Choquet Resilience: Power Grid Metric

• The paper demonstrates Choquet Resilience by combining risk-tail statistics using CVaR with expert-driven weighting through the Choquet integral.
• It quantifies resilience via five parameters—availability, robustness, brittleness, resistance, and resourcefulness—validated on a modified IEEE 123-bus system.
• Simulation results show that smart network enhancements markedly improve resilience scores, underlining the metric's practical impact on grid modernization.

Choquet Resilience is a quantitative metric developed to evaluate the resilience of power distribution systems to extreme events, particularly those characterized by low-probability and high-impact outcomes. The methodology integrates risk-tail statistics, specifically the Conditional Value-at-Risk (CVaR) of a set of distribution grid performance parameters, and aggregates them using the Choquet integral based on elicited priority weights. This framework supports nuanced investment and operational decision-making by accommodating both the severity of worst-case scenarios and expert-driven prioritization among multiple resilience dimensions (Poudyal et al., 2022).

1. Resilience Parameters and System Attributes

The formulation begins with the identification of five distinct parameters, each representing foundational attributes of a distribution grid’s resilience under extreme events. Let $N_C$ denote the number of weighted critical loads (CLs). The parameters are:

• Availability ($\mathcal{R}_\psi$): The fraction of time that critical loads remain energized.

$$\mathcal{R}_\psi = \frac{\sum_{i=1}^{N_C} T_U^i}{\sum_{i=1}^{N_C}\bigl(T_U^i+T_D^i\bigr)}$$

where $T_U^i$ and $T_D^i$ are the up- and down-times of load $i$.

• Robustness ($\mathcal{R}_\beta$): The complement of the maximum outage incidence, indicating the portion of CLs that remain connected at the event peak.

$$\theta_{\max} = \frac{N_C-\bar{N}_C}{N_C}, \quad \mathcal{R}_\beta = 1-\theta_{\max} = \frac{\bar{N}_C}{N_C}$$

where $\bar{N}_C$ is the number of CLs still connected at the event peak.

• Brittleness ($\mathcal{R}_\gamma$): Damage-weighted outage incidence, scaled by the damage percentage $D$ of the infrastructure.

$\mathcal{R}_\gamma = 100 \times \theta_{\max} D$

• Resistance ($\mathcal{R}_\xi$): The system’s capacity to withstand damage prior to disconnection, where $\sigma$ encapsulates event severity and $\Delta t_1$ is the damage-assessment delay.

$$\mathcal{R}_\xi = \frac{\sigma\sum_{i=1}^{N_C}T_{1,U}^i}{\theta_{\max}\,N_C\,\Delta t_1}$$

• Resourcefulness ($\mathcal{R}_\delta$): Quantifies the availability of restorative paths after an extreme event, accounting for network flexibility via switches, generators, and the number of post-event source-to-CL paths.

$\mathcal{R}_\delta = \frac{N_P}{(N_{SW}+N_S)\,N_C}$

Here $N_{SW}$, $N_S$, and $N_P$ are counts of tie switches, generators (including DERs), and simple source-to-CL paths, respectively.

These parameters holistically represent resilience-driven system dynamics in response to hazards.

2. Tail-Risk Quantification via CVaR

Each parameter $\mathcal{R}$ is modeled as a random variable governed by the intensity $I$ of the external event (e.g., wind speed) via the probability density $p(I)$. For risk assessment, the cumulative distribution function is

$$\Psi(\zeta) = \Pr[\mathcal{R} \leq \zeta] = \int_{\mathcal{R}(I)\leq\zeta} p(I) \, dI.$$

To focus on tail-risk, the Conditional Value-at-Risk at confidence level $\alpha$ is used:

$$\mathrm{VaR}_\alpha(\mathcal{R}) = \inf\{\zeta : \Psi(\zeta) \geq \alpha\},$$

$$\mathrm{CVaR}_\alpha(\mathcal{R}) = \frac{1}{1-\alpha} \int_{\mathcal{R}(I)\geq \mathrm{VaR}_\alpha} \mathcal{R}(I)\,p(I)\,dI.$$

For $\alpha = 0.95$, $\mathrm{CVaR}_{0.95}$ represents the expected performance in the worst 5% of scenarios, directly capturing resilience under extreme but plausible external stresses.

3. Aggregation with the Choquet Integral

Aggregation of the five CVaR-parameter values leverages the Choquet integral, a framework for combining scores under a non-additive (fuzzy) capacity function $\mu: 2^\Gamma \to [0,1]$, with $$\Gamma = \{\mathcal{R}_\psi, \mathcal{R}_\beta, \mathcal{R}_\gamma, \mathcal{R}_\xi, \mathcal{R}_\delta\}$$. The measure $\mu$ is monotonic with $\mu(\emptyset)=0$ and $\mu(\Gamma)=1$.

For a vector $f = (f_1, ..., f_5)$, with $f_i = \mathrm{CVaR}_\alpha(\mathcal{R}_i)$ sorted so $f_{(1)} \leq ... \leq f_{(5)}$ and $f_{(0)} = 0$, the discrete Choquet integral is computed as

$$\mathcal{C}_\mu(f) = \sum_{i=1}^5 \bigl(f_{(i)} - f_{(i-1)}\bigr) \mu\bigl(A_{(i)}\bigr)$$

where $A_{(i)} = \{\, j \mid f_j \geq f_{(i)} \,\}$.

Selection and interpretation of $\mu$ allow for operator- or expert-driven prioritization via "singleton" values $\mu(\{\mathcal{R}_i\})$. These are further refined using the Shapley index to account for the interaction effects among criteria.

4. Capacity Selection and Priority Modeling

The capacity $\mu$ is defined over all subsets of resilience criteria and encodes both importance and interaction. Singleton weights are elicited directly from domain experts or system operators to reflect their resilience investment or operational priorities. The general Sugeno $\lambda$-measure enables extension to all subsets, and the overall normalization $\mu(\Gamma) = 1$ governs final scaling.

The Shapley index

$$\eta_{\mathcal{R}} = \sum_{S\subseteq\Gamma\setminus\{\mathcal{R}\}} \frac{(n-|S|-1)!\,|S|!}{n!} \bigl[\mu(S\cup\{\mathcal{R}\})-\mu(S)\bigr]$$

yields an adjusted importance measure for each individual parameter that corrects for inter-criterion dependencies.

Different choices for $\mu$ correspond to distinct operational or investment philosophies. By adjusting $\mu$, an operator can emphasize, for example, fast restoration (availability/resourcefulness) over structural hardening (robustness/brittleness) without sacrificing overall resilience, as numerically demonstrated in simulation results.

5. Simulation Methodology and Numerical Analysis

The framework is empirically validated using a modified IEEE 123-bus distribution system subjected to simulated extreme wind events. The assessment workflow encompasses:

• Sampling wind intensities from a prescribed event PDF.
• Applying a fragility model to derive per-line outage probabilities as functions of wind speed.
• Monte-Carlo simulation (1,000 trials per scenario) to generate outage realizations and, for each, compute the five resilience parameters post-event.
• Collation across all trials and intensities to infer empirical distributions for each $\mathcal{R}_i$; extraction of $\mathrm{CVaR}_{0.95}$ for further aggregation.
• Comparison between a base network (without DERs or restoration capabilities) and a “smart” variant (with DERs and remote switches).

Results demonstrate that, at $\alpha=0.95$, key parameters such as availability ($\mathcal{R}_\psi$: 0.0112 base $\to$ 0.0193 smart) and resourcefulness ($\mathcal{R}_\delta$: 0.00005 base $\to$ 0.00314 smart) improve substantially with network enhancements. The Choquet resilience scores, computed across five distinct $\mu$-weightings (priority-cases), also uniformly favor the smart network (e.g., base 7.89 vs. smart 10.93 in Case IV).

These findings illustrate both the discriminatory power of the metric and its flexibility under varying operational preferences (Poudyal et al., 2022).

6. Interpretability, Trade-Offs, and Implications

The Choquet Resilience metric, through explicit modeling of expert-driven capacity functions and tail-risk quantification, provides system operators with a robust means to balance trade-offs between different resilience-enhancement measures. By modifying the weights in the capacity $\mu$, decision-makers can prioritize investments—such as favoring faster restoration capabilities over physical infrastructure fortification—depending on contextual constraints and performance targets.

A plausible implication is that, in systems where costlier investments deliver diminishing returns, reallocating focus toward operational adaptability (e.g., resourcefulness and availability) can yield resilience gains comparable to those achievable by high-investment network hardening. The flexibility of the Choquet framework ensures that resilience assessments adapt to both risk profile and stakeholder values without neglecting worst-case performance.

7. Broader Context and Significance

The Choquet Resilience metric extends beyond traditional additive aggregation schemes, enabling more nuanced and actionable resilience quantification in the face of correlated, multidimensional risks. The integration with CVaR ensures that low-probability, high-impact scenarios are directly and rigorously incorporated into resilience planning. The approach is particularly relevant in the context of power grid modernization, where distributed energy resources, advanced switching, and restoration protocols increasingly define the system’s capacity to withstand and adapt to extreme hazards. The adoption of this metric may inform both practical planning and future methodological research in resilience quantification (Poudyal et al., 2022).