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 maxi,j|θi(t)−θj(t)|, where θi is the ith phase angle such that the ensemble approaches complete position synchronization.