Real-Value Power-Voltage Formulations of, and Bounds for, Three-Wire Unbalanced Optimal Power Flow

Unbalanced optimal power flow refers to a class of optimization problems subject to the steady state physics of three-phase power grids with nonnegligible phase unbalance. Significant progress on this problem has been made on the mathematical modelling side of unbalanced OPF, however there is a lack of information on implementation aspects as well as data sets for benchmarking. One of the key problems is the lack of definitions of current and voltage bounds across different classes of representations of the power flow equations. Therefore, this tutorial-style paper summarizes the structural features of the unbalanced (optimal) power problem for three-phase systems. The resulting nonlinear complex-value matrix formulations are presented for both the bus injection and branch flow formulation frameworks, which typically cannot be implemented as-is in optimization toolboxes. Therefore, this paper also derives the equivalent real-value formulations, and discusses challenges related to the implementation in optimization modeling toolboxes. The derived formulations can be re-used easily for continuous and discrete optimization problems in distribution networks for a variety of operational and planning problems. Finally, bounds are derived for all variables involved, to further the development of benchmarks for unbalanced optimal power flow, where consensus on bound semantics is a pressing need. We believe benchmarks remain a cornerstone for the development and validation of scalable and reproducible optimization models and tools. The soundness of the derivations is confirmed through numerical experiments, validated w.r.t. OpenDSS for IEEE test feeders with 3x3 impedance matrices.