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

TL;DR

This paper investigates how damping influences the stability and bifurcation behavior of second-order dynamical systems, with a focus on power grid models, providing theoretical insights and conditions for stability changes.

Contribution

It offers a rigorous theoretical framework linking damping to hyperbolicity and stability, including matrix perturbation results and bifurcation conditions, applied to power system models.

Findings

Damping has a monotonic effect on the hyperbolicity of equilibrium points.

Necessary and sufficient conditions for hyperbolicity breakdown are established.

Hopf bifurcations can result from variations in damping.

Abstract

We consider a broad class of second-order dynamical systems and study the impact of damping as a system parameter on 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 the intuitive notion 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 the swing equation model which is the fundamental dynamical model of power systems, and demonstrate Hopf bifurcations resulting from damping variations.