Synchronized states of power grids and oscillator networks by convex optimization

Synchronization is essential for the operation of AC power systems: All generators in the power grid must rotate with fixed relative phases to enable a steady flow of electric power. Understanding the conditions for and the limitations of synchronization is of utmost practical importance. In this article, we propose a novel approach to compute and analyze the stable stationary states of a power grid or an oscillator network in terms of a convex optimization problem. This approach allows to systematically compute \emph{all} stable states where the phase difference across an edge does not exceed $\pi/2$.Furthermore, the optimization formulation allows to rigorously establish certain properties of synchronized states and to bound the error in the widely used linear power flow approximation.