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**Singular Hopf Bifurcations in DAE Models of Power Systems** ()

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.

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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 |

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