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TRA VISIONS 2016 project
is funded by
the European Union
A Cubic Function Model for
Railway Line Delay
Stability and robustness analyses of
railway timetables compare some given
primary delays with their effect on the
whole operation.
Estimation of delays in Railway can
be performed by microsimulation with
good accuracy, but also high resources
consumption.
A theoretical model that describes
the total delay as a function of the
primary delay analytically, allows to
reduce the computation of
microsimulation keeping its accuracy.
The theoretical model is validated on a real railway line in Denmark. This model will allow the reduction in computation resources needed
for stability and robustness analyses of railway operation. and
Future research will extend the model to railway networks formulate new stability indexes. The average timetable allowance and the
average buffer time, will be computed from the regressed cubic parabola measured via micro-simulation.
Fabrizio CerretoTechnical University of Denmark
Department of Transport
Motivation & Objectives
A primary delay is assigned to the first train at the first station.
The model propagates the primary delay through consecutive trains and sums up all the
delays on individual trains at each station to compute the total delay.
The total delay results in a composite polynomial function of the primary delay.
Methodology
Results
TRA VISIONS 2016 project
is funded by
the European Union
Research Outlook
Total delay on railway lines as a polynomial function of the primary delay given to one train.
Simple analytic formulation, closed form expression.
Fast analyses.
Strategic planning of railway operations.
Key Characteristics
Initial delay
• Primary delay to the first train, at the first station
Delay propagation model
• Timetable Allowance
• Buffer time
Individual train delay at each station
• Residual delay from previous station
• Hindrance from previous train
Total delay
• No recovery
• Partial recovery
• Full recovery
NUMERICAL EXAMPLE
• Number of stations ns = 8
• Number of trains v = 6
• Timetable allowance between
stations a = 2 min
• Buffer time between trains b = 6 min
0
500
1000
1500
2000
2500
3000
0 10 20 30 40 50 60 70 80
Nu
me
rica
l su
mm
atio
n -
To
tal d
ela
yd
[m
in]
Primary delay d [min]
Total delay measured on the line
STATION
1 2 3 4 5 6 7 8
A B C D E F G HTR
AIN
1 50 48 46 44 42 40 38 36
2 44 42 40 38 36 34 32 30
3 38 36 34 32 30 28 26 24
4 32 30 28 26 24 22 20 18
5 26 24 22 20 18 16 14 12
6 20 18 16 14 12 10 8 6
STATION
1 2 3 4 5 6 7 8
A B C D E F G H
TRA
IN
1 14 12 10 8 6 4 2 0
2 8 6 4 2 0 0 0 0
3 2 0 0 0 0 0 0 0
4 0 0 0 0 0 0 0 0
5 0 0 0 0 0 0 0 0
6 0 0 0 0 0 0 0 0
STATION
1 2 3 4 5 6 7 8
A B C D E F G H
TRA
IN
1 25 23 21 19 17 15 13 11
2 19 17 15 13 11 9 7 5
3 13 11 9 7 5 3 1 0
4 7 5 3 1 0 0 0 0
5 1 0 0 0 0 0 0 0
6 0 0 0 0 0 0 0 0
No recovery
Linear
Partial recovery
Quadratic
Full recovery
Cubic
Simplified individual delay as a function of
the primary delay Pd. Assumed uniform
timetable allowance a and buffer time b:
𝑑𝑖,𝑠 = 𝑃𝑑 − 𝑠 − 1 𝑎 − 𝑖 − 1 𝑏
The total delay function has three
segments depending on whether the delay is
recovered fully, partially, or not recovered
within the study range:
• Full recovery – cubic relation
• Partial recovery – quadratic relation
• No recovery – Linear relation
The segment differ in the summation domain
over the stations and the trains.
y = 0,0124x3 + 1,4733x2 + 3,3203x - 1,7214R² = 0,9999
y = 0,1515x3 - 0,6515x2 + 5,3479x - 2,1268R² = 0,997
0
20
40
60
80
100
120
140
160
180
200
0 1 2 3 4 5 6 7 8 9 10
Tota
l D
ela
y [m
in]
Primary Delay [min]
Total delay measured on the line after primary delay to either line A or E
A E Poly. (A) Poly. (E)
THE CASE STUDY
Suburban railway line in the
Greater Copenhagen Area
• High frequency
• Cyclic timetable
• No overtakes
• Non-uniform timetable allowance
and buffer times
• 2 different stopping patterns
• Uniform rolling stock
CUBIC Total delay for small Primary
delaysMicrosimulation: RMCon RailSys 7.9
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