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.
Highlights
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 a steady state flow pattern exists, no fixed point exists in the dynamical variables of phases due to geometrical constraints
We show that long elementary cycles, strong edge weights, and spatially homogeneous distribution of natural frequencies or power injections cause such networks to have more fixed points
Summary
Coupled oscillator models are ubiquitous in science and technology, describing the collective dynamics of various systems on micro- to macro-scale. If the coupling constant K exceeds a critical value Kc, a fraction of the oscillators start to synchronize in the sense that they rotate at the same angular velocity their natural frequencies differ. In this state of partial frequency locking, commonly referred to in the Kuramoto oscillator literature as “partial synchrony,” the phases of parts of the oscillators are ordered, but they are not strictly phase-locked, such that the phase difference of two synchronized oscillators ðhj À h‘Þ is generally small but not constant. The globally phaselocked states are the fixed points of the system For both the Kuramoto model and the power grid model, these states are given by the solutions of the transcendental equations. Partial synchronization in first and second order models was reviewed in Ref. 28
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