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A model H.V.D.C. link Riches, Robert Ian Dudley 1974

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A MODEL H.V.D.C. LINK by Robert Ian Dudley Riches B.Sc.(E.E.), U.M.I.S.T. 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of E l e c t r i c a l E n g i n eering We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1974 In presenting this thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of this thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of t t LC\N I t A t The University of B r i t i s h Columbia Vancouver 8, Canada Date ABSTRACT This t h e s i s presents the design and implementation of a p h y s i c a l model of a high v o l t a g e d.c. l i n k to the extent that the steady s t a t e mathematical model may be v a l i d a t e d . A co n v e n t i o n a l c o n t r o l s y s -tem was implemented us i n g an analog computer and f e a t u r i n g a new method f o r the d e t e c t i o n of i n v e r t e r e x t i n c t i o n angle. A commercial 'equal angle' f i r i n g c i r c u i t was found to be a major l i m i t a t i o n . An attempt was made to v a l i d a t e a steady s t a t e l i n e a r i z e d con-tinuous mathematical model usi n g a method o f a n a l y s i s devised by S i l j a k , an extension of the M i t r o v i c method. The o f t e n made l i n e a r continuous assumption was found to be v a l i d only f o r s m a l l bandwidth systems, i n which case t h e o r e t i c a l and experimental behaviours agree. i i i TABLE OF CONTENTS Page 1. I n t r o d u c t i o n . . . . . 1 2. Design Considerations f o r a D.C. L i n k 4 2.1 The Converter . 4 2.2 The Gri d F i r i n g C o n t r o l 8 2.3 Eeactive Power Compensation . 9 2.4 D.C. Transmission System 12 2.5 Operation and c o n t r o l of a D.C. L i n k 12 3. Construction and Te s t i n g of the Model 17 3.1 The Converter Bridge 17 3.2 Mathematical Representation 21 3.3 I n i t i a l Experimental T e s t i n g . 2 5 4. I n v e r t e r C o n t r o l 33 4.1 General P r i n c i p l e s of.Operation 33 4.2 A New Method of Detection of E x t i n c t i o n Angle . 3 5 4.3 Implementation of I 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 38 4.4 Mathematical Models 42 4.5 Transient c.e.a. C o n t r o l . . . 48 5. A n a l y s i s and Experimental R e s u l t s . . . . . . . 50 5.1 The Method of S i l j a k 50 5.2 A p p l i c a t i o n to I n v e r t e r System . . 52 6. R e c t i f i e r C o n t r o l 60 6.1 - Constant-a C o n t r o l . . . . . . . . 61 6.2 Constant Current C o n t r o l 63 6.3 Operation of D.C. L i n k 64 6.4 Other Forms of Converter C o n t r o l . . . . . . . . 64 7. D i s c u s s i o n and Conclusion 66 References . . 69 Appendices A Computation of C h a r a c t e r i s t i c Equation 72 B Computation of S i l j a k Curves 77 ACKNOWLEDGEMENT This t h e s i s i s dedicated to the l a t e Dr. Frank Noakes whose i d e a of a Trans-Canada H.V.D.C. Li n k may yet become a dream r e a l i z e d . I should l i k e to thank my other two s u p e r v i s o r s , Dr. B. K a b r i e l and Dr. H. Cheann f o r t h e i r encouragement and a d v i c e . S p e c i a l thanks are due to T i l l y Martens f o r her t y p i n g , Joe Hosie and Don Lumley f o r t h e i r proof r e a d i n g , and many f r i e n d s , e s p e c i a l l y Dr. H. Moussa who provided many i n t e r e s t i n g i n s i g h t s . I a l s o acknowledge a Research A s s i s t a n t s h i p 1968-70 and the forebearance of the Department of E l e c t r i c a l E ngineering. 1 . INTRODUCTION The economic f e a s i b i l i t y and t e c h n i c a l a p p l i c a b i l i t y o f h.v.d.c. t r a n s m i s s i o n i n c e r t a i n f i e l d s of a p p l i c a t i o n have now been e s t a b l i s h e d i n a number of h.v.d.c. schemes throughout the w o r l d . In 1 2 3 4 long d i s t a n c e b u l k power transmission. ' * , underground and submarine 5 6 12 cables ' and t r a n s f e r of power between systems of d i f f e r e n t frequency , h.v.d.c. t r a n s m i s s i o n has been used to great advantage. As o p e r a t i n g experience i n c r e a s e s , more r e f i n e d c o n t r o l s y s -terns ' provide the p o s s i b i l i t y f o r h.v.d.c. t r a n s m i s s i o n to move i n t o new f i e l d s of a p p l i c a t i o n : such as the use of h.v.d.c. as a s t a b i l i z i n g element i n p a r a l l e l a.c. - d.c. s y s t e m s ^ , and m u l t i - t e r m i n a l ^ d.c. systems which would allow more e f f e c t i v e use of economic generating cap-a c i t y , f o r example, to supply areas w i t h i n d i f f e r e n t time zones. M o d e l l i n g provides a scope f o r experimentation and development that the f u l l - s i z e d commercial i n s t a l l a t i o n cannot provide due to the high cost of c a p i t a l equipment and the sometimes unknown, p o s s i b l y des-t r u c t i v e tendencies of c e r t a i n t e s t s . E x i s t i n g p h y s i c a l models of 17 18 h.v.d.c. t r a n s m i s s i o n systems range i n s i z e from micro- ' through 19 medium-power to the f u l l s i z e experimental d.c. l i n k developed by a l e a d i n g manufacturer i n h.v.d.c. equipment. The d i s t i n c t i o n between p h y s i c a l and mathematical models i s 20 an important one : a s m a l l - s c a l e p h y s i c a l dynamic model, r e p l i c a t i n g , as f a r as p o s s i b l e , p e r - u n i t q u a n t i t i e s , time constants and behaviour of the f u l l - s i z e d model provides v a l u a b l e p r a c t i c a l experience that a mathematical model may be used to d e s c r i b e . On the o t h er hand, a s a t -i s f a c t o r y mathematical d e s c r i p t i o n allows measures o f performance, c e r -t a i n p r e d i c t i o n s of behaviour and f u r t h e r model refinements to be made according to p r e s c r i b e d techniques. A s i m i l a r approach i s adopted i n t h i s t h e s i s i n developing a s m a l l s c a l e p h y s i c a l model of a h.v.d.c. l i n k w i t h c o n v e n t i o n a l c o n t r o l systems. The model s c a l e was chosen to be compatible w i t h e x i s t i n g micro-machine l a b o r a t o r y models. The model l i n k was b u i l t , t e s t e d , and a mathematical model developed according to c e r t a i n assumptions. I n t h i s study one such assumption i s c r i t i c a l l y examined: t h a t the converter system may be represented mathematically as a l i n e a r continuous system. 24 One of the more important measures of system performance i s s t a b i l i t y , and the p a r t i c u l a r emphasis of t h i s study i s r e l a t i v e s t a b -i l i t y . The mathematical statements of s t a b i l i t y c r i t e r i a i n l i n e a r continuous systems are w e l l known; the a n a l y s i s may be d i v i d e d i n t o two c a t e g o r i e s : c l o s e d form s o l u t i o n s i n the a l g e b r a i c domain and the open form time domain s o l u t i o n of the d i f f e r e n t i a l equations. For t h i s study the former was chosen s i n c e the c l o s e d form s o l u t i o n f a c i l i t a t e s r e l a t i n g the p h y s i c a l system parameters to the 22 t h e o r e t i c a l s o l u t i o n s . Other s t u d i e s i n c l u d e E r i k s s o n et a l i a , apply-23 i n g Nyquist's c r i t e r i o n to a mathematical model, and Clade and Lacoste were able t o compare the a p p l i c a t i o n of Routh's c r i t e r i o n to the r e s u l t s obtained from a s m a l l experimental model. The mathematical model i n t h i s t h e s i s i s developed from a p r a c t i c a l model d.c. l i n k . The design c o n s i d e r a t i o n s f o r the p h y s i c a l model are o u t l i n e d i n Chapter 2, the c o n s t r u c t i o n and t e s t i n g i n Chap-t e r 3. A s t e a d y - s t a t e t h e o r e t i c a l model i s . d e r i v e d and analysed u s i n g 2A 23 26 a parameter plane technique devised by S i l j a k ' .' . The method of S i l j a k i s a g r a p h i c a l mapping procedure which allows s p e c i f i c s -plane contours to be mapped onto the parameter plane of two adjus tab le system parameters appearing i n the c o e f f i c i e n t s of the c h a r a c t e r i s t i c equat ion . This e f f e c t i v e l y allows the l o c a t i o n s of the roots of the c h a r a c t e r i s t i c equation to be confined w i t h i n these contours by appropr ia te adjustment o f the parameters. In a d d i t i o n , a method for computing the c o e f f i -c i ent s o f the c h a r a c t e r i s t i c equation i s developed by the author . Fol lowing the development o f the experimental model, the con-stant e x t i n c t i o n angle and constant current c o n t r o l l e r s f o r the i n v e r t e r are analyzed using S i l j a k ' s method. The parameter plane maps are used to e f f e c t parameter adjustment for best r e l a t i v e s t a b i l i t y and to i l l u s -t r a t e the v a l i d i t y of the l i n e a r continuous assumption. A new method for the detect ion of i n v e r t e r e x t i n c t i o n angle i s presented i n Chapter 4. 4 2. DESIGN CONSIDERATIONS FOR A D . C . LINK T h i s chapter descr ibes the components of a h . v . d . c . l i n k , shown i n F i g . 2.1, and t h e i r behaviour which the model must a c c u r a t e l y r e f l e c t . While the c o n t r o l system of a commercial d . c . l i n k encompasses many ref inements , the b a s i c c o n t r o l systems for steady s ta t e opera t ion are descr ibed here s ince they represent the minimum requirements for the normal r e g u l a t i o n of a l i n k . 0 • a.c. system synchronous compensation <jy-t a p - c h a n g i n g t r a n s f o r m e r smooth Ing choke a . c . harmonic f i l t e r s c o n t r o l s synch ronoi compensat us ( r\, ) Ion \ V r e c t i f i e r ntiu I n v e r t e r t a p - c h a n q I n g t r a n s f o r m e r a . c . harmonic f i l t e r s a . c . system 2 F i g . 2 . 1 . Schematic d . c . L i n k , showing each component 2 .1 The Converter The converter i s a s t a t i c swi tching device which, by swi tch ing appropriate segments of a three-phase a . c . supply , permits the convers ion of power from a l t e r n a t i n g to u n i d i r e c t i o n a l , and v i c e v e r s a . The q u a l i t y of the d i r e c t vo l tage p r o f i l e , c h a r a c t e r i s e d by the imposed r i p p l e , depends upon the number, connect ion, and f i r i n g delay of the p h a s e - c o n t r o l l e d r e c t i f y i n g elements used, and the number of phases of the supply . The most common connection used i n h . v . d . c . t ransmiss ion i s the s i x - p u l s e br idge c i r c u i t , F i g . 2.2a, which i s known to maximize va lve and t r a n s -former u t i l i z a t i o n . Other p o s s i b l e connections are covered e x t e n s i v e l y 28 29 i n Hancock and Schaeffer , Valve u t i l i z a t i o n i s based upon the p r o p o r t i o n of the c y c l e t h a t the valv e spends i n conduction, which a f f e c t s the r a t i n g , and the f r a c t i o n of the t o t a l output current c a r r i e d by each v a l v e when i n f u l l conduction. Transformer u t i l i z a t i o n i s based upon the r e q u i r e d r a t i n g s o f primary and secondary windings, as determined from the cur r e n t waveform i n each winding. ' * • P a i r s of b r i d g e u n i t s are connected i n s e r i e s and phase s h i f t e d w i t h respect to each other by 30° t o form a u n i t w i t h output r i p p l e o f 12 pulses per c y c l e . These 12-pulse u n i t s are themselves connected i n s e r i e s to produce the high v o l t a g e r e q u i r e d to make d.c. t r a n s m i s s i o n e c o n o m i c a l ^ . However, a s a t i s f a c t o r y r e p r e s e n t a t i o n of conver t e r behaviour i s obtained w i t h the use of a s i n g l e 6-pulse b r i d g e u n i t ^ . While mercury arc valves are used i n commercial i n s t a l l a t i o n s , i t i s i m p r a c t i c a l to use them on a s m a l l s c a l e model. The b r i d g e elements used i n t h i s model are 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 , s o l i d s t a t e devices which have the same o p e r a t i n g c h a r a c t e r i s t i c s as mercury arc r e c t i f i e r s - conductive i n forward d i r e c t i o n only and conduction may be blocked u n t i l a pulse i s d e l i v e r e d t o the gate. R e c t i f i e r waveforms are shown i n F i g . 2.2: the d i r e c t output v o l t a g e , V^j., i s the average value o f the v o l t a g e envelope shown i n b o l d o u t l i n e i n F i g . 2.2(b); the numbers r e f e r to the conducting v a l v e o f the b r i d g e c i r c u i t shown i n F i g . 2.2(a). Voltage c o n t r o l i s e f f e c t e d by v a r i a t i o n i n f i r i n g angle a; so that an i n c r e a s e i n a i n c r e a s e s the v o l t - t i m e area 'A', F i g . 2.2(b), r e s u l t i n g i n a decrease i n average v o l t a g e . A f u r t h e r v o l t drop, shown by v o l t - t i m e area 'B', i s caused by the commutation process: —nm^-reo forcer power flow t i : ier trans- -^ 4^ secondary vndg. O Z A s Z_J (a) r e c t i f i e r bridge c i r c u i t dr (b) r e c t i f i e r d i r e c t output waveform with respect to secondary neutral 1 I (c) a.c current waveform i n phase 'a' (d) commutating voltage of valve 1 Fi g 2.2 R e c t i f i e r Waveforms ' t ^ i' d.c source -6 power flow"" di (a) Inverter bridge c i r c u i t and secondary transformer winding 'dlo (b) Inverter d.c waveform with respect to secondary neutral (c) current waveform i n phase 'a' (d) Commutating voltage of valve 1 J i g 2.3 Inverter Waveforms i n h.v.d.c. converter o p e r a t i o n , commutation occurs n a t u r a l l y , i . e . current t r a n s f e r s to the anode of the next v a l v e because, at the i n s t a n t of f i r i n g , the p o t e n t i a l at the anode of the new v a l v e i s higher than that at the anode of the conducting v a l v e . Due to the leakage reactance L c per phase of the converter transformer, the e x t i n c t i o n of c u r r e n t i n the conducting v a l v e cannot occur i n s t a n t a n e o u s l y , but over a short p e r i o d of angular d u r a t i o n u°. During t h i s p e r i o d , two v a l v e s on the same si d e of the bridge are conducting, g i v i n g the appearance of a short c i r c u i t between phases. The e n t i r e v o l t a g e f o r each phase i s absorbed i n the leakage reactance, and the output v o l t a g e d u r i n g commutation i s the mean v o l t a g e between these two phases. The commutation v o l t a g e of a v a l v e i s the v o l t a g e between anode and cathode, shown i n F i g . 2.2(d) f o r v a l v e 1. I n v e r t e r waveforms are shown i n F i g . 2.3: the f i r i n g a ngle, advanced to ct>90°, minimum a f o r i n v e r t e r o p e r a t i o n , i s more c o n v e n i e n t l y known as the 'angle of advance' B = IT - a. From F i g . 2.3(d), the commutating v o l t a g e f o r v a l v e 1, i t may be seen t h a t the i n s t a n t of f i r i n g occurs towards the end of the p e r i o d of p o s i t i v e commutating v o l t a g e . For s u c c e s s f u l commutation, the preceding v a l v e 5 must be e x t i n g u i s h e d , T^, before the end of t h i s p e r i o d , T2, a f t e r which time commutation cannot be completed or i n i t i a t e d . A f t e r e x t i n c t i o n , the v a l v e must d e - i o n i z e before i t becomes non-conducting. The margin or e x t i n c t i o n angle 6, provides a s a f e t y margin between e x t i n c t i o n and end of p o s i t i v e commutating v o l t a g e to a l l o w f o r d e - i o n i z a t i o n and c o n t r o l system i n a c c u r a c i e s and assure s u c c e s s f u l commutation. 2.2 G r i d C o n t r o l F i r i n g C i r c u i t In the b a s i c bridge u n i t , the s i x valves are f i r e d i n the c o r r e c t sequence by gate pulses at nominal i n t e r v a l s of 60° e l e c t r i c a l , c o n t r o l l e d i n phase to produce the f i r i n g angle a . The o v e r a l l c o n t r o l system i s u s u a l l y d i v i d e d p h y s i c a l l y i n t o two p a r t s . (a) the pulse u n i t which forms and times the delay according t o the input s i g n a l from the (b) c o n t r o l l e r i . e . r e f e r e n c e , feedback and e r r o r a m p l i f i e r s . While the l a t e r chapters d e a l w i t h the c o n t r o l l e r s and the mathematical a n a l y s i s of behaviour, there are two b a s i c approaches to delay t i m i n g i n the design of p u l s e u n i t s : ( i ) 'Equal Angle', the c o n v e n t i o n a l scheme; where, i n the case of the 3-phase b r i d g e , s i x independent delay c i r c u i t s time the delay of each f i r i n g p u l s e from a f i x e d r e f e r e n c e p o i n t on the corresponding phase v o l t a g e . 13 ( i i ) 'Equal Space', a contemporary scheme invented by Ainsworth where each f i r i n g p u l s e i s timed at 60° e l e c t r i c a l a f t e r the preceding pulse f o r steady s t a t e . A change i n a i s made by momentarily p e r t u r b i n g the frequency of the 360 hz p u l s e t r a i n . I n e i t h e r case i t i s d e s i r a b l e t h a t the f i r i n g p u l s e i t s e l f has a d u r a t i o n equal to the conduction p e r i o d of the v a l v e . This f o r e s t a l l the s y n c h r o n i z a t i o n problem which may otherwise be encountered d u r i n g s t a r t - u p 9 2.3 Reactive Power Compensation The transformer secondary cu r r e n t waveform, as shown i n P i g . 2.2 i s decidedly n o n - s i n u s o i d a l i n nature; there are two e f f e c t s . ( i ) The i n j e c t i o n of a.c. harmonic c u r r e n t s and v o l t a g e s i n t o the a.c. system can have d e l e t e r i o u s e f f e c t s upon communication 32 c i r c u i t s , cause overheating i n h i g h l y i n d u c t i v e c i r c u i t s J 34 and, under c e r t a i n c o n d i t i o n s cause harmonic i n s t a b i l i t y . ( i i ) Even w i t h no phase delay the power f a c t o r i s n e c e s s a r i l y l e s s than u n i t y ; w i t h the i n t r o d u c t i o n of f i r i n g delay the power f a c t o r i s f u r t h e r reduced: r e a c t i v e power i s consumed, i . e . l a g g i n g r e a c t i v e volt-amps are absorbed. From i n s p e c t i o n of the r e l a t i v e phase displacement of the transformer secondary current w i t h respect to v o l t a g e , F i g s . 2.2(b),(c) i t i s evident that the r e c t i f i e r absorbs l a g g i n g VARs. . That the i n v e r t e r absorbs r e a c t i v e power from the r e c e i v i n g a.c. system may be shown thus: the voltage and c u r r e n t waveforms f o r one phase at the a.c. t e r m i n a l s of the i n v e r t e r are shown i n F i g . 2.4(b). By c o n s i d e r i n g the i n v e r t e r as an a.c. system lo a d element, defined i n F i g . 2.4(a), the power i n the l o a d may be w r i t t e n from the phasor diagram F i g . 2.4(c) S = V I * or S* = V*I S*= |v|c " j e ' J < * + 8 + * > - | v | • | i | e J ( 0 + *> = IVI • IXI{-cos 0 - j s i n 0} S - | v | • | l l { - c o s 0 + j s i n 0} = -P + jQ i . e . s u p p l y i n g r e a l power and a. load type b. secondary voltage & c. phasor diagram c i r c u i t element current waveforms Fi g 2.4 Power at the Inverter F i g . 2.5 i l l u s t r a t e s the r e a c t i v e power requirement of a r e c t i f i e r and i t s v a r i a t i o n w i t h f i r i n g angle a and commutating reactance X c r f o r a d i r e c t power output o f 1 p.u.. Typical** values f o r commutating reactance and f i r i n g angle are 0.15 p.u. and 20°; f o r which 1 p.u. output power r e q u i r e s 0.67 p.u. r e a c t i v e power F i g 2.5 Reactive Power Demand of R e c t i f i e r F o u r i e r a n a l y s i s r e s o l v e s the transformer primary c u r r e n t i n t o harmonics of number 6n + 1, [n = 1,2,....], and the peak fundamental 2 v^ 3 33 current to be — — . Changes i n f i r i n g angle and commutation angle r e s u l t i n s l i g h t changes i n amplitude and frequency content of the harmonics, and f i r i n g pulse t i m i n g e r r o r s b r i n g i n s m a l l amounts of non-t h e o r e t i c harmonics. However, Read-^ has shown that the amplitude of the fundamental i s e s s e n t i a l l y unchanged. In a n a l y t i c a l s t u d i e s , the a.c. current i s assumed to be s i n u s o i d a l w i t h constant amplitude over changes i n a. Tuned f i l t e r s provide a low impedance path t o ground f o r these harmonic c u r r e n t s , and attenuate the harmonic v o l t a g e s to provide an 33 acceptable l e v e l of d i s t o r t i o n of the fundamental . At fundamental frequency the tuned f i l t e r s appear c a p a c i t i v e , and provide some of the r e a c t i v e power r e q u i r e d by the converter. By c a r e f u l l y s e l e c t i n g the i n d u c t i v e and c a p a c i t i v e component v a l u e s , the f u l l load r e a c t i v e power 38 requirement may be met s o l e l y by the f i l t e r . F i g . 2.5 shows the v a r i a t i o n i n demand f o r r e a c t i v e power w i t h r e s p e c t to f i r i n g angle and commutating reactance. However, i n t h i s case the amount of r e a c t i v e power s u p p l i e d by the f i l t e r remains f i x e d , and the excess VAR at low converter load i s fed i n t o the a.c. system. I f t h i s causes e x c e s s i v e r e g u l a t i o n at the i n v e r t e r t e r m i n a l s , then the f i l t e r s i z e must be reduced and a d d i t i o n a l r e a c t i v e power compensation p r o v i d e d , e i t h e r by s e r i e s c a p a c i t o r s ^ or synchronous condensers Voltage r e g u l a t i o n at the converter depends upon the dynamic 38 c h a r a c t e r i s t i c s of the a.c. system . Hence i t may be seen t h a t the design of f i l t e r s and compensation equipment, and the op t i m a l balance 12 of r e a c t i v e power su p p l i e d by each depends upon the c h a r a c t e r i s t i c s of the a.c. system to which i t i s . appended. 2.4 D.C. Transmission System Depending upon the number of c o n v e r t e r s , there e x i s t s a choice of t r a n s m i s s i o n c o n f i g u r a t i o n s : one-pole, ground r e t u r n or two-pole. The one-pole c o n f i g u r a t i o n i s adequate f o r m o d e l l i n g most s i t u a t i o n s , and was used i n the present model. A l s o , the inductance of a d.c. l i n e , based upon the dimensions given f o r the Nelson R i v e r Line"'*"' i n d i c a t e s m a l l values of inductance f o r a model P i - s e c t i o n r e p r e s e n t i n g 100 m i l e s e c t i o n s . The valu e s were very s m a l l compared to the value of smoothing r e a c t o r inductance and were consequently ignored s i n c e the scope of t h i s study d i d not embrace the i n v e s t i g a t i o n of d.c. l i n e f a u l t s . 2.5 Operation and C o n t r o l of a D.C. L i n k This s e c t i o n describes the p r i n c i p l e s of o p e r a t i o n and requirements of a s u i t a b l e c o n t r o l system f o r a d.c. l i n k . The schematic diagram of the l i n k i s shown w i t h the b a s i c components of i t s c o n t r o l system i n F i g . 2.6. del tap-changing transformer a , B d.c transmission line -4) O [Rectifier] Converter 1 dr Converter 2 [inverter] 4> a , 8 margin -Q reference <3> tap-chan,\i ng transformer - O i ds2 Fig 2.6 Schematic Unipolar d.c link While d.c. l i n k s operate w i t h c o n s i d e r a b l e margin w i t h respect to the power l i m i t , the p o s s i b i l i t y does not e x i s t ^ , as i s normally the case w i t h a . c , of t r a n s f e r r i n g power f a r above the design economic l e v e l . This a r i s e s from the f a c t that present converter equipment cannot be overloaded - there are d e f i n i t e l i m i t s to the v o l t a g e and cu r r e n t h a n d l i n g c a p a b i l i t i e s of converters which, i f exceeded, r e s u l t i n the d e s t r u c t i o n of the v a l v e s . Hence a c e r t a i n amount of converter p r o t e c t i o n i s b u i l t i n t o the c o n t r o l system. The u s u a l mode of c o n t r o l f o r the r e c t i f i e r i s a constant current r e g u l a t o r , so that i f the c u r r e n t exceeds the s e t v a l u e , c o n t r o l a c t i o n i n c r e a s e s the f i r i n g angle, decreasing the d i r e c t output v o l t a g e , and thus preventing the output c u r r e n t r i s i n g above the s e t value.' On the other hand, the i n v e r t e r c o n t r o l must ensure t h a t the commutation and mercury vapour d e - i o n i z a t i o n i s complete before the commutation v o l t a g e becomes ne g a t i v e , F i g . 2.3(e). U s u a l l y the f i r i n g angle i s changed i n such a manner as to keep the e x t i n c t i o n angle 6 q constant. This c o n t r o l i s known as constant e x t i n c t i o n angle (c.e.a.) c o n t r o l . Fig 2.7 Control Characteristic of Converter The c o n t r o l c h a r a c t e r i s t i c w i t h these minimum requirements i n F i g . 2.7 c o n s i s t s o f three s e c t i o n s : (1) The n a t u r a l v o l t a g e c h a r a c t e r i s t i c of the r e c t i f i e r i s the r e g u l a t i o n c h a r a c t e r i s t i c of the converter w i t h f i r i n g angle a h e l d at i t s minimum v a l u e . The slop e de-pends upon the value of commutating reactance. (2) Constant e x t i n c t i o n angle (c.e.a.) c o n t r o l a l l o w s the i n v e r t e r t o operate at minimum angle of advance, a l l o w i n g f o r a s a f e t y margin. While i t does not p r o t e c t the i n -v e r t e r from o v e r - c u r r e n t s , i t does prevent commutation f a i l u r e . (3) Constant current (c.c.) c o n t r o l i s provided f o r both r e c t i f i e r and i n v e r t e r ; f o r the r e c t i f i e r , the c o n t r o l a c t i o n i n response to a r i s e i n l i n e current above a p r e -set reference I d s l ^ s t o f u r t h e r delay a, reducing the r e c t i f i e r output v o l t a g e and t h e r e f o r e c u r r e n t ; f o r the i n v e r t e r , should the l i n e current f a l l below then the angle of advance 3 i s made l a r g e r than t h a t r e q u i r e d to maintain constant e x t i n c t i o n angle. Constant power (c.p.) c o n t r o l i s an a l t e r n a t i v e mode of c o n t r o l f o r both r e c t i f i e r and i n v e r t e r c h a r a c t e r i s t i c (4) i n F i g . 2.7 i n which c.c. or c.e.a. c o n t r o l i s provided f o r o v e r r i d e . Constant B c o n t r o l [part 5 of the c h a r a c t e r i s t i c ] i s the a l t e r n a t i v e f o r c.e.a. c o n t r o l , being the n a t u r a l r e g u l a t i o n curve o f the i n v e r t e r . However, an i n v e r t e r i s r a r e l y operated on t h i s c h a r a c t e r i s t i c . Pig 2.8 Two-converter control characteristic For the two converter system, the cu r r e n t s e t t i n g 1^^. of converter 1, see F i g . 2.6, ope r a t i n g as a r e c t i f i e r , i s g r e a t e r than the current s e t t i n g 1^ 2 °^ converter 2, ope r a t i n g as an i n v e r t e r , by a sma l l current margin 1^ » t y p i c a l l y 20%. This margin s e t t i n g has t o be l a r g e enough to a l l o w s u f f i c i e n t o p e r a t i n g margin between the two constant current c h a r a c t e r i s t i c s , i n order to avoid simultaneous o p e r a t i o n of both. Such o p e r a t i o n i s c l e a r l y u n d e s i r a b l e and may a l s o be v e r y u n s t a b l e . I n F i g . 2.8, X-^  i s the ope r a t i n g p o i n t w i t h converter 1 [ r e c t i f i e r ] o p e r a t i n g i n c.c. mode and the i n v e r t e r o p e r a t i n g i n c.e.a. c o n t r o l mode. Converter 1 w i l l operate as a r e c t i f i e r and converter 2 as an i n v e r t e r as long as cur r e n t s e t t i n g 1^ ^ Is grea t e r than I ( j s 2 * ^ t n-e a > c « system a t the r e c t i f i e r changes so t h a t the n a t u r a l v o l t a g e c h a r a c t e r i s t i c f a l l s below the c.e.a. c o n t r o l c h a r a c t e r i s t i c of the i n v e r t e r , converter 1 w i l l s t i l l operate as a r e c t i f i e r , but the constant c u r r e n t c o n t r o l l e r of the i n v e r t e r w i l l be a c t i v a t e d , and the o p e r a t i n g p o i n t changes from X, to X 0. The power flow i n the l i n k w i l l not change d i r e c t i o n unless the sense of the current margin 1^ i s reversed. Each converter i s equipped w i t h a tap changing transformer to ensure that the system operates i n the p r e f e r r e d mode, i . e . , the r e c t i f i e r i n c c . c o n t r o l mode and the i n v e r t e r i n c.e.a. c o n t r o l mode. In p r a c t i c e the r e c t i f i e r tap changer i s operated to keep the f i r i n g a ngle, and hence r e a c t i v e power demand, w i t h i n s p e c i f i e d l i m i t s . The i n v e r t e r tap changer i s operated to keep the d.c. l i n e v o l t a g e a t a s p e c i f i e d l e v e l . The change i n operating p o i n t from X^ t o has r e s u l t e d i n the r e c t i f i e r f i r i n g angle reaching i t s l i m i t i n g [minimum] v a l u e of a. The oper a t i o n of the r e c t i f i e r tap changer }to i n c r e a s e the transformer secondary/primary t u r n s - r a t i o } r a i s e s the n a t u r a l v o l t a g e c h a r a c t e r i s t i c to above the c.e.a. c h a r a c t e r i s t i c of the i n v e r t e r , a l l o w i n g the op e r a t i n g p o i n t to r e v e r t to X^. I t can be seen that a change i n power t r a n s f e r , and t h e r e f o r e c u r r e n t , through the l i n k r e q u i r e s c a r e f u l c o - o r d i n a t i o n of both converter c o n t r o l systems i n order to mainta i n the curr e n t margin. - I n p r a c t i c e , a telecommunication l i n k i s maintained between the two s t a t i o n s , as i s a l s o some predetermined procedure t o prevent shutdown of the power l i n k i n the event of a communication f a i l u r e . T h i s chapter has described the components of a d.c. l i n k and i t s b a s i c mode of behaviour. The next chapter d e s c r i b e s the implementation of these components. F i l t e r s have not been i n s t a l l e d w i t h each converter s i n c e there e x i s t s no l a b o r a t o r y model power system w i t h which to operate one end of the d.c. l i n k . Each converter t h e r e f o r e , i s connected to the ' i n f i n i t e ' system. 3. CONSTRUCTION AND TESTING OF THE.MODEL This chapter describes i n d e t a i l the c o n s t r u c t i o n of the model and the i n i t i a l t e s t i n g of the components. A l s o i n c l u d e d are the power equations f o r the converter and the s t a r t - u p procedure f o r the l i n k . 3.1 The Converter Bridge Two converter b r i d g e s , each c o n s i s t i n g of one s i x - p u l s e u n i t , were b u l i t . The power r a t i n g o f up t o 4.5kw, maximum of 300v at 15A, allows c o m p a t i b i l i t y w i t h e x i s t i n g micro-machine equipment. For the converter b r i d g e elements, s c r s ' were used as t h e i r o p e rating c h a r a c t e r i s t i c s are s i m i l a r to mercury a r c r e c t i f i e r s or th y r a t r o n s . They a l s o have the advantage t h a t the forward v o l t drop w h i l e i n the conducting s t a t e i s s m a l l and t h e i r s w i t c h i n g time i s s h o r t . I n a d d i t i o n t h e i r s i z e i s s m a l l but the power d i s s i p a t i o n at the j u n c t i o n r e q u i r e s t h a t the s c r be mounted on a heat s i n k . The c i r c u i t c o n s i d e r a -t i o n s and design r a t i n g f o r a s p e c i f i e d b r i d g e r a t i n g are as f o l l o w s : 1. The curr e n t r a t i n g [average r e p e t i t i v e ] i s determined by the maximum allowable s c r o p e r a t i n g temperature. In the three-phase br i d g e c i r c u i t each element conducts f o r j u s t over one t h i r d of a c y c l e : hence f o r a maximum r a t i n g o f 15A the average r e p e t i t i v e current i s 5A, from which may be determined the heat s i n k requirements. 2. Peak reverse v o l t a g e . For the b r i d g e c i r c u i t which may be r e q u i r e d to d e l i v e r ^300v at a = 30° the peak l i n e - l i n e v o l t a g e i s T ^ r o TT/3 - 330V. cos 30 3. Peak forward v o l t a g e . The s c r may be turned on i n the absence of gate d r i v e by exceeding i t s forward breakdown v o l t a g e - t h i s could be as high as the peak reverse v o l t a g e . 4. High dv/dt. A r a p i d r i s e of v o l t a g e a p p l i e d between anode and cathode can t u r n the s c r on. This c o n d i t i o n e x i s t s at turn-on f o r a > 0 and t u r n o f f of a v a l v e and i s u s u a l l y absorbed i n damping c i r -c u i t s . 5. High d i / d t . C i r c u i t c o n d i t i o n s which a l l o w the r a t e of r i s e of current to be very r a p i d r e l a t i v e to the s c r turn-on time do not e x i s t due to the leakage reactance of the transformer. With the above c o n s i d e r a t i o n s taken i n t o account a s a f e t y -f a c t o r of at l e a s t 100% i n both c u r r e n t and v o l t a g e r a t i n g s was allowed. The s c r s e l e c t e d was the medium c u r r e n t g e n e r a l purpose s c r General E l e c t r i c C35S r a t e d at 35 Amps and 700 v o l t s . 43 Each s c r was p r o t e c t e d by a c u r r e n t l i m i t i n g fuse , which melts extremely r a p i d l y at h i g h c u r r e n t l e v e l s but does not i n t e r r u p t the current too q u i c k l y : a h i g h d i / d t can induce v o l t a g e t r a n s i e n t s which could damage other s c r ' s . For t h i s reason s c r fuses are r a t e d f o r v o l t a g e as w e l l as c u r r e n t , and t h e i r t o t a l c l e a r i n g I^T must not o 44 exceed the I T f a c t o r of the s c r . In a d d i t i o n a damping c i r c u i t i s provided f o r each s c r to reduce the dv/dt and overshoot of the v o l t a g e a t s c r t u r n - o f f . A de-t a i l e d a n a l y s i s of v a l v e damping c i r c u i t s i s given i n reference [9] and based on those r e s u l t s s i m i l a r p e r - u n i t values f o r the components of a simple R-C damping c i r c u i t were designed and connected i n p a r a l l e l w i t h each s c r . F i n a l l y a bypass s c r was connected across the t e r m i n a l s of each'converter. The bypass v a l v e i s used.in r e c t i f i c a t i o n and, at l a r g e values of a,energy from the load,which i s normally passed back to the a.c system, bypasses the converter and i s returned to the l o a d . The gate of the bypass s c r i s conti n u o u s l y enabled f o r r e c t i f i c a t i o n and d i s a b l e d f o r i n v e r s i o n . F i g . 3.1 shows the c i r c u i t diagram of the converter. The r e c t i f i e r negative t e r m i n a l of the converter was used as c o n t r o l ground and a cur r e n t shunt was connected on t h i s s i d e of the b r i d g e f o r a c u r -rent c o n t r o l s i g n a l . The i s o l a t i n g s w i t c h a l l o w s i n t e r r u p t i o n of the ra t e d d.c. c u r r e n t , i f necessary. 5A current shunt for current control signal F i g . 3.1 Converter b r i d g e c i r c u i t Each converter was s u p p l i e d by a power transformer whose f u n c t i o n i s to i s o l a t e the d.c. l i n k from the supply system and provide the necessary v o l t a g e c o n t r o l through automatic tap changing. At t h i s stage the on-load tap changer was not b u i l t as i t s use was not a n t i c i -pated, nor i t s o p e r a t i o n d e s i r e d i n the experimental s t e a d y - s t a t e s t a b -i l i t y study. The transformers, r a t e d at lOkVA, custom b u i l t by Hammond, inc o r p o r a t e a number of taps f o r the p r o v i s i o n of the tap changer, as i n F i g . 3.2. However, s i n c e some form of v o l t a g e c o n t r o l i s necessary to maintain or change an operating p o i n t , a three-phase v a r i a b l e a u t o t r a n s -former ( v a r i a c ) was provided at each converter. The v a r i a c a l s o served as a d d i t i o n a l p r o t e c t i o n , as a c c l i m a t i z i n g experiments could be c a r r i e d out a t r e l a t i v e l y low v o l t a g e s . The f i r i n g c i r c u i t r e q u i r e d to compute the f i r i n g delay was a commercially a v a i l a b l e 'equal angle' type [see chapter 2.2] c o n s i s t i n g of s i x independent delay elements which operate according to the magnetic a m p l i f i e r p r i n c i p l e of d e l a y i n g the s a t u r a t i o n of a s a t u r a b l e core by a d.c. c o n t r o l current: at the i n s t a n t of s a t u r a t i o n the sudden change i n p e r m e a b i l i t y of the core reduces the back emf i n the e x c i t i n g c o i l , r e s u l t i n g i n a sudden i n c r e a s e of c u r r e n t i n the e x c i t a t i o n c i r c u i t A f t e r a f u r t h e r process of shaping t h i s pulse i s d e l i v e r e d to the gate of the appropriate s c r . The f i r i n g c i r c u i t r e q u i r e d a three phase 240v i n p u t f o r e x c i t a t i o n . Since the l o c a l supply was 208 v o l t s the r e q u i r e d v o l t a g e was conveniently obtained u s i n g the combination of transformer windings shown i n F i g . 3.2 o b v i a t i n g the use of a separate transformer. 240 0-138 O-o o--O 256 taps 224 -O 190 -O102 supply to f i r i n g c i r c u i t a . c supply to conver te r n e u t r a l o o one phase of i s o l a t i n g transformer power v a r i a c * P i g 3.2 ' S i n g l e Phase Diagram of Transformer and Var i ac Connections 3.2 Mathematical Representat ion In the steady s ta te a n a l y s i s of h igh vo l tage converters the [28 29] fo l lowing assumptions are u s u a l l y made ' : ( i ) The a . c input vo l tage i s assumed to be s i n u s o i d a l and balanced i n both amplitude and phase. Only fundamental q u a n t i t i e s of vo l tage and current are cons idered . ( i i ) The e f f e c t of d . c . r i p p l e i s neg lec ted: t h i s assumes the presence of a p e r f e c t f i l t e r , u s u a l l y c h a r a c t e r i z e d by a smoothing choke of i n f i n i t e inductance and zero re-s i s tance placed at the terminals of the conver ter . ( i i i ) The r e s i s t a n c e of the transformer and a . c . system are n e g l i g i b l e . ( iy ) The valves are assumed to be per fec t c i r c u i t elements: the forward v o l t a g e drop of a conducting v a l v e and the reverse leakage current w h i l e b l o c k i n g are both con-s i d e r e d n e g l i g i b l e . The equations presented here describe the v o l t a g e and curr e n t r e l a t i o n s h i p s o f the two t e r m i n a l d.c. l i n k shown i n F i g . 2.6. These equations are used to determine system parameters such as commutating reactance, o v e r a l l e f f e c t i v e f i r i n g angle. The steady s t a t e equations are presented here without d e r i v a t i o n , which may be found i n any standard t e x t 1°. In the f o l l o w i n g equations the ' r ' and ' i ' s u b s c r i p t s denote r e c t i f i e r and i n v e r t e r q u a n t i t i e s r e s p e c t i v e l y . The v o l t a g e equation f o r the r e c t i f i e r i s given by V d r = -|-Vd o rIcosa + cos (ct + u r ) J (3.1) where = converter d i r e c t output v o l t a g e V(j 0 - converter 'no-load' v o l t a g e w i t h zero f i r i n g delay a = f i r i n g angle, measured from e a r l i e s t p o s s i b l e i n s t a n t that v a l v e could conduct. u r = the angle of commutation. The current equation of a r e c t i f i e r i s given by Ipcxh = 2 V d o r I c O S a * C ° s ( a + U r ) ] < 3 ' 2 ) where I d i s the d.c. l i n e c urrent X c r i s the commutating reactance per phase A d d i t i o n of equations (3.1) and (3.2) y i e l d the more popular v e r s i o n of the converter voltage equation: V d r = v d o r " s e t - | x c r I d (3.3) The two corresponding equations f o r the i n v e r t e r are v d i = y v d o i I c o s ^ - u ±) + cos BJ C3.4) | x c l I d = ^Vdo±[cos (B - u ±) - cos B] (3.5) whence V d i = V d o i cosB + | x c I d (3.6) A l t e r n a t i v e l y , s i n c e 6 = B-u-^  these equations may be w r i t t e n V d i = J V d o i f c o s < S + c o s ^ < 3- 7) f x c i x d = | v d o i Icosfi - cosB] (3.8) V d i = V d o i cos - ^  I d (3 .9) The d.c. l i n e current i s determined from the d i f f e r e n c e i n d i r e c t voltage at each end of the l i n e . T = V d i (3.10) K d c Where RJJQ i s the r e s i s t a n c e of the d.c. l i n e . In p r a c t i c e the inductance smoothing choke i s f i n i t e but the e f f e c t o f r i p p l e i s s t i l l n e g l e c t e d . The current equation (3.10) becomes more a c c u r a t e l y , i n transformed form I d • I"/ I (3.11) K d c + S L s m where L s m i s the value of inductance o f the smoothing choke. N e g l e c t i n g l o s s e s i n the co n v e r t e r , the t o t a l a.c. r e a l power i n p u t i s equal to the d.c. power output r e c t i f i e r : 3 I i V i costfi = V d r I d (3.12) 24 i n v e r t e r : 3I2V2 cosy^ - V<ji (3.13) where V and I are the fundamental v o l t a g e and current phasors. The phase angle between them i s given by Read-^5 u - s i n ( u ) cos (2a + u)  t a n * " s i n ( u ) s i n (2a + u) However the f o l l o w i n g more convenient approximate formula i s used e r r o r i n c u r r e d from u s i n g the approximation i s shown i n F i g . 3.5 (3.14) 11. the cos^2 = y [cosa + cos(a + u r ) ] cos^2 = "7 [cosg + cos 6] (3.15) (3.16) a v 2 , i 2 F i g 3.3 A.c. and d.c. Q u a n t i t i e s i n the L i n k 25 ,3.3 I n i t i a l Experimental T e s t i n g  The f i r i n g c i r c u i t s The f i r i n g c i r c u i t s were manufactured by F i r i n g C i r c u i t s I n c . , Model 613A372. Connection of the f i r i n g c i r c u i t to the- s c r c o n v e r t e r 46 as per manufacturer's i n s t r u c t i o n s r e s u l t e d i n the f o l l o w i n g d i s p o s i -t i o n of f i r i n g angle pulse and a-c phase v o l t a g e , F i g . 3.6(a). 26 \ / c \ / a \ / b b' X c' / 165 [ j \ a ' / \ [ end of p u l s e i s f i x e d a. As p e r a p p l i c a t i o n i \ \ • b. R e c t i f i c a t i o n \ c. I n v e r s i o n F i g 3.6 Phase V o l t a g e and F i r i n g P u l s e f o r SCR I where i t can be seen that 60° of phase c o n t r o l i s not used. A more e f f i c i e n t use of the f i r i n g pulse i s shown i n F i g 3.6(b) where the p u l s e s p e c i f i e d f o r v a l v e 6 i s used to t r i g g e r v a l v e 1. Since the end-of-pulse i s f i x e d , the maximum range of 165° does not allow the continuous t r a n s i t i o n from r e c t i f i c a t i o n to i n v e r s i o n . The f u l l range of the i n -v e r t e r i s p o s s i b l e by u s i n g the p u l s e s p e c i f i e d f o r v a l v e 3 to t r i g g e r v a l v e 1. The modified connection t a b l e i s shown i n Table 3.7. F i r i n g C i r c u i t gate number [46] used to f i r e s c r no. R e c t i f i c a t i o n I n v e r s i o n 1 2 3 4 3 6 5 6 1 4 5 2 5 2 3 6 1 4 Table 3.7 M o d i f i e d Connections The f i r i n g c i r c u i t s were provided w i t h 4 c o n t r o l windings of 100, 200, 500, 1000 Turns. The a m p l i f i e r s used to co n s t r u c t the con-t r o l system and d r i v e the f i r i n g c i r c u i t were those comprising The DONNER Analog Computer. Due to s a t u r a t i o n of these a m p l i f i e r s o n l y the two low current windings [500T, 1000T] provided c o n t r o l over the t o t a l p o s s i b l e range. A l s o , a r e s i s t o r connected i n s e r i e s w i t h each c o n t r o l winding increased the speed of response s p e c i f i e d as 0.004[ T • ] t 12 m sec. Although the f i r i n g c i r c u i t i s e s s e n t i a l l y a cu r r e n t c o n t r o l -l e d d e v i c e , i t i s more convenient f o r the purposes of a n a l y s i s t o meas-ure t r a n s f e r f u n c t i o n s as r a t i o s of two voltag e s o r radians to v o l t a g e . Thus the c o n t r o l v o l t a g e - f i r i n g angle c h a r a c t e r i s t i c s are shown i n F i g s . 3.8(a) and (b) f o r r e c t i f i c a t i o n and i n v e r s i o n r e s p e c t i v e l y f o r " each f i r i n g c i r c u i t according to the m o d i f i e d connection. I 2 3 4 5 I 2 3 4 5 6 control voltage c o n t r o l voltage R e c t i f i c a t i o n using f i r i n g c i r c u i t Inversion using f i r i n g c i r c u i t EE |653 EE 2250 Fi g 3.8 Control Voltage - F i r i n g Angle C h a r a c t e r i s t i c s W i t h i n the t y p i c a l o p e r a t i n g range of a[10° - 30°] and g[15° - 50°] the f i r i n g c i r c u i t s are l i n e a r . The incremen t a l gains f o r each low-current winding are shown i n t a b l e 3.9. R e c t i f i c a t i o n - I n v e r s i o n F i r i n g C i r c u i t 1000T 500T 1000T 500T 1 0.69 0 . 3 7 1 . 5 0 . 3 2 2 0.69 0 . 4 3 1 . 2 5 0 . 2 1 Table 3.9 Incremental Gains shown i n Radians per V o l t The frequency response of the f i r i n g c i r c u i t was measured u s i n g the arrangement show i n F i g . 3.10. o out f i l t e r G(s) = -8s/(l+.043s) z(l+.04)(l+. f iring <S circuit G in - O 4.0 msec or 86.4 at 60 hz positive half of bridge Fig 3.10 Measurement of Frequency Response The input frequency was v a r i e d from 1 t o 40hz and the amplitude and r e l a t i v e phase of the output was measured. From the o v e r a l l response was subtracted the measured frequency response of the f i l t e r , r e s u l t i n g i n the f o l l o w i n g amplitude and phase p l o t . S 45 <u 1-1 00 <u •o I OJ to .2 90 135 Frequency c/s 5 10 20 50 100 Phase \^G 3 ^ " Ln J i r i n g Angi .e a X \ \ \ ; -2 -4 -6 OJ 3 -8 i -10 =r -12 10 20 50 100 Fig 3.11 F i r i n g C i r c u i t Frequency Response - SOOT Winding With the gain f a l l i n g o f f a t 6db/octave a 1st order element i s suggested, but the phase f a l l s o f f to w e l l beyond 90°. By s u b t r a c t -i n g the f i r i n g angle of 4msec from the phase c h a r a c t e r i s t i c , a w e l l behaved 1st order l i n e a r element appears. T h i s suggests a t r a n s f e r G f u n c t i o n of F(s) = y, n n r i 0 w i t h a t r a n s p o r t l a g which w i l l be negl e c t -J.+U . U U O S ed f o r low-bandwidth systems but otherwise equal to the f i r i n g c u r c u i t delay angle: T -T = -Ts E G 1+0.008s a 2TT.60 IT/3 - 8 f o r r e c t i f i e r f o r • i n v e r t e r 2ir.60 G = i n c r e m e n t a l d.c. g a i n - see Table 3.9 F i g 3.12 Block Diagram of F i r i n g C i r c u i t The measurements made were estimated to be w i t h i n - 10% e r r o r , and w i t h i n t h i s margin, both f i r i n g c i r c u i t s were estimated to have the same frequency response f o r the 500T winding. Two other e f f e c t s : the f i r i n g angle changes w i t h a p p l i e d a.c.voltage, and due to d i s t o r t i o n caused by the v o l t a g e dents which appear at commutation a r e l a t i o n between the cu r r e n t and the change i n f i r i n g angle may be measured, F i g 3.12. F i g . 3.13 shows the change i n a at d i f f e r e n t values of a f o r changes i n a-c v o l t a g e ( u n d i s t o r t e d ) . a Q) F i g 3 . 1 3 T y p i c a l V a r i a t i o n of f i r i n g a n g l e w i t h change in c u r r e n t F i r i n g A n g l e degrees T y p i c a l V a r i a t i o n o f f i r i n g a n g l e w i t h phase v o l tag . i n p u t to f i r i n g c i r c u i t 31 Measurement of Transformer commutating reactance The leakage reactance of the transformer was measured by op-e r a t i n g the r e c t i f i e r w i t h a r e s i s t i v e l o a d . A p p l i c a t i o n of equation (3.2) y i e l d s a value of commutating reactance of 0.03 ft/phase. The commutating reactance may be i n c r e a s e d by the i n s e r t i o n of a d d i t i o n a l e x t e r n a l reactance, which may be p l a c e d on e i t h e r s i d e of the v a r i a c , F i g . 3.15. V a r i a c l : y <3L>-T r a n s f o r m e r X = 0 . 0 3 Y 2 - X , A d d i t i o n a l R e a c t a n c e a n d t h e i r V a l u e s F i g 3 . 1 5 I n s e r t i o n o f A d d i t i o n a l C o m m u t a t i n g R e a c t a n c e Inductors of nominal values 5, 10, 20 mH are a v a i l a b l e , each found to have a reactance of 1.4, 5, 9 ohms r e s p e c t i v e l y and n e g l i g i b l e r e s i s t a n c e . 3.4 D.C. L i n k Start-up Procedure In e n e r g i z i n g the l i n k , the e s s e n t i a l p r e c a u t i o n i s to f i r s t s e t the v o l t a g e at the i n v e r t e r to the nominal o p e r a t i n g v o l t a g e u s i n g the i n v e r t e r v a r i a c and w i t h the i n v e r t e r f i r i n g c i r c u i t d e l i v e r i n g pulses at around B = 25°. Then the r e c t i f i e r i s brought up to the o p e r a t i n g voltage so that the l i n k c u r r e n t s t a r t s from zero. The f i n a l o p e r ating p o i n t - d e s i r e d V ( j r , V ^ i , I Q , a, B - i s achieved through adjustments i n v a r i a c s and f i r i n g c i r c u i t c o n t r o l s . A change i n o p e r a t i n g v o l t a g e upwards i s i n i t i a t e d w i t h ad-justment of the i n v e r t e r v a r i a c , a change i n v o l t a g e downwards i s i n i t -i a t e d w i t h adjustment of the r e c t i f i e r v a r i a c , tending to keep the cur-rent low. And, i n the absence of the i n v e r t e r feedback c o n t r o l , the e x t i n c t i o n angle i s c a r e f u l l y monitored and adjusted i f i t f a l l s below 15°. Model users are advised to a c c l i m a t i z e to t h i s procedure u s i n g low v o l t a g e and current l e v e l s . This chapter has presented the b a s i c components of the p h y s i -c a l model and t h e i r mathematical behaviour. The omission of f i l t e r s was not regarded as d e t r i m e n t a l to the r e a l i s m of the model s i n c e no s a t i s f a c t o r y model a.c.system e x i s t s f o r t h e i r behaviour [ v o l t a g e r e g -u l a t i o n at the converter's a.c. t e r m i n a l s ] to be a p p r e c i a t e d . Instead the converters were operated u s i n g the l o c a l hydro system which e a s i l y accommodated the r e a c t i v e power requirements of the c o n v e r t e r s . The f o l l o w i n g chapters d e s c r i b e the converter c o n t r o l systems w i t h attendant s t u d i e s i n r e l a t i v e s t a b i l i t y . The i n v e r t e r c o n t r o l i s developed f i r s t as a p r o t e c t i v e measure. 4. INVERTER CONTROL The c o n t r o l l e d o p e r ation of the d.c. l i n k r e q u i r e s r e g u l a t i o n at both r e c t i f i e r and i n v e r t e r . Although the l i n k may be operated w i t h both s t a t i o n s on open loop, the r i s k i s great of commutation f a i l u r e at the i n v e r t e r , r e s u l t i n g i n e x c e s s i v e l y l a r g e c u r r e n t s . Operation at a l a r g e enough angle of 8 at the i n v e r t e r a l s o r e s u l t s i n a l a r g e component o f r e a c t i v e power. Hence the f i r s t p r i o r i t y i s the pr e v e n t i o n of i n v e r t e r comm-u t a t i o n f a i l u r e i . e . constant e x t i n c t i o n angle c o n t r o l . This chapter describes the r e a l i z a t i o n of the i 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 shown i n F i g . 2.7. The f i r s t s e c t i o n describes i n d e t a i l the p r i n c i p l e s o f i n v e r t e r c.e.a. c o n t r o l ; the second a new method of d e t e c t i o n o f ex-t i n c t i o n angle; the t h i r d s e c t i o n d e s c r ibes the p h y s i c a l components of the c o n t r o l system and the f o u r t h s e c t i o n d e r i v e s the mathematical model i n the form of the c h a r a c t e r i s t i c e q u a t i o n . 4.1 General P r i n c i p l e s of Operation The constant e x t i n c t i o n angle c o n t r o l i s important to the i n v e r t e r s t a t i o n i n that i t p r o t e c t s the i n v e r t e r from commutation f a i l u r e by advancing the f i r i n g angle B i n response to a decrease i n margin o r e x t i n c t i o n angle 6, see F i g . 4.1. The i n v e r t e r normally operates on a c h a r a c t e r i s t i c which keeps the margin angle constant at i t s lowest safe value to minimize r e a c t i v e power demand. The margin angle 6 i s maintained t y p i c a l l y 34 2 phase-neutral tage waveforms • F i g 4-1 Inverter Connnutation 47 at 15-20° to allow f o r v a l v e d e - i o n i z a t i o n and c o n t r o l system i n -a c c u r a c i e s before the commutation v o l t a g e turns n e g a t i v e a f t e r which i t becomes imp o s s i b l e to i n i t i a t e o r complete s u c c e s s f u l commutation. Given that the e x t i n c t i o n angle i s r e q u i r e d to remain con-s t a n t , the problem i s to a n t i c i p a t e the d u r a t i o n of the commutation p e r i o d , s i n c e , once the val v e i s t r i g g e r e d , there i s no c o n t r o l over the r e s u l t i n g e x t i n c t i o n angle, which becomes c r i t i c a l . The 'in -coming' v a l v e must reach f u l l conduction i n order to t u r n o f f the 'out-going 1 v a l v e . I f commutation i s not complete by i n s t a n t 'T' i n F i g . 4.1, the current w i l l s t a r t to commutate back to the o r i g i n a l v a l v e , r e s u l t i n g i n commutation f a i l u r e , e x c e s s i v e d.c. l i n e c u r r e n t and p o s s i b l e damage i f there i s no means of recovery. The i n v e r t e r c o n t r o l l e r , t h e r e f o r e , must respond to changes i n current and e x t i n c t i o n angle to produce the re q u i r e d angle of advance 8. For t h i s reason, the c.e.a. c o n t r o l i s r e f e r r e d to i n some l i t e r a t u r e as p r e d i c t i v e . There are two b a s i c approaches to the implementation o f c.e.a. 41,48 c o n t r o l : the f i r s t i s c a l l e d the "angle comparator method", ' where comparison of c o n t r o l s i g n a l s p r o p o r t i o n a l to <5 and u w i t h the d e s i r e d reference e x t i n c t i o n angle e f f e c t s adjustment of angle of advance 3. The second approach i s c a l l e d the " v o l t - i n t e g r a l comparator method",^ ^ '-^ where the c o n t r o l s i g n a l s are p r o p o r t i o n a l to the v o l t - t i m e areas a s s o c i a t e d w i t h commutation and a f t e r e x t i n c t i o n , areas.'B' and 'A' r e s p e c t i v e l y i n F i g . 4.1. In these reported cases [37,42,47,48,49] t l i e f i r i n g angle 3 i s computed from p r e d i c t e d (or known from l a s t f i r i n g ) values, of u and'3 and devices were developed to detect u and 6 s e p a r a t e l y . The next s e c t i o n describes a simple method, devised by the author, d e t e c t i n g the e x t i n c t i o n angle of each v a l v e d i r e c t l y , without the need f o r d e t e c t i n g two angles s e p a r a t e l y . The output of the d e t e c t o r i s a pulse of equal d u r a t i o n to the e x t i n c t i o n p e r i o d , which may then be used i n e i t h e r of the two approaches mentioned above. 4.2 A New Method of Detection of E x t i n c t i o n Angle The method c o n s i s t s of comparing the d i r e c t . v o l t a g e o f the i n v e r t e r w i t h the output of a diode 3-phase b r i d g e c i r c u i t connected to the same secondary winding of the i n v e r t e r transformer. This r e s u l t s i n an i n t e r e s t i n g compensation f o r the commutation v o l t a g e drop due t o the transformer leakage reactance. secondary winding . see Fig 4.5 2 diode bridge inverter bridge Pig 4.2 Inverter Extinction Angle Detector .see Fig 4.4 (a) inverter direct voltage waveform (b) phase ' a' waveform (c) phase 'b' waveform (d) phase 'o' waveform (e) output voltage waveform from -»- t diode bridge (a) waveform at positive terminal of diode bridge (b) waveform at positive terminal of inverter (o) the two voltage waveforms superimposed upon each other (d) diode bridge waveform with respect to the inverter positive terminal Fig 4.4 Pulse Waveform Produced by Subtraction of Inverter Waveform from Diode Bridge Waveform, shown for the Positive side of the Inverter (a) the waveforms of the inverter and diode bridge superimposed upon each other (b) diode bridge waveform with respect to the inverter negative terminal a l l shaded areas here are equal to the 'volt-time' area—•") dropped during commutation Fig 4.5 Pulse Waveform on the Negative Side of the Inverter Fig 4.3 Distortion of a.c. Waveforms by Inverter Commutation and Effect on Diode Bridge Output The de t e c t o r c i r c u i t , shown i n F i g . 4.2 c o n s i s t s of a t h r e e -phase diode b r i d g e which produces a r e c t i f i e d v o l t a g e waveform w i t h no phase c o n t r o l . I t i s connected to the i n v e r t e r , w i t h s i m i l a r p o l a r i t -i e s connected, v i a r e s i s t o r s R, which enable c u r r e n t to flo w through the diode b r i d g e . The e f f e c t i v e short c i r c u i t between two phases d u r i n g commu-t a t i o n r e s u l t s i n the impression of 'voltage dents', as shown i n F i g . 4.3(b-d). I f these same phase v o l t a g e s supply the diode b r i d g e , r e c t -i f i e d output has a waveform s i m i l a r to that shown i n F i g . 4.3(e). The 'dent' i n each pulse of the diode output waveform has the same v o l t - t i m e area as the i n v e r t e r commutation v o l t - d r o p . By m o n i t o r i n g the v o l t a g e on each s i d e of the diode b r i d g e w i t h respect to the vo l t a g e s on each s i d e of the i n v e r t e r , the two voltage s are e f f e c t i v e l y s u b t r a c t e d . The r e s u l t i s two t r a i n s of pu l s e s , shown i n F i g . 4.4(d) f o r the v o l t a g e p o s i t i v e s i d e of the b r i d g e , and F i g . 4.5(d) f o r the neg a t i v e s i d e of the b r i d g e . Each pulse has an angular d u r a t i o n i d e n t i c a l to the e x t i n c t i o n a n g l e , amp-l i t u d e equal to the instantaneous v a l u e of commutating v o l t a g e a t ex-t i n c t i o n , and v o l t - t i m e area equal to the v o l t - t i m e area l e f t a f t e r e x t i n c t i o n of each v a l v e to the end of p o s i t i v e commutating v o l t a g e . In the model, the p o s i t i v e pole of the i n v e r t e r was taken as c o n t r o l ground, and the pulses shown i n F i g . 4.5(b) were e x c l u s i v e -l y used. The a p p l i c a t i o n of the d e t e c t o r t o the i n v e r t e r c.e.a. con-t r o l system i s des c r i b e d i n the next s e c t i o n . 38 4.3 Implementation of the I 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 The i 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 , shown i n F i g . 2.7, i s reproduced i n F i g . 4.6: normal o p e r a t i o n i s w i t h c.e.a. c o n t r o l , but a constant current c o n t r o l i s u s u a l l y i n c l u d e d to prevent a break i n tr a n s m i s s i o n should the r e c t i f i e r v o l t a g e f a l l . Pig 4.6 Inverter Control Characteristic Constant E x t i n c t i o n Angle C o n t r o l A feedback c o n t r o l i s apparent where one of the two t r a i n s of margin angle pulses i s low-pass f i l t e r e d and compared to a r e f e r -ence to produce an e r r o r s i g n a l , a m p l i f i e d by the e r r o r a m p l i f i e r t o d r i v e the f i r i n g c i r c u i t . The d.c. output of the low-pass f i l t e r i s p r o p o r t i o n a l to the average v o l t - t i m e area of the p u l s e s , r e l a t e d to the margin angle, as i n F i g . 4.7, by • c c . control c.e.a. control A. E / " v - comm Pig 4.7 volt-time area of detector pulses wt=0 1 o v a v e = 3.2^ / - s i n ( u t ) . d(u)t) -8 3 = 2TT Ecomm t 1 ~ cos6] = ^ V j o i [1 ~ cos6] i . e . non l i n e a r And f o r s m a l l changes i n 6 about an o p e r a t i n g p o i n t A V a v e = t j AVrfoi (1 ~ cos6) + -| V d o i sinSAS] F(s) a l s o n o n l i n e a r , and where F(s) i s the t r a n s f e r f u n c t i o n of the f i l t e r . This n o n - l i n e a r i t y may be avoided by the a d d i t i o n of a zener diode, F i g . 4.7, whence V a v e a S. Although the diode s w i t c h - o f f produces a d e f i n i t e s l o p e , and due to balance e r r o r , each pulse has a d i f f e r e n t d u r a t i o n , l i n e a r i t y over the whole o p e r a t i n g range of 8 was found (8 > 10°). From p r a c t i -c a l experience i t was found that the f i l t e r was unnecessary, and i n the case of a s i n g l e pole f i l t e r , c o n t r i b u t e d to a p o t e n t i a l l y undes-i r a b l e t r a n s i e n t overshoot response to a step change i n margin. Since c.e.a. c o n t r o l i s e s s e n t i a l l y p r e d i c t i v e i n n a t u r e , and s i n c e i t s f u n c t i o n i n m a i n t a i n i n g minimum s a f e margin angle f o r i n v e r t e r p r o t e c t i o n , an a d d i t i o n a l f e a t u r e to the c.e.a. c o n t r o l l e r would be an asymmetric response to p o s i t i v e and n e g a t i v e d-c c u r r e n t t r a n s i e n t s : i n response to an i n c r e a s e i n c u r r e n t , and hence an i n -crease i n commutation angle and a decrease i n margin angle, a f a s t c o n t r o l a c t i o n to increase B i s d e s i r a b l e . Momentary overshoot c o u l d be t o l e r a t e d F i g . 4.8(a), so long as the time constant of i t s decay was s m a l l e r than the time constant provided by the smoothing choke i n 40 governing the rate of increase of l i n e c u r r e n t . However, i n response to a decrease i n current , l i t t l e or no overshoot i s r e q u i r e d , ra ther a slow c r i t i c a l l y damped response, F i g . 4 . 8 (b ) . Such a feature i s descr ibed at the end of the chapter , but not inc luded for a n a l y s i s . increase F i g 4.8 C.E.A. Controller Responses to Increases and Decreases i n Current Constant Current Contro l Inverter operat ion requires tha t , when the current f a l l s below a c e r t a i n v a l u e , the c o n t r o l mode changes to mainta in constant c u r r e n t . c o n t r o l , which i s normally 20% below the constant current s e t t i n g of the r e c t i f i e r . This i s achieved i n the model by p r o v i d i n g a d d i t i o n a l e r r o r s i g n a l when the current f a l l s below the current s e t t i n g I d s 2 » s n o w n i n F i g . 4 . 9 . a d d i t i o n a l e r r o r e„ t Jds2 decrease Ids2 Pig 4.9 A d d i t i o n a l Error S i g n a l when I< I . -ds2 The current s i g n a l p r o p o r t i o n a l to current below Id S2 a n <* then damped to V j ^ ^ when 1^ > l d s 2 : S u D t r a c t i ° n °f t h i s s i g n a l from the reference VT, gives an a d d i t i o n a l e r r o r s i g n a l p r o p o r t i o n a l to ( I d a o -ds2 S i l I d ) . The o v e r a l l model c o n t r o l system i s shown i n F i g . 4.10. Fig 4.10 Constant Extinction Angle and Constant Current Controls for Inverter 4.4 Mathematical Models This s e c t i o n d e r i v e s the mathematical model i n the form o f the l i n e a r i z e d perturbed equations. As o u t l i n e d i n the i n t r o d u c t i o n , the equations are w r i t t e n down to d e s c r i b e the p r a c t i c a l system: an operating p o i n t was e s t a b l i s h e d , the feedback loop i n t r o d u c e d and the d e t a i l s o f the o p e r a t i n g p o i n t form the b a s i s of the perturbed equations. As usual the Laplace transform i s used. The c h a r a c t e r i s t i c equation i s d e r i v e d f o r both the c.e.a. and c c . c o n t r o l f o r the i n v e r t e r which was s u p p l i e d by a power s u p p l y , and not the r e c t i f i e r . From the system equations w r i t t e n i n m a t r i x form the c h a r a c t e r i s t i c equation i s d e r i v e d as the determinant of the coef-f i c i e n t m a t r i x u s i n g an e f f i c i e n t computer program employing a n o v e l method i l l u s t r a t e d i n Appendix A. For the a p p l i c a t i o n of S i l j a k ' s method, des c r i b e d i n chapter 5, two a d j u s t a b l e parameters were chosen a r b i t r a r i l y as two parameters of the e r r o r a m p l i f i e r . The i n v e r t e r was operated such t h a t a s m a l l d i s t u r b a n c e d i d not r e s u l t i n a change i n c o n t r o l mode from c.e.a. to c c o r v i c e v e r s a . Constant E x t i n c t i o n Angle Co n t r o l The b l o c k diagram f o r c.e.a. c o n t r o l , d e r i v e d from the c i r c u i t shown i n F i g 4.10, i s shown i n F i g . 4.11. The G2 b l o c k a r i s e s from the i n t e r f e r e n c e w i t h f i r i n g angle by changes of d.c. l i n e c u r r e n t through d i s t o r t i o n of the a.c. l i n e v o l t a g e - see F i g . 3.13. A numerical constant has been assigned t o G2, even though a s m a l l time constant probably e x i s t s . 43 'ref K AVB 0.214 / (1 + Ts) (1 + .5s) 1 + .003s JO Al G2(s) G3(s) 9.6P. 46 G,(s) A(5 Fig 4.11 Block Diagram of c.e.a. C o n t r o l l e r In response to a s m a l l d i s t u r b a n c e , the l i n e a r i z e d Laplace transformed equations may be w r i t t e n : AVe = -9.6P n A6 (4.1) where P i i s the a c t u a l s e t t i n g of the potentiometer P-^  i n the c.e.a. feedback loop of F i g . 4.10 K (1 + T s ) ( l + 0.5s) AV e (4.2) where K and T are two a r b i t r a r y a d j u s t a b l e parameters used i n S i l j a k ' s a n a l y s i s . A P = 1 "I'oOSs A V e " G2<s>AId (4.3) u s i n g the 500-turn c o n t r o l winding, and G£ measured at the o p e r a t i n g p o i n t . An expression f o r G i ( s ) r e l a t i n g A8 and A6 may be d e r i v e d from the f o l l o w i n g power equations v d i T Vdi° t c o s < S + c o s^> = J v d i o ^ c o s 6 " • c o s ^ (4.4) (4.5) I d ( R d c + s L s m ) = V d r - V d i (4.6) With the a v a i l a b l e d.c. power supply d r i v i n g the i n v e r t e r , V d r i s assumed constant, and w i t h the i n v e r t e r feeding i n t o a l a r g e a.c. system, and without r e g u l a t i o n provided by f i l t e r s , V d o^, r e l a t e d to the a.c. system v o l t a g e , i s al s o assumed constant. The l i n e a r i z e d p e r turbated equations become 1. A V d i = - | v d i Q sin<5 AS - | v d i o s i n g Ag | v d i o sinS AS + | v d i o s i n g Ag | x c A I d = A V d i = A I d ( R d c + s L s m ) from which Rrlr + Xp + s L S T n Rdc ~ x c + s L s m as i n (4.7) (4.8) (4.9) (4.10) (4.11) A6 » G]_(s) Ag S i m i l a r l y , an e x p r e s s i o n f o r G3(s), r e l a t i n g A I d and Ag, may be derived from equations (4.7-9). However the approach used i n t h i s study was to w r i t e down the r e l e v a n t equations i n m a t r i x form and compute the determinant by a method developed by the author i n con-j u n c t i o n w i t h Dr. K a b r i e l [see A p p e n d i x A ] . In t h i s case the re l e v a n t equations are (4.1-3), (4.7-9) and, w r i t t e n i n m a t r i x form, appear as A l , AVE Va A8 . A6 AVj, A l j 1 9.6P1 -K (1+Ts)(1+0.5s) -0.214 1+0.008s k2(1+0.008s) i v d i o s i n e i v d i 0 s l n 6 " 1 - i v d l 0 s i n e i V d , 0 s i n 6 3Xc/n 1 dc sm AVe AVS AB A6 AV di Al A computer program, p r i m a r i l y intended for l a r g e r sparse m a t r i c e s , was used to compute the determinant, where some elements are to be re ta ined as a lgebra i c v a r i a b l e s and some are polynomials i n s . The program u t i l i z e s e f f i c i e n t sparse matr ix methods. The determinant of the c o e f f i c i e n t matrix i s known to be the c h a r a c t e r i s t i c equation of the system. Hence n A(s) = E a k s k = 0 k=0" (4.13) i s the c h a r a c t e r i s t i c equation governing the d e s c r i p t i o n o f system behaviour fo l lowing a smal l d is turbance from a steady opera t ing p o i n t . The i n v e r t e r and i t s c o n t r o l was operated alone us ing a d . c . power supply . In doing so , the f u l l range of i n v e r t e r behaviour may be i n v e s t i g a t e d without in t er f erence from the r e c t i f i e r c o n t r o l system. R d c Lsm 4 v d 10Sv 5.OA 19.5° F i g 4.12 I n v e r t e r O p e r a t i n g P o i n t l d i sv 40v hence V .. = 98v d i o k. in L = 1.2H X = 14(0.25) sm c o.88n a - 31.0 For I B A S E " 5 A becomes VBA S E = 100v, the c o e f f i c i e n t m a t r i x AVe Ave A3 AS AT-d AV 1. 6,3 -K 1+(0.5+T)s+0.5Ts 2 -0.214 1+0.008s 0.2+0.0016s 0.252 0.165 1. -0.252 0.165 0.044 1. 0.205+0.06s the determinant of which i s 0 = 5.77 + 5.47K +(3.92 + 2.04K + 5.77T)s +(0.526 +•3.92T)s 2 +(0.004 + 0.526T)s3 + 0 . 0 0 4 T S 4 (4.14) and t h i s i s the c h a r a c t e r i s t i c equation f o r the c.e.a. con-t r o l l e r . The next chapter describes the a p p l i c a t i o n of S i l j a k ' s method to the c h a r a c t e r i s t i c equation i n t h i s form. Constant current c o n t r o l The constant current c o n t r o l l e r i s a c t i v a t e d when the oper-a t i n g current f a l l s to the current corresponding to the refe r e n c e s e t -t i n g l d s 2 * n this case set at 7A. The b l o c k diagram f o r the constant current c o n t r o l i s shown i n F i g . 4.13. G, (S) 9 . 6 P , G2(s) 4 WB 0.2 ; 1 + .008s 5P, AB G3(s) AS G4(s) 0.16 At Fig 4.13 Block Diagram for Inverter Constant Current Control Mode The constant current feedback loop produces an e r r o r s i g n a l which tends to increase 8 when the c u r r e n t f a l l s : the i n c r e a s e i n 8 i s accompanied by an increase i n 6 which tends to cause a r e d u c t i o n i n 8 through the s t i l l o p e r a t i v e c.e.a. feedback loop. The r e d u c t i o n i n 8, and hence c u r r e n t , i n an op e r a t i n g region where the constant c u r r e n t c o n t r o l l e r i s a c t i v e c a l l s f o r an i n c r e a s e i n 8. Thus an e q u i l i b r i u m i s obtained between two a n t a g o n i s t i c c o n t r o l l o o p s , each w i t h t h e i r c o n trary a c t i o n upon 8. Gi(s) has been added to a l l o w independent ad-justment of gain and time constant of each system. In the m u l t i l o o p system, choosing two parameters becomes even more a r b i t r a r y : however, i t was decided to leave the c.e.a. c o n t r o l l e r w i t h a s a t i s f a c t o r y s e t t i n g and vary two parameters of G-^(s). G2(s) was s e t to 30 -y^ r- and Gi (s) was designated 1 + j s . Again (1+0.5s)(1+0.Is) i n s t e a d of d e r i v i n g expressions f o r G3(s) and G4(s) and reducing the block diagram to the o v e r a l l t r a n s f e r f u n c t i o n , the r e l e v a n t equations are w r i t t e n i n matrix form. In t h i s case the equations are those used f o r the c.e.a. case w i t h the f o l l o w i n g m o d i f i c a t i o n to eqn (4.1): AVe = - 9 .6P i A6 - 5 P 3 . A I d For the same operat ing point the c o e f f i c i e n t matrix becomes AVe AVB AB A6 A I . AV d i (4.15) 1+Ts 6.3(1+Ts) K 30 1+0.51s+0.05s2 0.214 1+0.008s 0.2+0.0016s 0.252 .0.165 1 -0.252 0.165 o.okk 1 0.205+0.06s the determinant of which i s 0 = 15.8 + 5.34K +(5.72 + 15.8T)s +(0.082 + 5.72T)s2 +(0.0056 + 0.082T)s3 +(0.00004 + 0 .0056T)s 4 + 0.0004Ts 5 (4.16) 4.5 Trans ient c . e . a . Contro l Due to the p r e d i c t i v e nature of c . e . a . c o n t r o l , i t i s usua l to compensate for a t rans i en t increase i n c u r r e n t : when the current i n c r e a s e s , the margin angle momentarily decreases , g r e a t l y i n c r e a s i n g the r i s k of commutation f a i l u r e . Hence a fas t t r a n s i e n t response i s d e s i r a b l e and a non-optimal value of 6 i s t o l era ted for a short d u r a -t i o n i f overshoot occurs . However, a decrease i n current and the mom-entary increase i n margin requires a well-damped response of the con-t r o l l e r to b r i n g the margin angle back to the steady s ta t e v a l u e . This asymmetrical c o n t r o l i s produced by a cu r r e n t d e r i v a t i v e s i g n a l which i s a c t i v e when the current i n c r e a s e s and blocked when the current decreases. The c i r c u i t i s shown i n F i g . 4.14. Fig 4.14 . Transient c.e.a. Circuit The output of A m p l i f i e r 1, zero at steady s t a t e , i s connected to the 1000-turn winding of the f i r i n g c i r c u i t and summation w i t h the c.e.a. c o n t r o l s i g n a l appears as the r e s u l t a n t m . m . f. i n the s a t u r -able core. The e f f e c t i v e time constant of t h i s c i r c u i t should be s i g -n i c a n t l y d i f f e r e n t than the e f f e c t i v e time constant of the c.e.a. con-t r o l loop. In t h i s chapter the i n v e r t e r c o n t r o l implemenation was p r e -sented together w i t h the mathematical d e s c r i p t i o n and d e r i v a t i o n of the c h a r a c t e r i s t i c equation s u i t a b l e f o r a n a l y s i s . u s i n g S i l j a k ' s method. The next chapter describes the a p p l i c a t i o n of t h i s method to the con-t r o l l e r i n i t s two modes of c o n t r o l : constant e x t i n c t i o n angle and constant c u r r e n t . . 5. ANALYSIS AND EXPERIMENTAL RESULTS This chapter describes the a p p l i c a t i o n of S i l j a k ' s method of parameter plane a n a l y s i s to the comparison of the p h y s i c a l model i n two ways: 1) by p l o t t i n g the s t a b i l i t y boundaries i n the parameter plane and determining the experimental s t a b i l i t y l i m i t by a d j u s t i n g the same two parameters i n each case. 2) By photographing the response at d i f f e r e n t c o n t r o l l e r op-e r a t i n g p o i n t s and comparing the response to the damping f a c t o r accord-i n g to the parameter plane p l o t . The method may then be a p p l i e d to determine the combination of parameters which produces best r e l a t i v e damping. 5.1 The Method of S i l j a k The S i l j a k method of a n a l y s i s 24,25,26 a p p l i e s to the manipu-l a t i o n of the roots of the c h a r a c t e r i s t i c equation by two a d j u s t a b l e system parameters through the c o e f f i c i e n t s of the c h a r a c t e r i s t i c equa-t i o n , w r i t t e n as n E a k s k = 0 (5.1) k=0 Where s = a + jco i s the complex v a r i a b l e and a k (k = 0,1,2—n), the r e a l c o e f f i c i e n t s , a r e l i n e a r f u n c t i o n s o f two a d j u s t a b l e system parameters a and 8: a k = abk + 8 c k + d k (5.2) F o l l o w i n g a s m a l l disturbance the system may execute o s c i l l a t i o n s o f decreasing magnitude i n r e t u r n i n g to the e q u i l i b r i u m p o s i t i o n . I t i s p o s s i b l e to define an undamped n a t u r a l frequency to n and damping f a c t o r £ f o r these o s c i l l a t i o n s 21 S u b s t i t u t i n g s = - u)n£ + ju^ V 1-r,2 (5.3) and a p p l y i n g the c o n d i t i o n that the summation of r e a l and imaginary p a r t s must go to zero independently, (5.1) may be w r i t t e n R = R I« n >C,a,6] = 0 I = I Iu)n,c,a,B] = 0 " (5.4) Equations (5.4) may be considered as two equations i n two unknowns a and B which may be s o l v e d a = f(u>n,c) B = g(u) n,C) (5.5) [provided t h a t the Jacobian J=J (R,l/a,B) e x i s t s and i s d i f f e r e n t from z e r o ] . From these equations i t i s p o s s i b l e to map s-plane contours, d e f i n e d by w n ,£ i n t o the parameter plane of a and B. For three cases, F i g . 5.1, the mapping f u n c t i o n s may be d e r i v e d a n a l y t i c a l l y : F i g 5.1 C o n t o u r s w h i c h n a y be Mapped I n t o Parameter P l a n e Numerical mapping of a r b r i t r a r y s-plane contours i s a l s o pos-s i b l e u s i n g the method described i n reference [27]. The r e s u l t of mapping a number of s-plane contours i n t o the parameter plane i s a d i r e c t g r a p h i c a l technique of r e l a t i n g the root l o c a t i o n s of the char -a c t e r i s t i c equation to two adjustable system parameters, such as g a i n , time constant e t c . . . The S i l j a k technique has the fo l l owing c h a r a c t e r i s t i c s : ( i ) there i s a d e f i n i t e advantage to the M i t r o v i c method i n that the adjustable parameters may appear i n any number o f c o e f f i c i e n t s o f the c h a r a c t e r i s t i c equat ion , ( i i ) the r e l a t i v e s t a b i l i t y , or other performance measure, may be d i r e c t l y r e l a t e d to a p a i r of system parameters , ( i i i ) two parameters may be s imultaneously ad jus ted , p r o v i d i n g a d e f i n i t e advantage over other a l g e b r a i c domain methods where on ly one system parameter i s a d j u s t a b l e , ( iv ) the method has been extended by S i l j a k ^ S s o that p a r a -meters may appear n o n - l i n e a r i y i n the c o e f f i c i e n t s o f the c h a r a c t e r i s t i c equat ion , and i s u s e f u l where two a d -j u s t a b l e parameters appear i n d i f f e r e n t c o n t r o l loops o f the system. 5.2 A p p l i c a t i o n to Inverter System The parameter plane mapping funct ions a = f ( t o n , £ ) , 6 = g ( u ) n , £ ) f o r the c h a r a c t e r i s t i c equation (A.14) are der ived a n a l y t i c a l l y i n Appendix B. From these funct ions i t i s p o s s i b l e to def ine areas o f absolute s t a b i l i t y by mapping the l e f t h a l f s -plane i n t o the parameter p lane , shown i n F i g . 5 .2 . . T Fig 5.2 Regions of Absolute S t a b i l i t y Segment 'A' corresponds to the imaginary a x i s o r s=0 l i n e and segments B and C r e s p e c t i v e l y to the r e a l root boundaries s=0 and s=°°. Shading of the curves emphasizes the regions of s t a b i l i t y : i n the d i r e c t i o n of i n c r e a s i n g shade on the l e f t o f the curve i f A<0 and on the r i g h t i f A>0 [see Appendix B ] . That these areas correspond to four roots w i t h negative r e a l p a r t s may be confirmed by a p p l y i n g the Routh-Hurwitz c r i t e r i o n : the c h a r a c t e r i s t i c equation at the p o i n t (50,0.1) becomes 3.5xl0" 6 s 4 + l x l O - 4 s 3 +8.3x10" 3 s 2 + 1.06s + 2.8 = 0 (5.6) The Routh-Hurwitz c o e f f i c i e n t a r r a y i s 2.8 8.4x10" 3 3.5x10-6 1.06 5 . 1 x l 0 _ 4 7.47xl0" 3 3r5xl0"6'. -l . O x l O - 7 0 There are no s i g n changes i n the l e f t hand column, i n d i c a t i n g t h a t a l l roots have negative r e a l p a r t s f o r values of K and T w i t h i n the shaded boundaries of F i g . 5.2. In the determination of the experimental s t a b i l i t y l i m i t the gain K was i n c r e a s e d f o r a number of values of T u n t i l u n stable o s c i l l a -t i o n occurred a f t e r a step i i i p u t . At each p o i n t where i n s t a b i l i t y o c c u r r e d , the undamped frequency of o s c i l l a t i o n was measured. By mark-i n g i n vvalues of o j n on the t h e o r e t i c a l curves and by p l o t t i n g more con-s t a n t - ? curves a more complete p i c t u r e o f t h e o r e t i c a l behaviour i s ob-t a i n e d as i n F i g . 5.3. The experimental s t a b i l i t y l i m i t i s shown and may be compared to the r,=o l i n e . The parameter plane map shows t h a t over a l a r g e range of gain K the system remains predominantly under-damped, a f a c t which corresponded w i t h p r a c t i c a l experience. A f u r t h e r comparison may be made by photographing the o s c i l l o -gram of the response at a number of o p e r a t i n g p o i n t s . F i g . 5.4 shows the photographed response at nine d i f f e r e n t o p e r a t i n g p o i n t s d e f i n e d i n F i g . 5.3. A f a i r l y c l o s e correspondence between t h e o r e t i c a l and e x p e r i -mental behaviours i s seen, and that the t r a n s i e n t response f o l l o w s a 55 £=0.5 ?=0.3 C=0.1 200ms/dlv 1. T=0.1 K=3 2. T=0.1 K=8 3. T=0.1 K=25 50ms/dlv 4. T=0.01 K=10 5. T=0.01 K=17 6. T=0.01 K=34 5 0 m s / d i v 7. T=0.005 K=13 8. T=0.005 K=24 9. T=0.005 K=48 F i g 5.4 Photographed Responses at the P o i n t s Indicated i n F i g 5.3 more p r e d i c t a b l e p a t t e r n as T i n c r e a s e s , o r oj n decreases. More s p e c i f -i c a l l y the t h e o r e t i c a l damping f a c t o r s f i n d c l o s e r agreement w i t h the a c t u a l response as c o n , or the system b andwidtti^ decreases. This would seem to suggest that the f a s t e r the response, i . e . the l a r g e r the band-width, the l e s s v a l i d l i n e a r continuous theory becomes, and hence the l e s s r e l i a b l e are p r e d i c t i o n s made by l i n e a r continuous techniques. I t should be emphasized t h a t the e r r o r a m p l i f i e r must pr o v i d e enough f i l t e r i n g of the discontinuous 6 feedback s i g n a l , s i n c e the time constants are e f f e c t i v e l y d i v i d e d by the c l o s e d loop g a i n . Thus f o r s m a l l time constants and l a r g e values of gain the system does not ex-h i b i t l e s s damping but imbalance o f the f i r i n g pulses due to inadequate f i l t e r i n g . An e r r o r a m p l i f i e r w i t h a s i n g l e pole d i d not e x h i b i t any correspondenca to the l i n e a r model: o s c i l l a t i o n o n l y at l a r g e values of T and unbalanced f i r i n g at lower values of T. Constant Current C o n t r o l ' ~ In a s i m i l a r manner the s t a b i l i t y regions f o r the constant current c o n t r o l l e r may be der i v e d f o r the c h a r a c t e r i s t i c equation (4.16), Fig 5 . 5 Stability Regions For Constant Current Control The constant damping curves f o r the hi g h e r order system are found to be somewhat confusing due to the f o l d - o v e r e f f e c t , i n d i c a t i n g the e x i s t e n c e of a number of complex r o o t s . In s e l e c t i n g values o f the v a r i a b l e parameters f o r best r e l a t i v e damping, the p a i r of roots w i t h the lower damping f a c t o r i s of primary i n t e r e s t : whatever the value of gain and time constant, the best t h a t can be achieved i s a l a r g e l y underdamped c o n t r o l , a f a c t which corresponded very c l o s e l y w i t h the p r a c t i c a l experience that c o n t r o l was extremely d i f f i c u l t to main-t a i n at a s t a b l e o perating p o i n t except at low values o f gain and slow response time. An a l t e r n a t i v e scheme f o r the a n a l y s i s of constant c u r r e n t i s shown i n F i g . 5.6 where the two a d j u s t a b l e parameters are the values o f gain i n the feedback loops and the value of 30 i s se t f o r the ga i n o f the e r r o r a m p l i f i e r . Fig 5.6 Alternate Scheme for Analysis The c h a r a c t e r i s t i c equation i n t h i s case i s 0 =0.05 + 0.26Ki + 0.53K2 + (0.037 + 0.097Ki)s + (0.0074Ki)s2 + 0.0005 s 3 + 3.6x10-4 s4 (5.6) The parameter plane curves are shown i n F i g . 5.7. S e t t i n g K£ = 0 and K]_ = 6.3 the p o i n t A i n F i g . 5.7 corresponds to the p o i n t K=30, T=0.1 i n F i g . 5.3,with a damping f a c t o r between 0 and 0.1. The i n t r o d u c t i o n of the a d d i t i o n a l feedback s i g n a l again produces an underdamped s y s t e m . u n t i l t h e v a l u e s o f gain are reduced. Any change i n must be cross-checked w i t h F i g . 5.3. F i g 5 .7 C h a r a c t e r i s t i c Curves For Eqn ( 5 . 6 ) 6. RECTIFIER CONTROL Under normal c o n d i t i o n s , the r e c t i f i e r operates on e i t h e r con-s t a n t current c o n t r o l or constant power c o n t r o l : . o v e r a l l c o n t r o l of the r e c t i f i e r o p e r ation i s maintained by f a s t e l e c t r o n i c c o n t r o l of f i r i n g angle a, responding to spontaneous f l u c t u a t i o n s w i t h i n the system, and the slower e l e c t r o m e c h a n i c a l on-load tap changer, which serves t o main-t a i n the f i r i n g angle w i t h i n p r e s c r i b e d o p e r a t i n g l i m i t s ^ " 4 > ^ . The upper l i m i t i s imposed to minimize r e a c t i v e power consumption at the r e c t i f i e r and the lower l i m i t of a provides a margin f o r a r a p i d i n c r e a s e i n power demand by a c o n t r o l only. This lower l i m i t i s u s u a l l y l a r g e r than another (lowest p o s s i b l e ) l i m i t (cx>o) which ensures the s i m u l t a n -eous f i r i n g and equal l o a d s h a r i n g of a number of anodes i n p a r a l l e l . I f , due to f l u c t u a t i o n s o f v o l t a g e at the sending end a.c. system, the f i r i n g angle i s brought to a l i m i t i n g v a l u e , then normal c o n t r o l a c t i o n . i s to b r i n g the tap changer i n t o o p e r a t i o n , a f t e r an a p p r o p r i a t e delay to accommodate t r a n s i e n t s . For example, i f the a.c. v o l t a g e f e l l to a new low v a l u e , then a i s taken past i t s lower l i m i t i n order to main-t a i n constant c u r r e n t or constant power. Tap changer a c t i o n to i n c r e a s e the secondary/primary turns r a t i o allows the e l e c t r o n i c c o n t r o l to b r i n g a back to i t s normal operating range. In the model, however, the tap changer was found to be unnec-essary and unusable at t h i s stage of development, and the turns r a t i o adjustment was accomplished manually using a three phase power v a r i a c at each converter s t a t i o n . The remainder of t h i s chapter presents the implementation of two r e c t i f i e r c o n t r o l s : constant - a and constant current . The l a s t s e c t i o n describes other forms of r e c t i f i e r c o n t r o l and suggests how they might be implemented. Again the DONNER Analog Computer was used f o r the c o n t r o l l e r . 6.1 Constant a c o n t r o l Advantageous use may be made here of the e x t i n c t i o n angle de-t e c t o r c i r c u i t described i n s e c t i o n 4.2. This same c i r c u i t may be used to detect the f i r i n g angle a and d e r i v e a feedback s i g n a l f o r the con-st a n t - a c o n t r o l shown i n F i g . 6.1. Reference E r r o r A p l i f i e r F i r i n g C i r c u i t power conver ter 3 s i g n a l diode r e c t i f i e r zener diode F i g 6.1 R e c t i f i e r Constant-o C o n t r o l The constant-a c o n t r o l l e r shown above d i f f e r s from the con-s t a n t e x t i n c t i o n angle c o n t r o l l e r shown i n F i g . 4.10 only i n the sense of the s i g n a l diode bridge r e c t i f i e r w i t h respect to the convert e r t e r -minals. The equations are s l i g h t l y d i f f e r e n t s i n c e the angle detected i s the f i r i n g angle i t s e l f and not some angle r e l a t e d to i t through the power c i r c u i t . From the bl o c k diagram i n F i g . 6.2 f i v e equations may be w r i t t e n : ref + \ A V e G^s) A V a -0.37 ; 1+0.008s K> Aa Al3 G 2(s) - K , Fig 6.2 ConstanC-a Block Diagram Three c o n t r o l equations: AVe = K]_ Aa (6.1) AVa = Gi(s) AVe (6.2) A a = 1 ^ 0 0 8 8 A V a ~ G 2 ( 5 ) A I d ( 6 ' 3 ) Two power equations Vdr " V d o r cosa - | x c I d (6.4) I d ( R d c + s L s m ) = V d v - V d i (6.5) 3 from which A V d r = - V d o r s i n a Aa - — X c A I d (6.6) assuming V d o r remains constant, and A I d ( R d c + s L s m ) = A V d r - A V d i (6.7) U n f o r t u n a t e l y does not remain constant, and s i n c e the r e c t i f i e r can-not be operated i n 'hvdc mode'.without a re g u l a t e d i n v e r t e r , these equations should be used i n con j u n c t i o n w i t h those d e s c r i b i n g the app-r o p r i a t e behaviour of the i n v e r t e r . In t h i s case, constant-a c o n t r o l a t the r e c t i f i e r i m p l i e s constant current mode at the i n v e r t e r . 6.2 Constant Current Control The constant c u r r e n t loop i s the primary r e g u l a t o r f o r the r e c t i f i e r . I t is,added to the constant-a c o n t r o l l e r w i t h a clamp to l i m i t the feedback s i g n a l when the cur r e n t f a l l s below the s e t t i n g I d s l * The c i r c u i t , shown i n F i g . 6.3, allows independent adjustment o f and I d s l * should be noted that a i s l i m i t e d at i t s lowest permis-s i b l e value and not at the two op e r a t i n g l i m i t s a t which the tap changer would operate. The block diagram i s shown i n F i g 6.4. i — i / W — l "Ref ds2 Ref Fig 6.3 Circuit for Rectifier Constant Current and Constant-a Controllers Equation (6.1) becomes AVe = K/jAa - K 2 A I d (6.8) 6.3 Operation o f the d.c. l i n k Operation of the d.c. l i n k i n two modes i s now p o s s i b l e : (1) R e c t i f i e r i n c.c. mode, i n v e r t e r i n c.e.a. mode. (2) R e c t i f i e r i n constant-a mode, i n v e r t e r i n c.c. mode. The o p e r a t i n g procedure i s to s t a r t up,the l i n k without regu-l a t i o n to a low current o p e r a t i n g p o i n t , then the c o n t r o l l e r s are a p p l i e d and the l i n k brought up to the r e q u i r e d o p e r a t i n g p o i n t . The o p e r a t i o n of the l i n k w i t h the c o n t r o l l e r s d e s c r i b e d i n t h i s t e x t was found to be s t a b l e but underdamped, thus r e q u i r i n g com-pensation. By usi n g s i n g l e p o l e e r r o r a m p l i f i e r s the l i n k a c q u i r e s con-s i d e r a b l e s t a b i l i t y , but l i n e a r continuous models do not apply. 6.4 Other Forms of Converter C o n t r o l Constant power c o n t r o l was e a s i l y implemented u s i n g the m u l t i -p l i e r a v a i l a b l e w i t h the DONNER analog computer. In a p p l i c a t i o n to the' model i t should be used i n conj u n c t i o n w i t h a constant c u r r e n t o v e r r i d e f o r overcurrent p r o t e c t i o n . 48 Another type o f c o n t r o l , suggested by Machida and Yoshida i s to c o n t r o l the frequency of a weak a.c. system by f a s t c o n t r o l of power i n a d.c. l i n k of comparable s i z e . Where r e a c t i v e power compensation i s accomplished by s w i t c h i n g c a p a c i t o r s , constant r e a c t i v e power c o n t r o l has been suggested by K a n n g e i s s e r ^ l , again f o r use w i t h weak a.c. systems. 65 F i g 6.4 Constant Current C o n t r o l l e r Block Diagram 7. DISCUSSION AND CONCLUSION This concludes the study. The p h y s i c a l model has been shown to s i m u late the stead y - s t a t e o p e r a t i o n of a d.c. l i n k when connected between two p o i n t s i n a st r o n g a.c. system. The c o n t r o l systems have been b u i l t u s i n g the d i f f e r e n t i a l a m p l i f i e r s o f an analog computer. Hence they can e a s i l y be changed to the mode of c o n t r o l r e q u i r e d by the experimenter, whatever the mode of c o n t r o l f o r the i n v e r t e r , i t must always be provided w i t h a constant/minimum e x t i n c t i o n angle o v e r r i d e f o r p r o t e c t i o n against commutation f a i l u r e . The novel e x t i n c t i o n angle d e t e c t o r , designed by the author, i s i d e a l l y s u i t e d to the model, and may without too much d i f f i c u l t y be a p p l i e d to a f u l l - s i z e d c o n v e r t e r . The only change from c.e.a. c o n t r o l f o r i n v e r s i o n to constant a f o r r e c t i f i -c a t i o n i s the change i n sense of the s i g n a l diode three-phase r e c t i f i e r connected across the terminals of the power.converter. Further a p p l i c a t i o n of the model to the study of generator -d.c. l i n k - i n f i n i t e bus, or p a r a l l e l a.c. - d.c. t r a n s m i s s i o n would r e q u i r e the a d d i t i o n of a.c. harmonic f i l t e r s . The p e r - u n i t r e a c t i v e power provided may then be v a r i e d by changing e i t h e r the components o f the f i l t e r or the r a t i n g of the l i n k . With respect to the mathematical d e s c r i p t i o n and i t s c o r r e s -pondence w i t h the p h y s i c a l model, the assumption t h a t the convert e r system i s l i n e a r and continuous i s v a l i d o n l y f o r s m a l l p e r t u r b a t i o n s and s m a l l system bandwidth. When both c o n d i t i o n s were f u l f i l l e d , then the a n a l y s i s u s i n g S i l j a k ' s method was found to be q u i t e a c c u r a t e . However i n the a n a l y s i s o f f a u l t response and t r a n s i e n t response, both assumptions must be d i s c a r d e d , as p e r t u r b a t i o n s are f a r from s m a l l and e f f e c t i v e c o n t r o l response must occur w i t h i n the p e r i o d of a few succes-s i v e f i r i n g s - ^ . A l s o , the contemporary trend i s to apply d.c. l i n k s i n s i t u a t i o n s where t h e i r high bandwidth c a p a b i l i t i e s are f u l l y r e a l i z e d , eg. p a r a l l e l a.c. - d.c. t r a n s m i s s i o n , weak a.c. systems. The a n a l y s i s has shown that the l i n e a r continuous assumption i s u n s u i t a b l e f o r such a p p l i c a t i o n s . F i g s . 5.3 and 5.4 i n d i c a t e that the damping curves r e f l e c t the system behaviour at slow frequency of response, becoming more i n a c -curate as the frequency i n c r e a s e s . In the a c t u a l p r a c t i c e of a p p l i c a t i o n , the S i l j a k method of s t a b i l i t y a n a l y s i s appears as r e s t r i c t e d as one-parameter methods f o r the r e l a t i v e s t a b i l i t y o p t i m i z a t i o n of a number of parameters i n a con-t r o l system. Since a l l parameters must be simultaneously a d j u s t e d , the two dimensional g r a p h i c a l technique becomes l a b o r i o u s when three or more parameters are r e q u i r e d to be adjusted. With respect to the f u r t h e r development of the model, the p h y s i c a l l i m i t a t i o n to the achievement of a f a s t s t a b l e response i s the f i r i n g c i r c u i t , which, i n order to m a i n t a i n balanced f i r i n g p u l s e s , r e q u i r e s the feedback s i g n a l to be low-pass f i l t e r e d , i n e v i t a b l y s l o w i n g response time. This 'equal-angle' f i r i n g c i r c u i t i s a l s o unbalanced by asym-m e t r i c a l phase c o n d i t i o n s , a f f e c t e d by commutation d i s t o r t i o n of phase voltages and s e n s i t i v e to changes i n supply v o l t a g e . Under s i n g l e -phase a.c. f a u l t c o n d i t i o n s , c o n t r o l i s sometimes maintained by asym-c m e t r i c a l f i r i n g o f the v a l v e s , not p o s s i b l e w i t h the present f i r i n g c i r c u i t s . Also the r a p i d r e v e r s a l of power flow i s not p o s s i b l e due to the r e s t r i c t e d range of f i r i n g p u l s e v a r i a t i o n . Since the f l e x i b i l -68 i t y of the d . c . l i n k i s the f l e x i b i l i t y of i t s e l e c t r o n i c c o n t r o l s y s -tem, the f i r i n g c i r c u i t , as an e s s e n t i a l par t of the c o n t r o l sys tem, i s seen as a major l i m i t a t i o n of the p h y s i c a l model. 69 REFERENCES 1. E.S. G r o i s s , A. V. Posse, V.E. T o u r e t s k i , "800kV D.C. Transmission System Stalingrad-Donbass", CIGRE Paper 414, 1960. 2. G.D. Brewer, E.M. Hunter, P.G. Engstrom, R.F. Stevens, " C e l i l o Converter S t a t i o n of the P a c i f i c h.v.d.c. I n t e r t i e " , IEEE Transac-t i o n s on Power Apparatus and Systems, V o l PAS-85, No 11, Nov 1966, pp 1116-1128. 3. K.L. Hurdle, C. MacGregor, T.E. Storey, "The S e l e c t i o n of ±450kV d.c. f o r the Transmission of Power from the Nelson R i v e r " , Engineer-i n g J o u r n a l Vol 50, No 10, Oct 1967, pp 17-20. 4. T.E. C a l v e r l e y , "Development i n D.C. Transmission", Transactions of the Sourth A f r i c a n I n s t i t u t e of E l e c t r i c a l Engineers, V o l 59, P a r t 7, J u l y 1967, pp 133-163. 5. P.J. C r o f t , H.M. E l l i s , "D.C. Transmission to Boost Vancouver I s l a n d Supply", E l e c t r i c a l World, May 31, 1965, pp 44-47. 6. B.J. Cory, "High Voltage D i r e c t Current Converters and Systems", MacDonald 1965, Chap. 8, pp 175-218. 7. I b i d , Chap 5, p 91. 8. I b i d , Chap 7, p 145. 9. I b i d , Chap 8, pp 165-173. 10. I b i d , Chap 4, pp 44-72. -11. I b i d , Chap 9, p 236. 12. S. Fukunda, I . Takei, "The Sukuma D.C. Frequency Converter P r o j e c t " , D i r e c t Current, V o l 7, 1964. 13. J.D. Ainsworth, "The Phase-locked 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 Converters", IEEE Transactions on Power Apparatus and Systems, V o l PAS-87, No 3, March 1968, pp 859-865. 14. A. Ekstrom, G. L i s s , "A Refined HVDC C o n t r o l System", IEEE Trans, on Power Apparatus and Systems, V o l PAS-89, May 1970, pp 723-732. 15. E. Uhlmann, " S t a b i l i s a t i o n of an AC L i n k by a P a r a l l e l DC L i n k " , D i r e c t Current, Aug 1964, pp 89-94. 16. J . J . Dougherty, "Operating C h a r a c t e r i s t i c s o f a Three-Terminal DC Transmission L i n e " , IEEE Summer Power Meeting, D a l l a s , Texas, June 22-27, 1969. 17. J.P. Bowles, B.J. Cory, "The Model hvdc Transmission System at I m p e r i a l C o l l e g e , London", D i r e c t Current, Sept 1963, pp 236-240. 70 18. J.D. Ainsworth, J.P. Bowles, "An HVDC Simulator S u i t a b l e f o r both Steady-State and Transient S t u d i e s " , IEE Proc. Conference No 22 on HVDC Transmission, 1966, Paper 25. 19. J . J . Dougherty, V. Caleca, "The EEI AC/DC Transmission Model", IEEE Trans, on Power Apparatus and Systems, V o l PAS-87, No 2, Feb 1968, pp 504-512. 20. G. Gordon, "System S i m u l a t i o n " , P r e n t i c e H a l l , 1969. 21. G.J. Thaler, R.G. Brown, " A n a l y s i s and Design of Feedback C o n t r o l Systems", McGraw-Hill 1960, Chap 4, p 105. 22. K. E r i k s s o n , G. L i s s , " S t a b i l i t y A n a l y s i s of the HVDC Transmission C o n t r o l System Using T h e o r e t i c a l l y C a l c u l a t e d Nyquist Diagrams", IEEE Trans, on Power Apparatus and Systems, V o l PAS-89, No 5,. May/ June 1970, pp 733-739. 23. J . Clade, A. Lacoste, " S i m p l i f i e d Study of The S t a b i l i t y o f Regula-t i o n of a D.C. L i n k " , D i r e c t Current, Feb 1967, p 9-22. 24. D.D. S i l j a k , " A n a l y s i s and Synthesis of Feedback C o n t r o l Systems i n the Parameter Plane, 1 - L i n e a r Continuous Systems", IEEE Trans. ( A p p l i c a t i o n s and I n d u s t r y ) , V o l 83, Nov 1964, p 449-458. 25. D.D. S i l j a k , " G e n e r a l i s a t i o n of the Parameter Plane Method", IEEE Trans, on Automatic C o n t r o l , V o l AC-11, No 1, Jan 1966, pp 63-70. 26. D.D. S i l j a k , "Non-Linear Systems", John Wiley & Sons, 1969. 27. I b i d Appendix B, pp 411-2. 28. N.N. Hancock, " E l e c t r i c Power U t i l i s a i o n " , Pitman 1967, pp 320-339. 29. J . Sc h a e f f e r , " R e c t i f i e r C i r c u i t s , John Wiley & Sons, 1968. 30. A.J. Wood, S.B. Gray, C. Concordia, "An Economic Study of D.C. versus A.C. Overhead Transmission", AIEE Trans, on Power, V o l 78, No 6, June 1959, pp 338-349. 31. L.R. Neiman and O t h e r s ( E d i t o r s ) , "D-C Transmission i n Power Systems", I s r a e l Program f o r S c i e n t i f i c T r a n s l a t i o n s , 1967, p 59. 32. C. Adamson, N.G. H i n g o r a n i , "High Voltage D i r e c t Current Power Trans-m i s s i o n " , Garraway, 1960, pp 159-166. 33. I b i d , Chap 10, p 143. 34. J.D. Ainsworth, "Harmonic I n s t a b i l i t y Between C o n t r o l l e d S t a t i c Converters and A.C. Networks", Proc IEE, V ol 114, No 7, J u l y 1967, pp 949-957. 35. J.C. Read, "The C a l c u l a t i o n of R e c t i f i e r and I n v e r t e r Performance C h a r a c t e r i s t i c s " , Proc IEE, V o l 92-11, pp 495-509. 71 36. D.E. F l e t c h e r , CD. C l a r k e , "A.C. F i l t e r s f o r the Nelson R i v e r Pro-j e c t " , "Manitoba Power Conference on EHV-DC, June 1971, Winnipeg, Manitoba. 37. J.P. Bowles, " C o n t r o l Systems f o r High Voltage D i r e c t Current Trans-m i s s i o n " , Spring Meeting, CEA, Vancouver 1971. 38. G.L. Brewer, CD. C l a r k e , A. G a v r i l o v i c , "Design Considerations f o r A.C F i l t e r s " , IEE Proc. on HVDC Transmission, 1966, Paper 56. 39. E. F r i e d l a n d e r , "Features of S t a t i c Reactive Power Supply f o r P o l y -phase Converters", Proc. IEE Conference on HVDC Transmission, 1966, Paper 54. 40. H.A. Calderbank, J.S. C o l l s , "Compensation f o r D.C. Transmission w i t h P a r t i c u l a r Reference to Synchronous Converters", Proc. IEE Conference on HVDC Transmission, 1966, Paper 53. 41. F. I l e c e t o , "D.C. l i n k and A.C. Network Combined Power Requirements", D i r e c t Current, May 1964, pp 147-155. 42. C. Adamson, N.G. H i n g o r a n i , " C o n t r o l of h.v.d.c. Converters", D i r e c t Current, June 1962, pp 147-155. 43. General E l e c t r i c Company SCR Manual, 1967 E d i t i o n . 44. I b i d , p 304. 45. F i r i n g C i r c u i t s I n c., "SCR C o n t r o l s " , B u l l e t i n 5001, Nov 1967. 46. Three-Phase F u l l Wave SCR F i r i n g C i r c u i t A p p l i c a t i o n Drawing 1241- . 3003C, F i r i n g C i r c u i t s I n c . , Connecticut. 47. N.G. H i n g o r a n i , P. Chadwick, "A New Constant E x t i n c t i o n Angle C o n t r o l f o r S t a t i c AC/DC/AC Converters", IEEE Trans, on Power Apparatus And Systems, V o l PAS-87, No 3, March 1968, pp 866-872. 48. T. Machida, Y. Yoshida, "A Method to Detect The D e - i o n i s a t i o n Margin Angle and to Prevent the Commutation F a i l u r e of an I n v e r t e r f o r D.C. Transmission", IEEE Trans, on Power Apparatus and Systems, V o l PAS-86, No 3, March 1967, pp 259-262. 49. J . A r r i l l a g a , G. Galanos, E.T. Powner, " D i r e c t D i g i t a l C o n t r o l of HVDC Converters", IEEE Trans, on Power Apparatus and Systems, V o l PAS-89, No 8, Nov/Dec 1970, pp 2056-2065. 50. L.A. Bateman, R.W. Haywood, R.F. Brooks, "Nelson R i v e r D.C. Trans-m i s s i o n P r o j e c t " , IEEE-EHV Transmission Conference Proceedings, 1968, Paper 18. 51. K.W. Kanngi-sser, H.D. L i p s , " C o n t r o l Methods to Improve the Reactive Power C h a r a c t e r i s t i c s of HVDC L i n k s " , IEEE Transactions on Power Apparatus and Systems, V o l PAS-89, No 6, J u l y 19 70, pp 1120-5 52. B.J. K a b r i e l , p e r s o n a l conversations APPENDIX A (a) Computation of C h a r a c t e r i s t i c Equation While the a lgebra ic d e s c r i p t i o n of the method i s lengthy and cumbersome, i t s s i m p l i c i t y i n p r i n c i p l e i s bes t i l l u s t r a t e d by example: consider the fo l lowing 3x3 matrix M = s z + K x s + 3 0 s + 1 s + 2 s + K 2 0 0 s + K i s + 4 where Kj_ and K 2 are a lgebra ic v a r i a b l e s . Expanding along the t h i r d row A = (s+l)(s+2) (s+Ki) + (s+4) (s+K 3) (s3+K l S+3) = s 4 + (5+Ki+K 2 ) S 3 + (6+5Kl+4K 2 +K 1 K 2 )s 2 + ( 1 4+3^+3^ + 4^^)3 + 2Ki + 12K 2 (4.1) The f i r s t step i n the computer program i s to compute the de t -erminant of numerical terms only i . e . determinant of M above wi th coef -f i c i e n t s of the a lgebra i c v a r i a b l e s omitted s 2 + 3 s + 2 0 A = 0 s s s + 1 0 s + 4 s(s+l)(s+2) + s(s 2+3)(s+4) s 4 + 5 s J + 6sz + 14s It i s noted that two a l g e b r a i c v a r i a b l e s occur i n three p o s i -t ions K i ( l , l ) , K i ( 2 , 3 ) , K 2 ( 2 , 2 ) . For each v a r i a b l e p o s i t i o n i n turn e l iminate a l l terms i n that row and column except the v a r i a b l e and i t s c o e f f i c i e n t . For KT i n p o s i t i o n (1,1) the mo d i f i e d determinant becomes 0 0 0 1 s s 1 0 0 s+4 K x ( s 3 + 4 s 2 ) For K i i n p o s i t i o n (2,3) s 2+3 s+1 s+2 0 l 0-0 <5> = K ^ s ^ + 3s + 2) For K 2 i n p o s i t i o n (2,2) s 2+3 0 1 0 % 2 = 0 — 0 = K 2 ( s 3 + 4 s 2 + 3s + 12) s+1 i 0 s+4 Next where 2 a l g e b r a i c v a r i a b l e s occur i n 2 p o s i t i o n s , each w i t h a d i f -f e r e n t row and column, the same r u l e a p p l i e s f o r each p o s i t i o n : For Ki i n M ( l , l ) and K 2 i n M(2,2) J K T K 2 K l s ) — p. -0 -0 6 s+4 K = K - j K ^ s ^ s ) For Ki i n M ( l , l ) and i n M(2,3) [ K L S ) 0 0 F j r 2 = K l K l 0 = 0 The p o s s i b i l i t i e s having been exhausted the determinant of the m a t r i x M i s A = A + BK]_ + C K 1 + D K 2 + E K L K 2 + F K l 2 = s 4 + 5 s 3 + 6 s 2 + 14s + Ki_s 3 + 5 K l s 2 + 3K]_s + 2K X + K 2 s 3 + 4 K 2 s 2 + 3K 2s + 12K2 + K ^ s 2 + 4K;jK 2s = s 4 + s 3 ( 5 + K x + K 2) + s 2 ( 6 + 5 K i + 4K 2 + K i K 2 ) + s(14 + 3Ki + 3K 2 + 4K]K 2) + 2Ki_ + 12K 2 which i s equal to equation (4.1). This p r i n c i p l e may be a p p l i e d t o matrices w i t h more than two a l g e b r a i c v a r i a b l e s and may be extended to the computation of completely a l g e b r a i c determinants. (b) Determinant program The program w r i t t e n i n FORTRAN H r e s i d e s i n the f i l e TJEFG: CHAREQN and i s invoked w i t h the f o l l o w i n g command: $RUN UEFG:CHAREQNH 5 = f i l e 1 6 = f i l e 2 7 = f i l e 3 where f i l e 1 i s the inp u t f i l e c o n t a i n i n g data i n the format (G20.7, 415) one data card o r l i n e i s r e q u i r e d f o r each term i n each m a t r i x element: eg. the term s 2 + 3Kls + 2 i n p o s i t i o n (1,1) would r e q u i r e 3 cards: where X i s the c o e f f i c i e n t ( I , J ) are the row and column of the term K i s the index which represents the power of s of which X i s the c o e f f i c i e n t s° « 1 s 1 = 2 s 2 = 3 e t c . L i s the number of the a l g e b r a i c v a r i a b l e a l l of which are denoted Kj_,. F i l e 2 i s the f i l e or device f o r the p o i n t out of the s o l u t i o n , the inpu t data i s echo p r i n t e d on u n i t 6. 3 i s the f i l e i n t o which data i s w r i t t e n f o r r eading by gram which computes the S i l j a k curves. the example given i n p a r t A, the inp u t data appears as 1 I . 1 1 2 1 2 .. .. 3 . 1 1 i 3 * • 1 1 3 u I 2 1 b - • 1 • 1 2 2 6 • • ? 2 2 7 d 2 1 2 8 2 3 2 9 2 3 1 1 10 • 3 1 1 11 X % J V 2 12 *• • 3 3 2 13 4 . 3 3 1 And the output, equal to (A.l ) and (A.2). CONSTANT TERM(S) 1 COEFFICIENTS OF S K l : K 2 : K'l: K2: K1 *K2: 2 , 0 0 0 0 0 0 0 0 0 1 2 , 0 0 0 0 0 0 0 0 1 4 , 0 0 0 0 0 0 0 0 3 , 0 0 0 0 0 0 0 0 0 - 3 , 0 0 0 0 0 0 0 0 0 4 , 0 0 0 0 0 0 0 0 0 d COEFFICIENTS OF S 3 COEFFICIENTS OF S COEFFICIENTS OF S 6,00000 0 000 K l : 5,000000000 K.a? 4,000000000 K1*K2: 1,000000000 5,000000000 Ki: l . o o o o o o o o o K ? : 1,000000000 1,000000000 APPENDIX B: COMPUTATION OF SILJAK CURVES Consider the c h a r a c t e r i s t i c equation f ( s ) = Z a v s K = o (B'.l) k=o K u s i n g the usual n o t a t i o n ; i f s i s expressed as s = -u) nC + j u n / 1 - 5 Z (B.2) where c u n i s the undamped n a t u r a l frequency, then i t has been shown that powers of s may be expressed c o n c i s e l y as s k - ( % k { ( - D k T k ( ? ) + j / 1 - ? z ( - l ) k + 1 Uk(C)} (B.3) where T^( C) and U k ( 5 ) may be obtained by the recurrence formulae T k + 1 ( C) -2 c T^?) + T k_ 1 ( ?) = 0 ( B. 4) U ^ (O - 2 ^ 0 + Vk_±(0 = 0 w i t h T 0(£) = 1, T!(C) =C, U 0 ( ? ) = 0 and U x(5) = 1 S u b s t i t u t i o n of equation (B.3) i n t o ( B . l ) and s e p a r a t i o n of r e a l and imaginaries which must both independently go to zero enables equation ( B . l ) to be r e w r i t t e n : ? ( - l ) k a k a ) n k T k ( 0 = 0 k=o (B.5) , k + l „ ... k Z ( - l ) K + i a k o ) n k U k(C)> / 1 - 5 = 0 Now, the c o e f f i c i e n t s a k of the c h a r a c t e r i s t i c equation 1 appear as l i n e a r f u n c t i o n s of v a r i a b l e system parameters a and g. a k = b k a + ck& + d k (B .6) 78 where e q u a t i o n s D.5 a r e t h e n a b l e to.be r e w r i t t e n a B i ( u n , 0 + eCi(oj n,C) + Dif c o n . O = 0 (B.7) aB 2(wn,C) + 8C2(a>n,C) + D^u^.C) = 0 B! - ? ( - l ) k b k W n k T k ( C ) B 2 = Z ( - l ) k + 1 b k t o n k / 1 u k ( ? ) k=o k=o n vk k„ n Cj_ - I ( - l ) k c^/TkCO C 2 = 2 ( - l ) k + 1 c k u n k / 1 U ka) K-O k=0 D l = J Q ( - l ) k d k n k T k ( ? ) ^ = z ( _ 1 ) k + l ^ ^ — ^ v o o (B.8) E q u a t i o n s B.6 may be s o l v e d f o r unknowns a and B. a = i l cl D2 _ C 2 D l 3 (B.9) 8 = i l B ^ - Bi D 2] where A = Bj_C2 ~ B 2 C 1 F o r t h e c h a r a c t e r i s t i c e q u a t i o n 4.13, t h e mapping f u n c t i o n s a = a(o) n?) , B= 8(w n?) a r e d e r i v e d a n a l y t i c a l l y BT = 0.547 T 0 ( ? ) - 0.203 u> nTl(?) B 2 =-0.547 U i ( ? ) + 0.203 oj nUi(?) Ci =-0.577 w nTi(C) + 0.363 u ) n 2 T 2 ( ? ) - 0.047 a>n3T 3(rJ+ 0.00036aj n 4T 4(?) C 2 = 0.577 ovules) " 0.363 u) n 2U 2(c) + 0.047 t U n 3 U 3 ( ^ ) - 0.00036aj n 4U 4(?) D x = 0.577 uJ nT 0 (rJ - 0.363 ^nT±(0 + 0.047 cu n 2T 2(<;)- 0.00036^313(0 D 2 = 0.577 w nU 0(?) + 0.363 u^U^e)'- 0.047 w n 2U 2(r,)+ 0.00036u) n 3U 3U) 

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