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
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.
Car Following Models Page 6
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.
Car Following Models Page 8
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
Car Following Models Page 9
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.
Car Following Models Page 10
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