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 T^ 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 T^ 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. 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 Q^ / 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 x^ „ 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 T^ 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 x^ ' . 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 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 = P^ D + VD 0 = P^Q + 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 * 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 P^ 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 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 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 / 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 °°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 <|) = (^,...,± = $ 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 , 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 u^ ,, 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» 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 , 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, 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) v^_ 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» 1^ 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 (v2R - 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 103x -.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 • ,:• 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 Controller 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 Controller 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 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. 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