Performance analysis of MILP based model predictive control algorithms for dynamic railway scheduling J´ anos Rudan 1 , Bart Kersbergen 2 , Ton van den Boom 2 and Katalin Hangos 3 1 Faculty of Information Technology, P´ azm´ any P´ eter Catholic University 2 Delft Center for Systems and Control, Delft University of Technology 3 Computer and Automation Research Institute, Hungarian Academy of Sciences E-mail: rudan.janos@itk.ppke.hu, b.kersbergen@tudelft.nl, a.j.j.vandenboom@tudelft.nl, hangos@sztaki.hu Abstract— In this paper we analyse the performance of solvers for Mixed Integer Linear Programming (MILP) prob- lems that appear from the model predictive control of railway networks. Our aim is to study techniques that reduce the amount of delay using dynamic traffic management by the rescheduling of trains. Due to the size of the emerging MILP problem and the given constraints on solution time, a thorough analysis of different MILP solution techniques was necessary. It has been proven that a significant speedup in the solution time can be achieved by the proper restructuring of the matrices of the MILP problem. The simulation results also confirm the effectiveness of the proposed control technique and the ability of this setup to analyse the most delay-sensitive trains in the network. I. I NTRODUCTION Considerable effort has been spent recently on the topic of delay management in a railway network via rescheduling trains. Delays can be caused by technical failures, accidents, weather conditions or other unexpected situations. Some of the delays can be handled by a stable and robust timetable [1] but in case of large delays, rerouting of the trains, reshuffling train orders or breaking connections can be necessary to minimize the effect of the disturbance. Various quantitative models have been developed in the past mostly based on mathematical programming methods. A comprehensive survey of these kinds of methods can be found in [2]. A delay-management problem handled as mixed- integer programming first appeared in [3] and more recently in [4]. In [5] a permutation-based methodology was proposed which uses max-plus algebra to derive a Mixed Integer Linear Programming (MILP) to find optimal rescheduling patterns [6]. In this paper the dynamic traffic management of railway networks is formulated as a Mixed Integer Linear Program- ming (MILP) problem [7] in a model predictive framework. The network model is constructed in a way that the the order of the trains using the critical resource (namely the tracks) is con- trolled by binary variables. During the optimization this set of binary variables is determined while minimizing the given cost function. The solution of these kinds of problems can cause serious computational difficulties since integer programming is known to be NP-hard generally. The proposed technique is capable of predicting a given network’s future behaviour in case of a cyclic timetable and find an optimal rescheduling of the trains to minimize the total delay in the network along the prediction horizon. The structure of this paper is the following: in Section II the scheduling problem is introduced - namely the problem of dynamic traffic management of a railway network. In Section III the simulation results can be found while in Section IV the conclusions and possible future works are summarized. II. FORMULATING THE MODEL Consider a railway network having a periodic timetable with cycle time T . In case of nominal operation the trains follow a pre-scheduled route repeated every T minutes where the order of the trains on the tracks is naturally defined by the schedule. The network can be deviated from the nominal operation by an emerging delay. We call the resulting schedule a perturbed mode. With every new schedule we can associate a perturbed mode. A possible new schedule can be represented with a set of constraints determining the new order of the trains on the tracks. The aim of the control method is to find a set of con- straints which describes a schedule having minimal deviation from the nominal schedule. In this section only a short review of the model will be discussed. More detailed explanation of the model can be found in [5], [8]. A. Modelling the trains and the tracks The operation of the schedule can be divided into a set of train runs, where each run starts with a departure and ends with an arrival. A number of train runs, performed by the same ’physical’ train will be denoted as a line. In the remainder of this paper we will simply refer to a ’train run’ as a ’train’. In this particular model we do not take into consideration the possible existence of parallel tracks between two stations which means that overtaking is possible only at stations. We also take the assumption that at the stations can host an unlimited number of trains and the order of departures from stations can be arbitrarily chosen. B. Building up the constraint set The departure and arrival times of the trains have to meet with the following constraints: • Time schedule constraint: a departure may not occur before its scheduled departure time, so we have to satisfy the timetable constraint d i (k) ≥ r d i (k) (1) 2013 European Control Conference (ECC) July 17-19, 2013, Zürich, Switzerland. 978-3-952-41734-8/©2013 EUCA 4562