Development of a Maintenance Option Model to Optimize Offshore Wind Farm Sustainment

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1 Dvlopm of Mic Opio Modl o Opimiz Offshor Wid Frm Susim Pr Sdbor, 1 Gilbr Hddd, 2 Amir Kshi-Pour 2 d Xi Li 2 Cr for Advcd Lif Cycl Egirig, Uivrsiy of Mryld, Collg Prk, MD, This ppr prss mhod h uss simulio-bsd rl opios lysis o drmi h vlu of wiig fr obiig rmiig usful lif (RUL) idicio from progosics d hlh mgm (PHM) srucurs wihi wid urbi. This mhodology poilly llows h hrshold for mic o b opimizd i rl-im bsd o h s of hlh of h urbi, h icipd fuur s of mic rsourcs, mic flxibiliy (curr d fuur), d mrk ucriis (.g., icipd fuur pric d dmd for rgy). CA C EH C f C M d R P V M W WT PR W X μ σ Nomclur = cos voidc = cos of rgy = cpciy fcor = cos of mic = discou fcor = cumuliv rvu = umbr of phs lyzd = im = mic vlu = umbr of im priods corrspodig o h mximum wiig im = urbi powr rig = Browi moio = h vlu of h quiy big simuld = drif = vric I. Iroducio ffshor wid frms r cpil isiv projcs whos coomic vibiliy dpds o my higs icludig: Oh wid rsourcs, h chology, h dph of h wr, h pric of rgy, d h succssful log-rm susim of h urbis. Susim is h cpciy of sysm o dur; d is ffcivss is usully msurd by ssssig h lif-cycl (or ol owrship) cos of h sysm. Susim icluds: rlibiliy, miibiliy, opriol logisics, cofigurio corol, obsolscc mgm d h biliy o (d opimum frqucy of) sysm upgrds d rfrshs. Susim, lso rfrrd o s oprio d mic (O&M) or oprio d suppor (O&S), is projcd o b h scod lrgs coribuor o h lif-cycl cos of offshor wid urbis, rprsig 17-28% of h ol lvlizd cos of offshor wid frms d mor for frms h r mor h 12 mils offshor (Rf. 1). Ths umbrs r oly sim, rliy migh b wors; wihou crful srgic plig, solid lif-cycl cos simios, d rlisic udrsdig of h susim rsourcs rquird, h frvor o isll wid urbis could b quickly rplcd by drlic wid frms. I Novmbr 2011, NurlNws.com simd h ovr 14,000 wid urbis hv b bdod i h US du o high rs of filur d high mic coss, (Rf. 2). 1 Profssor, Dprm of Mchicl Egirig, Collg Prk, MD. 2 Grdu Sud, Dprm of Mchicl Egirig, Collg Prk, MD 1 Amric Isiu of Arouics d Asrouics

2 Th prdicio d opimizio of mic civiis provids sigific opporuiy for wid rgy cos rducio h bfis ll skholdrs. Udrsdig h lif-cycl susim rquirms of offshor wid frms is r h hs rcivd fr lss io h wid urbi dsig, mufcurig, siig, c. II. Offshor Wid Frm Mic wih PHM Mic of offshor wid frms is chllgig bcus h rsourcs rquird o prform mic r xpsiv, o coiuously ccssibl, d h cpciy fcor for h urbis is vribl. For my yps of ifrsrucur-criicl sysms, cusomrs r rsiioig o buyig h vilbiliy of sysm hrough vilbiliy corcs, isd of cully buyig h sysm islf (Rf. 3). Avilbiliy is h biliy of srvic or sysm o b fuciol wh i is rqusd for us or oprio. Avilbiliy is fucio of sysm s rlibiliy (how of i fils) d miibiliy (how fficily i c b rsord wh i dos fil). Evluig vilbiliy rquirm is chllg for mufcurrs d supporrs of wid frms bcus drmiig wh i will cos o dlivr spcific lvl of vilbiliy is difficul. This ppr will discuss pproch o implmig mic-opimizd vilbiliy h will llow h vluio of mic opios for idividul urbis i h cox of wid frm mgd usig vilbiliy (or similr) prformc-bsd corc. Th spcific chllg ddrssd is mkig busiss cs for, d opimizig h us of, progosics d hlh mgm (PHM) d codiio-bsd mic (CBM) for drmiig which urbis o mii wh mic rsourcs r vilbl. PHM cosiss of chologis d mhods o ssss h rlibiliy of sysm i is cul lif-cycl codiios o drmi h dv of filur d h miigio of sysm risks (Rf. 4). PHM-bld ms h h sysm is isrumd wih ssors h msur ihr h virom srsss h sysm hs courd (idirc PHM), spcific physicl propris of h sysm (d-driv PHM), or boh. Th criicl subsysms i my of ody s wid urbis lrdy hv rry of such ssors. PHM is blr of vilbiliy-bsd corcs d poilly rducs lif-cycl cos. A rviw of codiio moiorig pplid o wid urbis is providd by Hmd l. (Rf. 5). I his ppr w will us rl-opios lysis o vlu opios o: mii ow, d wi d mii lr.. Th mhodology llows h ffciv hrshold for mic o b opimizd i rl-im bsd o h s of hlh of h urbis, h icipd fuur s of mic rsourcs, mic flxibiliy (curr d fuur), d mrk ucriis (.g., icipd fuur pric d dmd for rgy). III. Usig Rl Opios Alysis (ROA) o Vlu Mic Opios Wh omly is dcd i PHM-bld sysm (.g., wid urbi), d h rmiig usful lif (RUL) of h sysm is simd wih ssocid ucriis, h dcisio mkr hs mulipl choics clld opios, which c b xrcisd (or o xrcisd) o mg h hlh of h sysm. Rl Opios Alysis (ROA) is usd o sim h vlu h mic opios risig from h implmio of PHM. 3 Th vluio (quificio of h vlu) of h opios udr ucriy rprss sysm mgm opimizio problm. I h cs of wid frms, wh progosic-idicio is obid for urbi h dcisio-mkr hs h opio o ur h urbi off i ordr o void furhr dmg, modify h wid urbi s oprio i such wy s o rduc h lodig, or coiu orml oprio (i.., do ohig). Mrk risks d chicl risks r rlv o ROA dcisio mkig. As xmpl h pric of powr producd by wid urbis my b rgrdd s mrk risk bcus i is coig o h pric of ohr rgy sourcs. A xmpl of chicl risk is h risk ssocid wih h fficicy which h chology oprs,.g., cpciy fcor, which is h rio of h urbis cul oupu ovr som priod of im o is poil oupu. Th ivsm i implmig PHM wihi sysm is cosidrd h purchs pric of h opio. If RUL producd by h PHM rsuls i mic civiy prior o h filur of h sysm, h h opio is xrcisd; if h RUL is o cd upo prior o filur h h opio xpirs. Th fudml rdoff i mic problms wih progosics is fidig h im o prform mic h miimizs h combiio of rmiig usful lif h is hrow wy (which dcrss s you us up h RUL) d h risk of xpsiv uschduld mic (which icrss s you us up h RUL). For opios vluio w rs his s: fid h im o prform mic h mximizs h combiio of h rvu h c b grd by h sysm durig is RUL d h cos of uschduld filur h c b voidd. I Fig. 1, h logr h miir wis fr h progosic idicio o rplc filig pr, h mor rvu c b md wih h pr (i.., lss rmiig lif of h pr is hrow wy); howvr, h logr o wis fr progosic idicio, h lss cos 3 Th vluio of opios ssocid wih prformig mic hs b do i h ps,.g., Rfs 6-8, howvr, h us of mic opios whr h opio big purchsd is h iclusio of PHM wihi h sysm hs o pprd prviously. 2 Amric Isiu of Arouics d Asrouics

3 voidc is rlizd du o h icrsd risk of filur. Th sum of hs wo ffcs is h vlu of xrcisig h Bfi obid from ruig h sysm hrough h RUL + = Progosic idicio Ed of RUL Tim Progosic idicio Ed of RUL Tim Progosic idicio Ed of RUL Tim Figur 1. Vlu of wiig o mii. This simpl rprsio ssums h h svriy of h filur is cos bu h probbiliy of filur icrss s h RUL is usd up. This figur dos o ddrss ucriis i h RUL. opio, V M CA R (1) whr V M is h mic vlu, R is h rvu obid durig h RUL d CA is h cos voidd. I rliy, hr r my possibl phs (giv ucriis i boh rvu d cos voidc) h h vlu could k ovr im srig wh h RUL is forcsd by h PHM sysm. Th rvu d cos voidc h ppr i Fig. 1 r im-dpd ucri quiis h c b modld wih dymic forcsig modls such s gomric Browi moio. Th followig sochsic diffril quio is usd o propg ucriy wih im for h grio of h dymic quiis, dx X d X dw X dq (2) whr X is h vlu of h quiy big simuld im, μ is drif compo, σ is vric compo (lso clld shock or voliliy), W is Browi moio ( d ), d dq is idpd procss h rprss jump o zro. Figur 2 shows h cos voidc, rvu d vlu for pich corol mchism i wid urbi. I Fig. 2, h horizol xis rprss im i dys; im 0 progosic idicio is obid, d ucriy is propgd for 100 dys i o-dy im sps. Th cos voidc srs vlu of $6984 (60% of h cos of w pich corol mchism) wih μ = -0.6 d σ = Th dily rvu for h urbi is clculd usig, R 24WT C C (3) PR EH f Whr w hv ssumd 600KW urbi (WT PR ) wih cpciy fcor of C f = 0.33 d cos of rgy C EH = $0.05/KWh. From (3) h cumuliv rvu of h urbi srs $237 wih μ = 0.18 d σ = 0.5. Filly h vlu is h summio of h cos voidc d h cumuliv rvu. Th vlus for drif (μ) d voliliy (σ) r simd from h vriio i h powr curv for h rvu d from hisoricl d for cos voidc. I h cs show i Fig. 2, jump i h cos voidc 80% of h forcsd RUL is ssumd d rprss h risk of h sysm filig prior o rchig h forcsd RUL. No, giv cos voidc rprss poil collrl dmg. Usig h s of possibl vlu phs for h sysm s fucio of im, w wish o vlu h vlu of wiig o prform mic fr RUL simio is rcivd. To vlu h phs w firs d o drmi cos of mic (C M ) o compr h mic vlu (V M ) o, 3 Amric Isiu of Arouics d Asrouics

4 ) A (C 5000 c id 4000 v o A s o C ) (R u v R iv l u m u C ) M40000 (V lu V c i M Tim (dys) Tim (dys) Tim (dys) Tim (dys) Tim (dys) Tim (dys) Figur 2. Propgio of h propris i Figur 1 i im wih ucriis. I his xmpl, h RUL is 100 dys d 20 phs r show. No, h propris lso iclud jump o zro i h cos voidc 80% of h RUL. C M C C C (4) Filur Dowim Ply whr C Filur is h cos of sysm filur, C Dowim is h cos of dowim d C Ply is h ply cos. Th ply cos is h cos icurrd if h urbi s dowim rsuls i h wid frm o mig is rquird vilbiliy. Th ply my k h form of h cos icurrd by hvig o buy or produc rgy vi lriv ms o supplm h wid frm i ordr o m h vilbiliy rquirm. Th cos of mic rprss h cos of uschduld mic civiy, i.., ru o filur. Th vluio of h opio o wi for spcific priods of im is prformd usig simulio mhod clld ls-squrs Mo Crlo (LSM) dvlopd by Logsff d Schwrz (Rf. 9). 4 Th LSM pproch srs h d d ( slcd wiig im), d works bckwrds i im hrough h vlu phs. A ch possibl im h mic could b prformd, h lgorihm drmis h vlus of prformig mic (h opio s xrcis vlu) d h xpcd vlu of coiuig for ohr priod of im (o xrcisig h opio). Th xpcd vlu of coiuig is drmid usig ls-squrs fi of h vlu of xrcisig h curr im vrsus h diffrc bw h vlu of xrcisig h x im priod d C M discoud o h curr im. Th ls-squrs fi oly cosidrs phs h r i h moy h curr im sp, i.., V M >C M. This procss is rpd ll h wy bck o zro im (h progosic idicio poi). W h work forwrd i im from 0 o h wiig im i vry ph d sop h firs mic opporuiy i ch ph whr h curr vlu is grr h h coiuio vlu. Alyzig ch ph forwrd i im idifis h opimum soppig poi (prior o h slcd wiig im) for ch ph d h ph s ssocid vlu. Th bsic prmis of h lysis is h y possibl xrcis im (mic opporuiy), h xpcd pyoff from immdi xrcis is comprd gis h xpcd pyoff from coiuio wihou xrcisig, d h xrciss if h immdi pyoff is highr. This is prformd for ch ph sprly; howvr, ll phs r dd o ccurly vlu h xpcio vlus. Th possibl xrcis pois r drmid by h vilbiliy of mic rsourcs. Th ovrll vlu corrspodig o priculr wiig im is giv by, 1 Vlu P P W i 1 j 1 V M d P, j j (5) 4 Tuoril xmpls of h LSM lgorihm ppr i Logsff d Schwrz (Rf. 9) for Amric pu opio d i Hddd. l. (Rf. 10) for Amric cll opio. Th opio lyzd i his ppr is Amric cll opio. 4 Amric Isiu of Arouics d Asrouics

5 whr, P is h umbr of phs lyzd, W is h umbr of mic opporuiis (im priods) corrspodig o h mximum wiig im cosidrd, d j is h discou fcor corrspodig o h jh im priod, d V is h mic vlu corrspodig o h jh im priod of h Ph ph (o, V is oly M P, j M P, j o-zro h opimum xrcis poi i h ph). Equio (5) givs h vlu ssocid wih priculr wiig im, i.., i is h vlu of ch ph soppig is opimum poi up o h mximum wiig im cosidrd. Th lysis is rpd for wiig ims h rg from zro o h RUL o drmi h opimum mou of im o wi (h mximizs vlu) fr progosic idicio for priculr RUL. IV. A Cs Sudy Rsul This scio dmosrs h wi-o-mii opio o o urbi subsysm h is idicig rmiig usful lif, mor dild dscripio d discussio of his xmpl pprs i Hddd l. (Rf. 10). Ucriis r h hr of mic opios vluio. I fc, if hr ws o ucriy, i would b rivil o drmi h vlu of h opio. Icludig hs ucriis is crl lm of h lysis h cuss h mulipl sochsic phs fr progosic idicio s discussd i Scio III. This xmpl cs lyzs 100 phs, d ssums h C M is h cos of uschduld mic ($11,640, h cos of pich mchism i 2006 dollrs (Rf. 11)). Th xmpl cosidrs o filur mod wih rmiig usful lif forcsd o b 37 dys. Th cos voidc d rvu r ssumd o b s modld i Fig. 2 xcp h w ssum h 80% of h RUL (80% of 37 dys rhr h 100 dys show i Fig. 2). Th jump modls icrs i h risk of collrl dmg if h urbi fils, which rsuls i cos of filur h is lrgr h h cos of h filur of h subsysm udr cosidrio (h o h idicd RUL). A xmpl of collrl whr filur of o subsysm c ld o cos h is lrgr h h subsysm h fild is discussd by Rghb d Rghb (Rf. 12). Figur 3 shows h vlu of wiig o prform mic fr progosic idicio for h subsysm i h urbi udr cosidrio whr rmiig usful lif of 37 dys is ssumd. Th horizol xis o Fig. 3 is h wiig im; im 0 progosic idicio is obid. Th vricl xis o Fig. 3 is h vlu of h wiig opio. No h his is o h vlu of PHM (h vlu of PHM is br cpurd by ROI modl,.g., Rf. 13), bu rhr i is h ddiiol bfi from cpurig h upsid ffc of ucriis (such s h probbiliy of high wid spds). Figur 3 shows h vlu of wiig from 5 dys o 37 dys. Vlu of Wiig ($) Wiig Tim fr Progosic Idicio (dys) Figur 3. Vlu of wiig for 37 dy prdicd RUL. Th vlu of wiig i Fig. 3 is 0 for h firs 7 dys. This ms h wiig for 7 dys will o gr ddiiol vlu from PHM. This is du o h fc h h cumuliv rvu grd i 7 dys will o comps for h cos of filur. Howvr, fr 7 dys, h cumuliv rvu grd from ruig h urbi compss for h risk of filur; his is wh h vlu of wiig srs o icrs. Th vlu of wiig hs pk pproximly 25 dys, wih vlu of wiig slighly lrgr h $200. Th vlu from wiig is h 5 Amric Isiu of Arouics d Asrouics

6 ddiiol bfi h h dcisio-mkr obis from PHM (kowldg of h rmiig lif d ruig h sysm hrough h RUL). Th mgiud of h vlus i Fig. 3 my sm smll comprd o h sysm s cos. Howvr, his xmpl clculs h vlu of h wiig opio for o priculr subsysm xpricig o filur. I rliy, h PHM sysm will gr h wiig opio ovr h lifim of h urbi (my possibl filurs), i.., i is sgig opio. Morovr, hr r mor mic opios byod wiig h c poilly cr mor vlu. Filly his xmpl is for o urbi oly; i rliy, h urbi is pr of fl of urbis (i.., wid frm) d h vlu would b ggrgd for h lifim of h fl. V. Discussio d Coclusios Th vlu drmid usig h mhod dscribd i his ppr co b giv bcus w r rprsig h opporuiy of h upsid ffc grd from wiig. If h cos of filur is lrgr h h rvu, h wiig hs o vlu; i ohr words, w r o rprsig h diffrc i cos of filur d rvu, bu isd h ddiiol vlu w g from h opio (i.., wiig). No h h vlu i h cs sudy xmpl dcrss fr 25 dys (d vully gos o 0) bcus of h ssumpios md h h cos of filur icrss 80% of h RUL. Hd his ssumpio o b md (i.., if hr ws o ffc of collrl dmg o cos), h vlu of wiig would hv icrsig uil h d of h RUL. Wh h cos of mic d filur is vilbl from hisoricl d, his ssumpio c b rlxd d sims from cul d c b usd o dfi h disribuios h r ssumd i his xmpl. A pr isc cric ssssm of vlu is ssumd i his ppr, i.., w r lookig h vlu chd o priculr urbi subsysms bcus h lysis prsd hr dos o ccou for rvu h is rd fr h mic v. I h cs sudy providd i his ppr, w ssumd h mic could b prformd ch dy up o h d of h RUL; i rliy, spcilly for offshor wid frms, mic rsourcs will o b vilbl vry dy d hir vilbiliy will dpd o whr d ohr fcors. This will dfi s of spcific ds o which mic c occur. Th lgorihm for quifyig h vlu of h wi o mii opios s dmosrd hr provids h vlu obid from PHM h sysm-lvl. This mhodology c b xdd o h fl lvl (wid frm lvl) whr i is dsirbl o kow wh h bs im o mii is bsd o h vlu o h whol frm o h idividul urbi, i.., h opimum mic wh h frm is mgd vi vilbiliy corc. 5 Cosidr h followig problm: wh h mic rsourcs r vilbl, which of h urbis i wid frm should b miid? A idividul urbi s mic hrshold dpds o h curr s of h urbi, h xpcd fuur codiios for prformig mic (.g., wh will mic rsourcs b vilbl?), h s of hlh of h ohr urbis i h frm, d h mgm opios h r vilbl. Th hrshold is o cos; i chgs dpdig o ll h fcors lrdy miod. A xsio of h rl opios pproch dmosrd hr will llow vilbiliy rquirms o b mixd wih mrk d chicl risks o opimiz h mic hrsholds for offshor wid frms. No, h highs vluio (opimum) opio for idividul urbi will o cssrily b h highs vluio for h urbi i h cox of frm h mus m spcific vilbiliy rquird or b plizd. Th ply could k h form of hvig o bur col or oil o mk up for lcriciy h co b grd by h frm. Curr mic prcic for wid frms cosiss primrily of schduld mic, corrciv mic, d prviv mic, (Rfs. 5 d 16). Rsrchrs hv ddrssd h opimizio of mic for wid frms, howvr, h prvious work is o vilbiliy driv. PHM hs b show o b ffciv wy o pproch h hlh mgm wid frms. Howvr, drwbck of PHM is h i ssums h h hrshold for mic is cos. 5 Avilbiliy corcs (Rf. 3) r subs of oucom-bsd corcs whr h cusomr purchss oucom from sysm, rhr h purchsig h sysm. I h dfs commuiy (US d Europ) hs corcs r kow s Prformc-Bsd Logisics (PBL). Oucom-bsd corcs py for ffcivss (vilbiliy, rdiss d/or ohr prformc msurs) fixd r, pliz prformc shorcomigs, d/or wrd gis byod rg gols. Avilbiliy corcs r bcomig populr for susim-domid sysms such s ircrf, ships, d miliry sysms bcus hy forc h dsigr d mufcurr of h sysm o k rsposibiliy for h ir lif cycl of h sysm hus brkig h rdiiol modl of purchsig ss d h sprly purchsig mic. A rc IHS mrgig rgy rsrch rpor o Norh Amric wid powr (Rf. 14) ss: producio-bsd vilbiliy corcs - which i O&M compsio o projc oupu - r bcomig mor prvl d r icipd o bgi rplcig rdiiol upim vilbiliy gurs. From Rf. 15: io is shifig owrds smr rgy grids h icorpor disprsd gr lcriciy supplirs d w yps of cosumrs h mov from ll im powr corcs o rducd powr vilbiliy corcs lowr cos. 6 Amric Isiu of Arouics d Asrouics

7 Ackowldgms Th uhors would lik o hk h Niol Scic Foudio, Divisio of Dsig d Mufcurig Iovio (Gr Numbr: CMMI ), for hir suppor. Fudig for his work ws providd by Exlo for h dvcm of Mryld's offshor wid rgy d joily dmiisrd by MEA d MHEC, s pr of Mryld Offshor Wid Frm Igrd Rsrch (MOWFIR): Wid Rsourcs, Turbi Aromchics, Progosics d Susibiliy. Rfrcs 1 Hmilo, B., Offshor Wid O&M: Coss, Trds, d Srgis for h Comig Dcd, prsd AWEA Offshor Widpowr Cofrc & Exhibiio, Blimor, MD, Ocobr Bso, J., Gr Dbcl: Ts of Thousds of Abdod Wid Turbis Now Lir Amric Ldscp, NurlNws.com, hp:// 3 Cushwy, G., Avilbiliy Corcig, Dfc Mgm Jourl, Vol. 32 No. 1, 2006, pp Pch, M., Progosics d Hlh Mgm of Elcroics. Wily, 2008, Hobok, NJ. 5 Hmd, Z., Hog, Y. S., Cho, Y. M., Ah, S., d Sog, C. K., Codiio Moiorig d Ful Dcio of Wid Turbis d Rld Algorihms: A Rviw, Rwbl d Susibl Ergy Rviws, Vol. 13, 2009, pp Koid, Y., Kio, K., d Ab, M., Lif-Cycl Cos Alysis of Bridgs Whr h Rl Opios r Cosidrd, Procdigs of h Curr d Fuur Trds i Bridg Dsig, Cosrucio d Mic 2001, pp Millr, L., d Prk, C., Ecoomic Alysis i h Mic, Rpir, d Ovrhul Idusry: A Opios Approch, Th Egirig Ecoomis, Vol. 49, 2004, pp Ji, X., Li, L., d Ni, J., Opios Modl for Joi Producio d Prviv Mic Sysm, Iriol Jourl of Producio Ecoomics, Vol. 119, 2009, pp Logsff, F., d Schwrz, E., Vluig Amric Opios by Simulio: A Simpl Ls-Squrs Approch, Th Rviw of Ficil Sudis, Vol. 14, 2001, pp Hddd, G., Sdbor, P. A., d Pch, M. G., Usig Mic Opios o Mximiz h Bfis of Progosics for Wid Frms, o b publishd Wid Ergy. 11 Khrob, S., d Asgrpoor, S., Risk-Bsd Filur Mod d Effc Alysis for Wid Turbis, Norh Amric Powr Symposium (NAPS), Boso, MA, Augus Rghb, A., d Rghb, M., Wid Turbi Grbox Tchologis, Fudmls d Advcd Topics i Wid Powr, Vol. 8, 2011, pp Fldm, K., Jzouli, T., d Sdbor, P., A Mhodology for Drmiig h Rur o Ivsm Associd wih Progosics d Hlh Mgm, IEEE Trs. o Rlibiliy, Vol. 58, 2009, pp Rpor: O&M Spdig o Doubl o Nrly $6 Billio By 2025, Norh Amric Widpowr, July 10, 2012, hp:// 15 Byr, J., Hughs, K., Mriz, C., Hughs, M. A., Mlkwi, A., d Gugbro, G., A Urb Agd for h Nw Clim, UNFCCC COP-15, hp:// Hyrs, R. W., McGow, J. G., Mwll, K. L., d Syr, B. C., Codiio Moiorig d Progosis of Uiliy Scl Wid Rurbis, Ergy Mrils, Vol. 1 No. 3, 2006, pp Amric Isiu of Arouics d Asrouics