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

A modified differential evolution based solution technique for economic dispatch problems

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

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.
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:
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R. Balamurugan and S. Subramanian, An improved differential evolution based dynamic economic dispatch with nonsmooth fuel cost function,, J. Electr. Syst., 3 (2007), 151. Google Scholar

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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. Google Scholar

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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. Google Scholar

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[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. Google Scholar

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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. Google Scholar

[30]

F. Glover and M. Laguna, "Tabu Search,", Kluwer Academic Publishers, (1997). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[34]

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W. Huyer and A. Neumaier, Global optimization by multilevel coordinate search,, J. Glob. Optim., 14 (1999), 331. doi: 10.1023/A:1008382309369. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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, "LINDOGlobal: Solver Manual,", Lindo Systems, (2007). Google Scholar

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J. Liu and J. Lampinen, A fuzzy adaptive differential evolution algorithm,, Soft Comput., 9 (2005), 448. doi: 10.1007/s00500-004-0363-x. Google Scholar

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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. Google Scholar

[52]

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

H.-G. Beyer and H.-P. Schwefel, Evolution strategies: A comprehensive introduction,, J. Nat. Comput., 1 (2002), 3. doi: 10.1023/A:1015059928466. Google Scholar

[6]

P. Belotti, "Couenne: A User's Manual,", Available from: , (2009). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

A. Brooke, D. Kendrick, A. Meeraus and R. Raman, "GAMS: A User's Guide,", Release 2.25, (1992). Google Scholar

[11]

R. H. Byrd, J. Nocedal and R. A. Waltz, KNITRO: An integrated package for nonlinear optimization,, In, (2006), 35. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[21]

V. N. Dieu and W. Ongsakul, Augmented lagrange hopfield network for large scale economic dispatch,, Proc. Int. Symp. Electr. Electron. Eng., (2007), 19. Google Scholar

[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. Google Scholar

[23]

A. Drud, "CONOPT: Solver Manual,", ARKI Consulting and Development. Available from: , (). Google Scholar

[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. Google Scholar

[25]

R. Fletcher and S. Leyffer, Nonlinear programming without a penalty function,, Math. Program., 91 (2002), 239. doi: 10.1007/s101070100244. Google Scholar

[26]

R. Fourer, D. M. Gay and B. W. Kernighan, "AMPL: A Modeling Language for Mathematical Programming,", Boyd & Fraser Publishing Co., (1993). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[30]

F. Glover and M. Laguna, "Tabu Search,", Kluwer Academic Publishers, (1997). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[34]

J. H. Holland, "Adaptation in Natural and Artificial Systems,", University of Michigan Press, (1997). Google Scholar

[35]

W. Huyer and A. Neumaier, Global optimization by multilevel coordinate search,, J. Glob. Optim., 14 (1999), 331. doi: 10.1023/A:1008382309369. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[40]

J. Kennedy, R. C. Eberhart and Y. Shi, "Swarm Intelligence,", Morgan Kaufmann, (2001). Google Scholar

[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. Google Scholar

[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). Google Scholar

[43]

, "LINDOGlobal: Solver Manual,", Lindo Systems, (2007). Google Scholar

[44]

J. Liu and J. Lampinen, A fuzzy adaptive differential evolution algorithm,, Soft Comput., 9 (2005), 448. doi: 10.1007/s00500-004-0363-x. Google Scholar

[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. Google Scholar

[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. Google Scholar

[47]

B. A. Murtagh and M. A. Saunders, "MINOS 5.5 User's Guide, Report SOL 83-20R,", Dept. of Operations Research, (1998). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[52]

T. P. Runarsson and X. Yao, Constrained evolutionary optimization-the penalty function approach,, in, (2003), 87. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[57]

M. Tawarmalani and N. Sahinidis, "Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming,", Kluwer Academic Publishers, (2002). Google Scholar

[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. Google Scholar

[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. Google Scholar

[60]

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