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Nuclear Engineering and Design 241 (2011) 3985– 3995

Contents lists available at ScienceDirect

Nuclear Engineering and Design

j ourna l ho me page: www.elsev ier .com/ locate /nucengdes

probabilistic seismic risk assessment procedure for nuclear power plants: (II)pplication

in-Nan Huanga,∗, Andrew S. Whittakerb, Nicolas Lucoc

Department of Civil Engineering, National Taiwan University, 10617, TaiwanDept. of Civil, Structural and Environmental Engineering, State Univ. of New York at Buffalo, Buffalo, NY 14260, United StatesUnited States Geological Survey, P.O. Box 25046, MS 966, Denver, CO 80225, United States

r t i c l e i n f o

rticle history:eceived 16 November 2009eceived in revised form 23 June 2011ccepted 25 June 2011

a b s t r a c t

This paper presents the procedures and results of intensity- and time-based seismic risk assessments ofa sample nuclear power plant (NPP) to demonstrate the risk-assessment methodology proposed in itscompanion paper. The intensity-based assessments include three sets of sensitivity studies to identify the

impact of the following factors on the seismic vulnerability of the sample NPP, namely: (1) the descriptionof fragility curves for primary and secondary components of NPPs, (2) the number of simulations of NPPresponse required for risk assessment, and (3) the correlation in responses between NPP components.The time-based assessment is performed as a series of intensity-based assessments. The studies illustratethe utility of the response-based fragility curves and the inclusion of the correlation in the responses ofNPP components directly in the risk computation.

. Introduction

The companion paper, Huang et al. (2011), proposed a five-stepethodology for probabilistic seismic risk assessment of nuclear

ower plants (NPPs). The methodology assesses the annual fre-uency of unacceptable performance of a NPP using a time-basedpproach, where the seismic hazard is characterized using a haz-rd curve. A time-based assessment is performed as a series ofntensity-based assessments, where the probability of unacceptableerformance is estimated at each intensity level identified from theazard curve.

Step 1 of the methodology proposed in the companion papererforms plant system analysis to determine accident sequenceshat could contribute to the target unacceptable performance andevelops component fragility curves as a function of a structuralesponse parameter. Step 2 develops the seismic hazard curve(s)or the NPP site and selects and scales ground motions for eachntensity level. Step 3 identifies the distributions and correlationf all structural response parameters of Step 1 using nonlin-ar response-history analysis at each intensity level. Step 4 usesonte-Carlo-based procedures to generate a significant number of

esponse data that are statistically consistent with those of Step 3nd to assess the possible distribution of damage to structural andonstructural components of the NPP for each set of simulations.

∗ Corresponding author. Tel.: +886 2 3366 4325; fax: +886 2 2739 6752.E-mail address: ynhuang@ntu.edu.tw (Y.-N. Huang).

029-5493/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.nucengdes.2011.06.050

© 2011 Elsevier B.V. All rights reserved.

Step 5 computes the probabilities of unacceptable performance ateach intensity level and the annual frequency of unacceptable per-formance of the NPP subjected the seismic hazard of Step 2. Step 1is discussed in Sections 2 and 3.3; Steps 2 through 5 are identifiedin Section 4.

The methodology proposed in the companion paper differs inmany regards from methods used to date for the probabilistic riskassessment of NPPs, namely, the Zion method (Pickard, Lowe, andGarrick, Inc., et al. 1981) and the Seismic Safety Margin (SSM)method (Smith et al., 1981). Key differences include (1) the useof component fragility curves that are expressed in terms of seis-mic demand (e.g., story drift and peak floor acceleration) and noton ground-motion intensity (e.g., peak ground acceleration), and(2) procedures for both scaling earthquake ground motions andassessment of component damage.

A series of sensitivity studies were conducted and are summa-rized here to answer the following questions:

1. What is the impact of using a mean or median fragility curveinstead of a family of fragility curves for NPP components on theannual frequency of unacceptable performance of a NPP?

2. How many simulations of NPP response are required perintensity level to establish a reliable estimate of the seismic

vulnerability of a NPP?

3. The correlation in responses between NPP components isignored in the SSM procedure (see Section 2.2 and Eq. (4) of thecompanion paper) but included in the methodology proposed in

3986 Y.-N. Huang et al. / Nuclear Engineering a

iSomtbfts4

pincptrt

2

bdhtaamstpcmmib

ttros

Fig. 1. Cutaway view of the sample NPP reactor building.

the companion paper. What is the impact of the correlation onthe annual frequency of unacceptable performance of a NPP?

The sensitivity studies were conducted using a series ofntensity-based assessments of a sample NPP reactor building.ection 2 introduces the sample reactor building. The first halff Section 3 presents the assumed seismic hazard and groundotions (Section 3.1), response-history analyses (Section 3.2) and

he fault tree and fragility curves (Section 3.3) used in the intensity-ased assessments. Sections 3.4–3.6 present the sensitivity studiesor the three questions listed above, respectively. The observa-ions of Section 3 are applied to the time-based assessment of theample reactor building and the results are presented in Section.

Herein, we demonstrate the methodology presented in the com-anion paper by a project-independent study and investigate the

mpacts of (a) different descriptions of fragility curves, (b) theumber of simulations and (c) the correlation in responses of NPPomponents. A simple fault tree was used to define unacceptableerformance. HCLPF-based fragility curves were used to charac-erize the capacities of sample NPP components. Unidirectionalesponse-history analyses were performed to simplify the compu-ational effort.

. The sample reactor building

Fig. 1 presents a cutaway view of the sample NPP reactoruilding. A lumped-mass stick model of this reactor building waseveloped in the computer code SAP2000 (CSI, 2006) for response-istory analysis. The model, shown in Fig. 2a, is composed ofwo sticks: one representing the (external) containment structurend the other representing the internal structure. The two sticksre structurally independent and connected only at the base. Theechanical properties of the frame elements that compose each

tick were calculated from a 3D model of the reactor building. Theotal height of the containment structure is 59.5 m and its first modeeriod is approximately 0.2 s. The thickness of the post-tensionedoncrete cylindrical wall of the containment structure is approxi-ately 1.2 m. The height of the internal structure is 39 m. The firstode period of the internal structure in both horizontal directions

s approximately 0.14 s. The total weight (W) of the NPP reactoruilding is approximately 75,000 tons.

The seismic risk assessments performed in this study focus onhe secondary systems in the sample NPP reactor building since

he costs associated with analysis, design, construction, testing andegulatory approval of secondary systems can dominate the costf NPPs (Huang et al., 2008). Fig. 1 illustrates the distribution ofeveral important secondary systems in the sample reactor build-

nd Design 241 (2011) 3985– 3995

ing, including the reactor, steam generator and emergency coolantinjection (ECI) tank. These secondary systems are attached to theinternal structure and supported at elevations of 7 (Node 201), 18(Nodes 1006 and 1009) and 39 m (Nodes 215 and 216). Fig. 2b iden-tifies the node numbers assigned to the internal structure of thesample NPP.

The demand parameter used to quantify the seismic demandsand develop fragility curves for the secondary systems in the sam-ple NPP is Average Floor Spectral Acceleration over a frequencyrange from 5 to 33 Hz, termed AFSA hereafter, since the seismicdemand on secondary systems are typically characterized usingfloor response spectra and the frequencies of most secondary sys-tems are in the range of 5–33 Hz. This study focuses on secondarysystems supported at Nodes 201, 1009 and 216 of the sample NPP.The AFSA at a given node of the sample NPP for a response-historyanalysis is defined by the arithmetic mean of the 29 floor spectralordinates at frequencies of 5–33 Hz with increments of 1 Hz. Anydemand parameter could be used for a project-specific study butthe corresponding fragility curves would have to be constructed forthat parameter.

3. Intensity-based assessment

3.1. Seismic hazard and ground motions

Seventeen electrical utility companies intend to submit appli-cations to the USNRC for 30 new NPP licenses; 29 of the 30 plantsare to be located in the Central/Eastern United States (CEUS) (NEI,2007; Gunberg and Green, 2008). The sample NPP is assumed sitednear Richmond, Virginia, in the Eastern United States. The targetspectrum used for the intensity-based assessment is a UniformRisk Spectrum (URS) prepared for an Early Site Permit (ESP) reportfor a nearby NPP site. The thick line of Fig. 3a presents the URS,which was established using an approach similar to that describedin ASCE Standard 43-05 (ASCE 2005) for determining the DesignBasis Earthquake. The URS of Fig. 3a has significant spectral demandin the high frequency range: a range of importance to secondarysystems in NPPs.

Earthquake histories from large magnitude events in the CEUSare non-existent. The computer code “Strong Ground Motion Simu-lation (SGMS)” (Halldorsson, 2004) was used to generate CEUS-typeground motions for response-history analysis. The SGMS code isbased on the Specific Barrier Model, which provides a completeand self-consistent description of the heterogeneous earthquakefaulting process and can capture the rich high frequency con-tent in CEUS ground motions (Halldorsson and Papageorgiou,2005).

The SGMS code requires users to provide information for the sitecondition and the magnitude and distance for the scenario eventof interest to simulate ground motions. The site condition for thesample NPP was assumed to be hard rock based on the results ofthe geotechnical survey presented in the ESP report. A controllingevent with a moment magnitude of 5.3 and a distance of 7.5 kmwas determined by the deaggregation of the short-period seis-mic hazard for the sample NPP site at a mean annual frequency ofexceedance (MAFE) of 10−5 per the procedure of USNRC RegulatoryGuide 1.165 (USNRC, 1997). More information for the deaggrega-tion results for the sample NPP site can be found in Huang et al.(2008).

Eleven ground motions were generated using SGMS and scaledto the URS of Fig. 3a at a period of 0.14 s: the fundamental period

of the internal structure of the sample NPP. Fig. 3a also presentsthe response spectra for the 11 scaled SGMS ground motions andFig. 3b presents the median spectrum of the 11. The median spec-tral ordinates of the 11 scaled ground motions agree well with the

Y.-N. Huang et al. / Nuclear Engineering and Design 241 (2011) 3985– 3995 3987

e sam

Utga

Fgo

Fig. 2. Stick model for th

RS demands at periods smaller than 0.4 s and are smaller thanhe URS demands at periods greater than 0.4 s since the deaggre-ated seismic hazard (McGuire, 2004) at short and long periodsre dominated by different earthquake scenarios (i.e., magnitude-

a. URS and the spectra of the 11 scaled SGMS motions

32.521.510.50Period (sec)

0

0.5

1

1.5

Spe

ctra

l acc

eler

atio

n (g

)

URS

SGMS motions

32.521.510.50Period (sec)

0

0.2

0.4

0.6

0.8

1

Spe

ctra

l acc

eler

atio

n (g

)

URS

Median of SGMS motions

b. URS and the median spectrum of the 11 scaled SGMS motions

ig. 3. The URS for the sample NPP site and (a) the response spectra of the 11 SGMSround motions scaled to the URS at a period of 0.14 s and (b) the median spectrumf the 11 scaled SGMS ground motions.

ple NPP reactor building.

distance pairs). The 11 scaled ground motions were used in theresponse-history analysis for intensity-based assessment.

3.2. Response-history analysis

Nonlinear uni-directional response-history analyses of the sam-ple NPP subjected to the 11 scaled SGMS ground motions of Fig. 3ain the X and Y directions were performed for the intensity-basedassessment of the sample reactor building. The containment struc-ture was assumed to remain elastic because (a) the key secondarysystems identified in Fig. 1 are supported by the internal struc-ture; and (b) containment structures are designed for large internalpressures (up to 500 kPa) resulting from a postulated accident,and seismic loadings generally do not control their design. Bilinearshear hinges with 3% post-yield stiffness were assigned to all frameelements in the internal structure. The yield shear strengths of thebilinear shear hinges were estimated as 0.5

√f ′CAs (Wood, 1990),

where f ′C is the compressive strength of concrete with a value of

35 N/mm2 and As is the shear area of each internal-structure stickprovided by the supplier of the sample NPP. Demands (AFSA) atNodes 201, 1009 and 216 of the internal structure were computedsince the assessments presented herein focus on secondary systemsof the sample NPP.

The product of the response-history analysis described hereinis the 11 × 6 demand matrix of Table 1 for the AFSA at Nodes 201,1009 and 216 in the X and Y directions. The demand matrix is usedto compute the unacceptable performance of the sample NPP sub-jected to the ground motions of Fig. 3a and conduct the sensitivitystudies of Sections 3.4–3.6.

3.3. Assumed fault tree and fragility curves

In the studies reported here, the unacceptable performance ofthe sample NPP was defined using the fault tree of Fig. 4, wherethe unacceptable performance was defined as the failure of any ofthe secondary systems at Nodes 201, 1009 and 216 using an “OR”gate. Fragility curves for each basic component of the fault tree of

Fig. 4 were constructed using a HCLPF-based procedure based ondemands of Table 1 and dispersions from the studies of Kennedyand Ravindra (1984). We note that these fault trees and fragilitycurves are used solely to demonstrate the proposed procedure.

3988 Y.-N. Huang et al. / Nuclear Engineering and Design 241 (2011) 3985– 3995

Table 1AFSA at Nodes 201, 1009 and 216 for the intensity-based assessment of the sample NPP.

GM No. AFSA in the X direction (g) AFSA in the Y direction (g)

Node 201 Node 1009 Node 216 Node 201 Node 1009 Node 216

1 0.99 1.28 2.60 1.00 1.30 3.012 0.79 1.09 2.05 0.78 1.15 2.483 0.78 1.24 2.25 0.73 1.01 2.444 1.06 1.49 2.89 1.02 1.49 2.935 0.74 1.02 1.93 0.61 0.90 2.086 0.91 1.34 2.17 0.75 1.12 2.457 0.67 0.96 1.83 0.75 0.91 2.198 0.78 1.02 1.98 0.84 1.24 2.679 0.95 1.16 2.09 1.04 1.47 2.93

10 0.65 0.93 1.88 0.67 0.89 2.3211 1.03 1.28 2.34 1.18 1.54 3.24

Unacceptable Performance

Failu re of second ary syst ems at Node 201 in the X direction

OR

Fail ure of seconda ry systems at Nod e 2 01 i n the Y di recti on

Failu re of sec ondar y syst ems at Node 100 9 in t he X direction

Fail ure of secondary systems at Node 1009 in the Y directi on

Failu re of second ary syst ems at Node 21 6 in t he X direc tion

Fail ure of secondary systems at Node 216 in the Y direction

studies presented in this paper.

pe

G

wfQmarhtwt

p

�

wiToTnsa2

Table 2Logarithmic standard deviations for fragility curves of mechanical equipments(Kennedy and Ravindra, 1984).

Item ˇr ˇu

for these parameters.Eqs. (4) and (5) are used to characterize ˇr and ˇu for the devel-

opment of fragility curves and the dispersion in structural response

Fig. 4. The fault tree for the

The fragility curves used in the studies of this paper were com-uted using (1), which is a rearrangement of (2) and (3) of Huangt al. (2011):

C (a) = ˚

(ln a/a + ˚−1(Q )ˇu

ˇr

)(1)

here GC(a) is the fragility curve as a function of a, which is AFSAor this paper; is the standardized normal distribution function;

is defined in Fig. 2 of Huang et al. (2011); and a, ˇr and ˇu are theedian capacity of the secondary system and aleatory randomness

nd epistemic uncertainty in the capacity of the secondary system,espectively. The secondary system at a given node was assumed toave the same capacity (i.e., the same family of fragility curves) inhe X and Y directions and the HCLPF value of the secondary systemas assumed to be equal to the median AFSA demand of Table 1 at

he given node.The median AFSA demand of Table 1 at a given node was com-

uted using the following equation:

= exp

(1n

n∑t=1

ln yt

)(2)

here yi is the value of each AFSA in a column of Table 1 and ns the total number of the realizations in the column, namely, 11.he median AFSA demands for the second through seventh columnf Table 1 are 0.84, 1.16, 2.16, 0.84, 1.17 and 2.60 g, respectively.

he greater of the median values in the X and Y directions at aode was used to develop the fragility curves for the secondaryystems at the node; i.e., the HCLPF values of the secondary systemst Nodes 201, 1009 and 216 were assumed to be 0.84, 1.17 and.60 g, respectively.

Equipment capacity, ˇCE 0.10–0.18 0.22–0.32Equipment response, ˇRE 0.18–0.25 0.18–0.25Building response, ˇRS 0.20–0.32 0.18–0.33

The values of a were computed using (3)1:

a = HCLPF · e1.65(ˇr+ˇu) (3)

Kennedy and Ravindra (1984) used (4) and (5) to estimate ˇr andˇu for ground-motion-based fragility curves of mechanical equip-ments in a NPP:

ˇr = (ˇ2CE,r + ˇ2

RE,r + ˇ2RS,r)

1/2(4)

ˇu = (ˇ2CE,u + ˇ2

RE,u + ˇ2RS,u)

1/2(5)

where ˇCE,r and ˇCE,u are randomness and uncertainty in the capac-ity of an equipment, respectively; ˇRE,r and ˇRE,u are those in theresponse of the equipment; and ˇRS,r and ˇRS,u are those in theresponse of the structure that the equipment is attached to. Table 2presents the values recommended by Kennedy and Ravindra (1984)

1 HCLPF is the acronym for High Confidence of Low Probability of Failure andis defined in this study as the value of AFSA associated with 95% confidence of a5% probability of failure for a secondary system. The relationship presented hereinbetween the median capacity, a, and HCLPF of a secondary system is derived using(2) and (3) of Huang et al. (2011) with Q = 0.95 and GC(HCLPF) = 0.05.

Y.-N. Huang et al. / Nuclear Engineering and Design 241 (2011) 3985– 3995 3989

a. Node 201

b. Nod e 10 09

876543210AFSA (g )

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0.5

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ility

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a= 2.26 gβr= 0.26

βu= 0.34

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11109876543210(g)AFSA

0

0.25

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0.75

1

Pro

ba

bili

ty o

f fa

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a= 3.15 gβr= 0.26

βu= 0.34

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2520151050(g)AFSA

0

0.25

0.5

0.75

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ba

bili

ty o

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a= 7.0 gβr= 0.26

βu= 0.34

^

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u

Table 3The values of AFSA associated with a 1% and a 10% probability of failure for themedian fragility curves of Fig. 5.

Probability of failure AFSA (g)

Node 201 Node 1009 Node 216

c. Node 216

ig. 5. Fragility curves for the secondary systems at Nodes 201, 1009 and 216.

iven a ground motion intensity, namely, ˇRS,r and ˇRS,u, were set toero since the fragility curves used in this study are defined in termsf a response parameter rather than a ground-motion parameter.ach value of ˇCE,r, ˇRE,r, ˇCE,u and ˇRE,u used in the studies of thisaper is the middle point of the corresponding range recommended

n Table 2, namely, 0.14, 0.215, 0.27 and 0.215, respectively, and theesultant values of ˇr and ˇu are 0.26 and 0.34, respectively. The val-

es of a were computed using (3) to be 2.26, 3.15 and 7.0 g at Nodes01, 1009 and 216, respectively.

Fig. 5 presents the three families of fragility curves generatedsing (1) for the key secondary systems at Nodes 201, 1009 and 216.

1% 1.23 1.72 3.8210% 1.62 2.26 5.02

Each family of fragility curves includes 11 curves and each curve hasan equal probability of occurrence. For a set of a, ˇr and ˇu and avalue of Q, a single curve can be determined. For example, the boldsolid line in panel a of Fig. 5 was determined by a = 0.84 g, ˇr =0.26, ˇu = 0.34 and Q = 0.5 At a given node, the values from1/22 to 21/22 with increments of 1/11 were used in (1) for Q todevelop the 11 fragility curves in each family of curves in Fig. 5.

The appropriateness of the fragility curves of Fig. 5 is evaluatedusing ASCE/SEI 43-05 (ASCE 2005), where the seismic design of NPPstructural and nonstructural components aims to achieve both ofthe following performance criteria:

1. Less than a 1% probability of unacceptable performance for thedesign basis earthquake.

2. Less than a 10% probability of unacceptable performance forshaking equal to 150% of the design basis earthquake shaking.

Table 3 presents the values of AFSA associated with a 1% and a10% probability of failure for the median fragility curves of Fig. 5for Nodes 201, 1009 and 216. The AFSA demands at Nodes 201,1009 and 216 for the design basis earthquake can be representedby the medians of the AFSA values of Table 1 at Nodes 201, 1009 and216, namely, 0.84, 1.17 and 2.61 g, respectively. The median AFSAdemands of Table 1 are smaller than the AFSA values presentedin Table 3 for a 1% probability of failure; the 150% of the medianAFSA demands of Table 1 at Nodes 201, 1009 and 216 (i.e., 1.26,1.76 and 3.92 g, respectively) are also smaller than the AFSA valuespresented in Table 3 for a 10% probability of failure. The fragilitycurves of Fig. 5 are sufficient for the purpose of this study.

3.4. Sensitivity studies: fragility curves

The first set of sensitivity studies includes Analyses 1, 2 and 3ato investigate the impact of using alternate descriptions of fragilitycurves on the probability of unacceptable performance of the sam-ple NPP subjected to the seismic hazard of Fig. 3.

Analysis 1 involved four parts: 1a through 1d. Table 4 summa-rizes the key variables for 1a through 1d, namely, (a) the selectionof the fragility curves, (b) the number of fragility curves in a family,(c) the number of row vectors in the demand-parameter matrix,and (d) the number of trials used in the analysis.

For Analysis 1a, the procedure of Yang et al. (2009) (see adescription in Section 3.5 of Huang et al. (2011)) was used togenerate 200 sets of correlated AFSA values at Nodes 201, 1009and 216 in each of the X and Y directions using the underlyingdemand-parameter matrix of Table 1. The results formed a 200 × 6demand-parameter matrix. Fig. 9 of Huang et al. (2011) presentsthe sample distributions of AFSA in the X direction at Nodes 201and 1009 obtained from the 11 × 6 demand matrix of Table 1 andthe 200 × 6 matrix generated per Yang et al. (2009). Both the distri-butions and correlation of AFSA at Nodes 201 and 1009 in the 11 × 6matrix of Table 1 are preserved in the 200 × 6 matrix generated perYang et al. (2009).

For each AFSA value in a row vector of the 200 × 6 demand-parameter matrix, the damage state of a secondary system can bedetermined from the corresponding fragility curve using the fol-lowing steps: (1) a random number is generated using a generator

3990 Y.-N. Huang et al. / Nuclear Engineering and Design 241 (2011) 3985– 3995

Table 4Analyses 1, 2 and 3 for the intensity-based assessment of the sample NPP.

Analysis Fragility curve for each node Number of fragility curves in a family Number of row vectorsa Number of trials Medianb Meanc

Analysis 1 a Randomly selected 11 200 0.040 0.061b 11 2000 0.037 0.060c 21 200 2000 0.045 0.070d 201 200 0.040 0.079

Analysis 2 Median – 200 0.005 0.004Analysis 3 a Mean – 200 0.080 0.079

a

e from the 2000 trials.rformance from the 2000 trials.

tapontvst

uoionTdepbp2a

tAcAcft

Aosavaf1

iprTtffbt(

i

a. Analyses 1a and 1b

b. Analyses 1a, 1c and 1d

0.40.350.30.250.20.150.10.050

Probability of unacceptable performance

0

0.25

0.5

0.75

1

Cum

ulat

ive

dist

ribut

ion

func

tion

Analysis 1a

Analysis 1b

0.40.350.30.250.20.150.10.050

Probability of unacceptable performance

0

0.25

0.5

0.75

1

Cum

ulat

ive

dist

ribut

ion

func

tion

Analysis 1a

Analysis 1c

Analysis 1d

mean, Analysis 1a

mean, Analysis 1c

mean, Analysis 1d

Number of row vectors in the demand-parameter matrix.b Median of the 2000 realizations for the probability of unacceptable performancc Arithmetic mean of the 2000 realizations for the probability of unacceptable pe

hat produces uniformly distributed random numbers between 0nd 1; (2) if the generated number is smaller than or equal to therobability of failure identified from the fragility curve, the sec-ndary system is considered to have failed; and (3) if the generatedumber is greater than the probability of failure, the secondary sys-em is considered to have passed (i.e., safe). If any of the 6 AFSAalues in a row vector is associated with the failure of a secondaryystem, the performance of the NPP is considered unacceptable forhis row vector.

For Analysis 1a, the analysis described above was repeatedsing all 200 row vectors in the demand-parameter matrix andne fragility curve randomly selected from each of the three fam-lies of curves presented in Fig. 5. A realization for the probabilityf unacceptable performance was computed as the ratio of theumber of row vectors with unacceptable performance to 200.he analysis was repeated 2000 times using a newly generatedemand-parameter matrix and combination of fragility curves forach time to develop 2000 realizations for the distribution of therobability of unacceptable performance. The cumulative distri-ution function (CDF) of the 2000 realizations for Analysis 1a isresented in Fig. 6a using a solid line. To develop the curve, the000 realizations were sorted from smallest to largest and assigned

probability from 1/2000 to 1.0 in increments of 1/2000.To study the impact of the choice of number of row vectors on

he distribution of the probability of unacceptable performance,nalysis 1a was repeated but using demand-parameter matricesonsisting of 2000 row vectors and denoted Analysis 1b. Results ofnalysis 1b are presented in Fig. 6a using a dotted line. For thease studied herein, the increase in the number of row vectorsrom 200 to 2000 has negligible influence on the distribution ofhe realizations.

To study the impact of the number of fragility curves in a family,nalysis 1a (family of 11 curves) was repeated but using familiesf 21 (Analysis 1c) and 201 (Analysis 1d) curves for the secondaryystems at Nodes 201, 1009 and 216. The 21 and 201 fragility curvest a given node were developed using (1) and the correspondingalues of a, ˇr and ˇu used to develop the fragility curves of Fig. 5t the node. The values of Q for the 21 (201) fragility curves in aamily were 1/42 (1/402) to 41/42 (401/402) with increments of/21 (1/201).

Fig. 6b presents the CDFs and arithmetic means of the 2000 real-zations for Analyses 1a, 1c and 1d. The last two columns of Table 4resent the medians computed using (2) and arithmetic means,espectively, of the 2000 realizations for Analyses 1a through 1d.he increase in the number of fragility curves in a family from 11o 201 also increases the arithmetic mean of the 2000 realizationsrom 0.06 to about 0.079, which is used as the benchmark valueor the probability of unacceptable performance for the intensity-ased assessment of the sample NPP since a further increase of

he number of fragility curves in a family does not alter this value0.079).

If only one fragility curve, instead of a family of curves, is usedn the risk assessment for a secondary system, possible choices are

Fig. 6. Distributions of the probability of unacceptable performance for Analyses 1athrough 1d for the intensity-based assessment.

either the median or mean fragility curve. The median curve for afamily of fragility curves is defined by a lognormal distribution withmedian and dispersion of a and ˇr, respectively, such as the solidbold curves of Fig. 5, and the mean curve is defined by a lognor-mal distribution with median and dispersion of a and

√ˇ2

r + ˇ2u,

respectively (Reed and Kennedy, 1994). Fig. 7 presents both themedian and mean curves for the three families of fragility curvesof Fig. 5.

To study the impact of the use of median or mean fragility curveson the distribution of the probability of unacceptable performance,Analysis 1a was repeated but using the (a) median (Analysis 2), and

(b) mean (Analysis 3a) curves of Fig. 7 in each of the 2000 trials inlieu of randomly selected curves.

Fig. 8 presents the CDFs for the 2000 realizations of Analysis 2and Analysis 3a. Two curves were developed for Analysis 3a: one

Y.-N. Huang et al. / Nuclear Engineering and Design 241 (2011) 3985– 3995 3991

a. Node 201

b. Node 1009

c. Node 216

876543210

AFSA (g)

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failu

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Median

Mean

a= 2.26 gβr= 0.26βu= 0.34

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11109876543210

AFSA (g)

0

0.25

0.5

0.75

1

Pro

babi

lity

of fa

ilure

a= 3.15 gβr= 0.26βu= 0.34

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2520151050

AFSA (g)

0

0.25

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0.75

1

Pro

babi

lity

of fa

ilure

a= 7.0 gβr= 0.26

βu= 0.34

^

F1

elp(

ˇ

wtt

0.40.350.30.250.20.150.10.050

Probability of unacceptable performance

0

0.25

0.5

0.75

1

Cum

ulat

ive

dist

ribut

ion

func

tion

Ana lysis 2

Ana lysis 3a ,reali zation s

Ana lysis 3a ,

log nor mal dist.

the 2000 realizations for the benchmark Analysis 1d. The results

ig. 7. Median and mean fragility curves for the secondary systems at Nodes 201,009 and 216.

stablished by sorting the 2000 realizations, and one by fitting aognormal distribution to the data, where the median (�) and dis-ersion (ˇ) were estimated from the same 2000 realizations using2) and (6), respectively:

=√

1n − 1

∑n

t=1(ln yt − ln �)2 (6)

here yi is the value of each realization and n is the total number ofhe realizations. The goodness-of-fit test shows that the distribu-ion computed using the mean fragility curves is well represented

Fig. 8. Distributions of the probability of unacceptable performance for Analyses 2and 3a for the intensity-based assessment.

by a lognormal distribution. The curve for Analysis 2 was estab-lished by sorting the 2000 realizations.

The arithmetic means of the two sets of 2000 realizations forAnalysis 2 and Analysis 3a are 0.004 and 0.079, respectively (see thelast column of Table 4). This result is not surprising since the valuesof AFSA in the underlying demand-parameter matrix of Table 1are much smaller than the value of a for the corresponding familyof fragility curves, where the probability of failure for the mediancurve is smaller than that for the mean curve (see Fig. 7).

The use of mean and not median fragility curves is more appro-priate for risk assessment since both aleatory randomness andepistemic uncertainty is considered. Importantly, that use of meanfragility curves (Analysis 3) will provide an unbiased estimate of themean probability of unacceptable performance generated using afamily of fragility curves (the benchmark Analysis 1d).

3.5. Sensitivity studies: the number of row vectors in thedemand-parameter matrix

The results of Fig. 8 for Analysis 3a raise one question: canAnalysis 3a be performed for only one trial, instead of 2000, andreasonably estimate the benchmark value of 0.079? To address thisquestion, the confidence level for the realization from each trialof Analysis 3a to be within ±10% of its mean value was computedusing the difference between the ordinates of the lognormal CDF ofFig. 8 for Analysis 3a associated with the probabilities of unaccept-able performance of 0.071 (=0.9 × 0.079) and 0.087 (=1.1 × 0.079).For this example, the confidence level is 30%. The confidence levelcan be increased by reducing the dispersion in the realizations forthe probability of unacceptable performance; a reduction in the dis-persion can be achieved by increasing the number of row vectorsin the demand-parameter matrix.

To study the impact of the number of row vectors on the confi-dence level of the estimation using mean fragility curves, Analysis3a was repeated using 500, 1000, 1500, 2000, 3000 and 5000 rowvectors. These analyses were denoted Analyses 3b through 3g,respectively.

Fig. 9 presents the CDFs of the probability of unacceptable per-formance for Analyses 3a (200 rows), 3b (500 rows), 3c (1000 rows),3e (2000 rows) and 3g (5000 rows) using the lognormal distri-butions, where the median and dispersion (logarithmic standarddeviation) for each analysis were estimated from the 2000 corre-sponding realizations, together with the CDF and mean value of

for Analyses 3a through 3g all provide an unbiased estimate of themean value of the distribution for Analysis 1d but with differentdispersions.

3992 Y.-N. Huang et al. / Nuclear Engineering and Design 241 (2011) 3985– 3995

Table 5Statistics of Analysis 3 for the intensity-based assessment of the sample NPP.

Analysis 3a 3b 3c 3d 3e 3f 3g

Number of row vectors 200 500 1000 1500 2000 3000 5000Arithmetic mean 0.079 0.079 0.079 0.079 0.079 0.079 0.079Median (Eq. (2)) 0.077 0.078 0.079

Dispersion (Eq. (6)) 0.26 0.16 0.11

Confidence level 0.30 0.47 0.65

0.250.20.150.10.050Probability of unacceptable performance

0

0.25

0.5

0.75

1

Cum

ulat

ive

dist

ribut

ion

func

tion

Analysis3a

3b

3c

3e

3g

1d

mean, 1d

Fa

iladvf2

3c

iFaoryt

te

P

wdcraitisA

S

using a seismic hazard curve over a wide range of MAFE. Fig. 11presents the seismic hazard curve used in the time-based assess-ment, which was developed for a period of 0.14 s based on the

2 The scenario-based assessment was performed for a magnitude-distance pair of(5.3, 7.5 km) identified from the deaggregation of the short-period spectral demandat an annual frequency of exceedance of 10−5 for the sample NPP site per the pro-cedure of USNRC Regulatory Guide 1.165 (USNRC 1997). A logarithmic standarddeviation (ˇ) of 0.62 was identified using the attenuation relationship of Campbell(2003) for the spectral demand at a period of 0.14 second for the magnitude-distancepair and used to scale ground motions for the risk assessment of the sample reactorbuilding. This high value of for the spectral demand resulted in the significantdispersion in the functions of fR of Fig. 10 for the scenario-based assessment. Moreinformation for this scenario-based assessment can be found in Huang et al. (2008).

3

ig. 9. Distributions of the probability of unacceptable performance for Analyses 1dnd 3 for the intensity-based assessment.

Table 5 summarizes the statistics for Analyses 3a through 3g,ncluding the arithmetic mean, median, dispersion and confidenceevel of the 2000 realizations for each analysis. The values of meannd median are almost constant whereas the value of dispersionecreases and the confidence level increases as the number of rowectors increases. For this example, the demand-parameter matrixor assessment using mean fragility curves should consist of at least000 (5000) row vectors to achieve a confidence level of 80% (95%).

.6. Sensitivity studies: the correlation in responses of NPPomponents

To address question 3 of Section 1, the SSM procedure describedn Section 2 of Huang et al. (2011) together with the fault tree ofig. 4, the mean fragility curves of Fig. 7 and the responses of Table 1re used to compute the probability of unacceptable performancef the sample NPP subjected to the ground motions of Fig. 3a. Theesult is compared with the benchmark value obtained from Anal-sis 1d (0.079) to study the impact of this alternative process onhe probability of unacceptable performance.

The probability of occurrence for each of the six basic events inhe fault tree of Fig. 4 was computed using (7), which is (4) of Huangt al. (2011),

C =∫ ∞

0

GC (r)fR(r)dr (7)

here GC is the corresponding fragility curve and fR is a lognormalistribution with � and estimated from the ASFA values in theorresponding column of Table 1 using (2) and (6) of this paper,espectively. The probabilities of failure for the secondary systemst Nodes 201, 1009 and 216 are 0.016, 0.013 and 0.004, respectively,n the X direction and 0.018, 0.018 and 0.013 in the Y direction. Sincehe six basic events are assumed to be independent, the probabil-ty of the unacceptable performance can be computed using the

ix probabilities to be 0.079, equal to the benchmark value fromnalysis 1d.

The results presented above cannot be interpreted as that theSM and the proposed procedure will always generate the same

0.079 0.079 0.079 0.0790.088 0.078 0.063 0.0430.75 0.80 0.89 0.96

result. In the SSM procedure, the correlation in responses of NPPcomponents is not considered in the computation of the probabilityof component failure. In the proposed procedure, this correlationis preserved when the demand-parameter matrix is generated.Whether this correlation has a significant impact on the probabilityof unacceptable performance depends on the relationship betweenGC and fR.

The blue dotted lines of Fig. 10 present the distributions of ASFAat Nodes 201, 1009 and 216, respectively, in the X direction for theintensity-based assessment (fR), and the solid lines present the dis-tributions of the capacity of the secondary systems at Nodes 201,1009 and 216 (or, the first derivative of GC). In each panel of Fig. 10,the range of AFSA for fR is much narrower than that for the deriva-tive of GC and distributes at abscissas where the failure is less likelyto occur. For such a circumstance, the correlation of the responsesat the nodes has insignificant impact on the probability of failure.However, when the distribution of the response of a NPP compo-nent spans that of the capacity of the component, it is important toinclude the response correlation in the risk computation. The reddashed line in each panel of Fig. 10 presents fR estimated from theresults of a scenario-based assessment of the sample NPP2 and withthe range of AFSA much wider than the fR for the intensity-basedassessment. For the fR represented by the dashed lines of Fig. 10and the fault tree of Fig. 4, where the unacceptable performance isdefined as the occurrence of any of the six basic events, ignoringthe correlation of component responses is expected to produce ahigher probability of unacceptable performance.3 To confirm thishypothesis, the procedures of SSM and Analysis 3a were used toanalyze the ASFA values for the scenario-based assessment of thesample NPP. The probability of unacceptable performance for SSMis 0.67 and that for Analysis 3a is 0.52 (mean of 2000 realizations).

4. Time-based assessment

4.1. Seismic hazard (Step 2) and response-history analysis(Step 3)

A time-based assessment is performed as a series of intensity-based assessments with target spectral demands characterized

For a set of six AFSA values, if one value is small, there is a greater probability thatthe other five values are small if the correlation of AFSA is included in the analysis.The probability of unacceptable performance defined using the fault tree of Fig. 4 isthus smaller for the analysis including the correlation of AFSA than for the analysisignoring the correlation.

Y.-N. Huang et al. / Nuclear Engineering and Design 241 (2011) 3985– 3995 3993

a. Node 206

b. Nod e 10 09

c. Node 216

121086420AFSA (g)

0

1

2

3P

rob

ab

ility

de

nsi

ty f

un

ctio

n

Derivative of GCfR, Intensity-based

fR, Scena rio-based

121086420AFSA (g)

0

1

2

3

Pro

ba

bili

ty d

en

sity

fu

nc

tion

121086420AFSA (g)

0

1

2

3

Pro

ba

bili

ty d

en

sity

fu

nct

ion

Fs

rEc

nows1u

32.521.510.50

Earthquake intensity, e (g)

1E-006

1E-005

0.0001

0.001

0.01

Mea

n an

nual

freq

uenc

y of

exc

eeda

nce

0.21 0.54 0.87 1.2 1.53 1.86 2.19 2.52

ig. 10. Distributions of (1) capacities and (2) demands for the intensity- andcenario-based assessments for the secondary systems at Nodes 206, 1009 and 216.

esults of probabilistic seismic hazard analysis presented in theSP report. Huang et al. (2011) presents information on the hazardurve of Fig. 11.

Fig. 11 illustrates the computation of the target spectral ordi-ates for scaling ground motions for the time-based assessmentf the sample reactor building. The range of spectral acceleration

as selected as 0.05–2.68 g, where the lower- and upper-bound

pectral ordinates are associated with MAFEs of 1.43 × 10−3 and.0 × 10−6, respectively. The appropriateness of this range is eval-ated at the end of Section 4.2 using the results of the time-based

Fig. 11. Seismic hazard curve for the time-based assessment of the sample NPP.

assessment. The range of spectral acceleration was split into eightequal intervals and the midpoint value in each interval identifiesa target spectral demand for the scaling of ground motions. Theeight target spectral ordinates are identified in the figure by thesymbol “×” with values of 0.21, 0.54, 0.87, 1.2, 1.53, 1.86, 2.19 and2.52 g. The Mean Annual Frequency (MAF, ��H,i) associated witheach spectral interval is computed as the difference in the MAFEs(�H,i) at the boundaries of the interval and presented in the secondcolumn of Table 6.

Eight bins of ground motions, termed Bins T1 through T8, usedin the time-based assessment for the sample reactor building weredeveloped by amplitude scaling the eleven SGMS ground motionsdescribed in Section 3.1 to each of the eight target spectral ordinatesidentified in Fig. 11 at a period of 0.14 s. Unidirectional nonlinearresponse-history analyses were performed for the sample reactorbuilding subjected to the Bins T1 through T8 ground motions inthe X and Y directions. The response-history analysis for each binof ground motions produced an 11 × 6 demand-parameter matrix,where the number of rows is equal to the number of simulationsand the number of columns is determined by the number of basicevents identified in the lowest level of the fault tree of Fig. 4. Table 7presents the demand-parameter matrix for the time-based assess-ment of the sample reactor building subjected to the Bin T1 groundmotions.

4.2. Risk computation (Steps 4 and 5)

The sensitivity analyses of Sections 3.4 and 3.5 show that themean probability of unacceptable performance computed using afamily of fragility curves can be estimated without bias and withhigh confidence by analysis using mean fragility curves and thou-sands of row vectors. This procedure was used for the time-basedassessments of the sample reactor building.

For each underlying 11 × 6 demand-parameter matrix for BinsT1 through T8, five thousands of 1 × 6 row vectors were gener-ated per the procedure of Yang et al. (2009) for a confidence levelover 95% in the estimate of the mean probability of unacceptableperformance (see Table 5). A single value of the probability of unac-ceptable performance was computed for each ground-motion binof T1 through T8 using the mean fragility curves of Fig. 7 and the5000 row vectors for the corresponding ground-motion bin. Theproduct of the probability of unacceptable performance for eachground-motion bin (Pi) and the MAF associated with the bin (��i)

was computed; the summation of the eight products was definedas the mean annual frequency of unacceptable performance for thesample reactor building.

3994 Y.-N. Huang et al. / Nuclear Engineering and Design 241 (2011) 3985– 3995

Table 6Computation of �UP of the sample NPP.

Ground-motion bin ��H,i Probability of unacceptable performance, Pi ��H,i × Pi��H,i×Pi∑8

i=1��H,i×Pi

T1 1.35E-03 0 0 0.00T2 5.18E-05 0.16 8.29E-06 0.32T3 1.24E-05 0.71 8.80E-06 0.34T4 4.63E-06 0.96 4.44E-06 0.17T5 2.23E-06 0.99 2.21E-06 0.08T6 1.08E-06 1 1.08E-06 0.04T7 6.90E-07 1 6.90E-07 0.03T8 4.59E-07 1 4.59E-07 0.02Mean annual frequency of unacceptable

performance∑8

i=1��i × Pi

2.6E-05 –

Table 7AFSA at Nodes 201, 1009 and 216 for the time-based assessment of the sample NPP subjected to the Bin T1 ground motions.

GM No. AFSA in the X direction (g) AFSA in the Y direction (g)

Node 201 Node 1009 Node 216 Node 201 Node 1009 Node 216

1 0.45 0.58 1.19 0.45 0.59 1.382 0.36 0.50 0.93 0.35 0.53 1.133 0.36 0.57 1.03 0.33 0.46 1.114 0.49 0.68 1.32 0.47 0.68 1.345 0.34 0.47 0.88 0.28 0.41 0.956 0.41 0.61 0.99 0.34 0.51 1.127 0.31 0.44 0.84 0.34 0.42 1.008 0.36 0.47 0.90 0.39 0.57 1.22

5

6

7

fbTpcTtesre0tat

5

Nfi

ftaffi

dufor

9 0.43 0.53 0.910 0.30 0.42 0.811 0.47 0.58 1.0

Table 6 summarizes the computation of the mean annualrequency of unacceptable performance of the sample reactoruilding, including the values of ��i, Pi and ��i × Pi for Bins1 through T8 and the mean annual frequency of unacceptableerformance, 2.6 × 10−5. Table 6 shows that the probability of unac-eptable performance for Bin T1 is 0 and that for each of Bins T6,7 and T8 is 1. The last column in Table 6 presents the contribu-ion to the annual frequency of unacceptable performance fromach intensity level, which can be used to evaluate the range ofpectral acceleration selected for the time-based assessment. Theatio for Bin T1 is 0, which deems a lower-bound spectral accel-ration smaller than 0.05 g unnecessary. The ratio for Bin T8 is.02, which implies that the contribution for the spectral accelera-ions higher than the upper-bound value considered herein to thennual frequency of unacceptable performance is insignificant andhe upper-bound spectral acceleration used herein is sufficient.

. Conclusions

Procedures and results of seismic risk assessments for a samplePP reactor building are presented to provide the technical basis

or and to demonstrate the risk assessment methodology proposedn the companion paper. The key conclusions are:

The methodology of Huang et al. (2011) enables the use ofragility curves defined using structural response parameters forhe seismic risk assessment of NPPs. The methodology can bepplied to a time-based assessment to evaluate the mean annualrequency of unacceptable performance of a NPP and the per-ormance of a NPP subjected to earthquake shaking for a givenntensity or scenario.

The use of mean fragility curves for NPP components and aemand-parameter matrix of 1000+ row vectors can provide an

nbiased estimate of the mean probability of unacceptable per-ormance for the NPP with high confidence. The required numberf row vectors in a demand-parameter matrix is a function of theequired confidence level and the dispersion in the probability of

0.48 0.67 1.340.31 0.41 1.060.54 0.70 1.48

unacceptable performance. In this case, 2000 (5000) row vectorsare required to achieve a confidence level of 80% (95%).

The methodology of Huang et al. (2011) includes the correla-tion of the responses of NPP components in the risk computation.If the response of each NPP component (fR) distributes in a rangemuch narrower than the capacity of the component (GC), the corre-lation has an insignificant impact on the probability of unacceptableperformance and vice versa.

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