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.

In contemporary industrial applications, predictive models have been pivotal in bolstering production efficiency, product quality, scalability, and cost-effectiveness while promoting sustainability. These predictive models can be constructed solely based on domain-specific knowledge, exclusively on observational data, or by amalgamating both approaches. They are commonly referred to as Full-, Zero-, or Partial-knowledge-based predictive models, respectively. Full-knowledge based models are highly explainable, point-wise accurate, data efficient, computationally demanding in the prediction phase to achieve high accuracy. Zero-knowledge based models are poorly explainable, data hungry, highly accurate on average, computationally demanding in the construction phase but inexpensive the prediction phase. Partial-knowledge based models focus on taking the best of the two worlds. To maintain a focused scope and avoid redundancy with existing literature, our review will primarily delve into the key industrial applications within a narrow context, encompassing sectors such as extraction, chemical processes, manufacturing, transportation, energy, and construction. Our approach entails several steps. Initially, we conducted a meta-review to pinpoint gaps in previously published surveys on Full-, Zero-, and Partial-knowledge-based predictive models for industrial applications. Subsequently, we present a formal analysis of the subject matter, supplemented with illustrative examples to offer valuable insights. The core of our work comprises a review of existing research categorized by specific industrial applications. Finally, we outline the unresolved challenges and future prospects in this burgeoning field of research. We contend that our work serves as a valuable resource, catering to the needs of both young researchers seeking a solid foundation for their studies, industrial practitioners aiming to grasp core concepts and applications, and senior researchers seeking potential real-world applications for their findings.

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.