## 2 Spatiotemporal Grid Failures and Emergency Scenarios During Typhoons
### Abnormal States of Typical Grid-Source Equipment During Typhoons
#### 2.1.2 Abnormal States of Transmission Line During Typhoons
\[Pr_{k,i,q}^{\rm PC}(N)=Pr_{k,i,1}^{\rm P}(N)\cdot Pr_{k,i,2}^{\rm P}(N)\cdots Pr _{k,i,q}^{\rm P}(N) \tag{6}\]  
where the probability \(Pr_{k,i,t}^{\rm P}(N)\) of pole \(k\) of line \(i\) not collapsing in the time period \(t\) can be expressed by:  
\[Pr_{k,i,t}^{\rm P}(N)=e^{-\frac{D_{k,i,t}}{1-\left(\sum_{i=1}^{t-1}D_{k,i,q} \right)-D_{k,i,t}}},1\leq t\leq q \tag{7}\]  
Equation (7) reflects that the probability of not collapsing in the first \(j\) time periods is determined by the combined effect of the cumulative fatigue damage \(\sum_{i=1}^{t-1}D_{k,i,s}\) from theprevious \(t-1\) time periods and the one-time fatigue damage \(D_{k,i,t}\) in the time period \(t\). The greater the one-time fatigue damage and cumulative fatigue damage are, the smaller the probability \(P^{\text{P}}_{k,j,t}(N)\) of the pole not collapsing before time period \(t\). Based on Equations (7) and (27), the probability \(Pr^{\text{PC}}_{k,i,q}(N)\) of pole \(k\) of line \(i\) not collapsing in the first \(q\) time periods can be rewritten as:  
\[Pr^{\text{PC}}_{k,i,q}(N)=e^{\sum_{t=1}^{t}-\frac{D_{k,i,t}}{1-\left(\sum_{s=1 }^{t-1}D_{k,i,s}\right)-D_{k,i,t}}} \tag{8}\]  
Equation (8) shows that if the cumulative fatigue damage before period \(t-1\) is less than \(1\left(\left(\sum_{s=1}^{t-1}D_{k,i,s}\right)<1\right)\) and the one-time fatigue damage \(D_{k,i,t}\) at time period \(t\) is zero. The probability of pole \(k\) of line \(i\) before time period \(t\) is only affected by the first \(t-1\) time periods. \(\left(\sum_{s=1}^{t-1}D_{k,i,s}\right)=1\) means the cumulative fatigue damage over the first \(t\) periods reaches the threshold value \(1\), the probability of pole \(k\) of line \(i\) is \(1\) and the pole will definitely collapse for the first \(t\) periods. Based on (8), the probability \(Pr^{\text{PC}}_{k,i,q}(B)\) of pole \(k\) of line \(i\) collapsing before time period \(q\) is:  
\[Pr^{\text{P}}_{k,i,q}(B)=1-Pr^{\text{PC}}_{k,i,q}(N) \tag{9}\]  
The typhoon increases the wind load, which can exceed the conductor's endurance limit, leading to a breakage. In this section, the failure rate \(\lambda_{k,i,q}\) of the conductor \(k\) of line \(i\) at period \(q\) is used to calculate the probability of a failure occurring [27]:  
\[\lambda_{k,i,q}=\left\{ \begin{array}{cc}0,&W_{k,i,q}\in[0,W^{\text{norm}}_{ \text{max}})\\ \frac{(1-\lambda_{1})W_{k,i,q}+\lambda_{1}W_{\text{max}}-W^{\text{norm}}_{ \text{max}}}{W_{\text{max}}-W^{\text{norm}}_{\text{max}}},W_{k,i,q}\in[W^{ \text{norm}}_{\text{max}},W_{\text{max}})\\ 1,&W_{k,i,q}\in[W_{\text{max}},\infty)\end{array} \right. \tag{10}\]  
where \(W_{k,i,q}\) represents the wind pressure endured by conductor \(k\) of line \(i\) at period \(q\); \(W^{\text{norm}}_{\text{max}}\) represents the upper limit of normal wind pressure; \(W_{\text{max}}\) represents the upper limit of wind pressure; \(\lambda_{1}\) represents the failure rate when the wind pressure is equal to \(W^{\text{norm}}_{\text{max}}\). The probability \(R_{q}\) of no failure before time \(q\) can be written as:  
\[R_{k,i,q}=R_{k,i,q-1}-\lambda_{k,i,q-1}R_{k,i,q-1} \tag{11}\]  
Finally, the fault probability of \(Pr^{\text{c}}_{k,i,q}\) of conductor \(k\) of line \(i\) before time period \(q\) is:  
\[Pr^{\text{c}}_{k,i,q}(B)=1-R_{k,i,q} \tag{12}\]