Rail Zürich, 11.2.2009

Train scheduling based on speed profiles

© ETH Zürich | M. Fuchsberger

Martin Fuchsberger, ETH Zurich

RailZurich, 11. February 2009

D. Burkolter, G. Caimi, T. Herrmann, S. Roos, R. Wüst

2Rail Zürich, 11.2.2009 M.Fuchsberger / IFOR ETHZ martin.fuchsberger@ifor.math.ethz.ch

What is train scheduling?

INPUT: Train service intention (SI) Aggregated and detailed track topology of the

network Rolling stock with dynamic properties

OUTPUT: Conflict-free periodic train schedule fulfilling SI

3Rail Zürich, 11.2.2009 M.Fuchsberger / IFOR ETHZ martin.fuchsberger@ifor.math.ethz.ch

Two-level approach reduces complexity

Macro scheduling:

Find a timetable that fulfills trip

time, connection and macro level

safety requirements

Micro scheduling:

Find locally a conflict free

schedule, fulfilling detailed safety

requirements for a given macro

schedule

Focus of this talk

4Rail Zürich, 11.2.2009 M.Fuchsberger / IFOR ETHZ martin.fuchsberger@ifor.math.ethz.ch

Condensation vs. compensation zones

Condensation zone:• Main station area• Bottleneck• Maximum speed policy• Many routes

Compensation zone:• Regions connecting

main stations• Time reserves• Variable speed• Few routes

Portal:• Link between zones• Macroscopic draft

passing times Focus of this talk

5Rail Zürich, 11.2.2009 M.Fuchsberger / IFOR ETHZ martin.fuchsberger@ifor.math.ethz.ch

Micro scheduling in compensation zone

ZG LZ

Fixed speed profile

Fixed speed profile

t

Flexible speedand travel time

Entrance point

Exit point

6Rail Zürich, 11.2.2009 M.Fuchsberger / IFOR ETHZ martin.fuchsberger@ifor.math.ethz.ch

Micro train scheduling - Objectives

1. Conflict-free assignment of track paths to the

trains

2. Fulfill safety requirements on the micro level

3. Meeting portal (boundary) conditions

4. Additional quality criteria:

Energy, time reserve distribution

7Rail Zürich, 11.2.2009 M.Fuchsberger / IFOR ETHZ martin.fuchsberger@ifor.math.ethz.ch

Two step approach to micro scheduling

1. Track path generation

Apply two reasonable simplifications: Approximation of the continuous track path by a

finite chain of <location,time> points Represent the infinitely many track paths by a

representative finite set of track paths

2. Conflict-free track path assignment

8Rail Zürich, 11.2.2009 M.Fuchsberger / IFOR ETHZ martin.fuchsberger@ifor.math.ethz.ch

1. Trackpath generation

a) Enumerate meaningful route alternatives

b) Generate viable speed profiles for each route,

which: Are a versatile representation of the infinitely many

speed profiles Comply with maximum speed limits Obey dynamic train properties Meet portal boundary conditions

9Rail Zürich, 11.2.2009 M.Fuchsberger / IFOR ETHZ martin.fuchsberger@ifor.math.ethz.ch

2. Speed profile generation

a) Generate ®-speed profile

„drive as fast as it is allowed“

minimal travel time

b) Calculate time reserve based on ®-speedprofile

c) Generate several speed profiles by distributing

the time reserve among track sections

10Rail Zürich, 11.2.2009 M.Fuchsberger / IFOR ETHZ martin.fuchsberger@ifor.math.ethz.ch

11Rail Zürich, 11.2.2009 M.Fuchsberger / IFOR ETHZ martin.fuchsberger@ifor.math.ethz.ch

Time reserve is divided by a parameterN=6 parts and …

}

Example of speed profile generation

t

0 1 2 1 2s

®-speedprofile

Portaltime

... distributed over K track sections

12Rail Zürich, 11.2.2009 M.Fuchsberger / IFOR ETHZ martin.fuchsberger@ifor.math.ethz.ch

S1 Lucerne – Zug:- ®-profile- Track section split

13Rail Zürich, 11.2.2009 M.Fuchsberger / IFOR ETHZ martin.fuchsberger@ifor.math.ethz.ch

Optimisation model assigns a track path per train.

1. Resource tree conflict graph

2. Multicommodity flow

3. Constrain flow (conflict free)

4. Integer linear program

5. Optimise for a quality criteria

Models train dynamics and detailed safety system

Conflict-free trackpath assignment

14Rail Zürich, 11.2.2009 M.Fuchsberger / IFOR ETHZ martin.fuchsberger@ifor.math.ethz.ch

S1 Lucerne – Zug:- Min. energy consumption- Max. time reserve distribution desirability- Combination of both objectives

Remember:The optimal solution considers all trains, not only this train!

15Rail Zürich, 11.2.2009 M.Fuchsberger / IFOR ETHZ martin.fuchsberger@ifor.math.ethz.ch

Testcase

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Testcase – continued

1. Based on SBB 2008 timetable we derive a service

intention

2. Solve macroscopic timetable scheduling

Generates portal times

Schedule contains per hour and direction 2 intercity trains 1 interregio train 2 commuter trains

10 trains / hour

17Rail Zürich, 11.2.2009 M.Fuchsberger / IFOR ETHZ martin.fuchsberger@ifor.math.ethz.ch

Effects of parameters K and N

Too few track sections (K) lead to: Less variety of speed profiles Problem became infeasible

High granularity partitioning of time reserve (N): Improves objective value Increases memory consumption

Computation times < 30 s

After tuning parameters K and N, trains are swiftly

scheduled and comply with security standards.

18Rail Zürich, 11.2.2009 M.Fuchsberger / IFOR ETHZ martin.fuchsberger@ifor.math.ethz.ch

Outlook

Our current research focuses on

Application of this approach for rescheduling

Interaction between: Macro and micro level (2-level approach) Compensation and condensation zones

Possible contributions of Operations Research

(OR) to the field of railway rescheduling

19Rail Zürich, 11.2.2009 M.Fuchsberger / IFOR ETHZ martin.fuchsberger@ifor.math.ethz.ch 19

Thank You!

Time for questions!