Linear Quadratic Phasor Control of Unbalanced Distribution Networks

This paper investigates the feedback control challenge of tracking a voltage phasor target at a given node on an unbalanced three phase distribution network. We derive a quasi-steady state system model, modeling the progression of phasors as a martingale and describing the input sensitivity with the linearization of the unbalanced three phase power flow manifold for the Thevenin equivalent circuit. We establish the convexity of the stable linear feedback matrix region, and investigate the stability of Single Input Single Output Phasor Control (SISOPC) with gains chosen based on the reactance of the Thevenin equivalent circuit. To improve upon SISOPC, we propose the Linear Quadratic Phasor Controller (LQPC), which generates a multiple input multiple output linear feedback matrix which takes into account both the network resistance and the mutual impedances that couple the unbalanced three phase injections. We demonstrate the performance of the LQPC using OpenDSS simulations on an unbalanced three phase distribution network, demonstrating the effectiveness of the LQPC.