Post on 21-Dec-2015
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
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 3/30
The main goal in Rail Network Design is:
• To satisfy the transport needs of people.
Motivation
Demand Modelling on rail Lines, Stations and Trains
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 4/30
The main goal in Rail Network Design is:
• To satisfy the transport needs of people.
• To satisfy the transport needs of people, efficiently.
Motivation
Demand Modelling on rail Lines, Stations and Trains
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 5/30
The main goal in Rail Network Design is:
• To satisfy the transport needs of people.
• To satisfy the transport needs of people, efficiently.
• To satisfy the transport needs of people, efficiently,
complying with economical constraints.
Motivation
Demand Modelling on rail Lines, Stations and Trains
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 6/30
The main goal in Rail Network Design is:
• To satisfy the transport needs of people.
• To satisfy the transport needs of people, efficiently.
• To satisfy the transport needs of people, efficiently,
complying with economical constraints.
Motivation
FUNCTIONAL Rail Network FUNCTIONAL Rail services
Demand Modelling on rail Lines, Stations and Trains
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 7/30
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
Demand Modelling on rail Lines, Stations and Trains
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 8/30
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)
Demand Modelling on rail Lines, Stations and Trains
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 9/30
Train T(r,t,D,P)
Route r
Time period: t
Station: e
Line: l
Determining passengers demand
Demand Modelling on rail Lines, Stations and Trains
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 10/30
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)
Determining passengers demand
Demand Modelling on rail Lines, Stations and Trains
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 11/30
MADRID
BILBAO
VALENCIABARCELONA
Practical Case: Metropolitan rail lines
Demand Modelling on rail Lines, Stations and Trains
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 12/30
( )
( )
( )( )
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
Determining passengers demand
Demand Modelling on rail Lines, Stations and Trains
Methodology
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 13/30
( )
( )
( )( )
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
Determining passengers demand
Demand Modelling on rail Lines, Stations and Trains
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 14/30
( )
( )
( )( )
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 adjacent time Period
1, 1t h h
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
Determining passengers demand
Demand Modelling on rail Lines, Stations and Trains
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 15/30
• same Station
• same Line
• same day Type (labor day, holiday, saturday, holiday eve, friday,…)
• same time Period adjacent 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? ? ? ?
??
Determining passengers demand
Demand Modelling on rail Lines, Stations and Trains
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 16/30
Curve fitting
• same Station
• same Line
• same day Type (labor day, holiday, saturday, holiday eve, friday,…)
• same time Period adjacent time Period
To estimate a variable, statistics is used, through data corresponding to:
xx x
x x
eO
xx
Service timeh
xxx
xxxxxx
x? ? ? ?
??
Determining passengers demand
Demand Modelling on rail Lines, Stations and Trains
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 17/30
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
Bajan Polinomio O(4)
Suben Polinomio O(4)
Time period
Tra
velle
rs
• Data getting-on • Data getting-off
___ Curve fitting getting-on___ Curve fitting getting-off
Time period
Tra
velle
rs
• Data getting-on • Data getting-off
___ Curve fitting getting-on___ Curve fitting getting-off
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
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
• 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)
Determining passengers demand
Demand Modelling on rail Lines, Stations and Trains
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 18/30
Determining passengers demand
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)-
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 19/30
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.
Determining passengers demand
Demand Modelling on rail Lines, Stations and Trains
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 20/30
• Variable estimates:
– By Station:• Getting-on• Getting-off• Remaining
– By Service:• Occupancy
Variables are estimated independently, with different methodologies and error level.
Determining passengers demand
Demand Modelling on rail Lines, Stations and Trains
Independent Variable Estimates
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 21/30
Determining passengers demand
Demand Modelling on rail Lines, Stations and Trains
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 22/30
• 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”.
Determining passengers demand
Demand Modelling on rail Lines, Stations and Trains
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 23/30
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
Determining passengers demand
Demand Modelling on rail Lines, Stations and Trains
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 24/30
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”
Determining passengers demand
Demand Modelling on rail Lines, Stations and Trains
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 25/30
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”
Determining passengers demand
Demand Modelling on rail Lines, Stations and Trains
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 26/30
• 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:
Determining passengers demand
Demand Modelling on rail Lines, Stations and Trains
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 27/30
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
Determining passengers demand
Demand Modelling on rail Lines, Stations and Trains
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 28/30
% 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
Determining passengers demand
Demand Modelling on rail Lines, Stations and Trains
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 29/30
• 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
Demand Modelling on rail Lines, Stations and Trains
F. G. Benitez Transportation Engineering. University of Sevilla, Spain 30/30
Demand Modelling on rail Lines, Stations and Trains
That’s over !
Thanks for your attention