Joint Sensor Deployment and Physics-Informed Graph Transformer for Smart Grid Attack Detection

Abstract: This paper proposes a joint multi-objective optimization framework for strategic sensor placement in power systems to enhance attack detection. A novel physics-informed graph transformer network (PIGTN)-based detection model is proposed. Non-dominated sorting genetic algorithm-II (NSGA-II) jointly optimizes sensor locations and the PIGTN's detection performance, while considering practical constraints. The combinatorial space of feasible sensor placements is explored using NSGA-II, while concurrently training the proposed detector in a closed-loop setting. Compared to baseline sensor placement methods, the proposed framework consistently demonstrates robustness under sensor failures and improvements in detection performance in seven benchmark cases, including the 14, 30, IEEE-30, 39, 57, 118 and the 200 bus systems. By incorporating AC power flow constraints, the proposed PIGTN-based detection model generalizes well to unseen attacks and outperforms other graph network-based variants (topology-aware models), achieving improvements up to 37% in accuracy and 73% in detection rate, with a mean false alarms rate of 0.3%. In addition, optimized sensor layouts significantly improve the performance of power system state estimation, achieving a 61%--98% reduction in the average state error.

• The paper introduces a joint NSGA-II framework that co-optimizes phasor sensor placement and physics-informed graph transformer training under connectivity, coverage, redundancy, and sensor-count constraints.
• The PIGTN combines graph attention with AC power-flow residuals, achieving approximately 93.3% accuracy, 92.8% F1, 86.5% TPR, and a 0.23% mean false-positive rate across seven benchmark systems.
• The optimized layouts improve robustness to sensor failures and reduce state-estimation voltage errors, although nominal detection gains vary by grid topology and do not yet account for location-specific costs or targeted attacks.

This paper presents a joint multi-objective optimization framework that simultaneously determines phasor sensor placement and trains a physics-informed graph transformer network (PIGTN) for false data injection attack (FDIA) detection in power systems (2603.00085). The central premise is that data-driven detectors are fundamentally bounded by the representativeness of their input data, so the sensing infrastructure and the detection model should not be designed in isolation. The framework couples NSGA-II-based combinatorial search over sensor layouts with closed-loop training of the PIGTN detector, and is evaluated on seven benchmark systems (14, 30, IEEE-30, 39, 57, 118, and 200 bus cases).

Motivation and positioning in the literature

The authors identify two gaps. First, existing graph neural network (GNN)-based FDIA detectors assume full sensor coverage at all buses, which is unrealistic under budgetary constraints; the paper argues that detection performance is contingent on where measurements are collected, an interplay largely ignored by prior work. Second, prior sensor placement studies optimize either for topological or numerical observability—which adversaries can exploit to craft stealthy attacks—or for model-based detection schemes such as sparse subspace projection, but none integrate placement optimization directly with a learned detection model. Against this backdrop, the paper claims to be the first PIGTN-based cyber-attack detector for power systems, extending PINN ideas to graph transformer architectures.

Problem formulation

Sensor deployment is cast as constrained multi-objective optimization over a binary decision vector indicating sensor locations. The objectives minimize both the number of installed sensors and the detection model's loss, subject to weighted constraint violations covering connectivity (enforced via a virtual single-commodity flow formulation), $r$-hop coverage with criticality-weighted penalties, and redundancy ($R_{\min} = 2$ in experiments). NSGA-II handles the constraints through Deb's constraint-domination principle, treating violation as a lexicographic objective ahead of the two cost terms. Notably, the constraint tolerance $\varepsilon$ is not predefined; it emerges from the final Pareto front via a champion-selection rule that prefers feasible solutions, then minimal violation, then minimal summed objective.

A key algorithmic contribution is a hybrid node importance score combining normalized betweenness and eigenvector centrality (topological) with electrical betweenness centrality and electrical coupling degree derived from resistance distance (electrical). This score drives heuristic population seeding (top-$K$ selection plus greedy $r$-hop coverage), diversity-filtered random sampling via Hamming-distance thresholds, and biased mutation, improving the exploration–exploitation balance of the GA. The importance weights were fixed equally at 0.25 based on ablation results.

Physics-informed detection model

The PIGTN processes per-bus measurement vectors comprising voltage/current magnitudes and angles and active/reactive power injections. Its distinguishing feature is an AC power flow residual loss—penalizing deviations from Kirchhoff's laws relating $\hat{P}_i$ to $$\hat{V}_i\hat{I}_i\cos(\hat{\theta}_i - \hat{\delta}_i)$$ and analogously for reactive power—combined with the supervised data loss via coefficients $\lambda_{\mathrm{Data}} = 1$ and $\lambda_{\mathrm{Phy}} = 0.2$. Attacks tampering with local measurements while leaving global AC relations inconsistent produce large residuals that the network learns to separate from benign variability. The authors report that this inductive prior suppresses false positives while recovering true positive rate: the mean false positive rate across all cases is approximately 0.23%, essentially preserving the low FPR of the underlying graph models.

Detection performance

Benchmarked against GraphSAGE, GAT, GCN, graph diffusion networks, plain GTNs, PIGCN, and PIGDN across seven test systems, the PIGTN achieves average accuracy of about 0.933 and F1-score of about 0.928 with TPR around 0.865. Improvements relative to non-physics topology-aware baselines range up to 37% in accuracy and 73% in detection rate (both extremes occurring on the 118-bus case), with improvements of 2.23%–27.5% averaged across cases. Testing used unseen load redistribution attacks with load shift limited to $\tau \leq 20\%$, supporting a generalization claim beyond the random and general attack types seen during training.

Two caveats qualify these results. On the moderately complex 39-bus system most models perform near-identically, so gains there are marginal. Also, the plain GTN baseline performs poorly at scale (F1 of 0.472 on Case 200), suggesting the comparison partly reflects instability of the vanilla transformer variant rather than only the benefit of physics regularization; the strongest baselines remain GraphSAGE and PIGCN.

Placement benchmarking and robustness

Compared against greedy importance-score placement and the Yildiz–Abur sparse PMU placement method, the GA-PIGTN layouts produce more distributed sensors and yield nominal-detection improvements of 5.4%–7.3% TPR in Cases 30, IEEE-30, and 118—but only marginal or slightly negative gains in Cases 14, 39, 57, and 200. The paper attributes this heterogeneity honestly to structural characteristics: dense or highly observable networks render additional sensors redundant, whereas modular, heterogeneous topologies (Case 118) benefit most. An implication worth noting is that joint optimization guarantees no universal accuracy gain; its consistent advantage appears under stress.

Under simulated random sensor failures of up to 30% of buses, the optimized layouts dominate robustness metrics: robustness score improvement of 3.4% versus 1.3%–1.5% for benchmarks, average accuracy improvement of 4.8% versus 3%, average precision of 6.5% versus 3.2%–3.7%, and higher critical failure thresholds ($R_{\min} = 2$0 and 16 sensors for Cases 118 and 200). Thus even where nominal gains are absent, the co-optimized layouts degrade more gracefully—a practically meaningful property for adversarial settings.

State estimation side benefits

Using the optimized sensor layout with a weighted least squares estimator reduces mean voltage angle error by 77%–98% and voltage magnitude error by 61%–94% across test cases. This indicates the framework's outputs serve dual purposes in energy management systems rather than being specialized solely to attack detection.

Limitations and open questions

The paper concedes several limitations. Deployment costs vary by location yet are excluded from the objective function, so the "minimum number of sensors" proxy does not capture realistic budgeting. The evaluation assumes the grid is already observable under existing metering, meaning the framework supplements rather than establishes observability. Robustness testing covers uniform random failures, not targeted or coordinated failures, and the attack set excludes stealthy FDIAs crafted against the specific AC model embedded in the loss—an adversary aware of the physics regularizer could potentially design attacks satisfying Kirchhoff's laws while remaining malicious. Whether the equal weighting of centrality metrics transfers to other networks is settled only by ablation within these seven cases. Finally, adversarial training to harden the detector against sophisticated perturbations remains unexplored.

Conclusion

This work makes a concrete case for co-designing sensor infrastructure and learned detectors: embedding AC power flow constraints into a graph transformer yields strong generalization to unseen load redistribution attacks with near-zero false alarm rates, and NSGA-II-driven placement delivers layouts whose principal value lies in resilience under sensor failure and improved state estimation, even when nominal detection gains are topology-dependent. The main open questions concern cost-aware placement, adversarially robust physics priors, and performance against stealthy attacks tailored to the embedded physical model.