  Modern state and parameter estimations in power systems consist of two
stages: the outer problem of minimizing the mismatch between network
observation and prediction over the network parameters, and the inner problem
of predicting the system state for given values of the parameters. The standard
solution of the combined problem is iterative: (a) set the parameters, e.g. to
priors on the power line characteristics, (b) map input observation to
prediction of the output, (c) compute the mismatch between predicted and
observed output, (d) make a gradient descent step in the space of parameters to
minimize the mismatch, and loop back to (a). We show how modern Machine
Learning (ML), and specifically training guided by automatic differentiation,
allows to resolve the iterative loop more efficiently. Moreover, we extend the
scheme to the case of incomplete observations, where Phasor Measurement Units
(reporting real and reactive powers, voltage and phase) are available only at
the generators (PV buses), while loads (PQ buses) report (via SCADA controls)
only active and reactive powers. Considering it from the implementation
perspective, our methodology of resolving the parameter and state estimation
problem can be viewed as embedding of the Power Flow (PF) solver into the
training loop of the Machine Learning framework (PyTorch, in this study). We
argue that this embedding can help to resolve high-level optimization problems
in power system operations and planning.
