# Optimization of Stop Plan for Skip-Stop Operation on Suburban Railway Line
## 3 Modeling
### Train Timetable Calculations
#### 3.1.3 Overtaking Judgement and Adjustments  
Due to the express trains have fewer stop stations and run faster than local trains, express trains will catch up with the local trains when they are running on the same route. At this moment, the express trains need to overtake the local trains. The overtaking judgment condition can be defined as follows: if the arrival interval between two successive trains cannot satisfy the required safe interval \(t_{min}\) at the last station, the rear train needs to overtake the front train (as shown in Equation (7)).  
1. Overtaking judgement condition:  
\[t_{N}^{i-1,i}=a_{i,N}-a_{i-1,N}-I_{aa} \tag{7}\]where \(t_{N}^{i-1,i}\) is the value used to determine whether the interval between the arrival times of trains (\(i-1\)) and \(i\) at the last station (i.e., station \(N\)) satisfies the required \(t_{min}\) (\(t_{N}^{i-1,i}<0\) means that train \(i\) needs to overtake train (\(i-1\)); \(t_{N}^{i-1,i}\geq 0\) means that train \(i\) does not overtake train (\(i-1\)).  
2. Determination of the locations of the overtaking stations  
When overtaking is deemed essential, it is necessary to determine the locations of the overtaking station. Two scenarios require overtaking to occur (Figure 2).  
Scenario 1: When the departing interval between the local train and the behind express train at station \(j\) satisfies the safety requirement, however, the arrival interval at the station (\(j+1\)) is not satisfied, the express train needs to overtake the local train at station \(j\), as shown in Figure 2a. Whether the train \(i\) overtake the local train at station \(j\) is calculated by Equation (8).  
\[y_{i,j}=\left\{\begin{array}{rl}1,&tdd_{j}^{i-1,i}\geq x_{i,j}\cdot I_{dd}+ (1-x_{i,j})\cdot I_{dt}\;\;\text{and}\;\;\;ta_{i+1}^{i-1,i}<I_{at}\\ 0,&tdd_{j}^{i-1,i}\geq x_{i,j}\cdot I_{dd}+(1-x_{i,j})\cdot I_{dt}\;\;\text{and} \;\;ta_{j+1}^{i-1,i}\geq I_{at}\end{array}\right. \tag{8}\]  
where \(tdd_{j}^{i-1,i}\) is the departing interval between the two successive trains at station \(j\) and \(ta_{j+1}^{i-1,i}\) is the arrival interval between two trains at the station (\(j+1\)).  
Scenario 2: When the arrival interval between the local train and the behind express train at station \(j\) satisfies the safety requirement, however, the departing interval at station \(j\) is not satisfied, the express train needs to overtake the local train at station \(j\), as shown in Figure 2b. In this scenario, whether the train \(i\) overtake the local train at station \(j\) is calculated by Equation (9).  
\[y_{i,j}=\left\{\begin{array}{rl}1,&ta_{j}^{i-1,i}\geq x_{i,j}\cdot I_{aa}+( 1-x_{i,j})\cdot I_{dt}\;\;\text{and}\;\;\;tdd_{j}^{i-1,i}<x_{i,j}\cdot I_{dd}+ (1-x_{i,j})\cdot I_{dt}\\ 0,&ta_{j}^{i-1,i}\geq x_{i,j}\cdot I_{aa}+(1-x_{i,j})\cdot I_{dt}\;\;\text{and} \;\;tdd_{j}^{i-1,i}\geq x_{i,j}\cdot I_{dd}+(1-x_{i,j})\cdot I_{dt}\end{array}\right. \tag{9}\]  
3. Adjustment of the train timetable after an overtaking  
If overtaking occurs, the \(t_{z}\) of the overtaken train at the overtaking station needs to be extended. The departure time of the overtaken train at the overtaking station is changed to the original time at which overtake train passes through the overtaking station plus \(I_{dd}\). The arrival and departure times of the overtaken train at each subsequent station need to be recalculated based on the corresponding inter-station travel time and \(t_{z}\).  
The timetable for each subsequent train needs to be adjusted accordingly based on the \(t_{min}\) constraint. If the interval constraint is unsatisfied, then the absolute value of the difference between the required and actual intervals needs to be added to the current arrival and departure times. The arrival and departure times at each subsequent station need to be recalculated based on the corresponding inter-station travel time and \(t_{z}\).