5.4 5.4 8.4 8.4 8.4 12 12 12 12

Unit 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110

26

a 82.136 82.298 82.464 82.626 218.895 219.335 219.775 143.735 144.029 144.318 144.597 269.131 269.649 270.176 187.057 320.002 321.91 52.136 42.298 32.464 23.626 220 190 250 230 70 60 210 150 130 80 90 80 125 160 50 400 260 110 170 140 26.389 25.411 25.638 25.76 65 82 86 84 75 56 67 68 69 72

b 14.327 14.354 14.38 14.407 19 19.1 19.2 11.694 11.715 11.737 11.758 24 24.1 24.2 11.862 8.492 8.503 13.327 12.354 11.38 9.407 14 13.1 13.2 13.5 24 14.5 14.2 13.4 11.3 8.9 14.4 14.3 13.9 13.8 13.7 13.6 8.4 7.6 7.5 7.2 26.547 26.675 26.803 26.932 15.3 16 20.2 20.2 25.6 30.5 32.5 26 25.8 27

c 0.0098 0.0099 0.0092 0.0094 0.0072 0.0071 0.007 0.0066 0.0057 0.0058 0.0059 0.0036 0.0036 0.0036 0.0025 0.0029 0.003 0.0054 0.0055 0.0099 0.0031 0.0024 0.0023 0.0036 0.0049 0.0061 0.007 0.0088 0.0022 0.0048 0.0053 0.0021 0.0033 0.0034 0.0037 0.0066 0.0043 0.0022 0.0055 0.0032 0.0077 0.0353 0.0365 0.038 0.0384 0.021 0.023 0.024 0.035 0.034 0.037 0.039 0.035 0.028 0.026

P min 96 96 100 100 120 120 120 185 185 185 185 197 197 197 360 400 400 300 250 90 50 450 600 200 120 55 40 80 200 325 440 35 55 100 220 140 100 440 500 600 700 15 15 22 22 60 80 100 120 150 280 520 150 320 200

P max 25.2 25.2 35 35 45 45 45 54.3 54.3 54.3 54.3 70 70 70 150 160 160 60 50 30 12 160 150 50 20 10 12 20 50 80 120 10 20 20 40 30 40 100 100 100 200 3.6 3.6 4.4 4.4 10 10 20 20 40 40 50 30 40 20

2.1

x 10

5

CLPSO SLPSO BLPSO

Average fuel Cost

2.08 2.06 2.04 2.02 2 1.98 0

0.5

1 FES

1.5

2 x 10

5


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