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A converter model for the digital simulation of transients in AC/DC transmission systems Chiu, But-Chung 1980

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/ A C O N V E R T E R M O D E L F O R T H E D I G I T A L S I M U L A T I O N O F T R A N S I E N T S I N A C / D C T R A N S M I S S I O N S Y S T E M S ~ b y c 'BUT-CHUNGJOTIU B . E . E . , U n i v e r s i t y o f M i n n e s o t a , 1 9 7 4 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T OF THE R E Q U I R E M E N T S F O R THE D E G R E E OF M A S T E R OF A P P L I E D S C I E N C E i n THE F A C U L T Y OF G R A D U A T E S T U D I E S 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 We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d T H E 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 M a y , 1 9 8 0 (T) B u t - c h u n g C h i u , 1 9 8 0 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Brit ish Columbia, I agree that the Library shall make i t 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 representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of ZttCTWCAL EAJ^/AJEEA /AJGT The University of Brit ish Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date . i i ABSTRACT The successful application of HVDC transmission links requires correct predictions of the performance of the dc link and the ac system to which i t is interconnected. Whatever the system configuration, the steady-state, dynamic and transient behaviour of the associated dc and ac systems are mostly interdependent. To simulate these phenomena with d i g i t a l com-puters, converter stations must be modelled in more detail than as simple dc sources. This thesis discusses the development and implementation of a converter model which enables the converter bridge circuits to be repre-sented in detail and the valve ignition to be controlled in the constant current mode. The model has been added to the U.B.C. Electromagnetic Transients Program to permit simulations of the complete ac/dc system. It is used to analyze the harmonics during steady-state operation, and to compare the results with those obtained from conventional (approximate) formulae. In a t r a n s i e n t case, the new model gives closer agreement with f i e l d measurements than the simplified model used before. i i i T A B L E O F C O N T E N T S Page ABSTRACT i i . MABLE OF CONTENTS i i i LIST OF TABLES v LIST OF ILLUSTRATIONS v i ACKNOWLEDGEMENTS v i i 1. INTRODUCTION 1 2. MODELLING AN HVDC TRANSMISSION SYSTEM 4 2.1 Network Representation 4 2.2 The Converter 5 2.3 Converter Control C h a r a c t e r i s t i c s 8 3. IMPLEMENTATION OF THE CONVERTER MODEL 14 3.1 Subroutine VALCON 14 3.2 Limiter Representation 18 3.3 I n i t i a l i z a t i o n 23 4. HARMONIC ANALYSIS 30 4.1 Analyzing Method 30 4.2 Harmonic Contents . . . 32 4.3 Abnormal Harmonics 36 5. TRANSIENTS SIMULATION 39 5.1 Point to Point Operation 39 5.2 Three Terminal Operation 41 i v Page 6. CONCLUSIONS 46 REFERENCE 48 APPENDIX 1 Subroutine VALCON Programme L i s t i n g s 50 APPENDIX 2 Modifications i n the Transients Program 54 APPENDIX 3 Fourier Analysis Programme L i s t i n g s 59 APPENDIX 4 Line Parameters and Converter Station Parameters of the P a c i f i c HVDC I n t e r t i e 61 APPENDIX 5 Input L i s t i n g s of the Point to Point Operation Simulation 65 APPENDIX 6 Sample Data Deck of the Three-Terminal Operation Simulation 71 V L I S T O F T A B L E S T a b l e P a g e 4 . 1 C o m p a r i s o n o f c u r r e n t h a r m o n i c s o b t a i n e d f r o m t h e o r e t i c a l c a l c u l a t i o n s a n d f r o m F o u r i e r a n a l y s i s o f s i m u l a t e d c u r v e s 34 4 . 2 C o m p a r i s o n o f d c v o l t a g e h a r m o n i c s u n d e r b a l a n c e d o p e r a t i o n a n d i m b a l a n c e i n m a g n i t u d e a n d p h a s e i n o n e p h a s e o f t h e a c b u s 37 5 . 1 S t e a d y - s t a t e s y s t e m p a r a m e t e r s b e f o r e f a u l t 42 A 2 . 1 C h a n g e s i n t h e T r a n s i e n t s P r o g r a m f o r d a t a t r a n s f e r r e d t o s u b r o u t i n e V A L C O N 5 5 A 4 . 1 T r a n s m i s s i o n l i n e d a t a o f t h e P a c i f i c HVDC I n t e r t i e 63 v i L I S T O F I L L U S T R A T I O N S  Fig u r e Page •2.1 Basic HVDC l i n k 6 2.2 Basic hardware of a converter bridge 6 2.3 D i r e c t voltage (heavy l i n e s ) of bridge converter w i t h v a r i a t i o n of i g n i t i o n delay angle a 7 2.4 Constant current c o n t r o l l e r 8 2.5 C o n t r o l c h a r a c t e r i s t i c of converter. .'. .' 10 2.6 Two-converter c o n t r o l c h a r a c t e r i s t i c 10 2.7 R e l a t i o n s h i p between dc voltage Vd and r e g u l a t o r output eg 12 3.1 Flow chart of the s i m u l a t i o n . 17 3.2 C o n t r o l a m p l i f i e r c i r c u i t 18 3.3 Three operating regions of the c o n t r o l a m p l i f i e r 20 3.4 Response of the current r e g u l a t o r when subject to a saw-toothed disturbance 22 3.5 System set up f o r i n i t i a l i z a t i o n 24 3.6 Long run to v e r i f y steady s t a t e s o l u t i o n 26 3.7 Simulation w i t h dc s t a r t s at (a) a poin t of commutation; (b) middle of non-commutation i n t e r v a l 27 3.8 Simulation r e s u l t s w i t h e l e c t r o d e l i n e s t a r t s at zero i n i t i a l c o n d i t i o n 28 4.1 Simulated ac c h a r a c t e r i s t i c harmonic current f o r 6-pulse operation 35 4.2 Harmonic v o l t a g e on the dc s i d e of the r e c t i f i e r ? t e r m i n a l 38 5.1 Comparison between s i m p l i f i e d and d e t a i l e d converter station/, models f o r a f a u l t on the dc l i n e 40 5.2 System c o n f i g u r a t i o n f o r dc c i r c u i t breaker t e s t 42 5.3 Current through dc breaker f o l l o w i n g a c l o s e - i n f a u l t 43 5.4 Current through dc breaker f o l l o w i n g a remote f a u l t 44 A4.1 Main c i r c u i t of P a c i f i c HVDC I n t e r t i e 62 A4.2 Converter bridge at C e l i l o 63 A4.3 F i l t e r arrangement at C e l i l o 64 V i i ACKNOWLEDGEMENT I would like to express my sincere thanks to my supervisor, Dr. H.W. Dommel, for his inspiring guidance, unflagging patience and con-tinued encouragement during my period of graduate study. I am also deeply grateful to Dr. M.S. Davies for his interest and timely advice during the research work. The financial assistance and the invaluable discussions of the project with engineers of the System Engineering Division of B.C. Hydro and Power Authority are gratefully acknowledged. Special thanks are due to Mrs.K. Brindamour of the Department of Electrical Engineering for her proficient typing of this thesis. I am greatly indebted to my mother and the members of my family for their patience and encouragement throughout my university studies . 1 1 . I N T R O D U C T I O N Power system networks are subjected to d i f f e r e n t forms of d i s -turbances, such as switching operations, f a u l t s , l i g h t n i n g surges, and other intended or unintended manual interventions. The overvoltages and currents caused by these sudden changes i n c i r c u i t conditions may produce excessive. stresses on the system components which must be prevented, or at l e a s t l i m i t e d , to avoid p o t e n t i a l damages. More i n s u l a t i o n i s one way of prevent-ing excessive stresses, but there are strong economic reasons for keeping the system i n s u l a t i o n at i t s lowest possible l e v e l . This can be achieved only i f the transient phenomena are f u l l y analyzed. Extended from the idea of a-c c a l c u l a t i n g boards, the transient network analyzer (TNA) was developed f o r transient s t u d i e s 1 . It has been widely used by the u t i l i t y industry for over forty years. The system being studied i s represented by actual components of reduced s i z e and modified e l e c t r i c a l c h a r a c t e r i s t i c s . It i s p a r t i c u l a r l y useful i n modelling the frequency dependent c h a r a c t e r i s t i c s of the l i n e constants and i n those cases where the work i s of exploratory nature. However, the transmission l i n e s are approximated by cascaded u - c i r c u i t s , which impose an accuracy problem as far as high-frequency components of voltage and current are con-cerned. Another drawback of t h i s method i s the l i m i t e d access to the sophis-t i c a t e d equipment. The introduction of computers has extended the scope of the mathe-matical modelling i n solving engineering problems. D i g i t a l programming to-day o f f e r s a r e l a t i v e l y easy and f l e x i b l e a l t e r n a t i v e to the p h y s i c a l models in power system a n a l y s i s . In the l a t e 1960's, the Bonneville Power Adminis-2 tration (BPA) initiated the development of the Electromagnetic Transients Program (EMTP)2 which simulates the behaviour of the electrical system by using mathematical representations of the characteristics of the components and by solving a set of differential and algebraic matrix equations. The generalized access to computers, the f l e x i b i l i t y of the EMTP for running cases, and the relative ease with which program changes can be made are the main reasons of i t s widespread acceptance. Of course, there are many areas such as frequency-dependent characteristics, nonlinearity of surge diverters, magnetic saturation of transformers, etc., which need further improvements. A combination of physical models and dig i t a l simulation can be extremely powerful and the two approaches should be seen as being complementary rather than competitive 5. With the advent of the economic fe a s i b i l i t y and technical applica-b i l i t y of HVDC transmission, there is a need for an extension of the existing f a c i l i t i e s to carry out DC transient studies. The DC simulator uses the same control circuitry and a scaled reproduction of the commutation valves of the actual system 3' 4. When interconnected with the traditional a-c TNA, i t be-comes particularly valuable in development and evaluation of the control schemes. During the same period of time, considerable amount of work has been done in the f i e l d of digital simulation of HVDC systems. Additional features have also been implemented in the BPA EMTP for this purpose: a simplified model which represents the converter station as a current con-trolled dc voltage source 6 and a control system simulation package "TACS" (Transient Analysis of Control Systems)7 which can accept any arbitrary interconnection of a set of control system building blocks. Good results obtained by using these two models are shown in reference papers 8' 9. The UBC Transients Program (a simpler version of the BPA EMTP) :3 contains only the simplified model which is inadequate for representing a combined ac/dc system. "TACS" is very flexible and can be used for a wide range of applications. However, i t is relatively large and usually re-quires a f a i r l y long computation time. The major research effort of this thesis has been directed towards the development of a new converter model which can be interfaced with the UBC Transients Program for HVDC simulations. The model is complex enough to analyze transients in.ac/dc systems real i s -t i c a l l y enough, but i t is less flexible than "TACS" inasmuch as only one particular type of converter control can be handled. As the thesis proceeds, the implementation and testing of the model are described. New approaches in representing the limiters and the i n i t i a l i z a t i o n process are investigated. Then the program is applied to harmonic analysis and transient studies where the comparisons with results obtained by other methods and f i e l d measurements are also included. Final-ly, a summary of important results and conclusions is given. 4 2. MODELLING AN HVDC TRANSMISSION SYSTEM To provide accurate predictions on the preformance of an HVDC system under d i f f e r e n t operating conditions, the simulation method should be capable of representing the dc l i n k , the associated ac systems, the converters and the controls as w e l l . This chapter describes the main com-ponents of an HVDC system and t h e i r behaviour which the model must r e f l e c t c o r r e c t l y . 2.1 Network Representation The basic components of a dc l i n k between two ac systems are shown in F i g . 2.1. The basis f o r the simulation i s the UBC Electromagnetic Transients Program. For convenience, i t w i l l be referred to as the "Transients Program". I t i s a general purpose program designed to cal c u l a t e the instantaneous voltages and currents under any type of disturbance. The solut i o n methods and i t s a p p l i c a b i l i t i e s have been w e l l d e s c r i b e d 2 ' 1 0 ' 1 1 . The e x i s t i n g models i n the Transients Program are adequate to simulate most c i r c u i t elements i n both the ac and dc side networks. The dc transmission configuration can be monopolar, b i p o l a r , ground return or m e t a l l i c return. The transmission l i n e s are represented e i t h e r as multiphase IT c i r c u i t s or d i s t r i b u t e d constants. The ac and dc f i l t e r s which comprise branches tuned to d i f f e r e n t harmonic orders of the supply frequency are represented as lumped R, L, C elements according to t h e i r connections i n the system. Although zigzag winding, fork connection, polygon connection etc., are used i n some commercial i n s t a l l a t i o n s , the converter transformer can only be modelled with wye and delta connections as of now. The amount of d e t a i l i n the representation of the ac systems depends on the nature of the study. 5 T h e y may b e r e p r e s e n t e d w i t h d e t a i l e d g e n e r a t o r a n d l i n e m o d e l s , o r j u s t a s a c v o l t a g e s o u r c e s b e h i n d T h e v e n i n e q u i v a l e n t i m p e d a n c e s . 2 . 2 T h e C o n v e r t e r I t i s t h e p r e d o m i n a n t d e v i c e i n a n HVDC s y s t e m w h i c h t r a n s f o r m s a n a c v o l t a g e i n t o a d c v o l t a g e a n d v i c e v e r s a , c o n t r o l s t h e e x c h a n g e o f p o w e r b e t w e e n t h e t w o s y s t e m s , a n d l i m i t s t h e d i s t u r b a n c e s c a u s e d b y f a u l t s o c c u r r i n g o n e i t h e r s i d e o f t h e c o n v e r t i n g u n i t . T h e m o s t c o m m o n l y u s e d c i r c u i t i n HVDC t r a n s m i s s i o n i s t h e s i x - p u l s e b r i d g e . F i g . 2 . 2 s h o w s t h e b a s i c h a r d w a r e r e q u i r e m e n t . T h e v a l v e s c a n b e m e r c u r y - a r c v a l v e s o r s i l i -c o n c o n t r o l l e d r e c t i f i e r s . T h e v a l v e d a m p e r s a r e u s e d t o a v o i d s u d d e n j u m p s i n t h e i n v e r s e v o l t a g e a c r o s s t h e v a l v e s w h e n t h e y t u r n o f f ( d v / d t p r o t e c -t i o n ) , w h i l e t h e a n o d e d a m p e r s l i m i t t h e r a t e o f r i s e o f c u r r e n t d u r i n g v a l v e i g n i t i o n ( d i / d t p r o t e c t i o n ) . T h e p e r f o r m a n c e o f a c o n v e r t e r b r i d g e i s c o n t r o l l e d b y c h a n g i n g t h e f i r i n g a n g l e , a, o f t h e v a l v e s . F i g . 2 . 3 s h o w s how t h e d c v o l t a g e i s e f f e c t e d b y v a r i a t i o n s i n a. A s a e x c e e d s 9 0 ° , t h e b r i d g e c h a n g e s f r o m r e c t i f i c a t i o n o p e r a t i o n t o i n v e r s i o n o p e r a t i o n . To m i n i m i z e t h e g e n e r a t i o n o f h a r m o n i c s , t h e b r i d g e u n i t s a r e u s u a l l y p a i r e d w i t h w y e / w y e a n d d e l t a / w y e t r a n s f o r m e r s , w i t h a p h a s e s h i f t o f 3 0 ° b e t w e e n t h e m , t o f o r m a 1 2 - p u l s e u n i t . T h e s e 1 2 - p u l s e u n i t s a r e t h e m s e l v e s c o n n e c t e d i n s e r i e s t o p r o d u c e t h e r e q u i r e d t r a n s m i s s i o n v o l t a g e l e v e l . T o s i m u l a t e t h e o p e r a t i o n o f t h e v a l v e s , t h e e x i s t i n g m o d e l o f a d i o d e s w i t c h , w h i c h c o n d u c t s w h e n e v e r t h e a n o d e v o l t a g e i s h i g h e r t h a n t h e c a t h o d e v o l t a g e a n d i n t e r r u p t s a t c u r r e n t z e r o , i s m o d i f i e d s o t h a t t h e i g n i t i o n i s c o n t r o l l e d b y a n e x t e r n a l f i r i n g s i g n a l . T h e c h a r a c t e r i s t i c s o f 6 5 DC LINE AC SYSTEM I 5 -^JTTTT-AC SYSTEM II T 1 F i g . 2.1 Basic HVDC Link: 1 - AC f i l t e r s ; 2 - R e c t i f i e r s t a t i o n ; 3 - Inverter station; 4 - DC f i l t e r s ; 5 - DC reactors. F i g . 2.2 Basic Hardware of a converter bridge: 1 to 6 -Main valves; 7 - Bypass valve; 8 - Valve damper; 9 - Anode damper. / 7 x A A A A A A A M )L y. v V v V \ (a) a = 0 \ / V A A \ r ^ A /\ /\ /\ ' \ / \ _y. Y. Y v__y V { ^ u t (b) a = 30 \ X Y X x ' V Y Y O I x \ A i \ ' \ A ^ / ' v V v V V V v L >\ A A \ /V A A > A \ h \ A A A V V V v v7 V v / \ A. /\ / A j\ A -o)t (c) a = 90 -/ v Y y \ y \ • -7T 7 7 \ 7 T 7 / / \ / \ / \ / \ / \ / \ / \ / -OJt \ A K \ / h X v (d) a = 150 F i g . 2.3 Direct voltage (heavy l i n e s ) of bridge converter with v a r i a t i o n of i g n i t i o n delay angle a : (a),(b) - R e c t i f i c a t i o n ; (c) - Zero voltage; (d) - Inversion. 8 t h e c o n t r o l s y s t e m s a n d t h e g e n e r a t i o n o f t h e f i r i n g p u l s e s w i l l b e d e -s c r i b e d i n t h e n e x t s e c t i o n . 2 . 3 C o n v e r t e r C o n t r o l C h a r a c t e r i s t i c s T h e c o n t r o l l o o p o f r e c t i f i c a t i o n i s g e n e r a l l y a c o n s t a n t c u r r e n t 1 9 l o o p . T h e s c h e m a t i c d i a g r a m o f t h e b a s i c c o m p o n e n t s o f s u c h a c o n t r o l l e r i s s h o w n i n F i g . 2 . 4 . R e c e n t i n s t a l l a t i o n s c o n t a i n a d d i t i o n a l a n d m o r e s o p h i s t i c a t e d s c h e m e s , s u c h a s c o n s t a n t p o w e r o r d e r , d c p o w e r m o d u l a t i o n o r f r e q u e n c y m o d u l a t i o n f o r a c s y s t e m d a m p i n g , e t c . T h e s e l a t t e r a p p l i c a t i o n s a r e e x t e n s i o n s o f t h e f o r m e r a n d t h e v a r i a b l e s u n d e r c o n t r o l a r e u l t i m a t e l y c o n v e r t e d i n t o c h a n g e s o f c u r r e n t o r d e r . F o r 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 a n a l y s i s , t h e c o n s t a n t c u r r e n t c o n t r o l l e r p l a y s t h e m o s t s i g n i f i c a n t p a r t . T h e r e f o r e , t o s i m p l i f y t h e m o d e l l i n g , o n l y t h i s c o n t r o l s c h e m e i s i m p l e -m e n t e d i n t h e p r o g r a m . i nrmnr—-F i g . 2 . 4 C o n s t a n t C u r r e n t C o n t r o l l e r : I ^g c u r r e n t c o m m a n d ; I d l i n e c u r r e n t ; A l c u r r e n t m a r g i n ; G ( s ) r e g u l a t o r t r a n s f e r f u n c t i o n ; e a r e g u l a t o r o u t p u t ; e a max e d m i n l i m i t s o f r e g u l a t o r ; C o n . c o n v e r t e r ; F ' . P . f i r i n g p u l s e g e n e r a t i o n ; T r . t r a n s d u c t o r . 9 T h e c o n t r o l c h a r a c t e r i s t i c s may b e d i v i d e d i n t o t h r e e s e c t i o n s 1 3 ' 1 4 ( F i g . 2 . 5 ) : ( 1 ) C o n s t a n t m i n i m u m i g n i t i o n a n g l e ( C . I . A . ) c o n t r o l i s u s e d t o k e e p t h e r e c t i f i e r o p e r a t i n g w i t h i t s m i n i m u m p e r m i s s i b l e f i r i n g d e l a y a Q . T h e t h e o r e t i c a l m i n i m u m an e q u a l s z e r o , b u t i n p r a c t i c e i t i s k e p t t o b e t w e e n 1 0 ° t o 2 0 ° i n o r d e r t o m a i n t a i n a n a d e q u a t e m a r g i n f o r r a p i d i n c r e a s e s i n r e c t i f i e r v o l t a g e . T h e s l o p e o f t h e c u r v e i n t h i s s e c t i o n d e p e n d s u p o n t h e v a l u e o f c o m m u t a t i n g r e a c t a n c e ; t h e o u t p u t v o l t a g e o f t h e c o n v e r t e r i s g i v e n b y , V , = V , c o s a 0 — X I , ( 2 . 1 ) d do TT c d w h e r e = a v e r a g e c o n v e r t e r v o l t a g e , = c o n v e r t e r i d e a l n o - l o a d d i r e c t v o l t a g e , an = m i n i m u m d e l a y a n g l e , X ^ = c o m m u t a t i n g r e a c t a n c e p e r p h a s e , I , = d c l i n e c u r r e n t , d ( 2 ) C o n s t a n t C u r r e n t ( C C ) C o n t r o l i s u s e d t o p r o v i d e b o t h t h e r e c t i f i e r a n d i n v e r t e r w i t h t h e c o n t r o l a b i l i t y t o r e g u l a t e t h e l i n e c u r r e n t w h e n i t d o e s n o t a g r e e w i t h a s e t r e f e r e n c e I . T h e s l o p e o f t h e c u r v e d e p e n d s m a i n l y o n t h e o u t p u t o f t h e c u r r e n t r e g u l a t o r , a n d t h e o u t p u t v o l t a g e i s g i v e n b y , V , = V\, c o s a - X I , ( 2 . 2 ) d do ir c d w h e r e a = a n g l e o f d e l a y o f f i r i n g . ( 3 ) C o n s t a n t E x t i n c t i o n A n g l e ( C . E . A . ) C o n t r o l i s u s e d t o k e e p t h e i n v e r t e r o p e r a t i n g a t i t s m i n i m u m e x t i n c t i o n a d v a n c e a n g l e . T h e a n g l e m u s t b e k e p t s m a l l t o p r o v i d e h i g h p o w e r f a c t o r b u t l a r g e e n o u g h t o m a i n t a i n a s a f e m a r -g i n t o p r e v e n t c o m m u t a t i o n f a i l u r e s . T h e s l o p e o f t h e c u r v e a g a i n d e p e n d s o n t h e v a l u e o f t h e c o m m u t a t i n g r e a c t a n c e , a n d t h e o u t p u t v o l t a g e o f t h e c o n v e r t e r i s g i v e n b y 10 F i g . 2 . 5 C o n t r o l C h a r a c t e r i s t i c o f C o n v e r t e r : 1 - C . I . A . c o n t r o l ; 2 - C C . c o n t r o l ; 3 - C . E . A . c o n t r o l . C o n v e r t e r 1 ( r e c t i f i e r ) o n C C . c o n t r o l C C . c o n t r o l F i g . 2 . 6 T w o - c o n v e r t e r C o n t r o l C h a r a c t e r i s t i c . 11 V , = V J c o s y — X I , ( 2 . 3 ) d do n TT c d w h e r e Y = m i n i m u m e x t i n c t i o n a d v a n c e a n g l e , n F o r a t w o - c o n v e r t e r s y s t e m , t h e o p e r a t i n g p o i n t i s a t t h e i n t e r -s e c t i o n o f t h e two c o n t r o l c h a r a c t e r i s t i c s . I n F i g . 2 . 6 , t h e i n t e r s e c t i o n Xi b e t w e e n t h e s o l i d l i n e s i s t h e o p e r a t i n g p o i n t w h e r e c o n v e r t e r 1 i s a t c o n s t a n t c u r r e n t mode a n d c o n v e r t e r 2 a t c o n s t a n t e x t i n c t i o n . a n g l e m o d e , P o w e r i s t r a n s m i t t e d f r o m c o n v e r t e r 1 t o c o n v e r t e r 2 . I f a r e v e r s a l o f p o w e r t r a n s m i s s i o n i s n e c e s s a r y , t h e c h a r a c t e r i s t i c s a r e c h a n g e d t o t h o s e s h o w n b y t h e b r o k e n l i n e s a n d X 2 w i l l b e c o m e t h e o p e r a t i n g p o i n t . T h e d i f f e r e n c e b e t w e e n t h e c u r r e n t s e t t i n g s I , a n d I , o f t h e t w o c o n v e r t e r s d s i d s 2 i s c a l l e d t h e c u r r e n t m a r g i n , A l . T h i s m a r g i n , t y p i c a l l y 15% o f t h e r a t e d c u r r e n t , h a s t o b e l a r g e e n o u g h t o a v o i d t h e t w o c o n v e r t e r s f r o m o p e r a t i n g a t t h e s t e e p c o n s t a n t - c u r r e n t l i n e s s i m u l t a n e o u s l y . S u c h o p e r a t i o n i s u n d e s i r a b l e a n d may a l s o b e v e r y u n s t a b l e . T h e o p e r a t i n g mode o f a c o n v e r t e r i s d e t e r m i n e d b y t h e c o n t r o l a m p l i f i e r o u t p u t v o l t a g e e . A s s e e n f r o m F i g . 2 . 4 , e ( s ) = ( I , -T - A l ) G ( s ) ( 2 . 4 ) a d s d T h e p a r a m e t e r s o f t h e t r a n s f e r f u n c t i o n G ( s ) c a n b e o b t a i n e d f r o m t h e s t a t i o n o p e r a t i n g m a n u a l s 1 5 . T h e t r a n s f e r f u n c t i o n o f t h e c o n t r o l s c h e m e s t u d i e d i n t h i s t h e s i s i s a s e c o n d o r d e r o n e o f t h e f o r m K ( l + s T 2 ) G ( S ) = ( l + s T l ) ( l + s T 3 ) ( 2 ' 5 ) w h e r e K i s t h e s t a t i c g a i n a n d Ti, T 2 , T 3 a r e t i m e c o n s t a n t s . 12 T h e o t h e r s e t t i n g s d e p e n d o n t h e o p e r a t i n g c o n d i t i o n s o f t h e s y s t e m . D i f -f e r e n t a m p l i f i e r o u t p u t v o l t a g e s l e a d t h e c o n v e r t e r t o d i f f e r e n t o p e r a t i n g r e g i o n s : a ) i f e > e >e . , t h e c o n v e r t e r o p e r a t e s i n r e g i o n ' 2 ' a max a a m m ( s e e F i g . 2 . 5 ) , i . e . o n C C . C o n t r o l . b ) i f e <e . , t h e c o n v e r t e r o p e r a t e s i n r e g i o n ' 3 ' , a a mxn i . e . C E . A . C o n t r o l . c ) i f e >e > t h e c o n v e r t e r o p e r a t e s i n r e g i o n ' 1 ' , a a max i . e . C . I . A . C o n t r o l . A s s h o w n i n F i g . 2 . 7 , t h e d i r e c t v o l t a g e i s a f u n c t i o n o f a n d c a n b e d e s c r i b e d a s 6 : V , = k ! + k 2 e ( 2 . 6 ) d ^ a w h e r e k i = i n t e r c e p t , k 2 = , s l o p e o f t h e c o n v e r t e r c o n s t a n t c u r r e n t c o n t r o l c h a r a c t e r i s t i c . e a F i g . 2 . 7 R e l a t i o n s h i p b e t w e e n D . C . v o l t a g e a n d r e g u l a t o r o u t p u t e a 13 K n o w i n g t h e v a l u e o f a n d t h e o p e r a t i n g r e g i o n o f t h e c o n v e r t e r , t h e d e l a y a n g l e a c a n b e c a l c u l a t e d f r o m e q u a t i o n s 2 . 1 , 2 . 2 o r 2 . 3 . T h e n t h e p u l s e g e n e r a t o r d e t e r m i n e s t h e f i r i n g i n s t a n t f o r e a c h v a l v e . T h e r e a r e t w o b a s i c a p p r o a c h e s i n t h e d e s i g n o f p u l s e g e n e r a t o r s : 1 ) I n d i v i d u a l p h a s e c o n t r o l : S i x i n d e p e n d e n t d e l a y c i r c u i t s t i m e t h e f i r i n g p u l s e o f e a c h v a l v e . T h e d e l a y i s c a l c u l a t e d f r o m t h e e a r -l i e s t i n s t a n t a t w h i c h f i r i n g i s p o s s i b l e , t h a t i s , t h e i n s t a n t a t w h i c h t h e c o m m u t a t i n g v o l t a g e b e c o m e s p o s i t i v e . T h i s t y p e o f c o n -t r o l h a s t h e a d v a n t a g e t h a t i t c a n a c h i e v e t h e h i g h e s t p o s s i b l e d i r e c t v o l t a g e w i t h t h e a s y m m e t r y p r e v a i l i n g a t t h e t i m e . H o w e v e r , i t h a s a d r a w b a c k i n a s m u c h a s a b n o r m a l h a r m o n i c s , e s p e c i a l l y o f t h e t h i r d o r d e r , a p p e a r i n t h e c u r r e n t s . 2) E q u i d i s t a n t f i r i n g c o n t r o l ( o r ' e q u a l s p a c e ' c o n t r o l ) 1 6 : T h e r e i s a l w a y s o n e r e f e r e n c e p h a s e a t w h i c h t h e f i r i n g a n g l e i s d e t e r m i n e d a c c o r d i n g t o t h e c o n t r o l a m p l i f i e r o u t p u t . S t a r t i n g w i t h t h i s a n g l e , e a c h f i r i n g p u l s e i s t i m e d a t 6 0 ° e l e c t r i c a l a f t e r t h e p r e -c e d i n g p u l s e , t h a t i s , t h e v a l v e s a r e i g n i t e d a t e q u a l t i m e i n t e r -v a l s . A s a r e s u l t , a b n o r m a l h a r m o n i c s a r e s u p p r e s s e d b u t t h e r e s p o n s e i n t h e e v e n t o f a f a u l t i s s l o w e r . 14 3 . I M P L E M E N T A T I O N OF THE C O N V E R T E R MODEL T h i s c h a p t e r d e s c r i b e s t h e s o l u t i o n m e t h o d a n d t h e c o m p u t e r i m p l e -m e n t a t i o n o f t h e c o n v e r t e r m o d e l . A l s o i n c l u d e d i s a d i s c u s s i o n o f t h e t w o m a j o r p r o b l e m s e n c o u n t e r e d d u r i n g t h e i m p l e m e n t a t i o n . 3 . 1 S u b r o u t i n e V A L C O N To s i m u l a t e t h e c o n v e r t e r c o n t r o l o f t h e HVDC s y s t e m , a s u b r o u t i n e V A L C O N ( " V a l v e C o n t r o l " ) w a s d e v e l o p e d . T h e F O R T R A N l i s t i n g o f t h e p r o g r a m i s i n c l u d e d i n A p p e n d i x 1 . I n p u t R e q u i r e m e n t s T h e u s e r m u s t p r o v i d e t h e f o l l o w i n g i n f o r m a t i o n , i n a d d i t i o n t o t h e u s u a l i n p u t d a t a o f t h e T r a n s i e n t s P r o g r a m : 1 ) T h e c o e f f i c i e n t s o f t h e t r a n s f e r f u n c t i o n K , T i , T 2 , T3, 2) t h e o p e r a t i n g mode o f t h e c o n v e r t e r , i . e . r e c t i f i c a t i o n o n i n v e r s i o n , 3 ) t h e i n i t i a l o p e r a t i n g r e g i o n o n t h e c o n v e r t e r c o n t r o l c h a r a c t e r i s t i c , 4 ) t h e c o m m u t a t i n g r e a c t a n c e X ^ , 5) t h e e l e c t r i c a l f r e q u e n c y , 6) t h e c o n n e c t i o n o f t h e c o n v e r t e r t r a n s f o r m e r , i . e . w y e - w y e o n w y e - d e l t a , 7 ) i d e n t i f i c a t i o n o f a b r a n c h ( u s u a l l y a s w i t c h ) w h e r e t h e r e f e r e n c e d i r e c t c u r r e n t w i l l b e t a k e n f r o m , 8 ) t h r e e n o d e n a m e s o n t h e p r i m a r y s i d e o f t h e c o n v e r t e r t r a n s f o r m e r f r o m w h i c h t h e c o m m u t a t i o n v o l t a g e o f e a c h v a l v e w i l l b e c a l c u l a t e d , 9) t h e l i m i t s V , , V , . , t h a t a r e p l a c e d o n t h e e x c u r s i o n s o f t h e d max d mxn r e g u l a t o r a m p l i f i e r , 1 0 ) t h e p u l s e g e n e r a t o r t y p e . 1 5 N u m e r i c a l I n t e g r a t i o n T h e s - d o m a i n e q u a t i o n ( 2 . 4 ) c a n b e e x p r e s s e d a s t w o f i r s t o r d e r d i f f e r e n t i a l e q u a t i o n s : d e ( t ) X ( t ) = d t e ( t ) + T X ( t ) + P = K I ( t ) + K T 2 ( 3 . 1 ) w h e r e I ( t ) " X d s - V A I T = T1 + T 3 P = T1 • T 3 T h e s e e q u a t i o n s , a s d e s c r i b e d i n m o r e d e t a i l i n t h e n e x t s e c t i o n , r e p r e s e n t t h e c o n v e r t e r c o n t r o l l o o p w h e n t h e r e g u l a t o r a m p l i f i e r o p e r a t e s w i t h i n t h e l i m i t s . T h e y m u s t b e s o l v e d s i m u l t a n e o u s l y w i t h t h e r e s t o f t h e e x t e r n a l e l e c t r i c n e t w o r k . T h e n u m e r i c a l i n t e g r a t i o n f o r m u l a b a s e d o n t h e t r a p e z o i d a l r u l e w a s a d o p t e d i n t h e p r o g r a m . T h i s a l g o r i t h m w a s c h o s e n 5 f i r s t l y b e c a u s e i t i s t h e s i m p l e s t s e c o n d o r d e r i m p l i c i t m e t h o d a n d i s n u m e r i c a l l y s t a b l e f o r a n y s t e p s i z e A t 1 7 ; s e c o n d l y , i t s i m p l i f i e s t h e i n t e r -f a c i n g p r o b l e m a s t h e T r a n s i e n t s P r o g r a m u s e s t h e s a m e s t e p s i z e a n d i n t e g r a -t i o n t e c h n i q u e . A p p l y i n g t h e t r a p e z o i d a l r u l e t o e q u a t i o n ( 3 . 1 ) r e s u l t s i n t h e 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 X ( t ) + X ( t - A t ) = [ e Q ( t ) - e a ( t - A t ) ] | [ e a ( t ) + e a ( t - A t ) ] + | [ X ( t ) + X ( t - A t ) ] + | r - [ X ( t ) - X ( t - A t ) ] ( 3 . 2 ) T K = | [ I ( t ) + I ( t - A t ) ] + - | ^ [ I ( t ) - I ( t - A t ) ] R e a r r a n g i n g t e r m s , t h i s b e c o m e s e ( t ) = 4 x ( t ) + H I S T ( t - A t ) a A X ( t ) = j-; [ e o ( t ) - e a ( t - A t ) ] - X ( t - A t ) ( 3 . 3 ) 16 w h e r e A = 1 + 2T , 4 P A t ( A t ) z a n d t h e " h i s t o r y " o f t h e s t a t e o f t h e a m p l i f i e r H I S T ( t - A t ) = 2 K - B A I ( t - A t ) + A - 2 e ( t - A t ) + 4P X ( t - A t ) A a A - A t A f t e r t h e p r o g r a m e n t e r s t h e t i m e - s t e p l o o p , t h e i n s t a n t a n e o u s v a l u e s o f e ^ a n d X a r e c a l c u l a t e d a n d t h e h i s t o r y t e r m i s u p d a t e d a t e a c h s t e p . T h e n t h e d e l a y a n g l e s a n d t h e f i r i n g p u l s e s a r e d e t e r m i n e d b a s e d o n t h e e q u a t i o n s d e s c r i b e d i n t h e p r e v i o u s c h a p t e r . s e p a r a t e l y i n t h e T r a n s i e n t s P r o g r a m a n d i n t h e s u b r o u t i n e V A L C O N , r e s p e c -t i v e l y . V a r i a b l e s s u c h a s a c b u s v o l t a g e s , d c l i n e c u r r e n t s , v a l v e s t a t u s ( o p e n o r c l o s e ) e t c . , a r e p a s s e d f r o m t h e m a i n p r o g r a m t o V A L C O N , w h i c h , a f t e r m a n i p u l a t i o n t h r o u g h i t s l o g i c , r e t u r n s t h e a p p r o p r i a t e f i r i n g s i g n a l s t o c o n t i n u e t h e s i m u l a t i o n . M o d i f i c a t i o n s i n t h e T r a n s i e n t s P r o g r a m t o i n c o r p o r a t e t h i s i n t e r f a c e a r e d o c u m e n t e d i n A p p e n d i x 2 a n d a g e n e r a l f l o w c h a r t o f t h e s i m u l a t i o n a l g o r i t h m i s s h o w n i n F i g . 3 . 1 . T h e I n t e r f a c e T h e e l e c t r i c n e t w o r k a n d t h e c o n t r o l s y s t e m a r e s o l v e d 17 VALCON Update 60° Count! e =e a a max ^ "^peratingS ^ ICalculatd e Decrement , Yes< Group Count Generate Ign i t i o n Signal Decrement Valve Counter e *=e a a mxrt Update Delay Counter F i g . 3.1 Flow Chart of the Simulation. 18 3 . 2 L i m i t e r R e p r e s e n t a t i o n T h e c o n t r o l a m p l i f i e r o u t p u t i s u s u a l l y c l a m p e d a c c o r d i n g t o t h e m a x i m u m a n d m i n i m u m a n g l e o f d e l a y c o n s t r a i n t s o f t h e c o n v e r t e r s . T h i s t y p e o f l i m i t e r c r e a t e s a s p e c i a l p r o b l e m o f n o n l i n e a r i t y . T h e s o l u t i o n m e t h o d w h i c h h a s b e e n u s e d i n t h e e x i s t i n g s i m p l i f i e d m o d e l i s s u c h t h a t t h e d e -r i v a t i v e s o f t h e o u t p u t a r e s e t t o z e r o 7 w h e n a l i m i t i s r e a c h e d , w h i c h c h a n g e s t h e s t r u c t u r e 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 . To i n v e s t i g a t e t h e v a l i d i t y o f t h i s r e p r e s e n t a t i o n , t h e a c t u a l h a r d w a r e a r r a n g e m e n t o f t h e c o n -t r o l c i r c u i t m u s t b e a n a l y z e d . A s s u m i n g t h a t a l l t h e t r a n s d u c e r s a n d f i l t e r s a r e i d e a l , a n d n e g l e c t i n g a l l t h e p r o t e c t i v e c i r c u i t s a n d m o n i t o r i n g d e v i c e s , t h e c o n t r o l a m p l i f i e r c i r c u i t i s s h o w n i n F i g . 3 . 2 . T h e o v e r a l l r e l a t i o n s h i p b e t w e e n t h e i n p u t a n d o u t p u t v o l t a g e i s 1 8 : e . x Zo zT ( 3 . 4 ) w h e r e Z Q = s h o r t - c i r c u i t t r a n s f e r i m p e d a n c e f u n c t i o n o f t h e f e e d b a c k l o o p Z . = s h o r t - c i r c u i t t r a n s f e r i m p e d a n c e f u n c t i o n o f t h e i n p u t c i r c u i t , d s A l • r R i n R i n w -Z i r -a mm d l -Or-" R 2 -m— R 1 -kr-d 2 e 41 a max F i g . 3 . 2 C o n t r o l A m p l i f i e r C i r c u i t . A H i g h G a i n A m p l i f i e r ; C , R ' s C a p a c i t a n c e a n d R e s i s t a n c e o f t h e D a m p i n g C i r c u i t ; e . , e I n p u t & O u t p u t V o l t a g e s ; d l 5 d 2 c l a m p i n g d i o d e s . 19 T h e i n p u t t r a n s f e r i m p e d a n c e i s s i m p l y . R - n > a n d t h e c o m p l e x t r a n s f e r i m p e d a n c e o f t h e RC f e e d b a c k n e t w o r k i n t e r m s o f t h e L a p l a c e -t r a n s f o r m o p e r a t o r S i s i n t h e s a m e f o r m a s e q u a t i o n ( 2 . 5 ) w i t h t h e t i m e c o n s t a n t s h a v i n g t h e r e l a t i o n s h i p s g i v e n b y 1 9 T 2 = R i C j T i • T 3 = Ri R 2 Ci C 2 ( 3 . 5 ) I l + T 3 = R 2 C1 + R 2 C 2 + R 2 Ci T h e f u n c t i o n o f t h e c o n t r o l a m p l i f i e r c a n b e d i v i d e d i n t o t h r e e d i f f e r e n t o p e r a t i n g r e g i o n s . F i g . 3 . 3 s h o w s t h a t i n e a c h r e g i o n , t h e c u r r e n t r e l a t i o n s h i p s h o u l d b e d e s c r i b e d b y d i f f e r e n t d i f f e r e n t i a l e q u a t i o n s : a ) e . < e < e i . e . t h e a m p l i f i e r i s i n i t s a c t i v e r e g i o n . a mxn a a m a x e ^ v a r i e s a s t h e i n p u t c u r r e n t ±i c h a n g e s , t h e e q u a t i o n o f t h e s y s t e m e ( t ) = - Z 0 ( t ) M t ) ( 3 . 6 ) b ) e = e . i . e . t h e a m p l i f i e r i s c l a m p e d t o p r e v e n t i t f r o m g o i n g a a m i n n e g a t i v e o r l e s s t h a n m i n i m u m . T h e d i o d e d i i s c o n d u c t i n g a n d t h e o u t p u t i s k e p t a t e . . When t h e a m p l i f i e r i s d r i v e n t o t h e l i m i t , v a m m t h e c i r c u i t w i l l u n d e r g o a t r a n s i e n t p e r i o d w h i c h c a n b e d e s c r i b e d b y t h e e q u a t i o n e a m i n = Z ° ( t ) ' [ ± 2 ( t ) " ± l ( t ) ] < 3 - 7 > T h e a m p l i f i e r c o m e s o u t o f t h e c l a m p i n g c o n d i t i o n w h e n I b e c o m e s l e s s t h a n ( I • - A l ) . d s c ) e = e i . e . t h e a m p l i f i e r g o e s i n t o s a t u r a t i o n o r t o a a a max c l a m p e d v a l u e o f m a x i m u m o u t p u t . D i o d e d 2 i s c o n d u c t i n g a n d t h e e q u a -t i o n u n d e r t h i s c o n d i t i o n i s e a m a x = Z ° ( t ) ^ ( O " M O ] ( 3 . 8 ) T h e a m p l i f i e r c o m e s o u t o f t h i s r e g i o n w h e n I b e c o m e s g r e a t e r t h a n ds 20 Fig. 3.3 Three Operating Regions of the Control Amplifier: (a) -in active region; (b) - clamped to minimum value; (c) -clamped to maximum value. 21 T o s i m u l a t e t h e b e h a v i o u r o f t h e a m p l i f i e r a c c u r a t e l y , a l l t h e s e t h r e e e q u a t i o n s m u s t b e i n c l u d e d . E v e r y t i m e t h e s u b r o u t i n e i s c a l l e d , t h e o p e r a t i n g r e g i o n o f t h e a m p l i f i e r i s d e t e r m i n e d a n d t h e m a t h e m a t i c a l s o l u -t i o n w i l l b e s w i t c h e d f r o m o n e e q u a t i o n t o a n o t h e r w h e n e v e r t h e o p e r a t i n g r e g i o n i s c h a n g e d . T h e p r e v i o u s l y u s e d m e t h o d o f s e t t i n g t h e d e r i v a t i v e s e q u a l t o z e r o a t t h e l i m i t g i v e s e r r o n e o u s s i m u l a t i o n r e s u l t s , e s p e c i a l l y w h e n t h e s y s t e m i s s u b j e c t e d t o d i s t u r b a n c e s w h i c h d r i v e t h e a m p l i f i e r i n a n d o u t o f t h e l i m i t i n s h o r t t i m e i n t e r v a l s . To i l l u s t r a t e t h e d i f f e r e n c e b e t w e e n t h e t w o a p p r o a c h e s , a s i m p l e c o n t r o l s c h e m e , a s s h o w n i n F i g . 2 . 4 , w a s s i m u l a t e d w i t h r / o = 1 . 3 9 ( 1 + 4 . 4 s )  K J (1 + . 0 4 s ) ( 1 + . 0 1 0 3 s ) a n d I , = 9 0 0 A m p . , A l = - 1 0 0 A m p . e = 1 8 . 9 v o l t . d s a max T h e c o n v e r t e r o p e r a t e d i n t h e u p p e r l i m i t i n s t e a d y s t a t e a n d w a s s u b j e c t e d t o a s a w - t o o t h d i s t u r b a n c e . S i n c e I , - A l = 1 0 0 0 A m p . , t h e a m p l i f i e r i s e x -ds p e c t e d t o s t a y i n t h e u p p e r l i m i t u n t i l t h e i n p u t c u r r e n t e x c e e d s t h i s v a l u e . F i g . 3 . 4 ( a ) s h o w s t h a t t h e a m p l i f i e r j u m p s o u t o f t h e l i m i t p r e m a t u r e l y w i t h t h e p r e v i o u s l y u s e d a p p r o a c h , a n d t h u s c a n n o t r e t u r n t o t h e l i m i t a g a i n i n t h e s u b s e q u e n t f a l l - o f f o f t h e i n p u t c u r r e n t . T h e m e t h o d o f s o l v i n g d i f f e r -e n t d i f f e r e n t i a l e q u a t i o n s a c c o r d i n g t o t h e o p e r a t i n g r e g i o n s o f t h e a m p l i -f i e r g i v e s a m o r e a c c u r a t e s i m u l a t i o n . F i g . 3 . 4 ( b ) a l s o s h o w s t h a t t h e a m p l i f i e r r e s p o n d s d i f f e r e n t l y d e p e n d i n g o n w h e t h e r t h e a m p l i f i e r h a s s t a y e d i n t h e l i m i t l o n g e n o u g h t o r e a c h i t s s t e a d y s t a t e : A t A , t h e a m p l i f i e r j u m p s o u t o f t h e l i m i t a f t e r t h e d a m p i n g c i r c u i t r e a c h e s t h e s t e a d y s t a t e , w h i l e a t C , t h e d a m p i n g c i r c u i t i s s t i l l i n t h e t r a n s i e n t p r o c e s s a f t e r t h e a m p l i f i e r s w i t c h e s i n t o t h e l i m i t a t B . 22 0.0 0.08 0.16 0.2k 0.32 O.k Time (10 seconds) (a) First Approach : setting derivatives to zero when output hits the limit. 0.0 0.08 0.16 _ 0.24 0.32 O.k Time ( 1 0 _ 1 seconds) (b) Second Approach : solving different differential equations according to the operating regions of the amplifier. . 3.4 Response of the current regulator when subject to a saw-toothed disturbance : amplifier output; - - - - current input. 23 3 . 3 I n i t i a l i z a t i o n T y p i c a l t r a n s i e n t s t u d i e s s t a r t w i t h t h e e l e c t r i c n e t w o r k b e i n g i n i t s s t e a d y s t a t e c o n d i t i o n s . W h i l e i t i s a l w a y s p o s s i b l e t o r a m p t h e s i m u -l a t i o n u p t o s t e a d y s t a t e , t h i s n o t o n l y r e q u i r e s a d d i t i o n a l c o m p u t e r t i m e , b u t a l s o s k i l l o n t h e s i d e o f t h e u s e r . I t i s t h e r e f o r e d e s i r a b l e t o i n i -t i a l i z e t h e n e t w o r k v a r i a b l e s t o t h e i r s t e a d y - s t a t e v a l u e s a s c l o s e l y a s p o s s i b l e . T h e p r e s e n t v e r s i o n o f t h e T r a n s i e n t s P r o g r a m h a s a s u b r o u t i n e w h i c h a u t o m a t i c a l l y c a l c u l a t e s t h e a c s t e a d y s t a t e s o l u t i o n f o r a s i n g l e s o u r c e f r e q u e n c y . I n a d d i t i o n , i t h a s a n o p t i o n t o r e a d i n u s e r - s p e c i f i e d i n i t i a l d a t a , w h i c h w i l l o v e r r i d e a u t o m a t i c a l l y c o m p u t e d v a l u e s . T h e i n i -t i a l i z a t i o n p r o c e s s i n t h e s i m u l a t i o n o f a c / d c s y s t e m i s c o m p l i c a t e d b y t h e f a c t t h a t s u c h s y s t e m s h a v e t w o o r m o r e f u n d a m e n t a l f r e q u e n c i e s a s w e l l a s h a r m o n i c s , a n d t h a t t h e c o n v e r t e r v a l v e s c r e a t e a d i s c o n t i n u i t y b e t w e e n t h e a c a n d d c c o m p o n e n t s . N e i t h e r o p t i o n m e n t i o n e d i s t h e r e f o r e a d e q u a t e t o h a n d l e t h e i n i t i a l i z a t i o n a l o n e . To s o l v e t h e p r o b l e m , a c o m b i n a t i o n o f t h e t w o f e a t u r e s h a s b e e n u s e d . A t t i m e t < 0 , a t y p i c a l s y s t e m * , a s s h o w n i n F i g . 3 . 5 , c a n b e p a r t i -t i o n e d i n t o t h r e e s e p a r a t e g r o u p s : T h e p o w e r t r a n s m i t t i n g a c s y s t e m , t h e d c l i n k , a n d t h e p o w e r r e c e i v i n g a c s y s t e m . T h e c o n v e r t e r b r i d g e s s e r v e a s t h e b o u n d a r i e s , a n d a l l t h e v a l v e s a r e a s s u m e d i n o p e n s t a t e a t t < 0 . W i t h t h e a c n e t w o r k s d i s c o n n e c t e d f r o m t h e d c s y s t e m , d c v o l t a g e s ( s i m u l a t e d a s a c s o u r c e s o f v e r y l o w f r e q u e n c i e s a n d a m p l i t u d e s e q u a l t o t h e d c o p e r a t i n g v o l t a g e s ) a r e i n s e r t e d a t t h e t e r m i n a l s o f t h e d c l i n e . A C s o u r c e s a r e u s e d b e c a u s e t h e s t e a d y - s t a t e s u b r o u t i n e c a n o n l y h a n d l e a c s t e a d y - s t a t e s o l u t i o n s , s i n c e a t d c , a d m i t t a n c e s o f i n d u c t i v e b r a n c h e s w o u l d b e c o m e i n f i n i t e . P r a c --3 t i c e h a s s h o w n t h a t w i t h a f r e q u e n c y o f f = 1 0 H z , t h e a c s o l u t i o n i s s u f -f i c i e n t l y c l o s e t o t h e d c s o l u t i o n 1 4 , a n d r e a c t a n c e s OJL a n d s u s c e p t a n c e s u)C 24 /AC \ 2 [ S Y S T E M Y _ | j t = 0 t=o H E t=o T 6 DC L i n k t=0 ^ t-o 2 t=0 1 F i g . 3 . 5 S y s t e m s e t u p f o r i n i t i a l i z a t i o n : 1 - E q u i v a l e n t l o a d ; 2 - C o n v e r t e r t r a n s f o r m e r ; 3 - C o n v e r t e r s t a t i o n ; 4 - A C c u r r e n t s o u r c e w i t h v e r y l o w f r e q u e n c y ; 5 — DC f i l t e r s ; 6 - S m o o t h i n g r e a c t o r ; 7 - E l e c t r o d e l i n e . a r e s t i l l l a r g e e n o u g h t o a v o i d n u m e r i c a l p r o b l e m s . T h i s a p p r o a c h g i v e s g o o d d c s t e a d y s t a t e s o l u t i o n f o r t h e d o t t e d a r e a i n F i g . 3 . 5 , w i t h a l l t h e h a r -m o n i c c o m p o n e n t s n e g l e c t e d . To p r o p e r l y i n i t i a l i z e t h e a c p o r t i o n s o f t h e s y s t e m , t h e d c n e t -w o r k i s r e p r e s e n t e d a s a n e q u i v a l e n t l o a d c o n n e c t e d t o e a c h l e g o f t h e s e c o n d a r y w i n d i n g s o f t h e c o n v e r t e r t r a n s f o r m e r s . T h e i n i t i a l d c o p e r a t i n g c o n d i t i o n s , s u c h a s d c t e r m i n a l v o l t a g e s a n d c u r r e n t s , c a n b e o b t a i n e d f r o m l o a d f l o w s t u d i e s , a n d w i t h l o s s e s i n t h e c o n v e r t e r s t a t i o n s n e g l e c t e d , a c p o w e r e q u a l s d c p o w e r . T h e e q u i v a l e n t l o a d , w h i c h w i l l b e a p o s i t i v e o r n e g a t i v e i m p e d a n c e d e p e n d i n g o n w h e t h e r t h e c o n v e r t e r o p e r a t e s a s r e c t i f i e r o r i n v e r t e r , c a n b e e s t i m a t e d f r o m ( 3 . 9 ) e q 3 S * w h e r e V = t h e r m s l i n e - t o - g r o u n d a l t e r n a t i n g v o l t a g e S * = t h e c o n j u g a t e o f t h e a p p a r e n t p o w e r . W i t h t h i s l o a d c o n n e c t e d a t t < 0 , t h e i n i t i a l c o n d i t i o n s o f t h e t w o a c s y s t e m s a r e c o m p u t e d b y t h e s t e a d y - s t a t e s u b r o u t i n e . A t t = 0 , t h e e q u i v a l e n t l o a d s a r e s w i t c h e d o f f a n d t h e a c a n d d c n e t w o r k s a r e r e c o n n e c t e d t h r o u g h t h e v a l v e s . T h e d c s t e a d y - s t a t e i n i t i a l 25 conditions are set through the read-in option, and with subroutine VALCON, determining which valves must be ignited, the system is virtually at i t s normal operating state and ready for the transient simulation for t > 0. Fig. 3.6 shows a simulation over a time span of 75 ms for a case of a six-pulse converter operation with both ac and dc f i l t e r s . The system settles down very fast. Simulations with the order of dc and ac steady state solu-tions reversed were also studied and the results were basically identical. It is therefore reasonable to assume that harmonics can be ignored in the i n i t i a l i z a t i o n . Other researchers have found that ac/dc system simulations are very sensitive to wrong i n i t i a l values 1 2. One concern in the method de-scribed here is the behaviour of the valve dampers. Since the circuits in the converter bridge are disconnected during the computation of the steady-state solution, they are subjected to wrong i n i t i a l conditions when the simu-lation starts at t = 0. Two cases were simulated: one with the simulation starting at the middle of the non-commutation period, i.e. the valve dampers are quiescent; the other with the simulation starting at a point of commuta-tion, i.e. the valve dampers are in a transient state. The resulting curves in Fig. 3.7 show no differences after one sixth of a cycle. The exclusion of the valve dampers in the i n i t i a l i z a t i o n process does not seem to cause noticeable disturbances. However, improper i n i t i a l i z a t i o n on any part of the dc l i n e may give inaccurate results or even lead to wrong f i r i n g sequences. Fig. 3.8 illustrates the severity of this problem. The system studied previously is i n i t i a l i z e d in the same way except that the electrode line, which is simply modelled as a lumped R-L branch, was not included in the i n i t i a l i z a t i o n , i.e. i t had zero i n i t i a l values. The curves show an erroneous high voltage 2 6 — i — 30.0 — i r ~ 37.5 45.0 52.5 — i — 60.0 — i 1 67.5 7;75.0 t m s e c . 0.0 w & . g I D . 1 7.5 15.0 22.5 ( b ) — i — 7.5 0,0 15.0 22.5 30.0 37.5 45.0 ( c ) — r — 52.5 60.0 "6^5 ~75.0 t m s e c . o I—I •x co 30.0 37.5 45.0 675 •'0.0 7.5 15.0 22.5 ( d ) 52.5 60.0 75.0 t m s e c . F i g . 3 . 6 L o n g r u n t o v e r i f y s t e a d y s t a t e s o l u t i o n : ( a ) - R e c t i f i e r t e r m i n a l v o l t a g e ; ( b ) - DC l i n e v o l t a g e ; ( c ) - DC l i n e c u r r e n t ; ( d ) - E l e c t r o d e l i n e v o l t a g e . 27 k 8 12 Time ( m i l l i s e c o n d s ) 16 h 8 12 16 Time ( m i l l i s e c o n d s ) (b) F i g . 3.7 Simulation w i t h DC s t a r t s at (a) a poi n t of commutation; (b) middle of non-commutation i n t e r v a l . Fig. 3.8 Simulation results with electrode line starts at zero i n i t i a l conditions. 29 o s c i l l a t i o n on the e l e c t r o d e and an erroneous waveform at the converter t e r m i n a l . The s i m u l a t i o n does not s e t t l e to steady-state even a f t e r a few c y c l e s . Since most ac/dc system s i m u l a t i o n s c o n s i s t of a l a r g e number of components, the i n i t i a l i z a t i o n process described must be done w i t h e x t r a -ordinary care. I t i s b e l i e v e d that w i t h some program changes i n the steady-s t a t e subroutine to handle the s u p e r p o s i t i o n of s o l u t i o n s at d i f f e r e n t net-work frequencies, the whole i n i t i a l i z a t i o n process could be done automati-c a l l y . This i s beyond the scope of t h i s t h e s i s , and a p o s s i b i l i t y f o r f u t u r e improvements. 30 4 . HARMONIC A N A L Y S I S One a d v a n t a g e o f a d e t a i l c o n v e r t e r m o d e l i s i t s a b i l i t y t o s i m u -l a t e t h e i n t e r a c t i o n b e t w e e n t h e a c a n d d c s i d e s o f t h e n e t w o r k a c c u r a t e l y . T h i s i s i m p o r t a n t b o t h i n s t e a d y - s t a t e a n d t r a n s i e n t s a n a l y s i s . One m a j o r c r i t e r i o n i n HVDC s y s t e m d e s i g n i s i t s h a r m o n i c p e r f o r m a n c e . T h e a p p e a r a n c e o f u n d e s i r a b l e h a r m o n i c s 1 3 ' 2 0 may c a u s e o v e r h e a t i n g o f c a p a c i t o r s a n d g e n e r a t o r s , i n t e r f e r e n c e i n t e l e c o m m u n i c a t i o n c i r c u i t s , s h o r t e n e d l i f e o f i n c a n d e s c e n t l i g h t s , i n s t a b i l i t i e s o f t h e c o n v e r t e r c o n t r o l , a n d r e m o t e r e s o n a n c e s i n t h e a c s y s t e m . C o r r e c t i v e m e a s u r e s m u s t b e i m p o s e d t o e l i m i -n a t e t h e m a t t h e s o u r c e o r a t l e a s t t o r e d u c e t h e m t o p e r m i s s i b l e l e v e l s . A g o o d a m o u n t o f p u b l i s h e d l i t e r a t u r e d e s c r i b e s t h e d i f f e r e n t m e t h o d s f o r d e t e r m i n i n g t h e l e v e l o f h a r m o n i c g e n e r a t i o n w h i c h c a n b e a n t i c i p a t e d f r o m a c o n v e r t e r i n s t a l l a t i o n . T h e m a i n o b j e c t i v e o f t h i s c h a p t e r i s n o t t o d i s c u s s t h e s e l a b o r i o u s m a t h e m a t i c a l p r o c e d u r e s i n d e t a i l , b u t r a t h e r t o s h o w t h e u s e f u l n e s s o f t h e d i g i t a l t i m e - d o m a i n s i m u l a t i o n i n p r e d i c t i n g t h e h a r m o n i c b e h a v i o u r , a n d t o c o m p a r e t h e r e s u l t s w i t h t h o s e o b t a i n e d f r o m t h e t h e o r e t i c a l c a l c u l a t i o n s w h e r e a p p r o p r i a t e . 4 . 1 A n a l y z i n g M e t h o d A F o u r i e r A n a l y s i s P r o g r a m w a s u s e d t o a n a l y z e t h e s i m u l a t i o n r e -s u l t s . T h e o u t p u t q u a n t i t i e s o f t h e T r a n s i e n t s P r o g r a m ( n o d e v o l t a g e s , b r a n c h c u r r e n t s , b r a n c h v o l t a g e s , e t c . , a s a f u n c t i o n o f t i m e ) c a n b e s t o r e d i n a f i l e f o r f u r t h e r p r o c e s s i n g . O n c e t h e s i m u l a t i o n s e t t l e s i n t o s t e a d y -s t a t e , t h e v a l u e s w i t h i n o n e p e r i o d c a n b e e x p r e s s e d a s oo oo f ( x ) I a . c o s ( i x ) + .1 b . s i n ( i x ) ( 4 . 1 ) i = 0 1 1=0 1 w h e r e a . , b . a r e t h e c o e f f i c i e n t s o f t h e F o u r i e r s e r i e s . 1 x 31 T h e p r o g r a m , o r i g i n a l l y w r i t t e n b y H . W . D o m m e l 2 1 , r e a d s a l l t h e n d a t a p o i n t s w i t h i n a u s e r - s p e c i f i e d p e r i o d a n d c o m p u t e s t h e c o s i n e c o e f f i -c i e n t s a . . . . a a n d s i n e c o e f f i c i e n t s b . . . . b o f e q u a t i o n ( 4 . 1 ) a s w e l l o m o m a s t h e m a g n i t u d e o f w i t h C . = / a . 2 + b . z ( 4 . 2 ) 1 i x i = 0 , 1 , . . . . m m = — , w h e n n i s e v e n n , w h e n n i s o d d T h e r e s u l t i n g f i n i t e s e r i e s m m F ( x ) = E a . c o s ( i x ) + E b . s i n ( i x ) ( 4 . 3 ) i = 0 1 i = 0 1 w i l l p a s s t h r o u g h e a c h d a t a p o i n t o f t h e s p e c i f i e d p e r i o d , a n d p r o v i d e a s m o o t h i n t e r p o l a t i o n b e t w e e n p o i n t s w i t h t h e l e a s t n u m b e r o f h a r m o n i c s . A FORTRAN l i s t i n g o f t h e p r o g r a m i s s h o w n i n A p p e n d i x 3 . I n o r d e r t o o b t a i n h i g h e r o r d e r h a r m o n i c s , t h e t i m e s t e p f o r t h e s i m u l a t i o n m u s t b e c h o s e n i n s u c h a w a y t h a t t h e c o m p u t a t i o n c o s t d o e s n o t i n c r e a s e u n n e c e s s a r i l y o n o n e h a n d , a n d t h a t t h e r e s u l t s a r e s t i l l a c c u r a t e e n o u g h o n t h e o t h e r h a n d . E x p e r i e n c e h a s s h o w n t h a t f o r a n a l y z i n g c h a r a c -t e r i s t i c h a r m o n i c s , a s a m p l i n g r a t e o f 1 0 p o i n t s p e r c y c l e a t t h e h i g h e s t f r e q u e n c y o f i n t e r e s t w i l l g i v e r e a s o n a b l e s o l u t i o n s . I n s t u d y i n g i m b a l a n c e f a c t o r s , h o w e v e r , a s t e p l e n g t h o f l e s s t h a n o n e d e g r e e o f f u n d a m e n t a l f r e q u e n c y s h o u l d b e u s e d . 32 4 . 2 H a r m o n i c C o n t e n t s A c o n v e r t e r b r i d g e w i l l p r o d u c e t o some n o t i c e a b l e d e g r e e a l l t h e l o w e r o r d e r h a r m o n i c s . T h e s e h a r m o n i c c u r r e n t s a p p e a r i n b o t h t h e a c a n d d c n e t w o r k s , a n d may p e n e t r a t e i n t o t h e p o w e r s y s t e m f a r f r o m t h e c o n n e c t i o n p o i n t . U n d e r n o r m a l o p e r a t i n g c o n d i t i o n s , some h a r m o n i c s a r e p r e d o m i n a n t i n m a g n i t u d e . T h e y a r e u s u a l l y r e f e r r e d t o a s n o r m a l o r c h a r a c t e r i s t i c h a r -m o n i c s a n d a t t r a c t m o s t o f t h e a t t e n t i o n i n s y s t e m d e s i g n . T h e y a r e o f t h e f o l l o w i n g o r d e r : DC s i d e : : h = k p AC s i d e : : h = k p ± 1 w h e r e h = o r d e r o f a h a r m o n i c k = p u l s e n u m b e r p = 1 , 2 , 3 . . . a n y p o s i t i v e i n t e g e r B a s e d o n t h e a s s u m p t i o n s o f p e r f e c t l y s m o o t h e d d i r e c t c u r r e n t ( i . e . i n f i n i t e s m o o t h i n g i n d u c t a n c e o n t h e d c s i d e ) a n d s y m m e t r i c a l o p e r a t i o n o f t h e a c n e t w o r k , t h e a m p l i t u d e o f t h e c h a r a c t e r i s t i c a c s i d e c u r r e n t h a r m o n i c s 13 Ik. f o r o v e r l a p a n g l e s o f l e s s t h a n 6 0 ° may b e w r i t t e n a s ' r I l o F l \ - " h D " ( 4 ' 4 ) w h e r e IlQ = rms f u n d a m e n t a l a l t e r n a t i n g c u r r e n t w i t h n o o v e r l a p D = c o s a - c o s (a+u ) U =. o v e r l a p a n g l e Fl = [ A 2 + B 2 - 2AB c o s ( 2 a + u ) ] 2 A _ s i n [ ( h - 1 ) u / 2 ]  A h - 1 •n _ s i n [ ( h + 1 ) u / 2 ] - — • 33 a n d t h e rms v a l u e o f t h e h a r m o n i c s o f t h e d i r e c t v o l t a g e i s g i v e n b y V d h " V d o ^ ( 4 ' 5 ) w h e r e F 2 = [ C 2 + D 2 - 2CD c o s ( 2 a + u ) ] r - c o s [ ( h - 1 ) u / 2 ] ° " h - 1 n = c o s [ ( h + 1 ) u / 2 ] h+1 T h e s e e q u a t i o n s a r e f o r i d e a l i z e d c o n d i t i o n s , a n d t h e t r u e v a l u e w i l l v a r y f r o m t h e r e s u l t s o b t a i n e d w i t h t h e m d e p e n d i n g o n t h e o p e r a t i n g c o n d i t i o n o f t h e p a r t i c u l a r s y s t e m . I n t h e p a s t , p l a n n i n g e n g i n e e r s c o m -p u t e d t h e s e i d e a l h a r m o n i c l e v e l s a n d u s e d e m p i r i c a l v a l u e s b a s e d o n t h e c o n t r a c t o r ' s e x p e r i e n c e o f p r e v i o u s HVDC i n s t a l l a t i o n s t o m a k e m o d i f i c a -t i o n s 2 2 . A s t h i s d i s c u s s i o n p r o g r e s s e s , t h e p r a c t i c a l i t y o f t h e s i m p l i f y i n g a s s u m p t i o n s w i l l b e i n v e s t i g a t e d . T h e f i r s t q u e s t i o n a b l e a s s u m p t i o n i s t h e p e r f e c t s m o o t h n e s s o f d i r e c t c u r r e n t . I n r e a l i t y , t h e s m o o t h i n g r e a c t o r s i n HVDC i n s t a l l a t i o n s a r e t y p i c a l l y i n t h e r a n g e o f 0 . 5 t o 1 . 0 H . T a b l e 4 . 1 c o m p a r e s t h e h a r -m o n i c s c o n t a i n e d i n t h e s i m u l a t e d c u r r e n t c u r v e w i t h t w o t h e o r e t i c a l c a l c u -l a t i o n s , o n e b a s e d o n e q u a t i o n ( 4 . 4 ) a n d a n o t h e r d e r i v e d e a r l i e r b y B r o w n a n d S m i t h 2 3 who d e s c r i b e d t h e c u r r e n t w a v e s h a p e o n t h e a c s i d e o f t h e m e r -c u r y a r c r e c t i f i e r u n d e r p u r e r e s i s t i v e l o a d c o n d i t i o n a s : 3 . 3 0 8 I i = [ s i n 6 - 0 . 2 2 6 s i n 50 - 0 . 1 1 3 s i n 7 0 + 0 . 0 9 1 s i n 1 1 0 + 0 . 0 6 5 s i n 1 3 0 - 0 . 0 5 6 7 s i n 1 7 0 ] ( 4 . 6 ) T h e h a r m o n i c s e x t r a c t e d f r o m t h e d i g i t a l s i m u l a t i o n l i e i n b e t w e e n t h e two e x t r e m e s . F u r t h e r m o r e , a f i n i t e i n d u c t a n c e a l s o i m p l i e s t h a t t h e c u r r e n t w a v e s h a p e i s a f f e c t e d b y t h e n a t u r e o f t h e l o a d . S i m u l a t e d v a l u e s o f a c 34 ORDER OF HARMONIC MAGNITUDE OF HARMONIC CURRENT . Calcul a t i o n based on L=0 EMTP Simulation Results C a l c u l a t i o n based on L=°° L = ,5H L = IH 1 1.000 1.000 1.000 1.000 5 .226 .202 .198 .192 7 .113 .131 .135 .132 11 .091 .079 .080 .074 13 .065 .065 .067 .057 17 .0567 .0404 .0427 .0354 19 .0454 .0339 .0352 .0273 23 .0412 .0183 .0207 .0153 25 .0349 .0139 .0148 .0108 Table 4.1 Comparison of current harmonics obtained from t h e o r e t i c a l c a l c u l a t i o n s and from Fourier analysis of simulated curves. c h a r a c t e r i s t i c harmonic currents as a function of d i r e c t current are given i n F i g . 4.1. The overlap angle, which r e s u l t s from the commutating reactance, a f f e c t s the harmonic content as the loading condition changes. This should be taken into account i n the optimal design of the f i l t e r s , the converter transformers and the smoothing reactors. 35 36 4.3 Abnormal Harmonics The assumption of i d e a l balanced (symmetrical) operation i s also not r e a l i s t i c a l l y achievable. Asymmetrical conditions i n c l u d e 2 5 ' 2 6 f i r i n g angle errors from converter c o n t r o l , d i s t o r t i o n s of the ac bus voltage wave-forms, imbalances i n the ac system voltages and unequal transformer phase impedances. These imperfections r e s u l t i n the generation of non-characteris-t i c or abnormal harmonics. The e f f e c t of imbalances in the ac system v o l t -ages on the dc harmonics was studied. Table 4.2 contains the magnitude of the harmonic voltages on the dc side due to imbalance i n the voltage of one phase of the ac bus. The imbalances i n magnitude and phase angle are -5% of the nominal value. Results show that the c h a r a c t e r i s t i c harmonics do not change much but an appreciable amount of even harmonics are generated. These agree with the findings of Mathur and S h a r a f 2 4 . In p r a c t i c a l systems, a cer-t a i n degree of d i f f e r e n t imbalance factors i s i n e v i t a b l e . Most imbalances r e s u l t from an i n t e r a c t i o n among various phenomena. For example, one or more phases of the ac voltages can be depressed as a r e s u l t of a remote f a u l t , which w i l l lead to the generation of c e r t a i n abnormal harmonics. These har-monics w i l l propagate into the ac system, which i n turn may lead to voltage d i s t o r t i o n . Such cumulative e f f e c t s cannot be described by simple mathemati-cal formulations. The Transients Program i s capable of simulating cases with combinations of imbalance f a c t o r s . For example, a s i t u a t i o n where the sup-p l y i n g ac voltages are at 1.0/-90° , .98/151.2° f 1.02/31.2° p . u . , has been analyzed. F i g . 4.2 shows that the magnitudes of some non-characteristic harmonics are comparable to those of the c h a r a c t e r i s t i c ones i n t h i s case. There i s no doubt that t h i s simulation approach for obtaining non-c h a r a c t e r i s t i c harmonics o f f e r s a r e l a t i v e l y easy and economical way to gain a deeper and more systematic understanding of the causes and e f f e c t s i n the 37 HARMONIC ORDER R E C T I F I E R S I D E I N V E R T E R S I D E B a l a n c e d O p e r a t i o n I m b a l a n c e o f A C B a l a n c e d I m b a l a n c e o f AC M a g n i t u d e P h a s e O p e r a t i o n M a g n i t u d e P h a s e 1 . 0 0 0 0 . 0 0 1 7 . 0 0 1 8 . 0 0 0 9 . 0 0 0 7 . 0 0 2 1 2 . 0 0 3 3 . 0 2 0 8 . 0 3 7 5 . 0 0 1 9 . 0 1 7 2 . 0 3 8 1 3 . 0 0 1 5 . 0 0 1 9 . 0 0 2 0 . 0 0 0 8 . 0 0 0 8 . 0 0 0 5 4 . 0 0 0 7 . 0 0 8 4 . 0 1 4 8 . 0 0 0 8 . 0 0 5 9 . 0 0 8 1 5 . 0 0 0 7 . 0 0 1 5 . 0 0 0 4 . 0 0 2 2 . 0 0 2 2 . 0 0 1 6 6 . 1 4 7 9 . 1 3 9 4 . 1 4 6 3 . 1 5 1 9 . 1 4 6 9 . 1 5 2 2 7 . 0 0 2 5 . 0 0 3 4 . 0 0 2 3 . 0 0 4 6 . 0 0 4 4 . 0 0 4 8 8 . 0 0 0 7 . 0 1 0 4 . 0 1 9 1 . 0 0 2 4 . 0 1 1 5 . 0 1 7 6 9 . 0 0 1 9 . 0 0 2 7 . 0 0 1 4 . 0 0 0 5 . 0 0 0 4 . 0 0 0 1 1 0 . 0 0 1 0 . 0 0 6 5 . 0 1 0 6 . 0 0 0 4 . 0 0 3 1 . 0 0 5 6 1 1 . 0 0 1 6 . 0 0 3 4 . 0 0 1 8 . 0 0 0 4 . 0 0 0 4 . 0 0 0 3 12 . 0 4 7 2 . 0 4 1 3 . 0 4 4 6 . 0 4 3 9 . 0 3 9 5 . 0 4 4 0 1 3 . 0 0 1 7 . 0 0 2 8 . 0 0 2 1 . 0 0 0 4 . 0 0 0 2 . 0 0 1 3 1 4 . 0 0 1 2 . 0 0 7 2 . 0 1 1 3 . 0 0 1 1 . 0 0 5 2 . 0 0 8 6 1 5 . 0 0 1 5 . 0 0 2 9 . 0 0 0 9 . 0 0 0 6 . 0 0 0 5 . 0 0 0 3 T a b l e 4 . 2 C o m p a r i s o n o f d c v o l t a g e h a r m o n i c s u n d e r b a l a n c e d o p e r a t i o n a n d i m b a l a n c e i n m a g n i t u d e , a n d p h a s e i n o n e p h a s e o f t h e a c b u s . g e n e r a t i o n o f h a r m o n i c s o n t h e a c a n d d c s i d e o f c o n v e r t e r s . M o r e o v e r , c u r -r e n t a n d v o l t a g e l o a d i n g s i n e a c h e l e m e n t o f t h e f i l t e r a r m s a r e r e a d i l y a v a i l a b l e , w h i c h a r e v a l u a b l e t o t h e e n g i n e e r s who m u s t s e l e c t t h e p r o t e c -t i v e l e v e l o f t h e s e d e v i c e s . 38 3 H 10 12 IW 16 H a r m o n i c O r d e r 18 20 22 24 26 F i g . 4 . 2 H a r m o n i c v o l t a g e o n t h e d c s i d e o f t h e r e c t i f i e r t e r m i n a l : b a l a n c e o p e r a t i o n ; " i m b a l a n c e o p e r a t i o n ; x l e v e l o f c h a r a c t e r i s t i c h a r m o n i c s c a l c u l a t e d b y e q u a t i o n " ( 4 . 5 ) 39 5 . T R A N S I E N T S S I M U L A T I O N Two e x a m p l e s a r e p r e s e n t e d i n t h i s c h a p t e r t o t e s t t h e c a p a b i l i t y o f t h e c o n v e r t e r m o d e l i n t r a n s i e n t s s i m u l a t i o n s . P a r t o f t h e n e t w o r k f r o m t h e P a c i f i c HVDC I n t e r t i e S y s t e m i s u s e d . T h e t r a n s m i s s i o n l i n e d a t a a n d t h e c o n v e r t e r e q u i p m e n t l a y o u t o f t h e s y s t e m i s i n c l u d e d i n A p p e n d i x 4 . 5 . 1 P o i n t t o P o i n t O p e r a t i o n A s a p r e l i m i n a r y t e s t , a s i m p l e t w o t e r m i n a l c a s e w a s s t u d i e d . R e s u l t s u s i n g t h e s i m p l i f i e d m o d e l m e n t i o n e d i n C h a p t e r 1 w e r e u s e d a s a r e f e r e n c e . T h e s i m p l i f i e d m o d e l w a s d e v e l o p e d i n t h e e a r l y 1 9 7 0 ' s 2 7 f o r s t u d i e s c o n n e c t e d w i t h DC c i r c u i t b r e a k e r t e s t s . I t s i m u l a t e s t h e HVDC c o n v e r t e r s t a t i o n a s a d c v o l t a g e s o u r c e w h o s e a m p l i t u d e i s c o n t r o l l e d b y i t s c u r r e n t o u t p u t . W i t h t h i s s i m p l i f i e d m o d e l , t h e a c n e t w o r k s a n d t h e c o n -v e r t e r t r a n s f o r m e r s a r e t o t a l l y i g n o r e d a n d t h e v a l v e a n d a n o d e d a m p e r s a r e r e p r e s e n t e d a s a n R C - b r a n c h i n p a r a l l e l w i t h t h e d c s o u r c e . I n t h e d e t a i l e d c a s e , t h e s y s t e m c o n f i g u r a t i o n i s s i m i l a r t o F i g . 2 . 1 . T h e a c s i d e b e h i n d e a c h c o n v e r t e r i s r e p r e s e n t e d a s a c v o l t a g e s o u r c e s b e h i n d T h e v e n i n e q u i v a -l e n t i m p e d a n c e s , w h o s e v a l u e s c a n b e f o u n d f r o m t h e p o s i t i v e a n d z e r o s e -q u e n c e s h o r t - c i r c u i t M V A . T h e c o n v e r t e r t r a n s f o r m e r s a r e c o n n e c t e d a s w y e g r o u n d e d / w y e a n d t h e v a l v e a n d a n o d e d a m p e r s a r e r e p r e s e n t e d i n d e t a i l . T h e d c l i n e i s 8 5 0 m i l e s l o n g a n d o p e r a t e s i n m o n o p o l a r g r o u n d r e t u r n m o d e . A l i n e - t o - g r o u n d f a u l t w a s a p p l i e d 5 0 m i l e s f r o m t h e r e c t i f i e r e n d a t t i m e t = 1 0 m s e c . T h e v o l t a g e a n d c u r r e n t a t t h e r e c t i f i e r t e r m i n a l s a r e s h o w n i n F i g . 5 . 1 f o r b o t h t y p e s o f m o d e l s ( s e e A p p e n d i x 5 f o r t h e i n p u t l i s t i n g s o f b o t h r u n s ) . T h e a g r e e m e n t i s r e a s o n a b l y g o o d . T h e p h a s e s h i f t b e t w e e n t h e two c u r v e s may b e d u e t o t h e e x c l u s i o n o f a c c i r c u i t s a n d t h e l u m p e d 40 F i g . 5.1 Comparison between s i m p l i f i e d and d e t a i l e d converter s t a t i o n models f o r a f a u l t on the dc l i n e . S o l i d l i n e . = d e t a i l e d model; dotted l i n e = s i m p l i f i e d model. 41 representation of dampers i n the s i m p l i f i e d simulation. In such simple cases, the s i m p l i f i e d model does save computation time, while the d e t a i l e d model provides more in s i g h t into the stress on the conversion components and the i n t e r a c t i o n between the ac and dc sides . 5.2 Three-Terminal Operation To check the performance of a prototype dc c i r c u i t breaker i n the f i e l d , several tests were conducted on the P a c i f i c HVDC I n t e r t i e System 8* 1 5. The system was tapped as a three-terminal system with constant voltage r e c t i -f i e r control and independent constant current controls on the two i n v e r t e r s . The network set-up i s shown i n F i g . 5.2, and the system steady-state condi-tions before the staged f a u l t test are summarized i n Table 5.1. Two manually-initiated f a u l t s were applied, a c l o s e - i n f a u l t by operating an ac breaker which was connected to ground through a 5f2 r e s i s t o r , and a remote f a u l t by dropping a wire pendulum to short the pole conductor to a tower at 250 miles away from the breaker. Melvoid et a l . 1 5 made some comparisons between f i e l d tests and computer simulations using the s i m p l i -f i e d model. The same arrangement was simulated using the d e t a i l e d model. A sample data deck of the simulation i s included i n Appendix 6. The r e s u l t -ing curves were superimposed with the r e s u l t s given i n reference [15] . As seen i n F i g . 5.3 and 5.4, closer agreement with the f i e l d test measurements can be obtained using the d e t a i l e d model. The converters were operated with abnormally large valve voltage stresses r e s u l t i n g from the large commutation margins. T o t a l l y ignoring the e f f e c t s of the ac c i r c u i t s i n the s i m p l i f i e d simulation may cause the inaccurate p r e d i c t i o n of the current transfer among the terminals. Furthermore, the s l i g h t disagreement between the f i e l d tests and the d e t a i l e d simulation, p r i m a r i l y i n the form of high frequency o s c i l -42 5^' CELILO SYLMAR R e c t i f i e r j — - T r y -r d i n J — o r m -0_ FILTERS FILTERS -fY Y L _ _ ^ j r « v I n v e r t e r 8 50 miles Inverter /mnp—| Y Y * FILTERS I I —< 250 (-.miles \~ CIRCUIT BREAKER fault B fault A Fig. 5.2 System Configuration for DC Circuit Breaker Test FAULT CELILO RECTIFIER SYLMAR INVERTER CELILO INVERTER CLOSE-IN 69.2KV/630A 63KV/330A 53.2KV/350A REMOTE 70.5KV/558A 60KV/290A 50.5KV/3 06A TABLE 5.1 Steady-State System Parameters Before F a u l t ~i 1 1 1 1 1 1 r 1 10 2 0 3 0 4 0 5 0 60 '. 7 0 8 0 90 T i m e ( m i l l i s e c o n d s ) F i g . 5 . 4 C u r r e n t t h r o u g h d c b r e a k e r f o l l o w i n g a r e m o t e f a u l t . 45 l a t i o n s , may be a t t r i b u t e d to the i n a b i l i t y to represent the frequency de-pendence of l i n e parameters i n the UBC v e r s i o n of the Transients Program. 4 6 6 . C O N C L U S I O N S T h i s t h e s i s d e s c r i b e s t h e d e v e l o p m e n t o f a new c o n v e r t e r m o d e l w h i c h a l l o w s t h e u s e r t o r e p r e s e n t HVDC c o n v e r t e r v a l v e s a n d t h e i r a s s o -c i a t e d f i r i n g c o n t r o l s y s t e m s i n t h e U B C v e r s i o n o f 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 P r o g r a m . T h e m o d i f i e d p r o g r a m c a n b e u s e d t o s i m u l a t e i n t e r -a c t i o n s b e t w e e n t h e a c a n d d c s i d e a n d t h e b e h a v i o u r o f t h e c o n t r o l s y s t e m f o r a n y t y p e o f s t e a d y - s t a t e o r t r a n s i e n t o p e r a t i n g c o n d i t i o n . T h e new m o d e l t a k e s a c c o u n t o f t h e a c t u a l h a r d w a r e a r r a n g e m e n t o f t h e c o n t r o l a m p l i f i e r w h i c h l e a d s t o a m o r e a c c u r a t e r e p r e s e n t a t i o n o f t h e l i m i t e r s . E f f o r t s h a v e b e e n m a d e t o m i n i m i z e u n n e c e s s a r y c o m p u t a t i o n s i n t h e i n i t i a l i z a t i o n p r o c e s s . T h e m e t h o d d e v e l o p e d i n t h i s t h e s i s i s v e r y e c o n o m i c a l , b u t i t c o u l d b e i m p r o v e d a n d m a d e m o r e o r l e s s f o o l p r o o f i f t h e w h o l e p r o c e d u r e w e r e d o n e a u t o m a t i c a l l y . T h i s w o u l d b e a w o r t h w h i l e t o p i c f o r f u t u r e i n v e s t i g a t i o n s . T h e m o d e l w a s a p p l i e d t o h a r m o n i c s t e a d y - s t a t e a n a l y s i s a n d t r a n -s i e n t a n a l y s i s t o v e r i f y i t s c a p a b i l i t y . I n t h e l a t t e r a p p l i c a t i o n , s i m u -l a t i o n r e s u l t s came c l o s e t o f i e l d m e a s u r e m e n t s . H o w e v e r , i t s h o u l d b e r e a l i z e d t h a t t h e f r e q u e n c y - d e p e n d e n t c h a r a c t e r i s t i c s o f l i n e p a r a m e t e r s h a d b e e n n e g l e c t e d i n t h e s e s t u d i e s . C o n s i d e r i n g t h e c o m p l e x f r e q u e n c y s p e c t r a a s s o c i a t e d w i t h t h e h a r m o n i c s a n d t r a n s i e n t s , t h i s w o u l d d e f i n i t e l y c o n t r i b -u t e some i n a c c u r a c i e s i n t h e s i m u l a t i o n . 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 p a r a m e t e r s i s t h e s u b j e c t o f a n o n g o i n g P h . D . p r o j e c t i n t h i s D e p a r t m e n t , a n d t h e r e f o r e b e y o n d t h e s c o p e o f t h i s M . A . S c . t h e s i s . I n t e r m s o f c o m p l e x i t y , c a p a b i l i t y a n d f l e x i b i l i t y , t h i s m o d e l l i e s i n b e t w e e n t h e s i m p l i f i e d d c s o u r c e m o d e l a n d t h e ' T A C S ' p a c k a g e 7 . I t 47 i s adequate and p a r t i c u l a r l y u s e f u l f o r b a s i c s t u d i e s i n the research en-vironment of a u n i v e r s i t y . The model i s developed not as a s u b s t i t u t e to e i t h e r of the two e x i s t i n g features but as a supplement to these. 4 8 R E F E R E N C E S [ 1 ] A l l a n G r e e n w o o d , " E l e c t r i c a l T r a n s i e n t s i n P o w e r S y s t e m s " , W i l e y -I n t e r s c i e n c e , 1 9 7 1 . [ 2 ] H . W . D o m m e l , " 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 - a n d M u l t i p h a s e N e t w o r k s " , I E E E T r a n s , o n P A S , V o l . P A S - 8 8 , p p . 3 8 8 - 3 9 6 , A p r i l 1 9 6 9 . [ 3 ] J . E . H u d s o n , E . M . H u n t e r a n d D . D . W i l s o n , " E H V - D C S i m u l a t o r " , I E E E T r a n s , o n P A S , V o l . P A S - 8 5 , p p . 1 1 0 1 - 1 1 0 7 , N o v . 1 9 6 6 . [ 4 ] S . S a s a k i , H . M a t s u b a r a a n d D . J . M e l v o l d , " H i g h - A c c u r a c y A n a l y s i s o f F a u l t S u r g e b y D C - T N A o n HVDC T r a n s m i s s i o n L i n e s a n d C o m p a r i s o n w i t h F i e l d T e s t R e s u l t s " , P a p e r s u b m i t t e d t o 1 9 8 0 I E E E - P E S Summer M e e t i n g . [ 5] J . P . B i c k f o r d , N . M u l l i n e u x a n d J . R . R e e d , " C o m p u t a t i o n o f P o w e r -S y s t e m T r a n s i e n t s " , P e t e r P e r e g r i n u s L t d . , 197 6 . [ 6 ] H . W . D o m m e l , I . I . D o m m e l , " T r a n s i e n t s P r o g r a m U s e r ' s M a n u a l " , U . B . C . V a n c o u v e r , 1 9 7 6 . [ 7 ] L . D u b e , H . W . D o m m e l , " S i m u l a t i o n o f C o n t r o l S y s t e m s i n a n 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 w i t h T A C S " , I E E E P I C A C o n f e r e n c e , p p . 2 6 6 -2 7 1 , May 1 9 7 7 . [ 8 ] G . A . H o f m a n n , G . L . L a B a r b e r a , N . E . R e e d , L . A . S h i l l o n g , W . F . L o n g - a n d D . J . M e l v o l d , " F i e l d T e s t o f HVDC C i r c u i t B r e a k e r : L o a d B r e a k a n d F a u l t C l e a r i n g o n t h e P a c i f i c I n t e r t i e " , I E E E T r a n s , o n P A S , V o l . P A S -9 5 , p p . 8 2 9 - 8 3 8 , M a y / J u n e 197 6 . [ 9 ] D . J . M e l v o l d , P . C . Odam a n d J . J . V i t h a y a t h i l , " T r a n s i e n t O v e r v o l t a g e s o n a n HVDC B i p o l a r L i n e D u r i n g M o n o p o l a r L i n e F a u l t s " , I E E E T r a n s , o n P A S , V o l . P A S - 9 6 , p p . 5 9 1 - 6 0 1 , 1 9 7 7 . [ 1 0 ] H . W , Domme l a n d W . S . M e y e r , " C o m p u t a 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 " , P r o c . I E E E , V o l . 6 2 , p p . 9 8 3 - 9 9 3 , J u l y 1 9 7 4 . [ 1 1 ] H . W . D o m m e l , " N o n l i n e a r a n d T i m e - v a r y i n g E l e m e n t s i n D i g i t a l S i m u l a 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 " , P r o c . 1 9 7 1 P I C A C o n v e r e n c e , p p . 1 2 1 -1 2 7 , B o s t o n , M a s s . , M a y 1 9 7 1 . [ 1 2 ] J . A r r i l l a g a , J . G . C a m p o s B a r r o s a n d H . J . A l - K h a s h a l i , " D y n a m i c M o d e l -l i n g o f S i n g l e G e n e r a t o r s C o n n e c t e d t o HVDC C o n v e r t e r s " , I E E E T r a n s , o n P A S , V o l . P A S - 9 7 , p p . 1 0 1 8 - 1 0 2 9 , J u l y / A u g . 1 9 7 8 . [ 1 3 ] E . W . K i m b a r k , " D i r e c t C u r r e n t T r a n s m i s s i o n " , V o l . 1 , W i l e y - I n t e r s c i e n c e 1 9 7 1 . [ 1 4 ] E . U h l m a n n , " P o w e r T r a n s m i s s i o n b y D i r e c t C u r r e n t ' , S p r i n g e r - V e r l a g , 1 9 7 5 . 49 [ 1 5 ] J . M e l v o l d , P . R . S h o c k l e y , W . F . L o n g a n d N . G . H i n g o r a n i , " T h r e e T e r -m i n a l O p e r a t i o n o f t h e P a c i f i c HVDC I n t e r t i e f o r DC C i r c u i t B r e a k e r T e s t i n g " , I E E E T r a n s , o n P A S , V o l . P A S - 9 5 , p p . 1 2 8 7 - 1 2 9 6 , J u l y / A u g . 1 9 7 6 . [ 1 6 ] J . D . A i n s w o r t h , " T h e P h a s e - l o c k e d O s c i l l a t o r - A New C o n t r o l f o r C o n -t r o l l e d S t a t i c C o n v e r t e r s " , I E E E T r a n s , o n P A S , V o l . P A S - 8 7 , p p . 8 5 9 -8 6 5 , M a r c h 1 9 6 8 . [ 1 7 ] C . W . G e a r , " N u m e r i c a l I n i t i a l V a l u e P r o b l e m s i n O r d i n a r y D i f f e r e n t i a l E q u a t i o n s " , P r e n t i c e - H a l l I n c . , 1 9 7 1 . [ 1 8 ] W . J . K a r p l u s , W . W . S o r o k a , " A n a l o g M e t h o d s - C o m p u t a t i o n a n d S i m u l a -t i o n " , M c G r a w - H i l l , 1 9 5 9 . [ 1 9 ] F . R . B r a d l e y , R . M c C o y , " D r i f t l e s s D - C A m p l i f i e r " , E l e c t r o n i c s , V o l . 2 5 , p p . 1 4 4 - 1 4 8 , A p r i l 1 9 5 2 . [ 2 0 ] D . J . M e l v o l d , " P a c i f i c HVDC I n t e r t i e S y s t e m A C S i d e H a r m o n i c S t u d i e s " , I E E E T r a n s , o n P A S , V o l . P A S - 9 2 , p p . 6 9 0 - 7 0 1 , M a r c h / A p r i l 1 9 7 3 . [ 2 1 ] H . W . D o m m e l , " C a s e S t u d i e s o n 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 " , 1 9 7 8 . [ 2 2 ] J . J . V i t h a y a t h i l , D . J . M e l v o l d a n d V . M a d z a r e v i c , " C a l c u l a t i o n a n d M e a s u r e m e n t o f H a r m o n i c s o n t h e A C s i d e o f C o n v e r t e r S t a t i o n s o f t h e P a c i f i c HVDC I n t e r t i e " , P r o c . o n S y m p o s i u m o n HVDC P o w e r T r a n s m i s s i o n , p p . 2 6 9 - 3 2 4 , 1 9 7 6 . [ 2 3 ] H . D . B r o w n , J . J . S m i t h , " C u r r e n t a n d V o l t a g e W a v e S h a p e o f M e r c u r y A r c R e c t i f i e r s " , A I E E T r a n s . , V o l . 5 2 , p p . 9 7 3 - 9 8 6 , 1 9 3 3 . [ 2 4 ] R . M . M a t h u r a n d A . M . S h a r a f , " H a r m o n i c s o n t h e DC s i d e i n HVDC C o n v e r -s i o n " , I E E E T r a n s , o n P A S , V o l . P A S - 9 6 , p p . 1 6 3 1 - 1 6 3 8 , S e p t . / O c t . 1 9 7 7 . [ 2 5 ] T . S u b b a r a o a n d J . R e e v e , " H a r m o n i c s C a u s e d b y I m b a l a n c e d T r a n s f o r m e r I m p e d a n c e s a n d I m p e r f e c t T w e l v e - p u l s e O p e r a t i o n i n HVDC C o n v e r s i o n " , I E E E T r a n s , o n P A S , V o l . P A S - 9 5 , p p . 1 7 3 2 - 1 7 3 5 , S e p t . / O c t . 1 9 7 6 . [ 2 6 ] J . R e e v e a n d P . C . S . K r i s h n a y y a , " U n u s u a l C u r r e n t H a r m o n i c s A r i s i n g f r o m H i g h - V o l t a g e DC T r a n s m i s s i o n " , I E E E T r a n s , o n P A S , V o l . P A S - 8 7 , p p . 8 8 3 - 8 9 3 , M a r c h 1 9 6 8 . [ 2 7 ] W . F . L o n g , " A S t u d y o n Some S w i t c h i n g A s p e c t s o f a D o u b l e C i r c u i t HVDC T r a n s m i s s i o n L i n e " , I E E E T r a n s , o n P A S , V o l . P A S - 9 2 , p p . 7 3 4 - 7 4 1 , M a r c h / A p r i l 1 9 7 3 . 50 APPENDIX 1 Subroutinei VALCON! Programme L i s t i n g s 1 SUBROUTINE VALCON<ISTEP, DELTAT, KSWTCH. KP053) 2 COMMON /THYRIS/THYR < 51 )» THYR2 < 51). DCCUR(10) > DCK(10)> DCT1< 10)» 3 1 DCT2<10),DCT3<10), DCK1<10), DCK2<10), DCMIN<10), DCMAX<10). 4 2 ACVA<10),ACVB<10), ACVC(10), PHSHIF<10), 5 3 ACVAI<10),ACVBI(10). ACVCI(10),COMR<10), 6 4 DC INI<10)>DFREQ <10), ICON(51), IFIRE(51),NGROUP, IMODE <10) 7 5 NCDUT(IO) 8 DIMENSION DELAY <10 >,COUNT < 51),EALFA <10),EMAX < 10),CTIME <10). 9 1 EMIN(10),X(10),KP033<51),NMODE<10),CORDER<10), 10 2 CMARG(IO), XCURUO), XB<10) 11 DOUBLE PRECISION THYR,THYR2,DCCUR,DELTAT,DELAY,COUNT, DCK, DCK1. 12 ' " 1 DCK2, DCT1.DCT2, DCT3, DCMIN, DCMAX,DCINI, EALFA, DFREQ, EMAX, EMIN, 13 2 DELTA2,T,A,B, P, X,ACVA, ACVB, ACVC,PHSHIF, ACVAI, ACVBI, ACVCI, 14 3 CTIME,CORDER, CMARG, COMR, XCUR, XB 15 C ***** START CONTROL GROUP LOOP 16 DO 104 J=l,NGROUP 17 DCCUR < J)-DABS(OCCUR < J)) IS IF< ISTEP. GT. 0) GO TO 101 " 19 C ***** INITIALIZATION BEFORE ENTERS TIME STEP LOOP 20 . CORDER(J)=DCCUR < J) 21 CTIME(J)=0. DO 22 NMUDE<J)=1 23 I=IMODE(J) _ 24 IF(I ABS <I).GT. 3)NMODE < J)=2 25 IF <I. LT. O)NMODE(J)=—NMODE(J) 26 I MODE(J)=1ABS(I)-3*(IABS < NMODE < J))—1) 27 IF < J. GT. 1 ) GO TO 103 28 DO 102 1=1,KSWTCH 29 COUNT < I ) =0. DO 30 IFIRE(I)=0 31 102 CONTINUE 32 103 DELTA2=DELTAT/2. DO 33 DFREQ<J)=1.D0/<6.2831S53D0*DFREQ<U)) 34 T=DCT1< J)+DCT3 iJ) 35 P=DCT1<J)*DCT3<J) 36 A=l.D0+T/DELTA2+P/(DELTA2**2) 37 B= ( 1. D0+DCT2 < J) /DELTA2 ) *DCK ( J) . _:  33 XB<U)=2. DO/B 39 DCT1<J)=4. DO*P/A/DELTAT 40 P=DCINKJ) 41 IF(IMODE(J). EQ. 2)DC INI(J)=DCMAX(J) 42 IF(IMODE < J). EQ. 3)DC INI<J)=DCMIN<J) 43 EMAX<U)=<DCMAX<J)-DCK1<J))/DCK2(U) 44 EMIN<J)=<DCMIN<J)-DCK1<J))/DCK2<J) 45 X < J)=0. DO 46 T=DCCUR(J) 47 IF ( I MODE < U). GT. 1 ) T=P 48 CMARG < J)=CORDER(J) -T 49 XCUR < J)=CORDER < J)-CMARG < J)-DCCUR(J) 50 EALFA(J)=(DCINI(U)-DCKl(J))/DCK2(J) 51 DCT2<J)=1.DO-DCK<J)*XB<J) 52 IF <IMODE < U). GT. 1) XCUR<J)=EALFA<J)/DCK<J) 53 DCT3(J)-B/A 54 DCMAX(J)=2. DO*DCK<U)/A-DCT3<U) 55 DCK< J) = l . DO-2. DO/A 56 DCMIN(J)=DCMAX(J)*XCUR<J)+DCK<J)*EALFA(J) 57 GO TO 105 58 C ***** ON EACH TIME STEP 59 _C ***** CALCULATE AMPLIFIER OUTPUT & DETERMINE^OPERATING REGION 60 " 1 0 1 A=EALFA (J) 61 _ DCCUR < J) =CORDER (J) -DCCUR < J) -CMARG < J) 62 IF < I MODE (J). EQ. 2. AND. DCCUR ( J). GT. O. DO) GO TO 106 63 IF (I MODE ( J). EQ. 3. AND. DCCUR < J). LT. 0. DO) GO TO 107 64 EALFA< J) =DCT3< J) *DCCUR < J) +DCMIN( J) 65 XCUR(J)=DCCUR(J) 66 ' X < J) = (EALFA < J ) -A )/DELTA2-X ( J) 67 IF < EALFA <J). GT. EMAX< J) ) GO TO 106 68 '' IF <EALFA(J). LT.LMIN<J)) GO TO 107 69 IF<IMODE(J). GT. 1)WRITE(6, 300)J 70 300 FORMAT < ' CONVERTER NO. ',13, ' BACK OFF FROM LIMIT') 71 IM0DE(J)=1 72 GO TO 108 73 C ***** CONVERTER IN UPPER LIMIT 74 106 EALFA(J)=EMAX < J) 75 IF<IMODE<J). EQ. 1)WRITE<6,301)J 76 301 FORMAT < ' CONVERTER NO. ',13,' HITS UPPER LIMIT') 77 XCUR < J)=EALFA < J)*XB < J)+DCT2 < J)*XCUR < J) 78 IMODE(J)=2 79 X < J)=0. DO BO GO TO 108 81 C ***** CONVERTER IN LOWER LIMIT 82 107 EALFA(J)=EMIN< J) 83 IF<IMODE<J). EQ. 1)WRITE<6,302)J 84 302 FORMAT < ' CONVERTER NO. ',13,' HITS LOWER LIMIT') 85 XCUR (J)=EALFA < J)*XB < J)+DCT2(J)*XCUR(J) 86 IMODE(J)=3 87 X <J)=0. DO 88 103 DCMIN(J)=DCMAX < J)*XCUR(J)+DCK < J)*EALFA(J) + DCT1(J)*X< J) 89 105 IF(IABS<NMODE < J)) . EQ. 1. OR. I STEP. EQ. 0)GO TO 160 90 A=6.2831B53D0*DFREQ(J) 91 CTIME(J)=CTIME< J)+DELTAT 92 A=A-CTIME<J) 93 IF(A.GT.DELTA2) GO TO 104 94 CTIME < J)==0. DO 95 C ***** CALCULATING VALVE DELAY TIME 96 160 A=DCK1< J)+DCK2(J)*EALFA< J) 97 B=DABS < DCK.1< J)) 98 P=A/B 99 DELAY < J)=DFREQ <J)*DARCOS< P) 100 IF<NCOUT<J). EQ. O) GO TO 104 101 C ***** OUTPUT DELAY ON REQUEST 102 WRITE (6,11) J,DELAY(J) 103 11 FORMAT < IH , 'CONVERTER NO. ',I3,2X, 'DELAY IS',E15. 6, ' SEC. 104 104 CONTINUE 105 C ***** START VALVE COUNT LOOP 106 DO 100 I=1,KBWTCH 107 K:=IABS<KP0S3< I ) ) 108 IF (K. NE. 4) GO TO 135 10? J=ICON(I) 110 IF<J.LE.0)G0 TO 100 111 J=J/10 112 JJ=ICON <I)—J*10 113 IFIRE(I)=0 114 IF<PHSHIF<J). EQ. O. DO)GO TO 137 115 C ***** CALCULATE THE COMMUTATING VOLTAGES 116 C ***** CONVERTER TRANSFORMER WYE-DELTA CONNECTED 117 GD TO <141, 142, 143, 144, 145, 146),JJ 118 141 A=-l. DO*ACVC(J) 119 GO TO 117 _ _ 120 142 A=ACVB<J>' 52 121 GO TO 117 122 143 A=-l. DO*ACVA ( J j 123 GO TO 117 124 144 A=ACVC<J) 125 GO TO 117 126 145 A=-l. D 0 * A C V B < J ) " 127 GO TO 117 128 146 A=ACVA<J) 12? GO TO 117 130 C ***** CONVERTER TRANSFORMER WYE-WYE CONNECTED 131 137 GO T0( 1 11, 112, 113, 114, 115, 116), J J 132 111 A=ACVA(J)-ACVC(J) " " 133 GO TO 117 134 112 A=ACVB(J)-ACVC < J) 135 GO TO 117 136 113 A-ACV'S< J)-ACVA< J) 137 GO TO 117 138 114 A=ACVC(J)-ACVA(J) 139 GO TO 117 140 115 A=ACVC < J)-ACVB < J) 141 GO TO 117 142 116 A=ACVA < J)-ACVB(J) 143 117 IF(NMODE(J >. LT. 0)A=-A 144 IF < ISTEP. GT. 0) GO TO 118 145 IF <PHSHIF< J). EQ. 0. DO)GO TO 138 146 GOTO ( 151, 152, 153, 154. 155, 156), J J 147 151 P=-l.DO*ACVCI(J) 148 GO TO 120 149 152 P=ACVBI(J) 150 GO TO 120 151 153 P~ - l . DO*ACVAI< J) 152 GO TO 120 153 154 P=-ACVC.I(J) 154 GO TO 120 155 155 P=-l. DO*ACVBI < J) 156 GO TO 120 157 156 P~ACVAI < J) 158 GO TO 120 159 138 GOTO ( 121, 122, 123, 124, 125, 126), J J 160 121 P=ACVAI<J)-ACVCI<J) 161 GO TO 120 162 122 P=ACVBI<J)-ACVCI<J) 163 GO TO 120 164 123 P:=ACVBI (J)-ACVA I < J) 165 GO TO 120 166 124 P=ACVCI<J)-ACVAI<J) 167 GO TO 120 168 125 P=ACVCI< J)-ACVBI(J) 169 GO TO 120 170 126 P=ACVAI(J)-ACVBI(J) 171 120 IF<NMODE<J). LT. 0)P=-P 172 T=DATAN2(P, A)+l. 570796327D0 173 IF <T. LT. 0. DO) T=T+6. 2B3135307D0 174 IF<NMODE(J). GT. 0)GO TO 132 175 IF <T.GT.4.276056675D0) GO TO 100 176 GQ TO 131 177 132 IF<T. GT. 3. 441592654D0)GO TO 100 178 131 COUNT<I)=T*DFREQ<J) 179 GO TO 119 180 "c ***** GENERATION OF FIRING PULSES 118 IF <A. LE. 0. DO) GO TO 133 COUNT(I)=COUNT <I)+DELTAT 119 T=DELAY<U)-COUNT<I) IF < T. GE.DELTA2) GO TO 100 IFIREU)=1 133 COUNT (I )=0. DO GO TO 100 135 IFIRE<I)=0 100 CONTINUE RETURN END 54 A P P E N D I X 2 M o d i f i c a t i o n s i n t h e T r a n s i e n t s P r o g r a m To i n c o r p o r a t e t h e c o n v e r t e r m o d e l i n t o t h e U . B . C . 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 , m o d i f i c a t i o n s h a v e t o b e m a d e i n t h e p r o g r a m s o t h a t i n f o r m a t i o n may b e t r a n s f e r r e d b e t w e e n t h e m a i n p r o g r a m a n d t h e s u b r o u t i n e . T h e r e q u i r e d c h a n g e s a r e s h o w n i n t h e : F O R T R A N /I l i s t i n g s o f T a b l e A 2 - 1 . E x p l a n a t i o n o f C h a n g e s . F i l e l i n e n o . 5 3 - 5 9 . 2 , 9 7 . 2 - 9 8 3 9 3 - 3 9 5 , 4 5 0 - 4 5 1 3 9 9 - 4 2 9 , 5 7 1 1 2 9 6 . 0 6 - 1 2 9 6 . 9 1 6 8 0 . 2 - 1 6 8 1 1 6 8 7 - 1 6 8 7 . 2 1 7 2 3 , 1 7 9 5 1 7 2 6 - 1 7 2 8 1 7 6 4 . 2 , 1 7 9 5 . 2 1 7 9 6 1 7 7 6 . 2 1 7 6 7 , 1 8 2 3 1 8 2 0 2 0 5 3 . 0 5 - 2 0 5 3 . 6 C o m m e n t s New v a r i a b l e s f o r c o l l e c t i n g i n f o r m a t i o n r e q u i r e d b y V A L C O N . R e s e t v a r i a b l e s . I n p u t c o n t r o l p a r a m e t e r s . S t o r e n o d e v o l t a g e s f o r c o m m u t a t i o n v o l t a g e c a l c u -l a t i o n b e f o r e e n t e r i n g t h e t i m e s t e p l o o p . C a l l s u b r o u t i n e V A L C O N . C a l c u l a t e g r o u p n u m b e r . S t o r e v a l v e v o l t a g e a n d c u r r e n t . D e t e r m i n e t h e i g n i t i o n o f v a l v e s . A v o i d p r e m a t u r e b l o c k i n g a f t e r t h e v a l v e f i r e d w h i c h c a u s e s m a l l c u r r e n t o s c i l l a t i o n . S t o r e d c l i n e c u r r e n t . To m a k e t h e s w i t c h o p e n e x a c t l y a t t i m e t = 0 . R e s e t v a l v e v o l t a g e a n d c u r r e n t . R e s e t d c l i n e c u r r e n t . S t o r e n o d e v o l t a g e s f o r c o m m u t a t i o n v o l t a g e c a l c u -l a t i o n a t e a c h t i m e s t e p . 55 TABLE A2.1 Changes i n the Transients Program f o r data t r a n s f e r r e d to subrou- : .:. t i n e VALCON C=Change I= I n s e r t i o n DIMENSION KMS<650). YS<650) COMMON/HERMAN/ VOLTI (50), V0LTK<50), V0LK50), VIM(50) DIMENSION ADELAY(51) , XMAX(50), XOUT(100),POLAR(51), JJJ(380) FOLLOWING DIMENSION STATEMENT PERTAINS TO VALVE CONTROL PARAMETERS COMMON /THYRIS/THYR(51),THYR2(51),DCCUR(10),DCK(10),DCT1(10), 1 DCT2(10), DCT3(10), DCK1(10), DCK2<10), DCMIN(10), DCMAX(IO), 2 ACVA(10),ACVB(10), ACVC(10), PUSHIF(10), 3 ACVAI(10),ACVBI(10), ACVCI(10), COMR(10), 4 DCINI(10),DFREG(10),ICON(51),IFIRE(51),NGROUP,IMODE<10), 5 NCOUT(10) _ DOUBLE PRECISION THYR, THYR2, DCCUR,DCK,DCT1,DCT2,DCT3, DCK1, DCK2, 1 D.CMIN, DCMAX, DCINI, DFREQ, ACVA, ACVB, ACVC, PHSHIF, 2 ACVAI, ACVBI,ACVCI, COMR DOUBLE PRECISION COPT,C,R, CK, CREST,BUS, XOPT,TR,TIME,TSTART, TCLOSE, C: 51 52 53 53. 54 55 56 56. 57 57. 58 59 59. 60 96 C HISTORY OF DISTRIBUTED LINES IS CHANGED 97 LPAST=1250 T[97. 2 C CHANGE FOLLOWING STATEMENT IF DIMENSION OF CONTROL GROUP IS CHANGFD r?8 LGR0UP = 10 "390" KSWTCH=0 391 KCDNST=0 392 N4=0 p93 NGROUP=0 '394 DO 229 1 = 1, LBUS " ' 395 229 JJJ(I)=0 396 209 KSWTCH=KSWTCH-t-l 397 READ(5,77)IT2 398 77 FORMAT(12) 399 IF(IT2.NE.16)GO TO 199 400 IPRINT=16 " " " ; 401 IF(NGROUP. GT. LGROUP)G0 TO 9000 402 NGRDUP=NGR0UP+1 403 KSWTCH=KSWTCH-1 404 BACKSPACE 5 405 READ(5,89) N2, IMODE<NGROUP),DCK(NGROUP), DC INI(NGROUP), 406 1 OCT 1(NGROUP),DCT2(NGROUP),DCT3(NGROUP),DCCUR(NGROUP) 407 89 FORMAT ( 12, 6X, 12, 6E10. 6) -408 WRITE(6, 179) N2, IMODE(NGROUP),DCK(NGROUP), DC INI(NGKOUP), " 409 1 DCT1(NGROUP),DCT2<NGROUP >, DCT3(NGROUP),DCCUR(NGROUP) 410 179 -FORMAT( ' CONTROL PARAMETERS: '/ 1 X, 12, 1 OX, 14,6E14. 6) 411 READ(5, 159)DCK1(NGROUP)i DCK2(NGROUP),DCMIN(NGROUP), DCMAX<NGROUP), 412 1 DFREQ(NGROUP),NCOUT(NGROUP) 413 159 FORMAT(10X, 5E10. 6, 19X, II) 414 WRITE(6, 149)DCK1 (NGROUP), DCK2 (NGROUP ), DCMIN ( NGROUP ), DCMAX (.NGROUP ) 415 1 DFREQ(NGROUP) 416 149 FORMAT(IH ,16X,5E14.6) 417 READ(5, 139)(VOLT(K),K=l, 3),PHSHIF(NGROUP),COMR(NGROUP) 418 139 FORMAT(2X>3A6,2E10. 6) 419 WRITE(6, 129)(VOLT(K),K=l,3), PHSHIF(NGROUP),COMR<NGROUP) 420 129 FORMAT(10X,3<A6,2X),2E14. 6) 56 TABLE A2.1 (cont'd) 421 D O 239 J = l . 3 422 A=VOLT<J) 423 DO 249 1=2,NTOT 424 IF < A. EQ. BUS ( I) )GO TO 259 425 249 CONTINUE 425. 6 WRITE<6, 1042) A 425. 65 GO TO 239 425. 7 259 JJJ(I)=NGR0UP*10+J 425. 75 239 CONTINUE 426 GO TO 209 427 199 BACKSPACE 5 428 READ< 5, 35) IT2, BUS1, BUS2, BUS3, BUS4, CK1, A, JJ, J 429 35 FORMAT( 12, 2A6, 4E10. 6, 23X, 12, 11 ) 44S 449 [450 11*51 452 570 I [571 572 573 1295 1296 1296. 1296. 1296. 1296. 1296. 1296. 36 12V6. 42 1296. 48 1296. 54 1296. 6 1296.66 1296. 72 1296. 78 1296. 84 1296. 9 1297 06 12 18 24 3 ISOURC < KSWTCH)=0 ENERGY ( KSWTCH)=0. ODO THYR(KSWTCH)=0. DO THYR2 (KSWTCH) =0. DO IF (J. LT. 2) GO TO 224 CRIT(KSWTCH)=CK1 ICON(KSWTCH)=JJ GO TO 209 213 DO 214 1=2,NTOT 590 L=l DO 3711 1=2,NTOT JJ== J J J < I) IF(JJ. LE. 0) GO TO 3711 NJ=JJ/10 JJ^JJ-NJ*10 GO TO(3712,3713, 3714), J J 3712 ACVA(NJ)=E(I) ACVAI(NJ)=F(I) GO TO 3711 3713 ACVB(MJ)=E(I) ACVBI(NJ)=F(I) GO TO 3711 3714 ACVC(NJ)=E(I) ACVCI(NJ)=F(I) 3711 CONTINUE 11=0 BEGIN OF LOOP FOR ADMITTANCE 57 TABLE A 2 . 1 (cont'd)-1678 C BEGIN OF TIME-STEPS 1679 1000 KCOUNT=NV 1680 IF<KSWTCH. EQ. 0) GO TO 1009 -£[7680.2 . _ IF (NGROUP. LE. 0) GO TO 3722 11681 CALL VALCON(ISTEP, DELTAT, KSWTCH,KP0S3) 1682 C CHECKING SWITCH-POSITIONS FOR CHANGE 1683 3722 DO 1003 K=l,KSWTCH 1684 II=KP0S3<K> 1685 IT1=KP0SKK) 1_686 ICHECK=KP0S2<KJ 1687 JJ=ICON(K) L687. 2 IF(JJ. GT. 0)JJ=JJ/10 1688 I=IABS(II) 1721 2103 CK1=E(N2)-E(N1> I [1722 A= < CK 1+ENERGY (K ) ) #0. 5D0*P0LAR<K) 1723 THYR2<K)=CK1*P0LAR(K) 1724 ENERGY(K)=CK1 [1726 IF ( JJ. LE. 0) GO TO 2122 1 1727 IF< IFIRE<K). LE. 0>G0 TO 1002 1727.2 IF(ISTEP.EQ.0) GO TO 2123 4 2122 IF<A. LE. 0. DO) GO TO 1002 28 2123 1=3 1729 GO TO 2105 [1727. |l71763 2102 1=1 1764 2105 TCLOSE<K)=0. DO I [1764. 2 IF< I. EQ. 3) ADELAY (K ) =T+TOPEN < K > 1765 IF<TCL. GE. O. DO) TCL=T 1766 KOhJTRL-4 I[1767 THYR2<K)=0. DO 1768 WRITE<6,2107) TCL 1773 2108 IF'(T. LT. TCL ) GO TO 1002 1774 1=2 l[|775 IF< JJ. GT. 0) GO TO 2105 1776 IF < II. GT. 0 . AND. TOPEN < K ). GT. TMAX ) 1=0 I[1776. 2 IF(TCL. LT. O. DO. AND. TOPEN<K). LT. DELTA2)G0 TO 2110 , 1777 GO TO 2105 1773 2110 L=N2 1794 IF(I. EQ. 3) BUSl=-A*POLAR(K) ' 1795 THYR<K)=BUS1 1795.2 IF < I. EQ. 3. AND. ADELAY < K). GT. T) BUS 1 = 1. DO 1796 IF< I. EQ. 2. AND. JJ. GT. 0) DCCUR (JJ) =-A*POLAR <K) 1797 IF( ISSS. LE. 0) GO TO 2112 58 TABLE A2.1 (cont'd) 1818 2118 TCLOSE<K)=0. DO 1819 ' IF(I. EQ. 3) 1=5 1 5820 IF< I. EQ. 5. AND. JJ. GT. O) DCCUR <JJ)=0 DO 1821 ENERGY(K> =0. DO 1822 K 0 N T R L = 3 " "~ I 5B23 THYR(K)=0. DO 1B24 WRITE«6,2114) T ""2052 1200 K=K + IT2 2053 IF<K.LE.IBR) GO TO 1100 [2053.05 DO 1601 1=2, NTOT 2053. 1 J J=J J J < I) 2053. 15 IF( JJ. LE. 0)G0 TO 1601 2053. 2 NJ=JJ/10 2053. 25 JJ=JJ-NJ*10 2053.3 GO TO < 1602, 1603, 1604 ), J J 2053.35 1602 ACVA(NJ)=E<I) 2053. 4 GO TO 1601 2053.45 1603 ACVB(NJ)=E<I) 2053. 5 GO TO 1601 2053.55 1604 ACVC(NJ)=E<1) 2053.6 1601 CONTINUE 2054 IF<NV. EQ. O) GO TO 1202 59 APPENDIX 3 F o u r i e r A n a l y s i s Programme L i s t i n g s 1 ___C_FOURIER ANALYSIS PROGRAM FOR DISCRETE POINTS READ IN FROM DEVICE 4 2 DOUBLE PRECISION CTIME,GK2,X,TSTART,C1,GK1,SI,GK, AN, CP, AP, S, 3 1 BP, A, B,F, BUS,PAIR 4 "" DIMENSION A ( 1001), B <1001), F < 5000), TEXT(17), Y <100),BUS(100), 5 1 PAIR(200) 6 1 READ<5, 2)CTIME,TSTART, IOPT,MLIM, INUMB 7 2 F0RMAK2F10. 5, 12,213) _ _ 'B ' "' IF<INUMB. LT. 1)STOP " 9 REWIND 4 ~" • 10 READ ( 4) T E X T " ' ' " 11 READ<4)NT,DELTAT, IMAX, < BUS(I ), 1 = 1, NT) 12 READ < 4)L,NV, NI, < PAIR < I ), 1 = 1,L) 13 _ NT=0 _ _ _ 14 " "51 NT--NT+Y " " ' " 15 IF(NT.GT. 5000) GO TO 56 16 READ(4) K.KONTRL, ISTEP, (Y(T),1=1,K) 17 F < MT)=Y <INUMB) 18 IF(KONTRL. EQ. 1) GO TO 50 19 GO TO 51 _ _ 20 SO NSTART=TSTART/DELTAT 21 IF<(TSTART/DELTAT-NSTART). GT. 0. 5) NSTART=NSTART+1 22 N-CTIME/DELTAT 23 IF( (CT.TME/DELTAT-N). GT. 0. 5)N=N+1 24 MM=NSTART+N 25 BEGINT=NSTART*DELTAT 26 " " ""REALCY=N*DELTAT ~ " " " " " " " 27 K=l 28 NPLUS=NSTART+1 " ~' " ' "• 29 DO 52 I=NPLUS, MM 30 F<K)=F(I) 31 _ 52 K=K+1 32 WRITE(6, 5)INUMB, (TEXT<i), I = 1, 17)>N, (F(K),K~1,N) 33 5 FORMAT(35H1FOURIER ANALYSIS FOR OUTPUT NUMBER ,15, ' OF ',17A4, /23H 34 10REC0RD OF ORDINATES IN, 14, 19H EQUIDISTANT POINTS /(IH ,6F13. 7)> 35 UR T TE \' c>> 61 )REALCY> BEG I NT 36 61 FORMAT(//'CYCLE TIKE= ', FI 5. 10, 'SECOND ', ' ANALYSIS 37 1 STARTS AT ',F15. 10, 'SECOND' ) 38 AN--N 39 AN=AN/2. 0 40 J=AN+1. 6 " " 41 M=AN+.t. 0 42 IF ( ML IM. LT. 1 ) GO TO 62 .43 . . IF<MLIM. LT. M)IOPT=0 : . _.' .' 44 IF(MLIM. LT. M)M=MLIM 45 62 C1=DC0S(3. 141593/AN) 46 S1=DSIN(3. 141593/AN) 47 CP=1.0 48 SP=0. 49 L=0 50 10 L=L+1 51 IF(L. GT. M) GO TO 100 52 GK2=0. 53 GK1=0. 54 K=N .55 ..... . 20 GK=F(K)+CP*GK1#2. 0-GK2 . _ ._ 56 GK2=GK1 57 GK1=GK 58 K=K-I ;r 59 IF ( K . GT. 1 ) GO TO 20 60 AP=<F<1)+GKl*CP-GK2)/AN 60 61 BP=SP*GK1/AN __ 6 2 I F < L. EQ. 1 ) GO TO 3 0 6 3 , _ I F ( L . NE. J ) GO TO 4 0 6 4 •; BP=O." " _ • • ' • - / 6 5 3 0 AP=AP/2. 6 6 4 0 A ( L ) = A P 6 7 _ B ( L ) = B P ' ' _ '_ 6 8 A P = C 1 * C P - S 1 * S P 6 9 SP=C1*SP+S1*CP 7 0 CP=AP ' '_' 71 GO TO 1 0 " " 7 2 100 WRITE<6,101) 7 3 101 FORMAT(21HOFOURIER C O E F F I C I E N T S / 7 8 H HARMONIC _COS-COEFF. S I 7 4 " l N - C O E F F . MAGNITUDE M U L T I P L E OF FUNDAMENTAL > 7 5 CP=1.0/DSGRT<A<2)#*2+B<2)**2) 7 6 DO 1 1 0 K = l, M 7 7 L=K-1 7 8 AP=DSQRT<A C K ) **2+B < K ) * * 2 ) 7 9 BP=AP*CP 8 0 1 10 WRITE'< 6, 11 i ) L> A ( K ) , B ( K ) , AP» BP '" "' " 81 111 FORMAT ( I H , I 5, 2X, 4F1 5. 6 ) 8 2 1F« I OPT. LE. 0 ) GO TO 1 : 8 3 AN - N 8 4 AN=6.283186/AN 8 5 DO 2 1 0 K=1 , N 8 6 AP==K-1 ' " " ~" 8 7 AP=AP*AN 8 8 X=AP ' 8 9 SP-^A < 1 ) 9 0 DO 2 0 0 L=2, M 91 SP=SP+A<L)*DCOS<X)+B(L)*DSIN<X> 9 2 2 0 0 X^X+AP 9 3 2 1 0 F<K)=SP 9 4 W R I T E ( 6 , 2 1 1 ) ( F < K ) , K = l , N) 9 5 2 1 1 FORMAT < 4 7 H 0 0 R D I N A T E S RECOMPUTED WITH FOURIER C O E F F I C I E N T S / ( I H . 9 6 1 6F 1 8 . 7 ) ) 9 7 GO TO 1 _ '_ . ... . _ 9 8 56 WRITE(6,57) 9 9 57 FORMAT < 'COMPUTED P O I N T S ARE MORE THAN 5 0 0 0 , INCREASE THE DIMENSION 1 0 0 1 OF V A R I A B L E F AS PER NEED') 101 STOP 1 0 2 END 61 A P P E N D I X 4 L i n e P a r a m e t e r s a n d C o n v e r t e r S t a t i o n P a r a m e t e r s o f t h e P a c i f i c HVDC I n t e r t i e T h e P a c i f i c HVDC I n t e r t i e , w h i c h w a s c o m m i s s i o n e d o n M a y 2 1 , 1 9 7 0 , w a s t h e f i r s t o v e r h e a d HVDC t r a n s m i s s i o n s y s t e m i n s t a l l e d i n N o r t h A m e r i c a . F i g . A 4 . 1 s h o w s t h e m a i n c i r c u i t o f t h e i n t e r t i e . T h e n o r t h e r n s e c t i o n , i n -c l u d i n g t h e C e l i l o C o n v e r t e r S t a t i o n , " . i s o w n e d a n d o p e r a t e d b y t h e 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 ( B P A ) , w h i l e t h e s o u t h e r n s e c t i o n a n d t h e S y l m a r C o n -v e r t e r S t a t i o n i s p a r t l y o w n e d b u t o p e r a t e d b y t h e L o s A n g e l e s D e p a r t m e n t o f W a t e r a n d P o w e r ( L A D W P ) . T h e I n t e r t i e i s a b i p o l a r ( e a r t h r e t u r n ) s y s t e m w i t h DC r a t e d v o l t a g e o f ± 4 0 0 K V a n d t r a n s m i s s i o n p o w e r c a p a c i t y o f 1 4 4 0 MW. T h e d c t r a n s m i s s i o n l i n e i s 8 4 7 m i l e s l o n g a n d i t s l i n e d a t a i s s h o w n i n T a b l e A 4 - 1 . T h e r a t e d d i r e c t v o l t a g e o f t h e d c l i n e i s o b t a i n e d b y u s i n g s i x s e r i e s c o n n e c t e d , s i x - p u l s e , t h r e e p h a s e b r i d g e g r o u p s , e a c h r a t e d 1 3 3 K V , 1 8 0 0 A . T h e e q u i p m e n t l a y o u t s o f t h e C e l i l o a n d S y l m a r C o n v e r t e r S t a t i o n s a r e b a s i c a l l y t h e s a m e . T h e b r i d g e c i r c u i t a n d t h e f i l t e r a r r a n g e m e n t a t C e l i l o a r e s h o w n i n F i g . A 4 - 2 a n d F i g . A 4 - 3 , r e s p e c t i v e l y . f-D— <2H F I L T E R S S P F CAPAC ITORS) 2 LJ t-> (A U < X a O Y A I—^ , Y Y F I L T E R S Y Y POLE 4 ...... L T E R S F I L T E R S a P F CAPACITORS 5 + 400 KV, 848 M I L E H V D C L I N E "JL Y Y M N RICE F L A T S GROUND E L E C T R O O E Y A F L1 L' T T E E 9 S R S 3 IB > N o 5 O C E A N E L E C T R O O E ^ M L * H M H M A Y H M 2 Y Y L * H M A Y Y H F I L T E R S 8 P F C A P A C I T O R S F I L T E R S Q P F CAPAC ITORS r — 0 5 CELILO CONVERTER STN. POLE 9 SYLMAR CONVERTER STN. a > o CM oc s >-M (A in o z t-F l g . A4.1 MAIN CIRCUIT OF PACIFIC HVDC INTERTIE O N 63 P A R A M E T E R S M A I N L I N E GROUND W I R E N u m b e r o f s u b c o n d u c t o r s 2 1 S u b c o n d u c t o r d i a m e t e r 4 . 5 7 7 cm 1 . 1 1 0 cm C r o s s - s e c t i o n a l a r e a o f s u b c o n d u c t o r 1 1 7 1 mm 2 ( 2 3 1 2 MCM) 9 6 . 8 mm 2 B u n d l e s p a c i n g 4 5 . 7 cm N i c k n a m e a n d c o m p o s i t i o n o f s u b c o n d u c t o r T h r a s h e r A C S R 7 6 / 1 9 E N S N u m b e r o f c o n d u c t o r s 2 ( p o s . a n d n e g . ) 2 DC R e s i s t a n c e a t 2 5 ° C 0 . 0 1 2 5 f t / k m / p o l e T o t a l r e s i s t a n c e a t 1 8 0 0 A 1 9 . 3 ^ / p o l e A v e r a g e m a x . h e i g h t o f c o n d u c t o r s 2 4 . 8 4 m 3 3 . 8 3 m S a g 1 1 . 5 5 m 1 1 . 5 5 m A v e r a g e h e i g h t o f c o n d u c t o r s 1 8 . 4 5 m 2 6 . 1 3 m C o n d u c t o r s p a c i n g 1 2 . 1 9 m 5 . 4 9 m TABLE A4-1 Transmission l i n e data of the P a c i f i c HVDC I n t e r t i e 289 MVA •I'0.172 235 KV*l3VII3kv -2% 709 .1470 A F i g . A 4 . 2 C o n v e r t e r B r i d g e a t C e l i l o 6 4 ft B I G EDDY B U S I MILE O V C H M E A O l L l N E 33a 27.BMVAR *v—-1 201 282 MVAR ?44«M C 4 4 m H -0»Sf**=rl.33^F 093ft 50*2 Q Pf HP 13th l l t h i I I M I L E O V E R H E A D ) I L l U t BIG EDDY BUS F i g . A 4 . 3 F I L T E R ARRANGEMENT A T C E L I L O 65 A P P E N D I X 5 I n p u t . L i s t i n g s o f t h e P o i n t t o P o i n t O p e r a t i o n S i m u l a t i o n ( a ) S i m p l i f i e d M o d e l 1 CURRENT . CONTROL SOURCE SIMULATION 2 50. E-6 75. E-3 2 1 w VR VRT 500. 4 VRT CELEC' 6."3 "" 280. . 7 5 VR CELEC 900. 0 . 15 6 VR CELEC . 001 7 VI SELEC . 001 8 Vi" SELEC 900."0 . 15 9 VRT CCAP . 7 10 VRT RF 2. 5 11 RF CELEC 7. 12 RF CELEC lOO. 13 VIT VI 500. 14 VIT SELEC 6". 3 200.' . 7 " ' 15 VIT SCAP . 7 16 VIT ' IF 2. 5 17 IF SELEC 7. 18 IF SELEC 100. 19 CCAP 5. 20 SOAP 5. 21 CCAP CELEC . 01 22 SCAP SELEC . 01 23 CELEC . 43 22. 24 SELEC . 43 22. 25 CAN CGR 1. 26 SAN " SGR " " i . " ' ' " ' 2/ OCR CELEC I. SGR SELEC l . ""' 29 CGR . 06 30 SGR • 06 31 F SW 5. 00 32 -1VRT" F .02 2. 2 "0173 33 - I F VIT , 02 2- 2 . 0173 35 SW ' 1. E-2 1. 36 37 1 6VR 1 1. 39 133000. 4. 4 0 38 16CAN -143000. 14300. -120000 14 39 16SAN 3 1. 39 800. 0000 4. 6 0 40 16VI -148000. 14800. -117700 -1 41 42 VRT VR VIT VI 43 44 45 46 150. 700. _0. 0103 O. 0125 900. 66 (b) D e t a i l e d Model ._. l._.. .. ..PACIFIC HV_-DC. .INTERTIE , _ \ 2 40. V ; - 6 75. E~C i 1 1 0 3 GENAS li I GEA . 01 4 " OENBS I i IGEB G E N AS B I G E A * 5 GENCS B I G E C GEUAS B I G E A 6 B I G E A .86 114. 2. 46 7 B.IGEB.. . B I C E A a B I G E C B IGEA 9 B I G E A 1. 18 114. 1. 2 8 10 BIGEri BIGEA. 11 B I G E C BIGEA 12 B I G E A C E T A . 02 1. 56 .13 B I G E B CETB _ B IGEA CETA 14 B I G E C CETC B I G E A CETA 15 CETA CHPA 40. 16 CHPA C E T A 17 C E T B CHPB CETA CHPA I B CHPB r. 'f F: CHPA CETA 19 CETC CHPC CETA CHPA 20 C H F ' C CETC CKF'A CETA 21 CHPA 3. 9 22 CHPS5 CHPA 23 CHPC CHPA 24 CETA . 93 44. . 95 25 C E T B CETA 26 C!;TC CETA 27 CETA C 1 1 A GENAB 15 (GEA 23 CETB C .1 13 GEN A 3 h I GEA 29 CETC C1 1C GENAS DIOEA 30 C l I A . 82 44. 1. 33 31 C 1 1B . C l IA 32 c u e C J 1A 33 C i t A CT'XAP . c c 2. 07 34 CUB C T x n p .C11A CTXAP 35 C l 1C CTXCP Cl IA CTXAP 36 51CTXAP 99300. 37 52CTXAS .NS 4795J. 38 1CTXBP CTXAP 39 2CTXBB N S 40 1CTXCP CTXAP 41 2CTXCS NS 42 N S l . E + 10 43 .. . CTXAS CVDA . . 07. . 0343 _.. 44 CTXBS CVDB CTXAS CVDA 45 CTXCS c v o c C T X A S CVDA 46 CEL . CVDA 1200. 0. 0 T . 1 47 CEL CVDB CEL CVDA 48 CEL CVDC CEL CVDA 49 CAH_.. CVDA . CEL .... CVDA ... .. „ . 50 CAN CVDB CEL CVDA 51 C A N CVDC C E L CVDA 52 CVDA CCA 1000. 53 CCA CVDA . 25 54 CVDB CCB CVDA CCA .55 . .CCB ... OVDL< CCA . CVDA. . „ 56 CVDC CCC CVDA CCA 57 CCC CVDC CCA CVDA 58 CIR1 CCTH .01 59 C1R3 CCTH CIR1 CCTH 60 CIR5 CCTH CIR1 CCTH 23174. 61 6 2 6 3 6 4 6 4 . 6 4 . . 6 5 6 6 6 7 6 3 . 6 9 7 0 7 1 7 2 7 3 7 4 7 7 7 8 7 9 8 0 8 1 8 2 8 3 8 4 8 5 8 6 8 7 8 3 8 9 9 0 9 1 9 2 9 3 9 4 9 5 9 6 9 7 9 8 9 9 1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 1 0 5 1 0 6 1 0 7 1 0 3 1 0 ? 1 1 0 i l l 1 1 2 1 1 3 1 1 4 1 1 5 1 1 6 1 1 7 1 1 8 1 1 9 1 2 0 _ _ C E I _ C C T H C D 1 " . " ' C C A C E L S A N C D 3 C C B C D 5 C C C C D 4 C A N C D 6 C A N C D 2 C A N C A N C G R C G R 1 C E L E C C E L P O L E S P O L E S H P S C E L E C P O L E S S U R C S S U R C S -1 P O L E S ' ; -1 F A U L T S A N P O L E R P O L E R H P R S E L E C P O L E R S U R OR S U R C R P O L E R C A P C A P S T A S T B S T C L A LJ3 L C L A L 3 L C S 5 7 A S 5 7 B S 5 7 C . C C T H C E L C C A C D 1 C A N S C T H C C B C D 3 C C C C D 5 " C A N C D 4 C A N C D 6 C A N C D 2 C G R C E L E C J C V D A C C A -CCA_. CVDA 3 0 0 0 . 1. E 2 0 1. E 2 0 1. O C D 1 CCA CCA C D 1 CPt CCA CCA C D 1 C D 1 CCA CCA C D 1 cp.i. . CCA CCA C D 1 C D 1 CCA CCA C D 1 CAN CGR P O L E S C E L E C H P S C E L E C H P S SURCS" C E L E C - F A U L T P O L E R Fi S Y L S E L E C H P R S E L E C H P R S U R C R S E L E C C A P S Y L L A L B L C . 4 3 6. 3 1 0 0 . 1. 0 2 2 . 5 0 0 . O 2 8 0 . 0 6 0 0 0 7 2. 5 7 . O . 7 5 . G E N A S ' 3 I G E A . 0 2 . 0 2 C E L . P O L E S P O L E S H P S C E L E C P O L E S S U R C S S E N A S G E N A S 0 1 7 3 1 5 0 . 0 1 7 3 7 0 0 . P O L E S C E L E C HPS C E L E C H P S S U R C S B I G E A " B I G E A 1. O 1. 0 S 5 7 B S 5 7 C . STA, .. STA LA L A GENAS GElv'AS GENAS S 5 7 A S 5 7 A LA LA B I G E A B I G E A B I G E A S 5 7 A S 5 7 B S 5 7 C 5 1 3 A 5 5 7 A S 5 7 B 5 5 7 C 3 1 3 A S 1 3 B S 1 3 C 4 . 0 5. 7 ! 1 4 . ! 1 4 . 1. 3 2 6 7 GENAS GENAS GENAS S 5 7 A S 5 7 A 3 I G E A B I G E A B I G E A 1. 8 44. 1. 3 3 .121 S13J3 S13A 122 S13C S13A 123 S13A 2. 2 44. . 95 125 S13C S13A 126 S13A SHPA 34. 5 126. 2 . SW _ 5. 127 SHPA S13A ~ " 4. 3 128 SHPA 4. 5 128. i " CDI ';* CIRI ; '""/ .001 128. 11 CD3 CIR3 CDI CIRI 128. 12 CD5 CIR5 C D I C I R i 123. 13 CD4 CCA CDI CIRI 128. 14 CD6 CCB CDI CIRI 123. 15 CD2 CCC CDI CIRI 129 " 513B S H P B S 1 3 A S H P A 130 S H P B S 1 3 B S H P A S 1 3 A 1 3 1 S H P 3 S H P A 132 S 1 3 C S H P C S 1 3 A S H P A _ 133 S H P C S 1 3 C S H P A S 1 3 A 134 S H P C S H P A 135 S 1 3 A G E N A R G E N A S B I G E A 136 S 1 3 B G E N S R G E M A S B I G E A 137 S 1 3 C G E N C R G E N A S B I G E A 1 3 5 S T A 5 T X A P C t I A C T X A P 139 S T O S T X B P C l IA C T X A P 140 S T C S T X C P Cl IA C T X A P 141 1 S T X A P ' C T X A P 142 2 S T X A S NR 1 4 3 1 3 T . X B P C T X A P 144 . 2 S T X ' E 3 NR 145 1 5 T X C P ' C ' f X A P 146 2 S T X C 3 NR 147 NR U S 148 ST.XA.S S V D A Cl XAS C V D A 149 ST.X3 3 SV.D3 C T X A S C V D A 150 . . S T X C S . C T X A S C V D A 1 5 1 S A N S V D A C E L C V D A l f . 2 S A N S V D B C E L C V D A 1 5 3 S A N S V D C C E L . C V D A 1 5 4 S C T H S V D A C E L C V D A 1 5 5 S C T H 5VDJ1 C e l . C V D A 1 5 6 ... . S C T H S V D C .. cn..... C V D A 157 S I R 2 S C T H C I R I C C T H 158 S I R 6 S C T H C I R 1 C C T H 159 S I R 4 S C T H . C 1 f' 1 C C T H 160 S A N S D 1 C D l C C A 161 S D i BA;-i C C A C u t 162 S A N S D 3 . C D i CCA.. 1 6 3 S D 3 S A N C C A C D I 1 64 S A N snr; C D I C C A 1 6 5 S D S S A N C C A C D I 1 6 6 S V D A S D 4 C D I C C A 1 6 / 5 D 4 S V D A C C A C D i 1 6 3 S V D B SD6 C G I .. . C C A 169 S D 6 S V D B C C A C D I 170 S V D C S D 2 C D I C C A 171 SD.2 S V D C C C A CDI 1 7 1 . 0 3 CCA CCA 11 171. 0 6 CCB CCB 11 . 01 . 01 6 9 171. 171. 171. 171. 171. 171. 171. 172. 172. 172. 174 175 176 177 17B 179 180 181 182 183 134 136 187 1S3 139 190 191 191. 191 . 191 . 191. 191. 191. 191. 191. 191. 191. 19.1. 191. 191. 192 i92. 193 194 195 196 197 193 199 200 200. 200. 200. 200 200. 200. 200. 200. 200. 200. 0? 12 15 IS 3 6 9 _ 2 5 8 U2 04 06 03 1 23 26 29 32 35 5 ecc__. SVDA SVDB SVDC CCAI CCBI CCCI SVDA I SVDB I SVDC I SCTH SGR SGR 1SELEC -1 C IR1 -1CIR3 -1CIR5 -1CCA •-1CCB -1CCC -1SVDA •-1SVDB -1SVDC -1BIR4 -1SIR6 --1SIR2 CCAI I CCB 11 CCC I I SVDA I SVD3 I SVDC I 16 CCCI .L_ SVDA11 SVDB11 SVDC11 ^01 . 6f . 01 . o i " 87. 22 112. 94 87. 22 112. 94 87. 22 112. 94 -73. 24125. 14 -73. 24125. 14 -73. 24125. 14 SGR CAN SELEC CAN CGR CGR CGR CDl CD3 CD5 CD4_ CD6 CD2 SD1 SD3 SD5 SD4 SD6 SD2 CCAI CCB I CCCI ISVDAI ISVDBI ISVDCI 4 1. CELEC 0. 0 0. 0 0. 0 0. 0 o. 0 39 011A 16 STA POLES POLER FAULT 14GENAS 14GENES 14GENCS 14GENAR 14GENBR 14GENCR -14S000. CUB C11C -6 1. 39 -148000. STB, STC POLESS -1. POLERR -1. SW . 01 1918S0 191830 191830 177000 177000 177000 1)3 1 15 p 25 3 35 4 405 41 2CCTH 133000. 2CEL 133000. 2P0L.ES 133000. 2P0LESS133000. 2FAULT .130300. 2P0LERR1.17700. 2P0LER 117700. 2CAP 117700. 2SCTH 387. 2SGR 387. 133000. 14300. +735. 00 14800. 5 5 5 60. 60. 60. 60. 60. 60.. 4. 4 -120000. 7. 18996 4. 6 -117700. 7. 139V6 -60. 130. +60. 0 -AO. 180. +60. 0 1. E10 1. E10 1. E10 1. E10 1. E10 1. E10 O. 04 14 6000. 0. 03 -10000. .0103 60. . 0125 60. 900. 900. 200. 415 _2SELEC 387. 200. 42 2CELEC -387. 200. 425 2CGR -387. 200. 43 2CAN -387. 200. 45 2SYL 117700. 200. 5 2SAN 140350. 200. 52 3SAN SVDA 200. 525 3SAN SVDB 140350". 200. 53 3SAN SVDC 200. 55 3CEL POLES 900. 200 6 3P0LESBFAULT 900. 200. 65 3FAULT POLERR900. 200. 7 3P0LER CAP 900. 200. 71 3CAP SYL 900. 200. 72 3CAP 117700. 200. 73 3CELEC HPS 200. 74 3SELEC HPR 200. 75 3P0LE3 SURCS 133000. 200. 76 3P0L.ER SURCR 1 17700. 200. 77 3P0LE3 HPS 133000. 200. .78 3P0LER HPR 117700. 200. 79 3P0LE3 CELEC 133000. 200. 791 3CELEC -900. 200. 792 3SEl.EC 900. 200. 793 SCAN CGR -900. 200. 794 3CGR CELEC -900. 200. 795 3CGR 200. 796 3SCTH SGR 900. 200. 797 3SG3 SELEC 900. 200. 798 3SGP 200 3 3PQLER SELEC 117700. 200. a l 3SAN SYL -9C0. 200. 82 3CCTH CEL 900. 201 CEL SAN 202 203 71 APPENDIX 6 Sample Data Deck of the Three-Terminal Operation Simulation 1 PACIFIC HV-DC INTERTIE O 2 60. E-6 98. BE-3 -1 1 O 1 3 GENAS BIGEA .01 4 GENBS BIGEE GENAS BIGEA 5 GENCS BIGEC GENAS BIGEA 6 BIGEA - - .86 114. 2. 46 7 BIGEB BIGEA 8 BIGEC BIGEA 9 BIGEA 1. 18 114. 1. 28 10 BIGEB BIGEA 11 BIGEC BIGEA 12 BIGEA CETA .02 1.56 - - -13 BIGEB CETB BIGEA CETA 14 BIGEC CETC BIGEA CETA 15 CETA CHPA 40. 16 CHPA CETA 5. 5 17 CETB CHPB CETA CHPA 18 CHPB CETB CHPA CETA 19 CETC CHPC CETA CHPA 20 CHPC CETC CHPA CETA 21 CHPA 3. 9 22 CHPB CHPA 23 CHPC CHPA 24 CETA -. 93 44. . 95 25 CETB CETA 26 CETC CETA 27 CETA C11A GENAS BIGEA 28 CETB CUB GENAS BIGEA 29 CETC C11C GENAS BIGEA 30 CI 1A . 82 44. 1. 3 3 -31 CUB C11A 32 CUC CUA 33 CUA CTXAP .22 2.07 34 CUB CTXBP CUA CTXAP 35 CUC CTXCP CUA CTXAP 36 51CTXAP - " 99300. ' 37 52CTXAS NS 47951. 23174. 38 1CTXBP CTXAP 39 2CTXBS NS 40 1CTXCP CTXAP 41 2CTXCS NS 43 CTXAS CVDA .07 .8343 44 CTXBS CVDB CTXAS CVDA 45 CTXCS CVDC CTXAS CVDA 46 CEL CVDA 1200. O. O . 1 47 CEL CVDB CEL CVDA 48 CEL CVDC CEL CVDA 49 CAN CVDA CEL CVDA ^ " " ' 50 CAN CVDB CEL CVDA 51 CAN CVDC CEL CVDA 52 CVDA CCA 1000. 53 CCA CVDA .25 54 CVDB CCB CVDA CCA 55 CCB CVDB CCA CVDA - " 56 CVDC CCC CVDA CCA 57 CCC CVDC CCA CVDA 50 CIR1 CCTH .01 5? CIR3 CCTH CIR1 CCTH 60 CIR5 CCTH CIR1 CCTH 61 ~ - CEL CCTH* CVDA CCA • 72 62 CCTH C E L CCA CVDA 63 ...... CDI CCA 3 0 0 0 . -64 CCA CDI 1. O 6 7 CD3 CCB CDI CCA 6 8 CCB C D 3 CCA CDI 6 9 CDS CCC CDI CCA" - - -7 0 CCC CD5 CCA CD1 71 CD4 CAN CDI CCA 7 2 CAN CD4 CCA CDI 7 3 CD6 CAN CDI CCA 7 4 CAN CD6 CCA CDI 7 5 CD2 CAN CDI CCA 7 6 CAN CD2 CCA CDI 7 7 CAN CGR 1 .0 7 8 CGR C E L E C CAN CGR 7 9 CGR . 0 6 8 0 1 C E L E C . 4 3 22. 81 C E L P O L E S 500 . 0 8 1 . z> SW1 1 POLERR . 0 1 5 8 1 . 4 SW 1 1 SW1 1 I GENAS B I G E A 8 1 . 5 SW12 SW12I GENAS B I G E A 8 2 P O L E S C E L E C 6. 3 2 8 0 . . 7 8 3 P O L E S HPS 2. 5 84 HPS C E L E C 100. 8 5 C E L E C HPS 7. 0 8 6 P O L E S SURCS . 7 8 7 s u n c s 5. 8 8 SURCS C E L E C GENAS B I G E A 8 9 - 1 P 0 L E 3 3 F A U L T 3 . 0 1 5 4. 2 7 . 0 1 4 2 6 0 0 89 . 2 - 2 P 0 L E 4 F A U L T 4 . 0 2 1. 56 . 0 1 9 2 6 0 0 9 0 -1F A U L T 3 P 0 L E R ! . 0 1 5 4. 2 7 . 0 1 4 2 2 5 0 90 . 2 - 2 F A U L T 4 S W 1 2 . 0 2 1. 56 . 0 1 9 2 2 5 0 9.1 SAM S Y L C E L P O L E S 9 2 POLER S E L E C P O L E S C E L E C 92. 0 2 G E N A S B I G E A , GEN AS B I G E A 92. 04 GENB3 D IGEB, - G E N A S B I G E A 92. 0 6 G E N C S D I G F C , 'VSEMAS B I GEA 92. OS B IGEA* E I GEA 92. 1 B I G E B A 0 I GEA 92. 12 B I O E C A B I G E A 92. 122 B I G E A ; \ B I G E A 92. 124 D ICED* -1 B I G E A • • 92. 126 B I G E C , B I G E A 92. 14 B I GEA,' • iCETAA B I GEA C E T A ' 92. 16 B I C E r?,' ' i CETCA B I G E A C E T A 92. 18 B I G E C A C E T C A B I GEA C E T A 92. 2 C E T A A CHP A A C E T A CHPA 92. 2 2 C E T B A C H P B A C E T A CHPA 92. 24 C E T C A C H P C A C E T A CHPA 92. 2 6 CHP A A C E T A A CHPA C E T A 92. 2 8 C H P B A C E T B A CHPA C E T A 92. 3 OHPCA C E T C A CHPA C E T A 92. 3 2 Q jjp C H P A 92. 34 C H P B A CHPA " 92. 3 6 CHPCA CHPA 92. 3 3 C E T A A C E T A 92. 4 CETB.A C E T A 92. 4 2 C E T C A C E T A 92. 44 C E T A A C 1 1 A A C E T A Cl IA 92. 4 6 ~ C E T B A C U B A C E T A C 1 1 A " -92. 48 CETCA C11CA CETA CUA 92. 5 CUAA CUA - -92. 52 CUBA CI 1A 92. 54 CI ICA CUA 92. 56 CI 1AA CTXPA CI 1A CTXAP "92. 53 CUBA CTXPB CUA CTXAP 92. 6 CI ICA CTXPC CUA CTXAP 92. 62 1CTXPA CTXAP 92. 64 2CTXSA ANS 92. 66 1CTXPB CTXAP 92. 68 2CTXBB ANS 92 7 1CTXPC CTXAP . . . . . 92. 72 2CTXSC ANS 93 POLER HPR • POLES HPS 93. 1 ACTH AGR. CAN CGR 93. 15 AGR CGR 93 p AGR CELEC CAN CGR 93. 3 AAN AC EL GENAS BIGEA 93 35 ACEI. POLA CEL POLES 93. 4 POLA CELEC POLES CELEC 93. 45 POL A AMPS POLES HPS 93. 5 AHPS CELEC HPS CELEC 93 55 CELEC AHPS CELEC HPS 93. 6 POLA ASUR ^  POLES SURCS 93. 65 ASUR SURCS 93. 7 ASUR CELEC GENAS BTGEA 94 HPR SKI FC HPS CELEC 95 SELEC HPR CELEC HPS 96 POLER SURCR POLES SURCS 97 SURCR SURCS 98 SURCR SELEC GENAS B JGEA 99 POLER CAP GENAS BIGEA 100 CAP 100. c CTXSA AVDA CTXAS CVDA 100. 4 CTXSB AVDB CTXAS CVDA 100. 6 CTXSC AVDC CTXAS CVDA 101 CAP SYL 101. 05 AAN AVDA CEL CVDA 101. 1 AAN AVDB CEL CVDA 101. 15 AAN AVDC CEL. CVDA 101. 2 ACTH AVDA CEL CVDA 101. 25 ACTH AVDB CEL CVDA 101. 3 ACTH AVDC CEL CVDA 101. 35 AVDA ACCA CVDA CCA 101. 4 AVDB ACCB CVDA CCA 101. 45 AVDC ACCC CVDA CCA 101. 5 ACC A AVDA CCA CVDA 101. 55 ACCB AVDB CCA " CVDA 101. 6 ACCC AVDC CCA CVDA 101. 65 ACIR4 ACTH CIR1 CCTH 101. 67 ACIR6 ACTH C1R1 CCTH 101. 6? AC IR 2 ACTH C T.-R 1 CCTH 101. 71 AAN A0D1 CD1 CCA 101. 73 AAN ACD3 CD1 CCA 101. 75 AAN ACD5 CD1 CCA 101. 77 ACCA ACD4 CD1 CCA 101. 79 ACCB ACD6 CD1 CCA 101. 81 ACCC AC.02 CD1 CCA 101. B3 ACD1 AAN CCA CD1 101. 85 - ACD3 AAN CCA CD1 --101. 87 ACD5 AAN CCA CD1 101. 8? ACD4 ACCA CCA CD1 101. ?1 ACD6 ACCB CCA CD1 101. 93 ACD2 ACCC CCA CD1 10? STA LA 1. 0 103 STD LB STA LA — - • 104 STC LC STA LA 105 LA "706 106 LB LA 107 LC LA 10S LA S57A GENAS BIGEA 107 LB S57B GENAS BIGEA • - - •: • 110 LC S5 / C GENAS BIGEA 1 1 1 S57A 4. 0 "214. 1. 32 112 S57B S57A 113 S57C 5 37A 114 5 57A 5. 7 214. . 67 115 S^7B S57A 116 S57C S57A 1 17 S57A S i 3A GENAS BIGEA 11S 3570 S13B GENAS BIGEA 11? S57C S.t. 3C GENAS BIGEA 120 313A 1. 8 44. 1. 33 121 S1 3B S13A 122 S13C S13A 123 S13A 2. 2 44. . 95 124 S13B S13A 125 S13C S13A 126 S13A SHPA 34. 5 127 SW 5. . . . 123 SHI"'A S13A 4. 0 129 SUP A 4. 5 136 SI 3D SHPB S13A SHPA 137 SMPB S13B SHPA S13A 138 SHPB SHPA 139 51 3C C!-!PC S13A SHPA 140 Si-if'C S13C SHPA S13A 141 SHPC SI-IP A 142 S13A GENAR GENAS BIGEA 143 SI 3D GEN3R GENAS BIGEA 144 313C OENCR GENAS 3IGEA 145 STA STXAP CUA CTXAP - - • -146 STB STXBP CUA CTXAP 147 STC STXCP CUA CTXAP 143 1STXAP CTXAP 149 :?-TV >:.••"; MR 150 15TXBP. CTXAP _ 151 2STXBS NR 152 1STXCP CTXAP 153 2STXCS NR 155 STXAS SVDA CTXAS CVDA 156 STXBS SVDB CTXAS CVDA 157 STXCS SVDC CTXAS CVDA 158 SAN SVDA CEL " CVDA -159 SAN SVDB CEL CVDA 160 SAW SVDC CEL CVDA 161 SCTH SVDA CEL CVDA 162 SCTH SVDB CEL CVDA .163 SCTH SVDC CEL CVDA 164 SIR2 SCTH CIR1 CCTH — — 165 SIR6 SCTH CIRI CCTH 166 • SIR4 SCTH CIRI CCTH 167 SAN SD1 CDI CCA 168 SD1 SAN CCA CDI 169 SAN SD3 CDI CCA 170 . ... _ . - SD3 SAN CCA CDI . . . . 171 SAN SDS CDI CCA 172 SD5 SAN CCA CDI 173 SVDA SD4 CDI CCA 174 SD4 SVDA CCA CDI 175 SVDB SD6 CDI CCA 176 SD6 SVDB CCA CDI .... _ 177 SVDC SD2 CDI CCA 178 SD2 SVDC CCA CDI 191 SCTH SGR CAN CGR 192 SGR 5ELEC CAN CGR 193 SGR CGR 194 1SELEC CELEC 194. 05 CCAI 59. 93 296. 194. 1 CCBI 59. 93 296. 194. 15 CCCI 59. 93 296. 194 •n c SVDAI -90 7 627. 194. 25 SVDB I -90 7 627. 194. 3 SVDC I -90 7 627. 194. 35 ACC AI -99 3 618. 194. 4 ACCBI -99 3 610. 194. 45 ACCCI -99 3 618. 194. 5 CCA CCAI I GENAS BIGEA 194. 55 CCB CCB I I GENAS BIGEA 194. 6 CCC CCCI I GENAS 3 I GEA 194. 65 ACCA ACCAI IGENAS •B I GEA 194. 7 ACCB ACCBIIGENAS BIGEA 194. 75 ACCC ACCCIIGENAS BIGEA 194. 8 SVDA SVDAI I GENAS BIGEA 194. 85 SVD3 SVDB11 GENAS BIGEA 194. 9 SVDC SVDC11GENAS BIGEA 195 196 -1CIR1 CDI 0. 0 197 -1CIR3 CD3 198 -1CIR5 CD5 199 -1CCA CD4 200 . . . --1CCB ' CD6 201 -1CCC CD2 0. 0 202 --1SVDA SD1 203 -1SVDB SD3 0. 0 204 -1SVDC SDS 0. 0 205 -1SIR1 SD4 0. 0 206 -15IR6 SD6 207 -1SIR2 SD2 207. 005 CCAI I CCAI -1. 207. 01 CCB I I CCD I -1. 207. 015 CCCI I CCC I -1'. 207. 02 SVDAT ISVDAI -1. 207. 025 SVDB!ISVDBI -1. 207. 03 SVDCI ISVDCI — 1 207. 035 ACCA I IACCAI -1. 207. 04 ACCBIIACCBI -1. 207. 045 ACCCIIACCCI -1. 207. 1 -1ACCA ACD1 207. 2 " --1ACCB ACD3 — 76 207. 3 -1ACCC ACD5 207. 4 -1AC1R4 ACD4 " " ' . . . . . . . . . _ . . ' * ***" 207. 5 -1ACIR6 ACD6 207. 6 -1ACIR2 ACD2 214 1.6 2 1. 39 800. 000 4. 6 0. 03 . 0125 700. 215 . -143000. 14300. -70000.0 79000. 0 60. 216 CUA CUB CUC 7. 18796 217 16 -1 1.39 -58000. 4.4 0. 04 . 0103 370. 218 -158000. 14800. -80000.0 -10000. 60. 219 STA STB STC 7. 189V6 219. *-> 16 -1 1. 37 -52500. 4. 6 0. 03 . 0125 370 219. 4 -162000. 15300. -120000. - I O O O O . - 60." — 219. 6 C 11A A CUBA CUCA 7. 13996 220 POLES POLESS -1. 1. 5 221 POLER POLERR -1. 1. 5 221. 2 P0LE4 POLA -1. 1. 5 221. 4 SW 111 SW12I -1 1. 5 2 2 2 SW12 SW " .01 1: 5 • - — 223 224 14GENAS 1S7770. 60. -50. -1. 225 14GENB3 187770. 60. -170. -1. 2 2 6 14GENCS 187770. 60. +70. 0 -1. 227 14GEMAR 177000. 60. -50. -1. 228 14GENBR 177000. 60. ' 170. — — • • • • • -1. 229 14GENCR 177000. 60. +70. 0 -1. 230 231 2CCTH 70000. 232 2CEL 70000. 233 2P0LES 70000. 234 2P0L.ES: 370000. - - - - — •- • -235 2FAUL.r362370. 235. 05 2FAULT456400. 235. 1 2P0LE4 52500. 235. 15 2P0LA 52500. 235. 2 2ACEL 52500. 235. 25 2SW11 58000. • — - - - • • •• 235. 255 2SW.12I 53000. 235. 27 2SW11 I 50000. 235. 3 2SW12 58000. 235. 35 2AAN 52500. 235. 4 2 ACTH -132. 235. 45 2AGR -132. - • — — •• • - - - — — - — — 236 2P0LERR53000. 237 2P0LER 50000. 238 2CAP 58000. 239 2SCTH 132. 24 0 2SGR 132. 241 2SELEC 132. - - • • • - . . . . 242 2CELEC -132. 243 2CGR -132. 244 2CAN -132. 245 2SYL 58000. 246 2SAN 140350. 247 3SAN SVDA — -248 3SAN SVDB 249 3SAN SVDC -250 3CEL POLES 640. 251 3P0LESSFAULT3640. -640. 251. 2 3P0LE4 FAULT4-370. 370. 252 3FAULT3P0LERR640. ., — -640. 252. 2 3FAULT4SW12 -370. 253 3P0LER CAP 370. -254 3CAP SYL 370. 255 3CAP 58000. 256 3CELEC HPS 257 3SELEC HPR • • - - • 258 3P0LES SURCS 70000. 258. 1 3P0LA ASUR 52500. 258. 2 3P0LA AHPS 52500. 258. 3 3PDLA CELEC 52500. 258. 4 3CELEC AHPS 259 3P0LER SURCR • - - - 58000. 260 3P0LES HPS 70000. 261 3P0LER HPR 58000. 262 3P0LES CELEC 70000. 263 3CELEC -370. 264 3SELEC 370. 265 3CAN CGR -640. 266. 3CGR CELEC -640. 266*. 1 3ACEL POLA -370. 266. 2 3ACTH AGR 370. 266. 3 3AGR CELEC 370. 266. 4 3AGR 267 3CGR 263 3SCTH SGR 370. 269 3SGR SELEC 370. 270 3SGR 271 3P0LER SELEC 50000. 271. 1 3SW1 1 POLERR -370. 271. p 3SURCS • -271. 3 3SURCR 271. 4 3ASUR 272 3SAN SYL -370. 273 3CCTH CEL 640. 274 SWH P0LE4 POLES 275 276 

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