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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.
The functions of many networked systems in physics, biology, or engineering rely on a coordinated or synchronized dynamics of their constituents. In power grids for example, all generators must run at the same frequency and their phases need to lock to guarantee a steady power flow. Here, we analyze the existence and multitude of states exhibiting this phase locking behaviour. Focusing on edge and cycle flows instead of the nodal phases, we derive rigorous results on the existence and number of such states. Generally, multiple phase-locked states coexist in networks with edges capable of carrying high flows, long elementary cycles, and a homogeneous spatial distribution of natural frequencies or power injections. Utilizing concepts from the graph theory, we derive scaling relations for the number of such states in plane embedded networks. We also offer an algorithm to systematically compute all phase-locked states, both stable and unstable.
