# Power Grid Reliability Evaluation Considering Wind Farm Cyber Security and Ramping Events
## 4 Attack Scenarios
### Long-Term Imperfect Attack of Wind Turbine  
Since wind power equipment is designed for lightness and efficiency but is often fragile, the attacker probably launches an IPA to damage the physical states of wind turbines. An IPA is characterized by strong concealment, long duration, and uncertain damage. The impact of an IPA is similar to the Stuxnet attack in the uranium hexafluoride centrifuges in Iran's Natanz facility [25]. Investigation [26] and review [27] presented the failure statistics of a wind turbine, indicating that most wind turbine failures are caused by the failure of gearboxes and their bearings. Running in critical speed or overloading the operation of a wind turbine can accelerate the fatigue of the mechanical part and even destroy the WT. Excessive load and overheating are the most frequent reason for the premature fatigue of a \begin{table}
\begin{tabular}{c c c c c c} \hline
**DC** & **0 \textless{} \(t\leq 7\)** & **7 \textless{} \(t\leq 28\)** & **28 \textless{} \(t\leq 84\)** & **84 \textless{} \(t\leq 365\)** & **PM** \\ \hline Weak & 0.245 & 0.088 & 0.235 & 0.254 & 0.010 \\ Normal & 0.245 & 0.070 & 0.176 & 0.190 & 0.008 \\ Good & 0.020 & 0.044 & 0.029 & 0.032 & 0.004 \\ Perfect & 0.007 & 0.088 & 0 & 0 & 0.001 \\ \hline \end{tabular} DC is the defense capability of power system, and PM is the annual mean probability of a cyber-attack.  
\end{table} 
Table 2: Probabilities of a cyber-attack. WT bearing. For example, when the temperature is in excess of 400 \({}^{\circ}\)F, the ring and ball materials are annealed [28]. The hardness of the bearing and the state of the lubricant then deteriorate. Since bearing wear is the most common cause of a WT fault, the impact of an IPA can be quantified by the estimation of the WT bearing state. In this paper, the wear process of a bearing infected was modeled as a continuous degenerative process that obeys the Gamma distribution [29]. The Markov chain [5] was utilized to estimate the states of bearing in different stages. Combining the physical state and service time of bearing, the failure rates of a WT can be further estimated for system reliability evaluation. Table 3 presents the wearing conditions of a WT bearing, where \(\Delta\text{max}=R_{0}-R_{min}\). \(R_{0}\) is the size of new bearing. \(R_{min}\) is the minimum size allowed for the bearing to operate. Suppose the set of the bearing states is \(S_{i}\) at time \(t_{i}\). The time domain state of bearing is \(x(t)\) after a short time \(\Delta t\). The change value of time domain state is \(y(\Delta t)\), and the according state of bearing changes to \(S_{j}\). The transition probability from \(S_{i}\) to \(S_{j}\) is formulated as Equation (6). The probabilities of bearing states can be obtained by iterations of the Markov process as shown in Equation (7). \[P_{ij}(t)=P[\zeta_{j-1}\leq x(t)+y(\Delta t_{i})\leq\zeta_{j-1}]\zeta_{i-1} \leq x(t_{i})\leq\zeta_{i}]=\frac{\int_{\zeta_{i-1}}^{\zeta_{i}-X(t)}f[y( \Delta t_{i})]\cdot f[x(t)]dydx}{\int_{\zeta_{i-1}}^{\zeta_{i}}f[x(t)]dx} \tag{6}\]  
\[S_{n}=S_{0}P^{(n)}=S_{0}P^{n} \tag{7}\]  
where \(f(x)\) is the probability density function of the state increment. \(\zeta_{i}\) and \(\zeta_{j}\) are the state top and bottom limitation, where \(i=1\), \(2\), \(\dots\)\(n\); \(j=i\), \(i+1\), \(\dots\)\(i+n\). \(S_{0}\) is the initial state of bearing. \(P\) is the transition matrix, and \(n\) is the number of state transitions. To evaluate the consequence of an IPA, the acceleration fatigue of a bearing is measured by its equivalent operation time (hours). The impact of an IPA is transformed to a time calculation problem. When the sum of normal working hours and the extra equivalent time caused by an attack exceed the boundary of state transition, the transfer times \"\(n\)\" should increase in the Markov process, as shown in Equation (8). Then, the WT's remaining life can be estimated as the duration between the current state and the failure state.