ANALYSIS APPROACHES OF SSO BASED ON LDAE MODEL

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1 ANAYI APPROAE OF O AED ON DAE MODE hg Yu, Yx N, or Mbr, IEEE Th Uvrsty o og Kog og Kog, h [email protected] Abstrct A lrz rtl lgbrc qutos DAE) ol or both g-lyss colx torqu coct T) clculto o owr syst subsychroous oscllto O) s roos ths r. Th DAE ol or O stuy s rv t rst. rlz g-lyss T clculto bs o th DAE ol r rst thrtr. Th ctvss o th roos DAE ol bs O lyss rochs r vr through coutr tst rsults. Th DAEol-bs O stuy roch suggst ths r vs th wy or O stuy wth VD trssso /or FAT vcs wthout ltg thr lgbrc vrbls, whch c obt or vlubl orto. Kywors: O, olx torqu coct, rlz gvlu, DAE INTRODUTION ubsychroous Oscllto O) s colx stblty ssu owr systs thr hv b lot o rsrch oututs o O rct cs []. Th rlry stus o O cocr ly bout ucto grtor ct uc by srs costo cctor. I rly 97s, srs costo stult lctrochcl torsol trcto, whch cus svr torsol osclltos g o turb-grtor shts, ttrct grt ttto rsrch work. Ths l to w ro o O stuy. I lt 97 s, VD trssso syst cus O ws rst obsrv brought bout -th stus o vc-t O. Th lcto o FAT tchology hs root th stuy o vc-t O v or sgctly [2~4]. I orr to stuy O, vrous thos r us [5, 6]. Aog th t sulto s th ost rct wy, whch EMT ols o owr syst lts wth ult-ss srg sht cts st-by-st urcl tgrto thos r us to gt t-voluto o vrous vrbls O. A stct vtg o t sulto tho s tht t llows tl olg o owr syst lts wth syst olrty clu. Thror t s vry usul or th stuy o trst torqu lyg ct th xto o O g cotrol rorc ur lrg sturbcs. owvr, usg EMT ols or O stuy s t cosug or ts xtrly sll t st sclly wh thr r VD trssso /or FAT vcs th stu owr syst. ss, t sulto s oor hyscl trsrcy cult to rvl O rcls. ll sgl stblty stuy s othr tho wly us or O stuy, whr th owr syst ol s lrz rou th syst orto ot covrt to str or o x& A x A s th syst stt-sc trx). Eg-lyss s th couct to gt -sght o syst O chrctrstcs. Eg-lyss s ccurt tho whch c x th cts o syst cotrol rtrs o g th torsol ol oscllto. Wth th hl o tur lr cotrol thors, vc O g cotrol c b rlz. owvr, rl owr syst wth VD trssso /or FAT vcs, t s t-cosug rvg syst trx A wth ll th lgbrc vrbls lt. Morovr, tr ructo o lgbrc vrbls, t y b covt to couct crt gvlu sstvty lyss bs o A. Wh th stu owr syst s rthr colct wth hghr orrs, both t sulto g-lyss thos y hv cults lyss wthout syst ructo. I rct yrs rqucy-scg bs thos bco couttolly ttrctv bcus thy c rov syst g orto ovr th tr subsychroous rqucy b [7], lthough thy ght ot b so rcs s g-lyss tho. Aog th th colx torqu coct T) tho or sy torqu r ut s tho) s ost sutbl to lrg-scl owr syst O lyss wth th cblty o showg clrly th syst lctrc g cts ovr O rqucy b. I ths r, lrz rtl lgbrc qutos DAE) ol or both g-lyss T clculto o O s roos. I th DAE roch, th DAEs o vul owr syst coots r rv rst, whch ct s bulg blocks o ovrll syst ol. Th syst ol c th b stblsh quckly sly ccorg to th twork toology. Ths tho s vry sutbl or O stuy o owr systs wth VD trssso /or FAT vcs wthout th o ltg thr lgbrc vrbls. rlz g-lyss tchqu [8] c b rctly l to th DAE ol. Th rsults c rov usul orto o obsrvblty cotrollblty o lgbrc vrbls o vul O os, whch s vry usul cotrol sg. Furthror, th rqucy-scg T lyss tho c b sly rlz bs o th DAE ol. Th stblty o torsol os bs o thr g torqu cocts s lso scuss. Th ctvss o th roos DAE ol th rlvt lyss rochs s vr wth coutr tst rsults. Th DAE-bs O stuy ol ts corrsog lyss rochs v th wy to clu VD trssso 5th P, g, August 25 sso 34, Pr 4, Pg

2 FAT vc ols O stuy gt sght bout thr cts o O. Th r s rrg s ollows. Th DAE ol or O s rv scto 2. Th DAE-olbs grlz g-lyss or O s rst scto 3. Th T clculto tho bs o th DAE ol s scrb scto 4. Flly coutr tst rsults r gv scto 5 wth coclusos rw scto 6. 2 DAE MODE FOR O TUDY Powr systs c b xrss by st o olr rtl lgbrc qutos DAE): x& x, y) gx, y) whr x: th vctor o stt vrbls; y: th vctor o lgbrc vrbls. rzg th bov qutos rou th orto ot, w hv: & x A x y ) x D y g g whr A,,, D clcult t x y x y th qulbru ot; : ortor to rrst rturb vlus. Fro ), w c lt ll lgbrc vrbls y D - xsts, whch ls to lrz orry rtl qutos ODE) ol stt sc: x& A x 2) whr syst trx A A - D -. Equto 2) s wly us or covtol g-lyss s to bov. owvr, to lt ll th lgbrc vrbls rl owr systs wth VD trssso or/ FAT vcs ght b t cosug. Morovr th t o D - ks A ubl to b xrss by syst rtrs xlctly, whch, ur crt crcustcs, crss th culty gvlu sstvty lyss. I ths r, th syst subtrcs A,,, D wll b kt, g-lyss or O stuy wll b bs o th syst DAE ol ) rctly. Powr syst coot ols or O stuy c b ou [9]. Atr lrzto o coot ols t orto ot, w c gt DAE ols wth lgbrc vrbls rsrv or vrous coots s ollows. Tk th turb sht syst o sychroous ch s xl. uos th sht o turb grtor c b xrss s sx-ss srg syst, ts DAE ol s : stt vrbls; Y: lgbrc vrbls): & A TYT 3) whr 6, 6 ) T : gl s crts o th sx-ss sht; Y T T : lctrogtc torqu o grtor ss For slcty, th s govror s ycs s glct), whch c b xrss s ucto o grtor wg currts lux lkgs -q coorts DAE ort): T DTY DTYT 4) whr ψ, ψ q, ψ, ψ D, ψ g, ψ Q ) T : grtor wg lux lkgs -q coorts ssug grtor hs sx wgs, Y wth Y, q,, D, g, Q ) T : grtor wg currts -q coorts. To k Y T 4) xlctly syst ol s goo or utur T clculto ovr subsychroous rqucy rg through rqucy-scg. lrly, w c obt DAEs or lctrogtc crcut o chs, xctto systs, trsorrs, trssso ls, los, shut cctors, tc. I th grtor DAE ol, w rsrv th lgbrc vrbls o grtor trl voltgs currts -q coorts Y u & Y ), thr coo vrbls x-y sychroous coorts Y xy) u & Y xy) ). c, th lgbrc qutos or coorts trsorto btw -q x-y c b rctly clu syst ol s show 5): s cos x q q y 5) cos s whr c b u or to ot grtor trl voltg or currt rsctvly; : rotor gl o grtorss whch s th lg gl o ts q-xs w.r.t. th sychroous coort x-xs. Kg 5) DAE ol s bcl to st trc o grtor wth c twork. Equto 5) or grtor trl voltgs jcto currts coorts trsorto c b rwrtt to: xy) xy) DY D Y 6) xy) xy) u DuuYu Duu Yu A th ccorg to th twork toology, th busl ct trcs c b obt. Assug NA, NT, N, D N D N r th ct trcs o twork buss w.r.t. trssso ls, trsorr brchs, lo brchs, grtor brchs, shut cctor brchs rsctvly, bus jcto currt blc qutos bs o K wll b: NA A NT T N 7) xy ) ) D NY D N Y whr A : trssso l currt; T : trsorr brch currt; : lo brch currt; Y xy) : grtor trl currt; Y ) : shut cctor brch currt. Atr cobg ll th vul DAEs togthr, th tr syst DAE ol or O stuy tks th or: & A 8) D Y whr th lgbrc qutos clu th grtor lctrogtc torqu coutto qutos s 4)); th grtor wg lux lkg qutos; th coorts trsorto qutos or grtor currts voltgs s 5) 6)); bus jcto currt blc qutos s 7)). A br scusso o 8) s s ollows. 5th P, g, August 25 sso 34, Pr 4, Pg 2

3 ) Mtrcs A,,, D r xtrly srs. I A D, th o-zro block-trcs r ly o th gol ost o th r lso vry srs. c srs trx tchqus c b us to couct glyss. Th coutto work c crs otcbly, whch s sgct to O stuy o rl owr systs. ) Th DAE orulto roch brks ow th olg ssu to two rts, o s lt-olg rt othr s twork toology tgrto rt. Thror, w wt to stuy th cts o VD trssso /or FAT vcs o O, w c just sly thr vul DAE ols to 8); th oy th corrsog bus jcto currt blc quto 7) to clu thr jcto currt cts). Th rocss s sl rct. ) Th vc grlz g-lyss tho c b rctly l to th DAE ol. Th corrsog grlz gvctors c rov vry usul orto o obsrvblty cotrollblty o both stt lgbrc vrbls w.r.t crt O o. Ths s sgct tur o th suggst tho, whch c rov sght to syst O chrctrstcs s bcl to cotrollr sg or O g s scto 3 or tls). v) Th suggst DAE ol orulto c vo scrcg th hyscl structur trsrcy u to th lto o lgbrc vrbls trtol ODE olg. c syst rtrs c b show xlctly th DAE ol, t s covt to k gvlu sstvty lyss w.r.t syst rtr rturbto. v) Kg th rtl ortor /t th DAE ol, th sttg jω Ω s O rqucy b), w c quckly cout th T T / t rqucy Ω. o rqucy-scg lyss c b sly ror bs o th DAE ol s scto 4 or tls). 3 ENERAIZED EIEN-ANAYI AED ON TE DAE MODE For th trtol g-lyss o 8), w usully lt lgbrc vrbls Y gt: & A 9) whr A ) A - D - ssug D - xsts. lrly, A ght ot b srs. Its hyscl trsrcy y b lost u to th lto o lgbrc vrbls. Egvlu rght & lt gvctor coutto bs o 9) s gv ): tλi A ) ) T λ A, λ A whr I s tty trx; r th rght lt gvctors wth rsct to th th gvlu λ ) o A. For DAE ol 8), w shll couct grlz gvlu gvctor clculto usg grlz chur coosto tho [8]. For th suggst DAE ol 8), w : A T ; A ~ Y : xt syst trx; D I ~ whr r th ubr o th stt vrbls th ubr o th lgbrc vrbls Y), w c rwrt 8) to: ~ ~ T& AT ) Th soluto o quto ) s quvlt to solv th grlz gvlu robl: ~ ~ λ T AT 2) I ~ s osgulr, w c covrt th robl to ~ ~ str g-robl A) T λt to solv th grlz gvlus o 2). owvr, or our cs, ~ ~ s sgulr os ot xst. A vc gvlu coutto tchqu,.. grlz chur coosto tho [8], s ot, through whch th vrs clculto o ~ c b vo th orgl syst gvlus c b clcult ctly. c ths s l tho to solv th scl grlz g-robl wth rk-ct ~. Accorg to grlz chur coosto or r o colx squr trcs ~, A ~ ) o s so l l), t s rov tht thr xsts r o utry trcs Q l l) Z l l) Q Q I, Z Z I ) such tht: ~ QZ ˆ ~, QAZ Aˆ 3) whr both Ĥ Â r ur trgulr trcs. t-ultlyg 2) by Q w hv: Q ~ ~ λ Z Z) T QA Z Z) T 4) Accorg to 3), quto 4) wll b T ZT ): ˆ ˆ λ T AT 5) Equto 5) s quvlt to solv hoogous lr qutos λ ˆ Aˆ ) T. Th suct cssry coto or th xstc o o-zro T s: t λ ˆ Aˆ ) 6) c both Ĥ Â r ur trgulr trcs, 6) c b wrtt s: l t λ ˆ Aˆ ) λˆ Aˆ ) whr Ĥ Â : th th gol lt o Ĥ Â rsctvly. Thror λ c b solv sly s: λ ˆ / ˆ ˆ A ) 7) c ~ s rk ct or our cs, w c ssu ~ th ort o: I ~ ε whr ε g ε ε ε ); ε >,, : tsl t costt tur. 5th P, g, August 25 sso 34, Pr 4, Pg 3

4 Th tr chur coosto, thr wll b so gol lts Ĥ o Ĥ wth xtrly sll vlus, whch corrso to ε ~ l to t gvlu λ 7). Ths c sly b t s cttous gvlus sur. Atr tru gvlu λ,,) clculto, ts corrsog grlz rght lt) gvctor ~ ~ ) c b clcult by: ~~ ~~ ~ ~ ~ T ~ λ A, λ A 8) whr ~ ~,, ) s ) ). It s sy to rov: -D -, - D - ) T It s wll kow tht, ) rovs th orto o obsrvblty cotrollblty o stt vrbls o o λ. lrly, ) rovs th slr orto o lgbrc vrbls Y o o λ. Th orto s vry usul or sgg cotrollrs or O g. rlz chur coosto c b us to ly th colt g-lyss bs o th DAE ol rctly. owvr, or vry lrg owr systs, th colt g-lyss s cult to lt wthout syst ructo. Altrtvly, by tkg vtgs o owr syst srsty, so trtv thos [] r us to cout th syst crtcl gvlus quckly lso bs o th DAE ol. Ths thos srv th uthor to stuy urthror o how to solv th torsol os lrg owr systs. c th DAE ol rsrvs ll lgbrc vrbls syst structur, t s lso vry covt to couct gvlu sstvty []. 4 OMPE TORQUE OEFFIIENT AUATION AED ON DAE MODE Th rqucy-scg T clculto s wly us or O stuy, whch c rov T orto ovr th tr subsychroous rqucy rg tct ottl O rsk. Th T clculto orul c b rv ro th suggst DAE Mol, whch s gv blow. 4. Prcl o T lculto s o DAE Mol Fro O thory [9], w kow tht or crt ch hr t s th stu ch), ts srg-ss sht ol c b xrss s torsol-ocoul ol s gur ) wh chcl g s glct. Wh rorly slct trx s us to covrt th ult-ss sht ol to olsc wth govror ycs gor T ), ch ol-sc ss xrcs coo lctrogtc torqu T o th grtor th grtor ss rotor gl hs th rlto: N N: totl ubr o sss o th sht; surscrt : ol sc vrbl),.. s gur ): M 2 K N T ~ N), 9) whr M : ol rtl costt sc) o th -th sht o; K : ol srg costt u torqu/r) o th -th sht o; T : u; : th syst rqucy bs r/sc; rottg ss gl or ): r; /t wth t sc. It s clr ro 9) gur tht oly T o th stu ch wll trc th sht ol ol-sc to th rg syst, th lttr c b s s IO syst wth ut, outut T rt ovrll trsr ucto o ). s o bov lyss gur, T o th stu ch c b work out sly bs o th suggst DAE ol. It shoul b otc tht th trsr ucto o ) hs clu th cts o othr ch sht ycs, ll ch lctrc crcut trsts wth xctr cts clu) othr lts twork trsts ccurtly. T Th stu chs sht qtos M T K T M ) K T T N M N K N N Othr chs sht qtos All chs lctrc crcut uqtos clug th stu ch) All othr lts twork qutos N Th rg syst w.r.t th stu sht Fgur : T clculto bs o DAE ol 4.2 Procur o T lculto Th rocur o clcultg th lctrcl T o th stu ch bs o DAE ol s s ollows. ) Tkg o th srg-ss sht qutos 3) o th stu ch ro 8), w c gt k th rtl ortor /t, t sc): A A D Y 2) whr :subst-vctor o stt vrbls xclusv o th stu ch. 5th P, g, August 25 sso 34, Pr 4, Pg 4

5 r).u.) r th gl s crts o grtor-ss o th stu ch, whch ct s th ut sgl to th rg syst s to bov. It shoul b otc tht lctrogtc toqu lgbrc quto 4) s kt th syst ol, whch s clu 2). ) shoul stsy th rlto whr s th syst rqucy bs r/sc), hc w c sly 2) urthr obt: A A D Y ) Rrrg bov quto to: A I A 2) D Y Thus w c cosr tht Y r lct lr uctos o clug ortor ot th to: K, Y K Y 22) v) t jω 2) whr Ω s crt rqucy o trst r/sc wth subsychroous rqucy b). I w hv ov th lt Y T corrsog to th lctrogtc torqu T o th stu ch) rlvt quto to th lst quto 2), t s sy to solv or t s s ucto through oly orwr uss lto o 2), th obt colx coct 23) s th rqur T. T Ω K Ω) j D Ω) 23) jω whr K : lctrcl sychrozg torqu coct; D : lctrcl g torqu coct. rs trx tchqu c b l T clculto. Th lctrcl g torqu coct D vrs wth syst toology, rtr th orto ot. For colct owr systs wth VD trssso /or FAT vcs, th vlu o D clus thr cts s wll. 4.3 tblty Jugt or Torsol Mo s o scto 4. 9), w ssu tht th chcl g o th -th ol D s lry kow sy tr ro crt tsts or so othr thos), w c k stblty jugt or th -th torsol o o rqucy Ω bs o D clcult D Ω) ccorg to th ollowg ruls: I D D, th torsol o s crtcl. I D D >, th torsol o s stbl. I D D <, th torsol o s ustbl. I ths r, th tho suggst [2] s ot to clcult ol chcl g torqu cocts roxtly. Th gvlus o th stu srg-ss sht lo r clcult rst.. gvlus or quto 3) wth Y T ). Th ol chcl g coct c b clcult by: D -2σ M, whr σ s Rλ ) o th -th o λ. W lso ssu tht ) tr coctg to th syst, th rl torsol oscllto rqucy Ω) s clos to ts turl rqucy, whch s kow ror to T clculto; ) th ol trcto og rt torsol os o th stu ch c b glct so tht ch t w c stuy oly o rtculr torsol o. o rrks r rst s ollows. ) I w ssu th torsol rqucy s uchg tr coctg th torsol sht syst wth th rg syst, tht s w c sly us D clcult ro 23)) t Ω to jug th stblty o th -th torsol o. Th rt o vrgc or cy) o th torsol oscllto c b roxtly wgh by th vlu o D D Ω))/2M ) whr M s th ol rt costt). ) owvr, th rqucy o th torsol o wll ctully vt ro ts turl rqucy to sll xtt wth cts o rl K D. c, wh stuyg th -th torsol o, w c ssu th xu rqucy sht s ζ r/sc. Th w c chck th vlu o D D wth th rqucy b Ω ζ, ζ ), D D s lwys gtv ostv), th torsol o s ustbl stbl). Whrs D D chgs ts olrty or th oc wth th stu rqucy b, w c ot bsolutly cor th stblty o th torsol o. ) Th syst s ssu to b stbl coost g torqus D D ) wth th vcty o ll sht torsol rqucs r ostv. v) Although T lyss tho s ot s ccurt s g-lyss or t sulto, w ust t tht t s stll ctv wy to rct th stblty o th torsol os. Esclly t c b l O stuy or vry lrg-scl owr systs wth VD or/ FAT wh g-vlu t sulto r cult to ly wthout syst ructo. 5 OMPUTER TUDY REUT A two-ch-t-bus owr syst s us or tst s gur 2). Th syst t s gv x. Th sht o s tk ro th IEEE FM th sht o 2 hs our sss. Th ol-sc sht rtrs r lst tbl bor coctg grtors to th syst. Ur th coto o srs cctor costg 8% o th l rctc 2 r coct to th syst wth th owr outut o 6MW 52MW rsctvly, coutr tsts r couct. 2 R T T T 2 R T 2 R Fgur 2: Two-ch-t-bus owr syst o-l gr Y R Y 5th P, g, August 25 sso 34, Pr 4, Pg 5

6 Torsol Mo M K D u) r/sc) z) sc) u/r) Tbl : Mol qutts o sht systs o & 2 rlz g-lyss bs o DAE ol s ror tr th grtors coct to th syst wth tst rsults gv tbl 2. Fro tbl 2 w c s tht th torsol os λ λ 3 o r ustbl, th torsol o λ 2 o 2 s ustbl s wll. I to, wh corg torsol o oscllto rqucs show tbl ) tbl 2 Iλ )), w c s tht tr coctg to th syst th sht torsol rqucy sht ro thr orgl turl oscllto rqucs s vry sll. Torsol Mo Rλ )σ /sc) Iλ ) r/sc) /2π z) λ λ λ λ λ λ λ λ Tbl 2: Eg-lyss rsults At th s orto ot, th roos DAEol-bs T clculto s lso couct cor wth g-lyss rsults. Th clcult K D o & 2 w.r.t scg rqucy Ω r show gurs 3 ) & b) rsctvly oly th rqucy rg o 9~9 r/sc s lott sc th g torqu coct D rs to b clos to ovr th rgs o ~9 9~ r/sc). Th thr vrtcl sh-ot ls ro lt to rght gur 3 ) corrso to th turl rqucs o torsol os, 2 & 3 o rsctvly, th o vrtcl shot l gur 3 b) corrsos to th turl rqucy o torsol o o 2. Th torsol stblty o wll b scuss rst. It c b obsrv ro gur 3 ) tht th vcts o torsol os, 2 & 3 o, th syst xhbts ll gtv lctrc g D < ), whch ssus tht th torsol os, 2 & 3 o r ottl cts o ustbl os. owvr, th cocluso hs to b tr corg D Ω) wth th ol-sc chcl g torqu coct D. As to th torsol stblty o 2, t c b s ro gur 3 b) th vcts o torsol os o 2, th syst xhbts clrly gtv lctrc g D. c, torsol o o 2 s th ottl ct o ustbl o. T.u.) D K Ω r/sc T.u.) ) T o Fgur 3: T clculto rsults Ω r/sc b) T o 2 It shoul b otc tht wh clcultg th T o, both K D wll hv rstc chg t th torsol oscllto rqucs o 2 vc vrs. I th torsol oscllto rqucs o 2 r ot clos to ch othr, t s sbl to jug o ch s torsol stblty ccorg to ts T clculto rsults. owvr, wh y torsol oscllto rqucs o 2 r vry clos to ch othr, t wll b cult to jug th stblty o corrsog o oly through T clculto. D u) Torsol Mo M s s D u) Ω bs D D 2M /sc) Tbl 3: Nurcl rsults o T orto Th coost cts o D D Ω) r gv tbl 3. All D D Ω), clug rtl costts M hv b rrr to th coo syst bs o MVA. Fro tbl 3 w c s tht D D Ω)) o torsol os & 3 o r gtv, hc ths os r ustbl. Whrs torsol o 2 o s ctully stbl bcus o ts ostv D D Ω)). A torsol o o 2 s ustbl. Furthror, wh corg roxt D D 2M cy coct) tbl 3 w.r.t Rλ ) tbl 2, t c b obsrv tht or torsol os ~5 o torsol os 2~3 o 2 wth rltvly sllr Rλ ), th cy cocts obt through T clculto tho g-lyss r qut clos to ch othr. For torsol o o 2 wth rltvly bggr Rλ ), stll th stblty jugt v T clculto s cosstt to th g-lyss. Although th cy coct hs rltvly lrgr vto ro tht o g-lyss, t s wth 5%. Actully usg colx rqucy scg trto, or ccurt cy coct c b obt or o o 2 t th cost o PU t s tst by th uthors. D K 5th P, g, August 25 sso 34, Pr 4, Pg 6

7 6 ONUION I ths r, DAE ol or owr syst O stuy s roos. Th syst DAE ol c b stblsh ctly through two sts wth lt DAE ols s bulg blocks. rlz glyss c b rctly l to th DAE ol or O stuy. Th rsults c rov usul orto o obsrvblty cotrollblty o lgbrc vrbls o vul O os. Furthror, th rqucy-scg T clculto c b sly rlz bs o th DAE ol, whch s usul or stblty jugt o torsol os. Th coutr tst rsults show clrly th ctvss o th roos DAE ol th rlvt lyss. Th T clculto rsults r cor vorbly wth tht ro th grlz g-lyss. Th DAE-bs O stuy ol ts corrsog lyss thos v th wy to clu VD trssso FAT vc ols O stuy gt sght bout thr cts o O. AKNOWEDMENT Th rsrch s suort by NF o h No. 5337), Ntol Ky sc Rsrch cl Fu, P.R. h 24279), R o og Kog ovrt, to who th uthors woul lk to xrss thr scr ckowlgts. APPENDI Th two-ch-t-bus owr syst t s s ollows. Th grtor t u o ch MVA bs, t costts sc.) rt MVA7, rt kv22.) t: r., l.3,.79,.69,.35, q.7, q.228, q.2 4.3, τ.32, τ q.85, τ q.5. τ 2 rt MVA6, rt kv22.) t: r.45, l.4,.65,.25, q.59, q.46, q.2 4.5, τ.4, τ q.55, τ q.9. τ.2, rg-ss sht t : s s IEEE FM ol [3] sl-g costt o ch ss s suos to b. u wth utul g btw jct sss glct). 2: our-ss sht wth ts sht t s ollows u o ch MVA bs): Irtl costts sc): M.8785, M P.2235, M IP.4263, M P.. rg costts u): K P 8.852, K PIP 6.358, K IPP l-g costts u utul g btw jct sss s glct): D D D D.. P IP P Th twork c u vlu o MVA bs) R.2,.2, R.4,.4, T T T 2 T 2.52,.54, RY.4, Y R.3. REFERENE [] IEEE ott. Th 4 th ult to blogrhy or R, IEEE Trs. O PWR, 997 [2] N.Rostkol, R.J.Pwko, t l. ubsychroous Torsol Itrctos wth ttc VAR ostors Iluc o VD. IEEE Trs. O Powr ysts, 99, 6): [3] Plotto,.A..; co, A.; og, W.F.; Ers, A.- A.; Ict o T otrol Mthoologs o ubsychroous Osclltos. IEEE Trs. O Powr Dlvry, 23, 8): [4] W o, Zhg Y, Dg subsychroous oscllto usg UPF- FAT vc, Powr yst Tchology, 22. Procgs. Powro 22. Itrtol orc o, Volu: 4, 3-7 Oct. 22 Pgs: vol.4 [5] IEEE ott. Rrs u to ubsychroous Rsoc, IEEE Trs. O Powr ysts, 992, 7):5 57 [6] IEEE ott Rort, orso o R lcultos Tst Rsults, IEEE Trs. O Powr ysts, Vol. 4, No., , Fbrury [7] y I. M., A Novl Aroch to th Torsol Itrctos Elctrcl Dg o th ychroous Mch, Prt-I: Thory, Prt-II: Alcto to rbtrry twork, IEEE Trsctos o Powr Artus ysts, Vol. PA-, No., , 982 [8] Arso, E., Z.,. scho,. lckor, J. Dl, J. Dogrr, J. Du roz, A. rbu,. rlg, A. Mcky, D. ors, APAK Usr s u, Thr Eto, IAM, Phllh, 999 [9] P. M. Arso,.. Agrwl, J. E. V Nss. ubsychroous Rsoc Powr ysts, IEEE Prss, Nw York, 99 []Irov thoologs or th clculto o crtcl gvlus sll sgl stblty lyss, Agls,.; ly, A.; IEEE Trsctos o Powr ysts, Volu, Issu 3, Aug. 996 Pgs):29 27 []Fsbl gvlu sstvty or lrg owr systs,, T.; IEEE Trsctos o Powr ysts, Volu 8, Issu 2, My 993 Pgs): [2]T. r, N. Kobysh, Dvlot o A Dg Alyss Progr or Mult-rtor Powr ysts, IEEE Trsctos o Powr ysts, Vol. 9, No. 4: [3]IEEE R Workg rou. Frst chrk Mol or outr ulto o R, IEEE Trs. O PA, 977, 965): th P, g, August 25 sso 34, Pr 4, Pg 7