Multistability in lossy power grids and oscillator networks

TL;DR

This paper introduces a systematic method to analyze the existence and multiplicity of steady states in lossy power grids and oscillator networks, revealing mechanisms of multistability that can impact grid stability.

Contribution

It provides an analytical approach to construct solutions of load-flow equations with Ohmic losses, extending understanding beyond special cases.

Findings

Explicit steady states for elementary networks

Identification of mechanisms leading to multistability

Applicability to coupled oscillator models

Abstract

The stable operation of the electric power grid relies on a precisely synchronized state of all generators and machines. All machines rotate at exactly the same frequency with fixed phase differences, leading to steady power flows throughout the grid. Whether such a steady state exists for a given network is of eminent practical importance. The loss of a steady state typically leads to power outages up to a complete blackout. But also the existence of multiple steady states is undesirable, as it can lead to sudden transitions, circulating flows and eventually also to power outages. Steady states are typically calculated numerically, but this approach gives only limited insight into the existence and (non-)uniqueness of steady states. Analytic results are available only for special network configuration, in particular for grids with negligible Ohmic losses or radial networks without any loops. In this article, we introduce a method to systematically construct the solutions of the real power load-flow equations in the presence of Ohmic losses. We calculate the steady states explicitly for elementary networks demonstrating different mechanisms leading to multistability. Our results also apply to models of coupled oscillators which are widely used in theoretical physics and mathematical biology.