Invertible Neural Network for Consistent State Estimation in Distribution Grid with Unobservability

State estimation (SE) serves as the basis for monitoring and control, but the performance is challenged when both prior knowledge and observability are limited due to increasing distribution system extension and renewable penetration, typically on system edges. To solve the problem, machine learning approaches have been recently applied to approximate the mapping from measurements to system states for SE. However, such direct approximation of the inverse system process lacks physical consistency with the forward underlying model (i.e., power flow equations), leading to inaccurate or even physically infeasible SE solutions. Thus, we propose a two-way learning method by designing tractably invertible paths in structural neural networks, which build a perfectly matched forward -inverse system model to estimate states. For the physical consistency of learning, we make the most of prior power system knowledge to compensate for information loss against unobservability and contract feasible SE solutions in the inverse learning process. Specifically, the designs are threefold to regularize SE performance: embedding interpretable power flow basis functional, regularizing dimensional homogeneity, and complementing latent variables. We test the SE performance of invertible learning via extensive simulations on IEEE test systems and a utility distribution grid. Numerical results show high accuracy, degradability of the data -driven model, and robustness to data quality issues.