## 2 Resilience Assessment Framework Considering Critical Loads
### Restoration of Critical and Non-Critical Loads  
The restoration of outage components begins after the wind speeds drop to a safe level for outdoor repair work. It is the usual practice to restore the critical loads before restoring the non-critical loads. The restoration methodology presented in this paper extends the hierarchical restoration model proposed in Reference [28] by prioritizing the critical loads, while recognizing the dependent repair events in assessing the overall restoration time. The restoration time depends on the number of failed components and the sequence of component repair. The component repair sequence further depends on the interdependency of the failed components and the number of available repair crews. Failed components are restored either sequentially or simultaneously, depending on whether they result from dependent or independent outages, respectively. The hierarchy-based load repair model assigns component outages resulting from dependent faults to separate repair hierarchies and addresses them sequentially. In contrast, components that have failed independently are grouped within the same hierarchy, enabling simultaneous repairs. It is assumed that each outage task is assigned to a single crew unit consisting of a few repair personnel, required equipment, and necessary replacement components. Thismodel optimizes the repair process to restore power to affected customers as quickly as possible, aiming to minimize the number of disconnected load points at each simulation step. The outage components are evaluated based on their ability to reconnect the maximum number of system elements upon repair. The component with the highest restoration impact is assigned the highest priority. This process is iteratively applied to establish an optimal repair priority order for both critical and non-critical loads. The resilience trapezoid in Figure 2 illustrates how system performance changes in regards to time chronology during the event-progress phase and the post-event phase. As the event unfolds, load points begin disconnecting and reach their maximum status at the end of the event-progress phase. During the post-event phase, restoration strategies are employed to swiftly reconnect disconnected load points. The total restoration time can be calculated using Equations (5)-(11), considering a combination of four possible repair scenarios resulting from load prioritization. Equations (5) and (6) represent the condition where the number of available repair crews, denoted as \(N_{cr}\), is fewer than both the critical outage tasks \(N_{cl}\) and the non-critical outage tasks \(N_{ncl}\). The repair time for each set of outage components can be calculated using Equation (5), with the total repair time being the sum of both repair times, as outlined in Equation (6). \[r_{x}=\min\left\{max\left\{\sum\limits_{k_{e}=1}^{\left\lceil\frac{N_{cr}}{N_{ rel}}\right\rceil}\sum\limits_{h=1}^{m}t_{h_{j}}\right\}_{for\ j=1\ to\ N_{cr}}\right\}_{for\ p=1\ to\ \left(N_{pr}\right)_{x}} \tag{5}\]  
\[r_{tot}=r_{cl}+r_{ncl} \tag{6}\]  
where \(x\) represents the set of loads (either critical or non-critical), \(t_{hj}\) is the time to repair a component within an outage task assigned to crew \(j\) and has a dependent hierarchy \(h\), and \(m\) is the number of dependent repair hierarchy tasks within an outage task. \((N_{pr})_{x}\) gives the permutations of the task sequences for the respective sets and is determined as \(\left(N_{pr}\right)_{cl}=N_{cl}!\) and \(\left(N_{pr}\right)_{ncl}=N_{ncl}!\). Equations (7) and (8) evaluate the repair time when the number of crew members exceeds the number of critical outage tasks but is still fewer than the total outage tasks. Initially, each critical outage task is assigned to a crew unit. At this condition, a number of non-critical outage tasks can also be addressed simultaneously alongside the critical  
Figure 2: Resilience trapezoid showing different phases of an extreme event. components by some of the surplus crew units.