leak localization in open water channels
DESCRIPTION
Leak Localization in open water Channels. Workshop on irrigation channels and related problems. N.Bedjaoui, E.Weyer and G. Bastin. Nadia Bedjaoui. Outline. Problem statement Objective of this work Leak localization methods Application Conclusion. Outline. Problem statement - PowerPoint PPT PresentationTRANSCRIPT
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
• Problem statement
• Objective of this work
• Leak localization methods
• 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
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
•Outline Problem statement Objective Methods Application
Types of leaks in irrigation channelsProblem statement
7
Types of leaks in irrigation channels
• Unpredicted offtakes Affect the farmer offtakes
•Outline Problem statement Objective Methods Application
Problem statement
8
• Important to
– Detect the presence of the leak– Estimate the size of the leak– Localize the position of the leak
•Outline Problem statement Objective Methods Application
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
• Problem statement
• Objective of this work
• Leak localization methods
• 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
•Outline Problem statement Objective Methods Application
12
Outline
• Problem statement• Objective of this work
• Leak localization methods
– 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)(),(
•OutlineProblem statementObjective Methods (1)ApplicationConclusion
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
•OutlineProblem statementObjective Methods (1)ApplicationConclusion
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
•OutlineProblem statementObjective Methods (1)ApplicationConclusion
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
•OutlineProblem statementObjective Methods (2)ApplicationConclusion
0 0 0 0ˆ ˆ,A A Q Q
18
Method (2): using a bank of Observers
• Observer structure
• Boundary conditions
ˆ ˆˆ ˆ ˆ ˆ ˆ( , ) ( , , )
ˆ ˆt x
A AF A Q f A Q w
Q Q
•OutlineProblem statementObjective Methods (2)ApplicationConclusion
)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
•OutlineProblem statementObjective Methods (2)ApplicationConclusion
Method (2):Observer
20
• Observer design
1) Linearized model around an equilibrium
-Deviations from the equilibrium
-Linearized model
•OutlineProblem statementObjective Methods (2)ApplicationConclusion
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
•OutlineProblem statementObjective Methods (2)ApplicationConclusion
AA
q
aˆ
ˆ
ˆ
ˆ0w
wWq
aB
q
aC
q
axt ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
www ˆˆ
AA
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
•OutlineProblem statementObjective Methods (2)ApplicationConclusion
0
0
0
0
AA
q
a
),(
)0,(
),(
)0,(
Lt
t
Ltq
tq
0
0
0
0
ˆ
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
Method (2): using a bank of Observers
• 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
•OutlineProblem statementObjective Methods (2)ApplicationConclusion
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
• Leak localization methods
– 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
Outline• Introduction
– Problem statement– Objective of this work
• Leak localization methods
– 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
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
Outline• Introduction
– Problem statement– Objective of this work
• Leak localization methods
– 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
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)