Study of Reliability with Mixed Standby Components
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1 Avll hp://pvudu/ Appl Appl Mh IN: Vol 7, Iu D, pp Applon nd Appld Mh: An Innonl Jounl AAM udy of Rlly wh Mxd ndy Coponn M A El-D nd A N ly Dpn of Mh Tn Unvy, Egyp ld@yhooo Rvd: Jun, ; Apd: p 7, A Th pp dl wh h lly h of wo dffn y onfguon wh xd ndy nlud old nd w ndy oponn Th flu of h py nd w ndy oponn ud o follow h Wull duon Th p duon of h v xponnlly dud Moov, w wll dv h n -o-flu, nd h dy- vlly fo pl of l y of wo py oponn, wo w ndy oponn, nd on old ndy oponn, whn h flu nd p onn Kywod: Rlly, Avlly, T vyng flu, ndy oponn, Mkov Mhod MC No: 6K; 6; 6J Inoduon y lly nd y vlly hv wdly n udd u of h pv n pow pln, nufung y, nd ndul y Mnnng hgh o qud lvl of lly nd/o vlly ofn n nl qu In h pp, w ond h nufung y o h pow pln o y wh xd ndy nlud old nd wo ndy oponn A ndy oponn lld 'old ndy' f flu zo Th ndy oponn fd 'w ndy' whn h flu nonzo nd 67
2 AAM: Inn J, Vol 7, Iu D 67 l hn h flu of py oponn y, w, nd old oponn n ondd o pl Th pn udy dff fo p wok n h pn novl hodology o dgn y onfguon nvolvng nd xd ndy oponn Th lly h of y wh M opng hn, w ndy p nd R pn wh xponnl flu nd xponnl p duon w nvgd y Wng nd vzln 99 nvn nd Gopln 97 udd on on- un opng hn wh gnl lf duon, w W ndy wh xponnl flu nd xponnl p duon d on only on upon, nly, h y fl whn no p vll o pl h fld opng hn ud hv n foud on ung h h -o-p follow n xponnl duon [ Ydvll l ; Chn l 6] Th lly nd vlly h of wo dffn y onfguon udd y [El-d nd El-hny 7] El-hny l 9 dvd h lly nd vlly h of h dffn y onfguon wh w ndy oponn nd pl v on El-hny dud h opl y fo y wh xd ndy oponn In h pp, w gong o udy h dffn y onfguon of nd xd ndy oponn Confguon, opd d on h lly In ddon, fo onfguon, whh pl, w gong o dvlop h xpl xpon fo h n -o-flu MTTF nd h dy- vlly A T, nd o lul h o/nf o C / B d on ud nul vlu gvn o h y p, wll o h o oponn Eon of h -p Wull duon Th hzd funon of oponn followng -p Wull duon n dd y: h, wh θ l p, nd η hp p Th lklhood funon fo gn h y Fnu nd Booh 997 : L f * R [ ] *
3 67 M El-D nd A ly L Th pl dvv of h nul log of h lklhood funon : ] [ L, L,, nd uung h ul fo quon, no, hn w hv: L,,, nd h 5 Fo nong, odd flu fo nd fo, wh h xu fo nong, h nu of h fl fo Whn ll
4 AAM: Inn J, Vol 7, Iu D 675,,,, vll, h d opl; opl d pl of gh onnng fo Ou pl nvgon ugg h hoong: v l h Copu of h p θ nd η fo nu of flu Th poof of h followng ul pnd n h ppndx Dpon of h y Fo h k of duon, w ond h qun of MW pow pln W lo u h gno vll n un of oh nd 5 MW ndy gno lwy ny n of flu W u h h wh pf Wng nd Kuo, W lo u h h whov fo w ndy oponn o py oponn, fo old ndy oponn o w ndy oponn, fo flu o p, o fo p o old ndy oponn o py oponn f h y ho nnnou y oponn nd w ndy oponn n ondd o pl y Wng l 6 nd X l Eh of h py oponn fl ndpndnly of h of h oh nd h -dpndn flu wh p Whnv on of h py oponn fl, w ndy ov no opon f ny vll, nd old ndy pu on w ndy f ny vll, w now u h whn w ndy ov no py oponn, flu h wll h of h py oponn, nd whn old ndy ov no w ndy, flu h wll h of w ndy W u h h of h vll w ndy oponn fl ndpndnly of h of ll h oh nd h -dpndn flu wh p Whnv py oponn o w ndy oponn fl, dly pd n h od of kdown wh -o-p, whh xponnlly dud wh p On oponn pd, " good nw", no h fld y nv pd Th followng onfguon ondd: Th f onfguon l y of on py MW oponn, on w ndy MW oponn, nd on old ndy MW oponn Th ond onfguon l y of wo py 5MW oponn, on w ndy 5MW oponn, nd on old ndy 5MW oponn Th hd onfguon pl : of wo py 5MW oponn, wo w nd y 5MW oponn, nd on old ndy 5MW oponn, wh onn flu,,nd onn p µ
5 676 M El-D nd A ly Th Rlly of h y j Th poly,fo j=,,, n vwd ul of olvng of fou f od dffnl quon gvn y h followng dny: d d j j j j j j 6 wh h non fo j o Clulon fo Confguon Fo onfguon, l h poly h xly oponn wokng, If w l dno h poly ow vo, hn h nl ondon fo h pol : j,,,,,, 7 Th y- quon fo Mkov odl whh h of h f-od dffnl quon gvn y Q Th non x Q fo lly odng o onfguon gvn y: Q W wll k h x Q nd dl h ow nd olun fo h ong Th nw x lld
6 AAM: Inn J, Vol 7, Iu D 677 W n w h y n h fo:, wh, To olv quon wh h nl ondon 9 Mulplyng oh d of quon y d hn w hv: d d d, And, hn, d Ou pol now how o dn h vlu of d o u h d D, wh
7 67 M El-D nd A ly d D, nd dx x dx x, Th hod of olvng w follow gv u h vlu of D y h followng lon D D I D, wh I h dny x of nk nd,, h p ond fo h oluon of h followng y:,, 5, 6 wh,, h h oo of h x D h oo ond fo h h quon g of h x D gvn y: g 7 By olvng quon 6, w hv,, 9
8 AAM: Inn J, Vol 7, Iu D 679 Now w n on h vlu of D fo quon nd on h vlu of qud pol fo quon whh :, wh,, Th y lly funon of onfguon : R n fo quon, on n on R, wh / / / 6 9 6, / /
9 6 M El-D nd A ly / / nd fo , 5, Clulon fo Confguon L h poly h xly oponn wokng, If w l dno h poly ow vo, hn h nl ondon fo h pol :,,,,,, Th non x Q fo lly odng o onfguon gvn y: Q W wll k h x Q nd dl h ow nd olun fo h ong Th nw x lld
10 AAM: Inn J, Vol 7, Iu D 6 W n w h y n h fo:, wh, To olv quon wh h nl ondon W wll olv quon wh h d of h hod ud n h pvou on nd hn: d D nd dx x dx x,, h vlu of D wll gvn y h lon, whh : D D I D, 5
11 6 M El-D nd A ly wh I h dny x of nk nd,, h p ond fo h oluon of h followng y:, 6, 7, wh,, h h oo of h x D h oo ond fo h h quon g of h x D gvn y: g 9 By olvng quon 6, w hv,, nd Now w n on h vlu of D fo quon 5 nd on h vlu of qud pol fo quon whh :
12 AAM: Inn J, Vol 7, Iu D 6 wh,, Th y lly funon of onfguon : R n fo quon, on n on R, wh / / / 6 9 6, / / / / nd 6 6 fo
13 6 M El-D nd A ly , 5, Clulon fo Confguon : Mn o flu of h y: Fo onfguon, l h poly h xly 5 oponn wokng, If w l dno h poly ow vo, hn h nl ondon fo h pol :,,,,,,,,, wh h non x Q fo lly odng o onfguon gvn y: Q To vlu h nn oluon oo oplx Thfo, w wll oulv n lulng h MTTF Thfo, w wll k h npo x of Q nd dl h ow nd olun fo h ong Th nw x lld A A 5 Th xpd o h n ong luld fo:
14 AAM: Inn J, Vol 7, Iu D 65 A T E ong, 6 MTTF T E ong Avlly Anly of h y: Fo h vlly of onfguon, w wll u h nl ondon nl ondon fo h pol fo quon : Th dffnl quon fo n xpd : Th dy vlly n ond ung h followng podu In h dy, h dvv of h pol o zo Th llow u o lul h dy pol wh: A T, nd, Q, o, n h x fo:
15 66 M El-D nd A ly 9 Ung h followng nolzon ondon: W uu n ny on of h dundn ow n 9 o yld oluon of povd h dy- pol n h vlly Th xpl xpon fo T A gvn y: ] 7 ]/[ [ A T Co/Bnf Ro: Th noon of o-nf nly pl n pnpl W u h z-popoonl o fo h py oponn, w ndy oponn, nd old ndy oponn, pvly, hown n l wh h w lul h o fo onfguon I ulz h o/nf o / B C n o nk lnv, l: C = h o fo h onfguon, B = h MTTF of onfguon,
16 AAM: Inn J, Vol 7, Iu D 67 o B = h A of onfguon T Tl Th z-popoonl o fo h oponn Coponn Co n $ y 5MW 5E+6 W ndy 5MW E+6 Cold ndy 5MW E+6 Th o fo onfguon wh h wo py oponn, wo w ndy oponn, nd on old ndy oponn= $E+6 A nul lluon povdd y ondng h followng p: 6, 5, Gvn h vlu, w n lul fo onfguon : o/mttf=9e+6 o / =5E+6 A T 5 Conluon W hv povdd n h pp, h lly of wo onfguon, whn h oponn hv -dpndn flu nd onn p By opng h R n oh onfguon, w n h n h f onfguon h lly hgh hn lly n ond onfguon hown n Fgu Moov, fo nul ul fo h o/nf u hv n ond fo onfguon pl, w hv povdd y hodology o dvlop h n o y flu nd h dy- vlly of y wh xd ndy oponn By opng h MTTF nd h A T, w n dw onluon h h n o y flu nd h dy- vlly gnfnly povd y ddng old ndy oponn Nul ul fo h o /nf u hv n ond fo h onfguon gv ll o/mttf hn h o/mttf y onfguon wo py 5MW oponn, on w ndy 5MW oponn, nd on old ndy 5MW oponn, nd h onfguon gv ll o / A T hn o / A T y onfguon on py MW oponn, on w ndy MW oponn, nd on old ndy MW oponn, on py MW oponn, on w ndy MW oponn, nd wo old ndy
17 6 M El-D nd A ly MW oponn, nd on py MW oponn, wo w ndy MW oponn, nd on old ndy MW oponn R R R Fgu : Rlonhp wn R nd T h n Confguon, Thn fo quon, w hv: Appndx nd v A lh v v v A Ung quon A nd A o on : v / v A v /
18 AAM: Inn J, Vol 7, Iu D 69 Th ppoxon plf o Equon A povd quk ppoxon o nd v n ud n nl of fo v MLE oun Tl Copung of h p nd fo nu of flu Od flu Fo =,,, v / V / Rfn Chn, Y, K, J C, nd L, L 6 Aypo onfdn l fo pfon u of pl y wh pf v on, Counon n : ulon nd Copuon, 5: - El-d, K M, nd El-hny M 7 On h vluon of lly nd vlly h of wo dffn y, Infoon nd Mngn n, : - 9 hny M, Rhd, A M, nd Gh, D M 9 Th opl y fo y wh w ndy oponn pl v on, kn Jounl of Opon Rh, 5: -7 hny M Th opl y fo y wh xd ndy oponn, Jounl of Quly Mnnn Engnng, 6: 9- Fnu, N R 997 Unqun of xu lklhood o of h -p wull duon, IEEE Tn Rlly, 6: 5-55 Kuo-ung, Wng nd Chng-Chng, Kuo Co nd ol nly of y wh xd ndy oponn, Appl Mh Modg, : 6- Kuo-ung, Wng, Wn-L, Dong nd Jyh-Bn, K 6 Copon of lly nd h vlly wn fou y wh w ndy oponn nd ndy whng flu, Appld Mh nd Copuon,: - Mng, FC 99 On opon of MTBF of fou dundn y, Jounl of Molon Rlly, : nvn, K, nd Gopln, MN 97 ol nly of wo- un y wh w ndy nd ngl p fly, Jounl of Opon Rh, : 7-7 Wng, K, nd vzln B D 99 Rlly of y wh w ndy nd pn, Jounl of Molon Rlly, 95: 9-6
19 69 M El-D nd A ly X, M, Yun-hn, D nd K oh Copung y Rlly: Modl nd Anly, Kluw Ad, Nw Yok Ydvll, V, Boh, M, nd Bkk, A Aypo onfdn l fo h dy vlly of wo-un plll y wh ppon fo h p fly A-f Jounl of Opon Rh, 9: 9-56
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