## 4 Conclusions  
The present study presented a comprehensive framework for applying PINNs to state estimation in unbalanced distribution power systems. By embedding physical constraints--specifically current consistency derived from power flow equations--directly into the training process, the proposed method effectively integrates the strengths of data-driven learning and physical modeling. Evaluation across four IEEE benchmark feeders (13, 34, 37, and 123 buses) demonstrates that the PINN consistently outperforms baseline neural networks trained without physical supervision. Specifically, the model achieves up to a 97.9% reduction in current RMSE and up to a 66.2% reduction in voltage MAE, while also exhibiting up to a 95% improvement under 5% noise and over a 93% improvement in current estimation under 20% input dropout, showcasing its robustness under realistic, imperfect conditions.  
In addition to its accuracy, the PINN demonstrates rapid convergence--reducing both voltage and current errors significantly within the first 20 epochs--and provides fast inference during deployment through a single forward pass. These attributes make it highly suitable for real-time applications in distribution system monitoring.  
Future research directions include incorporating prior knowledge about load and zero-injection buses, either through masking mechanisms or structured supervision, to further improve prediction quality in poorly observable areas. Additional enhancements may come from dynamic or adaptive loss weighting schemes, which allow the network to balance data fidelity and physics consistency during training. Another promising direction involves extending the framework to jointly estimate network topology and line impedances by treating the admittance matrix as a learnable parameter, provided that appropriate sparsity constraints and structural priors are introduced. Furthermore, future work may explore architectures that directly predict both voltages and currents, rather than reconstructing currents from voltages via physical constraints. Such formulations could leverage richer supervision signals and potentially yield further improvements in performance.  
The proposed PINN approach offers a scalable, interpretable, and high-performing solution for distribution system state estimation, paving the way toward more intelligent, resilient, and data-efficient energy networks. However, it is important to note that the current study is based entirely on synthetic datasets generated through OpenDSS simulations. Although these datasets were designed to reflect realistic operational variability, the introduced noise and observability constraints do not fully capture the complexities of actual measurement noise and field conditions. As such, the conclusions regarding real-world applicability should be interpreted in the context of simulated environments. Future work will aim to validate the framework using real measurement data to further assess its practical deployment potential.  
Conceptualization, P.I. and S.P.; methodology, P.I.; software, S.P.; writing--original draft preparation, P.I.; writing--review and editing, A.S.; supervision, A.S. and E.K.; project administration, D.R. All authors have read and agreed to the published version of the manuscript.  
This research was supported by the PANNEL (Physics-aware neural networks and edge processing for low voltage systems) project, which has received funding from the European Union under the call oc1\(-\)2024\(-\)TIS\(-\)01, issued and implemented by the ENFIELD project, under the grant agreement No. 101120657.  
Not applicable.  
The raw data supporting the conclusions of this article will be made available by the authors on request.