Current-Voltage Formulation of the Unbalanced Optimal Power Flow Problem

State estimation and optimal control approaches in power systems use a variety of sensors, to enable smarter distribution grid management. Such approaches can be cast as optimization models subject to the power flow physics. Kirchhoff’s circuit laws are linear equations in current and voltage phasors for known branch impedance matrices, but nonlinear in variable spaces such as power-voltage. Nevertheless, minimum voltage magnitude constraints, and nonlinear models representing load and generator behavior in the complex power variable space, make the overall feasible set nonlinear and nonconvex. The resulting complex-value equations in current and voltage phasors are presented, as well as their real-value equivalents in rectangular coordinates. The derived formulations and their implementation are validated through numerical experiments w.r.t. Matpower and OpenDSS. An implementation is made publicly available as part of the PowerModels.jl and PowerModelsDistribution.jl toolboxes in the Julia programming language. Numerical illustrations are provided for power flow and optimal power flow problem types using the formulation developed in this paper, evaluated through a nonlinear programming solver.