Singular Hopf Bifurcations in DAE Models of Power Systems
Wieslaw Marszalek, Zdzislaw Trzaska
DOI: 10.4236/epe.2011.31001   PDF    HTML     5,457 Downloads   10,562 Views   Citations


We investigate an important relationship that exists between the Hopf bifurcation in the singularly perturbed nonlinear power systems and the singularity induced bifurcations (SIBs) in the corresponding different- tial-algebraic equations (DAEs). In a generic case, the SIB phenomenon in a system of DAEs signals Hopf bifurcation in the singularly perturbed systems of ODEs. The analysis is based on the linear matrix pencil theory and polynomials with parameter dependent coefficients. A few numerical examples are included.

Share and Cite:

W. Marszalek and Z. Trzaska, "Singular Hopf Bifurcations in DAE Models of Power Systems," Energy and Power Engineering, Vol. 3 No. 1, 2011, pp. 1-8. doi: 10.4236/epe.2011.31001.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] W. Marszalek and Z.W. Trzaska, “Singularity-Induced Bifurcations in Electrical Power Systems,” IEEE Transactions on Power Systems, Vol. 20, No. 1, 2005, pp. 312- 320. doi:10.1109/TPWRS.2004.841244
[2] H. G. Kwatny, R. F. Fischl and C. O. Nwankpa, “Local Bifurcation in Power Systems: Theory, Computation, and Applications,” Proceeding of IEEE, Vol. 83, No. 11, 1995, pp. 1456-1483. doi:10.1109/5.481630
[3] H. G. Kwatny, A. K. Pasrija and L. Y. Bahar, “Static Bifurcations in Electric Power Networks: Loss of Steady- state Stability and Voltage Collapse,” IEEE Transactions on Circuits and Systems, Vol. CAS-33, No. 10, 1986, pp. 981-991.
[4] H. G. Kwatny and X.-M. Yu, “Energy Analysis of Load-Induced Flutter Instability in Classical Models of Electric Power Networks,” IEEE Transactions on Cir- cuits and Systems, Vol.36, No.12, 1989, pp. 1544-1557.
[5] S. Ayasun, C. O. Nwankpa and G. G. Kwatny, “Compu- tation of Singular and Singularly Induced Bifurcation Points of Differential-Algebraic Power System Model,” IEEE Transactions on Circuits and Systems I, Vol. 51, No. 8, 2004, pp. 15251538.
[6] D. J. Hill and I. M. Y. Mareels, “Stability Theory for Dif- ferential/Algebraic Systems with Application to Power System,” IEEE Transactions on Circuits and Systems, Vol. CAS-37, No. 11, 1990, pp. 1416-1423. doi:10.1109/ 31.62415
[7] C. A. Canizares, N. Mithulananthan, F. Milano and J. Reeve, “Linear Performance Indices to Predict Oscilla- tory Stability Problems in Power Systems,” IEEE Trans- actions on Power System, Vol. 19, No. 2, 2004, pp. 1104- 1114. doi:10.1109/TPWRS.2003.821460
[8] I. Dobson, J. Zhang, S. Greene, H. Engdahl and P. W. Sauer, “Is Strong Modal Resonance a Precursor to Power System Oscillations?” IEEE Transactions on Circuits and Systems, Vol. 48, No. 3, 2001, pp. 340-349.
[9] V. Vekatasubrumanian, H. Schattler and J. Zaborszky, “A Stability Theory of Large Differential Algebraic Systems: A Taxonomy,” Report SSM 9201 — Part I, Washington University, St. Louis, 1992.
[10] V. Vekatasubrumanian, H. Schattler and J. Zaborszky, “Analysis of Local Bifurcation Mechanisms in Large Dif- ferential-Algebraic Systems such as the Power System,” Proceedings of 32nd Conference on Decision and Con- trol, San Antonio, December 1993, pp. 3727-3733.
[11] R. E. Beardmore and R. Laister, “The Flow of a Differen- tial-Algebraic Equation near a Singular Equilibrium,” SIAM Journal on Matrix Analysis, Vol. 24, No. 1, 2002, pp. 106-120. doi:10.1137/S0895479800378660
[12] R. E. Beardmore, “The Singularity-Induced Bifurcation and Its Kronecker Normal Form,” SIAM Journal on Ma- trix Analysis, Vol. 23, No. 1, 2001, pp. 1-12.
[13] L. J. Yang and Y. Tang, “An Improved Version of the Singularity Induced Bifurcation Theorem,” IEEE Trans- actions on Automatic Control, Vol. AC-46, No.9, 2001, pp. 1483-1486. doi:10.1109/9.948482
[14] W. Marszalek and S. L. Campbell, “DAEs Arising from Traveling Wave Solutions of PDEs II,” Computers and Mathematics with Applications, Vol. 37, 1999, pp. 15-34. doi:10.1016/S0898-1221 (98)00238-7
[15] R. E. Beardmore, “Double Singularity-Induced Bifurca- tion Points and Singular Hopf Bifurcations,” Dynamics and Stability of Systems, Vol. 15, No. 4, 2000, pp. 319-342.
[16] R. Riaza, “On the Singularity-Induced Bifurcation Theo- rem,” IEEE Transactions on Automatic Control, Vol. AC-47, No. 9, 2002, pp. 1520-1523. doi:10.1109/TAC. 2002.802757
[17] S. L. Campbell and W. Marszalek, “Mixed Symbolic- numerical Computations with General DAEs: An Appli- cations Case Study,” Numerical Algorithms, Vol. 19, No. 1-4, 1998, pp. 85-94. -doi:10.1023/A:1019106507166
[18] S. L. Campbell and W. Marszalek, “DAEs Arising from Traveling Wave Solutions of PDEs I,” Journal of Com- putational Applied Mathematics, Vol. 82, No. 1-2, 1997, pp. 41-58. doi:10.1016/S0377-0427(97)00084-8
[19] K. L. Praprost and K. A. Loparo, “An Energy Function Method for Determiningvoltage Collapse during a Power System Transient,” IEEE Transactions on Circuits Sys- tem, Vol. CAS-41, No. 11, 1994, pp. 635-651.
[20] M. M. Begovic and A. G. Phadke, “Voltage Stability as- Sessment of a Reduced State Vector,” IEEE Transactions on Circuits System, Vol. 5, No. 1, 1990, pp. 198-203. doi: 10.1109/59.49106
[21] R. Riaza, “Double SIB Points in Differential-Algebraic Systems,” IEEE Transactions on Automatic Control, Vol. AC-48, No. 9, 2003, pp. 1625-1629. doi:10.1109/TAC. 2003.817002
[22] Yang Lijun and Zeng Xianwu, “Stability of Singular Hopf Bifurcations,” Journal of Differential Equations, Vol. 206, No. 1, 2004, pp. 30-54. doi:10.1016/j.jde.2004. 08.002
[23] R. Riaza, S. L. Campbell and W. Marszalek, “On Singular Equilibria of Index-1 DAEs,” Circuits, Systems, and Signal Processing, Vol. 19, No. 2, 2000, pp. 131-157. doi: 10.1007/BF01212467
[24] P. Lancaster and M. Tismenetsky, “The Theory of Matri- ces,” London Academic Press, London, 1985, pp. 454- 474
[25] J. Guckenheimer, M. Myers and B. Strumfels, “Comput- ing Hopf Bifurcations,” SIAM Journal on Numerical Analysis, Vol. 34, No. 1, 1997, pp. 1-2. doi:10.1137/S00 36142993253461

Copyright © 2024 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.