A Bivariate Chance Constraint of Wind Sources for Multi-Objective Dispatching


The economic emission dispatch (EED) problem minimizes two competing objective functions, fuel cost and emission, while satisfying several equality and inequality constraints. Since the availability of wind power (WP) is highly dependent on the weather conditions, the inclusion of a significant amount of WP into EED will result in additional constraints to accommodate the intermittent nature of the output. In this paper, a new correlated bivariate Weibull probability distribution model is proposed to analytically remove the assumption that the total WP is characterized by a single random variable. This probability distribution is used as chance constraint. The inclusion of the probability distribution of stochastic WP in the EED problem is defined as the here-and-now strategy. Non-dominated sorting genetic algorithm built in MATLAB is used to handle the EED problem as a multi-objective optimization problem. A 69-bus ten-unit test system with non-smooth cost function is used to test the effectiveness of the proposed model.

Share and Cite:

M. Elshahed, H. Zeineldin and M. Elmarsfawy, "A Bivariate Chance Constraint of Wind Sources for Multi-Objective Dispatching," Smart Grid and Renewable Energy, Vol. 4 No. 4, 2013, pp. 325-332. doi: 10.4236/sgre.2013.44039.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] http://www.wwindea.org/home/index.php
[2] P. Zhang and S. T. Lee, “Probabilistic Load Flow Computation Using the Method of Combined Cumulants and Gram-Charlier Expansion,” IEEE Transactions on Power Systems, Vol. 19, No. 1, 2004, pp. 676-682. doi:10.1109/TPWRS.2003.818743
[3] A. Schellenberg, W. Rosehart and J. Aguado, “CumulantBased Probabilistic Optimal Power Flow (P-OPF) with Gaussian and Gamma Distributions,” IEEE Transactions on Power Systems, Vol. 20, No. 2, 2005, pp. 773-781. doi:10.1109/TPWRS.2005.846184
[4] A. Schellenberg, W. Rosehart and J. Aguado, “Introduction to Cumulant-Based Probabilistic Optimal Power Flow (P-OPF),” IEEE Transactions on Power Systems, Vol. 20, No. 2, 2005, pp. 1184-1186. doi:10.1109/TPWRS.2005.846188
[5] D. Villanueva, A. Feijóo and J. Luis Pazos, “Probabilistic Load Flow Considering Correlation between Generation, Loads and Wind Power,” Smart Grid and Renewable Energy, Vol. 2, No. 1, 2011, pp. 12-20. doi:10.4236/sgre.2011.21002
[6] Q. Fu, D. C. Yu and J. Ghorai, “Probabilistic Load Flow Analysis for Power Systems with Multi-correlated Wind Source,” Power and Energy Society General Meeting, San Diego, 24-29 July 2011, pp. 1-6. doi:10.1109/PES.2011.6038992
[7] H. Yang and B. Zou, “The Point Estimate Method Using Third-Order Polynomial Normal Transformation Technique to Solve Probabilistic Power Flow With Correlated Wind Source and Load,” Asia-Pacific Power and Energy Engineering Conference (APPEEC), Shanghai, 27-29 March 2012, pp. 1-4. doi:10.1109/APPEEC.2012.6307479
[8] X. Liu, “Economic Load Dispatch Constrained by Wind Power Availability: A Wait-and-See Approach,” IEEE Transactions on Smart Grid, Vol. 1, No. 3, 2010, pp. 347355. dio:10.1109/TSG.2010.2057458
[9] X. Liu and W. Xu, “Economic Load Dispatch Constrained by WP Availability: A Here-and-Now Approach,” IEEE Transactions on Sustainable Energy, Vol. 1, No. 1, 2010, pp. 2-9. doi:10.1109/TSTE.2010.2044817
[10] J. Hetzer and D. C. Yu, “An Economic Dispatch Model Incorporating Wind Power,” IEEE Transactions on Energy Conversion, Vol. 23, No. 2, 2008, pp. 603-611. doi:10.1109/TEC.2007.914171
[11] D. Villanueva, A. Feijóo and J. Pazos, “Simulation of Correlated Wind Speed Data for Economic Dispatch Evaluation,” IEEE Transactions on Sustainable Energy, Vol. 3, No. 1, 2012, pp. 142-149. doi:10.1109/TSTE.2011.2165861
[12] Y. Fang, D. Zhao, M. Ke, X. Zhao, C. Herbert and K. Wong, “Quantum-Inspired Particle Swarm Optimization for Power System Operations Considering Wind Power Uncertainty and Carbon Tax in Australia,” IEEE Transactions on Industrial Informatics, Vol. 8, No. 4, 2012, pp. 880-888. doi:10.1109/TII.2012.2210431
[13] G. S. Piperagkas, A. G. Anastasiadis and N. D. Hatziargyriou, “Stochastic PSO-Based Heat and Power Dispatch under Environmental Constraints Incorporating CHP and Wind Power Units,” Electric Power Systems Research, Vol. 81, No. 1, 2011, pp. 209-218. doi:10.1016/j.epsr.2010.08.009
[14] X. Liu, W. Xu and C. C. Huang, “Economic Load Dispatch with Stochastic Wind Power: Model and Solutions,” Transmission and Distribution Conference and Exposition, New Orleans, 19-22 April 2010, pp. 1-7. doi:10.1109/TDC.2010.5484550
[15] M. Elshahed, M. Elmarsfawy and H. Zeineldin, “A Here-and-Now Stochastic Economic Dispatch with NonSmooth Fuel Cost Function and Emission Constraint,” The Online Journal on Electronics and Electrical Engineering (OJEEE), Vol. 3, No. 4, 2011, pp. 484-489.
[16] M. Elshahed, M. Elmarsfawy and H. Zeineldin, “Dynamic Economic Dispatch Constrained by Wind Power Weibull Distribution: A Here-and-Now Strategy,” World Academy of Science, Engineering and Technology Journal, Vol. 56, 2011, pp. 384-389.
[17] M. Elshahed, M. Elmarsfawy and H. Zeineldin, “A New Economic Dispatch Constrained by Correlated Weibull Probability Distribution Model for Wind Power,” IEEE PES Conference on Innovative Smart Grid TechnologiesME, Jeddah, 17-20 December 2011, pp. 1-6. doi:10.1109/ISGT-MidEast.2011.6220811
[18] X. Liu and W. Xu, “Minimum Emission Dispatch Constrained by Stochastic Wind Power Availability and Cost,” IEEE Transactions on Power Systems, Vol. 25, No. 3, 2010, pp. 1705-1713. doi:10.1109/TPWRS.2010.2042085
[19] M. A. Abido, “Environmental/Economic Power Dispatch Using Multi-Objective Evolutionary Algorithms,” IEEE Transactions on Power Systems, Vol. 18, No. 4, 2003, pp. 1529-1537. doi:10.1109/TPWRS.2003.818693
[20] J. R. Birge and F. Louveaux, “Introduction to Stochastic Programming,” 2nd Edition, Springer, Berlin, 2011.
[21] M. Lefebvre, “Applied Stochastic Processes,” Springer Inc., Berlin, 2007.
[22] A. Papoulis, “Probability, Random Variables and Stochastic Processes,” 3rd Edition, McGraw-Hill Inc., Boston, 1991.
[23] T. A. Saafan, S. H. Moharram, M. I. Gad and S. K. Allah, “A Multi-Objective Optimization Approach to Groundwater Management Using Genetic Algorithm,” International Journal of Water Resources and Environmental Engineering, Vol. 3, No. 7, 2011, pp. 139-149.
[24] M. Basu, “Dynamic Economic Emission Dispatch Using Non-Dominated Sorting Genetic Algorithm-II,” Electrical Power and Energy Systems Journal, Vol. 30, No. 2, 2008, pp. 140-149. doi:10.1016/j.ijepes.2007.06.009
[25] K. Deb, “Multi-Objective Optimization Using Evolutionary Algorithms,” John Wiley & Sons, Ltd., Chichester, 2001.

Copyright © 2023 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.