Scholarly Communications Project

Application of Bifurcation Theory to Subsynchronous Resonance in Power Systems


Ahmad M. Harb

Dissertation submitted to the Faculty of the Virginia Tech in partial fulfillment of the requirements for the degree of



Electrical Eng.


Ali H. Nayfeh, Chair
Lamine M. Mili, Co-Chairman
Arun G. Phadke
H. F. Van Landingham
Dean T. Mook

December 16, 1996
Blacksburg, Virginia


A bifurcation analysis is used to investigate the complex dynamics of two heavily loaded single-machine-infinite-busbar power systems modeling the characteristics of the BOARDMAN generator with respect to the rest of the North-Western American Power System and the CHOLLA$\#$ generator with respect to the SOWARO station. In the BOARDMAN system, we show that there are three Hopf bifurcations at practical compensation values, while in the CHOLLA$\#4$ system, we show that there is only one Hopf bifurcation. The results show that asthe compensation level increases, the operating condition loses stability with a complex conjugate pair of eigenvalues of the Jacobian matrix crossing transversely from the left- to the right-half of the complex plane, signifying a Hopf bifurcation. As a result, the power system oscillates subsynchronously with a small limit-cycle attractor. As the compensation level increases, the limit cycle grows and then loses stability via a secondary Hopf bifurcation, resulting in the creation of a two-period quasiperiodic subsynchronous oscillation, a two-torus attractor. On further increases of the compensation level, the quasiperiodic attractor collides with its basin boundary, resulting in the destruction of the attractor and its basin boundary in a bluesky catastrophe. Consequently, there are no bounded motions. When a damper winding is placed either along the q-axis, or d-axis, or both axes of the BOARDMAN system and the machine saturation is considered in the CHOLLA$\#4$ system, the study shows that,there is only one Hopf bifurcation and it occurs at a much lower level of compensation, indicating that the damper windings and the machine saturation destabilize the system by inducing subsynchronous resonance. Finally, we investigate the effect of linear and nonlinear controllers on mitigating subsynchronous resonance in the CHOLLA$\#4$ system . The study shows that the linear controller increases the compensation level at which subsynchronous resonance occurs and the nonlinear controller does not affect the location and type of the Hopf bifurcation, but it reduces the amplitude of the limit cycle born as a result of the Hopf bifurcation.

Full text (PDF) 2,253,570 Bytes

The author grants to Virginia Tech or its agents the right to archive and display their thesis or dissertation in whole or in part in the University Libraries in all forms of media, now or hereafter known. The author retains all proprietary rights, such as patent rights. The author also retains the right to use in future works (such as articles or books) all or part of this thesis or dissertation.
[ETD main page] [Search ETDs][] [SCP home page] [library home page]

Send Suggestions or Comments to