# Random Violation Risk Degree Based Service Channel Routing Mechanism in Smart Grid
## 3 SCVRD Model
### SCVRD Model  
As shown in Figure 3, \(SCVRD\) which represents the probability of service channel violation risk \(P_{SCVRD}\) can be expressed by Equation (8):  
\[P_{SCVRD}=P(A_{APA}(t)<A_{Thr}) \tag{8}\]  
However, \(SCVRD\) is hard to be accurately quantified and tracked due to the fluctuations of \(A_{APA}(t)\). Thus, we introduce cumulative repair time TR and corresponding repair time threshold \(T_{Thr}\) in the statistical period to describe the service channel violation risk.  
\[TR=\sum_{i=0}^{+\infty}TTR_{i} \tag{9}\]  
Figure 3: The integral varying curve of \(A_{APA}(t)\).  
Figure 2: The varying curve of \(A_{APA}(t)\) with \(TBF_{k}\).  
\[T_{Thr}=(1-A_{Thr})\times\sum\limits_{i=0}^{+\infty}\left(TBF_{i}+TTR_{i}\right) \tag{10}\]  
Then according to Equation (4), \(SCVRD\) can be quantified by Equation (11).  
\[\begin{array}{l}P_{SCVRD}=P(A_{APA}(t)<A_{Thr})\\ =P\left(\left(1-\frac{\sum\limits_{i=0}^{+\infty}TTR_{i}}{\sum\limits_{i=0}^{+ \infty}\left(TBF_{i}+TTR_{i}\right)}\right)<A_{Thr}\right)=P\left(\frac{\sum \limits_{i=0}^{+\infty}TTR_{i}}{\sum\limits_{i=0}^{+\infty}\left(TBF_{i}+TTR_{ i}\right)}>1-A_{Thr}\right)\\ =P\left(\sum\limits_{i=0}^{+\infty}TTR_{i}>(1-A_{Thr})\times\sum\limits_{i=0}^{+ \infty}\left(TBF_{i}+TTR_{i}\right)\right)=P(TR>T_{Thr})\end{array} \tag{11}\]  
Thus, the probability of service channel violation risk based on availability is converted to the probability of cumulative repair time exceeding the corresponding repair time threshold in the statistical period. The equivalence relationship is shown in Figure 4.  
Next, the characteristics of \(SCVRD\) will be further analyzed. By further analyzing Equation (4), we can find that \(A_{APA}(t)\) is co-determined by \(FN\) of the service channel and each failure \(TTR_{i}\). According to [23], \(TTR\) of \(v_{i}\) and \(e_{ij}\) is independent and they all obey the log-normal distribution as expressed by Equation (12). Therefore, TR subjects to the joint distribution of \(\sum\limits_{i=1}^{\infty}TTR_{i}\), which means that SFE is a random process and \(P_{SCVRD}\) is a random probability model. Assume that \(FN\) of service channel obeys the Poisson distribution with average arrival rate \(\lambda\), as expressed by Equation (13). Thus, the service channel violation risk caused by the \(k\)th failure \(P_{SCVRD}(X=k,TR_{k}>T_{Thr})\) during the statistical period can be expressed by Equation (14):  
\[f_{i}(t)=\frac{1}{\sqrt{2\pi}\cdot\sigma_{i}\cdot t}e^{-\left(\ln\left(t\right) -\delta_{i}\right)^{2}/2\sigma_{i}^{2}}, \tag{12}\]  
\[P_{F}(X=k)=\frac{e^{-\lambda}\lambda^{k}}{k!},\lambda\in(0,1),k=1,2,\cdots \tag{13}\]  
\[P_{SCVRD}(TR_{X}>T_{Thr},X=k)=P_{F}(X=k)\cdot P_{R}\left(\sum\limits_{i=1}^{k} TTR_{i}>T_{Thr}\right) \tag{14}\]  
Because the failure occurrences of \(v_{i}\) and \(e_{ij}\) are independent, according to the conditional probability Equation and the n-fold convolution Equation, Equation (14) can be expanded to Equation (15):  
Figure 4: Availability curve comparison chart.  
to \(A_{SPA}\), \(P_{SCVRD}\) is more accurate to distinguish the differences of \(A_{APA}(t)\) among the channels. Thus \(P_{SCVRD}\) is more suitable in the quantization process of SFE.