Synchronization and transient stability in power grids based on łojasiewicz inequalities

In this paper we are concerned with a second-order Kuramoto-type model and its synchronization problem. This study was motivated by its significant relation to the power grid systems. The power grids in industry have become increasingly complex and, more and more, power disturbances could be drawn into the system due to the highly stochastic renewable power sources. This poses an increasing challenge for us to investigate its transient stability. In the very recent literature Dörfler, 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 another important problem is the region of attraction of a synchronized solution. The main purpose of this work is to address the transient stability problem and find a region of attraction for a class of stable synchronous states. This trapping region is explicitly expressed in the parameters of the system. Our key insight is to exploit the gradient inequality and the łojasiewicz exponent of the potential function for the oscillators on networks, which reveal a fundamental relation between the potential and its gradient.