Station capacity and stability of train operations

Transcription

1 Station capacity and stability of train operations LA. Hansen Faculty of Civil Engineering and Geo Sciences, Delft University of Technology, The Netherlands Abstract The capacity of railway stations is determined by the characteristics of the tracks, platforms, crossovers, signals, rolling stock, and the timetable. For design purposes the expected track occupation can be estimated on the basis of deterministic routes, train speeds, travel times, stop times and frequencies. Stochastic impacts on arrival and departure times generally intend to be compensated by travel time supplements and buffer times. The amount of train waiting time due to conflicting claims of routes at a station in case of delays is estimated by means of both queuing theory and a max-plus algebra approach. The principle characteristics, as well as shortcomings of these analytical models in order to assess station capacity and stability of train services are discussed. A comparison of the calculation results concerning the estimated waiting time, occupation of different track sections and available buffer time within the station of Den Hague HS shows significant differences that emphasise the need to validate the analytical approaches by detailed analysis of train detection data and suitable simulation programs. 1 Introduction The punctuality of train operations in networks depends mainly on technical failures, track occupation and buffer times between train paths. The delays of trains at conflict points of the network increase with the frequency and duration of occupation of these critical route sections by different lines and type of trains as the probability of simultaneous route claims and track occupation is growing. The conflicts between different route claims at merging and crossing of tracks

2 o i A Computers in Railways VII reduce the railway capacity and reliability of operations compared to open tracks where it is influenced only by the variety of train dynamics and the characteristics of the signal system. At stations the processing of trains is additionally hampered by deceleration, dwell time, acceleration and transfer between connecting trains. The design of station tracks is governed by the aim to minimise the number of tracks, switches and crossings, maximise the flexibility of operations and optimise the track occupation. So far, there are no clear design standards concerning the optimal degree of occupation of routes, switches, crossings and platform tracks. Which buffer times are necessary to assure stable operations in station areas? In the following two recent Dutch studies regarding the capacity of railway stations and the stability of operations are described and the results in the case of the station Den Hague HS are compared with each other. Finally, some conclusions are made concerning the further development of a more suitable approach to determine the design standards of station tracks. 2 Queuing theory approach Trains in stations are considered as units to be processed by different channels where the switches, crossings and platforms represent the locks. The conflict points are characterised by one access channel followed by one or two exits, two entries followed by one exit or both two entries and exits. The service times of each channel are given by the approach, occupation and release time of the trains at the individual track sections that depend on the actual speed, braking rate, route length, train length, and switch times. The conventional queuing theory is only applicable for servers with single access channels, e.g. for trains approaching a signal or negotiate a facing point. The train queues in front of a merge or crossing of two tracks, however, are in first instance interpreted as if they were only a single one representing a certain train mix. The queuing model is given by the triple A/B/s where A expresses the distribution of train arrival times, B the distribution of service times (i.e. blocking times), and s the number of servers. As no more than one train can pass a conflict area at the same time, s is always equal to one. A is assumed to be stochastic with the arrival time being a negative exponential distribution which corresponds to a Poisson distribution of the train headway [1]. The negative exponential distribution of arrival and departure times of delayed trains seems to be confirmed by recent statistical analysis of train detection data at the station of Eindhoven/The Netherlands [2]. The distribution of the service times B is, so far, unknown and will be investigated further on the basis of empirical data. In case of an exponential distribution of arrival times and service times the queuing model M/M/l is known by:

3 Computers in Railways I'll g j j w- I- ^ rn ^" ^1^ ^ // : service time "' p : track occupation time = < 1 X : headway "*. Steady state exists if the degree of track occupation is less than 1. This is valid for a time period of a whole day or a longer peak period, but it might not be given during shorter time periods. In this case the trains in the queue would not arrive independently from other which means that the queuing model in theory cannot not be applied. For reasons of simplicity the interdependency of train movements in stations in peak intervals for the present is assumed to be insignificant. The application of queuing theory to railway operations in stations has been developed first by Wakob [3]. His approach is based on any general probability distribution of the headway G and a determination of the deviation of the variance coefficients relative to the M/M//1 model by means of linear regression. The expected waiting times per route section consisting of at least one switch or crossing that can be occupied only by a single train at the same time (,,Teilfahrstrassenknoten") are calculated for a given train schedule in order to identify the route section with the maximal occupation time. The station track capacity per train schedule is then estimated by means of extrapolation of the sum of the waiting time in the whole station area until a certain total time per day is reached. The standard value for good quality of passenger train operations of 130 min is derived from practical experience of Deutsche Bahn AG [4]. However, empirical evidence or analytical proof of being optimal in terms of service quality and efficiency is still lacking. The approach of Wakob has been applied to the scheduled train operations at the station Den Hague HS by De Kort []. This station consists of 2 tracks in each direction with centre platforms and an at-grade crossing of the track leading to the station Den Hague CS (figure 1). fhmutrecwbincfchorst toanctgfdorncs Y//////////A / /=\ % to The Hague CS LLJ Figure 1: Track layout and route sections of the station Den Hague HS

4 o 12 Computers in Railways VII The station is served daily by about 00 trains that consist of 1C, IR, Regional and very few freight trains. The heaviest loaded route section crossing n where 248 trains pass daily generates an occupation of 33 % and an estimated waiting time of min (table 1). The occupation of the switches and crossings varies only between 20 and 33 % while the scheduled average train headway is ranging between 6 and 1 min. It can be seen clearly that the crossover n 9-10 where all the trains running in the direction of Amsterdam merge is producing rather long waiting times because of the long service time. This is due to the limited speed of the switches of 0 km/h for trains using the curved track. The amount of estimated daily waiting time of about 130 min at the critical route sections indicate that the expected quality of operations of the timetable with 00 trains a day seems still to be very good according to the standard of the Deutsche Bahn. The maximal queue length is there defined as: /ff=0,2j7.e-a.?p (2) p representing the share of passenger trains. The influence of the quality of operations on the queue length is expressed by a factor of 0., 1.0 or 1. for very good, satisfactory and unsatisfactory level of quality [6]. Table 1: Operations characteristics of route sections of Den Hague HS, timetable 1996/97 [source: De Kort, 1999] Route Number of Number Average Average Average Track Waiting section trains/day of lines frequency headway service time occupation time perh [min] [min] p [min] ,2 8,0,0,0 9,2 6,2 6,0 6,2 7,0,0 9, 2 7, 14,4 14,4, 8 9, 7 11,6 9, 7 9, 7 11,3 2,3 2,3 3,1 3,7 *1,9 2,6 3, 2,2 2,9 2,9 0,2 0,30 0,21 0,2 0,33 0,27 0,30 0,23 0,30 0,26 66,8 116,4 2,6 72,7 139,8 103,4 111,7 67,8 11,6 76,3 However, it represents only an estimate of a certain timetable and does not reflect the impact of primary delays that the trains approaching to the station introduce to the local track network. For that reason, a detailed statistical analysis of train delays, dwell times and track occupation times at several Dutch railway stations, based on train detection data, is currently taking place.

5 Computers in Railways VII g j 3 3 Max-plus algebra approach In recent years a new mathematical method named m ax-plus algebra has been applied successfully to model cyclic railway timetables on a network scale [7, 8, 9, 10]. The connection of line services at transfer stations can be modelled by means of simple travel time matrices and unconventional analytical functions for the addition and selection of the maximum matrix element respectively. The eigenvalue of the max-plus matrix represents the duration of the critical circuit of a network timetable. The difference between the eigenvalue and the cycle time of the timetable (e.g. 1 or 20 min for Regional and 30 or 60 min for 1C trains) is a measure for the buffer time between arrivals and departures of the trains at the transfer stations of the network. A recently developed tool for computer-aided conversion and evaluation of railway timetables by means of max-plus algebra is described by Soto y Koelemeijer et al. [1 1]. The location of the critical circuit in a network is given by the maximum average cycle time of the circuits between the transfer stations which are calculated by dividing the sum of the travel and transfer times by the number of links. The critical circuit, however, changes when the frequency of trains of a line is modified. The time period until damping out of a primary delay in a strongly connected network can be estimated in case of deterministic travel times simply by dividing the delay time through the stability margin: AT : stability margin of timetable TC : cycle time of timetable A : eigenvalue of the max-plus travel time matrix. The modelling of train operations in stations by max-plus algebra has been shown by van Egmond [12]. The estimated block times of the scheduled trains at the different route sections are piled up and consequently the eigenvalue of the max-plus matrix of block times and transfer times is calculated. These times indicate the degree of occupation of the resources of a station (figure 2). The remaining stability margin compared to the cycle time of the timetable is the buffer time for compensating delays. The propagation of primary delays within the station track area can be estimated by calculating the amount and location of secondary delays on the basis of the max-plus matrix: Td : fading out time p /: primary delay.

6 814 Utg -, Asd. Asd - Asd - Gvt < Gvc - Computers in Railways VII, C.A. Brebbia J.Allan, R.J. Hill, G. Sciutto & S. Sone (Editors) Computers in Railways VII i 12, \ w /, / ' \ / ' 9 ' 10 \/ /s \S y-' y- \ \ \ / i / 1 Figure 2: Station layout and track resources of the station Den Hague HS Asd = Amsterdam Gvc = Den Haag Centraal Rtd = Rotterdam Utg = Utrecht freight station The critical circuit of the station track network in Den Hague HS for the same timetable as investigated by De Kort is affecting nearly every resource while the eigenvalue of a peak hour interval according to the max-plus approach is only 3 min. The degree of occupation of the different resources varies a lot between less than % and a maximum of 27 % (figure 3). The location of the heaviest loaded track sections coincides with the calculations of De Kort, but the amount of buffer time estimated by means of the max-plus approach might be significantly less than indicated for the individual route sections by De Kort due to the strong interconnection of the resources. Rtd Rtd Rid Rtd I I platform tracks H other tracks resource Figure 3: Occupation of track resources of the station Den Hague HS, timetable 1996/97 [source: van Egmond, 1999]

7 Computers in Railways Conclusions The different analytical approaches by means of queuing theory and m ax-plus algebra in order to determine the capacity and stability of operations in railway stations lead broadly to similar results regarding the location of bottlenecks and the occupation of route sections. However, significant differences remain with regard to the amount of buffer time and the ability of the track network to compensate for delays. The determination of an optimal degree of occupation of track resources in stations needs to be further supported by more comprehensive empirical research on the basis of real train detection data and validation of analytical approaches by means of realistic simulation of train processing on signal controlled routes in stations. Acknowledgement This publication is a result of the research programme Seamless Multimodal Mobility (SMM), carried out within the TRAIL Research School for Transport, Infrastructure and Logistics, and financed by Delft University of Technology. References [1] SchwanhauBer, W. (1974) Die Bemessung der Pufferzeiten im Fahrplangefiige der Eisenbahn, Veroffentlichungen Verkehrswissenschaftliches Institut RWTH Aachen, H. 20 [2] Goverde, R.M.P., Hansen, LA. (2000) TNV-Prepare: Analysis of Dutch railway operations based on train detection data, in: Proc. Computers in Railways VII, Computational Mechanics Publ./WIT Press, Southampton [3] Wakob, H. (198) Ableitung eines generellen Wartemodells zur Ermittlung der planmabigen Wartezeit im Eisenbahnbetrieb unter besonderer B erticks ichtigung der Aspekte Leistungsfahigkeit und Anlagenbelastung, Veroffentlichungen Verkehrswissenschatliches Institut RWTH Aachen, H. 36 [4] Sitzmann, E., Eilers. W. (1990) Betriebliche Untersuchung von Eisenbahnknoten, Eisenbahntechnische Rundschau, 39(11), pp [] De Kort, A., Heidergott, B., van Egmond, R.J., Hooghiemstra, G. (1999) Train Movement Analysis at Railway Stations. Procedures & Evaluation of Wakob 's Approach, TRAIL Studies in Transport Series, S99/1, Delft [6] SchwanhauBer, W. (1994) Theoretical Background of the Program Family SLS, in: J.F. Mortelmans (ed.) Proc. F' Erasmus-network conference on Transportation and Traffic Engineering, Kerkrade, pp [7] Braker, J.G. (1993) Algorithms and application in Timed Discrete Event Systems, PhD thesis, TU Delft [8] Subiono (1997) On large scale Max-Plus Algebra models in railway systems, Proc. 3 TRAIL year congress, part 2, Den Haag/Scheveningen

8 g j ft Computers in Railways VII [9] Egmond, R.J. van (1998) Propagation of delays in public transport, paper for Euro Conference on Transportation, Goteborg [10] Goverde, R.M.P. (1998) The Max-Plus Algebra approach to railway timetable design, in: Computers in Railways VI, Computational Mechanics Publ./WIT Press, Southampton, pp [11] Soto y Koelemeijer, G., lounoussov, A.R., van Egmond, R.J., Goverde, R.M.P. (2000) PETER, a performance evaluator for railway timetables, in: Proc. Computers in Railways VII, Computational Mechanics Publ./WIT Press, Southampton [12] Van Egmond, R.J. (1999) Railway Capacity Assessment an Algebraic Approach, TRAIL Studies in Transportation Science Series, N S99/2, Delft