On the synchronization of the Kuramoto-type model of oscillators with lossy couplings

We consider the problem of synchronization of coupled oscillators in a Kuramoto-type model with lossy couplings. Kuramoto models have been used to gain insight on the stability of power networks which are usually nonlinear and involve large scale interconnections. Such models commonly assume lossless couplings and Lyapunov functions have predominantly been employed to prove stability. However, coupling conductances can impact synchronization. We therefore consider a more advanced Kuramoto model that includes coupling conductances, and is characterized by nonhomogeneous coupling weights and noncomplete coupling graphs. Lyapunov analysis once such coupling conductances and aforementioned properties are included becomes nontrivial and more conventional energy-like Lyapunov functions are not applicable or are conservative. Small-signal analysis has been performed for such models, but due to the fact that we have convergence to a manifold, stability analysis via a linearization is on its own inconclusive for the nonlinear model. In this paper, we provide a formal derivation using centre manifold theory that if a particular condition on the equilibrium point associated with the coupling conductances and susceptances holds, then the synchronization manifold for the nonlinear system considered is asymptotically stable. Our analysis is demonstrated with simulations.