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Linear optical stabilization and representation of multi-machine power systems Moussa, Hamdy Aly Mohammed 1971

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LINEAR OPTIMAL STABILIZATION AND REPRESENTATION OF MULTI-MACHINE POWER SYSTEMS by • HAMDY ALY MOHAMMED MOUSSA B.Sc, Ain Shams University, Egypt, 1965 M.Sc, Ain Shams University, Egypt, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY' i n the Department of E l e c t r i c a l Engineering We accept this thesis as conforming to the required standard Research Supervisor Members of the Committee Head of the Department Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF"BRITISH COLUMBIA July, 1971 In present ing th i s thes is in pa r t i a l f u l f i lmen t o f the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ib ra ry sha l l make it f r ee l y ava i l ab le for reference and study. I fu r ther agree that permission for extens ive copying of th i s thes i s fo r s cho la r l y purposes may be granted by the Head of my Department or by his representat ives . It i s understood that copying or pub l i ca t i on o f th i s thes i s f o r f i nanc i a l gain sha l l not be allowed without my wr i t ten permiss ion. Department of The Un iver s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date AlAA^j 3 , 197/ ABSTRACT Linear optimal regulators have been designed for power system s t a b i l i z a t i o n by introducing control signals to voltage regulators and/or governors. A new technique i s developed i n this thesis to determine the state weighting matrix Q of the regulator performance function with a dominant eigenvalue s h i f t of the closed loop optimal system. The technique i s used to investigate the s t a b i l i z a t i o n of a t y p i c a l one-machine i n f i n i t e system and a multi-machine system with different s t a b i l i z a t i o n schemes. The objective i s to find the best way to s t a b i l i z e a power system. An optimally sensitive c o n t r o l l e r i s also developed to offset the effects of the changing system operating conditions on the e f f o r t of the s t a b i l i z i n g signal. The controller automatically adjusts i t s gains so that i t always provides the system with the optimum s t a b i l i z i n g signal. A new multi- machine state variable formulation, necessary for these studies,' i s developed. I t requires minimum computations and retains a l l the parameter information for s e n s i t i v i t y studies. An exact representation of synchronous machines i s investigated and test methods are suggested for the determination of exact c i r c u i t parameters. i i TABLE OF CONTENTS / • Page ABSTRACT / 1 1 TABLE OF CONTENTS ..../. i i : L - LIST OF TABLES .' v i LIST OF ILLUSTRATIONS v i i ACKNOWLEDGMENT v i i i NOMENCLATURE . i x 1. INTRODUCTION 1 2. EXACT EQUIVALENT CIRCUITS AND PARAMETERS OF SYNCHRONOUS MACHINES 4 2.1. d-Axis Exact Equivalent. C i r c u i t s ^ 2.2. q-Axis Exact Equivalent C i r c u i t s 8 2.3. C i r c u i t Parameters i n Terms of Conventional Parameters........ 11 2.4. Extra Tests to Determine T̂  and x" • 13 D do 2.4.1. Determination of T Q From a Varying S l i p Test. 13 2.4.2. Determination of T Q From Decaying Current Test 14 2.4.3. Determination of xV 16 do 2.5. Laboratory Test Results .' 17 3. STATE VARIABLE EQUATIONS OF MULTI-MACHINE POWER SYSTEMS 19 3.1. Terminal Voltages and Currents : 19 3.2. Nonlinear Machine Equations 21 3.3. Linearized Machine Equations..... 23 3.4. Exciter and Voltage Regulator System 25 3.5. Torque Equations 26 3.6. Governor-hydraulic System 28 i i i Page 3.7. State Equations 29 3.8. M u l t i - M a c h i n e System w i t h an I n f i n i t e Bus..... 29 3.9. S i m p l i f i c a t i o n of Power System Dynamics 31 / / 4. OPTIMAL LINEAR REGULATOR DESIGN WITH DOMINANT EIGENVALUE 'SHIFT 34 4.1. L i n e a r Optimal Regulator Problem. 34 4.2. Eigenvalue S h i f t P o l i c y . . . . 35 4.3. The S h i f t .. 36 4.4. Determination of Aq 37 4.5. S e n s i t i v i t y C o e f f i c i e n t s X,q 37 4.6. A l g o r i t h m 40 5. OPTIMAL POWER SYSTEM STABILIZATION THROUGH EXCITATION AND/OR GOVERNOR CONTROL '. . 41 5.1. System Data 41 5.2. Case 1: u £ C o n t r o l 43 5.3. Case 2a: u_ C o n t r o l , w i t h Dashpot i 44 G 5.4. Case 2b: u' C o n t r o l , without pashpot 45 G 5.5. Case 3: u„ Plus u' C o n t r o l 46 E (j 5.6. Nonlinear Tests v 47 6. OPTIMAL STABILIZATION OF A MULTI-MACHINE SYSTEM. 51 6.1. System Data and D e s c r i p t i o n 51 6.2. Case 1: One Machine Optimal E x c i t a t i o n C o n t r o l u„ T 54 EI 6.3. Case 2: M u l t i - O p t i r t a l C o n t r o l l e r s u ^ 55 6.4. Case 3: Approximated One Machine Optimal Design 57 6.5. Case 4: Subsystems Optima]. Design • 58 Page 6.6. Nonlinear Tests 59 7. OPTIMUM STABILIZATION OF POWER SYSTEMS OVER WIDE RANGE OPERATING CONDITIONS 63 7.1. Opt i m a l l y S e n s i t i v e L i n e a r Regulator Design 63 7.2. S e n s i t i v i t y Equations of the L i n e a r i z e d Power System.... 66 7.3. Optimally S e n s i t i v e S t a b i l i z a t i o n of a Pox^er System. . 71 8. CONCLUSIONS 81 APPENDIX A 84 APPENDIX B ... 86 REFERENCES 90 LIST OF TABLES TABLE PAGE 3-1 Eigenvalues of the Typical One Machine I n f i n i t e System of Different Modelling 33 7-1 Controller Gains for u^g and u* at Different Operating Conditions .: 77 7-2 Dominant Eigenvalues of the System with the Different , ... . Controllers 78 v i LIST OF ILLUSTRATIONS FIGURE £AGE 2-1 General d-Axis C i r c u i t s . . . . 7 2-2 Simplified d-Axis C i r c u i t 7 2-3 General q-Axis Equivalent C i r c u i t 9 2-4 q-Axis C i r c u i t , (x ' - X ) . 10 qQ q 2-5 q-Axis C i r c u i t , (XqQ ~ x^) . . . . 10 2-6 Determination of T̂  from S l i p Test 14 2-7 Connection for the Decaying Current Test 14 2- 8 Resolving Decaying Current into Two Components 16 3- 1 Components of V i n dq and DQ coordinates 20 3-2 A Typical Exciter-Voltage Regulator System 25 3- 3 A Typical Governor-Hydraulic System 28 4- 1 Algorithm to Determine Q with Dominant Eigenvalue S h i f t . 40 5- 1 A Typical One-Machine I n f i n i t e System 42 5- 2 Nonlinear Test Results 50 6- 1 A Typical Four-Machine Power System 52 6- 2 Nonlinear Tests of the Multi-Machine System 62 7- 1 Structures of Nominal and Optimally Sensitive Controllers 66 7-2 Speed and Torque Angle Gains for the Controllers 79 7-3 Nonlinear Test Results 80 v i i ACKNOWLEDGMENT I wish to express my most grateful thanks and deepest gratitude to Dr. Y.N. Yu, supervisor Of th i s project, for his continued i n t e r e s t , encouragement and guidance during the research work and xwriting of th i s thesis. I also wish to thank Dr. E.V. Bohn, Dr. M.S. DaVies and Dr. H.R. Chinn for reading the draft, and for the i r valuable comments. The proof reading of the f i n a l draft by Mr. B. P r i o r i s duly appreciated. Thanks are due to Miss Linda Morris for typing t h i s thesis. The f i n a n c i a l support from the National Research Council and the University of B r i t i s h Columbia i s g r a t e f u l l y acknowledged. I am grateful to my wife Zainab for her encouragement throughout my graduate program. v i i i NOMENCLATURE General A system matrix B control matrix Y state vector u control vector Q p o s i t i v e semi-definite symmetric matrix, weighting matrix of Y R pos i t i v e d e f i n i t e symmetric matrix, weighting matrix of u q vector, diagonal elements of Q K R i c c a t i matrix G closed loop system matrix A=£+jri eigenvalue vector of G A. s e n s i t i v i t y vector of the eigenvalue A. w.r.t. <i x,q J G 1 S s e n s i t i v i t y matrix M composite matrix as defined i n (4.18). A,X,V eigenvalue vector, eigenvector matrices of M and M' o subscript denoting i n i t i a l condition Y time derivative of Y * superscript denoting conjugate ' or T superscripts denoting transpose A p r e f i x denoting a li n e a r i z e d variable [ ] diagonal matrix with elements of each machine 0)^ synchronous angular v e l o c i t y : 377 rad/s p d i f f e r e n t i a l operator i x suffices a,d,q armature a-phase, d-axis, and q-axis windings suffices F,D,Q rotor f i e l d , d-axis damper and q-axis damper windings s e n s i t i v i t y of matrix A with respect to parameter q. System parameters (P.U., except as indicated) Y„, network node admittance matrix N Ẑ  network node impedance matrix Z network node impedance matrix i n i n d i v i d u a l machine m r coordinates r+jx t i e - l i n e impedance G+jB terminal load admittance R's,r's winding resistances i n ti, and per unit X's,x's s e l f and mutual reactances i n ti, and per unit L's s e l f and mutual inductances, H Z armature base ohm, ti n x ^ x ^ x ^ d-axis synchronous, transient and subtransient reactances x" newly defined open f i e l d d-axis subtransient reactance do J r x ,x" q-axis synchronous, and subtransient reactances q q T'T" short c i r c u i t d-axis transient and subtransient time a a constants, s T' , T" open c i r c u i t d-axis transient and subtransient time do do constants, s d-axis damper winding time constant, s T" open c i r c u i t q-axis subtransient time constant, s qo r i . exciter amplifier gain exciter amplifier time constant, s exciter time constant, s governor permanent droop governor temporary droop gate actuator time constant, s dashpot time constant, s hydraulic turbine gate time constant, s water time constant, s i n e r t i a constant damping c o e f f i c i e n t les (P.U., except as indicated) - . optimal e x c i t a t i o n s i g n a l , one-machine i n f i n i t e system conventional e x c i t a t i o n control signal optimal governor control signals with and without dashpot one-machine optimal excitation control, multi-machine system multi-machine optimal ex c i t a t i o n controls, multi-machine system optimally sensitive excitation control f l u x linkages, currents, voltages torque angle, radians angular v e l o c i t y , e l e c t r i c a l rad/s exciter regulator voltage gate actuator signal dashpot feedback signal x i g gate movement h hydraulic head t ,t mechanical, e l e c t r i c a l torques m e v , i machine voltages and currents i n common coordinates n n v , i machine voltages and currents i n i n d i v i d u a l coordinates m m V ,I voltage and current matrices with diagonal elements m m v and i of each machine m m U =U ,+iU s e n s i t i v i t y matrix of v with respect to <$.' m md mq m v i n f i n i t e bus voltage o v generator terminal voltage P+jQ generator output power V's,v's applied voltages i n V, and per unit U's,u's ro t a t i o n a l voltages i n V, and per unit I ' s , i ' s currents i n A, and per unit V ,1 base armature voltage, current n n V̂ .̂ jV-^ tV^T, base f i e l d , D-winding and Q-winding voltages rJ3 DD QB Ipg, I^gj Iqg base f i e l d , D-winding and Q-winding currents. x i i 1. INTRODUCTION The s t a b i l i z a t i o n of power systems has become increasingly impor- tant because of the increase i n the size of power systems, the number of interconnections, the voltage l e v e l , the number of large generating units, and the introduction of fast-response excitation systems and dc transmission l i n e s . Much attention has been focussed recently on the application of control signals to the excitation system for s t a b i l i z a t i o n , or s t a b i l i t y control, to improve the a b i l i t y of a power system to return to i t s synchronous operating equilibrium after 1 2 3 a disturbance. These signals can be derived from shaft speed ' ' , terminal frequency^'^'^, or terminal power'''8. They are used to o f f - set the voltage regulator reference i n the transient period with the object of producing p o s i t i v e damping torques on the synchronous machine shaft In view of the fast development of control theory, more work must be done to explore the p o s s i b i l i t y of deriving better methods and techniques for power system s t a b i l i z a t i o n . Optimal l i n e a r regulators 11 12 are designed and quadratic performance functions are chosen ' There are many problems unsolved. Four of them are mentioned below. The f i r s t i s that i n the optimal state regulator design, the choice of the weighting matrix Q associated with the' performance function i s based e n t i r e l y upon past experience or guessing. Therefore, the designed controller i s not necessarily the best. The second problem i s the normal controller i s designed for only one p a r t i c u l a r operating condition, and this condition cannot be estimated p r i o r to a disturbance. Can an optimal controller be designed to cope with the wide range operating condition? The t h i r d problem i s the multi-machine dynamics formulation. The problem i s not how to obtain a set of state equations but how to avoid the large number of high-order matrix inversions and how to re t a i n a l l the parameter information for s e n s i t i v i t y investigations. F i n a l l y there i s the problem of exact representation of synchronous machines and how to determine the c i r c u i t parameters from simple f i e l d tests. This must be done i n order to obtain an accurate evaluation of system dynamic behaviour during and after a disturbance. This thesis provides some answers to the problems mentioned above.. In Chapter 2 the exact equivalent c i r c u i t s for the synchronous machines are derived from the MKS voltage equations and by the use of per unit systems. Simple f i e l d tests to determine the exact machine parameters are then suggested. The multi-machine state equations are derived i n chapter 3 by r e l a t i n g the transmission network algebraic equations to i n d i v i d u a l machine dq coordinates. Detailed representation of excitation and governor systems i s presented. The one machine i n f i n i t e bus system i s only a special case of the multi-machine system. Dynamic approxi- mation of the formulation i s then discussed. A new technique for the design of optimal regulators i s developed i n Chapter 4. The choice of the state weighting matrix elements of Q of the performance function i s related to the movements of the dominant eigenvalues of the closed loop system. The dominant eigenvalues are shifted to the l e f t on the complex plane within the p r a c t i c a l l i m i t s of the cont r o l l e r . - . The technique 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 in f i n i t e - b u s system. Various s t a b i l i z a t i o n schemes are investigated. Optimal excitation and/or governor controls are compared with conventional e x c i t a t i o n control. The objective i s to find the best way to s t a b i l i z e a pox^er system. Some of the s t a b i l i z a t i o n techniques for the one-machine inf i n i t e - b u s system are further developed for multi-machine system i n Chapter 6. Although the one-machine design i s more often than not the only case considered, no more d i f f i c u l t y i s involved i n the formulation or computation for multi-machine systems. Several schemes are investigated, multi-machines x^ith multi optimal controllers or with one optimal co n t r o l l e r as compared x^ith multi-machines with i n d i v i d u a l optimal controllers or an equivalent one-machine with one optimal controller. An answer to the wide range operating condition problem i s given i n Chapter 7. An optimally sensitive c o n t r o l l e r i s developed which provides s t a b i l i z a t i o n for a power system which departs widely from normal operation conditions. A comparison i s then made of the optimally sensitive control design x^ith other nominal designs. 2. EXACT EQUIVALENT CIRCUITS AND PARAMETERS OF SYNCHRONOUS MACHINES For s t a b i l i t y studies of large power systems, accurate re- presentation of the synchronous machine i s required. As pointed out 14 by Canay , the conventional equivalent c i r c u i t s for synchronous machines do not give accurate computed f i e l d voltage and current values. He suggested several c i r c u i t s and showed good agreement between his test and calculated r e s u l t s . His c i r c u i t parameters were calculated from design. Questions ar i s e : How to determine accurate c i r c u i t parameters from simple f i e l d tests and how to choose the equivalent c i r c u i t s . The c i r c u i t s are not unique because of different base v o l t s , base amperes and c i r c u i t elements. In t h i s chapter exact equivalent c i r c u i t s for synchronous machines are derived from voltage equations i n MRS units. Some constraints are then imposed so that the equivalent c i r c u i t s w i l l lead to the simplest form. A systematic procedure i s then developed to determine these c i r c u i t parameters from simple f i e l d tests. 2.1. d-Axis Exact Equivalent C i r c u i t s Applying Park's transformation, the d-axis voltage equations of a synchronous machine i n MKS unit can be written i n the form V u d R R_ X, X ap X _ d a r aD 3 T ^ a X F X I 2 ^ 3 XDF h I , d T D_ . 15 The X-matrix i s not symmetric .• Here a l l X's are reactances of (2.1 single-phase e x c i t a t i o n except X^ which i s of three-phase e x c i t a t i o n . The numerical c o e f f i c i e n t 3/2, and hence the asymmetry of the m a t r i x , r e s u l t s from the a,b and c three-phase e x c i t a t i o n on the s t a t o r and the F or D single-phase e x c i t a t i o n on the r o t o r . The m a t r i x form i t s e l f suggests that the per u n i t reactances must, and voltages and currents may, be defined as f o l l o w s X „ ; • x dF aF V X DB dD aD V n x _ = X DB FD "FD V FB 3 I n 3 I n 'Fd = ^ F a - * ^ XDd = ^ D a - * ^ r B UD DF = ̂ F "FB F V. DB d V R a V FB r F ~ h V, FB FB XD = XD; DB V DB I. DB V DB (2.2 a n h = V ^ B iD = V ^ B v. = V./V d d n VF=.V VFB The m a t r i x of x's i s not n e c e s s a r i l y r e c i p r o c a l . To make i t r e c i p r o c a l " ^ , the f o l l o w i n g c o n s t r a i n t s must be imposed, 6 —V I = V I = V I 2 n n FB FB DB DB (2.3) res u l t i n g i n XdF XFd Z 4 ] ' XdD XDd Z 4 } n n n n - 1 XFD/ IFB, D̂B. XFD XDF 3 Z I I n n n = 2 XF r T'FB, 2 2 ̂ /SB.2 X d Z ' X F 3 Z 4 ; ' XD • 3 Z 4 ; n n n n n (2.4) = \ _ 2 R F TFB 2 r a Z ' r F 3 Z 4 ; n n n V ^ B ^ ' rD 3 Z 4 ) n n Z = V /I n n n The d-axis voltage equations can be written now, i n per unit, as vd" ud = r +px, a r d P X d F P XdD V F p X F d r F + p X F P XFD _ 0 - P XDd P XDF V P X D • A (2.5) One of the general d-axis equivalent c i r c u i t s corresponding to (2.5) i s as shown i n Fig.2-1, which reduces to Fig. 2-2, the simplest form, i f one sets XFd = XDd = XFD ( 2 - 6 ) Note that x ^ , x ^ and x m are no longer leakage reactances. They are defined as Xd£ X d XFd' XF£ ~ X F XFd' XM ~ XD " XFD. (2.7) 7 . — V \ A - W — r - r r p — — - o ro n —OnfT^VV*—I Fig. 2-1 General d-Axis C i r c u i t s J J f f \ ^ V V N . Fig. 2-2 Simplified d-Axis C i r c u i t The following information, although not needed i n the determination of parameters from f i e l d t e s t s , i s useful i n design. From (2.6) the current ratios of (2.4) can be determined as follows 1FB 3 XaD 2 X. (2.8) FD •""DB 3 XaF 2 X. (2.9) FD Substituting (2.8) and (2.9) into (2.4) and the results into (2.7) the c i r c u i t parameters of Fig. 2-2 can be expressed i n terms of winding parameters as follows 8 3 XaF XaD n n TD = 3 faD /F XaD XaF XF£ 2 X_ 4 X_n Z J FD n FD n (2.10) XD£ - 2 X F D 4 n X F D " Z aDs _ 3 XaF XaD XFD XDd XFd 2 Z x R n JFD v x = _a = _3 ̂  /*AD>2 r a Z ' r F 2 Z ' rD n n TD 3 V j F , 2 2 Z n XFD 2.2. q-Axis Exact Equivalent C i r c u i t s The q-axis voltage equations for a synchronous machine i n MKS unit are as follows r- -i "v - u q q = R a + P X q XaQ I q 0 _2XQa XQ A (2.11) The X-matrix i s again not symmetric"'""'. While X^ i s a reactance of three-phase e x c i t a t i o n , X , X and X are of single-phase ex c i t a t i o n . The matrix form suggests the following d e f i n i t i o n s of per unit reactances, voltages and currents XqQ = XaQ V V n > XQq= ^ V W X q " X q V V n > XQ = XQ W r = R I /V , r_ = R I n p/V n_ a a n n Q Q QB OB i =1/1 , i = . I . / I _ , v = V /V , u = U /V q q n ' Q Q QB q q n' q q n (2.12) 9 / 16 To make the x-matrix reciprocal , the following constraint must be imposed / resulting i n — V I = V I 2 n n QB QB XqQ ~ XQq " Z h } ' X q " Z H* M n n n v = i Q̂ / p j . 2 = 2 _Q,_QBs2 XQ 3 Z ' ' Q 3 Z 1 ' n n n n The per unit q-axis voltage equation now can be written as r + px px _ i a q qQ v - u = q q 0 (2.13) (2.14) (2.15) p XQq rQ + p XQ, The general q-axis equivalent c i r c u i t corresponding to (2.15) i s as Fig. 2-3. A A A *ql - O T P Ql '0 A A A 'Q Fig. 2-3 General q-Axis Equivalent C i r c u i t s where x = x - x , xr„ = x - x (2.16) qH q qQ QJ6 Q Qq Although x ̂  of Fig. 2-3 exactly represents the mutual reactance and x . and x̂ „ the leakage reactances, mathematically, qJi QJi however, the branch reactance x ̂  can be set equal to x q or x^ resulting i n two s i m p l i f i e d equivalent c i r c u i t s , Figs. 2-4 and 2-5 respectively. -A/\A—rirpL- Fig. 2-4 q-Axis C i r c u i t s ( XqQ = V Fig. 3-5 q-Axis C i r c u i t ( XqQ = V The parameters of these two c i r c u i t s can be easily determined from f i e l d tests. They can also be expressed i n terms of winding parameters: Fig. ,2-4,, x q Q = x q From (2-14) one has I X n aQ Hence XQ " XQq R ( 3 X a Q X a Q ^ * K \ Q 3 Z n X a Q (2.17) Fig. 2-5, x q Q = x Q From (2.14) one has ""•QB _ 3 XaQ n 2 X, 11 The s o l u t i o n s o f t h e c i r c u i t p a r a m e t e r s a r e x ,. = x - / x 2 X a X d ( x d X d o ) X d X d o ( x d ^c? where and d * I C x d - x J o ) - ( x ' - x J ) d d d do ( x d - X d o ) - ( x d - X d ) r„ = F (u T' x ,-x' o do d d 1 r X d " X d £ ) 2 •D u T" x ' - x " o do d d Next, i f T^ i s s e p a r a t e l y d e t e r m i n e d , we have a n o t h e r e q u a t i o n i n s t e a d o f ( 2 . 2 1 ) . The s o l u t i o n s a r e (2.23) and (2.22) =  X d X d i l X D d X d X d £ ' XFi x , - x i X D d d d ^ X d X d £ ^ X d / n 00-. x_. = ; ri (2.23) d • d 2 1 X D d U o T D " V r D ( 2 ' 2 4 ) x d . = x d - J^-^^a-^Y^- <2'22a) D do The c u r r e n t r a t i o I ^ / I o f (2.4) can nov7 be d e t e r m i n e d , b u t n o t FB n 1^ / I s i n c e t h e r e i s no way t o measure b e c a u s e o f t h e s h o r t c i r c u i t . DB n D The v o l t a g e r a t i o V„_/V can t h e n be d e t e r m i n e d from ( 2 . 3 ) . r JJ n The q - c i r c u i t p a r a m e t e r s can be e a s i l y d e t e r m i n e d . F o r F i g . 2-4 we have q XqQ + XQ£ ° q ° Q Q £ q Q / The solutions are „ 2 / X X ^ X ! X ~ ~ X , X ~ — 11 i r^ — r r i t l It ^ Z . Z O / qQ q ' Q£ x -x Q u) T x -x q q o qo q q For Fig. 2-5 we have' x" = x . , (j T" r . = x . , x = x . + x „ (2.27) q qit ' o qo Q qQ q qQ qH The solutions are x . = x" , x• n = x =x -x" , r_ = — ^ r r (x -xV) (2.28) ql q qQ Q q q 0 % q 0 ^ ^ 2.4. Extra Tests to Determine T^ and x'1 . _ D do Two test methods are suggested to determine T^ and one to determine x'' . A l l methods were tested i n the laboratory, do 2.4.1. Determination of T̂  from a Varying Slip Test The rotor i s driven at various speeds. Positive sequence voltages are applied to the armature winding with the f i e l d open. From phase voltage-current r a t i o equivalent reactances x^Cs) and x^(s) are approximately determined. Replacing r ^ by r^/s i n Fig.2.2, the imaginary part of the c i r c u i t impedance i s a function of s l i p ' s as follows x 2 x X d ( s ) = X d " ^p-f T X , U 2 ( 2 ' 2 9 ) or . j 1 X° (~)2 + -TT (2.30) x,-x,(s) 2 s 2 d C XdD XD . • ' XdD Fig. 2-6 Determination of from S l i p Test which can be plotted as Fig.-2-6 for the determination of T^. An accurate value of x^, from open and short c i r c u i t t e s t s , must be used for the calculations. ' •2.4.2. Determination of T from Decaying Current Test Fig. 2-7 Connection for the Decaying Current Test 19 Kaminosono and Uyeda's i n d i c i a l response method i s modified to determine T . Since a clear step voltage i s hard to obtain, a decaying current i s used instead. Apply a constant cur 15 to one phase winding i n the d-axis position and then suddenly short c i r c u i t the armature terminals with the switch Sw i n Fig. 2-7. The rheostat protects the power supply. The voltage equations for Fig. 2-7 i n Laplace transform are 0 I 0 r +sL sL -a d dD sL dD r^+sL, " i (s)" a • - L d LdD i ao -Vs>. -LdD LD - o (2.31) where i i s the i n i t i a l current i n the armature winding, ao The solution of I (s) can be written i n a convenient form a s+ T' V s ) = — T - ^ ( s + f ) ( s + i ) 3 0 1 2 (2.32) where TD " ^ H d 5 ' T1 T2 = V i > T l + T 2 = V T D (2.32a) and TD " V r D > T d = L d / r a > 4d = L D d / L d L D (2.32b) I (s) of (2.32) can be resolved into two components a . I a ( s ) X10 . X20 + 4 . 1 4 . 1 s+ — s+ — 1 2 (2.33) and i t can be shown that the i n i t i a l component current r a t i o i T T -T' 10 = Jl_ 1 D i x T'-T 20 1 D 2 (2.34) From T^, T 1 + 1^ of (2.32a) and (2.34), the following solutions are ob tained 16 T d = ( i 1 0 T l + i20 T2 ) / iao> TD = T l V T d > TD = < 1 1 0 T 2 + 1 2 0 T l ) / i a o < 2 ' 3 5 ' T^, T^, i^Q and i^Q are determined from a semilog plot as Fig. 2-8. / The T, value from (35) should be checked with d. • / . (2.36) Fig. 2-8 Resolving Decaying Current into Two Components 2.4.3. Determination of x" „ , , •• . . - , ^ 0 20 Dalton and Cameron's method to determine x\j i s adapted to determine x̂ ' . The rotor remains stationary and the f i e l d winding i s open-circuited. Single phase Voltage of rated frequency i s applied to each of., a pair of stator terminals i n turn, leaving - the t h i r d terminal open. Three such tests are performed Xi/ith the rotor position fixed throughout the test'. The armature voltage and current and the f i e l d voltage are recorded i n each test. Let the single-phase reactance X be a function of 6, the angular position of the rotor X = K + M cos 26 (2.38) and l e t the voltage-current r a t i o of the three tests be A, B, and C. It can be shown that ^' / v - A + B + C /' K = , , (2.39a) and M = J(B-K) 2 + ^~- 2 The open f i e l d d-axis subtransient reactance i s then given by (2.39b) The plus sign should be used i f the largest measured reactance, A, B or C, and the largest measured f i e l d voltage occur i n the same test. 2.5 Laboratory Test Results The methods thus developed were applied to a small synchronous machine to determine the c i r c u i t parameters. From IEEE test code the following d-axis parameters are determined. r = 0.72Q, x, = 16.2n, x' = 2.74Q, x" = 2.42 ^ a a a a T' = 0.27s, T" = 0.027s do do The per unit values can be obtained when the base ohm Z i s chosen. n From extra tests the following are determined A T D = 0.049 s (varying s l i p test) B T n = 0.055 s (decaying current test) C x^'o = 8.18 Q (adapted Dalton and Cameron) The computed results of d - c i r c u i t parameters i n ohms are as follows XDd XD£ r F rD A 15.8 0.40 2.75 14.6 0.182 1.66 B 15.5 0.68 2.38 10.9 0.176 1.28 C 15.9 0.33 2.84 15.5 . . 0.184 1.76 / / / The discrepancy i n results of B i s attributed to the d i f f i c u l t y of resolving the decaying current into components. The f i e l d resistance Rp i s 70ft and the current and voltage ratios are I F B / I n = 0.0625 , V F B/V n = 24 For the q-axis x = 9.71 ft, x" = 7.2..ft q q are determined by conventional methods and T" = 0.0165 s qo by a decaying current method sim i l a r to Fig. 2-7. The computed results of q-axis parameters Fig. 4 x . = 9.71 ft, x0„ = 27.8ft , r A = 6.05 ft qQ QSL Q Fig. 5 x q £ = 7.2 ft , x q Q = 2.51 ft, r ^ = 0.407 ft 3. STATE VARIABLE EQUATIONS OF MULTI-MACHINE POWER SYSTEMS21 / In s t a b i l i z a t i o n studies of large interconnected'multi- machine power systems, the system dynamics must be expressed i n the 22 state variable form Y = AY + Bu. Laughton suggested a method of building the A matrix from matrix elemination of algebraic and 23 2 A d i f f e r e n t i a l equations. U n d r i l l ' proposed to b u i l d up the A matrix from i n d i v i d u a l system submatrices. U n d r i l l ' s method requires a matrix inversion of mn x mn for m machines each described by n-th order equations. The system parameters are not retained i n the f i n a l formulation. This i s also the case i n Laughton's formulation. In t h i s chapter a new multi-machine formulation i s proposed. The main objective i s to reduce the number of matrix inversions and to keep them of low order. A l l the system parameters are retained i n the f i n a l formulation making i t convenient for s e n s i t i v i t y and control studies. The synchronous machine parameters are based on an exact equivalent c i r c u i t , and can.be determined from f i e l d tests as described i n chapter 2. 3.1. Terminal Voltages and Currents Let the i n d i v i d u a l synchronous machine rotating coordinates be d and q and the common rotating coordinates of the complete system be D and Q. Let the terminal voltages and currents of a l l machines i n dq coordinates be a vector v and a vector i and those i n DQ coordinates m . m be a vector V and a vector i respectively, and l e t the phase r e l a t i o n of the k-th machine x^ith respect to the two coordinate systems be as i n Fig. 3-1. / / Fig. 3-1 Components of i n dq and DQ Coordinates Then we have for the k-th machine VNk e VimV XNk Z > " Xmk and for a group of m machines (3.1) The transmission system i s usually considered as a s t a t i c network i n s t a b i l i t y and control studies, i . T = YT1v.T N N N (3.2) Substituting (3.2) into (3.3) x<re have v = Z i m mm where and Zm = [ e - j 6 i ] Z N [ e j 6 J ] m Z = Y (3.3) (3.4) (3.5) (3.6) 21 Note that the highest order matrix inversion required i n the formulation i s Expanded we have where ~ vd' R - X m m _ 1 d / - ( V L qJ X R L . m m J i L qJ ZN = RN + Z m R + jX m J m (3.7) (3.8) R m(i,3) x m(i,j> R ^ i . j ) - X N ( i , j ) cos6 . . sin<5.. 13 J (3.9) 6 . . = 6. - 6. (3.9a) 3.2. Nonlinear Machine Equations The synchronous machine equations are as follows, the i - t h machine For V F " P^F + r F S where d r d a d e q v =p^ - r i + i b i K q q a q e d 0 = P̂ D + VD 0 = P̂Q + VQ " V " XF XFd XFD i p • *d XdF X d XdD - : Ld 0) o - XDF XDd - j (3.10) (3.10a) 22 1_ 0) qQ - i q (3.10b) Note that - i ^ and' - i ^ are used i n the synchronous generator equations. Actually a l l the notations of (3.10) should be given a s u f f i x "i" for the i - t h machine, except for p and <i) which are common to a l l machines. The s u f f i x i s dropped for c l a r i t y . I t i s also intended that the same equations be used for the description of multi-machine systems. In such a case a l l the v's, i ' s and i|> * s of (3.10), become column vectors, and x's and r's, diagonal matrices. These statements apply also to the rest of the chapter. The current solutions of (3.10a) and (3.10b) have the form - l Y. FF dF DF Fd dd Dd FD dD DD - V ^d (3.11a) and Q qq - - V (3.11b) Note that the solution of currents from (3.10) for in d i v i d u a l machines does not involve equations of other machines. The Y matrices of (3.11) are not the inverses of the x matrices of (3.10). I f equal per unit mutual reactances are used, the elements of the Y matrices of (3.11a) of i n d i v i d u a l machines can be determined d i r e c t l y from the d-axis exact equivalent c i r c u i t of Fig. 2-2 using the well-known star-mesh relations i n network analysis. Substituting i ^ and i of (3.11) into (3.7), and the results into v, and v of (3.10), we have d q "*d" = • V " R Y d F " R Y d d w +X Y —RY X Y e m qq dD m qQ -X Y,_ -0) -X Y,, -RY m dF e m dd qq -X Y -RY n m dD qQJ * [*p; V * q . V V where (3.12) R = Re Z + [r ] m a (3.12a) Substituting !„, i n and i of (3.11) into v^, v =0, and v =0 of (3.10), r 1) Q r D Q we have - r Y '\b - F FF ^F r F Y F d ^ d " r Y - TJJ + v F FD VD F P̂ D = - r Y •\b -D DF VF rD YDd^d " r Y 't D DD VD (3.13) P'̂Q = Q Qq \ R Q Y Q Q ' * Q Thus the transmission l i n e r e l a t i o n (3.7) at the machine terminals has been included i n the nonlinear state form of machine equations (3.12) and (3.13). 3.3. Linearized Machine Equations When equation (3.4) i s l i n e a r i z e d , i t has three terms, Av = Z A i + jZ [A6.]i - j[A6.]v (3.14) m m m J m . i m J i m ' which can be written as Av = Z Ai + jU A6' m m m m (3.14a) where U = Z I - V m mm m (3.14b) 24 Note that A6, i and V are column vectors and [A<S], I and V are m m m m diagonal matrices. Since Vm * V d + j V ^ = ^ + j ^ = ^ + ^ m rm the voltage equations v^ and .v of (3.10) can be written as v = pib m ^rm [r ] i + j [to ]i> a m J e rm After l i n e a r i z a t i o n and making use of (3.14a), we have P A ^ ' m [ Z +(r )] A i -j[co ]AUJ - j |> ] Ato +jU A<5 m a -m J e rm J rm e J m (3.15) (3.16) Expanded-and with the substitution of i , and i from (3.11) we have d q Ail), Aiji -RY dF -RY dd co +X Y -RY,^ X Y e m qq dD m qQ -X Y,_ - t o -X Y,, -RY m dF e .E I dd qq -X Y -RY m dD qQ J [A^ p, A^d, A^q, A^ D,Ai{) Q] + -U mq A6 Aw ^ e Umd "'*d ] Equation (3.13), after l i n e a r i z a t i o n , becomes P ^ F = - r F Y F p.A^ F - r F Y p d . A ^ - r ^ - A ^ + Av p PA^D = " rD V A * F ~ rD YDd" A*d " V W ^ D PA*Q = " rQ YQq* A V rQ.VA*Q (3.17a) (3.17b) Equations (3.17a) and (3.17b) are the lin e a r i z e d multi-machine equations 'i i n state variable form. 25 3.4. Exciter and Voltage Regulator System Fig. 2 shows the block diagram of a t y p i c a l exciter voltage regulator system ^6 \ KA- r i VF ) l+TAs 1+TEs Fig. 3-2 A Typical Exciter-Voltage Regulator System The corresponding state equations are PAVF = - Av p 4- |- Av R 1 KA KA P A v R = - - Av R - - Av + - u E A A A Since v Av = v,Av, + v Av t t d d q q then from (3.14a) Av, d R m -X " m + -u mq Av X _ m R mj Ai . q_ U A md. A6 (3.18a) (3.18b) (3.19) (3.20) Substituting A i d and A i q of a l i n e a r i z e d (3.11) into (3.20), the results into (3.19), and the results into (3.18b), we have pAv R = A(7,1)A* F + A(7,2)Ai|jd + A(7,3)A<fq + A(7,4)A^ D 1 K A + A(7,5)A.j; - —- Av + A(7,8)A<5 + „ ^ A TA L (3.21) where A(7,l) = MY d F , A(7,2) = MY^ , A(7,4) = MY^ A(7,3) = NYqc. , A(7,5) = NY q Q , KA / A(7,8) = -[«r—KV U , - V, U ), ' A V t q m q K K M =, [-A-](V R + V X ), N = [-^-](V R - V X ) (3.21a) T.v dm q m l.v q m dm A t • A t KA Note that [- ], V,, and V are diagonal matrices b u i l t up from the data T.v d q A t of i n d i v i d u a l machines. So far we have eight state variable sets i n the order of ( V V ' V ^D' V V V 6 ) 3.5. Torque Equations The li n e a r i z e d torque equation i n MKS may be written pA<5 = Aw (3.22) pAw = T C A t - AT - AT ] (3.23) v m J m e D Now i f A u e ' s unit i s changed from MKS to per unit, and per unit mechanical torque At and e l e c t r i c a l torque At are used i n the formulation, m e (3.23) becomes w Aw T Aw p n e = -J l ( t At - Dt — -At ) (3.23a) pp J o m o • w e o where pp i s the number of pole p a i r s , ŵ  the base e l e c t r i c a l rad/s, T^ the base torque of the complete system, t T the base operating torque of an i n d i v i d u a l primemover, and 27 A A TD j Aw D = t T ' w (3.23b) o n o Thus we have .0) Aw pAco = ~ - (t At - At - Dt —-) (3.24) e 2 H o m e o t o o where 1 W W 9 H = £ j(_°_S.)z/p (3.24a) 2 pp n and i s the base power of the system. Note that co co = 120-n- rad/s (3.24b) o n Thus to = 1 i f re a l time i s chosen as the base of computation. Other- wise a l l time constants and H must be mult i p l i e d by c o ^ . Now since At = A ( i K i - i> i . ) (3.25) e d q q d and At = g + 1.5h (3.26) m b for a hydraulic system, substituting i ^ and i from (3.11) into (3.25) and the results into (3.24), we have p A t o e = A ( 9 , l ) A i p F + A(9,2 ) A ^ d + A(9,3)AiJ) + A(9 ,4) A ^ D 2 2 to D to t to t + A < 9 ' 5 > % " 25" V U e + ."^21 8 + ! " ¥ h ( 3 ' 2 7 ) where 2 to A(9,l) - [*q] Y d F 2 A(9,2) = ( [ I q ] + [^ q]Y d d) 2 to A(9,3) = I § ( [ ^ ] Y q q + [ l d ] ) to2 (3.27a) A(9,4) - - 2 J H * q ] Y d D 2 to A(9,5) = 2 i [^ d]Y q Q 28 (3.27b) The complete state v a r i a b l e sets are / <ipF»^d» iq> iP-Qy i>q> V F » V R » 5> <*>e> a> a f> g» h ) (3.28) in c l u d i n g governor actuator s i g n a l a and feedback a^ as i n F i g . 3-3. 3.6. Governor-hydraulic System F i g . 3-3 shows the block diagram of a t y p i c a l governor- hydraulic system I 1+Trs I F i g . 3-3 A Ty p i c a l Governor-Hydraulic System The corresponding state equations are pa T 3 ~ T a f 1_ C a 1 . 1 — = ~ Ato - = u co 1 e T G o a a PS, = - Pg = T a T f co T a r a o a 1 Acoe - — u G a (3.29) 3.7. State Equations There are altogether 13 sets of state variables, (3/28). Ea-ch set i s an m-vector for an m-machine system. Equations' (3.17a), (' (3.17b), (3.18a), (3.21), (3.22), (3.27) .and (3.29) are the complete sets of the system state equations. They are assembled into a matrix equation form as Y = A Y + Bu, Y = A(«,F ^ ^ q ijijj ^ u = ( u £ u a ) , R (3.30) <5 to a a„ g h), (3.30a) (3.30b) B = K 0 0 0 0 0 0 "A 0 0 0 0 0 0 O ' O O O O O O 0 0 z i Hi. T a T a 0 0 (3.30c) and A i s given as equation (3.30d) including (3.21a) and (3.27a) as the a u x i l i a r y equations. It i s obvious that any other type of exciter and governor systems can be easily incorporated with the rest of the state equations. 3.8. Multi-machine System with an I n f i n i t e Bus For the study of m machines with an i n f i n i t e bus, the. matrix equations (3.4) can be partitioned as V m = " z mm Z m°° i m V CO Z . °°m Z oooo i oo (3.31) Linearization of (3.31) can be written as ' rF YFF " r F Y F d 0 " rF YFD 0 •RY._, -RY,, w +X' Y -RY.,. X I . 0 dF dd e m qq dD m qQ 'XmYdF " We" Xm Ydd - R l -X Y.„ -RY. qq m dD qQ •r Y - r Y D DF D Dd ~ rD YDD 0 -r Y • Q Qq 0 0 -VQQ o o - I / T E I / T E A(7,l) A(7,2) A(7,3) A(7,4) "A(7,5) 0 0 0 . 0 0 0 0 A(9,l) A(9,2) A(9,3) A(9,4) A(9,5) 0 -1/T, mq > q ] U md 0 A(7,8) 2H" 2. "0*0 2H 4H. . l / U o T a -,/Ta -l/T -1 * t t' 0 a v t a r a l/T -2/T -l/T 2/T -2/T. (330d) "Av " m "z mm z m°° "Ai " m + i Av 00 Z . °°m Z COCO A i Z Z mm m°° Z Z cofrl cooo I 0 m V 0 m 0 v / 0 1 . / A<5 m AS (3 Note that I and V are diagonal matrices with i and v as- diagonal m m m m ° elements r e s p e c t i v e l y . Since f o r an i n f i n i t e bus we have Av = 0 00 A6 = 0 oo Substituting (3.33) in t o (3.32) and eliminating A i ^ r e s u l t s i n Av. = Z A i + j U A<5 m m m m m where U = Z I - V , m mm m and Z = Z - Z Z /Z m mm m<x> °°m °°°° (3.33a) (3.33b) (3.14a) (3.34) The l i n e a r i z e d state equations of the multi-machine system with and without an i n f i n i t e bus have exactly the same form. But we have to eliminate the i n f i n i t e bus when the network impedance matrix i s expressed i n machine's dq Coordinates, (3.34). '3.9. S i m p l i f i c a t i o n of Power System Dynamics For System analysis and design purposes i t i s usually d e s i r a b l e to simplfy the dynamic d e s c r i p t i o n of the system. Numerical approaches 25 26 of approximating high order systems by low order systems are a v a i l a b l e ' The p r i n c i p l e involved i s to r e t a i n only the dominant eigenvalues of the exact system i n the reduced model. The i n d i v i d u a l system parametric values, however, are completely l o s t during the process of numerical approximation. The s i m p l i f i c a t i o n of power system dynamics i s diff e r e n t i n nature. It i s governed mainly by the degree of accuracy of describing the flux linkage variations of the synchronous machine windings. Three diff e r e n t approximations are suggested A: complete description for the system, 7th order syn- chronous machine, f i r s t order voltage regulator and 4th order governor. B: neglecting damper winding flux linkage v a r i a t i o n s , i . e . P^ D = P * p = 0 C: neglecting damper and armature flux linkage variations P^D = P^Q = 0, and 'P*d = P*q = °» C': The same s i m p l i f i c a t i o n as i n model C, except that the system has no governor representation. The s i m p l i f i c a t i o n can be easily implemented on the high . order system equations (3.30) using matrix elimination technique. The lin e a r i z e d state form equations of a multi-machine power system with 5th order synchronous machine, model B, with second order voltage regulator and exciter system are given i n appendix A. From the numerical example of a t y p i c a l one machine i n f i n i t e system, Fig. 5-1, i t i s found that the dominant eigenvalues d i f f e r very l i t t l e from each other i n the diff e r e n t s i m p l i f i c a t i o n methods. Table 3-1 shows the eigenvalues of the t y p i c a l one machine i n f i n i t e system of diff e r e n t modelling. Although there are dynamic couplings among a l l system state variables, roughly, the model Eigenvalues #1 #2 #3 u A .165+J4.69 -15.2,-3.99 -14.8,-2.24, -1.15,-.034 -847+J3151,-26.1,-12.4 B .229±j4.67 -16.9,-3.76 -15.1,-2.23, -1.15,-.034 -486+11857 / C .234+J4.67 -16.9,-3.77 -15.1,-2.23, -1.15,-.034 C* .178+J4.77 -16.9,-3.68 Table 3-1 Eigenvalues of the Typical One Machine I n f i n i t e System of Different Modelling 4 column eigenvalues correspond to the mechanical system, the voltage regulator and excitation system, the governor system, and the synchronous machine armature and damper windings respectively. Here Column #1. gives the dominant eigenvalues. - 4. OPTIMAL LINEAR REGULATOR DESIGN WITH DOMINANT EIGENVALUE SHIFT 2 7 Optimal l i n e a r regulators have been designed for power 11 12 28 systemstabilization ' and for frequency control . . The performance function J must be chosen i n the quadratic form, J = I • /°°(Y'QY + u'Ru)dt (4.1) I o The choice of the weighting matrix Q of (4.1) i s e n t i r e l y l e f t to experience and guessing u n t i l satisfactory results are obtained. In t h i s chapter a new method i s developed to determine Q in'conjunction with the dominant eigenvalue s h i f t of the closed loop system as far as the p r a c t i c a l controllers permit. For the eigen- value s h i f t of an n-th order system, i t i s found that i t i s s u f f i c i e n t to adjust the n diagonal elements of the Q matrix alone without the need of changing the off-diagonal elements. This also leaves out the change i n R elements which decide the r e l a t i v e strength of the different control signals and can be l e f t e n t i r e l y to economical and p r a c t i c a l considerations. 4.1. Linear Optimal Regulator Problem The l i n e a r optimal regulator problem may be formulated as follows. Consider the l i n e a r i z e d system state equations Y* = AY + Bu . (4.2) Find the optimal control which minimizes the chosen quadratic performance function of (4.1) subject to the system dynamics constraint (4.2). The 29 optimal control i s given by 35 u = -R~1B' K Y (4.3) and the R i c c a t i matrix K s a t i s f i e s the nonlinear matrix algebraic equation KA + A'K - K B R 1B'K = -Q (4.4) With u decided,the closed loop system equations become Y = GY (4.5) where G = A - BR""1B'K (4.6) Thus the eigenvalues of the closed loop system G depend upon the s e l e c t i o n of Q for J i n (4.1). Consequently the designed optimal co n t r o l l e r i s not necessarily the best since Q i s a r b i t r a r i l y chosen. On the other hand i f Q i s adjus ted.constantly and simultaneously with the dominant eigenvalue s h i f t of the closed loop system, the results w i l l be. the best. 4.2. Eigenvalue Shift Policy The s h i f t i s r e s t r i c t e d to the r e a l part and to the l e f t . Let a l l the eigenvalues of G be ordered as a vector always with the eigenvalue with the largest r e a l part as the f i r s t ' element, A^, and the rest i n decreasing order of magnitude. A three-point s h i f t policy i s established to avoid unnecessary and undesired large change in Aq which may result i n impractical controller gains 1. Assign a negative r e a l s h i f t e to the most dominant eigenvalue A only. 2. Keep a l l negative movements of less dominant eigenvalues, e.g., those having negative r e a l parts up to f i v e or ten 36 times that of X^, within e and damp out a l l p o s i t i v e movements to the right to avoid the i r to and fro motion. 3. Relax the movements of the remaining eigenvalues to avoid unusually large co n t r o l l e r gains. 4.3. The Shift Let the incremental change i n an eigenvalue X^ res u l t i n g from the change i n the diagonal elements of the weighting matrix Q, written as a vector q, be AX. = X! Aq (4.7) l x,q Z1 since for a conjugate eigenvalue pair A. - X* + 1 (4.8) the i r s e n s i t i v i t y c o e f f i c i e n t s are also conjugate X. = X* (4.9) x,q l+l,q Therefore the increments AX. = AX* (4.10) I i + l There are, i n general, k re a l eigenvalues and (n-k)/2 conjugate eigenvalue pairs of the n-th order closed loop system G, and only (n+k)/2 independent eigenvalues need to be considered i n the s h i f t i n g process. Let the number be p. Let the p-eigenvaliie vector s h i f t be AX =X,q Aq (4.11) and l e t them be separated into real and imaginary parts AX = AC + j An (4.12) Then the rea l part-of AX may be Written .37 • A £ - S • Aq (4.13) where • ' S ^ Real ( A , q ) (4.,14) 4.4. Determination of Aq Let the number of dominant eigenvalues be m. Since A^ cannot be s h i f t e d alone, l e t a weighted t o t a l r e a l s h i f t of the m dominant eigenvalues be z = e ^ e d ) + e 2Ac;(2) + ... + emAs(m) (4.15) From (4.13) We have E - 4> * Aq (4.16) where and <|) = (^,...,<!..,...,tj)^' (4.16a) <j>± = $ 1 S ( l , i ) + B 2 S ( 2 , i ) + ...+e mS(m,i) (4.16b) The 3's are p o s i t i v e numbers s a t i s f y i n g the s h i f t p o l i c y p o i n t two. To make E negat i v e , Aq i s moved i n the d i r e c t i o n of the steepest descent, Aq « -k<f> , k > 0 , . (4.17) The step s i z e k i s so determined that i t w i l l have a negative s h i f t f o r the most dominant eigenvalue A^. 4.5. S e n s i t i v i t y C o e f f i c i e n t s A,q 30 Although Chen and Shen gave two algorithms to compute A,q t h e i r method r e q u i r e s many computations and l a r g e computer storage. A new s e n s i t i v i t y formula for A,q i s developed i n th i s section. The computation of A,q and the solution of the R i c c a t i matrix K w i l l be much s i m p l i f i e d through an eigenvector matrix X of a composite matrix M; M = A -Q -BR~1B' -A' (4.18) 31 32 The composite matrix M has the following properties ' 1. The 2n eigenvalues of M are symmetrically located with respect to both r e a l and imaginary axes of the complex plane. Let the eigenvalue vector A of M be partitioned as A = [ A r A I I ] ' (4.19) where A has negative r e a l parts and A ^ has po s i t i v e r e a l parts. Then we have A I I = " A I (4.20) 2. The eigenvalues with the negative r e a l parts of M are the same eigenvalues of the optimal closed loop system G, i. e . A T = (X.,...^.,...,* )' (4.21) i i l n 3. The solution of the R i c c a t i matrix equation (4.4) i s -1 K - X I IX I where X' X, I I I X I I X IV (4.22) (4.22a) i s the eigenvector matrix of M, and the f i r s t column of the eigenvector matrix X corresponds to the stable eigenvalues A . 39 4. The eigenvector matrix of M' may be Written V = X I V X I I ~ X I I I ~ X I (4.23) Let an eigenvector of the stable eigenvalue X_̂  of M be X. - ( X l. , x m ) ' and that of M' be (4.24). V i = ( X I V i . • - X I I I i ) ' 33 Following Faddeev and Faddeeva , we have (4.25) AX. = ~ V! AM X. l . C. I I l (4.26) where C = V!X. l i i (4.26a) Since i n our case We s h a l l have A M = '•-AQ l] A X i = C T X i l l l A Q X I i I For the diagonal changes i n Q We write where and AX. = X! . Aq l i,q i,q i , q l ' i,q2' i , q j i,qn (4.27) (4.28) (4.29) (4.30) X i , q j - = cT XIIIi.«> X I i ^ (4.31) where n C i = * [ X l V i ^ ) X I i ( J ) " X I I I i ^ ) X T T . i ( J ) ] 3=1 (4 4.6. Algorithm The algorithm f o r the design of l i n e a r optimal regulators with dominant eigenvalue s h i f t i s summarized i n F i g . 4-1. o AO A AND X OF M CHECK CONTROLLER CAINS Aq F i g . 4-1 Algorithm to Determine Q with Dominant Eigenvalue S h i f t 1. Start with a small a r b i t r a r y Q. 2. Find the eigenvalues A and eigenvectors X of the composite matrix M. 32 3. Calculate K from the stable eigenvectors of X and check the c o n t r o l l e r gains at each s h i f t . 44. Find Aq from the s e n s i t i v i t y c o e f f i c i e i n t s A,q. 5. Update Q and repeat the process u n t i l a s a t i s f a c t o r y eigen- value s h i f t i s made or u n t i l the p r a c t i c a l c o n t r o l l e r ' s l i m i t i s reached. 5. OPTIMAL POWER SYSTEM STABILIZATION THROUGH EXCITATION AND/OR GOVERNOR CONTROL27 In t h i s chapter the l i n e a r optimal regulator design technique developed i n the previous chapter i s applied to the optimal s t a b i l i z a t i o n of a t y p i c a l one machine-infinite system, F i g . 5-1. Three d i f f e r e n t optimal s t a b i l i z a t i o n schemes are investigated, the f i r s t with an optimal e x c i t a t i o n c o n t r o l û ,, the second with optimal governor controls U-, and u' , with and without the dash-pot, and the t h i r d with u plus u^ c o n t r o l . The l i n e a r optimal s t a b i l i z i n g signals thus obtained are tested on a high order nonlinear model of the system with d e t a i l e d d e s c r i p t i o n . It i s found from the'test r e s u l t s that the optimal controls are more e f f e c t i v e than conventional e x c i t a t i o n c o n t r o l , that the optimal governor control with the dash-pot removed i s j u s t as good as the optimal e x c i t a t i o n c o n t r o l , and that the Optimal u„ plus u' cont r o l i s the best E G way to s t a b i l i z e a power system. 5.1.' System Data A t y p i c a l one machine-infinite system as shown i n F i g . 5-1 i s chosen for t h i s study. The re g u l a t o r - e x c i t e r and governor-hydraulic systems are shown i n Pig. 3--2 and Pig. 3-3 re s p e c t i v e l y . / Fig. 5-1 A Typical One-Machine I n f i n i t e System The system data are as follows r X G B V 0 V t P 0 H . D -.034 .997 .249 .262 1.02 1.05 .952 4.63 0 *d x' x d x" x d X q x" q rpll do rj-. 11 qo TD .973 .190 .133 .55 .216 .0436 .0939 .13 KA TA TE h a *t T a T r T g 50 .05 .003 .182 .05 .25 .02 4.8 .50 controller constraints are, exciter amplifier l i m i t s (p.u.) 8.83 and -7, dash-pot signal l i m i t s + .025 p.u.,- governor gate speed l i m i t .1 p.u./sec, excitation control l i m i t s + .12 p.u. and governor control l i m i t s + .0.15 p.u. For the design the synchronous machine i s described as a t h i r d order system with p» <S, to)' as the state variable vector. Thi i s done by neglecting the f l u x linkage variations i n the armature and damper windings. 5.2. Case 1: u 'Control E The system has an optimal e x c i t a t i o n c o n t r o l u . The time constant T of a s o l i d state e x c i t e r i s neglected and the voltage E regulator of F i g . 3-2 i s approximated as a f i r s t order system. For the data given, the per unit l i n e a r state equations for the complete system are where • = -.196 1.0 -1.39 -.003 + "o' -50.9 -20. 87.0 -2.4 * V F 1 A 6 0 0 0 1 *6 0 • . Aco -2.94 0 -22.6 -.008 a 0). 0 u = 1000 u^ ex E (5 (5 The optimal control s i g n a l u i s found as E (-.099Ai|;„ - .004Av„ - .62A6 + O.lAu) The f i n a l value of the diagonal elements of Q are (2524 , 0 , 913.6 , 23865) The co n t r o l T^eighting R i s unity. The eigenvalues of the i n i t i a l svstem without u„ control are E (.178 + J4.77 , -3.68 , -16.9) and the eigenvalues of the f i n a l system with u„ c o n t r o l are (-2.07 + J4.9 , -3.85 , -16.7)' Thus the dominant eigenvalue p a i r s are s h i f t e d from (.178+J4.77) to (-2.07 + j4.9) / 5.3. Case 2a: Control, With Dash-pot ^ / The system has an optimal governor control u . The 4-th order governor hydraulic system i s as Fig! 3-3 and the voltage regulator i s approximated as a f i r s t order system. For the data given the per unit l i n e a r system state equations for the complete system are Y = AY + Bu where A = Y = [ AUJ F' A v F* A<5, AGO, a, a f , g, h]' u = u G B = [0, 0, 0, o, -50, -12.5, 0 > 0]' -.196 1.0 -1. 39 -.003 0 0 0 . '0 -50.9 -20 87. 0 -2.4 0 0 0 0 0 0 0 1 0 0 0 0 -2.94 0 -22. 6 -.008 0 0 38.8 58.2 0 0 0 -.133 -2.5. -50 0 0 0 0 0 -.033 -.625 -12.7 0 0 • 0 0 0 0 2 0 -2 0 0 0 0 0 -4 0 4 -12.5 (5.2) (5.2a) (5.2b) (5.2c) (5.2d) and R i s set R (5.2e) The optimal control signal u i s found as 45 (.0255A4»F + .0012Av + .126A6 - .0254Aw + .08a - .112 a f - . 3g - .4h) The f i n a l values of the diagonal elements of Q are (.56, 4.8 .116, 6.8, l O " 4 .034, .0019, .52 , 0 ) The eigenvalues of the i n i t i a l system without u control are (.23 + J4.67, -3.77, -16.9, -.034, -1.149, -2.23, -15) and the eigenvalues of the f i n a l system with u„ .control are (-1.35 + j4.9, -4.1, -16.8, -.049, -1.2, -1.6, -15) Thus the most dominant eigenvalue pairs are shifted from (.23+J4.67). to (-1.35 + j4.9) The eigenvalue -0.034, corresponding to a large time constant of the dashpot, has slow response to system disturbance and does not affect the e a r l i e r part of system s t a b i l i t y . 5.4. Case 2b: u l control, without dashpot The dashpot i s removed from Fig. 3-3 for this study. Neglecting the actuator time constant T the governor transfer function can be a ° written as 1/(a + T's) where T' = aT . For the data given the equations g g g for the complete system are 46 = F A<S • Aw g n -.196 -50.9 0 •2.94 0 0 1.0 -20 0 0 0 0 -1.39 87 0 -22.6 0 0 -.003 -2.4 1 -.008 -.1 .21 0 0 0 38.1 -2 0 0 0 % + 0 Av F 0 A<5 0 Aw 0 g -40 h 80 (5.3) and R i s set R = 1 (5.3a) The optimal control signal u' i s found as (.00628Aib + .0002AV_-+ .0238A6 - .01620Aw r r -.0216g - .1 h) The f i n a l values of the diagonal elements of Q are 10~ 4(0, 0, .0063, 1.83, 31.1, 0) The eigenvalues of the i n i t i a l system without u' control are (.715 + J4.35, -4.3, -16.8, -.89, -2.8) and the eigenvalues of the f i n a l system with u' control are (-3.7 + j4.9, -3.27, -16.8, -1.18, -2.12) Thus the dominant eigenvalue pairs are shifted from (.715 + J4.35) to (-3.7 + j4.9) 5.5. Case 3: u^ Plus M.\ Control The system under study i s the same as that of case 2b except i t has both u„ and u' control signals. The l a s t term of the system equations becomes [° 1 0 0 0 0 0 0 0 ]' [ UE] (5.4) 0 0 0 0 -40 80 where R = 1 0 3 [ J ; 6 Q ] (5.4a) and i s chosen to coordinate the e f f o r t of e x c i t a t i o n and governor c o n t r o l s i g n a l s . The optimal u„ and u' co n t r o l signals are found r e s p e c t i v e l y as (-.047AiJv " -002AV-, - .319A6 + 0.05Aw + . 779g + .78h), r r (.005Aif^ +:0002Av_, + .025A6 - .0127ALO - .045g - .094h) r r The f i n a l values of the diagonal elements of Q are (1.42, 0, .0859, 25.8, 82.28, .025) The eigenvalues of the i n i t i a l system without u„ and u' co n t r o l are the same as those of case 2b and the eigenvalues of the f i n a l system with û , and u' co n t r o l are (-4.13 + J5.33, -3.6, -16.79, -.997, -1.66) Thus the dominant eigenvalue p a i r s are s h i f t e d from (.715 + j4.35) to (-4.13 + J5.33) 5.6. Nonlinear Tests A l l the optimal s t a b i l i z i n g s i g n a l s thus obtained are tested on the same system of F i g . 5-1 but described by high order nonlinear d i f f e r e n t i a l equations with the synchronous machine as a 7-th order system (ij ^ , ij> , ifjp, ijjp, !pQ> 6, to), e x c i t a t i o n and governor systems r e s p e c t i v e l y as Figs. 3-2 and 3-3 with c o n t r o l l e r c o n s t r a i n t s . A conventional e x c i t a t i o n control as designed i n reference 12 .04s Aw (5.5) EC l+.5s using the speed deviation signal i s also included for comparison. The system disturbance for the tests i s as follows: a three- phase fau l t occurs at one of the system buses and the faulted l i n e i s isolated at 5 cycles followed by a system restoration at 30 cycles. The results are summarized i n Fig. 5-2. The system responses for the system with conventional and the optimal excitation controls are displayed on the l e f t column of the figures, and the system responses for the system with the optimal governor, and the optimal governor and exc i t a t i o n controls are displayed on the right column of the figures. • From the r e s u l t s , i t i s observed that: 1. Although the e f f o r t of the optimal excitation control signal u„ i s smaller than that of the conventional excitation signal .hi u„^, the system with u„ control i s much more stable. 2. The optimal governor control signal u' for the governor with- er out dashpot provides more damping for the sytem than that with a dashpot. 3. The optimal excitation and governor signals, u and u', when coordinated, provide the best means for s t a b i l i z i n g a power system, i . e . , more damping with less e f f o r t than either u^ or u' control. In other words, for the same amount of e f f o r t , the optimal u plus u' control has the a b i l i t y to s t a b i l i z e the system under more severe fa u l t conditions. 4< 0.003 0 . 5 0 . 5 1 .0 1 .5 2 . 0 T I M E (SECONDS) (e) 2 . 5 1.0 1.5 2 . 0 TIME (SECONDS) 2 0 . 0 N 15.0 1 0 . 0 4 9 5 0 CE -3 0 jj - 5 . 0 - 1 0 . 0 - 1 5 . 0 I I , 1 0 ( . 5 2 . 0 TIME (SECONDS) 0.010 0.020 £0.000 S O . 020 o •> j0.0<10 • a. So.060 - -0.060 - \ -a 3 . 0 1.0 1.5 TIME (SECONDS) (ll 0 . 5 1.0 1:5 2 . 0 TIME (SECONDS) (gi 0 . 5 1.0 1.5 2 . 0 TIME (SECONDS) (l>) Continued. 1.5 2.0 TIME tSECONDS) (il 1.5 2.0 TIME (SECONDS) 1.0 1.5 2.0 TIME (SECONDS) (i) SO.005 1.0 1.5 2.0 TIME (SECONDS) If I Fig. 5.2 Nonlinear Test Results u = 0, unstabilized system u „ n 5 conventional excitation control Ug , optimal excitation control u , optimal governor control with dashpot u' , optimal governor control without dashpot u plus u' control 6. OPTIMAL STABILIZATION OF A MULTI-MACHINE SYSTEM' The optimal line a r regulator design technique of determining the weighting matrix Q i n conjunction with the dominant eigenvalue s h i f t developed i n chapter 4, i s applied to the optimal s t a b i l i z a t i o n of a multi-machine system. Two systems are investigated, the f i r s t with a one machine optimal c o n t r o l l e r , u ,- and the second with a multi-machine h i optimal c o n t r o l l e r s , u ^ . Each design i s given a nonlinear test on the same multi-machine system. I t i s found that the multi-machine system with a one machine optimal controller u ^ , designed for the multi machine system i s better than a one machine optimal c o n t r o l l e r , u , designed for the same system but approximated as a one machine-infinite system, and that the multi-machine system with a multi-machine optimal c o n t r o l l e r , u , i s better s t i l l than the multi-machine system with the one machine optimal c o n t r o l l e r , u j , designed for multi-machine system. 6.1. System Data and Description The system under study, Fig. 6-1, i s the same as that of reference 12, consisting of one thermo plant (#1), two hydro plants (#2 and #3), and an i n f i n i t e system equivalent (#4). Fig. 6-1 Typical Four-Machine Power System (Admittances i n p.u. on 1000 MVA) The system data are as follox^s Plant r X T H D a d d d q q do do qo D #1 .0019 1.53 .29 .17 1.51 .17 4 .029 .029 .116 2.31 0 #2 .0023 .88 .33 .22 .53 .29 8 .022 .044 .077 . 3.4 0 #3 .0025 .97 .19 .13 .55 .216 7.76 .044 .094 .131 4.63 0 K, T. T R V V A A E F Rtnax Rrnin 13 .21 .15 .129 4.5 0 45 .07 .5 .237 3.5 -3.5 50 .02 .003 .12 8.8 -7 The operating conditions from load flow studies are Plant P (MW) o Q (MVA) 0 V t o(p.u.) 6(deg.) / / l 26.5 37 1.04 -10.7 #2 518 -31 1.025 11.8 n 1582 -49.6 1.03 25 #4 410 49.3 1.06 0 For the design each plant i s modelled as a fourth-order system (i/jp, Vp, 6, tii), a third-order synchronous machine plus a f i r s t - o r d e r exciter-regulator system. The l i n e a r i z e d system equations are written as • Y l = " A l l A12 A13 \ " + B ' U l " • Y2 A21 A22 A23 Y2 u2 Y 3 . . A31 A32 A 3 3 . Y 3 - U 3 . (6. For the data given the numerical values of the A and B matrices are 11 13 = -.922 1 -.266 -.009 -2.75 -2.78 -1.36 -.037 0 0 0 1 -4.95 0 -55.5 -.039 .072 0 -.25 .003 .46 0 2.8 -.02 0 0 0 0 .924 0 17.5 .02 12 21 .024 0 -.087 .158 0 1.11 0 0 0 .222 0 8.17 .021 0 .121 003 -1.1 0 -1.62 015 0 0 0 0 -2.43 0 1.37 034 0 004 22 -.21 -1.9 0 •3.1 1 -1. 0 0 -1.6 9.3 0 -56 -.005 -.12 1 .032 A 23 .06 0 .46 .002 -1 0 1.49 -.04 0 0 0 0 .12 0 29.8 -.028 31 33 -.002 0 .083 0 .011 0 .22 0 -6.78 0 -10.1 -.09 A32 = -2.1 0 1.7 -.123 0 0 0 0 0 0 0 0 -1.24 0 .498 -.017 .-,07 0 6.37 -.011 -.197 1 -1. 2 003~ -54.4 -20 70.1 -2.37 0 0 0 1 -3.4 -21 -.017 0 36.1 0 0 0 0 0 0 0 0 0 78.9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1000 0 0 The eigenvalues of the unstabilized multi-machine system are -.013 + J7.8 -1.85 + j l . 3 5 (#1) -.018 + j7.4 -1 + J1.3 (#2) + .177 -3.84 , + J3.98 -16.6 (#3) Although there are dynamic couplings among a l l three plants, roughly, the three column eigenvalues correspond to three plants respectively. Also the the f i r s t row eigenvalues of each column correspond to the plant dynamics, 6 and to. 6.2. Case 1: One Machine Optimal Excitation Control u EI Since i t i s found from the eigenvalue analysis of the unstabilized multi-machine system that plant //3 i s unstable, a one-machine optimal excitation control, u„ T, i s designed for plant #3 i n order to s t a b i l i z e the multi-machine system. In the design, of course, a l l system dynamics are included. The diagonal elements of the weighting matrix, Q, determined -3 from the dominant eigenvalue s h i f t are the l i s t e d Values times' 10 , for R=l. A V F AS Aco plant #1 .011 .018 .348 19.6 plant #2 .023 .536 .284 18.3 plant #3 0 0 .022 .523 u„ T for plant #3 are A^p Av p A6 Aco plant #1 .0172 -.0128 .88 -.04 plant #2 -.0345 -.0109 -.28 -.14 plant #3 -.154 -.0066 -.878 .18 The eigenvalues of the f i n a l multi-machine system are -1.17 .+ J7.86 -1.77 + j l . 3 6 (#1) -.3 + J7.86 -1.02 + j l . 2 5 (#2) -1.88 + J3.55 -3.6 , -16.6 (#3) Thus the f i r s t txro eigenvalues of the l a s t column are shifted from +.177 + J3.98 to -1.88 + j 3.55, indicating great improvement i n damping of plant #3. The control s i g n a l , u„ T, also improves the damping of plants #1 and #2. 6.3. Case 2: Multi-Optimal Controllers u„„ One would expect that a multi-machine system with multi-optimal controllers w i l l be better s t a b i l i z e d than the system with only one optimal c o n t r o l l e r . This i s studied as case 2. The multi-optimal controllers are designed, of course, simultaneously considering a l l machine dynamics. 56 The diagonal elements of 0 determined from the dominant eigenvalue -3 s h i f t are the l i s t e d values times 10 , A<S AGO plant #1 .145 .001 2.64 97.2 plant #2 4.65 3.36 3.11 93.6 plant #3 .1 .0007 4.02 88.2 The weighting matrix elements of three plant controls which give the best results are R = diag (1, 2, 10) of the three control signals are: UEM(#1) Av p A<5 ACQ plant #1 -1.06 -.029 -.639 •S.18 plant #2 -.052 -.0039 -.588 .0313 plant #3 -.073 -.0026 .127 .137 UEM(#2) A ^ F Av F A<S Ato plant #1 .0084 -.00427 .539 .0218 plant #2 -.069 -.0399 -.826 -.132 plant #3 -.0406 -.00146 -.0097 -.10 EM(#3) AIJJF Av p A 6 AGO plant #1 -.00569 -.0072 .38 -.0225 plant #2 -.00832 -.00369 .0718 -.0516 plant #3 -.1123 -.00497 -.718 .1156 The eigenvalues of the f i n a l multi-machine system are -1.01 + J7.64 -1.94 + jl.099 -.448 + J7.89 -1.7 , -2.74 -2.05 + J4.04 -3.03 , -16.65 (#1) (#2) (#3) There i s no doubt that a multi-machine system with multi-optimal c o n t r o l l e r s , u ^ , i s better than the system with a one plant optimal c o n t r o l l e r , . 6.4. Case 3: Approximated One-Machine Optimal Design For comparison, the u optimal excitation control signal of the same power system as cases 1 and 2, but approximated as a one-machine i n f i n i t e system as i n Chapter 5, i s recorded here... The control signal u„ = - . 0 9 9 A i J ; - . 0 0 4 A v T , - . 6 2 0 A 6 + . lAw E F F was designed for plant #3 as the one-machine and i n f i n i t e system. When thi s signal i s applied to plant //3 of the multi-machine system the eigenvalue are .084 + J7.46 -.1 + j7. -1.46 + j l . 1 5 -.63 + j l . 5 1 -2.88 , -15.9 -3.3 + j4.5 (#1) (//2) (#3) When these results are compared with the eigenvalues of the unstabilized multi-machine system, i t i s found that the u control signal does improve the damping of plant //3, but not much of plant #1 or #2. 58 6.5. Case 4: Subsystems Optimal Design One would be curious to know what would happen i f a l l plants had ind i v i d u a l u„ control-designs. This i s to say .that a l l the dynamic b coupling of the three plants, off-diagonal elements of the A matrix i n (6-1), w i l l be neglected and the in d i v i d u a l optimal controllers are designed from -' — - - - Y l ~ A11 Y1 + b l U l Y2 " A22 Y2 + b 2 U 2 (6.2) Y 3 = A33 Y3 + b 3 U 3 respectively. Applying the dominant eigenvalue s h i f t technique, the -3 ind i v i d u a l weighting 0 matrices are the l i s t e d values times 10 ; for R = 1 i n each design, A * F A6 Aco plant #1 0 0 2.296 72.88 plant #2 -4 .7x10 .8xl0~ 2 2.219 69.38 plant #3 .19 0 .549 12.6 The gains of the in d i v i d u a l optimal controllers are A i J J F A<5 Aco u, (#1) -.0738 -.0231 -1.059 -.2018 u E (#2) -.0446 -.0182 -.838 -.228 uE(//3) -.075 -.0035 -.455 .071 The eigenvalues of the i n d i v i d u a l closed loop systems are (#1) (#2) (#3) -.58 + J7.52 -1.1 + j l . 1 6 -.418 + J7.48 -1.868 + j l . 2 5 -1.9 + J4.766 -3.23, -16.6 Next the eigenvalues of the multi-machine system are: With u„(//l) alone -.36 + J7.42 -1.73 + j l +.014 + J7.84 -.93 + j l . 4 -.11 + J3.96 -3.86, -16.62 With u^ (#2) alone -.04 + J7.45 -1.89 + j l . 2 9 -.44 + j7.8 -.956 + j l . 1 5 -.097 + j4 -3.84, -16.62 With u„ (#3) alone hi -.064 + J7.46 -1.47 + 11.25 -.079 + J7.83 -.778 + j l . 5 9 -2.33 + J4.08 -3.24, -16.71 With a l l three u E's -.419 + J7.58 -1.55 + jl.136 -.463 + J7.843 +.169 , -2.25 -2.53 + 74.47 -2.95, -16.7 Although the i n d i v i d u a l optimal c o n t r o l l e r provides good damping to the i n d i v i d u a l plant, the e f f e c t s on other plants are unpredictable. 6.6. Nonlinear Tests The optimal s t a b i l i z i n g signals thus obtained are tested on the same system of Fig. 6-1 but described by high order nonlinear d i f f e r e n t i a l equations including the controller's constraints. The system disturbance for the tests i s the same one used i n the previous chapter. The test results are summarized i n Fig. 6-2. From the results the following i s observed 1. - u„ c o n t r o l l e r , designed for the system approximated as one- E machine i n f i n i t e system, case #3, provides the required damping to plant #3 but not much to other plants. 2. û -j. c o n t r o l l e r , case #1, provides damping to each plant i n the system, allowing the co n t r o l l e r to s t a b i l i z e the system for wider f a u l t locations than the case with u„. E 3. u„,, c o n t r o l l e r s , case #2, provide the best s t a b i l i z a t i o n for EM . the whole system with less e f f o r t than the case with u„ or u„ T. 4. The s i m p l i f i e d subsystems controllers f a i l to s t a b i l i z e the system. TERMINAL VOLTRGE (PUJ • I,, DELTA IDEG) SPEED DEVIATION (PU) t o 1/1 ; AND 3 1.3 :.S 2.0 TIME ISECON0S) 3.0 0.C8 0.04 -0.00 -0.04 - -o.os - -0.12 -0.16 ; AND J 'A // ^ / \ 0.5 1.0 1.5 2.0 TIME (SECONDS) ! z.s 3.0 4.0 0.5 1.0 l.S 2.0 Tlt£ '.CECCNDSl i 0.5 — i 1 1— 1.0 l.S 2.0 TIKE (SECONDS) J e 2 ) 2.5 3. 0.5 1.0 1.5 2.0 TIME '.SECONDS J (d3) 2.5 3.0 :.0 1.5 2.0 T I M E : 'isEcoros: fe3) 2.S 2 (plant #1) (plant #2) (plant #3) Fig. 6.2 Nonlinear Tests of the Multimachine System (a three-phase fau l t disturbance) 2. with multi-optimal controllers u £ ^ 4. with three individual^optimal controllers 1. with one optimal control u^j on plant #3 3. with.u p on plant #3; approximated one machine-infinite system 7. OPTIMUM STABILIZATION OF POWER SYSTEMS / OVER WIDE RANGE OPERATING CONDITIONS34 / I Nominal system operating conditions were assumed i n chapters 5 and 6 for the design of the optimal s t a b i l i z i n g signals. In r e a l power systems the operating conditions are not constant but subject to the load demands over the system. The question arises: How can we design an optimal controller for the power system sensitive to and good for the wide range of operating condtions? In an attempt to answer this question, an optimally sensitive co n t r o l l e r i s developed i n this chapter. The con t r o l l e r i s capable of adjusting i t s e f f o r t i n such a manner that optimum system s t a b i l i z a t i o n can always be achieved over the wide range operating conditions.. The sensitive c o ntroller thus designed under such conditions i s tested on the nonlinear model. The results are compared with that of the system with a nominal controller.. 7.1. Optimally Sensitive Linear Regulator Design Constructing a co n t r o l l e r x./hich preserves optimality for a nonlinear control system i n spite of i t s parameter variations has been 3 *5 36 37 the object of several recent publications '" ' . The synthesis of lin e a r optimally sensitive controllers by means of perturbation of the R i c c a t i equation (A.4) i s dealt with i n this chapter. Let the l i n e a r i z e d system equations be 6. Y = A(q) Y + Bu (7.1) where q i s a vector contains the m changable parameters of the/system. / / For a quadratic performance function / 1 00 J = — / (Y'QY + u'Ru)dt, (4.1) the optimal control law i s u* = -R - 1 B' K(q) Y (7.2) where K(q) s a t i s f i e s the R i c c a t i matrix equation, K(q)A(q) + A'(q)K(q) - K(q)BR _ 1B*K(q) = -Q (7.3) In conventional regulator design the controller i s computed for nominal values of the plant parameters q u = -R _ 1B'K(q )Y , (7.4) o o for a constant K(q^) • This w i l l be referred to as the nominal optimal control hereafter. But this becomes impractical for system over wide range operating condition^.. It .implies that i t i s necessary to recompute K for a large number of sets of the plant parameters q, and the implementation of u* under every operating condition. To approximate the control lav; of (7.2), an optimally sensitive control u g i s introduced. This control u g tends to track the new optimum of J whenver there i s a v a r i a t i o n i n q. The f i r s t order approximation u s i s written as m u . = -R~ 1B ,[K(q ) + .E K Aq.]Y (7.5) ' s i no i = l q. a. x The R i c c a t i s e n s i t i v i t y matrices K are obtained from the d i f f e r e n t i a t i o n v of (7.3) w.r.t. q̂ ^ where and K G + G'K = -C, (7.6a) q. q. 1 i i G = A - BR^B'K, / (7.6b) C = KA + A* K (7.6c) 1 q. q. l i The second order approximation u g may be written as i m - m u „ = -R B'[K(q ) + .ZnK Aq. + ̂  ?- • K Aq.Aq.]Y (7.7) s2 V*V 1=1 q± 4 i 2 i j q ±q 4 i H j where the s e n s i t i v i t y matrices K are computed from equations (7.6), ^ i and K from q .q . 1 J K G + G'K = -C„ (7.8a) q.q, q.q, 2 where C = K G +G'K + K A +A'K + 2 q.q. q. q. q.q. q.q. M i Hj " i H i H i + K A + A' K (7.8b) q.q. q.q. and G = A - BR_1B'K (7.8c) q. q. q. J J 3 Equation (7.8a) i s obtained by d i f f e r e n t i a t i n g (7.6a) with respect to q^. Other matrices of equations (7.6) and (7.8) are computed for q = q^. The procedure can be extended to obtain a c o n t r o l l e r with higher order approximation by adding more Taylor series terms. Hoxjever, i t w i l l become increasingly d i f f i c u l t to implement the high order controller with a large number of changeable parameters. The structures of the nominal controller and the optimally sensitive controller of the f i r s t order approximation are shown i n Fig. 7-1 a and b respectively. / SYSTEM SYSTEM Y'= A(q)Y+ Bu R B K Y . A(q)Y+ flu -I ' R 8 K. R-'B'K a. Nominal optimal regulator b. Optimally sensitive regulator Fig. 7-1 Structures of Nominal and Optimally Sensitive Controllers The R i c c a t i s e n s i t i v i t y matrices, necessary for the optimal sensitive regulator design, must be computed from the Lyapunov matrix equations of (7.6) and (7.8). A new technique i s developed to solve these equations and i s given i n appendix B. The computational ef f o r t i s much reduced by the use of the known eigensystem of the closed loop matrix G. 7.2. S e n s i t i v i t y Equations of the Linearized Power System For the design of optimally sensitive controllers i t i s necessary to compute the system s e n s i t i v i t y matrices A^. This section deals with the derivations of these s e n s i t i v i t y matrices for a general multi-machine power system. There are i n general (4n-l) variables that affect the steady state operating condition for an n machine power system, three terminal conditions for each machine and (n-1) angular differences between net- work buses. The operating conditions are expressed i n terms of i^, i , v , and S which give the simplest s e n s i t i v i t y expressions. Referring to the multi-machine equations of chapter 3. the deviations of Z of i m (3.5), U of (3.14b), M and N of (3.21a), >, and i|> of (3.10) for m a q varying operating conditions are as follows AZ = jZ [A5] - j[A6]Z (7.9) m m m The r e a l and imaginary parts respectively are and Next, AR = -X [AS] + [A6]X (7.9a) m m m AX = R [LSI - [A6]R (7.9b) m m . m AU = Z AI + AZ I - AV (7.10) m m m mm m using (7.9) and (3.14b), U can be written as m AU = Z AI + jU [A6] - j[A6]U - AV ' (7.11) m m m m m m The r e a l and imaginary parts respectively are AU , = R AI, - X AI - U [A6] + [A6]U -AV,, (7.11a) md m d m q mq mq d and AU = X AI, + R AI + U ,[A6] - [Ao^U , - AV (7.11b) mq m d m q md md q Note that V = V, + jV , and I = I, + j l . They are diagonal matrices m d q m d q with v and i vector elements of each machine as the diagonal matrix m m elements. Next, ' AM = I^—]{V,AR m + V AX + A 2 A V R - [^.]AV,X + l.v d m q m v dm I dm A t t v f c + A 2AV X - [ — ^ H V R} (7.12) v̂_ q m 2 q m t V t KA V d 2 AN = [ ~ ] { V A R - VAX + R ' A V R m + I.v q m d m v q m -SL£LlAv v _ r - i L ^ A V y - tJLA> + Hy^AV X - C ]̂ .X  [-^]AV,R } / (7.13) 2 am v dm I d m / V t t v t / - V d V q V d Note that [ — 9 ] > [~—]» etc. are matrices consisting of diagonal V t elements computed from data of i n d i v i d u a l machines. F i n a l l y the armature flu x linkage variations from the normal steady state operating conditions are as follows, From (3.10) we have Ai|>, = — ( A v + r Ai ) (7.14) d co q a q o From (3.10b) we have x = - -3- Ai ' (7.15) q • a, q In the case of a one machine i n f i n i t e system, a l l matrices become scalars and the s e n s i t i v i t y equations (7.9) through (7.13) reduce to AZ = AR + jAX = 0 (7.16) m m J m u» AU = RAi - (X + x )Ai (7.17) md d m q q AU = X A i , + R Ai - Av (7.18) mq m d m q q KA 2 AM = {( v R - v.v X )(x Ai - r Ai.) 3 w q m d q m/ q q a &J A t + (v 2 X - v v R )Av } (7.19) dm d q m q KA 2 AN = - ^ r {(v, R ' + v.v X )Av ^ 3 d m d q m q A. t - ( v 2 X + v ,v R ) (x Ai - r Ai,)} (7.20) q m d q m q q a d J 69 The system s e n s i t i v i t y matrices A , q=(i,,i ,v )', for a one machine q a q q 7 i n f i n i t e system of the 5th order synchronous machine model equations of / appendix B are as follows. / 0 . 0 0 0 0 0 0 0 0 0 0 0 -X m 0 0 0 0 0 0 R 0 0 0 0 0 0 0 0 Y' "dF M-' Y' N.. Y'' x^.qq. 0 0 A. (5,6) d 0 0 0 0 0 0 0 0 0 0 2 % 2H 0. 0 0 V c 2H (7.21) 0 0 i df q o o o i dd q 0 w x w x 0 q v' ° ^CY' -Y» 1̂ 2H dF 2H K dd qqJ 0 0 N. Y' l q qq to r o a 2H qq 0 0 0 0 0 0 0 0 0 0 0 0 Y 1 0 0 m •(X +x ) m q' A± (5,6) q o o -x _£L o 0. 0 0 (7.22) v +x i,+2r i q q d a q 2H D J / 70 v 0 •0 0 v dF q 0 0 0 0 0 M Y ^ M v dd q 0 0 0 0 0 N Y' V q M 0 to _° Y' 2H qq 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 A v (5,6) q o o to 0 _1 i) o 0 0 0 - V D 2H (7.23) where M i d -K. r (v 2R - v,v X )/T AV 3 A a v q m d q m A t N. = K. r (v 2X + v.v R )/T Av 3 i ^ A a q m d q m A t (7.24) K 2 V = V t " d M ~ V Q - — T ~ r a v t and M. = K.x (v2R - v,v X )/T V 3 l A q q m d q m A t q . N. = -K Ax (v 2X + v.v R )/T AV 3 l A q q m d q m A t (7.25) and A. (5,6) q KA v U + v.v U , i A V t d m q?. v2 q M = K. (v 2X - -v'v R )/T.V3 v A d m d q m A t q N = Kk(y% + v j V X )/T Av 3 V A d m d q m A t q (7,26) K A v d K A v ( 5 ' 6 ) T~v~ .3 ^ vd"md ' v d V m q q A t T.v H H A (v?U J + v,v U ) A t Although the system s e n s i t i v i t y matrices are derived i n terms of the variations A i ^ , A i ^ , and Av^, i t i s always possible to relate these variations to another measurable set through a nonsingular transformation. For example, " A i d " = T" 1 AP Ai AQ q Av Av L q L t J (7.27) where T = VA - T- XA d a d V + r i q a q V d —- r v t a v + x i , q q d -(v + X i ) d q q v d - — X v t 1 V V t (7.27a) 7.3. Optimally Sensitive S t a b i l i z a t i o n of a Power System The one machine i n f i n i t e system of Fig. 5-1 i s chosen for this study. The synchronous machine i s described as a 5th order system with ijjp, lf^* t 6, and 00 as the state variables, appendix A. The voltage regulator i s approximated as a f i r s t order system by neglecting T p for the s o l i d state exciter system. Nominal system operating conditions are i n p.u.; 72 P = .952 , 0 = .015, and v = 1.05 o o to (7.28) The per unit l i n e a r state equations for the system at these nominal operating conditions are where A = o Y = A Y + B u 0 Y = AO1 F» *d» V v 6, to) B = (0 0 0 1 0 0)' data given i n chapter 5, u = 1000 u„ E -.660 8.55 0 1 0 44.9 -723 1230 0 59.9 153 -2848 -250 0 -497 418 6736 -368 -20 1125 0 0 0 0 0 5.95 62.7 86.6 0 0 0 -.449 -.954 0 1 0 (7.29) (7.29a) (7.29b) (7.29c) (7.29d) The technique of determining Q developed i n chapter 4 i s applied to the nominal optimal regulater design of the system. With the weighting factor for control chosen as R = 1, Q i s found to be Q = diag.(0 1.55 16.3 0 737.4 19084) (7.30) The Riccati matrix i s 1.79 .215 .08 .078 12.2 -2.48 .215 .04 -.003 .008 .863 -.549 .08 -.003 .015 .005 1.1 .114 .078 .008 .005 .004 .6 -.08 12.2 .863 1.1 .6 133 -6.5 2.48 -.549 .114 -.08 -6.5 7.78 The nominal optimal c o n t r o l through e x c i t a t i o n i s UE0 = (~- 0 7 8 - - 0 0 8 - - 0 0 5 - - 0 0 4 -' 6 - 0 8) Y (7.32) The system s e n s i t i v i t y matrices, of equations (7.21), (7.22), and (7.23), are computed at the nominal operating conditions. Their values are given i n equations (7.33), (7.34), and (7.35). To check the computation of A^ matrices, the system matrix A i s computed from the l i n e a r i z e d equations (A.3d) and from the s e n s i t i v i t y equation A = A + A Aq. A good agreement between both methods i s r e a l i z e d over o q a wide range of system operating conditions. 0 0 0 0 0 0 0 0 0 0 -470 0 0 0 0 0 138 0 -.17 2.75 6 0 202 0 0 0 0 0 0 0 0 0 41 0 0 0 (7.33) I t 0 0 0 0 0 0 0 0 0 0 - 1 3 7 - . 5 5 0 0 0 0 - 6 7 8 - . 0 0 3 31 - 5 0 5 - 1 1 0 3 0 1 4 4 5 0 0 0 0 0 0 0 - 7 . 3 77 . 2 2 0 0 0 - 0 0 0 0 0 — i 0 0 0 0 0 377 0 0 0 0 0 0 - 1 - 2 7 4 3 1 9 4 3 0 - 1 4 0 n 0 0 0 0 0 0 . 0 0 74 0 0 0 ( 7 . 3 4 ) ( 7 . 3 5 ) The R i c c a t i s e n s i t i v i t y matrices are obtained by solving the Lyapunov matrix equations ( 7 . 6 ) using the frequency domain technique developed i n appendix B. These matrices are K. l . 1 0 3 x 1 . 2 2 7 . 0 0 9 . 0 4 4 - 1 0 . 4 - 1 . 8 5 . 2 2 7 . 0 4 9 . 0 0 2 . 01 - . 6 0 6 - . 3 9 6 . 0 0 9 . 0 0 2 - . 0 0 2 0 - . 8 3 3 - . 0 9 4 . 0 4 4 . 01 0 . 0 0 2 - . 4 6 9 - . 0 9 2 - 1 0 . 4 - . 6 0 6 - . 8 3 3 - . 4 6 9 - 1 9 6 7 . 1 1 - 1 . 8 5 - . 3 9 6 - . 0 9 4 - . 0 9 2 7 . 1 1 1 . 3 2 ( 7 . 3 6 ) / 75 K. * 10 x 1 1.49 .074 .045 .056 13.4 ,333 .074 -.013 -.003 .002 1.14 .292. .045 -.003 .015 .004 .756 .161 .056 .002 .004 .002 .578 .046 13.4 1.14 .756 .578 112 -3.73 .333 .292 .161 .046 -3.73 -4.54 (7.37) v 10 3x -.654 -.03 -.022 -.031 13.1 1.51 -.03 .04 -.025 -.004 .836 -.267 -.022 -.025 .013 0 1.23 .242 -.031 -.004 0 -.001 .613 .085 13.1 .836 1.23 .613 241 -6.61 1.51 -.267 .242 .085 -6.61 -.918 (7.38) The f i r s t order optimally s e n s i t i v e e x c i t a t i o n c o n t r o l , equation (7.5), i s then designed u ^ = (-.078 -.008 -.004 -.6 .08) Y + + 10 ( A i , , A i Av ) d q. q o o o -44 -9.7 -56 -1.5 31 4.2 -.07 -1.9 470 92 •3.7 -2.4 -580 -46 -.34 1.3 -613 -85 (7.39) The control can be expressed i n terms of AP , AQ , and Av instead of o o t o A i , , A i , and Av d q ' ( o o q Q through the transformation m a t r i x T, equation (7.27), T = .446 1.17 .814 .953 -.895 .399 -.001 .234 .905 The r e s u l t s are UES (7.40) .078 -.008 -. 005 -.004 -.6 .08) Y + " -77 -8 9 -2. 6 -3.3 50 "51 Y AQ ,A v. ) o t o -10 -6 1. 1 -.43 468 73 108 15 1. 4 4.6 -928 -172 (7.41) For comparison the c o n t r o l l e r gains of the optimal s i g n a l ug, b equation (7.2), f o r d i f f e r e n t o p e r a t i n g c o n d i t i o n s are computed and compared - w i t h the r e s u l t a n t gains of the o p t i m a l l y s e n s i t i v e c o n t r o l l e r u Eg, i n t a b l e 7-1. The speed and torque angle gains f o r both s i g n a l s are p l o t t e d i n f i g u r e 7-2. I t i s c l e a r that the o p t i m a l l y s e n s i t i v e c o n t r o l l e r u^^ gains adjust themselves to cover the wide range o p e r a t i n g c o n d i t i o n s and to match the absolute optimal c o n t r o l l e r s u* gains. The dominant eigenvalues f o r the system w i t h the d i f f e r e n t c o n t r o l l e r s at d i f f e r e n t o p erating c o n d i t i o n s are given i n t a b l e 7-2. While a r e d u c t i o n of s t a b i l i t y of the system i s observed when i t departs from the nominal o p e r a t i n g c o n d i t i o n , the o p t i m a l l y s e n s i t i v e c o n t r o l l e r u„ 0 provides b e t t e r . bb r e s u l t s than the nominal optimal c o n t r o l l e r u E Q . Although u* provides the best s t a b i l i t y , i t i s i m p r a c t i c a l to implement as s t a t e d b e f o r e , on the other hand there i s no d i f f i c u l t y to implement , i t i s j u s t as good as u* except f o r the worst operating c o n d i t i o n (P = 1.25, Q = .45, & o o v = 1.05) • system operating c o n d i t i o n s i\ =1.05) o P =1.25 0 Q =.45 o P =1.2 0 Q =.34 o P =1.15 o Q =.25 0 P =.952 Nominal Q =.015 0 P =.7 0 Q =-.15 0 P =.'5 0 Q =-.225 0 P =.3 0 Q0=-.256 3 UES 128 160 117 134 107 117 80.1 80.1 56.3 61 44.9 53.9 41.3 49.9 UES -376 -434 -479 -603 -682 -691 -648 0 0 o • ,<o U f -148 . -316 -420 -603 -660 -664 -661 T-t X CO a •H cd OC >:• UES U£ -4.7 -5.18 -4.49 -4.77 -4.29 -4.45 -3.57 -3.57 -2.6 -2.75 -1.9 -2.1 -1.2 -1.5 Co nt ro ll er  UES UE* -5.18 -4.57 -5.19 -4.88 -5.17 -5 -4.96 -4.96 -4.4 ' -4.3 -3.9 -3.56 -3.15 -2.5 Co nt ro ll er  UES -13 -11.9 -10.9 -7.74 -4.5 , -2.5 -.96 u i -16.7 -13.9 -11.9 -7.74 -5.1 -3.8 -2.9 UES -104 -117 -99.5 -107 -94.9 -99 -77.8 -77.8 -55.8 -58.8 -38.3 -45.3 -21.6 -31 . Table 7-1 C o n t r o l l e r Gains For u„_ and u* Eb E ' at D i f f e r e n t Operating Conditions 78 Table 7-2 Dominant Eigenvalues of the System with the D i f f e r e n t C o n t r o l l e r s Operating Conditions v . = 1-05 P 0 o o UE0 UES A / i i 1.25 .45 .717+J2.86 -4.8 -16.8 1.49 -4+J4.3 -17.16 .137 -4+J3.8 -17 -2.1 -3.2+J3.1 -16.9 1.2 .34 .56+J3.47 -4.5 -16.9 .467 -3.56+J4.37 -17.15 -1.1 -3.3+J3.9 -17 -2.6 -2.7+J3.7 -16.9 1.15 .25 .44+J3.9 -4.2 -16.9 -.449 -3.1+J4.43 -17.1 -2 -2.8+J4.1 -17 -2.7 -2.5+J4.1 -16.97 Nominal .952 .015 .17+J4.8 -3.6 -16.9 -1.98+J4.99 -2.89 -16.96 -1.98+J4.99 -2.89 -16.96 -1.98+J4.99 -2.89 -16.96 .7 -J.5 .023+J5.2 -3.4 -16.8 -1.39+J5.3 -4.4 -16.6 -1.6+J5.3 -2.7 -16.9 -1.56+J5.3 -2.96 -16.89 .5 -.225 -.02+J5.27 -3.4 -16.8 -1.04+J5.3 -5.57 -16.1 -1.28+J5.4 -2.6 -16.97 -1.22+J5.37 -3.1 -16.8 .3 -.256 -.023+J5.25 -3.4 -16.7 -.66+J5.2 -6.8 -15.56 -.83+J5.3 -2.7 -17 -.83+J5.3 -3.3 -16.7 P0 (PU.) Fig. 7-2 Speed and Torque Angle Gains for the Controllers (1) u E Q (2) u E g (3) u* Both controllers u_ n and u„„ are tested on the nonlinear model EO ES of the system on two operating conditions, - P q = .952, Q = .015, v t =1.05 (Nominal) (7.28) o and P =1.2, Q =.34, v =1.05 (7.42) o ' o t o The system disturbance i s the same as i n chapter 5. The test results are summarized i n Fig. 7-3. While the optimally sensitive controller u„„ maintains system s t a b i l i t y for the operating conditions of Eb (7.42), the nominal controller u A f a i l s to do so. 80 0 .0035- 0 .0030- a .0025- 3 D. 0 0020- g 0 0015- 0 0010 - o 0 0005- a n -0 0000-tn -0 0005- •0 0010- 0 0015 n 0.5 1^0 1.5 2.0 TIME (SECONDS) 2.5 3.0 lO.O-i 0.05 0.00 0.5 1.0 1.5 2.0 TIME (SECONDS) lb) 2.5 3.0 E?-0.05 y-o.io -0.15 — i 1 — i 1— 0.5 1.0 1.5 2.0 TIME (SECBNOS1 (c) 0.10 _ 0.05 0.5 1.0 1.5 2.0 TIME ISECONDS) Id) .3.0 0.5 1.0 1.5 2.0 TIME ISECONDS) 1*) Fig. 7-3 Nonlinear Test Results (1) Nominal operating conditions, U^Q or u E g (2) Nominal optimal control u E Q , for P = 1.2 (3) Optimally sensitive control u E g , for P = 1.2 8. CONCLUSIONS An exact representation of synchronous machines i s presented / and a step by step derivation of the exact equivalent c i r c u i t given i n Chapter 2. I t i s found that an extra test with the IEEE test code i s needed to determine the d - c i r c u i t synchronous machine parameters. Three different methods are suggested, a varying s l i p test or a decaying current test to determine the D-damper time constant T^,o'r an adaptation of Dalton and Cameron's method to determine the newly defined open f i e l d d-axis subtransient reactance x1.1 . No extra test i s needed to determine do the q - c i r c u i t parameters. A l l three methods gave close results i n laboratory tests. A new multi-machine state variable formulation i s presented i n Chapter 3. The largest matrix inversion i s the nodal admittance matrix Y . A l l system parameters are retained i n the f i n a l formulation, convenient for s e n s i t i v i t y studies. Systems with an i n f i n i t e bus are also considered. The results have the same form as that of multi-machine systems without an i n f i n i t e bus. Dynamic s i m p l i f i c a t i o n of power systems i s discussed. I t i s found from a numerical example that conventional s i m p l i f i c a t i o n i n power system engineering retains the most dominant eigenvalues of the system. A new technique for the design of optimal lin e a r regulators i s developed i n Chapter 4. The Weighting matrix Q of the regulator per- formance function i s determined i n conjunction with the dominant eigen- value s h i f t of the closed loop system. The eigenvalue s e n s i t i v i t i e s of the optimal closed loop system with respect to the Q elements are expressed i n terms of the same eigenvector matrix of the composite matrix M of equation (4.18), which i s required for computing the R i c c a t i matrix K. ^ / / Applying the technique developed i n Chapter 4, the optimal s t a b i l i z a t i o n of a one machine i n f i n i t e system i s investigated i n Chapter 5. Three different methods of s t a b i l i z a t i o n are considered, through ex c i t a t i o n , through the governor, or through both as compared with the conventional s t a b i l i z a t i o n through excitation control. I t i s found that optimal s t a b i l i z a t i o n through excitation i s more effec t i v e than conventional excitation s t a b i l i z a t i o n , that optimal s t a b i l i z a t i o n through a governor without dashpot i s better than that through a governor With a dashpot, and that optimal s t a b i l i z a t i o n through both excitation and governor without dashpot i s the best of a l l . In Chapter 6, the s t a b i l i z a t i o n of multi-machine systems i s investigated again using the technique developed i n Chapter 4. Several cases are considered. I t i s found that a multi-machine system with multi-machine optimal controller u E^, i s better than the multi-machine system with only one optimal c o n t r o l l e r , u„ T, which i s i n turn better than the multi-machine system with the approximated one machine i n f i n i t e system controller U g . I t i s also found that although the in d i v i d u a l optimal controller designs are effe c t i v e i n providing damping to i n d i v i d u a l machines, thei r effects on other machines are unpredictable. Therefore the dynamic coupling of the multi-machine system must always be included i n optimal controller design. The optimal controllers i n Chapters 5 and 6 are a l l for nominal system operating conditions. Since the operating conditions of a re a l system change from time to time, the controllers so far designed are not adequate for varying operating conditions. In an attempt to face this challenge an optimally sensitive controller i s designed i n Chapter 7. I t i s found that the newly developed optimally sensitive controller can adjust i t s e l f to s t a b i l i z e a power system over a wide range of operating conditions and the optimum s t a b i l i z a t i o n i s always achieved. A new method to solve the Lyapunov type matrix equation necessary for the design i s also developed. Although the techniques have been tested on the detailed non- li n e a r mathematical model of the systems, i t i s highly desirable to implement them on a re a l poxver system. Other problems remain to be solved. One i s to develop test methods to determine exact parameters of synchronous machines with additional rotor c i r c u i t s . Another problem i s how to obtain better approximate representation for system loads and i n f i n i t e systems for power system dynamic studies. F i n a l l y there i s the challenging problem of nonlinear optimal s t a b i l i z a t i o n , which needs more investigation to make i t p r a c t i c a l . 84 APPENDIX A MULTI-MACHINE STATE FORM EQUATIONS FOR 5th ORDER SYNCHRONOUS MACHINE MODEL / ( For a 5th order synchronous machine model, the damper flu x linkage variations are neglected, i . e . pAtf»D =0, pA ^ = 0, (A.l) (A. 2) implementing (A.l) and (A.l) into (3.30) and elimenating A i j ^ and AIJJQ from the r e s u l t s , system equations become, Y* = AY + Bu, Y * A(*_ * v v. F r d Tq F R u = u E» B = [0 0 0 0 K. 6 co)', 0 0]', (A. 3) (A.3a) (A.3b) (A.3c) A A = " rF YFF " R YdF m dF MY' dF 0 A(7,l) -r Y' 0 F Fd -RY* co +X Y' dd e m qq. -co -X Y' -RY' e m dd qq MY' dd NY' 0 0 A(7,2) A(7,3) I 0 0 -1 T„ 0 0 0 0 0 1_ T E TA 0 -U mq md 0 -[* d] A(5,6) 0 0 o o 2H (A. 3d) /' 85 where K A(5,6) = h—"](V,U - V I I ) i . v d mq q md / A t / / ' "2 ' A ( 7 5 l ) = - ^§ ^ Y-p / 2 CO A(7,2) = - -r£ i|. (Y' - Y» ) 2H q dd qq 2 A(7,3) - - ^ { * d ( Y j d - Y^) + W YFF YFF YFD YDD YDF % * d / x F X d YdF " YdF " YdD YDD YDF = " % X d F / x F X d ( A * 3 e ) YFd " YFd YFD YDD YDd." % X F d / x F X d Ydd Ydd YdD YDD YDd w o / x d Y 1 = Y - Y . Y"J Y. = co /x qq qq qQ QQ Qq o q M and N are as given i n (3.21a). The governor equations can be easily incorporated into (A.3) i f required. APPENDIX B FREQUENCY DOMAIN SOLUTION OF LYAPUNOV MATRIX EQUATION A new method for solving the Lyapunov matrix equation i n the frequency domain i s proposed. The highest matrix order used i n the computation i s the same as the system matrix and no matrix inversion.is required. Two algorithms are given,-the f i r s t uses the Leverrier algorithm and the second uses the eigensystem of the system matrix. The equation i s usually of the form A TK + KA = -Q (B.l) where A i s the system matrix, K the matrix to be solved and Q a p o s i t i v e semi-definite symmetric matrix. • Equation (B.l) consists of e s s e n t i a l l y n(n+l)/2 l i n e a r equations for an n-order sytem. The equation can be expanded as N k = q (B.2) and solved d i r e c t l y . Since for a stable system T K = Ja e A t Q e A t dt (B.3) which has f i n i t e value, the i n t e g r a l can be approximated as a series 38 39 summation and evaluated i t e r a t i v e l y ' . Transformation approaches are also r e p o r t e d ^ ^ . Solutions are obtained after (B.l) i s reduced to a special form. In what follows, the method of frequency domain solution of (B.l) w i l l be presented. Applying Parseval's theorem (B.3) becomes K = ~~r f ,:P(s)ds (B.4) 2ITJ -j« 87 where t(s) = (-si - A 1)"" 1 Q(sl - A)""1 (B.5) K can thus be evaluated from the residue theorem. Let ( s i - A ) " 1 = R(s)/g(s) (B.6) where R(s) = I s 1 1  + R.s" 2 + .. . + R.s n 1 - 1 + ... + R . 1 l n-1 g(s) = de t ( s l - A) = s 1 1 - h n s n ^ - ... -h.s 1 1 ^ - ... - h to 1 l n i = 1, 2, n <B'7> The matrix c o e f f i c i e n t s R_̂  of the adjoint matrix polynomial R(s) and the scalar c o e f f i c i e n t s h. of the c h a r a c t e r i s t i c equation g(s) can be 33 determined simultaneously by.Leverrier's algorithm , h. = ~ trace [A.], R. = A. - h.I i i 1 I ' I l i A = A , A. = AR. n 1 i i - i (B.8) (B.9) substituting (B.6) into (B.5) gives g(-s) g(s) which can be written as n C. n D. i 1=1 i where C. and D. are residue matrices of F(s) i n the l e f t and right half i i complex planes respectively. I t i s assumed that are d i s t i n c t . Let g'(s) = d • g(s)/ds. Then ' R T ( - X . ) R ( X . ) or c. = < s-x.) • F(S)| S = = X i = ^ r r x Q p ^ CB . i D 1 " 1 1 R T(-X ) Q R(X.) C = 1 — r - i r - (B.12) 1 2X. IT (XT-XT)" Applying the residue theorem one has n K = E C . ( B . 1 3 ) i = l 1 Since C ± + 1 = C* ( B . U ) for conjugate-pair roots, X , = X*. For a system with m conjugate pair i + 1 i roots and I r e a l roots, m n K = 2 E Real C„ + E C. ( B . 1 5 ) • i 2 r - l . o .-i J i = l j=2m+l J The residue matrices C_̂  can be computed also from the eigenvalues and eigenvectors of the system. Since R / v n R(X.) — T ^ Y = E , w \ N ( B . 1 6 ) g(s) j = 1 g'(Xj)(s-Xj) 4 3 and Morgan has shown that R(A.) x,v T = - n f . (B.17) J J g (Xj) T where and v^ are the normalized j - t h eigenvectors of A and A respectively, equation (11) may be written as n R T(X.) R(X ±) °i = g ^ X j X - x " ^ ) Q g^TxT)" T n v X = - E - r ^ - Q X. V. J=1'3 i =-VA.X TQx.v T- < B' 1 8 ) 1 1 1 where /' A. = diag[X. + X. , X. + X_, ..., X. + X ]• / (B.19) 1 l 1 i 2 l n / and X, V are eigenvector matrices of columns of X_̂  and v_̂ , respectively; 90 REFERENCES 1. 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 the Peace River System", IEEE Transactions, Vol. PAS-85, pp. 586-600, June 1966. 1 2. P.L. Dandeno, A.N. Karas, K.R. McClymont and W. Watson, "Effect of High- Speed R e c t i f i e r E xcitation Systems on Generator S t a b i l i t y Limits", IEEE Transactions, Vol. PAS-87, pp. 190-201, January 1968. 3. O.W. Hanson, C.J. Goodwin, and P.L. Dandeno, "Influence of Excitation 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. 4. F.R. S c h l e i f , G.E. Martin, and R.R. Angell, "Damping of System O s c i l l a t i o n s with a Hydrogenerating Unit", IEEE Transactions, Vol. PAS-86, pp. 438-442, A p r i l 1967. 5. F.R. S c h l e i f , H.D. Humlins, G.E. Martin, and E.E. Hattan, "Excitation Control to Improve Powerline S t a b i l i t y " , IEEE Transactions, Vol. PAS-87, pp. 1426-1434, June 1968. 6. F.R. S c h l e i f , H.D. Hunkins, E.E. Hattan, and W.B. Gish, "Control of Rotating Exciters for Power System Damping-Pilot Applications and Experience", IEEE Transactions, Vol. PAS-88, pp. 1259-1266, August 1969. 7. R.M. Shier, and A.L. Blythe, "F i e l d Tests of Dynamic S t a b i l i t y Using a S t a b i l i z i n g Signal and Computer Program V e r i f i c a t i o n " , IEEE Transactions, Vol. PAS-87, pp. 315-322, February 1968. 8. R.T. Byerly, F.W. Keay, and J.W. Skooglund, "Damping of Power O s c i l l a t i o n s i n Salient Pole Machines With St a t i c Exciters", IEEE Transactions, Vol. PAS-89, pp. 1009-1021, July/August 1970 91 9. F.P. deMello and C. Concordia, "Concepts of Synchronous Machine S t a b i l i t y as Affected by E x c i t a t i o n Control", IEEE Transactions, Vol. PAS-88, pp. 316-329, A p r i l 1969. / 10. P.C. Krause and J.N. Towle, "Synchronous Machine Damping by ; E x c i t a t i o n Control with D i r e c t and Quadrature Axis F i e l d Windings", IEEE Transactions, Vol. PAS-88, pp. 1266-1274, August 1969. 11. Y.N. Yu, K. Vongsuriya, and L.N. Wedman, "Application of an Optimal Control Theory to a Power System", IEEE Transactions, V ol. PAS-89, pp. 55-62, January 1970. 12. Y.N. Yu and C. Siggers, " S t a b i l i z a t i o n and Optimal Control Signals f o r a Power System", presented at IEEE Summer Power Meeting, paper No. 70TP531-PWR. 13. Yao-nan Yu and H.A.M. Moussa, "Experimental Determination of Exact Equivalent C i r c u i t Parameters of Synchronous Machines", 1971 IEEE Winter Power Meeting, paper No. 71 TP 63-PWR. 14. I.M. Canay, "Causes of Discrepancies on C a l c u l a t i o n of Rotor Quantities and Exact Equivalent Diagrams of the Synchronous Machine." IEEE Transactions, PAS-88 No. 7, July 1969, pp. 1114-1120. 15. C. Concordia, "Synchronous Machines" (book), John Wiley, 1951, eqts. 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Sannutx, "Synthesis of Optimally Sensitive Systems", Proceedings of the IEEE, Vol. 56, pp. 1318-1324, August 1968. 38. Davison, E.J. and Man, F.T.: "The Numerical Solution of A'Q+QA = -C", IEEE Trans., 1968, AC-13, pp. 448-449. 39. Wedman, L.N. and Yu, Y.N.: "Computation Techniques for the S t a b i l i z a t i o n and Optimization of High Order Systems", IEEE PICA Conference P r o c , 1969, pp. 324-343. 40. Barnett, S. and Storey, C.: " S t a b i l i t y Analysis of Constant Linear Systems by Lyapunov's Second Method", Electron. L e t t . , 1966, 2, pp. 165-166. 41. Power, H.M.: "Further Comments on the Lyapunov Matrix Equation", Electron. L e t t . , 1967, 3, pp. 153-154. 42. M o l i n a r i , B.P.: "Algebraic Solution of Matrix Linear Equations i n Control Theory", Proc. IEE, 1969, 116-10, pp. 1748-1754. 43. Morgan, B.S., "Computational Procedure for S e n s i t i v i t y of an Eigenvalue", Electron. L e t t . , 1966, 2, pp. 197-198.

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