UNIVERSITÀ DI PISA DIPARTIMENTO DI INGEGNERIA MECCANICA, NUCLEARE E DELLA PRODUZIONE VIA DIOTISALVI 2, PISA

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1 G r u p p o R I c e r c a N u c le a r e S a n P I e r o a G r a d o N u c l e a r a n d I n d u s r I a l E n g I n e e r I n g UNIVERSIÀ DI PISA DIPARIMENO DI INGEGNERIA MECCANICA NUCLEARE E DELLA PRODUZIONE VIA DIOISALVI PISA A Novel Approach for Inpu and Oupu Uncerainy Quanificaion in Nuclear Sysem- hermal-hydraulics A. Peruzzi A. Kovonyuk F. D Auria V&V 2013 ASME Verificaion and Validaion Symposium Plane Hollywood Resor Las Vegas Nevada May

2 CONENS 1. FOREWORD 2. A DEERMINISIC S&U MEHODS FOR BEPU ANALYSIS he Bases and he Srucure of he Mehod A Demonsraive Applicaion: he Blowdown Paradigm Problem he Deerminisic Sensiiviy Analysis Procedures FSAP ASAP he Uncerainy Evaluaion by he Daa Adusmen and Assimilaion DAA Main Resuls from he Demonsraive Applicaion 3. CONCLUSIONS 2/29

3 FOREWORD Hisorically deerminisic and saisical mehods for sensiiviy and uncerainy analysis have been developed separaely Each class of mehods have limiaions ha canno be overcome unless by furher developmens wihin heir respecive paradigms he srenghs of he deerminisic and saisical mehods can be combined o eliminae mos of heir respecive limiaions; his will lead o he developmen of new hybrid mehods 3/29

4 FOREWORD Sensiiviy and Uncerainy analysis are needed respecively o idenify he main sources of errors and o quanify he errors; Sensiiviy and uncerainy analysis procedures can be eiher local or global in scope he obecive of local analysis is o analyze he behavior of he sysem response locally around a chosen poin or raecory in he combined phase space of parameers and sae variables; he obecive of global analysis is o deermine all of he sysem's criical poins in he combined phase space formed by he parameers and dependen sae variables and subsequenly analyze hese criical poins by local sensiiviy and uncerainy analysis. 4/29

5 FOREWORD he mehods for local and global S&U analysis are based on eiher deerminisic or saisical procedures Saisical mehods for uncerainy and sensiiviy analysis firs commence wih he uncerainy analysis sage and only subsequenly proceed o he sensiiviy analysis sage; his pah is he exac reverse of he concepual pah underlying he deerminisic mehods for sensiiviy and uncerainy analysis where he sensiiviies are deermined prior o using hem for uncerainy analysis 5/29

6 CASUALIDAD MEHOD HE BASES AND HE SRUCURE OF HE MEHOD CASUALIDAD: Code wih he capabiliy of Adoin Sensiiviy and Uncerainy AnaLysis by Inernal Daa Adusmen and Assimilaion Fully Deerminisic Mehod based on: Advanced Sensiiviy ools ASAP/FSAP and GASAP for derivaion of he Local and Global Sensiiviy Marix S R i and for he evaluaion of he compuaional covariance Marix C R by he momenum propagaion equaion or sandwich rule; A Daabase of Experimens; A mehod DAA based on Bayesian Approach for updaing he a priori PDF of he inpu parameer and responses R wih he available experimen M likelihood observaion for geing a poseriori Improved Evaluaion IE of inpu parameer IE and responses R IE and respecive covariances; A Saus Approach for collecing and soring he a poseriori evaluaions. 6/29

7 CASUALIDAD MEHOD HE BASES AND HE SRUCURE OF HE MEHOD S E F I F P h e n o m e n a o rie n e d S y s e m o rie n e d B a y e s ia n M o d e l a n d M a x im u m lik e lih o o d C O D E S IM U L A IO N S & U o o ls R e d u c e d Im p ro v e d P a r a m e e r s R e s p o n s e s C o v a r ia n c e M a rix M o d e ls In p u P a r a m e e r s a n d C o v a r ia n c e M a rix S e n s iiv iy B a y e s ia n M o d e l a nd M a x im u m L ik e lih o o d C A S U A L ID A D 7/29

8 Driving Quaniies Developmen Phase: Derivaion of Improved Esimaion of and C IF calc. UMAE b a Inpu Deck d B IC SS CASUALIDAD MEHOD g Selecion of Responses B IC PP C O D E = { } SS o Experimenal Daa IF M C M e f Sandard Process Numerical ools Prior inpus PP N O D Prior Calculaions Poserior Calculaions i c ASAP FSAP DIREC s GASAP H-CODE CODE C O D E k Daabase S Y S Y B IC IE P P IE C O D E IE SS h R B I C PP C O D E C = { } l B IC IE P P IE C O D E IE C C C S S C S S C m C Covariance Calculaed Response C R = S C S q r RANSIEN IDENIFICAION & Similariy Check n p DAA Improved Esimaion IE R C R IE B IC IE P P IE C O D E IE SS B IC IE P P IE C O D E IE C C C S S 8/29

9 CASUALIDAD MEHOD RANSIEN IDENIFICAION e m p e r a u r e K Applicaion Phase: Use of Improved Esimaion of and C for NPP UA NPP calc. UMAE a k ASAP FSAP DIREC GASAP d m l S Y S Y n Covariance Calculaed Response C R IE = S C IE S o c Inpu Deck B IC SS R IE Selecion of Responses and Driving Quaniies C la d d in g e m p e ra u re s Low er Band U pper Band PP N O D b H-CODE CODE d e f g R i im e s C B IC S S IE C P P IE B IC IE P P IE C O D E IE SS h C CODE IE 9/29

10 Pressure bar P r e s s u r e b a r CASUALIDAD MEHOD Gas & Wall emperaure K em peraure K A DEMONSRAIVE APPLICAION: HE BLOWDOWN PARADIGM PROBLEM 1 P P k 1 h A W F k R N 0 S o u V n 1 n n n n n n n n n n n n n N S n n k h A W R n F n ou V P P w n 1 n 1 n 1 x n n 1 n 1 n N 0 0 n 1 2 h 0 x K w n 1 n 1 n n 2 n 1 w n 1 n 1 n n 1 n 2 J 2 J 1 J A e n v J J x x K w x w 1 J 1 h J n 2 he SIM-CODE Blowdown of gas fro a pressurized vessel aking ino accoun he vessel wall hea conducion Abou 300 inpu parameers including gas properies Validaion agains experimen performed a he Imperial College C a lc u la e d g a s - C a lc E x p e rim e n wall - C alc - Inernal Surface g a s : E x p - B a n d s w a ll: E x p - B a n d s im e s im e s 10/29

11 11/29 SENSIIVIY ANALYSIS PROCEDURES FSAP ASAP J J N N N N d d N d dp NL NL NL NL NL u u u u u NL ; ; R k P F W A P h k V N ou J S N ; ; P R P F P W A P h V k N ou J S N ; P h x K N N w x N w env J J A J J w J h x K N ; J HE MAHEMAICAL FORMULAION AND IMPLEMENAION IN FORRAN Derivaion of he Sensiiviy Marix S by: 1. HE FSAP Variaional angen Sysem or FSS 2. HE ASAP Variaional Adoin Sensiiviy Sysem or VASS CASUALIDAD MEHOD

12 CASUALIDAD MEHOD SENSIIVIY ANALYSIS PROCEDURES FSAP ASAP HE SOFWARE S IM -C O D E C a lc u la io n N L u +1 u = 0 S O L U IO N u R E S P O N S E R = R u D e r iv a iv e s a s e p N N N N ; ; ; P θ θ J N N N N ; ; ; P θ θ J h h h h ; ; ; P θ θ N N N A 0 J D e r iv a iv e s a s e p N N N N ; ; ; P θ θ N N N N ; ; ; P θ θ h h h h ; ; ; P θ θ N N h ; ; 0 J J N N N A 1 2 N i i i 0 J ; h A i D e r iv a iv e s o f h e R e s p o n s e s R u R ; D e r iv a iv e s o f F = f R = F P θ...θ ; d = 0 F i 0 J D D D S M F S A P A S A P L is o f P a ra m e e rs S E N S I IV I Y P R O F IL E S S 12/29

13 Pressure Pa W all em peraure # 1 K CASUALIDAD MEHOD G a s e m p e r a u r e K H C H i W /m 2.K SENSIIVIY ANALYSIS PROCEDURES FSAP ASAP HE COMPARISON OF SENSIIVIY RESULS 0.0E E E+04 BRUE - k=1% FSAP - k=1% ASAP - k=1% DIR - k=1% 1.0E E E +00 B R U E - k = 1 % F S A P - k = 1 % A S A P - k = 1 % D IR - k = 1 % Variaion of 1% of sysem parameer k Cp/Cv of gas E E E E+04 Gas Pressure -1.0E E +00 Gas emperaure -6.0E E -02 ime s 0.0E E E E -02 Wall emperaure #1-2.0E E E E E E -0 1 im e s B R U E - k = 1 % F S A P - k = 1 % A S A P - k = 1 % D IR - k = 1 % HC inernal side -8.0E E E E E E E E -01 B R U E - k = 1 % F S A P - k = 1 % 0.0 E A S A P - k = 1 % D IR - k = 1 % -2.0 E -0 1 im e s im e s 13/29

14 Pressure Pa W all em peraure # 1 K CASUALIDAD MEHOD Gas emperaure K H C H i W /m 2.K SENSIIVIY ANALYSIS PROCEDURES FSAP ASAP HE COMPARISON OF SENSIIVIY RESULS 1.2E E E+04 BRUE - Di=1% FSAP - Di=1% ASAP - Di=1% DIR - Di=1% 1.5E E E-01 Variaion of 1% of sysem parameer Di inernal vessel wall diameer 0.0E E E E+04 Gas Pressure -1.0E E E+00 BRUE - Di=1% FSAP - Di=1% ASAP - Di=1% DIR - Di=1% -2.5E E E+00 VERIFICAION PROCESS: -3.5E E E E -0 2 ime s OK 1.4 E B R U E - D i= 1 % 4.0 E -0 2 F S A P - D i= 1 % 1.2 E A S A P - D i= 1 % D IR - D i= 1 % 2.0 E E Gas emperaure ime s B R U E - D i= 1 % F S A P - D i= 1 % A S A P - D i= 1 % D IR - D i= 1 % 0.0 E E E E E -0 2 Wall emperaure #1 4.0 E E E E E -0 1 im e s 0.0 E E E E -0 1 HC inernal side im e s 14/29

15 D e r iv a iv e o f P r e s s u r e D P % /D a lp h a % CASUALIDAD MEHOD D e r iv a iv e o f e m p e r a u r e D % /D a lp h a % SENSIIVIY ANALYSIS PROCEDURES FSAP ASAP COMPARISON OF HE PERCENUAL VARIAION OF SYSEM RESPONSES RESPEC O PERCENUAL VARIAION OF PARAMEERS A S A P - D i A S A P - H A S A P - C d A S A P - c A S A P - d A S A P - k F S A P - D i F S A P - H F S A P - C d F S A P - c F S A P - d F S A P - k R: Gas Pressure Di E -01 A S A P - S A S A P - g A S A P - o w a ll-5 F S A P - S F S A P - g F S A P - o w a ll H 1.2E -01 R: Gas emperaure d 0.0 k -2.0 Cd E -01 c 8.0E -02 RELEVANCE FOR ACCURAELY 6.0E -02 o w a ll-5 RANKING HE MOS SENSIBLE 4.0E -02 PARAMEERS g E im e s S 0.0E E im e s 15/29

16 CASUALIDAD MEHOD SENSIIVIY ANALYSIS PROCEDURES FSAP ASAP SANDWICH RULE MOMEN PROPAGAION EQUAION Covariance of Inpu Parameer i a ime ν and Inpu Parameer a ime μ C i P D F i P D F S Sensiiviy Coefficien of Response R a ime C P D F ν respec wih he Inpu Parameer a ime μ P D F i i i S R i Inpu Covariance Marix C N 11 C C C S N 2 1 C C C S S N 1 N 2 N N N 1 C C C S Sensiiviy Marix N 11 C C C S N C C C S S 0 0 C S S S 11 N 1 N 2 N N N 1 N 2 N N C C C S S S 0 Compuaional Covariance of Responses: C R S C S C S C S S C S R * * * * 16/29

17 d P / d a [P a /a ] CASUALIDAD MEHOD d P / d a x s a [P a ] SENSIIVIY ANALYSIS PROCEDURES FSAP ASAP COMPUAIONAL UNCERAINY BANDS SANDWICH RULE AND CONRIBUIONS im e = 8 0 s 5.0 E im e = 9 5 s 3.0E E E E E +08 a = N μ _ 4 -P 3 a = N μ _ 4 -P 4 a = N μ _ 5 -P 3 a = N μ _ 5 -P E E E E a = R -1.0 E E +00 a = l W a = N μ _ 5 -P E E E E E E E E +08 = Ar a = N μ _ 4 -P 2 a = N μ _ 5 -P 2 im e = 8 0 s im e = 9 5 s -7.0 E E a = C d im e = 8 0 s im e = 9 5 s -3.0E E a = Ar -9.0 E P a r a m e e r # -3.0 E P i P a r a m e e r # P i R: Gas Pressure 17/29

18 d / d a [K /a ] CASUALIDAD MEHOD d P / d a x s a [P a ] SENSIIVIY ANALYSIS PROCEDURES FSAP ASAP COMPUAIONAL UNCERAINY BANDS SANDWICH RULE AND CONRIBUIONS 3.0 E a = A r im e = 8 0 s im e = 9 5 s 1.0 E E E E E a = R -2.0 E E E E E E E E E E a = λ W a = N μ _ 4 -P 2 a = N μ _ 5 -P 3 a = N μ _ 4 -P 3 a = N μ _ 5 -P E E a = C d im e = 8 0 s im e = 9 5 s -1.0 E E P a r a m e e r # P a r a m e e r # i i R: Gas emperaure 18/29

19 P r e s s u s e M P a CASUALIDAD MEHOD G a s e m p e r a u r e K SENSIIVIY ANALYSIS PROCEDURES FSAP ASAP COMPUAIONAL UNCERAINY BANDS SANDWICH RULE AND CONRIBUIONS L o w e r U n c e ra in y B a n d N o m in a l U p p e r U n c e ra in y B a n d C R S C S R: Gas Pressure L o w e r U n c e ra in y B a n d N o m in a l U p p e r U n c e ra in y B a n d im e s R: Gas emperaure im e s 19/29

20 P? 1? 2? 3? 4? 5? 6 CASUALIDAD MEHOD? 7? 8? 9? 10? 11 M F o u hn ha im e s SENSIIVIY ANALYSIS PROCEDURES FSAP ASAP COMPUAIONAL COVARIANCE MARIX EXAMPLE OF A DIAGONAL BLOCK MARIX OF HE COMPUED COVARIANCE RESPONSES EXAMPLE OF BLOCK MARIX CORRELAION BEWEEN SOME SYSEMS RESPONSES A DIFFEREN IME SEPS ha s hn F o u M? 11? 10? 9? 8? 7? 6? 5? 4? 3? 2? % P im e s 20/29

21 CASUALIDAD MEHOD HE UNCERAINY EVALUAION BY DAA ADJUSMEN AND ASSIMILAION DAA Prior Informaion R C C R New Informaion Experimen daa M C M Poserior Informaion R IE IE C IE C R IE 21/29

22 CASUALIDAD MEHOD HE UNCERAINY EVALUAION BY DAA ADJUSMEN AND ASSIMILAION DAA 22/29

23 CASUALIDAD MEHOD HE UNCERAINY EVALUAION BY DAA ADJUSMEN AND ASSIMILAION DAA = 23/29

24 CASUALIDAD MEHOD HE UNCERAINY EVALUAION BY DAA ADJUSMEN AND ASSIMILAION DAA = 24/29

25 CASUALIDAD MEHOD HE UNCERAINY EVALUAION BY DAA ADJUSMEN AND ASSIMILAION DAA = 25/29

26 CASUALIDAD MEHOD HE UNCERAINY EVALUAION BY DAA ADJUSMEN AND ASSIMILAION DAA HE SOFWARE S IM -C O D E C a lc u la io n N L u +1 u = 0 S O L U IO N u R E S P O N S E R = R u D D D S M F S A P A S A P L is o f P a r a m e e r s S E N S I IV I Y P R O F IL E S S In p u C o v a r ia n c e M a r ix C P ro p a g a io n o f E rro r C = S C S R E x p e r im e n a l m e a s u r e m e n M C M DAA IM P R O V E D E S IM A IO N R IE C IE R IE C IE 26/29

27 Pressure Pa CASUALIDAD MEHOD HE UNCERAINY EVALUAION BY DAA ADJUSMEN AND ASSIMILAION DAA HE IMPROVED ESIMAION RESULS R: Gas Pressure 3.0 CALC - LUB CALC - Reference CALC - UUB EXP Improved Esimae - LUB Improved Esimae - Reference Improved Esimae - UUB ime s 27/29

28 emperaure K CASUALIDAD MEHOD HE UNCERAINY EVALUAION BY DAA ADJUSMEN AND ASSIMILAION DAA HE IMPROVED ESIMAION RESULS CALC - LUB CALC - Reference CALC - UUB EXP Improved Esimae - LUB Improved Esimae - Reference Improved Esimae - UUB R: Gas emperaure ime s 28/29

29 CONCLUSIONS he CASUALIDAD mehodology is a comprehensive approach for uilizing quanified uncerainies arising from SEFs and IFs in he process of calibraing complex compuer models for he applicaion o NPP ransien scenarios. he model validaion and calibraion mehodology makes exensive use of he concep of sensiiviy analysis procedures FSAP and ASAP. he developed CASUALIDAD mehodology consiues a maor sep forward wih respec o he generally used exper udgmen and saisical mehods as permis: a o esablish he uncerainies of any parameer characerizing he sysem based on a fully mahemaical approach where he experimenal evidences play he maor role and b o calculae an improved esimae of he compued response and relaive improved i.e. reduced uncerainy 29/29