Chapter 6. Small Signal Stability Analysis

Transcription

1 Chapter 6 Small Sgal Stablty Aalyss Small-sgal stablty s the ablty of the system to be sychrosm whe subjected to small dsturbaces. he dsturbaces ca be swtchg of small loads, geerators or trasmsso le trppg etc. he small-sgal stablty ca lead to oscllatos the system. here are three modes of oscllatos due to small dsturbaces. Local modes of oscllatos are due to a sgle geerator or group of geerators oscllatg agast the rest of the system.. Itra-plat modes of oscllatos are due to oscllatos amog the geerators the same plat. he typcal frequeces of oscllatos of local ad tra plat modes are the rage of Hz to Hz. 3. Iter area modes of oscllatos are due to a group of geerator oe area oscllatg together agast aother group of geerators a dfferet area. he typcal frequecy rage of ter area mode of oscllatos are 0. Hz to 0.8 Hz. Local, tra-plat ad ter-area are electromechacal modes of oscllatos whch the rotor agle ad the speed of the geerators oscllate. Apart from electromechacal modes of oscllatos there ca be cotrol modes, due to lack of proper tug of cotroller lke voltage regulator of a exctato system, ad torsoal mode of oscllatos due to oscllatos the turbe-geerator shafts. he ma dea behd small sgal stablty aalyss s that the power system, whch s olear, for small dsturbaces ca be learzed aroud the steady state operatg pot as the system wll be oscllatg a small area aroud the steady state operatg pot. Oce the olear system s learzed aroud a operatg pot lear cotrol theory ca be appled to assess the stablty of the system. Before uderstadg how to asses small sgal stablty of a system some terms used cotrol theory should be defed. 6.

2 6. Fudametal Cocepts of Stablty of a Dyamc System Ay dyamc system ca be represeted as x () t f x(), t u() t (6.) yt () g xt (), ut () (6.) Where, x() t s called as a state varable, ut () s the cotrol put, yt () s the system output. f xt (), ut (), (), () g x t u t are the fuctos depedet o the state varable ad the cotrol put. For a mult-varable, multple-put multple-output (MIMO) lear tme-varat system equatos (6.) ad (6.) ca be wrtte as X () t AX() t BU() t (6.3) Y() t CX() t DU () t (6.4) I equatos (6.3) ad (6.4), X () t s called as the state vector, U () t s the cotrol put vector, Y () t s the output vector, A s the state matrx, B s the put matrx, C s the output matrx ad D s the feed forward matrx. State varables are the mmum umber of varables a system that ca completely defe the dyamc behavour of that system. If X () t s defed as x () t x() t X() t x () t (6.5) he the state varables defe the behavour of the system. State space s the -dmesoal space created by the state varables. Equatos (6.3) ad (6.4) together are called as state space represetato of a dyamc system. he soluto of equatos (6.3) ad (6.4) ca be obtaed usg Laplace trasforms. Applyg Laplace trasforms to equatos (6.3) ad (6.4), assumg a tal codto X (0), we get 6.

3 sx() s X(0) AX() s BU () s (6.6) Y() s CX() s DU () s (6.7) o further smplfcato I A adj s X() s sia X(0) BU() s X(0) BU() s si A (6.8) Y() s C sia X(0) BU() s DU () s (6.9) Suppose the tal codto s take as zero ad assume there s o feed forward from the put to the output the for a sgle-put sgle-output system (SISO) Y() s adj si A CsIA BC B U() s si A (6.0) Equato (6.0) shows the trasfer fucto betwee the output Y( s ) ad the put U() s. he deomator equato (6.0) s a th order polyomal equato called as characterstc equato ad the roots of the equato are called as ege values. he roots of the characterstc equato or the ege values ca be computed by fdg the solutos of the equato gve equato (6.) sia 0 (6.) 6. Ege Propertes he ege values correspodg to the state matrx A have certa propertes. Let the ege values be represeted as,,..., the for a th ege value A (6.) 0 A I (6.3) 6.3

4 s called as the rght ege vector correspodg to the ege value. Smlarly a left ege vector correspodg to the ege value ca be defed as A (6.4) 0 A I (6.5) he left ad rght ege vectors correspodg to dfferet ege values are orthogoal whch meas that j 0 j (6.6) C (6.7) Where, C s a costat. Usually, equato (6.7) ca be ormalzed so that the maxmum value wll be ad hece (6.8) Let, the rght ad left ege vector matrces be defed as Φ [ ] Ψ [ ] (6.9) From equatos (6.6) to (6.8), ad the ege vector matrces defed equato (6.9) the followg expresso ca be wrtte ΨΦ I (6.0) Ψ Φ (6.) Let, a dmesoal dagoal matrx of ege values be defed as 6.4

5 0 Λ (6.) 0 he, from equato (6.) AΦ ΦΛ (6.3) Pre multplyg equato (6.3) wth Φ lead to Λ Φ AΦΨAΦ (6.4) ΨΦΛ,, are of sze ad are called as modal matrces. 6.3 Stablty of Homogeous or Uforced System o uderstad the stablty of a system let us take a homogeous or uforced system that s there s o exteral put to the system ad the respose s a atural respose. Hece, the dyamc system gve equato (6.3) the cotrol put vector s zero ad the state space represetato of the system s gve as X () t AX() t (6.5) he stablty of a system ca be better uderstood through the use of modal matrces rather tha the state matrx A. hs s because the state matrx s ot a dagoal matrx ad hece cross couplg terms betwee states wll be preset. Let a ew state vector be defed as X() t ΦZ () t (6.6) Substtutg equato (6.6) equato (6.5) lead to ΦZ () t AΦZ() t (6.7) 6.5

6 Z () t Φ AΦZ() t ΨAΦZ() t ΛZ() t (6.8) Sce, the matrx Λ s a dagoal matrx wth ege values as ts elemets, each dyamc state ca be represeted terms of a sgle state varable leadg to decouplg of states. hs ca be represeted as z ( t) z ( t) for,,... (6.9) he soluto of equato (6.9), wth tal codto take as z (0) for,,..., s gve as z ( t) e z (0) for,,... (6.30) From equato (6.30) the stablty of the system ca be assessed as followg []:. If all the ege values are o the left half of s -plae that s the real part of the ege values should be havg a egatve value the the system s stable. From equato (6.8) t ca be see that wth the ege values o the left sde of s - plae as tme moves towards fte the state varables wll reach ther tal codtos makg the system stable. hs s called as asymptotc stablty.. If at least oe of the ege values s o the rght sde of s -plae that s the real part of the ege value s postve the the system s ustable. Wth at least oe ege value wth postve real part t ca be observed from equato (6.30) that the state varable ca ever come back to ts tal codto. hs s called as aperodc stablty. 3. he ege values of a system wth a real state matrx have complex cojugate pars of ege values. If at least oe of the complex cojugate par les o the magary axs of the s -plae the the system becomes oscllatory. hs s called as oscllatory stablty. 4. If at least oe ege value s o the org the the system stablty caot be assessed. 6.6

7 From equato (6.6), the tal value of the orgal state vector terms of the ew state vector ca be wrtte as X(0) ΦZ (0) (6.3) Z(0) Φ X(0) ΨX (0) (6.3) z (0) X (0) (6.33) Hece, X() t ΦZ() t z() t z() t z() t z () t z () t z () t z () t (6.34) Sce, the ew state varables Z () t are drectly related to ege values ad each of the ew sate varables correspods to exactly oe ege value, t ca be observed from equato (6.3) that the rght ege vector correspods to the actvty of the state varables X ( t) th mode. For example the state varable x j actvty th mode s gve by the elemet ad the drecto of ts actvty s gve by the agle of j. Smlarly, j show the cotrbuto of the state varables X ( t) a th mode whch ca be observed from equatos (6.30) ad (6.33). For example j shows the cotrbuto of the state varable x j the th mode. he total effect of actvty ad cotrbuto of a state varable x j the th mode s the gve as j j. Hece, the combed effect of actvty ad cotrbuto of a state varable a partcular mode ca be represeted as a matrx called as partcpato factor. he partcpato factor matrx s gve as 6.7

8 P P P P (6.35) P for,,... modes (6.36) As the rght ad left egevectors are ormalzed the sum of the elemets of the partcpato factor matrx correspodg to a th mode wll be k P k. Smlarly, the sum of the elemets of the partcpato factor matrx correspodg to a wll be Pk. th k state 6.4 Algorthm to Fd Ege Values ad Ege Vectors I 960 G. C. Fraces [] has proposed a algorthm amed as QR to fd the ege values ad ege vectors. hough may modfcatos have bee suggested to ths algorthm the uderlyg prcple s same. Here QR algorthm wll be dscussed. Before dscussg QR algorthm some terms eed to be defed. Utary matrx: a utary matrx U s a orthogoal matrx wth the property UU I f U s real ad wth same sze as U. UU * I f U s complex, where I s a detty matrx Smlarty trasformato: If a matrx A be trasformed by a trasformato matrx - such that A B ad f the ege values of the matrces A, B are same the trasformato s called as smlarty trasformato ad the matrces A, B are called as smlar matrces. Hesseberg Matrx: A matrx G s sad to be a Hesseberg matrx f the elemets below the sub-dagoal are zeros. Householder s trasformato ca be used to trasform ay matrx A to ts Hesseberg form G. he matrces A, G are smlar. 6.8

9 he QR algorthm covergece s mproved f stead of takg the matrx A, whose ege values eed to be computed, ts Hesseberg form G s cosdered. Schur form: A symmetrcal matrx A wll have a utary matrx U such that the trasformato, U AU S, lead to a matrx S Schur form. he matrx S s a tragular matrx. he matrces A, S are smlar ad hece the ege values of both the matrces wll be same. If matrx A s real the ts Schur form S s also real ad f matrx A s complex the S s also complex. Ege values of a matrx Schur form: Let A be a symmetrcal matrx ad let t be trasformed to a Schur form S through smlar trasformato. he matrx S s of the form S S S S 0 S S S S 0 0 S S S (6.37) he ege values of A wll be S S S3 S 0 S S3 S det [ AI] det [ SI] det 0 0 S33 S 3 0 (6.38) S or S S S33 S 0 (6.39) Whch meas that the ege values of the matrx A are othg but the dagoal elemets of the matrx S. It has to be oted that f the matrx A s ot symmetrcal the ts Schur form S wll ot be a tragular matrx but may cota blocks alog the prcpal dagoal whose ege values wll gve a complex 6.9

10 cojugate par of ege values. QR method smply s a method to trasform a gve matrx A to ts Schur form S so that the ege values wll be smply the dagoal or block dagoal elemets of S. Usually t s solved two steps. Step : rasform a symmetrcal/usymmetrcal matrx A to Hesseberg form G through householder s trasformato. Step : Use QR method, based o stable Gram-Schmdt, to covert the matrx G to a Schur form S. he ege values of A wll the be dagoal elemets of S f A s symmetrcal ad f A s usymmetrcal the the ege values wll be the ege values of the block dagoal elemets of S. Step : rasform a symmetrcal/usymmetrcal matrx A to Hesseberg form G Let the matrx A be represeted as a a a a a a a a A a a a a a a a a (6.40) Defe, vectors X, Y as 0 0 a X X a 3, Y sg( a) 0 0 a (6.4) Wth X, Y defed as equato (6.4) a ut vector U ca be formed as U X X Y Y (6.4) 6.0

11 Householder s trasformato ca be defed o the bass of the ut vector U as H I UU (6.43) Now apply the trasformato matrx H to the matrx A such that A HAH (6.44) A wll be of the form a a a a 3 a a a a 3 A 0 a a a a a a 3 where, a s a j th th row j colum elemet of A defed by the trasformato. After trasformato t ca be see that the frst colum elemets below the secod row are zero. Now defe X, Y such that X a, Y sg( a ) X, U a X Y X Y (6.45) H I UU (6.46) A HAH (6.47) A wll be of the form 6.

12 A a a a a a a a a 0 a a a 0 0 a a 0 0 a a Repeat ths process for tmes ad at the ed of the A has the Hesseberg form th terato the orgal matrx G a a a a a a a a 0 a a a A 0 0 a a 43 4 a Step : Use QR method, based o stable Gram-Schmdt, to covert the matrx G to a Schur form S. Let each colum of the matrx G be cosdered as a depedet vector ad represeted as 3 G W W W W (6.48) Where, a a a a a a W 0, W a, W 3 a a (6.49) 6.

13 Each of these depedet vectors W, W, W3, W ca be expressed terms of learly depedet orthogoal ut vectors. Let oe of the orthogoal ut vectors be chose as Q W (6.50) W so that W r Q, r W (6.5) I a smlar way each of the vector W, W, W3, W ca be wrtte as W r Q, W r Q r Q W r Q r Q r Q W r Q r Q r Q r Q 3 3 (6.5) Here, r to r are scalars. Sce, r, Q are kow rest of the ukows ca be computed as, keepg md that sce Q,... Q are orthogoal Q Q 0f jadq Q Q, j r j W Q j (6.53) Q r W r (6.54) j j Q W j j j r r Q (6.55) 6.3

14 Let, r r r3r 0 r r3 r R 0 0 r33 r r 3 (6.56) Q Q Q Q Q (6.57) Hece, 3 G W W W W QR (6.58) I equato (6.58), Q s a utary matrx ad R s a upper tragular matrx. he method followed so far decomposg a gve matrx G to product of a utary matrx Q ad a upper tragular matrx R s called as stable Gram-Schmdt method. G. C. Fraces has used QR decomposto to trasform a gve matrx G to Schur form. G. C. Fraces has chaged the order of multplcato of QR, equato (6.59) ad the defed a ew matrx G as G RQQ GQ (6.59) Applyg QR decomposto to matrx G obta ew matrces Q, R the G QR (6.60) Smlar to equato (6.59) defe ew matrx G as G RQ QGQ QQGQQ (6.6) 6.4

15 Repeat ths teratve process tll the dfferece betwee the determats of ay two cosecutve terato G matrx values falls below certa value say that s G G. If the algorthm coverges at th terato the G R Q Q G Q Q Q Q Q G QQ Q Q (6.6) he matrx Q Q Q Q G wll be Schur form. Let QQ Q Q the. he matrx s also a utary matrx as the product of utary matrces s also a utary matrx. Hece, matrx s the trasformato matrx whch trasforms the Hesseberg matrx G to Schur form S as gve equato (6.63) SG G (6.63) For a gve real symmetrcal matrx A the ege values are gve by the dagoal elemets of the matrx S gve equato (6.63). If the matrx A s real usymmetrcal or complex the the ege values are gve by the ege values of the block dagoal elemets alog the prcpal dagoal of matrx S. Ege vector Oce the ege values are computed the the rght ad left ege vectors ca be computed. If,,, are the ege values of the matrx A the 0 Λ (6.64) 0 AΦ ΦΛ (6.65) Where, Φ [ ] s the rght egevector matrx. For a th ege value 6.5

16 0 A I A (6.66) Let, a E a a a (6.67) a F (6.68) a Where, a j s a elemet of the matrx A. he let the frst elemet of th rght ege vector, that s gve as be arbtrarly take as. Now the th rght ege vector s E F (6.69) o get a ormalzed rght ege vector dvde equato (6.60) by. he procedure ca be used to fd other rght ege vectors. o fd the left ege vector matrx Ψ smply take the verse of the rght ege vector matrx that s Ψ Φ (6.70) 6.5 Learzg a Nolear System he modal or ege value aalyss ca be doe for a dyamc system whch s lear. However, a olear system caot be expressed state space form ad hece ege value aalyss caot be used for a dyamc system whch s olear. Oe way of overcomg ths problem s to learze the olear system aroud a 6.6

17 small rego of ts steady state operatg pot. hs ca be demostrated as followg. Let a olear dyamc system be represeted as X f X, U (6.7) Where, X s a vector of dyamc varables of the system. U s the vector of puts ad f, XU s a olear fucto of both X ad U. Let the steady state codto values of X, U be X 0, U 0. If the system dyamc varables ad the puts are perturbed by a small value X, U from the steady state operatg pot the from equato (6.7) X X f X X, U U (6.7) Expadg the rght had sde of equato (6.7) as aylor seres wth secod ad hgher order terms eglected lead to f f X 0 X f X0, U0 X U (6.73) X U XX0 XX0 UU0 UU0 Sce, X f X, U 0 0 0, equato (6.73) ca be wrtte as f f X X U (6.74) X U XX0 XX0 UU0 UU0 Let, A f f, X B U, the XX0 XX0 UU0 UU0 X AX BU (6.75) he system gve (6.75) s a lear system ad the stablty of t s decded by the ege values of the state matrx A. hs process of covertg a olear system 6.7

18 gve (6.7) to a lear system for a small dsturbace aroud the steady state operatg pot gve (6.75) s called as learzato. 6.6 Small Sgal Stablty Aalyss of Sgle Mache Coected to Ifte Bus Let us cosder a sgle mache coected to a fte bus as show Fg. 6.. ( I ji ) e j d q jx e Re ( ) j j Vd jvq e Ve t j0 Ve Fg. 6.: Sgle mache coected to fte bus he effectve mpedace betwee the geerator termal ad the fte bus s j0 Re jx e. he fte bus voltage s Ve. he geerator termal voltage ad curret dq0 -axs s gve as Vd jvq ad Id jiq. he teral rotor agle s. Let the put mechacal torque M be costat. Hece, turbe ad speed goveror dyamcs eed ot be cosdered. Let the geerator be represeted by.0 flux decay model. he DAE of flux decay model, excludg excter dyamcs, are gve as deq do q ( d d ) d fd E X X I E (6.76) dt d base (6.77) dt H d m e D base (6.78) dt base Vd RsId XqIq (6.79) 6.8

19 V R I X I E (6.80) q s q d d q Where, Real X X I je I ji e q d q q d q EI X X II q q q d q d * From the Fg. 6., the geerator curret ca be wrtte as ( I ji ) e d q j j0 j ( V ) d jvq e Ve R e jx e (6.8) O further smplfcato ad separatg the real ad magary parts from (6.8), the followg expressos ca be obtaed. RI X I V V (6.8) e d e q d s XI RI V V cos (6.83) e d e q q Let the steady state operatg pot values of E q Efd Id IqVdV q be represeted by E E I I V V q0 0 fd0 d0 q0 d0 q0. Learzg equatos (6.79)-(6.80) ad (6.8)-(6.83) for a small perturbato the parameters aroud the steady state operatg pot leads to, eglectg the stator resstace, V 0 0 d X q Id V q X d 0 I q Eq Vd R Id V cos e Xe 0 V X R I V s q e e q 0 (6.84) (6.85) 6.9

20 Sce, the left had sde of the equatos (6.84) ad (6.85) are same the rght had sdes ca be equated as 0 X 0 q Id R d cos e X V ei 0 X 0 I q E I q Xe R e q V s d 0 (6.86) Solvg for Id, Iq from equato (6.86) lead to q X q s e Xd R e 0 e e q e d I 0 d Re Xe X q V cos 0 I E V Re Xe X q V cos0 0 V s 0 E q e qs 0 e cos0 Xe Xq e d cos0 e s 0 e Re Xe XqXe Xd Xe Xd Re R X X X X V X X RV V X X RV R E q Now learzg the dfferetal equatos (6.76) to (6.78) lead to (6.87) ( Xd Xd ) do Eq 0 0 Id Xq XdI base base d0 Iq base Xq XdI q0 q0 0 H H q0 E q do 0 0 base I D E H H 0 do E fd 0 0 m base 0 H (6.88) Substtutg (6.87) (6.88) ad smplfyg leads to 6.0

21 4 0 0 E q 3 do E do q do E fd m base base base base D 0 H H H H (6.89) Where, s 0 Iq0V Xd Xq Xe Xqs 0 Recos0 cos 0 e d q d0 e q0 Xd XdXe Xq e e q e X d V Xd Xd e q e R e Xe Xq Xe Xd V Xd Xq Id Eq Xe Xd R e Iq Re Xe Xq Xe Xd Xd Xq Xe Xq Re Xe Xq Xe Xd R X X I R E R X X X X X s R cos Re Xe Xq Xe Xd (6.90) A smplfed fast actg statc excter s ow cosdered whch s represeted by a frst order cotrol block wth a ga A ad tme costat A as show Fg. 6.. V t V ref + - A s A E fd Fg. 6. A smplfed fast actg statc excter From Fg. 6. the followg dfferetal equato represetg the dyamcs of the excter ca be wrtte as 6.

22 de fd A Efd A( Vref Vt) (6.9) dt Learzg equato (6.9) gves defd A Efd A( Vref Vt) (6.9) dt he termal voltage ca be wrtte terms of the dq 0 parameters as V V V (6.93) t d q Learzg equato (6.93) V V 0 q0 d V V d0 q0vd Vt Vd Vq V V t Vt Vt Vt q (6.94) Substtutg (6.87) (6.84) lead to V 0 0 d 0 X Re Xe X q q V cos 0 Vq X 0 Eq V s E d Xe Xd R e 0 q (6.95) Substtutg (6.95) (6.94) ad rearragg lead to V E (6.96) t 5 6 q 6.

23 Where, 5 Re Xe Xq Xe Xd Vd 0 X q RV e s 0 V X e X d 0 Vt V V cos X RV cos V X X s q0 d e 0 e q 0 t V V V 6 X R X X X R V V V e Xe Xq Xe Xd d 0 q0 q0 q e d e q t t t Equato (6.9) ca be substtuted (6.89), wth the learzed termal voltage defed (6.96), ad ca be wrtte as do do E do q E q V base 0 base base base D 0 H H H H E E fd fd A 0 A6 A5 0 A A A A ref m (6.97) he costats 6 are called as Heffro-Phllps costats [3] ad the state space model gve (6.97) s called as Heffro-Phllps model [3]. It ca be observed from (6.97) that the olear SMIB system has bee learzed ad ow represeted state space wth the state varables beg Eq,,, E fd. he stablty of the system ow depeds o the ege values of the state matrx. he state space represetato of the system gve (6.97) ca be represeted as a cotrol block dagram as show Fg

24 Fg. 6.3 Heffro-Phllps Model [3] of a SMIB system Covertg equato (6.97) to Laplace doma lead to the followg set of equatos. se () s E () s () s E () s 4 q q fd 3 do do do s() s () s base base base s() s m() s () s Eq() s D() s H H H se () s E () s () s E () s V () s A A fd fd 5 6 q ref A A A (6.98) Smplfyg equato (6.98) E () s () s E () s q s3do s3do E () s () s E () s V () s A A fd 5 6 q ref sa sa fd (6.99) (6.00) () s s () s (6.0) 6.4

25 base base base s () s m() s () s Eq() s Ds () s (6.0) H H H Assumg V () s =0 (6.00), equato (6.00) ca be substtuted (6.99) whch gves ref A 4 s A5 3 Eq () s () s s s 3 do A A 6 3 (6.03) he learzed electrcal torque, e, expresso ca be wrtte as E I E I X X I I X X I I e q q q q q d q d q d d q I d IqEq Xq Xd Iq Eq Xq Xd Id I q (6.04) Substtutg equato (6.87) (6.04), gves E (6.05) e q Covertg (6.05) to Laplace doma ad substtutg (6.03) gves H( s) A A 4 s 5 3 e () s () s s3do sa A6 3 H() s () s (6.06) Substtutg (6.06) (6.0) lead to a secod order equato, assumg the put torque ( s) =0, m base base s Ds H() s () s 0 H H (6.07) 6.5

26 he small sgal oscllatos are typcally the rage of 0. Hz to 3 Hz. I ths rego of frequecy terest the trasfer fucto H( s ) ca be splt to two compoets as H( s j) A A j j 3do j A A63 4 A A 5 3 j A63 3doA j3do A 4 A5 j4a A A6 doa jdo 3 3 (6.08) Neglectg the effect of the costat 4 H( j) A 5 A A6 doa jdo 3 3 S A 5 A 6 doa 3 A A6 doa do 3 3 A A 5do 3 j A A6 do A do 3 3 ( j) j e S d S d d (6.09) (6.0) 6.6

27 he electrcal torque has two compoets sychrosg ad dampg. Sychrosg torque s phase wth the chage rotor agle. he dampg S torque s quadrature to the rotor agle or phase wth the chage the speed. Expressg (6.09) s -doma as H() s S sd (6.) Substtutg (6.) (6.07) base base s Dds s () s 0 H H (6.) Equato (6.) represets a damped secod order uforced system. If the dampg coeffcet D s ot cosdered the the system becomes udamped system wth atural frequecy of oscllatos beg d base j s (6.3) H As ca be uderstood from (6.3), wthout dampg the system wll have sustaed oscllatos wth the frequecy of oscllatos as defed (6.3). If the sychrosg torque becomes zero for a forced system that s M 0 the the system wll become aperodcally ustable that s the rotor agle keeps o creasg ad ultmately lead to loss of sychrosm. I case the dampg coeffcet D becomes egatve due to system codtos the there wll be oscllatory stablty that s the peak of the oscllatos keep o creasg leadg to loss of sychrosm. If both the sychrosg ad dampg torque are postve for a gve system the after the dsturbace the system wll settle dow to a steady state operatg pot wth oscllatos beg damped out. he costat 5 becomes mportat whle assessg the stablty. d 6.7

28 s d A 5 A 6 doa 3 A A6 doa do 3 3 A A 5do 3 A A6 doa do 3 3 (6.4) From (6.4) t ca be observed that f 5 becomes egatve the for a practcal system the sychrosg coeffcet s becomes postve ad creases as compared to 5 beg postve. Hece, wth a egatve 5 the sychrosg torque mproves. But, wth 5 beg egatve t ca be see that the dampg coeffcet d becomes egatve leadg to egatve dampg torque ad hece oscllatory stablty. hs effect of egatve dampg torque due to egatve 5 gets proouced f the excter has a very hgh ga A. I a practcal system 5 ca become egatve a heavly loaded case ad a hgh ga excter ca lead to system stablty ssues [3], [4]. he physcal uderstadg of ths s that a hgh ga excter ca buld up the termal voltage rapdly after a dsturbace mprovg the sychrosg torque but at the same tme t has a egatve dampg, due to ts hgh ga t amplfes a very small error termal voltage ad wthout eough system dampg ca lead to oscllatos. 6.7 Power System Stablzer (PSS) A devce called as power system stablzer [5], [6] ad [7] s used for overcomg the egatve dampg effect of the hgh ga excter. he power system stablzer acts as a supplemetary cotroller to the exctato system. Iputs to the power system stablzer ca be chage frequecy, speed, power or a combato of these. he output s a voltage sgal troduced the exctato system to cotrol the output of the excter. he basc dea of the power system stablzer s to troduce a 6.8

29 pure dampg term (6.) so as to couter the egatve dampg effect of the excter. Let PSS have a trasfer fucto Gs ( ) ad the chage the speed be the put to the (PSS). he output of the PSS be VPSS supplemetary sgal to the excter referece hece. he output of PSS s added as a defd A Efd A( Vref VPSS Vt) (6.5) dt How the PSS output aalysed by fdg the trasfer fucto betwee e VPSS effects the sychrozg ad dampg torque ca be VPSS ad the electrcal torque. o get the trasfer fucto let us assume 0, 0, the from (6.99) ad applyg Laplace trasform to (6.00), the followg expressos ca be wrtte as V ref E () s Efd() s s 3 q 3 do E () s E () s V () s A 6 A fd q PSS sa sa (6.6) (6.7) Substtutg (6.7) (6.6) ad the fdg the trasfer fucto betwee Eq () s ad V ( s) PSS E () s V () s 3 A q s3do sa 3 A6 PSS (6.8) Substtutg (6.8) (6.05) E () s V () s 3 A e q PSS s3do sa 3 A6 (6.9) Equato (6.9) ca be approxmated as 6.9

30 E () s V () s A e q PSS do A6 s sa 3 A6 (6.0) For a hgh ga excter A6 hece 3 e V PSS s 6 do s sa A6 () (6.) Sce, PSS has a trasfer fucto Gs () wth put, (6.) ca be wrtte as e G s 6 do s sa A6 () (6.) I order for the PSS to cotrbute a pure dampg term throughout the frequecy of terest (6.) the trasfer fucto betwee e ad should be do Gs () PSS s s A6 A, where PSS s the ga of PSS, so that s (6.3) e PSS PSS 6 6 Wth the torque defed as (6.3), equato (6.) ca be approxmated ad wrtte as base base s Dd PSS s s () s 0 H 6 H (6.4) 6.30

31 Hece, PSS mproves the dampg of system by provdg a postve dampg torque. Sce, troducg zeros or poles aloe s ot possble a practcal method s to take PSS trasfer fucto as Gs () s s W PSS sw s (6.5) Where, PSS s the ga. W s the washout flter tme costat., are the lead-lag etwork tme costats. s the umber of lead-lag etwork blocks. he block dagram of the PSS s show Fg Wash out Lead-Lag Ga flter etwork sw s VPSS PSS s W s Fg. 6.4: Power System Stablzer (PSS) block dagram. he fucto of washout flter wll be dscussed later. he procedure for fdg the parameters PSS,, s as followg. Let, G EX () s 3 s s A 3 do A 3 A 6 (6.6) Fd the atural frequecy of oscllatos of the system wthout the effect of dampg due to ay other parameter. For ths a costat flux lkage ca be cosdered so that E q s zero, hece e. Also the dampg coeffcet D s assumed zero. Wth these assumptos (6.) ca be wrtte as base s () s 0 H (6.7) 6.3

32 base j (6.8) H At the atural frequecy fd the tme costat of lead-lag etwork such that the lag due to G ( s ) s compesated that s EX G( j ) G ( j ) 0 (6.9) EX For a hgh value of washout tme costat the washout flter wll ot cotrbute ay agle to the trasfer fucto of PSS hece for j j G ( j ) 0 (6.30) EX Oe of the lead-lag tme costats ca be arbtrarly chose the the other tme costat ca be derved from (6.30). If lag of G ( j ) s hgh the multple leadlag blocks ca be used. Sce, the PSS should provde dampg over a rage of small sgal oscllato frequeces (0. Hz to 3 Hz) stead of tug the lead-lag tme costat at a partcular frequecy they ca be tued such a way that over all lag compesato s at least betwee 0 to 45. he dustry practce for choosg the ga PSS s to crease the ga tll the system becomes ustable, say take oe thrd of * PSS. EX * PSS ad the he power system stablzer should ot operate for steady state chages the speed, frequecy or power. It should oly operate oly durg trasets. I order to make PSS operate oly durg trasets washout flter s used. Washout flter acts lke a hgh-pass flter allowg those sgals whch are above the cut-off frequecy. he washout flter tme costat s chose so that the sgal wth lowest frequecy of terest ca be passed through. For washout flter tme costat of 0 s sgals of frequecy 0. Hz ad above ca be passed. he stead state chages ca be cosdered as DC sgals ad hece wll be blocked by the washout flter. W 6.3

33 Let the output of the washout flter be block. Let the output of the lead-lag block be equatos of the PSS ca be wrtte as VWF ad ths s the put to the lead-lag VPSS the the learzed dyamc V WF VWF PSS (6.3) W V V V V (6.3) PSS PSS WF WF From (6.97) the rate of chage of speed s kow ad ca be substtuted (6.3). Smlarly V WF ca be substtuted (6.3) hece, the followg expresso ca be derved E q base base base V base ref V WF PSS PSS DPSS PSS H H H W E fd H m VWF V PSS (6.33) E q base base base V PSS PSS PSS DPSS 0 H H H W E fd VWF V PSS V base ref 0 PSS H m (6.34) he state space model cludg the PSS dyamcs s gve as 6.33

34 do do do E q Eq base base base D H H H A6 A5 E fd A A E fd A V WF base base base V WF PSS PSS DPSS 0 0 V PSS H H H V W PSS base base base PSS PSS DPSS 0 H H H W base 0 H A Vref 0 A m base 0 PSS H base 0 PSS H (6.35) 6.8 Small Sgal Stablty Aalyss of Mult-Mache System he small sgal stablty of a sgle mache coected to a fte bus has bee dscussed the prevous secto from whch the effect of exctato system o stablty was uderstood. Power system stablzer whch s used to couter the egatve dampg ducg effect of exctato system was explaed. Now the small sgal stablty for a mult-mache system cludg the power system stablzer at each geerator wll be explaed. Let each geerator be modelled by a sub-traset model alog wth a power system stablzer ad stator algebrac equatos. he etwork s represeted by real ad reactve power balace equatos. he DAE of a geeral power system are gve as followg 6.34

35 Dfferetal equatos (for,,..., g, geerator) " deq ( Xd Xd ) do ( d d ) d q ( d ls ) d d dt ( Xd Xls ) X X I E X X I E q E fd (6.36) " d d do Eq ( Xd Xls ) Id d (6.37) dt E X X I E X X I " de ( Xq X ) d q qo d ( q q ) q d ( q ls ) q q dt ( Xq Xls ) (6.38) " d q qo Ed ( Xq Xls ) Iq q dt (6.39) d base dt (6.40) H d D m e base base (6.4) dt de fd E VR E SE ( Efd ) Efd (6.4) dt dv V R E V V (6.43) R A F A R A F fd A ref t dt F dr dt F F RF E fd (6.44) F base V WF VWF PSS m e D base H (6.45) W V V V D base PSS PSS WF PSS m e base W H (6.46) d M HP RH HPRH RH M PCH PSV dt CH CH (6.47) dpch CH PCH PSV dt (6.48) dpsv SV PSV Pref dt RD base (6.49) 6.35

36 Stator algebrac equatos ( X X ) ( X X ) V cos( ) R I X I E 0 " " " d ls d d s q d d q d ( Xd Xls ) ( Xd Xls ) ( X X ) ( X X ) V s( ) R I X I E 0 " " " q ls q q s d q q d q ( Xq Xls ) ( Xq Xls ) (6.50) (6.5) Network equatos For,,,..., g, geerator buses IVs( ) IVcos( ) P( V) VVYcos 0 (6.5) d q D j j j j j IVcos( ) IVs( ) Q ( V) VVYs 0 (6.53) d q D j j j j j For,,,...,, for load buses g g D j j j j j P ( V) VV Y cos 0 (6.54) D j j j j j Q ( V) VV Y s 0 (6.55) he dfferetal equatos (6.36) to (6.49) ca be represeted as X f X, I, I,, V, U (6.56) d q g g Where, ( g umber of geerators ad umber of buses) X Eqd Ed q VR Efd RF M PSV PCH VFW V PSS,..., U Vref P ref g..., V..., g g V V g l...,... g V l V V g g 6.36

37 he stator equatos (6.50) ad (6.5) ca be wrtte as g X, I, I,, V 0,,..., (6.57) d q g g g he etwork power balace equatos at the geerator buses (6.5) ad (6.53) ca be wrtte as h X, I, I,, V,, V 0,,..., (6.58) d q g g l l g he etwork power balace equatos at the load buses (6.54) ad (6.55) ca be wrtte as k, V,, V 0,,..., (6.59) g g l l g he fuctos f, ghk,, are o lear fuctos. Let the steady state operatg pot of the system be represeted by X0 Id_0 Iq_0U 0 for,..., g ad the etwork voltages ad agles tal codto are represeted as g0vg0l0v l0. Learzg (6.56) aroud the tal operatg codtos, that s the partal dervatve are evaluated at the tal codto, lead to A B I dq W B g f f I f d f V f g f 00 X I Id Iq q g V g U l X X U Vl A,44 X,4 B,4 Idq, B 0 g 4 g V V W U,4 g g l 4 (6.60) 6.37

38 Each of the matrx sze (6.60) s gve as a subscrpt to the correspodg matrx. If the learzed equato (6.60) s formulated for all the geerators ad the equatos are put together the X A X B I,4g4g 4g,4g g dq,g 0 B V V W U 4 g g,4gg g g l 4gg g (6.6) Where,...,...,... g g dq dq dqg X X X U U U I I I Smlarly, learzg (6.57) C C D g g g I g d g V g g X X I Id I q q g V g l Vl C,4 X,4 C, Idq, D, 0 g g g Vg l V l 0 (6.6) learzed equato (6.57) of all the geerator together ca be wrtte as C, 0 0 g 4 g X 4 g C, g g Idq, g D, g g g g g Vg l V l (6.63) Smlarly, learzg (6.58) ad (6.59) lead to 6.38

39 C C 3 4 D D3 g h h I h d h h h h l X 0 X I Id Iq q g Vg l V l V g Vl C3,4 X,4 C4, Idq, D, D V V 0,,..., g 3, g g g l l g or (6.64) C X C I D D 3, 4 4 4,,, 3, V V 0 g g g g g dq g g g g g l l g g (6.65) D5 D6 g k k k k l 0 g Vg l Vl Vg Vl D5, D 0,..., g 6, g g Vg l Vl g or (6.66) D D 0 5, g g 6, g g g Vg l V l (6.67) Equatos (6.6), (6.63), (6.65) ad (6.67) ca be wrtte as 00 X AX BIdq B g Vg l V WU (6.68) 0CX CIdq D0 0 g Vg l V l (6.69) 0 C3X C4Idq D D 3 g Vg l V l (6.70) 0 D4 D 5 g Vg l V l (6.7) From (6.7), the load bus voltages ad agles ca be elmated as l Vl D5 D 4 g V g (6.7) 6.39

40 Let, D D D D D (6.73) Substtutg (6.73)-(6.7) (6.70) ad (6.69), ad elmatg the geerator bus voltages, agles, drect axs curret ad quadrature axs curret gves I dq C D C X g C4 D 6 C 3 V g (6.74) Substtute (6.74) (6.68), I dq X AX B B g WU V g Asys C D C A X WU A B B X W U C4 D 6 C 3 sys (6.75) Equatos (6.75) s state space form wth the state trasto matrx A sys, put matrx W, state vector X ad put vector U. he small sgal stablty of the mult-mache system ca ow be kow by fdg the ege values of the state trasto matrx A sys [7]. Sce, the rotor agle of ay sychroous mache should be defed wth respect to a referece, oe state varable correspodg to the rotor agle of a geerator whch s take as referece becomes redudat hece there wll be oe zero or a slghtly postve ege value correspodg to the redudacy of the referece agle. 6.40

41 6.9 Sub-Sychroous resoace he sub-sychroous resoace pheomeo [8] was frst observed 975 Mohave, Calfora where the turbe-geerator shaft has faled due to torsoal oscllatos teractg wth the sub-sychroous oscllatos, produced because of seres compesato trasmsso le. o uderstad ths pheomeo let there be a sgle mache coected to fte bus. Let the geerator be represeted by a subtraset voltage " E behd a sub-traset reactace betwee the geerator termal ad the fte bus be Re jx Fg " X. Let the effectve mpedace e. he system s show " X X e Re X C " E V Fg. 6.5: Seres capactor compesated SMIB system. Let the system be compesated by a seres capactor wth capactace C as show Fg Whe a fault or a dsturbace occurs the there wll be a traset the curret wth a off set ad ths off set oscllate at a atural frequecy of ( case there s o seres capactor the the off set would be DC) LC X c f fs Hz, X s X c rad / s X (6.76) Where, f s s the sychroous frequecy ad " X X Xe. hese oscllatos at the frequecy f the stator curret of the geerator wll duce a slp frequecy, f s f voltages the rotor crcuts ad there by producg slp frequecy torque. If 6.4

42 f s a sub-sychroous frequecy the, sce the rotor s rotatg at sychroous speed the steady state, the slp s ( f f ) / f wll be egatve. hs codto s s smlar to ducto motor rug at super sychroous speed wth a egatve slp. X ef Ref R r s s Fg. 6.6: Approxmate equvalet crcut of the sychroous geerator wth sub sychroous frequecy stator currets he approxmate equvalet crcut of the sychroous geerator wth subsychroous frequecy stator currets s show Fg he magetsg ad core loss compoet brach s gored. he R ef s the summato of the trasmsso le resstace, stator resstace, rotor resstace referred to stator. Smlarly X ef summato of the trasmsso le reactace, stator leakage reactace, rotor leakage reactace referred to stator. Sce the slp s egatve the varable rotor resstace becomes egatve leadg to self exctato the crcut ad the oscllatos wll keep creasg reachg uacceptable levels. Before dscussg the teracto of subsychroous oscllatos, due to ducto motor effect, wth the torsoal oscllatos the torsoal oscllatos wll be brefly dscussed. 6.0 orsoal oscllatos If two rotatg masses are coected together by a o rgd shaft the the two eds of the shaft are relatvely dsplaced wth respect to each ad wll have a agle of dsplacemet due to whch the shaft gets twsted. he torque delvered through the shaft s drectly proportoal to the relatve agle of dsplacemet betwee the two eds of the shaft. I case the relatve agle of dsplacemet betwee the two eds of the shaft oscllates the that s called as torsoal oscllatos [9], [0]. o uderstad 6.4

43 the basc pheomeo of torsoal oscllatos let a geerator be cosdered coected to a turbe, wth sgle mass, through a shaft as show Fg urbe Shaft Geerator rotor H H M Fg. 6.7: urbe-geerator system Let the geerator ad turbe erta costats be H, H. he relatve agle of dsplacemet betwee the two eds of shaft s. he stffess of the shaft s represeted by. he torque delvered through the shaft to the geerator rotor s ca be wrtte as. he equato of moto of the two masses that s turbe ad geerator H d d d D D M (6.77) s dt dt dt H d d d (6.78) D D e s dt dt dt Where, D s the dampg due to shaft ad D, D are the dampg costats of the turbe ad geerator rotor. Multply (6.77) by H ad (6.78) by H ow subtract both the equatos, eglectg the effect of dampg due to dampg coeffcets D, D, s HH d d H H D H H dt dt H H H H M e (6.79) 6.43

44 Learzg (6.79) ad assumg the put mechacal torque s costat leads to d H H d H H s s D s e dt HH dt HH H d dt d dt e (6.80) Equato (6.80) shows that t represets a secod order damped system wth atural frequecy of oscllatos beg. hese oscllatos are called as torsoal t oscllatos. If electrcal torque dsturbace s of the form e cos( t ), the the geeral soluto of the equato (6.80) s gve as t t cos t s t e e cos t P (6.8) Where,, are costats depedg o the tal codtos. P, are gve as ta 4 P (6.8) he torque at the shaft due to the electrcal torque dsturbace s gve as shaft (6.83) It ca be observed from (6.8) that f the appled electrcal torque dsturbace s udrectoal that s 0 the P. If the appled electrcal torque dsturbace has a frequecy close to the atural frequecy of the shaft that s the P. he atural frequecy of oscllatos of the shaft s the rage of 0 Hz to 40 Hz. Now f the frequecy of the electrcal torque dsturbace s equal to or greater tha the system frequecy the P s. he value of P wll be least the 6.44

45 case whe the electrcal torque dsturbace s equal to the shaft atural frequecy of oscllatos ad hece has the maxmum effect o the shaft torque as gve (6.83) leadg to a sever stress the shaft. Apart from ths f the electrcal torque dsturbace has egatve dampg, usually s qute hgh ad the secod term (6.8) damps out fast, the the electrcal torque dsturbace keeps creasg magtude causg extreme stress the shaft. hough there s shaft dampg t s very low ad does ot help ths stuato. hough the pheomeo of torsoal oscllatos was explaed usg two masses a actual system lke thermal geeratg system there wll be HP turbe, multple IP turbe, LP turbe, geerator rotor ad f the geerator has DC excter the rotor mass of that excter. here wll be torsoal modes of oscllatos f dfferet masses are coected together by a shaft. If the torque produced by the curret the rotor of the sychroous mache due to the sub-sychroous oscllatos the etwork has a frequecy that s close to the atural frequecy of torsoal oscllatos the t wll lead to very hgh shaft torque shaft. If sub-sychroous oscllatos are self exctg type the ths problem becomes eve more severe ad may ultmately lead to falure of the shaft. A way of avodg ths s to choose the seres compesator such that the atural frequecy of oscllatos of LC crcut does ot produce sub-sychroous oscllatos whch ca excte the torsoal modes. he other way s to have eough mpedace the etwork to couter the egatve resstace due to ducto motor effect ad there by dampg the sub-sychroous oscllatos so as to prevet exctg the torsoal modes. 6.45

46 Example Problems E. Fd the ege values, ege vectors ad partcpato factor of the system represeted by state space model ad whose state matrx s gve as é ù A = ê ë3 4 ú û Sol: he characterstc equato s gve as ( li A) - = 0 æl 0 ö é ù é ù ç ê - ú = ( l-)( l-4) - 6= l -5l- = 0 çê0 lú ê3 4ú çèë û ë ûø (E.) he soluto of the characterstc equato sl = , l = Sce, oe of the ege values s postve the system s ustable. he rght ege vector matrx Φ s defed as AΦ= ΦΛ (E.) Where, é ù Λ = (E.3) ê ë ú û Let the rght ege vector matrx be represeted as éf F ù = ê F F ú ë û Φ (E.4) he, éf ù é0ù - = êf ú ê ë0ú ë û û [ l ] I A (E.5) 6.46

47 éf ù é0ù - = êf ú ê ë0ú ë û û [ l ] I A (E.6) Now, [ F F ] s the rght ege vector correspodg to the ege value l =- ad [ ] F F s the rght ege vector correspodg to the ege valuel = From equato (E.5) ad (E.6), the followg expresso ca be wrtte.373f + F = 0 (E.7) 3F F = 0 (E.8) Equatos (E.7)-(E.8) are homogeeous equatos wth a trval soluto of [ ] F F = [0 0]. hs s because the equatos are ot depedet, that s oe of the equatos ca be wrtte as a multple of the other equato. ypcally, a th order homogeeous equato has ( - ) depedet equatos. Hece, for a o trval soluto let us assume the value of F = ad from oe of the equatos (E.7) or (E.8) we ca obtaf. Hece, éf ù é ù = êf ú ê ë ú ë û û (E.9) Sce, the soluto s ot uque t s better to ormalze the rght ege vector hece éf ù é ù é ù é ù = = = ê ú ê 0.686ú.8 ê 0.686ú ê ú ëf û é ù ë- û ë- û ë- û ê ë-0.686ú û (E.0) Smlarly, the ormalzed rght ege vector correspodg to the ege value l = s gve as 6.47

48 éf ù é0.460ù = êf ú ê ë0.9094ú ë û û (E.) Hece, é ù Φ = (E.) ê ë ú û he left ege vector ca be defed as - é Ψ= Φ = ù (E.3) ê ë ú û Each row of the matrx Ψ s a left ege vector correspodg to the respectve ege value, that s frst row s a left ege vector correspodg to the ege value l = he partcpato factor of the states the th mode s gve as éf Y ù FY P =, =,, F êë Y úû (E.4) he partcpato factor matrx s gve as P = [ ] (E.5) P P P he partcpato factor ca ow be computed as éfy FY ù é ù = = êfy F ú ê Y ë ú ë û û P (E.6) 6.48

49 E. Fd the ege values, ege vectors ad partcpato factor of the matrx gve below usg QR method. é 4 9 7ù 0-4 A = 3 7 ê0-3ú ë û Sol: he frst step usg QR method to fd the ege values s to covert the matrx A to Hesseberg form. Defed two colum matrces XY, ad ut vector U as é0ù é 0 ù é 0 ù.44, sg( A(,)) X - X= Y=- = 0 0 ê0ú 0 ê 0 ú ë û êë úû ë û (E.) U é 0 ù X-Y = = X-Y ê 0 ú ë û (E.) Defg Householders trasformato H ad applyg the trasformato to the matrx A lead to H IUU é ù = = ê ú ë û (E.3) A H AH (E.4) 6.49

50 It ca be observed from the trasformed matrx A that the rows below the subdagoal of the frst colum have become zero that s 3 rd ad 4 th row, st colum elemets have become zero. After repeatg ths for all colums we fally get Hesseberg form of matrx A as s gve as é ù A H = (E.5) ê ú ë û Secod step s to apply QR method to the matrx Hesseberg form, gve (E.5), so as to covert t to Schur form. From equatos (6.48) to (6.57) the matrx QR, for the frst terato ca be obtaed as é ù Q = (E.6) ê ú ë û é ù R = (E.7) ê ú ë û It ca be observed that A H = QR. Defe a ew matrx A RQ (E.8) H = Fd QR, for the ew matrx defed (E.8) repeat ths process tll the dfferece betwee the cosecutve terato QR, matrces s less tha a specfed value. he teratve process coverges after teratos. he matrx Schur form s gve as 6.50

51 é ù A Schur = (E.9) ê ú ë û he ege values are gve as é ù j Λ = j ê ú ë û (E.0) he left ad rght ege vectors ca be computed from equatos (6.67) to (6.69) ad s gve (E. ) é ù j j Φ = j j ê j j ú ë û (E.) - Ψ= Φ (E.) Partcpato factor matrx, ca be computed as é ù P = Ψ Φ = (E.3) ê ú ë û 6.5

52 E3. A two area test system cosstg of four geerators ad eleve buses s show the fgure E3. below G G4 G 3 G3 Fg. E3.: Sgle le dagram of two area test system [7] he geerators G, G are area- ad geerators G3, G4 are area-. All the four geerators are of same ratg that s three-phase 60 Hz, 0 kv ad 900 MVA. he trasmsso le parameters are defed o the base of 30 kv, 00 MVA. he system base s take as 00 MVA for load flow aalyss. At bus-7 ad bus-9 two capactors of 00 MVAr ad 350 MVAr are coected, respectvely. he bus data s gve able E3., le data s gve able E3., le flow data s gve able E3.3. A et real power of 00 MW s beg trasferred from area- to area-, whch s from bus-7 to bus-8, as ca be observed from able E3.3. he geerators parameters o 900 MVA, 0 kv bases are gve below. " " s ls d d d d0 d0 " " q = q = q = q0 = q0 = = R = 0, X = 0., X =.8, X = 0.3, X = 0.5, = 8 s, = 0.03s X.7, X 0.55, X 0.4, 0.4 s, 0.05 s, H 6.5 s he statc hgh ga excters are used at all the four geerators ad the parameters are gve as = 00, = 0.0. A R 6.5

53 able E3.: Bus data GENERAION (pu) LOAD (pu) ANGLE BUS VOLAGE (pu) (degrees) REAL REACIVE REAL REACIVE able E3.: Le data (pu) From Bus o Bus Resstace Reactace Le chargg

54 able E3.3: Le flow data LINE FLOWS LINE FROM BUS O BUS REAL REACIVE (a) Fd the ege values of the system ad commet o the system stablty. (b) I case power system stablzers are used at all the four geerators the fd the ege values ad commet o the effect of power system stablzer o the system. he power system stablzer parameters are gve as 50, 0 s, 0.05s, 0.0 s, 3.0 s, 5.4 s PSS = W = = = 3= 4 = Sol: Wth the gve system data frst the geerators are talzed usg equatos (5.76) to (5.89). he the system s learzed aroud the tal operatg pot. Each geerator s represeted by a 7 th order model,. geerator model alog wth frst order excter model. he state vector of a th geerator s gve as D X = é d, w, Ed, Eq, yd, yq, E ù êd D D D D D D fdú, =,,3,4 ë û (E3.) Sce, there are four geerators there wll be 8 states. he state matrx of the learzed system ca be computed as gve equato (6.75). he sze of the state matrx s 8 8 ad hece there wll be 8 ege values. he ege values ca be computed from the state matrx as explaed secto 6.4 usg QR method. 6.54

55 he ege values of the system are gve able E3.4. Each ege model alog wth ts frequecy of oscllatos ad dampg rato are gve able E3. 4. It ca be observed from able E3.4 that there s oe ege mode wth slghtly postve value ad a secod ege value very close to zero. he frst zero ege value s because there s o referece for the geerator agles ad hece the agles are ot uque. Secod zero ege value s because of the assumpto that the geerator electrcal torque s ot effected by the speed varatos. able E3.4: Ege values of the system wthout PSS Frequecy Dampg No Ege Value (Hz) rato j j j j j j j j j j j j j j he ege mode 7 ad 8 are complex cojugate pars wth a very low dampg rato of ad frequecy of oscllato of Hz. hese two modes are ter area modes. he ormalzed partcpato factor of the states mode 7 are gve Fg. 6.55

56 E3.5. It ca be observed that the rotor agle ad speed of geerator G, G, G3 ad G4 are partcpatg maxmum ths mode. Fg. E3.5: Partcpato factor of state ege mode 7 Fg. E3. 6: Mode shape of the ege mode 7 he elemets, correspodg to the agles of geerators G, G, G3 ad G4, of the rght ege vector of the ege mode 7 are plotted Fg. E3. 6. It ca be observed that 6.56

57 the geerators G, G are oe drecto ad geerators G3, G4 are almost opposte drecto. From ths t ca be uderstood that ege modes 7 ad 8 are due to the ter area oscllatos betwee geerator G, G area- agast geerator G3, G4 area-. (b) able E3. 5: Ege values of the system wth PSS Frequecy Dampg No Ege Value (Hz) rato

58 he ege values of the system wth PSS at all the four geerators s gve able E3. 5. Each PSS s represeted by three dyamc equatos. Hece, the total umber of state wll be 40. he ege mode 5 ad 6, wth dampg ad frequecy Hz, are ter area modes. he partcpato of state ege mode 5 s gve Fg. E3. 7. Fg. E3. 7: Partcpato of the states ege mode 5 It ca be observed that the states agle ad speed of geerator G, G, G3 ad G4 are partcpatg maxmum. Hece, t ca be sad that wth PSS coected at the geerators G, G, G3 ad G4 the dampg of the ter area mode has creased from to 0.057, whch s a very sgfcat mprovemet. 6.58

59 Refereces. A. M. Lyapuov, Stablty of Moto, Eglsh traslato, Academc press Ic., J.G.F. Fracs, he QR rasformato-a utary aalogue to the LR trasformato, he computer Joural, Part I, Vol. 4, pp. 65-7, 96; Part II, pp , W. G. Heffro ad R. A. Phllps, Effects of moder amplydye voltage regulators uderexcted operato of large turbe geerators, AIEE ras., PAS-7, pp , August F. P. DeMello ad C. Cocorda, Cocepts of sychroous mache stablty as affected by exctato cotrol, IEEE ras., PAS-88, pp , Aprl E. V. Larse ad D. A. Swa, Applyg power system stablzers, Part I: geeral cocepts, Part II: Performace objectves ad tug cocepts, part III: practcal cosderatos, IEEE Power Appar. Syst., PAS-00, Vol., pp , December P. udur, M. le, G. J. Rogers ad M. S. Zywo, Applcato of power system stablzer for ehacemet of overall system stablty, IEEE ras., Vol. PWRS-4, pp , May P. udur, G. J. Rogers, D. Y. Wog, L. Wag, ad M. G. Lauby, A comprehesve computer program for small sgal stablty aalyss of power systems, IEEE ras., Vol. PWRS-5, pp , November C. Cocorda, J. B. ce, ad C. E. J. Bowler, Sub-sychroous torques o geeratg uts feedg seres-capactor compesated les, Amerca Power Coferece, Chcago, May 8-0, M. C. Jackso, S. D. Umas, R. D. Dulop, S. H. Horowtz, ad A. C. Parkh, urbe-geerator shaft torques ad Fatgue: part I-smulato methods ad fatgue aalyss; Part II-Impact of system dsturbace ad hgh speed reclosure, IEEE ras., Vol. PAS-98, No. 6, pp , November D.N. Walker, S. L. Adams ad R. J. Placek, orsoal vbrato ad fatgue of turbe-geerator shafts, IEEE ras., Vol. PAS-00, No., pp , November