## 2 Resilience Assessment Framework Considering Critical Loads
### Resilience Metrics for Performance Evaluation  
The MCS framework chronologically simulates the system performance to determine the critical and non-critical load point indices and the system indices. The expected outage duration (EOD) is the measurement of the duration of the outage that a customer can expect, given that the system succumbed to an extreme event, and is calculated at each load point using Equation (12). The load point EOD for the critical and non-critical loads is then obtained using Equations (13) and (14), respectively, by grouping the respective \(EOD_{lp}\). The system EOD is obtained from Equation (15). \[EOD_{lp}=\frac{\sum_{s=1}^{N_{int}}(OD_{lp})_{s}}{N_{sim}^{c}} \tag{12}\]  
\[EOD_{cl}=\frac{\sum_{1}^{N_{cl}}EOD_{lp_{cl}}}{N_{cl}} \tag{13}\]\[EOD_{ncl}=\frac{\sum_{1}^{N_{ncl}}EOD_{lp_{ncl}}}{N_{ncl}} \tag{14}\]  
\[EOD_{sys}=\frac{\sum_{1}^{N_{lp}}\left(EOD_{lp}\right)_{n}}{N_{lp}} \tag{15}\]  
where \(N_{sim}^{\times}\) is the number of simulations in which a outage occurs at the load point. The expected energy not supplied (EENS) is the expected energy curtailed due to an extreme event. This index can be calculated at each load point using Equation (16) and can be aggregated using Equation (17) to obtain the system index. The EENS for critical and non-critical loads is obtained from Equations (18) and (19). \[EENS_{lp}=\frac{\sum_{s=1}^{N_{sim}}ENS_{lp}(s)}{N_{sim}} \tag{16}\]  
\[EENS_{sys}=\sum_{1}^{N_{lp}}EENS_{lp} \tag{17}\]  
\[EENS_{cl}=\frac{N_{cl}}{1}EENS_{lp_{cl}} \tag{18}\]  
\[EENS_{ncl}=\sum_{1}^{N_{ncl}}EENS_{lp_{ncl}} \tag{19}\]  
where \(ENS_{lp}\) is the energy not supplied for a load point. The EENS index is useful to convert the impact of extreme events into monetary value to determine the system outage cost. However, unlike EOD, the system EENS cannot be used to compare different systems unless it is normalized by the system capacity. This work proposes two new resilience measurement metrics, DLH and NDLH. The DLH is the expected sum of the outage durations at all the load points, considering all probable durations and the intensity of extreme weather events. The DLH is a system index, and it considers the fact that the system does not succumb to all of the events and therefore, incorporates the system's ability to withstand them. The DLH index captures the total number of customer-hours impacted by an outage, making it a particularly important metric for system operators. It quantifies overall system impact by capturing both the scale and duration of disruptions and serves as a valuable input for estimating economic losses, assessing customer-hour losses. DLH is calculated by Equation (20). \[DLH=\frac{\sum_{s=1}^{N_{sim}}\left(\sum_{n=1}^{N_{lp}}OD_{lp}\right)_{s}}{N_{ sim}} \tag{20}\]  
where \(OD_{lp}\) is the outage duration of a load point, \(N_{sim}\) is the total number of simulated event scenarios, and \(N_{lp}\) is the number of load points. The NDLH is the expected outage duration per load point and is obtained from Equation (21) by dividing the DLH by the total number of load points in the system. It should be noted that NDLH and system-EOD both measure the average outage duration per load point per extreme event. The difference, however, is that EOD only considers the extreme events that cause outages, whereas NDLH considers the likelihood of all extreme events and the ability of the system infrastructure to withstand the events. \[NDLH=\frac{DLH_{sys}}{N_{lp}} \tag{21}\]The relevance and effectiveness of these resilience metrics are demonstrated through a series of case studies that assess system performance under different investment strategies. The overall simulation is repeated until the stopping criterion is met. This paper uses the convergence of the mean estimate of the load point EOD values as the stopping criterion with a threshold \(\epsilon\) of 3%. Equations (22)-(24) determine the stopping criterion.