  Distribution networks are usually multiphase and radial. To facilitate power
flow computation and optimization, two semidefinite programming (SDP)
relaxations of the optimal power flow problem and a linear approximation of the
power flow are proposed. We prove that the first SDP relaxation is exact if and
only if the second one is exact. Case studies show that the second SDP
relaxation is numerically exact and that the linear approximation obtains
voltages within 0.0016 per unit of their true values for the IEEE 13, 34, 37,
123-bus networks and a real-world 2065-bus network.
