Relationship between Delay Time and Gamma Process Models

Transcription

1 A ublicion of CHEMICAL ENGINEERING TRANSACTIONS VOL. 33, 23 Gues Ediors: Enrico Zio, Piero Brldi Coyrigh 23, AIDIC Servizi S.r.l., ISBN ; ISSN The Ilin Associion of Chemicl Engineering Online : DOI:.333/CET3334 Relionshi beween Dely Time nd Gmm Process Models Renyn Jing Fculy of Auomoive nd Mechnicl Engineering, Chngsh Universiy of Science nd Technology, 96, Secion II, Wn Ji Li Nn Lu, Chngsh, Hunn 44, Chin jing@csus.edu.cn The dely-ime conce nd gmm rocess model hve been widely lied in CBM seings. This er revels close relionshi beween hese wo models. Th is, given one of hem, he oher cn be roximely deermined. This nure is very useful in relibiliy modeling nd minennce decision nlysis bsed on he dely ime model or gmm rocess model becuse some nlysis is esier for one of hem hn for he oher. The mehod o mke muul conversions beween hese wo models is develoed nd illusred by hree exmles. The oenil licions nd ossible exensions of he mehod re lso discussed.. Inroducion Condiion-bsed minennce (CBM) hs been widely used o chieve high relibiliy for criicl comonens or mjor filure modes in degrding sysem. The bsic ide is o secify he degrdion level by condiion monioring or insecions so h he comonen cn be revenively reired or relced before i fils. In simle cses, he CBM decision is direcly bsed on he observed se. In his cse, he degrdion nd filure models re needed for oimizing he lrm limi, filure limi or/nd he insecion scheme (Jing, 2). Two yicl models used in his conex re he dely ime model nd he gmm rocess model. Similr o he conce of P-F inervl of RCM (Moubry, 992), where P mens oenil filure nd F mens funcionl filure, he dely ime conce divides he degrdion rocess ino norml nd defecive hses. The norml hse is he ime inervl from he use beginning o defec iniiion; nd he defecive hse is he ime eriod from defec iniiion o filure nd is ermed s dely ime. The dely ime conce is lied o minennce by imlemening n insecion scheme o check wheher he iem is defecive or no. If defec is found, he iem is revenively reired or relced. In such wy, he minennce cion cn be rrnged in relively imely wy nd he oerionl relibiliy ges imroved. Okumur (997) resens mehod o uilize he dely-ime model for deermining he ime oins of insecion for deerioring sysem under condiion-bsed minennce. Wng (28) resens review of he dely ime insecion models. Jing (22) develos n insecion imeliness mesure nd shows h here exiss unique insecion inervl where he imeliness mesure chieves is minimum. This inervl is clled he imeliness-bsed oiml insecion inervl. Wng (22) overviews he recen dvnces in dely-ime-bsed minennce modelling wih focus on new mehodologicl develomens nd indusril licions. The gmm rocess hs been exensively used o model degrdion rocesses due o wer (Grll e l, 22), corrosion (Yun nd Pndey, 28), nd so on; nd is suied o reresen he slow, coninuous nd rogressive degrdion rocess (Abdelbki e l., 22). Noorwijk (29) resens deiled lierure survey of he licion of gmm rocesses in minennce. While lying he gmm rocess model o CBM, wo degrdion limis (i.e., lrm nd filure hresholds) re usully re-secified. Referring o Figure, he inersecion oin beween he degrdion curve nd he lrm limi cn be viewed s he P oin of RCM nd he inersecion oin beween he degrdion curve nd he filure limi cn be viewed s he F oin. In erms of he dely ime conce, he ime inervl wih he degrdion level being Plese cie his ricle s: Jing R., 23, Relionshi beween dely ime nd gmm rocess models, Chemicl Engineering Trnscions, 33, 9-24 DOI:.333/CET3334 9

2 smller hn he lrm limi cn be viewed s he norml hse nd he ime inervl wih he degrdion level being beween he lrm nd filure limis cn be viewed s he defecive hse. There re differences beween he dely ime model nd he gmm rocess model. For he dely ime model, he defec is hysiclly idenifible nd usully resuls in hrd filure; nd he defec sus is reresened in yes-or-no wy wihou quniive informion. Once he insecion scheme is secified, he minennce decision is srighforwrd. The led ime (i.e., he ime inervl beween he idenificion of defec nd he ime o filure) my be shor, deending on he disribuion of dely ime nd he ime when he defec is found. For he gmm rocess model, he filure is defined sus nd usully ssocied wih funcionl filure or sof filure. The degrdion level is quniively mesured nd he iem wih he degrdion beween he wo limis is no necessry o hve hysiclly observble defec. Once he insecion scheme, lrm nd filure limis re secified, he minennce decision is lso srighforwrd, nd he led ime cn be conrolled by roriely djusing he lrm limi. Filure limi Alrm limi g() h() P F y() y() Figure : Links beween P-F inervl, gmm rocess nd dely ime In his er, we will furher exmine he relionshi beween he dely ime nd gmm rocess models, nd show how hey re muully convered. The usefulness of such muul conversions is obvious. For exmle, if one wns o fi gmm rocess model bu he d vilble is only enough for fiing dely ime model, i will be ossible o derive he gmm rocess model from he fied dely ime model. Anoher oenil licion is o uilize he exising resuls from one model o nlyze he oher model if i is more difficul o nlyze he ler model. Such n exmle is o oimize he insecion inervl of CBM wih he gmm degrdion rocess using he dely-ime model. This is becuse he relionshi beween iem filure nd insecion inervl cn be exlicily modelled by he dely-ime model (see Wng, 22). The er is orgnized s follows. Secion 2 resens he deils of he wo models. The conversion mehod beween he wo models is resened nd illusred in Secion 3. The er is concluded in Secion Dely ime nd gmm rocess models 2. Dely ime model Consider single iem wih mjor filure mode (e.g., figue). The filure rocess cn be reresened by he dely ime conce. Le U nd Z denoe he iniil nd dely imes, resecively. Assume h U nd Z re muully indeenden nd heir disribuions re known nd denoed s Gu ( ) nd H( z ), resecively. Time o filure is rndom vrible given by T = U + Z. Le F () denoe he disribuion of T. Given Gu ( ) nd H( z ), F () cn be derived hrough convoluion oerion. To void he convoluion oerion, simle wy o ge F () is o use simulion. Firs, rndomly genere lrge smle of T ( = U + Z ), nd hen fi he emiricl disribuion of T obined from he rndom smle o n rorie disribuion funcion. We illusre he roch s follows. Exmle : Assume h G ( ) = W ( ; 2.5,) nd H ( z) = Gmm( z;2,), where Wβ (;, η ) is Weibull disribuion wih he she rmeer β nd scle rmeer η, nd Gmm(;,) z s b is gmm disribuion wih he she rmeer s nd scle rmeer b. 2

3 Rndomly genere smle of T wih smle size 5, nd fi he rndom smle o he Weibull, norml, lognorml nd gmm disribuions, resecively. I is found h he Weibull disribuion wih β = 2.86 nd η = 2.9 rovides he bes fi. Figure 2 shows he emiricl nd fied CDFs, which re close o ech oher..8 F() Figure 2: Emiricl nd fied CDFs for Exmle 2.2 Sionry gmm rocess model Suose h non-decresing degrdion quniy Y follows he sionry gmm rocess. In given ime, Y () is rndom vrible nd follows he gmm disribuion wih she rmeer s = nd scle rmeer b. As such, he CDF of Y () is given by Gmm( y;, b ). An imorn roery of he sionry gmm rocess is h incremen Δ Y is indeenden nd hs he disribuion Gmm( Δy; Δ, b). Referring o Figure, le y nd y denoe he lrm nd filure limis. The ime o he lrm limi (similr o U in he dely ime model) is rndom vrible nd hs he disribuion funcion given by F () = Gmm( y ;, b) () The ime o he filure limi (similr o T in he dely ime model) is rndom vrible nd hs he disribuion funcion given by F () = Gmm( y ;, b) (2) The durion h Y is beween he lrm nd filure limis (similr o Z in he dely ime model) is rndom vrible nd hs he disribuion funcion given by F ( z) = Gmm( y y ; z, b) z (3) 2.3 Discussion. The dely ime model cn be fully secified by wo disribuionl models nd hence usully hs four rmeers. Given he disribuions of wo of U, Z nd T, he oher disribuion cn be derived. Bu he derivion is usully mhemiclly inrcble. 2. The sionry gmm rocess model lso hs four rmeers (, b ). The disribuions of he hree rndom vribles (similr o U, Z nd T ) re given direcly by () hrough (3), which cn be esily evlued using sredshee rogrm such s Microsof Excel. 3. Conversions beween he wo models 3. Conver he dely ime model ino he gmm rocess model Suose h he dely ime model is given by H( z ) nd F (). The roch is similr for oher combinions. Le =.(.).99, z = H ( ), = F ( ) (4) 2

4 The roblem is o find he rmeers se (, b, y ) so h SSE chieves is minimum: {[ ( )] [ z( )] } =. SSE = F + F z (5) where F () nd ( ) Fz z re given by (2) nd (3), resecively. Exmle 2: H( z ) nd F () re he sme s hose given in Exmle. Using he bove roch, we hve (, b ) = (.6,.28, , 7.98) (6) Idelly, Fz( z) = F ( ) =. Th is, he los of Fz ( z ) nd F ( ) versus re srigh line hrough he oins (, ) nd (, ). Figure 3 shows he los of F ( z ) nd F ( ) versus. As seen, he fied gmm rocess model roughly mees he desired roery. z.8 CDFs F z (z) F () Figure 3: Goodness of fi of he gmm rocess model for Exmle 2 I is difficul o derive he disribuion of U from H( z ) nd F () bu he disribuion of U cn be esily obined from he fied gmm rocess model. Using he rmeers given by (6) o (), G () cn be roximed by F (), which cn be furher roximed by he Weibull disribuion W ( ; 2.8,.84). Figure 4 shows he rue disribuion W ( ;2.5,) nd is roximions. As seen, he roximions re firly close o he rue disribuion. This illusres he usefulness of he conversion..8 G() W(;2.5,) G(), F () Figure 4: Plos of W ( ;2.5,), G () nd F () for Exmle 2 22

5 3.2 Conver he gmm rocess model o he dely ime model The conversion is srighforwrd by leing F() = F (), G () = F(), H( z) = F ( z) (7) z As illusred erlier, hese disribuions cn be furher roximed by common disribuions such s he Weibull, gmm nd lognorml disribuions using he les squre mehod. Exmle 3: Assume h he rmeers se of he gmm rocess model is (, b ) = (.6,., 7., 8.) (8) The roblem is o find he rmeers of he dely ime model. Wihou loss of generliy, ssume h T follows he Weibull disribuion nd Z follows he gmm disribuion. Using he bove roch, we obined he rmeers of he dely ime model: ( β,η, s, b ) = (3.38, 5.59,.98,.29) 9) I is execed h he lo of Fz ( z ) versus H( z ) or F () versus F () is srigh line hrough he oins (, ) nd (, ). Figure 5 shows hese los. As seen, he fied dely ime model mees he desired roery well. For given dely ime model, he oiml decision models nd heir soluions (e.g., oimizion of insecion schemes) hve been well documened in he lierure (Wng, 28). This imlies h convering he gmm rocess model ino he dely ime model cn fcilie he minennce decision nlysis..8 CDFs F z (z) F () F(),H(z) Figure 5: Goodness of fi of he dely ime model for Exmle 3 4. Conclusions In his er, we hve shown he links nd differences beween he dely ime nd gmm rocess models, nd resened he roch o conver one model ino he oher model. The roch hs been illusred by hree exmles. The roch resened in his er is oenilly useful for relibiliy modeling nd minennce decision nlysis. Two ossible licions re s follows:. To simlify he CBM decision oimizion. For exmle, he oimizion of he lrm limi nd insecion scheme of CBM cn be conduced in he frmework of he dely-ime model. 2. To simlify he sre r invenory conrol nlysis in CBM seing. This is becuse he ime o ordering sre r is similr o U of he dely-ime conce nd he rndom led ime is similr o he dely ime (e.g., see Segersed, 994). As such, he sre r invenory conrol nlysis cn be lso conduced in he frmework of he dely-ime model. 23

6 The mehod cn be exended in he following severl direcions:. For he gmm rocess model in CBM seing, oic for fuure sudy is o ke he minennce led ime ino ccoun for deermining he lrm limi. 2. i is nurl o exend he mehod o oher sochsic rocesses such s he Wiener rocess (e.g., see Lindqvis nd Skogsrud, 29) rher hn he gmm rocess. 3. I is more rcicl o exend he mehod o he non-sionry gmm rocess (e.g., see Nicoli e l., 29) rher hn he sionry gmm rocess. 4. For he dely ime model, oic for fuure sudy is o exend he roch o he cse where U nd H re correled. Acknowledgemen This reserch ws suored by he Nionl Nurl Science Foundion (grn 7726). References Abdelbki N., Bouli E., Gceb M., Beyeb M., 22, Anlysis of he redicive models of oil nd gs rnsorion ducs' rehbiliion. Chemicl Engineering Trnscions, 29, Grll A., Berenguer C., Dieulle L., 22, A condiion-bsed minennce olicy for sochsiclly deerioring sysems. Relibiliy Engineering nd Sysem Sfey, 76, Jing R., 2, Oimizion of lrm hreshold nd sequenil insecion scheme. Relibiliy Engineering nd Sysem Sfey, 95, Jing R., 22, A Timeliness-Bsed oiml insecion inervl ssocied wih he dely ime model, 22 Prognosics & sysem helh mngemen conference, 23 My - 25 My 22, Beijing, -5. Lindqvis B.H., Skogsrud G., 29, Modelling of deenden comeing risks by firs ssge imes of Wiener rocesses. IIE Trnscions, 4(), Okumur S., 997, An insecion olicy for deerioring rocesses using dely-ime conce. Inernionl Trnscions in Oerionl Reserch, 4(5 6), Moubry J., 992, Relibiliy-Cenred Minennce. Indusril Press, New York, he USA. Nicoli R.P., Frenk J.B.G., Dekker R., 29, Modelling nd oimizing imerfec minennce of coings on seel srucures. Srucurl Sfey, 3(3), Segersed A., 994, Invenory conrol wih vriion in led imes, esecilly when demnd is inermien. Inernionl Journl of Producion Economics, 35( 3), Vn Noorwijk J.M., 29, A survey of he licion of gmm rocesses in minennce. Relibiliy Engineering nd Sysem Sfey, 94, 2 2. Wng W., 28, Dely ime modelling, in: Comlex Sysem Minennce Hndbook, Eds. Murhy DNP., Kobbcy AKS., Sringer, Germny. Wng W., 22, An overview of he recen dvnces in dely-ime-bsed minennce modelling. Relibiliy Engineering nd Sysem Sfey, 6, Yun X.X., Pndey M.D., Bickel G.A., 28, A robbilisic model of wll hinning in CANDU feeders due o flow-ccelered corrosion. Nucler Engineering nd Design, 238(),