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It has been recently proven that the semidefinite programming (SDP) relaxation of the optimal power flow problem over radial networks is exact under technical conditions such as not including generation lower bounds or allowing load over-satisfaction. In this paper, we investigate the situation where generation lower bounds are present. We show that even for a 2-bus 1-generator system, the SDP relaxation can have all possible approximation outcomes, that is 1) SDP relaxation may be exact, 2) SDP relaxation may be inexact, or 3) SDP relaxation may be feasible while the optimal power flow (OPF) instance may be infeasible. We provide a complete characterization of when these three approximation outcomes occur and an analytical expression of the resulting optimality gap for this 2-bus system. In order to facilitate further research, we design a library of instances over radial networks in which the SDP relaxation has positive optimality gap. Finally, we propose valid inequalities and variable bound tightening techniques that significantly improve the computational performance of a global optimization solver. Our work demonstrates the need of developing efficient global optimization methods for the solution of OPF even in the simple but fundamental case of radial networks.

Inexactness of SDP Relaxation and Valid Inequalities for Optimal Power Flow