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Modelling and parameter estimation of unknown large power system dynamics El-Sharkawi, Mohamed Ali Ahmed Ali 1977

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MODELLING AND PARAMETER ESTIMATION OF UNKNOWN LARGE POWER SYSTEM DYNAMICS b y Mohamed Ali Ahmed All EL7/SHARKAWI B.Sc. Cairo High Institute of Technology A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of Electrical Engineering We accept this thesis as conforming to the required standard • THE UNIVERSITY OF BRITISH COLUMBIA June 1977 © M.A.A.A. El-Sharkawi, 1977 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements f an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree tha the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree tha t permiss ion fo r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying o r p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed without my w r i t t e n p e r m i s s i o n . Department of The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 ABSTRACT The current practice in modelling unknown large power systems is reviewed in Chapter 1, and the inadequate representation by constant volt-age and constant frequency is discussed. To determine the unknown large power system dynamics, estimation must be used. A complete model for estimation, including known and unknown power systems, is derived in Chapter 2 and the mathematical formulation in Chapter 3. Chapter 4 presents the estimation algorithm, data and results. It is found that the estimated unknown system parameter values are unique, independent of various operating conditions. Conclusions are drawn in Chapter 5 . i i CONTENTS Page A b s t r a c t i i C o n t e n t s i i i Acknowledgement i v 1. I n t r o d u c t i o n 1 2 . M o d e l l i n g t h e Sy s tem f o r E s t i m a t i o n S t u d i e s 4 2.1 M o d e l l i n g t h e Known Sy s tem 7 2.2 M o d e l l i n g t he Unknown Sys tem 10 2 .3 The Comp le te B l o c k D i a g r a m and t he S t a t e E q u a t i o n s 14 2 .4 The I n i t i a l V a l u e s 20 3. E s t i m a t i o n o f Unknown Sy s tem P a r a m e t e r s 23 3.1 M a t h e m a t i c a l F o r m u l a t i o n , 23 3.2 E v a l u a t i o n o f t h e I n t e g r a l s 25 3.3 P a r t i a l D e r i v a t i v e s o f A ( a ) and H(a ) 25 4. A l g o r i t h m , D a t a and R e s u l t s 35 4 .1 A l g o r i t h m 35 4.2 D a t a 36 4 . 3 R e s u l t s 36 5. C o n c l u s i o n s 52 6. R e f e r e n c e s 53 i i i ACKNOWLEDGEMENT I wish to extend my sincere thanks to my supervisors, Dr. Y.N. Yu and Dr. M.D. Wvong, for t h e i r guidance, advice and encouragement during my period of study. The f i n a n c i a l support from the Univ e r s i t y of B r i t i s h Columbia and the National Research Council of Canada i s g r a t e f u l l y acknowledged. My thanks also to Ms. Mary-Ellen Flanagan for typing the th e s i s . I am greatly indebted to my parents f o r t h e i r encouragement throughout my post-graduate work, and f i n a l l y , I must give c r e d i t to my wife,Fatma, who through her s a c r i f i c e s has made i t possible f or me to complete t h i s work. i v 1 CHAPTER 1 INTRODUCTION Much work has already been done on the modelling of power systems so that the computer models may represent the actual system with nearly the same,characteristics. These computer models are widely used i n the study of power system dynamics and c o n t r o l . For example, Dandeno, Hauth and Schulz [1] have reported on the accuracy of power system s t a b i l i t y simula-tions, as af f e c t e d by the degree of d e t a i l s i n the representation of the synchronous generators and the data used. Another study was also recorded [2]. Although much work has been done on modelling power systems, l i t t l e has been done on modelling the unknown large power system of neigh-boring areas. An assumption i s usually made that an unknown large power system may be represented by an i n f i n i t e - b u s with constant voltage and constant frequency. For example, deMello and Concordia [3] used t h e i r model with an i n f i n i t e - b u s for the supplemental e x c i t a t i o n c o n t r o l design. El-Sherbiny and Mehta [4] made a synchronous machine s t a b i l i t y study for d i f f e r e n t loading and power factors using the deMello and Concordia model which includes an i n f i n i t e - b u s . Yu, et a l . [5-9] used various synchronous machine models for the l i n e a r optimal c o n t r o l design. Again an i n f i n i t e -bus was used f o r the representation of a neighboring unknown large power system. The assumption of constant voltage* and constant frequency of an unknown large power system, however, i s not always v a l i d . In t h i s thesis a better representation for the unknown large power system than the i n f i n -ite-bus w i l l be developed, which may be c a l l e d the "Dynamic I n f i n i t e Sys-tem Equivalent". Also included w i l l be an estimation method for the 2 determination of the parameters of the unknown system. A review of the l i t e r a t u r e shows that there are many estimation techniques. Examples are: a) the Least-Squares method [10-11] by which the output e r r o r cost function of the calculated and measured variables i n a quadratic form i s minimized, b) the Maximum Li k e l i h o o d method [11] when there i s no p r i o r information a v a i l a b l e regarding the p r o b a b i l i s t i c des-c r i p t i o n of the parameters to be estimated, c) the Bayes' Estimation theory [11] by which the Bayes r i s k function representing the expected cost of an error i n estimation i s minimized, d) the Linear Minimum Variance method [11] by which the trace of the variance of the erro r function i s minimized, e) the Kalman F i l t e r technique [11-12] i n which a zero mean white noise i s con-sidered, and f) the Instrumental Variable method [13] by which the parameter set can be estimated from an array of l i n e a r algebraic equations, a set of observations, and a set of unobservable noise terms. Estimation techniques began to f i n d useful a p p l i c a t i o n i n power system studies very recently. Examples are as follows. Stanton [14] e s t i -mated turboalternator t r a n s f e r functions by using normal operating data. Wong and Polak [13] i d e n t i f i e d a l i n e a r d i s c r e t e time system by using the instrumental v a r i a b l e method. Schweppe, e t . a l . [15-17] developed s t a t i c state estimation of power systems. Debs and Larson [18] used a dynamic estimator to track the state of a power system. M i l l e r and Lewis [12] used a Kalman f i l t e r f o r power system state estimation. Debs [19] estimated external network equivalents from i n t e r n a l system data. Handschin, et a l . [20] accounted f o r bad data i n power system estimation. Ueda, et a l . [21] estimated the transient state of a power system by a discre t e nonlinear observer. Lee and Tan [10] used a weighted least-squares estimator f o r synchronous machine studies. Takata, et a l . [22] developed an i t e r a t i v e 3 sequential observer f or estimating the transient state of a power system. Ueda, et a l . [ 2 3 ] estimated the transient state of a multi-machine power system by an extended l i n e a r observer. This t h e s i s i s mainly concerned with the estimation of dynamics of unknown large power systems. I t i s found that the weighted-least-squares -method without noise s t a t i s t i c s gives the most s a t i s f a c t o r y r e s u l t s . The estimation model w i l l be developed i n Chapter 2. The estimation technique w i l l be presented i n Chapter 3, and the algorithm, data and r e s u l t s i n Chapter 4. 4 CHAPTER 2 MODELLING THE SYSTEM FOR ESTIMATION STUDIES The modelling of a power system made up of a generator, a l o c a l load, and a transmission l i n e interconnected to other generators has been discussed previously by many authors [3,6,26].In t h e i r models, as i n most other models, the dynamics of the unknox<rn neighboring generators are neglec-ted and simply approximated by an " i n f i n i t e - b u s " with constant voltage and constant frequency. In fac t neither the voltage nor the frequency of the " i n f i n i t e bus" are constant. Therefore, a bet t e r model i s needed to rep-resent the e f f e c t of changes i n voltage and frequency of the unknown sys-tem. F i g . 2-1 shows the system to be studied, c o n s i s t i n g of the known system ( i ) and the unknown dynamic i n f i n i t e system ( j ) , interconnected by a t i e l i n e . There i s also a l o c a l load. F i g . 2-1 Interconnected System The known system ( i ) i s represented by an equivalent s a l i e n t - p o l e synchronous generator with voltage regulator, which may include e x c i t a t i o n c o n t r o l . The unknown i n f i n i t e system ( j ) i s represented by a c y l i n d r i c a l rotor synchronous machine, which has only one synchronous reactance. F i g . 2-2 Phasor diagram of the system 6 The phasor diagram of the two machines is shown in Fig. 2-2 where and represent the direct and quadrature axes of the system, res-pectively, and D_j and Q_j are the corresponding axes of the j' * 1 system. In this analysis we shall consider the flow of current in the transmission line to be from i to j , which means that the j*"*1 system is th taking electric energy from the i system or is motoring. This is shown clearly in Fig. 2-1. If I changes its direction, both systems wil l supply electric energy to the local load. The analysis consists of three parts. 1. The modification to the deMello-Concordia model [3] to f i t the inter-connected system under study. 2. The derivation of a simple model to represent the unknown system. 3. To find a state variable representation for the complete system suitable for estimating the unknown parameters. The torque equations in linearized form for the two machines are MJ A co. + D. A a. = AT - AT . i i i 1 mi ei (2-1) = AT - AT M, A w. + D. A w. A m j U J - e j 5 J 3 J where M: Inertia constant to: Mechanical angular velocity D: Damping coefficient T : Mechanical torque m T f i: Electric torque We shall consider that represents the damping of the whole j*"*1 system, not only the damping of the synchronous machine, but also a l l the electric damping derived from stabilization. The true mechanical speed is represented by co, and the per unit mechanical speed co is equal to ^/^^ 7 where m i s the base mechanical angular v e l o c i t y . Note that to '= to/to , = to /to , (2-2) mb e eb where to i s the e l e c t r i c a l angular v e l o c i t y and to , i s the base e l e c t r i c a l e a J eb angular v e l o c i t y which i s 377 rad/sec. for a 60 Hz system. Therefore, to = 377 to (2-3) e F i g . 2-3 i s a block diagram representation of equations (2-1) and (2-3). *5r> + 1 AW J77 7 A S ) MS - f D s F i g . 2-3 Block diagram of the mechanical loop 2.1 Modelling the Known System From F i g . 2-1 we have ZI. = (1 + ZY) V - V l t c where Z: Transmission l i n e impedance Y : L o c a l load admittance Terminal voltage of system i Dynamic i n f i n i t e bus voltage Output current of system i V. V th (2-4) In the D_^  - coordinates of the i system, the current compon-ents can be written as I q i i i L 2 i V E q i Z* -X, R, s i n 6.-l cos 6^ -1 (2-5) 8 where I. = I , . + j i . x d i J q i 1 ^ , 1 ^ : The d and q components E\: An i n t e r n a l voltage; F i g . 2-2 The magnitude of phasor V z R + jX Y = G + jB + ZY = A l + J A2 R l = R " A2 X d R2 = R - A„ X 2 q X l = X + A, X' 1 d X2 z 2 l = X + A. X i q = R l R2 + X l X2 Y l i = (X2 A ± - R 2 A2)/Z2± Y 2 i = (Xx A 2 + Rx A^/Z2 A l l the above are defining equations. The l i n e a r i z e d form of (2-5), taking i n t o account that V q i s not constant, i s r A * , . - | d i = " Y i i -A E* + q i - f d i " A 6 . + l - g d i " A 4 1 ,J q i ^Y2 J - f q i where r f d i i q i z 2 x„ R, -R. X, s i n 6 . -1 0 Lcos 6 . -I 1 0 s d i ' q i " -a 2 r R- X„ r s i n 6 . -i l O - cos <5. J l O 9 and S. = I n i t i a l value of the torque angle of system i . 10 We can now express the following equations i n terms of A E ^ , A 6. and A V T = E! . I . + (X - X') I,. I . e qx qx q d d i q i ci + S Tp r± - EfD1 t td tq td q i q V ^ = E • " 1 A - XA tq q i d i d E _ . = X . V./r. fDx ad f f ( X d " Xd> X d i (2-7) (2-8) (2-9 a) (2-9 b) (2-9 c) (2-10) where E' . ,th T! x El f D i An i n t e r n a l voltage of i " " system F i e l d time constant; x / ( ^ r £ ) f' o f Equivalent f i e l d voltage Stator to rotor mutual reactance Let us f i n d the e l e c t r i c torque A T £ i of the i u " system f i r s t . L i n e a r i z a t i o n of equation (2-7) with s u b s t i t u t i o n of values of A I and X ad .th A I . from equation (2-6) gives A T = K,. A 6 . + K_. A E ' . + C , A V T.1 2 i q i (2-11) where - K l i - r O - i K 2 i = 1 - C l - . 0. I '. + qxo q i 2 i L g q i L l i 'dx E' + (X - X') I,, " qio q d dio (X - X') I . q d qxo S i m i l a r l y , l i n e a r i z a t i o n and s u b s t i t u t i o n of the f i e l d voltage equation (2-8) gives 10 A Eq l - K 3 i ( A E f D i " K 4 i A 6 i " C2 A V 1 ( 1 + S T i hl> ( 2 " 1 2 ) where gives K 3. - 1/(1 + ( X D - Xj) Y 1 ± ) K 4 i - ( X d " X d ) f d i C2 = ( X d " Xd> 8 d i F i n a l l y , the l i n e a r i z a t i o n of equation (2-9 a) with s u b s t i t u t i o n A V = K A 6. +K-. A E'. + C „ A V t 5 i l 6 i q i 3 o (2-13) where K 5 i n K 6 i V to r f • Y 2 i Lg q i " x d f d i (1 - X ' Y U ) " X d 8 d l J rv,. x-dio q L V qio The block diagram of the known system shown i n F i g . 2-4 represents the equations (2-1), (2-11), (2-12) and (2-13). Notice that A V q appears at a l l the summing points. 2.2 Modelling the Unknovn System As mentioned before, we s h a l l consider the unknown dynamic i n f i n i t e system as a large synchronous machine with i n e r t i a and damping. For s i m p l i c -i t y we s h a l l consider the synchronous machine to have a c y l i n d r i c a l rotor. This w i l l s i m p l i f y the model and decrease the number of unknown parameters to be estimated by one. The number of d i f f e r e n t i a l equations also can be reduced from four to three i f the secondary e f f e c t s of A 6. and A E'. on 3 3 qj A E f D j > a 1 1^ the time delay due to e x c i t a t i o n and voltage regulator i n s i d e th the j system, are neglected. From F i g . 2-2 we have I , . = - I . s i n (•/>".< -<(>.), I . = I t COS (i>. - <j>.) dj j J j qj 3 3 3 r i g . 2-4 Block diagram f o r the known sys tem 12 = - V s i n 6. t j V t q j = V cos 6 . t J V o d j = - V s i n i,. o J V . oqjj = V cos i>. T. z J = V - V t o (2-14) The l a s t equation can be written as •R -X-i where where LX r I,.-i R-l U l qj r - s i n 6 . -| 3 L cos 6* . -I J r V odj L V (2-15) oqj V o d j " = r 0 i E* . -qj r° X.-i J - V . -oqj . I . 3 0 -r i dj L - I . J qj (2-16) X. 3 Synchronous reactance Transient reactance An i n t e r n a l voltage of the j w " machine Substituting equation (2-16) in t o equation (2-15) we have X'. 3 E' . qj .th - v - I . -qj X e X + X. 3 X' = e z 2 = 3 X + X'. 3 R 2 + X e X' e v - -x /z 2 e 3 ? 2 J " -R/Z2 V . r -R E'. qj 7 2 j ux' X R-r s i n 6 L cos 6 . J (2-17) L i n e a r i z a t i o n of equation (2-17) gives 13 dj = - v A E' . + qj - v A 6. + J A V t -A I .-qj - f . -qj L g q ^ (2-18) where r-f dj f . J qj -v to i-X R e z2 Z j L R -X ' J t- s i n 6 jo L- cos 6 . -I jo * " g d j - ' s q j -R X - i e R s i n 6 . ~\ jo cos 6 . -1 jo V and 6*. are the i n i t i a l values of the terminal voltage magnitude and to 3 o torque angle, respectively. th The per unit e l e c t r i c torque of the j system may be calculated from T . = V , . I , . + V . 1 . ej odj dj oqi qj (2-19) where V ,. and V . were given i n equation (2-16). Therefore the e l e c t r i c odj oqj torque can be expressed i n the form T . = E'. I . - (X. - X!) I,. I . ej qj qj J J dj qj (2-20) which can be l i n e a r i z e d A T ej K. . A 6 . + K„ . A E 'j. + B 1 A V l j J 2j qj 1 t (2-21) where -Kir r- 0 -= 1 _B . _ 0 . I . + qjo r f . - f d j qj Y -Y 2j " l j L g qj - g d j J E' . - (X. - X'.) I -, qjo J J djo 1 (X. - X'.) I . J j J qjo 14 Similar to (2-8) we have (1 + ST!) E \ = E _ . + (X. - X!) -I.. L i n e a r i z a t i o n of t h i s equation gives A E \ = K„. (K. . A <S . + B_ A V ) / (1 + ST'. K„.) qj 3j 4] j 2 t ] 3] (2-22) (2-23) where K3. - 1/(1 - (X. - X-) Yy) hi - ffj - XP hi B„ = (X. - x!) g,. 2 3 3 dj We have assumed a constant f i e l d voltage, i . e . , A equals zero. F i n a l l y the i n f i n i t e system voltage i s 2 2 2 V = V ,. + V . o odj oqj (2-24) where- V Q (j.. and V Q ^ J were given i n (2-16). L i n e a r i z a t i o n of equation (2-24) gives where A V = K c. A 6. +K,. A E ' . + B A V. o 5j 3 6j qj 3 t (2-25) -B3 J r f V qj Y23 x ' £,. -(1 + !• Ty) r-V x . i odj j °qj From equations (2-21) and (2-23), a block diagram representation of the dynamic i n f i n i t e system may be drawn as F i g . 2-5. 2.3 The Complete Block Diagram and the State Equations The complete block diagram i s shown i n F i g . 2-6. Let the state equations of the known generator and the unknown dynamic i n f i n i t e system be written as X = A X + P V '4ft im 1 1 ' , t. - 1 + SljK3j B-7 F i g . 2-5 Block diagram of the unknown systein 16 where Let and Y = H X X: the state v a r i a b l e vector A: the state v a r i a b l e matrix V: the distrubance P: the disturbance input vector Y: the output vector H: the observation vector X = [A 6\, A to., A E,_., A E* A 6., A co., A E * . ] T x' i ' fDx' q i ' 2 3 qj Y = A V From equations (2-13) and (2-25) the values of A V t and A can be solved A V = (JSL A 6. + K,, A E' + C . L , H . + C . L , A E' .) / F. t 5 i i 6 i q i 3 5 ] j 3 63 q j ' 3 (2-26) A V = (K,_. A 6. + K,. H ' . + B . L . H , + B , L , A E* .) / F 0 o 5 j j 6 j qj 3 5 i i 3 61 q i ' 3 where F = 1 - B C„ j 3 J becomes From F i g . 2-4 and assuming A T equals zero, the torque equation m A to, = -(K.. . A <5 + K„ A E' . + C- A V ) / (M, S + D j (2-27) i l x i 2 i q i 1 o i i c l 1 Substituting the value of A V q from (2-26) into (2-27) and rearranging terms, we have A co = a_- A 5. + a 0 0 A co. + a„. A E 1 . + a._ A 6. + a„-, A E'. 21 x 22 x 24 q i 25 i 27 qj where = - ( K ^ + C± ^ K^/Fj / M. a 2 2 = -Di/Mi a24 " " ( K 2 i + C l B 3 K 6 i / F 3 } 1 \ (2-28) F i g . 2-6 The complete block diagram 18 a 2 5 - - C l K 5 . / M . F 3 a 2 7 = " C l K 6 j 1 M i F 3 From F i g . 2-4 we have A E f M = - K ± A V t / ( 1 + S T ± ) ( 2 - 29 ) S u b s t i t u t i n g t he v a l u e o f A V f r o m (2 -26 ) i n t o e q u a t i o n ( 2 -29 ) and r e -a r r a n g i n g t e r m s , we have A E f M = a 3 1 A 6 i + a 3 3 A E f D i + a 3 4 A E q i + a 3 5 A &i + a. A E ' ( 2 -30 ) 37 q j where a a a . A l s o f r o m F i g . 2-4 t h e i n t e r n a l v o l t a g e may be w r i t t e n A E q i " K 3 i ( A E f D i " K 4 i A 6 i " C 2 A V I ( 1 + ST1 K 3 i > ( 2 - 3 1 ) S u b s t i t u t i n g t h e v a l u e o f A V q i n t o ( 2 - 3 1 ) , we have A E* . = a... A 6. + a . _ A . + a . . A E 1 . + a.,. A 6-: q i 41 i 43 f D i 44 q i 45 J + a 4 7 A E q j ( 2 ~ 3 2 ) where a^ = - ( K 4 . + C 2 ^^.f^) / T ' a 4 3 = 1 7 TI a 4 4 = " ( 1 / K 3 i + C 2 B 3 K 6 i / F 3 ) 1 T l 31 = "K i K 5 i 1 T i F 3 33 = -1 / T . 34 = - K . l K 6 i / T i F 3 35 = "K i C 3 K 5 . / T. X F 3 37 = - K . X C 3 K 6 j 1 T. X F 3 a 4 5 - " C 2 K 5 j 1 T x F 3 = - C 0 / T! F . a 47 " 2 ~6j ' From F i g . 2-6 t he s t a t e v a r i a b l e s o f t h e unknown s y s t e m can be s i m i l a r i l y d e t e r m i n e d A to. = -(K. . A 6. + K_. A E ' . + A V ) / (M. S + D.) ( 2 -33 ) J l j J 2 j q j 1 t j j A E ' . = K Q . (K. . A 6 . +' B_ A V ) / (1 + ST'. K „ . ) ( 2 -34 ) q j 3 j 4 j 3 2 t ] 3] 19 Substituting the value of A V from (2-26) into equations (2-33) and (2-34), we have A co . 3 where And = a %1 l64 l65 l66 l67 61 A 6. + a,. A E'. + a r c A 6. + a,, A to. + a,n A E 1 64 *65 66 '67 " B l K 5 i I M j F 3 " B l K 6 i ' M j F 3 - ( K l j + B l C 3 K 5 . / F 3 ) /M. -D. / M. J J -(K_. + B. C, K,./F„) / M. 2j i 3 6j y j qj (2-35) A E' . qj a,, A 6. + a_,. A E'. + a_ c A 6. + a „ A E'. 71 i 74 qx 75 j 77 qj (2-36) where l71 l74 l75 l77 B 2 K 5. / T' F 3 B2 K 6 i / T a F 3 (K 4. + B 2 C 3 K 5./F 3) / T« -(1/K 3. - B 2 C 3 K 6./F 3) / T' For the purpose of estimation, the terminal voltage. A. V of the th i system i s chosen as the v a r i a b l e to be measured, which does s a t i s f y the observation condition A V = h A 6. + h. A E'. + h A 6. + h A E'. (2-37) t I x 4 q x 5 j 7 q j where h^ to h^ can be found d i r e c t l y from equation (2-26). Hence, the state v a r i a b l e representation f o r the o v e r a l l system i s as follows X = A X + P V Or r-A 6. -X r 0 377 0 0 0 0 o - - A 6. - i X A toi • a21 a22 0 a24 a25 0 a27 A to. X A E f D i 1 a 3 i 0 a33 a34 a35 0 a37 A E f D i A • * V = a41 0 a43 a44 a45 0 a47 A E* . qi A 6 . 3 0 0 0 0 0 377 0 A 6. J A to. J a61 0 0 a64 a65 a66 a67 A to. J -A E' . . qj L a 7 1 0 0 a74 a75 0 a 7 7 " - A E' -qi + P V 20 And Y = H X or A V = [h, 0 0 h. h_ 0 h i t 1 4 5 7 rA 6 i " A CO . X A E f M A E' q i A 6 . 3 A co. 3 _ A E* . . qj Some system matrix elements are already known for any given oper-ating condition. The others are functions of the unknown parameters X^, X'., M., D. and T! to be estimated. 3 3 3 3 2.4 The I n i t i a l Values From the computation of the constants i n the previous sections, the i n i t i a l values of the machines must be known. From the phasor diagram i n the c l a s s i c books on A.C. machines F i g . 2-7, the i n t e r n a l phase angle i>. can be found f i r s t and then the current and voltage components can be xo found as follows. ' i o = cos -1 io V t o (2-38) xo = tan , I j X cos (j). -1 i o q r i o V„ + 1 . X s i n <f>. to xo q xo (2-39) I,. = I. s i n (i> + d>. ) dxo xo i o Txo I . = 1 . cos 0/. +<!>.) qxo xo xo xo (2-40) (2-41) V. . = V^ . s i n i>. tdo to xo (2-42) V t = V t cos i'. tqo to xo (2-43) 21 F i g . 2-7 Phasor diagram of system i E' . = V + I j . qio tqo dio d (2-44) V = (1 + Y Z ) V - I . Z o v ' to i o = Vdo + J V qo , V 1 _ao 6. = 90 - tan i o V do th S i m i l a r l y , f o r the j machine we have • -1 T O cp. = cos J — jo V I. o jo , .1. X. cos d>. i. = t a n " 1 v 1° .1 v -1° . j o V - I. X. s m <j). o j o j I. = I. - Y V. jo jo i o to (2-45) (2-46) (2-47) (2-48) (2-49) 22 I,. = - I . s i n (i. - <f). ) (2-50) djo jo v r j o Y j o ' I . = I. cos (i. - <j). ) (2-51) qjo jo j o Y j o ' V = -V s i n (2-52) odj o j o V . = V cos i. (2-53) oqj o jo E' . = V - I j . X'. (2-54) qjo oq djo j E. = V - I j . X. (2-55) jo oq djo j 6 . = i>. + 6 . - i (2-56) j o j o i o i o 23 CHAPTER 3 ESTIMATION OF UNKNOWN SYSTEM PARAMETERS In t h i s chapter, the method of least-squares estimation and i t s a p p l i c a t i o n to the model developed i n Chapter 2 w i l l be presented. In this method, the values of some of the parameters i n the mathematical model of the system are adjusted u n t i l the calculated values, which are a func-tio n of the state v a r i a b l e s , agree with those measured on the actual sys-tem. In other words, i f y i s the measured value of the ac t u a l system and m y i s the calculated value from the mathematical model, where both of them c are time varying, then the method i s to f i n d the values of the parameters i n the mathematical model which minimize the difference between y and y m^ J c throughout a given time period. 3.1 Mathematical Formulation As shown i n Chapter 2, the state v a r i a b l e vector of the model i s X = [A <5., A co., A E _ . , A E* A 6., A co., A E ' . ] T , i i fDx' q i ' 2 2 12 and the parameter vector i s a = [X., X'., M., D., T ! ] T 2 2 2 2 2 The model i s described by X(a) = A(a) X(a) + P V (3-1) y (a) = H(a) X(a) (3-2) Note that not a l l the matrix elements depend d i r e c t l y or i n d i r e c t l y upon the parameter vector a. As i n optimal control theory a should be chosen to decrease a cost index J of the form l t f T J< a) = f t 1 e (<*) R e(o) dt (3-3) o 24 where e(a) = y - y (a) (3-4) ym c t : I n i t i a l time o t^: F i n a l time R : P o s i t i v e value Expanding y c ( a ) about a given i n i t i a l value a Q i n a Taylor ser i e s and neglecting the higher order terms, we get 9 y . ( a ) y c ( a ) a, y c ( a d ) + ( ^ ) Aa (3-5) a o where A a = a - a o From equations (3-3) and (3-5) l \ 9 y c ( a ) T J(a) = \ t / [ ( y m - y c ( a Q ) - (-f-^) Aa) R o a o 3 y (a) ( y m " y„(a ) - ( a°„ ) Aa)] dt (3-6) m c o d a a o J(a) i s minimized with respect to Aa 9 J(a) = n 3 A a (3-7) Hence tl 3 y (a) T t f 3 y (a) T 3 y (a) A o , !* t /• "-irr-> R < y n - W > ] d t / t / " - r j -> R ( " iV"> ] d t " a o (3-8) a o a o o The parameter vector ct i s then updated a, => a + k Aa (3-9) 1 o where k i s the step s i z e or gain value and has to be chosen according to ce r t a i n c r i t e r i o n . 25 3.2 Evaluation of the Integrals P a r i t a l d i f f e r e n t i a t i o n of equation (3-2) with respect to a gives d a o d a d a o a a a o o o where X(a)^ ^ e £ o u n ( j f r o m equation (3-1) as follows d a a o It <¥^> - «•„> <%Fh + <¥^> a a o o where 3P , 8V , 3 x(a)> _ . fwm 8a 8a 1 8 a ; ~ U ^ ; a , t o o A l l the p a r t i a l derivatives of A(a) and H(a) with respect to a w i l l be de-ri v e d i n section 3-3. With equations (3-10) and (3-11) and assuming we have R and t ^ , Aa can be found from equation (3-8). 3.3 P a r t i a l Derivatives of A(a) and H(q) A(a) and H(a) are function of Kj , B s and Cs which are i n turn functions of f s , gs> ys and z^ . To f i n d those p a r t i a l derivatives some current, voltage and phase angle p a r t i a l derivatives must be found f i r s t . F i g . 3-1 shows the phasor diagram of the dynamic i n f i n i t e - s y s t e m discussed i n Chapter 2. From the diagram we have 6\ = TJj + y (3-13) where v i s constant as long as V , V , and I. are constants. Therefore ' ° o' t ' j 8 6 . 8 - / . _ _ J - = (3-14) 8 a 8 a v The p a r t i a l derivatives of voltage V q and current I_. with respect to a are zero because these are d i r e c t l y measurable values. However, this statement does not apply to t h e i r d and q components with respect to X.. 26 8 V o 0 (3-18) 27 3 I. 3 = 9 a 0 (3-19) 3 a = 0; except 3 V ... odj 3 X. J X. I = - i . a + - J —iL> q3 j (3-20) 9 V . 9 a = 0; except 3 V . oqj 3 X. 3 I 2 (3-21) di = 0; except 3 I., dj (3-22) 9 a 3 X. 3 E. 3 I . _J3L1 _ 3 a 0; except 3 I . 3 X. J I . I , . E. 3 (3-23) Next we have 3 Y 3 Y 9 Y X X ' T T 1 = = 0 e x c e P t T x ! 1 = 7 ~ » <3~24> 3 3 9 Y X 2 •ri?- - fk ( 3 - 2 5 ) 11 3 (3-26) (3-27) ! _ l 2 i 1 5 9 X 3 " Z 4 3 3 Y„. R X ?J_ = e 8 X3 2 4 3 With the p a r t i a l derivatives obtained so f a r , we may obtain 9 f d . 3 f . = -rf- = 0; except 3 f V I . X' . v 3 = — r (X cos 6. - R s i n 6.) 1 (R cos 6 . + X s i n 6.) + s i n .6 . ] 9 X z2 E e j j z2 3 e 3 3 3 3 (3-28) 9 f V X -—-£1 = , e (R cos 6. + X s i n 6.) (3-29) 9 X z4 3 e 3 3 28 V I . X' ~ [-jiR- (X* s i n 5. + R cos 6.) +4 (x* c o s 6 • ~ R s i n 5 •)! z2 E. e j j z2 e j j 3 3 (3-30) 9 f . V X = - -4 [-4 (X' cos 6. - R s i n 6.) - cos 6.] (3-31) 9 X z2 L z2 e j 2 3 3 3 3 g 9 g . — — 1 = ^ 3 = 0: except 9 a 9 a ' F I.. X' AT [ (R cos 6 . + X s i n 6.) + (R s i n 6. - X cos 6.) + cos Z 2 Ej ^ 6 3 Z 2 3 * 3 3 3 (3-32) 3 g,. X ^-—1 = - ( R s i n S . - X cos 6.) (3-33) 3 X' 4 3 e j - I . X' Ar f-^ 3- (X' cos 5. - R s i n 5.) % (X' s i n 5 . + R cos 6.)] 2 E e 3 j ' 2 e 3 3 Zj J j (3-34) 9 g . , X . VV = AT [--I (X' s i n 6. + R cos 6.) + s i n 6.] (3-35) 9 X z2 z2 e 3 3 3 3 3 Next we have 9 K 9 a = 0, m = l , ...6; except 9 K . 9 E'. 3 f . 9 f . 9 X qj 3 X. q 3 9 X 3 3 d3 3 X j 9 1,. 3 f , . 9 1 . •^L + I + f 21) - f I + f qj 9 X j qj 9 X.. dj 9 X ^ qj dj - f j . I . (3-36) dj qi 9 K . 9 E'. 9 f 9 f . i-L = f £LL + E t 92. _ ( x - X') (I 3 1 9 X! rqj 9 X'. qj 9 X'. ^Aj V Udj 3 X! 3 H J 3 1 3 3 f + I . ^—If) + f . I + f ,. I . (3-37) q3 9 XV q3 d3 d3 q3 29 3 E', 9 Y 9.1. 9 1 . Y 21 + E • =3- + 31 _ f x - YM (Y ^ J -2j 9 X. qj 3 X. 3 X. ^ j V U l j 3 X. 3 H J 3 3 3 3Y., 3 1,. 3 Y + I ~* + Y — + I ——=^ - Y I - Y I qj 3 X j L2j 3 X.. ""dj 3 X ; *2j ""dj * l j qj (3-38) 3 E* . 3 Y 3 Y 3 Y 0. + I 2j dj 3X!' dj 2j qj l j (3-39) K33 ( Y l j + ( X j " X ? *-T& (3-40) K3j <" Ylj + <Xj " Xj> ^ (3-41) f d j + <Xj - Xj> 3 X. " f d j + <Xj " X ? !3 f t (3-42) (3-43) 3 f 3 V =^  (x! v . —|l+x: f.. -jSfll-v .. f . V q ] oqj 3 X . j dj 3 X odj qj - X. V 3 f . 9 V , . JLL _ x. f 2 i L j odj 3 X, ~j qj 3 X j ) (3=44) (X! V 3 f,. 3 f . — ^ + V , fAA - X, V —If) (3-45) V„ v j oqj 3 Xj oqj ^dj ' j odj 3 Xj 3 V oqj 3 Y, r ( ( 1 + x j V 3 x . o J + v . x ! oqj j 3 X . V Y odj *2j - X. V 3 Y_. 3 V 2j_ _ odj, j "odj 3 Xj j 2j 3 X j ; (3-46) 0, m = 1, 2, 3; except 3 E* . .3 g . 3 1 . K — a i + E « qj - (x - X M fs 2 1 gqj 3 X + *qj 3 X ^ X j V t g d j 3 X *J J J 3 g , . 3 1 , . 3 g . q j 3 X j g q j 3 X.. + X d j 3 X ^ g q j ldj ~ g d j X q j 3 E ' . 3 g . 3 g . 8qj 3 X ' + E q j 3 x! ( X j Xj> ( I d j 3 X ' J J J + \ j + «dj X q j + g q j * d j 8 g d i g d j + ( X j " X ? -g, 4 + (x, - xl) 3 g d i ' d j ^ j 3 X ' i 3 g , . 3V . 3 g . - ± (x' v —31 + X ' g °q j - v X —31 V D ^ j oq j 3 X + X j g d j 3 X V o d j X j 3 X . J J J v J . g - X g o d i l ) o d j S q j j S q j 3 X ; i 9 g d i 3 gqJ F ; ( X j V o q j T x 7 + V o q j g d j " V odj X j T X T ) 31 Next we have f o r the matrix A(a) the following d e r i v a t i v e s . 3 a, 11 = 9 a = 0, i , j = 1, 7; except 1*21 C 1 K 5 1 3 E 3 x 3 8 °24 C l K61 3 B 3 3X. " " M i F 2 H . »-55) a 2 5 ^ 8 K 5 j K 5 j . c 3 3 B 3 ~ T ~ ^ SYY + v ." • a v ) (3-56) 9 X M. F„ v 9XX. F„ " 3 X. 3 x 3 j 3 j -?-7 - _JL_ _ / 6j 6j 3 3, 3 X. M. F 9 X. + F, 3 X.* (3-57) J x 3 j 3 j 3 a K. K . C_ 3 B, 31 _ x 5x 3 3 3 X, 2 9~1T (3-58) T i F 3 J 9 a34 K i K 6 i C3 9 B 3 "~3~^" T F2 F X 7 C 3 " 5 9 ) i 3 2 3 a,. K. C„ 3 K_. K_. C 0 3 B. 3 5 = _ i 3 , 5j_ 5,1 3 3. 9 X. T. F 9 X. F 0 9 X / (.3-bU) J i 3 J 3 9 a37 9 X. J K. C_ 3 L . . 1 3 . 6j_. T. F„ 9 X.' i 3 i + K, . C 0 3 3_ 63 3 3. F„ 3 X.J (3-61) 3 a 41 9 X. K 5 i C2 9 B 3 TI F2. 9 X j x 3 J (3-62) K 61 C 2 ! i 3 C„ 3 L . K_. C, 9 B, T| F 3 V 8 Xj F 3 8 X ^ F 3 s 8 X 8 K... K.. . C„ 8 B_ £l + 6 J 3 3 ) 8 X. M. F„ C8 X. F„ 8 X / J 3 3 3 3 K 6 i M. F 3 8 B •8 X 1 , C 3 B l 8^3 8 X. , 3 K.. . C- 8 K_. 3 B . _ J L r i l + _3 ( 5j_ 1 M. L 8 X. F 0 1 8 X. 5j 8 X. 3 3 3 3 3 K_. B. C, 8 B, + 5 3 1 3 I)! F 0 3 X / 1 3 J 8 K 8 K, 8 B, + K, M. 8 X. F 0 v 1 8 X. 6j 8 X. 3 . 3 3 3 3 K, . B- C_ 8 B„ + _ 6 j _ L _ j 3 3 } ] F 3 3 X j K_. 8 B. B„ C„ 8 B, 5 i , 2_ 2 3 3_s T*. F„ l3 X. F. 8 X / J 3 3 3 3 (3-63) (3-64) (3-65) (3-66) (3-67) (3-68) (3-69) (3-70) 33 3 a,, K , . 3 B , B_ C„ 3 B , ZA = 6 1 f 2 3 1) (3-71) 3 X . T! F 0 3^ X . F_ 3 X / v ' 3 3 3 1 3 3 3 a,_ , 3 K , , C , 3 L , 3 B , JJ1 . J, r l l i U l 3 ( B 1151+K H i 3 X . T! 1 3 X . F 0 V 2 3 X . 5j 3 X . 3 3 3 3 3 1 K , . B 0 C , 3 B , + 2 3 -^x3-)] (3-72) 3 3 3 a__ 3 K C 3 K 3 B — — - 4 [4 ^4 L+ ir 3-(B„-r4l + K. —^ 3 X . T'. l__2 3 X . F Q v"2 3 X . 6j 3 X . 3 3 K 3 j 3 3 3 3 K , . B . C , 3 B„ S i m i l a r r e s u l t s may be ob ta ined f o r ^g^x^ * 3 A l s o g^*- = - ^ , k = 1, 4, 5, 6 and 7 (3-74) 3 3 9 a66 = __1 3D. ~ M. 3 3 9 a7k a7k (3-75) j - ^ = - ^ , K = 1, 4, 5 and 7 (3-76) 3 3 F i n a l l y f o r the v e c t o r H ( a ) , the d e r i v a t i v e s are as f o l l o w s 3 h . - — - = 0 , i = 1, . . . , 7; except a ct 34 9 h 9 X. J K 5 i C3 8 B 3 ,2 . 9 X. (3-77) 9 h. K, . C Q 9 B_ 4_ _ 6 i 3 3_ 9 X. Jl 9 X. (3-78) 9 h_ C_ 9 K_. K C. 9 B 1 = i f 5,1 5.1 3 3. 9 X. F„ v 9 X. F Q 9 X / J 3 j 3 j (3-79) 3 L C Q 9 K , . K,. C. 9 B„ L = _1 / 6j_ _ 6 j _ 3 . 3, (3-80) 9 X. F / 3 X, F„ 9 X / V ' 3 3 3 3 3 Similar results may be obtained for ^g^x^ j 35 CHAPTER 4 ALGORITHM, DATA AND RESULTS In t h i s chapter, the algorithm for computation i s developed. Numerical examples are included, and the r e s u l t s are presented. 4.1 Algorithm A flow chart of the estimation algorithm i s shown i n F i g . 4.1. There are three loops. In the f i r s t loop the two d i f f e r e n t i a l equations (3-1) and (3-11) are solved by the Runge-Kutta method, and the two i n t e -grations i n equation (3-8) are calculated using the trapezoidal method of i n t e g r a t i o n . A l l c a l c u l a t i o n s are made with the time step At of 0.05 or 0.1 seconds. The value of R i s chosen to be 10. Being s c a l a r i t makes no difference whether R i s chosen to be 10 or 100. In the second loop, the estimated parameter values are updated. The step s i z e factor k_^  must be chosen c a r e f u l l y , an excessively large value may cause to overshoot and diverge, whereas a small value may cause very slow convergence. In the program k_^  i s chosen as follows 1. I f |Act ./a.| > £, k. = k. l a./Aa . l X 1 X 1 1 X X 1 2. I f l A O j / a J « ? k i = k2 where t; i s chosen to be 0.3, k^ i n the range of 0.10 to 0.45 arid k^ i n the neighborhood of 1.0 The t h i r d loop i t e r a t e s the estimation u n t i l a l l the unknown parameter values have converged, with the maximum number of i t e r a t i o n s r e s t r i c t e d to 25. j B>ad data | Set I t e ra t i on count to zero L L - C Ca lcu la te constants ot known system: K M . . . . K . , . C , . C 2 . C 3 Ca lcu la te constants of unknown system: K I T ••• K 6 1 ' B 1 ' B 2 ' B 3  Ca lcu la te matr ices : Set numerator and denominator of cost Index to zero: JNITM - 0, JDKN - 0 I Set converged and t o t a l parameter counts to zero, ICON - 0, I T O T - 0 Increase t o t a l parameter count by one, ITOT - ITOT + 1 No Increase converged parameter f:ount Terra - JCON + 1 Increase iteratJorJ count IT • IT t 4-1 Flow chart of the estimation algorithm. 37 4.2 Data The system has the following data. Generator, e x c i t e r and voltage regulator of the known system i n per unit 0.9 P. X, = 1.0 d D. = 10.0 x V = 1.05 t X' = 0.1 d T! = 7.8 x T. = 0.05 x X = 0 . 6 q M. = 5.0 x K. = 20.0 x 2. Transmission l i n e R = 0.04 X = 0.5 3. AV i s pre-recorded as F i g . 4-2. A AT of 20% for 0.1 second i s applied t ni to disturb the system dynamically. 4. System operating conditions Generator l o c a l load P. X V t PF G PF A l 0.9 1.05 1.0 0.5 0.9 (lag) A2 0.9 1.05 0.9(lead) 0.5 0.9 (lag) A3 0.9 1.05 0.9 (lag) 0.5 0.9 (lag) B l 0.9 1.05 0.9 (lag) 0.35 0.9 (lag) B2 0.9 1.05 019 (lag) 0.25 0.9 (lag) B3 0.9 1.05 0.9 (lag) 0.15 0.9 (lag) Table 4-1 Operating Conditions 4.3 Results The number of i t e r a t i o n s to converge on the unknown parameters for various cases l i s t e d i n Table 4 r l i s shown i n Table 4 -2 . 38 -0.8E-03] Fig. A-2 AV due to disturbance 39 I n i t i a l Guess (Per unit of true value) A l A2 A3 B l B2 B3 3.0 13 10 12 8 7 7 2.0 10 9 6 5 6 6 0.6 17 21 15 18 11 12 * nonuniform — — 14 11 7 7 Table 4-2 Number of it e r a t i o n s for various cases. Nonuniform guess X. = 2.0, X! = J 3 0.5, M. = J 2.15, D. 1 = 3.0, T'. J Typical results are shown i n Fig. 4-3 to 4-14. Regardless of the generator output, l o c a l load and i n i t i a l guess, the estimated unknown system parameters always converge to the same values, X. = 0.576549 J X! = 0.414053 J M. = 9.26075 J D. = 25.91 J T'. = 5.273 J Details are as follows. The effect of i n i t i a l guess For the same operating condition B2, comparison of the figures on pp. 48-51 shows that (a) Convergence i s faster for an overguessing than an underguessing; (b) None of the uniform and nonuniform guesses takes more than 12 it e r a t i o n s for the estimation to converge. The effect of l o c a l load For various l o c a l load, otherwise the same, operating conditions, B3/B2/B1/A3, comparison of the figures on 40 pp. 52/48/47/43 shows that (a) Convergence i s generally f a s t e r with the increase of l o c a l load; (b) None of the cases takes more than 6 i t e r a t i o n s f o r the estimation to converge. The e f f e c t of generator power fa c t o r For various generator power fac t o r s , otherwise the same, operating conditions, A1/A2/A3/A3', comparison of the figures on pp. 41-44 shows that (a) Convergency i s generally f a s t e r for a lagging power factor than a leading power factor; (b) None of the cases takes more than 12 i t e r a t i o n s f o r the estimation to converge. 44 (per unft)\ i 1 1 1 1 1 — — i 1 1 1 / 2 3 4 5 6 7 8 S 70 Iteration no F i g . 4-6 Parameter estimation resul ts for case A3 (non-uniform guess 1) F i g . 4 - 7 Parameter estimation r e s u l t s for case A 3 (non-uniform guess 2 ) F i g . 4-9 Parameter estimation r e s u l t s for case. B l ! , 1 1 I I / 2 3 4 5 Iteration no. F i e . 4-11 Paramter estimation r e s u l t s f o r case B2 51 F i g . 4-13 Parameter estimation r e s u l t s for case B2 oC , (per unit) 5 2 2.0 Iteration no. 53 C H A P T E R 5 C O N C L U S I O N S A n u n k n o w n d y n a m i c i n f i n i t e s y s t e m m o d e l h a s b e e n d e v e l o p e d i n C h a p t e r 2 f o r p a r a m e t e r e s t i m a t i o n . T h e m o d e l o f t h e l o c a l k n o w n s y s t e m m u s t b e a d a p t e d a c c o r d i n g l y . T h e m a t h e m a t i c a l f o r m u l a t i o n f o r t h e e s t i m a t i o n o f t h e u n k n o w n s y s t e m p a r a m e t e r s , a n d t h e n e c e s s a r y e q u a t i o n s h a v e b e e n d e v e l o p e d i n C h a p t e r 3. T h e e s t i m a t i o n a l g o r i t h m , t h e d a t a u s e d a n d t h e r e s u l t s o b t a i n e d h a v e b e e n p r e s e n t e d i n C h a p t e r 4. I t i s f o u n d t h a t ( a ) r e g a r d l e s s o f t h e g e n e r a t o r o u t p u t , t h e l o c a l l o a d a n d t h e i n i t i a l g u e s s e s , a l l t h e u n k n o w n p a r a m e t e r s a l w a y s c o n v e r g e t o t h e s a m e v a l u e s ; ( b ) f o r t h e s a m e p e r c e n t a g e o f g u e s s o f f t h e t r u e v a l u e s , c o n -v e r g e n c e w i t h o v e r g u e s s i n g i s f a s t e r t h a n t h a t w i t h u n d e r -g u e s s i n g ; ( c ) a l a r g e r l o c a l l o a d r e s u l t s i n f a s t e r c o n v e r g e n c e t h a n a s m a l l e r l o a d . T h e t e c h n i q u e d e v e l o p e d i s v e r y u s e f u l f o r d y n a m i c s s t u d i e s o f l a r g e p o w e r s y s t e m s . 54 REFERENCES 1. P.L. Dandeno, R.L. Hauth and R.P. Schulz, " E f f e c t s of Synchronous Machine Modeling i n Large Scale System Studies", IEEE Trans, on PAS, Vol. PAS-92, pp. 574-582, Mar/April 1973. 2. G.E. Dawson,. 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Hancock , " M a t r i x A n a l y s i s o f E l e c t r i c a l M a c h i n e r y " , Pergamon P r e s s , M a c m i l l a n , New Y o r k , 1964. 3 1 . K i m b a r k , " P o w e r Sy s tem S t a b i l i t y " , V o l . I l l , J ohn W i l e y , New Y o r k , 1948. 32 . C. B i r d ( e d } ) , " U . B . C . M a t r i x , A Gu ide t o S o l v i n g M a t r i x P r o b l e m s " , Comput ing C e n t r e , U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r , Canada, September 1976. 33 . C M . L e e , " The S o l u t i o n o f O r d i n a r y D i f f e r e n t i a l E q u a t i o n s by a S i m p l e R u n g e - K u t t a M e t h o d " , Comput ing C e n t r e , U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r , Canada , J a n u a r y 1969. 34. A . E . F i t z g e r a l d and C. K i n g s l e y , " E l e c t r i c M a c h i n e r y " , M c G r a w - H i l l , New Y o r k , 1961 . 

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