# Investigation of Railway Network Capacity by Means of Dynamic Flows
## 1 Introduction  
Ensuring climate-friendly and sustainable mobility requires strong traffic growth on the railways. However, in many places, numerous line sections and nodes are already operating at high capacity today. Therefore, the railway infrastructure must be optimally used, upgraded, and expanded. Future bottlenecks in the network must be detected at an early stage to derive expansion and new construction measures. Railway capacity studies are a means of identifying existing bottlenecks on the one hand and making a statement about feasible train numbers before the realization of a measure on the other hand. Capacity analyses are thus a key component for the planning and dimensioning of railway infrastructure. Depending on the investigated problem, different capacity calculation methods are applied. The internationally widely applied UIC Code 406 [1] introduced by the International Union of Railways (UIC) determines the capacity usage of a specific timetable and is mostly limited to lines [2]. Simulation methods are also based on a specific timetable while being mainly used for the identification of bottlenecks, the assessment of timetable stability and thus for the comparison of different timetable or infrastructure options [3]. They model interactions between different infrastructure elements and are hence also suitable to analyze railway networks. Since they rely on microscopic infrastructure data and are therefore very complex and computationally time-consuming, the network size that can be examined is limited. Analytical methods offer a possibility for capacity determination that is independent of a concrete timetable. These methods are mainly used for the assessment of different infrastructure and operating program variants and have short computing times due to being based on rough information on the operating program instead of concrete schedules [4]. Analytical methods are hence especially suitable for long-term infrastructure planning, which is the application area this paper focuses on. Recognized analytical capacity calculation methods, however, determine capacities for individual building blocks such as line sections, track groups, and route nodes separately, as well as according to different quality standards. As a result, the respective capacities are not comparable with one another and do not sufficiently consider interactions between the single elements. However, since the capacity of an individual element changes depending on the surrounding elements, it is of particular interest for long-term infrastructure planning to make statements about the interaction of elements in larger contiguous areas. Simulations as well as queueing theory, which is used as the main tool in analytical capacity assessments, quickly reach their computational limits when considering networks. Therefore, most alternative approaches to capacity determination in networks make use of other techniques such as very macroscopic mathematical model descriptions or the formulation of optimization models that are solved numerically. Even though these models are more suitable for network descriptions than models based on queueing theory, their complexity also increases rapidly. For this reason, current optimization approaches are largely based on strong simplifications or consider only the details that are relevant regarding the specifically investigated problem. In this paper, an optimization model that aims at determining the usable line capacities in a railway network under consideration of influences by surrounding infrastructure elements and of railway-specific details is presented. These details include, for example, the modeling of minimum headway times on lines and route conflicts in nodes. Furthermore, the model ensures that the load on each line can maintain given quality values through the application of the STRELE-formula [5; 6]. This practice guarantees that the developed model is in line with state-of-the-art methods in Germany and therefore prepares for a potential practical implementation.