ECE 422 Power System Operatons & Plannng 2 Synhronous Mahne Moelng Sprng 219 Instrutor: Ka Sun 1
Outlne 2.1 Moelng of synhronous generators for Stablty Stues Synhronous Mahne Moelng Smplfe Moels for Stablty Stues Materals: Saaat s Chapter 8 Kunur s Chapters 3 5 2.2 Moelng of loas 2
Synhronous Generators Salent-pole rotor Stator el wnng Roun rotor f=pn/12 Armature wnng el urrent 3
Types of Rotors Salent pole rotors Have onentrate wnngs on poles an non unform ar gap Short axal length an large ameter to extrat the maxmum power from a waterfall On hyraul turbnes operate at low spees <18 rpm (havng a large number of poles from 4 to 6) Have a surrel age wnngs (amper wnngs) embee n the pole faes to help amp out spee osllatons Cylnral/roun rotors strbute wnng an unform ar gap Large axal length an small ameter to lmt the entrfugal fores Steam an gas turbnes, operate at hgh spees, typally 18 36 rpm (4 or 2 pole) Ey n the sol steal rotor gves ampng effets 7% of large synhronous generators (15 to 15MVA) 2 poles salent-pole rotor Roun rotor generator uner onstruton (Soure: rspower.om an emarl.blogspot.om) 4
Stator an Rotor Wnngs Armature wnngs: a a, b b an wnngs Rotor wnngs: el wnngs el wnng proues a flux on the s the splaement of -axs from the rotatng referene axs Referene axs axs. amper wnngs Two amortsseur/amper wnngs an respetvely on an axes, whh oul be atual wnngs or effetve parts of the sol steal rotor. or a roun rotor mahne, for symmetry, onser two wnngs on eah axs by ang a seon amper wnng G G to the axs Total number of wnngs: Salent pole: 3+3 (a, b,,,, ) Roun rotor: 3+4 (a, b,,,,, G) Rotor poston Conser a referene frame rotatng synhronously wth an -axes at spee r (assume to be along wth the axs of a-a at t= n the fgure) s the splaement of -axs from the axs of a-a ret or axs t r 2 uarature or axs r G G ANSI/IEEE stanar 1-1977 efnes the -axs to lea the -axs by 9 5
Voltage an lux Euatons (Salent pole mahne) Moel wnngs as a group of magnetally ouple ruts wth nutanes epenng on R ea Ra a a e b R b b b e R e R t e R e R eab Rab ab ψab e R t ψ e e = e = R R a R b a R b R e a e e b a laa lab la la la l a a b l ba lbb lb lb lb l b b l l l l l l a b la lb l l l l l l l l l l a b l a lb l l l l ψ L L ab SS SR ab ψ LRS LRR Stator self nutanes (l aa, l bb, l ) Stator mutual nutanes (l ab, l b, l a ) Stator to rotor mutual nutanes (l a, l b, l a ) Rotor self nutanes (l, l, l ) Rotor mutual nutanes (l, l, l ) A man objetve of synhronous mahne moelng s to fn a mnmum set of onstants by whh voltage an flux euatons an be smplfe wth an aeptable level of auray. 6
Varatons of Self an Mutual Inutanes N x Eah wnng s ether statonary (on the stator) or rotatng wth the magnet (on the rotor): x, y { a, b,,,, } l xy =N x N y P xy = l yx Stator self/mutual nutanes (e.g. l aa an l ab ): P xy between stator wnngs s a pero funton of wth a pero 18 o ue to pero ar gap hanges; assume a snusoal funton: P xy P +P 2 os2(+) l xy =l +l 2 os2(+) N y (Symmetr) a laa lab la la la l a a b l ba lbb lb lb lb l b b L SS L SR l l l l l l a b la lb l l l l L RS L RR l l l l l l a b l a lb l l l l P xy s the permeane of the mutual flux path manly nfluene by the ar gap. Stator to Rotor Mutual Inutanes (e.g. l a ): P xy between stator an rotor wnngs s almost onstant; however, ther effetve ouplng,.e. N x N y, s a pero funton of wth pero 36 o ; the flux leakage an be gnore (l =). N x N y N os(+) l xy =l 1 os(+) Rotor Inutanes are all onstant usng a referene revolvng wth the rotor 7
Stator self nutanes (l aa, l bb, l ) an Mutual Inutanes (l ab, l b, l a ) l aa s eual to the rato of flux lnkng phase a wnng to urrent a, wth zero urrents n all other ruts (maxmum P aa at = o or 18 o ) l aa =L s + L m os2 l bb =L s + L m os2(-2/3) l =L s + L m os2(+2/3) L s >L m b a = o l ab < sne wnngs a an b have 12 o (>9 o ) splaement (maxmum P ab at = 3 or 15 ) l ab = -M s -L m os2(+/6) = l ba b a b a l b = -M s -L m os2(-/2) = l b l a = -M s -L m os2(+5/6) = l a = -3 o S N =15 o N S M s L s /2 8
Stator to Rotor Mutual Inutanes (L SR : l a l b l l a l b l l a l b l ) The rotor sees a onstant permeane f negletng varatons n the ar gap ue to stator slots b () (, ) a When the flux lnkng a stator wnng an a rotor wnng reahes the maxmum when they algns wth eah other an s when they are splae by 9 o (maxmum N a N at = o ) axs l a = l a =M os l b = l b =M os(-2/3) l = l =M os(+2/3) l a = l a = M os l b = l b = M os(-2/3) l = l = M os(+2/3) axs l a = l a = -M sn l b = l b = -M sn(-2/3) l = l = -M sn(+2/3) What f we efne -axs laggng -axs by 9 o? (no negatve sgns) 9
or Salent pole Rotors or roun rotors, whh of the urves wll be fferent? 1
Rotor Inutanes (L RR : l, l, l, l, l, l ) They are all onstant Rotor self nutanes l L l L l L Rotor mutual nutanes l = l M R l = l = l = l = 11
Summary ψab L SS LSR ab ψ L L RS RR L RS = L T SR L RR L M R M R L L urther thnkng: L RR s onstant beause t s n a referene frame rotatng wth the rotor L SS an L SR are (t) epenent ue to varatons of N x N y or P xy ause by the rotaton of the rotor relatve to the stator = - L RS () ab + L RR L RS () ab = - +L RR LH se = [L RS () P -1 ()][P () ab ] L RS Only L SR () s expltly epenent of 12
Park s () Transformaton P ab k k k k os k os( 2 / 3) k os( 2 / 3) k sn ksn( 2 / 3) k sn( 2 / 3) a b or balane steay-state ontons: a = I m sn s t b = I m sn( s t -2/3) = I m sn( s t + 2/3) = r t+-/2, r s = k I m sn( s t-) 3/2 = k I m sn(/2-) 3/2 = -k I m os( s t-) 3/2= -k I m os (/2-) 3/2 = All onstant urrents! If we expet the transformaton to be power nvarant: T P 3 e a a e b b e e 1 1 ( ) T P e ab ab P -T P -1 =I P -1 = P T or P T P=I (orthogonal matrx) k =k = an k = 1/ 2 1/ 2 1/ 2 2 P os os( 2 / 3) os( 2 / 3) 3 sn sn( 2 / 3) sn( 2 / 3) P 1 P What f we efne -axs laggng -axs by 9 o? e P P T T 1 =e +e +e 1/ 2 os sn 2 1/ 2 os( 2 /3) sn( 2 /3) 3 1/ 2 os( 2 /3) sn( 2 /3) 13
ervng lux Euatons 1 ψ P ψab ψab P ψ ψ Pψa b ψ I ψ ψ I ψ 1 1 ψab L SS LSR ab P ψ LSS LSR P ψ LRS LRR I ψ LRS LR R I 1 ψ ψ L SS LSR P L SS LSR P ψ LRS L RS RR RR ψ I L L I L L km km L km km L MR km MR L km L 3 L L 2 M L L M L 2 3 L Ls Ms Lm k 3 / 2 2 s s s s m L L km km km L MR km MR L L km km L 14
ervng Voltage Euatons 1 e eab P e Pea b e I e eab Rab ab ψab e R t ψ P e R P e e I t P ψ 1 1 1 ab Ie R I I ψ 1 P Rab P 1 P P ψ I R I t I ψ 1 e R p P P ψ ψ t e R ψ t ψ P P P P P P t t 1 1 1 r P P t 1 1/ 2 1/ 2 1/ 2 sn os 2r 2 2 os os( ) os( ) 2 2 sn( ) os( ) 3 3 3 3 3 2 2 sn sn( ) sn( ) 2 2 sn( ) os( ) 3 3 3 3 r ( rl ) ( ) rkm r rl ( ) rkm rkm Note: P s NOT onstant r 1 1 15
ervng Voltage Euatons (ont ) Assume R ab =R a I e Ra L e Ra r L rkm L km km e rl Ra rkm rkm L km e R km L M R t e R km M R L e R km L e Ra L e Ra rl rkm L km km e R km L MR R km MR L t e rl rkm rkm Ra L km R km L ( rl) ( ) rkm = r rl ( ) rkm rkm r e Ra L e Ra L km km r e R km L M R R km M R L t e Ra L km r R km L 16