Workshop
ANalysis and COntrol on
NETworks:
trends and perspectives
Padova, March 9-11, 2016
Optimization based control of networks of discretized
PDEs: application to road traffic management
Paola GOATIN (Inria Sophia Antipolis Méditerranée)
March 9-11, 2016
Associated Team “ORESTE”
ORESTE (Optimal RErouting Strategies for Traffic mangEment) is an
associated team between Inria project-team OPALE and the Connected
Corridors project at UC Berkeley.
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Paola Goatin (PI)
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Maria Laura Delle Monache
Paola GOATIN - Inria
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Alexandre Bayen (PI)
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Jack Reilly
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Samitha Samaranayake
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Walid Krichene
ANCONET (March 9-11, 2016) - 2
ORESTE project goals
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Optimize traffic flow in corridors
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ramp metering
re-routing strategies
Modeling approach:
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macroscopic traffic flow models
discrete adjoint method for gradient computation
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 3
SUMMARY
1. Macroscopic models for road traffic
2. Discretized system
3. Quick review of the adjoint method
4. Numerical results
5. Application to optimal re-routing
6. Conclusion and perspectives
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 4
1. Macroscopic models for road traffic
Outline of the talk
1. Macroscopic models for road traffic
2. Discretized system
3. Quick review of the adjoint method
4. Numerical results
5. Application to optimal re-routing
6. Conclusion and perspectives
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 5
1. Macroscopic models for road traffic
LWR model
[Lighthill-Whitham ’55, Richards ’56]
Non-linear transport equation: PDE for mass conservation
∂t ρ + ∂x f (ρ) = 0
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x ∈ R, t > 0
ρ ∈ [0, ρmax ] mean traffic density
f (ρ) = ρv (ρ) flux function
Empirical flux-density relation: fundamental diagram
f (ρ)
f (ρ)
f max
f max
f max
ρmax −ρcr
vf
0
ρcr
Greenshield ’35
Paola GOATIN - Inria
ρmax
ρ
ρcr
ρmax ρ
Newell-Daganzo
ANCONET (March 9-11, 2016) - 6
1. Macroscopic models for road traffic
Extension to networks
m incoming arcs
n outgoing arcs
junction
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LWR on networks:
[Holden-Risebro, 1995; Coclite-Garavello-Piccoli, 2005; Garavello-Piccoli, 2006]
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LWR on each road
Optimization problem at the junction
Modeling of junctions with a buffer:
[Herty-Lebacque-Moutari, 2009; Garavello-Goatin, 2012; Bressan-Nguyen, 2015]
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Junction described with one or more buffers
Suitable for optimization and Nash equilibrium problems
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 7
1. Macroscopic models for road traffic
Extension to networks
m incoming arcs
n outgoing arcs
junction
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LWR on networks:
[Holden-Risebro, 1995; Coclite-Garavello-Piccoli, 2005; Garavello-Piccoli, 2006]
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LWR on each road
Optimization problem at the junction
Modeling of junctions with a buffer:
[Herty-Lebacque-Moutari, 2009; Garavello-Goatin, 2012; Bressan-Nguyen, 2015]
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Junction described with one or more buffers
Suitable for optimization and Nash equilibrium problems
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 7
1. Macroscopic models for road traffic
A model for ramp metering
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Two incoming links:
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Upstream mainline I1 =] − ∞, 0[
Onramp R1
I1
R1
Paola GOATIN - Inria
Two outgoing links:
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J
Downstream mainline I2 =]0, +∞[
Offramp R2
I2
R2
ANCONET (March 9-11, 2016) - 8
1. Macroscopic models for road traffic
A model for ramp metering
Coupled PDE-ODE model:
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Classical LWR on each mainline I1 , I2
∂t ρ + ∂x f (ρ) = 0,
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(t, x ) ∈ R+ × Ii ,
Dynamics of the onramp described by an ODE (buffer)
dl(t)
= Fin (t) − γr1 (t),
dt
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t ∈ R+ ,
l(t) ∈ [0, +∞[ length of the onramp queue
Fin (t) flux entering the onramp
γr1 (t) flux leaving the onramp (through the junction)
Coupled at junction with an LP-optimization problem
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 9
1. Macroscopic models for road traffic
A model for ramp metering
Coupled PDE-ODE model:
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Classical LWR on each mainline I1 , I2
∂t ρ + ∂x f (ρ) = 0,
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(t, x ) ∈ R+ × Ii ,
Dynamics of the onramp described by an ODE (buffer)
dl(t)
= Fin (t) − γr1 (t),
dt
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t ∈ R+ ,
l(t) ∈ [0, +∞[ length of the onramp queue
Fin (t) flux entering the onramp
γr1 (t) flux leaving the onramp (through the junction)
Coupled at junction with an LP-optimization problem
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 9
1. Macroscopic models for road traffic
A model for ramp metering
Coupled PDE-ODE model:
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Classical LWR on each mainline I1 , I2
∂t ρ + ∂x f (ρ) = 0,
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(t, x ) ∈ R+ × Ii ,
Dynamics of the onramp described by an ODE (buffer)
dl(t)
= Fin (t) − γr1 (t),
dt
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t ∈ R+ ,
l(t) ∈ [0, +∞[ length of the onramp queue
Fin (t) flux entering the onramp
γr1 (t) flux leaving the onramp (through the junction)
Coupled at junction with an LP-optimization problem
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 9
1. Macroscopic models for road traffic
Junction solver: Demand & Supply
δ(ρ) , f (ρ)
δ(ρ1 ) =
f (ρ1 ) if 0 ≤ ρ1 < ρcr ,
f max if ρcr ≤ ρ1 ≤ 1,
f max (ρ)
ρ
d(Fin , l) =
max
θ γr1
if l(t) > 0,
max
min (Fin (t), θ γr1
) if l(t) = 0,
θ ∈ [0, 1] control parameter
σ(ρ) , f (ρ)
σ(ρ2 ) =
f max if 0 ≤ ρ2 ≤ ρcr ,
f (ρ2 ) if ρcr < ρ2 ≤ 1,
f max (ρ)
ρ
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 10
1. Macroscopic models for road traffic
Junction solver: flux maximization
1. Mass conservation: f (ρ1 (t, 0−)) + γr1 (t) = f (ρ2 (t, 0+)) + γr2 (t)
2. f (ρ2 (t, 0+)) maximum subject to 1. and
f (ρ2 (t, 0+)) = min (1 − β)δ(ρ1 (t, 0−)) + d(Fin (t), l(t)), σ(ρ2 (t, 0+))
3. Right of way parameter P ∈ ]0, 1[ to ensure uniqueness:
f (ρ2 (t, 0+)) = P f1 (ρ(t, 0−)) + (1 − P) γr 1
4. Offramp treated as a sink (infinite capacity)
5. No flux from the onramp to the offramp is allowed
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 11
1. Macroscopic models for road traffic
Riemann solver: feasible set
To find a solution of the problem we solve an LP-optimization problem
Γ1
1. Define the spaces of the incoming
fluxes
2. Consider the demands
3. Trace the supply line
4. The feasible set is given by Ω
Γr1
Different situations can occur:
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Demand limited case
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Supply limited case
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 12
1. Macroscopic models for road traffic
Riemann solver: feasible set
To find a solution of the problem we solve an LP-optimization problem
Γ1
1. Define the spaces of the incoming
fluxes
δ(ρ1 )
2. Consider the demands
3. Trace the supply line
4. The feasible set is given by Ω
d(Fin , l̄) Γr1
Different situations can occur:
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Demand limited case
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Supply limited case
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 12
1. Macroscopic models for road traffic
Riemann solver: feasible set
To find a solution of the problem we solve an LP-optimization problem
Γ1
1. Define the spaces of the incoming
fluxes
δ(ρ1 )
2. Consider the demands
Γ1 (1 − β) + Γr1 = Γ2
3. Trace the supply line
4. The feasible set is given by Ω
d(Fin , l̄) Γr1
Different situations can occur:
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Demand limited case
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Supply limited case
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 12
1. Macroscopic models for road traffic
Riemann solver: feasible set
To find a solution of the problem we solve an LP-optimization problem
Γ1
1. Define the spaces of the incoming
fluxes
δ(ρ1 )
2. Consider the demands
Ω
Γ1 (1 − β) + Γr1 = Γ2
3. Trace the supply line
4. The feasible set is given by Ω
d(Fin , l̄) Γr1
Different situations can occur:
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Demand limited case
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Supply limited case
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 12
1. Macroscopic models for road traffic
Riemann solver: feasible set
To find a solution of the problem we solve an LP-optimization problem
Γ1
1. Define the spaces of the incoming
fluxes
δ(ρ1 )
2. Consider the demands
3. Trace the supply line
4. The feasible set is given by Ω
d(Fin , l̄) Γr1
Different situations can occur:
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Demand limited case
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Supply limited case
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 12
1. Macroscopic models for road traffic
Riemann solver: feasible set
To find a solution of the problem we solve an LP-optimization problem
Γ1
demand limited
δ(ρ1 )
1. Define the spaces of the incoming
fluxes
2. Consider the demands
3. Trace the supply line
4. The feasible set is given by Ω
d(Fin , l̄) Γr1
Different situations can occur:
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Demand limited case
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Supply limited case
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 12
1. Macroscopic models for road traffic
Riemann solver: feasible set
To find a solution of the problem we solve an LP-optimization problem
Γ1
1. Define the spaces of the incoming
fluxes
δ(ρ1 )
2. Consider the demands
3. Trace the supply line
supply limited
4. The feasible set is given by Ω
d(Fin , l̄) Γr1
Different situations can occur:
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Demand limited case
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Supply limited case
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 12
1. Macroscopic models for road traffic
Riemann solver: optimal point
Demand limited case
The optimal point Q is the point of maximal demands
Γ1
Q
δ(ρ1 )
Γ1 (1 − β) + Γr1 = Γ2
Ω
d(Fin , l̄) Γr1
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 13
1. Macroscopic models for road traffic
Riemann solver: optimal point
Supply limited case
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We introduce the right of way parameter
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We set optimal point Q to be the point of intersection
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Different situations can occur depending on the value of the intersection
point
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Q∈Ω
Q∈
/Ω
Γ1
Γ2
δ(ρ1 )
d(Fin , l̄)
Paola GOATIN - Inria
Γr1
ANCONET (March 9-11, 2016) - 14
1. Macroscopic models for road traffic
Riemann solver: optimal point
Supply limited case
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We introduce the right of way parameter
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We set optimal point Q to be the point of intersection
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Different situations can occur depending on the value of the intersection
point
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Q∈Ω
Q∈
/Ω
Γ1
Γ2
δ(ρ1 )
Γ1 =
d(Fin , l̄)
Paola GOATIN - Inria
P
1−P Γr1
Γr1
ANCONET (March 9-11, 2016) - 14
1. Macroscopic models for road traffic
Riemann solver: optimal point
Supply limited case
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We introduce the right of way parameter
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We set optimal point Q to be the point of intersection
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Different situations can occur depending on the value of the intersection
point
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Q∈Ω
Q∈
/Ω
Γ1
Γ2
δ(ρ1 )
Γ1 =
P
1−P Γr1
Q
d(Fin , l̄)
Paola GOATIN - Inria
Γr1
ANCONET (March 9-11, 2016) - 14
1. Macroscopic models for road traffic
Riemann solver: optimal point
Supply limited case
I
We introduce the right of way parameter
I
We set optimal point Q to be the point of intersection
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Different situations can occur depending on the value of the intersection
point
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Q∈Ω
Q∈
/Ω
Γ1
Γ2
Γ2
δ(ρ1 )
Γ1 =
P
1−P Γr1
Q
d(Fin , l̄)
Paola GOATIN - Inria
Γr1
ANCONET (March 9-11, 2016) - 14
1. Macroscopic models for road traffic
Riemann solver: optimal point
Supply limited case
I
We introduce the right of way parameter
I
We set optimal point Q to be the point of intersection
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Different situations can occur depending on the value of the intersection
point
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Q∈Ω
Q∈
/Ω
Γ1
Γ2
δ(ρ1 )
Γ1 =
P
1−P Γr1
Q
d(Fin , l̄)
Paola GOATIN - Inria
Γr1
ANCONET (March 9-11, 2016) - 14
1. Macroscopic models for road traffic
Riemann solver: optimal point
Supply limited case
I
We introduce the right of way parameter
I
We set optimal point Q to be the point of intersection
I
Different situations can occur depending on the value of the intersection
point
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Q∈Ω
Q∈
/ Ω −→ Optimal point: S
Γ1
Γ2
δ(ρ1 )
S
Γ1 =
P
1−P Γr1
Q
d(Fin , l̄)
Paola GOATIN - Inria
Γr1
ANCONET (March 9-11, 2016) - 14
1. Macroscopic models for road traffic
Difference with
[Coclite-Garavello-Piccoli, 2005]
model
The traffic distribution across the junction is given by the distribution matrix A:
1−β 1
A=
β
0
Γ1
Γ1
δ(ρ1,0 )
Γ1 =
δ(ρ1,0 )
P
1−P γr 1
Q
Q
σ(ρ2,0 ) = (1 − β)γ1 + γr 1
d(Fin , l̄)
Paola GOATIN - Inria
Γr 1
σ(ρ2,0 ) = (1 − β)γ1 + γr 1
d(Fin , l̄)
Γr 1
ANCONET (March 9-11, 2016) - 15
1. Macroscopic models for road traffic
Riemann solver: theorem
Theorem
Delle Monache& al., SIAP, 2014
Fix P ∈ ]0, 1[. For every ρ1,0 , ρ2,0 ∈ [0, 1] and l0 ∈ [0, +∞[, there exists a
unique admissible solution (ρ1 (t, x ), ρ2 (t, x ), l(t)) satisfying the priority
(possibly in an approximate way). Moreover, for a.e. t > 0, it holds
(ρ1 (t, 0−), ρ2 (t, 0+)) = Rl(t) (ρ1 (t, 0−), ρ2 (t, 0+))
t
ρ̄1
ρ̄2
ρ̂1
ρ̂2
t̄
ρ1,0
ρ2,0
0
Paola GOATIN - Inria
x
ANCONET (March 9-11, 2016) - 16
1. Macroscopic models for road traffic
Modified Godunov scheme
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At some ∆t n , we might have multiple shocks exiting the junction at t̄
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We divide the time step ∆t n = (t n , t n+1 ) into two sub-intervals
∆ta = (t n , t̄) and ∆tb = (t̄, t n+1 )
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We solve in one time step two different Riemann Problems at the junction
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For ∆ta : Classical Godunov flux update
ρn+1
J
ρn+1
0
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∆ta
(Γ̂1 − F (ρnJ−1 , ρnJ ))
∆x
∆ta
(F (ρn0 , ρn1 ) − Γ̂2 )
= ρn0 −
∆x
= ρnJ −
For ∆tb : Modified flux update
ρn+1
J
ρn+1
0
Paola GOATIN - Inria
∆tb t̄
Γ̂1 − F (ρnJ−1 , ρt̄J )
∆x
∆tb t̄
F (ρt̄0 , ρn1 ) − Γ̂t̄2
= ρ0 −
∆x
= ρt̄J −
ANCONET (March 9-11, 2016) - 17
1. Macroscopic models for road traffic
Numerical simulations
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 18
2. Discretized system
Outline of the talk
1. Macroscopic models for road traffic
2. Discretized system
3. Quick review of the adjoint method
4. Numerical results
5. Application to optimal re-routing
6. Conclusion and perspectives
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 19
2. Discretized system
Ramp metering discretized system H(~
ρ, ~u ) = 0
fi in (k)
Di (k)
i
on-ramp i
fi out (k)
ri (k)
in
fi+1
(k)
i
i +1
out
fi+1
(k)
...
βi (k) fi out (k)
block i
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Dynamics and junctions solutions based on the model described earlier
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Piecewise affine system
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Control parameter ui is a constraint on the ramp inflow ri (k)
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 20
2. Discretized system
Ramp metering discretized system H(~
ρ, ~u ) = 0
Lower triangular forward system H(~
ρ, ~u ) = 0:
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hi,k = ρi (k) − ρi (k − 1) − {flux update eq. for cell i and time step k}
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H system of concatenated hi,k
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H lower triangular
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Very efficient to solve H T x = b
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ANCONET (March 9-11, 2016) - 21
2. Discretized system
Lower triangular forward system H(~
ρ, ~u ) = 0
Figure:
Paola GOATIN - Inria
∂H
∂~
ρ
matrix
ANCONET (March 9-11, 2016) - 22
3. Quick review of the adjoint method
Outline of the talk
1. Macroscopic models for road traffic
2. Discretized system
3. Quick review of the adjoint method
4. Numerical results
5. Application to optimal re-routing
6. Conclusion and perspectives
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 23
3. Quick review of the adjoint method
Optimization of a PDE-constrained system
Optimization problem
minimize~u∈U
subject to
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J(~
ρ, ~u )
H(~
ρ, ~u ) = 0
ρ
~ ∈ X ⊆ RNT : state variables
~u ∈ U ⊆ RMT : control variables
Gradient descent: How to compute the gradient
∂J ∂~
ρ ∂J
+
?
∂~
ρ ∂~u
∂~u
On trajectories, H(~
ρ, ~u ) = 0 constant, thus
ρ ∂H
∂H ∂~
+
=0
∂~
ρ ∂~u
∂~u
Paola GOATIN - Inria
O(T 2 N 2 M)
ANCONET (March 9-11, 2016) - 24
3. Quick review of the adjoint method
Discrete adjoint method
Consider the Lagrangian
L(~
ρ, ~u , λ) = J(~
ρ, ~u ) + λT H(~
ρ, ~u )
thus ∇~u L = ∇~u J:
∂J ∂~
ρ
∂H
∂H ∂~
ρ
∂J
+
+ λT
+
∂~u
∂~
ρ ∂~u
∂~u
∂~
ρ ∂~u
∂J
∂~
ρ
∂H
∂J
∂H
=
+ λT
+
+ λT
∂~u
∂~u
∂~
ρ
∂~u ∂~u
∇u L(~
ρ, ~u , λ) =
Compute λ s.t.
∂H T
∂J T
λ=−
∂~
ρ
∂~
ρ
O(T 2 N 2 )
Then
∇~u J(~
ρ, ~u ) = ∇~u L(~
ρ, ~u , λ) =
Paola GOATIN - Inria
∂J
∂H
+ λT
∂~u
∂~u
O(TN + TM)
ANCONET (March 9-11, 2016) - 25
3. Quick review of the adjoint method
Exploiting system structure
The structure of the forward system influences the efficiency adjoint system
solving
Solving for λ
∂J T
∂H T
λ=−
∂~
ρ
∂~
ρ
Since
∂H
∂~
ρ
is lower triangular,
∂H T
∂~
ρ
is an upper triangular matrix
=⇒ The adjoint system can be solved efficiently using backward substitution
Complexity reduction from O(T 2 NM) to O(NT + MT )
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 26
3. Quick review of the adjoint method
Exploiting system structure
The structure of the forward system influences the efficiency adjoint system
solving
Solving for λ
∂J T
∂H T
λ=−
∂~
ρ
∂~
ρ
Since
∂H
∂~
ρ
is lower triangular,
∂H T
∂~
ρ
is an upper triangular matrix
=⇒ The adjoint system can be solved efficiently using backward substitution
Complexity reduction from O(T 2 NM) to O(NT + MT )
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 26
3. Quick review of the adjoint method
Optimization algorithm
Algorithm 1 Gradient descent loop
Pick initial control ~uinit ∈ Uad
while not converged do
ρ
~ = forwardSim(~u , IC , BC )
λ = adjointSln(~
ρ, ~u )
∇u J =
λT ∂H
∂~
u
+
∂J
∂~
u
solve for state trajectory (forward system)
solve for adjoint parameters (adjoint system)
compute the gradient (search direction)
~u ← ~u − α∇u J, α ∈ (0, 1) s.t. ~u ∈ Uad update the control ~u (update step)
end while
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 27
3. Quick review of the adjoint method
Remarks on descrete adjoint
Strengths:
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Requires only one solution of adjoint system, independent of dim(~u )
(unlike finite differences or sensitivity analysis).
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Extends existing simulation code
General technique, can be applied in very different settings
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Weaknesses:
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Optimization of the discretized problem, rather than approximation of the
optimal solution of the continuous problem
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No proof of convergence
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 28
4. Numerical results
Outline of the talk
1. Macroscopic models for road traffic
2. Discretized system
3. Quick review of the adjoint method
4. Numerical results
5. Application to optimal re-routing
6. Conclusion and perspectives
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 29
4. Numerical results
Numerical Results: case study
Figure: I15 South, San Diego: 31 km
N = 125 links and M = 9 onramps
T = 1800 time-steps
∆t = 4 seconds (120 minutes time interval)
reduced congestion c̄ = 100(1 − cc /cnc ), with c = max{TTT − VMT /vmax , 0}
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 30
4. Numerical results
Numerical Results
Density and queue lengths without control
Density and queue difference with control
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 31
4. Numerical results
Model Predictive Control
Performance under noisy input data: MPC loop
I initial conditions at time t and boundary fluxes on Th (noisy inputs)
I optimal control policy on Th
I forward simulation on Tu ≤ Th using optimal controls and exact initial and
boundary data
I t → t + Tu
3.5
4.0
3.0
3.5
3.0
2.5
2.5
2.0
2.0
1.5
1.5
1.0
1.0
0.5
0.0
0.5
Adjoint
Adjoint w/ Noise
Alinea
Alinea w/ Noise
Adjoint
Alinea
Reduced Congestion (%)
Reduced Congestion (%)
Comparison with Alinea (local feedback control without boundary conditions)
0.0
10−2
10−1
100
101
102
Noise factor (-)
Figure: Congestion reduction and noise robustness
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 32
4. Numerical results
Running time
234
232
231
230
Convergence
Finite differences
Adjoint
Total travel time (veh-s)
233
229
228
227
226
10−1
100
101
102
103
Running time (ms)
Simulation time
Reduced Congestion (%)
3.5
3.0
2.5
2.0
1.5
1.0
Adjoint
Alinea
0.5
0.0
0
50
100
150
200
250
300
350
400
450
Running time (seconds)
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 33
5. Application to optimal re-routing
Outline of the talk
1. Macroscopic models for road traffic
2. Discretized system
3. Quick review of the adjoint method
4. Numerical results
5. Application to optimal re-routing
6. Conclusion and perspectives
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 34
5. Application to optimal re-routing
Application to optimal re-routing
System Optimal Dynamic Traffic Assignment problem with Partial Control:
Multi-commodity flow:
X
ρi (k) =
ρi,c (k)
c∈C
accounting for compliant cc ∈ CC and non-compliant cn users
I
I
full Lagrangian paths known for the controllable agents
knowledge of the aggregate split ratios for the non-controllable (selfish)
agents.
Goal: Control compliant users to optimize traffic flow
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 35
5. Application to optimal re-routing
Application to optimal re-routing
System Optimal Dynamic Traffic Assignment problem with Partial Control:
Multi-commodity flow:
X
ρi (k) =
ρi,c (k)
c∈C
accounting for compliant cc ∈ CC and non-compliant cn users
I
I
full Lagrangian paths known for the controllable agents
knowledge of the aggregate split ratios for the non-controllable (selfish)
agents.
Goal: Control compliant users to optimize traffic flow
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 35
5. Application to optimal re-routing
Numerical study: synthetic network
I
I
[Ziliaskopoulos, 2001]
Normal conditions: link capacity between cell 3 and cell 4 = 6
Optimal total travel time: 178
Incident between cell 3 and cell 4: capacity goes to 0
Optimal total travel time: 211 (otherwise 244)
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 36
5. Application to optimal re-routing
Synthetic network: partial control
Total travel time reduction as a function of the percentage of vehicles that are
rerouted:
almost optimal allocation can be achieved by controlling ∼60% of the demand
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 37
5. Application to optimal re-routing
Numerical study: real case
Figure: I210 with parallel arterial route, Arcadia (13 km)
N = 24 links
1 hour time-horizon
∆t = 30 seconds
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 38
5. Application to optimal re-routing
Numerical Results
50% capacity drop between min 10 and min 30
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 39
5. Application to optimal re-routing
Numerical Results
Arterial capacity used:
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 40
6. Conclusion and perspectives
Outline of the talk
1. Macroscopic models for road traffic
2. Discretized system
3. Quick review of the adjoint method
4. Numerical results
5. Application to optimal re-routing
6. Conclusion and perspectives
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 41
6. Conclusion and perspectives
In summary
Results:
I
Definition of appropriate junction solvers for ramp metering and
multi-commodity
I
Formulation of the corresponding finite horizon optimal control problems
I
Gradient computation with O(NT + MT ) complexity
I
Numerical tests showing the benefits on real field applications
Possible extensions:
I
Variable speed limit, maximal queue length, ...
I
Selfish agents response (Stackelberg games)
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 42
6. Conclusion and perspectives
References
I
M.L. Delle Monache, J. Reilly, S. Samaranayake, W. Krichene, P.Goatin
and A. Bayen. A PDE-ODE model for a junction with ramp buffer, SIAM
J. Appl. Math., 74(1) (2014), 22-39.
I
J. Reilly, S. Samaranayake, M.L. Delle Monache, W. Krichene, P. Goatin
and A. Bayen. Adjoint-based optimization on a network of discretized
scalar conservation law PDEs with applications to coordinated ramp
metering, J. Optim. Theory Appl., 167(2) (2015), 733-760.
I
S. Samaranayake, W. Krichene, J. Reilly, M.L. Delle Monache, P. Goatin
and A. Bayen. Discrete-time system optimal dynamic traffic assignment
(SO-DTA) with partial control for horizontal queuing networks, in review.
Thank you for your attention!
Paola GOATIN - Inria
ANCONET (March 9-11, 2016) - 43