# Optimization of Stop Plan for Skip-Stop Operation on Suburban Railway Line
## 3 Modeling
### Train Timetable Calculations
#### 3.1.1 Calculation of the Arrival and Departure Times of Trains at Each Station  
1. Departure intervals  
A train schedule should be formulated to satisfy the passenger flow demand on the route. The minimum operating frequency depends on the maximum sectional passenger volume during peak hours as well as the capacity and the passenger load factor of the trains. Then, the interval at which trains depart from the origin station can be calculated using Equation (1):  
\[T=\frac{3600}{f_{1}+f_{2}}=3600/\frac{q_{jmax}}{D\cdot\eta_{max}} \tag{1}\]  
where \(f_{1}\) and \(f_{2}\) are the departure frequencies of local and express trains during the morning peak period, respectively, \(q_{jmax}\) is the maximum sectional inter-station passenger volume during the morning peak period (unit: persons), \(D\) is the capacity of the trains (unit: persons/train), and \(\eta_{max}\) is the maximum passenger load factor of the trains. 2. Inter-station travel times  
The operation of a train between two adjacent stations is simplified to the following process: It accelerates when departing from a station, then travels at a constant speed, and finally decelerates when entering the next station. A train's travel time between two adjacent stations consists of the time when it is uniformly accelerating, the time when it is traveling at a constant speed, and the time when it is uniformly decelerating. For an express train, because it skips some stations, the uniform acceleration time and the uniform deceleration time or both should be removed if it does not stop at one or both \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Definition** \\ \hline \(y_{i,j}\) & Whether train \(i\) overtakes the preceding local train at station \(j\) (\(y_{i,j}=1\) means that train \(i\) overtakes the preceding local train; \(y_{i,j}=0\) means that train \(i\) does not overtake the preceding local train) \\ \hline \(I_{aa}\) & Minimum interval between two successive trains arriving at the same station (seconds, s) \\ \hline \(I_{dd}\) & Minimum interval between two successive trains departing from the same station (s) \\ \hline \(I_{da}\) & Minimum interval between one train departing from a station and the next train arriving at the same station (s) \\ \hline \(I_{la}\) & Minimum interval between a train passing through a station without stopping and the next train arriving at the same station (s) \\ \hline \(I_{at}\) & Minimum interval between one train arriving at a station and the next train passing through the same station without stopping (s) \\ \hline \(I_{dt}\) & Minimum interval between one train departing from a station and the next train passing through the same station without stopping (s) \\ \hline \(T_{R}\) & Duration of the study period (s) \\ \hline \(T_{C}\) & Duration of one cycle (s) \\ \hline \(od_{rs}\) & Passenger OD volume from station \(r\) to station \(s\) during the study period (persons) \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._stations of a segment. The travel time of train \(i\) in inter-station segment \(j\) is calculated using Equation (2):  
\[t_{i,j}=\frac{L_{j}-x_{i,j}\cdot\frac{p^{2}}{2a_{1}}-x_{i,j+1}\cdot\frac{p^{2}}{2 a_{2}}}{v}+x_{i,j}\cdot\frac{v}{a_{1}}+x_{i,j+1}\cdot\frac{v}{a_{2}}=\frac{L_{j}}{v}+x_ {i,j}\cdot\frac{v}{2a_{1}}+x_{i,j+1}\cdot\frac{v}{2a_{2}} \tag{2}\]  
where \(L_{j}\) is the distance between stations \(j\) and \((j+1)\) (meters, m), \(v\) is the speed at which train \(i\) travels between the stations (km/h), and \(a_{1}\) and \(a_{2}\) are the acceleration and deceleration rates of train \(i\), respectively (m/s\({}^{2}\)). 3. Arrival and departure times of trains at each station  
Based on the above-calculated departure interval and inter-station travel times combined with \(t_{2}\), the arrival and departure times of each train at each station can be calculated using Equations (3)-(5). The arrival and departure times of an express train at a station where it does not stop are the same.