# Investigation of Railway Network Capacity by Means of Dynamic Flows
## 4 Exemplary Application to a Sample Network  
### Definition of the Sample Network and the Train Routes  
To validate the developed optimization model, it is tested using a sample network that is based on a real network in North Rhine-Westphalia, Germany. The core network consists of ten vertices connected by eleven line sections, each of which is depicted by two arcs, as described in Section 3.1. The linking lines contain eight virtual vertices connected to the core network by ten line sections (see Figure 3). Apart from two operationally relevant junctions (vertices 3 and 4 in Figure 3), all vertices of the core network correspond to stations, whereas smaller junctions are neglected in favor of computability. Most of the line sections are double-track line sections, but the network also comprises two short single-track line sections, so that all aspects covered by  
Figure 3: Investigated sample network comprising 18 vertices and 42 arcs in total. The evaluated core network is shown in black, with solid arcs referring to double-track and dashed arcs indicating single-track line sections. The linking lines (also containing the virtual nodes) are depicted in gray color [27]. the model can be tested in the validation process. The distances covered by the different line sections lie between three and fifteen kilometers. To apply the model to the sample network, it is first necessary to distinguish the different route numbers and, based thereon, to determine the sets of scheduled trains \(K_{sch}\) and of potentially to be planned trains \(K_{pot}\). In the presented case, we consider 13 local passenger and three freight train route numbers in each of the two directions. All the considered route numbers run on at least one arc of the core network. The set of scheduled trains is defined over a specified number of trains per route number and hour. Here, the freight train route numbers are included in \(K_{sch}\) once per hour and direction and the passenger route numbers are included according to their frequency in the current timetable. Note that the exact schedule itself is not fixed and only frequencies are extracted from the timetable. The defined trains of a route number can be distributed arbitrarily by the model within the hourly slices of the time horizon according to their frequency. In the set of potentially to be planned trains \(K_{pot}\), only trains of the freight train route numbers are considered, as this is a realistic scenario in the actual practical application. Usually, timetables for passenger trains are fixed and additional (short-term) slots saturating the timetable are then mostly allocated to freight trains. For all passenger and freight train route numbers, the departure frequencies and the assigned routes through the sample network of Figure 3 are listed in Table 2.  
With the route numbers defined, it is possible to determine the basic load on the individual arcs of the sample network shown in Figure 3. The resulting numbers of trains are listed in Table 3. \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Route Number** & **Frequency (per Hour and Direction)** & **Route** \\ \hline P1E, P1W & Once & 18–9–10–5–14 \\ \hline P2N, P2S, & Once & 15–5–4–3–2 \\ \hline P3N, P3S, P8N, P8S & Once & 16–7–8–9–10 \\ \hline P4N, P4S & Twice & 14–4–3–2–12 \\ \hline P5E, P5W & Once & 11–1–10–5–14 \\ \hline P6E, P6W, P11E, P11W & Once & 10–1–2–13 \\ \hline P7E, P7W & Three times & 10–5–14 \\ \hline P9E, P9W & Once & 18–9–10–1–2–13 \\ \hline P10E, P10W & Once & 18–9–10 \\ \hline P12E, P12W & Once & 17–7–6–5–14 \\ \hline P13E, P13W & Once & 7–6–5 \\ \hline F1N, F1S & Once (+ potentially to be planned trains) & 16–7–8–9–10–1–11 \\ \hline F2N, F2S & Once (+ potentially to be planned trains) & 15–5–4–3–13 \\ \hline F3E, F3W & Once (+ potentially to be planned trains) & 18–9–10–1–2–13 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Route numbers of passenger (P) and freight (F) trains with their departure frequencies and assigned routes through the sample network.