Railway disruption management

Transcription

1 Railway disruption management Delft Center for Systems and Control

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3 Railway disruption management For the degree of Master of Science in Systems and Control at Delft University of Technology November 30, 2015 Faculty of Mechanical, Maritime and Materials Engineering (3mE) Delft University of Technology

4 Copyright Delft Center for Systems and Control (DCSC) All rights reserved.

5 Abstract Railway networks are susceptible to disturbances and disruptions caused by failures in the infrastructure and rolling stock which lead to delays and inconvenience for passengers. Disruption management is currently performed manually by dispatchers and can be improved by using dispatching support systems that can calculate optimal solutions using a model and information on the current condition of the network. In this thesis, an existing macroscopic constraint model for railway networks is extended to include the properties of a disruption. It is found that a disruption can be modelled in a constraint framework by an assignment problem that allocates arriving trains to planned departures. The assignment constraint model is further extended to include rolling stock actions that can be performed at stations to manage the disruption. The constraint model is applied in a mixed integer linear programming problem (MILP) that describes the rescheduling of railway traffic during a disruption. The objective of the MILP aims at minimizing the sum of delays and the number of cancelled trains in the network. A Matlab program is written to simulate disruption scenarios for a case study considering a full blockade at a track section in the south-west of the Dutch railway network. New feasible timetables for a horizon of two hours considering all trains in the network could be found within a minute. A model predictive control (MPC) scheme was developed and implemented in Matlab that can calculate and apply control actions on-line taking into account new delay information from the network. The resulting model predictive controller was used in the simulation of delay scenarios in combination with the disruption from the case study. Solutions for every instance could be found within half a minute, thereby proving the potential of a MPC approach for the management of railway networks during disturbances and disruptions.

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7 Table of Contents List of Figures List of Tables Acknowledgements vii ix xi 1 Introduction Relevance Literature Research goals Thesis structure Modelling railway networks Model for uncontrolled operation Train separation Coupling and connection constraints Control actions Feasibility Mixed Integer Linear Programming Conclusion Case study Disruption scenario Cancelling trains Modelling definitions and assumptions Program Conclusion

8 iv Table of Contents 4 Disruption model Modelling the train lines Timetable constraints Running time constraints Continuity constraints Headway constraints Assignments Case study Results Conclusions Capacity constraints Allocating trains to platforms Ordering trains at platforms Case study Results Conclusions Rolling stock limitations Shunting actions More assignments Resuming the nominal timetable Case study Results Conclusions Distributed model predictive control Model predictive control Case study Creating time-plan Delay scenario Creating subset Building constraint sets Solving the model Updating the time-plan Results Conclusions Conclusions and recommendations Conclusions Recommendations

9 Table of Contents v A List of train lines 81 B Matlab program chapter 4 83 C Matlab program chapters 5 and 6 97 D Matlab program chapter Bibliography 125 Glossary 129

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11 List of Figures 1-1 Microscopic, mesoscopic and macroscopic model of a station, line and junction Directed graph G = (V, E) showing a line which consists of four consecutive train runs from station A to final destination E Directed graph G = (V, E) showing consecutive train runs performed by two trains that use the same resources. The headway order is fixed by four edges between the departure and arrival events of both trains Alternative graph G = (V, F A) showing consecutive train runs performed by two trains that use the same resources. The headway order is determined by picking the green or orange pairs of alternative arcs Main lines of the Dutch railway network. The disrupted track section is shown in red, trains running on the white tracks can be cancelled. Rot=Rotterdam, Dor=Dordrecht, Lzw=Lage-Zwaluwe, Rsd=Roosendaal, Bre=Breda and Ehv=Eindhoven Outtake of a time-plan containing all train runs for a scenario from the case study Remaining train runs of an affected line during a disruption between stations D and E Time distance graph showing a disruption between stations D and E Directed graph corresponding to the time-distance graph shown above Den-Haag-Venlo for a disruption of 80 minutes that leads to a balanced case Den-Haag-Venlo for a disruption of 100 minutes that leads to an unbalanced case Two options for ordering the activities on the platforms Time-distance diagram for the line Den-Haag - Venlo considering a disruption of 60 minutes Station Dordrecht Station Lage-Zwaluwe Platform schedules corresponding to the timetable of figure 5-2 above Assignment problem at one side of a disruption

12 viii LIST OF FIGURES 6-2 Possible solution with deployment of new rolling stock at station C Assignments at every station where trains can be cancelled or turned Ordering of train runs: past (black), in progress (blue), in disruption window (green), after disruption window (red) Result for disruption scenario 101 (table 6-4) for train line IC Model predictive control scheme for control of a railway network Time plan for train line IC1900 for a disruption starting at minute 115 and ending at minute Subset from the time-plan for train line IC1900 in figure 7-2 for a prediction horizon of 60 minutes Solution for the subset shown in figure Ordering of the railway network using 4 zones Box plots showing run times, sum of arrival delays and maximum arrival delay for 60 simulated scenarios Time-distance graph of train line IC1900 for a simulation including a disruption of 60 minutes and a delay scenario

13 List of Tables 1-1 Control actions, network properties and methods used in every chapter Lines crossing the disrupted tracks and lines ending at stations adjacent to the disruption Assignment problem (left) and possible solution (right) Unbalanced assignment problem (left) and possible solution (right) Average number of constraints and variables for the 30 scenarios Results for 30 scenarios of a disruption with a duration of 60 minutes Assigning activities to platforms (left) and possible solution (right) Structure of sequencing variables Different solution to the assignment problem in table Number of constraints and variables for the simulation Two scenarios for an assignment problem at a station with extra rolling stock events Availability of rolling stock actions at the stations where train runs can be cancelled Average numbers of constraints and variables for the 30 simulations disruption scenarios for a disruption of 60 minutes and a horizon of 150 minutes Scenario data on cancellations and rolling stock actions (left table) and average scenario data on constraints and variables (right) A-1 All train lines and frequencies from the Dutch railways timetable

14 x LIST OF TABLES

15 Acknowledgements I want to thank Ton van den Boom and Bart Kersbergen for supervising me during this project. Their enthusiasm and input on the topic proved to be valuable and have led to some interesting results. To my parents and brother, your wisdom and support have kept me going, look how far i have come. Delft, University of Technology November 30, 2015

16 xii Acknowledgements

17 Luctor et emergo

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19 Chapter 1 Introduction 1-1 Relevance Railway networks form an important part of the public transport system in many countries and passengers expect trains to be punctual for their travels. Delays caused by problems in the network lead to inconvenience for the passengers and must be prevented where possible and otherwise minimized. Problems that can occur can be roughly divided into two categories being disturbances and disruptions. Disturbances are defined as small perturbations that can cause delays and thereby influence the departure and arrival times of multiple trains in the network. They can be caused by for example minor defects in rolling stock or by a personnel no-show at stations. The effects of disturbances can be minimized by relatively small control actions such as reordering trains at stations. Disruptions on the other hand, such as a partial or full track blockade at a certain track section, lead to a large decrease in network capacity and require drastic actions such as the cancellation and turning of trains to prevent the propagation of large delays throughout the network. The current approach undertaken by dispatchers in the case of disturbances or disruptions is based on experience and simple but robust dispatching rules such as first come first served (FCFS). They monitor the railway traffic and if it starts to deviate from its nominal behaviour they will intervene by reordering, turning, cancelling or possibly rerouting trains to minimize the adverse effects for passengers. This process is further complicated in the case of large problems such as disruptions because these also require the rescheduling of resource duties such as rolling stock and personnel schedules. In these cases they make use of contingency plans and emergency timetables to manage the traffic. This manual dispatching process takes time and often leads to suboptimal results as only a limited amount of options can be reviewed before a decision must be made. Research in the field of on-line dispatching for railway networks is therefore focusing on the development of dispatching support systems that can quickly model specific situations, calculate optimal solutions on-line and advise on the control actions to implement. The optimization solvers that are used for this purpose must calculate solutions to complex scheduling problems

20 2 Introduction because of the combinatorial nature of rescheduling timetables, rolling stock and personnel rosters. These problems remain challenging because of their complexity, scale and considering that solutions must be provided on-line and therefore more research is needed before such systems can be applied in real networks and trusted upon by the dispatchers. The work performed in this thesis focuses on the modelling and control problem of railway networks suffering from a disruption. Although much research has already gone into the delay management of networks there has not been much attention yet for large network problems such as disruptions that occur less frequently but have a large impact on system performance when they do occur. Therefore in this thesis a constraint model is developed that is able to represent all essential characteristics of a disruption allowing for the calculation of new feasible timetables and corresponding control actions to manage it. The performance of the model is evaluated by means of a case study simulating a disruption on a part of the Dutch railway network. 1-2 Literature Research on the rescheduling of railway traffic focuses on station areas, parts of a network but also entire networks. Depending on the research goal, railway networks can be modelled on different scales with varying levels of detail. The most frequently used models can be classified as follows [17](see figure 1-1): 1. Microscopic models: a highly detailed model including at least all block sections and switch locations in the network under consideration. Models can be used for the scheduling and (re)routing in station areas or larger networks as in [11] but become limited as the number of variables increases quickly. Computational complexity becomes an issue for large instances. Results from optimizations are the allocation of trains to block sections and are close to the requirements for implementation by dispatchers. 2. Macroscopic models: the most abstract representation possible where stations are nodes and lines are links between them. No further details such as block sections or signalling are provided. Station areas are not modelled or only represented by macroscopic parameters such as capacity. This type of model is suited for global scheduling and routing problems as in [20] and [37]. Results from macrosopic models can be departure and arrival times at stations and possible routes. Train speed profiles, switch positions, signalling and routing at stations must be determined and checked for feasibility before arriving at a complete result suitable for dispatching. 3. Mesoscopic models: this model type includes elements from both the microscopic and macroscopic models: certain parts of the network such as stations can be represented with more detail whereas other parts such as the tracks between stations are modelled without further details and remain single links as in the macroscopic model. Mesoscopic models therefore have the advantage of providing more detailed results compared to macroscopic models for certain areas of the infrastructure.

21 1-2 Literature 3 Figure 1-1: Microscopic, mesoscopic and macroscopic model of a station, line and junction When considering rescheduling for disruptions one can distinguish two scenarios: 1. Partial blockade: the number of tracks that is blocked at a certain track section is smaller than the total number of tracks at this section. Trains will still be able to pass the disruption but balancing trains on the remaining track(s) will be necessary in getting passengers to travel in both directions. 2. Full blockade: means that there is no traffic possible between stations at each end of the disruption. In literature on disruption management microscopic as well as macroscopic models are used and the developed methods are often applied to case studies simulating partial or full blockades to calculate new feasible timetables. An overview is provided on the publications that are most relevant and which will be built on in this work. In [1] the problem of optimally incorporating scheduled and unscheduled track maintenance in timetables is considered. In this case the unscheduled track maintenance could also be considered as being a disruption. An algorithm is developed for determining a possession plan that assigns possessors to certain track elements for a certain time window. A local search heuristic called problem space search is used for the scheduling of the possessor actions. The algorithm is applied for a case study with an instance of 50 trains operating on a network spanning 480 km in Queensland, Australia. Results of the optimization show a reduction of total delay of 20% for the scheduled maintenance and 10% for the unscheduled maintenance compared to a simulation using current timetabling methods. The ROMA algorithm in [11] operates on a microscopic level including all block sections from the part of the network that is considered. In [9] and [13] the ROMA algorithm is expanded

22 4 Introduction to include local rerouting strategies in case of unavailable block sections in station areas or in between stations due to disruptions. Several scenarios of unavailable blocks sections in station areas and also between stations are generated. For small instances the problem could be solved to optimality. However, cancelling or turning of trains were not included and effects of the found timetable on the rest of the network were not considered. In [10] a case study is performed applying ROMA on a severely disrupted network. The situation of a single blocked track on a double track line is considered and the frequencies of the train lines on this part of the network are manually reduced. Remaining train lines are globally rerouted via an alternative route. The ROMA system is applied on the timetable with the revised train frequencies to determine an optimal timetable for this specific rerouting strategy. The algorithm is able to find feasible solutions for a time horizon of an hour. In [26] a conflict detection and resolution algorithm is developed for a single track line layout with bi-directional traffic. A macroscopic model is applied defining stations as nodes where trains can overtake and lines in between as single track where no overtaking can take place. Disruptions are defined as time slots in which a track between two stations cannot be used. The rescheduling actions that can be performed are the re-timing and reordering of trains. The objective function to be minimized is the weighted sum of the difference between the actual arrival time at the destination and the scheduled arrival time at the destination for all trains. The MILP is solved for four different disruptions for a system with 6 trains and 5 stations using the GLPK solver. All instances could be solved to optimality within a second. More advanced approaches to control disruptions have not been part of research until very recently by [25] and [37]. In [25] a constraint optimization problem is developed directed towards partial and full track blockades. Since every disruption is unique, the emergency timetables currently in use will most likely not be optimal in every situation. The underlying idea in their research is therefore to adjust the nominal timetable to provide an optimal train service during the disruption. The adjusted timetable must deviate as little as possible from the nominal one to minimize passenger inconvenience. The research focuses on finding a stable cyclic timetable during a disruption that utilizes the available infrastructure optimally. Only the phase in which the disruption persists is considered and transitional effects into and after the disruption are neglected. For a partial and full blockade two separate integer programs are formulated based on an alternative graph similar to the one used by [11]. The objective function aims at minimizing the number of cancelled trains and delays as well as balancing the number of trains in both directions and in time. The algorithms are applied on two real-world cases of the Dutch railways. The timetables from the optimizations are compared to the emergency timetables from NS. With an allowable delay of zero minutes, the same timetable as the emergency timetable is found. However, when allowing a delay of five minutes for all trains major differences start to occur as less trains need to be cancelled and the frequencies of the train lines increase. The work of [25] is further extended in [37] to include the first and third transitional phases in the optimization, creating an algorithm that provides a timetable for the entire disruption process. The algorithm is based on a macroscopic model similar to the one used in [20] and [25]. The resulting algorithm is used for a computational test on a busy part of the Dutch railway network. A maximum allowable departure delay for trains that will depart after the start of the disruption is defined to be 0, 3, 5 or 10 minutes depending on the simulation that is performed. Different disruption scenarios are generated to simulate the effectiveness of the

23 1-3 Research goals 5 algorithm. For all of the track sections 30 scenarios of full and partial blockades were created for a 2 hour disruption after which the timetable should return to normal after an hour. It was found that feasible solutions could be found within 90 seconds assuming an allowable delay of 5 minutes. Better solutions with less cancelled trains are found with an allowable delay of 10 minutes at a cost of a higher total delay of the system. 1-3 Research goals This thesis will build on the work from [25] and [37] with the key difference that disruption scenarios will be solved for an entire network instead of a partial one to ensure that the resulting timetables are feasible for the entire network. Besides the difference in scale there will be no limitation on the maximum departure or arrival delay that may occur as a result of the rescheduling. It will lead to a more generic model with the drawback that the scheduling problems will become more complex. For the modelling of the network the macroscopic constraint model defined in [20], that includes rescheduling actions for disturbances, will be used. The research question can now be formulated as follows: How can the macroscopic constraint model from [20] be extended to include disruption scenarios and how can the resulting model be used in the calculation of feasible timetables and a control policy for dispatchers during disturbances and disruptions? To answer this question the problem is broken down into the following goals: Developing the constraint sets necessary to model a disruption. Augmenting the existing constraint model with the new constraint sets. Creating a Matlab program that composes the disruption model and can calculate a new feasible timetable when a disruption occurs in the network. The resulting timetables can be used by dispatchers as a reference for managing the railway traffic. Applying the disruption model in a model predictive control (MPC) scheme to calculate feasible timetables on-line taking into account disturbances in the network as well as a disruption to provide control actions that can be implemented directly. The timetables and corresponding control actions are the end product of an optimization process that aims at minimizing the inconvenience for passengers that is caused by disturbances and disruptions. The general goal must be the minimization of the delays in the network and the number of trains that must be cancelled because of the disruption but the exact objective is ultimately up to the railway operators to determine. Because of this, the results presented provide no exact solution thereby explaining the title of this thesis.

24 6 Introduction 1-4 Thesis structure The remaining chapters of this report are structured as follows: Chapter two introduces the macroscopic model from [20] that will be used for the modelling of a railway network without a disruption. In addition the method used for optimizing the scheduling problem is discussed. In chapter three the case study is presented at which the developed disruption model will be applied. In chapter four the constraints sets are presented that will model the train movements during a disruption by adding the options for cancelling and turning trains to the model. The disruption model is applied to the case study and results are discussed. Chapter five focuses on a microscopic constraint model for station schedules to correctly represent the capacity limitations at stations during a disruption. The developed station model is added to the disruption model from chapter four and results are presented. In chapter six the disruption model from chapter four is further developed to include shunting yards and rolling stock limitations. In chapter seven the developed model without the station capacity constraints is implemented in a MPC-scheme for the on-line calculation of control actions for disturbance and disruption management. The final chapter will provide conclusions and recommendations about the research performed. An overview of the network properties and methods that are applied in every chapter is shown in table 1-1. Table 1-1: Control actions, network properties and methods used in every chapter. Chapter Cancelling trains Turning trains Station capacity Rolling stock DMPC

25 Chapter 2 Modelling railway networks A railway network can be modelled as a discrete event system having a number of jobs that must be executed on a limited number of resources. The jobs are the train runs and dwell periods and they are performed on the tracks between the stations and the platforms at the stations which are the resources. The dynamics of the system are determined by synchronization: an event such as a departure or arrival can only start if all related preliminary events have been completed. This chapter focuses on the methodology of modelling railway networks and presents the macroscopic model and optimization method as used in [20]. The model will be extended in later chapters and the optimization method will be used to calculate solutions to the scheduling problems. 2-1 Model for uncontrolled operation The macroscopic model consists of a number of train lines that are defined as a series of consecutive train runs performed by the same train. From a macroscopic perspective, these train runs can all be represented by two parameters which are the arrival and departure times. Between train runs the train dwells at each station in order for passengers to get in and out. The train runs that together form one train line are directly related to each other via these dwell times at stations. The above mentioned can be conveniently depicted by a directed graph G = (V, E) having vertices V and edges E. The vertices correspond to the departure and arrival events of the train runs and the edges to the running times and dwell times in between these events. An example of a directed graph for a train line consisting of four train runs is shown in figure 2-1 below. The red nodes indicate the departure events of every train run and the blue nodes the corresponding arrival events. Departures and arrivals belonging to the same train run are connected via arcs that represent the running times. Arrival and successive departure events are related via arcs that represent the dwell times at stations. In this case the train line connects stations A and E via three intermediate stations. When a train has reached its final destination (station E) it is turned to return back to station A. It is clear from this figure how all departure and arrival events are subject to synchronization constraints as they cannot start before the previous events have finished and the process

26 8 Modelling railway networks Figure 2-1: Directed graph G = (V, E) showing a line which consists of four consecutive train runs from station A to final destination E. times have been respected. This system behaviour can be modelled with linear inequality constraints that describe the relations between the events. The constraints necessary for representing the line in figure 2-1 are timetable, continuity and running time constraints: 1. Timetable constraints restrict the departure and arrival events to only take place at or after the scheduled times: d i r di i = 1, 2,..., 8 (2-1) a i r ai i = 1, 2,..., 8 with d i, a i the departure and arrival times of train run i and r di, r ai departure and arrival times of train run i according to the timetable. the scheduled 2. Running time constraints separate departure and arrival events belonging to the same train run by a minimum running time the train needs to traverse the track between two stations: a i d i + τ ri i = 1, 2,..., 8 (2-2) with a i, d i the arrival and departure times of train run i and τ ri the corresponding running time. The difference between the scheduled departure and arrival time of a train run is often larger than the pure running time τ r r ai r di τ ri (2-3) thereby incorporating a recovery time in the run, enabling a train to make up for small delays. 3. Continuity or dwell time constraints relate arrival and departure events of different train runs to each other and correspond to the dwell events at stations: d i+1 a i + τ di,i+1 i = 1, 2,..., 7 (2-4)

27 2-1 Model for uncontrolled operation 9 with d i+1 the departure of train run i + 1, a i the arrival of train run i and τ di,i+1 the dwell time at the station. Similar to the running times the dwell times in the timetable are the absolute minimum wait times at stations and the difference r di+1 r ai τ di,i+1 (2-5) can be larger than the dwell time providing a buffer for trains to make up for earlier incurred delays Train separation The timetable, running time and continuity constraints are able to model a single train line in a further empty network. In reality multiple trains are present and can use the same tracks between stations simultaneously. To ensure a safe distance between these trains the tracks in the real network are divided into block sections with signals at their start points. If a train occupies a block section the signal at the start of this section turns red and no subsequent train can enter the block section. This principle of safe train separation could be incorporated into the model by including all block sections but it would lead to a very large number of constraints as the model shifts from macroscopic to microscopic. However, train separation can be conveniently modelled in a different way with the use of headway constraints as shown in figure 2-2. Figure 2-2: Directed graph G = (V, E) showing consecutive train runs performed by two trains that use the same resources. The headway order is fixed by four edges between the departure and arrival events of both trains. The directed graph shows two trains that pass consecutively through stations A and B and occupy the same tracks between stations. During normal operation train one is in front of train two and leaves stations A and B first. To ensure safe separation between the two the second train can only depart at both stations after train one has left and a specified headway time for train one has elapsed. This is indicated by the two arcs between the departures of both trains at both stations. Although this method does not physically separate the trains at all times by a block section as in the real network, the trains are separated in time. Depending on the train type and its dynamics (regional or interregional), trains may have different running times between the stations in the model. This is the case because smaller

28 10 Modelling railway networks stations at which trains cannot overtake are not explicitly included in the macroscopic model but only in the running times. A regional train might therefore have multiple stops between two modelled stations leading to a longer running time. The departure headway constraints described earlier are therefore not sufficient in all cases as a faster interregional train might catch up with a slower regional train on the same track. To ensure that trains cannot overtake each other on the same track also arrival headway constraints are included in the model. In the case of figure 2-2 these are depicted by the two arcs between the arrival events of both trains. The departure and arrival headway constraints between trains running on the same tracks take the following form: d k d l + τ hl,k,d a k a l + τ hl,k,a (2-6) with d k, d l, a k, a l the departure and arrival events of train runs k and l utilizing the same track, τ hl,k,d the headway time between these departures and τ hl,k,a the headway time between the arrivals. The headway times between two consecutive departures and arrivals are not necessarily the same but in this work it is assumed that they are. The headway constraints in (2-6) must be established between all train runs that utilize the same track at some point in time Coupling and connection constraints A number of train lines in the Dutch network combine their rolling stock during a few train runs at which they use the same tracks. Rolling stock of both lines is combined at the station prior to these train runs and divided afterwards. This behaviour can be modelled with coupling constraints that state that the departure and arrival events of both train lines during the combined train runs must be the same. Consider two combined train runs e and f then the coupling constraints become: d e = d f a e = a f (2-7) These equality constraints force the train runs e and f of two different lines to take place at the same time. Trains from different lines can wait for each other at stations to provide an efficient transfer for passengers. These transfers can be modelled with connection constraints that establish relations between the arrival and departure events of two trains at the same station. Connection constraints are therefore similar to continuity constraints: d l a k + τ cl,k (2-8) with l and k the indices of train runs for two different trains that will connect with a minimum connection time τ cl,k.

29 2-2 Control actions Control actions With the constraint sets presented in section 2-1 a complete macroscopic model of a railway network can be constructed that includes all train runs that are planned in the timetable. This constraint model can be used to simulate the system behaviour under changing conditions such as different running times but is very limited as no control actions can be applied. To control the network when it starts to deviate from the timetable because of disturbances and disruptions it is necessary to have the possibility of changing train orders, turn or cancel trains. To this end the constraint model must be adapted to include these options. The main principle that enables the implementation of control actions in the constraint model is that the inequality constraints presented in section 2-1 can be rendered ineffective. Consider the relation between two continuous variables x, z R x z (2-9) stating that parameter x must always be equal or greater than parameter z. This dependence between x and z can be removed by introducing ɛ = and stating x z + ɛ (2-10) which ensures that the inequality will always be fulfilled for any values x and z. This concept can be extended to controlling the sign of the inequality constraint in (2-9) with the control variable b ɛ {0, ɛ} and its adjoint: bɛ = ɛ if b ɛ = 0 bɛ = 0 if b ɛ = ɛ (2-11) The sign in (2-9) can be switched by composing two constraints x z + b ɛ z x + b ɛ (2-12) of which only one will cause a direct relation between parameters x and z at any time because of b ɛ and its adjoint in the equations. A practical implementation of this principle however will require another formulation and can be done by defining a binary control variable b {0, 1} and a large negative constant β. The function of control variable b ɛ and its adjoint can now be replicated by: b ɛ βb bɛ β(1 b) (2-13) and rewriting (2-12) with this new formulation leads to x z + βb z x + β(1 b) with β x, β z (2-14) This representation does require prior knowledge on the range of values x and z take to ensure that the control action by binary variable b will have the same effect as b ɛ in (2-12).

30 12 Modelling railway networks However this condition poses no limitations for the application in the constraint model that is developed. The continuous parameters between which inequality relations are changed are the continuous arrival and departure events a, d that represent the train runs and these times must always satisfy the timetable constraints in (2-1): d i r di 0 a i r ai 0 i T i T (2-15) with the set T containing all train runs in the network. Because all d i, a i 0 the condition for the correct functioning of (2-14) is satisfied by setting β max{d 1, a 1, d 2, a 2,..., d n, a n } (2-16) with n the number of train runs that will be performed in the network. Although the departure and arrival times in (2-16) are not known beforehand it is possible to estimate their maximum possible values depending on the maximum delays that can occur in the network. The principle of changing the constraint conditions with the use of binary control variables can be applied on the constraint sets presented in section 2-1 to allow control actions be performed on the network. Two control actions will be discussed being the breaking of continuity constraints and changing the order of train runs on a track. Equation 2-17 shows a continuity constraint between the arrival event of train run i and consecutive departure event of train run i + 1 of the same train. Parameter β is a large constant value and parameter b {0, 1} is a binary control variable. d i+1 a i + τ d + βb (2-17) Equation 2-17 can take two forms depending on the value of b {0, 1}: d i+1 a i + τ d if b = 0 d i+1 a i + τ d + β if b = 1 (2-18) If b = 0 the constraint will be "active" meaning that the start time of d i+1 is directly related to arrival event a i and if b = 1 the constraint is "inactive" as the relation between d i+1 and a i is lost. The departure d i+1 can now take place before the arrival it was earlier constraint to. The physical meaning of this control action is the breaking of a continuity constraint and could be used in case train run i is heavily delayed. To prevent the further propagation of this delay the train corresponding to this train run is cancelled after arrival and new rolling stock (from a shunting yard) will be deployed for train run i + 1 which can then depart on time. The principle of turning constraints on and off with binary control variables can therefore be used to manipulate the constraint set allowing the railway traffic to deviate from its nominal operation to compensate for disturbances or disruptions. A second control action that can be performed is changing the order at which trains traverse the same track. It can be used in case a train A is delayed and the standard headway order would cause further delays as consecutive trains normally departing after train A would have to wait for it to arrive, depart again and be given a sufficient headway. Changing the order requires an adaptation to the fixed headway constraints defined in (2-6). Consider two trains performing train runs k and l in consecutive order on the same track. Changing the order at which these trains traverse the track can be done as follows:

31 2-2 Control actions 13 d k d l + τ hk,l + βu d l d k + τ hl,k + β(1 u) a k a l + τ hk,l + βu a l a k + τ hl,k + β(1 u) (2-19) with variable u {0, 1} a binary control variable that switches the order of the train runs depending on its value. If u = 0 train run l will depart and arrive first as the second and fourth constraints are rendered inactive. Otherwise, when u = 1 train run k will depart and arrive first because the first and third constraints are now inactive and the second and fourth active. Figure 2-3 shows the alternative graph G = (V, F A) corresponding to (2-19). V is the set of vertices representing the events, F the set of fixed arcs corresponding to running and dwell times and A the set of alternative arcs modelling the processing order of the trains. Setting u = 0 corresponds to choosing the orange arcs for the headway order and setting u = 1 the green arcs. Figure 2-3: Alternative graph G = (V, F A) showing consecutive train runs performed by two trains that use the same resources. The headway order is determined by picking the green or orange pairs of alternative arcs. If one wants to change the order between three train runs or more and allow all possible orders between these trains it is necessary to include constraint sets shown in (2-19) that relate all trains to each other. In the case of three trains utilizing the same track for train runs {d 1, a 1 }, {d 2, a 2 } and {d 3, a 3 } the required departure headway constraint set should take the following form (the arrival constraint set is found by replacing d 1, d 2, d 3 with a 1, a 2, a 3 ): d 1 max{d 2 + τ h1,2 + β(1 u 1 ), d 3 + τ h1,3 + β(1 u 2 )} d 2 max{d 1 + τ h2,1 + βu 1, d 3 + τ h2,3 + β(1 u 3 )} d 3 max{d 1 + τ h3,1 + βu 2, d 2 + τ h3,2 + βu 3 } (2-20) with u 1, u 2, u 3 {0, 1}. Equation 2-20 clearly shows the structure that is needed: all departure events are related to each other by two constraints that both share the same binary control variable. Only one of two constraints can be "active" at the same time thereby fixing the order of the two events. Rewriting (2-20) leads to the separate constraints:

32 14 Modelling railway networks d 1 d 2 + τ h1,2 + β(1 u 1 ) d 1 d 3 + τ h1,3 + β(1 u 2 ) d 2 d 1 + τ h2,1 + βu 1 d 2 d 3 + τ h2,3 + β(1 u 3 ) (2-21) d 3 d 1 + τ h3,1 + βu 2 d 3 d 2 + τ h3,2 + βu 3 and to ensure no overtaking can take place a set with the identical structure must be included relating all arrival events to each other: a 1 a 2 + τ h1,2 + β(1 u 1 ) a 1 a 3 + τ h1,3 + β(1 u 2 ) a 2 a 1 + τ h2,1 + βu 1 a 2 a 3 + τ h2,3 + β(1 u 3 ) (2-22) a 3 a 1 + τ h3,1 + βu 2 a 3 a 2 + τ h3,2 + βu 3 Breaking or connecting events through continuity constraints as shown in (2-18) as well as changing the order of trains by switching the headway constraints are the most important control actions that are needed when managing railway traffic during a disruption. The concepts presented in this section will be used to develop a complete constraint model that also allows the turning and cancelling of trains in order to find a feasible timetable Feasibility The macroscopic model that is presented in this chapter simplifies the network to leave only the essential elements. The effects of block sections are approximated by headway constraints, small stations are not explicitly modelled and large stations are only nodes in the model because station areas are not included. Because of this limited detail level it is important to note that schedules that are found with this model are not directly implementable in the real network. Conflicts on the tracks between stations can be ruled out by the headway constraints but the same cannot be said about the train movements in the station areas. As these are not explicitly modelled a result from the model might be that multiple trains will arrive or depart at the same time at a station. This could lead to conflicts as they might need the same resources in the station area to reach their tracks. This limitation can be overcome by introducing headway constraints between all trains sharing the same infrastructure at station areas but this will not be done in this work to limit the model complexity. 2-3 Mixed Integer Linear Programming The constraint model that describes the relations between all discrete events and allows changes with the use of binary control variables can be used to solve the dispatching problem

33 2-3 Mixed Integer Linear Programming 15 of finding a new feasible timetable. To achieve this, the dispatching problem is formulated as a Mixed Integer Linear programming problem (MILP) that can be optimized to a certain objective. A linear objective function describes the relative importance between events and control variables and the constraint model describes the relations between the variables that must be satisfied to find a feasible solution. The constraint model consists of continuous variables which are the discrete events and integer variables which are the binary control variables. The constraints can be rewritten into standard form and together with the objective function the MILP becomes: minimize c T z (2-23) subject to Az v with c a constant weight vector and [ ] T z = d 1.. d n a 1.. a n b 1.. b k u 1.. u l (2-24) the variable vector consisting of the continuous departure and arrival events and binary control variables. The objective that must be minimized depends on the requirements and priorities of the railway operators and can therefore be subject to discussion. In general a good objective is to minimize the train delays in the network which can be achieved by weighting all departure and arrival events in (2-24). Weighting these events in proportion to the number of passengers present during every train run can further improve results as it is more important to minimize delays for full trains than for empty ones. This last policy corresponds to minimizing the sum of passenger delays in the network and would require an estimation of passenger numbers in trains based on historical or real-time data. In this work all train runs will be equally weighted assuming all trains to contain an equal number of passengers and the objective thereby corresponds to minimizing the sum of all delays in the network. In chapter 4 the constraint model is extended to include disruptions and this includes the option of cancelling trains. At that point the objective function is again reviewed and weights for the cancellation variables are determined to find a balance between delays in the network and the number of trains that is cancelled. The scheduling problem in (2-23) that must be solved belongs to the class of NP-hard problems [3] and a consequence is that the computation time will scale exponentially with the number of binary variables in worst case scenarios. For large instances it is therefore generally not possible to find an optimal solution in a limited amount of time. Thus from a computational perspective there is a clear trade-off between the detail level a model can have and the size of the modelled network in order to find optimal solutions. If one wants to calculate schedules for a small part of the network then it can be modelled with a high level of detail but the model for an entire network with this level of detail would become too large. This is another reason why a macroscopic approach is used for the modelling of the network. The problem can be optimized with the application of open-source or commercial solvers such as CPLEX or Gurobi. These programs use branch and bound methods in combination with linear programming techniques such as the simplex method and interior-point methods to work through the search space and find the optimal solution. In [20] MILP problems are created for minimizing secondary delays due to disturbances in the Dutch network by reordering trains at stations. Three solvers being GLPK, Gurobi and CPLEX were used

34 16 Modelling railway networks to calculate solutions for the implicit model as well as reduced explicit models and it was found that Gurobi and CPLEX performed equally well with the shortest calculation times for the implicit constraint models. In this work very similar implicit MILP models will be developed and therefore the Gurobi optimizer will be used in this work to solve the constraint optimization problems. All optimizations are performed on a computer with an Intel i7-4500u processor and 8 GB of memory running Ubuntu Linux The constraint model and objective function are composed in Matlab r2015a and solved with Gurobi 6.4 via its Matlab interface [16]. 2-4 Conclusion This chapter started with an explanation of the methodology that can be used to model railway networks in a constraint framework and how control actions can be performed by expanding the framework with the use of binary control variables. The resulting constraint set can be applied in a MILP for solving a scheduling problem to a certain objective by using optimization software such as Gurobi.

35 Chapter 3 Case study In this chapter the case study is presented that will be used to simulate disruption scenarios. An overview is provided on the network that is considered, the location of the disruption, the train lines that will be affected by the disruption and the modelling assumptions. Afterwards, the method used by the Matlab program to compose the constraints for the optimization problem is discussed. 3-1 Disruption scenario The constraint optimization problem that is created to calculate a new timetable in case of a disruption will be applied at a case study for the Dutch railway network. The case study considers a disruption on the track section between the stations Lage-Zwaluwe and Dordrecht which is part of one of three train routes from north to south. Moreover, trains that travel between these stations pass over the Moerdijk bridge at which disruptions often occur due to adverse weather conditions. Figure 3-1 shows the main lines of the Dutch network that is considered in this work as well as the disrupted section in red. At station Dordrecht trains of all interregional and regional lines dwell whereas station Lage-Zwaluwe is a small station at which only regional trains dwell. Directly south to Lage-Zwaluwe is a junction for trains travelling to and from the stations Breda and Roosendaal. The disruption at the bridge is a full blockade meaning that both tracks are unavailable and trains from the south cannot travel further than station Lage-Zwaluwe. Although normally only regional trains stop here, in the case of a full blockade it will also be the endpoint of the interregional lines as passengers will travel by bus from this station to station Dordrecht during the disruption. The trains will be turned at Lage-Zwaluwe and return to their starting destination. Trains arriving from the north will have station Dordrecht as their endpoint and will be turned here as passengers will continue their trip to Lage-Zwaluwe by bus. The turning of trains at the stations Dordrecht and Lage-Zwaluwe will lead to local deviations from the timetable that can cause secondary delays for the rest of the network. The case study aims at calculating a new timetable for the disruption period that is feasible for the entire network and therefore all train lines running

36 18 Case study Disrupted track section Tracks where trains can be cancelled Rot Rsd Dor Lzw Bre Ehv Figure 3-1: Main lines of the Dutch railway network. The disrupted track section is shown in red, trains running on the white tracks can be cancelled. Rot=Rotterdam, Dor=Dordrecht, Lzw=Lage-Zwaluwe, Rsd=Roosendaal, Bre=Breda and Ehv=Eindhoven.

37 3-2 Cancelling trains 19 Table 3-1: disruption Lines crossing the disrupted tracks and lines ending at stations adjacent to the Line Departure Destination Times/hour Blocked IC1900 Den Haag Venlo 2 y IC2151 Amsterdam CS Vlissingen 2 y IC2249 Amsterdam CS Dordrecht 2 n SPR5000 Den Haag CS Breda 2 y SPR5100 Den Haag CS Roosendaal 2 y IC9240INT Roosendaal Amsterdam CS 1 y in the network are taken into account. The timetable that is used is taken from part of the timetable from 2011 and consists of all train lines that run during the afternoon of a weekday. During rush hours in the morning and evening some lines run at higher frequencies and some extra lines are introduced to temporarily increase the transport capacity of the network but these additions are not taken into account. The train lines that are directly affected (must be turned or cancelled) because of the full blockade at the defined location as well as the lines that have the stations Dordrecht or Lage-Zwaluwe as their normal final destination are listed in table 3-1. All lines in table 3-1 run twice every hour in both directions except for the international line IC9240 that runs to and from Brussels but has Roosendaal as its first/last destination in the Netherlands. This international line is not considered in this case study as it would not be turned around in case of a disruption but rerouted instead to reach its destination. 3-2 Cancelling trains Due to capacity limitations at the turning stations Dordrecht and Lage-Zwaluwe or because trains cannot be turned for a return trip, trains of affected lines might need to be cancelled before reaching their final destination. The current approach by manual dispatchers is often to partially cancel trains instead of cancelling them completely to minimize the inconvenience for the passengers. In the case study the same approach will be used and in a certain area around the disruption trains of the affected lines will have the option of being cancelled. Further from the disruption trains from these lines must keep running just as trains from all other lines. In figure 3-1 the network is shown depicting in white the tracks where trains can be cancelled and in red the tracks on which trains cannot. Certain lines perform multiple consecutive train runs on white track sections and if a train of these lines is cancelled then all these consecutive runs are cancelled together.

38 20 Case study 3-3 Modelling definitions and assumptions The following definitions and assumptions are made when modelling the system: 1. The end time of the disruption is known in advance. In reality the end time of the disruption may be unknown until the cause has been found. In that case an infinite disruption time can be assumed and a feasible timetable must be calculated without an end time. When the end time is known, the timetable must be recalculated once more to transition the traffic back towards the nominal timetable. 2. All train runs taking place before the start of the disruption are assumed to be on time. In reality, delays might be presented in the network at the start of the disruption. 3. The train runs that make up the lines from the timetable are all allocated to a certain track number. Although there are two or more tracks between all stations considered in this work, it is not possible for trains to change tracks. Train runs must always take place on the track they are scheduled to according to the timetable. 4. Rolling stock cannot be exchanged between trains from different lines but can be split or combined for specific train runs according to the timetable. 5. Consecutive train runs of directly affected lines can be cancelled on the white tracks as depicted in figure 3-1. Trains can therefore be cancelled at the stations Rotterdam, Roosendaal, Breda and Eindhoven. Train runs of these lines on other tracks than white tracks cannot be cancelled and must always continue. 6. Trains running on the same tracks must hold a safe distance to each other which is enforced by headway constraints. The standard headway times between all trains is three minutes. 7. Trains that are turned at stations Dordrecht and Lage-Zwaluwe will have a minimum turning time of five minutes before their departure. 8. Trains arriving at stations are able to reach every platform from every track. 9. Rolling stock of all train lines can arrive at all platforms of the stations adjacent to the disruption. 10. Although rerouting trains might be an option in some situations, it is not considered in this work and therefore the international train IC9240 is not taken into account in the problem. 11. Train runs taking place at the blocked tracks at the start of the disruption continue their run as normal. Planned departures on these tracks after the start of the disruption are cancelled. 12. The objective of the optimization is the minimization of the sum of delays and number of cancelled trains in the network. Some additional assumptions will be presented in the forthcoming chapters as the model is further developed.

39 3-4 Program Program To determine schedules for the case study presented in section 3-1, a program is written in Matlab that will compose the constraint sets and objective function that forms the input for the Gurobi solver. After a solution is found the results are processed and plotted by the program. The equality and inequality constraint sets that are generated for the model relate departure and arrival events to each other that must be clearly distinguished to end up with a correct and feasible model. To do this a time-plan is created that contains all planned train runs according to the timetable from a start time up to a certain horizon. In this work a part of the Dutch timetable from 2011 is used that includes all trains running in the afternoon of a normal weekday. Train lines that only run during rush hours in the morning and evening are excluded from this timetable although it is possible to add them to accurately model train traffic during these times of day. The train runs that are included for a certain scenario are stored in a matrix that includes all the specifics to model them correctly. An outtake of a time-plan generated for the case study for a certain disruption scenario is shown in figure 3-2. Every row stores information on one train run, the most important columns store the following data: 1. First column: all train runs have a unique index which is used to relate them to each other, these indices are stored in the first column. 2. Column 2 contains the train line number. In the figure train runs are shown of train line IC1900 which corresponds to the interregional line from Den-Haag to Venlo and back. 3. Column 4 contain the track number of the track the train run traverses. 4. Column 5 contains the direction of the train of the line. Value 1 means from start to final destination and value 2 vice versa. 5. Column 7 contains the scheduled departure times. 6. Column 8 contains the scheduled arrival times. 7. Column 9 contains the running times. 8. Column 10 has value -1 if the previous train run took place in the previous cycle. 9. Column 11 contains the dwell times of the train between the departure of the train run and the arrival of the previous train run. 10. Columns 12 and 13 store information on train runs from different lines that are combined. 11. Column 15 stores timestamps for every train run. -4 means that it lies in the past. -3 that it is in progress, -2 that it lies in the future but takes place during the disruption, -1 that it lies in the future and takes place after the disruption has ended. 12. Column 16 contains the indices for the cancellation variables. Certain train runs can be cancelled and every one of these receives a unique cancellation variable.

40 22 Case study Figure 3-2: Outtake of a time-plan containing all train runs for a scenario from the case study.

41 3-5 Conclusion Column 17 contains value 1 for train runs that have a continuity constraint with the previous train run. This data is used to calculate the set of all continuity constraints in the model. 14. Columns 18 and 20 contain the calculated departure and arrival times that are found by the solver. 15. Columns 19 and 21 contain the delays that are incurred by the train runs, these are the differences between the scheduled times in columns 7 and 8 and the calculated times in columns 18 and 20. The plan is generated as follows: 1. A train line is selected from the timetable. 2. The line spans a number of cycles which is determined from column 10. Every switch to a new cycle indicates another train present on the line. 3. For every train present on the line a list of train runs is generated according to the timetable from the start of the disruption up to the defined time horizon. 4. Trains of affected lines utilize the blocked tracks. The list of train runs is checked if one of the blocked tracks is traversed during the disruption period. If it is the train run is removed from the list. This effectively partitions the list into two sections, a set before the disruption is reached and a set after. Indices of removed train runs are stored for later reference. 5. The list with all planned train runs of the train up to the defined time horizon is added to the time-plan. Indices on the first and last train run of this train is stored for later reference. This process is repeated for all train lines in the timetable and the result is a list containing all planned train runs of all trains present in the network during the disruption period. Information on the train runs that belong to the same train as well as information on removed train runs is stored so that it can be used later to generate the constraint sets correctly. 3-5 Conclusion The optimization framework that is developed in this thesis will be applied on a case study to determine its effectiveness. The specifics of the case study as well as the modelling assumptions were explained. Finally the Matlab program that has been written was introduced and the way the information from the timetable is managed was explained.

42 24 Case study

43 Chapter 4 Disruption model The presence of a disruption on a certain track section has direct consequences for the train lines that use it to reach all their intermediate destinations and their final destination. If the disruption leads to a full blockade these trains must be cancelled or turned at stations prior to the disruption. This chapter will start with an explanation of the constraint sets that model the train lines during the disruption. Then the constraints that allow the turning of trains at the two stations adjacent to the disruption will be introduced. The chapter will conclude with the results for the case study at which the developed model is applied. 4-1 Modelling the train lines Before the constraint sets can be determined some additional assumptions must be made: 1. Besides the end time also the start time of the disruption is known in advance. 2. At the stations Rotterdam, Roosendaal, Breda and Eindhoven an infinite amount of rolling stock of all train types is available such that rolling stock is not considered to be a limitation. Moreover, an unlimited number of rolling stock can be shunted to the shunting yards of these stations in the case trains are cancelled. The following sets are defined for building the constraints: 1. set T contains the indices i for all train runs {d i, a i } that are stored in the time-plan introduced in section All train runs in T can be divided into two sets depending on the train line they are part of (see appendix A): set L 1 consists of all train runs that are part of lines that are not directly affected by the disruption, trains of these lines can always run from their start point to their final destination. They might suffer from secondary delays if they share tracks with trains from affected lines that must be rescheduled.

44 26 Disruption model set L 2 consists of all train runs from lines that are directly affected by the disruption because they have train runs on the blocked tracks. The lines IC1900, IC2151, SPR5000 and SPR5100 from table 3-1 belong to this set. Train runs on the blocked tracks cannot take place and trains must be cancelled or turned at the stations adjacent to the disruption. 3. subset C y T contains the indices i for all train runs {d i, a i } from set T that are allocated a cancellation variable and can be cancelled because they utilize one of the tracks from the white track sections in figure subset C n T contains the indices i for all train runs {d i, a i } from set T that cannot be cancelled. 5. set N contains all tracks z = 1,..., n in the network. 6. sets I z contain the indices i of all train runs {d i, a i } taking place on track z = 1,..., n. 7. sets A s contain the indices i of all train runs {d i, a i } with an arrival event at station s. 8. sets D s contain the indices i of all train runs {d i, a i } with a departure event at stations s. 9. set S contains all stations s at which an assignment problem will be solved (stations adjacent to the disruption). 10. sets P s contain all platform numbers at station s. Although the lines in set L 1 are not directly affected by the disruption they might suffer from delays caused by the turning or cancelling actions that must be performed on trains from set L 2. As a result it might be necessary to perform rescheduling actions on train runs of trains in lines L 1 to minimize the adverse effects of the disruption on these events. Train runs that are part of lines in this set are not allowed to be cancelled and control actions are limited to the reordering of trains at stations to minimize secondary delays caused by the disruption. During the disruption, the train lines in set L 2 are effectively cut into two parts that both end Figure 4-1: Remaining train runs of an affected line during a disruption between stations D and E

45 4-1 Modelling the train lines 27 at the stations adjacent to the blocked tracks. During the blockade trains from the affected lines can still travel from their start and end points to the two stations adjacent to the blocked track and back. Figure 4-1 shows multiple instances of an affected line and the consecutive events that can still take place. The red train runs as well as the related planned dwell times in the red box cannot take place due to a disruption between stations D and E. The blue arcs indicate the train runs and related dwell events adjacent to the red train runs that can be cancelled. The black arcs indicate train runs and dwell events that cannot be cancelled and always will take place. If a train run is cancelled all related constraints are rendered inactive as explained in chapter 2. These are the continuity, running time and headway constraints and this ensures that in the model the domains of the variables of non-cancelled train runs are no longer restricted by the variables of the cancelled train runs. Timetable constraints for all departure and arrival events always remain but they pose no restrictions on other variables. The earlier stated assumption on the availability of rolling stock guarantees that the departure events at stations C and F away from the disruption will always take place. If a blue train run with a continuity constraint to a black train run is cancelled, new rolling stock will be deployed at these stations to take over from the cancelled train. This process is not explicitly modelled and it is assumed that enough rolling stock is available at all times to do this. What follows is an explanation of the timetable, running time, continuity and headway constraints that must be composed for all train runs in sets L 1 and L 2 to model the full lines in L 1 and the partial lines in set L Timetable constraints The timetable constraints for sets L 1 and L 2 are identical for all arrival and departure events. They ensure that these events can only take place at or after the scheduled times as stated in the timetable. As shown in (2-1) every train run has a departure and arrival timetable constraint and these can be gathered into: d i r di a i r ai i T (4-1) Certain train runs in set L 2 have the option of being cancelled and in this case all constraints corresponding to these departure and arrival variables will be rendered ineffective except for their timetable constraints. As a result, these departure and arrival times will be minimized by the solver to match the scheduled timetable times. Unique cancellation variables will identify which train runs can be cancelled as will be shown later on in this chapter Running time constraints The running time constraints of train runs in subset C n are identical to (2-2): a i d i + τ ri i C n (4-2) with τ ri the minimum running time of train run i. Train runs in subset C y can be cancelled, requiring an additional term in the running time constraint. The event pairs {d i, a i } with i C y will be assigned a binary cancellation variable and the constraint set becomes: a i d i + τ ri + βc i i C y (4-3)

46 28 Disruption model with c i {0, 1}. If the train run is cancelled, the cancellation variable is set to one rendering the constraint ineffective Continuity constraints Consecutive train runs are connected by means of continuity constraints as presented in (2-4). The continuity constraints between all train runs in set L 1 and all train runs in set L 2 that cannot be cancelled have the form d i+1 a i + τ di,i+1 i C n, i + 1 C n (4-4) with a i the arrival of the train in run i and d i+1 the departure of the same train after the dwell time τ di,i+1 has passed. Train runs in set C y L 2 can be cancelled and if they share continuity constraints with train runs in set C n these constraints must have the option of being rendered ineffective. These constraints correspond to the blue arcs at the stations C and F in figure 4-1. Therefore all continuity constraints between train runs in C y and C n become d i+1 a i + τ di,i+1 + βc i i C y, i + 1 C n d i+1 a i + τ di,i+1 + βc i+1 i + 1 C y, i C n (4-5) with c {0, 1}. The index of the cancellation variable c depends on the index of the train run that can be cancelled. If the arrival belongs to the train run that can be cancelled then c a with a = i and otherwise if the departure d i+1 belongs to a train run that can be cancelled then c a with a = i Headway constraints As described in chapter 2, all train runs are linked to a certain track number z = 1,..., n. Although there are two or more tracks between all stations in the network it is not possible for train runs to be rerouted via another track. The headway constraints are created per track and ensure a safe distance between the trains running on them. A train run that cannot be cancelled will always have headway constraints relating it to the other train runs on the same track. On the contrary, a train run that can be cancelled will need headway constraints that can be rendered ineffective to ensure that it poses no further restriction on the track resource when it is cancelled. Otherwise the cancelled run would still be allocated time on this track that cannot be used by other train runs. The headway constraints are build per track in the same way as shown in (2-20) such that all train orders on a track are possible to achieve. Train runs on the same tracks with planned departures that lie far apart are not likely to be reordered unless very large delays occur in the network. Because of this, it is possible to reduce the set of headway constraints by removing ordering options that will likely be of no importance to decrease the computational burden. This reduction of the search-space is however not further considered in this work. It is important to note that train runs on certain tracks can be a mix of train runs that can be cancelled and ones that cannot. Therefore, let us consider an example of three train runs {d 1, a 1 }, {d 2, a 2 } and {d 3, a 3 } with r d1 < r d2 < r d3 using the same track of which the first two can be cancelled by setting their respective binary cancellation variables c 1 {0, 1} and

47 4-1 Modelling the train lines 29 c 2 {0, 1} to 1. The expression representing all necessary constraints for ordering these trains will look as follows (replace d 1, d 2, d 3 with a 1, a 2, a 3 for the arrival headway constraints): d 1 max{τ h12 + β(1 u 12 + c 1 + c 2 ) + d 2, τ h13 + β(1 u 13 + c 1 ) + d 3 } d 2 max{τ h21 + β(u 12 + c 1 + c 2 ) + d 1, τ h23 + β(1 u 23 + c 2 ) + d 3 } d 3 max{τ h31 + β(u 13 + c 1 ) + d 1, τ h32 + β(u 23 + c 2 ) + d 2 } (4-6) with parameters u ij {0, 1} binary control variables that facilitate the reordering of departure events. The events d 1 and d 2 that are part of the first two train runs can be cancelled and headway constraints corresponding to them must be rendered ineffective if that is the case. Therefore their corresponding binary cancellation variables are included into entries containing these events. Equation 4-6 can be converted into the following headway constraints: d 1 τ h12 + β(1 u 12 + c 1 + c 2 ) + d 2 d 1 τ h13 + β(1 u 13 + c 1 ) + d 3 d 2 τ h21 + β(u 12 + c 1 + c 2 ) + d 1 d 2 τ h23 + β(1 u 23 + c 2 ) + d 3 (4-7) d 3 τ h31 + β(u 13 + c 1 ) + d 1 d 3 τ h32 + β(u 23 + c 2 ) + d 2 Based on the previous example, the headway constraints that are generated between all departure events and between all arrival events in the network can be generalized into: d k τ hkl + β(1 u kl + c k + c l ) + d l d l τ hlk + β(u kl + c k + c l ) + d k a k τ hkl + β(1 u kl + c k + c l ) + a l a l τ hlk + β(u kl + c k + c l ) + a k d k τ hkl + β(1 u kl + c k ) + d l d l τ hlk + β(u kl + c k ) + d k a k τ hkl + β(1 u kl + c k ) + a l a l τ hlk + β(u kl + c k ) + a k d k τ hkl + β(1 u kl + c l ) + d l d l τ hlk + β(u lk + c l ) + d k a k τ hkl + β(1 u kl + c l ) + a l a l τ hlk + β(u lk + c l ) + a k d k τ hkl + β(1 u kl ) + d l d l τ hlk + β(u lk ) + d k a k τ hkl + β(1 u kl ) + a l a l τ hlk + β(u lk ) + a k k C y, l C y k C y, l C n k C n, l C y k C n, l C n k, l : d l > d k k, l I z, z N (4-8)

48 30 Disruption model 4-2 Assignments The timetable, running time, continuity and headway constraints presented in section 4-1 are able to describe all train runs of the full lines in set L 1 and partial lines in set L 2. However, with the current description the partial lines end at the stations adjacent to the disruption as there are no continuity constraints connecting the arrival events with further departures. After all, trains cannot traverse the blocked tracks as long as the disruption persists. Moreover, the planned and still feasible departures from these stations have no physical rolling stock as they would normally have when trains arrive from the tracks that are now disrupted (see figure 4-1). Therefore it is necessary to perform a coupling between the set of arriving events (with rolling stock) to the set of planned and still feasible departure events that have no rolling stock. The coupling must be performed in such a way that as many planned train runs can be continued during the disruption with a minimal amount of delay. Figure 4-2: Time distance graph showing a disruption between stations D and E. The problem with this approach is that the affected lines were never planned to end at the stations adjacent to the disruption. Although trains of these lines will normally pass through these stations from both directions, the times at which they do are not directly related. Moreover the number of affected trains travelling in both directions during the disruption might not necessarily be equal as in figure 4-1. This figure shows two trains in both directions that cannot continue due to the disruption. Depending on when the disruption starts and ends there might be an imbalance in the number of trains travelling towards the disruption and from it. In that case not all trains can be turned at the stations adjacent to the disruption and some must be cancelled at an earlier moment. Coupling the arrival and departure events at these stations is therefore not trivial and must be incorporated in the constraint model in such a way that a feasible solution can be found.

49 4-2 Assignments 31 Figure 4-3: Directed graph corresponding to the time-distance graph shown above. Figure 4-2 shows a time-distance graph of an affected train line that traverses the disrupted tracks between stations D and E from both directions periodically. The orange horizontal lines indicate the start and end times of the disruption which are known in advance. The red lines indicate the train runs that cannot take place due to the disruption. The blue lines are the incoming and outgoing train runs at stations D and E during the disruption that can take place but have the option of being cancelled. In this figure an unbalanced situation is shown in which three trains travelling from station A to K are affected but only two travelling trains from station K to A. Trains with a departure time larger than the end of the disruption can continue their train run across the disrupted tracks as normal. The directed graph corresponding to the case in figure 4-2 including a possible solution is shown in figure 4-3. An assignment must be made to connect the incoming with the outgoing events at stations D and E. A solution approach for this assignment problem can be to create continuity constraints at these stations that connect partial lines to each other. That way the rolling stock arriving from a partial line can again be deployed onto a returning partial line. The orange arcs in figure 4-3 relate the arrivals and departures via continuity constraints, thereby turning two trains at both stations. Because of the imbalance one incoming train at station D must be cancelled at station C and one planned train run from station E to station F must be cancelled. The solution shown in the figure is one of many solutions for this problem that can differ in quality as certain assignments might cause more delays and more cancellations than others. Ideally the assignments are performed in a way that the turned trains can depart on time and the number of cancelled train runs is minimized. However, depending on the moment the disruption starts and ends this might not be entirely possible. With the assignment problem being non-trivial, the continuity constraints created at the stations adjacent to the disruption must be turned on or off depending on which assignments lead to the least cancellations and delays in the network. The assignment problem that must be solved requires that every arrival is assigned to one departure and vice versa. The decision process can be depicted in matrix form as shown in

50 32 Disruption model table 4-1. Here an assignment problem is shown having three arrivals that must be assigned to three departures. One of the possible solutions to the problem is shown in the right table. Table 4-1: Assignment problem (left) and possible solution (right) a 1 a 2 a 3 d 1 b 1 b 2 b 3 d 2 b 4 b 5 b 6 d 3 b 7 b 8 b 9 a 1 a 2 a 3 d d d The binary variables b i {0, 1}, i = 1..9 determine which departure events are connected to which arrival events. As stated earlier, an arrival can only be assigned to one departure and vice versa meaning that the sum of every row and column in the tables 4-1 must be one. This condition can be rewritten into a constraint set and added to the constraint model of the network. Weights can be assigned to the binary variables in the objective function as certain assignments maybe preferred over others. The constraint set for this assignment problem having three arrivals and three departures becomes [7]: b 1 + b 4 + b 7 = 1 b 1 + b 2 + b 3 = 1 b 2 + b 5 + b 8 = 1 b 4 + b 5 + b 6 = 1 b 3 + b 6 + b 9 = 1 b 7 + b 8 + b 9 = 1 (4-9) The current form of the assignment problem only allows for the matching of an equal number of arrivals a i and departures d i. Depending on the start time of the disruption an imbalance of trains is likely to occur as was shown earlier in figure 4-2. Because of the imbalance the number of arrivals and departures used can differ, something the current form of the assignment problem does not account for. In the case of an imbalance, two situations can be distinguished: 1. More arrivals than planned departures: some incoming train runs must be cancelled as the corresponding trains cannot be turned and continued. 2. More departures than arrivals: some planned and outgoing train runs must be cancelled as there is not enough rolling stock for them to take place. To ensure a feasible solution of the problem in these cases, arrival and departure events have the option of being cancelled. Corresponding columns or rows in table 4-1 will then remain empty as there is no assignment. Consider the following unbalanced assignment problem (and possible solution) with three arrivals and two departures: Table 4-2: Unbalanced assignment problem (left) and possible solution (right) a 1 a 2 a 3 d 4 b 41 b 42 b 43 d 5 b 51 b 52 b 53 a 1 a 2 a 3 d d

51 4-2 Assignments 33 with the corresponding constraints: b 41 + b 51 + c 1 = 1 b 42 + b 52 + c 2 = 1 b 43 + b 53 + c 3 = 1 b 41 + b 42 + b 43 + c 4 = 1 b 51 + b 52 + b 53 + c 5 = 1 (4-10) with the binary cancellation variables c i {0, 1}, i = 1, 2,..., 5 for every arrival and departure event. Since at most two arrivals can be assigned to two departures, one arrival must be cancelled which is now possible by setting its corresponding cancellation variable to one. By representing the assignment problem with these constraints the solver is forced to couple or cancel arrival and departure events. These assignments must be translated into continuity constraints that connect the corresponding arrival and departure variables at the stations adjacent to the disruption. The continuity constraints represent the dwell time and turning of the train before departing in the direction it came from. The required set of continuity constraints for the unbalanced assignment problem in table 4-2 can be conveniently written as d 4 max{τ d41 + β(1 b 41 ) + a 1, τ d42 + β(1 b 42 ) + a 2, τ d43 + β(1 b 43 ) + a 3 } d 5 max{τ d51 + β(1 b 51 ) + a 1, τ d52 + β(1 b 52 ) + a 2, τ d53 + β(1 b 53 ) + a 3 } (4-11) The structure of (4-11) clearly resembles that of table 4-2 and the entries in the two statements show how the continuity constraints function: parameter β ensures that the constraints with binary variables set to zero are rendered ineffective whilst the variables set to one "activate" a continuity constraint stating the relation between an arrival and departure. Writing out expression 4-11 leads to the following separate continuity constraints d 4 τ d41 + β(1 b 41 ) + a 1 d 5 τ d51 + β(1 b 51 ) + a 1 d 4 τ d42 + β(1 b 42 ) + a 2 d 5 τ d52 + β(1 b 52 ) + a 2 d 4 τ d43 + β(1 b 43 ) + a 3 d 5 τ d53 + β(1 b 53 ) + a 3 (4-12) The constraints for the assignment problems at the stations adjacent to the disruption can now be generalized to: d i τ di,j + β(1 b ij ) + a j b i,j + c i = 1 i D s j A s b i,j + c j = 1 j A s i D s i D s, j A s s S (4-13) The first equation in the set 4-13 represents the continuity constraints and the second and third the assignment constraints.

52 34 Disruption model 4-3 Case study With the introduction of the constraint sets in section 4-1 and the constraint model in section 4-2 for solving assignment problems a new timetable can be calculated for disruption scenarios considering a full blockade. A program is written in Matlab that composes the constraints and objective function to recalculate the timetable for the case study presented in chapter 2. The Gurobi solver will calculate the optimal solution to the constraint optimization problem. The program runs the following functions in sequence to build the constraints (see appendix B for a detailed explanation of these functions): 1. Function sitspecific.m defines all parameters that are specific to the case study such as the train lines that must be included in the optimization and the tracks that are disrupted. 2. Function timetable.m loads the timetable for all train lines in the network and adds additional information such as the rolling stock types of every train run. 3. Function dagplan.m creates the time plan consisting of all train runs taking place in the time window of interest. Train runs on the disrupted tracks during the disruption are removed and cancellation variables are allocated to train runs that can be cancelled. 4. Function constraintbuild.m creates the constraint sets for the timetable, running time, headway and continuity constraints of the partial and full lines and the constraints belonging to the two assignment problems that must be solved. 5. Function matbuild.m collects all inequality and equality constraint sets generated by function constraintbuild.m and combines them into one constraint matrix. 6. Function gursolve.m loads the constraint matrix from function matbuild.m and defines the objective function that must be minimized. Further solver parameters are defined and the solver is invoked to minimize the objective function. 7. Function procout.m processes the results from the Gurobi solver. The solution vector consisting of all continuous variables and binary control variables is split into smaller vectors and colors are allocated to all train runs for plotting purposes. 8. Function visualisation.m plots all train runs against their calculated times from the solver for single and multiple train lines. The variable vector z from (2-24) is constructed from the continuous departure and arrival event pairs {d, a}, binary cancellation variables c {0, 1}, binary assignment variables b {0, 1} and binary headway order variables u {0, 1}: [ d 1.. d i a 1.. a i c 1.. c j b 1.. b k u 1.. u l ] T (4-14) with weight vector c [ ] (4-15)

53 4-4 Results 35 that determines the relative importance between delays and cancelling events. The cancellation variables receive a large weight to indicate that small delays in the system are more desirable than the cancellation of trains. Only in the case that the assignment problems are unbalanced or if feasible assignments would lead to large delays are the cancellation of trains allowed. If the passenger delay would be the objective for minimization the unique cancellation variables should receive weights that reflect the delays that the passengers in the cancelled train would incur. In this case, all cancellation variables receive the same weight and these are not directly related to passenger numbers in trains. The binary assignment and headway control variables receive no weight at this point but could be used later to indicate that certain assignments or headway orders are more desirable than others due to the practical implementation of the calculated schedule for example. The structure of the inequality and equality constraint matrices become A timetable A runningtime A continuity A continuity,lagezwaluwe A continuity,dordrecht A headway z v timetable v runningtime v continuity v continuity,lagezwaluwe v continuity,dordrecht v headway (4-16) [ Aassign,lagezwaluwe ] [ vassign,lagezwaluwe ] A assign,dordrecht z = v assign,dordrecht (4-17) which can be further rewritten into standard form (2-23) to be solved by Gurobi. 4-4 Results The program is used to calculate new timetables for two disruption scenarios to illustrate how the assignment constraints model the disruption. All train lines in the timetable take part in the optimization. Train lines that are directly affected are turned or cancelled and unaffected train lines may be rescheduled as a consequence of the disruption. Figure 4-4 shows the time-distance graph for the affected train line IC1900 for a disruption of 80 minutes starting at minute 120 and ending at minute 200. Figure 4-5 shows the solution for the same train line but for a disruption with a duration of 100 minutes, starting at minute 100 and ending at minute 200. Every train has its own color in the time-distance graph to indicate the paths the rolling stock take and where they are turned or cancelled. The start and end times of the disruptions are shown by the horizontal dotted red lines. The colored dashed lines indicate the planned train paths according to the timetable. The black dashed lines indicate train runs that are cancelled. Figure 4-4 shows a disruption that leads to a balanced situation in which three trains on both sides cannot continue their trip because of the unavailability of the tracks between the stations Lage-Zwaluwe and Dordrecht. At station Dordrecht these three trains are turned and because of planned departures times that are later than the arrivals these three trains can start their return trip on time. At station Lage-Zwaluwe the trains arriving from the south part of the network are also turned and must dwell for five minutes before they can depart. Because the arrival times of these trains are systematically larger

54 36 Disruption model Time [minutes] DenHaagCSDenHaagHS Delft Aansl Rotterdam Dordrecht LageZwaluwe Breda Tilburg Boxtel Eindhoven Deurne Venlo Station Figure 4-4: Den-Haag-Venlo for a disruption of 80 minutes that leads to a balanced case Time [minutes] DenHaagCSDenHaagHS Delft Aansl Rotterdam Dordrecht LageZwaluwe Breda Tilburg Boxtel Eindhoven Deurne Venlo Station Figure 4-5: Den-Haag-Venlo for a disruption of 100 minutes that leads to an unbalanced case.

55 4-4 Results 37 than the planned departures there is a mismatch that leads to delays. All trains depart again towards Breda with a delay of 13 minutes. Because of the slack that is incorporated in the running times and dwell times, these trains can make up for the delay and arrive almost on time at their final destination. In this scenario all trains are turned and a delay is chosen over the cancellation of trains. Tuning the weight of the cancellation parameters in the objective function can be used to determine when cancelling a train should be chosen over a (large) delay. Figure 4-5 shows a disruption that leads to an unbalanced case for the assignment problems that must be solved at the stations adjacent to the disruption. Three trains from the north should traverse the disrupted tracks during the disruption whereas there are four from the south. Because of this mismatch, only three trains are turned at Dordrecht and Lage-Zwaluwe and one train run on both sides is cancelled. The solution that is found is optimal as the assignments do not lead to delays in the network. Other solutions to the assignment problem would lead to the cancellation of two train runs but also a mismatch between arrivals and departures which would cause delayed departures and thus a larger value for the objective function. Now the program is used to calculate solutions to a range of scenarios: 30 simulations are performed all considering a disruption with a duration of 60 minutes with the disruption in the first simulation starting at minute 100. The end time of every simulation is 60 minutes after the end time of the disruption. The moment the disruption starts is shifted by a minute for every scenario. Because the Dutch timetable is cyclic with a period of 30 minutes the results for the remaining half hour will be very similar to these 30 scenarios and are therefore omitted. Table 4-3 shows the average number of constraints and variables for the 30 simulations. Table 4-4 shows the results of the optimizations with in the columns: 1. t start : the start time of the disruption in minutes. 2. t solve : the calculation time of the Gurobi solver in seconds. 3. D: the sum of departure delays of all train runs in minutes. 4. L 1 : the sum of departure delays of train runs in set L 1 in minutes. 5. max L 1 : the maximum departure delay occurring in set L 1 in minutes. 6. L 2 : the sum of departure delays in set L 2 in minutes. 7. max L 2 : the maximum of the departure delays in set L 2 in minutes. 8. C: the number of cancelled train runs in set L 1. The results indicate how much influence the start time of the disruption has on the delays and the number of cancellations in the network. Depending on the start time, the assignment problems on both sides of the disruption can be balanced or unbalanced and there can be a mismatch between arrivals and departures or no mismatch. The table shows that the scenarios starting from minute and are the same with identical delays and no cancellations. They all correspond to a balanced case in which all trains can be turned and none need to be cancelled. Because of a mismatch, some of the

56 38 Disruption model Table 4-3: Average number of constraints and variables for the 30 scenarios number of constraints continuous variables 7481 cancellation variables 20 assignment variables 15 reordering variables Table 4-4: Results for 30 scenarios of a disruption with a duration of 60 minutes. t start t solve D DL1 max D L1 DL2 max D L2 C

57 4-5 Conclusions 39 turned trains will incur a delay leading to a total departure delay in set L 2 of 74 minutes. A disruption starting at minute 107 or 122 leads to an unbalanced case for which two train runs must be cancelled to find a feasible solution. All other train runs can be turned without a mismatch leading to zero departure delays in set L 2. The scenarios from minute 108 to 121 are very similar with an identical sum of departure delays for all trains in set L 1 and L 2, with the difference being between the number of cancelled train runs. Depending on the start time of the disruption train runs for certain series in set L 2 are cancelled because of an unbalanced number of arrivals and departures of that series during the disruption. Finally it can be concluded that the adjustments made at the disruption area to find a feasible timetable does not lead to delays in any train line from set L 1 for any of the 30 scenarios. This can be attributed to the fact that not many lines from L 1 utilize the same tracks as the affected lines in L 2 and the disruption is therefore fairly isolated from the rest of the network. 4-5 Conclusions This chapter presented the main elements for calculating new timetables in the case of a disruption. The modelling of the partial lines affected by the disruption and full lines has been discussed. Timetable, running time, continuity and headway constraints were presented that together form a complete model of the network with the exception of the disrupted area. Subsequently the concept of assigning arrival events to departure events with the use of equality constraints was introduced. The equality constraints model the decision process and in their turn control continuity constraints that connect the arrivals and departures via an inequality relation. A program was written in Matlab that constructs the constraint model as presented to calculate new timetables for the case study. Finally, results were shown and discussed for 30 scenarios for a disruption of 60 minutes. It was found that cancelling and turning of trains at the disrupted area does not have any impact on the performance of the rest of the network. Optimal solutions for all simulations could be found within half a minute and the current model can therefore be used for the on-line calculation of new timetables.

58 40 Disruption model

59 Chapter 5 Capacity constraints In the previous chapter constraint sets were introduced for modelling partial and full train lines and the turning of trains at stations adjacent to the disruption. Because of the choice for a macroscopic model, it is implicitly assumed that these stations have enough capacity to process all the arrivals and departures. During normal operation the timetable is followed which is feasible at all stations meaning that trains can arrive at their planned times as there is always a platform available. However, this may not necessarily be true in the case of a disruption as the station usage at the two stations adjacent to the disruption will then be completely different. Depending on the mismatch between arrivals and departures during the disruption the dwell times may be long resulting in a shortage of platforms. The assignment problem that is solved for the two stations in the previous chapter only accounts for the turning of trains but does not include the station capacity limitations. There might be situations in which an assignment cannot be made due to the fact that at that moment all platforms are occupied. In this chapter the constraints are presented that can be used to solve the station scheduling problem taking into account the available resources at the station. The first section will introduce the constraints for allocating trains to different platforms. In the second section the constraints needed for ordering the trains on the platforms are discussed. 5-1 Allocating trains to platforms The stations Dordrecht and Lage-Zwaluwe both have a certain amount of resources which are the platforms where the trains can arrive and dwell. In this work it is assumed that every track at these stations has a platform next to it and that all arriving trains can reach every platform. It is also assumed that every rolling stock type can dwell at every platform. This might not always be the case as some platforms are designed for regional trains that are shorter than interregional trains. The assignment problem that is solved in chapter 4 couples arrivals to departures at these stations but does not allocate the coupled arrival departure sets to a platform. The constraint optimization problem must therefore be expanded with constraints that model the allocation of trains to platforms in such a way that only one train occupies a platform at the time.

60 42 Capacity constraints The assignment of an arrival event a u and departure event d v at a station with a turnaround/dwell time τ d leads to an activity E = (a u, d v ). This activity must be assigned to a platform and during the activity no other activity can be assigned to the same platform. Only after the activity has ended and a headway time is respected a new activity can be allocated to the platform. The allocation of trains at a station is therefore a disjunctive and non-preemptive scheduling problem that is NP-hard [3]. Each resource can execute only one activity at the time and activities cannot be interrupted. Allocating each activity to a platform can be modelled by adapting the assignment constraint set presented in (4-13). Assigning a specific arrival to a specific departure event can now be done by multiple binary control variables instead of only one. All assignment variables belonging to the same arrival departure assignment represent the assignment to a different platform. If a station has n platforms every assignment option must have n binary assignment variables to ensure that every activity can take place at every platform. For an assignment problem with three arrivals and departures and a station with two platforms the set of assignment variables and a possible solution are shown in table 5-1. The left table shows an Table 5-1: Assigning activities to platforms (left) and possible solution (right) a 1 a 2 a 3 d 4 b 1 41, b2 41 b 1 42, b2 42 b 1 43, b2 43 d 5 b 1 51, b2 51 b 1 52, b2 52 b 1 53, b2 53 d 6 b 1 61, b2 61 b 1 62, b2 62 b 1 63, b2 63 a 1 a 2 a 3 d 4 1, 0 0, 0 0, 0 d 5 0, 0 0, 1 0, 0 d 6 0, 0 0, 0 1, 0 assignment problem with all necessary binary assignment variables b p i,j with i, j the departure and arrival indices and p the platform number to which the assignment is made. To satisfy the conditions that an arrival is only assigned to one departure and vice versa, the following equality constraints must be respected [7]: Columns Rows b b b b b b 2 61 = 1 b b b b b b 2 43 = 1 b b b b b b 2 62 = 1 b b b b b b 2 53 = 1 b b b b b b 2 63 = 1 b b b b b b 2 63 = 1 (5-1) This type of assignment problem can again be extended to unbalanced cases by including cancellation variables for all departure and arrival events in the set of equality constraints (5-1). The set of continuity constraints belonging to this assignment problem becomes: d 4 max{τ d41 + β(1 b 1 41 b 2 41) + a 1, τ d42 + β(1 b 1 42 b 2 42) + a 2, τ d43 + β(1 b 1 43 b 2 43) + a 3 } d 5 max{τ d51 + β(1 b 1 51 b 2 51) + a 1, τ d52 + β(1 b 1 52 b 2 52) + a 2, τ d53 + β(1 b 1 53 b 2 53) + a 3 } d 6 max{τ d61 + β(1 b 1 61 b 2 61) + a 1, τ d62 + β(1 b 1 62 b 2 62) + a 2, τ d63 + β(1 b 1 63 b 2 63) + a 3 } (5-2) The entries in (5-2) indicate that all continuity constraints can now be turned on or off by one of two binary control variables. Only one of every pair {b 1 i,j, b2 i,j } can become one at the time.

61 5-2 Ordering trains at platforms 43 If both remain zero, constant β ensures that the constraint is turned off. The expression can be further rewritten into the constraints: d 4 τ d41 + β(1 b 1 41 b 2 41) + a 1 d 4 τ d42 + β(1 b 1 42 b 2 42) + a 2 d 4 τ d43 + β(1 b 1 43 b 2 43) + a 3 d 5 τ d51 + β(1 b 1 51 b 2 51) + a 1 d 5 τ d52 + β(1 b 1 52 b 2 52) + a 2 (5-3) d 5 τ d53 + β(1 b 1 53 b 2 53) + a 3 d 6 τ d61 + β(1 b 1 61 b 2 61) + a 1 d 6 τ d62 + β(1 b 1 62 b 2 62) + a 2 d 6 τ d63 + β(1 b 1 63 b 2 63) + a 3 The equality constraints (5-1) including cancellation variables for all events become: Rows 3 2 b k 4n + c 4 = 1 n=1 k=1 3 2 n=1 k=1 3 2 n=1 k=1 b k 5n + c 5 = 1 b k 6n + c 6 = 1 Columns 6 2 b k n1 + c 1 = 1 n=4 k=1 6 2 n=4 k=1 6 2 n=4 k=1 b k n2 + c 2 = 1 b k n3 + c 3 = 1 (5-4) The constraints for the assignment part of the station scheduling problem can now be generalized to: d i τ di,j + β(1 b p i,j ) + a j i D s, j A s p P s b p i,j + c i = 1 i D s s S (5-5) j A s p P s b p i,j + c j = 1 j A s p P s i D s The first equation in (5-5) represents the set of continuity constraints and the second and third the summations of binary variables over the rows and columns of the assignment matrix. 5-2 Ordering trains at platforms The constraints in (5-5) model the allocation of trains to platforms. Since only one train can occupy a platform at the time, an order must be established between all activities that are assigned to the same platform. Headway times between the departure of an ending

62 44 Capacity constraints activity and the arrival of a new activity must ensure that trains can enter and leave the platform without conflicts. Although the required constraint set is more complicated, this concept clearly resembles the headway constraints for ordering trains that traverse the same track. The mixed integer constraint model for multi-machine allocation as presented in [28] is adapted and used to build the constraints that are needed to separate the trains on the platforms. The method is explained by using the example and its possible solution shown in table 5-1. In this example three assignments are made: {a 1, d 4 }, {a 2, d 5 } and {a 3, d 6 }. Activities {a 1, d 1 } and {a 3, d 6 } are allocated to platform two and activity {a 2, d 5 } is allocated to platform one. The activities that take place on the same platform {a 1, d 1 } and {a 3, d 6 } are ordered with the use of binary sequencing variables y i,j {0, 1}. There are two options as shown in figure 5-1, first the red activity and then the blue one and vice versa. The green activity {a 2, d 5 } is allocated to platform one and no order between other activities needs to be established. The ordering problem starts with listing all headway constraints between the Figure 5-1: Two options for ordering the activities on the platforms red, blue and green events just as in the headway problem for trains running on the same track. The headway constraints for ordering the red, blue and green activities on one platform become: a 3 d 4 + τ h + β(1 y 43 ) a 2 d 4 + τ h + β(1 y 42 ) a 2 d 6 + τ h + β(1 y 62 ) a 1 d 6 + τ h + β(1 y 61 ) a 1 d 5 + τ h + β(1 y 51 ) a 3 d 5 + τ h + β(1 y 53 ) (5-6) with y i,j {0, 1}, i D s, j A s binary sequencing variables that determine the order between two activities. The sequencing variables establish an order between a specific arrival and departure event. For the example the structure of these variables is shown in table 5-2. Table 5-2: Structure of sequencing variables d 4 d 5 d 6 a 1 y 41 y 51 y 61 a 2 y 42 y 52 y 62 a 3 y 43 y 53 y 63 Because the three assignments {a 1, d 4 }, {a 2, d 5 }, {a 3, d 6 } are already fixed in this example the sequencing variables y 41, y 52, y 63 are not used.

63 5-2 Ordering trains at platforms 45 To establish an order between the two activities that are on the same track one of the pair sequencing variables {y 43, y 61 } must be one when both events are allocated to the same track. To force this decision the following constraints are introduced: b b 1 63 y 61 y 43 1 b b 1 52 y 51 y 42 1 b b 1 63 y 62 y 53 1 b b y 61 + y 43 3 b b y 51 + y 42 3 b b y 62 + y 53 3 (5-7) If the activities are allocated to different tracks no order needs to be established and it must be ensured that both of the headway constraints between these activities are turned off by setting y v,w = y w,v = 0. This can be done by introducing a third constraint set: b b y 61 + y 43 2 b b y 51 + y 42 2 b b y 62 + y 53 2 b b y 61 + y 43 2 b b y 51 + y 42 2 b b y 62 + y 53 2 (5-8) which must be added for all combinations of allocating the activities to different platforms. The constraints shown in (5-8) are only the ones relevant to the solution of the assignment problem from the example. Because the pairs {a 1, d 4 } and {a 3, d 6 } are allocated to the same platform, the constraints in the first column of (5-7) force a decision between the pair {y 61, y 43 } resulting in one of two headway orders in column one of (5-6) as shown in figure 5-1. Because activity {a 2, d 5 } is on a different platform, the constraints in the second and third column of (5-8) ensure that the pairs {y 51, y 42 }, {y 62, y 53 } are all zero. As a consequence, no headway order is forced between activity {a 2, d 5 } and the other activities as can be seen in the second and third columns of (5-6). The constraint sets in (5-6), (5-7) and (5-8) form the basis for fixing train orders on platforms. The example above explains how the constraints can establish an order between three known activities. The additional problem in the case of the platform assignment problem is that the activity pairs {a u, d v } are not known in advance. If the assignment problem is solved in a different way, other pairs of sequencing variables are needed to fix headways between activities on the same track. Because the binary sequencing pairs y a,b, y c,d as used in the constraints in (5-7) and (5-8) are not known in advance, the constraint set in (5-8) must be adapted to ensure that these constraints pose no limitations to the solution space of the sequencing variables. This can be seen by considering a different solution to the assignment problem in table 5-1 with the assignments made on the anti-diagonal as shown in table 5-3. For this so- Table 5-3: Different solution to the assignment problem in table 5-1 a 1 a 2 a 3 d 4 0, 0 0, 0 1, 0 d 5 0, 0 0, 1 0, 0 d 6 0, 1 0, 0 0, 0 lution the relevant pairs of sequencing variables become (see table 5-2): {y 51, y 62 }, {y 41, y 63 } and {y 42, y 53 } instead of {y 42, y 51 }, {y 43, y 61 } and {y 53, y 62 }. Every sequencing variable of every pair must be able to become zero or one independent of the assignment solution that is chosen. The current form of the constraint set in (5-8) does not allow this and the sequencing

64 46 Capacity constraints constraints belonging to one assignment solution can restrict the solution-space for the sequencing variables belonging to another assignment solution. Constraint set (5-8) is therefore adapted to β(2 b 1 41 b 2 63) + y 61 + y 43 0 β(2 b 1 41 b 2 52) + y 51 + y 42 0 β(2 b 1 52 b 2 63) + y 62 + y 53 0 β(2 b 2 41 b 1 63) + y 61 + y 43 0 β(2 b 2 41 b 1 52) + y 51 + y 42 0 β(2 b 2 52 b 1 63) + y 62 + y 53 0 (5-9) which ensures that the constraints belonging to a certain assignment solution are only "active" when both assignment variables are one. If one of them or both are zero these constraints do not restrict the solution space for the sequencing variables. Because of the unknown assignments also all possible headways orders must be included. Moreover, cancellation variables must be included in case an event can be cancelled. The complete headway constraint set for the example considering unknown assignments can be written in the form: a 1 max{τ h14 + β(1 y 41 + b b c 1 + c 4 ) + d 4, τ h15 + β(1 y 51 + b b c 1 + c 5 ) + d 5 a 2 max{τ h24 + β(1 y 42 + b b c 2 + c 4 ) + d 4, τ h25 + β(1 y 52 + b b c 2 + c 5 ) + d 5 a 3 max{τ h34 + β(1 y 43 + b b c 3 + c 4 ) + d 4, τ h35 + β(1 y 53 + b b c 3 + c 5 ) + d 5 τ h16 + β(1 y 61 + b b c 1 + c 6 ) + d 6 } τ h26 + β(1 y 62 + b b c 2 + c 6 ) + d 6 } τ h36 + β(1 y 63 + b b c 3 + c 6 ) + d 6 } (5-10) Every entry in (5-10) corresponds to a headway constraint that is paired with another headway constraint in the case two activities are on the same track. If a certain arrival is assigned to a certain departure, the corresponding constraint in (5-10) is turned off by the binary assignment variable b p i,j as the arrival and departure event belong to the same train. If an arrival or departure is cancelled all corresponding headway constraints are turned off. The station ordering constraint set can now be generalized into: a j τ h + β(1 y ij + p P s b p ij + c j + c i ) + d i i D s, j A s b k ab + b k cd y cb y ad 1 a, c D s b, d A s b > a d c k P s b k ab + b k cd + y cb + y ad 3 a, c D s b, d A s b > a d c k P s β(2 b u ab b v cd) + y cb + y ad 0 a, c D s b, d A s b > a d c u, v P s u v (5-11) The complete station scheduling problem can now be built for an assignment problem having a set departures D s, arrivals A s and platforms P s by building the constraint sets shown in (5-5) and (5-11). s S 5-3 Case study The constraint station scheduling model developed in the previous sections can be used to calculate a feasible platform schedule while solving the assignment problem to determine which trains are turned or cancelled. The assignment model from (4-13) is therefore switched for the station scheduling model in (5-5) and (5-11) to solve assignments and allocate trains

65 5-4 Results 47 to platforms. The remaining constraint sets of the disruption model defined in chapter 4 remain the same. At the two stations adjacent to the disruption an assignment problem will be solved including the station scheduling constraints. The Matlab program is extended with the function assign_mma.m that will compose the station capacity constraints (see appendix C). It is invoked by the function Constraintbuild.m when composing the constraint sets for the disruption model. The complete inequality and equality constraint matrices for the problem become: A timetable v timetable A runningtime v runningtime A continuity v continuity A continuity,lagezwaluwe v continuity,lagezwaluwe A continuity,dordrecht A z v continuity,dordrecht headway v headway A headway,lagezwaluwe v headway,lagezwaluwe A headway,dordrecht v headway,dordrecht A sequencing,lagezwaluwe v sequencing,lagezwaluwe A sequencing,dordrecht v sequencing,dordrecht (5-12) [ Aassign,lagezwaluwe ] [ vassign,lagezwaluwe ] A assign,dordrecht z = v assign,dordrecht (5-13) The headway and sequencing constraints at Lage-Zwaluwe and Dordrecht are generated with (5-11). The continuity and assignment constraints at both stations are generated with (5-5) and the remaining constraint sets are identical to those used in chapter 4. The objective function changes because of the addition of sequencing variables y ij in the constraint sets and becomes: [ ] T d 1.. d i a 1.. a i c 1.. c j b 1.. b k u 1.. u l y 1.. y m (5-14) with weight vector c [ ] (5-15) with the same weights for the continuous and cancellation variables as used in section 4-3 and with zero weights for the binary decision variables b, u, y {0, 1}. 5-4 Results The advantage of including the station scheduling model is that eventual capacity limitations can be accounted for. If the amount of available platforms poses a limitation to the number of trains that can be turned or cancelled at the stations adjacent to the disruption then different solutions should be found compared to the model from chapter 4. The expected result would be more cancelled trains instead of turned trains because of the capacity limitations. The calculation of an exact station schedule for the stations Dordrecht and Lage-Zwaluwe means that all arriving trains at these stations must be taken into account and not only

66 48 Capacity constraints trains from the affected lines. Because of this one more train line must be included in the assignment problem, being the train line IC2249, (see table 3-1) having station Dordrecht as its final destination. There are no additional train lines that have station Lage-Zwaluwe as their endpoint. An optimization is performed for a scenario having a horizon of 150 minutes with a disruption starting at minute t ds = 103 and ending 60 minutes later at minute t de = 163. Details on the size of the optimization problem are shown in table 5-4. Figure 5-2 shows a time-distance Table 5-4: Number of constraints and variables for the simulation number of constraints continuous variables 7980 cancellation variables 52 assignment variables 874 reordering variables sequencing variables 725 diagram of all tracks on the route from stations Den-Haag CS to Venlo. All train lines that utilize tracks on this route are depicted in the figure, including all affected lines in set L 2. Trains that overtake each other because their lines are crossing are running on track sections that have four tracks between stations. Train runs taking place between minute 0 and minute 100 lie in the past and are shown in black, they do not participate in the optimization. Assignments between arrival and departure events are also shown in black. Figures 5-5 depict the platform schedules corresponding to the solution in figure 5-2 resulting from the constraint capacity model. Figure 5-3 shows the schedule for the six platforms at station Dordrecht and figure 5-4 the schedule for the four platforms at station Lage-Zwaluwe. The line colors in figure 5-2 correspond to the colors used for the platform schedules in figures 5-5. For both stations a feasible schedule is found with no dwell periods overlapping each other. The minimal platform headway time of three minutes is respected between the train dwells at all platforms. It can be noticed from the platform schedules that the assignments that are solved are not limited to the disruption but continue up to 240 minutes for the schedule of station Dordrecht. This is because a transition period is included after the end of the disruption during which trains can still be cancelled or turned. The method used to transition back towards the nominal timetable is explained in section 6-3. The time the solver needed to find the optimal solution for the scenario in figure 5-2 was approximately 18 minutes which is an indication that the disruption model has become very complex by switching the assignment model for the generic station capacity model. Multiple scenarios were generated in the same fashion as in section 4-4 by shifting the start time of the disruption, but only in a few cases an optimal solution could be found. For most scenarios the solver ran out of memory before finding an optimal solution. The reason for this is the number of headway and sequencing constraints that is needed to model the station scheduling problem. For every conceivable headway order and combination of assignments on platforms the corresponding headway and sequencing constraints in (5-11) must be generated. Depending on the duration of the disruption the number of constraints will therefore quickly increase to a point where it is not practical anymore to apply this model in its current form. The generic form of the model however does leave room for improvement as the search space

67 5-4 Results Time [minutes] DenHaagCS DenHaagHS Delft Aansl Rotterdam Dordrecht LageZwaluwe Breda Tilburg Boxtel Eindhoven Deurne Venlo Station Figure 5-2: Time-distance diagram for the line Den-Haag - Venlo considering a disruption of 60 minutes Platform number 4 3 Platform number Time [minutes] Figure 5-3: Station Dordrecht Time [minutes] Figure 5-4: Station Lage-Zwaluwe Figure 5-5: Platform schedules corresponding to the timetable of figure 5-2 above.

68 50 Capacity constraints can be significantly reduced by removing all scheduling options that are not relevant. number of ways to reduce the size of the problem are: A The capacity model includes all possible headway orders between all trains arriving at the station. The number of headway constraints can be reduced by taking into account the planned arrival times of trains. Headway orders between trains with arrival times that lie far apart are unlikely to change and these can therefore be fixed. Remove assignment options between arrivals and departures that lie far apart in time and will only be relevant in case of large delays in the network. Fixing arrivals to specific platforms to limit the number of allocation options of activities to different platforms. The above-mentioned options can reduce the search-space by composing a constraint model that will much better conform to the specific disruption scenario and include only the scheduling options that are relevant. Besides reducing the search-space the optimization times might also significantly decrease by applying a different objective function. The current objective only applies a penalty on the continuous and cancellation variables. No weights are added to the assignment and sequencing variables that are directly related to the station scheduling problem. Because of this decision there are many different scheduling solutions that will lead to the same objective value. By adding positive/negative weights to unfavorable/favorable assignment and allocation solutions the solver will be able to determine an optimal solution more quickly. 5-5 Conclusions In this chapter a constraint model has been developed that extends the assignment model from chapter 4 with constraint sets for the allocation of trains to platforms. To ensure that a platform can only be occupied by one train at the time additional headway and sequencing constraints were introduced that separate the train dwells with a minimum dwell time of three minutes. The resulting mesoscopic model is able to model all train runs in the network and in addition the capacity limitations at the stations adjacent to the disruption. The drawback of including the generic microscopic constraint model for the stations adjacent to the disruption is the large increase of constraints and binary control variables which lead to a complex optimization problem that could not be solved to optimality for all instances. The model complexity leads to long computation times making the current model not practical for implementation in real time. However, the generic form of the model allows for a reduction of the search space by minimizing the amount of headway constraints and sequencing variables.

69 Chapter 6 Rolling stock limitations In the previous chapters new timetables were calculated assuming that at the stations Eindhoven, Roosendaal, Breda and Rotterdam enough rolling stock is available to take over lines of which earlier train runs are cancelled because of the disruption. This is not an unreasonable assumption as the large stations in the network have spare rolling stock available at all times. However to ensure that possible limitations to the available rolling stock are taken into account it is necessary to include available rolling stock at stations in the constraint model. This chapter will build on the constraint model developed in chapter 4 to incorporate rolling stock actions so that new rolling stock can only be deployed if it is available at the station. Extended assignment problems including the rolling stock actions will be composed for all stations at which trains can be cancelled or turned being Rotterdam, Dordrecht, Lage-Zwaluwe, Breda, Eindhoven and Roosendaal. 6-1 Shunting actions The program of chapter 4 incorporates assignment constraints that connect arrival events with departure events to find a feasible timetable during the disruption. If the number of arrivals is unequal to the number of departures the solution was to cancel train runs at an earlier station, a process which is shown in figure 6-1. Station A borders the disrupted area and therefore the two incoming trains must be turned or cancelled. Cancelling the train runs between stations A and C can only be done in pairs by setting the respective binary cancellation variable c 1, c 2, c 3 {0, 1} to value one. Train runs from station C to station D and beyond are guaranteed to take place, even if earlier train runs are cancelled. In this case one of two planned departures at station A must be cancelled thereby introducing new rolling stock at station C that will continue the line towards station D. A solution to the situation is shown in figure 6-2 where new rolling stock takes over the second line that is cancelled between stations A and C. In the current constraint model rolling stock that "disappears" or "appears" in this way is not accounted for. Stations at which trains can be cancelled can therefore be seen as sources for an infinite supply and storage of rolling stock.

70 52 Rolling stock limitations Figure 6-1: Assignment problem at one side of a disruption. Figure 6-2: Possible solution with deployment of new rolling stock at station C. In reality most large stations have a shunting yard with a limited number of rolling stock available to deploy. Smaller stations may have a small shunting yard without excess rolling stock and it is uncommon to shunt trains at these locations during the day because there is no personnel present. To account for the specific conditions of every station the assignment constraints from (4-13) are expanded to include the actions of deploying a new rolling stock piece and shunting an excess rolling stock piece to the shunting yard. To this end new sets of arrival events A srs and departure events D srs are introduced at every station where trains can be cancelled. The set of arrival events A srs corresponds to rolling stock that is in the shunting yard of the station and is available to be assigned to a planned departure. If it is used for a departure the rolling stock must be introduced at the station five minutes before the departure. The departures in set D srs correspond to train runs from a station to a shunting yard and facilitate the shunting of an arrived train that is cancelled and will not continue. The arrival and departure events in sets A srs, D srs are for rolling stock actions performed at stations and are only constraint by d rsi 0 a rsi 0 i D srs i A srs } s S (6-1) with s the station at which the rolling stock is available. It is assumed that the shunting

71 6-1 Shunting actions 53 operation can take place without interference with other traffic and therefore no headway constraints are used for these events. The assignment problem at stations where trains can be cancelled must now be extended with these extra arrival and departure events as shown in table 6-1. Here two assignment problems are shown, one having two arrivals and three departures and the other one vice versa. Both assignment problems receive one extra arrival event from the set A srs and departure event from set D srs to include the possibilities to introduce and remove rolling stock at the station. Table 6-1: Two scenarios for an assignment problem at a station with extra rolling stock events a 1 a 2 a rs d 4 b 41 b 42 b 4rs d 5 b 51 b 52 b 5rs d 6 b 61 b 62 b 6rs d rs b rs1 b rs2 a 1 a 2 a 3 a rs d 4 b 41 b 42 b 43 b 4rs d 5 b 51 b 52 b 53 b 5rs d rs b rs1 b rs2 b rs3 The row equality constraints belonging to both of these situations are Left table Right table b 41 + b 42 + b 4rs + c 4 = 1 b 41 + b 42 + b 43 + b 4rs + c 4 = 1 b 51 + b 52 + b 5rs + c 5 = 1 b 51 + b 52 + b 53 + b 5rs + c 5 = 1 b 61 + b 62 + b 6rs + c 6 = 1 b 61 + b 62 + b 63 + c drs = 1 b rs + b rs2 + c drs = 1 (6-2) and column equality constraints: Left table Right table b 41 + b 51 + b 61 + b rs1 + c 1 = 1 b 41 + b 51 + b rs1 + c 1 = 1 b 42 + b 52 + b 62 + b rs2 + c 2 = 1 b 42 + b 52 + b rs2 + c 2 = 1 b 4rs + b 5rs + b 6rs + c ars = 1 b 43 + b 53 + b rs3 + c 3 = 1 b 4rs + b 5rs + c ars = 1 (6-3) The situation in the left table requires the deployment of a new rolling stock piece to ensure that all planned departures can take place. The extra event a rs can be used to connect the remaining departure whereas the extra event d rs can be cancelled as no train needs to be cancelled and taken to the shunting yard at this station. In the right table the situation is the other way around with more arrivals than departures. In this case one arriving train must be cancelled at this station and stored in the shunting yard. To do this one of three arrivals must be connected to the extra departure event d rs. The extra arrival event a rs must be cancelled as all remaining arrivals can be connected to planned departures. Weights can be assigned to the binary assignment variables that correspond to shunting actions to indicate that it is undesirable to perform shunting actions. The solver will then

72 54 Rolling stock limitations only perform shunting actions if no other feasible solution with an equal or lower objective can be found. In case of an imbalance in the number of arrivals and departures new rolling stock must be deployed but it can also be favorable in case a train has incurred large delays to prevent further propagation of these delays through the network. 6-2 More assignments The shunting actions introduced by extending the assignment problem in the previous section allow for the deployment and storage of rolling stock at stations. Currently only two assignment problems are solved for the stations bordering the disrupted tracks with the goal of connecting arrivals and departures. The function of these assignments is now expanded to manage all rolling stock present at these stations. To manage the amount of rolling stock present in the rest of the network assignments must be solved at all stations where rolling stock can be cancelled or introduced. Figure 6-3 shows an example of this for a disruption between stations B and C. Besides the two assignment problems that are modelled at the Figure 6-3: Assignments at every station where trains can be cancelled or turned. stations adjacent to the disruption (B and C), now also the traffic at other stations can be modelled with assignments instead of fixed continuity constraints. In this case stations A and D are also modelled by two assignment problems as can be seen in the figure. The result is a constraint model that includes multiple assignment problems, one for each station at which rolling stock changes can be made. Arriving trains at these stations cannot only be turned but also taken to the shunting yard. New rolling stock can be introduced to take over departures at these stations. A number of the assignment options at stations A, B, C and D are shown by the orange arcs. Solving assignment problems at other stations than the two bordering the disruption leads to the situation where arrival and/or departure events may not have the option of being cancelled. This can be seen in figure 6-3 at stations A and D. Departures and arrival corresponding to black train runs cannot be cancelled and must be assigned to find a feasible solution. Taking this into account and the addition of rolling stock actions the assignment