The interconnection of quadratic droop voltage controllers is a Lotka-Volterra system: implications for stability analysis

TL;DR

This paper models power network voltage dynamics with quadratic droop controllers as a Lotka-Volterra system, providing new insights into stability conditions and equilibrium behavior in nonlinear positive systems.

Contribution

It establishes the Lotka-Volterra framework for quadratic droop voltage controllers and analyzes stability and equilibrium properties under various assumptions.

Findings

Voltage dynamics form a Lotka-Volterra system

Proved uniform ultimate boundedness of the system

Established conditions for asymptotic stability of equilibrium

Abstract

This paper studies the stability of voltage dynamics for a power network in which nodal voltages are controlled by means of quadratic droop controllers with nonlinear AC reactive power as inputs. We show that the voltage dynamics is a Lotka-Volterra system, which is a class of nonlinear positive systems. We study the stability of the closed-loop system by proving a uniform ultimate boundedness result and investigating conditions under which the network is cooperative. We then restrict to study the stability of voltage dynamics under a decoupling assumption (i.e., zero relative angles). We analyze the existence and uniqueness of the equilibrium in the interior of the positive orthant for the system and prove an asymptotic stability result.