Annals of Nuclear Energy 38 (2011) 2575–2580

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Annals of Nuclear Energy

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Reliability data update method for emergency diesel generator of Daya BayNuclear Power Plant

Muhammad Zubair ⇑, Zhang ZhijianCollege of Nuclear Science and Technology, Harbin Engineering University, 145 Nantong Street, Nangang District, Harbin, Heilongjiang 150001, PR China

a r t i c l e i n f o

Article history:Received 26 May 2011Received in revised form 9 July 2011Accepted 15 July 2011Available online 1 September 2011

Keywords:Bayes’ methodCalculation of parametersClassical methodFailure rate estimationUpdating reliability data of EDG

0306-4549/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.anucene.2011.07.020

⇑ Corresponding author.E-mail addresses: zubairheu@gmail.com (M. Zubair

(Z. Zhijian).

a b s t r a c t

In the field of Living Probabilistic Safety Assessment (LPSA) the reliability data updating is an importantfactor. In risk analysis equipment failure data is needed to estimate the frequencies of events contributingto risk posed by a facility. Five years data of emergency diesel generator (EDG) of Daya Bay Nuclear PowerPlant (NPP) has been studied in this paper. The data updating process has been done by using two meth-ods, i.e., the classical method and Bayesian method. The aim of using these methods is to calculate theoperational failure rate (k) and demand failure probability (p). The results show that the operational fail-ure rate is 1.7E�3 per hour and the demand failure probability is 2.4E�2 demand per day for Daya BayNPP. By comparing the results obtain from classical and Bayesian methods with EDF (Electric De France)it is concluded that the design and construction of Daya Bay NPP is very different than EDF therefore thereliability parameters used in Daya Bay NPP is based on the classical method.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The internal events of nuclear power plant are complex andinclude equipment maintenance, equipment damage, etc. Theseevents will affect the probability of the current risk level of thesystem as well as the reliability of the equipment parametervalues, so such kind of events will serve as an important basis forsystematic analysis and calculation.

Reliability is not a deterministic parameter it is expressed asprobability in terms of likelihood functions which shows thatsystem is operational. The exponential reliability expression isgiven by;

RðtÞ ¼ expð�ktÞ ð1Þ

where R(t) is probability that item will operate with out failure forthe time period t and k is item failure rate (Lakner et al., 1985). Innuclear power plants the equipment status changes due to mainte-nance or failure. So in order to calculate reliability data we use his-torical information about the equipment states and then useBayesian theory for updating. General reliability database storethe reliability data of Probabilistic Safety Assessment (PSA) equip-ment including failure rate and Mean Time To Repair (MTTR).

Daya Bay NPP is a two-unit PWR NPP (900 MWe each), importedfrom France and began its commercial operation since 1994 inChina (Xuhong et al., 2007). The research related to PSA model

ll rights reserved.

), zhangzhijian@hrbeu.edu.cn

development and risk monitor tools had been finished in the past.Due to current changes in NPP some questions arises about theupdating of reliability data for every system and component. So inthis paper we focus our research on EDG of Daya Bay NPP and up-date failure rates with the help of Bayes’ theorem. For this purposewe have developed a computer based program named Update Datafor Emergency Diesel Generator (UDEDG). This program use genericand specific data for EDG and update important reliability data likeshape parameter (a), scale parameter (b) and failure rate easily.

There are different methods for reliability data updating asshown in Fig. 1. The first one is that the operational data shouldfirst reduced and put into reliability math models. These modelsinclude reliability block diagrams (RBDs) or fault tree diagrams(FTDs). Reliability block diagram (RBD) is a graphical analysis tech-nique, which expresses the concerned system as connections of anumber of components in accordance with their logical relationof reliability. Series connections represent logic ‘‘and’’ of compo-nents, and parallel connections represent logic ‘‘or’’, while combi-nations of series and parallel connections represent voting logic(Gua and Yang, 2007). Reliability block diagrams are frequentlyused to model the effect of item failures on system performance.It often corresponds to the physical arrangement of items in thesystem. Fault tree diagrams is the most common technique to eval-uate reliability of a system, these are graphical representations ofvarious faults that will result in an undesired event. They allow avariety of qualitative and quantitative analysis. Fault Tree Analysis(FTA) provides a logic model of basic causes leading to top event.FTA provides prioritization to lead top event. In nuclear industryFTA can be used in updating basic event data, maintenance and

Reduced Statistically evaluated

Bayesian updating

Direct apply Reliability math models

Operational data

Beta posterior distribution

Beta prior distribution

Correlation coefficient

analysis

Multiple regression

test

Reliability data update

Reliability block diagram

Fault tree diagram

F-test

Fig. 1. Different method for calculation of reliability data.

2576 M. Zubair, Z. Zhijian / Annals of Nuclear Energy 38 (2011) 2575–2580

estimation of failure rates. The second method is to apply statisti-cal techniques such as multiple regression test, F-test, and correla-tion coefficient analysis to identify those variables that impactsystem and component failure rate. Multiple regression is a statis-tical technique that allows us to predict someone’s score on onevariable on the basis of their scores on several other variables. Iftwo variables are correlated, then knowing the score on one vari-able will allow you to predict the score on the other variable. Thestronger the correlation, the closer the scores will fall to the regres-sion line and therefore the more accurate the prediction. Multipleregression is simply an extension of this principle, where we pre-dict one variable on the basis of several other variables. F-test is de-rived from a scientist name (Ronald A. Fisher). F-test can be used totest the equality of two population variances. Suppose a researcherwants to test whether or not two independent samples have beendrawn from normal populations with the same variability. In thiscase, the researcher uses the F-test to do this study. The F-test

Reliability analysis module

Reliability data update module

Preventive replacement module

Equipment availability

event

Expert opinion

Basic efile

Schedulequery

Event report

Importance degree

Fig. 2. Six modules for reli

can also be used to know whether there is any homogeneity be-tween the two independent estimates of the population variance.Correlation coefficient analysis is a measure of the relation be-tween two or more variables. The measurement scales used shouldbe at least interval scales, but other correlation coefficients areavailable to handle other types of data. Correlation coefficientscan range from�1.00 to +1.00. The value of�1.00 represents a per-fect negative correlation while a value of +1.00 represents a perfectpositive correlation. The third and fourth methods are the Classicaland Bayesian statistics that are adopted in this paper.

In PSA equipment state changes particularly in relation toequipment failure and maintenance that will affect the reliabilitydata values. The function of reliability data update module is to re-new original reliability data and provision of equipment unavail-ability for the system reliability module. So that we can reflectthe impact on the system risk level that change in equipment reli-ability parameters as shown in Fig. 2.

preventive replacement cycle

Fault tree analysis module

vent frequency of fault tree

Graphic modeling module

Output module

CDF tImportance

Risk curveSchedule

Current risk

ET/FT model

ability data updating.

Table 3Classical calculation of reliability parameters.

Name of reliability parameter Value of parameters

Operational failure rate 1/579.37 = 1.7E�3 per hourDemand failure probability 7/290 = 2.4E�2 demand per day

M. Zubair, Z. Zhijian / Annals of Nuclear Energy 38 (2011) 2575–2580 2577

2. Classical method for the calculation of reliability parameters

To estimate reliability parameters there are two approaches oneis Bayesian method and other is Classical method. In order to cal-culate event frequency (k), the classical estimate method usedMaximum Likelihood estimate (MLE). The MLE of k is given by;

k ¼ x=t ð2Þ

where x is observed number of events and t is observed time period.In classical estimate approach the probability of a random event isdefined as long term fraction of times that the event would occur ina large number of trials. The primary use of classical approach is toexamine data and decide which model is useful (NUREG/CR 2300,1983). In a classical analysis, knowledge and expertise also play arole, but formally, in general serving only as aids in choosing prob-ability models and relevant data. For example, data obtained undernormal operating conditions may or may not be applicable to acci-dent conditions. An understanding of the situation is needed to re-solve this question. Once such questions are resolved, a classicalanalysis lets the data ‘‘speak for themselves.’’ The users of a classicalanalysis must be aware that limited data can lead to impreciseestimates.

2.1. EDG parameters calculation based on classical method

The data for EDG considered here has been collected from Jan-uary 1997 to December 2001 as shown in Table 1. The equipmentfailure data is sample from EFS (Experience Feedback System).Each nuclear power generating units of diesel generator systemconsists of two identical entities separate and independent seriesA (LHP) and series B (LHQ) component, each diesel generator setsand related auxiliary equipment installed in a Separated factories.In case of electricity loss, EDG supplies 6.6 kev power to both A andB series.

Each diesel generator set includes the following equipment:

(i) Two diesel engines and its immediate installation ofequipment.

(ii) A generator and the excitation and protection equipment.

Table 1Five years EDG data.

Time Operational time (h) Failure time Start time

Unit 1 Unit 2 Demandfailure

Operationalfailure

1997 187.5 1 0 761998 99 1 1 551999 48.22 42.88 3 0 522000 44.65 46.32 2 0 482001 62.95 47.85 0 0 59

Total 579.37 7 1 290

Table 2Summary of EDG data.

Name Data

Equipment type 1LHP001AP,1LHQ001AP, 2LHP001AP,2LHQ001AP

Data collection time 1997/1/1-2001/12/31Total operation time 579.37 hTotal start time 290 timesTotal operational failure

time1

Total demand failure time 7

(iii) Auxiliary systems: fuel system, lubricating system, enginecooling and preheating system, air starting system combus-tion air and engine exhaust systems, ventilation systems anddiesel engine plant instrumentation measurement and con-trol equipment.

Table 2 shows the summary of EDG data and Table 3 gives thecalculated values of operational failure rate and demand failureprobability.

3. Bayesian method for update parameters

To estimate parameters for Probabilistic Risk Analysis (PRA) theBayesian approach works better for two reasons. First the datafrom reliable equipments are typically spare with few or even zeroobserved failure. The Bayesian approach provides a mechanism forincorporating such information as prior belief. Second the Bayesianframe work allows straightforward propagation of basic eventuncertainty through a logical model to produce an uncertainty onthe frequency of the undesirable end state.

The first step in Bayesian parameter estimation is identificationof the parameters to be estimated. The second step is developmentof a prior distribution that appropriately quantifies the analyst’sstate of knowledge concerning the unknown parameters. The thirdstep is collection of evidence and construction of an appropriatelikelihood function. The fourth and final step is derivation of theposterior distribution using Bayes’ theorem.

3.1. Bayes’ theorem and beta distribution

In order to calculate demand failure probability for EDG thelikelihood function is defined by binomial distribution. The proba-bility of a failure on demand is denoted by p, a unit less quantity.The data consist of k failures in n demands, with 0 6 k 6 n and pis between 0 and 1. Before the data are generated the number offailures is random and denoted by X, so the probability of x failuresin n demands is (Zubair et al., 2011);

PrðX ¼ k=pÞ ¼ n!

k!ðn� kÞ! pkð1� pÞn�k ð3Þ

The conjugate family of prior distribution for binomial data is fam-ily of beta distribution. The beta distribution with parameters a andb has density function;

f ðpriorÞ ¼ Cðaþ bÞCðaÞCðbÞp

a�1ð1� pÞb�1 ð4Þ

This equation can also be written as;

f ðpriorÞ / pa�1ð1� pÞb�1

According to Bayes’ theorem the posterior distribution is related toprior distribution by;

f ðpostÞ / Prðx=pÞf ðpriorÞ

Put Eqs. (3) and (4) into Bayes’ theorem;

f ðp=kÞ ¼n!

k!ðn�kÞ! ðPÞkð1� PÞn�k CðaþbÞ

CðaÞCðbÞ ðPÞa�1ð1� PÞb�1

R 10

n!k!ðn�kÞ! ðPÞ

kð1� PÞn�k CðaþbÞCðaÞCðbÞ ðPÞ

a�1ð1� PÞb�1dp

After canceling the constants from numerator and denominator;

Table 6Reliability prior parameters for EDG defined by EDF.

Name of parameter Reliability parameter Error factor

Operational failure rate 7.7E�3 1.4Demand failure probability 3.4E�3 1.4

Table 5Bayes’ estimate results (demand failure probability).

Posterior mean value 4.3E�3 demand per dayPosterior distribution Beta distributionPosterior parameter a 3.0E1Posterior parameter b 7.0E+3Failure rate (k) 3.27E�3

2578 M. Zubair, Z. Zhijian / Annals of Nuclear Energy 38 (2011) 2575–2580

f ðp=kÞ ¼ ðPÞkþa�1ð1� PÞn�kþb�1

R 10 ðPÞ

kþa�1ð1� PÞn�kþb�1 dp

As;

CðaÞCðbÞCðaþ bÞ ¼

Z 1

0xa�1ð1� xÞb�1 dx

So above equation becomes;

f ðp=kÞ ¼ ðPÞkþa�1ð1� PÞn�kþb�1

Cðaþ kÞCðn� kþ bÞ=Cðaþ nþ bÞ

f ðp=kÞ ¼ Cðaþ nþ bÞðPÞkþa�1ð1� PÞn�kþb�1

Cðaþ kÞCðn� kþ bÞ

f ðp=kÞ / Pkþa�1ð1� PÞn�kþb�1

This means;

apost ¼ kþ aprior ð5Þ

bpost ¼ n� kþ bprior ð6Þ

Fig. 3. Comparison of operational failure rate.

Fig. 4. Comparison of demand failure probability.

3.2. Bayes’ theorem and gamma distribution

To calculate operational failure rate, Poisson likelihood functioninstead binomial and gamma distribution in place of beta distribu-tion has been selected. The Poisson distribution is given by;

PrðX ¼ x=kÞ ¼ e�ktðktÞx

x!ð7Þ

The conjugate family of prior distribution for Poisson data is familyof gamma distribution. The gamma distribution with parameters aand b has density function;

f ðpriorÞ ¼ ba

ða� 1Þ! ka�1e�kb ð8Þ

Above equation can be written as;

f ðpriorÞ / ka�1e�kb

So with the help of Eqs. (7) and (8) and by using Bayes’ theorem asin previous calculation we can write as;

fpost / e�ktðktÞx

x!ka�1e�kb

By simplifying;

fpost / kðxþaÞ�1e�kðtþbÞ

Hence posterior gamma distribution becomes;

apost ¼ xþ aprior ð9Þ

bpost ¼ t þ bprior ð10Þ

3.3. Calculation of EDG failure rate with Bayes’ method

The posterior values obtained from Eqs. (5), (6), (9), and (10) arethe main tools to update EDG data as well as to make UDEDG. The

Table 4Bayes’ estimate results (operational failure rate).

Posterior mean value 6.7E�3 per hourPosterior distribution Gamma distributionPosterior parameter a 2.3E+1Posterior parameter b 3.5E+3Failure rate (k) 6.28E�3

Fig. 5. Effect of a on k in case of beta distribution.

Fig. 6. Effect of a on k in case of gamma distribution.

M. Zubair, Z. Zhijian / Annals of Nuclear Energy 38 (2011) 2575–2580 2579

mean value and failure rate obtained from following formulas.Tables 4 and 5 describes the estimation of operational failure rateand demand failure probability with the help of Gamma and Betadistribution respectively. While in Table 6 EDG failure rate obtainfrom EDF has been shown.

Mean ¼ l ¼ aaþ b

kpost ¼ aprior=bpost

Enter

Input size anof arra

Mean davalue =

Betaprior=0.5Betapost=n-k+betaprior

Mean=alphapost/ alphapost+betapostVariance=alphapost*betapost/

(alphapost+betapost+1)(alphapost+betapost)2

Lembdapost=alphaprior/betapost

Print alphapost, betapost, mean,

variance, lembdapost

Exit

Yes

Put value of n,k,alpha

prior

Fig. 7. Algorithm for calcu

4. Types of failures and failure rate estimation

A failure is defined as the loss of the ability of an item, a com-ponent or a system to perform its required function (IAEA, 1992).An item or component failure is always defined in relation to thesystem in which the item or component resides. There are differenttypes of failure among them for EDG we have consider time relatedfailure and demand related failure. A failure occurrence (e.g. perhour) of a component or system which is in the running or func-tioning mode of operation, is known as time related failure.

Failure rate or failure per hour ¼ Number of failuresOperating time

A failure on demand is defined as a failure occurrence of a com-ponent or system when it is demanded to start operation or tochange its present state.

Failure probability or failure per demand ¼ Number of failuresNmber of demands

From Tables 4–6 we have calculate operational failure rate anddemand failure probability. For the EDG of Daya Bay NPP these fail-ures rates are calculated from EDF, Bayesian method and Classicalmethod. Due to change in design and construction the data from

d value y

ta 0

Beta prior=alphaprior/mean data valueBetapost=t+betaprior

Mean=alphapost/betapostVariance=(alphapost/betapost)2

Lembdapost=alphaprior/betapost

Print alphapost, betapost, mean,

variance, lembdapost

No

Put value of n,k,alpha

prior,years

lations of parameters.

2580 M. Zubair, Z. Zhijian / Annals of Nuclear Energy 38 (2011) 2575–2580

EDF and Bayesian method is not suitable for Daya Bay NPP, so re-sults obtained from Classical method has been selected. A compar-ison of data from these sources can be seen in Figs. 3 and 4.

By increasing the value of Shape parameter (a) the failure rate(k) increases both in beta and gamma distributions as shown inFigs. 5 and 6.

5. Algorithm for UDEDG (Update Data for Emergency DieselGenerator)

To calculate reliability parameters easily a computer based pro-gram named UDEDG has been designed which works in same timefor both beta and gamma distribution. With the help of this pro-gram reliability parameters can be calculated easily. In start of cal-culation process we have to give generic data values then programwill calculate mean value. If this mean value is zero then programwill calculate parameters with the help of beta distribution but ifnot so then program automatically works for gamma distribution.There is also another situation in which there is no generic dataand only success or failures rate are available, in such case programwill prefer to works with the help of beta distribution.

So with the help of UDEDG we can calculate operational and de-mand failure probability at the same time. Fig. 7 represents thealgorithm for the calculation of parameters in any possible situa-tion by using the tools of gamma and beta distribution.

6. Conclusion

In this paper three types of data of EDG for Daya Bay NPP hasbeen considered and updated with a computer based programUDEDG. Data has been selected from EDF, classical and Bayesianmethods. Operational failure rate and demand failure probabilityhas been calculated. Failures rates obtain from EDF and Bayesianmethod are close to each other but results from classical methodare different. On applying these three types of results in DayaBay NPP it is conclude that classical results (1.7E�3 per hour and2.4E�2 demand per day) are well matched with construction andenvironment of Daya Bay NPP.

References

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NUREG/CR 2300, 1983. Human Reliability Analysis, vol. I (Chapter 4).Xuhong, H. et al., 2007. Maintenance risk management in Daya Bay nuclear power

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