A Boltzmann-type kinetic approach to traffic flow on road ...acesar/slidesanconet/Tosin.pdf · A...

A Boltzmann-type kinetic approach to traffic flowon road networks

Andrea Tosin

Department of Mathematical Sciences “G. L. Lagrange”Politecnico di Torino, Italy

Analysis and Control on Networks: Trends and PerspectivesUniversity of Padua

9th-11th March 2016

Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks

From Real Road Networks. . .

Usually one describes a portion of a real network1

Possibly one distinguishes between main and minor roads

Dynamics at junctions are the key point of the description of traffic flowson networks

Well established theory for macroscopic models (Garavello and Piccoli,2006 [4])Here we refer to the theory for kinetic models proposed in Fermo andTosin, 2015 [3]

1But Caramia et al., 2010 [1] simulated the whole road network of Salerno

. . . To Oriented Graphs

The edges of the graph are the roads, with their orientation

Vertexes of type • are the junctions transmission conditions

Vertexes of type � are access/exit points boundary conditions

Network structure described by the incidence matrix of the graph:

{−1 if road r enters junction s (incoming road)

1 if road r leaves junction s (outgoing road)

0 if road r and junction s are not incident

The Model on Single Roads (Fermo and Tosin, 2013 [2])

Network composed by R roads indexed by r = 1, . . . , R

Finite length of each road ⇒ finite number of cells i = 1, . . . , m:

m⋃i=1

Iri = [0, m), Iri1 ∩ Iri2 = ∅ ∀ i1 6= i2, |Iri | = 1

Speed lattice (the same in each road): vj = j−1n−1

, j = 1, . . . , n

Kinetic distribution function for road r: frij = frij(t) : [0, T ]→ [0, 1]

Density, flux in the ith cell of road r: ρri =n∑j=1

frij , qri =

n∑j=1

vjfrij

Evolution equation:

dfrijdt

+ vj(Φri,i+1f

rij − Φri−1,if

ri−1,j

)= Qij ,

Φri,i+1 =

min{ρri , 1−ρri+1}

ρrin∑j=1

Qij = 0 ∀ i

m⋃i=1

, j = 1, . . . , n

frij , qri =

n∑j=1

vjfrij

Evolution equation:

dfrijdt

+ vj(Φri,i+1f

rij − Φri−1,if

ri−1,j

)= Qij ,

Φri,i+1 =

ρrin∑j=1

Qij = 0 ∀ i

m⋃i=1

, j = 1, . . . , n

frij , qri =

n∑j=1

vjfrij

Evolution equation:

dfrijdt

+ vj(Φri,i+1f

rij − Φri−1,if

ri−1,j

)= Qij ,

Φri,i+1 =

ρrin∑j=1

Qij = 0 ∀ i

m⋃i=1

, j = 1, . . . , n

frij , qri =

n∑j=1

vjfrij

Evolution equation:

dfrijdt

+ vj(Φri,i+1f

rij − Φri−1,if

ri−1,j

)= Qij ,

Φri,i+1 =

ρrin∑j=1

Qij = 0 ∀ i

m⋃i=1

, j = 1, . . . , n

frij , qri =

n∑j=1

vjfrij

Evolution equation:

dfrijdt

+ vj(Φri,i+1f

rij − Φri−1,if

ri−1,j

)= Qij ,

Φri,i+1 =

ρrin∑j=1

Qij = 0 ∀ i

m⋃i=1

, j = 1, . . . , n

frij , qri =

n∑j=1

vjfrij

Evolution equation:

dfrijdt

+ vj(Φri,i+1f

rij − Φri−1,if

ri−1,j

)= Qij ,

Φri,i+1 =

ρrin∑j=1

Qij = 0 ∀ i

The Collisional Operator

The right-hand side

Qij = Q[fr, fr](t, Iri , vj) :=

n∑k=1

n∑h=1

Pjhkfrihf

rik − ρri frij

describes, in average, microscopic binary interactions among the vehiclescausing speed variations

The termPjhk := P(vh → vj | vk)

is the probability that a vehicle with pre-interaction speed vh switches tothe speed vj after interacting with a leading vehicle with speed vk

A possible model of the interactions is e.g., (see Puppo, Semplice,

Tosin, Visconti, 2016 [5])

Pjhk = ρri δj,min{h, k} + (1− ρri )δj,min{h+1, n}

The right-hand side

n∑k=1

n∑h=1

Pjhkfrihf

rik − ρri frij

The right-hand side

n∑k=1

n∑h=1

Pjhkfrihf

rik − ρri frij

Boundary Conditions at Access Points

At an access point consider the equation for i = 1 (first cell):

dfr1jdt

+ vj(Φr1,2f

r1j − Φr0,1f

)= Q1j

{fr0j}nj=1 must be prescribed (the cell I0 does not exist) speeddistribution of the incoming vehicles

The flux limiter Φr0,1 at the road entrance is automatically defined:

Φr0,1 =min{ρr0, 1− ρr1}

ρr0with ρr0 =

n∑j=1

dfr1jdt

+ vj(Φr1,2f

r1j − Φr0,1f

)= Q1j

Φr0,1 =min{ρr0, 1− ρr1}

ρr0with ρr0 =

n∑j=1

dfr1jdt

+ vj(Φr1,2f

r1j − Φr0,1f

)= Q1j

Φr0,1 =min{ρr0, 1− ρr1}

ρr0with ρr0 =

n∑j=1

dfr1jdt

+ vj(Φr1,2f

r1j − Φr0,1f

)= Q1j

Φr0,1 =min{ρr0, 1− ρr1}

ρr0with ρr0 =

n∑j=1

Boundary Conditions at Exit Points

At an exit point consider the equation for i = m (last cell):

dfrmjdt

+ vj(Φrm,m+1f

rmj − Φrm−1,mf

rm−1,j

)= Qmj

Φrm,m+1 must be prescribed (the cell Im+1 does not exist) trafficconditions downstream

Typical conditions:Φrm,m+1 = 1 vehicles can freely leave the network from road r

Φrm,m+1 = 0 vehicles cannot leave the network from road r

dfrmjdt

+ vj(Φrm,m+1f

rmj − Φrm−1,mf

rm−1,j

)= Qmj

dfrmjdt

+ vj(Φrm,m+1f

rmj − Φrm−1,mf

rm−1,j

)= Qmj

dfrmjdt

+ vj(Φrm,m+1f

rmj − Φrm−1,mf

rm−1,j

)= Qmj

The 1-2 Junction

Im1 I 1

The 1-2 Junction: Mass Conservation

Im1 I 1

Write the kinetic equation in the cellsaround the junction (I1m, I21 , I31 )

Sum over j = 1, . . . , n each equation:dρ1mdt

+ Φ1m,m+1q

1m − Φ1

m−1,mq1m−1 = 0

dρ21dt

+ Φ21,2q

21 − Φ2

0,1q20 = 0

dρ31dt

+ Φ31,2q

31 − Φ3

0,1q30 = 0

Sum term by term:

(ρ1m + ρ21 + ρ31

)= Φ1

m−1,mq1m−1 − Φ2

1,2q21 − Φ3

1,2q31

0,1q20 + Φ3

0,1q30 − Φ1

m,m+1q1m

)Mass conservation through the junction requires:

Φ1m,m+1q

1m = Φ2

0,1q20 + Φ3

0,1q30 {f2

0j}nj=1, {f30j}nj=1, Φ1

Im1 I 1

+ Φ1m,m+1q

1m − Φ1

m−1,mq1m−1 = 0

dρ21dt

+ Φ21,2q

21 − Φ2

0,1q20 = 0

dρ31dt

+ Φ31,2q

31 − Φ3

0,1q30 = 0

Sum term by term:

(ρ1m + ρ21 + ρ31

)= Φ1

m−1,mq1m−1 − Φ2

1,2q21 − Φ3

1,2q31

0,1q20 + Φ3

0,1q30 − Φ1

m,m+1q1m

Φ1m,m+1q

1m = Φ2

0,1q20 + Φ3

0,1q30 {f2

0j}nj=1, {f30j}nj=1, Φ1

Im1 I 1

+ Φ1m,m+1q

1m − Φ1

m−1,mq1m−1 = 0

dρ21dt

+ Φ21,2q

21 − Φ2

0,1q20 = 0

dρ31dt

+ Φ31,2q

31 − Φ3

0,1q30 = 0

Sum term by term:

(ρ1m + ρ21 + ρ31

)= Φ1

m−1,mq1m−1 − Φ2

1,2q21 − Φ3

1,2q31

0,1q20 + Φ3

0,1q30 − Φ1

m,m+1q1m

Φ1m,m+1q

1m = Φ2

0,1q20 + Φ3

0,1q30 {f2

0j}nj=1, {f30j}nj=1, Φ1

Im1 I 1

+ Φ1m,m+1q

1m − Φ1

m−1,mq1m−1 = 0

dρ21dt

+ Φ21,2q

21 − Φ2

0,1q20 = 0

dρ31dt

+ Φ31,2q

31 − Φ3

0,1q30 = 0

Sum term by term:

(ρ1m + ρ21 + ρ31

)= Φ1

m−1,mq1m−1 − Φ2

1,2q21 − Φ3

1,2q31

0,1q20 + Φ3

0,1q30 − Φ1

m,m+1q1m

Φ1m,m+1q

1m = Φ2

0,1q20 + Φ3

0,1q30 {f2

0j}nj=1, {f30j}nj=1, Φ1

Im1 I 1

+ Φ1m,m+1q

1m − Φ1

m−1,mq1m−1 = 0

dρ21dt

+ Φ21,2q

21 − Φ2

0,1q20 = 0

dρ31dt

+ Φ31,2q

31 − Φ3

0,1q30 = 0

Sum term by term:

(ρ1m + ρ21 + ρ31

)= Φ1

m−1,mq1m−1 − Φ2

1,2q21 − Φ3

1,2q31

0,1q20 + Φ3

0,1q30 − Φ1

m,m+1q1m

Φ1m,m+1q

1m = Φ2

0,1q20 + Φ3

0,1q30 {f2

0j}nj=1, {f30j}nj=1, Φ1

Im1 I 1

+ Φ1m,m+1q

1m − Φ1

m−1,mq1m−1 = 0

dρ21dt

+ Φ21,2q

21 − Φ2

0,1q20 = 0

dρ31dt

+ Φ31,2q

31 − Φ3

0,1q30 = 0

Sum term by term:

(ρ1m + ρ21 + ρ31

)= Φ1

m−1,mq1m−1 − Φ2

1,2q21 − Φ3

1,2q31

0,1q20 + Φ3

0,1q30 − Φ1

m,m+1q1m

Φ1m,m+1q

1m = Φ2

0,1q20 + Φ3

0,1q30 {f2

0j}nj=1, {f30j}nj=1, Φ1

The 1-2 Junction: The Flux Distribution Rule

Assume that there exists a flux distribution coefficient a ∈ [0, 1] suchthat

q20 = aq1m, q30 = (1− a)q1m

Rewriting in terms of the kinetic distribution functions yields:

n∑j=1

vj(f20j − af1

n∑j=1

vj(f30j − (1− a)f1

This guides the choice of the speed distributions through the junction (notunique in general):

f20j =

{0 if j = 1

af1mj if j ≥ 2

, f30j =

{0 if j = 1

(1− a)f1mj if j ≥ 2

Finally, from mass conservation: Φ1m,m+1 = aΦ2

0,1 + (1− a)Φ30,1

q20 = aq1m, q30 = (1− a)q1m

n∑j=1

vj(f20j − af1

n∑j=1

vj(f30j − (1− a)f1

f20j =

{0 if j = 1

af1mj if j ≥ 2

, f30j =

{0 if j = 1

(1− a)f1mj if j ≥ 2

0,1 + (1− a)Φ30,1

q20 = aq1m, q30 = (1− a)q1m

n∑j=1

vj(f20j − af1

n∑j=1

vj(f30j − (1− a)f1

f20j =

{0 if j = 1

af1mj if j ≥ 2

, f30j =

{0 if j = 1

(1− a)f1mj if j ≥ 2

0,1 + (1− a)Φ30,1

q20 = aq1m, q30 = (1− a)q1m

n∑j=1

vj(f20j − af1

n∑j=1

vj(f30j − (1− a)f1

f20j =

{0 if j = 1

af1mj if j ≥ 2

, f30j =

{0 if j = 1

(1− a)f1mj if j ≥ 2

0,1 + (1− a)Φ30,1

The 2-1 Junction

Write the kinetic equation in the cellsaround the junction (I1m, I2m, I31 )

+ Φ1m,m+1q

1m − Φ1

m−1,mq1m−1 = 0

dρ2mdt

+ Φ2m,m+1q

2m − Φ2

m−1,mq2m−1 = 0

dρ31dt

+ Φ31,2q

31 − Φ3

0,1q30 = 0

Sum term by term:

(ρ1m + ρ2m + ρ31

)= Φ1

m−1,mq1m−1 + Φ2

m−1,mq2m − Φ3

1,2q31

0,1q30 − Φ1

m,m+1q1m − Φ2

m,m+1q2m

Φ1m,m+1q

1m + Φ2

m,m+1q2m = Φ3

0,1q30 {f3

0j}nj=1, Φ1m,m+1, Φ2

+ Φ1m,m+1q

1m − Φ1

m−1,mq1m−1 = 0

dρ2mdt

+ Φ2m,m+1q

2m − Φ2

m−1,mq2m−1 = 0

dρ31dt

+ Φ31,2q

31 − Φ3

0,1q30 = 0

Sum term by term:

(ρ1m + ρ2m + ρ31

)= Φ1

m−1,mq1m−1 + Φ2

m−1,mq2m − Φ3

1,2q31

0,1q30 − Φ1

m,m+1q1m − Φ2

m,m+1q2m

Φ1m,m+1q

1m + Φ2

m,m+1q2m = Φ3

0,1q30 {f3

0j}nj=1, Φ1m,m+1, Φ2

+ Φ1m,m+1q

1m − Φ1

m−1,mq1m−1 = 0

dρ2mdt

+ Φ2m,m+1q

2m − Φ2

m−1,mq2m−1 = 0

dρ31dt

+ Φ31,2q

31 − Φ3

0,1q30 = 0

Sum term by term:

(ρ1m + ρ2m + ρ31

)= Φ1

m−1,mq1m−1 + Φ2

m−1,mq2m − Φ3

1,2q31

0,1q30 − Φ1

m,m+1q1m − Φ2

m,m+1q2m

Φ1m,m+1q

1m + Φ2

m,m+1q2m = Φ3

0,1q30 {f3

0j}nj=1, Φ1m,m+1, Φ2

+ Φ1m,m+1q

1m − Φ1

m−1,mq1m−1 = 0

dρ2mdt

+ Φ2m,m+1q

2m − Φ2

m−1,mq2m−1 = 0

dρ31dt

+ Φ31,2q

31 − Φ3

0,1q30 = 0

Sum term by term:

(ρ1m + ρ2m + ρ31

)= Φ1

m−1,mq1m−1 + Φ2

m−1,mq2m − Φ3

1,2q31

0,1q30 − Φ1

m,m+1q1m − Φ2

m,m+1q2m

Φ1m,m+1q

1m + Φ2

m,m+1q2m = Φ3

0,1q30 {f3

0j}nj=1, Φ1m,m+1, Φ2

+ Φ1m,m+1q

1m − Φ1

m−1,mq1m−1 = 0

dρ2mdt

+ Φ2m,m+1q

2m − Φ2

m−1,mq2m−1 = 0

dρ31dt

+ Φ31,2q

31 − Φ3

0,1q30 = 0

Sum term by term:

(ρ1m + ρ2m + ρ31

)= Φ1

m−1,mq1m−1 + Φ2

m−1,mq2m − Φ3

1,2q31

0,1q30 − Φ1

m,m+1q1m − Φ2

m,m+1q2m

Φ1m,m+1q

1m + Φ2

m,m+1q2m = Φ3

0,1q30 {f3

0j}nj=1, Φ1m,m+1, Φ2

+ Φ1m,m+1q

1m − Φ1

m−1,mq1m−1 = 0

dρ2mdt

+ Φ2m,m+1q

2m − Φ2

m−1,mq2m−1 = 0

dρ31dt

+ Φ31,2q

31 − Φ3

0,1q30 = 0

Sum term by term:

(ρ1m + ρ2m + ρ31

)= Φ1

m−1,mq1m−1 + Φ2

m−1,mq2m − Φ3

1,2q31

0,1q30 − Φ1

m,m+1q1m − Φ2

m,m+1q2m

Φ1m,m+1q

1m + Φ2

m,m+1q2m = Φ3

0,1q30 {f3

0j}nj=1, Φ1m,m+1, Φ2

The 2-1 Junction: The Priority Rule

Assume that priority is given to road r = 1 and let p ∈ [0, 1] be a givenflux threshold (to be determined).

If q1m + q2m ≤ p then q30 = q1m + q2m

If q1m + q2m > p then q30 = q1m and Φ2m,m+1 = 0.

n∑j=1

vj(f30j − f1

mj − f2mj

)= 0 if q1m + q2m ≤ p

n∑j=1

vj(f30j − f1

)= 0 if q1m + q2m > p

Choice of the speed distributions (not unique in general):

f30j =

0 if j = 1

f1mj + f2

mj if j ≥ 2 and q1m + q2m ≤ pf1mj if j ≥ 2 and q1m + q2m > p

Massconservation

⇒ Φ1m,m+1 = Φ3

0,1 , Φ2m,m+1 =

0,1 if q1m + q2m ≤ p0 if q1m + q2m > p

n∑j=1

vj(f30j − f1

mj − f2mj

)= 0 if q1m + q2m ≤ p

n∑j=1

vj(f30j − f1

)= 0 if q1m + q2m > p

f30j =

0 if j = 1

f1mj + f2

Massconservation

⇒ Φ1m,m+1 = Φ3

0,1 , Φ2m,m+1 =

0,1 if q1m + q2m ≤ p0 if q1m + q2m > p

n∑j=1

vj(f30j − f1

mj − f2mj

)= 0 if q1m + q2m ≤ p

n∑j=1

vj(f30j − f1

)= 0 if q1m + q2m > p

f30j =

0 if j = 1

f1mj + f2

Massconservation

⇒ Φ1m,m+1 = Φ3

0,1 , Φ2m,m+1 =

0,1 if q1m + q2m ≤ p0 if q1m + q2m > p

n∑j=1

vj(f30j − f1

mj − f2mj

)= 0 if q1m + q2m ≤ p

n∑j=1

vj(f30j − f1

)= 0 if q1m + q2m > p

f30j =

0 if j = 1

f1mj + f2

Massconservation

⇒ Φ1m,m+1 = Φ3

0,1 , Φ2m,m+1 =

0,1 if q1m + q2m ≤ p0 if q1m + q2m > p

The 2-1 Junction: Determination of the Flux Threshold p

When the two incoming fluxes q1m, q2m merge, the total outgoing densityρ30 must not exceed the admissible maximum density:

ρ30 =

n∑j=1

f30j (by definition)

=n∑j=2

f30j =

n∑j=2

(f1mj + f2

)(because f3

01 = 0)

n∑j=1

vj(f1mj + f2

)(because v1 = 0 < v2 ≤ vj ∀ j ≥ 2)

=q1m + q2m

v2≤ p

Therefore ρ30 ≤ 1 is guaranteed if p ≤ v2, for instance

p = v2

ρ30 =

n∑j=1

=n∑j=2

f30j =

n∑j=2

(f1mj + f2

)(because f3

01 = 0)

n∑j=1

vj(f1mj + f2

)(because v1 = 0 < v2 ≤ vj ∀ j ≥ 2)

=q1m + q2m

v2≤ p

p = v2

ρ30 =

n∑j=1

=n∑j=2

f30j =

n∑j=2

(f1mj + f2

)(because f3

01 = 0)

n∑j=1

vj(f1mj + f2

)(because v1 = 0 < v2 ≤ vj ∀ j ≥ 2)

=q1m + q2m

v2≤ p

p = v2

ρ30 =

n∑j=1

=n∑j=2

f30j =

n∑j=2

(f1mj + f2

)(because f3

01 = 0)

n∑j=1

vj(f1mj + f2

)(because v1 = 0 < v2 ≤ vj ∀ j ≥ 2)

=q1m + q2m

v2≤ p

p = v2

ρ30 =

n∑j=1

=n∑j=2

f30j =

n∑j=2

(f1mj + f2

)(because f3

01 = 0)

n∑j=1

vj(f1mj + f2

)(because v1 = 0 < v2 ≤ vj ∀ j ≥ 2)

=q1m + q2m

v2≤ p

p = v2

ρ30 =

n∑j=1

=n∑j=2

f30j =

n∑j=2

(f1mj + f2

)(because f3

01 = 0)

n∑j=1

vj(f1mj + f2

)(because v1 = 0 < v2 ≤ vj ∀ j ≥ 2)

=q1m + q2m

v2≤ p

p = v2

References

[1] M. Caramia, C. D’Apice, B. Piccoli, and A. Sgalambro.Fluidsim: A car traffic simulation prototype based on fluid dynamic.Algorithms, 3(3):294–310, 2010.

[2] L. Fermo and A. Tosin.A fully-discrete-state kinetic theory approach to modeling vehicular traffic.SIAM J. Appl. Math., 73(4):1533–1556, 2013.

[3] L. Fermo and A. Tosin.A fully-discrete-state kinetic theory approach to traffic flow on roadnetworks.Math. Models Methods Appl. Sci., 25(3):423–461, 2015.

[4] M. Garavello and B. Piccoli.Traffic Flow on Networks – Conservation Laws Models.AIMS Series on Applied Mathematics. American Institute of MathematicalSciences (AIMS), Springfield, MO, 2006.

[5] G. Puppo, M. Semplice, A. Tosin, and G. Visconti.Kinetic models for traffic flow resulting in a reduced space of microscopicvelocities.In preparation, 2016.

Case Study: The Traffic Circle

Simulation

Traffic circle with different types of pri-ority at junctions À, Â:

Case 1: normal priority to thecirculating flow

at À: road 8 has right-of-wayover road 1at Â: road 4 has right-of-wayover road 5

Case 2: inverted priority to theincoming flow

Case 3: normal priority at junctionÀ, inverted at Â

Simulation

A Boltzmann-type kinetic approach to traffic flow on road ...acesar/slidesanconet/Tosin.pdf · A...

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