Computing Modes of Instability of Parameterized Nonlinear Systems for Vulnerability Assessment
Jinghan Wang, Michael W. Fisher
TL;DR
The paper tackles vulnerability assessment in parameterized nonlinear systems by identifying the nonlinear mode of instability without explicitly locating the Controlling Unstable Equilibrium Point (CUEP). It develops a Jacobian-averaging framework that, as boundary parameters are approached from within the recovery region, yields an average Jacobian $F(p)$ that converges to $A(p^*)=\partial f/\partial x(x^u(p^*),p^*)$, whose unique unstable eigenvalue and eigenvector give the mode of instability. A discrete-time numerical algorithm using $\hat F(p)$ and a carefully defined time horizon $j(p)$ provides provable convergence to the true mode as the integration step $h\to 0$ and as $p\to p^*$. The method is validated on a damped pendulum and the IEEE 9-Bus power system, revealing non-obvious contributions to instability (e.g., which generator dominates) and remaining robust in high-dimensional parameter spaces. This yields a scalable, reliable tool for stability-vulnerability analysis and controller design in complex engineering systems.
Abstract
Engineered systems naturally experience large disturbances that can disrupt desired operation because the system may fail to recover to a stable equilibrium point. It is valuable to determine the mechanism of instability when the system is subject to a particular finite-time disturbance, because this information can be used to improve vulnerability detection, and to design controllers to reduce vulnerability. Often there exists a particular unstable equilibrium point on the region of attraction boundary of the stable equilibrium point such that the unstable eigenvector of the Jacobian at this unstable equilibrium point represents the mode of instability for the disturbance. Unfortunately, it is challenging to find this mode of instability, especially in high dimensional systems, because it is computationally intractable to obtain this particular unstable equilibrium point. This paper develops a novel algorithm for numerically computing the mode of instability for parameterized nonlinear systems without identifying the particular unstable equilibrium point, resulting in a computationally efficient method. The key idea is to first consider the setting where the system recovers, and to average the Jacobian along the system trajectory from the post-disturbance state up until the Jacobian becomes stable. As the system approaches inability to recover, the averaged Jacobians converge to the Jacobian at the particular unstable equilibrium point, and can be used to extract the unstable eigenvector representing the mode of instability. Convergence guarantees are provided for computing the mode of instability, both for the theoretical setting in continuous time, and for the proposed algorithm which relies on numerical integration. Numerical examples illustrate the successful application of the method to identify the mechanism of instability in power systems subject to temporary short circuits.