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Multi-mode stabilization of torsional oscillations in single and multi-machine systems using excitation… Yan, Andrew 1982

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MULTI-MODE STABILIZATION OF TORSIONAL OSCILLATIONS IN SINGLE AND MULTI-MACHINE SYSTEMS USING EXCITATION CONTROL by Andrew Yan B.S.E.E., University of Texas at Arlington, 1977 M.A.Sc, University of B r i t i s h Columbia, 1979 ' A_ THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of E l e c t r i c a l Engineering) We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February , 1982 Andrew Yan, 1982 In presenting t h i s 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 a v a i l a b l e f o r 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 h i s or her representatives. I t i s understood that copying or pub l i c a t i o n of t h i s 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 O f F.lftr-.rri r a l F . n g i n P P r i n g The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 D a t e March 30, 1982 D F - f i ( 7 / 7 9 1 ABSTRACT Subsynchronous Resonance (SSR) phenomena i n a t h e r m a l - e l e c t r i c power system with series-capacitor-compensated transmission l i n e s may cause damaging t o r s i o n a l o s c i l l a t i o n s i n the shaft of the turbine-generator. This t h e s i s deals with a wide-range multi-mode s t a b i l i z a t i o n of single-machine and multi-machine SSR systems using output feedback e x c i t a t i o n c o n t r o l . Chapter 1 summarizes the SSR countermeasures to date. Chapter 2 presents a u n i f i e d electro-mechanical model f o r SSR studies, i l l u s t r a t e s the t o r s i o n a l i n t e r -a c t i o n between the e l e c t r i c a l and mechanical systems, and demonstrates that multi-mass representation of the turbine-generator must be used f o r SSR studies. For the c o n t r o l design, a reduced order model i s d e s i r a b l e . For the model reduction, modal a n a l y s i s i s applied to i d e n t i f y the e x c i t a b l e t o r s i o n a l modes, and a mass-spring equivalencing technique to r e t a i n only the unstable modes i s developed i n Chapter 3. Using the reduced order one-: machine models, l i n e a r optimal e x c i t a t i o n c o n t r o l s are designed i n Chapter 4. The c o n t r o l s are fur t h e r s i m p l i f i e d by examining the eigenvalue s e n s i t i v i t y , and the r e s u l t s are tested on the l i n e a r and nonlinear f u l l models. In Chapter 5, the s t a b i l i z a t i o n technique i s further extended and applied to a two-machine system and a three-4nachine system. The s t a b i l i z e r s s t i l l can be designed one machine at a time using a one-machine equivalent f o r each machine by r e t a i n i n g only the path with the strongest i n t e r a c t i n g current and the c r i t i c a l e l e c t r i c a l resonance frequency as seen by the machine. To coordinate a l l c o n t r o l l e r s f o r the e n t i r e system, an i t e r a t i v e process i s developed. The c o n t r o l l e r s thus designed are tested on l i n e a r and nonlinear f u l l models. From both eigenvalue a n a l y s i s and nonlinear dynamic performance t e s t s of the one-machine, tx<«-machine, and three-machine systems, a conclu-sion i s drawn i n Chapter 6 that the e x c i t a t i o n c o n t r o l s thus designed by the i i methods developed i a t h i s t h e s i s can e f f e c t i v e l y s t a b i l i z e single-machine and multi-machine SSR, systems over a wide range of capacitor compensation. i i i TABLE OF CONTENTS Page ABSTRACT .. i TABLE OF CONTENTS i i i LIST OF TABLES v i LIST OF ILLUSTRATIONS v i i ACKNOWLEDGEMENT i x NOMENCLATURE .. •. x 1. INTRODUCTION 1. 1.1 Subsynchronous Resonance 1 1.2 Count ermea sur es to SSR 3 1.3 Previous Works on E x c i t a t i o n Control of SSR 5 1.4 Scope of the Thesis 7 2. MODELLING POWER SYSTEMS FCR SSR STUDIES 8 2.1 Introduction 8 2.2 The Mechanical System l n 2.3 The E l e c t r i c a l System 1 3 2.4 Complete System Equations f or Single and Multi-Machine Systems 16 2.5 The T o r s i o n a l I n t e r a c t i o n 19 3. MODEL REDUCTION OF A F0WER SYSTEM FOR SSR STUDIES 22 3.1 Introduction 22 3.2 To r s i o n a l Resonance and the Unstable Modes 23 3.2.1 Natural Frequencies and Mode Shapes of the 23 Mass-spring System 23 3.2.2 Damping and the Resonant Peak 26 3.2.3 Resonant Peaks of the Six Mass-spring System .... 27 3.2.4 I d e n t i f i c a t i o n of the Unstable T o r s i o n a l Modes ... 30 i v Page 3„2„5 Modal Resonance Peaks and Unstable Modes 31 3.2.6 Other Uses of Modal Analysis i n SSR Studies ... 31 3.3 Equivalent Mass-spring System 32 3.3.1 Mode Frequencies and the Adjustment of the S t i f f n e s s Constant 32 3.3.2 Mode Shapes of the O r i g i n a l and Equivalent Systems .... 35 3.3.3 Eigenvalues of the O r i g i n a l and Reduced Order Models . 39 4. EXCITATION CONTROL DESIGN 43 4.1 Introduction 43 4.2 Linear Optimal Control 44 4.3 Eigenvalue S e n s i t i v i t y 46 4.4 Reduced Order C o n t r o l l e r v i a Eigenvalue S e n s i t i v i t y A n alysis 47 4.5 Examples of the Con t r o l l e r Design 48 4.6 The Output Feedback Control 51 4.7 Eigenvalue Analysis of the S i m p l i f i e d C o n t r o l l e r s 53 4.8 Dynamic Performance Test Using Nonlinear Model 53 4.9 The Control Signal 65 5. MULTI-MACHINE SSR STUDIES 67 5.1 Introduction 67 5.2 A Two-machine and a Three-machine System 68 5.3 Prelimary Studies of the Two-machine System 72 5.4 Prelimary Studies of the Three-machine System 80 5.5 C o n t r o l l e r Design Considerations of Multi-machine System 83 5.6 C o n t r o l l e r s Design and Test of the Two-machine System.. 85 5.6.1 Testing of C o n t r o l l e r s Using Eigenvalue Analysis 88 v Page 5.6.2 Dynamic Performance Test of the Two-machine System 90 5.7 C o n t r o l l e r s Design and Test of the Three-machine System 99 5.7.1 S e n s i t i v i t y Studies and Choice of Weighting Elements i n [ Q ] 100 5.7.2 Dynamic Performance Test of the Three-machine System 107 5.8 Concluding Remarks of the Multi-machine SSR Studies ... 120 6. CONCLUSION . 121 REFERENCE 123 APPENDIX I 126 APPENDIX I I 127 APPENDIX I I I 129 v i LIST OF TABLES Table Page 3.1 Modal resonance peaks of the s i x mass-spring system .... 29 3.2 Magnitude of the resonant peaks of each mass using approximate modal a n a l y s i s , 29 3.3 Unstable modes of the SSR system . 30 3.4-3.6 Eigenvalues of various SSR model at various system conditions , 40-42 4.1 T y p i c a l value of the eigenvalue s h i f t due to i n d i v i d u a l state feedback 49 4.2-4.4 T y p i c a l mechanical modes of the system with and without c o n t r o l at various operating conditions ................ 54-56 5.1 Summary of the components and number of s t a t e i n the two-machine system 69 5.2 Summary of the components and number of state i n the three-machine system 71 5.3 Various machine operating conditions i n the three-machine system i 71 5.4 Unstable modes of the two-machine system 72 5.5 T y p i c a l mechanical modes of the two-machine system 74 5.6 T y p i c a l mechanical modes of the twonnachine system using d i f f e r e n t models f o r the i n v e s t i g a t i o n of ' t o r s i o n a l i n t e r a c t i o n between machines , . 75 5.7 Unstable modes of the three-machine system 80 5.8 T y p i c a l mechanical modes of the three-machine system ... 81 5.9 T y p i c a l mechanical modes of the three-machine system using d i f f e r e n t models f o r the i n v e s t i g a t i o n of t o r s i o n a l i n t e r a c t i o n between machines ' 82 5.10 Testing sequence f o r the two-machine system 88 5.11 Ty p i c a l mechanical modes of the two-machine system without and with co n t r o l 89 5.12 T y p i c a l mechanical modes of the three-machine system with c o n t r o l 102 v i i LIST OF ILLUSTRATIONS Figure Page 1.1 Photographs of the damaged generator, Mohave No. 1 ..... 2 2.1 Power system model for SSR studies 9 2.2 Modelling of the mass-spring system i n the v i c i n i t y of the i - t h r o t a t i o n a l mass 11 2.3 The transmission system 13 2.4 Component of I i n dq and DQ coordinates 17 2.5 Power v a r i a t i o n of the one-machine i n f i n i t e - b u s system with the mass-spring system modelled i n d e t a i l 20 2.6 Power v a r i a t i o n of the one-machine i n f i n i t e - b u s system with the mass-spring system lumped into one mass ....... 21 2.7 Power v a r i a t i o n of the onennachine i n f i n i t e - b u s system with the mass-spring system lumped•into one mass and a constant negative resistance inserted i n the l i n e ...... 21 3.1 Mode shapes of the s i x mass-spring system 24 3.2 A t o r s i o n a l mass-spring system 26 3.3 A damped s i x mass-spring system with u n i t y s i n u s o i d a l f o r c i n g torque 27 3.4 V a r i a t i o n of the l e a s t square error e^ v s . the v a r i -a t i o n of K i t 5 of the f i v e mass equivalent 34 3.5 V a r i a t i o n of the l e a s t square error e^ v s . the v a r i -a t i o n of K 2 3 of the four mass equivalent 34 3.6 Mode shapes of the o r i g i n a l and equivalent systems ..... 38 4.1 Dynamic responses of AI^ and A I ^ 51 4.2 Dynamic responses of the power system without c o n t r o l 57-60 4.3 Dynamic responses of the poxjer system with c o n t r o l 61-64 4.4 The c o n t r o l s i g n a l 66 4.5 The frequency sprectrum of the c o n t r o l s i g n a l 66 5.1 A two-machine power system 68 5.2 .A three-machine power system 70 v l i i Figure Page 5.3-5.6 V a r i a t i o n of the r e a l part of the unstable t o r s i o n a l modes of the two-machine system as capacitor compen-sation of the other l i n e changes 76-79 5.7 I t e r a t i v e scheme f o r adapting c o n t r o l l e r into the o r i g i n a l system 84 5.8 Two subsystems r e s u l t i n g from the two-machine system .. 85 5.9 Mode shapes of the o r i g i n a l and equivalent systems f o r machine 2 i n the two-machine system 87 5.10 The two-machine system subjected to disturbance • 90 5.11-5.12 T y p i c a l dynamic responses of the two-machine system . without co n t r o l 91-94 5.13-5.14 T y p i c a l dynamic responses of the two-machine system with c o n t r o l .....v 95-98 5.15 V a r i a t i o n of mechanical damping as weighting elements;-- .• of the damper winding currents change' 4.. ........... 103-104 5.16 V a r i a t i o n of mechanical damping as weighting elements of the st a t o r currents change 105-106 5.17 The three-macbine system subjected to disturbance . .... 107 5.18-5.20 T y p i c a l responses of the three-machine system without c o n t r o l , ...108-113 5.21-5.23 T y p i c a l responses of the three-machine system with c o n t r o l 114-119 i x ACKNOWLEDGEMENT I would l i k e to express my g r a t e f u l thanks and deepest gratitude to Dr. Yao-nan Yu and Dr. H. W. Dommel, supervisors of t h i s project, f o r t h e i r continued i n t e r e s t , encouragement and guidance during the research work of t h i s t h e s i s . Valuable advice from Dr. S. G. Hutton and Dr. M. D. Wvong during the course of t h i s study i s g r a t e f u l l y acknowledged. I am g r a t e f u l to the department of E l e c t r i c a l Engineering of the U n i v e r s i t y of B r i t i s h Columbia f o r providing me with excellent typing f a c i l i t i e s . The f i n a n c i a l support from the Natural Science and Engineering Research Council of Canada i s g r a t e f u l l y acknowledged. NOMENCLATURE General A system matrix B c o n t r o l matrix u c o n t r o l vector x state vector of the system A Q symmetric p o s i t i v e semi-definite weighting matrix R symmetric p o s i t i v e d e f i n i t e weighting matrix K R i c c a t i matrix M composite matrix as defined i n (4.7) A eigenvalue matrix of M A c system matrix of the c o n t r o l l e d system F feedback matrix as defined i n (4.21) X eigenvalue of the system X,V eigenvector matrix of A c and transpose of A £ r e s p e c t i v e l y x time d e r i v a t i v e of x 8 p a r t i a l d i f f e r e n t i a l operator s Laplace tramsform operator 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 o subscript denoting i n i t i a l c ondition T superscript denoting transpose -1 superscript denoting inverse j complex operator /-1 Mass-spring system M i n e r t i a constant K shaft s t i f f n e s s constant D damping c o e f f i c i e n t 9 rotor angular displacement i n radian to rotor speed i n per u n i t o>Q synchronous speed which i s one per unit co, base speed or 377 radian/second b q modal angular displacement O^ mode eigenvector matrix of the mass-spring system Synchronous Machine I instantaneous value of current V instantaneous value of voltage ¥ f l u x - l i n k a g e R r e s i s t a n c e X reactance 6 torque angle T g e l e c t r i c torque I terminal current V terminal voltage P^ + jQ^ generator output power d,q subscript denoting d i r e c t - and quadrature-axis stator quant i t i e s f subscript denoting f i e l d c i r c u i t q u a n t i t i e s kd,kq subscripts denoting d i r e c t - and quadrature-axis damper qua n t i t i e s a - subscript denoting armature phase qu a n t i t i e s i (=1,2,3) subscript"denoting to which machine the quantity belong Transmission l i n e X^.-R . reactance and r e s i s t a n c e of the i - t h transformer t i ' t i X^i'P^Li reactance and r e s i s t a n c e of the i - t h transmission l i n e X . reactance of the capacitor i n the i - t h transmission l i n e c l c subscript denoting q u a n t i t i e s associate with capacitor E x c i t e r and Voltage Regulator gain of the i - t h regulator T ^ time constant of the i - t h regulator T„. time constant of the i - t h e x c i t e r E i V ._ reference voltage re f Governor and Steam Turbine System K actuator gain T^,T2 actuator time constants T^ servomotor time constant a actuator s i g n a l power at gate o u t l e t T™, steam chest time constant T ^ reheater time constant T cross-over time constant F high pressure turbine power f r a c t i o n HP Fjp medium pressure turbine power f r a c t i o n F^p^ low pressure turbine A power f r a c t i o n F low pressure turbine B power f r a c t i o n LPB T^p bigh pressure turbine torque T^p medium pressure torque T ,T low pressure turbine torque L r A Lr/B 1 1. INTRODUCTION 1.1 Subsynchronous Resonance The a p p l i c a t i o n of s e r i e s capacitors to increase the power tra n s f e r c a p a b i l i t y of the transmission system i s the best a l t e r n a t i v e to cope with the ever-increasing e l e c t r i c power damand, the u n a v a i b i l i t y of generation s i t e s to b u i l d thermal e l e c t r i c power plant at heavy load centers, and the d i f f i c u l t i e s i n obtaining the r i g h t of way to b u i l d new transmission l i n e s due to envirnomental and economical considerations. However, the series-capacitor-compensated transmission l i n e w i l l cause e l e c t r i c a l resonance at c e r t a i n frequencies which may e x c i t e the mechan-i c a l mode o s c i l l a t i o n s of the steam turbine and generator mass-spring system r e s u l t i n g i n shaft damage and other detrimental e f f e c t to the power system. The term "Subsynchronous Resonance (SSR)" has been used to d e s i g -nate the o s c i l l a t i n g phenomenon of the e l e c t r i c a l and mechanical v a r i a b l e s associated with turbine-generators connected to transmission systems wit.h s e r i e s capacitor compensation. T y p i c a l example of the damaging e f f e c t due to SSR i s i l l u s t r a t e d i n Figure 1.1 ; there were two shaft f a i l u r e s at Mohave generating s t a t i o n i n 1970 and 1971 [ 1 ]. Despite the hazards of SSR, u t i l i t i e s s t i l l favor the use of s e r i e s capacitors to increase the power tr a n s f e r c a p a b i l i t y . In order to overcome the problems caused by SSR, extensive e f f o r t has been made i n analyzing the two shaft f a i l u r e s . Problems are i d e n t i f i e d as the induc-t i o n generator e f f e c t , t o r s i o n a l i n t e r a c t i o n , and t r a n s i e n t torques [ 2 ]. SSR phenomenon may occur i n two d i f f e r e n t forms : the steady state SSR which i s the r e s u l t of induction generator e f f e c t and t o r s i o n a l (a) (b) Figure 1.1 Photographs of (a) Damaged c o l l e c t o r , Mohave No. 1 (b) Cross s e c t i o n from damaged shaft * P i c t u r e s are taken from reference [ 3 ]. 3 i n t e r a c t i o n , and the transient SSR which involves the transient torques on segments of the turbine-generator shaft caused by f a u l t or switching operation i n the e l e c t r i c a l system. 1.2 Countermeasures to SSR In the past decade, many countermeasures to SSR problems are proposed. Some s i g n i f i c a n t ones are as follows: 1) S t a t i c blocking f i l t e r [ 2,4 ] High q u a l i t y f a c t o r blocking f i l t e r s are inserted per-phase i n between the high voltage winding of the step-up transformer and the ne u t r a l to block e l e c t r i c a l resonance currents at the c r i t i c a l frequencies corresponding to the t o r s i o n a l modes of the turbine-generator mass-spring system. When p e r f e c t l y tuned, the f i l t e r s w i l l block the currents at c r i -t i c a l frequencies completely. But the f i l t e r s may be detuned due to the change i n system frequency a f t e r a disturbance or the change i n ambient temperature which a f f e c t s the f i l t e r ' s parametric values. A large space i s required to i n s t a l l the f i l t e r s and the basic i n s u l a t i o n l e v e l of the step-up transformer must be increased; i t i s an expensive device. 2) Dynamic f i l t e r [ 5 ] An a c t i v e device which generates a voltage i n s e r i e s with the generator to n u l l i f y the subsynchronous voltage generated by any o s c i l l a t -ing motion of the r o t o r , thereby preventing s e l f e x c i t a t i o n due to the t o r s i o n a l i n t e r a c t i o n . It i s unaffected by the system frequency and the number of se r i e s capacitors i n s e r v i c e . But i t i s quite c o s t l y because of the complex c o n t r o l system and the requirement of an i s o l a t e d power source. 4 3) Dynamic s t a b i l i z e r [ 6,7 ] The device c o n s i s t s of t h y r i s t o r c o n t r o l l e d shunt reactors connected to the synchronous machine terminal. Control of SSR i s achieved by modulating the t h y r i s t o r switch f i r i n g angles to c o n t r o l the r e a c t i v e power consumed by the re a c t o r s . 4) Bypassing the s e r i e s capacitor a) The s e r i e s capacitor i s bypassed with the a i d of a dual gap which flashes at a lower current l e v e l to l i m i t the transient torque build-up, and the gap i s reset each time to a higher l e v e l allowing a current decay to the l e v e l f o r successful r e i n s e r t i o n of the capacitor. The dual gap f l a s h i n g scheme can reduce the tr a n s i e n t torque s i g n i f i c a n t -l y i n . b) In another scheme, c a l l e d the NGH-SSR scheme [8,9 ], a t h y r i s t o r p a i r i n s e r i e s with a r e s i s t o r i s inserted across the capacitor. By f i r i n g the t h y r i s t o r s by some co n t r o l scheme, the capacitor's charges are d i s s i p a t e d through the r e s i s t o r . However, t h i s device cannot be used to s t a b i l i z e the SSR system over a wide range of capacitor compen-sation [ 8 ] . 5) Supplemental e x c i t a t i o n c o n t r o l By t h i s method, si g n a l obtained from a properly designed con-t r o l l e r i s used to modulate the output of the e x c i t a t i o n i n response to the t o r s i o n a l o s c i l l a t i o n s of the turbine-generator, and hence provides adequate damping to the SSR system. Although the shaft f a i l u r e i ncidents are ten years o l d , many ongoing researches are s t i l l focused on SSR , i n d i c a t i n g that power engineers are s t i l l searching f o r more e f f e c t i v e and l e s s expensive means to overcome the problem. Of a l l the proposed countermeasures of SSR, the supplemental e x c i t a t i o n c o n t r o l seems to be the l e a s t expensive means which needs furt h e r i n v e s t i g a t i o n . 1.3 Previous Works of E x c i t a t ion Control of SSR E x c i t a t i o n c o n t r o l of SSR involves the use of a c o n t r o l s i g n a l to modulate the output of the e x c i t a t i o n system to enhance the damping of t o r s i o n a l modes of the generator mass-spring system. Many c o n t r o l schemes have been proposed [ 10-18 ], but i t i s very d e s i r a b l e to have a c o n t r o l l e r which can s t a b i l i z e multi-mode SSR o s c i l l a t i o n s i n a multi-machine power system over a wide range of capacitor compensation and operating c o n d i t i o n s . The major approaches of the c o n t r o l l e r design are two: . 1) Transfer function approach j 10-16 ] A l i n e a r i z e d model i s used for the c o n t r o l l e r design and i t i s mainly based upon the phase compensation concept; a t r a n s f e r function representing the c o n t r o l l e r i s assumed and the parameters of the c o n t r o l l e r are chosen such that i t can s t a b i l i z e the system. Since the generator mass-spring system representation plays a very important r o l e i n SSR studies, the c o n t r o l design based on a one lumped-mass generator model can only suppress the e l e c t r i c a l resonance phenonmena [ 10-13 ], and cannot reduce the t o r s i o n a l i n t e r a c t i o n between the e l e c t r i c a l and mechanical systems [ 12 ] . Unified electro-mechanical system model i s also used i n e x c i t -a t i o n c o n t r o l design [ 14-16 ], but the effectiveness of the c o n t r o l l e r s are v e r i f i e d on the l i n e a r i z e d model [ 14 ], and neglecting the e x c i t e r voltage c e i l i n g l i m i t s f 15,16 ]. There i s no evidence that those c o n t r o l -l e r s can s t a b i l i z e the SSR system over a wide range of operating conditions and capacitor compensation. 6 2) Linear Optimal E x c i t a t i o n Control For the c o n t r o l design, a u n i f i e d e l e c t r i c a l and mechanical model i s developed [ 17 ]. A l i n e a r i z e d model i s used for the c o n t r o l design, based upon modern con t r o l laws, and using a l i n e a r combination of feedback s i g n a l s which c o l l e c t i v e l y ensures proper damping to a l l t o r s i o n a l modes of the system [ 17,18 ]. Although a p p l i c a t i o n of l i n e a r optimal e x c i t a t i o n c o n t r o l (L0EC) to power system dynamic s t a b i l i t y c o n t r o l i s well documented [ 19-21 ], i t i s r e l a t i v e l y new i n using L0EC for SSR s t a b i l i z a t i o n . To apply L0EC to SSR problem, Yu,Wvong,and Tse had shown as the f i r s t step the f e a s i b i l i t y of l i n e a r s t a b i l i z a t i o n of SSR which i s not optimal [ 17 ] . The work was continued by Yan,Wvong, and Yu [ 18 ] to develop L0EC of SSR. The c o n t r o l was tested on both l i n e a r and nonlinear f u l l models. The r e s u l t s indicated that the LOEC can e f f e c t i v e l y s t a b i l i z e the SSR system over a wide, range of capacitor compensation and operating conditions. But the design s t i l l requires improvement, e s p e c i a l l y i n two aspects 1) The order of the model i n f 17,18 ] s h a l l be f u r t h e r reduced f o r the c o n t r o l l e r design and the c o n t r o l l e r designed must be s i m p l i f i e d to the extent that only a minimum number of measurable output feedback s i g n a l s are required to implement the c o n t r o l l e r . Techniques of furth e r reduction of the model and s i m p l i f i c a t i o n of the f i n a l con-t r o l must therefore be developed. 2) The e x c i t a t i o n c o n t r o l of SSR must be a p p l i c a b l e not only to the one-machine system but also to a multi-machine system. There i s dynamic i n t e r a c t i o n between machines i n a multi-machine SSR system, which may be divided into two categories: 7 a) The dynamic i n t e r a c t i o n of the low frequency o s c i l l a t i o n s between machines f 22 ]. b) The i n t e r a c t i o n between t h e ' t o r s i o n a l modes of the mass-spring system of d i f f e r e n t machines [ 23 ]. 1.4 Scope of the Thesis This t h e s i s deals with the output feedback l i n e a r optimal e x c i t a t i o n c o n t r o l of one-machine and multi-machine SSR systems. Chapter two r e c a p i t u l a t e s a l l the basic equations of power system models f or SSR studies. The t o r s i o n a l i n t e r a c t i o n e f f e c t between the e l e c t r i c a l and mechanical systems i s i l l u s t r a t e d . In Chapter three, modal a n a l y s i s i s applied to the generator mass-spring system and a mass-spring equiva-le n c i n g technique [ 24 ] f o r model reduction i s developed. In Chapter four, procedures of l i n e a r optimal e x c i t a t i o n c o n t r o l design are given, and the designed c o n t r o l l e r i s implemented to the f u l l model f or both eigenvalue a n a l y s i s and nonlinear dynamic performance t e s t s . In Chapter f i v e , the multi-machine SSR problem i s examined, one machine equivalents of the multi-machine system are developed, and the e x c i t a t i o n controls designed f o r a two-machine and a threennachine power system are imple-mented on the f u l l models f o r eigenvalue analysis and nonlinear dynamic performance t e s t s . Summary of a l l the important fin d i n g s i s given and a conclusion i s drawn i n Chapter s i x . 8 2. MODELLING POWER SYSTEMS FOR SSR STUDIES 2.1 Introduction To accurately simulate the transient and dynamic behaviour of a power system, a proper and adequate model must be chosen. In con-ve n t i o n a l power system s t a b i l i t y studies f o r which the low frequency o s c i l l a t i o n s (.5- 2 Hz.) i s of the main concern, the generator and turbine shaft s t i f f n e s s , the amortisseur winding e f f e c t , and the arma-ture and network transient may be neglected. However, f o r SSR studies, the emphasis i n modelling system components i s d i f f e r e n t . In order to account for the t o r s i o n a l o s c i l l a t i o n s of the mechanical mass-spring system and the t o r s i o n a l i n t e r a c t i o n between e l e c t r i c a l and mechanical systems,those f a c t o r s which are not important i n conventional s t a b i l i t y studies can no longer be neglected. In the f i r s t part of t h i s chapter, a summary of equations describing the power system for SSR studies i s given. The complete model f o r a one machine system i s shown in Figure 2.1, which c o n s i s t s of the turbine generator mass-spring system, the speed governor .and the turbine torque [ 2 5 ] , the generator and e x c i t a t i o n [ 2 6 ] , and the capacitor compensated transmission l i n e . In the second part of t h i s chapter, the complete system state equations are obtained. I t i s a p p l i c a b l e to both s i n g l e machine and multi-machine systems. Towards the end of t h i s chapter, the e f f e c t of t o r s i o n a l i n t e r a c t i o n between mechanical and e l e c t r i c a l systems i s i l l u s t r a t e d . _ Turbine_ tQJqn_e_ J 1 LPB Turbine Generator Mass-spring System T ^ ^ ^ ^ I - | (—HHP V V — T P W-i n f i n i t e Ibus Transmission Line rfif_ E Generator Dynamics Kma 1 + T. 7Z 1 + sT i E min Voltage Regulator and Exciter Figure 2.1 Power system model f o r SSR studies >J3 10 2.2 The Mechanical System Consider the torques df the steam t u r b i n e s f i r s t , they may be w r i t t e n as HP 'it* T GV CH T C H T R P (2.1) IP IP F H P T E H H P X T IP RH (2.2) LPA LP A F I P T C O I P Tco L P A (2.3) LPB where GV CH RH CO HP IP "LPA LPB H^P IP LPA LPB LPB  TLPA L P A power at gate o u t l e t steam chest time constant reheater time constant cross-over time constant h i g h pressure t u r b i n e power f r a c t i o n medium pressure t u r b i n e power f r a c t i o n low pressure t u r b i n e A power f r a c t i o n low pressure t u r b i n e B power f r a c t i o n h i g h pressure t u r b i n e torque medium pressure t u r b i n e torque low pressure t u r b i n e A torque low pressure t u r b i n e B torque (2.4) Consider the mass-spring system next. Assume t h a t there i s one hig h pressure t u r b i n e (HP), one medium pressure t u r b i n e ( I P ) , two low pressure t u r b i n e s (LPA,LPB), one generator (Gen), and one e x c i t e r ( E x ) , a l l 11 on one shaft as shown i n Figure 2.1. Although a more accurate mass-spring model i s a v a i l a b l e by modelling shaft and masses i n f i n i t e sections f 34 ], a l i n e a r mass-spring model i s recommended by an IEEE committee report for SSR a n a l y s i s f 35 ] . According to [35 ], i t i s assumed that (a) There are six lumped masses, each with i t s i n e r t i a constant. (b) The shaft between any two masses behaves l i k e a l i n e a r t o r s i o n a l spring with n e g l i g i b l e mass. (c) There i s mechanical damping to each r o t a t i n g mass, although i t i s very d i f f i c u l t to determine [ 36 ]. Figure 2.2 i l l u s t r a t e s the various t o r s i o n a l forces experienced by the i t h element on the mass-spring system. A p o s i t i v e t o r s i o n a l torque K i i + l ^ i + l " 6 i ^ ° n t h e r l S n t > a negative torque -K^_^ ±(.Q ± - ® ±_{) on the l e f t ; and an external torque T.,a p o s i t i v e a c c e l e r a t i n g torque M to , and a negative damping torque -D.^ o n t n e mass i t s e l f . A general equation of motion of the i r o t o r i s as follows M.to. = T. - D.w. + K. l i where M. 3 6 . K i . i + i i . i + l s i+1 - 9.) - K. . .(9 . - G . ,) i - r (2.5) .th i n e r t i a constant of the i u " r o t a t i o n a l mass angular displacement of the i t n r o t a t i o n a l mass t o r s i o n a l s t i f f n e s s constant of the shaft between i and i+l*"^ 1 r o t a t i o n a l mass t h damping D. Figure 2.2 Modelling of the ma^ss-spring system i n the v i c i n i t y of the i r o t a t i o n a l mass Applying equation (2.5) to the s i x mass-spring system as shown i n Figure 2.1, twelve d i f f e r e n t i a l equations are obtained : High pressure turbine Med ium pressure turbine 2 K12 fi K12 D l THP M7 82 " M — 61 " i£ Ul + t o „ = c o b ( U l - (0 O ) K. 12 fl ,K12 +K23, fi , .^ 23 M7 91 - ( M 2 } 62 + M 2 °3 c o b ( « 2 - t o o ) D T 2 . IP iq*2 + *r <2.6) (2.7) (2.8) (2.9) Low pressure turbine A = K23 fl r K23 +K34.  3 M T 62 ~ ( M „ > 93 + M T ^ D3 '. TLPA (2.10) M ^ 3 + ^ -e 3 - c o b ( «o 3 - <oo ) (2.11) Low pressure turbine B D 4 + TLPB U7% + 1 V (2.12) Generator 0. = 03, ( CO. - 10 ) 4 b 4 o 10 - *45 e r ^ - 5 6 ) 6 + e Dc T 5 e (2.13) (2.14) E x c i t e r 6 = c o , ( t o - t o ) D O 6 M T 6 - e6 " 57 w6 6 6 6 (2.15) (2.16) 6 C = U K ( ~ " ) 6 b 6 o (2.17) where 6 : e l e c t r i c a l angular displacement i n e l e c t r i c a l radian which i s equal to the mechanical radian f o r a" two-pole machine. OK : speed of the i * " * 1 r o t o r i n per u n i t . to_o: synchronous speed which i s one per u n i t . to. : base speed or 377 radian/second b 6 : mechanical angular displacement i n radian. T : e l e c t r i c torque across the a i r gap i n per u n i t . 13 written Consider a speed governor next. The state equations may be GV K 1 ~t ( " r e f " u > " " f i ~ ( P „ r - P ) T„ GV o (2.18) (2.19) where K J r e f 3 < P < P 'GV . - GV - GV min max actuator gain actuator time constant servomotor time constant actuator s i g n a l generator speed reference speed i n i t i a l power reference 2.3 The E l e c t r i c a l System A transmission system between any two buses i s shown i n Figure 2.3, where Rfc i s the transformer r e s i s t a n c e , X t i s i t s reactance, R^ the transmission l i n e r e s i s t a n c e , X^ the l i n e reactance, and X c the capacitor reactance. For a one machine i n f i n i t e system, V denotes the generator terminal voltage, V o the i n f i n t e bus voltage, and V c the voltage across the capacitor. o v. R. A A A / A A / V X Figure .2.3 The transmission system 14 The general voltage equation becomes t V t W e = [ R 1 [ X t ] Phase + ^ ^ [ \ ] Phase + [ V c W s e + [ V 1 < 2 - 2 ° ) o phase In Park's coordinates and f o r a balanced three-phase operation the d and q components of the generator terminal voltage become V d - < > h - ( \ + *L > X q + ( \ + *L > X d + V c d + Vod ( 2 - 2 1 ) X + X, V q - < > *q + < \ + *L > T d + < \ + \ > \ + V + V (2.22) cq oq and the two i n f i n i t e bus voltage components are V , = V sin6 od o o V = V cosS oq o o (2.23) where V q i s the magnitude of the i n f i n i t e bus voltage, and 6 q i s the angle between machine terminal and the i n f i n i t e bus. For a multi-machine system, V q i s not the i n f i n i t e bus voltage anymore. The capacitor voltages i n the d-q coordinate become V C D = V + co X I j (2.24) to, cq c d V C Q -V , + to X I (2.25) 0)^ . cd c q 15 Equations for e x c i t e r and voltage regulator are hi ti \ = 4t( V r e f " \ + U E > < 2' 2 7> A V t = 7 V d + V q (2.28) V < V < V R . - R - R min - max where Ug i s the supplementary c o n t r o l s i g n a l K i s the voltage regulator gain i s the voltage regulator time constant T„ i s the e x c i t e r time constant hi E^p i s the e x c i t e r output voltage V and V n are the regulator c e i l i n g voltage K . R mxn max Previous work [ 18 ] found that f i v e winding generator model with one damper winding on each of the d,q-axis i s s u f f i c i e n t . The voltage equations of synchronous generator are [37,38] • V. = — - -0)4' - R I, (2.29) d o^ q a d V = —9- + OJ'F, - R I (2.30) q 0)^ d a q V f = + Rf I f (2.31) 0 = R ^ I ^ (2.32) 0), Icq kq 16 and the e l e c t r i c torque equation i n per u n i t i s T = T, I - y I e d q q d where the f l u x linkage (2.34) Xmd Xmd * q -X q X mq I q * f - Xmd X f Xmd x f *kd Xmd Xmd X k d x k d kq J ... -X mq kq J Aq, (2.35) 2.4 Complete System Equations f or Single and Multi-machine Systems The o r i g i n a l nonlinear equations are used f o r time domain simulation , and the equations are l i n e a r i z e d f o r eigenvalue a n a l y s i s . The l i n e a r i z e d e l e c t r i c a l system equations may be written i n the matrix form as follows Hence where [ B ] [ X ^ ] = [ C ] [ ^  ] + [ D ] [ X I I ] (2.36) t x n i - = n v i + t A i i , n n x i i 1 ( 2 - 3 7 ) -1 ] = [ B ] ] (2.38) (2.39) [ Xj. ] and" [ Xj.^. ^  ] represent the i n t e r a c t i o n between the mechanical system and the e l e c t r i c a l system which can be combined with the l i n e a r i z e d mechinical system equations to give h,I VII ^I.I E^I.II v r ) i x ) II (2 .AO) where [ Xj. ] = [ Acox , A 9 i ,Aw 2 , A 9 2 ,Aco3 , A 9 3 .Aw^ ,A9^ ,Au),A5 ,Ato6 , A 9 6 , ,T (2.41) A a ' A P G V ' A T H P ' A T I P ' A T L P A ^ [ X I I ] = [ A I d , A I q , A I f , A I k d , A I k q f A V c d f A V c q , A V R , A E f d ] T (2.42) Equa t ion (2.40) may a l s o be w r i t t e n compactly as [ X ] = [ A ] [ X ] (2.43) For a m u l t i - m a c h i n e system where more than one machine i s i n v o l v e d , the i n d i v i d u a l machine c o o r d i n a t e (d,q) may be r e f e r r e d to common r e f e r e n c e frame (D,Q) as f o l l o w s : q - a x i s Q-axis d - a x i s D-ax i s F i g u r e 2.4 Component of I i n dq and DQ c o o r d i n a t e s 18 where cos0 -sin9 sinG cos9 Linearized, . 4 I Q . cos9 Q -sinBo s i n 0Q, cos0o Therefore . ' V cos0 g -sin0Q sin9o cos0 Q Inversely, I cos© sin0 -sin9 cos0 (2.44) f AI, ] f d + AI <3'. I I q J " ( l d o S i n 9 ° + I q o C O S ® ° ) i ( I , cos9 n - I sin9n) do u qo • u A8 (2.45) f AI^ 1 d 0 + AI I q J - ( I d o s i n 9 0 + I q o c o s 9 0 ) ( I d o c o s 9 0 - I _ s i n 9 0 ) qo A9 (2.46) f I 1 D I ) l Q J (2.47) L i n e a r i z e d f 1 d AI I q J cos9 0 s i n 9 0 -sin9g cos9g Therefore d AI q . cos9 0 sin9g -sin9 0 cos9g r + • \<\ f + / . V Do - ( I D o c o s 9 0 + I Q o s i n 9 0 ) A6 (2.48) ( I Q o c o s 9 0 - I D o s i n 9 0 ) - ( I D o c o s 8 0 + I n o s i n 0 o ) Qo A 9 (2.49) For the multi-machine formulation, i t i s convenient to write T x T - = t x I 1 I 1 "II ' X I 2 > X I 3 ' '•'» X I i ] Y = f y y y i r H P i i 2 ' i i 3 ; x l l i ] (2.50) where X.,. , X ,.. are the mechanical and e l e c t r i c a l state v a r i a b l e s of I i * I I I machine i , r e s p e c t i v e l y . 2.5 The Torsional Interaction When an e l e c t r i c a l resonance occurs i n the e l e c t r i c a l system (i n c l u d i n g the generator stator and transmission l i n e ) at a subsynchronous frequency f i t w i l l i n t e r a c t with the rotor and induce a p u l s a t i n g torque at the frequency of ( 60 - f ), which becomes a f o r c i n g torque to the mass-spring system. I f the frequency of the o s c i l l a t i n g torque equals to a t o r s i o n a l modal frequency f , the e l e c t r i c a l resonance and the p a r t i -c u l a r t o r s i o n a l mode xtfill be mutually excited, and a voltage w i l l be induced i n stator winding at frequency f g = 60-f . Thus the t o r s i o n a l i n t e r a c t i o n looks l i k e a negative resistance to the e l e c t r i c a l system and negative damping to the t o r s i o n a l system under these conditions. To i l l u s t r a t e the e f f e c t of t o r s i o n a l i n t e r a c t i o n , the one machine i n f i n i t e bus system as shown in Figure 2.1 i s tuned so that SSR w i l l occur. The v a r i a t i o n of e l e c t r i c power which i s approximately equal to the e l e c t r i c torque i n the per unit system, as shown i n Figure 2.5, roughly c o n s i s t s of two components: a low frequency o s c i l l a t i o n i n the range of 1 - 2 Hz. , and a higher frequency o s c i l l a t i o n . In t h i s p a r t i c u l a r case, the subsynchronous torque increases with time. The r e s u l t s of Figure 2.5 also can be synthesized as follows : F i r s t l e t the system be modified by lumping the s i x t o r s i o n a l masses into one. The power v a r i a t i o n s for the same system disturbance are shown as Figure 2.6. The responses as shown i n Figure 2.5 and 2.6 are the same within 0.5 second but there i s no subsynchronous torque component i n 20 Figure 2.6 because of the modelling. A constant negative resistance i s then inserted i n the t r a n s -mission l i n e of the above modified system. The system response for the same disturbance i s shown i n Figure 2.7. Although the mass-spring system model has been s i m p l i f i e d , the subsynchronous torque component i s substan-t i a l . The concept that the t o r s i o n a l system looks l i k e a negative r e s i s -tance to the e l e c t r i c a l system i s further v e r i f i e d . Of course, a s t a t i c negative r e s i s t a n c e representation of the t o r s i o n a l i n t e r a c t i o n i s over-s i m p l i f i e d i n the SSR studies. Figure 2.5 Power v a r i a t i o n of the one-machine i n f i n i t e bus system with the mass-spring system modelled i n d e t a i l . i i r 2.0 3.0 TIME (SEC) r 4.0 5.0 Figure -2.6 Power v a r i a t i o n of the one-machine i n f i n i t e bus system with the mass-spring system lumped in t o one mass. o:c 1 i I r I rw V \ o " 0.0 1.0 , , [— 2.0 3.0 TIME (5EC) — i — 4.0 5.0 Figure 2.7 Power v a r i a t i o n of the one-machine i n f i n i t e bus system with the mass-spring system lumped into one mass and a constant negative r e s i s t a n c e inserted i n the transmission l i n e . 22 3. MODEL REDUCTION OF A POWER SYSTEM FOR SSR STUDIES 3.1 Introduction The requirement of including the t o r s i o n a l mass-spring system the generator , and transmission system i n one si n g l e u n i f i e d model f or SSR studies so that the t o r s i o n a l i n t e r a c t i o n of the e l e c t r i c a l and mechanical systems w i l l be automatically included i n e v i t a b l y r e s u l t s i n a very high order system. For instance, the f u l l model of a one-machine i n f i n i t e - b u s system f o r SSR studies developed i n the l a s t chapter i s of 26th order. For some SSR studies, e s p e c i a l l y f o r the co n t r o l design, a reduced order model i s very d e s i r a b l e . For an e x c i t a t i o n c o n t r o l design, the steam turbine torque and governor equations together with a small time constant of the exciter can be neglected. The order of the model i s reduced from 26th to 20th. However, further order reduction i s s t i l l necessary. Anew technique of obtaining a reduced order equivalent mass-spring system i s developed i n t h i s chapter by r e t a i n i n g the unstable t o r s i o n a l modes alone without changing t h e i r o s c i l l a t i n g frequencies. The f i r s t step i s to determine the r e l a t i v e i n s t a b i l i t y among the unstable modes using modal a n a l y s i s . The second step i s to r e t a i n only the unstable t o r s i o n a l modes. F i n a l l y , the eigenvalues of the o r i g i n a l model and the reduced order models are compared. 23 3.2 T o r s i o n a l Resonance and the Unstable Modes The p r e v a i l i n g techniques i n SSR s t a b i l i t y studies are either the frequency scanning method together with the t o r s i o n a l i n t e r a c t i o n equations [ 27 1 t o determine the s t a b i l i t y of the t o r s i o n a l modes one at a time, or apply e i t h e r eigenvalue a n a l y s i s or Nyguist c r i t e r i a [ 28 ] to the u n i f i e d electro-mechanical power system. Without the a n a l y s i s , no one could c o n f i d e n t l y predict which mode i s more vulnerable to t o r -s i o n a l o s c i l l a t i o n s than others. In t h i s section, a technique to i d e n t i f y the e x c i t a b l e t o r s i o n a l modes, or the mode which i s vulnerable to t o r s i o n a l i n t e r a c t i o n , and to determine the r e l a t i v e i n s t a b i l i t y of the unstable modes i s presented. The technique requires no information about the e l e c t r i c a l network, and i s based on the modal a n a l y s i s of the t o r s i o n a l mass-spring system alone. 3.2.1 Natural Frequencies and Mode Shapes of the Mass-spring System The n a t u r a l frequencies and the mode shapes of the turbine generator mass-spring system can be found as f o l l o w s : Consider an unforced and undamped mass-spring system as shown on top of Figure 3.1. The system can be described by a set of second order d i f f e r e n t i a l equations i n matrix form as follows 1 [ M ] [ e ] + [ K ] [ e ] = o (3.1) GO b where t M ] = diag [ Mj, M2, M3, M^, M5, M5 ] (3.2) [ 9 ] = T (3.3) 24 F i g u r e 3.1 Mode Shapes of the s i x mass-spring system 25 [K] = K 1 2 _ K 1 2 0 0 0 0 - K 1 2 (K 1 2+ K 2 3 ) -K 2 3 0 0 0 0 -K 2 3 (K 2 3+ K 3 t f) - K 3 4 0 0 0 0 (K 3 t t+ K^s) - K 4 5 0 0 0 0 - K 4 5 ( K 4 5 + K 5 6 ) - K56 0 0 0 0 -K 5 6 K 5 6 (3.4) where the subscripts 1,2,3,...,6 correspond to HP, IP, LPA, LPB, Gen, Ex r e s p e c t i v e l y . Assume that a l l masses o s c i l l a t e at a resonant frequency to^ , such that 6 . = X. sin( to t + a ) x 1 m i = 1,2, (3.5) S u b s t i t u t i n g equation ( 3.5) into (3.1) gives , . , 2 or where b ,-1 [ M ] 1 [ K ] [ X ] = 0 [ M ] • k [ K ] [ X ] = X [ X ] m A = t o 2 / to, m m b (3.6) (3.7) (3.8) Therefore X can be obtained by solving (3.7) and mode frequency to m can be calculated from m /~X t oT m b m = 0,1,...,5 (3.9) There i s an eigenvector X^ corresponding to each eigenvalue X^ , which, when normalized with respect to the element with the la r g e s t magnitude gives, the mode shape of that p a r t i c u l a r mode. The mode frequencies and the mode shapes of the mass-spring system are given i n Figure 3.1. A mode shape g i v e s the r e l a t i v e displacement of each spring-mass during normal mode v i b r a t i o n ( when one p a r t i c u l a r mode i s e x c i t e d ) , but i t g i v e s no inf o r m a t i o n about which mode i s more uns t a b l e than the ot h e r s . An a l t e r n a t i v e method w i l l be introduced to overcome t h i s d i f f i c u l t y i n the subsequent s e c t i o n s . 3.2.2 Damping and the Resonant Peak constant M, a s t i f f n e s s constant K, a damping c o e f f i c i e n t D, and a f o r c i n g f u n c t i o n T sincot Figure 3.2 shows a t o r s i o n a l mass-spring system w i t h an i n e r t i a o D zvFigure 3.2 A t o r s i o n a l mass-spring system The equation of motion of the system i s T = M 8 + D0 + K9 (3.10) For a s o l u t i o n G = X s i n ( cot - <f> ) , we have T s i n cot o ( K - Mco2 ) s i n ( uit - < ( ) ) + Deo cos( cot — <{> ) (3.11) X In phasor n o t a t i o n T o = ( K - Mco2 ) + j Deo (3.12) X 27 Hence T X = • (3.13) ( K - Mo2 ) + j Deo and the phase angle, <j> i s given by tan- *} = ^ — (3.14) (K - Ma>2) When the frequency of the f o r c i n g function equals : the n a t u r a l frequency of the system, the r e a l part of the denominator of (3.13) vanishes and the equation becomes x - -~^r (3*15> During resonance, the amplitude X i s d i r e c t l y p roportional to the applied force and inversely proportional to the system damping. There-f o r e , a large applied force together with a small system damping w i l l r e s u l t i n a l a r g e resonant peak. 3.2.3 Resonant Peaks of the Six Mass-spring System s i n tut 1 4 HP IP LPA LPE Gen Ex Figure 3.3 A damped s i x mass-spring system with u n i t y s i n u s o i d a l f o r c i n g torque. Consider a damped mass-spring system with u n i t y s i n u s o i d a l f o r c i n g torque s i n cot applied on the generator rotor as shown i n Figure 3.3. The matrix equation in per u n i t describing the dynamical behaviour of the system becomes - J ^ _ ; p M ] •[ 8 ] H i — [ D ] [ 9 ] + [- K ] [ 9 ] = [ T ] (3.16) 28 where [ M ] and [ K ] re s p e c t i v e l y are the i n e r t i a constant and s t i f f n e s s c o e f f i c i e n t matrices as shown i n equations (3.2) and (3.4), and [ D ] = diag [ D L , D 2 , D 3 , D 4 , D 5 , D 6 ] [ T ] = [ 0 0 0 0 s i n tot 0 ] T Let the mechanical angular displacements • [ 0 ] be transformed into modal angular displacements [ q ] by the eigenvector matrix ['Q m o^ e]» from the eigenvalue a n a l y s i s of the undamped system re ] - [ Q ^ H q ] '(3.17) S u b s t i t u t i n g (3.17) into (3.16) and premultiplying the whole equation by [ ^ o d e ^ > w e h a v e — [ M . ] [ q ] + — [ D , ] [ q ] + [ K . ] [ q- ] to^ modeJ 1 M J eo, mode ^ J mode ^ where (3.19) [ M , ] = [ Q j ] T [ M ] [ Q , ] modeJ L xmode 1 L xmodeJ [ K , ] = [ Q , ] T [ K ] [ Q , ] mode ^mode 1 xmode mode mode 1 Tnode [ T ] = [ Q , ] T [ T ] mode mode Note that [ T , ] indicates the contr i b u t i o n of the applied force on mode each mode of v i b r a t i o n ; both [ M , ] and I K' , .] are diagonal matrices ' mode mode because [ M ] and [ K ] are symmetrical. Neglecting the off diagonal elements of [ D m o < j e l » equation 29 (3.19) becomes s i x second order d i f f e r e n t i a l equations each of which corresponds to (3.10). Applying (3.15), the ca l c u l a t e d magnitude of the resonant peaks f o r various modes for the uni t y f o r c i n g function s i n cot of Figure 3.3 are shown in Table 3.1 . Mul t i p l y i n g the modal resonance peaks by i t s corresponding mode shapes as shown i n Figure 3.1, gives the r e l a t i v e magnitude of the 'angular displacement' of each mass at the resonant frequencies. Results are shown i n Table 3.2. Table 3.1 Modal resonance peaks of the mass- j -spring system j mode 1 6.47 mode 2 1.00 mode 3 3.075 mode 4 4.56 mode 5 0.035 Table 3.2 Magnitude of the resonant peaks of each mass using approximate modal an a l y s i s HP IP LPA LPB Gen Ex mode 1 4.97 3.73 2.20 0.72 2.39 6.47 mode 2 0.125 0.07 0.017 0.05 0.04 1.00 mode 3 3.075 1.04 0.70 0.29 0.51 0.77 mode 4 3.94 0.20 2.29 4.56 2.83 1.72 mode 5 .0.027 0.035 0.004 0.0007 0.0002 0.00003 30 3.2.4 I d e n t i f i c a t i o n of the Unstable Torsional Modes Neglecting the o f f diagonal elements of [ n m 0 £ j e l > - equation (3.18) becomes s i x second order d i f f e r e n t i a l equations, each corresponds to one mode of v i b r a t i o n . Excluding mode 0, we s h a l l have co, mode.^i uj, mode.^i mode.^i mode, b I b I I i , i=l,..,5 (3.20) when the un i t y f o r c i n g torque sincot i s applied to the system, i t i s found that i s t h i s p a r t i c u l a r study • Tmode ] = f °- 3 7 3» 0.0374, 0.166, 0.6205, 0.0045 ] In the mean time l.D . ] = [ 0.22, 0.102, 0.127, 0.253, 0.163 1 • mode . When these r e s u l t s are compared with those from eigenvalue a n a l y s i s as shown i n Table 3.3 Table 3.3 " ~ " Mode Frequency Comp en sat ion Eigenvalue 0 1 - 2 Hz below 30% 1 15.7 Hz above 70% +1.7178+J102.16 4 32.3 Hz at 50%,60% +0.7094±j203.32 3 25.5 Hz at 40%,50%,60%,70% +0.5595+J161.00 there i s an i n d i c a t i o n that i f ( d mode t , ) < 0 mode (3.21) the p a r t i c u l a r t o r s i o n a l mode i s vulnerable to i n s t a b i l i t y i n t h i s p a r t i c u l a r study. 31 3.2.5 Modal Resonance Peaks and Unstable Modes The modal resonance peaks i n Table 3.1 in d i c a t e the r e l a t i v e i n s t a b i l i t y among the t o r s i o n a l modes. A large modal resonance peak means a l a r g e amplitude of v i b r a t i o n , r e s u l t i n g i n large negative damping due to the t o r s i o n a l i n t e r a c t i o n , and v i c e versa. In a d d i t i o n a l to the si x mass-spring system, several other systems are investigated using both eigenvalue a n a l y s i s of the u n i f i e d electro-mechanical model and modal resonance peak a n a l y s i s of the mass-spring system. A l l r e s u l t s suggested that the modal resonance peaks can be used as an index to determine the r e l a t i v e i n s t a b i l i t y among the un-stable modes. In t h i s p a r t i c u l a r case, mode 1 w i l l be the most unstable one followed by mode 4 and mode 3 i n that order. 3.2.6 Other uses of modal analysis, i n SSR Studies S t a b i l i t y of a SSR system also depends upon the conditions of the e l e c t r i c a l system. However, the modal an a l y s i s i s a very u s e f u l f i r s t step to i d e n t i f y the ex c i t a b l e modes so that the range of frequency scan-ning can be narrowed, and the number of equations can be reduced. With the e x c i t a b l e modes i d e n t i f i e d , one can also construct equivalent mass-spring system by r e t a i n i n g only the unstable modes alone, which w i l l be presented i n the next section. 32 3.3 Equivalent Mass-spring system As shown i n Table 3.3, there are only three t o r s i o n a l modes i n the study system. A low order equivalent mass-spring system may be obtained by r e t a i n i n g only the unstable modes as follow: 1) Mode 2 in Figure 3.6a i s roughly equal to the exc i t e r mass swings against the rest of the mass-spring system which can be eliminated according to the ongoing a n a l y s i s . Combining the generator (Gen) and the ex c i t e r (Ex) masses together, eliminating mode 2, and keeping a l l other natural frequencies by adjusting the s t i f f n e s s constant K 4 5 betvreen the low pressure turbine B (LPB) and the generator (Gen) , r e s u l t i n an equivalent f i v e mass-spring system. 2) The same procedure i s applied to the high pressure turbine (HP) and the medium pressure turbine (IP). Combining the two masses and elimin a t i n g mode 5 by adjusting the s t i f f n e s s constant K23 between the medium pressure turbine (IP) and the low pressure turbine A (LPA) r e s u l t i n an equivalent four mass-spring system. 3) F i n a l l y , the procedure i s applied to the two low pressure turbines (LPA) and (LPB). Combining the two masses and elminating mode 1 by adjusting both s t i f f n e s s constant K 23 and K^ 5 but one at a time, r e s u l t i n an equivalent three mass-spring system. 3.3.1 Mode Frequencies and the Adjustment of the S t i f f n e s s Constant Mode frequencies of the mass-spring system depend upon the i n e r t i a constants and the s t i f f n e s s constants. To r e t a i n other natural frequencies a f t e r combining the two neighbouring masses into one, the . st i f f n e s s - c o n s t a n t s must be v a r i e d , which can be found as follows: 33 1) The i n i t i a l eigenvalues are found from the c h a r a c t e r i s t i c equation of the s i x mass-spring system, each corresponds to a second order equa-t i o n , and together they may be written i n matrix form as I M I ] - [ M K ] | = 0 (3.22) where [ M ] i s the i n e r t i a constant matrix of appropriate dimension [ K ] i s the spring c o e f f i c i e n t matrix [ I ] i s the i d e n t i t y matrix X = - to2/to, , to, = 377 rad/sec. D D 2) To f i n d a reduced order mass-spring system, one of the s t i f f n e s s constant, say K „ , i s treated as an. unknown. By s u b s t i t u t i n g the known X's already obtained from step (1) one at a time, an average K ' i s obtained. 3) The average K value i s substituted into equation (3.22) of the reduced order system, and eigenvalues are r e c a l c u l a t e d . The value i s then adjusted according to the s e n s i t i v i t y c o e f f i c i e n t , 3wV(8K.j) , u n t i l the l e a s t square error of the natural frequencies of the equivalent reduced order system i s the minimum. The l e a s t square error n . e = £ ( to. - t o . Y (3.23) i=l desired retained versus the v a r i a t i o n of K i ^ of the equivalent fi v e , mass-spring system • and K 2 3 of the four mass-spring system are shown i n Figure 3.4 and 3.5 r e s p e c t i v e l y . 34 2.OH 70.858 71.858 72.858 73 .858 K 45 Figure 3.4 V a r i a t i o n of the l e a s t square error e v s . the", v a r i a t i o n of K 4 5 of the f i v e mass equivalent. 23.5 24.0 24.5 25.0 25.5 K, 2 3 Figure 3.5 V a r i a t i o n of the least square e r r o r e vs. the v a r i a t i o n of K 2 3 of the four mass equivalent. 3.3.2 Mode Shapes of the O r i g i n a l and Equivalent Systems 35 The eigenvalues of a system are unique, but the eigenvectors are not and can be normalized or m u l t i p l i e d by any non-zero s c a l a r . A l -though an equivalent reduced order mass-spring system r e t a i n s a l l the dominant eigenvalues of the o r i g i n a l system, the v a l i d i t y remains to be proven: a reduced order model of the equivalent system s h a l l have almost the same dominant eigenvector of the o r i g i n a l system. In a d d i t i o n , the modal displacement of M Q e n s h a l l be close to that of the o r i g i n a l system so that the t o r s i o n a l i n t e r a c t i o n between the e l e c t r i c a l and mechanical system can be accurately accounted f o r . In t h i s section, the normalized eigenvectors of the o r i g i n a l and reduced order mass-spring systems are compared. Excluding mode 0, by which a l l the masses swing i n unison, the eigenvectors of the o r i g i n a l system before normalization are ' -1 .0428 -1.5584 -2.2820 +0.5914 -0.6161 -0 .7833 -0.9168 -0.4041 -0.0299 +0.7825 -o .4595 -0.2128 +0.2713 -0.3442 -G.0886 +0 .1499 +0.5601 +0.1127 +0.6847 +0.0165 +0 .5007 +0.5297 -0.1960 -0.4248 -0.0350 +1 .3420 -14.182 +0.2982 +0.2579 +0.0007 (3.24) Afte r normalization, they become -0.7770 +0 .1099 1 +0 .864 -0 .7870 -0.5840 +0 .0650 +0.3420 -0 .0440 1 -0.3420 +0 .0150 -0.22 90 -0 .5030 -0 .1130 +0.1120 -0 .0400 -0.0950 1 +0 .0210 +0.3730 -0 .0370 +0.1660 -0 .6210 -0 .0045 1 1 -0.2530 +0 .3770 +0 .0009 (3.25) A f t e r eliminating mode 2, the eigenvectors cf the reduced order f i v e mass equivalent system by r e t a i n i n g mode frequencies 15.94 Hz., 25.46 Hz., 32.28 Hz., and 47.46 Hz., before normalization are +1.0642 +0.9543 +0.6536 +0.6161 +0.7916 +0.3310 -0.0332 -0.7825 +0.4532 -0.2135 -0.3805 +0.0887 -0.1719 -0.1009 +0.7573 -0.0166 -0.5087 +0.1467 -0.4414 +0.0034 J Scaling the modal eigenvectors with respect to the c normalization ( e.g., x column 1. . " J - J ™ column 3, 1 x column 4 ) gives , -1.0428 -1.1810 +0.5914 +0.6161 -0.7757 -0.4096 -0.0300 -0.7825 -0.4441 +0.2642 -0.3443 +0.088? +0.1684 +0.1249 +0.6852 -0.0166 +0.4985 -0.1816 -0.3994 +0.0034 Normalized with respect to the o r i g i n a l system y i e l d • -0.7770 1 +0.8630 -0.7870 -0.5780 +0.3470 -0.0400 1 -0.3310 -0.2340 -0.5020 -0.1130 +0.1250 -0.1060 1 +0.0210 +0.3710 +0.1540 -0.5830 -0.0045 (3.26) i o 0-5914 x column I, T,- ^  X '..0.6536 (3.27) (3.28) After eliminating mode 5, by r e t a i n i n g the mode frequencies 15.82 Hz., 25.49 Kz., and 32.47 Hz., the eigenvectors of the four mass equivalent system before normalization are 37 -0.9534 -0.4477 +0.1785 +0.5174 -0.8973 +0.3383 +0.1617 -0.2231 +C.3436 -0.4378 +0.7580 -0.4340 ( 3 . 2 9 ) Scaling the eigenvectors with respect to the o r i g i n a l system before 0.2713 -i . . , 0.4595 . . normalization ( e.g., n ,, -,v x column 1 0.6847 0.7580" 0.4477 x column 3 ) we have 0.3383 x column 2. -0.9785 -0.4594 +0.1832 +0.5269 -0.7196 +0.3103 +0.2713 -0.3955 +0.1297 +0.6847 -0.1869 -0.3920 (3.30) Normalization y i e l d s -0.7290 -0.3420 +0.1360 +0.3930 +0.6090 -0.2300 -0.1090 +0.1970 +0.4530 -0.5780 1 -0.5730 (3.31) Using the r e s u l t s i n (3.25), (3.28), and (3.31), the mode shapes of the o r i g i n a l , the f i v e mass equivalent, and the four mass equivalent systems are shown in Figure 3.6a, 3.6b, and 3.6c r e s p e c t i v e l y . Note that the mode shapes of the equivalent systems are close to those of the o r i g i n a l system. Figure 3.6 Mode Shapes of the o r i g i n a l and Equivalent Systems co 39 3 . 3 . 3 E igenva lues o f . t h e o r i g i n a l and reduced order models A f t e r r e d u c i n g the order of the o r i g i n a l system model from 26 th to 20 by n e g l e c t i n g t u r b i n e torque and governor e q u a t i o n s , e t c . , the order i s f u r t h e r reduced u s i n g mass-spr ing e q u i v a l e n c i n g t echn ique developed i n 3 . 3 , r e s u l t i n g i n a 16th or a 14th o r d e r mode l . The 14th order model , however, i s v a l i d up to 70% c a p a c i t o r compensation because mode 1 i s not cons idered i n the model . E igenva lues of the 2 6 t h , 1 6 t h , and 14th order models over a wide range o f c a p a c i t o r compensation are examined. T y p i c a l v a l u e s a re shown i n Table 3.4 to 3 . 6 . The 16th o rder model r e t a i n s a l l the dominant e igenva lues over a wide c a p a c i t o r range , and the 14th o rder model a l s o r e t a i n s most of the dominant p r o p e r t i e s except f o r v e r y h i g h compensat ion. 40 Table 3.4 Eigenvalues of various order SSR model at 30% capacitor compensation for P = 0.9 p.u. at 0.9 power fa c t o r lagging and V =1.0 p.u. 26th order model reduced 16th order model reduced 14th order model Mechanical modes -0.l818±j298.18 -0.4938±j203.59 -0.2513±jl60.64 -0.6705±jl27.02 -0.2810±j99.136 -0.0031±j8.4105 -0.4046±j204.71 -0.2069±jl60.05 -0.2289±j99.826 -0.0399±j8.4284 -0.2723±j200.37 -0.4052+J157.16 -0.0229±j8.4469 Turbine and Governor -0.1418 -4.5826 -3.1056 -4.6721±j0.5722 Stator and Network -7.0419+j542.94 -5.5469±j210.33 -7.0444±j542.93 -5.6283±j210.34 -7.044 0±j 54 2.93 -5.6596±j210.69 Machine rotor -8.6469 -31.876 -2.0220 -8.5228 -32.611 -2.2563 -8.5180 -32.627 -2.2569 E x c i t e r and voltage regulator -499.97 -101.94 -101.57 -101.56 41 Table 3.5 Eigenvalues of various order SSR model at 50% coapcitbr compen-sation f o r P = 0.9 p.u. at 0.9 power factor lagging and V =1.0 p.u. 26th order model reduced 16th order model reduced 14th order model Mechanical modes -0.l818±j298.18 +0.1237±j202.88 +0.2603±jl61.38 -0.6829±jl27.06 -0.3524±j99.345 -0.2327±j9.4692 +0.1033±j204.01 +0.2341±jl61.72 -0.3053*3100.05 -0.2566±j9.4905 +0.0167±j200.32 +1.0147±jl60.11 -0.2408±j9.5172 Turbine and governor -0.1418 -3.8415 -3.5122 -4 ,8239±j0.2945 Stator and Network -7.0969+J591.27 -6.1493±jl61.74 -7.0987±j591.27 -6.0584±jl61.72 -7.0986±j591.27 -6.9236±j159.65 Machine rotor -8.2324 -32.776 -1.9458 -8.3289 -32.776 -1 .9458 -8.2283 -33.475 -2.1670 E x c i t e r and voltage regulator -499.97 -101.76 -101.76 -101.44 42 Table. 3 .6 E igenva lues of v a r i o u s order SSR model at 80% c a p a c i t o r compensation f o r P = 0 .9 p.-u. at 0.9 power f a c t o r l a g g i n g and V = 1.0 p . u . e L 26th order model reduced 16th order model reduced 14th o rder model Mechan ica l modes - 0 . l 8 l 8 ± j 2 9 8 . 1 8 + 0 . 0 1 3 4 ± j 2 0 2 . 9 1 - 0 . 0 9 6 7 ± j l 6 0 . 5 2 - 0 . 5 9 9 8 ± j l 2 6 . 9 5 + 1 . 7 1 7 8 ± j ' l 0 2 . 1 7 - 0 . 7 6 1 9 ± j l l . 6 6 2 + 0 . 0 1 2 5 ± j 2 0 4 . 0 4 - 0 . 0 4 9 6 ± j l 5 9 . 9 3 + 2 . 1 9 9 9 ± j l 0 2 . 7 8 - 0 . 7 6 6 9 ± j l l . 6 8 2 -0.04 0 9 ± j 2 0 0 . 3 3 + 0 . 2 9 8 l ± j l 5 6 . 5 9 - 0 . 7 5 7 2 ± j l l . 7 3 7 Turb ine and governor -0.1419 - 3 . 4 7 8 4 ± j 0 . 5 9 6 0 -4 . 9841± j0 .07 92 S t a t o r and Network - 7 . 1 7 1 0 ± j 6 4 8 . 0 8 - 6 . 7 6 5 4 ± j l 0 3 . 0 2 - 7 . 1 5 2 3 ± j 6 4 8 . 0 8 - 7 . 1 3 2 8 ± j l 0 3 . 0 6 - 7 . 1 5 2 3 ± j 6 4 8 . 0 8 - 5 . 0 8 4 8 ± j l 0 6 . 2 6 Machine r o t o r -7.6295 -35.049 -1 .7807 -7.5898 -35.594 -1.9481 -7.5873 -35.613 -1.9489 E x c i t e r and v o l t a g e r e g u l a t o r -499.97 -101.43 -101.15 -101.15 43 4. EXCITATION CONTROL DESIGN 4.1 Introduction The main objective of the c o n t r o l l e r design i s to s t a b i l i z e a l l unstable modes over a wide range of power and capacitor compensation with minimum number of feedback s i g n a l s . For the multi-mode s t a b i l i z a t i o n of t o r s i o n a l o s c i l l a t i o n s , the phase compensation power system s t a b i l i z e r with s i n g l e s i g n a l input i s i n -adequate for the narrow frequency band s e n s i t i v i t y . I t may also have d e t r i -mental e f f e c t s on other t o r s i o n a l modes [ 17 ] . M u l t i p l e loop lead-lag compensation e x c i t a t i o n c o n t r o l has also been designed [ 14 ], but remains to be improved. In t h i s chapter, the state regulator problem of c o n t r o l theory i s applied, and the l i n e a r optimal e x c i t a t i o n c o n t r o l i s designed for the m u l t i p l e t o r s i o n a l mode s t a b i l i z a t i o n . I t i s a l i n e a r combination of many system feedback s i g n a l s . Instead of the phase compensation, the l i n e a r combination of feedback signals according to the c o n t r o l law c o l l e c t i v e l y ensures proper damping for a l l t o r s i o n a l modes. In engineering p r a c t i c e , i t i s also d e s i r a b l e to have the mini-mum number of feedback signals which can be e a s i l y measured. Due to the complexity of the SSR problem, the order of the system model i s u s u a l l y very high and the l i n e a r optimal co n t r o l designed u s u a l l y r e q u i r e s a large number of feedback signals [ 17,18]. Suboptimal co n t r o l alqorithms are avaiable in the c o n t r o l l i t e r a t u r e [ 29,30 ], but heavy computation i s involved. In t h i s chapter, a d i f f e r e n t suboptimal e x c i t a t i o n c o n t r o l design technique i s presented. The l i n e a r optimal e x c i t a t i o n c o n t r o l of SSR i s 44 designed i n the usual way, but the c o n t r o l l e r i s s i m p l i f i e d by r e j e c t i n g some feedback si g n a l s which have the l e a s t e f f e c t on the system damping, determined from an eigenvalue s e n s i t i v i t y a n a l y s i s . Formulation w i l l be given, suboptimal e x c i t a t i o n c o n t r o l w i l l be designedj and the r e s u l t s of both eigenvalue a n a l y s i s and time domain simulation of the SSR c o n t r o l of a power system w i l l be presented. . 4.2 Linear Optimal Control For the l i n e a r optimal e x c i t a t i o n c o n t r o l design, l e t the state equation be, [ x ] = [ A ] [ x ] + [ B ] [ u ] (4.1) and a cost index be chosen as J = \ f ( [ x ] T [ Q ] [ x ] + [ u ] T [ R ][ u 1 )dt (4.2) where [ x ] i s the state v a r i a b l e vector, [ u ] the c o n t r o l vector, [ A ] the system matrix, [ B ] a con t r o l matrix, [ Q ] a p o s i t i v e semi-d e f i n i t e weighting matrix, and [ R ] a p o s i t i v e d e f i n i t e weighting matrix. A Hamiltonian i s formed by appending (4.1) to (4.2) H = ~ ([x] T[Q] [x] + [u] T[R] [u])+ [p] T( [A] [x]+ [B] [u] ) (4.3) where [p] i s the costate vector or Lagrange m u l t i p l i e r s , and the optimal c o n t r o l can be found from 3H/9u , r e s u l t i n g i n [ u ] = - [ R ] _ 1 t B ] T [ p ] (4.4) Let [ p ] = [ K ][ x ] ((4.5) and assume a time-invariant system, [ K ] must s a t i f y the following matrix R i c c a t i equation [K] [A] + [A] T[K] - [K] [B] [ R ] _ 1 [ B ] T [ K ] + [Q] = 0 (4.6) 45 From the state equation of [x] and the co-state equation of [p] , a composite system matrix [ M ] becomes [ M ] = [A] [Q] -IB] [ R ] - : L [ B ] T - [ A l ' (4.7) There are 2n eigenvalues of matrix [ M ] f o r an n-th order system, and the eigenvalues are symmetrically d i s t r i b u t e d on the r i g h t and the l e f t parts of the complex plane. Let the eigenvalue matrix be [ A ] A. 'II and the corresponding eigenvector matrix be (4.8). [ X ] *I *11 X. I l l IV (4.9) The R i c c a t i matrix [ K ] may be computed from [ K ] = [ X t I] [ X [ ] -1 (4.10) where [ A ] c o n s t i t u t e s the n eigenvalues of [ M ] on the l e f t hand side of the complex plane, which are the eigenvalues of the closed loop system matrix [ A ], as the closed loop state equations may be written [ x ] = [ A ] [ x ] + [ B ] [ u ] = ( [A] - [B] [ R ] _ 1 [ B ] T [ K ] )[x] (4.11) = [ A ][ x ] c 46 4.3 Eigenvalue S e n s i t i v i t y [ 31,32 ] Consider the c o n t r o l l e d system matrix A . For the i - t h c eigenvalue X^ and eigenvector , we have A X. = X. X. (4.12) c 1 x 1 For the i - t h eigenvalue X. and the eigenvector V. of the transposed A , X X c we have A T V. = V. X. (4.13) C X X X Taking the p a r t i a l d e r i v a t i v e of both sides of equation (4.12) with respect to a system parameter a gives 3A 3X 3X. &X. - X4 + A ( T - i ) = X„ ( ) + ( r ~ ) X, (4.14) 3a i c v 8a • i " 3a v 3a •- - ,^ Premultiplying both sides of equation (4.14) by Vj r e s u l t s i n 3X. 3X. 3X. Vt ( ~ ) X. + vj A ( - r - i ) = v! X . ( ) + (, - r - i ) v! X. (4.15) j 3a I j c 3a j x 3a 3a j x But V? A = V? X. (4.16) J c 3 3 Therefore equation (4.15) becomes T 8 A c 3 X i T v! ( - ~ ) X. = ( ^ - i ) V. X, (4.17) x 3a l 3a x x or T dkc 3X. v; ( ) x x x 3a x 9 a ( X. , V. ) (4.18) T where ( X . , V. ) equals to V. . X. , according to (4.12) and (4.16) x x x x 47 * 0 1 = 3 V t X . < J 1 (4.19) 0 1 4 j 4.4 Reduced Order Co n t r o l l e r v i a Eigenvalue S e n s i t i v i t y Analysis [24] -1 T Let the feedback matrix [BR B K ] of (4.11) be simply written as [ F ] , and l e t AV fo the voltage regulator be the l a s t state R v a r i a b l e . The closed loop system matrix becomes [ A c] = [ A - F ] where [ F ] I 0 ] f „ i f i f n l , , nk,... ., nn (4.20) (4 .21) n i ' Therefore, the eigenvalue s h i f t of the c o n t r o l l e d system i s due to the change i n the l a s t row of [ F ]. Since not a l l the feedback elements , f^ i=l,2,..,k,..,n , contribute substantial damping to the system, those which have r e l a t i v e l y small c o n t r i b u t i o n may be neglected, which w i l l not a f f e c t the o v e r a l l performance of the c o n t r o l l e d system. Since 3A 3f nk therefore (4.18) becomes 3A . 1.. [ 0 ] 0 0. . . ,-1,, 3f nk • X i • V i (4.22) (4 .23) k n where X_^  i s the k-th element of the eigenvector X^ , and ' V\ i s the n-th element of the eigenvector V\ 48 When d e l e t i n g a feedback element f , A f n k = - f R k , therefore X k . AX. = * ' * f . (4.24) 1 (X. , V.) nk By examining AX^ , i=l,2,....,n , one can decide which feedback elements of [ P ] i n (4.21) can be deleted. The t o t a l e f f e c t of d e l e t i n g some feedback elements can be c a l -culated from , ' \ t , ° I a. , v1) fna «- 2 5 > t o t a l d x * l where f n ( j ' s a r e those feedback co n t r o l elements being eliminated. Of course, one has to check whether Re ( X. + AX. ) < 0 i=l,2,....,n (4.26) 1 ^ o t a l i n order to have a stable system, where X^ , i=l,2,...,n , are the eigen-value of the co n t r o l l e d system without deleting any feedback s i g n a l s . 4.5 Examples of the Co n t r o l l e r Design Two examples of the SSR c o n t r o l design are given i n t h i s sec-t i o n using the reduced order one-machine i n f i n i t e - b u s models developed i n Chapter 3. For the 14th order model, the state v a r i a b l e s are [ x ] = [ A W l p , A e i p,Aa ) L p B , A 6 L p B,Aco , A 6,AI d,AI q,AI f,AI k d, AI .AV ,,AV ,AVD ] T (4.27) kq- cd' cq R For the e x c i t a t i o n c o n t r o l design, f B '] become a vector with . only one non-zero element K\/T, associated with AV„ . A A R 49 From e a r l i e r experience [ 18 ], the weighting matrices are chosen as [ Q ] = diag [ 5000,50,55000,50,3000,25;1000,1000, 0,5,5,1,1,0 ] [ R ] = l (4.28) Afte r design, the con t r o l i s substituted into the o r i g i n a l 26th order f u l l model whose state v a r i a b l e s are [ x I = [ Acoi ,A6i ,At02 JAe2 , A a)3 , A e 3 , A t O [ t , A e i + , A i o ; ,A9 , Aco G , A9 6 , A a , AP , A T H p , A T I p , A T J J p A ; A I d , A I q , A I f , A I k d , A I k q , A V c d , A V c q , V E f d ] (4.29) which corresponds to equations (2.41) and (2.42). Eigenvalue s e n s i t i v i t y technique i s then applied. T y p i c a l eigenvalue s h i f t due to i n d i v i d u a l state feedback i s shown i n Table 4.1. Table 4.1 Ty p i c a l value of the eigenvalue s h i f t due to i n d i v i d u a l state feedback Mechanical modes of the f u l l order c o n t r o l l e d system Net eigenvalue s h i f t due to the d e l e t i o n of f.^ g -0.1836 ± J298.18 -0.6618 ± J203.32 -0.1949 ± j160.68 -0.6980 ± j127.08 -0.2649 ± J98.889 -3.1529 ± J4.8785 0.0 +0.1182 + J0.0641 -0.1228 + J0.1891 -0.0033 + j0.0334 -0.1053 + jO.2758 -0.5931 + jO.2181 It i s found that the state feedback of Aw , A0 T„ , A9 T T 1 A , Aw , AI. , IP ' IP LPA kq ' AV , , AV do not have s i g n i f i c a n t e f f e c t on the eigenvalues and hence cd cq may be deleted, r e s u l t i n g i n a 7th order c o n t r o l l e r as follows 50 I L = 918.63ALO T T % T, - 46.159A6 + 214.58AI,, - 23.06AI E LPB d q -200.67AI F - 1 9 8 . 6 7 A I K D - 0.0558AV R (4.30) With s i m i l i a r procedures, an e x c i t a t i o n c o n t r o l i s also designed using the 16th order model. The state v a r i a b l e s of the model are AT ,AI ,AI,,AI..,AI. ,AV , 5AV ,AV_ ] , (4.31) d q f kd kq c d 5 cq R the weighting matrices are chosen as [ Q ] = diag [ 500,0,10,1,600000,10,500,10;800,500, 0,0.5,0.5,1,1,0 ] (4.32) r R i = i and the c o n t r o l U i s found to be t, = 54.896A6 T T 1 > + 788.26AtoT - 11.187A6 + 188.79A1, E LPA LPB LPB d -9.725AI - 175.93AI £ + 8.4749AI., - O.OAOAAV^ (4.33) q f kd R The operating conditions f o r which the above c o n t r o l l e r s are designed are e l e c t r i c a l power = 0.9 per unit power f a c t o r = 0.9 lagging terminal voltage = 1 . 0 per unit capacitor compensation = 60% 51 4.6 The Output Feedback Control Although the state v a r i a b l e s Aw LPB A5 , and AcoT_A of LP A the mass-spring system can be processed by the t o r s i o n a l s t r e s s analyzer [ 33 ], e l e c t r i c a l v a r i a b l e s A I d , AI A I ^ can not be measured, and must be expressed i n terms of measurable v a r i a b l e s . It i s found that A f k q i s r e l a t i v e l y small A<F kq and AI kq (4.34) X 3 ^ 1 AI X k q * Since \ q / \ q i s close to 1 f o r t h i s p a r t i c u l a r case, the dynamic responses of A I q and AI as shown i n Figure 4.1 are almost i d e n t i c a l T I M E ( S EC ) 5.0 —'o o ° . met o C 3 i i i r 1 1 r 1.0 2.0 3.0 T IME ( S E C ) 4.0 Figure 4.1 Dynamic responses of AI and q AI 5.0 kq 52 Other equations are I - Y I • d q q d Y,I, -VI ( 4 . 3 5 ) d d q q I 2 + I 2 which are l i n e a r i z e d f o r the SSR studies. Therefore AI, , AI , AI, , . d ' q ' kd ' AI^ can be expressed i n terms of the output v a r i a b l e s AP^ , AQ e , AI , AI^ according to the following r e l a t i o n fAP 1 e f (X -X,)I ((X -X - —3. ) T +E,. ) I X , q d qo q d X^ do fdo qo md ^ f d o ^ d V do to X 2 -(2x -^3 . ) r q X kq q ° I / T qo to ^do^md 1 x A qo md ^do^md AI AI, AI, Lkd(4.36) Applying (4.36), the s i m p l i f i e d c o n t r o l l e r of the 14th order model design of (4.30) becomes uV, = 918 .63Aio T T 1 T , - 4 6 . 1 5 9 A 6 - 1 4 5 . 4 6 A P - 68 .665AQ E LPB e e + 160.88AI - 1.9982AI - 0.0558AVW (4.37) which has seven measurable feedback s i g n a l s , and that of the 16th order model design of (4.33) becomes U_ = 54.896A6 T D A + 788.28AwT__ - 11.187A6T__ - 118.85AP E LPA LPB LPB e -64..489AQ + 137 .89AI - 1.0512AIr - 0.0404AVT, e t f R (4.38) which lias eight measurable feedback signals. 53 4.7 Eigenvalue Analysis of the Si m p l i f i e d -Controllers The e x c i t a t i o n c o ntrols of the 16th order model and 14th order model design are substituted into the l i n e a r i z e d 2 6th order f u l l model f o r eigenvalue a n a l y s i s , f or a wide range compensation of X^/X^ from 10% to 80%, and for three operating conditions of P £ equal to 0.5, 0.9, and 1.25 per u n i t . The system with either one of these two con t r o l s under a l l conditions are stable. T y p i c a l r e s u l t s are given i n Tables 4.2, 4.3, and 4.4 . Note that the cont r o l designed for the 14th order model not only s t a b i l i z e s mode 3 and mode 4, but also provides damping to mode 1 which i s not included i n the model f o r the design. 4.8 Dynamic Performance Test using Nonlinear Model The s i m p l i f i e d c o n t r o l l e r s are also substituted into the nonlinear 26th order f u l l model derived i n Chapter 2 f o r dynamic perform-ance t e s t . A pulsed torque disturbance of 20% f o r 0.2 second i s assumed for the system at various compensation and operating conditions. T y p i c a l responses of the system without and with co n t r o l of the 14th order model design, at 0.9 per unit generator load and 50% capacitor compensation, are shown in Figures 4.2 and 4.3 r e s p e c t i v e l y . Although some responses of the system without c o n t r o l are unstable, responses of the system with c o n t r o l are a l l s t a b l e . Note that a l l responses are in per un i t except the torque angle which i s i n degree. The speed response i s the deviation from i t s steady s t a t e value. 54 Table 4.2 Typ i c a l mechanical modes of the system with and without c o n t r o l normal load ( P = 0.9 p.u., Q e = 0.4359 p.u., V £ = 1.0 p.u. ) Without the With a control With a c o n t r o l Compensation e x c i t a t i o n designed for designed f o r control the 16th order the 14th order -0.1818±j298.18 -0.18l9±j298.18 -0.1818+J298.18 -0.4938±j203.60 -1.2264±j201.18 -0.8842±j201.52 30% -0.2513±jl60.64 -0.3396+J160.39 -0.2672±jl60.45 -0.6705±jl27.02 -0.7310±jl27.15 -0.6927±jl27.09 -0.2811±j99.136 -0.7864±j99.290 -0.4608±j99.295 -0.0031±j8.4l05 -1.3534±j5.1525 -1.3517±j6.3035 -0.18l8±j298.18 -0.1819±j298.18 -0.18l8±j298.18 +0.1237±j202.87 -0.4237±j203.56 -0.4554+J203.44 +0.2603±jl61.38 -0.6495±jl60.11 -0.4192±jl60.36 50% -0.6828±jl27.05 -0.7958±jl27.23 -0.7322±jl27.16 -0.3528±j99.345 -1.0005+J99.479 -0.5894±j99.490 -0.2327±j9.4692 -1.7884±j6.0755 -1.6931±j7.1144 -0.18l8±j298.18 -0.1819±j298.18 -0.1818±j298.18 +0.0612±j202.88 -0.4547±j203.32 -0.4526±j203.22 +0.0908±jl60.45 -0.5202±jl61.06 -0.3562±j160.72 60% -0.6947±jl27.23 -0.8516±jl27.37 -0.7674±jl27.25 '-0.4l62±j99.567 -1.1792±j99.678 -0.6892±j99.674 -0.3773±jl0.099 -2.0708±j6.5702 -1.9159±j7.5326 -0.1818+J298.18 -0.1819±j298.l9 -0.18l8±j298.18 -0.0134±j202.89 -0.5034±j203.17 -0.4788+J203.09 -0.0967±jl60.52 -0.3236±jl60.68 -0.2693±jl60.58 80% -0.5998±jl26.94 -0.2165±jl27.36 -0.2864+J127.16 +1.7l78±jl02.17 -1.1561±jl01.12 -0.6186±jl00.85 -0.7619+jll.662 -2.8558±j7.5796 -2.5196±j8.2862 55 Table 4.3 Ty p i c a l mechanical modes of the system with and without control at l i g h t load ( P g = 0.5 p.u., Q g = 0.4359 p.u., V t = 1.0 p.u. ) Compensation Without the E x c i t a t i o n control With a con t r o l designed for the 16th order With a control designed f o r the 14th order 30% -0.18l8±j298.18 -0.1624±j203.39 -0.2079±jl60.64 -0.6629±jl27.02 -0.2l78±j99.147 -0.4313±j8.4705 -0.1819±j298.18 -0.6431±j203.39 -0.2565±jl60.45 -0.7198±jl27.11 -0.6132±j99.198 -1.4887±j5.8854 -0.18l9±j298.18 -0.4097+J201.66 -0.2110±jl60.50 -0.6883ijl27.07 -0.3883±j99.227 -1.3734±j6.4l30 50% -0.1818ij298.18 +0.0503±j202.88 +0.3037±jl60.06 -0.6665±jl27.05 -0.2474±j99.356 -0.6297±j9.4819 -0.1819±j298.18 -0.5279±j203.59 -0. 3 5 7 3 i j l 6 0 . i l -0.7922±jl27.16 -0.7957±j99.309 -1.9004±j6.6909 -0.1819±j298.18 -0.5680±j203.47 -0.1995+J160.34 -0.7302±jl27.12 -0.4974ij99.38l -1.7489±j7.1128 60% -0.1818±j298.18' +0.0138±j202.88 -0.0050±j160.42 -0.6586±jl27.11 -0.2655±j99.553 -0.7548±jl0.082 -0.18l9±j298.18 -0.4872±j203.39 -0.7074±jl60.90 -0.8700±jl27.28 -0.9639±j99.425 -2.1637±j7.1201 ~0.18l9±j298.18 -0.4972±j203.38 -0.4829±jl60.56 -0.7748ijl27.20 -0.5871±j99.520 -1.992 6±j9.4656 80% -0.1818+J298.18 -0.0135+J202.91 -0.1284±jl60.52 -0.6624±jl26.94 +1.7579±jl01.46 -1.0877±j11.568 -0.18l8±j298.18 -0.4812±j203.26 -0.3631±jl60.6l -0.2603±jl27.42 -1.3391±jl00.52 -2.9051±j7.9656 -0.1819±j298.18 ,-0.4728±j203.17 -0.3139±jl60.59 -0.3162±jl27.19 -0.6494ijl00.47 -2.6895±j8.0386 56 Table 4.4 Ty p i c a l mechanical modes of the system with and without c o n t r o l at heavy load ( P = 1.25 p.u.,Q = 0.4359 p.u., V = 1.0 p.u. ) Compensation Without the e x c i t a t i o n c o n t r o l With a c o n t r o l designed f o r the 16th order With a c o n t r o l designed f o r the 14th order 30% -0.1818+J298.18 -0.8632±j203.72 -0.2907±jl60.63 -0.6774±jl27.02 -0.3384±j99.104 +0.4213±j8.1027 -0.18l9±j298.18 -1.5884±j201.03 -0.3915±jl60.33 -0.7369±jl27.17 -0.8755±j99.329 -1.4984±j4.7420 -0.18l9±j298.18 -1.1821±j201.42 -0.3036ijl60.42 -0.6952+j127.11 -0.4942ij99.318 -I.5920ij6.4917 50% -0.1818±j298.18 +0.1881±j202.89 +0.2020±jl61.68 -0.6981±jl27.06 -0.4477±j99.3l8 +0.1360±j9.2235 -0.1819±j298.18 -0.3394±j203.55 -0.8504±jl60.10 -0.7978±jl27.26 -1.1084±j99.570 -1.8497±j5.9294 -0.1819ij298.l9 -0.3682ij203.44 -0.5762ijl60.37 -0.7343ijl27.17 -0.6348ij99.54l -1.8564ij7.5255 60% -0.1818ij298.18 +0.1031±j202.89 +0.1726±jl60.49 -0.7293+J127.13 -0.5537±j99.546 -0 . 04 02±j9.8917 -0.1819±j298.18 -0.4l75±j203.29 -0.4076±jl61.16 -0.8409±jl27.4l -1.2920+J99.823 -2.1070±j6.5627 -0.18l9ij298.18 -0.4098ij203.19 -0.2738ijl60.83 -0.7656ijl27.28 -0.7407ij99.753 -2.0503ij8.0835 80% -0.1818+J298.18 +0.0373+J202.91 -0.0687ijl60.52 -0.5800+J126.95 +1.6687+J102 .81 -0.4977±jll .552 -0.1819±j298.18 -0.5021+J203.12 -0.2926±jl60.67 -0.1939ijl27.33 -0.9919±jl01.47 -2.8669±j7.9327 -0.1819ij298.18 -0.4692ij203.05 -0.2361ijl60.58 -0.2693ijl27.14 -0.5752ijl01.08 -2.6189ij9.2738 VT (P.U.) (X1CT1 ) ! ANGLE(GEN) DEG 9.8 9.9 10.0 10.1 50.0 55.0 60.0 S5.0 70.0 75.0 80.0 58 Figure 4.2 (continued ) 59 in o 0.0 1 .0 2.0 3.0 4.0 . 5.0 T I M E ( S E C ) Figure 4.2 ( continued ) in T I M E ( S E C ) Figure 4.2 (continued ) 62 Figure 4.3 ( continued ) 63 CM 0.0 ] .0 T 2.0 3.0 T I M E ( S E C ) 4.0 1 5.0-0.0 1 .0 2.0 3.0 T I M E ( S E C ) 4.0 5.0 Figure 4.3 ( continued ) 64 Figure 4 .3 ( continued ) 65 4.9 The Control Signal The c o n t r o l signal U of the 14th order model design and i t s frequecy spectrum from Fourier analysis are shown i n Figures 4.4 and 4.5 r e s p e c t i v e l y . There are four s i g n i f i c a n t peaks at 1.15 Hz , 15.7 Hz , 25.6 Hz , and 32.3 Hz in the spectrum, which correspond to mode 0, mode 1, mode 3, and mode 4 of the generator mass-spring system r e s p e c t i v e l y . The s t a b i l i z i n g e f f e c t s on a l l mode of o s c i l l a t i o n s of t h i s design are well coordinated. Although the output feedback e x c i t a t i o n c o n t r o l of SSR proves very e f f e c t i v e f o r a sing l e machine system, i t s effectiveness on m u l t i -machine system i s s t i l l unknown. The in v e s t i g a t i o n of SSR co n t r o l of multi-machine system with e x c i t a t i o n control w i l l be continued i n the subsequent chapter. 0.0 2.0 3.0 T I M E ( S E C ) Figure 4.4 The c o n t r o l s i g n a l O •. C Q O C D o ' L J Q C D — » o Q_<=> o o 0.0 I I i i r 10.0 20.0 30.0 FREQUENCY (HZ) 40.0 Figure 4.5 The frequency spectrum of the c o n t r o l s i g n a l 50 5 MULTI-MACHINE SSR STUDIES 67 5.1 Introduction The SSR l i t e r a t u r e s so f a r are dealing with the t o r s i o n a l o s c i l l a t i o n s and counter measures of s i n g l e generating unit possibly f or two reasons: one, i t has been considered as a l o c a l problem, two, i t i s d i f f i c u l t to deal with a very high order multi-machine system. In t h i s chapter, the multi-machine SSR problem w i l l be ex-amined, and e x c i t a t i o n c o n t r o l of SSR w i l l be developed. Two working examples w i l l be given: a two-machine system and a three-machine system. Two f a c t o r s make SSR studies i n a multi-machine system d i f f e r e n t from that of a s i n g l e machine system. F i r s t , more than one e l e c t r i c a l resonance frequency may exist f o r the series-capacitor-compensated m u l t i p l e transmis-sion l i n e s . Second, the t o r s i o n a l i n t e r a c t i o n of the mass-spring systems and the dynamic i n t e r a c t i o n of the low frequency o s c i l l a t i o n s between mac-hines may e x i s t . Therefore, the strategy of c o n t r o l l e r design depends very much upon the degree of i n t e r a c t i o n between machines. The procedures of the multi-machine SSR studies are as follows: f i r s t , the system i s given an eigenvalue analysis to f i n d the e f f e c t of other capacitor-compensated l i n e s not d i r e c t l y connected to a p a r t i c u l a r machine on the t o r s i o n a l modes of that machine. Second, the t o r s i o n a l i n t e r a c t i n g e f f e c t s of other machines on the i n d i v i d u a l systems are examined. Thi r d , e x c i t a t i o n c o n t r o l s of SSR are designed, using the techniques deve-loped in the previous chapters, for the machines with unstable t o r s i o n a l modes. F i n a l l y , the system with c o n t r o l l e r i s evaluated using eigenvalue a n a l y s i s and. computer simulation test using the nonlinear power system model. 68 5.2 A Two-machine System and a Three-machine System A two-machine system f o r the SSR stu.dies i s shown i n Figure 5.1 with components l i s t e d i n Table 5.1 and data given in Appendix I I . As f o r the operating conditions, the conditions of machine 1 are fi x e d ( i . e . P ^ = 0.9 p.u. , P.F. = 0.9 lagging , = 1.0 p.u. ) and these of machine 2 vary with the compensation l e v e l ( 10% - 80% ) . The operating conditions of a l l buses other than the terminal bus of machine 1 are c a l c u l a t e d f o r various compensation l e v e l s . Machine 1 I n f i n i t e bus >W l i n e 2 - A V v -l i n e 4 load Machine 2 Figure 5.1 A two-machine power system. 69 Table 5.1 Summary of the components and number of state i n the two-machine system system component No. of state Machine 1 six mass-spring system 12 five-winding generator model 5 second order e x c i t a t i o n system 2 Machine 2 f i v e mass-spring system 10 five-winding generator model 5 second order e x c i t a t i o n system 2 Transmission system Line No. capacitor compensated 6 1 2 3 and 4 yes yes no Total No. of state 42 A three-machine system f o r SSR studies i s shown i n Figure 5.2 with components l i s t e d i n Table 5.2 and data given i n Apprendix I I I . Three base cases are studied: 1) 50% compensation f o r a l l l i n e s , except no compensation f o r l i n e 7. 2) 60% compensation for a l l l i n e s , except no compensation f o r l i n e 7. 3) The compensation of l i n e 1 and 5 i s 40%, that of l i n e s 3,4,6,and 8 i s 70%, and that of l i n e 9 i s 35%. No compensation f o r l i n e 7. In a d d i t i o n to the three base case studies, l i n e compensation i s also varied one at a time in the range of 30% to 70% for further studies. Operating conditions of the three base case studies are given i n Table 5.3 . Machine 1 CVVTX l i n e 8 l i n e 6 l i n e 3 p. ine 7 l i n e 9 l i n e 4 l i n e 1 Mac l i n e 2 l i n e 5 Machine 3 T x •5H-F i g u r e 5.2 A three-^machine power system. 71 Table 5.2 Summary of the components and number of state i n the three-machine system system component No. of state Machine 1 six mass-spring system 12 five-winding generator model 5 second order e x c i t a t i o n system 2 Machine 2 f i v e mass-spring system 10 five-winding generator model 5 second order e x c i t a t i o n system 2 Machine 3 four mass-spring system 8 five-winding generator model 5 second ore Ier e x c i t a t i o n system 2 Transmission system Line No. capacitor compensated 28 1 to 6 7 8 to 9 yes no yes Total No. of state 79 Table 5.3 Various machine operating conditions i n the three-machine system Machine 1 Machine 2 Machine 3 base case 1 P e 0.9 0.9068 0.5 % 0.3359 0.3408 0.2744 V t 1.0 1-3.12° 1.0 / o l 1.0 /-13.60 base case 2 P e 0.9 0.9056 0.5 9 e 0.3071 0.3196 0.2602 V t 1.0 /-2_._38° 1.0 ( o l 1.0 /-ll.48° base case 3 P e 0.9 0.9086 0.5 % 0.3076 0.3799 0.2627 V t 1.0 /-4.56° i . o / o l 1.0 / - l l .32° 72 5.3 Prelimary Study of the two-machine System From f u l l model eigenvalue analysis of the two-machine system over a wide range of capacitor compensation, f i v e unstable modes of the system are i d e n t i f i e d and given i n Table 5.4. . Typical mechanical modes of the system are shown i n Table 5.5 . The e f f e c t of capacitor compensa-t i o n on the t o r s i o n a l modes of the machine not d i r e c t l y connected to that transmission l i n e i s also investigated. T y p i c a l r e s u l t s as manifested by the v a r i a t i o n of r e a l parts of the eigenvalues of p a r t i c u l a r t o r s i o n a l modes are shown i n Figures 5.3 through 5.6 . The natural frequencies also change s l i g h t l y . Therefore, f o r a multi-machine system with multiple capacitor-compensated-lines, more than one condition at which SSR may occur. Table 5.4 Unstable modes of the two-machine system Machine 1 mode 3 (25.5 Hz.) mode 4 (32.3 Hz.) Machine 2 mode 0 ( 1-2 Hz.) mode 2 (24.0 Hz.) mode 3 (30.2 Hz.) Inter a c t i o n between machines' t o r s i o n a l modes i s obviously an important f a c t o r to be considered i n the SSR c o n t r o l l e r design. I f the in t e r a c t i o n between machines i s s i g n i f i c a n t it. must be considered, other-wise, the c o n t r o l l e r can be designed one machine at a time. The t o r s i o n a l i n t e r a c t i o n between machines are investigated as follows: 1) Find the eigenvalues from the f u l l model of the system. 73 2) Find the eigenvalues of the reduced order models that the mass-spring system of one machine i s represented i n d e t a i l and the mass-spring system of the other i s lumped into one mass. 3) Compare the r e s u l t s of steps 1) and 2). Any s i g n i f i c a n t change i n the t o r s i o n a l modes indicates the existence of i n t e r a c t i o n . Otherwise, the i n t e r a c t i o n between machines i s i n s i g n i f i c a n t . T y p i c a l r e s u l t s f o r the two-inachine system are shown i n Table 5.6 . Although there are s l i g h t changes i n the t o r s i o n a l modes between steps 1) and 2) e s p e c i a l l y when two mode frequencies are cl o s e , these i n t e r a c t i o n s are not strong enough to make the unstable modes stable or v i c e versa. Therefore, the interaction, between the two machines' t o r -s i o n a l modes i s i n s i g n i f i c a n t . Table 5.5 Ty p i c a l mechanical modes of the two-machine system Line compensation Machine 1 Machine 2 Line 1 70% Line 2 70% -1 .2096±j298.18 +0.1503+J203.02 -0.2213±jl60.46 -0.7152±jl27.08 -0.6302±j99.416 -0.9949±jl0.242 -0.1215±j276.41 +0.0862±jl89.96 +0.3546+J151.55 -0.2503+J102.42 -0.1677ij6.8353 Line 1 70% Line 2 30% -1.2224±j298.18 +0.1073+J202.87 +0.1379±jl60.88 -0.7183ljl27.08 -0.6550±j99.429 -0.6239±jl0.422 -0.1215+J276.41 +0.1396+J189.81 -0.2186±jl51.65 -0.2048±jl02.26 +0.3318±j6.6557 Line 1 50% Line 2 50% -1.2066±j298.18 +0.0610±j202.91 -0.5470±jl60.75 -0.7039+J99.204 -0.6625±j9.3108 -0.1215+J276.41 +0.1352±jl89.97 -C.1927±jl51.52 -0.0119±j6.5236 Line 1 50% Line 2 7 0% -1 .1997±j298.18 -0.0383 ±j 2 03.38 -0.4321 ±jl60.95 -0.7027 ±jl27.04 -0.5727 ±j99.211 -0.9671±j9.3253 -0.1215±j276.41 -0.0467±jl89.98 -0.0619±jl51 .68 -0.2159±jl02.27 -C.1467 ±j6.6456 75 Table 5.6 T y p i c a l mechanical modes of the two-machine system using d i f f e r e n t models f or the i n v e s t i g a t i o n of t o r s i o n a l i n t e r a c t i o n between machines Ml : d e t a i l M2 : d e t a i l Ml : d e t a i l M2 t one mass Ml : one mass M2 : d e t a i l Line 1 60% Machine (Ml) 1 -1.2143±j298.18 +0.1296±j202.87 -0.5487±jl61.02 -0.7096±jl27.05 -0.6088±j99.282 -0.6064±j9.7300 -1.2143±j298.18 +0.1303±j202.87 -0.5472±jl60.01 -0.7095+J127.05 -0.6l79ij99.299 -0.6064tj9.730 -0.5022ij9.7802 Line 2 40% Machine (M2) 2 -0;12I5±j276.4l -0.0295±jl89.95 -0.1843+J151.52 -0.1887±jl02.18 -0.0831±j6.4809 -0.0799±j6.5O63 -0.1215ij276.41 -0.0296ijl89.95 -0.1850ijl51.52 -0.1797ijl02.16 -0.0623ij6.4929 Line 1 70% Machine (Ml) 1 -1.2194±j298.18 +0.0714±j202.87 +0.1326+J160.77 -0.7167±jl27.08 -0.6448±j99.411 -0.6822±jl0.288 -1.2914+j298.18 +0.0718+j202.87 +0.1228ijl60.78 -0.7167ijl27.07 -0.6612+J99.438 -0.6944ijl0.313 -0.5805ijl0.351 Line 2 40% Machine (M2) 2 -0.1215±j276.41 -0.0509±jl89.95 -0.1115±jl51.62 -0.1970+J102.25 -0.1094+J6.7660 -0.1065+J6.7949 -0.1215ij276.41 -0.0509ijl89.96 -0.1058+J151.96 -0.l807ijl02.23 -0.0885ij6.7772 76 | 1 I 1 1 1 1 1 I 1 T 1 1 < I I 10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80 L i n e 2 % compensation L i n e 2 % compensation Figure 5.3 V a r i a t i o n of the r e a l part of mode 3 (160 rad/sec.) of machine 1 ( v e r t i c a l a x i s ) as l i n e 2's c a p a c i t o r changes and l i n e l ' s c a p a c i t o r kept constant at (a) 10%, (b) 20%, (c) 30%, (d) 40%, (e) 50%, ( f ) 60%, (g) 70%, (h) 80% compensation. 7 7 1 1 1 1 1 ., . -.2-1 .3 ,1J X ( e ) \ ^ 5 -6. "6. (f) 6 l 6 2 T J Line 2 % compensation 30 5fj io To 80 10 20 30 if 0 50 60 70 80 Line 2 % compensation Figure 5.4 V a r i a t i o n of the r e a l part of mode 4 (203 rad/sec ) of machine 1 ( v e r t i c a l axis ) as l i n e 2's capacitor changes and l i n e l ' s capacitor kept constant at (a) 10%, (b) 20%, (c) 30%, (d) 4 0%, (e) 50%, (f) 60% (g) 70%, (h) 80% compensation. 78 i i i i i i 1 1 i r 1 1 1 1 1 1 10 20 30 U0 50 60 70 80 10 20 30 UO 50 60 70 80 L i n e 1 % compensation L i n e 1 % compensation Figure 5.5 V a r i a t i o n of the r e a l part of mode 2 (151 rad/sec) of machine 2 ( v e r t i c a l a x i s ) as l i n e l ' s compen-s a t i o n changes and l i n e 2's c a p a c i t o r kept constant at (a) 10%, (b) 20%, (c) 30%, (d) 40%, (e) 50%, ( f ) 60%, (g) 70%, (h) 80% compensation. 79 ( 1 1 1 1 1 1 i 10 20 30 40 50 60 70 80 Line 1 % compensation •2-1 , 1 1 r — 1 1 1 I 10 20 30 kO 50 60 70 80 Line 1 % compensation Figure 5.6 V a r i a t i o n of the r e a l part of mode 3 (190 rad/sec.) of machine 2 ( v e r t i c a l axis) as l i n e l ' s capaciotr changes and l i n e 2's capacitor kept constant at (a) 10%, (b) 20%, (c) 30%, (d) 40%, (e) 50%, (f) 60%, (g) 70%, (h) 80% compensation. 80 5.4 Preliminary Study of the Three-machine System Again from the f u l l model eigenvalue analysis of the three-machine system over a wide range of capacitor compensation, s i x unstable modes of the system are i d e n t i f i e d , and they are given i n Table 5.7 . Typ i c a l mechanical modes of the system are given i n Table 5.8 . The i n t e r a c t i n g e f f e c t of t o r s i o n a l modes f o r the system i s also investigated using the same procedures as described i n section 5.3 . The r e s u l t s are shown i n Table 5.9 , which also suggest that i n t e r a c t i o n between machine's t o r s i o n a l modes i s i n s i g n i f i c a n t . Table 5.7 Unstable modes of the three-machine system Machine 1 mode 1 (15.7 Hz.) mode 4 (32.3 Hz.) Machine 2 mode 0 ( 1-2 Hz.) mode 1 (16.2 Hz.) mode 2 (24.0 Hz.) mode 3 (30.2 Hz .) Machine 3 none 81 Table 5.8 Typi c a l mechanical modes of the three-machine system Line compensation Machine 1 Machine 2. Machine 3 Line 1 60% Line 2 70% Line 3-9 60% (except 7) -0.18l8±j298.18 +0.2653±j203.29 -0.2753+J160.70 -0.6612±jl27.04 -0.2455±j99.456 -1.7915±jl0..363 -0.1214±j276.41 -0.0644±jl89.99 +0.0832±jl51.81 +0.2785±jl02.65 +1.310UJ10.175 -0.1336±j353.25 -0.1336±jl90.17 -0.7337±jl67.73 -3.2331+J17.474 Line 1-8 50% (except 7) Line 9 30% -0.1818±j298.18 -0.2359+J203.20 -0.2283±jl60.66 -0.6773±jl27.02 -0.2707±j99.224 -1.1930±j9.8449 -0.1215±j276.41 +0.1086±jl89.99 -0.0544±jl51.58 -0.3096±jl02.66 +0.6526±j8.8284 -0.1344±j353.24 -0.2284±jl90.11 -0.7334±jl67.73 -3.1623±jl7.262 Line 1-3 50% Line 4 30% Line 5-9 50% (except 7) -0.1818±j298.18 +0.0608±j203.34 -0.3752±jl60.63 -0.6427±jl26.99 +0.1309±j99.502 -0.9892±jl0.472 -0.1214±j276.41 -0.0549±jl89.84 +0.1253±jl51.93 -0.4555±jl02.56 +0.2684+J8.7070 -0.1334±j353.24 -0.1776+J190.28 -0.7326±jl 67.73 -3.3444±jl7.589 Line 1,5 40% Line 9 35% Line 2-4,6,8 70% -0.1818±j298.18 +0.265?±j203.29 -0.2753±jl60.70 -0.6612±jl27.04 -0.2455±j99.456 -1 .7915±jl0.363 -0.1214±j276.41 -O.C644±jl89.99 +0.0832±jl51.81 +0.2785±jlO2.65 +1.3101±jl0.175 -0.1336+J353.24 -0.1631±jl90.17 -0.7337+J167.73 -3.2331±jl7.474 Table 5.9 Typical mechanical modes of the three-machine system using different models for the investigation of torsional interaction between machines Ml : detail M2 : detail M3 : detail Ml : one mass M2 : detail M3 : detail Ml : detail M2 : one mass M3 : detail Ml : detail M2 : detail M3 : one mass Ml : one mass M2 t one mass M3 : detail Ml : one mass M2 : detail M3 : one mass Ml : detail M2 : one mass M3 : one mass Machine 1 (Ml) -0.182±j298.2 +0.265±j203.3 -0.275+J160.7 -0.661±j.l27.0 -0.245±j99.46 -1 ,792±j10.36 -1.818±jl0.4l -0.182±j298.2 +0.253±j203.3 -0.270±jl60.7 -0.661±jl27.0 -0.412±j99.45 -1.805±jl0.40 -0.182±j298.2 +0.266±j203.3 -0.275±jl60.7 -0.661±jl27.0 -0.245±j99.46 -1.792±jl0.36 -1.834±jl0.45 -1.818±jl0.41 -0.182±j298.2 +0.253±j203.3 -0.270±jl60.7 -0.661+J127.0 -0.411±j99.45 -1.805+J10.40 Machine 3 (M3)| Machine 2 (M2) -0.121+J276.4 -0.064±j189.9 +0.083±j151 .8 +0.278±jl02.6 +1.310±jl0.18 -0.121±j276.4 -0.067+J189.9 +0.080+J151.8 +0.414±jl02.6 +1.327±jl0.20 +1.331±jl0.25 -0.121±j276.4 -0.013±jl90.0 +0.083±jl51.8 +0.278±jl02.6 +1.310±jl0.18 +1..350±jl0.27 -0.121±j276.4 -0.0l4±jl90.0 +0,081±jl51.8 +0.413±jl02.6 +1 .327±jl0.20 +1.332±jl0.25 Machine 3 (M3)| Machine 2 (M2) -0.134±j353.2 -0.163+J190.2 -0.734±jl67.7 -3.233±jl7.47 -0.134±j353.2 -0.161+J190.2 -0.734±jl67.7 -3.233±jl7.47 -0.134±j353.2 -0.216±jl90.1 -0.734±jl67,7 -3.234±jl7.48 -3.248±jl7,51 -0.134±j353,2 -0.216±jl90.1 -0.734±jl67.7 -3.235±jl7.47 -3.248±jl7.51 -3 ,24 9+j .17.51 83 5.5 C o n t r o l l e r Design Considerations of Multi-machine SSR System Although the c o n t r o l l e r design using the f u l l model, which accounts f o r a l l natural frequencies i n the e l e c t r i c a l system and t o r s i o n a l i n t e r a c t i o n between machines, could be the best, i t often r e s u l t s i n a high order c o n t r o l l e r . Since the t o r s i o n a l i n t e r a c t i o n between machines according to the foregoing studies i s i n s i g n i f i c a n t , an a l t e r n a t i v e method using a reduced order model for the design i s developed. The o r i -g i n a l system i s divided into several one-machine systems, and the c o n t r o l l e r i s designed one at a time. Two general steps are as f o l l o w s : 1) Choose an appropriate one-machine system model by r e t a i n i n g one of the transmission l i n e s d i r e c t l y connected to the machine, which has the l a r g e s t steady current and hence the strongest t o r s i o n a l i n t e r a c t i n g e f f e c t between the e l e c t r i c a l and mechanical systems. 2) Modify the l i n e by adding some impedance so that the c r i t i c a l e l e c t r i -c a l frequency f o r the l i n e with compensation as viewed from the machine w i l l not change. When the multi-machine system i s divided into several one-machine systems, the dynamic i n t e r a c t i o n between machines has been neglected. But these i n t e r a c t i o n s depend very much upon the t i e l i n e s between machines. For a strong t i e l i n e , strong i n t e r a c t i o n may e x i s t . Therefore, the con-t r o l l e r s designed f o r i n d i v i d u a l one-machine systems must be coordinated. An i t e r a t i v e scheme as shown in Figure 5.7 may be applied to adapt the designed c o n t r o l l e r s so that a l l dampings of the mechanical modes in the system are coordinated. 84 S e n s i t i v i t y studies] of the weighting matrix Q Design c o n t r o l f o r Machine l j with a one-machine model Design c o n t r o l f o r Machine 2 using a one-machine model Test the c o n t r o l l e r s on the o r i g i n a l system No Continue to the other machine' ! with the s i m i l a r process J I — — — — —. — — — — — —I > loop one c Stop 3 Figure 5.7 I t e r a t i v e scheme f o r adapting c o n t r o l l e r into the o r i g i n a l system. 85 5.6 C o n t r o l l e r s Design and Test of the Two-Machine System The two-machine system i s divided into two one-machine i n f i n i t e - b u s system models f or the c o n t r o l l e r design as follo w s : 1) The natural frequencies of eit h e r one of the two transmission l i n e s f o r a wide range of capacitor compensation are determined by s e t t i n g the capacitor compensation of the other l i n e to zero. 2) The re s t of the system i s replaced by an equivalent impedance, Xg^ or Xg,, as shown i n Figure 5.8(a) or (b) r e s p e c t i v e l y , such that each system w i l l have the same e l e c t r i c a l n atural frequencies as determined i n step 1). l i n e 1 l i n e 2 Machine 1 Machine 2 (a) (b) Figure 5.8 Two subsystems res u l t e d from the two-machine system. For machine 1 of Figure 5.8(a), X ^ i s found to be 0.1 p.u. . Since the operating conditions, the machine and transmission l i n e para-meters of the system given i n Appendix I I are almost the same as that of the one-machine system studied i n the previous chapters, the same control-l e r i n Equation (4.37) xcrill be applied without change. For machine 2 of Figure 5.8(b), X ^ i s 0.11 p.u. , and the average operating conditions are P e 2 = 0.8 p.u. , Q e 2 = 0.4 p.u. , V t 2 = 0.9 p.u. Applying the mass-spring equivalencing technique developed i n Chapter 3, the mass-spring system of machine 2 i s reduced to a three-mass equivalent 86 r e t a i n i n g only modes 2 and 3. The mode shapes of the o r i g i n a l and the equivalent system are shown i n Figure 5.9. Including the e l e c t r i c a l sys-tem, a reduced 14th order model i s obtained. The state v a r i a b l e s are T x 2 ] = T A a . I p 2 , A e i p 2 , A a . ^ B 2 > A e L p B 2 , A i ) G e r i 2 , A 6 G e n 2 , A I d 2 , A I q 2 , A l f 2 ' A l k d 2 ' A l k q 2 ' A V c d 2 ' A V c q 2 ' A V R 2 ] (5.1) Using the 14th order model, an e x c i t a t i o n c o n t r o l i s designed using the l i n e a r optimal c o n t r o l laws and eigenvalue s e n s i t i v i t y technique developed i n Chapter 4, r e s u l t i n g an 8th order c o n t r o l l e r . The weighting matrices are [ R ] = 1 I 0 ] = d i a g [ 5000,50,50000,50,100,25;100,100, 0,50,50,1,1,0 ] (5.2) The state feedback c o n t r o l a f t e r the eigenvalue s e n s i t i v i t y a n alysis r e s u l t s i n U E 2 = 633.82Aco L p B 2 + 35,559A9 L p B 2 - 56.804A6 G e n 2 + 56.63AI d 2 + 12.453AI _ - 56.089AT „ - 52.01AI.,- - 0.214AV- (5.3) q2 f2 kd2 R2 Applying (4.37), the state feedback c o n t r o l of (5.3), i n terms of output v a r i a b l e s , becomes U E 2 = 633.83Au L p B 2 + 35.559A9 L p R 2 - 56.804A5 G e n 2 - 17.481AP e 2 -27.287AQ - 4.0792AI + 29.981AI - 0.214AV (5.4) (1) (2 ) 1 1 HP J IP t r i i i i J u LP A 1 1 1 J Gen 1 1 0 •1J 1-1 mode 1 (16.2 Hz.) mode 2 (24.0 Hz .) mode 3 (30.2 Kz.) mode 4 (44.0 Hz.) (a) Five mass-spring system 0 - I J 1 0 0 (b) Four mass-spring system 1 0 -1. (c) Three mass-spring system Figure 5.9 Mode shapes of the o r i g i n a l and equivalent systems f o r machine 2 i n the two-machine system. oo 88 5.6.1 Testing of Co n t r o l l e r s Using Eigenvalue Analysis After the c o n t r o l l e r design, eigenvalues of the two-machine system are analyzed i n the sequence as shown in Table 5.10 . Table 5.10 Testing sequence for the two-machine system Machine 1 Machine 2 1 with co n t r o l without c o n t r o l 2 without c o n t r o l with c o n t r o l 3 with co n t r o l with c o n t r o l The f i r s t two t e s t s are used to examine the e f f e c t of the c o n t r o l l e r designed f or one. machine on the other machine, t y p i c a l r e s u l t s are shown i n columns 2 and 3 of Table 5.11 . I t was found that the damping provided by the c o n t r o l l e r to the t o r s i o n a l modes of the other machine i s i n s i g n i f i c a n t . However, the e f f e c t of the c o n t r o l l e r on mode 0 of the other machine i s n o t i c a b l e . It may give p o s i t i v e or negative damping to the other machine, depending on the operating conditions. With both c o n t r o l l e r s applied to machines 1 and 2, a l l un-stable modes are s t a b i l i z e d over the e n t i r e range of capacitor compen-sat i o n . T y p i c a l r e s u l t s are shown i n column 4 of Table 5.11 . Note that the' c o n t r o l l e r s provide substantial amount of damping to mode 1 of both machines even though they are not considered i n the c o n t r o l l e r design. Although the e f f e c t of c o n t r o l l e r on mode 0 of the other machine i s n o t i c a b l e , i t i s not s i g n i f i c a n t due to the weak t i e l i n e s between the machines. Therefore, each con t r o l can be separately designed and the i t e r a t i v e scheme presented in f i g u r e 5.7 i s not necessary. j 89 Table 5.11 • Ty p i c a l mechanical modes of the two-machine system without and with co n t r o l Ml without M2 without Ml with M2 without Ml without M2 with Ml with M2 with Line 1 50% , Line 2 30% Machine 1 (Ml) -1.2128±j298.18 40.4186±j202.99 -0.5580±jl60.73 -0.7047±jl27.03 -0.5844±j99.199 -0.4836±j9.2996 -1.2129±j298.18 -0.934l±j203.25 -0.5254+J160.51 -0.7329+J127.10 -0.7476+J99.349 -1.9182±j7.9942 -1.2128±j298.18 +0.3826±j203.00 -0.5551±jl60.73 -0.7047±jl27.03 -0.5813±j99.203 -0.5765±j9.4304 -1.2129±j298..I8 -0.9107±j203.19 -0.5249±jl60.51 -0.7328±jl27.10 -0.7429±j99.351 -2.2889±j8.5922 Line 1 50% , Line 2 30% Machine. 2 (M2) -0.1214±j276.41 -0.0318±jl90.02 -0.1690±jl51.46 -0.1711±jl02.11 -0.0261±j6.3218 -0.1214±j276.41 -0.0621±jl90.02 -0.1665+J151.46 -0.1700±jl02.11 -0.2053±j5.7251 -0.1223±j276.41 -0.4476±jl89.17 -0.1992+J151.06 -0.6437±jl02.06 -2.4631±j5.8497 -0.1223±j276.41 -0.3403±jl89.15 -0.2045±jl51.07 -0.6434±jl02.23 _2.4449±j4.7145 Line 1 70% , Line 2 '70% Machine 1 (Ml) -1.2096±j298.18 +0.1503±j203.02 -0.2213±jl60.46 -0.7152±jl27,08 -0.6302±j99.416 -0.9949±jl0.246 -1.2096±j298.18 -0.4857±j203.03 -0.4882±jl60.52 -0.7699±jl27.19 -0.8320±j99.550 -3.0374±j9.0538 -1.2096±j298.18 +0.1422±j203.01 -0.2893±jl60.53 -0.7116±jl27.11 -0.6115±j99.439 -0.9926±jl0.613 ~1.2098±j298.18 -0.4790±j203.01 -0.5114±jl60.52 -0.7673±jl27.21 -0.8113±j99.565 -2.7396±j9.9798 Line 1 70% , Line 2 '70% Machine 2 (M2) -0.1215±j276.4l +0.0862±jl89.96 +0.3546±jl51.51 -0.2503±jl02.42 -0.1677±j6.8352 -0.1215+J276.41 +0.0779±j189.95 +0.2397±jl51.80 -0.2336±jl02.41 -0.0079±j6.0831 -0.1221±j276.41 -1.2334±jl89.43 -0.897 6±jl51.13 -1.0867±jl02.52 -2.3147±j6.8506 -0.1221±j276.41 -1.1587±j189.43 -1.0465±jl50.77 -1.0463±jl02.55 -2 .2604±j5.4211 T 90 5.6.2 Dynamic Performance Test of the Two-machine System The c o n t r o l l e r designs i n (4.38) and (5.4) are substituted into the nonlinear two-machine system model f o r dynamic performance t e s t . A three-phase f a u l t f o r 0.075 second i s assumed at the load bus as shown i n Figure 5.10. Ty p i c a l responses of the system with 50% compensation f o r both l i n e s 1 and 2 without co n t r o l are shown i n Figures 5.11 and 5.12, and those with c o n t r o l i n Figures 5.13 and 5.14 r e s p e c t i v e l y . For the system without c o n t r o l , most responses of the machines are either o s c i l l a t o r y or unstable. However, a l l responses of the system with e x c i t a t i o n c o n t r o l are stable and a l l o s c i l l a t i o n s are damped out within 5 seconds. -TP W-Machine 1 - W -Vv Ih I n f i n i t e bus -/JT* vv- e -W TiV-"•I c l o s e t=0 1 open t=0.075 sec. "achine 2 ( ^ Figure 5.10 The two-machine system subjected to disturbance. 16 .7 o VT (P.U.) SPEED (P.U.) (XIO"1 ) 0.8 0.85 0.9 0.95 1.0 1.05 1.1 -0 .06 - 0 . 0 4 - 0 . 0 2 0.0 0.02 0.04 0.06 £6 94 Figure 5.12 (ccmt inued) CO o a T o " O r— •o a 1 UJfsj L U a 0_o CD I o o . I 0.0. 0.0 1 .0 2.0 T IME 3.0 ( SEC) i — 4.0 5.0 T 1 1 r— 2.0 3.0 T I M E ( SEC ) 4.0 5.0 O o ' Q_ co a ' in a ' 1.0 1.0 2.0 3.0 4.0 5.0 0,.0 T IME ( SEC) Figure 5.13 Typ i c a l responses of machine 1 in the two-machine system with control, 2.0 3.0 T IME ( SEC ) 4.0 5.0 «3 96 0.0 1 1 r- T 2.0 3.0 TIME (SEC) 5.0 CD CD I CQ Q_CNJ LU ZDCD C3 • O t -co a CD a 0.0 ;!IJ! i! I' hi I II 1 .0 2.0 TIME 3.0 (SEC) 4.0 5.0 Figure 5.13 (continued) 98 Figure 5.14 (continued) 99 5.7 C o n t r o l l e r Design and Test of the Three-machine System S i n c e a l l the mechanica l modes of machine 3 are s t a b l e , the c o n t r o l l e r d e s i g n w i l l be focused on machines 1 and 2 . Two one-machine models f o r machines 1 and 2 , by r e t a i n i n g the s t ronges t t o r s i o n a l i n t e r -a c t i o n p a t h and the c r i t i c a l e l e c t r i c a l frequency f o r each model a re chosen as f o l l o w s : 1) Among the t h r e e t r a n s m i s s i o n l i n e s 1, 3 , and 6 connected to machine 1, o n l y l i n e 6 i s r e t a i n e d because i t has the l a r g e s t per u n i t c u r r e n t i n d i c a t i n g the s t ronges t i n t e r a c t i o n p a t h . The r e s t of the system can be r e p l a c e d by an e q u i v a l e n t reactance of 0.07 p . u . . 2) For machine 2 , o n l y l i n e 2 i s r e t a i n e d and the r e s t of the system i s r e p l a c e d by an e q u i v a l e n t reac tance of 0 .1 p . u . . A g a i n , the same c o n t r o l l e r of (A.37) i s used f o r machine 1, because the o p e r a t i n g c o n d i t i o n s , the machine and t r a n s m i s s i o n l i n e p a r a -meters f o r machine 1 are almost the same as those of the system p r e v i o u s l y s t u d i e d . The o p e r a t i n g c o n d i t i o n s f o r machine 2 are P e 2 = 0 > 9 0 6 P- u-> °- e2 = 0 , 3 4 1 p , u - ' V t 2 = 1 , 0 P ' U * Us ing the 14th order model of (5.1) together %tfith the techniques develpoed i n Chapter 4 , an 8 t h order e x c i t a t i o n c o n t r o l f o r SSR i s designed by choos ing the same w e i g h t i n g m a t r i c e s as shown i n ( 5 . 2 ) . When both c o n t r o l l e r s a re a p p l i e d to the three-machine system, a l l u n s t a b l e t o r s i o n a l modes i n the system are s t a b i l i z e d over the e n t i r e range of the p r e s c r i b e d o p e r a t i n g c o n d i t i o n s . However, mode 0 of machine 2 remains u n s t a b l e due to the inadequacy of the one-machine model by which 100 the dynamic i n t e r a c t i o n between machines i s neglected. But either one of the two c o n t r o l l e r s can be adapted also to s t a b i l i z e mode 0 using the i t e r a t i v e scheme as shown in Figure 5.7 . 5.7.1 S e n s i t i v i t y Studies and Choice of Weighting Elements i n [Q] Dynamic i n t e r a c t i o n between machines i s transmitted through the e l e c t r i c a l network by the l i n e current. Therefore, proper choice of weighting elements i n conjunction with current i s important to enhance the mode 0 damping in a multi-machine system. For the three-machine system, the c o n t r o l l e r f o r machine 2, o r i g i n a l l y based on a one-machine model, i s adapted by studying the s e n s i t i v i t y of mechanical damping with respect to the weighting elements of AI. AI, „, AI,„, AI AI, ,„ and AI, _ are included because kd2' kq2' d2 5 q2 kd2 kq2 they a f f e c t the s e l f damping of machine 2 which i n turn a f f e c t the dynamic i n t e r a c t i o n between machines. Keeping other weighting elements of [Q] in (5.2) constant, the e f f e c t of the weighting elements of A I ^ ^ a n c * ^1^2 on the mechanical modes of machine 2 i s shown in Figure 5.15 and that of A I J O and AI . in Figure 5.16. d2 q2 Two observations are as foll o w s : 1) As the weighting elements Q I k d 2 a n c * ^Ikq2 i n c r e a s e ( more penalty on the deviation of damper winding currents are imposed ), mode 0 damping of machine 2 increases and that of machine 1 decreases. Damping of machine 2's t o r s i o n a l modes also decrease, but at a much slower r a t e . 2) As the weighting elements a n ^ ^Iq2 decrease ( penalty on the d e v i a t i o n of stator currents i s reduced ) , mode 0 damping of machine 2 increases and that of machine 1 decreases. Damping of the t o r s i o n a l modes of machine 2 remains f a i r l y constant. 101 S i n c e the w e i g h t i n g elements of the damper winding c u r r e n t s a f f e c t the damping of machine 2 ' s t o r s i o n a l modes, they must be chosen such t h a t a l l mechanica l modes i n the three-machine system have rea son-a b l e p o s i t i v e damping. In t h i s case a n c l ^Iq2 ° ^ .2) a re adapted to 50 , Qjkd2 a n c * ^Ikq2 a < ^ a P t e c * t o ^^0 . The r e s u l t i n g w e i g h t i n g m a t r i c e a re [ R ] = 1 [ Q ] = d i a g [ 5000 ,50 ,50000 ,50 ,100 ,25 ;50 ,50 ,0 (5.3) 1 0 0 , 1 0 0 , 1 , 1 , 0 ] and the des igned s t a t e feedback c o n t r o l a f t e r s i m p l i f i c a t i o n from e i g e n -v a l u e s e n s i t i v i t y a n a l y s i s becomes U E 2 = 549.26Au> L p B 2 + 6 7 . 1 l 8 A 8 L p B 2 - 8 0 . 1 7 2 A 6 G e n 2 + 4 7 . 7 6 A I d 2 +12.289AI _ - 4 8 . 4 7 8 A I , - - 41 .527AI. , 0 - 0.249AV D „ (5 .4) q2 f2 kdz Rz A p p l y i n g ( 4 . 3 6 ) , the c o n t r o l of (5 .4) i n terms of output v a r i a b l e s becomes U £ 2 = 5 4 9 . 2 6 A u ) L p B 2 + 67 .118Ae L p B 2 - 8 0 . 1 7 2 A S G e n 2 - U . t W l A P ^ -12 .196AQ e 2 - 6 . 9 5 A I f 2 + 99 .296AI f c 2 - 0 . 2 4 9 A V r 2 (5 .5) W i t h c o n t r o l l e r of (4.37) on machine 1 and (5 .5) on machine 2 of the three-machine system, a l l u n s t a b l e modes i n the system are s t a b i - . l i z e d over the e n t i r e range of the p r e s c r i b e d o p e r a t i n g c o n d i t i o n s . T y p i c a l r e s u l t s are shown i n Table 5.12 . 102 Table 5.12 Typi c a l mechanical modes of the three-machine with c o n t r o l Line compensation Machine 1 Machine 2 Machine 3 Line 1 60% Line 2 70% Line 3-9 60% (except 7) -0.1820±j298.18 -1.0612±j202.58 -0.3l41±jl60.49 -0.7058±jl27.16 -0.5637±j99.549 -1.987l±jl0.923 -0.1222±j276.41 -0.6101±jl89.31 -0.3091±jl51.04 -0.6588±jl03.29 -1.3621±j5.4707 -0.1334±j353.24 -0.2016±jl90.16 -0.7333±jl67.73 -3.3618±jl7.607 Line 1-8 50% (except 7) Line 9 30% -0.1820±j298.18 -0.6224±j202.15 -0.2893±jl60.48 -0.7071±jl27.15 -0.5635±j99.507 -2.1187±j9.8608 -0.1221±j276.41 -0.4 904±jl89.45 -0.2752±jl51.12 -0.9266±jl02.71 -1.0805±j5.2511 -0.1337±.j353.24 -0.2366±jl90.09 -0.7334±jl67.73 -3.1780±jl7.277 Line 1-3 50% Line 4 30% Line 5-9 50% (except 7) -0.1820+J298.18 -0.6799+J202.25 -0.3017±jl60.31 -0.6629±jl27.14 -0.5368±j99.606 -2.1820±j9.8276 -0.1221±j276.41 -0.3588±jl89.41 -0.3372±jl51.36 -0.9538±jl02.42 -1.1479±j5.2261 -0.1335±j353.24 -0.1876±jl90.05 -0.7313±jl67.73 -3.2170±jl7.439 Line 1,5 40% Line 9 35% Line 2-4,6,8 70% -0.1820±j298.18 -1.5122±j202.49 -0.3139±jl60.46 -0.6738±jl27.17 -0.5616±j99.859 -1,9833±j11.218 -0.1221±j276.41 -0.5766±jl89.39 -0.3l47±jl51.17 -0.2140±jl02.85 -I.5807±j5.6033 -0.1336±j353.24 -0.2310±jl90.l4 -0.7336±jl67.73 -3.2409±jl7.489 103 F i g u r e "5.15 V a r i a t i o n of mechanical damping as weighting elements Q I > d 2 and Q 2 change w h i l e Q I d 2 and Q I q 2 constant at 50. 104 C V ! ° . • — 1 „ Figure 5.15 (continued) 105 Figure 5.16 V a r i a t i o n of mechanical damping as weighting elements Q T 1„ and Q „ change w h i l e Q T 1 ,„ and Q . _ constant at 50. Id2 Iq2 Ikdz Ikq2 106 Figure 5.16 (continued) 107 5.7.2 Dynamic Performance Test of the Three-machine System The dynamic performance of the three-machine system without and with e x c i t a t i o n c o n t r o l i s tested using the nonlinear system, model. A r e s i s t i v e load i s switched into the system at bus 8 f o r 0.075 second as shown i n Figure 5.17 , so that the bus voltage w i l l drop 20% . T y p i -c a l responses of the system without co n t r o l are shown i n Figures' 5.18 through 5.20, and those with c o n t r o l i n Figures 5.21 through 5.23 r e -s p e c t i v e l y . The l i n e compensation of the system i s 50% f o r a l l l i n e s , except f o r l i n e 7 and l i n e 2 which has no compensation and 70% compen-sation r e s p e c t i v e l y . For the system without c o n t r o l , some responses of machines 1, 2, and 3 are unstable, but the responses of a l l machines are stable f o r the system with the e x c i t a t i o n c o n t r o l . Machine 1 bus 8 Machine 2 0 c l o s e t=0 * 0 I n f i n i t e bus open t=0.075 t-*KE) Machine 3 Figure 5.17 The three-machine system subjected to disturbance. 109 to • 0.0 1.0 2.0 3.0 4.0 5.0 T I M E ( S E C ) Figure 5.18 (continued) Figure 5.19 (continued) Figure 5.21 (continued) 117 0.0 1 .0 2.0 3.0 TIME (SEC) 4.0 5.0 CO Cxi" •sr U j o CD I CQ Q_co L U ID CM O • CD • -co NT o ifl Ii I 0.0 2.0 3.0 TIME (SEC) 4.0 5.0 Figure 5.22 (continued) 8TT f i g u r e 5.23 (continued) 120 5.8 Concluding Remarks for Multi-machine SSR Studies From the foregoing SSR studies of the two-machine and three-machine systems, general conclusions are as follows: 1) In a multi-machine system with mul t i p l e capacitor-compensated trans-mission l i n e s , there i s more than one condition at which SSR may occur. 2) I n t e r a c t i o n between t o r s i o n a l modes of d i f f e r e n t machines has no s i g n i f i c a n t e f f e c t on SSR s t a b i l i t y . 3) To apply the s i m p l i f i e d output feedback e x c i t a t i o n c o n t r o l design technique developed for the one-machine system to a multi-machine system, a one-machine equivalent including the strongest transmission t i e and the c r i t i c a l e l e c t r i c a l resonance frequency must be derived f o r each machine in the multi-machine system. When the c o n t r o l l e r s thus designed are applied to the multi-machine system, some adaptation may be required, which can be achieved using an i t e r a t i v e process. 4) Both eigenvalue a n a l y s i s and dynamic performance test using nonlinear f u l l models prove that the e x c i t a t i o n c o n t r o l designed according to the procedures presented in t h i s chapter i s very e f f e c t i v e for m u l t i -machine multi-mode s t a b i l i z a t i o n of the SSR. 121 6. CONCLUSION S e v e r a l u s e f u l model r e d u c t i o n , e q u i v a l e n c i n g , and c o n t r o l s i m p l i f i c a t i o n techniques have been developed and many i n t e r e s t i n g r e s u l t s a r e found i n t h i s t h e s i s s tudy . A f t e r p r e s e n t i n g a u n i f i e d e l e c t r i c a l and mechan ica l model f o r the SSR s t u d i e s i n Chapter 2 , i t i s shown i n s e c t i o n 2 .5 t h a t , a l t h o u g h the n e g a t i v e r e s i s t a n c e concept i s u s e f u l to e x p l a i n t h e t o r s i o n a l i n t e r -a c t i o n between the e l e c t r i c a l and mechanica l systems, the lumped mass r e -p r e s e n t a t i o n of the t u r b i n e - g e n e r a t o r i s not s u f f i c i e n t , and the m u l t i -mass - spr ing system must be used f o r SSR s t u d i e s . For the SSR c o n t r o l d e s i g n , the e x c i t a b l e t o r s i o n a l modes are i d e n t i f i e d from modal a n a l y s i s i n Chapter 3 . A mass - spr ing e q u i v a l e n c i n g t e c h n i q u e i s then developed f o r order r e d u c t i o n by r e t a i n i n g o n l y t h e u n s t a b l e t o r s i o n a l modes at c e r t a i n f r e q u e n c i e s , r e s u l t i n g i n a l o w e r o r d e r mass - spr ing system. Based on the l i n e a r o p t i m a l c o n t r o l laws and w i t h the reduced 14th and 16th o r d e r models o f the o r i g i n a l 2 6 t h o r d e r system, l i n e a r o p t i m a l e x c i t a t i o n c o n t r o l l e r s a re des igned i n Chapter A . The c o n t r o l l e r s a re f u r t h e r s i m p l i f i e d from a s e n s i t i v i t y a n a l y s i s by d e l e t i n g some l e s s s e n s i t i v i t y feedback s i g n a l s , and the f i n a l c o n t r o l l e r s employ the system output s i g n a l s as the feedback. Both e igenva lue a n a l y s i s of the l i n e a r i z e d f u l l model and the computer s imula ted dynamic performance t e s t u s i n g n o n -l i n e a r f u l l model i n d i c a t e that the l i n e a r o p t i m a l e x c i t a t i o n c o n t r o l thus des igned i s e f f e c t i v e i n p r o v i d i n g damping to a l l t o r s i o n a l modes of t h e system over a wide range of c a p a c i t o r compensation and o p e r a t i n g c o n d i t i o n s . The s t a b i l i z a t i o n technique i s f u r t h e r extended and a p p l i e d to a two-machine and a three-machine SSR systems i n Chapter 5 . I t i s found 122 that the SSR s t a b i l i z e r s s t i l l can be designed f o r one machine at a time but coordination may be required a f t e r a l l c o n t r o l l e r s are implemented. For the i n d i v i d u a l machine SSR c o n t r o l l e r design, however, i t i s neces-sary to derive a one-machine equivalent f o r each i n d i v i d u a l machine by r e t a i n i n g the strongest t i e of the machine, which has the largest cur-rent, to the remaining system, an equivalent reactance i s adapted so that the e l e c t r i c a l resonance frequency which a f f e c t s the vulnerable t o r s i o n a l mode of the machine i s retaine d . For the coordination of the damping provided by a l l c o n t r o l l e r s f o r the e n t i r e system, an i t e r a t i v e process i s developed. Extensive eigenvalue a n a l y s i s and nonlinear com-puter simulation t e s t s again i n d i c a t e that l i n e a r optimal e x c i t a t i o n c o n t r o l s thus designed are very e f f e c t i v e f o r the SSR c o n t r o l . The l i n e a r optimal e x c i t a t i o n developed i n the t h e s i s probably provides the most e f f e c t i v e and l e a s t expensive means to s t a b i l i z e the SSR of one-machine as well as multi-machine systems. 123 REFERENCE I 1 ] M.C.Hall and D.A.Hodges,"Experience with 500Kv Subsynchronous Resonance and Resulting Turbine Generator Shaft Damage at Mohave Generating S t a t i o n " , IEEE P u b l i c a t i o n 76 CH1066-0-PWR, pp. 22-29, 1976. [ 2 ] R.G.Farmer,A.L.Schwalb, and E l i Katz,"Navajo Project Report on Subsynchronous Resonance Analysis and Solution", IEEE Trans, on PAS,Vol PAS-96, No. 4 July/August 1977 pp. 1226-1232. [ 3 ] G.A. Fischer,R.Quay,R.L.Winchester, Discussion on paper "Subsyn-chronous Resonance i n Series Compensated.Transmission Line",IEEE Trans, on PAS,Vol PAS-92,No.4,Sept/Oct.,1973 pp.1655-1657 [ 4 ] C.E.J.Bowler,D.H.Baker,N.A.Mincer,and P.R.Vandiwer,"Operation and Test of Navajo SSR Prote c t i v e Equipment", IEEE Trans, on PAS, Vol PAS-97, July/August 1978, ppl030-1035. [ 5 ] L.A.Kilgore,D.G.Ramey,and W.H.South,"Dynamic F i l t e r and other Solutions to the Subsynchronous Resonance Problems",Proceeding of American Power Conference, Vol 37,1975, pp. 923-929. [ 6 ] 0. Wasynczuk,"Damping of Subsynchronous Resonance using Reactive Power Control" IEEE Trans.on PAS, Vol PAS-100, No. 3,March 1981, pp. 1096-1103. [ 7 ] D.G.Ramey,D.S.Kimmel,J.W.Dorney,and F.H.Kroening, "Dynamic S t a b i -l i z e r V e r i f i c a t i o n Tests at the San Juan S t a t i o n " , IEEE P u b l i c a t i o n 81 TH0086-9-PWR 1981, pp. 34-42 . [ 8 ] R.A.Hedin,K.B.Stump,and N.G.Hingornai,"A New Scheme for Subsyn-chronous Resonance Damping of Torsional O s c i l l a t i o n s and Tran-sient Torque - Part I I , Performance", IEEE Trans, on PAS,Vol. PAS-100, No. 4, A p r i l 1981, pp. 1856-1863. [ 9 ] N.G.Hingorani,R.A.Hedin,K.B,Stump,and Bharal Bhargava,"Evalu-l a t i o n of NGH Damping Scheme Applied to Mohave Generator", IEEE P u b l i c a t i o n 81 TH0086-9-PWR 1981, pp. 70-80. [ 10 ] 0.Saito,H.Mukae, and K.Murotani,"Suppression of Self Excited O s c i l l a t i o n s i n Series-capacitor-compensated Transmission Lines by E x c i t a t i o n Control of Synchronous Machine",IEEE Trans, on PAS, Vol PAS 94, Sept/Oct. 1975, pp 1777-1788. [ 11 ] H.M.AHamdan and F.M.Hughes,"Excitation C o n t r o l l e r Design f o r Damping of Self Excited O s c i l l a t i o n s i n Series Compensated Lines", Paper A78 565-4, IEEE PES Summer Meeting, Los Angeles, July 1978. 124 [ 12 ] K.Murotani and M.Asano,"Subsynchronous Resonance O s c i l l a t i o n s of Series-Compensated Transmission System and t h e i r Suppressions" E l e c t r i c a l Engineering in Japan,Vol. 96,No. 6, 1976, pp. 113-121. [ 13 ] M.M.Sartavi and B.T.Ooi, " F i e l d E x c i t a t i o n Suppression of Sub-synchronous Resonance by Real and Reactive Power Signal Feedback", Paper A 78 602-5, IEEE PES Summer Meeting, Los Angeles, J u l y 1978. [ 14 ] A.M.El-Serafi and A.A.Shaltout,"Control of Subsynchronous Resonance O s c i l l a t i o n s by Multi-loop E x c i t a t i o n C o n t r o l l e r " , Paper A79 076-1, IEEE PES Winter Meeting, New York C i t y , Feb. 1979. [ 15 ] A.A.Fouad and K.T.Khu,"Damping of Torsional O s c i l l a t i o n s i n Power Systems with Series-compensated Lines", IEEE Trans, on PAS, V o l . PAS 97, No. 3, May/June 1978, pp. 774-751. [ 16 ] F.M.Hughes and H.M.A.Hamdan,"Damping of Subsynchronous Resonance in Series-compensated Systems by E x c i t a t i o n Control", INT. J . Control, Vol 31, No. 1, 1980, pp.63-77. [ 17 ] Yao-nan Yu,M.D.Wvong,and K.R.Tse, "Multi-Mode Wide-range Subsyn-chronous Resonance S t a b i l i z a t i o n " , Paper A78 554-8, IEEE PES Summer Meeting, Los Angeles, July 1978. T 18 ] A.Yan,M.D.Wvong,and Yao-nan Yu,"Excitation Control of Torsional O s c i l l a t i o n s " , Paper A79 505-9, IEEE PES Summer Meeting, Vancouver, B.C. , July 1979. I 19 ] Yao-nan Yu,K.Vongsuriya,and L.N.Wedman,"Application of an Optimal Control Theory to a Power System", IEEE Trans, on PAS, Vol. PAS 89, No. 1, Jan/Feb. 1970, pp. 55-62. [ 20 ] H.A.M.Moussa and Yao-nan Yu, "Optimal Power System S t a b i l i z a t i o n through e x c i t a t i o n and/or Governor Control",.IEEE Trans, on PAS, V o l . PAS 91, No. 3, May/June 1972, pp. 1166-1174. I 21 ] Habibullah and Yao-nan Yu, "Physical Realizable Wide Power Range Optimal C o n t r o l l e r s f o r Power Systems", IEEE Trans, on PAS, V o l . PAS 93, No. 5, Sept/Oct. 1974, pp. 1498-1506. J 22 ] H.A.M. Moussa and Y.N.Yu, "Dynamic i n t e r a c t i o n of Multi-Machine Power System and E x c i t a t i o n Control", IEEE Trans, on PAS, V o l . PAS-94, August, 1975,pp. 1150-1158. I 23 ] J . M . U n d r i l l and T .E.Kostyniak,"Subsynchronous O s c i l l a t i o n s Part -I-Comprehensive System S t a b i l i t y A n a l y s i s " , IEEE Trans, on PAS, V o l . PAS-95, No. 4, July/August 1976, pp. 1446-1454. [ 24 ] A.Yan and Yao-nan Yu,"Multi-Mode S t a b i l i z a t i o n of Torsional O s c i l -l a t i o n s Using Output Feedback E x c i t a t i o n Control", Paper 81 SM 383-9, IEEE PES Summer Meeting, Portland, July 1981. To be published i n IEEE Trans, on Power Apparatus and Systems. 125 t 25 ] IEEE Committee Report,"Dynamic Models f o r Steam and Hydro Turbines i n Power System Studies", IEEE Trans, on PAS, V o l . PAS-92, Nov/Dec, 1973, pp. 1904-1915. [ 26 ] IEEE Committee Report, "Computer Representation of E x c i t a t i o n Systems", IEEE Trans, on PAS, Vol PAS-87, June/July 1968, pp. 1460-1470. [ 27 ] B.L.Agrawal and R.G.Farmer," Use of Frequency Scanning Techniques for Subsynchronous Resonance Analysis ",IEEE Trans, on PAS, V o l . PAS-98 ,No. 2, March/April 1979, pp. 341-349. r 28 ] J.M.Undrill and F.P.DeMello, "Subsynchronous O s c i l l a t i o n s Part 2- Shaft-System Dynamic Interactions", IEEE Trans, on PAS, Vol. PAS-95, No. 4, July/August 1976, pp. 1150-1158. [ 29 ] W.S.Levine and. M. Athans," On the Determination of the. Optimal Constant Output Feedback Gains for Linear M u l t i v a r i a b l e s Systems", IEEE Trans, on Automatic Control, V o l . AC-15, No.1, Feb. 1970, pp. 44-48. [ 30 ] N.Munro and A.Vardulakis," P o l e - s h i f t i n g using Output Feedback ", Int. J . Control, V o l . 18, No. 6, 1973, pp. 1267-1273. [31 ] J.E.Van Ness,J.M.Boyle, and F.P.Imad," S e n s i t i v i t i e s of Large, M u l t i p l e Loop Control System", IEEE Trans, on Autumatic Control, March 1965, pp.308-315. [ 32 ] L.N.Wedman and Yao-nan Yu," Computation Techniques for the S t a b i l i z a t i o n and Optimization of High Order Power Systems ", The S i x t h PICA Conference Proceedings, May 1969, pp. 324-343. [ 33 ] Jan Stein and Horst F i c k , " The Torsional Stress Analyzer for Continously Monitoring Turbine-Generators ", IEEE Trans, on PAS, Vo l . PAS-99, No. 2, March/April 1980, pp. 703-710. [ 34 ] D.M.Triezenber / ' C h a r a c t e r i s t i c Frequencies and Mode Shapes For Turbogenerator Shaft Torsional V i b r a t i o n s " , Paper A 78 809-6, IEEE/ASME/ASCE Joint Power Generation Conference, D a l l a s , TX, Sept. 10-13, 1978. [ 35 ] IEEE Committee Report, " F i r s t Benchmark Model for Computer Simulation of Subsynchronous Resonance", IEEE Transactions on PAS, PAS-96, Sept./Oct. , pp. 1565-1572. [ 36 ] D.N.Walker,A.L.Schwalb,"Results of Subsynchronous Resonance Test at Navajo", IEEE P u b l i c a t i o n 76 CHI066-0-PWR, pp.37-45, 1Q76. [ 37 ] Charles Concordia,"Synchronous Machines - Theory and Performance", John Wiley and Son, Inc., New York, 1951. [ 38 ] Gerald Manchur, David C. Lee, M.E. Coultes,J.D.A G r i f f i n , and Wilfred Watson,"Generator Models Established by Frequency Response Tests on A 555 MVA Machine" IEEE Trans, on PAS, Sept./Oct.., 1972,pp. 2077-2083. APPENDIX I SYSTEM DATA FOR THE ONE MACHINE SYSTEM A l l the data are i n per unit based on 500KV and 900 MVA except the time constant which i s i n second. Synchronous Machine Parameters x d = 1.79 X f = 1.6999 R f " 0.00105 Xmd =- 1.66 X k d - 1.6657 R k d = 0.00371 X q = 1.71 kq 1.6531 V 0.00491 X mq 1.58 R a 0.0015 Mass-spring System ^ P * I P = 0.185794 0.311178 1.717340 K12 = ^ 3 =  K34 =  K45 = K56 = 19.303 34.929 52.038 D. . = l l 1=1,2,. 0.1 A P B = 1.768430 70.858 ..., 6 MGen = 1.736990 2.8220 "fix — 0.068433 Turbine and Governing System K g = 25 T l = 0.2 T2 = 0.0 T 3 = 0.3 T CH 0.3 TRH " 7.0 T CO = 0.2 FHP = 0.3 F I P - 0.26 F LPA = 0.22 F LPB 0.22 PGV " ±0.1 Exc i t er and [ Voltage Re gulator K A = 50 TA = 0.01 T "E 0.002 Exc i t er voltage c e i l i n g l i m i t s = ±7.0 Transmission l i n e Parameters X t = 0.14 R t = 0.01 h * 0.56 0.02 X var c ied from .0.056 - 0.448(10% - 80%) 127 APPENDIX II SYSTEM DATA FOR THE TWO MACHINE SYSTEM A2.1 Machine 1 Synchronous Machine Parameters S i m i l i a r to that of the machine i n Appendix I Mass-spring System S i m i l i a r to that of the machine i n Appendix I Turbine Torque D i s t r i b u t i o n FHP1 = °' 3  FLPB1= ° ' 2 2 F i p l = 0.26 FLPA1= ° * 2 2 E x c i t e r and Voltage Regulator K ' = 5 0 T 0.01 Al " "Al E x c i t e r voltage c e i l i n g l i m i t s = ±7.0 T_. = 0.002 hi A2.2 Machine 2 Synchronous Machine Parameters Xd2 = 1.82 X f 2 •= 1.92 R f 2 = 0.0067 Xmd2 = 1.65 X kd2 1.76 Rkd2 = . 0.0043 x o = 1.73 Xkq2 = 2.05 \q2 = 0.0089 X = mqz 1.59 R 0 -a2 0.0015 128 Mass-spring System M^ IPZ = 0.248 = 0.464 K 1 2 = 21.8 K 2 3 = 48.4 K 4 =74.6 M = 2.38 1=1,2,..,5 *LPB2 MGen2= 1 J l K. = 62.3 4 5 E x c i t e r voltage c e i l i n g l i m i t s = ±7.0 A2.3 Transmission System Transformer Turbine Torque D i s t r i b u t i o n F H p 2 = 0.3 F i p 2 = 0.26 F L p A 2 = 0.22 FLPB2 = ° ' 2 2 E x c i t e r and Voltage Regulator KA2 = 5 0 TA2 = ° ' 0 1 TE2 - ° ' ° 0 5 Xfc. = 0.14 R . = 0.01 t l t i X f c 2 = 0.1 R t 2 = 0.01 Transmission Line Line 1 X ^ = 0.42 ^ 1 = ° - 0 2 X varied from 0.042 to 0.336 (10% - 80%) Line 2 X L 2 = 0.4 R^ = 0.01 X c 2 varied from 0.04 to 0.32 (10% - 80%) Line 3 X ^ = 0.2 R j 3 = 0.01 Line 4 X Y. = 0.28 R . = 0.05 L4 L4 Load ^load" 0 , 9 129 APPENDIX I I I SYSTEM DATA FOR THE THREE MACHINE SYSTEM A3.1 Machine 1 A l l parameters are s i m i l i a r to those given i n A2.1 of Appendix I I . A3.2 Machine 2 A l l parameters are s i m i l i a r to those given i n A2.2 of Appendix I I A3.3 Machine 3 Synchronous Machine Parameters Xd3 " 1.78 X f 3 = 1.7781 R f 3 - 0.00109 Xmd3 = 1.6845 Xkd3 = 1.7368 \d3 = 0.0117 X Q = q3 1.7067 \ q 3 = 1.6409 t> = kq3 0.0151 X Q = mq3 1.6063 R i = a3 0.00357 Mass-spring System "HP3 ° 0.262 K 12 47.48 " l P 3 " 0.525 v -61.85 D. . = l l i=l,2 0.1 M Gen3 1.85 4.51 " R X 3 = 0.0595 Turbine Torque D i s t r i b u t i o n "HP3 0.4 F IP3 0.6 E x c i t e r and Voltage Regulator KA3 = 50 TA3 = 0.02 XE3 " 0.002 Exc i t e r voltage c e i l i n g l i m i t s = ±7.0 130 A3 .4 Transmission System Transformer X u = 0.14 P.tl = 0.01 X t 2 = 0.14 R t 2 = 0.01 X ^ = 0.14 R t 3 = 0.01 Transmission Line Line 1 X ^ = 0.47 R^ = 0.035 X , varied from 30% to 70% compensation c l Line 2 X ^ = 0.4 R^  = 0.02 X c 2 varied from 30% to 70% compensation Line 3 X ^ = 0.14 = 0.01 X c 3 varied from 30% to 70% compensation Line 4 X ^ = 0.39 R^ = 0.03 X ^ varied from 30% to 70% compensation Line 5 X ^ = 0.34 ^ 5 = 0.025 X^^ varied from 30% to 70% compensation Line 6 = 0.51 'R^ = 0.04 c g varied from 30% to 70% compensation X Line 7 ^ = 0.11 R^ ? = 0.01 No capacitor compensation Line 8 X T O = 0.3 F, 0 = 0.02 Lo Lo X^g varied from 30% to 70% compensation Line 9 X g = 0.32 = 0.024 

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