Resilience Revisited: A Multidimensional Framework Derived from Realistic Attack Scenarios

1. Problem Statement

Modern power systems are increasingly vulnerable to High-Impact, Low-Probability (HILP) events, particularly coordinated cyberattacks targeting inverter-based resources (IBRs) like solar PV. Existing resilience assessment frameworks suffer from three critical limitations:

• Single-Dimensionality: Most metrics focus on a single aspect (e.g., load curtailment or frequency deviation), failing to capture the multifaceted nature of modern threats.
• Independence Assumption: Current models treat system dimensions (physical, cyber, operational) as independent, ignoring cross-dimensional coupling effects. This leads to significant underestimation of degradation (up to 65% in some studies).
• Linear Evolution: Geometric models (e.g., resilience triangles) often assume linear performance recovery, which deviates from the non-linear, cascading behaviors observed in real systems.

The authors argue that accurate vulnerability characterization requires a framework that simultaneously captures degradation across multiple interacting domains and distinguishes between internal system dynamics and external contextual amplifiers.

2. Methodology: The Multidimensional Resilience Index (MDRI)

The paper proposes the MDRI, a framework that decomposes system degradation into five interacting dimensions and aggregates them using a calibrated multiplicative interaction term.

A. Five Resilience Dimensions

The framework defines five sub-indices ($D_k$):

1. Physical ($D_{phy}$): Quantifies direct generation loss (specifically disconnected PV capacity).
2. Operational ($D_{op}$): Captures dynamic response degradation, including Rate of Change of Frequency (RoCoF), performance loss severity, and loss of generator synchronism.
3. Digital-Cyber ($D_{cyb}$): Measures the loss of observability and controllability due to cyber compromises.
4. Climatic ($D_{clim}$): An exogenous factor representing environmental stressors (e.g., extreme cold, snowstorms) that amplify demand or reduce generation.
5. Regulatory ($D_{reg}$): An exogenous factor quantifying institutional vulnerabilities (e.g., lack of multi-factor authentication, unpatched firmware) based on a reference framework (IEC 62443/NERC CIP).

B. Mathematical Formulation

The MDRI is calculated in two stages:

1. Endogenous Coupling ($M$):
   The three internal dimensions (Physical, Operational, Cyber) are combined using a convex combination that separates independent contributions from coupled interactions:
   $$M = (1 - \gamma) \underbrace{\frac{1}{|K_{sim}|} \sum D_k}_{\text{Independent Sum}} + \gamma \underbrace{\left( \prod D_k \right)^{1/|K_{sim}|}}_{\text{Coupled Term}}$$

   • $\gamma$ is a coupling coefficient calibrated from simulation data.
   • The product term ensures that the coupling effect activates only when all dimensions are simultaneously compromised, mimicking cascading failure mechanisms.

2. Exogenous Amplification:
   The core index $M$ is amplified by external factors:
   $$MDRI = M \times \prod_{j \in \{clim, reg\}} (1 + D_j)$$
   This structure allows the index to reflect how environmental and regulatory contexts exacerbate internal system failures.

3. Case Study and Validation

The framework was validated on the IEEE 39-bus New England test system (modified to include 14.1% inverter-based PV capacity) using MATLAB/Simulink. The study simulated two attack scenarios inspired by a hypothetical December 2025 coordinated cyberattack on Polish energy infrastructure:

• Scenario A (Baseline): A single-plant control attack (Bus 33) causing 190 MW loss. Operators retained full observability and control of the rest of the grid.
• Scenario B (Multi-Vector): A coordinated attack replicating the Polish incident tactics:
  • Cyber: Disruption of communication to six PV plants (loss of observability/controllability).
  • Physical: Forced disconnection of 1,115 MW (74.3% of total PV).
  • Exogenous: Simultaneous 25% load increase, sub-zero temperatures, and snowstorms.

4. Key Results

Dynamic Response

• Scenario A: The system remained stable. Frequency nadir was 60.323 Hz, and the system recovered within 1.45 seconds.
• Scenario B: The system collapsed. The multi-vector attack caused a loss of synchronism among generators. The frequency nadir dropped to 59.776 Hz, RoCoF increased by 61.8%, and the system failed to recover, leading to a total collapse at $t=14.22$s.

MDRI Analysis

• Amplification Factor: The observed resilience loss in Scenario B was 5.6 times higher than what would be expected if dimensions were independent and additive.
• Coupling Contribution: The calibrated coupling coefficient ($\gamma = 0.381$) revealed that simultaneous failures in Physical, Operational, and Cyber dimensions contributed an additional 60.6% to the degradation score through endogenous coupling.
• Exogenous Impact: Climatic and regulatory factors further amplified the total vulnerability by 84%.
• Final Scores:
  • Scenario A MDRI: 0.142
  • Scenario B MDRI: 1.470

5. Key Contributions

1. Multidimensional Decomposition: Introduction of a framework that explicitly separates and integrates five distinct dimensions of resilience (Physical, Operational, Digital-Cyber, Climatic, Regulatory).
2. Coupling Mechanism: Development of a convex combination formula with a calibrated coupling coefficient ($\gamma$) that mathematically captures the non-linear, multiplicative effects of simultaneous dimensional failures.
3. Endogenous/Exogenous Distinction: A hierarchical structure that differentiates between internal cascading failures and external contextual amplifiers (weather, policy gaps).
4. Realistic Validation: Demonstration that single-dimensional metrics fail to predict system collapse in multi-vector attack scenarios, whereas MDRI correctly quantifies the catastrophic amplification of risk.

6. Significance

This paper fundamentally shifts the paradigm of power system resilience assessment. It proves that resilience is not merely the sum of its parts; the interaction between cyber, physical, and operational domains creates a "super-linear" degradation effect that traditional metrics miss.

The findings have critical implications for:

• Grid Planning: Utilities must account for cross-domain dependencies (e.g., how a cyber breach prevents the mitigation of a physical fault).
• Policy & Regulation: The inclusion of a regulatory dimension highlights that institutional weaknesses (e.g., poor segmentation, default credentials) are not just administrative issues but direct multipliers of physical system risk.
• Defense Strategy: It underscores the necessity of defending the "cyber-physical" interface, as the loss of observability/controllability prevents operators from arresting cascading failures, turning a manageable disturbance into a total system collapse.