Sufficient Conditions for Exact Semi-definite Relaxation of Optimal Power Flow in Unbalanced Multiphase Radial Networks

This paper proves that in an unbalanced multi-phase network with a tree topology, the semi-definite 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.