  To perform any meaningful optimization task, power distribution operators
need to know the topology and line impedances of their electric networks.
Nevertheless, distribution grids currently lack a comprehensive metering
infrastructure. Although smart inverters are widely used for control purposes,
they have been recently advocated as the means for an active data acquisition
paradigm: Reading the voltage deviations induced by intentionally perturbing
inverter injections, the system operator can potentially recover the electric
grid topology. Adopting inverter probing for feeder processing, a suite of
graph-based topology identification algorithms is developed here. If the grid
is probed at all leaf nodes but voltage data are metered at all nodes, the
entire feeder topology can be successfully recovered. When voltage data are
collected only at probing buses, the operator can find a reduced feeder
featuring key properties and similarities to the actual feeder. To handle
modeling inaccuracies and load non-stationarity, noisy probing data need to be
preprocessed. If the suggested guidelines on the magnitude and duration of
probing are followed, the recoverability guarantees carry over from the
noiseless to the noisy setup with high probability.
