Reference Point Based TR-PSO for Multi-Objective Environmental/Economic Dispatch

Affiliation(s)

Department of Mathematics, Faculty of Science, Qassim University, Buraydah, KSA.

Department of Basic Engineering Science, Faculty of Engineering, Menoufiya University, Shibin El-Kom, Egypt.

Department of Mathematics, Faculty of Science, Qassim University, Buraydah, KSA.

Department of Basic Engineering Science, Faculty of Engineering, Menoufiya University, Shibin El-Kom, Egypt.

ABSTRACT

A reference point based multi-objective optimization using a combination between trust region (TR) algorithm and particle swarm optimization (PSO) to solve the multi-objective environmental/economic dispatch (EED) problem is presented in this paper. The EED problem is handled by Reference Point Interactive Approach. One of the main advantages of the proposed approach is integrating the merits of both TR and PSO, where TR has provided the initial set (close to the Pareto set as possible and the reference point of the decision maker) followed by PSO to improve the quality of the solutions and get all the points on the Pareto frontier. The performance of the proposed algorithm is tested on standard IEEE 30-bus 6-genrator test system and is compared with conventional methods. The results demonstrate the capabilities of the proposed approach to generate true and well-distributed Pareto-optimal non-dominated solutions in one single run. The comparison with the classical methods demonstrates the superiority of the proposed approach and confirms its potential to solve the multi-objective EED problem.

Cite this paper

A. El-Sawy, Z. Hendawy and M. El-Shorbagy, "Reference Point Based TR-PSO for Multi-Objective Environmental/Economic Dispatch,"*Applied Mathematics*, Vol. 4 No. 5, 2013, pp. 803-813. doi: 10.4236/am.2013.45110.

A. El-Sawy, Z. Hendawy and M. El-Shorbagy, "Reference Point Based TR-PSO for Multi-Objective Environmental/Economic Dispatch,"

References

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[33] J. Dennis, M. El-Alem and K. Williamson, “A Trust Region Approach to Nonlinear Systems of Equalities and Inequalities,” SIAM Journal on Optimization, Vol. 9, No. 2, 1999, pp. 291-315. doi:10.1137/S1052623494276208

[34] S. K. Hwang, K. Koo and J. S. Lee, “Homogeneous Particle Swarm Optimizer for Multi-Objective Optimization Problem,” ICGST International Journal on Artificial Intelligence and Machine Learning, 2006.

[35] W. F. Abd-El-Wahed, A. A. Mousa and M. A. El-Shorbagy, “Integrating Particle Swarm Optimization with Genetic Algorithms for Solving Nonlinear Optimization Problems,” Journal of Computational and Applied Mathematics, Vol. 235, No. 5, 2011, pp. 1446-1453. doi:10.1016/j.cam.2010.08.030

[36] A. A. Mousa, M. A. El-Shorbagy and W. F. Abd El-Wahed, “Local Search Based Hybrid Particle Swarm Optimization for Multiobjective Optimization,” International Journal of Swarm and Evolutionary Computation, Vol. 3, 2012, pp. 1-14. doi:10.1016/j.swevo.2011.11.005

[37] A. A. Mousa and M. A. El-Shorbagy, “Enhanced Particle Swarm Optimization Based Local Search for Reactive Power Compensation Problem,” Applied Mathematics, Vol. 3, No. 10A, 2012, pp. 1276-1284.

[38] R. Fletcher, “Practical Methods of Optimization,” 2nd Edition, John Wiley and Sons, Chichester, 1987.

[39] M. S. Osman, M. A. Abo-Sinna and A. A. Mousa, “An ε Dominance-Based Multiobjective Genetic Algorithm for Economic Emission Load Dispatch Optimization Problem,” Electric Power Systems Research, Vol. 79, No. 11, 2009, pp. 1561-1567. doi:10.1016/j.epsr.2009.06.003

[40] R. Ah King, H. Rughooputh and K. Deb, “Evolutionary Multi-Objective Environmental/Economic Dispatch: Stochastic versus Deterministic Approaches,” KanGAL Report No. 2004019, 2004.

[1] S. F. Brodesky and R. W. Hahn, “Assessing the Influence of Power Pools on Emission Constrained Economic Dispatch,” IEEE Transactions on Power Systems, Vol. 1, No. 1, 1986, pp. 57-62. doi:10.1109/TPWRS.1986.4334844

[2] A. Farag, S. Al-Baiyat and T. C. Cheng, “Economic Load Dispatch Multiobjective Optimization Procedures Using Linear Programming Techniques,” IEEE Transactions on Power Systems, Vol. 10, No. 2, 1995, pp. 731-738. doi:10.1109/59.387910

[3] C. S. Chang, K. P. Wong and B. Fan, “Security-Con strained Multiobjective Generation Dispatch Using Bicriterion Global Optimization,” IEE Proceedings Generation, Transmission & Distribution, Vol. 142, No. 4, 1995, pp. 406-414. doi:10.1049/ip-gtd:19951806

[4] J. X. Xu, C. S. Chang and X. W. Wang, “Constrained Multiobjective Global Optimization of Longitudinal Interconnected Power System by Genetic Algorithm,” IEE Proceedings Generation, Transmission & Distribution, Vol. 143, No. 5, 1996, pp. 435-446. doi:10.1049/ip-gtd:19960418

[5] J. Zahavi and L. Eisenberg, “Economic-Environmental Power Dispatch,” IEEE Transactions on Systems, Man, and Cybernetics SMC, Vol. 5, No. 5, 1985, pp. 485-489. doi:10.1109/TSMC.1975.5408370

[6] Y. T. Hsiao, H. D. Chiang, C. C. Liu and Y. L. Chen, “A Computer Package for Optimal Multi-Objective VAR Planning in Large Scale Power Systems,” IEEE Transactions on Power Systems, Vol. 9, No. 2, 1994, pp. 668-676. doi:10.1109/59.317676

[7] B. S. Kermanshahi, Y. Wu, K. Yasuda and R. Yokoyama, “Environmental Marginal Cost Evaluation by Non-Inferiority Surface,” IEEE Transactions on Power Systems, Vol. 5, No. 4, 1990, pp. 1151-1159. doi:10.1109/59.99365

[8] M. A. Abido, “A novel Multiobjective Evolutionary Algorithm for Environmental/Economic Power Dispatch,” Electric Power Systems Research, Vol. 65, No. 1, 2003, pp. 71-81. doi:10.1016/S0378-7796(02)00221-3

[9] M. A. Abido, “A Niched Pareto Genetic Algorithm for Multiobjective Environmental/Economic Dispatch,” Electric Power Systems Research, Vol. 25, No. 2, 2003, pp. 97-105. doi:10.1016/S0142-0615(02)00027-3

[10] M. A. Abido, “Environmental/Economic Power Dispatch Using Multiobjective Evolutionary Algorithms,” IEEE Transactions on Power Systems, Vol. 18, No. 4, 2003, pp. 1529-1537. doi:10.1109/TPWRS.2003.818693

[11] K. Deb, “Multi-Objective Optimization Using Evolution ary Algorithms,” Wiley, New York, 2001.

[12] C. M. Fonsecam and P. J. Fleming, “An Overview of Evolutionary Algorithms in Multiobjective Optimization,” Evolution Computing, Vol. 3, No. 1, 1995, pp. 1-16. doi:10.1162/evco.1995.3.1.1

[13] F. Wang, K. Zhang, C. Wang and L. Wang, “A Variant of Trust-Region Methods for Unconstrained Optimization,” Applied Mathematics and Computation, Vol. 203, No. 1, 2008, pp. 297-307. doi:10.1016/j.amc.2008.04.049

[14] M. Ahookhosh, K. Amini and M. R. Peyghami, “A Non monotone Trust-Region Line Search Method for Large Scale Unconstrained Optimization,” Applied Mathematical Modelling, Vol. 36, No. 1, 2012, pp. 478-487. doi:10.1016/j.apm.2011.07.021

[15] M. Ahookhosh and K. Amini, “A Nonmonotone, Trust Region Method with Adaptive Radius for Unconstrained Optimization Problems,” Computers & Mathematics with Applications, Vol. 60, No. 3, 2010, pp. 411-422. doi:10.1016/j.camwa.2010.04.034

[16] Z. Shi and J. Guo, “A New Trust Region Method for Un constrained Optimization,” Journal of Computational and Applied Mathematics, Vol. 213, No. 2, 2008, pp. 509-520. doi:10.1016/j.cam.2007.01.027

[17] J. Zhang, K. Zhang and S. Qu, “A Nonmonotone Adaptive Trust Region Method for Unconstrained Optimization Based on Conic Model,” Applied Mathematics and Computation, Vol. 217, No. 8, 2010, pp. 4265-4273. doi:10.1016/j.amc.2010.10.043

[18] B. El-Sobky, “A Global Convergence Theory for an Active Trust Region Algorithm for Solving the General Nonlinear Programming Problem,” Applied Mathematics and Computation, Vol. 144, No. 1, 2003, pp. 127-157. doi:10.1016/S0096-3003(02)00397-1

[19] Y. Ji, K. Zhang, S. Qu and Y. Zhou, “A Trust-Region Method by Active-Set Strategy for General Nonlinear Optimization,” Computers & Mathematics with Applications, Vol. 54, No. 2, 2007, pp. 229-241. doi:10.1016/j.camwa.2007.02.003

[20] S. Kim and J. Ryu, “A Trust-Region Algorithm for Bi Objective Stochastic Optimization,” Procedia Computer Science, Vol. 4, 2011, pp. 1422-1430. doi:10.1016/j.procs.2011.04.153

[21] B. El-Sobky, “An Active-Set Trust-Region Algorithm for Solving Constrained Multi-Objective Optimization Problem,” American Mathematical Society, Vol. 6, 2012, pp. 1599-1612.

[22] J. Kennedy, R. C. Eberhart and Y. Shi, “Swarm Intelli gence,” Morgan Kaufmann, San Francisco, 2001.

[23] M. R. Sierra and C. C. Coello, “Multi-Objective Particle Swarm Optimizers: A Survey of the State-of-the-Art,” International Journal of Computational Intelligence Research, Vol. 2, No. 3, 2006, pp. 287-308.

[24] D. Hazarika and P. K. Bordoloi, “Modi?ed Loss Co-efficients in the Determination of Optimum Generation Scheduling,” IEEE Proceedings, Vol. 138, No. 2, 1991, pp. 166-172.

[25] W. Y. Ng, “Generalized Generation Distribution Factors for Power System Security Evaluations,” IEEE Transactions on Power Apparatus and Systems, Vol. 100, No. 3, 1981, pp. 1001-1005. doi:10.1109/TPAS.1981.316635

[26] C. A. C. Coello, “An Updated Survey of GA-Based Multiobjective Optimization Techniques,” ACM Computing Surveys, Vol. 32, No. 2, 2000, pp. 109-143. doi:10.1145/358923.358929

[27] V. Pareto, “Cours d’économie Politique, Volume I and II,” F. Rouge, Lausanne, 1896, p. 97.

[28] K. Miettinen, “Nonlinear Multiobjective Optimization,” Kluwer Academic Publishers, Boston, 1999.

[29] A. M. J. Skulimowski, “Classification and Properties of Dominating Points in Vector Optimization,” Mathemati cal Methods of Operations Research, Vol. 58, 1989, pp. 99-112.

[30] R. Byrd, “Robust Trust Region methods for Nonlinearly Constrained Optimization,” A Talk Presented at the Second SIAM Conference on Optimization, Houston, 1987.

[31] E. Omojokun, “Trust-Region Strategies for Optimization with Nonlinear Equality and Inequality Constraints,” Ph.D. Thesis, Department of Computer Science, University of Colorado, Boulder, 1989, pp. 57-87.

[32] M. El-Alem, “A Robust Trust-Region Algorithm with a Non-Monotonic Penalty Parameter Scheme for Con strained Optimization,” SIAM Journal on Optimization, Vol. 5, No. 2, 1995, pp. 348-378. doi:10.1137/0805018

[33] J. Dennis, M. El-Alem and K. Williamson, “A Trust Region Approach to Nonlinear Systems of Equalities and Inequalities,” SIAM Journal on Optimization, Vol. 9, No. 2, 1999, pp. 291-315. doi:10.1137/S1052623494276208

[34] S. K. Hwang, K. Koo and J. S. Lee, “Homogeneous Particle Swarm Optimizer for Multi-Objective Optimization Problem,” ICGST International Journal on Artificial Intelligence and Machine Learning, 2006.

[35] W. F. Abd-El-Wahed, A. A. Mousa and M. A. El-Shorbagy, “Integrating Particle Swarm Optimization with Genetic Algorithms for Solving Nonlinear Optimization Problems,” Journal of Computational and Applied Mathematics, Vol. 235, No. 5, 2011, pp. 1446-1453. doi:10.1016/j.cam.2010.08.030

[36] A. A. Mousa, M. A. El-Shorbagy and W. F. Abd El-Wahed, “Local Search Based Hybrid Particle Swarm Optimization for Multiobjective Optimization,” International Journal of Swarm and Evolutionary Computation, Vol. 3, 2012, pp. 1-14. doi:10.1016/j.swevo.2011.11.005

[37] A. A. Mousa and M. A. El-Shorbagy, “Enhanced Particle Swarm Optimization Based Local Search for Reactive Power Compensation Problem,” Applied Mathematics, Vol. 3, No. 10A, 2012, pp. 1276-1284.

[38] R. Fletcher, “Practical Methods of Optimization,” 2nd Edition, John Wiley and Sons, Chichester, 1987.

[39] M. S. Osman, M. A. Abo-Sinna and A. A. Mousa, “An ε Dominance-Based Multiobjective Genetic Algorithm for Economic Emission Load Dispatch Optimization Problem,” Electric Power Systems Research, Vol. 79, No. 11, 2009, pp. 1561-1567. doi:10.1016/j.epsr.2009.06.003

[40] R. Ah King, H. Rughooputh and K. Deb, “Evolutionary Multi-Objective Environmental/Economic Dispatch: Stochastic versus Deterministic Approaches,” KanGAL Report No. 2004019, 2004.