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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
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
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
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
. . . 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:
Isr =
{−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
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
. . . 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:
Isr =
{−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
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
. . . 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:
Isr =
{−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
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
. . . 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:
Isr =
{−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
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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}
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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}
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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}
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
Boundary Conditions at Access Points
At an access point consider the equation for i = 1 (first cell):
dfr1jdt
+ vj(Φr1,2f
r1j − Φr0,1f
r0j
)= 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
fr0j
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
Boundary Conditions at Access Points
At an access point consider the equation for i = 1 (first cell):
dfr1jdt
+ vj(Φr1,2f
r1j − Φr0,1f
r0j
)= 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
fr0j
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
Boundary Conditions at Access Points
At an access point consider the equation for i = 1 (first cell):
dfr1jdt
+ vj(Φr1,2f
r1j − Φr0,1f
r0j
)= 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
fr0j
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
Boundary Conditions at Access Points
At an access point consider the equation for i = 1 (first cell):
dfr1jdt
+ vj(Φr1,2f
r1j − Φr0,1f
r0j
)= 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
fr0j
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
The 1-2 Junction
r = 2
r = 3
r = 1
Im1 I 1
2
I1 3
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
The 1-2 Junction: Mass Conservation
r = 2
r = 3
r = 1
Im1 I 1
2
I1 3
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:
d
dt
(ρ1m + ρ21 + ρ31
)= Φ1
m−1,mq1m−1 − Φ2
1,2q21 − Φ3
1,2q31
+(Φ2
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
m,m+1
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
The 1-2 Junction: Mass Conservation
r = 2
r = 3
r = 1
Im1 I 1
2
I1 3
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:
d
dt
(ρ1m + ρ21 + ρ31
)= Φ1
m−1,mq1m−1 − Φ2
1,2q21 − Φ3
1,2q31
+(Φ2
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
m,m+1
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
The 1-2 Junction: Mass Conservation
r = 2
r = 3
r = 1
Im1 I 1
2
I1 3
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:
d
dt
(ρ1m + ρ21 + ρ31
)= Φ1
m−1,mq1m−1 − Φ2
1,2q21 − Φ3
1,2q31
+(Φ2
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
m,m+1
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
The 1-2 Junction: Mass Conservation
r = 2
r = 3
r = 1
Im1 I 1
2
I1 3
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:
d
dt
(ρ1m + ρ21 + ρ31
)= Φ1
m−1,mq1m−1 − Φ2
1,2q21 − Φ3
1,2q31
+(Φ2
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
m,m+1
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
The 1-2 Junction: Mass Conservation
r = 2
r = 3
r = 1
Im1 I 1
2
I1 3
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:
d
dt
(ρ1m + ρ21 + ρ31
)= Φ1
m−1,mq1m−1 − Φ2
1,2q21 − Φ3
1,2q31
+(Φ2
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
m,m+1
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
The 1-2 Junction: Mass Conservation
r = 2
r = 3
r = 1
Im1 I 1
2
I1 3
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:
d
dt
(ρ1m + ρ21 + ρ31
)= Φ1
m−1,mq1m−1 − Φ2
1,2q21 − Φ3
1,2q31
+(Φ2
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
m,m+1
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
mj
)= 0,
n∑j=1
vj(f30j − (1− a)f1
mj
)= 0
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
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
mj
)= 0,
n∑j=1
vj(f30j − (1− a)f1
mj
)= 0
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
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
mj
)= 0,
n∑j=1
vj(f30j − (1− a)f1
mj
)= 0
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
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
mj
)= 0,
n∑j=1
vj(f30j − (1− a)f1
mj
)= 0
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
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
The 2-1 Junction
r = 2
r = 1
r = 3
I13
Im 1
I m2
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
The 2-1 Junction: Mass Conservation
r = 2
r = 1
r = 3
I13
Im 1
I m2
Write the kinetic equation in the cellsaround the junction (I1m, I2m, I31 )
Sum over j = 1, . . . , n each equation:dρ1mdt
+ 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:
d
dt
(ρ1m + ρ2m + ρ31
)= Φ1
m−1,mq1m−1 + Φ2
m−1,mq2m − Φ3
1,2q31
+(Φ3
0,1q30 − Φ1
m,m+1q1m − Φ2
m,m+1q2m
)Mass conservation through the junction requires:
Φ1m,m+1q
1m + Φ2
m,m+1q2m = Φ3
0,1q30 {f3
0j}nj=1, Φ1m,m+1, Φ2
m,m+1
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
The 2-1 Junction: Mass Conservation
r = 2
r = 1
r = 3
I13
Im 1
I m2
Write the kinetic equation in the cellsaround the junction (I1m, I2m, I31 )
Sum over j = 1, . . . , n each equation:dρ1mdt
+ 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:
d
dt
(ρ1m + ρ2m + ρ31
)= Φ1
m−1,mq1m−1 + Φ2
m−1,mq2m − Φ3
1,2q31
+(Φ3
0,1q30 − Φ1
m,m+1q1m − Φ2
m,m+1q2m
)Mass conservation through the junction requires:
Φ1m,m+1q
1m + Φ2
m,m+1q2m = Φ3
0,1q30 {f3
0j}nj=1, Φ1m,m+1, Φ2
m,m+1
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
The 2-1 Junction: Mass Conservation
r = 2
r = 1
r = 3
I13
Im 1
I m2
Write the kinetic equation in the cellsaround the junction (I1m, I2m, I31 )
Sum over j = 1, . . . , n each equation:dρ1mdt
+ 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:
d
dt
(ρ1m + ρ2m + ρ31
)= Φ1
m−1,mq1m−1 + Φ2
m−1,mq2m − Φ3
1,2q31
+(Φ3
0,1q30 − Φ1
m,m+1q1m − Φ2
m,m+1q2m
)Mass conservation through the junction requires:
Φ1m,m+1q
1m + Φ2
m,m+1q2m = Φ3
0,1q30 {f3
0j}nj=1, Φ1m,m+1, Φ2
m,m+1
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
The 2-1 Junction: Mass Conservation
r = 2
r = 1
r = 3
I13
Im 1
I m2
Write the kinetic equation in the cellsaround the junction (I1m, I2m, I31 )
Sum over j = 1, . . . , n each equation:dρ1mdt
+ 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:
d
dt
(ρ1m + ρ2m + ρ31
)= Φ1
m−1,mq1m−1 + Φ2
m−1,mq2m − Φ3
1,2q31
+(Φ3
0,1q30 − Φ1
m,m+1q1m − Φ2
m,m+1q2m
)Mass conservation through the junction requires:
Φ1m,m+1q
1m + Φ2
m,m+1q2m = Φ3
0,1q30 {f3
0j}nj=1, Φ1m,m+1, Φ2
m,m+1
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
The 2-1 Junction: Mass Conservation
r = 2
r = 1
r = 3
I13
Im 1
I m2
Write the kinetic equation in the cellsaround the junction (I1m, I2m, I31 )
Sum over j = 1, . . . , n each equation:dρ1mdt
+ 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:
d
dt
(ρ1m + ρ2m + ρ31
)= Φ1
m−1,mq1m−1 + Φ2
m−1,mq2m − Φ3
1,2q31
+(Φ3
0,1q30 − Φ1
m,m+1q1m − Φ2
m,m+1q2m
)Mass conservation through the junction requires:
Φ1m,m+1q
1m + Φ2
m,m+1q2m = Φ3
0,1q30 {f3
0j}nj=1, Φ1m,m+1, Φ2
m,m+1
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
The 2-1 Junction: Mass Conservation
r = 2
r = 1
r = 3
I13
Im 1
I m2
Write the kinetic equation in the cellsaround the junction (I1m, I2m, I31 )
Sum over j = 1, . . . , n each equation:dρ1mdt
+ 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:
d
dt
(ρ1m + ρ2m + ρ31
)= Φ1
m−1,mq1m−1 + Φ2
m−1,mq2m − Φ3
1,2q31
+(Φ3
0,1q30 − Φ1
m,m+1q1m − Φ2
m,m+1q2m
)Mass conservation through the junction requires:
Φ1m,m+1q
1m + Φ2
m,m+1q2m = Φ3
0,1q30 {f3
0j}nj=1, Φ1m,m+1, Φ2
m,m+1
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
mj
)= 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 =
{Φ3
0,1 if q1m + q2m ≤ p0 if q1m + q2m > p
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
mj
)= 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 =
{Φ3
0,1 if q1m + q2m ≤ p0 if q1m + q2m > p
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
mj
)= 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 =
{Φ3
0,1 if q1m + q2m ≤ p0 if q1m + q2m > p
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
mj
)= 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 =
{Φ3
0,1 if q1m + q2m ≤ p0 if q1m + q2m > p
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
mj
)(because f3
01 = 0)
≤ 1
v2
n∑j=1
vj(f1mj + f2
mj
)(because v1 = 0 < v2 ≤ vj ∀ j ≥ 2)
=q1m + q2m
v2≤ p
v2
Therefore ρ30 ≤ 1 is guaranteed if p ≤ v2, for instance
p = v2
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
mj
)(because f3
01 = 0)
≤ 1
v2
n∑j=1
vj(f1mj + f2
mj
)(because v1 = 0 < v2 ≤ vj ∀ j ≥ 2)
=q1m + q2m
v2≤ p
v2
Therefore ρ30 ≤ 1 is guaranteed if p ≤ v2, for instance
p = v2
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
mj
)(because f3
01 = 0)
≤ 1
v2
n∑j=1
vj(f1mj + f2
mj
)(because v1 = 0 < v2 ≤ vj ∀ j ≥ 2)
=q1m + q2m
v2≤ p
v2
Therefore ρ30 ≤ 1 is guaranteed if p ≤ v2, for instance
p = v2
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
mj
)(because f3
01 = 0)
≤ 1
v2
n∑j=1
vj(f1mj + f2
mj
)(because v1 = 0 < v2 ≤ vj ∀ j ≥ 2)
=q1m + q2m
v2≤ p
v2
Therefore ρ30 ≤ 1 is guaranteed if p ≤ v2, for instance
p = v2
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
mj
)(because f3
01 = 0)
≤ 1
v2
n∑j=1
vj(f1mj + f2
mj
)(because v1 = 0 < v2 ≤ vj ∀ j ≥ 2)
=q1m + q2m
v2≤ p
v2
Therefore ρ30 ≤ 1 is guaranteed if p ≤ v2, for instance
p = v2
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
mj
)(because f3
01 = 0)
≤ 1
v2
n∑j=1
vj(f1mj + f2
mj
)(because v1 = 0 < v2 ≤ vj ∀ j ≥ 2)
=q1m + q2m
v2≤ p
v2
Therefore ρ30 ≤ 1 is guaranteed if p ≤ v2, for instance
p = v2
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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.
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
at À: road 1 has right-of-wayover road 8at Â: road 5 has right-of-wayover road 4
Case 3: normal priority at junctionÀ, inverted at Â
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
at À: road 1 has right-of-wayover road 8at Â: road 5 has right-of-wayover road 4
Case 3: normal priority at junctionÀ, inverted at Â
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks
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
at À: road 1 has right-of-wayover road 8at Â: road 5 has right-of-wayover road 4
Case 3: normal priority at junctionÀ, inverted at Â
Andrea Tosin A Boltzmann-type kinetic approach to traffic flow on road networks