# A Memetic Approach for Improving Minimum Cost of Economic Load Dispatch Problems

Authors: Jinho Kim; Chang Seob Kim; Zong Woo Geem
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

Economic load dispatch problem is a popular optimization problem in electrical power system field, which has been so far tackled by various mathematical and metaheuristic approaches including Lagrangian relaxation, branch and bound method, genetic algorithm, tabu search, particle swarm optimization, harmony search, and Taguchi method. On top of these techniques, this study proposes a novel memetic algorithm scheme combining metaheuristic algorithm and gradient-based technique to find better solutions for an economic load dispatch problem with valve-point loading. Because metaheuristic algorithms have the strength in global search and gradient-based techniques have the strength in local search, the combination approach obtains better results than those of any single approach. A bench-mark example of 40 generating-unit economic load dispatch problem demonstrates that the memetic approach can further improve the existing best solutions from the literature.


---

## Body

## 1. Introduction

Economic load dispatch (ELD) problem is a classical form of optimization problems and has been one of the most important decision-making processes in the operation of electrical power systems. The total system-wide generation cost is generally defined as the objective function of ELD problem. The equality and inequality power system constraints are embedded in ELD formulation, such as power balance and generation limits of each generating unit’s output capacity. ELD problem has been thought of as a mathematically complex and highly nonlinear optimization problem, especially in larger systems. For many decades, many algorithms have been presented to solve the optimization problem of ELD. First, conventional deterministic approaches which resort to mathematical gradient information have been developed to obtain the minimum cost of ELD problem. To overcome the limitations of those deterministic algorithms in real-system applications which are basically associated with the simplification of mathematical formulation, a variety of evolutionary frameworks that employ metaheuristic computational intelligence have explored their capabilities to search optimal solution of ELD problem with little abbreviation of original formulation.Many gradient-based and enumeration-based deterministic approaches such as Lagrangian relaxation and linear, nonlinear, and dynamic programming techniques have been applied to find the optimal solution of ELD problem [1–7]. They are suitable, however, only when the problem satisfies certain conditions (complex calculus-based gradient should be derived or formulation can be approximated in terms of linearity and/or convexity), and therefore this mathematical requirement has limited the extensive applications of these deterministic algorithms in real ELD problem. In addition, extensive search algorithms which explore all the solution space such as dynamic programming [2, 3], therefore, have been employed in ELD problem to tackle the issue of the simplification in the mathematical formulation of ELD. Due to the so-called “curse of dimensionality” and local optimality, however, these approaches hardly give solutions with higher level of satisfaction in the real-world applications.For recent decades, therefore, a variety of artificial intelligence approaches which are dependent upon heuristic and stochastic search scheme have been intensively adopted in ELD problem. Many studies have aimed to overcome the shortcomings of the conventional deterministic algorithms and to investigate the efficiency and applicability of the algorithms in ELD problem. For example, genetic-based different types of algorithms (GA) [8–15], a variety of evolutionary programming-based algorithms (EP) [16–19], particle swarm optimization (PSO) with its variations [20–28], tabu search (TS) [29], Taguchi method [30], bioinspired optimization algorithms [31–34], harmony search (HS) [35, 36], and hybrid methods combining two or more metaheuristic algorithms [37–42] have been applied to explore the capability of stochastic artificial intelligence algorithms to solve ELD problem so far.Up until now, we can see that several well-developed metaheuristic algorithms have shown a higher level of applicability in ELD problem with fine performance [43–53]. In spite of the acceptable level of performance in some algorithms of ELD problem, we have found that a memetic approach for ELD problem to combine the well-developed evolutionary frameworks with gradient-based local search algorithm can provide an opportunity for better solutions. Through this memetic combination, it can be seen that the minimum cost of ELD problem can be further improved because metaheuristic algorithm’s weak local search performance is reinforced by calculus-based method and calculus-based method’s weak global search performance is reinforced by metaheuristic algorithm.This paper is organized as follows. Mathematical formulation of ELD problems and recent metaheuristic algorithms to solve ELD problem are described in Section2. In Section 3, memetic implementations for those algorithms are explored to integrate gradient-based local search into the evolutionary frameworks for the solution of ELD. Simulation results and discussions on the algorithm proposed are provided in Section 4, followed by conclusion and future investigation in Section 5.


## 2. Problem Description: Economic Load Dispatch with Valve-Point Loading

The ELD problem can be stated as to determine the optimal set of individual generating units’ generation outputs minimizing the objective function(s) as well as satisfying both the equality and inequality constrains. The ELD can, therefore, be mathematically formulated as a continuous variable optimization problem. The objective function to be minimized can be defined as the system-wide generation cost across all the generators. Equality constraint of ELD is a power balancing equation where total power supply of all the generators is equal to the total system demand plus system loss. In addition, individual generators’ generation output should be in-between its minimum and maximum generation capacity, and this condition is imposed as the inequality constraint for each generator’s output in ELD problem. The mathematical formulation of the generic ELD problem can, therefore, be described as follows:(1)
Min
⁡
C
=
∑
i
∈
I
F
i
(
P
i
)
=
∑
i
∈
I
⌊
a
i
+
b
i
P
i
+
c
i
P
i
2
⌋
subject
to
∑
i
∈
I
P
i
=
D
+
L
a
a
a
a
a
a
a
a
P
i
_
≤
P
i
≤
P
i
¯
,
where C is system-wide total generating units’ cost function; F
i is generation cost function of generating unit-i; P
i is generation output of generating unit-i; a
i, b
i, and  c
i are cost coefficients of generating unit-i; Iis index set of generating units in the system; D is total system demand; L is total system loss; P
i
_, P
i
¯ are minimum and maximum generation output of generating unit-i.The system loss in the transmission lines,L, can be presented as a quadratic form of generating units’ output generations as follows:
(2)
L
=
∑
i
∈
I
∑
j
∈
I
P
i
B
i
j
P
j
+
∑
i
∈
I
P
i
B
0
i
+
B
00
,
where B
i
j is the i
jth element of loss coefficient matrix, B
0
i is ith element of loss coefficient vector, and B
00 is ith element of loss coefficient constant.Moreover, more practical consideration of ELD problem requires an inclusion of valve-point loading effects in ELD problem, and the mathematical formulation of ELD with valve-point loading can be rephrased as follows:(3)
Min
⁡
C
=
∑
i
∈
I
F
i
(
P
i
)
=
∑
i
∈
I
⌊
a
i
+
b
i
P
i
+
c
i
P
i
2
+
|
e
i
·
sin
(
f
i
·
(
P
i
_
-
P
i
)
)
|
⌋
,
where e
i and f
i are nonsmooth fuel cost coefficients of generating unit-i.Many gradient-based and enumeration-based deterministic approaches have been applied to find the optimal solution of ELD problem, and they are suitable only when the problem satisfies certain conditions. Extensive search algorithms have been employed in ELD problem to tackle the issue of the simplification in the mathematical formulation of ELD. These approaches hardly give solutions with higher level of satisfaction in the real-world applications due to the curse of dimensionality and local optimality. For recent decades, many artificial intelligence approaches have been intensively adopted in ELD problem to investigate the efficiency and applicability of the algorithms in ELD problem. In ELD problem, although several algorithms for solving economic dispatch are well developed up until now, we have found that memetic crossover between gradient-based algorithm and metaheuristic methodology can provide opportunity of better solutions for ELD. The memetic algorithm is based on the characteristic of the meme in human culture, which is the basic unit of knowledge that can be modified, combined with other ones, and generating new ones to propagate in the communities [63]. Under the optimization problems like ELD, the memetic algorithm considers complex structures of conventional metaheuristic algorithms and simple gradient-based optimization operators, where the interactions of evolutionary and local search algorithm lead to an efficient solution method that is able to solve ELD problem more effectively. In this study, therefore, a memetic approach to combine stochastic metaheuristic algorithms with gradient-based local search is presented to explore the opportunity of better solution for ELD problem.Many metaheuristic approaches using population-based evolutionary frameworks have been applied in ELD problem in order to find the optimal solution. The population-based metaheuristic algorithms for ELD primarily resort to iterated procedures for initializing, competing, and updating of population in reaching optimal solution(s), and, therefore, the general solution steps of the metaheuristic algorithms for ELD can be described as shown in Algorithm1.Algorithm 1:The general solution steps of the metaheuristic algorithms for ELD.
Step  1. Initialization of the population for the parent generation
Step  2. Iterations
Step  2.1. Creation of the population for the new generation
Step  2.2. Competition among populations
Step  2.3. Population update
Step  3. TerminationDepending upon the specific individual evolutionary framework in a metaheuristic algorithm, the initialization, competition, and update of population process for ELD in the algorithm can vary. Some algorithms employ a random initialization for the parent population, while the others attempt to use a better quality one as a starting population by using a refined heuristic. In the iterated competing and updating processes for ELD, the respective metaheuristic algorithms adopt their own computational intelligence operators based upon their functional advantages over other ones. For example, genetic and evolutionary upgrade operators are used in genetic and evolutionary algorithms. Social patterns in the behaviors of animals are modeled in particle swarm optimization, honey bee mating, and firefly algorithm. Also, physical phenomenon-based population improvement is explored in harmony search, tabu search, and simulated annealing. As for the termination, the iterated processes in the algorithms stop based upon their own termination criteria, which are typically defined as the maximum number of iterations and/or minimum extent of the improvement of the fitness value(s) for ELD.In recent ELD studies using metaheuristic algorithms, several biogeography-based and musical improvisation methodologies are presented to find the optimal solution of ELD problem [33, 34, 36, 64]. In [33, 34], a firefly algorithm (FA) and a honey bee mating optimization (HBMO) algorithm have been developed to solve nonconvex ELD problem with valve loading effect, respectively, and they have shown a higher level of performance than conventional metaheuristic algorithms. In addition, a variation of pattern search (PS) methods and a population-variance harmony search (HS) algorithm have also been presented to solve the constrained ELD problem, in [36, 64], respectively. Two algorithms have also demonstrated their better capabilities in ELD applications than existing relevant approaches.


## 3. Memetic Implementations for ELD Problem

Memetic computing is a branch in computer science which regards complicate structures as the combination of heterogeneous operators, named memes, whose evolutionary interactions contribute to intelligent structures for problem solving [65, 66].For recent decades, memetic algorithms have been widely applied in the large-scale complex optimization problems. The problem of balance between global and local search, that is, balance between computational intelligence and gradient-based search algorithm, has been explored under a multiobjective optimization setting in [67, 68]. In [69], a memetic algorithm is presented to analyze the evolutionary artificial neural network for training of a medical application. For a large-scale combinatorial optimization problem, a parallel memetic algorithm with selective local search is proposed in [70]. In [71–73], memetic algorithms are applied to solve scheduling and planning problems such as vehicle routing and path planning. A fast adaptive memetic algorithm for a design of controller of engineering drives is presented in [74], and a compact memetic differential algorithm has been developed for a robot control in [75]. In [76], a differential evolution-based hybrid algorithm is provided to solve the optimization problem in aerodynamic design.In order to explore the opportunity of better solution in ELD problem using the memetic combination proposed in this study, we have basically considered three metaheuristic algorithms: harmony search (HS), firefly algorithm (FA), and honey bee mating optimization (HBMO), which show the highest level of performances among all the existing relevant population-based metaheuristic approaches in the most recent ELD problem literatures [33, 34, 36]. HS algorithm was originally inspired by the improvisation process of Jazz musicians. The algorithm uses a novel stochastic derivative which utilizes the experiences of musicians in Jazz improvisation and can be applicable to optimization problem. Instead of the gradient-based inclination information of an objective function, the stochastic derivative of the algorithm gives a probability to be selected for each value of a decision variable. In [36], the evolution of the population-variance over successive generations in HS was analyzed and the algorithm showed that it can easily take care of solving nonconvex ELD problems along with different constraints. The FA was based on the idealized behavior of the flashing characteristics of fireflies such that the brightness or light intensity of a firefly is affected or determined by the landscape of the objective function to be optimized. In [33], a FA for solving nonsmooth economic dispatch with various constraints was presented and the results are compared with recent ELD solution methods. The HBMO algorithm is based upon the social characteristics of the honey bee in working together in a highly structured social order. In [34], a method to improve the mating process of HBMO combining the HBMO with a chaotic local search was proposed to overcome the disadvantage of the conventional algorithm that may miss the optimum and provide a near optimum solution in a limited runtime period. Table 1 shows the minimum cost of ELD problem using the most recent three metaheuristic algorithms in [33, 34, 36] (i.e., HS, FA, and HBMO), with the same test system of 40 generating units.Table 1
Minimum costs of ELD problem from three different algorithms.
Number
Generation (MW)
Cost ($)
HS
FA
HMBO
HS
FA
HMBO
1
110.8312
110.8099
110.8018
925.6191
925.2642
925.1296
2
110.8234
110.8059
110.8000
925.4891
925.1976
925.0998
3
97.3999
97.4023
97.3999
1190.5485
1190.5949
1190.5485
4
179.7331
179.7332
179.7331
2143.5503
2143.5524
2143.5503
5
88.9932
92.7070
87.7998
726.2998
787.1211
706.5003
6
140.0000
140.0000
140.0000
1596.4643
1596.4643
1596.4643
7
259.6006
259.6004
259.5997
2612.9019
2612.8983
2612.8846
8
284.6143
284.6004
284.5997
2780.1031
2779.8502
2779.8368
9
284.5997
284.6004
284.5997
2798.2312
2798.2440
2798.2303
10
130.0000
130.0028
130.0000
2502.0650
2502.1290
2502.0650
11
168.7998
168.8008
94.0000
2959.4585
2959.4811
1893.3054
12
168.7998
168.8008
94.0000
2977.4549
2977.4776
1908.1668
13
214.7598
214.7606
214.7598
3792.0703
3792.0901
3792.0700
14
394.2794
304.5204
394.2794
6414.8612
5149.7187
6414.8604
15
304.5191
394.2801
394.2794
5171.1965
6436.6047
6436.5863
16
394.2794
394.2801
394.2794
6436.5870
6436.6047
6436.5863
17
489.2794
489.2801
489.2794
5296.7114
5296.7265
5296.7107
18
489.2794
489.2801
489.2794
5288.7658
5288.7809
5288.7652
19
511.2796
511.2817
511.2794
5540.9342
5540.9797
5540.9292
20
511.2794
511.2817
511.2794
5540.9099
5540.9597
5540.9092
21
523.2867
523.2793
523.2794
5071.4381
5071.2898
5071.2897
22
523.2836
523.2793
523.2807
5071.3753
5071.2898
5071.3156
23
523.2794
523.2832
523.2794
5057.2237
5057.3002
5057.2231
24
523.2794
523.2832
523.2794
5057.2237
5057.3002
5057.2231
25
523.2794
523.2793
523.2794
5275.0891
5275.0886
5275.0885
26
523.2794
523.2793
523.2794
5275.0891
5275.0886
5275.0885
27
10.0000
10.0000
10.0000
1140.5240
1140.5240
1140.5240
28
10.0000
10.0000
10.0000
1140.5240
1140.5240
1140.5240
29
10.0000
10.0000
10.0000
1140.5240
1140.5240
1140.5240
30
91.2184
87.8008
87.7999
762.9889
706.5149
706.5001
31
190.0000
189.9989
190.0000
1643.9913
1643.9869
1643.9913
32
190.0000
189.9989
190.0000
1643.9913
1643.9869
1643.9913
33
190.0000
189.9989
190.0000
1643.9913
1643.9869
1643.9913
34
164.9179
164.8036
164.8015
1587.5968
1585.6100
1585.5733
35
164.8672
164.8036
194.3928
1541.0193
1539.9348
1985.3690
36
164.8786
164.8036
200.0000
1541.2137
1539.9348
2043.7270
37
110.0000
110.0000
110.0000
1220.1661
1220.1661
1220.1661
38
110.0000
110.0000
110.0000
1220.1661
1220.1661
1220.1661
39
110.0000
110.0000
110.0000
1220.1661
1220.1661
1220.1661
40
511.2795
511.2794
511.2794
5540.9320
5540.9299
5540.9292
Sum
10500.0000
10500.0000
10500.0000
121415.4560
121415.0522
121412.5704Based upon the result in Table1, therefore, we have proposed a memetic combination of the most recent HS, FA, and HBMO algorithms with gradient-based local search operator to improve the minimum cost of the ELD problem. The memetic approach to metaheuristic algorithms for ELD problem combines a local search operator with a metaheuristic algorithm, and it is a basically population-based metaheuristic which enables an evolutionary framework to search local solutions in a gradient manner within a certain generation cycle. As the memetic approach blends together a population-based computational intelligence algorithm and local search method, the memetic algorithm for ELD problem can provide a better opportunity for the optimal solution by combining two individual robust operators, which have higher capability of global search and local search, in the solution space, respectively. Therefore, the general structure of the memetic algorithm for ELD can be rephrased as shown in Algorithm 2 to mimetically associate gradient-based search algorithm with conventional population-based metaheuristic.Algorithm 2:The general structure of the memetic approach to metaheuristic for ELD.
Step  1. Initialization of the population for the parent generation
Step  2. Iterations
Step  2.1. Creation of the population for the new generation
Step  2.2. Competition among population
Step  2.3. Population update
Step  3.
Gradient-based local search
Step  4. Termination


## 4. Gradient-Based Local Search Algorithm

For the gradient-based local search algorithm, this study adopted BFGS (Broyden-Fletcher-Goldfarb-Shanno) method [77], which can solve nonlinear optimization problems by eliminating technical constraints. The solution steps of the BFGS method are as follows.Step 1.
Identify an initial feasible solution,x
k
=
0
=
(
x
1
0
,
x
2
0
,
…
,
x
n
0
).Step 2.
Calculate the searching direction,d
k.Step 3.
Calculate a new solution,x
k
+
1
=
x
k
+
β
k
d
k, where β
k is the step size which minimizes f
(
x
k
+
β
k
d
k
).Step 4.
If convergence criterion is not satisfied, setk
=
k
+
1 and go to Step 2.The searching direction in Step2 can be obtained by various methods such as steepest descent, conjugate gradient, quasi-Newton, and Newton methods. Out of various approaches, this study chose the quasi-Newton method, named BFGS, because it is one of the most powerful techniques. The BFGS method emulates the inverse Hessian matrix instead of directly calculating it as follows:(4a)
d
k
+
1
=
G
k
+
1
∇
f
(
x
k
+
1
)
,
(4b)
∇
f
(
x
)
=
[
∂
f
∂
x
]
=
(
∂
f
∂
x
1
,
∂
f
∂
x
2
,
…
,
∂
f
∂
x
n
)
T
,
(4c)
G
k
+
1
=
G
k
+
(
1
+
(
Y
k
)
T
G
k
Y
k
(
Y
k
)
T
S
k
)
S
k
(
S
k
)
T
(
S
k
)
T
Y
k
-
Y
k
(
S
k
)
T
G
k
+
G
k
S
k
(
Y
k
)
T
(
Y
k
)
T
S
k
,
(4d)
Y
k
+
1
=
∇
f
(
x
k
+
1
)
-
∇
f
(
x
k
)
,
(4e)
S
k
+
1
=
x
k
+
1
-
x
k
.The convergence criterion in this study is the relative error as follows:(5)
|
f
(
x
k
)
-
f
(
x
k
+
1
)
f
(
x
k
)
|
<
ε
.The BFGS method, which is a calculus-based technique, has the advantages over metaheuristic algorithms in terms of(
1
) fast convergence (it takes less than one second in this study), (
2
) good local search performance, and (
3
) having no algorithm parameter requirement (e.g., crossover rate in GA and harmony memory considering rate in HS). However, it also has the disadvantages over metaheuristic algorithms in terms of (
1
) less global search performance and even (
2
) divergence [77–79].


## 5. Numerical Experiments and Discussions

### 5.1. Test System

The numerical test system used to explore the applicability of memetic algorithms in ELD problem in this study consists of forty generating units with valve-point loading effects, and the total system demand is 10,500 MW [18]. Table 2 shows the data of the forty generating units with valve-point for the test system. In order to compare the performance proposed in this paper to those of existing metaheuristic only algorithms, we have adopted the same test system that is used in previous studies [18].Table 2
Generating units data for test system.
Number
P
_
P
¯
a
b
c
e
f
1
36
114
0.0069
6.73
94.705
100
0.084
2
36
114
0.0069
6.73
94.705
100
0.084
3
60
120
0.02028
7.07
309.54
100
0.084
4
80
190
0.00942
8.18
369.03
150
0.063
5
47
97
0.0114
5.35
148.89
120
0.077
6
68
140
0.01142
8.05
222.33
100
0.084
7
110
300
0.00357
8.03
287.71
200
0.042
8
135
300
0.00492
6.99
391.98
200
0.042
9
135
300
0.00573
6.6
455.76
200
0.042
10
130
300
0.00605
12.9
722.82
200
0.042
11
94
375
0.00515
12.9
635.2
200
0.042
12
94
375
0.00569
12.8
654.69
200
0.042
13
125
500
0.00421
12.5
913.4
300
0.035
14
125
500
0.00752
8.84
1760.4
300
0.035
15
125
500
0.00708
9.15
1728.3
300
0.035
16
125
500
0.00708
9.15
1728.3
300
0.035
17
220
500
0.00313
7.97
647.85
300
0.035
18
220
500
0.00313
7.95
649.69
300
0.035
19
242
550
0.00313
7.97
647.83
300
0.035
20
242
550
0.00313
7.97
647.81
300
0.035
21
254
550
0.00298
6.63
785.96
300
0.035
22
254
550
0.00298
6.63
785.96
300
0.035
23
254
550
0.00284
6.66
794.53
300
0.035
24
254
550
0.00284
6.66
794.53
300
0.035
25
254
550
0.00277
7.1
801.32
300
0.035
26
254
550
0.00277
7.1
801.32
300
0.035
27
10
150
0.52124
3.33
1055.1
120
0.077
28
10
150
0.52124
3.33
1055.1
120
0.077
29
10
150
0.52124
3.33
1055.1
120
0.077
30
47
97
0.0114
5.35
148.89
120
0.077
31
60
190
0.0016
6.43
222.92
150
0.063
32
60
190
0.0016
6.43
222.92
150
0.063
33
60
190
0.0016
6.43
222.92
150
0.063
34
90
200
0.0001
8.95
107.87
200
0.042
35
90
200
0.0001
8.62
116.58
200
0.042
36
90
200
0.0001
8.62
116.58
200
0.042
37
25
110
0.0161
5.88
307.45
80
0.098
38
25
110
0.0161
5.88
307.45
80
0.098
39
25
110
0.0161
5.88
307.45
80
0.098
40
242
550
0.00313
7.97
647.83
300
0.035


### 5.2. Test Case 1: Memetic Implementation with Harmony Search

First, we have implemented a memetic approach using HS algorithm. The best minimum generation cost of ELD for the given test system using HS reported until now is $121,415.4560 [36], and each generator’s output generation at the optimal point is presented in Table 3. The better minimum cost of ELD using a memetic approach to HS (M_HS) proposed in this study, however, can be obtained as $121,415.4525, and the results are also provided in Table 3.Table 3
Results for Test Case 1.
Number
Generation (MW)
Cost ($)
HS [36]
M_HS
HS [36]
M_HS
1
110.8312
110.8322
925.6191
925.6363
2
110.8234
110.8234
925.4891
925.4891
3
97.3999
97.3999
1190.5485
1190.5484
4
179.7331
179.7331
2143.5503
2143.5503
5
88.9932
88.9932
726.2998
726.2998
6
140.0000
140.0000
1596.4643
1596.4643
7
259.6006
259.6005
2612.9019
2612.9009
8
284.6143
284.6142
2780.1031
2780.1021
9
284.5997
284.5996
2798.2312
2798.2303
10
130.0000
130.0000
2502.0650
2502.0650
11
168.7998
168.7998
2959.4585
2959.4585
12
168.7998
168.7998
2977.4549
2977.4549
13
214.7598
214.7598
3792.0703
3792.0699
14
394.2794
394.2793
6414.8612
6414.8603
15
304.5191
304.5195
5171.1965
5171.1976
16
394.2794
394.2793
6436.5870
6436.5862
17
489.2794
489.2794
5296.7114
5296.7108
18
489.2794
489.2794
5288.7658
5288.7652
19
511.2796
511.2794
5540.9342
5540.9303
20
511.2794
511.2794
5540.9099
5540.9092
21
523.2867
523.2866
5071.4381
5071.4355
22
523.2836
523.2835
5071.3753
5071.3728
23
523.2794
523.2793
5057.2237
5057.2231
24
523.2794
523.2793
5057.2237
5057.2231
25
523.2794
523.2793
5275.0891
5275.0885
26
523.2794
523.2793
5275.0891
5275.0885
27
10.0000
10.0000
1140.5240
1140.5240
28
10.0000
10.0000
1140.5240
1140.5240
29
10.0000
10.0000
1140.5240
1140.5240
30
91.2184
91.2184
762.9889
762.9891
31
190.0000
190.0000
1643.9913
1643.9913
32
190.0000
190.0000
1643.9913
1643.9913
33
190.0000
190.0000
1643.9913
1643.9913
34
164.9179
164.9179
1587.5968
1587.5964
35
164.8672
164.8672
1541.0193
1541.0191
36
164.8786
164.8786
1541.2137
1541.2135
37
110.0000
110.0000
1220.1661
1220.1661
38
110.0000
110.0000
1220.1661
1220.1661
39
110.0000
110.0000
1220.1661
1220.1661
40
511.2795
511.2793
5540.9320
5540.9292
Sum
10500.0000
10500.0000
121415.4560
121415.4525


### 5.3. Test Case 2: Memetic Implementation with Firefly Algorithm

As a second experiment, we have carried out a memetic implementation using FA. In [33], the best minimum generation cost of ELD for the same test system using FA has been reported as $121,415.0522, and each generator’s output generation at this point is given in Table 4. However, the better minimum cost of ELD can be obtained using a memetic approach to FA (M_FA) presented in this paper as $121,414.9137, and the results are also provided in Table 4.Table 4
Results for Test Case 2.
Number
Generation (MW)
Cost ($)
FA [33]
M_FA
FA [33]
M_FA
1
110.8099
110.8341
925.2642
925.6675
2
110.8059
110.8059
925.1976
925.1976
3
97.4023
97.3999
1190.5949
1190.5484
4
179.7332
179.7330
2143.5524
2143.5501
5
92.7070
92.7077
787.1211
787.1319
6
140.0000
140.0000
1596.4643
1596.4643
7
259.6004
259.5996
2612.8983
2612.8845
8
284.6004
284.5997
2779.8502
2779.8377
9
284.6004
284.5996
2798.2440
2798.2303
10
130.0028
130.0000
2502.1290
2502.0650
11
168.8008
168.7998
2959.4811
2959.4583
12
168.8008
168.7999
2977.4776
2977.4566
13
214.7606
214.7598
3792.0901
3792.0700
14
304.5204
304.5195
5149.7187
5149.6989
15
394.2801
394.2792
6436.6047
6436.5857
16
394.2801
394.2792
6436.6047
6436.5857
17
489.2801
489.2799
5296.7265
5296.7230
18
489.2801
489.2799
5288.7809
5288.7772
19
511.2817
511.2793
5540.9797
5540.9291
20
511.2817
511.2793
5540.9597
5540.9091
21
523.2793
523.2793
5071.2898
5071.2898
22
523.2793
523.2793
5071.2898
5071.2898
23
523.2832
523.2794
5057.3002
5057.2231
24
523.2832
523.2794
5057.3002
5057.2231
25
523.2793
523.2793
5275.0886
5275.0886
26
523.2793
523.2793
5275.0886
5275.0886
27
10.0000
10.0000
1140.5240
1140.5240
28
10.0000
10.0000
1140.5240
1140.5240
29
10.0000
10.0000
1140.5240
1140.5240
30
87.8008
87.8009
706.5149
706.5162
31
189.9989
190.0000
1643.9869
1643.9913
32
189.9989
190.0000
1643.9869
1643.9913
33
189.9989
190.0000
1643.9869
1643.9913
34
164.8036
164.8028
1585.6100
1585.5961
35
164.8036
164.8032
1539.9348
1539.9274
36
164.8036
164.8032
1539.9348
1539.9274
37
110.0000
110.0000
1220.1661
1220.1661
38
110.0000
110.0000
1220.1661
1220.1661
39
110.0000
110.0000
1220.1661
1220.1661
40
511.2794
511.2789
5540.9299
5540.9289
Sum
10500.0000
10500.0000
121415.0522
121414.9137


### 5.4. Test Case 3: Memetic Implementation with Honey Bee Mating Optimization

HBMO algorithm is selected as a final experiment for the memetic implementation in this study. Up until now, the best minimum generation cost of ELD for the same test system using HBMO has been reported as $121,412.5704 [34], and each generator’s output generation at this point is presented in Table 5. Using a memetic combination to HBMO (M_HBMO) proposed in this paper, the better minimum cost of the same ELD problem can be obtained as $121,414.5702, and the results are presented in Table 5.Table 5
Results for Test Case 3.
Number
Generation (MW)
Cost ($)
HBMO [34]
M_HBMO
HBMO [34]
M_HBMO
1
110.8018
110.8014
925.1296
925.1218
2
110.8000
110.8000
925.0998
925.0998
3
97.3999
97.3999
1190.5485
1190.5489
4
179.7331
179.7331
2143.5503
2143.5507
5
87.7998
87.7999
706.5003
706.5002
6
140.0000
140.0000
1596.4643
1596.4643
7
259.5997
259.5997
2612.8846
2612.8850
8
284.5997
284.5997
2779.8368
2779.8372
9
284.5997
284.5997
2798.2303
2798.2307
10
130.0000
130.0000
2502.0650
2502.0650
11
94.0000
94.0000
1893.3054
1893.3054
12
94.0000
94.0000
1908.1668
1908.1668
13
214.7598
214.7598
3792.0700
3792.0702
14
394.2794
394.2794
6414.8604
6414.8606
15
394.2794
394.2794
6436.5863
6436.5865
16
394.2794
394.2794
6436.5863
6436.5865
17
489.2794
489.2794
5296.7107
5296.7112
18
489.2794
489.2794
5288.7652
5288.7656
19
511.2794
511.2794
5540.9292
5540.9296
20
511.2794
511.2794
5540.9092
5540.9096
21
523.2794
523.2794
5071.2897
5071.2902
22
523.2807
523.2806
5071.3156
5071.3153
23
523.2794
523.2794
5057.2231
5057.2236
24
523.2794
523.2794
5057.2231
5057.2236
25
523.2794
523.2794
5275.0885
5275.0890
26
523.2794
523.2794
5275.0885
5275.0890
27
10.0000
10.0000
1140.5240
1140.5240
28
10.0000
10.0000
1140.5240
1140.5240
29
10.0000
10.0000
1140.5240
1140.5240
30
87.7999
87.7999
706.5001
706.5005
31
190.0000
190.0000
1643.9913
1643.9913
32
190.0000
190.0000
1643.9913
1643.9913
33
190.0000
190.0000
1643.9913
1643.9913
34
164.8015
164.8015
1585.5733
1585.5732
35
194.3928
194.3928
1985.3690
1985.3693
36
200.0000
200.0000
2043.7270
2043.7270
37
110.0000
110.0000
1220.1661
1220.1661
38
110.0000
110.0000
1220.1661
1220.1661
39
110.0000
110.0000
1220.1661
1220.1661
40
511.2794
511.2794
5540.9292
5540.9296
Sum
10500.0000
10500.0000
121412.5704
121412.5702


## 5.1. Test System

The numerical test system used to explore the applicability of memetic algorithms in ELD problem in this study consists of forty generating units with valve-point loading effects, and the total system demand is 10,500 MW [18]. Table 2 shows the data of the forty generating units with valve-point for the test system. In order to compare the performance proposed in this paper to those of existing metaheuristic only algorithms, we have adopted the same test system that is used in previous studies [18].Table 2
Generating units data for test system.
Number
P
_
P
¯
a
b
c
e
f
1
36
114
0.0069
6.73
94.705
100
0.084
2
36
114
0.0069
6.73
94.705
100
0.084
3
60
120
0.02028
7.07
309.54
100
0.084
4
80
190
0.00942
8.18
369.03
150
0.063
5
47
97
0.0114
5.35
148.89
120
0.077
6
68
140
0.01142
8.05
222.33
100
0.084
7
110
300
0.00357
8.03
287.71
200
0.042
8
135
300
0.00492
6.99
391.98
200
0.042
9
135
300
0.00573
6.6
455.76
200
0.042
10
130
300
0.00605
12.9
722.82
200
0.042
11
94
375
0.00515
12.9
635.2
200
0.042
12
94
375
0.00569
12.8
654.69
200
0.042
13
125
500
0.00421
12.5
913.4
300
0.035
14
125
500
0.00752
8.84
1760.4
300
0.035
15
125
500
0.00708
9.15
1728.3
300
0.035
16
125
500
0.00708
9.15
1728.3
300
0.035
17
220
500
0.00313
7.97
647.85
300
0.035
18
220
500
0.00313
7.95
649.69
300
0.035
19
242
550
0.00313
7.97
647.83
300
0.035
20
242
550
0.00313
7.97
647.81
300
0.035
21
254
550
0.00298
6.63
785.96
300
0.035
22
254
550
0.00298
6.63
785.96
300
0.035
23
254
550
0.00284
6.66
794.53
300
0.035
24
254
550
0.00284
6.66
794.53
300
0.035
25
254
550
0.00277
7.1
801.32
300
0.035
26
254
550
0.00277
7.1
801.32
300
0.035
27
10
150
0.52124
3.33
1055.1
120
0.077
28
10
150
0.52124
3.33
1055.1
120
0.077
29
10
150
0.52124
3.33
1055.1
120
0.077
30
47
97
0.0114
5.35
148.89
120
0.077
31
60
190
0.0016
6.43
222.92
150
0.063
32
60
190
0.0016
6.43
222.92
150
0.063
33
60
190
0.0016
6.43
222.92
150
0.063
34
90
200
0.0001
8.95
107.87
200
0.042
35
90
200
0.0001
8.62
116.58
200
0.042
36
90
200
0.0001
8.62
116.58
200
0.042
37
25
110
0.0161
5.88
307.45
80
0.098
38
25
110
0.0161
5.88
307.45
80
0.098
39
25
110
0.0161
5.88
307.45
80
0.098
40
242
550
0.00313
7.97
647.83
300
0.035


## 5.2. Test Case 1: Memetic Implementation with Harmony Search

First, we have implemented a memetic approach using HS algorithm. The best minimum generation cost of ELD for the given test system using HS reported until now is $121,415.4560 [36], and each generator’s output generation at the optimal point is presented in Table 3. The better minimum cost of ELD using a memetic approach to HS (M_HS) proposed in this study, however, can be obtained as $121,415.4525, and the results are also provided in Table 3.Table 3
Results for Test Case 1.
Number
Generation (MW)
Cost ($)
HS [36]
M_HS
HS [36]
M_HS
1
110.8312
110.8322
925.6191
925.6363
2
110.8234
110.8234
925.4891
925.4891
3
97.3999
97.3999
1190.5485
1190.5484
4
179.7331
179.7331
2143.5503
2143.5503
5
88.9932
88.9932
726.2998
726.2998
6
140.0000
140.0000
1596.4643
1596.4643
7
259.6006
259.6005
2612.9019
2612.9009
8
284.6143
284.6142
2780.1031
2780.1021
9
284.5997
284.5996
2798.2312
2798.2303
10
130.0000
130.0000
2502.0650
2502.0650
11
168.7998
168.7998
2959.4585
2959.4585
12
168.7998
168.7998
2977.4549
2977.4549
13
214.7598
214.7598
3792.0703
3792.0699
14
394.2794
394.2793
6414.8612
6414.8603
15
304.5191
304.5195
5171.1965
5171.1976
16
394.2794
394.2793
6436.5870
6436.5862
17
489.2794
489.2794
5296.7114
5296.7108
18
489.2794
489.2794
5288.7658
5288.7652
19
511.2796
511.2794
5540.9342
5540.9303
20
511.2794
511.2794
5540.9099
5540.9092
21
523.2867
523.2866
5071.4381
5071.4355
22
523.2836
523.2835
5071.3753
5071.3728
23
523.2794
523.2793
5057.2237
5057.2231
24
523.2794
523.2793
5057.2237
5057.2231
25
523.2794
523.2793
5275.0891
5275.0885
26
523.2794
523.2793
5275.0891
5275.0885
27
10.0000
10.0000
1140.5240
1140.5240
28
10.0000
10.0000
1140.5240
1140.5240
29
10.0000
10.0000
1140.5240
1140.5240
30
91.2184
91.2184
762.9889
762.9891
31
190.0000
190.0000
1643.9913
1643.9913
32
190.0000
190.0000
1643.9913
1643.9913
33
190.0000
190.0000
1643.9913
1643.9913
34
164.9179
164.9179
1587.5968
1587.5964
35
164.8672
164.8672
1541.0193
1541.0191
36
164.8786
164.8786
1541.2137
1541.2135
37
110.0000
110.0000
1220.1661
1220.1661
38
110.0000
110.0000
1220.1661
1220.1661
39
110.0000
110.0000
1220.1661
1220.1661
40
511.2795
511.2793
5540.9320
5540.9292
Sum
10500.0000
10500.0000
121415.4560
121415.4525


## 5.3. Test Case 2: Memetic Implementation with Firefly Algorithm

As a second experiment, we have carried out a memetic implementation using FA. In [33], the best minimum generation cost of ELD for the same test system using FA has been reported as $121,415.0522, and each generator’s output generation at this point is given in Table 4. However, the better minimum cost of ELD can be obtained using a memetic approach to FA (M_FA) presented in this paper as $121,414.9137, and the results are also provided in Table 4.Table 4
Results for Test Case 2.
Number
Generation (MW)
Cost ($)
FA [33]
M_FA
FA [33]
M_FA
1
110.8099
110.8341
925.2642
925.6675
2
110.8059
110.8059
925.1976
925.1976
3
97.4023
97.3999
1190.5949
1190.5484
4
179.7332
179.7330
2143.5524
2143.5501
5
92.7070
92.7077
787.1211
787.1319
6
140.0000
140.0000
1596.4643
1596.4643
7
259.6004
259.5996
2612.8983
2612.8845
8
284.6004
284.5997
2779.8502
2779.8377
9
284.6004
284.5996
2798.2440
2798.2303
10
130.0028
130.0000
2502.1290
2502.0650
11
168.8008
168.7998
2959.4811
2959.4583
12
168.8008
168.7999
2977.4776
2977.4566
13
214.7606
214.7598
3792.0901
3792.0700
14
304.5204
304.5195
5149.7187
5149.6989
15
394.2801
394.2792
6436.6047
6436.5857
16
394.2801
394.2792
6436.6047
6436.5857
17
489.2801
489.2799
5296.7265
5296.7230
18
489.2801
489.2799
5288.7809
5288.7772
19
511.2817
511.2793
5540.9797
5540.9291
20
511.2817
511.2793
5540.9597
5540.9091
21
523.2793
523.2793
5071.2898
5071.2898
22
523.2793
523.2793
5071.2898
5071.2898
23
523.2832
523.2794
5057.3002
5057.2231
24
523.2832
523.2794
5057.3002
5057.2231
25
523.2793
523.2793
5275.0886
5275.0886
26
523.2793
523.2793
5275.0886
5275.0886
27
10.0000
10.0000
1140.5240
1140.5240
28
10.0000
10.0000
1140.5240
1140.5240
29
10.0000
10.0000
1140.5240
1140.5240
30
87.8008
87.8009
706.5149
706.5162
31
189.9989
190.0000
1643.9869
1643.9913
32
189.9989
190.0000
1643.9869
1643.9913
33
189.9989
190.0000
1643.9869
1643.9913
34
164.8036
164.8028
1585.6100
1585.5961
35
164.8036
164.8032
1539.9348
1539.9274
36
164.8036
164.8032
1539.9348
1539.9274
37
110.0000
110.0000
1220.1661
1220.1661
38
110.0000
110.0000
1220.1661
1220.1661
39
110.0000
110.0000
1220.1661
1220.1661
40
511.2794
511.2789
5540.9299
5540.9289
Sum
10500.0000
10500.0000
121415.0522
121414.9137


## 5.4. Test Case 3: Memetic Implementation with Honey Bee Mating Optimization

HBMO algorithm is selected as a final experiment for the memetic implementation in this study. Up until now, the best minimum generation cost of ELD for the same test system using HBMO has been reported as $121,412.5704 [34], and each generator’s output generation at this point is presented in Table 5. Using a memetic combination to HBMO (M_HBMO) proposed in this paper, the better minimum cost of the same ELD problem can be obtained as $121,414.5702, and the results are presented in Table 5.Table 5
Results for Test Case 3.
Number
Generation (MW)
Cost ($)
HBMO [34]
M_HBMO
HBMO [34]
M_HBMO
1
110.8018
110.8014
925.1296
925.1218
2
110.8000
110.8000
925.0998
925.0998
3
97.3999
97.3999
1190.5485
1190.5489
4
179.7331
179.7331
2143.5503
2143.5507
5
87.7998
87.7999
706.5003
706.5002
6
140.0000
140.0000
1596.4643
1596.4643
7
259.5997
259.5997
2612.8846
2612.8850
8
284.5997
284.5997
2779.8368
2779.8372
9
284.5997
284.5997
2798.2303
2798.2307
10
130.0000
130.0000
2502.0650
2502.0650
11
94.0000
94.0000
1893.3054
1893.3054
12
94.0000
94.0000
1908.1668
1908.1668
13
214.7598
214.7598
3792.0700
3792.0702
14
394.2794
394.2794
6414.8604
6414.8606
15
394.2794
394.2794
6436.5863
6436.5865
16
394.2794
394.2794
6436.5863
6436.5865
17
489.2794
489.2794
5296.7107
5296.7112
18
489.2794
489.2794
5288.7652
5288.7656
19
511.2794
511.2794
5540.9292
5540.9296
20
511.2794
511.2794
5540.9092
5540.9096
21
523.2794
523.2794
5071.2897
5071.2902
22
523.2807
523.2806
5071.3156
5071.3153
23
523.2794
523.2794
5057.2231
5057.2236
24
523.2794
523.2794
5057.2231
5057.2236
25
523.2794
523.2794
5275.0885
5275.0890
26
523.2794
523.2794
5275.0885
5275.0890
27
10.0000
10.0000
1140.5240
1140.5240
28
10.0000
10.0000
1140.5240
1140.5240
29
10.0000
10.0000
1140.5240
1140.5240
30
87.7999
87.7999
706.5001
706.5005
31
190.0000
190.0000
1643.9913
1643.9913
32
190.0000
190.0000
1643.9913
1643.9913
33
190.0000
190.0000
1643.9913
1643.9913
34
164.8015
164.8015
1585.5733
1585.5732
35
194.3928
194.3928
1985.3690
1985.3693
36
200.0000
200.0000
2043.7270
2043.7270
37
110.0000
110.0000
1220.1661
1220.1661
38
110.0000
110.0000
1220.1661
1220.1661
39
110.0000
110.0000
1220.1661
1220.1661
40
511.2794
511.2794
5540.9292
5540.9296
Sum
10500.0000
10500.0000
121412.5704
121412.5702


## 6. Conclusions

In summary, this study introduced a memetic scheme of metaheuristic algorithm and gradient-based technique and then applied it to a popular benchmark problem of 40 generating-unit ELD problem with valve-point loading, obtaining better solutions than other solutions ever found in the literature. As shown in Table6, the result ($121415.4560) of HS was further improved into $121415.4525 as shown in bold font; that ($121415.0522) of FA was further improved into $121414.9137 as shown in bold font; and that ($121412.5704) of HBMO was further improved into $121412.5702 as shown in bold font. We have extended the simulation comparisons against the other recent metaheuristic approaches, and it is also listed in Table 6. We also hope that this table and corresponding references become a good literature survey for future researches.Table 6
Test results and comparison.
Algorithm
Minimum cost ($)
Chaotic differential evolution and quadratic programming [54]
121,741.9700
PSO [55]
121,735.4700
Hybrid differential evolution [56]
121,698.5100
New PSO [57]
121,664.4300
Antipredatory PSO [55]
121,663.5200
Self-organizing hierarchical PSO [58]
121,501.1400
Biography-based optimization [59]
121,479.5000
GA-PS-SQP [60]
121,458,0000
Bacterial foraging [61]
121,423.6300
Harmony search (HS) [36]
121,415.4560
Memetic harmony search (this study)
121,415.4525
Pattern search [62]
121,415.1400
Firefly algorithm (FA) [33]
121,415.0522
Memetic firefly algorithm (this study)
121,414.9137
Honey bee mating optimization (HBMO) [34]
121,412.5704
Memetic HBMO (this study)
121,412.5702While metaheuristic algorithms perform well in global search, they do not perform well in local search. On the contrary, while gradient-based techniques perform well in local search, they do not perform well in global search. Thus, the memetic approach in this study can be mutually complementary to obtain better solutions than either metaheuristic-only solution or gradient-based-only solution. Not only this ELD problem, but also hydrologic flood model calibration had better results using the memetic approach [79].With this successful approach, we would like to explore more complex real-world ELD problems as well as other optimization problems. The proposed method can be applied to improve the simulation results of existing approaches [80, 81] in the future.



---

*Source: 906028-2014-02-23.xml*