Complete position synchronization in the power grid system

We present a sufficient condition for synchronization of power grid model when the moment of inertia is zero. Unlike the Kuramoto model, the power grid model contains nonlinear terms of the first- and second-order derivatives. We assume that the natural frequencies are all the same and larger than coupling strength K > 0. These conditions allow us to use a Lyapunov functional and a priori estimate method. From the Lyapunov functional, we can obtain the exponential decay result for the phase difference max(i,j)vertical bar theta(i)(t) - theta(j)(t)vertical bar, where theta(i) is the ith phase angle such that the ensemble approaches complete position synchronization. (C) 2018 Elsevier Ltd. All rights reserved.