Cycle flows and multistabilty in oscillatory networks: an overview

TL;DR

This paper reviews the analysis of phase-locked states in oscillatory networks, emphasizing cycle flows, and introduces methods to compute multiple stable states with implications for power grid stability.

Contribution

It provides a rigorous framework for understanding multistability in oscillatory networks using cycle flows and offers an algorithm to identify multiple phase-locked states.

Findings

Multiple phase-locked states coexist in certain network conditions.

Strong edges and long cycles promote multistability.

The algorithm effectively computes various stable states.

Abstract

The functions of many networked systems in physics, biology or engineering rely on a coordinated or synchronized dynamics of its constituents. In power grids for example, all generators must synchronize and run at the same frequency and their phases need to appoximately lock to guarantee a steady power flow. Here, we analyze the existence and multitude of such phase-locked states. Focusing on edge and cycle flows instead of the nodal phases we derive rigorous results on the existence and number of such states. Generally, multiple phase-locked states coexist in networks with strong edges, long elementary cycles and a homogeneous distribution of natural frequencies or power injections, respectively. We offer an algorithm to systematically compute multiple phase- locked states and demonstrate some surprising dynamical consequences of multistability.