# Investigation of Railway Network Capacity by Means of Dynamic Flows
## 5 Conclusions and Discussion  
In this paper, an approach to calculate a network-wide capacity was developed. The approach is based on existing optimization methods and considers characteristic capacity values for lines that are determined using established procedures from railway operations science. This ensures that the developed approach is as consistent as possible with existing procedures and quality standards. The derived model maximizes the number of trains in the considered network while ensuring that railway-specific constraints are met. These constraints include, for example, minimum headway times and hourly capacity limits on lines, route conflicts between arriving and departing trains, the number of available tracks, as well as permissible dwell time intervals for trains in stations. As the input of the optimization model, an individual travel time can be specified for each train on each arc of the network; thereby, different train characteristics as well as the maximum permissible speeds of the line sections are reflected in the model. Figure 6: Upper subnetwork (**left** side) and lower subnetwork (**right** side) after saturation in separate analyses. The additional freight train route numbers F1, F2 and F3 are depicted in green, dark red and blue color, respectively. Red numbers on the arcs highlight saturated line sections after optimization [27]. In addition to the total number of trains determined for the entire network under consideration, the number of trains that can be operated can also be evaluated in relation to individual lines of the network. In this way, the train runs determined for the individual routes can be compared with the capacities that were entered into the model as input parameters. The application of the model to a sample network illustrated how the model can be used to examine the extent to which the capacities theoretically available on the individual line sections can actually be used, taking network effects into account. The amount by which the usable capacity is smaller also depends on the network size and the utilization of the network at its edges. Therefore, the network boundaries for such a network analysis should be carefully chosen to minimize the interactions that are neglected by the network selection. The exemplary application of the model is intended to show in a comprehensible way that certain sections of a railway network may already be at their capacity limit, while there is still residual capacity on other sections. Such an investigation can therefore be used, especially with regard to economic aspects, to identify crucial bottlenecks in a network and to derive targeted investment decisions that can increase the capacity of the overall system. The presented study should be considered as a preliminary investigation. Further research is required regarding, for instance, the following aspects: The input parameters of the model are based on the train mix that is given by the initially scheduled trains. However, the train mix actually changes as soon as additional freight trains are added to the network. It should therefore be investigated whether it is possible either to adapt the input parameters in each iteration, or to not exclusively add freight trains but to increase the number of trains in such a way that the initially defined train mix is maintained. Another possibility to extend the model is the inclusion of stochastic travel times to consider deviations from ideal conditions such as adverse weather conditions or time restrictions due to track construction and maintenance. Furthermore, travel times of freight trains should depend on whether a freight train stops or not. One approach would be to define a variable for each freight train that ensures that in the event of an unscheduled stop at a node, the required travel times on the adjacent arcs are increased for this train to adequately model braking and acceleration processes. One more drawback of the presented approach is the rather rough modeling of nodes.