A multi-objective optimization framework for a second order traffic flow model on a junction

A multi-objective optimization framework
for a second order traffic flow model on a junction
Guillaume Costeseque
collaboration with Paola Goatin, Simone Göttlich and Oliver Kolb
Inria Sophia-Antipolis Méditerranée
ACUMES meeting
Sophia-Antipolis – July 04, 2017
G. Costeseque Second order models on junctions Valbonne, July 04, 2017 1 / 46

Motivation
Traﬃc ﬂows on a network
Nice network [Google maps, Jul. 3, 2017]

Motivation
Traﬃc ﬂows on a network
Road network ≡ graph made of edges and vertices

Motivation
Outline
1 Introduction
2 Some background
3 Riemann solver

Introduction
Outline
1 Introduction
2 Some background
3 Riemann solver

Introduction
ARZ model on a junction
(i)
(j)
(ρi, wi)
(ρj, wj)

Introduction
ARZ model on a junction
(continued)
ARZ model [1, 10] on each branch (i)



∂tρi + ∂x (ρi vi ) = 0,
∂t (ρi wi ) + ∂x (ρi vi wi ) = 0,
wi := vi + pi (ρi )
(1)
Coupling conditions needed to ensure conservation of
Mass ﬂow q = ρv
Momentum ﬂow qw = ρvw
through the junction

Introduction
Problem statement
Why ARZ model?

Introduction
Problem statement
Why ARZ model?
To reproduce the capacity drop phenomenon
(+ control thanks to variable speed limits and/or ramp metering)
What are we looking for?

Introduction
Problem statement
Why ARZ model?
To reproduce the capacity drop phenomenon
(+ control thanks to variable speed limits and/or ramp metering)
What are we looking for?
Well-posedness of Riemann solvers at the junction

Some background
Outline
1 Introduction
2 Some background
Basics
Computation of the supply for second order models
3 Riemann solver

Some background Basics
Common assumptions
(A1) Conservation of the fluxes:
n
i=1
ρi vi
=:qi
=
n+m
j=n+1
ρj vj
=:qj
(A2) Fixed assignment coefficients:
∃ (αji )i,j ∈ [0, 1], s.t.
n+m
j=n+1
αji = 1 and qj =
n
i=1
αji qi
(A3) Bounds on the fluxes
0 ≤ qi ≤ ∆i , i = 1, . . . , n,
0 ≤ qj ≤ Σj , j = n + 1, . . . , n + m,
∆i demand and Σj supply

Common assumptions
(A4) Maximization of the total incoming ﬂuxes:
max
n
i=1
qi

Common assumptions
(A4) Maximization of the total incoming ﬂuxes:
max
n
i=1
qi
Literature:
ARZ model
Garavello-Piccoli [4]
Herty-Rascle [7]
Herty-Moutari-Rascle [6]
Haut-Bastin [5]
Phase Transition model
Colombo, Goatin, Piccoli [2]
Garavello, Marcellini [3]
Engineering community: Lebacque’s works [9, 8]

Common assumptions
(A4) Multi-objective optimization of the incoming ﬂuxes:
max (q1, . . . , qn)
and for any ﬁxed P = (P1, . . . , Pn) such that Pi ∈]0, 1[ and
n
i=1 Pi = 1, the ratio
ql
n
i=1 qi
is the closest to Pl

Some background Computation of the supply for second order models
Demand and supply
ρ
ρ → ρ(w − pi(ρ))
Di(ρ, w)
Si(ρ, w)
ρv
ρv
ρ
σi(w)
ρv
ρ
σi(w)
σi(w)

Some background Computation of the supply for second order models
Downstream density perceived by upstream traﬃc
= wl − pr(˜ρr)
ρv
{w = wl}
ρ
{w = wr}
ρr˜ρr
(ρr, wr)
vr = wr − pr(ρr)
(ρl, wl)
x
The velocity is conserved through a contact discontinuity!

Riemann solver
Outline
1 Introduction
2 Some background
3 Riemann solver
General 1-to-m diverge
2-to-1 merge

Riemann solver General 1-to-m diverge
RS for a 1-to-m diverge (m ≥ 1)
qm+1
q1
q2
(1)
(2)
(m + 1)

Initial states ((ρ1,0, v1,0), (ρ2,0, v2,0) . . . , (ρm+1,0, vm+1,0))
Multi-optimization ≡ optimization of the total through-ﬂow
max
Ω1×m
q1
Set of admissible states
Ω1×m := q1 ∈ R
0 ≤ q1 ≤ ∆1
0 ≤ qj = αj1q1 ≤ Σj , ∀j

Solution for a 1-to-m diverge
q1 = min ∆1, min
j=2,...,m+1
1
αj1
Σj ,
qj = αj1q1, ∀j = 2, . . . , m + 1
with
∆1 = D1 (ρ1,0, w1)
Σj = Sj p−1
j (max{0, w1 − v2,0}), w1 , ∀j = 2, . . . , m + 1

Example of a 1 × 1 junction
(1) (2)
(ρ1, w1) (ρ2, w2)
q1 = q2 = min {D1(ρ1,0, w1) , S2( ˜ρ2, w1)}
where
˜ρ2 = p−1
2 (max{0, w1 − v2,0})

Riemann solver 2-to-1 merge
Case of a 2 × 1 merge
(2)
(3)
q3
q1
(1)
q2
q3 = q1 + q2
+ initial conditions
((ρ1,0, v1,0), (ρ2,0, v2,0), (ρ3,0, v3,0))

Example of a 2 × 1 merge
(continued)
Multi-objective optimization problem
max
Ω2×1
(q1, q2)
with
Ω2×1 =



(q1, q2) ∈ R2
0 ≤ q1 ≤ ∆1,
0 ≤ q2 ≤ ∆2,
0 ≤ q3 = q1 + q2 ≤ Σ3(q1, q2)



∆i = Di (ρi,0, wi ), i = 1, 2
Σ3(q1, q2) = S3 (˜ρ3, ˜w)

Example of a 2 × 1 merge
(continued)
Multi-objective optimization problem
max
Ω2×1
(q1, q2)
with
Ω2×1 =



(q1, q2) ∈ R2
0 ≤ q1 ≤ ∆1,
0 ≤ q2 ≤ ∆2,
0 ≤ q3 = q1 + q2 ≤ Σ3(q1, q2)



∆i = Di (ρi,0, wi ), i = 1, 2
Σ3(q1, q2) = S3 (˜ρ3, ˜w)
˜w =
q1
q1 + q2
w1 +
q2
q1 + q2
w2
˜ρ3 = p−1
3 (max{0, ˜w − v3,0})

First property
Proposition (Convexity of the feasible set)
The set of admissible states Ω2×1 is non-empty and convex.

First property
Proposition (Convexity of the feasible set)
The set of admissible states Ω2×1 is non-empty and convex.
Sketch of the proof:
(0, 0) ∈ Ω2×1
Classical convexity proof: take two points on the boundary of Ω2×1
and show that a convex combination of these two points still belongs
to the set

Analysis of the supply
Assume ∃z ∈ [0, 1] such that
q1 = z(q1 + q2)
q2 = (1 − z)(q1 + q2)
Deﬁne
˜Σ3(z) = Σ3 (q1, q2)
and set
∆w = w1 − w2

˜Σ3(z)
q2(z)
q1(z)
q1
0 ˜Σ3(z)
1 − z
z
q2
˜Σ3 − q1 − q2 = 0
−1

Local optima
˜Σ3 − q1 − q2 = 0
q1
0
q2
1 − P∗
P∗∗
If ∆w > 0
P∗
local minimum for
z → q1(z) = z ˜Σ3(z)
P∗∗
local maximum for
z → q2(z) = (1 − z)˜Σ3(z)
If ∆w < 0
P∗
is a local maximum for
z → q1(z) = z ˜Σ3(z)
P∗∗
is a local minimum for
p → q2(z) = (1 − z)˜Σ3(z)

Riemann solver for the 2-to-1 merge
Algorithm
Consider given pressure functions pi (ρ) and initial conditions.
1 Fix a priority ratio P ∈]0, 1[
2 Compute
F(P) = min
∆1
P
,
∆2
1 − P
, ˜Σ3(P)
with
˜Σ3(P) = S3(ρP , ˜wP )
and ˜wP = Pw1 + (1 − P)w2 and ρP = p−1
3 (max{0, ˜wP − v3,0})
3 Distinguish the diﬀerent cases

Set
˜q1 = P ˜Σ3(P) and ˜q2 = (1 − P)˜Σ3(P)
q∗
1 = P∗ ˜Σ3(P∗
) and q∗
2 = (1 − P∗
)˜Σ3(P∗
)
1 If
∆w = 0, or
∆w < 0 and P ≤ P∗
, or
∆w > 0 and P ≥ P∗∗
,
(2)
we choose
q1 = min {∆1, max {˜q1, Σ3(q1, q2) − q2}} ,
q2 = min {∆2, max {˜q2, Σ3(q1, q2) − q1}} .
(3)

2 If
∆w < 0 and P ≥ P∗
,
then
q2 = min {∆2, max {q∗
2, Σ3(q1, q2) − q1}} . (4)
Computation of q1:
1 If
F(P) = ˜Σ3(P) and q∗
2 ≤ ∆2,
we apply
q1 = min{q∗
1 , ∆1}. (5)
2 Otherwise
q1 = min {∆1, max {˜q1, Σ3(q1, q2) − q2}} . (6)
3 The case ∆w > 0 and P ≤ P∗∗ treated analogously to case 2.

Easy case ∆w = 0
Σ3 is a constant
q2
q1
q1 = Σ3 − q2
Σ3 − ∆2
∆1
Σ3 − ∆1 ∆2
(1 − P)Σ3
P
1 − P
PΣ3

Subcase 1: F(P) =
∆1
P
∆1
∆2
∆2
1 − P
P
0
q2
q1
Σ3 = q1 + q2

Subcase 1: F(P) =
∆1
P
q1 = ∆1
q2 = min ∆2, max (1 − P)˜Σ3(P), Σ3(∆1, q2) − ∆1
∆1
∆2
∆2
1 − P
P
0
q2
q1
Σ3 = q1 + q2

Subcase 2: F(P) =
∆2
1 − P
1 − P
P
∆2
∆1
0
q2
q1
Σ3 = q1 + q2
∆1

Subcase 2: F(P) =
∆2
1 − P
q1 = min ∆1, max P ˜Σ3(P), Σ3(q1, ∆2) − ∆2
q2 = ∆2
1 − P
P
∆2
∆1
0
q2
q1
Σ3 = q1 + q2
∆1

Subcase 3: F(P) = ˜Σ3(P) and P ≤ P∗
0 ∆1
∆2
q2
q1
Σ3 = q1 + q2
P ˜Σ3(P)
(1 − P)˜Σ3(P)
1 − P
P
1 − P∗
P∗
Pareto front

Subcase 3: F(P) = ˜Σ3(P) and P ≤ P∗
q1 = P ˜Σ3(P)
q2 = (1 − P)˜Σ3(P)
0 ∆1
∆2
q2
q1
Σ3 = q1 + q2
P ˜Σ3(P)
(1 − P)˜Σ3(P)
1 − P
P
1 − P∗
P∗
Pareto front

Subcase 4: F(P) = ˜Σ3(P) and P ≥ P∗
(a) If q∗
1 ≤ ∆1 and q∗
2 ≤ ∆2
∆2
Pareto front
0
q2
q1
Σ3 = q1 + q2
(1 − P∗
)˜Σ3(P∗
)
P∗ ˜Σ3(P∗
)∆1
1 − P
P
1 − P∗
P∗

(a) If q∗
1 ≤ ∆1 and q∗
2 ≤ ∆2
q1 = q∗
1 = P∗ ˜Σ3(P∗)
q2 = q∗
2 = (1 − P∗)˜Σ3(P∗)
∆2
Pareto front
0
q2
q1
Σ3 = q1 + q2
(1 − P∗
)˜Σ3(P∗
)
P∗ ˜Σ3(P∗
)∆1
1 − P
P
1 − P∗
P∗

(b) If q∗
1 ≤ ∆1 and q∗
2 > ∆2
1 − P∗
P∗
∆2
0
q2
q1
Σ3 = q1 + q2
∆1
1 − P
P

(b) If q∗
1 ≤ ∆1 and q∗
2 > ∆2
q1 = Σ3(q1, ∆2) − ∆2
q2 = ∆2
1 − P∗
P∗
∆2
0
q2
q1
Σ3 = q1 + q2
∆1
1 − P
P

(c) If q∗
1 > ∆1 and ¯q2 ≤ ∆2
∆2
∆10
q2
q1
Σ3 = q1 + q2
1 − P
P
1 − P∗
P∗
Pareto front

(c) If q∗
1 > ∆1 and ¯q2 ≤ ∆2
q1 = ∆1
q2 = ¯q2 solution of q2 = Σ3(∆1, q2) − ∆1 and q2 ≥ q∗
2
∆2
∆10
q2
q1
Σ3 = q1 + q2
1 − P
P
1 − P∗
P∗
Pareto front

(d) If q∗
1 > ∆1 and ¯q2 ≥ ∆2 ≥ q2
∆1
∆2
0
q2
q1
Σ3 = q1 + q2
1 − P
P
1 − P∗
P∗

(d) If q∗
1 > ∆1 and ¯q2 ≥ ∆2 ≥ q2
q1 = ∆1
q2 = ∆2
∆1
∆2
0
q2
q1
Σ3 = q1 + q2
1 − P
P
1 − P∗
P∗

(e) If q∗
1 > ∆1 and ∆2 < q2
Σ3 = q1 + q2
∆2
0
q2
q11 − P
P
1 − P∗
P∗
∆1

(e) If q∗
1 > ∆1 and ∆2 < q2
q1 = Σ3(q1, ∆2) − ∆2
q2 = ∆2
Σ3 = q1 + q2
∆2
0
q2
q11 − P
P
1 − P∗
P∗
∆1

References
Some references I
A. Aw and M. Rascle, Resurrection of “second order” models of traffic flow,
SIAM journal on applied mathematics, 60 (2000), pp. 916–938.
R. M. Colombo, P. Goatin, and B. Piccoli, Road networks with phase
transitions, Journal of Hyperbolic Differential Equations, 7 (2010), pp. 85–106.
M. Garavello and F. Marcellini, The riemann problem at a junction for a
phase transition traffic model, Preprint, (2016).
M. Garavello and B. Piccoli, Traffic flow on a road network using the
Aw–Rascle model, Communications in Partial Differential Equations, 31 (2006),
pp. 243–275.
B. Haut and G. Bastin, A second order model of road junctions in fluid models
of traffic networks, Networks and Heterogeneous Media, 2 (2007), pp. 227–253.
M. Herty, S. Moutari, and M. Rascle, Optimization criteria for modelling
intersections of vehicular traffic flow, Networks and Heterogeneous Media, 1
(2006), pp. 275–294.

References
Some references II
M. Herty and M. Rascle, Coupling conditions for a class of second-order
models for traffic flow, SIAM Journal on mathematical analysis, 38 (2006),
pp. 595–616.
J. Lebacque, S. Mammar, and H. Haj-Salem, An intersection model based
on the GSOM model, in Proceedings of the 17th World Congress, The
International Federation of Automatic Control, Seoul, Korea, 2008, pp. 7148–7153.
J.-P. Lebacque, H. Haj-Salem, and S. Mammar, Second order traffic flow
modeling: supply-demand analysis of the inhomogeneous Riemann problem and of
boundary conditions, Proceedings of the 10th Euro Working Group on
Transportation (EWGT), 3 (2005).
H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior,
Transportation Research Part B: Methodological, 36 (2002), pp. 275–290.

References
Thanks for your attention
guillaume.costeseque@inria.fr

A multi-objective optimization framework for a second order traffic flow model on a junction

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