Reliability Investigation of Series-Parallel and Components of Power System using Interval Type-2 Fuzzy Set Theory

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1 Iteratoal Joural of Iovatve Research Scece, Egeerg ad Techology (A ISO 3297: 2007 Certfed Orgazato) Relablty Ivestgato of Seres-Parallel ad Compoets of Power System usg Iterval Type-2 Fuzzy Set Theory Sharma Aurag 1, Jha Maoj 2, Quresh M.F. 3 Ph.D. Research Scholar, Departmet of Computer Scece & Egeerg, Dr.C.V.R.U. Blaspur, Ida 1 Assocate Professor, Departmet of Appled Mathematcs, Rugta Egeerg College, Rapur, Ida 2 Assocate Professor, Departmet of Electrcal Egeerg, Govt. Polytechc College, Dhamtar, Ida 3 ABSTRACT: Fuzzy set based methods have bee proved to be effectve hadlg may types of ucertates dfferet felds, cludg relablty egeerg. Ths paper presets a ew approach o fuzzy type-2 relablty, based o the use of type-2 FOU as membershp fucto. Cosderg experts deas ad by askg operators lgustc varables, a rule base s desged to determe the level of relablty of each compoet. The outputs of the preseted model are type-2 fuzzy sets represetg the relablty levels of compoets. After determg the level of relablty of each compoet, the relablty of the composed system ca be determed by usg t-orm ad s-orm fuctos. The system ca be parallel, seres, parallel- seres or seres-parallel. I the preset paper the probablstc cosderato of basc evets s replaced by possbltes, thereby leadg to fuzzy fault tree aalyss. Tragular ad trapezodal type-2 fuzzy umbers are used to represet the falure possblty of basc evets. The falure possblty of a basc evet wll be assged more tha oe type-2 fuzzy umbers by dfferet experts uder varous operatg codtos. The proposed techques are dscussed ad llustrated by takg a example of a Thermal power plat. KEYWORDS: Fuzzy relablty of seres parallel compoets; terval type-2 Fuzzy sets, Fuzzy Fault tree aalyss (FTA). I.INTRODUCTION Fault tree aalyss (FTA) seems to be a very effectve tool to predct probablty of hazard, resultg from sequeces ad combatos of faults ad falure evets. A fault tree s a logcal ad graphcal descrpto of varous combatos of falure evets. To depct a fault tree, frst we determe the hazards ad the look for the evets causg ths hazard. I covetoal FTA based o a probablstc approach the basc evets are represeted by the probabltes (crsp umbers). However for the system lke uclear power plats, space shuttles, clcal applaces etc., where avalable data are suffcet for statstcal ferece (Jackso et al., 1981), t s ofte very dffcult to estmate precse falure rates of the basc evets. For such systems t s therefore urealstc to assume a crsp umber (classcal) for dfferet basc evets. Zadeh (1965) suggested a paradgm shft from a theory of total deal ad affrmato to a theory of gradg to gve ew cocept of fuzzy set. Taaka ad Sger (1983, 1990) the used fuzzy set theory to replace crsp umbers by fuzzy umbers for better estmato of possblty of top evet FTA. (Suresh et al. 1996) used a method based o α-cuts to deal wth FTA, treatg the falure possblty as tragular ad trapezodal fuzzy umbers. Accurate falure data s a crucal requremet for relablty assessmet. I may stuatos, where huma judgmet, evaluato ad decso-makg are mportat, falure data may ot be corrected accurately. It mght sometme requre lgustc terms to express data value (Padey et al. 2007). But f more tha oe fuzzy umber s assged to a partcular evet the radom selecto of ay of these fuzzy umbers to determe the falure possblty of Copyrght to IJIRSET

2 Iteratoal Joural of Iovatve Research Scece, Egeerg ad Techology (A ISO 3297: 2007 Certfed Orgazato) ths evet s ot realstc. Ths work proposes a method to obta a sgle fuzzy umber havg least varace wth the fuzzy umbers assged to that partcular evet. I FTA the cocept of mportace may be used to make some vtal modfcatos the desgg of system. (Furuta et al. 1984) proposed the cocept of fuzzy mportace usg max-m fuzzy operator ad fuzzy tegral. (Pa et al. 1988) developed a model for computg the mportace measure of basc evets usg varace mportace measure. Mote-Carlo smulato s geerally used the determato of varace mportace measure eve though computg process s tme takg ths method. Thus for a very complex system havg large umber of compoets, the whole procedure has to be repeated aga ad aga, thus ot sutable for the fuzzy approach. (Suresh et al. 1996) proposed aother method to evaluate a mportace measure called fuzzy mportace measure (FIM).For effectve evaluato of the mportace dex of each basc evets, we have troduced a comparatvely easer method to calculate fuzzy mportace dex (FII), based o rakg of fuzzy umbers ad Hammg dstace. The proposed methods are demostrated by takg a example of uclear power plat. I real system, the formato s accuracy ad supposed to lgustc represetato, the estmato of precse values of probablty becomes very dffcult may cases. I order to hadle the suffcet formato, the type-2 fuzzy approach s used to evaluate the falure rate status. Sger preseted a type-2 fuzzy set approach for fault tree ad the relablty aalyss whch the relatve frequeces of the basc evets are cosdered as fuzzy umbers. Poted out that there are two fudametal assumptos the covetoal relablty theory,.e. (a)bary state assumptos: the system s precsely defed as fuctog or falg. (b) Probablty assumptos: the system behavor s fully characterzed the cotext of probablty measures. However, because of the accuracy ad ucertates of data, the estmato of precse values of probablty becomes very dffcult may systems. (Ca et al. 1993) preseted the followg two assumptos: (a) Fuzzy-State assumpto: the meag of the system falure ca t be precsely defed a reasoable way. At ay tme the system may be oe of the followg two states: fuzzy success state or fuzzy falure state.(b) Possblty assumpto: the system behavor ca be fully characterzed the co text of possblty measures. (Ca et al. 1993) preseted the followg three forms of fuzzy relablty theores. () Profust relablty theory, based o the probablty assumpto ad fuzzy-state assumpto. ()Posbst relablty theory, based o the possblty assumpto ad bary-state assumpto. () Posfust relablty theory, based o the possblty assumpto ad the fuzzy-state assumpto. (Cheg ad Mo 1994) used terval of cofdece for aalyzg the fuzzy system relablty. (Che 1994) preseted a ew method for aalyzg the fuzzy system relablty usg fuzzy umber arthmetc operatos ad used smplfed fuzzy arthmetc operatos rather tha complcated terval fuzzy arthmetc operatos of fuzzy umbers or the complcated exteded algebrac fuzzy umbers. (Che 1994) preseted a ew method for fuzzy system relablty aalyss based o fuzzy tme seres ad the α-cuts arthmetc operatos of fuzzy umbers. So far, the lterature, arthmetc operatos betwee same types of vague sets are dscussed. Also to aalyze the fuzzy system relablty, t s assumed that the relablty of all compoets of a system follows the same membershp fuctos. However, practcal problems, such type of stuatos rarely occurs. Therefore, t s eed of a method by whch we ca also fd the fuzzy relablty of systems havg compoets followg dfferet type of type-2 membershp fuctos. To llustrate the above approach the fuzzy relablty of seres, parallel, parallel-seres ad seres-parallel systems all cosstg of four compoets has bee evaluated usg the proposed algorthm. II.RELATED WORK The relablty of a system ca be determed o the bass of tests or the acqusto of operatoal data. However, due to the ucertaty ad accuracy of ths data, the estmato of precse values of probabltes s very dffcult may systems (e.g. power system, electrcal mache, hardware etc., Hammer (2001), El-Hawary (2000)). The bass for ths approach s costtuted by the fudametal works o fuzzy set theory of Zadeh (1978), Dubos ad Prade (1980), Zmmerma (1986) ad other. The theory of fuzzy relablty was proposed ad developmet by several authors, Ca, We ad Zhag (1991, 1993); Ca (1996); Che, Mo (1993); Hammer (2001); El-Hawary (2000), Osawa, Kacprzyk (1995); Utk, Gurov (1995). The recet collecto of papers by Osawa ad Kacprzyk (1995), gave 654 I.M. ALIEV, Z. KARA may dfferet approach for fuzzy relablty. Accordg to Ca, We ad Copyrght to IJIRSET

3 Iteratoal Joural of Iovatve Research Scece, Egeerg ad Techology (A ISO 3297: 2007 Certfed Orgazato) Zhag (1991, 1993); Ca (1996) varous form of fuzzy relablty theores, cludg profust relablty theory Dobos, Prade (1980); Ca, We ad Zhag (1993); Ca (1996); Che, Mo (1993); Hammer (2001); El-Hawary (2000); Utk, Gurov(1995), posbst relablty theory, Ca, We ad Zhag (1991, 1993) ad posfust relablty theory, ca be cosdered by takg ew assumptos, such as the possblty assumpto, or the fuzzy state assumpto, place of the probablty assumpto or the bary state assumpto. Che [14] aalyzed the fuzzy system relablty usg vague set theory. The values of the membershp ad o-membershp of a elemet, a vague set, are represeted by a real umber [0, 1]. Ca, We ad Zhag (1993) preseted a fuzzy set based approach to falure rate ad relablty aalyss, where profust falure rate s defed the cotext of statstcs. El-Nawary (2000) preseted models for fuzzy power system relablty aalyss, where the falure rate of a system s represeted by a tragular fuzzy umber. The work of Jerry M.Medel ad Felog Lu (2007) o Super-Expoetal Covergece of the Kark Medel Algorthms for Computg the Cetrod of a Iterval Type-2 Fuzzy Set s a well-recogzed work the feld. Desg of Iterval Type-2 Fuzzy Logc Based Power System Stablzer (Imam Robad, ad Bedy Kharsma 2008) has suffcet materals as a referece work. Jua R. Castro ad Oscar Castllo (2007) worked o Iterval Type-2 Fuzzy Logc for Itellget Cotrol Applcatos. Also Jerry M. Medel ad Robert I.Bob Joh (2002) preseted, how Type-2 Fuzzy Sets Made Smple. Mamda ( 1974) developed the method to apply the fuzzy algorthm for smple cotrol of dyamc plat. Quresh (2003) publshed hs work o power system relablty problems, cotrol problems ad protecto problems. Quresh (2004) hs Ph.D. thess took the project work of Relablty of uclear plats usg fuzzy logc trasformato. R.R.Yager(2000) reported a valuable formato o fuzzy subsets of type-2 decso. N.N.Kark ad J.M. Medel worked o terval type-2 fuzzy logc systems ad reported hs fdgs IEEE Trasactos, fuzzy systems. III. FUZZY NUMBERS AND ARITHMETIC OPERATIONS FUZZY OPERATORS Usg algebrac operatos o fuzzy umbers (tragular or trapezodal) we ca obta fuzzy operators FNOT, ANDF ad ORF correspodg to Boolea operators NOT, AND ad OR respectvely as follows. () If a fuzzy evet s represeted by a possblty fucto pi the geeralzed Boolea operator NOT to be deoted by FNOT ad defed as:(for tragular fuzzy umbers) FNOT P = 1 P = 1 a 1, a 2, a 3 = 1 a 3, 1 a 2, 1 a 1 () If p1, p2,.. p are the possblty fuctos of basc evets ad py be the same for resultg evet. The the fuzzy operators ANDF ad ORF are defed the followg maer: p y = ANDF P 1, P 2, P 3.. P = p, (a) where deotes the fuzzy multplcato ad py be the possblty of resultg evet. Let p s are represeted by tragular fuzzy umbers (a1, a2, a3),,2, the p y = ANDF P 1, P 2, P 3.. P = a 1, a 2, a 3 = a 1, a 2, a 3 (b) Let p s are represeted by tragular fuzzy umbers (a1, a2, a3),,2, the p y = ORF P 1, P 2, P 3.. P = 1 1 a 1, a 2, a 3 = 1 1 a 3, 1 a 2, 1 a 1 = 1 1 a 3, 1 a 2, 1 a 1 = 1 1 a 1, 1 1 a 2, 1 1 a 3 Usg Zadeh s Exteso Prcple, the membershp grades for uo, tersecto ad complemet of type-2 fuzzy sets A ad B have bee defed as follows: Uo: A B μ A B x = μ A x μ B x Copyrght to IJIRSET

4 Iteratoal Joural of Iovatve Research Scece, Egeerg ad Techology Itersecto: A B μ A B x = μ A x μ B x Complemet: A μ A x = 1 μ A x (A ISO 3297: 2007 Certfed Orgazato) INTERVAL TYPE-2 FUZZY SETS: Fg.1 Iterval Type-2 Fuzzy Sets The upper MF s abbrevated to UMF, ad lower MF s abbrevated to LMF. A over-bar o the T2 MF deotes the former, ad a uder-bar o the T2 MF deotes the latter. The LMF ad UMF play very mportat roles all calculatos volvg IT2 FSs. A embedded set (also called a embedded T1 FS) s a fucto that les wth or o the FOU. Two other examples of embedded sets are the LMF ad UMF. A short arrow labeled "1" s show alog the embedded T1 FS. Whe t s cluded wth the embedded T1 FS, the result s a embedded T2 FS. Because the thrd dmeso of a geeral T2 FS s rrelevat for a IT2 FS, t s uecessary to carry alog the equal ut secodary grades. The FOU says t all for a IT2 FS. For a cotuous FOU (.e. a completely flled- FOU) there are a ucoutable umber of embedded sets. Do t' worry, though, because such sets wll oly be used for theoretcal dervatos, ad ever for computato. If both the prmary ad secodary varable axes are dscretzed, the there wll be a coutable umber of embedded sets, but there could stll be a astroomcal umber of them. Aga, do't worry because such sets wll oly be used for theoretcal dervatos, ad ever for computato Observe that a embedded set looks lke a wavy slce that cuts through the FOU. 1 μ A x,. μ A x x X d A = 1 FOU A = 1 A j = j=1 A e 1 x X μ A x, μ A x The 1/FOU(x) otato s a shorthad otato. It meas the secodary grades equal 1 at all pots the FOU. A B 1 x X μ A x μ B x, μ A x μ B x A B 1 μ A x μ B x, μ A x μ B x x X A = 1 1 μ A x, 1 μ A x x X TRIANGULAR INTERVAL TYPE 2 FUZZY SET A tragular terval type-2 fuzzy A set over the uverse of dscourse X show Fg. 2 may be deoted A =< a 1, a 2, a 3 ; μ A x, a 1, a 2, a 3 ; v A x >. The membershp fucto s deoted as μ A x. Copyrght to IJIRSET

5 Iteratoal Joural of Iovatve Research Scece, Egeerg ad Techology (A ISO 3297: 2007 Certfed Orgazato) Fg. 2 Tragular terval type-2 fuzzy Number IV. FUZZY IMPORTANCE I FTA we have observed that each basc evet play dfferet role the occurrece of top evet, whch fers that the basc evets are of dfferet mportace. Thus a crtcal aalyss of the mportace of dfferet basc evets may help makg a proper sequece of ther mportace. O mprovg the relablty of the evet havg greater mportace, oe ca mprove the relablty of the system. The fuzzy mportace of ay evet s always calculated the form of fuzzy mportace dex (FII). Ths FII may be evaluated by rakg fuzzy umbers (PT PT) for,2,3. Here PT ad PT deote the possblty of absolute occurrece of top evet ad the possblty of occurrece of top evet absece of basc evet respectvely. I our aalyss we have used less complcated ad very sgfcat method for rakg of fuzzy umbers. To rak the fuzzy umbers (PT PT) s for, 2,..., frst of all we have to fd MAX (PT PT),2, where the MAX operator o fuzzy umbers s defed as below. MAX A 1, A 1, A 1,. A z = sup m A 1 x 1, A 2 x 2,. A x z=max x 1, x 2, x Where A1, A2, A3 A are dfferet fuzzy umbers. Takg the MAX of gve fuzzy umbers, we try to get the dstace of all these fuzzy umbers from ther MAX wth the help of Hammg dstace formula [0,1] d H A, B = A x B x dx Betwee two fuzzy umbers A ad B. The dstace of these fuzzy umbers (PT PT) for =1, 2 from ther MAX decdes the rak of fuzzy umbers (PT PT). Smaller the dstace of fuzzy umber (PT PT) from MAX (PT PT),,2, comparso to dstace of PT PT2 from MAX (PT PT) mples that fuzzy umber (PT PT) s greater tha (PT PT). It cocludes that the fuzzy mportace dex (FII) may be defed form of dstace of PT from PT.e. 1 FII = 1 + Dstace of fuzzy umber P T P T from ther MAX V. RELIABILITY ANALYSIS OF SERIES AND PARALLEL COMPONENTS I ths secto, takg the relablty of each compoet to be a tragular terval type-2 fuzzy set we have evolved a fuzzy relablty evaluato techque for seres ad parallel systems. Let us cosder a system cosstg of compoets, the terval type-2 fuzzy R j sets j=1,2,3., are take to represet the relablty of each compoet. If the compoets are coected as a seres system as show Fg.3, the relablty R s of the seres system s defed as follows: R 1 R 2 R 2.. R R s = j=1 R j =< a 1j, a 2j, a 3j j=1 j=1 j=1 Fg. 3 Systems Seres : m μ A j=1 j x, a 1j, a 2j, a 3j j=1 j=1 j=1 : max j=1 v A j x > Copyrght to IJIRSET

6 ----- ISSN: Iteratoal Joural of Iovatve Research Scece, Egeerg ad Techology (A ISO 3297: 2007 Certfed Orgazato) If the compoets are supposed to be parallel as show Fg.4, the relablty R p of the parallel system ca be defed by usg the expresso R 1 R 2 R 3 R p = 1 1 R 1 = < 1 1 a 1 1 a 2, 1 1 a 2, 1 1 a 3 R Fg. 4 Parallel System, 1 1 a 3 : max j=1 v A j x > : m μ A j=1 j x, 1 1 a 1, 1 PARALLEL-SERIES SYSTEM Cosder a parallel-seres system cosstg of m braches coected parallel ad each brach cotas compoets as show Fg.5. The fuzzy relablty RPS= 1 Ө m k=1 (1 Ө Rk)) of the parallel-seres system show Fg.5 ca be evaluated usg the algorthm proposed secto 3 for multplcato ad subtracto. where Rk represets the relablty of the th compoet at kth brach. R 11 R 12 R 13 R 1 R 21 R 22 R 23 R 2 R m1 R m2 R m3 R m Fg.5. SERIES-PARALLEL Parallel-Seres System SYSTEM Cosder a seres-parallel system cosstg of stages coected seres ad each stage cotas m compoets as show Fg.6. The fuzzy relablty RSP= m (1 Ө (1 Ө Rk)) of the seres-parallel system k=1 show Fg.6 ca be evaluated usg the algorthm proposed secto 3 for multplcato ad subtracto. where Rk represets the relablty of the th compoet at kth stage. Copyrght to IJIRSET

7 Iteratoal Joural of Iovatve Research Scece, Egeerg ad Techology (A ISO 3297: 2007 Certfed Orgazato) R 11 R 12 R 1 R 21 R 22 R 2 R 31 R 32 R 3 R m1 R m2 R m Fg.6 Seres-Parallel System VI. FUZZY FAULT TREE OF THERMAL POWER PLANT Top Evet SP X Y A B C D F H 0 J 0 I 0 L 0 P 0 Q 0 G 0 I 0 K 0 E 0 Fg.7 Fault tree of Nuclear Power Plat Copyrght to IJIRSET

8 Iteratoal Joural of Iovatve Research Scece, Egeerg ad Techology (A ISO 3297: 2007 Certfed Orgazato) A Thermal power plat model s cosdered. Develop a fault tree for the desred evet (.e. top evet): Each fault evet the fault tree dagram s cosdered as Tragular Iterval Type-2 Fuzzy Number (TIT2FN) & Trapezodal Iterval Type-2 Fuzzy Number (TrIT2FN). Usg seres parallel compoets formula ad fuzzy fault tree Fg.7, the relablty of thermal power plat ca be vestgated as below. Steam pressure release (low) s assumed to be a hazard ad treated as the top evet fault tree aalyss. The Steam pressure release may be caused due to the occurrece of some evets, ad these evets may aga occur due to some other evets as show Fg.7 The terval type-2 set operatos correspodg to ths Fuzzy fault tree s gve below: SP=X Y, X=A B, Y=C D E 0, A= F G 0, F=P 0 Q 0, B=H 0 I 0, C=J 0 K 0, D=I 0 L 0, where SP deotes Steam Pressure Low turbe L 0 = hgh premse temperature Y = uwated shaft vbrato A= physcal damage to the rotor B = thermal damage to the rotor C = alarm system fals D = stress preset E 0 = breach of physcal boudary F=mechacal damage to the turbe G 0 = explosve damage to the boler P 0 = safety sgal fals Q 0 =power fals H 0 = suffcet thermal geerato I 0 = Safety Valve ot workg J 0 =cotrol crcut of alarm system fals K 0 = sesor fals X = formato of corroso product due to turbe blade corroso O replacg the Boolea operators wth fuzzy logc operators FNOT, ORF ad ANF, we get the possblty of the top evet form of a fuzzy umber. It s also assumed that each basc evet s fuzzzfed by assgg three fuzzy umbers to each basc evet followg the decso of three experts. The Tragular ad trapezodal fuzzy umbers assged to these basc evets are lsted Table 1 ad Table 2. Table 1. Tragular Iterval Type-2 Fuzzy Numbers (Basc Evets) Evet E 0 Expert a 1(UMF,LMF) a 2(UMF,LMF) a 3(UMF,LMF) , , , , , , , , ,0.032 Evet G 0 Expert a 1(UMF,LMF) a 2(UMF,LMF) a 3(UMF,LMF) , , , , , , , , ,0.046 Evet P 0 Expert a 1(UMF,LMF) a 2(UMF,LMF) a 3(UMF,LMF) , , , , , , , , ,0.078 Evet J 0 Expert a 1(UMF,LMF) a 2(UMF,LMF) a 3(UMF,LMF) , , , , , , , , ,0.056 Copyrght to IJIRSET

9 Iteratoal Joural of Iovatve Research Scece, Egeerg ad Techology (A ISO 3297: 2007 Certfed Orgazato) Evet L 0 Expert a 1(UMF,LMF) a 2(UMF,LMF) a 3(UMF,LMF) , , , , , , , , ,0.26 Table 2. Trapezodal Fuzzy Number (Basc Evets) Evet Q 0 Expert a 1(UMF,LMF) a 2(UMF,LMF) a 3(UMF,LMF) a 4(UMF,LMF) , , , , , , , , , , , ,0.068 Evet H 0 Expert a 1(UMF,LMF) a 2(UMF,LMF) a 3(UMF,LMF) a 4(UMF,LMF) , , , , , , , , , , , ,0.052 Evet I 0 Expert a 1(UMF,LMF) a 2(UMF,LMF) a 3(UMF,LMF) a 4(UMF,LMF) , , , , , , , , , , , ,0.034 Evet K 0 Expert a 1(UMF,LMF) a 2(UMF,LMF) a 3(UMF,LMF) a 4(UMF,LMF) , , , , , , , , , , , ,0.39 Usg the approxmato method dscussed earler, a sgle fuzzy umber s obtaed by whch suts wth all the three experts decso for each basc evet. The tragular fuzzy umbers thus obtaed for each basc evet are lsted Table 3. Table 3. Approxmated Tragular Fuzzy Numbers (Basc Evets) Basc Evet a 1(UMF,LMF) a 2(UMF,LMF) a 3(UMF,LMF) E , , ,.0238 G , , ,.056 P , , ,.080 J 0.044, , ,.062 L 0.110, , ,.242 Copyrght to IJIRSET

10 Iteratoal Joural of Iovatve Research Scece, Egeerg ad Techology (A ISO 3297: 2007 Certfed Orgazato) Table 4. Approxmated Trapezodal Fuzzy Numbers (Basc Evets) Basc Evet a 1(UMF,LMF) a 2(UMF,LMF) a 3(UMF,LMF) a 4(UMF,LMF) Q 0.058, , , ,.078 H 0.022, , , ,.045 I 0.152, , , ,.220 K 0.192, , ,.278 FII OF BASIC EVENTS IN THERMAL POWER PLANT The fuzzy mportace of each basc evet ca be obtaed the form of fuzzy mportace dex (FII) for all basc evets. We calculate the possblty of top evet RR usg fuzzy operators ad possbltes of basc evets. The possblty of top evet s resulted as trapezodal fuzzy umber ([.044,.048], [.054,.058], [.056,.060, [.064,.068]) gve by the followg expresso. x f.044 x.054 P T= 1 f.054 x x.012 f.056 x.064 x f.048 x.058 PT= 1 f.058 x x.012 f.060 x.068 Here PT s for dfferet evets = E0, G0, P0, J0, L0, Q0, H0, I0 ad K0 obtaed as tragular ad trapezodal fuzzy umbers are lsted Table 5. Table 5. Possblty of Top Evet absece of dfferet basc evets Evet () Possblty of top evet absece of evet, ( P T ) (UMF,LMF) E 0 ([.044,.046], [.052,.056], [.054,.060], [.064,.068]) G 0 ([.006,.008], [.012,.016],.014,.018], [.016,020]) P 0 ([.042,.046], [.052,.056], [.053,.057],[.062,.066]) J 0 ([.044,.048], [.054,.058], [.056,.06],[.068,.072]) L 0 ([.045,.049], [.055,.059], [.057,.061],[.069,.073]) Q 0 ([.040,.044],[.052,.056], [.054,.058],[.063,.067]) H 0 ([.042,.046], [.052,.056],[.060,.064]) I 0 ([.043,.047], [.053,.057],[.061,.065]) K 0 ([.046,.050], [.056,.060], [.058,.062],[.070,.074]) Frst we evaluate P T P T, P T P T ad the MAX of these fuzzy evets for,2,. The dstace of the fuzzy umbers P T P T, P T P T from ther MAX s obtaed by usg Hammg dstace formula. Fuzzy mportace dex (FII) s thus obtaed by the followg expresso. P T P T = [(PT PT ) +(P T P T )]/2 1 FII = 1 + Dstace of fuzzy umber P T P T from ther MAX Table 6. Fuzzy Importace Idex of basc evets Evet G 0 I 0 E 0 Q 0 P 0 L 0 K 0 H 0 J 0 FII () Copyrght to IJIRSET

11 Iteratoal Joural of Iovatve Research Scece, Egeerg ad Techology (A ISO 3297: 2007 Certfed Orgazato) VII.DISCUSSION The techques developed ths paper are demostrated by takg the example of a Thermal power plat, ad the followg results are draw: All basc evets E 0, Q 0, I 0 etc are assged tragular/ trapezodal terval type-2 fuzzy umbers (TIT2FN)/(TrIT2FN) by dfferet experts uder prescrbed codto. Usg techque developed ths paper, a sgle terval type-2 fuzzy umber for each basc evet s obtaed, that tues fe wth all experts judgmets. I may stuatos, where the falure possblty of dfferet basc evets s collected from dfferet experts uder varous operatg codtos. It s more useful to adopt ths realstc approach to get a sgle terval type-2 fuzzy umber for ths purpose. Applyg the method developed our study for the mportace of basc evets, the fuzzy mportace dex (FII) of basc evets s calculated. The basc evets are lsted Table 6, accordace wth the descedg order of ther FII. It s observed that the basc evet G o s of hgher sestvty (greater mportace) comparso to other succeedg evets. Takg ote of FII of basc evets lsted Table 6, t s cocluded that we should emphasze o basc evet G o rather tha other succeedg evets I o, E o etc. to mprove the relablty of the Thermal power plat. VIII.CONCLUSION Ths paper presets relablty vestgato of Seres-Parallel ad Compoets of thermal power plat usg Iterval Type-2 Fuzzy Set Theory. Applyg the method developed our study for the mportace of basc evets, the fuzzy mportace dex (FII) of basc evets s calculated. The basc evets are lsted Table 6, accordace wth the descedg order of ther FII. It s observed that the basc evet G o s of hgher sestvty (greater mportace) comparso to other succeedg evets. Takg ote of FII of basc evets lsted Table 6, t s cocluded that we should emphasze o basc evet G o rather tha other succeedg evets I o, E o etc. to mprove the relablty of the Thermal power plat. Also arthmetc operato of proposed TIT2FN/TrIT2FN s evaluated. Here, a method to aalyze system relablty whch s based o terval type-2 fuzzy set theory has bee preseted, where the compoets of the system are represeted by TIT2FN/TrIT2FN fuzzy umber. By performg the techques developed here early desg phase of varous complex systems lke uclear power plat, potetal defceces ca be detfed ad averted to precptate the occurrece of varous basc evets causg the happeg of top evet. REFERENCES 1. Attaassov, K.T. (1986). Itutostc fuzzy sets. Fuzzy Sets ad Systems, Vol. 20, pp Attaassov, K.T. (1989). More o tutostc fuzzy sets, Fuzzy Sets ad Systems Vol. 33, Issue 1, Attaassov, K.T. (1999). Itutostc Fuzzy Sets, Physca-Verlag, Hedelberg, New York. 4. Bowles, J.B. ad Palaez, C.E. (1995). Applcato of fuzzy logc to relablty egeerg, Proceedg of the IEEE, Vol.83, No. 3, pp Ca, K. Y., We C. Y. ad Zhag, M. L. (1993). Fuzzy states as a bass for a theory of fuzzy relablty, Mcroelectroc Relablty, Vol. 33, pp Ca, K. Y., We, C. Y. ad Zhag, M. L. (1993). Fuzzy varables as a bass for a theory of fuzzy relablty the possblty cotext, Fuzzy Sets ad Systems, Vol. 42, pp Ca, K.Y. (1996). System falure ad fuzzy methodology: A troductory overvew. Fuzzy Sets ad Systems, Vol. 83, pp Che, S. M. (1994). Fuzzy system relablty aalyss usg fuzzy umber arthmetc operatos, Fuzzy Sets ad Systems, Vol. 64, pp De, S.K., Bswas R. ad Roy, A.R. (2000). Some operatos o tutostc fuzzy sets, Fuzzy Sets ad Systems, Vol. 114, pp De, S.K., Bswas R. ad Roy, A.R. (2001). A applcato of tutostc fuzzy sets medcal dagoss, Fuzzy Sets ad Systems, Vol. 117, pp Deschrjver, G. ad Kerre, E.E. (2002). O the relatoshp betwee tutostc fuzzy sets ad some other extesos of fuzzy set theory, Joural of Fuzzy Mathematcs, Vol. 10, Issue 3, pp Mtchell, H. B. (2004). Rakg-Itutostc Fuzzy Numbers, Iteratoal Joural of Ucertaty, Fuzzess ad Kowledge-Based Systems, Vol. 12, Issue 3, pp Copyrght to IJIRSET

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