Redundancy System Fault Sampling Under Imperfect Maintenance

Transcription

1 A publcao of CHEMICAL EGIEERIG TRASACTIOS VOL. 33, 03 Gues Edors: Erco Zo, Pero Barald Copyrgh 03, AIDIC Servz S.r.l., ISB ; ISS The Iala Assocao of Chemcal Egeerg Ole a: DOI: /CET33377 Redudacy Sysem aul Samplg Uder Imperfec Maeace Chao Zhag*, Lha Guo, Bopg Xao, Ru Kag School of Relably ad Sysems Egeerg, Bejg Uversy of Aeroaucs ad Asroaucs, Bejg 009, Cha Whe coducg smulao for evaluag complex sysem relably or avalably, mehod of radom samplg s appled o smulae faul occaso of complex sysem. Curre research of faul samplg geerally assumes maeace acvy resores sysems o good-as-ew ad bad-as-old whou cosderg parly good suao. However, sysem usually keeps s fuco hrough mperfec maeace ha resores sysem o parly good : he sysem afer maeace wll o as good as ew, bu youger. A sysem faul radom samplg mehod uder codo of mperfec maeace mus be preseed order o ehace smulao model credbly ad correcess. Whe relably srucure of sysem s he form of redudacy, he faul samplg mehod uder codo of mperfec maeace s dffcul because o oly redudacy sysem faul me s expoeal dsrbuo bu also ha he sae raso process of redudacy sysem s hard o model ad seady sae probably of each compoe he redudacy sysem s hard o deerme.so hs paper, a geerally reparable parallel sysem wh o-decal compoes: whch s a commo form of redudacy s cosdered, he a faul radom samplg mehod for hs kd of redudacy sysem uder codo of mperfec maeace accordg o moe carlo smulao prcple s preseed. The characerscs of hs mehod meoed above s ha Markov cha embedded wh hs mehod s employed o model he sae raso process of redudacy sysem ad o deerme seady sae probably of each compoe he sysem. aul occasos of redudacy sysem uder mperfec maeace ca be smulaed ad faul compoes he sysem ca be deermed va he faul samplg mehod. These are ovel corbuos made hs paper. ally, a umercal case usg faul samplg mehod for a redudacy sysem uder mperfec maeace s gve ad he valdy ad feasbly of faul samplg mehod s verfed.. Iroduco Whe coducg smulao for evaluag redudacy sysem relably or avalably, mehod of radom samplg s appled o smulae faul occaso of sysem. Redudacy sysem faul samplg ad deermao are referred o smulag he faul occaso accordg o radom umbers ad he sysem faul dsrbuo fuco ad deerme he compoe ha malfucoed accordg o he sysem srucure ad compoe falure raes whe sysem dsrbuo fuco s kow. Maeace effec clude perfec effec, o-perfec effec ad mperfec effec o a compoe falure. A perfec effec ca resore he compoe o good-as-ew, a o-effec o bad-as-old, ad a mperfec effec o parly good. Sce s usually assumed ha maeace effec s parly good realy, so he compoe faul samplg wh mperfec maeace mehod s more applcable o egeerg sysem. A he curre sage, sudy maly focus o he mperfec maeace model, o referrg o he faul samplg mehod uder mperfec maeace. Imperfec maeace model s dscussed deal by Doye ad Gaudo (004. Jao ad Wag(006 prese a faul samplg mehod o arplae sysem whose lfe s expoeal dsrbuo. Su(006 ad Zhao(006 suppose falure rae of sysem s mevara, however, he sysem relably srucure s smply cosdered as seres,faul samplg of sysem wh redudacy relably srucure uder mperfec maeace s cosdered. Guo ad Kag(007 sudes he faul samplg mehod of seres sysem for evaluag sysem avalably, bu he lfe of each Please ce hs arcle as: Zhag C., Guo L., Xao B., Kag R., 03, Redudacy sysem faul samplg uder mperfec maeace, Chemcal Egeerg Trasacos, 33, DOI: /CET

2 compoe s cosdered as expoeal dsrbuo. Zo(00 coducs he sysem faul samplg wh he cosderao ha compoes sysem are o-decal, however, he lfe dsrbuo ype s all he same. I hs paper, radom faul samplg mehod for sysem ha has redudacy relably srucure uder mperfec maeace s researched accordg o Moe Carlo smulao prcple. The ovely of he mehod sems from he facs ha o oly he maeace s mperfec, bu also he falure rae of compoes sysem s me-vara ad he sysem relably srucure s redudacy. O he bass meoed above, whe coducg faul samplg for redudacy sysem, deermao of faul compoes becomes dffcul, Markov sae raso process s used o solve hs problem.va he faul samplg mehod, faul occasos of complex sysem uder mperfec maeace ca be smulaed ad faul compoes he redudacy sysem ca be deermed.. Compoe faul samplg uder mperfec maeace Maeace effecs ca be perfec, o-perfec ad mperfec. Tme-vara fucos of compoe falure rae correspodg o he maeace effecs are dffere. alure rae of compoe should suae bewee he level of good-as-ew ad bad-as-old afer mperfec maeace, f compoe faul me are expressed as (, compoe falure rae whou maeace s f (.alure rae afer mperfec maeace s r(, acual age of compoe afer he - h mperfec maeace a me of s: ( α Z T, 0,,. ( where α s he recovery coeffce,0<α <. alure rae of compoe s ( ( ( α r f Z f T ( Process of compoe faul samplg s show as follows: uco of compoes uder mperfec maeace cao resore o as good as ew. Because faul dsrbuo fuco of he compoe s moooously creasg ad he value of faul dsrbuo fuco of compoe a equals o, so equaly (3 ca be obaed: < (3 alure rae fuco of compoe afer -h mperfec maeace s show Eq (4 ( α r f (4 Based o he relably heory, he relaoshp bewee falure rae fuco ad relably fuco ca be defed as Eq (5: Suppose p ( α + ( ( α p p ( α dl d l r d d ( α p [( ] α,aul samplg Eq (6 ca be obaed from Eq (4 ad Eq (5: where p s a radom umber from 0 o, ca be calculaed by - ad p accordg o equao (6. (5 (6 058

3 3. Sysem faul samplg uder mperfec maeace 3. Redudacy Sysem Model The basc assumpos of redudacy sysem model are The complex sysem ca be dvded o p layers, compoes each layer are cossed parallel. Compoe s cossed of several sub-compoes meoed relaoshp above he ex layer Suppose sysem faul s evoked by oe sub-compoe faul for hs compoe s cossed by subcompoes parallel. Correcve maeace resores sysem or compoes o parly good codo. 3. aul samplg mehod of redudacy sysem Whe coducg faul samplg for redudacy sysem, he Markov mehod s used, s referred o smulae he faul me of sysem by faul dsrbuo fuco ad Markov cha. The compoes malfuco ca be deermed va combg faul me of he sysem wh radom umber group ad aalyzg falure raes relao or faul sae probably of compoes each layer. The correspodg ree of faul samplg mehod s show gure Accordg o he faul samplg mehod gve las chaper, he sysem faul me ca obaed by equao group (7: ( η ( η ( ( α... α ( η ( ( α α Suppose a compoe s combed by sub-compoes parallel, sub-compoes he subcompoes malfuco ca be deermed accordg o samplg resul obaed by gve radom umbers ad faul probably a he me of. The faul sae of parallel sysem ca be expressed as {R/, f(, M-,r(M, he sae of parallel sysem s expressed by R or, dcaes he sae ha sysem have fauls, R dcaes he sae ha sysem does have fauls, umbers bracke afer f are he subscrps markg he faul compoes he parallel sysem, umbers bracke afer r are he subscrps markg he relable compoes, o he bass above, he faul sae raso dagram of parallel compoe wh dffere sub-compoes ca be show gure : I he seady sae, sysem sae probably ca be obaed by he followg rules: rae of raso from hs sae o oher saes mulply he probably of hs sae equals o sum raes of raso from oher saes o hs sae mulply he probably of he correspodg sae. The correspodg faul sae probably equao group of parallel compoe ca be show equao group(8, he above equao group, sum of sysem faul sae probably equals o, ca be expressed Eq (9. A me of, accordg o equao group (8 ad Eq (9, equao of sysem faul sae probably ca be obaed, Eq (0 s for he sae ha here s o compoe, Eq ( s for he sae ha here are M- faul compoes he parallel sysem. φ (7 ( λ λ ( 3 λ 3( λ ( λ M( M λ M ( λ ( M λ ( λ ( λ ( { ( a { R, f ( φ, r(,... ( b R, f(, M, r( M ( C { R, f(,, r( φ φ gure Sae raso dagram of parallel compoe wh sub-compoes 059

4 There are faul saes for complex sysem combed by compoes parallel. Suppose sum of faul k sae probably of he frs k saes s Pk p.if he gve radom umber s bewee ( P, P, + 0 he parallel sysem he -h sae, faul sub-compoe ca be deermed by he subscrp. ( P{, (, (,... ( P Rfφ r λ { Rf, (, r(,..., +,... M P P P M + P{ Rf, (.. M, rm (, M+... M + + P{, (..,, (... Rf M rm... P{, f(,..., r( P{ R, f(,...,,, r( λ( φ + {, (,.., (... + λ( { λ, (.., (,... ( + + R f M r M R f M r M { R, f(,.. M, r( M... λ M ( (8 P ( (9 ( ( η ( α λ( λ α P{ R, f ( φ, r(,... ( ( ( ( η α + + α λ λ λ P{ P P { R, f ( φ, r(,... { R, f ( φ, r(,... ( { R, f ( φ, r,... ( η ( ( α M M λ ( α R, f (,... M, r( M... M M P ( η ( ( α M λ α M (0 ( 4. umercal Example Suppose a redudacy sysem s desged as wo layers. The sysem s cosruced by hree LRUs combed parallel show gure.alure rae fucos of sysem ad LRUs ca be show Table : Table : alure dsrbuo of sysem, LRU ad LRU Sysem LRU LRU LRU3 Dsrbuo Type Webull Webull Exp Exp Shape Parameer m 3, 0 m, 0 λ λ 3 alure Rae uco f ( 3 f ( f ( f ( 3 alure uco Dsrbuo ( e 3 ( e ( e 3 ( e 060

5 Sysem LRU LRU LRU3 gure Redudacy sysem srucure The samplg resul of sysem faul me ca be obaed accordg o Equao group (7,suppose he sysem s uder mperfec maeace ad α 0.5, he samplg resul ca be show Table 3: Accordg o radom umber group (,, 3, 4, he rao of compoe falure rae ad sae raso probably for he hrd layer, faul compoes hs sysem ca be deermed usg op-dow mehod, where repar rae 3,. The sae probably of compoe show Table.4 ca be obaed by faul compoes ls whch s show g.4 ca be obaed by he equaos (-(4 gve he 3rd chaper. Table : Sysem faul me Sep Radom umber aul Tme Combao of complex sysem samplg resul s show Table ad Table 4.The resul of Table s sysem faul samplg me, he resul of Table 4 s he compoes malfuco deermed by faul sae probably of compoes coeced parallel. or he sysem s oly combed by hree parallel LRUs, so whe he sysem malfucos, he hree LRUs mus have fauls, however, faul compoes deermed by he Markov cha ca be he oal LRUs, whe he wo resuls coradcs, he faul me ca be rue smulao me o be he pu daa, so hs faul me s elmaed, oly he faul me whch ca coordae he sysem faul ad he hree LRUs faul ca be he pu daa, so he sysem faul me mus go over he coformy es. Accordg o he mehod, faul compoes of redudacy sysem are deermed ad he resul whch go over coformy es s show Table 4: Table 3: aul sae probably of parallel compoe STEP Probably P P P 3 P 4 P 5 P 6 P 7 P P dcaes P {R, f(φ,r(,,3, P dcaes P {R, f(,r(,3, P 3 dcaes P {R, f(,r(,3, P 4 dcaes P {R, f(3,r(, P 5 dcaes P {, f(,r(3, P 6 dcaes P {, f(,3,r(, P 7 dcaes P {R, f(,3,r(, P 3 dcaes P {, f(,,3,r(φ 06

6 Table 4: Real faul me Tme Radom umber aul compoe True resul L,L,L L L,L,L L,L L,L,L3 5. Cocluso I order o crease he credbly of smulao model whe redudacy sysem relably ad avalably are evaluaed. hs paper preses a faul samplg mehod for redudacy sysem uder mperfec maeace. Whe coducg faul samplg for he redudacy sysem, Markov sae raso process s embedded he mehod o deerme he faul producs. Based o he faul samplg mehod meoed above, faul occasos of redudacy sysem uder mperfec maeace are smulaed ad faul compoes he sysem are deermed, he valdy ad feasbly of faul samplg mehod s verfed he umercal example. or adapg o k ou of sysem, he faul samplg mehodology ca be exeded uder mperfec maeace fuure research work. Ackowledgeme Ths research s suppored by he aoal aural Scece oudao of Cha (Gra o.6043, o , o Refereces Doye L., Gaudo O., 004, Classes of mperfec repar models based o reduco of falure esy or vrual age, Relably Egeerg ad Sysem Safey, 84, Guo L.H., Kag R., 007, Base operaoal u correcve maeace process modellg ad smulao, Joural of BeJg Uversy of Aeroaucs ad Asroaucs, 33, 7-3. Jao J., Wag Z.L., 006, Operaoal avalably demosrao of mlary arplae based o smulao, Joural of BeJg Uversy of Aeroaucs ad Asroaucs, 3, -6. Marseguerra M., Zo E., Cad., 00, Based Moe Carlo uavalably aalyss for sysems wh me depede falure raes, Relably Egeerg ad Sysem Safey, 76, -7. Su C., Wag S., Xu Y., 006, Dyamc relably of Hydraulc sysem based o moe carlo smulao, Joural of Souheas Uversy, 53, Zhao B.H., Wag Y., 006, Research o Model ad Smulao for he Mlary Arcraf Avalably, Joural of he Academy of Equpme Commad & Techology, 7,