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