  Distributed renewable generation, elastic loads, and purposeful manipulation
of meter readings challenge the monitoring and control of today's power systems
(PS). In this context, to maintain a comprehensive view of the system in real
time, fast and robust state estimation (SE) methods are urgently needed.
Conventional PSSE solvers typically entail minimizing a nonlinear and nonconvex
least-squares by e.g., the workhorse Gauss-Newton method. Those iterative
solvers however, are sensitive to initialization and may get stuck in local
minima. To overcome these hurdles and inspired by recent image denoising
techniques, this paper advocates a learnable regularization term for PSSE that
uses a deep neural network (DNN) prior. For the resultant regularized PSSE
problem, a "Gauss-Newton-like" alternating minimization solver is first
developed. To accommodate real-time monitoring, a novel end-to-end DNN is
constructed by unrolling the proposed alternating minimization solver.
Interestingly, the power network topology can be easily incorporated into the
DNN by designing a graph neural network (GNN) based prior. To further endow the
physics-based DNN with robustness against bad data, an adversarial DNN training
method is discussed. Numerical tests using real load data on the IEEE $118$-bus
benchmark system showcase the improved estimation and robustness performance of
the proposed scheme compared with several state-of-the-art alternatives.
