A modified differential evolution based solution technique for economic dispatch problems

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October 2012, 8(4): 1017-1038. doi: 10.3934/jimo.2012.8.1017

A modified differential evolution based solution technique for economic dispatch problems

Md. Abul Kalam Azad ^1, and Edite M.G.P. Fernandes ^1,

1.	Algoritmi R&D Centre, School of Engineering, University of Minho, 4710-057 Braga, Portugal, Portugal

Received July 2011 Revised May 2012 Published September 2012

Abstract
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Economic dispatch (ED) plays one of the major roles in power generation systems. The objective of economic dispatch problem is to find the optimal combination of power dispatches from different power generating units in a given time period to minimize the total generation cost while satisfying the specified constraints. Due to valve-point loading effects the objective function becomes nondifferentiable and has many local minima in the solution space. Traditional methods may fail to reach the global solution of ED problems. Most of the existing stochastic methods try to make the solution feasible or penalize an infeasible solution with penalty function method. However, to find the appropriate penalty parameter is not an easy task. Differential evolution is a population-based heuristic approach that has been shown to be very efficient to solve global optimization problems with simple bounds. In this paper, we propose a modified differential evolution based solution technique along with a tournament selection that makes pair-wise comparison among feasible and infeasible solutions based on the degree of constraint violation for economic dispatch problems. We reformulate the nonsmooth objective function to a smooth one and add nonlinear inequality constraints to original ED problems. We consider five ED problems and compare the obtained results with existing standard deterministic NLP solvers as well as with other stochastic techniques available in literature.

Keywords: constraint violation, Economic dispatch, valve-point loading effects, tournament selection., generator ramp-rate, differential evolution.

Mathematics Subject Classification: 90C30, 90C56, 90C59, 90C9.

Citation: Md. Abul Kalam Azad, Edite M.G.P. Fernandes. A modified differential evolution based solution technique for economic dispatch problems. Journal of Industrial & Management Optimization, 2012, 8 (4) : 1017-1038. doi: 10.3934/jimo.2012.8.1017

References:

[1]	M. M. Ali, A recursive topographical differential evolution algorithm for potential energy minimization,, J. Ind. Manag. Optim., 6 (2010), 29. doi: 10.3934/jimo.2010.6.29.
[2]	P. Attaviriyanupap, H. Kita, E. Tanaka and J. Hasegawa, A hybrid EP and SQP for dynamic economic dispatch with nonsmooth fuel cost function,, IEEE Trans. Power Syst., 17 (2002), 411. doi: 10.1109/TPWRS.2002.1007911.
[3]	R. Balamurugan and S. Subramanian, Differential evolution-based dynamic economic dispatch of generating units with valve-point effects,, Electr. Power Compon. Syst., 36 (2008), 828. doi: 10.1080/15325000801911427.
[4]	R. Balamurugan and S. Subramanian, An improved differential evolution based dynamic economic dispatch with nonsmooth fuel cost function,, J. Electr. Syst., 3 (2007), 151.
[5]	H.-G. Beyer and H.-P. Schwefel, Evolution strategies: A comprehensive introduction,, J. Nat. Comput., 1 (2002), 3. doi: 10.1023/A:1015059928466.
[6]	P. Belotti, "Couenne: A User's Manual,", Available from: , (2009).
[7]	S. I. Birbil and S. C. Fang, An electromagnetism-like mechanism for global optimization,, J. Glob. Optim., 25 (2003), 263. doi: 10.1023/A:1022452626305.
[8]	E. G. Birgin, C. A. Floudas and J. M. Martínez, Global minimization using an augmented Lagrangian method with variable lower-level constraints,, Math. Program. Ser. A, 125 (2010), 139. doi: 10.1007/s10107-009-0264-y.
[9]	J. Brest, S. Greiner, B. Bošković, M. Mernik and V. Žumer, Self-adapting control parameters in differential evolution: a comparative study on numerical benchmark problems,, IEEE Trans. Evol. Comput., 10 (2006), 646. doi: 10.1109/TEVC.2006.872133.
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[11]	R. H. Byrd, J. Nocedal and R. A. Waltz, KNITRO: An integrated package for nonlinear optimization,, In, (2006), 35.
[12]	C.-L. Chen, Non-convex economic dispatch: a direct search approach,, Energy Convers. Manage., 48 (2007), 219. doi: 10.1016/j.enconman.2006.04.010.
[13]	P. H. Chen and H. C. Chang, Large-scale economic dispatch by genetic algorithm,, IEEE Trans. Power Syst., 10 (1995), 1919. doi: 10.1109/59.476058.
[14]	J.-P. Chiou, Variable scaling hybrid differential evolution for large-scale economic dispatch problems,, Electr. Power Syst. Res., 77 (2007), 212. doi: 10.1016/j.epsr.2006.02.013.
[15]	A. R. Conn, N. I. M. Gould and Ph. L. Toint, A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds,, SIAM J. Numer. Anal., 28 (1991), 545. doi: 10.1137/0728030.
[16]	D. B. Das and C. Patvardhan, Solution of economic load dispatch using real coded hybrid stochastic search,, Int. J. Electr. Power Energy Syst., 21 (1999), 165. doi: 10.1016/S0142-0615(98)00036-2.
[17]	S. Das and P. N. Suganthan, Differential evolution: A survey of the state-of-the-art,, IEEE Trans. Evol. Comput., 15 (2011), 4. doi: 10.1109/TEVC.2010.2059031.
[18]	S. Das, A. Abraham, U. K. Chakraborty and A. Konar, Differential evolution using a neighborhood-based mutation operator,, IEEE Trans. Evol. Comput., 13 (2009), 526. doi: 10.1109/TEVC.2008.2009457.
[19]	K. Deb, Scope of stationary multi-objective evolutionary optimization: A case study on a hydro-thermal power dispatch problem,, J. Glob. Optim., 41 (2008), 479. doi: 10.1007/s10898-007-9261-y.
[20]	K. Deb, An efficient constraint handling method for genetic algorithms,, Comput. Meth. Appl. Mech. Eng., 186 (2000), 311. doi: 10.1016/S0045-7825(99)00389-8.
[21]	V. N. Dieu and W. Ongsakul, Augmented lagrange hopfield network for large scale economic dispatch,, Proc. Int. Symp. Electr. Electron. Eng., (2007), 19.
[22]	M. Dorigo, V. Maniezzo and A. Colorni, The ant system: optimization by a colony of cooperating agents,, IEEE Trans. Syst. Man Cybern., 26 (1996), 29. doi: 10.1109/3477.484436.
[23]	A. Drud, "CONOPT: Solver Manual,", ARKI Consulting and Development. Available from: , ().
[24]	D. E. Finkel and C. T. Kelley, Additive scaling and the DIRECT algorithm,, J. Glob. Optim., 36 (2006), 597. doi: 10.1007/s10898-006-9029-9.
[25]	R. Fletcher and S. Leyffer, Nonlinear programming without a penalty function,, Math. Program., 91 (2002), 239. doi: 10.1007/s101070100244.
[26]	R. Fourer, D. M. Gay and B. W. Kernighan, "AMPL: A Modeling Language for Mathematical Programming,", Boyd & Fraser Publishing Co., (1993).
[27]	Z.-L Gaing, Particle swarm optimization to solving the economic dispatch considering the generators constraints,, IEEE Trans. Power Syst., 18 (2003), 1187. doi: 10.1109/TPWRS.2003.814889.
[28]	Z. W. Geem, J.-H. Kim and G. V. Loganathan, A new heuristic optimization algorithm: harmony search,, Simulation, 76 (2001), 60. doi: 10.1177/003754970107600201.
[29]	Ph. E. Gill, W. Murray and M. A. Saunders, SNOPT: An SQP algorithm for large-scale constrained optimization,, SIAM Rev., 47 (2005), 99. doi: 10.1137/S0036144504446096.
[30]	F. Glover and M. Laguna, "Tabu Search,", Kluwer Academic Publishers, (1997).
[31]	G. P. Granelli and M. Montagna, Security-constrained economic dispatch using dual quadratic programming,, Electr. Power Syst. Res., 56 (2000), 71. doi: 10.1016/S0378-7796(00)00097-3.
[32]	D. He, F. Wang and Z. Mao, A hybrid genetic algorithm approach based on differential evolution for economic dispatch with valve-point effect,, Electr. Power Energy Syst., 30 (2008), 31. doi: 10.1016/j.ijepes.2007.06.023.
[33]	K. S. Hindi and A. R. Ab-Ghani, Dynamic economic dispatch for large-scale power systems: a Lagrangian relaxation approach,, Electr. Power Syst. Res., 13 (1991), 51. doi: 10.1016/0142-0615(91)90018-Q.
[34]	J. H. Holland, "Adaptation in Natural and Artificial Systems,", University of Michigan Press, (1997).
[35]	W. Huyer and A. Neumaier, Global optimization by multilevel coordinate search,, J. Glob. Optim., 14 (1999), 331. doi: 10.1023/A:1008382309369.
[36]	R. A. Jabr, A. H. Coonick and B. J. Cory, A homogeneous linear programming algorithm for the security constrained economic dispatch problem,, IEEE Trans. Power Syst., 15 (2000), 930. doi: 10.1109/59.871715.
[37]	P. Kaelo and M. M. Ali, A numerical study of some modified differential evolution algorithms,, Eur. J. Oper. Res., 169 (2006), 1176. doi: 10.1016/j.ejor.2004.08.047.
[38]	D. Karaboga and B. Basturk, A powerful and efficient algorithm for numerical function optimization: Artificial bee colony (ABC) algorithm,, J. Glob. Optim., 39 (2007), 459. doi: 10.1007/s10898-007-9149-x.
[39]	A. A. El-Keib, H. Ma and J. L. Hart, Environmentally constrained economic dispatch using the Lagrangian relaxation method,, IEEE Trans. Power Syst., 9 (1994), 1723. doi: 10.1109/59.331423.
[40]	J. Kennedy, R. C. Eberhart and Y. Shi, "Swarm Intelligence,", Morgan Kaufmann, (2001).
[41]	S. Kirkpatrick, C. D. Gelatt Jr. and M. P. Vecchi, Optimization by simulated annealing,, Science, 220 (1983), 671. doi: 10.1126/science.220.4598.671.
[42]	J. J. Liang, T. P. Runarsson, E. Mezura-Montes, M. Clerc, P. N. Suganthan, C. A. Coello Coello and K. Deb, "Problem Definitions and Evaluation Criteria for the CEC 2006 Special Session on Constrained Real-Parameter Optimization,", Technical Report, (2006).
[43]	, "LINDOGlobal: Solver Manual,", Lindo Systems, (2007).
[44]	J. Liu and J. Lampinen, A fuzzy adaptive differential evolution algorithm,, Soft Comput., 9 (2005), 448. doi: 10.1007/s00500-004-0363-x.
[45]	R. Mallipeddi, P. N. Suganthan, Q. K. Pan and M. F. Tasgetiren, Differential evolution algorithm with ensemble of parameters and mutation strategies,, Appl. Soft Comput., 11 (2011), 1679. doi: 10.1016/j.asoc.2010.04.024.
[46]	R. Mallipeddi and P. N. Suganthan, Ensemble of constraint handling techniques,, IEEE Trans. Evol. Comput., 14 (2010), 561. doi: 10.1109/TEVC.2009.2033582.
[47]	B. A. Murtagh and M. A. Saunders, "MINOS 5.5 User's Guide, Report SOL 83-20R,", Dept. of Operations Research, (1998).
[48]	C. K. Panigrahi, P. K. Chattopadhyay, R. N. Chakrabarti and M. Basu, Simulated annealing technique for dynamic economic dispatch,, Electr. Power Compon. Syst., 34 (2006), 577. doi: 10.1080/15325000500360843.
[49]	J.-B. Park, K.-S. Lee, J.-R. Shin and K. Y. Lee, A particle swarm optimization for economic dispatch with nonsmooth cost functions,, IEEE Trans. Power Syst., 20 (2005), 34. doi: 10.1109/TPWRS.2004.831275.
[50]	Y.-M. Park, J. R. Won and J. B. Park, New approach to economic load dispatch based on improved evolutionary programming,, Eng. Intell. Syst. Electr. Eng. Commun., 6 (1998), 103.
[51]	A. K. Qin, V. L. Huang and P. N. Suganthan, Differential evolution algorithm with strategy adaptation for global numerical optimization,, IEEE Trans. Evol. Comput., 13 (2009), 398. doi: 10.1109/TEVC.2008.927706.
[52]	T. P. Runarsson and X. Yao, Constrained evolutionary optimization-the penalty function approach,, in, (2003), 87.
[53]	N. Sinha, R. Chakrabarti and P. K. Chattopadhyay, Evolutionary programming techniques for economic load dispatch,, IEEE Trans. Evol. Comput., 7 (2003), 83. doi: 10.1109/TEVC.2002.806788.
[54]	R. Storn and K. Price, Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces,, J. Glob. Optim., 11 (1997), 341. doi: 10.1023/A:1008202821328.
[55]	C.-T. Su and W.-T. Tyen, A genetic algorithm approach employing floating point representation for economic dispatch of electric power systems,, Proc. Int. Congr. Model. Simul., (1997), 1444.
[56]	S. Takriti and B. Krasenbrink, A decomposition approach for the fuel-constrained economic power-dispatch problem,, Eur. J. Oper. Res., 112 (1999), 460. doi: 10.1016/S0377-2217(98)00131-3.
[57]	M. Tawarmalani and N. Sahinidis, "Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming,", Kluwer Academic Publishers, (2002).
[58]	R. J. Vanderbei and D. F. Shanno, An interior-point algorithm for nonconvex nonlinear programming,, Comput. Optim. Appl., 13 (1999), 231. doi: 10.1023/A:1008677427361.
[59]	T. A. A. Victorie and A. E. Jeyakumar, Hybrid PSO-SQP for economic dispatch with valve-point effect,, Electr. Power Syst. Res., 71 (2004), 51. doi: 10.1016/j.epsr.2003.12.017.
[60]	A. Wächter and L. T. Biegler, On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming,, Math. Program., 106 (2006), 25. doi: 10.1007/s10107-004-0559-y.
[61]	D. C. Walters and G. B. Sheble, Genetic algorithm solution of economic dispatch with valve-point loadings,, IEEE Trans. Power Syst., 8 (1993), 1325. doi: 10.1109/59.260861.
[62]	K. P. Wong and Y. W. Wong, Genetic and genetic/simulated-annealing approaches to economic dispatch,, IEE Proc. Gener. Transm. Distrib., 141 (1994), 507. doi: 10.1049/ip-gtd:19941354.
[63]	K. P. Wong and C. C. Fung, Simulated annealing based economic dispatch algorithm,, IEE Proc. Gener. Transm. Distrib., 140 (1993), 509.
[64]	H.-T. Yang, P.-C. Yang and C.-L. Huang, Evolutionary programming based economic dispatch for units with non-smooth fuel cost functions,, IEEE Trans. Power Syst., 11 (1996), 112. doi: 10.1109/59.485992.
[65]	J. Zhang and A. C. Sanderson, JADE: Adaptive differential evolution with optional external archive,, IEEE Trans. Evol. Comput., 13 (2009), 945. doi: 10.1109/TEVC.2009.2014613.

show all references

References:

[1]	M. M. Ali, A recursive topographical differential evolution algorithm for potential energy minimization,, J. Ind. Manag. Optim., 6 (2010), 29. doi: 10.3934/jimo.2010.6.29.
[2]	P. Attaviriyanupap, H. Kita, E. Tanaka and J. Hasegawa, A hybrid EP and SQP for dynamic economic dispatch with nonsmooth fuel cost function,, IEEE Trans. Power Syst., 17 (2002), 411. doi: 10.1109/TPWRS.2002.1007911.
[3]	R. Balamurugan and S. Subramanian, Differential evolution-based dynamic economic dispatch of generating units with valve-point effects,, Electr. Power Compon. Syst., 36 (2008), 828. doi: 10.1080/15325000801911427.
[4]	R. Balamurugan and S. Subramanian, An improved differential evolution based dynamic economic dispatch with nonsmooth fuel cost function,, J. Electr. Syst., 3 (2007), 151.
[5]	H.-G. Beyer and H.-P. Schwefel, Evolution strategies: A comprehensive introduction,, J. Nat. Comput., 1 (2002), 3. doi: 10.1023/A:1015059928466.
[6]	P. Belotti, "Couenne: A User's Manual,", Available from: , (2009).
[7]	S. I. Birbil and S. C. Fang, An electromagnetism-like mechanism for global optimization,, J. Glob. Optim., 25 (2003), 263. doi: 10.1023/A:1022452626305.
[8]	E. G. Birgin, C. A. Floudas and J. M. Martínez, Global minimization using an augmented Lagrangian method with variable lower-level constraints,, Math. Program. Ser. A, 125 (2010), 139. doi: 10.1007/s10107-009-0264-y.
[9]	J. Brest, S. Greiner, B. Bošković, M. Mernik and V. Žumer, Self-adapting control parameters in differential evolution: a comparative study on numerical benchmark problems,, IEEE Trans. Evol. Comput., 10 (2006), 646. doi: 10.1109/TEVC.2006.872133.
[10]	A. Brooke, D. Kendrick, A. Meeraus and R. Raman, "GAMS: A User's Guide,", Release 2.25, (1992).
[11]	R. H. Byrd, J. Nocedal and R. A. Waltz, KNITRO: An integrated package for nonlinear optimization,, In, (2006), 35.
[12]	C.-L. Chen, Non-convex economic dispatch: a direct search approach,, Energy Convers. Manage., 48 (2007), 219. doi: 10.1016/j.enconman.2006.04.010.
[13]	P. H. Chen and H. C. Chang, Large-scale economic dispatch by genetic algorithm,, IEEE Trans. Power Syst., 10 (1995), 1919. doi: 10.1109/59.476058.
[14]	J.-P. Chiou, Variable scaling hybrid differential evolution for large-scale economic dispatch problems,, Electr. Power Syst. Res., 77 (2007), 212. doi: 10.1016/j.epsr.2006.02.013.
[15]	A. R. Conn, N. I. M. Gould and Ph. L. Toint, A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds,, SIAM J. Numer. Anal., 28 (1991), 545. doi: 10.1137/0728030.
[16]	D. B. Das and C. Patvardhan, Solution of economic load dispatch using real coded hybrid stochastic search,, Int. J. Electr. Power Energy Syst., 21 (1999), 165. doi: 10.1016/S0142-0615(98)00036-2.
[17]	S. Das and P. N. Suganthan, Differential evolution: A survey of the state-of-the-art,, IEEE Trans. Evol. Comput., 15 (2011), 4. doi: 10.1109/TEVC.2010.2059031.
[18]	S. Das, A. Abraham, U. K. Chakraborty and A. Konar, Differential evolution using a neighborhood-based mutation operator,, IEEE Trans. Evol. Comput., 13 (2009), 526. doi: 10.1109/TEVC.2008.2009457.
[19]	K. Deb, Scope of stationary multi-objective evolutionary optimization: A case study on a hydro-thermal power dispatch problem,, J. Glob. Optim., 41 (2008), 479. doi: 10.1007/s10898-007-9261-y.
[20]	K. Deb, An efficient constraint handling method for genetic algorithms,, Comput. Meth. Appl. Mech. Eng., 186 (2000), 311. doi: 10.1016/S0045-7825(99)00389-8.
[21]	V. N. Dieu and W. Ongsakul, Augmented lagrange hopfield network for large scale economic dispatch,, Proc. Int. Symp. Electr. Electron. Eng., (2007), 19.
[22]	M. Dorigo, V. Maniezzo and A. Colorni, The ant system: optimization by a colony of cooperating agents,, IEEE Trans. Syst. Man Cybern., 26 (1996), 29. doi: 10.1109/3477.484436.
[23]	A. Drud, "CONOPT: Solver Manual,", ARKI Consulting and Development. Available from: , ().
[24]	D. E. Finkel and C. T. Kelley, Additive scaling and the DIRECT algorithm,, J. Glob. Optim., 36 (2006), 597. doi: 10.1007/s10898-006-9029-9.
[25]	R. Fletcher and S. Leyffer, Nonlinear programming without a penalty function,, Math. Program., 91 (2002), 239. doi: 10.1007/s101070100244.
[26]	R. Fourer, D. M. Gay and B. W. Kernighan, "AMPL: A Modeling Language for Mathematical Programming,", Boyd & Fraser Publishing Co., (1993).
[27]	Z.-L Gaing, Particle swarm optimization to solving the economic dispatch considering the generators constraints,, IEEE Trans. Power Syst., 18 (2003), 1187. doi: 10.1109/TPWRS.2003.814889.
[28]	Z. W. Geem, J.-H. Kim and G. V. Loganathan, A new heuristic optimization algorithm: harmony search,, Simulation, 76 (2001), 60. doi: 10.1177/003754970107600201.
[29]	Ph. E. Gill, W. Murray and M. A. Saunders, SNOPT: An SQP algorithm for large-scale constrained optimization,, SIAM Rev., 47 (2005), 99. doi: 10.1137/S0036144504446096.
[30]	F. Glover and M. Laguna, "Tabu Search,", Kluwer Academic Publishers, (1997).
[31]	G. P. Granelli and M. Montagna, Security-constrained economic dispatch using dual quadratic programming,, Electr. Power Syst. Res., 56 (2000), 71. doi: 10.1016/S0378-7796(00)00097-3.
[32]	D. He, F. Wang and Z. Mao, A hybrid genetic algorithm approach based on differential evolution for economic dispatch with valve-point effect,, Electr. Power Energy Syst., 30 (2008), 31. doi: 10.1016/j.ijepes.2007.06.023.
[33]	K. S. Hindi and A. R. Ab-Ghani, Dynamic economic dispatch for large-scale power systems: a Lagrangian relaxation approach,, Electr. Power Syst. Res., 13 (1991), 51. doi: 10.1016/0142-0615(91)90018-Q.
[34]	J. H. Holland, "Adaptation in Natural and Artificial Systems,", University of Michigan Press, (1997).
[35]	W. Huyer and A. Neumaier, Global optimization by multilevel coordinate search,, J. Glob. Optim., 14 (1999), 331. doi: 10.1023/A:1008382309369.
[36]	R. A. Jabr, A. H. Coonick and B. J. Cory, A homogeneous linear programming algorithm for the security constrained economic dispatch problem,, IEEE Trans. Power Syst., 15 (2000), 930. doi: 10.1109/59.871715.
[37]	P. Kaelo and M. M. Ali, A numerical study of some modified differential evolution algorithms,, Eur. J. Oper. Res., 169 (2006), 1176. doi: 10.1016/j.ejor.2004.08.047.
[38]	D. Karaboga and B. Basturk, A powerful and efficient algorithm for numerical function optimization: Artificial bee colony (ABC) algorithm,, J. Glob. Optim., 39 (2007), 459. doi: 10.1007/s10898-007-9149-x.
[39]	A. A. El-Keib, H. Ma and J. L. Hart, Environmentally constrained economic dispatch using the Lagrangian relaxation method,, IEEE Trans. Power Syst., 9 (1994), 1723. doi: 10.1109/59.331423.
[40]	J. Kennedy, R. C. Eberhart and Y. Shi, "Swarm Intelligence,", Morgan Kaufmann, (2001).
[41]	S. Kirkpatrick, C. D. Gelatt Jr. and M. P. Vecchi, Optimization by simulated annealing,, Science, 220 (1983), 671. doi: 10.1126/science.220.4598.671.
[42]	J. J. Liang, T. P. Runarsson, E. Mezura-Montes, M. Clerc, P. N. Suganthan, C. A. Coello Coello and K. Deb, "Problem Definitions and Evaluation Criteria for the CEC 2006 Special Session on Constrained Real-Parameter Optimization,", Technical Report, (2006).
[43]	, "LINDOGlobal: Solver Manual,", Lindo Systems, (2007).
[44]	J. Liu and J. Lampinen, A fuzzy adaptive differential evolution algorithm,, Soft Comput., 9 (2005), 448. doi: 10.1007/s00500-004-0363-x.
[45]	R. Mallipeddi, P. N. Suganthan, Q. K. Pan and M. F. Tasgetiren, Differential evolution algorithm with ensemble of parameters and mutation strategies,, Appl. Soft Comput., 11 (2011), 1679. doi: 10.1016/j.asoc.2010.04.024.
[46]	R. Mallipeddi and P. N. Suganthan, Ensemble of constraint handling techniques,, IEEE Trans. Evol. Comput., 14 (2010), 561. doi: 10.1109/TEVC.2009.2033582.
[47]	B. A. Murtagh and M. A. Saunders, "MINOS 5.5 User's Guide, Report SOL 83-20R,", Dept. of Operations Research, (1998).
[48]	C. K. Panigrahi, P. K. Chattopadhyay, R. N. Chakrabarti and M. Basu, Simulated annealing technique for dynamic economic dispatch,, Electr. Power Compon. Syst., 34 (2006), 577. doi: 10.1080/15325000500360843.
[49]	J.-B. Park, K.-S. Lee, J.-R. Shin and K. Y. Lee, A particle swarm optimization for economic dispatch with nonsmooth cost functions,, IEEE Trans. Power Syst., 20 (2005), 34. doi: 10.1109/TPWRS.2004.831275.
[50]	Y.-M. Park, J. R. Won and J. B. Park, New approach to economic load dispatch based on improved evolutionary programming,, Eng. Intell. Syst. Electr. Eng. Commun., 6 (1998), 103.
[51]	A. K. Qin, V. L. Huang and P. N. Suganthan, Differential evolution algorithm with strategy adaptation for global numerical optimization,, IEEE Trans. Evol. Comput., 13 (2009), 398. doi: 10.1109/TEVC.2008.927706.
[52]	T. P. Runarsson and X. Yao, Constrained evolutionary optimization-the penalty function approach,, in, (2003), 87.
[53]	N. Sinha, R. Chakrabarti and P. K. Chattopadhyay, Evolutionary programming techniques for economic load dispatch,, IEEE Trans. Evol. Comput., 7 (2003), 83. doi: 10.1109/TEVC.2002.806788.
[54]	R. Storn and K. Price, Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces,, J. Glob. Optim., 11 (1997), 341. doi: 10.1023/A:1008202821328.
[55]	C.-T. Su and W.-T. Tyen, A genetic algorithm approach employing floating point representation for economic dispatch of electric power systems,, Proc. Int. Congr. Model. Simul., (1997), 1444.
[56]	S. Takriti and B. Krasenbrink, A decomposition approach for the fuel-constrained economic power-dispatch problem,, Eur. J. Oper. Res., 112 (1999), 460. doi: 10.1016/S0377-2217(98)00131-3.
[57]	M. Tawarmalani and N. Sahinidis, "Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming,", Kluwer Academic Publishers, (2002).
[58]	R. J. Vanderbei and D. F. Shanno, An interior-point algorithm for nonconvex nonlinear programming,, Comput. Optim. Appl., 13 (1999), 231. doi: 10.1023/A:1008677427361.
[59]	T. A. A. Victorie and A. E. Jeyakumar, Hybrid PSO-SQP for economic dispatch with valve-point effect,, Electr. Power Syst. Res., 71 (2004), 51. doi: 10.1016/j.epsr.2003.12.017.
[60]	A. Wächter and L. T. Biegler, On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming,, Math. Program., 106 (2006), 25. doi: 10.1007/s10107-004-0559-y.
[61]	D. C. Walters and G. B. Sheble, Genetic algorithm solution of economic dispatch with valve-point loadings,, IEEE Trans. Power Syst., 8 (1993), 1325. doi: 10.1109/59.260861.
[62]	K. P. Wong and Y. W. Wong, Genetic and genetic/simulated-annealing approaches to economic dispatch,, IEE Proc. Gener. Transm. Distrib., 141 (1994), 507. doi: 10.1049/ip-gtd:19941354.
[63]	K. P. Wong and C. C. Fung, Simulated annealing based economic dispatch algorithm,, IEE Proc. Gener. Transm. Distrib., 140 (1993), 509.
[64]	H.-T. Yang, P.-C. Yang and C.-L. Huang, Evolutionary programming based economic dispatch for units with non-smooth fuel cost functions,, IEEE Trans. Power Syst., 11 (1996), 112. doi: 10.1109/59.485992.
[65]	J. Zhang and A. C. Sanderson, JADE: Adaptive differential evolution with optional external archive,, IEEE Trans. Evol. Comput., 13 (2009), 945. doi: 10.1109/TEVC.2009.2014613.

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