  We consider the synchronization problem of swing equations, a second-order
Kuramoto-type model, on connected networks with inhomogeneous dampings. This
was largely motivated by its relevance to the dynamics of power grids. We focus
on the estimate of the region of attraction of synchronous states which is a
central problem in the transient stability of power grids. In the recent
literature, D\"{o}rfler, Chertkov, and Bullo [Proc. Natl. Acad. Sci. USA, 110
(2013), pp. 2005-2010] found a condition for the synchronization in smart
grids. They pointed out that the region of attraction is an important unsolved
problem. In [SIAM J. Control Optim., 52 (2014), pp. 2482-2511], only a special
case was considered where the oscillators have homogeneous dampings and the
underlying graph has a diameter less than or equal to 2. There the analysis
heavily relies on these assumptions; however, they are too strict compared to
the real power networks. In this paper, we continue the study and derive an
estimate on the region of attraction of phase-locked states for lossless power
grids on connected graphs with inhomogeneous dampings. Our main strategy is
based on the gradient-like formulation and energy estimate. We refine the
assumptions by constructing a new energy functional which enables us to
consider such general settings.
