Optimal Net Load Flattening in Unbalanced Distribution Systems via Rank-Penalized Semidefinite Programming

A formidable challenge that hinders the widespread adoption of renewable energy sources is the potential mismatch between their intermittent supply and the fluctuating demand. This necessitates proper coordination to moderate temporal net load variations while reducing costly curtailment of renewable energy production. By capturing the physical and security constraints of unbalanced distribution systems, this paper formulates a problem to manage various fleets of commercial- and residential-scale distributed energy resources (DERs), i.e., photovoltaics (PVs), deferrable loads (DLs), electric vehicles (EVs), and thermostatically-controlled loads (TCLs). A multi-phase distribution system expanded on the relaxed power flow constraints is considered to account for network awareness. The proposed objective is to minimize hour-to-hour fluctuations of the net load variable, reduce solar energy curtailment, and prioritize preferred EV state of charge and indoor temperature. This objective, however, renders the convex relaxation inexact, wherein positive-semidefinite (PSD) matrices are higher than rank-1. To overcome this issue and therefore enhance the reliability of the solution, we propose to tighten the relaxation constraints via appending the trace of the power flow PSD matrices to the objective function. Multiple case studies on the IEEE 13-bus feeder demonstrate the effectiveness of the proposed problem to optimize the load profile and yield exact solutions.