  We consider the problem of characterizing and assessing the voltage stability
in power distribution networks. Different from previous formulations, we
consider the branch-flow parametrization of the power system state, which is
particularly effective for radial networks. Our approach to the voltage
stability problem is based on a local, approximate, yet highly accurate
characterization of the determinant of the power flow Jacobian. Our determinant
approximation allows us to construct a voltage stability index that can be
computed in a fully scalable and distributed fashion. We provide an upper bound
on the approximation error, and we show how the proposed index outperforms
other voltage indices that have been recently proposed in the literature.
