Parameter Estimation of a Reliability Model of Demand ...
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Research ArticleParameter Estimation of a Reliability Model ofDemand-Caused and Standby-Related Failures of SafetyComponents Exposed to Degradation by Demand Stress andAgeing That Undergo Imperfect Maintenance
S. Martorell,1,2 P. Martorell,1,2 A. I. SΓ‘nchez,2,3 R. Mullor,4 and I. MartΓ³n1,2
1Department of Chemical and Nuclear Engineering, Universitat Politecnica de Valencia, Valencia, Spain2MEDASEGI, Valencia, Spain3Department of Statistics and Operational Research, Universitat Politecnica de Valencia, Valencia, Spain4Department of Mathematics, Universidad de Alicante, Alicante, Spain
Correspondence should be addressed to S. Martorell; [email protected]
Received 7 August 2017; Accepted 5 November 2017; Published 18 December 2017
Academic Editor: Giovanni Falsone
Copyright Β© 2017 S. Martorell et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
One can find many reliability, availability, and maintainability (RAM) models proposed in the literature. However, such modelsbecome more complex day after day, as there is an attempt to capture equipment performance in a more realistic way, suchas, explicitly addressing the effect of component ageing and degradation, surveillance activities, and corrective and preventivemaintenance policies.Then, there is a need to fit the bestmodel to real data by estimating themodel parameters using an appropriatetool. This problem is not easy to solve in some cases since the number of parameters is large and the available data is scarce. Thispaper considers two main failure models commonly adopted to represent the probability of failure on demand (PFD) of safetyequipment: (1) by demand-caused and (2) standby-related failures. It proposes a maximum likelihood estimation (MLE) approachfor parameter estimation of a reliability model of demand-caused and standby-related failures of safety components exposed todegradation by demand stress and ageing that undergo imperfect maintenance. The case study considers real failure, test, andmaintenance data for a typical motor-operated valve in a nuclear power plant. The results of the parameters estimation and theadoption of the best model are discussed.
1. Introduction
The safety of nuclear power plants (NPPs) depends on theavailability of safety-related components that are normally onstandby and only operate in the case of a true demand.Thesecomponents typically have two main types of failure modesthat contribute to the probability of failure on demand:
(a) by demand-caused failure, associated with a demandfailure probability (π),
(b) standby-related failure, associated with a standbyhazard function (β).
Both are generally associated with constant values in astandard Probabilistic Risk Assessment (PRA) models, that
is, π0 and β0, respectively. Such parameters are associatedprobability density functions in PRA, which are tailoredbased on a priori generic probability distribution function,for example, exponential, lognormal, Weibull, and beta,depending on the particular sort of component, for example,motor-driven pump and motor-operated valve. A Bayesianapproach is used to combine such generic probability densityfunctions with plant specific failure data for each particularcomponent [1β4].
However, both failure modes are often affected by degra-dation such as demand-related stress and ageing, whichcause the component to degrade with chronological timeand ultimately to fail. Maintenance and test activities areperformed to control degradation and the unreliability and
HindawiMathematical Problems in EngineeringVolume 2017, Article ID 7042453, 11 pageshttps://doi.org/10.1155/2017/7042453
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unavailability of such components, although this has bothpositive and negative effects. Thus, different approaches havebeen proposed in the literature to model time-dependent πand β that take into account such effects in an either implicitor explicit way.
Samanta et al. [5, 6] proposed a well-organized foun-dation to account for ageing and the positive and adverseeffects of testing components in modelling demand failureprobability and standby hazard function models. However,this model does not take into account the positive effect ofmaintenance activities as a function of their effectiveness inmanaging component degradation due to demand-inducedstress and ageing.
As regards the standby-related failure mode, Martorell etal. [7] provide an age-dependent reliability model associatedonly with standby-related failures which explicitly takes intoaccount the effect of equipment ageing and the positiveand negative effects of maintenance activities founded onimperfect maintenance modelling. Mullor [8] proposes anapproach for parameter estimation of such a sort of imperfectmaintenance models. Marton et al. [9] propose an approachto modelling the unavailability of safety-related compo-nents associated with standby-related failures that explicitlyaddresses all aspects of the effect of ageing, maintenanceeffectiveness, and test efficiency.Other authors have proposedalternative approaches to modelling the effect of ageing andtest and maintenance activities [10β13].
As regards the demand-caused failure mode, this prob-ability of a safety component is normally considered tobe mainly affected by demand-induced stress, for example,due to true demands, proof tests, and others. The demand-induced stress is therefore modelled with a stochastic degra-dation jump in [14, 15]. These studies consider that randomshocks occur according to a Nonhomogeneous Poisson Pro-cess, leading to the immediate failure of the component. Shinet al. [16] propose an age-dependent model that considers,among others, the effect of βtest stressβ and maintenanceeffects. In general, previous studies have found that thedemand failure probability should be considered as a functionnot only of the number of tests but also of the effectivenessof maintenance activities. Thus, recently, Martorell et al. [17]have proposed a new reliability model for the demand failureprobability that explicitly addresses all aspects of the effect ofdemand-induced stress, maintenance effectiveness, and testefficiency.
In this context, the objective of this paper focuses onfitting the bestmodel to represent the real operation of safety-related equipment, dealing with the problem of estimating asignificant number of parameters considering a small amountof data. With this aim, a methodology of parameters esti-mation and model selection is developed. This methodologyallows the joint estimation of reliability and maintenancerelated parameters as well as obtaining ameasure of goodnessof fit to select the best imperfect maintenance model for eachfailure mode. This study considers a standby-related failuremodel assuming linear ageing and a demand-caused failuresmodel assuming test-induced stress. In addition, it considersimperfect maintenance adopting Proportional Age Setbackand Proportional Age Reduction for preventive maintenance
modelling. Then, maximum likelihood estimation (MLE)using a direct search algorithm based on the Nelder-MeadSimplex (NMS) method is used to estimate maintenanceeffectiveness and ageing rate simultaneously. A practicaland realistic case study is included facing the parametersestimation of a typical motor-operated valve in a nuclearpower plant. Additionally, how the estimates obtained canbe used, for example, in the planning of maintenance andsurveillance test activities with the aim of minimizing equip-ment unavailability, is shown.
The rest of this paper is organized as follows: Section 2introduces briefly the demand failure probability model andthe standby-related failure model that addresses componentdegradation because of demand-induced stress and ageing,respectively, and the positive effect of imperfect preventivemaintenance. Section 3 describes the parameter estimationmethod used to fit plant data to reliability models introducedin the previous section. Section 4 describes a case studyinvolving a motor-operated valve of a pressurized waterreactor nuclear power plant. Lastly, Section 5 presents theconcluding remarks.
2. Reliability Models underImperfect Maintenance
In this paper the models presented by Martorell et al. [7, 17]have been selected to model the standby hazard function andthe demand failure probability, respectively. In the follow-ing subsections, both models are briefly described and theexpressions involved in the parameters estimation andmodelselection are obtained under the following assumptions:
(1) Time-directed preventive maintenance effect whichdepends on its effectiveness. The effectiveness isrepresented by an imperfect maintenance modelwith parameter π, ranging in the interval [0, 1] andadopting either Proportional Age Setback (PAS) orProportional Age Reduction (PAR) model.
(2) Corrective maintenance with minimal repair. Thatis, repairing failures do not improve the age ofequipment.Therefore, for correctivemaintenance, weadopt the Bad As Old (BAO) model.
(3) A linear ageing model which is selected to model thestandby hazard function.
(4) Test-caused stress which is the only degradationmechanism considered to model the demand failureprobability.
2.1. Reliability Model of Standby-Related Failures. In the con-text of safety-related equipment of NPP, the most frequentlyused function in reliability analysis is the hazard function.The standby hazard function of equipment depends on itsage, which is a function of the chronological time elapsedsince its installation and the effectiveness of the maintenanceactivities performed on it. So, an age-dependent hazard
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function model, in period π after the maintenance numberπ β 1, can be expressed as [7]
βπ (π‘) = β (π€π (π‘)) + β0 π€ β₯ π€+πβ1, (1)
where β0 is the initial hazard function of the equipment andπ€+πβ1 is the age of the equipment immediately after the (πβ1)maintenance activity.
Adopting a linear model for hazard function, the expres-sion for the age-dependent hazard function after the mainte-nance numberπ β 1 can be written as
βπ (π‘) = πΌ β π€π (π‘) + β0 π€ β₯ π€+πβ1, (2)
where πΌ is the linear ageing rate and
π€π (π‘) = π€+πβ1 + (π‘ β π‘πβ1) π‘ β₯ π‘πβ1 (3)
with π‘πβ1 being the time in which the equipment undertakesthe (π β 1)-maintenance activity.
The cumulative hazard function in the periodπ, after themaintenance number π β 1, can be obtained by integrationfrom the hazard function given by equation (2) as
π»π (π‘) = πΌ2 (π€π (π‘))2 + β0π€π (π‘) . (4)
The age of the component immediately after the main-tenance number π β 1, π€+πβ1, and, therefore, the hazardfunction and the cumulative hazard function depend on themodel of imperfect maintenance selected (PAS or PAR).In the following subsections, the particularization of theprevious equations to PAS or PAR model is presented.
2.1.1. Proportional Age Setback Model. In the PAS approach,each maintenance activity is assumed to shift the origin oftime from which the age of the equipment is evaluated.The PAS model considers that maintenance activities reduceproportionally to a factor π, the age the equipment hadimmediately before it enters inmaintenance. If π = 0, the PASmodel simply reduces to the BAO situation, whereas π = 1corresponds to the Good As New (GAN) situation. Thus,this model is a natural generalization of both GAN and BAOmodels in order to account for imperfect maintenance. Con-sidering PAS approach the age of the equipment immediatelyafter the (π β 1)maintenance activity is given by [7]
π€+πβ1 = π‘ β πβ2βπ=0
(1 β π)π ππ‘πβπβ1 π‘ β₯ π‘πβ1. (5)
Replacing the expressions corresponding to π€π(π‘) and π€+πβ1given by (3) and (5), respectively, into (2) the expression forthe induced hazard function becomes
βπ (π‘) = πΌ[π‘ β πβ2βπ=0
(1 β π)π ππ‘πβπβ1] + β0. (6)
In a similar way, the cumulative hazard function, π»π(π‘), inthe period π, can be obtained by replacing (3) and (5) into(4) obtaining
π»π (π‘) = πΌ2 [π‘ β πβ2βπ=0
(1 β π)π ππ‘πβπβ1]2
+ β0 [π‘ β πβ2βπ=0
(1 β π)π ππ‘πβπβ1] .(7)
2.1.2. Proportional Age Reduction Model. In the PAR ap-proach, each maintenance activity is assumed to reduceproportionally the age gained from the previous main-tenance. Thus, while the PAS model considers that eachmaintenance activity reduces the total equipment age, thePARmodel assumes that maintenance only reduces a portionof the equipment age, the one gained from the previousmaintenance, keeping the rest unaffected. The PAR modelconsiders that maintenance reduces the age gained betweentwo consecutive maintenance activities by a factor π. Again,one can realize that if π = 0, the PAR model simply reducesto BAO, whereas if π = 1 it reduces to GAN.
According to the above conditions, the age of the equip-ment in instant π‘ of period π, after the (π β 1)-maintenanceactivity using the PAR model, is given by
π€+πβ1 = π‘ β ππ‘πβ1 π‘ β₯ π‘πβ1. (8)
Using a similar process as the one described for the PASmodel, but adopting (8) instead of (5), it is possible to derivethe expression for the hazard function and the cumulativehazard function of imperfect maintenance at instant π‘, underthe PAR approach
βπ (π‘) = πΌ (π‘ β ππ‘πβ1) + β0π»π (π‘) = πΌ2 (π‘ β ππ‘πβ1)2 + β0 (π‘ β ππ‘πβ1) . (9)
2.2. Reliability Model of Failures by Demand. The demandfailure probability of a component, which is normally instandby and ready to perform a safety function on demand,depends on the number of demands performed on thecomponent, which are often associated with performingsurveillance tests. In addition, it is necessary to considerthe positive effect that the preventive maintenance activitiesperformed on the equipment have on the degradation factorand, therefore, on demand probability failure.
A time-dependent demand failure probability model thataddresses the demand-induced stress and the effect of π β 1maintenance activities can be formulated for the periodπ asfollows [17]:
ππ (π‘) = π0 + π0 β ππ (π‘) (10)
with π0 being the residual demand failure probability andππ(π‘) being the degradation function.Assuming, the same degradation factor, π1, for all types
of demands, the evolution of the degradation function in the
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period numberπ, that is, betweenmaintenanceπβ1 andπ,can be expressed as
ππ (π‘) = π+πβ1 + π1 β β π‘ β π‘πβ1π β , (11)
where π+πβ1 is the degradation function immediately aftermaintenanceπ β 1 which depends on the selected imperfectmaintenance model (PAS or PAR) and βπ₯β is the floorfunction that gives the largest integer less than or equal to π₯,which returns the number of tests performed in the interval[π‘πβ1, π‘] that are performed with periodicity π.
Time-dependent evolution of the cumulative demandfailure probability,π·π(π‘), over the periodπ, can be obtainedby adding the cumulative distribution function in the π β 1maintenance to the demand probability functions in each testperformed over the periodπ. Generally,π·π(π‘) does not havea closed-form expression.
In the following subsections, the particularization ofequations ππ(π‘) and π·π(π‘) for the PAS or PAR model ispresented.
2.2.1. Proportional Age Setback Model. If a PAS modelis considered, the degradation function after maintenancenumber π β 1 assuming preventive maintenance activitiesare performed on a regular basis with constant maintenanceinterval given byπ can be formulated by [17]
π+πβ1 = π1 β ππ β πβ2βπ=0
(1 β π)π+1
= π1 β ππ β (1 β π)πβ2βπ=0
(1 β π)π .(12)
Substituting (11) and (12) into (10) the function of demandfailure probability for the periodπ can be obtained as
ππ (π‘)= π0(1 β βπ‘πβ1π βπ1πβ2β
π=1
(1 β π)π + βπ‘ β π‘πβ1π βπ1) . (13)
The distribution function of the cumulative demandfailure probability, π·π(π‘), in the period π, after the (π β 1)-maintenance activity, can be obtained, as it is mentionedabove, by summing the distribution function immediatelyafter the (π β 1) maintenance activity and the probabilityfunctions in the tests performed between the π β 1 main-tenance and π‘ to yield:
π·π (π‘) = π0(1 + βπ‘πβ1π ββ π1(πβ1β
π=1
((βπ‘πβ1π β + 1) π + 1) (1 β π)π+1βπ
+ (π + 1) (βπ‘πβ1/πβ + 1)2 )) + π0(1
+ βπ‘ β π‘πβ1π β βπ‘πβ1π βπ1πβ1βπ=1
(1 β π)π
+ β(π‘ β π‘πβ1) /πβ (β(π‘ β π‘πβ1) /πβ + 1)2 π1) .(14)
2.2.2. Proportional Age Reduction Model. In the PARapproach the degradation function immediately aftermaintenance number π assuming preventive maintenanceactivities are performed on a regular basis with constantmaintenance interval,π, is given by
π+πβ1 = π1 β (1 β π) β ππ β (π β 1) . (15)
Using an analogous procedure as the one described for thePAS model, a time-dependent model for the demand failureprobability can be obtained substituting (15) into (10) and (11)to yield
ππ (π‘)= π0 (1 + πβπ‘πβ1π βπ1 (1 β π) + βπ‘ β π‘πβ1π βπ1) . (16)
In addition, the cumulative demand failure probability,π·π(π‘), considering a PAR model is given by
π·π (π‘) = π0 (1 + (π β 1) βπ‘πβ1/πβ (βπ‘πβ1/πβ + 1)2 π1+ ((βπ‘πβ1π β + 1) π (π β 1)2 β βπ‘πβ1π β (π β 1))β β π‘πβ1π βπ1 (1 β π)) + π0 (1 + βπ‘ β π‘πβ1π β βπ‘πβ1π ββ (π β 1) π1 (1 β π)+ β(π‘ β π‘πβ1) /πβ (β(π‘ β π‘πβ1) /πβ + 1)2 π1) .
(17)
3. Methodology of Parameters Estimation andModel Selection
Manymethods for parameter estimation of reliability modelshave been proposed in the literature, such as the maximumlikelihood, methods of moments, and Bayesian estimators.In this paper, the maximum likelihood estimation methodhas been selected to estimate the parameters of the reliabilitymodels presented in Section 2. For a given model and a setof observed data, the likelihood function πΏ is the product ofprobabilities of the observed data as a function of the modelparameters. It can be applied to reliability and imperfectmaintenance models for standby-related failures and fordemand-caused failures. Thus, the likelihood function for
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standby-related failures, πΏ1(π), and the likelihood functionfor demand-caused failures, πΏ2(π), can be formulated as
πΏ1 (π | model, observed data)= β
failuresβ (π‘) β
maintenanceexp [βπ» (π‘)]
πΏ2 (π | model, observed data)= β
failures
π (π‘)1 β π· (π‘) βmaintenance
(1 β π· (π‘)) .(18)
The maximum likelihood estimation (MLE) method pro-vides estimators, called maximum likelihood estimators, ofparameters involved in reliability and maintenance models.The maximum likelihood estimations of these parametersare those values which make the likelihood function aslarge as possible, that is, which maximize the probabilityof the observed data. Since the natural logarithm is anincreasing function, the likelihood function and its logarithmachieve their maximum at the same values of their objectiveparameters. For computational purpose it is preferable tomaximize the log likelihood function. By maximizing theexpressions corresponding to logπΏ(π), the maximum like-lihood estimators of the objective parameters are obtained.In this paper, the Nelder-Mead Simplex [18, 19] algorithm is
used to maximize the likelihood functions for each proposedmodel.
The maximum likelihood estimation method provides,in addition to the parameter estimates, information on itsvariability through the Fisher information matrix, whichis defined as the opposite of the partial second derivativematrix, that is, the opposite of its Hessian. So, for the set ofestimated parameters the variance-covariance matrix as theinverse of the information matrix divided by the sample sizecan be obtained.
In particular, taking advantage of the asymptotic normal-ity of the maximum likelihood estimation, if the sample sizeis large enough, we can obtain the standard deviations of theparameter estimation as the square root of the main diago-nal of the variance-covariance matrix to obtain confidenceintervals for each of the parameters, as well as informationon the relationship between the parameters through theircovariance.
3.1. Likelihood Function for Standby-Related Failures, πΏ1(π).Let ππ,π be the number of standby-related failures of com-ponent π, during the maintenance period π which occur attimes ππ,π,1, ππ,π,2, ππ,π,3, . . ., and let π‘π,π be the chronologicaltime for theπ-maintenance in component π. The likelihoodfunction for π identical components of equipment underimperfect preventive maintenance is given by
πΏ1 (π) = πβπ=1
{{{ππ+1βπ=1
[[ππ,πβπ=1
βπ,π (ππ,π,π) β exp(βππβπ=1
π»π,π (π‘π,π) βπ»ππ+1 (π‘βπ))]]}}} , (19)
where π is the vector of unknown parameters, (πΌ, π). For eachcomponent π, ππ is the number of preventive maintenanceactivities performed during the observation period π‘βπ, withβπ,π(π) and π»π,π(π‘) being the induced hazard function and
the cumulative hazard function in periodπ, respectively, andπ»ππ+1(π‘βπ) the cumulative hazard function in censoring timeπ‘βπ.The log likelihood function is given by
log πΏ1 (π) = πβπ=1
[[ππ+1βπ=1
ππ,πβπ=1
log (βπ,π (ππ,π,π)) β ππβπ=1
π»π,π (π‘π,π) βπ»ππ+1 (π‘βπ)]] . (20)
Equation (21) must be particularized depending on theimperfect maintenance model considered. If a PAS imper-fect maintenance model is considered, the expressions cor-responding to βπ,π(ππ,π,π), π»π,π(π‘π,π), and π»ππ+1(π‘βπ) areobtained from (6) evaluated in the failure times and (7)evaluated in the preventive maintenance activities times andcensure time:
βπ,π (ππ,π,π)= πΌ(ππ,π,π β πβ2β
π=0
(1 β π)π ππ‘π,πβπβ1) + β0 (21)
π»π,π (π‘π,π)= πΌ2 (π‘π,π β πβ2β
π=0
(1 β π)π ππ‘π,πβπβ1)2
+ β0ππβπ=1
(π‘π,π β πβ2βπ=0
(1 β π)π ππ‘π,πβπβ1)(22)
π»ππ+1 (π‘βπ)= πΌ2 (π‘βπ β
ππβ2βπ=0
(1 β π)π ππ‘π,πβπβ1)2
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+ β0(π‘βπ βππβ2βπ=0
(1 β π)π ππ‘π,πβπβ1) .(23)
In the case of PAR imperfect maintenance model, theexpressions corresponding to the failure rate, βπ,π(ππ,π,π), andthe cumulative failure rates, π»π,π(π‘π,π) and π»ππ+1(π‘βπ), areobtained from (9) as
βπ,π (ππ,π,π) = πΌ (ππ,π,π β ππ‘π,πβ1) + β0π»π,π (π‘π,π) = πΌ2 (π‘π,π β ππ‘π,πβ1)2
+ β0 (π‘π,π β ππ‘π,πβ1)π»ππ+1 (π‘βπ) = πΌ2 (π‘βπ β ππ‘π,ππ)2 + β0 (π‘βπ β ππ‘π,ππ) .
(24)
3.2. Likelihood Function for Demand-Caused Failures, πΏ2(π).In the same way as in the previous section, let ππ,π bethe number of demand-caused failures of component π,during the maintenance period π which occur at timesππ,π,1, ππ,π,2, ππ,π,3, . . ., and let π‘π,π be the chronological timefor the π-maintenance in component π. The log likelihoodfunction for π identical components of equipment underimperfect preventive maintenance is given by
logπΏ2 (π) = πβπ=1
[[ππ+1βπ=1
ππ,πβπ=1
log( ππ,π (ππ,π,π)1 β π·π,π (ππ,π,π))
+ ππβπ=1
log (1 β π·π,π (π‘π,π))
+ log (1 β π·ππ+1 (π‘βπ))]] .
(25)
The probability function, ππ,π(ππ,π,π), and the cumulativeprobability functions, π·π,π(π‘π,π) and π·ππ+1(π‘βπ), depend onthe imperfect maintenance model considered. In the case ofa PAS model these functions are obtained from (14) and (15)as
ππ,π+1 (ππ,π+1,π) = π0(1 + βπ‘π,ππ βπ1 πβπ=1
(1 β π)π
+ βππ,π+1,π β π‘π,ππ βπ1) .π·π,π+1 (π‘π,π+1) = π0(1 + βπ‘π,ππ β
β π1( πβπ=1
((βπ‘π,ππ β + 1) π + 1) (1 β π)π+1βπ
+ (π + 1) (βπ‘π,π/πβ + 1)2 ))
π·π,π+1 (π‘βπ) = π·π,π (π‘π,π) + π0(1 + βπ‘βπ β π‘π,ππ ββ βπ‘π,ππ βπ1 πβ
π=1
(1 β π)π
+ β(π‘βπ β π‘π,π) /πβ (β(π‘βπ β π‘π,π) /πβ + 1)2 π1)
(26)
If a PAR maintenance model adopted the expressionscorresponding to demand failure probability and cumulativedemand failure probability is obtained by particularizing (16)in ππ,π,π and (17) in π‘π,π and π‘βπ,ππ,π+1 (ππ,π+1,π) = π0 (1 + πβπ‘π,ππ βπ1 (1 β π)
+ βππ,π+1,π β π‘π,ππ βπ1)π·π,π+1 (π‘π,π+1) = π0(1 + πβπ‘π,π/πβ (βπ‘π,π/πβ + 1)
2β π1 + ((βπ‘π,ππ β + 1) π (π + 1)2 β βπ‘π,ππ βπ)β βπ‘π,ππ βπ1 (1 β π))
π·π,π+1,π (ππ,π+1,π) = π·π,π (π‘π,π) + π0(1+ βππ,π+1,π β π‘π,ππ ββπ‘π,ππ βππ1 (1 β π)+ β(ππ,π+1,π β π‘π,π) /πβ (β(ππ,π+1,π β π‘π,π) /πβ + 1)
2β π1) .
(27)
4. Case Study
This section encompasses the estimation of the parametersassociated with the reliability models presented in Section 2for a motor-operated valve (MOV) of a nuclear power plant.The parameters are estimated and the reliability modelsthat best fit the plant data are selected using the methodspresented in Section 3. Then, the estimates obtained are usedto predict the performance of the MOV as a function of testand maintenance intervals. In particular, the MOV averageunreliability contribution of each failure mode and the totalMOV unavailability are computed and plotted as a functionof maintenance and test intervals for a 10-year horizon.
4.1. Historical Maintenance and Testing Data. Historical fail-ure, maintenance, and test data have been collected from a
Mathematical Problems in Engineering 7
Table 1: Failure data collected for two identical motor-operated valves of a nuclear power plant.
Failure time [h] Equipment Failure cause Failure mode384 MOV2 Motor circuit breakers degraded Standby23472 MOV1 Fail to open. Limit switches fail Demand24336 MOV1 Fail to open. Limit switches fail Demand27024 MOV2 Fail to open. Limit switches fail Demand56424 MOV1 Motor circuit breakers burned Standby94512 MOV1 Fail to open. Limit switches fail Demand94584 MOV1 Fail to open.Throttle misadjusted Demand
Table 2: MLEs of parameters of the reliability model of standby related failures under PAS and PAR models.
β0 [hβ1] πΌ [hβ2] π [-] πΏPAS model 5.860πΈ β 06 3.424πΈ β 10 Β± 1.0798π β 10 0.716 Β± 0.084 5.202πΈ β 10PAR model 5.860πΈ β 06 5.793πΈ β 10 Β± 1.757πΈ β 10 0.995 Β± 0.012 2.215πΈ β 10
Table 3: MLEs of parameters of the reliability model of demand caused failures under PAS and PAR models.
π0 [-] π1 [-] π [-] πΏPAS model 6.420πΈ β 03 5.415πΈ β 3 Β± 1.127πΈ β 03 0.886 Β± 0.084 1.136πΈ β 18PAR model 6.420πΈ β 03 1.141πΈ β 3 Β± 1.999πΈ β 4 0.719 Β± 0.1 1.776πΈ β 20nuclear power plant for two identical motor-operated safetyvalves. The data set contains all the failures, preventivemaintenance, and surveillance test activities registered duringan observation period of 27 years.
Table 1 shows the failure times of the two MOVsstudied obtained from the plant operational data. Table 1provides also a brief description of the corresponding failurecause and failure mode. The failures have been classified aseither standby-related or demand-caused failure taking intoaccount the information available for the failure cause.
A total of 432 surveillance tests and 17 preventive mainte-nance tasks were performed onMOV1, distributed uniformlywith periodicity 22 and 572 days, respectively, along the 27-year period analysed. In addition, a total of 424 surveillancetests and 18 preventive maintenance tasks were performedonMOV2, distributed uniformly with periodicity 22 and 528days, respectively, within the same period.
4.2. Results of the Maximum Likelihood Estimation. Thissection presents the results of the joint estimation of theeffectiveness ofmaintenance, π, and the reliability parameters,πΌ for standby-related failures and π1 for demand-causedfailures, under PAS and PAR imperfect maintenance modelsusing the plant data introduced in the previous section. Themodel that provides the best fit is identified for each of thetwo failure modes.
The maximum likelihood estimations of parameters π,πΌ, and π1 are obtained maximizing the log likelihood func-tions given by (20) for standby-related failures and (25)for demand-caused failures using the Nelder-Mead Simplexalgorithm. Table 2 gives MLEs of parameters corresponding
to reliability model of standby-related failures consideringPAS and PAR imperfect maintenance models, the doubleof the standard deviations, 2π, which are obtained fromthe Fisher information matrix, and the values of likelihoodfunctions πΏ. Table 3 shows the same information for the caseof the reliability model of demand-caused failures.
The best model for standby-related failures and demand-caused failures is the PAS model in both cases since itprovides the higher value for the likelihood function shownin Tables 2 and 3, respectively. So that, the reliability modelthat considers PAS imperfectmaintenance is selected for bothfailure modes with the value of the corresponding modelparameters given in Tables 2 and 3.
4.3. Average Unreliability Contribution as a Function ofMaintenance and Test Intervals. The average unreliabilitycontribution to the unavailability of a component normally instandby over its renewal period can be formulated as follows[9, 17]:
π’π = π’π ,π + π’π ,π·, (28)
where π’π ,π is the standby-related unreliability contributionand π’π ,π· is the demand-caused unreliability contribution.
On one hand, adopting the PAS model to representthe behavior of the imperfect maintenance for the standby-related failures of the component according to the results inthe previous section, π’π ,π is given by [9]
π’π ,π β 12 (β0 + 12πΌπ(2 β ππ ))π. (29)
8 Mathematical Problems in Engineering
uR,D (M = 5 years)uR,D (M = 3 years)uR,D (M = 1.5 years)uR,S (M = 1.5 years)
uR,S (M = 3 years)uR,S (M = 5 years)
1.00e β 02
2.00e β 02
3.00e β 02
4.00e β 02
5.00e β 02
Una
vaila
bilit
y u
(-)
1460 2190 2920 3650 4380 730Test interval T (h)
Figure 1: π’π ,π and π’π ,π· as a function of the test interval for differentmaintenance periods.
On the other hand, adopting the PAS model to representthe behavior of the imperfect maintenance for demand-caused failures of the component according to the results inthe previous section, π’π ,π· is given by [17]
π’π ,π· = π0 + 12π0 β π1 β ππ β (2 β ππ ) . (30)
Figure 1 shows the evolution of π’π ,π and π’π ,π· as afunction of the test interval, regarding different preventivemaintenance intervals for a 10-year horizon renewal period.It can be seen that π’π ,π increases significantly for high π andπ values. Nevertheless, the effect of maintenance is positivefor both unreliability contributions.Moreover, an increase ontest frequency between maintenances, that is, low π values,has a very negative effect on π’π ,π· for very low π intervals.
In addition, Figure 1 shows confidence intervals for thevalues predicted for the unreliability contributions π’π ,π andπ’π ,π· for different couples π and π. One can realize largeconfidence intervals exist, which even increase with π andπ because of the RAM model and the uncertainty in theestimation of the model parameters shown in Tables 2 and3.
4.4. Average Unavailability as a Function of Maintenance andTest Intervals Regarding Unreliability and Downtime Effects.In accordance with [9], the averaged unavailability of acomponent is the sum of the average unreliability contri-bution and the unavailability contributions due to detecteddowntimes for performing testing andmaintenance activitieswith the plant at power, which can be formulated as follows:
π’ = π’π + π’π + π’π + π’πΆ + π’π, (31)
where π’π represents the unavailability contribution due totesting, π’π is the unavailability contribution due to perform-ing preventivemaintenance, π’πΆ is the unavailability contribu-tion due to performing corrective maintenance conditionalto detecting a failure during a previous test, and π’π is thecontribution due to replacement of the equipment, if any.
1460 2190 2920 3650 4380 730Test interval T (h)
1.00e β 02
2.00e β 02
3.00e β 02
4.00e β 02
5.00e β 02
Una
vaila
bilit
y u
(-)
uR,S (M = 1.5 years)uR,S (M = 3 years)uR,S (M = 5 years)uR,D + uT + uM (M = 1.5 years)uR,D + uT + uM (M = 3 years)uR,D + uT + uM (M = 5 years)
Figure 2: π’π ,π and π’π ,π· + π’π + π’π as a function of the test intervalfor different maintenance periods.
These downtime contributions can be evaluated using thefollowing equations [9]:
π’π = ππ (32)
π’π = πΏπ (33)
π’πΆ = 1ππβπ (34)
π’π = πRP
, (35)
where π is the downtime for testing, πΏ is the downtime forpreventive maintenance, π is the downtime for correctivemaintenance or repair, and π is the downtime for replacementor renewal.
For the sake of simplicity, the last two contributions, π’πΆand π’π, are not included in the sensitivity analysis due toboth are negligible as compared with the downtime effect ofpreventive maintenance and testing activities. Therefore, theaveraged unavailability of the component is given by
π’ = π’π ,π + π’π ,π· + π’π + π’π. (36)
Figure 2 shows the evolution of π’π ,π versus π’π ,π· + π’π +π’π as a function of the test interval considering differentpreventive maintenance intervals for a 10 years horizonrenewal period. The term π’π ,π allows quantifying the benefitof developing test and maintenance activities on the totalcomponent unavailability while the sum of contributionsπ’π ,π· + π’π + π’π represents their negative effect.
Figure 2 shows confidence intervals for the values pre-dicted for the unavailability contributions for different cou-ples π and π. Again, large confidence intervals exist, whicheven increase with π and π because of the RAM model andthe uncertainty in the estimation of the model parametersshown in Tables 2 and 3.
Mathematical Problems in Engineering 9
438043800 365035040 2920
Test interval T (h)Maintenance interval M (h)
219026280146017520
73013140
1.00e β 02
2.00e β 02
3.00e β 02
4.00e β 02
5.00e β 02
Una
vaila
bilit
y u
(-)
1.00e β 02
2.00e β 02
3.00e β 02
4.00e β 02
5.00e β 02
Una
vaila
bilit
y u
(-)
1.00e β 02
2.00e β 02
3.00e β 02
4.00e β 02
5.00e β 02
Una
vaila
bilit
y u
(-)
17520 26280 35040 4380013140Maintenance interval M(h)
1460 2190 2920 3650 4380730Test interval T (h)
365035040
Figure 3: Unavailability for different maintenance and test intervals under PAS model.
Substituting (29), (30), (31), and (32) into (36) yields thefollowing formulation of the total average unavailability of acomponent:
π’ = 12 (β0 + 12πΌπ(2 β ππ ))π + π0+ 12π0π1ππ (2 β ππ ) + ππ + πΏπ. (37)
The last study involves the analysis of the total averageunavailability of the component as a function of the couple{π, π} for a 10-year horizon, which is shown in Figure 3.The highest values of π’ are reached adopting the highestmaintenance and test intervals. The main contributor tothe total unavailability, π’ (see (37)), is the standby-relatedunreliability contribution given by equation (29) as it can beseen in Figure 2. This explains the direct and proportionaldependence between π’ and π and π. Nevertheless, thesum of the demand unreliability contribution and downtimeeffects considered, that is, downtime effect of preventivemaintenance and testing activities, become more relevant forvery low π values. This fact is appreciated in Figure 2 too.
5. Concluding Remarks
This paper presents a methodology of parameters estimationand model selection for safety-related equipment. In theliterature, complex reliability, availability, and maintainabil-ity (RAM) models have been proposed with the aim ofcapturing equipment performance in a more realistic way,such as explicitly addressing the effect of component ageing
and degradation, surveillance activities, and corrective andpreventive maintenance policies. A major challenge for theadoption of the newmodels in practice is to estimate reliabil-ity and maintenance parameters with the aim of selecting thebest model for describing the real operation of safety-relatedequipment.
Then, there is a need to fit the best model to real data byestimating the model parameters using an appropriate tool,which could be a problem in some cases because the numberof parameters is large and the available data is scarce.Thismayhave great influence in the confidence intervals of the valuesfound for the model parameters that better fit the data.
The paper considers a standby-related failure modelassuming linear ageing and a demand-caused failures modelassuming test-induced stress. In addition, it considers imper-fect maintenance adopting Proportional Age Setback andProportional Age Reduction for preventive maintenancemodelling. Maximum likelihood estimation (MLE) using adirect search algorithm based on the Nelder-Mead Simplex(NMS) method is used to estimate maintenance effectivenessand ageing rate simultaneously.
A practical and realistic case study is included facing theparameters estimation of a typical motor-operated valve in anuclear power plant.The case study considers real failure, test,and maintenance data for a typical motor-operated valve in anuclear power plant.The results of the parameters estimationinclude confidence intervals and the selection of the bestmodel.
Equipment RAM is quantified based on the best modelfitted to make the impact of such an estimation in a testing
10 Mathematical Problems in Engineering
and maintenance-planning context clear. Thus, the results ofsuch a predictive model may help to plan in a more efficientway the test andmaintenance program,which should provideappropriate balance among the different contributions to theunavailability of the MOV, with the aim of minimizing itsunavailability assuring a low level of unreliability.
However, the effect of the uncertainties introduced in theestimation of the model parameters, because of the avail-ability of scarce data, can jeopardize the decision-making.Thus, the example of application shows large confidenceintervals for the unreliability and unavailability contributionsfor different π and π couples, which even increase with πandπ because of the RAMmodel and the uncertainty in theestimation of the model parameters shown in Tables 2 and 3.
It can therefore be concluded that estimating the param-eters and, consequently, fitting these models, it is possibleto manage in a more efficient way the test and maintenanceprogram, by providing appropriate balance among the differ-ent contributions to the unreliability and unavailability of thecomponent. However, there is a need to increase the data setused to reduce the uncertainty in the decision-making.
Acronyms and Notations
BAO: Bad As OldGAN: Good As NewMLE: Maximum likelihood estimationMOV: Motor-operated valveNPP: Nuclear power plantPAR: Proportional Age ReductionPAS: Proportional Age SetbackPFD: Probability of failure on demandPRA: Probabilistic Risk AssessmentRAM: Reliability, availability, and maintainabilityπ0: Residual demand failure probabilityππ(π‘): Time-dependent demand failure
probability for the periodππ·π(π‘): Cumulative demand failure probability inthe periodππ+π: Degradation function of the componentimmediately after maintenanceπππ(π‘): Degradation function of the componentassociated with demand-related stress forthe periodπβ0: Residual standby-related hazard functionπ»π(π‘): Cumulative hazard function in the periodππΏ1(π): Likelihood function for π identicalcomponents of an equipment underimperfect preventive maintenance forstandby time-related hazard functionπΏ2(π): Likelihood function for π identicalcomponents of an equipment underimperfect preventive maintenance fordemand failure probabilityπ: Preventive maintenance numberππ: Preventive maintenance intervalπ(π‘): Cumulative number of demands at time π‘
π1: Test degradation factor associated withdemand failuresππ,π: Number of failures of component π duringthe maintenance periodπ
RP: Replacement interval (overhaulmaintenance)π‘: Chronological timeπ‘π: Time at which the component undertakesthe maintenance numberπ β 1π: Test intervalπ’: Averaged unavailability of a componentπ’πΆ: Averaged unavailability contribution dueto performing corrective maintenanceπ’π: Averaged unavailability contribution dueto performing preventive maintenanceπ’π: Averaged unavailability contribution dueto replacement or component renewalπ’π : Unreliability contribution to thecomponent averaged unavailability overthe component useful lifeπ’π ,π·: Demand-caused unreliability contributionπ’π ,π: Standby-related unreliability contributionπ’π: Averaged unavailability contribution dueto testingπ€+πβ1: Age of the component immediately afterthe maintenanceπ β 1π€π(π‘): Age of the component in the periodππΌ: Linear ageing rateπΏ: Downtime for preventive maintenanceπ: Preventive maintenance effectivenessπ: Downtime for testingπ: Downtime for corrective maintenance orrepairπ: Downtime for replacement or renewalπ: Standard deviationπ: Failure times.
Conflicts of Interest
Thementioned received funding in βAcknowledgmentsβ didnot lead to any conflicts of interest regarding the publicationof this manuscript. There are no other possible conflicts ofinterest in the manuscript.
Acknowledgments
The authors are grateful to the Spanish Ministry of Scienceand Innovation for the financial support received (ResearchProject ENE2016-80401-R) and the doctoral scholarshipawarded (BES-2014-067602). The study also received finan-cial support from the Spanish Research Agency and theEuropean Regional Development Fund.
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