  This paper develops a branch-flow based optimal power flow (OPF) problem for
multi-phase distribution networks that allows for tap selection of wye,
closed-delta, and open-delta step-voltage regulators (SVRs). SVRs are assumed
ideal and their taps are represented by continuous decision variables. To
tackle the non-linearity, the branch-flow semidefinite programming framework of
traditional OPF is expanded to accommodate SVR edges. Three types of
non-convexity are addressed: (a) rank-1 constraints on non-SVR edges, (b)
nonlinear equality constraints on SVR power flows and taps, and (c) trilinear
equalities on SVR voltages and taps. Leveraging a practical phase-separation
assumption on the SVR secondary voltage, novel McCormick relaxations are
provided for (c) and certain rank-1 constraints of (a), while dropping the
rest. A linear relaxation based on conservation of power is used in place of
(b). Numerical simulations on standard distribution test feeders corroborate
the merits of the proposed convex formulation.
