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arxiv: 2507.05874 · v1 · submitted 2025-07-08 · 💻 cs.LG · cs.SY· eess.SY

Robust Power System State Estimation using Physics-Informed Neural Networks

Solon Falas , Markos Asprou , Charalambos Konstantinou , Maria K. Michael This is my paper

Pith reviewed 2026-05-19 06:11 UTC · model grok-4.3

classification 💻 cs.LG cs.SYeess.SY
keywords physics-informed neural networkspower system state estimationrobustnessdata manipulation attackspower-flow equationsmachine learning for gridscybersecurity in power systems
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The pith

Embedding power-flow equations into neural networks boosts state estimation accuracy in power systems by up to 83 percent on unseen data and 93 percent during attacks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a hybrid method that trains neural networks for power system state estimation by adding the governing power-flow equations directly to the loss function. This approach seeks to deliver more accurate voltage and angle estimates than conventional neural networks, both in normal operation and when measurements are faulty or deliberately altered. A sympathetic reader would care because reliable real-time state estimation underpins grid stability, especially as power systems incorporate more variable renewables and face rising cyber threats. The reported gains on held-out data and entirely new networks suggest the physics constraints help the model generalize and resist manipulation at critical points.

Core claim

By embedding the nonlinear power-flow equations into the neural-network loss function, the resulting physics-informed network produces state estimates whose accuracy exceeds that of an otherwise identical neural network by up to 83 percent on unseen subsets of the training distribution, 65 percent on entirely new system topologies, and 93 percent when a critical bus is subjected to data manipulation.

What carries the argument

Physics-informed neural network whose loss combines conventional data-fitting terms with the power-flow equations that relate bus voltages, angles, and injections.

If this is right

  • Accuracy remains higher on measurement patterns never encountered in training.
  • Performance advantage persists when the test system has a different topology from any training example.
  • Estimation error stays lower than a standard network when a critical bus measurement is falsified.
  • The same embedding yields gains under both normal operation and simple fault conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could reduce the number of sensors required for acceptable accuracy in real-time monitoring.
  • Similar physics embedding might improve state estimation in other networked physical systems such as gas pipelines or water networks.
  • Integration with existing SCADA systems could provide an additional layer of attack detection without extra hardware.
  • The approach may scale to larger transmission networks if the embedded equations are solved efficiently inside the loss.

Load-bearing premise

The power-flow equations are embedded exactly and without approximation error that would break under realistic attack or fault conditions.

What would settle it

Remove the physics loss term entirely and retrain; if the accuracy advantage over the plain neural network disappears on the same test sets and attack scenarios, the central claim is falsified.

Figures

Figures reproduced from arXiv: 2507.05874 by Charalambos Konstantinou, Maria K. Michael, Markos Asprou, Solon Falas.

Figure 1
Figure 1. Figure 1: Model training procedure for the power system state estimation use case. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Scenario 1 (IEEE 14-bus System) - MAE for steady-state conditions using varying λd, λp, λc weights for data, physics, and constants, respectively, in the loss function. TABLE I: Computational Costs. Power System Training Time (s) Inference Time (ms) IEEE 14 34.375 0.03946 IEEE 118 39.709 0.03479 I. All measurements are derived from a single NVIDIA RTX 4000 Ada Generation GPU. B. Test cases and results A se… view at source ↗
Figure 3
Figure 3. Figure 3: Scenario 2 - Testing datasets with varying generator shutdowns. Models trained on a bus 2 shutdown are tested with load spikes at equal (S2.1) and double the initial shutdown magnitude (S2.2) [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Scenario 3 - Testing datasets with varying generation increases. Models trained on a bus 2 ramp-up are tested with scenarios simulating load drops at equal (S3.1) and double the initial increase (S3.2) [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Scenario 4 - Testing datasets for 3-phase fault, with different per-unit resistance and locations. Graphs represent the testing set, plus Scenario S4.1 to S4.5 from left to right. of the model to unknown states of the system [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Scenario 5 - Testing datasets, showing the local and global accuracy of the models during Data Manipulation Attacks. can act as a filter against such erroneous data. Such filtering properties enhance continuous and accurate state monitoring capabilities, even under data communication disruptions. To evaluate the robustness of the IEEE 14-bus system’s state estimation against data manipulation attacks, we t… view at source ↗
Figure 9
Figure 9. Figure 9: Scenario 8 - Testing datasets for 3-phase fault. Graphs represent the testing set, plus Scenario S8.1 [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Scenario 9 - Testing datasets showing the local and global accuracy of the models during Data Manipulation Attacks. the increasing data manipulation magnitude on bus 54. The PINN shows considerable improvements of up to ∼ 76.11% in accurately predicting the system-wide state. 3) Experimental Results Discussion: The PINN has been comprehensively tested in a large variety of datasets and has shown adequate … view at source ↗
Figure 8
Figure 8. Figure 8: Scenario 7 - Testing datasets with generator shutdown. Models trained on a bus 26 shutdown are tested with load spikes at equal (S7.1) the initial shutdown magnitude. different location, bus 33. The PINN generates more accurate results when tested against both the unknown subset of the training dataset and the completely unknown dataset, with a ∼ 37.67% improvement on the testing dataset. Finally, Scenario… view at source ↗
read the original abstract

Modern power systems face significant challenges in state estimation and real-time monitoring, particularly regarding response speed and accuracy under faulty conditions or cyber-attacks. This paper proposes a hybrid approach using physics-informed neural networks (PINNs) to enhance the accuracy and robustness, of power system state estimation. By embedding physical laws into the neural network architecture, PINNs improve estimation accuracy for transmission grid applications under both normal and faulty conditions, while also showing potential in addressing security concerns such as data manipulation attacks. Experimental results show that the proposed approach outperforms traditional machine learning models, achieving up to 83% higher accuracy on unseen subsets of the training dataset and 65% better performance on entirely new, unrelated datasets. Experiments also show that during a data manipulation attack against a critical bus in a system, the PINN can be up to 93% more accurate than an equivalent neural network.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. This paper proposes a hybrid physics-informed neural network (PINN) approach for power system state estimation. Physical laws in the form of power-flow equations are embedded as a residual term in the neural network loss function to improve accuracy and robustness under normal operation, faults, and data manipulation attacks. The central empirical claims are that the method outperforms traditional machine learning models by up to 83% on unseen subsets of the training dataset, 65% on entirely new unrelated datasets, and 93% during a data manipulation attack on a critical bus.

Significance. If the reported gains can be reproduced with full methodological transparency, the work would demonstrate a practical benefit of soft physics constraints for generalization and attack resilience in power-grid monitoring. The approach follows the standard PINN template and does not introduce new theoretical machinery, so its primary value would lie in the empirical validation on realistic grid models and attack scenarios.

major comments (2)
  1. [§4] §4 (PINN formulation) and the loss-function description: the embedding of the power-flow equations is presented only as an additive residual penalty without specifying whether the weighting hyperparameter is fixed or tuned, whether the AC equations are used in full nonlinear form or linearized, and whether any hard constraints or projection steps are applied. This detail is load-bearing for the 93% attack-robustness claim, because a standard soft penalty permits the network to trade off small nonzero physics residuals against fitting corrupted measurements.
  2. [§5.3] §5.3 (attack experiments) and Table reporting attack results: the 93% accuracy improvement is stated without the exact attack model (magnitude and location of manipulated measurements), the architecture and regularization details of the “equivalent neural network” baseline, error bars, or statistical significance tests. Without these controls it is impossible to attribute the gain specifically to the physics term rather than to generic regularization or outlier handling.
minor comments (2)
  1. [Abstract] Abstract: the accuracy metric underlying the 83%, 65%, and 93% figures is never defined (MSE, MAE, voltage-angle error, etc.).
  2. [§2] Notation: the symbols for voltage magnitudes, angles, and power injections are introduced without a consistent table of definitions or reference to standard power-system notation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which highlight areas where additional methodological detail will strengthen the paper. We respond point-by-point to the major comments and commit to revisions that improve transparency without altering the core claims or experimental design.

read point-by-point responses

  1. Referee: §4 (PINN formulation) and the loss-function description: the embedding of the power-flow equations is presented only as an additive residual penalty without specifying whether the weighting hyperparameter is fixed or tuned, whether the AC equations are used in full nonlinear form or linearized, and whether any hard constraints or projection steps are applied. This detail is load-bearing for the 93% attack-robustness claim, because a standard soft penalty permits the network to trade off small nonzero physics residuals against fitting corrupted measurements.

    Authors: We agree that these implementation details were insufficiently specified. In the revised manuscript we will expand Section 4 to state that the weighting hyperparameter is tuned via cross-validation on a validation split, that the residual term employs the full nonlinear AC power-flow equations, and that the formulation uses only a soft penalty with no hard constraints or projection steps. We will also report the specific tuned value and include a short sensitivity study showing how performance varies with the weight, thereby supporting the robustness results under data manipulation. revision: yes

  2. Referee: §5.3 (attack experiments) and Table reporting attack results: the 93% accuracy improvement is stated without the exact attack model (magnitude and location of manipulated measurements), the architecture and regularization details of the “equivalent neural network” baseline, error bars, or statistical significance tests. Without these controls it is impossible to attribute the gain specifically to the physics term rather than to generic regularization or outlier handling.

    Authors: We concur that the attack section requires greater specificity to allow readers to attribute improvements to the physics term. The revised version will specify the attack model (magnitude and exact bus location of the manipulated measurements), confirm that the baseline neural network uses an identical architecture and regularization schedule except for the absence of the physics residual, report standard deviations across repeated runs with different random seeds, and add statistical significance tests (e.g., paired t-tests) comparing the two models. These additions will be placed in Section 5.3 and the associated table. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the proposed PINN method or experimental claims

full rationale

The paper presents a hybrid PINN for power system state estimation by embedding power-flow equations as a residual term in the network loss, following the standard PINN construction of data loss plus physics loss. Performance claims (83% higher accuracy on unseen subsets, 65% on new datasets, 93% under data manipulation attack) are presented as direct experimental outcomes from comparisons against baseline neural networks on specific test cases, not as quantities derived by algebraic reduction or by fitting parameters that are then relabeled as predictions. No equations or steps in the described approach reduce the robustness improvements to self-definitions, fitted inputs called predictions, or load-bearing self-citations; the physical embedding is an independent modeling choice whose effectiveness is assessed externally via held-out data and attack simulations. The derivation chain therefore remains self-contained and does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that standard neural-network training plus a physics residual term produces robust estimates. No new entities are introduced. Hyperparameters of the network and weighting of the physics term are free parameters whose values are not reported.

free parameters (2)
  • physics-loss weighting coefficient
    Controls the trade-off between data fit and satisfaction of power-flow equations; value not stated in abstract.
  • neural-network architecture hyperparameters
    Layer count, width, activation functions, and optimizer settings are chosen but not disclosed.
axioms (2)
  • domain assumption Power-flow equations can be evaluated pointwise inside the neural-network loss without significant discretization error.
    Invoked when the abstract states that physical laws are embedded into the architecture.
  • domain assumption Training and test distributions are sufficiently similar for the reported generalization claims to hold.
    Required for the 65% improvement on entirely new datasets.

pith-pipeline@v0.9.0 · 5690 in / 1473 out tokens · 40186 ms · 2026-05-19T06:11:53.895841+00:00 · methodology

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Reference graph

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