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Targeted Power System Frequency Attack via the Selection of Maliciously Controlled Inverters

Xiangyu Zou, Betelihem Ashebo, Daniel K. Molzahn

arXiv:2608.28533v1eess.SY

TL;DR

The paper asks which limited subset of IBRs an adversary should compromise to destabilize designated synchronous generators. It formulates this selection and control problem through unstable-mode design and MIQCP optimization, then proposes ranking heuristics. On WSCC 179-bus and ACTIVSg500 cases, the study finds that many device combinations are ineffective, while selected groups can induce unstable target oscillations; the analysis remains limited by its linearization after generator disconnection.

  • Problem

    The paper addresses how an adversary can select a limited subset of IBRs to destabilize predetermined target generators when compromising additional devices requires time and resources.

  • Method

    The paper jointly selects compromised IBRs and malicious controls by designing an unstable eigenstructure, formulating an MIQCP, and applying ranking-based heuristics.

  • Results

    Nonlinear simulations on WSCC 179-bus and ACTIVSg500 show unstable oscillations in target generators, while many compromised-device combinations cannot successfully execute an attack.

  • Takeaways & Limitations

    Attack severity depends on the selected devices, unstable eigenvalue, target set, and number of compromised IBRs.

  • Takeaways & Limitations

    The linearization may break down after generator disconnections, and post-disconnection dynamics are left for future work.

Abstract

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This paper studies how an adversary can execute a power system frequency attack by choosing the most effective subset of inverter-based resources (IBRs) as malicious control nodes. During the attack, the adversary controls the attacking devices to destabilize a group of designated target generators. The attack is designed by introducing an unstable oscillatory mode whose eigenvector has large components at the target generators and small components at the compromised IBRs. We formalize the optimal attacker selection problem and present an equivalent mixed-integer quadratically constrained program (MIQCP). To address this combinatorial nonconvex problem, we develop two heuristic algorithms by introducing a ranking metric. The attack process is demonstrated on a modified WSCC 179-bus system, with results verified through nonlinear dynamic simulations. We show that the proposed heuristics find optimal attacking sets in a majority of evaluated scenarios while significantly reducing the required computational time. Scenarios from the ACTIVSg500 system are used to further support our results. Finally, we discuss how selecting different eigenvalues, targets, and numbers of compromised devices impact the attack's severity.

I. INTRODUCTION

The paper addresses the cybersecurity risk created by increasingly connected IBRs and formulates the unresolved problem of selecting which compromised devices can most effectively destabilize designated generators.

  • Motivation: IBRs expand the power-grid attack surface through digital controllers, communication interfaces, and configurable software settings.Unauthorized manipulation can affect measurements, control logic, or setpoints.
  • Motivation: Coordinated compromise offers more control actions than a single-device breach, but each additional device requires time, access, and resources.This creates a resource-allocation problem for adversaries seeking high-impact devices.
  • Research Gap: Prior malicious-control studies assumed the compromised-device set was fixed, leaving device selection as an unaddressed degree of freedom.The paper identifies this as a combinatorial selection problem.
  • Research Gap: The proposed attack selects IBRs to maximize destabilization of designated synchronous-generator targets using small-signal eigenvalue and eigenvector placement.The objective favors large modal participation at targets and limited participation at compromised IBRs.
  • Research Gap: Unlike controllability-oriented actuator selection, the attacker seeks one unstable mode with large target participation rather than broad system controllability.This distinguishes the paper’s objective from general input-placement methods.

B. Proposed Framework and Contributions

The paper combines detailed generator–IBR dynamic modeling with optimization-based attacker selection, MIQCP reformulation, and ranking heuristics, and evaluates these methods on two test systems.

  • Framework and Contributions: The paper formalizes selection of compromised IBRs and their malicious controls as a combinatorial optimization problem.The objective is to maximize destabilization of a prescribed target-generator set.
  • Evaluation: The methods are demonstrated on the WSCC 179-bus and ACTIVSg500 test cases, with nonlinear dynamical simulations showing unstable target-generator oscillations.The case studies evaluate whether compromised devices can induce the intended instability.
  • Framework and Contributions: An equivalent mixed-integer quadratically constrained program enables the attacker-selection problem to be addressed with state-of-the-art solvers.The MIQCP is presented as a reformulation of the optimization model.
  • Framework and Contributions: Two ranking-based heuristics are developed to reduce the computational burden caused by combinatorial complexity.The paper also presents exhaustive search, mixed-integer optimization, and a row-search heuristic.

B. Linearized System Model

The nonlinear differential-algebraic power-system model is linearized around a power-flow equilibrium to obtain a local state-space approximation for subsequent eigenvalue analysis.

  • Linearization: Linearization defines state and algebraic deviations from the equilibrium operating point as x ∈R8g+2h and y ∈R2g+2b.These variables represent changes in dynamic and algebraic quantities from their steady-state values.
  • Linearization: A first-order approximation produces a linear system whose block matrices contain coefficients from the linearized differential and algebraic equations.The approximation is constructed around a power-flow solution.

A11 A12 A21 A22

The block-matrix linearization is reduced by eliminating algebraic variables, yielding a square system matrix whose eigenvalues determine small-disturbance stability.

  • A11 A12 A21 A22: The block matrices A11, A12, A21, and A22 contain coefficients from the linearized differential and algebraic equations.The algebraic variables are then eliminated using Kron reduction, or a Schur complement.
  • A11 A12 A21 A22: Kron reduction produces an n = 8g+2h square matrix A describing system dynamics around the power-flow solution.This reduced matrix is the basis for the subsequent stability analysis.
  • A11 A12 A21 A22: The eigenvalues of A determine small-disturbance stability, and the attack seeks to move an initially stable system toward instability.The initial condition assumes all eigenvalues have negative real parts.

III. OPTIMIZATION FORMULATION

The paper formulates limited-effort malicious control as selecting IBRs and designing feedback to place an unstable mode that emphasizes designated generator targets. The resulting selection problem is combinatorial and difficult because device-selection variables interact with continuous eigenvector variables.

  • A. Malicious Control Threat Model: The threat model selects m < h IBRs to destabilize a predetermined group of synchronous-generator targets under limited compromise resources.The adversary injects malicious reference signals through the selected devices.
  • A. Malicious Control Threat Model: The state-feedback formulation assumes perfect state information, while local-data implementations can use linear state observers.The paper focuses on state-feedback design despite this assumption.
  • A. Malicious Control Threat Model: The attack analysis assumes linearization remains valid until disconnection, attacking IBR controls do not saturate, and the unstable eigenvalue is chosen separately.These assumptions define the operating scope of the threat model.
  • B. Attacker Selection Problem: An unstable eigenvalue and eigenvector are selected so target states have large components relative to compromised IBR states.This eigenvector assignment shapes transient behavior toward the designated targets.
  • B. Attacker Selection Problem: The attacker maximizes the ratio of target-state participation to attacker-state participation in the unstable mode.A larger ratio indicates more severe target oscillations relative to the compromised devices.
  • B. Attacker Selection Problem: For a fixed eigenvalue, the eigenvector must satisfy the closed-loop eigenvalue equation, and a compatible gain matrix can then be constructed.The gain matrix follows from a full set of independent eigenvectors.
  • B. Attacker Selection Problem: The matrix B encodes which IBR states are compromised, replacing the previously fixed attacking-device set with a decision variable.Its binary-continuous cross-terms substantially increase problem complexity.
  • B. Attacker Selection Problem: Brute-force enumeration evaluates every m-device combination but scales poorly as the number of IBRs and selected devices grows.Each candidate set requires solving the fixed-B optimization.

C. MIQCP Attacker Selection

The nonlinear attacker-selection formulation is reformulated as an equivalent MIQCP by encoding compromised states with binary variables and replacing binary-continuous products with mixed-integer constraints. The normalization constraint remains nonconvex, except in the single-target case.

  • C. MIQCP Attacker Selection: The original formulation is difficult for mixed-integer solvers because it has a rational objective and cross-terms in an equality constraint.An equivalent MIQCP makes the problem compatible with solvers such as Gurobi.
  • C. MIQCP Attacker Selection: Binary variables x identify compromised states, enforce exactly m selected devices, and restrict selections to IBR frequency states.The subspace condition requires unselected entries of (A − ˜λI)v to be zero.
  • C. MIQCP Attacker Selection: The scale-invariant eigenvector objective is normalized by fixing the relevant magnitude, while BB⊤ is replaced with X = diag(x).This removes the denominator and exposes the attacker-selection variables.
  • C. MIQCP Attacker Selection: Auxiliary variables and big-M inequalities replace each binary-continuous product with a linear objective and mixed-integer convex constraints.The auxiliary variable represents the squared eigenvector magnitude when a state is compromised and zero otherwise.
  • C. MIQCP Attacker Selection: The MIQCP retains normalization as its sole nonconvex constraint.With one target machine, that constraint is resolved and the problem becomes mixed-integer convex.
  • C. MIQCP Attacker Selection: Big-M constants are not uniquely determined because eigenvector values are otherwise unbounded, but target-state normalization provides reasonable guidance.The attackers seek large target-state magnitudes, informing the choice of M.
  • C. MIQCP Attacker Selection: For larger systems with many candidate IBRs, both the MIQCP and brute-force search may be computationally impractical.This motivates a heuristic approach based on selecting a single attacker.

IV. ATTACK CAPABILITY RANKING HEURISTIC

The heuristic approach uses the single-attacker case to quantify each IBR’s independent attacking capability, then extends that ranking to the general multi-attacker selection problem.

  • IV. ATTACK CAPABILITY RANKING HEURISTIC: The single-attacker case reduces selection to comparing ratios, which motivates a ranking-based heuristic for selecting multiple attackers.The general heuristic ranks IBRs by their capability when acting independently.

A. Single Attacker Scenario

When only one attacker is allowed, algebraic elimination reduces the selection problem to comparing diagonal-term ratios, yielding an efficient capability metric for each IBR.

  • A. Single Attacker Scenario: With m = 1, the attack input matrix becomes a binary vector and the malicious signal becomes a complex scalar.The formulation is then reduced to a single-device selection problem.
  • A. Single Attacker Scenario: Substituting the eigenvector expression eliminates the continuous variables, leaving binary selection variables constrained to standard basis vectors.The target-state selector is represented through C.
  • A. Single Attacker Scenario: The optimal single-attacker ratio is found efficiently by comparing h matrix-diagonal ratios.This avoids enumerating combinations of multiple devices.
  • A. Single Attacker Scenario: The metric z_i = ( ˆA^HCC⊤ˆA)_ii / ˆA_ii measures how effectively IBR i can execute a malicious attack alone.Higher values identify devices with greater independent attacking capability.

B. The Row-Search Heuristic

The row-search heuristic refines individually ranked attackers through iterative swaps, improving the objective until convergence while avoiding exhaustive enumeration. It is computationally efficient but not guaranteed to find the optimal attacking set because attacker interactions may be missed.

  • Single-best initialization: The “single best” heuristic selects the m IBRs with the strongest individual attack performance.The resulting set is denoted C1.
  • Limitation: The heuristic may be sub-optimal because individually effective IBRs are not necessarily optimal collectively.The single-best formulation omits interactions represented by off-diagonal terms, so partial search can refine its initial candidate set.
  • Iterative refinement: The row-search heuristic starts from C1 and repeatedly swaps selected devices with unselected devices when the swap improves the objective.After evaluating all swaps, it retains the best improvement and repeats the process until no further swap occurs.
  • Convergence and efficiency: The algorithm converges because its objective value monotonically increases at each accepted iteration.It evaluates far fewer combinations than exhaustive brute-force search or the MIQCP when h and m are large.

V. CASE STUDY

The case study evaluates attacker selection and attack formulation on modified WSCC 179-bus and ACTIVSg500 systems. In the WSCC example, the MIQCP identifies a three-IBR attack that produces substantially larger target-generator participation and growing target oscillations, while the authors note a limitation after generator removal.

  • Test systems: ACTIVSg500 provides supporting evaluation using a 500-bus South Carolina system with 60 synchronous generators and 30 vulnerable IBRs.Its dynamic parameters are set within typical ranges.
  • Example attack: For target states T = {57,58}, unstable eigenvalue ˜λ = 0.2 + j5.12, and m = 3, the MIQCP selects attackers C = {262,263,264}.The target and attacker locations are illustrated in Figure 1.
  • Example attack: 6.357 is the optimal objective, meaning target-machine participation is 6.357 times greater than attacker participation.The corresponding target and attacker eigenvector components are reported in Table I.
  • Example attack: The nonlinear DAE simulation produces growing oscillations near 5.12 rad/s, with target swings much larger than attacker swings.The target frequency states and attacker frequency states are shown separately in Figure 2.
  • Scope boundary: The linearized model may not accurately capture dynamics after generator removal, and continued attack evolution remains future work.Generator disconnection changes the original nonlinear system.
  • Attacking-set comparison: The optimal three-device set outperforms all other combinations, while most groups achieve a target-to-attacker ratio below 1.The histogram indicates that attacker selection directly determines whether the designated targets can be destabilized effectively.

C. Evaluation of Attacker Selection Heuristics

The evaluation compares exhaustive, MIQCP, and ranking-based heuristic approaches for selecting attacking IBRs, then examines how eigenvalue and compromise choices affect attack severity. The heuristics generally preserve solution quality while reducing computational burden, but target effects can differ across eigenvalue choices.

  • Heuristic evaluation: The single best heuristic identifies the optimal attacking set in a majority of evaluated scenarios and performs well when it does not.
  • Heuristic evaluation: For larger m, brute-force search slows significantly because it checks more combinations, whereas heuristics consider only a small subset and take about the same time.
  • Heuristic evaluation: The MIQCP solves the problem optimally but is much slower than brute search because its n-dimensional formulation cannot exploit the m-dimensional generalized eigenvalue reduction when n ≫ m.
  • Eigenvalue and target selection: The attack objective peaks near Im(˜λ) = j5.12 for targets {57,58}, where both target states contribute substantially and the original system has a mode near j5.14.
  • Eigenvalue and target selection: For targets {53,58}, the objective peaks at j5.14 but mainly affects state 58, while a smaller peak at j5.69 affects only state 53.
  • Eigenvalue and target selection: Increasing Re(˜λ) decreases the optimal objective, exposing a trade-off between unstable growth rate and target participation relative to attackers.
  • Eigenvalue and target selection: Compromising more IBRs increases attack potency with diminishing returns, producing negligible improvement beyond three devices in this case.

VI. CONCLUSION

The paper formulates compromised-IBR selection and malicious control as a combinatorial optimization problem, presenting solver-based and heuristic approaches. Results across WSCC 179-bus and ACTIVSg500 systems show that attack success depends strongly on device selection and scenario choices, while the formulation retains computational and modeling limitations.

  • VI. CONCLUSION: The study selects compromised IBRs and their malicious controls to destabilize prescribed target generators, using a combinatorial optimization formulation.An alternative MIQCP formulation and a row-search heuristic are also presented.
  • VI. CONCLUSION: Many combinations of compromised devices cannot successfully execute the attack, underscoring the importance of selecting potent attacking groups.The attack-selection process is demonstrated on the WSCC 179-bus network, with supplementary results for ACTIVSg500.
  • VI. CONCLUSION: Attack outcomes are influenced by the selected unstable eigenvalue, target set, and number of compromised IBRs.
  • VI. CONCLUSION: The MIQCP retains a nonconvex normalization constraint whose removal could simplify the problem and improve computational time.Relaxation or decomposition approaches are identified as future directions for resolving this constraint.
  • VI. CONCLUSION: The linearized analysis may break down after substantial system changes, such as synchronous-generator disconnections caused by an attack.Future work would examine post-disconnection dynamics to model more sophisticated attacks.
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