  The nonlinear programming (NLP) problem to solve distribution-level optimal
power flow (D-OPF) poses convergence issues and does not scale well for
unbalanced distribution systems. The existing scalable D-OPF algorithms either
use approximations that are not valid for an unbalanced power distribution
system or apply relaxation techniques to the nonlinear power flow equations
that do not guarantee a feasible power flow solution. In this paper, we propose
scalable D-OPF algorithms that simultaneously achieve optimal and feasible
solutions by solving multiple iterations of approximate, or relaxed, D-OPF
subproblems of low complexity. The first algorithm is based on a successive
linear approximation of the nonlinear power flow equations around the current
operating point, where the D-OPF solution is obtained by solving multiple
iterations of a linear programming (LP) problem. The second algorithm is based
on the relaxation of the nonlinear power flow equations as conic constraints
together with directional constraints, which achieves optimal and feasible
solutions over multiple iterations of a second-order cone programming (SOCP)
problem. It is demonstrated that the proposed algorithms are able to reach an
optimal and feasible solution while significantly reducing the computation time
as compared to an equivalent NLP D-OPF model for the same distribution system.
