Power Economic Dispatch Using a Hybrid Genetic Algorithm
nary encoding [4]. The efficiency of the GA is increased as there is no need to convert chromosomes to the binary type, less memory is required, there is no loss in precision by discretization to binary or other values, and there is greater freedom to use different genetic operators. For the real valued representation, the kth chromosome C k can be defined as follows:
T. Yalcinoz, H. Altun Author Affiliation: Department of Electrical and Electronic Engineering, Nigde University, Nigde, Turkey. Abstract: This letter outlines a hybrid genetic algorithm (HGA) for solving the economic dispatch problem. The algorithm incorporates the solution produced by an improved Hopfield neural network (NN) [1] as a part of its initial population. Elitism, arithmetic crossover, and mutation are used in the GAs to generate successive sets of possible operating policies. The technique improves the quality of the solution and reduces the computation time, and is compared with the classical optimization technique, an improved Hopfield NN approach (IHN) [1], a fuzzy logic controlled GA (FLCGA) [2], and an improved GA (IGA) [3]. Keywords: Economic dispatch, HGA, Hopfield neural networks, arithmetic crossover Introduction: Hopfield NNs perform fast and efficient local searches while guaranteeing convergence to feasible solutions, but they are unable to perform efficient global searches. On the other hand, GAs perform powerful global searches, but their long computation times limit them when solving large-scale problems. To overcome these difficulties hybrid approaches may be considered. GAs are probabilistic heuristic procedures that have been studied to solve the power optimization problems. An FLCGA has been applied to environmental/economic dispatch by Song et al. [2]. Song and Chou [3] proposed an improved GA (IGA) that is a combination strategy involving local search algorithms and GA. The validity of a fuzzy logic controlled GA [2] and an improved GA [3] is illustrated in an economic dispatch problem with a six-unit system. Hopfield NNs using the energy function approach have been applied to several optimization problems. Yalcinoz and Short [1] demonstrated an improved Hopfield NN to solve the constrained ED problem that takes into account transmission capacity constraints. The proposed method reduces computation time significantly for large systems up to 120 units, although the operation costs obtained by the proposed method are slightly higher than the conventional technique. In this letter, the IHN is used to produce initial populations of the GA in order to reduce computing expenses. Elitism, arithmetic crossover that defines a linear combination of two chromosomes [4] and mutations, is used in the GAs to generate successive sets of possible operating policies. HGA: The basic GA creates an initial random population. The proposed HGA incorporates the solution produced by an IHN [1] as part of its initial population, however. The IHN approach [1] uses a new mapping technique for Hopfield NN for solving quadratic programming problems. Quadratic programming problems are subject to a number of inequality and equality constraints that can be handled by adding corresponding terms to the energy function. An efficient simulation algorithm has been used to solve the dynamic equations of the Hopfield NN where the time step has been calculated in such a manner as to reduce computing expenses. Although binary representation is usually applied to power optimization problems, in this letter we use the real valued representation scheme for solution. The use of real valued representation in the GA has a number of advantages in numerical function optimization over bi-
C k = [ Pk 1 , Pk 2 ,…, Pkn ] k = 1, 2,…, popsize
(1)
where popsize means population size and Pki is the generation power of the i-th unit at k-th chromosome. Reproduction involves the creation of new offspring from the mating of two selected parents or mating pairs. It is thought that the crossover operator is mainly responsible for the global search property of the GA. An arithmetic crossover operator was introduced into the GA [4]. We used an arithmetic crossover operator that defines a linear combination of two chromosomes. Two chromosomes, selected randomly for gen crossover, C igen and C gen + 1 and j , may produce two offspring, C i gen C j + 1, which is a linear combination of their parents, i.e., C igen + 1 = a ⋅ C igen + (1 − a ) ⋅ C gen j +1 = (1 − a ) ⋅ C igen + a ⋅ C gen C gen j j
(2)
where a is a random number in range of [0, 1]. The mutation operator is used to inject new genetic material into the population and it is applied to each new structure individually. A given mutation involves randomly altering each gene with a small probability. We generate a random real value, which makes a random change in the mth element of the chromosome selected randomly. An elitist GA search is used and guarantees that the best solution obtained so far in the search is retained and used in the following generation, and thereby ensuring no good solution already found can be lost in the search process. Economic Dispatch Using HGA: The initial populations of GA are created by an IHN [1]. For mapping of the economic dispatch problem, a new mapping technique described in [1] is used. After creating initial populations, the GA steps, which have been described in the previous section, are implemented for the solution of the economic dispatch problem. In the economic dispatch problem, the unit power output is used as the main decision variable, and each unit’s loading range is represented by a real number. The representation takes care of the unit minimum and maximum loading limits since the real representation is made to cover only the values between the limits. The main objective of economic dispatch is to minimize fuel costs while satisfying constraints such as the power balance equation. In order to produce two offspring, an arithmetic crossover operator is used. After the crossover is completed, mutation is performed. For the mutation step, a random real value makes a random change in the mth element of the chromosome. After mutation, all constraints are checked whether violated or not. If the solution has at least one constraint violated, a new random real value is used for finding a new value of the mth element of the chromosome. Then, the best solution obtained in the search so far is retained and used in the next generation. The GA process repeats until the specified maximum number of generations is reached. Simulation Results of Economic Dispatch: The proposed method has been applied to two test systems; a power system with six units and
Table 1. Results of CM, FLCGA, IGA, IHN, and HGA for a six-unit system Unit 2 (MW)
Unit 3 (MW)
CM
Methods
1800.0
Load (MW)
Unit 1 (MW) 248.00
217.72 7
5.18
588.04
335.53
335.53
16579.33
FLCGA [2]
1800.0
250.49
215.43
109.92
572.84
325.66
325.66
16585.85
IGA [3]
1800.0
248.07
217.73
75.30
587.70
335.60
335.60
16579.33
IHN [1]
1800.0
248.08
217.74
75.18
587.90
335.55
335.55
16579.33
HGA
1800.0
248.14
217.74
75.20
587.80
335.56
335.56
16579.33
IEEE Power Engineering Review, March 2001
Unit 4 (MW)
Unit 5 (MW)
Unit 6 (MW)
Cost ($/h)
59
Table 2. Results obtained by CM, FLCGA, IHN, and HGA Methods
Load (MW)
CM
Cost ($/h)
Load (MW)
8227.09
FLCGA
800.0
Cost ($/h) 11477.09
8231.03
1200.0
11480.03
IHN
8228.05
11477.20
HGA
8227.09
11477.09
Table 3. Results obtained by CM, IGA, IHN, and HGA Methods
Load (MW)
CM
Cost ($/h)
Load (MW)
14169.54
IGA
1520.0
14169.54
Cost ($/h) 20465.24
2238.0
20470.48
IHN
14169.54
20465.44
HGA
14169.54
20465.24
Table 4. Simulation Results for a 20-unit system Load (MW)
Cost ($/h) HGA
Error % of IHN / HGA
CM
IHN [1]
3150
46026.9
46460.84
46027.01
0. 94/0.0002
3800
52852.36
53207.7
52852.49
0.67/0.0002
4600
63401.27
63791.72
63401.76
0. 61/0.0007
a 20-unit system, respectively. The cost function of each unit is chosen as a quadratic function for the test systems. The proposed method is compared with the classical optimization technique (CM), which is based on the sequential quadratic programming method, an FLCGA [2], an IGA [3], and an IHN [1]. The classical optimization program was written using the Matlab optimization toolbox and the proposed method was also implemented in Matlab. The first test system has six units and details of this test system are obtained from [2]. In Table 1, the results of the proposed method (HGA) are compared with the results of the classical method (CM), FLCGA, IGA, and IHN when the load demand is 1800 MW. It is seen that there is a negligible difference in the optimal values between the many methods. The FLCGA produced a higher operation cost than other methods. Table 2 presents the results of HGA, CM, IHN, and FLCGA when the load demands are 800 MW and 1200 MW. The FLCGA produced the highest operation cost and the obtained operation costs by the HGA and the CM are smaller than the IHN, respectively. Table 3 gives a comparison of the economic dispatch results of CM, IGA, IHN, and HGA for 1520 MW and 2238 MW of load demand. It can be seen that the hybrid genetic algorithm achieved lower operation costs for different load demands, in contrast, the IGA and the IHN produced higher operation costs for some loading conditions. To demonstrate the efficiency and the robustness of the hybrid genetic algorithm, a 20-unit system is considered. The results of the proposed method are shown in Table 4 and compared against the results of the CM and the IHN for three load levels. The operation cost obtained by the IHN is about 0.6% more than the HGA, in contrast, the obtained operation cost by the HGA is slightly higher than the CM. The execution time of the proposed method for the 20-unit system is about 4 s; in contrast, the classical technique takes about 9 s. Conclusions: In this letter, a new approach to the economic dispatch problem using a hybrid GA has been presented. The proposed hybrid GA incorporates the solution produced by an IHN [1] as a part of its initial 60
population in order to reduce the computing expenses. The real valued representation scheme, arithmetic crossover, mutation, and elitism are used in the GAs to generate successive sets of possible operating policies. In the case studies, the proposed method is applied to the economic dispatch problem with six generators and 20 generators. Test results have shown the proposed method can provide better solutions than the classical optimization technique, a fuzzy logic controlled genetic algorithm [2], an improved genetic algorithm [3], and an improved Hopfield NN approach [1]. For a six-unit system, the IGA and the IHN produced higher operation costs than the proposed method for some loading conditions. The operation cost obtained by the IHN is about 0.6% more than the HGA for a 20-unit system. The proposed technique improves the quality of the solution and reduces the computation time. References: [1] T. Yalcinoz and M.J. Short, “Neural networks approach for solving economic dispatch problem with transmission capacity constraints,” IEEE Trans. Power Syst., vol. 13, pp. 307-313, 1998. [2] Y.H. Song, et. al, “Environmental/economic dispatch using fuzzy logic controlled genetic algorithms,” IEE Proc. Gener. Transm. Distrib., vol. 144, no. 4, pp. 377-382, July 1997. [3] Y.H. Song and C.S.V. Chou, “Advanced engineered- conditioning genetic approach to power economic dispatch,” IEE Proc. Gener. Transm. Distrib., vol. 144, no. 3, pp. 285-292, May 1997. [4] Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, 2nd ed. Berlin: Springer-Verlag, 1994. Copyright Statement: ISSN 0282-1724/01/$10.00 2001 IEEE. Manuscript received 3 August 2000. This paper is published herein in its entirety.
Power Engineering Letters Call for Short Papers Power Engineering Letters are short technical papers on new results, discoveries, and developments in areas of interest to PES members. Original and significant contributions in all fields of power engineering are invited: ● Applications ● Case studies ● Research. Of specific interest are contributions defining emerging problems and special needs in specific areas. Peer Review: All Power Engineering Letters are peer reviewed under the direction of the Power Engineering Letters editorial board. Manuscript Preparation and Submission: Papers intended for publication in the Power Engineering Letters section of IEEE Power Engineering Review magazine are limited to 2,500 words, all inclusive; if tables, figures, or equations are included, subtract 75 words per column inch from the word count. Submit text (MS Word, or WP) files and graphic files (TIFF) via E-mail or disks, followed by a hard copy (on 216 x 280 mm paper) and signed IEEE Copyright form. Direct your submissions and queries to M.E. El-Hawary, DalTech, Dalhousie University, P.O. Box 1000, (courier address: 1360 Barrington Street, Building A , Room A-217), Halifax, NS B3J 2X4, Canada, +1 902 494 6198 or +1 902 494 6199, fax +1 902 429 3011, e-mail
[email protected].
IEEE Power Engineering Review, March 2001
