lable at ScienceDirect

Energy 35 (2010) 1799–1804

Contents lists avai

Energy

journal homepage: www.elsevier .com/locate/energy

Safety assessment for the passive system of the nuclear power plants(NPPs) using safety margin estimation

Tae-Ho Woo*, Un-Chul LeeDepartment of Nuclear Engineering, Seoul National University, Gwanak 599, Gwanak-ro, Gwanak-gu, Seoul 151-742, Republic of Korea

a r t i c l e i n f o

Article history:Received 31 May 2009Received in revised form20 December 2009Accepted 22 December 2009Available online 13 January 2010

Keywords:Probabilistic safety assessmentVery high temperature reactorPassive systemAnticipated transient without scram

* Corresponding author. Tel.: þ82 2 880 8337; fax:E-mail address: thw@snu.ac.kr (T.-H. Woo).

0360-5442/$ – see front matter � 2009 Elsevier Ltd.doi:10.1016/j.energy.2009.12.034

a b s t r a c t

The probabilistic safety assessment (PSA) for gas-cooled nuclear power plants has been investigatedwhere the operational data are deficient, because there is not any commercial gas-cooled nuclear powerplant. Therefore, it is necessary to use the statistical data for the basic event constructions. Severalestimations for the safety margin are introduced for the quantification of the failure frequency in thebasic event, which is made by the concept of the impact and affordability. Trend of probability of failure(TPF) and fuzzy converter (FC) are introduced using the safety margin, which shows the simplified andeasy configurations for the event characteristics. The mass flow rate in the natural circulation is studiedfor the modeling. The potential energy in the gravity, the temperature and pressure in the heatconduction, and the heat transfer rate in the internal stored energy are also investigated. The values inthe probability set are compared with those of the fuzzy set modeling. Non-linearity of the safety marginis expressed by the fuzziness of the membership function. This artificial intelligence analysis of the fuzzyset could enhance the reliability of the system comparing to the probabilistic analysis.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The very high temperature reactor (VHTR) type of the gas-coolednuclear power plant has been developed for the commercialpurpose. However, there is not the generic data for the probabilisticsafety assessment (PSA) in the passive system like the naturalcirculation of the long-term cooling. Although the commercialreactor is proposed as the gas-turbine-modular helium reactor (GT-MHR) of the General Atomics for VHTR, it is still under construction.The decay heat removal is done as the natural circulation in thedesign basis accident (DBA). The particular characteristics of thedecay heat removal in VHTR is that the ability of the heat transfer isonly related to the fuel temperature, which is different from thecommercialized nuclear power plants (NPPs) of the pressurizedwater reactor (PWR). So, it is important that the statistical simulationis effective to construct the safety assessment in the passive system.The design for the nuclear fuel utilizes the TRIstructural-ISOtropic(TRISO) fuel used in present high temperature gas-cooled reactordesigns [1]. It also utilizes the concept of the passive safety as used inseveral previous designs [2]. There are some references regardingthe VHTR [3–6].

þ82 2 889 2688.

All rights reserved.

The safety margin is described by the impact-affordability algo-rithm. The impact means the load of the event by the interestedsystem and the affordability is the capacity of the event. Previously,there was a similar concept [7], which had no versatile comparisonsfor the interested component of the passive system. This paper wouldlike to show how to treat the safety margin. The anticipated transientwithout scram (ATWS) of the VHTR is a model for the passive systemusing the impact-affordability method. The data of DBA are from theGT-MHR of the General Atomics. The specification of the GT-MHR isgiven (Table 1). The DBA was constructed by the Korea Atomic EnergyResearch Institute (KAERI) and the Idaho National Laboratory (INL),which is based on the license procedures of the Fort Saint Vrain (FSV)Reactor [8]. The passive system is the major characteristics of theVHTR instead of the LWR. Especially, the natural circulation is thevery important system in the VHTR, because there is no active pumpin the safety system. Although the passive system exists in the LWR,the key issue of this paper focuses on the VHTR.

Using the statistical analysis, there are several kinds of the resultevaluations suggested which are used in the fuzzy set theory. This isdifferent from the conventional probabilistic analysis. The ‘repre-sentative value method’, ‘interval value method’, and ‘center ofgravity method’ are introduced for the evaluations of the resultusing the fuzzy set theory.

Section 2 explains the algorithm of the study. The calculation forthe study is described in Section 3. Section 4 makes the results ofthe study. There are some conclusions in Section 5.

Table 2Procedures of safety assessment.

Procedures Contents

1. Basic dataconstruction

� Safety margin construction using Trend ofProbability of Failure (TPF)� Membership number construction using

Fuzzy Converter (FC)2. Tree � Event/fault tree construction3. Propagation � Data quantification4. Analysis � Difference error analysis

� Uncertainty analysis using fuzzyconfidence interval

Nomenclature

x variableI(x) impact functionA(x) affordability functionsm(x) safety margin sm(x) ¼ I(x) – A(x)Nn nominal valueNI actual valuemA,mI mean of NA,NI

sA2,sI

2 variance of NA,NI

T.-H. Woo, U.-C. Lee / Energy 35 (2010) 1799–18041800

2. Method

The procedure of the study is done by the impact-afford-ability algorithm for the ATWS in VHTR (Table 2). There is therelationship between impact and affordability in the case of thenormal distribution (Fig. 1) and the fuzzy distribution (Fig. 2).The distance between two graphs in each case shows the safetymargin. This function can show the distributions of mass flowrate, potential energy, temperature, pressure, and hear transferrate in passive system. The impact is the load of the event. Theaffordability means the capacity of the event. Namely, theaffordability can show the maximum margin of the event. As it isexplained, the difference between affordability and impact is thesafety margin of the event. The longer difference has the highersafety margin. The probabilities of failure of several variables(Fig. 3) [9] are shown, which are modified from the case of themass flow rate. The main object of this paper is the comparisonof safety margin between the probability function of the prob-ability set and the membership function of the fuzzy set.Therefore, the normal distribution is exampled as a function ofthe probability set. There is not any special reason whya symmetric probability distribution with thin tails is moreappropriate than a skewed distribution with fat tails (Fig. 1). Thegeneral comparison between the probability function of theprobability set and the membership function of the fuzzy set isdiscussed.

3. Calculation

The simulation for the PSA in the natural circulation is per-formed using linear and non-linear statistical data. The linearprobabilistic distribution and the non-probabilistic fuzzy distribu-tion are used. In the probabilistic distribution, the normal distri-bution is considered [10]. Otherwise, the fuzzy set distribution isconsidered as the membership algorithm [11], which is modified inthis paper for much more reasonable analysis.

Table 1Specification of the GT-MHR.

Parameter Value

Reactor power (MWt) 600Tin/Tout (�C) 491/850Reactor pressure (bars) 70Power density (W/cc) w5Reactor mass flow rate (kg/s) 320Effective core height (m) 7.93Core diameter (m) 2.63 ID/4.83 ODNumber of fuel blocks/pebbles 1020Bypass flow fraction (%) 10–15

Using the definition of safety margin, one can find as follows,

smðxÞ > 0 for safe functionssmðxÞ ¼ 0 at limit statesmðxÞ < 0 for mission failure

(3.1)

Therefore, as Burgazzi postulated [4],

Prf ¼ PrðI � A < 0Þ ¼ZZ

I�A�0

fIðIÞfAðAÞdIdA (3.2)

Using a normal distribution (Table 2), from standard normaltable and if M is a safety margin, mM/sM > 2.33 (F(Z) < 10�2).

ðmA � mIÞ=�

s2A þ s2

I

�1=2> 2:33 (3.3)

For the applications of the modeled system, the probability offailure is shown in the event/fault tree. So, there are several vari-ables for the special cases. As one can calculate, if mI ¼ 10 kg/s, sA ¼2, sI ¼ 2, then, mA > 16.6 kg/s. In the similar way, other kind of thebasic event distribution is constructed using the fuzzy set (Table 3),where the m2�m1 is the safety margin. In case of the triangular form,the distribution of failure frequency can be obtained (Table 3). Then,

mA > mI þ�����

1A� 1

B

����� (3.4)

If mI ¼ 10 kg/s, A ¼ 2, B ¼ 3, then, mA > 10.2 kg/s. So, themaximum safety margin is shown as 2

ffiffiffi5pþ 2

ffiffiffiffiffiffi10p

=2 ¼ffiffiffi5pþ

ffiffiffiffiffiffi10p

(in the membership number ¼ 1.0).There are some other variables (Table 4). The 6 cases for the

modeling are investigated, which are based on the mass flow rate ofthe natural circulation. The other physical variables are quantifiedfor the probability of failure (Fig. 3). Namely, in the probability set,the safety margin is the distance between the probability values ofthe failure in two events. Otherwise, in the fuzzy set, the safetymargin is the distance between the points in the slopes of twomembership functions. These values are made by the linear changewith the mass flow rate, where the mean and standard deviation

Fig. 1. I-A algorithm by normal distribution.

Fig. 2. I-A algorithm by statistical distribution (Fuzzy set-Triangular).

0.00 0.02 0.04 0.06 0.08 0.10

12

14

16

18

20

Qu

an

tity

Probability of Failure

Nat.Cir.(MassFlowRate),

HeatCond.(Press.),

Int Stor. Eng.(Heat Tr. Ra.)

Trend of Prob. of Failure (TPF)

Fig. 3. Trend of Probability of Failure (TPF) using of comparison between probability offailure and quantity Natural circulation (Mass flow rate,mA ¼ 10 kg/s), Heat conduction(Pressure,mA ¼ 10 MPa), Internal stored energy (Heat transfer rate,mA ¼ 10 kw/s).

Table 3Safety margin by statistical distributions.

Probability set-Normal distribution Fuzzy set-Triangulardistribution

F1ðzÞ ¼1

s1ffiffiffiffiffiffiffi2pp e

�ðz� m1Þ2

s22

F2ðzÞ ¼1

s2ffiffiffiffiffiffiffi2pp e

�ðz� m2Þ2

2s22

m2 � m1 ¼ ð�2 ln½F1ðzÞffiffiffiffiffiffiffi2pp

s1�s21Þ

1=2

� ð�2 ln½F2ðzÞffiffiffiffiffiffiffi2pp

s2�s22Þ

1=2

F1ðzÞ ¼ �Ajðz� m1Þj þ 1F2ðzÞ ¼ �Bjðz� m2Þj þ 1

m2 � m1 ¼ j1A½1� F1ðzÞ j � j1B�

½1� F2ðzÞ����� ¼

�����

1A� 1

B

�

�½F1ðzÞ � F2ðzÞ�����

Table 4Probability of failure vs. membership number for a safety margin of 0.2.

Probabilityof failure

Membershipnumber

Mass flow rate, heat conduction(Pressure), internal stored energy(heat transfer rate)

0.002 0.030

Gravity (potential energy) 0.002 0.030Heat conduction (temperature) 0.002 0.030

0.1 1 100.0

0.2

0.4

0.6

0.8

1.0Fuzzy Converter (FC)

Mem

bers

hip

Num

ber

Safety Margin

Relationship Line

Fig. 4. Fuzzy Converter (FC) using comparison between membership number andsafety margin.

Table 6Key points of the characteristics between probability set and fuzzy set.

Probability set Fuzzy set-Triangle

Function Probability function Membership functionRepresentative

valueMean Membership number

Unique value Standard deviation Line slopeSafety margin Distance between

meansDistance between membershipfunctions

Immovabilityof function

Movable by themean value

Fixed function, shape changedby the line slope

Table 5Probability of failure vs. membership number for a safety margin of 0.2 withphysical value.

Probability function Membership function

Physical value(kg/s) 6.600 0.200Non-dimension value 0.002 0.030

T.-H. Woo, U.-C. Lee / Energy 35 (2010) 1799–1804 1801

are used for the failure frequency construction of the basic event. Inthe case of the fuzzy set modeling, the membership function isused, where the maximum membership value of the function is 1.0.So, the safety margin is the distance between the mean values intwo normal distributions for the probabilistic calculation. Other-wise, the safety margin is the distance between the sidelines in themembership distributions for the fuzzy calculation. The geometricconfiguration decides the slope of the diagram. In the fuzzy case,the frequency of event success is changed by the proportionalvalues of the safety margin for the probability value. Namely, themaximum value of the frequency of event success hasthe membership number of 1.0 (Fig. 4). The safety margin is 0.2 andthe membership number is 0.030 which is seen as the arrow lines(Fig. 4). This is shown as the comparison (Table 5). The fuzzyconverter (FC) is constructed for the simplified descriptions for theprobability of failure using safety margin. FC is used in the case ofthe triangular form of the fuzzy calculations. The membershipnumber is changed from 0.0 to 1.0. The safety margin value is therelativistic quantity without any unit. Some key points of thecharacteristics between probability set and fuzzy set are shown(Table 6). The data propagation is done using the safety margin ineach event distribution.

a

b

Initiating Event Response to Initiating Event

SCS Fail to

Start

Pre-

Turbine

Trip

Flow

Coastdown

& Power Eq.

Recriticality Long-term

Conduction

Sequ-

ence

Num-

ber

Event Sequence

Frequency

(Rx-yr)

Remarks

Fig. 5. Event/fault tree for VHTR (a) Event tree, (b) Fault tree.

T.-H. Woo, U.-C. Lee / Energy 35 (2010) 1799–18041802

Table 7Modified event likelihood of occurrence based on SECY-93-092.

Event Frequency of Occurrence

Possible events >10�2/Plant-yearNon-possible events 10�2–10�4/Plant-yearExtremely non-possible events 10�4–10�6/Plant-yearVery rare events <10�6/Plant-year

Fig. 6. Configuration for uncertainty analysis of failure frequency (Fuzzy set-Triangular).

Table 9Uncertainty analysis of failure frequency (Fuzzy set-Triangular) (Rx-yr).

Event Failure frequency Uncertainty (at membership number of 0,1; 0.9)

1 7.5 � 10�2 (7.5 � 10�3, 1.4 � 10�1; 6.5 � 10�2, 8.3 � 10�2)2 3.0 � 10�7 (3.0 � 10�8, 5.7 � 10�7; 2.7 � 10�7, 3.3 � 10�7)3 7.5 � 10�8 (7.5 � 10�9, 1.4 � 10�7; 6.5 � 10�8, 8.3 � 10�8)4 1.8 � 10�10 (1.8 � 10�11, 3.4 � 10�10; 1.6 � 10�10, 2.0 � 10�10)5 1.4 � 10�8 (1.4 � 10�9, 2.7 � 10�8; 1.3 � 10�8, 1.5 � 10�8)6 4.0 � 10�9 (4.0 � 10�10, 8.0 � 10�9; 4.0 � 10�9, 4.0 � 10�9)

T.-H. Woo, U.-C. Lee / Energy 35 (2010) 1799–1804 1803

4. Results and discussions

For the passive system in ATWS of the VHTR, the statisticalexpression for the safety margin is used for the PSA. The proba-bilistic and non-linear fuzzy calculations are used for the massflow rate of the system. Using the impact-affordability algorithm,the safety margin is manipulated. The event/fault tree is shown in(Fig. 5) [12]. The red color cases are events of the natural circu-lation of the long-term cooling. For the non-passive part of theATWS, the modified data are from the SECY-93-092 (Table 7) [13].The quantifications of the propagation using the impact-afford-ability algorithm are given in (Table 8). For some other variables,when the probability value and the membership number (Table 3)are used, the final failure frequencies are same. The propagationsof the fuzzy set distribution have lower values than those of theprobability set in the case of passive system, cases #1, #2, #3, and#4. So, the values are much more conservative than conventionalmethod. The events 5 and 6 (Table 8) are not related to the passivesystem of the natural circulation. Just the events from case #1 tocase #4 are related to the passive system of the natural circulation,where the propagations of the fuzzy set distribution have lowervalues than those of the probability set in the case of passivesystem. In the results, there are the difference errors between thevalues of the probability set and the fuzzy set (Table 8), which areobtained as {(failure freq. of prob. dist. – failure freq. of fuzzydist.)/(failure freq. of prob. dist.) � 100}. The uncertainty analysisis done using the fuzzy confidence interval (Fig. 6) and thequantifications are listed (Table 9) [14]. The interval values aredone in the membership number of 0.1 and 0.9, which is theanalysis using fuzzy set theory like the confidence level in theprobability set calculations. Namely, the 0.9 of the membershipnumber is assumed as the 90% of the probability values, becausethis may be considered as the confidence interval of the singlevalue. The comparisons of 6 cases are obtained using this fuzzyconfidence interval. The double line arrows show the fuzzyconfidence interval values (Fig. 6).

The TPF shows the trend of probability of failure using ofcomparison between the probability of failure and the interestedquantity in the defined mean value (Fig. 3). This shows thesimplified configurations of the trend of functional failure in theseveral variables, which is constructed by the safety marginconcepts. The FC is used for finding the membership number usingthe safety margin instead of the probability of the failure (Fig. 4).The safety margin is converted to the membership number easilycomparing to the case of the probability value of the failure,because the membership function is constructed by the slope.

Table 8Failure frequency of event for ATWS (Rx-yr).

Event Probability set-Nor.distribution

Fuzzy set-Triangledistribution

Differenceerror (%)

1 2.5 � 100 7.5 � 10�2 97.02 5.0 � 10�3 3.0 � 10�7 99.93 2.5 � 10�6 7.5 � 10�8 97.04 5.0 � 10�9 1.8 � 10�10 96.45 1.4 � 10�8 1.4 � 10�8 0.06 4.0 � 10�9 4.0 � 10�9 0.0

There are 3 kinds of evaluations for the result [15]. The ‘repre-sentative value method’ is the numerical value in the interestedmembership number which is also called ‘Yager’s value method’.The result is obtained by the average of the values in the interestedmembership number. This is explained in equation (4.1).

KðVÞ ¼

Xn

i¼1

ai þ bi

2ðbi � aiÞ

Xn

i¼1

ðbi � aiÞ(4.1)

If 0 � a1 � b1 � a2 � b2 �. � an � bn � 1

V ¼ U{ai � x � bi}, U means the summation of the elements.The second method is the ‘interval value method’, where the

solutions are the interval values in the maximum membershipnumber. The last one is the ‘center of gravity method’. This isexplained in equation (4.2).

x ¼

Zf ðxÞ � xdxZ

f ðxÞdx(4.2)

where, f(x): Membership function, x: Probability variable.So, the central value is obtained. As the results are shown

(Table 10), the values are same in these methods; representative

Table 10Analysis of failure frequency (Fuzzy set-Triangular) (Rx-yr).

Event Representativevalue method

Interval value method(at m ¼ 1.0)

Center of gravitymethod

1 7.5 � 10�2 7.5 � 10�2 7.5 � 10�2

2 3.0 � 10�7 3.0 � 10�7 3.0 � 10�7

3 7.5 � 10�8 7.5 � 10�8 7.5 � 10�8

4 1.8 � 10�10 1.8 � 10�10 1.8 � 10�10

5 1.4 � 10�8 1.4 � 10�8 1.4 � 10�8

6 4.0 � 10�9 4.0 � 10�9 4.0 � 10�9

T.-H. Woo, U.-C. Lee / Energy 35 (2010) 1799–18041804

value method, center of gravity method, and interval value method,because the fuzzy function is the isosceles triangle. If the trianglehas scalene, three values could be different.

5. Conclusions

The passive system of the NPPs is examined for the PSA by thenew algorithm. This study concludes the probability set algorithmcould be substituted with the non-linear fuzzy set algorithm. Theconventional mean and standard deviation are changed to theanalysis of the fuzzy membership function. If the other geometryfor the membership function is considered, other values are usedlike the radius of the circular form of the membership function.Some metric should be done using the safety margin. The easierexpression could be constructed for the fuzzy calculation in theconstruction of the failure frequency of the basic events. Theparticular meanings of this study are as follows;

� The impact-affordability algorithm of the event is introducedfor the PSA of the passive system.� The fuzziness of the membership function expresses the non-

linearity of the safety margin.� Several variables are tested using the safety margin as TPF and FC.� For the failure frequency of the rare event like the passive

system in VHTR, the safety margin is introduced for the dataquantification of the statistical variables (mass flow rate andother variables).� The probabilistic distribution is compared with the non-linear

fuzzy distribution.� The restrictions of the probabilistic distributions are modified

to the simpler ways using the geometrical expressions.� A newly developed PSA algorithm can be applied to the license

construction.� The impact-affordability algorithm can be applied to active as

well as passive systems.

The quantity of the physical variables can be expressed aslinguistic performance of the operator in the fuzzy set distribution.The human error could be reduced due to the human orientedalgorithm of the theory in some active systems, because the fuzzyset theory is related to the linguistic expression of the operator.Some other kinds of the complex algorithms like the neuralnetwork or chaos theory could be used for the data quantification inPSA. Using the safety margin, the thermal-hydraulic variables aretransformed from the probability function to the fuzzy member-ship faction, which means that physical phenomena can beexpressed by the non-linear artificial intelligence algorithm like the

fuzzy set. Newly introduced factors as TPF and FC could be used forthe other system applications like the thermal-hydraulic variables,which are described above. So, all variables of the system can beanalyzed by the safety margin modified factors for the safetyassessment as well as the system analyses. It expresses one of theprobability distributions. Any kinds of probability distributionscould be described for the comparison with the membershipfunction. The simpler calculations can be done in the transformednon-linear function as it is seen above in the case of the PSA. Forexample, the heat transfer rate and the temperature can beobtained in the interested accident by the non-liner algorithm.

Acknowledgements

Authors thank Dr. S. J. Han in Korea Atomic Energy ResearchInstitute (KAERI) in Korea for his research discussions. It alsothanked for the financial support from the Ministry of Education,Science and Technology (MEST).

References

[1] Lohnet GH, Nabielek H, Schenk W. Teh fuel element of the HTR-Modules,a prerequisite of an inherently safe reactor. In: Proceedings of the 10thinternational conference on HTGR’s. Paper II. 24. San Diego, 1988.

[2] Weisbrodt IA. Inherently Safety Design Features of the HTR-Module, IAEA-CN-48/125. HTGR, The Modular-High-termperature Gas-cooled Reactor (MHTGR)in the U.S., Doe-HTGR-87–088;2003.

[3] Goodjohn AJ. Summary of gas-cooled reactor programs. Energy 1991;16:79–106.

[4] An S, Hayashi T. Studies on the concepts of HTGR plants viable in Japan part I:outline of the activities. Energy 1991;16:137–41.

[5] An S, Hayashi T. Studies on the concepts of HTGR plants viable in Japan part II:discussion of the results. Energy 1991;16:143–53.

[6] Nickel H, Bodmann E, Seehafer HJ. The materials program for the HTR in thefrg: integrity concept, status of the development of high temperature mate-rials and design codes. Energy 1991;16:221–42.

[7] Bianchi F, Burgazzi L, D’auria F, Galassi GM, Ricotti ME, Oriani L. Evaluation ofthe reliability of a passive system. In: International conference of nuclearenergy in central Europe, 2001.

[8] Korea Atomic Energy Research Institute. Experiment Database:I-NERI FinalProject Technical Report. Korea; 2007.

[9] Nayak AK, Sinha RK. Role of passive system in advanced reactor. Progress inNuclear Energy 2007;49:486–98.

[10] Burgazzi L. Thermal-hydraulic passive system reliability-based designapproach. Reliability Engineering & System Safety 2007;92:1250–7.

[11] Zio E, Baraldi P. Sensitivity analysis and fuzzy modeling for passive systemsreliability assessment. Annals of Nuclear Energy 2003;31:277–301.

[12] USNRC. Regulatory effectiveness of the anticipated transient without scramrule, NUREG-1780. USA; 2003.

[13] USNRC. SECY-93-092, PRA attachment 4. USNRC Commission Papers. USA;1993.

[14] Brown LD, Cai TT, Dasgupta A. Interval estimation for a binomial proportion.Statistical Science 2001;16:101–33.

[15] Zimmermann HJ. Fuzzy set theory and its applications. USA: Kluwer AcademicPublishers; 2001.