The Impact of Damping in Second-Order Dynamical Systems with Applications to Power Grid Stability

Abstract

We consider a broad class of second-order dynamical systems and study the role of damping as a system parameter in the stability, hyperbolicity, and bifurcation in such systems. We prove a monotonic effect of damping on the hyperbolicity of the equilibrium points of the corresponding first-order system. This provides a rigorous formulation and theoretical justification for some intuitive notions that damping increases stability. To establish this result, we prove a matrix perturbation result for complex symmetric matrices with positive semidefinite perturbations to their imaginary parts, which may be of independent interest. Furthermore, we establish necessary and sufficient conditions for the breakdown of hyperbolicity of the first-order system under damping variations in terms of observability of a pair of matrices relating damping, inertia, and Jacobian matrices, and propose sufficient conditions for Hopf bifurcation resulting from such hyperbolicity breakdown. The developed theory has significant applications in the stability of electric power systems, which are one of the most complex and important engineering systems. In particular, we characterize the impact of damping on the hyperbolicity of swing equations, which is the fundamental dynamical model of power systems, and demonstrate Hopf bifurcations resulting from damping variations.

A Distributed Scheme for Stability Assessment in Large-Scale Structure-Preserving Models via Singular Perturbation

“…Much effort has been devoted to understanding the stability properties of this network-reduced model, e.g. studying its small-signal stability [3], hyperbolicity and bifurcation [4], phase portrait [5], constructing energy functions and Lyapunov functions [6], and using direct methods [7].…”