Flow-PIN: A Two-Stage Power-Flow-Guided Method for System-Wide Multivariate Profile Inpainting in Distribution Networks

High-quality system measurement data is critical for power distribution system operation. As deep generative models (e.g., GAN, Diffusion, etc.) have been widely studied to solve the missing data restoration problem to enhance the data quality, their results may look "realistic" but not sufficiently "accurate" due to lacking physical guarantees. To address this limitation, a two-stage physics-guided framework, Flow-PIN, is proposed in this paper for system-wide multivariate profile inpainting. The first stage employs a conditional flow matching model, conditioned on topological and correlation graphs, to generate candidate values. A physical penalty is integrated into the loss function to constrain the generative vector field based on grid physical laws. The second stage introduces a topology-aware power-flow-guided refiner that utilizes Laplacian positional encoding to inject topology information into node embeddings. By coupling alternating current power flow equations with a differentiable correlation alignment mechanism, this refiner further corrects numerical deviations. Evaluations on an active distribution network dataset benchmark the proposed framework against ten representative baselines. The results show that Flow-PIN achieves high-fidelity profile inpainting across three dimensions: maximizing numerical accuracy, capturing temporal fluctuation, and preserving spatial topological correlations.