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.

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.

Description of dynamics on a single road Description of dynamics on a single road

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.

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

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

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

Road networks Road networks

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

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.

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

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

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

(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

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.

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

Dynamics at a nodeDynamics at a nodeThree possible cases for RS at (i, j).

Assumption: presence of queues on roads.

RS1RS2 RS3

Formulation of an optimal control problem Formulation of an optimal control problem

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.

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.

Empty queues: the nesting phenomenonEmpty queues: the nesting phenomenon

1 20

ij i j i jV V VN N N

Nesting equation!!

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

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:

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

Dynamics of needle variations Dynamics of needle variations

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

Variational equationsVariational equations

For to be optimal, we require that:

The tangent vector v satisfies the following equations:

For t < while:

Continuous dynamic:

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

Discrete dynamics of needle Discrete dynamics of needle variationsvariations

, , ,V ijij

Vt 1RS

ˆijVO

ˆijHO

ijVO

ijHO ˆ ˆ,

ij ijV HP O O

Some preliminary numerical results Some preliminary numerical results

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

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.

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

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

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'

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.