d dt d d dt dt Also recall that by Taylor series, / 2 (enables use of sin instead of cos-see p.27 of A&F) dsin

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1 Learzato of the Swg Equato We wll cover sectos ad begg of Secto 3.3 these otes. 1. Sgle mache-fte bus case Cosder a sgle mache coected to a fte bus, as show Fg. 1 below. E y1 V=1./_ Fg. 1 The admttace matrx s gve by Y Y Y Y 1 Y y1 y (1) 1 Let s assume the mache s modeled by the swg equato wth dampg, gve by H d d m y e m y M s( ) where M E V Y1 Y 1 Y1 1 1 / (eables use of s stead of cos-see p.7 of A&F) Now let the agle δ chage by a small amout. The d d d d, (3) Also recall that by Taylor seres, ds x s x s x x s x x s x (cos x ) x (4) dx x () 1

2 The we also see that s( ) s( ) s( ) (cos( )) (5) (Eqt. 3.3) Applyg (3) to the left-had-sde of () ad (5) to the rght-hadsde of (), we obta H d d s( m M ) But Therefore, H d [s( ) (cos( )) ] m M m M s( ) M (cos( )) (6) m M s( ) d M (cos( )) Or H d d (cos( )) M (8) Now defe S M cos( ) (9) What s t? To aswer ths questo, observe: e M s( ) (1) de M cos( ) d (11) de d M cos( ) (1) (7)

3 Therefore, S d d e M cos( ) (13) S s called the sychrozg power coeffcet. I regards to early swg stablty (whch s a olear pheomea), the larger S s, the more stable wll be the geerator for a gve dsturbace. Ths s true because S dcates the slope of the power-agle curve, ad the hgher ths slope, the more deceleratg eergy s avalable to the mache for a gve fault. Ths dea s llustrated Fg.. Fg. But let s see what t meas for small sgal stablty, whch s characterzed by the egevalues (roots) of the system dfferetal equato trasformed to the s-doma through Lalace trasforms. Substtutg (13) to (8) results H d d S (14) Takg the Lalace trasform (assumg all tal coos are ), we obta 3

4 H s ( s) s ( s) 4 ( s) S (15) Elmatg δ(s), we obta the system s characterstc equato: H s s S (16) Solvg usg the quadratc formula, we get s S (Eqt. 3.7) 1 H (17) ullg ω/h out of the radcal, we have 8 H s S (18) (Eqt. 3.8) We ca make some observatos about (18), as follows: 1. No dampg: If =, the H s 8 S S H, (19a) or s S H (19b) a. Observe (19b) that f S>, (a) ay respose to a small dsturbace wll be oscllatory, ad (b) the oscllatory frequecy becomes lower as H becomes larger. b. Observe (19a) that f S<, the s S H () ad ay respose s ustable.

5 Fgures 3, 4 llustrate, for both stuatos S>, S<, respectvely, the pole (egevalue) locatos the s-plae ad the operatg pot locato o the power-agle curve. Im{s} X e S> {s} X Im{s} Fg. 3: S> e δ S> δ X X {s} δ δ Fg. 4: S< I Fg. 3, the oscllatory system s characterzed by purely magary poles (left) ad a stable operatg pot (rght). I Fg. 4, the ustable system s characterzed by the RH-pole (left) ad a ustable equlbrum pot (rght). 5

6 . Wth dampg: If, the 8 H s S (18) Let s look at the most postve root (ad so we wll use + sg before the radcal, ad we esure the cotrbuto from the secod term sde the radcal s postve,.e., S<) ad ask what are the coos uder whch t ca be the rght-half-plae, that s: s 6 8 S H 8 H S 8 H? S 8 H S 8? S H??? The above relato must be true. Because the above relato s depedet of dampg, we coclude that f S<, the system must be ustable, depedet of how much dampg exsts.. Mult-mache case (Secto 3.4) (We wll come back to sectos 3. ad 3.3.1) call that for a geerator coected to a fte bus, we foud that the swg equato s

7 H d d m e (1) where e M s( ) M E V Y1 Y 1 Y1 1 Lettg ad learzg, we fd that H d d S where d e S M cos( ) d Let s ow cosder the mult-mache system assumg: Classcal models Network reduced to oly teral geerator odes For geerator, we have that the swg equato s H d d m e where e E G EE Y cos( ) 1 E G 1 Y cos( ) () (13) (3) (4) 7

8 where δ=δ-δ. I (4), all voltages E, E, ad all Y-bus elemets Y are magtudes. Now let s cosder a small chage the agle of mache : δ=δ+ δ. The left-had-sde of (4) s precsely as the case of the sgle geerator vs. fte by s case. But what happeed to the rght-hadsde? Now the rght-had-sde s, by (3), m e m s uaffected by + δ, but e s affected by t. call δ=δ-δ. We cosder a small chage rotor agle at geerator. To be more geeral, we also allow a small chage geerator. However, geerator wll ot chage as a result of the geerator chage; they are are depedet chages ad we could ust as well have oly oe of them. δ=δ+ δ δ=δ+ δ callg that e E G EE Y cos( ) 1 (5) we eed to see what happes to the cos term for the small chage agle. We kow from trgoometry that cos( x y) s xs y cos x cos The cos( ) s s cos cos (6) Applcato of (6) to (5) yelds:. y 8

9 e E E G G 1 1 Y B s s Y s G cos cos cos (7) (Eqt. 3.1) Now we eed to learze the cosδ ad sδ terms usg δ=δ+ δ. From Taylor seres wth frst order term oly, s s( ) s cos (8) cos cos( ) cos s (9) Substtutg (8) ad (9) to (7), we get e E G EE B s cos 1 Now collect terms δ: G cos s (3) 9

10 e E G 1 1 B B B s cos cos m 1 (31a) call that the rght-had-sde of the swg equato s m-e. Equato (31a) ca be rewrtte the as (31b) 1 G G G e m 1 therefore the swg equato (3), whch s H d d m e becomes H d d B cos s s cos G s 1 B cos G s (31b) (3) (3) efe everythg sde the expresso wth the summato of (3), except δ, as S, that s

11 S EE B cos G s (33) The (3) becomes H d d S (34) Gve the mechacal power s costat, the rght-had-sde of (34) gves the egatve of the chage electrc power out of the mache due to the small chages δ, that s e S (35) (Eqt. 3.3) What s S? We aswer ths questo by observg that the power flowg from geerator teral ode to geerator teral ode s EE B s G cos (36) fferetatg, we get EE B cos G s (37) Evaluatg at δ, we get B cos G s S (38) (Eqt. 3.4) Note that f bus s the fte bus, eglectg resstace, we have: B S cos whch s the same as the sychrozg power coeffcet the fte bus case (we called t S). We wll look at multmache systems, but before we do, we cosder somethg very mportat respose to load chages...

12 Oe last ssue: what s the dfferece betwee sychrozg power coeffcet, geerato shft factor (GSF) ad power trasfer dstrbuto factor? We aswer ths here. Sychrozg power coeffcet (SC): S EE B cos G s (a1) Observe that the SC gves o chage flow o crcut {,} wth respect to o a chage agular separato across {,}. Geerato shft factor (GSF): { b} t{ b}, (b1) allocato olcy Observe that the GSF gves o chage flow across ay brach b wth respect to o a chage ecto at bus, subect to a reallocato polcy (.e., how the bus chage ecto s compesated). ower trasfer dstrbuto factor (TF) for 1-bus ecto chage: {} b TF { b}, (c) allocato olcy The TF for 1-bus ecto chage s the same as the GSF. ower trasfer dstrbuto factor for -bus ecto chage: TF { },, { } { } (c) allocato olcy allocato olcy The upshot of the above s that relatg SC to GSF s eough to relate SC to TF. We relate SC to GSF as follows: 1

13 From (a1), we wrte that S S (a) From (b1), we wrte that {} b t{ b}, { b} t{ b}, allocato olcy allocato olcy allocato olcy (b) Now 1. Cosder our power system s experecg coos such that the agular separato betwee buses ad s δ.. Le b s termated by buses ad,.e., b {,}. 3. We make a chage ected power at bus equal to Δ compesated by a reallocato polcy where a equal ad opposte chage, Δ, s made at bus. The (a) ad (b) are equvalet: t allocato S { } { }, allocato b b olcy: olcy: That s: t S { }, allocato b olcy: whch shows us that allocato olcy: t { b}, allocato S olcy: allocato olcy: 13