ECE 692 Avance Topics on Power System Stability 2 Power System Moeling Spring 2016 Instructor: Kai Sun 1
Outline Moeling of synchronous generators for Stability Stuies Moeling of loas Moeling of frequency an voltage regulation systems Materials Kunur s Chapters 3 5 an 7 9 Chapter 12 of Power System Analysis (3 r E.) by Saaat 2
2.1 Moeling of Synchronous Generators for Stability Stuies 3
Synchronous Generators Salient-pole rotor Cylinrical/roun rotor Fiel wining Armature wining Stator 16 poles salient-pole rotor (12 MW) Roun rotor generator uner construction 4 (Source: http://emarlc.blogspot.com)
Stator an Rotor Winings Armature winings: a a, b b an c c winings Rotor winings: Fiel winings Fiel wining F F prouces a flux on the axis. Damper winings Two amper winings D D an Q Q respectively on an q axes For a roun rotor machine, consier a secon amper wining G G on the q axis (two winings on each axis) Consier a reference frame rotating synchronously with an q-axes at spee r (assume to be along with the axis of a-a at t=0) is the isplacement of -axis from the axis of a-a is the isplacement of q-axis from the rotating reference axis Reference axis Direct or axis r Total number of winings: Salient pole: 3+3 Roun rotor: 3+4 t r 2 Quarature or q axis ANSI/IEEE stanar 100-1977 efines the q-axis to lea the -axis by 90 0 5
Voltage an Flux Equations (Salient pole machine) Moel winings as a group of magnetically couple circuits with inuctances epening on R F ea Ra 0 0 0 0 0 ia a e b 0 R 0 0 0 0 i b b b e c 0 0 Rc 0 0 0 i c c ef 0 0 0 RF 0 0 if t F e 0 0 0 0 0 RD 0 D i D D e 0 0 0 0 0 0 R Q Q iq Q eabc Rabc 0 iabc ψabc FDQ 0 e R FDQ ifdq t ψ FDQ e F e D =0 e Q =0 R D F D R a Q R Q b a R b R c c e a e c e b l l l l l l i a aa ab ac af ad aq a b lba l i bb lbc lbf lbd l bq b l l l l l l i c ca cb cc cf cd cq c F lfa lfb lfc lff lfd lfq if l l l l l l i D Da Db Dc DF DD DQ D l Q Qa lqb lqc lqf lqd lqq iq ψ L L i abc SS SR abc ψ FDQ LRS LRR i FDQ Stator self inuctances (l aa, l bb, l cc ) Stator mutual inuctances (l ab, l bc, l ac ) Stator to rotor mutual inuctances (l af, l bd, l aq ) Rotor self inuctances (l FF, l DD, l QQ ) Rotor mutual inuctances (l FD, l DQ, l FQ ) A main objective of synchronous machine moeling is to fin constants for simplification of voltage an flux equations 6
Self an Mutual Inuctances Each of winings 1 & 2 is stationary (on the stator) or rotating (on the rotor) N 1 N 2 l 12 =N 1 N 2 P 12 = l 21 (Symmetric) P 12 - permeance of the mutual flux path (mainly influence by the air gap) Stator self/muual inuctances (e.g. l aa an l ab ): P 12 between stator winings is a function of an reaches its maximum twice per cycle P 12 P 0 +P 2 cos2(+) l 12 =l 0 +l 2 cos2(+) a laa lab lac laf lad laq ia b lba lbb lbc lbf lbd l i bq b L SS L SR c lca lcb lcc lcf lcd l cq i c F lfa lfb lfc lff lfd lfq if L RS L RR l l l l l l i D Da Db Dc DF DD DQ D l Q Qa lqb lqc lqf lqd lqq i Q Stator to Rotor Mutual Inuctances (e.g. l af ): P 12 between stator an rotor winings is almost constant but N 1 N 2 is a function of an reaches its maximum once per cycle; flux leakage can be ignore (l 0 =0) N 1 N 2 N 0 cos(+) l 12 =l 1 cos(+) Rotor Inuctances are all constant F R 0 R D 0 0 0 Q Using a reference revolving with the rotor will lea to a constant inuctance matrix 7
Park s Transformation i 0q Pi abc 1/ 2 1/ 2 1/ 2 P 2 / 3 cos cos( 2 / 3) cos( 2 / 3) sin sin( 2 / 3) sin( 2 / 3) ψabc L SS LSR iabc ψ L L i FDQ RS RR FDQ ψ Pψ 0q abc 0 L0 0 0 0 0 0 i 0 0 L kmf kmd 0 0 i F 0 kmf LF MR 0 0 i F D 0 kmd MR LD 0 0 i D q 0 0 0 0 Lq km Q i q 0 0 0 0 km Q Q L Q i Q L L M 0 s 2 L L M 3 L 2 L L M 3 L 2 s s m q s s m k 3/2 s eabc Rabc 0 iabc ψabc FDQ 0 e R FDQ ifdq t ψ FDQ e Pe 0q abc e0 Ra 0 0 0 0 0 i0 L0 0 0 0 0 0 i0 0 e 0 R 0 0 0 0 i a 0 L km F km 0 0 i D r q e F 0 0 RF 0 0 0 i F 0 km F LF M R 0 0 i F 0 0 0 0 0 RD 0 0 id 0 km D M R LD 0 0 i t D 0 e 0 0 0 0 Ra 0 i q 0 0 0 0 L q q km Q i q r 0 0 0 0 0 0 RQ i 0 0 0 0 km Q Q L Q i 0 Q 8
Per Unit Representation Using the machine ratings as the base values e s base (V) peak value of rate line-to-neutral voltage i s base (A) peak value of rate line current f base (Hz) rate frequency S 3 base (VA) = e s base i s base Z s base ( ) =e s base /i s base L s base (H) =Z s base / base base (elec. ra/s) =2f base t base (s) =1/ base =1/(2f base ) s base (Wbturns) =L s base i s base= e s base / base T base (Nm) = s base i s base L a -L aq base per unit system: assume all per unit mutual inuctances between the stator an rotor circuits are all equal to in -axis or in q-axis 2 2 p.u. If f=f base 0 L0 0 0 0 0 0 i0 0 Ll La 0 La La 0 i q 0 0 Ll Laq 0 0 L aq i q F 0 La 0 LF MR 0 if D 0 La 0 MR LD 0 i D Q 0 0 Laq 0 0 LQ iq Base q 0 F D Q 1 f base 2 e s base S 3 base 3 i s base i F base i D base i Q base i Fbase, i Dbase an i Qbase enable a symmetric per-unit inuctance matrix 9
Equivalent Circuits e0 Ra 0 0 0 0 0 i0 L0 0 0 0 0 0 i0 0 e 0 R 0 0 0 0 i a 0 Ll La La La 0 0 i r q e F 0 0 RF 0 0 0 i F 0 La LF MR 0 0 i F 0 p 0 0 0 0 RD 0 0 id 0 La MR LD 0 0 i D 0 e 0 0 0 0 Ra 0 i q 0 0 0 0 L q l Laq L aq i q r 0 0 0 0 0 0 RQ i 0 0 0 0 L Q aq L Q i 0 Q 0 L0 0 0 0 0 0 i0 0 Ll La La La 0 0 i F 0 La LF M R 0 0 i F p p D 0 L a M R L D 0 0 i D q 0 0 0 0 Ll Laq L aq i q Q 0 0 0 0 Laq LQ iq (ifferential operator p=/t) They follow Faraay s law i D +i F e 10
Equivalent Circuits with Multiple Damper Winings M R -L a 0 Usually ignore L 1 =L D - M R L f =L F -M R R f =R F R 1 =R D e f =e F L 1q =L Q L aq L 2q =L G L aq R 1q =R Q R 2q =R G EPRI Report EL-1424-V2, Determination of Synchronous Machine Stability Stuy Constants, Volume 2, 1980 11
L a =K s L au K s = at / at0 = at /( at + I ) = I 0 / I 12
Steay state Analysis All erivatives are zero: p r =0 r =1 an L=X in p.u. p f =0 e f = R f i f p 1 =0 i 1 =0 = -L i +L a i f p 1q =0 i 1q =0 q = -L q i q p =0 e = r L q i q -R a i = X q i q -R a i p q =0 e q = - r L i + r L a i f -R a i q = -X i +X a i f -R a i q Terminal voltage & current phasors: =e +je q =i +ji q = (R a + jx q where =j[x a i f -(X -X q )i ] If saliency is neglecte X =X q =X s (synchronous reactance) E q =X a i f 13
Computing per unit steay state values 1p.u. r Active an Reactive Powers S EI e e t * t ( e j e )( i j i ) q q ( ei ei) j( ei ei) qq q q R i r q a R i q r a q P ei ei T R i i 2 2 t q q r e a( q) Q e i e i t q q Air-gap torque (or electric torque) T i i P R i i 2 2 e q q = t a( q) 14
Sub transient an Transient Analysis Following a isturbance, currents are inuce in rotor circuits. Some of these inuce rotor currents ecay more rapily than others. Sub transient parameters: influencing rapily ecaying (cycles) components Transient parameters: influencing the slowly ecaying (secons) components Synchronous parameters: influencing sustaine (steay state) components 15
Short circuit an open circuit time constants Consier the -axis network Short-circuit time constant Instantaneous change on Delaye change on i ( through Open-circuit time constant 0 Instantaneous change on i Delaye change on ( through ) ) L () s L 1 s ( s) L (1 st)(1 st i 1 s e 0 0 ) f (1 st 0)(1 st 0 L ) Time constant or 0 equals the ivision of the total inuctance an resistance (L/R) with the effective circuit 16
Transient an sub transient parameters + p 1 - + p f - + p 1q - R 1 >>R f L f /R f >>L 1 /R 1 axis circuit L 1q /R 1q >>L 2q /R 2q q axis circuit Consiere rotor winings Time constant (open circuit) Time constant (short circuit) Inuctance (Reactance) L (s) an L q (s) T 0 = 8.07(s) T = Only fiel Wining + // A the amper wining Only 1 st amper wining A the 2 n amper wining T 0 = T q0 = + T q0 = 0.03(s) // 1.00(s) 0.07(s) T = // // T q = T q = L = L l +L a //L f L = L l +L a //L f //L 1 L q = L L q = l +L aq //L 1q L l +L aq //L 1q //L 2q 0.30(pu) 0.23(pu) 0.65(pu) 0.25(pu) // // // // Base on the parameters of Kunur s Example 3.2 Note: time constants are all in p.u. To be converte to secons, they have to be multiplie by t base =1/ base (i.e. 1/377 for 60Hz). 17
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Synchronous, Transient an Subtransient Inuctances L ( s) L (1 st )(1 st (1 st )(1 st 0 0 ) ) Uner steay state conition: s=0 (t) L (0)=L (-axis synchronous inuctance) During a rapi transient: s TT L L ( ) L L L L L a f 1 l T 0T 0 LaLf LaL1 LfL1 (-axis sub-transient inuctance) Without the amper wining :s>>1/t an 1/T 0 but << 1/T an 1/T 0 T L L ( ) L L L L a f l T 0 La Lf (-axis transient inuctance) 19
X X q X q X X q X >X l T 0 > T > >T 0 > T T q0 > T q > >T q0 > T q 20
Swing equations 2 2H T - 2 m Te KDr t 0 (s) ( r ) 2H Tm Te -KDr t 1 0 t = r Some references efine M=2H, calle the mechanical starting time, i.e. the time require for rate torque to accelerate the rotor from stanstill to rate spee 21
State Space Representation of a Synchronous Machine So far, we moele all critical ynamics about a synchronous machine: State variables (px): stator an rotor voltages, currents or flux linkages swing equations (rotor angle an spee) Time constants: Inertia: 2H Sub transient an transient time constants, e.g. T 0 an T 0 Other parameters Stator an rotor self or mutual inuctances an resistances Rotor mechanical torque T m an stator electromagnetic torque T e 22
State Space Moel on a Salient pole Machine Consier 5 winings:, q, F (f), D (1) an Q (1q) Voltage an flux equations: e RiL iωψ Ψ Li t e0 i0 0 0 e i r q e i f f f 0 e, i, Ψ, ωωψ 0 i1 1 0 e i q q q r 0 i 0 1q 1q Swing equations: t 1 Ψ ( RL Ω) Ψe r 2 H Tm Te Tm ( iq qi) t r 0 r 1 t Define state vector x=[ f 1 q 1q r ] T Thus, the state space moel: x f( x, e, T, e, e ) f m q e f an T m are usually known but e an e q are relate to its loaing conitions (the gri), so algebraic power flow equations shoul be introuce. The gri moel is a set of Differential Algebraic Equations (DAEs) 23
Neglect of Stator p terms 24
Simplifie Moels [ f 1 q 1q r ] T or [ f 1 q 1q 2q r ] T Let p =p q =0 an r =1 pu [ f 1 1q r ] T or [ f 1 1q 2q r ] T p 1 Ψ( RL Ω) Ψe 2H p T T p 1 r r m e = q = -L q i q Inertia 2H ~ p r Transient T 0 ~ p f T q0 ~ p 1q Sub-transient T 0 ~p 1 T q0 ~ p 2q Let p 1 =p 1q =0 (neglect amper winings) [ f r ] T = = -L i q Inertia 2H ~ p r Transient T 0 ~ p f Constant flux linkages [ r ] T (classic moel) Inertia 2H ~ p r 25
E q = f L a /(L a +L f ) E f =e f L a /R f T 0 ~pe q ( f ) T 0 ~p ( 1 ) T q0 ~ p q ( 1q ) Source: J. Weber, Description of Machine Moels GENROU, GENSAL, GENTPF an GENTPJ, PowerWorl, Oct 2015 26
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Phasor iagram =jx a i f 28
Classic Moel Eliminate the ifferential equations on flux linkages (swing equations are the only ifferential equations left) Assume X =X q 2H p T T p 1 r r m e E E ( R jx ) I t a t E is constant an can be estimate by computing its pre-isturbance value EE ( R jx) I t0 a t0 29
Comparison of PSS/E Generator Moels 30