# Investigation of Railway Network Capacity by Means of Dynamic Flows
## 2 Literature Review
In addition, each train is assigned an earliest departure time and a latest arrival time in a discrete time horizon. Further binary variables are used to depict whether a train reaches its destination within the specified time horizon or not. As part of the model's constraints, it is required that a certain minimum percentage of trains arrive at their destination on time. Finally, it is compulsory that freight trains are only scheduled up to a certain maximum percentage of the train mix. Similar to the previously described approach, one main drawback of the method is that it is only applicable to railway lines and does not extend to more complex and branched networks. Since requests for freight train paths are often received at very short notice, a model with which as many train paths for freight traffic as possible can be planned ad hoc into an existing timetable is developed in [21]. A larger network area is investigated, which is modeled using a time-expanded network. The resulting optimized timetable can then be used to determine a maximum number of trains. The results are strongly dependent on the existing timetable. An investigation based on the same methodology is presented in [22]. Here, too, the usable residual capacities in an existing timetable are determined for a network. First, the number of trains is maximized, and, in a further step, the travel times of the trains are minimized. The capacity of the network can also be inferred from the optimized train journeys. It should be noted, though, that the results are only valid for the respective input timetable, meaning that the method is not suitable for planning processes. In [23], the authors concentrate on the differences between freight and passenger traffic from another perspective. They especially investigate the effect of traffic separation on capacity. Therefore, they optimize train flows, i.e., routes are not fixed and minimize different objective functions such as travel time or energy intensity. Another capacity study that takes infrastructure as well as vehicle circulation into account using a hybrid time-space network is given in [24]. Capacity is defined as maximal transportation, which does include the number of passengers instead of absolute train numbers with the result of a saturated timetable. The approach described in [25] focuses on the differences in train properties such as different speed or stopping patterns. It uses multiobjective optimization to create a saturated timetable whereas the application of the solutions in practice requires further research according to the authors. Also [26] includes different properties of trains. For fixed sets of trains, the time that is needed for all of them to pass the network is calculated. Hereby, different sequences within the set are considered and capacity of a set is defined by the time in which a certain percentage of all sequences can pass the network. This hence makes the approach timetable independent with the limitation that the sets are fixed beforehand. In summary, the methods currently used in practice to determine capacity are very much geared to the subdivision of the railway network into its single elements. The capacity parameters used to derive the number of trains that can be operated differ for the individual elements. To include interactions between elements in capacity assessments, several studies, a selection of which is described above, present approaches that consider longer line sections or the network as a whole. However, these methods often rely on many assumptions and were mostly used to investigate specific problems. For example, some of the methods are limited to sequences of line sections and do not extend to branched networks. Others are based and very much dependent on a specific input timetable or relate to a capacity definition that is based on passenger numbers. They are therefore only suitable to a limited extent as a method for calculating a generally defined capacity in the context of long-term infrastructure planning.