  The topology of a power grid affects its dynamic operation and settlement in
the electricity market. Real-time topology identification can enable faster
control action following an emergency scenario like failure of a line. This
article discusses a graphical model framework for topology estimation in bulk
power grids (both loopy transmission and radial distribution) using
measurements of voltage collected from the grid nodes. The graphical model for
the probability distribution of nodal voltages in linear power flow models is
shown to include additional edges along with the operational edges in the true
grid. Our proposed estimation algorithms first learn the graphical model and
subsequently extract the operational edges using either thresholding or a
neighborhood counting scheme. For grid topologies containing no three-node
cycles (two buses do not share a common neighbor), we prove that an exact
extraction of the operational topology is theoretically guaranteed. This
includes a majority of distribution grids that have radial topologies. For
grids that include cycles of length three, we provide sufficient conditions
that ensure existence of algorithms for exact reconstruction. In particular,
for grids with constant impedance per unit length and uniform injection
covariances, this observation leads to conditions on geographical placement of
the buses. The performance of algorithms is demonstrated in test case
simulations.
