Saiic Prrin Saiic 10-2014 Uing Dgradaion Modl o A Pilin Lif Shiyao Liu Iowa Sa Univriy William Q. Mkr Iowa Sa Univriy, wqmkr@iaa.du Follow hi and addiional work a: h://lib.dr.iaa.du/a_la_rrin Par of h Saiic and Probabiliy Common Rcommndd Ciaion Liu, Shiyao and Mkr, William Q., "Uing Dgradaion Modl o A Pilin Lif" (2014). Saiic Prrin. 127. h://lib.dr.iaa.du/a_la_rrin/127 Thi Aricl i brough o you for fr and on acc by h Saiic a Iowa Sa Univriy Digial Roiory. I ha bn accd for incluion in Saiic Prrin by an auhorizd adminiraor of Iowa Sa Univriy Digial Roiory. For mor informaion, la conac digir@iaa.du.
Uing Dgradaion Modl o A Pilin Lif Abrac Longiudinal incion of hickn a aricular locaion along a ilin rovid uful informaion o a h lifim of h ilin. In alicaion wih diffrn mchanim of corroion roc, w hav obrvd variou y of gnral dgradaion ah. W rn wo alicaion of fiing a dgradaion modl o dcrib h corroion iniiaion and growh bhavior in h ilin. W u a Bayian aroach for aramr imaion for h dgradaion modl. Th failur-im and rmaining lifim diribuion ar drivd from h dgradaion modl, and w comu Bayian ima and crdibl inrval of h failurim and rmaining lifim diribuion for boh individual gmn and an nir ilin circui. Kyword Bayian, longiudinal daa, ilin rliabiliy, rmaining lif Dicilin Saiic and Probabiliy Commn Thi rrin wa ublihd a Shiyao Liu and William Q. Mkr, "Uing Dgradaion Modl o A Pilin Lif". Thi aricl i availabl a Iowa Sa Univriy Digial Roiory: h://lib.dr.iaa.du/a_la_rrin/127
Uing Dgradaion Modl o A Pilin Lif Shiyao Liu, William Q. Mkr Darmn of Saiic Iowa Sa Univriy Ocobr 8, 2014 Abrac Longiudinal incion of hickn a aricular locaion along a ilin rovid uful informaion o a h lifim of h ilin. In alicaion wih diffrn mchanim of corroion roc, w hav obrvd variou y of gnral dgradaion ah. W rn wo alicaion of fiing a dgradaion modl o dcrib h corroion iniiaion and growh bhavior in h ilin. W u a Bayian aroach for aramr imaion for h dgradaion modl. Th failur-im and rmaining lifim diribuion ar drivd from h dgradaion modl, and w comu Bayian ima and crdibl inrval of h failur-im and rmaining lifim diribuion for boh individual gmn and an nir ilin circui. Ky Word: Bayian; longiudinal daa; ilin rliabiliy, rmaining lif. 1
1 Inroducion 1.1 Moivaion and Puro Rad maur of wall hickn acro im a amld locaion along a ilin circui can b ud o valua h rliabiliy of a ilin. Dgradaion modl for longiudinal incion of h ilin hickn can b ud o dcrib ilin corroion bhavior, ima h lifim diribuion of ilin comonn, and rdic h rmaining lifim of a ilin circui. Thr ar wo diffrn uro for uch analy: (1) imaing h lif im cumulaiv diribuion funcion (cdf) of ilin gmn o rovid informaion ha can b ud o lan h conrucion of fuur ilin and (2) o ima h rmaining lif of an xiing ilin curcui. Dnding on dgradaion and corroion mchanim, diffrn aiical modl and mhod ar ndd o analyz ilin daa. In hi ar, w analyz hickn daa from wo diffrn ilin and roo dgradaion modl for ach alicaion. In om dgradaion modl, i i comuaionally challnging o ima aramr uing h radiional liklihood-bad mhod. Bayian mhod wih aroria rior diribuion rovid an alrnaiv aroach for imaing aramr of a comlicad dgradaion modl. In addiion, valuaion of h failur im and rmaining lifim diribuion i alo comuaionally faibl and fficin whn uing h Bayian mhod. 1.2 Pilin Daa Figur 1 how im lo of longiudinal ilin daa from Circui G in Faciliy 3. Daa wr obaind from a aml hickn maurmn locaion (TML). For h fir wo incion, only 12 TML wr ud. Subqunly, a rcivd rik of failur incrad, an addiional 76 TML wr ud. Som of h TML corrond o lbow and h ohr corrond o raigh i. For ach TML, h hickn wa maurd a four diffrn quadran locad a h 0, 90, 180, and 270 dgr oiion (o, righ, boom, and lf for a horizonal ilin). Th lin joining h oin rrn h dgradaion ah of h diffrn combinaion of locaion and quadran. Th fir incion wa rformd on 2
Fbruary 11, 1995, a numbr of yar afr h ilin had bn inalld. 0.12 0.14 0.16 0.18 0.22 0.24 0.26 Da Wall Thickn 11Fb95 10Fb2000 11May2001 12Jul2001 18Oc2001 2Ar2002 20Jan2003 Figur 1: Tim lo for ilin daa from Circui G in Faciliy 3. Th cond ilin daa i from a diffrn faciliy. Figur 2 dilay im lo for h ilin daa from Circui Q in Faciliy 1. Th daa coni of hickn valu a 33 TML and ach TML wa maurd a 4 im. Thr comonn y of h ilin in hi daa ar lbow, raigh i, and. In hi faciliy, h fir maurmn wa akn a h ilin inallaion da. Th im lo indica ha h original hickn vary from TML o TML. Alo, h i ar gnrally hickr han h lbow and raigh i. 1.3 Rlad Work Dgradaion modl ar ofn ud o a rliabiliy of indurial roduc. Lu and Mkr (1993) illura ha undr om iml dgradaion ah modl, hr can b a clod-form xrion for h failur im cdf. Char 13 of Mkr and Ecobar (1998) giv a gnral inroducion o dgradaion modl and dcrib h rlaionhi bwn 3
Wall Thickn 0.35 0.40 0.45 0.50 0.55 Ar00 Nov00 Jul01 Da Jan04 Figur 2: Tim lo for ilin daa from Circui Q in Faciliy 1. h dgradaion and failur-im analyi mhod of imaing a im-o-failur diribuion. Char 8 of Hamada al. (2008) rovid an ovrviw of Bayian dgradaion modl and u vral xaml o illura how o ima aramr of a dgradaion modl. Nlon (2009) dicu a modl for dfc iniiaion and growh ovr im and u maximum liklihood o ima aramr in h modl. Shikh, Boah, and Hann (1990) analyz daa from war injcion ilin ym and u h Wibull diribuion o modl h im-o-fir-lak. Pandy (1998) u a robabiliy modl o ima h lifim diribuion of a ilin bfor and afr rair du o h mal lo. 1.4 Ovrviw Th r of hi ar i organizd a follow. Scion 2 roo a dgradaion modl for ilin daa from Circui G in Faciliy 3 and u h Bayian aroach o ima h aramr in h dgradaion modl. Scion 3 driv failur im and rmaining lifim diribuion for h circui and comu h Bayian ima and h corronding 4
crdibl inrval. Scion 4 analyz ilin daa from Circui Q in Faciliy 1. A dgradaion modl i rood o dcrib h corroion iniiaion and growh bhavior obrvd in hi ilin. Scion 5 valua h failur im diribuion and rdic h rmaining lifim diribuion of Circui Q in Faciliy 1. In ordr o udy h daa ndd for imabiliy, Scion 6 analyz imulad daa for a ingl circui having mor han on incion afr corroion iniiaion. Scion 7 conain h concluding rmark and ara for fuur rarch. 2 Modling Pilin Daa from Circui G in Faciliy 3 In hi cion, w focu on h analyi of h ilin daa from Circui G in Faciliy 3 hown in Figur 1. W roo a dgradaion modl and Bayian imaion wih diffu rior diribuion o ima h aramr of h dgradaion modl. 2.1 Dgradaion Modl for Pilin Daa from Circui G in Faciliy 3 L Y ik dno h ilin hickn a im k for TML i (i = 1, 2,..., 88; k = 1, 2,..., 7). W aum ha h dgradaion ah of Circui G in Faciliy 3 i linar wih rc o incion im and ha h form Y ik = y 0 β 1i ( k 0 ) + ɛ ik (1) whr β 1i i 1 im h corroion ra a locaion i and ɛ ik i h maurmn rror rm. Hr y 0 i h original hickn a inallaion im 0. Scifically, h original hickn y 0 i 0.25 inch and h inallaion im 0 i Fbruary 12, 1990. Th rci da of inallaion and bginning-u wr no availabl and hi da wa obaind by xraolaing backward in im. Bcau h corroion ra dfind a h hickn chang r yar vari from locaion o locaion and could only b ngaiv, β 1i in h dgradaion modl (1) i a oiiv random variabl. To guaran a oiiv β 1i, w aum ha β 1i ha a lognormal diribuion [i.., β 1i Lognormal (µ β1, σβ 2 1 )] and ha h maurmn rror i ɛ ik NOR (0, σɛ 2 ). Thu h aramr in h dgradaion modl (1) ar: θ = (µ β1, σ β1, σ ɛ ). 5
2.2 Bayian Eimaion of h Paramr in h Dgradaion Modl Bayian imaion wih h u of diffu rior informaion i cloly rlad o liklihood imaion (wih a fla rior, h Bayian join orior diribuion i roorional o h liklihood). Bayian mhod rovid a convnin alrnaiv for imaing h aramr in h dgradaion modl, aricularly bcau w nd o mak infrnc on comlicad funcion of h modl aramr. For h xaml, w u a normal diribuion wih man zro and a larg varianc [i.., NOR (0, 10 3 )] a h rior diribuion for h aramr µ β1. Th rior diribuion for σ β1 and σ ɛ ar Uniform (0, 5). W obain a larg numbr of draw from h join orior diribuion of h dgradaion modl aramr uing Markov Chain Mon Carlo (MCMC) imlmnd in OnBUGS. Tabl 1 rn marginal orior diribuion ummari for h aramr in θ, including h man and 95% crdibl inrval. Figur 3 how h im lo of h fid hickn valu for Circui G in Faciliy 3 wih a 10-yar xraolaion afr h la incion in January 20, 2003. 95% Crdibl Inrval Paramr Porior Man Porior Sd. Dv. Lowr Ur µ β1 11.62 0.06263 11.75 11.50 σ β1 0.5753 0.04743 0.4910 0.6768 σ ɛ 6045 2.537E 4 5575 6574 Tabl 1: Marginal orior diribuion ummari of h dgradaion modl aramr ima for ilin daa from Circui G in Faciliy 3 uing h dgradaion modl (1). 2.3 Saiical Modl for Diffrn Quadran In Scion 2.1 and 2.2, w aumd ha h corroion ra of diffrn quadran from h am locaion follow h am diribuion. In non-vrical i, howvr, h corroion ra of locaion in h ur quadran migh b xcd o diffr from ha in h lowr quadran a h am TML. Th dgradaion modl in hi cion aum ha man of 6
0.25 Fid Wall Thickn Fir Incion La Incion Criical Lvl 12Fb90 11Fb95 10Fb2000 18Oc2001 20Jan2003 1Jan2004 1Jan2006 1Jan2007 1Jan2008 1Jan2009 1Jan2010 1Jan2011 1Jan2012 1Jan2013 Figur 3: Tim lo howing h fid hickn valu for h ilin daa from Circui G in Faciliy 3 uing h dgradaion modl (1). h logarihm of h corroion ra vary from quadran o quadran. Auming ha h circui wih iniial hickn 0.25 inch wa inalld on Fbruary 12, 1990, h dgradaion modl i Y ij k = y 0 β 1ij ( k 0 ) + ɛ ij k (2) whr β 1ij i h corroion ra of quadran j a TML i (i = 1, 2,..., 22; k = 1, 2,..., 7; j = 1,..., 4) and ɛ ij k, a bfor, i h maurmn rror rm. Similar o modl (1), β 1ij i alo oiiv in modl (2). W aum ha β 1ij ha a lognormal diribuion [i.., β 1ij Lognormal (µ β1j, σβ 2 1 )] and ɛ ij k NOR (0, σɛ 2 ). Th aramr in modl (2) ar: θ = (µ β11, µ β12, µ β13, µ β14, σ β1, σ ɛ ). Th Bayian mhod i again ud o ima θ. Tabl 2 rn marginal orior diribuion ummari for h aramr in θ, including h man and 95% crdibl inrval. Figur 4 how h im lo of h fid hickn valu for diffrn quadran of hi circui. 7
95% Crdibl Inrval Paramr Porior Man Porior Sd. Dv. Lowr Ur µ β11 11.31 0.1182 11.54 11.08 µ β12 11.67 0.1179 11.90 11.45 µ β13 11.59 0.1167 11.82 11.36 µ β14 11.92 0.1219 12.16 11.68 σ β1 0.5416 0.04551 0.4619 0.6377 σ ɛ 6033 2.556E 4 5556 6551 Tabl 2: Marginal orior diribuion ummari of h dgradaion modl aramr ima for ilin daa from Circui G in Faciliy 3 uing h dgradaion modl (2). Th dvianc informaion cririon (DIC) (dfind in Glman al. 2003 on ag 182-184), a maur of modl goodn-of-fi and comlxiy, i ud for h Bayian modl comarion. Th valu of DIC for modl (1) and (2) ar 2574.0 and 2633.0, rcivly. Bcau modl (2) ha an imoranly mallr DIC han modl (1), w can conclud ha hr i a quadran ffc. 3 Modl Rlaing Dgradaion and Failur in Circui G of Faciliy 3 3.1 Bayian Eimaion of h Failur Tim Diribuion Th dgradaion ah ovr im i D = D(, θ). Th failur of an individual gmn in a ilin i aid o hav hand whn h rmaining ilin hickn i l han h criical lvl D f ( inch in our xaml). Thi i known a a of failur dfiniion and uch criical lvl ar drmind hrough nginring judgmn a h hickn blow which hr i rik of a lak. Bcau β 1ij Lognormal (µ β1j, σβ 2 1 ) in modl (2), h failur im cumulaiv diribuion funcion (cdf) F () of individual gmn in a oulaion of 8
Quadran 1 (o) Quadran 2 (righ) Fid Wall Thickn 0.25 Fid Wall Thickn 0.25 12Fb90 11Fb95 10Fb2000 Da 18Oc2001 20Jan2003 1Jan2004 1Jan2006 1Jan2007 1Jan2008 1Jan2009 1Jan2010 1Jan2011 1Jan2012 1Jan2013 12Fb90 11Fb95 10Fb2000 Da 18Oc2001 20Jan2003 1Jan2004 1Jan2006 1Jan2007 1Jan2008 1Jan2009 1Jan2010 1Jan2011 1Jan2012 1Jan2013 Quadran 3 (boom) Quadran 4 (lf) Fid Wall Thickn 0.25 Fid Wall Thickn 0.25 12Fb90 11Fb95 10Fb2000 Da 18Oc2001 20Jan2003 1Jan2004 1Jan2006 1Jan2007 1Jan2008 1Jan2009 1Jan2010 1Jan2011 1Jan2012 1Jan2013 12Fb90 11Fb95 10Fb2000 Da 18Oc2001 20Jan2003 1Jan2004 1Jan2006 1Jan2007 1Jan2008 1Jan2009 1Jan2010 1Jan2011 1Jan2012 1Jan2013 Figur 4: Tim lo howing h fid hickn valu for diffrn quadran of ilin daa from Circui G in Faciliy 3 uing h dgradaion modl (2). gmn of quadran j in h ilin can b xrd in a clod form: F () = Pr(D() D f ) = Pr(y 0 β 1ij ( k 0 ) ) ( = Pr β 1ij ) ( ) log() log(k 0 ) µ β1j = 1 Φ nor k 0 σ β1 ) = Φ nor ( log(k 0 ) log() + µ β1j σ β1 whr Φ nor i h andard normal cdf.. (3) Th failur im diribuion, a a funcion of h dgradaion aramr, can b valuad imly by uing h Bayian aroach. For ach draw from h join orior di- 9
ribuion, on can valua F () in (3) o obain a corronding draw from h marginal orior diribuion of failur im cdf. Tabl 2 and Figur 4 ugg ha h corroion ra of quadran 1 from h ur quadran i h larg among h four diffrn quadran. Figur 5 (a) dilay h ima of h failur im cdf wih wo-idd 95% and 80% crdibl inrval for h ilin daa from quadran 1 of Circui G in Faciliy 3. On can alo obain h corronding failur im cdf lo for ohr quadran. Bu wih h larg corroion ra, h failur im lo for quadran 1 i h mo imiic. Th failur im cdf in (3) i an ima of h cdf for an individual ilin gmn. Alhough h rimary inr i o ima h lifim of a ilin viwd a a ri ym of many gmn, h lif im cdf of an individual ilin gmn rovid uful informaion o lan h conrucion of fuur ilin. (a) (b).2.1 cdf Eima 95% Confidnc Inrval 80% Confidnc Inrval.2.1 cdf Eima 95% Confidnc Inrval 80% Confidnc Inrval.05.05 Fracion Failing.02.01.005.002 Fracion Failing.02.01.005.002.001.0005.001.0005.0001.00005.0001.00001 4 6 8 10 14 18 22 28 Yar Afr Pilin Inallaion (Fb 1990) 0.50 2.00 5.00 2 Yar Afr January 2003 Figur 5: Dgradaion modl ima of (a) failur im cdf (yar afr ilin inallaion) and (b) rmaining lifim cdf (yar afr h la incion c ) wih wo-idd 95% and 80% crdibl inrval on h lognormal ar for ilin daa from quadran 1 of Circui G in Faciliy 3. 10
3.2 Prdicion of h Rmaining Lif of h Currn Circui In h ilin alicaion, h rmaining lif of a aricular gmn of a circui i an imoran quaniy for aing h lifim of h ilin. Th cdf of h rmaining lifim F RM () condiional on urviving unil h la incion im (January 2003) i whr c F RM () = Pr(T T > c ) = F (; θ) F ( c; θ), c (4) 1 F ( c ; θ) i h la incion im and F () i h failur im diribuion drivd in Scion 3.1. A bfor, valuaing (4) a orior draw rovid ima and h corronding crdibl inrval of h rmaining lifim cdf. Figur 5 (b) how h orior ima of h rmaining lifim cdf afr h la incion in January 2003 wih 95% and 80% crdibl inrval. In h ilin alicaion, i i of gra inr o ima mall quanil of h minimum rmaining lifim of h oulaion. To do hi, on nd o xraola furhr ino h ail of h rmaining lif diribuion imad for a givn gmn. Tyically a TML gmn i abou on foo long. Suo ha h nir ilin lngh ha M gmn of hi lngh. Thn h cdf of h minimum rmaining lif among all of h M gmn along h ilin can b xrd a F M () = Pr[T min ] = 1 [1 F RM ()] M (5) whr F RM () i h rmaining lifim cdf for a ingl gmn. If on wan o conrol F M (), uch ha F M () = Pr[T min ] =, hn on would choo h hrhold o b = F 1 M (), h quanil of h diribuion of h minimum T min among h M ilin gmn. Th ranlaion o h adjud quanil in rm of h rmaining lifim cdf F RM () i a follow: ( ) = F 1 1 M () = FRM 1 (1 ) 1 M. (6) Thi indica ha quanil of h minimum rmaining lifim diribuion of h oulaion of M gmn corrond o h 1 (1 ) 1/M quanil of h rmaining lifim cdf for ach gmn. Figur 6 how h orior dniy of 0.1, 0.2, 0.3, and 0.4 quanil of h minimum rmaining lifim diribuion wih h oulaion iz M = 100 uing h 11
dgradaion modl (1) and (2) rcivly. Modl (2) i mor conrvaiv han modl (1) a i gnra h mallr quanil ima. Quanil of h Porior Minimum Rmaining Lifim Diribuion 0.4 0.3 0.2 0.1 0 1 2 3 4 Yar Afr January 2003 Modl (1) Modl (2) Figur 6: Porior dniy of h 0.1, 0.2, 0.3 and 0.4 quanil of h minimum rmaining lifim diribuion (yar inc h la incion im c : January 2003) wih h oulaion iz M = 100 of ilin daa from Circui G in Faciliy 3 uing h dgradaion modl (1) and (2). Th mall quanil ima ugg ha h Circui G in Faciliy 3 could hav lakag rik wihin on yar afr h la incion. On hould ay clor anion o hi circui. Carful xaminaion, mor frqun incion a mor TML, or rirmn/rlacmn of h ilin would roc again h unxcd ilin lakag. 12
4 Modling Pilin Daa from Circui Q in Faciliy 1 Figur 7 i a rlli lo for h ilin daa from Circui Q in Faciliy 1. Each anl of h rlli lo corrond o hickn maurmn for a cific TML. Th rlli lo ugg an inring ilin corroion roc. For xaml, in TML #1, #2, and #3, hr i no dcabl hickn lo in h fir hr incion. Significan hickn lo, howvr, wr dcd a h forh incion im. Thi ugg ha h corroion roc wa iniiad bwn h hird and forh incion im. A om TML (.g., TML #12, #13, and #33), h corroion aar no o hav iniiad bfor h la incion im. Wall Thickn 0.35 0.45 0.55 0.35 0.45 0.55 0.35 0.45 0.55 8 1 9 2 10 3 11 4 12 5 13 6 14 7 1Ar00 1Nov00 1Jul01 1Ar00 1Nov00 1Jul01 1Ar00 1Nov00 1Jul01 1Ar00 1Nov00 1Jul01 1Ar00 1Nov00 1Jul01 1Ar00 1Nov00 1Jul01 29 22 15 30 23 16 31 24 17 32 25 18 33 26 19 27 20 28 21 0.35 0.45 0.55 0.35 0.45 0.55 Ar2000 Nov2000 Jul2001 Jan2004 Da Figur 7: Trlli lo for ilin daa from Circui Q in Faciliy 1. 4.1 Dgradaion Modl for Corroion Iniiaion and Growh W aum ha afr h corroion iniiaion, h corroion ra i conan for a aricular locaion, bu may diffr from locaion o locaion. W roo a dgradaion modl wih a random corroion iniiaion im and random corroion ra o dcrib h ovrall corroion 13
iniiaion and growh roc. Th dgradaion modl for h ilin hickn Y ij a im j for h TML i (i = 1, 2,..., 33; j = 1, 2,..., 4) i: Y 0i + ɛ ij for j < T Ii Y ij = (7) Y 0i β 1i ( j T Ii ) + ɛ ij for j T Ii. In hi modl, Y ij dno h hickn maurmn for TML i a im j. Y 0i i h original hickn of TML i. Bcau h diribuion of h original hickn dnd on h comonn y of h TML (lbow,, or raigh i), w aum ha h iniial maurmn Y 0i ha a normal diribuion wih diffrn man bu a common andard dviaion: If h TML i an lbow, w aum ha Y 0i NOR (µ y0lbow, σy 2 0 ); If h TML i a i, w aum ha Y 0i NOR (µ y0i, σy 2 0 ); If h TML i a, w aum ha Y 0i NOR (µ y0, σy 2 0 ). β 1i i h corroion ra for TML i and w aum ha β 1i Lognormal (µ β1, σβ 2 1 ) or β 1i Wibull (ν β1, λ β1 ). T Ii i h corroion iniiaion im a TML i and w aum ha T Ii Lognormal (µ TI, σt 2 I ). ɛ ij i h maurmn rror and w aum ha ɛ ij NOR (0, σɛ 2 ). j i h im whn h maurmn j wa akn. Th modl aramr ar: θ = (µ y0lbow, µ y0i, µ y0, σ y0, µ β1, σ β1, µ TI, σ TI, σ ɛ ) for h lognormal corroion ra. Whn h corroion ra ha a Wibull diribuion, h modl aramr ar: θ = (µ y0lbow, µ y0i, µ y0, σ y0, ν β1, λ β1, µ TI, σ TI, σ ɛ ). 4.2 Bayian Eimaion of h Paramr in h Dgradaion Modl In addiion o h modl, w nd o cify rior diribuion for h aramr in h dgradaion modl (7). Glman (2006) rovidd gnral uggion for chooing ror 14
rior diribuion for varianc aramr in h hirarchical modl. W u h following diffu rior diribuion for h andard dviaion σ y0, σ β1, σ TI, and σ ɛ : σ y0 Uniform (10 5, 5), σ TI Uniform (10 5, 10), σ β1 Uniform (10 5, 5), σ ɛ Uniform (10 5, 0.25). Th fac ha ilin daa of Circui Q in Faciliy 1 ha no mor han on incion afr h corroion iniiaion rul in difficuly idnifying h corroion ra and iniiaion im in h dgradaion modl. Tha i, for a givn TML wih vidnc of an iniiaion, w canno diinguih bwn an iniiaion clo o h fourh incion and a larg (in abolu valu) corroion ra and an iniiaion im clo o h hird incion im and a mallr corroion ra. According o h knowldg from xr in h ilin alicaion, h mdian corroion ra for h TML hould no xcd 0.022 inch r yar. Thu w cify a omwha informaiv rior diribuion for h mdian of corroion ra for TML β 1i0.5 ha imli an ur bound on h corroion ra: β 1i0.5 Uniform (10 6, 0.022). Rgarding h rior diribuion for h aramr µ y0lbow, µ y0i, µ y0, and µ TI, w u h following rior by cifying h lowr and ur bound of h uniform diribuion: µ y0lbow Uniform (0.4, 0.47), µ y0i Uniform (0.4, 0.47), µ y0 Uniform (0.5, 0.62), µ TI Uniform (9.31, 10 6 ). Th lowr bound of h uniform diribuion for µ TI i drmind by h aumion ha h corroion iniiaion can only occur afr h inallaion da. Similarly, if h corroion ra ha a Wibull diribuion, w cify h am indndn rior diribuion for 15
h aramr σ y0, σ TI, σ ɛ, µ y0lbow, µ y0i, µ y0, and µ TI. For h Wibull corroion ra diribuion, w cify h rior diribuion in rm of ν β1, h Wibull ha aramr and β 1i0.5, h mdian of corroion ra for TML. Th following ar rior for h ha aramr ν β1 and β 1i0.5 in h Wibull diribuion: whr λ β1 = log (2)/β ν β 1 1i 0.5 ν β1 Uniform (1.5, 5), β 1i0.5 Uniform (10 6, 0.022), i h alrnaiv cond aramr ud in h OnBUGS aramrizaion of h Wibull diribuion. Tabl 3 and 4 rn h orior diribuion ummari of aramr in h dgradaion modl uing lognormal and Wibull corroion ra rcivly. Figur 8 and 9 how h rlli lo of h fid hickn valu for Circui Q in Faciliy 1 uing lognormal and Wibull corroion ra wih 10-yar of xraolaion afr h la incion on January 1, 2004. 95% Crdibl Inrval Paramr Porior Man Porior Sd. Dv. Lowr Ur µ y0lbow 0.4379 4035 0.4301 0.4461 µ y0i 0.4315 3448 0.4246 0.4382 µ y0 0.5215 6226 0.5093 0.5336 σ y0 0.01342 192 0.01022 0.01767 β 1i0.5 0.01815 3171 9919 0.02187 σ β1 0.5902 0.3129 0.1877 1.391 µ TI 9.411 0.0122 9.389 9.438 σ TI 0.04408 0.01536 0.01804 0.07968 σ ɛ 4693 4.005E 4 3973 5552 Tabl 3: Marginal orior diribuion ummari of h aramr in h dgradaion modl wih a lognormal corroion ra for ilin daa from Circui Q in Faciliy 1. Th dvianc informaion cririon (DIC) i again ud for h Bayian modl comarion. DIC valu for modl wih lognormal and Wibull corroion ra ar 1513.0 and 1016.0, rcivly. Th modl uing h lognormal diribuion for h corroion ra ha 16
95% Crdibl Inrval Paramr Porior Man Porior Sd. Dv. Lowr Ur µ y0lbow 0.4380 4059 0.4299 0.4458 µ y0i 0.4314 3478 0.4245 0.4383 µ y0 0.5215 6039 0.5096 0.5333 σ y0 0.01347 1942 0.01027 0.01797 ν β1 2.890 0.9342 1.556 4.816 β 1i0.5 0.0197 1908 0.01488 0.02192 µ TI 9.412 0.01221 9.392 9.441 σ TI 0.04744 0.01431 0.02657 0.08333 σ ɛ 4715 4.016E 4 4007 5566 Tabl 4: Marginal orior diribuion ummari of h aramr in h dgradaion modl wih Wibull corroion ra for ilin daa from Circui Q in Faciliy 1. a mallr DIC, indicaing a br fi. Figur 10 and 11 how h box lo of aml from h marginal orior diribuion of corroion ra and iniiaion im for ach TML in Circui Q uing h lognormal corroion ra. Th lo indica ha for h TML whr ilin corroion aar no o hav iniiad bfor h la incion im, h orior diribuion of h iniiaion im ar righ kwd. Figur 12 comar lo of h marginal orior diribuion of h iniiaion im for TML wih vidnc of iniiaion and wihou iniiaion bfor h la incion. Th lo how ha h marginal orior diribuion of h iniiaion im for h TML wihou iniiaion ar hifd o h righ, righ kwd, and clo o ach ohr. 17
Fid Wall Thickn 0 0.1.30.40.5 0 0.1.30.40.5 0 0.1.30.40.5 8 1 9 2 10 3 11 4 12 5 13 6 14 7 1Jan12 1Jan14 1Jan12 1Jan14 1Jan12 1Jan14 1Jan12 1Jan14 1Jan12 1Jan14 1Jan12 1Jan14 29 22 15 30 23 16 31 24 17 32 25 18 33 26 19 27 20 28 21 0 0.1.30.40.5 0 0.1.30.40.5 Ar2000 Nov2000 Jul2001 Jan2004 Da Figur 8: Trlli lo of h fid hickn valu for Circui Q in Faciliy 1 uing h lognormal corroion ra diribuion. Th dod lin indica xraolaion. 5 Modl Rlaing Dgradaion and Failur for Circui Q in Faciliy 1 5.1 Bayian Evaluaion of h Failur Tim Diribuion A in h analyi of h ilin daa from Circui G in Faciliy 3, hr ar wo main uro for uing h dgradaion modl. Th fir i o a h lifim cdf of individual ilin comonn or gmn. Th cond i o rdic h rmaining lifim of h nir circui. Th dgradaion ah ovr im i D = D(, θ). A of failur i dfind o b h im a which h rmaining ilin hickn i l han 20% of h man of h hickn a h inallaion da. Suo ha T I Lognormal (µ TI, σt 2 I ), Y 0 NOR (µ y0lbow, σy 2 0 ), Y 0 NOR (µ y0i, σy 2 0 ), Y 0 NOR (µ y0, σy 2 0 ), and β 1 Lognormal (µ β1, σβ 2 1 ). Thn h cdf giving h roorion of ilin gmn ha hav a of failur a a funcion of oraing 18
Fid Wall Thickn 0 0.1.30.40.5 0 0.1.30.40.5 0 0.1.30.40.5 8 1 9 2 10 3 11 4 12 5 13 6 14 7 1Jan12 1Jan14 1Jan12 1Jan14 1Jan12 1Jan14 1Jan12 1Jan14 1Jan12 1Jan14 1Jan12 1Jan14 29 22 15 30 23 16 31 24 17 32 25 18 33 26 19 27 20 28 21 0 0.1.30.40.5 0 0.1.30.40.5 Ar2000 Nov2000 Jul2001 Jan2004 Da Figur 9: Trlli lo of h fid hickn valu for Circui Q in Faciliy 1 uing h Wibull corroion ra diribuion. Th dod lin indica xraolaion. im i F () = Pr(D() D f ) = Pr(Y 0 β 1 ( T I )I( T I ) D f ) = Pr(Y 0 β 1 ( T I ) D f TI ) + Pr(Y 0 D f < TI ) = Pr(Y 0 β 1 ( T I ) D f TI ) + Pr(Y 0 D f ) Pr( < T I ) 1 = φ NOR (z y0 ) 1 φ NOR (z TI ) 1 φ(z β )dy 0 dt I dβ 1 σ y0 T I σ TI β 1 σ β1 y 0 β 1 ( T I ) D f, T I + Φ NOR ( Df µ y0 σ y0 ) [1 Φ NOR (z TI )], (8) whr z y0 = (y 0 µ y0 )/σ y0, z TI = (log(t I ) µ TI )/σ TI, and z β = (log(β 1 ) µ β1 )/σ β1. Whn h corroion ra ha a lognormal diribuion, φ(z β ) = φ NOR (z β ) i h andard (µ = 0, σ = 1) normal robabiliy dniy funcion (df). Whn h corroion ra ha a Wibull diribuion, φ(z β ) = φ SEV (z β ) = x(z β x(z β )) i h andardizd mall xrm valu df. Bcau F () in (8) do no hav a clod form, h Mon Carlo imulaion 19
1 2 3 4 5 6 7 8 9 11 13 15 17 19 21 23 25 27 29 31 33 0.25 Corroion Ra: β 1i TML Numbr Figur 10: Saml from h marginal orior diribuion of h lognormal corroion ra for ach TML in Circui Q in Faciliy 1. mhod dcribd in Scion 13.5.3 of Mkr and Ecobar (1998) i ud o valua failur im cdf, uing 1,000 imulaion rial for h valuaion. Figur 13 how failur im cdf for lbow, i and uing normal, lognormal and Wibull corroion ra. Th lo ugg ha in h dgradaion modl (7), h lognormal corroion ra rovid h mo conrvaiv rul comard wih h ohr wo modl aumion on h corroion ra. Figur 14 how failur im cdf for lbow, i and gmn uing h lognormal corroion ra diribuion wih wo-idd 95% and 80% crdibl inrval. 5.2 Prdicaion of h Rmaining Lif of h Currn Circui Figur 15 comar h rmaining lifim cdf wih normal, lognormal and Wibull corroion ra for lbow, i and. Th lo ugg ha a lognormal diribuion for h corroion ra in h dgradaion modl (7) rovid h mo conrvaiv ima of rmaining lif. Thi i du o h long ur ail of h lognormal diribuion. Figur 16 20
1 2 3 4 5 6 7 8 9 11 13 15 17 19 21 23 25 27 29 31 33 Iniiaion Tim: T Ii 1Ar2000 1Nov2000 1Jul2001 1Jan2004 1Jan2006 1Jan2008 1Jan2010 1Jan2012 1Jan2014 TML Numbr Figur 11: Saml from h marginal orior diribuion of h corroion iniiaion im for ach TML in Circui Q in Faciliy 1 uing a lognormal diribuion o dcrib corroion ra. how ima of h rmaining lifim cdf uing h lognormal corroion ra in h dgradaion modl (7) and h corronding wo-idd 95% and 80% crdibl inrval. A in Scion 3.2, w ar rimarily inrd in imaing mall quanil of h minimum rmaining lifim cdf for Circui Q in Faciliy 1. Figur 17 how h orior dniy of 0.1, 0.2, 0.3, and 0.4 quanil of h minimum rmaining lifim diribuion from h dgradaion modl (7) wih h oulaion iz M = 100 uing h lognormal diribuion for corroion ra. Th largr quanil ima for h comonn indica ha hav a longr rmaining lifim. Th rul ar conin wih wha w obrvd rviouly in Figur 14 and 16. 21
15 TML wih Iniiaion TML wihou Iniiaion Dniy 10 05 00 1Ar00 1Nov00 1Jul01 Da Figur 12: Porior dnii of h iniiaion im for ach TML in Circui Q in Faciliy 1. 6 Effc of Addiional Incion on Idnifiabiliy In Scion 4.2, in h analyi of h ilin daa from h Circui Q in Faciliy 1, w ud a modraly informaiv rior diribuion o dcrib rior knowldg abou h mdian of h corroion ra, allviaing h idnifiabiliy roblm ha wa caud by having no mor han on incion afr any of h obrvd corroion iniiaion vn. Th rul of ha analyi howd a larg amoun of uncrainy in rdicion of rmaining lif. In h acual alicaion, h ownr of h ilin would hav o wai unil afr h nx incion im o obain mor rci ima of rmaining lif wihou uing informaiv rior informaion. To inviga hi idnifiabiliy roblm, in hi cion, w imula daa from modl (7) uch ha hr i mor han on incion afr an iniiaion (i.., daa ha i imilar o ho from Circui Q in Faciliy 1 bu wih addiional incion a fuur im). W coninu o u a lognormal corroion ra diribuion. Figur 18 dilay h im lo for h imulad ilin daa from a ingl circui wih 33 TML and hr comonn: 22
TML Comonn: Elbow TML Comonn: Pi Normal Corroion Ra Lognormal Corroion Ra Wibull Corroion Ra Normal Corroion Ra Lognormal Corroion Ra Wibull Corroion Ra Probabiliy Probabiliy 0 5 10 15 20 25 Yar Afr Inallaion 0 5 10 15 20 25 Yar Afr Inallaion TML Comonn: T Normal Corroion Ra Lognormal Corroion Ra Wibull Corroion Ra Probabiliy 0 5 10 15 20 25 Yar Afr Inallaion Figur 13: Dgradaion modl ima of failur im cdf for ilin comonn from Circui Q in Faciliy 1 comaring normal, lognormal and Wibull corroion ra diribuion in h dgradaion modl (7). lbow, raigh i and i. Corroion wa maurd a ach TML a 5 im. W u h am diffu rior diribuion ud in Scion 4.2 for all aramr xc for h mdian of h corroion ra β 1i0.5, i = 1, 2,..., 33. Bcau hr i mor han on incion afr h corroion iniiaion in h imulad daa, h idnifiabiliy roblm no longr xi. Thrfor, rahr han rric h ur bound of h rior diribuion of β 1i0.5 o 0.022, w can rlax h ur bound o roviding a diffu rior for β 1i0.5 [i.. β 1i0.5 Uniform (10 6, )]. For h imulad daa, h Bayian aramr ima ar clo o h ru aramr valu from which h daa wr imulad. Figur 19 how h rlli lo of h fid 23
TML Comonn: Elbow TML Comonn: Pi Probabiliy Probabiliy 0 5 10 15 20 25 Yar Afr Inallaion 0 5 10 15 20 25 Yar Afr Inallaion TML Comonn: T Probabiliy 0 5 10 15 20 25 Yar Afr Inallaion Figur 14: Dgradaion modl ima (h cnr lin) of failur im cdf for ilin comonn from Circui Q in Faciliy 1 wih h lognormal corroion ra diribuion in h dgradaion modl (7) and wo-idd 95% and 80% crdibl inrval. hickn valu for h imulad ilin daa uing h diffu rior diribuion. A in Scion 5, w ud h Mon Carlo imulaion mhod o valua h marginal orior diribuion of h failur im cdf a chon oin in im. Figur 20 how h failur im cdf for h imulad ilin daa of a ingl circui, uing h diffu rior. Comard wih h rul in Figur 14 for h ilin daa from Circui Q in Faciliy 1, h crdibl inrval in Figur 20 ar much narrowr. Th raon i ha in h imulad daa w hav mor incion afr h corroion iniiaion. Thu, h idnifiabiliy roblm ha caud h wid inrval i no longr rn. From a racical rciv, having vral incion ha occur afr an iniiaion im rovid a much mor ffciv imaion of 24
TML Comonn: Elbow TML Comonn: Pi Probabiliy Probabiliy Normal Corroion Ra Lognormal Corroion Ra Wibull Corroion Ra Normal Corroion Ra Lognormal Corroion Ra Wibull Corroion Ra 0 5 10 15 20 0 5 10 15 20 Yar Afr h La Incion Yar Afr h La Incion TML Comonn: T Probabiliy Normal Corroion Ra Lognormal Corroion Ra Wibull Corroion Ra 0 5 10 15 20 Yar Afr h La Incion Figur 15: Dgradaion modl ima of rmaining lifim cdf for ilin comonn from Circui Q in Faciliy 1 comaring normal, lognormal and Wibull corroion ra diribuion in h dgradaion modl (7). ilin gmn lifim diribuion. 7 Concluding Rmark and Ara for Fuur Rarch In hi ar, w dvlod dgradaion modl o dcrib h ilin corroion bhavior for wo aricular ilin daa. Th Bayian aroach wih aroria rior diribuion i uful for imaing aramr in h dgradaion modl. Th Bayian mhod, a an alrnaiv o h liklihood aroach, rovid a convnin mhod o ima and comu crdibl bound for funcion of h dgradaion modl aramr, vn whn a 25
TML Comonn: Elbow TML Comonn: Pi Probabiliy Probabiliy 0 5 10 15 20 Yar Afr h La Incion 0 5 10 15 20 Yar Afr h La Incion TML Comonn: T Probabiliy 0 5 10 15 20 Yar Afr h La Incion Figur 16: Dgradaion modl ima (h cnr lin) of rmaining lifim cdf for ilin comonn from Circui Q in Faciliy 1 wih h lognormal corroion ra diribuion in h dgradaion modl (7) and wo-idd 95% and 80% crdibl inrval. clod-form xrion of h funcion do no xi. A imulaion udy in Liu, Mkr, and Nordman (2014) how ha h inrval hav frquni covrag robabilii ha ar clo o h nominal crdibl lvl. Th failur im and h rmaining lifim diribuion and mall quanil ima of h minimum rmaining lifim diribuion rovid uful informaion o valua of h lif of a ilin. Thr rmain, howvr, a numbr of ara for fuur rarch. Th includ: In h dgradaion modl for corroion iniiaion and growh, lanning mhod ( Scion 9.6 of Hamada al. 2008) could b dvlod o choo an aroria numbr of incion afr h corroion iniiaion o obain mor rci ima of h failur 26
Quanil of h Porior Minimum Rmaining Lifim Diribuion 0.4 0.3 0.2 0.1 0 5 10 15 Elbow Pi T Yar Afr January 2004 Figur 17: Porior dniy of h 0.1, 0.2, 0.3, and 0.4 quanil of h minimum rmaining lifim diribuion (yar afr h la incion im c : January 2004) wih h oulaion iz M = 100 uing h lognormal corroion ra diribuion of ilin daa from Circui Q in Faciliy 1. im diribuion. Th modl wih linar dgradaion ah and h conan corroion ra can b xndd o h modl having nonlinar rlaionhi bwn ilin hickn and im. Each ilin circui wihin a faciliy, viwd a a ri ym of many gmn, could b conidrd a a comonn in a larg comlx ym of circui. In om alicaion, h lif im of uch a ilin ym could b imoran. In om ilin alicaion, i may b oibl o obain dynamic covaria informaion uch a mraur, flow, and y of marial. Th dgradaion modl could hn b gnralizd by incororaing hi dynamic covaria informaion ino h modling and 27
0.1 0.2 0.3 0.4 0.5 0.6 0.7 Da Wall Thickn Ar00 Jan04 Jan06 Jan08 Jan10 Figur 18: Tim lo for h imulad ilin daa from a ingl circui wih 33 TML. analyi. Rfrnc [1] Glman, A., Carlin, J. B., Srn, H. S., and Rubin, D. B. (2003). Bayian Daa Analyi, cond diion. London: Chaman and Hall. [2] Glman, A. (2006). Prior diribuion for varianc aramr in hirarchical modl. Bayian Analyi, 1: 515-533. [3] Hamada, M. S., Wilon, A., R, C. S. and Marz, H. (2008). Bayian Rliabiliy. Nw York: Sringr. [4] Liu, J., Mkr, W. Q. and Nordman, D. J. (2014). Th numbr of MCMC iraion ndd o comu Bayian crdibl inrval. Prrin, Darmn of Saiic, Iowa Sa Univriy. 28
0.2 0.3 0.4 0.5 0.6 Fid Wall Thickn 8 0.2 0.3 0.4 0.5 0.6 9 10 11 12 13 14 0.2 0.3 0.4 0.5 0.6 1 2 3 4 5 6 7 1Ar00 1Ar00 1Ar00 1Ar00 1Ar00 1Ar00 29 30 31 32 33 22 15 23 16 24 17 25 18 26 19 27 20 28 21 0.2 0.3 0.4 0.5 0.6 0.2 0.3 0.4 0.5 0.6 Ar2000 Jan2004 Jan2006 Jan2008 Jan2010 Da Figur 19: Trlli lo of h fid hickn valu for h imulad ilin daa in a ingl circui wih 33 TML uing h diffu rior diribuion. [5] Lu, C. J. and Mkr, W. Q. (1993). Uing dgradaion maur o ima a im-ofailur diribuion. Tchnomric, 35: 161-174. [6] Mkr, W. Q. and Ecobar, L. A. (1998). Saiical Mhod for Rliabiliy Daa. Nw York: John Wily and Son. [7] Nlon, W. B. (2009). Dfc iniiaion, growh, and failur a gnral aiical modl and daa analy. In M. Nikulin, N. Limnio, N. Balakrihnan, W. Kahl, and C. Hubr-Carol (Ed.), Advanc in Dgradaion Modling: Alicaion o Rliabiliy, Survival Analyi, and Financ (Saiic for Indury and Tchnology),. 27-37. Boon: Birkhur. [8] Pandy, M. D. (1998). Probabiliic modl for condiion amn of oil and ga ilin. NDT & E Inrnaional, 31: 349-358. [9] Shikh, A. K., Boah, J. K. and Hann, D. A. (1990). Saiical modling of iing corroion and ilin rliabiliy. Corroion, 46: 190-197. 29
TML Comonn: Elbow TML Comonn: Pi cdf Eima 95% Crdibl Inrval 80% Crdibl Inrval cdf Eima 95% Crdibl Inrval 80% Crdibl Inrval Probabiliy Probabiliy 0 5 10 15 20 Yar Afr Inallaion 0 5 10 15 20 Yar Afr Inallaion TML Comonn: T cdf Eima 95% Crdibl Inrval 80% Crdibl Inrval Probabiliy 0 5 10 15 20 Yar Afr Inallaion Figur 20: Dgradaion modl ima of failur im cdf for h imulad ilin daa in a ingl circui wih 33 TML uing h lognormal corroion ra diribuion and diffu rior in h dgradaion modl (7) and wo-idd 95% and 80% crdibl inrval. 30