## 2 Spatiotemporal Grid Failures and Emergency Scenarios During Typhoons
### Evolution of Spatiotemporal Grid Failures and Emergency Scenarios During Typhoons  
Based on (12)-(25) in Section 2.1, we can calculate the fault probability \(Pr_{i,j}(B)\) of line \(i\) before time period \(j\). In consideration of the whole prediction period \(T\), the fault probability matrix is expressed by:  
\[\Pr(B)=\left[\begin{array}{cccc}\Pr_{1,1}(B)&\Pr_{1,2}(B)&\cdots&\Pr_{1,T} (B)\\ \Pr_{2,1}(B)&\Pr_{2,2}(B)&\cdots&\Pr_{2,T}(B)\\ \vdots&\vdots&\cdots&\vdots\\ \Pr_{L,1}(B)&\Pr_{L,2}(B)&\cdots&\Pr_{L,T}(B)\end{array}\right] \tag{13}\]  
Based on (1)-(3), we can obtain the wind power of wind farm \(i\) at time period \(j\). Figure 1 presents the flowchart of the abnormal spatiotemporal distribution of the power grid during typhoons.  
The spatiotemporal distribution of transmission lines during typhoons is depicted by a probability matrix. A common method of constructing the emergency scenarios using the probability matrix is to simulate the sufficient numbers of times a line breakage accident occurs. Nevertheless, the large sample size leads to the computational burdens on those small probability events which rarely happen. The probability threshold exists to curb the inclination of the decision maker to overvalue these low-probability threats and to deter the uneconomic over-suppression. In this paper, the probability threshold is used to construct the emergency scenarios. Let \(\mu_{i,t}\in\left\{0,1\right\}\) represent the on-off status of line \(i\) at period \(t\); the on-off status of all affected lines during typhoons can be written as a matrix  
Figure 1: Flowchart of the abnormal spatiotemporal distribution of the power grid during typhoons.  
form \(\mu\). Suppose the probability threshold is \(Pr_{0}\), \(t_{0}\) represents the period in which the fault probability of line \(i\) exceeds \(Pr_{0}\) for the first time:  
\[t_{0}=F_{ij}(Pr_{0}) \tag{14}\]  
where \(F_{ij}\) represents the mapping from \(Pr_{0}\) to \(t_{0}\). Then, the sequential on-off statuses of line \(ij\) during typhoons are determined as:  
\[\mu_{ij,t}=\left\{\begin{array}{l l l}1,&if&t_{0}\
otin\mathcal{N}_{T}\\ 0,&if&t_{0}\in\mathcal{N}_{T}\quad\text{and}\quad t\geq t_{0}\\ 1,&if&t_{0}\in\mathcal{N}_{T}\quad\text{and}\quad t\leq t_{0}\end{array}\right. \tag{15}\]  
where \(\mathcal{N}_{T}\) represents the set of time periods during typhoons. Equation (15) shows that when the fault probability of line \(i\) never reaches the threshold \(Pr_{0}\) during typhoons (i.e., \(t_{0}\
otin\mathcal{N}_{T}\)), the status of line \(ij\) is always regarded as on-status (\(\mu_{ij,t}=1\)). When the fault probability of line \(ij\) reaches \(Pr_{0}\) at period \(t_{0}\), the status of line \(ij\) after period \(t_{0}\) is regarded as off-status (\(\mu_{ij,t}=0\)). In other words, the fault probability is high enough to trigger the breakage; since timely repair is not considered, the off-status lasts for the remaining periods \(t\geq t_{0}\). The status of line \(ij\) before period \(t_{0}\) is regarded as on-status (\(\mu_{ij,t}=1\)) since the fault probability before period \(t_{0}\) is not big enough to trigger the breakage. In practice, the decision maker can set different \(Pr_{0}\) to reflect the flexibility of their choice of the threshold. Each \(Pr_{0}\) will generate one emergency scenario based on (15). Suppose there are \(R\) choices of \(Pr_{0}\), then \(R\) emergency scenarios are established.