on the modeling of vehicular...

ON THE MODELING OF VEHICULAR TRAFFIC

ON THE MODELING OF VEHICULAR

TRAFFIC

Ph.D. CourseComplex Systems in Engineering Sciences

Luisa FermoDipartimento di Matematica, Politecnico di Torino

[email protected]


The system to be modeled: Vehicular Traffic along a one-way road. A first

description.

“The width of a traffic shock only encompasses a few vehicles ”

“ A fluid particle responds to stimuli from the front and frombehind, but a car is an anisotropic particle that mostly responds tofrontal stimuli ”

“ Unlike molecules, vehicles have personalities (e.g., aggressive andtimid) that remain unchanged by motion”

C.F. Daganzo

Requiem for second-order fluid approximation of traffic flow ,

Transp. Res., 29B (1995), 277/286


The first step towards modeling: the choice of the scale of representation.

◮ Microscopic scale: Each vehicle is individually followed.The model writes as

xi = ai [t, {xk}Nk=1, {xk}

Nk=1], i = 1, ...,N

where xi is the scalar position of the i-th vehicle,t is the time,xi is the velocity,ai is a function describing the acceleration.

These models are not competitive from a computational pointof view.



◮ Macroscopic scale: Each vehicle is not individually followed.An example is given by

∂ρ

∂t+

∂q

∂x= 0

where ρ is the density,q is the flux.

Some of these models recover missing information fromexperimental observation in steady flow conditions.



◮ Kinetic Scale

1. Each vehicle is identified at time t∗ by◮ its position x

∗

∈ Dx∗ = [0, L] with L > 0 the length of theroad;

◮ its velocity v∗

∈ Dv∗ = [0, Vmax ] with Vmax > 0 the maximumspeed allowed along the road.

Remark: In the following we will consider the dimensionlessquantities:

x =x∗

L∈ Dx ≡ [0, 1], v =

v∗

Vmax

∈ Dv ≡ [0, 1],L

Vmax

t∗ = t.

The set {x , v} defines the microscopic state of the vehicle.


The first step towards modeling: the choice of the scale representation.

2. In order to describe the overall system a distribution function

f = f (t, x , v) : [0, Tmax ] × Dx × Dv → R+

is introduced.

f (t, x , v)dxdv represents the infinitesimal number of vehiclesthat at time t are located in [x , x + dx ] and travel with aspeed belonging to [v , v + dv ].


The first step towards modeling: the choice of the scale representation.

3. The model writes as:

∂f

∂t+ v

∂f

∂x= J[f ]

where J describes the interactions among vehicles.

4. One can compute macroscopic quantities like the density

ρ(t, x) =

∫

Dv

f (t, x , v)dv ,

and the flux

q(t, x) =

∫

Dv

v f (t, x , v)dv .


Modeling the vehicular traffic at the Kinetic Scale

Remark on the Kinetic Approach

Problem:Like the macroscopic models, the kinetic approach is based on acontinuum hyphothesis.

⇓

This assumption is not phisically satisfied by cars along a road.

⇓

A Possible Solution: Discrete velocity kinetic models.


Discrete velocity kinetic models: the basic idea

1. Introduce in Dv = [0, 1] a grid

Iv = {v1, v2, ..., vn−1, vn}

with v1 ≡ 0,vn ≡ 1,

vi < vi+1, ∀i = 1, ..., n − 1.



2. The overall system is now described by

f (t, x , v) =n

∑

i=1

fi (t, x)δvi(v)

where the n functions fi (t, x) = [0, Tmax ] × Dx → R+ are the

distribution functions of the speed classes.

fi (t, x)dx denotes the infinitesimal number of vehicles havingspeed vi that at time t are in [x , x + dx ].



3. The model writes as:

∂fi∂t

+ vi∂fi∂x

= Ji [f], ∀i = 1, ..., n

where f = (f1, ...fn) and Ji describes the interactions amongvehicles.

4. One can compute macroscopic quantities like the density

ρ(t, x) =n

∑

i=1

fi (t, x)

and the flux

q(t, x) =n

∑

i=1

vi fi (t, x).


Discrete velocity kinetic model: the interaction term Ji

The kinetic models describe the interactions by appealing to thefollowing guidelines:

1. Cars are regarded as points, their dimensions are negligible;

2. Interactions are binary. More precisely we will call

A. Candidate vehicle: the vehicle that change its state;B. Field vehicle: the vehicle that causes such a change;C. Test vehicle: an ideal vehicle of the system whose microscopic

state is targeted by a hypothetical observer;


Kinetic discrete velocity model: the interaction term Ji

3. Interactions modify by themselves only the velocity of thevehicles, not their positions;

4. Vehicles are anisotropic particles;

5. Interactions are conservative, in the sense that they preservethe total number of vehicles of the system. Thus the operatorJi [f] is required to satisfy

n∑

i=1

Ji [f] = 0.



If the interactions are such that 1 − 5 are fulfilled then

Ji [f] = Gi [f, f] − fiLi [f],

where

◮ Gi [f, f] is the i-th gain operator giving the amount per unittime that get the velocity vi ;

◮ Li [f] is the i-th loss operator giving the amount of vehicles perunit time that lose the velocity vi .



The interactions among vehicles are described in a stochastic way.Then,

◮ A table of games Aihk is introduced such that

Aihk ≥ 0,

n∑

i=1

Aihk = 1, ∀i = 1, ..., n.

◮ Moreover, an encounter rate ηhk is defined.


Kinetic discrete velocity model: a look at the literature

V. Coscia, M. Delitala and P. FrascaOn the mathematical theory of vehicular traffic flow II. Discrete velocity kinetic

models ,Int. J. Non-Linear Mech., 42(3) (2007), 411-421.

M. Delitala and A. TosinMathematical modeling of vehicular traffic: a discrete kinetic theory approach ,Math. Models Methods Appl. Sci., 17 (2007), 901-932.

C. Bianca and E. CosciaOn the coupling of steady and adaptive velocity grids in vehicular traffic

modelling,

24(2) (2011), 149-155.

A. Bellouquid, E. De Angelis and L. FermoTowards the modeling of vehicular traffic as a complex system: a kinetic

approach,Math. Models Methods Appl. Sci., (2012), to appear


Kinetic discrete velocity models with Local Interactions

The models write as:

∂fi∂t

+ vi∂fi∂x

= Ji [f], ∀i = 1, ..., n

where

Ji [f] =n

∑

h,k=1

ηhkAihk fhfk − fi

n∑

k=1

ηik fk

V. Coscia, M. Delitala and P. FrascaOn the mathematical theory of vehicular traffic flow II. Discrete velocity kinetic

models ,Int. J. Non-Linear Mech., 42(3) (2007), 411-421.

C. Bianca and E. CosciaOn the coupling of steady and adaptive velocity grids in vehicular traffic

modelling,

24(2) (2011), 149-155.


Kinetic discrete velocity models with non local interactions

◮ One define Interaction length ξ the distance between theinteracting field and the candidate vehicle.

◮ If x is the position of the candidate vehicle and ξ is the lengthinteraction then one can define the interaction interval orvisibility zone as Iξ(x) = [x , x + ξ].

◮ One introduce a weight function w(x , y) weigthing theinteraction over the visibility zone.

M. Delitala and A. TosinMathematical modeling of vehicular traffic: a discrete kinetic theory approach ,Math. Models Methods Appl. Sci., 17 (2007), 901-932.


approach,Math. Models Methods Appl. Sci., (2012), to appear


Kinetic discrete velocity models with non local interactions

The models write as

∂fi∂t

+ vi∂fi∂x

= Ji [f], ∀i = 1, ..., n (1)

where

Ji [f] =n

∑

h,k=1

∫

Iξ

ηhk(t, y)Aihk(t, y)fh(t, x)fk(t, y)w(x , y)dy

− fi (t, x)

n∑

k=1

∫

Iξ

ηik(t, y)fk(t, y)w(x , y)dy .

M. Delitala and A. Tosin

Mathematical modeling of vehicular traffic: a discrete kinetic theory approach ,Math. Models Methods Appl. Sci., 17 (2007), 901-932.

A. Bellouquid, E. De Angelis and L. Fermo

Towards the modeling of vehicular traffic as a complex system: a kinetic approach,Math. Models Methods Appl. Sci., (2012), to appear.


A Kinetic discrete velocity model

All the terms appearing in the equations are modelled taking intoaccount the Traffic Phase, namely, traffic states having specificempirical spatiotemporal features. These features are specific onlyto a single traffic phase. It is characterized by a certain set ofstatistical properties of traffic variables (density, mean velocity,flux).


approach,Math. Models Methods Appl. Sci., (2012), to appear.



Classically there are two traffic phase:

◮ Free flow where vehicles are able to change a lane and topass. The maximum density achievable under free flow iscalled critical density ρc .

◮ Congested flow occurs when the vehicle density is highenough. Here the speed is lower than the lowest speed in freeflow.



Kerner identify three traffic phase:

◮ Free flow (F) where vehicles are able to change a lane and topass. The maximum density achievable under free flow iscalled critical density ρc .

◮ Congested flow occurs when the vehicle density is highenough. Here the speed is lower than the lowest speed in freeflow.

◮ Syncronized flow (S) Kerner describes synchronized flow asthe phase at which vehicles are accelerating to meet free flowtraffic.

◮ Wide moving jam (J) When vehicles move up the highwaythrough bottlenecks. The minimum density achieved undercongestion is called jam density ρj .

B. S. KernerThe Physics of Traffic,Springer 2004.



Now, we come back to equation (1) and model terms ηhk and Aihk .

◮ At first, we introduce a parameter α identifying theenvironmental conditions:

α = α0 +ρc

ρcmax(1 − α0),

where α0 is the minimum value of α identified by experiments,ρc is the critical density,ρcmax is the maximum critical density.

◮ Then we define the encounter rate η = ηhk = 1 + αρ2, ∀h, k .





◮ We give a table of games according to the three traffic phase.In FREE FLOW (0 ≤ ρ ≤ ρc)

Aihk =

1, i = n;

0, otherwise.(2)





◮ In WIDE MOVING JAM (ρj ≤ ρ ≤ 1)

Aihk =

1, i = 1;

0, otherwise.(3)





◮ In SYNCRONIZED FLOW (ρc < ρ < ρj)

A. Interaction with faster vehicles.

Aihk =

1 − α (ρj + ρc − ρ), i = h;

α(i − h)

1k

∑

i=h+1

1

i − h

(ρj + ρc − ρ), i = h + 1, ..., k ;

0, otherwise.





B. Interaction with slower vehicles.

Aihk =

α (ρj + ρc − ρ), i = h;

[1 − α (ρj + ρc − ρ)](h − i)

h−1∑

i=k

(h − i)

, i = k , ..., h − 1;

0, otherwise.





C. Interaction with equally fast vehicles.

Aihk =

(1 − α)(h − i)

h−1∑

i=1

(h − i)

(1 − ρj − ρc + ρ), i = 1, ...h − 1;

α + (1 − 2α)(ρs + ρc − ρ), i = h;

α 1(i − h)

1n

∑

i=h+1

1

i − h

(ρj + ρc − ρ), i = h + 1, ..., n.





C. Interaction with equally fast vehicles.

Ai11 =

1 − α(ρj + ρc − ρ), i = 1;

α(ρj + ρc − ρ), i = 2;

0, otherwise.

Ainn =

α (1 − ρj − ρc + ρ), i = n − 1;

1 − α (1 − ρj − ρc + ρ), i = n;

0, otherwise.

A. Bellouquid, E. De Angelis and L. Fermo

Towards the modeling of vehicular traffic as a complex system: a kinetic approach,Math. Models Methods Appl. Sci., (2012), to appear.



Only for the sake of simplicity here we have consideredthe microscopic state {x , v}.

In the following paper



the microscopic state {x , v , u} is considered where u is anadditional variable denoting the quality of the micro-systemdriver-vehicle.


The non stationary problem

◮ We study the evolution of the system.The solution will depend on the time and mathematically themodel will be described by

1. Integro-differential equations;2. Initial conditions;3. Boundary conditions.


Example: Formation of Clustering

The model:

∂fi∂t

+ vi∂fi∂x

= Ji [f]

fn(0, x) = 100 sin2 (10π(x − 0.2)(x − 0.3)), x ∈ [0.2, 0.3],

fn−1(0, x) = 50 sin2 (10π(x − 0.5)(x − 0.6)), x ∈ [0, 5, 0.6]

fi (0, x) = 0, ∀i = 1, ..., n − 2, ∀x ∈ Dx

fi (t, 0) = fi (t, 1) ∀i = 1, ..., n


Example: Formation of Clustering in bad road conditions

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

ρ

t=0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

ρ

t=1.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

ρ

t=1.14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

ρt=1.40


Example: Formation of Clustering in optimal road conditions

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

ρ

t=0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

ρ

t=1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

ρ

t=1.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

ρ

t=2.19


The spatially homogeneous problem

The model writes as

dfijdt

= η

n∑

h,k=1

Aihk fh(t) fk(t) − fi (t)

n∑

k=1

fk(t)

,

fi (0) = f 0i ,


Velocity Diagram

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ρ

ξ

α=0.95

α=0.7

α=0.5

α=0.3

Figure: Velocity diagram: mean velocity ξ versus density ρ


Fundamental Diagram

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

ρ

q

α=0.95

α=0.7

α=0.5

α=0.3

Figure: Fundamental diagram: flux q versus density ρ

on the modeling of vehicular...

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