  Optimal Power Flow (OPF) is an important tool used to coordinate assets in
electric power systems to ensure customer voltages are within pre-defined
tolerances and to improve distribution system operations. While convex
relaxations of Optimal Power Flow (OPF) problems have been proposed for both
balanced and unbalanced networks, these approaches do not provide universal
convexity guarantees and scale inefficiently as network size and the number of
constraints increase. To address these issues, we have recently explored a
novel linearized OPF formulation for unbalanced distribution systems. Our
approach is made possible in part by approximating the ratio of voltages across
phases throughout the network. In this work, we generalize the previous
formulation to allow arbitrary complex numbers to approximate voltage ratios at
different locations in the feeder. Furthermore, we continue the analysis of the
linearized OPF via comparison to results obtained through convex relaxations
and Semi-Definite Programming (SDP). Simulations of IEEE test feeders show that
the proposed formulation produces control decisions that closely approximate
those obtained via SDP relaxations. In a specific case, we show that the
linearized OPF is capable of solving certain problems which cannot be
formulated as SDPs.
