## 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  
The elementary components of a transmission line include the conductor and the pole. The pole or tower is a tall vertical structure to support the conductor which transmits electrical energy. Suppose that transmission line \(i\) is composed of \(M\) poles and \(M-1\) interconnected conductors, since the collapse of any pole or conductor will lead to the disconnection of transmission line \(i\), the fault probability \(Pr_{i,t}(B)\) of line \(i\) before time period \(t\) can be written by [26]:  
\[Pr_{i,t}(B)=\max\left\{ \begin{array}{c}\left[1-\left(1-Pr_{1,i,t}^{\rm P}(B) \right)\cdots\left(1-Pr_{M,i,t}^{\rm P}(B)\right)\right]\\ \left[1-\left(1-Pr_{1,i,t}^{\rm c}(B)\right)\cdots\left(1-Pr_{M-1,i,t}^{\rm c }(B)\right)\right]\end{array} \right\} \tag{4}\]  
where \(Pr_{k,i,t}^{\rm c}(B)\) represents the fault probability of conductor \(k\) of line \(i\) before time period \(t\); \(Pr_{k,i,t}^{\rm P}(B)\) represents the fault probability of pole \(k\) of line \(i\) before time period \(t\). Equation (25) shows that the fault probability of a transmission line is dependent upon the fault probability of the conductors and poles. The fault probability of the pole is related to the cumulative fatigue damage during typhoons. According to the Palmgren-Miner linear fatigue damage criterion, the cumulative fatigue damage \(D\) of a pole can be obtained by summing the fatigue damage over specific time periods. When there is no fatigue damage \(D=0\), the fault probability is zero; when the fatigue damage reaches the threshold value \(D=1\), the fault probability is one and the collapse of the pole is an inevitable event. If the damage value is between zero and one \(0<D<1\), the pole will have a certain probability of experiencing a collapse. The fatigue damage of a pole during time period \(t\) exhibits an exponential relationship with the wind load during that period, where the wind load is proportional to the square of the wind speed. The fatigue damage \(D_{k,i,t}\) of conductor \(k\) of line \(i\) at time period \(t\) can be expressed by:  
\[D_{k,i,t}=\left\{ \begin{array}{ll}0,&v_{k,i,t}\in[0,v_{\rm cr})\\ ae^{\rm p}v_{k,i}^{2},&v_{k,t}\in[v_{\rm cr},v_{\rm m})\\ 1,&v_{k,i,t}\in[v_{\rm m},\infty)\end{array} \right. \tag{5}\]  
where \(v_{k,i,t}\) represents the perceived wind speed of conductor \(k\) of line \(i\) at time period \(t\); \(a\) and \(b\) represent the fatigue coefficient. \(v_{\rm cr}\) represent the threshold wind speed at which the typhoon begins to trigger fatigue in the pole. \(v_{\rm m}\) represents the maximum wind speed corresponding to the one-time damage to the pole, without considering cumulative effects over time. Equation (26) indicates that the one-time damage to the pole is zero when the perceived wind speed is below the threshold wind speed; while the one-time damage to the pole will definitely cause collapse if the perceived wind speed is beyond the maximum. Equation (26) formulates the one-time damage for a specific time period. In fact, whether a pole collapses depends on the cumulative fatigue damage over multiple time periods. As for the first \(q\) time periods, if the cumulative fatigue damage generated in time period \(t\) (\(t\leq q\)) from the previous \(t-1\) time periods does not lead to a collapse, then the pole will not experience a collapse in the the first \(q\) time periods. Therefore, the probability \(Pr_{k,i,q}^{\rm PC}(N)\) of pole \(k\) of line \(i\) not collapsing in the first \(q\) time periods can be expressed as the product of the probabilities of not collapsing from period \(1\) to \(q\).