Topological equivalence of a structure-preserving power network model and a non-uniform Kuramoto model of coupled oscillators

We study synchronization in the classic structure-preserving power network model proposed by Bergen and Hill. We find that, locally near the synchronization manifold, the phase and frequency dynamics of the power network model are topologically conjugate to the phase dynamics of a non-uniform Kuramoto model together with decoupled and stable frequency dynamics. This topological conjugacy implies the equivalence of local synchronization in power networks and in non-uniform Kuramoto oscillators. Hence, we can harness the results available for Kuramoto oscillators to analyze synchronization in power networks. We establish necessary and sufficient conditions for phase synchronization, sufficient conditions for frequency synchronization, and necessary and sufficient conditions for frequency synchronization with a uniform topology. These conditions also extend the results known for the classic first-order Kuramoto model and second-order consensus protocols. Our conditions all share a common physical interpretation: the non-uniformity between real power injections has be compensated by sufficiently strong coupling.