A Reliability-based Opportunistic Predictive Maintenance Model for k-out-of-n Deteriorating Systems

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1 A blicatio of CHEMICAL ENGINEERING TRANSACTIONS VOL Gest Editors: Erico Zio Piero Baraldi Coyright 203 AIDIC Servizi S.r.l. ISBN ; ISSN The Italia Associatio of Chemical Egieerig Olie at: DOI: 0.33/CET3383 A Reliability-based Oortistic Predictive Maiteace Model for k-ot-of- Deterioratig Systems Ta K. Hyh* a Ae Barros a Christohe Béreger b a Uiversité de Techologie de Troyes ICD & STMR UMR CNRS re Marie Crie CS Troyes cedex Frace b Gisa-lab CNRS Greoble Istitte of Techology re des Mathématiqes BP Sait Marti d Hères cedex Frace ta.hyh@tt.fr Motivated by the effectiveess of coditio-based maiteace (CBM for sigle-it systems we develo i this aer a oortistic redictive maiteace model for a k-ot-of- deterioratig system. This model reflects the joit effects of ecoomic deedece ad CBM decisio o the maiteace cost. Ulike most existig CBM models whose maiteace decisio-makig relies directly o a coditio idex of degradatio level we base the decisio o the reliability of each comoet comted coditioally o its degradatio level. This makes the model robst with resect to (w.r.t. measremet errors more efficiet ad easier to otimize comared to a corresodig degradatio-based maiteace model.. Itrodctio With the dissemiatio of coditio moitorig techiqes CBM is owadays a very romisig aroach to imrove the drability reliability ad safety of idstrial systems ad hece to brig ot cometitive advatages. Over the last few decades may CBM models have bee develoed ad sccessflly alied for sigle-it systems or comoets (Jardie et al However i ractice a idstrial system sally cosists of several comoets which may stochastically ecoomically ad/or strctrally deed o each other (Cho et Parlar 99. Becase of these iteractios the otimal maiteace strategy of mlti-it systems is ot a simle jxtaositio of otimal strategies of their comoets (Castaier et al As a coseqece meros CBM models existig i the literatre caot be roerly alied for mlti-it systems. This aer therefore aims to develo a ew CBM model for a k- ot-of- deterioratig system i the cotext of ecoomic deedece amog comoets. By defiitio ecoomic deedece imlies that cost savig ca be eared whe several comoets are maitaied together rather tha searately (Dekker et al Obviosly cosiderig joitly both ecoomic deedece ad CBM decisio ca lead to sigificat savig i maiteace cost. However the literatre of sch maiteace models is somewhat limited. Geerally these models are orieted toward two mai aroaches amely groig maiteace ad oortistic maiteace (Dekker et al I the former re CBM strategies are firstly alied for comoets to determie their otimal maiteace times ad the ecoomic deedece amog comoets is take ito accot by groig these otimal times (see e.g. (Bovard et al. 20. I the latter maiteace decisios sally follow cotrol-limit rle with i additio to roer revetive maiteace thresholds of comoets oe or several oortistic thresholds itrodced to exloit the ecoomic deedece (see e.g. (Tia ad Liao 20. Comared to the latter the former does ot well characterize the joit effects of CBM ad ecoomic deedece becase of two relatively ideedet stages of maiteace decisios. This is why we focs oly o oortistic CBM models i the reset aer. Moreover like most existig models whose maiteace decisio-makig relies directly o a coditio idex of degradatio level we base the decisio o the reliability of each comoet comted coditioally o its degradatio level. We will show i followig that sig coditioal reliability istead of degradatio level ca make the model more efficiet robst w.r.t. measremet errors ad easier to otimize. Please cite this article as: Hyh K.T. Barros A. Bereger C. 203 A reliability-based oortistic redictive maiteace model for k-ot-of- deterioratig systems Chemical Egieerig Trasactios DOI: 0.33/CET

2 The remaider of the aer is orgaized as follows. Sectio 2 is devoted to describe the system ad to comte the reliability of comoet ad system. I Sectio 3 we reset a reliability-based oortistic redictive maiteace strategy ad the associated cost model. The erformace ad robstess of this strategy are aalyzed i Sectio 4 by comarig with a degradatio-based oortistic CBM strategy. Fially we coclde the aer ad give some ersectives i Sectio System modelig ad reliability aalysis We cosider a k-ot-of- system whose comoets degrade stochastically ideedetly ad gradally over time. To describe sch a system oe sally relies o lifetime models (Wag These models are however disadvataged i characterizig the system behavior becase they caot reflect roerly the itermediates states (i.e. degradatio state of system. To avoid this drawback we base the system modelig o stochastic rocesses. This will hels s to comte more fiely the comoet ad system reliabilities. I the followig for a easier aalysis we divide the system ito comoet level ad system level. 2. Comoet level Cosiderig the i-th comoet i = 2 of the system its accmlated degradatio level at time t ca be smmarized by a scalar radom variable X it. Withot ay maiteace actio this variable evolves as a cotios-time mootoically icreasig stochastic rocess { Xit } t 0 with X i0 = 0. As a case stdy a homogeeos gamma rocess with shae arameter αi ad scale arameter β i is sed to describe the degradatio ath { Xit } t 0 (see also (va Noortwijk 2009 for a thorogh review o the se of gamma rocess i maiteace modelig. As sch the desity fctios of the degradatio icremet Xit Xis betwee two times s ad t 0 s t is give by αi( t s αi( t s βix f i α ( i t s β = β i i xi e { xi 0} Γ ( αi ( t s where Γ ( α = (0 α + e d is the Eler gamma fctio. The behavior of sch a degradatio ath deeds closely o the cole of arameters ( αi βi ad its average degradatio rate ad variace are give by mi = αi βi ad σ 2 2 i = α i β i resectively. The comoet i fails as soo as its degradatio level exceeds a critical refixed threshold L i its failre time is the exressed by τ i = if { t Xi t Li}. Ths let Rτ i Xi t( xi be its coditioal reliability at time give its degradatio level at time t X = x it ca be comted as follows (va Noortwijk 2009 ( i i i i τ i X x i t i = P τi Xi t = xi = { xi< Li} Γ( αi ( t it i ( Γ( α ( t β ( L x R (2 where Γ ( α x = ( x α + e ddeotes the icomlete gamma fctio. 2.2 System level The cosidered system has k-ot-of-:f strctre i.e. it fails if ad oly if at least k of the comoets fail. Let τ s be the system failre time it is eqal to k-th smallest vale amog the set of comoet failre times { τ τ 2 τ }. Ths the coditioal reliability of the system Rτ ( : : sx x t at time give the comoets degradatio level at time t x: ( x x 2 x is comted as (David ad Nagaraja 2003 k j τ X x: = P τs X: t = x: τ X xi Rτ X xi = R ( ( R ( ( (3 s : t il il t l il il t j j = 0 C l= l= j+ where Rτ ( i X x i t i is give from (2 C j icldig C =!( j!( j! terms deotes the sm over all ermtatios { τi τi2 τi } of the set of idices { 2 } for which i < i 2 < < i j ad ij+ < i j+ 2 < < i. 3. Reliability-based oortistic redictive maiteace We roose i this sectio a oortistic redictive maiteace strategy for the aforemetioed k-otof-:f deterioratig system. For this strategy maiteace decisio-makig is ot based o a classical coditio idex of degradatio level bt o the coditioal reliability give from Sectio 2. Its erformace is evalated throgh a mathematical cost model develoed o the basis of semi-regeerative theory. j l ( 494

3 3. Maiteace assmtios We assme that the degradatio of each comoet i the system is hidde ad the comoet failre is ot-self-aocig. This reqires the isectio oeratios o the total system to reveal the degradatio level ad the failre/workig state of comoets. The isectio is assmed istataeos erfect odestrctive ad is icrred a cost ci. Two maiteace oeratios are available o each comoet: a revetive relacemet (cost c ci ad a corrective relacemet (cost cc c. Each relacemet oeratio ts the comoet back i the as-good-as-ew state. Moreover a relacemet (either revetive or corrective ca oly be istataeosly erformed at isectio times. Therefore a system dowtime may aear withi two sccessive isectio times hece a additioal cost is icrred from the system failre time til the ext relacemet time at a cost rate cd. Associated with a relacemet oeratios sch as dismatlig ad reassembly of the system sedig a maiteace team to the site etc. are eeded ad they icr a set- cost c s. Ths ecoomic deedece amog comoets exists ad oe wold like erform maiteace joitly o several comoets to save set- costs. 3.2 Maiteace strategy We aim at a oortistic maiteace strategy reresetig both CBM decisio ad ecoomic deedece. The CBM decisio is itegrated i the strategy throgh a re revetive relacemet threshold R of the comoet coditioal reliability while the ecoomic deedece is take ito accot by a oortistic revetive relacemet threshold R. More i detail a eriodic scheme with eriod ΔT is sed to isect the health state of comoets/system. At a isectio time Tm = m Δ T m = 2 we relace correctively the failed comoets ad revetively still srvivig comoets if their coditioal reliability at the ext isectio time give the actal detected degradatio level is less tha a threshold R (i.e. Rτ i Xi ( T Tm m +ΔT xi R for the comoet i i = 2. Whe a comoet is relaced (either revetively or correctively this is the oortity to relace other srvivig oes. More recisely if a relacemet is schedled o the comoet i we also oortistically relaced the comoet j i if its coditioal reliability at the ext isectio time give the actal detected degradatio level is less tha a threshold R (i.e. R < Rτ j X j Tm ( Tm +ΔT xj R. Ths for this strategy ΔT R ad R are the decisio arameters to be otimized. To ehace the imortace of the decisio arameters we call this strategy ( Δ TR R. Based o the ( Δ TR R strategy a degradatio-based oortistic maiteace strategy amely ( Δ TZ Z ca be easily give by relacig resectively the thresholds R ad R by degradatio thresholds Z ad Z. Oe ca remark that the ( Δ TZ Z strategy is less advatageos tha ( Δ TR R strategy whe the system is heterogeeos. I fact i a heterogeeos system a same reliability threshold for all comoets ca corresod to differet degradatio levels of differet comoets. Makig a maiteace decisio based o a reliability level is therefore more flexible tha based o a same degradatio level for all comoets. As a reslt the ( Δ TR R strategy is geerally more rofitable tha ( Δ TR R strategy. Of corse oe ca atrally thik abot a strategy with varios degradatio thresholds for differet comoets. However sch a strategy seems ifeasible i ractical alicatios becase it is relatively comlex. Moreover otimizig the strategy is really itractable becase of a large mber of decisio arameters. The ( Δ TR R strategy whose mber of decisio arameters is ivariat for all size of system is mch easier to be otimized. 3.3 Mathematical cost model To assess the erformace of the roosed maiteace strategy we focs o a widely sed asymtotic criterio which is the log-r exected maiteace cost rate of the overall system (Béreger 2008 C ( E C t = lim t t where Ct ( deotes accmlated maiteace cost of the system to time t. Accordig to the strctre of the ( ΔTR R strategy Ct ( ca be give as Ct ( = c N ( t c ( h N ( t + ( c + c N ( t + ( c + c N ( t + c W( t (5 i Is s G h s i Prev c s i Corr d h= 2 i= i= where NIs( t is the isectios mber i [0t] N iprev ( t ad N icorr ( t are resectively the mber of revetive relacemets (i.e. either oortistic relacemets or re revetive relacemets ad corrective relacemets of the comoet i i [0t] ideedetly o oeratios erformed o other comoets NGh ( t is the mber of gros for which h comoets 2 h are relaced at the same time i [0t] ad W( t deotes the system dowtime i i [0t]. (4 495

4 A classical way to aalytically evalate (4 is to se the reewal-reward theorem to exress the cost rate C o a reewal cycle of system (Tijms However i the reset case owig to very log reewal cycle i which too may scearios of maitaied system evoltio may aear alyig this aroach is almost imossible. To overcome the roblem we ca resort to semi-regeerative theory to exress C o a semi-reewal cycle (Béreger Sch a cycle is mch shorter ad cotais simler maiteace scearios comared to the reewal cycle ad as a reslt it allows comtig more easily. I fact accordig to the ( Δ TR R strategy after a isectio at time Tm m = 2 the evoltio of every comoet ad hece of the system deeds oly o the revealed degradatio levels. So let s deote { X : t } t 0 the mltivariate rocess reresetig the evoltio of the maitaied system ( X : t = ( X t X 2 t X t ad X it is the degradatio level of the i-th maitaied comoet at time t it is a semi-regeerative rocess. The discrete time rocess describig the maitaied system states at isectio times { Y: m } m where Y : m = X : T m is the a embedded Markov chai of { X : t } t 0 with statioary law π = π( x :. Ths alyig the semi-regeerative roerties of the maitaied system exressio (4 ca be rewritte as (Béreger 2008 ( Eπ C( T E [ ] E [ ] EC t C = lim = = ci Eπ NIs( T cs ( h π NG h( T t t π T π T E h= 2 + ( c + cs Eπ Ni Prev ( T + ( c + cs Eπ Ni Corr ( T + cd Eπ W ( T i= i= (6 where T is the first isectio time ad Eπ is the exectatio w.r.t. the statioary law π of { Y: m } m. Comtig the exectatio qatities i (6 is qite comlicated bt classical ad we omit it here becase of limit mber of ages. Istead we show the feasibility of (6 by givig a examle o a 2-comoets series system. The followig set of arameters is sed: α = 2.25 β = 0.75 α 2 = β 2 = L = 0 L 2 = 0 ci = 0 cs = 5 c = cc = 00ad cd = 25. Figre illstrates the statioary law π ad the shae of the cost rate whe oe of decisio arameters is fixed at its otimal vale. I fact for the chose data set otimal decisio arameters are R ot = 7 R ot = 0.87 ad Δ Tot = 2.4 which corresod to the otimal cost rate ot =.692. R ot = 7 R ot = 0.87 ΔT ot = π (x x R ot = 7 ΔT ot = x x ΔT R R ot = 0.87 ΔT ot = 2.4 R ot = 0.87 R ot = ΔT 2 R R R 0.8 Figre : Shae of the statioary law π ad the log-r exected maiteace cost rate comoets series system der ( Δ TR R strategy of a 2-496

5 4. Performace ad robstess aalysis This sectio aims to aalyze more deely the erformace ad the robstess of the ( Δ TR R strategy throgh sesitivity stdies. For a simler aalysis the 2-comoets series system i Sectio 3.3 is resed bt its characteristic arameters or maiteace costs ca be sccessively chaged deedig o differet case stdies. The ( Δ TZ Z strategy rereseted i Sectio 3.2 is sed as a bechmark. 4. Sesitivity to the maiteace costs We vary resectively oe of itervetio costs (i.e. ci cs cad cd fixe the others ad we ivestigate the evoltio of otimal cost rate of both ( Δ TR R strategy ad ( ΔTZ Z strategy. Figre 2 shows the reslts. We ca remark that the otimal cost rate of ( Δ TR R strategy is always lower tha the oe of ( ΔTZ Z strategy. This affirms the advatage of the coditioal reliability idex over the degradatio level i CBM decisio-makig i the cosidered case. However this advatage is decreased for smaller isectio cost ad dowtime cost rate or for higher set cost ad revetive relacemet cost c i c s c c d Figre 2: Evoltio of otimal maiteace cost rate w.r.t. the variatio of itervetio costs 4.2 Sesitivity to the system characteristics Here the degradatio seed m ad variace σ 2 of comoet are resectively varied other arameters are fixed ad we observe the otimal cost rate evoltio of the cosidered strategies. The ( ΔT R R strategy is still more rofitable tha the ( ΔTZ Z strategy (see Figre 3. However both strategies give almost the same rofit for a qasi-homogeeos system (see left figre below or for a heterogeeos system whose degradatio variaces of differet comoets are too differet (see right figre below m 2 = α 2 /β 2 = 34 σ 2 2 = α 2 /β 2 2 = m 2 = α 2 /β 2 = σ 2 2 = α 2 /β 2 2 = m = α /β 27m σ 2 2 = α /β Figre 3: Evoltio of otimal maiteace cost rate w.r.t. the variatio of degradatio seed ad variace 4.3 Sesitivity to the errors made o maiteace strategy decisio arameters I may ractical alicatios the estimatio of system arameters may be imrecise becase of oisy measremets or the lack of data. This ca lead to errors i determiig the otimal decisio arameters of a maiteace strategy ad hece a loss i its erformace. This sectio therefore aims to examie the imact of sch a error o the erformace of ( Δ TR R strategy. For this ed we comte its 497

6 maiteace cost rate whe the decisio arameters vary arod theirs otimal vales ad we comare with the otimal cost rate of ( Δ TZ Z strategy. Figre 4 shows the reslts. Obviosly the erformace.4.2 Low degradatio variace: σ = α /β 2 = 3 R Error o R (R.6.5 High degradatio variace: σ = α /β 2 = 6 R Error o R (R ΔTZ Z C ot.4 ΔTZ Z C ot ΔTR R C ot.2. ΔTR R C ot.2 -% -24% -8% -2% -6% 0% 6% 2% ε -6% -2% -8% -4% 0% 4% 7% ε Figre 4: Evoltio of otimal maiteace cost rate w.r.t. the variatio of decisio arameters error of the ( Δ TR R strategy decreases w.r.t. decisio arameters error. However it remais better tha the ( Δ TZ Z strategy whe the error is ot too high. This shows the robstess of the ( Δ TR R w.r.t. measremet errors. Moreover the strategy is more robst for smaller variace of degradatio rocess. 5. Coclsio ad ersectives We roose i this aer a reliability-based oortistic redictive maiteace model which ca characterize both asects of CBM decisio ad ecoomic deedece amog comoets of system. The merical reslts show that the roosed maiteace model is sitable for a k-ot-of- deterioratig system ad that sig coditioal reliability istead of degradatio level i maiteace decisio-makig ca make the model simle rofitable robst w.r.t. measremet errors ad easy to otimize. Desite these advatages the roosed maiteace strategy is still a semi-blid strategy becase the global system health state is ot take ito accot i maiteace decisios. The ftre work therefore aims to develo a reliability-based oortistic maiteace strategy cosiderig the global system health. Moreover for a comlete stdy other tyes of iteractios amog comoets (i.e. stochastic deedece ad strctral deedece shold be itrodced ito the model. Refereces Béreger C O the mathematical coditio-based maiteace modellig for cotiosly deterioratig systems Iteratioal Joral of Materials ad Strctral Reliability Bovard K. Arts A. Béreger C. Cocqemot V. 20 Coditio-based dyamic maiteace oeratios laig & groig: Alicatio to commercial heavy vehicles Reliability Egieerig & System Safety Castaier B. Grall A. Béreger C A coditio-based maiteace olicy with o-eriodic isectios for a two-it series system Reliability Egieerig & System Safety Cho D.I. Parlar M. 99 A srvey of maiteace models for mlti-it systems Eroea Joral of Oeratioal Research David H.A. Nagaraja H.N Order statistics 3 rd ed Joh Wiley & Sos New Jersey Uited States. Dekker R. Wildema R.E. Va der Dy Shcote F.A. 997 A review of mlti-comoet maiteace models with ecoomic deedece Mathematical Methods of Oeratios Research Jardie A.K.S. Li D. Bajevic D A review o machiery diagostics ad rogostics imlemetig coditio-based maiteace Mechaical Systems & Sigal Processig Tia Z. Liao H. 20 Coditio-based maiteace otimizatio for mlti-comoet systems sig roortioal hazards model Reliability Egieerig & System Safety Tijms H.C A first corse i stochastic models Joh Wiley & Sos West Sssex Eglad. va Noortwijk J.M A srvey of gamma rocesses i maiteace Reliability Egieerig & System Safety Wag H A srvey of maiteace olicies of deterioratig systems Eroea Joral of Oeratioal research