Part 2 - Modeling traffic flow
engineering models
microscopic models
kinetic models
macroscopic models
D. Helbing, A. Hennecke, and V. Shvetsov, Micro- and macro-simulation of freeway
traffic. Math. Computer Modelling 35 (2002).
N. Bellomo, M. Delitala, V. Coscia, On the mathematical theory of vehicular traffic flow
I. Fluid dynamic and kinetic modelling. Math. Models Appl. Sci. 12 (2002).
M. Garavello and B. Piccoli, Traffic Flow on Networks. Conservation Laws Models.
AIMS Series on Applied Mathematics, Springfield, Mo., 2006.
Alberto Bressan (Penn State)
Scalar Conservation Laws
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A delay model
(T. Friesz et al., 1993)
X (t) = number of cars on a road at time t
If a new car enters at time t,
it will exit at time t + D(X (t))
0
L
D(X ) = delay = total time needed to travel along the road
depends only on the total number of cars at the time of entrance
Alberto Bressan (Penn State)
Scalar Conservation Laws
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An ODE model
(D. Merchant and G. Nemhauser, 1978)
X (t) = total number of cars on a road at time t
u(t) = incoming flux
g(X(t)) = outgoing flux
Ẋ (t) = u(t)
g (X (t))
u
conservation equation
g(X)
X
0
L
L = length of road,
⇢ ⇡
g (X ) = ⇢ v (⇢) =
Alberto Bressan (Penn State)
X
= density of cars
L
⇣X ⌘
X
·v
L
L
Scalar Conservation Laws
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L
0
Models favored by engineers:
simple to use, do not require knowledge of PDEs (or even ODEs)
easy to compute, also on a large network of roads
become accurate when the road is partitioned into short subintervals
Alberto Bressan (Penn State)
Scalar Conservation Laws
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Microscopic models
vi
x i+1(t)
x i(t)
vi−1
x i−1(t)
xi (t) = position of the i-th car
vi (t) = velocity of the i-th car
i = 1, . . . , N
Goal: describe the position and velocity of each car,
writing a large system of ODEs
Alberto Bressan (Penn State)
Scalar Conservation Laws
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Car following models
vi
x i+1(t)
vi−1
x i(t)
x i−1(t)
Acceleration of i-th car depends on:
its speed: vi
speed of car in front: vi
1
distance from car in front: xi
8
< ẋi
:
v̇i
1
xi
= vi
i = 1, . . . , N
= a(vi , vi
Alberto Bressan (Penn State)
1,
xi
1
xi )
Scalar Conservation Laws
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Microscopic intelligent driver model
i-th driver
⇢
(Helbing & al., 2002)
accelerates, up to the maximum speed v̄
decelerates, to keep a safe distance from the car in front
vi 2 [0, v̄ ]
v̄ = maximum speed allowed on the road
a = maximum acceleration

v̇i = a · 1
s i = xi
1
⇣v ⌘
i
v̄
a·
✓
s ⇤ (vi ,
si
vi )
◆2
xi = actual gap from vehicle in front
si⇤ = desired gap
Alberto Bressan (Penn State)
Scalar Conservation Laws
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Desired gap from the vehicle in front
vi
x i+1(t)
si⇤
vi = vi
vi
=
1
x i(t)
0
+
1
r
vi−1
x i−1(t)
vi
vi vi
+ Tvi + p
= desired gap
v̄
2 ab
= speed di↵erence with car in front
0
= jam distance (bumper to bumper)
1
= velocity adjustment of jam distance
T = safe time headway
b = comfortable deceleration
Alberto Bressan (Penn State)
Scalar Conservation Laws
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Equilibrium traffic
Assume: all cars have the same speed, constant in time.
Choose 0 = 1 = 0, = 1
h
v̇i = a · 1
vi i
v̄
a·
✓
s ⇤ (vi ,
si
vi )
◆2
= 0
Equilibrium gap from vehicle in front
h
se (v ) = s ⇤ (v , 0) · 1
Equilibrium velocity:
=)
ve = Ve (⇢)
Alberto Bressan (Penn State)
vi i
v̄
s2
ve (s) =
2v̄ T 2
⇢⇡s
1
Scalar Conservation Laws
1/2
1+
r
4T 2 v̄ 2
s2
!
= macroscopic density
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Statistical (kinetic) description
f = f (t, x, V )
statistical distribution of position and velocity of vehicles
f (t, x, V ) dxdV =
number of vehicles which at time t
are in the phase domain [x, x + dx] ⇥ [V , V + dV ]
local density:
average velocity:
Alberto Bressan (Penn State)
⇢(t, x) =
v (t, x) =
Z
1
f (t, x, V ) dV
0
1
⇢(t, x)
Scalar Conservation Laws
Z
1
0
V · f (t, x, V ) dV
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Evolution of the distribution function
@f
@f
@f
+V
+ a(t, x)
= Q[f , ⇢]
@t
@x
@V
a(t, x) = acceleration
(may depend on the entire distribution f )
Q(f , ⇢) models a trend to equilibrium (as for BGK model in kinetic theory)
⇣
Q = cr (⇢) · fe (V , ⇢)
f (t, x, V )
⌘
cr = relaxation rate
Alberto Bressan (Penn State)
Scalar Conservation Laws
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A conservation law model
(M. Lighthill and G. Witham, 1955)
ρ = density of cars
a
t= time,
b
x= space variable along road,
flux:
x
⇢ = ⇢(t, x) = density of cars
= [number of cars crossing the point x per unit time]
= [density] ⇥ [velocity] = ⇢ · v
v = V (⇢)
⇥
⇤
⇢t + ⇢ V (⇢) x = 0
Alberto Bressan (Penn State)
Scalar Conservation Laws
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Flux function
Assume: ⇢ 7! ⇢V (⇢) is concave
V 0 (⇢) < 0 ,
vmax
2V 0 (⇢) + ⇢V 00 (⇢) < 0
M
v(ρ)
0
Alberto Bressan (Penn State)
1
ρ
0
Scalar Conservation Laws
ρ*
1
ρ
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Characteristics vs. car trajectories
t
flux
ρV(ρ)
0
x
density
ρ
[⇢V (⇢)]0 = V (⇢) + ⇢V 0 (⇢) < V (⇢)
characteristic speed
<
speed of cars
Weak solutions can have upward shocks
ρ(t,x)
x
Alberto Bressan (Penn State)
Scalar Conservation Laws
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Adding a viscosity ?
⇥
⇤
⇢t + ⇢ V (⇢) x = 0
 ⇣
⇢t + ⇢ V (⇢)
e↵ective velocity of cars: v = V (⇢)
( = "⇢xx )
"
⇢x ⌘
⇢
"
= 0
x
⇢x
⇢
can be negative, at the beginning of a queue
ρ(t,x)
x
Alberto Bressan (Penn State)
Scalar Conservation Laws
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Second order models
v = V (⇢) =) velocity is instantly adjusted to the density
Models with acceleration
8
< ⇢t + (⇢v )x
:
vt + v v x
8
< ⇢t + (⇢v )x
:
v t + v vx
= 0
a = acceleration
= a(⇢, v , ⇢x )
= 0
=
1
⌧ (V (⇢)
v)
p 0 (⇢)
⇢
(Payne - Witham, 1971)
⇢x
[relaxation] + [pressure term]
Alberto Bressan (Penn State)
Scalar Conservation Laws
p=⇢ ,
>0
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C. Daganzo, Requiem for second-order fluid approximation to traffic flow, 1995
8
<
:
⇢t + (⇢v )x
vt + v v x +
p 0 (⇢)
⇢
✓ ◆ ✓
⇢t
v
+
vt
p 0 (⇢)/⇢
⇢x
⇢
v
= 0
=
1
⌧ (V (⇢)
v)
◆✓ ◆
✓ ◆
⇢x
0
=
vx
0
eingenvalues = characteristic speeds: v ±
t
p
p 0 (⇢)
domain of the perturbation
x(t) = car position
x
Wrong predictions:
• negative speeds
• perturbations travel faster than the speed of cars
Alberto Bressan (Penn State)
Scalar Conservation Laws
64 / 117
A. Aw, M. Rascle, Resurrection of second-order models of traffic flow, 2000
Idea: replace the partial derivative of the pressure @x p with the
convective derivative (@t + v @x )p
8
<
:
@t ⇢ + @x (⇢v ) = 0
(Aw - Rascle)
@t (v + p(⇢)) + v @x (v + p(⇢)) = 0
✓ ◆ ✓
⇢t
v
+
vt
0 v
strictly hyperbolic for ⇢ > 0,
eigenvalues:
1
Alberto Bressan (Penn State)
=v
⇢
⇢p 0 (⇢)
◆✓ ◆
✓ ◆
⇢x
0
=
vx
0
positive speed: v + p(⇢)
⇢p 0 (⇢),
2
0
=v
Scalar Conservation Laws
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Properties of the Aw-Rascle model
• system is strictly hyperbolic (away from vacuum)
• the density ⇢ and the velocity v remain bounded and non-negative
• characteristic speeds (= eigenvalues) are smaller than car speed
=) drivers are not influenced by what happens behind them.
• maximum speed of cars on an empty road depends on initial data
Alberto Bressan (Penn State)
Scalar Conservation Laws
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An improved model
(R. M. Colombo, 2002)
⇢
Aw - Rascle:
@t ⇢ + @x (v ⇢) = 0
@t q + @x (vq) = 0
q = v ⇢ + ⇢p(⇢) = “momentum”
Colombo:
⇢
@t ⇢ + @x (v ⇢) = 0
@t q + @x (v (q qmax )) = 0
v =
⇢max = maximum density
✓
1
⇢
1
⇢max
◆
q
qmax = “maximum momentum”
=) velocity can vanish only when ⇢ = ⇢max ,
and remains uniformly bounded
Alberto Bressan (Penn State)
Scalar Conservation Laws
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Concluding remarks
Number of vehicles on a road << number of molecules in a gas
microscopic models (solving an ODE for each car) are within
computational reach
kinetic models and macroscopic models are realistic on longer
stretches of road, for densities away from vacuum
optimization problems, dependence of solution on parameters, are
better understood by studying macroscopic models
Simple ODE models, delay models are popular among engineers.
Scalar conservation laws are OK.
Kinetic models, second order models, are a hard sell.
Alberto Bressan (Penn State)
Scalar Conservation Laws
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