Cycle flows and multistability in oscillatory networks

Abstract

We study multistability in phase locked states in networks of phase oscillators under both Kuramoto dynamics and swing equation dynamics-a popular model for studying coarse-scale dynamics of an electrical AC power grid. We first establish the existence of geometrically frustrated states in such systems-where although a steady state flow pattern exists, no fixed point exists in the dynamical variables of phases due to geometrical constraints. We then describe the stable fixed points of the system with phase differences along each edge not exceeding π/2 in terms of cycle flows-constant flows along each simple cycle-as opposed to phase angles or flows. The cycle flow formalism allows us to compute tight upper and lower bounds to the number of fixed points in ring networks. We show that long elementary cycles, strong edge weights, and spatially homogeneous distribution of natural frequencies (for the Kuramoto model) or power injections (for the oscillator model for power grids) cause such networks to have more fixed points. We generalize some of these bounds to arbitrary planar topologies and derive scaling relations in the limit of large capacity and large cycle lengths, which we show to be quite accurate by numerical computation. Finally, we present an algorithm to compute all phase locked states-both stable and unstable-for planar networks.

Multistability in lossy power grids and oscillator networks

“…In this article, we studied solutions to the real power load-flow equations in AC transmission grids of general topology with a special focus on the impact of Ohmic losses. Extending our previous work 19 , we constructed an analytical method for computing all load flow solutions, both stable and unstable ones. We demonstrated how to explicitly compute all steady states in two elementary test topologies: a 4-node tree and a 3-node ring.…”

“…We briefly review the analysis of the lossless case to introduce the fundamentals of our approach as well as some notation and methodology. This review mostly follows 19 , but provides some additional examples and results.…”

“…Fortunately, both steps can be operationalized in terms of the cycles of the network 19 . Let start constructing the solution space of Eq.…”

“…In particular for plane networks the winding vector is unique, i.e. two steady states with same winding vector are identical 19 . We summarize these results in the following theorem.…”

“…The situation is more involved for networks of intermediate sparsity such as power transmission grids, which can give rise to multistability 11,19,[21][22][23][24][25] . The existence of multiple steady states in meshed networks can be traced back to the existence of cycle flows, which do not affect the power balance at any node in the grid.…”