Synchronization of nonuniform Kuramoto oscillators for power grids with general connectivity and dampings

We consider the synchronization problem of swing equations, a second-order nonuniform Kuramoto model, with general connectivity and dampings. This is motivated by its relevance to the dynamics of power grids. As an important topic in power grids, people have been paying special attention to the transient stability which concerns the system's ability to reach an acceptable synchronism after a transient disturbance. For this concern, an important problem is to determine whether the post-fault state is located in the basin of attraction of synchronous states (sync basin). Recently this issue is becoming more and more challenging since the highly stochastic renewable power sources exert more transient disturbances on the power grids with increasing size and complexity. In Dörfler et al (2013 Proc. Natl Acad. Sci. USA 110 2005-10), it was pointed out that the sync basin is an important unsolved problem. In Li et al (2014 SIAM J. Control Optim. 52 2482-511), an explicit estimate on the region of attraction of coupled oscillators with homogenous damping for network with diameter less than two was obtained. However, it turns out that these assumptions are too restrictive in many real situations. The purpose of this work is to study the emergence of synchronization and give an estimate for sync basin for the nonuniform Kurmaoto model on connected graphs with general dampings, which is the most general setting for a connected power grid. Our strategy is based on the gradient-like formulation and energy estimate.