Multistability and anomalies in oscillator models of lossy power grids

Abstract

The analysis of dissipatively coupled oscillators is a challenging problem with high stakes in actual applications, such as large scale physical systems. Many standard mathematical methods are not applicable to such systems, due to the lack of symmetry of the network induced by dissipative couplings. Here we emphasize that the synchronization of coupled oscillators can be equivalently interpreted as the problem of flow distribution over a network. Based on these equivalent interpretations, we demonstrate a close correspondence between multiple stable synchronous states and winding cells in systems of dissipatively coupled oscillators. The recently introduced notion of winding cells, associated to a graph, forms a natural winding partition of the n-torus and capture essential characteristics of synchronous states in lossless systems. Leveraging the winding partition of the n-torus, we provide algorithms to compute the synchronous solutions of general networks of coupled oscillators. Furthermore, we identify three paradoxical behaviors of lossy networked systems, to be contrasted with the behavior lossless systems. Namely, we show that loop flows and dissipation can increase the system's transfer capacity, and that dissipation can promote multistability.