Quantum-behaved Particle Swarm Optimization for Power Economic Dispatch Problem of Units with Multiple Fuel Option

Chao Lung Chiang


This paper presents a quantum-behaved particle swarm optimization (QPSO) with a multiple updating (MU) for solving the power economic dispatch problem (PEDP) of generators with multiple fuel options (MFOs). The QPSO assists the proposed method efficaciously find and precisely search. The MU helps the proposed method prevent deforming the augmented Lagrange function (ALF) and caused difficultly in searching optimal solution. The proposed approach combines the QPSO and the MU that has benefits of adopting a widespread area of punishment parameters and a small-size population. The proposed algorithm has been demonstrated on a practical ten generating units system; every one unit is composed of two or three fuel changes. The entire fuel price got by the proposed QPSO-MU has been competed with former studies for validating its efficacy. Compared achievements clearly express that the presented method is an effective alternative for resolving PEDP of units with MFOs in the realistic operations of power system.


Augmented Lagrange Function; Economic Dispatch; MFOs, Particle Swarm Optimization.

Full Text:



A. J. Wood and B. F. Wollenberg, Power Generation Operation and Control, 2nd ed, Wiley, New York, 1996.

A. Chakrabarti and S. Halder, Power System Analysis Operation and Control, 3rd Edition, PHI, New Delhi, 2010.

R. R. Shoults and M.M. Mead, “Optimal estimation of piece-wise linear incremental cost curves for EDC,” IEEE Trans. Power Appar. Syst. PAS-103, vol. 6, pp. 1432–1438, 1984.

C. E. Lin and G. L. Viviani, “Hierarchical economic dispatch for piecewise quadratic cost functions,” IEEE Trans. Power App. Syst., PAS-103, no. 6, pp. 1170-1175, 1984.

J. H. Park, Y. S. Kim, I. K. Eom, and K. Y. Lee, “Economic load dispatch for piecewise quadratic cost function using Hopfield neural network,” IEEE Trans. Power Syst., vol. 8, no. 3, pp. 1030–1038, 1993.

S. C. Lee and Y. H. Kim, “An enhanced Lagrangian neural network for the ELD problems with piecewise quadratic cost functions and nonlinear constraints,” Electr. Power Syst. Res., vol. 60, no. 3, pp. 167–177, 2002.

K. Y. Lee, A. Sode-Yome, and J. H. Park, “Adaptive Hopfield neural networks for economic load dispatch,” IEEE Trans. Power Syst., vol. 13, no. 2, pp. 519–526, May 1998.

K. Y. Lee, F. M. Nuroglu, and A. Sode-Yome, “Real power optimization with load flow using adaptive Hopfield neural network,” Eng. Intell. Syst., vol. 8, no. 1, pp. 53-58, 2000.

N. Amjady and H. Nasiri-Rad, “Solution of nonconvex and nonsmooth economic dispatch by a new adaptive real coded genetic algorithm,” Expert Syst. Appl., vol. 37, no. 7, pp. 5239–5245, 2010.

C. L. Chiang and C. T. Su, “Adaptive-improved genetic algorithm for the economic dispatch of units with multiple fuel options,” Cybern. Syst.: An Internat. J., vol. 36, no. 7, pp. 687–704, 2005.

T. Jayabarathi, K. Jayaprakash, D. N. Jeyakumar, and T. Raghunathan, “Evolutionary programming techniques for different kinds of economic dispatch problems,” Electr. Power Syst. Res., vol.73, no. 2, pp. 169–176, 2005.

Y. M. Park, J. R. Wong, and J. B. Park, “A new approach to economic load dispatch based on improved evolutionary programming,” Eng. Intell. Syst. Elect. Eng. Commun., vol. 6, no. 2, pp. 103–110, 1998.

N. Noman and H. Iba, “Differential evolution for economic load dispatch problems,” Electr. Power Syst. Res., vol. 78, no. 8, pp. 1322–1331, 2008.

J. B. Park, K. S. Lee, and K. W. Lee, “A particle swarm optimization for economic dispatch with nonsmooth cost function,” IEEE Trans. Power Syst., vol. 12, no. 1, pp.34–42, 2005.

W. M. Lin, F. S. Cheng, and M. T. Tsay, “Nonconvex economic dispatch by integrated artificial intelligence,” IEEE Trans. Power Syst., vol. 16, no. 2, pp. 307–311,2001.

S. Baskar, P. Subbaraj, and M. V. C. Rao, “Hybrid real coded genetic algorithm solution to economic dispatch problem,” Comput. Electr. Eng., vol. 29, no. 3, pp. 407–419, 2003.

R. Balamurugan and S. Subramanian, “Hybrid integer coded differential evolution-dynamic programming approach for economic load dispatch with multiple fuel options,” Energy Convers. Manag., vol. 49, no. 4, pp. 608–614, 2008.

N. J. Singh, J. S. Dhillon, D. P. Kothari, "Synergic predator-prey optimization for economic thermal power dispatch problem," Applied Soft Computing, vol. 43, pp. 298–311, 2016.

K. Vaisakha and A. Srinivasa Reddy, "MSFLA/ GHS/ SFLA-GHS/ SDE algorithms for economic dispatch problem considering multiple fuels and valve point loadings," Applied Soft Computing, vol. 13, pp. 4281–4291, 2013.

Vo Ngoc Dieu and Peter Schegner, "Augmented Lagrange Hopfield network initialized by quadratic programming for economic dispatch with piecewise quadratic cost functions and prohibited zones," Applied Soft Computing, vol. 13, pp. 292–301, 2013.

A. K. Barisal, "Dynamic search space squeezing strategy based intelligent algorithm solutions to economic dispatch with multiple fuels," Electrical Power and Energy Systems, vol.45, pp. 50–59, 2013.

Vo Ngoc Dieu and W. Ongsakul, "Economic dispatch with multiple fuel types by enhanced augmented Lagrange Hopfield network," Applied Energy, vol. 91, pp. 281–289, 2012.

R. Anandhakumar and S. Subramanian, "Economic Dispatch with Multiple Fuel Options Using CCF," Energy and Power Engineering, vol. 3, pp. 113-119, 2011.

T. Niknam, H. D. Mojarrad, H. Z. Meymand, and B. B. Firouzi, "A new honey bee mating optimization algorithm for non-smooth economic dispatch," Energy, vol. 36, pp. 896-908, 2011.

R. C. Eberhart and Y. Shi, “A modified swarm optimizer,” In: Proceedings of the 1998 IEEE international conference of evolutionary computation, pp. 1945–1950, 1998.

J. Kennedy, and R. Eberhart, “A new optimizer using particle swarm theory,” Proceedings of the Sixth International Symposium on Micro Machine and Human Science, pp. 39-43, 1995.

Y. Wang, Y. Yang, “Particle swarm with equilibrium strategy of selection for multi-objective optimization,” European Journal of Operational Research, vol. 200, issue 1, pp. 187-197, Jan. 2010.

J. Chang, “A robust adaptive array beamformer using particle swarm optimization for space-time code division multiple access systems,” Inf. Sci., vol. 278, pp. 174–186, 2014.

A. Ouyang, K. Li, T. K. Truong, A. Sallam, and E. H. M. Sha, “Hybrid particle swarm optimization for parameter estimation of Muskingum model,” Neural Comput. & Applic., vol. 25, issue. 7–8, pp. 1785–1799, 2014.

A. Ouyang, Z. Tang, X. Zhou, Y. Xu, G. Pan, and K. Li, “Parallel hybrid PSO with CUDA for LD heat conduction equation,” Comput. Fluids, vol. 110, pp. 198–210, 2015.

J. Zhao, M. Lin, D. Xu, L. Hao, and W. Zhang, “Vector control of a hybrid axial field flux-switching permanent magnet machine based on particle swarm optimization,” IEEE Trans. Magn., vol. 51, issue11, Nov. 2015.

P. Regulski, D. Vilchis-Rodriguez, S. Djurovic, and V. Terzija, “Estimation of composite load model parameters using an improved particle swarm optimization method,” IEEE Trans. Power Deliv., vol. 30, no. 2, pp. 553–560, 2015.

B. Mohammadi-Ivatloo, M. Moradi-Dalvand and A. Rabiee, “Combined heat and power economic dispatch problem solution using particle swarm optimization with time varying acceleration coefficients,” Electric Power Syst. Res., vol. 95, pp. 9–18, 2013.

F. Bergh, “An Analysis of Particle Swarm Optimizers,” Ph.D. thesis, University of Pretoria, 2001.

F. Bergh, A. Engelbrecht, “A new locally convergent particle swarm optimizer,” IEEE Int. Conf. Syst. Man Cybern. Vol. 3, 2002.

J. Sun, B. Feng, W. Xu, “Particle swarm optimization with particles having quantum behavior,” in: Proceedings of the 2004 Congress on Evolutionary Computation, CEC2004, pp. 326–331, 2004.

J. Sun, X. Xu, V. Palade, W. Fang, C. Lai, and W. Xu, “Convergence analysis and improvements of quantum-behaved particle swarm optimization,” Inf. Sci., vol. 193, no. 15, pp. 81–103, 2012.

J. Sun, W. B. Xu, and B. Feng, “Adaptive parameter control for quantum-behaved particle swarm optimization on individual level,” In: Proceedings of the 2005 IEEE international conference on systems, man and cybernetics, vol. 4, pp. 3049–3054, 2005.

F. S. Levin, “An introduction to quantum theory,” Cambridge: Cambridge University Press, 2002.

T. Liu, L. C. Jiao, W. P. Ma, and R. H. Shang, “Quantum-behaved particle swarm optimization with collaborative attractors for nonlinear numerical problems,” Commun. Nonlinear Sci. Numer. Simulat., vol. 44, pp. 167–183, 2017.

Y. Y Li, X. Bai, L. C Jiao, and Y. Xue, “Partitioned-cooperative quantum-behaved particle swarm optimization based on multilevel thresholding applied to medical image segmentation,” Applied Soft Computing, vol. 56, pp. 345–356, 2017.

Z. k. Feng, W. J Niu, and C. T Cheng, “Multi-objective quantum-behaved particle swarm optimization for economic environmental hydrothermal energy system scheduling,” Energy, vol. 131, pp. 165-178, 2017.

T. Liu, L. C Jiao, W. P Ma, J. J Ma, and R. H Shang, “Cultural quantum-behaved particle swarm optimization for environmental/economic dispatch,” Applied Soft Computing, vol. 48, pp. 597–611, 2016.

W. Fang, J. Sun, H. H Chen, and X. J. Wu, “A decentralized quantum-inspired particle swarm optimization algorithm with cellular structured population,” Information Sciences, vol. 330, pp. 19–48, 2016.

M. Clerc and J. Kennedy, “The particle swarm: explosion, stability and convergence in a multi-dimensional complex space,” IEEE Trans Evol. Comput., vol. 6, pp.58–73, 2002.

Z. Michalewicz and M. Schoenauer, “Evolutionary algorithms for constrained parameter optimization problems,” Evolutionary Computation, vol. 4, no. 1, pp.1-32, 1996.

M. J. D. Powell, “Algorithms for nonlinear constraints that use Lagrangian function,” Math. Programming, vol. 14, pp 224-248, 1978.

C. L. Chiang, C. T. Su, and F. S. Wang, “Augmented Lagrangian method for evolutionary optimization of mixed-integer nonlinear constrained problems,” International Mathematics Journal, vol.2, no. 2, pp. 119-154, 2002.

DOI: http://dx.doi.org/10.24018/ejers.2017.2.12.492


  • There are currently no refbacks.

Copyright (c) 2017 Chao Lung Chiang