A general phase transition model for vehicular traffic
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
A general phase transition model for vehicular traffic Sébastien Blandin
May 4th 2009
1/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Macroscopic models of traffic
◮
How to model traffic flow on a stretch of highway ?
2/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Macroscopic models of traffic
◮ ◮
How to model traffic flow on a stretch of highway ? Assumptions:
2/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Macroscopic models of traffic
◮ ◮
How to model traffic flow on a stretch of highway ? Assumptions: ◮
One lane
2/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Macroscopic models of traffic
x
◮ ◮
ρ(x, t)
How to model traffic flow on a stretch of highway ? Assumptions: ◮ ◮
One lane Continuum approximation
2/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Macroscopic models of traffic
x
◮ ◮
ρ(x, t)
How to model traffic flow on a stretch of highway ? Assumptions: ◮ ◮ ◮
One lane Continuum approximation No ramp
2/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Scalar macroscopic models
◮
Models of traffic based on mass conservation. ∂ρ ∂q(ρ) + =0 ∂t ∂x
3/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Scalar macroscopic models
◮
Models of traffic based on mass conservation. ∂ρ ∂q(ρ) + =0 ∂t ∂x
◮
In integral form, with N(t) the number of cars on [0, L] at t, and q(x, t) the flux of cars at (x, t): ∂N (t) = q(0, t) − q(L, t) ∂t
3/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Scalar macroscopic models
◮
Models of traffic based on mass conservation. ∂ρ ∂q(ρ) + =0 ∂t ∂x
◮
Knowledge of the ‘fundamental diagram’, which is an empirical relation between the flux and the density.
3/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
The fundamental diagram Classical fundamental diagrams include: ◮
Greenshields q(ρ) = ρ V
1−
ρ R
Q(ρ)
R/2
R
ρ
4/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
The fundamental diagram Classical fundamental diagrams include: ◮
Newell-Daganzo q(ρ) =
( ρV
ρV ρc −R
if ρ ≤ ρc if ρ > ρc
(ρ − R)
Q(ρ)
ρc
R
ρ
4/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
The fundamental diagram Classical fundamental diagrams include: ◮
Greenberg q(ρ) = ρ v0 log
◮
ρmax ρ
Papageorgiou q(ρ) = ρ vmax
1 exp − a
ρ ρc
a
4/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Real speed-density relation Q (veh/hr ) 3600
2000
0
◮ ◮
0
100
225
ρ(veh/mile)
Has different behaviors in free-flow and congestion Is not single-valued in congestion
5/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
6/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Model definition
◮
Two different modes for the dynamics ∂ (t ρ + ∂x (ρ vf (ρ)) = 0 ∂t ρ + ∂x (ρ vc (ρ, q)) = 0 ∂t q + ∂x ((q − q ∗ ) vc (ρ, q)) = 0
in free-flow in congestion
7/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Model definition
◮
◮
Two different modes for the dynamics ∂ (t ρ + ∂x (ρ vf (ρ)) = 0 ∂t ρ + ∂x (ρ vc (ρ, q)) = 0 ∂t q + ∂x ((q − q ∗ ) vc (ρ, q)) = 0
in free-flow in congestion
A set-valued velocity function in congestion ρ ρ q vf (ρ) = 1 − V and vc (ρ, q) = 1 − R R ρ
7/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Improved features of this 2X2 model
◮
Characteristics speeds lower than vehicle speeds ⇒ the model is anisotropic
◮
Vehicles stop only at maximal density
8/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Limitations of the 2X2 phase transition model ρv
Congestion Free − flow
R ρ
But ◮
◮ ◮
Solution of the discretized PDE constructed through wavefront-tracking is complex Model not customizable Free-flow speed is not constant
9/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
10/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
A general 2X2 phase transition model ◮
Constant free-flow speed v = vf (ρ) := V in free-flow
11/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
A general 2X2 phase transition model ◮
Constant free-flow speed v = vf (ρ) := V in free-flow
◮
Connection of the free-flow and congested phases ρv
Congestion
Free − flow
11/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
A general 2X2 phase transition model ◮
Constant free-flow speed v = vf (ρ) := V in free-flow
◮
Connection of the free-flow and congested phases
◮
Introduction of the notion of perturbation, q v = vc (ρ, q) := vc (ρ, 0) (1 + q) in congestion. in free-flow (Ωf ) ∂ (t ρ + ∂x (ρ v ) = 0 ∂t ρ + ∂x (ρ v ) = 0 in congestion (Ωc ) ∂t q + ∂x (q v ) = 0
11/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Physical and mathematical constraints ◮
Positivity of speed 1+q ≥0
12/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Physical and mathematical constraints ◮
Positivity of speed 1+q ≥0
◮
Strict hyperbolicity of the system in congestion ∀ (ρ, q) ∈ Ωc
ρ ∂ρ vc (ρ, 0) + q (vc (ρ, 0) + ρ ∂ρ vc (ρ, 0)) 6= 0
12/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Physical and mathematical constraints ◮
Positivity of speed 1+q ≥0
◮
Strict hyperbolicity of the system in congestion ∀ (ρ, q) ∈ Ωc
◮
ρ ∂ρ vc (ρ, 0) + q (vc (ρ, 0) + ρ ∂ρ vc (ρ, 0)) 6= 0
Shape of Lax-curves 2 ρ (2 ∂ρ vc (ρ, 0) + ρ ∂ρρ vc (ρ)) 2 + q (2 vc (ρ, 0) + 4 ρ ∂ρ vc (ρ, 0) + ρ2 ∂ρρ vc (ρ, 0))
is identically zero or has only one zero and is increasing. 12/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Instantiation ◮
Newell-Daganzo Q(ρ)
Q(ρ)
ρ
ρ
q ∈ [q− , q+ ] where q− ≥ −1
13/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Instantiation ◮
Greenshields Q(ρ)
Q(ρ)
ρ
q ∈ [q− , q+ ] where q− ≥
ρ
R 2 ρc − 3 R
13/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Instantiation ◮
Non-concave flux Q(ρ)
Q(ρ)
ρ
ρ
Bounds on both q− and q+
13/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Specific features of the 2X2 phase transition model ◮
Modelization of different types of congestion ◮
◮
Wide moving traffic jams (backwards-moving jams ⇒ usual first order congestion) Synchronized traffic flow (forward-moving discontinuities with the same speed on both sides)
◮
Riemann problem solved by two different waves, one which has negative speed and one which has the speed of vehicles
◮
‘Big shocks’ from classical LWR equation are phase transition in this framework
◮
Possibility of integrating both density measurements and speed measurements 14/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Example ◮
Riemann problem with datum: ( uleft = (ρA , vA ) uright = (ρB , vB )
Q(ρ)
B
A ρ 15/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Example ◮
Backward-moving discontinuity (shockwave) between A′ and B ′ Q(ρ)
B B′ A′
A ρ
15/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Example ◮
Backward-moving discontinuity (shockwave) between A and C followed by a forward moving discontinuity (contact discontinuity) between C and B Q(ρ)
B
C
A ρ
15/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
16/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Dataset
◮
NGSIM dataset recorded on Highway 101 in Los Angeles
◮
Car trajectories available every 0.1 seconds
◮
45 minutes long
◮
0.4 miles
17/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Comparison of triangular model and its extension
Error metric Eu =
R T R x1 0
ku(t, x) − uc (t, x)k1 dxdt R T R x1 0 x0 ku(t, x)k1 dxdt
x0
Newell-Daganzo N-D phase transition
Eρ 17.8% 15.0%
Ev 23.5% 22.1%
18/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Future work
◮
Comparison of travel-time estimations for different models on the Mobile Century dataset
◮
Define an optimal shape for the clouds of points in congestion
19/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Questions ?
20/20
A general phase transition model for vehicular traffic Sébastien Blandin
May 4th 2009
1/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Macroscopic models of traffic
◮
How to model traffic flow on a stretch of highway ?
2/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Macroscopic models of traffic
◮ ◮
How to model traffic flow on a stretch of highway ? Assumptions:
2/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Macroscopic models of traffic
◮ ◮
How to model traffic flow on a stretch of highway ? Assumptions: ◮
One lane
2/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Macroscopic models of traffic
x
◮ ◮
ρ(x, t)
How to model traffic flow on a stretch of highway ? Assumptions: ◮ ◮
One lane Continuum approximation
2/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Macroscopic models of traffic
x
◮ ◮
ρ(x, t)
How to model traffic flow on a stretch of highway ? Assumptions: ◮ ◮ ◮
One lane Continuum approximation No ramp
2/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Scalar macroscopic models
◮
Models of traffic based on mass conservation. ∂ρ ∂q(ρ) + =0 ∂t ∂x
3/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Scalar macroscopic models
◮
Models of traffic based on mass conservation. ∂ρ ∂q(ρ) + =0 ∂t ∂x
◮
In integral form, with N(t) the number of cars on [0, L] at t, and q(x, t) the flux of cars at (x, t): ∂N (t) = q(0, t) − q(L, t) ∂t
3/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Scalar macroscopic models
◮
Models of traffic based on mass conservation. ∂ρ ∂q(ρ) + =0 ∂t ∂x
◮
Knowledge of the ‘fundamental diagram’, which is an empirical relation between the flux and the density.
3/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
The fundamental diagram Classical fundamental diagrams include: ◮
Greenshields q(ρ) = ρ V
1−
ρ R
Q(ρ)
R/2
R
ρ
4/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
The fundamental diagram Classical fundamental diagrams include: ◮
Newell-Daganzo q(ρ) =
( ρV
ρV ρc −R
if ρ ≤ ρc if ρ > ρc
(ρ − R)
Q(ρ)
ρc
R
ρ
4/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
The fundamental diagram Classical fundamental diagrams include: ◮
Greenberg q(ρ) = ρ v0 log
◮
ρmax ρ
Papageorgiou q(ρ) = ρ vmax
1 exp − a
ρ ρc
a
4/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Real speed-density relation Q (veh/hr ) 3600
2000
0
◮ ◮
0
100
225
ρ(veh/mile)
Has different behaviors in free-flow and congestion Is not single-valued in congestion
5/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
6/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Model definition
◮
Two different modes for the dynamics ∂ (t ρ + ∂x (ρ vf (ρ)) = 0 ∂t ρ + ∂x (ρ vc (ρ, q)) = 0 ∂t q + ∂x ((q − q ∗ ) vc (ρ, q)) = 0
in free-flow in congestion
7/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Model definition
◮
◮
Two different modes for the dynamics ∂ (t ρ + ∂x (ρ vf (ρ)) = 0 ∂t ρ + ∂x (ρ vc (ρ, q)) = 0 ∂t q + ∂x ((q − q ∗ ) vc (ρ, q)) = 0
in free-flow in congestion
A set-valued velocity function in congestion ρ ρ q vf (ρ) = 1 − V and vc (ρ, q) = 1 − R R ρ
7/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Improved features of this 2X2 model
◮
Characteristics speeds lower than vehicle speeds ⇒ the model is anisotropic
◮
Vehicles stop only at maximal density
8/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Limitations of the 2X2 phase transition model ρv
Congestion Free − flow
R ρ
But ◮
◮ ◮
Solution of the discretized PDE constructed through wavefront-tracking is complex Model not customizable Free-flow speed is not constant
9/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
10/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
A general 2X2 phase transition model ◮
Constant free-flow speed v = vf (ρ) := V in free-flow
11/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
A general 2X2 phase transition model ◮
Constant free-flow speed v = vf (ρ) := V in free-flow
◮
Connection of the free-flow and congested phases ρv
Congestion
Free − flow
11/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
A general 2X2 phase transition model ◮
Constant free-flow speed v = vf (ρ) := V in free-flow
◮
Connection of the free-flow and congested phases
◮
Introduction of the notion of perturbation, q v = vc (ρ, q) := vc (ρ, 0) (1 + q) in congestion. in free-flow (Ωf ) ∂ (t ρ + ∂x (ρ v ) = 0 ∂t ρ + ∂x (ρ v ) = 0 in congestion (Ωc ) ∂t q + ∂x (q v ) = 0
11/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Physical and mathematical constraints ◮
Positivity of speed 1+q ≥0
12/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Physical and mathematical constraints ◮
Positivity of speed 1+q ≥0
◮
Strict hyperbolicity of the system in congestion ∀ (ρ, q) ∈ Ωc
ρ ∂ρ vc (ρ, 0) + q (vc (ρ, 0) + ρ ∂ρ vc (ρ, 0)) 6= 0
12/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Physical and mathematical constraints ◮
Positivity of speed 1+q ≥0
◮
Strict hyperbolicity of the system in congestion ∀ (ρ, q) ∈ Ωc
◮
ρ ∂ρ vc (ρ, 0) + q (vc (ρ, 0) + ρ ∂ρ vc (ρ, 0)) 6= 0
Shape of Lax-curves 2 ρ (2 ∂ρ vc (ρ, 0) + ρ ∂ρρ vc (ρ)) 2 + q (2 vc (ρ, 0) + 4 ρ ∂ρ vc (ρ, 0) + ρ2 ∂ρρ vc (ρ, 0))
is identically zero or has only one zero and is increasing. 12/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Instantiation ◮
Newell-Daganzo Q(ρ)
Q(ρ)
ρ
ρ
q ∈ [q− , q+ ] where q− ≥ −1
13/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Instantiation ◮
Greenshields Q(ρ)
Q(ρ)
ρ
q ∈ [q− , q+ ] where q− ≥
ρ
R 2 ρc − 3 R
13/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Instantiation ◮
Non-concave flux Q(ρ)
Q(ρ)
ρ
ρ
Bounds on both q− and q+
13/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Specific features of the 2X2 phase transition model ◮
Modelization of different types of congestion ◮
◮
Wide moving traffic jams (backwards-moving jams ⇒ usual first order congestion) Synchronized traffic flow (forward-moving discontinuities with the same speed on both sides)
◮
Riemann problem solved by two different waves, one which has negative speed and one which has the speed of vehicles
◮
‘Big shocks’ from classical LWR equation are phase transition in this framework
◮
Possibility of integrating both density measurements and speed measurements 14/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Example ◮
Riemann problem with datum: ( uleft = (ρA , vA ) uright = (ρB , vB )
Q(ρ)
B
A ρ 15/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Example ◮
Backward-moving discontinuity (shockwave) between A′ and B ′ Q(ρ)
B B′ A′
A ρ
15/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Example ◮
Backward-moving discontinuity (shockwave) between A and C followed by a forward moving discontinuity (contact discontinuity) between C and B Q(ρ)
B
C
A ρ
15/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
16/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Dataset
◮
NGSIM dataset recorded on Highway 101 in Los Angeles
◮
Car trajectories available every 0.1 seconds
◮
45 minutes long
◮
0.4 miles
17/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Comparison of triangular model and its extension
Error metric Eu =
R T R x1 0
ku(t, x) − uc (t, x)k1 dxdt R T R x1 0 x0 ku(t, x)k1 dxdt
x0
Newell-Daganzo N-D phase transition
Eρ 17.8% 15.0%
Ev 23.5% 22.1%
18/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Future work
◮
Comparison of travel-time estimations for different models on the Mobile Century dataset
◮
Define an optimal shape for the clouds of points in congestion
19/20
Motivation The Colombo 2X2 phase transition model A class of phase transition models Model accuracy
Questions ?
20/20
