Time-varying Projected Dynamical Systems with Applications to Feedback Optimization of Power Systems

TL;DR

This paper develops a framework for analyzing time-varying projected dynamical systems, especially in non-convex, non-smooth contexts like power system optimization, providing conditions for solution existence and practical implications.

Contribution

It generalizes projected dynamical systems to time-varying, possibly non-regular domains, establishing existence conditions for solutions in non-smooth, non-convex settings.

Findings

Existence of Krasovskii solutions under bounded domain contraction.

Applicable to non-regular, piecewise differentiable feasible sets.

Insights for designing feedback optimization in power systems.

Abstract

This paper is concerned with the study of continuous-time, non-smooth dynamical systems which arise in the context of time-varying non-convex optimization problems, as for example the feedback-based optimization of power systems. We generalize the notion of projected dynamical systems to time-varying, possibly non-regular, domains and derive conditions for the existence of so-called Krasovskii solutions. The key insight is that for trajectories to exist, informally, the time-varying domain can only contract at a bounded rate whereas it may expand discontinuously. This condition is met, in particular, by feasible sets delimited via piecewise differentiable functions under appropriate constraint qualifications. To illustrate the necessity and usefulness of such a general framework, we consider a simple yet insightful power system example, and we discuss the implications of the proposed conditions for the design of feedback optimization schemes.