Power Network Dynamics on Graphons

TL;DR

This paper develops a continuum limit model for power grid dynamics on complex networks using graphons, analyzes stability of synchronized states, and highlights potential instability in small-world networks due to topological effects.

Contribution

It introduces a continuum limit integral equation for large power networks on graphons and analyzes the stability of synchronization, revealing topological influences on stability.

Findings

Existence of a well-posed continuum limit for large networks.

Linear stability of synchronized solutions is established.

Small-world network topology can induce near-instability in finite time.

Abstract

Power grids are undergoing major changes from a few large producers to smart grids build upon renewable energies. Mathematical models for power grid dynamics have to be adapted to capture, when dynamic nodes can achieve synchronization to a common grid frequency on complex network topologies. In this paper we study a second-order rotator model in the large network limit. We merge the recent theory of random graph limits for complex small-world networks with approaches to first-order systems on graphons. We prove that there exists a well-posed continuum limit integral equation approximating the large finite-dimensional case power grid network dynamics. Then we analyse the linear stability of synchronized solutions and prove linear stability. However, on small-world networks we demonstrate that there are topological parameters moving the spectrum arbitrarily close to the imaginary axis leading to potential instability on finite time scales.