Optimal Power Flow Pursuit via Feedback-based Safe Gradient Flow

This paper considers the problem of controlling inverter-interfaced distributed energy resources (DERs) in a distribution grid to solve an AC optimal power flow (OPF) problem in real time. The AC OPF includes voltage constraints, and seeks to minimize costs associated with the economic operation, power losses, or the power curtailment from renewables. We develop an online feedback optimization method to drive the DERs' power setpoints to solutions of an AC OPF problem based only on voltage measurements (and without requiring measurements of the power consumption of non-controllable assets). The proposed method - grounded on the theory of control barrier functions - is based on a continuous approximation of the projected gradient flow, appropriately modified to accommodate measurements from the power network. We provide results in terms of local exponential stability, and assess the robustness to errors in the measurements and in the system Jacobian matrix. We show that the proposed method ensures anytime satisfaction of the voltage constraints when no model and measurement errors are present; if these errors are present and are small, the voltage violation is practically negligible. We also discuss extensions of the framework to virtual power plant setups and to cases where constraints on power flows and currents must be enforced. Numerical experiments on a 93-bus distribution system and with realistic load and production profiles show a superior performance in terms of voltage regulation relative to existing methods.