Rescheduling through stop-skipping in dense railway systems

Transportation Research Part C 79 (2017) 73–84

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Rescheduling through stop-skipping in dense railway systems q Estelle Altazin a,b,⇑, Stéphane Dauzère-Pérès b,c, François Ramond a, Sabine Tréfond a a

SNCF, Innovation and Research Direction, 40 avenue des Terroirs de France, F-75611 Paris Cedex 12, France Ecole des Mines de Saint-Etienne, Department of Manufacturing Sciences and Logistics, CMP, LIMOS UMR CNRS 6158, 880 avenue de Mimet, F-13541 Gardanne, France c BI Norwegian Business School, Department of Accounting, Auditing and Business Analytics, Nydalsveien 37, 0484 Oslo, Norway b

a r t i c l e

i n f o

Article history: Received 29 April 2016 Received in revised form 17 March 2017 Accepted 18 March 2017

Keywords: Rescheduling Integer Linear programming Macroscopic modelling Real time Dense railway system

a b s t r a c t Based on the analysis of the railway system in the Paris region in France, this paper presents a rescheduling problem in which stops on train lines can be skipped and services are retimed to recover when limited disturbances occur. Indeed, in such mass transit systems, minor disturbances tend to propagate and generate larger delays through the shared use of resources, if no action is quickly taken. An integrated Integer Linear Programming model is presented whose objective function minimizes both the recovery time and the waiting time of passengers. Additional criteria related to the weighted number of train stops that are skipped are included in the objective function. Rolling-stock constraints are also taken into account to propose a feasible plan. Computational experiments on real data are conducted to show the impact of rescheduling decisions depending on key parameters such as the duration of the disturbances and the minimal turning time between trains. The trade-off between the different criteria in the objective function is also illustrated and discussed. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction and industrial context About 18% of the French population live in the Paris region, which only covers 2% of the French territory. Every day, 8.3 million of trips are performed on the public transportation system in the Paris region. SNCF Transilien is a major operator of Paris suburban trains. Each day, Transilien must carry over 3.2 million commuters in 6200 trains on 1300 km of tracks. The number of passengers is continuously increasing (3% each year since 2000). To cope with this rise, trains have been added, up to 32 trains per hour run on the busiest parts of the infrastructure. Some lines are expected to reach a capacity crisis by 2020. Operating the Transilien rapid transit system is thus challenging on a daily basis. An unexpected large number of passengers boarding at a station or a minor technical problem can create small delays during running, dwell or turning times. These small delays are difficult to predict and, because of the saturation of the network, time buffers are not sufficient to absorb them. They can rapidly accumulate along lines and propagate to other delays, causing larger delays and degrading the quality of service offered to passengers. Transilien combines the characteristics of regular trains and subways: (1) Tracks are shared with high-speed, regional and freight trains; drivers and rolling-stock are shared between Transilien lines; and several services (trains with different

q

This article belongs to the Virtual Special Issue on ‘‘Integr Rail Optimization”.

⇑ Corresponding author at: SNCF, Innovation and Research Direction, 40 avenue des Terroirs de France, F-75611 Paris Cedex 12, France. E-mail addresses: [email protected] (E. Altazin), [email protected] (S. Dauzère-Pérès), [email protected] (F. Ramond), sabine. [email protected] (S. Tréfond). http://dx.doi.org/10.1016/j.trc.2017.03.012 0968-090X/Ó 2017 Elsevier Ltd. All rights reserved.

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stopping pattern) exist on each line. Yet (2) the frequency is very high (less than 3 min on certain lines) and passengers connections do not need to be considered. Operational decisions like skipping a train stop or cancelling a whole train can thus be taken without severely impacting the quality of service, even though these decisions must be taken as early as possible to inform all operational actors and passengers. Such decisions are sometimes hard to make because of the multiplicity and heterogeneity of actors involved in the Paris rapid transit system. First, in agreement with the Paris transportation authority, Transilien takes into account precise performance criteria, for instance: The number of passengers delayed on each line and each branch and the number of trains actually operated compared to the number of trains planned. Other criteria concern the quality of service offered to passengers and the passenger information system, especially in case of disturbances. These criteria are associated with a bonus/penalty system associated to Transilien’s performances. Secondly, the main goal of the infrastructure manager is that each train runs on time and respects its scheduled path. Thirdly, Transilien passengers expect frequent and rapid trains, they want comfortable trips with reliable service and information, whatever the train path, the rolling-stock unit or the driver. In addition, many actors are involved in traffic management within Transilien, and have different objectives: Rollingstock management, driver management, passenger information operators, etc. This makes the decision-making process complex and adds to the difficulty of implementing decisions. Finally, Transilien operators need to find, at all times, the right balance between quality of service and performance. They need to constantly maintain a good quality of service for passengers and to fulfil the requirements of the transportation authority, while seeking performance, to minimize delays of trains, and to minimize costs. It is often hard to take realtime decisions that satisfy all the stakeholders simultaneously. For example, skipping stops on a delayed train may reduce the delay and avoid its propagation to the next train, but it affects the quality of service for passengers that wanted to board at the skipped stations and a penalty is incurred for delaying them. For most decisions, a trade-off has to be found between (1) penalizing a limited number of passengers and maybe paying a penalty or (2) risking to propagate the delay to the next train with potentially more passengers impacted and an even higher penalty to pay. Operators need to constantly have all criteria and constraints in mind to anticipate the potential impacts of each decision. In real-time, Transilien operators need to quickly determine the best actions from a system-wide perspective. In this work, we develop a real-time integrated rescheduling model that proposes stops to be skipped and a new timetable for trains in order to minimize both delay propagation and the waiting time of passengers. Moreover, we include both train rescheduling and rolling-stock constraints as turning times tend to propagate delays very quickly in dense systems. The next section gives an overview of existing operational policies and rescheduling models for railway operations, as well as various approaches for bus or metro traffic. Section 3 presents the problem characteristics and various assumptions. Section 4 introduces a mathematical model in which train stops can be skipped. Section 5 presents the Transilien system characteristics, numerical results on real data obtained with the model, and discusses how to manage multiples criteria in the objective function. Section 6 presents results with multiple initial delays, associated to industrial instances. Finally, conclusions and directions for future work are provided in Section 7.

2. Literature review Real-time traffic management consists in supervising the traffic and adjusting timetable, rolling-stock and driver schedules to prevent and reduce delays caused by incidents. Incidents can be either small perturbations called disturbances, or larger perturbations, called disruptions (Cacchiani et al., 2014). In this paper, we focus on disturbances that cause delays of a few minutes. In recent years, Cacchiani et al. (2014), Corman and Meng (2015) and Toletti et al. (2015) reviewed several approaches that have been developed to propose automatic actions for railway traffic management. Different problems are usually defined, coping with different aspects of railway traffic. Each problem considers specific actions to cope with incidents, and tries to optimize a specific type of objective. The train dispatching problem, also referred to as train path rescheduling or conflict prediction and resolution, consists in adjusting a timetable that has become infeasible because of disturbances or disruptions (Hansen and Pachl, 2008). Train routes, orders, timetable or speeds can be modified in order to achieve a feasible timetable. The objective is to improve the performance of the railway system by minimizing train delays and recovering the original timetable. Meng and Zhou (2014) propose to minimize the total deviation time from the original timetable, while Samà et al. (2016) choose to minimize the total consecutive delays. However, some approaches consider a passenger-based objective: Sato et al. (2013) propose a rescheduling model including train reordering and retiming (adjustments of the timetable) that minimizes the inconvenience of passengers (waiting time, travelling time and number of transfers). Caimi et al. (2012) propose a dispatching model that reroutes and re-platforms (changes in the platform assignment) trains in complex station areas, and maximizes the passenger satisfaction (maintained connections, schedule all trains and delays). In our problem, the layout of tracks does not allow for overtaking, thus trains cannot be rerouted or reordered. Reservicing actions, that consist in modifying the traffic plan (adding or skipping stops, cancelling trains, short-turning trains) can be included in train dispatching approaches. Sato et al. (2013) modify the train type in order to make an express train stop at a station that it should not serve. Veelenturf et al. (2016) propose partial or whole cancellations of trains in addition to rerouting and retiming for handling

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large disruptions. Following Hansen and Pachl (2008), other reservicing actions such as adding stops or extending a scheduled stop can be used to compute conflict free routes. This paper focuses on retiming and reservicing actions. Schöbel (2001) studies the delay management problem, that aims at determining which passenger connections should be kept and which should be dropped in a delayed situation. Objective functions are related to passengers, such as Dollevoet et al. (2015), who minimize the total delay of passengers. Gatto et al. (2007) propose a family of online algorithms to solve the delay management problem. They decide in which station the train will wait for delayed passengers from other trains, without knowing if other delayed passengers will enter the system. The objective is to minimize the total passenger delay on the train line. Recently, Corman et al. (2016) proposed an integrated approach for the microscopic delay management problem: Trains are re-ordered and re-timed while passengers are re-routed through the network. Their approach minimizes the total travel time of passengers, combining conflict resolution and control actions on passenger connections, at a microscopic level. Our approach does not consider passenger connections since we focus on rapid transit systems with high frequencies. The rolling-stock rescheduling problem consists in adjusting rolling-stock assignments and deciding whether spare units are to be used. This step is usually carried out once the timetable has been adjusted, in order to adapt the rolling-stock allocation to the new timetable. It can be necessary in case of big disruptions: Kroon et al. (2015) propose to reschedule the rolling-stock in case of a 3-h blockage. In case of small disruptions, Cadarso et al. (2013) propose to reschedule the rolling-stock allocation as well as the timetable in case of disruptions in rapid transit networks, and Almodóvar and García-Ródenas (2013) describe a vehicle rescheduling approach to cope with an unexpected peak of demand on one line by using vehicles assigned to other lines. The usual objective is related to carriage kilometers and number of shunting movements (Nielsen et al., 2012), yet Kroon et al. (2015) and Almodóvar and García-Ródenas (2013) also include a passenger-oriented objective. As explained in Cadarso et al. (2013), computing a new timetable without considering rolling-stock constraints might produce an infeasible plan. Due to short turning times in rapid transit systems, rollingstock is a critical resource. We present an integrated approach of train rescheduling with rolling-stock constraints to cope with small disturbances. Finally, the crew rescheduling problem is the process of adjusting the assignments of drivers to trains. This step is usually performed when the timetable and the rolling-stock have been rescheduled. The aim is usually to cover as many train services as possible, while minimizing costs and deviation from the original schedule (Potthoff et al., 2010). Our approach assumes that drivers operate the same rolling-stock unit during the rescheduling horizon, that is limited to a few hours. The above mentioned approaches are on conventional railway systems, whereas the Transilien system has a high frequency of trains, short distances between stations and short turning times for rolling-stock. Besides, the infrastructure does not permit overtaking or rerouting of trains. We will thus combine retiming with reservicing decisions, as commonly used in subway or bus traffic. Rescheduling in public transport systems is often referred to as real time control strategies, and allows to cope with perturbations as well. The constraints are different, as the infrastrucutre is simpler and passengers do not take a specific train. Reservicing actions are mostly used in public transit, and the objective is usually passenger focused. Eberlein et al. (1999) classify these control strategies into three categories: Station control, inter-station control and others. Station control consists of holding trains at station and station-skipping (Wilson et al., 1992). The second category includes speed control, traffic signal pre-emption, etc., while the third category includes strategies such as adding vehicles or splitting trains. Holding a vehicle at a station is used to even out the headways between its preceding and following vehicles and thus to reduce the waiting time of passengers. Different station-skipping strategies exist: Deadheading a vehicle consists in running empty from a terminal through a number of stations. Headway with the preceding train can be reduced, along with the waiting time of passengers at stations beyond the deadhead segment. Yet it increases the waiting time for passengers at the skipped stations. Expressing a vehicle is similar to deadheading except that expressing can start at any station and the vehicle generally does not run empty, passengers must thus be notified. Short-turning a vehicle consists in skipping the last stations of the line and turning the vehicle at an earlier station. Headways and therefore the waiting time in high demand zones can be reduced. Eberlein et al. (1999) propose mathematical formulations for the real-time holding problem, deadheading problem and expressing problem. The aim is to minimize the waiting time of passengers. Each of the strategies is tested alone first, and then combinations of strategies are tested, with best results on a Boston light rail line (MTBA Green Line) where 76% of the vehicles are controlled. This paper shows that frequent control actions are needed for maximum effectiveness, and that control should be a continuous process. O’Dell and Wilson (1999) propose mixed integer programming formulations when holding and short-turning are allowed on a light rail line. Results on the MTBA Red Line show that the waiting time is reduced by 15–25% for a 10-min delay and up to 40% for a 20-min delay. Short-turning is efficient when the time required to execute the short-turn is significantly shorter than the blockage time and the number of skipped stations is small, so that a large number of passengers will benefit from the short-turn. Cortés et al. (2011) present an integrated model combining short-turning and deadheading for a single bus line. It takes into account the demand and proposes that some vehicles perform short cycles to serve the most loaded stations and increases the frequency in this segment. The waiting and in-vehicle times of passengers are minimized, as well as the operator cost.

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Canca et al. (2014) propose a model capable of inserting special short-turning services in case of disruptions causing peaks of demand at certain stations. The average waiting time is reduced by up to around 30%. Such a strategy needs reserve rolling-stock units as well as capacity on the line. The real-time stop-skipping problem is first studied by Sun and Hickman (2005), as a real-time strategy than can be applied to all vehicles, even those already dispatched from the terminal. They propose a formulation of two stop-skipping policies: Expressing the vehicle over a whole segment and dropping off the passengers destined for stops in the skipping segment at the last stop before the skipping segment, or allowing some passengers to alight at stops in the skipped segment if their destination is in this segment. The capacity of vehicles is considered and the model includes assumptions of random distributions of passenger boardings and alightings. The aim is to minimize the waiting time of passengers and the number of passengers that are forced to get off when applying the first strategy. It appears that the second strategy outperforms the first one, and is simpler to implement in a real-time manner with limited passenger cost. Because missed connections increase the waiting time and frustration of passengers, Nesheli and Ceder (2015) propose a mathematical model combining holding, stop-skipping and short-turning that minimizes total passenger travel time and maximizes direct transfers. They use an agent-based simulator (each trip time is recorded) to validate the optimization results on real-life scenarios. Tests on the Auckland bus network show that the combination of the three strategies reduce the average travel time by up to 5% and increase the direct transfers by up to 153%. Furthermore, it is shown that, in every case, the total travel time is increased by less than 1.5%. Gao et al. (2016) present a model that defines skip-stop patterns to recover from a disruption on an over-crowded metro line. Services are thus adapted by skipping some stops to speed up the circulation and limit the number of stranded passengers. This approach minimizes both the total travel time of services and the number of passengers waiting in stations, through a fine modelling of passenger flows. Most approaches for railway rescheduling consider microscopic models with detailed infrastructure, as it is needed when considering rerouting decisions (Samà et al., 2016; Pellegrini et al., 2014; Meng and Zhou, 2014; Caimi et al., 2012). Some papers propose macroscopic models, for instance at station level, to cope with large networks and lower the computational times (Kecman et al., 2013; Krasemann, 2012). Reservicing approaches in metro or buses, that focus on passengers and do not consider vehicle routing, propose macroscopic models at the station level (Gao et al., 2016; Eberlein et al., 1999; O’Dell and Wilson, 1999; Sun and Hickman, 2005). Our work combines retiming and reservicing trains with a high frequency through stop-skipping and rolling-stock constraints. We aim at optimizing both the performance of the railway system and the quality of service for passengers. Considering Transilien constraints and the actions we want to implement, we chose to develop a macroscopic model. The next section will describe the problem and our assumptions. 3. Problem description In this work, we consider the problem of real-time rescheduling for rapid transit railway systems. In case of disturbances, a new transportation plan is proposed to minimize the impact of disturbances. Our assumptions are detailed below. We consider a double-track rapid transit network and each track is operated in a single direction. In each served station, there is one platform for each direction and the trains cannot overtake each other. Some lines consist of several branches. The capacity of terminal stations is not considered as a constraint, and thus is infinite. The frequency of trains on each line is between 3 and 15 min during peak hours, and lower during off-peak periods. A limited rescheduling horizon (1.5–2.5 h of circulation in the experiments) is considered. In case of small disturbances, typically causing less than 10 min of delay, we determine which train stops to skip and plan a new timetable for trains, based on the original transportation plan. The running and dwell times of trains in the original schedule are considered as minimal values for train operations. Headway constraints between trains running on the same track (i.e. in the same direction), are considered as well. The rolling-stock schedule is considered through minimal turning time constraints at terminal stations, between trains using the same rolling-stock unit. We assume that drivers operate the same rolling-stock unit during the rescheduling horizon. Data on origin-destination trips of passengers are used, with a fixed number of passengers boarding and alighting each train at each station. The capacity of trains is not considered. Our primary objective is to recover the original timetable as fast as possible, while minimizing the waiting time of passengers. When a stop is skipped, the train will reduce its delay by the planned dwell time and an extra time parameter corresponding to the saved braking and acceleration times. The number of skipped stops is also minimized, for two reasons: To limit the deviation from the original plan and thus reduce the workload of operators when implementing the proposed changes, and to maintain a satisfying quality of service for passengers. 4. Mathematical modelling We develop an Integer Linear Programming model, where stops can be skipped to minimize the impact of disturbances. We model railway operations at macroscopic level as a directed graph. Set N is the set of nodes corresponding to events of trains: Departures from a station, arrivals at a station or transits through a station. Arcs of the graph can correspond to two

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types of links between events: (1) Operations, particularly running between stations, dwelling at a station or turning at a terminal station or (2) headway constraints between two trains running consecutively on the same infrastructure. An example is illustrated in Fig. 1. Trains t1 and t 3 are consecutively running from station A to station D, passing through station B and stopping at station C. Train t 2 runs in the opposite direction, from station D to station A, with the same stopping pattern. Trains t1 and t 2 are using the same rolling-stock unit. Plain arcs correspond to run and dwell time constraints between two consecutive events e and e0 of a train. Thus, de1 e2 ; de2 e3 , and de4 e5 , are minimal running times between events e1 and e2 ; e2 and e3 , and e4 and e5 respectively. Similarly, de3 e4 is the minimal dwelling time at station C between events e3 and e4 . The dotted arc represents the turning time of the rolling-stock unit between arrival of train t1 at its terminal Station D, represented by the event e5 , and the departure of t 2 from Station D, represented by event e6 . Hence, de5 e6 corresponds to the minimal turning time required between e5 and e6 to turn the rolling-stock unit. The dashed arcs represent headway constraints between trains t1 and t3 that run in the same direction. As there is only one track per direction, and one platform per direction in each station, only one train can dwell at a time. Thus, train t 3 cannot arrive at Station C (event e13 ) before train t1 has left the station (event e4 ) since there is a minimal headway time he4 e13 . 4.1. Notations Let us consider the following sets: N SN DN A E S L

Set of nodes representing events e of a train at a location in the original schedule (departure, arrival or passing) Set of events associated to train stops that can be skipped Set of events representing departures of a train from a station Set of pairs ðe; e0 Þ of consecutive events (run or dwell) of a train Set of pairs ðe; e0 Þ of final and first events of two trains using the same rolling-stock unit Set of pairs ðe; e0 Þ of events of two consecutive trains on the same track Set of pairs ðe; e0 Þ of arrival events at the same station and of trains running in the same direction, with e scheduled before e0 . This set will be used to compute the waiting time of passengers

The following parameters are defined: se dee0 hee0 sav ed dee0 we pe

Time at which event e was originally scheduled Minimal duration of the running, dwelling or turning operation between events e and e0 such that ðe; e0 Þ 2 A or ðe; e0 Þ 2 E Minimal headway time between events e and e0 of consecutive trains on the same infrastructure section Time saved for a train between events e 2 SN and its following event e0 if stop e is skipped, corresponding to the theoretical dwell time and saved braking and acceleration times Penalty associated to skipping the stop at e Number of passengers alighting or boarding at stop e. This parameter is used as a penalty for the extra waiting time of passengers

Station A Station B e1 e2 de1 e2 de2 e3

t1

t3

de4 e5

Station D e5

he4 e13

he2 e12 t2

Station C e3 de e e4 3 4

de5 e6

e10

e9

e8

e7

e6

e11

e12

e13

e14

e15

Fig. 1. Graph representation of 3 trains and 4 stations.

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The following variables are considered: Pe De Se RT WT e;e0

Rescheduled time of event e, i.e. time at which event e is re-timed 1, if event e 2 N is late compared to its original schedule (i.e. Pe > se ), and 0 otherwise 1, if stop e 2 SN is skipped, and 0 otherwise Recovery time, i.e. maximum time at which rescheduled times of events are different from scheduled times Waiting Time for passengers between scheduled stops e and e0 , such that ðe; e0 Þ 2 L

4.2. Objective function Two main aspects are optimized in the model: The performance of the railway system and the quality of service for passengers. The system’s performance is considered through minimizing the recovery time RT, associated to the duration of the perturbation. If the system cannot recover within the time horizon, then RT is larger than the time horizon, but the solution remains feasible. The quality of service is modeled through the weighted number of skipped stops, and the weighted total passenger waiting time. The number of skipped stops also models the deviation from the original schedule, and thus the difficulty of implementing the new plan. The waiting time of passengers is computed between each pair ðei ; ej Þ of scheduled stops at the same station and in the same direction. Its weight represents the number of passengers boarding or alighting trains at event ej and at events corresponding to stops scheduled between ei and ej at the same station and in the same direction (i.e. such that ðei ; eÞ 2 L and ðe; ej Þ 2 L). The sum of delays and the number of delayed events are also minimized as secondary criteria, to ensure the consistency of the solution.

min

aRT þ b

X

X

ðpej þ

ðei ;ej Þ2L

pe ÞWT ei ej þ c

e2SN ðei ;eÞ2L ðe;ej Þ2L

X

X X we Se þ d ðPe  se Þ þ g De :

e2SN

e2N

e2N

The five criteria are combined in the objective function as a weighted sum. They are treated in a lexicographic order as the coefficients a; b; c; d and g are tuned such as there cannot be any compensation between the criteria. This is done by estimating the largest value that each criterion can take. 4.3. An Integer Linear Programming (ILP) model The mathematical model M ss to reschedule trains in dense areas by skipping stops is given below:

X

min aRT þ b

ðpej þ

ðei ;ej Þ2L

X

pe ÞWT ei ej þ c

X

X X we Se þ d De þ g ðP e  se Þ

e2SN

e2SN ðei ;eÞ2L ðe;ej Þ2L

e2N

ð1Þ

e2N

subject to

8ðe; e0 Þ 2 A s:t: e R SN

Pe0 P Pe þ dee0 Pe0 P Pe þ dee0 

sav ed dee0 Se

ð2Þ

8ðe; e0 Þ 2 A s:t: e 2 SN

ð3Þ

Pe0 P Pe þ dee0

8ðe; e0 Þ 2 E

ð4Þ

Pe0 P Pe þ hee0

8ðe; e0 Þ 2 S

ð5Þ

P e P se

8e 2 DN

ð6Þ

8e 2 N ;

De P ðPe  se Þ=M 1 ; RT P se De ;

8e 2 N ;

WT ei ;ej P Pej  Pei 

X

ð7Þ ð8Þ ð1  Se ÞM2

8ðei ; ej Þ 2 L;

ð9Þ

e2SN ðei ;eÞ2Lðe;ej Þ2L

De 2 f0; 1g;

8e 2 N ;

ð10Þ

Se 2 f0; 1g;

8e 2 SN :

ð11Þ

where M 1 , resp. M 2 , is a sufficiently large coefficient to prevent constraints (7), resp. (9), from being binding when Pe is equal to se , resp. Se is equal to 1. The objective function (1) is explained in Section 4.2. Constraints (2)–(5) ensure the minimal times between consecutive events:

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 Running or dwell time between two consecutive events that cannot be skipped (2);  Dwell time between events that can be skipped or saved time if stop e is skipped (3);  Minimal turning time between arrival at terminal stations and departure from terminal stations of two trains using the same rolling-stock unit (4);  Minimal safe headway time between two consecutive trains on the same infrastructure segment (5). Constraints (6) ensure that no train can leave from a station earlier than its scheduled departure time for the quality of service. Constraints (7) define the variables De characterizing delayed events. Constraints (8) determine the time to recover the original schedule. Waiting time variables are calculated in Constraints (9) as follows: Each pair ðei ; ej Þ 2 L represents two stops at the same station and in the same direction, with ei scheduled before ej . For each pair ðei ; ej Þ 2 L, variable WT ei ej models the waiting time of passengers between scheduled stops ei and ej . If there is no stop between ei and ej , whether in the scheduled timetable or as the result of intermediary stops being skipped, the waiting time is the duration between ei and ej , i.e. WT ei ej ¼ P ej  P ei . If there is one or more scheduled stops between ei and ej , the waiting time depends on whether these stops are skipped or not. If at least one of them is not skipped, the waiting time between ei and ej has no meaning since another train will board the station between ei and ej , thus WT ei ej ¼ 0. This model has been tested with various scenarios, and computational results are presented and discussed in the next sections. 5. Computational experiments 5.1. Transilien network specification The Transilien network consists of 13 train lines. Five of them are crossing Paris (RER lines), along a north-south axis or an east-west axis. The other eight lines start from one large train station in Paris, and travel towards the suburbs. Some lines share the infrastructure with other SNCF long distance trains (mostly regional and intercity trains). Most of the lines consist of several branches, and each line has its own characteristics (platform height, length, rolling stock, train protection system, etc.). Lines consist of double tracks, and tracks are operated in a single direction, although most of them are equipped to be operated in the opposite direction in case of large disruptions. As lines are double-tracked, there is only one possibility for trains paths, and re-routing is not possible. Some stations are equipped with a third central track, allowing trains to turn. Yet these tracks are not used to change the order of trains, except in case of large disruptions. Frequencies widely vary depending on the type of lines (crossing Paris or not) and the distance from Paris. Furthest stations have the lowest frequency (two trains per hour) while, in central Paris stations, the frequency is about two minutes during peak hours. Thus, the impact of an operation control action depends on its location: Skipping a stop in Paris has almost no impact on the quality of service, while skipping a stop further from Paris can result in passengers waiting for one hour or more. Transilien trains are most of the time composed of a locomotive and a set of coaches or two coupled units that operate as a pair for the entire day. Here, each train is considered as one rolling-stock unit. 5.2. Test instances We performed several tests on real SNCF Transilien instances, on a line where 6 stations are served in addition to the terminal ones. Trains are running during morning or evening peak hours, with a frequency of one train every 10 min in both directions. The instances cover between 1.5 and 2.5 h of circulation. Some trains are using the same rolling-stock units, with turning times of at least 9.5 min. We set the model parameters as follows (recall that weights a; b; c; d and g are established by estimating the maximum value of the associated criterion, see Section 4.2):  The minimal headway time is 4 min,  The penalty associated with deleting one stop is 10 (we ¼ 10; 8e),  The saved braking and acceleration times are equal to sav ed

dee0 ¼ dee0 þ 30s;  a ¼ 10,

8e 2 SN ),

 b ¼ 105 ,  c ¼ 1,  d ¼ 106 ,  g ¼ 101 ,  M 1 ¼ 35; 000, which is larger than the horizon (in seconds),  M 2 ¼ 35; 000, which is larger than the horizon (in seconds).

30 s

added

to

the

saved

dwell

time

(i.e.

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A single perturbation is created on one of the first trains of the instance, at different stations given the instance. Three different sets of experiments are run for several values of the duration of the initial perturbation:  In Section 5.3, we test different minimal turning times and stop-skipping strategies: No stop-skipping authorized, stopskipping authorized for trains that have not departed their first station within a certain time period after the perturbation, and no constraints on stop-skipping.  In Section 5.4, we fix the number of skipped stops between 0 and the optimal number determined in Section 5.3 plus one. We study the balance between different criteria of the objective function.  In Section 5.5, the number of skipped stops is again fixed, and the recovery time is set as a constraint instead of being minimized in the objective function. We test which stops are skipped when minimizing the waiting time of passengers with a larger weight in the objective function. The ILP model M ss is solved using the standard solver IBM ILOG CPLEX 12.6.2, and all instances are solved to optimality within less than 1 s of computational time. 5.3. Impact of minimal turning time and stop-skipping strategy In this first set of experiments, we ran several scenarios by varying different parameters:  Initial disturbance of 3, 5, 7 or 10 min;  Minimal turning time of 8 or 9.5 min;  Stop-skipping strategy: no stop-skipping (noskip), authorized for trains that depart after the perturbation plus a given time period set to 10 min, which is the current situation in Transilien (after10), authorized for all train stops after the perturbation (allstops). Table 1 summarizes the average numerical results for five instances. Each scenario is characterized by an initial delay, a minimal turning time and a stop-skipping strategy. For each scenario, the average values of the following indicators are provided: The recovery time, i.e. the duration of the perturbation, in minutes; The extra waiting time of passengers, i.e. the percentage of increased waiting time induced by the stops that are skipped, compared with strategy (noskip);

Table 1 Average numerical results on five instances, with varying initial delays, minimal turning times and stop-skipping strategies. Initial delay (min)

Minimal turning time (min)

3

8

5

7

10

Stop skipping strategy

Recovery time (min)

Extra waiting time (%)

Skipped stops

Delayed events

Total delay (min)

noskip after10 allstops

16 16 4

0 0 4

0 0 1.6

14 14 4

43 43 8

9,5

noskip after10 allstops

96 80 71

0 2 3

0 1.4 2

58 39 34

68 56 40

8

noskip after10 allstops

30 23 9

0 1 10

0 0.8 3.2

23 18 9

84 74 28

9,5

noskip after10 allstops

96 81 74

0 2 6

0 1.8 2.8

58 40 36

109 93 71

8

noskip after10 allstops

43 32 14

0 3 13

0 1.6 4.2

39 32 17

152 132 62

9,5

noskip after10 allstops

105 86 75

0 4 8

0 2.6 3

87 66 58

194 168 127

8

noskip after10 allstops

59 42 23

0 8 25

0 4 6

54 42 28

296 249 120

9,5

noskip after10 allstops

106 89 80

0 4 11

0 2.2 4

103 81 78

392 344 275

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The number of skipped stops; The number of delayed events; The total delay of all delayed events, in minutes. Table 1 shows that, by deleting some stops, the duration of the perturbation can be strongly reduced, on average by 47% between strategies (noskip) and (allstops). The number of delayed events and the total delay are also strongly reduced, showing that the propagation of the perturbation can quickly be stopped. The duration of the initial perturbation is an important factor. The longer it is, the more the delay will be propagated to other trains. This will happen either through headway constraints, as trains are running at a 10-min frequency with a minimal headway of 4 min, or through minimal turning time constraints, when trains are using the same rolling-stock unit. The minimal turning time parameter shows a significant impact on the total delay and the number of skipped stops, especially with strategy (after10). Indeed, the longer the minimal turning time, the greater the chance the delay will be propagated to other trains, on which stops will then be skipped. For instance, when the minimal turning time is increased from 8 min to 9.5 min with an initial 7-min delay and strategy (after10), on average 1 additional stop is skipped, twice more events are delayed and the total delay increases by 27%. The reduction of the recovery time is also much larger when the minimal turning time is short: 44% of reduction on average with a 8-min minimal turning time, but only 21% with a 9.5-min minimal turning time. Being able to reduce the minimal turning time during operations is thus a very important factor for a fast recovery of the system. The stop-skipping strategy (after10), allowing stop-skipping only on trains that depart 10 min after the initial perturbation occurs, corresponds to the current situation at Transilien. This rule is set to ensure that stops will not be skipped on trains already boarding, thus the passenger information can be adjusted in time. Results show that, even if this strategy is better than strategy (noskip) with no stop-skipping, the impact of a perturbation can be strongly reduced if trains stops can be skipped directly after the perturbation occurs (strategy (allstops)). On average, the total delay is reduced by about 43% when it is compared to strategy (after10) (up to 81% for a 3-min delay with 8-min minimal turning time). Additional tests have been conducted for strategy (after10) with a period of 5 or 20 min to skip stops after the perturbation occurs, and the results are similar to those obtained with a period of 10 min. Thus, more flexible and reactive operations are essential to significantly limit the impact of a perturbation. Yet, strategy (after10) allows the perturbation to be reduced using the buffer times available in turning times or in headways, and less stops need to be deleted than in strategy (allstops). Also, the impact on the waiting time of passengers is reduced. Indeed, the waiting time of passengers remains almost the same between strategies (noskip) and (after10) because a limited number of stops are skipped, but it is increased when using strategy (allstops). The larger the number of skipped stops, the longer the passengers will wait: Up to 25% extra waiting time is caused by skipping 6 stops for a 10-min delay with 8-min turning time and strategy (allstops). Fig. 2 shows the time-space diagram of a test instance with a 7-min initial delay, 8-min minimal turning time and strategy (after10). The dashed lines correspond to the original schedule and the plain lines to the proposed timetable. The thick line shows the train T delay on which the delay is created. The model proposes to skip 4 stops to recover the delay, they are represented by dots on the diagram. Two trains are delayed by train T delay because of headway and turning time constraints, and the model delays one train running before train T delay , in order to even out the headways and thus the waiting time of passengers. 5.4. Impact of the number of skipped stops We performed additional tests to study the impact of each additional skipped stop. The minimal turning time is set to 8 min and the stop-skipping strategy (after10) is applied. We fixed the number of skipped stops as a constraint, from 0 to the optimal number (see Section 5.3) plus one. Table 2 summarizes these results on one of the previous instances. The previous results are in bold. It shows that for each initial delay, the optimal solution corresponds to the best recovery time, but not necessarily the best waiting time. The tests with a 10-min initial delay show that more retiming is performed when moving from 7 to 8 skipped stops. Thus, the waiting time of passengers is improved, but the number of delayed events and the sum of delays increase. A balance needs to be found between the performance of the system, i.e. the recovery time, and the waiting time of passengers. 5.5. Waiting time of passengers as primary criterion In the third series of experiments, the recovery time and the number of skipped stops are set as constraints. The initial delay is set to 7 min, the minimal turning time to 8 min, and the stop-skipping strategy (after10) is used. The number of stops to skip is set to 4, as determined in Section 5.3, and different margins are used to relax the optimal recovery time RT opt . More precisely, a new parameter RecoveryMargin is introduced, and the recovery time is no longer considered in the objective function, but is constrained in the model as follows: RT 6 RT opt þ Recov eryMargin. The results on the same instance as in Section 5.4 are summed up in Table 3. Note that there is a threshold of the recovery margin that allows the waiting time of passengers to be significantly reduced. In this case, when allowing at least 40 min of

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Fig. 2. Time-space diagram of results with 7-min delay and 4 skipped stops.

Table 2 Numerical results with a fixed number of skipped stops for one instance. Initial delay (min)

Fixed number of skipped stops

Recovery time (min)

Extra waiting time (%)

Delayed events

Total delay (min)

5

0 1 2 3

55 55 37 37

0.0 0.5 2.3 3.4

36 40 25 26

117 147 91 91

7

0 1 2 3 4

55 55 55 40 40

0.0 0.5 1.2 3.7 4.8

51 55 57 42 43

219 245 253 173 173

10

0 1 2 3 4 5 6 7 8

65 65 65 61 55 55 55 47 47

0.0 0.5 1.2 2.5 3.6 4.0 4.9 11.1 10.8

72 72 72 69 77 77 75 71 75

414 415 416 382 468 463 465 401 449

relaxation, the waiting time of passengers reaches a minimum value that cannot be improved with a larger relaxation of the recovery time: The frequencies between trains are adjusted. These tests illustrate the results when the focus is set on passengers instead of train delays: By retiming many trains, we optimize the frequency for passengers, but we create delays on many trains that could have run on time. A trade-off between the two aspects needs to be made. 6. Additional experiments with multiple delays In this section, we present results of tests conducted on industrial instances: The same real data as for the previous tests are used, but we did not create a delay on one train. Instead, real initial delays of trains at a specific rescheduling date are considered.

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E. Altazin et al. / Transportation Research Part C 79 (2017) 73–84 Table 3 Numerical results with relaxation of recovery time. Recovery margin (min)

Recovery time (min)

Extra waiting time (%)

Delayed events

Total delay (min)

0 5 10 20 30 40 50 60 70 80 90

46 51 56 66 76 86 96 106 116 126 136

0 6 4 9 10 12 12 12 12 12 12

59 86 64 102 74 70 70 70 70 70 70

226 352 238 573 442 270 270 270 270 270 270

Table 4 Numerical results on industrial instances with multiple delays. Sum of initial delays (min)

Trains initially delayed

4:23

2

5:49

Stop skipping strategy

Recovery time (min)

Extra waiting time (%)

Skipped stops

Delayed events

Total delay (min)

noskip after10 allstops

20 20 6

0 0 7

0 0 4

53 53 36

86 86 46

2

noskip after10 allstops

60 39 21

0 1 1

0 1 1

93 79 72

294 290 281

6:55

3

noskip after10 allstops

28 28 10

0 0 15

0 0 6

85 85 56

165 165 116

9:30

3

noskip after10 allstops

46 25 9

0 1 9

0 1 3

93 79 53

189 173 144

12:11

5

noskip after10 allstops

50 29 9

0 3 8

0 1 5

115 97 65

296 286 259

For each instance, we specify in Table 4 the sum of delays of all running trains at this rescheduling date along with the number of trains delayed at this date. The model then includes these multiple delays and proposes a solution. Tests with an 8-min minimal turning time and the three stop-skipping strategies have been conducted and the results are summed up in Table 4. As in Table 1, we see that the stop-skipping strategy has a strong impact on the propagation of delays, and that our approach helps to recover from delays much faster by skipping stops. 7. Conclusion and perspectives Based on the analysis of the situation in the dense railway system of the Paris area, this paper studies the problem of minimizing the recovery time when disturbances occur by skipping stops. Additional criteria are considered, in particular the waiting time of passengers. An Integer Linear Programming model is proposed which takes into account constraints related to running times, dwell times, turning times and headways. Numerical experiments on various test instances based on real data have been performed, showing that even a limited number of skipped stops can help to significantly reduce the recovery time. Different scenarios were analyzed to study the impact of key parameters such as the minimal turning time, and also the impact of each skipped stop. Tests with multiple delays have also been performed using industrial instances with real delays, showing the relevance of our approach to recover from limited disturbances. Various research avenues will be pursued in the future. Other types of decisions could be included. In Transilien real-time traffic management, operators also use train cancellations as they free capacity on tracks, rolling-stock units and drivers. However, cancelling a whole train is more penalizing for commuters than skipping stops. Moreover, the planning of rolling-stock and drivers must be explicitly managed. Also, other criteria related to passengers could be proposed and analyzed, in particular to better dynamically manage the flow of passengers. The ILP model cannot easily include all decisions and criteria, or a dynamic management of passenger flows while being solved in real-time. We are thus developing an alternative iterative approach based on local search and simulation of decision scenarios. Acknowledgements This work has been partially financed by the ANRT (Association Nationale de la Recherche et de la Technologie) through the PhD number 2014/1195 with CIFRE funds and a cooperation contract between SNCF and ARMINES.

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