ZESZYTY NAUKOWE NR 10(82) AKADEMII MORSKIEJ W SZCZECINIE. Probabilistic Analysis of Marine Binary Technical Systems Represented by Boolean Models

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1 ISSN ZSZYTY NAUKOW NR 08 AKADMII MORSKIJ W SZCZCINI IV MI DZYNARODOWA KONFRNCJA NAUKOWO-TCHNICZNA X L O - S H I robablstc Aalyss of Mare Bary Techcal Systems Represeted by Boolea Models Key words: logcal model, logcal operators, fault tree aalyss, Boolea rules Basc relatos useful the reducto of Boolea models of techcal systems have bee preseted. lemetary logcal gates used a coheret fault tree ad ther probablstc evaluatos have bee poted out. A mare system sea water coolg system has bee aalyzed wth the use of the preseted methodology. Aalza probablstycza morskch dwustaowych systemów techczych reprezetowaych modelam boolowskm S owa kluczowe: model logczy, operatory logcze, aalza drzewa ezdato c, zasady logk Boole'a rzedstawoo podstawowe zale o c przydate przy redukcj boolowskch model systemów techczych. Wyszczególoo podstawowe operatory logcze wykorzystywae w koheretych drzewach ezdato c ch aalze probablstyczej. rzyk adowy system okr towy system ch odzea wod morsk zosta przeaalzoway z u ycem przedstawoej metodolog. 6

2 Itroducto For a bary model of the system the form of a fault tree, usually the frst step the depedablty aalyss s to detfy mmal cut sets of the aalysed system. The process of searchg for mmal cut sets ad path sets s based o the applcato of Boolea algebra rules to the bary equato whch represets a gve fault tree model [, 3, 4]. The detfcato of mmal cut sets for a gve fault tree requres:. Coverso of fault tree to equvalet the form of Boolea formulas set logcal model.. Determato of the top evet wth the use of Boolea algebra by tracg of tree from bottom to top or from top to bottom.. Reducto of logcal models Let, ad 3 represet ay logcal evets, a empty set, a full set, ad set s a complemetato of set. Basc rules of Boolea algebra for these symbols are preseted below. These rules are used for the reducto ad trasformato of Boolea equatos, whch represet the fault tree model. The most mportat formulas for fault tree evaluatos are: Commutatve Law: Assocatve Law: Idempotet Law: Law of Absorpto:

3 robablstc Aalyss of Mare Bary Techcal Systems Dstrbutve Law: Complemetato: De Morgas Theorem: Other relatos: ' ' ' ' ' 3 ' ' ' 4 5 ' ' ' ' 6 '. Logcal operators I the classcal fault tree aalyss basc kds of logcal gates are used,.e. uo ad tersecto operators. The structure modelled by meas of these operators s always a coheret tree. If for the buldg of a fault tree also the egato operator s used or complex gates wth teral egato, the the tree may, but does ot have to be a coheret fault tree. These kds of systems are very rare ad wll ot be aalysed ths paper. The gate OR represets the uo of put evets. If put evets are deoted as,,, ad the gate output as, the logcal represetato of OR gate operato s gve as: or,, A output evet s geerated from OR gate whe there does exst at least oe put evet. The probablty of the output evet geerato from OR gate wth two put evets ad wth the probabltes of occurrece, respectvely, ad, accordg to probablstc rules s as follows: 63

4 8 Accordg to the sets theory rules: f evets ad are mutually exclusve, whch meas = 0, the: 9 f put evets ad are depedet, whch meas =, the: 0 f evet s completely depedet o evet, whch meas =, the: For all cases, f occurrece of two put evets the same tme s eglected, the probablty ca be estmated accordg to ths formula: For low probabltes of put evets less tha 0. ad for depedet put evets, the probablty of output geerato ca be estmated wth the relatve error less tha 0. wth the use of the rare evet approxmato: 3 For OR gate wth depedet put evets, the probablty of the output evet geerato s gve by ocare equato: j j k k j j j 64

5 robablstc Aalyss of Mare Bary Techcal Systems If the occurrece of two or more put evets the same tme s eglected, ths equato wll be reduced to the rare evet approxmato form: 5 For completely depedet evets formula 6 s useful. I practce ths equato s used for systems wth partly depedet assocated put evets: [ ] 6 The gate AND represets a tersecto of put evets. If put evets are deoted as,,, ad the gate output as, the logcal represetato of the AND gate operato has ths form: ad,, A output evet s geerated from the AND gate whe there do exst all of the put evets. The probablty of output evet geerato from the AND gate wth two put evets ad wth the probabltes ad, respectvely, accordg to the probablstc rules s expressed as: 8 Accordg to the relevat sets theory rules: f put evets ad are depedet, = ad =, the: 9 f put evets ad are ot depedet ad >, the: 30 f evet s completely depedet o evet, whch meas =, the: 3 65

6 For the AND gate wth depedet put evets, the probablty of output evet geerato has ths form: 3 The votg gate represets the logcal operato whch geerates a output evet whe ther exsts at least k out of all puts to the gate. Ths operato s also called K-out-of-N gate. Ths gate ca be represeted the form of AND ad OR gates combato wth artfcally etered termedate evets []. The votg gate s logcally a uo of all possble k-elemets tersectos of put evets to. The umber of all combatos s :! C k 33 k k! k! The probablty of output evet geerato through the votg gate wth parameters k, ad put evets,, s gve by ths formula: ZS m k m k m k m L j L {... }: l... l [ L ] Case study The preseted methodology s applcable for tme depedet ad costat probablty models. A case study for selected mare power plat systems stalled oboard offshore mult support vessel s show below. Oe of these systems: ma power plat eges sea water coolg system s preseted Fg.. The descrpto of all compoets ad respectve evets for the bary model fault tree s show Table. The fault tree s preseted Fgure. Table also defes the evets the bary model. The aalyss was carred out o the bass of the costructed trees takg the calculated falure measures as put data, momets of falures take from the real system. The aalyss was doe by meas of CARA-FaultTree computer code from Sydvest Software. The calculatos bascally amed at the estmato of the uavalablty of a selected mare power plat system. 66

7 robablstc Aalyss of Mare Bary Techcal Systems Fg.. Dagram of the aalyzed sea water coolg system Rys.. Schemat aalzowaego systemu ch odzea wod morsk Table Symbol Descrpto of system compoets ad aalyzed evets the system Ops elemetów systemu oraz aalzowaych w systeme zdarze Compoet ame Type vet descrpto arameter VL VLS Bottom sea chest valve o ort Bottom sea chest valve o Stbd O demad O demad Valve faled closed posto Valve faled closed posto Value 67

8 Table cotued VL VLS VH VHS VO VOS VS VSS VS VSS VD VDS VD VDS VC VCS VC VCS S Bottom sea chest valve o ort Bottom sea chest valve o Stbd Hgh sea chest valve ort Hgh sea chest valve Stbd Outlet valve ort Outlet valve Stbd Sucto valve of pump o ort Sucto valve of pump o Stbd Sucto valve of pump o ort Sucto valve of pump o Stbd Delvery valve of pump o ort Delvery valve of pump o Stbd Delvery valve of pump o ort Delvery valve of pump o Stbd Cooler let valve ort Cooler let valve Stbd Cooler outlet valve ort Cooler outlet valve Stbd Sea water pump o ort actve pump Sea water pump o Stbd actve pump O demad O demad O demad O demad O demad O demad O demad O demad O demad O demad O demad O demad O demad O demad O demad O demad O demad O demad No reparable No reparable Valve faled closed posto Valve faled closed posto Valve faled closed posto Valve faled closed posto Valve faled closed posto Valve faled closed posto Valve faled closed posto Valve faled closed posto Valve faled closed posto Valve faled closed posto Valve faled closed posto Valve faled closed posto Valve faled closed posto Valve faled closed posto Valve faled closed posto Valve faled closed posto Valve faled closed posto Valve faled closed posto Falure durg startg/rug Falure durg startg/rug [falure/h] [falure/h] 68

9 robablstc Aalyss of Mare Bary Techcal Systems Table cotued S F FS F FS C CS Sea water pump o ort stadby pump Sea water pump o Stbd stadby pump Sucto flter o ort Sucto flter o Stbd Sucto flter o ort Sucto flter o Stbd Cetral cooler ort Cetral cooler Stbd O demad O demad No reparable No reparable No reparable No reparable No reparable No reparable Start o demad faled Start o demad faled Flter clogged Flter clogged Flter clogged Flter clogged Cooler clogged / seals damaged Cooler clogged / seals damaged [falure/h] [falure/h] [falure/h] [falure/h] [falure/h] [falure/h] e e e e e e e e006 Fg.. Fault tree for the aalyzed sea water coolg system for crossover valves closed posto Rys.. Drzewo ezdato c dla aalzowaego systemu ch odzea wod morsk przy zaworach rozdzelaj cych w po o eu zamk tym Besdes, aalyses were performed wth exact RAC calculato for a tme depedet evet model expoetal dstrbuto. The operato of the ege room s accordg to the thrd D class. Ths meas that crossover valves 69

10 betwee the ege rooms are closed, whle the ma electrc buses are dvded. Some falures at the begg of the observato have bee smulated, e.g. oe straer was clogged F, two straers were clogged oe each ege room F, FS, oe sea chest was out of operato VL, VL. The results of the aalyss of 4300 hours smulato are show Table ad Fg. 3 ad 4. Uavalablty characterstcs of the aalyzed system wth some compoets the dow state at the start of operato Charakterystyk egotowo c dla aalzowaego systemu z ektórym elemetam ezdatym w chwl rozpocz ca pracy Compoets that faled at the start of observato t oe F F, FS VL, VL Table The preseted characterstcs show that wth the hgher umber of faled compoets the system, the uavalablty fucto values are also hgher. Fal coclusos The preseted methodology s applcable to coheret fault trees wth bary logcal operators. The operato of preseted gates s depedet of tme. The statc fault trees use these operators combato wth prmary evets represeted by the costat probablty of evet occurrece. If evets or gates are tme depedet, the bult fault tree s called a dyamc fault tree. The group of tme depedet logcal operators.e. spare gates hot, warm, cold, prorty AND gate, fuctoal depedecy gate etc. are ot preseted here. 70

11 robablstc Aalyss of Mare Bary Techcal Systems Fg. 3. Uavalablty of the aalyzed system for some straers beg out of operato at the start Rys. 3. Negotowo aalzowaego systemu dla ektórych fltrów ezdatych w chwl rozpocz ca pracy Fg. 4. Uavalablty of the aalyzed system for oe sea chest beg out of operato at the start Rys. 4. Negotowo aalzowaego systemu dla jedego kosza ssawego ezdatego w chwl rozpocz ca pracy The applcato of tme-depedet models offers a fuller descrpto of the system behavour durg ts operato tha a classcal model, whch has bee show Fg. 3 ad 4. These have bee preparedo the bass of prevously preseted values of relablty characterstcs ad the gve fault tree model. 7

12 The preseted method gves a coveet aalyss of system depedablty e.g. uavalablty characterstcs of the system at selected sutable values measures of the charactersed evets faults the techcal system. The classcal bary model s very fast computg, so t s ofte used a prelmary relablty aalyss. Refereces. Chybowsk L., Auxlary stallatos fault tree model for operato aalyss of vessels power plat ut, a 00, KGTU, Kalgrad, 00, pp Chybowsk L., Wykorzystae bramk k-z- w aalze pracy s ow okr towej, a 00, KGTU, Kalgrad, 00, pp Chybowsk L., Matuszak Z., odstawy aalzy jako cowej lo cowej metody drzewa ezdato c, Zeszyty Naukowe r 73 Akadem Morskej w Szczece, xplo-shp 004, Szczec 004, pp Chybowsk L., rzyk ad modelu logczego wybraej struktury systemu s ow okr towej w aalze drzew uszkodze, KGTU, Kalgrad 004, pp Matuszak Z., Surma T., Applcato of the fault tree ad elemets of the Boole algebra estmatg of relablty of power plat ege room stallatos, Scetfc Coferece Trasport Systems geerg, Secto 3 Operato, Mateace ad Relablty of Trasport Systems, Warszawa, 995, pp Matuszak Z., Surma T., Drzewo uszkodze elemety algebry Boole'a jako sposób ocey ezawodo c dagozowaa stalacj s ow okr towej, Matera y XVI Sesj Naukowej Okr towców, Szczec Dzwówek 994. Cz II, Wyd. Stocz Szczec skej, Szczec 994, pp Recezet prof. dr hab.. Meczys aw Ha Wp y o do redakcj w lutym 006 r. Adresy Autorów mgr. Leszek Chybowsk dr hab.. Zbgew Matuszak, prof. AM Akadema Morska w Szczece ul. Wa y Chrobrego -, Szczec 7 Vew publcato stats