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o, Gw

Passive systemAnticipated transient without scram

sesare

essamaby(FC)

(VHTRvelopeneric dsive sg. Alth

(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 [36].

Using the statistical analysis, there are several kinds of the resultevaluations suggestedwhich 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.

* Corresponding author. Tel.: 82 2 880 8337; fax: 82 2 889 2688.

Contents lists availab

Ener

.e ls

Energy 35 (2010) 17991804E-mail address: thw@snu.ac.kr (T.-H. Woo).reactor 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

using the impact-affordability method. The data of DBA are from theGT-MHR of the General Atomics. The specication of the GT-MHR isgiven (Table 1). TheDBAwas constructed by the KoreaAtomic 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.1. Introduction

The very high temperature reactornuclear power plant has been depurpose. However, there is not the gesafety assessment (PSA) in the pascirculation of the long-term coolin0360-5442/$ see front matter 2009 Elsevier Ltd.doi:10.1016/j.energy.2009.12.034the 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 articial intelligence analysis of the fuzzyset could enhance the reliability of the system comparing to the probabilistic analysis.

2009 Elsevier Ltd. All rights reserved.

) type of the gas-cooledd for the commercialata for the probabilisticystem like the naturalough the commercial

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 paperwouldlike to showhow to treat the safetymargin. The anticipated transientwithout scram (ATWS) of the VHTR is amodel for the passive systemProbabilistic safety assessmentVery high temperature reactorfor 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 inKeywords:easy congurations for the event characteristics. The mass ow rate in the natural circulation is studiedSafety assessment for the passive system(NPPs) using safety margin estimation

Tae-Ho Woo*, Un-Chul LeeDepartment of Nuclear Engineering, Seoul National University, Gwanak 599, Gwanak-r

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

a b s t r a c t

The probabilistic safety aswhere the operational dataplant. Therefore, it is necestimations for the safetybasic event, which is made(TPF) and fuzzy converter

journal homepage: wwwAll rights reserved.of the nuclear power plants

anak-gu, Seoul 151-742, Republic of Korea

sment (PSA) for gas-cooled nuclear power plants has been investigateddecient, because there is not any commercial gas-cooled nuclear powerry to use the statistical data for the basic event constructions. Severalrgin are introduced for the quantication of the failure frequency in thethe concept of the impact and affordability. Trend of probability of failureare introduced using the safety margin, which shows the simplied and

le at ScienceDirect

gy

evier .com/locate/energy

margin. This function can show the distributions of mass ow

Table 2Procedures of safety assessment.

Procedures Contents

1. Basic dataconstruction

Safety margin construction using Trend ofProbability of Failure (TPF)

Membership number construction usingFuzzy Converter (FC)

2. Tree Event/fault tree construction3. Propagation Data quantication4. Analysis Difference error analysis

Uncertainty analysis using fuzzycondence interval

Nomenclature

x variableI(x) impact functionA(x) affordability functionsm(x) safety margin sm(x) I(x) A(x)Nn nominal valueNI actual valuemA,mI mean of NA,NIsA2,sI

2 variance of NA,NI

T.-H. Woo, U.-C. Lee / Energy 35 (2010) 179918041800rate, 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 modied from the case of themass ow 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 is2. 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 safetyconsidered as the membership algorithm [11], which is modied inthis paper for much more reasonable analysis.

Table 1Specication of the GT-MHR.

Parameter Value

Reactor power (MWt) 600Tin/Tout (C) 491/850Reactor pressure (bars) 70Power density (W/cc) w5Reactor mass ow rate (kg/s) 320Effective core height (m) 7.93Core diameter (m) 2.63 ID/4.83 ODNumber of fuel blocks/pebbles 1020Bypass ow fraction (%) 1015Using the denition of safety margin, one can nd as follows,

smx > 0 for safe functionssmx 0 at limit statesmx < 0 for mission failure

(3.1)

Therefore, as Burgazzi postulated [4],

Prf PrI A < 0 ZZIA0

fIIfAAdIdA (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).

mA mI=s2A s2I

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 10 kg/s, sA 2, sI 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 m2m1 is the safety margin. In case of the triangular form,the distribution of failure frequency can be obtained (Table 3). Then,

mA > mI 1A 1B

(3.4)If mI 10 kg/s, A 2, B 3, then, mA > 10.2 kg/s. So, the

maximum safety margin is shown as 25

p 2

10

p=2

5

p

10

p

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

modeling are investigated, which are based on themass ow rate ofthe natural circulation. The other physical variables are quantiedfor 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 ow rate, where the mean and standard deviationFig. 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 ow rate,mA 10 kg/s), Heat conduction(Pressure,mA 10 MPa), Internal stored energy (Heat transfer rate,mA 10 kw/s).

Table 3Safety margin by statistical distributions.

Probability set-Normal distribution Fuzzy set-Triangulardistribution

F1z 1

s12p

p ez m1

2

s22

F2z 1

s22p

p ez m2

2

2s22

m2 m1 2 lnF1z2p

ps1s211=2

2 lnF2z2p

ps2s221=2

F1z Ajz m1j 1F2z Bjz m2j 1m2 m1 j1A1 F1z j j1B

1 F2z

1A 1B

F1z F2z

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

Probabilityof failure

Membershipnumber

Mass ow 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

T.-H. Woo, U.-C. Lee / Energy0.1 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 and

35 (2010) 17991804 1801are used for the failure frequency construction of the basic event. Inthe case of the fuzzy set modeling, the membership function isused, where themaximummembership 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 geometricconguration 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 simplied 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.

safety 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

ab

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) 179918041802

4. Results and discussions

If 0 a1 b1 a2 b2 . an bn 1V 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

Zf x xdxZf xdx

(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 7Modied event likelihood of occurrence based on SECY-93-092.

Event Frequency of Occurrence

Possible events >102/Plant-yearNon-possible events 102104/Plant-yearExtremely non-possible events 104106/Plant-yearVery rare events

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. 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 modied factors for the safetyassessment as well as th...