) 1 i -> 7 DYNAMIC POWER SYSTEM MODELLING AND LINEAR OPTIMAL STABILIZATION DESIGN USING A CANONICAL FORM by BAVADEEN S. HABIBULLAH B.E. (Hons.), University of Madras, Madras, India, 1963 M.Sc. (Engg.), University of Madras, Madras, India, 1966 M.A.Sc, University of Toronto, Toronto, Canada, 1971 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Electrical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1973 In presenting t h i s thesis i n p a r t i a l f ulfilment 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 freely available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. I t i s understood that copying or publication 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 of HWacY^ctd £ncj±rxx^^^ The University of B r i t i s h Columbia Vancouver 8, Canada Date ABSTRACT The l i n e a r optimal s t a b i l i z a t i o n of power systems has become a very active area of research i n recent years. Implementation of an optimal control scheme usually requires the measurement of a l l state v a r i a b l e s , some of which are not a c c e s s i b l e . Dynamic estimators may be used to estimate immeasurable s t a t e s . But the addition of dynamic estimator makes the o v e r a l l control scheme more complex and unduly sen-s i t i v e to disturbances and changes i n parameters. There i s another pro-blem with the optimal control design, i . e . the choice of the performance index matrices Q and R. In this thesis a f a i r l y accurate synchronous machine model i n terms of e a s i l y measurable state v a r i a b l e s i s developed. By neglecting the short l i v e d armature t r a n s i e n t s , the best dynamic model of a syn-chronous machine with torque angle, speed, e l e c t r i c output power, t e r -minal voltage or current and f i e l d voltage or current as the state v a r -i a b l e s i s derived. The model i s used for supplemental e x c i t a t i o n c o n t r o l and l i n e a r optimal control designs. For the former not only the mechanical mode but also the e l e c t r i c a l mode o s c i l l a t i o n s are considered. For the l a t t e r the system equations are transformed i n t o a canonical form and the optimal control thus designed i s a function of the weighing matrices Q and R of the cost f u n c t i o n . Chapter 2 i s devoted to the dyn-amic modelling of a synchronous machine, Chapter 3 to the supplemental e x c i t a t i o n control and Chapter 4 to the development of design techniques f o r the l i n e a r optimal c o n t r o l . Numerical examples are given i n Chapters 5 and 6 f o r both s i n g l e machine and multi-machine power systems. The nonlinear t e s t r e s u l t s of power systems i n d i c a t e that transient responses can be greatly improved by the l i n e a r optimal control schemes developed In t h i s t h e s i s . i l TABLE OF CONTENTS AB STL ACT . . . i 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 H i s t o r i c a l Remarks . . . " 1 1.2 The Scope of the Present Thesis 2 2. DYNAMIC MODELLING OF SYNCHRONOUS MACHINES 2.1 Synchronous Machine Equations i n MKS 5 2.2 Synchronous Machine Equations i n per-unit • 7 2.3 Flux Linkage Model or iJj-Model of the Machine 8 2.4 ifj-Model of Reduced Order 10 2.5 Voltage Model of V-Model of the Synchronous Machine . 12 2.6 V~Model of A l t e r n a t i v e Form 14 2.7 E f f e c t of Olive's Assumption on Modelling 17 2.8 E f f e c t of Transmission Line and Local Load 19 2.9 E f f e c t of Machine Loadings 19 2.10 Voltage-Current-Power or V-I-P Model of the Machine . 21 2.11 State Equations and Transformation Matrix of V-I-P Model 23 3. SUPPLEMENTAL EXCITATION CONTROL DESIGN 3.1 Introduction 28 3.2 E x c i t a t i o n System and E f f e c t of Voltage Regulator Gains 29 3.3 Compensation Network f o r S t a b i l i z a t i o n Signals . . . . 31 3.4 Parameter Optimization 33 3.5 Eigenvalue Searches 34 3.6 Time Response of the System 37 i i i Page 4. OPTIMAL CONTROL DESIGN BASED ON EQUATIONS IN CANONICAL FORM 4.1 Introduction 39 4.2 Canonical Form of System Equations 40 4.3 Transformation Matrix of the Canonical Form 42 4.4 Optimal Control Design with Canonical Form 43 4.5 Optimal C o n t r o l l e r 46 5. PHYSICALLY REALIZABLE OPTIMAL EXCITATION AND GOVERNOR CONTROLLERS 5.1 Introduction 48 5.2 System Models f o r Optimal E x c i t a t i o n Control Design . 48 5.3 Eigenvalue Movements with the Optimal E x c i t a t i o n Control 51 5.4 Nonlinear Test Results of Optimal E x c i t a t i o n Control . 54 5.5 System Models f o r Optimal Governor Control Design . . 56 5.6 Eigenvalue Movement with the Optimal Governor Control 60 5.7 Optimal Governor Control Supplementing Optimal Ex-c i t a t i o n Control 62 5.8 Wide Power Range Operating Conditions . . . . . . . . . . 64 6. MULTI-MACHINE LINEAR OPTIMAL CONTROL DESIGN 6.1 Introduction 69 6.2 Summary of Single Machien Dynamics 69 6.3 Aggregated Multi-Machine Equations i n Individual Coordinates . . . . . . 70 6.4 m-Machine Equations i n Common Coordinates 74 6.5 T o t a l System Equations 76 6.6 Multi-Machine Model i n Terms of Measurable State Variables 77 6.7 Modified Control 78 6.8 V-I-P Model i n a Canonical Form 79 6.9 System Studied . . . . . 79 6.10 Multi-Machine Optimal Control Design 82 6.11 Single Machine Design of Multi-Machine System . . . . 85 6.12 Nonlinear Test 88 7. CONCLUSIONS 93 i v 3?ag<5 APPENDICES A. Derivation of Sti-te '-.quai:ion, X = AX 'r BU 96 33. I n i t i a l Value of the System Variables . . . . . . . . . 101 'REFERENCES 104 v LIST OF TABLES Table Page I Data f or the Models 16 II Eigenvalues of Various Synchronous Machine models . . . 18 III Influence of Transmission System on the System Eigen-values 20 IV Influence of l o c a l Load on the System Eigenvalues . . . 20 V V a r i a t i o n of Eigenvalues with Load 21 VI Eigenvalues of One Machine System 22 VII Dominant Eigenvalue Movement due to Optimal E x c i t a t i o n Control ' 53 VIII Eigenvalues of the T o t a l System 59 IX Dominant Eigenvalue Movement due to Optimal Governor Control 61&62 X Further Movement of Dominant Eigenvalue due to Supplemental Governor Control . . . . 62 XI Eigenvalues of the System with D i f f e r e n t C o n t r o l l e r s . 66 XII Machine Data 80 XIII E x c i t e r Data 80 XIV Terminal Conditions 81 XV Dominant Eigenvalue Movement 83 XVI Dominant Eigenvalue Movement 84 vi LIST OF ILLUSTRATIONS Figure Page 2.1 A 2-axia Synchronous Machine in Park's .q Coordinate, .. 6 2.2 One Machine and Infinite Power System . . . . . . . . . 17 3.1 Block Diagram of the Excitation System 29 3.2 Dominant Eigenvalue Move cf the System vith Gain .. . 31 3.3 Block Diagram of the Compensation Netwo: k 32 3.4 Dominant Eigenvalue move, with Single Stabilizing Signal without Compensation . . . 35 3.5 Dominant Eigenvalue Move, with Single Stabilizing Signal with Compensation . 35 3.6 Dominant Eigenvalue Move, with Single Stabilizing Signal with Lead Compensation for Aw and Lag Compensation for -AP and -Ai f 36 3.7 Dominant Eigenvalue Move, with. Two Compensated Signals 36 3.3 Time AespouBe uf fciie System Lo a S L C J J J J J _ O uUi/uciuce . . . 38 3.9 Time Response of the System to an Impulse Disturbance . 38 5.1 Root Loci of the System with Optimal Excitation Control 52 5.2 Nonlinear Power Plant with Linear Optimal Controller . 54 5.3 Nonlinear Dynamic Responses with Different Controls . . 55 5.4 Speed Governor and Hydroturbine System 56 5.5 Root Loci of the System with Optimal Governor Control . 61 5.6 Nonlinear Dynamic Responses with Optimal Controllers . 63 5.7 Dominant Eigenvalue Loci of the System with Various Control Designs 65 5.8 Nonlinear Dynamic Responses of the system With Different Loadings 67 6.1 Transformation of Reference Frame . 73 6.2 System Studied 81 v i i Figure Page 6.3 Nonlinear Dynamic Responses . . . c . 90,91&92 B . l Phaser Diagram of a Synchronous Machine. 101 v i i i ACKNOWLEDGEMENT I wish to express my deepest gratitude to my supervisor, Dr. Y.N. Yu f o r h i s valuable assistance, constant encouragement and keen i n t e r e s t i n my graduate program and the preparation of this t h e s i s . I would also l i k e to thank Dr. H.R. Cheann, Dr. H.W. Dommel, Dr. M.P. Beddoes, Dr. A.C. Soudack and Dr. V.J. Modi for reading the dr a f t , and making valuable comments. The f i n a n c i a l support received from the Commonwealth Schol-arship and Fellowship Administration and the National Research Council of Canada i s g r a t e f u l l y acknowledged. Special thanks are due to Miss Norma Duggan f o r her e x c e l l e n t work i n typing the f i n a l d r a f t of the t h e s i s , Mr. A l MacKenzie f o r d r a f t i n g .the .diagrams., -Mr.. ..flerb -31-a.ck ,fox-.his ..ass-is.feance .in photo-graphic work and fellow graduate students i n the Power Group f o r the discussions which I had with them. I s i n c e r e l y wish to thank my wife, Mustheri f o r the coopera-t i o n , understanding and encouragement given during the e n t i r e period of the study. ix NOMENCLATURE atate vector of flux-linkage model state vector of voltage - current - power model state vector of canonical model system matrix of X-model system matrix of Z-model system matrix of Y-model cont r o l matrix of X-model control matrix of Z-model control matrix of Y-model control vector 4- , . „ „ „ . £ „ „ „ „ 4-J *- A. c u i o x u i ma l _ J L V J L l transformation matrix f o r Y-model p o s i t i v e d e f i n i t e symmetric matrix whose elements weigh the contributions of each state i n the cost func-t i o n p o s i t i v e d e f i n i t e symmetric matrix whose elements weigh the contributions of each c o n t r o l motion i n the cost: function axis transformation r e l a t i n g the machine reference frame component voltages to the network reference com-ponent voltages eigenvalue time, s complex operator, x p(A) specl.r;.! radius of A Synchronous Machine i instantaneous value of current v instantaneous vr.lue of voltage i|> flux-linkage R resistance x reactance 6 torque angle, rad. to angular v e l o c i t y , rad./s K i n e r t i a constant T e l e c t r i c torque e . T mechanical torque m v generator terminal voltage JL generator uermiual current P+jQ generator output power Transmission Network R +jX equivalent s e r i e s impedance of transmission system G +jB equivalent shunt admittance of transmission system V Q i n f i n i t e bus voltage Voltage Regulator and E x c i t e r K. regulator gain T, regulator time constant, s K„ e x c i t e r gain hj T„ e x c i t e r time constant, s h K„ gain constant of s t a b i l i z i n g c i r c u i t r T- time constant of s t a b i l i z i n g c i r c u i t , s r x i u s.abiM.^ins s i g n a l u(Z) s t a b i l i z i n g function E r rc- ference voltage re i " . V„ s t a b i l i z i n g s i g n a l r Speed Governor and Hydroturbine Tp actuator time constant, s T dashpot time constant, s T_ gate servomotor time constant, 5 G T water s t a r t i n g time constant, s W a permanent speed droop c o e f f i c i e n t 6j. temporary speed droop c o e f f i c i e n t . g change i n gate opening h change i n h y d r a u l i c head a change i n actuator s i g n a l b temporary droop output s i g n a l AT change i n mechanical torque m Subscripts d, q d i r e c t - and quadrature-axis s t a t o r q u a n t i t i e s f f i e l d c i r c u i t q u antities D, Q d i r e c t - and quadrature-axis damper qua n t i t i e s M quantities r e f e r i n g to machine reference frame N quantities r e f e r i n g to network reference frame a armature phase qu a n t i t i e s x i i Supercripts -1 t P r e f i x A P synchronous machine tr a n s i e n t quantities synchrono\is machine subtransient q u a n t i t i e s inverse of a matrix transpose of a matrix a l i n e a r i z e d quantity a d i f f e r e n t i a l operator x i i i 1. INTRODUCTION 1.1 H i s t o r i c a l Remarks The design, transmission, dispatching, operation and c o n t r o l of a modern power system o f f e r s many challenging problems. The most important one among them i s probably the s t a b i l i z a t i o n of a power system e s p e c i a l l y under transient conditions. The advent of high speed d i g i t a l computers and the development of new concepts i n c o n t r o l theory have encouraged the power system engineers to consider the machines and t h e i r c o n t r o l loops i n great d e t a i l under almost any complex c o n d i t i o n s 1 ^ , 16 17 28 ' ' f o r the s t a b i l i z a t i o n study so that the system can be s t a b i l i z e d and i t s dynamic behaviour can be predicted with great accuracy. The remoteness of hydraulic generation from load centres, the i n t r o -'duction 'of-fas t-actin-g high ..gain s o l i d s^ate'excitation.systems, .and the wide-spread interconnection of large power systems not only increase the s i z e and complexity, but also create new problems of dynamic i n t e r -a c t i o n r e s u l t i n g i n severe mechanical mode o s c i l l a t i o n . I t did not take too long to f i n d that some type of s t a b i l i z i n g s i g n a l s must be supplied to improve the p o s i t i v e damping of the system i n order to dampen out the system o s c i l l a t i o n 1 ^ ' 1 1 * 1 " ^ . Among them the lead-lag compensated speed or power s t a b i l i z i n g s i g n a l , fed through the e x c i t a t i o n system, and known as the supplemental e x c i t a t i o n c o n t r o l was found to be very e f f e c t i v e i n s t a b i l i z i n g the system under the transient disturbances. The s i g n a l s are usually derived from a lower order synchronous machine model and a t h e o r e t i c a l basis has been 13 f i r m l y established by DeMello and Concordia Only i n recent years the l i n e a r optimal c o n t r o l technique 2 has provided an alternative for the stabilization of power systems. It . , . 30,31,32,33,34 _ . . has become a very active research area . I n the analysis, an optimal control scheme is obtained by minimizing a performance index of a quadratic form, subject to the constraint of the system's linear dynamics. Implementation of "such a control scheme usually re-quires the measurement of a l l state variables of the system, some of which may not be attainable. An alternative i s to design a dynamic 37 estimator using Kalman f i l t e r or Luenberger's observer but i t i s im-practical and uneconomical in some cases. There i s another problem in linear optimal control design, that i s , the proper choice of the weighting matrices Q and R of the cost 34 function. Moussa and Yu developed a technique of choosing Q with the dominant eigenvalue shift of the closed loop system but very complicated -to :appiy. This 'thesis'-provides a completely <-d»if feren>t *way-«of-determin- •-ing Q and R and they are easy to apply. 1..2 The Scope of the Present Thesis This thesis is intended to present an analytical method specially suitable to d i g i t a l computation and at the same time the con-t r o l scheme designed can be easily implemented for actual power systems. As for the stabilizing signal feedback input, only the easily measur-able state variables are used. The optimal control scheme designed depends largely on the mathematical model used for the synchronous machine and control systems. 30 32 3^ Some controller designs are based on simplified low order models ' ' 33 36 and others on more elaborate ones ' . Since the synchronous machine 3 is the "centre figure" of the power system s t a b i l i t y studies, the following questions arise: How accurate shall be the model? Which 3 v a r i a b l e s s h a l l be chosen as the state v a r i a b l e s , f l u x - l i n k a g e s , currents or other measurable variables? Chapter 2 i s completely devoted to the modelling of a synchronous machine f o r the s t a b i l i t y study. Various mathematical models of the machine with varying degrees of complexity are i n v e s t i g a t e d . As the r e s u l t of t h i s i n v e s t i g a t i o n a f a i r l y accurate yet not unnecessarily complicated f i f t h order synchronous machine model i n terms of the e a s i l y measurable state v a r i a b l e s i s developed. The next chapter, Chapter 3, deals with the conventional sup-plemental e x c i t a t i o n control design using the f i f t h order model. The lead-lag compensation of such a design was based on the n a t u r a l mechani-c a l mode o s c i l l a t i o n without considering damping. With the s i g n a l implemented, the o s c i l l a t i o n frequency changes which are usually found from an eigenvalue analysis. This i n d i c a t e s that a more accurate model ..and-.a .dif;f-er.ent,<appjoaGh,.xonsldering .-both mechanical-and . e l e c t r i c a l o s c i l l a t i o n s must be used for the c o n t r o l l e r design. There i s also a problem whether a s i n g l e v a r i a b l e or a combination s h a l l be used f o r the s t a b i l i z i n g s i g n a l input and which gives better r e s u l t s ? The answers to these problems are found i n t h i s chapter. For the l i n e a r optimal c o n t r o l design the p h y s i c a l l y r e a l i z -able state v a r i a b l e model developed i n Chapter 2 i s further transformed i n t o a canonical form which enables one to determine the weighting matrices Q and R of the cost function by a simple procedure. The analy-t i c a l d e t a i l s are given i n Chapter 4. The canonical form design technique developed i n Chapter 4 i s then applied i n Chapter 5, to s t a b i l i z e a t y p i c a l one machine-infinite power system. Both optimal e x c i t a t i o n control and optimal governor control are designed. Transient response tests on the nonlinear 4 model of the system prove the e f f e c t i v e n e s s of the l i n e a r c : n t r r l design. Since, there i s one degree of freedom, the Q/R r a t i o , le:'t ir. this design technique, an eigenvalue search technique i s developed and implemented, so that i t i s p o s s i b l e to design a s i n g l e optimal con-t r o l which can s t a b i l i z e a power system under wide power range operating conditions. Chapter 6 further applies the c o n t r o l design techniques-, developed i n Chapters 4 and 5 to the interconnected multi-machine power system. Several c o n t r o l schemes are developed. I t i s found i n general that a s i n g l e optimal c o n t r o l can be designed to s t a b i l i z e the e n t i r e power system. 5 2. DYNAMIC MODELLING OF SYNCHRONOUS MACHINES 2.1 Synchronous Machine Equations i n MKS For the dynamic study of a power system, the f i r s t step i s to obtain a good mathematical model f o r the system, f a i r l y accurate yet not unnecessarily complicated. The cont r o l schemes obtained from the model must be r e a l i z a b l e . In th i s chapter various models of synchronous machines are derived and a f a i r l y accurate model of the f i f t h order i s developed i n terms of e a s i l y measurable state v a r i a b l e s . The model w i l l be used f o r the supplemental and l i n e a r optimal s t a b i l i z a t i o n designs of a power system. Park's coordinates are used to describe the nonlinear d i f f e r -e n t i a l equations of a synchronous machine 1'* i ,~ >' ^ . F i g . 2.1 shows the -.fewOT.a-xd.-s...model x>f -...sy.n,ch.rono.us -machine...with .one damper winding in" each axis. The generator current convention commonly used i n the power industry i s adopted. A l l equations i n MKS units are as follows: Voltage Equations d r r d r q a d q q d a q v f = p * f + Rf i f (2.1) 0 " P *D + *D S 0 = P *Q + RQ \ Flux Linkage Equations d d d af f aD D 1 = - L i H . L q q q aQ Q 6 * f = " I L a f ±d + h l ± f + L f D 4 D D 2 LaD 4 d + L f D * f + LD S — L rt i , + i . 2 aQ q Q Q (2.2) d-AXIS q-AXIS- rHW\ Figure 2.1 A 2-axis synchronous machine i n Park's dq coordinates Mechanical Equations Tm= Te+^ f *** • Te"f <*d Wd* ( 2 ' 3 ) to where T mechanical torque, T e l e c t r i c torque, H the i n e r t i a constant, m e G the machine r a t i n g , and U q the e l e c t r i c synchronous speed a l l i n MKS units, The mutual inductances between the rotor and s t a t o r c i r c u i t s of eqn. (2.2) are not r e c i p r o c a l . Although this does not cause any d i f f i c u l t y i n a n a l y s i s , i t becomes an obstacle f o r simulation s t u d i e s . There are a number of ways of obtaining r e c i p r o c a l mutual inductances. The simplest procedure i s to modify the rotor currents as w e l l as the 7 rotor parameters by choosing a common volt-ampere base f o r a l l c i r c u i t s i n c l u d i n g the three-phase on the s t a t o r as a whole. Again, there are a number of independent choices f o r the rotor base currents. Two of them are noteworthy. The f i r s t i s c a l l e d the x ^ per-unit system by which a base rotor current i s chosen so that an equal per-unit mutual inductance r e s u l t s i n each axis f o r a l l rotor and s t a t o r windings. This system i s frequently used i n the power industry. The second may be r e f e r r e d to as the per-unit system for which the base e x c i t e r output voltage i s chosen so that i t produces the rated generated voltage on the air-gap l i n e of the machine. In t h i s thesis the x ^ per-unit system i s adopted. 2.2 Synchronous Machine Equations i n Per-Unit Following Park's convention the " f l u x linkages" are defined from the per-unit reactance instead of inductance as follows: |, = - x, i , + x , i , + x , 1 r d d d ad f ad D ii = - x i + x i _ q q q aq Q f ad d f f ad D / 0 , s (2.4) *D = _ X a d i d + X a d * f + : * D *D \ = " X a q S + XQ \ In f a c t , those " f l u x linkages" are "per-unit voltages". Thus the voltage equations may be w r i t t e n 1 I U I T, • v, = — p iK i> - R i , d ai r r d to q a d o o v = — p f + —— iK - R i (2.5) q a) r q ui d a q o o v f = ^ p *f + V f o 8 0 = t P * D + V D o 0 - ? V V Q o and the torque equation becomes P 2 6 = I I ( Tm-V' Te • Vq - Vd (2'6) In the above equations a l l quantities are expressed i n per-unit except t i n s e c , 6 i n rad., and w i n rad./sec. 2.3 Flux Linkage Model or ^-Model of the Machine Flux linkage models are frequently used i n power system studies 1"' s 20 33 5A ' ' f o r which the f l u x linkages are chosen as state v a r i a b l e s . The voltage and torque equations can be w r i t t e n e i t h e r i n c i r c u i t parameters l i k e those i n eqns. (2.4) or i n Park's parameters which are more f a m i l i a r "-to engineers. "Park'-s- 'parameters 'can ;be; -defined '-in"terms "of c i r c u i t parameters of the machine as follows: A , -x, = x „ + x , d al ad 2 , A ad x' = x, d d x^ . x*" . ( x £ + x_, - 2x ,) „ A _ ad f D ad X d X d 2 xf *D ~ X ad X = X . + X q ax. aq 2 A X x" = x„ ~ (2-7) ,2 q x Q A x f 1 ( x d *as)' do a) R. a) R. (x,-x' ,) o r o f d a 2 . , v2 ,, = Xf VXa d = 1 ^Xd~Xax/ do " % ^ ' ^ <xj-xd) 9 X" = y qo ID R i ( y ^ ' oo R (x -x" ) o Q q q The synchronous machine equations i n terms of Park's parameters are derived i n this thesis as follows: pS = co poo = 2 H (T -T ), T = i - i> i , m e e d q q d piK = oo v. - —f <X'd " *V f o v f T' x" (x' -x ..) do d - d al ^ a 9 ? * A <Xd"Xa£> + x (*d-*aP d (x'-x" d ) j , (VXa £) (tXd) xa £ , ^ ^ d ^ l ^ a P , + 1 *D + _ — *d T' x'!(x'-x do d d al T' x" (x'-x ) do d d al ( xd" xy X a * D " Tdo < W > X d x d , , (xd~xaP , * f - T T ~ x T *D + T" x" *d do d do d x qo q *Q + T^V qo q ( 2 . 8 ) pi|) = oo •&> +oo (v, + R i . ) r d q o d a d pii = - coil), + oo (v + R i ) r r q r d o ^ q a q ± _ ( x d - X a ^ d X d ( xd- Xa£ } ( xd- Xa£ } , , <xd-xy q x" (x -x ) n q q a«. . la X Here, the current of damper c i r c u i t s i ^ and i ^ and f i e l d c i r c u i t i ^ are eliminated from equation ( 2 . 4 ) , ( 2 . 5 ) and ( 2 . 6 ) and the results are given i n the state variable form as ( 2 . 8 ) • 10 2.4 ^-Model of Reduced Order The synchronous machine, represented by a seventh order model includes the e f f e c t of speed v a r i a t i o n , damper c i r c u i t s and s t a t o r trans-i e n t s . This d e t a i l e d machine formulation necessitates the elaborate, representation of an external transmission system. A large set of i n t e g r o - d i f f e r e n t i a l equations can be w r i t t e n f o r the transmission l i n e s , transformers and loads. The number of equations r a p i d l y increases with the complexity of the power system and makes the computation i m p r a c t i c a l . In p r a c t i c a l s t a b i l i t y studies, s i m p l i f y i n g assumptions are usually made by i n c l u d i n g only the s i g n i f i c a n t p ortion of the dynamics of the system. The r e s u l t s thus obtained are s u f f i c i e n t l y accurate for engineering purposes. e l e c t r i c transients of an external system and the v a r i a t i o n of i t s para-meters due to changes i n frequency and voltage l e v e l . A furt h e r assump-t i o n frequently made i s to neglect the e f f e c t of transformer voltages piK and pifi of the armature on transient currents and torques. Thus, we s h a l l have a 5th order synchronous machine and the approximation i s e s p e c i a l l y h e l p f u l i n multimachine studies. Since the v a r i a t i o n i n speed i s small, i t s e f f e c t upon the generated voltage can also be ne-glected. The re s u l t a n t equations derived from (2.4), (2.5) and (2.6) are summarized. An assumption usually made i n such a study i s to neglect the q to (T -T ), T = [• m e e ^d^aP ( x d " x a l ) <xd~xa*> *Q h + ^ q " ^ *d \ 11 do (xd - XaP' Tdo (xd"Xa£}' <xd-*a*> (xd-Xa^ . 4 do d ax. <Vxd> " T" (x,-x .) * f T" do d ail do (x'-x.) a ai6 r p t l do (2.9) rjpll qo Tqo < which can be wri t t e n as X = f (X, U, i ) (2.10) where X = [<5, a), if^, ij) , ij> ] . t l The d and q component currents and voltages at the machine terminal can be expressed i n terms of state v a r i a b l e s ifi's, <5 and to. To do t h i s , we have two more equations. v R -x x" R 0 0 0 0 0 0 <xd-xa*> (xd-xp <*d-x;? (xd-XaP(xd-Xa^ (xd_XaP (Xn"Xn> 3 9_ (2.11) X or v + P i = S X (2.12) 12 2.5 Voltage Model or V-Mod'el of the Synchronous Machine Based on the x , - per-unit system, the following voltages au are defined. ^Xd~XcP Xad E£j ~ 7—~ T T\ w . v = — — v.. f d ( xd" Xa£ ) do o f Rf f , ( xd "xd} , Xad , q (x d-x a £) f x f f e" = , , s ^ = — K (2.13) ^ ( xd- Xa£ } *D (x - x " ) X e" = _ q q ,h = _ -59. ^ d < V « 4 > *Q XQ *Q and they are called by Young^ and Olive^ as the p.u. f i e l d voltage, p.u. f i e l d flux linkages, p.u. d-axis damper flux linkages, and p.u. q-axis damper flux linkages respectively. They are i n fact a l l p.u. voltages. 7th Order Voltage Model Using those newly defined voltages, a V-model can be derived from eqns. (2.4)-(2. 6) . The V-model i s similar to \J;-model but a physical interpretation can be given to those variables as in reference 6. The resultant equations for the 7-th order model are: p6 = to - u) o pa) =-£ ( T - T ) , T = i - 4> 1. V 2H m e e d q q d E £ J (x' - x") (x,-x „) (x"-x .) , _ fd d d r d al' d al' , . , P e q T' ~ T' x" (x'-x J lXa£ (x'-x ) + X d (x'-x") J 6 q n do do d d al d al d d , Xal ( xd" Xd } „ . ( xd- Xd } ( xd" XaP + T' x" (x'-x .) eq + T' x" (x'-x .) ^d do d d al do d d al 13 x - x" x . X^XP D e- = 5 d al , d_ „ d d P Sq T" x" (x'-x .) eq x" T" 6q T" x" Wd H do d d al ^ d do do d x (x -x") (2.14) P ed T" x" d T" x" Vq qo q qo q. pi|i . -• to il; + to (v, + R i,) r Yd q o d a d pili = -co ii i , + co (v + R i ) r q d o q a q • - < x r x o ) e " ^ U II / T \ C 1 II II Xd ( xd " Xa £} q Xd Xd e" \ \ x" x" q q 5th Order Voltage Model Neglecting piK and pi|i terms, introducing the newly defined q voltages, and eliminating i ^ , i ^ , i ^ , ^ and iji from eqns. (2.4)-(2.6) the 5th order V-model becomes, i ( v * d ) ( * d - x d ) (, d-»y E f d pe' = - -=7i— [1+ 5 ] e + , , , — r - e + = j — ^ do (x' - x .) ^ do d al n do Cl SLJO (xd-x')(xg - x a t) Tdo <Xd - XaP ^ (2.15) „ (Xd-XP , _J_ „ ( Xd- X? . ^ = Tdo <*d"V> ~ Tdo ~ Tdo ^ qo (x -x") qo 14 where e^> e^ and e^ were defined i n eqn. (2.13). Eqn. (2.15) may be written as X = f (X, U, i ) Also v v + R -x" a q R 0 0 0 1 o o , , a\ 1 0 ( xd- XaP (2.16) X (2.17) Again eqn. (2.17) may be written as v + P i = S X (2.18) 2.6 V-Model of A l t e r n a t i v e Form A l t e r n a t i v e l y , a modified V-model can be derived based on the E^^ per unit system. In such a case, a l l the mutual inductances are r e -c i p r o c a l but they are not equal between two sets of windings, e..'g. x f £ x aD" x af f d q x • e" --92 (2.19) D d x^ r Olive's assumption that "any t r a n s i e n t change i n the s t a t o r currents w i l l be r e f l e c t e d i n i t i a l l y i n the damper windings and w i l l not a f f e c t the f i e l d " , i s adopted and a simpler expression f or x^' i s obtained. Thus we have xaD X f D *D X a f 15 x' = x, -2 af d d x f f X , = X 2 2 X a f *D + X f XaD ~ 2 x a f X f D XaD d d 2 X f ^ x f D 2 x ^ x , - — (2.20) d XD % R D ( x d " *d> *D T d o <V*d> Modified 7-th Order Voltage Model Using Olive's assumption and voltages defined i n the per unit system, the modified seventh order voltage model can be derived from (2.1), (2.2) and (2.3) as follows: di p 2 6 = — - (T -T ), T = e" i + e" i - (x"-x") i i v 2R m e " e q q d q v d q' d q pe' ! M ( V X d > , A ( xd-Xd> q T' T' (x'-x") q T' (x'-x") q n do do v d d n do v d d n e' (x' - x") x' p e " = _9_ + _ d <L_ _ __d „ P e q T" T" x" *d x" T" 6 q ^ do do d d do n x (x - x") P e d T" x" d T" x" ^q qo q qo q pW> = w TJ) + ai ( v , + R i , ) r d q o d a d p i | ; = a ) t | » + 0 3 ( v + R i ) (2.21) r r q r o q a q e" * d ±d"+'^ ' " xT *d d d e" * i = _ _d _ 13. q x" x" q q 16 Modified 5-th Order Voltage Model Neglecting the transients i n the armature, the modified 5-th order voltage model can be derived from eqns. (2.1)-(2.3). p 26 = T J§ (T -T ) , T = e" i + e" 1 , - (x" - x") i , i r 2H m e ' e q q d d d q' d q "" I " " Tdo ( xd-v e' e" o e " = 0. - -9_ a T11 T" q do do e1 + q ( vv ... + !« T' (x'-x") q T', (2.22) do v d d' do ^d- Xd> n i t do 1 ( X n " X " ) D e " = i _ e i i + 9, 3_ , T" ~d qo qo v. v. + R -x R 0 0 0 0 1 0 0 0 1 0 (2.23) Note that the f l u x linkage and the voltage models require eleven parameters f o r t h e i r representations whereas the modified voltage model requires only ten parameters for i t s representation (Table I) because of the approximation. The problem of measuring the constant x i s eliminated by this assumption. 3X> Parameters R a Xa£ X d x d T' do rp IT do X q x" q r p l l qo H Flux linkage/voltage 5th and 7th order X X X X X X X X X X X Modified voltage model 5th and 7th order X X X X X X X X X X 3rd order model X X X X X X Table I Data for the Models 17 2.7 Effect of Olive's Assumption on Modelling A typical one machine and i n f i n i t e power system of Fig. 2.2 is taken for this study. The development of the state model for a complete system is given in Appendix A. The calculation of i n i t i a l values of the system i s included in Appendix B. The dynamic models are described in terms of machine parameters calculated from acceptance tests. In order to test the validity of Olive's assumption, eigenvalues of those models are compared. The eigenvalues are given in Table I I . The syn-chronous machine of the same order has identical eigenvalues either with a ^-model or a V-model. P+jQ X . R = .00 a .973 x' = .190 Q T' = 7.765 do G = .249 R = -.034 X R X . ,133 x = .550 9 x" = .216 x = .05 q ax-do x = .997 044 T" = .094 H = 4.63 B = .262 qo v. 1.05 P - .952 Q = .015 Fig. 2.2 One Machine and Infinite Power System It i s found from the results that Olive's assumption has the effect of pushing the most dominant eigenvalues to the right without affecting the rest of the eigenvalues. The error incurred by this 18 assumption is about 7% for this particular example. The present IEEE Test Code does not include a definition for the armature, leakage reactance x „ , consequently, Olive's assumption ax. finds wide applications in synchronous machine modelling. Since Canay's 8 9 design procedure and Yu and Moussa's test procedure are available to determine this constant, there is no more d i f f i c u l t ) 7 in forming an exact model for a synchronous machine although the definition for x remains ajo to be standardized. Table II Eigenvalues of various synchronous machine models State variables Flux Linkage/Voltage Model Modified Voltag e Model 7th order 5th order 7th order 5th order -.0142 -.0142 -.0133 -.0133 6 ,OJ -.1684 + J4.77 -.1619 + J4.78 -.1572 + J4.79 -.1506 + J4.79 *Q <EV -12.86 -12.88 -12.86 -12.88 * D ( EV -23.75 -23.76 -25.22 -25.24 -840.5 + J3153 -840.4 + J3153 19 2.8 E f f e c t of Transmission Line and Local Load The l a r g e s t eigenvalue of the system belongs to the two st a t o r f l u x linkage equations and P^ , which are l a r g e l y a f f e c t e d by the ex-t e r n a l system connected to the machine. Decreasing the transmission l i n e impedance pushes the la r g e s t eigenvalues of the system c l o s e r to the imaginary axis but i t also increases the p o s i t i v e damping of the mecha-n i c a l mode o s c i l l a t i o n s . In the case of very short transmission l i n e s the eigenvalues due to IJJ^ and are comparable with that due to damper c i r c u i t s , which are the next l a r g e s t eigenvalues of the group. When the l i n e i s compensated, t h i s compensation improves the t o t a l damping of the system. Those r e s u l t s are shown i n Table I I I . The power factor of l o c a l load has very s l i g h t influence on the dominant eigenvalues and the r e s u l t s are shown i n Table IV. 2.9 E f f e c t of Machine Loadings The e f f e c t of machine loadings on the eigenvalues of the system has been in v e s t i g a t e d and the r e s u l t s obtained are given i n Table V. The modes due to s t a t o r and damper c i r c u i t s are i n s e n s i t i v e to load v a r i a t i o n s but there are considerable changes i n the modes associated with the mechanical and f i e l d c i r c u i t s . The active power generation of the machine improves the damping whereas the large inductive re a c t i v e power reduces the synchronizing torque of the system. 20 Table III Influence of transmission system on the system eigenvalues R = .05 X = .5 B = G = 0 State variables single circuit transmis-sion double circuit triple circuit single ct. plus 50% compensa-tion double ct. plus 50% compensa-tion treble circuit plus 50% com-pensation 6,co, ij>f -.144 -.223 -.268 -.223 -.297 -.334 -.42+ -.79 + -1.06+ -. 78 + -1.23 + -1.48 + J7.12 J8.58 j.926 J8.86 J9.26 j i o . i -15.02 -16.97 -18.14 -16.93 -18.93 -19.95 -24.82 -26.07 -26.95 -26.00 -27.57 -28.51 -124.2+ -64.95+ -44.6+ -122.3+ -62.74+ -42.48+ J1496 ' J939 J753 J938 J658 J565 Table IV Influence of local load on the system eigenvalues R = .05 X = .50 State G = .25 G = .25 G = .25 G = .50 G = .50 G = .50 variables B = -.05 B = -.15 B = .05 B = -.05 B = -.15 B = .05 6,co,4)f -.150 -.156 -.145 -.157 -.162 -.152 -.43+ -.45+ -.40+ -.43 + -.45 + -.41 + J7.20 J7.31 J7.09 J7.25 J7.36 J7.14 * Q -15.12 -15.24 -14.99 -15.18 -15.30 -15.05 -24.85 -24.93; -24.78 -24.85 -24.92 -24.77 d q -247 + -226 + -271 + -356 + -327 + -389 + J1428 J1383 J1476 J1361 J1324 J1400 21 Table V V a r i a t i o n of eigenvalues with load v = 1.05 State variables 1 P = 1.25 Q = .45 2 P = 1.20 Q = .34 3 P = 1.15 Q = .25 4 P = .452 Q = .015 Nominal Load 5 P = .1 Q = 0.15 6 P = .5 Q = -.225 -.525+ -.304 + -.230 + -.168 + -.199 + -.246 + J2.24 j 3 . l l J3.67 J4.78 J5.28 J5.39 .701 .286 .148 -.014 -.092 -.136 *Q -12.89 -12.91 -12.92 -12.86 -12.69 -12.53 -23.73 -23.73 -23.73 -23.75 -23.78 -23.80 -840 + j 3153 -840+ j 3153 -840 +j 3153 -840+ j 3153 -840 +j 3153 -840 + j 3153 2.10 Voltage-Current-Power or V-I-P Model of the Machine The i p - and V-models have one common shortcoming. They both include state variables which are not d i r e c t l y measurable. The choice of state v a r i a b l e i s not unique as exp l o i t e d by many authors"*^ ' . Although f l u x linkages are the n a t u r a l candidates f o r the state v a r i a b l e s of Park's equations of synchronous machines, any seven independent v a r i -ables can be the state v a r i a b l e s . There are four e l e c t r i c a l v a r i a b l e s which can be measured instantaneously: terminal voltage, terminal current, f i e l d current, and e l e c t r i c power output. But only three of them are independent which can be chosen as state v a r i a b l e s . This implies that i f the state v a r i a b l e s are r e s t r i c t e d to measurable v a r i a b l e s i n c l u d i n g AS and Aco, the highest 22 order of synchronous machine model attainable i s five instead of seven. This i s mainly because of the two short-circuited damper windings in the d- and q-axes. What is the difference i n accuracy between a f i f t h and seventh order model of a synchronous machine? From the eigenvalue analysis of a typical one machine system (Fig. 2,2) without control, i t was found that the eigenvalues of the stator windings are by far the largest of them a l l which suggests that a f a i r l y accurate synchronous machine model can be obtained by neglecting pij^ and p ^ terms. Furthermore, since the step-size in d i g i t a l computation required to find the response of the system is determined by the spectral radius p(A) the modulus of the largest eigenvalue of A of the system, the lower order model i s very desirable for the computation. For Runge-Kutta Fourth Order Approxima-tion, the maximum step-size should be limited to l / p ( A ) . The reduction in computation time is considerable for the f i f t h order model over the seventh order model. Table VI Eigenvalues of one machine system Circuits 3rd Order Model 5th Order Model 7th Order Model f i e l d and mechanical -.0144 -.0906+ J4.74 -.0142 -.1619+J4.78 -.0142 -.1684+J4.77 q-axis damper -12.88 -12.86 d axis damper -23.76 -23.75 stator -840.5 + J3153 In addition further reduction of the order can be achieved by neglecting p i ^ and P*|/Q terms or the damper circuits. A simple third order synchronous machine model w i l l result. This means that a source of positive damping to the system i s ignored. According to Table VI the 23 damping of the mechanical mode o s c i l l a t i o n i s reduced by 40% of the higher order models by i t s e l f and the 3rd order model i s not accurate. Therefore a reasonably accurate synchronous machine model i s the f i f t h order one. It i s clear from the Table VI that the 6, to and ip^ modes are the most important ones, and the next s i g n i f i c a n t ones are due to damper c i r c u i t s . 2.11 State Equations and Transformation Matrix of V-I-P Model Henceforth we s h a l l choose the 5th order V-I-P model f o r the control studies and the 5th order \p-model for the nonlinear t e s t s . We s h a l l also designate the ip-model as the X-model and the V-I-P model as the Z-model. The state variables of X-model are ,t X = [A6 AGO Aijif Ai|>D Ai^] (2.24) There are three sets of state v a r i a b l e which can be used f or the f i f t h order Z-model. .1. Z = [AS Ato Av AP A i f ] t 2. Z = [A 6 Ato Av A i t A i f ] (2.25) 3. Z = [A6 Afo Ai AP Ai ]' They are a l l equivalent. Let the f i r s t set be chosen as the Z-variables. They are r e l a t e d to the X-variables as follows: A6 AGO Aipf (2.26) AIJJ^ A6 Ato Av t AP A if » — 1 0 0 0 0 0 1 0 0 0 m31 0 m 3 3 m34 m35 m41 0 m43 m44 "45 m51 0 m53 m54 m 5 5 24 where the transformation matrix elements m's can be found as follows: Terminal Current i , i Ai = -A A i ^ + -A A i t l t d i t q From equation (A.15), we have A i = [~ -A] C X (2.27) 1 \ \ Terminal voltage v v Av = — A V J + - 1 Av t v t d v t q From equations (A.10), (A.11) and (A.15) Av t = m 3 l A6 + m 3 3 A ^ F + m^ A ^ + ra^ A^ Q (2.28) Thus we have m 3 l = [ C21 ( v d X q " V q V " S i ( V d R a + V q X d ) ] / v t m33 = I C23 ( v d X q - Va + kf> " C13 < vd Ra + V a X ' d ) ] / v t m„. = [ C 0 / (v,x" - v R + v /k.) - C,. (v,R + v x")]/v 34 24 d q q a q 5 14 d a q d t m„, = [C 0, (v,x" - v R ) - C.. (v.R + v x" - v,/k,)]/v 35 25 d q q a 15 d a q d d 5' t E l e c t r i c Power AP = v, A i n + v A i + i . i v . + i Av d d q q d d q q Again from equations (A.10), (A.11) and (A.15) AP = m, , A<5 + m.„ t^>c + m. . AIJJ + m. _ Ai|». (2.29) 41 43 f 44 D 45 Q ra4l = V d + 0 5 V q + *d ( ° 5 X q ~ a 6 V " \ < V d + a 5 V m43 = ° 2 [ ( k 6 v d + k 5 V q } + h ( k 5 x q - k 6 V + \ ( 1- k6 xd " W ] 25 m45 = % [ 1 q ( k 5 X d + W " ± d ( 1 + k 7 X q " k 5 V " <k5vd + k 7 V q ) ] m44 = °3 m43 / a2' 05 = k3 k7 + k 4 k 5 J ° c = k^ k c + k/. K '3 5 4 6 F i e l d Current A i f = ( x d " X a J l ) A i f = m51 A 6 + m53 A * f + m54 A^D + m55 A*Q where (2.30) m 5 l = °2 C l l ( x d " X a £ ) mrn = a 0 C,0 ( X j - x_„) + ——-—7 [1 + 7.—J "53 "2 "13 v d a l ' (x,-x 0) d a l ( x d - x d ) m54 = °2 C14 ( x d " X a £ ) " °3 (x'-x ) d al mc, = a 0 C. c (x ,-x ,) 55 2 15 d al < Xd " Xal>' where the elements of matrix C are given i n equation (A.14) In compact form, eqn. (2.26) becomes: Z = M X (2.31) where z's are the weighted sum of the components of x's. Next, consider the state equation X = AX + BU and l e t the transformed equation be Z = F Z + G U (2.32) Then we have F = MAM-1 = 0 1 0 0 0 f21 f22 £23 f24 f25 f31 f32 f33 f34 f35 f41 f42 f43 f44 f45 f51 f52 f53 f54 f55 (2.33) 26 and G = MB = 0 0 0 a, f36 f46 0 "56 0 (2.34) The transformation does not alter the eigenvalues nor the responses of the linear system which can be proved as follows: 1. invariant eigenvalues The characteristic equation of the transformed model is -1 , -1 |F - XI| = |MAM - AMM| = 0 (2.35) = |M| IA-AI||M 1| or | A - XI | =0 which i s the characteristic equation of the original system. 2. identical dynamic responses The response of the transformed system can be written as t Z(t) = e F t Z Q + j e ± V L w GU(t) dt F ( t - T ) substituting Z = MX, the equation (2.26) becomes: (2.36) -1 X (t> - MX + / ^ ^ W ) ^ o Expanding the exponential term in Taylor's series and assuming conver-gence, i t can be shown that At „ , r e A ( t k" T ) BU(T) dx (2.37) X(t) = e A t X o + / The transformation matrix is well-conditioned indicating that the solu-tions are not very sensitive to small changes in the system parameters. The f i f t h order V-I-P model of synchronous machine in terms 27 of measurable state variables w i l l be used i n succeeding chapters for the supplemental and optimal c o n t r o l designs. 28 3. SUPPLEMENTAL EXCITATION CONTROL DESIGN 3.1 Introduction The s t a b i l i z a t i o n of power systems with supplementary e x c i t a -tion c o n t r o l has received more attention i n recent years than ever , , 10,11,12,13,14,15,16 c . -, . • , , before . The r o l e of the power plant i s not only to generate e l e c t r i c i t y at prescribed voltage and frequency and make i t ava i l a b l e to consumers but also to ensure the best performance of the system under abnormal conditions. The design of supplemental e x c i t a t i o n c o n t r o l f or i n d i v i d u a l plant involves the s e l e c t i o n of the best s t a b i l i -zing s i g n a l and the lead-lag compensation with proper gains and time constants so that the s i g n a l provides the best p o s i t i v e damping for the system. However, i t i s found that when a conventional supplemental e x c i t a t i o n control designed from the mechanical n a t u r a l o s c i l l a t i o n frequency i s implemented, the mechanical mode o s c i l l a t i o n as w e l l as other e l e c t r i c a l mode o s c i l l a t i o n s w i l l be a l t e r e d . This suggests that for a' good c o n t r o l l e r design a more accurate model which includes e l e c -t r i c a l modes must be used. The objective of t h i s chapter i s to i n v e s t i g a t e various output sig n a l s and t h e i r combinations, with or without compensation, used i n supplemental e x c i t a t i o n c o n t r o l for improving the system damping. The f i f t h order model developed i n Chapter 2 i s used for the design. It i s found that the signals derived from the speed, f i e l d current and power are very e f f e c t i v e whereas the signals derived from the torque angle, terminal current, and terminal voltage are not s u i t a b l e f o r the s t a b i l i -zation of a power system (Fig. 2.2,P. 17). 29 3.2 E x c i t a t i o n System and E f f e c t of Voltage Regulator Gains The type 1 computer representation of the e x c i t a t i o n system 19 recommended by IEEE Power Generation Committee i s adapted as i n F i g . 3.1 for the e x c i t a t i o n control design. Eref KF P 1+TfP KA VR ^Rmax UTAP -^vRmin 1+T£P •fd K A =50 T A = .05 T_ = .003 K = 0.03 E F T F = 1 V R M A X = + 8 ' 8 V R M I N = - 7 - ° F i g . 3.1 Block-diagram of the e x c i t a t i o n system The equations of the e x c i t a t i o n and voltage regulator system are K K. K, R _ A V _ _ A V _ J R + ^ U , + T. t T A F T • T. 1 T. r e f . A A A A A K F E f d " V F T F T E T F Ef d -E f d TE + ! i VR K VE V R (3.1) When the e x c i t a t i o n system eqns. (3.1) are l i n e a r i z e d and combined with (2.32) which i s derived from (2.26) the order of the system i s increased by three.and the augmented system equations are 30 AS 0 1 0 0 0 0 0 0 AS * Ato f21 0 f23 f24 f25 0 0 0 Ato Avt f31 f32 f33 f34 f35 0 0 f38 Avt AP f41 f42 f43 f44 f45 0 0 f48 AP A * f f51 f52 f53 f54 f55 0 0 f58 • A 1 f 0 0 " KA / TA 0 0 -1/TA " KA / TA 0 A VR % 0 0 0 0 0 K F V T F T E "KF/TFTE AVF 0 0 0 0 0 K E / T E 0 -1/T E A E f c + 0 0 0 a 1 0 0 0 0 0 0 KA / TA ° 0 0 0 0 t AT ] t m - (3.2). which may be written in compact form as Z = F Z + G U (3.3) The effect of voltage regulator parameters on. system s t a b i l i t y i s investigated from the eigenvalues of the system matrix F of (3.3). The introduction of a large gain to the voltage regulator has the effect of pushing the most dominant eigenvalue (upper, Fig. 3.2) to the right of the complex plane, hence making the system more vulnerable to disturbances. Fig. 3.2 plots the l o c i of the dominant eigenvalues with a variable gain K . A high voltage regulator gain provides some synchronizing torque but the requirement is the damping torque whir.h must rely on the supplementary control. The el e c t r i c a l modes (lower, Fig. 3.2) are well damped due to the high gain of the exciter. It is found that the time constant T^ has l i t t l e influence on the s t a b i l i t y of the system. In general, the introduction of continuously-acting, high-gain 31 5 » Im.(\) 5.0 30 50 .4.0 \ .1.0 Re.M -2.0 -75 -7.0 -.5 .5 F i g . 3.2 Dominant eigenvalue move of the system with gain (number in d i c a t e s gain K.) and fast -response e x c i t a t i o n systems and long EHV transmission l i n e s are the factors among others which cause severe system frequency 13 14 o s c i l l a t i o n , thus aggravating the problem of system damping ' . The p o s i t i v e dampings of the generator f i e l d and damper c i r c u i t s , the load, and the prime-mover are not s u f f i c i e n t to overcome the negative damping of the system. Consequently, a supplementary c o n t r o l scheme must be designed. 3.3 Compensation Network for S t a b i l i z a t i o n Signals The compensation network i n c l u d i n g a washout block f o r the s t a b i l i z i n g s i g n a l i s shown i n F i g . 3.3. This further increases the order of the system by three. The a d d i t i o n a l state equations are U l 1 1 1 1 pa, = pu(Z) - — + (—- — ) u_ + ( — - — ) u p u 2 = pu(Z) - ^ + ( i - - £-) u 3 (3.4) 32 3 pu, = pu(Z) - — 1 PT, u3 (l+pT3)T2 u2 (1+pT5)T4 1+P^ 0+pT2)T3 (1+pT4)T5 F i g . 3.3 Block diagram of the compensation network. Since i n general u(Z) = K x A6.+ K 2 Aco + K 3 Avfc + AP + K 5 A i f (3.5) where K's are s t a b i l i z i n g s i g n a l gains, pu(Z) = [K x K 2 K 3 K 4 K ]-[pA6 pAco pAvfc pAP p A i f ] t (3.6) S u b s t i t u t i n g f o r the derivatives pA<5, pAco, e t c . from eqn. (3.2), eqn. (3.6) becomes (3.7) pu(Z) = K 0 1 0 0 0 0 0 0 f21 0 f23 f24 f25 0 0 0 f31 f32 f33 f34 f35 0 0 f38 hi f42 f43 f44 f45 0 0 f48 f51 f52 f53 f54 f55 0 0 f58 [A<5 Aco Av AP A i , AV AV^ , AE, , ] t + K.a, AT (3.7) t f R F f d 2 1 m Thus i n the f i n a l form Z = F Z + G U (3.8) the matrices and var i a b l e s are Z = [A5 Aco Av AP A i , AV AV_ AE,, u, u, u.] 1" t i K c i d X / j 33 G = [0 a± O O O O O O K ^ ] ' 5 °1 = C l ) o / ^ 2 H ) U = AT m F = F (K^, I ^ , K^, K^, K,., T^, T^, T^, T^, Tj.) The problem i s to optimize the s t a b i l i z i n g gains and time con-stants so that the most dominant eigenvalues of the system w i l l be s h i f t e d to the l e f t on the complex plane as much as pos s i b l e i n order to obtain a f a s t and stable response of the system. 3.4 Parameter Optimization This can be accomplished by parameter optimization techniques which can be stated as follows: Find K's and T's so as to: min. max. tR g. \V(F), i = 1, 2, ..'.] (3.9) The best eigenvalue d i s t r i b u t i o n of the system i s obtained by searching along each parameter space so that an eigenvalue function n f(X) = II [R . X.(F)] i s minimized. The algorithm may be summarized i = l 6 1 as follows: i ) Assign nominal values f o r the T-parameters; i i ) S t a r t the search at the o r i g i n and make one-dimensional searches one at a time i n a l l d i r e c t i o n s of K-parameter space so that the function f(X) i s minimized; i i i ) A f t e r reaching the minimum i n K-parameter space, switch over the search to the T-parameter space u n t i l a new minimum of f(X) i s attained; iv) Go to step i i and repeat the procedure u n t i l s a t i s f a c t o r y answers are found. 34 3.5 Eigenvalue Searches The objective of parameter optimization can be accomplished by simple eigenvalue searches. For that the same system equation (3.8) i s used. F i g . 3.4 shows the dominant eigenvalue move of the system with si n g l e s t a b i l i z i n g s i g n a l without compensation. I t reveals that both -AP and - A i ^ signals can be used to s t a b i l i z e the system and each one has an optimum point i f the gain i s increased too f a r . The p o s i t i v e AP and Aij_ increase the p o s i t i v e synchronizing torque and reduce the p o s i t i v e damping of the system but negative AP and A i ^ do exactly the converse. This suggests that a small l a g compensation i s necessary f o r -AP and - A i ^ to improve the damping torque as w e l l as synchronizing torque of the system. The speed s i g n a l as i t i s has a detrimental e f f e c t on s t a b i l i t y . This suggests that i t d e f i n i t e l y needs lead compensation. The other three s t a b i l i z i n g s i g n a l s A6, Av^ and A i f c are found to be i n e f f e c t i v e f o r the system studied and r e s u l t s are not included i n F i g . 3.4. F i g . 3.5 shows the dominant eigenvalue move of the system with s i n g l e s t a b i l i z i n g s i g n a l . AP and A i ^ with the lead compensation network of F i g . 3.3 seem i n e f f e c t i v e . On the other hand, i f the sign of AP and A i ^ i s reversed and a l a g instead of le a d compensation i s used, the s t a b i l i z a t i o n becomes very e f f e c t i v e as i l l u s t r a t e d i n F i g . 3.6. F i g . 3.7 shows the dominant eigenvalue move of the system with two compensated s i g n a l s , Aco and -AP or Aco and - A i ^ , both very e f f e c t i v e . In some studies,' the regulator must be represented by two time F i g 3.4 Dominant eigenvalue move, with s i n g l e s t a b i l i z i n g s i g n a l without compensation (number indicates gains) Im.\ F i g . 3.5 Dominant eigenvalue move, with s i n g l e s t a b i l i z i n g s i g n a l with compensation. F i g . 3.7 Dominant eigenvalue move, with two compensated s i g n a l s . 37 constants rather than one and the inherent phase l a g contributed by the e x c i t a t i o n system would be very l a r g e . In such case -AP and - A i f may not require compensation. 3.6 Time Response of the System F i g . 3.8 shows the time responses AS, A to and AP of the system to a step input torque disturbance. Four cases are studied. 1. No s t a b i l i z i n g s i g n a l ; 2. With compensated Ato; 3. With compensated Ato and -AP; 4. With compensated Aco and - A i ^ ; I t i s observed that the response curves of case 2 are s l i g h t l y o s c i l -l a t o r y . Although cases 3 and 4 have appreciable overshoot i n the f i r s t swing, they give b e t t e r r e s u l t s i n the subsequent swings. F i g . 3.9 shows the time respones of AS, Ato and AP of the sys-tem to an impulse disturbance i n the input torque. The comments made to F i g . 3.8 i s also v a l i d to F i g . 3.9. The following conclusions may be drawn. 1. For s i n g l e input s i g n a l s AS, Avfc and A i f c are unsuitable f o r the s t a b i l i z a t i o n of power systems. Of the remaining input s i g n a l s , Ato, -AP and - A i ^ are e f f e c t i v e when compensated. 2. For . combined compensated input s i g n a l s , Ato and -AP or Ato and - A i ^ always gives b e t t e r r e s u l t s than that of the s i n g l e input s i g n a l , thus improving the system damping. 3. A combined compensated Ato and uncompensated -AP contributes the best damping to the system. 38 Fi g . 3.8 Time responses of the system to a step disturbance 1. No s t a b i l i z i n g s i g n a l 2. With compensated Aw 3. With compensated Aco and -AP 4. With compensated Aco and - A i 1 F i g . 3.9 Time responses of the system to an impulse disturbance 1. uncomp. AP 2. uncomp. AP + comp. Aco 3. uncomp. A i ^ 4. uncomp. A i , + comp. Aco 39 4. OPTIMAL CONTROL DESIGN BASED ON EQUATIONS IN CANONICAL FORM 4.1 Introduction In t h i s chapter a new l i n e a r optimal c o n t r o l design technique i s proposed. The system equations are transformed i n t o a canonical form and the optimal c o n t r o l l e r thus designed i s a function of the weigh-t i n g matrices Q and R of the cost function. As a r e s u l t , the eigenvalue search technqiue can be applied and the c o n t r o l l e r designed w i l l most e f f e c t i v e l y s t a b i l i z e a power system under more severe disturbance con-d i t i o n than conventional l i n e a r optimal c o n t r o l design. Although the optimal c o n t r o l of power system dynamics has be-i 31,32,16,33,34 v . , T T , 30 ,. ^ . come popular since Yu, Vongsuriya and wedman f i r s t i n -troduced the subject, i t has not yet been used i n any e l e c t r i c power system. The main c r i t i s m was that those state v a r i a b l e s used i n t h e i r design required f o r the c o n t r o l l e r were not d i r e c t l y measurable. Luen-37 berger's observer can be constructed to estimate the immeasurable states but the ad d i t i o n of a dynamic observer of high-order system makes the o v e r a l l c o n t r o l scheme more complex and unduly s e n s i t i v e to d i s t u r -bances and changes i n system parameters. An a l t e r n a t i v e i s to design 26 a suboptimal c o n t r o l l e r using only d i r e c t l y measurable states but i t w i l l never be as good as a f u l l - s t a t e optimal c o n t r o l . In t h i s chapter, an optimal c o n t r o l l e r i n terms of d i r e c t l y measurable state v a r i a b l e s i s designed. In other words, the c o n t r o l l e r designed i s p h y s i c a l l y r e a l i z a b l e and can be e a s i l y mechanized. There i s another problem connected with the optimal control design, i . e . how to s e l e c t the cost function or the performance index 40 matrices Q and R? Once these matrices are chosen, the design procedure i s straightforward and the r e s u l t i s unique. In the past, the choice of those matrices r e l i e d mainly on i n t u i t i o n and engineering experiences. 34 Moussa and Yu developed a method to determine Q based on the dominant eigenvalue s h i f t but l e f t R to the d i s c r e t i o n of engineers. In t h i s chapter a d i f f e r e n t and a more general method i s developed. 4.2 Canonical Form of System Equations A system equation can be transformed i n t o the phase-variable 23 28 39 canonical form ' ' which i s very convenient for the optimal c o n t r o l design. Consider a s i m i l a r i t y transformation f o r the Z-model (2.31). Let Z = T Y Then we have Y where F Y + G U o o F = T 1 F T o G = T o (4.1) (4.2) (4.3) (4.4) Let F q and G q have the following p a r t i c u l a r form, F = o 0 1 0 0 0 0 1 0 0 0 0 0 • a l "a2 _a3 v l -a (4.5) G = [0 0 o 0 1]' (4.6) 4 1 Being a s i m i l a r i t y transformation the matrix F q possesses the same pro-p e r t i e s as F but i t i s e a s i e r to manipulate. Consider the a elements f i r s t . Since the eigenvalues do not change i n a system with the s i m i l a r i t y transformation, we s h a l l have J A I ^ - F J = (A - A N ) (A - A,) (A - A N ) = 0 ( 4 . 7 ) and IAI -F I = A N + a A N _ 1 + a . A N " 2 + . . . . + a = 0 ( 4 . 8 ) 1 nn o 1 n n - l 1 Assuming d i s t i n c t roots A^, k^, ... A N f o r the system, the a elements are determined from ( 4 . 7 ) and ( 4 . 8 ) as follows. n a = - t r . (F) = -I A. n v .. l i = l V - l = h X 2 + V 3 + + V l X n ( 4 ' 9 ) a = -(A,A 0Ao + A,A„A, + ... +A 0A .X ) n-2 1 2 3 1 2 4 n-2 n - l n n ct = - ( - l ) n det (F) = ( - l ) n n A 1=1 . Thus an algorithm f o r determining these c o e f f i c i e n t s can be w r i t t e n . b l l = \ b._ =b. , - i = 2 , 3, n i l x-1,1 x a = -b -n n l b U = x j b i , j - r k = n + j " 1 b i j = b i - l , j + X i + j - l b i , j - l ' i - 2, ...k, J - 2, 3, ...(n-l) °K = ( ~ 1 ) J b k j a, = ( - 1 ) " A b . 1 n l,n-l 42 4.3 Transformation Matrix of the Canonical Form Consider the transformation matrix T next. Let i t be w r i t t e n T = [ T r T 2, T 3, Tfl] (4.11) where or T., T „ , ... T are column ve c t o r s . From (4.3) we have 1 <d n [ T1 ' T2 T3 ' •'•» Tn] Fo = F [ T1 ' T2 ' T3 ' Tn] ( 4'1 2 ) 1 - 0 , 1 , -a 9T + T , -a T + T „ , -a T + T ,] I n z n ± j n 2. n n n - l = IFT 1 9 FT 2, FT 3, F T J (4.13) Thus T n, T „ , ... T can be determined from a c o e f f i c i e n t s . From (4.4), I I n we have T G = G o or (4.14) T = G n Hence, we have an extra equation for the determination of T^, T 2, . . . T^ Eqns. (4.13) and (4.14) may be w r i t t e n as T = G I I (4.15) T . = FT ... + a ... G, i = 1, 2, ... (n-l) n - i n-i+1 n-i+1 ' » » \ / and T = FT, + a,G = 0 (4.16) o i l The best way of computing T-, T-, ... T i s to use the recursive formula 1 2 n (4.15) and to derive a formula from (4.16) to correct the errors ac-cumulated i n the computation. In general FT 1 + a G f 0 (4.17) 43 near the end of computation. Let the computed r e s u l t s be T^, T 2, and l e t the corrected answers be T^, T^, .... In other words l e t FT, + a.G = 0, FT, + a G = T (4.18) 1 1 1 1 • o and l e t the errors be n. = T. - T. (4.19) i I i Then we s h a l l have n, = -F 1 (4.20) l o Next FT 2 + a 2G = T l 5 FT 2 + c^G = 1 (4.21) Therefore n 2 = F" 1 T l l ru _ 1 n. '(4.22) In general \ = - F _ 1 V \ = F _ 1 V i T = T + N , T. = T. + n. I l l 1 1 X i = 2, 3, ... (4.23) I t i s found that only corrections f o r T^ and T 2 are necessary. When the matrix T i s not w e l l behaved i t can be conditioned by diagonal matrices to obtain an i n v e r s e . 4.4 Optimal Control Design with Canonical Form For mathematical convenience, the system dynamics described by the canonical form w i l l be used f o r the optimal c o n t r o l design. A quadratic cost function CO J = -| J (Y F C Q QY + U t R U) dt (4.24) 44 i s chosen as usual where Q = q I , R = r ^o nn which corresponds to Q = q ( X T 1 1 ) - 1 for the Z-model Here, I i s an i d e n t i t y matrix of order n and q and r are s c a l a r s . By nn introducing a co-state v a r i a b l e p Q , the system dynamics (4.1) and the cost function (4.24) are appended to form a Hamiltonian H = - i - Y t Q Y + i - U t R U + p t [ F Y + GU] 2 ^o 2 r o o o Applying Pontryagin's maximum p r i n c i p l e we have Y = H = F Y + G U p o o r o -H = -Q Y - F p Y o o o (4.25) (4.26) (4.27) and 0 = H = R U + G p (4.28) U o o Solving U from (4.28) and s u b s t i t u t i n g the r e s u l t i n t o (4.26) we have Y P. o - q l ± G G f c r o o -F Y P. (4.29) nn o which are the combined state and co-state equations f or the system. Eqn. (4.29) i s a system of 2n l i n e a r d i f f e r e n t i a l equations. The c h a r a c t e r i s t i c equation of (4.29) may be wri t t e n (XI - F ) nn o q i nn ±-G G f c r o o (XI + F ) nn o = 0 (4.30) i n which each element i s a square matrix of order n f o r an nth order system except f or q, r and X. Assuming |XI n - F^j ^ 0 and e l i m i n a t i n g 45 ql , we have nn (XI - F ) nn o — G G fc r ° o (XI + F ) nn o 5- (XI - F )~1 G G t r nn o o o = 0 or I (XI - F ) (XI + F C) - 1 G G t| =0 1 nn o nn o r o o 1 Expansion and rearrangement result in IXI - F I IXI + P 'I + ( - l ) n a = 0 (4.31) nn (4.32) or r n ^n-1 s n - 2 , [ X + a X + a .. X + n n-l + a,J[X - a X + 1 n ..+ ( - l )n a j or + ( - l ) n ± = 0 Un - (a 2 - 2a .) y11"1 + (a 2 - 2a _ a + 2a „) Pn 2 n n - l ' v n-l n-2 n n-3 + ( - l ) n ( a ^ + = 0 (4.33) where V - X Equation (4.33) contains the characteristic roots of the optimal and the adjoint systems. The eigenvalues of the combined system are symmetric with respect to both real and imaginary axis of the com-plex plane and only the eigenvalues with negative real part . need 34 to be considered for the optimal control system . Since a l l a's are known, one can plot the root l o c i by varying q/r to search for the best optimal control design. Also a restriction on the movement of imaginary part of X may be placed i f desirable. 46 4.5 Optimal C o n t r o l l e r The optimal c o n t r o l found from (4.28) may be written as u = - i f 1 G t p o r o = -R _ 1 G ' K Y (4.34) o o where the R i c c a t i matrix K i s defined by p = KoY and can be c a l c u l a t e d o . J r o 38 from the eigenvectors of (4.29) or from the nonlinear algebraic equa-t -1 t 30 32 t i o n F K + K F +Q - K G R G K = 0 ' ; but i t i s not necessary 0 0 0 0 0 0 0 o o to c a l c u l a t e K by the present method. The c o n t r o l can be defined as o u = -S Y (4.35) o where the row vector i s given by S o = R ~ l G o t K o = [h*2 ••• * J ( 4 - 3 6 ) •The" problem now "reduces* to 'finding"""S . 'Sinces,we"''have from the •eharac-t e r i s t i c equation of the open loop system, i AI - F I = Xn + a A11"1 + a . X n~ 2 +...+a„ X + a, = 0 (4.37) 1 nn o' n n - l 2 1 from the closed loop system, IXI - (F - G S )I = Xn + (a + 0 ) X n - 1 + ( a . + 0 .) X n" 2 + 1 nn o o o 1 n n n - l n - l ... (a 2 + 02)X + + 0^ = 0 (4.38) and from the desired eigenvalues of the root l o c i p l o t of the system, (X - \ ) (X - A ) ... (X - X ) J . Z n = Xn + a X n _ 1 + a , X n" 2 + ... + a0X + a, = 0 (4.39). n n - l 2 1 where a's are c a l c u l a b l e , then i t i s p o s s i b l e to evaluate 0± = a± - a±1 i = 1, 2, ... n (4.40) 47 The optimal c o n t r o l u can be determined once g's are known. F i n a l l y , since the Z-model i s the model with measurable state v a r i a b l e s , the optimal control thus designed from the canonical form s h a l l be transformed back to the Z-model f o r actual implementation. The optimal c o n t r o l i s given by U = -S Y o « -S T _ 1 Z (4.41) o Thus, a r e a l i z a b l e l i n e a r time-invariant feedback co n t r o l system i s de-signed. There i s no s o l u t i o n f o r the optimal co n t r o l when the matrix T i s singular i . e . j T} = 0. But det (T) = d e t { [ F n _ 1 G , F n _ 2 G , ... FG,G]•[F"_1 G q , F*~ 2 G q , ... F ^ , G j " 1 } - det { c o n t r o l l a b i l i t y matrix of Z-model}"det{[controllability matrix of Y-model]" 1} (4.42) The system described by the state equation Z = FZ + GU w i l l be completely c o n t r o l l a b l e i f the c o n t r o l l a b i l i t y matrix has rank n. Therefore, a unique nonsingular transformation T and hence an optimal c o n t r o l l e r U always e x i s t f o r a c o n t r o l l a b l e system. I t should be noted that from a computational viewpoint, the procedure i s quite p r a c t i c a l . The main computing time expended i s f o r the canonical form and the transformation matrix. 48 5. PHYSICALLY REALIZABLE OPTIMAL EXCITATION AND GOVERNOR CONTROLLERS 5. .1. Introduction Based on the l i n e a r optimal design technique developed i n Chapter 4 using e a s i l y measurable state v a r i a b l e s , l i n e a r optimal e x c i -t a t i o n and governor controls are developed i n t h i s chapter f o r the stab-i l i z a t i o n of power systems. The c o n t r o l l e r requires neither a state e s t i -mator nor an observer f o r i t s implementation. For the design, a f i f t h order synchronous machine model developed i n Chapter 2 i s used. Trans-i e n t response of the power system based on the nonlinear model and a large disturbance demonstrate the effectiveness of the optimal control design. One wonders: Can one design a s i n g l e optimal c o n t r o l for a system operating ever wide power range conditions? Since the ordinary optimal control designed for c e r t a i n power conditions i s inadequate f o r 35 other operating points, Moussa and Yu developed an optimally s e n s i -t i v e c o n t r o l l e r with an a d d i t i o n a l feedback loop to cope with such con-36 d i t i o n s . Stromotich and Fleming also made some suggestions on adaptive c o n t r o l . In this chapter, i t i s shown that with the assistance of eigen-value searches, i t i s possible to design a s i n g l e optimal c o n t r o l without a d d i t i o n a l feedback loop, which can s t a b i l i z e the system under wide power range operating conditions. 5.2 System Models for Optimal E x c i t a t i o n Control Design The power system to be studied consists of a s i n g l e machine connected to an i n f i n i t e bus through transmission network as i n F i g . 2.2. The synchronous machine i s represented by the f i f t h order model. Assuming a s o l i d state e x c i t e r of n e g l i g i b l e time constant and one time constant 49 voltage regulator we have P A E f d = 1_ T. AE fa A « . A T=— Av + — u, T A t T. 3 A A (5.1) + This r a i s e s the order of the complete power system equations to s i x . The present section i s confined to the optimal e x c i t a t i o n c o n t r o l . The governor and the prime mover dynamics w i l l be considered i n the succeeding s e c t i o n . The system data were given i n Section 2.7 of Chapter 2. X-Model or Flux Linkage Model For the data given the state equation of the model can be w r i t t e n r 0 0 0 0 0 1000 (5.2) or X = A X + B U Z-Model or Voltage-Current-Power Model There are l i n e a r r e l a t i o n s between the terminal voltage v , the e l e c t r i c power P, and the f i e l d current i ^ , which can be used as the state v a r i a b l e s instead of unmeasurable quantities ib^, \b^ , and The transformation was given i n eqn. (2.26) of Chapter 2. For the data given, the numerical r e s u l t s are A6 0 1 0 0 0 0 AS • Aco -25.63 0 -25 -20.2 -5.67 0 Aco -0.06 0 -.45 0.32 .01 0.15 • Ai|»f _ -2.78 0 18.2 -23.6 .36 0 A*D % -1.72 0 -.45 -0.36 -12.9 0 A E f d 75.1 0 -422 -340 231 -20 A E f d — _ — 50 A6 1 0 0 0 0 0 AS Ato 0 1 0 0 0 0 Aco Av t -.075 0 .42 .34 -.23 0 • AP .63 0 .61 .50 .14 0 % A i f .41 0 2.93 -2.14 -.05 0 _ A V 0 0 0 0 0 1 (5.3) or or Z = M X With the transformation, eqn. (5.2) becomes Z = F Z + G U — — AS 0 0 0 0 0 0 AS 0 • Ato 0 0 0 -40.7 0 0 Ato 0 Avt -5.31 -.08 -11.5 4.34 2.83 .06 • Av t + 0 AP 1.14 .63 1.92 -6.85 4.12 .09 AP 0 A i f 5.29 .41 13.2 14.4 -18.6 .44 A 1 f 0 A E f d _ 0 0 -1000 0 0 -20 _ A E f d _ 1000 — Z-Model in a Canonical Form or Y-Model u. (5.4) For the optimal excitation control design, i t i s desirable to 23 28 39 have the system equation written in a phase variable canonical form ' ' For the system studied the a coefficients are 89xl0 4 24xl0 4 60xl0 3 l l x l O 3 1144 u6 57 The characteristic equation of the optimal system and the adjoint becomes: 51 6 2 5 2 ' " ' ' A " 2 3 y - (cig -2a,.) y + (a^ ^a^a^Kia^) y - (a^ - 2a3a,-+2a2ag-2a^) y 2 2 2 2 + ( a 3 -2a 2a^+2a 1a 5) y - ( a 2 ^ a ^ a ^ y + + q/r = 0 (5.5) where _ ,2 y - A . Thus by varying q/r, the c h a r a c t e r i s t i c roots of the augmented closed loop system may be a l t e r e d . 5.3 Eigenvalue Movements with the ^Optimal E x c i t a t i o n Control U n r e s t r i c t e d Movement St a r t i n g the root l o c i p l o t with the open loop system, the eigenvalues move l i k e F i g . 5.1 as q/r increases. There should be s i x curves (conjugate mechanical modes A^ and A 2 > conjugate f i e l d and voltage regulator A„ and A,, Q-damper A,, and D-damper A,) for a 6th order system, j 4 i> o but only four of them are p l o t t e d because two of them are symmetric w.r.t. the r e a l axis. It i s found that the inc r e a s i n g q/r can push eigen-values of the mechanical mode to the l e f t to a c e r t a i n extent. Further push w i l l mainly increase the mechanical mode o s c i l l a t i o n frequency. The la r g e s t eigenvalue (Ag) i s i n s e n s i t i v e to optimal c o n t r o l . R e s t r i c t e d Movement R e s t r i c t i o n may be placed on the eigenvalue movement so that they are allowed to move only to the l e f t , with changes only i n the r e a l p a rt and not i n the imaginary part. Furthermore, the r e a l part changes can be r e s t r i c t e d to the mechanical mode eigenvalues alone. I t i s i n t e r e s t i n g to f i n d that the r e s t r i c t e d eigenvalue movement does not require a q/r value as large as the u n r e s t r i c t e d eigenvalue movement. The elements of S take the following form, o 52 h / 2 2 N = - G ) = (d 1 - a ) = (a 1 - a ) a 3 + a 2 06 \ ^ 2 2, = (a 1 - a ) A4 + A 3 S6 . 2 2, = (a 1 - G ) + A4 P6 h = 2 (CX-L + a) where X2 = o + ju i n i t i a l values, X^ , X2 = + jco a l = X 3 X 4 A5 A 6 A2 = -(A 3 A4 A5 + *4 X5 X6 + X3 X4 X6 + X 3 X4 V 3 3 = (A 3 A4 + A 3 X5 + X 3 X6 + X 4 X5 + X/f X6 + X5 X6) A 4 - - a 3 + x4 + x5 + x6) (5.6) *5 48x10 11 — • H --27.5 -27.0 * q/r=162xw' \ O \ 4Bx10 22x10"'\ 4x10^ • 13.5 -13.0 * -9.5 -9.0 -8.5 4—-* -4.0 lm.(\) 80 7.0 6-0 5J0 0 4.0 1.0 . 0.0 .3.0 -2.0 -10 0 Re/X) Fig. 5.1 Root l o c i of the system with optimal excitation control 53 The eigenvalues of the optimal system and the controllers are summarized in Table VII. For the same amount of system damping, the optimal control obtained with the restricted eigenvalue movement requires less gains than the control obtained with the unrestricted eigenvalue movement. Table VII Dominant eigenvalue movement due to optimal excitation control Eigenvalues of open loop system q/r = 0 0.203+J4.99, -8.465+J5.26, -13.2. -27.3 Unrestricted A move optimal excitation controller u^ q/r = 2.10 1 1 -0.994+J5.17 q/r = lOxlO 1 1 -1.900+J5.70 q/r = 30-xlO11 -2.634+J6.51 q/r = lOxlO 1 2 -3.308+J7.77 .033, -.304, -.060, -.003]Z .034, -.544, -.114, -.005]Z .232, -.740, -.172, -.007]Z .850, -.835, -.255, -.01-]Z Restricted A move optimal excitation controller u, q/r = 1.83x10 -.994+J4.99 [ .046, .036, .066, -.307, -.057, -.002]Z q/r = 7.59xl04 -1.900+J4.99 [ .114, .071, .108, -.549, -.101, -.004]Z q/r = 16.36xl04 -2.634jJ4.99 [ .190, .102, .134, -.752, -.138, -.006]Z q/r = 29.1xl0 4 -3.308+J4.99 [ .275, .134, .160, -.943, -.172, -.007]Z Z = [A6 Aco Av AP A i , AE_] t f fd J The performance index matrix is 54 9.2 .06 19.25 -13.91 .39 -.003 .06 .00 .13 -.10 .003 .00 19.25 .13 40.37 -29.17 .83 -.006 -13.91 -.10 -29.17 21.08 -.60 .004 .39 .003 .83 -.60 .017 .00 -.003 0 -.006 .004 0 0 5.4 Nonlinear Test Results of Optimal Excitation Control LINEAR CONTROL U ft > NONLINEAR PLANT Fig. 5.2 Nonlinear Power Plant with linear optimal controller The optimal controls thus designed are substituted into the nonlinear system equation X = f(X,U), X(0) = X 0 (5.7) for a dynamic response test. The optimal control becomes U = -S Q T _ 1 M (X-X Q). The nonlinear model was described in Section 2.4 of Chapter 2. A 5th order synchronous machine model but with voltage regulator and linear optimal control is considered. The voltage regulator ceiling was set at +8.3 and -7.0 as shown in Fig. 3.1. Although the closed loop system with linear optimal control is always stable, the resulting 55 0.3 -es o.2 -0.2 -0.3 -0.9 •0.6 SiJO.3 -r0.3 -L0.6 0.5 1.0 1.5 2.0 2.5 TIME (SECONDS) 3.0 3.5 -0.9 0.3 : -0.2 --0.3 0.5 1.0 1.5 2.0 2.5 3.0 TIME I SECONDS I 0.5 1.0 1.5 2.0 2.5 3.0 TIME ISECONDS) 3.5 „ 0. ne. -j0.03 -0.03 r0.06 -0.09 --0.12 6.0 9: 4.0 0.5 1.0 1.5 2.0 2.5 TIME ISECONDS) 3.0 3.5 -6.0 U.12 .0.09 /0.05: ,0.03 d 0 • L0.03 a-0.06: 0.5 1.0 1.5 2.0 2.5' 3.0 3.5 TIME (SECONDS) -0.09-I A , 1/ \> I f v • i * i * 0 0 5 l.o i s 2,o 2 S i 0 3 5 F i g . 5.3 Nonlinear dynamic responses with d i f f e r e n t controls 1. open loop system 2. system with the optimal c o n t r o l from u n r e s t r i c t e d eigenvalue move 3. system with the optimal control from r e s t r i c t e d eigenvalue move 56 dynamic response may or may not be satisfactory. A typical 3-phase fault is assumed on one of the double-circuit transmission lines for .08s, followed by a single circuit transmission before complete line restoration at 0.50s. The results obtained are plotted in Fig. 5.3. From the test results i t is found that 1. The optimal control can effectively stabilize the system subject to such, a fault disturbance in less than 2s; 2. The optimal control designed with the restricted eigenvalue provides better results than that with the unrestricted movement which produces larger overshoot in elec t r i c a l response than the former. 3. The optimal control can be designed to produce any amount of positive damping and synchronizing torque to the system. 5.5 System Models for Optimal Governor Control Design The block diagram of the speed regulating systems with hydro-turbine prime mover is shown in Fig. 5.4 -AOJ (ii) GATE SERVOMOTOR (iii) HYDROTURBINE permanent droop compen. (i) ACTUATOR Tp = -02 T R=4.8 T G - .50 6 t = .25 a = .05 p.u. Fig. 5.4 Speed governor and hydroturbine system. 57 The above model i s s i m i l a r to those used i n references 30, 34 and 43. The governor and the prime-mover dynamic equations are h - - | - h - ( a - g ) f - , AT m W G a -1.5 h + g Aco co u„ (5.8) T„co P o <S a <5 , . 6 A t / t , 1 v , t Aco - — a - ( — + — ) b - — P P R I Z-Model or Voltage-Current-Power Model The transformation given i n Section 2.11 of Chapter 2 i s augmented to include the governor state v a r i a b l e s . The same M-matrix elements are used. The r e s u l t i n g equations are i n the following form: AY 0 0 0 0 0 0 0 0 0 0 • Aco 0 0 0 -40.7 0 0 61.1 40.7 0 0 Av t -5.31 -.08 -11.5 4.34 2.83 .06 0 0 0 0 Ap 1.14 .63 1.92 -6.85 4.12 .09 0 0 0 0 A i f = 5.29 .41 13.2 14.4 -18.6 .44 0 0 0 0 A E f d 0 0 -1000 0 0 -20 0 0 0 0 h 0 0 0 0 0 0 -1.25 4 -4.0 0 g 0 0 0 0 0 0 0 -2 2 0 a 0 -1.33 0 0 0 0 .0 0 -2.5 50 b 0 -.033 0 0 0 0 0 0 -.63 -12 [A6 Aco Av t AP A i f A E f d h g a b ] t + [0 0 0 0 0 0 0 0 50 12.5]'. (-u2) (5.9) The state v a r i a b l e s 'g' and 'a' can be measured with high degree of pre-c i s i o n . The water head h can be measured with reaonsable accuracy (1/2% 58 or b e t t e r ) . Although valve or gate p o s i t i o n feedback i s used i n speed governor c o n t r o l , the modern trend i s to use machine v a r i a b l e s - power, voltage, current e t c . as s t a b i l i z i n g s i g n a l s . The eqn. (5.9) i s almost i l l - c o n d i t i o n e d . In other words, there are two v a r i a b l e s which are l i n e a r l y dependent. Consider equations of the actuator and the temporary droop compensation. • = _ 1 Aw a _ b ^2. a T c o ~ T a T T P o P P P /x \ - 1 A O J 0 A J . 1 s. u 1 « a In p r a c t i c a l systems, 2.5 $ T R $ 25 and .02 $ T p $ .05 Therefore, •^r1 » which in d i c a t e s that P R b = 6 a (t) (5.10) Computational i n s t a b i l i t y due to i l l - c o n d i t i o n i n g i n system modelling can be avoided by using the r e l a t i o n (5.10) which reduces the system order by one. The eigenvalues of the system with and without temporary droop compensation dynamics are shown i n table V I I I . The eigenvalue due to temporary droop compensation (-.0342) should not be confused with the most dominant eigenvalue of the system. Whenever there i s a l i n e a r dependency i n system equations, then t h i s w i l l contribute to a redundant eigenvalue at or near, the o r i g i n . Such an eigenvalue must be discarded before applying control techniques. Circuits nearly i l l conditioned system Well-conditioned system 10th order model 9th order model Mechanical (A^, A ) .2507 + J4.89 •2543 + J4.89 Field and regulator < V V -8.4783 + J5.26 -8.4788 + J5.26 Q-axis damper ^ -13.123 -13.120 D-axis damper A^ -27.301 -27.301 prime mover A^ , -1.1430 -1.1267 servo motor Ag -2.2958 -2.323 actuator ^ -15.080 -14.905 Temp. droop compen. -.0342 Table VIII Eigenvalues of the total system 60 Z-Model i n a Canonical Form or Y-M6del For the system studied the c o e f f i c i e n t s of the c h a r a c t e r i s t i c equation are a l a2 a 3 "4 • a5 a6 a7 a 8 °9 33.3xl0 6 54.9xl0 6 30xl0 6 85.2xl0 5 19.2xl0 5 32.2xl0 4 . 34.9xl0 3 2236 75.2 The c h a r a c t e r i s t i c equation of the optimal and i t s adjoint can be written as u 9 - ( a g 2 - 2a g) u 8 + ( a g 2 - 2 a ? a 9 + 2ag) y 7 . . . ( a 2 + q/r) = 0 (5.11) where ,2 y = X . 5.6 Eigenvalue Movement with the Optimal Governor Control U n r e s t r i c t e d Movement The foot l o c i p l o t of the system with optimal governor c o n t r o l i s shown i n F i g . 5.5. The eigenvalues X^ (D-damper), X,. (Q-damper), X^ and X^ ( f i e l d and voltage r e g u l a t o r ) , and Xg (actuator) are not a f -fected by the c o n t r o l . Modes due to the prime mover and the gate servomoter are very s e n s i t i v e to the optimal c o n t r o l whereas the most dominant mech-a n i c a l modes X^ and \^ are not as s e n s i t i v e as i n the ease of optimal e x c i t a t i o n c o n t r o l . I f the desired system damping i s to be produced i n the system with optimal governor c o n t r o l , i t can be achieved only with large feedback s i g n a l s . R e s t r i c t e d Movement A r e s t r i c t i o n may be placed i n the movement of the system eigenvalues that they be allowed to move to the l e f t of the complex plane without a f f e c t i n g the o s c i l l a t i o n frequency. As i n the optimal 61 x X • i • -27.0 < -15.0 * 5 x X 1/ =50x10 4 x 7 0 76 - x -0 ^ - 5 . 0 Im f\) 6.0 54 % * 0 4.0 3.0 2.0 1.0 -14.0 -130 * -9.0 -8. -4.0 -3.0 -2.0 -1.0 Fig. 5.5 Root l o c i of the system with optimal governor control ''excitation^control "design, cthe -restricted eigenvalue movement does not require q/r value as large as that of the unrestricted eigenvalue movement. The results obtained are given in Table IX. Table IX Dominant eigenvalue movement due to optimal governor control Eigenvalues of open loop system (q/r = 0) -.254+J4.89 -1.127, -2.323, -8.479 + j5.26 unrestricted X move q/r = lOxlO 1 5 -.544+J5.02 -2.47+J2.17 q/r = 500xl0 1 5 -1.424+J6.15 -4.922+J3.06 optimal governor control u 2 [-.03, -.017, -.014, .114, .02, .0009, -.008, .496, 106]Z [-.36, -/07, .143, .07, .0023, .004, 2.2 6.36, .195]Z 62 Table IX Cont'd res tricted X move optimal governor control q/r = .53xl0 1 5 -.504+J4.89 [ -.001, - .01, -.024, .098, .019, .0008, -1.28, -2.47 -.187, -.09, .036]Z q/r = 20xl0 1 5 -2.504+J4.89 [ -.17, -. 074, -.053, .444, .079, .0032, -2.48, 03.67 -.439, .566, .164]Z q/r = 47.8xl0 1 5 -3.254jj4.89 [ -.309, - .110, -.038, .579, .10, .004, -2.93, -4.12 -.356, 1.245, .212]Z Z t = [A6. Aco Av AP A i , AE,, h g a] 1 t f fd 6 J 5.7 Optimal Governor Control Supplementing Optimal Excitation Control The optimal governor control i s designed to supplement the optimal excitation control. The eigenvalues of the optimal system are given in Table X. Table X Further movement of Dominant eigenvalue due to supplemental governor control Original eigenvalues of the system with optimal excitation control u^ -3.386 + j4.15 -1.027, -3.451, -7.906 +j5.16 Restricted X move q/r = 10.4xl0 1 5 [-.026, -.003, -.059, .057, .003, 0, -3.807 + j4.77, -1.63, -3.85 .001, .338, .044]Z -8.08 + J5.10 q/r = 46.4xl0 1 5 [-.093, -.011, -.163, .163, .007, -4.20 + J5.37, -2.23, -4.24 .003, .141 .977, -.088]Z -8.25 + J5.10 Z = [A6 Aco Av AP A i f AE f d h g a] 63 0.3 S-o.i • O.OB -0.3 0.03 O.0.02 -0.5 1.0 1.5 2.0 2.5 3.0 TIME (SECONDS! 3.5 g-0.01 -0.02 • -0.03 T^TTP^ TmT 0 0.5 1.0 1.5 2.0 2.5 3.0 tlME (SECONDS) 3.5 0.06 ,0.05 0.02 •0.01 ,-0.04 --"-0.07 0. IP 0 5 10 K5 2.0 2.5 3.0 tlME ISECONDS) 3.5 0.015 • ME (SECONDS) _0.09 => 50.06 3E o o0.03 z 5 0 -0.06 • -0.09 ^ . . . iJ A-"' • 1/ ME ISECONOSI 0.5 1 .0 1.5 2.0 2.5 3.0 3.5 TIME (SECONDS) Fig. 5.6 Nonlinear dynamic responses with optimal c o n t r o l l e r s 1. open loop system 2. system with optimal governor c o n t r o l u 2 3. system with optimal e x c i t a t i o n c o n t r o l A. system with optimal governor c o n t r o l supplementing optimal e x c i t a t i o n c o n t r o l 64 The dynamic performance of the same power system with 3-phase fault disturbance i s investigated. The temporary droop compensation dynamic i s included. The results obtained are plotted in Fig. 5.6. I t is observed that 1. Both of these controls give improved nonlinear dynamic responses of the system, however the optimal excitation control i s better than the optimal governor control. 2. By controlling the exciter as well as the governor, excellent elec-t r i c a l as well as mechanical responses can be obtained. Again, the combined control response i s comparable with that of a single excitation control. 3. Neglecting the dynamics of temporary droop compensation does not introduce any error. 5.8 Wide Power Range Operating Conditions From the results obtained, i t seems that there i s considerable freedom for the choice of q/r. It i s decided to use this freedom more meaningfully by designing an optimal controller which w i l l cover the wide power range operating conditions. It i s immediately apparent that there i s no d i f f i c u l t y at a l l for a light load operation with a controller designed for f u l l load operation. But there w i l l be a.difficulty for heavy load with large inductive reactive power because of insufficient synchronizing torque and damping torque in the system. Therefore, the following procedure, may be established to search for an optimal controller which w i l l stabilize the system over wide power range operating conditions 1. Continue the restricted eigenvalue optimal control design from the point of f u l l load operating condition; 65 2. Test the control on heavy load c o n d i t i o n , e.g. P = 1.25 p.u. e t c . to see i f a l l the eigenvalues have negative r e a l p a r t s , which w i l l guarantee a f a i r l y stable system; 3. I f not, decrease or increase q/r while r e s t r i c t i n g the eigenvalue movement to the mechanical mode" ones, and only to the l e f t of the complex plane. Repeat the computation u n t i l a s a t i s f a c t o r y r e s u l t has been found. 4. Carry out a nonlinear test f o r the f i n a l design. The dominant eigenvalue l o c i of the system with various control designs are p l o t t e d i n F i g . 5.7 and some d e t a i l s are given i n Table XI. Re.(\) F i g . 5.7 Dominant eigenvalue l o c i of the system with various control designs vt = 1.05 1 2 3 4 5 6 7 P = 1.25 1.2 1.15 0.952 0.70 0.500 0.300 Q = 0.45 0.34 0.25 0.015 -0.15 -0.225 -0.256 Control P - ° - 5 0 , - l . O S Q = -0.225 P = 0.952 . v = 1.05 Q = 0.015 P = 1 ' 2 5 - v - 1 . 0 5 Q = 0.45 No co n t r o l -0.232 + J5.40, -12.61 -8.236 + J5.37, -27.43 0.203 +j4.99, -13.20 -8.465 + J5.26, -27.3 0.686 + J3.37, -13.60 -8.757 + J5.68, -27.3 Design I -1.747 + j5.54, -12.60 -10.11 + j 7.78, -27.7 -3.308 + J4.99, -13.2 -8.465 + J5.26, -27.3 0.4, -t-10.5, -12.96 -6.853 + J6.94, -27.20 Design II -1.335 + J5.49, -12.60 -9.277 + J6.88, -27.6 -2.010 + J4.99, -13.2 -8.465 + J5.26, -27.3 0.004, -6.11, -13.40 -7.347 + J5.6, -27.20 Design I I I -1.020 + 5.48, -12.60 -8.860 + j6.35, -27.50 -1.250 + J4.99, -13.2 -8.465 + J5.26, -27.3 -1.463 + j l . 2 4 , -13.5 -8.119 + J5.44, • -27.3 Table XI Eigenvalues of the system with d i f f e r e n t c o n t r o l l e r s 67 0.3 5: 0.2 5 0.1 B-0.1 • -0.2 -0.3 _0.06 0.5 1.0 1.5 2.0 2.5 3.0 3.5 TIME (SECONDS! tf°-°B|| -0.09 -0.12 _0.09 0.5 1.0 1.5 2.0 2.5 3.0 3.5 TIME ISECONDS) -0.06 -0.09 0.5 1.0 1.5 2.0 2.5 3.0 3.5 tlME (SECOND;) n.3 ' i I). 2 £ 0.1 -£ o B-0.1 -0.2 -0.3 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 TIME (SECONDS) „0.06 J0.03 ,r0.06-, L0.09 -0.12 0.5 1.0 1.5 2.0 2.5 TIME (SECONDSI 3.0 3.5 _ 0.09 | 0.«H o CO . t, 0.03^ -0.03--0.06--0.09 0.5 1.0 1.5 2.0 2.5 TIME (SECONDS) 3.0 3.5 a) C o n t r o l l e r designed based on f u l l load condition b) C o n t r o l l e r designed f o r wide power loadings F i g . 5.8 Nonlinear dynamic responses of the system with d i f f e r e n t loadings 1. P-= 1.23 Q,=0.35 2. P = 0.952 Q = 0.015 3. P = 0.30 Q = -0.256 68 Nonlinear t e s t r e s u l t s of the system under the same 3-phase f a u l t disturbance of F i g . 5.3 but with load conditions s l i g h t l y modified form Table XI i s shown i n F i g . 5.8. It i s found that while the optimal control of the wide power loading design can s t a b i l i z e the system i n l e s s than 2s, there i s no d i f f i c u l t y at a l l for e i t h e r control to s t a b i l i z e the system at l i g h t loads. At heavy load and with large inductive reactive power, the optimal c o n t r o l l e r based on f u l l load condition completely destroys the system synchronizing torque whereas the c o n t r o l l e r designed for wide power range loading does not a f f e c t i t so severely. 69 6. MULTI-MACHINE LINEAR OPTIMAL CONTROL DESIGN 6.1 Introduction I t has been shown i n Chapter 5 that a l i n e a r optimal feedback c o n t r o l l e r can be designed to s t a b i l i z e a c o n t r o l l a b l e system using only d i r e c t l y measurable state v a r i a b l e s . This technique w i l l be ex-tended i n t h i s chapter to interconnected multi-machine power systems. The transformation of system equations i n t o the phase v a r i a b l e canonical form f o r a s i n g l e input system i s unique but not uniform f o r a m u l t i -machine system. Therefore, the design procedure for a s i n g l e machine co n t r o l cannot be r e a d i l y applied and must be modified f o r a m u l t i -machine co n t r o l design. The objectives of t h i s chapter are to obtain a generalized mathematical model for the interconnected multi-machine power system and to design an optimal c o n t r o l l e r f or the system through a canonical form. 6.2 Summary of Single Machine Dynamics Let the nonlinear equations of a s i n g l e machine power system with two time constant voltage regulator - e x c i t e r be summarized as follows. ^ = f± a v \> ± v v x ) (6.1) Also V l + P l \ = S l X l ( 6 , 2 ) Retaining only the f i r s t term of Taylor s e r i e s , l i n e a r i z e d equations become: X, = Dn Xn + H. A i . + B n U- (6.3) 70 A v l + P l A i l = S l X l (6.4) where X± = [A§ 1 Au^ Aijj f l Ai|' D l A\p AV R 1 A E f d l ] > s t a t e vector A i ^ = [ A i ^ ^ Ai components of machine terminal current Av^ = [ A v ^ A vql-'> components of machine terminal voltage = [ 0 0 0 0 0 K A/T A 0] P l = S l = R -x x" R d a 0 0 0 0 -a. 0 0 4 0 0 a 2 a 3 0 0 0 and U l - U l which i s the s t a b i l i z i n g s i g n a l input to the regulator. 6•3 Aggregated Multi-Machine Equations i n I n d i v i d u a l Coordinates S i m i l a r l y , f o r an m-machine system, m sets of equations can be combined i n t o a s i n g l e set of equation as follows. X = f (X, U, i M , v M ) v M + P i M = S X (6.5) (6.6) The l i n e a r i z e d equations become 71 X m D l ° 0 D„ 0 0 B 1 0 0 B, 0 0 0 0 D m X, X, m Hi 0 0 H„ 0 0 0 0 m J U. U L_ m 0 0 H m 1 Ai„ 2 • A i m + (6.7) or compactly as ~ A v l ~ _ P 1 0 . . . 0 A i l S l 0 . . . 0 X l Av 2 0 P2 0 A ± 2 0 S2 * " . 0 X2 « + • • • • = . a • • • Av m 0 0 . . . p A i m 0 0 . . . s m X m X = D X + H Ai„ + B U M Av„ + P Ai., = S X M M (6.8) (6.9) (6.10) where D, H, B, P and S are block diagonal matrices. External Connection of the m-machine System The m-machines of a power system are interconnected through the transmission system at the machine terminals as follows. i 72 h = y " l l V l + y l 2 V2 + • • • + \ i 2 = y 2 1 v, + y 2 2 v 2 + . . . + y 2 m v m (6.11) I = y , V, + y . V 0 + m ml 1 m2 2 + y V mm m where the y's are the nodal admittance elements. The m-complex equations can be rewritten i n the r e a l form with 2m order as follows. Si 811 " b l l 812 " b12 * ' ' 8 l m - b l m " _ V D l " V b l l 811 b12 812 b l m 8 l m VQ1 iD2 g21 _ b 2 1 822 _ b 2 2 •82m _ b2m VD2 1Q2 b21 S21 b22 822 b2m b2m VQ2 "W gml " bml 8m2 bm2 6mm -b mm v Dm bml 8ml bm2 bm2 b mm 8mm VQm or i n compact form, i = Y v N N N (6.12) where v ^ v ^ d i r e c t - and quadrature-axis components of nodal voltages i"Di' i ^ d i r e c t - and quadrature-axis components of nodal currents y i 3 = 8 i j + j b i j V i " V D i + * V Q i Note that i n eqn. (6.11), only the machine terminal voltage nodes are 73 F i g . 6.1 Transformation of Reference Frame 74 retained and the other voltage nodes which are not connected to the m-machines are eliminated When an i n f i n i t e bus i s also considered i n conjunction with 49 the m-machine-node system, eqn. (6.12) can be augmented accordingly — = i — CO _ — — Y CO VN Y 2 _ V _ ° ° _ Therefore i . T = Y.T v„ + Y v N N N o o c o (6.13) 6.4 m-Machine Equations i n Common Coordinates 49 Sim i l a r to Yu-Moussa's work , the equations of synchronous machines expressed w.r.t. i n d i v i d u a l dq coordinates can be transformed i n t o a common D^ coordinates which i s r o t a t i n g at the synchronous speed of the complete system l i k e F i g . 6.1 where 6^ varies during the transient period. The transformation can be written as 'dl dm qm cos6^ sin6^ -sin6^ cos6^ COS62 sinS2 -sinS2 cos6 2 cos 6 sin6 m m -sin<5 cos 6 m m V . Dl Dm or v M = T(5) v N 75 where the matrix T(6) i s a block diagonal matrix. This i s a co-ordinate transformation i n the Euclidean space which i s a r o t a t i o n of rectangular coordinates about a f i x e d o r i g i n . Since i t i s an orthogonal t r a n s f o r -mation where the magnitudes of v . and i . are preserved, the machine and t i t i network component currents are r e l a t e d by i M = T(6) i N (6.15) Sub s t i t u t i n g (6.14) and (6.15) i n (6.13), the following i s obtained. Si " YM VM + T ( 6 ) V c ( 6 ' 1 6 ) where and where Y M = T(5) Y N T f c(6) v = Y v , constant C o o C o L i n e a r i z a t i o n of eqn. (6.16) r e s u l t s i n A 1M = YM [ A VM " A 6 VMI ] + M SlI ( 6 - 1 ? ) Y M = T(6.) Y Tfc (6 ) M o N o Sa " ^ q l ' ""Silj ±q2' _ ± d 2 | ,'SHU' " ^ ^ " - T ( 6o> <V Si VMI = [\V " V d l ! Vq2' " Vd2 1 . -!Vqm' ^dm 1' - 5 <V VM 76 AS = AS. A6\ AS, AS, ' A6 m AS m T(S) = p T(S) 6.5 T o t a l System Equations The nonlinear equations of the multi-machine system can be writ-ten as i M = [I„ + Y M P] 1 [Y S X + T(S) v ] M 2m M M c (6.18) (6.19) X = f (X, U, i M , v M ) (6.20) At each step of the numerical i n t e g r a t i o n of equation (6.20), the current i w and voltage v„ must be solved f i r s t . M M The l i n e a r i z e d equation becomes: X = A X + B U (6.21) where and A «= D + H[I„ + Y P] 1 S 2m m S X - Y M [S X - AS v M I ] + AS i M I The formulation of the multi-machine dynamic model presented here i s general and i t only requires the inverse of a (2m x 2m) matrix where m 20 47 49 i s the number of machines included i n the system ' ' 77 6.6 Multi-Machine Model In Terms of Measurable State Variables The state eqn. (6.21) can now be transformed into measurable state model of the system by the coordinate transformation. The trans-formation for a single machine becomes: - - — — AS. 1 0 0 0 0 0 0 AS Ato 0 1 0 0 0 0 0 Ato Av t 0 0 ° 2Vq/ vt o.v /v 3 q t - V d/ vt 0 0 AP 0 0 ° 2iq cr 0i 3 q " V d 0 0 % Mf = 0 0 -f f3 ( xd" xd ) / ( xd' xaA ) 0 0 0 • A VR 0 0 0 0 0 1 0 A VR A Ef d 0 0 0 0 0 0 1 _ A Ef d _ + 0 0 (R v,+x"v ) a d d q v v,-R 1,-x'li d a d d q a2 ( xd~ Xa£ ) 0 0 0 0 (x"v -R v ) q d a q v +x"i ,-R i q q d a q 0 0 0 Ai Ai or Z l = M l X l + M l A ± l (6.22) where " ^ a S ? . ( xd~ Xd } (x"-x„0) + (x"-x„0) °3 d ar d aV 78 For an m machine system eqn. (6.22) becomes: Z = M X + M A i M (6.23) Substituting for A i ^ from (6.18), the resultant transformation can be written as Z = M X (6.24) where M « M + M (I_ +' Y P)~ S 2m M The V-I-P model becomes: Z = F Z + G U (6.25) whe re F = M A M _ 1 and G = M B These two forms (6.21) and (6.25) can be transformed from one to the other by the transformation (6.24). 6.7 Modified Control The single machine control technique cannot be directly applied to multi-machine system because the system includes numerous controls. In order to extend the single machine control technique to a multi-machine system, the following modification i s necessary. Let the con-t r o l be defined as U = [ u r u 2, ... u j 1 1 = [e l f E 2 , ... e j 1 1 U T i.e. U = E U T (6.26) where at least one of the elements of E i s non-zero. Carrying out the indicated modification in eqn. (6.25) we obtain the 79 following: where Z = F Z + G E U T (6.27) U. = e. l i 1 i T and also, the choice of is arbitrary. Therefore, there is a good deal of choice for a multi-machine control. The one to be examined in this chapter i s a single optimal feedback control which stabilizes the whole system. 6.8 V-I-P Model in a Canonical Form The optimal control designed from the phase-variable canonical form is extended to the multi-machine system. The design procedure was developed in Chapter 4. For the multi-machine control design, eqn. (6.27), rather than eqn. (2.25) w i l l be used. The state eqn. (6.27) has now been transformed into the following form. Y = F Y + G U_ (6.28) o o T where and F = T _ 1 F T o G = T _ 1 G E o Z = T Y From section (4.3) the optimal controller can be written as U T - -S T _ 1 Z (6.29) 6.9 System Studied The system studied is shown in Fig. 6.2 They are one thermal and two hydroplants. The 4th machine represents the i n f i n i t e system. Plant R a x a * X d X q x " q T' , do IT7lt do qo H ' Base quantities #1 Ther-mal .0019 .0673 1.53 .29 .17 1.51 .17 4.0 .029 .029. 2.31 360 MVA, 13.8 KV #2 Hydro .0023 .1709 .876 .324 .215 .53 .29 8.0 .022 .044 3.40 503 MVA, 13.8 KV #3 Hydro .0025 .0496 .973 .19 .133 .55 .216 7.76 .0436 .0938 4.63 1673 MVA, 13.8 KV Table XII Machine Data Unit KA TA h T E V RMAX. V RMIN. Type #1 13 .21 1 .15 4.5 0 amplidyne exciter #2 45 . .07 1 .50 3.5 -3.5 magnetic amplifier #3 50 .05 1 .003 8.8 -7.0 static exciter Table XIII Exciter Data 81 The data f o r the machines and voltage r e g u l a t o r - e x c i t a t i o n systems are 16 49 summarized i n Tables XII and XIII r e s p e c t i v e l y ' F i g . 6.2 System studied (Admittances i n p.u. on 1000 MVA base) l a b l e XIV Terminal conditions. Plant P M.W. o Q M.V.A. o v p.u. to 6 deg. #1 26.5 37 1.04 -10.69 #2 518 -31.5 1.025 11.83 #3 1582 -49.6 1.03 25.13 #4 410 49.3 1.06 0 82 The admittance matrices are: .4257 2.038 .0923 -.5313 .1293 -.7169 2.038 .4257 .5313 .0923 .7169 .1293 .0923 -.5313 .1121 1.226 .0628 -.4745 .5313 .0923 -1.226 .1121 .4745 .0628 .1293 -.7169 .0628 -,4745 .4218 1.475 .7169 .1293 .4745 .0628 -1.475 .4218 .1782 -.7998 .7998 .1782 .0666 -.3250 .3250 .0666 .0926 -.6508 .6508 .0926 6.10 Multi-Machine Optimal Control Design Case I Full-State Variable Design The synchronous machine dynamics are represented by the f i f t h order model and voltage regulator-excitation system by the second order model. The largest eigenvalue due to static exciter of machine 3 is eliminated by a matrix reduction. The resultant 20th order state model is used for the optimal controller design. The design procedure was developed in Chapter 4 and the results obtained are given in Table XV. Case II Control Design with State Variables of Only One Machine in Detail Although a l l states can be measured accurately, i t i s unecono-mical to telemeter a l l of them. A special control i s considered here which has only limited state feedback from the other machines. The machine 3 i s represented by the 6th order model and the rest are represented Optimal controller Dominant Eigenvalues machine 1 machine 2 machine 3 1. No control -0.549 + J7.89 -0.372 + J6.72 0.195 + J4.12 -2.004 + jl.99 -0.838 + J2.77 -8.310 + J5.29 2. q/r = 14.1x10 u = [-.01 .02 .17 -.04 -.03 .001 .002]Z1 -1.046 + J7.88 -0.957 + J7.73 -0.605 + J4.12 + [.14 -.05 -.08 .29 .06 0.001 -.002]Z2 + [-.03 .07 -.16 -.23 -.11 -.004]Z3 -2.054 + jl.99 -1.088 + J2.77 -8.302 + J5.29 3. q/r = 50x10 Uj = [.46 .12 .91 -.29 -.08 -.0004 -.004]Z1 -1.522 + J5.84 -1.606 + J7.77 -1.398 + J4.12 + [-.17 -.24 -.98 .04 -.18 -.005 -.03]Z2 + [.26 .21 -.14 -.65 -.22 -.009]Z.3 -2.103 + jl.99 -1.341 + J2.77 -8.252 + J5.30 40 4. q/r = 32x10 = [5.2 .27 .21 -.4 -.31 -.023 -.16]Z"1 -2.398 + J7.906 -2.977 + J7.68 -2.987 + J4.12 + [-6.5 -.67 -9.8 .6 2.1 -.01 -.1]Z 2 + [1.8 .7 -3.2 -.34 .72 -.02]Z3 -2.202 + jl.98 -1.844 + j2.77 -7.764 + J5.40 Table XV Dominant Eigenvalue Movement Optimal Control Dominant Eigenvalues machine 1 machine 2 machine 3 1. No control -0.043 +j7.89 -1.818 + jl.58 -0.014 + j 7.65 -1.062 + J2.53 0.226 + J4.16 -8.294 + J5.32 21 2. q/r = 4x10 u = [.003 .054 1.621 .004^ + [.019 -.10 -.616 -.01]Z2 + [.324'.103 .69 -.57 -.13 -.005]Z -0.543 + j 7.89 -1.868 + jl.58 -0.613 + J7.65 -1.312 + J2.30 -1.026 + J4.16 -8.295 + J5.32 3. q/r - 13.4xl0 2 1 u n = [-.357 .087 10.48 .004]Z]L + [.229 -.231 -5.46 -.036]Z2 + [1.27 .248 2.71 -1.51 -.25 -.01]Z -1.047 + J7.89 -1.917 + jl.58 -1.208 + j 7.65 -1.562 + J2.53 -1.826 + J4.16 -8.297 + J5.31 Z 1 t = [A61 Av t l AE f d l] Z 2 t - [M2 Aco2 Av t 2 AE f d 2] Z3fc - [A63 Aco3 Av t 3 AP3 A l f 3 AE f d 3] Table XVI Dominant eigenvalue movement 85 by the 4th order model. The optimal controller obtained is tested with the higher order model of the system. The controller stabilizes the higher order model but i t introduces oscillations in the system modes which are neglected (Table XVI). 6.11 Single Machine Design of Multi-Machine System An optimal controller i s designed for one machine by represen-ting other machine as an i n f i n i t e bus, i f larger or as a constant im-pedance i f smaller. The advantage of the procedure i s the saving of extensive system telemetering. Machine 1 Since the capacity of machine 1 i s the smallest, for the design of a controller for machine 1, machines 2^and 3 are treated as i n f i n i t e busses. The eigenvalues of machine 1 obtained from a single machine design and a multi-machine design are compared. The model acur-rately describes the dynamics of machine 1 and any error due to the above assumption in modelling machine 1 i s negligible. The eigenvalues of machine 1 from the single machine system are: -0.598 + j7.79, -2.042 + j l . 94, -8.256, -46.95, -167.7 The eigenvalues of machine 1 form the multi-machine model are -0.549 + j7.89, -2.004 + jl.98, -8.226, -46.54, -164.1 The controller u^ i s designed and tested with the multi-machine model of the system. The controller improves the damping of machine 1 alone. The controller i s u = .79 AS. -.64 Au. + .87 Av + .81 AP. + .04 Ai,. - .05 AV_, -X X X ZX X IX KX .034 AE £ J 1 rdX The eigenvalues of the multi-machine system with u control are 86 machine 1 machine 2 machine 3 -1.675 + J7.57 -0.367 +j7.91 0.186 +j3.26 -1.786 +j3.95 -1.167 +j2.43 -8.219 +j5.36 Machine 2 Since the capacity of machine 2 i s between that of machine 1 and 3, for the optimal control design, machine 1 i s replaced by a con-stant impedance and machine 3 i s treated as an i n f i n i t e bus. Again, the model i s reasonably accurrate i n describing the dynamics of machine 2. The eigenvalues of machine 2 from the s i n g l e machine model are -.324+J7.58, -.732 + J2.66, -14.89, -29.82, -59.94 The eigenvalues from the- multi-machine model are -.371 + J7.72, -.838+J2.77, -14.88, -29.78, -53.15 The optimum c o n t r o l l e r i s u 2 = -.8A6 - .2 76Aco2 - 2.64Av f c 2 - .85AP2 - . 82Ai f -.016AV R l - .137AE f d 2 The eigenvalues of the system with u 2 c o n t r o l are machine 1 machine 2 machine 3 -.578+J7.75 -1.552 +J7.82 -.082 + J3.95 -2.18+J2.17 -4.552 +j2.51 -8.104 +j5.19 -8.12 -14.53 The c o n t r o l l e r improves the damping of machines 2 and 3 but the improve-ment i s not s u f f i c i e n t f o r machine 3. Machine 3 Since the capacity of machine 3 i s the l a r g e s t , i n the optimal co n t r o l design f or machine 3, machines 1 and 2 are replaced by constant impedances. The control i s 87 u„ = .21AS„ + .lllAw- + .401Av _ - .54AP„ - .166Ai._ - .007AE,,o 3 3 3 t3 3 r J f d j The eigenvalues of the multi-machine system with u^ con t r o l are machine 1 machine 2 machine 3 -.615+J7.90 -.805+J7.42 -2.55+J4.96 -1.995 + 99 -.715+J2.74 -8.975 + j3.55 -7.53 -14.85 The c o n t r o l l e r improves the damping of a l l machines. Co n t r o l l e r u plus u Combination of controls thus designed i s also applied to the system. The eigenvalues of the system with u^ and u 2 controls are machine 2 machine 2 machine 3 -1.378 +j8.11 -1.84+J7.08 -.023+J2.95 -2.606 + j4.43 -4.361 +j2.14 -7.989 +j5.22 -8.886 -14.51 The c o n t r o l l e r s improve the damping f o r a l l machines but the improvent i s not s u f f i c i e n t f o r machine 3. Also, they reduce the synchronizing torque of machine 3. Co n t r o l l e r u^ plus u^ The c o n t r o l l e r s do not improve the damping of machine 2. machine 1 machine 2 machine 3 -1.899 + j8.10 -.539+J7.83 -4.134 +j3.92 -1.238 +j2.39 -.370+J2.97 -8.110 + j3.94 Co n t r o l l e r plus u^ The c o n t r o l l e r improves the damping for machines 2 and 3. Since there i s no severe dynamic i n t e r a c t i o n between c o n t r o l l e r s u 2 88 and u^ and machine 1, this leads to a simple control scheme for multi-machine power system. machine 1 machine 2 machine 3 -.643+J7.77 -2.045 + j8.09 -2.109 +j3.57 -2.234 + j2.25 -6.22 + J3.63 -8.125 + jl.703 Controllers u^, u^ and u^ When these controllers are applied together simultaneously, dynamic interaction between them are observed which can be seen from the eigenvalue analysis of the system. machine 1 machine 2 machine 3 -1.353 +j8.11 -2.287 + j8.18 -.600+J2.77 -3.731 +j2.40 -6.626 + j5.03 -6.678 + jl.10 Control u^ acting with other controls reduces the damping as well as the synchronizing torque of the system. 6.12 Nonlinear Test Optimal controllers thus designed are given nonlinear tests. The disturbance here i s the same as the one assumed in Chapter 5. The results obtained are plotted in Fig. 6.3. The following conclusions can be drawn, 1. The optimal control based on single machine design for machine 3 provides enough damping for machine 3 alone whereas the multi-machine design provides damping for a l l machines. This can be seen from-the" elec t r i c a l responses of machines 1 and 2. 2. The controls on machines 2 and 3 which are based on single machine design stabilizes the system very effectively. There i s no dynamic interaction between controls u„ and u . 89 Introducing a l i m i t e r i n the s t a b i l i z i n g s i g n a l of machine 2 redu a large overshoot i n the terminal voltage. 9 0 0.05 q _ : 0 CL! UJ0.05 o <x I— _J o =--0.10 fcfe-0.15 .r0.20 L0.25 -0.30 0.5 1.0 1.5 2.0 2.5 TIME (SECONDS! 3.0 3.5 1.5 2.0 2.5 TIME (SECONDS) 0.5 1.0 1.5 2.0 2.5 TIME (SEC0N0S) 3.0 3.5 a) Machine 1 b) Machine 2 92 0.9 0.7 -£ 0.5 UJ i 0.3 | I 0.) -0.3 - 0 . 5 u 8 1.2 0.11 - i _0.08 -UJO 05 -o > 0 02 .0) ,.r0.04-"-0.07-0.5 1.0 1.5 2.0 2.5 3.0 3.5 tlKE (SECONDS) -0.10 0.5 1.0 1.5 2.0 2.5 3.0 3.5 TIME (SECONDS) 0.5 1.0 1.5 2.0 2.5 3.0 3.5 TIKE (SECONDS) 1. Open Loop System 2. System with single machine control on machine 3 3. System with single machine control on machines 2 and 3 4. System with multi-machine design control on machine 3 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 c) Machine 3 "HE (SECONDS! .,0.08 0-5 1.0 1.5 2.0 2.5 3 0 3 5 TIME (SECONDS) Fig. 6.3 Nonlinear Dynamic Responses 93 7. CONCLUSIONS From the dynamic modelling of a synchronous machine by examining i t s eigenvalues, i t reveals that the f i f t h order synchronous machine model, with torque angle, speed, e l e c t r i c power output, terminal voltage or current, and f i e l d voltage or current as the state v a r i a b l e s , i s the best dynamic model f o r the s t a b i l i t y study. A l l the state v a r i -ables are d i r e c t l y measurable and no dynamic estimator i . e . e i t h e r Kalman's f i l t e r or Luenberger's observer i s required. The model i s reasonably accurate and there i s no r e a l advantage i n using a higher order synchronous machine model f o r the s t a b i l i t y c o n t r o l l e r design. With the omission of the eigenvalues remote from the imaginary axis or the short l i v e d armature t r a n s i e n t s , the computation time f o r nonlinear dynamic responses i s considerably reduced. Using the model, new supplementary e x c i t a t i o n s t a b i l i z i n g s i g n a l s are designed. Eigenvalues of not only the mechanical mode but also the e l e c t r i c a l modes are searched. I t i s found that a combined compensated Ato and uncompensated -AP provide the best damping f o r the system under study. For the l i n e a r optimal c o n t r o l design, i t i s convenient to have the system equations transformed to a canonical form. A simple method to determine the weighing matrix Q and R i s developed. With the a i d of eigenvalue searches, optimal e x c i t a t i o n controls can-be designed which can s t a b i l i z e a power system under f a u l t y conditions, not only at f u l l and l i g h t load but also at heavy load. I t i s found that eigenvalue move r e s t r i c t e d only to the l e f t gives even b e t t e r r e s u l t s with l e s s over shoot and smaller gains than an u n r e s t r i c t e d 94 eigenvalue move. However, the optimal e x c i t a t i o n c o n t r o l i s more e f -f e c t i v e than the optimal governor c o n t r o l . The canonical state equation and eigenvalue search techniques f o r power system s t a b i l i z a t i o n are further applied to the interconnected multi-machine system. Several c o n t r o l schemes are developed. Single machine con t r o l and multi-machine co n t r o l schemes are compared. I t i s noted that the case with two s i n g l e machine controls on machines 2 and 3 of separate design without considering multi-machine dynamics i s more e f f e c t i v e than the s i n g l e machine con t r o l on machine 3 considering the multi-machine dynamics. A l l c o n t r o l s t r a t e g i e s thus designed are tested on the non-l i n e a r power system model of d i g i t a l simulation. The power group at the U n i v e r s i t y of B r i t i s h Columbia has also developed a dynamic test "•model i n the - research -labo-ratory ,and.'has-,been .s.uc.ces.sfully .testing the s t a b i l i z a t i o n schemes of various designs. The l i n e a r optimal s t a b i l i z a t i o n scheme implemented at the moment uses the speed deviation, the e l e c t r i c power output, the terminal voltage and the f i e l d voltage as the state v a r i a b l e s feedback. There i s no d i f f i c u l t y at a l l of the instrumentation. I t seems that time has ripened that power system engineers should design and t e s t those optimal c o n t r o l s t r a t e g i e s on large power system. The contributions of this thesis may be summarized as follows. 1. the development of a reasonably accurate f i f t h order synchronous machine model i n terms of e a s i l y measurable state v a r i a b l e s ; 2. a systematic approach of s e l e c t i n g the best supplementary s t a b i l i -zing signals considering both mechanical and e l e c t r i c a l o s c i l l a t i o n s ; 3. the development of a canonical state v a r i a b l e form f o r the l i n e a r 95 optimal co n t r o l design and the optimal e x c i t a t i o n c o n t r o l design with eigenvalue searches which can s t a b i l i z e a power system under wide power range operating conditions; the development of s t a b i l i z a t i o n schemes of s i n g l e or multi-machine controls which can s t a b i l i z e the interconnected multi-machine large power systems. 96 APPENDIX A ' DERIVATION OF STATE EQUATION, X = AX + BU A.l 5th Order Flux Linkage Model The linearized equations of a synchronous machine dynamics can be written as: where X = DX + HAi + BU (A.l) X = [AS Ato Ai|if A I / J d AiJ;^]t (A.2) Ai = [Ai, Ai ] d q U = [AE AT j 1 Id m (A. 3) (A. 4) and D = 0 1 0 0 0 0 0 d23 d24 d25 o o d 3 3 d 3 4 0 0 0 d43 d44 ° 0 0 0 0 d 55 J (A.5) 23 33 -°la2 V d24 = " a l a 3 V d25 " a l C T 4 *d 1 Tdo a (x -x') [1+ 3 d d ] f ( x d " xa£ ) a3 ( x d " 3 4 Tdo <xd " Xa*> d43 " < V * d ) / T d V < V x a * > » d44 = - 1 / Tdo' d 55 = qo al = "o / 2 H> °2 = <V xd> <x2-xa£)/<xd-xa£),^d-xai) °3 * < x d " X d ) / ( x d " XaP> V ( x q " x q ' ) / ^ q " « a £) 97 H = 0 h21 h31 h41 ° o h 2 2 o o h 5 2 (A.6) h n - -0-3 [(xj-xj) i q - y 4 ] , h 4 1 - - ( x d - x a i ) / T d o h22 = "°1 I^q'-^ *d + *f°2 + V 3 ]> h52 " -<VXa*)/Tqo h31 = - ( x d " *a*> ^ d-V^do'^d-V 0 0 0 a, (xd"XaP T' . (x.-x') do d d 0 0 0 0 0 (A. 7) The terminal currents and voltages of the synchronous machine can be expressed in terms of state variables in the form of Av + P -Ai = S X (A. 8) A.2 Representation of External System The transmission system external to the generator bus consists of an equivalent series impedance and a shunt admittance as in Fig. 2.2. At the generator bus Let t d J q v = v, + j v t d J q v = v sinS + i v cos<5 o o J o In terms of d,q quantities the above equation can be written as 98 (1 + RG - XB) -(XG + RB) (XG + RB) (1 + RG - XB) v. v L q v s i n 6 o v cos6 . o R -X X R l q (A.9) Solving f o r the machine component voltages we obtain the following: v, = v (K.sin6 + K„cos6) + (K..R + K„X) i , + (-K..X + K.R) i (A.10) d o X • J. X 2 d X 2 q v = v (-K0sin<5 + K.cosS) +(-K„R + K-X) i , + (K 0X + K,R) i q o 2 1 ' K 2 1 • d 2 1 ' q where (A.11) K x = (1 + RG - XB)/D , K 2 = (XG + RB)/D^ D][ = (1 + RG - XB) 2. + (XG + RB) 2 A f t e r l i n e a r i z a t i o n (A.10) and (A.11) become Av, = (K..R + K.X) A i , + (K 0R - K,X) Ai + v (K-CosS - K„sin6)A6 (A.12) d X z d 2 X q o X 2 Av = (K,X— K_R) A i , + (K.R + K„X) A i - v (K„cos6 + K nsin6)A6 (A.13) q 1 2 ' d l 2 q o v 2 1 From equations (A.8) and (A.13) Ai Ai K 5K 3 + K 6K 4 0 o 2K 6 a 3 K 6 - a ^ K..K- + K_K. 0 a„K, a_Kc -a.K_ 7 3 5 4 2 5 3 5 4 7 (A.14) i . e . A i = C X where (A.15) K„ = v (K„sin6 - K.cos6), j o 2 I K, = v (K 0cos6 + K,sin6) o / 1 JL = (K R + K X + R )/D0, 5 1 I a 2 K 6 = ( K l X - K 2 R + x ^ ) / D 2 K ? = (K 2R - K^X - x J ) / D 2 , D„ = (K^R + K 2X + Ra) - (K 2R K^x (K^X - K 2R + x p 99 From eqns. (A.l) and (A.15) the state equation can be written in stan-dard form: X = A X + B U (A.16) where A = D + H C (A.17) A.3 7th Order Flux Linkage Model The seventh order model of one machine and i n f i n i t e power system can be written in standard form as — — A6 0 0 0 0 0 0 0 A6 0 0 Ato 0 0 a23 a24 a25 a26 a27 Ato 0 °1 a i f 0 0 a33 a34 0 0 a37 A1JJf b31 0 = 0 0 a43 a44 0 0 a 47 A*D + 0 0 AV 0 0 0 0 a55 a56 0 0 0 A*q a61 -*d a63 a64 a65 a66 a67 At|) q 0 0 a71 *q a73 a74 a75 a76 a77 0 0 AE fd AT m or (A.18) X <= A X + B U (A.19) where _ V q 23 x" (x,-x 0)(x'-x .) d d al d al *24 xd ( xd " *ai? o.* (x -x") a = _ i q q q a25 x" (x -x .) ' q q al a27 = ^ l ( 1 q + l ^ ) * a26 = °1 ( 1d + ^ l33 (xd - *;> TdoXd ( xd"xai) A (xd-XaP + x. <xd"xa£> d(xd-xS> 100 a77 a34 = " a l ^ a J / Tdo Xd ' a37= <V xa£>< xa^a£ ) / T ao Xd <xd-xa£> a43 = Xa£ ( xd- Xd ) / Tdo xd <Xd"Xa£>> a44 = " xd / Tdo xd a47 = < Xd- Xa£ ) / Tdo Xd » a55 = "Y^o Xq a56 = ( x q " Xa£ ) / Tqo Xq' a 6 l = " " o W 0 8 6 + K l S ± n 6 ) MoL2 (xd-*d> "oL2 ( x d - X d } 6 3 " Xd <xd"xa£> <xd~xa£> ' = " x d <xd"xa£> a65 = u o L l <V X? / XS C W > a66 = - % L l / x q L2 a6 7 = _ w0 C1 - ^ ir)» a ? 1 = wo V q (K COS6 - K 2sin6) d a M o L l ( x d ~ Xa£ ) ( xd - Xd> »oLl <*d - X'cP 7 3 " Xd <xd - xa£><xd " xa£> ' ^ = x d <xd " Xa£> a75 = %L2 ( V x q ) / x q <V*a£>' a76 = "o ( 1 " L2 / xq> " W o L l / x d ' L l = K1 R + ^2X + Ra' L2 = ~K1X + K2 R b31 = ( xd- Xa£ ) / Tdo ' 101 APPENDIX B INITIAL VALUE OF THE SYSTEM VARIABLES Given v , P and Q at the generator terminal the i n i t i a l value of system variables can be found as follows, -1 Q -}> = tan power factor angle t v cos<j> , terminal current Vc/ q - AXIS d-AXIS F i g . B . l Phasor diagram of a synchronous machine When a synchronous machine i s i n a steady-state condition, the equations which describe the machine performance can be represented by the phasor diagram, F i g . B . l . From the phasor diagram, the following r e l a t i o n s h i p can be w r i t t e n . v, = x i - R i , , d q q a d v = E,, - x , i , - R i q f d d d . a q Also v, = v sing , d t i d = l t s i n (B+$) , V = V COS0 q t i = i cos .(B+$) q t 102 Thus we have x costi - R sind> Q a tang = v t -— + R cos* + x sin<f> i t a Y q The following i n i t i a l values can be ca l c u l a t e d , v, = v t sing • , v = v cosB d t q t i d = i t s i n (g+<j>) , i = ± t cos (g+<j>) E T = E,, = v + x . i , + R i I fd q d d a q i K = v + R i = E T - x , i , r d q a q 1 d d *q = "Vq = " ( V d + R a i d ) i f = Ej/C^-x^)' 2 ( x d ~ X a £ ) % " ' " ^ d " ^ *d + (x.-x') ±f d d *D = ( x d - X a ^ ( i f " i d > = -(x -x .) i rQ q al q B.1 C a l c u l a t i o n of Torque Angle and I n f i n i t e Bus Voltage From eqn. (A.9) v sinS = (1 + R G - X BW,, - (X G + R B)v - R i , + X i o d q d q v cos6 = (X G + R B)v J + (1 + R G - X B)v - X i , - R i o d ' q d q Thus (1 + R G - X B)v, - (X G + R B)v - R i , + X I t a n 6 = _ i _g - _ i i (X G + R B)v, + (1 + R G - X B)v - X i , - R i d q d q v = [(1 + R G - X B)v, - (X G + B R)v - R i . + X l ]/sin6 o d q d q i f there i s a l o c a l load and 103 tan6 = — d — 9 - , v = _ J * < V'q ~ d ~ q ° s i n 6 i f there i s no l o c a l load at the machine te r m i n a l . 104 REFERENCES 1. C. Concordia, "Synchronous Machines", (book), New York, John Wiley & Sons, Inc., New York, 1951. 2. B. Adkins, "The General Theory of E l e c t r i c a l Machines", (book), John Wiley and Sons, Inc., 1959. 3. W.A. Lewis, "The P r i n c i p l e s of Synchronous Machines", (book), IIT, Chicago, I l l i n o i s , 1954. 4. R.H. Park, "Two-Reaction Theory of Synchronous Machines-Generalized Method of Analysis - Part I", AIEE Transactions, V o l . 48, p. 716, 1929. 5. R.H. Park, "Two-Reaction Theory of Synchronous Machines - Generalized Method of Analysis - Part I I " , AIEE Transactions, V o l . 52, p. 352, 1933. 6. C.C. Young, "The Synchronous Machine", IEEE T u t o r i a l Course on Modern Concepts of Power System Dynamics, 1970. 7. D.W. Olive, " D i g i t a l Simulation of Synchronous Machine Transients", IEEE Transactions, V ol. PAS-87, No. 8, August 1968, pp. 1669-1675. 8. I.M. Canay, "Causes of Discrepancies on C a l c u l a t i o n of Rotor Quan-t i t i e s and exact Equivalent Diagrams of the Synchronous Machine", IEEE Transactions, Vol. PAS-88, No. 7, July 1969, pp. 1114-1120. 9. Y.N. Yu and H.A.M. Moussa, "Experimental Determination of Exact Equivalent C i r c u i t Parameters of Synchronous Machines, IEEE Trans-actions, V o l . PAS-90, No. 6, November/December 1971, pp. 2555-2560. 10. W. Watson, " S t a t i c Exciters and S t a b i l i z i n g Signals", presented to Power System Planning and Operation Section, CEA, Toronto, A p r i l 1968. 11. F.R. S c h l e i f , H.D. Hunkins, G.E. Martin and E.E. Hattan, " E x c i t a t i o n Control to Improve Powerline S t a b i l i t y " , IEEE Transactions, V o l . PAS-87, pp. 1426-1434, June 1968. 12. F.R. S c h l e i f , H.D. Hunkins, E.E. Hattan, and W.B. Gish, "Control of Rotating Exciters f o r Power System Damping: P i l o t A p plications and Experience", IEEE Transactions, V o l . PAS-88, No. 8, August 1969. 13. F.P. deMello and C. Concordia, "Concepts of Synchronous Machine as Affec t e d by E x c i t a t i o n Control", IEEE Trans. (Power Apparatus.and Systems), Vol. 88, pp. 316-329, A p r i l 1969. 14. H.M. E l l i s , J.E. Hardy, A.L. Blythe and J.W. Skooglund, "Dynamic S t a b i l i t y of Peace River Transmission System", IEEE Transactions PAS, V o l . 85, pp. 586-600, June 1966. 105 15. P. Kundur, "The Design of Synchronous.Generator E x c i t a t i o n Control Using the State Space Approach", National Conference on Automatic Control, Cosponsored by NRC Associate Committee on Automatic Control and the University of Waterloo, August 1970. 16. Y.N. Yu and C. Siggers, " S t a b i l i z a t i o n and Optimal Control Signals for a Power System", IEEE Transactions, PAS V o l . 90, pp. 1469-1481, July/August 1971. 17. M.K. El-Sherbiny and A.A. Fouad, " D i g i t a l Study of Dynamic I n s t a b i l i t y for Inter-Connected Power Systems", paper 70-670, presented at the IEEE Summer Power Meeting & EHV Conference, Los Angeles, C a l i f o r n i a , July 12-17, 1970. 18. E.J. Davison, "An Automatic Way of Finding Optimal Control Systems for Large M u l t i v a r i a b l e Plants", IFAC Tokyo Symposium, pp. 367-372, August 1965. 19. IEEE Committee Report, "Computer Representation of E x c i t a t i o n Systems", IEEE Transactions, PAS Vol. 87, pp. 1460-1464, June 1968. 20. J.M. U n d r i l l , "Dynamic S t a b i l i t y C a l c u l a t i o n s f or an A r b i t r a r y Number of Inter-Connected Synchronous Machines", IEEE Transactions, PAS V o l . 87, pp. 835-844, March 1968. 21. B. Habibullah, "Suppression of the DC Short C i r c u i t Component Using the DQO Representation.of the System", M.A.Sc. Thesis, U n i v e r s i t y of Toronto, Toronto, January 1971. 22. O.W. Hanson, C.J. Goodwin and P.L. Dandeno, "Influence of E x c i t a t i o n and Speed Control Parameters i n S t a b i l i z i n g Intersystem O s c i l l a t i o n s " , IEEE Transactions, Vol. PAS-87, pp. 1306-1313, May 1968. 23. F.M. Brasch, J r . , and J.B. Pearson, "Pole Placement Using Dynamic Compensators", IEEE Transactions on Automatic Control, Vol. AC-15, No. 1, February 1970, pp. 34-43. 24. D.G. Luenberger, "Canonical Forms f o r Linear M u l t i v a r i a b l e Systems", IEEE Transactions on Automatic Control, short paper, pp. 290-293, June 1967. 25. J.D. Simon and S.K. M i t t e r , "A Theory of Modal Control", Information and Control 13, pp. 316-353, 1968. 26. R.L. Kosut, "Suboptimal Control of Linear Time-Invariant Systems Subject to Control Structure Constraints", IEEE Transactions on Automatic Control, V ol. AC-15, No. 5, October .1970, pp.557-563. 27. W.S. Levine and M. Athans, "On the Determination of the Optimal Con-stant Output Feedback Gains for Linear M u l t i v a r i a b l e Systems", IEEE Transactions on Automatic Control, V o l . AC-15, No. 15, pp. 44-48, February 1970. 106 28. E.J. Davison, "On Pole Assignment i n Linear Systems with Incomplete State Feedback", IEEE Transactions on Automatic Control V o l . AC-15, No. 3, pp. 348-351, June 1970. 29. A.P. Sage, "Optimum Systems Control", (book), P r e n t i c e - H a l l , Inc. Englewood C l i f f s , N.J., 1968. 30. Y.N. Yu, K. Vongsuriya and L.N. Wedman, "Application of an Optimal Control Theory to a Power System", IEEE Transactions PAS V o l . 89, pp. 55-62, January 1970. 31. C.E. Fosha J r . , and 0.1. Elgerd, "The Megawatt-frequency Control Problem: A New Approach v i a Optimal Control Theory", IEEE Transactions PAS, Vol. 89, pp. 563-577, A p r i l 1970. 32. J.H. Anderson, "The Control of a Synchronous Machine Using Optimal Control Theory", Proceedings of the IEEE, V o l . 59, pp. 25-35, January 1971. 33. E.J. Davison and N.S. Rau, "The Optimal Output Feedback Control of a Synchronous Machine", IEEE Transactions PAS, V o l . 90, pp. 2123-2134, September/October 1971. 34. H.A.M. Moussa and Y.N. 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 Transactions PAS, V o l . 91, pp. 1166-1174, May/June 1972. 35. H.A.M. Moussa and Y.N. Yu, "Optimal S t a b i l i z a t i o n of Tower Systems Over Wide Range Operating Conditions", presented at IEEE Summer' Power Meeting, paper No. C72-459-6, San Francisco, C a l i f o r n i a , July 9-14, 1972. 36. F.L. Stromotich and R.J. Fleming, "Generator Damping Enhancement Using Suboptimal Control Methods", presented at IEEE Summer Power Meeting, paper No. C72-472-9, San Francisco, C a l., July 9-14, 1972. 37. D.G. Luenberger, "Observers f o r M u l t i v a r i a b l e Systems", IEEE Trans-actions AC, Vol. 14, pp. 380-384, August 1969. 38. James E, Potter, "Matrix Quadratic Solutions", SIAM Journal of Applied Math. Vol. 14, No. 3, pp. 496-506, May 1966. 39. K. Ogata, "State Space Analysis of Control Systems", (book), P r e n t i c e -H a l l , Inc., N.J., 1967. 40. Y.N. Yu and B. Habibullah, "Improving Supplemental E x c i t a t i o n Control Design Using Accurate Model", presented at IEEE Winter Power Meeting, paper No. C73-219-3, New York, January 28 to February 2, 1973. 41. S. Narayana Iyer and B.J. Cory, "Optimum Control of a Turbo-generator in c l u d i n g an Ex c i t e r and Governor", IEEE Transactions, V o l . PAS-90, No. 5, pp. 2142-2149, September/October 1971. 42. S. Narayana Iyer, and B.J. Cory, "Optimization of Turbo-generator 107 Transient Performance By D i f f e r e n t i a l Dynamic Programming", IEEE Transactions, V o l . PAS-90, No. 5, pp. 2149-2159, September/October 1971. 43. L.M. Hovey, "Optimum Adjustment of Governors i n Hydro Generating Stations", The Engineering Journal, pp. 64-71, November 1960. 44. H.A.M. Moussa and Y.N. Yu, "Improving Power System Damping Through Supplemental Governor Control", presented at the IEEE Summer Meeting, San Francisco, Cal. July 9-14, 1972. 45. IEEE Committee Report, "Dynamic Models f o r Steam and Hydro Turbines i n Power System Studies", presented at the IEEE Winter Meeting, New York, N.Y., January 28-February 2, 1973. 46. D.G. Taylor, "Analysis of Synchronous Machines Connected to Power System Networks", The I n s t i t u t i o n of E l e c t r i c a l Engineers, Monograph No. 5265, pp. 606-610, July 1962. 47. M.A. Laughton, "Matrix Analysis of Dynamic S t a b i l i t y i n Synchronous Multimachine Systems", Proc. IEE, Vol. 113, No. 2, February 1966, pp. 325-336. 48. K. Prabhashankar and W. Janischewskyj, " D i g i t a l Simulation of M u l t i -machine Power Systems for S t a b i l i t y Studies", IEEE Transactions, V o l . PAS-87, No. 1, January 1968, pp. 73-80. •49. Y-'.N, Yu and-HfAVM. ,Moxis-s,a;'v^!OpX''±mal''StablTiz'a't'ion of a "Multi-machine System", IEEE Transactions V o l . PAS-91, No. 3, May/June 1972, pp. 1174-1182. 50. D.K. Faddeev and V.N. Faddeeva, "Computational Methods of Linear Algebra", (book), Freeman, 1963. 51. V.N. Sujeer and B. Habibullah, "Application of Runge-Kutta Fourth Order Approximation Method to the Study of Transient S t a b i l i t y of Salien t Pole Generator", Journal of the I n s t i t u t i o n of Engineers (India), V o l. 48, No. 2, pt. E L . l , October 1967, pp. 122-140. 52. V.N. Sujeer and B. Habibullah, " A p p l i c a t i o n of Milne's P r e d i c t o r -Corrector Method to the Study of Transient S t a b i l i t y " , Journal of the I n s t i t u t i o n of Engineers (India), V o l . 48, No. 12, p t . EL.6, August 1968, pp. 970-980. 53. S.N. Talukdar, " I t e r a t i v e Multistep Methods for Transient S t a b i l i t y Studies", IEEE Transactions, V o l . PAS-90, pp. 96-102, January/ February 1971. 54. Y.N. Yu, "The Torque Tensor of the General Machine", IEEE, V o l . PAS-83, pp. 623-629, February 1963.
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Dynamic power system modelling and linear optimal stabilization design using a canonical form Habibullah, Bavadeen S. 1973
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Title | Dynamic power system modelling and linear optimal stabilization design using a canonical form |
Creator |
Habibullah, Bavadeen S. |
Publisher | University of British Columbia |
Date Issued | 1973 |
Description | The linear optimal stabilization of power systems has become a very active area of research in recent years. Implementation of an optimal control scheme usually requires the measurement of all state variables, some of which are not accessible. Dynamic estimators may be used to estimate immeasurable states. But the addition of dynamic estimator makes the overall control scheme more complex and unduly sensitive to disturbances and changes in parameters. There is another problem with the optimal control design, i.e. the choice of the performance index matrices Q and R. In this thesis a fairly accurate synchronous machine model in terms of easily measurable state variables is developed. By neglecting the short lived armature transients, the best dynamic model of a synchronous machine with torque angle, speed, electric output power, terminal voltage or current and field voltage or current as the state variables is derived. The model is used for supplemental excitation control and linear optimal control designs. For the former not only the mechanical mode but also the electrical mode oscillations are considered. For the latter the system equations are transformed into a canonical form and the optimal control thus designed is a function of the weighing matrices Q and R of the cost function. Chapter 2 is devoted to the dynamic modelling of a synchronous machine, Chapter 3 to the supplemental excitation control and Chapter 4 to the development of design techniques for the linear optimal control. Numerical examples are given in Chapters 5 and 6 for both single machine and multi-machine power systems. The nonlinear test results of power systems indicate that transient responses can be greatly improved by the linear optimal control schemes developed in this thesis. |
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Thesis/Dissertation |
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Text |
Language | eng |
Date Available | 2011-03-03 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0093066 |
URI | http://hdl.handle.net/2429/31961 |
Degree |
Doctor of Philosophy - PhD |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
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Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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