Distribution Grid Optimal Power Flow (D-OPF): Modeling, Analysis, and Benchmarking

In the power distribution systems, optimal power flow (D-OPF) is formulated as a non-convex and non-linear programming (NLP) problem. Convex relaxation and linear approximation models have been increasingly adopted to achieve computational efficiency for D-OPF. Despite the benefits of scalability and global optimality, each method is based on certain assumptions, performs differently, and may lead to solutions that are physically not meaningful. In this context, this work numerically evaluates the relative performance of second-order cone programming (SOCP), semi-definite programming (SDP), and linear programming (LP) formulations of D-OPF in terms of their feasibility, optimality, and scalability with respect to NLP-based formulations. We also compare the bus injection (in bus voltage and current variables) and branch flow (in active and reactive power flow variables) based on NLP formulations. The performance is evaluated using small (123-node), medium (730-node), and large (2522-node) sized distribution feeders. Case studies, which are backed up by visualization of the analytical models for the solution space to the extent possible, show that (1) the feasibility and exactness of relaxed D-OPF formulations depend upon the problem type, (2) some NLP formulations are computationally more tractable than others, (3) different NLP formulations can converge to different local solutions, and (4) the approximate linear model may underestimate or overestimate the cost function (depending upon the problem type) and may lead to AC-infeasible solutions.