  We consider the problem of quantifying and assessing the steady-state voltage
stability in radial distribution networks. Our approach to the voltage
stability problem is based on a local, approximate, and yet highly accurate
characterization of the determinant of the Jacobian of the power flow equations
parameterized according to the branch-flow model. The proposed determinant
approximation allows us to construct a voltage stability index that can be
computed in a fully distributed or in a hierarchical fashion, resulting in a
scalable approach to the assessment of steady-state voltage stability. Finally,
we provide upper bounds for the approximation error and we numerically validate
the quality and the robustness of the proposed approximation with the IEEE
123-bus test feeder.
