Journal of Rail Transport Planning & Management
RECIFE-SAT: A MILP-based algorithm for the railway saturation problem
Introduction
Measuring the capacity of a railway infrastructure is a task that has been challenging the academic and industrial community for a long time. It is commonly agreed that the theoretical capacity is the theoretical maximum number of trains that can utilize the infrastructure over time, and is an ideal level that only occurs when critical portions of rail are saturated (i.e., it is not possible to add any further train path). However, this definition does not really identify the conditions under which the maximum number of trains should be planned. Indeed, different specifications of these conditions have been considered in the literature and in the practice (see, e.g., UIC (2012)), bringing to different capacity measurement strategies. Analytical methods exist (see, e.g., Burdett and Kozan (2006) and Bevrani et al. (2015)), in which formulas are defined to quantify the capacity utilization of a generic train in terms of space and time. Then, through these formulas, the space characterizing the infrastructure and the time corresponding to the time horizon of interest are filled with as many generic trains as possible. Some approaches (see, e.g., Florio and Mussone (1995) and Mussone and Wolfler Calvo (2013)) include some probabilistic relation between generic trains, and are hence referred as probabilistic. Further approaches focus on the impact of operational issues on capacity. Specifically, Medeossi et al. (2011) suggest that, when assessing capacity and producing a timetable, the planning of train paths shall take into account different train driving strategies. Then, Goverde et al. (2013) measure capacity in various infrastructure conditions, changing for example the characteristics of the signalling system, in case of traffic perturbations managed through the microscopic train dispatching system named ROMA. Finally, some approaches deal with the so called saturation problem. In this problem, a specific timetable (possibly empty) is considered, as well as a set of saturation trains whose planning may be of interest. As many saturation trains as possible are to be added to the timetable, possibly following some predefined saturation strategies (e.g., giving higher priority to some types of trains or fixing constraints on the frequency of some other types). The peculiarity of the saturation problem is that the result returned is a detailed timetable if all the relevant constraints are considered in the model.
The focus of this study is the quantification of capacity through the solution of the saturation problem. In the literature, different approaches have been proposed to tackle this problem, although they mostly neglect the actual microscopic characteristics of the infrastructure or are capable of solving rather small instances. Apart from rich German literature on capacity analysis and simulation (e.g., Schwanhäußer (1994)) the first work on the saturation problem we know is the one by Hachemane (1997). The latter proposes and tests the CAPRES method: the author models the stations and junctions as nodes and the lines as edges of a graph, and applies constraint programming techniques to solve the problem. Constraints as the minimum headway times, the maximum number of trains which can simultaneously be present at a station, and the respect of correspondences are imposed. A following relevant paper concerning the same problem is the one by Delorme et al. (2001). The authors propose two heuristics capable of saturating junctions considering all their microscopic details. The first heuristic is based on a constraint programming model which is solved using a greedy algorithm. The second one identifies the saturation problem as a unicost set packing problem and its solution is performed through a GRASP (greedy randomized adaptive search procedure) algorithm. The effectiveness of the heuristics is tested on a rather small, though quite congested, junction. Ingolotti et al. (2004) present another approach to saturate a microscopically represented infrastructure. The problem studied is slightly different in its formulation, since it concerns the addition of specific trains to a timetable. Indeed, this can be seen as a particular variant of the saturation problem. A sequential algorithm is designed to add trains progressively to the timetable, and it is applied to instances representing a Spanish line. In the approach proposed by Burdett and Kozan (2006), the capacity is quantified considering a fixed proportion between the number of trains of different types which shall be added to the timetable. Also the proportion of trains travelling in the same direction with respect to the overall traffic is fixed in advance. Thanks to the a priori knowledge of the running time associated to each train and of the minimum headway times between pairs of trains, the capacity of lines and junctions is computed. Then, through the solution of a linear programming formulation, line and junction capacities are combined. The resulting capacity is given by the number of trains travelling along each line and crossing each junction, number which is not constrained to be integer. In a following paper (Burdett and Kozan, 2009), the same authors propose a model based on the hybrid job shop scheduling formulation with time window constraints. To solve this problem, a constructive algorithm and meta-heuristic scheduling technique that operate upon a disjunctive graph model of train operations are utilized. In the paper, despite the movement toward a microscopic infrastructure representation, several aspects, as the interlocking system, are neglected. Abril et al. (2008) present an analysis of how different factors, as train speeds or signal positions, impact on capacity thanks to the decision-support tool named MOM (Modulo Optimizador de Mallas). In this tool, a module supports the solution of the saturation problem. Some infrastructure characteristics are considered at a microscopic level, although not in full detail. Considering an infrastructure with particular characteristics, Tan and Yalcinkaya (2014) saturate a double parallel rail transit line with a mixed-integer programming model. The model adds trains to a timetable, with the objective of obtaining higher frequencies in some special sections and special time periods to cope with large passenger volumes. Finally, Meirich and Nieβen (2016) propose a linear model for the saturation problem. The model considers capacity as known, and the best combination of trains for filling this capacity is to be found. The capacity must be estimated a priori, with methods based for example on queuing theory, as the one reported by Nieβen (2014).
In this paper we propose RECIFE-SAT, which is an optimization algorithm that has recently been implemented within the RECIFE platform. RECIFE is a decision support platform under development at IFSTTAR (France) since the early 2000's. The acronym stays for REcherche sur la Capacité des Infrastructures FErroviaires, i.e., research on the capacity of railway infrastructures (Rodriguez et al., 2007). This platform includes several optimization and human-machine interaction tools aimed at aiding decisions in railway planning and management. Among these tools, RECIFE-SAT is a mixed-integer linear programming (MILP) based algorithm to deal with the railway infrastructure saturation problem. The version of RECIFE-SAT proposed here does not consider any rolling-stock re-utilization constraints. Hence, we neglect the need to guarantee, e.g., some turn-around time between trains' arrival and departure times. This follows the other approaches proposed in the literature. Indeed, the consideration of the additional constraints would be possible through the introduction of further variables and constraints. To solve the saturation problem, RECIFE-SAT models the infrastructure at the microscopic level and implements the route-lock sectional-release interlocking system (Theeg et al., 2009), taking into account all the relevant operational constraints existing in the practice. Thanks to this microscopic representation, it ensures the feasibility of the saturated timetable: in nominal conditions, all trains can be run without the emergence of conflicts, that is, without having trains encountering restrictive signals. Moreover, this is ensured while fully exploiting capacity, since the minimum headway times between pairs of consecutive trains depend on the specific blocking time stairways (Pachl, 2014) representing the trains. For doing so, a precise representation of the interlocking system (signal positions, number of aspects, etc.), the track (slope, curve, speed limit, etc.) and the rolling-stock (acceleration capabilities, mass, length, etc.) is crucial. In the algorithm, we set a time limit after which the best feasible solution found is returned. Remark that, since the decisions are whether to add or not saturation trains to the existing timetable, and if so, which ones and when, a first feasible solution always corresponds to the existing timetable itself. To cope with instances representing very long time horizons, in the algorithm the time horizon to be dealt with is split in time windows tackled sequentially: when the first window is saturated, the algorithm passes to the second one considering the saturation trains already included as non-modifiable, and so on. In the version presented in this paper, RECIFE-SAT performs a bi-objective optimization: on the one hand, it maximizes the number of saturation trains scheduled; on the other hand, the number of freight ones. This bi-objective optimization is performed by implementing an iterative constraint method. We first determine the optimum (or the best solution we can find within a given computational time) with respect to each objective when considered alone. Then, we maximize for the total number of saturation trains imposing the schedule of a given number of freight ones: this number is varied iteratively to build an approximation of the Pareto front.
In addition to the novel formulation proposed for the saturation problem, a main contribution of this paper is the original representation of the infrastructure introduced. It is based on the concept of section including one or more track-circuits and preserving all the characteristics of the microscopic representation while bringing far less computational burden to the decision process. This original representation allows tackling much larger instances than what had been previously done in the literature.
In the experimental analysis, we apply RECIFE-SAT to saturate a portion of the line between Paris and Le Havre, in France. The infrastructure considered is about 100 km long, including fourteen stations and one junction. Along this line, a dense mixed traffic is present in the timetable. We consider two time horizons, at peak time and at peak plus off-peak time, and we show that saturated timetables can be found in a computational time in line with the capacity quantification needs.
The rest of the paper is organized as follows. Section 2 describes the way the railway infrastructure is modelled. Section 3 details RECIE-SAT. Then, Section 4 presents the experimental analysis performed and Section 5 illustrates the main conclusions that can be drawn on account of the obtained results, and gives some hints for future research.
Section snippets
Modeling of the railway infrastructure
The railway infrastructure is composed of sets of tracks which can be traversed by trains, where a safety distance is assured through signals.
The basic element in the infrastructure is the track-circuit. A track-circuit is a portion of track on which the presence of a train is detected by an electrical device. Sequences of track-circuits are typically grouped into block sections, which start and end with a light signal. Several block sections can share track-circuits, for example in presence of
RECIFE-SAT
RECIFE-SAT is a MILP-based algorithm to solve the saturation problem. It consists in the solution of a MILP formulation through a commercial solver, stopping the computation as soon as the optimality of a solution is proven or after the elapse of a fixed computational time. In the latter case, the best feasible solution found is returned.
The size of the instance to be tackled depends on both the infrastructure and the time horizon, which identifies the number of timetable trains to be
Experimental analysis
To assess the applicability of RECIFE-SAT, we consider a portion of the Paris-Le Havre line in France. This is a line characterized by dense mixed traffic: freight, high speed and conventional passenger trains share the infrastructure along the day. The 100 km line portion considered goes from Mantes-Station to Malaunay le Houlme, including sixteen stations and a junction, in addition to the access zones to two important yards. It is shown in Fig. 5.
For this network, we consider the timetable
Conclusions
In this paper we have proposed RECIFE-SAT. It is a MILP-based algorithm to quantify the capacity of a railway infrastructure by saturating it, while considering an initial (possibly empty) timetable to be preserved. The peculiarity of RECIFE-SAT is that it can be applied to rather large infrastructures, representing networks with several stations, still representing these infrastructures microscopically. With this representation, we can actually measure the whole existing capacity and impose
Acknowledgments
The work presented in this papers is part of the SIGIFret project supported by MEDDE, the French Ministry of Ecology, Sustainable Development and Energy, through the PREDIT (Programme de Recherche Et D'Innovation dans les Transports terrestres).
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