Power Network Dynamics on Graphons

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.