  Distribution grids refer to the part of the power grid that delivers
electricity from substations to the loads. Structurally a distribution grid is
operated in one of several radial/tree-like topologies that are derived from an
original loopy grid graph by opening switches on some lines. Due to limited
presence of real-time switch monitoring devices, the operating structure needs
to be estimated indirectly. This paper presents a new learning algorithm that
uses only nodal voltage measurements to determine the operational radial
structure. The algorithm is based on the key result stating that the correct
operating structure is the optimal solution of the minimum-weight spanning tree
problem over the original loopy graph where weights on all permissible
edges/lines (open or closed) is the variance of nodal voltage difference at the
edge ends. Compared to existing work, this spanning tree based approach has
significantly lower complexity as it does not require information on line
parameters. Further, a modified learning algorithm is developed for cases when
the input voltage measurements are limited to only a subset of the total grid
nodes. Performance of the algorithms (with and without missing data) is
demonstrated by experiments on test cases.
