Linearized Three-Phase Optimal Power Flow Models for Distribution Grids with Voltage Unbalance

One of the major challenges to efficiently solving an optimal power flow (OPF) problem for large-scale, unbalanced distribution grids with high penetration of distributed energy resources is computation time. Although various convex relaxations of the OPF problems have been proposed for distribution grids, they do not scale well for large networks or guarantee solutions that are AC feasible. Many relaxations and linear approximations make limiting assumptions on the relationship between phase voltages, making them unsuitable for analysis of voltage unbalance. In this paper, we utilize three linearized formulations that do not make such assumptions to reduce the computational complexity. These linearizations, which are based on first-order Taylor expansion, fixed-point equation and forward-backward sweep, can either directly replace the AC power flow constraints for an approximate solution or be incorporated in an iterative, successive approximation scheme to yield AC feasible solutions upon convergence. For our analysis, we test the scalability and solution quality of the proposed strategies using a large 1833-bus taxonomic feeder. For all three linearizations, the successive approximation scheme converges to high quality, AC feasible solutions with significant improvement in computation time. The best solutions are obtained with the first-order Taylor expansion, but the iterative method based on forward-backward sweep is significantly faster. We further observe that the three linearizations provide better approximation accuracy than Lin3DistFlow when used to directly replace the AC power flow constraints.