Hysteretic transitions in the Kuramoto model with inertia

TL;DR

This paper investigates how inertia affects the synchronization transition in the Kuramoto model, revealing hysteresis, cluster formation, and oscillations, with implications for power grid stability.

Contribution

It extends mean field theory to analyze inertia-induced hysteretic transitions and cluster coexistence in the Kuramoto model with both fully coupled and diluted networks.

Findings

Hysteretic transition occurs for large inertia.

Clusters of synchronized oscillators coexist during transition.

Application to power grids shows emergence of quasi-periodic oscillations.

Abstract

We report finite size numerical investigations and mean field analysis of a Kuramoto model with inertia for fully coupled and diluted systems. In particular, we examine for a Gaussian distribution of the frequencies the transition from incoherence to coherence for increasingly large system size and inertia. For sufficiently large inertia the transition is hysteretic and within the hysteretic region clusters of locked oscillators of various sizes and different levels of synchronization coexist. A modification of the mean field theory developed by Tanaka, Lichtenberg, and Oishi [Physica D, 100 (1997) 279] allows to derive the synchronization profile associated to each of these clusters. We have also investigated numerically the limits of existence of the coherent and of the incoherent solutions. The minimal coupling required to observe the coherent state is largely independent of the system size and it saturates to a constant value already for moderately large inertia values. The incoherent state is observable up to a critical coupling whose value saturates for large inertia and for finite system sizes, while in the thermodinamic limit this critical value diverges proportionally to the mass. By increasing the inertia the transition becomes more complex, and the synchronization occurs via the emergence of clusters of whirling oscillators. The presence of these groups of coherently drifting oscillators induces oscillations in the order parameter. We have shown that the transition remains hysteretic even for randomly diluted networks up to a level of connectivity corresponding to few links per oscillator. Finally, an application to the Italian high-voltage power grid is reported, which reveals the emergence of quasi-periodic oscillations in the order parameter due to the simultaneous presence of many competing whirling clusters.