  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.
