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 gradientbased technique to find better solutions for an economic load dispatch problem with valvepoint loading. Because metaheuristic algorithms have the strength in global search and gradientbased techniques have the strength in local search, the combination approach obtains better results than those of any single approach. A benchmark example of 40 generatingunit economic load dispatch problem demonstrates that the memetic approach can further improve the existing best solutions from the literature.
Economic load dispatch (ELD) problem is a classical form of optimization problems and has been one of the most important decisionmaking processes in the operation of electrical power systems. The total systemwide 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 realsystem 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 gradientbased and enumerationbased deterministic approaches such as Lagrangian relaxation and linear, nonlinear, and dynamic programming techniques have been applied to find the optimal solution of ELD problem [
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, geneticbased different types of algorithms (GA) [
Up until now, we can see that several welldeveloped metaheuristic algorithms have shown a higher level of applicability in ELD problem with fine performance [
This paper is organized as follows. Mathematical formulation of ELD problems and recent metaheuristic algorithms to solve ELD problem are described in Section
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 systemwide 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 inbetween 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:
The system loss in the transmission lines,
Moreover, more practical consideration of ELD problem requires an inclusion of valvepoint loading effects in ELD problem, and the mathematical formulation of ELD with valvepoint loading can be rephrased as follows:
Many gradientbased and enumerationbased 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 realworld 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 gradientbased 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 [
Many metaheuristic approaches using populationbased evolutionary frameworks have been applied in ELD problem in order to find the optimal solution. The populationbased 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 Algorithm
Depending 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 phenomenonbased 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 biogeographybased and musical improvisation methodologies are presented to find the optimal solution of 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 [
For recent decades, memetic algorithms have been widely applied in the largescale complex optimization problems. The problem of balance between global and local search, that is, balance between computational intelligence and gradientbased search algorithm, has been explored under a multiobjective optimization setting in [
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 populationbased metaheuristic approaches in the most recent ELD problem literatures [
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  121412.5704 
Based upon the result in Table
For the gradientbased local search algorithm, this study adopted BFGS (BroydenFletcherGoldfarbShanno) method [
Identify an initial feasible solution,
Calculate the searching direction,
Calculate a new solution,
If convergence criterion is not satisfied, set
The searching direction in Step
The convergence criterion in this study is the relative error as follows:
The BFGS method, which is a calculusbased technique, has the advantages over metaheuristic algorithms in terms of
The numerical test system used to explore the applicability of memetic algorithms in ELD problem in this study consists of forty generating units with valvepoint loading effects, and the total system demand is 10,500 MW [
Generating units data for test system.
Number 








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 
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 $121,415.4560 [
Results for Test Case 1.
Number  Generation (MW)  Cost ($)  

HS [ 
M_HS  HS [ 
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 
As a second experiment, we have carried out a memetic implementation using FA. In [
Results for Test Case 2.
Number  Generation (MW)  Cost ($)  

FA [ 
M_FA  FA [ 
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 
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 $121,412.5704 [
Results for Test Case 3.
Number  Generation (MW)  Cost ($)  

HBMO [ 
M_HBMO  HBMO [ 
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 
In summary, this study introduced a memetic scheme of metaheuristic algorithm and gradientbased technique and then applied it to a popular benchmark problem of 40 generatingunit ELD problem with valvepoint loading, obtaining better solutions than other solutions ever found in the literature. As shown in Table
Test results and comparison.
Algorithm  Minimum cost ($) 

Chaotic differential evolution and quadratic programming [ 
121,741.9700 
PSO [ 
121,735.4700 
Hybrid differential evolution [ 
121,698.5100 
New PSO [ 
121,664.4300 
Antipredatory PSO [ 
121,663.5200 
Selforganizing hierarchical PSO [ 
121,501.1400 
Biographybased optimization [ 
121,479.5000 
GAPSSQP [ 
121,458,0000 
Bacterial foraging [ 
121,423.6300 
Harmony search (HS) [ 
121,415.4560 
Memetic harmony search (this study) 

Pattern search [ 
121,415.1400 
Firefly algorithm (FA) [ 
121,415.0522 
Memetic firefly algorithm (this study) 

Honey bee mating optimization (HBMO) [ 
121,412.5704 
Memetic HBMO (this study) 

While metaheuristic algorithms perform well in global search, they do not perform well in local search. On the contrary, while gradientbased 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 metaheuristiconly solution or gradientbasedonly solution. Not only this ELD problem, but also hydrologic flood model calibration had better results using the memetic approach [
With this successful approach, we would like to explore more complex realworld ELD problems as well as other optimization problems. The proposed method can be applied to improve the simulation results of existing approaches [
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the Gachon University Research Fund of 2013 (GCU2013R194).