15.3.2007 Train Scheduling in a Main Station Area © ETH Zürich | M. Fuchsberger Martin Fuchsberger...

15.3.2007

Train Scheduling in a Main Station Area

© ETH Zürich | M. Fuchsberger

Martin Fuchsberger

Master thesis, Final Presentation

Zurich, March 15. 2007

215.03.2007 M.Fuchsberger / D-INFK ETHZ / [email protected]

Outline

Introduction

Train Routing Three Conceptual Models Conflict Modeling Results

Train Scheduling Model Results

Outlook


Introduction

Goal: Satisfy customers demands by finding

suitable conflict-free timetables

Restrictions: Topology, Rolling stock, service

requirements


Network Density

Local LayerGlobal Layer

Bottlenecks: Main Station Areas


Example Main Station Area: Bern

Radius of about 6 km

500 switches

6 main directionsOlten

Neuchatel

Fribourg

Bienne

Bern

Belp

Thun


Example Main Station Area: Bern


Two Problems in Main Station Areas

1. Train Routing

Input: Topology, Rolling Stock, Departure

Times at portals/platforms

Output: Train Routings

2. Train Scheduling

Input: Topology, Rolling Stock, Departure Time

Windows at portals/platforms

Output: Conflict-free Timetable


Train Routing:Three conceptual models

Conflict Graph

Tree Conflict Graph

Resource Tree Conflict Graph


Train Routing Model: Conflict Graph1

: Each routing of a train corresponds to a node

in the conflict graph.

: Conflict Edges model conflicts between two

routings (nodes).

: As one train can use only one routing, the

routings of a train form a clique.

1Zwaneveld et al. 1997


Train Routing: Conflict GraphSolution Approach: Independent Set with Cardinality equal to number of trains

Train 1

Train 2

Train 3


Conflict Graph – Mathematical Model

Only one nodefor each train

Only non-connectednodes


Conflict Graph – Solving Problems

Finding exact solutions for larger problem instances

took too much time1.

The heuristic attempt (Randomized FPI2) fails to find

solutions for big instances.

How can the model (structure) be improved?

1Zwaneveld et al. 1997

2Fixed Point Iteration, Herrmann, Burkolter et al. 2005


Improvement: Include Local Topology

0 2 2

31 3 4 4 7 7

5 5 6 6

9

8 8 10

0 23 4

5 6

7 8 10

8 10

5 6

7 8 10

8 10

7 8 10

A

B

C

D

E

31 47 9

5 6 7 9

Y

Z

1 Conflict

B C D

Y Z

A E

6 Conflicts

Conflict Graph:

Conflict


Train Routing Model: Tree Conflict Graph1

: For each tuple (train,time,topology-

element,velocity) a node is created.

: Red edges model conflicts between two

tuples (nodes).

: Flow edges model the routings from origin to

destination.

1Herrmann and Caimi 2005


Tree Conflict Graph - Solution Approach: Multi Commodity Flow

1. Add Sources and Sinks

2. Assign Variables xij to the flow edges

3. Flow equal to 1 from source to destination

4. Conflict Constraints

5

6

7

8

10

8

10

8

10

8

10

8

10

7 7

6

5

4

3

2

0 1

3

4

5

7

6

9

7

9

x00

S0 S1

S0 S1


Tree Conflict Graph – Mathematical Model


Allocation of a Resource

A resource is composition of track elements.

A resource can be allocated by trains for a

closed time interval („allocation time interval“).

A conflict exists, if a resource is allocated by

more than one train at the same time.

track section switch crossing single slip


Allocation Time Intervals

How are the allocation time intervals of the different

trains determined?

An allocation time interval consists of: Occupation time

Minimal braking time: Based on track signals

Additional security related times (track switch, reaction...)


Conflict Modeling for a Resource

TimeTime Interval where the Resource is occupied by a train

Conflicts between two trains

Grouped conflicts between several trains = „Cliques“


How to gather the conflicts into cliques?

Time

B

C

FD

IG

E

J

H

A A‘

A1 B1 A2 B2 C1 D1 E1 F1 D2 C2 G1 H1 G2 F2 E2 I1 J1 H2 I2 A‘1J2

Minimum number of cliques to cover all the edges in the corresponding circular interval graph


Resulting Train Routing Model: Resource Tree Conflict Graph

0

1 7

2

3 3

2 8

9

4 6

5 5 5

64

5 5

6

10

11 15

8

12

13 13

9

14

2

17

14

13

18

1

2

3

4 6

5 5

0

1

2

3

4

Each Resource has its set of colored cliques.


Resource Tree Conflict Graph – Mathematical Model


Results for the Train Routing Problem

Scenario # nodes # conflictBern Min Average Max Construction Solving

West 2003 12 000 700 2 12 130 <1 <1East 2003 46 500 3400 2 50 590 1 2West 2020 15 000 900 2 35 300 <1 <1East 2020 58 000 4300 2 60 1100 2 2

time [s]Clique size

Scenario # conflict # conflictBern TCG CG Construction Solving Construction Solving

West 2003 15 500 70 000 4 <1 1 5East 2003 85 500 740 000 50 4 7 15West 2020 56 500 300 000 10 2 3 7East 2020 1 110 000 7 100 000 930 180 80 2400

FPI time [s]TCG time [s]

Conflict Graph (CG&FPI) / Tree Conflict Graph (TCG)

Resource Tree Conflict Graph

Reduced #Conflicts! Better CPU time!

Cause: Strong clique constraints!


Train Scheduling

Discretise the time windows to create several start times

for each train (Puls 90 SBB Project).

Each train has then a set of Resource Tree Conflict

Graphs with distinct and selectable starting times.

Solve the train scheduling problem using the same

algorithms.

Train Routing

Input: Topology, Rolling Stock,

Departure Times at portals/platform

Output: Train Routings

Train Scheduling

Input: Topology, Rolling Stock, Departure Time Windows at

portals/platforms

Output:Conflict-free Timetable


Train Scheduling – Prefered Start Times

Some start times may be preferred over others.

Incorporate this idea in the model by using

weights in the objective function:Train 1

Start time T1 Start time T2


Results for the Train Scheduling Problem

in Bern East 2003#Start Times #nodes #constraints

Min Average Max Construction Solving

1 175 000 3 900 2 44 851 3 32 350 000 7 900 2 46 851 4 54 700 000 14 500 2 54 851 6 106 1 050 000 22 300 2 68 1068 9 188 1 400 000 27 200 2 74 1068 11 27

10 1 750 000 34 800 2 80 1068 14 35

Clique Size time[s]

Ad

din

g m

ore

Sta

rt T

imes

Now bigger problem size compared to the train

routing problem.

Still fast computable.


Results for the Train Scheduling Problem

Bern East 2003

05

1015

2025

3035

40

0 500000 1000000 1500000 2000000

# Nodes

Tim

e [

s]

Solving Time Construction Time (suboptimal)


Example Clique Size DistributionsBern East 2003

One start time for each trainAverage Clique Size = 44

Ten selectable start times for each trainAverage Clique Size = 80


Outlook

Further enhance the model (English track

switchs, more complex resources)

Extend testing on other main station areas

besides Bern (data from SBB is required)

Check performance on track regions between

stations1

Interaction with the global layer

1 Collaboration with SBB and sma (D. Burkolter)


Thank you for your attention!


Scheduling: Connecting Global Layer to Local Layer

On the global layer, the train scheduling problem is

usually modeled as a periodic event scheduling problem

(PESP)

The solution of a PESP provides input data for the

discussed train routing algorithms

A modified PESP could support time windows and hence

serve as an input for (local) train scheduling algorithms1

1Topic of the Master Thesis of Kaspar Schüpbach

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