An Improved Differential Evolution and Its Industrial Application

Affiliation(s)

Centre for Signal Processing, Department of Electronic and Information Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong, China.

Centre for Health Technologies, Faculty of Engineering and IT, University of Technology Sydney, Sydney, Australia..

Centre for Signal Processing, Department of Electronic and Information Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong, China.

Centre for Health Technologies, Faculty of Engineering and IT, University of Technology Sydney, Sydney, Australia..

ABSTRACT

In this paper, an improved Differential Evolution (DE) that incorporates double wavelet-based operations is proposed to solve the Economic Load Dispatch (ELD) problem. The double wavelet mutations are applied in order to enhance DE in exploring the solution space more effectively for better solution quality and stability. The first stage of wavelet operation is embedded in the DE mutation operation, in which the scaling factor is governed by a wavelet function. In the second stage, a wavelet-based mutation operation is embedded in the DE crossover operation. The trial population vectors are modified by the wavelet function. A suite of benchmark test functions is employed to evaluate the performance of the proposed DE in different problems. The result shows empirically that the proposed method out-performs signifycantly the conventional methods in terms of convergence speed, solution quality and solution stability. Then the proposed method is applied to the Economic Load Dispatch with Valve-Point Loading (ELD-VPL) problem, which is a process to share the power demand among the online generators in a power system for minimum fuel cost. Two different conditions of the ELD problem have been tested in this paper. It is observed that the proposed method gives satisfactory optimal costs when compared with the other techniques in the literature.

In this paper, an improved Differential Evolution (DE) that incorporates double wavelet-based operations is proposed to solve the Economic Load Dispatch (ELD) problem. The double wavelet mutations are applied in order to enhance DE in exploring the solution space more effectively for better solution quality and stability. The first stage of wavelet operation is embedded in the DE mutation operation, in which the scaling factor is governed by a wavelet function. In the second stage, a wavelet-based mutation operation is embedded in the DE crossover operation. The trial population vectors are modified by the wavelet function. A suite of benchmark test functions is employed to evaluate the performance of the proposed DE in different problems. The result shows empirically that the proposed method out-performs signifycantly the conventional methods in terms of convergence speed, solution quality and solution stability. Then the proposed method is applied to the Economic Load Dispatch with Valve-Point Loading (ELD-VPL) problem, which is a process to share the power demand among the online generators in a power system for minimum fuel cost. Two different conditions of the ELD problem have been tested in this paper. It is observed that the proposed method gives satisfactory optimal costs when compared with the other techniques in the literature.

Cite this paper

J. Lai, F. Leung, S. Ling and E. Shi, "An Improved Differential Evolution and Its Industrial Application,"*Journal of Intelligent Learning Systems and Applications*, Vol. 4 No. 2, 2012, pp. 81-97. doi: 10.4236/jilsa.2012.42008.

J. Lai, F. Leung, S. Ling and E. Shi, "An Improved Differential Evolution and Its Industrial Application,"

References

[1] R. J. M. Vaessens, E. H. L. Aarts and J. K. Lenstra, “A Local Search Template,” Proceedings of Parallel Problem Solving Nature, Brussels, 28-30 September 1992, pp. 65-74.

[2] J. B. Park, K. S. Lee and K. Y. Lee, “A Particle Swarm Optimization for Economic Dispatch with Non-Smooth Cost Functions,” IEEE Transactions on Power Systems, Vol. 20, No. 1, 2005, pp. 34-49. doi:10.1109/TPWRS.2004.831275

[3] T. A. A. Victoire and A. E. Jeyakumar, “Hybrid PSOSQP for Economic Dispatch with Valve-Point Effect,” Electrical Power Systems Research, Vol. 71, No. 1, 2004, pp. 51-59. doi:10.1016/j.epsr.2003.12.017

[4] L. S. Coelho and V. C. Mariani, “Combining of Chaotic Differential Evolution and Quadratic Programming for Economic Dispatch Optimization with Valve-Point Effect,” IEEE Transactions on Power Systems, Vol. 21, No. 2, 2006, pp. 989-995.

[5] R. Storn and K. Price, “Differential Evolution—A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces,” Journal of Global Optimization, Vol. 11, No. 4, 1997, pp. 341-359. doi:10.1023/A:1008202821328

[6] L. Fogel, “Evolutionary Programming in Perspective: The Top-Down View,” In: Computational Intelligence: Imitating Life, IEEE Press, Piscataway, 1994, pp. 135-146.

[7] D. Goldberg, “Genetic Algorithms in Search Optimization and Machine Learning,” Addison-Wesley, Boston, 1989.

[8] X. Yao and Y. Liu, “Evolutionary Programming Made Faster,” IEEE Transactions Evolutionary Computation, Vol. 3, No. 2, 1999, pp. 82-102. doi:10.1109/4235.771163

[9] Y. Ao and H. Chi, “Differential Evolution Using Opposite Point for Global Numerical Optimization,” Journal of Intelligent Learning Systems and Applications, Vol. 4, No. 1, 2012, pp. 1-19. doi:10.4236/jilsa.2012.41001

[10] S. Paterlini and T. Krink, “High Performance Clustering with Differential Evolution,” 2004. http://dsp.szu.edu.cn/dsp2006/research/areas/t08/papers/Clustering/03.High%20performance%20clustering%20with %20differential%20evolution.pdf

[11] J. H. van Sickel, K. Y. Lee and J. S. Heo, “Differential Evolution and Its Applications to Power Plant Control,” Proceedings of Intelligent Systems Applications to Power System, Kaohsiung, 5-8 November 2007, pp. 1-6.

[12] B. Babu and R. Angira, “Optimization of Non-Linear Functions Using Evolutionary Computation,” Proceedings of 12th ISME International Conference on Mechanical Engineering, India, 2001, pp. 153-157.

[13] A. Qing, “Dynamic Differential Evolution Strategy and Applications in Electromagnetic Inverse Scattering Problems,” IEEE Transactions on Geoscience and Remote Sensing, Vol. 44, No. 1, 2006, pp. 116-125. doi:10.1109/TGRS.2005.859347

[14] R. Eberhart and J. Kennedy, “A New Optimizer Using Particle Swarm Theory,” Proceedings of 6th International Symposium on Micro Machine and Human Science, Nagoya, 4-6 October 1995, pp. 39-43. doi:10.1109/MHS.1995.494215

[15] J. Vesterstroem and R. Thomsen, “A Comparative Study of Differential Evolution, Particle Swarm Optimization, and Evolutionary Algorithms on Numerical Benchmark Problems,” Congress on Evolutionary Computation, Portland, 19-23 June 2004, pp. 1980-1987.

[16] I. Daubechies, “Ten Lectures on Wavelets,” Society for Industrial and Applied Mathematics, Philadelphia, 1992. doi:10.1137/1.9781611970104

[17] Z. Michalewicz, “Genetic Algorithm + Data Structures = Evolution Programs,” 2nd Edition, Springer-Verlag, New York, 1994.

[18] A. Neubauer, “A Theoretical Analysis of the Non-Uniform Mutation Operator for the Modified Genetic Algorithm,” IEEE International Conference on Evolutionary Computation, Indianapolis, 13-16 April 1997, pp. 93-96.

[19] J. Brest, S. Greiner, B. Bo?kovi?, M. Mernik and V. ?umer, “Self Adapting Control Parameters in Differential Evolution: A Comparative Study on Numerical Benchmark Problems,” IEEE Transactions on Evolutionary Computation, Vol. 10, No. 6, 2006, pp. 646-657. doi:10.1109/TEVC.2006.872133

[20] Y. Ao and H. Chi, “Experimental Study on Differential Evolution Strategies,” Proceedings of Global Congress on Intelligent Systems, Anqing, 19-24 May 2009, pp. 1924. doi:10.1109/GCIS.2009.31

[21] H. Y. Fan and J. Lampinen, “A Trigonometric Mutation Operation to Differential Evolution,” Journal of Global Optimization, Vol. 27, No. 1, 2003, pp. 105-129. doi:10.1023/A:1024653025686

[22] M. M. Ali, C. Khompatraporn and Z. B. Zabinsky, “A Numerical Evaluation of Several Stochastic Algorithms on Selected Continuous Global Optimization Test Problems,” Journal of Global Optimization, Vol. 31, No. 4, 2005, pp. 635-672. doi:10.1007/s10898-004-9972-2

[23] U. K. Chakraborty, “Advances in Differential Evolution,” Springer, Heidelberg, 2008.

[24] S. Rahnamayan, H. R. Tizhoosh and M. M. A. Salama, “Opposition-Based Differential Evolution,” IEEE Transactions on Evolutionary Computation, Vol. 12, No. 1, 2008, pp. 64-79. doi:10.1109/TEVC.2007.894200

[25] P. H. Chen and H. C. Chang, “Large-Scale Economic Dispatch by Genetic Algorithms,” IEEE Transactions on Power Systems, Vol. 10, No. 1, 1995, pp. 117-124.

[26] S. H. Ling, H. C. C. Iu, K. Y. Chan, H. K. Lam, C. W. Yeung and F. H. F. Leung, “Hybrid Particle Swarm Optimization with Wavelet Mutation and Its Industrial Applications,” IEEE Transactions on Systems, Man, and Cybernetics, Part B, Vol. 38, No. 3, 2008, pp. 743-763.

[27] A. I. Selvakumar and K. Thanushkodi, “A New Particle Swarm Optimization Solution to Non Convex Economic Dispatch Problems,” IEEE Transactions on Power Systems, Vol. 22, No. 1, 2007, pp. 42-50. doi:10.1109/TPWRS.2006.889132

[1] R. J. M. Vaessens, E. H. L. Aarts and J. K. Lenstra, “A Local Search Template,” Proceedings of Parallel Problem Solving Nature, Brussels, 28-30 September 1992, pp. 65-74.

[2] J. B. Park, K. S. Lee and K. Y. Lee, “A Particle Swarm Optimization for Economic Dispatch with Non-Smooth Cost Functions,” IEEE Transactions on Power Systems, Vol. 20, No. 1, 2005, pp. 34-49. doi:10.1109/TPWRS.2004.831275

[3] T. A. A. Victoire and A. E. Jeyakumar, “Hybrid PSOSQP for Economic Dispatch with Valve-Point Effect,” Electrical Power Systems Research, Vol. 71, No. 1, 2004, pp. 51-59. doi:10.1016/j.epsr.2003.12.017

[4] L. S. Coelho and V. C. Mariani, “Combining of Chaotic Differential Evolution and Quadratic Programming for Economic Dispatch Optimization with Valve-Point Effect,” IEEE Transactions on Power Systems, Vol. 21, No. 2, 2006, pp. 989-995.

[5] R. Storn and K. Price, “Differential Evolution—A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces,” Journal of Global Optimization, Vol. 11, No. 4, 1997, pp. 341-359. doi:10.1023/A:1008202821328

[6] L. Fogel, “Evolutionary Programming in Perspective: The Top-Down View,” In: Computational Intelligence: Imitating Life, IEEE Press, Piscataway, 1994, pp. 135-146.

[7] D. Goldberg, “Genetic Algorithms in Search Optimization and Machine Learning,” Addison-Wesley, Boston, 1989.

[8] X. Yao and Y. Liu, “Evolutionary Programming Made Faster,” IEEE Transactions Evolutionary Computation, Vol. 3, No. 2, 1999, pp. 82-102. doi:10.1109/4235.771163

[9] Y. Ao and H. Chi, “Differential Evolution Using Opposite Point for Global Numerical Optimization,” Journal of Intelligent Learning Systems and Applications, Vol. 4, No. 1, 2012, pp. 1-19. doi:10.4236/jilsa.2012.41001

[10] S. Paterlini and T. Krink, “High Performance Clustering with Differential Evolution,” 2004. http://dsp.szu.edu.cn/dsp2006/research/areas/t08/papers/Clustering/03.High%20performance%20clustering%20with %20differential%20evolution.pdf

[11] J. H. van Sickel, K. Y. Lee and J. S. Heo, “Differential Evolution and Its Applications to Power Plant Control,” Proceedings of Intelligent Systems Applications to Power System, Kaohsiung, 5-8 November 2007, pp. 1-6.

[12] B. Babu and R. Angira, “Optimization of Non-Linear Functions Using Evolutionary Computation,” Proceedings of 12th ISME International Conference on Mechanical Engineering, India, 2001, pp. 153-157.

[13] A. Qing, “Dynamic Differential Evolution Strategy and Applications in Electromagnetic Inverse Scattering Problems,” IEEE Transactions on Geoscience and Remote Sensing, Vol. 44, No. 1, 2006, pp. 116-125. doi:10.1109/TGRS.2005.859347

[14] R. Eberhart and J. Kennedy, “A New Optimizer Using Particle Swarm Theory,” Proceedings of 6th International Symposium on Micro Machine and Human Science, Nagoya, 4-6 October 1995, pp. 39-43. doi:10.1109/MHS.1995.494215

[15] J. Vesterstroem and R. Thomsen, “A Comparative Study of Differential Evolution, Particle Swarm Optimization, and Evolutionary Algorithms on Numerical Benchmark Problems,” Congress on Evolutionary Computation, Portland, 19-23 June 2004, pp. 1980-1987.

[16] I. Daubechies, “Ten Lectures on Wavelets,” Society for Industrial and Applied Mathematics, Philadelphia, 1992. doi:10.1137/1.9781611970104

[17] Z. Michalewicz, “Genetic Algorithm + Data Structures = Evolution Programs,” 2nd Edition, Springer-Verlag, New York, 1994.

[18] A. Neubauer, “A Theoretical Analysis of the Non-Uniform Mutation Operator for the Modified Genetic Algorithm,” IEEE International Conference on Evolutionary Computation, Indianapolis, 13-16 April 1997, pp. 93-96.

[19] J. Brest, S. Greiner, B. Bo?kovi?, M. Mernik and V. ?umer, “Self Adapting Control Parameters in Differential Evolution: A Comparative Study on Numerical Benchmark Problems,” IEEE Transactions on Evolutionary Computation, Vol. 10, No. 6, 2006, pp. 646-657. doi:10.1109/TEVC.2006.872133

[20] Y. Ao and H. Chi, “Experimental Study on Differential Evolution Strategies,” Proceedings of Global Congress on Intelligent Systems, Anqing, 19-24 May 2009, pp. 1924. doi:10.1109/GCIS.2009.31

[21] H. Y. Fan and J. Lampinen, “A Trigonometric Mutation Operation to Differential Evolution,” Journal of Global Optimization, Vol. 27, No. 1, 2003, pp. 105-129. doi:10.1023/A:1024653025686

[22] M. M. Ali, C. Khompatraporn and Z. B. Zabinsky, “A Numerical Evaluation of Several Stochastic Algorithms on Selected Continuous Global Optimization Test Problems,” Journal of Global Optimization, Vol. 31, No. 4, 2005, pp. 635-672. doi:10.1007/s10898-004-9972-2

[23] U. K. Chakraborty, “Advances in Differential Evolution,” Springer, Heidelberg, 2008.

[24] S. Rahnamayan, H. R. Tizhoosh and M. M. A. Salama, “Opposition-Based Differential Evolution,” IEEE Transactions on Evolutionary Computation, Vol. 12, No. 1, 2008, pp. 64-79. doi:10.1109/TEVC.2007.894200

[25] P. H. Chen and H. C. Chang, “Large-Scale Economic Dispatch by Genetic Algorithms,” IEEE Transactions on Power Systems, Vol. 10, No. 1, 1995, pp. 117-124.

[26] S. H. Ling, H. C. C. Iu, K. Y. Chan, H. K. Lam, C. W. Yeung and F. H. F. Leung, “Hybrid Particle Swarm Optimization with Wavelet Mutation and Its Industrial Applications,” IEEE Transactions on Systems, Man, and Cybernetics, Part B, Vol. 38, No. 3, 2008, pp. 743-763.

[27] A. I. Selvakumar and K. Thanushkodi, “A New Particle Swarm Optimization Solution to Non Convex Economic Dispatch Problems,” IEEE Transactions on Power Systems, Vol. 22, No. 1, 2007, pp. 42-50. doi:10.1109/TPWRS.2006.889132