Enhanced Particle Swarm Optimization Approach for Solving the Non-Convex Optimal Power Flow

An enhanced particle swarm optimization algorithm (PSO) is presented in this work to solve the non-convex OPF problem that has both discrete and continuous optimization variables. The objective functions considered are the conventional quadratic function and the augmented quadratic function. The latter model presents non-differentiable and non-convex regions that challenge most gradient-based optimization algorithms. The optimization variables to be optimized are the generator real power outputs and voltage magnitudes, discrete transformer tap settings, and discrete reactive power injections due to capacitor banks. The set of equality constraints taken into account are the power flow equations while the inequality ones are the limits of the real and reactive power of the generators, voltage magnitude at each bus, transformer tap settings, and capacitor banks reactive power injections. The proposed algorithm combines PSO with Newton-Raphson algorithm to minimize the fuel cost function. The IEEE 30-bus system with six generating units is used to test the proposed algorithm. Several cases were investigated to test and validate the consistency of detecting optimal or near optimal solution for each objective. Results are compared to solutions obtained using sequential quadratic programming and Genetic Algorithms.

Authors:

• M. R. AlRashidi
• M. F. AlHajri
• M. E. El-Hawary

Keywords:

• Particle Swarm Optimization
• Optimal Power Flow
• Economic Dispatch.

References:

[1] R. B. Squires, "Economic dispatch of generation directly from power
system voltages and admittances," AIEE Transactions on Power
Apparatus and Systems, vol. PAS-79, no. 3, pp. 1235-1245, 1961.
[2] J. D. Weber and T. J. Overbye, "An individual welfare maximization
algorithm for electricity markets," IEEE Transactions on Power
Systems, vol. 17, no. 3, pp. 590-596, 2002.
[3] Y. He, Y. H. Song, and X. F. Wang, "Bidding strategies based on bid
sensitivities in generation auction markets," IEE Proceedings-
Generation, transmission and Distribution, vol. 149, no. 1, pp. 21-26,
2002.
[4] Y. Valle, G. K. Venayagamoorthy, S. Mohagheghi, J. Hernandez, and R.
G. Harley, "Particle Swarm Optimization: Basic Concepts, Variants and
Applications in Power Systems," IEEE Transactions on Evolutionary
Computation, vol. 12, no. 2, pp. 171-195, 2008.
[5] M. R. AlRashidi and M. E. El-Hawary, "Applications of computational
intelligence techniques for solving the revived optimal power flow
problem," Electric Power Systems Research, vol. 79, no. 4, pp. 694-702,
2009.
[6] C. Wang, H. Yuan, Z. Huang, J. Zhang, and C. Sun, "A modified
particle swarm optimization algorithm and its application in optimal
power flow problem," Proceedings of 2005 International Conference on
Machine Learning and Cybernetics., vol. 5, pp. 2885-2889, Guangzhou,
China, 2005.
[7] R. Ma, P. Wang, H. Yang, and G. Hu, "Environmental/Economic
Transaction Planning Using Multiobjective Particle Swarm Optimization
and Non-Stationary Multi-Stage Assignment Penalty Function," 2005
IEEE/PES Transmission and Distribution Conference and Exhibition:
Asia and Pacific, pp. 1-6, Dalian, China, 2005.
[8] B. Zhao, C. X. Guo, and Y. J. Cao, "Improved particle swarm
optimization algorithm for OPF problems," IEEE/PES Power Systems
Conference and Exposition, pp. 233-238, New York, USA, 2004.
[9] P. N. Biskas, N. P. Ziogos, A. Tellidou, C. E. Zoumas, A. G. Bakirtzis,
and V. Petridis, "Comparison of two metaheuristics with mathematical
programming methods for the solution of OPF," IEE Proceedings-
Generation, transmission and Distribution, vol. 153, no. 1, pp. 16-24,
2006.
[10] S. He, J. Y. Wen, E. Prempain, Q. H. Wu, J. Fitch, and S. Mann, "An
improved particle swarm optimization for optimal power flow,"
International Conference on Power System Technology, vol. 2, pp.
1633-1637, The Pan Pacific, Singapore, 2004.
[11] M. A. Abido, "Optimal power flow using particle swarm optimization,"
International Journal of Electrical Power & Energy Systems, vol. 24, no.
7, pp. 563-571, 2002.
[12] D. C. Walters and G. B. Sheble, "Genetic algorithm solution of
economic dispatch with valve point loading," IEEE Transactions on
Power Systems, vol. 8, no. 3, pp. 1325-1332, 1993.
[13] R. Eberhart and J. Kennedy, "A new optimizer using particle swarm
theory," Proceedings of the Sixth International Symposium on Micro
Machine and Human Science, pp. 39-43, Nagoya, Japan, 1995.
[14] H. Xiaohui, S. Yuhui, and R. Eberhart, "Recent advances in particle
swarm," Proceedings of 2004 Congress on Evolutionary Computation,
vol. 1, pp. 90-97, 2004.
[15] R. C. Eberhart and Y. Shi, "Guest Editorial Special Issue on Particle
Swarm Optimization," IEEE Transactions on Evolutionary Computation,
vol. 8, no. 3, pp. 201-203, 2004.
[16] J. Kennedy and R. C. Eberhart, Swarm Intelligence. San Francisco:
Morgan Kaufmann, 2001.
[17] Y. Shi and R. Eberhart, "A modified particle swarm optimizer," IEEE
World Congress on Computational Intelligence, pp. 69-73, Alaska,
USA, 1998.
[18] G. Coath and S. K. Halgamuge, "A comparison of constraint-handling
methods for the application of particle swarm optimization to
constrained nonlinear optimization problems," The 2003 Congress on
Evolutionary Computation, vol. 4, pp. 2419-2425, Canberra, Australia,
2003.
[19] M. R. AlRashidi and M. E. El-Hawary, "Hybrid Particle Swarm
Optimization Approach for Solving the Discrete OPF Problem
Considering the Valve Loading Effects," IEEE Transactions on Power
Systems, vol. 22, no. 4, pp. 2030-2038, 2007.