1. 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

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

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

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

5. Rail Zürich, 11.2.2009 M.Fuchsberger / IFOR ETHZmartin.fuchsberger@ifor.math.ethz.ch 5 Micro scheduling in compensation zone ZG LZ Fixed speed profile Fixed speed profile t Flexible speed and travel time Entrance point Exit point

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

7. Rail Zürich, 11.2.2009 M.Fuchsberger / IFOR ETHZmartin.fuchsberger@ifor.math.ethz.ch 7 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 points  Represent the infinitely many track paths by a representative finite set of track paths 2. Conflict-free track path assignment

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

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

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

11. Rail Zürich, 11.2.2009 M.Fuchsberger / IFOR ETHZmartin.fuchsberger@ifor.math.ethz.ch 11 Time reserve is divided by a parameter N=6 parts and … } Example of speed profile generation t 01212 s ® -speedprofile Portaltime... distributed over K track sections

12. Rail Zürich, 11.2.2009 M.Fuchsberger / IFOR ETHZmartin.fuchsberger@ifor.math.ethz.ch 12 S1 Lucerne – Zug: - ® -profile - Track section split

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

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

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

16. Rail Zürich, 11.2.2009 M.Fuchsberger / IFOR ETHZmartin.fuchsberger@ifor.math.ethz.ch 16 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

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

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

19. Rail Zürich, 11.2.2009 M.Fuchsberger / IFOR ETHZmartin.fuchsberger@ifor.math.ethz.ch 19 Thank You! Time for questions!