Leak Localization in open water Channels Nadia Bedjaoui Workshop on irrigation channels and related problems N.Bedjaoui, E.Weyer and G. Bastin

Leak Localization in

open water Channels

Nadia Bedjaoui

Workshop on irrigation channels and related problems

N.Bedjaoui, E.Weyer and G. Bastin

2

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

5

•Outline Problem statement Objective Methods Application

Types of leaks in irrigation channelsProblem statement

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

6

• Failures in the civil engineering:Affect an escape gate


Types of leaks in irrigation channelsProblem statement

7

Types of leaks in irrigation channels

• Unpredicted offtakes Affect the farmer offtakes


Problem statement

8

• Important to

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


Problem statement

9

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

11

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


12

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

13

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

14

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

17

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

18


• 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

19

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

20

• 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

22

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

23

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

24

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

26

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

27

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

28

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

29

• Scenario

Pool 5

Gate 4Gate 5

pxL

yxL

Offtake

qx0

px0

qxL

yx0

dxL

Section=35

30

• 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

32

Observer convergence from different initial conditions

33









• Simulated data

• Real data

• Conclusion

34

Localization scheme (method 1)

35

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

38

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

39

• 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

42

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

43

Results on simulated data

44

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

45

Conclusion (2/2)

• Both methods give similar results

• Leak localization is too sensitive:

• Model uncertainty

• Offset on measurments

• Time Starting detection

• Feedback control

46

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)

Documents

Leak Localization in open water Channels Nadia Bedjaoui Workshop on irrigation channels and related problems N.Bedjaoui, E.Weyer and G. Bastin