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
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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
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(IHS) [30], modified artificial bee colony (MABC) [31], incremental artificial bee colony (IABC)
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[32], ramp rate biogeography based optimization (RRBBO) [33], dynamic non-dominated sorting
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biogeography-based optimization (Dy-NSBBO) [34], multi-strategy ensemble biogeography-based
35
optimization (MsEBBO) [35], and modified group search optimizer (MGSO) [36].
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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)
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[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].
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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.
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(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
59
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
63
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
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78
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
84
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
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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
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3.1. Particle swarm optimization
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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
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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
NA means the data are not available in the literature.
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
NA means the data are not available in the literature.
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
NA means the data are not available in the literature.
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
Figure 5: Convergence characteristics for test system 2
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
NA means the data are not available in the literature.
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
NA means the data are not available in the literature.
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
Figure 6: Convergence characteristics for test system 3
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
Figure 7: Convergence characteristics for test system 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
(2) Table S2: The system data of test system 2 (15-unit system)
343
(3) Table S3: The system data of test system 3 (20-unit system)
344
(4) Table S4: The system data of test system 4 (38-unit system)
345
(5) Table S5: The system data of test system 5 (110-unit system)
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
Figure 8: Convergence characteristics for test system 5
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