a brief course on demand modelling in railway mode

Transportation EngineeringHigher Technical School of Engineering, University of Sevilla, Spain

Fall School on Robust Network Design and Delay Management, Sevilla, November 2007 1/30

A Brief COURSE on A Brief COURSE on DEMAND MODELLING in RAILWAY MODEDEMAND MODELLING in RAILWAY MODE

Francisco G. Benitez

F. G. Benitez Transportation Engineering. University of Sevilla, Spain 2/30

Demand Modelling on rail Lines, Stations and Trains

• Motivation for modelling of passengers demand• Determining passengers demand

Contents


The main goal in Rail Network Design is:

• To satisfy the transport needs of people.

Motivation





• To satisfy the transport needs of people, efficiently.

Motivation






• To satisfy the transport needs of people, efficiently,

complying with economical constraints.

Motivation






• To satisfy the transport needs of people, efficiently,

complying with economical constraints.

Motivation

FUNCTIONAL Rail Network FUNCTIONAL Rail services



In order to reach a Functional design of Rail (transport) Services the following tasks are needed:

• Determining passenger demand (travellers demand):

- actual (by measuring, by estimating)- prospective (by estimating)

• Planning lines in function of demand and available budget (optimizing lines layout).

• Designing services in function of demand, lines and available budget (trains, vehicles).

Motivation



Determining passengers demand

• Determining actual passengers demand by measuring:

- Indirect measures- Ticketing (aggregated values, passes, travelcards).

- Direct measures- travellers counting by manual survey

(reduced sample, expensive, distorting)- travellers counting by automatic counters

(some are aggregated)



Train T(r,t,D,P)

Route r

Time period: t

Station: e

Line: l




15

15

travellers getting-on O (data known)

Trains monitorized: ( , , , ) travellers getting-off F (data known)

travellers staying R (data known)Services ( , , , )

Non-monitorized Train

Ce

C CeClhej

T r t D P

T r t D P

15

travellers getting-on O (?)s: ( , , , )

travellers getting-off F (?)e

e

T r t D P

same day type

same station ,F O , F high correlation for days of same type

same line

same time period

C Ce e e eO

Train T(r,t,D,P)

Path r

Time period: t

Station: e

Line: l

(by measuring)

(by estimating)




MADRID

BILBAO

VALENCIABARCELONA

Practical Case: Metropolitan rail lines



( )

( )

( )( )

Ce

NMC he

O h

O hNMC h

( )

( )

( )( )

Ce

NMC he

F h

F hNMC h

• same Station

• same Line

• same day Type (labor day, holiday, saturday, holiday eve, friday,…)

• same time Period

To estimate a variable, statistics is used, through data corresponding to:

xx x

x x

eO

xx

Service time

n measures

h

Direct

estimate

xxx

xxxxxx

x



Methodology


( )

( )

( )( )

Ce

NMC he

O h

O hNMC h

( )

( )

( )( )

Ce

NMC he

F h

F hNMC h

• same Station

• same Line


• same time Period


xx x

x x

eO

xx

Service time

n measures

h

Direct

estimate

xxx

xxxxxx

x




( )

( )

( )( )

Ce

NMC he

O h

O hNMC h

( )

( )

( )( )

Ce

NMC he

F h

F hNMC h

• same Station

• same Line


• same time Period adjacent time Period

1, 1t h h


xx x

x x

eO

xx

Service time

n measures

h

Direct

estimate

xxx

xxxxxx

x




• same Station

• same Line




xx x

x x

eO

xx

Service time

n measures

h

Direct

estimate

xxx

xxxxxx

x? ? ? ?

??




Curve fitting

• same Station

• same Line




xx x

x x

eO

xx

Service timeh

xxx

xxxxxx

x? ? ? ?

??




R2 = 0.8267

R2 = 0.2785

-100

0

100

200

300

400

500

600

700

0 10 20 30 40 50 60 70 80 90 100

Tramo horario

Nº

de

Via

jero

s

Bajan

Suben

Bajan Polinomio O(6)

Suben Polinomio O(6)

Time period

Tra

velle

rs

• Data getting-on • Data getting-off

___ Curve fitting getting-on___ Curve fitting getting-off

R2 = 0.6332

R2 = 0.1521

-100

0

100

200

300

400

500

600

700

0 10 20 30 40 50 60 70 80 90 100

Tramo horario

Nº

de

Via

jero

s

Bajan

Suben



Time period

Tra

velle

rs



Time period

Tra

velle

rs



Estación: 18002 Línea: C1 Día: L

R2 = 0.7561

R2 = 0.2073

-200

-100

0

100

200

300

400

500

600

700

19 29 39 49 59 69 79 89 99

Tramo Horario

Nº

de

Via

jero

s Bajan

suben



Time period

Tra

velle

rs



• same Station, same Line (Madrid, C1, path 0), same day Type (L)

• same Station, same Line (Madrid, C1, path 0), same day Type (L)

Polynomial O(4)

Polynomial O(4) Polynomial O(5)

Polynomial O(5)





Demand Modelling on rail Lines, Stations and Trainsp

asse

ng

ers

Curve fitting with variable time interval lengthVariable: Getting-off

-polynomial O(5)-

pas

sen

ge

rs

Curve fitting with variable time interval lengthVariable: Getting-off-polynomial O(10)-


Get

ting-

on

Time period

Polynomial O(5) r2=0.83Chebyshev O(5) r2=0.85Chebyshev O(20) r2=0.87

Other curve fitting

• R. Kelly (2007). “The generation of profiles by formulae”. Traffic Engineering & Control, Vol. 48, No.8, 368-371, 2007.




• Variable estimates:

– By Station:• Getting-on• Getting-off• Remaining

– By Service:• Occupancy

Variables are estimated independently, with different methodologies and error level.



Independent Variable Estimates


• The approach does not complete the estimates for all time period.

• Some “mismatching” might arise in the independent estimating of variables “getting-on”, “getting-off” and “remains”.




eO

eF

eR

1eR

0

0

0

e

e

e

e e f

e e o

e e r

F f err

O o err

R r err

“Station” variables




eO

eF

1eR eR

1O

1 0F

1 1R O

0NEO

NEF

0NER

1

1

1

,

e e e e

e e e e

e r e o e r e f

e R O R F

r err o err r err f err

Balance for stations

NE equations

1NER

Balance equilibrium for “stations”




2 equations

1 1

1

1

e

e

NE NE

c e ee e

NE

c e oe

NE

c e fe

Occupancy C c err O F

c err o err

c err f err

1O

1 0F

1 1R O

0NEO

NEF

0NER 1NER eR

eO

eF

1eR

Balance equilibrium for “train-service”




• A weighting function is applied to each variable attending to its reliability (error level):

• We look for the minimum deviation w.r.t. estimates, in particular for most reliable variables.

(var) (variable error level)w weight

(var) for variables with an error level w

2var

vars

2 2 2 2

(var)

( ) ( ) ( ) ( )e e ee o e f e r c

e

of w err

w O err w F err w R err w C err

Objective function to minimize:




Mathematical scheme of the optimization problem :

2 2 2 2Minimize ( ) ( ) ( ) ( )e e ee o e f e r c

e

w O err w F err w R err w C err

0

0 ,

0

e

e

e

e b

e s

e q

b err

s err e

q err

1

1

e

e

NE

c e oe

NE

c e fe

c err o err

c err f err

s.t.11 ,

e e e ee r e o e r e fr err o err r err f err e




% Trains monitorized Estimate scheme ON OFF Rem. Occ.

20% VOID 0 0 0 0

20% +/- 1 time interval 429 429 429 52

20% Chebyshev 0 0 0 0

20% Direct 1181 1181 1181 178

15% VOID 3 4 0 0

15% +/- 1 time interval 705 705 705 71

15% Chebyshev 12 11 15 3

15% Direct 883 883 883 156

10% VOID 44 15 0 0

10% +/- 1 time interval 1023 1023 1023 0

10% Chebyshev 83 112 127 100

10% Direct 453 453 453 130

5% VOID 160 183 0 0

5% +/- 1 time interval 736 736 736 111

5% Chebyshev 646 623 806 53

5% Direct 61 61 61 66




• Optimum prediction with a 20 % of train-services monitorized.

• Maximum explotation of the available data.

• An error minimazing scheme (Quadratic optimization) consistent among independent variable w.r.t the variable reliability.

Conclusions




That’s over !

Thanks for your attention

a brief course on demand modelling in railway mode

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