Reliability model of a component equipped with PHM capabilities

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Reliability model of a component equipped with PHM capabilities

Michele Compare, Luca Bellani, Enrico Zio

To cite this version:

Michele Compare, Luca Bellani, Enrico Zio. Reliability model of a component equipped with PHM capabilities. Reliability Engineering and System Safety, Elsevier, 2017, 168, pp.4-11.

�10.1016/j.ress.2017.05.024�. �hal-01652191�

ContentslistsavailableatScienceDirect

Reliability Engineering and System Safety

journalhomepage:www.elsevier.com/locate/ress

Reliability model of a component equipped with PHM capabilities

Michele Compare

, Luca Bellani

, Enrico Zio

aEnergy Department, Politecnico di Milano, Italy

bAramis S.r.l., Italy

cChair on Systems Science and the Energetic challenge, Foundation Electricité de France at CentraleSupélec, France

a r t i c le i n f o

Keywords:

PHM metrics Reliability Particle Filtering Monte Carlo Simulation

a b s t r a ct

Weproposeananalytic,time-variantmodelthatconservativelyevaluatestheincreaseinreliabilityachievable whenacomponentisequippedwithaPrognosticsandHealthManagementsystemofknownperformancemetrics.

Thereliabilitymodelbuildsonmetricsofliteratureandisapplicabletodifferentindustrialcontexts.Asimulated casestudyconcerningcrackpropagationinamechanicalcomponentisconsideredtovalidatetheproposedmodel.

© 2017ElsevierLtd.Allrightsreserved.

1. Introduction

Inthelastdecade,PrognosticsandHealthManagement(PHM)has oftenbeenproposedasaneffectivetechnologytorespondtothereli- abilitychallengesposedbythemodernsafety-criticalcomponentsand systems(e.g.,nuclearpowerplants,oil&gasassets,etc.),inwhichfail- urescanresultnotonlyinsignificantcosts,butalsoinlife-threatening consequencessuchasexplosionsandnaturaldisasters.

PHMallows inprinciple monitoringthesystem healthcondition, predictingitsRemainingUsefulLife(RUL)and,ultimately,preventing catastrophicfailures[1–5].However,inpracticeitisimportanttoknow whicharethereliabilityandavailabilityofacomponentorsystem.In thisrespect,totheauthors’ bestknowledgeamodelingframeworkthat allowstranslatingthePHMcontributionintothecomponentorsystem reliabilityisstilllacking.

AfewworkshaveattemptedtoevaluatetheinfluenceofPHMon systemLifeCycleCost(LCC, [6–11]),lookingat theeconomicbene- fitsofPHMintermsofincreaseofcomponentorsystemavailability.

Ontheotherhand,forsafety-criticalapplicationsPHMisexpectedto mainlyincreasethecomponentorsystemreliability(ratherthanavail- ability).PHMhelpsavoidingover-estimationsoftheactualcomponent RUL,whichmayleadtoaccidentswithpossibleconsequencesonthe asset,theenvironmentandthepublic.

ToevaluatetheaddedvalueofthePHMtechnologyonsystemre- liability, itis necessarytocharacterizethe performanceofthe PHM adopted.Inthisrespect,avarietyofperformancemetricsandindica- torshavebeenintroducedfordetection(i.e.,therecognitionofade- viationfromthenormaloperatingconditionscausingsuchdeviation, e.g.,[8,12]),diagnostics(i.e.,thecharacterizationoftheabnormalstate, e.g.,[13])andprognostics,(i.e.,thepredictionoftheevolutionofthe

∗Corresponding author at: Energy Department, Politecnico di Milano, Italy.

E-mail address: enrico.zio@polimi.it (E. Zio).

abnormalstateuptofailure,e.g.,[2,14,15]).Theoriginalcontribution ofthisworkistoproposeageneralmodelinganddecisionframework forlinkingPHMmetricsofliteraturetothecomponentreliability.This frameworkalsoallowsaccountingforthedecisioncriterionadoptedfor maintenance(overhaul),whichheavilydependsontheriskattitudeof thedecisionmaker.

Theproposedreliabilitymodelisvalidatedbywayofasimulated casestudyconcerningthecrackpropagationinamechanicalcompo- nent,whichrequirestoestimatethevaluesoftherelevantPHMmetrics.

Although various definitions of performance metrics exist in the PHMliterature,adetailedproceduretoestimatetheirvaluesisstilllack- ing,apartfromafewmetricssuchastheMTTF[16].Forthis,afurther originalcontributionofourworkistheMonteCarlo(MC)procedure proposedtoestimatetheperformancemetricsencodedinthedeveloped reliabilitymodel.

Theremainderofthepaperisorganizedasfollows:Section2briefly introducesthegeneralframework;inSection3,theimpactofaPHMtool onsystemreliabilityismodeled;Section4illustratesasimulatedcase studyconcerning thecrack propagationina mechanicalcomponent;

Section5validatesthedevelopedmodelbywayofthesimulatedcase study;Section6concludesthework.

2. Modelingframework

We considera degrading component, whose degradation state is monitoredeveryΔtunitsoftimewithrespecttoacontinuousindicator variable(Fig.1).Thedegradationprocessisstochasticforthedegrada- tionstateandtwothresholdsareconsidered:thedetectionthreshold, whichmainlydependsonthecharacteristicsoftheinstrumentusedfor monitoringthedegradationvariable(forexample,consideringthatthe instrumentisnotcapableofdetectingthedegradationstateforvalues

http://dx.doi.org/10.1016/j.ress.2017.05.024 Available online 10 May 2017

0951-8320/© 2017 Elsevier Ltd. All rights reserved.

M. Compare et al. Reliability Engineering and System Safety 168 (2017) 4–11

Nomenclature

𝜆 Timewindowmodifier,suchthat𝑡𝜆=𝑇𝑝𝑟+𝜆(𝑇𝑓−𝑇𝑝𝑟); 𝜆∈[0,1]

𝜆^∗ Timefromwhichthevaluesoftheperformancemetrics areestimated

T_d Timeinstantatwhichthesystemreachesthedetection threshold

T_f Timeinstantatwhichthesystemreachesthefailure threshold

T_𝜙 Lengthofthetimeinterval𝑇_𝑓−𝑇_𝑑 𝑓_𝑇𝑑 pdfoftimeT_d

𝑓_𝑇𝜙 pdfofT_𝜙 𝑓_𝑇𝑓 pdfofT_f

Δt TimeintervalbetweentwosuccessiveRemainingUse- fulLife(RUL)predictions

DTD DetectionTimeDelay,𝑇_𝑝𝑟−𝑇_𝑑

f_DTD probabilitydensityfunction(pdf)ofDTD 𝑃_𝜆^𝛼 𝛼−𝜆performance

⌊x⌋ Integerpartofx;thatis,𝑛≤𝑥<𝑛+1,𝑥∈ℝ,𝑛∈ℕ N NumberofmaximumRULpredictionsbeforefailure k^∗ Indexofthefirsttimechannelatwhichthedecisionto

removethesystemfromoperationcanbetaken h^∗ Indexofthefirsttimechannelatwhichamissingalarm

isrisky

R_𝜆 UncertainpredictedRULattimeindicatedby𝜆 Y_𝜆 PointsummarizingtheuncertaintyinR_𝜆(e.g.,mean,

median,10thpercentile,etc.) 𝑅𝑈𝐿^∗_𝜆 ActualRULatthetimeindicatedby𝜆 T_pr TimeofthefirstRULprediction FP Falsepositives

FN Falsenegatives

m EmpiricalestimateofmetricM

𝑓_𝑅𝜆 pdfofthepredictedRULatthetimewindowindicated by𝜆

(𝜇,𝜎²) Normaldistributionwithmean𝜇andvariance𝜎²

(𝑎,𝑏) Uniformdistributionbetweenaandb

belowsuchthreshold),andthefailurethreshold,abovewhichthecom- ponentdoesnotfunctionanymoreor,morepractically,mustbemain- tainedorreplacedforavoidingacatastrophicfailure.

The uncertaintyin the time instant T_d at which the component reachesthefirstthresholdisdescribedbytheprobabilitydensityfunc- tion(pdf)𝑓_𝑇𝑑.Ifnoactionistaken,thecomponentcontinuesitsdegrad- inguptofailureoccurringattimeT_f;itsuncertaintyisdescribedbypdf 𝑓_𝑇𝑓.Finally,wealsoconsidertherandomvariable𝑇_𝜙=𝑇_𝑓−𝑇_𝑑,whose pdfis𝑓_𝑇𝜙.

Realistically,itisassumedthatdetectionisnotperfect.Thus,metrics ofliteratureareexploitedtocharacterizethedetectionperformance.In thisrespect,thefollowingtwoarewidelyusedinpractice:falsepos- itiveprobability(i.e.,theprobabilityoftriggeringunduealarms)and falsenegativeprobability(i.e.,theprobabilityofmissingalarmwhen required)[8]).Inaddition,DetectionTimeDelay(DTD,[12])isade- tectionmetricwhichmeasurestheintervalfromthetimewhenthede- tectabledegradationstateisreachedbythecomponentuptoitsdetec- tion.Weusethisperformancemetric,duetotwomainreasons:onone hand,DTDisviewedasafalsenegativeindicatorwhichdepends on time(i.e.,alarmsaremissinguptoDTD);ontheotherhand,theDTD valuesaredependentonthedetectionalgorithmsettings,whichcanbe adjustedsothatthefalsepositiveprobabilityisnegligibleintheinital partofthecomponentlife[12].Thisway,themodeldevelopmentissim- plified.Toberealistic,weassumethatDTDisaffectedbyuncertainty, whosepdfisf_DTD(𝛿).

Fig. 1. Model setting description; ℎ = 4 , 𝛼= 0 . 1 , 𝑁 ¹= 11 and 𝑁 ²= 19 .

Inthis setting,thePHM systemstarts topredict theRULat time 𝑇_𝑝𝑟=(⌊^𝑇𝑑+_Δ^{𝐷𝑇 𝐷}_𝑡 ⌋+1)Δ𝑡,where⌊○⌋indicatestheintegerpartofitsar- gument.ThenumberofpredictionsthatthePHMcanperformbefore failureis𝑁=⌊^𝑇𝑓_Δ⁻_𝑡^𝑇𝑝𝑟⌋.Fromnowon,itisassumedthatthesystemac- tuallyfailsattime𝑇_𝑝𝑟+𝑁Δ𝑡,insteadofT_f;thesmallerΔt,thesmaller theapproximation.

Notice that wehave assumed,forsimplicity, that theconsidered componentisaffectedbyasinglefailuremode,sothatwedonothave theneedoftacklingtheissueofembeddingdiagnosticmetricsintothe reliabilitymodel,andofconsideringallscenariosoriginatingfromdeci- sionsbasedonerroneousdiagnosesofthefailuremode.Suchdiagnostic issueisleftforthefutureresearchwork.

Finally, notice also that, in practice, both detection and failure thresholdsmaynot beeasily determined.Forexample,in helicopter applications, PHMsystems(alsocalledHealth andUsage Monitoring System,HUMS)aremainlybasedonvibrationmonitoringtoinferthe equipment health [17–19]; thus,there is no simplewaytodefinea thresholddirectlyrelatedtofailure.Similarchallengesareencountered inthepackagingindustry,wherethefailureconditionsofcomponents maynotbepreciselyknown[20].Nonetheless,theapproachproposed inthepresentworkappliestoanysystem,providedthatsomecriterion todefinethethresholdsexists.Thedefinitionofsuchcriterionisoutof thescopeofthiswork,whereweassumethattheDecisionMaker(DM) hasalreadydefinedathresholdcoherentwithhis/herobjective.

3. Reliabilitymodel

InthisSection,weillustratethemathematicalmodeldevelopedto evaluatetheincreaseinsystemreliabilitybroughtbyaPHMsystem.

WeassumethatthePHM-equippedcomponentisstoppedwhenthe (100−𝛽)thpercentile(e.g.,100−90=10th)ofthecurrentlypredicted RULpdfissmallerthanh ·Δt:thelargerthevalueof𝛽,thesmaller thevalueofthepredictedRULpercentile,themorerisk-aversethedeci- sion.Similarly,thelargerthevalueofh,themorecautiousthedecision maker.

Tosethand𝛽 inrealindustrialapplications,itshouldbekeptin mindthatthevalueofhstronglydependsonthetimerequiredtosafely removethecomponentfromoperation(e.g.,timerequiredforsystem shutdown),whereas𝛽relatestotheriskassociatedtothefailure(e.g., 𝛽=5isaveryconservative value,suitableforsafetycriticalapplica- tion).TohelptheDMtosethand𝛽wecanusetheproposedreliability modelina‘reverse’ way,tofindthecombinationsofvaluesofhand𝛽 thatallowmeetingthesystemreliabilityrequirements,alsotakinginto accounttheconsideredPHMperformancevalues.Furthermore,wecan evaluatethesensitivityofthecomponentreliabilityvaluetotheselected applicablevaluesofhand𝛽,tofindthesettingswhicharelesssensitive tothepossiblevariabilityofthemetricsduetotheuncertaintyintheir estimations.

Toevaluatetheprobabilityofremovingthesystemfromoperation before failure,we needtoconsideratime-variantprognostic perfor- 5

Fig. 2. 𝑃 _𝜆^𝛼description.

manceindexandlinkittotheprobabilityofbeinginstoppingcondi- tions.

Amongtheprognosticmetricsavailableintheliterature[14,15],the mostsuitableisthe𝛼−𝜆performanceindex,𝑃_𝜆^𝛼,whichisatime-variant accuracyindicatorrangingin[0,1];thisallowsustogive𝑃_𝜆^𝛼aprob- abilisticinterpretation.Variousdefinitionsof𝑃_𝜆^𝛼 havebeenproposed intheliterature[14,15],referringtoeitherpoint-wiseorpdfRULpre- dictions.Inthiswork,wegivethefollowingdefinition,derivedfrom [14](Fig.2).

Considertheindicatorvariable:

Π^𝛼_𝜆= {

1, if𝑓_𝑅𝜆|^𝛼_𝛼^𝜆+− 𝜆 ≥𝛽 0, else

(1)

where𝑓_𝑅𝜆isthepdfoftheRULR_𝜆predictedattime𝑡_𝜆=𝑇_𝑝𝑟+𝜆(𝑇_𝑓− 𝑇_𝑝𝑟),𝜆∈[0,1],whereas𝛼isauser-definedparameterwhichindicates therequiredtolerancearoundthevalueofRUL^∗(e.g.,𝛼∈[0.05,0.2]).

Then,𝑃_𝜆^𝛼isthemeanvalueofΠ^𝛼_𝜆,i.e.,𝑃_𝜆^𝛼=𝔼[Π^𝛼_𝜆].

Namely,duringthetestcampaignof thealgorithm,in whichthe valueoftheprognosticperformancemetricsarecomputed,thealgo- rithmisrunontheworkingsystemanaslargeaspossiblenumberof times.Then,atanytrial,Π^𝛼_𝜆issetto1iftheRULpdfpredictedatt_𝜆has anarealargerthan𝛽between𝛼_𝜆⁻=(1−𝛼)𝑅𝑈𝐿^∗_𝜆and𝛼⁺_𝜆 =(1+𝛼)𝑅𝑈𝐿^∗_𝜆, being𝑅𝑈𝐿^∗_𝜆theactualRULattimet_𝜆,i.e.,thetimeuptoreachingthe failurethresholdorthethresholdabovewhich amaintenanceaction mustbeperformed,depending ontheapplication(Fig.2).TheRUL^∗ valueisexactlyknownattheendofeverytrial.

𝑃_𝜆^𝛼is,then,practicallygivenbytheestimate𝑝^𝛼_𝜆,whichiscalculated byaveragingthevaluesΠ^𝛼_𝜆gatheredfromdifferenttrialsofthePHM toolatasmanyaspossibleinstantst_𝜆.Thelargerthevalueof𝑃_𝜆^𝛼,the betterthePHM systemprediction capability.Formore detailson𝑝^𝛼_𝜆 computation,seeSection5.2.

NoticethatwhenΠ^𝛼_𝜆=0,noinferencecanbemadeaboutthevalue oftheuncertainRULprediction:oneonlyknowsthattheareaoverlap- ping[𝛼⁻_𝜆,𝛼_𝜆⁺]issmallerthan𝛽,withnofurtherinformationabouteither theactualextentofthisoverlappingortheportionofprobabilitymass locatedbelow𝛼⁻_𝜆,above𝛼⁺_𝜆 orinanin-betweenposition.

NoticealsothatwhenΠ^𝛼_𝜆=1,thentheinterval[(1−𝛼)𝑅𝑈𝐿^∗_𝜆,(1+ 𝛼)𝑅𝑈𝐿^∗_𝜆]is the2-sided𝛽confidence intervalofthefailuretimepre- dictedattimet_𝜆.However,forthepredictionmetricstobeapplicable forsupportingrisk-aversedecisionmaking,weneedtorefertoanup- perbound oftheprobability of over-estimatingtheRUL(i.e.,of not stoppingthecomponent),ratherthantoa2-sidedconfidenceinterval.

Tocopewiththissituation,wecombine𝑃_𝜆^𝛼withthefalsepositiveand falsenegativemetrics[15],whicharetime-variantindexesdefinedas, respectively:

𝐹𝑁𝜆=𝔼[Φ𝑁𝜆],Φ𝑁𝜆=

{1, ifΥ_𝜆−𝑅𝑈𝐿^∗_𝜆>𝑑^{𝑡ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑}_𝜆

0, else (2)

𝐹𝑃_𝜆=𝔼[Φ𝑃_𝜆],Φ𝑃_𝜆=

{1, ifΥ_𝜆−𝑅𝑈𝐿^∗_𝜆<−𝑑^{𝑡ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑}_𝜆

0, else (3)

Fig. 3. Regions partitioning the time horizon and examples of possible RUL predictions.

whereY_𝜆isapointestimateofthepredictedRULdistribution(e.g.,the mean,themedianoranyotherpercentileofR_𝜆,etc.)and𝑑_𝜆^{𝑡ℎ𝑟𝑒𝑠𝑜𝑙𝑑} isa user-definedthresholdvalue,whichdependsonthePHMapplication.

Proceedingexactlyinthesamewayasthatof𝑝^𝛼_𝜆,wewillconsiderthe estimatesfn_𝜆andfp_𝜆ofFN_𝜆andFP_𝜆,respectively,whicharegivenby thecorrespondingempiricalaveragesofΦN_𝜆andΦP_𝜆overtheavailable numberoftesttrials,achievedthroughanalgorithmtestcampaign.As mentionedabove,noticethatwhenperformingaPHMtest,RUL^∗isex- actlyknownattheendofeverytrial.Thisvalueis,then,usedtoestimate ΦN_𝜆,ΦP_𝜆andtheothervariablesofthemodel,asshowninSection5.2. Inoursetting,Y_𝜆isthe(1−𝛽)thpercentileof𝑓_𝑅𝜆 and𝑑_𝜆^{𝑡ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑}= 𝛼×𝑅𝑈𝐿^∗_𝜆.Then,FP_𝜆measurestheaverageportionoftimesinwhich Y_𝜆isbelow𝛼⁻_𝜆 =(1−𝛼)𝑅𝑈𝐿^∗_𝜆and,thus,itbecomesanindicatorofhow muchconservativeourPHMpredictionsareattimet_𝜆.Similarly,FN_𝜆 indicatestheriskinessofthePHMalgorithm.

Basedontheseconsiderations,wecanbuildthereliabilitymodelofa PHM-equippedcomponentwithestimatedvalues𝑝^𝛼_𝜆,𝑓𝑛𝜆,𝑓𝑝𝜆ofmetrics 𝑃_𝜆^𝛼,𝐹𝑁_𝜆,𝐹𝑃_𝜆,respectively.Todothis,wedividethetimehorizoninto threeregions(Fig.3):

1. Theregioninproximityoffailure,whichisdefinedbythetime indexes𝑘≥𝑁−ℎsuchthat(1+𝛼)𝑅𝑈𝐿^∗_𝜆≤ℎΔ𝑡,where𝑅𝑈𝐿^∗_𝜆= (𝑁−ℎ)Δ𝑡.Thisisthesameask≥h^∗,whereℎ^∗=⌊𝑁− ^ℎ

1+𝛼⌋.Ge- ometrically,thisregioncorrespondstotimevaluesontherightof theintersectionbetweentheerrorupperboundline(1+𝛼)𝑅𝑈𝐿^∗_𝜆 andthehorizontallinepositionedat𝑅𝑈𝐿=ℎΔ𝑡(Fig.3).

2. Thesaferegion,whichisindicatedbytimeinstantsk<k^∗,where k^∗geometricallycorrespondstothepredictionmostproximalto theintersectionbetween thepredictionerrorlowerbound line (1−𝛼)𝑅𝑈𝐿^∗_𝜆andthehorizontallineat𝑅𝑈𝐿=ℎΔ𝑡(Fig.3).

3. Thein-betweenregion,identifiedbyk^∗≤k<h^∗.

Withrespect toregion1, wecan notethattohavea failure,the alarmisrequiredtobe missingh^∗consecutivetimes.Now,ifΠ^𝛼_𝜆=1, thenthealarmistriggeredandthecomponentfailureisavoided.On thecontrary,ifΠ^𝛼_𝜆=0,thenecessaryconditiontonotactivatethealarm isthattheRULisover-estimated.Thissituationoccurswithprobability (1−𝑃_𝜆^𝛼)𝐹𝑁_𝜆≃ (1−𝑝^𝛼_𝜆)𝑓𝑛_𝜆.Weassumethatthisprobabilityvaluealso describestheuncertaintyinhavingmissingalarms;thisisaverycon- servativeassumptions:thecloserthecurrenttimetofailure,thelarger

M. Compare et al. Reliability Engineering and System Safety 168 (2017) 4–11

theover-estimationerrorrequiredtonottriggerthealarm(predictions mustbeabovethehΔtthreshold(seeFig.3)).

Withrespecttothesaferegion,wefirstnotethatwhicheverthevalue ofΠ^𝛼_𝜆,anover-estimationof𝑅𝑈𝐿^∗_𝜆leadstonotstoppingthesystembe- forefailure.Thisdoesnotentailanyriskofmissingstops.Onthecon- trary,anunder-estimationoftheRULcouldleadtocomponentstop.In therisk-aversesettingwearedealingwith,theanticipatedmaintenance isbeneficialforsystemreliability,asitavoidscomponentfailure.For this,weconservativelyassumethat inthisleft-mostregionthePHM systemneverstopsthecomponent.

Finally,withrespecttothein-betweentimehorizonregion,torig- orouslyderivetheprobabilityofnotstoppingthesystem,wehaveto giveaccounttothefactthatsomeextremecasesmayoccur,whereeven ifΠ^𝛼_𝜆=1,the1−𝛽probabilitymassand,thus,the(1−𝛽)thpercentile, ispositionedabovehΔt.Forexample,Fig.3showsthesituationwhere 𝑡_𝜆1=(𝑁−ℎ)Δ𝑡andallthe𝛽massisconcentratedbetween𝑅𝑈𝐿^∗_𝜆=ℎΔ𝑡 and𝛼_𝜆⁺.Inthiscase,PHMwillnotadvicetostopthecomponentat𝑡𝜆1. Thus,weconservativelyassumethatinthisregionthecomponentdoes notundergoamaintenanceactionaslongasΠ^𝛼_𝜆=1.

Onthecontrary,whenΠ^𝛼_𝜆=0,whichoccurswithprobability(1− 𝑃_𝜆^𝛼),thefollowingthreepossiblesituationscanoccur:

• The(1−𝛽)thpercentile,Y_𝜆,issmallerthan(1−𝛼)𝑅𝑈𝐿^∗_𝜆.Inthissitu- ation,whichoccurswithprobability(1−𝑃_𝜆^𝛼)𝐹𝑃_𝜆,evenifweconser- vativelyassumethatthe(1−𝛽)thpercentiletakesthelargestpossi- blevalue(i.e.,Υ_𝜆=(1−𝛼)𝑅𝑈𝐿^∗_𝜆),thecomponentisstoppedasthis timeissmallerthanhΔt.

• Withprobability(1−𝑃_𝜆^𝛼)𝐹𝑁𝜆,Y_𝜆willbeabove(1+𝛼)𝑅𝑈𝐿^∗_𝜆.Inthis situation,wewillnotstopthecomponent.

• Withprobability 1−𝐹𝑁_𝜆−𝐹𝑃_𝜆 wearein thesituationin which thepredictedRULvalueisbetween[𝛼_𝜆⁻𝛼_𝜆⁺].Tobeconservative,we assumethatalsointhiscasewedonotremovethecomponentfrom operation.

Toconclude,aconservativeestimationofthestopprobabilityinthe timewindow[𝑇_𝑝𝑟+𝑘^∗Δ𝑡,𝑇_𝑝𝑟+ℎ^∗Δ𝑡]is(1−𝑃_𝜆^𝛼)𝐹𝑃_𝜆≃ (1−𝑝^𝛼_𝜆)𝑓𝑝_𝜆.

Fig.1brieflysummarizestheconsiderationsproposedabove.Two differenttrialsofthesamePHM-equippedcomponentareplottedover time,whichareindicatedwithsuperscript1(continuousline) and2 (dashedline).𝑇_𝑝𝑟¹ and𝑇_𝑝𝑟² indicate thecorrespondingfirst prediction times,whereask^∗1andk^∗2representthefirsttimeinstantswherethe systemcanbestoppedwithprobabilities(1−𝑃_𝜆^𝛼)𝐹𝑃_𝜆;h^∗1andh^∗2are thefirsttimeindexesfromwhichthesystemisstoppedwithprobability 𝑃^𝛼_𝑘

𝑁

+(1−𝑃^𝛼_𝑘

𝑁

)(1−𝐹𝑁𝑘

𝑁).Finally,𝑇_𝑓¹ and𝑇_𝑓² representthelastpossi- blepredictioninstantsbeforefailureandareconsideredasfailuretimes withinourframework.

Basedontheconsiderationsabove,itisnowpossibletocomputethe unreliabilityU(t)attimet,whichisheredefinedastheprobabilityof reachingthefailurethresholdbeforet:

𝑈(𝑡)=ℙ(𝑇_𝑓≤𝑡 ∩ systemnotstoppedbefore𝑡;𝛼,𝛽,Δ𝑡,ℎ,𝑓𝑛,𝑓𝑝,𝑝^𝛼_𝜆)

=ℙ(𝑇_𝑓≤𝑡|systemnotstoppedbefore𝑡;𝛼,𝛽,Δ𝑡,ℎ,𝑓𝑛,𝑓𝑝,𝑝^𝛼_𝜆)

×ℙ(systemnotstoppedbefore𝑡;𝛼,𝛽,Δ𝑡,ℎ,𝑓𝑛,𝑓𝑝,𝑝^𝛼_𝜆)

where𝛼,𝛽,Δ𝑡,ℎ,𝑓𝑛,𝑓𝑝,𝑝^𝛼_𝜆explicitlyindicatethedependenceoftheun- reliabilityvalueontheparametersdeterminingtheperformanceofthe PHMsystem.

Noticethatthereareseveraldefinitionsof reliability[21].Differ- entlyfromthe‘traditional’ definitions,inwhichtheunreliabilityisthe CDFofthefailuretimeand,thus,ittendstooneastincreases(i.e.,the componentwillalwaysfail,[21,22]),inthiscasewearecompelledto consider

𝑡lim→∞𝑈(𝑡)=ℙ(systemnotstoppedbefore𝑡;𝛼,𝛽,Δ𝑡,ℎ,𝑓𝑛,𝑓𝑝,𝑝^𝛼_𝜆)≤1 Thedifferenceisduetothefactthatifthecomponentisremovedfrom operationbeforefailure,thenitsfailuretimewillno-longerexistand

K(t, F ailed) K(t, Remov

ed)

Fig. 4. Three-state system.

the‘traditional’ definitionsarenolongerapplicable.Thatis,thePHM- equippedcomponentcanbeframedasathree-statesystem,thepossible statesbeing:Working,FailedandRemoved(Fig.4),inwhichU(t)repre- sentstheprobabilityofhavingatransitionfromWorkingtoFailedbe- foretimet.Accordingtothisview,wederiveU(t)fromtheprobabilistic transport kernelK(t,Failed|t′, s′),which isdefined astheprobability density thatthecomponentmakesthenexttransitionbetweentand 𝑡+𝑑𝑡towardstateFailed[23],providedthattheprevioustransitionhas occurredattimet′andhatthesystemhadenteredinstates′.However, inourcaseweassumethatthecomponentalwaysstartsat𝑡=0instate Working.Forthis,wewillindicatethekernelasK(t,Failed),withoutthe conditioningevent.

TocalculateK(t,Failed),wefirstfirstcalculatethefailuretransporta- tionkernelgivenarealization𝛿fromf_DTD:

𝐾(𝑡,𝐹𝑎𝑖𝑙𝑒𝑑|𝛿;𝛼,𝛽,Δ𝑡,ℎ,𝑓𝑛,𝑓𝑝,𝑝^𝛼_𝜆)

=∫

𝑡

𝑡−𝛿𝑓𝑇_𝑑(𝜏)𝑓𝑇_𝜙(𝑡−𝜏)𝑑𝜏+

∫

𝑡−𝛿

0 𝑓𝑇_𝑑(𝜏)𝑓𝑇_𝜙(𝑡−𝜏)

ℎ∏^∗−1 𝑘=𝑘^∗

[1−(1−𝑝^𝛼_𝑘

𝑁

)𝑓𝑝𝑘 𝑁]

𝑁−1∏

𝑘=ℎ^∗

[(1−𝑝^𝛼_𝑘

𝑁

)𝑓𝑛𝑘

𝑁]𝑑𝜏 (4)

Inotherwords,itisassumedthatafailureoccurswhenoneoutofthe followingconditionsissatisfied,whicharerepresentedbythefirstand thesecondaddendofEq.(4),respectively:

1. ThecomponentfailsbeforePHMalertsthedetectionthreshold (detectionerror);thismayhappenincasethecomponentfails abruptly.

2. PHMcorrectlydetects,withdetectiondelay𝛿,thatthedegrada- tionhasreachedthedetectionthresholdbut,then,over-estimates theactualfailuretimeT_f(prognosticerror);thishappensafter 𝑇_𝑝𝑟+𝑘^∗Δ𝑡(i.e.,thefirstpredictioninstantwherethestopping decisionshouldbemade),withprobability1−(1−𝑝^𝛼_𝑘

𝑁

)𝑓𝑝𝑘 𝑁 and withprobability(1−𝑝^𝛼_𝑘

𝑁

)𝑓𝑛𝑘

𝑁 from𝑇_𝑝𝑟+ℎ^∗Δ𝑡on.

Toremove thedependencefrom 𝛿,we integrateEq.(4)over the distributionofDTD:

𝐾(𝑡,𝐹𝑎𝑖𝑙𝑒𝑑;𝛼,𝛽,Δ𝑡,ℎ,𝑓𝑛,𝑓𝑝,𝑝^𝛼_𝜆)

=∫

∞

0 𝐾(𝑡,𝐹𝑎𝑖𝑙𝑒𝑑|𝛿;𝛼,𝛽,Δ𝑡,ℎ,𝑓𝑛,𝑓𝑝,𝑝^𝛼_𝜆)𝑓_{𝐷𝑇 𝐷}(𝛿)𝑑𝛿 (5) Generallyspeaking,theintegralofK(t,Failed)overthetimeinterval [t₁,t₂]givestheprobabilityof failurein thattimespan[23]. Then, 7

Fig. 5. Crack propagation process: example

Eq.(5)allowsestimatingthecomponentunreliabilityas:

𝑈(𝑡)=

∫

𝑡

0 𝐾(𝜏,𝐹𝑎𝑖𝑙𝑒𝑑;𝛼,𝛽,Δ𝑡,ℎ,𝑓𝑛,𝑓𝑝,𝑝^𝛼_𝜆)𝑑𝜏 (6) Finally,noticethatthedevelopedmodelallowsconsideringalsothecase wherethereisnoadvantageinremovingthecomponentfromoperation thelasth instants:in thiscase,stopping thecomponent inthethird region(region1inFig.3)isequivalenttohavingafailure.

4. Casestudy

InthisSection,weillustratetheapplicationofthemodelingframe- workdevelopedtoacomponentaffectedbyfatiguedegradation,de- scribedbytheParisErdogan(PE)model([2,24],Fig.5):

1. Thecracklengthx_ireachesthefirstthreshold,𝑥=1mm,accord- ingtothefollowingequation:

𝑥_𝑖+1=𝑥_𝑖+𝑎×𝑒^𝜔1𝑖

where 𝑎=0.003 is the growth speed parameter and 𝜔¹_𝑖 ∼

(−0.625,1.5)modelstheuncertaintyinthespeedvalues.The uncertaintyinthearrivaltimeat𝑥=1isdescribedbypdf𝑓_𝑇𝑑(𝑡). 2. Thecrack lengthreachesthefailurethreshold𝑥=100mmac-

cordingtothefollowingequation:

𝑥_𝑖+1=𝑥_𝑖+𝐶×𝑒^𝜔2𝑖(𝜂√ 𝑥_𝑖)^𝑛

where𝐶=0.005and𝑛=1.3areparametersrelatedtothecom- ponentmaterialproperties,andaredeterminedbyexperimental tests;𝜂=1isaconstantrelatedtothecharacteristicsoftheload andthepositionofthecrackand𝜔²_𝑖∼(0,1)describestheun- certaintyinthecrack growthspeedvalues.Theuncertaintyin thearrivaltimeat𝑥=100isdescribedbypdf𝑓_𝑇𝑓(𝑡).

Thenumericalvaluesaretakenfrom[2].

5. Validationofthereliabilitymodel

TheaimofthisSectionistovalidatethereliabilitymodeldeveloped inSection3bywayofthecasestudypresentedabove.Todothis,we carryoutthefollowingsteps,whicharedetailedinthenextSections:

• Choosetheprognosticanddetectionalgorithmsthatareassumedto beimplementedinthePHMsystem.

• Estimatetheperformancevaluesfp_𝜆,fn_𝜆and𝑝^𝛼_𝜆.

• EstimatetheintegralinEq.(6).

• Estimatetheunreliabilityinthe‘on-line’ setting,inwhichthecrack propagationissimulatedtogetherwiththeselectedprognosticand detectionalgorithms,andwiththedecisionsbasedontheiroutcomes aswell.

5.1. Algorithms

The prognostic algorithm we rely on is Particle Filtering (PF, [25,26]),whichhasbeen establishedasthede-factostateof theart infailureprognostics[27].Briefly,atanytimeinstantPFestimatesthe pdfofthedegradationstateofthecomponent(i.e.,itscrackdepthin ourcase)withasetofweightedparticles,whichconstituteaprobabil- itymassfunction(pmf).Whenameasureofthecrackdepthisacquired, suchpmfisadjustedinaBayesianperspective,sothattheweightsre- latedtoparticleswhichareneartheacquireddataareaugmented.

ThePFalgorithmchosenforourapplicationisthesameasthatused in [1];itrelies onasimplifiedapproachforpredictingtheevolution ofthecrack,whichdoesnotgivefullaccounttotheuncertaintyinthe particleevolution[1].Certainly,morerefinedversionsofPFcouldbe consideredtoimprovetheprognosticperformance,butthisisoutofthe scopeofthiswork:ouraimistocheckwhetherthemodeldevelopedin Section3providesconservativeestimatesofthecomponentreliability foragivensetof performancevaluesfp_𝜆,fn_𝜆,and𝑝^𝛼_𝜆,whicheverthe prognosticalgorithmis.

AsmentionedinSection2,ourmodelmainlyfocusesonprognostics.

Thus,weassumethattheuncertaintyinDTDisalreadyknownandit is describedbyanormaldistribution,which forthesimulationsthat follows,isarbitrarilytakentohavemean5andstandarddeviation1, inarbitraryunits.Then,inthesimulations,thedegradationisdetected toreachthedetectionthresholdatatimeT_pr,whichisonaverage5 timeunitslargerthanT_dandthevariabilityofthisdelayisgivenbythe standarddeviationof1unit.Finally,withrespecttothemaintenance policysettings,weassumeℎ=1,𝛽=40andΔ𝑡=30inarbitraryunits:

largervaluesofhorsmallervaluesof𝛽wouldresultinreliabilityvalues verycloseto1,which donotallowafairvalidationof theproposed modelingframework.

5.2. Performanceestimation

ToestimatethevaluesoftheperformancemetricsFP_𝜆,FN_𝜆and𝑃_𝜆^𝛼, weimplementthefollowingMCprocedure:

1. SimulatethecrackpropagationmechanismtofindT_f,T_d,theN predictioninstantsateveryΔttimeandthecorrespondingcrack lengths.Inparticular,T_pr isobtainedbyaddingasamplefrom

(5,1)toT_d,whereas𝑇_𝑓=𝑅𝑈𝐿^∗at𝜆=0.Thegatheredvalues ofT_fandT_darealsousedtoderive𝑓_𝑇𝜙and𝑓_𝑇𝑑,respectively,at step3.

2. Ateverypredictioninstantt_𝜆,𝜆=_𝑇^{𝑡𝜆−𝑇𝑝𝑟}

𝑓−𝑇_𝑝𝑟,runthePFalgorithmto estimatethecurrentcracklengthandthepdf𝑓_𝑅𝜆ofthepredicted RULR_𝜆.Onthisbasis,useEqs.(1)–(3)tocalculatethevaluesof ΦP_𝜆,ΦN_𝜆andΠ^𝛼_𝜆using𝑓𝑅_𝜆and𝑅𝑈𝐿^∗_𝜆=𝑇𝑓−𝑡𝜆.Inthisrespect, Fig.6,showsthehistogramsof ^𝑘_𝑁^∗and^ℎ_𝑁^∗ over𝜆asderivedfrom thesimulationof 15,000MonteCarlo trialsof crackdegrada- tion:itcanbeseenthatinalmost90%ofthetrials,^𝑘∗

𝑁 ≥0.9and

M. Compare et al. Reliability Engineering and System Safety 168 (2017) 4–11

Fig. 6. ^𝑘_𝑁^∗and ^ℎ_𝑁^∗vs 𝜆.

Fig. 7. Example of degradation evolutions and computation of 𝜆related to prediction instants.

ℎ^∗

𝑁 ≥0.95.ThisimpliesthatwecanavoidcalculatingΦP_𝜆,ΦN_𝜆, andΠ^𝛼_𝜆 forall valuesof𝜆;rather,wecan reducetherangeof interestat𝜆 >𝜆^∗,forsomeappropriatevalueof𝜆^∗.

3. Oncesteps1and2aresimulatedalargenumberoftimesandthe correspondingvaluesofΦP_𝜆,ΦN_𝜆,andΠ^𝛼_𝜆arecollected,divide [𝜆^∗,1)inIintervalsofthesamelength[𝜆𝑖,𝜆𝑖+1),𝜆0=𝜆^∗,𝜆𝐼=1; Ishouldbesmallenoughthatintervals[𝜆_𝑖,𝜆_𝑖+1)donotcontain multiplepredictioninstantsofthesameMCtrial.Derivealso𝑓_𝑇𝑑 and𝑓𝑇_𝜙.

4. Foreachinterval[𝜆_𝑖,𝜆_𝑖+1),computetheaverage ofthe values ofΦP_𝜆,ΦN_𝜆,andΠ^𝛼_𝜆gatheredatthetimeinstant𝜆∈ [𝜆_𝑖,𝜆_𝑖+1); thisprovidestheestimatesfp_𝜆,fn_𝜆,and𝑝^𝛼_𝜆,whicharestep-wise functionsovertheidentifiedIintervals.

Fig.7and8provideanexampleofthedescribedprocedurefor3MC trials,inwhich𝜆^∗=0.1.Thedegradationpathsaresimulatedovertime (Fig.7)andthecorrespondingvaluesofinterestarecollected.Fig.7also reportsforeverydegradationpaththe𝜆valuescorresponding tothe predictioninstants,which dependonthedurationof thecomponent life.Then,Fig.8partitionstheinterval[0.1;1)inintervalsoflength 0.06,whichcontainatmostonepredictioninstantofthesametrial.In thisrespect,asimpleruletoselectthemaximum𝜆intervallengthisto selectthemaximumnumberN_mofpredictioninstantsinasingletrial;

then,themaximum𝜆intervallengthis ¹

𝑁𝑚.

Fig. 8. Computation of the performance metric values of Fig. 7 .

Fig.9showstheresultsoftheproceduredetailedaboveforthecase studyillustratedinSection4,startingfrom𝜆≥𝜆^∗=0.45.Inparticular, twodifferentlengthvaluesofthe[𝜆_𝑖,𝜆_𝑖+1)intervalshavebeenconsid- ered:0.05(Fig.9a)and0.005(Fig.9b).Inbothcases,wecheckedthat everyintervalcontainsatmostonepredictioninstantofthesametrial, although forsomesimulatedtrialsome intervalsdonot containany prediction(seeFig.8).This causesthenoisybehaviorofthemetrics 9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

λ

metricvalue

p^α_λ fpλ

fnλ

(a) Interval length 0.05

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

λ

metricvalue

p^αλ

fpλ

fnλ

(b) Interval length 0.005

Fig. 9. Experimental metrics values.

inFig.9b,asthenarrowertheinterval[𝜆_𝑖,𝜆_𝑖+1),thesmallerthecorre- spondingnumberofgatheredvaluesofΦP_𝜆,ΦN_𝜆andΠ^𝛼_𝜆overtheMonte Carlotrials,thelargertheMCerroraffectingtheaveragedvalues.

FromtheanalysisofFig.9,wecannoticethatboththefpandfn valuesincreaseover𝜆,exceptfor𝜆≥0.95.Thisisduetofactthatwhen 𝜆≃1,𝑅𝑈𝐿^∗=𝑁(1−𝜆)Δ𝑡≃ 0;hence,ononesidethechancesofhaving 𝜙𝑝=1reduces,whereasontheothersidethereisnopossibilitytohave 𝜙𝑛=0.Thisentailsthatfp_𝜆andfn_𝜆tendtoconvergeto0and1,respec- tivelyas𝜆→1.Moreover,fpisalwayslargerthanfn,exceptwhen𝜆

≥0.95.Thiscanbeeasilyexplainedrememberingthatwearetracing apercentileoftheRUL,whichfavorsthefalsepositivealarms.Forthe samereason,𝑝^𝛼_𝜆tendstoconvergeto0inthelastpartofthecomponent lifecycle:whenthecomponentisapproachingitsfailuretime,theRUL estimationsbecomemoreprecise;then,trackingapercentileinsteadof theRULmedianintroduces abiasthatimpactsonthepredictionac- curacy(seeFig.2).Noticethatthisbehaviordoesnot contradictthe presentedmodel:predictionsdoneatfailuretimearenotconsidered,as thecomponentisalwaysassumedtofailatNΔt,whichimpliesthatthe largestpossiblevalueof𝜆=^(𝑁−1)

𝑁 <1. 5.3. Componentunreliabilityestimation

ToestimatethecomponentunreliabilitybasedonEqs.(4)–(6),the followingprocedure,derivedfrom[23],hasbeenimplemented:

• Divide the time horizon in J time intervals of length Δt, [0,𝑡1),[𝑡1,𝑡2),…,[𝑡_𝐽−1,𝑡_𝐽],andassociateacountertoeveryinterval, whoseinitialvalueissetto0.

• Sample𝛿∼f_DTD;thisway,wecanestimatetheKernelinEq.(4), whichisconditionalonDTD.

• ComputethefirstaddendumofEq.(4)byMonteCarlo,evaluating theintegralcorrespondingtotheundetectedfailureprobability:for eachfailuretimet_j,wesampleT_prfrom(𝑡_𝑗−𝛿,𝑡_𝑗),i.e.,auniform distributionbetween𝑡𝑗−𝛿andt_j(seeforcedsimulationin[23]).

• ComputethesecondintegralofEq.(4),similarlytothepreviousone exceptthatT_prmustbesampledfrom(0,𝑡_𝑗−𝛿).Then,k^∗,h^∗and Narecomputed,andthevaluesoftheperformancemetricsobtained areusedtocompleteEq.(4).

• Estimatek(t_j,Failed)oftheintegralinEq.(5)byapplyingMCmethod [23].

• Estimatetheunreliabilityattimet_j (Eq.(6)),bysummingallthe failurecontributionsontherightoft_j:

𝑢(𝑡𝑗)≃ Δ𝑡

∑𝑗 𝑖=1

[𝑘(𝑡𝑗,𝐹𝑎𝑖𝑙𝑒𝑑)] 𝑗=1,…,𝐽

5.4. Estimationofthe‘on-line’ unreliability

TheonlineunreliabilityisestimatedthroughtheMCprocedurede- velopedin[1].Briefly,thetimehorizonispartitionedintime-channels

0 100 200 300 400 500 600 700 800 900 1000 1100

0 0.04 0.08 0.12 0.16

time

unreliability

offline online

Fig. 10. Comparison between ‘on-line ’ and ‘off-line ’ unreliabilities.

oflengthΔtunitsoftime.Thecrackgrowthprocessissimulatedover timetogether withDTDtocomputeT_d. IfT_pr ≥T_f,theunreliability countersassociatedtothechannelsfromT_ftotheendofthetimewin- dowaresetto1;otherwise,theempiricalpdf𝑓𝑅_𝜆isestimatedeveryΔt unitsoftimebymeansoftheParticleFiltering.Then,ateachpredic- tiontimet_𝜆,ifthepredicted𝛽thpercentileof𝑓_𝑅𝜆isbeforethenexthth inspectiontime,thenthecomponentisremovedfromoperation,other- wiseitcontinuestowork.Thetrialsimulationcontinuesuntileitherthe componentfailsorisremovedfromoperation:intheformercase,the unreliabilitycountersassociatedtothechannelsfromT_ftotheendof thetimewindowaresetto1;otherwisetheyaresetto0.Finally,the onlineunreliabilityateveryΔtisestimatedastheaverageovermany MCsimulationtrialsoftheaccumulatedcountervalues.Asmentioned before,weexpectthattheofflineunreliabilitycurveisalwaysabovethe onlineone,aswehavebuiltamodelwhichunder-estimatesthesafety benefitofaPHMsystem.

5.5. Results

Fig.10showsthetwounreliabilitycurvesobtainedusingthetwo methodsdescribedabove.ThebarsinFig.10representthe68%two- sidedconfidenceintervaloftheMCsimulationerror,bothintheon-line andoff-linesetting.FromtheanalysisoftheFigure,itseemsfairtosay thattheproposedreliabilitymodelisaccurate,asthetwocurvesare closetoeachother.Noticethatthedifferencebetweenthetwocurves increaseswithtime,meaningthatthere arenopredictioninstantsat whichourmodelover-estimatesthecomponentstoppingprobability.

6. Conclusion

Inthiswork,wehavepresentedanovelgeneralframeworktocom- putethereliabilityofaPHM-equippedcomponent.Themodelingframe- workproposedappliestosafetycriticalcomponentsandrisk-aversecon- texts(e.g.,applicationsofthenuclear,aerospace,oilandgasindustries),