TIME DELAY APPROACH TO THE MODELING OF FLUID NETWORKS

Title INTRODUCTION Flow dynamics Riemann Invariants Method of Characteristics Time Delay Model Network Model

TIME DELAY APPROACH TO THE

MODELING OF FLUID NETWORKS

David NovellaEmmanuel Witrant

Olivier Sename

GIPSA LAB, FRANCEDELSYS Workshop

20-22 November 2013

1 / 50


Table of Contents.

INTRODUCTIONDifferent ApproachesObjective

Flow dynamics

Riemann Invariants

Method of Characteristics

Time Delay Model

Network Model

2 / 50


Outline


Flow dynamics

Riemann Invariants


Time Delay Model

Network Model

3 / 50


Fluid Networks

Fluid network systems appear in different areas

Figure 1 : Mine Ventilation Systems Figure 2 : Gas and WaterDistribution Lines

4 / 50


Figure 3 : Traffic Flow

Figure 4 : Blood flow

5 / 50


They main difficulties of deal with this class of systems are

◮ High order nonlinear dynamics

◮ Complex interconnected flows

◮ Transport phenomena

6 / 50


Different Approaches

Lumped Parameter Model

◮ Modeling of pipes as lumped parameters

◮ Use of approximations of incompressible Navier-Stokesequation

◮ Network modeled using Kirchhoff’s laws

◮ Analogies with RL non-linear circuits

7 / 50



Lumped Parameter Model

◮ Modeling of pipes as lumped parameters

◮ Use of approximations of incompressible Navier-Stokesequation

◮ Network modeled using Kirchhoff’s laws

◮ Analogies with RL non-linear circuits

◮ [Petrov et al., 1992]

◮ [HL et al., 1997]

◮ [Hu et al., 2003]

◮ [Koroleva and Krstic, 2005]

8 / 50



Boundary Feedback Control

◮ Modeling by means of partial differential equations

◮ Riemann invariants transformation

◮ Boundary control techniques

9 / 50



Boundary Feedback Control

◮ Modeling by means of partial differential equations

◮ Riemann invariants transformation

◮ Boundary control techniques

◮ [de Halleux et al., 2003], [Halleux, 2004]

◮ [Prieur et al., 2008]

◮ [Bastin et al., 2008]

◮ [Gugat and M., 2011], [Gugat et al., 2011]

10 / 50



Time-delay modeling

◮ Model for large convective flows

◮ Transport properties involved in the flow model

◮ Parameter estimation of the transport coefficient

◮ Using some appropriate physical hypotheses

◮ A mathematical equivalence is then obtained between thedistributed model and a time-delay system

11 / 50



Time-delay modeling

◮ Model for large convective flows

◮ Transport properties involved in the flow model

◮ Parameter estimation of the transport coefficient

◮ Using some appropriate physical hypotheses

◮ A mathematical equivalence is then obtained between thedistributed model and a time-delay system

◮ [Witrant and Marchand, 2008]

◮ [Witrant and Niculescu, 2010]

◮ [Bradu et al., 2010]

12 / 50


Objective

Aims

Figure 5 : Fluid Flow Network

◮ To improve the classical lumped parameter model.◮ To obtain a dynamic model from the physic properties◮ To introduce the transport phenomena as a time-delay

13 / 50


Objective

14 / 50


Outline


Flow dynamics

Riemann Invariants


Time Delay Model

Network Model

15 / 50


Navier-Stokes Equations1

∂

∂t

ρρ~vρE

+ ~∇ ·

ρ~v

ρ~v ⊗ ~v + pI − τ

ρ~vH − τ · ~v − k ~∇T

=

0~fe

Wf + qH

(1)

◮ ρ is the density,

◮ ~v is the velocity vector,

◮ E is the total energy,

◮ p is the pressure,

◮ τ is the stress tensor,

◮ H is the total enthalpy,

◮ k is the coefficient ofthermal conductivity,

◮ T is the temperature,

◮ ~fe is the external forcevector,

◮ qH is the heat source.1[Hirsch, 2007]

16 / 50


Euler Equations 2

∂

∂t

ρρVVV

ρE

+ ~∇ ·

ρVVV

ρVVV T ⊗ VVV + pIII

ρVVV H

=

00q̇

(2)

◮ ρ is the density,

◮ VVV is the velocity,

◮ ρVVV is the moment,

◮ P is the pressure,

◮ E is the energy,

◮ H is the total enthalpy,

◮ q̇ rate of heat addition,

◮ ⊗ is a tensor product.

2[Toro, 2009]17 / 50


Isothermal Euler Equations

◮ A common model for gas flow in pipes

◮ The temperature is considered constant

◮ [Gugat and M., 2011], [Gugat et al., 2011]

◮ Pressure is obtained from a equation of state:

p = p(ρ) ≡ a2ρ, (3)

where a is a non zero constant propagation speed of sound,[Toro, 2009].

18 / 50


We define:

a =

√

ZRT

Mg

(4)

◮ Z is the natural gas compressibility factor

◮ R the universal gas constant

◮ T the absolute gas temperature

◮ Mg the gas molecular weight

19 / 50


The isothermal Euler equations for a single pipe are defined by:

◮ Mass conservation

∂ρ

∂t+

∂q

∂x= 0 (5)

◮ Momentum conservation

∂q

∂t+

∂

∂x

(

q2

ρ+ a2ρ

)

= −fgq | q |

2Dρ(6)

◮ fg is the friction factor,

◮ D is the diameter of the pipe.

20 / 50


Outline


Flow dynamics

Riemann Invariants


Time Delay Model

Network Model

21 / 50


Definition

Consider the hyperbolic systems described as follows

∂U

∂t+ A(U)

∂U

∂x= 0 (7)

x ∈ [0, L], t ∈ [0, T ]

The system (7) can be transformed into a system of coupledtransport equations

∂ξi(x , t)

∂t+ λi(ξ(x , t))

∂ξi (x , t)

∂x= 0 for i = 1 · · · , n. (8)

22 / 50


Figure 6 : Representation of the characteristic curves

dx

dt= λi(ξ(x , t)). (9)

Since dξi/dt = 0 along the characteristic curve, it follows that ξi isconstant (or invariant) along the characteristic curve.

23 / 50


Riemann Invariants for the Isothermal Euler Equations

We can express the equations (5) and (6) as follows

∂U

∂t+

∂F (U)

∂x= D(U), (10)

with U(x , t) = [ρ, q]. The Jacobian of the flux matrix F (U(x , t))is

A(U) =

(

0 1

a2 − q2

ρ2 2qρ

)

. (11)

24 / 50


Eigenvalues and Eigenvectors

The eigenvalues of the Jacobian matrix A(U) are

λ1, 2 =q

ρ± a. (12)

And the right eigenvectors are

K1 =

[

1qρ

− a

]

K2 =

[

1qρ+ a

]

(13)

25 / 50


Diagonal System

Then we obtain the following transformation of the system (10):

∂ξ

∂t+ Λ(ξ)

∂ξ

∂x= S(ξ), (14)

where

Λ(ξ) =

[

− ξ1+ξ22 + a 0

0 − ξ1+ξ22 − a

]

and the source term

S(ξ) = −fg

8D(ξ1 + ξ2)|ξ1 + ξ2|

(

11

)

26 / 50


Riemann Invariant

The respective Riemann invariant for this conservation system are

ξ1,2(ρ, q) = −q

ρ∓ a ln(ρ) (15)

We can express the original variables ρ and q in terms of theRiemann invariant as

ρ = exp

(

ξ2 − ξ1

2a

)

, (16)

q =ξ1 + ξ2

2exp

(

ξ2 − ξ1

2a

)

27 / 50


Outline


Flow dynamics

Riemann Invariants


Time Delay Model

Network Model

28 / 50


Definition

Allows to solve linear, quasilinear and nonlinear first-order PDEs.E.g. for the first order linear equation:

a(x , y)ux + b(x , y)uy = c(x , y) (17)

◮ Suppose we can find a solution u(xy). Consider the graph ofthis function given for

S.= {(x , y , u(x , y))}

◮ If u is a solution of (17), we know that at each point (x , y),then

(a(x , y), b(x , y), c(x , y)) · (ux (x , y), uy (x , y), −1) = 0. (18)

29 / 50


◮ Then the normal to the surface S = {(x , y , u(x , y))} at thepoint (x , y , u(x , y)) is given byN(x , y) = (ux (x , y), uy (x , y), −1).

30 / 50



To construct a curve C (the integral orcharacteristic curve) parameterized by s suchthat it is tangent to(a(x(s), y(s)), b(x(s), y(s)), c(x(s), y(s)))at each point (x , y , z)

31 / 50



To construct a curve C (the integral orcharacteristic curve) parameterized by s suchthat it is tangent to(a(x(s), y(s)), b(x(s), y(s)), c(x(s), y(s)))at each point (x , y , z)

⇒ In particular, the curve C = {(x(s), y(s), u(x(s), y(s))} willsatisfy the following system of ODEs:

dx

ds= a(x(s), y(s))

dy

ds= b(x(s), y(s))

dz

ds= c(x(s), y(s))

32 / 50


Outline


Flow dynamics

Riemann Invariants


Time Delay Model

Network Model

33 / 50


Assumptions

Remark 1Note that the method of characteristics can not be applied directly

to the PDE system (14) due to the coupled term in the source

S(ξ) = −fg

8D(ξ1 + ξ2)|ξ1 + ξ2|

(

11

)

Remark 2As a start point, let us consider the characteristic velocities of the

hyperbolic system λ1 and λ2 as constant parameters.

34 / 50


In order to handle with this term it is possible to approximate thePDE system (14) as follows

∂

∂t

[

ξ1

ξ2

]

+

[

λ1 00 λ2

]

∂

∂x

[

ξ1

ξ2

]

=

[

−α 00 −α

] [

ξ1

ξ2

]

+α

[

−ξ̄2

−ξ̄1

]

(19)With

◮ α = fg4D

◮ ξ̄i represents the averaged value of the respective wave.

35 / 50


Then we can solve separately each PDE, for ξ1 we have

dt

ds1= 1 t0 = 0

dx

ds1= λ1 x0 = r

dz1

ds1= −α(z1 + ξ̄2) z0 = φ1(r)

Solving the system of ODEs we obtain

t = s

x = sλ1 + r ⇔ r = x − tλ1

z1 (s) = −ξ̄2 + e−αsφ1(r)

36 / 50


Then, it is possible to obtain the following expression

ξ1(L, t) = −ξ̄2 + e−αtξ1(0, t −

L

λ1), (20)

Similarly for the second wave

ξ2(0, t) = −ξ̄1 + e−αtξ1(L, t −

L

λ2), (21)

37 / 50


Figure 7 : Wave Propagation

38 / 50


Outline


Flow dynamics

Riemann Invariants


Time Delay Model

Network Model

39 / 50


Figure 8 : Network Example

AIM

◮ To consider each node as a finite control volume.◮ To apply conservation fundamentals for each wave in each

node.◮ To obtain a time-delay model of the network in terms of the

propagation waves. 40 / 50


Dynamic Equations

In general, we can obtain the model from the following principle

ξ̇Ni (t) =

∑

inflows −∑

outflows (22)

Then, for the wave ξ1 in the node N we have the followingdynamics

ξ̇N1 (t) =

∑

i=inflows

β(Xi ,N)ξXi

1 (t−h(Xi ,N)1 )−ξ2

Xi −∑

j=outflow

ξNj

1 (t) (23)

41 / 50


And for the wave ξ2 we have

ξ̇N2 (t) =

∑

j=inflows

β(Yj ,N)ξYj

1 (t−h(Yj,N)2 )−ξ1

Yj −∑

i=outflow

ξNi

1 (t) (24)

with

◮ β = e−αt

◮ The superscript (Xi , N) points to the coefficient in the linebetween te node Xi and the node N

◮ The time delay h1,2 = L(Xi ,N)/λ1,2.

42 / 50


Conclusion

◮ We present a time delay model for the flow through a pipebased on the isothermal Euler equations

◮ Some physical assumptions were done in order to simplify thesolutions

◮ Conservation laws yield to delayed differential equations modelof the network system

43 / 50


Further work

◮ Validation of the model and comparison with differentmodeling approaches

◮ Lumped parameter model◮ Computational Fluid Dynamics (CFD)

◮ Solution for the hyperbolic coupled quasilinear system◮ Time varying characteristic velocities◮ Coupled nonlinear source term

◮ Design of a feedback control strategy for the network system◮ Decentralized control◮ LPV approach

44 / 50


Final Goal

45 / 50


THANKS FOR YOURATTENTION

46 / 50


Bastin, G., Coron, J.-M., and d’Andrea Novel, B. (2008).Using hyperbolic systems of balance laws for modeling, controland stability analysis of physical networks.17th IFAC World Congress, Workshop on Complex Embedded

and Networked Control Systems.

Bradu, B., Gayet, P., Niculescu, S.-I., and Witrant, E. (2010).Modeling of the very low pressure helium flow in the lhccryogenic distribution line after a quench.Cryogenics, 50(2):71 – 77.

de Halleux, J., Prieur, C., Coron, J.-M., d’Andrea Novel, B.,and Bastin, G. (2003).Boundary feedback control in networks of open channels.Automatica, 39(8):1365–1376.

Gugat, M., Herty, M., and Shcleper, V. (2011).

47 / 50


Flow control in gas networks: Exact controllability to a givendemand.Mathematical Methods in the Applied Sciences, 34:745–757.

Gugat, M. and M., D. (2011).Time-delayed boundary feedback stabilization of theisothermal euler equations with friction.Mathematical Control and Related Fields, 1(4):469–491.

Halleux, J. (2004).Boundary control of quasi-linear hyperbolic initial

boundary-value problems.PhD thesis, Universitï¿1

2 catholique de Louvain.

Hirsch, C. (2007).Numerical Computation of Internal and External Flows: The

Fundamentals of Computational Fluid Dynamics.Elsevier, 2nd edition.

48 / 50


HL, H., JM, M., RV, R., and Mine, W. Y. (1997).Mine Ventilation and Air Conditioning.Wiley: New York, 3rd edition.

Hu, Y., Koroleva, O. I., and Krstic, M. (2003).Nonlinear control of mine ventilation networks.Systems & Control Letters, 49(4):239 – 254.

Koroleva, O. I. and Krstic, M. (2005).Averaging analysis of periodically forced fluid networs.Automatica, 41(1):129 – 135.

Petrov, N., Shishkin, M., Dmitriev, V., and Shadrin, V.(1992).Modeling mine aerology problems.Journal of Mining Science, 28(2):185–191.

Prieur, C., Winkin, J., and Bastin, G. (2008).

49 / 50


Robust boundary control of systems of conservation laws.Mathematics of Control, Signals and Systems, 20:173–197.

Toro, E. F. (2009).Riemann Solvers and Numerical Methods for Fluid Dynamics.Third edition.

Witrant, E. and Marchand, N. (2008).Mathematical Problems in Engineering, Aerospace and

Sciences, chapter Modeling and Feedback Control for Air FlowRegulation in Deep Pits.Cambridge Scientific Publishers.

Witrant, E. and Niculescu, S.-I. (2010).Modeling and control of large convective flows withtime-delays.Mathematics in Engineering, Science and Aerospace,1(2):191–205.

50 / 50

TIME DELAY APPROACH TO THE MODELING OF FLUID NETWORKS

Documents