Load Equations. So let s look at a single machine connected to an infinite bus, as illustrated in Fig. 1 below.

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1 oa Euatons Thoughout all of chapt 4, ou focus s on th machn tslf, thfo w wll only pfom a y smpl tatmnt of th ntwok n o to s a complt mol. W o that h, but alz that w wll tun to ths ssu n Chapt 9. So lt s look at a sngl machn connct to an nfnt bus, as llustat n g. blow. g. om KV, w ha c b a U c b a U c b a c b a U U,,,, Now us Pak s tansfomaton to obtan:

2 P P, PU P PU P P, () TEM TEM 2 TEM W woul lk to xpss an as a functon of stat aabls (th cunts fo th cunt mol o th flux lnkags fo th flux lnkag mol). t s cons ach tm. TEM: P,,, So what s? A goo assumpton fo puposs of stablty assssmnt s that thy a a st of balanc oltags hang ms alu of V,..,, a 2V cos( t ),, b 2V cos( t 2), c 2V cos( t 2) Ht th abo wth Pak s tansfomaton matx to obtan:, P, V sn( ) cos( ) wh, as w ha pously sn, whch mpls that. t t (4.5) An so w s that th balanc AC oltags tansfom to a st of C oltags, as w ha obs bfo. TEM2: Ths on s asy as t s alay wttn n tms of th cunts. 2

3 TEM: W must b a lttl caful h. It s tmptng to us P. But s ths tu? t s back up an call that P Takng th at of th lft-han-s, w obtan: P P (2) An ths pos that But w know that P PP P. P Isolatng th fst tm on th ght sults n PP callng that tm s obtan tm: P PP P, an usng ths n (2) sults n P, w multpl th abo by to You may call now that n Scton 4.4 (nots on machts, pp. 22-2) that w foun P P So tm bcoms P O

4 4 P Substtuton of ou tms, 2, an back nto. () sults n V ) cos( ) sn( (4.49) Now w n to ncopoat ths nto ou stat-spac mol. W ha th ffnt mols. A. Cunt stat-spac mol B. lux-lnkag-stat-spac mol wth λ A an λ A (fo molng satuaton) C. lux-lnkag-stat-spac mol wth λ A an λ A lmnat (an so wthout th ablty to molng satuaton) I ha han-wttn nots wh I wnt though th tals of ths fo mols (A) an (C), although I not nclu th -wnng. I want to o that but ha ust not ha tm to o t. An so I smply po th sults fo th mol wthout th -wnng. A. Cunt stat-spac mol (S scton 4..2) call that th cunt stat-spac mol s

5 5 ) ( m T N (4. ) wh th submatcs a gn by ; N Y Y M M M M ; Incopoatng nto ou loa uatons,. (4.49), nto ou statspac cunt mol, (4. ), sults n

6 K sn ˆ ˆ + ˆ N ˆ K cos k M k M k M Tm wh th matcs wth th hats abo thm,.., xactly as th unhat- sons abo, xcpt that Wh you s, plac t wth Wh you s, plac t wth Wh you s, plac t wth Not that: K= V (not th sam K as us n th satuaton nots), an γ=δ-α. (4.54) ˆ, ˆ, Nˆ You txt maks a usful mak (pg. 7) n sayng that, Th systm scb by (4.54) s now n th fom of T x f ( x, u, t), wh x [ ]. (an, of cous, ), a Th functon f s a nonlna functon of th stat aabls an t, an u contans th systm ng functons, whch a an T m. Th loang ffct of th tansmsson ln s ncopoat n th matcs ˆ, ˆ, Nˆ. Th nfnt bus oltag V appas n th tms Ksnγ an Kcosγ. Not also that ths latt tms a not ng functons, but ath nonlna functons of th stat aabl δ. 6

7 C. lux-lnkag-stat-spac mol wth λ A, λ A lmnat (so wthout ablty to molng satuaton) (S scton 4.. of txt). call th stat-spac mol of. (4.8) Wthout -wnng: M M Tm M M M M 2 2 (4.8) Wth -wnng: M M M l l M M M l l M M M ll ll l l M M M 2 2 ll T m (4.8 ) W s w n to ncopoat th loa uatons, (4.49), though th, tms.ths uatons a pat h fo connnc: 7

8 V sn( ) (4.49) cos( ) How, ths tm w n th loa uatons n tms of flux lnkags. Ths taks som wok, whch I ha on n tal hanwttn nots (wll b happy to po f you want thm). Ths sults n ts. 4.57, 4.58 n you txt, pat h. ˆ ˆ ˆ M M V sn (4.57) ˆ ˆ M M M M V cos (4.58) Not n ths two uatons that th a sal at tms an so w cannot clanly us ths uatons to smply plac th ats on λ an λ n th flux-lnkag stat-spac mol (w w abl to o so wth th cunt stat-spac mol). ath, w ha to cat a p-multpl matx T such that T x C x wh 8

9 An T, C, an a gn by x Not w n to nclu λ h, an thn augmnt th T, C, an matcs accongly. M M (4.6) T ˆ ˆ ˆ M M C ˆ ˆ M M M M M M 2 2 (4.6) 9

10 V sn V cos (4.62) Tm / Thn w can p-multpl both ss by T - to obtan x T Cx T (4.6) Euaton (4.6) scbs th complt systm of ntst to us at ths pont,.., th systm of g. at th bgnnng of ths nots. To us t, w n th ntal stats x() whch a foun by solng T x Cx, a x C wh cto pos systm loang nfomaton. Thn, f w ptub th systm by sttng, fo xampl, V = fo a fw cycls, thn th spons can b obtan by solng. (4.6) usng numcal ntgaton.