Process modelling and optimization aid

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 aidModel validation and prediction error FONTEIX Christian Professor of Chemical Engineering Polytechnical National Institute of Lorraine Chemical Engineering Sciences Laboratory

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

4. Model validation and prediction errorValidation 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 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

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

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

8. 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)

9. Model validation and prediction errorValidation tests • Example : Modelling of polymer blend Young modulus

10. Model validation and prediction errorValidation 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 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

12. 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

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

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

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

16. 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

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

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

19. Model validation and prediction errorPrediction 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. 1 000 1,0% 0,9% Masse de Pu Incertitude relative sur le Pu 0,8% 800 0,7% 0,6% Masse de Pu dans le cycle (tonne) Incertitude relative 600 0,5% 0,4% 0,3% 400 0,2% 0,1% 200 0,0% 0 10 20 30 40 50 60 Date (année) Model validation and prediction errorPrediction error determination • Total plutonium quantity in circulation in the cycle and its associated uncertainty (%) :

21. 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