# A Hybrid Particle Swarm Optimization Algorithm for the Economic Dispatch Problem

Keywords:
Economic Dispatch, Transmission Losses, Particle Swarm Optimization, Hybridization

### Abstract

This article proposes a hybrid global-local algorithm - Hybrid Particle Swarm Optimization (HPSO) - applied to solve the Economic Dispatch (ED) problem. The HPSO algorithm combines the classical Particle Swarm Optimization (PSO) with the Conjugate Gradient (CG) non-linear optimizing method, included in the optimizing tool within MathCAD commercial software product. The global optimizer is the PSO algorithm, and the local one is the CG method. Two variants including the CG within the PSO, which are analyzed, called HPSO-RC (randomly controlled) and HPSO-RU (randomly uncontrolled). Both PSO and CG methods are easy to implement and together help reaching the best solution. The HPSO algorithm’s ability to avoid premature convergence and provide a stabile solution is tested on three systems consisting of 6, 13 and 38 thermal generating units. The HPSO algorithm’s efficiency in solving the ED problem is shown through a comparison with several other recently published algorithms.### References

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[6] W. Sa-ngiamvibool, S. Pothiya and I. Ngamroo, “Multiple tabu search algorithm for economic dispatch problem considering valve-point effects”, Electric Power and Energy Systems, Vol.33, No.4, pp. 846–854, 2011.

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[18] N. M. Jothi Swaroopan1 , P. Somasundaram, “A Novel Combined Economic and Emission Dispatch Control by Hybrid Particle Swarm Optimization Technique”, Majlesi Journal of Electrical Engineering, Vol.4, No.2, 2010.

[19] K.T. Chaturvedi, M. Pandit and L. Srivastava, “Particle swarm optimization with time varying acceleration coefficients for non-convex economic

power dispatch”, Electrical Power and Energy Systems, Vol.31, No.6, pp. 249-257, 2009.

[20] A. Bhattacharya , P.K. Chattopadhyay, “Hybrid differential evolution with biogeography-based optimization for solution of economic load dispatch”, IEEE Transactions On Power Systems, Vol.25, No.4, pp. 1955-1964, 2010.

[21] T.A.A. Victoire , A.E. Jeyakumar, “Hybrid PSO–SQP for economic dispatch with valve-point effect”, Electric Power System Research, Vol.71, No.1, pp. 51–59, 2004.

[22] D.K. He, F.L. Wang and Z.Z. Mao, “A hybrid genetic algorithm approach based on differential evolution for economic dispatch with valve-point effect”, Electrical Power and Energy Systems, Vol. 30, No.1, pp. 31-38, 2008.

[23] D.K. He, F.L. Wang and Z.Z. Mao, “Hybrid genetic algorithm for economic dispatch with valve-point effect”, Electric Power System Research, Vol.78, No.4, pp. 626–633, 2008.

[24] S. Khamsawang , S. Jiriwibhakorn, “DSPSO-TSA for economic dispatch problem with nonsmooth and noncontinuous cost functions”, Energy Conversion and Management, Vol.51, No.4, pp. 365–375, 2010.

[25] G. Xiong, D. Shi and X. Duan, “Multi-strategy ensemble biogeography-based optimization for economic dispatch problems”, Applied Energy, Vol. 111, pp. 801-811, 2013.

[26] A. Qteish , M. Hamdan, “Hybrid particle swarm and conjugate gradient optimization algorithm”, Advances in Swarm Intelligence (Lecture Notes in Computer Science), Part 1, pp. 582-588, 2010.

[27] S. Chen, T. Mei, M. Luo and X. Yang, “Identification of nonlinear system based on a new hybrid gradient-based PSO algorithm”, in Proc. 2007 International Conference on Information Acquisition, ICIA, Seogwipo-si, pp. 265-268.

[28] A.H. Kayhan, H. Ceylan and M.T. Ayvaz, Gurarslan, G. “PSOLVER: A new hybrid particle swarm optimization algorithm for solving continuous optimization problems”, Expert Systems with Applications, Vol.37, No.10, pp. 6798-6808, 2010.

[29] M.T. Ayvaz, A.H. Kayhan, H. Ceylan and G. Gurarslan, “Hybridizing the harmony search algorithm with a spreadsheet “solver” for solving continuous engineering optimization problems”, Engineering Optimization, Vol.41, No.12, pp. 1119-1144, 2009.

[30] J. Kennedy, R.C. Eberhart, “Particle swarm optimization”, in Proc. 1995 of the IEEE international conference on neural networks, Perth, Australia. Piscataway, NJ: IEEE, pp. 1942–1948.

[31] Y. Shi and R.C. Eberhart, “A modified particle swarm optimizer”, in Proc. 1998 of the IEEE conference on evolutionary computation, Anchorage, Piscataway, NJ: IEEE Press, pp. 69–73.

[32] Y. Shi , R.C. Eberhart, “Empirical study of particle swarm optimization”, in Proc. 1999 of the IEEE congress on evolutionary computation,

Majlesi Journal of Electrical Engineering Vol. 9, No. 1, March 2015

53

Piscataway, NJ. Piscataway, NJ: IEEE Press, pp. 1945–1950.

[33] M. Fesanghary, M. Mahdavi, M. Minary-Jolandan and Y. Alizadeh, “Hybridizing harmony search algorithm with sequential quadratic programming for engineering optimization problems”, Computer Methods in Applied Mechanics and Engineering, Vol. 197(33-40), pp. 3080–3091, 2008.

[34] Z-L. Gaing, “Particle swarm optimization to solving the economic dispatch considering the generator constraints”, IEEE Transaction on Power System, Vol.18, No.3, pp. 1187–1195, 2003.

[35] Z-L. Gaing, “Closure to “Discussion of “Particle swarm optimization to solving the economic dispatch considering the generator constraints””, IEEE Transactions on Power Systems, Vol.19, No.4, pp. 2122–2123, 2004.

[36] M. Pourakbari-Kasmaei , M. Rashidi-Nejad, “An effortless hybrid method to solve economic load dispatch problem in power systems”, Energy Conversion and Management, Vol.52(8-9), pp. 2854-2860, 2011.

[37] A.Y. Saber, S. Chakraborty, S.M. Abdur Razzak and T. Senjyu, “Optimization of economic load dispatch of higher order general cost polynomials and its sensitivity using modified particle swarm optimization”, Electric Power Systems Research, Vol.79, No.1, pp. 98-106, 2009.

[38] E. Sayedi, M.M. Farsangi, M. Barati and K.Y. Lee, “A modified shuffled frog leaping algorithm for

nonconvex economic dispatch problem”, in Proc. 2012 IEEE Power and Energy Society General Meeting, San Diego, CA, pp. 1-8.

[39] L. Wang, L.P. Li, “An effective differential harmony search algorithm for the solving non-convex economic load dispatch problems”, Electrical Power and Energy Systems, Vol. 44, No.1, pp. 832-843, 2013.

[40] M. Moradi-Dalvand, B. Mohammadi-Ivatloo, A. Najafi and A. Rabiee, “Continuous quick group search optimizer for solving non-convex economic dispatch problems”, Electric Power Systems Research, Vol. 93, pp. 93-105, 2012.

[41] J. Cai, Q. Li, L. Li, H. Peng and Y. Yang, “A hybrid CPSO–SQP method for economic dispatch considering the valve-point effects”, Energy Conversion and Management, Vol.53, No.1, pp. 175-181, 2012.

[42] J. Cai, Q. Li, L. Li, H. Peng and Y. Yang, “A hybrid FCASO-SQP method for solving the economic dispatch problems with valve-point effects”, Energy, Vol. 38, No.1, pp. 346–353, 2012.

[43] A.S. Reddy, K. Vaisakh, “Shuffled differential evolution for large scale economic dispatch”, Electric Power Systems Research, Vol. 96, pp. 237-245, 2013.

[44] M. Sydulu, “A very fast and effective noniterative ‘‘Lamda Logic Based’’ algorithm for economic dispatch of thermal units”, in Proc. 1999 IEEE region 10 conference TENCON, Cheju Island, Vol.2, pp. 1434–1437.

[2] L. Papageorgiou and E. Fraga, “A mixed integer quadratic programming formulation for the economic dispatch of generators with prohibited operating zones”, Electric Power System Research, Vol.77, No.10, pp. 1292–1296, 2007.

[3] D. Travers, R. Kaye, “Dynamic dispatch by constructive dynamic programming”, IEEE Transaction on Power System, Vol.13, No.1, pp. 72-78, 1998.

[4] C.L. Chiang, “Genetic-based algorithm for power economic load dispatch”, IET Generation, Transmission and Distribution, Vol.1, No.2, pp. 261–269, 2007.

[5] D.C. Walters, G.B. Sheble, “Genetic algorithm solution of economic dispatch with valve-point loadings”, IEEE Transaction on Power System, Vol.8, No.3, pp. 1325–1331, 1993.

[6] W. Sa-ngiamvibool, S. Pothiya and I. Ngamroo, “Multiple tabu search algorithm for economic dispatch problem considering valve-point effects”, Electric Power and Energy Systems, Vol.33, No.4, pp. 846–854, 2011.

Majlesi Journal of Electrical Engineering Vol. 9, No. 1, March 2015

52

[7] C.T. Sun, C.T. Lin, “New approach with a Hopfield modeling framework to economic dispatch”, IEEE Transaction on Power System, Vol.15, No.2, pp. 541-545, 2000.

[8] N. Sinha, R. Chakrabarti and P.K. Chattopadhyay, “Evolutionary programming techniques for economic load dispatch”, IEEE Transactions on Evolutionary Computation, Vol.7, No.1, pp. 83–94, 2003.

[9] A. Pereira-Neto, C. Unsihuay and O.R. Saavedra, “Efficient evolutionary strategy optimisation procedure to solve the nonconvex economic dispatch problem with generator constraints”, IEE Proceedings Generation, Transmission and Distribution, Vol.152, No.5 pp. 653–660, 2005.

[10] N. Noman, H. Iba, “Differential evolution for economic load dispatch problems”, Electric Power System Research, Vol.78, No.3, pp. 1322–1331, 2008.

[11] T. Niknam, “A new fuzzy adaptive hybrid particle swarm optimization algorithm for non-linear, non-smooth and non-convex economic dispatch problem”, Applied Energy, Vol. 87, No.1 pp. 327-339, 2010.

[12] R.V. Pandi, B.K. Panigrahi, R.C. Bansal, S.Das and A. Mohapatra, “Economic load dispatch using hybrid swarm intelligence based harmony search algorithm”, Electric Power Components and Systems, Vol.39, No.8, pp. 751–767, 2011.

[13] M. Fesanghary, M.M. Ardehali, “A novel meta-heuristic optimization methodology for solving various types of economic dispatch problem”, Energy, Vol.34, No.6, pp. 757-766, 2009.

[14] S. Pothiya, I. Ngamroo and W. Kongprawechnon, “Ant colony optimisation for economic dispatch problem with non-smooth cost functions”, Electrical Power and Energy Systems, Vol.32, No.5, pp. 478–487, 2010.

[15] J. Cai, X. Ma, L. Li and H. Peng, “Chaotic particle swarm optimization for economic dispatch considering the generator constraints”, Energy Conversion and Management, Vol.48, No.2, pp. 645–653, 2007.

[16] S. Özyön, D. Aydin, “Incremental artificial bee colony with local search to economic dispatch problem with ramp rate limits and prohibited operating zones”, Energy Conversion and Management, Vol. 65, No.1, pp. 397-407, 2013.

[17] J.B. Park, Y.W. Jeong, J.R. Shin and K. Y Lee, “An improved particle swarm optimization for nonconvex economic dispatch problems”, IEEE Transaction on Power System, Vol. 25, No.1,pp. 156–166, 2010.

[18] N. M. Jothi Swaroopan1 , P. Somasundaram, “A Novel Combined Economic and Emission Dispatch Control by Hybrid Particle Swarm Optimization Technique”, Majlesi Journal of Electrical Engineering, Vol.4, No.2, 2010.

[19] K.T. Chaturvedi, M. Pandit and L. Srivastava, “Particle swarm optimization with time varying acceleration coefficients for non-convex economic

power dispatch”, Electrical Power and Energy Systems, Vol.31, No.6, pp. 249-257, 2009.

[20] A. Bhattacharya , P.K. Chattopadhyay, “Hybrid differential evolution with biogeography-based optimization for solution of economic load dispatch”, IEEE Transactions On Power Systems, Vol.25, No.4, pp. 1955-1964, 2010.

[21] T.A.A. Victoire , A.E. Jeyakumar, “Hybrid PSO–SQP for economic dispatch with valve-point effect”, Electric Power System Research, Vol.71, No.1, pp. 51–59, 2004.

[22] D.K. He, F.L. Wang and Z.Z. Mao, “A hybrid genetic algorithm approach based on differential evolution for economic dispatch with valve-point effect”, Electrical Power and Energy Systems, Vol. 30, No.1, pp. 31-38, 2008.

[23] D.K. He, F.L. Wang and Z.Z. Mao, “Hybrid genetic algorithm for economic dispatch with valve-point effect”, Electric Power System Research, Vol.78, No.4, pp. 626–633, 2008.

[24] S. Khamsawang , S. Jiriwibhakorn, “DSPSO-TSA for economic dispatch problem with nonsmooth and noncontinuous cost functions”, Energy Conversion and Management, Vol.51, No.4, pp. 365–375, 2010.

[25] G. Xiong, D. Shi and X. Duan, “Multi-strategy ensemble biogeography-based optimization for economic dispatch problems”, Applied Energy, Vol. 111, pp. 801-811, 2013.

[26] A. Qteish , M. Hamdan, “Hybrid particle swarm and conjugate gradient optimization algorithm”, Advances in Swarm Intelligence (Lecture Notes in Computer Science), Part 1, pp. 582-588, 2010.

[27] S. Chen, T. Mei, M. Luo and X. Yang, “Identification of nonlinear system based on a new hybrid gradient-based PSO algorithm”, in Proc. 2007 International Conference on Information Acquisition, ICIA, Seogwipo-si, pp. 265-268.

[28] A.H. Kayhan, H. Ceylan and M.T. Ayvaz, Gurarslan, G. “PSOLVER: A new hybrid particle swarm optimization algorithm for solving continuous optimization problems”, Expert Systems with Applications, Vol.37, No.10, pp. 6798-6808, 2010.

[29] M.T. Ayvaz, A.H. Kayhan, H. Ceylan and G. Gurarslan, “Hybridizing the harmony search algorithm with a spreadsheet “solver” for solving continuous engineering optimization problems”, Engineering Optimization, Vol.41, No.12, pp. 1119-1144, 2009.

[30] J. Kennedy, R.C. Eberhart, “Particle swarm optimization”, in Proc. 1995 of the IEEE international conference on neural networks, Perth, Australia. Piscataway, NJ: IEEE, pp. 1942–1948.

[31] Y. Shi and R.C. Eberhart, “A modified particle swarm optimizer”, in Proc. 1998 of the IEEE conference on evolutionary computation, Anchorage, Piscataway, NJ: IEEE Press, pp. 69–73.

[32] Y. Shi , R.C. Eberhart, “Empirical study of particle swarm optimization”, in Proc. 1999 of the IEEE congress on evolutionary computation,

Majlesi Journal of Electrical Engineering Vol. 9, No. 1, March 2015

53

Piscataway, NJ. Piscataway, NJ: IEEE Press, pp. 1945–1950.

[33] M. Fesanghary, M. Mahdavi, M. Minary-Jolandan and Y. Alizadeh, “Hybridizing harmony search algorithm with sequential quadratic programming for engineering optimization problems”, Computer Methods in Applied Mechanics and Engineering, Vol. 197(33-40), pp. 3080–3091, 2008.

[34] Z-L. Gaing, “Particle swarm optimization to solving the economic dispatch considering the generator constraints”, IEEE Transaction on Power System, Vol.18, No.3, pp. 1187–1195, 2003.

[35] Z-L. Gaing, “Closure to “Discussion of “Particle swarm optimization to solving the economic dispatch considering the generator constraints””, IEEE Transactions on Power Systems, Vol.19, No.4, pp. 2122–2123, 2004.

[36] M. Pourakbari-Kasmaei , M. Rashidi-Nejad, “An effortless hybrid method to solve economic load dispatch problem in power systems”, Energy Conversion and Management, Vol.52(8-9), pp. 2854-2860, 2011.

[37] A.Y. Saber, S. Chakraborty, S.M. Abdur Razzak and T. Senjyu, “Optimization of economic load dispatch of higher order general cost polynomials and its sensitivity using modified particle swarm optimization”, Electric Power Systems Research, Vol.79, No.1, pp. 98-106, 2009.

[38] E. Sayedi, M.M. Farsangi, M. Barati and K.Y. Lee, “A modified shuffled frog leaping algorithm for

nonconvex economic dispatch problem”, in Proc. 2012 IEEE Power and Energy Society General Meeting, San Diego, CA, pp. 1-8.

[39] L. Wang, L.P. Li, “An effective differential harmony search algorithm for the solving non-convex economic load dispatch problems”, Electrical Power and Energy Systems, Vol. 44, No.1, pp. 832-843, 2013.

[40] M. Moradi-Dalvand, B. Mohammadi-Ivatloo, A. Najafi and A. Rabiee, “Continuous quick group search optimizer for solving non-convex economic dispatch problems”, Electric Power Systems Research, Vol. 93, pp. 93-105, 2012.

[41] J. Cai, Q. Li, L. Li, H. Peng and Y. Yang, “A hybrid CPSO–SQP method for economic dispatch considering the valve-point effects”, Energy Conversion and Management, Vol.53, No.1, pp. 175-181, 2012.

[42] J. Cai, Q. Li, L. Li, H. Peng and Y. Yang, “A hybrid FCASO-SQP method for solving the economic dispatch problems with valve-point effects”, Energy, Vol. 38, No.1, pp. 346–353, 2012.

[43] A.S. Reddy, K. Vaisakh, “Shuffled differential evolution for large scale economic dispatch”, Electric Power Systems Research, Vol. 96, pp. 237-245, 2013.

[44] M. Sydulu, “A very fast and effective noniterative ‘‘Lamda Logic Based’’ algorithm for economic dispatch of thermal units”, in Proc. 1999 IEEE region 10 conference TENCON, Cheju Island, Vol.2, pp. 1434–1437.

Published

2014-09-19

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