Extended radial Distribution ACOPF Model: Retrieving Exactness Via Convex Iteration

Internalizing the distribution system's fundamental components into the alternating-current optimal power flow (ACOPF) problem is of essence for solution quality and physical viability of control setpoints. In this manuscript, we formulate a problem to minimize the voltage deviations from the nominal value subject to physical constraints of a comprehensive distribution ACOPF model. The model encompasses a mixture of wye and delta loads, distributed energy resources (DERs), and step-voltage regulators (SVRs). We expand upon the branch flow model with non-convex constraints capturing primary-to-secondary SVR voltage relationships as well as rank-one constraints belonging to power flow and delta-connected net injections. Relaxing the constraints renders a semidefinite program (SDP) whose AC feasibility depends on the solution exactness (proximity of all positive semidefinite (PSD) matrices to being rank 1). For the underlying model, three sources of inexactness are identified: (i) the non-monotonic voltage-positioning (VP) objective function, (ii) the relaxed delta-connected load and DER constraints, and (iii) the relaxed constraints for SVRs with continuous and non-uniformly-operated tap positions. We propose to ultimately circumvent this rank conundrum via the application of convex iteration whereby the inexact solution initializes a sequence of rank-constrained problems. The correlation among the previous components allows the convergence to rank-1 solutions. Case studies on three IEEE distribution feeders evince the merits of the proposed problem.