1. Process modelling and optimization aid FONTEIX Christian Professor of Chemical Engineering Polytechnical National Institute of Lorraine Chemical Engineering Sciences Laboratory

2. Process modelling and optimization aid Model validation and prediction error FONTEIX Christian Professor of Chemical Engineering Polytechnical National Institute of Lorraine Chemical Engineering Sciences Laboratory

3. Model validation and prediction error Validation tests Variance of replication error : Variance of validation error : Variance of identification error :

4. Model validation and prediction error Validation tests Choice of experiments for parametric identification : By optimality criteria of experimental design With additional experiments in order to have the total number of freedom degree > 3 Choice of replication experiments : Different to identification experiments Minimum of 4 measurements for each component Choice of validation experiments : Different to identification and replication experiments About the number of identification experiments / 3

5. Model validation and prediction error Validation tests Number of experiments for parametric identification : n j for component j Number of replication experiments : n Rj (component j) Number of validation experiments : n Vj (component j) Measurement error modelling : To calculate the variance of y j separately for each different operating condition To plot the variance versus the average of y j (logarithmic) and see the slope of the curve

6. Model validation and prediction error Validation tests Figure of variance versus average (logarithmic scales) Average Variance Multiplicative errors Additive errors

7. Model validation and prediction error Validation tests Fisher Snedecor test for identification - replication comparizon : Fisher Snedecor test for validation - replication comparizon :

8. Model validation and prediction error Validation 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)

9. Model validation and prediction error Validation tests Example : Modelling of polymer blend Young modulus ratioValueFreedomMiniMaxi Validation /Identification 1.677(6,11)0.1853.88 Replication/Identification 1.467(3,11)0.074.63 Validation/ Replication 1.144(6,3)0.151514.73

10. Model validation and prediction error Validation tests Example : Modelling of polymer blend Young modulus DNLR model for the prediction of the stress–strain responses of the blends

11. Model validation and prediction error Prediction 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

12. Model validation and prediction error Prediction 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

13. Model validation and prediction error Prediction error determination The truth is given by : A limited expansion give :

14. Model validation and prediction error Prediction error determination The propagation error model is : This one corresponds to the real propagation error :

15. Model validation and prediction error Prediction error determination Finally the propagation error model become :

16. Model validation and prediction error Prediction 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

17. Model validation and prediction error Prediction error determination Example : uncertainty propagation in a nuclear fuel cycle (electricity production plant) Uranium naturel Uranium minig enrichissement Enrichment fabrication combustible Fuel fabrication Parc UOX REP UOX retraitement Reprocessing DéchetsWaste Uranium Plutonium Uranium appauvri Depleted Uranium fabrication combustible Fuel fabrication Parc UOX REP MOX

18. Model validation and prediction error Prediction error determination Complex model : 1 000 000 equations U 236 92 U 238 92 Pu 238 94 Pu 239 94 Pu 240 94 Pu 241 94 Pu 242 94 Am 241 95 Np 237 93 Am 243 95 Cm 243 96 Cm 244 96 Cm 245 96 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,  + ce  n,2n Pseudo n,2n n,  n,2n n,  n,  +  - +  -  15 a  163j U 235 92 Pseudo Cm 242 96 Pseudo nature MOX fuel

19. Model validation and prediction error Prediction error determination PWR UOX (3.2% in U235) : number = 47 feeding =1/4 PWR MOX (6% in Pu) : number = 7 feeding =1/3 Others common specifications : fuel mass = 100 tons specific power = 38 w/g

20. Model validation and prediction error Prediction error determination Total plutonium quantity in circulation in the cycle and its associated uncertainty (%) :

21. Model validation and prediction error Prediction 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