# Collision Risk Evaluation and Verification of GNSS-Based Train Integrity Detection
## 3 Collision Risk Evaluation of Train Integrity Detection
\(\sigma_{G}=\sqrt{\sigma_{D}^{2}+\sigma_{\Theta}^{2}-2\rho_{D\Theta}\sigma_{D \Theta}}\) is the standard deviation. Thus, the probability of failure is  
\[R_{\Theta}=\int\limits_{-\infty}^{0}f_{G}(G)dG=1-\Phi(\beta)=\Phi(-\beta) \tag{4}\]  
where  
\(\beta_{i}=\frac{\mu_{i}^{G}}{\sigma_{i}^{G}}=\frac{\mu_{i}^{R}-\mu_{i}^{S}}{ \sigma_{i}^{K}+\sigma_{i}^{G}}\) is defined as the safety index. Values of \(\beta\) for typical values of \(G(\cdot)\) are shown in [18]. \(\Phi(\cdot)\) is the standard normal cumulative distribution function. Kalman filter (KF) is employed to solve the Bayesian filtering problem and obtain the state estimation results in GNSS-based train position estimation. Variation of PL is determined by relevant information from the Kalman filtering process. Besides, an efficient way is to project the test statistic to the position domain by using the uncertainty of position estimation. The first step for identifying PL is to carry out the state estimation by fusing information with the system model and sensor measurement. Assume that \(h_{k}\) denotes the state vector at instant \(k\) that describes the dynamic state of a train moving along the track; the system model for fusion is:  
\[ \begin{array}{c}h_{k}=A_{kt}h_{k-1}+\eta\\ y_{k}=B_{k}h_{k}+w\end{array}  \tag{5}\]  
where \(A_{k}\) is the active transition matrix at time \(k\); \(B_{k}\) is the active emission matrix at time \(k\). \(\eta\) and \(w\) are the independent Gaussian transitions and observation noises, and \(y_{k}\) is the observation vector with instant sensor measurement. As the definition of filtering state, the horizontal position uncertainty of PL can be calculated as  
\[\text{PL}=\gamma_{k}\cdot\sigma_{k} \tag{6}\]  
Figure 5: The distributed structure of the train integrity detection. where \(\gamma_{k}\) is the factor determined to reflect the probability of missed detection of PL, and the value \(\gamma_{k}\) = 6.18 indicates a probability of missed detection of \(5\times 10^{-9}\)[19]. \(\sigma_{k}\) is the is the estimation residual in KF [14]. For GNSS-based train localization, the probability of failure, risk of GNSS integrity monitoring, which indicates the positioning error exceeding PL, can be calculated as:  
\[R^{f}=\Phi\bigg{(}\frac{-\mathrm{PL}-\mu}{\sigma}\bigg{)}+1-\Phi\bigg{(}\frac {\mathrm{PL}-\mu}{\sigma}\bigg{)} \tag{7}\]  
where \(\mu\) and \(\sigma\) are the mean and variance of the position estimation error \(\hat{h}_{k}\). Then we can work out the safety risk of train integrity. As shown in Figure 5, there are two different situations in train integrity monitoring:  
* \(L_{e}<L_{p}<L_{d}\) The detection threshold \(L_{d}\) is lower than the measured train length \(L_{e}\) and protection length \(L_{p}\). Then, \(R_{k}=\mathrm{Pr}\{G_{1}(\Theta)<0\cup G_{2}(\Theta)<0\}=\mathrm{Pr}\{G_{2}( \Theta)<0\}\) and the PFD can be derived as: \[\mathrm{PFD}_{k}=\mathrm{Pr}\{G_{2}(\Theta)<0\}=R^{f}\times R_{\Theta_{2}}\] (8)
* \(L_{e}<L_{d}<L_{p}\) The detection threshold \(L_{d}\) is bigger than the measured train length \(L_{e}\) and lower than the protection length \(L_{p}\). Then \(R_{k}=\mathrm{Pr}\{G_{1}(\Theta)<0\cup G_{2}(\Theta)<0\}=\mathrm{Pr}\{G_{1}( \Theta)<0\}\), and the PFD can be derived as: \[\mathrm{PFD}_{k}=\mathrm{Pr}\{G_{1}(\Theta)<0\}=R_{\Theta_{1}}\] (9)  
If the train parting is still not detected in the limited safe time, a rear-end collision will occur. Then we can get the safety risk for collision in GNSS-based train integrity detection:  
\[\mathrm{PFH}=\frac{3600}{m}\sum_{k=1}^{m}\mathrm{PFD}_{k} \tag{10}\]  
where \(m\) is the limited safe time. In the train control system, the time interval between the adjacent two trains is greater than the train braking time, as illustrated in Table 3[20]. There are differences in the train coasting time and braking coefficient between full-service braking and emergency braking.