Workshop on “Irrigation Channels and Related Problems” Variation of permeability parameters
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Transcript of Workshop on “Irrigation Channels and Related Problems” Variation of permeability parameters
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Workshop on Workshop on ““Irrigation Channels and Related Irrigation Channels and Related
Problems”Problems” Variation of permeability parametersVariation of permeability parameters
in Barcelona networksin Barcelona networks
Workshop on Workshop on ““Irrigation Channels and Related Irrigation Channels and Related
Problems”Problems” Variation of permeability parametersVariation of permeability parameters
in Barcelona networksin Barcelona networks
Department of Information Engineering and Applied Mathematics
October, 2°, 2008
University of Salerno
Luigi RaritàJoint work with:
Ciro D’Apice, Dirk Helbing,
Benedetto Piccoli.
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Organization of the presentation
A model for car traffic on a single road. Dynamics at nodes. Formulation of an optimal control problem. Simulations of queues on roads.
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Description of dynamics on a single road Description of dynamics on a single road
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a b
Dynamics on roadsDynamics on roads
L
Length of the road: L;
Congested part of length: l;
Free part: L – l
lL – l
Slope at low densities : V0 ;
Slope at high densities: c.
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a b
Dynamics on roads Dynamics on roads
Ll
L – l
max
1c
T
Incoming flow
A t O t
A t
O t Outgoing flow
1
0 max
1Q̂ T
V
T: safe time.
ˆ , if ,ˆ0
, if .
Q l t L
A t A t LO t l t L
c
0
max
, if 0,ˆ0
ˆ , if 0 .
LA t N
VO t O t t
Q N N
t
Maximal flux
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a b
The permeability parameterThe permeability parameter
L
0t
If the permeability parameter is zero, traffic is stopped (outgoing flow equal to zero).
If the permeability parameter is one, traffic can flow and the outflow can depend either on queues on the road or the arrival flow.
1t
We can study some situation of traffic when the permeability is among zero and one. The permeability can control traffic flows!!
2008
2006
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a b
Traffic jams modelled by a DDETraffic jams modelled by a DDE
L
The number of delayed vehicles can be expressed by the following DDE (Delayed Differential Equation):
0
LN A t O t
V
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Road networks Road networks
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Barcelona networks A Barcelona network is seen as a finite collection of roads (arcsarcs), meeting at some junctions (nodesnodes). Every road has not a linear shape.
0 0
ˆ ˆ;
;
.
k
k
Q Q
V V
different lengths
Assumptions
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Helbing model for Barcelona networks
1ij ijV Ht t
, ,node
row i, column j
i j
:
:
vertical road;
horizontal road;
ij
ij
V
H
, :
, :ij ij
ij ij
V H
V H
A A
O O
inflows;
outflows.
Dynamics on roads are solved by the Helbing model.
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For every road an initial data.Boundary data for roads with infinite endpoints.
For inner roads of the network, solving dynamics at nodes is fundamental!
Boundary data!! The
arrival flow…
Boundary data!! The
arrival flow…
Junctions It is necessary solving
dynamics at road junctions
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Riemann Solver (RS)Riemann Solver (RS)
A RS for the node (i, j) is a map that allows
to obtain a solution for the 4 – tuple
1, 1, , , .
i j ij ij ijV H V HA A O O
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Rule ARule ADistribution of TrafficDistribution of Traffic
(A) Some coefficients are introduced in order to describe the preferences of drivers. Such coefficients indicate the distribution of traffic from incoming to outgoing roads. For this reason, it is necessary to define a traffic Distribution Matrix
such that 1,..., ; 1,...,
,m nji j n n m i n
C
1
0 1, 1, 1,..., , 1,..., .n m
ji jij n
i n j n n m
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(A) Some coefficients are introduced in order to describe the preferences of drivers. Such coefficients indicate the distribution of traffic from incoming to outgoing roads. For this reason, it is necessary to define a traffic Distribution Matrix
,,...,1;,...,1 nmnimnnjjiA
.,...,1,,...,1,1,101
mnnjnimn
njjiji
1
is the percentage by which cars
arrive from the incoming road i and
take the outgoing road j.
ji
Rule ARule ADistribution of TrafficDistribution of Traffic
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Rule BRule BMaximization of the fluxMaximization of the flux (B) Assuming that (A) (A) holds, drivers choose destination so
as to obtain the maximization of the flux.
No one can stop in front of the traffic junction without crossing it.
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Dynamics at a nodeDynamics at a nodeAssumption: one lane.Solution for the junction:
1 1ij ij
ij ij
C
1ij ij ijH ij V ij HA O O
1(1 ) (1 )
i j ij ijV ij V ij HA O O
(A)(A)(A)(A)
1
1
max
ˆ0
ˆ0
ˆ0
ˆ0 1 1
ij ij
ij ij
ij ij
ij ij ij
ij ij i j
V H
V V
H H
ij V ij H H
ij V ij H V
O O
O O
O O
O O A
O O A
ˆijVO
ˆijHO
ijVO
ijHO
PP * *,ij ijV HP O O
(B)(B)(B)(B)
1, 1, , ,
i j ij ij ijV H V HA A O O
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Dynamics at a nodeDynamics at a nodeThree possible cases for RS at (i, j).
Assumption: presence of queues on roads.
RS1RS2 RS3
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Formulation of an optimal control problem Formulation of an optimal control problem
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Optimization and controlOptimization and controlfor Barcelona networksfor Barcelona networks
Dynamics in form of a control system: the state is the number of delayed vehicles, the
control is the permeability.
Presence of delayed permeabilities.
Extra variable.
U = set of controls; R = set of roads.
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Not empty queuesNot empty queues
A non linear control system, with delayed controls, given by
permeabilities.
In this case, RS for the node (i, j) depends only on controls
(permeabilities) and not on the state.
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Empty queues: the nesting phenomenonEmpty queues: the nesting phenomenon
1 20
ij i j i jV V VN N N
Nesting equation!!
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A hybrid approachA hybrid approachThe evolution of y and do not depend only on dynamics at (i, j).
ijVNijVA
ijVO
1i jV
1i jH
1i j
1i j
1i jVO
1i jHO
N
depends on and .
ijVA is described by RS at (i, j) by:
1i jVO depend on:
1i jHO
ijVN1i jVN
1ijHN
1 1i jHN
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A hybrid approachA hybrid approachTo describe the whole dynamics at (i, j), we define the logic variables
ijVas follows:
For , the definition is similar. ijH
A complete hybrid dynamic for the node (i, j) can be described by the following equation:
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A hybrid approachA hybrid approachThe dynamic of a control parameter (or a distribution coefficient or ) influences the dynamic (which is of continuous type) of the couple (A, O) through RS. Dynamics of (A, O) determine a continuous dynamic of both (A, O) through RS and . The dynamic of implies a continuous dynamic of y and a discrete dynamic, through the logic variable , of the couple (A, O).
N N
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Dynamics of needle variations Dynamics of needle variations
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Needle variation and variational Needle variation and variational equationsequations
Let be a Lebesgue point for . For , we can define a family of controls in this way;
* * *, ,t f x 0,1
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Variational equationsVariational equations
For to be optimal, we require that:
The tangent vector v satisfies the following equations:
For t < while:
Continuous dynamic:
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Discrete dynamics of needle Discrete dynamics of needle variationsvariationsConsider a time interval [0, T] and a Lebesgue point . 0,T
Notice that: .
, , ,V ijij
Vt
, , ,V ijij
O Vt O
11, , ,
V i ji jA Vt A
, , ,H ijij
O Ht O
11, , ,
H ijijA Ht A
ijVO
ijHO1i jVA
1ijHA
RS
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Discrete dynamics of needle Discrete dynamics of needle variationsvariations
, , ,V ijij
Vt 1RS
ˆijVO
ˆijHO
ijVO
ijHO ˆ ˆ,
ij ijV HP O O
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Some preliminary numerical results Some preliminary numerical results
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Preliminary simulationsPreliminary simulations
ijH
1
2 4
3
,i j
1ijH
ijV
1i jV
6ijVL 5
ijHL
14
i jVL
1
3ijHL
0.7ij 0.7ij
0 2V 2c
max 1
1 1
0 0 0 0 0ij ij i j i jV H V VN N N N
0.5ijVA t
0.3ijHA t
200T
0.01h
max
0
1ˆ 1.1 1 1 1
2 2
Q
c V
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SimulationsSimulations
ijH
1
2 4
3
,i j
1ijH
ijV
1i jV 0
0.5
1
1.5
2
0 50 100 150 200
'O1.dat'
0
0.5
1
1.5
2
0 50 100 150 200
'O2.dat'
Jump of implies jumps of O
ijVO
ijHO
1ijV t 0
ijH t
Period of wave: 15.
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0
0.5
1
1.5
2
0 50 100 150 200
'A4.dat'
0
0.5
1
1.5
2
0 50 100 150 200
'A3.dat'
SimulationsSimulations
ijH
1
2 4
3
,i j
1ijH
ijV
1i jV
O influences dynamics at the node (i, j). Hence, we have variations of A.
1i jVA
1ijHA
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QueuesQueues
ijH
1
2 4
3
,i j
1ijH
ijV
1i jV
Problems of saturation!!! Congested roads!
0
1
2
3
4
5
6
7
8
0 50 100 150 200
'coda1.dat'
ijVN
0
1
2
3
4
5
6
7
8
0 50 100 150 200
'coda3.dat'
1i jVN
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QueuesQueues
ijH
1
2 4
3
,i j
1ijH
ijV
1i jV
ijVN
1i jVN
1
ij ijV H
0.3; 0.5;ij ijV H
0
1
2
3
4
5
6
0 50 100 150 200
'coda1.dat'
0
1
2
3
4
5
6
0 50 100 150 200
'coda3.dat'
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Some referencesSome referencesRarità L., D’Apice C., Piccoli B., Helbing D.,
Control of urban network flows through variation of permeability parameters,
Preprint D.I.I.M.A.
D. Helbing, J. Siegmeier and S. Lammer, Self-organized network flows, NHM, 2, 2007, no. 2, 193 – 210..
D. Helbing, S. Lammer and J.-P. Lebacque, Self-organized control of irregular or perturbed network traffic, in C. Deissenberg and R. F. Hartl (eds.), Optimal Control and Dynamic Games, Springer, Dordrecht, 2005, pp. 239 – 274.