  Distribution grids constitute complex networks of lines often times
reconfigured to minimize losses, balance loads, alleviate faults, or for
maintenance purposes. Topology monitoring becomes a critical task for optimal
grid scheduling. While synchrophasor installations are limited in low-voltage
grids, utilities have an abundance of smart meter data at their disposal. In
this context, a statistical learning framework is put forth for verifying
single-phase grid structures using non-synchronized voltage data. The related
maximum likelihood task boils down to minimizing a non-convex function over a
non-convex set. The function involves the sample voltage covariance matrix and
the feasible set is relaxed to its convex hull. Asymptotically in the number of
data, the actual topology yields the global minimizer of the original and the
relaxed problems. Under a simplified data model, the function turns out to be
convex, thus offering optimality guarantees. Prior information on line statuses
is also incorporated via a maximum a-posteriori approach. The formulated tasks
are tackled using solvers having complementary strengths. Numerical tests using
real data on benchmark feeders demonstrate that reliable topology estimates can
be acquired even with a few smart meter data, while the non-convex schemes
exhibit superior line verification performance at the expense of additional
computational time.
