ESTIMATIONS OF RESIDUAL LIFETIME OF ALTERNATING PROCESS. COMMON APPROACH TO ESTIMATIONS OF RESIDUAL LIFETIME

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1 Srucural relably. The heory and pracce Chumakov I.A., Chepurko V.A., Anonov A.V. ESTIMATIONS OF RESIDUAL LIFETIME OF ALTERNATING PROCESS. COMMON APPROACH TO ESTIMATIONS OF RESIDUAL LIFETIME The paper descrbes a generalzed mehod of esmang resdual lfeme for an objec wh an arbrary operang sraegy. Asympoc and non-asympoc assessmens of facors convergng n he long-erm objec operaon have been obaned. The man dea of he developed approach s o brng down an arbrary renewal process o a well-known alernang renewal process. The mehod makes possble o predc he resdual lfeme of equpmen a he operaonal sage, and can be used n esmang resource characerscs of nuclear power plan (NPP) equpmen. Keywords: resdual lfeme, alernang process, operang sraegy. Inroducon Nowadays, he so-called elemenary model of repar s used n mos cases when carryng ou a probablsc analyss of relably characerscs. Afer each falure, a sysem s brough no sound condon n neglgbly shor me and mmedaely comes back no operaon. However, he funconng of modern echncal sysems, as a rule, represens a more complcaed process characerzed by perodcal or consan conrol of fauls, schemes of fal deecon, emergency and renewal works, ec. In addon, should be noed ha when dealng wh esmaon or predcon of a resdual resource, everyhng generally bols down no o calculaons of durably characerscs bu o analyss of relably parameers, such as falure rae, probably of falure-free operaon (PFO), avalably facor, and based on he resuls, a concluson s hen made abou he echncal sae of an objec. To produce a more accurae sudy of equpmen servce lfe, s necessary o develop mehods for esmang durably characerscs per se. Also, s necessary o ake no accoun feaures and modes of equpmen funconng ha can consderably nfluence relably as seen from experence. As parameers of durably, resource and servce lfe are one of he basc conceps of he relably heory. Predcon of resource of objecs a he operaonal sage s hen of specal mporance. Subjec o esmaon s he resdual resource whch defnes possble duraon of he operaon of an objec from he gven momen of me and up o he momen when he parameer of echncal sae acheves s lm value. 65

2 The presen work presens mehods of sascal esmaon of resdual resource whch allow us o ake no accoun he peculares of funconng and manenance of elemens as par of complex echncal sysems.. Non-asympoc esmaons of average resdual me for alernang process There are a number of publcaons n whch a lo of aenon s pad o such resource characerscs as drec resdual me (DRT), reverse resdual me (RRT), as well as average drec resdual me (ADRT) and average reverse resdual me (ARRT). The sudy [], for example, consders how o defne resdual operang me (ADRT) for non-recoverable equpmen. In he majory of scenfc publcaons [2, 3] on hs ssue, he defnon of DRT and ADRT (and, accordngly, formulas and calculaons) are provded for he case of smple renewal process ( s supposed ha me of renewal can be negleced). So, s necessary o develop mehods for esmang resdual me for more complcaed alernang processes. A smple process of renewal represens a sequence of mes o falure of an objec under consderaon, wh he me of renewal se o zero. In hs case, ADRT s an expecaon of he remaned operang me of sysem o a nex falure, sarng from he me pon n whch he sysem was effcen [2]: V( ) =Μ ( τ ) I{ τ <τ }; + + () where τ s he me pon of -h falure, I {A} s funcon-ndcaor of he argumen valdy whch s equal o one, f A s rue, and o zero f no. Le s exend he concep DRT for he case of any renewal process. In he work [5], ADRT s reaed as an expecaon of he remaned operang me of sysem o a nex falure, sarng from he me pon n whch he sysem was effcen: V( ) =Μ ( τ ) I{ ν <τ }; + + (2) Here ν and τ are me pons of he -h renewal and -h falure, ν =, ν τ + ν +. I should be noed ha n he formula (2) here s a condon of he objec beng effcen a he me pon. Le s show he general approach o oban esmaons of resdual me for he sraegy of funconng descrbed by means of alernang process of objec renewal. Such a process s n deal nvesgaed n he heory of renewal [2]. I s consdered ha a he nal me pon = an objec s n sound condon. The objec funcons ll he momen of falure τ, hen he emergency renewal s carred ou ll he momen of renewal ν. Afer renewal he objec connues o work ll he nex momen of falure, and hen here s renewal and ranson no effcen sae. Random varables and β are duraon of he -h nerval of avalably and unavalably respecvely. Such a cycle s reproduced agan ll he chosen momen of me. The gven process s represened n fgure : 66

3 Fgure. Alernang process of renewal The formula (2) for he gven process can be wren down n he form of: V( ) =Μ ( ν + ) I{ ν <ν + }; + + (3) Alernang process s a specal case examned n [5], and based on works [4, 5], s enough jus o oban non-asympoc esmaon of ADRT n he form of negraed Volerra equaon of he second knd n mages of he Laplace ransform: G ( p) V( p) = ; (4) f ( p) f ( p) β where G M P ( p) ( p) = ; p And n space of orgnals (herenafer he symbol * desgnaes operaon of negraed convoluon): where G () s a free member of he equaon (5), such ha: V( ) = G( ) + ( V* f * fβ)( ); (5) ( ) (6) G () = xg (; x) dx = xf + x dx = P () x dx = M P () x dx = M + F () x dx; where f (), f β () s he densy of durably dsrbuon of he -h nerval of avalably and unavalably respecvely. Noe ha such characersc as reverse resdual me (RRT) s poorly suded. By defnon RRT s a me from he momen of he las falure ll he momen of me. We shall show how o deduce nonasympoc esmaon for an average reverse resdual me (ARRT) smlar o (5). We shall desgnae ARRT as R () smlar o [4-5], so we shall oban: R() =Μ ( ν) I{ ν <ν + } = ϕ(); + (7) 67

4 where ϕ ( ) =Μ[( ν ) I{ ν < ν + }] = ( y) I{ y < y + x} f ( x) f ( y) dxdy = + ν = ( y) f ( y) f ( x) dxdy = ( y) f ( y) P ( y) dy = ( f * A )( ); ν ν ν y where A ()=xp (x); (8) Le s apply he Laplace ransform o φ (, so we wll ge he followng: ϕ ( p) = fν ( p) A( p); The Laplace represenaon for (7) wll look as follows: ϕ ν ν R( p) = ( p) = f ( p) A ( p) = A ( p) f ( p); I should be noed ha ν = ( +β ), =,2,..., ν =. j j j= Hence f ν ( p) = ; f ( p) f ( p) β We oban R( p) = A ( p) + R( p) f ( p) f ( p); β Movng o orgnals, we shall wre down he requred expresson of ADRT: R() = A () + ( R* f * f )() = P () + ( R* f * f )(). (9) β β By analogy wh ARRT and ADRT, we shall nroduce and nvesgae new resource characerscs: W () s average drec resdual me of unavalably (ADRTU) he expecaon of remaned me of objec unavalably before he nex renewal, from he me pon n whch he sysem s n unavalable sae. 68

5 Q () s average reverse resdual me of unavalably (ARRTU) he expecaon of objec unavalably me from he momen of he las falure ll he momen of me n whch he sysem s unavalable. Fgure 2. ADRTU and ARRTU Smlarly o (3) and (7), formulas for ADRTU and ARRTU can be wren down n he followng form: W( ) =Μ ( ν ) I{ τ < τ + β }; Q( ) =Μ ( τ ) I{ τ < τ + β }; Le s oban non-asympoc esmaon for W () smlarly o he way how he esmaon of ADRT has been obaned above: W( p) = Gβ( p) fτ ( p); where G β (p) s he Laplace mage of he funcon G β (): G ( ) = Mβ P ( x) dx; () β β I should be noced ha τ = + ( + β ); j j j= f p f p f p f p τ ( ) = ( ) ( ( ) ( )) ; β ( p) fτ ( p) = f ( ) ( ( ) ( )) ; p f p fβ p = f( p) fβ( p) f 69

6 Fnal esmaon n space of mages: W( p) = G ( p) f ( p) + W( p) f ( p) f ( p); β β In space of orgnals: W( ) = ( Gβ * f)( ) + ( W* f * fβ)( ). () The formula () can be wren down n some oher form. Applyng Laplace s ransformaon o (), we have: Mβ Pβ ( p) Gβ ( p) = ; (2) p Then: Mβ Pβ ( p) f( p) f( p) Gβ( p) f( p) = f( p) = Mβ Pβ( p) ; p p p Movng no he space of orgnals: ( G * f )( ) = Mβ F ( ) ( P * F )( ); (3) β β The formula () becomes as he followng: W( ) = Mβ F ( ) ( P * F )( ) + ( W* f * f )( ). (4) β β Le s more brefly dwell upon he concluson of non-asympoc esmaon for ARRTU: Q( p) = Aβ( p) fτ ( p); where A β (p) s an mage of funcon A β () = xp β (x); The requred equaon n mages: And afer ranson o orgnals: Q( p) = A ( p) f ( p) + Q( p) f ( p) f ( p); β β 7

7 Q( ) = ( A * f )( ) + ( Q* f * f )( ). (5) β β Noe : All he four obaned equaons (4, 7, 4, 5) of non-asympoc esmaons of nvesgaed resource characerscs are Volerra equaons of he second knd wh varous free members, bu wh he same kernel he funcon (f * f β )() represenng he negraed convoluon of dsrbuon denses of objec avalably and unavalably nervals. In some cases s convenen o nroduce a random varable ω = + β descrbng duraon of one cycle of nvesgaed objec work and s relaed densy of dsrbuon f ω () = (f * f β )( whch furher s consdered as he kernel of negral equaons. 2. Asympoc esmaons of average resdual me for alernang process Le s show how o oban asympoc esmaon for alernang process of ADRT: The frs mehod: In he lm (6), we shall pass n space of mages akng no accoun (4): (6) G( p) M P( p) lm V( ) = lm V( p) p= lm p[ ] = lm ; f ( p) f ( p) f ( p) f ( p) p p p β β Noe ha: p p p lm( M P ( p)) = M lm e P ( ) d = M P ( ) d = ; p p p lm f ( p) = lm e f ( ) d = f ( ) d = ; To ge rd of uncerany of he knd, we shall apply L Hôpal s rule: ( M P ( p)) P ( p) lm V( p) p= lm = lm ; ( f ( p) f ( p)) f ( p) f ( p) + f ( p) f ( p) p p p β β β Noe ha: 2 p ( M) + D lm P ( p) = lm e P( ) d P( ) d ( ); p p = = 2 7

8 f p p = e f d f d M p p = = lm ( ) lm ( ) ( ) ; Fnal esmaon for ADRT has he followng form: (7) The second mehod: An alernang process can be consdered as generalzaon of classcal process of renewal [2], hen we shall oban asympoc esmaon for ADRT: where P s he probably of any momen of me fallng no he nerval of avalably, V s he asympoc esmaon of ADRT for he classcal process conssng of a sequence of me beween falures { }. Furher we have: P M = ; M + Mβ Accordng o [3], an asympoc esmaon for classcal process: V 2 2 M ( M) + D = = ; 2M 2M where M 2 s he 2-nd me momen of a random varable. The fnal esmaon akes he followng form: The gven expresson compleely concdes wh he earler receved esmaon (7). Thus, he problem of asympoc esmaon for ADRT s generally reduced o an esmaon of he momens of random varables duraon of nervals of objec avalably and unavalably. By smlar reasonng, we can oban: (8) (9) 72

9 3. Esmaons of resdual me for an arbrary process In any sysem s possble o carry ou echncal renewal acons. Such acons can be carred ou a momens apponed n advance and a random momens. Types of consdered sraegy reflec presence of varous prevenve manenance ypes (for example, scheduled/correcve, emergency/prevenve), conrol, and varous ypes of falures, renewal, and oher varous condons of operaon. All aforesad generaes a huge amoun of equpmen manenance sraeges and even more of mahemacal models for he descrpon of occurrng evens. Process of renewal for an arbrary manenance sraegy can be generally reduced o alernang process. By desgnang as and β he general duraons of objec avalably and unavalably nervals respecvely, we shall receve he renewal process smlar o ha n Fgure. To apply he descrbed mahemacal ools, s necessary o descrbe dsrbuons of random varables and β. As an example, le s consder one sraegy of objec funconng ypcally me n pracce. The sysem s deemed o be avalable a he nal momen of me, and he facor of sysem avalably a he pon = s equal o one. The sysem foresees scheduled prevenve manenance whch s carred ou perodcally n he me nerval Т (he perod of prevenve manenance) and lass for he random me η m (see Fgure 2). If before he momen of scheduled prevenve manenance here s a falure durng he casual momen of me ξ, hen he emergency renewal of he sysem s carred ou whose casual me s equal o η f. Afer sysem emergency renewal here s a re-plannng of he momen of prevenve manenance ha s he curren perod of prevenve manenance moves o he rgh a he me nerval ξ + η f. Afer renewal he sysem connues o work ll he momen ξ, f here was a falure or ll he nex momen of prevenve manenance Т f a falure dd no occur. Furher here s a correspondng renewal and a smlar cycle of work sars over agan. The descrbed sraegy of funconng s presened n Fgure 3: Fgure 3. The sraegy whch akes no accoun he bul-n conrol wh prevenve manenance ξ means he momens of falure, τ means me nervals from he begnnng of sysem operaon ll he momen of he nex falure of sysem, and ν means me nervals from he begnnng of sysem operaon ll he end of he nex recovery. Accordng o [6] he momens of renewal and falure are defned as follows: τ = ν + T I{ ξ > T} + ξi{ ξ < T} + I{ ξ j < T} ( ξ j + ηf ) + I{ ξ j > T} ( T + ηm); j= ν = ν + I{ ξ < T} ( ξ + ηf ) + I{ ξ > T} ( T + ηm) + I{ ξ j < T} ( ξ j + ηf ) + I{ ξ j > T} ( T + ηm); j= 73

10 Hence, he -h nerval of avalably can be presened as { } { } τ ν ξ ξ ξ = = T I > T + I < T ; Smlarly, we shall oban he expresson for he -h nerval of unavalably: { } { }; β = ν τ = η I ξ > T + η I ξ < T m f Accordng o (8-9), n order o oban asympoc esmaons of ADRT, s necessary o fnd he frs and second cenral momens of random varables and β. For non-asympoc esmaons, we generally need dsrbuon funcons and denses of random varables of mean mes beween falures and me of renewal. Le s nvesgae random varables andβ. Noe ha Hence Fξ ( ), < T F ( ) = ;, T T M = P ( ) d; (2) To oban M 2, we shall wre down he dsrbuon funcon n he form of: where H (x) s he Heavsde funcon, hen: ξ F ( x) = H( T x) F ( x) + H( x T); (2) ξ f ( x) = F ( x) = H( T x) f ( x) δ( T x) F ( x) + δ( x T); (22) ξ ξ where δ (x) s he Drac dela funcon. So, we oban: M x f() x dx x H ( T x) fξ() x dx x δ( T x) Fξ() x dx x δ( x T ) dx = = + = T x fξ( x) dx T Fξ( T ) T ; (23) = + The mahemacal expecaon of he value β s { } { } { } { } Mβ = M( η I ξ > T ) + M( η I ξ < T ) = Mη P ξ > T ) + Mη P ξ < T = m f m f = Mη ( F ( T)) + Mη F ( T); (24) mr ξ fr ξ 74

11 To compue Mβ 2, we shall fnd he funcon of dsrbuon of β: ( ) ({ } { }) ( ) { } { } Fβ ( x) =Ρ β < x =Ρ β < x ξ T +Ρ β < x ξ < T = ( ) =Ρ ({ ηm < x} { ξ T} ) +Ρ { ηf < x} { ξ < T} = ( ) F ( x ) F ( T ) + F ( x ) F ( T ); ηm ξ ηf ξ Hence ( ) f ( x ) = f ( x ) F ( T ) + f ( x ) F ( T ); β η ξ η ξ m f β ( ξ ) m ξ f Mβ = x f ( x) dx= F ( T) Mη + F ( T) Mη ; (25) Consderng (2), (23), (24), (25), he asympoc esmaons become suable for numercal calculaons: R x f ( ) ( ) 2 ξ x dx T Fξ T + T M = V = = ; 2( M + Mβ) 2( P ( ) d+ Mη ( F ( T)) + Mη F ( T)) асимп асимп T T ξ m ξ f ξ W 2 2 ( Fξ( T) ) Mηm + Fξ( T) Mηf 2 M β = Q = = ; 2( M + Mβ) 2( P ( ) d+ Mη ( F ( T)) + Mη F ( T)) асимп асимп T ξ m ξ f ξ To oban non-asympoc esmaons of characerscs suded n hs paper, s necessary o solve negral equaons (4, 7, 4, 5). Accordng o Noe, a kernel of he equaons n all four cases s he funcon f ω (), he densy of dsrbuon of a random varable ω. For he sraegy under consderaon, we have: ξ + ηf, ξ < T ω = ; T + ηm, ξ T Le s fnd he funcon of dsrbuon of hs random varable: { } { } Fω ( ) = P( ω < ) = P( I ξ T ( T + ηm) + I ξ < T ( ξ + ηf ) < ) = = P( η < T; ξ > T) + P( η < ξ; ξ < T) = m f 75

12 T ^T x ^T ( ) = f () x f () y dydx + f () x f () y dydx = F ( T ) F ( T ) + f () x F ( x) dx; T ξ η ξ η ξ η ξ η m m m f We shall oban he requred densy of dsrbuon by dfferenaon: df () f () = = F ( T ) f ( T ) + f () x f ( x) dx; ^T ω ω ( ξ ) ηm d ξ η f (26) Nex we shall fnd free members of he negral equaons. For ADRT he free member of he equaon s defned (6), he funcon of dsrbuon and he expecaon of he random varable are defned n (2), (2). For ARRT he free member of he equaon s defned n (8). For ADRTU looks lke n (3). For ARRTU s necessary o fnd (A β * f ) (). Ths problem becomes complcaed by he fac ha n he found formula (22) here s a Drac dela funcon, hus makng he correc numercal negraon f (x) mpossble. Consderng (22) and pary of dela funcon, we shall wre down he densy of dsrbuon n he followng form: where C( x) = H( T x) f ( x); f ( x) = H( T x) f ( x) + δ( x T) P ( x) = C( x) + δ( x T) P ( x); ξ ξ ξ ξ Then ( A * f )( ) = A ( x) C( x) dx + A ( x) δ( x T ) P ( x) dx = β β β ξ = A β ( x) C( x) dx+ A β ( T) P ξ ( T) H( T); (27) where A ( ) = ( F ( )) = P ( ); β β β I s already possble o calculae numercally a free member under he formula (27). Thus, we have found a kernel (26) and free members (6, 8, 3, 27) of negral equaons (4, 7, 4, 5) respecvely, n he form ha s suable for numercal soluon. 76

13 4. An example of calculaon Le s compue asympoc and non-asympoc esmaons of ADRT, ARRT, ADRTU, and ARRTU for he sraegy of objec funconng descrbed n Secon 3. The perod of prevenve manenance wll be se o 7 arbrary uns (a.u.) of me. As he law of dsrbuon for me o falure and scheduled prevenve manenance, we shall ake a wo parameer Webull dsrbuon, and for emergency renewal we shall ake a gamma dsrbuon. Parameers of he dsrbuon law are specfed n Table : Table Parameers of he law of dsrbuon of random varables Random varable Desgnaon Parameer of form, a.u. Parameer of scale, a.u. Tme o falure ξ 5 6 Duraon of scheduled prevenve manenance Duraon of emergency renewal η m 2 2 η f 3 5 Fgures 4-5 represen asympoc and non-asympoc esmaons of ADRT and ARRT, ADRTU and ARRTU respecvely. The gven fgures show convergence of asympoc (doed lne) and non-asympoc (connuous lne) esmaons: Fgure 4. Graph of behavor of esmaons of ADRT and ARRT expecaons 77

14 Fgure 5. Graph of behavor of esmaons of ADRTU and ARRTU expecaons 5. Concluson We have developed a generalzed mehod for calculang he followng resource characerscs for an objec wh a random sraegy of funconng: ) Average drec resdual me; 2) Average reverse resdual me; 3) Average drec resdual me of unavalably; 4) Average reverse resdual me of unavalably. We have obaned boh asympoc and non-asympoc esmaons of he gven parameers. The analyss of esmaons shows ha he dfference beween asympoc and non-asympoc esmaons a he nal me nerval can be a sgnfcan quany, whereas n case of long-erm funconng of an objec he gven esmaons converge. The mehod can be appled for esmang resource characerscs of NPP equpmen, ncludng echncal sysems wh perodcal or consan conrol of fauls, schemes of fal deecon, emergency and renewal works, ec. nheren n hem. The mehod allows us o predc a resdual resource of objecs a he sage of operaon ha defnes possble duraon of he operaon of an objec from he gven momen of me and up o he momen when he parameer of echncal sae acheves s lm value. 78

15 References. Chn-Dew La, Mn Xe. Sochasc ageng and dependence for relably. Sprnger, 26. p Baehel F., Franken P. Relably and manenance servce. The mahemacal approach М: Rado and communcaon, 988. p Venzel E.S., Ovcharov L.A. The heory of random processes and her engneerng applcaons. М: Nauka, 99. p Chepurko V.A. Characerscs of sysem dependably n vew of heerogeney of falure flows. Dagnoscs and predcon of complex sysems sae. The collecon of proceedngs 7. Obnnsk: INPE. p Sokolov S.V. Esmaon of resdual resource of power reacor facly subsysems of he frs block reacor RBMK- of he Smolensk aomc power saon // News of hgh schools. Nuclear power. 3, 29. Pp Dagaev A.V. Research and developmen of non-asympoc mehods n he analyss of relably of elemens and power reacor facly subsysems n vew of he conrol and prevenve manenance: The hess of Cand.Tech.Sc. / Dagaev A.V.: INPE. Obnnsk, 23. p