# Collision Risk Evaluation and Verification of GNSS-Based Train Integrity Detection
## 5 Simulation and Results
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Scenarios**} & \multicolumn{3}{c}{**Open Area**} & \multicolumn{3}{c}{**Forest**} \\ \hline \multicolumn{2}{c}{Percentage} & \multicolumn{3}{c}{98\%} & \multicolumn{3}{c}{2\%} \\ \hline \multirow{2}{*}{States} & \multirow{2}{*}{Upstate} & \multicolumn{2}{c}{Degraded} & Fault & \multirow{2}{*}{Upstate} & \multirow{2}{*}{Upstate} & \multirow{2}{*}{\
\begin{tabular}{c} Degraded \\ State \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} Fault \\ State \\ \end{tabular} } \\ \hline \multicolumn{2}{c}{Percentage} & 95.78\% & 3.26\% & 0.96\% & 73.69 & 7.92\% & 18.39\% \\ \hline \multirow{2}{*}{Positioning error} & \(\mu\)(m) & 6.84 & 16.84 & 26.84 & 8.96 & 18.96 & 28.96 \\ \cline{2-8}  & \(\sigma\)(m) & 24.11 & 24.11 & 24.11 & 29.71 & 29.71 & 29.71 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Statistics of GNSS positioning errors. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Transition** & **Mean Time (s)** & **Corresponding in CPN Model** \\ \hline \multicolumn{2}{c}{upstate to degraded state transition} & 15.31 & UP\_DE \\ \multicolumn{2}{c}{degrad state to upstate transition} & 2.46 & DE\_UP \\ \multicolumn{2}{c}{upstate to faulty state transition} & 20.79 & UP\_FA \\ \multicolumn{2}{c}{faulty state to upstate transition} & 5.91 & FA\_UP \\ \multicolumn{2}{c}{degraded state to faulty state transition} & 2.88 & DE\_FA \\ \multicolumn{2}{c}{faulty state to degraded state transition} & 6.71 & FA\_DE \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Distribution of Each Transition and Parameters. The model is divided into two layers; the first layer (see Figure 10) includes EoT and HoT in open area scenarios and forest scenarios. As shown in Figure 11, the second layer described each scenario in detail; parameterization was done in this layer (EoT_Open_Area, HoT_Open_Area, EoT_Forest, and HoT_Forest share the same model structure in Figure 11, different parameters according to Table 4). Relative time and distributions characteristics are involved in the second layer; position error data can be extracted from up_data, degraded_data, and faulty_data. Figure 10: First layer CPN model of scenario in detail. Figure 9: Test scenarios and transition. The off-line data positioning errors (see Figure 12) aligned to simulation time from CPN tools are imported into MATLAB and integrated with the train motion simulation. The train motion simulation generates kinematic data (i.e., position, velocity, and acceleration) of both HoT and EoT. Train parting events are so rare that they do not provide enough information for statistical analysis, and we cannot have a true experiment yet. To take into account each parameter subject to failure, train parting processes are simulated with different relative distances, velocities, and accelerations over a mission time to provide an interval of time-dependent probabilities for the train integrity detection. Hence in this Monte Carlo process, the system states evolve, while the operating context remains the same and experiences all the possible GNSS localization and train operation scenarios and states. Figure 11: Second layer CPN model of scenario in detail. The autonomous positioning error of offline data corresponding to the simulation time from the CPN tool (see Figure 5) are imported into MATLAB and combined with the simulation data of train motion characteristics to generate kinematic data (i.e., position, speed, and acceleration) for evaluating hot and EoT. Due to the severity and contingency of train separation accidents, they cannot provide enough measurement information for a statistical analysis, and it is difficult to carry out corresponding field tests. In order to consider the parameters of each safety failure model, different relative distances, speeds, and accelerations are used to simulate the train separation process in order to provide the time-dependent probability interval of train integrity detection.