Calibration Techniques for Looped Water Distribution Systems

1
Calibration Techniques for Looped Water
Distribution Systems
First Annual Water Distribution Modeling
Symposium Perugia (Italy), May 2003
Tullio Tucciarelli1, Alessandra Bascià2
1
2
2
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.

3
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).

4
Simulation model
Provides the relationship between state
variables, parameters and control variables.
Example
5
Simulation model
Provides the relationship between state
variables, parameters and control variables.
Example
6
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
• . for the corresponding operating conditions, the
  computed values of the state variables have to be
  similar to the measured ones
• . 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.

7
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

Model 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
8
Computational error
For continuous problems, the optimization
algorithms can locate only local minima, but we
are looking for the global one!
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.
9
Dominio della funzione f(x1,x2)
convesso
non convesso
10
Funzione f(x)
11
Esempio di funzione obiettivo convessa con
dominio di ricerca convesso.
12
Esempio di funzione obiettivo convessa con
dominio di ricerca non convesso.
13
Esempio di funzione obiettivo non convessa con
dominio di ricerca convesso.
14
Minimization of the measurement error
(Maximum Likelihood Theory)
Hypothesis the measurement error has a normal
distribution, with known variance and zero mean.
Result the optimum unknown parameter vector
(Z) is the one minimizing the opposite of the
Likelihood Function Sp
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.
15
OSSERVAZIONI
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.
MATRICE DI COVARIANZA
varianza di Hi
covarianza di Hi ed Hj
Ipotesi la varianza è pari allerrore di misura
medio dello strumento e la covarianza è nulla
(misure incorrelate)
16
Measurement error of the optimal parameters
An inferior limit is the Minimum Variance Bound
(MVB)
F is the Fisher matrix, defined as follows
17
Measurement error of the optimal parameters
18
Two parameter case
It can be shown that the axes of the ellipsis are
proportional to the eigenvalues of the inverse of
the Fisher matrix.
19
Non-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 e1 and e2 producing
the same pressure heads h1 and h2, even if e3 is
known and the number of equations is equal to the
number of unknowns.
20
D-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.
21
Effect of the measurement number
First order approximation for the Fisher matrix
Sensitivity matrix
The size of the element of F grows along with the
number of measurements. Because of this the error
decreases.
22
Effect of the parameter number Example
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.
23
Choice of the number of parameters
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.
24
ZONAZIONE
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 lerrore di
modello !!
25
Choice of the number of parameters
The number of parameters has to be chosen in
order to obtain a compromise between
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.
26
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

27
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).

28
Choice of the measurement point

• . Evaluate the D value reduction for every
  available measurement point
• . Choose the measurement point with the greatest
  D value reduction.
• Is it possible to evaluate the D value before
  taking the measurement?

29
Choice of the measurement point
The answer is yes
30
Valve regulation
Parameters P, q
31
Valve regulation
32
Valve regulation

• The valve regulation can be optimized before
  taking the new measurement set

33
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.

34
Calibration procedure
Measurement of water heads and flow rates
Maximization of OF2 and estimation of the optimal
resistances of the valves
t 1
Minimization of -Sp estimation of parameter
vector Z
t t 1
35
Application to a field case

• Oreto-Stazione area of Palermo water
  distribution network
• 90 nodes
• 105 pipes

36
Numerical validation
37
Sources of error
Computational error
Model error
Measurement error
38
An other way
Minimize the number of parameters while keeping
satisfied some fixed constraint. e.g., at the
pressure measurement nodes
39
An other way
This can be obtained by casting the minimization
problem in the following form
40
The transformation matrix
If both the decision variables of link j are
zero, the link j belongs to the same resistance
zone of the upstream link k.
41
An other way

• 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 ror r- will be
  zero for each link.

42
Test case
14 links, 14 internal nodes with known
topographical elevation and flow demand, one
source node, with known constant total head.
43
Test case
44
Test case
45
Case 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.

46
Two 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

47
Two 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.

48
Two 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

• The measurements should be ordered according to
  their effect on the flux distribution.

49
Test case
18 links, 12 internal nodes with known
topographical elevation and flow demand, one
source node, with known constant total head.
50
Test case
51
Test case
52
Laboratory 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

53
Measurement points
Air differential manometers
Pressure transducers
Mercury differential manometers
54
Resistance of the equivalent link
Real scheme two parallel real links
Equivalent scheme a unique equivalent link
55
Lab results
R 1615 m-6 s2
R 5653 m-6 s2
56
Lab results
57
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

58
Field case

• 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

59
Localizzazione ottimale delle misure in
funzione delle future condizioni di esercizio
xjx(p1,,pp)
60
1) Effettuare la calibrazione dai dati noti nelle
condizioni di esercizio disponibili
2) Se uno o più degli autovalori dellinverso
della matrice di covarianza dei parametri sono
nulli, ridurre il numero dei parametri e
ricalcolare linverso della matrice
3) Calcolare la matrice di sensitività nei punti
delle misure disponibili o possibili future
61
4) Valutare lincertezza nei punti critici delle
future condizioni di esercizio
Attenzione possiamo valutare le varianze dei
parametri senza effettuare le misure con
lapprossimazione del primo ordine della matrice
di Fisher 5) Selezionare la misura che produce la
massima riduzione dellincertezza
62
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.

63
!
Thank you
?
Any question