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

Sufficient Conditions for Exact Semidefinite Relaxation of Optimal Power Flow in Unbalanced Multiphase Radial Networks