Power System Nonlinearity Modal Interaction by the Normal Forms of Vector Fields

  • Zhang, Jing (Huazhong University of Science and Technology(HUST), China) ;
  • Wen, J.Y. (electrical engineering at HUST, China) ;
  • Cheng, S.J. (electrical engineering at HUST, China)
  • Published : 2008.03.01


Because of the robust nonlinear characteristics appearing in today's modern power system, a strong interaction exists between the angle stability and the voltage stability, which were conventionally studied insularly. However, as the power system is a complex unified system, angle instability always happens in conjunction with voltage instability. The authors propose a novel method to analyze this type of stability problem. In the proposed method, the theory of normal forms of vector fields is utilized to treat the auxiliary dynamic system. By use of this method, the interaction between response modes caused by the nonlinearity of the power system can be analyzed. Consequently, the eigenvalue analysis method is extended to cope with performance analysis of the power system with heavy nonlinearity. The effectiveness of the proposed methodology is verified on a 3-bus power system.


  1. K R Padiyar, S S Rao. Dynamic analysis of small signal voltage instability decoupled from angle instability. Electrical Power & Energy Systems, 1996, 18(7): 445-452
  2. C D Vournas, P W Sauer, M A Pai. Relationships between voltage and angle stability of power systems. Electrical Power & Energy Systems, 1996, 18(8): 493-500
  3. K. R. Padiyar, K. Bhaskar. An integrated analysis of voltage and angle stability of a three node power system. Electrical Power and Energy Systems, 2002, 24(6): 489-501
  4. T. J. Overbye, M. A. Pai, P. W. Sauer. Some aspects of the energy function approach to angle and voltage stability analysis in power systems. In: Proceedings of the 31st Conference on Decision and Control. Tucson (Artzons): 1992, 2941-2946
  5. Felix F. Wu, Liu Chen-ching. Characterization of power system small disturbance stability with models incorporating voltage variation. IEEE Trans. on Circuits and Systems, 1986, 33(4): 406-417
  6. Prabha Kundur, John Paserba, Venkat Ajjarapu. Definition and Classification of Power System Stability. IEEE Trans. on Power Systems, 2004, 19(2): 1387-1340
  7. Charles Concordia. Dynamic Performance and Security of Interconnected Systems. IEEE Power Engineering Review, 1992, 12(3): 11-14
  8. N. Kshatriya, U. D. Annakkage, A.M.Gole, I.T.Fernando. Improving the accuracy of normal form analysis. IEEE Transactions on Power Systems, 2005, 20(1): 286-293
  9. Irma Martinez, A.R. Messina, E.Barocio. Perturbation analysis of power systems: effects of second- and third-order nonlinear terms on system dynamic behavior. Electric Power Systems Research, 2004, 71(2): 159-167
  10. Y. Ni, V. Vittal, W. Kliemann. System separation mechanism in neighborhood of relevant type-n UEP using the normal form of vector field. IEEE Trans. on power systems, 1998, 145(2): 139-144
  11. Chih-ming Lin, V. Vittal, W. Kliemann. Investigation of modal interaction and its effects on control performance in stressed power systems using normal forms of vector fields. IEEE Trans. on power systems, 1996, 11(2): 781-787
  12. Li Ying-hui, Zhang Bao-hui, Li Meng. Study on electrical power system stability boundary. Proceedings of the CSEE, 2003, 22(3): 72-77
  13. Xu Dong-jie, He Renmu, Wang Peng, Hu Guoqiang. Analysis of the inter-area mode oscillations using normal form method. Proceedings of IEEE International Conference on Electric Utility Deregulation, Restructuring and Power Technologies, 2004, Vol. 1: 146-150
  14. E. Barocio, A. R. Messina. Application of perturbation methods to the analysis of low frequency inter-area oscillations. Power Engineering Society Summer Meeting, 2000, vol. 3: 1845-1850
  15. R. B. L. Guedes, A.C.P. Martins, L. F. C. Alberto, N. G. Bretas. An extended energy function for voltage collapse analysis considering voltage dependent load models. Power Tech Conference Proceedings, 2003, Vol. 1: 6 pp
  16. Renato B. L. Guedes, Luis F. C. Alberto, and Newton G. Bretas. Power system low-voltage solutions using an auxiliary gradient system for voltage collapse purposes. IEEE Trans. on Power Systems, 2005, 20 (3): 1528-1537

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