# Train Service Design for Rail Transit Cross-Line Operation Applying Virtual Coupling
## 2 Problem Description  
### Problem Statement  
This study examines the train service design of CO systems during a specified study period. The system comprises two tracks and three lines, as illustrated in Figure 1. We use line \(l\) to denote local lines and cross-line to represent inter-line connections. Line 1 includes \(n_{1}\) stations, and line 2 includes \(n_{2}\) stations. We use \(l\), \(i\) to represent the \(i\)-th station on line \(l\). In addition, \(1\), \(b\), and \(2\), \(a\) are turn-back stations for cross-line trains, respectively. \(1\), \(p/2\), \(n_{2}\) indicates a transfer station of three lines. There are three train types on line 1, line 2, and the cross-line, namely, train 1, train 2, and train 3, respectively. The spatial distribution of three lines divides two tracks into five zones, denoted by set \(Z\) = {I, II, III, IV, V}. In the CO system, zones I, III, and V are non-collinear zones; zones II and IV are collinear zones. Among them, line 1 and the cross-line are collinear in zone II, and line 2 and the cross-line are collinear in zone IV. Train service design represents a social transportation resource-allocation problem. Focusing solely on minimizing passenger travel costs and operation costs may result in an inequitable distribution of train capacity. Especially in the oversaturation state of CO systems, inequitable distribution primarily exists in two aspects: (1) The passenger travel demand is high in collinear zones but relatively low in non-collinear zones. Consequently, train capacity in non-collinear zones may be underutilized when using uniform train compositions. (2) In collinear zones, local line passengers can choose between local and cross-line trains. During peak hours, local line passengers may occupy cross-line train resources, resulting in insufficient capacity on cross-line trains and a crowded travel environment. Therefore, this study aims to address the inequitable distribution of train capacity in CO systems. \begin{table}
\begin{tabular}{c c c c c} \hline
**Literature** & **Line Type** & **Group Type** & **Equity** & **Objective Function** \\ \hline [PERSON] et al. [11] & S & SG & TTE & Min. PTC, OC, TTE \\ [PERSON] et al. [9] & S & SG & TTE & Min. PTC, TTE \\ [PERSON] et al. [19] & S & VC & TCE & Min. PTC, OC, TCE \\ [PERSON] et al. [12] & Y & SG & TTE & Min. PTC, OC \\ [PERSON] et al. [4] & Y & MG & TCE & Min. PTC, OC, TCE \\ This paper & Y & VC & TCE & Min. PTC, OC, TCE \\ \hline \end{tabular} Note: S: Single line; Y: “Y”-type; SG: Single grouping; MG: Multiple grouping; VC: Virtual Coupling; TTE: Travel time equity; TCE: Train capacity equity; PTC: Passenger travel cost OC: Operation cost.  
\end{table} 
Table 1: Comparison of this paper with previous papers. Figure 2 illustrates the VC process considered in this study. The blue, red, and green trains represent train 1, train 2, and train 3, respectively. The number of train groups can be flexibly changed during operation by coupling/decoupling activities on different lines. Generally, train 2 and train 3 operate in VC configuration in zone IV, while train 1 and train 3 operate in VC configuration in zone II. Taking direction as an example, train 2, after completing service in zone V, is required to operate in zone IV in VC configuration with train 3 and perform a virtual decoupling operation at the terminal of zone IV. Train 1, after completing service in zone I, is required to operate in zone II in VC configuration with train 3, and virtual decoupled at the terminal of zone II. The downward direction follows a similar pattern, with train 1 operating in zone II in VC configuration with train 3, and train 2 operating in zone IV in VC configuration with train 3. VC technology is used in CO systems. While the number of carriages for each line remains fixed, trains from different lines can VC in collinear zones and virtually decouple in non-collinear zones.