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Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Calibration Techniques Calibration Techniques for Looped Water Distribution Systemsfor Looped Water Distribution Systems
Tullio TucciarelliTullio Tucciarelli11, Alessandra Bascià, Alessandra Bascià22
First Annual Water Distribution Modeling SymposiumFirst Annual Water Distribution Modeling SymposiumPerugia (Italy), May 2003Perugia (Italy), May 2003
1
2
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Summary Summary
• Definition of the calibration problem;• Maximum Likelihood (ML) theory for the minimization of the
measurement error;• Calibration problem is ‘sick’: instability and nonuniqueness;• Choice of the measurement location by means of the D-optimality
criterion;• Selection of several set of measurements for different operating
conditions;• ML failure: an other way;• The other way for tree networks;• The other way for looped networks;• The other way: lab and field experiments.
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Preliminary definitions: Preliminary definitions:
• State Variables: time-variable flow data (pressure, velocity, concentration);
• Parameters: time-constant network data (pipe roughness, pipe length, node topographical elevation);
• Control Variables: network data that can be controlled by the water network manager (valve opening, pumping conditions).
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
iMj i
1 i i Nj=1
ij j i
H - H - Q - PS = 0 i=1,...,N
R H - HiM
ai i i ij ij ij
j=1
Q = (H - z ) D θ L2
o 2 i ii i o
H -z S = S sin
2 h
i i o0< H -z < h
oi i S = S i i oH -z > h
i S = 0 i iH -z < 0
if
if
if
Simulation modelSimulation model
Example:
Provides the relationship between state variables, parameters and control variables.
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
ijoij ij 5
ij
L R = R V +0.0826
D
Simulation modelSimulation model
Example:
2
ij
ij
= 4 log3.71 D
Provides the relationship between state variables, parameters and control variables.
iMj i
1 i i Nj=1
ij j i
H - H - Q - PS = 0 i=1,...,N
R H - H
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Calibration Calibration
• Definition: estimation of the unknown simulation model parameters.
• First criterion for judging the estimation quality: the estimated parameter values should be close to the “real” ones.
• If no parameter measure is possible:1. for the corresponding operating conditions, the computed values of the state variables have to be similar to the measured ones;
2. the computed values of the state variables have to be similar to the real ones also for operating conditions different from those corresponding to the measurements.
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Model Error
Estimation error Estimation error
It is the difference between the “real” parameter values and the estimated ones.
According to the sources of error, it can be partitioned as follows:
Error
=Measurement Error Computational Error
due to the simplifi- cations adopted in the
simulation model
due to the incorrect location of the global minimum
due to the incorrect measurement of the state variables
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Computational errorComputational error
For continuous problems, the optimization algorithms can locate only local minima, but we are looking for the global one!
F
parameterLocal minimum
Local minimum
Local and global
minimum
A sufficient condition for the existence of a unique local minimum: the convexity of both the objective function and the parameter domain. It is not our case.
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Dominio della funzione f(x1,x2)
x2
x1
x2
x1
convesso non convesso
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Funzione f(x)
f(x) f(x)
convessa non convessa
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Esempio di funzione obiettivo convessa con dominio di ricerca convesso.
limite del dominio di ricerca
isolinea della funzione obiettivo
parametro 1
parametro 2
30
20
10
Minimo locale
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Esempio di funzione obiettivo convessa con dominio di ricerca non convesso.
parametro 1
parametro 2
Minimo locale
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Esempio di funzione obiettivo non convessa con dominio di ricerca convesso.
5040
30
20
parametro 1
parametro 2
Minimo locale
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Minimization of the measurement errorMinimization of the measurement error (Maximum Likelihood Theory)(Maximum Likelihood Theory)
Hypothesis: the measurement error has a normal distribution, with known variance and zero mean.
1* *t
p HS = - H H C H H
H: Vector of the state variables computed by the simulation model (at measurement points);H*: Vector of the measured state variables;CH: Covariance matrix of the state variables at the measurement
points.
Result: the “optimum” unknown parameter vector (Z) is the one minimizing the opposite of the Likelihood Function Sp:
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
OSSERVAZIONIOSSERVAZIONI
MATRICE DI COVARIANZA
n
HH
limC
n
jii
niiH
1
2
La funzione di verosimiglianza è funzione dei parametri attraverso il vettore H.Il valore “ottimo” della funzione Sp può considerarsi come unica misura della vicinanza delle variabili misurate con quelle calcolate attraverso i parametri “ottimi”, per le correnti condizioni di esercizio.
varianza di Hi
n
HHHH
limC
n
jjjii
nijH
1 covarianza di Hi ed Hj
Ipotesi: la varianza è pari all’errore di misura medio dello strumento e la covarianza è nulla (misure incorrelate)
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Measurement error of the optimal parametersMeasurement error of the optimal parameters
An inferior limit is the Minimum Variance Bound (MVB):
1
iMVB = ii
F
2
1
2p
iji j
SF E
p p
F is the Fisher matrix, defined as follows:
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
-Sp
Sp
1p2p
measurement 2measurement 1
parameter
Measurement error of the optimal parametersMeasurement error of the optimal parameters
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Two parameter caseTwo parameter case
Parameter 1
Parameter 2
-Sp = constant
It can be shown that the axes of the ellipsis are proportional to the eigenvalues of the inverse of the Fisher matrix.
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Non-uniqueness caseNon-uniqueness case
It occurs when an axis of the ellipsis degenerates to infinity and one or more parameters are undetermined.
There is an infinite number of linear combinations of parameters 1 and 2 producing the same pressure heads h1 and h2, even if 3 is known and the number of equations is equal to the number of unknowns.
h1 h2
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
D-optimalityD-optimality
A common measure of the uncertainty of the estimated parameters is the determinant of the inverse of the Fisher matrix
The optimality criterion should be better defined according to the management quality criterion.
1
PN
ii
D
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
1
M M
t
N H N
F J C J
Sensitivity matrix1 1
1
1
P
M
M M
P
N
N
N N
N
H H
p p
H H
p p
J NM : number of measurementsNP : number of parameters
P P P M M M M PN N N N N N N N
The size of the element of F grows along with the number of measurements. Because of this the error decreases.
Effect of the measurement number Effect of the measurement number
First order approximation for the Fisher matrix:
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
If only one parameter for the roughness in the links is used, the sensitivity of the piezometric head at the downstream node is equal to the sum of the sensitivities of the same head with respect to each roughness coefficient.The size of the elements of the sensitivity matrix grows and the error becomes smaller when the number of parameters is reduced.
Effect of the parameter numberEffect of the parameter number Example: Example:
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
The Minimum Variance Bound can be viewed as a measure of the stability of the estimated parameters with respect to variations of the operating conditions.
The Likelihood Function is a measure of the ability of the parameters to reproduce the state variables at the present operating conditions.
-Sp MVB
NM =NP Number of measurement
Choice of the number of parameters Choice of the number of parameters
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
ZONAZIONEZONAZIONE
Il numero di parametri presenti nel modello di simulazione (centinaia o migliaia) è di gran lunga eccedente il valore ottimale.
Esempio: le scabrezze di ogni condotta, le domande ad ogni nodo.
Zonazione: scelta di un numero limitato di parametri di calcolo, legati ai parametri originali da relazioni note.
Esempio: una scabrezza unica per tutte le condotte di materiale e diametro uguale, una domanda ai nodi proporzionale al numero di utenze allacciate.
Attenzione: così facendo aumentiamo l’errore di modello !!
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
The number of parameters has to be chosen in order to obtain a compromise between
Choice of the number of parameters Choice of the number of parameters
the accuracy of prediction for the state variables at the present operating conditions
and the reliability of the estimated parameters to
be used in different operating conditions.
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
The Inverse problem is ‘sick’The Inverse problem is ‘sick’
• The location of the global maximum for the Likelihood Function is difficult to find;
• The estimated parameters are very sensitive to the errors in the measured state variables;
• The estimated parameters are very sensitive to the model errors.
The problem is said to be ill-posed
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
The Inverse problem is ‘sick’The Inverse problem is ‘sick’
What can we do?
• Choose the measurement location according to the highest sensitivity of the variable to the unknown parameters, or to the maximum reduction of the D value;
• Zone the parameters;
• Take different sets of measurements at different operating conditions (e.g. acting on the valves).
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Choice of the measurement point Choice of the measurement point
1. Evaluate the D value reduction for every available measurement point;
2. Choose the measurement point with the greatest D value reduction.
Is it possible to evaluate the D value before taking the measurement?
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
1
iMVB = ii
F
Choice of the measurement point Choice of the measurement point
1 1
1
1
P
M
M M
P
N
N
N N
N
H H
p p
H H
p p
J
1
M M
t
N H N
F J C JThe answer is yes:
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
iMj i
1 i i Nj=1
ij j i
H - H - Q - PS = 0 i=1,...,N
R H - HiM
ai i i ij ij ij
j=1
Q = (H - z ) D θ L2
o 2 i ii i o
H -z S = S sin
2 h
oi i S = S
i S = 0
i i o0< H -z < h
i i oH -z > h
i iH -z < 0
if
if
if
ijoij ij 5
ij
L R = R V +0.0826
D
2
ij
ij
= 4 log3.71 D
Valve regulation Valve regulation
L = 1000 m D = 0.50 m = 1.0E-04 m
23
tank
50.0m
10.0m
H = 10.0mH = 29.48mH = 48.96m
Parameters: P,
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Valve regulation Valve regulation
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Valve regulation Valve regulation
• The valve regulation can be optimized before taking the new measurement set:
iMj i
1 i i Nj=1
ij j i
H - H - Q - PS = 0 i=1,...,N
R H - H ijo
ij ij 5ij
L R = R V +0.0826
D
*, ,
21 *
,1
H
o H
Nt j t j
h Ntj
t ii
H HMaxOF Min w
H
R
t: set index
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Valve regulation Valve regulation
• The feasible domain of the above optimization problem is not convex: a discrete-variable global optimization algorithm should be used, such as Simulated Annealing or Genetic Algorithms.
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Calibration procedureCalibration procedure
Stop
True
Open valves
Measurement of water heads and
flow rates
Minimization of -Sp: estimation of parameter vector
Z
Start
t =1
Maximization of OF2 and
estimation of the optimal
resistances of the valves
Falset >1,
μ1tt ZZ
t = t +1
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Application to a field case Application to a field case Loss zone / roughness zone
S ource
• Oreto-Stazione area of Palermo water
distribution network
• 90 nodes
• 105 pipes
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Numerical validation Numerical validation
0,0 6,0 12,0 18,0 24,0
1
2
3
4
5
6
7zone
loss/load
final estimate
first estimate
"true"
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Sources of error Sources of error
Model errorComputational error
Measurement error
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
An other way An other way
Minimize the number of parameters while keeping satisfied some fixed constraint.
e.g., at the pressure measurement nodes:
* *j j j j jh h h R
R links the head loss in the pipe connecting nodes i and j to the flow rate in the link by means of the relationship 2
ij ij ij ijH L R q
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
This can be obtained by casting the minimization problem in the following form:
1
n
i i ii
Min C r r
r
s.t.
: decision variables, two for each link;
,i ir r
: positive coefficient, usually equal to 1;
iC
: fixed value of tolerance between computed and measured state variables;
j
: transformation matrix.I
An other way An other way
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
The transformation matrixThe transformation matrix
1
n
k i i kii
R r r I
1 24
83
7
5
611
9
12
10 9 1 1 3 3 6 6 9 9R r r r r r r r r
6 1 1 3 3 6 6R r r r r r r
If both the decision variables of link j are zero, the link j belongs to the same resistance zone of the upstream link k.
1 0 0 0 0 0 0 0 0 0 0 0
1 1 0 0 0 0 0 0 0 0 0 0
1 0 1 0 0 0 0 0 0 0 0 0
1 1 0 1 0 0 0 0 0 0 0 0
1 1 0 0 1 0 0 0 0 0 0 0
1 0 1 0 0 1 0 0 0 0 0 0
1 0 1 0 0 0 1 0 0 0 0 0
1 0 1 0 0 0 0 1 0 0 0 0
1 0 1 0 0 1 0 0 1 0 0 0
1 0 1 0 0 1 0 0 0 1 0 0
1 0 1 0 0 0 0 1 0 0 1 0
1 0 1 0 0 0 0 1 0 0 0 1
I
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Some remarks:
• In a not looped network the problem is linear;
•At the solution, most of the decision variables will be zero, depending on both the tolerance value and the number of measurements (i.e. constraints);
• At the optimal solution, either r+or r- will be zero for each link.
An other way An other way
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Test case Test case
1
3
4
5
67
8
9
10
1112
13
14
10
1
2
34
5
67
8
9
11
1213
14
15
12
5
Link label
Node label
2
14 links,
14 internal nodes with known topographical elevation and flow demand,
one source node, with known constant total head.
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Test case Test case
0
24
6
8
1012
14
0 2 4 6 8 10 12 14 16
Num
ber
of z
ones
= 5 m
= 10 m
= 1 m
= 0.1 m
2 64 8 10 120
Number of measurement points NM
14 16
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Test caseTest case
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
0 2 4 6 8 10 12 14 16
= 5 m
= 10 m
= 1 m
= 0.1 m
2 64 8 10 120
Number of measurement points NM
14 16
Para
met
er a
vera
ge u
ncer
tain
ty (
m-12
s4)
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Case of looped networksCase of looped networks
• In looped networks, several paths exist between each link and the source node;
• A single path is assigned to each link. The connections between different paths should be located in nodes where a resistance discontinuity is known to exist;
• The ensemble of all the single paths forms an open auxiliary network.
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Two step solution - First stepTwo step solution - First step
•A LP problem is solved assuming a first order approximation of the simulated heads as function of the decision variables and of their gradients computed for given flow distribution:
r,rMinF
s.t.
isi
mi HH i =1,…,NM,
minii RR i =1,…,NL,
sR(i)
sL(i) H H i =1,…,NC
0r,0r ii i =1,…,NL,
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Two step solution - Second stepTwo step solution - Second step
• The flow distribution corresponding to the optimal decision variables of the previous LP problem is used to update the gradients of the decision variables.
•If convergence is achieved in a feasible point, you can prove that this is a local minimum of the original problem.
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
• The measurements should be ordered according to their effect on the flux distribution.
Two step solutionTwo step solution
• The convergence is guaranteed if a first feasible point exists;
• The procedure can be applied iteratively, introducing the measurement constraints one after the other;
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Test case Test case
54
2 3
7
8
1
6
9 10
11 12 13 14
15 16 17
182 31
6 75
10 119
4
12
8
13
12
5
Link label
Node label
18 links,
12 internal nodes with known topographical elevation and flow demand,
one source node, with known constant total head.
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Test case Test case
0
2
4
6
8
10
12
14
16
Num
ber
of z
ones
= 1 m
= 5 m
= 0.5 m
= 0.1 m
2 64 8 10 120
Number of measurement points NM
14
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Test caseTest case
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
0 2 4 6 8 10 12 14
= 5 m
2 64 8 10 120
Number of measurement points NM
14
= 0.5 m
= 1 m
= 0.1 mPa
ram
eter
ave
rage
unc
erta
inty
(m-
12s4
)
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
From the pump
To the tank
a
b
c
d
e
f
h kig
j lm
Laboratory testLaboratory test
A looped network at the Laboratory of Hydraulics of the University of Catania, Italy. The valves have been regulated in order to obtain an equivalent scheme with two resistance zones:
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Measurement pointsMeasurement points
a
i
j
d gPressure transducers
Mercury differential manometers
Air differential manometers
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Resistance of the equivalent link Resistance of the equivalent link
Real scheme: two parallel real links
2 ,1jqL
HR
2
jj
inj
i n1
2
Equivalent scheme: a unique equivalent link
i nk 2k jq q2 4
jink
k k
RHR
L q
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Lab results Lab results
a
i
j
d g
R = 1615 m-6 s2
R = 5653 m-6 s2
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Lab resultsLab results
0
1000
2000
3000
4000
5000
6000
7000
0 20 40 60 80 100
links n.3,4,5,6,10
links n.7,9
links n.1,2,8
Res
ista
nce
valu
es in
link
s (m
-6s2
)
Number of iterations NIT
25 50 750 100
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Field case Field case
• 115 nodes • 148 links• High Density Polyethylene• One source point, with total
head of 54 m above the sea level• Three pressure meters, located at
nodes 1, 47 e 57• Lower bound of Resistances:
Colebrook-White formula, assuming a roughness coefficient of 0.01 millimeters
• 34 cutted links
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Field case Field case
1
10
100
• 22 resistance values • 4 macrozones:
from 1.45 to 1.50 s2m-6
from 15.00 to 20.00 s2m-6
from 43.00 to 45.00 s2m-6
from 60.00 to 63.00 s2m-6
• the results are in good agreement with the known diameter values
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Localizzazione ottimale delle misureLocalizzazione ottimale delle misure
in funzione delle future condizioni di esercizioin funzione delle future condizioni di esercizio
jp1,…,pp)
p
ii
i
jjj dp
pE
1
p
ii
i
jj pVar
pVar
1
2
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
1) Effettuare la calibrazione dai dati noti nelle condizioni di esercizio disponibili
2) Se uno o più degli autovalori dell’inverso della matrice di covarianza dei parametri sono nulli, ridurre il numero dei parametri e ricalcolare l’inverso della matrice
3) Calcolare la matrice di sensitività nei punti delle misure disponibili o possibili future
p
ξJ
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
4) Valutare l’incertezza nei punti critici delle future condizioni di esercizio
Attenzione: possiamo valutare le varianze dei parametri senza effettuare le misure con l’approssimazione del primo ordine della matrice di Fisher
5) Selezionare la misura che produce la massima riduzione dell’incertezza
p
ii
i
jj pVar
pVar
1
2
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
Conclusions Conclusions
•Calibration can not be sold, but is necessary to sell good results;
•A lot of uncertainty still exists on the optimal location, type and amount of measurements to be carried out for a model calibration.
Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Hydraulic Engineering and Environmental Applications, University of Palermo, Italy. Department of Mechanics and Materials, Department of Mechanics and Materials, MediterraneanMediterranean University of Reggio Calabria, Italy. University of Reggio Calabria, Italy.
MatMec
??!!Thank youThank you
Any questionAny question