Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Design of a dual-excited synchronous machine and development of stabilization techniques Dick, Eugene Peter 1973

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1973_A7 D52_5.pdf [ 4.25MB ]
Metadata
JSON: 831-1.0101428.json
JSON-LD: 831-1.0101428-ld.json
RDF/XML (Pretty): 831-1.0101428-rdf.xml
RDF/JSON: 831-1.0101428-rdf.json
Turtle: 831-1.0101428-turtle.txt
N-Triples: 831-1.0101428-rdf-ntriples.txt
Original Record: 831-1.0101428-source.json
Full Text
831-1.0101428-fulltext.txt
Citation
831-1.0101428.ris

Full Text

DESIGN OF A DUAL-EXCITED SYNCHRONOUS MACHINE AND DEVELOPMENT OF STABILIZATION TECHNIQUES by Eugene Peter Dick B.A.Sc, Un i v e r s i t y of Waterloo, 1971 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 Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH August 1973 COLUMBIA In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a llowed w ithout my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8 , Canada ABSTRACT A dual-excited synchronous machine i s designed and constructed. A power system i s simulated using the machine and a medium length trans-mission l i n e joined to the laboratory bus energized from B.C. Hydro which i s considered the i n f i n i t e bus. The e f f e c t of supplemental e x c i -t a t i o n c o n t r o l on power system s t a b i l i t y i s inv e s t i g a t e d . Feedback c o n t r o l i s used i n both the d i r e c t and quadrature f i e l d s . Torque angle, speed, power and terminal current s i g n a l s are compared i n t h e i r damping action. The response i s also c a l c u l a t e d from a ninth order mathematical model. I t i s found that the shaft speed i s the best supplemental s i g n a l and that the d i r e c t and quadrature f i e l d s are equally e f f e c t i v e i n dynamically c o n t r o l l i n g the machine when operating near rated output. TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v LIST OF ILLUSTRATIONS v i ACKNOWLEDGEMENT v i i NOMENCLATURE . . . . • • • v i i i 1. INTRODUCTION . . . 1 2. DUAL FIELD SYNCHRONOUS MACHINE DESIGN AND PARAMETER TESTS . . 3 2.1 Tamper Set 3 2.2 Pole and Yoke Design of the Synchronous Machine . . . . 3 2.3 Field Windings 6 2.4 Pole Shoe 9 2.5 Parameter Tests of the Machine Set 10 3. INSTRUMENTATION AND CONTROL PANELS 13 3.1 Torque Angle Transducer 13 3.2 Speed Transducer 15 3.3 Terminal Voltage, Current and Power Transducers . . . . 19 3.4 Control Signal Compensation 22 3.5 Exciter and Field Time Constant Modification 23 3.6 One-line Diagram of the Model Power System 24 4. EXCITATION CONTROL SCHEMES FOR ENHANCING SYSTEM DAMPING . . . 26 4.1 The System Under Study 26 4.2 Phase Plane Analysis 28 4.3 Torque Angle Feedback 31 4.4 Speed Feedback 34 4.5 Power Feedback 36 4.6 Terminal Current Feedback 37 4.7 Comparison of Schemes 39 i i i Page 5. BASIC EQUATIONS OF THE DUAL-EXCITED SYNCHRONOUS MACHINE . . . 42 5.1 Voltage and Flux Linkage Equations 42 5.2 Torque Equations of the Machine 45 5.3 Transmission Line Equations 46 5.4 Open Loop System Equations 47 5.5 C a l c u l a t i o n of Operating Point . 50 5.6 Voltage Regulator Equations 51 5.7 Supplemental Feedback Schemes for Damping . 52 5.8 Closed Loop System Equations 53 6. -CONCLUSIONS 56 REFERENCES 58 i v LIST OF TABLES Table Page 2.1 Ratings of Rotor Parameters of the Induction Machine . . 3 2.2 Machine Test Parameters 12 .2.3 Terminal Voltage Harmonics 12 4.1 Operating Conditions 28 4.2 Voltage Regulator E f f e c t on Mechanical Mode 28 4.3 Gain and Phase S h i f t of Transfer Functions 31 4.4 Mechanical Mode of Torque Angle Feedback 32 4.5 Mechanical Mode of Speed Signal Feedback 34 4.6 Mechanical Mode of Power Feedback 36 4.7 Mechanical Mode of Terminal Current Feedback 37 4.8 Number of Transient Swings Before Damping 39 4.9 D and Q F i e l d Dynamic Torque Control 41 4.10 Terminal Voltage Swing per Radian Shaft Swing 41 5.1 Per Unit Base Quantities of the Machine 44 5.2 Per Unit Machine Parameters 44 5.3 Compensation Scheme Parameter Values 53 v LIST OF ILLUSTRATIONS Figure Page 2.1 Dimensions of Magnetic C i r c u i t 5 2.2 Pole Piece f o r Dual F i e l d Machine 8 2.3 F i e l d and Damper Windings 8 2.4 Dual F i e l d Machine Before Assembly 11 2.5 Laboratory Setup and Author 11 3.1 Torque Angle Transducer Transfer C h a r a c t e r i s t i c . . . . 14 3.2 Block Diagram of Torque Angle Transducer 15 3.3 Wiring Schematic of Torque Angle Transducer 16 3.4 Speed Transducer Transfer C h a r a c t e r i s t i c 17 3.5 Block Diagram of Speed Transducer . . . . . 18 3.6 Wiring Schematic of Speed Transducer 20 3.7 Voltage Transducer Transfer C h a r a c t e r i s t i c 21 3.8 Current Transducer Transfer C h a r a c t e r i s t i c 21 3.9 Power Transducer Transfer C h a r a c t e r i s t i c 22 3.10 Time Constant M o d i f i c a t i o n C i r c u i t 23 3.11 Power System One Line Diagram 25 4.1 Block Diagram of the Voltage Regulator 27 4.2 Block Diagram of the Phase Plane Model 29 4.3 Torque Angle Signal Compensation 33 4.4 Speed Signal Compensation • 35 4.5 Power and Terminal Current Compensation 38 4.6 Comparison of Feedback Source and Axis 40 5.1 Transmission Line Sketch and Phasor Diagram 46 v i ACKNOWLEDGEMENT I am g r a t e f u l to a l l the people who as s i s t e d i n the completion of t h i s t h e s i s . My supervisor, Dr. Y.N. Yu, p a t i e n t l y worked through the experimental problems as they developed and provided invaluable d i r e c t i o n during the w r i t i n g period. I also wish to thank Dr. H. Cheann and B. Habibullah for t h e i r h e l p f u l comments while reviewing the d r a f t . The machining and f a b r i c a t i o n of the generator were done by Derrik Daines and h i s exc e l l e n t workmanship resulted i n a flawless operating record. My appreciation to Mr. H.T. Walters for ordering the e l e c t r o n i c components and A l MacKenzie for l o c a t i n g instruments and mis-cellaneous parts. The photographs were done by Mr. H.H. Black and the typing by Norma Duggan. F i n a l l y , I wish to thank fellow graduate students and other f a c u l t y members for the encouragement and experience shared during the l a s t two years. v i i NOMENCLATURE Note: Upper case symbols i n d i c a t e MKS u n i t s ; lower case symbols i n d i c a t e per unit q u a n t i t i e s . General P d e r i v a t i v e operator, d/dt d sub s c r i p t denoting i n i t i a l condition A p r e f i x denoting a l i n e a r i z e d v a r i a b l e coo synchronous speed, 377 rad/s Synchronous Machine b,g,r,x transmission l i n e parameters D,d n a t u r a l damping I , , i , , I , i D and Q axis armature current d d q q ^ T j T j j i ^ j j T j r > i i r D a n d Q axis f i e l d current fd f d fq fq 1 ^ , i magnitude of armature current I C T , I , base f i e l d and armature current rb nb J rotor i n e r t i a P, P e l e c t r i c a l power e Q reactive power R , r armature resistance a a R,j,r^,,R-* , r . D and Q axis f i e l d resistance fd f d fq fq T , T , t e l e c t r i c a l and mechanical torque e' m m T, base torque b T ' , T ' D and Q axis open c i r c u i t transient time constant do qo U,, u,, U . u D and Q axis generated voltage d d q q V,, v., V , v D and Q axis terminal voltage d' d q q v i i i V,. ,,v,.,.V.. ,v e D and Q axis f i e l d voltage fd f d fq fq V , v magnitude of terminal voltage V , v i n f i n i t e bus voltage o o V „ , V , base f i e l d and armature voltage fb nb X ,,,X, 3,x , ,X - ,X£ ,x D and Q axis mutual reactance afd' fad md' a f q ' faq' mq X,, x,. X , x D and Q axis synchronous reactance d d q q X,- ,x ,X ' ,x 1 D and Q axis transient reactance d . d q q X^oX.j.X,, ,x r D and Q axis f i e l d reactance f r d rd rrq rq Z~ , Z , base f i e l d and armature impedance fb nb <5 torque angle 6 power factor f,. iK, '¥ , <i> D and Q axis f l u x linkage d' r d q q ^ f d ' ^ f q ^ a n d ^ a x l s f i e l d f l u x linkage u shaft speed Control System and E x c i t e r i , i , phase A,B l i n e current a b i f i e l d current K e x c i t e r gain Kp^ voltage regulator and supplemental control gain K^ , transducer gain L ^ , L ^ ' conventional and modified f i e l d inductance L search c o i l inductance s M f i e l d to search c o i l mutual inductance R , R ' conventional and modified f i e l d resistance i x R f i e l d current sampling r e s i s t o r x T j 1 modified f i e l d time constant T, ...T,. regulator time constants 1 6 u supplemental input s i g n a l v , v, l i n e terminal voltage ac be v^ e x c i t e r input voltage V__„ voltage regulator reference x 1. INTRODUCTION There has been widespread i n t e r e s t i n the use of dual f i e l d synchronous machines having two independent windings on the r o t o r . In 1962, Hamdi-Sepen proposed that a second winding be temporarily energized during severe disturbances giving improved transient s t a b i l i t y [1], Two years l a t e r , Botvinnik described how such a machine could be operated at d i f f e r e n t s l i p speeds, e l i m i n a t i n g the problem of synchronous s t a -b i l i t y [2]. Between 1968 and 19 70 a large number of papers examined the t h e o r e t i c a l advantages of dual axis machines. Takata confirmed that a two-axis machine i s more e f f e c t i v e i n c o n t r o l l i n g bus voltage f l u c t u a t i o n • than a s i n g l e - a x i s machine [3]. In a second paper he applied the maximum p r i n c i p l e to derive a bang-bang optimal co n t r o l of dual f i e l d voltage to e f f e c t i v e l y suppress machine hunting [4]. Soper and Fagg showed that torque and r e a c t i v e requirements can be separately c o n t r o l l e d , that machine s t a b i l i t y i s not reduced at leading power fa c t o r and that the speed of response to transients i s twice that of a conventional machine [5]. Robinson extended t h i s a nalysis to prove that the dual f i e l d machine has no transient s a l i e n c y r e s u l t i n g i n good t r a n s i e n t s t a b i l i t y performance [6]. Krause and Towle compared the damping c h a r a c t e r i s t i c s of a t w o - f i e l d machine to a o n e - f i e l d machine at varying torque angles and emphasized the use of proper c o n t r o l l e r compensation [7]. More re c e n t l y , the emphasis i n Canada has s h i f t e d to control design f o r the dual f i e l d generator. Subramaniam and Malik at Calgary used dynamic optimization techniques to determine the optimum control f o r a seventh order model [8]. E l - S e r a f i and Badr i n Saskatoon have 1 2 chosen e x c i t a t i o n control parameters which enable a machine to develop maximum poss i b l e capacitive power [9]. This thesis compares the r e l a t i v e effectiveness of torque angle, speed, e l e c t r i c a l power and terminal current feedback i n dynami-c a l l y damping machine o s c i l l a t i o n s . For each s i g n a l the proper compen-s a t i o n i s designed and the output i s applied to the d i r e c t and quadrature f i e l d windings i n turn. There i s always a voltage regulator used on the d i r e c t f i e l d . The quadrature f i e l d e x c i t a t i o n i s zero i n steady state operation. Other studies have demonstrated the superior charac-t e r i s t i c s of the quadrature winding when the machine i s operating at low power and high ca p a c i t i v e load. Here other differences between the windings are i n v e s t i g a t e d by operating at high power and small r e a c t i v e load.• The study i n this thesis i s conducted on both a laboratory micromachine and a mathematical model of the nin t h order. The micromachine i s a modified wound rotor induction motor with the wound rotor acting as the synchronous machine armature. The two f i e l d windings are located on the s t a t o r and t h e i r design i s described i n Chapter 2. The design of torque angle and speed sensing c i r c u i t s are discussed i n Chapter 3. 3 2. DUAL FIELD SYNCHRONOUS MACHINE DESIGN AND PARAMETER TESTS 2.1 Tamper Set The d u a l - f i e l d synchronous machine i s adapted from the induc-t i o n machine of the Tamper Set used for undergraduate l a b o r a t o r i e s . The dc machine of the set i s used to simulate a constant torque turbine and the synchronous machine to provide torque angle and frequency s i g n a l s . The induction motor i s modified into a synchronous generator by designing a new s t a t o r with f i e l d s on both d i r e c t (D) and quadrature (Q) axes but using the e x i s t i n g wound rotor f o r the armature. This i n -verted arrangement has the advantage that the r o t a t i n g member can be read-i l y used without any change and more winding space i s a v a i l a b l e for the s t a t o r f i e l d s . Table 2.1 l i s t s the ratings and rotor parameters of the i n -duction motor as supplied by the manufacturer. Table 2.1 Ratings of Rotor Parameters of the Induction Machine Horsepower 3.0 Poles 4 Line voltage 125 V Current 11.0 A Resistance/phase at 75°C .18°. 2.2 Pole and Yoke Design of the Synchronous Machine S a l i e n t poles are chosen for the design as d i s t r i b u t e d wind-ings are too complex to be constructed with u n i v e r s i t y f a c i l i t i e s . An Diameter 13.89 cm Length 9.52 cm A i r gap .038 cm Pole f l u x 536 K.G. 4 optimum r a t i o of i r o n to copper may e x i s t but i t i s very d i f f i c u l t to ca l c u l a t e due to the n o n - l i n e a r i t y of saturation i n i r o n . Good r e s u l t s are obtained by l i m i t i n g the maximum f l u x density i n i r o n to 15 K i l o -gauss (K.G.). With rotor s l o t s covering about 50 per cent of the tooth p i t c h , t h i s c r i t e r i o n i s met by choosing the a i r gap f l u x density to be seven K.G. Thus the f l u x per pole i s approximately determined by the p h y s i c a l dimensions of the rotor as follows: t>_ = - B = 231 K i l o - l i n e (2.1) P TT g 4 where B i s the maximum f l u x density i n the a i r gap, D i s the rotor diameter, and I the lamination length. Terminal voltage, which i s proportional to f l u x per pole, can now be ca l c u l a t e d from Table 2.1 and Equation 2.1. V = - 2 - 3 - i - = 31.1 V (2.2) phase 536 J ' V K ' Thus an autotransformer with r a t i o 1:4 i s required to step up the v o l -tage to 120/208 v o l t s of the laboratory supply. 2 The cross section of the pole i s chosen as 18.5 cm from consideration of the allowable f l u x density i n i r o n and assuming 20% leakage f l u x between pole shoes. The s t a t o r yoke f l u x density i s 12 K.G. 2 and with f l u x d i v i d i n g on entering the yoke, an area of only 11.6 cm i s required for each set of poles. The a x i a l length of the yoke i s l i m i t e d to 15 cm by the distance between e x i s t i n g bearing supports, r e s u l t i n g i n a yoke thickness of 1.5 cm. Figure 2.1 gives d e t a i l s of the dimensions of magnetic c i r c u i t f o r one pole. Ah a i r gap s i z e of one millimeter i s chosen. I t i s a n t i c i -pated that approximately 70% of the no load e x c i t a t i o n ampere-turn w i l l 5 72 - /6.0 »-S C A L E : DIMENSIONS I N C M Figure 2.1 Dimensions of Magnetic C i r c u i t be consumed i n t h i s region. The ampere-turn loss for each component of the magnetic c i r c u i t i s the product of f l u x path length and magnetic f i e l d i n t e n s i t y , H. For the a i r gap, H equals B divided by y , where Bg i s the maximum f l u x density and \iQ the p e r m i t t i v i t y of free space. 6 Due to the r o t o r s l o t opening, the e f f e c t i v e a i r gap length i s increased by a f a c t o r of K equal to 1.045 [10]. The calculated ampere-turn loss i s 573. For the i r o n part of the c i r c u i t , H i s determined from the B-H curves a v a i l a b l e i n reference [10]. Sheet s t e e l punchings are used on the r o t o r . Although the f l u x density i n the teeth i s high, the tooth length i s only one centimeter, r e s u l t i n g i n a los s of nine ampere-turn. As shown i n Figure 2.1, the f l u x divides on entering the rotor yoke, producing a f l u x density of 7.3 K.G. Neglecting the s h a f t , the rotor yoke length i s 5.0 cm with a loss of seven ampere-turn. The pole i s made of mild s t e e l and a high f l u x density of 15 K.G. i s chosen to provide maximum space for the f i e l d windings. With a pole length of 7.5 cm the ampere-turn loss i s 127. The s t a t o r yoke i s made of the same material with a magnetic c i r c u i t length per pole of 12.7 cm. The f l u x density i s lower than i n the pole r e s u l t i n g i n 115 ampere-turn. The t o t a l ampere-turn per pole i s 914 allowing 10 per cent f o r armature reacti o n . 2.3 F i e l d Windings The winding area depends on the required e x c i t a t i o n , the maxi-mum allowable i n s u l a t i o n temperature and the a b i l i t y of the winding -2 to d i s s i p a t e heat. In t h i s design, a current density of 250 amp cm 2 i s used, r e q u i r i n g a copper cross section of 3.7 cm and a winding area 2 of about 6.1 cm . The winding length i s chosen as 3.6 cm, allowing space f o r the pole shoe, damper winding search c o i l , and i n s u l a t i o n . The two sets of f i e l d windings and poles are placed at opposite ends of the s t a t o r , each set occupying 7.5 cm. The space remaining for the 7 pole body i s 2.8 cm. Thus the pole width i n the tangential d i r e c t i o n i s 7.2 cm to provide the required pole area. The four corners of the pole body are b e v e l l e d by 1.3 cm and .7 cm to reduce the winding turn length. One of the eight poles i s shown i n Figure 2.2. The mean turn length (MTL), calculated from the geometry of the winding, i s 26.4 cm. The design winding resistance (R w) i s one ohm per pole so that the e x c i t e r a m p l i f i e r i s loaded by four ohms. The conductor cross s e c t i o n (A c) i s given by: /pA MTL 2 A = / — ^ = 1.4 x 10 cm (2.3) c ./ R V w where p i s the conductor r e s i s t i v i t y and A^ the t o t a l copper cross sec-t i o n . Wire s i z e AWG #16 i s used with a diameter of .14 cm i n c l u d i n g i n s u l a t i o n . The number of turns per pole i s decided by the winding space, 1.7 cm by 3.6 cm, and wire s i z e with 25 turns per layer and 12 layers r e s u l t i n g i n 300 turns. To provide the 914 ampere-turns requires a f i e l d current of about 3.0 amps and a voltage drop of 12 v o l t s f or the four poles. Number 16 wire can handle considerably higher currents temporarily, allowing forced e x c i t a t i o n of several per u n i t . Each wind-ing i s subdivided i n t o two c o i l s , as i n Figure 2.3, so that i t can be e a s i l y shaped to f i t the s t a t o r yoke. To simulate the f i e l d time constant of a large machine, a search c o i l of 75 turns of AWG #30 wire i s embedded between the two main windings and connected to the e x c i t e r a m p l i f i e r . A l l windings are wrap-ped i n S c o t c h N o . 27 e l e c t r i c a l tape and sealed with a c r y l i c spray. Connections are soldered and i n s u l a t e d with heat shrinkable tubing. Figure 2.3 F i e l d and Damper Windings 2.4 Pole Shoe The pole shoe design determines the a i r gap f l u x d i s t r i b u t i o n which i n turn a f f e c t s the harmonic content of the output voltage. Two dimensions are v a r i a b l e . The f i r s t i s the a x i a l shoe length. With D and Q poles adjacent at the midpoint of the machine, leakage f l u x i s minimized by c l i p p i n g the shoe corners, leaving a gap of 1.6 cm. As shown i n Figure 2.2, the shoe narrows away from the centerline with armature voltage decreased accordingly. The second v a r i a b l e dimension i s the a i r gap length. For a ro t o r of radius R , a curved shoe surface of radius R and a minimum a i r r s gap of <$o, the a i r gap length at an angle 0 from the centerline can be des-cri b e d by R 2 R 6 6(6) = (R ~ - ^-°-) (1 - cos6) + 6 (2.4) r R s R s , ° The f l u x density and hence the induced voltage vary i n v e r s e l y with 6(9) as follows E(6) = E 1 " C max 1-C * cos9 where R 2 R 6 T> _ _J1_ _ r ° r R R C = ^ ^ (2.5) R R 6 R — £-£ + 6 r R R o s s Computational techniques are used to approximate the i d e a l cosine curve with the e f f e c t of shoe c l i p p i n g and Equation (2.5) combined. The shoe surface radius R i s then c a l c u l a t e d as 7.28 cm. s Laminated structure of the pole i s imp r a c t i c a l i n t h i s case so seven .16 cm s l i t s are machined i n t o the shoe surface to reduce 10 eddy current l o s s e s . Shoulders are provided to support the windings. The dampers, which appear i n Figure 2.3, are punched from .16 cm sheet aluminum. An exploded view of the completed machine i s shown i n Figure 2.4 with the laboratory setup i n Figure 2.5. 2.5 Parameter Tests of the Machine Set Figure 2.6 shows the open and short c i r c u i t test r e s u l t s with D and Q f i e l d e x c i t a t i o n . The reactances X,, X , X ,,, X ^ are deter-^ d q afd afq mined d i r e c t l y . The transient reactances X ' and X ' are measured by d q the I.E.E.E. Sudden Short C i r c u i t method while time constants T. , do' and T t are obtained from the F i e l d Short C i r c u i t method. I n e r t i a i s q° c a l c u l a t e d from the frequency of shaft o s c i l l a t i o n s when i t i s loaded with a known spring constant. Natural damping i s mainly due to changes i n counter EMF with speed i n the dc d r i v i n g motor. It i s determined from the change i n shaft speed with removal of a known e l e c t r i c a l load on the generator. These r e s u l t s are given i n Table 2.2. Harmonics i n output voltage are measured with a tunable v o l t -meter. The 3rd, 11th, and 13th are l a r g e r than one per cent as shown i n Table 2.3. The t h i r d harmonic i s due to pole shaping while the 11th and 13th are caused by the rotor s l o t opening, 12 s l o t s per pole. The damper windings attenuate the l a t t e r under f u l l load. The e f f e c t of the rotor s l o t s could be reduced with a larger a i r gap, but the machine r a t i n g would be decreased. It i s assumed that these high frequency harmonics do not influence the transient and dynamic s t a b i l i t y study because the natural mechanical o s c i l l a t i o n frequency of the machine i s about 1.9 Hz. Figure 2.4 Dual Field Machine Before Assembly Figure 2.5 Laboratory Setup and Author 12 Table 2.2 Machine Test Parameters X, 2.14 a X 1.98 fi d q X ' 1.20 fi X ' 1.20 fi d q X £ , 11.1 fi X ,, 10.0 fi afd afd T„ ' 2.3 sec T , 2.3 sec do qo 2 J .040 nt.m.sec D .053 nt.m.sec Table 2.3 Terminal Voltage Harmonics (Per Cent) No Load D No Load Q F u l l Load.D F u l l Load Q 3rd 4.5 4.0 9.0 12.0 11th 6.4 6.5 1.9 3.0 13th 2.5 1.7 1.5 .9 13 3. INSTRUMENTATION AND CONTROL PANELS Supplemental e x c i t a t i o n control requires accurate, low noise and f a s t responding transducers. In t h i s chapter, devices for measuring torque angle, speed, terminal voltage, current and e l e c t r i c a l power are developed. Also described are means for s i g n a l compensation and a high speed e l e c t r o n i c e x c i t e r . 3.1 Torque Angle Transducer The transducer provides a dc output s i g n a l proportional to the torque angle between the generator Q axis and the i n f i n i t e bus of the system represented by the laboratory supply. The s i g n i f i c a n t features of the transducer are: 1. l i n e a r c h a r a c t e r i s t i c i n the range 0 - 2ir radian 2. noise of 360 Hz under .03 radian 3. time, l a g l e s s than. 20 msec. The l i n e a r i t y i s checked within the steady state s t a b i l i t y l i m i t of the machine. The true torque angle i s measured on a dual channel o s c i l l o -scope from the zero crossing of voltages of the o r i g i n a l synchronous machine and the laboratory supply. Results between 0 and 60 degrees are recorded i n Figure 3.1. Resolution i s l i m i t e d by d r i f t i n torque angle caused by power supply disturbances feeding the drive motor. The transducer operates as follows. The o r i g i n a l synchronous machine of the Tamper Set converts shaft p o s i t i o n into three-phase induced voltages. These signals and the three bus voltages of the l a b -oratory supply are reduced and applied to s i x comparators which sense zero crossings and output 0 - 5 v o l t l o g i c l e v e l s . Figure 3.2 describes the l o g i c f o r one of three phases. A hexinverter integrated c i r c u i t makes ava i l a b l e the l o g i c a l 14 OUTPUT DC VOLTS A /2 -e -o * 0 20 H-0 60 Figure 3.1 Torque Angle Transducer Transfer C h a r a c t e r i s t i c complements a and a' from inputs a and a'. The four signals enter two Reset-Set (RS) f l i p - f l o p s with outputs u,v,w and x summed i n the analog sense to Z. The pulses i n Z are constant i n amplitude with width equal to input voltage phase d i f f e r e n c e . Inclusion of a l l three phases gives 360 pulses per second. A 12 dB per octave low pass f i l t e r with cu t o f f at 60 Hz smoothes the output. The complete wiring schematic i s given i n Figure 3.3. 15 3.2 Speed Transducer Relatively large swings in torque angle are reflected in only small changes in shaft speed; thus requiring a speed transducer of high sensitivity. Conventional tachometer generators and digital methods 16 Figure 3.3 Wiring Schematic of Torque Angle Transducer 17 use heavy f i l t e r i n g to reduce the noise. The transducer designed for th i s model i s a frequency demodulator driven from the same three phase 60 Hz a l t e r n a t o r used f o r torque angle measurement. The transducer has the following features: 1. l i n e a r c h a r a c t e r i s t i c i n the range 377 +3.7 radian per second. 2. noise of 360 Hz under .1 radian per second. 3. time lage l e s s than 20 msec. An accurate true measure of deviation from synchronous speed i s obtained by timing the c y c l i n g of the phase transducer through 360 degrees. This check on the l i n e a r i t y of the speed transducer i s given by Figure 3.4. OUTPUT DC VOLTS -3 -a - / 0 I 2 3 fad-Sec' 1 Figure 3.4 Speed Transducer Transfer C h a r a c t e r i s t i c 18 The a l t e r n a t o r frequency i s demodulated using a Wien Bridge c i r c u i t with notch tuned to 60 Hz. The block diagram for one of three phases i s given by Figure 3.5. 1 T R I G G E R S H R P T HHRMOMIC PET A L T E R N A T O R F H - T G R W I S " M B R I D G E . A M P U P I G H S W I T C H F I L T E R Figure 3.5 Block Diagram of Speed Transducer At the center frequency, the s e r i e s impedance R^ and equals the p a r a l l e l impedance R 2 and C^, and the bridge output i s t h e o r e t i c a l l y zero. At other frequencies, the impedance i s no longer equal and an ac output p r o p o r t i o n a l to deviation r e s u l t s . The output phase, r e l a t i v e to the bridge input, leads by 90 degrees above center frequency and lags by 90 degrees below. The F i e l d E f f e c t T r a n s i s t o r analog transmission switch i s c o n t r o l l e d by the input s i g n a l r e s u l t i n g i n phase s e n s i t i v e r e c t i f i c a t i o n . Above 60 Hz, the transmission switch output i s p o s i t i v e dc and below 60 Hz negative dc. 19 Harmonics of the input s i g n a l pass through the Wien Bridge and may overload the following a m p l i f i e r . This i s prevented with a preliminary f i l t e r adjusted to resonate at 60 Hz. The Q i s l i m i t e d to maintain the time l a g under 20 msec. Each of three phases uses the above c i r c u i t r y to achieve a r i p p l e i n the output of 360 Hz which i s removed i n the f i n a l low pass f i l t e r . The transducer output i s the product of a l t e r n a t o r amplitude and frequency, both p r o p o r t i o n a l to shaft speed. The e r r o r caused by t h i s n o n - l i n e a r i t y i s minor since the absolute speed change i s small. The complete wi r i n g schematic i s given i n Figure 3.6. 3.3 Terminal Voltage, Current and Power Transducers For the voltage sensing, three p o t e n t i a l transformers are connected l i n e to l i n e with secondaries r e c t i f i e d f u l l wave r e s u l t i n g i n 360 Hz r i p p l e . The reference voltage i s subtracted and the analog d i f f e r e n c e s i g n a l i s f i l t e r e d by a 12 dB per octave low pass f i l t e r with 60 Hz c u t o f f . Test r e s u l t s of l i n e a r i t y and s e n s i t i v i t y are shown i n Figure 3.7. For the current sensing, three transformers of 1:20 r a t i o are i n s e r t e d i n the l i n e . The secondaries are connected to r e s i s t o r s which drop 20 v o l t s with 10 amps on the primary. R e c t i f i c a t i o n and f i l t e r i n g are s i m i l a r to the voltage transducer with test r e s u l t s shown i n Figure 3.8. Instantaneous e l e c t r i c a l power with no zero sequence i s P = v i + v. i , (3.1) e ac a be b Two integrated c i r c u i t m u l t i p l i e r s generate the power s i g n a l from voltage and current transformer inputs. The l i n e a r i t y as checked with 20 Figure 3.6 Wiring Schematic of Speed Transducer 21 OUTPUT DC ~ 1/OZ.TS 26 28 30 32 3V- R M S V L ~ h Figure 3.7 Voltage Transducer Transfer C h a r a c t e r i s t i c OUTPUT OC VOLTS S L 0 P E : If I, / /.OOft RMS 8 i 0 4: — 1 1 1 1 1 1—**— o ^ s IZ RMSR' Figure 3.8 Current Transducer Transfer C h a r a c t e r i s t i c 22 OUTPUT DC VOLTS s -9- -0 SLOPE • - 0 2 £ r l / 1.00 W/PH X n 5*-0 160 3Z0 Figure 3.9 Power Transducer Transfer C h a r a c t e r i s t i c a wattmeter i s confirmed by Figure 3.9. 3.4 Control S i g n a l Compensation Adjacent to the transducers i s an analog panel with ten operational amplifiers and a patching board to simulate various phase and gain compensation schemes. Ten feedback s i g n a l s , c o n t r o l l e d by c a l i -brated potentiometers, can be combined with p o s i t i v e or negative p o l a r i t y to summing busses, feeding the d i r e c t and quadrature f i e l d s . Ten-turn reference potentiometers allow s e t t i n g of the dc operating point. Provision i s made f o r the e x c i t e r c e i l i n g and time lag. The f i n a l control signals 23 for the two f i e l d s are displayed on meters. 3.5 E x c i t e r and F i e l d Time Constant M o d i f i c a t i o n For the e x c i t e r a Phase Linear "400" watt RMS stereo music a m p l i f i e r i s used. I t i s modified f o r dc operation as a dual channel, b i p o l a r , programmable supply. The output r a t i n g i s 60 v o l t s at 4 amps but can be increased to 10 amps with forced a i r cooling. The c i r c u i t of Figure 3.10 i s used to a r t i f i c i a l l y increase the f i e l d time constant of the generator by a fac t o r of ten. A search c o i l with mutual inductance M i s coupled to the f i e l d winding with r e -sistance Rj- and inductance L^. F i e l d current, i ^ , passes through the sampling r e s i s t o r R . The voltages on L and R are attenuated by X S X potentiometers P.. and P 0 and combined with input voltage, v., at the V Figure 3.10 Time Constant M o d i f i c a t i o n C i r c u i t 24 high impedance input of the e x c i t e r with gain K. The r e l a t i o n s h i p between v . and i c i s x f R f + R L d i V i " ( P 2 R x + — K " ^ ^ + <P1 M + /> dT" ( 3' 2 ) The f i r s t term represents the new equivalent resistance and the second term inductance. The e x c i t e r gain K i s very large so the modified time constant i s V * » f * P2 \ C 3 ' 3 > Armature c i r c u i t s see the modified time constant through mutual coupling with the search c o i l . The method i s inherently stable as a c t u a l f i e l d r e s i s t a n c e R^, which changes with heating, has l i t t l e e f f e c t on Equation 3.3. 3.6 One-line Diagram of the Model Power System Figure 3.11 gives a one-line diagram of the complete power system. The dc machine f i e l d rheostat R^ controls shaft speed for syn-chronization and torque once the generator i s synchronized. The arma-ture voltage i s adjusted by which influences the change i n torque f o r a change i n speed. Pulses of torque may be generated by a time delay relay S^ and v a r i e d by R y These dynamic system disturbances are reproducible. The operating point of the dual f i e l d generator i s set by the voltmeter, ammeter and wattmeter and the control signals are a v a i l -able from the transducers. Switch S^, with time delay r e c l o s i n g , i s provided f o r transient s t a b i l i t y studies. The transmission l i n e consists of 20 mH of lumped inductance with an extra 5 mH a v a i l a b l e when S^ i s open. A bank of three 1 KVA s i n g l e phase transformers, connected A - Y, 25 steps up the voltage with 1:4 r a t i o . The d e l t a primary allows a two-pole re l a y f o r S 2. The laboratory voltage supply i s three phase 120/208 v o l t s from B.C. Hydro. For the s t a b i l i t y experiments, a Sanborn two pen hot-wire recorder of 30 Hz frequency response i s uied to measure the phase re -l a t i o n s h i p between feedback s i g n a l s . This i s backed up by a dual-beam storage o s c i l l o s c o p e . For the short c i r c u i t t e s t s , the Honeywell U l t r a - v i o l e t recorder capable of 3 KHz response i s used. F I E L D TRANSDUCERS T o ' N f r , N I T E : BUS GENERATOR. Figure 3.11 Power System One Line Diagram 26 4. EXCITATION CONTROL SCHEMES FOR ENHANCING SYSTEM DAMPING This chapter develops an i n t u i t i v e understanding of the sup-plemental e x c i t a t i o n control design for increasing the mechanical damp-ing of the machine. Torque angle, speed, power and terminal current s i g n a l s are fed back to both d i r e c t and quadrature f i e l d s . Laboratory test and eigenvalue analysis r e s u l t s are given. The system eigenvalues are computed from equations derived i n Chapter 5. 4.1 The System Under Study A s i n g l e machine i s connected to an i n f i n i t e bus through a medium length transmission l i n e of constant impedance. The prime mover normally supplies constant torque with no governor e f f e c t s included. Disturbances may be created i n shaft torque for dynamic response studies. There i s only one voltage regulator which i s used i n the D f i e l d . The Q f i e l d i s adjusted to be zero under steady state conditions. A s t a b i l i z e r loop, s i m i l a r to that used i n reference [11], i s modelled with an operational a m p l i f i e r . The e x c i t e r c e i l i n g voltage i s + 6 per unit i n each a x i s . The voltage transducer and am p l i f i e r add a small time l a g r e s u l t i n g i n a voltage regulator.of the second order. The loop gain through the regulator and machine i s 390 Rp^ where Kp^ i s a potentiometer s e t t i n g which can be vari e d between 0. and 0.7. Figure 4.1 d e t a i l s the regulator block diagram. The system operating point i s chosen as follows. The i n f i n i t e bus voltage i s f i x e d at one per u n i t . The r e a l power output of the system i s set at 960 Watt where the machine hunting i s a problem. The 27 corresponding torque angle, f o r the given transmission l i n e , i s close to 45 degrees. At t h i s angle, D and Q f i e l d s are about equally e f f e c t i v e \ 3 9 0 KP» I I-K0D7S + 6 P U / — / ~&pu S 1 + 1.0 s Figure 4.1 Block Diagram of the Voltage Regulator i n c o n t r o l l i n g the system dynamics [7]. A s l i g h t l y leading reactive power i s scheduled by adjusting the reference voltage of Figure 4.1. The precise operating values are l i s t e d i n Table 4.1. The voltage regulator improves the e l e c t r i c transient but not the dynamic s t a b i l i t y of the system. Table 4.2 gives test and eigenvalue r e s u l t s f or various regulator gains, . The damping c o e f f i c i e n t , a, i s the inverse of the time constant of shaft o s c i l l a t i o n s and i s negative f o r stable systems. The damped frequency, u^, i s i n radians per second. An increase i n indicates l a r g e r synchronizing torque. The d i f f e r e n c e between t e s t and eigenvalue r e s u l t s i s mainly due 28 Table 4.1 Operating Conditions, Per Unit P .90 v 1.01 Q -.077 i .91 v 1.00 xc, 1.96 o f d 6„ 43° i , 0.0 o fq Table 4.2 Voltage Regulator E f f e c t on Mechanical Mode Gain Test Results Eigenvalue Results a W d a "d 0. -.65 11.9 -.48 10.6 .2 .24 11.9 .25 10.9 .3 .52 . 12.5 .39 11.1 .5 .82 12.9 .42 11.6 to errors i n model and measurements. In p a r t i c u l a r , the feedback gain Kp^ i s c a l c u l a t e d through many parameters. For the following supple-mentary c o n t r o l schemes, Kp^ i s held constant at .3. The loop gain of 120 i s t y p i c a l f o r modern voltage regulators and the system i s unstable without further damping feedback. 4.2 Phase Plane Analysis In deMello and Concordia's analysis [12] the block diagram trans-f e r functions are evaluated at the mechanical resonant frequency. The 29 torque i s resolved into damping and synchronizing components. In t h i s study 6, oi, P and i are a v a i l a b l e as transducer J ' e t voltages which appear on the lower r i g h t part of Figure 4.2. Their e f f e c t s on system damping are compared. The s i g n a l s are obtained from Figure 4.2 Block Diagram of the Phase Plane Model t r a n s f e r functions, G^,(s) , m u l t i p l y i n g the instantaneous torque angle 6. An assumption i s made that the closed loop e f f e c t on f i e l d voltage i s small and may be neglected. The feedback s i g n a l s are compensated by i n d i v i d u a l blocks G v(s) and are applied to both D and Q f i e l d windings. The purpose i s 30 to b r i n g the supplemental e l e c t r i c a l torque i n phase with the speed providing a p o s i t i v e damping. Note that each f i e l d t r a n s f e r function, Gp(s), produces a gain and phase s h i f t on the input s i g n a l . The system with the voltage regulator but not the supplemental c o n t r o l i s modelled by the upper part of Figure 4.2 r e s u l t i n g from the second order system of standard form ( s 2 + 2a s + m 2 ) 6 = 0 (4.1) o o The a Q and ui are constants measured at the operating point of the system. The t r a n s f e r functions G(s) and G„(s) are experimentally T r determined at the mechanical o s c i l l a t i o n frequency as follows. The system i s made marginally stable with a speed s i g n a l roughly compensated and fed to the Q f i e l d . A s i g n a l generator, adjusted to the mechanical resonance, i s also connected to the Q f i e l d to maintain machine o s c i l -l a t i o n s at a constant amplitude. The magnitude r e l a t i o n between and V , V-r, , V. i s measured by a dual channel o s c i l l o s c o p e . Phase to Pe i t J d i f f e r e n c e i s determined by the Lissajous method. ^ x6(s) ^ S known f r o m the torque angle transducer s e n s i t i v i t y described i n Chapter 3, assum-ing no phase s h i f t due to the small time constants. Then Gj. w( s) > ^ x P ( s ) ' and G_T, . can be c a l c u l a t e d with r e s u l t s l i s t e d i n Table 4.3. TI(s) At mechanical resonance, the e l e c t r i c a l torque produced by the o s c i l l a t o r i s 180 degrees out of phase with speed. The angle between o s c i l l a t o r voltage and shaft speed can be d i r e c t l y measured on the o s c i l l o s c o p e . Thus the phase s h i f t of the t r a n s f e r functions Gp.^^ and G„ . N can be determined. S i m i l a r l y the magnitude of G„,. N and Fq(s) J ° Fd(s) G„ , . can be c a l c u l a t e d from the voltage required to maintain the Fq(s) 6 H measured speed swing and the damping c o e f f i c i e n t a . 31 Table 4.3 Gain and Phase Shift of Transfer Functions G^, 1.3/-40 Rad/Volt Sec 2 G_ 29.8/90 Volt/Rad rd ' Tu G F q 1.5^90 " G^ 14.3^_ G T 6 9.55/0_Volt/Rad G x 23.9/0 " The D f i e l d has less phase lag than the Q f i e l d because of the negative feedback of the voltage regulator. The speed leads torque angle by 90 degrees as expected. Power and terminal current are in phase with torque angle since AP = — cos <5 A6 (4.2) x s when V and E are held constant and Al = A P . (4.3) V cos0 when V cos9 i s held constant. In the following sections, the compensation G,,., „ is designed for each feedback scheme. X(s) 6 4.3 Torque Angle Feedback A direct connection of V^ to negative V^ signal appears use-f u l . Hopefully, the f i e l d lag brings el e c t r i c a l torque in phase with speed for positive damping. However, an experimental t r i a l reveals that synchronization is lost for the following reason. A decrease in f i e l d current tends to increase the measured torque angle. When inverted and amplified by the proposed feedback scheme,, the dynamic interaction aggravates the decrease in f i e l d current, leading to an in s t a b i l i t y of the system. The same situation holds for the direct axis f i e l d . A solution i s found by ro l l i n g off feedback gain, as on a 32 Bode p l o t , immediately below mechanical resonant frequency. Such a high pass f i l t e r contributes unavoidable phase lead at resonant frequency but can be t o l e r a t e d f or the Q f i e l d . A d d i t i o n a l l a g i s necessary i n the D axis. The f i l t e r schematic and phasor sketch are shown i n Figure 4.3. The compensation scheme has the following transfer function and phasor value at mechanical resonant frequency. .2 3.1 K i 6 4 ( 4 . 4 ) ; GXd ( s> = -.14 s-(l+.2s) (l+.07s) • G x d < s ) -.20 K s" Vs) = ~777~z3. ' G x q ( s ) S=JCd' = 4.4 (4.5) (l+.2s)' Test and eigenvalue r e s u l t s are given i n Table 4.4 for various feedback gains Kp 2. Table 4.4 Mechanical Mode of Torque Angle Feedback Gain Test Results Eigenvalue Results d.s KP2 a W d a a' "d D 0.0 .52 12.5 .39 .39 11.1 rt .2 -.17 12.6 .06 .06 10.9 l l .4 -.95 12.0 -.32 -.32 10.7 I I .7 -4.0 7.0 -1.2 -1.2 10.2 Q .1 .08 12.4 .05 .05 10.9 I I .2 -.28 12.5 -.36 -:36 10.6 I I .5 -2.2 12.6 -2.2 -1.5 10.2 33 200K r v V A — 200 K 70 K r-vVW—? (a) F i l t e r Schematic (b) Phasor Sketch Figure 4.3 Torque Angle Signal Compensation Good mechanical damping i s achieved but low frequency e l e c t r i c a l i n -s t a b i l i t y r e s u l t s f or large gain. This i s ind i c a t e d by the r e a l part of the most dominant eigenvalue, a', which becomes more p o s i t i v e than 34 the mechanical a. 4.4 Speed Feedback Speed feedback requires lead compensation equal to the f i e l d l a g . In the Q axis t h i s amounts to 90 degrees and at l e a s t two ampli-f i e r stages are necessary. High frequency noise tends to be amplified more than the speed s i g n a l so a preliminary low pass f i l t e r i s used. For the smaller lag of the D f i e l d , t h i s f i l t e r i n g i s made heavier to reduce the o v e r a l l phase lead. The schematic and phasor sketch are shown i n Figure 4.4. The t r a n s f e r function and phasor value at mechanical resonant frequency are 2 (l+.07s) (l+.04s) 0056 s , - " =1-23 K V 6 6 Gxd< s> - , ; 9 ' G x d ( s ) s=ju p 2 ^ - ^ (4.6) d .016 K s 2 G (s) = , G (a) =1.76 /lU X q • (l+.02s) (l+.04s) 2 X q 3 J a )d (4-7) Test and eigenvalue r e s u l t s are given i n Table 4.5. Table 4.5 Mechanical Mode of Speed Signal Feedback Gain Test Results Eigenvalue Results Axis a "d a a' "d D 0.0 .52 12.5 .39 .39 11.1 I I .2 -.53 12.6 -.23 -.23 11.0 I I .4 -2.1 13.6 -1.4 -1.4 10.0 I I .7 -5.0 -2.3 -2.3 9.6 35 Gain Test Results Eigenvalue Results .2 -.73 12.5 .86 .86 10.4 -.4 -2.5 12.5 -2.2 -2.2 8.9 .7 -4.0 - 12. -8.6 r -vVW - T 1.0 20K r - W W - i 2.0 20K rvWv-i a.o (b) Phasor Sketch Figure 4.4 Speed Signal Compensation 36 Very good mechanical damping i s achieved without the problem of low frequency i n s t a b i l i t y . 4.5 Power Feedback In Equation 4.2, e l e c t r i c a l power i s shown to be i n phase with torque angle. The low frequency i n s t a b i l i t y problem experienced with the torque angle scheme i s reduced since steady state e l e c t r i c a l power i s independent of f i e l d current. Negative Vp g s i g n a l i s fed to the Q f i e l d and the 90 degree lag brings torque i n phase with speed. A s i n g l e low pass stage increases the D f i e l d lag to the same amount. A washout block removes any dc component from the power s i g n a l . The schematic and phasor sketch are shown i n Figure 4.5. The t r a n s f e r function and phasor value at mechanical reson-ant frequency are -7 Kp 2 s G X d ( s ) (1+1.0s). (l+.07s) ' ~Xdv > G V J ( s ) • = 5- 0 K„„ / l 2 0 . S = 3 W d V 1 (4.8) "10 K p 2 s s=jto = 10. (1+1.0s) Test and eigenvalue r e s u l t s are given i n Table 4.6. Table 4.6 Mechanical Mode of Power Feedback Kp2Z 180 (4.9) Gain Test Results Eigenvalue Results Axis a W d a a' "d D 0.0 .52 12.5 .39 .39 11.1 I I .1 -.51 12.6 -.76 -.76 10.9 I I .2 -1.9 13.6 -2.2 -1.04 10.2 I I .3 -5.0 -8.0 -1.02 12.5 37 Gain Test Results .1 -1.07 12.5 .2 -2.0 13.8 .4 -11. Eigenvalue Results -4.4 -.88 11.6 -9.8 -.87 12.8 -20. -.84 13.1 Mechanical damping i s good but low frequency i n s t a b i l i t y l i m i t s the us e f u l gain. 4.6 Terminal Current Feedback For small s i g n a l conditions, terminal current behaves l i k e power and the compensation scheme of the previous s e c t i o n i s duplicated. Current feedback may be useful during•transients since f a u l t current increases while power decreases from normal l e v e l s . The proper com-b i n a t i o n should enhance transient s t a b i l i t y . Test and eigenvalue r e s u l t s f o r current, feedback are given i n Table 4.7. Table 4.7 Mechanical Mode of Terminal Current Feedback Gain . Test Results Eigenvalue Results Axis ^ 2 a U d a a' W d D 0.0 .52 12.5 . .39 .39 11.1 ti .1 -1.1 12.6 -1.2 -.73 10.9 I I .15 -2.1 13.0 -2.4 -.70 10.9 I I .2 -2.3 13.6 -3.8 -.67 11.2 Q .1 -1.3 12.5 -1.7 -1.1 11.2 I I .2 -1.9 12.5 -4.0 -1.1 10.8 • I .4 -11. - -9.4 -1.0 8.3 As with power feedback, \ *ood damping i s achieved but gain i s 1: 38 2.M .5 a) F i l t e r Schematic b) Power Phasor Sketch 5— £> Vfc c) Terminal Current Phasor Sketch Figure 4.5 Power and Terminal Current Compensation 39 by low frequency i n s t a b i l i t y . 4.7 Comparison of Schemes With any of the previous feedback s i g n a l s , c r i t i c a l damping can be achieved f o r the mechanical o s c i l l a t i o n s , i f e l e c t r i c a l o s c i l -l a t i o n s at other frequencies are not considered. An optimum feedback gain i s r e a l i z e d when e l e c t r i c a l poles instead of mechanical poles become the most dominant. Figure 4.6 p l o t s negative a' against Kp^ f o r the various schemes. The transient performance cannot be judged by eigenvalue analysis alone. Test r e s u l t s reveal that complex f i l t e r i n g schemes delay the i n i t i a t i o n of damping of the system. Table 4.8 l i s t s the number of shaft swings before damping takes e f f e c t . Table 4.8 Number of Transient Swings Before Damping Axis Feedback Source 6 CO P e \ D 1 1 1/2 1/2 Q 1 1/2 1/2 1/2 The dual f i e l d generator has a better performance i n the low power, high leading power f a c t o r region as mentioned i n the i n t r o d u c t i o n . The s e n s i t i v i t y of each f i e l d i n c o n t r o l l i n g dynamic torque i s tested f o r various torque angles with'results i n Table 4.9. The D axis supple-mental torque c o n t r o l i s more e f f e c t i v e at large torque angles whereas the Q axis c o n t r o l i s better near zero output power. However, s t a b i -l i t y i s not u s u a l l y a great problem at l i g h t loads. Another b e n e f i t of dual axis control i s better terminal 40 -.5-Figure 4.6 Comparison of Feedback Source and Axis voltage r e g u l a t i o n during swings. The induced voltage, which i s r e l a t e d to d i r e c t f i e l d current, may be strongly affected by D axis damping schemes. Meanwhile, the Q f i e l d f l u x can change by s i g n i f i c a n t amounts near the zero operating point without a f f e c t i n g the t o t a l f i e l d f l u x magnitude. Table 4.10 describes the change i n terminal voltage per radian shaft swing f o r various feedback schemes. The r e s u l t s show the 41 superiority of the Q f i e l d in improving voltage regulation. Table 4.9 D and Q Field Dynamic Torque Control T Torque Angle Normalized Sensitivity (rr-) f (degrees) D axis Q axis -26. 0. .82 -13. .22 .85 0. .61 1.00 13. .68 .67 26. .93 .70 42. 1.27 .74 Table 4.10 Terminal Voltage Swing per Radian Shaft Swing S cheme Axis None 6 co P i e t D 6.8 11.4 12.3 12.7 9.1 Q 6.8 5.8 8.6 6.1 5.7 42 BASIC EQUATIONS OF THE DUAL-EXCITED MACHINE A fourth order f l u x linkage model derived from b a s i c synchron-ous machine equations i s used for the study. Including the feedback c o n t r o l schemes, the r e s u l t i n g system equations are l i n e a r i z e d and the system matrix i s found. Eigenvalue r e s u l t s are compared with the tested r e s u l t s of Chapter Four. 5.1 Voltage and Flux Linkage Equations The following assumptions are made i n t h i s a n a l y s i s : 1. Q axis leading. 2. Constant voltage due to speed. 3. No s a t u r a t i o n . 4. Armature transients neglected. 5. Damper c i r c u i t s neglected. Applying Park's transformation, the D axis equations i n MKS units are: 'fd V u d R f d 0 R + us 3 y _ y f f d 2 fad X a f d X d \ I fd - I , d / "— -(5.1) where U*d =» ui ¥ (5.1a) and U^ i s the speed voltage and the f l u x linkage, both i n MKS u n i t s . When an i n v a r i a n t base power i s chosen for both armature and f i e l d c i r -c u i t s , the X matrix w i l l be symmetric. Thus the f l u x linkage becomes i n per unit fd X f d Xmd x md r f d - i , (5.2) 43 Equation (5.1) i s now v fd V d " U d r f d ° 0 "fd - l + 2-co o 'fd (5.3) The Q axis c i r c u i t equations are of s i m i l a r form. For a leading Q axis the generated voltages are u. — ^ co q o (5.4) u = H iK q co d o In view of the assumptions made, co equals CO q and P^* Vi>^ can be neglected. Equation (5.3) and (5.4) can be written i n state form where and co fd 'fq " r f d ° 0 - r "fd "fq 'fd 'fq 'fd rq x f d xmd Xmd X d x r x fq mq x x mq q L f d - l "fq -r 0 a + -v. (5.5) (5.6) (5.7) The per unit parameter values are calculated from the base qu a n t i t i e s l i s t e d i n Table 5.1 and measurements obtained i n Chapter 2. 44 Results are given in Table 5.2. V nb nb nb •fb 'fb Jfb Table 5.1 Per Unit Base Quantities of the Machine 45.3 V peak 15.6 A peak 2.91 Q 1.76 A 603 V 342 fi defined by i n f i n i t e bus voltage defined by machine rating defined by short circuit ratio defined by power invariance do Table 5.2 Per Unit Machine Parameters .062 .74 .41 2.3 sec T ' qo .68 .41 2.3 sec The mutual reactances are derived from the open circuit ratio; X a f d Ifd md V nb x « X a f q I f b mq V .61 .55 nb field inductance from the transient reactance: 2 md f d xd" X'd = 1.13 x 2 x = _ E 9 L fq x -x1 q q = 1.12 (5.8) (5.9) 45 and f i e l d resistance from the open c i r c u i t f i e l d time constant: = .0013 •fd 377 T' do (5.10) -^V = .0013 "fq 377 T qo 5.2 Torque Equations of the Machine In MKS u n i t s , the e l e c t r i c a l torque The r e s u l t i n g shaft a c c e l e r a t i o n with torques at e l e c t r i c a l angular v e l o -c i t y pco = i (T - | I n ^ + -fr I i|» - Deo) (5.12) J m 2 q d z d q The base torque chosen T = 1 I n b V n b (5 13) T b 2 377 1 J ; Equation (5.12) i n per unit and l i n e a r form becomes pto = K. (t - i iK - i iK + 1, + ±,i> - dio) (5.14) * j m qo rd q rdo do rq d rqo where 3 I V nb nb _ 7 1 , 2 377 D _ n i n K. = — - 71., d = — — = .010 (5.14a) j 2 377 J 3 I V , J nv nb A l l v a r i a b l e s are understood to be incremental q u a n t i t i e s . The second d i f f e r e n t i a l equation r e l a t e s angular v e l o c i t y i n radians per second and torque angle i n radians p 6 = to (5.15) 46 5.3 Transmission Line Equations The s i n g l e l i n e diagram of Chapter 3 i s modelled by a p i equi-valent with one l e g absorbed i n the i n f i n i t e bus as i n Figure 5.1. With transformer losses included i n b and g, short c i r c u i t and open c i r c u i t t ests define the parameter values which are given i n per u n i t . l-fc [GEN T AAA—OTTTV-r x VI . X = . 2 0 r~ = .04-8 b = -- OOIS Cj-.OOIT Figure 5.1 Transmission Line Sketch and Phasor Diagram By in s p e c t i o n V t ~ V o i = (B + ib) V + ; : t K B J ' t r + j x (5.16) D i v i d i n g i n t o components along the D and Q axis (1+rg-xb) -(rb+xg) (rb + xg) (1+rg-xb) v d _ v sin6 o + r -x V v cos<$ o x r i q (5.17) 47 The matrix equation i s inverted to solve f o r v^, v^ v K l K2 -K 2 K 1 v sinS o v cos 6 o + -K 2r+K l X)(K 2x+K 1r) (5.18) where K 1-xb+rg A xg+rb A 1 ' 2 5.4 Open Loop System Equations A = (1-xb+rg) 2 + (xg+rb) 2 (5.18a) The machine equations are of fourth order, two from Equation (5.5) and one each from Equations (5.14) and (5il5) with st a t e s : ^ f d ' ^ f q ' W ' ^" A u x i l i a r y equations (5.6), (5.7) and (5.18) are combined to eliminate u ) J , i L , v , , v . r d r q d' q where ~*fd" •*d *£<. _ V >d" -X f d md 0 0 md '22 "42 X22 = X d " K 2 r + K 1 X x„. = r + K x + K n r 24 a 2 1 x. „ = - r -K„x - K,r 42 a 2 1 x. , = x^ - K r + K x 44 q 2 1 0 0 x X fq mq v sin6 o v cos6 o X, 24 mq "44 " f d ^ q - i q J (5.19) (5.20) 48 i n compact notation ip* = X i (5.19a) where Equation (5.20) i s l i n e a r i z e d •'d H 6 (5.21) mi 8 2 6 g.. = (-K0 cos 6 - K, s i n 6 ) v °1 - 2 o 1 o o g 0 = (-K. cos 6 + K_ s i n 6 ) v 2 1 o 2 o o (5.21a) The sub s c r i p t 'o' i n d i c a t e s the i n i t i a l value of the v a r i a b l e . The LHS of Equation (5.19) may also be defined by the state X *fd *'d . •fq 0 0 1 0 0 0 0 0 0 S i 32 J 'fd 'fq (5.22) or jfc.1 = Q x (5.22a) Now i can be r e l a t e d to x through Equations (5.19) and (5.22) jL = X Q x The f l u x linkages ij;^ and are found through Equation (5.6) (5.23) "*d~ — Xmd X d 0 0 i (5.24) A 0 0 X mq X q or <J> = R i (5.24a) 49 The terminal voltage components from Equation (5.7) V , "o -1 O r 0 0 ' d i + a i V 1 0 0 0 O r L qJ a (5.25) or v = S ip + T i (5.25a) The linearized terminal parameters v f c, i t > P g are required for analyzing feedback schemes or -V t \ = p e r V do V to do V + V . to qo v + 0 0 0 "do to L to 0 -v, 0 -v do qo i (5.26) t = U v + V i (5.26a) The state equations can be expressed from Equations (5.5), (5.14) and (5.15) p x = 0 0 0 0 0 0 0 0 0 0 0 0 -d 0 1 0 x + -co rc, 0 o f d 0 0 0 0 -w r . 0 o rq 0 -K.iLi 0 K.iK j rqo j rdo 0 0 0 0 i + 0 0 -K.i 3 q° o o -K.I, J do (5.27) 50 or p x = C x + D i _ + E j ^ (5.27a) Equations (5.23) and (5.24) allow e l i m i n a t i o n of i_ and The system 'A' matrix i s found p x = A x (5.28) where A = C + D X 1 Q + ER X _ 1 Q (5.28a) The matrix m u l t i p l i c a t i o n i s done with Fortran subroutines. 5.5 C a l c u l a t i o n of Operating Point The operating point i s usually defined by P Q , Q q, and v but i n t h i s study v , i and v are used because the i n f i n i t e bus voltage i s J to to o f i x e d . Equation (5.16) i s rewritten as i t o ^ t o <*' + J b'> (5.29) where g' + j b ' = g + jb + r+jx (5.29a) Equation (5.29) may be represented by a t r i a n g l e of phasors with y the angle between i and v (g* + b ' ) . A p p l i c a t i o n of the cosine law gives cos -1 to ( g '2 + b' 2) + i 2 Q v 2, 2 r +x 2 i  - v A ' 2 + b ' 2 to to V 6 (5.30) From the d e f i n i t i o n of y, the power factor and complex power are . b ' 3 o = -tan -, - Y P = v i cos 6 o to to o Q = v i s i n 8 o to to o (5.31) 51 Assuming v lags the machine Q axis with angle 8, the terminal components are v, = v s i n 8 do to v = v cos 8 1° t 0 (5.32) ido = \o S i n ( e + 9 o ) From Equations ( 5 . 6 ) and ( 5 . 7 ) v, = -x i . + x i - r i , (5.33) do mq f qo q qo a do With Q f i e l d e x c i t a t i o n zero i n the steady state, i . vanishes. S u b s t i -fqo t uting Equation (5.32) i n (5.33) and applying the trigonometric expansion formulas -1 = tan - r sine + x cos9 a o q o  v / i + r cos6 + x sine to to a o q o ( 5 . 3 4 ) The terminal components of Equation ( 5 . 3 2 ) are evaluated, followed by S Q and V q from Equation ( 5 . 1 7 ) . i f i ^ and are obtained from Equation ( 5 . 7 ) followed by i f ^ 0 > ^ f j 0 a n c * ^ f q Q f r o m Equation ( 5 . 6 ) . 5 .6 Voltage Regulator Equations The block diagram of the voltage regulator i n Figure 4 .1 i s modelled by two tra n s f e r functions x l 390 K p i (-Vfc + V r e f ) 1 + .007 s ( 5 . 3 5 ) = (1 + s)  X2 (1 + 2s) X l where x 2 i s summed with any D axis supplemental s i g n a l and applied to 52 the D f i e l d winding. Equation (5.35) i s modified to incremental per u n i t q u a n t i t i e s with open loop gain through the machine held constant -:88 hi \ X In state form 1 1 + .007 s = (1 + s)  X2 (1 + 2s) 1 X l - 8 8 K P l V t (5.36) P X l rp rp ll ll (5.37) x„ , T„ .88 K p i v t 2 , 1 2. P xo = ~ + - ^ - ( l - ^ - ) x 9 T T T - 1 T T A3 3 1 1 3 where 11 = .007 , T 2 = 1.0, T 3 = 2.0 (5.37a) 5.7 Supplemental Feedback Schemes for Damping A l l of the feedback compensation schemes are^modelled by the generalized tra n s f e r function K T KP2 g 2  Gx(s) ( i + T . s ) ( l + T c s ) ( l + T-s) (5.38) H J D The transducer gain Y^, includes the s c a l i n g e f f e c t i n converting to per unit while remains the potentiometer s e t t i n g between 0. and .7. The parameter values for each scheme are given i n Table 5.3. The t r a n s f e r function forms three state equations - x 3 K T K p 2 u P X 3 = T7 +  1*4 -A- | ^ KP2 U  P ' T 5 ~ V 5 T4 T5 53 x x x K u 5 T CT„ T. T T T. T c T, ^'^J 6 5 6 4 5 6 4 5 6 Table 5.3 Compensation Scheme Parameter Values Source Axis T T4 T5 T6 <5 D -.0026 .07 .2 .2 fi Q r.0026 .007 .2 .2 CO D .000060 .07 .04 .04 CO Q .000060 .02 .04 .04 p e D -.36 .07 1.0 1.0 P e Q -.36 .007 1.0 1.0 \ D -.165 .07 1.0 1.0 \ Q -.165 .007 1.0 1.0 where u is the source <5, co, P e or I and x,. is the voltage applied to the D or Q f i e l d winding. 5.8 Closed Loop System Equations The state vector is of ninth order (^.^J ij>f , co, 5, x 1, x 2, x 3, and Equation (5.27) is modified to include the closed loop P x = C x + D i + E 1 + G 1 (5.40) where for power feedback to the Q f i e l d , for example 54 C = 0 0 0 0 0 to o 0 0 0 0 0 0 0 0 0 0 0 to c 0 0 -d 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 T l 0 0 0 0 0 0 0 0 c f 3 - 1 2 ) T T 1 3 -1 T3 0 0 0 0 0 0 0 0 0 -1 T4 0 0 0 0 0 0 0 0 -1 T4 T5 -1 T5 0 0 0 0 0 0 0 -1 -1 -1 T4 T5 T6 T T l5 6 T6 G = 0 0 0 0 .88 Kp3 88 Kp^T 12 T T ll 3 0 0 0 0 0 0 0 0 0 K T ^2 4 K T ^2 T T 4 5 *T ^2  T4 T5 T6 (5.42) 55 Matrices D and E are unchanged from the open loop case. Equations (5.23), (5.24) and (5.25) are substituted in (5.40) to obtain the closed-loop system matrix A = G + {D + ER + G[U(SR + T) + V]} X _ 1 Q (5.43) The equation i s evaluated and the eigenvalues found using available Fortran subroutines. The results are compared with the experimental tests as described i n Chapter 4. 56 6. CONCLUSIONS From laboratory test r e s u l t s and eigenvalue analysis of the dual f i e l d generator, the following conclusions are formed. 1. When operating close to rated output, the d i r e c t and quadrature f i e l d s are about equally e f f e c t i v e i n dynamically c o n t r o l l i n g the machine o s c i l l a t i o n s . At torque angles near zero, the quadrature f i e l d i s superior. 2. Compensated speed s i g n a l provides the best damping f o r e i t h e r f i e l d . No low frequency e l e c t r i c a l i n s t a b i l i t y appears f o r large feedback gains. 3. Power and terminal current signals provide good damping with very simple compensation. Large gains r e s u l t i n low frequency e l e c t r i c a l i n s t a b i l i t y . 4. Torque angle s i g n a l requires complex f i l t e r i n g f o r f a i r damping. The method i s very s e n s i t i v e to low frequency e l e c t r i c a l i n s t a -b i l i t y . 5. Complex f i l t e r i n g or compensation schemes are undesirable. The i n i t i a t i o n of damping i s delayed. Lead compensation amplifies noise while l a g compensation increases the tendancy to low f r e -quency i n s t a b i l i t y . 6. Terminal voltage regulation i s less s e n s i t i v e to shaft swings with dual f i e l d c o n t r o l . Conclusions 1 and 6 agree with r e s u l t s i n the l i t e r a t u r e [3, 5 , 7 ] . The r e s u l t s of th i s thesis may be extended i n the following areas: 57 1. Operation at constant torque angle f o r improved transient s t a b i l i t y as i n reference [5]. P r o v i s i o n i s made on the model f o r transient l i n e switching. 2. Study of a quadrature f i e l d with l e s s turns. A major impedi-ment to s t a b i l i t y i s the large f i e l d l a g . Since the quadrature f i e l d does not need to supply continuous ampere-turns, i t s - -turns and inductance may be reduced. The f i e l d constant i s e a s i l y changed on the model. 3. A more accurate block diagram with f l u x linkage parameters would improve the accuracy of design i n Chapter 4. The f l u x linkage model i n Chapter 5 i s e a s i l y modified to s i x t h or eighth order machine equations f o r more accurate r e s u l t s . The pro-blem i s the measurement of the sub transient machine parameters as damper winding resistance i s r e l a t i v e l y high i n the laboratory model. The laboratory synchronous machine model could be improved with the following changes: 1. Reduce 11th and 13th harmonics by rounding the shoe t i p s to a radius of .32 cm. 2. Make the dc drive motor torque more constant with speed. The model damping i s presently too large due to t h i s f a c t o r . 3. Provide greater cooling on the e x c i t e r a m p l i f i e r to extend the range of voltage c e i l i n g . 4. Find and correct the cause of small d r i f t s i n torque angle. Change i n brush resistance i s suspected. The noise makes r e l a t i v e l y large t e s t swings necessary when checking small s i g n a l dynamic r e s u l t s . Errors are then caused by the f i e l d voltage reaching the upper l i m i t . DO REFERENCES 1. Hamdi-Sepen, C , "Process for increasing the transient s t a b i l i t y power limits on ac transmission systems - Pt. 1", CIGRE, 1962, paper 305. 2. Botvinnik, M.M., "Asynchronized synchronous machine", (Pergamon, 1964). 3. Takata, S., "Compensation of bus voltage fluctuation by means of optimal control of synchronous machine excitation", J. Inst. Elec. Engrs. Japan, Vol. 88, No. 12, 1968, p. 42. 4. Takata, S., Ohta, E., and Naganuma, Y,, "Suppression of hunting in two-axis synchronous machines by control of f i e l d voltage", J. Inst. Elec. Engrs. Japan, Vol. 88, No. 2, 1968, p. 28. 5. Soper, J.A., Fagg, A.R., "Divided-winding-rotor synchronous generator", Proc. IEE, Vol. 116, No. 1, p. 113. 6. Robinson, R.B., "Transient equivalent circuit of the divided-winding-rotor synchronous machine", Proc. IEE, Vol. 117, No. 3, p. 552. 7. Krause, P.C. and Towle, J.N., "Synchronous machine damping by exci-tation control with direct and quadrature axis f i e l d windings", IEEE PAS, Vol. 88, No. 8, 1969, p. 1266. 8. Subramaniam, P. and Malik, O.P., "Closed loop optimization of power systems with two-axis excitation control", IEEE Winter Meeting, 1972. 9. El-Serafi, A.M. and Badr, M.A., "Choice of the excitation system parameters for maximum possible capacitive power loading of dual-excited synchronous generators", IEEE Winter Meeting, 1973. 10. Kuhlmann, J.H., "Design of e l e c t r i c a l apparatus", Wiley, 1950, p. 67 and Appdx. 1. 11. Yu, Y.N. and Habibullah, B., "Improving supplemental excitation control design using accurate model", IEEE Winter Meeting, 1973. 12. de Mello, F.P. and Concordia, C , "Concepts of synchronous machine st a b i l i t y as affected by excitation control", IEEE PAS, Ap r i l , 1969, p. 316. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0101428/manifest

Comment

Related Items