Predogled

ISSN 0039-2480

UDC 519.2:621.039.58

Review scientific paper (1.02)

Dolo�anje odzivne povr�ine z optimalnim statisti�nim cenilnikom

Response Surface Generation with Optimal Statistical Estimator

Andrej Pro�ek - Borut Mavko

Na podro�ju jedrske tehnike se odzivna povr�ina uporablja za re�evanje problemov, povezanih z jedrsko varnostjo. Glavni namen te �tudije je bil zasnovati orodje za avtomatsko dolo�anje odzivne povr�ine zapletenih in nelinearnih pojavov ter ga preskusiti pri dolo�anju odzivne povr�ine za najvi�jo temperaturo sraj�ke med malo izlivno nezgodo v jedrski elektrarni.

Za dolo�anje odzivne povr�ine smo uporabili optimalni statisti�ni cenilnik, ki smo ga priredili za uporabo v ve�dimenzionalnem prostoru. V postopek smo vgradili dva statisti�na kazalca, ki povesta, kako to�no smo napovedali posamezne to�ke in mo�nost nastavljanja �irine Gaussove krivulje. Za preskus delovanja optimalnega statisti�nega cenilnika smo uporabili rezultate male izlivne nezgode 59 razli�nih primerov.

Uporaba optimalnega statisti�nega cenilca za dolo�anje odzivne povr�ine je pokazala vrsto prednosti pred regresijsko analizo, ki se ve�inoma uporablja v svetu. Rezultati so pokazali, da z optimalnim statisti�nim cenilcem lahko dovolj natan�no napovemo odzivno povr�ino za najvi�jo temperaturo sraj�ke, kar z regresijsko analizo ni bilo mogo�e. �e ve�, z avtomatiziranim dolo�anjem odzivne povr�ine se je odprla �iroka mo�nost uporabe optimalnega statisti�nega cenilnika za oceno negotovosti poljubne vrste �asovno odvisnih pojavov in nezgod.

In the field of nuclear engineering the response surface is used to solve some problems related to nuclear safety. The main purpose of the study was to develop a tool suitable for response surface generation of complex and non-linear phenomena and to demonstrate its applicability for the response surface generation of peak cladding temperature during a small-break loss-ofcoolant accident in a nuclear power plant.

The optimal statistical estimator, adapted for use in multi-dimensional space, was used for response surface generation. For assessing the adequacy and predictive capability of the optimal statistical estimator two statistics were built in, and the possibility to set the width of the Gaussian curve. The performance of the optimal statistical estimator was tested with the results from 59 different calculations of the small-break loss-of-coolant accident.

The application of the optimal statistical estimator shows several advantages when compared to the more commonly used regression analysis. The results showed that the response surface for the peak cladding temperature was adequately predicted by the optimal statistical estimator but not with regression analysis. Furthermore, an ability to automate the response surface generation provides the possibility of using the optimal statistical estimator for an uncertainty evaluation of any kind of time dependent phenomena and transients.

Na podro�ju jedrske tehnike se odzivna povr�ina uporablja za re�evanje razli�nih problemov, povezanih z jedrsko varnostjo. Odzivna povr�ina nadomesti termohidravli�ni ra�unalni�ki program, ko za statisti�no analizo potrebujemo na tiso�e izra�unov. Obi�ajno so odzivno povr�ino ra�unali za eno samo vrednost, npr. najvi�jo temperaturo sraj�ke

0 INTRODUCTION

In the field of nuclear engineering the re-sponse surface is used to solve various problems re-lated to nuclear safety. When thousands of complex computer code runs are needed for statistical analy-sis, the response surface is used to replace the computer code. Usually the response surface is generated for a single value parameter, for example: peak

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A. Pro�ek - B. Mavko: Dolo�anje odzivne povr�ine - Response Surface Generation

goriva, najni�ji nivo v sredici reaktorja ali najve�ji tlak sistema.

Uporaba odzivne povr�ine na podro�ju jedrske tehnike se v svetu zve�uje. �e leta 1989 so v ZDA za izra�un negotovosti najvi�je temperature sraj�ke med veliko izlivno nezgodo uporabili odzivno povr�ino �1]. Za dolo�anje odzivne povr�ine so uporabili regresijsko analizo (prilagajanje s polinomi). Leta 1990 so, ponovno v ZDA, uporabili odzivno povr�ino za dolo�itev nastavitvene to�ke, pri kateri se odpirajo varnostni ventili na sekundarni strani jedrske elektrarne. Za prilagajanje z regresijsko analizo so uporabili linearne in kvadratne �lene. Novo uporabo odzivne povr�ine je zaslediti v letu 1992 za veliko in malo izlivno nezgodo izlivno nezgodo (�2] do �4]). Leta 1996 je bila odzivna povr�ina uporabljena za dolo�itev negotovosti najvi�je temperature sraj�ke med veliko izlivno nezgodo �5]. Odzivno povr�ino uporablja za svoje prera�une negotovosti najvi�je temperature sraj�ke od leta 1996 dalje tudi podjetje Westinghouse �6].

�eprav je �e bilo opravljenega dosti dela, je vsem �tudijam skupno to, da lahko uporabijo odzivno povr�ino le za eno vrednost parametra, ne pa tudi za �asovni potek. Glavni namen na�ega dela je bil samodejno dolo�iti odzivno povr�ino za poljubne diskretne to�ke, ki popisujejo �asovni potek parametra. To je pomembno pri uporabi postopkov, ki za ocenjevanje negotovosti uporabljajo odzivno povr�ino, saj regresijska analiza ni uporabna za vse vrste parametrov in s tem scenarijev, za katere bi �eleli oceniti negotovost.

1 DOLO�ANJE ODZIVNE POVR�INE

Da bi lahko statisti�no sklepali o negotovosti rezultatov, sama analiza ob�utljivosti ni zadostna, ker je �tevilo izra�unov premajhno. Ker zaradi omejenih ra�unalni�kih zmogljivosti ni mogo�e izvesti ve� tiso� izra�unov, se iz dolo�enega �tevila izra�unov dolo�i odzivno povr�ino in s postopkom Monte Carlo naklju�no spreminja vhodne parametre. Na podlagi ve� deset tiso� vrednosti, dobljenimi z odzivno povr�ino, se da statisti�no sklepati o negotovosti rezultata. Do zdaj se je v svetu za dolo�anje odzivne povr�ine najpogosteje uporabljala regresijska metoda (�1] do �5]).

Regresijska metoda je primerna predvsem za opis pojavov, pri katerih je odvisnost prete�no linearna. Stopnja polinomov dolo�a �tevilo potrebnih prera�unov s termohidravli�nim programom, samo dolo�anje odzivne povr�ine je zamudno in neprimerno za ra�unalni�ko avtomatizacijo.

Zaradi na�tetih omejitev, predvsem nezmo�nosti obravnavanja mo�no nelinearnih pojavov, regresijska metoda npr. ni bila primerna za uporabo na mali izlivni nezgodi. Poskusili smo z linearno interpolacijo, ki je bila uporabna le za dve

cladding temperature, lowest reactor core level or peak system pressure.

The response surface is increasingly used in the field of nuclear engineering. In 1989 the response surface was used in the USA for the uncertainty evaluation of the peak cladding temperature �1]. Regression analysis (polynomial fit) was used for the response surface generation. In 1990, again in the USA, the re-sponse surface was developed for the determination of high-pressure setpoints for the safety valves on the secondary side of a nuclear power plant. Linear and quadratic terms were used for a polynomial fit. In 1992, other applications of the response surface were in a large-break and a small-break loss-of-coolant accident (�2] to �4]). In 1996, the response surface was used to evaluate uncertainties in the peak cladding temperature during a large-break loss-of-coolant accident. Westinghouse have also begun to use the response surface to evaluate uncertainties in the peak cladding temperature since 1996.

Although much work has already been done, in all the studies so far the response surfaces devel-oped could only be used for single-value parameters, with no possibility for continuous-valued parameters. The main purpose of this work was to automate the response surface generation for any set of discrete points, characterising the time trend of the parameter. This is very important for application of uncer-tainty methods, which use the response surface, be-cause regression analysis is not applicable to all kinds of parameters and scenaria to be evaluated for un-certainty.

1 RESPONSE SURFACE GENERATION

In order to statistically quantify the uncer-tainties, the number of calculations in the sensitivity analysis is too small. Because computer capabilities introduce a limit of the thousand calculations, the practice is to develop the response surface from se-lected calculations and with the Monte Carlo method, randomly select input parameters. The uncertainty is evaluated on the basis of ten thousand values, pre-dicted by the response surface. In the past, the re-sponse surface was generated mostly with regression analysis (�1] to �5]).

Regression analysis is most suitable for a description of parameters with mostly linear behav-iour. The order of the polynomial determines the number of calculations to be performed with the thermalhydraulic code, the response surface generation is time consuming and not adequate for computer automation.

Because of these limitations, especially the capability of the response surface to describe highly non-linear phenomena, regression analysis was not applicable to a small-break loss-of-coolant accident. Therefore, we tried with a linear interpolation method

A. Pro�ek - B. Mavko: Dolo�anje odzivne povr�ine - Response Surface Generation

neodvisni spremenljivki �7]. Linearna interpolacija torej ni re�itev za ve�je �tevilo parametrov.

Da bi lahko dolo�ili negotovost izra�una male izlivne nezgode, smo zato potrebovali novo orodje. Postopek optimalnega statisti�nega cenilnika, uporabljen �e leta 1992 �2] se je izkazal za primernega, �e ga posplo�imo za uporabo v ve�dimenzionalnem prostoru. Postopek optimalnega statisti�nega cenilnika namre� ni odvisen od �tevila izhodnih parametrov, uporablja zelo nelinearne funkcije in je preprost za ra�unalni�ko uporabo.

2 POSTOPEK OPTIMALNEGA STATISTI�NEGA CENILNIKA

Za dolo�itev odzivne povr�ine smo uporabili postopek optimalnega statisti�nega cenilca (OSC), ki je bil formuliran in uporabljen pri modeliranju ultrazvoka �8]. Na podro�ju jedrske tehnike smo postopek prvi� uporabili leta 1992 za primerjavo rezultatov regresijske analize. Uporabili smo osnovni niz ena�b iz �8]. V letu 1998 smo jih dodatno posplo�ili za ve�dimenzijski primer uporabe (�9] in �10]).

Odzivno povr�ino dolo�imo iz znanih izra�unanih ali merjenih vrednosti. Predpostavimo, da parameter, ki ga �elimo nadomestiti z odzivno povr�ino, lahko opi�emo s to�kami x, ki sestavljajo popoln vektor X=(x1, x2,...,x). Ta vektor lahko sestavimo iz dveh delnih vektorjev G=(x1, x2,...,xM; 0) in H=(0; xM+1, xM+2,...,x). Simbol 0 ozna�uje podatke, ki niso podani, I pa �tevilo vhodnih in izhodnih to�k. Popoln vektor lahko zapi�emo kot sestav:

using two independent input parameters �7]. However, the linear interpolation does not have a solution when we have more than two input parameters. The need for a new tool was identified for the uncertainty evaluation of small-break loss-of-cool-ant results. The optimal statistical estimator developed and applied in 1992 seems to be correct if adapted for use in the multidimensional space. Namely, the optimal statistical estimator is independent of the number of output parameters, it uses highly non-linear functions and is simple for a computer application.

2 OPTIMAL STATISTICAL ESTIMATOR METHOD

For the response surface generation the optimal statistical estimator (OSC) that was formulated and used in the modelling of ultrasonic data �8], was used. In the nuclear field the method was for the first time adopted in 1992 for the comparison of OSC with regression analysis. The basic equations derived in �8] were used. In 1998, the method was further improved for multi-dimensional space (�9] and �10]).

The response surface is predicted from the calculated or the measured values. In the following we assume that the parameter under consideration can be characterised by a finite number of data x, which are represented by the complete data vector X=(x1, x2,...,x). The complete data vector is often composed of two partial data vectors G=(x1, x2,...,xM; 0) and H=( 0; xM+1, xM+2,...,xI). Here the symbol 0 indicates the corresponding data, which are not specified, and I the number of input and output parameters. The complete data vector can then be expressed by a composition:

X = G®H = (x1,x2,...,xM,xM+1,x

..., xI )

V na�em primeru vektor G pomeni M vhodnih vrednosti (npr. vhodni negotovi parametri), H pa (I-M) izhodnih vrednosti (npr. najvi�jo temperaturo sraj�ke in najve�ji tlak sistema).

Za optimalni statisti�ni cenilnik HO so v �8] izpeljali izraz, ki predstavlja linearno kombinacijo izra�unanih ali izmerjenih vrednosti H in koeficientov C:

In our case, the vector G represents M input data points (for example, the values of input parameters) and H represents (I-M) output data points (for example, the peak cladding temperature and peak system pressure).

The optimal statistical estimator HO , derived in ref. �8], is expressed as a linear combination of given values H and coefficients C :

H0(G) = N CnHn

da(G -Gn)

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A. Pro�ek - B. Mavko: Dolo�anje odzivne povr�ine - Response Surface Generation

Koeficienti C so merilo podobnosti med danim vektorjem G in vektorjem vhodnih parametrov G za n-ti izra�un. Pri tem je pribli�ek funkcije d Gaussova funkcija:

The coefficient C represents a measure of the similarity between a given vector of input data G and the vector of input data G, for n-th calculation. The approxi-mation of the d function is the Gaussian function:

da ( G-Gn ) = n

2p ■ s

Gn=(xn1,xn2,...,xn

Sedaj se pojavi problem, kako najbolje izbrati �irino si. Glavni namen glajenja je raztegniti vpliv posameznih merjenih to�k v njihovo okolico, npr. do sosednjih to�k. �e �elimo z N vzorci pokriti prostor pribli�no enakomerno, potem moramo za si v i-ti dimenziji izbrati :

si= Si f,

kjer je Si razlika med najve�jo in najmanj�o vrednostjo iz mno�ice vhodnih podatkov xni (n=1, 2,...,N) za i-to dimenzijo, Ni pa je �tevilo korakov med to�kami v isti dimenziji. Faktor f je faktor za �irino Gaussove krivulje, ki ga izberemo na podlagi prej opravljenega testa optimalnega statisti�nega cenilnika. S faktorjem za �irino Gaussove krivulje se nastavi prispevek posameznih to�k h kon�nemu rezultatu.

Z uporabo izpeljanega optimalnega statisti�nega cenilnika H0 lahko kon�ni vektor zapi�emo kot:

The problem now appears as how best to select the width si. The main purpose of smoothing is to stretch the influence of a particular input data point into its surroundings, e.g., approximately to its neighbours. If we want to cover the volume by sam-ples uniformly, we define si for the dimension i:

i = 1,2,...,M

where Si is the distance between the minimum and maximum value of the set of input data points xni (n=1, 2,...,N) in the i-th dimension and Ni is the number of intervals between data points in the i-th dimension. The factor f, for the width of the Gaussian curve, is to be selected by the user, based on the desired and previously tested performance of the optimal statistical estimator. The contribution of each data point to the final output parameter estimation can be adjusted by this factor.

Using the derived optimal statistical estimator H0 , the complete estimated vector can be defined as:

y = (g®h0

Funkcijo Y(X) je treba modelirati z ra�unalni�kim programom. Na vektor Y vplivajo vhodne vrednosti, ki jih neposredno priredimo izhodu, medtem ko komplementarne vrednosti dolo�imo z optimalnim statisti�nim cenilnikom H0(G) . Ena izmed najpomembnej�ih lastnosti tega cenilca je, da vsebuje zelo nelinearne funkcije C. Dobljeni sistem je tako v bistvu nelinearen, �eprav je vektor H0 izra�en kot linearna kombinacija vrednosti H in koeficientov C .

Za natan�nost prileganja odzivne povr�ine izra�unanim ali izmerjenim to�kam smo uporabili srednji kvadratni pogre�ek za m-ti parameter:

The function Y(X) is modelled by the computer code. Vector Y is influenced by input parameters, directly transferred to the output, while the complementary values are determined by the optimal statistical estimator H0(G) . One of the most important characteristics of this estimator is that involves the highly non-linear function C .

n For assessing the adequacy and predictive capability of the optimal statistical estimator, the root mean square error for m-th parameter was used:

in koeficient dolo�itve R2:

Y(x -x )2

/ ,\xnm n=1

x avg,m )2

; m=(M+1), (M+2),...,I

and coefficient of determination R2: ; m=(M+1), (M+2),...,I

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A. Pro�ek - B. Mavko: Dolo�anje odzivne povr�ine - Response Surface Generation

Pri tem je xm n-ta, s programom izra�unana vrednost za m-ti izhodni parameter, xest m n-ta vrednost, ocenjena z optimalnim statisti�nim cenilnikom za m-ti izhodni parameter in xavgm povpre�na vrednost, s programom izra�unanih N to�k za m-ti izhodni parameter. Ujemanje med izra�unanimi vrednostmi in vrednostmi, ki jih napove optimalni statisti�ni cenilnik, je najbolj�e, ko se srednji kvadratni pogre�ek manj�a proti vrednosti RMS = 0 in ko se koeficient dolo�itve bli�a vrednosti '= 1.

Da bi Rm zra�unali vrednost izhodnih parametrov, vrednost vhodnih parametrov (x1,x2,...,xM) vsaki� naklju�no spreminjamo (ali jih vnesemo) in potem z optimalnim statisti�nim cenilnikom ocenimo izhodno vrednost z uporabo ena�b ((2) do (5)). Pri vsaki oceni izra�unamo nove vrednosti funkcij C, medtem ko so vrednosti H izra�unane to�ke.

�e primerjamo OSC z regresijsko analizo, vidimo, da OSC ni odvisen od �tevila podanih to�k za opis odzivne povr�ine, medtem ko je pri regresijski analizi to �tevilo to�k odvisno od reda polinoma. �e primerjamo naravo pojavov, potem je prednost OSC pred regresijsko analizo v tem, da z OSC lahko popi�emo nelinearne in zapletene funkcijske odvisnosti, z regresijsko analizo pa ne. Nenazadnje je OSC algoritem zelo primeren za ra�unalni�ko uporabo, pri regresijski analizi pa se obi�ajno uporabljajo statisti�ni paketi.

3 IZRA�UN NAJVI�JE

TEMPERATURE SRAJ�KE ZA MALO

IZLIVNO NEZGODO

Da bi pokazali delovanje postopka OSC, smo uporabili rezultate analize ob�utljivosti najvi�je temperature sraj�ke na izbrane vhodne parametre med malo izlivno nezgodo �11]. Za analizo male izlivne nezgode je bil uporabljen termohidravli�ni ra�unalni�ki program RELAP5/MOD3.2. Uporabljeni vhodni model za program RELAP5/ MOD3.2 je bil model dvozan�ne tla�novodne jedrske elektrarne. Ko smo si izbrali scenarij in elektrarno, smo identificirali pomembne pojave, ki vplivajo na najvi�jo temperaturo sraj�ke in jih razvrstili po pomembnosti za jedrsko varnost. Pojavom smo pripisali parametre, s katerimi se jih da popisati v ra�unalni�kem programu RELAP5. Izbrani parametri (normirani) so bili enofazni izto�ni koeficient (SDC), dvofazni izto�ni koeficient (TPDC), toplotna prestopnost (HTC), medfazno trenje (IDC) in cepitveni dele� zaostale toplote (FPYF). Te parametre smo v analizi ob�utljivosti spreminjali in dobili razli�ne izra�unane vrednosti najvi�je temperature sraj�ke (PCTRELAP5), prikazane v preglednici 1. Za te izra�unane najvi�je temperature sraj�ke smo dolo�ili odzivno povr�ino, ki smo jo potrebovali za statisti�no dolo�itev negotovosti z uporabo postopka Monte Carlo.

Here xnm is the n-th code calculated value of the m-th output parameter, xestn,m is n-th estimated value with the optimal statistical estimator and xavg,m is the mean of the N code calculated values of the m-th output parameter. The predictive capability of the optimal statistical estimator, assessing with the two proposed statistics, is perfect when RMSm=0 and Rm2 = 1.

To produce output results the values of the input parameters (x1,x2,...,xM) were randomly sampled (or input by the user) each time and then the corre-sponding unknown output values were estimated by the optimal statistical estimator using Eqs. ((2) to (5)). Each time a new coefficient Cn is calculated, while the values of Hn are calculated points obtained by computer code.

When comparing the OSC to the regression analysis, we see that for the regression analy-sis with the polynomial the amount of data is pre-scribed to describe the response surface while in the OSE more data mean more information that can be extracted. When comparing the nature of the phenomena, the advantage of OSC with respect to the regression analysis, is in its ability to predict very complex and highly non-linear functions. Fi-nally, the algorithm for OSC is suitable for computer automation, while for regression analysis statis-tical packages are used.

3 CALCULATION OF PEAK CLADDING

TEMPERATURE FOR SMALL-BREAK

LOSS-OF-COOLANT ACCIDENT

To demonstrate the OSC method, the results of the sensitivity analysis of the peak cladding temperature on selected input parameters during the small-break loss-of-coolant accident were used �11]. For the small-break loss-of-coolant accident analy-sis the RELAP5/MOD3.2 thermal-hydraulic computer code was used. The input model for the RELAP5/ MOD3.2 code was a model of a two-loop pressu-rised water reactor. After the scenario selection, all the important phenomena influencing on the peak cladding temperature were identified and ranked by their importance in nuclear safety. Then key RELAP5 code parameters were selected to represent the im-portant phenomena. The selected parameters, all normalised, were subcooled discharge coefficient (SDC), two-phase discharge coefficient (TPDC), heat transfer coefficient (HTC), interphase drag coefficient (IDC), and fission product yield factor (FPYF). In the sensitivity analysis these parameters were varied to obtain the corresponding calculated peak cladding temperatures (PCTRELAP5), which are shown in Table 1. For the calculated peak cladding temperatures the response surface was generated and sampled with the Monte Carlo method to generate an approximate distribution that characterises the uncertainty.

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A. Pro�ek - B. Mavko: Dolo�anje odzivne povr�ine - Response Surface Generation

Preglednica 1. Izra�unana in napovedana najvi�ja temperatura sraj�ke v odvisnosti od vhodnih parametrov Table 1. Calculated and predicted peak cladding temperature as a function of input parameters

A. Pro�ek - B. Mavko: Dolo�anje odzivne povr�ine - Response Surface Generation

4 REZULTATI DOLO�ANJA ODZIVNE

POVR�INE ZA NAJVI�JO TEMPERATURO

Preglednica 1 ka�e izra�unane vrednosti najvi�je temperature sraj�ke s programom RELAP5/ MOD3.2 (PCT AP5) in napovedane vrednosti najvi�je temperature sraj�ke z optimalnim statisti�nim cenilnikom (PCTOSC) v odvisnosti od petih vhodnih parametrov (SDC, TPDC, HTC, IDC, FPYF) 59 primerov. V na�em primeru vhodni vektor G (n = 1, 2,...,59) sestavlja pet vhodnih parametrov, n izhodni vektor (H, n = 1, 2,...,59) pa ena komponenta, tj. najvi�ja temperatura sraj�ke PCTRELAP. Faktor �irine za Gaussovo krivuljo f se izbere po kriteriju, da vpliv izra�unanih to�k raz�irimo do npr. sosednje to�ke in da je pri tem to�nost prilagajanja �e zadosti velika (R2 > 0,95). V na�em primeru je za izbrani f = 0,25 imel srednji kvadratni pogre�ek vrednost 1,55 K in koeficient dolo�itve R2 vrednost 0,97. Z vstavitvijo vrednosti H, izra�unanih s programom RELAP5/ MOD3.2, v ena�bo (2) lahko napovemo najvi�je temperature sraj�ke za poljubno kombinacijo vhodnih parametrov G znotraj podanih meja, �e koeficiente C izra�unamo po ena�bi (3). Za nove izbrane vrednosti vhodnih parametrov G se dolo�ijo novi koeficienti C .

Naslednje vpra�anje, ki se nam zastavi je n. kak�ne so vrednosti odzivne povr�ine v to�kah, za katere nimamo podanih izra�unanih vrednosti? Ker je obravnavana odzivna povr�ina petdimenzionalna, ni mogo�a grafi�na predstavitev na eni sliki. Slike od 1 do 5 ka�ejo krivulje, pri katerih v osnovnem primeru spreminjamo samo en parameter. S kro�cem so ozna�ene to�ke, izra�unane s programom RELAP5/ MOD3.2, ki smo jih med seboj povezali s �rtkano pik�asto linijo. Slike ka�ejo tudi vpliv faktorja za �irino Gaussove krivulje na odzivno povr�ino, dobljeno z OSC. Na slikah je prikazana tudi krivulja, dobljena z regresijsko analizo, ki je ozna�ena z �regr.�.

Za dolo�itev negotovosti najvi�je temperature sraj�ke smo naklju�no spreminjali vhodne podatke in z OSC izra�unali najvi�je temperature sraj�ke. Ta postopek smo ponovili 100000 krat, in za rezultat dobili, da obstaja 95% verjetnost, da najvi�ja temperatura sraj�ke ne bo presegla 1157 K, srednja vrednost zna�a 1085 K, medtem ko je negotovost razlika med 95. odstotkom in srednjo vrednostjo, in zna�a 72 K. Srednja vrednost najvi�je temperature sraj�ke z njej pripadajo�o negotovostjo je precej pod dopustno mejo 1478 K.

5 RAZPRAVA

Pravih vrednosti najvi�jih temperatur sraj�k med izra�unanimi to�kami ne poznamo. En mo�en pribli�ek je linearna odvisnost. OSC kot pribli�ek potegne krivuljo s prevojem, gledano v eni izmeri. Manj�i ko je faktor f, bolj stopni�ast je prevoj. �elimo

4 RESULTS OF RESPONSE SURFACE DEVELOPMENT FOR PEAK CLADDING TEMPERATURE

Table 1 shows the values of the calculated peak cladding temperatures (PCTRELAP5) and the pre-dicted peak cladding temperatures (PCTOSC) by OSC as a function of five input parameters for 59 cases. In our case, the input vector Gn (n = 1, 2,...,59) are five input parameters and the output vector Hn (n = 1, 2,...,59) has only one value, i.e. PCTRELAP5. The factor for the width of the Gaussian curve (f) is se-lected on the basis of criteria to stretch the influ-ence of a particular input data point into its sur-roundings, e.g. approximately to its neighbours and that the accuracy of fit is adequate (R2 > 0.95). In our case the root mean square error was equal to 1.55 K and the coefficient of determination was 0.97 for f = 0.25. By inserting the values for Hn, which were calculated with RELAP5/MOD3.2, into eq. (2) we can predict the peak cladding temperatures for any combination of input parameters G within parameter boundaries, if coefficients Cn are calcu-lated by eq. (3).

The next question is, what are the values of the response surface in the points where the cal-culated values are not given? Because the con-structed response surface is five dimensional, visual presentation for all dimensions is not possible. Fig-ures 1 to 5 show the response surface with one parameter varied in base case calculation. The points calculated by the RELAP5/MOD3.2 code are marked with circles, connected with dash-dotted lines. In the figures is shown the influence of the factor for the width of the Gaussian curve on the response surface predicted by OSC. In the figures is also shown the regression curve which is labelled with �regr.�.

To evaluate uncertainties in the peak clad-ding temperatures, input parameters were randomly selected by the Monte Carlo method and the peak cladding temperatures were estimated by OSC. This procedure was repeated 100000 times, and the result indicates that there is the 95% probability that the peak cladding temperature will not exceed 1157 K. The mean value is expected to be 1085 K, and the uncertainty of PCT is the difference between 95% and mean value, which is 72 K. The peak cladding temperature with its uncertainty is well above the cri-terion, 1478 K.

5 DISCUSSION

The peak cladding temperatures between the calculated points are unknown values. One possible fit is a linear dependence. The OSC fit is an inflected curve in the one-dimensional case. The smaller is the factor f, the closer is the inflected curve to the

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A. Pro�ek - B. Mavko: Dolo�anje odzivne povr�ine - Response Surface Generation

si sicer �imbolj blage prevoje, vendar je to odvisno od narave pojavov. Te�e, ko jih je popisati, manj�i faktor za �irino Gaussove krivulje moramo uporabiti, da �e dobimo zadovoljivo ujemanje v podanih izra�unanih to�kah. Posebej bi �eleli opozoriti, da so pribli�ki na slikah 1 do 5 rezultat petdimenzionalne odzivne povr�ine. Ker odzivno povr�ino potrebujemo za integracijo Monte Carlo, za napoved verjetnosti zadostuje optimalni statisti�ni cenilnik s stopni�asto funkcijo.

1160 r---------------------------------

step function. We wanted smooth curves but smooth-ness is dependent on the nature of the phenomenon. The more complex is the phenomenon, the smaller is the factor for the width of Gaussian curve, which is needed to adequately fit the calculated points. It is worth noting that the fits in Figures 1 to 5 are the result of a five-dimensional response surface. Because the response surface is needed for the Monte Carlo integration, the robust response surface with step function transitions still satisfies for the probability evaluation.

1140 1120 1100 1080 1060

Sl. 1. Najvi�ja temperatura sraj�ke v odvisnosti od enofaznega izto�nega koeficienta Fig. 1. Peak cladding temperature as a function of subcooled discharge coefficient

1200 1160 1120

1080 1040 1000

Sl. 2. Najvi�ja temperatura sraj�ke v odvisnosti od dvofaznega izto�nega koeficienta Fig. 2. Peak cladding temperature as a function of two phase discharge coefficient

1120 1110 1100 1090 1080 1070

Sl. 3. Najvi�ja temperatura sraj�ke v odvisnosti od toplotne prestopnosti Fig. 3. Peak cladding temperature as a function of heat transfer coefficient

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A. Pro�ek - B. Mavko: Dolo�anje odzivne povr�ine - Response Surface Generation

1081 1079 1077 1075 1073 1071 1069 1067 1065

Sl. 4. Najvi�ja temperatura sraj�ke v odvisnosti od medfaznega trenja Fig. 4. Peak cladding temperature as a function of interphase drag coefficient

Sl. 5. Najvi�ja temperatura sraj�ke v odvisnosti od cepitvenega dele�a zaostale toplote Fig. 5. Peak cladding temperature as a function of fission product yield factor

V na�em primeru optimalni statisti�ni cenilnik skupaj s postopkom Monte Carlo uporabimo za dolo�anje negotovosti termohidravli�nih ra�unalni�kih programov. Dobra lastnost OSC je, da je funkcija med dvema poznanima to�kama monotona. To tudi pomeni, da bodo vse vrednosti le�ale med najvi�jo in najni�jo izra�unano vrednostjo s programom RELAP5/MOD3.2. Druga dobra lastnost je, da se da dolo�anje odzivne povr�ine avtomatizirati. Za dolo�itev odzivne povr�ine ni predpisano �tevilo potrebnih to�k. Ve� to�k ko imamo, ve�je je zaupanje v rezultate. S primerno izbiro faktorja za �irino Gaussove krivulje lahko dose�emo �eleno to�nost ujemanja napovedanih to�k s podanimi izra�unanimi to�kami za �e tako zapletene odvisnosti. Z zo�evanjem �irine Gaussove krivulje so prehodi med podanimi izra�unanimi to�kami vedno bolj stopni�asti, zato optimalni statisti�ni cenilnik deluje bolj grobo. V takem primeru je priporo�ljivo pove�ati �tevilo izra�unanih to�k.

Prirejeni optimalni statisti�ni cenilnik za potrebe ra�unanja negotovosti termohidravli�nih ra�unalni�kih

In our case, the optimal statistical estimator with the Monte Carlo method is used for the uncer-tainty evaluation of thermal-hydraulic computer codes. One good characteristic of the OSC is that between the two code-calculated values, the function is monotonic. This also means that all the values pre-dicted by OSC will be between the minimum and maximum RELAP5/MOD3.2 code calculated value. A second valuable characteristic of the OSC is that the response surface generation can be automated. The number of calculated points is not prescribed, but the higher the number of calculated points, the higher the confidence level. With the proper selection of the width of the Gaussian curve, the desired accuracy of fit in the calculated values can be obtained for very com-plex phenomena. By decreasing the width of the Gaussian curve the transitions between the points are increasingly stepwise, as a result, the response sur-face performance is crude. In such cases, it is recom-mended that more calculated points are provided.

6 CONCLUSION

The adapted optimal statistical estimator for the uncertainty evaluation of the thermal-hydraulic

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A. Pro�ek - B. Mavko: Dolo�anje odzivne povr�ine - Response Surface Generation

programov je pokazal, da se da dolo�iti odzivno povr�ino za pojave in procese z zapleteno in nelinearno odvisnostjo. �e ve�, dolo�anje odzivne povr�ine z OSC se da z ra�unalni�kim programom avtomatizirati.

Avtomatiziran postopek dolo�anja odzivne povr�ine zelo raz�iri podro�je uporabe metod za dolo�anje negotovosti prera�unov s termo-hidravli�nimi programi od izlivnih nezgod na poljubne nezgode, kar do zdaj ni bilo mogo�e.

codes showed that the response surface can be de-veloped for complex and non-linear phenomena and processes. Furthermore, the response surface generation can be automated.

The automated procedure for response sur-face generation extends the use of uncertainty evaluation methods for thermal-hydraulic codes from a loss-of-coolant accident for which the uncertainty was evaluated for any accident.

7 LITERATURA 7 REFERENCES

�I] Boyack, B.E et al. (1990) Quantifying reactor safety margin parts 1 to 6. Nuclear engineering and design, Amsterdam, 119, 1-117.

�2] Pro�ek, A., B. Mavko, A. Stritar (1992) Ocena velike izlivne nezgode v jedrski elektrarni z analizo negotovosti. Zbornik Kuhljevi dnevi �92, Slovensko dru�tvo za mehaniko, Ljubljana, 178-185.

�3] Mavko, B., A. Stritar, A. Pro�ek (1993) Aplication of code scaling, applicability and uncertainty methodology to large break LOCA analysis of two-loop PWE. Nuclear engineering and design, Amsterdam, 143, 95-109.

�4] Ortiz, M.G., L.S. Ghan (1992) Uncertainty analysis of minimum vessel liquid inventory during a small-break LOCA in a B&W plant - an application of the CSAU methodology using the RELAP5/MOD3 computer code. NUREG/CR-5818, EGG.2665, Idaho National Laboratory.

�5] Haskin, E.F., Bevan, B.D., C. Ding (1996) Efficient uncertainty analyses using fast probability integration. Nuclear engineering and design, Amsterdam, 166, 225-248.

�6] Rombouts, D., Denil, D., Simon, C, C Matthys (1998) Westinghouse advanced safety analysis technology for plant power upratings. Proc. of nuclear energy in central Europe �98, Nuclear Society of Slovenia, Ljubljana, 439-444.

�7] Mavko, B., A. Pro�ek (1997) Peak cladding temperature response surface generation based on simulations of a small-break loss-of-coolant accident scenario. Proc. of 4th regional meeting: Nuclear energy in central Europe, Nuclear Society of Slovenia, Ljubljana, 605-612.

�8] Grabec, I., W. Sachse (1991) Automatic modeling of physical phenomena: Applicaton to ultrasonic data. J.Appl. Phys., 69(9), 6233-6244.

�9] Pro�ek, A. (1998) Ocena negotovosti realisti�nih simulacij potekov nezgod v jedrskih elektrarnah. Doktorska disertacija, univerza v Ljubljani, Fakultete za matematiko in fiziko, Ljubljana.

�10] Pro�ek, A., B. Mavko (1999) Evaluating Code Uncertainty - II: An optimal statistical estimator method to evaluate the uncertainties of calculated time trends. Nuclear Technology, La Grange Park, Vol. 126, 186-195.

�II] Pro�ek, A., B. Mavko (1999) Evaluating Code Uncertainty - I: Using the CSAU method for uncertainty analysis of a two-loop PWR SBLOCA, Nuclear Technology, La Grange Park, Vol. 126, 170-185.

Naslov avtorjev: dr. Andrej Pro�ek

profdr. Borut Mavko In�titut Jo�ef Stefan Jamova 39 1000 Ljubljana

Authors� Address: Dr. Andrej Pro�ek

Prof.Dr. Borut Mavko Jo�ef Stefan Institute Jamova 39 1000 Ljubljana, Slovenia

Prejeto: Received:

21.4.1999

Sprejeto: Accepted:

29.2.2000

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