We are at the cusp of a historical transformation of our power systems into a more sustainable, dynamic, intelligent, and distributed form with hundreds of millions of intelligent distributed energy resources. The optimal power flow (OPF) problem underlies numerous system operation and planning applications. It is a nonconvex problem and computationally hard because of the nonlinear power flow equations. In this talk, I will describe two approaches to deal with this nonconvexity.

We first summarize main results on solving the OPF through semidefinite relaxations. These algorithms, as well as almost all traditional OPF algorithms, are offline and therefore not suitable for real-time control and optimization of distributed energy resources. Moreover, the grid implicitly solves power flow equations in real-time at scale for free.

This motivates the second approach that explicitly exploits the network as a power flow solver to carry out part of our optimization algorithm. This approach naturally adapts to evolving network conditions. Specifically, we present an algorithm that adapts controllable devices and interacts continuously with the grid which computes a power flow solution given a control action. Collectively these devices and the grid implement a gradient projection algorithm in real time. We characterize optimality and tracking properties of the algorithm. We apply this idea to a unified frequency controller at a fast timescale that integrates primary frequency regulation, secondary frequency regulation, and congestion management. We prove sufficient conditions under which the algorithm converges to a global optimum.