department of hydraulic engineering and environmental applications, university of palermo, italy....

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.

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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


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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.


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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).


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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.


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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


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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.


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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


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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.


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Dominio della funzione f(x1,x2)

x2

x1

x2

x1

convesso non convesso


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Funzione f(x)

f(x) f(x)

convessa non convessa


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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


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Esempio di funzione obiettivo convessa con dominio di ricerca non convesso.

parametro 1

parametro 2

Minimo locale


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Esempio di funzione obiettivo non convessa con dominio di ricerca convesso.

5040

30

20

parametro 1

parametro 2

Minimo locale


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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:


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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)


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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:


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-Sp

Sp

1p2p

measurement 2measurement 1

parameter

Measurement error of the optimal parametersMeasurement error of the optimal parameters


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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.


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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


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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


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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:


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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:


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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


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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 !!


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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.


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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


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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).


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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?


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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:


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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,


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• 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


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• 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.


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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


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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


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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"


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Sources of error Sources of error

Model errorComputational error

Measurement error


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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


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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



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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


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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.



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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.


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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


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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


14 16

Para

met

er a

vera

ge u

ncer

tain

ty (

m-12

s4)


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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.


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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,


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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.


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• 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;


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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.


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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


14


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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


14

= 0.5 m

= 1 m

= 0.1 mPa

ram

eter

ave

rage

unc

erta

inty

(m-

12s4

)


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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:


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Measurement pointsMeasurement points

a

i

j

d gPressure transducers

Mercury differential manometers

Air differential manometers


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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


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Lab results Lab results

a

i

j

d g

R = 1615 m-6 s2

R = 5653 m-6 s2


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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


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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


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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


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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


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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


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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


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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.


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??!!Thank youThank you

Any questionAny question

department of hydraulic engineering and environmental applications, university of palermo, italy....

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