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Local assessment of transient stability for generator tripping Mihirig, Ali Mohamed 1984

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LOCAL ASSESSMENT OF TRANSIENT STABILITY FOR GENERATOR TRIPPING by ALI MOHAMED MIHIRIG B . S c , U n i v e r s i t y o f T r i p o l i , 1978 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL ENGINEERING We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA March 1984 © A l i Mohamed M i h i r i g , 1984 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 o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t 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 r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f 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 o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t 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 g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department o f Zl^ct^cd S^fl• r«j*A<* The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date DE-6 (3/81) i i ABSTRACT The l o c a l v a r i a b l e s o f a synchronous g e n e r a t o r a r e i n v e s t i -g a ted as p o s s i b l e i n d i c a t o r s o f the t r a n s i e n t s t a b i l i t y o f the g e n e r a t o r . A number o f case s t u d i e s a r e c a r r i e d o u t on t h r e e t e s t systems t o f i n d out which g e n e r a t o r v a r i a b l e o r v a r i a b l e s can be used. By comparing the c o n v e n t i o n a l swing c u r v e o f each g e n e r a t o r w i t h i t s l o c a l a c c e l e r a t i o n curve d u r i n g the same t r a n s i e n t p e r i o d , a new c r i t e r i o n f o r t r a n s i e n t s t a b i l i t y assessment i s d e v e l o p e d t o r e p l a c e e x i s t i n g g e n e r a t o r t r i p p i n g schemes t h a t r e l y upon p r e - d e t e r m i n e d c o n t i n g e n c y s t u d i e s . The new c r i t e r i o n i s based upon the b e h a v i o r o f the a c c e l e r a t i o n o f the g e n e r a t o r f o l l o w i n g a d i s t u r b a n c e t o the system. No measurements from any o t h e r machine i n the system are r e q u i r e d t o a s s e s s the s t a b i l i t y o f the g e n e r a t o r . i i i TABLE OF CONTENTS Page ABSTRACT i TABLE OF CONTENTS i i LIST OF TABLES i v LIST OF ILLUSTRATIONS v ACKNOWLEDGEMENT v i i CHAPTER 1 INTRODUCTION 1 CHAPTER 2 TRANSIENT STABILITY ANALYSIS 5 2.1 I n t r o d u c t i o n 5 2.2 Assessment o f T r a n s i e n t S t a b i l i t y 7 2.2.1 The C o n v e n t i o n a l Method 7 2.2.2 The Equal Area C r i t e r i o n 8 2.2.3 Energy F u n c t i o n Method 13 2.2.4 L o c a l Assessment 19 2.3 S t a b i l i t y C o n t r o l s 20 2.3.1 Dynamic B r a k i n g 21 2.3.2 High Speed C i r c u i t B r e a k e r R e c l o s i n g .. 22 2.3.3 S e r i e s C a p a c i t o r s 23 2.3.4 F a s t V a l v e A c t i o n 23 2.3.5 Independent P o l e O p e r a t i o n o f C i r c u i t B r e a k e r s 24 2.3.6 G e n e r a t o r T r i p p i n g 25 CHAPTER 3 SYSTEM MODELS AND TEST SYSTEMS 27 3.1 C l a s s i c a l Model o f Synchronous Machines 27 3.2 System R e p r e s e n t a t i o n 28 3.3 The Three T e s t Systems 30 3.3.1 The Three Machine System 30 3.3.2 The Four Machine System 30 3.3.3 The F i v e Machine System 34 CHAPTER 4 STUDIES OF GENERATOR VARIABLES AVAILABLE FOR LOCAL STABILITY ASSESSMENT 37 4.1 , A c c e l e r a t i n g Power 37 4.2 Speed and A c c e l e r a t i o n 40 4.3 A c c e l e r a t i o n S t a b i l i t y L i m i t s 41 4.4 Case S t u d i e s 47 iv Page 4.4.1 The Three Machine System 47 4 .4 .2 The Four Machine System Case S t u d i e s .. 64 4 .4 .3 The Five Machine System Case Studi e s 64 4.5 Case S t u d i e s D i s c u s s i o n 68 4.6 T r a n s i e n t S t a b i l i t y C r i t e r i o n Based on A c c e l e r a t o n Measurements ... 70 CHAPTER 5 CONCLUSION AND FUTURE WORK 74 REFERENCES 77 V LIST OF TABLES P a 9 e Table 3.1 Generator data and i n i t i a l c o n d i t i o n s of the three-machine system 32 Table 3.2 Generator data and i n i t i a l c o n d i t i o n s of the four-machine system 32 Table 3.3 Transmission l i n e parameters and loads of the five-machine system 36 Table 3 .4 Generator data and i n i t i a l c o n d i t i o n s of the five-machine system 36 v i LIST OF ILLUSTRATIONS P a 9 e Figure 2.1 S i n g l e machine i n f i n i t e bus 11 Figure 2.2 The equal area c r i t e r i o n 11 Figure 2.3 Comparison of equal area c r i t e r i o n and energy method 18 Figure 3.1 C l a s s i c a l model of synchronous machine 28 Figure 3.2 The three-machine system 31 Figure 3.3 The four-machine system 33 Figure 3.4 The five-machine system 35 Figure 4.1 A c c e l e r a t i n g power and a c c e l e r a t i o n ........ 39 Figure 4.2 Speed d e v i a t i o n of machine 4 f o r d i f f e r e n t f a u l t l o c a t i o n s 42 Figure 4.3 Swing curves and a c c e l e r a t i o n 43 Figure 4.4 A c c e l e r a t i o n of machine 4 f o r d i f f e r e n t f a u l t l o c a t i o n s 44 Figure 4.5 A c c e l e r a t i o n of machine 2 f o r d i f f e r e n t f a u l t l o c a t i o n s 46 Figure 4.6 A c c e l e r a t i o n of machine 3 f o r d i f f e r e n t f a u l t l o c a t i o n s 46 Figure 4.7 The three-machine system case s t u d i e s 48 Figure 4.8 Swing curve and a c c e l e r a t i o n of machine 1 f o r a f a u l t at bus 4 49 Figure 4.9 Swing curve and a c c e l e r a t i o n of machine 2 f o r a f a u l t at bus 4 50 Figure 4.10 Swing curve and a c c e l e r a t i o n of machine 3 f o r a f a u l t at bus 4 • > • 51 Figure 4.11 Swing curve and a c c e l e r a i t o n of machine 2 f o r a f a u l t at bus 7 53 Figure 4.12 Swing curve and a c c e l e r a t i o n of machine 3 f o r a f a u l t at bus 7 54 v i i Page F i g u r e 4.13 Swing c u r v e and a c c e l e r a t i o n o f machine 2 f o r a f a u l t a t bus 9 55 F i g u r e 4.14 Swing c u r v e and a c c e l e r a t i o n o f machine 3 f o r a f a u l t a t bus 9 56 F i g u r e 4.15 Swing c u r v e and a c c e l e r a t i o n o f machine 2 f o r a f a u l t a t bus 5 • 58 F i g u r e 4.16 Swing c u r v e and a c c e l e r a t i o n o f machine 3 f o r a f a u l t a t bus 5 59 F i g u r e 4.17 Swing c u r v e and a c c e l e r a t i o n o f machine 2 f o r a f a u l t a t bus 6 60 F i g u r e 4.18 Swing c u r v e and a c c e l e r a t i o n o f machine 2 f o r a f a u l t a t bus 6 61 F i g u r e 4.19 Swing c u r v e and a c c e l e r a t i o n o f machine 2 f o r a f a u l t a t bus 8 62 F i g u r e 4.20 Swing c u r v e and a c c e l e r a t i o n o f machine 3 f o r a f a u l t a t bus 8 63 F i g u r e 4.21 Swing c u r v e and a c c e l e r a t i o n o f machine 4 f o r a f a u l t a t bus 10 65 F i g u r e 4.22 Swing c u r v e and a c c e l e r a t i o n o f machine 2 f o r a f a u l t a t bus 10 66 F i g u r e 4.23 Swing c u r v e and a c c e l e r a t i o n o f machine 3 f o r a f a u l t a t bus 10 67 F i g u r e 4.24 Swing c u r v e and a c c e l e r a t i o n o f machine 5 f o r a f a u l t a t bus 11 69 F i g u r e 4.25 Rate o f change o f a c c e l e r a t i o n 73 ACKNOWLEDGEMENT I w i s h t o e x p r e s s my s i n c e r e s t g r a t i t u d e t o my s u p e r v i s o r D r . M. D. Wvong f o r a l l h i s h e l p and u n d e r -s t a n d i n g d u r i n g t h e p r e p a r a t i o n o f t h i s t h e s i s . Acknowledgement i s due t o Mr. K. P o k r a n d t , o f B.C. H y d r o , f o r o r i g i n a l l y s u g g e s t i n g t h e t o p i c o f t h i s t h e s i s i n d i s c u s s i o n s w i t h D r . M. D. Wvong , and f o r i n f o r m a t i o n on t h e p r o b l e m . I am a l s o g r a t e f u l t o my c o l l e a g u e B. G a r r e t f o r p r o o f r e a d i n g t h e m a n u s c r i p t . I am i n d e b t e d t o t h e De p a r t m e n t o f E l e c t r i c a l E n g i n e e r i n g f o r t h e t e a c h i n g a s s i s t a n t s h i p p r o v i d e d . T hanks a r e due t o my w i f e Najwa f o r h e r c o n t i n u o u s e n c o u r a g e m e n t t h r o u g h o u t my g r a d u a t e p r o g r a m . CHAPTER 1 1 INTRODUCTION When power system e n g i n e e r s use the term ' s t a b i l i t y * t h e y mean the p r o p e r t y which ens u r e s t h a t the power systems w i l l r emain i n e q u i l i b r i u m through normal and abnormal o p e r a t i n g c o n d i t i o n s . As power systems grow l a r g e r and more complex, s t a b i l i t y s t u d i e s become v e r y i m p o r t a n t . W i t h the e v e r i n c r e a s i n g demand f o r e l e c t r i c a l energy and dependence on an u n i n t e r r u p t e d s u p p l y , the a s s o c i a t e d r e q u i r e m e n t o f h i g h r e l i a b i l i t y d i c t a t e s t h a t power systems be d e s i g n e d t o m a i n t a i n s t a b i l i t y under s p e c i f i c d i s t u r b a n c e s , c o n s i s t e n t w i t h economy. The problem o f s t a b i l i t y a r i s e s when the system i s d i s t u r b e d . The n a t u r e o r magnitude o f the d i s t u r b a n c e g r e a t l y a f f e c t s the s t a b i l i t y o f the power system. I f the d i s t u r b a n c e i s l a r g e , then the o s c i l l a t o r y t r a n s i e n t s t h a t o c c u r w i l l a l s o be l a r g e . The q u e s t i o n o f whether the power system w i l l s e t t l e t o a new s t a b l e o p e r a t i n g s t a t e , o r whether i t w i l l l o s e s y n c h r o n i s m then becomes i m p o r t a n t . T h i s i s known as the t r a n s i e n t s t a b i l i t y problem. G e n e r a t o r s t h a t a r e l i k e l y t o l o s e s y n c h r o n i s m under c e r t a i n d i s t u r b a n c e s must be d i s c o n n e c t e d from the power system t o p r e v e n t system breakup o r g e n e r a t o r damage. T h i s needs a complete a n a l y s i s ' o f the s t a b i l i t y problem t o know which g e n e r a t o r s must be d i s c o n n e c t e d from the power system f o r a c e r t a i n d i s t u r b a n c e i n o r d e r t o m a i n t a i n s t a b i l i t y . 2 The c o n v e n t i o n a l method used t o a n a l y z e the t r a n s i e n t s t a b i l i t y problem i s the time s o l u t i o n o f the d i f f e r e n t i a l e q u a t i o n s u s i n g a d i g i t a l computer. Then, based on c o n s i d e r a -t i o n o f v a r i o u s l a r g e d i s t u r b a n c e s on the power system, the t r a n s i e n t s t a b i l i t y problem i s s o l v e d ( s p e c i f i c a l l y ) and the r e s u l t s can be used t o d e s i g n the power system s w i t c h g e a r and t o a d j u s t the p r o t e c t i v e r e l a y s f o r these s p e c i f i c d i s t u r b a n c e s . The r e s u l t s o f such t r a n s i e n t s t a b i l i t y s t u d i e s f o r a power system can be s t o r e d i n an o n - l i n e computer a t the c e n t r a l c o n t r o l room. S u i t a b l e a c t i o n can be t a k e n a u t o m a t i c a l l y when any o f the s p e c i f i e d d i s t u r b a n c e s o c c u r s . T h i s method i s v e r y r i g i d because i t i s o n l y v a l i d f o r s p e c i f i c d i s t u r b a n c e s and i s based upon p r e - d e t e r m i n e d r e s u l t s . U s u a l l y the most se v e r e c a s e s o f d i s t u r b a n c e s o n l y a r e c o n s i d e r e d , so i n c o r r e c t d e c i s i o n s may be made i n some i n s t a n c e s . P r e s e n t day p r a c t i c e o f g e n e r a t o r t r i p p i n g i s based upon t h e s e c o n t i n g e n c y s t u d i e s . The c o n v e n t i o n a l method used i n s t a b i l i t y s t u d i e s i s e x p e n s i v e and time consuming because a l a r g e number o f d i f f e r e n t i a l e q u a t i o n s must be s o l v e d f o r each d i s t u r b a n c e c o n s i d e r e d . A l s o , the c o n v e n t i o n a l p r o t e c t i o n system a g a i n s t i n s t a b i l i t y r e l i e s h e a v i l y on p r e - d e t e r m i n e d r e s u l t s and r e q u i r e s e x t e n s i v e h i g h - r e l i a b i l i t y communications equipment t o t r a n s m i t the commands f o r t r i p p i n g l i n e s , g e n e r a t o r s , e t c . There have been at t e m p t s a t s o l v i n g the t r a n s i e n t s t a b i l i t y problem d i r e c t l y u s i n g Lyapunov f u n c t i o n s and c o n t r o l t h e o r i e s [ 1 , 2 ] , b u t the r e s u l t s o b t a i n e d are u s u a l l y t o o c o n s e r v a t i v e . 3 A simple and l e s s expensive s o l u t i o n i s s t i l l needed. The best s o l u t i o n to t h i s problem i s probably to r e l y upon the measurements of v a r i a b l e s a v a i l a b l e l o c a l l y at each generator and to assess i t s t r a n s i e n t s t a b i l i t y based upon these l o c a l measurements. This w i l l not r e q u i r e p r i o r d i g i t a l computer s t u d i e s and the expensive dedicated communication equipment p r e s e n t l y used. There are two generator t r i p p i n g schemes that do not use the conventional technique. The f i r s t one i s a power swing r e l a y i n s t a l l e d a t the northern t e r m i n a l of the Ontario Hydro 500 KV transmission l i n e [ 3 ] . The b a s i c idea of t h i s r e l a y i s to use measurements of the instantaneous a c c e l e r a t i n g power w i t h the equal area c r i t e r i o n to estimate the power angle and to compare i t w i t h the c r i t i c a l c l e a r i n g angle of the protected generator. A t r i p s i g n a l i s produced i f t h i s angle i s exceeded. Of course, t h i s scheme depends on the assumption of a one-m a c h i n e - i n f i n i t e - b u s system which may be v a l i d only f o r s p e c i a l cases. However, the technique does look a t l o c a l v a r i a b l e s of an i n d i v i d u a l generator. The second generation t r i p p i n g scheme i s that used a t C o l d s t r i p - Montana [4]. The scheme r e l i e s on speed and a c c e l e r a t i o n measurements to p r e d i c t the t r a n s i e n t s t a b i l i t y of the protected generator. Mini-computers are used to process the measured information instantaneously and the t r i p s i g n a l i s issued f o r a s p e c i a l c o n f i g u r a t i o n of the a c c e l e r a t i o n curve o n l y . This scheme was designed f o r a l i m i t e d protected area 4 of the Montana system and f o r two g e n e r a t o r s i n the system o n l y . The scheme cannot be a p p l i e d g e n e r a l l y u n l e s s a new c r i t e r i o n f o r t r a n s i e n t s t a b i l i t y i s developed based on the a c c e l e r a t i o n and speed measurements. The g o a l of t h i s t h e s i s p r o j e c t has been to search f o r an a l t e r n a t i v e method to s o l v e the t r a n s i e n t s t a b i l i t y problem using l o c a l measurements. A proposed approach i s presented i n t h i s t h e s i s . Chapter 2 w i l l review the p r e s e n t methods used to s o l v e the problem of t r a n s i e n t s t a b i l i t y and w i l l i n t r o d u c e the proposed method of l o c a l assessment of t r a n s i e n t s t a b i l i t y . The system models and t e s t systems used i n the proposed method w i l l be presented i n Chapter 3. Chapter 4 p r e s e n t s the r e s u l t s o f a number of s t u d i e s of d i f f e r e n t t r a n s i e n t cases to i n v e s t i g a t e the v a r i o u s l o c a l measurements a v a i l a b l e f o r s t a b i l i t y a s s e s s -ments. Chapter 5 summarizes the r e s u l t s and suggestions f o r f u t u r e work. 5 CHAPTER 2 TRANSIENT STABILITY ANALYSIS 2.1 I n t r o d u c t i o n S t a b i l i t y o f a generator i m p l i e s t h a t i t w i l l remain in synchronism with the r e s t of the power system. T r a n s i e n t s t a b i l i t y r e f e r s to the amount o f power t h a t can be t r a n s m i t t e d with s t a b i l i t y when the power system i s subjected to a l a r g e d i s t u r b a n c e such as a sudden change i n l o a d or g e n e r a t i o n , s w i t c h i n g o p e r a t i o n s , or f a u l t s with subsequent c i r c u i t i s o l a t i o n . Such a l a r g e d i s t u r b a n c e c r e a t e s a power imbalance between supply and demand i n the system. T h i s imbalance a c t u a l l y takes p l a c e a t each generator s h a f t . The mismatch between the mechanical power input and the e l e c t r i c a l power output ( n e g l e c t i n g l o s s e s ) a c c e l e r a t e s or d e c e l e r a t e s the g e n e r a t o r [ 5 ] . In the case of a f a u l t on the high v o l t a g e s i d e , a t the beginning of the d i s t u r b a n c e the generator r o t o r s are at t h e i r p r e - f a u l t steady s t a t e o p e r a t i n g c o n d i t i o n . During the f a u l t p e r i o d there i s excess mechanical power which i s converted to k i n e t i c energy, causing the r o t o r s to speed up. When the f a u l t i s c l e a r e d , most of the k i n e t i c energy produced during the f a u l t must be absorbed o r transformed i n t o p o t e n t i a l energy i n order to maintain s t a b i l i t y . I f the system a f t e r the f a u l t i s not capable of absorbing t h i s k i n e t i c energy then the r o t o r s w i l l continue a c c e l e r a t i n g (or d e c e l e r a t i n g ) u n t i l they l o s e 6 synchronism. The s i t u a t i o n d u r i n g and a f t e r the f a u l t i s more complicated i n the case o f a l a r g e i n t e r c o n n e c t e d power system, because the power imbalance i n v o l v e s groups of g e n e r a t o r s . Some generators w i l l a c c e l e r a t e and some w i l l d e c e l e r a t e , a t d i f f e r e n t r a t e s depending upon t h e i r i n e r t i a c o nstants and o p e r a t i n g c o n d i t i o n s , i . e . , there i s an energy interchange between a c c e l e r a t i n g and d e c e l e r a t i n g r o t o r s a f t e r the f a u l t i s c l e a r e d . The exchange of energy c o n t i n u e s u n t i l a new s t a b l e o p e r a t i n g c o n d i t i o n i s reached. Otherwise some h i g h l y a c c e l e r a t e d g e n e r a t o r s may p u l l - o u t and cause o t h e r generators to l o s e synchronism. These generators have to be t r i p p e d before they p u l l - o u t some oth e r generators and cause the system to break up. Loss of synchronism must be prevented or c o n t r o l l e d , because i t has a d i s t u r b i n g e f f e c t on v o l t a g e s , frequency and power, and i t may cause s e r i o u s damage to generators which are the most expensive element i n any power system [ 6 ] . The gene r a t o r s which tend to l o s e synchronism should be t r i p p e d and subsequenty brought back to synchronism before any s e r i o u s damage o c c u r s . While t h i s can be done r e a d i l y with gas and water t u r b i n e g e n e r a t o r s , steam t u r b i n e generators r e q u i r e many hours to r e b u i l d steam so t h a t the o p e r a t o r has to shed l o a d to compensate f o r l o s s o f these g e n e r a t o r s . Loss o f synchronism may a l s o cause some p r o t e c t i v e r e l a y s to operate f a l s e l y and t r i p the c i r c u i t breakers o f un f a u l t e d l i n e s . 7 2.2 Assessment of T r a n s i e n t S t a b i l i t y ; S i nce i n t e r c o n n e c t e d power systems have been recognized and e s t a b l i s h e d , power engineers have worked c o n t i n u o u s l y to s o l v e the problem o f t r a n s i e n t s t a b i l i t y . But the problem i s f a r from being completely s o l v e d i n s p i t e of the v a s t amount of work done. An e a s i l y reached c o n c l u s i o n i s t h a t the problem i s so complex when f u l l y t r e a t e d t h a t an exact answer i s almost hopeless to o b t a i n . In s p i t e o f the f a c t t h a t the d i g i t a l computer i s being used, d e t a i l e d s t a b i l i t y s t u d i e s are very time consuming s i n c e the problem i n v o l v e s many n o n - l i n e a r and d e t a i l e d machine equ a t i o n s . Some of the methods t h a t have been used i n a s s e s s i n g t r a n s i e n t s t a b i l i t y w i l l now be d i s c u s s e d . 2.2.1 The Conventipnal Method T h i s method r e l i e s on the d i g i t a l computer to analyze the t r a n s i e n t behavior o f the i n t e r c o n n e c t e d synchronous machines. The synchronous machines are d e s c r i b e d by a s e t o f d i f f e r e n t i a l e q u a t i o n s . A time s o l u t i o n i s o b t a i n e d , s t a r t i n g with the p r e -t r a n s i e n t c o n d i t i o n and continues u n t i l each synchronous machine can be shown to maintain o r l o s e synchronism. T r a n s i e n t s t a b i l i t y s t u d i e s are r o u t i n e l y conducted on a power system. The major o b j e c t i v e of each study i s to a s c e r t a i n whether the e x i s t i n g (or planned) switchgear and network arrangements are adequate f o r the system to withstand a pre -s c r i b e d s e t of d i s t u r b a n c e s without l o s s o f synchronism. The 8 r e s u l t s of the t r a n s i e n t s t a b i l i t y s t u d i e s are the main t o o l f o r s e t t i n g the p r o t e c t i o n equipment i n o r d e r to p r o t e c t each synchronous machine from l o s i n g synchronism. The numerical method o f s t a b i l i t y a n a l y s i s i s v e r y r e l i a b l e and has been widely used and accepted by the power i n d u s t r y , but there are c e r t a i n s i t u a t i o n s i n the day to day o p e r a t i o n of the power system, where an o p e r a t o r has to decide q u i c k l y the t r a n s i e n t s t a b i l i t y of the machines. These s i t u a t i o n s could a r i s e due to c e r t a i n unforseen circumstances, l i k e equipment breakdown, c i r c u i t breaker f a i l u r e and m u l t i p l e d i s t u r b a n c e s i n the system. These s i t u a t i o n s need a quick and r e l i a b l e d e c i s i o n . Furthermore the c o n v e n t i o n a l technique c o n s i s t s of n u m e r i c a l l y i n t e g r a t i n g a l a r g e number o f d i f f e r e n t i a l equations f o r each f a u l t case [ 7 ] . A number o f repeat s i m u l a t i o n s are thus r e q u i r e d . Hence, i n terms of computational c o s t using the d i g i t a l computer, t h i s method i s expensive and time consuming. 2.2 .2 The Equal Area C r i t e r i o n The equal area c r i t e r i o n i s a v a l u a b l e conceptual t o o l f o r the a n a l y s i s o f power system s t a b i l i t y . I t i s p a r t i c u l a r l y u s e f u l f o r the study of t r a n s i e n t s t a b i l i t y / and i n v i s u a l i z i n g the behavior of the synchronous machine during the t r a n s i e n t . But t h i s c r i t e r i o n can only be a p p l i e d to a single-machine-i n f i n i t e - b u s system or to l a r g e r systems t h a t have been reduced to two e q u i v a l e n t machines [5,6]. For each generator the f o l l o w i n g swing equation may be M - P - P » P pu (2 .Jl m e a r at 2H where M • — » Inertia constant «»>r = r a t e d synchronous speed 6 = generator r o t o r angle P m = mechanical power input P e = e l e c t r i c a l power output P f l = a c c e l e r a t i n g power. M u l t i p l y i n g both s i d e s of equation (2.1) by d6_ we get d t u d 25 dS m p d6 M 2 * dt a dt dt 1 M d(d6/dt) „ p d5 2 M dt a dt Now m u l t i p l y i n g both s i d e s by d t and i n t e g r a t i n g j x 2 6 2P • / T T " O dt / J — D 5 o where * o i s the r o t o r angle before the d i s t u r b a n c e . Before and a f t e r the d i s t u r b a n c e d6_ = 0 s i n c e the system i s i n steady d t s t a t e . 10 Now c o n s i d e r the case of a s i n g l e machine connected to an i n f i n i t e bus as shown i n F i g . 2 . 1 . Consider a three-phase f a u l t on the t r a n s m i s s i o n l i n e CD followed by the simultaneous opening of c i r c u i t breakers C and D a t steady s t a t e E » v e " ~ i q ~ 8 l n 6o (2.3) - P s i n 5 max o Where E = generator v o l t a g e V = i n f i n i t e bus v o l t a g e *0 = steady s t a t e r o t o r p o s i t i o n X l = reactance between E and V Pe = e l e c t r i c a l power d e l i v e r e d from the ge n e r a t o r . The power d e l i v e r e d d u r i n g the f a u l t i s - r. P s i n 6 (2 L\ e 1 max K*-'*) and the power a f t e r the f a u l t i s c l e a r e d i s - r_ P s i n 6 (2.5) e 2 max v 7 where the c o n s t a n t s r ^ and r 2 r e p r e s e n t the change i n the e f f e c t i v e inductance between the machine and the i n f i n i t e - b u s . The power b e f o r e , d u r i n g , and a f t e r the f a u l t curves are shown i n F i g u r e 2 . 2 . For the machine to be s t a b l e i n t h i s case, equation (2.2) has to be s a t i s f i e d , i . e . , the a l g e b r a i c sum of the areas A l and A2 has to be zero,' or A l must be l e s s than or equal t o A2 . The area A l r e p r e s e n t s the k i n e t i c energy gained by the generator r o t o r d u r i n g the f a u l t , and area A2 r e p r e s e n t s the 11 A hO-e -CH K3-c D : • i n f i n i t e bus Fig 2 . 1 Single machine Infinite-bus Power curve 1 pre-fault power 'curve 2 post-fault power curve 3 during-fault power Fig 2 . 2 The equal area criterion Angle 12 p o t e n t i a l energy t h a t can be s t o r e d a f t e r the f a u l t i s c l e a r e d a t § c . I f a l l the k i n e t i c energy can be converted to p o t e n t i a l energy f o r a r o t o r angle 6_< 6max, then the system w i l l be s t a b l e , otherwise the e x t r a k i n e t i c energy w i l l b u i l d up the r o t o r ' s a c c e l e r a t i o n i n c r e a s i n g 6 and causing i n s t a b i l i t y . T h i s b r i e f p r e s e n t a t i o n of the equal area c r i t e r i o n shows that the c r i t e r i o n i s u s e f u l and very simple to c a r r y out f o r t r a n s i e n t s t a b i l t y , g i v e n the p r e - f a u l t steady s t a t e c o n d i t i o n . The c r i t e r i o n i s fundamentally i n c o r r e c t because i t equates the product o f power and angle t o energy. I f the speed of the synchronous machine were con s t a n t d u r i n g the t r a n s i e n t p e r i o d , the c r i t e r i o n would be e x a c t , but i t i s used to analyze machine performance when the angle i s v a r y i n g . Since the angle v a r i e s , however, the speed cannot be e x a c t l y c o n s t a n t . The e r r o r i s however very s m a l l . Even the most dynamic generators r e q u i r e s a second to advance 360° r e l a t i v e to the r e s t o f the system [4] i . e . , they are t u r n i n g a t an average angular v e l o c i t y of 61 Hz r a t h e r than 60 Hz. The e r r o r i s t h e r e f o r e maximum o f 1 p a r t i n 60 which i s small when compared to other e r r o r s i n troduced i n the s t a b i l i t y study. Although the c r i t e r i o n g i v e s very v a l u a b l e r e s u l t s about t r a n s i e n t s t a b i l i t y , i t cannot be a p p l i e d to the multi-machine power systems to analyze the behavior of each generator because many generators are in v o l v e d i n the energy i n t e r c h a n g e . 13 2.2.3 Energy Function Method T h i s method has been developed on the b a s i s o f using the energy f u n c t i o n as a Lyapunov f u n c t i o n to f i n d the s t a b i l i t y r e g i o n f o r the power system [2,8]. C o n s i d e r a b l e e f f o r t has been devoted to Lyapunov methods f o r power system s t a b i l i t y a n a l y s i s i n the l a s t decade. A s u b s t a n t i a l p a r t of the e f f o r t has i n v o l v e d the search f o r b e t t e r Lyapunov f u n c t i o n s , i . e . , a Lyapunov f u n c t i o n t h a t e i t h e r g i v e s l a r g e r r e g i o n s of s t a b i l i t y i n s t a t e space or i s v a l i d f o r more complex system models [ 9 ] . The t r a n s i e n t energy f u n c t i o n c o n t a i n s both k i n e t i c and p o t e n t i a l terms. The system k i n e t i c energy, a s s o c i a t e d with the r e l a t i v e motion of machine r o t o r s , i s f o r m a l l y independent o f the network. The system p o t e n t i a l energy, a s s o c i a t e d with the p o t e n t i a l energy of network elements and machine r o t o r s , i s always d e f i n e d f o r the post f a u l t system. The p r i n c i p a l idea of the energy f u n c t i o n method [10] i s t h a t t r a n s i e n t s t a b i l i t y can, f o r a g i v e n contingency, be determined d i r e c t l y by comparing the t o t a l system energy which i s gained d u r i n g the f a u l t - o n p e r i o d , with a c e r t a i n c r i t i c a l p o t e n t i a l energy. For a two-machine system t h i s c r i t i c a l energy i s uniquely d e f i n e d and the d i r e c t a n a l y s i s i s e q u i v a l e n t to the equal area c r i t e r i o n . For a system w i t h three or more machines the d i r e c t a n a l y s i s becomes more d i f f i c u l t , because of the f a c t t h at most of the machine r o t o r s are i n v o l v e d i n the energy interchange. Formulation i n terms of an i n e r t i a l c e n t r e , or sometimes c a l l e d c e n t r e - o f -angle, overcomes the problem o f r e f e r e n c e . In the i n e r t i a l 14 c e n t r e f o r m u l a t i o n , the equations d e s c r i b i n g the motion of the synchronous machines are formulated w i t h r e s p e c t to a f i c t i t i o u s i n e r t i a l c e n t r e [11]. The importance of t h i s f o r m u l a t i o n l i e s i n c l e a r l y f o c u s i n g on the motion t h a t tends to separate one o r more generators from the r e s t of the system. Consider the c l a s s i c a l model of the power system. The equation of motion of any machine u n i t i n a power system i s : d 26. M m p - P (2.6) d t 2 1 e i where P e i - j E ± Ej Y ± j cos< (2.7) P I " Pml " E i G l i <2'8> Pm^ = mechanical power i n p u t G i i = r e a l p a r t of the d r i v i n g p o i n t admittance f o r the i n t e r n a l generator node Y^j = t r a n s f e r admittance between node i and j E^,Ej = c o n s t a n t v o l t a g e behind t r a n s i e n t reactance = g e n e r a t o r r o t o r angle = moment of i n e r t i a c o n s t a n t = 2Hi/Wr Wr = r a t e d synchronous speed. Equations 2.3 and 2 .4 were w r i t t e n with r e s p e c t to an a r b i t r a r y synchronously r o t a t i n g frame o f r e f e r e n c e . 15 For the i n e r t i a l c e n t r e , d e f i n e : n n 6o - I T I , M i 6 i • M t " I , M i <2'9> t i»i i=i , n M~ I \ « ± (2.10) 0) ° " t i-1 then the motion of the i n e r t i a l c e n t r e i s given by n Vo ' I . P i " P e i " PcoA (2.11) i=l where <$ Q , C D 0 a r e fc^e p o s i t i o n and speed of i n e r t i a l c e n t r e , M f ct i s the i n e r t i a constant o f the i n e r t i a l c e n t e r . The ge n e r a t o r s ' angles and speeds with r e s p e c t to the i n e r t i a l c e n t r e are d e f i n e d by: the equation of motion of the i n d i v i d u a l machines with r e s p e c t to the i n e r t i a l c e n t r e become [11]. MA - P i " P e i ' ( V V P c o l ( 2' 1 3> M u l t i p l y equation (2ol3) by and form the sum then I [\\ - Pi + \t + < W P c o i 1 S i ( 2 * 1 4 ) i=l I n t e g r a t i n g (2.14) with r e s p e c t to time using as a lower l i m i t t = t s , where 6 ( t s ) = 6 i s t h e s t e a < 3 v s t a t e c o n d i t i o n (pre-i s f a u l t c o n d i t i o n ) , we get the energy f u n c t i o n V i V l ' i »t ' V i - 6 ! . ' + }.. E i E 3 hi I «'(V 6 i + V d e t 3*1 1 S e. o is) + (M./M ) / P C Q l d9 1 , 1 - 1, 2, n 6 i 8 16 The energy f u n c t i o n of equation (2.15) can be p h y s i c a l l y e x p l a i n e d as f o l l o w s : 1. The f i r s t term r e p r e s e n t s the change i n k i n e t i c energy and p o t e n t i a l energy due to the motion o f the r o t o r o f machine i 2. The second term r e p r e s e n t s the change i n p o t e n t i a l energy due to the change i n r o t o r p o s i t i o n between e i s and e i 3. The t h i r d term r e p r e s e n t s the change i n p o t e n t i a l energy due to the power flow from node i to j and from node j to i 4 . The f o u r t h term r e p r e s e n t s the change i n p o t e n t i a l energy due to the i t h machine c o n t r i b u t i o n to the a c c e l e r a t i o n o f the c e n t r e of i n e r t i a (COI). Equation (2.15) can be w r i t t e n as = k i n e t i c energy + p o t e n t i a l energy V - v + V (2.16) v i KEi PEi the t o t a l energy f u n c t i o n of the system i s v - I v± - I ( y m + v R 1 ) (2.17) 1=1 i-1 f o r s t a b i l i t y » zero n n °r lml VKE1 " lml VPE1 T o t a l k i n e t i c energy = t o t a l p o t e n t i a l energy. 17 Analogy with Equal Area C r i t e r i o n The energy f u n c t i o n method i s s i m i l a r t o the equal area c r i t e r i o n . Both methods compare the k i n e t i c energy with the p o t e n t i a l energy d u r i n g and a f t e r the f a u l t . F i g u r e 2.3 shows two p l o t s with the same a b s c i s s a [11] . The upper p l o t i l l u s t r a t e s the f a m i l i a r equal area c r i t e r i o n i n which the c r i t i c a l c l e a r i n g angle i s d e f i n e d by the e q u a l i t y of areas A l and A2. The lower p l o t i l l u s t r a t e s the t r a n s i e n t energy method which can be used to s p e c i f y the c r i t i c a l angle i n terms of p o t e n t i a l and k i n e t i c energy as shown i n Fig u r e (2.3). PE ( 6 U ) i s the maximum value o f p o t e n t i a l energy t h a t occurs a t 6 U . I t p r o v i d e s a measure of the energy-absorbing c a p a c i t y of the system, and i s c a l l e d the c r i t i c a l energy. In the t r a n s i e n t energy method, the excess k i n e t i c energy, which c o n t r i b u t e s to i n s t a b i l i t y d u r i n g the f a u l t - o n p e r i o d , i s added to the p o t e n t i a l energy a t the corresponding angle c o o r d i n a t e . T h i s g i v e s the t o t a l energy a t c l e a r i n g . The t o t a l energy a t c l e a r i n g i s compared with the value o f c r i t i c a l energy. The system i s s t a b l e when the t o t a l energy i s l e s s than or equal to the c r i t i c a l energy. The c r i t i c a l c l e a r i n g angle i s d e f i n e d when the t o t a l energy a t c l e a r i n g j u s t becomes equal to the c r i t i c a l energy. For a system w i t h three or more machines, the energy f u n c t i o n method becomes more d i f f i c u l t to use because the c r i t i c a l energy cannot be d e f i n e d . However, r e c e n t work [12] has been done based on the f a c t that f o r a l a r g e power system Pig 2.3 Comparison of equal area criterion and energy method 19 o n l y a few machines are important f o r any p a r t i c u l a r f a u l t l o c a t i o n . The energy f u n c t i o n method can then be a p p l i e d s u c c e s s f u l l y to those c r i t i c a l machines to assess t h e i r s t a b i l i t y . But f o r many f a u l t l o c a t i o n s , t h i s method cannot be a p p l i e d because many machines c o n t r i b u t e to the f a u l t energy exchange. The energy f u n c t i o n method can be used to assess the o v e r a l l t r a n s i e n t s t a b i l i t y of the power systems by using the t o t a l k i n e t i c and p o t e n t i a l e n e r g i e s o f a l l the machines, or a c r i t i c a l group of machines. 2.2.4 L o c a l Assessment L o c a l assessment of t r a n s i e n t s t a b i l i t y r e f e r s to the monitoring of some important q u a n t i t i e s of a generator l o c a l l y t h a t w i l l help to i n d i c a t e the t r a n s i e n t s t a b i l i t y of t h a t g e n e r a t o r . Montana Power Company developed [4] a d i g i t a l l y c o n t r o l l e d d e v i c e f o r generator t r i p p i n g i n June 1979. The idea of t h i s d e v i c e i s t o p r e d i c t whether the generator i s s t a b l e or not and then i s s u e a t r i p s i g n a l to shed the generator when i t i s u n s t a b l e . The d e v i c e r e l i e s on speed and a c c e l e r a t i o n measurements u s i n g a toothed wheel b o l t e d to the generator's r o t o r and counting the number of t e e t h which pass i n some p e r i o d of time. I t was p o s s i b l e to p r e d i c t when the generator was going to be.unstable and to t r i p . i t a t the r i g h t time. The d e v i c e has been f u n c t i o n i n g very w e l l s i n c e i t was i n s t a l l e d and there has been no f a l s e t r i p s . However, the c r i t e r i o n used to 20 determine s t a b i l i t y i s very s p e c i a l t o the Montana Power System because two u n i t s o f the system were to be c o n t r o l l e d to l i m i t t h e i r output power and the system i s a r a d i a l system, with the two u n i t s connected through the t r a n s m i s s i o n l i n e s to the r e s t o f the l a r g e r western U.S. system. There are some f a c t s about the behavior of the synchronous machines d u r i n g t r a n s i e n t d i s t u r b a n c e s which suggest the p o s s i b i l i t y o f a s s e s s i n g s t a b i l i t y . F i r s t , an imbalance between mechanical input and e l e c t r i c a l output power takes p l a c e i n the a i r g a p between r o t o r and s t a t o r o f the machine. Second, the r o t o r undergoes instantaneous a c c e l e r a t i o n o r d e c e l e r a t i o n on the occurrence o f a f a u l t . T h i r d , i n the case of i n s t a b i l -i t y , p ole s l i p p i n g takes p l a c e between r o t o r and s t a t o r l o c a l l y . T h e r e f o r e , there i s s i g n i f i c a n t i n f o r m a t i o n imbedded i n the genera t o r v a r i a b l e s r e g a r d i n g the d i s t u r b a n c e and the s t a b i l i t y o f the ge n e r a t o r . For the case of steady s t a t e s t a b i l i t y a group of Japanese s c i e n t i s t s were able to monitor and measure the a i r gap f l u x [13], and they designed a l o c a l automatic s t a b i l i t y p r e d i c -t i o n d e v i c e (ASPAC) [14]. T h i s d e v i c e has fu n c t i o n e d v e r y w e l l . In t h i s t h e s i s we w i l l use the generator l o c a l v a r i a b l e s t o get s i g n i f i c a n t i n f o r m a t i o n f o r t r a n s i e n t s t a b i l i t y a s s e s s -ment. 2.3 S t a b i l i t y C o n t r o l s T h i s s e c t i o n i s a quick survey o f some methods used i n 21 the power i n d u s t r y to c o n t r o l and improve t r a n s i e n t s t a b i l i t y . These methods are known i n the l i t e r a t u r e as supplementary c o n t r o l s o f t r a n s i e n t s t a b i l i t y to d i s t i n g u i s h them from primary and continuous c o n t r o l s such as the speed governer and the e x c i t a t i o n system c o n t r o l [15]. Supplementary c o n t r o l s are a p p l i e d o n l y under c e r t a i n severe c o n d i t i o n s to maintain s t a b i l i t y . Most turbogenerators are c o n t r o l l e d s u c c e s s f u l l y with a combination of primary and supplementary c o n t r o l s . 2.3.1 Dynamic Braking Dynamic brakes are r e s i s t i v e elements t h a t are switched o n - l i n e to absorb power. G e n e r a l l y l o c a t e d near g e n e r a t o r s , they are switched on i n case of f a u l t s t o d i s s i p a t e excess r o t o r energy. Since the excess energy i s consumed by the braking r e s i s t o r , the t r a n s m i s s i o n system i s r e l i e v e d from having to t r a n s m i t the energy t o the load and so i t reduces the a c c e l e r a t i o n o f the generator. T h i s method has been s u c c e s s f u l l y used i n some power systems. However, there are some l i m i t a t i o n s to the use o f dynamic brakes such as the f o l l o w i n g [15,16]. 1. P i c k i n g the proper s i z e o f the r e s i s t a n c e f o r s e v e r a l d i s t u r b a n c e c o n t i n g e n c i e s can be a problem [17]. 2. The maximum s i z e o f the braking r e s i s t a n c e i s r e a l i z e d when the r e s i s t a n c e i s equal to the machine t r a n s i e n t r e a c t a n c e . 22 3. The braking r e s i s t a n c e i s most needed when <? i s l a r g e , and the g e n e r a t o r i s a c c e l e r a t i n g . However, under these c o n d i t i o n s the generator's t e r m i n a l v o l t a g e i s low, and when the brake i s a p p l i e d the v o l t a g e goes even lower. 2.3.2 High _Speed C i r c u i t Breaker R e c l o s i n g When a f a u l t occurs a transmision line , the ci r c u i t breakers a t each end of the l i n e w i l l open to i s o l a t e the f a u l t from the system, remain open f o r a s p e c i f i e d time, and then r e c l o s e . I f the t r a n s m i s s i o n l i n e f a u l t has been c l e a r e d , then the c i r c u i t breakers remain c l o s e d and the t r a n s m i s i o n system r e t u r n s to i t s p r e f a u l t c o n d i t i o n . I f the f a u l t s t i l l e x i s t s , the c i r c u i t breakers w i l l open and l o c k o u t . The procedure o f opening and f i n a l c l o s i n g must take p l a c e before any generator i n the system reaches i t s c r i t i c a l c l e a r i n g a n g l e . T h i s method i s used because approximatley 80% o f t r a n s m i s s i o n l i n e f a u l t s are t r a n s i e n t i n nature [18]. That i s , i f the l i n e i s de-energized f o r a s h o r t time the f a u l t a r c w i l l d e i o n i z e and the i n t e g r i t y of the i n s u l a t i o n system w i l l be r e - e s t a b l i s h e d . T y p i c a l l y , the f a u l t a r c can d e i o n i z e i n approximately 12 c y c l e s . The advantage o f t h i s method i s t h a t i t keeps g e n e r a t i n g u n i t s o n - l i n e f o r t r a n s i e n t t r a n s m i s s i o n l i n e f a u l t s and i t a l s o minimizes the number of outages that the generator e x p e r i e n c e s i n i t s l i f e t i m e . However, i t has some s e r i o u s disadvantages as a consequence o f u n s u c c e s s f u l r e c l o s u r e [15,19,20] , as follows ; 23 1. A second major t r a n s i e n t could be a p p l i e d to the s h a f t b e f o r e the i n i t i a l o s c i l l a t i o n s have damped out, and t h i s may damage the s h a f t . 2. E x i s t e n c e o f two s u c c e s s i v e v o l t a g e d i p s . 3. Increased duty on c i r c u i t b r e a k e r s . 4. Increased damage a t f a u l t l o c a t i o n s . 5. P o s s i b l e i n s t a b i l i t y . 2.3.3 S e r i e s C a p a c i t o r s S e r i e s c a p a c i t o r s are used to i n c r e a s e the power t r a n s f e r c a p a c i t y [21] by compensating f o r the i n d u c t i v e reactance of the t r a n s m i s s i o n l i n e s . S e r i e s c a p a c i t o r s are good f o r improving steady s t a t e s t a b i l i t y , but i t can pr e s e n t some problems under t r a n s i e n t c o n d i t i o n s : 1. When a f a u l t o c c u r s , p r o t e c t i v e d e v i c e s may bypass s e r i e s c a p a c i t o r s i n the f a u l t e d and nearby l i n e s thus removing them from s e r v i c e . 2. Subsynchronous resonance may damage machine s h a f t s [22]. 3. S e r i e s c a p a c i t o r s are expensive compared t o oth e r methods of improving t r a n s i e n t s t a b i l i t y . 2.3.4 F a s t V a l v e A c t i o n F a s t v a l v e a c t i o n [23] i s the r a p i d c l o s i n g o f the gen e r a t o r ' s steam v a l v e s f o l l o w i n g a t r a n s i e n t d i s t u r b a n c e . Mechanical i n p u t t o the generator i s thus reduced and t h i s w i l l reduce the a c c e l e r a t i o n . T h i s method can maintain t r a n s i e n t 24 s t a b i l i t y i n many cases, but i t needs good p r e d i c t i o n of the p o s t - f a u l t network i n order to reopen the v a l v e s a t a c e r t a i n l e v e l of power, otherwise a second t r a n s i e n t may occur and cause l o s s of synchronism. A l s o when the v a l v e s are c l o s e d the p r e s s u r e and the temperature i n the b o i l e r w i l l i n c r e a s e . The b o i l e r cannot withstand t h i s p r e s s u r e and temperature i n c r e a s e f o r more than 10 minutes, so quick a c t i o n i s r e q u i r e d , e i t h e r reopening the v a l v e s or bypassing the p r e s s u r e . Although f a s t v a l v i n g can do n o t h i n g to prevent steady s t a t e i n s t a b i l i t y , i t i s a c c e p t a b l e and used by some power companies [24]. F u r t h e r -more, s t u d i e s are r e q u i r e d and f i e l d t e s t s may be necessary to e v a l u a t e f a s t v a l v i n g and to determine t h e i r e f f e c t on: second swing i n s t a b i l i t y , steam pressure and temperature, s h a f t and v a l v e s t r e s s e s , and t r a n s i e n t s i n steam supply systems. 2.3.5 Independent P o l e Operation of C i r c u i t Breakers Independent p o l e o p e r a t i o n o f a c i r c u i t breaker r e f e r s t o the mechanism by which the three phases of the breaker are c l o s e d or opened independently o f each o t h e r [25]. The f a i l u r e o f any one phase does not a u t o m a t i c a l l y prevent any o f the two remaining phases from proper o p e r a t i o n . However, f o r a three phase f a u l t , the three phases are s i m u l t a n e o u s l y a c t i v a t e d f o r o p e r a t i o n by the same r e l a y i n g scheme. The three phases are m e c h a n i c a l l y independent, such that the mechanical f a i l u r e of any one p o l e does not prevent o p e r a t i o n of the remaining p o l e s [15]. T h i s method helps maintain system s t a b i l i t y by 25 q u i c k l y c l e a r i n g o r reducing the s e v e r i t y of multiphase f a u l t s . Independent p o l e o p e r a t i o n i s used a t l o c a t i o n s where the d e s i g n c r i t e r i o n i s to guard a g a i n s t a three phase f a u l t c o i n c i d e n t w i t h breaker f a i l u r e . S u c c e s s f u l independent p o l e o p e r a t i o n o f a f a i l e d breaker w i l l reduce a three phase f a u l t to s i n g l e l i n e - t o - g r o u n d f a u l t ( i f one pole o f the breaker i s s t u c k ) , o r to a double l i n e to ground f a u l t ( i f two p o l e s of the breaker are s t u c k ) . T h i s method can i n c r e a s e the c r i t i c a l c l e a r i n g time by as much as two to f i v e c y c l e s [26]. Independent pole o p e r a t i o n i s easy to i n s t a l l ; the only a d d i t i o n a l complexity i s to p r o v i d e a separate t r i p c o i l f o r each pole (most EHV breakers are equipped with separate pole mechanisms). 2.3.6 Generator Tripping^ Generator t r i p p i n g i s a form o f energy c o n t r o l much l i k e f a s t v a l v i n g or dynamic b r a k i n g . I f generator t r i p p i n g i s a p p l i e d i t improves both t r a n s i e n t and steady s t a t e s t a b i l i t y . O r i g i n a l l y the method was c o n f i n e d to hydro generators and was a means of p r o v i d i n g t r a n s i e n t s t a b i l i t y f o r remote g e n e r a t i o n . Recently generator t r i p p i n g has been extended to c e r t a i n thermal g e n e r a t i o n . Generator t r i p p i n g can be i n i t i a t e d from a t r a n s f e r t r i p scheme, a r r a n g i n g the breakers at the power p l a n t or by a l o c a l t r i p scheme by measuring the generator q u a n t i t i e s l o c a l l y [ 4 ] . Schemes have been used where the generator load i s maintained connected to the u n i t a f t e r t r i p p i n g and the u n i t i s 26 r a p i d l y r e l o a d e d a f t e r the d i s t u r b a n c e . G e n e r a l l y the u n i t can be r e s y n c h r o n i z e d t o the system and f u l l l o a d r e s t o r e d i n about 15-30 minutes [ 1 5 ] . The most common t e c h n i q u e used t o t r i p g e n e r a t o r s i s o u t -o f - s t e p r e l a y i n g w h i c h i s b a s i c a l l y a d i s t a n c e scheme. The t r i p s i g n a l depends on the v o l t a g e measurements a t the i n v o l v e d l o c a t i o n s and then the s i g n a l has t o be t r a n s m i t t e d to the g e n e r a t o r l o c a t i o n t o t r i p the s y n c h r o n i z i n g b r e a k e r s . T h i s t e c h n i q u e cannot p r o v i d e t o t a l p r o t e c t i o n u n l e s s s i g n a l s from e v e r y s i g n i f i c a n t s t a t i o n a r e t r a n s m i t t e d . These s i g n a l s would have t o be p e r m i s s i v e l y c o n t r o l l e d by the g e n e r a t o r o u t p u t and l i n e f l o w s . The r e s u l t i n g l o g i c and communication system used would have t o be as complex as the power systems t o p r o v i d e t o t a l p r o t e c t i o n . However, the s i g n i f i c a n t i n f o r m a t i o n f o r g e n e r a t o r t r i p p i n g i s a l r e a d y a t the g e n e r a t o r l o c a t i o n . I t i s imbedded i n the g e n e r a t o r speed, a c c e l e r a t i o n and a n g l e . These w i l l be f u r t h e r d i s c u s s e d i n Chapter 4 . 2 7 CHAPTER 3 SYSTEM MODELS AND TEST SYSTEMS 3.1 C l a s s i c a l Model of Synchronous Machines The induced v o l t a g e i n the s t a t o r windings o f a synchronous machine can be considered as having two components; a component E^ t h a t corresponds to the f l u x l i n k i n g the main f i e l d winding, and a component E2 t h a t c o u n t e r a c t s the armature r e a c t i o n . E2 can change i n s t a n t a n e o u s l y because i t corresponds to c u r r e n t s i n the armature, but E^ cannot change i n s t a n t a n e o u s l y ( u n t i l the e x c i t e r a c t s ) . When a sudden change occurs i n the network, the f l u x l i n k a g e i n the a f f e c t e d synchronous machine w i l l not change and hence E^ cannot change. Currents w i l l be produced i n the armature. Hence oth e r c u r r e n t s w i l l be induced i n the v a r i o u s r o t o r c i r c u i t s to keep t h i s f l u x l i n k a g e c o n s t a n t . Both the armature and r o t o r c u r r e n t s w i l l u s u a l l y have AC and DC components as r e q u i r e d to match the ampere-turns o f v a r i o u s coupled c o i l s . The f l u x w i l l decay a c c o r d i n g to the e f f e c t i v e time c o n s t a n t . Under no-load c o n d i t i o n s t h i s i s on the order o f s e v e r a l seconds w h i l e under load i t i s on the order of one second. From the above d i s c u s s i o n we can see t h a t f o r a p e r i o d o f one second the f l u x l i n k a g e and hence can be considered c o n s t a n t . U s u a l l y e x c i t e r s do not respond f a s t enough i n a p e r i o d o f one second o r l e s s . T h i s p e r i o d i s o f t e n considered 28 adequate f o r determining the t r a n s i e n t s t a b i l i t y of the synchronous machine. The main f i e l d winding f l u x i s almost the same as a f i c t i t i o u s f l u x t h a t would c r e a t e EMF behind the machine d i r e c t a x i s reactance. T h i s model i s c a l l e d the c l a s s i -c a l model o f synchronous machine. I t i s a v o l t a g e source behind the d i r e c t t r a n s i e n t reactance X'^ as shownt i n F i g 3.1 X d '7JTP E and 6 Q can be determined V I A ( ] V | 0 ' ° from the i n i t i a l c o n d i t i o n s *• | 0 o i . e . , p r e t r a n s i e n t c o n d i t i o n s F i g 3.1 During the t r a n s i e n t the magnitude E i s considered constant while the angle 6 i s considered as the angle between the r o t o r p o s i t i o n and the t e r m i n a l v o l t a g e . The machine output power w i l l be a f f e c t e d by the change i n the r o t o r p o s i t i o n and any changes i n the impedance seen by the machine t e r m i n a l s . However, u n t i l the speed changes to the p o i n t where i t i s sensed and c o r r e c t e d by the governor, the change i n the output power w i l l come from the s t o r e d energy i n the r o t a t i n g masses. The important parameters here are the k i n e t i c energy i n MW.S/MW u s u a l l y c a l l e d H, or the machine i n e r t i a c o n s t a n t . 3.2 System R e p r e s e n t a t i o n Since we are concerned about the t r a n s i e n t s t a b i l i t y i n the f i r s t swing, i t i s reasonable to c o n s i d e r the c l a s s i c a l 29 model of the synchronous machine and of the power system which can be summarized i n the f o l l o w i n g assumptions: 1. Constant v o l t a g e behind d i r e c t t r a n s i e n t reactance model Q f o r the synchronous machine i s v a l i d i n the p e r i o d o f the f i r s t swing. 2. The mechanical r o t o r angle ( r o t o r p o s i t i o n ) of a machine c o i n c i d e s with the angle o f the v o l t a g e behind the d i r e c t t r a n s i e n t r e a c t a n c e . 3. Mechanical power input to each generator i s constant. 4. The t r a n s m i s s i o n network i s modeled by steady s t a t e e q u a t i o n s . 5. Damping or s y n c h r o n i z i n g power i s n e g l i g i b l e . 6. Loads are represented by cons t a n t p a s s i v e impedances f o r the c l a s s i c a l model being c o n s i d e r e d . The equations o f motion f o r each machine i n a power system a r e : M..u » P. - P M (3.1) i i 1 ei 2 n Where Pgi= G ± ±+ Z (B s i n ^ - f i ^ - K ^ c o s t f ^ )) J#i-p . = mechanical power input G^JL = d r i v i n g p o i n t conductance E^,Ej = co n s t a n t v o l t a g e behind t r a n s i e n t reactance e i » j = gener a t o r r o t o r angle BJLJ = t r a n s f e r susceptance between nodes i and j G j = t r a n s f e r conductance between nodes i and j 30 oj = speed d e v i a t i o n with r e s p e c t to synchronous speed OJ r r = i n e r t i a constant (M^ = ) r 3.3 The Three T e s t Systems In t h i s p r o j e c t a d e t a i l e d i n v e s t i g a t i o n o f i n d i v i d u a l machine speed and a c c e l e r a t i o n i s c a r r i e d out f o r three phase f a u l t s i n d i f f e r e n t l o c a t i o n s i n each system. Three systems were s t u d i e d : a three-generator system, a fo u r - g e n e r a t o r system and a f i v e - g e n e r a t o r system. 3.3.1 The Three Machine System T h i s t e s t system i s the w e l l known nine-bus, t h r e e -machine, t h r e e - l o a d system widely used i n the l i t e r a t u r e and o f t e n r e f e r r e d to as the V7SCC system. The system i s shown i n Fig u r e 3.2. A l l the impedances are i n per u n i t on a 100-MVA base. Machines 2 and 3 are steam turbogenerators with r a t e d speed o f 3600 rev/min, while machine 1 i s a hydro generator with r a t e d speed o f 180 rev/min. The generator data and the i n i t i a l o p e r a t i n g c o n d i t i o n s are g i v e n i n T a b l e 3.1. 3.3.2 The Four Machine System T h i s system i s the same as the three-machine system except f o r the f o l l o w i n g changes: 32 Table 3.1 Generator data and i n i t i a l c o n d i t i o n s of the three-machine system Generator d a t a I n i t i a l c o n d i t i o n s Gen. No. H (MW/MVA) x ' d (pu) P *mox (pu) E (pu) (degree) 1 23.64 .0608 2.269 1.096 6.95 2 6.40 .1198 1.6 1.102 13.49 3 3.01 .1813 1.0 1.024 8.21 Table 3.2 Generator data and i n i t i a l c o n d i t i o n s of the four-machine system Generator data i n i t i a l " cond i iIons Gen. No. H (MV7/MVA) ^ d (pu) p *mo. (pu) E (pu) (degree) 1 23.64 .0608 2.269 1.0967 6.95 2 6.4 .1198 1.6 1.102 13.49 3 3.01 .1813 1.0 1.1125 8.21 4 6.4 .1198 1.6 1.074 24.9 33 34 The r a t i n g o f the t r a n s m i s s i o n network was changed from 230 KV to 161 KV; the R and X values of the l i n e s i n pu remain the same. - A f o u r t h generator was connected t o the o r i g i n a l network through a step-up transformer and a double c i r c u i t t r a n s m i s s i o n l i n e (161 KV) to bus 8. The new gener a t o r has the same r a t i n g as generator 2. The system i s shown i n Figure 3.3 and the i n i t i a l o p e r a t i n g c o n d i t i o n s are g i v e n i n Table 3.2. 3.3.3 The Five-Machine System T h i s system i s shown i n F i g u r e 3.4. The t r a n s m i s s i o n l i n e s and lo a d data are gi v e n i n Table 3.3. The data are g i v e n i n per u n i t on the 10,000 MVA and 500 KV bases. The machine i n i t i a l c o n d i t i o n s and parameters are g i v e n i n Table 3.4. 35 36 Table 3.3 Transmission l i n e parameters and loads of the five-machine t e s t system. Transmission l i n e (impedances) Load (Admittance) pu Line R(pu) X(pu) Load No G(pu) B (pu) 7-8 .12 2.397 LI .0098 -.0049 7-6 .03 .597 L2 .0192 -.0092 9-6 .032 .639 L3 1.088 -.5271 10-11 .15 2.996 L4 .6598 -.3195 10-6 .186 3.71 L5 1.269 -.6144 11-6 .24 4.79 Table 3.3 Generator data and i n i t i a l c o n d i t i o n s o f the five-machine system. Generator data I n i t i a l c o n d i t i o n s Gen No. M (pu) X ' d (pu) P m ? P u ) E (pu) (degree) 1 .46 3.2 .0498 .981 37.6 2 1.1 1.6 .069 1.032 41.24 3 7.41 .25 1.138 1.171 40.9 4 .28 .799 .6998 1.397 37.6 5 .32' .89 1.269 2.08 31.8 37 CHAPTER 4 STUDIES OF GENERATOR VARIABLES AVAILABLE FOR LOCAL STABILITY ASSESSMENT In t h i s chapter we w i l l d i s c u s s the s t u d i e s c a r r i e d out to i n v e s t i g a t e the s u i t a b i l i t y of a v a i l a b l e machine v a r i a b l e s f o r a s s e s s i n g t r a n s i e n t s t a b i l i t y . The angle swing curves w i l l be used to d i s t i n g u i s h between s t a b l e and unstable cases, then a comparison w i l l be made with l o c a l v a r i a b l e curves to assess the t r a n s i e n t s t a b i l i t y . The l o c a l v a r i a b l e s that w i l l be i n v e s t i g a t e d are angle, speed, a c c e l e r a t i o n and r a t e of change of a c c e l e r a t i o n . 4.1 Acce1e ra t ing Powe r A c c e l e r a t i n g power i s the d i f f e r e n c e between the mechanical power in p u t and the e l e c t r i c a l power output dur i n g the t r a n s i e n t p e r i o d . R e c a l l i n g the swing equation DT2 m e Where M i s the i n e r t i a c onstant, S i s the r o t o r a n g l e, P m i s the mechanical power i n p u t , P e i s the e l e c t r i c a l power output and P a i s the a c c e l e r a t i n g power. P a i s one of the most important q u a n t i t i e s a t the generator t e r m i n a l d u r i n g t r a n s i e n t s . Because the value o f P a can g i v e a v a l u a b l e i n d i c a t i o n of the s e v e r i t y of the f a u l t and the s t a b i l i t y c o n d i t i o n of the g e n e r a t o r . 38 Since M i s approximately constant f o r any generator undergoing small speed d e v i a t i o n s , t h e r e f o r e the a c c e l e r a t i n g power i s d i r e c t l y p r o p o r t i o n a l to the a c c e l e r a t i o n . P r a c t i c a l l y both a c c e l e r a t i n g power and a c c e l e r a t i o n can be measured l o c a l l y a t the generator s i t e [3,4], A l s o both a c c e l e r a t i n g power and a c c e l e r a t i o n have the same waveshape, the o n l y d i f f e r e n c e between them i s the manner of measurement, where a c c e l e r a t i o n can be measured m e c h a n i c a l l y [4] and the a c c e l e r a t i n g power can be measured e l e c t r i c a l l y [3] . F i g u r e 4.1 shows both a c c e l e r a t i n g power and a c c e l e r a t i o n f o r a three-phase f a u l t at bus 10 of the four-machine system (see Chapter 3) c l e a r e d at .1 second, f o r machine 4. The a c c e l e r a t i n g power curve o f F i g 4.1 can represent the k i n e t i c and p o t e n t i a l energy, where the area under the curve before c l e a r i n g r e p r e s e n t s the power gained d u r i n g the f a u l t which a c c e l e r a t e s the machine r o t o r . The n e g a t i v e area under the curve a f t e r c l e a r i n g r e p r e s e n t s the power absorbed by the system a f t e r f a u l t c l e a r i n g . Machine 4 i s the o n l y machine i n the four-machine system t h a t may l o s e synchronism f o r a f a u l t at bus-10. I t can be considered as a s i n g l e - m a c h i n e - i n f i n i t e - b u s system, or the c r i t i c a l machine [11] f o r a f a u l t a t bus-10. Therefore the area A^ and A 2 under the a c c e l e r a t i n g power o f F i g 4.1 can be compared t o f i n d the s t a b i l i t y o f machine 4 ( f o r 4* machine 4 to be s t a b l e Aj^ must be l e s s than or equal to A 2 ) . I f we move the f a u l t from bus-10 to bus-8 then i t i s very hard to apply the equal area c r i t e r i o n , because machine 4 i s not 39 4-Nach.systeni(f at 10) i i i i l I I I I I L Time (sec) (a) n.D 0.4 i 1 r 0.8 U Time (sec) (o) F i g 4 .1 (a) a c c e l e r a t i n g pwer , (b) a c c e l e r a t i o n t 0 f machine A f o r a f a u l t a t bus 10 c l e a r e d at .Is 40 the o n l y machine that may l o s e synchronism. In t h i s case machine 3 and machine 2 may a l s o l o s e synchronism. The equal area c r i t e r i o n can be a p p l i e d to the whole system to f i n d g l o b a l system s t a b i l i t y but i t cannot be a p p l i e d t o a s i n g l e machine i n an i n t e r c o n n e c t e d system. Therefore we need to know which machine l o s e s synchronism f i r s t and which machine we should t r i p f i r s t to maintain system s t a b i l i t y i f p o s s i b l e . These q u e s t i o n s cannot be answered when using a g l o b a l s t a b i l i t y c r i t e r i o n . The answer i s to search f o r another c r i t e r i o n based upon a n a l y z i n g and t r a c k i n g each machine v a r i a b l e i n d i v i d u a l l y . 4'2 Speed and A c c e l e r a t i o n I t has been found [10] t h a t a l l o f the f a u l t k i n e t i c energy c o n t r i b u t i n g to i n s t a b i l i t y i s imbedded i n the uns t a b l e machine. C r i t i c a l machines are easy to d e f i n e [10] f o r a c e r t a i n f a u l t i n any power system, then t r a n s i e n t s t a b i l i t y can be p r e d i c t e d f o r those c r i t i c a l machines by using the two-machine e q u i v a l e n t technique [12]. R e c a l l i n g equation (2.15) f o r energy f u n c t i o n o f an i n d i v i d u a l machine (2.15) For the c r i t i c a l machine of a c e r t a i n f a u l t the f i r s t term of equa t i o n (2.15) i s the most e f f e c t i v e [10] term when c a l c u l a t i n g the i n d i v i d u a l machine energy V*. T h i s shows the importance 41 of speed d e v i a t i o n i n t r a n s i e n t s t a b i l i t y o f the c r i t i c a l machine d u r i n g and a f t e r a f a u l t . The f i r s t term i n equation (2.15) d u r i n g the f a u l t p e r i o d r e p r e s e n t s the k i n e t i c energy i n j e c t e d to the system and a f t e r the f a u l t i s c l e a r e d i t re p r e s e n t s the p o t e n t i a l energy being absorbed by the system. T r a c k i n g the speed curves f o r d i f f e r e n t f a u l t l o c a t i o n s shows a slow response to the f a u l t a p p l i e d . F i g 4.2 shows speed d e v i a t i o n s f o r machine 4 f o r d i f f e r e n t f a u l t l o c a t i o n s . The a c c e l e r a t i o n s i g n a l can be picked up from the speed s i g n a l [4] i n s t a n t a n e o u s l y i n a very s h o r t p e r i o d o f time. The a c c e l e r a t i o n o f the c r i t i c a l machine f o r a c e r t a i n f a u l t shows an instantaneous response f o r any f a u l t and f o r any mismatch that takes p l a c e i n the system, F i g 4.3 shows the a c c e l e r a t i o n and the swing curve of machine-4 f o r a f a u l t a t bus-10 , the r a p i d response of the a c c e l e r a t i o n o ccurs much e a r l i e r than the f i r s t peak of the f i r s t swing. 4.3 A c c e l e r a t i o n S t a b i l i t y " L i m i t s " When p l o t t i n g a c c e l e r a t i o n v e r s u s time f o r d i f f e r e n t f a u l t l o c a t i o n s i t i s n o t i c e d t h a t the f i r s t peak of the a c c e l e r a t i o n curves i s d i f f e r e n t from one l o c a t i o n to another depending on f a u l t s e v e r i t y , i . e . , the most severe f a u l t r e s u l t s i n a maximum f i r s t peak of a c c e l e r a t i o n . F i g 4.4 shows the a c c e l e r a t i o n f o r ' d i f f e r e n t f a u l t l o c a t i o n s f o r machine 4 . When the f i r s t peak of the a c c e l e r a t i o n exceeds a c e r t a i n maximum l i m i t i t i s found t h a t the machine l o s e s synchronism r i g h t away. 42 4-flach. system I i \ \ I I I 1 1 L Timelsec) F i g 4.2 Fig 4.2 speed deviation of machine A f o r d i f f e r e n t f a u l t locations. i CD 4-HfiCH.SYSTEH(F RT I0JC=.1S) i i i i i — i — i — i — i — i — i D(4-1) D(3-t) D(2-l) 0.0 i—i 1 1 r 0.41 0.83 U5 1.66 TinE(SEC) n — r U5 (a) in. tn I J I I I L J L J L — d(4-l) - ficce.4 i 1 1 — l 1 1 J 1 I i i „ 0.0 0.16 0.33 aS 0.66 0.83 1JJ Timelsec) (b) Fig 4.3 (a) swing curve of machines 4»3 &nd 2 , (b) swing curve and acceleration of machine 4. For afault at bus 10 cleared at .Is. 44 4 - M a c h . s y s t e m » I I J I 1 J L D.26 0.4 0.53 Time (sec) F i g 4.4 Acceleration of machine 4 for d i f f e r e n t f a u l t locations. 45 Also f o r some f a u l t l o c a t i o n s f a r away from the machine i t i s found t h a t the machine never l o s e s synchronism even f o r a su s t a i n e d f a u l t , i . e . , there i s a minimum value o f the a c c e l e r a t i o n ( k i n e t i c energy) t h a t c o n t r i b u t e s to i n s t a b i l i t y . From the above d i s c u s s i o n we can d e f i n e two important a c c e l e r a t i o n l i m i t s . A maximum l i m i t can be d e f i n e d as the maximum value of the f i r s t peak of a c c e l e r a t o n f o r any f a u l t f o r which the machine can c r i t i c a l l y m a i n t a i n synchronism, i . e . , i f the f i r s t peak exceeds the maximum l i m i t the machine w i l l l o s e synchronism. Second, a minimum l i m i t , can be d e f i n e d as the minimum value o f the f i r s t peak above which the machine could l o s e synchronism, i . e . , i f the f i r s t peak of the a c c e l e r a t i o n f o r any f a u l t i s l e s s than t h i s minimum v a l u e , the machine w i l l never l o s e synchronism. For any f a u l t l o c a t i o n i t i s found that each machine has some constant f i r s t peak of a c c e l e r a t i o n r e g a r d l e s s of the f a u l t d u r a t i o n , t h a t i s because t h i s peak depends on the machine o p e r a t i n g c o n d i t i o n s . F i g 4.4 shows the d i f f e r e n t l e v e l s of a c c e l e r a i t o n f o r d i f f e r e n t f a u l t l o c a t i o n s f o r machine 4, the lowest l e v e l o f the f i r s t peak o f a c c e l e r a t i o n can be considered as the minimum l i m i t , while the maximum l e v e l o f the f i r s t peak o f a c c e l e r a i o n i s the maximum l i m i t f o r machine 4. F i g 4.5 and F i g 4.6 show the a c c e l e r a t i o n l i m i t s f o r machine 2 and machine 3 f o r d i f f e r e n t f a u l t l o c a t i o n s i n the 4-machine system. Since the a c c e l e r a t i o n has the p r o p e r t y o f showing the f a u l t s e v e r i t y b e f o r e the f a u l t i s c l e a r e d , t h e r e f o r e i t w i l l 4 6 4 - d a c h . system J L 0.26 0.4 Time (sec) f i g 4.5 o in t 0 D " o'.2S 1 0.5 0.75 W Time(sec) F i g 4.6 I.2S 1.5 A c c e l e r a t i o n f o r d i f f e r e n t f a u l t l o c a t i o n s of machines 2 and 3 . 47 l i k e l y r e v e a l the s t r e n g t h of the power system a f t e r the f a u l t i s c l e a r e d . In o t h e r words, the a c c e l e r a t i o n may i n d i c a t e whether the machine i s s t a b l e or not. S e v e r a l case s t u d i e s have been c a r r i e d out to demonstrate the importance of a c c e l e r a t i o n and to come up with v a l u a b l e i n f o r m a t i o n about the s t a b i l i t y of the d i s t u r b e d machines. 4.4 Case S t u d i e s 4.4.1 The Three-machine System A number of cases were s t u d i e d w i t h a three-phase f a u l t a p p l i e d a t d i f f e r e n t l o c a t i o n s i n the three-machine system as shown i n F i g 4.7. Swing curves and a c c e l e r a t i o n o f each machine f o r each case were p l o t t e d and the s t a b l e and unstable cases were determined from the swing c u r v e s . For each f a u l t l o c a t i o n the c r i t i c a l c l e a r i n g time was found and then the c r i t i c a l l y s t a b l e curves of a c c e l e r a t i o n were p l o t t e d with the swing curve on the same f i g u r e . A l s o the c r i t i c a l l y u n s table curves were p l o t t e d f o r each machine. Case 1: With a three phase f a u l t on l i n e 5-4 near bus-4, the c r i t i c a l c l e a r i n g time i s .3 sec, the f a u l t being c l e a r e d by d i s c o n n e c t i n g l i n e 5-4. F i g u r e s 4.8 to 4.10 show the swing curve and the a c c e l e r a t i o n of the machines 1, 2 and 3 f o r s t a b l e and unstable cases. For the s t a b l e cases the f a u l t i s c l e a r e d a t .3 seconds, and f o r the unstable cases the f a u l t i s c l e a r e d 48 8-IS). u u CE I CM in ro. i 3-riach.system(f at 4) J i i i i i i i i i 49 — A n g l e D i f f . — — A c c e l e r a t i o n 0.0 -I 1 1 1 1 — I 1 1 1 1 — 0.33 0.66 1JD 1.33 1.66 2.0 Time(sec) 8-in CE in. to i T3 in 8. i J I L (a) J I L J L J _JL Angle D i f f . A c c e l e r a t i o n I 1 1 1 1 1 1 1 I I i 0.0 0.33 0.66 1JJ 1.33 1.66 2.0 Time(sec) (b) F i g 4.8 A n g l e d i f f . and a c c e l e r a t i o n o f machine 1 f o r a f a u l t a t bus 4; (a) s t a b l e case (Tc=.30 s e c ) . (b) u n s t a b l e case (Tc=.31 s e c ) . 3-nach.systemCf a l 4) g'_l ' » ' I I I 1 1 1 L 50 in. o> ( N o u QZ a «-> o . n in I v ^  Angle D i f f . \ — —— A c c e l e r a t i o n i i 0.0 8-in INI rg • fN " cj u CC i*> in. to -a in 5?. i ~ T — 0.33 J I I Angle D i f f . — — A c c e l e r a t i o n r-7-—| 1 1 1 1 1 1 1 1 1 I 0.0 0.33 0.66 1.0 1.33 1.66 2.0 Timelsec) (b) F i g 4.9 Angle d i f f . and a c c e l e r a t i o n of machine 2 f o r a f a u l t at bus 4 ; (a) s t a b l e case (Tc=.30 s e c ) . (b) unstable case (Tc=,31 s e c ) . 51 3-riach.systent(f a t 4) i * ^  Angle D i f f . (a) 8-in no «_> w> in_J 10 l in 8 i — Angle D i f f . Acceleration 1.66 2.0 0.0 < U 3 ' 0 £ 6 1-0 - «-33 Time(sec) (b) Fig 4.10 Angle d i f f . and acceleration of machine 3 for a fault at btfs 4 ; (a) stable case(Tc=.30 sec). (b) unstable case (Tc=.31 sec). 52 at .31 seconds. The peak of the a c c e l e r a t i o n a f t e r c l e a r i n g i s lower than the f i r s t peak b e f o r e c l e a r i n g f o r the s t a b l e cases a t t c = .3 seconds f o r a l l the three machines. I t can be noted that machine 1, which i s the c l o s e s t machine to the f a u l t , i s not the c r i t i c a l machine, because i t a c c e l e r a t e d v ery s l o w l y to t h i s f a u l t and a l s o most of the k i n e t i c energy has been generated by machines 2 and 3. So machine 1 w i l l not be t r i p p e d f o r such a case. For the unstable case a t t c = .31 seconds the peak of the a c c e l e r a t i o n a f t e r c l e a r i n g i s higher than the peak before c l e a r i n g f o r a l l the machines, i t means th a t a l l the three machines are a c c e l e r a t i n g more a f t e r c l e a r i n g and the system breaks up. Case 2: A three-phase f a u l t on l i n e 7-8 near bus 7, c l e a r e d a t .11 seconds f o r the s t a b l e case and .12 seconds f o r the un s t a b l e case. F i g 4.11 and F i g 4.12 show the swing curve and a c c e l e r a -t i o n f o r machines 2 and 3 f o r the s t a b l e and unstable c a s e s . Case 3: A three-phase f a u l t on l i n e 9-8 near bus-9. The f a u l t c l e a r e d by d i s c o n n e c t i n g l i n e 9-8. The c r i t i c a l c l e a r i n g time f o r t h i s f a u l t i s .41 seconds. F i g u r e s 4*-13 and 4.14 show the swing curve and a c c e l e r a t i o n f o r machines 2 and 3. For the s t a b l e case the 53 R-o. CD CN u u cr o in I ro. 3-nach.system(f a t 7) i I I I I t I I L • s Angle D i f f . / \ / v 1 • Acceleration / \ / \ / \ . / / \ I 1 1 1— 0.0 0.33 D.6B i 1 1 1 1 1 r 1.0 1.33 1.66 Time(sec) (a) 2.0 CM u u CE in U) fM*_J I CM in .^ ro. in i J L J L J I I L Angle D i f f . Acceleration 1 1 1 1 1 1 1 1 1 1 1 0.0 0.33 0.66 Ul 1.33 1.66 2.0 Time (sec) F i g 4.11 Angle d i f f . and acceleration of machine 2 fo r a fault at bus 7 ; (a) stable case (Tc=.ll sec). (b) unstable case (Tc=.12 sec). 3-f1ach.system(f a t 7) R-| 1 1 I 1 1 I I t t i i - Angle D i f f . Accleration 2.4 (a) 5-cn u u CX IO T CO " "° = (M o I J I L Angle D i f f . Acceleration m/ — l 1 1 1 1 1 1 1 1 1 1— 0.0 0.4 0.8 U 1.6 2.0 2.4 Time(sec) (b) Pig 4.12 Angle d i f f . and acceleration of machine 2 for f a u l t at bus "7 ; (a) stable case (Tc=.ll sec). (b) unstable case (Tc=.12 sec). o o o-U u CC o u> co-in I CN TO m. 3-nach. sys tem(f at 9) j i i i i i i i i ' 55 - ^ ^  Angle D i f f . Acceleration \ \ \ i 1 1 r Q.D 0.33 n r 0.66 IJ) 1.33 Time(sec3 (a) i — i — i r S-in CM u u CC J L 1.66 2.0 J L Angle D i f f . • Acceleration 0.0 i 1 r 0.33 ~I—I—I—I—I—I—I—I— 0.66 101 1.33 1.66 2.0 Time(sec) (b) F i g 4.13 Angle d i f f . and acceleration of machine 2 fo r a fault at b u s 9 » (a) stable case (Tc=.41 sec). ( b ) unstable case (Tc=.42 sec). 8-co u u H cc a m T ro 3-Mach.systein(f at 9) j i i i i i t i i i 56 ---- Angle Diff. • Acceleration \ 8 • i — i — r 0.0 0.33 ID-CO " CJ u CE o u) o'. CD I CO 8 l 1 1 1 1 1 1 1 1— 0.66 10 1.33 1.66 2.0 Time (secj (a) » i i i 1 I L I I I L_ / Angle Di f f . '- Acceleration —i 1 1 1 1 1 r 0.0 0.4 0.8 12 Time(sec) (b) ~ i — i — i — r ~ 1.6 2.0 2.4 Pig 4.14 Angle,diff. and acceleration of machine 3 for a fault at bus 9 ; (a) stable case (Tc=.41 sec). (b) unstable case (Tc=.42 sec). 57 f a u l t i s c l e a r e d a t .41 seconds and f o r the unstable case t c = .42 seconds. S i m i l a r r e s u l t s are seen i n the f o l l o w i n g cases: Case 4: A three-phase f a u l t on l i n e 7-5 near bus-5 i s c l e a r e d by d i s c o n n e c t i n g l i n e 7-5. See F i g 4.15 and F i g 4.16. Case 5: A three-phase f a u l t on l i n e 9-6 near bus-6, i s c l e a r e d by d i s c o n n e c t i n g l i n e 9-6. See F i g 4.17 and F i g 4.18. Case 6: A three-phase f a u l t on l i n e 7-8 near bus-8, i s c l e a r e d by d i s c o n n e c t i n g l i n e 7-8. See F i g 4.19 and 4.20. 58 3-nach. sys tem(f at 5) j [ i l l I I 1 1 1 L i 1 1 1 r—T 0.0 0.33 0.66 1.0 Time(sec) (a) D a-in i I L i — i — r 1.33 1.66 J I L 2.0 Angle Diff. — — Acceleration i 1 1 1 „ ! * ' , i ' U 3 ' U S 2.0 0.0 0.33 Time(sec) (b) Fig 4.15 Angle j i i f f . and acceleration of machine 2 for a fault at bus 5 ; (a) stable case (Tc=.21 sec). (b) unstable case (Tc=.22 sec). 59 3-nach. sys tem(f a t 5) g U 1 1 1 L in r-" «T> t_> u CE in'. ID i r o ' " a in I N . J 1 I L J L Angle Diff. — — Acceleration r i l 1 1 1— 00 0.33 0.66 in CO o ID i co " "O in r\J. rN I J I I I L (a) i J I I Angle Di f f . — - Acceleration —i i 1 1 1 1 1 1 1 1 r— 0.0 0.33 0.66 UD 1.33 1.66 2.0 Time(sec) (b) Fig 4.16 Angle diff.and acceleration of machine 3 for a fault at bus 5 ; (a) stable case (Tc=.21 sec). (b) unstable case (Tc=.22 sec). 60 3-nach . sy s teml f at 6) 0.0 JL33 0 £ 6 UJ Timelsec) 0.0 0.33 0.66 U 1.33 1.66 2.0 Timelsec) (b) F i g 4.17 Angle'diff. and acceleration of machine 2 for a fault at bus 6 » (a) stable case (Tc=.51 sec). (b) unstable case (Tc=.52 sec) 3-ttach.system(f a t 6) J I I I 1 « 1 1 L 61 i — i 1 — i — i — i — i r 0.33 0.66 UO 1.33 Timelsec) 8-tn r-' i J I I L (a) J I I 1 J L Angle Diff. Acceleration l 1 1 1 1 1 1 1 1 1 I 0.0 0.33 0.66 UO 1.33 1.66 2.0 Timelsec) (b) Fig 4.18 Angle d i f f . and acceleration of machine 3 for a fault' at bus 6 ; (a) stable case (Tc=.51 sec). (b) unstable case (Tc=.52 sec). 3-nach. sys tem(f a t 8) i i i i i — i — i — J — 1 — L 62 0.0 0.33 DJ56 t0 1.33 Tiae(sec) (b) 1.66 2.0 Pig 4.19 Angle d i f f . and acceleration of machine 2 for a fault at bus 8 ; (a) stable case(Tc=.29 sec ). (b) unstable case (Tc=.30 sec) 3-Mach.system(f at 8) g I I I I I L_ I I 1 t I « 63 I 0.66 t0 Tine(sec) (a) i r 1.33 n r 1.66 2.0 i — i 1 — i — i — i — i 1 — i — i — i — r 0.0 0.33 036 10 1.33 1.66 2.0 Time(sec) (b) Fig 4.20 Angle d i f f . and acceleration of machine 3 for a fault at bus 8 ; (a) stable case (Tc=.29 sec). (b) unstable case (Tc=.30 sec). 64 4.4.2 The Four-machine System'Case S t u d i e s The four-machine system i s the same as the three-machine system except f o r a f o u r t h machine connected to bus-4 through a double c i r c u i t t r a n s m i s s i o n l i n e . A lo a d flow study was c a r r i e d out to f i n d the i n i t i a l o p e r a t i n g c o n d i t i o n s of the machines. A three-phase f a u l t was considered a t each bus of the system, and by l o o k i n g a t the a c c e l e r a t i o n response of each machine i t i s c l e a r t h at the a c c e l e r a t i o n of a machine i n d i c a t e s whether i t i s s t a b l e or not. A number of s t u d i e s were c a r r i e d out on the system. A t y p i c a l case of a three-phase f a u l t near bus-10 w i l l be cons i d e r e d here. The f a u l t i s c l e a r e d by d i s c o n n e c t i n g the f a u l t e d l i n e between bus-8 and bus-10. The other three machines i n the system never l o s e t h e i r s t a b i l i t y even i f machine-4 l o s e s s t a b i l i t y . T h i s means t h a t most of the k i n e t i c energy produced by the machine r o t o r s came from the r o t o r o f machine-4. In t h i s case, a l s o , the f i r s t peak of a c c e l e r a t i o n o f machines 1, 2 and 3 i s l e s s than the minimum l i m i t o f a c c e l e r a t i o n . F i g 4.21 to F i g 4.23 show the behavior of machines 4, 3 and 2 f o r s t a b l e and unstable cases of machine 4. 4.4.3 The Fiv e Machine System Case S t u d i e s Two cases'are considered f o r t h i s system (see F i g 3.3), Case 1; a three-phase f a u l t near bus-11, Case 2; a three-phase f a u l t near bus-6. The c r i t i c a l machine f o r both cases i s 4-Mach. sys ten(F a t 10) i i i I I I I L 1 1 L 65 Angle Diff. Acceleration 8-1 I L u u CC I r o . I Tiae(sec) (a) i _ i i ' ' i ' ' Angle Diff. — — Acceleration —I 1 1 1 1 ! 1 1 1 1 1 0.0 0.33 D.66 10 1.33 1.66 2.0 Tine(sec) (b) Fig 4.21 Angle d i f f . and acceleration of machine 4 for a fault at bus 10 ; (a) stable case(Tc=.45 sec). (b) unstable case (Tc = .46 sec). a . ro fN u u d I CM C O . I I O . 4-nach . sys tem(f at 10) • i i i i i i i i i Angle d i f f . N Acceleration 0.D to fN o CC « fN ro. I 10 0.25 l I I D.5 i 1 r 0.75 Time(sec) 1.0 (a) I I I L i 1 r 1.25 J L Angle d i f f . Acceleration 1.5 • — r 1 1 1 1 1 1 i i i i 0.0 0.25 0.5 0.75 1.0 1.25 1.5 Time(sec) (b) Fi g 4.22 Angle d i f f . and acceleration of machine 2 for a f a u l t at bus 10; (a) stable (Tc=.45 sec). (b) stable (Tc=.46 sec). Note that the acceleration i s lower than the min. l i m i t . 67 u CC I C3 I o to. 4-Mach.systeni(F at 10) I ' l l ' J I I L Angle d i f f . A c c e l e r a t i o n D.O 0.25 I 0.5 to m o u CC £2 rft. i J 1 f I i 1 r 0.75 Timelsec) (a) J I L 1.0 i r 1.25 1.5 J L Angle d i f f . A c c e l e r t i o n —i 1 1 1 1 1 1 1 1 l l 0.0 0.25 0.5 D.75 1.0 1.25 1.5 Time (sec) (b) F i g 4.23 Angle d i f f . and a c c e l e r a t i o n o f machine 2 f o r a f a u l * a t bus 10 ; (a) and (b) are s t a b l e f o r Tc=.45t.46 sec. 68 machine 5, the ot h e r four machines d i d not show any s i g n f i i c a n t response to both f a u l t s (because each machine i s l o c a l l y l o a d e d ) . The behavior of machine 5 i s shown i n F i g 4.24. 4*5 Case S t u d i e s D i s c u s s i o n From the cases s t u d i e d , f o r any f a u l t considered i t i s found that i f the f i r s t peak of a c c e l e r a t i o n l i e s between the p r e v i o u s l y d e f i n e d maximum and minimum l e v e l s of a c c e l e r a t i o n , then the machine may l o s e synchronism unless the r e s t of the system i s capable of handling the power flow a f t e r f a u l t c l e a r i n g . When we examine c l o s e l y the a c c e l e r a t i o n curves f o r the s t a b l e cases we n o t i c e t h a t the peak of the a c c e l e r a t i o n a f t e r f a u l t c l e a r i n g i s l e s s than the f i r s t peak before c l e a r i n g . On the other hand we see the o p p o s i t e f o r the unstable cases, where the peak o f the a c c e l e r a t i o n a f t e r f a u l t c l e a r i n g i s always hi g h e r than the peak o f a c c e l e r a i t o n before f a u l t c l e a r i n g . For any d i s t u r b e d machine i n the t e s t e d systems, i f the a c c e l e r a t i o n i s hig h e r than the minimum l i m i t , then i t i n d i c a t e s t h a t the machine i s one of the c r i t i c a l machines f o r a s p e c i f i c d i s t u r b a n c e . The a c c e l e r a t i o n response i s d i f f e r e n t f o r each machine i n a power system when i t i s subjected to a f a u l t . Some nachines are a c c e l e r a t i n g and some d e c e l e r a t i n g . U s u a l l y the machines which are c l o s e to the f a u l t l o c a t i o n are a c c e l e r a t i n g and most probably one or more o f these machines may l o s e cu (_i u ^ = u> g". QJ cn -CT o o in. i 5 - M a c h i n e system j » i i i i i i i i i / N Angle Diff. Acceleration 0.0 0.26 - i — r 0.53 I 0.8 T - i — i 1 r 1.06 1.33 1.6 Timelsec) (b) m . CM m r—1 QJ i-1 U o U J g _ | cu cn -c CC o LO _ (VI LO J L » » » I i ' Angle Di f f . Acceleration I 1 1 1 1 1 1 1 1 1 1— 0.0 0.26 0.53 0.8 , 1.06 1.33 1.6 Time (sec) (b) 4.24 Angle difference and acceleration of machine 5 for a fai at bus 11 ; (a) stable case (Tc « . 4 7 sec). (b) unstable case (Tea .48 sec). 70 synchronism. A l s o when the f a u l t i s c l e a r e d , most of the a c c e l e r a t i n g machines d e c e l e r a t e (depending upon the system c o n f i g u r a t i o n a f t e r f a u l t c l e a r i n g and the remaining l o a d s ) . I f a machine i s s t a b l e then the maximum a c c e l e r a t i o n a f t e r c l e a r i n g should be l e s s than t h a t before c l e a r i n g , which means that t h i s machine i s converging toward a new s t a b l e o p e r a t i n g p o i n t . For an unstable machine, the peak of the a c c e l e r a t i o n a f t e r c l e a r i n g i s higher than that before c l e a r i n g which means t h a t t h i s machine i s a c c e l e r a t i n g higher a f t e r c l e a r i n g and d i v e r g i n g from s t a b i l i t y . Sometimes c l e a r i n g the f a u l t causes a second severe d i s t u r b a n c e t h a t may break up the system. In t h i s case the machines continue to a c c e l e r a t e o r d e c e l e r a t e u n t i l they l o s e synchronism. T h i s can be monitored by l o o k i n g a t t h e i r a c c e l e r a t i o n curves. T h i s happens o n l y i f the r e s t o f the system i s not capable o f absorbing the f a u l t k i n e t i c energy and meeting the demand of the new o p e r a t i n g c o n d i t i o n s . 4.6 T r a n s i e n t S t a b i l i t y C r i t e r i o n Based on A c c e l e r a t i o n Measurements T h i s c r i t e r i o n r e l i e s on the instantaneous measurements of the a c c e l e r a t i o n o f the synchronous machine d u r i n g the t r a n s i e n t p e r i o d . The instantaneous a c c e l e r a t i o n can be measured by keeping t r a c k of time and sensing speed by monitoring the passage o f te e t h on a toothed wheel b o l t e d to the machine's r o t o r [ 4 ] . Then the speed and the a c c e l e r a t i o n can be 71 c a l c u l a t e d by the use of s p e c i a l l y designed micro- computers. Very good measurements have been achieved using such a system by the Montana Power Company. The procedure f o r t r a n s i e n t s t a b i l i t y assessment using t h i s c r i t e r i o n i s o u t l i n e d i n the f o l l o w i n g s t e p s : 1. Find the maximum and minimum l i m i t s o f a c c e l e r a i t o n t o i n d i c a t e the s e v e r i t y o f the f a u l t f o r each machine i n the power system, then a d j u s t the t r i p p i n g system t o : (a) ignore the a c c e l e r a t i o n s i g n a l i f the f i r s t peak i s l e s s than the minimum l i m i t , because nothing s e r i o u s has o c c u r r e d , i . e . , the machine i s s t a b l e . (b) r e l e a s e immediately a t r i p s i g n a l to the r e l a y t o d i s c o n n e c t the generator i f the f i r s t peak of the a c c e l e r a t i o n exceeds the maximum l i m i t . (c) ready (but not r e l e a s e ) a t r i p p i n g s i g n a l f o r t r a n s m i s s i o n to the r e l a y i f the f i r s t peak of the a c c e l e r a t i o n passes the minimum l i m i t but does not exceed the maximum l i m i t . 2. V7hen the f a u l t i s c l e a r e d ( t h i s i s c l e a r l y i n d i c a t e d by the r a t e o f change of a c c e l e r a t i o n ) and the peak o f the a c c e l e r a t i o n a f t e r c l e a r i n g i s higher than that before c l e a r i n g , then a t r i p s i g n a l must be issued t o t r i p the generator as i t i s going to be u n s t a b l e . But i f the peak o f the a c c e l e r a t i o n a f t e r c l e a r i n g the f a u l t i s lower than t h a t b e f o r e the f a u l t c l e a r i n g then the generator i s s t a b l e and the t r i p s i g n a l must be b l o c k e d . The c r i t e r i o n needs three important measurements to d e c i d e the t r a n s i e n t s t a b i l i t y : f i r s t a c c e l e r a t i o n peak, f a u l t c l e a r i n g i n d i c a t i o n , and a c c e l e r a t i o n peak a f t e r f a u l t c l e a r i n g . The t o t a l time to issue a t r i p should be of the o r d e r o f one second f o r case (c) of step 1. But f o r cases (a) and (b) the d e c i s i o n to i s s u e a t r i p s i g n a l must occur immediately a f t e r the f a u l t o c c u r s . The r a t e of change of a c c e l e r a t i o n can be used to i n d i c a t e the occurrence and c l e a r i n g of any f a u l t because i t shows two sharp jumps f o r these two i n c i d e n t s , as shown i n F i g 4.25. There are some s p e c i a l cases where some machines i n the f a u l t e d system are d e c e l e r a t i n g i n s t e a d of a c c e l e r a t i n g . The c r i t e r i o n i s a p p l i c a b l e to these cases a l s o , but u s u a l l y those machines are not the c r i t i c a l machines f o r such cases, and u s u a l l y do not c o n t r i b u t e to i n s t a b i l i t y o f the power systems. 73 4 - m a c h i n e s y s t e m I I I I I I L Time (sec) Fig 4.25 Rate of change of acceleration of machine 4 for a fault at bus 8, the two jerks at fault and clearing can be used to detect both events. 74 CHAPTER 5 CONCLUSION AND FUTURE WORK T r a n s i e n t s t a b i l i t y o f power systems i s c o n v e n t i o n a l l y s t u d i e d many years i n advance by the use o f d i g i t a l computer programs. The main f a c t o r used i n d e c i d i n g the t r a n s i e n t s t a b i l i t y of any machine i s i t s angle d i f f e r e n c e s with r e s p e c t to other machines i n the system. I t i s c l e a r from the i n v e s t i g a t i o n s d i s c u s s e d i n Chapter 4 t h a t the l o c a l a c c e l e r a t i o n o f each machine can g i v e the e s s e n t i a l i n f o r m a t i o n needed to assess the t r a n s i e n t s t a b i l i t y o f t h a t machine. The i n f o r m a t i o n provided by the l o c a l a c c e l e r a t i o n o f a machine, about i t s t r a n s i e n t s t a b i l i t y i s s u f f i c i e n t and d e f i n i t e f o r the t e s t e d systems, j u s t l i k e the angle d i f f e r e n c e between machines. The t r a n s i e n t s t a b i l i t y c r i t e r i o n developed i n Chapter 4 i s simple and p r a c t i c a l f o r l o c a l l y a s s e s s i n g t r a n s i e n t s t a b i l i t y . G e n e r a l l y the b e n e f i t s o f t h i s new c r i t e r i o n can be summarized as f o l l o w s : 1. The c r i t e r i o n can be used f o r automatic s w i t c h i n g o f some s t a b i l i t y c o n t r o l s such as dynamic brakes, shunt c a p a c i t o r s and f a s t v a l v i n g , where the instantaneous a c c e l e r a t i o n p r o v i d e s v a l u a b l e i n f o r m a t i o n about the s e v e r i t y o f the f a u l t . 2. A g r e a t d e a l o f time and money can be saved s i n c e l a r g e 75 system s t u d i e s need n o t be made f o r d i f f e r e n t c o n t i n g e n c i e s . 3. The c r i t e r i o n works r e g a r d l e s s o f how many d i s t u r b a n c e s a r e t a k i n g p l a c e and what changes have o c c u r r e d i n t h e t r a n s m i s s i o n system a t t h e same t i m e , b e c a u s e i t depends o n l y upon l o c a l a c c e l e r a t i o n measured a t each machine. 4. S i n c e no i n f o r m a t i o n i s needed from o t h e r machines i n t h e system t o a s s e s s the s t a b i l i t y o f a p a r t i c u l a r machine, c o n t r o l o f t h e machines c a n be d e c e n t r a l i z e d . 5. The c r i t e r i o n d e a l s w i t h r e a l q u a n t i t i e s and hence w i t h the r e a l s i t u a t i o n o f any machine i n t h e sy s t e m d u r i n g t r a n s i e n t s . T h i s w i l l p r e v e n t any e r r o r s i n the c h o i c e o f t r i p p i n g . O n l y the u n s t a b l e machine o r machines w i l l be t r i p p e d . F u t u r e Work Based on t h i s new c r i t e r i o n o f t r a n s i e n t s t a b i l i t y we c a n t h i n k o f new a u t o m a t i c p r o t e c t i o n systems f o r l a r g e g e n e r a t o r s t o p r e v e n t i n s t a b i l i t y o f power s y s t e m s . T h i s p r o t e c t i o n s y s t e m can be used t o s w i t c h on some s t a b i l i t y c o n t r o l s t o m a i n t a i n s t a b i l i t y o r t o d i s c o n n e c t t h e u n s t a b l e g e n e r a t o r s from t h e r e s t o f the power s y s t e m . F u t u r e work c o u l d i n v e s t i g a t e t h e f o l l o w i n g : 1. D e s i g n o f ' a d e v i c e t o measure the i n s t a n t a n e o u s a c c e l e r a t i o n q u i c k l y and a c c u r a t e l y . 2. D e s i g n o f a m i c r o - c o m p u t e r t h a t can p r o c e s s the i n f o r m a t i o n from the me a s u r i n g d e v i c e and p r o v i d e the r i g h t d e c i s i o n i n a r e a s o n a b l e t i m e a b o u t t h e s t a t e o f s t a b i l i t y o f t h e d i s t u r b e d machine. 3. D e s i g n o f a r e l a y i n g system, w i t h b l o c k i n g a b i l i t y , t o e n s u r e system s t a b i l i t y . Such a p r o t e c t i o n system w i l l go a l o n g way t o s o l v e the t r a n s i e n t s t a b i l i t y p r o b l e m , and t o g i v e more r e l i a b l e and c o n t i n u o u s power s e r v i c e . 77 References 1. G l e s s , G.E. " D i r e c t method of lyapunov a p p l i e d to t r a n s i e n t power system s t a b i l i t y " IEEE Trans. PAS-85, Feb. 1966, pp. 164-179. 2. P a i , M.A., Mohan, M.A. and Rao, J.G. "Power system t r a n s i e n t s t a b i l i t y r e g i o n s using popov's method" IEEE Trans. PAS-89, May/June 1970, pp. 788-794. 3. Brown, R.D., McClymont, K.R. "A power swing r e l a y f o r p r e d i c t i n g g e n e r a t i o n i n s t a b i l i t y " IEEE Trans. PAS, March 1965, pp. 219-224. 4. J . F . J o l l e y " D i g i t a l l y c o n t r o l l e d g enerator t r i p p i n g enchances s t a b i l i t y performance of Montana Power Transm i s s i o n System" Montana Power and L i g h t . 5. Crany, S.B. Power System S t a b i l i t y . V o l . I . V o l . I I . New York: John Wiley & Sons, Inc., 1948. 6. Kimbark, E.W. Power System S t a b i l i t y . V o l . I. New York: John Wiley & Sons, Inc., 1948. 7. Anderson, P.M. and Fouad, A.A. Power System C o n t r o l and S t a b i l i t y . V o l . I. Ames, Iowa, Iowa S t a t e U n i v e r s i t y P r e s s , 1977. 8. E l - A b i a d , A.H. and Nagappan, K. " T r a n s i e n t s t a b i l i t y r e g i o n s of multimachine power systems" IEEE. Trans. PAS-85, Feb. 1966, pp. 158-168. 9. Athay, T., Podmore, R. and V i r m a n i , S. "A p r a c t i c a l method f o r d i r e c t a n a l y s i s o f t r a n s i e n t s t a b i l i t y " . IEEE. Trans. PAS-98, M a r c h / A p r i l 1979, pp. 573-584. 10. F o u a d , A . A . , Stanton, S.E., Mamandur, K.R. and Kruempel, K.C. "Contingency a n a l y s i s using the t r a n s i e n t energy margin technique". Paper 81 SM 397-9. IEEE PES Summer Meeting, P o r t l a n d , 1981. 11. V i t t a l , V. "Power system t r a n s i e n t s t a b i l i t y u s ing the c r i t i c a l energy of i n d i v i d u a l machines", Ph.D. T h e s i s , Iowa S t a t e U n i v e r s i t y , Ames, Iowa, 1982. 12. F o u a d , A o A . , V i t t a l , V. "Power system response to a l a r g e d i s t u r b a n c e energy a s s o c i a t e d w i t h system s e p a r a t i o n " IEEE PES Summer Meeting, 1983. 78 13. Uemosono, Matsuki, Okada, "An a l y s i s of the step-out process of a three phase synchronous machine by a i r gap f l u x " JIEE B100-1, 1980, Japan. 14. Uemosono, Okada, Matsuki, Yamada, Yokokawa, Moriyasu, "Development and t e s t i n g of an automatic s t a b i l i t y p r e d i c t i o n and c o n t r o l (ASPAC) f o r a synchronous generator by a i r gap f l u x " , IEEE Trans. PAS. Paper 81SM 462-1, J u l y 1981. 15. IEEE. Committee Report "A d e s c r i p t i o n of d i s c r e t e supplementary c o n t r o l s f o r s t a b i l i t y " , IEEE. Trans. PAS-97, Jan/Feb 1978, pp. 149-165. 16. Kimbark, E.W. "Improvement of power system s t a b i l i t y by changes i n network", IEEE Trans., V o l . PAS-88, May 1969, pp. 773-781. 17. Shelton, M i t t e s l s l a d t , Winkleman, B e l l e r b y " B o n n e v i l l e power adminstration 1400 MW braking r e s i s t o r " , IEEE. Trans. V o l . PAS-94, March 1975, pp. 602-611. 18. Westinghouse, E l e c t r i c a l Transmission and D i s t r i b u t i o n Reference Book, U.S.A. 19. Cogswell, S.S. and others "Generator s h a f t torques r e s u l t i n g from operation of EHV breakers", IEEE Paper C-74-087-3, Winter Power Meeting, N.Y., Jan. 1974. 20. A b o l i n s , A. and others " E f f e c t of c l e a r i n g short c i r c u i t s and automatic r e c l o s i n g on t o r s i o n a l and l i f e expenditures of turbine-generator s h a f t s " , IEEE Trans., V o l . PAS-95, Jan. 1976, pp. 14-25. 21. Kimbark, E.W. "Improvement of power system s t a b i l i t y by switched s e r i e s c a p a c i t o r s " , IEEE. Trans. V o l . PAS-85, Feb. 1966, pp. 180-188. 22. IEEE. Committee Report, "A b i b l i o g r a p h y f o r the study of subsynchronous resonance between r o t a t i n g machines and power systems" IEEE Trans., V o l . PAS-95, Jan. 1976, pp. 216-218. 23. Cushing, E.W. and others "Fast v a l v i n g as an a i d to power system t r a n s i e n t s t a b i l i t y and prompt resynchronizing and r a p i d reload a f t e r f u l l load r e j e c t i o n " , IEEE. Trans. V o l . PAS-91, Aug.' 1972, pp. 1624-1636. 24. Park, R.H. "Fast turbing v a l v i n g " , IEEE Trans., V o l . PAS-92, June 1973, pp. 1065-1073. 79 25. Kimbark, E.W., "Bibl i o g r a p h y on s i n g l e pole s w i t c h i n g " , IEEE. Trans. V o l . PAS-94, May 1975, pp. 1072-1078. 26. Holland, D.R. and others "Conemaugh p r o j e c t : new concept f o r 500 KV system p r o t e c t i o n " , IEEE. Trans. V o l . PAS-90, Jan. 1971, pp. 1-10. 

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