System Ageing Assessment in Energy Supply Security Model

Transcription

1 Sytm Agg Amt Ergy Suppy Scurty Mod Juoza AUGUTIS Dpartmt of Mathmatc ad Stattc Vytauta Magu Uvrty Kaua Lthuaa Rčarda KRIKŠTOLAITIS Dpartmt of Mathmatc ad Stattc Vytauta Magu Uvrty Kaua Lthuaa ad Iga ŢUTAUTAITĖ-ŠEPUTIENĖ Laboratory of Nucar Itaato Safty Lthuaa Ergy Ittut Kaua Lthuaa ABSTRACT Th tudy prt a agorthm dvopd for th amt ad updatg tmat of th paramtr mathmatca mod of dvc or ytm agg proc (that charactrzd by a ag faur rat) wth rpct to pror formato ad obtad obrvato (faur data) Th propod agorthm bad o modfd appcato of Baya approach (BA) Th papr prt om mthod for forcatg th rmag ftm wth rpct to th aowd rk marg ug th obtad updatd mathmatca mod of ag faur rat Th dvopd mthodoogy ud for th modg of tchca dturbac rgy curty aay mod [] Kyword: agg ag faur rat Baya approach INTRODUCTION Scurty of rgy uppy ha rcty gad mportac o th pocy agda du to th growg dpdc of dutrazd coom o rgy coumpto ad th ad frqucy of uppy drupto Scurty of rgy uppy a vry mportat fd of atoa curty vry coutry It cud mg covro ad traportato of prmary rgy ourc grato dtrbuto ad uppy of rgy fuctog of fratructur cur f of octy from tchca coomca oco-potca ad vromta pot of vw Ergy curty v [] maurd by a ytm of dcator Accordg to th ytm th rut of ach caro muato ar trafrrd to umrca vau of curty v Scurty maur tgra ad covr th mot mportat tchca coomca vromta ad potca tra of rgy uppy [] Th rarch work focud o ag th rabty of tchca ytm fac of agg phoma Rabty or ffccy of th ytm ca chag ( b dag) bcau of t agg Th agg of th ytm whch coud b udrtood a a gra proc whch th charactrtc of compot ytm ad tructur ("dvc") graduay chag wth tm or u vtuay ad to dgradato of matra ubjctd to rvc codto ad coud cau a rducto compot ad ytm afty marg Uuay om dvc or ytm ca work afy for a ogr tm tha t dotd th tchca pcfcato Morovr th rpacmt of dvc (or om parat part) xpv coty ( mag of th tm) or coud b ratd to ta rk I grg maagmt t mportat to dtrm th maxmum rmag ftm wth th aowd rk v Th proc of ytm agg ca b dfd by varou charactrtc that dpd o tm ( u or othr factor) Th papr aayz th ca wh agg proc charactrzd by a ag faur rat that dpd o tm For th outo of th dcud probm th foowg tak ar gog to b aayzd: Etmato of th momt at whch th agg proc tart (or dtrmg that agg proc ha arady tartd) Dvopmt of a agorthm for obtag ad updatg th tmat of paramtr th mathmatca mod of agg proc wth rpct to pror formato ad w obrvato Forcat of afty durato of th ytm oprato ad th rmag ftm For tac dtrmato of pot ad trva tmat of th tca tm momt rpct of th aowd tca vau of ag faur rat I th papr a chm of modfd appcato of Baya approach (BA) that utab for updatg radom paramtr tmat th mathmatca mod of o-tatoary proc a agg prtd SYSTEM AGEING PROCESS CHARACTERIZED BY INCREASING FAILURE RATE Dvc / ytm oprato tm ca b dvdd to thr part (Fgur ): bur- prod (wh faur rat dag); prod of ufu f (charactrzd by cotat faur rat); ad war-out (or o cad agg) prod (wh faur rat ag): at th tm momt t t tart th tca vau of faur rat markd rachd at th tm momt t Fgur Th bathtub curv Hypothtc faur rat vru tm

2 It vdt that th dpdabty of th codrd ytm / dvc / oftwar dag th thrd agg prod bcau of mor ad mor frqut faur So t mportat to dtrm whthr th agg prod ha tartd for th aayzd ytm / dvc; to dvop a mathmatca mod for th ytm / dvc agg prod that ab mag pot ad trva forcat of th rmag ftm of th dvc ( faur rat do ot xcd th aowd tca vau ) Th tmato of th momt at whch th agg proc tart Thr ar om tt for dtrmg whthr dt vt a proc hav a trd: Lapac tt or o cad ctrod tt [6] vro tt [4 8] two-c tt [3] Faur rat trd updatg Aum that th ytm agg proc ha arady tartd ad t charactrzd by a ag faur rat Somtm th approxmat faur rat dpdc o tm (or othr factor) for th partcuar group of dvc kow advac Howvr pror formato ca ad to om utat thu th paramtr of th dpdc ar aumd a radom varab Lftm dtrbuto of th codrg ytm ctd accordg to th form of faur rat trd Uuay th appd ftm dtrbuto ar charactrzd by ag faur rat ( ) or bath-tub ( U ) hap faur rat fucto: Wbu Brbaum-Saudr ( ); Grazd Modfd Wbu Expotatd Wbu Addtv Wbu Modfd Wbu Modfd Wbu Exto ( ad U) tc I rabty / dpdabty aay of tchca dvc tchca ytm or v oftwar o of th mot popuar dtrbuto of faur data Poo dtrbuto (wth paramtr ) Th dtrbuto ca b ud ca of ocotat faur rat: xcpt that Poo mod paramtr rpacd wth (t) fucto of tm t or ay othr that dpd o o (or mor) factor() I th ca t dfd a a mathmatcay mp mod of ftm Apart from that th trd curv of ag faur rat coud b ay rpct of th aayzd ytm or dvc Th xt tp to vauat th paramtr ftm dtrbuto Th maxmum khood tmato (MLE) qut popuar tattca mthod ud for provdg tmat for th mod paramtr Th mtato of th mthod that thr o pobty to u pror formato about radom paramtr; t tmat ar obtad ug oy tattca data If th dtrbuto of th mod paramtr ar kow thy ar vovd to th mod through th aw of tota probabty; th ca th obtad mod compx ad t uag qut compcatd Th mthod that aow vovg tattca data ad pror formato about dtrbuto ad tmat of radom paramtr Baya approach; ug th mthod th obtad potror dtrbuto (paramtr tmat a w) coud b updatd by w avaab tattca data Modfd appcato of Baya approach: Commoy Baya approach appd to updat th tmatd paramtr of tatoary proc wh mor tattca formato bcom avaab I ca of o-tatoary proc aay th avaab tattca data ca ot b ud to updat th charactrtc of th prvou prod bcau t rprt th othr tat of th ytm For aayzg otatoary proc th rqurd formato th foowg: dtrbuto of tattca data; form of th trd of ytm dyamc dbg charactrtc ξ (a a fucto of om factor F F r ad paramtr θ θ : f(θ θ F F r ) for xamp t xpota poyoma ar tc Th xpctd vau of radom otatoary charactrtc ξ atf th rqurmt Eξ = f(θ θ F F r ) () Not that paramtr of dtrbuto of tattca data / obrvato ar xprd rpct of q () mut b atfd BA appd to updat radom ukow paramtr θ θ of th fucto dfd by q () I rpct of th utaty of pror formato th maurmt of o ad th paramtr θ θ of th codrd mod coud b aumd a radom dpdt varab wth pror kow dtrbuto (othrw oformatv for tac uform dtrbuto ca b ud a pror; ot that thr pror pdf ar p (x ) = ) Aumg that th dtrbuto of tattca data y = m ao kow th khood fucto L( ) atf q () th potror mutdmoa dty fucto obtad by th appcato of Baya formua for th formato ( x x y y ) () R R p ( x ) L( y y p ( u ) L( y y x x ) u u )du du = m R rag (t of a pob va of paramtr θ = Som compcato ar practca computato of th tgra that appar th domator of Bay formua Som mpfcato for th probm ar pob (of tac ug cojugatd par of pror dty fucto ad khood fucto = avodac of umrca cacuato of tgra gv covt appcato of BA) Dpdg o th khood fucto om pror dtrbuto ca away ad to th potror dtrbuto whch ha th am fuctoa form a th pror dtrbuto for tac Norma khood ad to th cojugat potror Norma dtrbuto Th tattca proprty ratd to th o-cad cojugat par of pror dtrbuto ad khood [5] Ug th cojugat par th ma ad varac a w a othr paramtr ca b ay tmatd for th potror dtrbuto Ug a cojugat par of khood ad pror umrca rror ad covt agorthm for updatg th dtrbuto ca b dvopd [7] Baya pot tmat (xpctd vau of potror dtrbuto) of paramtr θ ˆ x ( x x x y y ) (3) R R R = Th aymptotc of Baya pot tmat: Th ma tro for vauatg th utaty of th Baya tmat th aay of t varac I gra t dffcut to rarch t bcau th varac xprd a a tgra Oy om ca of th covrgc of Baya tmat ar prtd (wh cojugat par [3] ar ud) For tac ca of th pror Gamma pdf ad Poo khood th xpctd vau ad varac ar b y b y b b E E Var a a a a a

3 If E cot Var ; rat of covrgc hr amout of tattca data Not that a ca of cojugat par th varac of Baya pot tmat dag wth covrgc rat ot owr tha / Th commo covrgc of Baya pot tmat that accuracy of updatd tmat of radom paramtr t a op probm (f a ca th varac of updatd tmat dag) Baya trva tmat: I ytm rabty aay th tmat of mod paramtr ar ot away uffct: t cary to obta thr cofdc trva (wth a gv cofdc v) a w It cary to cotruct a modfd aymmtrc cofdc trva f th bgg (or th dg) of th trva ha a hghr mportac for rabty aay For tac aymmtrc trva a ufu trva tmat for Th dfto of th d of trva a foow: Dfto Aymmtrc cofdc trva of th ukow tru vau of θ for a gv gfcac v α db trva x ˆ ˆ ] that atf th quat [ x P ( x P( x ) c P( ( c) c For tac th aymmtrc cofdc trva of Norma radom varab ( N(μ σ)) [μ u cα σ; μ + u ( c)α σ] hr u quat of tadard Norma dtrbuto Prformg th utaty aay t mportat to cotruct th arrowt cofdc trva of th codrd radom paramtr For tac f th radom paramtr th momt at whch th ytm agg tart t forma dfto may b t a foow: Dfto Crdb trva of th ukow tru vau of θ for a gv gfcac v α th arrowt cofdc trva xˆ ; xˆ ] : [ x x argm x wh Δx dfd ˆ x ( x P x x ) Not Crdb trva potror (obtad by BA) pdf ud for th cacuato Qut oft compx cacuato of th db trva mt thr appcabty practc (for tac ra tm dco bad agorthm that mut b covt ot rqurg o much tm for prformg th cacuato Som apct of BA appcato: Bhavor of th mod (for tac ar or xpota tc) do ot chag wh appyg BA BA updat jut tmat th paramtr th cho mod BA aow updatg th tmat of a paramtr th mod wth a g w obtad obrvato Th mthod do ot dmad to coct obrvato Iformato about th obrvato corporatd to th dtrbuto of mod radom paramtr through th khood fucto Modfd appcato of BA for NHPP (ohomogou Poo proc) data: Aum that dvc faur umbr pr tm ut (markd a k) foow Poo dtrbuto wth a tm dpdt paramtr λ = λ( θ) trprtd a tm ad paramtr θ radom varab wth kow pdf Th th potror pdf obtad ug th obrvato of th ytm faur data ad Baya approach x k k ) ( ( ( ( du (4) Ca Aum that th faur rat trd ar λ( θ) = θ th cojugat pror pdf of radom paramtr θ Gamma Ga(a b) Potror pdf Gamma Ga(a b ) a w wth paramtr a a b b pot tmat of radom paramtr θ k Baya b ˆ (5) a wth varac Var b ˆ k ( a ) I ca of + th obrvato k + th paramtr of a w potror pdf coud b ay rcacuatd a + = a + ( + ) ad b + = b + k + tmat of θ a w Ca (umrca xamp) Aum that faur rat trd ( xpota ) (6) ad pror pdf of radom paramtr θ xpota wth paramtr μ Baya pot tmat of radom paramtr θ (ot: paramtr θ aumd to b kow) ˆ x x x x x x x wth varac x x ( x ˆ ) ˆ Var A umrca xprmt wa prformd to utrat th covrgc ( Fgur ) of Baya pot tmat to tru vau Faur umbr k = th th trva of tm muatd by Poo dtrbuto wth paramtr * ) (θ = 5 θ = ) Th umrca xprmt ( paramtr θ aumd a radom varab (ca : wth a pror o-formatv dtrbuto t dty fucto cotat; ca : wth a pror formatv dtrbuto) For th ca a xpota dtrbuto wth paramtr μ = 3 wa cho a a pror kow formatv dtrbuto A atratv mthod for th tmato of th paramtr th mod wth kow trd fucto at quar mthod Accordg to th obtad rut th um of rror quar of BA (wth o-formatv pror dtrbuto) approxmaty % bggr tha BA (wth pror xpota dtrbuto); th um of rror quar of LSM approxmaty twc bggr tha BA (wth pror xpota dtrbuto) Obvouy f a t of obrvato qut bg a mthod gv qut prc tmat of th paramtr BA powr th combato of

4 pror formato ad obrvato (khood a w) th ca of jut fw obrvato ( th bgg) wth xpota dtrbuto a pror pdf (dah ) wth o-formatv pror pdf (od ) tru vau of paramtr θ = Fgur Baya pot tmat of θ ( = ) Radom paramtr th trd of faur rat rpacd wth th updatd Baya pot tmat (xpctd va I th futur aay t coud b ud to dvop a mathmatca mthod for th amt of th momt at whch th ag faur rat woud xcd th aowd faur rat Forcat of th rmag ftm rpct of ag faur rat I th commo ca a aumpto mad that trd fucto of ag faur rat dpd o paramtr t ) (7) ( tca vau of ag faur rat dfd ( tchca pcfcato or dtrmd by xprt) for aayzd dvc or ytm; t corrpodg tm momt t rmag ftm Th cto aay th probm of vauatg th rmag ftm of a dvc ad propo two agorthm I Th xpctd vau of th rmag ftm t ca b tmatd ovg th quato ( ˆ ˆ tkr ) (8) hr ˆ Baya pot tmat of paramtr θ = obtad by formua (3) Fgur 3 Iag faur rat th prod of agg II a) Faur rat (t) (dfd by (7) quato) a fucto of radom varab() θ θ t probabty dty fucto (pdf) obtad ug th potror pdf of θ = ad traformato formua (om ca ar prtd Tab ) Itrva tmat: aymmtrc or db Tab Probabty dty fucto f(y k k ) of faur rat (t) Expro of (t) ( t tm ) Radom paramtr Pdf of faur rat (t) ( t tm ) λ(t) = θ t θ y f ( y) p t t t ( t) y (ot: θ θ f ( y) p t y kow) t Not: ) = x k k ) potror pdf of radom paramtr dfd by formua (4) For th gv fxd tm momt t * th probabty that th ag faur rat woud ot xcd th tca vau coud b ay cacuatd P( y t t ) f ( y t t )dy (9) * II b) O othr had t rvat to vauat th rmag ftm of th aayzd dvc or ytm wth rpct to th aowd rk v xprd by th tca vau of faur rat Th ma vau of th rmag ftm ot prc ough I fact th ftm t a fucto of radom varab() θ θ dfd by quato (mpct form) ( t ) () Pdf of t obtad ug th potror pdf of θ = ad traformato formua (om ca ar prtd Tab ) Tab Probabty dty fucto g(z k k ) of th rmag ftm t Trd Lftm t a fucto fucto of Pdf of rmag ftm t (t) radom ( t tm ) varab λ(t) = θ t radom t g z ( ) p paramtr θ z z t ( t) radom paramtr θ t g( z) p z z (ot: θ kow) Not: ) = x k k ) potror pdf of radom paramtr dfd by formua (4) Pdf g(z) that cota formato about th obrvato ad pror dtrbuto of radom paramtr gv pot ad trva tmat of ftm t I th pot of trva tmat th cotructo of aymmtrc (wth mor attto gv to th bgg Fgur 3) or db trva ha mor advatag tha caca cofdc trva wth quata For tac db trva th hortt of a cofdc trva ad cota mor probab vau of th rmag ftm 3 RESULTS AND CONCLUSIONS Th agorthm for th tmato of th tm momt at whch th agg proc of dvc or ytm tart wa dvopd (t wa bad o ttg of paramtrc hypoth wth rpct to faur data) *

5 Th papr prtd th agorthm for obtag (ad updatg) th tmat of radom paramtr th mathmatca mod of charactrtc that db ytm dvc or ubtac agg: th bhavor (trd fucto dpdt o om factor ad paramtr()) of th aayzd charactrtc prory pror kow Th propod agorthm bad o th modfd appcato of Baya approach I th papr th utrato of th dvopd agorthm appcabty wa prtd by a umrca xprmt: ca wh agg proc charactrzd by a ag faur rat wth pror kow trd; th covrgc of Baya pot tmat of th paramtr of faur rat trd fucto wa dmotratd; th obtad rut wr compard wth th o cacuatd ug at quar mthod (LSM): th um of rror quar of LSM wa approxmaty twc bggr tha th um of rror quar of BA Th agorthm for th amt pot ad trva tmat of th dvc or ytm rmag ftm ( forcat of afty durato of th dvc or ytm oprato) wr prtd ca wh th aowd rk v wa dtrmd ug tca vau of ag faur rat Th dvopd mthodoogy ud for tchca dturbac modg rgy curty aay mod ACKNOWLEDGMENTS Th rarch wa fudd by a grat (No ATE-8/ ad ATE-/) from th Rarch Couc of Lthuaa 4 REFERENCES [] Augut J Matuzė V Krkštoat R Pčuytė S Norvaša E 8 Aay of curty of rgy uppy amt mthod Ergtka Vo 54(4) -9 [] Augut J Krkštoat R Pčuytė S Kotatavčūtė I Sutaab Dvopmt ad Ergy Scurty Lv Aftr Igaa NPP Shutdow Tchoogca ad Ecoomc Dvopmt of Ecoomy Vo 7() 5- [3] Atwood C Crova O Patrk M Rodoov A 7 Mod ad data ud for ag th agg of ytm tructur ad compot (Europa twork o u of probabtc afty amt (PSA) for vauato of agg ffct to th afty of rgy fact) EUR 483 EN Ptt: EC DG JRC Ittut for Ergy [4] Bdat J S Pro A G 986 Radom data: aay ad maurmt procdur Nw York: Wy [5] Brardo J M Smth A F M 3 Baya thory Joh Wy & So [6] Bro A 7 Rabty Egrg Thory ad Practc ISBN thdSprgr Br Hdbrg Nw York [7] Lttwood B Popov P Strg L Amt of th Rabty of Faut-Torat Softwar: a Baya Approach Proc 9th Itratoa Cofrc o Computr Safty Rabty ad Scurty SAFECOMP' Rottrdam th Nthrad Sprgr [8] Rodoova A Atwood C L Krchtgr C Patrk M 8 Dmotrato of tattca approach to dtfy compot agg by opratoa data aay A ca tudy for th agg PSA twork Rabty Egrg ad Sytm Safty vo