  Optimal power flow (OPF) is considered for microgrids, with the objective of
minimizing either the power distribution losses, or, the cost of power drawn
from the substation and supplied by distributed generation (DG) units, while
effecting voltage regulation. The microgrid is unbalanced, due to unequal loads
in each phase and non-equilateral conductor spacings on the distribution lines.
Similar to OPF formulations for balanced systems, the considered OPF problem is
nonconvex. Nevertheless, a semidefinite programming (SDP) relaxation technique
is advocated to obtain a convex problem solvable in polynomial-time complexity.
Enticingly, numerical tests demonstrate the ability of the proposed method to
attain the globally optimal solution of the original nonconvex OPF. To ensure
scalability with respect to the number of nodes, robustness to isolated
communication outages, and data privacy and integrity, the proposed SDP is
solved in a distributed fashion by resorting to the alternating direction
method of multipliers. The resulting algorithm entails iterative
message-passing among groups of consumers and guarantees faster convergence
compared to competing alternatives
