  This paper proves that in an unbalanced multi-phase network with a tree
topology, the semidefinite programming relaxation of optimal power flow
problems is exact when critical buses are not adjacent to each other. Here a
critical bus either contributes directly to the cost function or is where an
injection constraint is tight at optimality. Our result generalizes a
sufficient condition for exact relaxation in single-phase tree networks to tree
networks with arbitrary number of phases.
