Sensitivities. A flowgate is a circuit or set of circuits that interconnect different regions of a network that can be limiting under some condition.

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Magnus Hodge
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1 Snstvts. Introducton Opraton of th Eastrn Intrconncton rls on usng th socalld Intrchang Dstruton Calculator (IDC []. hs s an ntrnt-accssd systm that ntrfacs wth OASIS and allows mart partcpants and ntwor oprators to ffcntly, ut approxmatly, dtrmn th chang n MW flow on a flowgat gvn a st of changs n MW us njctons. A flowgat s a crcut or st of crcuts that ntrconnct dffrnt rgons of a ntwor that can lmtng undr som condton. h IDC dos not rprsnt uss ut rathr rprsnts control aras, and thr ar aout of thm n th astrn ntrconncton. hrfor th flowgats mostly rprsnt ntrconnctons twn ths control aras; howvr, a flowgat may also ntrnal to a sngl control ara as wll. For purposs of th IDC, a control ara s a us, and th flowgats ar ntrconnctons twn th uss. On of th most mportant uss of th IDC s n th coordnaton of ransmsson Loadng Rlf (LR actons. LR procdurs ar n plac to gud oprators n mtgatng flows that xcd opratonal scurty lmts. LR lvls, summarzd n al [] hav n dfnd that corrspond to dffrnt typs of actons that may tan for whch curtalmnts must mad. Whn a LR lvl 5 s dclard, all ongong transactons ncludng thos wth frm transmsson srvc ar sujct to curtalmnt. What w dsr to otan, thn, s an xprsson for computng th chang n flow on a ranch n a ntwor for a gvn chang n MW us njcton.

2 Systm Scur Scurty Lmt Volaton Systm Scur Scurty Lmt Volaton Systm Scur LR Lvl al : Summary of LR Lvls [] RELIAILIY COORDIAOR Acton otfy RELIAILIY COORDIAORS of potntal OERAIG SECURIY LIMI volatons Hold IERCHAGE RASACIOS at currnt lvls to prvnt OERAIG SECURIY LIMI volatons a Rallocaton ransactons usng on-frm ont-to- ont ransmsson Srvc ar curtald to allow ransactons usng hghr prorty ont-to-ont ransmsson Srvc Curtal ransactons usng on-frm ont-to-ont ransmsson Srvc to mtgat Opratng Scurty Lmt Volaton Commnts Of thos transactons at or aov th CURAILME HRESHOLD, only thos undr xstng ransmsson Srvc rsrvatons wll allowd to contnu, and only to th lvl xstng at th tm of th hold. ransactons usng Frm ont-to-ont ransmsson Srvc ar not hld. S Scton.. Curtalmnt follows ransmsson Srvc prorts. Hghr prorty transactons ar nald to start y th REALLOCAIO procss. S Scton.. Curtalmnt follows ransmsson Srvc prorts. hr ar spcal consdratons for handlng ransactons usng Frm ont-to-ont ransmsson Srvc. S Scton.. Rconfgur transmsson systm to allow ransactons usng Frm ont-to-ont ransmsson Srvc to contnu 5a 5 Rallocaton ransactons usng Frm ont-to-ont ransmsson Srvc ar curtald (pro rata to allow nw ransactons usng Frm ont-to-ont ransmsson Srvc to gn (pro rata. Curtal ransactons usng Frm ont-to-ont ransmsson Srvc to mtgat Opratng Scurty Lmt Volaton hr may or may not an OERAIG SECURIY LIMI volaton. hr ar spcal consdratons for handlng ransactons usng Frm ont-to-ont ransmsson Srvc. S Scton.5. Attmpts to accommodat all ransactons usng Frm ont-to-ont ransmsson Srvc, though at a rducd ( pro rata lvl. ro forma tarff also rqurs curtalmnt / REALLOCAIO on pro rata ass wth twor Intgraton ransmsson Srvc and atv Load. S Scton.6. ro forma tarff rqurs curtalmnt on pro rata ass wth twor Intgraton ransmsson Srvc and atv Load. S Scton.7. 6 Emrgncy Acton Could nclud dmand-sd managmnt, rdspatch, voltag rductons, ntrruptl and frm load shddng. S Scton.8. LR Concludd Rstor transactons. S Scton.9.

3 LR Lv a 5a 5 6 Rs Crtra ransacton crtra RELIAILIY COORD IMMIECE Stat Acton Fors possl condton rsultng n volaton Expctd to approach, s approachng, SOL Expctd to approach s approachng, SOL Exstng or mmnnt SOL volaton or wll occur on lmnt rmoval Exstng or mmnnt SOL volaton At SOL, no furthr rconfg possl Exstng or mmnnt SOL volaton or on wll occur on lmnt rmoval, no furthr rconfg possl Exstng SOL volaton or on wll occur upon lmnt rmoval Scur Inscur or aout to Inscur or aout to Scur Inscur or aout to Inscur or aout to Som non-frm ptp at or aov curtalmnt thrs - holds, hghr prorty ptp rsrvaton approvd Som non-frm ptp at or aov thr curtalmnt thrsholds. All non-frm ptp at or aov curtalmnt thrsholds curtald; xacton rqust for prvously arrangd frm xmsson srvc. All non-frm ptp at or aov curtalmnt thrsholds curtald. otfy Hold Rallocat Hold and Curtal Hold and Rconfgur Rallocat Curtal Emrgncy Acton Commnts ot > mnuts for gong to hghr lvls so xactons may mad asd on prorty. Curtalmnts mad at top of hour. Hold on nonfrm; Curtalmnts mad mmdatly. Hold on nonfrm. Curtalmnts mad at top of upcomng hour. Curtalmnts mad mmdatly. Could nclud rdspatch, rconfguraton, voltag rductons, ntrruptl and frm load shddng.

4 . Calculaton of Gnraton Shft Factors h dsrd quantty for computng th chang n flow on a ranch for a gvn chang n gnraton s rfrrd to as th gnraton shft factor and wll dnotd y t,. It gvs th fracton of a chang n njcton at us that appars on ranch. h owr ransfr Dstruton Factor (DF s a gnralzaton of th gnraton shft factor. hs quantty s not only usful for th IDC, t s also usful n otanng fast (ut approxmat answrs for many othr dffrnt nds of plannng and opratng prolms. hs calculaton of gnraton shft factors s rlatvly straghtforward asd on what w hav don usng th DC powr flow modl. Rcall th DC powr flow quatons and th corrspondng matrx rlaton for flows across ranchs. ( ( D A ( Invrtng q ( ylds: ( Susttuton of ( nto ( ylds: ( D A ( As w hav dfnd n th nots on DC owrflow: s th vctor of ranch flows. It has dmnson of M x, whr M s th numr of ranchs. ranchs ar ordrd artrarly, ut whatvr ordr s chosn must also usd n D and A. D s an M x M matrx havng non-dagonal lmnts of zros; th dagonal lmnt n poston row, column contans th ngatv of th suscptanc of th th ranch. A s th M x (- nod-arc ncdnc matrx.

5 5 s th DC powr flow matrx of dmnson (-x(-, whr s th numr of uss n th ntwor, otand from th Y-us as follows:. Rplac dagonal lmnt wth th sum of th nondagonal lmnts n row. Altrnatvly, sutract (th shunt trm from, and multply y -.. Multply all off-dagonals y -.. Rmov row and column. s th vctor of nodal njctons for uss,, h calculaton of q. ( provds th flows on all crcuts gvn th njctons at all uss. us ths s not what w want. What w want s th chang n flow on all crcuts gvn a chang n njcton at on us. Hr s a chang n njcton vctor, : (5 h chang n crcut flows can thn xprssd as A D A D A D A D ( ( ( ( (6

6 6 ow lt th vctor all zros xcpt for th lmnt corrspondng to th th us, and assgn ths us an njcton chang of. (7 hn M M A D t t t t (,,,, (8 Quston: Dos th aov quaton mply that th njcton s changd at only on us? Explan. Dfnton: h gnraton shft factor t, s dfnd as olcy on Rallocat, t hs s dnotd as a l n W&W txt (s q... Exampl : W consdr an xampl usd n th DCowrFlowEquatons nots, llustratd low. Comput th gnraton shft factors for all ranchs corrspondng to an ncras n us njcton and a dcras n us njcton.

7 g =pu g =pu y =-j y =-j y =-j d =pu y =-j y =-j g =pu d =pu Fg. : Four-us ntwor usd n xampl t, - t, - t, - t, - t5, ot that th aov gnraton shft factors ar for a doul shft. 7

8 You can thn of t l ths. A gnraton shft factor for ranch, us would t, and anothr gnraton shft factor for ranch, us j would t,j. If w hav an njcton ncras at us of and an njcton ncras at us j of j (ngatv, thn t, t, jj (9 Incras, Dcras Dcras j, Incras hrfor, f =- j, thn t, t, j ( ot that onc s otand, thn t must addd to th orgnal flow on ranch to gt th rsultng total flow followng th gnraton shft,.., ˆ h last quaton s th sam as q. (. n W&W txt.. Gnraton Shft Factors wth Dstrutd Slac Equaton (8 shows how to comput th gnraton shft factors for th cas whn a sngl spcfd slac us corrsponds to us. Exampl aov shows how to comput th gnraton shft factors for th cas whn a sngl spcfd slac us corrsponds to som othr us n th ntwor (not th us corrspondng to th rfrnc y way of omsson from ts corrspondng row and column n th matrx. What w ar ntrstd n hr s computaton of gnraton shft factors for th cas whn w would l to dstrut th slac, or th compnsaton, throughout th ntwor. h y crtron to gud ths s that th lmnts n th nodal njcton vctor should corrspond to th prcntag of dsrd compnsaton for ach us. 8

9 9 hs crtron s llustratd low: us (,,,, M M c c c A D t t t t ( whr c c c ( s th allocaton dsrd for th rfrnc us. On way to dstrut th slac s to dstrut qually to all uss. In ths cas, c ( whr w us - n th dnomnator caus on us, us, s th us for whch th computaton s ng mad (and thrfor c =. If w us (, thn w can susttut nto ( to otan: ( c c c

10 Exampl : Usng th systm from Exampl aov, comput gnraton shft factors for all ranchs corrspondng to an ncras n us njcton, whn th slac s qually dstrutd to all uss. t - t t t t,all,all,all,all 5,all It s of ntrst to compar th answr from th xampl whr th slac was dstrutd ntrly to us and th xampl whr th slac was dstrutd to all uss...

11 t,.5 t,all.66 t.75, t,all.5 t,.65 t,all.999 t,.5 t,all.666 t5,.5 t. 5,all Clarly th assumpton on slac dstruton s mportant! hr ar othr ways to dstrut th slac. For xampl, w may dstrut th slac qually to all gnraton uss. Or w may dstrut th slac qually to all load uss. Or w may dstrut th slac to all gnraton uss n proporton to th MVA ratng of th gnraton that s locatd thr (ths approach conforms st to ralty, as w wll s whn w study AGC.. Gnraton Shft Factor Matrx Gvn a spcfd slac dstruton, w may comput a matrx of gnraton shft factors accordng to t t, t, t M, t, t t t,,,, t, t, t, t, tm tm tm tm,,,,

12 ( D A M M M ( ( ( c, c, c, ( c, c, c c c c c,,,,, c c c c c,,,,, c c c c c c,,,,,, c c c c c,,,,, h last matrx on th rght s calld th rallocaton matrx and s th matrx for all consdrd {njcton changs wth corrspondng rallocaton polcs}, whr an lmnt c, s th prcnt allocaton to us whn an njcton of. s mad at us. For xampl, frst column c,, =,, provds th rallocaton polcy at all uss =,, whn an njcton of. s mad at us. scond column c,, =,, provds th rallocaton polcy at all uss =,, whn an njcton of. s mad at us. last column c, provds th rallocaton polcy at all uss =,, whn an njcton of. s mad at us. h numr of columns n th rallocaton matrx s qual to th numr of consdrd {njcton changs wth corrspondng rallocaton polcs}. In th rallocaton matrx, th sum of all lmnts n a column (corrspondng to th njcton of. ng mad at us s th ngatv of th allocaton mad to us (a gnralzaton of q. (, as follows: c c,,...,,

13 h formulaton aov assums that w dsr gnraton shft factors for vry ranch (a row of and vry us (a column of. h frst column of s for a shft at us, whch s th us for whch a column and a row ar dltd from th matrx. Howvr, w nd not nclud vry ranch. hr may som ranchs that w now from xprnc wll nvr ovrload, or thr may polcy that rqurs a partcular applcaton to only montor crtan ranchs. h lattr s th cas for ERC s IDC dscrd at th gnnng of ths documnt. Exampl : Lt s comput th -matrx for Exampl. W assum a dstrutd slac us, whr, c, =-/. hrfor Rmmr: ach column s th st of shft factors for a unt ncras n njcton (gnraton at a crtan us. Column s whn th njcton at us s ncrasd (thr s no n that column caus that s th on corrspondng to th us that was dltd n th matrx. Column s whn th njcton at us s ncrasd, and so on...

14 h on-ln dagram s shown low to facltat undrstandng of th rlaton twn ncrasd njcton at us (whch dntfs a partcular column and how ranch flows ar affctd (th lmnts n that partcular column. g =pu g =pu 5 d =pu g =pu d =pu For xampl, th frst column ndcats that f w ncras us njcton y pu, w gt. flowng ovr ranchs,, 5, wth pu on ranchs,. On cauton: It s possl to otan a complt powr flow calculaton usng th shft factors. hat s t, * whr s th total njcton at us. In ths cas, howvr, th shft factors t,, dfnd accordng to It s ntrstng that w gt pu flowng ovr ranchs and. h rason s du to ntwor symmtry (all ranch mpdancs hav Z=j. pu. hs symmtry s most clarly undrstood y usng suprposton. gn y applyng. pu njcton at us, -. pu njcton at us, and comput th flows. hn apply. pu njcton at us, -. njcton at us, and comput th flows. hn apply. pu at us, -. pu njcton at us, and comput th flows. hn add th sts of flows for ach ranch, and on wll osrv th xact canclaton of th thr flows n ranch, and th thr flows on ranch.

15 t, Rallocaton olcy must computd accordng to a consstnt rallocaton polcy. hus, w should not comput (* aov usng th -matrx valus asd on th dstrutd slac. If w dd, for our us systm: For column, "us " = us. It mans that a unt chang at us gts compnsatd y a -/ unt chang at uss,,. For column, "us " = us. It mans that a unt chang at us gts compnsatd y a -/ unt chang at uss,,. And so on. hs wll rsult n a alancd dspatch (powr alanc wll satsfd, ut t wll a dffrnt dspatch than what was ntndd. As a smpl xampl, try a thr us systm havng njctons -6,,, at uss,, and, rspctvly. Wth "c"=.5, thn w wll gt th followng dstruton: us us us Each row hr shows njctd us n old and th allocaton to othr uss asd on c= hs last row shows total njcton at ach us from th calculatons,.., th rsultng us dstruton from applyng th "dst slac" as, whch dos not match wth th ntndd dstruton of -6,,. hus, you wll gt a dffrnt st of flows. 5

16 So, for ths xampl, n computng th "" matrx, th "c" matrx (that post multpls DA - for our us systm should appar as whch mans usng us # as th slac n ach cas, that s, Effcnt computaton of GSFs In th prvous dscusson, t was assumd that w would al to comput ( -,.., that th numr of nods would not too larg, whch can th cas undr som approxmatons such as thos mad y th IDC []. Howvr, t s also common that ths s not th cas,.., that w may want to otan GSFs for a systm whr th numr of nods s vry larg. In such a cas, on can otan th GSFs wthout matrx nvrson ut only for on shft at a tm, va ( ( D A (5 Equaton ( s solvd for θ va LU factorzaton for a gvn, and thn th rsultng θ s usd n (5 to otan th ln flow shfts n. Exampl : Rpat xampl, whch s to otan th GSF for all ranchs corrspondng to an ncras n us njcton and a dcras n us njcton. 6

17 7 Usng LU factorzaton: U L L L ow us acwards/forwards susttuton to otan θ, rsultng n ( /6 * (. 5 / *.5 ( U w w w w w w Lw

18 Ux w.5 x.5 x x x. x x.. * ( * ( And ths gvs our angl changs as ow w can us q. (5 to otan ( D A whch s n agrmnt wth th rsult of xampl Ln outag dstruton factors h ln outag dstruton factor (LODF s drvd n your txt (Appndx A, pp.-. h LODFs ar lnar stmats of th rato: chang n flow on crcut l du to outag of crcut, dnotd y Δf l, to pr-contngncy flow on crcut, dnotd y f. 8

19 In othr words, t provds th fracton of pr-contngncy flow on crcut that appars on crcut l followng outag of crcut, and s gvn y d l, = Δf l / f (6 It s thn clar that th chang n flow on crcut l du to th outag of crcut s gvn y Δf l = d l, f (7 h drvaton n th txt s lngthy; w wll not go through t hr. o undrstand th rsult, w dfn a matrx X such that X ( (8 hs mans that X (9 hn w dfn anothr matrx X such that t s th sam as X xcpt w appnd anothr row at th top and anothr column to th lft, corrspondng to th rfrnc us (assumd us # njcton and angl, as shown low: X X ( h ln outag dstruton factor d l, = Δf l / f corrspondng to th addtonal flow on ranch from outag of ranch l s thn gvn y 9

20 d x X n X jn X m X jm x, x X nn X mm X nm ( In (, x and x l ar th ractancs of outag ranch and rmanng ranch l, rspctvly; m and n ar us numrs trmnatng ranch ; and j ar us numrs trmnatng ranch l. hrfor, X n s th lmnt of X n row, column n. X jn s th lmnt of X n row j, column n. X m s th lmnt of X n row, column m. X jm s th lmnt of X n row j, column m. X nm s th lmnt of X n row n, column m. X nn s th lmnt of X n row n, column n. X mm s th lmnt of X n row m, column m. 7. A computatonally ffcnt mthod to otan LODFs A sgnfcant prolm wth W&W s mthod of otanng th LODFs s that t rqurs X=( -, and f th systm s vry larg, thn nvrtng th matrx can a computatonally ntns prolm. W provd anothr mthod n ths scton. Our tratmnt s adaptd from []. Lt s rconsdr our famlar -us, 5-ranch xampl prolm.

21 g =pu g =pu 5 d =pu g =pu d =pu h matrx for ths systm s What happns to f w los th crcut # (from us to us? W could r-dvlop th nw from th on-ln dagram as w ar accustomd to dong now. Anothr way s to dscrn how th crcut # affcts th matrx, n that t wll affct xactly lmnts, as ndcatd wth th undrlns low, corrspondng to lmnts n us numrd postons (,, (,, (,, and (,. Rcallng that all ranch admttancs of our ntwor ar j, what would ths four lmnts f ranch # (twn uss and wr not thr?

22 out What s th dffrnc twn and out? out otc that th lmnts n ar all multpls of =-,.., otc that th aov matrx can xprssd as From ths smpl llustraton, w can s a gnralzaton, that whnvr w rmov a ranch twn uss and j, wth corrspondng matrx lmnt, th matrx wll chang as ndcatd low. j j out ( Inconsstncy: In Scton 6., w usd (,j to ndcat trmnals of th crcut to loadd (l and (m,n to ndcat trmnals of th crcut to outagd (. In th dvlopmnt of ths scton, th nomnclatur on trmnal numr has n rvrsd,.., (,j coms th trmnals of th crcut to outagd ( and (m,n coms th trmnals of th crcut to loadd (l.

23 whr s th suscptanc of ranch -j. W us nstad of n ordr to nsur w hav a dfnd trm vn whn or j ar th swng us. otc that wll always ngatv. h prvous rlaton may xprssd as j j ( If w dfn j ( thn ( coms (5 Cauton: h dsgnatd postons n th row and column vctors corrspond to uss and j,.., thy ar not th th and j th postons.

24 Spcal cas: If th ranch to outagd s connctd to th swng us (n our cas, t s us #, thn, f =, f j=, j j From (, and usng (5, w hav that out (6 hrfor th post-contngncy matrx can xprssd as out (7 From (, w rcall th DC powr flow rlaton as ( If, whn w rmov th ranch connctd twn uss and j, th angls chang y θ, thn th nw (post-contngncy angls wll θ+ θ, and ( coms out ( (8 Susttutng (7 nto (8, w otan ( (9 W can solv for th nw angls accordng to (

25 W do not sm to hav mad much progrss, caus w stll hav to ta an nvrs Howvr, thr s a sgnfcant nft to wrtng th nw matrx n th way that w hav wrttn t, and that nft coms apparnt f w larn a crtan matrx rlaton. hs rlaton s gnrally rfrrd to as a lmma. Matrx Invrson Lmma (MIL: Assum s a nonsngular n n matrx, and lt c and d n M matrcs wth M<n. hn: ( M cd c I d c d whr I (M s th M M dntty matrx. W nglct th proof ut mnton that t s provd n [, p. ] y smply multplyng th rght-hand-sd of MIL y th xprsson nsd th racts of th lft-hand-sd, and showng that th product s th n n dntty matrx. W also mnton that MIL s drvd n [, pp. 8-]. It may not vry ovous at ths pont that MIL wll hlp us, snc w s dffrnt nvrss on th rght-hand-sd of MIL. Lt s apply MIL to th nvrtd trm of ( to s what happns. Osrvng that w can dfn d w can apply MIL accordng to c ( 5

26 ( M I ( On of th nvrss on th rght-hand-sd can addrssd rght away, howvr, y dntfyng th dmnsonalty of th xprsson nsd th rght-hand-sd racts, [I (M +d - c]. Osrvng from th MIL that M s th numr of columns n c and d, and notng from ( that n our cas, c and d hav only M= column, w s that what s nsd th rght-hand-sd racts s a scalar quantty! So that nvrs w can ta, and accordngly, w can xprss ( as: ( ullng out th scalar multplr from whr t appars n oth th numrator and dnomnator, w hav ( ow w can solat to only on apparanc n th xprsson y dvdng top and ottom y t, rsultng n: (5 What w hav just xprssd n (5 s th nvrtd trm on th rght-hand-sd of (, rpatd low for convnnc: ( Susttutng (5 nto (, w otan: 6

27 7 (6 Dstrutng th njcton vctor rsults n (7 ut θ= -, and thrfor w can rplac th corrspondng xprssons n oth rght-hand-sd trms to otan: (8 W can smplfy a lttl mor y nvstgatng θ n th numrator. hs would : j n j j (9 Susttutng (9 nto (8 rsults n:

28 ( j ( ow w hav only two nvrss lft. Intrstngly, thy oth prmultply. hat s, w osrv that oth nvrss appar n -, an n vctor. Quston: sds nvrtng -, how mght w valuat ths trm? Advc: Whn you don t now how to valuat somthng, just nam t. hn, f thngs don t gt ttr rght away, you can at last mov on wth a sort of ndcator of whr your prolm ls. So lt s nam ths n vctor as g,.., g ( ot sur f that hlps much ut t dos ndcat that g ( Equaton ( should stmulat a vry good da wthn your mnd. Snc w vry wll now and, w can otan g through LU factorzaton. Dong so wll gv us vrythng w nd to valuat (, whch, whn w susttut g for -, coms: ( j g g ( On last small chang should mad to (, and that s to rcognz that th trm n th dnomnator g can xprssd as 8

29 g hrfor, ( coms j ( g g g g g ( j g j n j g g g j ( (5 Inconsstncy: Rcall th not on p., whch ndcats an nconsstncy n nomnclatur wth Sc 6.. ow what s th LODF? Rcall th dfnton of th LODF s d l, = Δf l / f (6 whr w rcall that dsgnats th outagd crcut, trmnatd y uss and j; l dsgnats th crcut for whch w want to comput th nw flow, trmnatd y uss m and n. Frst, lt s xprss th dnomnator of (6 f, whch s f ( j (6 ow lt s xprss th numrator of (6 Δf l, whch s f l mn( m n mnmn (7 ut not that θ n (7 can xprssd usng th scond trm of (5,.., 9

30 ( g Susttutng (8 nto (7 rsults n ( j fl mnmn g ( g g j It s hlpful at ths pont to rarrang (9 accordng to j g j g ( ( (8 (9 j fl mn mng ( g g j (5 W rcognz n (5 that f ( j (5 and mng gm gn (5 Susttutng (5 and (5 nto (5 rsults n f( gm gn f l mn ( g g j (5 So (5 can usd to otan th chang n flow on crcut l (trmnatd y uss m and n du to outag of crcut (trmnatd y uss and j. o gt th LODF, w dvd (5 y f, rsultng n

31 d f f l ( g m l, mn ( g g j (5 h approach, thn, to usng (5, s to factorz nto th L and U factors onc. hn, for ach contngncy =,, C, (pr (, w us forward and acwards sustaton to otan th vctor g. h LODFs for vry ranch l (trmnatd y uss m and n, ar thn computd from (5. Exampl 5: Consdr our -us, 5-ranch xampl prolm agan. Comput g for a ln - outag. hn us t to comput th post-contngncy flow on crcut -. Soluton: Rcall (: g whr s gvn y: and s gvn y j And so our quaton s: g g g rformng LU dcomposton, w otan g n

32 L 5 6 U.5. otc that th aov factors nd computd only on tm; thy may susquntly appld to otan th g-vctor for outag of any crcut. In ths cas, w ar ntrstd n outag of th ln from us to us, thrfor w wrt L w U g w 5.5 w w.5 w w. 6 w.5 w g.5 g.75. g. g.5 g.5.5 g hn w can comput th LODF for th crcut - aftr outag of crcut -: d l, f f ( g.5.75 l ( g mn ( gm ( g g g. gn g j (.5.5 (.75.5 If crcut - has flow of.5, thn th chang n flow on crcut -, followng outag of crcut - coms f l d l, f.*.5.8 If th pr-contngncy flow on crcut - was.5, thn ow lt s chc t wth th DC powr flow.

33 y =-j y =-j y =-j y =-j y =-j g=pu d=pu d=pu g=pu g=pu 5 g=pu d=pu d=pu g=pu g=pu Wth all lns n w otan A D ( Osrv that th flow on crcut - s.5 pu. Wth crcut - out, w otan:

34 Also us W&W s mthod. Exampl 6: For outag of ranch connctd to swng. Do t thr ways as n Exampl 6: a. Us th aov mthod. Us DC flow wth and wthout outag; c. Us W&W s mthod. Rfrncs: [] J. Mdna, Intrchang Dstruton Calculator (IDC for ransmsson Congston Managmnt: Implmntaton and Challngs, prsntaton at th SERC IA Mtng, May 6, 8, avalal at 6am/prsntat/mdna_oat_dc_psrc_a_may8.pdf.

35 [] orth Amrcan Elctrc Rlalty Councl (ERC Opratng Manual, Appndx 9C, May,, avalal at [] A. Ds, Modrn owr Systms Control and Opraton, Kluwr, 988. [] A. Montcll, Stat stmaton n lctrc powr systms, a gnralzd approach, Kluwr,

External Equivalent. EE 521 Analysis of Power Systems. Chen-Ching Liu, Boeing Distinguished Professor Washington State University

External Equivalent. EE 521 Analysis of Power Systems. Chen-Ching Liu, Boeing Distinguished Professor Washington State University xtrnal quvalnt 5 Analyss of Powr Systms Chn-Chng Lu, ong Dstngushd Profssor Washngton Stat Unvrsty XTRNAL UALNT ach powr systm (ara) s part of an ntrconnctd systm. Montorng dvcs ar nstalld and data ar