Perturbations of Weakly Resonant Power System Electromechanical Modes

Abstract

Abstract-It is not uncommon for oscillatory electric power system modes to move close to a resonance in which eigenvalues coincide. In a weak resonance, the modes are decoupled and the eigenvalues do not interact. We analyze general perturbations of a weak resonance and find two distinct behaviors, including interactions near strong resonances in which the eigenvalues quickly change direction. The possible perturbations are illustrated with interactions between electromechanical modes in a 4-generator power system. Some of the interactions are similar to subsynchronous resonance and can lead to oscillatory instability.

Modifying eigenvalue interactions near weak resonance

“…In particular, the condition for coincident and resonant eigenvalues is µ = 0. Moreover, the relative direction of the eigenvalues is ∠(λ 1 −λ 2 ) = 1 2 ∠µ+kπ, for some integer k. Dobson [5] shows that close to a weak resonance, the locus t → µ(t) for real t describes to second order a parabola in the complex plane passing near the origin. In this paper, we propose a method to modify the eigenvalue movement.…”

“…Suppose that the two eigenvalues resonate weakly for some t, say t = 0. Dobson [5] shows that this implies µ t (0) = µ(0) = 0, where the subscript denotes differentiation. A weak resonance is thus a root of µ of multiplicity at least two.…”

“…It is known that there are two types of perturbations from a weak resonance : type A and type B [5]. In type A the eigenvalues do not pass each other as the parameter is varied and in type B, the eigenvalues pass each other.…”

“…As parameters vary, the power system modes vary and it is not uncommon for two complex eigenvalues to pass near resonance. For a comprehensive review up to 2000 see [3] and for more recent work see [6,7,4,5]. Only a subset of oscillatory power system problems involve resonant eigenvalues and there are * Vincent Auvray is a visiting scholar at the University of Wisconsin and a F.N.R.S.…”

“…Similar interactions cause subsynchronous resonance instabilities [4] and instabilities in models of interarea oscillations [5]. Let δ be the maximum separation between the eigenvalues in damping (maximum difference in their real parts) as the parameter t is changed.…”