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The dynamic stability of voltage-regulated and speed-governed synchronous machines in power systems. 1964

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THE DYNAMIC STABILITY OP VOLTAGE-REGULATED AND SPEED- GOVERNED SYNCHRONOUS MACHINES IN POWER SYSTEMS by KHIEN VONGSURIYA B.E., Chulalongkorn U n i v e r s i t y , 1960 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE In the Department of E l e c t r i c a l Engineering We accept t h i s t h e s i s as conforming to the standards required from candidates f o r the degree of Master of Applied Science Members of the Department of E l e c t r i c a l Engineering The U n i v e r s i t y of B r i t i s h Columbia SEPTEMBER 1964 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study, I f u r t h e r agree that per- m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i - c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of ^-/UAyn<^i^ ' The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada Date V4 <> *t ABSTRACT The dynamic s t a b i l i t y of a power system has been of constant i n t e r e s t f o r many y e a r s . Among the many c o n t r i b u t o r s are Concordia, H e f f r o n , P h i l l i p s , and Messerle. In most s t u d i e s , the t i e l i n e r e s i s t a n c e and the s a l i e n c y e f f e c t were n e g l e c t e d . An attempt has been made i n t h i s t h e s i s to study the dynamic s t a b i l i t y of the power system i n the small i n c l u d i n g a l l important e f f e c t s such as t i e l i n e r e s i s t a n c e , reactance, s a l i e n c y , v o l t a g e r e g u l a t o r , s t a b i l i z e r time constants and gains, and governor time constants and gai n . Two d i f f e r e n t methods f o r studying s t a b i l i t y have been applied„ namely, the Routh-Hurwitz c r i t e r i o n and the D — p a r t i t i o n method. The important r e s u l t s found i n the study are the s i g n i f i c a n t e f f e c t s of t i e l i n e r e s i s t a n c e and s a l i e n c y upon the s t a b i l i t y l i m i t from Routh-Hurwitz c r i t e r i o n s t u d i e s and upon the s t a b i l i t y r e g i o n from s t u d i e s using the D - p a r t i t i o n method. A l s o , s i g n i f i c a n t are the e f f e c t s of the machine short c i r c u i t ratio-, :"the governor, the e x c i t e r and s t a b i l i z e r time constants and gains, and t h e i r c o o r d i n a t i o n . i ACKNOWLEDGEMENT The author would l i k e to express h i s sinc e r e g r a t i t u d e to h i s s u p e r v i s i n g p r o f e s s o r , Dr. T. N. Yu who guided and i n s p i r e d him throughout the course of the r e s e a r c h . The author a l s o wishes to thank members of the Department of E l e c t r i c a l E n g i n e e r i n g e s p e c i a l l y Dr. F. Noakes and Dr. A. D. Moore f o r t h e i r generous a s s i s t a n c e and encouragement. The encouragement of Dr. B. Binson of Th a i l a n d i s a l s o g r a t e f u l l y acknowledged* The author i s indebted to the Columbo P l a n A d m i n i s t r a t i o n i n Canada f o r the s c h o l a r s h i p granted to him during 1962 to 1964 and to the N a t i o n a l Research C o u n c i l of Canada f o r f i n a n c i a l support of the r e s e a r c h . TABLE OP CONTENTS Page A b s t r a c t *o.oso.eo.eo.o...oo««ooo. o ooooooooooooooe 1 T ab l e o f Contents . . o . o o . e o . . • « < > » . • ,» a * a . « o « . L i s t Of I l l u s t r a t i o n s ooooo.«*«o»ee»oooooo.ooo«oo« v AcknOWled-gement « « « a a o a o'o o a .«oo.eo«»ooooooooooooooo v i i 1» I n t r o d u c t i o n • a s a a a o o o e•a »*a•*• aa o o o a o o e s a o » o s 1 2. Review of General Theories ».:*«• • o o. o o o o <>.o a.. 5 2#1 C l a s s i c a l Synchronous Machine Theory .... 5 2»2 S t a b i l i t y of a Dynamieal System ......... 8 2.2.1 S t a b i l i t y i n the small ........... 8 2.2.2 S t a b i l i t y of d system w i t h l i n e a r i z e d d i f f e r e n t i a l equations 8 2.2.3 The Routh-Hurwitz s t a b i l i t y c r i t e r i a «.*aaaa•*aaa............. 9 2.2.4 D - p a r t i t i o n method f o r two r e a l par am e t e r s ...... ......oooo........ 10 3» D e r i v a t i o n of System Equations f o r the Dynamic Stab i i i t y S t u d i e s .....»«•.«•«..«...........•. 13 3*1 The System Under Study . . o . . . . . . . . . . . . . 13 3*2 System Eq u a t i o n .o«s.*«eae>aa«o<>o<>...<>....o 14 3.2.1 The synchronous machine .......... 14 3.2.2 The txe l i n e aaa«a*aaa..o.o.os..oa 15 3.2.3 The v o l t a g e r e g u l a t o r and the g O V e r no P ..aaaaa a* a'aaoooooo.ooooaa 15 3*3 L i n e a r i z a t i o n of The System Equations ... 16 3»4 The C h a r a c t e r i s t i c E quation of the System 17 3»5 The Routh-Hurwitz Determinants of the C h a r a c t e r i s t i c E quation o . o ........ o 20 3«6 The D - p a r t i t i o n Equations *•............. 21 4, System Parameters and I n i t i a l C o n d i t i o n s of Synchronous Machines . . o . . . . . . . . . . . . . . . . . . . . . . 23 i i i i. Page 4.1 System Parameters •«..••«•«•»« o « . o ... ... « . 23 Synchronous Machines »«««• « . . « . » o <>. . ... . o 23 Voltage Regulator S e t t i n g s •.•••»»«.•«•.. 23 Governor Settings.*••«•••••••«»».»».•««.o 23 4.2 I n i t i a l C o nditions of Salient--pole Synchronous Machine «•• ....... ........... 24 5. : Dynamic S t a b i l i t y Studies Using Routh-Hurwitz C r i t e r i o n .............................. » 27 5«1 E f f e c t of S a l i e n c y on the Dynamic S t ab i i i t y . . . . . . . . . . . . . . . . . . . . . . . . . . . a . . . 27 5.2 E f f e c t of the Short C i r c u i t Ratio of a Synchronous Machine . . . o . . . . . . . . . . . . . . . . . 29 5*3 E f f e c t of the Tie Line Impedance ........ 31 5.3.1 E f f e c t of the l i n e r e s i s t a n c e .... 31 5.3.2 The e f f e c t of the 1 l i n e re ac tance o «-« . . » . . . « ^ . e . . . . . . . . . . . 31 5.4 E f f e c t of E x c i t e r Time Constant ......... 34 5.5 E f f e c t of the Governor Time Constants on the Dynamic S t a b i l i t y L i m i t ............. 34 5.5.1 Governor w i t h one time constant 34 5.5.2 Governor w i t h two l i n e constants .. 37 6. Dynamic S t a b i l i t y Studies Using the D- ^ p a r t i t i o n Method «. . . . . « » « . « « . « « . « . o . . . . o . . . . . . 40 6.1 The E f f e c t of S a l i e n c y on the S t a b i l i t y BoUIlcLclPi GS « a o o o e « o 0 e 9 « » « * « « o o o « a a o a a « a o o 40 6*2 The E f f e c t of the T i e Line Resistance and Reactance a « a o o o « « « » « « O i « « ' » * a « o « « o « e o « « 0 » 42 6.3 N e g l e c t i n g Both the Tie Line Resistance and S a l i e n c y i n a Round-rptor Machine ... 44 6*4 The E f f e c t of Short C i r c u i t Ratio 44 6.5 The E f f e c t of the E x c i t e r Time Constant . 44 6*6 The E f f e c t of the S t a b i l i z e r Time Constant * B « O « O O « « « « O O . « « « * « « « « » O O O O O O O O « O 4T 6*7 The E f f e c t of the Governor Time Constants 47 Page 7. Conclusions ..«•«• ...... o ............ 52 APPENDIX I . Symbols and U n i t s 54 APPENDIX I I . Voltage Regulator and Speed Governor T r a n s f e r Functions ........ 57 APPENDIX I I I . D e r i v a t i o n of the C h a r a c t e r i s t i c E q uation ............ 59 APPENDIX IV. D e r i v a t i o n of the D - p a r t i t i o n Equations ... . o « » . . . . . . . . . . . . . . . . 0 . . 62 REFERENCES . . . . . . . . . . . . . . . . o o o . . . . . . . . . . . . . . . . . . . . 64 i v LIST OP ILLUSTRATIONS Figure P a g e 1—1 Synchronous Machine Models .............. 6 3»1 Schematic of System Under Study .... 13 3-2 Voltage Phasor Diagram of the I n f i n i t e Bus and a Synchronous Machine 13 5-1 S a l i e n c y E f f e c t on Power L i m i t s without Governor ................................ 28 5-2 S a l i e n c y E f f e c t on Power L i m i t s with a One-time-constant Governor .............. 28 5-3 S a l i e n c y E f f e c t on Power L i m i t s with a Two-time-constant Governor 30 5-4 Short C i r c u i t Ratio E f f e c t on Power L i m i t s .................................. 30 5-5 T i e Li n e Resistance E f f e c t on Power L i m i t s .................................. 32 5—6 Tie Line Resistance E f f e c t on Power L i m i t at U n i t y Power F a c t o r ................... 32 5-7 Tie Line Reactance E f f e c t o n Power L i m i t s 33 5—8 Tie Line Reactance E f f e c t on Power L i m i t s at U n i t y Power F a c t o r «................. . 33 5—9 E x c i t e r Time Constant E f f e c t on Power 5—10 E x c i t e r Time Constant and Gain E f f e c t on Power L i m i t s .............o .............. 35 5—11 E f f e c t of the Qne-time-constant Governor on Power Limxts . * « • « . . . . . . . . . . . . . . o o o . . . 36 5—12 E f f e c t of the Gain of a One-t time-constant Governor on Power L i m i t s ................ ^ 36 5—13 E f f e c t of the T w o - t i m e - i C o n s t a n t Governor on Power L i m i t s 38 5— 14 E f f e c t of the Gain of a Two—time~*constant Governor on Power L i m i t s 38 6- 1 S a l i e n c y E f f e c t on the S t a b i l i t y Region . 41 6—2 S a l i e n c y E f f e c t on the S t a b i l i t y Region f o r V a r i o u s Power L i m i t s ................ 41 v F i g u r e Page 6-3 Tie ; L i n e R e s i s t a n c e E f f e c t on the S t a b i l i t y R e gion •.«.«••«••• •» 43 6-4 T i e L i n e Reactance E f f e c t on the S t a b i l i t y R e g ion ..•.......••••«..«........ a...... 43 6-5 E f f e c t of N e g l e c t i n g T i e L i n e R e s i s t a n c e and S a l i e n c y on the S t a b i l i t y Region ... 45 6-6 E f f e c t of S h o r t C i r c u i t R a t i o on the S t a b i l i t y R e gion 45 6-7 E x c i t e r Time Constant E f f e c t on the S t a b i l i t y R e gion of a Round-rotor Machine 46 6—8 E x c i t e r Time Constant E f f e c t on the S t a b i l i t y R e gion of a S a l i e n t - p o l e Machine . . . . . . . . a o . . . . . . * . . « « . . . . . . . . . . . 46 6-9 S t a b i l i z e r Time Constant E f f e c t on the S t a b i l i t y Region of a Round-rotor Machine 48 6-10 S t a b i l i z e r Time Constant E f f e c t on the S t a b i l i t y Region of a S a l i e n t - p o l e Machine 48 6—11 E f f e c t of the Two-time - r C b n s t a n t Governor on the S t a b i l i t y Region of a Round-rotor Machine ........................ 49 6-12 E f f e c t of the Two-time-constant Governor on the S t a b i l i t y R e gion of a S a l i e n t - p o l e Machine •• .......»«»..... 49 6—13 E f f e c t of the One-time-constant Governor on. the S t a b i l i t y Region .................... 51 I V o l t a g e R e g u l a t o r ....................... 59 v i 1. INTRODUCTION Since the i n t r o d u c t i o n of the c o n t i n u o u s l y - a c t i n g v o l t a g e r e g u l a t o r , both theory and p r a c t i c e have shown that a v o l t a g e - r e g u l a t e d generator has a g r e a t e r steady s t a t e s t a b i l i t y l i m i t i n a power s y s t e m . ^ ^ ^ Normally, the steady s t a t e s t a b i l i t y l i m i t of a generator i n a power system i s d e f i n e d as i t s maximum steady s t a t e l o a d l i m i t (2lY • without f a l l i n g out of synchronism. v ' Por an unregulated generator, t h i s l i m i t depends on the machine and the system (21 22 23) parameters alone. ' ' For a machine with a c o n t i n u o u s l y - a c t i n g v o l t a g e r e g u l a t o r and governor, negative feedback i s i n t r o d u c e d so t h a t the power and v o l t a g e d i f f e r e n c e s , which may cause machine i n s t a b i l i t y , are c o n s t a n t l y monitored and c o r r e c t e d before the machine becomes u n s t a b l e . In t h i s way, the machine l o a d can be i n c r e a s e d beyond the o r d i n a r y steady sta t e l i m i t to a new power l i m i t c a l l e d the dynamic s t a b i l i t y l i m i t . The dynamic s t a b i l i t y l i m i t of a generator i n a jpower system p r i m a r i l y depends upon the parameters i n the feedback loops i n t r o d u c e d by the c o n t i n u o u s l y — a c t i n g v o l t a g e r e g u l a t o r and governor. However, the machine parameters and the-"nature of the system to which the machine i s connected are a l s o important. In the development of a modern power system, the a b i l i t y of the synchronous generator to operate i n the dynamic s t a b i l i t y r e g i o n becomes more and more important. This i s due to the f a c t t h a t many synchronous generators have to operate with almost u n i t y power f a c t o r or even l e a d i n g power f a c t o r s . The steady sta t e 2 s t a b i l i t y l i m i t o f a n u n d e r — e x c i t e d s y n c h r o n o u s g e n e r a t o r , h o w e v e r , i s s m a l l u n l e s s i t i s o p e r a t e d i n t h e d y n a m i c s t a b i l i t y r e g i o n w i t h t h e a i d o f a v o l t a g e r e g u l a t o r . T h e t e c h n i c a l s o u n d n e s s o f s u c h a n o p e r a t i o n h a s b e e n c o n f i r m e d i n b o t h t h e o r y a n d p r a c t i c e . ^ » ^ » 1 3 ) T o t h e t h e o r e t i c a l a n a l y s i s o f t h e d y n a m i c s t a b i l i t y o f a r e g u l a t e d g e n e r a t o r , s t a n d a r d c o n t r o l t h e o r y c a n b e a p p l i e d . ^ * ^ * ^ T h i s r e q u i r e s a c o m p l e t e d e s c r i p t i o n o f t h e s y s t e m w i t h d i f f e r e n t i a l e q u a t i o n s . T h i s i s u s u a l l y c o n s i d e r e d t e d i o u s b e c a u s e h i g h o r d e r d i f f e r e n t i a l e q u a t i o n s w i t h c o m p l i c a t e d c o e f f i c i e n t s a r e i n v o l v e d . S i m p l i f i c a t i o n o f t h e p r o b l e m i s u s u a l l y m a d e b y u s i n g o n l y o n e r e a c t a n c e a n d o n e v o l t a g e s o u r c e f o r t h e s y n c h r o n o u s ( 1 7 ) m a c h i n e . T h e r e s u l t s o b t a i n e d , n e v e r t h e l e s s , a r e n o t a c c u r a t e , b e c a u s e m a n y i m p o r t a n t c h a n g e s l i k e v o l t a g e a n d c u r r e n t t r a n s i e n t s , a n d v o l t a g e r e g u l a t o r a n d g o v e r n o r a c t i o n s a r e n o t (4) i n c l u d e d i n t h e e q u a t i o n s . (26) A s f o r s y n c h r o n o u s m a c h i n e s , P a r k ' s e q u a t i o n s c a n a l w a y s b e u s e d f o r a c c u r a t e i n v e s t i g a t i o n s , s i n c e a l l i m p o r t a n t m a c h i n e p a r a m e t e r s a r e i n c l u d e d . W o r k h a s b e e n d o n e i n t h i s d i r e c t i o n s i n c e 1 9 4 4 w h e n C o n c o r d i a ^ ^ d i s c u s s e d a n d p o i n t e d o u t t h e i m p o r t a n t e f f e c t s o f v o l t a g e r e g u l a t i o n w i t h r e s p e c t t o s t a b i l i t y l i m i t . P a r k ' s e q u a t i o n s w e r e u s e d a n d t h e s t a b i l i t y l i m i t o f a r o u n d - r o t o r m a c h i n e o p e r a t i n g a t u n i t y p o w e r f a c t o r w a s s t u d i e d . C o n c o r d i a ' s w o r k w a s f o l l o w e d b y s e v e r a l o t h e r s . A m o n g ( 3 ) t h e m w e r e H e f f r o n a n d P h i l l i p s w h o g a v e a m e t h o d , a l s o b a s e d o n P a r k ' s e q u a t i o n s , f o r t h e i n v e s t i g a t i o n o f a r o u n d - r o t o r generator operating at any power factor. The dynamic s t a b i l i t y l i m i t s i n the leading power factor region were established with the aid of an analogue computer. Analogue computer m e t h o d s ^ ' ^ ' ^ w e r e also developed by Messerle and Bruck, ̂ ""^ and Aldred and (7) Shackshaft i n their dynamic s t a b i l i t y studies. Messerle and Bruck had gone a step further i n taking the prime mover control into consideration • Nyquist c r i t e r i o n was also employed to determine the s t a b i l i t y l i m i t of a round-rotor generator. (8) Messerle's work was further extended by Goodwinv ' who, using a network analyser, determined the external impedance and the i n i t i a l conditions for the s t a b i l i t y study of a water wheel generator. Reports from the C.I.G.R.E. ( 1 0' 1 : L » 1 2^ indicated that similar approaches to the problem of dynamic s t a b i l i t y studies hav been done i n the U.S.S.R. i n the past two decades. The two- reaction theory for the synchronous machine has also been employed. The previous work of Heffron and P h i l l i p s , and, also that of Messerle have been extended i n t h i s thesis. A new method for the study of a salient-pole machine operating at any power factor has been established. The method, includes ,. the round-rotor machine as a special case. The effects of saliency i n the round-rotor machine, and the t i e - l i n e resistance^ which were normally neglected, are taken into consideration. Voltage regulator and governor effects i n the under-excited region are investigated. The well known Routh-Hurwitz c r i t e r i o n i s used to determine the s t a b i l i t y l i m i t s . The D-partition 4 (24) method i s used for the investigation of the effect of two p a r t i c u l a r parameters on s t a b i l i t y . A d i g i t a l computer i s used for the computations. 5 2. REVIEW OP GENERAL THEORIES 2.1 C l a s s i c a l Synchronous Machine Theory(22.23,26,27) When the equations d e s c r i b i n g the performance of a synchronous machine are w r i t t e n i n phase co-ordinates, they become l i n e a r equations w i t h time-varying c o e f f i c i e n t s because the mutual inductances between the s t a t o r and r o t o r c i r c u i t s are p e r i o d i c functions of r o t o r angular p o s i t i o n . In the case of a s a l i e n t - p o l e machine the mutual inductances between s t a t o r phases are also p e r i o d i c f u n c t i o n s of r o t o r angular p o s i t i o n . This causes d i f f i c u l t i e s i n a n a l y s i s . (26) Park introduced the w e l l known d—q transformation w i t h the f o l l o w i n g assumptions: 1. The s t a t o r windings are s i n u s o i d a l l y d i s t r i b u t e d around the a i r gap. - 2. The e f f e c t of the s l o t s on the a i r gap f l u x i s not appreciable w i t h v a r i a t i o n i n the r o t o r angle. 3. S a t u r a t i o n e f f e c t s are neglected. (27) Park's transformation matrix, jnodified by Yu , has the f o l l o w i n g forms cos © K a K b /2 = Y3 K c 1 1 1 - s i n 9 cos (0 - 120) - s i n (© - 120) cos (© + 120) - s i n (0 + 120) K d K K 0 (2-1) where K , K, , K represent v a r i a b l e s such as armature c u r r e n t s , £L D C f l u x l i n k a g e s or" v o l t a g e s i n phase co-ordinates and , , the corresponding v a r i a b l e s i n the d i r e c t - , quadrature-and zero - a x i s c o o r d i n a t e s . When the synchronous machine equations are transformed by the t r a n s f o r m a t i o n matrix ( 2 — l ) from the phase coordinates to the d-q-o coordinates, the time-varying c o e f f i c i e n t s of the inductances are e l i m i n a t e d . "» The t r a n s f o r m a t i o n , i n e f f e c t , r e s o l v e s the three phase windings of the machine i n t o a two-phase model with the d i r e c t — and the quadrature—axis windings, and i n a d d i t i o n to t h a t a zero a x i s winding which can be analysed s e p a r a t e l y . d—axis d-axis q—axxs q-axis (a) Synchronous Machine i n Phase Coordinates (b) Synchronous Machine i n Park's Coordinates F i g . 2 - 1 Synchronous Machine Models Park's equations i n the 7 operator form can be w r i t t e n : v d = pxHi " \& a d V q - r i a q (2-2) V 0 = pw - r i •^o a o _ SLEI V _ tt v f d X d ( p ) • a> . d tt q ( 2 - 3 ) T 0 X = - -2- i tt 0 where x d(l + -C>) (1 + T*dp) x d(p) = (1 + < o p ) ( l + r d o p ) X ( l + -£V) X n ^ ) = " - 9 7 ~ ~ (2-4) q (1 + T* P x qo^ G(p) H p d + f d 0 P ) ( l + ^ o p ) Por d e t a i l s of symbols see Appendix I. A d d i t i o n a l equations r e l a t i n g the machine term i n a l c o n d i t i o n s are 2 2 ^ 2 t d q q * d q ( 2 _ 5 ) 2 T = Jp 0 +^pO + T m r e .4, 'ar;-'v\-.J^v;^»k'--:^fe..;=: •'- 8 where i s the terminal voltage and T e, the electromechanical energy conversion torque. 2.2 S t a b i l i t y of a Dynamical System 2.2.1 S t a b i l i t y i n the small The system equations d e s c r i b i n g the performance of a r e g u l a t i n g process are u s u a l l y nonlinear d i f f e r e n t i a l equations. However, i n the study of a synchronous machine i n a power system, a small d e v i a t i o n from an i n i t i a l e q u i l i b r i u m p o i n t i s u s u a l l y considered. Liapounoff has shown^^* 2"^ that the s t a b i l i t y of a non- l i n e a r system under small d e v i a t i o n s from e q u i l i b r i u m ^ i§ 0> completely determined by the l i n e a r l y perturbated system equations which are the f i r s t v a r i a t i o n of the d i f f e r e n t i a l equations. There are also c r i t i c a l cases when the method cannot be a p p l i e d . Then e i t h e r a l l l i n e a r p e r t u r b a t i o n terms v a n i s h or the c h a r a c t e r i s t i c roots of the l i n e a r i z e d system equations possess a zero root or a p a i r (or p a i r s ) of purely imaginary r o o t s . In power system dynamic s t a b i l i t y s t u d i e s , however, i t has been assumed that the r e g u l a t i n g system i s s t i l l at the s t a b i l i t y boundary when these cases a r i s e . 2 . 2 . 2 S t a b i l i t y of a system wi t h l i n e a r i z e d d i f f e r e n t i a l equations The s t a b i l i t y of a l i n e a r d i f f e r e n t i a l equation i s determined by the homogeneous d i f f e r e n t i a l equation t h d e s c r i b i n g the fr e e motion of the system. For a system of n order the equation of the f o l l o w i n g form w i l l be obtained: .n ,-n-l ao tt S ( t ) + a l ̂ d t " S ( t ) + — — • a n S ( t ) = 0 (2-5) The s o l u t i o n of which i s w e l l known as n S(t) = A.e 1 i = 1 where are the roots of the c h a r a c t e r i s t i c equation of the form n , n-1 . a QP + a^p + •>.••. ah = 0 (2-6) The s t a b i l i t y of the system i s then determined by the r e a l part of Zc whose nature can be i n v e s t i g a t e d by a number of c r i t e r i a , w h i c h make i t p o s s i b l e to judge the s t a b i l i t y without recourse to the computation of Ẑ ,. 2.2.3 The Routh-Hurwitz S t a b i l i t y C r i t e r i o n In order to ensure t h a t a l l roots of the equation (2—6) have negative r e a l p a r t s , i t i s necessary and s u f f i c i e n t t h a t , w i t h a Q greater than z e r o , a l l the diagonal determinants of the array constructed from the c o e f f i c i e n t s of (2—6), a l a 3 a_ « • • 0 j a a h a. o 2 4 * 6 0 . • • a n (2-7) be greater than zero; i,e« a l a 3 a 0 a4 >̂ 0, .... 10 A = n a. a^ • • . 0 a, 0 0 . . . a n (2-8) (24) 2.2.4 D-.partition method f o r two r e a l parameters Vhen the e f f e c t of .tvo r e a l parameters on ,th.e4 s t a b i l i t y of a r e g u l a t i n g system i s to be considered and i f the two parameters can be expressed e x p l i c i t l y i n the c h a r a c t e r i s t i c equation, the s t a b i l i t y boundaries w i t h respect to the two parameters can be determined. Let the c h a r a c t e r i s t i c equation of the system be expressed i n the form *"P(p) + l f i ( p ) + B(p) = 0 (2 - 9 ) where f and X. are the two r e a l parameters under c o n s i d e r a t i o n and Q(p), P(p) and R(p) are polynomials w i t h c o e f f i c i e n t s that are independent of C and Z « Por a stable system a l l the zeroes of the c h a r a c t e r i s t i c equation must be i n the l e f t h a l f z-plane. I f the values of if and z are allowed to vary continuously so that at l e a s t one of the zero l o c a t i o n s moves towards and f i n a l l y reaches the 11 imaginary a x i s , the system i s on the s t a b i l i t y l i m i t and becomes o s c i l l a t o r y . The locus of points i n the {ti-z) plane representing such a s i t u a t i o n i s c a l l e d the s t a b i l i t y boundary. to the case of a l l the zeroes of the c h a r a c t e r i t i c equation l y i n g i n the l e f t h a l f z-plane. The other side of the boundary i s the unstable region corresponding to the case where at l e a s t one of the zeroes of the c h a r a c t e r i s t i c equation i s i n the r i g h t h a l f z-plane. Therefore, i f a l l the po s s i b l e l o c i which correspond to the j — a x i s zeroes of the c h a r a c t e r i s t i c equation are found i n the (if — z) plane, the s t a b i l i t y boundary can be determined from the l o c i . The l o c i would d i y i d e the (tf-z) plane i n t o s t a b l e and unstable regions provided that stable regions e x i s t . To f i n d the l o c i of the s t a b i l i t y boundary i n the (ti-z) plane, l e t p = jw i n the c h a r a c t e r i s t i c equation, Let P, Q and R f u r t h e r be separated i n t o r e a l and imaginary p a r t s : One side of the boundary i s the stable region which corresponds r B ( j w ) + ^ Q ( j t t ) + B(3 W) = 0 (2-10) R(j«) P(j«) Q(j») P^w) + j P 2 ( « 0 Q n(») + 3'Q2(«) Rx(a>) + jR2(a>) or tfP^a) +ZQ1(<a) + R-^tt) = 0 sP2(a>) + tQ2(a>) + R 2(«) = 0 (2-12) 12 ( 2 - 1 3 ) ( 2 - 1 4 ) N o n t r i v i a l s o l u t i o n s ofif andZ e x i s t when A £ 0 . I f A = 0 the n o n t r i v i a l s o l u t i o n s do not e x i s t which means that there i s no p o s s i b l e point i n the (jT—T) plane which gives an imaginary- a x i s r o o t . By v a r y i n g a> from —oo to ©o a l l the l o c i of the j - a x i s c h a r a c t e r i s t i c root can be obtained. In the computation i t i s s u f f i c i e n t to vary w from 0 to co since both ifandr are the r a t i o s of odd polynomials. Note t h a t P ^ * and R^ are even, and P 2 , Q^f &2 a r e polynomials i n tt* St r a i g h t l i n e l o c i a r e obtained by s u b s t i t u t i n g tt = 0 and to = o o i n t o equation ( 2 - 1 2 ) because they s a t i s f y i t . These l i n e s i n the (f-z) plane represent the c h a r a c t e r i s t i c roots j o andjoo and can, t h e r e f o r e , be p o s s i b l e s t a b i l i t y boundaries. A f t e r the l o c i are obtained the stable and unstable regions can be i d e n t i f i e d by some other s t a b i l i t y C r i t e r i o n * w i i e u c e where Z = T -R^t t ) P 1 ( « ) -R2(<o) P 2 ( » ) A = Q1(o>) -R^tt) Q2(a>) -R2(o>) Q 1 ( « ) P 1(w) Q 2(») P 2(») 3. DERIVATION OP SYSTEM EQUATIONS FOR THE DYNAMIC STABILITY STUDIES 3*1 The System Under Study A power system normally c o n s i s t s of a number of synchronous machines supplying power to the system at various p o i n t s * For the dynamic s t a b i l i t y study of one p a r t i c u l a r synchronous machine of the system, the remaining parts of the system are normally considered as an i n f i n i t e bus to which the machine i s connected through an external impedance* The synchronous machine of the system under study i s c o n t r o l l e d by a voltage r e g u l a t o r and a speed governor, both of the cont i n u o u s l y — a c t i n g type. A schematic diagram of the system i s shown i n F i g . 3-1. oil relay system ~<SH fawrrtar vaWe •turbine "speed sensing element. rekttnee e x c i t e r ffBTHT) generator ^ M O l t a g t , regulator t ie . Iine —TnSOT^—wvv. ^ injmite —TjtTOfX—VWfV- F i g . 3-1 Schematic of System Under Study E F i g , 3-2 Voltage Phasor Diagram of the I n f i n i t e Bus and a Synchronous Generator 14 3*2 System Equations 3.2.1 The synchronous machine Park's equations w i l l be a p p l i e d to the dynamic s t a b i l i t y study. In order to f i n d the major e f f e c t s of the synchronous machine on s t a b i l i t y of the system, the f o l l o w i n g a d d i t i o n a l assumptions w i l l be made: 1. The induced v o l t a g e s due to the change of f l u x l i n k a g e s pvj/̂  and p\|/̂  i n the d and q axes can be ne g l e c t e d because they are small compared to the speed v o l t a g e s ©U^ and ©ty, due to the cross e x c i t a t i o n s , a 2. The armature r e s i s t a n c e v o l t a g e drop can be ne g l e c t e d because i t i s s m a l l . 3. The time constants T . ' , • r f * 7"> T* and T_ , being d 7 do ^q ^qo Dl r e l a t i v e l y s m a l l , w i l l a l so be n e g l e c t e d . With the above assumptions, v d = - v P e Y d <o v f d <o d x (p) V = - i q w q T e = V d " V ^ q ^ x-,(p)= x, ~t " (1 + *d>> x (p) = x q G(p) = ad Hp(l + t d 0 P ) 15 The other r e l e v a n t equations are T = Jp© + oc p© + T m r r g v t 2 - v 4 2 + v 4 2 (3-2) 3.2.2 The t i e l i n e An assumption w i l l be made f o r the e x t e r n a l impedance, namely t h a t i t s r e s i s t a n c e and reactance elements are constant. Two more equations are obtained from the steady s t a t e phasor diagram, F i g . 3-2, v., = v s i n i 5 + f i , - x i d o : d q (3-3) v = v cosS + x i n + r i q o d q 3.2.3 The v o l t a g e r e g u l a t o r and the governor Since a small change i n machine t e r m i n a l v o l t a g e causes a small change i n the f i e l d e x c i t a t i o n , the t r a n s f e r f u n c t i o n of a v o l t a g e r e g u l a t o r can be w r i t t e n g e n e r a l l y as A V - P A g(p) = (3-4) A s i m i l a r t r a n s f e r f u n c t i o n can be w r i t t e n f o r a speed governor, , AT f(p) =1$ (3-5) A small change i n speed or frequency causes a small change i n prime mover i n p u t . The d e t a i l s of the t r a n s f e r f u n c t i o n s are given i n Appendix I I , 3.3 L i n e a r i z a t i o n of the System Equations For the dynamic s t a b i l i t y study only small changes of vai'iaus v a r i a b l e s around an i n i t i a l o p e r a t i n g p o i n t w i l l be taken i n t o c o n s i d e r a t i o n * Hence s t a b i l i t y can be determined from a set of l i n e a r i z e d equations of the o r i g i n a l system equations. The frequency of the i n f i n i t e bus i s considered f i x e d The instantaneous angular displacement,., v e l o c i t y and a c c e l e r a t i o n of the- machine, r e s p e c t i v e l y , can be w r i t t e n as © = .© o + & p© = to + p£ , where p © Q = « and p © = p 8 Then the dynamic s t a b i l i t y of the system can be d e f i n e d by whether or not 6" goes to i n f i n i t y w i t h time a f t e r a small l o d i s t u r b a n c e . From equations (3—l) through ( 3 - 5 ) , . t h e f o l l o w i n g l i n e a r i z e d system equations are obtained? A V j = -V pA<5 + x A i d qo q q Av =. V, A<5 + x, ( p ) A i , + G(p)Av„, q do d r d f d Av = g(p)Av i d x AT = f'(p)pA£ 17 Av, = Av = 6>AT Av^ v cos rS A6 + r A i , - x Ai o o d q -v s i n S A6 + x A i , + r A i o o d q i 3 Av,, + i Av + v, A i , + v A i - T pA6 do d qo q do d qo q eo Jp A6 + ocpAd + AT ^ Av, + Av = v' Av, + v' Av v t o d v t o * do"'d ' qo q By d e f i n i n g h(p) £ g(p)G(p) (3-8) J(p) A j 2 p + ( a - - j a)p - f ( p ) , where f ( p ) = p f ( p ) , and e l i m i n a t i n g AT m, ATg * Av^ from the system equations, the f o l l o w i n g f i v e homogeneous equations are obtained: 1 0 - r X -v cos 6 0 0 A v d 0 1 -X .—r v s i n <5 0 0 Av q 1 0 0 ~\ V - h ( P ) v ; o - h ( p ) v ^ + l x d ( p ) 0 A i q ido i qo Vdo V qo a>j(p) A<§ •= 0 (3-9) 3«4 The C h a r a c t e r i s t i c Equation of the System This i s obtainable from the c h a r a c t e r i s t i c determinant of the above matrix equation. The c h a r a c t e r i s t i c determinant 18 where A = J(p)[h(p)A ]_+ x d ( p ) A 2 + A, h(p) [ A 4P + A 5 + x d ( p ) A 6p + A y + JA gp + Aql (3-10) A. = -tt v, rx - v ( r + x + x x ) 1 do q qo q A n = « fx + x 2 I q A 0 = ttIr2+ x 2+ x x 3 I q A. = Y -v' ( v , x + v r + i (r +x ) + 4 qo doj do qo qo ') < J v ^ r - v x + i , „ ( r 2 + x 2). qoj^do" ' qo" ' "do A c = v sinS 5 o o v., l rx do qo q + v cos S o o — v ' -(v, (x+x ) + r ( v + i , x )) qo \ do q qo do q'J v ' x (v, + i x) - v ' r + x(v + i J A ) l do q do qo qo\ do qo do / A, = Y (v - i , x + i r ) 6 qo qo do qo A„ =-v sinS i (x+x ) + v cos S (v - i , x + i r) 7 o o qo q o o qo do q qo A = Y A 8 do + v { 2 2 (r +x )+ xx _ qo - v ^ r + v x - x, (r + x ) do qo do + i , r x do q A Q = v s i n 8 9 o o v, (x+x ) + v r + i . , r x do q qo do q "V cos 5 o o -v, r + v x + i n x x do qo do q Since the t r a n s f e r f u n c t i o n of the v o l t a g e r e g u l a t o r - H B ( 1 + 7 P ) g (p) = { 1 + ( Te + V s ) p + V s ? 2 ] (3-11) and the t r a n s f e r f u n c t i o n of the speed governor —u " Tl f ( p ) = m (1+ T j p H l + ^ p ) 19 or f ( p ) (i+rp ) ( n-Tp) (3-12) J( p ) and h(p) f u n c t i o n s become o T u p J(p) = J p 2 + (a - - § £ ) P + h(p) = - x ad ( l + ^ p ) ( l + r 2 p ) A (1 + < 0 P ) | + r e ^ 2 ) , / A ad where u, = u, —^— , *e ^e Rp Therefore, the c h a r a c t e r i s t i c determinant A = J p 2 + ( a „ ^ P _ ) p + a p (i+r l P)(i+r 2p) -M-e (1+t P ) A X ( 1 + rd>H 1 + ( r e + ^ > + r e V ' x ( 1 + ? P ) A + — a — + A. ( 1 + ^ o p ) ( i + ^ o p ) { i + ( r e + r / s ) p + T e r p ' A 4P+ A 5 d < i + rd>> A6p+ A ? A gp+ A q (3-14) From equation (3—14), the c h a r a c t e r i s t i c equation i s obtained? 20 7 6 5 4 3 2 a o P + a l P + a 2 P + a 3 P + a 4 P + a 5 P + a 6 p + a 7 = 0 (3-15) where a = T J B. o a 1 a, = T B + T, J B, 1 a 4 d 1 a, = T B_+ T J B + T,B . - jx A, J ^ n a 5 c 1 d 4 r e 1 1 2 s a- = a. = a r = a, a. T a V T b J B l + T c B 4 + T d B 5 + > W e V - ^ ( J A . V B 6 r l T 2 T s ) T b B 4 + T c V T d B 3 + ^ { V V V s * + B 2 T e * s V E ( J W V I W V E ) + J B I T b V T c B 3 + B 2 ( W s > ~ A i r s ) - , ; { J A 1 + A 5 T E + B 6 T F ) + B 4 T b V % ( B 2 " ^'eV - ^ < A 5 V B 6 ) + B 5 B 3 " ^e A5 For d e t a i l s of the d e r i v a t i o n ^ see Appendix I I I . 3.5 The Routh-Hurwitz Determinants of the C h a r a c t e r i s t i c Equation Routh-Hurwitz Determinants can be w r i t t e n as a ] *< 0 0 0 0 0 a, a, a ] a 0 0 0 0 a„ a a. a, a-, a 0 0 a„ a. a, a 4 a. a. a. 0 0 0 0 a, 7 0 0 0 0 a, a,- a 0 0 0 a, a, 0 a- a P a. (3-16) A 6 = 21 S e p a r a t e l y w r i t t e n , they are A l = a i A 2 = a ] L a 2 - a Q a 3 = SL^&2 ~~ a i ^ o i ' w ^ e r e Ao = a l a 4 ~ a 0 a 5 A 4 = a 4 A 3 + A 2 ( a l a 6 " a 0 a 7 " a 2 a 5 ) + a 0 a 5 A 0 A 5 = a 5 A 4 ~ a 3 a 6 A 3 + A 2 ^ a 2 a 3 a 7 + a l a 6 a 5 " a l a 4 a ? ) ~ a 0 a 3 a 7 A 0 -a-^ (a^ a^-a^a^.) 2 a^A^— a^ayA^+ a^a^ayA^*^ A^^a^a^a^— a 2^7 a O a 6 a 5 ^ a 7 ' 2 2 + a 2 a i a jAQ+ a 0 a 7 - ( a 1 a 6 - a ^ ^ ) A ? = a ? A 6 (3-17) 3.6 The D - p a r t i t i o n Equations To f i n d the e f f e c t of r e l a t i v e values of two r e a l parameters on s t a b i l i t y , the Routh—Hurwitz C r i t e r i o n can also be used. The procedure, however, i s r a t h e r t e d i o u s . When the c h a r a c t e r i s t i c equation of the system can be expressed e x p l i c i t l y as the f u n c t i o n of two r e a l parameters, the D - p a r t i t i o n method can be a p p l i e d . The c h a r a c t e r i s t i c equation of the system under study by the D — p a r t i t i o n method can be f i r s t w r i t t e n as equation (2—9) and then separated i n t o r e a l and imaginary p a r t s as equation (2-12). In terms of the \i and \i , which are of p a r t i c u l a r e s i n t e r e s t , equation (2-12) becomes (/ P (tt) - j/ Q (tt) + Rn (tt) = 0 , s 1 e 1 1 (3-18) (x P.(tt) - (/ Q„(tt) + R.(tt) = 0 22 The s t a b i l i t y boundaries can be determined by s o l v i n g t h i s equation and by v a r y i n g tt from zero to i n f i n i t y . D e t a i l e d d e r i v a t i o n of equation (3-18) i s g i v e n i n Appendix I I I . 23 4. SYSTEM PARAMETERS AND INITIAL CONDITIONS OF SYNCHRONOUS MACHINES 4.1 System Parameters The system under study has parameters w i t h the f o l l o w i n g v a l u e s . Synchronous Machines. Synchronous machines of v a r i o u s ratings w i l l behave i n e x a c t l y the same way as long as the per u n i t v a l u e s of the machine parameters are the same. In the a n a l y s i s the d i r e c t - and quadrature—axis reactances of the synchronous machine and the t i e l i n e impedance w i l l be v a r i e d except f ° r the f o l l o w i n g parameters which w i l l be kept constant: J = 120 TIH = 4000 rd0 = 2000 p.u. T d = 600 p.u. a = 3.0 p.u. Voltage Regulator S e t t i n g s . The vo l t a g e r e g u l a t o r s e t t i n g s are assumed, unless otherwise s p e c i f i e d , to be normally U = 800 p.u. e r = 800 p.u. s 30 *e r s Governor S e t t i n g s . Two d i f f e r e n t governor t r a n s f e r f u n c t i o n s are cons i d e r e d , u n l e s s otherwise s p e c i f i e d . For a governor w i t h one time constant, the g a i n and the time constant r e s p e c t i v e l y are ,: 24 T ^ = 377 p.u. t = 0 p.u. For a governor with two time constants u = 5 and b rm T^= T 2= 300 p.u. are assumed. 4.2 I n i t i a l C o n d i t i o n s of Salient-frOte Synchronous Machine A new method f o r f i n d i n g i n i t i a l c o n d i t i o n s , i ^ 0 > "^qo' V d o ' •V * M*-, , "V » v„, and 8 f o r a s a l i e n t - p o l e synchronous qo do' qo ido o machine w i l l be developed. Consider the steady s t a t e . From Park's equations ( 3 - l ) to (3—3) we have v„ = x i (4-1 do q qo v = ^ v„, - x, i , (4-2 qo R p fdo d do P = v, i , + v i (4-3 do do qo qo Q = v i , - v, i (4-4 * qo do do qo v, = v s i n S + r i , - x i (4-5 do o o do qo v = v cos 6 + x i , + r i (4—6 qo o o ,do qo v 2 , = v 2 + v 2 (4-7 to do qo In these equations v, . v , i , , i , v , A ~ , v are ^ do* qo' do qo' o uo fdo unknown while V^. Q > P a n d Q are g i v e n . There are 7 equations w i t h 7 unknowns. From (4-3) and (4-1) P = i , i x + v i (4-8) do qo q qo qo From (4-4) and (4-1) .2 Q = v l , — x 1 qo do q qo 25 (4-9) From (4-7) and (4-1) L2 ~ x2 ±2 V q o ~ \l to q qo (4-10) From (4-9) 2 Q + x i ± = _ a—ap_ (4 - n ) d 0 V q o From (4-11) and (4-8) / .2 \ P =  ( Q * X q 1 q° x i + v i (4-12) Y ^ o q qo qo qo From (4-12) and (4-10) (Q + x i 2 ) p = q x i + / v 2 - n> 9 .o q. q.° / t o (4-13) 2 .2 /~2 _ x2 ±2 F to X q qo x i i q qo qo Hence P v, to I qo V = X 1 do q qo (4-14) (4-15) (4-16) 2 2 qo y to do • 2 Q + x i „ . * a — a ° . (4-17) •̂do v Q ° R (v - x , i , J - 2 - (4-18) V f d o " q o "d~do' x consequently ad V = -v (4-19) do do V = v (4-20) qo qo 26 Next from (4-5) and (4-6) v = \ ( r A ~ r 1* + x 1 ) 2 + (v - x i — r i ) 2 o V do do qo qo do qo (4-21) SQ = arc tan (^ d o~ I fet * fe) (4-22) qo do qo A l l the i n i t i a l c o n d i t i o n s have thus been found. The method i s general and a p p l i c a b l e to a s a l i e n t - p o l e machine as w e l l as a round-rotor machine, the l a t t e r being a s p e c i a l case of the former. The e x t e r n a l t i e l i n e reactance,and r e s i s t a n c e have been i n c l u d e d . 27 5. DYNAMIC STABILITY STUDIES USING ROUTH-HURWITZ CRITERION 5.1 E f f e c t of S a l i e n c y on the Dynamic S t a b i l i t y The synchronous s a l i e n c y i n a round-rotor machine has (14-15) u s u a l l y been assumed to be zero, but tests on a c t u a l machines show that some s a l i e n c y i s prese n t . Thus the assumption of non— s a l i e n c y f o r round—rotor machines i s not v a l i d . In the f o l l o w i n g , three cases w i l l be s t u d i e d . The f i r s t case, x = x ,corresponds to an i d e a l round-rotor machine; the second d q case, x = 0.85 x , a round—rotor machine wi t h s a l i e n c y e f f e c t d q considered; and the t h i r d case, x,= 0.6 x , corresponds to a ' d. q t y p i c a l s a l i e n t - p o l e machine* F i g . 5-1 shows the e f f e c t of r o t o r s a l i e n c y on the dynamic s t a b i l i t y of the system under u n i t y and l e a d i n g power f a c t o r o p e r a t i o n . The a c t i o n of the speed governor i s d i s - counted by s e t t i n g fj, = 0, while the v o l t a g e r e g u l a t o r has the normal s e t t i n g s . Curve 1 i n d i c a t e s the dynamic s t a b i l i t y l i m i t of the i d e a l round-rotor machine w i t h x ='x = 1.0, p.u., curve 2 the d q s t a b i l i t y l i m i t of the round—rotor machine with a s a l i e n c y e f f e c t considered (x = 1, x ..= 0.85), and curve 3 the s t a b i l i t y d q l i m i t of the s a l i e n t - p o l e machine (x,= 1, x = 0.6). I t can be d q seen t h a t the r o t o r s a l i e n c y i n c r e a s e s the dynamic s t a b i l i t y l i m i t , e s p e c i a l l y under l e a d i n g power f a c t o r o p e r a t i o n . The f a m i l y of curves shown i n do t t e d l i n e s corresponds to the system without t i e l i n e r e s i s t a n c e (r=0), and those shown i n s o l i d l i n e s corresponds to the system w i t h the t i e l i n e r e s i s t a n c e i n c l u d e d . 2 8 0.5 y) u 2.0 P-U. Power Limits without Governor P 0.5 1.0 1.5 2.0 PU. with a one - time - constant Governor S i m i l a r computations are repeated w i t h governor e f f e c t i n c l u d e d , and the r e s u l t s are summarized i n P i g . 5-2 and 5—3. In P i g . 5-2 the speed governor of the system has one time constant, while i n F i g , 5—3 i t has two time constants. The same c o n c l u s i o n can be drawn t h a t the s a l i e n c y e f f e c t of a synchronous machine i n c r e a s e s the dynamic s t a b i l i t y l i m i t . 5.2 E f f e c t of the Short C i r c u i t R a tio of a Synchronous Machine The short c i r c u i t r a t i o of a synchronous machine of modern design i s approximately equal to — . A high short x d c i r c u i t r a t i o machine costs more than one w i t h a low short c i r c u i t r a t i o . Machines designed w i t h low short c i r c u i t r a t i o s have l i m i t e d s t a b i l i t y l i m i t i f i t i s operated without a modern v o l t a g e r e g u l a t o r , but the s t a b i l i t y l i m i t w i l l be i n c r e a s e d c o n s i d e r a b l y by u s i n g a c o n t i n u o u s l y - a c t i n g v o l t a g e r e g u l a t o r . F i g . 5—4 shows the e f f e c t of the short c i r c u i t r a t i o on ;the s t a b i l i t y l i m i t f o r an u n d e r — e x c i t e d machine. The dotted l i n e s i n d i c a t e the power l i m i t of the machine with v o l t a g e r e g u l a t o r set at \i'e= 0.01 and Z^= 3000 to approximate a system without a c o n t i n u o u s l y - a c t i n g r e g u l a t o r , and the s o l i d l i n e s are those f o r a system with a c o n t i n u o u s l y - a c t i n g v o l t a g e r e g u l a t o r . For example, when the short c i r c u i t r a t i o i s equal to •—^ the machine without a v o l t a g e r e g u l a t o r can h a r d l y be loaded beyond 0.9 p.u. power at u n i t y p . f . When the c o n t i n u o u s l y - a c t i n g v o l t a g e r e g u l a t o r i s used the l i m i t can be i n c r e a s e d to as high as 1.35 p.u. at the same p . f . i n the dynamic s t a b i l i t y 0.5 1.0 15 2.o p.u. 30 r =0.1 X = 1 P.U. Me - 30,Ms = 5,Mm = 5 7e =Zs =800 P.U. Zi =Z 2 =300 P.U. 1. Xot - 1, X, =1 2. Xrf * 1, Xq = 0-85 3 U = Xq = 0-6 F i ^ 5 - 3 Soliency Ejfect on. Power Limits'- with a Two- time - constant Governor. Ro\ 5 -.4 Short Circuit Ratio Effect on Power Limits 31 r e g i o n . 5.3 E f f e c t of the Tie Line Impedance 5.3.1 E f f e c t of the l i n e r e s i s t a n c e I t i s observed from F i g . 5-1, 5-2 and 5-3 that the t i e l i n e r e s i s t a n c e has a d i s t i n c t e f f e c t on the dynamic s t a b i l i t y l i m i t . Larger s t a b i l i t y l i m i t s are observed when the t i e l i n e r e s i s t a n c e i s i n c l u d e d i n the computations. The s t u d i e s are extended by i n c r e a s i n g the r e s i s t a n c e value from 0 to 0.3 p.u. while the reactance i s kept constant at 1.0 p.u. In a l l cases the v o l t a g e r e g u l a t o r and governor are at normal s e t t i n g s . The r e s u l t s are shown i n F i g . 5-5 and 5-6. Cl o s e r examination of F i g . 5-6 f u r t h e r r e v e a l s the e f f e c t of the l i n e r e s i s t a n c e on s t a b i l i t y l i m i t at u n i t y power f a c t o r and w i t h d i f f e r e n t degrees of s a l i e n c y . 5.3.2 The e f f e c t of the l i n e reactance F i g . 5-7 and 5-8 show the r e s u l t s of s t a b i l i t y s t u d i e s by v a r y i n g the value of t i e l i n e reactance. The d i f f e r e n c e between the s t a b i l i t y l i m i t w i t h l i n e r e s i s t a n c e i n c l u d e d and that without is. i n s i g n i f i c a n t , when the . , t i e l i n e reactance i s comparatively s m a l l . But i t becomes s i g n i - f i c a n t when the reactance i s l a r g e . F i g . 5-8 e s p e c i a l l y shows "the r e s u l t s of s t u d i e s f o r u n i t y power f a c t o r . I t a l s o r e v e a l s t h a t the smaller the t i e l i n e reactance, the l a r g e r i s the s t a b i l i t y l i m i t . 32 i.o 2.0 r.u \ 1 ' 2 / 3 / X i = 1, X q = 0 . 8 5 P.U. ^ *30,Ms- S,Mm*S ?e.*-7* = 8 0 0 P.U / = Z 2 = 3 0 0 P.U. ( i - r r o.o X--1.0 2. f = 0.1 3. f • 0.3 rip. 5-5 Tie Line Resistance Effect on. Power Limits. 1. X i = 1, Xq. = 1 2. X i = 1 , X q = 0-8 3. X i - 1, X q «= 0.6 P . f . = 1,0 0.25 0.5 0-75 10 r Fi£L 5-6 Tie Line Resistance Effect on Power Limits at Unity Power Factor. 33 34 5,4 E f f e c t of E x c i t e r Time Constant The range of conventional e x c i t e r time constants v a r i e s from about 0.5 to 3 seconds. P i g . 5-9 shows the e f f e c t of v a r y i n g e x c i t e r time constant while the other parameters of the r e g u l a t o r and governor are kept constant. I t can be seen that f o r the p a r t i c u l a r g a i n l/ e= 30 the s t a b i l i t y l i m i t tends to inc r e a s e w i t h i n c r e a s i n g e x c i t e r time constant. F i g . 5-10 shows the r e s u l t s of two other s t u d i e s . By v a r y i n g the v o l t a g e - r e g u l a t o r time Z& and keeping the g a i n \i = 30, the in c r e a s e i n the value of Z seems to have a r e e favourable e f f e c t on s t a b i l i t y l i m i t . However, the maximum l i m i t depends on both (/ and ZQ» F i g . 5-10 a l s o shows t h a t f o r a f i x e d Z^ of 800 p.u. the maximum power l i m i t occurs i n the r e g i o n of low v o l t a g e - r e g u l a t o r g a i n . 5.5 E f f e c t of the Governor Time Constants on the Dynamic S t a b i l i t y L i m i t 5.5.1 Governor w i t h one time constant When the speed governor has a s i n g l e time constant, the s t a b i l i t y l i m i t of a synchronous generator can be improved c o n s i d e r a b l y by making the time constant s m a l l . F i g . 5-11 shows the s t a b i l i t y l i m i t of the system under u n i t y and l e a d i n g power f a c t o r o p e r a t i o n as a f f e c t e d by the value of the governor time constant. The s o l i d l i n e s are f o r the round-rotor machine and the dotted l i n e s are f o r the s a l i e n t - p o l e machine. Both sets, of curves show that f a s t a c t i n g governors are d e s i r a b l e f o r the improvement of the system dynamic s t a b i l i t y . 35 5 - 1 0 Exciter Time Constant a n d & a i n Effect on Power L i m i t s . 36 05 1.0 1.5 2.0 P.U. X « 1, f - 0,1 P.U. \\ Mi = 3 0 , Ms =5 \ \ \ Mm = 25 ~frZe = Zs = 800 P.U ' / " / i. ?i = too , z 2 =0 l. 7, = 3 7 7 , 3 2 =0 3. Zi = 2000, Zt - 0 = -I , X<j =• 0 6 Xi = 1 , * q = 1 Ro\ 5-11 Effect of tKe One - time-constant Governor on. Power Limits. 5-12 Effect of the Gain of a One-time-constant Crovernor on Power Limits. 37 F i g . 5-12 shows the e f f e c t of governor gains on the s t a b i l i t y l i m i t . I t can be seen t h a t the l a r g e r the gain the l a r g e r i s the s t a b i l i t y l i m i t . For a governor with a l a r g e time constant i t w i l l have l i t t l e e f f e c t on the s t a b i l i t y l i m i t . This can be seen by comparing curve 1 of F i g . 5-1 with curve 1 of F i g . 5-11. 5.5.2 Governor with two time constants When the speed governor has two equal time constants i t s e f f e c t on the system dynamic s t a b i l i t y changes. The improve- ment i n the s t a b i l i t y l i m i t i s v e r y small even when the governor time constants are decreased from 1000 to as low as 100 p.u. Moreover,there i s a value of governor time constant, between zero and the hi g h e s t p o s s i b l e v a l u e , t h a t gives the sm a l l e s t s t a b i l i t y l i m i t . Curve 3 ( s o l i d l i n e ) i n F i g . 5-13 shows t h a t the lowest s t a b i l i t y l i m i t , among the four computations, occurs f o r a governor time constant value of 200 p.u. While the s o l i d l i n e s i n F i g . 5-13 i n d i c a t e the s t a b i l i t y l i m i t of the round-rotor synchronous machine , the dotted l i n e s i n the same f i g u r e i n d i c a t e those of the s a l i e n t - p o l e machine. Very l i t t l e improvement i n s t a b i l i t y l i m i t i s obtained by reducing the governor time constant of a s a l i e n t - p o l e machine from 1000 p.u. t o - z e r o . F i g . 5—14 shows the s l i g h t decrease obtained i n s t e a d of an expected i n c r e a s e of s t a b i l i t y l i m i t w i t h an i n c r e a s e i n governor g a i n . This i s c o n t r a d i c t o r y to the r e s u l t s obtained from the study of the system w i t h a governor w i t h one time constant. The above c o n c l u s i o n s suggest that the governor e f f e c t on the dynamic s t a b i l i t y l i m i t must be i n c l u d e d i f the governor Ft^ . 5713 Effect of the Two - time-constant G-overnor on. rower Limits. 0.5 i.o 1.5 2.0 p. u. 1 2 \ |v x = i,r=o.i p.u. II Ml ^ JO , Ms * 5 jf 7e = 7s - 800 P U . Zi = Za s 300 P U . X<M,X,=1 J l .^mHO Xel=1 , *« '0 .6 |2 . JMn* 5 -f.O 2.0 Fip .̂ 5-14 Effect of the Sain o} a Two-time constant Governor on Power Limits. i s represented by a s i n g l e time constant. On the other hand, the e f f e c t of the governor i s r a t h e r small and could t h e r e f o r e be n e g l e c t e d i f the governor i s represented by two equal time c o n s t a n t s . 40 6. DYNAMIC STABILITY STUDIES USING THE D-PARTITION METHOD Por a given operating load c o n d i t i o n , the boundary-of the dynamic s t a b i l i t y of a system can be found i n the (|x -u. ) plane by using the D - p a r t i t i o n Method. An <c value ranging from 0 to 0.02 i s found to be s u f f i c i e n t f o r f i n d i n g a l l the r e s u l t s of p r a c t i c a l i n t e r e s t . U s u a l l y (0 = 0 to 0.001 y i e l d s the boundary of maximum permissible \i f and <o = 0.001 to 0.011 y i e l d s the boundary of the maximum s permissible The studies are c a r r i e d out on the system by varying system operating conditions and also some of the system parameters. In general, the s t a b i l i t y r egion of i n t e r e s t i n each study i s shown w i t h two boundaries i n the (\x - j / ) plane. The 6 S u p p e r — l e f t boundary w i l l be r e f e r r e d to hereafter as the high-p.' s bound and the l o w e r - r i g h t boundary as the h i g h - ^ bound. 6.1 The E f f e c t of Saliency on the S t a b i l i t y Boundaries F i g . 6-1 shows the e f f e c t of s a l i e n c y on the s t a b i l i t y boundaries i n the {\i - |/ ) plane. The governor a c t i o n i s excluded by s e t t i n g \i = 0. Curves 1, 2 and 3 correspond to x = ^ m q 0.6, 0.85, and 1.0 r e s p e c t i v e l y while x i s kept constant at d u n i t y . I t i s observed that f o r the p o r t i o n of the curves shown, the presence of s a l i e n c y increases the s t a b i l i t y region at the high-p/ bound, but decreases i t i n the high-j/ bound. However, i n the very low \i region the presence of s a l i e n c y decreases the 41 42 s t a b i l i t y region at the high-u^ bound. Curves 3 and 2 i n P i g . 6—1, which correspond r e s p e c t i v e l y to the s t a b i l i t y r egion of a round-rotor machine wi t h and without r o t o r s a l i e n c y , c l e a r l y i n d i c a t e that the s a l i e n c y y i e l d s a r a t h e r p e s s i m i s t i c r e s u l t f o r the s t a b i l i t y region at the high-|/ e bound and a r a t h e r o p t i m i s t i c r e s u l t f o r the s t a b i l i t y r e g i o n at the high-[/ bound. s F i g . 6-2 shows th a t f o r the same power l i m i t a s a l i e n t - pole machine must be operated w i t h a higher [/ gain than f o r the round-rotor machine. 6.2 The E f f e c t of the Tie Line Resistance and Reactance F i g . 6-3 shows the e f f e c t of the t i e l i n e r e s i s t a n c e on the s t a b i l i t y region of the system. The l i n e reactance i s set at 1.0 p.u., and the — r a t i o i s v a r i e d . The s t a b i l i t y region i s increased considerably by an increase i n the ^ r a t i o from zero. By comparing curves 1 and 2 i n F i g . 6-3 , i t can be seen that a very p e s s i m i s t i c r e s u l t i n the s t a b i l i t y r egion would be obtained i f the t i e l i n e r e s i s t a n c e i s completely ignored. This i s r a t h e r misleading and might r e s u l t i n an inappropriate c o o r d i n a t i o n of the voltage r e g u l a t o r parameters. Curves 1 and 2 of F i g . 6—4 show the e f f e c t of t i e l i n e reactance on the s t a b i l i t y r e g i o n . I t can be seen that the decrease i n the t i e l i n e reactance gives a l a r g e r s t a b i l i t y r e g i o n .  44 6.3 Neglecting Both the Tie Line Resistance and the Saliency i n a Round-jiotor Machine F i g . 6-5 shows the e f f e c t of both t i e l i n e r e s i s t a n c e and s a l i e n c y on the s t a b i l i t y region of a round-rotor machine operating at u n i t y power f a c t o r f o r various power value s . I t can be seen that the neglect of both l i n e r e s i s t a n c e and s a l i e n c y gives a smaller s t a b i l i t y r egion at the h i g h - ^ bound*, For the s t a b i l i t y r egion at the high-[/ bound, the e f f e c t of n e g l e c t i n g both l i n e r e s i s t a n c e and s a l i e n c y becomes very great when x, i s large or the short c i r c u i t r a t i o i s s m a l l . This can d be seen from F i g . 6-6. 6*4 The E f f e c t of Short C i r c u i t Ratio F i g . 6-6 also shows that the s t a b i l i t y region at the h i g h — b o u n d decreases w i t h short c i r c u i t r a t i o . As f o r the s t a b i l i t y region at the high—u' bound, the decrease i n short s c i r c u i t r a t i o u s u a l l y increases the s t a b i l i t y r e g i o n . The s o l i d l i n e s show the s t a b i l i t y r egion of the round-rotor machine w i t h x a s a l i e n c y —̂ - = 0.85, and the dotted l i n e s are those of the x d round—rotor machine w i t h s a l i e n c y and t i e l i n e r e s i s t a n c e neglected. 6.5 The E f f e c t of the E x c i t e r Time Constant F i g . 6-7 shows the s t a b i l i t y region of a round-rotor machine, x,= x , operating at u n i t y power f a c t o r subjected to d q the v a r i a t i o n of £ .. Noticeable e f f e c t s are observed only i n the s t a b i l i t y r egion at the high-p,^ bound, e s p e c i a l l y f o r low \i v a l u e s . 45 1 . 0 P . U . j Xi - 1 , X , » 1 L r - 0 1 I r = 0 . 0 p . 1 . 4 5 P . U . p - 1 . 5 5 p.u. ? - 1 . 6 5 P . U . F i o ^ 6 - 5 E f f e c t o f N e o l e c t i n o Tts L i n e R e s i s t a n c e a n d S a l i e n c y o n t h e o r a b i l i t v ty Re^on . S . C . R » 1 S . C . R • l / l . l S . C . R - 1 / i . s . 0 , j £ i . 1 . 0 ' ' X = 1 - 0 . 0 . 1 , ^ 1 = 0 . 8 1 9 . 1 . 4 5 P . U . . - 1 . 5 , t , - 3 2 • 3 0 0 P . U . Zj-eoo P.u. Fjo\ 6-6 Effect of Short Circuit Ratio on Region . t h e S t a b i l i t y rip 6-a Exciter Time Constant Effect on trie Stability Reoijn of a Salient-pole Machine. 47 S i m i l a r computations are repeated f o r a s a l i e n t - p o l e machine i n s t e a d of the round—rotor machine. The r e s u l t s are given i n P i g . 6-8 which shows that the same conclusion can be drawn - f o r both types of machine, namely * f o r low \x values the s smaller the e x c i t e r time constant, the smaller the s t a b i l i t y region at the high-|/ bound/ , 6.6 The E f f e c t of the S t a b i l i z e r Time Constant The decrease i n T u s u a l l y r e s u l t s i n an increase i n s the s t a b i l i t y r e g i o n . The s t a b i l i t y region at the h i g h - ^ bound and the s t a b i l i t y region at the high-p,' bound increase s as T decreases which can be seen from F i g . 6-9 and 6-10. How— s ever, f o r a s a l i e n t - p o l e machine, x^= 1 and x^= 0.6, a decrease i n z m a y cause a decrease i n the s t a b i l i t y region at the high— s \i bound f o r low \i v a l u e s , as i s seen i n F i g . 6-10. 6.7 The E f f e c t of the Governor Time Constants F i g . 6-11 shows the e f f e c t of a governor w i t h two time constants on the s t a b i l i t y region of a round-rotor machine, x^= xq= 1»0. I t can be seen that the two-time-constant governor a f f e c t s the s t a b i l i t y region at the high-u^ bound but not the high— \i bound. s For high \x v a l u e s , the smaller the governor time constants, the l a r g e r would be the s t a b i l i t y region at the high—(/ bound. However, there e x i s t s a case, T, = T0= 200 p.u., that y i e l d s the smallest s t a b i l i t y r e g i o n . The above conclusion a p p l i e s also to the case of a s a l i e n t - pole machine, x,= 1, x = 0.6, as seen i n F i g . 6-12. 48 49 100 » TO 00 SO fo / 10 f\ f X « « 1 , X q ' 1 P . U . x . i , r - 0 1 Te - Z i • 8 0 0 P . U . . 5 1. 7, = 7 , . 3 0 P .U a 7 < I 4 1 2 i / i / ^2 if ii ^ 2 . 7, - C j - 7 0 4 J . t l = t , o TUU 4 . 7 , = 7 t - 2 0 0 5. 7 t = ?» - 6 0 0 P - 1 .45 P . U . P . f . . 1 hi 5 « 7 S • 10 MC Fig. 6 - 1 1 Effect of the Two - time - constant G-overnor on the Stability Report of a Round - rotor Machine . too 90 70 to 40 30 10 10 f e 7 * 9 4 t t < ft x . 1 , r - 0 1 Zt - ? » . eoo p.u. Mm . 5 3> 1. ? , • 7 , - 3 0 P.U J , 2 . 3 . 4 <? 2 . 7 , - Z z - 70 .A _ 3 . 7 , - Zt - 2 0 0 4 . 7 , = 7 , = 6 0 0 P • 1 . 4 5 P.f - 1 iii /» rl/ I' ij f 0 7 • 0 10 Mi. Ft^ 6 • 12 Effect of the Two - time - constant Governor on the .Stability flexion of a Salient-pole Machine. 50 The e f f e c t of a governor w i t h one time c o n s t a n t on the s t a b i l i t y r e g i o n i s shown i n F i g . 6-13. The time c o n s t a n t changes the s t a b i l i t y r e g i o n c o n s i d e r a b l y . A f a s t e r governor u s u a l l y g i v e s a l a r g e r s t a b i l i t y r e g i o n a t the high-p,' bound, except f o r low yf- where a v e r y f a s t a c t i n g governor may cause a s m a l l e r s s t a b i l i t y r e g i o n a t the high—\x bound. 00 » 60 7o U Wft 40 30 20 10 2 ~V / / Xd • X< • 1 P.U. x - 1 , r - o . i p .u . > ^ Ze »Z« -flOO P U . 7 /̂m = 25 1. Zi - o = e2 2. TA = 100, Z , = 0 3. 7 i = 377 , Z2 - 0 , 4. Ti = 2000,Z,- 0 P = 1.45 P U . P.f =1 8 7 6 S 4 3 2 1 / • / 1 7 * J i » 6 7 a e ID ., 1 0o looo F"Lo\ 6 -13 Effect of the One-time - constant Governor on tke Stability Region . 7. CONCLUSIONS 52 Prom the r e s u l t s of the studies i n t h i s t h e s i s , the p r i n c i p a l conclusions that can be drawn are as f o l l o w s : 1. I t has been found from Routh-Hurwitz C r i t e r i o n study that f o r f i x e d s e t t i n g s of c o n t r o l parameters the s a l i e n c y of a synchronous machine considerably increases the dynamic s t a b i l i t y l i m i t of a power system and from the D- p a r t i t i o n study t h a t f o r a f i x e d power l i m i t the s a l i e n c y of a synchronous machine increases the s t a b i l i t y region at the h i g h — b o u n d except f o r low [i v a l u e s , but decreases i t at the 6 S high— j i ' bound considerably. s 2. The short c i r c u i t r a t i o of a synchronous machine 1 _ has e f f e c t s s i m i l a r to that of x , • The l a r g e r the short c i r c u i t d r a t i o , the l a r g e r i s the dynamic s t a b i l i t y l i m i t . I t also increases the s t a b i l i t y region at the high-|/ e bound and decreases i t at the high-[/ bound. s 3. The t i e l i n e impedance has a s i g n i f i c a n t e f f e c t upon the dynamic s t a b i l i t y . Both the s t a b i l i t y l i m i t i n the Routh—Hurwitz C r i t e r i o n study and the s t a b i l i t y region i n the D — p a r t i t i o n study increase w i t h the t i e l i n e r e s i s t a n c e , but decrease w i t h an increase i n the l i n e reactance. For a f i x e d r a t i o of —, the e f f e c t of t i e l i n e r e s i s t a n c e on s t a b i l i t y l i m i t i s r a ther small f o r a small t i e l i n e reactance but becomes l a r g e r f o r a l a r g e r t i e l i n e reactance. 4. Both the s t a b i l i t y l i m i t and the s t a b i l i t y region are g r e a t l y reduced i f the s a l i e n c y and the t i e l i n e r e s i s t a n c e are neglected. A c t u a l l y a l a r g e r ^ rather than a smal l e r , \i , 53 g a i n i s p e r m i s s i b l e f o r the machine when the s a l i e n c y and the t i e l i n e r e s i s t a n c e are c o n s i d e r e d * 5. I t has been found t h a t a l a r g e r e x c i t e r time c o n s t a n t gives a l a r g e r s t a b i l i t y l i m i t f o r a p a r t i c u l a r s e t t i n g of param e t e r s , o r a l a r g e r s t a b i l i t y r e g i o n f o r a p a r t i c u l a r power. However, i n o r d e r t o o b t a i n a maximum s t a b i l i t y l i m i t , the time constants and the g a i n of an e x c i t e r must be a p p r o p r i a t e l y coordinated. 6. A decrease i n the s t a b i l i z e r time c o n s t a n t i n c r e a s e s the s t a b i l i t y r e g i o n a t the high-|/ e bound and a l s o a t the h i g h — a bound. However, f o r low \L v a l u e s , d e c r e a s i n g the s s s t a b i l i z e r time c o n s t a n t may r e s u l t i n a decrease of the s t a b i l i t y r e g i o n a t the high—u.^ bound. 7. The governor has c o m p a r a t i v e l y l e s s e f f e c t on the dynamic s t a b i l i t y . F o r a governor w i t h two equ a l time c o n s t a n t s , i t s e f f e c t on the dynamic s t a b i l i t y l i m i t i s s m a l l and i n some cases i t can even decrease the s t a b i l i t y l i m i t . F o r a governor w i t h one time c o n s t a n t , i t s e f f e c t on s t a b i l i t y i s c o m p a r a t i v e l y l a r g e • F o r high-jx' v a l u e s a t the high-[/ bound, the s m a l l e r the governor time c o n s t a n t , the l a r g e r i s the s t a b i l i t y r e g i o n . F o r the low \i p a r t of the high-|/ bound, however, a s m a l l t i m e — c o n s t a n t governor may decrease the s t a b i l i t y r e g i o n . On the o t h e r hand, the governor time c o n s t a n t has l i t t l e e f f e c t on the s t a b i l i t y r e g i o n a t the high-|/ bound. s The s t u d y i n t h i s t h e s i s has been c o n f i n e d t o the dynamic s t a b i l i t y i n the s m a l l w i t h o u t c o n s i d e r i n g n o n l i n e a r i t i e s . The dynamic s t a b i l i t y i n the l a r g e of a power system i n c l u d i n g non- l i n e a r i t i e s i s l e f t f o r f u t u r e s t u d i e s . 54 APPENDIX I. SYMBOLS AND UNITS Subscript o denotes an i n i t i a l condition Prefix A denotes a small change about the i n i t i a l operating value p = ̂ £ = time derivative i ^ # i ^ = armature currents i n d— and q-axes respectively v,,v = armature voltages i n d— and q-axes respectively = armature terminal voltage v ^ = applied voltage i n f i e l d winding V q = i n f i n i t e bus-bar voltage ^d'^q = a r m a " k u r e f l u x linkages i n d- and q-axes respectively E x , —§— v_, = armature open c i r c u i t voltage x^»x^ = synchronous reactance i n d- and q-axes respectively x , = mutual reactance between the stator and rotor i n aa d—axis x = t i e - l i n e reactance' between the generator and the bus—bar Rp = f i e l d winding resistance r = armature winding resistance i n d- or q-axis c i r c u i t r = t i e - l i n e resistance between the generator and the bus—bar 8 = power angle T m = mechanical input torque to the rotor T = energy conversion torque e P = real power output of the machine Q = reactive power output of the machine H = i n e r t i a constant J = moment of i n e r t i a 55 a = damping c o e f f i c i e n t © = instantaneous angular p o s i t i o n of r o t o r *p0 = angular v e l o c i t y of machine # =. rated angular v e l o c i t y f = rated system frequency 7/ = d i r e c t - a x i s t r a n s i e n t o p e n - c i r c u i t time constant do X^ = d i r e c t - a x i s t r a n s i e n t s h o r t - c i r c u i t time constant Mo X? - d i r e c t - a x i s subtransient s h o r t - c i r c u i t time constant r ^ o = quadrature-axis subtransient o p e n - c i r c u i t time constant quadrature-axis subtransient s h o r t - c i r c u i t time constant = d i r e c t - a x i s damper leakage time constant T = e x c i t e r time constant e = s t a b i l i z e r time constant ^1*^2 = g ° v e r n o r time constants H = e x c i t e r gain 6 X f i r = converter gain H = a m p l i f i e r gain £t • p, , = s t a b i l i z e r gain He = H e x ^ r = r e g u l a t o r g f t i n ii = —§— \i = over a l l r e g u l a t o r gain e *p e / , A H_ = 1 + r i o v r l c + = over a l l s t a b i l i z e r gain Throughout the t h e s i s a l l c a l c u l a t i o n s are made using the per- u n i t system based on M.K.S. u n i t s . 56 The u n i t of time i s 1 radian; at f = 60 Hz, 1 second = 2nf = 377 r a d i a n s . Moment of i n e r t i a J = 4n;fH, and p.u. power and p.u. torque are numerically i d e n t i c a l , i . e . P = T^Q p.u. 57 APPENDIX I I . VOLTAGE REGULATOR AND SPEED GOVERNOR TRANSFER FUNCTIONS a. Voltage Regulator Transfer F u n c t i o n ^ ' ^ F i g . I . Voltage Regulator The voltage r e g u l a t i n g system under c o n s i d e r a t i o n i s shown i n F i g . I . I t has the f o l l o w i n g p a r t s : 1. A means of measuring the terminal voltage e r r o r . 2. A device to convert the voltage e r r o r to a s u i t a b l e s i g n a l ( A v r ) . The converter may be defined by an a m p l i f i c a t i o n f a c t o r (|i ) and zero time constant. 3. A s t a t i c or r o t a t i n g a m p l i f i e r w i t h a large gain (ji ) and a small time constant ("C ). 4. An e x c i t e r supplying the f i e l d current to a synchronous machine. 5. A s t a b i l i z i n g transformer feeding back i n t o the re g u l a t o r c i r c u i t a s i g n a l p r o p o r t i o n a l to the rate of change of the f i e l d c u r r e n t . 58 Since Av =-u Av,, converter r r r t F (i Ay Av = —-—-$ amplifier a 1 + *a P Av, = Si S | Av„,, s t a b i l i z e r and |X Av., = , 6 X (Av - Av ), f i e l d r a l + r p a D e the voltage regulator transfer function g ( P ) £ A v f d Q i + y y A V t d+VfHVV^^^eV] where u; = l l LX [X , IX = u, , u <e r e x r a r r r s " s v e x b. Speed Governor Transfer Function^** The speed governor system, including the time lags due to steam or water and some intermediate actuating member can be specified by two time constants, and one gain. Thus the transfer function of the governor / / v A ~ % f(p) = (rlP+i)(r2p+i) If one j, of the time constants can be neglected the transfer function becomes (i+^p) 59 APPENDIX I I I . DERIVATION OP THE CHARACTERISTIC EQUATION Prom equation ( 3 - 1 4 ) , the following characteristic equation i s obtained 2 i ' ( l + ^ 0 P ) ( l + \P)(l+^p){l+-(VVs>P + TeV J A = p(P) ( i + r l P ) (i+ ^p) (i+ ̂ p ) ( i + r i P ) (I+T 2P){i+ ( z e + r n s)p + 2 jh( P ) + ( 1 < p ) ( l + 7 l p ) ( l + V H 1 + ( r e + y s ) p n e T s P 2 ) ( V + A 9 ) = 6 To expand, let V V d V A 3 r d o W d + A 3 B 3 = V d + A 9 T B 4 = V d V A 8 r d o + B l ( a " TT> + B 2 J B 5 = ( A 6 + A 7 r d ) K d + A 8 + A9 T d o + B 2 ( a " ^ ^ V * ^ + A 4 T .= T T T T a 1 2 e s 60 Nov |= p 7T aJB 1 + p 6 +P +P J(p )< l+y> ( l + V ) ( 1 + ^ T a { B l ( a - ¥ ) + J B 2 ; T a B 2 ( a - VVTdiVa - ¥ ^ B 2 K J V i V s T T T.JB.+T {B. (a So.) + J B J + T,B„(a - -££•) b 1 cl 1 a 7 2) d 2 w ' T - a A1 (jT_+(a ££•) 7 r j . \ l e s ^ e l ( E » ' 1 2 s) T T T. (B. (a - -££) + JB-1 + T B_(a - -££) + H {B. ( r + r u' )+B o r r ) M l tt 2J c 2 « ml 1 e s r s y 2 e sj +P T eo •) + JB, tt j 1 Next consider h(p)(i+r d y op)(i+r 1p)(i +T 2p)|i+(r e+r ̂ s) P+rr sp 2 f ( A 4 P + A 5 ) - ^ ( P V I ¥ S + P 3 ^ A 4 T E + A 5 T l T 2 7 s ) + P 2 ( A 4 V A 5 T E ) + P ( A 4 + A 5 T P ) 61 ( l + r ^ ) ( i + r 1 p ) ( i + r 2 p ) { l + ( T e + r ^ J p +r T s p 2 | x d ( P ) ( A 6 P + A ? ) + P 4 | T a A ? + T cA 6 d + T D ( A 6 + A 7 T D )} + P 3 ( T bW- . T c<VVd> + V r ) + P 2 | T b ( A 6 + A ^ ) + T ^ * A 6r d j = P 6 T a A8 T do + P 5{Ta<V A9 Tdo ) + TdVdo) + P 4 { T a A 9 + W d o + T d ( V A9rdo)} + P 3 { T b A8 r d o + . V V V i o * + 'V? ) + P 2 {T CA 9 + T b(A 8 + A 9r d;) + AgT^J + P {T bA 9 + A8+ A 9 r D Q J + A 9 Hence the f i n a l form o f the c h a r a c t e r i s t i c e q u a t i o n i s w r i t t e n as e q u a t i o n ( 3 - 1 5 ) • APPENDIX IV. DERIVATION OF THE D-PARTITION EQUATIONS Let T =" T r« + rz + c 1 2 e s 1 2 e equation (3-15) becomes .6 _ ... 9 P J B i r i V s * r { J B l . « V T2 ) s + B 4 T l T 2 T s ] + M J B l T 3 + B 4 ( T l + T 2 ) r s + V l T 2 T s j + P 3 { B 3 r i r 2 r s + V s + W W ^mVs) + P 2 | B 3 ( r 1 + r 2 ) T : s + B 5 r g + % B 2 T S ) + P B 3 T 3 P 5 j V i V . + W V l V s ) + P 3 { T E B 6 + V W l V s ) +P 2 {T E A 5 + T P B 6 + J A 1 + % A 1 X G j + p (T P A 5 + B 6 + + A ? p 7 T A J B 1 + P 6 { T A B 4 + T E J B l r e ] + P 5 { T A B 5 + T ^ * T ^ t ] + P 4 { T A B 3 + T C B 4 + T E B 5 T E + T P J B 1 + » r * f e * s ) + P 3 { t / c B 5 + T E B 3 V W B 1 J + % ( B l + B2 T s ) r e} + P 2 ( T C B 3 + T P B 5 + B 4 + % ( B 1 + B 2 T E ) ) +p(T PB 3 + B 5 % B 2 ) • + B, = 0 The c h a r a c t e r i s t i c equation can be expressed as j / P(p) - |/Q(p) + R(p) = 0 Let p = j«, we have U , S {P 1 («) + j P 2 ( « ) J - n / e | ^ ( « ) + d Q 2 ( » ) J +{B1(»)' + j E 2 ( « o ) J = 0 where 63 T1 (») P2(») A W . + < ° V B l V B 4 ( r i + r 2 ) r s ' V l V ^ - « 2 ( B 3 ( V r 2 ) X s + B 5 r s + % B2 T s ) • 5{jB 1(r 1+r 2)T 8.+ B ^ r ^ ] + B 5 ( V r 2 ) T a + ^ { B 3 T l V s • ~4^s T 7 + B,r + • B3T3 T_A_+ B + p, A , F 5 6 rm 1 «4{Vki+ V l V s j - * 2 { T E V T F B 6 + J A 1 + %Vs) + A 5 " ^ i V s " TT3{TEB6+ V W l V . ] + * ^ 6 { T a V W e ) + t t 4 { T a B 3 + V V V 5 V V B 1 +H B z r 1 - « 2 ( T ' B - + T B + B + jx (B + B T ) ' rm 1 e s ] I c 3 F 5 4 m i 2 e V T A J B 1 + «. 5(T AB 5 + T ' 0 JB 1 + T E B 4 C E ] " »3{T'oB5 + W e + V 4 + J V " » ( V V s ) r , + " ( TP B3 + B 5 + »m B 2 64 REFERENCES 1. 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