Stability-constrained Optimization for Nonlinear Systems based on Convex Lyapunov Functions

TL;DR

This paper introduces a scalable convex Lyapunov function-based framework for optimizing nonlinear systems with stability constraints, avoiding large discretization schemes and enabling efficient solutions for power system stability.

Contribution

It develops a novel stability certification method using convex Lyapunov functions and LaSalle's invariance principle, specifically tailored for Lur'e and quasi-polynomial nonlinearities.

Findings

Successfully applied to a 3-generator power network.

Provides a scalable alternative to discretization-based methods.

Enables stability-constrained optimization with fewer algebraic constraints.

Abstract

This paper presents a novel scalable framework to solve the optimization of a nonlinear system with differential algebraic equation (DAE) constraints that enforce the asymptotic stability of the underlying dynamic model with respect to certain disturbances. Existing solution approaches to analogous DAE-constrained problems are based on discretization of DAE system into a large set of nonlinear algebraic equations representing the time-marching schemes. These approaches are not scalable to large size models. The proposed framework, based on LaSalle's invariance principle, uses convex Lyapunov functions to develop a novel stability certificate which consists of a limited number of algebraic constraints. We develop specific algorithms for two major types of nonlinearities, namely Lur'e, and quasi-polynomial systems. Quadratic and convex-sum-of-square Lyapunov functions are constructed for the Lur'e-type and quasi-polynomial systems respectively. A numerical experiment is performed on a 3-generator power network to obtain a solution for transient-stability-constrained optimal power flow.