Process modelling and optimization aid FONTEIX Christian Professor of Chemical Engineering Polytechnical National Institute of Lorraine Chemical Engineering Sciences Laboratory
Jan 24, 2016
Process modelling and optimization aid
FONTEIX ChristianProfessor of Chemical Engineering
Polytechnical National Institute of Lorraine
Chemical Engineering Sciences Laboratory
Process modelling and optimization aid
Model validation and prediction error
FONTEIX ChristianProfessor of Chemical Engineering
Polytechnical National Institute of Lorraine
Chemical Engineering Sciences Laboratory
Model validation and prediction errorValidation tests
• Variance of replication error :
• Variance of validation error :
• Variance of identification error :
ˆ V Vj 1
nVj
ykj ˆ y j xkj ,ˆ 2
k1
nVj
ˆ V UBj
ˆ V Rj 1
nRj 1ykj
1
nRj
y pj
p1
n Rj
2
k1
n Rj
Model validation and prediction errorValidation tests
• Choice of experiments for parametric identification :By optimality criteria of experimental designWith additional experiments in order to have the total number of freedom degree > 3
• Choice of replication experiments :Different to identification experimentsMinimum of 4 measurements for each component
• Choice of validation experiments :Different to identification and replication experimentsAbout the number of identification experiments / 3
Model validation and prediction errorValidation tests
• Number of experiments for parametric identification :
nj for component j
• Number of replication experiments : nRj (component j)
• Number of validation experiments : nVj (component j)
• Measurement error modelling :
To calculate the variance of yj separately for each different operating condition
To plot the variance versus the average of yj (logarithmic) and see the slope of the curve
Model validation and prediction errorValidation tests
• Figure of variance versus average (logarithmic scales)
Average
Variance Multiplicative errors
Additive errors
Model validation and prediction errorValidation tests
• Fisher Snedecor test for identification - replication comparizon :
• Fisher Snedecor test for validation - replication comparizon :
1
F nRj 1,n j
nm
nm n m 1
ˆ V UBj
n j
nm
nm n m 1 ˆ V Rj
nRj 1
F
n j
nm
nm n m 1 ,nRj 1
1
F nRj 1,nVj
ˆ V Vj
nVj
ˆ V Rj
nRj 1
F nVj,nRj 1
Model validation and prediction errorValidation tests
• Fisher Snedecor test for validation - identification comparizon :
• If the 3 tests are true we cannot said that the model is not validated (we consider that the model is validated in default of)
1
F
n j
nm
nm n m 1 ,nVj
ˆ V Vj
nVj
ˆ V UBj
n j
nm
nm n m 1
F nVj ,n j
nm
nm n m 1
Model validation and prediction errorValidation tests
• Example : Modelling of polymer blend Young modulus
ratio Value Freedom Mini Maxi
Validation /Identification1.677 (6,11) 0.185 3.88
Replication/Identification1.467 (3,11) 0.07 4.63
Validation/ Replication1.144 (6,3) 0.1515 14.73
21
22
F
1 2,n n 0.05 2 11/ ,F n n 0.05 1 2,F n n
Model validation and prediction errorValidation tests
• Example : Modelling of polymer blend Young modulus
0
10
20
30
40
50
60
70
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Strain
Str
ess
(M
Pa)
Experiment number 21Simulation
0
5
10
15
20
25
30
35
40
45
50
0 0.1 0.2 0.3 0.4 0.5 0.6
Strain
Str
ess
(MP
a)
Experiment number 22Simulation
DNLR model for the prediction of the stress–strain responses of the blends
Model validation and prediction errorPrediction error determination
• Hypothesis : the prediction error of the model is mainly due to the estimation error on the parameters
• Case of static model : the prediction error is
˜ y ij ˆ y j x ij,ˆ ˆ y j x ij,
d˜ y ij ˆ y j x ij,
d Sij d
Variance ˜ y ij A ˜ y ij2 Sij
T ˆ A ˆ ˆ T
Sij
ˆ where A ˆ ˆ T
is the var iance matrix of
Model validation and prediction errorPrediction error determination
• The parameters variance matrix is estimated from the confidence domain determination by evolutionary algorithm (set of solutions)
• is the sensitivity (sensitivity of the prediction to the parameters values)
• Case of dynamic model : X is the state vector
Sij
0ˆˆ
ˆ,ˆ,ˆˆ
0
tatXX
dtuXfXd
Model validation and prediction errorPrediction error determination
• The truth is given by :
• A limited expansion give :
erroreltherepresentsd
ddtuXfdX
mod
,,
XXXerrorionlinearizattheisd
dddtuu
fdt
fdtX
X
fXd
dtuXfddtuuXXfXdXddX
XXX
~ˆ
~~~~
ˆ,ˆ,ˆ~ˆ,~ˆ,
~ˆ~ˆ
ˆˆˆ
Model validation and prediction errorPrediction error determination
• The propagation error model is :
• This one corresponds to the real propagation error :
euGSXFX ~~~~0
dddtuu
fdt
fedt
X
f
dtuGX
fdtS
X
fdtXF
X
fXd
deudGdSXdFXd
XXX
XXX
~~
~~~~
~~~~
ˆˆˆ
ˆˆ0
ˆ
0
Model validation and prediction errorPrediction error determination
• Finally the propagation error model become :
00
00
00
0
ˆ
ˆˆ
ˆˆ
ˆ
tate
tatG
tatS
tatIF
ddedtX
fde
dtu
fGdt
X
fdG
dtf
SdtX
fdS
FdtX
fdF
X
XX
XX
X
Model validation and prediction errorPrediction error determination
• F is the transition matrix
• S is the sensitivity matrix to parameters
• G is the sensitivity matrix to inputs
• e is a residual error
Model validation and prediction errorPrediction error determination
• Example : uncertainty propagation in a nuclear fuel cycle (electricity production plant)
Uranium
naturel
Uranium
minigenrichissementEnrichment
fabrication
combustible
Fuel
fabricationParc UOXREPUOX
retraitementReprocessing DéchetsWaste
UraniumUranium
PlutoniumPlutonium
Uranium
appauvri
Depleted
Uranium
fabrication
combustible
Fuel fabrication Parc UOX
REPMOX
Model validation and prediction errorPrediction error determination
• Complex model :1 000 000 equations
U23692
U23892 Pu238
94
Pu23994
Pu24094
Pu24194
Pu24294
Am24195
Np23793
Am24395 Cm243
96
Cm24496
Cm24596
n,
n,2n
n,
n,2n
n,
n,2n
n,
n,2n
n,
n,2n
n,
n,2n
n,
n,2n
n,
n,2n
n, + -
n,2n + -
n, + -
-
n, + -
n, + -
n, + -n, + ce
n,2n
Pseudo
n,2n
n,2n
n,2n
n,
n,2n
n,
n, + - + -
15 a
163j
U23592
Pseudo
Pseudo
Cm24296
Pseudo
Pseudo
Pseudo
nature
MOX fuel
Model validation and prediction errorPrediction error determination
• PWR UOX (3.2% in U235) :number = 47feeding =1/4
• PWR MOX (6% in Pu) :number = 7feeding =1/3
• Others common specifications :fuel mass = 100 tonsspecific power = 38 w/g
Model validation and prediction errorPrediction error determination
• Total plutonium quantity in circulation in the cycle and its associated uncertainty (%) :
200
400
600
800
1 000
0 10 20 30 40 50 60
Date (année)
Ma
ss
e d
e P
u d
an
s l
e c
yc
le (
ton
ne
)
0,0%
0,1%
0,2%
0,3%
0,4%
0,5%
0,6%
0,7%
0,8%
0,9%
1,0%
Inc
ert
itu
de
re
lati
ve
Masse de Pu
Incertitude relative sur le Pu
Model validation and prediction errorPrediction error determination
• Risk due to uncertainty on radioactive materials storage : undetectable misappropriation of plutonium or others radioactive materials (terrorism risk)
• The models used for uncertainty calculations seem well adapted to our fuel cycle code and to be a relative fast means of obtaining uncertainties