Lecture on Car Following Model: Introduction and Analysis of Stability

1. Microscopic Traffic Theory:
Car Following Models
Prof. Gaetano Fusco
gaetano.fusco@uniroma1.it
http://gaetanofusco.site.uniroma1.it/
Academic Year 2017-2018, Spring Semester
Course of Traffic Engineering and ITS

2. Microscopic Traffic Theory
 Reproduce the behavior of each single driver
 Significant driving tasks affecting traffic
performances:
 Car following
 Lane changing/Overtaking
 Other relevant issues concern road safety
 Approaches to microscopic traffic dynamics:
 Analytical  Differential equation
 Simulation  Software programs
 Artificial intelligence  Black-box heuristics
Car Following Models Page 2

3. Car Following Drivers’ Behaviour
Car Following Models Page 3

4. Modelling Car Following Behaviour
Car Following Models Page 4

5. Car Following Model
 Dynamic Car Following Model
 Assumptions:
 One-way One-lane traffic stream
 Lane changing not allowed.
 Relevant issues:
 Stability:
 local (a single pair of vehicles);
 asymptotic (the whole traffic stream);
 Stationary state
 Experiments and model calibration
Car Following Models Page 5

6. Conceptual Framework
Response = sensitivity x stimulus
 Response = acceleration, that driver controls
acting on acceleration or braking pedals
 Sensitivity = function that equals the stimulus
function to the control function.
 Stimulus = relative speed (u)
 Driver’s tasks:
 Follow the preceding vehicle: u≈0;
 Avoid collision: collision time tc=s(t)/u as large as
possible.
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7. Car Following Models
Different formulations
 Driver’s reaction occurs after a reaction time T.
 Model by Chandler et al. (1958):  =cost
 Model General Motors (Gazis, Herman, Rothery, 1963):
 Different values of l and m provide steady-state different
models.
 Other models: Gipps, Wiedeman
l =
al,m vn+1 t +T( )[ ]
l
sn+1[ ]
m
dvn+1 t+T( )
dt
= l vn t( )-vn+1 t( )[ ]
Page 7

8. System stability
 Stable system:
after a small perturbation, it
comes back to its initial state.
 Unstable system:
after a small perturbation, it gets
away from its initial state
indefinitely.
 Asymptotically stable system:
after an infinite time all solutions
of the system tend to the same
value.
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9. Example of unstable traffic stream
50
100
150
200
250
300
350
400
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
t (s)
x(m)
n =1
n =3 n =5 n=7 n =9
Collisione
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10. Study of local stability
 To determine if a system is locally stable we
integrate the response function dvn+1/dt.
 A system is locally stable if the general
integral of the homogeneous differential
equation is a decreasing function or it is an
oscillatory dumped function.
 The integral of the homogeneous equation
describes the natural characteristics of the
system, independently of external stresses, if
any.
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11. Car Following Models
Study of local stability (2)
 Basic Model:
 Let’s be:  =c (necessary to solve the integral),
=t+T (to simplify Taylor expansion)
 by Taylor series expansion we get a 2nd order
differential equation with constant coefficients.
 Homogenous equation:
dvn+1 t+T( )
dt
= l vn t( )-vn+1 t( )[ ]
Page 11
1
2
cT2 d2
vn+1
dJ 2
+ 1-cT( )
dvn+1
dJ
+cvn+1
= 0

12. Car Following Models
Study of local stability (3)
 The general integral of the 2nd order differential
equation with constant coefficient is
 Where m1 and m2 coefficients are the roots of
characteristic 2nd degree algebraic equation:
 The shape of the integral depends on the sign of
the discriminant :
m1,2
= -
1-cT( )
cT2
±
1-cT( )
cT2
æ
è
ç
ç
ö
ø
÷
÷
2
-
2
T2
= b ± D =
b ±g, if D ³ 0
b ±ig, if D < 0
ì
í
ï
îï
Page 12
vn+1
J( )= c1
e
m1J
+c2
e
m2J
cT2
2
m2
+ 1-cT( )m+c = 0

13. Car Following Models
Non oscillatory decreasing solution
 If 0<cT ≤ 0,414
 ∆>0  m1 and m2
real numbers <0
Since cT<1, is:
γ<0,β< 0
 vn(t) decreasing
non oscillatory
function
 Stable system
0 2 4 6 8 10
0
0.4
0.8
1.2
1.6
2
t
v
  
 21
211
mm
n ececv 
Page 13

14. Car Following Models
Oscillatory dumped solution
 If 0,414<cT < 1
 ∆<0  m1 and m2
complex numbers
 Since 0<cT<1, is:
β<0 
vn(t) dumped
oscillatory function
 Stable system
  
 ii
n ececv 
  211
0 10 20 30 40 50
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
t
v
eβt
v
Page 14

15. Car Following Models
Oscillatory unstable solution
 If cT > 1
 ∆<0  m1 and m2
complex numbers
 Since cT>1:β>0 
vn(t) oscillatory
function with
increasing amplitude
 Unstable system 0 2 4 6 8 10
-8
-6
-4
-2
0
2
4
6
8
t
v
eβt
  
 ii
n ececv 
  211
Page 15

16. Car Following Models
Analogy with a mechanical system
 Mass–Spring–Damper system: Dynamic of a
mass m subject to a stress force f(t) attached
to a spring and a viscous damper:
 Analogies:
 cT 2=>m [mass]
 (1−cT) =>μ [viscous coefficient]
 c =>k [spring constant]
 tfkxxxm   
Page 16

17. Car Following Models
Analogy with an RLC circuit
 Electrical circuit consisting of a resistor R, an
inductor L, and a capacitor C with a voltage
source f
 Analogies:
 cT2=>L [inductance]
 (1−cT) =>R [resistance]
 c =>1/C [inverse of capacitance]
 
t
tf
i
Ct
i
R
t
i
L
d
d1
d
d
d
d
2
2

Page 17

18. Car Following Models
Suggested reading:
Page 18
US Federal Highway Administration
(1996). Traffic Flow Theory.
Chapter 4: Car Following Models
by Richard W. Rothery
https://www.fhwa.dot.gov/publications/research/operatio
ns/tft/chap4.pdf