Generating Complete Bifurcation Diagrams Using a Dynamic Environment Particle Swarm Optimization Algorithm

Barrera, Julio; Flores, Juan J.; Fuerte-Esquivel, Claudio

doi:https://doi.org/10.1155/2008/745694

Journal of Artificial Evolution and Applications

On this page

Abstract References Copyright Related Articles

Special Issue

Particle Swarms: The Second Decade

View this Special Issue

Research Article | Open Access

Volume 2008 | Article ID 745694 | https://doi.org/10.1155/2008/745694

Generating Complete Bifurcation Diagrams Using a Dynamic Environment Particle Swarm Optimization Algorithm

Julio Barrera,¹Juan J. Flores,¹and Claudio Fuerte-Esquivel¹

Academic Editor: Jim Kennedy

Received20 Jul 2007

Accepted28 Nov 2007

Published31 Dec 2007

Abstract

A dynamic system is represented as a set of equations that specify how variables change over time. The equations in the system specify how to compute the new values of the state variables as a function of their current values and the values of the control parameters. If those parameters change beyond certain values, the system exhibits qualitative changes in its behavior. Those qualitative changes are called bifurcations, and the values for the parameters where those changes occur are called bifurcation points. In this contribution, we present an application of particle swarm optimization methods for dynamic environments for plotting bifurcation diagrams used in the analysis of dynamical systems. The use of particle swarm optimization methods presents various advantages over traditional methods.

References

E. H. Abed, J. C. Alexander, H. Wang, A. M. A. Hamdan, and H. C. Lee, “Dynamic bifurcations in a power system model exhibiting voltage collapse,” Tech. Research Report, Systems Research Center, Maryland University, College Park, Md, USA, February 1992.
View at: Google Scholar
E. H. Abed and P. P. Varaiya, “Nonlinear oscillations in power systems,” International Journal of Electrical Power and Energy System, vol. 6, no. 1, pp. 37–43, 1984.
View at: Publisher Site | Google Scholar
V. Ajjarapu and B. Lee, “Bifurcation theory and its application to nonlinear dynamical phenomena in an electrical power system,” IEEE Transactions on Power Systems, vol. 7, no. 1, pp. 424–431, 1992.
View at: Publisher Site | Google Scholar
J. Barrera and J. J. Flores, “Multimodal optimization by decomposition of the search space in regions,” in Proceedings of the 7th International Conference on Intelligent System Design and Applications (ISDA '07), pp. 863–868, Rio de Janeiro, Brazil, October 2007.
View at: Google Scholar
C. A. Canizares, “On bifurcations, voltage collapse and load modeling,” IEEE Transactions on Power Systems, vol. 10, no. 1, pp. 512–522, 1995.
View at: Publisher Site | Google Scholar
H. D. Chiang, T. P. Connen, and A. J. lueck, “Bifurcation and chaos in electric power systems,” Journal of the Franklin Institute, vol. 33, no. 6, pp. 1001–1036, 1994.
View at: Publisher Site | Google Scholar
M. Clerc and J. Kennedy, “The particle swarm-explosion, stability, and convergence in a multidimensional complex space,” IEEE Transactions on Evolutionary Computation, vol. 6, no. 1, pp. 58–73, 2002.
View at: Publisher Site | Google Scholar
T. V. Cutsem and C. Vournas, Voltage Stability of Electric Power Systems, Power Electronics and Power System Series, Springer, New York, NY, USA, 1998.
View at: Google Scholar
D. J. Hill, “Special issue on nonlinear phenomena in power systems: theory and practical impli,” Proceedings of the IEEE, vol. 83, no. 11, p. 1439, 1995.
View at: Publisher Site | Google Scholar
E. Doedel, A. Champneys, T. Fairgrieve, Y. Kuznetsov, B. Sandstede, and X. Wang, “Auto97: continuation and bifurcation software for ordinary differential equations,” 1997, User's Guide, Concordia University, Montreal, Canada.
View at: Google Scholar
B. Ermentrout, 2006, Xppaut5.96—the differential equations tool.
W. J. F. Govaerts, Numerical Methods for Bifurcations of Dynamic Equilibria, SIAM, Philadelphia, Pa, USA, 2000.
View at: Google Scholar
IEEE Working Group, “Voltage stability of power systems: concepts, analytical tools, and industry experience,” Tech. Rep. IEEE 90 TH 0358-2-PWR, IEEE Power Systems Engineering Committee, New York, NY, USA, 1990.
View at: Google Scholar
J. S. T. Thorp, I. Dobson, and H. D. Chiang, “A model of voltage collapse in electric power systems,” in Proceedings of the 27th IEEE Conference on Decision and Control (CDC '88), vol. 3, pp. 2104–2109, Austin, Tex, USA, December 1988.
View at: Publisher Site | Google Scholar
K. Walve, “Modeling of power system components at severe disturbances,” in Proceedings of the 15th International Conference on Large High Voltage Electric Systems, pp. 38–18, Ljubljana, Slowenia, August 1986.
View at: Google Scholar
J. Kennedy and R. C. Eberhart, “Particle swarm optimization,” in Proceedings of IEEE International Conference on Neural Networks (ICNN '95), pp. 1942–1948, Perth, Australia, December 1995.
View at: Google Scholar
J. Kennedy, R. C. Eberhart, and Y. Shi, Swarm Intelligence, Morgan Kaufmann, San Francisco, Calif, USA, 2001.
View at: Google Scholar
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York, NY, USA, 1998.
View at: Google Scholar
H. Kwatny, A. Pasrija, and L. Bahar, “Static bifurcations in electric power networks: loss of steady-state stability and voltage collapse,” IEEE Transactions on Circuits and Systems, vol. 33, no. 10, pp. 981–991, 1986.
View at: Publisher Site | Google Scholar
X. Li, J. Branke, and T. Blackwell, “Particle swarm with speciation and adaptation in a dynamic environment,” in Proceedings of the 8th Annual Conference Genetic and Evolutionary Computation (GECCO '06), vol. 1, pp. 51–58, New York, NY, USA, July 2006.
View at: Publisher Site | Google Scholar
J. -P. Li, M. E. Balazs, G. T. Parks, and P. J. Clarkson, “A species conserving genetic algorithm for multimodal function optimization,” Evolutionary Computation, vol. 10, no. 3, pp. 207–234, 2002.
View at: Publisher Site | Google Scholar
S. W. Mahfoud, Niching methods for genetic algorithms, Ph.D. thesis, Genetic Algorithms Laboratory, University of Illinois at Uraban-Champaign, Urbana, Ill, USA, 1995.
View at: Google Scholar
A. Petrowski, “A new selection operator dedicated to speciation,” in Proceedings of the 7th International Conference on Genetic Algorithms (ICGA '97), T. Bäck, Ed., July, pp. 144–151, Morgan Kaufmann, San Francisco, Calif, USA, 1997.
View at: Google Scholar
S. Richter and R. DeCarlo, “Continuationmethods: theory and applications,” IEEE Transactions on Circuits and Systems, vol. 30, no. 6, pp. 347–352, 1983.
View at: Publisher Site | Google Scholar
R. Seydel, Practical Bifurcation and Stability Analysis From Equilibrium to Chaos, Springer, New York, NY, USA, 2nd edition, 1994.
View at: Google Scholar
S. H. Strogatz and S. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering, Perseus Books, New York, NY, USA, 2000.
View at: Google Scholar

Copyright

Copyright © 2008 Julio Barrera et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

282

Downloads

0

Citations