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A model for the synchronous machine using frequency response measurements Bacalao, Nelson Jose 1987

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A MODEL FOR THE SYNCHRONOUS MACHINE USING FREQUENCY RESPONSE MEASUREMENTS By NELSON JOSE BACALAO E l e c . Eng . (Hons . )> U n i v e r s i d a d Simon B o l i v a r , V e n e z u e l a , 1979 Master E n g . , Rensse laer P o l y t e c h n i c I n s t i t u t e , New Y o r k , 1980 A THESIS SUBMITTED IN THE REQUIREMENTS DOCTOR OF PARTIAL FULFILLMENT OF FOR THE DEGREE OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department of E l e c t r i c a l E n g i n e e r i n g ) We accept t h i s t h e s i s as conforming to the r e q u i r e d s tandard THE UNIVERSITY OF BRITISH COLUMBIA August , 1987 (c) Nelson J . B a c a l a o , 1987 4 6 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6(3/81) ABSTRACT In t h i s d i s s e r t a t i o n a new model f o r the s y n c h r o n o u s machine i s p r e s e n t e d . T h i s mod e l , based on n o n - s t a n d a r d t e s t d a t a , a l l o w s f o r the a p p r o p r i a t e m o d e l l i n g o f the f r e q u e n c y d e p e n d e n t b e h a v i o u r o f t h e d a m p e r w i n d i n g s . T h e n o n - s t a n d a r d t e s t d a t a c o n s i s t of f r e q u e n c y r e s p o n s e s , e i t h e r measured or c a l c u l a t e d . The form of t h e s e r e s p o n s e s w i l l a u t o m a t i c a l l y d e t e r m i n e t h e o r d e r o f t h e r e s u l t i n g m o d e l . S a t u r a t i o n e f f e c t s i n t h e s y n c h r o n o u s m a c h i n e a r e a l s o model led wi th t h i s new method. T h e m o d e l was s u c c e s s f u l l y t e s t e d i n b o t h an e l e c t r o m a g n e t i c t r a n s i e n t s program (EMTP) and i n a s t a b i l i t y p r o g r a m . I t was f o u n d t h a t when f r e q u e n c y r e s p o n s e measurements a r e used d i r e c t l y , the model i s more a c c u r a t e than when u s i n g the s t a n d a r d d a t a from the m a n u f a c t u r e r or da ta e s t i m a t e d to match a p p r o x i m a t e l y the f r e q u e n c y response measurements . I t was a l s o a s c e r t a i n e d t h a t t h i s model c o u l d be used to speed up the s o l u t i o n i n a s t a b i l i t y p r o g r a m , both by a l l o w i n g the user to match the o r d e r of the model to the r e q u i r e d a c c u r a c y depending on the event and i n t e g r a t i o n s t e p , and by m o d i f y i n g the i n p u t f r e q u e n c y r e s p o n s e da ta to m i n i m i z e t h e d i s c r e t i z a t i o n e r r o r made when u s i n g l a r g e i n t e g r a t i o n s t e p s . - i i i -TABLE OF CONTENTS PAGE ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v i i LIST OF ILLUSTRATIONS v i i i LIST OF MAJOR SYMBOLS x i i i ACKNOWLEDGEMENTS x v i i INTRODUCTION 1 OBJECTIVES AND PHILOSOPHICAL PRECEPTS 4 CHAPTER 1 : BASIC THEORY OF THE SYNCHRONOUS MACHINE 5 1.1 P h y s i c a l D e s c r i p t i o n of the Synchronous Machine 5 1.2 D i f f e r e n t i a l E q u a t i o n s of the Synchronous Machine 7 1.3 E q u i v a l e n t C i r c u i t s of the Synchronous Machine 13 1.4 S o l u t i o n of the Machine D i f f e r e n t i a l E q u a t i o n s i n the Frequency Domain 15 1.5 Steady S t a t e E v a l u a t i o n of the Synchronous Machine 23 1.5.1 P o s i t i v e sequence 23 1.5.2 Negat ive sequence 27 1 . 5 . 3 . Zero sequence 30 CHAPTER 2 : BASIC THEORY OF THE NEW MODEL 31 2 . 1 . I n t r o d u c t i o n 31 2.2 T r a n s f o r m a t i o n of the Frequency Domain E q u a t i o n s i n t o the Time Domain 31 2.3 Approx imat ion by R a t i o n a l F u n c t i o n s 37 2 .3 .1 B r i e f D e s c r i p t i o n of the Approx imat ion Method 37 - i v -PAGE 2.4 C o r r e c t i o n of the F u n c t i o n s to be Approximated 40 2 .4 .1 I n c o r p o r a t i o n of S a t u r a t i o n E f f e c t s 40 2 .4 .2 Approx imat ion of Curves 47 2 . 4 . 2 . 1 Approx imat ion of X j ( s ) and X^(s) 47 2 . 4 . 2 . 2 Approx imat ion of G(s) 48 2 . 4 . 2 . 3 Approx imat ion of F l ( s ) to F6 ( s ) 51 2.5 Run Time Reduced Models and Compensation of Numer ica l E r r o r s 57 2 .5 .1 Reduct ion of the Order of the Model 58 2 .5 .2 E v a l u a t i o n of the E r r o r i n the Frequency Domain and I n t r o d u c t i o n of C o r r e c t i n g Po le s 59 CHAPTER 3 : INCLUSION OF NONLINEARITIES 68 3.1 I n t r o d u c t i o n 68 3.2 Method 1 f o r the C o n s i d e r a t i o n of S a t u r a t i o n 68 3.3 Method 2 f o r the C o n s i d e r a t i o n of S a t u r a t i o n 71 3 .3 .1 G e n e r a l D e s c r i p t i o n of the Method 72 3 .3 .2 E q u a t i o n s of the Model 75 CHAPTER 4 : IMPLEMENTATION OF THE MODEL IN AN ELECTROMAGNETIC TRANSIENTS PROGRAM 82 4.1 I n t r o d u c t i o n 82 4.2 Genera l D e s c r i p t i o n of the E l e c t r o m a g n e t i c T r a n s i e n t s Program Used (EMTP) 82 4.3 Implementat ion of Method 1 for the C o n s i d e r a t i o n of S a t u r a t i o n 83 4.4 Implementat ion of Method 2 f o r the C o n s i d e r a t i o n of S a t u r a t i o n 87 4.5 R e s u l t s 91 V PAGE 4 .5 .1 V a l i d a t i o n of Method 1 91 4 . 5 . 2 E f f e c t s of Us ing D i f f e r e n t Input Data 96 4 . 5 . 3 E v a l u a t i o n of the U s e f u l n e s s of the Proposed Method 104 4 .5 .4 G e n e r a l O b s e r v a t i o n s on the Numerica l Behav iour of the Method 111 4 . 5 . 5 C o n c l u s i o n s 111 CHAPTER 5 : IMPLEMENTATION IN A STABILITY PROGRAM 113 5.1 I n t r o d u c t i o n 113 5.2 Genera l D e s c r i p t i o n of the S t a b i l i t y Program 113 5.3 D e s c r i p t i o n of the Implementat ion 115 5.4 R e s u l t s of the Implementat ion 117 5.5 Usage of the Model f or Speeding up a S t a b i l i t y Program 120 5.6 E v a l u a t i o n of the Impact of the Trans former Terms 125 5.7 C o n c l u s i o n s 128 CHAPTER 6 : CONCLUSIONS 135 R E F E R E N C E S 139 APPENDIX 1 : THE RECURSIVE CONVOLUTION TECHNIQUE 141 A l . l C o n v o l u t i o n wi th an E x p o n e n t i a l . 141 A1.2 C o n v o l u t i o n wi th an Impulse Response 143 APPENDIX 2 : BLOCK DIAGRAMS OF EXCITERS AND GOVERNORS USED IN STABILITY SIMULATIONS 145 A2.1 E x c i t e r 145 A2.2 Governor 146 - v i -PAGE APPENDIX 3: DESCRIPTION OF THE STAND-STILL FREQUENCY RESPONSE METHODS FOR THE EVALUATION OF X d ( s ) , X (s) AND G(s) 147 A3.1 Measurement of X^(s) 147 A3.2 Measurement of X (s) 148 q A3.3 Measurement of G(s) 149 APPENDIX 4 : DEVELOPMENT OF THE INTEGRATION EQUATIONS FOR THE METHOD 2 FOR THE CONSIDERATION OF SATURATION 150 APPENDIX 5 : CONSIDERATION OF UNEQUAL FLUX LINKAGES USING AN EQUIVALENT CIRCUIT 158 APPENDIX 6 : EFFECT OF SATURATION ON THE MACHINE TIME CONSTANTS 162 - v i i LIST OF TABLES TABLE PAGE 5.1 R e s u l t s o b t a i n e d by t e s t i n g the model wi th d i f f e r e n t i n t e g r a t i o n s teps A t . 124 A . l E f f e c t of s a t u r a t i o n i n Guri u n i t 7 to 10. 162 A.2 E f f e c t of s a t u r a t i o n i n a f o s s i l - f i r e d u n i t . 162 - v i i i -LIST OF ILLUSTRATIONS  FIGURE PAGE 1.1 P h y s i c a l r e p r e s e n t a t i o n of the synchronous machine 6 1.2 R e l a t i o n s h i p between the d and q - a x i s magnetomotive f o r c e s 6 1.3 Schematic r e p r e s e n t a t i o n of the f l u x l i n k a g e path between the windings i n the d - a x i s 16 1.4 E q u i v a l e n t c i r c u i t s of the synchronous machine 16 1.5 E q u i v a l e n t c i r c u i t s i n the frequency domain 20 1.6 Steady s t a t e phasor diagram with s a t u r a t i o n 20 2.1 Method f o r a l l o c a t i n g the po les and zeros from a Bode p l o t 39 2 .2-A1 Approx imat ion of F l to F3 f o r O n t a r i o Hydro generator module of the f u n c t i o n s 41 2 .2-A2 Approx imat ion of F l to F3 for O n t a r i o Hydro generator angle of the f u n c t i o n s 41 2 .2-B1 Approx imat ion of F4 to F6 f o r O n t a r i o Hydro generator module of the f u n c t i o n s 42 2 .2-B2 Approx imat ion of F4 to F6 f o r O n t a r i o Hydro generator angle of the f u n c t i o n s 42 2.3 L i n e a r i z a t i o n of the o p e n - c i r c u i t s a t u r a t i o n curve 44 2 .4 -A X j ( s ) and X ^ ( s ) f o r d i f f e r e n t s a t u r a t i o n segments 49 2 .4 -B A s s o c i a t e d f u n c t i o n s to X^(s) and X (s) f o r d i f f e r e n t s a t u r a t i o n segments ^ 49 2 .4 - C Method f o r e v a l u a t i n g an a p p r o x i m a t i o n f o r the s a t u r a t i o n segment i from i - 1 50 2 . 5 - A F u n c t i o n G(s) f o r d i f f e r e n t s a t u r a t i o n segments 52 2 .5 -B A s s o c i a t e d f u n c t i o n s to G(s) f or d i f f e r e n t s a t u r a t i o n segments 52 2.6 A s s o c i a t e d f u n c t i o n s to F l ( s ) to F6(s ) f o r d i f f e r e n t s a t u r a t i o n segments 55 2 . 7 - A Study of reduced order a p p r o x i m a t i o n s : f u n c t i o n F l ( s ) 55 2 .7 -B Study of reduced order a p p r o x i m a t i o n s : f u n c t i o n F2( s ) 56 - i x -FIGURE . PAGE 2 . 7 - C Study of reduced order a p p r o x i m a t i o n s : f u n c t i o n F3( s ) 56 2 . 8 - A Study of reduced order a p p r o x i m a t i o n s : f u n c t i o n F4(s ) 61 2 .8 -B Study of reduced order a p p r o x i m a t i o n s : f u n c t i o n F5(s ) 61 2 . 8 - C Study of reduced order a p p r o x i m a t i o n s : f u n c t i o n F6( s ) 62 2 . 9 - A Study of the e f f e c t of the t r a n s f o r m e r terms: f u n c t i o n F l ( s ) 62 2 .9 -B Study of the e f f e c t of the t r a n s f o r m e r terms: f u n c t i o n F2( s ) 65 2 . 9 - C Study of the e f f e c t of the t r a n s f o r m e r terms: f u n c t i o n F3( s ) 65 2 .10 - A Study of the e f f e c t of the t r a n s f o r m e r terms: f u n c t i o n F4( s ) 66 2 .10-B Study of the e f f e c t of the t r a n s f o r m e r terms: f u n c t i o n F5( s ) 66 2 .10 -C Study of the e f f e c t of the t r a n s f o r m e r terms: f u n c t i o n F6( s ) 67 3.1 L i n e a r i z a t i o n of the s a t u r a t i o n curve 70 3.2 E q u i v a l e n t c i r c u i t of the synchronous machine f o r method 2 f o r the c o n s i d e r a t i o n of s a t u r a t i o n 73 3 . 2 - A D-axi s 73 3 . 2 - B Q - a x i s 74 4.1 Flow diagram f o r the implementat ion of method 1 i n the EMTP 88 4.2 Flow diagram for the implementat ion of method 2 i n the EMTP 90 4.3 C i r c u i t and machine data used f o r t e s t i n g the model 92 4.4 Comparison between methods 1 and 2 for s a t u r a t i o n : f i e l d c u r r e n t 93 4.5 Comparison between methods 1 and 2 f o r s a t u r a t i o n : power angle 93 - X -FIGURE PAGE 4.6 Comparison between methods 1 and 2 f o r s a t u r a t i o n : e l e c t r i c a l power output 94 4.7 Comparison between methods 1 and 2 f o r s a t u r a t i o n : v o l t a g e i n the d and q a x i s 94 4.8 Comparison between methods 1 and 2 f o r s a t u r a t i o n : c u r r e n t i n the d and q a x i s 95 4.9 Comparison between methods 1 and 2 for s a t u r a t i o n : angu lar speed 95 4 .10-A E l e c t r i c a l power a f t e r opening a l i n e : m a n u f a c t u r e r ' s data 97 4 .10-B E l e c t r i c a l power a f t e r opening a l i n e : data e s t imated by O n t a r i o Hydro from SSFR 97 4.11 Comparison between d i f f e r e n t i n p u t data : a c t i v e power 98 4 .12-A F i e l d c u r r e n t a f t e r opening a l i n e : m a n u f a c t u r e r ' s data 100 4 .12-B F i e l d c u r r e n t a f t e r opening a l i n e : data e s t imated by O n t a r i o Hydro from SSFR 100 4.13 Comparison between d i f f e r e n t i n p u t d a t a : f i e l d c u r r e n t 101 4.14 Comparison between d i f f e r e n t i n p u t d a t a : f i e l d c u r r e n t ( m a n u f a c t u r e r ' s parameters ) 102 4.15 Comparison between d i f f e r e n t i n p u t d a t a : power ang le 102 4.16 Comparison between d i f f e r e n t input d a t a : a n g u l a r speed 103 4.17 Frequency response f o r Lambton genera tor u s i n g d i f f e r e n t i n p u t d a t a : Xd(s) 106 4.18 Frequency response f o r Lambton genera tor us ing d i f f e r e n t i n p u t d a t a : Xq(s ) 106 4.19 Frequency response f o r Lambton g e n e r a t o r us ing d i f f e r e n t i n p u t d a t a : G(s) 107 4.20 Frequency response f o r Nant i coke g e n e r a t o r u s i n g d i f f e r e n t i n p u t d a t a : Xd(s) 107 4.21 Frequency response f o r Nant icoke genera tor u s i n g d i f f e r e n t i n p u t d a t a : Xq(s) 108 - x i FIGURE PAGE 4.22 Frequency response for Nant icoke genera tor u s i n g d i f f e r e n t i n p u t d a t a : G(s) 108 4.23 Comparison between d i f f e r e n t i n p u t data f o r Nant i coke u n i t : e l e c t r i c a l torque 109 4.24 Comparison between d i f f e r e n t i n p u t data f o r Nant i coke u n i t : power angle 109 4.25 Comparison between d i f f e r e n t i n p u t data f o r Nant icoke u n i t : c u r r e n t d and q a x i s 110 5.1 B a s i c a l g o r i t h m of PSS/ED 114 5 .1 -A E q u i v a l e n t c i r c u i t f o r the m o d e l l i n g of the machine 118 5.2 Flow diagram f o r the implementat ion of the model i n PSS/ED 119 5 .3 -A F u n c t i o n s used f o r the v a l i d a t i o n of the s t a b i l i t y program : d a x i s 121 5 .3 -B F u n c t i o n s used f o r the v a l i d a t i o n of the s t a b i l i t y program : q a x i s 121 5.4 V a l i d a t i o n of the method used i n PSS /ED: power angle 122 5.5 V a l i d a t i o n of the method used i n PSS /ED: mechanica l and e l e c t r i c a l power 122 5.6 V a l i d a t i o n of the method used i n PSS /ED: f i e l d v o l t a g e ( E f ( J ) and c u r r e n t ( I f d ) 123 5.7 V a l i d a t i o n of the method used i n PSS /ED: t e r m i n a l c u r r e n t and v o l t a g e 123 5.8 T e s t of the reduced order model wi th and without c o r r e c t i o n : power angle 126 5.9 Tes t of the reduced order model wi th and wi thout c o r r e c t i o n : e l e c t r i c a l power 126 5.10 Tes t of the reduced order model wi th and wi thout c o r r e c t i o n : f i e l d c u r r e n t I c . 127 t a 5.11 Tes t of the reduced order model wi th and wi thout c o r r e c t i o n : t e r m i n a l c u r r e n t and v o l t a g e 127 5.12 E v a l u a t i o n of the e f f e c t of t r a n s f o r m e r terms: power angle 129 - x i i FIGURE PAGE 5.13 E v a l u a t i o n of the e f f e c t of t r a n s f o r m e r terms: e l e c t r i c a l power 129 5.14 E v a l u a t i o n of the e f f e c t of t r a n s f o r m e r terms: v o l t a g e i n the d and q - a x i s 130 5.15 E v a l u a t i o n of the e f f e c t of t r a n s f o r m e r terms: c u r r e n t i n the d and q - a x i s 130 5.16 E v a l u a t i o n of the e f f e c t of t r a n s f o r m e r terms: back-swing i n the power angle 131 5.17 E f f e c t of u s i n g a l a r g e i n t e g r a t i o n step i n a s p e c i a l l y des igned e x c i t e r model 134 A2.1 E x c i t e r b lock diagram 145 A2.2 Governor b lock diagram 146 A5.1 E q u i v a l e n t c i r c u i t for the d - a x i s t a k i n g i n t o account unequal f l u x l i n k a g e s 159 A5.2 E q u i v a l e n t c i r c u i t wi thout the s e r i e s branch 159 - x i i i -LIST OF MAJOR SYMBOLS SYMBOLS C. i n t e g r a t i o n cons tant from the i m p l i c i t c o n v o l u t i o n , number i e 2.718281828 D mechanica l damping c o e f f i c i e n t d s u b s c r i p t denot ing the d i r e c t a x i s f f requency (Hz) f r a t e d frequency ( 60 Hz i n examples) F l ( s ) - F6 ( s ) f u n c t i o n s used to model the machine G(s) s t a t o r to f i e l d t r a n s f e r f u n c t i o n H^(t) term that i s dependent on past va lues I c u r r e n t I , s t a t o r phase c u r r e n t s a, b, c 1^ d i r e c t a x i s s t a t o r c u r r e n t I^j f i e l d c u r r e n t 1^^ d i r e c t a x i s damper winding c u r r e n t 1^ quadrature a x i s damper winding c u r r e n t I quadrature a x i s s t a t o r c u r r e n t I machine t e r m i n a l c u r r e n t j complex o p e r a t o r -1 J moment of i n e r t i a L i n d u c t a n c e L . . s e l f - i n d u c t a n c e of winding i 1 1 6 L . . mutual i n d u c t a n c e between windings i J and j L j d i r e c t a x i s s t a t o r to r o t o r mutual induc tance - x i v -quadrature a x i s s t a t o r to r o t o r mutual induc tance d i r e c t a x i s synchronous induc tance d i r e c t a x i s t r a n s i e n t i n d u c t a n c e d i r e c t a x i s s u b t r a n s i e n t i n d u c t a n c e d i r e c t a x i s o p e r a t i o n a l i n d u c t a n c e . f i e l d leakage i n d u c t a n c e f i e l d s e l f - i n d u c t a n c e g - c o i l l eakage i n d u c t a n c e d i r e c t a x i s damper winding leakage induc tance d i r e c t a x i s damper winding s e l f induc tance s t a t o r leakage i n d u c t a n c e quadrature a x i s damper winding leakage i nductance quadrature a x i s damper winding s e l f -induc tance q u a d r a t u r e a x i s synchronous i n d u c t a n c e quadrature a x i s o p e r a t i o n a l i n d u c t a n c e a po le s u b s c r i p t denot ing the quadrature a x i s s t a t o r winding r e s i s t a n c e per phase f i e l d winding r e s i s t a n c e d - a x i s damper winding r e s i s t a n c e q - a x i s damper winding r e s i s t a n c e L a p l a c e o p e r a t o r L a p l a c e t r a n s f o r m a t i o n P a r k ' s t r a n s f o r m a t i o n - X V -e l e c t r i c a l torque mechanica l torque d - a x i s t r a n s i e n t s h o r t - c i r c u i t time cons tant d - a x i s s u b t r a n s i e n t s h o r t - c i r c u i t time cons tant d - a x i s t r a n s i e n t o p e n - c i r c u i t time cons tant d - a x i s s u b t r a n s i e n t o p e n - c i r c u i t time cons tant s t a t o r p h a s e - t o - n e u t r a l v o l t a g e d i r e c t a x i s s t a t o r v o l t a g e f i e l d v o l t a g e q u a d r a t u r e a x i s s t a t o r v o l t a g e machine t e r m i n a l v o l t a g e p o s i t i o n of the r o t o r r e l a t i v e to a f i r e f e r e n c e smal l change machine power angle or load angle i n t e g r a t i o n step s i z e angu lar speed r a t e d angu lar speed . magnetic f l u x f l u x which l i n k s the s t a t o r d - a x i s wi ndi ng f l u x which l i n k s the s t a t o r q - a x i s wi nd i ng f l u x which l i n k s the f i e l d winding f l u x which l i n k s the damper d - a x i s wi ndi ng - XVI -^. f l u x which l i n k s t h e damper q - a x i s w i n d i n g TT 3. 1415926 - x v i i -ACKNOWLEDGEMENTS I would l i k e to express my s i n c e r e g r a t i t u d e to a l l those p e r s o n s who i n one way or a n o t h e r h e l p e d me t h r o u g h o u t t h i s work and p a r t i c u l a r l y , To Dr H.W. Dommel, f or h i s i n v a l u a b l e h e l p , d i r e c t i on,and a d v i c e . To Dr Jose M a r t i , f o r i n t r o d u c i n g me to the b e a u t i f u l f i e l d o f f r e q u e n c y - d e p e n d e n c e m o d e l l i n g . To L u i s M a r t i , f o r h i s v a l u a b l e s u g g e s t i o n s and d i s c u s s i ons . To my wife Paloma, f o r her he lp i n many aspec t s of t h i s p r o j e c t , but most of a l l f o r her love and p a t i e n c e . To my f a t h e r and Mae, f o r p r o o f r e a d i n g the o r i g i n a l m a n u s c r i p t . To E l e c t r i f i c a c i o n d e l Caron i ( EDELCA ) , f or t h e i r s u p p o r t . To Joaqu in Da S i l v a , f o r g i v i n g me the t ime , the t r u s t , and the means to f i n i s h t h i s work. To The Fundac ion Gran M a r i s c a l de Ayacucho, f or t h e i r f i n a n c i a l s u p p o r t . To my mother and m o t h e r - i n - l a w , f o r h e l p i n g me put the whole t h i n g t o g e t h e r . To my s i s t e r Mercedes , f or drawing those b e a u t i f u l d iagrams . - 1 -1) INTRODUCTION In r e c e n t y e a r s , the e l e c t r i c power i n d u s t r y has become one o f the most t e c h n o l o g i c a l l y complex e n t e r p r i s e s of our t i m e . In t h i s i n d u s t r y , the r o l e of the computer ranges from the v e r y f i r s t s t a g e s o f the d e s i g n of e l e c t r i c ne tworks up to the o p e r a t i o n of e x i s t i n g systems. T h e r e f o r e , t h e a r t o f d e v e l o p i n g adequate computer models f o r the d i f f e r e n t components i n the ne twork and f o r t h e i r i n t e r a c t i o n s h a s o c c u p i e d t h e m i n d s and h e a r t s of many e n g i n e e r s i n t h e p a s t . The r e s u l t s o f t h i s e f f o r t a r e the very s o p h i s t i c a t e d a n a l y i c a l t o o l s which we have today . In t h i s d i s s e r t a t i o n , our o b j e c t i v e i s t o a d v a n c e the s t a t e - o f - t h e - a r t i n m o d e l l i n g one o f t h e most i m p o r t a n t components of the network, namely the synchronous machine. T h e s y n c h r o n o u s m a c h i n e w i t h i t s d y n a m i c s , h a s t r a d i t i o n a l l y been m o d e l l e d o n l y i n s t a b i l i t y s i m u l a t i o n s , where the o b j e c t i v e i s to d e t e r m i n e whether the network can o r c a n n o t w i t h s t a n d a p e r t u r b a t i o n w i t h o u t i t s m a c h i n e s l o s i n g s y n c h r o n i s m among t h e m s e l v e s , as w e l l as to e v a l u a t e the e f f e c t o f d i f f e r e n t c o n t r o l sys tems and s t r a t e g i e s . In t h e s e s i m u l a t i o n s , the t ime e l a p s e d of the event s t u d i e d i s g e n e r a l l y of a few s e c o n d s and t h e n e t w o r k i s assumed to r e m a i n b a s i c a l l y a t n o m i n a l f r e q u e n c y . T h e m i n i m u m i n t e g r a t i o n s tep that i s g e n e r a l l y used i s h a l f a c y c l e . C o n s e q u e n t 1 y , f o r t h i s type of s i m u l a t i o n s , one should t r y to have models v a l i d f o r a f requency range from 0 to 10 or 15 - 2 -Hz at most. A n o t h e r t y p e o f a n a l y s i s , n a m e l y t h e a n a l y s i s o f e l e c t r o m a g n e t i c t r a n s i e n t s , has r e c e n t l y become more and more i m p o r t a n t . As the sys tem v o l t a g e s have i n c r e a s e d above the 230 KV l e v e l , i n s u l a t i o n d e s i g n has become d e p e n d e n t on o v e r v o l t a g e s which are genera ted i n t e r n a l l y , as f o r example , d u r i n g s w i t c h i n g and i n i t i a t i o n o f f a u l t s . In lower v o l t a g e s y s t e m s , p r o t e c t i o n a g a i n s t l i g h t n i n g i s t h e p r e d o m i n a n t c o n c e r n . As the phenomena to be a n a l y z e d g e n e r a l l y i n v o l v e t r a v e l l i n g waves and are no l o n g e r b a l a n c e d , the network must be m o d e l l e d as a t h r e e - p h a s e s y s t e m and the l i n e s must be r e p r e s e n t e d as d i s t r i b u t e d - p a r a r a e t e r e l ements . The f r e q u e n c i e s o f i n t e r e s t i n t r a n s i e n t s t u d i e s a r e u s u a l l y i n the o r d e r of KHz or h i g h e r and the time span under s t u d y i s u s u a l l y i n the o r d e r of a few c y c l e s . T h e r e f o r e i n t h i s t y p e o f s i m u l a t i o n , the mach ine can be m o d e l l e d as a v o l t a g e s o u r c e b e h i n d an i m p e d a n c e , s i n c e the d y n a m i c s of f l u x changes a r e too slow to come i n t o the p i c t u r e . But some phenomena which f a l l somewhere i n between t h o s e s t u d i e d i n t r a n s i e n t s i m u l a t i o n s a n d s t a b i l i t y s i m u l a t i o n s h a v e p r o d u c e d t h e n e e d f o r i n c l u d i n g f u l l m o d e l s o f t h e s y n c h r o n o u s machine i n t r a n s i e n t s i m u l a t i o n p r o g r a m s . One of s u c h phenomenon i s the s u b s y n c h r o n o u s r e s o n a n c e , i n which t h e r e a r e v e r y l i g h t l y damped o s c i l l a t i o n s i n the m a c h i n e ' s s h a f t , i n d u c e d by the e l e c t r i c a l system to which the machine i s c o n n e c t e d . O t h e r examples i n w h i c h the m a c h i n e must be m o d e l l e d a c c u r a t e l y are the e v a l u a t i o n of the s e c o n d a r y arc - 3 -c u r r e n t d u r i n g s i n g l e - p h a s e r e c l o s i n g , and l o a d r e j e c t i o n s t u d i e s . In t h e t y p e s of s t u d i e s i n d i c a t e d a b o v e , t h e m a c h i n e model s h o u l d be as a c c u r a t e as p o s s i b l e , a n d , g e n e r a l l y , i t must be v a l i d f o r the frequency range from 0 to 100 Hz. In s u m m a r y , i t i s i m p o r t a n t to be a b l e t o m o d e l t h e synchronous machine and the a c c u r a c y of the models needed i s a f u n c t i o n of the type of s tudy to be u n d e r t a k e n . T h e r e f o r e , i t i s d e s i r a b l e to have a m a c h i n e model w h i c h i s f l e x i b l e e n o u g h t o a d a p t i t s a c c u r a c y to t h e t y p e o f s t u d y to be u n d e r t a k e n , and thus a v o i d u n d e r - m o d e l l i n g or o v e r - m o d e l l i n g the machine . An i m p o r t a n t c o n c e r n when m o d e l l i n g any d e v i c e i s the d a t a , and i t has been r e c o g n i z e d t h a t the s o - c a l l e d "s tandard d a t a " f o r s y n c h r o n o u s m a c h i n e o f t e n y i e l d u n s a t i s f a c t o r y r e s u l t s [ 1 , 2 , 3 ] . C o n s e q u e n t l y , a s i g n i f i c a n t e f f o r t has been made by t h e i n d u s t r y t o d e v e l o p more a d e q u a t e t e s t i n g t e c h n i q u e s , among w h i c h , the most s u c c e s s f u l one seems to be the frequency response measurements. In t h i s d i s s e r t a t i o n , a model w i l l be p r e s e n t e d which uses e s s e n t i a l l y the same p r i n c i p l e s t h a t were s u c c e s s f u l l y used i n m o d e l l i n g the f r e q u e n c y - d e p e n d e n c e o f t r a n s m i s s i o n l i n e s p a r a m e t e r s [ 5 ] . With t h i s t e c h n i q u e , f r e q u e n c y r e s p o n s e s are u s e d t o s y n t h e s i z e an impulse response w h i c h c a n be c o n v o l u t e d n u m e r i c a l l y w i t h an i n p u t t o f i n d t h e t i m e r e s p o n s e o f the o u t p u t . I t w i l l be shown t h a t t h i s a p p r o a c h - A -not o n l y u t i l i z e s the b e s t d a t a a v a i l a b l e f o r the phenomena be ing s t u d i e d , i n c l u d i n g f requency response measurements, but i t makes i t p o s s i b l e to c h a n g e the d e t a i l i n w h i c h the machine i s to be m o d e l l e d . 2) OBJECTIVES AND PHILOSOPHICAL PRECEPTS As m e n t i o n e d i n the I n t r o d u c t i o n , the o b j e c t i v e of t h i s d i s s e r t a t i o n i s t o make a c o n t r i b u t i o n i n t h e a r e a o f s y n c h r o n o u s m a c h i n e m o d e l l i n g . T h i s i s done f a c i n g two a s p e c t s of the p r o b l e m ; namely , the need f o r a f a s t e r mode l , i n which the machine i s on ly mode l l ed i n as much d e t a i l as i s r e q u i r e d , and the need f o r an a c c u r a t e m o d e l , i n which the best data a v a i l a b l e i s u sed . T h r o u g h o u t t h i s d i s s e r t a t i o n , t h e f o l l o w i n g g e n e r a l p h i l o s o p h i c a l p r e c e p t s were f o l l o w e d : * The product of t h i s r e s e a r c h p r o j e c t shou ld he lp to s o l v e p r a c t i c a l problems . * The m o d e l s h o u l d be u n d e r s t a n d a b l e and e a s y to u s e . * T h e n u m e r i c a l m e t h o d s t o be u s e d s h o u l d be f a s t and r e l i a b l e . * T h e r e s u l t s o f t h e r e s e a r c h s h o u l d be e a s y t o implement i n e x i s t i n g programs. - 5 -CHAPTER 1 BASIC THEORY OF THE SYNCHRONOUS MACHINE 1.1) P h y s i c a l D e s c r i p t i o n of the Synchronous Machine A s y n c h r o n o u s m a c h i n e c o n s i s t s e s s e n t i a l l y o f two e l e m e n t s , the r o t o r and the s t a t o r ( s e e f i g . 1 . 1 ) . In the s t a t o r or a r m a t u r e , t h e r e are t h r e e w i n d i n g s (phases a , b and c ) d i s p l a c e d 120 degrees a p a r t . These w i n d i n g s , when fed wi th a ba lanced set of c u r r e n t s , (the machine is assumed "ideal" [24]) i = I s i n ( w t ) (a) a m i . = I sin(u>t - 1 2 0 ° ) (b) D in i = I sin(ujt + 1 2 0 ° ) (c ) c m 1.1 p r o d u c e a r o t a t i n g m a g n e t o m o t i v e f o r c e MMFa w i t h c o n s t a n t a m p l i t u d e i n the a i r gap, whose a n g u l a r speed i s the same as that of the c u r r e n t s . On the r o t o r t h e r e i s one w i n d i n g , known as t h e f i e l d w i n d i n g , which when fed w i t h dc c u r r e n t , produces a c o n s t a n t m a g n e t o m o t i v e f o r c e MMFf s t a t i o n a r y w i t h r e s p e c t to the r o t o r . I f the r o t o r s p i n s at an a n g u l a r speed u) equal to to , as i s t h e c a s e i n t h e s y n c h r o n o u s m a c h i n e s , t h e m a g n e t i c f l u x e s produced by these two MMF w i l l become s u p e r i m p o s e d . I f the f l u x due to the r o t o r l e a d s the f l u x due to the s t a t o r , t h e n we have a g e n e r a t o r o p e r a t i o n , i . e . , t h e r o t o r f i e l d p u l l s t h e s t a t o r f i e l d and power i s t r a n s f e r r e d f rom the - 6 -FIGURE 1 . 1 : P h y s i c a l r e p r e s e n t a t i o n of the synchronous machine B ^ a * ^ b ' c M a g n e t i c f i e l d i n t h e a , b , a n d c w i n d i n g s , p u l s a t i n g i n t i m e a t co a n d c o n s t a n t i n d i r e c t i o n . B B r . a 1 n : R e s u l t i n g f i e l d l- B ^ + B c o n s t a n t t ime ancf r o t a t i n g a t : E x c i t a t i o n f i e l d e t o t h e f i e l d w i n d i n g , c o n s t a n t i n t ime r o t a t i n g at to d u FIGURE 1 . 2 : R e l a t i o n s h i p between the d and q a x i s  m a K i i e c o m o t i ve f o r c e s Vd "S 4' oils - 7 -former to the l a t t e r . C o n v e r s e l y , i f the s t a t o r f l u x l eads the r o t o r f l u x then we have motor o p e r a t i o n . 1.2) D i f f e r e n t i a l E q u a t i o n s of the Synchronous Machine In o r d e r to w r i t e the se t of d i f f e r e n t i a l e q u a t i o n s which c h a r a c t e r i z e the s y n c h r o n o u s m a c h i n e , i t i s a d v i s a b l e to c o n s i d e r i t as a set of m u t u a l l y c o u p l e d w i n d i n g s , much l i k e a t r a n s f o r m e r . T h e f o l l o w i n g s e t of d i f f e r e n t i a l e q u a t i o n s must be s a t i s f i e d f o r t h e s t a t o r w i n d i n g s , p h a s e s a , b and c : v = - r i - d \J> (a) a — J a v b - " r H " 4-*b ( b ) dt v = - r i - d 4> (c ) c C d i C 1.2 where the c u r r e n t i s d e f i n e d p o s i t i v e l e a v i n g the s t a t o r . For the winding i n the r o t o r or f i e l d w i n d i n g , we have: v f = - r i f - d_J, 1 1 dt 1 1.3 where a g a i n , the c u r r e n t i s d e f i n e d p o s i t i v e l e a v i n g the f i e l d w i n d i n g . In a l l the se e q u a t i o n s v^ i s the v o l t a g e a p p l i e d to the w i n d i n g i , i ^ i s t h e c u r r e n t t h r o u g h i t and ty^ i s t h e f l u x which l i n k s i t . T h i s f l u x i s g iven by: * a - L a a *a + L a b d b + L a c * c + L a f * f ( a ) *b = L b a 4 a + L b b i b + L b c d c + L b f * f ( b ) - 8 -*c * L c a *a + L c b "b + L c c *c + L c f * f <c> * f = L f a "a + L f b "b + L f c ic + L f f - f ( d ) 1.4 where L . . i s the s e 1 f - i n d u c t a n c e o f w i n d i n g i and L . . i s n i J the mutual i n d u c t a n c e between windings i and j . T h e s e e q u a t i o n s f u l l y d e s c r i b e t h e b e h a v i o u r o f the s y n c h r o n o u s m a c h i n e w i t h t h e e x c e p t i o n o f t h e d a m p e r w i n d i n g s , which w i l l be i n t r o d u c e d l a t e r . The d i f f e r e n t i a l e q u a t i o n s a r e e x t r e m e l y d i f f i c u l t to s o l v e as they s t a n d , s i n c e i n a l l o f them, the m u t u a l and s e 1 f - i n d u c t a n c e s a r e ( w i t h t h e e x c e p t i o n o f L f £ ) f u n c t i o n s o f t h e p o s i t i o n o f t h e r o t o r and h e n c e , t i m e . T h i s c a n be v e r i f i e d i f we c o n s i d e r t h a t depending on the r o t o r p o s i t i o n (see f i g . 1 .1 ) , the r e l u c t a n c e of the d i f f e r e n t magnet ic paths v a r i e s from a maximum, when the r o t o r i s a t r i g h t a n g l e to the p a t h , to a minimum, when r o t o r and path are a l i g n e d . So, we have f o r the s e l f i n d u c t a n c e s : (a) (b) (c) 1.5 L aa = L + L cos s m (26) L b b = L + L cos s m (2 6 - 2 4 0 ° ) L cc = L + L cos s m (2 6 + 2 4 0 ° ) e mutua l s : ab = L, = - M ba s - L m cos (29 + L ac = L = - M ca s - L m cos (26 + L. be = L , = - M cb s - L m cos (26 -L a f = M f cos(20) L b f = cos(28 - 1 2 0 ° ) L c f = M f cos(26 + 1 2 0 ° ) )°) . „ o (a) (b) (c) (d) (e) ( f ) - 9 -wi th OJ t + 6 ( 8 ) 1.6 T h i s p r o b l e m i s s o l v e d i n the l i t e r a t u r e by d e f i n i n g two f i c t i t i o u s w i n d i n g s d and q , s t a t i o n a r y w i t h r e s p e c t to the r o t o r , whose added e f f e c t i s to p r o d u c e a r o t a t i n g MMF e q u a l to t h a t p r o d u c e d by the o r i g i n a l s t a t o r w i n d i n g s (see f i g . 1 . 2 ) . The t r a n s f o r m a t i o n tha t p e r m i t s the c o n v e r s i o n of the s t a t o r c u r r e n t s and v o l t a g e s i n t o the a p p r o p r i a t e d and q q u a n t i t i e s i s c a l l e d P a r k ' s T r a n s f o r m a t i o n . I t s b a s i s can be d e d u c e d f r o m g e o m e t r i c a l c o n s i d e r a t i o n s i n f i g u r e 1 . 2 . In o r d e r to make the t r a n s f o r m a t i o n r e v e r s i b l e and to a l l o w f o r the p o s s i b i l i t y o f r e c o v e r i n g the o r i g i n a l phase q u a n t i t i e s , a t h i r d w i n d i n g h a s t o be i n c l u d e d . T h i s a d d i t i o n a l w i n d i n g i s i s o l a t e d f r o m t h e f i c t i t i o u s w i n d i n g s d and q , and c o r r e s p o n d s to t h e z e r o s e q u e n c e i n symmetr i ca l components. The r e q u i r e m e n t s i n d i c a t e d above do not u n i q u e l y d e f i n e the t r a n s f o r m a t i o n ; hence , t h e r e are s e v e r a l ways i n which i t can be f o r m u l a t e d . The one chosen here i s power i n v a r i a n t and i t s e t s the q u a d r a t u r e a x i s "q" l a g g i n g 9 0 ° w i t h r e s p e c t to the d i r e c t a x i s "d" [ 6 ] , T h i s t r a n s f o r m a t i o n i s g iven by [T] = K c o s ( B ) c o s ( B - 1 2 0 ° ) c o s ( $ + 1 2 0 ° ) s i n ( B ) s i n ( B - 1 2 0 ° ) s i n ( B + 1 2 0 ° ) 1//2 1//2" 1/ y/T (a) where K = • 2 / 3 B = 6 + wt + 90 ( (b) (c) - l o -rn - i = [T] (d) t ransposed 1.7 so tha t [V dqo ] = [T] [V abc ] (a) [ i dqo ] = [T] [ i abc (b) 1.8 Wi th t h i s t r a n s f o r m a t i o n , i t i s p o s s i b l e to c o n v e r t e q s . 1.2 (a) to 1.2 (c ) to t h e i r e q u i v a l e n t i n " d , q , 0 " , as the new set of v a r i a b l e s i s known. So f a r i t has been assumed t h a t the o n l y c u r r e n t i n the r o t o r i s t h e f i e l d c u r r e n t . T h i s a s s u m p t i o n i s n o t n e c e s s a r i l y v a l i d d u r i n g t r a n s i e n t s , when t h e r e c o u l d be i n d u c e d c u r r e n t s i n a l l c l o s e d c i r c u i t s i n the r o t o r . These c l o s e d c i r c u i t s are the s q u i r r e l - c a g e bars ( i n s t a l l e d i n the r o t o r to a i d t h e d a m p i n g o f o s c i l l a t i o n s ) and t h e eddy c u r r e n t paths i n the i r o n . The i n d u c e d c u r r e n t s a r e n o r m a l l y t a k e n i n t o a c c o u n t by t h e i n c l u s i o n o f two e q u i v a l e n t w i n d i n g s , c a l l e d damper w i n d i n g s , a l i g n e d a l o n g the d and q a x i s . In c o n v e n t i o n a l models , the damper w ind ings are g e n e r a l l y r e p r e s e n t e d , w i t h r e l a t i v e s u c c e s s , by a number o f c o n s t a n t p a r a m e t e r c o i l s c o n n e c t e d i n p a r a l l e l [ 1 , 4 , 7 ] , However , as n o t e d i n [ 3 ] , i n l a r g e r m a c h i n e s w i t h m a s s i v e i r o n r o t o r s , s i g n i f i c a n t d i s c r e p a n c i e s may o c c u r between r e s u l t s from f i e l d t e s t s and those p r e d i c t e d by these models . The r e a s o n f o r t h e s e d i s c r e p a n c i e s i s t h a t the a c t u a l i n d u c t a n c e s a n d r e s i s t a n c e s o f the damper w i n d i n g s a r e a f f e c t e d by the s k i n e f f e c t ; t h e r e f o r e , t h e i r v a l u e i s not - 11 -c o n s t a n t , but i s a f u n c t i o n of the f r e q u e n c y c o n t e n t of the t r a n s i e n t . T h i s e f f e c t i s most n o t i c e a b l e i n l a r g e s o l i d i r o n r o t o r m a c h i n e s , b u t i t i s a l w a y s p r e s e n t e v e n i n s a l i e n t po le machines . In t h i s d e r i v a t i o n , the parameters a s s o c i a t e d w i t h these w i n d i n g s w i l l be w r i t t e n a s L . , a n d r , , f o r t h e ° k d k d i n d u c t a n c e and r e s i s t a n c e of the damper w i n d i n g s i n the d a x i s , and and f o r the q a x i s . The model d e v e l o p e d i n t h i s d i s s e r t a t i o n accounts f o r the f requency-dependence of the damper w i n d i n g s , but i t assumes t h a t on ly the r e s i s t a n c e and the l eakage p a r t of the s e l f i n d u c t a n c e of these windings are f u n c t i o n s o f the f r e q u e n c y . T h i s a s sumpt ion i s n e c e s s a r y f o r the c o n s i d e r a t i o n o f s a t u r a t i o n , but i t i s a r e a s o n a b l e o n e , s i n c e t h e s k i n e f f e c t o n l y a f f e c t s t h e i n t e r n a l d i s t r i b u t i o n of c u r r e n t and f l u x . For the d e r i v a t i o n of the new m o d e l , i t i s u s e f u l to w r i t e t h e d i f f e r e n t i a l e q u a t i o n s of the s y n c h r o n o u s machine a s : v , = - r i , - d <K - 0) (a) d a d "dt ° q v r i - d TJ> + w ij;, a q . 77 q o d q   ^ (b) - v f = - r f i f - d _ j K (c) r r dt 0 = - r, , i . , - d_4v A <d) kd kd kd 0 = - r. i , - d ik (e) kq kq ^ r k q - v = - r i - L o d i ( f ) o o o r z o dt K 9 - 12 -where *d - L d *d + M d f * f + M dkd J k d ( a ) 4> = L i + M . i , (b) q q q qkq kq *f " L f * f + M d f *d + M f k d *kd ( C ) \ d - L k d *kd + M d k d *d + M f k d d f ( d ) 1.10 I n t h e s e e q u a t i o n s , i t was n e c e s s a r y to n e g l e c t t h e f r e q u e n c y - d e p e n d e n c e of the damper w i n d i n g s , but i t w i l l be r e c o v e r e d l a t e r when the e q u a t i o n s a r e t r a n s f o r m e d to the f r e q u e n c y domain . A l s o , i n e q u a t i o n s 1.9 (a) and 1.9 ( b ) , i t was assumed that the r o t o r speed i s cons tant OJ = OJ . T h i s r o i s a r e a s o n a b l e a s s u m p t i o n and i t i s g e n e r a l l y a c c e p t e d i n the m o d e l l i n g of the synchronous machine. To c o m p l e t e t h i s s e t of d i f f e r e n t i a l e q u a t i o n s , the m e c h a n i c a l system must be i n c l u d e d . T h i s system i s d e s c r i b e d by the swing e q u a t i o n : J df 9 6 + Du = Te - Tm d t 2 1.11 where J : Moment of i n e r t i a D : M e c h a n i c a l damping c o e f f i c i e n t w : Angu lar speed Te : E l e c t r i c a l Torque Tm : M e c h a n i c a l torque (5 : Load ang le as de f i n ed i n f i g . 1.2 - 13 -1.3) E q u i v a l e n t C i r c u i t s of the Synchronous Machine To b e t t e r u n d e r s t a n d the d i f f e r e n t i a l e q u a t i o n s of the synchronous machine and to ease t h e i r m a n i p u l a t i o n , i t i s u s e f u l to r e p r e s e n t them as two e q u i v a l e n t c i r c u i t s t h a t s u m m a r i z e t h e r o t o r d y n a m i c s i n t h e d a n d q a x i s . The f i r s t s tep i n d e v e l o p i n g these e q u i v a l e n t c i r c u i t s i s to t r a n s f o r m e q s . 1 . 9 , 1.10 and 1.11 to an a p p r o p r i a t e p e r -u n i t s y s t e m . The p e r - u n i t s y s t e m c h o s e n h e r e [8] uses the mach ine r a t i n g as the base v a l u e s f o r the s t a t o r ' s d and q w i n d i n g s . From t h e s e base v a l u e s , t h e c o r r e s p o n d i n g base v a l u e s f o r the r o t o r are found i n a c c o r d a n c e w i t h the t u r n s r a t i o b e t w e e n t h e r o t o r and t h e s t a t o r . T h i s s e l e c t i o n i n s u r e s t h a t i n p e r - u n i t , a l l the mutual i n d u c t a n c e s between the w ind ings i n one a x i s are equa l among t h e m s e l v e s , p r o v i d e d t h a t the magnet ic f l u x t h a t l i n k s two windings a l s o l i n k s the t h i r d . In p r a c t i c e t h i s i s only an a p p r o x i m a t i o n , as there i s some l e a k a g e i n the a i r gap and the damper and f i e l d wind ing t e n d t o be more c l o s e l y c o u p l e d among t h e m s e l v e s [ 7 ] , C o n s e q u e n t l y , the mutua l i n d u c t a n c e s between t h e s e w i n d i n g s s h o u l d be s l i g h t l y h i g h e r ( s e e f i g 1 . 3 ) . To t a k e t h i s d i f f e r e n c e i n t o a c c o u n t , an a d d i t i o n a l l e a k a g e i n d u c t a n c e 1 ^ £ i s i n c l u d e d i n the m o d e l . T h i s i n d u c t a n c e i s added to t h e d - a x i s common m u t u a l i n d u c t a n c e L , i n o r d e r to a d i n c r e a s e the mutual i n d u c t a n c e between the r o t o r w i n d i n g s , as i n d i c a t e d below. U s i n g the p e r - u n i t system sugges ted i n [ 8 ] , we have f o r - 14 -the s e l f and mutual i n d u c t a n c e s of the d - a x i s : , . M , = M . , = L , (a) dkd df ad M P 1 . = L . + 1. f ( b ) fkd ad kf L d = L a d + X a ( ° L k d " L a d + X k f + 2 k d ( d ) L f " L a d + X k f + V ( f ) 1.12 where 1 i s t h e l e a k a g e i n d u c t a n c e of t h e i w i n d i n g s . For the q - a x i s , we have i n p e r - u n i t : M . - L (a) qkq aq L = L + 1 (b) q aq a L. = L + 1 , (c ) kq aq kq 1.13 E q u a t i o n s 1.9 remain the same i n per—unit , but w i th : ^d - X a *d + L a d ^ f + *kd + V ( 3 ) * f * * f j f + L a d ( i f + d k d + i d ) + *fk ( i f + i k d ) ( b ) * k d - *kd ±kd + L a d ( i f + *kd + V + X f k ( i f + ' k d * ( c ) \ " X a \ + L a q ( 1 k q + V ^ 4>, = 1, i , + L ( i , + i ) ( e ) kq kq kq aq kq q ' 1.14 From e q u a t i o n s 1 .12 to 1 . 1 4 , t o g e t h e r w i t h t h e r o t o r e q u a t i o n s 1.9 ( c ) to 1.9 ( e ) , i t i s p o s s i b l e to d e r i v e the e q u i v a l e n t c i r c u i t f o r the "q" and "d" a x i s shown i n f i g u r e s 1.4a and 1 . 4 b , where the f r e q u e n c y - d e p e n d e n c e o f the damper w i n d i n g p a r a m e t e r s has been s t r e s s e d by w r i t i n g them as f u n c t i o n s of w ; i . e . , the c i r c u i t i s on ly f u l l y d e f i n e d for a g iven w . These two e q u i v a l e n t c i r c u i t s , t o g e t h e r w i t h the s t a t o r - 15 -e q u a t i o n s 1.9 ( a ) and 1.9 (b) and the swing e q u a t i o n 1 .11 , p r o v i d e enough i n f o r m a t i o n f o r the m o d e l l i n g of the machine . 1.4) S o l u t i o n of the Machine D i f f e r e n t i a l E q u a t i o n s in the  Frequency Domain The 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 d e v e l o p e d i n the p r e v i o u s s e c t i o n i g n o r e d the f r e q u e n c y - d e p e n d e n t behav iour of the damper w i n d i n g s , but t h i s can be e a s i l y r e c o v e r e d i f we r e a l i z e t h a t t h e e q u i v a l e n t c i r c u i t s a r e s t i l l v a l i d i f 1 , , , r . j , 1. a n d r , a r e n o t c o n s t a n t b u t a r e k d k d k q k q f u n c t i o n s o f f r e q u e n c y ( t h i s was s t r e s s e d i n f i g 1.4 by w r i t i n g t h e s e p a r a m e t e r s as a f u n c t i o n ofco) . However the r e s u l t i n g d i f f e r e n t i a l e q u a t i o n s are e x t r e m e l y d i f f i c u l t to s o l v e a s t h e y a r e . One m e t h o d o f s o l v i n g t h e m i s to a p p r o x i m a t e t h e damper w i n d i n g s as a g r o u p o f c o n s t a n t p a r a m e t e r c o i l s c o n n e c t e d i n p a r a l l e l , whose c o m b i n e d f r e q u e n c y b e h a v i o u r i s s i m i l a r to t h a t o f the o r i g i n a l w i n d i n g s . In a r e c e n t w o r k by O n t a r i o H y d r o [ 4 ] , t h e p a r a m e t e r s o f an e q u i v a l e n t c i r c u i t , which c o n t a i n e d two or t h r e e of t h e s e w i n d i n g s i n p a r a l l e l , were found by a d j u s t i n g t h e f r e q u e n c y b e h a v i o u r o f t h e m o d e l t o t h e m e a s u r e d f requency re sponse of the machine . T h i s g e n e r a l i d e a of us ing i d e n t i f i c a t i o n t e c h n i q u e s led to the method p r e s e n t e d i n t h i s d i s s e r t a t i o n , but as w i l l be shown, i n t h i s method, there i s no need f o r c h o o s i n g a g i v e n number of c o i l s i n p a r a l l e l , s i n c e the o r d e r of the model can be v a r i e d , depend ing on the case to be ana lyzed and on the response to be matched. - 16 -FIGURE 1.3 : Schematic r e p r e s e n t a t i o n of the f l u x l i n k a g e  path between the windings i n the d - a x i s . otr gap L ad FIGURE 1.4 : E q u i v a l e n t c i r c u i t s o f the s y n c h r o n o u s machine 1 L a d.td 1*1, f M * k d ( w ) 'kd<w> k<jcto) F i e . 1.4a F i g 1.4b - 17 -A f i r s t s t e p i n d e v e l o p i n g the new model i s to use the L a p l a c e t r a n s f o r m a t i o n i n t h e d i f f e r e n t i a l e q u a t i o n s p r e s e n t e d i n t h e p r e v i o u s p a r a g r a p h . T h i s can be e a s i l y a c h i e v e d when the t ime v a r i a b l e s i n v o l v e d have z e r o i n i t i a l c o n d i t i o n s . I f they are not z e r o , the i n i t i a l c o n d i t i o n s w i l l a p p e a r as p a r t o f t h e s o l u t i o n i n the f r e q u e n c y d o m a i n . A g e n e r a l way to a c h i e v e z e r o i n i t i a l c o n d i t i o n s i s to d e f i n e a new set of v a r i a b l e s ftg(t) as f o l l o w s : L e t g ( t ) a n d g Q ( t ) be t h e i n i t i a l s t e a d y s t a t e v a l u e s of the v a r i a b l e g, i n p o s i t i v e and n e g a t i v e sequence s y m m e t r i c a l c o m p o n e n t s , and l e t g ( t ) be i t s v a l u e at any g i v e n time t a f t e r the p e r t u r b a t i o n . T h e n , i f we d e f i n e A g ( t ) as Ag ( t ) = g ( t ) - g o + ( t ) - g o _ ( t ) (a) then Ag ( t ) = 0 at t=0, (b) 1.15 which i s the r e q u i r e m e n t we want the new v a r i a b l e to f u l f i l l . U s i n g t h i s change of v a r i a b l e s i n the e q u a t i o n s g i v e n i n the p r e v i o u s p a r a g r a p h , i t i s e v i d e n t that the form of the e q u a t i o n s i s n o t a l t e r e d . T h u s , u s i n g t h e L a p l a c e t r a n s f o r m a t i o n i n the v a r i a b l e s and e q u a t i o n s w h i c h d e f i n e t h e e q u i v a l e n t c i r c u i t s o f f i g u r e 1 . 4 , we can f i n d the c o r r e s p o n d i n g e q u i v a l e n t c i r c u i t s i n the f requency or L a p l a c e domain ( see f i g . 1-5 ) . In t h e s e c i r c u i t s c a p i t a l l e t t e r s are used to denote the transformed v a r i a b l e s i . e . : G(s) = L{ Ag ( t ) } . 1.16 - 1 8 -One of the main advantages o f t r a n s f o r m i n g the v a r i a b l e s to the f r e q u e n c y domain i s t h a t a l g e b r a i c m a n i p u l a t i o n s can be used to d e t e r m i n e r e l a t i o n s h i p s between the i m p o r t a n t v a r i a b l e s . S o , from the c i r c u i t i n f i g . 1 . 5 , i t i s p o s s i b l e to d e r i v e , a f t e r some a l g e b r a , the f o l l o w i n g r e l a t i o n s h i p s : \ ^ s > l l d ( s ) = 0 - G<S> V S> ( b ) U) Q ¥ q ( s ) = X q ( g ) J q C 8 ) ( c ) 1 . 1 7 and u s i n g s u p e r p o s i t i o n i n eqs . 1 . 1 7 (a) and 1 . 1 7 ( b ) , we get w o ¥ d ( s ) = X d ( s ) I d ( s ) + G(s) V f ( s ) 1 . 1 8 I n t h e s e e q u a t i o n s , X d ( s ) a n d X ^  ( s ) a r e t h e o p e r a t i o n a l impedances of the synchronous machine and G(s ) i s the t r a n s f e r f u n c t i o n between the r o t o r and the s t a t o r . These f u n c t i o n s can be w r i t t e n i n terms of the p a r a m e t e r s o f the c i r c u i t s i n f i g . 1 . 5 , a n d , as shown i n [ 7 ] , t h e y c a n be e x p r e s s e d , i n terms of the s t a n d a r d t e s t d a t a , as i n d i c a t e d below: ( 1 + s T , ' ) ( 1 + s T ") X . ( s ) = w L , (a) d (1 + s T d o ' > + S T d o " ) ( 1 + s T , ,) w L J 6 ( s ) = ^ (b) ( 1 + s T d Q ' ) ( 1 + s T d Q " ) r f - 19 -(1 + s T ) (1 + s T ) X ( s ) = 3 9 OJ L (c ) q ( 1 + S T ' ) ( 1 + S T " ) ° Q v qo ' v qo 1.19 B u t what i s m o r e i m p o r t a n t , t h e y c a n be m e a s u r e d d i r e c t l y , as shown i n Appendix 3, without any major a s s u m p t i o n . T h e r e f o r e , these o p e r a t i o n a l i n d u c t a n c e s and t r a n s f e r f u n c t i o n s a r e not n e c e s s a r i l y t i e d to any e q u i v a l e n t c i r c u i t , and when they are m e a s u r e d , they can i n c l u d e o t h e r e f f e c t s not c o n s i d e r e d so f a r , f o r example , the s k i n e f f e c t s i n the f i e l d w i n d i n g . E q u a t i o n s 1.17 ( c ) and 1.18 summarize a l l r o t o r dynamics a s f u n c t i o n s o f I ( j ( s ) , I ( s ) a n d t h e f i e l d v o l t a g e V ^ ( s ) . To c o m p l e t e t h e d e s c r i p t i o n o f t h e m a c h i n e i n the f r e q u e n c y domain , the s t a t o r d i f f e r e n t i a l e q u a t i o n s 1.9 (a) and 1.9 (b ) must be c o n s i d e r e d . H e n c e , t r a n s f o r m i n g t h e s e e q u a t i o n s to the f r e q u e n c y domain and c o m b i n i n g the r e s u l t w i t h e q u a t i o n s 1 . 1 7 a n d 1 . 1 8 , we g e t t h e f o l l o w i n g e x p r e s s i o n s f o r the v o l t a g e i n terms of the c u r r e n t i n the s t a t o r and the f i e l d v o l t a g e V ^ ( s ) : vH ( s ) = - (s_ X . ( a ) + r ) I (a) - X (a) I (a) + s_G(s) V (s ) OJ Q Q OJ , N o o (a) V (a) = - (s_ X (a) + r ) I (a) + X d ( a ) I j ( s ) + G(s) V f ( a ) (b) 1 . 2 0 These e q u a t i o n s must be s o l v e d t o g e t h e r w i t h the network to f i n d the machine t e r m i n a l v o l t a g e and c u r r e n t , b u t , a s the v o l t a g e i s g e n e r a l l y d e t e r m i n e d by the e x t e r n a l n e t w o r k , i t - 20 -FIGURE 1.5 : E q u i v a l e n t C i r c u i t s i n the frequency domain s L kf % ( 5 ) s L ad kd(s) "kd <s> sit kq« sL aq r k q ^ s ) s F i g . 1.5a F i g l - 5 b FIGURE 1.6 : Steady s t a t e phasor diagram wi th s a t u r a t i o n 4-AXI8 - 21 -i s b e t t e r to r e f o r m u l a t e them as: I d ( s ) = F l ( s ) V d ( s ) + F 2 ( s ) V q ( s ) + F 3 ( S ) V f ( s ) (a) I q ( s ) = F 4 ( s ) V d ( s ) + F 5 ( s ) V q ( s ) + F 6 ( s ) V f ( s ) (b) 1.21 where: F l ( s ) = F3(s ) = F4( s ) = F 5 ( s ) = F6(s ) = - ( (s) + r ) co ^ o ( j L _ X (s) + r ) (_s_X (s) + r ) + X (s ) X ( s ) (a) X (s) F 2 ( s ) = 9-(_s_ X d ( s ) + r ) (_s_X (s) + r ) + X (s ) X d ( s ) W OJ 4 4 ,, N o o (b) " [ ( _ s i 2 + 1 } X a ( s ) + - § - r ] G ( s ) (i) 4 CO o o ( _ § _ X , ( s ) + r ) ( _ s X (s) + r ) + X (s) X . ( s ) CO OJ 4 4 (c) o o - V s> (_s_ X d ( s ) ' + r ) (_s_X (s) + r ) + X (s) X d ( s ) co co 4 4 (d) o o v " X (s) + r ) d CO o (_s_ X (s) + r ) ( j s X (s) + r ) + X (s) X d ( s ) U o u o ( e ) r G(s) (_s_ X d ( s ) + r ) ( _ s X (s) + r ) + X (s ) X d ( s ) % % ( f ) 1.22 T h e s e e q u a t i o n s a r e the s o l u t i o n of t h e s y n c h r o n o u s - 22 -m a c h i n e i n t h e f r e q u e n c y d o m a i n , w i t h o u t a n y m a j o r s i m p l i f y i n g a s s u m p t i o n . In the f o l l o w i n g c h a p t e r s , a procedure w i l l be d e v e l o p e d to t r a n s f o r m t h e s e e q u a t i o n s to the t ime domain and to i n c l u d e a new a s p e c t not ye t m e n t i o n e d : the s a t u r a t i o n . B e f o r e we can f i n d the s o l u t i o n i n the time domain, i t i s i m p o r t a n t t o know t h e i n i t i a l c o n d i t i o n s f r o m w h i c h t h e t r a n s i e n t s t a r t e d . T h e r e f o r e i n the n e x t s e c t i o n of t h i s c h a p t e r , a p r o c e d u r e to e v a l u a t e these i n i t i a l c o n d i t i o n s i s p r e s e n t e d . - 23 -1.5) Steady S ta te E v a l u a t i o n of the Synchronous Machine Most s i m u l a t i o n s s t a r t from a s t e a d y s t a t e c o n d i t i o n or o p e r a t i n g p o i n t . T h i s i n i t i a l c o n d i t i o n c o u l d be a b a l a n c e d o p e r a t i o n , i n w h i c h a r e p r e s e n t o n l y p o s i t i v e s e q u e n c e c u r r e n t s and v o l t a g e s , or an u n b a l a n c e d o p e r a t i o n , i n which are p r e s e n t p o s i t i v e , n e g a t i v e , and zero sequence v a r i a b l e s . I n t h i s s e c t i o n , f o r m u 1 ae f o r t h e e v a l u a t i o n o f t h e s e components w i l l be d e v e l o p e d so t h a t they w i l l comply w i t h the methods and assumptions to be used a f t e r w a r d s . 1 .5 .1 ) P o s i t i v e Sequence F o r the p o s i t i v e s e q u e n c e , the machine sees the network as a b a l a n c e d s e t of c u r r e n t s and v o l t a g e s which c r e a t e s a r o t a t i n g f i e l d i n the same d i r e c t i o n and speed as i t s r o t o r . T h e r e f o r e , the v o l t a g e s and c u r r e n t s i n the e q u i v a l e n t d and q w i n d i n g s must be d c . T h i s can be p r o v e d by u s i n g P a r k ' s t r a n s f o r m a t i o n eq . 1.7 i n : cos(cot + 9) = /T Vrms cos(ojt + 9 - 1 2 0 ° ) c o s ( u t + 9 + 1 2 0 ° ) 1.23 thus y i e l d i ng: V , V = V = /3~ Vrms cos( Vrms cos( 0 - 6 - 9 0 ° ) (a) (b) (c) 1.24 - 24 -w h e r e i t c a n be o b s e r v e d t h a t V , a n d V a r e d q c o n s t a n t v a l u e s . From t h e s e e q u a t i o n s , the f o l l o w i n g phasor r e l a t i o n s h i p can be w r i t t e n : - ( j 6 ) V + j V , = / 3 V„ e" q J d t where V = V = I Vrras | e ( j 6 * 1.25 t a 1 1 S i m i l a r l y f o r the c u r r e n t we have : I + j I , = L e ~ ( j S ) q J d t where I = I = | I r m s l e ( j a } 1.26 t a 1 1 As t h e d and q - a x i s v a l u e s a r e d c , t h e d i f f e r e n t i a l equat ions of the machine reduce t o : V , = - r I . - X I (a) d d q q V = - r I + X . I . + X . V f / r f (b) q q d d ad r r 1.27 These e q u a t i o n s , t o g e t h e r w i t h the phasor r e l a t i o n s h i p s g i v e n b e f o r e , can be used to e v a l u a t e the m a c h i n e ' s i n i t i a l c o n d i t i o n s . B u t , i f s a t u r a t i o n i s to be t a k e n i n t o a c c o u n t , a d d i t i o n a l c o n s i d e r a t i o n s must be made. In t h i s d i s s e r t a t i o n , s a t u r a t i o n i s t a k e n i n t o a c c o u n t by a p p r o x i m a t i n g t h e s a t u r a t i o n curve u s i n g s t r a i g h t l i n e segments (see 2 .4 .1 f i g . 2 . 1 ) , so t h a t f o r any g i v e n segment or s a t u r a t i o n zone , the mutual f l u x i s g iven by : i p , = ( X , . I , . + E . ) / O J (a) rmd v a d i mdi oi o - 25 -i p = ( X . I . + E . ) / w (b) mq aqi mqi 0 1 o 1.28 w h e r e X , . and X . a r e t h e s a t u r a t e d v a l u e s o f X , and a d i a q i ad X ^ c o r r e s p o n d i n g to the i*"*1 s egment , E o i i s the v a l u e o f xfj , w h e r e t h e i ^ s e g m e n t c u t s t h e Y - a x i s , a n d I , md b * md a n d I a r e t h e m a g n e t i z i n g c u r r e n t s , w h i c h , i n t h e mq & o » balanced steady s t a t e , are g iven by I , . = I . + I , (a) mdi d f I . - I (b) mqi q 1.29 For the u n s a t u r a t e d case , e q u a t i o n s 1.27 can be w r i t t e n as V = - r I + X, I . + X , ( I . + I , ) (a) q q 1 d ad v d f V , = - r I , - X. I - X I (b) d d l q a q q 1.30 w h e r e we i d e n t i f y X j ( I j + I r ) a s OJ , a n d X a d d f ' o md a q I as OJ i> h e n c e ; so we c a n r e - w r i t e e q u a t i o n s 1 . 3 0 q o m q n as V = - r I + X, I . + OJ ijj (a) q q l d o m d V, = - r I , - X, I - OJ 4» (b) d d 1 q o mq 1.31 These e q u a t i o n s must be v a l i d r e g a r d l e s s of the s a t u r a t i o n of the machine; t h e r e f o r e , r e p l a c i n g these e q u a t i o n s i n e q . 1.25 and u s i n g eq . 1.26, we can w r i t e : V , = - r I - X, L + E n e j 5 (a) t t 1 t p where E | = OJ / i f ; 2 , +4>ln I /3" (b) p 1 o md mq 1.32 - 26 -T h e s e f o r m u l a e a l l o w u s , g i v e n t h e m a c h i n e ' s t e r m i n a l c o n d i t i o n s , t o e v a l u a t e , w h i c h i s p r o p o r t i o n a l to the net f l u x i n the m a c h i n e . T h e r e f o r e the s a t u r a t i o n segment in w h i c h t h e m a c h i n e i s o p e r a t i n g i s p e r f e c t l y d e t e r m i n e d . Once t h i s i s known, we can w r i t e , f o r any segment i n p a r t i c u l a r V t " ~ r I t " j X q I t + ' -E . + X . I . ( X , - X ) I . 0 1 ad f d q d + q t ^ yj - j E .] e j 6 e l —22. •3 1.33 where: X , = K. X , ad I ado X = K. X aq I aqo K.. = s a t u r a t i o n f a c t o r of segment i E q u a t i o n 1.33 can be used to draw the phasor d i a g r a m i n f i g u r e 1 .6 , from which i t i s p o s s i b l e to show t h a t : E f = E p 2 cos(6") - ( X d - X q ) I d / /T (a) -1 X a X t + V t S i n < t > 6' = tan A ( 9 ) (b) r I t + V coscp X_ I , + V s i n cb s i n 6 E p 2 - £ (c ) , E . / / J 6" = s i n " 1 ( — ° ^ ) (d) P2 6 = a + 6' + 6" (e) 4> = 0 - a ( f ) 1.34 From the v a r i a b l e s i n these e q u a t i o n s , a l l the c u r r e n t s , - 27 -1 ^ , I a n d I j , a s w e l l a s t h e v o l t a g e s , c a n be e v a l u a t e d by: x f - ( E f f3 - E o . ) / x a d (a xd - / T l t sin(rx - 6 ) (b) I = q vT I C O S ( 0 6 - 6 ) (c) v f - r f T f (d) v„ - /3" V sin( 9 -6 ) (e) v = q /3~ V cos( 6 - 6 ) (f) 1.35 These e q u a t i o n s complete the set of i n i t i a l c o n d i t i o n s for the p o s i t i v e sequence. 1 .5 .2) Negat ive Sequence D u r i n g u n b a l a n c e d o p e r a t i o n , t h e r e are n e g a t i v e sequence c u r r e n t s and v o l t a g e s i n the machine or mach ines a f f e c t e d . The p u r p o s e o f t h i s s e c t i o n i s to show the e q u a t i o n s t h a t r e l a t e the machine w i t h the network i n the n e g a t i v e sequence . The n e g a t i v e sequence i s c h a r a c t e r i z e d by a b a l a n c e d set of c u r r e n t s and v o l t a g e s s i m i l a r to the p o s i t i v e s e q u e n c e , but where phase b l e a d s phase a and phase c l a g s . T h e r e f o r e , i t w i l l s e t a r o t a t i n g f i e l d i n the a i r gap t h a t moves i n a d i r e c t i o n o p p o s i t e to the r o t o r ' s . v = /2~ Vrms cos( DJ t +6 ) (a) v u = f l Vrms c o s ( o ) t +6 + 1 2 0 ° ) (b) b v = f l Vrms c o s ( w t +6 - 1 2 0 ° ) (c) c 1.36 - 28 -T h e s e n e g a t i v e s e q u e n c e v o l t a g e s and c u r r e n t s , when t r a n s f o r m e d to dqo u s i n g P a r k ' s t r a n s f o r m a t i o n , w i l l produce s i n u s o i d a l v a r i a b l e s w i t h t w i c e t h e n o m i n a l f r e q u e n c y . T h e r e f o r e i t i s c o n v e n i e n t at t h i s s t a g e to r e p r e s e n t these v a r i a b l e s as p h a s o r s , wi th the f o l l o w i n g d e f i n i t i o n : V. = Re {v. e ( j 2 W c>} 1.37 where i s the p h a s o r a s s o c i a t e d w i t h v ^ ( t ) . S o , we h a v e , for and V , u s i n g P a r k ' s t r a n s f o r m a t i o n i n e q . 1.36: V d ( t ) = /3~ Vrms c o s ( 2 u t + 6 + 6 - 9 0 ° ) (a) V q ( t ) - /3~ Vrms c o s ( 2 u t + 6 + 6 ) (b) 1.38 which can be w r i t t e n u s i n g eq . 1.37 as : V„ - / T V r m s e J ( 6 + 6 + 9 ° 0 ) (a) d V = vT Vrms e j ( 6 + 6 } (b) where we have t h a t : V d " J V q ( c > 1.39 T h i s p h a s o r f o r m u l a t i o n a l l o w s us to use d i r e c t l y the s o l u t i o n o f t h e d i f f e r e n t i a l e q u a t i o n s i n t h e f r e q u e n c y domain. So, making s = 2 OJ j i n eq . 1 . 2 0 , we have: V d = - ( 2 j X d ( j 2co) + r ) I d - X q ( j 2 u ) I q + 2 j G(j 2 u ) V f ( j 2u) (a) V q = - ( 2 j X q ( j 2a)) + r ) I q + X q ( j 20)) l d + G ( j 2o>) V f ( j 2o)) ( b ) 1.40 - 29 -where V f (2 u> j ) = 0 . From t h e s e e q u a t i o n s , the p h a s o r r e l a t i o n s h i p s i n e q . 1 . 3 9 , and P a r k ' s t r a n s f o r m a t i o n , i t i s p o s s i b l e to f i n d the f o l l o w i n g e x p r e s s i o n s f o r the machine n e g a t i v e sequence t e r m i n a l v o l t a g e and c u r r e n t : V Q = f~T Irms/2 [ ( A d - A q ) c o s ( 3 w t + 26 + a ) + ( B d - B q ) s i n ( 3 w t + 2 6 + a ) - ( A d + A q ) cos( w t + a ) + ( B d + B q ) s i n( w t + a ) ] (a) where A d = R e { X d ( j 2u>)} + Im{X (J 2co)} - 2 Im{X d ( j 2a))} (b) A q = R e { X q ( j 2a,)} + I m { X d ( j 2u)J - 2 Im{X q ( j 2u)J (c) B d = 2 R e { X d ( j 2o))} - R e { X q ( J 2a))} (d) B q = 2 Re{X q ( j 2u )J - R e { X d ( j 2u>)} (e) 1.41 In t h i s e q u a t i o n , i f we assume t h a t X d ( 2 o j ) = X q ( 2 w ) , w h i c h i s g e n e r a l l y t r u e , t h e n we c a n n e g l e c t t h e t h i r d harmonic and w r i t e : v = >^ Irms/2 [ - ( A , + A ) cos( oo t + a ) 3 Q (J + ( B d + B ) sin(u) t + a ) ] 1 .42 which can be w r i t t e n u s i n g phasors as : V t = - 1/2 [ A d + A q + j ( B d + B q ) ] l t 1.43 T h e r e f o r e i n t h e n e g a t i v e s e q u e n c e t h e m a c h i n e c a n be - 30 -model led as an impedance g iven by: Z 2 = - 1/2 [ A d + A q + j ( B d + B q ) ] which can be reduced t o : Z 2 = r + [ X d ( 2 0) j ) + X q ( 2 to j ) ] / 2 1.44 1.45 I f ^ d ( s ) a n d X q ( s ) a r e k n o w n , t h e n Z 2 c a n be e v a l u a t e d and t h e m o d e l t h u s p r o d u c e d f o r t h e n e g a t i v e s e q u e n c e i s c o n s i s t e n t w i t h t h e i n f o r m a t i o n u s e d f o r m o d e l l i n g the machine i n the time l o o p . 1 .5 .3) Zero Sequence There i s a d i r e c t r e l a t i o n s h i p between the z e r o sequence va lues from the symmetr i ca l components t r a n s f o r m a t i o n : v = ( v + v, + v ) / 3 osc a b c 1.46 and the zero sequence va lues from P a r k ' s t r a n s f o r m a t i o n : v = ( v + v, + v ) / /~3~ op a b c 1.47 T h e r e f o r e , the d i f f e r e n t i a l e q u a t i o n of the "o" w i n d i n g can be used d i r e c t l y to model the machine i n t h i s sequence: V = - r I - j X I n o o o o 1 .48 I f t h e m e a s u r e d v a l u e o f L i s no t k n o w n , i t can be o ' assumed equal to the s t a t o r leakage i n d u c t a n c e . - 31 -CHAPTER 2  BASIC THEORY OF THE NEW MODEL 2.1) I n t r o d u c t i o n The o b j e c t i v e o f t h i s c h a p t e r i s to c o v e r the g e n e r a l p r i n c i p l e s o f the more e l a b o r a t e models to be d e s c r i b e d i n the f o l l o w i n g c h a p t e r s . A l s o i n t h i s c h a p t e r , the t h e o r y and assumpt ions behind the most i m p o r t a n t a u x i l i a r y programs w i l l be g i v e n . 2.2) T r a n s f o r m a t i o n of The Frequency Domain E q u a t i o n s i n t o . the Time Domain In Chapter 1, the e q u a t i o n s o f the synchronous machine i n the f r e q u e n c y domain were d e v e l o p e d w i t h o u t any a s s u m p t i o n w i t h r e s p e c t to the number of w i n d i n g s needed to a c c u r a t e l y r e p r e s e n t the f r e q u e n c y - d e p e n d e n t b e h a v i o u r of the "damper w i n d i n g s " . B u t , i n o r d e r f o r the model to be p r a c t i c a l , these e q u a t i o n s i n t h e f r e q u e n c y o r L a p l a c e d o m a i n must be t rans formed to the t ime domain. T h i s w i l l be the t o p i c of the f o l l o w i n g d i s c u s s i o n . When s a t u r a t i o n i s not t a k e n i n t o a c c o u n t , or when the machine can be assumed to remain i n the same l i n e a r segment i n t h e s a t u r a t i o n c u r v e , the m a c h i n e i s d e s c r i b e d by a u n i q u e s e t o f e q u a t i o n s i n t h e f r e q u e n c y d o m a i n ( e q . - 32 -1.21 ) . These e q u a t i o n s can be t r a n s f o r m e d to the time domain u s i n g t h e i n v e r s e L a p l a c e t r a n s f o r m a t i o n , y i e l d i n g the f o l l o w i n g e q u a t i o n s : A i d = L ' ^ F K s ) } * A v d ( t ) + L _ 1 { F 2 ( s ) } * A v q ( t ) + L _ 1 { F 3 ( s ) } * 7 a v f ( t ) (a) Ai = L - 1 { F A ( s ) } * A v . ( t ) + L ' ^ F S C s ) } * A v ( t ) q a q + L _ 1 ( F 6 ( s ) } * A V f ( t ) (b) 2.1 w h e r e A i A j Av a n d Av , a r e t h e v a r i a t i o n s o f t h e a • q • q d c o r r e s p o n d i n g v a r i a b l e s w i t h r e s p e c t t o i t s i n i t i a l c o n d i t i o n s , (see e q . 1.15) and s tands for the c o n v o l u t i o n of the two v a r i a b l e s . To p e r f o r m the c o n v o l u t i o n s i n d i c a t e d i n the e q u a t i o n s a b o v e , a method s i m i l a r to the one employed by M a r t i i n [ 5 ] , was u s e d . T h i s m e t h o d i s b a s e d on t h e f a c t t h a t t h e c o n v o l u t i o n of any a r b i t r a r y f u n c t i o n o f t i m e g ( t ) w i t h an e x p o n e n t i a l can be found n u m e r i c a l l y u s i n g a r e c u r s i v e formula (see Appendix 1 ) : S ( t ) = k e " p t * g ( t ) = S ( t ) = b S( t - A t ) + c g ( t ) + d g ( t - A t ) 2.2 where b , c and d are c o n s t a n t s . T h e r e f o r e , i f f u n c t i o n s F l ( s ) to F6 ( s ) cou ld be approximated by: n K . . F j ( s ) = K + I U j = 1 , 2 , 3 . . . 6 ° i = l (s + P i j ) 2.3 - 33 -whose i n v e r s e t r a n s f o r m i s a sum of e x p o n e n t i a l s : L _ 1 { F j ( s ) } cr K 6 ( t ) + I K i j e " ( p i j t ) ° i = l 2.4 where 6 ( t ) i s t h e i m p u l s e o r D i r a c ' s d e l t a , t h e n we can u s e t h i s " r e c u r s i v e c o n v o l u t i o n m e t h o d " to e v a l u a t e t h e c o n v o l u t i o n s i n e q s . 2.1 and t r a n s f o r m t h e s e e q u a t i o n s i n t o a l g e b r a i c e q u a t i o n s . The r e s u l t i n g e q u a t i o n s r e l a t e t h e v a l u e s o f t h e c u r r e n t s and v o l t a g e s a t any g i v e n t i m e t ( t n = n A t ) , among t h e m s e l v e s , and w i t h t h e v a l u e s t h e s e v a r i a b l e s assumed i n the p r e v i o u s time s t e p s : A V ' n * = C l A v d ( t n > + C 2 + C 3 " A v f ( t n > + H l ( t n ) + H 2 ( t n ) + H 3 ( t n ) (a) A - i q ( t n ) = C 4 A v d ( t n ) + C 5 A v q ( t n ) + C 6 A v f ( t n ) + H A ( t n ) + H 5 ( t n ) + H 6 ( t n ) (b) where to are c o n s t a n t s g i v e n by and H ^ ( t n ) to H ^ ( t n ) are the past h i s t o r y t e r m s : W = J i ^ j ( t n " A t ) + b i j S i j ^ „ " A t ) ( d ) " b i j S i j ( t n " A t ) + C i j ^ t n > A v k ( t n > + d . .Av. (t - At) (e) i j k v n 2.5 In the r e m a i n i n g of t h i s d i s s e r t a t i o n , a n d , f o r the sake of s i m p l i c i t y , whenever an i n t e g r a t i o n e q u a t i o n l i k e eq . 2.5 i s - 34 -w r i t t e n , " t " w i l l s t a n d f o r t h e d i s c r e t e v a r i a b l e " t n " . The two e q u a t i o n s g i v e n above r e p r e s e n t the s o l u t i o n f o r t h e d and q a x i s w i n d i n g s ; t h e r e f o r e , t o f i n i s h the e l e c t r i c a l m o d e l , i t i s n e c e s s a r y to take the z e r o sequence i n t o a c c o u n t . In o r d e r to do s o , t h e d i f f e r e n t i a l e q u a t i o n 1.9 ( f ) must be s o l v e d and t h i s can be done u s i n g any n u m e r i c a l t e c h n i q u e . In t h i s d i s s e r t a t i o n , t h e t r a p e z o i d a l r u l e [9] was c h o s e n b e c a u s e i t i s h i g h l y s t a b l e , and i t p r o d u c e s an i m p l i c i t i n t e g r a t i o n formula much l i k e e q u a t i o n s 2 .5 : A o M = C o V o ( t ) + H o ( t ) ( a ) C = -1 / ( r + 2 L / At) (b) o o o where H = C (r - 2 L / A t ) i ( t - A t ) + C v (t - A t ) (c ) o o o o o o o 2.6 E q u a t i o n s 2 .5 to 2 .6 must be s o l v e d a t e v e r y t ime s t e p wi th the network e q u a t i o n s ; t h e r e f o r e , t h e s e e q u a t i o n s need to be t r a n s f o r m e d t o p h a s e q u a n t i t i e s a , b , c u s i n g P a r k ' s t r a n s f o r m a t i o n . L a t e r , i n c h a p t e r 4 , s p e c i f i c formulae w i l l be g i v e n , but f o r now i t i s i m p o r t a n t to r e a l i z e t h a t i n o r d e r to u s e t h i s t r a n s f o r m a t i o n i t i s n e c e s s a r y to know the p o s i t i o n o f the r o t o r at each t ime s t e p , i.e. 8 ( t ) = w t + 6 + IT/2 , h e n c e , t h e swing e q u a t i o n must be s o l v e d ( E q . 1 .11 ) . In the d e v e l o p e d p r o g r a m s , the swing e q u a t i o n i s s o l v e d u s i n g a p r e d i c t o r - c o r r e c t o r a p p r o a c h . In t h i s m e t h o d , the v a l u e s f o r 8 ( t ) and w ( t ) are p r e d i c t e d u s i n g D a h l ' s f o r m u l a - 35 -[ 9 ] , the e l e c t r i c a l system i s s o l v e d , and then the v a l u e s of t h e p r e d i c t e d v a r i a b l e s a r e r e c a l c u l a t e d u s i n g t h e t r a p e z o i d a l r u l e a s t h e c o r r e c t o r . I f t h e r e i s no c o n v e r g e n c e , the e l e c t r i c a l system i s r e c a l c u l a t e d , u s i n g the most recen t v a l u e s , u n t i l the convergence i s r e a c h e d . Once t h e e l e c t r o m e c h a n i c a l s e t of e q u a t i o n s i s s o l v e d a n d i , , i , v . , a n d v a r e k n o w n , i t i s p o s s i b l e to d q d q r e v a l u a t e the mutual f l u x i n the d and q a x i s as w e l l as the c u r r e n t s i n the r o t o r c i r c u i t s i n the f o l l o w i n g way: The i n v e r s e t r a n s f o r m a t i o n s o f e q s . 1.17 ( c ) and 1.18 g i v e s : A ^ q ( t ) = L - 1 { X q ( s ) } * A i q ( t ) 7 OJ q (a) A * d ( t ) = ( L _ 1 { X d ( s ) } * A i d ( t ) + L - 1 { G ( s ) } A V f ( t ) ) / * q (b) 2.7 where a g a i n " * " means the c o n v o l u t i o n of the two v a r i a b l e s . S o , i f L ( s ) , L d ( s ) and G ( s ) a r e a l s o a p p r o x i m a t e d by r a t i o n a l f u n c t i o n s , then the same p r o c e d u r e o u t l i n e d b e f o r e can be u s e d , t h u s , t r a n s f o r m i n g e q u a t i o n s 2.7 i n t o a l g e b r a i c d i f f e r e n c e e q u a t i o n s much l i k e e q u a t i o n s 2.5 : A ^ q ( t ) = CH>q A i q ( t ) + H * q ( t ) (a) AiJ» d ( t ) = C ^ q A i d ( t ) + HiC d ( t ) (b) 2.8 w h e r e H ^  ( t ) a n d H il» ( t ) a r e t h e c o r r e s p o n d i n g p a s t h i s t o r y terms . T h e s e e q u a t i o n s c a n be u s e d t o e v a l u a t e ij> . and 4> H d q u s i n g i , , i and v f w h i c h a r e k n o w n , a n d t h e n t h e m u t u a l - 36 -f l u x can be found by *md = * d ( t ) ' X a (a) (b) 2.9 T o f i n d t h e f i e l d c u r r e n t i ^ , t h e d i f f e r e n t i a l e q u a t i o n 1.9 ( c ) must be s o l v e d . T h i s d i f f e r e n t i a l e q u a t i o n can be w r i t t e n as : where a l l the v a r i a b l e s a r e known except the f i e l d c u r r e n t . H e r e a g a i n , i t i s p o s s i b l e t o s o l v e f o r i ^ ( t ) u s i n g any n u m e r i c a l t e c h n i q u e . I n t h i s d i s s e r t a t i o n , t h i s e q u a t i o n was s o l v e d u s i n g t h e L a p l a c e t r a n s f o r m a t i o n a n d t h e r e c u r s i v e c o n v o l u t i o n , p r o d u c i n g the f o l l o w i n g i n t e g r a t i o n f o r m u l a e . From eq . 2 .10: v f ( t ) = r f i f ( t ) + l f d i f ( t ) + L a d d i m d ( t ) 2.10 i f ( t ) = ad md CO) i o e - ( r f / l f ) t _ Lad + md ( t ) 2.11 i f we l e t S f ( t ) = C f ( v f ( t ) + _ r _ ^ L a d i m d ( t ) ) + H f ( t ) 2.12 - 37 -be an a p p r o x i m a t i o n to the c o n v o l u t i o n above, then i f ( t ) = S f ( t ) + tt To 2 . 1 3 where: 1 . d < t > = ( * m d ( t ) - E d o 1 " o >  L a d 2.14 I t i s i m p o r t a n t to n o t i c e t h a t i n e q u a t i o n 2 .10 the f a c t was i g n o r e d t h a t i n some c a s e s , the m u t u a l f l u x t h a t l i n k s t h e s t a t o r i s n o t t h e same one t h a t l i n k s t h e r o t o r , as i n d i c a t e d i n [7] and i n C h a p t e r 1. These e q u a t i o n s can take t h i s e f f e c t i n t o a c c o u n t i f , i n s t e a d of u s i n g the a c t u a l p a r a m e t e r s f o r t h e f i e l d w i n d i n g , t h e s e p a r a m e t e r s a r e m o d i f i e d a c c o r d i n g to the p r o c e d u r e i n d i c a t e d i n Appendix 5. 2.3) Approx imat ion by R a t i o n a l F u n c t i o n s In the p r e v i o u s s e c t i o n . i t became obv ious tha t one of the most i m p o r t a n t r e q u i r e m e n t s of the method p r e s e n t e d i s tha t t h e o p e r a t i o n a l i m p e d a n c e s X ^  ( s ) a n d X ( s ) , a n d t h e f u n c t i o n s G ( s ) and F l ( s ) to F 6 ( s ) , are to be a p p r o x i m a t e d by r a t i o n a l f u n c t i o n s . In t h i s s e c t i o n , the p r o c e d u r e used f o r the a p p r o x i m a t i o n of these f u n c t i o n s i s p r e s e n t e d , as w e l l as the main m o d i f i c a t i o n s of the o r i g i n a l M a r t i ' s method t h a t had to be i n t r o d u c e d i n o r d e r to a d a p t t h i s method to the m o d e l l i n g of synchronous machine. 2 . 3 . 1 ) B r i e f d e s c r i p t i o n of the a p p r o x i m a t i o n method. The a p p r o x i m a t i o n method used i n t h i s d i s s e r t a t i o n to - 38 -s y n t h e s i z e a r a t i o n a l f u n c t i o n which " f i t s " a g iven frequency b e h a v i o u r ( i . e . , so that t h e i r r e sponse i s the same w i t h i n a t o l e r a n c e ) was d e v e l o p e d by M a r t i i n h i s work [ 5 ] , T h i s method i s used here w i t h o u t any major m o d i f i c a t i o n and the main d i f f e r e n c e s w i t h r e s p e c t of the o r i g i n a l method a r e i n the p r e - p r o c e s s i n g and the p o s t - p r o c e s s i n g of the f u n c t i o n s , before and a f t e r the f i t t i n g . The M a r t i ' s method to d e t e r m i n e the p o l e s and z e r o s tha t make t h e a p p r o x i m a t i n g r a t i o n a l f u n c t i o n i s b a s e d on the c o n s t r u c t i o n o f a Bode p l o t w h i c h r o u g h l y m a t c h e s t h e f u n c t i o n to be a p p r o x i m a t e d . From t h i s p l o t , a f i r s t e s t imate of the va lues of the poles and zeros can be d e r i v e d . S u b s e q u e n t l y , the p o l e s and z e r o s are moved i n f r e q u e n c y , so t h a t t h e e r r o r between t h e i r a s s o c i a t e d f u n c t i o n and the a c t u a l f u n c t i o n to be a p p r o x i m a t e d i s m i n i m i z e d . T h i s e r r o r i s then compared w i t h the d e s i r e d t o l e r a n c e and i f i t i s not a c c e p t a b l e , t h e n t h e p r o g r a m g o e s b a c k t o t h e B o d e c o n s t r u c t i o n a l g o r i t h m to f i n d a new Bode p l o t which more c l o s e l y matches the o r i g i n a l f u n c t i o n . In f i g u r e 2 . 1 , the p r o c e d u r e f o l l o w e d i n the c o n s t r u c t i o n of the Bode p l o t can be o b s e r v e d . The M a r t i ' s m e t h o d d e s c r i b e d a b o v e g i v e s v e r y good r e s u l t s i f the f u n c t i o n to be approximated i s "minimum phase" , i . e . a l l i t s z e r o s are i n the l e f t hand s i d e of the c o m p l e x p l a n e , so t h a t i t s phase t a k e s the minimum v a l u e a s s o c i a t e d w i t h a g i v e n m a g n i t u d e . F o r a good f i t t i n g , the f u n c t i o n should a l s o be as smooth as p o s s i b l e . T h i s l a s t FIGURE 2.1 : Method f o r a l l o c a t i n g the po le s and zeros from Bode p l o t F i r s t attempt to p o l e - z e r o a l l o c a t i o n a f t e r which the l o c a l e r r o r i s e v a l u a t e d and i f l a r g e r than a g iven t o l e r a n c e f u r t h e r s u b d i v i -s i o n c o n t i n u e s . Second s u b d i v i s i o n of Zone I I Error - 40 -r e q u i r e m e n t i s n o t f u l f i l l e d f o r f u n c t i o n s F l to F6 ( see f i g . 2 . 2 ) , and, t h e r e f o r e , they have to be m a n i p u l a t e d before they can be a p p r o x i m a t e d . In the next s e c t i o n , t h i s aspect of the method w i l l be d i s c u s s e d . 2 .4) C o r r e c t i o n of the F u n c t i o n s to be Approximated In t h i s s e c t i o n , the s e v e r a l c o r r e c t i o n s of the measured o r e v a l u a t e d f u n c t i o n s t h a t c h a r a c t e r i z e t h e m a c h i n e a r e p r e s e n t e d . Some of these c o r r e c t i o n s are n e c e s s a r y , as i t was m e n t i o n e d b e f o r e , t o make F l ( s ) t o F 6 ( s ) as s m o o t h as p o s s i b l e . H o w e v e r , a n o t h e r i m p o r t a n t c o r r e c t i o n p r e s e n t e d here i s the i n c l u s i o n of s a t u r a t i o n e f f e c t s . 2 . 4 . 1 ) I n c o r p o r a t i o n of S a t u r a t i o n E f f e c t s In the model d e v e l o p e d so f a r , s a t u r a t i o n was n e g l e c t e d , o r i t was i n d i c a t e d t h a t i t i s p o s s i b l e to a s s o c i a t e t h i s m o d e l to a l i n e a r s e g m e n t i n t h e s a t u r a t i o n c u r v e . No e x p l a n a t i o n o f t h i s and the u n d e r l y i n g a s s u m p t i o n s has been g i v e n , t h e r e f o r e i n t h i s s e c t i o n , t h i s a s p e c t w i l l be addressed. I n t h i s d i s s e r t a t i o n t h e f o l l o w i n g a s s u m p t i o n s i n c o n n e c t i o n with s a t u r a t i o n are made: a ) O n l y t h e m u t u a l i n d u c t a n c e s L , a n d L 3 a d a q s a t u r a t e , i . e . , the l e a k a g e path are not a f f e c t e d by s a t u r a t i on. b) The q u a d r a t u r e i n d u c t a n c e L does no t s a t u r a t e ' ^ a q i n t h e c a s e o f t h e s a l i e n t p o l e m a c h i n e s and i t - 41 -70 601 50 401 40 I F I D U R E 2 .2 -A1 : A P P R O X I M A T I O N O F F l TO F3 F O R O N T A R I O H. D E N M O D U L E O F T H E F U N C T I O N S *0 r3 351 301 251 201 151 30. 20 10 I -10 I 301 101 -20 + -30 + 01 -40 J -- i — i i 11 m i 1—i i 11 m i 1—i i 11 m i 1—i t » i n n 1 — t i l l i n O r i g i n a l -5 + -50 + Oj. -10J -60 o'.ooi1 1 '""o'.oio1 1 '""fe'.ioo1 1 '""I'.ooo1 1 ' "" I ID FRO tHZ) FI DURE 2 . 2 - A 2 « APPROXIMATION OF F l TO F3 FOR ONTARIO H . DEN ANOLE OF THE FUNCTIONS -604-- 1 2 0 -180. -240 -300+ -360 FRO (HZ) - 42 -FIGURE 2.2-&1 i APPROXIMATION OF F4 TO F6 FOR ONTARIO H. DEN MODULE OF THE FUNCTIONS FRO (HZ) FIGURE 2.2-B2 : APPROXIMATION OF F4 TO F6 FOR ONTARIO H. DEN ANGLE OF THE FUNCTIONS FRO (HZ) - 43 -s a t u r a t e s i n the same amount as L , i n round r o t o r a d machi nes . c ) T h e s a t u r a t i o n c u r v e i s i n d e p e n d e n t o f t h e l o a d . T h e r e f o r e i t i s p e r m i s s i b l e to use the open c i r c u i t s a t u r a t i o n c u r v e to e s t i m a t e the s a t u r a t e d va lues of L , and L ad aq I f these a s s u m p t i o n s are a c c e p t e d , t h e n , i n the proposed method, the open c i r c u i t s a t u r a t i o n curve i s l i n e a r i z e d i n up t o f i v e s e g m e n t s and t h e c o r r e s p o n d i n g v a l u e s of L , a r e ad i d e n t i f i e d as the s l o p e s of the segments (see f i g . 2 . 3 ) . The c o r r e s p o n d i n g v a l u e s o f L g q a r e found u s i n g a s s u m p t i o n (b) so that : L unsat L sat = — — L .sat aq T ad n L .unsat ad 2.15 O n c e t h e v a l u e s o f L , and L a r e known f o r e a c h ad a q s a t u r a t i o n s e g m e n t i , t h e m e a s u r e d f u n c t i o n s L d ( s ) and L ( s ) c a n be c o r r e c t e d t o p r o d u c e t h e c o r r e s p o n d i n g f u n c t i o n s to each segment as f o l l o w s : L d . ( s ) = l a + ( 1 / L a d . - l / L a d o + l / ( L d o ( s ) - l a ) ) " 1 (a) L q . ( s ) - l a + ( 1 / L a q . - l / L a q o + l / ( L q o ( s ) - l a ) ) ~ l (b) 2.16 w h e r e i=o c o r r e s p o n d s t o t h e u n s a t u r a t e d m a c h i n e . T h i s f o r m u l a was d e r i v e d from the c i r c u i t s i n f i g u r e 1.5 and the d e f i n i t i o n of L , ( s ) and L ( s ) . d v q F o r G ( s ) , t h e r e i s no e x a c t f o r m u l a u n l e s s one knew the FIGURE 2.3 : L i n e a r i z a t i o n of the o p e n - c i r c u i t s a t u r a t i o n curve - 45 -e q u i v a l e n t c i r c u i t p a r a m e t e r s , but i t i s p o s s i b l e to d e r i v e an a p p r o x i m a t e c o r r e c t i o n f o r m u l a i n the f o l l o w i n g way : O b s e r v e t h a t t h e e q u a t i o n s f o r X d ( s ) and X q ( s ) ( s e e e q . 1 .19 and r e f e r e n c e s [4 ] and [7 ] ) m a i n t a i n the same s t r u c t u r e f o r t h e c a s e s where the damper w i n d i n g s a r e not m o d e l l e d at a l l , or when they a r e m o d e l l e d by o n e , two or t h r e e w i n d i n g s i n p a r a l l e l . So i t can be i n f e r r e d t h a t i n g e n e r a l , the f o l l o w i n g s t r u c t u r e h o l d s : n ( i + s T ) i = i 0 3 X d ( s ) = Xd (a) .n" ( 1 + s T d o i > i = l 1 = 1 X a d G(s) = ^ _ (b) n ( i + s T. .) . , doi i = l r f 2.17 where T d . j , T ^ d ; . = Short c i r c u i t time cons tant T . . = Open c i r c u i t time c o n s t a n t s doi v T h i s o b s e r v a t i o n was a l s o noted by I.M. Canay i n a recen t work [10 ] . In e q u a t i o n s 2 . 1 7 , a s i d e from L , , the open c i r c u i t t ime a d ' r c o n s t a n t s , must be by t h e i r d e f i n i t i o n , the most a f f e c t e d by s a t u r a t i o n . T h i s can be v e r i f i e d by a n a l y z i n g the e x p r e s s i o n s a v a i l a b l e i n t h e l i t e r a t u r e f o r t h e s e c o n s t a n t s [ 3 , 7 ] . - 46 -In A p p e n d i x 6 , t h i s a s s e r t i o n was v e r i f i e d u s i n g the parameters of two d i f f e r e n t machines . I f t h e o b s e r v a t i o n and a s s u m p t i o n s s t a t e d a b o v e a r e a c c e p t e d , we have: n it (1 + s T . . ) I T t \ • i d o 1 ° T L d i ( s ) m V L L , ( s ) n L , * ( 1 + S T d o i ) ' j i = l J (a) and n JT (1 + s T k d ) L . . ( s ) L , L , . i = l L , . G ( S ) O - d - i — - a i l E = G ( S ) . L , ( s ) L j . L , n r , ^ d 0 d j a d o JT (1 + s T , . ) f (b) i = l d o 1 thus L , . ( s ) L , L , . G ( s ) . * G ( s ) o -aJ i£ _ad,L ( c ) L d o ( s ) L d j L ado 2.18 w h e r e t h e s u b s c r i p t " j " i m p l i e s t h a t t h e f u n c t i o n i s a s s o c i a t e d with s a t u r a t i o n segment " j " . T h i s e q u a t i o n can be used to e s t i m a t e the v a l u e o f G ( s ) a f f e c t e d by s a t u r a t i o n ( G ( s ) j ) f r o m m e a s u r e d u n s a t u r a t e d va lues ( G ( S ) Q ) . O n c e L , . ( s ) , L . ( s ) and G . ( s ) a r e k n ow n f o r e a c h s egment " j " i n t h e open c i r c u i t s a t u r a t i o n c u r v e , t h e i r c o r r e s p o n d i n g t r a n s f e r f u n c t i o n s F l ( s ) to F6 ( s ) can be e v a l u a t e d . From t h e s e f u n c t i o n s , i t i s p o s s i b l e to g e n e r a t e the c o r r e s p o n d i n g model u s i n g the f i t t i n g procedure d e s c r i b e d below. - 47 -2 .4 .2 ) A p p r o x i m a t i o n of Curves In o r d e r to u s e t h e i m p l i c i t c o n v o l u t i o n t e c h n i q u e d e s c r i b e d i n A p p e n d i x 1, the f u n c t i o n s X ^ ( s ) ,X ( s ) ,G( s ) and F l ( s ) to F 6 ( s ) c o r r e s p o n d i n g to each s a t u r a t i o n segment have to be a p p r o x i m a t e d by r a t i o n a l f u n c t i o n s . The g e n e r a l a p p r o x i m a t i o n method used i s M a r t i ' s ( s ee s e c t i o n 2 . 3 . 1 ) , w i t h the f o l l o w i n g m o d i f i c a t i o n i n the p r e - p r o c e s s i n g and p o s t - p r o c e s s i n g of the f u n c t i o n s : 2 . 4 . 2 . 1 ) Approx imat ion of X^(s) and X q ( s ) F i g u r e 2 . 4 a shows t h a t X q ( s ) and X ^ ( s ) a r e s m o o t h f u n c t i o n s of f r e q u e n c y and t h e r e f o r e they can be approximated d i r e c t l y . T h i s a p p r o x i m a t i o n ,however , can be more e f f i c i e n t l y a c h i e v e d i f we no te t h a t the c u r v e s c o r r e s p o n d i n g to these f u n c t i o n s have the same shape for the d i f f e r e n t s a t u r a t i o n segments ( see f i g u r e 2 . 4 a ) . S i n c e t h i s b e h a v i o u r was a l s o observed i n the r e s t of the c u r v e s to be a p p r o x i m a t e d , i t was d e c i d e d to e x p l o i t i t , i n order to reduce the l a r g e amount of c o m p u t i n g t i m e t h a t had to be d e v o t e d to t h e f i t t i n g p r o c e d u r e . So i n s t e a d o f a p p r o x i m a t i n g X , . ( s ) and X . ( s ) c o r r e s p o n d i n g to s a t u r a t i o n zone i , the f o l l o w i n g a s s o c i a t e d curves are approximated (see f i g u r e 2 . 4 b ) . (X . (s . ) / X . . ( s . ) + s) do min ' d i x m m ' A X . ( s ) = X , . ( s ) u u " J (a) d 0 3 (s X . . ( s ) / X , (s ) + 1) v d i v max do max (X (s . ) / X . ( s . ) + s) AX (s) = X . (a ) 4 0 m 3 n ^ m 3 n (b) q q 3 (s X . ( s ) /X (s ) + 1) v q i v m a x y qo v max' 2.19 - 48 -where: s . = minimum va lue of s (ito) i n the i n p u t data min r s = maximum va lue of s i n the i n p u t data max T h i s t r a n s f o r m a t i o n f o r c e s a l l t h e c u r v e s t o be a p p r o x i m a t e d t o a common v a l u e a t t h e e x t r e m e o f t h e f r e q u e n c y r a n g e . T h i s f a c t i s used i n a new a u x i l i a r y f i t t i n g program i n which the v a l u e s of po les and z e r o s , found f o r the a p p r o x i m a t i o n o f s a t u r a t i o n segment 1 i n t h e main f i t t i n g p r o g r a m , are d i s p l a c e d towards the new c u r v e , c o r r e s p o n d i n g to s a t u r a t i o n segment 2 (see f i g u r e 2 . 4 c ) . The e r r o r between the c u r v e c o r r e s p o n d i n g to the new p o l e s and z e r o s and the t a r g e t c u r v e i s then e v a l u a t e d . I f i t i s l e s s than a g i v e n t o l e r a n c e , the a p p r o x i m a t i o n i s a c c e p t e d , or e l s e the new p o l e s and z e r o s a r e s h i f t e d a g a i n , f o l l o w i n g t h e same p r o c e d u r e , u n t i l c o n v e r g e n c e i s o b t a i n e d . T h i s method was f o u n d t o be v e r y f a s t and i t u s u a l l y t a k e s one o r two s h i f t i n g i t e r a t i o n s u n t i l the e r r o r i s s m a l l enough , but i n s p e c i a l c a s e s a f t e r 5 or 6 i t e r a t i o n s , the e r r o r c e a s e s to decrease and the program branches out to the o r i g i n a l f i t t i n g program. O n c e A X , . ( s ) and AX . ( s ) a r e a p p r o x i m a t e d , X , . ( s ) d i q i t r d ] and X j ( s ) then a r e found by s o l v i n g e q u a t i o n s 2 .19 f o r the c o r r e s p o n d i n g f u n c t i o n s . 2 . 4 . 2 . 2 ) A p p r o x i m a t i o n of G(s) The t r a n s f e r f u n c t i o n G(s ) goes to zero as the f r e q u e n c y goes to i n f i n i t y , a s can be observed i n f i g u r e 2 . 5 a . T h i s type o f b e h a v i o u r p r e c l u d e s the use of the method o u t l i n e d above . - 49 -FIOURE 2 . 4 f l : XOtS) AND XO(S) FOR DIFFERENT SATURATION SEGMENTS 1Q. 2 0 . 4 - 1 6 . 2 . . 1 2 . » - 1 0 . . 2 0 . . . 1 4 - • - 4 -18- -8"-- 2 2 . - - 1 2 - 2 6 - - 1 6 - " - 3 0 - - - 2 0 -)—i i 11 m i -1—i i 11 m i 1—i i 11 m i S e g m e n t 1 S e g m e n t 2 S e g m e n t 3 -1 1 I I I l l l l 1 I I I I III 3.001' 1 ""'o'.oio' 1 1 ""o'-ioo1 1 '""i.ooo1 1 1 "" I 'D 1 1 FRO (HZ) FIDURE 2.4B « ASSOCIATED FUNCTIONS TO XO(S) AND XOIS) FOR DIFFERENT SATURATION SEGMENTS 10 2 0 . / 1 6 -12 - 2 -10 m 4 0 - --13 - 2 2 - 1 2 -25 -30 - 1 6 -1 l l l l l l l l 1 l l l l l l l l 1 l l l l l l l l 1 1 I I I l l l l 1 1 I I I III S e g m e n t 1 S e g m e n t 2 S e g m e n t 3 S e g m e n t 1 S e g m e n t 2 S e g m e n t 3 •"o'.ooi1 1 ""'b'.oio' 1 "'"b'.ioo1 1 ' " " I ' . D O O 1 1 '""l'o 1 1 1 1 1 FRQ (HZ) - 50 -FIGURE 2 . 4 - C : Method f o r e v a l u a t i n g an a p p r o x i m a t i o n f o r the s a t u r a t i o n segment i from i - 1 (<»b) Av«rog« valu« of P, and Z| - 51 -B u t , i f we o b s e r v e the d i f f e r e n t f o r m u l a e a v a i l a b l e i n the l i t e r a t u r e f o r G ( s ) , i t becomes e v i d e n t t h a t i n G ( s ) , the o r d e r of the d e n o m i n a t o r i s g r e a t e r than t h e n u m e r a t o r i n one. T h e r e f o r e i f we m u l t i p l y G(s) by (s+k) ,where k s tands f o r any c o n s t a n t , then the o r d e r of the numerator in the new f u n c t i o n i s the same as that of the denominator . The t o t a l c o r r e c t i o n , i n c l u d i n g s a t u r a t i o n , used i n t h i s d i s s e r t a t i o n i s as f o l l o w s : ( G „ ( s m , „ ) / G , ( s m , r , ) + s) do AG(s) = G(s) ' ° m l " 1 m i i r (1 + T » s) (s G . ( s ) / G o ( s ) + 1) v a. max' v max' 2.20 T h e r e s u l t i n g c u r v e s c a n be o b s e r v e d i n f i g u r e 2 . 5 b . Once t h e s e c u r v e s a r e a p p r o x i m a t e d i n t h e same way as A X ^ ( s ) and A X ( s ) , t h e c o r r e s p o n d i n g t r a n s f e r f u n c t i o n f o r G ( s ) a r e found by s o l v i n g e q . 2 . 2 0 . I t i s i m p o r t a n t to n o t i c e t h a t i n e q . 2 . 2 0 , the a c t u a l v a l u e o f T , " i s not ^ do i m p o r t a n t , as t h i s t e r m i s o n l y u s e d to a i d the f i t t i n g p r o c e d u r e . 2 . A . 2 . 3 ) A p p r o x i m a t i o n of F l ( s ) to F6(s ) I t was m e n t i o n e d b e f o r e t h a t f o r a good f i t t i n g the f u n c t i o n s to be a p p r o x i m a t e d must be as smooth as p o s s i b l e ; however, t h i s i s not the case wi th F l ( s ) to F 6 ( s ) , as can be o b s e r v e d i n f i g u r e 2 . 2 . T h e r e f o r e the f i r s t s t e p i n the a p p r o x i m a t i o n of these curves i s to make them smooth. An e x a m i n a t i o n of these f u n c t i o n s shows t h a t they have a -52 -FIGURE 2.SA : FUNCTION G(S) FOR OIFFERENT SATURATION SEGMENTS  I | I I I Mil I I I I I l l l l I I I I I l l l l I I I I I l l l l I I I I I III FRO (HZ) FIGURE 2.SB : ASSOCIATED FUNCTIONS TO G(S) FOR DIFFERENT SATURATION SEGMENTS FRO (HZ) - 53 -complex p o l e at 0 J q w h i c h i s the r a t e d a n g u l a r f r e q u e n c y of t h e m a c h i n e . T h i s c a n be v e r i f i e d by n o t i n g t h a t t h e denominator of these f u n c t i o n s can be arranged i n t o : X ^ s ) + X (s) . r 2 2 • d v ' q v ; , 2 . . . s + 1 r o i s + co ( 1 + X . ( s ) X (s) ° ° X , ( s ) X (s) a q d q 2.21 where the v a l u e s of X , ( s ) and X (s) are s l owly d e c r e a s i n g and d q are very c l o s e to the s u b t r a n s i e n t v a l u e s i n the v i c i n i t y of co^ . With t h i s a p p r o x i m a t i o n around to , the f o l l o w i n g r e l a t i o n s h i p o a p p l i e s f o r a l l p r a c t i c a l cases : 2 r 1 >> X " X " d q 2.22 T h e r e f o r e , eq . 2.21 can be approximated by: X , " + X_" s " + 2 ( l M2 a — r w s + w x „ x „ o o d q 2.23 which has two complex c o n j u g a t e d r o o t s : ( s + a + j OJ ) and ( s + a - j o j ) , w i t h : f _ _ a u (a) o u>« / 1 - ( C ) 2 u)0 ( b> ( X , " + X " ) « - d q < c ) 2 V V 2.24 - 54 -To e l i m i n a t e the e f f e c t of t h i s complex p o l e i n F l ( s ) to F 6 ( s ) , these f u n c t i o n s are m u l t i p l i e d by : (s X d / U ) Q + r ) (s X q / a)o + r ) + X d X q (a) where X = x.(a> ) * X " ( b ) d d o d $ = X (a, ) * X " <c> q q o q 2.25 As we a l r e a d y have the a p p r o x i m a t i o n s f o r the numerator o f F l ( s ) t o F 6 ( s ) , a d d i t i o n a l s a v i n g i n computer t ime can be o b t a i n e d , s i n c e o n l y the denominator has to be a p p r o x i m a t e d . The a c t u a l c u r v e to be a p p r o x i m a t e d i s g i v e n by ( see f i g 2 .6) : AAF(s) = (s X . / O J + r ) (s X / O J + r ) + X , X n - i 2 3 2 i 9 SCF(s ) (s X d ( s ) / O J q + r ) (s X (s) / O J q + r ) + X d ( s ) X q ( s ) - AF( s ) SCF(s ) 2.26 where S C F ( s ) i s the f a c t o r i n t r o d u c e d to make the f u n c t i o n s the same at the ex treme of the f r e q u e n c y range o f i n t e r e s t so t h a t i t i s p o s s i b l e t o t a k e a d v a n t a g e o f t h e method d e s c r i b e d i n s e c t i o n 2 . 4 . 2 . 1 . SCF(s ) i s g iven by: (s + AF (s . ) / AF. (s . )) SCF(s ) = ° m 3 n 1 m 3 n <S A V s m a x > / A V s m a x > + 2.27 O n c e t h e a p p r o x i m a t i o n o f A A F ( s ) i s f o u n d , t h e - 55 -FIGURE 2.6 : ASSOCIATED FUNCTIONS TO FUS1 TO F61S) FOR DIFFERENT SATURATION SEGMENTS FRO (HZ) FIGURE 2.7-A I STUOY OF REDUCED ORDER APPROXIMATIONS FUNCTION F U S ) FRQ (HZ) - 56 -FIGURE 2 . 7 - B » STUDY OF REDUCED ORDERftPPROXI MAT IONS FUNCTION F 2 ( S ) 40 l l l l Mill l l l l l l l l l l l l l Mill l l l l Mill I I I I j i lt 36-L 32 28 J . 24 20 164-12 + 8 + o.oo7"TT",lb'.oiol """b'.ioo1 """I'.ooo1 """lb 1 1 1 1 1 1 1 1 FRO IHZ) FIGURE 2 . 7 - C J STUDY OF REDUCED ORDER APPROXIMATIONS r FUNCTION F31S) so - | | i 11 mi i i i i i IIII i i i 11 IIII I i i 11 IIII i i i i-wt 601. 40 201 -20 N o t -C o r r e c t e d O r i g i n a l - 3 0 ' — 1 1 '""b'.oio1 1 '""b'.ioo1 1 ""T.ooo1 1 ' " " I ' D 1 ' " " " ,001 FRO (HZ1 - 57 -c o r r e s p o n d i n g a p p r o x i m a t i o n s of F l ( s ) to F 6 ( s ) a r e o b t a i n e d as i n d i c a t e d below: F l ( s ) = - F ( s ) (s X (s ) / OJ q + r ) (a) F2 ( s ) = F ( s ) X (s ) (b) F3(s ) = - F ( s ) [ ( si + 1 ) X (s) + s_ r ] G(s) (c) OJ 4 0) o o F4(s ) = - F ( s ) X d ( s ) (d) F5(s ) = - F ( s ) (s X d ( s ) / 0 J Q + r ) (e) F6(s ) = F ( s ) r G(s) ( f ) where: AAF(s) / SCF(s ) F ( s ) = K A ; A A (8> (s X . / a) + r ) (s X / OJ + r ) + X , X d o q o d q 2.28 and X ( s ) , G ( s ) , A F F ( s ) , and X d ( s ) s t a n d f o r t h e i r r a t i o n a l a p p r o x i m a t i o n s . 2.5) Run Time Reduced Models and Compensation of Numer ica l  E r r o r s A v e r y i m p o r t a n t f e a t u r e o f the m o d e l l i n g method j u s t d e s c r i b e d i s t h a t i t a l l o w s us to a d j u s t the d e t a i l of the model to the problem to be s o l v e d , a n d , as i t was r e a l i z e d by M a r t i [ 1 1 ] , i t i s p o s s i b l e to compensate up to some e x t e n t for the n u m e r i c a l e r r o r i n c u r r e d due to d i s c r e t i z a t i o n . In t h i s s e c t i o n both a spec t s w i l l be d i s c u s s e d . - 58 -2 . 5 . 1 ) Reduct ion of the order of the model A method n o r m a l l y e m p l o y e d i n t h e l i t e r a t u r e [12] to reduce the order of the models f o r the synchronous machine i s to i g n o r e the d e r i v a t i v e s of the f l u x or t r a n s f o r m e r terms i n t h e e q u a t i o n s f o r t h e s t a t o r c i r c u i t s 1 . 9 ( a ) and 1 .9 ( b ) . T h i s a p p r o x i m a t i o n can l e a d to l a r g e e r r o r s i n t h e f r e q u e n c i e s above one to two H e r t z (see f i g . 2.9), e s p e c i a l l y i n EMTP s i m u l a t i o n s , though to a l e s s e r extent i n s t a b i l i t y s i m u l a t i o n s . N e v e r t h e l e s s , i t does not seem r i g h t to use , f o r e x a m p l e , a t i m e s t e p of h a l f a c y c l e , w h i c h i s t y p i c a l i n s t a b i l i t y s i m u l a t i o n s , and t h a t can p o t e n t i a l l y make the m o d e l v a l i d t o up t o 15 Hz ( 6 0 Hz r a t e d f r e q u e n c y ) and at the same time use a model v a l i d only up to 1 or 2 Hz . The method proposed here p r e s e r v e s the d e r i v a t i v e s of t h e f l u x and the f u l l f r e q u e n c y range i s a p p r o x i m a t e d u s i n g the method o u t l i n e d i n the p r e v i o u s s e c t i o n , thus p r o d u c i n g f o r each of the r e l e v a n t f u n c t i o n s , a r a t i o n a l e q u i v a l e n t . T h i s e q u i v a l e n t , when e x p a n d e d i n p a r t i a l f r a c t i o n s , have the f o l l o w i n g g e n e r a l form: K K* m K i F i ( s ) = + + I s + P s + P„ i = l ( s + P ) C C 3 2.29 w h e r e K £ and K c a r e c o m p l e x c o n j u g a t e d a n d some t e r m s m i g h t o r m i g h t n o t be p r e s e n t , d e p e n d i n g on t h e a c t u a l f u n c t i o n approximated by F^(s). . Once these a p p r o x i m a t i o n s are o b t a i n e d they can be used i n many s i m u l a t i o n s t h a t r e q u i r e d i f f e r e n t d e g r e e s of - 59 -a c c u r a c y . T h e r e f o r e , b e f o r e t h e s t a r t i n g o f a s p e c i f i c s i m u l a t i o n , t h e o r d e r o f the mode l i s r e d u c e d , u s i n g the f o l l o w i n g a p p r o x i m a t i o n to equat ion 2.29 : n K. m K. K k F . (s) - I - + I — 2 - + — — + — | -1 i = l (s + p. ) i - 1 P i P c P c 2.30 w h e r e " n " i s c h o s e n so t h a t a l l the p o l e s i n t h e s e c o n d s u m m a t i o n a r e g r e a t e r t h a n t h e maximum f r e q u e n c y up to which the model should be v a l i d (Fmax). One problem w i t h the method o u t l i n e d above i s , as can be seen i n f i g u r e 2 . 7 , t h a t i n some c a s e s , i t a l s o i n t r o d u c e s an e r r o r a t f r e q u e n c i e s l e s s t h a n F m a x . T h e r e f o r e some a d d i t i o n a l c o r r e c t i o n s a r e n e e d e d . A n o t h e r p r o b l e m i s the a p p r o p r i a t e s e l e c t i o n o f Fmax. Both a s p e c t s w i l l be c o v e r e d in the next s e c t i o n . 2 . 5 . 2 ) E v a l u a t i o n of the E r r o r i n the Frequency Domain and  I n t r o d u c t i o n of C o r r e c t i n g P o l e s . U s i n g the Z - t r a n s f o r m (see r e f . [ 1 3 ] ) , i t i s p o s s i b l e to e v a l u a t e the e r r o r c o m m i t t e d i n the f r e q u e n c y r a n g e due to t h e use of a g i v e n n u m e r i c a l method and i n t e g r a t i o n s t e p A t . So, l e t F ( s ) be a t r a n s f e r f u n c t i o n g iven by: V(s ) n K F ( s ) = = I -I ( s ) i = l s + p. 2.31 then the a s s o c i a t e d f u n c t i o n F ( z ) , i m p l i c i t whenever there i s n u m e r i c a l i n t e g r a t i o n , can be e v a l u a t e d as f o l l o w s : - 60 -U s i n g the n u m e r i c a l c o n v o l u t i o n o u t l i n e d i n Appendix 1, we can w r i t e : v i ( t ) = b i v . ( t - A t ) + c i i ( t ) + d i i ( t - A t ) (a) where m , v ( t ) = E v . ( t ) = L MVCs)} (b) i = l 1 i ( t ) = L _ 1 { I ( s ) } (c) 2.32 and b . t c .. and d a r e . g i v e n i n A p p e n d i x 1. T r a n s f o r m i n g equat ion 2.7 to the Z domain, we have: V. .(z) = bi z " 1 V. . (z) + c i I ( z ) + d i z " 1 I ( z ) 2.33 so m V(z ) . £ V i ( z ) m z c + d F ( z ) = 1 = 1 = I — i 1 — I ( z ) I ( z ) i = l z - b. 2.34 In f i g u r e s 2 .7 to 2 . 8 , a c o m p a r i s o n between F ( s ) , c u r v e marked as O r i g i n a l , and F ( z ) , curve marked as Not C o r r e c t e d , i s m a d e , f r o m w h i c h t h e f o l l o w i n g o b s e r v a t i o n s c a n be d e r i v e d : i - The f u n c t i o n F ( z ) i s m e a n i n g l e s s f o r f r e q u e n c i e s a b o v e Fn = 0 . 5 / A t , where Fn i s known as the Nyqu i s t f r e q u e n c y , i i - T h e r e a r e some e r r o r s b e f o r e t h e N y q u i s t frequency which degrade the model . T h e r e f o r e a good c h o i c e f o r Fmax i n the r e d u c t i o n of the o r d e r o f t h e model w o u l d be t h i s N y q u i s t f r e q u e n c y . T h i s - 61 -40 FIGURE 2.8-fl : STUDY OF REDUCED ORDER WROX IMRT10NS FUNCTION F41S) - 1 — 1 1 ) 1 l l l l 1 — l l l l l l l l 1—I I I I l l l l 1—I I I I l l l l 1—I I I I III 36 32 28 .. 24 20 16 --12 8 --4 --Not Corrected Corrected 0 1 •' 1 1 " " b ' . o i o 1 1 1 " " b ' . i o o 1 1 ' " " I ' . O O O 1 1 " " T o 1 1 """ 0.001 FRO <H2) IGURE 2.8-B : STUDY OF REDUCED ORDER APPROXIMflTIONS FUNCTION F51S) 40 "I 1 I I I l l l l 1 1 I I I l l l l 1 1 I I I l l l l 1 1 I I I l l l l 1 1 I I I III 30. 20 10 -10 --20 -30---40" > - " ' • o o i 1 ' " " ' b ' . o i o ' ' " " ' b ' . i o o ' ' '""I'.ooo1 ' ' " " l ' o 1 ' m i " FRO (HZ) - 62 -FIGURE 2 . 8 - C : STUDY OF REDUCED ORDER APPROXIMAT IONS FUNCTION F6(S) AOL I I I i i i in i i i l i nn i i i n u n i i i MINI I I ^ F r t t r r 3 0 l _ 20 10 - l o r --20 - 3 0 f Not Corrected -40 Original -5°'.ooi' 1 1 ""b'.oio1 1 ""'b'.ioo1 1 1 " " I ' . O O O 1 1 ' " " l b ' 1 1 " m FRO (HZ) FIGURE 2 . 9 - A ; STUDY OF THE EFFECT OF THE TRANSFORMER TERMS FUNCTION F U S ) 40 -1—I I I I llll 1 — l l l l llll 1 — l l l l llll 1 — l l l l llll 1—I I I I HI 3 0 1 20l 10 0 . -10+--20' -30 -40 -50 Corrected No t r e n s f . I I I I I "b', 0 1 0' ' 1 " "b 1.-— 1 1 1 1 1 1 1 1 1 — 1 1 1 1 1 1 I M I III 1 0 0 I t o 00" FRO (HZ) .001 - 63 -f r e q u e n c y was a d o p t e d a s a d e f a u l t v a l u e i n t h i s d i s s e r t a t i o n . In o r d e r to r e d u c e t h e e r r o r i n t r o d u c e d b o t h by the r e d u c t i o n i n the number o f p o l e s and by the d i s c r e t i z a t i o n , s o m e t h i n g has t o be t r a d e d : what i s p r o p o s e d h e r e i s to i n c r e a s e t h e o r d e r o f the model i n o n e , by i n t r o d u c i n g a c o r r e c t i n g term, g iven by : k' k' p' 2.35 F c ( s ) = s +  The p a r a m e t e r s of t h i s c o r r e c t i n g term are s e l e c t e d so tha t the e r r o r 1 i n the n e i g h b o u r h o o d of Fn i s m i n i m i z e d . In t h i s r e s p e c t , s e v e r a l methods f o r the s e l e c t i o n of K' and p' were t r i e d , and the one that gave the best r e s u l t s i s as f o l l o w s : i - The pole p' i s chosen to be equal to 2 nt Fmax. i i - A f i r s t guess at the va lue of K' i s o b t a i n e d by making F ( z ) p lus c o r r e c t i n g term, equal to F ( s ) at u) = 2 TT Fmax. T h i s i s ach ieved e x a c t l y max J i f : k' o ( Z - b) = (F( to ) - Fo( z )) p'Z A t — max 7 v m a x " Y , * ,., . ( 1 - b ) ( 1 - z ) max' where: i Wraax At z = e max b = e-P' A t Fo(Zmax) = O r i g i n a l f u n c t i o n without compensat ion . 2.36 - 64 -But as K ' must be r e a l , some e r r o r i s a c c e p t e d and the a b s o l u t e v a l u e of the va lue above i s taken w i t h i t s s i gn equal to the r e a l p a r t , i i i - The e r r o r between F ( s ) and F ( z ) i s e v a l u a t e d from two d e c a d e s b e l o w Fmax up to t h i s v a l u e and t h e n K' i s i n c r e a s e d or d e c r e a s e d i n s u c c e s s i v e s t e p s u n t i l a minimum e r r o r i s found or u n t i l i t becomes l e s s than a gi ven t o l e r a n c e . In f i g u r e s 2 . 7 to 2 . 8 , i t c a n be o b s e r v e d t h a t t h i s c o r r e c t i o n method i s q u i t e s a t i s f a c t o r y , s i n c e the e r r o r has been reduced s i g n i f i c a n t l y w i t h o u t i n c r e a s i n g s i g n i f i c a n t l y e i t h e r the o r d e r of the model or the o v e r a l l c o m p u t a t i o n t i m e . In f i g u r e s 2 . 9 to 2 . 1 0 , the e f f e c t s of n e g l e c t i n g the t r a n s f o r m e r terms are shown, as w e l l as how they can be taken i n t o a c c o u n t f o r the f r e q u e n c y range where the model s h o u l d be v a l i d . - 65 -I F IOURE 2 . 9 — B : S T U P Y O F T H E E F F E C T O F T H E T R A N S F O R M E R T E R M S F U N C T I O N F 2 ( S ) 40 i I I I I IIII 1—IIII IIII 1—IIII IIII 1—IIII mi 1—i i 11 III 36 + 32 I 28 4-24 20 J. 16 + 12 f O r i g i n a l 4 + o r j Ti l I I I Q ' > 0 1 0 0 i - 1 0 0 "I'.ooo' 1 '""lb 1 1 M " " o.oor F R O ( H Z ) IF I C U R E 2 . 9 - C : 5 T W f O F T H E E F F E C T O F T H E T R A N S F O R M E R T E R M S F U N C T I O N F 3 1 S ) 80 - I — l l l l llll 1 — l l l l llll 1 — l l l l llll 1 — l l l l llll 1—I I I I I 604-40 20 No t r a n s f . O r i g i n a l -20 -3oL„.i i i " " 0 i , 0 1 0 ' ' '""b'.ioo 1 1 1 ""I'.ooo 1 1 '""lb 1 1 1 001 F R O ( H Z ) - 66 -40 FIGURE 2.10-fl •.5T0DY O F T U E E f f t C T OF THE TRANSFORMER TERMS FUNCTION F4IS) -I—I I I I l l l l 1—I I I I l l l l 1 — l l l l l l l l 1 — l l l l l l l l 1—I I I I III 36 + 32 28 24 20 J . 16 + 12 + O r i g i n a l Not Corrected-No t r a n s f . Corrected o.ooi1 1 '""b'.oio' 1 '""b'.ioo1 ' '""I'.ooo1 1 ""To 1 1 ''"" FRO (HZ) |FIGURE 2.10-B:STUO* Of THE. EFFECT OF THE TRANSFORMER TERMS FUNCTION F51S1 * ° | I I I I I l l l l I I I I I l l l l I I I I I l l l l I I I I I l l l l I I I l - r t t t t 30+ 2 0 f 10+ •10+ •20+ •30+ Corrected No t r a n s f . •401* •5 y.ooi1 1 ""'b'.oio' 1 '""b'.ioo1 ' ""'I'.ooo' ' ""To 1 1 1 1 1 1 1 1 FRQ (HZ) - 67 -- 68 -CHAPTER 3  INCLUSION OF NONLINEARITIES 3.1) I n t r o d u c t i on In t h i s c h a p t e r , t h e o v e r a l l m o d e l l i n g procedure presented i n the p r e v i o u s c h a p t e r s i s r e v i s e d i n o r d e r to i n c l u d e the s a t u r a t i o n o f t h e s y n c h r o n o u s m a c h i n e . Two m e t h o d s a r e proposed h e r e . The f i r s t one i s v a l i d for the m a j o r i t y o f the s t u d i e s n o r m a l l y made i n the a n a l y s i s of power sys tems , where i t i s assumed t h a t the v o l t a g e does not v a r y s i g n i f i c a n t l y , f o r a l o n g t i m e , from the s teady s t a t e v a l u e s . T h e r e f o r e the v a r i a t i o n s i n t h e f l u x a r e l i m i t e d . The s e c o n d method was d e v e l o p e d f o r s p e c i a l c a s e s , f o r e x a m p l e , s u s t a i n e d s h o r t c i r c u i t c o n d i t i o n s , where the machine can go from a s a t u r a t e d s t a t e to a comple t e ly u n s a t u r a t e d one. 3.2 ) Method 1 for the C o n s i d e r a t i o n of S a t u r a t i o n In most power system s t u d i e s , t h e v o l t a g e do not vary very much f o r very l o n g from the s teady s t a t e v a l u e s . F o r example, d u r i n g the o c c u r r e n c e of a f a u l t , the v o l t a g e s throughout the system might be very low, but t h i s event g e n e r a l l y l a s t s only a few c y c l e s (2 to 5 c y c l e s ) , d u r i n g which the f l u x e s i n the m a c h i n e do not have t i m e to c h a n g e ( t h e y v a r y w i t h t i m e c o n s t a n t s i n the o r d e r of s e c o n d s ) . When the f a u l t i s removed from the s y s t e m , t h e r e i s a t r a n s i e n t , but i n a l l p r a c t i c a l s y s t e m s , t h e v a r i a t i o n s i n the v o l t a g e s a f t e r w a r d are l i m i t e d . - 69 -F o r e x a m p l e , a d r o p i n t h e v o l t a g e o f 0 . 8 p u . i s n o t a c c e p t e d . T h e r e f o r e , i f t h e c h a n g e s i n t h e v o l t a g e a r e l i m i t e d , then the changes i n the f l u x i n the machine are a l s o l i m i t e d . T h i s can be v e r i f i e d i f we c o n s i d e r e q u a t i o n s 1.9 (a) and 1.9 ( b ) , from which i t i s p o s s i b l e to w r i t e t h a t : V d ( t ) « - OJ ^q (a) V ( t ) = OJ i>d (b) 3.1 because OJ 4> >> "j^'and the machine r e s i s t a n c e i s very low. C o n s e q u e n t l y , i f t h e f l u x e s ty^ and \p can be assumed t o r e m a i n c l o s e t o t h e i r s t e a d y s t a t e v a l u e s , t h e n the m a c h i n e c a n be c o n s i d e r e d to r e m a i n i n t h e same l i n e a r segment i n t h e s a t u r a t i o n c u r v e t h r o u g h o u t the t r a n s i e n t under study ( the concept of l i n e a r i z i n g the s a t u r a t i o n curve was i n t r o d u c e d i n C h a p t e r 2 ) . In r e f e r e n c e [ 1 4 ] , R . G . H a r l e y et a l . , show e x p e r i m e n t a l support f o r t h i s c o n c l u s i o n . F i n a l l y , i f the machine can be assumed to r e m a i n i n the same s a t u r a t i o n s e g m e n t , t h e n t h e f o r m u l a e d e v e l o p e d i n C h a p t e r 2 a p p l y t o t h i s c a s e , p r o v i d e d t h a t t h e t r a n s f e r f u n c t i o n s F l ( s ) t o F 6 ( s ) i n e q . 2 . 1 , a n d L ( s ) , L^Cs) and G(s ) i n e q . 2 .7 c o r r e s p o n d to the l i n e a r segment i n which the machine was o p e r a t i n g when the event s t a r t e d . I t w i l l be p r o v e d i n C h a p t e r 4 t h a t t h i s method g i v e s very good r e s u l t s f o r a l l v a r i a b l e s wi th the e x c e p t i o n o f the m a g n e t i z i n g c u r r e n t , w h i c h , as i n d i c a t e d i n f i g u r e 3 . 1 , could have a l a r g e e r r o r . T h i s e r r o r can be t r a c e d back to an e r r o r - 70 -FIGURE 3.1 : L i n e a r i z a t i o n of the s a t u r a t i o n curve < t Linearized models^X*/ * d(t). / ' ! Error In 1 f Im - 71 -i n t h e f i e l d c u r r e n t , but f o r t u n a t e l y i t i s p o s s i b l e to compensate f o r i t i n the f o l l o w i n g way: From the h y p o t h e s i s o u t l i n e d above, we have t h a t the f l u x i n t h e d - a x i s a n d t h e c u r r e n t i ^( t ) h a v e a c c u r a t e v a l u e s ; c o n s e q u e n t l y , the m u t u a l f l u x ^ , ( t ) ( s e e e q . 2 . 9 ) md must a l s o be a c c u r a t e . T h e r e f o r e , i f we use t h i s mutual f l u x to e v a l u a t e the m a g n e t i z i n g c u r r e n t (see e q . 2 . 1 4 ) , w i t h the v a l u e s o f X , . and E . c o r r e s p o n d i n g to t h e s e g m e n t i n a d i o i ° w h i c h t h e m a c h i n e s h o u l d be o p e r a t i n g ( segment 1 i n the example of f i g u r e 3 . 1 ) , then t h i s m a g n e t i z i n g c u r r e n t can be used t o g e t h e r w i t h t h e a p p l i e d v o l t a g e v ^ ( t ) t o e v a l u a t e the a p p r o p r i a t e v a l u e o f the f i e l d c u r r e n t (see eqs . 2.11 to 2 . 1 3 ) . T h i s method f o r the e v a l u a t i o n of i ^ ( t ) , as w e l l as the o v e r a l l p r o c e d u r e , g i v e s very good r e s u l t s , as w i l l be shown in Chapter 4. 3 .3) Method 2 f o r the C o n s i d e r a t i o n of S a t u r a t i o n T h i s method s h o u l d be used whenever the b a s i c as sumpt ion of method 1 r e g a r d i n g the v a r i a t i o n of the machine t e r m i n a l v o l t a g e does not a p p l y , as i t i s i n the case o f a s u s t a i n e d s h o r t c i r c u i t . In t h i s c o n d i t i o n , g e n e r a l l y t h e c u r r e n t c i r c u l a t i o n i n the machine i s r a t h e r h i g h ; c o n s e q u e n t l y , to a v o i d the p r o b l e m s i n d i c a t e d i n [ 1 6 ] , i t i s b e t t e r to use s t a n d a r d s h o r t c i r c u i t d a t a i n s t e a d of f r e q u e n c y r e s p o n s e m e a s u r e m e n t s b e c a u s e t h e f o r m e r i s o b t a i n e d d u r i n g h i g h c u r r e n t s i t u a t i o n s . H e n c e , t h i s model was deve loped to be ab le - 72 -to use o n l y t h i s type o f d a t a , w h i c h , when complemented wi th a u x i l i a r y t e s t s , g i v e s enough i n f o r m a t i o n to have one c o n s t a n t p a r a m e t e r damper w i n d i n g i n the d - a x i s and a maximum of two damper w i n d i n g s i n the q - a x i s . The second damper wind ing i n the q - a x i s i s known as the g - c o i 1 and r e p r e s e n t s the deep f l o w i n g eddy c u r r e n t s i n the r o t o r . In f i g u r e 3 . 2 , the v a r i a b l e s employed are d e f i n e d . They c o u l d r e p r e s e n t e i t h e r the a c t u a l p a r a m e t e r s o f the machine or the parameters c o r r e s p o n d i n g to C a n a y ' s e q u i v a l e n t c i r c u i t when the a i r gap f l u x l e a k a g e i s to be t a k e n i n t o a c c o u n t (see Appendi x 5 ) . 3 . 3 . 1 ) G e n e r a l D e s c r i p t i o n of the Method. The g e n e r a l i d e a of the method p r e s e n t e d i n t h i s s e c t i o n i s to d e v e l o p a model s i m i l a r to the one p r e s e n t e d i n the p r e v i o u s c h a p t e r s , but m o d i f i e d i n such a way tha t i t can be r e s t a r t e d a t any t ime d u r i n g the s i m u l a t i o n . T h i s f e a t u r e w i l l e n a b l e the model to s w i t c h from one s a t u r a t i o n segment i n t o another whenever the f l u x goes above or below the s a t u r a t i o n s e g m e n t i n w h i c h t h e m a c h i n e was o p e r a t i n g . In o r d e r to a c h i e v e t h i s , the model must i n c l u d e as p a r t o f i t , the v a l u e s of a l l the c u r r e n t s i n the machine a t the t ime when i t i s t o be r e s t a r t e d ( t h i s t ime w i l l be c a l l e d t=0 i n the r e s t of the c h a p t e r ) . The reason f o r t h i s i s that i f a l l the c u r r e n t s are known, then i t can be proved tha t the s t a t e of the machine i s u n i q u e l y d e f i n e d . - 73 -FIGURE 3.2 : E q u i v a l e n t c i r c u i t of the synchronous machine  f o r method 2 f o r the c o n s i d e r a t i o n of s a t u r a t i o n F i g u r e 3 . 2 - A : d - a x i s 1 = S t a t o r leakage i n d u c t a n c e . L> a d = d - a x i s m a g n e t i z i n g i n d u c t a n c e . l j ^ d = Damper winding leakage i n d u c t a n c e . = Damper winding r e s i s t a n c e . 1^ = F i e l d winding leakage i n d u c t a n c e . r f = F i e l d winding r e s i s t a n c e . - 74 -F i g u r e 3 -2 -B : q - a x i s 1 = S t a t o r l eakage i n d u c t a n c e . 3 L = q - a x i s m a g n e t i z i n g i n d u c t a n c e , aq 1 = G - c o i l or deeper damper winding leakage i n d u c t a n c e . r = G - c o i l r e s i s t a n c e , g l k q = Damper winding leakage i n d u c t a n c e , r, = Damper winding r e s i s t a n c e . - 75 -3 .3 .2 ) E q u a t i o n s of the model In o r d e r to d e v e l o p the a p p r o p r i a t e s e t of e q u a t i o n s , f i r s t c o n s i d e r t h e d i f f e r e n t i a l e q u a t i o n 1.9 ( c ) ( The e q u a t i o n s w i l l be d e v e l o p e d f o r the d - a x i s as the q - a x i s f o l l o w s by analogy ) . T h i s e q u a t i o n can be t r a n s f o r m e d to the f r e q u e n c y or L a p l a c e domain , k e e p i n g the i n i t i a l c o n d i t i o n s of the v a r i a b l e s as p a r t o f the model , by u s i n g the f o l l o w i n g mathematical theorem [17] : L{d_x(t)} = s X(s ) - x(0) dt 3.2 where: X(s ) = L{ x ( t ) } x ( 0 ) = The v a l u e o f x ( t ) a t t ime e q u a l z e r o , wh ich c o u l d be e i t h e r at the s t a r t of the s i m u l a t i o n , or a t the t ime when t h e r e i s a s w i t c h from one segment i n t o the o t h e r . So, u s ing t h i s theorem i n equat ion 1.9 ( c ) , we get : V f ( s ) = R f I f ( s ) + s L a d . d d ( s ) + I k d ( s ) + I f ( s ) ) + s l f I f ( s ) - ^ f o (a) where ^fo = ( i k d ( 0 ) + i d ( 0 ) + i f ( 0 ) ) L a d . + l f i f ( 0 ) (b) and f o r eq . 1.9 ( d ) , we get: 3.3 0 - Rkd + S Ladi <VS> + W 8 * + If< 8^> + S *kd Ikd ( s ) " *kdo ( a ) - 76 -where *kdo - + V°> + V ° » Ladi + hd W 0 ' <b> 3.4 E q u a t i o n s 3 .3 and 3 . 4 can be u s e d t o w r i t e I ^ ( s ) and I k d ( s ) as f u n c t i o n s of I d ( s ) and V ^ ( s ) , a n d , b y r e p l a c i n g t h e r e s u l t s f o r I ^ ( s ) a n d I ^ d ( s ) i n e q u a t i o n 3 . 5 f o r * d ( s ) ; * d ( s ) = ( I k d ( s ) + I d ( s ) + i f ( s ) ) L a d . + l a I Q ( s ) + E d o i / K S ) 3.5 i t i s p o s s i b l e to o b t a i n , a f t e r a r a t h e r l e n g t h y p r o c e d u r e , the f o l l o w i n g e q u a t i o n f o r I J J , ( S ) : u o ^ d ( s ) = X d ( s ) I d ( s ) + G(s) [ V f ( s ) + w o H>fQ + H(s) ^ k d Q ] + E d o . / s (a) 3.6 where (1 + T • s) (1 + T " s) X (s ) = S S x ( a ) a (1 + T ' s) (1 + T , " s) a do ' v do ' (1 + T, , s) X . G(s) = ^ - S i (b) (1 + T d o « s) (1 + T d o » s) r f (1 + T s) X H(s) = i -al (C) (1 + T d Q ' s) (1 + T d Q " s) r k d T k d = J k d / r k d Tf = V r f ( d ) - 77 -The t ime c o n s t a n t s i n the e q u a t i o n s above r e l a t e to the machine parameters (see f i g 3 .2) i n the f o l l o w i n g way : V j _ 1 ( h + Lad V L d , *kd + Lad V L d ) , V J 2 r f + r k d 1 A + Lad V L d Xkd + Lad V L d * / T , ^ 2 + (Lad — ) ' (a) 2 r f r k d r f r k d L d T J ') 1 Ir + L j 1. . + L . do I ^ _ f ad + kd ad ^ + T d o " i _ 2 r f r k d / 1 1, + L , L . + 1. . * L , 2 / f ad ad kd , ad , , N v _ _ + (b) 2 r c r. , r c r, , f kd f kd 3.8 In e q u a t i o n s 3 . 8 the p o s i t i v e s i g n c o r r e s p o n d s to the t r a n s i e n t t i m e c o n s t a n t s T ' and T , ' and t h e n e g a t i v e d do ° s i g n to t h e s u b t r a n s i e n t t i m e c o n s t a n t s T , " and T , " . & d o d In a c o m p l e t e l y ana logous way, the e q u a t i o n s c o r r e s p o n d i n g to the q - a x i s can be f o u n d , and they are g i v e n below f o r the sake of completeness : oo u» (s ) = X (s ) I (s) + J ( s ) \\i + K(s ) i>, + E . / s (a) o r q q q 8 ° kqo qoi v ' where •go " ( l kq ( 0 ) + V°> + y ° » Laqi + l g Y<>) (b) *kqo " < V 0 ) + i g ( 0 ) + Laqi + Hq <C> - 78 -and (1 + T ' s) (1 + T " s) X ( s ) 9 — 9 — q (1 + T ' s) (1 + T " s) qo qo (d) J ( s ) = + T kq S> (1 + T ' s) (1 + T " s) qo v qo ' _aq_ ( e ) K(s) = (1 + T ' s) (1 + T " s) qo ' v qo ' ± 1 kq (f) T. = 1 , / r. kq kq kq T = 1 Ix 8 8 8 (8) T •") 1 1 + L 1 / L 1. + L . 1 / L q L _ ( 8 aq a q + kg ad a q ) + T " ( 2 r r, q J 8 kq 1 / l + L , 1 / L 1 , + L 1 / L / g ad a q^  kq aq a r8 rkq rg rkq (Laq - ^ ) 2 Lq (h) T ') 1 1 + L 1. + L 1° L _ ( _g a i + ^3 §J. ) + T „ 2 r qo -> g rkq 1 + L L + 1 , 4 L ' _S SS. _ _Jjg kg. + aq kq rc r, f kq ( i ) - 79 -E q u a t i o n s 3 .6 (a ) and 3 . 9 ( a ) summar ize a l l t h e r o t o r d y n a m i c s . To c o m p l e t e the m o d e l , we have to c o n s i d e r the s t a t o r e q u a t i o n s 1 . 9 ( a ) a n d 1 . 9 ( b ) , w h i c h , when t r a n s f o r m e d to the L a p l a c e d o m a i n , and k e e p i n g the i n i t i a l c o n d i t i o n s as par t of the model , g i v e : V d ( s ) = - r I d ( s ) - s hj, d(s) - E d Q . / s] - ^ d Q - u) 0 ^ q (s) (a) V q ( s ) = - r I q ( s ) - s U q ( s ) - E q o . / s] - ^ q o + <V d ( s ) (b) 3.10 By r e p l a c i n g eqs . 3.6 (a) and 3.9 (a) i n equat ions 3.10 (a) and 3.10 ( b ) , we get : V d ( s ) = - _ s _ X d ( s ) I d ( s ) - _ s _ V g h ( s ) + ^ d Q - r I d ( s ) - X (s ) I (s) - V ., ( s ) - E / s (a) q v / q v / j k v / qo v V (s) = - s X (s ) I ( s ) - s V . . (s) + i> - r I ( s ) q v ' q q jk qo q o o + X d ( s ) I d ( s ) - V g h ( s ) - E d Q / s (b) where *do " Lad <V°> + M 0 * + W 0 " + l a V 0 ) ( c ) V g h ( s ) = G(s) ( V f ( s ) + * f Q ) + H(s) ^ k d Q (d) V j k ( s ) = J ( s ) ^ g o + K(s ) ^ k q ( 0 ) (e) i> = L , ( i (0) + i (0) + i , (0) ) + 1 i (0) ( f ) qo ad q g kq a q 3.11 A g a i n , a s was e x p l a i n e d i n Chapter 2, i t i s b e t t e r to have e q u a t i o n s 3.11 (a) and 3.11 (b) w r i t t e n as c u r r e n t s i n terms - 80 -o f t h e v o l t a g e s , so f o r t h e s e e q u a t i o n s i t can be shown that I d ( s ) = F l ( s ) [ V ^ + V ] + F2( s ) .[V + _ s _ V k ] w o + F 3 ( s ) [ V (a) + * f Q + ^ + S l f \dQ) (a) r k d + S 1kd I q ( s ) = F4( s ) [ V ^ + V j k ] + F5 ( s ) [ V q ; | j + _ s _ V J k ] o r f + s l f + F 6 ( s ) [ V f ( s ) + * f o + ; i ^ k d 0 ] (b) 3.12 r. + s 1, , kd kd where F l ( s ) to F6 ( s ) were d e f i n e d i n Chapter 2, and V , ( s ) = V (s ) - E , . / s - if> (a) q\jr ' q doi qo v V , , ( s ) = V (s ) - E . / s - i> . (b) dijr ' q qo i y do v ' 3.13 From e q u a t i o n s 3 .12 (a) and 3 .12 ( b ) , the c o r r e s p o n d i n g d i f f e r e n c e e q u a t i o n s f o r the i n t e g r a t i o n i n the t ime domain can be o b t a i n e d u s i n g the same o v e r a l l procedure i n d i c a t e d i n C h a p t e r 2, i . e . , F l ( s ) to F 6 ( s ) are a p p r o x i m a t e d by r a t i o n a l f u n c t i o n s and t h e n t h e n u m e r i c a l c o n v o l u t i o n i s u s e d to o b t a i n the d i f f e r e n c e e q u a t i o n s . In Appendix 4, the d e t a i l s of the d e r i v a t i o n of these d i f f e r e n c e e q u a t i o n s a r e c a r r i e d out , thus produc ing the f o l l o w i n g set of e q u a t i o n s : - 81 -i d ( t ) = C1 v d ( t ) + C 2 v ( t ) + C 3 v f ( t ) + H j C t ) + H 2 ( t ) + H 3 ( t ) (a) i q ( t ) = ° 4 V d ( t ) + C 5 v q ( t ) + C 6 v f ( t ) + H A ( t ) + H 5 ( t ) + H 6 ( t ) (a) 3.14 where C , to C , a r e c o n s t a n t s d e f i n e d i n A p p e n d i x 4 ( e q s . o f p a s t v a l u e s and known f u n c t i o n s of t i m e ( see A p p e n d i x 4 e q u a t i o n s A4.19 (c) t o A4.19 (e) and A4.20 (c) to A4.20 ( e ) ) . These e q u a t i o n s are a n a l o g o u s to e q u a t i o n s 2 .5 f o r model 1, w i t h the d i f f e r e n c e t h a t i n t h i s c a s e , t h e t erms H ^ ( t ) to H ^ ( t ) do not o n l y depend on past v a l u e s o f the v a r i a b l e s i n v o l v e d , but a l s o on some known f u n c t i o n s of t i m e . These f u n c t i o n s , as w e l l as the i n i t i a l c o n d i t i o n s o f the terms S . . ( t ) , d e p e n d on t h e v a l u e s t h a t t h e c u r r e n t s i n t h e d i f f e r e n t machine windings had at time equal to z e r o . A 4 . 1 9 ( b ) , A 4 . 2 0 ( b ) ) , and H j ( t ) to H A ( t ) a r e f u n c t i o n s - 82 -CHAPTER 4  IMPLEMENTATION OF THE MODEL IN  AN ELECTROMAGNETIC TRANSIENTS PROGRAM A . 1 ) I n t r o d u c t i o n In t h i s c h a p t e r the model deve loped i n t h i s d i s s e r t a t i o n i s i m p l e m e n t e d i n an e l e c t r o m a g n e t i c t r a n s i e n t s p r o g r a m ( E M T P ) . T h i s p r o g r a m i s u s e d f o r t h e v a l i d a t i o n o f the a p p r o x i m a t e method f o r the c o n s i d e r a t i o n of s a t u r a t i o n , as d i s c u s s e d i n Chapter 3. A t t h e end o f t h i s c h a p t e r , t h e e f f e c t s o f u s i n g d i f f e r e n t types of i n p u t data are a l s o a n a l y z e d . 4 . 2 ) G e n e r a l D e s c r i p t i o n of the E l e c t r o m a g n e t i c T r a n s i e n t s  Program Used (EMTP) The e l e c t r o m a g n e t i c t r a n s i e n t s program s e l e c t e d i n t h i s d i s s e r t a t i o n f o r t h e i m p l e m e n t a t i o n o f t h e m o d e l was o r i g i n a l l y deve loped by H.W. Dommel [15] and enhanced by many c o n t r i b u t o r s a f t e r w a r d s . T h i s program, the EMTP, i s nowadays one o f the most u s e f u l t o o l s f o r the e v a l u a t i o n o f s u r g e phenomena i n e l e c t r i c power systems. The s i m u l a t i o n o f t h e n e t w o r k i n EMTP i s p e r f o r m e d i n g r e a t d e t a i l i n a t h r e e p h a s e b a s i s , a n d , i n i t , i t i s p o s s i b l e to model l i n e a r and non l i n e a r e l e m e n t s , as w e l l as d i s t r i b u t e d parameters l i n e s and c a b l e s . In g e n e r a l , t h e d i f f e r e n t i a l e q u a t i o n s o f most EMTP models are s o l v e d u s i n g i m p l i c i t i n t e g r a t i o n t e c h n i q u e s , thus c o n v e r t i n g t h e s e d i f f e r e n t i a l e q u a t i o n s i n t o e q u i v a l e n t - 83 -d i f f e r e n c e e q u a t i o n s which are s o l v e d t o g e t h e r wi th the r e s t of the e q u a t i o n s of the network. T h e r e a r e two ways i n which a new model can be added to the EMTP: e i t h e r by programming the new e q u a t i o n s d i r e c t l y i n t o the s o l u t i o n a l g o r i t h m of EMTP, or by u s i n g a new method d e s c r i b e d i n [ 1 8 ] . In t h i s l a t e r method, a t each t ime s t e p , t h e e x t e r n a l n e t w o r k i s r e d u c e d t o an n - p h a s e T h e v e n i n equi v a l e n t : v l v o l v 2 = Z T h + v o2 • • v n • V on 4.1 which i s s o l v e d t o g e t h e r w i t h the e q u a t i o n s of the new model . T h i s l a s t method was adopted i n t h i s d i s s e r t a t i o n , s i n c e i t r e d u c e s c o n s i d e r a b l y t h e amount of p r o g r a m m i n g to be done . 4 .3) Implementat ion of Method 1 for the C o n s i d e r a t i o n of S a t u r a t i o n In Chapter 2, i t was shown t h a t i t was p o s s i b l e to d e r i v e a l g e b r a i c e q u a t i o n s which r e l a t e the v o l t a g e s and c u r r e n t s i n the dqo r e f e r e n c e frame at any g i v e n t ime t = n A t , among t h e m s e l v e s , and w i t h the v a l u e s t h a t t h e s e v a r i a b l e s had in t h e p r e v i o u s t i m e s t e p s ( see e q s . 2 . 5 and 2 . 6 ) . Then from - 84 -where equat ions i t i s p o s s i b l e to wri t e : i d ( t ) C l C 2 0 v d ( t ) 1 q ( t ) = c 4 c 5 0 v q ( t ) 0 0 c o v 0 ( t ) v f ( t ) H d ( t ) H q ( t ) H c ( t ) H d ( t ) = H l ( s ) + H 2 ( s ) + H 3 ( s ) + i d s s ( t ) - C l v d s g ( t ) " C 2 V d s s ( t ) - C 3 V f s s ( t ) H ( t ) = H A ( s ) + H 5 ( s ) + H 6 ( s ) + i ( t ) - C 4 v d g s ( t ) q s i - C r- V , 5 dss ( t ) - Cr v , ( t ) v / 6 f s s (a) (b) (c) 4.2 and i n the d - a x i s i n the q - a x i s i n the d - a x i s i n the q - a x i s dss q ss v , dss qss 3 c V ^ g g = I n i t i a l f i e l d v o l t a g e In t h e e q u a t i o n s a b o v e , t h e c o n s t a n t s to and the c o n s t a n t s u s e d i n t h e e v a l u a t i o n o f H j ( t ) t o H ^ ( t ) c o r r e s p o n d t o t h e m o d e l a s s o c i a t e d w i t h t h e s a t u r a t i o n segment i n which the machine i s o p e r a t i n g . I t i s i m p o r t a n t to mention here t h a t the i n i t i a l c o n d i t i o n s shown above have , i n the most g e n e r a l c a s e , p o s i t i v e , n e g a t i v e , and zero sequence components hence, they are steady s t a t e f u n c t i o n s of t ime . F o r the i n t e r f a c i n g w i t h t h e e x t e r n a l n e t w o r k , i t was - 8 5 -i n d i c a t e d b e f o r e t h a t EMTP w i l l r e d u c e t h i s n e t w o r k to a T h e v e n i n e q u i v a l e n t , w h i c h w i l l have to be s o l v e d t o g e t h e r w i t h e q u a t i o n 4.2 . T h e r e f o r e , e i t h e r the T h e v e n i n e q u i v a l e n t i s t r a n s f o r m e d to dqo or e q u a t i o n 4.2 has to be t r a n s f o r m e d to phase q u a n t i t i e s a b c . The l a t t e r approach was chosen h e r e . So, u s i n g P a r k ' s t r a n s f o r m a t i o n i n equat ion 4.2,we get : i W • [ T ] _ 1 [ c i , 5 ] [ T ] [ v a b J + m"1 tS.ei V f + [ H d q Q ] 4 .3 which can be w r i t t e n as - [ C E ] [ V a b J + t C E 3 j 6 ] vf + [ H a b c ] (a) where [CE] = [ T ] " 1 [ C 1 > 5 ] [T] (b) [ C E 3 f 6 ] - [ T ] " 1 [ C 3 f 6 ] ( O t H a b c ] - [ T I ' 1 [ H d q o ] (d) [T] = P a r k ' s t r a n s f o r m a t i o n d e f i n e d i n eq 1.7a (e) 4.4 In o r d e r to speed up the c a l c u l a t i o n s , e q u a t i o n s 4.1 and 4 . 4 were c o m b i n e d i n t o o n e , and the r e s u l t i n g e q u a t i o n i s s o l v e d i n the program by Gauss e l i m i n a t i o n w i t h o u t p i v o t i n g [ 1 9 ] . The f i e l d v o l t a g e v ^ ( t ) i n e q u a t i o n 4 . 4 i s e i t h e r a c o n s t a n t or i s g i v e n by the e x c i t e r d i f f e r e n t i a l e q u a t i o n s , which can be s o l v e d by any s tandard i n t e g r a t i o n t e c h n i q u e . An i m p o r t a n t aspec t i n the s o l u t i o n of these e q u a t i o n s i s t h a t i n o r d e r to use P a r k ' s t r a n s f o r m a t i o n i n e q . 4 . 3 , i t i s n e c e s s a r y to know the a c t u a l p o s i t i o n of the r o t o r 6 = OJ t + 6 + TT / 2 . T h e r e f o r e the swing or m e c h a n i c a l e q u a t i o n must be - 86 -s o l v e d . In t h i s d i s s e r t a t i o n the p r e d i c t o r - c o r r e c t o r approach was c h o s e n , s o , b e f o r e f o r m i n g e q u a t i o n 4 .3 a t any g i v e n t ime s t e p , t h e v a l u e o f $ ( t ) i s p r e d i c t e d u s i n g D a h l ' s [ 9 ] f o r m u l a : * t 2 B(t ) = 2 g(t - At) - 3 ( t - 2At) + [Pm(t - At ) J co(t - At) - P e l ( t - A t ) ] (a) oj(t) = 2 [ g ( t ) - B ( t - At)] / At - to(t - A t ) (b) 4.5 where Pm(t - A t ) = M e c h a n i c a l power i n p u t i n p r e v i o u s time s t e p . P e l ( t - A t ) = E l e c t r i c a l power output i n p r e v i o u s time step . With the v a l u e of $ ( t ) known, the e l e c t r i c a l e q u a t i o n i s s o l v e d f o r Vabc and I a b c , and a new v a l u e of the e l e c t r i c a l power o u t p u t P e l ( t ) i s f o u n d . W i t h t h i s v a l u e , 3 ( t ) i s c o r r e c t e d us ing the t r a p e z o i d a l r u l e : 6 ( t ) = - C P e l ( t ) + a ( t ) (a) where 6 t 2 C = (b) 2 (2 J + D At) oj(t) L t a ( t ) = 3(t - A t ) + [ 2 J oj(t - At) + D At u ] 2 J + D A t ° A t 2 [Pm(t) + Pm(t - At) - P e l ( t - At ) ] 2 (2 J + D At) iu ( t - At) (c) 4.6 - 87 -and a new v a l u e o f B ( t ) i s f o u n d u s i n g eq 4 . 5 b . I f the d i f f e r e n c e b e t w e e n t h e s e c o r r e c t e d v a r i a b l e s a n d t h e p r e d i c t e d ones i s l e s s than a g i v e n t o l e r a n c e , the s o l u t i o n i s a c c e p t e d , o t h e r w i s e , a new i t e r a t i o n i s p e r f o r m e d u s i n g t h e most r e c e n t v a l u e s of o)(t) and 6 ( t ) . In g e n e r a l , the p r e d i c t e d v a l u e i s a c c u r a t e e n o u g h and no i t e r a t i o n i s n e e d e d , u n l e s s t h e r e i s a change i n the n e t w o r k . In t h i s c a s e , the number of i t e r a t i o n s depends on the s e v e r i t y o f the change, but two or three i t e r a t i o n s are t y p i c a l . Once the s o l u t i o n i s o b t a i n e d , the v o l t a g e s and c u r r e n t s i n d , q , o are found from the c o r r e s p o n d i n g abc v a r i a b l e s , and t h e p a s t h i s t o r y t e r m s H j ( t ) t o H & ( t ) a r e e v a l u a t e d a c c o r d i n g to e q s . 2.5 and the p roced u re i n d i c a t e d i n Appendix 1. I n f i g u r e 4 . 1 , t h e p r o c e d u r e d e s c r i b e d a b o v e i s r e p r e s e n t e d as a schemat ic f low d i a g r a m . F o r the e v a l u a t i o n o f the f i e l d c u r r e n t the f o r m u l a e g iven i n Chapter 2 ( eqs . 2.11 to 2.14) can be used . 4 . 4 ) I m p l e m e n t a t i o n of Method 2 f o r the C o n s i d e r a t i o n of  S a t u r a t i on Once method 1 has been programmed, the i m p l e m e n t a t i o n of method 2 i s v e r y easy to c a r r y o u t , s i n c e t h e form of the c o r r e s p o n d i n g i n t e g r a t i n g e q u a t i o n s ( eqs . 3.14 (a) and 3.14 (b ) ) i s the same as b e f o r e . Hence, e q . 4 .2 and the subsequent m e t h o d a l s o a p p l y h e r e , b u t w i t h t h e f o l l o w i n g - 88 -FIGURE A.1 Flow diagram f o r the implementa t ion of method  1 1-n tke EMTP FIND THE INITIAL CONDITIONS OF THE NETWORK AND THE MACHINE FIND [Vabc ] , [Zthev] FROM THE CONDITIONS OF THE EXTERNAL NETWORK UPDATE THE PAST HISTORY VECTORS H . ( t ) OF THE MECHANICAL SYSTEM AND FIND NEW VALUES OF B & OJ USING 3 ( t ) FIND THE EQUATION IN ABC FOR THE MACHINE FROM DQO AND FIND [Vabc] & [Iabc] SOLVING SIMULTANEOUSLY WITH THE THEVENIN FIND THE ELECTRICAL TORQUE AND USING THE TRAPEZOIDAL RULE FIND B ( t ) AND w ( t ) e pre = 8 c o r r = to pre c o r r ~no ^3 pre - 3 c o r r < TOLERANCE yes UPDATE THE PAST HISTORY VECTORS FOR THE MACHINE t= t + At -x FIND THE CURRENTS IN THE FIELD AND DAMPER WIND-DINGS AND WRITE THE RESULTS FOR THIS TIME STEP no < t > TEND > es - 8 9 -modi f i c a t i ons : a) B e f o r e the i n t e g r a t i o n p r o c e d u r e s t a r t s , or when t h e r e i s s w i t c h from one s a t u r a t i o n segment i n t o a n o t h e r , t h e m o d e l must be r e i n i t i a l i z e d by s e t t i n g a l l t h e S , . ( t ) t o S , . ( t ) , V . . ( t ) a n d V . , ( t ) t o t h e i r c o r r e s p o n d i n g i n i t i a l v a l u e s , a c c o r d i n g t o t h e e q u a t i o n s i n Appendix 4. b) The terms H ^ ( t ) and H ^ ( t ) are g i v e n by: H d ( t ) = H j ( t ) + H 2 ( t ) + H 3 ( t ) (a) H q ( t ) = H A ( t ) + H 5 ( t ) + H 6 ( t ) (b) 4.7 where H ^ ( t ) t o H ^ ( t ) a r e a l s o g i v e n i n A p p e n d i x 4. c ) Once t h e r e s u l t i n g e q u a t i o n s a r e s o l v e d and 1^ and I a r e k n o w n , t h e c u r r e n t s i n a l l w i n d i n g s must be q 5 e v a l u a t e d . T h e c u r r e n t i n t h e f i e l d w i n d i n g i s e v a l u a t e d i n the same way as b e f o r e , and t h e c u r r e n t f o r t h e d a m p e r w i n d i n g i n t h e d - a x i s i s e a s i l y e v a l u a t e d , a s t h e m u t u a l c u r r e n t i s k n o w n , f r o m : 1 k d ( t ) = 1 m d ( t ) " V 0 " V 1 ^ 4 . 8 F o r t h e q - a x i s , s i m i l a r e x p r e s s i o n s can be f o u n d f o r t h e c u r r e n t s i n t h e g - c o i l ( s e e s e c t i o n 3 . 3 ) and damper w i n d i n g . F i g u r e 4 . 2 i s a s c h e m a t i c f l o w d i a g r a m o f t h e p r o c e d u r e j u s t p r e s e n t e d . - 90 -FIGURE A .2 : Flow diagram for the implementat ion of method 2 i n the EMTP FIND THE INITIAL CONDITIONS OF THE NETWORK AND THE MACHINE i : FIND [Vabc ] , [Zthev] FROM THE CONDITIONS OF THE EXTERNAL NETWORK t IF t=0 OR THERE IS A SWITCH TO ANOTHER SATURA-TION SEGMENT, REINITIALIZE ALL S i j ( t ) , v ( t ) and v .. ( t ) ACCORDING TO APPENDIX A J UPDATE THE PAST HISTORY VECTORS H . ( t ) OF THE MECHANICAL SYSTEM AND FIND NEW VALUES OF 3 & O J >\ -USING 6 ( t ) FIND THE EQUATION IN ABC FOR THE MACHINE FROM DQO AND FIND [Vabc] & [Iabc] SOLVING SIMULTANEOUSLY WITH THE THEVENIN \  : FIND THE E L E C T R I C A L TORQUE AND USING THE TRAPEZOIDAL RULE FIND 3 ( t ) AND u)(t) 8pre = S c o r r OJ = 10 pre •n -< Bpre - 3 c o r r < TOLERANCE t= t + At T > c o r r yes UPDATE THE PAST HISTOI U VECTORS FOR THE MACHINE FIND THE CURRENTS IN THE FIELD AND DAMPER WIND-DINGS AND WRITE THE RESULTS FOR THIS TIME STEP • n c r / - 4 — s ^ t > TEND )• ^Jkjes (END) - 91 -4.5 ) R e s u l t s 4 . 5 . 1 ) V a l i d a t i o n of Method 1 T h e a s s u m p t i o n t h a t t h e s a t u r a t i o n c u r v e c a n be l i n e a r i z e d around one o p e r a t i n g p o i n t , w i t h o u t a s i g n i f i c a n t l o s s of a c c u r a c y ( see s e c t i o n 3 . 2 ) , c o n s t i t u t e s one of the most r e l e v a n t s t a t e m e n t s made i n t h i s d i s s e r t a t i o n . T h i s a s s u m p t i o n was v e r i f i e d by s o l v i n g t h e t r a n s i e n t a f t e r a s h o r t c i r c u i t , u s i n g b o t h m e t h o d s one ( l i n e a r ) and two ( n o n l i n e a r ) . The s y s t e m d a t a and machine d a t a i s g i v e n i n f i g u r e 4 . 3 , and the f r e q u e n c y re sponse c u r v e s f o r s a t u r a t i o n segment 1 a r e i n f i g u r e 2 . 2 , a l o n g w i t h the c o r r e s p o n d i n g a p p r o x i m a t i o n s . The s h o r t c i r c u i t i s a p p l i e d at a t i m e o f 10 ms . and c l e a r e d a t 167 ms. ( 10 c y c l e s ) l a t e r by o p e n i n g the a s s o c i a t e d l i n e . T h i s l i n e i s r e c l o s e d 500 ms. l a t e r i n order to keep s t a b i l i t y . T h i s t e s t i s c o n s i d e r e d to be v e r y d e m a n d i n g on t h e l i n e a r i z a t i o n p r o c e d u r e , as the s h o r t c i r c u i t r e m a i n s on the s y s t e m f o r a v e r y l o n g p e r i o d ( t w i c e t h e a v e r a g e c l e a r i ng t i me) . The r e s u l t s of t h i s s i m u l a t i o n a r e i n f i g u r e s 4 .4 to 4 . 9 , where i t can be observed tha t i n g e n e r a l the match ing i s very g o o d b o t h i n m a g n i t u d e and i n f r e q u e n c y . T h e b i g g e s t d i f f e r e n c e s occur i n the f requency as t ime approaches 5 s e c . , but n e v e r t h e l e s s , i t i s c o n s i d e r e d t h a t t h i s e r r o r i n the frequency i s low enough f o r any p r a c t i c a l purpose . = 92 -FIGURE 4.3 : C i r c u i t and machine data used f o r t e s t i n g the model . GENERATOR DATA PARAM. 0. HYDRO MANUFAC. T d , T d ' T d o „ x d o c 2.0130 0.2866 0.2837 0.7674 0.0049 5.3903 0.0049 0.2789 1.9700 0. 2700 0.2150 0.5838 0.0249 4.3000 0.0310 PARAM. r q tl ^ q n ^q iq o , . 0. HYDRO MANUFAC 1. 9170 1.867 0. 5734 0.473 0. 2777 0.213 0. 1286 0.997 0. 0043 0.039 0. 4408 0.560 0. 0086 0.061 0. 15503 0.160 SYSTEM DATA a) T r a n s f o r m e r b) Theven in e q u i v a l e n t c) T r a n s m i s s i o n l i n e X rji — 0.2222 X t h « = 0.2970 r t h ' = 0.0416 X = + 0.5840 r = + 0.1002 0.0348 X = 0 1.9241 r = o 0.4626 B = 0 0.0243 A l l r e a c t a n c e and r e s i s t a n c e are i n 555.5 MVA base . - 93 -[f J CURE 4 .4 i COMPARISON BETWEEN METHODS 1 AND 2 FOR S A T U R A T I O N F I E L D CURRENT =F I I 2 . 7 -2 . 5 ' 2 . 3 + 2 . 1 1 . 9 + 1 . 7 1 . 3 1.1 0.9 M e t h o d 1 Method 2 0.7 100 -I — J I J . 1 . 1 o'.500 l'.OOO ltsOO 2'.000 2'.S00 3'.000 3'.500 4'.000 4'.500 5".000 TIME ( S E C . ) IF IOURE 4.5 : COMPARISON BETWEEN METHODS 1 AND 2 FOR S A T U R A T I O N , POWER ANCLE I I 90 J -Hethod 2 20 + 10 + J — J J — J — J — J 0*. 500 l'.OOO 1*7500 2'.000 2".500 S'.OOO 3'.500 4'.000 4'.500 5'.000 TIME ( S E C . ) - 94 -1.0 I F I G U R E 4.6 « C O M P A R I S O N B E T W E E N M E T H O D S 1BND 2 F O R S A T U R A T I O N E L E C T R I C A L POWER O U T P U T I I I I I I 0.9 O.S 0.7 0.6 0.5 0.4 I 0.3 + 0.2 I 0.1 + 0.0 o ' . s o o l'.OOO l'.SOO 2 !.000 2'.S00 a ' .ooo 3'.S00 V.OOO 4*7soo 5^000 TIME (SEC.) rf- •+-0.2 [ F I G U R E 4 . 7 » C O M P A R I S O N B E T W E E N M E T H O D S 1 A N D 2 F O R S A T U R A T I O N V O L T A G E I N T H E 0 A N O 0 A X I S 0.0 J_ -0.2 -0.4 I -0.6 -0.8 -1.0 -1.2 -1.4 J . -1.6 + -1.8 Method 1 o ' . s o o I ' .OOO I ' . S O O 2 ' .ooo 2 ' . 5 o o a ' . o o o a ' . s o o 4 ' . 0 0 0 4 ' . S 0 0 5 . 0 0 0 TIME (SEC) -95 -1.0 0.7 0.4 0.1 -0.2 -0.5 °- >- -0.6 -1.1 -1.4 -1.7 0.5 I F I C U R E 4.8 » C O M P A R I S O N B E T W E E N M E T H O D S 1 R N D 2 F O R S A T U R A T I O N C U R R E N T I N T H E D A N D 0 A X I S 0.2 _L -0.1 -0.4 J . -0.7 -1.0 J . - -1.3 + -1.6 + -1.9 + -2.2 + -2.0 -2.5 I ) +• J — J 0.500 1.000 l.SOO 2 .000 2 .500 3.000 3 .500 4 .000 4.500 5 .000 T I M E ( S E C . ) 382 381 380 | F I C U R E 4.9 t C O M P A R I S O N B E T W E E N M E T H O D S 1 A N D 2 F O R S A T U R A T I O N A N G U L A R S P E E D =F= 379 J . 378 377 376 _L 375 + 374 J_ 373 + 272 =F =F =F =F Method 1 o'.500 l'.OOO l'.500 2'.000 ^-SOO s'.OOO 3J.500 4'.000 4'.500 5.000 T I M E ( S E C . ) - 96 -4 . 5 . 2 ) E f f e c t s of Us ing D i f f e r e n t Input Data The case s e l e c t e d f o r t h i s e v a l u a t i o n c o r r e s p o n d s to the f i e l d t e s t per formed by O n t a r i o Hydro ( O . H . ) on a g e n e r a t o r a t Lambton [ 4 ] , In t h i s t e s t , o n e l i n e i n the system i n f i g u r e 4 . 3 was o p e n e d (no f a u l t p r e s e n t ) and t h e t r a n s i e n t was r e c o r d e d . T h r e e t y p e s o f d a t a were u s e d : the m a n u f a c t u r e r ' s s t a n d a r d data ( g i v e n i n f i g u r e 4 . 3 ) , the parameters e s t imated by O n t a r i o H y d r o ( O . H . ) from f r e q u e n c y r e s p o n s e t e s t ( see f i g u r e 4 . 3 ) , and t h e a c t u a l m e a s u r e d f r e q u e n c y r e s p o n s e c u r v e s ( g i v e n i n f i g u r e 2 .2) a l o n g w i t h the a s s o c i a t e d curves F l ( s ) to F 6 ( s ) and the c o r r e s p o n d i n g a p p r o x i m a t i o n s . One to one c o n c o r d a n c e w i t h O . H . r e s u l t s were not e x p e c t e d , due to t h e l a c k o f i n f o r m a t i o n a b o u t t h e e x t e r n a l s y s t e m . N e v e r t h e l e s s , the c u r v e s o b t a i n e d a r e v e r y s i m i l a r to the ones measured and the o b s e r v a t i o n s t h a t can be d e r i v e d from the O . H . t e s t are a l s o supported here : a) A c t i v e power: In f i g u r e 4 .10 the r e s u l t s o b t a i n e d by O . H . are r e p r o d u c e d . We o b s e r v e i n t h i s f i g u r e t h a t the s i m u l a t i o n o b t a i n e d by u s i n g t h e m a n u f a c t u r e r ' s s t a n d a r d d a t a h a s a d i f f e r e n t f r e q u e n c y t h a n t h e m e a s u r e d c u r v e and i t s a m p l i t u d e i s b i g g e r i n t h e f i r s t s w i n g and a b o u t t h e same i n t h e subsequent ones . The s i m u l a t i o n u s i n g parameters e s t i m a t e d by O n t a r i o Hydro from S t a n d S t i l l F r e q u e n c y Response t e s t s has t h e same f r e q u e n c y as t h e m e a s u r e d c u r v e b u t a b i g g e r ampli tude . In f i g u r e 4 .11 the r e s u l t s o b t a i n e d u s i n g the program - 97 -•150 TIME ( S E C . ) FIGURE 4 .10 -B E l e c t r i c a l power a f t e r opening a l i n e  data e s t imated by 0. Hydro from SSFR 100 50 /V~ U . H . - SSFR r / v • / \ \ * \J / t - 50 -100 -150 TIME ( S E C . ) - 98 -0.8 FIGURE 4.11 : COMPARISON BETWEEN DIFFERENT INPUT DflTfl: ACTIVE POWER 0.7 + . 0.6 X 0.5 0.4 + 0.3 v^ //I SSFR - " " v ~ \ * _ O . H . - SSFR o'.50O l'.OOO l'.500 2.000 2.500 3.000 3.500 4.000 4.500 5'.000 TIME (SEC.) - 99 -p r e s e n t e d i n t h i s d i s s e r t a t i o n (method 1) are shown. There we o b s e r v e t h a t t h e c u r v e c o r r e s p o n d i n g to t h e c a s e where f r e q u e n c y response t e s t data were used d i r e c t l y has the same f r e q u e n c y as the one o b t a i n e d u s i n g the p a r a m e t e r e s t i m a t e d by O . H . ; however , i t s a m p l i t u d e i s s m a l l e r and i s about the same as t h a t o b t a i n e d u s i n g the m a n u f a c t u r e r ' s s t a n d a r d d a t a . T h e r e f o r e i t i s s e n s i b l e to expect tha t t h i s case s h o u l d have had a b e t t e r c o n c o r d a n c e w i t h the measured r e s u l t s had the system been model led p r o p e r l y , b) F i e l d c u r r e n t : The o t h e r v a r i a b l e r e p o r t e d by O . H . i n t h e i r r e s u l t s i s the f i e l d c u r r e n t . The r e s u l t s i n d i c a t e t h a t the p r e d i c t e d curve u s i n g t h e m a n u f a c t u r e r ' s s t a n d a r d d a t a ( s e e f i g u r e A . 1 2 ) damps out q u i c k e r than the a c t u a l r e sponse and i t s f r e q u e n c y i s h i g h e r . When u s i n g p a r a m e t e r s e s t i m a t e d from S S F R , the f r e q u e n c y i s c o r r e c t but the a m p l i t u d e i s somehow h i g h e r . In f i g u r e A . 1 3 the r e s u l t s u s i n g the program d e v e l o p e d in t h i s d i s s e r t a t i o n a r e s u m m a r i z e d . The r e l a t i v e b e h a v i o u r of the r e s u l t s u s i n g the m a n u f a c t u r e r ' s s t a n d a r d data and parameters e s t i m a t e d from SSFR a r e r e p e a t e d h e r e . The case where SSFR d a t a were used d i r e c t l y d i d not improve the r e s u l t s o b t a i n e d u s i n g t h e p a r a m e t e r s e s t i m a t e d by O . H . , but t h i s i s not r e a l l y s u r p r i s i n g , as we had to r e l y on c i r c u i t p a r a m e t e r s , as p o i n t e d out i n s e c t i o n 2 . 2 , e q u a t i o n s 2.11 to 2 . 1 A . These parameters are i n t h i s case those e s t i m a t e d by O . H . , and , i n g e n e r a l , they have to be measured. An a d d i t i o n a l t e s t was run 100 -FIGURE A .12-A : F i e l d c u r r e n t a f t e r opening a l i n e : m a n u f a c t u r e r ' s data 1 1 O r i | \ \ n a l / V ( J k \ S t a n d a r d 0.0 1.0 i o 3.0 TIME ( S E C . ) FIGURE A.12 - B : F i e l d c u r r e n t a f t e r opening a l i n e ; data e s t imated by O n t a r i o Hydro from SSFR 7JO j- O . H . - SSFR / *v v— O r i gi n a l 2.0 3.0 «.0 TIME ( S E C . ) - 101 -FIGURE 4.13 : COMPARISON BETWEEN DIFFERENT INPUT DflTfl: FIELD CURRENT 1.94-1.2 --1.1 0.500 1 .000 1.500 2.000 2'.500 3*.000 3'. 500 V.000 4'.500 5' TIME (SEC.) - 102 -2 . 0 1.9. 1 . 8 1.7 1 . 6 => 1.5 1 . 4 I 1 . 3 + 1 . 2 1 . 1 + 1 . 0 FIGURE 4.14 i COMPARISON BETWEEN DIFFERENT INPUT OATA FIELD CURRENT « MANUFACTURER'S PARAMETERS) O . H . - SSFR J „ „ .1 J J I J J 0-500 .000 1.500 2.000 2.500 3.000 3.500 4 .000 4 .500 5.000 TIME (SEC.) 9 0 85 + 8 0 50 + 45 4 0 FIGURE 4.15 : COMPARISON BETWEEN DIFFERENT INPUT DATAJ POWER ANGLE I I I I O . H . - SSFR J — .1 — .1 — J 0'.500 l'.OOO l'.SOO 2 ' .000 2'.500 s'.OOO 3 U 0 O 4 ' . 0 0 0 4 !.S00 s'.OOO TIME (SEC.) - 103 -- 10A -u s i n g the m a n u f a c t u r e r ' s d a t a f o r the f i e l d w i n d i n g . I t was o b s e r v e d t h a t the d a m p i n g of t h e o s c i l l a t i o n s i n c r e a s e d s l i g h t l y , making the r e s u l t i n g c u r v e c l o s e r to the r e s p o n s e o b t a i n e d u s i n g e s t imated parameters (see f i g u r e A . 1 A ) . In f i g u r e s A.15 and A . 1 6 , the r e s u l t s f o r the power angle and a n g u l a r speed are shown. They p r e s e n t the same r e l a t i v e behav iour that was observed i n the e l e c t r i c a l power. A . 5 . 3 ) E v a l u a t i o n of the U s e f u l n e s s of the P r o p o s e d Method . In t h e p r e v i o u s s e c t i o n , i t was shown t h a t t h e r e a r e n o t i c e a b l e d i f f e r e n c e s i n the s i m u l a t i o n s o b t a i n e d u s i n g d i f f e r e n t types of i n p u t d a t a : m a n u f a c t u r e r ' s , e s t i m a t e d , and f r e q u e n c y r e s p o n s e s , and i t was d e m o n s t r a t e d t h a t t h i s l a s t type g ive s r e s u l t s that are c l o s e r to f i e l d t e s t s . The example used to d e m o n s t r a t e t h i s , was chosen because i t was w e l l - d o c u m e n t e d and i t i l l u s t r a t e d the s e n s i t i v i t y of t h e m e t h o d u s e d , a l t h o u g h i t c o u l d be a r g u e d t h a t t h e d i f f e r e n c e s o b s e r v e d a r e h a r d l y w o r t h t h e t r o u b l e o f m o d e l l i n g a c c u r a t e l y the f r e q u e n c y r e s p o n s e s . The r e a s o n for the s m a l l d i f f e r e n c e s can be found i n f i g s . A . 1 7 - A . 1 9 , where t h e f r e q u e n c y r e s p o n s e s o b t a i n e d u s i n g the t h r e e t y p e s of d a t a are shown. In these f i g u r e s . i t can be o b s e r v e d t h a t the r e s p o n s e s f o r t h e d - a x i s a r e v e r y c l o s e t o g e t h e r i n the lower f r e q u e n c y r a n g e , which i s the range i m p o r t a n t f o r the e v e n t under s t u d y . As the d - a x i s d a t a i s p r e c i s e l y the one which m o s t l y govern the t r a n s i e n t b e h a v i o u r of the m a c h i n e , the smal l d i f f e r e n c e s can be e x p l a i n e d . The d i f f e r e n c e s i n the q - a x i s are somehow more n o t i c e a b l e , - 105 -e s p e c i a l l y f o r t h e m a n u f a c t u r e r ' s d a t a , b u t , as t h e s e d i f f e r e n c e s a f f e c t m o s t l y the i n t e r c h a n g e of r e a c t i v e power, t h e i r e f f e c t s a r e l e s s n o t i c e a b l e i n t h e t i m e r e s p o n s e . U n f o r t u n a t e l y , f o r the method proposed by O n t a r i o H y d r o , t h i s i s not a lways the c a s e . In t h i s s e c t i o n , a n o t h e r machine i s a n a l y z e d w h i c h c o u l d not be a c c u r a t e l y m o d e l l e d , u s i n g n e i t h e r the m a n u f a c t u r e r ' s d a t a nor the d a t a e s t i m a t e d by O n t a r i o H y d r o . In f i g s . 4 . 2 0 - 4 . 2 2 , a c o m p a r i s o n i s made between the m a c h i n e ' s a c t u a l f r e q u e n c y r e s p o n s e s (which are r e p r e s e n t e d e x a c t l y by t h e new m o d e l ) and t h e r e s p o n s e s o b t a i n e d u s i n g b o t h the m a n u f a c t u r e r ' s d a t a and t h e d a t a e s t i m a t e d by O n t a r i o H y d r o . As can be s e e n , the m a t c h i n g of the r e s p o n s e s i s q u i t e u n a c c e p t a b l e both i n the d and q a x i s . In o r d e r to i l l u s t r a t e the e f f e c t t h a t these d i f f e r e n c e s have i n t h e t i m e r e s p o n s e , i t was d e c i d e d to r u n a q u i t e common but s p e c i a l l y s e v e r e c a s e , i n which a s i n g l e l i n e to ground f a u l t was a p p l i e d a t the r e c e i v i n g end of one o f the l i n e s i n f i g u r e 4 . 3 . The f a u l t was c l e a r e d f i v e c y c l e s l a t e r . F i g u r e s 4 .23 to 4.26 show the r e s u l t s of the s i m u l a t i o n where i t i s e v i d e n t tha t both the a m p l i t u d e and the f r e q u e n c i e s of the o s c i l l a t i o n s are a f f e c t e d by the poor a p p r o x i m a t i o n in t h e f r e q u e n c y d o m a i n . T h e s e e r r o r s a r e a v o i d e d c o m p l e t e l y us ing the method proposed i n t h i s t h e s i s . - 106 -FIGURE 4-17 : FREQUENCY RESPONSE FOR LRMBTON GENERATOR USING DIFFERENT INPUT ORTR i X01S) 15 - 1 — l l l l l l l l 1 — l l l l l l l l 1 — l l l l l l l l 1 — l l l l l l l l 1—I I I I III 12--9.. - 3 - --6 • " -12 O.H.- SSFR - i s j — l l l l Hfrl 0 1 Q l — l l I I iifci 1 0 Q l 1 I I I 11^ 1 1 l l l l 11^ 1 1 I I I III 0.001 F R Q ( H Z ) 15 12 FIGURE 4..18 : FREQUENCY RESPONSE FOR LRMBTON GENERRTOR USING DIFFERENT INPUT DRTR t XO(S) - 1 — l l l l l l l l 1—I I I I l l l l 1 — l l l l l l l l 1—("I I I l l l l 1—I I I I I 6 - i -3 •--6 Standard -12 -15 1—i i, 11 ny 0 1 Q i — i i * * I ' Q | . — I I I I m i — i — i 111 m u — i — i 111 in G . O O I 100 FRQ ( H Z ) - 107 -80 FIGURE 4.. 19 : FREQUENCY RESPONSE FOR LAMBTON GENERATOR USING DIFFERENT INPUT DATA: GlS) - i — I I I I m i 1—i i 1 1 I I I I 1 — I I I I I I I I 1—i i 1 1 m i 1—i i i I I I I 70.. 60 50 ,. 40 30 20.. 10 -10.-- S t a n d a r d -20 o'.ooi1 1 "'"b'.oio1 1 ' " " b ' . i o o 1 ' 1 1 " ' " l b 1 ' • " " " FRO (HZ) 15 FIGURE 4.20 : FREQUENCY RESPONSE FOR NANTICOKE GENERATOR USING DIFFERENT INPUT DATA : XO(S) " I — ' I I I llll 1 — l l l l llll 1 l l l l llll 1 — l l l l llll 1 — I I I I I I I 12--9.. 6- -3.. 0. O . H . - S S F R -6 - 9 " -12 SSFR S t a n d a r d 1 3oW ' '""b'.oio1 ' ""'b'.ioo' ' 1 ' — 1 1 1 '"U 1 11 FRQ (HZ) - 108 -15 12 FIGURE 4.21 : FREOUENCY RESPONSE FOR NRNTICOKE GENERATOR USING DIFFERENT INPUT DflTfl i XOtS) - 1 — l l l l l l l l 1 — l l l l l l l l 1 — l l l l l l l l 1—I I I I l l l l 1—I I I I III 3 O . H . - S S F R -9 --- 1 2 S t a n d a r 15 „ „ , l l l l l MM 0 1 Q I l l l l l l l j l 1 0 Q I l l l l IIM I l l l l 1 1 ^ 1 1 I I I III 0.001 FRQ (HZ) 8 0 7 0 6 0 5 0 4 0 3 0 2 0 - -1 0 " FIGURE 4^22 i FREOUENCY RESPONSE FOR NANTICOKE GENERATOR USING DIFFERENT INPUT DATA: CIS) -1—I I I I l l l l 1 — l l l l IIM 1 — l l l l l l l l 1 — l l l l l l l l 1—I I I I III 0. - 1 0 - -- 2 Q o . o o i ' 1 " " ' b ' . o i o 1 1 ' " " b ' . i o o 1 1 ' " " I ' 1 1 " " ' l b FRQ (HZ) - 109 -[FIGURE 4.23 i COMPARISON BETWEEN DIFFERENT INPUT DATA FOR NANTICOKE UNIT « ELECTRICAL TORQUE 1.3 1.2 1.1 Standard I 0.500 1.000 1.500 2.000 2.500 3.000 3.500 4.000 4.500 &.000 TIME ( SEC.) 100 |FIGUR£ 4*24 i COMPARISON BETWEEN DIFFERENT INPUT DATA FOR NANTICOKE UNIT : POWER ANGLE I I 96 + 92 -L 64 60 Standard ' A \ \ vV7 M / r \\ ^ ! V / SSFR V> SSFR O.H.- SSFR + 0.500 1.000 1.500 2.000 2.500 3.000 3.500 4.000 4.500 5-000 TIME ( S E C . ) - 110 -2.2 I -1.0 FIGURE 4.25 i COMPARISON BETWEEN D I F F E R E N T INPUT DATA FOR NANTICOKE UNIT : CURRENT 0 AND 0 AXIS 1.9 1.6 1.3 1.0 . 0 . 7 0 . 4 -1.3 + -1.6 -1.9 -2.2 + -2.5 + -2.8 - 0 . 1 - 0 . 2 - 3 . l t - 3 . 4 + - 0 . 5 -0.8 - 3 . 7 -4.0 - I l l -4 . 5 . 4 ) G e n e r a l O b s e r v a t i o n s on the N u m e r i c a l B e h a v i o u r of  the Method. In g e n e r a l , t h e method employed f o r the i n t e g r a t i o n of the model i s v e r y s t a b l e n u m e r i c a l l y , and t h i s was v e r i f i e d by u s i n g l a r g e i n t e g r a t i o n s t e p s . However, i t i s not n e c e s s a r y to c o r r e c t f o r d i s c r e t i z a t i o n e r r o r s , s i n c e the maximum i n t e g r a t i o n s t e p t h a t c a n be u s e d i s i m p o s e d by t h e a p p r o p r i a t e 60 H z . m o d e l l i n g o f the network and not by the machine. 4 . 5 . 5 ) C o n c l u s i ons F r o m t h e r e s u l t s p r e s e n t e d a b o v e , i t i s p o s s i b l e to d e r i v e the f o l l o w i n g c o n c l u s i o n s : a) The l i n e a r i z a t i o n method i s s u f f i c i e n t l y a c c u r a t e f o r most p r a c t i c a l c a s e s , i n c l u d i n g c l e a r i n g o f f a u l t e d l i n e s . b ) T h e r e a r e d i f f e r e n c e s i n t h e s i m u l a t i o n o b t a i n e d u s i n g d i f f e r e n t t y p e s o f i n p u t d a t a , b o t h i n t h e a m p l i t u d e and i n the f r e q u e n c y of the o s c i l l a t i o n s . From t h e r e s u l t s o b t a i n e d by O n t a r i o H y d r o and the r e s u l t s r e a c h e d by u s i n g t h e p r o g r a m d e v e l o p e d i n t h i s d i s s e r t a t i o n , i t can be s t a t e d t h a t i n g e n e r a l the s t a t o r c u r r e n t and v o l t a g e s , as w e l l as the power, are c l o s e r to the a c t u a l v a r i a b l e s i n the cases when f r e q u e n c y response d a t a i s u s e d . T h e r e f o r e , i t can be i n f e r r e d t h a t i n g e n e r a l the d e v e l o p e d model s h o u l d g i v e more a c c u r a t e r e s u l t s when t h i s type of data i s used d i r e c t l y . - 112 -F o r t h e c a l c u l a t i o n o f t h e f i e l d c u r r e n t t h e mode l d e v e l o p e d d i d not i m p r o v e the s i m u l a t i o n s u s i n g t h e d a t a e s t i m a t e d by O n t a r i o H y d r o f r o m f r e q u e n c y response t e s t s , but the r e s u l t s are undoubted ly good, and the p r o b l e m of e s t i m a t i n g p a r a m e t e r s from t h i s t y p e of data i s o b v i a t e d . The mode l i s n u m e r i c a l l y s t a b l e and d o e s n o t i m p o s e any l i m i t a t i o n on t h e s i z e o f the i n t e g r a t i o n s t e p . F i n a l l y , i t c a n be s a i d t h a t t h e mode l p r e s e n t e d i n t h i s d i s s e r t a t i o n a c c u r a t e l y r e p r e s e n t s the s y n c h r o n o u s m a c h i n e , g i v i n g the u s e r t h e f l e x i b i l i t y o f u s i n g the best data a v a i l a b l e . - 113 -CHAPTER 5 IMPLEMENTATION IN A STABILITY PROGRAM 5.1) I n t r o d u c t i on In p r e v i o u s c h a p t e r s , a new model was d e v e l o p e d f o r the synchronous machine which proved to be very s t a b l e , a n d , as i t was p o i n t e d o u t i n C h a p t e r 2, w i t h i t , i t i s p o s s i b l e to e v a l u a t e and c o r r e c t f o r the d i s c r e t i z a t i o n e r r o r due to the use of l a r g e i n t e g r a t i o n s t e p s . T h e r e f o r e t h i s m o d e l i s i d e a l l y s u i t e d f o r s t a b i l i t y s i m u l a t i o n s where the speed of the c a l c u l a t i o n s i s a lmost as important as the a c c u r a c y . F o r the i m p l e m e n t a t i o n i n a s t a b i l i t y p r o g r a m , method 1 ( s e e s e c t i o n 4 . 3 ) was a d o p t e d f o r t h e c o n s i d e r a t i o n of s a t u r a t i o n , s i n c e t h e a s s u m p t i o n s b e h i n d t h i s method a r e always t r u e i n t h i s type o f s i m u l a t i o n s . 5.2 ) G e n e r a l D e s c r i p t i o n of the S t a b i l i t y Program F o r t h e i m p l e m e n t a t i o n o f t h e m o d e l , t h e s t a b i l i t y p r o g r a m P S S / E D , E D E L C A ' s v e r s i o n o f t h e P S S / 2 p a c k a g e d e v e l o p e d by Power T e c h n o l o g i e s I n c o r p o r a t e d , was u s e d . T h i s p r o g r a m u s e s e x p l i c i t i n t e g r a t i o n ( R u n g e - K u t t a ) , w h e r e a s t h e m o d e l d e v e l o p e d i n t h i s d i s s e r t a t i o n u s e s i m p l i c i t i n t e g r a t i o n . T h i s s i t u a t i o n f o r c e d a d e t a i l e d study of the program. The b a s i c a l g o r i t h m of PSS/ED i s i n d i c a t e d i n f i g u r e 5 . 1 , - 114 -FIGURE 5.1 : B a s i c a l g o r i t h m of PSS/ED CALCULATE THE INTERNAL VOLTAGES FOR THE GENERATORS, FIELD VOLTAGES AND MECHANICAL POWERS. CALCULATE THE CURRENT INJECTIONS VECTOR [I] USING THE LATEST VALUE OF THE VOLTAGE. [ I ] = f ( V i - 1 ) SOLVE THE NETWORK EQUATION: [Y] [ V . ] = [I ] -no- V. - V. , < T o l e r . 1 i - l yes PERFORM THE NUMERICAL INTEGRATION OF THE MODELS (RUNGE-KUTTA) t = t + A t •no- t > Tpause yes - 115 -where i t can be o b s e r v e d t h a t i n t h i s p r o g r a m , the network c o n d i t i o n i s found at each i n t e g r a t i o n s tep by u s i n g downward o p e r a t i o n and back s u b s t i t u t i o n i n the e q u a t i o n : [Y] [V] = [I ] where [ I ] i s the c u r r e n t i n j e c t i o n a t each n o d e , [ V ] i s the v o l t a g e a n d , [Y] i s the a d m i t a n c e m a t r i x . T h i s l a s t m a t r i x has to be formed and t r i a n g u 1 a r i z e d b e f o r e the i n t e g r a t i o n procedure can s t a r t . When t h e r e a r e n o n - l i n e a r l o a d s i n t h e n e t w o r k , f o r example , c o n s t a n t power l o a d s , the c u r r e n t i n j e c t i o n s [ I ] are a f u n c t i o n o f the v o l t a g e and t h e r e f o r e the e q u a t i o n above must be s o l v e d i t e r a t i v e l y . 5.3 D e s c r i p t i o n of the Implementat ion As was p o i n t e d out b e f o r e , i n the program P S S / E D , some i t e r a t i o n s are n e c e s s a r y whenever t h e r e are n o n - l i n e a r l o a d s . T h i s f a c t can be used for the i m p l e m e n t a t i o n o f the mode l , by u s i n g a method s i m i l a r to the one d e s c r i b e d by Dommel and Sato [ 9 ] , i n w h i c h a f r i n g i n g c u r r e n t A l s a l i e n t i s used to take i n t o a c c o u n t t h e v o l t a g e d e p e n d e n t p a r t o f the e q u i v a l e n t c u r r e n t i n j e c t i o n s used to model the machine . A f i r s t s tep to c o n s i d e r i n d e v e l o p i n g t h e e q u a t i o n s u s e d to m o d e l t h e m a c h i n e i s t h a t , i n s t a b i l i t y s i m u l a t i o n s , the ne twork i s assumed to be i n a q u a s i - s t a t i o n a r y s t a t e . T h e r e f o r e phasors c a n be u s e d , and the f o l l o w i n g e x p r e s s i o n s a r e v a l i d f o r r e l a t i n g d q o v a r i a b l e s w i t h a b c ( s e e e q . 1 . 2 5 a n d 1 .26) : - 116 -/3 I f c e _ j 6 = i q ( t ) + j i d ( t ) (a) 3^ V t e " j 6 = v q ( t ) + j v d ( t ) (b) 5.1 From the e q u a t i o n s d e s c r i b e d i n C h a p t e r 4 ( e q . 4 . 2 ) , c o r r e s p o n d i n g t o the model where s a t u r a t i o n i s c o n s i d e r e d a p p r o x i m a t e l y (method 1 ) , we can w r i t e : i ( t ) + j i d ( t ) = ( Cj v d ( t ) + C 2 v ( t ) + EDO ) j + C 4 v d ( t ) + C 5 v ( t ) + EQO (a) where EDO = C 3 v f ( t ) + H d ( t ) (b) EQO = C 6 v f ( t ) + H q ( t ) (c ) 5.2 So , u s i n g 5.1 (a) and 5.1 (b) i n e q u a t i o n 5.2 ( a ) , we can f i n d the f o l l o w i n g equat ions : J t " Ym V t + lm <a> Ym - " ( C l " C 4 J > <b> where I = - ( j EDO + EQO + Al . . ^ ) e J ' 6 e l + AI . . ^ (c) m 3 s a l i e n t s a l i e n t A l = (( C 9 + C , ) j + C , - C , ) vn (d) s a l i e n t 2 k' J 5 1 q 5.3 T h e s e e q u a t i o n s can be a s s o c i a t e d w i t h t h e e q u i v a l e n t c i r c u i t shown i n f i g u r e 5 .1a , where i t can be seen t h a t in - 117 -o r d e r to model the m a c h i n e , t h e term Y must be added to the m d i a g o n a l e l ement of t h e m a t r i x [Y] t h a t c o r r e s p o n d s to the m a c h i n e ' s bus b a r and t h e t e r m I must be a d d e d t o the m c o r r e s p o n d i n g term i n the c u r r e n t i n j e c t i o n v e c t o r [ I ] . The m a c h i n e ' s e q u a t i o n s are then s o l v e d s i m u l t a n e o u s l y w i t h the e q u a t i o n s o f the n e t w o r k . In g e n e r a l i t e r a t i o n s a r e needed s i n c e the terra A l s a l i e n t i s a f u n c t i o n o f the v o l t a g e , and t h e r e f o r e i t must be e s t i m a t e d a t e v e r y i t e r a t i o n , u s i n g the p r e v i o u s v a l u e of the v o l t a g e , u n t i l t h e r e i s convergence . In f i g u r e 5 . 2 , a f low diagram of the o v e r a l l a l g o r i t h m i s shown. In t h i s f low d i a g r a m , t h e r e are two i t e r a t i o n l o o p s . The f i r s t one c o r r e s p o n d s to A l s a l i e n t and the second one to t h e use o f a p r e d i c t o r - c o r r e c t o r a p p r o a c h f o r s o l v i n g the swing e q u a t i o n as i n d i c a t e d i n Chapter 4. 5 .4) R e s u l t s of the Implementat ion T h e a c c u r a c y o f t h e method o u t l i n e d i n t h e p r e v i o u s s e c t i o n was e s t a b l i s h e d by r u n n i n g twice the same t e s t c a s e , once w i t h t h i s method and a s e c o n d t i m e u s i n g a s t a n d a r d model t h a t was a l r e a d y i n P S S / E D . T h i s l a t t e r model a l l o w e d f o r t h e c o n s i d e r a t i o n o f two damper w i n d i n g s and a f u l l m o d e l l i n g of s a t u r a t i o n . E x c i t e r a n d g o v e r n o r m o d e l s w e r e i n c l u d e d i n t h e s i m u l a t i o n u s i n g b o t h mode l s ( s ee A p p e n d i x 7 f o r c o m p l e t e d e s c r i p t i o n of these m o d e l s ) . - 118 -FIGURE 5 . 1 - A : E q u i v a l e n t c i r c u i t f o r the m o d e l l i n g of the machine REST OF THE NETWORK - 119 -FIGURE 5.2 Flow diagram f o r the implementa t ion of the model i n PSS/ED • no-•no-CALCULATE THE FIELD VOLTAGE AND MECHAN-ICAL POWER. PREDICT THE VALUE OF THE LOAD ANGLE 6 ( t ) , UPDATE THE TERMS EDO AND EQO FOR ALL GENERATORS FIND A l s a l i e n t AND FIND Imac. FOR ALL GENERATORS. FIND THE VECTOR [I ] = f ( V j _ i ) SOLVE THE NETWORK EQUATION [Y] [ V . ] = [I ] < V . - V . 1 " < T O L E R . l > res 6 i ( t ) - 6 i - l ( t ) < TOLER yes t = t . + A t > -Tpause - 120 -In o r d e r to make the r e s u l t s c o m p a r a b l e , the e q u i v a l e n t c i r c u i t c o r r e s p o n d i n g to the s t a n d a r d PSS/ED model was used t o g e n e r a t e the i n p u t f r e q u e n c y r e s p o n s e d a t a f o r the new mode l ( f i g 5 . 3 ) . I t i s i n t e r e s t i n g t o o b s e r v e t h a t t h e s e c u r v e s do no t h a v e a p o l e a t OJ = 2 IT f b e c a u s e i n the P S S / E D model the t r a n s f o r m e r terms are i g n o r e d . The r e s u l t s of a t h r e e phase f a u l t at the remote bus , c l e a r e d a t 5 c y c l e s by o p e n i n g the l i n e , a r e shown i n f i g u r e s 5 .4 to 5 . 7 . From t h e s e r e s u l t s i t c a n be c o n c l u d e d t h a t t h e m e t h o d a c t u a l l y r e p r e s e n t s t h e b e h a v i o u r i n t h e t i m e d o m a i n of the machine whose t r a n s f e r f u n c t i o n s a r e g i v e n ( o b t a i n e d i n t h i s c a s e f r o m s t a n d a r d d a t a ) , a n d , h e r e a g a i n , i t i s c o n f i r m e d t h a t t h e m o d e l l i n g o f s a t u r a t i o n i s a c c u r a t e enough. I t was a l s o o b s e r v e d d u r i n g t h e s i m u l a t i o n s t h a t the method under t r i a l d i d have a very good c o n v e r g e n c y , a n d , i n g e n e r a l , d i d n o t r e q u i r e a d d i t i o n a l i t e r a t i o n s o t h e r than those a l r e a d y needed f o r n o n l i n e a r l o a d s . 5 .5) Usage of the Model f or Speeding up the s o l u t i o n i n a  S t a b i l i t y Program C o n s i d e r a b l e a d v a n t a g e s can be d e r i v e d from the use of t h e mode l d e v e l o p e d i n t h i s d i s s e r t a t i o n by u s i n g l a r g e i n t e g r a t i o n s t e p s , w i t h c o r r e c t i o n s i n the f r e q u e n c y domain in chosen to minimize the d i s c r e t i z a t i o n e r r o r (see s e c t i o n 2 . 3 C h a p t e r 2 ) . In t a b l e 5 . 1 , t h e r e s u l t s o f s e v e r a l -121 -FIGURE 5. 3- A : FUNCTIONS USED FOR VALIDATION OF THE STABIL1TT FRDORftM : D-AXIS 80 f 26 f 0 -i 1—| | | i nil 1—I l I I llll 1—1 I I I 1111 1—I I l l l i n 1—l l l I 111 70 50-1 20 18 f 60f 164- -10 I 14 + 4 0 f 124- 20 ± -.30+ ~ 10+ 84- -30 + l O f 64-Of 4 - 10+ 24 -40 + -204- 04- -50 AF3(s) „ — i — i i i 111ii _ - i—i i i i IIII i—i i i i u n 1—i i i I I I I L 1—i i i i r11 0. OCl 0. 010 0. 100 \ 10 FRO (HZ) tFI&URE 5.3- S : FUNCTIONS USED FOR VALIDATION OF THE STH3ILITT PRODRRM ; C-AX15 4 0 4 - OA 30 + 20 18 + 20+ -10-f 16 + 10 + 14 + 0+ -20+ 12 + -io+ f i i o + - 1 — I I I I u n 1—i I I I n i l 1—I I i i IIII 1—I I I i n n 1—I i M i n -20+ -30 -30 + 6 + -40+ -40 + 4 + -504 -60+ -50 + 0 0. 001 1 1 1 ""b'.oio1 1 1 1 1 "b'.ioo' 1 " " T ' ' 1 ""l'o ' ' " " " FRO (HZ) - 1-22 -IFJOURE 5.4 i VALIDATION OF THE METHOD USED IN PSS/ED POWER ANGLE 2 0 . 1 6 l ° 15 J . 12 + 11 10 +- J ,., J + + +-0.500 1.000 1.500 2.000 2.500 3.000 3.500 4.000 4.500 5.000 TIME ISEC) FIGURE 5.5 i VALIDATION OF THE METHOD USED IN PSS/ED MECHANICAL AND ELECTRICAL POWER 0 . 3 0 0.27 + 0 . 2 4 + 0 . 2 1 + 0 . 1 8 + 0 . 1 5 + 0.12 V 0.09 0.06 T 0.03 0.00 I I I I I I I I PSS/ed P. »ech J .1 „ . .1 J 4 -0.500 1.000 1.500 2.000 2.500 3-000 3.500 4.000 4.500 5 TIME (SEC) 000 - 123 -FIGURE 5.6 i VALIDATION OF THE METHOD USED IN PSS/ED FIELD VOLTAGE IEFD) AND CURRENT UFDI 1.75 1.71 1.67 1.63 1 . 5 9 1 . 5 5 1 . 5 1 1 .47 1 . 4 3 1 . 3 9 1 . 3 5 0.50 2.0 1.8 1.6--1 . 4 1 . 2 . . 1 . 0 - -0 . 8 - " 0 . 6 0 . 4 " 0 . 2 0 . 0 I I E f d ( t ) New New rr———r-o'.500 l'.OOO l'.500 2'.000 2'.500 3'.000 3'.500 4 .000 4'.500 5.000 TIME ISEC1 0.45 - 1.07. 0.40.. 1.04 0.35.. 1.01.. 0.30 FIGURE 5.7 : VALIDATION OF THE METHOD USED IN PSS/ED TERMINAL CURRENT AND VOLTAGE 1.10. 0.98 0 . 2 5 - - - 0 . 9 5 . . 0 . 2 0 - 0 . 9 2 . 0 . 1 5 - 0 . 8 9 " 0 . 1 0 - - 0 . 8 6 0 . 0 5 - " 0 . 8 3 " 0 . 0 0 . - 0 . 8 0 I I PSS/ed *t(t> -New , PSS/ed o'.500 l'.OOO l'.SOO 2'.000 ^.SOO s'.OOO 3'.500 V.DOO 4'.500 5.000 TIME I SEC) TABLE 5 . 1 : R e s u l t s obta ined by t e s t i n g the model wi th  d i f f e r e n t i n t e g r a t i o n s teps A t . CASE A t ( sec ) C . P . U . ( sec ) TIME % OBSERVATIONS BASE 0,00833 257.64 100.0 An i n t e g r a t i o n s tep commonly used i n s t a b i l i t y was used INCREASE IN 5 THE A t 0.0415 56.86 22.06 There i s no d i f f e r -ence wi th the base case INCREASE IN 10 THE A t 0.0833 28. 26 10.96 There are l a r g e e r r o r s SAME CASE WITH CORREC. 0.0833 36.49 14.16 There are s m a l l e r r o r s - 125 -exper iments u s i n g d i f f e r e n t i n t e g r a t i o n s t e p s are summarized. In t h i s t a b l e i t can be o b s e r v e d t h a t r e d u c t i o n s up to 80 % of the computer time were o b t a i n e d , but these s a v i n g s c o u l d be s l i g h t l y l e s s f o r a system l a r g e r than the one modeled. These e x p e r i m e n t s a l s o d emon s t ra t e the u s e f u l n e s s o f the c o r r e c t i o n o f the d i s c r e t i z a t i o n e r r o r . F i g u r e 5 .8 to 5.11 show a c o m p a r i s o n of the r e s u l t s o b t a i n e d u s i n g a very l a r g e i n t e g r a t i o n s t e p ( 5 C y c l e s ) . I t can be o b s e r v e d t h a t the e r r o r s are kept s m a l l o n l y i n the case where c o r r e c t i o n s were made. 5 .6) E v a l u a t i o n of the Impact of the T r a n s f o r m e r Terms I t was ment ioned b e f o r e that i n s t a b i l i t y s i m u l a t i o n s the d e r i v a t i v e s o f the f l u x w i t h r e s p e c t to t i m e a r e n o r m a l l y n e g l e c t e d , and t h i s can cause l a r g e e r r o r s i n the f r e q u e n c y domain ( see f i g u r e 2 .10 ) . In t h i s s e c t i o n , the e f f e c t s t h a t t h e s e d e r i v a t i v e s o r t r a n s f o r m e r t e r m s h a v e on a t i m e s i m u l a t i o n a r e e v a l u a t e d by r u n n i n g t h e same c a s e i n a s t a b i l i t y p r o g r a m , b o t h w i t h c o r r e c t i o n f o r t h e s e t r a n s f o r m e r t e r m s as i n d i c a t e d i n f i g 2 . 1 0 a n d w i t h o u t c o r r e c t i o n . T h e r e s u l t s o f a s i m u l a t i o n u s i n g t h e e l e c t r o m a g n e t i c t r a n s i e n t s program EMTP a r e a l s o i n c l u d e d as a r e f e r e n c e o n l y . In f i g u r e s 5 .12 to 5 . 1 5 , the r e s u l t s of t h i s s i m u l a t i o n a r e s h o w n , a n d we c a n o f f e r f r o m t h e m t h e f o l l o w i n g o b s e r v a t i o n s : a ) The c a s e w i t h o u t the t r a n s f o r m e r t e r m s has an i n i t i a l -126 -20 [FIGURE 5.8 : TEST OF THE REDUCEO ORDER MODEL WITH AND WITHOUT CORRECTION ; POWER ANGLE 1 9 + 1 8 4-1 7 + 1 6 J . 1 5 + U+, 1 3 1 2 + 11 Hot Correc ted 10 d'.SOOT.000 1.500 2.000 2-500 3.000 3.500 4.000 4.500 5.000 TIME I SEC) 0.30. FIGURE 5.9 : TEST OF THE REDUCED ORDER MODEL WITH AND WITHOUT CORRECTION ; ELETRICAL POWER 0.27 J . 0.24 0 . 2 1 J -0 . 1 8 0 . 1 5 J . 0 . 1 2 - 4 -0.09 + 0.06 + 0.O3 + 0.00 I I Mot Corrected 4 - +• 0.500 1-000 1.500 2-000 2-500 3.000 3.500 4.000 4.500 5 TIME I SEC) 000 -127 -FIGURE 5.10 i TEST OF THE REDUCED ORDER MODEL WITH AND WITHOUT CORRECTION ; FIELD CURRENT IIFD) 1.50. 1.48-. 1.46.. 1.44.. 1.42 = 1 . 4 0 . . \ 1.38--1.36 1.34 - -1.32 " " 1.30 C o r r e c t e d O r i g i n a l 4- 4- 4 - 4-O.SOO 1.000 1.500 2.000 2.500 3.000 3.500 4.000 4.500 5-000 T I M E I S E C ) FIGURE 5.11 : TEST OF THE REDUCED ORDER MODEL WITH AND WITHOUT CORRECTION ; TERMINAL CURR. t VOLT. 0 . 5 0 X 1 .10 . 0 .45 - 1 . 0 7 - -0 .40 0.15 0.10 0.05 0.00 l.oi -- jr 0.35-- 1 010.30 - 0 . 9 8 . . 0.25-" 0 . 9 5 - -0.20-" " 0.92 --0.89 0 .86 - -0.83 0.80 O r i g i n a l .1 .1 J — J O'.SOO I'.OOO l'.SOO 2'.000 2'.500 s'.OOO s'.SOO 4'.000 4'.500 5.000 T I M E l S E C ) - 128 -l a r g e r o v e r s h o o t t h a n i n t h e o t h e r two c a s e s ( EMTP and the case w i t h the t r a n s f o r m e r t e r m s ) , w h i c h both have a b o u t t h e s a m e s i z e . H o w e v e r , i n t h e l o n g r u n , t h e c a s e s w i t h and w i t h o u t t r a n s f o r m e r t e r m s t e n d to e q u a l i z e t h e i r magnitude and f r e q u e n c y . T h i s r e s u l t i s not u n e x p e c t e d , b e c a u s e as t i m e g o e s o n , t h e r e i s l o w e r harmonic c o n t e n t i n the t r a n s i e n t . In the end both r e s u l t s are lower than the E M T P ' s . b ) T h e r e s u l t s f r o m E M T P h a v e s o m e i n i t i a l f a s t o s c i l l a t i o n s t h a t a r e n o t r e p r e s e n t e d w i t h t h e o t h e r m o d e l s . A l s o , i n t h i s s i m u l a t i o n , t h e p o w e r a n g l e d e c r e a s e d b e f o r e s t a r t i n g to go up (see f i g u r e 5 . 1 6 ) . T h i s d i p i n t h e power a n g l e c a u s e d t h e s l i g h t d i f f e r e n c e i n p h a s e s between t h i s s i m u l a t i o n and the o t h e r t w o . T h i s b e h a v i o u r can be s t u d i e d a n a l y t i c a l l y [ 2 0 ] , and i t can be t r a c e d back to a L - C - R t r a n s i e n t i n the l i n e t h a t i s not r e p r e s e n t e d i n s t a b i l i t y s i m u l a t i o n s . From the o b s e r v a t i o n s above, i t can be conc luded that the i n c l u s i o n of the t r a n s f o r m e r terms i n the low frequency range has very l i t t l e impact i n the time domain s o l u t i o n . But s i n c e the r e s u l t s are s l i g h t l y b e t t e r wi th the t r a n s f o r m e r terms, and s i n c e t h e i r i n c l u s i o n does not make the model any more c o m p l i c a t e d , they shou ld be r e t a i n e d . 5 .7) C o n c l u s i o n s In t h i s c h a p t e r , we have d i s c u s s e d a way i n w h i c h the new m o d e l can be i n c o r p o r a t e d i n t o a s t a b i l i t y p r o g r a m . I t s 129 -FIGURE 5.13 EVALUATION OF THE EFFECT OF TRANSFORMER TERMS: ELECTRICAL POWER TIME ( sec ) - 130 -FIGURE 5.14 EVALUATION OF THE EFFECT OF TRANSFORMER TERMS: VOLTAGE IN THE D AND Q AXIS 1 l - 1 1- • 1 1 1 1 1 < 1 1 ( , , . With t r a n s f o r m e r terms ^ / Without t r a n s f o r m e r terms j f l t f f n - v ( t > 1 1 1 1 1 1 l l l l 1 1 1 _L. . ' _ 1 - 1 TIME ( sec. ) FIGURE 5.15 EVALUATION OF THE EFFECT OF TRANSFORMER TERMS CURRENT IN THE D AND Q AXIS TIME ( sec. ) - 131 -FIGURE 5.16 EVALUATION OF THE EFFECT OF TRANSFORMER TERMS BACKSWING 1 1 i 1 1 1 1 1— IN 1-THE POWER — i 1 — ANGLE — ' 1 1 f 1 — • — i — , — — 10.0 iO.O • }0.0 «0T5 ?0~0 soTo ToTo fcTo 99^ 0 100.0 110.0 1J0.0 1)0.0 140.0 J'K.0 IM.O 170.1 160.0 110.0 TIME ( m i l i - s e c ) - 132 -a c c u r a c y has been p r o p e r l y a s s e r t e d by the c o m p a r i s o n o f the r e s u l t s o b t a i n e d a g a i n s t the o u t p u t o f a s t a n d a r d model i n which s a t u r a t i o n was f u l l y taken i n t o a c c o u n t . The advantages of the new mode l , b e s i d e s the o b v i o u s one o f b e i n g a b l e to use f r e q u e n c y r e s p o n s e d a t a d i r e c t l y , were e x p l o r e d and t h e y c a n be s u m m a r i z e d i n t h e n u m e r i c a l s t a b i l i t y o f the model and the easy way i n which the i n p u t data ( f r e q u e n c y re sponse ) can be handled i n o r d e r to min imize the d i s c r e t i z a t i o n e r r o r made when u s i n g l a r g e i n t e g r a t i o n s t e p s . These improvements were used to speed up the s o l u t i o n i n a s t a b i l i t y program. When the program was t e s t e d , i t produced a r e d u c t i o n of 85.4 % i n the C . P.U. t i m e . T h e r e f o r e i t can be c o n c l u d e d t h a t the new model i s not o n l y more a c c u r a t e i n t h a t the best data a v a i l a b l e can be used for the m o d e l l i n g of t h e m a c h i n e , b u t i s s i g n i f i c a n t l y f a s t e r t h a n s t a n d a r d models . B e f o r e f i n i s h i n g t h e s e c o n c l u s i o n s , i t i s i m p o r t a n t to m e n t i o n t h a t i n o r d e r to take f u l l a d v a n t a g e o f the use of l a r g e i n t e g r a t i o n t i m e s t e p s , i t i s i m p o r t a n t to d e v e l o p a p p r o p r i a t e models f o r the e x c i t e r and o t h e r c o n t r o l s which can have very s m a l l t ime c o n s t a n t s . A p r o m i s i n g way to model t h e s e c o n t r o l i s to use the t r a p e z o i d a l r u l e , complemented by some l o g i c tha t makes changes i n v a r i a b l e s i n s t a n t a n e o u s l y when these changes occur wi th smal l time c o n s t a n t s . Dommel and Sato worked on t h i s aspect as par t of the " E x p e r i m e n t a l T r a n s i e n t S t a b i l i t y Program" [ 9 ] . Some of t h e i r r e s u l t s are - 133 -r e p r o d u c e d i n f i g u r e 5 . 1 7 . In t h i s f i g u r e , we o b s e r v e t h a t the a p p r o x i m a t e model w i t h l a r g e i n t e g r a t i o n s t e p s r e p r o d u c e the r e l e v a n t b e h a v i o u r of the e x c i t e r . In any c a s e , i f doubts p e r s i s t about the r e l e v a n c e of the s m a l l t ime c o n s t a n t s , the program can a lways be used to narrow the s tudy down to a few c r i t i c a l c a s e s , t h a t can be a n a l y s e d more c a r e f u l l y w i t h a s m a l l e r time s t e p . - 134 -FIGURE 5 .17: E f f e c t of us ing a l a r g e i n t e g r a t i o n step i n a s p e c i a l l y des igned e x c i t e r model . F i g u r e reproduced from r e f . [ 2 3 ] , - 135 -CHAPTER 6  CONCLUSIONS In t h i s d i s s e r t a t i o n , the s t a t e - o f - t h e - a r t i n synchronous mach ine m o d e l l i n g was r e v i s e d and new mode l s were p r o p o s e d f o r b o t h s t a b i l i t y a n d e l e c t r o m a g n e t i c t r a n s i e n t s s i m u l a t i o n s . The fundamental c h a r a c t e r i s t i c s o f these models , as w e l l as t h e i r main advantages and l i m i t a t i o n s , are p o i n t e d out below: Model 1 a ) T h i s m o d e l c a n u s e f r e q u e n c y r e s p o n s e m e a s u r e m e n t s d i r e c t l y , and , t h e r e f o r e , the frequency dependent behav iour o f the damper w i n d i n g s i s m o d e l l e d a c c u r a t e l y w i t h o u t a s suming any g i v e n number of c o n s t a n t p a r a m e t e r w i n d i n g s in p a r a l l e l . b) For the c o n s i d e r a t i o n of s a t u r a t i o n , a new method was d e v e l o p e d i n which the s a t u r a t i o n curve i s l i n e a r i z e d a b o u t one o p e r a t i n g p o i n t ( c a l l e d m e t h o d 1 i n t h i s d i s s e r t a t i o n ) . I t was d e m o n s t r a t e d t h a t i f a c o r r e c t i n g term i s i n c l u d e d f o r the f i e l d c u r r e n t , the e r r o r i n a l l v a r i a b l e s becomes very s m a l l f o r a l l p r a c t i c a l s i t u a t i o n s , e x c e p t some s p e c i a l c a s e s , s u c h a s a s u s t a i n e d s h o r t c i r c u i t , where i t c o u l d have l a r g e e r r o r s . c ) In t h i s m o d e l , t h e f a c t t h a t t h e r o t o r w i n d i n g s a r e more c l o s e l y coup led among themse lves than wi th the s t a t o r i s t a k e n i n t o a c c o u n t , u s i n g an e q u i v a l e n t c i r c u i t proposed by Canay i n [ 3 ] , - 136 -d ) A v e r i f i c a t i o n o f t h e e f f e c t o f u s i n g d i f f e r e n t i n p u t da ta was c a r r i e d out u s i n g the r e s u l t s from a f i e l d t e s t p e r f o r m e d by O n t a r i o H y d r o . I t was f o u n d t h a t i f t h e f r e q u e n c y r e s p o n s e d a t a i s u s e d d i r e c t l y , t h e e r r o r between the measured and s i m u l a t e d r e s p o n s e i s m i n i m i z e d . Model 2 a ) T h i s model was d e v e l o p e d f o r t h o s e c a s e s i n w h i c h the a s s u m p t i o n s i m p l i c i t i n model 1 c o n c e r n i n g s a t u r a t i o n are not v a l i d ( f o r example s u s t a i n e d s h o r t c i r c u i t c o n d i t i o n s ) . I t a l s o p r o v i d e d the f i r s t e x p e r i m e n t a l e v i d e n c e t h a t l ed to the d e v e l o p m e n t o f model 1. In model 2 s a t u r a t i o n i s f u l l y taken i n t o account by s w i t c h i n g from one s a t u r a t i o n segment i n t o another whenever i t i s n e c e s s a r y . b) T h i s model can only use standard s h o r t c i r c u i t t e s t data f o r the i n p u t , but t h i s i s not seen as a r e a l l i m i t a t i o n , because , as p o i n t e d out i n C h a p t e r 2, the type of s t u d i e s i n which t h i s m o d e l i s l i k e l y t o be n e e d e d u s u a l l y i m p l i e s a s u s t a i n e d h i g h c u r r e n t i n the m a c h i n e t h a t makes t h e s e types of data the best c h o i c e . c ) The main a d v a n t a g e s of t h i s model over the s t a n d a r d ones a r e i t s n u m e r i c a l s t a b i l i t y and the c o n s i d e r a t i o n o f the i r o n gap l e a k a g e . A n o t h e r p r a c t i c a l a d v a n t a g e i s t h a t a u s e r who i s f a m i l i a r w i t h model 1 can e a s i l y a d a p t h i s data to t h i s model . - 137 -Model 3 a ) T h i s m o d e l was d e v e l o p e d f r o m m o d e l 1 t o be u s e d i n s t a b i l i t y s i m u l a t i o n s , s i n c e the a s s u m p t i o n s behind model 1 a r e a l w a y s t r u e i n t h e s e s i m u l a t i o n s . T h e r e f o r e , the model can use e i t h e r the frequency r e s p o n s e da.ta as the i n p u t , o r t h e b e s t t y p e o f d a t a a v a i l a b l e . b) As i n s t a b i l i t y s i m u l a t i o n s , t h e speed of the c a l c u l a t i o n s i s v e r y i m p o r t a n t . C o n s i d e r a b l e a d v a n t a g e s were d e r i v e d from t h i s model by m a n i p u l a t i n g the i n p u t da ta i n o r d e r t o r e d u c e the o r d e r of the mode l and by c o r r e c t i n g the d i s c r e t i z a t i o n e r r o r i n c u r r e d by t h e u s e o f l a r g e i n t e g r a t i o n s t e p s . T h i s m a n i p u l a t i o n , t o g e t h e r w i t h the h i g h n u m e r i c a l s t a b i l i t y o f t h e m o d e l , a l l o w s f o r a r e d u c t i o n of more t h a n 80 % i n the C . P . U . t i m e w i t h o u t s i g n i f i c a n t l y d e t e r i o r a t i n g the r e s p o n s e of t h e m o d e l . B u t , as p o i n t e d o u t i n C h a p t e r 5 i t i s n e c e s s a r y to d e v e l o p a p p r o p r i a t e m o d e l s f o r the e x c i t e r b e f o r e f u l l advantage can be d e r i v e d from the use o f l a r g e i n t e g r a t i o n s t e p s . S u m m a r i z i n g , i n t h i s d i s s e r t a t i o n a new way f o r m o d e l l i n g t h e s y n c h r o n o u s m a c h i n e was p r e s e n t e d w h i c h h a s many a d v a n t a g e s over t r a d i t i o n a l m e t h o d s , as p o i n t e d out a b o v e . The o v e r a l l m o d e l l i n g t e c h n i q u e was i n v e s t i g a t e d i n d i f f e r e n t c o n t e x t s and a p p l i c a t i o n s , d e m o n s t r a t i n g w i t h o u t any doubt the r e l e v a n c e of i t s advantages . I n c o n n e c t i o n w i t h f u t u r e r e s e a r c h , an i m p o r t a n t - 138 -i m p r o v e m e n t to t h e method p r o p o s e d i n t h i s d i s s e r t a t i o n would be i t s m o d i f i c a t i o n so t h a t o n l i n e f r e q u e n c y r e s p o n s e measurements [16] can be i n c o r p o r a t e d i n o r d e r to complement t h e s t a n d s t i l l m e a s u r e m e n t s , a n d , h e n c e , o v e r c o m e the problem of the low c u r r e n t s d u r i n g i t s e v a l u a t i o n . A n o t h e r i n t e r e s t i n g c o n s i d e r a t i o n f o r f u t u r e r e s e a r c h w o u l d be t h e e v a l u a t i o n o f d y n a m i c e q u i v a l e n t s , f o r r e d u c i n g l a r g e p a r t s of the network i n s t a b i l i t y or t r a n s i e n t s t u d i e s . In t h i s r e s e a r c h , a p p r o p r i a t e f r e q u e n c y f u n c t i o n s must be d e f i n e d and e v a l u a t e d i n a way t h a t i s e q u i v a l e n t to t h e c h a r a c t e r i s t i c f u n c t i o n s o f a s i n g l e s y n c h r o n o u s machine. - 139 -REFERENCES [ I ] O l i v e , D . W . , " M o d e l l i n g Synchronous Mach ines f o r D i g i t a l S t u d i e s " , IEEE t u t o r i a l , 1 9 8 0 . [2] K u n d u r , P . , Dandeno, P . L . , " V a 1 i d a t i o n of T u r b o g e n e r a t o r S t a b i l i t y Models by Comparison wi th Power System T e s t s " , IEEE T r a n s . , PAS-100, pp. 1637-1646, A p r i l , 1981. [3] C a n a y , I . M . , "Causes o f D i s c r e p a n c i e s on C a l c u l a t i o n of R o t o r Q u a n t i t i e s and E x a c t E q u i v a l e n t D iagrams of the Synchronous M a c h i n e " , I E E E T r a n s . , PAS-88 , pp. 1114-1120 , J u l y , 1969. [4 ] O n t a r i o H y d r o , " D e t e r m i n a t i o n o f S y n c h r o n o u s M a c h i n e S t a b i l i t y S t u d y C o n s t a n t s " , Volume 2, E P R I , E L - 1 4 2 4 , 1980. [5] M a r t i , J . R . , " A c c u r a t e M o d e l l i n g o f F r e q u e n c y - D e p e n d e n t T r a n s m i s s i o n L i n e s " , IEEE T r a n s . , P A S - 1 0 1 , pp. 147-155, J a n u a r y , 1982. [6] I E E E W o r k i n g Group R e p o r t , "Recommended P h a s o r D iagram F o r S y n c h r o n o u s M a c h i n e s " , I E E E T r a n s . , P A S - 8 8 , p p . 1593-1610, November, 1969. [7] A d k i n s . B . , H a r l e y R . , "The G e n e r a l Theory o f A l t e r n a t i n g C u r r e n t Machines" , Chapman and H a l l , London, 1975. [8] Anderson , P . , Fouad, A . , "Power System C o n t r o l and S t a b i l i t y " , Iowa S t a t e U n i v e r s i t y P r e s s , Iowa, 1977. [9] Dommel, H . W . , S a t o , N . , "Fast T r a n s i e n t S t a b i l i t y S o l u t i o n s " . I E E E T r a n s . , P A S - 9 1 , pp . 1643-1650 , J u l y / August , 1972. [10] Canay, I . M . , " I d e n t i f i c a t i o n and D e t e r m i n a t i o n of Synchronous Machine Parameters" , Brown B o v e r i Review, V o l 71, J u n e / J u l y , 1984. [ I I ] M a r t i , J . R . ,"Work i n p r o g r e s s at the U n i v e r s i t y of B r i t i s h Columbia under the O n t a r i o Hydro G r a n t " , Dep. of E l e c t r i c a l E n g i n e e r i n g , U . B . C . , 1985. [12] K r a u s e , P . C . , N o z a r i , F . , S k v a r e n i n a , T . L . , O l i v e , D . W . , "The Theory of N e g l e c t i n g S t a t o r T r a n s i e n t s " , IEEE T r a n s . , P A S - 9 8 , pp. 141-148, J a n u a r y / F e b r u a r y , 1979. [13] Kuo, B . C . , " A u t o m a t i c C o n t r o l Sys t ems" , P r e n t i c e - H a l l , New J e r s e y , 1975. - 140 -[14] H a r l e y , R . G . , L imebeer , D . J . M , C h i r r i c u z z i , E . " C o m p a r a t i v e Study of S a t u r a t i o n Methods i n Synchronous Machine Mode l s" , IEE P r o c , V o l 127, J a n u a r y , 1980. [ 1 5 ] D o m m e l , H . W . , " D i g i t a l C o m p u t e r S o l u t i o n o f E l e c t r o m a g n e t i c T r a n s i e n t s i n S i n g l e and M u l t i - P h a s e Networks", IEEE T r a n s . , PAS-88 , pp . 388-399, A p r i l , 1 9 6 9 . [16] Dandeno, P . L . , K u n d u r , P . , P o r a y , A . T . , Zeim E l - D i n , H.M " A d a p t a t i o n and V a l i d a t i o n of T u r b o g e n e r a t o r M o d e l P a r a m e t e r s t h r o u g h on L i n e F r e q u e n c y R e s p o n s e Measurements", IEEE T r a n s . , PAS-100, A p r i l , 1981. [17] K r e i d e r , K u l l e r , O s t b e r g . " E c u a c i o n e s D i f e r e n c i a l e s " , Fondo E d u c a t i v o I n t e r a m e r i c a n o S. A . , 1975. [18] Dommmel, H . W . , Dommel, I . I . , " T r a n s i e n t s Program U s e r ' s M a n u a l " , D e 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 , U n i v e r s i t y of B r i t i s h C o l u m b i a , Vancouver B . C . , R e v i s i o n of F e b . 1982. [ 1 9 ] S t a g g , E l - A b i a d , " C o m p u t e r M e t h o d s i n Power S y s t e m A n a l y s i s " , M c G r a w - H i l l , 1968. [20] B a c a l a o , N . J . . " S t u d y of T r a n s i e n t Torques i n Synchronous M a c h i n e s F o l l o w i n g F a u l t I n i t i a l i z a t i o n " , D e p . o f E l e c t r i c a l E n g i n e e r i n g , U n i v e r s i t y of B r i t i s h C o l u m b i a , 1984. [21] S c h u l z . R . P . , J o n e s , W . D . , E w a r t , D . N . , " D y n a m i c Mode l s of T u r b i n e G e n e r a t o r s D e r i v e d from S o l i d R o t o r E q u i v a l e n t C i r c u i t s " , IEEE T r a n s . , PAS-92 , M a y / J u n e , 1973. [22] Dandeno, P . L . , P o r a y , A . T . , "Development of D e t a i l e d T u r b o g e n e r a t o r E q u i v a l e n t C i r c u i t s f r o m S t a n d s t i l l F r e q u e n c y Response M e a s u r e m e n t s " , I E E E T r a n s , P A S - 1 0 0 , A p r i l , 1981. [23] Dommel, H . W . , "WSCC C a s e s S o l v e d w i t h an E x p e r i m e n t a l T r a n s i e n t S t a b i l i t y P r o g r a m " , B o n n e v i l l e P o w e r A d m i n i s t r a t i o n , 1972. [24] I . E . E . E . S tandard D i c t i o n a r y of E l e c t r i c a l and E l e c t r o n i c s Terms. Second E d i t i o n , p u b l i s h e d by I . E . E . E . W h i l e y - I n t e r -s c i e n c e , New Y o r k , 1979. - 141 -APPENDIX 1  THE RECURSIVE CONVOLUTION TECHNIQUE A l . l ) C o n v o l u t i o n with an E x p o n e n t i a l L e t f u n c t i o n g ( t ) be a f u n c t i o n c o n t i n u o u s i n t and l e t - P t f ( t ) be such t h a t f ( t ) = k e U ( t ) , w h e r e k and P might be complex and U ( t ) i s the u n i t s t e p . Then the c o n v o l u t i o n of these two f u n c t i o n s i s g iven by : CO S ( t ) = g ( t ) * f ( t ) = g ( t - u ) k e" P u U(u) du A l . 1 which can be w r i t t e n as : S ( t ) = j g ( t - u ) k e" P u du •"0 A1.2 I t i s i m p o r t a n t to note t h a t f ( t ) i n t h i s c o n t e x t o n l y has meaning f o r t > 0, a n d , f o r t h i s reason i t was m u l t i p l i e d by a u n i t s t e p . F r o m e q u a t i o n A 1 . 2 , we can w r i t e , by m a k i n g t i m e a d i s c r e t e v a r i a b l e ( t = n At) : S ( t - At) = JO -P u g ( t - A t - u ) k e du Al .3 which can be r e a r r a n g e d i n t o : r p A t set - A t ) -l e t t i n g v = A t + u -P v g ( t - v ) k e dv At Al .4 - 142 -S ( t ) = Now i f we w r i t e equat ion ( A l . l ) as At 0 -P u g ( t - u ) k e du + g ( t - u ) k e P u du At Al .5 we n o t i c e t h a t the second term i n A 1 . 5 i s g iven by A 1 . 4 , so S ( t ) = At 0 g(t - u) K e~ P u du + e" P A t S ( t - At) A1.6 and i f we assume t h a t A t i s s m a l l enough so that g ( t ) can be assumed to vary l i n e a r l y d u r i n g i t , then 8( t - u) * - A t > t - *<0 u + g ( t ) A1.7 w h i c h e n a b l e s us to s o l v e the i n t e g r a l i n A 1 . 6 and o b t a i n S ( t ) * b S( t - A t ) + c g ( t ) + d g(t - A t ) where Al .8 b = e -P At h = (1 - b) P At c = k (1 - h) P d = - k (b - h) P which g i v e s S ( t ) as a f u n c t i o n of the c u r r e n t v a l u e of g ( t ) and the past h i s t o r y , i . e . , t h e v a l u e s t h a t g ( t ) and S ( t ) had in p r e v i o u s time s t e p s . - 143 -A1 .2 ) C o n v o l u t i o n with a Impulse Response In t h e d e v e l o p m e n t o f t h e m o d e l f o r t h e s y n c h r o n o u s m a c h i n e , the f o l l o w i n g t y p i c a l t r a n s f e r f u n c t i o n was found : K K n k. F ( s ) £ + £ — + z (1 + s P ) ( 1 + s P ) i = l ( 1 + s P . ) c c y v 1 ' Al .9 * w h e r e K a n d P a r e c o m p l e x n u m b e r s , a n d K a n d c c v ' c P c are t h e i r complex c o n j u g a t e s . T h i s f u n c t i o n , when t rans formed to the time domain, g i v e s : - P t * - p t n - P . t f ( t ) = K e C + K e C + £ K. e 3 c c . , 1 i = l A l .10 The c o n v o l u t i o n o f f ( t ) w i t h g ( t ) can be found term by term us ing e q . A 1 . 8 , so : S C j C t ) = B c S C ; l ( t - At ) + C c g ( t ) + D c g ( t - A t ) S c 2 ( t ) = B* S C ; l ( t - At ) + C* g ( t ) + D*g(t - At) S i ( t ) = B. S i ( t - At) + C. ( t ) + D . g ( t - At) Al . 11 where S c ^ ( t ) and S c 2 ( t ) a r e t h e c o n v o l u t i o n s of g ( t ) w i t h the two complex conjuga te p o l e s , a n d , h e n c e , complex conjuga te t h e m s e l v e s . S i i s the c o n v o l u t i o n o f g ( t ) w i t h the i - t h term i n equat i on A l . 1 0 . Adding these terms t o g e t h e r , we o b t a i n - 144 -S ( t ) = f ( t ) * g ( t ) - C g ( t ) + H( t ) Al .12 where C = 2 Re{ C ) + I C " "i Al .13 i - 1 1 and H ( t ) i s f u n c t i o n of the past h i s t o r y : n H( t ) = ( 2 Re{ D } + Z D . ) g ( t - At) + 1 = 1 n 2 Re{ B c S c j ( t - At) } + Z B j S i ( t - At ) i = 1 A1.14 T h i s e q u a t i o n can be used to f i n d the c o n v o l u t i o n of f ( t ) with g ( t ) r e c u r s i v e l y . - 145 -APPENDIX 2 BLOCK DIAGRAMS OF EXCITERS AND GOVERNORS USED IN  STABILITY SIMULATIONS A2.1) E x c i t e r The b lock diagram of the e x c i t e r model used f o r t e s t i n g the model i n s t a b i l i t y s i m u l a t i o n s can be observed i n f i g u r e A 2 . 1 . T h i s b lock diagram corresponds to an s t a t i c e x c i t e r . FIGURE A2.1 : E x c i t e r b lock diagram STATIC EXCITER (SCRX) V r e f Reference v o l t a g e V t Machine t e r m i n a l v o l t a g e P ss : Output from power system s t a b i l i z e r T a : Lead time cons tant i n f i l t e r T b : Lag time cons tant i n f i l t e r K e : E x c i t e r ga in T e : E x c i t e r time cons tant E max : Maximum output from e x c i t e r E . min : Minimum output from e x c i t e r E f d : E x c i t a t i o n v o l t a g e r e f e r r e d to s t a t o r - 146 -A2.2 ) Governor F i g u r e A2.2 below i s the b lock diagram of the governor model used i n t h i s d i s s e r t a t i o n . I t corresponds to a governor for a h y d r a u l i c g e n e r a t o r , s i n c e t h i s type of machine was used i n the t e s t s . FIGURE A 2 . 2 : Governor b lock digram HYDRO TURBINE GOVERNOR (HYGOV) W r e f - ^ W \ 1 1 1 + T f s 3 A t —K^)—*P mech l n l D tur ~i W w : Machine speed tur T u r b i ne damping R : Permanent droop q n l : No load flow r . Temporary droop W , : r e f Reference speed T r T f : Governor time cons tant : F i l t e r time cons tant P , : Mechan ica l output mech Aj_: T u r b i n e Gain T g : Servo time cons tant T w : Water time c o n s t a n t G max G . mi n : Maximum gate l i m i t : Minimum gate l i m i t - 147 -APPENDIX 3 DESCRIPTION OF THE STAND-STILL FREQUENCY RESPONSE  METHODS FOR THE EVALUATION OF X D ( S ) . XQ(S) AND G(S) In t h i s Appendix we w i l l b r i e f l y o u t l i n e a method that can be used to e v a l u a t e the o p e r a t i o n a l inpedances of the synchronous machine ^ ( s ) and * q ( s ) a s w e l l as the t r a n s f e r f u n c t i o n G ( s ) . T h i s method i s based on a s e r i e s of t e s t s which have the f o l l o w i n g g e n e r a l c h a r a c t e r i s t i c s : a) The machine i n a l l of them i s at s t a n d - s t i l l , with i t s r o t o r l o c k e d i n a g i v e n p o s i t i o n . b) The f u n c t i o n to be found i s e v a l u a t e d as a r a t i o between the output and s i n u s o i d a l i n p u t (whose frequency i s v a r i e d ) . For t h i s c a l c u l a t i o n , a frequency response a n a l y z e r can be u s e d . In the f o l l o w i n g s e c t i o n s , more d e t a i l w i l l be g iven for the d i f f e r e n t t e s t s to be per formed . A3 .1 ) Measurement of X^(s) In t h i s t e s t , the r o t o r i s a l i g n e d so that i t s magnetic a x i s c o i n c i d e s wi th that of phase a ( B = 0 ° i n e q . 1 . 7a ) , and the i n p u t V (s ) i s a p p l i e d so t h a t : s 2 1 V V V b " V V a 3 s c 3 s - 148 -and I = - I I, = 1 = 1 / 2 a s b c s (b) A3.1 Us ing P a r k ' s t r a n s f o r m a t i o n ( Eq 1.7a) i n the e q u a t i o n s above, we get : ./r V , = V - V V = 0 V = 0 d 2 s 9 ° (a) and X..-/T I 1 = 0 1 = 0 (b) A3.2 In t h i s t e s t the f i e l d winding i s s h o r t c i r c u i t e d (V^ = 0) , so from equat ion 1.20 ( a ) , i t i s p o s s i b l e to prove that when the machine i s at a s t a n d - s t i l l , X^(s) i s g iven by: V (s ) w 0 X H ( s ) = - ( — + r ) — I d ( s ) s A3.3 T h e r e f o r e , s i n c e we know V^(s) and V^(s) from equat ion A 3 . 2 , the equat ion above can be used f o r f i n d i n g X ^ ( s ) . A3 .2 ) Measurement of X ( s ) For t h i s measurement, we have to d i s p l a c e the r o t o r 9 0 ° wi th r e s p e c t to the magnetic a x i s of phase a ( 8 = 9 0 ° ) and app ly the v o l t a g e V g i n the same way as b e f o r e . In t h i s - 149 -V = / - V V , = 0 V = 0 q 2 s " ° and 3 I = / - I I . = 0 1 = 0 q 2 s d ° (a) (b) A3.4 For the e v a l u a t i o n of X ( s ) , we can prove t h a t : V (s) OJ X q ( S ) = - ( - a — + r ) - a As we know V^(s) and V q ( s ) from e q u a t i o n A 3 . 4 , the e q u a t i o n A3.5 can be used for f i n d i n g X q ( s ) . A3.3 ) Measurement of G(s) From the e q u i v a l e n t c i r c u i t i n f i g u r e 1 . 5 a , i t i s p o s s i b l e to prove that : I f ( s ) s G(s) = I d ( s ) A3.6 T h e r e f o r e , i f i n the same se t -up used f o r the e v a l u a t i o n of X^Cs), I^(s) i s a l s o measured, i t i s p o s s i b l e to f i n d G(s) from the f o l l o w i n g r e l a t i o n s h i p : 1 I f ( s ) G(s) = - — s I . ( s ) a A3.7 Note that i n a l l the formulae g iven i n t h i s Appendix the v a r i a b l e s must be p e r - u n i t i z e d u s i n g the a p p r o p r i a t e f a c t o r s , a c c o r d i n g to the per u n i t system chosen (see r e f [ 8 ] ) . - 150 -APPENDIX 4 DEVELOPMENT OF THE INTEGRATION EQUATIONS FOR  METHOD 2 FOR THE CONSIDERATION OF SATURATION ' In C h a p t e r 3 , i t was p r o v e n t h a t f o r t h e c a s e when s t a n d a r d d a t a a r e u s e d and t h e i n i t i a l c o n d i t i o n s a r e r e t a i n e d as p a r t of the model i n the L a p l a c e d o m a i n , the f o l l o w i n g e q u a t i o n s can be d e r i v e d f o r the c u r r e n t i n the d and q a x i s : i d ( t ) = L _ 1 { F l ( s ) } * ( v d 4 , ( t ) + v j k ( t ) ) + + L _ 1 { F2 ( s ) } * ( v ( t ) + L _ 1 { s V j k ( s ) } ) + -1 -1 R f + s 1. + L { F3(s ) } * L { V (s ) + 4* + — i L f , } f f ° R. . + s 1. . k d o kd kd i q ( t ) = L _ 1 { F4(s ) } * ( v d l ( > ( t ) + v j k ( t ) ) + + L _ 1 { F5(s ) } * ( v ( t ) + L _ 1 { s _ V j k ( s ) } ) + w o + L-'i F6( s ) } * L - ^ V f ( s ) + * f o + R f + S 1 1 ^ k d o ) R. . + s 1. , kd kd A4.1 These e x p r e s s i o n s a l l o w to r e s t a r t the i n t e g r a t i o n p r o c e d u r e a t any g i v e n t i m e d u r i n g t h e s i m u l a t i o n , as l o n g as the c u r r e n t and v o l t a g e s in a l l the w i n d i n g s i n the machine are known. As i s e v i d e n t from equat ions A 4 . 1 , the n u m e r i c a l c o n v o l u t i o n method o u t l i n e d i n A p p e n d i x 1 c a n n o t be used - 151 -here w i t h o u t p e r f o r m i n g some m a n i p u l a t i o n of these equat ions So c o n s i d e r the term: L " 1 ! F l ( s ) } * ( L ~ * { V # ( s ) } + L _ 1 { V . k ( s ) } ) = i d l ( t ) i n equat ions A4.1 where : L _ 1 < V d*< s > > * V d ^ ( t ) = V d < f c > + E q o " *do fii^) and L " 1 ! V . k ( s ) } - v j k ( t ) - L'h J ( s ) ^ g Q + K ( s ) ^ k q o } A4.2 Us ing eq . 3 .9e , we can w r i t e f o r the f i r s t term i n V j k ( t ) : ( 1 + s T. ) X V . . . (s) = J ( s ) i|> = ^ 23- ^ J k l 8 0 (1 + s T 1 1) (1 + s T ') R 8 ° qo qo g A4.3 which can be trans formed i n t o : v . , . ( t ) = K. e ~ ( t / T q o " ) + K 9 ^ e ' ^ ^ q o ^ j k l v 1 r g o 2 go where K l K 2 ( 1 / T . - 1/T ") X T. v kq qo ; aq kq ( 1 / T ' - 1/T ") T " T • R qo qo qo qo g ( 1 / T . - 1/T ') X T, kq q o aq kq ( 1 / T " - 1/T ') T " T ' R qo qo qo qo g A4.4 - 152 -and f o r the second term: (1 + s T ) X V (s ) = K(s ) * k = 8 \ J k Z k q ° (1 + s T ") (1 + s T •) R, k q o qo qo ' kq i t can be w r i t t e n as v . . , ( t ) = K , ty. e " ( t / T q o M ) + K, ip. e ' ^ ^ q o ' 5 j k 2 v 3 kqo ^ A kqo M where ( 1 / T - 1/T ") X T K = 8 q ° 3J3 S 3 ( 1 / T » - 1/T ") T " T 1 R, qo q° qo qo kq ( 1 / T - 1/T ') X T K = - s q ° ag s A ( 1 / T " - 1/T ' ) T " T ' R, qo qo qo qo kq So we can w r i t e for V . , ( t ) : j k v ' v . , ( t ) (K. ^ + K„ % ) e " ( t / T d o M ) jk 1 qo 3 kqo + ( K , ^ + K . ik ) e - ( t / t d o ' ) 2 go A kqo AA.5 AA.6 AA.7 So we have f o r i j j C t ) i d l ( t ) = j K H e ~ ( P l i C ) * ( vd(t) + E i = l + ' jk** " " * d o . V i i e _ ( P l 1 ° J 1 = 1 A A . 8 where ^^.^(t) i s a known f u n c t i o n o f t i m e g i v e n by e q u a t i o n A A . 7 . , a n d K ^  a n d P j ^ c o r r e s p o n d t o t h e f r a c t i o n a l e x p a n s i o n of F j ( s ) (see e q s . 2.2 to 2 . A ) . - 153 -Now u s i n g the i m p l i c i t c o n v o l u t i o n t e c h n i q u e i n A 4 . 8 , we can w r i t e : ^ 1 ^ - j x S l i ( t ) " ^ d o . ^ K l i e " ( ? l i ° wi th S H ( t ) = c H v d ( t ) + c u [ E q o . + v . k ( t ) ] + b u S l i ( t - At) + d ^ [ v d ( t - At) + E q o . + v . k ( t - A t ) J A 4 . 9 where B^.. and D j . . a r e t h e same c o n s t a n t s d e s c r i b e d i n A p p e n d i x 1. T h i s e q u a t i o n can be used d i r e c t l y i n the i n t e g r a t i o n p r o c e d u r e , b u t , a s i t was neces sary to make time a d i s c r e t e v a r i a b l e ( t = n A t ) i n o r d e r t o p e r f o r m t h e n r c o n v o l u t i o n i n eqs A 4 . 8 , f u r t h e r s i m p l i f i c a t i o n c a n be obta ined i f we a c c e p t : " *do K l i e " ( ? l i ° " S ' l i ( t ) - h i S ' l i ^ " A t > where S' . (0) = - K, . , l i v ' l i d o A 4 . 1 0 w h i c h has the same l e v e l o f e r r o r as e q u a t i o n A 4 . 9 . Then equat ion A 4 . 9 can be r e w r i t t e n as : S H ( t ) = c n v d ( t ) + c H ( E q o . + v j k ( t ) ) + b H S H ( t ) + d l i ( v d ( t " A t ) + E q o + v j k ^ " A t ) ) A 4 . 1 1 where S ^ ( t ) , r e d e f i n e d such t h a t a t t ime e q u a l to z e r o or a t the t ime of r e i n i t i a l i z a t i o n of the m o d e l , i s g i v e n by: S. . (0) = - K l i ^do 11 - 154 -F i n a l l y , i t i s i n t e r e s t i n g t o n o t e t h a t f o r s p e e d i n g up c a l c u l a t i o n s , V j ^ ( t ) i n e q u a t i o n A 4 . 1 0 can be e v a l u a t e d a p p r o x i m a t e l y with the f o l l o w i n g e x p r e s s i o n : where v j k ( t ) = S j k l ( t ) + S j k 2 ( t ) and S j k l ( t ) = bj S j k l ( t - At) S j k 2 ( t ) = b 2 S j k 2 ( t - At) b x = e - ( A t / T q o " ) ^ m e - ( A t / T q o ' ) S j k l ( O ) = KI ^go + K3 Hqo Sjk2(0 ) = K2 ^go + K4 ^kqo A4.12 U s i n g the same procedure and assumpt ions g i v e n above , we can prove f o r the second term i n equat ion A 4 . 1 , i d 2 ( t ) = L - 1 { F 2 ( s ) } * L - V ^ s ) - ^ _ / s + _s_ V j k ( s ) } A4. 13 that i t can be t rans formed i n t o : m i ( t ) = z S 2 i ( t ) Q Z « i = l where S 2 i ( t ) = c 2 i v q ( t ) + c 2 . [ v g j k ( t ) - Edo ] + b 2 . S 2 i ( t - A t ) + d 2 . [ v ( t - A t ) - Edo + v g j k ( t - A-t) ] -155 -and K l + K 2 „ K 3 + K 4 S 2 I ( 0 ) = ( — i £_ * g o + _ i t - * k q o . * q o ) K , w OJ ° o V j k ( t > = - ( K l + K 3 \ q o ) e - ( t / T q o M ) T to T " 0) qo o qo o - ( h * g o + _ ! i < * k q o ) e - ^ / T q o ' ) T' ' OJ T ' OJ qo o x qo o A4 F i n a l l y f o r the t h i r d term: i d 3 ( t ) = L ! { F3 ( s ) } * L - 1 { V . ( s ) + *fo f l f d / T f + s) 2 k d ( 1 / T k d + S> kqo } A4 we can wri te i = l A4 where S 3 i ( t ) = C 3 i v f ^ ) + c 3 i v f g l ( t ) + b 3 i s3i^ ~ At> + + d 3 . [ v f ( t - At ) + v f g l ( t - A t ) ] and S 3 I ( 0 ) = K 3 . ( "'fo + - i l — ^kdo ) 2 k d v f g l ( t ) - ( 1 / T f - 1 / T k d ) *kdo e - t / T k d " A4 - 156 -From t h e c a l c u l a t i o n s a b o v e , w e can now w r i t e f o r i , ( t ) m i d ( t ) = i d l ( t ) + i d 2 ( t ) + i d 3 ( t ) - z s u ( t ) + z s 2 1 ( t ) 3=1 1=1 m + * S 3 i ( ° A4.18 3=1 which can be w r i t t e n as : i d ( t ) = C1 v d ( t ) + C 2 v q ( t ) + C 3 v f ( t ) + H x ( t ) + H 2 ( t ) where: + H 3 ( t ) m 1 C. = y c . . C 0 = v c 0 . C 0 = y c~. 1 . A .. 11 2 . L , 2i 3 . ^ , 3 i 1=1 1=1 i= l I l j ( t ) = C1 [Eqo + v j k ( t ) ] + _Z b x . S j . ( t - A t ) + ( 1^ d j j ) [ v d ( t - At) + Eqo + v k ( t - A t ) ] m H 2 ( t ) = C 2 [v . k ( t ) - Edo)] + i b 2 S 2 . ( t - At) J i = 1 m + ( E i d 2 . ) [ v q ( t - A t ) - Edo + v s j k ( t - A t ) ] H 3 ( t ) = C , [ v f g l ( t ) ] + ^ b , . S 3 . ( t - A t ) + d 3 . ) [ v f ( t - At) + v f g l ( t - At) ] A4.19 T h i s e q u a t i o n can be used to model the mach ine , t o g e t h e r w i t h the c o r r e s p o n d i n g e q u a t i o n f o r the q a x i s , w h i c h i s g iven by: V ° = c* V d ( t ) + ° 5 V ° + °6 V f ( t ) +H 4 ( t ) + H 5 ( t ) + H 6 ( t ) - 157 -where o p q C , = z c , . C , = Z c c . C , » Z c , . 4 . , 4 i 5 . , 5z 6 . . 6 1 i = l 1=1 1=1 H 4 ( t ) = C 4 [ E q o + v j k ( t ) ] + Z b 4 . S 4 . ( t - A t ) + (_Z^ d^ . ) [ v d ( t - A t ) + Eqo + v k ( t - A t ) ] ! 5 ( t ) = C 5 [ v s j k ( t ) - Edo)] + _ Z i b 5 . S 5 . ( t - At) i = l P + ( ^ d 5 . ) [ v q ( t - At) - Edo + v s j k ( t - At) ] H 6 < f c ) = C 6 f v f g l ( t ) ] + b 6 i S6i<* ~ A t > + d 6 i > [ v f ( t - At) + v f g l ( t - At) ] S 4 i ( 0 ) = - K 4 i ^ d 0 S 5 . ( 0 ) = K 5 i ( —1 2 - V + - 2 & *kqo - *qo ) ' 5 i w o 1 % w o S ^ ( 0 ) = K f t 4 ( * f o + ^kdo) A 4 . 2 0 6 i 61 n 1 k d - 158 -APPENDIX 5 CONSIDERATION OF UNEQUAL FLUX LINKAGES USING AN  EQUIVALENT CIRCUIT In C h a p t e r 1, i t was m e n t i o n e d t h a t i n some cases i t i s i m p o r t a n t to c o n s i d e r t h a t the f l u x t h a t l i n k s the w i n d i n g s i n the r o t o r i s n o t e x a c t l y the same one t h a t l i n k s the s t a t o r , due to some l e a k a g e i n the i r o n gap. T h i s f a c t can be t a k e n i n t o a c c o u n t w i t h t h e e q u i v a l e n t c i r c u i t shown i n f i g u r e A l ; h o w e v e r , t h e s e r i e s b r a n c h X r c r e p r e s e n t s an a d d i t i o n a l c o m p l i c a t i o n i n the s o l u t i o n o f t h i s network. I . M . C a n a y p r o p o s e d i n [ 3 ] to t r a n s f o r m t h i s c i r c u i t i n t o an e q u i v a l e n t c i r c u i t ( see f i g u r e A2 ) w i t h the same form as the t r a d i t i o n a l one , t h u s e n a b l i n g the use o f the f o r m u l a e a l r e a d y developed f o r t h i s c a s e . The p a r a m e t e r s i n the e q u i v a l e n t c i r c u i t r e l a t e to the o r i g i n a l one by : 1 1 1 x d - x c = • + and K = X ~ X - t X i X X j c 1 ad rc ad A5. 1 I n t h i s c i r c u i t i d ( t ) i s t h e same as i n t h e o r i g i n a l c i r c u i t , b u t t h e v a r i a b l e s i n t h e r o t o r c i r c u i t s * * * ( i ^ ( t ) , v ^ ( t ) a n d i ^ ^ ( t ) . ) a r e a s s o c i a t e d ones . The r e l a t i o n s h i p between these v a l u e s and the o r i g i n a l ones can be o b t a i n e d i f we o b s e r v e t h a t the f l u x t h a t l i n k s t h e d - a x i s ^d must be the same i n b o t h c i r c u i t s . T h i s i s - 159 -FIGURE A. .1 : E q u i v a l e n t c i r c u i t f o r the d - a x i s t a k i n g  i n t o account unequal f l u x l i n k a g e s FIGURE A .2 : E q u i v a l e n t c i r c u i t without the s e r i e s branch - 160 -s a t i s f i e d i f : . * = X a d . = 1 kd ^ kd ^ dk dc . * ^ad * X , r K dc v f * = K v f A5.2 T h e r e f o r e , c a r e must be t a k e n when u s i n g t h i s e q u i v a l e n t c i r c u i t i n o r d e r to c o n v e r t back to the o r i g i n a l q u a n t i t i e s before w r i t i n g the r e s u l t of the s i m u l a t i o n . A n o t h e r m a t t e r of c o n c e r n , when u s i n g t h i s e q u i v a l e n t c i r c u i t , i s t h e c o n s i d e r a t i o n of s a t u r a t i o n , b e c a u s e the p a r a m e t e r s i n the c i r c u i t change w i t h the v a l u e o f X . 3 Q T h e r e f o r e o n c e t h e d e c i s i o n f o r c h a n g i n g f r o m one s a t u r a t i o n segment i n t o a n o t h e r i s made ( u s i n g \b . = \b , T m d 1 d - 1 i j ( t ) ) , t h e c u r r e n t i n t h e new c i r c u i t m u s t be S Q e v a l u a t e d from the c u r r e n t i n the o l d one , c o n s i d e r i n g that t h e r e must be c o n t i n u i t y i n the c u r r e n t i n the o r i g i n a l c i r c u i t . So i f we are s w i t c h i n g from segment 1 i n t o 2 , then: 1kdl = 3 k d 2 3 f l = 1 £2 so * * * * K l * k d l = K 2 1 k d 2 K l 3 f l = K 2 ^'fl A5.3 w h i c h g i v e s us t h e f o l l o w i n g r e l a t i o n s h i p b e t w e e n t h e v a r i a b l e s i n the e q u i v a l e n t c i r c u i t s tha t must be m a i n t a i n e d - 161 -when s w i t c h i n g . 1 k d 2 i k d l * f 2 * K 2 i f 2 A5.4 F i n a l l y , i t i s i n t e r e s t i n g to observe tha t c h a n g i n g from one s a t u r a t i o n s e g m e n t i n t o a n o t h e r was done u s i n g the mutua l f l u x t h a t l i n k s the s t a t o r and the r o t o r a c c o r d i n g to t h e o r i g i n a l c i r c u i t . T h i s was done i n t h i s way to be c o n s i s t e n t w i t h the b a s i c a s s u m p t i o n t h a t the s a t u r a t i o n of the leakage i n d u c t a n c e s c o u l d be n e g l e c t e d . - 162 -APPENDIX 6 EFFECT OF SATURATION ON THE MACHINE TIME CONSTANTS In t h i s append ix , the e f f e c t s of s a t u r a t i o n i n some of the c o n s t a n t s that i n f l u e n c e L d ( s ) f o r two very d i f f e r e n t machines are shown. TABLE 1: EFFECT OF SATURATION IN GURI UNIT 7 TO 10 Lad A % Tdo' A% Tdo" A% T d ' A% Td" A% 1.035 00 0.828 20 0.621 40 0.414 60 9.120 00.00 7.576 16.92 6.032 33.85 4.488 50.78 0.0500 0.00 0.0489 2.18 0.0473 5.49 0.0444 11.07 2.723 0.000 2.154 0.864 2.125 2.220 2.072 4.633 0.3285 0.000 0.0326 0.599 0.0323 1.556 0.0318 3.330 A% = Percentage of change wi th r e s p e c t to the u n s a t u r a t e d case TABLE2: EFEECT OF SATURATION IN A FOSSIL-FIRED UNIT* Lad A% Tdo' A% Tdo" A% Td ' A% Td" A% 1.590 00 1.272 20 0.954 40 0.636 60 5.900 00.00 4.820 18.30 3.740 36.60 2.661 54.91 0.0330 0.000 0.0327 0.924 0.0322 2.381 0.0313 5.021 0.850 0.000 0.845 0.654 0.836 1.699 0.819 3.635 0.0249 0.000 0.0249 0.249 0.0248 0.655 0.0245 1.429 A% = Percentage of change wi th r e s p e c t to the u n s a t u r a t e d case * See r e f e r e n c e [8] These t a b l e s c l e a r l y show that f o r both a h y d r a u l i c u n i t and a thermal u n i t , s a t u r a t i o n a f f e c t s most ly the open c i r c u i t time c o n s t a n t s and thus c o n f i r m the assumptions made i n Chapter 2. 

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