Modern power grids are becoming increasingly complex with the integration of heterogeneous distributed energy resources, underscoring the need for accurate and efficient Power Flow Analysis to ensure stability, reliability, and market operations. Existing methods generally rely on iterative numerical techniques (INT) or machine learning (ML). While INT is physically consistent and highly accurate, it can be computationally expensive and vulnerable to slow or non-convergence. ML methods offer faster solutions but often require extensive data, suffer from limited extrapolation capabilities, and lack physical consistency. Physics-informed ML (PIML) bridges these gaps by embedding domain knowledge before, during, and after training. However, current PIML approaches typically do not leverage this full range of opportunities. In this paper, we propose a novel PIML framework for Power Flow Analysis that integrates physical insights at all three stages (pre, in, and post-processing) to achieve superior accuracy and efficiency. Notably, we introduce a new post-processing technique that partitions the power network into its mesh and radial components: the mesh portion is handled via PIML, while the radial portion is efficiently solved with a convex optimization approach informed by the PIML outputs. This approach is efficient with radial topologies, especially in power distribution networks where the radial part is predominant. Experiments on realistic power networks demonstrate that our method outperforms state-of-the-art approaches in both accuracy and computational performance.

The last decade has seen significant changes in the power grid complexity due to the increased integration of multiple heterogeneous distributed energy resources. Accurate and fast power flow analysis tools have then become essential to guarantee grid stability, reliable operation, strategic planning, and market strategies. State-of-the-art approaches to power flow analysis are based on iterative numerical techniques which exhibit high accuracy but slow-, or even no-, convergence. For this reason, researchers have investigated the use of data-driven techniques that, while exhibiting lower accuracy with respect to iterative numerical ones, have the advantage of being extremely fast. To address the lack of accuracy, physics-informed data-driven techniques, i.e., techniques that leverage both the data and domain knowledge to generate simultaneously fast and accurate models, have been proposed. Nevertheless, these works exhibit two main limitations: i) they do not fully leverage the physical knowledge, and ii) they do not fairly compare the different approaches. In this paper, we propose a novel physics informed data-driven model able to address both limitations by fully leveraging the physical knowledge into the data-driven, i.e., constraining the model and augmenting the available data, and proposing a framework able to fairly compare the different approaches proving the actual effectiveness of the proposal. Results on the IEEE 57 realistic power network will support the proposal.