## 2 Resilience Assessment Framework Considering Critical Loads  
This paper presents an evaluation framework that utilizes a sequential Monte Carlo simulation to assess the resilience of an active distribution system, considering the criticality of the loads. Figure 1 provides a process map outlining the overall methodology. The first step involves modeling the distribution system, where each component under consideration is modeled using graph theory in MATLAB 2023a. The individual components in the network are modeled as nodes, and the connection between them as edges. Next, the methodology incorporates the modeling of an extreme event. Since a distribution system cannot be resilient against all types of extreme incidents, the evaluation must be tailored to a specific extreme event. To develop this methodology, the framework evaluates the system's resilience in response to extreme wind conditions. The key input characteristics used in the model are wind speed and event duration, which are estimated from historical extreme wind data. The duration of a wind event is modeled using an exponential distribution, whereas wind speed for the duration of the event is modeled using a Weibull distribution. The next step in the methodology, as shown in Figure 1, involves assessing damage to the network components resulting from the wind event during each simulation iteration. The framework considers the structural fragility functions of the distribution poles and lines to evaluate their vulnerability to wind events, as given by Equations (1) and (2), respectively. \[FP_{p}=a*e^{b(0.6213*w)} \tag{1}\]  
\[FP_{l}=c*(0.6213*w)^{d} \tag{2}\]  
where \(a=0.0001\), \(b=0.0421\), \(c=8\times 10^{-12}\), and \(d=5.1\) are empirically determined tuning parameters, as reported in Reference [29], and \(w\) represents the wind speed in km/h. A depth-first search algorithm [30] is employed to traverse the graph structure of the network and assess the damage status of each component. For each component, a uniformly distributed random number, r-U(0,1), is compared to the failure probability derived from its fragility function. If the failure probability exceeds the random number, the component is classified as failed. In the event of a pole failure, all adjacent lines connected to that pole are also considered failed to account for dependent failures. This consideration of dependent failure in the framework provides a realistic approach to the resilience evaluation. Following an extreme event, the network divides into multiple  
Figure 1: Methodology to evaluate distribution system resilience considering critical and non-critical loads. segments, each transitioning into one of three possible states: (i) a grid-connected state, (ii) an islanded microgrid state, or (iii) a failed state. When an extreme event impacts the system and a portion of the network becomes isolated from the main grid, the DER management system within the microgrid takes over and supplies power to the connected loads. The DER considered in this study are wind turbine generators (WTGs) and battery energy storage (BESS) units. While conventional sources like diesel generators offer more dependable grid support during extreme events, the inclusion of WTGs and BESS in this analysis reflects the global shift toward low-carbon, sustainable energy solutions. This framework builds on the mixed-integer linear programming-based optimization model from Reference [28] to include the critical loads in the microgrid operation. The model is developed to prioritize the supply of critical loads in a microgrid before supplying the remaining loads. The objective function of the optimization problem is to minimize the overall load curtailment in the microgrid, as defined in Equation (3). The constraint to the function in Equation (4) prioritizes the critical loads such that non-critical loads can only be supplied by the DER once all critical loads are supplied in the microgrid.