Download - Leak Localization in open water Channels

Leak Localization in

open water Channels

Nadia Bedjaoui

Workshop on irrigation channels and related problems

N.Bedjaoui, E.Weyer and G. Bastin

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Outline

• Problem statement

• Objective of this work

• Leak localization methods

• Application

• Conclusion

3

Outline




• Application

• Conclusion

4

• Irrigation channel = supply water to users for irrigation purposes

• Supply done with less water losses possible

• Manual control large water losses

• Automatic control minimizes these losses

• Additional water losses due to the presence of leaks

• Leak =wasted water left definitively from

the channel

Problem statement•Outline Problem statement Objective Methods Application

http://gis-rci.montpellier.cemagref.fr/album_gignac/a_tronc_commun/slides/TC_fin_rijo_xl.html

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•Outline Problem statement Objective Methods Application

Types of leaks in irrigation channelsProblem statement

• Failures in the civil engineering: Affect the walls of the channel

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• Failures in the civil engineering:Affect an escape gate


Types of leaks in irrigation channelsProblem statement

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Types of leaks in irrigation channels

• Unpredicted offtakes Affect the farmer offtakes


Problem statement

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• Important to

– Detect the presence of the leak– Estimate the size of the leak– Localize the position of the leak


Problem statement

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Leak Detection + Estimation(E. Weyer& G. Bastin 2008)– Based on mass-balance model

– Idea :Do the measurements check the model?

– CUSUM algorithm: quick detection+ no faulse alarm

– Impossible leak localization

( 1)k

g z k

))()(()()1(

)1( 2/32/3 khcrkhct

kykykz outin

wzw

zw

0

00

Problem statement•Outline Problem statement Objective Methods Application

10

Outline




• Application

• Conclusion

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Objective of this work

• Interest: leak localization

– Leak is already detected and estimated by CUSUM algorithm (Weyer & Bastin 2008)

• Investigatation of two methods

– Model used: Saint Venant model as Hyperbolic Partial Differential Equations PDE

– Method (1) bank of Nonlinear Saint-Venant models– Method (2) bank of Nonlinear Observers


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Outline

• Problem statement• Objective of this work


– Method (1) using a bank of pure models• Modelling: Saint Venant is hyperbolic PDE

– Method (2) using a bank on observers• Observer objective• Observer structure• Observer Design

• Application

• Conclusion

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Method (1): Modelling

•OutlineProblem statementObjective Methods (1)ApplicationConclusion

x=Lx=0

Pool

Upstream Gate

DownstreamGate

PL(t)

Y(t,L)

Leak

Q(t,0)

P0(t)

Q(t,L)

Y(t,0)

w

x

L

xl

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Method (1): Modelling• Saint Venant Equations

•

• Boundary conditions (x=0 & x=L)(=Gate equations)

• Overshot gate

• Offtake

),()()(),()),(

),((),(

),(),(),(2

xtwA

QkSSgAAYxtgA

xtA

xtQxtQ

xtwxtQxtA

wfxxt

xt

0,1,3/42

22

wkRA

QnS wf

pyhchQ ,2/3

otherwise

xxforwxxwxtw ll 0)(),(


15

Method (1):Modelling

Two coupled quasi-linear Hyperbolic PDE

• subcritical flow

( , ) ( , , )t x

A AF A Q f A Q w

Q Q

A

Q

A

QYgA

QAFA

210

),(2

2

wA

QkSSgA

wQAfwf

1

)(

0),,(

0),(

0),(

YgAA

QQA

YgAA

QQA

A

A


16

• Initial Conditions (in t=0)

• Boundary Conditions (in x=0 & x=L)

),,(),( wQAfQ

AQAF

Q

Axt

),0()(0 xAxA ),0()(0 xQxQ

)0,(tQ ),( LtQ


Method (1):Modelling

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Method (2):Observer

Method (2): using a bank of Observers

Objective of the observer:

• From any Initial Conditions (t=0)

• Using the only measurements Y(t,0) & Y(t,L)

• The estimation error converges to zero


0 0 0 0ˆ ˆ,A A Q Q

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• Observer structure

• Boundary conditions

ˆ ˆˆ ˆ ˆ ˆ ˆ( , ) ( , , )

ˆ ˆt x

A AF A Q f A Q w

Q Q


)0,(ˆ tQ ),(ˆ LtQ

Method (2):Observer

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Method (2): using a bank of Observers• Observer design

1) Linearized model

2) Formulating the estimation problem as a control problem

3) Using the results on boundary control to determine the boundary conditions of the observer that achieves good estimation


Method (2):Observer

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• Observer design

1) Linearized model around an equilibrium

-Deviations from the equilibrium

-Linearized model


QQ

AA

q

a

0w

Wwq

aB

q

aC

q

axt

),,(),,,(),,( ),( wQAfWwQAfBBAFC wQA

www

Method (2):Observer

),( QA

21

• Observer design

1) Linearized observer around an equilibrium

-Deviations from the equilibrium

-Linearized observer

Estimation error


QQ

AA

q

aˆ

ˆ

ˆ

ˆ0w

wWq

aB

q

aC

q

axt ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

www ˆˆ

QQ

AA

qq

aa

e

e

q

a

ˆ

ˆ

ˆ

ˆ

Method (2):Observer

),( QA

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2) Formulating the estimation problem as a control problem

-Control objective: regulate the deviations to 0 using boundary inputs

-Estimation problem: regulate the estimation error to 0 using the boundary output errors


0

0

0

0

QQ

AA

q

a

),(

)0,(

),(

)0,(

Lt

t

Ltq

tq

0

0

0

0

ˆ

qq

âa

e

e

q

a

),(

)0,(

),(

)0,(

Lte

te

Lte

te

a

a

Method (2):Observer

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Summary on boundary control of Saint Venant equations

0( ,0) ( ,0)

( , ) ( , )L

t k t

t L k t L

( ) 0t xx

( ) ( ) ( )t xx B x W x w

-Linear case + non-homogenous terms

-Linear case +non-homogenous terms [ Bastin et al 2008]

small enough for Saint Venant Subcritical flow

B

( , ) ( , , )t x h w

-Quasi-linear case +non-homogenous terms [ Prieur et al 2008]

small enough & sufficiently small'(0) 0, (0)h h

B

w

10 Lkk( , ) 0

( , ) 0t

t

t x

t x

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observer design based on characteristic method

0 0 0 0 0ˆ ˆ ˆ ˆ( , , ), ( , , )x x x x xL L xL xL xLQ f Q A A Q f Q A A

0 1Lk k

( , ) 0

( , ) 0t

t

e t x

e t x

wxt Weeq

eaB

eq

eaC

eq

ea

weext eW

e

eB

e

e

e

e

25


• Initial Conditions (t=0)

• Boundary Conditions (x=0 & x=L)

ˆ ˆˆ ˆ ˆ ˆ ˆ( , ) ( , , )

ˆ ˆt x

A AF A Q f A Q w

Q Q

0 0 0 0ˆ ˆ,A A Q Q

)))0,(ˆ())0,(((1

1

)0,(

)0,(

)0,(ˆ)0,(ˆ 0 tAtA

k

k

tA

tQ

tA

tQ

L

))),(ˆ()),(((1

1

),(

),(

),(ˆ),(ˆ

LtALtAk

k

LtA

LtQ

LtA

LtQ

L

L

,10 LkkA

YgA

AA


Method (2):Observer

)))0,(())0,((( tAtAA

Q

A

Q )))0,(())0,((( tAtAA

Q

A

Q

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Localization scheme

1,{ :min ( )}l j j j

j Nx x J x

1,ˆ ˆ{ :min ( )}l j j j

j Nx x J x

• Method 1

• Method 2

NjxkYkYxLkYLkYxJjT

Tkjjjjj ,1,)),0,()0,(()),,(),(()(

0

22

NjxkYkYxLkYLkYxJjT

Tkjjjjj ,1,))ˆ,0,(ˆ)0,(())ˆ,,(ˆ),(()ˆ(

0

22

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Outline• Introduction

– Problem statement– Objective of this work


– Method based on models

– Method based on observers

• Application of the 2 methods

– Description of the system of application

– Results and observations with

• Simulated data

• Real data

• Conclusion

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Application of the 2 methods

• Description of the system of application

Gate 6Gate 5Gate 4Gate 3Gate 2Gate 1

Topview of Coly 6

Farm Farm

L=943m, delay=5mn,

Silde slope=2

Bottom width=1.80m

Gate width=1.91m

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• Scenario

Pool 5

Gate 4Gate 5

pxL

yxL

Offtake

qx0

px0

qxL

yx0

dxL

Section=35

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• Scenario

Application on simulated data

Pool 5

Gate 4Gate 5

pxL

yxL

Offtake

qx0

px0

qxL

yx0

dxL

Section=35

310 50 100 150 200 250 300 350 400

-0.2

0

0.2

0.4

0.6

0.8

time [min]

upst

ream

esi

mat

ion

erro

r [m

]

Error estimation for k0 =-0.1and different observer gains

0 50 100 150 200 250 300 350 400-0.2

0

0.2

0.4

0.6

0.8

time [min]

dow

nstre

am e

stim

atio

n er

ror [

m]

Error estimation for k0 =-0.1 and different observer gains

kL=-0.5

kL=-0.1

kL=0

kL=-0.5

kL=-0.1

kL=0

Observer convergence: using different gains

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Observer convergence from different initial conditions

33









• Simulated data

• Real data

• Conclusion

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Localization scheme (method 1)

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Localization scheme (method 2)

36

Results on simulated data

H1H2

37

Localization results on simulated data

0 5 10 15 20 25 30 35 40 45 500

1

x 10-4

sections

Cost fu

nction

model

Observer

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0 10 20 30 40 500.02

0.03

0.04

0.05

0.06

0.07

0.08

section

Cost fu

nction

Subject to a variation of 50% of n

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• Scenario

Application on real data

Pool 5

Gate 4Gate 5

pxL

yxL

Offtake

qx0

px0

qxL

yx0

dxL

40









• Simulated data

• Real data

• Conclusion

41

Results on Real data

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Localization scheme

1,{ :min ( )}l j j j

j Nx x J x

1,ˆ ˆ{ :min ( )}l j j j

j Nx x J x

• Method 1

• Method 2

NjxkYkYxLkYLkYxJjT

Tkjjjjj ,1,)),0,()0,(()),,(),(()(

0

22

NjxkYkYxLkYLkYxJjT

Tkjjjjj ,1,))ˆ,0,(ˆ)0,(())ˆ,,(ˆ),(()ˆ(

0

22

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Results on simulated data

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Conclusion Objective: Leak localizationInvestigate two methods for leak localizationMethod (1) based on pure modelsMethod (2) based on observers

Design of observer: - Characteristic method

- The estimation problem is written as boundary control problem for the linearized system

-Convergence of the observer can be fixed by the gains

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Conclusion (2/2)

• Both methods give similar results

• Leak localization is too sensitive:

• Model uncertainty

• Offset on measurments

• Time Starting detection

• Feedback control

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2) Réconciliation de données globale

Appliquée à un bief avec retards discrétisés :

Filtre de Kalman détection de prélèvements+défauts

Combinaison locale -globale Distinction défaut -prélèvement

3) Observateurs à entrées inconnues et H

Cas général des systèmes à retards• Retards dans l’état et les entrées • Retards variants dans le temps

Méthode testée avec succès sur le canal de Gignac

Conclusion (2/2)