# Investigation of Railway Network Capacity by Means of Dynamic Flows
## 3 Optimization Model for Determining a Network-Wide Capacity
### Input Sets and Parameters of the Model
#### 3.2.3 Time-Expanded Network  
Before the model can be formulated in detail, the time component is first added by initializing the time-expanded network \(G^{T}=\left(V^{T},A^{T}\right)\), which represents the investigated network. The time horizon \(T\), which is needed to construct the time-expanded network, is split into discrete time steps \(t\in\{0,1,\ldots,T\}\). Some of the constraints of the model presented in Section 3.3.3 require that minute steps are used for the discrete time steps and that the selected time horizon covers a full hour period.  
For each time step there is a copy of the original set of vertices \(V\) such that a vertex in the time-expanded network is given as tuple \((i,t)\in V^{T}\), which represents station (or junction) \(i\in V\) at time \(t\in\{0,1,\ldots,T\}\). Furthermore, for every vertex \(i\in V\) there is a source \((i,\alpha)\), in which a train starting at node \(i\) begins its journey through the network, and a sink \((i,\omega)\), in which  
parked on a siding in station 3. The green train no. 4 (k = 4) enters the core network at station 4 at time step 0 and continues its journey towards station 3 right away. In station 3, it has a dwell time of one time step before it departs for station 2 where it directly leaves the core network towards 1.  
Note that in this graphical example, route conflicts and minimum headway times are not considered. For example, if arc \((2,3)\) is assigned an average minimum headway time of \(\overline{h}_{23}=3\) and a minimum required buffer time of \(b_{23}=1\), the model uses appropriate constraints to ensure that in a feasible solution there is a minimum time distance of four time steps between two train journeys on the arc \((2,3)\). So, in this graphical example, it would not be possible for train no. 2 (k = 2) to follow train no. 1 (k = 1) on the line section between stations 2 and 3 directly in the next time step.  
Table 1 summarizes all parameters that are used in the developed model.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Meaning** \\ \hline \(G\) & Railway network (\(G=(V,A)=G_{c}\cup G_{l}\)) \\ \hline \(G_{c}\) & Investigated core network (\(G_{c}=(V_{c},A_{c})\)) \\ \hline \(G_{l}\) & Linking lines (\(G_{l}=(V_{l},A_{l})\)) \\ \hline \(G^{T}\) & Time-expanded network (\(G^{T}=\left(V^{T},A^{T}\right)\)) \\ \hline \(V\), \(V_{c}\), \(V_{l}\), \(V^{T}\) & Set of vertices in \(G\), \(G_{c}\), \(G_{l}\), \(G^{T}\) \\ \hline \(A\), \(A_{c}\), \(A_{l}\), \(A^{T}\) & Set of arcs in \(G\), \(G_{c}\), \(G_{l}\), \(G^{T}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Overview of the sets and parameters used in the model.  
Figure 2: The time-expanded network corresponding to the railway network in Figure 1 with time horizon \(T=6\). Travel arcs are defined according to their assigned travel times and depicted with solid lines. Holdover arcs plotted with dashed lines connect the time copies of the core network’s vertices. Auxiliary arcs are illustrated with dotted lines.