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Adversarially-Informed Node Criticality Identification in Power Grid Measurements
Koto Omiloli, Olugbenga Moses Anubi
TL;DR
Adversarial measurement corruption motivates identifying which power-grid sensor nodes are most consequential, a gap left by predominantly structural or steady-state methods. The paper combines structured stealthy FDIA generation, H-infinity resilient estimation, and coalition-based permutation ranking, then evaluates the approach on the IEEE 14-bus system. Adversarially identified nodes cause larger frequency, voltage-angle, and net-power deviations than random selections.
Problem
Existing node-criticality methods primarily rely on structural or steady-state analyses and do not explicitly account for adversarial effects on system behavior.
Method
The framework generates bounded, stealthy and effective measurement attacks against an H-infinity resilient estimator and ranks nodes by coalition-based marginal contributions estimated through permutation sampling.
Results
Generator frequency and network power deviations increased by approximately 50% under criticality-based selection, while overall system output increased by about 5%; generator angle deviation was nonzero for critical selection and absent for random selection.
Takeaways & Limitations
Adversarially ranked measurement nodes induce greater system-performance degradation than randomly selected nodes, supporting adversarial considerations in grid vulnerability assessment.
Takeaways & Limitations
Exact coalition evaluation requires n! permutations, so large systems require reduced permutation sampling that trades computational scalability against ranking accuracy.
Abstract
from arXiv · showhide
Power grid state estimation relies on sensor measurements that are increasingly vulnerable to adversarial corruption in cyberphysical environments, potentially leading to significant deviations in system observations. This motivates the need to identify critical measurement nodes whose compromise results in the most severe system-level impact. However, existing node criticality methods primarily rely on structural or steady-state analyses and do not explicitly account for adversarial effects on system behavior. To address this gap, this paper proposes an adversarially informed framework for identifying critical measurement nodes in linearized power systems. Within this framework, a structured attack generation mechanism is developed to construct stealthy and effective false data injection attacks (FDIAs) against an H-infinity resilient state estimator. Node criticality is then evaluated using coalition-based marginal contributions of compromised sensor subsets, estimated via permutation sampling over a prescribed set of admissible nodes, with the resulting importance scores mapped to the corresponding physical buses. Simulation results on the IEEE 14-bus system show that adversarially identified nodes induce larger deviations in frequency, voltage angle, and net power compared to randomly selected nodes, demonstrating the effectiveness of the proposed framework.
I. Introduction
Power-grid measurement corruption creates a need for critical-node identification that explicitly links adversarial attacks to state-estimation and system impacts. The paper proposes an adversarial framework integrating dynamics, resilient estimation, structured attacks, and coalition-based node ranking.
- Motivation: Cyber–physical integration improves observability but exposes measurement-based estimation and control to adversarial manipulation.Localized sensor corruption can propagate through system dynamics, degrading estimation and potentially causing incorrect control actions or instability.
- Research gap: Existing methods mainly use topology, steady-state sensitivity, or predefined dynamic scenarios without unifying these perspectives or quantifying estimation effects.They also typically model adversarial effects at the system level rather than linking measurement corruption to node-level criticality.
- Proposed framework: The proposed framework evaluates node importance through stealthy measurement attacks against an H∞ state estimator and coalition-based marginal contributions.A structured attack-generation scheme produces bounded perturbations, while the marginal-contribution metric ranks sensor nodes by impact.
- Contributions: The framework unifies linearized power-system dynamics, resilient state estimation, and adversarial measurement corruption for node criticality assessment.Its vulnerability metric measures marginal contributions from compromised sensor subsets under adversarial conditions.
- Validation: IEEE 14-bus simulations show adversarially ranked nodes produce substantially greater dynamic-performance degradation than randomly selected nodes.The paper presents this as comprehensive simulation validation of the proposed criticality assessment.
II. Notation
The notation defines real-valued spaces, scalar, vector, and matrix conventions, sensor-node indexing, row-restricted matrices, Gaussian notation, and the LeakyReLU activation.
- Basic notation: R and R^n denote the real numbers and n-dimensional real-valued vectors, respectively.Scalars use lowercase letters, vectors use bold lowercase letters with entries x_i, and matrices use uppercase letters.
- Distribution notation: N(0, I) denotes a Gaussian distribution with zero mean and identity covariance matrix.The same notation passage also defines the zero vector, all-ones vector, and identity matrix conventions.
- Activation function: LeakyReLU is defined piecewise as x for x ≥ 0 and αx for x < 0, with 0 < α ≪ 1.This specifies a small nonzero negative-side slope.
III. Grid Network Modelling
The grid model is a linearized networked dynamical system formed from generator swing and power-flow equations around a steady-state operating point under small-signal and DC power-flow assumptions. Its states describe generator angles, frequency deviations, and load-bus voltage angles, while inputs and outputs represent power quantities and measurements.
- Model formulation: The model linearizes generator swing and power-flow equations around a steady-state operating point under small-signal and DC power-flow assumptions.It is adapted from prior work and extended to include dynamic load models.
- States: The state vector contains generator rotor angles, generator frequency deviations, and load-bus voltage angles.Its dimension is given as 2n_g + n_l.
- Inputs: The input vector represents mechanical power injections and active loads.These inputs correspond to generator mechanical power and load demand components.
- Outputs and attacks: The outputs include measurements and net power injections, with measurements subject to corruption by measurement attacks.The passage identifies net power injections as p_net ∈ R^n_b and attacked measurements as y_a.
- Physical parameters: M, D_g, and D_l denote generator inertia, generator damping, and load damping matrices, respectively.The matrices are defined diagonally using generator and load parameters.
- Network representation: The network is represented by the Laplacian L = diag(B1) − B, with generator buses ordered before load buses.B is the network susceptance matrix.
IV. Resilient Network State Estimation
The section develops a discrete-time Luenberger observer with an H∞ performance requirement that attenuates measurement-attack effects on estimation error. Observer design uses a dissipation inequality, matrix reformulation, Schur complement, and optimization to recover the observer gain.
- The continuous-time grid dynamics are discretized with sampling period Ts, giving à = I + ATs and B̃ = BTs.
- A Luenberger-type observer estimates grid states, with L designated as the observer gain to be designed.
- The estimation-error dynamics are designed so attack input yᵃ_k is attenuated in the H∞ sense, providing robustness against worst-case measurement attacks.
- The H∞ requirement is imposed through a quadratic Lyapunov candidate and a dissipation inequality, which produce a quadratic form in the estimation error and attack input.
- Block matrices are factorized using Q = PL, then a Schur complement converts the negativity condition into an equivalent matrix constraint for observer design.
- The observer gain is recovered from the optimization variables as L = P^-1Q.
V. Attack Generation Modelling
The attack-generation model constructs sparse, bounded measurement attacks under effectiveness and stealthiness constraints. A generator, discriminator networks, a physics-based grid model, and an attack policy produce feasible per-node attack signals for criticality assessment.
- Structured attack generation models FDIAs as sparse attacks injected into system measurements, with feasible attacks defined for criticality assessment.
- Effectiveness and stealthiness are evaluated using corresponding metrics with thresholds τe and τs.
- The framework restricts attack injection to admissible measurement nodes and measures resulting degradation in state-estimation performance under effectiveness and stealth criteria.
- Six components comprise attack generation: a generator, effectiveness and stealthiness discriminators, a physics-based grid model, an attack policy, and a resilient H∞ estimator.
- Real-time measurements guide discriminator training, while the discriminators supervise the generator to produce feasible attacks.
- Per-node attack signals are parameterized by start time, duration, time span, and magnitude sampled within prescribed admissible ranges.
VI. Node Criticality Assessment
Node criticality is estimated from the marginal vulnerability contributions of compromised sensor coalitions. Permutation sampling averages these contributions into sensor scores that are mapped to physical buses.
- Permutations of admissible nodes generate coalitions incrementally by adding one node at a time.
- For node j, the marginal contribution is computed as v(S)−v(S\{j}), where v(S) measures vulnerability from compromising sensor subset S.
- Coalition vulnerability v(S) selects the maximum impact e_k among generated attack samples satisfying the stealthiness threshold s_k ≤ τs.
- The criticality score ϕ_j averages node j’s marginal contributions across sampled permutations.
- The resulting sensor-importance ranking is mapped to corresponding physical buses for system-level vulnerability assessment.
- Algorithm 1 uses trained generator and discriminators, generator samples, attack-policy parameters, a stealthiness threshold, admissible-node count, and permutation count as inputs.
VII. Simulation Results
The IEEE 14-bus simulations evaluate attack generation and node criticality, then compare system responses under criticality-based and random node selection. Criticality-based attacks produce larger deviations across measured grid signals.
- Test system: The framework was evaluated on a linearized dynamic model of the IEEE 14-bus power system.Network parameters were used to construct the system Laplacian and associated state-space representation.
- Attack generation: Post-training attacks were stealthy and effective on a 25% sample subset, with stealthiness below τs = 0.04 and effectiveness above τe = 0.64.These thresholds were used to assess the generated attack samples.
- Node criticality: The top-ranked sensor nodes were 8, 3, 4, 24, 7, 22, 23, and 5, indicating unequal criticality across nodes.The scores were computed from sampled permutation-based marginal contributions and mapped to buses.
- Dynamic responses: Attacks targeting critical nodes caused larger grid angle, frequency, and net power deviations than attacks targeting randomly selected nodes.The comparison used response trajectories shown for critical-node and random-node attacks.
- Degradation comparison: Approximately 50% increases occurred for generator frequency and network power deviations, while overall system output ∥y∥ increased by about 5%.For generator angle deviations, random selection produced no impact whereas critical selection produced a nonzero deviation.
- Bus mapping: The five most critical buses were 3, 6, 2, 1, and 8.Bus-level criticality was obtained by linearly combining associated measurement-node values and ranking the resulting scores.
A. Scalability Considerations
The framework’s scalability is constrained by the combinatorial growth of coalition permutations as admissible measurement nodes increase. Reduced permutation sampling and related efficiency strategies trade computational scalability against ranking accuracy.
- Exact evaluation requires n! coalition permutations, which rapidly becomes computationally intractable as the number of admissible nodes grows.
- Reduced permutation sampling evaluates only M ≪ n! permutations to approximate node marginal contributions and lower computational burden.
- Heuristic node pre-screening, coalition size constraints, and parallelized implementation can further improve efficiency for larger cyber–physical grid networks.
VIII. Conclusion
The paper presents an adversarially informed framework that integrates network dynamics, structured attack generation, and resilience-aware state estimation for critical measurement-node identification. Its results show that adversarially identified nodes cause greater system-performance degradation than random selections, while the simulation code is publicly available.
- The framework jointly integrates network dynamics, structured attack generation, and resilience-aware state estimation to identify critical measurement nodes.
- Unlike topology- or flow-based ranking, the approach accounts for worst-case measurement corruption under stealth and effectiveness constraints.
- Adversarially identified nodes induce higher system-performance degradation than randomly selected nodes, supporting adversarial considerations in grid vulnerability assessment.
- The source code used to generate the simulation results is publicly available at the stated GitHub repository.