  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 two-bus
one-generator system, the SDP relaxation can have all possible approximation
outcomes, that is (1) SDP relaxation may be exact or (2) SDP relaxation may be
inexact or (3) SDP relaxation may be feasible while the 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 two-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.
