# A Novel Method for Economic Dispatch with Across Neighborhood Search: A Case Study in a Provincial Power Grid, China

Authors: Guojiang Xiong; Jing Zhang; Xufeng Yuan; Dongyuan Shi; Yu He; Yao Yao; Gonggui Chen
Journal: Complexity
 (2018)

---

## Abstract

Economic dispatch (ED) is of cardinal significance for the power system operation. It is mathematically a typical complex nonlinear multivariable strongly coupled optimization problem with equality and inequality constraints, especially considering the valve-point effects. In order to effectively solve the problem, a simple yet very young and efficient population-based algorithm named across neighborhood search (ANS) is implemented in this paper. In ANS, a group of individuals collaboratively navigate through the search space for obtaining the optimal solution by simultaneously searching the neighborhoods of multiple superior solutions. Four benchmark test cases with diverse complexities and characteristics are firstly employed to comprehensively verify the feasibility and effectiveness of ANS. The experimental and comparison results fully demonstrate the superiority of ANS in terms of the final solution quality, convergence speed, robustness, and statistics. In addition, the sensitivities of ANS to variations of population size and across-search degree are studied. Furthermore, ANS is applied to a practical provincial power grid of China. All the comparison results consistently indicate that ANS is highly competitive and can be used as a promising alternative for ED problems.


---

## Body

## 1. Introduction

Economic dispatch (ED), playing an important role in the power system operation and planning, has received significant attention in recent years. The purpose of ED is to schedule the committed generating unit outputs so as to simultaneously minimize the operating cost and meet the load demand of a power system while satisfying all the equality and inequality constraints [1]. Traditionally, an approximate quadratic function is utilized to make the mathematical formulation of ED problem convex to reduce the computational difficulty. However, in practice, on one hand, the multi-valve steam turbines make the input–output curves of generators inherently present highly nonlinear characteristic. On the other hand, faults in the machines themselves or the associated auxiliaries prohibit generators from operating in some zones [2]. Therefore, the solution space of ED problem with the presence of valve-point effects and prohibited operating zones is highly nonlinear and discontinuous, making the optimization hard to be tractable. In this context, conventional solution methods including linear programming, Lagrange relaxation, nonlinear programming, quadratic programming, dynamic programming, and interior point method are likely to encounter dire difficulties and challenges mainly due to their heavy imposition of various restrictions such as continuity, convexity, and differentiability on the objective functions, and high sensitivity to the initial values of involved optimized variables.As a promising alternative to the conventional solution methods, metaheuristic methods for ED problems have attracted considerable attention recently. They have no strict requirements on the form of optimization problems and can avoid the influences of the initial condition sensitivity and gradient information. Up to now, the successfully implemented metaheuristic methods include simulated annealing [3], genetic algorithm [4, 5], particle swarm optimization [6–8], differential evolution [9, 10], artificial bee colony [11, 12], harmony search [13–17], biogeography-based optimization [18–22], teaching-learning-based optimization [23–25], firefly algorithm [26], crisscross optimization algorithm [27, 28], bat algorithm [29], grey wolf optimizer [30, 31], cuckoo search [32–34], ant lion optimizer [35], exchange market algorithm [36], symbiotic organisms search [37, 38], backtracking search algorithm [39, 40], interior search algorithm [41], whale optimization algorithm [42], mine blast algorithm [43], and hybrid methods [44–56].The abovementioned metaheuristic methods have verified their efficacy in solving the ED problems. Regardless of the achieved promising results, the no free lunch theorem [57] indicates that there is no specific method which can be adopted as a gold standard for all kinds of optimization problems. Namely, there is no single universal superior method that, theoretically, always performs best in solving the ED problems. Therefore, there are still some possibilities to attempt new ones to provide more alternatives, which inspires the authors to apply a recently developed metaheuristic method named across neighborhood search (ANS) [58] to obtain high quality solutions for the ED problems.As a simple yet versatile metaheuristic method, ANS is motivated by two common straightforward assumptions existing in different population-based algorithms: that searching around a superior solution has a higher probability to find another better solution and that high-quality solutions possess good solution components. In this context, ANS, following the law of parsimony, attempts to simultaneously search across the neighborhoods of multiple superior solutions to get as many potential good solution components as possible. The merits of ANS are its simple structure, ease of implementation, and strong robustness. In this paper, ANS is employed for the ED problems. The main contributions of this work are as follows.(1) Four benchmark test cases with diverse complexities and characteristics are firstly used to verify the feasibility and effectiveness of ANS comprehensively. The superior performance of ANS is experimentally verified by comparing with four popular population-based algorithms and some recently proposed ED solution methods.(2) The sensitivities of ANS to variations of population size and across-search degree are empirically investigated.(3) ANS is finally applied to a practical provincial power grid of China. Its performance is further verified. In addition, the experimental results reflect the necessity and importance of the power construction policy of “replacing small power plants with large ones” in China.The remainder of this paper is organized as follows. Section2 briefly introduces the mathematical formulation of ED problems. In Section 3, ANS is described. Next, in Section 4, the flowchart of ANS in solving the ED problems is illustrated. In Section 5, four benchmark test cases are employed to verify ANS. ANS is then applied to a practical provincial power grid of China in Section 6. Finally, Section 7 is devoted to conclusions and future work.


## 2. Problem Formulation

### 2.1. Objective Function

The mathematical model of ED can be formulated as follows [59]:(1)
m
i
n
C
P
=
∑
i
=
1
N
g
F
i
P
i
,
P
=
P
1
,
P
2
,
…
,
P
N
g
∈
R
N
g
s
.
t
.
h
j
P
=
0
,
j
=
1,2
,
…
,
m
g
j
P
≤
0
,
j
=
1,2
,
…
,
qwhere C
(
P
) is the total generation cost (in $/h), N
g is the number of operating generators, P
i is the active power output of thei-th generator (in MW), i
=
1,2
,
…
,
N
g, F
i
(
P
i
) is the generation cost function of thei-th generator (in $/h), i
=
1,2
,
…
,
N
g, m and q are the number of equality constraints and inequality constraints, respectively, h
j
(
P
) is thej-th equality constraint, j
=
1,2
,
…
,
m, and g
j
(
P
) is thej-th inequality constraint, j
=
1,2
,
…
,
q.The objective function of traditional ED problem is approximately formulated as follows [1, 2, 4]: (2)
F
i
P
i
=
a
i
+
b
i
P
i
+
c
i
P
i
2where a
i, b
i, and c
i are cost coefficients of thei-th generator.In practice, modelling valve-point effects is necessary and can be formulated as follows [60]:(3)
F
i
P
i
=
a
i
+
b
i
P
i
+
c
i
P
i
2
+
e
i
·
s
i
n
f
i
×
P
i
,
m
i
n
-
P
iwhere e
i and f
i are valve-point effects coefficients of thei-th generator and P
i
,
m
i
n is the minimum active power generation limit of thei-th generator (in MW).


### 2.2. Equality and Inequality Constraints

#### 2.2.1. Active Power Balance Constraint

The total active generated power must be equal to the sum of the total system demand (P
D) and the total transmission network loss (P
L): (4)
∑
i
=
1
N
g
P
i
=
P
D
+
P
Lwhere P
L is commonly calculated using the following B-coefficient method [9]:(5)
P
L
=
∑
i
=
1
N
g
∑
j
=
1
N
g
P
i
B
i
j
P
j
+
∑
i
=
1
N
g
B
0
i
P
i
+
B
00where B
i
j, B
0
i, and B
00 are loss coefficients.


#### 2.2.2. Generation Capacity Constraints

The active power output of each generator should be within its minimum and maximum limits:(6)
P
i
,
m
i
n
≤
P
i
≤
P
i
,
m
a
xwhere P
i
,
m
a
x is the maximum active power generation limit of thei-th generator (in MW).


#### 2.2.3. Ramp Rate Limits Constraints

The adjustment of active power output of each generator should be in an acceptable range:(7)
P
i
-
P
i
p
r
≤
U
R
i
P
i
p
r
-
P
i
≤
D
R
iwhere P
i
p
r is the previous active power output of thei-th generator and U
R
i and D
R
i are the up-ramp and down-ramp limits of thei-th generator, respectively.


#### 2.2.4. Prohibited Operating Zones Constraints

Generators should avoid operating in prohibited zones [2]:(8)
P
i
∈
P
i
,
m
i
n
≤
P
i
≤
P
i
,
1
L
P
i
,
k
-
1
U
≤
P
i
≤
P
i
,
k
L
,
k
=
2,3
,
…
,
p
z
i
P
i
,
p
z
i
U
≤
P
i
≤
P
i
,
m
a
xwhere p
z
i is the number of prohibited operating zones of thei-th generator and P
i
,
k
L and P
i
,
k
U are the lower bound and upper bound of thek-th prohibited zone of thei-th generator, respectively.


## 2.1. Objective Function

The mathematical model of ED can be formulated as follows [59]:(1)
m
i
n
C
P
=
∑
i
=
1
N
g
F
i
P
i
,
P
=
P
1
,
P
2
,
…
,
P
N
g
∈
R
N
g
s
.
t
.
h
j
P
=
0
,
j
=
1,2
,
…
,
m
g
j
P
≤
0
,
j
=
1,2
,
…
,
qwhere C
(
P
) is the total generation cost (in $/h), N
g is the number of operating generators, P
i is the active power output of thei-th generator (in MW), i
=
1,2
,
…
,
N
g, F
i
(
P
i
) is the generation cost function of thei-th generator (in $/h), i
=
1,2
,
…
,
N
g, m and q are the number of equality constraints and inequality constraints, respectively, h
j
(
P
) is thej-th equality constraint, j
=
1,2
,
…
,
m, and g
j
(
P
) is thej-th inequality constraint, j
=
1,2
,
…
,
q.The objective function of traditional ED problem is approximately formulated as follows [1, 2, 4]: (2)
F
i
P
i
=
a
i
+
b
i
P
i
+
c
i
P
i
2where a
i, b
i, and c
i are cost coefficients of thei-th generator.In practice, modelling valve-point effects is necessary and can be formulated as follows [60]:(3)
F
i
P
i
=
a
i
+
b
i
P
i
+
c
i
P
i
2
+
e
i
·
s
i
n
f
i
×
P
i
,
m
i
n
-
P
iwhere e
i and f
i are valve-point effects coefficients of thei-th generator and P
i
,
m
i
n is the minimum active power generation limit of thei-th generator (in MW).


## 2.2. Equality and Inequality Constraints

### 2.2.1. Active Power Balance Constraint

The total active generated power must be equal to the sum of the total system demand (P
D) and the total transmission network loss (P
L): (4)
∑
i
=
1
N
g
P
i
=
P
D
+
P
Lwhere P
L is commonly calculated using the following B-coefficient method [9]:(5)
P
L
=
∑
i
=
1
N
g
∑
j
=
1
N
g
P
i
B
i
j
P
j
+
∑
i
=
1
N
g
B
0
i
P
i
+
B
00where B
i
j, B
0
i, and B
00 are loss coefficients.


### 2.2.2. Generation Capacity Constraints

The active power output of each generator should be within its minimum and maximum limits:(6)
P
i
,
m
i
n
≤
P
i
≤
P
i
,
m
a
xwhere P
i
,
m
a
x is the maximum active power generation limit of thei-th generator (in MW).


### 2.2.3. Ramp Rate Limits Constraints

The adjustment of active power output of each generator should be in an acceptable range:(7)
P
i
-
P
i
p
r
≤
U
R
i
P
i
p
r
-
P
i
≤
D
R
iwhere P
i
p
r is the previous active power output of thei-th generator and U
R
i and D
R
i are the up-ramp and down-ramp limits of thei-th generator, respectively.


### 2.2.4. Prohibited Operating Zones Constraints

Generators should avoid operating in prohibited zones [2]:(8)
P
i
∈
P
i
,
m
i
n
≤
P
i
≤
P
i
,
1
L
P
i
,
k
-
1
U
≤
P
i
≤
P
i
,
k
L
,
k
=
2,3
,
…
,
p
z
i
P
i
,
p
z
i
U
≤
P
i
≤
P
i
,
m
a
xwhere p
z
i is the number of prohibited operating zones of thei-th generator and P
i
,
k
L and P
i
,
k
U are the lower bound and upper bound of thek-th prohibited zone of thei-th generator, respectively.


## 2.2.1. Active Power Balance Constraint

The total active generated power must be equal to the sum of the total system demand (P
D) and the total transmission network loss (P
L): (4)
∑
i
=
1
N
g
P
i
=
P
D
+
P
Lwhere P
L is commonly calculated using the following B-coefficient method [9]:(5)
P
L
=
∑
i
=
1
N
g
∑
j
=
1
N
g
P
i
B
i
j
P
j
+
∑
i
=
1
N
g
B
0
i
P
i
+
B
00where B
i
j, B
0
i, and B
00 are loss coefficients.


## 2.2.2. Generation Capacity Constraints

The active power output of each generator should be within its minimum and maximum limits:(6)
P
i
,
m
i
n
≤
P
i
≤
P
i
,
m
a
xwhere P
i
,
m
a
x is the maximum active power generation limit of thei-th generator (in MW).


## 2.2.3. Ramp Rate Limits Constraints

The adjustment of active power output of each generator should be in an acceptable range:(7)
P
i
-
P
i
p
r
≤
U
R
i
P
i
p
r
-
P
i
≤
D
R
iwhere P
i
p
r is the previous active power output of thei-th generator and U
R
i and D
R
i are the up-ramp and down-ramp limits of thei-th generator, respectively.


## 2.2.4. Prohibited Operating Zones Constraints

Generators should avoid operating in prohibited zones [2]:(8)
P
i
∈
P
i
,
m
i
n
≤
P
i
≤
P
i
,
1
L
P
i
,
k
-
1
U
≤
P
i
≤
P
i
,
k
L
,
k
=
2,3
,
…
,
p
z
i
P
i
,
p
z
i
U
≤
P
i
≤
P
i
,
m
a
xwhere p
z
i is the number of prohibited operating zones of thei-th generator and P
i
,
k
L and P
i
,
k
U are the lower bound and upper bound of thek-th prohibited zone of thei-th generator, respectively.


## 3. Across Neighborhood Search

ANS is a very young population-based algorithm proposed by Wu [58] in 2016. ANS, following the law of parsimony and showing good performance compared with other methods [61], attempts to simultaneously search across the neighborhoods of multiple superior solutions to achieve as many potential good solution components as possible. At the same time, it needs to dynamically maintain and update the superior solutions to guarantee the advancement and convergence of the population. The main difference between ANS and other population-based algorithms is that other algorithms mainly utilize some operations such as crossover and mutation to generate new solutions, whereas ANS directly searches across the neighborhoods of multiple superior solutions to produce new solutions.Like other population-based algorithms, ANS starts with a population ofp
s individuals X
=
[
X
1
,
X
2
,
…
,
X
p
s
] representing the potential solutions. Each individual X
i
=
[
x
i
,
1
,
x
i
,
2
,
…
,
x
i
,
D
] (i
=
1,2
,
…
,
p
s) consists of D variables and it is initialized as(9)
x
i
,
d
=
l
d
+
r
a
n
d
0,1
×
u
d
-
l
dwhere d
=
1,2
,
…
,
D, rand(0,1) is a uniformly distributed random real number in (0,1), and l
d and u
d are the lower bound and upper bound of thed-th dimension, respectively.In ANS, a group of individuals collaboratively navigate through the search space for obtaining the optimal solution. Each individual searches across the neighborhoods of multiple superior solutions. These superior solutions, being archived in a collectionS
=
(
S
1
,
S
2
,
…
,
S
c
) where c is the cardinality and is generally set to be the population size p
s, are directly derived from the individuals’ best positions found so far. The searching strategy is as follows:(10)
x
i
,
d
=
s
i
,
d
+
G
0
,
σ
2
×
s
i
,
d
-
x
i
,
d
i
f
d
∉
O
s
r
,
d
+
G
0
,
σ
2
×
s
r
,
d
-
x
i
,
d
e
l
s
ewhere O is a pool used to record the randomly selected n (n is called across-search degree, 1
≤
n
≤
D) dimensions for individual i, r (≠
i) is a randomly superior solution selected from the superior solution collection S, and G
(
0
,
σ
2
) is a Gaussian random value with mean zero and standard deviation σ which is usually set to be 0.5.It can be seen from (10) that each individual, on one hand, searches across the neighborhood of the individual’s best position achieved so far. On the other hand, it also simultaneously searches across the neighborhoods of other individuals’ best positions found so far. After updating individual X
i, i
=
1,2
,
…
,
p
s, its own superior solution S
i will be replaced by X
i if X
i has a better fitness value.The main procedure of ANS is given in Algorithm1. It can be seen that the individuals are guided by multiple superior solutions and the structure, following the law of parsimony, is very simple, making the implementation easy.Algorithm 1:The main procedure of ANS.
(1) Generate a random initial population X
(2) Evaluate the fitness for each individual
(3) Set X to be the superior solutions S
=
(
S
1
,
S
2
,
…
,
S
c
)
(4) Initialize the iteration counter t
=
1
(5)
While the stopping condition is not satisfied do
(6)
for
i
=
1 to p
s
do
(7) Generate a pool O to record the randomly selected n
(1
≤
n
≤
D) dimensions for individual i
(8)
for
d
=
1 to D
do
(9)
if
d
∉
O
then
(10)
x
i
,
d
=
s
i
,
d
+
G
(
0
,
σ
2
)
×
|
s
i
,
d
-
x
i
,
d
|
(11)
else
(12) Select a random superior solution r (r
≠
i) from S
(13)
x
i
,
d
=
s
r
,
d
+
G
(
0
,
σ
2
)
×
|
s
r
,
d
-
x
i
,
d
|
(14)
end if
(15) Evaluate the fitness for individual X
i
(16)
if
X
i is better than S
i
then
(17) Replace S
i with X
i
(18)
end if
(19)
end for
(20)
end for
(21)
t
=
t
+
1
(22)
End while


## 4. Implementing ANS for Solving ED Problem

The flowchart of implementing ANS for solving ED problem is depicted in Figure1. The main steps are as follows.Figure 1
Flowchart of implementing ANS for solving ED problem.Step 1.
Initialize a random population using (9).Step 2.
Handle the quality and inequality constraints using the following strategy [18, 59]:(1)
Truncatex
i
,
d
t according to(11)
x
i
,
d
t
=
P
d
,
m
i
n
,
i
f
x
i
,
d
t
<
P
d
,
m
i
n
P
d
,
m
a
x
,
i
f
x
i
,
d
t
>
P
d
,
m
a
xwhere i
=
1,2
,
…
,
p
s, d
=
1,2
,
…
,
N
g, and t
=
1,2
,
…
,
t
m
a
x; t
m
a
x is the maximum number of allowed iterations.(2)
For the prohibited operating zones constraints, ifx
i
,
d
t locates in thek-th prohibited operating zone, i.e., P
d
,
k
L
<
x
i
,
d
t
<
P
d
,
k
U, k
=
2,3
,
…
,
p
z
d, it is truncated to the closest boundary of thek-th prohibited operating zone as follows:(12)
x
i
,
d
t
=
P
d
,
k
L
,
i
f
P
d
,
k
L
<
x
i
,
d
t
<
P
d
,
k
L
+
P
d
,
k
U
2
P
d
,
k
U
,
i
f
P
d
,
k
L
+
P
d
,
k
U
2
≤
x
i
,
d
t
<
P
d
,
k
U
(3)
Calculate the corresponding transmission network lossP
L
(
i
).
(4)
Calculate the amount of the active power balance violationP
V
(
i
):(13)
P
V
i
=
∑
d
=
1
N
g
x
i
,
d
t
-
P
D
+
P
L
i
(5)
If|
P
V
(
i
)
|
=
0, go to Step (7). Otherwise, randomly select a generator r (r
≠
d) that has not been selected before and then use (14) to eliminate the power violation:(14)
x
i
,
r
t
=
x
i
,
r
t
-
m
i
n
P
V
i
,
x
i
,
r
t
-
P
r
,
m
i
n
,
i
f
P
V
i
>
0
x
i
,
r
t
+
m
i
n
P
V
i
,
P
r
,
m
a
x
-
x
i
,
r
t
,
i
f
P
V
i
<
0
(6)
Go back to Step (2) to handle ther-th generator.
(7)
Handle the next individual.Step 3.
Evaluate the fitness for each individual.Step 4.
Replace the superior solutionS
i with X
i
t if X
i
t has a better fitness value.Step 5.
Update the position of each individual.Step 6.
Go to Step2 to handle constraints if the stopping condition is not satisfied. Otherwise, stop the ANS and output the obtained results. In this study, the maximum number of fitness evaluations (Max_FEs) is used as the stopping condition.


## 5. Experimental Results on Benchmark Test Cases

In this section, we employ four benchmark test cases with different characteristics to verify the proposed ED method. These cases are described as follows.Case I.
A 13-generator system considering valve-point effects [62].Case II.
A 15-generator system considering ramp rate limits, prohibited operating zones, and transmission network losses [63].Case III.
A 38-generator system without considering valve-point effects [15].Case IV.
A 40-generator system considering valve-point effects [62].In order to validate the feasibility and effectiveness of the proposed ED method, four popular population-based algorithms, i.e., biogeography-based optimization (BBO) [64], competitive swarm optimizer (CSO) [65], differential evolution (DE) [66], and particle swarm optimization (PSO) [67], are employed for comparison. Their involved parameters are listed in Table 1. For the following experiments, p
s and Max_FEs are, respectively, set to be 10 and 1
×
1
0
3
×
N
g for the test cases without considering the valve-point effects, whereas they are 40 and 1
×
1
0
4
×
N
g for the test cases with the valve-point effects unless a change is mentioned. 50 independent runs are conducted to eliminate contingency. All experiments are executed in MATLAB 2017b.Table 1
Parameter settings of different algorithms.
Algorithm
Parameters settings
BBO
I = E = 1.0, m
m
a
x = 0.05
DE
F = 0.5, C
R = 0.9
CSO
φ = 0.1
PSO
w = 0.9→0.5, c
1 = c
2 = 2.0
ANS
n = 1, σ = 0.5


### 5.1. Experimental Results and Comparison

#### 5.1.1. Solution Quality

The experimental results of Cases I to IV are summarized in Tables2–5, respectively. The results include the minimum (Min), maximum (Max), mean costs, and standard deviation (Std Dev). These tables also list some recently proposed ED solution methods’ reported results for comparison.Table 2
Comparison of simulation results ($/h) for 13-generator system (Case III).
Method
Min.
Max.
Mean
Std Dev
FCASO-SQP [1]
17964.08
NA
18001.96
NA
TSARGA [4]
17963.94
18089.61
17974.31
3.18
DE [10]
17963.83
17975.36
17965.48
NA
ABC [11]
17963.86
17995.11
17987.22
NA
IHSWM [14]
17963.83
18041.3456
17976.4750
25.6079
aBBOmDE [20]
17963.8521
17975.0552
17967.3560
NA
TLBO [24]
18115
NA
NA
NA
CTLBO [25]
17972.81
18159.34
18013.38
43.2
FA [26]
17963.83
18168.80
18029.16
148.542
CBA [29]
17963.8339
17995.2256
17965.4889
6.8473
GWO [30]
18051.11
NA
NA
NA
GA–PS–QSP [44]
17964
NA
18199
NA
HMAPSO [48]
17969.31
17969.31
17969.31
NA
CE-SQP [54]
17963.85
NA
17965.97
NA
ANS-SDP [68]
17964
17985
17973
NA
BFO [69]
17974.48
17997.12
18018.75
NA
RTO [70]
17969.8024
18204.6303
18056.9358
NA
CMSFLA [71]
18294.9953
18297.3126
18295.7308
NA
OIWO [72]
17963.83
NA
NA
NA
IA_EDA [73]
17961.4331
18052.3155
17980.1898
21.6666
NSS [74]
17976.9512
17976.9512
17976.9512
NA
MSOS [75]
17963.8292
17963.8292
17963.8292
6.80E-12
QOSLTLBO [76]
18421.1718
NA
NA
NA
BBO
17973.1883
18090.6042
17993.9942
27.0827
CSO
17972.8619
18136.4236
18073.9003
56.1710
DE
17972.8105
18075.5913
17993.5153
38.8380
PSO
17981.3621
18312.9831
18127.7452
85.8468
ANS
17963.9031
17973.4437
17969.1487
3.0015
NA means not available in the corresponding literature.Table 3
Comparison of simulation results ($/h) for 15-generator system (Case II).
Method
Min.
Max.
Mean
Std Dev
MPSO [8]
32738.4177
NA
NA
NA
PVHS [13]
32780.00
NA
32892.463
35.331
IHSWM [14]
32693.1304
32721.3988
32699.5168
4.6937
FA [26]
32704.5
33175.0
32856.1
147.1702
ICS [32]
32706.7358
32752.5183
32715.4669
NA
EMA [36]
32704.4503
32704.4506
32704.4504
NA
BSA [39]
32704.4504
32704.5816
32704.4721
0.0280
GAAPI [46]
32732.95
32756.01
32735.06
NA
DSPSO-TSA [47]
32715.06
32730.39
32724.63
8.40
HHS [50]
32692.8571
NA
32740.1885
24.6697
ABF-NM [51]
32784.5024
NA
32976.81
85.7743
ANS-SDP [68]
32693
32693
32693
NA
RTO [70]
32701.81
32715.18
32704.53
5.07
AIS [77]
32854
32892
32873.25
10.8079
MVMO [78]
32704.47
32709.32
32706.65
NA
SCKF-PSO [79]
32725.8431
NA
NA
NA
SFS [80]
32702
NA
NA
NA
BBO
32692.3988
32693.7153
32692.6927
0.2592
CSO
32692.3972
32875.3601
32718.8607
38.0358
DE
32692.3966
32755.3193
32699.1724
12.9351
PSO
32692.3991
32781.9374
32701.0786
13.8044
ANS
32692.3962
32692.4016
32692.3973
0.0009Table 4
Comparison of simulation results ($/h) for 38-generator system (Case III).
Method
Min.
Max.
Mean
Std Dev
New-PSO [6]
9516448.312
NA
NA
NA
PSO_TVAC [6]
9500448.307
NA
NA
NA
HS [15]
9419960
9427466
9421056
NA
HHS [15]
9417325
9417466
9417336
NA
BBO(Model 6) [22]
9417557.25
9419854.237
9418394.069
NA
GWO [30]
9419270.188
9421100
9419978.978
NA
BBO [81]
9417633.6376
NA
NA
NA
DE/BBO [81]
9417235.7864
NA
NA
NA
BBO
9417446.0804
9418959.2333
9418143.6500
390.1769
CSO
9417307.5544
9417745.6891
9417494.7251
103.1130
DE
9417235.7977
9417235.8465
9417235.8184
0.0136
PSO
9426679.1431
9461023.8147
9441014.5256
9278.1375
ANS
9417235.7864
9417235.7864
9417235.7864
8.4314E-09Table 5
Comparison of simulation results ($/h) for 40-generator system (Case IV).
Method
Min.
Max.
Mean
Std Dev
FCASO-SQP [1]
121456.98
NA
122026.21
NA
TSARGA [4]
121463.07
124296.54
122928.31
NA
DE [10]
121416.29
121431.47
121422.72
NA
ABC [11]
121441.03
122023.77
121995.82
NA
IABC-LS [12]
121488.7636
121582.2525
121526.0333
NA
PVHS [13]
121415.4560
NA
121567.0292
94.1498
IHSWM [14]
121416.2652
121855.5521
121553.4208
90.1271
THS [16]
121425.15
NA
121528.65
NA
NTHS [17]
121412.7374
NA
121549.95
NA
BBO [19]
121426.953
121688.6634
121508.0325
NA
aBBOmDE [20]
121414.8734
121568.3254
121487.8532
NA
BBO(Model 5) [22]
121418.70
121954.4
121635.4
NA
TLBO [24]
12996
NA
NA
NA
CTLBO [25]
121553.83
122116.18
121790.23
150
FA [26]
121415.05
121416.57
121424.56
1.784
CBA [29]
121412.5468
121436.15
121418.9826
1.611
EMA [36]
121412.5355
121426.1548
121417.1328
NA
BSA [40]
121415.6139
121524.9577
121474.8823
NA
GA–PS–SQP [44]
121458
NA
122039
NA
HMAPSO [48]
121586.90
121586.90
121586.90
NA
FAPSO-NM [49]
121418.3
121419.8
121418.803
NA
HHS [50]
121415.5922
NA
121615.8544
114.5632
ABF-NM [51]
121423.6379
NA
121814.9465
124.8756
CE-SQP [54]
121412.88
NA
121423.65
NA
ANS-DSP [68]
12141E+05
12187E+05
12185E+05
NA
OIWO [72]
121412.54
NA
NA
NA
IA-EDA [73]
121436.9729
121648.4401
122492.7018
182.5274
NSS [74]
122186.9048
122186.9048
122186.9048
NA
MSOS [75]
121412.5355
121412.5355
121412.5355
2.47E-11
NGPSO [82]
121513.4808
122697.7672
122065.1193
267
C-GRASP-SaDE [83]
121414.621
122245.696
121736.025
166.896
BBO
121669.9298
122446.1176
121865.0181
150.4038
CSO
121440.8040
122092.4350
121637.1868
141.5265
DE
121420.9029
121618.6298
121509.0360
42.6215
PSO
122636.1149
125429.5349
123836.7368
603.6246
ANS
121412.6226
121472.9213
121427.7107
13.6539Case II is a canonical traditional ED problem with a smooth and continuous solution space. Although the solution space of Case III is discontinued by the prohibited operating zones, its objective function is also quadratic. Therefore, both cases are typical multiconstraint unimodal optimization problems which have high requirement on the solution methods’ local searching ability. It can be observed from Tables3 and 4 that ANS outperforms BBO, CSO, DE, and PSO on both cases in terms of minimum, maximum, and mean costs. In addition, ANS is also better than other reported solution methods. The comparisons demonstrate that ANS is able to search local range meticulously and thereby possesses good exploitation ability.Cases I and IV, whose solution spaces are highly nonconvex due to the valve-point effects, are multiconstraint multimodal optimization problems. The number of local minima increases at an exponential rate with the problem scale. They demand a lot on the solution methods’ global searching ability. The experimental results tabulated in Tables2 and 5 consistently indicate that ANS is significantly better than BBO, CSO, DE, and PSO in both cases in terms of minimum, maximum, and mean costs. Furthermore, the larger the system scale, the more significant the superiority. Additionally, ANS also can achieve better or highly competitive results compared with other reported solution methods. The comparisons fully conclude that ANS is with the capability of breaking away from the local minima and locating the global or near-global range. Namely, ANS has good exploration ability.The obtained optimal dispatching schedules for these four cases are presented in Tables6–9, respectively.Table 6
Best output power for 13-generator system (Case III).
Generator
BBO
CSO
DE
PSO
ANS
P
1
628.1947
628.3185
628.3185
538.5587
628.3187
P
2
223.4227
223.8651
297.5488
299.5838
223.2845
P
3
298.6788
298.0833
224.3995
224.8434
149.0866
P
4
60.0000
60.0000
60.0000
109.8666
60.0000
P
5
109.8650
60.0000
109.8666
60.0000
109.8661
P
6
60.0000
60.0000
60.0000
60.0000
109.8615
P
7
60.0000
60.0000
109.8666
109.8666
109.8603
P
8
60.0000
109.8665
60.0000
60.0004
109.8603
P
9
109.8388
109.8665
60.0000
109.8666
109.8620
P
10
40.0000
40.0000
40.0000
40.0063
40.0000
P
11
40.0000
40.0000
40.0000
40.0062
40.0000
P
12
55.0000
55.0000
55.0000
92.4002
55.0000
P
13
55.0000
55.0000
55.0000
55.0012
55.0000
T
P
a (MW)
1800.0000
1800.0000
1800.0000
1800.0000
1800.0000
T
C
b ($/h)
17973.1883
17972.8619
17972.8105
17981.3621
17963.9031
T
P
a denotes the total power output.
T
C
b denotes the total generation cost.Table 7
Best output power for 15-generator system (Case II).
Generator
BBO
CSO
DE
PSO
ANS
P
1
455.0000
455.0000
455.0000
455.0000
455.0000
P
2
380.0000
380.0000
380.0000
380.0000
380.0000
P
3
130.0000
130.0000
130.0000
130.0000
130.0000
P
4
130.0000
130.0000
130.0000
130.0000
130.0000
P
5
170.0000
170.0000
170.0000
170.0000
170.0000
P
6
460.0000
460.0000
460.0000
460.0000
460.0000
P
7
430.0000
430.0000
430.0000
430.0000
430.0000
P
8
69.9471
68.8487
69.7626
70.4928
69.4768
P
9
59.6421
60.7332
59.8226
59.0961
60.1079
P
10
159.9943
160.0000
160.0000
159.9999
160.0000
P
11
80.0000
80.0000
80.0000
80.0000
80.0000
P
12
80.0000
80.0000
80.0000
80.0000
80.0000
P
13
25.0000
25.0000
25.0000
25.0000
25.0000
P
14
15.0020
15.0000
15.0006
15.0005
15.0000
P
15
15.0011
15.0000
15.0001
15.0000
15.0000
T
L
a (MW)
29.5866
29.5819
29.5859
29.5893
29.5847
TP (MW)
2659.5866
2659.5819
2659.5859
2659.5893
2659.5847
TC ($/h)
32692.3988
32692.3972
32692.3966
32692.3991
32692.3962
T
L
a denotes the total transmission network loss.Table 8
Best output power for 38-generator system (Case III).
Generator
BBO
CSO
DE
PSO
ANS
P
1
417.4699
431.4435
426.6061
423.2786
426.6061
P
2
428.6549
432.7667
426.6061
447.2694
426.6061
P
3
430.7266
432.4754
429.6632
377.7218
429.6631
P
4
429.5193
426.7217
429.6632
391.1059
429.6632
P
5
438.8225
425.0443
429.6632
371.5138
429.6632
P
6
436.7947
426.7682
429.6632
417.4801
429.6631
P
7
424.1592
429.8489
429.6632
468.3885
429.6632
P
8
427.7913
433.0439
429.6632
376.9151
429.6631
P
9
114.0000
114.0000
114.0000
137.9410
114.0000
P
10
114.1613
114.0001
114.0000
155.5833
114.0000
P
11
114.7729
120.6554
119.7680
142.6568
119.7681
P
12
127.0026
128.0361
127.0728
187.8007
127.0729
P
13
110.0000
110.0000
110.0000
110.0000
110.0000
P
14
90.0025
90.0000
90.0000
90.0000
90.0000
P
15
82.0000
82.0000
82.0000
82.0000
82.0000
P
16
120.0000
120.0000
120.0000
120.0000
120.0000
P
17
160.1416
158.7822
159.5980
161.4797
159.5980
P
18
65.0008
65.0000
65.0000
65.0000
65.0000
P
19
65.0000
65.0000
65.0000
65.0000
65.0000
P
20
272.0000
272.0000
272.0000
271.9999
272.0000
P
21
272.0000
272.0000
272.0000
272.0000
272.0000
P
22
259.3185
260.0000
260.0000
260.0000
260.0000
P23
130.9561
129.4275
130.6486
138.0732
130.6486
P
24
10.0000
10.0000
10.0000
10.0002
10.0000
P
25
116.3424
108.7907
113.3050
110.9243
113.3050
P
26
90.6858
87.3194
88.0669
88.8202
88.0669
P
27
35.0000
36.1978
37.5051
39.1707
37.5051
P
28
20.0000
20.0000
20.0000
20.0000
20.0000
P
29
20.0000
20.0000
20.0000
20.0000
20.0000
P
30
20.0005
20.0000
20.0000
20.0000
20.0000
P
31
20.0000
20.0000
20.0000
20.0000
20.0000
P
32
20.0000
20.0000
20.0000
20.0000
20.0000
P
33
25.0000
25.0000
25.0000
25.0000
25.0000
P
34
18.0000
18.0000
18.0000
18.0000
18.0000
P
35
8.0000
8.0000
8.0000
8.0000
8.0000
P
36
25.0000
25.0000
25.0000
25.0000
25.0000
P
37
21.6764
21.5606
21.7821
21.3068
21.7821
P
38
20.0000
21.1175
21.0622
20.5701
21.0622
TP (MW)
6000.0000
6000.0000
6000.0000
6000.0000
6000.0000
TC ($/h)
9417446.0804
9417307.5544
9417235.7977
9426679.1431
9417235.7864Table 9
Best output power for 40-generator system (Case IV).
Generator
BBO
CSO
DE
PSO
ANS
P
1
112.2570
111.5065
110.8699
111.7714
110.8020
P
2
114.0000
112.0271
110.8626
75.1768
110.8004
P
3
101.5985
97.3999
97.3999
97.4000
97.4001
P
4
179.8608
179.7331
179.7331
129.8666
179.7330
P
5
97.0000
89.0477
97.0000
93.8081
87.8004
P
6
140.0000
140.0000
140.0000
140.0000
140.0000
P
7
260.2515
300.0000
259.5997
260.1252
259.5998
P
8
284.8846
284.6023
284.5997
285.0625
284.6000
P
9
285.6941
284.5997
284.5999
287.7589
284.5997
P
10
131.2062
130.0000
130.0000
204.7998
130.0000
P
11
168.4760
168.7998
168.7998
168.7998
94.0000
P
12
168.7683
94.0000
94.0000
243.5997
94.0000
P
13
215.2662
214.7598
214.7598
304.5196
214.7598
P
14
393.1850
394.2794
394.2794
394.2794
394.2794
P
15
306.7599
394.2794
394.2794
394.2794
394.2796
P
16
305.2877
304.5196
304.5196
304.5196
394.2793
P
17
488.9168
489.2794
489.2794
489.2794
489.2795
P
18
489.2538
489.2794
489.2794
489.2794
489.2793
P
19
511.3833
511.2794
511.2794
511.2794
511.2794
P
20
512.0211
511.2794
511.2794
511.2794
511.2795
P
21
523.1479
523.2794
523.2794
433.7901
523.2793
P
22
524.9892
523.2794
523.2794
523.4705
523.2794
P23
523.5749
523.2794
523.2794
523.2794
523.2793
P
24
524.6906
523.2794
523.2794
523.2794
523.2798
P
25
523.6776
523.2794
523.2794
523.2939
523.2794
P
26
526.3109
523.2794
523.2794
523.2794
523.2794
P
27
10.0000
10.0000
10.0000
10.0000
10.0000
P
28
10.0000
10.0000
10.0000
10.0000
10.0000
P
29
10.0000
10.0000
10.0000
10.0000
10.0000
P
30
93.4661
88.0999
87.8236
88.2486
87.8000
P
31
190.0000
190.0000
190.0000
190.0000
190.0000
P
32
190.0000
190.0000
190.0000
161.4680
189.9998
P
33
190.0000
190.0000
190.0000
161.6225
190.0000
P
34
199.8577
165.1100
164.8006
165.0999
164.7997
P
35
168.7132
165.1627
200.0000
168.0133
199.9752
P
36
184.9895
200.0000
200.0000
165.3862
194.4181
P
37
110.0000
110.0000
110.0000
91.6051
110.0000
P
38
109.1741
110.0000
110.0000
110.0000
110.0000
P
39
110.0000
110.0000
110.0000
110.0000
110.0000
P
40
511.3378
511.2794
511.2794
511.2794
511.2794
TP (MW)
10500.0000
10500.0000
10500.0000
10500.0000
10500.0000
TC ($/h)
121669.9298
121440.8040
121420.9029
122636.1149
121412.6226In conclusion, the abovementioned comparison results sufficiently demonstrate that ANS is able to achieve a strong equilibrium between the local exploitation and global exploration.


#### 5.1.2. Convergence Property

The convergence curves of the mean costs for the four cases are plotted in Figures2–5, respectively. It can be seen that although CSO and DE are slightly faster than ANS at the very beginning, both methods are quickly trapped into local search later and thus suffer from prematurity. The convergence speed of BBO and PSO, especially the latter, is slow. ANS can consistently improve the solution quality and converge towards the global optima throughout the whole evolutionary process in all cases especially in Cases I and IV, which, from another perspective, indicates that ANS possesses better exploration and exploitation abilities of jumping out of local search and finding a more promising searching direction.Figure 2
Convergence curves for the 13-generator system (Case I).Figure 3
Convergence curves for the 15-generator system (Case II).Figure 4
Convergence curves for the 38-generator system (Case III).Figure 5
Convergence curves for the 40-generator system (Case IV).


#### 5.1.3. Robustness

Since population-based algorithms use random numbers to initialize the population and employ randomization procedures to promote the search process, randomness is inevitable. In this context, it may be inappropriate to comprehensively assess their performance just through one single run. Thus, a number of independent runs with different initial populations can be used to measure their stability and consistency, i.e., robustness. The standard deviation results provided in Tables2–5 clearly illustrate that the recorded values of ANS are significantly smaller than those of BBO, CSO, DE, and PSO. Moreover, they are also very competitive with those of other recently proposed ED problem solution methods. The comparisons indicate that ANS has strong robustness and it can achieve a relatively stable optimal result in each trial. In addition, through careful observation, we can see that the standard deviation values of Cases I and IV are bigger than those of Cases II and III. This is because the valve-point effects make the ED problems exhibit highly multimodal characteristic and the solution methods are more likely trapped into different local minima in different trials. Therefore, the valve-point effects are more challenging for solution methods.


#### 5.1.4. Statistical Analysis

A nonparametric statistical test called Wilcoxon’s rank sum test for independent samples is employed to compare the significance differences between ANS and its competitors. The results based on the Wilcoxon’s rank sum test at a 0.05 confidence level are summarized in Table10. The mark “†” symbolizes that ANS is statistically better than its competitor. It can be seen that ANS performs significantly better than BBO, CSO, DE, and PSO in all cases, meaning that ANS is capable of obtaining overall higher quality of the final solutions than the other four popular population-based algorithms.Table 10
Statistical analysis results based on Wilcoxon’s rank sum test.
ANS vs.
BBO
CSO
DE
PSO
Case III
†
†
†
†
Case II
†
†
†
†
Case III
†
†
†
†
Case IV
†
†
†
†


### 5.2. Influence of Population Size

Choosing an appropriate population size is always critical for population-based algorithms in solving different problems. In this subsection, an experiment is conducted to investigate the sensitivity of ANS to variations in population size. The population sizep
s is set as 10 to 100 with interval 10. The results are shown in Figure 6.Figure 6
Influence of population size. (a) Case I. (b) Case II. (c) Case III. (d) Case IV.
(a)
(b)
(c)
(d)The following can be seen: (i) For the unimodal ED problems, i.e., Cases II and III, the smaller the population size, the better the performance ANS yields. (ii) For the multimodal ED problems, i.e., Cases I and IV, the bigger or smaller the population size, the worse the performance ANS obtains, and ANS can achieve the best results inp
s
=
40. The reason might be that, for the unimodal ED problems, it is relatively easy for ANS to find the correct searching direction and there is no need for exploring the entire solution space. Therefore, a small population can swarm towards the global optimum easily. While for the multimodal ED problems, on one hand, a small population tends to converge very fast before fully exploring the entire solution space, thus resulting in prematurity. On the other hand, although a large population can increase the population diversity significantly, the distribution of individuals is sparse and the probability of finding the correct searching direction is sharply reduced. In addition, a large population size will consume a large number of fitness evaluations in each iteration, which is not proper for computationally expensive problems. In general, for the traditional ED problems without considering the valve-point effects, a small population size is recommended, whereas for the nonconvex ED problems, it is safe to set a moderate population size.


### 5.3. Influence of the Across-Search Degree

The across-search degreen in ANS is utilized to control the amount of information deriving from other superior solutions. In this subsection, an experiment is conducted to investigate the influence of n on the performance of ANS. The experimental results in different across-search degrees are presented in Figure 7.Figure 7
Influence of across-search degree. (a) Case I. (b) Case II. (c) Case III. (d) Case IV.
(a)
(b)
(c)
(d)It can be observed that the best values ofn for all cases are all 1. Besides, for the unimodal ED problems, the bigger the value n is, almost the worse the performance of ANS gets. For the multimodal ED problems, a temperate value of n will considerably deteriorate ANS though a bigger value is also not good for ANS. The reason might be that, for both unimodal and multimodal ED problems, bigger values of n will damage individuals’ solution components vastly in each iteration, which is not conducive to the consistency of convergence. In addition, for the multimodal ED problems, although a more bigger value of n can fully recombine each individual to maintain the population diversity, it will make individuals trap into different local optima frequently and thus slow down the convergence considerably. Generally, the recommendation value of n for different ED problems is 1.


## 5.1. Experimental Results and Comparison

### 5.1.1. Solution Quality

The experimental results of Cases I to IV are summarized in Tables2–5, respectively. The results include the minimum (Min), maximum (Max), mean costs, and standard deviation (Std Dev). These tables also list some recently proposed ED solution methods’ reported results for comparison.Table 2
Comparison of simulation results ($/h) for 13-generator system (Case III).
Method
Min.
Max.
Mean
Std Dev
FCASO-SQP [1]
17964.08
NA
18001.96
NA
TSARGA [4]
17963.94
18089.61
17974.31
3.18
DE [10]
17963.83
17975.36
17965.48
NA
ABC [11]
17963.86
17995.11
17987.22
NA
IHSWM [14]
17963.83
18041.3456
17976.4750
25.6079
aBBOmDE [20]
17963.8521
17975.0552
17967.3560
NA
TLBO [24]
18115
NA
NA
NA
CTLBO [25]
17972.81
18159.34
18013.38
43.2
FA [26]
17963.83
18168.80
18029.16
148.542
CBA [29]
17963.8339
17995.2256
17965.4889
6.8473
GWO [30]
18051.11
NA
NA
NA
GA–PS–QSP [44]
17964
NA
18199
NA
HMAPSO [48]
17969.31
17969.31
17969.31
NA
CE-SQP [54]
17963.85
NA
17965.97
NA
ANS-SDP [68]
17964
17985
17973
NA
BFO [69]
17974.48
17997.12
18018.75
NA
RTO [70]
17969.8024
18204.6303
18056.9358
NA
CMSFLA [71]
18294.9953
18297.3126
18295.7308
NA
OIWO [72]
17963.83
NA
NA
NA
IA_EDA [73]
17961.4331
18052.3155
17980.1898
21.6666
NSS [74]
17976.9512
17976.9512
17976.9512
NA
MSOS [75]
17963.8292
17963.8292
17963.8292
6.80E-12
QOSLTLBO [76]
18421.1718
NA
NA
NA
BBO
17973.1883
18090.6042
17993.9942
27.0827
CSO
17972.8619
18136.4236
18073.9003
56.1710
DE
17972.8105
18075.5913
17993.5153
38.8380
PSO
17981.3621
18312.9831
18127.7452
85.8468
ANS
17963.9031
17973.4437
17969.1487
3.0015
NA means not available in the corresponding literature.Table 3
Comparison of simulation results ($/h) for 15-generator system (Case II).
Method
Min.
Max.
Mean
Std Dev
MPSO [8]
32738.4177
NA
NA
NA
PVHS [13]
32780.00
NA
32892.463
35.331
IHSWM [14]
32693.1304
32721.3988
32699.5168
4.6937
FA [26]
32704.5
33175.0
32856.1
147.1702
ICS [32]
32706.7358
32752.5183
32715.4669
NA
EMA [36]
32704.4503
32704.4506
32704.4504
NA
BSA [39]
32704.4504
32704.5816
32704.4721
0.0280
GAAPI [46]
32732.95
32756.01
32735.06
NA
DSPSO-TSA [47]
32715.06
32730.39
32724.63
8.40
HHS [50]
32692.8571
NA
32740.1885
24.6697
ABF-NM [51]
32784.5024
NA
32976.81
85.7743
ANS-SDP [68]
32693
32693
32693
NA
RTO [70]
32701.81
32715.18
32704.53
5.07
AIS [77]
32854
32892
32873.25
10.8079
MVMO [78]
32704.47
32709.32
32706.65
NA
SCKF-PSO [79]
32725.8431
NA
NA
NA
SFS [80]
32702
NA
NA
NA
BBO
32692.3988
32693.7153
32692.6927
0.2592
CSO
32692.3972
32875.3601
32718.8607
38.0358
DE
32692.3966
32755.3193
32699.1724
12.9351
PSO
32692.3991
32781.9374
32701.0786
13.8044
ANS
32692.3962
32692.4016
32692.3973
0.0009Table 4
Comparison of simulation results ($/h) for 38-generator system (Case III).
Method
Min.
Max.
Mean
Std Dev
New-PSO [6]
9516448.312
NA
NA
NA
PSO_TVAC [6]
9500448.307
NA
NA
NA
HS [15]
9419960
9427466
9421056
NA
HHS [15]
9417325
9417466
9417336
NA
BBO(Model 6) [22]
9417557.25
9419854.237
9418394.069
NA
GWO [30]
9419270.188
9421100
9419978.978
NA
BBO [81]
9417633.6376
NA
NA
NA
DE/BBO [81]
9417235.7864
NA
NA
NA
BBO
9417446.0804
9418959.2333
9418143.6500
390.1769
CSO
9417307.5544
9417745.6891
9417494.7251
103.1130
DE
9417235.7977
9417235.8465
9417235.8184
0.0136
PSO
9426679.1431
9461023.8147
9441014.5256
9278.1375
ANS
9417235.7864
9417235.7864
9417235.7864
8.4314E-09Table 5
Comparison of simulation results ($/h) for 40-generator system (Case IV).
Method
Min.
Max.
Mean
Std Dev
FCASO-SQP [1]
121456.98
NA
122026.21
NA
TSARGA [4]
121463.07
124296.54
122928.31
NA
DE [10]
121416.29
121431.47
121422.72
NA
ABC [11]
121441.03
122023.77
121995.82
NA
IABC-LS [12]
121488.7636
121582.2525
121526.0333
NA
PVHS [13]
121415.4560
NA
121567.0292
94.1498
IHSWM [14]
121416.2652
121855.5521
121553.4208
90.1271
THS [16]
121425.15
NA
121528.65
NA
NTHS [17]
121412.7374
NA
121549.95
NA
BBO [19]
121426.953
121688.6634
121508.0325
NA
aBBOmDE [20]
121414.8734
121568.3254
121487.8532
NA
BBO(Model 5) [22]
121418.70
121954.4
121635.4
NA
TLBO [24]
12996
NA
NA
NA
CTLBO [25]
121553.83
122116.18
121790.23
150
FA [26]
121415.05
121416.57
121424.56
1.784
CBA [29]
121412.5468
121436.15
121418.9826
1.611
EMA [36]
121412.5355
121426.1548
121417.1328
NA
BSA [40]
121415.6139
121524.9577
121474.8823
NA
GA–PS–SQP [44]
121458
NA
122039
NA
HMAPSO [48]
121586.90
121586.90
121586.90
NA
FAPSO-NM [49]
121418.3
121419.8
121418.803
NA
HHS [50]
121415.5922
NA
121615.8544
114.5632
ABF-NM [51]
121423.6379
NA
121814.9465
124.8756
CE-SQP [54]
121412.88
NA
121423.65
NA
ANS-DSP [68]
12141E+05
12187E+05
12185E+05
NA
OIWO [72]
121412.54
NA
NA
NA
IA-EDA [73]
121436.9729
121648.4401
122492.7018
182.5274
NSS [74]
122186.9048
122186.9048
122186.9048
NA
MSOS [75]
121412.5355
121412.5355
121412.5355
2.47E-11
NGPSO [82]
121513.4808
122697.7672
122065.1193
267
C-GRASP-SaDE [83]
121414.621
122245.696
121736.025
166.896
BBO
121669.9298
122446.1176
121865.0181
150.4038
CSO
121440.8040
122092.4350
121637.1868
141.5265
DE
121420.9029
121618.6298
121509.0360
42.6215
PSO
122636.1149
125429.5349
123836.7368
603.6246
ANS
121412.6226
121472.9213
121427.7107
13.6539Case II is a canonical traditional ED problem with a smooth and continuous solution space. Although the solution space of Case III is discontinued by the prohibited operating zones, its objective function is also quadratic. Therefore, both cases are typical multiconstraint unimodal optimization problems which have high requirement on the solution methods’ local searching ability. It can be observed from Tables3 and 4 that ANS outperforms BBO, CSO, DE, and PSO on both cases in terms of minimum, maximum, and mean costs. In addition, ANS is also better than other reported solution methods. The comparisons demonstrate that ANS is able to search local range meticulously and thereby possesses good exploitation ability.Cases I and IV, whose solution spaces are highly nonconvex due to the valve-point effects, are multiconstraint multimodal optimization problems. The number of local minima increases at an exponential rate with the problem scale. They demand a lot on the solution methods’ global searching ability. The experimental results tabulated in Tables2 and 5 consistently indicate that ANS is significantly better than BBO, CSO, DE, and PSO in both cases in terms of minimum, maximum, and mean costs. Furthermore, the larger the system scale, the more significant the superiority. Additionally, ANS also can achieve better or highly competitive results compared with other reported solution methods. The comparisons fully conclude that ANS is with the capability of breaking away from the local minima and locating the global or near-global range. Namely, ANS has good exploration ability.The obtained optimal dispatching schedules for these four cases are presented in Tables6–9, respectively.Table 6
Best output power for 13-generator system (Case III).
Generator
BBO
CSO
DE
PSO
ANS
P
1
628.1947
628.3185
628.3185
538.5587
628.3187
P
2
223.4227
223.8651
297.5488
299.5838
223.2845
P
3
298.6788
298.0833
224.3995
224.8434
149.0866
P
4
60.0000
60.0000
60.0000
109.8666
60.0000
P
5
109.8650
60.0000
109.8666
60.0000
109.8661
P
6
60.0000
60.0000
60.0000
60.0000
109.8615
P
7
60.0000
60.0000
109.8666
109.8666
109.8603
P
8
60.0000
109.8665
60.0000
60.0004
109.8603
P
9
109.8388
109.8665
60.0000
109.8666
109.8620
P
10
40.0000
40.0000
40.0000
40.0063
40.0000
P
11
40.0000
40.0000
40.0000
40.0062
40.0000
P
12
55.0000
55.0000
55.0000
92.4002
55.0000
P
13
55.0000
55.0000
55.0000
55.0012
55.0000
T
P
a (MW)
1800.0000
1800.0000
1800.0000
1800.0000
1800.0000
T
C
b ($/h)
17973.1883
17972.8619
17972.8105
17981.3621
17963.9031
T
P
a denotes the total power output.
T
C
b denotes the total generation cost.Table 7
Best output power for 15-generator system (Case II).
Generator
BBO
CSO
DE
PSO
ANS
P
1
455.0000
455.0000
455.0000
455.0000
455.0000
P
2
380.0000
380.0000
380.0000
380.0000
380.0000
P
3
130.0000
130.0000
130.0000
130.0000
130.0000
P
4
130.0000
130.0000
130.0000
130.0000
130.0000
P
5
170.0000
170.0000
170.0000
170.0000
170.0000
P
6
460.0000
460.0000
460.0000
460.0000
460.0000
P
7
430.0000
430.0000
430.0000
430.0000
430.0000
P
8
69.9471
68.8487
69.7626
70.4928
69.4768
P
9
59.6421
60.7332
59.8226
59.0961
60.1079
P
10
159.9943
160.0000
160.0000
159.9999
160.0000
P
11
80.0000
80.0000
80.0000
80.0000
80.0000
P
12
80.0000
80.0000
80.0000
80.0000
80.0000
P
13
25.0000
25.0000
25.0000
25.0000
25.0000
P
14
15.0020
15.0000
15.0006
15.0005
15.0000
P
15
15.0011
15.0000
15.0001
15.0000
15.0000
T
L
a (MW)
29.5866
29.5819
29.5859
29.5893
29.5847
TP (MW)
2659.5866
2659.5819
2659.5859
2659.5893
2659.5847
TC ($/h)
32692.3988
32692.3972
32692.3966
32692.3991
32692.3962
T
L
a denotes the total transmission network loss.Table 8
Best output power for 38-generator system (Case III).
Generator
BBO
CSO
DE
PSO
ANS
P
1
417.4699
431.4435
426.6061
423.2786
426.6061
P
2
428.6549
432.7667
426.6061
447.2694
426.6061
P
3
430.7266
432.4754
429.6632
377.7218
429.6631
P
4
429.5193
426.7217
429.6632
391.1059
429.6632
P
5
438.8225
425.0443
429.6632
371.5138
429.6632
P
6
436.7947
426.7682
429.6632
417.4801
429.6631
P
7
424.1592
429.8489
429.6632
468.3885
429.6632
P
8
427.7913
433.0439
429.6632
376.9151
429.6631
P
9
114.0000
114.0000
114.0000
137.9410
114.0000
P
10
114.1613
114.0001
114.0000
155.5833
114.0000
P
11
114.7729
120.6554
119.7680
142.6568
119.7681
P
12
127.0026
128.0361
127.0728
187.8007
127.0729
P
13
110.0000
110.0000
110.0000
110.0000
110.0000
P
14
90.0025
90.0000
90.0000
90.0000
90.0000
P
15
82.0000
82.0000
82.0000
82.0000
82.0000
P
16
120.0000
120.0000
120.0000
120.0000
120.0000
P
17
160.1416
158.7822
159.5980
161.4797
159.5980
P
18
65.0008
65.0000
65.0000
65.0000
65.0000
P
19
65.0000
65.0000
65.0000
65.0000
65.0000
P
20
272.0000
272.0000
272.0000
271.9999
272.0000
P
21
272.0000
272.0000
272.0000
272.0000
272.0000
P
22
259.3185
260.0000
260.0000
260.0000
260.0000
P23
130.9561
129.4275
130.6486
138.0732
130.6486
P
24
10.0000
10.0000
10.0000
10.0002
10.0000
P
25
116.3424
108.7907
113.3050
110.9243
113.3050
P
26
90.6858
87.3194
88.0669
88.8202
88.0669
P
27
35.0000
36.1978
37.5051
39.1707
37.5051
P
28
20.0000
20.0000
20.0000
20.0000
20.0000
P
29
20.0000
20.0000
20.0000
20.0000
20.0000
P
30
20.0005
20.0000
20.0000
20.0000
20.0000
P
31
20.0000
20.0000
20.0000
20.0000
20.0000
P
32
20.0000
20.0000
20.0000
20.0000
20.0000
P
33
25.0000
25.0000
25.0000
25.0000
25.0000
P
34
18.0000
18.0000
18.0000
18.0000
18.0000
P
35
8.0000
8.0000
8.0000
8.0000
8.0000
P
36
25.0000
25.0000
25.0000
25.0000
25.0000
P
37
21.6764
21.5606
21.7821
21.3068
21.7821
P
38
20.0000
21.1175
21.0622
20.5701
21.0622
TP (MW)
6000.0000
6000.0000
6000.0000
6000.0000
6000.0000
TC ($/h)
9417446.0804
9417307.5544
9417235.7977
9426679.1431
9417235.7864Table 9
Best output power for 40-generator system (Case IV).
Generator
BBO
CSO
DE
PSO
ANS
P
1
112.2570
111.5065
110.8699
111.7714
110.8020
P
2
114.0000
112.0271
110.8626
75.1768
110.8004
P
3
101.5985
97.3999
97.3999
97.4000
97.4001
P
4
179.8608
179.7331
179.7331
129.8666
179.7330
P
5
97.0000
89.0477
97.0000
93.8081
87.8004
P
6
140.0000
140.0000
140.0000
140.0000
140.0000
P
7
260.2515
300.0000
259.5997
260.1252
259.5998
P
8
284.8846
284.6023
284.5997
285.0625
284.6000
P
9
285.6941
284.5997
284.5999
287.7589
284.5997
P
10
131.2062
130.0000
130.0000
204.7998
130.0000
P
11
168.4760
168.7998
168.7998
168.7998
94.0000
P
12
168.7683
94.0000
94.0000
243.5997
94.0000
P
13
215.2662
214.7598
214.7598
304.5196
214.7598
P
14
393.1850
394.2794
394.2794
394.2794
394.2794
P
15
306.7599
394.2794
394.2794
394.2794
394.2796
P
16
305.2877
304.5196
304.5196
304.5196
394.2793
P
17
488.9168
489.2794
489.2794
489.2794
489.2795
P
18
489.2538
489.2794
489.2794
489.2794
489.2793
P
19
511.3833
511.2794
511.2794
511.2794
511.2794
P
20
512.0211
511.2794
511.2794
511.2794
511.2795
P
21
523.1479
523.2794
523.2794
433.7901
523.2793
P
22
524.9892
523.2794
523.2794
523.4705
523.2794
P23
523.5749
523.2794
523.2794
523.2794
523.2793
P
24
524.6906
523.2794
523.2794
523.2794
523.2798
P
25
523.6776
523.2794
523.2794
523.2939
523.2794
P
26
526.3109
523.2794
523.2794
523.2794
523.2794
P
27
10.0000
10.0000
10.0000
10.0000
10.0000
P
28
10.0000
10.0000
10.0000
10.0000
10.0000
P
29
10.0000
10.0000
10.0000
10.0000
10.0000
P
30
93.4661
88.0999
87.8236
88.2486
87.8000
P
31
190.0000
190.0000
190.0000
190.0000
190.0000
P
32
190.0000
190.0000
190.0000
161.4680
189.9998
P
33
190.0000
190.0000
190.0000
161.6225
190.0000
P
34
199.8577
165.1100
164.8006
165.0999
164.7997
P
35
168.7132
165.1627
200.0000
168.0133
199.9752
P
36
184.9895
200.0000
200.0000
165.3862
194.4181
P
37
110.0000
110.0000
110.0000
91.6051
110.0000
P
38
109.1741
110.0000
110.0000
110.0000
110.0000
P
39
110.0000
110.0000
110.0000
110.0000
110.0000
P
40
511.3378
511.2794
511.2794
511.2794
511.2794
TP (MW)
10500.0000
10500.0000
10500.0000
10500.0000
10500.0000
TC ($/h)
121669.9298
121440.8040
121420.9029
122636.1149
121412.6226In conclusion, the abovementioned comparison results sufficiently demonstrate that ANS is able to achieve a strong equilibrium between the local exploitation and global exploration.


### 5.1.2. Convergence Property

The convergence curves of the mean costs for the four cases are plotted in Figures2–5, respectively. It can be seen that although CSO and DE are slightly faster than ANS at the very beginning, both methods are quickly trapped into local search later and thus suffer from prematurity. The convergence speed of BBO and PSO, especially the latter, is slow. ANS can consistently improve the solution quality and converge towards the global optima throughout the whole evolutionary process in all cases especially in Cases I and IV, which, from another perspective, indicates that ANS possesses better exploration and exploitation abilities of jumping out of local search and finding a more promising searching direction.Figure 2
Convergence curves for the 13-generator system (Case I).Figure 3
Convergence curves for the 15-generator system (Case II).Figure 4
Convergence curves for the 38-generator system (Case III).Figure 5
Convergence curves for the 40-generator system (Case IV).


### 5.1.3. Robustness

Since population-based algorithms use random numbers to initialize the population and employ randomization procedures to promote the search process, randomness is inevitable. In this context, it may be inappropriate to comprehensively assess their performance just through one single run. Thus, a number of independent runs with different initial populations can be used to measure their stability and consistency, i.e., robustness. The standard deviation results provided in Tables2–5 clearly illustrate that the recorded values of ANS are significantly smaller than those of BBO, CSO, DE, and PSO. Moreover, they are also very competitive with those of other recently proposed ED problem solution methods. The comparisons indicate that ANS has strong robustness and it can achieve a relatively stable optimal result in each trial. In addition, through careful observation, we can see that the standard deviation values of Cases I and IV are bigger than those of Cases II and III. This is because the valve-point effects make the ED problems exhibit highly multimodal characteristic and the solution methods are more likely trapped into different local minima in different trials. Therefore, the valve-point effects are more challenging for solution methods.


### 5.1.4. Statistical Analysis

A nonparametric statistical test called Wilcoxon’s rank sum test for independent samples is employed to compare the significance differences between ANS and its competitors. The results based on the Wilcoxon’s rank sum test at a 0.05 confidence level are summarized in Table10. The mark “†” symbolizes that ANS is statistically better than its competitor. It can be seen that ANS performs significantly better than BBO, CSO, DE, and PSO in all cases, meaning that ANS is capable of obtaining overall higher quality of the final solutions than the other four popular population-based algorithms.Table 10
Statistical analysis results based on Wilcoxon’s rank sum test.
ANS vs.
BBO
CSO
DE
PSO
Case III
†
†
†
†
Case II
†
†
†
†
Case III
†
†
†
†
Case IV
†
†
†
†


## 5.1.1. Solution Quality

The experimental results of Cases I to IV are summarized in Tables2–5, respectively. The results include the minimum (Min), maximum (Max), mean costs, and standard deviation (Std Dev). These tables also list some recently proposed ED solution methods’ reported results for comparison.Table 2
Comparison of simulation results ($/h) for 13-generator system (Case III).
Method
Min.
Max.
Mean
Std Dev
FCASO-SQP [1]
17964.08
NA
18001.96
NA
TSARGA [4]
17963.94
18089.61
17974.31
3.18
DE [10]
17963.83
17975.36
17965.48
NA
ABC [11]
17963.86
17995.11
17987.22
NA
IHSWM [14]
17963.83
18041.3456
17976.4750
25.6079
aBBOmDE [20]
17963.8521
17975.0552
17967.3560
NA
TLBO [24]
18115
NA
NA
NA
CTLBO [25]
17972.81
18159.34
18013.38
43.2
FA [26]
17963.83
18168.80
18029.16
148.542
CBA [29]
17963.8339
17995.2256
17965.4889
6.8473
GWO [30]
18051.11
NA
NA
NA
GA–PS–QSP [44]
17964
NA
18199
NA
HMAPSO [48]
17969.31
17969.31
17969.31
NA
CE-SQP [54]
17963.85
NA
17965.97
NA
ANS-SDP [68]
17964
17985
17973
NA
BFO [69]
17974.48
17997.12
18018.75
NA
RTO [70]
17969.8024
18204.6303
18056.9358
NA
CMSFLA [71]
18294.9953
18297.3126
18295.7308
NA
OIWO [72]
17963.83
NA
NA
NA
IA_EDA [73]
17961.4331
18052.3155
17980.1898
21.6666
NSS [74]
17976.9512
17976.9512
17976.9512
NA
MSOS [75]
17963.8292
17963.8292
17963.8292
6.80E-12
QOSLTLBO [76]
18421.1718
NA
NA
NA
BBO
17973.1883
18090.6042
17993.9942
27.0827
CSO
17972.8619
18136.4236
18073.9003
56.1710
DE
17972.8105
18075.5913
17993.5153
38.8380
PSO
17981.3621
18312.9831
18127.7452
85.8468
ANS
17963.9031
17973.4437
17969.1487
3.0015
NA means not available in the corresponding literature.Table 3
Comparison of simulation results ($/h) for 15-generator system (Case II).
Method
Min.
Max.
Mean
Std Dev
MPSO [8]
32738.4177
NA
NA
NA
PVHS [13]
32780.00
NA
32892.463
35.331
IHSWM [14]
32693.1304
32721.3988
32699.5168
4.6937
FA [26]
32704.5
33175.0
32856.1
147.1702
ICS [32]
32706.7358
32752.5183
32715.4669
NA
EMA [36]
32704.4503
32704.4506
32704.4504
NA
BSA [39]
32704.4504
32704.5816
32704.4721
0.0280
GAAPI [46]
32732.95
32756.01
32735.06
NA
DSPSO-TSA [47]
32715.06
32730.39
32724.63
8.40
HHS [50]
32692.8571
NA
32740.1885
24.6697
ABF-NM [51]
32784.5024
NA
32976.81
85.7743
ANS-SDP [68]
32693
32693
32693
NA
RTO [70]
32701.81
32715.18
32704.53
5.07
AIS [77]
32854
32892
32873.25
10.8079
MVMO [78]
32704.47
32709.32
32706.65
NA
SCKF-PSO [79]
32725.8431
NA
NA
NA
SFS [80]
32702
NA
NA
NA
BBO
32692.3988
32693.7153
32692.6927
0.2592
CSO
32692.3972
32875.3601
32718.8607
38.0358
DE
32692.3966
32755.3193
32699.1724
12.9351
PSO
32692.3991
32781.9374
32701.0786
13.8044
ANS
32692.3962
32692.4016
32692.3973
0.0009Table 4
Comparison of simulation results ($/h) for 38-generator system (Case III).
Method
Min.
Max.
Mean
Std Dev
New-PSO [6]
9516448.312
NA
NA
NA
PSO_TVAC [6]
9500448.307
NA
NA
NA
HS [15]
9419960
9427466
9421056
NA
HHS [15]
9417325
9417466
9417336
NA
BBO(Model 6) [22]
9417557.25
9419854.237
9418394.069
NA
GWO [30]
9419270.188
9421100
9419978.978
NA
BBO [81]
9417633.6376
NA
NA
NA
DE/BBO [81]
9417235.7864
NA
NA
NA
BBO
9417446.0804
9418959.2333
9418143.6500
390.1769
CSO
9417307.5544
9417745.6891
9417494.7251
103.1130
DE
9417235.7977
9417235.8465
9417235.8184
0.0136
PSO
9426679.1431
9461023.8147
9441014.5256
9278.1375
ANS
9417235.7864
9417235.7864
9417235.7864
8.4314E-09Table 5
Comparison of simulation results ($/h) for 40-generator system (Case IV).
Method
Min.
Max.
Mean
Std Dev
FCASO-SQP [1]
121456.98
NA
122026.21
NA
TSARGA [4]
121463.07
124296.54
122928.31
NA
DE [10]
121416.29
121431.47
121422.72
NA
ABC [11]
121441.03
122023.77
121995.82
NA
IABC-LS [12]
121488.7636
121582.2525
121526.0333
NA
PVHS [13]
121415.4560
NA
121567.0292
94.1498
IHSWM [14]
121416.2652
121855.5521
121553.4208
90.1271
THS [16]
121425.15
NA
121528.65
NA
NTHS [17]
121412.7374
NA
121549.95
NA
BBO [19]
121426.953
121688.6634
121508.0325
NA
aBBOmDE [20]
121414.8734
121568.3254
121487.8532
NA
BBO(Model 5) [22]
121418.70
121954.4
121635.4
NA
TLBO [24]
12996
NA
NA
NA
CTLBO [25]
121553.83
122116.18
121790.23
150
FA [26]
121415.05
121416.57
121424.56
1.784
CBA [29]
121412.5468
121436.15
121418.9826
1.611
EMA [36]
121412.5355
121426.1548
121417.1328
NA
BSA [40]
121415.6139
121524.9577
121474.8823
NA
GA–PS–SQP [44]
121458
NA
122039
NA
HMAPSO [48]
121586.90
121586.90
121586.90
NA
FAPSO-NM [49]
121418.3
121419.8
121418.803
NA
HHS [50]
121415.5922
NA
121615.8544
114.5632
ABF-NM [51]
121423.6379
NA
121814.9465
124.8756
CE-SQP [54]
121412.88
NA
121423.65
NA
ANS-DSP [68]
12141E+05
12187E+05
12185E+05
NA
OIWO [72]
121412.54
NA
NA
NA
IA-EDA [73]
121436.9729
121648.4401
122492.7018
182.5274
NSS [74]
122186.9048
122186.9048
122186.9048
NA
MSOS [75]
121412.5355
121412.5355
121412.5355
2.47E-11
NGPSO [82]
121513.4808
122697.7672
122065.1193
267
C-GRASP-SaDE [83]
121414.621
122245.696
121736.025
166.896
BBO
121669.9298
122446.1176
121865.0181
150.4038
CSO
121440.8040
122092.4350
121637.1868
141.5265
DE
121420.9029
121618.6298
121509.0360
42.6215
PSO
122636.1149
125429.5349
123836.7368
603.6246
ANS
121412.6226
121472.9213
121427.7107
13.6539Case II is a canonical traditional ED problem with a smooth and continuous solution space. Although the solution space of Case III is discontinued by the prohibited operating zones, its objective function is also quadratic. Therefore, both cases are typical multiconstraint unimodal optimization problems which have high requirement on the solution methods’ local searching ability. It can be observed from Tables3 and 4 that ANS outperforms BBO, CSO, DE, and PSO on both cases in terms of minimum, maximum, and mean costs. In addition, ANS is also better than other reported solution methods. The comparisons demonstrate that ANS is able to search local range meticulously and thereby possesses good exploitation ability.Cases I and IV, whose solution spaces are highly nonconvex due to the valve-point effects, are multiconstraint multimodal optimization problems. The number of local minima increases at an exponential rate with the problem scale. They demand a lot on the solution methods’ global searching ability. The experimental results tabulated in Tables2 and 5 consistently indicate that ANS is significantly better than BBO, CSO, DE, and PSO in both cases in terms of minimum, maximum, and mean costs. Furthermore, the larger the system scale, the more significant the superiority. Additionally, ANS also can achieve better or highly competitive results compared with other reported solution methods. The comparisons fully conclude that ANS is with the capability of breaking away from the local minima and locating the global or near-global range. Namely, ANS has good exploration ability.The obtained optimal dispatching schedules for these four cases are presented in Tables6–9, respectively.Table 6
Best output power for 13-generator system (Case III).
Generator
BBO
CSO
DE
PSO
ANS
P
1
628.1947
628.3185
628.3185
538.5587
628.3187
P
2
223.4227
223.8651
297.5488
299.5838
223.2845
P
3
298.6788
298.0833
224.3995
224.8434
149.0866
P
4
60.0000
60.0000
60.0000
109.8666
60.0000
P
5
109.8650
60.0000
109.8666
60.0000
109.8661
P
6
60.0000
60.0000
60.0000
60.0000
109.8615
P
7
60.0000
60.0000
109.8666
109.8666
109.8603
P
8
60.0000
109.8665
60.0000
60.0004
109.8603
P
9
109.8388
109.8665
60.0000
109.8666
109.8620
P
10
40.0000
40.0000
40.0000
40.0063
40.0000
P
11
40.0000
40.0000
40.0000
40.0062
40.0000
P
12
55.0000
55.0000
55.0000
92.4002
55.0000
P
13
55.0000
55.0000
55.0000
55.0012
55.0000
T
P
a (MW)
1800.0000
1800.0000
1800.0000
1800.0000
1800.0000
T
C
b ($/h)
17973.1883
17972.8619
17972.8105
17981.3621
17963.9031
T
P
a denotes the total power output.
T
C
b denotes the total generation cost.Table 7
Best output power for 15-generator system (Case II).
Generator
BBO
CSO
DE
PSO
ANS
P
1
455.0000
455.0000
455.0000
455.0000
455.0000
P
2
380.0000
380.0000
380.0000
380.0000
380.0000
P
3
130.0000
130.0000
130.0000
130.0000
130.0000
P
4
130.0000
130.0000
130.0000
130.0000
130.0000
P
5
170.0000
170.0000
170.0000
170.0000
170.0000
P
6
460.0000
460.0000
460.0000
460.0000
460.0000
P
7
430.0000
430.0000
430.0000
430.0000
430.0000
P
8
69.9471
68.8487
69.7626
70.4928
69.4768
P
9
59.6421
60.7332
59.8226
59.0961
60.1079
P
10
159.9943
160.0000
160.0000
159.9999
160.0000
P
11
80.0000
80.0000
80.0000
80.0000
80.0000
P
12
80.0000
80.0000
80.0000
80.0000
80.0000
P
13
25.0000
25.0000
25.0000
25.0000
25.0000
P
14
15.0020
15.0000
15.0006
15.0005
15.0000
P
15
15.0011
15.0000
15.0001
15.0000
15.0000
T
L
a (MW)
29.5866
29.5819
29.5859
29.5893
29.5847
TP (MW)
2659.5866
2659.5819
2659.5859
2659.5893
2659.5847
TC ($/h)
32692.3988
32692.3972
32692.3966
32692.3991
32692.3962
T
L
a denotes the total transmission network loss.Table 8
Best output power for 38-generator system (Case III).
Generator
BBO
CSO
DE
PSO
ANS
P
1
417.4699
431.4435
426.6061
423.2786
426.6061
P
2
428.6549
432.7667
426.6061
447.2694
426.6061
P
3
430.7266
432.4754
429.6632
377.7218
429.6631
P
4
429.5193
426.7217
429.6632
391.1059
429.6632
P
5
438.8225
425.0443
429.6632
371.5138
429.6632
P
6
436.7947
426.7682
429.6632
417.4801
429.6631
P
7
424.1592
429.8489
429.6632
468.3885
429.6632
P
8
427.7913
433.0439
429.6632
376.9151
429.6631
P
9
114.0000
114.0000
114.0000
137.9410
114.0000
P
10
114.1613
114.0001
114.0000
155.5833
114.0000
P
11
114.7729
120.6554
119.7680
142.6568
119.7681
P
12
127.0026
128.0361
127.0728
187.8007
127.0729
P
13
110.0000
110.0000
110.0000
110.0000
110.0000
P
14
90.0025
90.0000
90.0000
90.0000
90.0000
P
15
82.0000
82.0000
82.0000
82.0000
82.0000
P
16
120.0000
120.0000
120.0000
120.0000
120.0000
P
17
160.1416
158.7822
159.5980
161.4797
159.5980
P
18
65.0008
65.0000
65.0000
65.0000
65.0000
P
19
65.0000
65.0000
65.0000
65.0000
65.0000
P
20
272.0000
272.0000
272.0000
271.9999
272.0000
P
21
272.0000
272.0000
272.0000
272.0000
272.0000
P
22
259.3185
260.0000
260.0000
260.0000
260.0000
P23
130.9561
129.4275
130.6486
138.0732
130.6486
P
24
10.0000
10.0000
10.0000
10.0002
10.0000
P
25
116.3424
108.7907
113.3050
110.9243
113.3050
P
26
90.6858
87.3194
88.0669
88.8202
88.0669
P
27
35.0000
36.1978
37.5051
39.1707
37.5051
P
28
20.0000
20.0000
20.0000
20.0000
20.0000
P
29
20.0000
20.0000
20.0000
20.0000
20.0000
P
30
20.0005
20.0000
20.0000
20.0000
20.0000
P
31
20.0000
20.0000
20.0000
20.0000
20.0000
P
32
20.0000
20.0000
20.0000
20.0000
20.0000
P
33
25.0000
25.0000
25.0000
25.0000
25.0000
P
34
18.0000
18.0000
18.0000
18.0000
18.0000
P
35
8.0000
8.0000
8.0000
8.0000
8.0000
P
36
25.0000
25.0000
25.0000
25.0000
25.0000
P
37
21.6764
21.5606
21.7821
21.3068
21.7821
P
38
20.0000
21.1175
21.0622
20.5701
21.0622
TP (MW)
6000.0000
6000.0000
6000.0000
6000.0000
6000.0000
TC ($/h)
9417446.0804
9417307.5544
9417235.7977
9426679.1431
9417235.7864Table 9
Best output power for 40-generator system (Case IV).
Generator
BBO
CSO
DE
PSO
ANS
P
1
112.2570
111.5065
110.8699
111.7714
110.8020
P
2
114.0000
112.0271
110.8626
75.1768
110.8004
P
3
101.5985
97.3999
97.3999
97.4000
97.4001
P
4
179.8608
179.7331
179.7331
129.8666
179.7330
P
5
97.0000
89.0477
97.0000
93.8081
87.8004
P
6
140.0000
140.0000
140.0000
140.0000
140.0000
P
7
260.2515
300.0000
259.5997
260.1252
259.5998
P
8
284.8846
284.6023
284.5997
285.0625
284.6000
P
9
285.6941
284.5997
284.5999
287.7589
284.5997
P
10
131.2062
130.0000
130.0000
204.7998
130.0000
P
11
168.4760
168.7998
168.7998
168.7998
94.0000
P
12
168.7683
94.0000
94.0000
243.5997
94.0000
P
13
215.2662
214.7598
214.7598
304.5196
214.7598
P
14
393.1850
394.2794
394.2794
394.2794
394.2794
P
15
306.7599
394.2794
394.2794
394.2794
394.2796
P
16
305.2877
304.5196
304.5196
304.5196
394.2793
P
17
488.9168
489.2794
489.2794
489.2794
489.2795
P
18
489.2538
489.2794
489.2794
489.2794
489.2793
P
19
511.3833
511.2794
511.2794
511.2794
511.2794
P
20
512.0211
511.2794
511.2794
511.2794
511.2795
P
21
523.1479
523.2794
523.2794
433.7901
523.2793
P
22
524.9892
523.2794
523.2794
523.4705
523.2794
P23
523.5749
523.2794
523.2794
523.2794
523.2793
P
24
524.6906
523.2794
523.2794
523.2794
523.2798
P
25
523.6776
523.2794
523.2794
523.2939
523.2794
P
26
526.3109
523.2794
523.2794
523.2794
523.2794
P
27
10.0000
10.0000
10.0000
10.0000
10.0000
P
28
10.0000
10.0000
10.0000
10.0000
10.0000
P
29
10.0000
10.0000
10.0000
10.0000
10.0000
P
30
93.4661
88.0999
87.8236
88.2486
87.8000
P
31
190.0000
190.0000
190.0000
190.0000
190.0000
P
32
190.0000
190.0000
190.0000
161.4680
189.9998
P
33
190.0000
190.0000
190.0000
161.6225
190.0000
P
34
199.8577
165.1100
164.8006
165.0999
164.7997
P
35
168.7132
165.1627
200.0000
168.0133
199.9752
P
36
184.9895
200.0000
200.0000
165.3862
194.4181
P
37
110.0000
110.0000
110.0000
91.6051
110.0000
P
38
109.1741
110.0000
110.0000
110.0000
110.0000
P
39
110.0000
110.0000
110.0000
110.0000
110.0000
P
40
511.3378
511.2794
511.2794
511.2794
511.2794
TP (MW)
10500.0000
10500.0000
10500.0000
10500.0000
10500.0000
TC ($/h)
121669.9298
121440.8040
121420.9029
122636.1149
121412.6226In conclusion, the abovementioned comparison results sufficiently demonstrate that ANS is able to achieve a strong equilibrium between the local exploitation and global exploration.


## 5.1.2. Convergence Property

The convergence curves of the mean costs for the four cases are plotted in Figures2–5, respectively. It can be seen that although CSO and DE are slightly faster than ANS at the very beginning, both methods are quickly trapped into local search later and thus suffer from prematurity. The convergence speed of BBO and PSO, especially the latter, is slow. ANS can consistently improve the solution quality and converge towards the global optima throughout the whole evolutionary process in all cases especially in Cases I and IV, which, from another perspective, indicates that ANS possesses better exploration and exploitation abilities of jumping out of local search and finding a more promising searching direction.Figure 2
Convergence curves for the 13-generator system (Case I).Figure 3
Convergence curves for the 15-generator system (Case II).Figure 4
Convergence curves for the 38-generator system (Case III).Figure 5
Convergence curves for the 40-generator system (Case IV).


## 5.1.3. Robustness

Since population-based algorithms use random numbers to initialize the population and employ randomization procedures to promote the search process, randomness is inevitable. In this context, it may be inappropriate to comprehensively assess their performance just through one single run. Thus, a number of independent runs with different initial populations can be used to measure their stability and consistency, i.e., robustness. The standard deviation results provided in Tables2–5 clearly illustrate that the recorded values of ANS are significantly smaller than those of BBO, CSO, DE, and PSO. Moreover, they are also very competitive with those of other recently proposed ED problem solution methods. The comparisons indicate that ANS has strong robustness and it can achieve a relatively stable optimal result in each trial. In addition, through careful observation, we can see that the standard deviation values of Cases I and IV are bigger than those of Cases II and III. This is because the valve-point effects make the ED problems exhibit highly multimodal characteristic and the solution methods are more likely trapped into different local minima in different trials. Therefore, the valve-point effects are more challenging for solution methods.


## 5.1.4. Statistical Analysis

A nonparametric statistical test called Wilcoxon’s rank sum test for independent samples is employed to compare the significance differences between ANS and its competitors. The results based on the Wilcoxon’s rank sum test at a 0.05 confidence level are summarized in Table10. The mark “†” symbolizes that ANS is statistically better than its competitor. It can be seen that ANS performs significantly better than BBO, CSO, DE, and PSO in all cases, meaning that ANS is capable of obtaining overall higher quality of the final solutions than the other four popular population-based algorithms.Table 10
Statistical analysis results based on Wilcoxon’s rank sum test.
ANS vs.
BBO
CSO
DE
PSO
Case III
†
†
†
†
Case II
†
†
†
†
Case III
†
†
†
†
Case IV
†
†
†
†


## 5.2. Influence of Population Size

Choosing an appropriate population size is always critical for population-based algorithms in solving different problems. In this subsection, an experiment is conducted to investigate the sensitivity of ANS to variations in population size. The population sizep
s is set as 10 to 100 with interval 10. The results are shown in Figure 6.Figure 6
Influence of population size. (a) Case I. (b) Case II. (c) Case III. (d) Case IV.
(a)
(b)
(c)
(d)The following can be seen: (i) For the unimodal ED problems, i.e., Cases II and III, the smaller the population size, the better the performance ANS yields. (ii) For the multimodal ED problems, i.e., Cases I and IV, the bigger or smaller the population size, the worse the performance ANS obtains, and ANS can achieve the best results inp
s
=
40. The reason might be that, for the unimodal ED problems, it is relatively easy for ANS to find the correct searching direction and there is no need for exploring the entire solution space. Therefore, a small population can swarm towards the global optimum easily. While for the multimodal ED problems, on one hand, a small population tends to converge very fast before fully exploring the entire solution space, thus resulting in prematurity. On the other hand, although a large population can increase the population diversity significantly, the distribution of individuals is sparse and the probability of finding the correct searching direction is sharply reduced. In addition, a large population size will consume a large number of fitness evaluations in each iteration, which is not proper for computationally expensive problems. In general, for the traditional ED problems without considering the valve-point effects, a small population size is recommended, whereas for the nonconvex ED problems, it is safe to set a moderate population size.


## 5.3. Influence of the Across-Search Degree

The across-search degreen in ANS is utilized to control the amount of information deriving from other superior solutions. In this subsection, an experiment is conducted to investigate the influence of n on the performance of ANS. The experimental results in different across-search degrees are presented in Figure 7.Figure 7
Influence of across-search degree. (a) Case I. (b) Case II. (c) Case III. (d) Case IV.
(a)
(b)
(c)
(d)It can be observed that the best values ofn for all cases are all 1. Besides, for the unimodal ED problems, the bigger the value n is, almost the worse the performance of ANS gets. For the multimodal ED problems, a temperate value of n will considerably deteriorate ANS though a bigger value is also not good for ANS. The reason might be that, for both unimodal and multimodal ED problems, bigger values of n will damage individuals’ solution components vastly in each iteration, which is not conducive to the consistency of convergence. In addition, for the multimodal ED problems, although a more bigger value of n can fully recombine each individual to maintain the population diversity, it will make individuals trap into different local optima frequently and thus slow down the convergence considerably. Generally, the recommendation value of n for different ED problems is 1.


## 6. Application to a Practical Provincial Power Grid of China

In the previous section, the feasibility and effectiveness of ANS in solving ED problems are comprehensively validated on four benchmark test cases. In this section, ANS is applied to a practical power grid of China. This system is more large-scale and has 46 operating generators which contain four different rated capacities, i.e., 150MW (#1~#2), 200MW (#3~#8), 300MW (#9~#38), and 600MW (#39~#46). The load demand is 10048MW. It is worth pointing out that the ED objective of this system is minimization of the total coal consumption rather than the total generation cost. The main reasons are twofold. On one hand, the price of coal fluctuates frequently. On the other hand, utilizing the coal consumption instead of coal cost to measure the generation efficiency of generators is more intuitive and reasonable. The objective function is a canonical traditional ED problem with a smooth and continuous solution space. The experimental results are tabulated in Table11. It can be seen that ANS is better than the other involved methods in terms of the minimum, maximum, and mean values of the total coal consumption. The statistical results based on Wilcoxon’s rank sum test also further confirm the conclusion. With respect to the standard deviation, the value of ANS is significantly smaller than those of compared methods, indicating that ANS is the most robust method among the five methods. In addition, the convergence curves in Figure 8 show that CSO converges the fastest in the early phase, followed by DE, ANS, PSO, and BBO. However, both CSO and DE stagnate quickly and are surpassed by ANS after about 7500 FEs. BBO and PSO are very slow during the whole evolutionary process. ANS is able to converge to the optimal solution.Table 11
Comparison of simulation results (t/h).
Method
Min.
Max.
Mean
Std Dev
Statistical result
BBO
3357.2587
3358.3753
3357.6023
0.2373
†
CSO
3356.9538
3363.8324
3357.5218
1.6223
†
DE
3356.9533
3357.2138
3357.0257
0.1061
†
PSO
3357.1167
3366.9071
3359.2455
2.4471
†
ANS
3356.9528
3356.9533
3356.9529
9.9122E-05
†Figure 8
Convergence curves for the practical provincial power grid of China.In order to learn the operation performance of each generator, the detailed load rate and coal consumption values are presented in Figure9. It is worth noting that the 150MW generators (#1~#2) are combined heat and power (CHP) generators whose peak regulation depth must not exceed 30%; namely, the adjusting range is from 70% to 100% of the rated capacity. For the other generators, their peak regulation depths are all 50%. It can be seen from Figure 9 that the operation performance indicators of different generators vary from one to another. The mean values of load rate and coal consumption of various capacity generators are summarized in Table 12. It is obvious that, except the 150MW generator, the larger the rated capacity of the generator, the higher the load rate, and the lower the coal consumption. For the 150MW generator, the load rate equals the allowed minimum limit, meaning that its operation performance is poor. In addition, although its load rate is greater than those of the 300MW and 600MW generators, its coal consumption is significantly higher, which indicates that its energy conservation is inefficient. The abovementioned results fully reflect the necessity and importance of the power construction policy of “replacing small power plants with large ones” in China.Table 12
Operation performance of various capacity generators.
Performance indicator
Generator capacity
150MW
200MW
300MW
600MW
Mean load rate (%)
70.00
58.73
65.61
67.28
Mean coal consumption (g/kWh)
373.95
391.77
337.64
321.73Figure 9
Operation performance of all generators.


## 7. Conclusions and Future Work

In this paper, a simple yet very young and efficient population-based algorithm named across neighborhood search (ANS) is applied to ED problems. Four benchmark test cases with diverse complexities and characteristics are firstly employed to comprehensively verify the feasibility and effectiveness of ANS. The experimental results demonstrate that ANS is able to effectively coordinate the local exploitation and global exploration. It significantly outperforms four popular population-based algorithms (BBO, DE, CSO, and PSO) in terms of the final solution quality, convergence speed, robustness, and statistics. ANS is also capable of achieving better or competitive results compared with some recently proposed ED solution methods. In addition, the sensitivities of ANS to variations of population size and across-search degree are investigated. The experimental results indicate that a small population size is recommended for the traditional ED problems, whereas a moderate population size is relatively safe for the nonconvex ED problems with valve-point effects. For both convex and nonconvex ED problems, the across-search degree with value 1 is appropriate. In addition to the benchmark test cases, ANS is further applied to a practical provincial power grid of China. The experimental results strongly verify ANS once again and fully reflect the necessity and importance of the power construction policy of “replacing small power plants with large ones” in China. In conclusion, ANS can be used as an efficient and reliable alternative for the ED problems.ANS is a young and promising population-based algorithm. In future work, we will employ some advanced learning strategies such as orthogonal learning and oppositional learning to further enhance its performance and then apply it to solve other power system optimization problems such as combined economic and emission dispatch and optimal power flow.



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*Source: 2591341-2018-11-04.xml*