S T A B I L I Z A T I O N AND O P T I M I Z A T I O N OF A POWER WITH S E N S I T I V I T Y SYSTEM CONSIDERATIONS by LEONARD N I C K O L A U S B.Sc, University A THESIS SUBMITTED of IN MASTER in Alberta, PARTIAL REQUIREMENTS FOR Ve accept this of Head the the of of as conforming to standard, Department of the Electrical . THE UNIVERSITY Department Engineering OF B R I T I S H November, \ SCIENCE Committee Members THE OF Supervisor Members of OF Engineering thesis required Research FULFILMENT Department Electrical 1964 THE DEGREE OF A P P L I E D the WEDMAN 1968 COLUMBIA the 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 of the r e q u i r e m e n t s an a d v a n c e d d e g r e e a t t h 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 , 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 r e f e r e n c e I further agree that permission f o r s c h o l a r l y p u r p o s e s may by h i s r e p r e s e n t a t i v e s . The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a Date Columbia thesis or publication g a i n s h a l l n o t be a l l o w e d w i t h o u t of that Study. Department It i s u n d e r s t o o d t h a t c o p y i n g o r permission. Department and copying of this be g r a n t e d by t h e Head o f my of this thesis for f i n a n c i a l written for extensive I agree for my ABSTRACT An i n v e s t i g a t i o n and design of high stabilization, controller solving is made order parameter problems some systems. and p a r a m e t e r these into The aspects of problems the analysis treated are optimization, computation of sensitivity. The developed are applied to a methods 9^ an system optimal order linearized shift technique f,or power system. To used. both stabilize Eigensystem the change new eigensystem parameter used a has in cost an method for that for change used analysis taken. system for form. and G r u v e r ' s with a approximation of the is by the initially sensitivity study is sensitivity function made using in For is order the is This to large setting to computation shifting a steps minimize of method The method time developed method the an approximation matrix. Finally, after subsequently recursive Ricatti is determine applied that successive fast to eigensystem parameter a method determination. new method eigenvalue stable. applied method This conjunction initiated the accuracy quadratic Puri and is A correction may be of in each c a n be the eigenvalue required improving controller, is an optimization procedure solving lation change been made. functional optimal system, sensitivity parameter the in the developed calcuto ensure response for simultaneous reduces computation (9) time the of significantly investigation parameters. plied to the over of The design the time response results of conventional of the a suboptimal ii method thus sensitivity to sensitivity study controller. a enabling large are number then ap- TABLE OF CONTENTS Page ABSTRACT (ii) TABLE (iii) OF CONTENTS LIST OF T A B L E S (v) LIST OF (vi) ILLUSTRATIONS ACKNOWLEDGEMENT . . NOMENCLATURE (viii) 1. INTRODUCTION 2. EIGENSYSTEM S E N S I T I V I T Y SYSTEM 2.1 2.2 2.3 2.4 2.5 3. 4. 4.3 4.4 Problem Formulation Eigensystem S e n s i t i v i t y Analysis An E i g e n v a l u e S h i f t i n g M e t h o d An E x t e n d e d C o l l a r a n d J a h n C o r r e c t i o n M e t h o d . A N u m e r i c a l Example A P P L I E D TO 20 FOR 20 21 22 23 A 26 Problem Formulation A R e c u r s i v e Method f o r O b t a i n i n g Successive Approximations A c c u r a c y of the R e c u r s i v e Method A Numerical Example MULTIVARIABLE 5.2 5.3 CONTROLLER SYSTEM ANALYSIS 6 7 10 12 14 PARAMETER Introduction . ... An E i g e n s y s t e m Form o f t h e C o s t F u n c t i o n a l . . . Performance Function M i n i m i z a t i o n A N u m e r i c a l Example . SENSITIVITY 5.1 A P P L I E D TO 6 COMPUTATION OF AN O P T I M A L 4.1 4.2 6. ANALYSIS STABILIZATION H I G H ORDER 5. 1 EIGENSYSTEM ANALYSIS OPTIMIZATION 3.1 3.2 3.3 3.4 (vii) OF T H E 26 the TIME RESPONSE 28 30 31 OF SYSTEMS 39 S i m u l t a n e o u s C o m p u t a t i o n of Time Response Sensitivities t o a L a r g e Number o f P a r a m e t e r s . D e r i v a t i o n of the Computation A l g o r i t h m A Numerical Example CONCLUSIONS 40 44 46 56 iii Page APPENDIX A A.l A.2 A.3 A.4 APPENDIX B FORMATION OF THE COEFFICIENT MATRIX A OF EQUATION ( 2 . 1 ) Third O r d e r M a c h i n e and T i e L i n e Equations Governor and H y d r a u l i c O p e r a t o r Equations Voltage Regulator-Exciter Equations Summary o f S y s t e m E q u a t i o n s INITIAL REFERENCES 59 ..... 62 63 64 OPERATING CONDITIONS OF A POWER SYSTEM APPENDIX C 59 ,66 FLETCHER AND POWELL'S DESCENT METHOD . 68 69 iv LIST OF TABLES Table Page 2.1 System Parameters 2.2 Eigensystem Shift 4.1 15 Through Parameter Adjustment 18 Error 32 of K Approximation v LIST OF ILLUSTRATIONS Figure 4.1 Page Governor Transfer Operator . Function and Hydraulic 33 4.2 Voltage 4.3 Original 4.4 System Responses for Q = d i a g [l 4.5 System Responses for Q = d i a g [lO 5.1 Sensitivity of S to System Operating Conditions 5.2 Sensitivity 5.3 S-Response Without S-Response with 5.4 of 5.5 Regulator-Exciter System of & to l] . . . 36 10 1 1 1 1 1 1 l ] . 37 Parameters Control, and Initial 48 Gains Over Control, 49 Variation Over of P q .. Suboptimal Control, Over Variation P o Control, Optimal Over Control, Variation Over of Q .. q 52 53 *o S-Response with Suboptimal Control. Over Variation Q *o 5.9 S-Response 5.10 S-Response w i t h Optimal 5.11 S-Response w i t h Suboptimal to 52 Variation Q v. 51 Variation o S-Response w i t h of 1 1 1 1 1 1 1 35 51 5.7 of i Controller Optimal S-Response W i t h o u t 5.8 33 Responses 5.6 of Function P S-Response w i t h of Transfer Without . . . . . . A.l Governor A.2 Voltage Transfer Control, Over Control, Function Regulator-Exciter Variation Over Control, vi v ^ «« 0 54 Variation Over and H y d r a u l i c Transfer of 53 Variation Operator.. Function 55 63 64 ACKNOWLEDGEMENT I wish for his the research continued many h e l p f u l Dr. draft encouragement a n d M r . G. valuable the British Also, the material i n this project, during I have had thesis with Dawson. suggestions. b y my c o l l e a g u e s of this and guidance thesis. a r e d u e t o D r . M. S. D a v i e s In a d d i t i o n , the support and of this d i s c u s s i o n s about for offering final interest, work and w r i t i n g K. V o n g s u r i y a Thanks and t o t h a n k D r . Y. N. Y u , s u p e r v i s o r i s duly from Columbia Telephone f o r reading The p r o o f the reading manuscript ofthe appreciated. the National Research Council Company i s g r a t e f u l l y a c - knowledged. I owe a l a r g e d e b i t patience o f g r a t i t u d e t o my w i f e and c o n t i n u i n g support Doris for her t h r o u g h o u t my p o s t g r a d u a t e vii work. NOMENCLATURE General A nxn system matrix B nxm u m-dimensional control b n-dimensional input u scalar \^ eigenvalue control matrix vector vector input T _ _ i ' X X, T i V V X , w y y o f A and A Vu\ eigenvectors respectively e i g e n v e c t o r m a t r i c e s composed o f columns v ^ , i = 1, . . . n , r e s p e c t i v e l y initial o f x. a n d o f system states state variables of y q parameter A, = J cost Q a positive d e f i n i t e nxn state V a positive d e f i n i t e mxm K nxn R i c a t t i R inverse g(s) c h a r a c t e r i s t i c polynomial h^ coefficients R(s) adjoint vector functional variable control signal weighting matrix weighting matrix matrix of discrete time i n t e r v a l £ct of matrix A of g(s) matrix polynomial of A 2 p^ c o e f f i c i e n t s of g(s) z(t) n-vector w(t) (2n-l)-vector l\ 1 X and V approximate n-vector Q o f A and A prefix of state variables of state denoting of variables a linearized viii (5.33) of (5.34) variable subscript p d/dt, System H field x d-axis x^ operator mutual d and q - a x i s ; reactance transient tie G+jB shunt T^k d-axis damping c i r c u i t I^JQ d-axis transient line Tqo ^ -r^j 1 d-axis ^ a n < a n c ^ ^ q.- ( i a x transient l - a £g hydraulic ^ dashpot Y water x i at time generator short gate actuator Y <-ex exciter time voltage regulator time control signal H generator D damping time and time constant time constant circuit time short time constants constant circuit time constants constant constant constant inertia constant actuator time constant loop gain constant coefficient voltage regulator gain ^2 voltage regulator speed or governor permanent droop ^ governor temporary droop feedback ix bus terminal open c i r c u i t time infinite constant gate governor i rotor constant -rj •^•^ generator open c i r c u i t turbine time between subtransient s and reactance subtransient s stator reactance impedance admittance tdo between synchronous R+jX a condition resistance d-axis Tq operating resistance x\ a 'Cd' initial Parameters R^ , ad an t i me d e r i v a t i v e armature a x^, denoting System P, Variables Q real and r e a c t i v e energy conversion mechanical generator torque CO generator angular rated *d' ^q ' d ' q ' ^ d V V ' angular H[ linkages to ^ * anc 1~ 'ax: terminal V infinite bus g proportional field exciter s radians/second radians/second currents, voltages and flux respectively armature v respectively radians v e l o c i t y , 377 v^ f d ' """f d ' ^ f d in velocity in voltage V of generator generator angle V p j j Q power torque input ^ ,1 output to f i e l d current voltage voltage voltage, control ^/p field flux Vp field voltage g p. u. gate h p. u. h y d r a u l i c a gate a„ governor current voltage linkage proportional proportional movement actuator and f l u x head signal feedback signal x to t o LfJ^^ v ^ linkages respectively 1 INTRODUCTION 1. In the problems the of to steady high power order be system overcome. state systems. The systems techniques but they systems the power ous manipulation calculation are thus costs system size attempt to and is and optimal by derived These in apply equations reduce approximati-ons the the or to problem. a and apply to In number the of method thesis, analysis and of large a laboriof Methods equations by computation accuracy, methods variable system Since this analysis costs. the in of process economize the analysing require computation efficient the the study usually in of single the which compromise systematic to methods difficult control for extending prohibitive either them to many problem are algebraic solve the techniques economization f i r s t . theory these still established essentially some are sensitivity field. to to is the develop trol some of there difficult result required required are of often introducing One stability, multivariable in design, it of one is reducing should intended from modern design of con- power systems. The first stability. theory for to large is usually requirement Although described The method by best system and m u l t i v a r i a b l e form eigenvalue for achieving Van Ness et. by y of a = methods system systems. set of In system operation available these are in not suited modern a n a l y s i s , differential For is control the equations in system state A y (l.l) an stability and are stability, determining analysis. al. satisfactory classical determining variable for stability unstable preferably Laughton^"^ with this system by suggest formulation a method parameter a is is required adjustments. stabilization 2• method which involves the eigenvalues with the eigenvalues and , are used value a to positive determine system eigenvalues was were stability The are difficulty by the parameter parts. and successfully found the real eigenvectors sensitivity. large calculating In their methods required since the the of latter finding handled method sensitivities by of to Van both former determine the of eigen- eigensystem Ness et. transforming of al.. A to The an upper (2) Hessenberg form eigenvectors values and were using then into the QR t r a n s f o r m f o u n d by A X = XX A V = XV X = back of Francis. substitution of The the eigen- (1.2) m and A where Once the the eigensystem real part into account than one a new of an more eigensystem that eigensystem an tivity is ously calculated algebraic These q. in new techniques c o u l d be eigenvalue can each change be that found. and to shifted To allow parameter can However, change. be found take more generalized. Laughton from the (1.3) Aq and ^lE/iq Since is the and Fadeev terms parameter eigensystem eigenvalue determining sensitivity form after • • • X^~J by VE . sensitivity Laughtonf"^ eigenvector for the method eigensystem a parameter eigenvector Both the •• • 2 negatively this approximate = X unstable required sensitivity denotes to one change, A E where E obtained, eigenvalue than parameter suggests was diagj^ only are The eigensystem adapted in and Chapter the were previ- the calculation derived results sensi- was adjustment, additional and F a d e e v a ^ " ^ the eigensystem sensitivity a parameter equations. of the required. required given in ^A/^q. 2 to the stabilization 3 of a power are system. computed mate the new sensitivity is not an extended often system i s used. the and problem making settling parameter available approximate shift, is costly response time. structural is highly This they mainly this engineer response the i s faced of This oscillatory Although are For changes. requires technique. literature, dynamic eigensystem developed. a l r e a d y i n oj>eration, the the approxi- sensitivity is required. method To the eigensystem correction improving sensitivities formulation. eigenvalue correction optimization i n the an Jahn any slow eigenvector a matrix a of system and then system i f the extremely after I f the enough Collar without required a eigensystem a power with eigenvalue s i m u l t a n e o u s l y from accurate For Both or a may be has an application of there are applicable many to methods low (6) order systems. minimize a q u a d r a t i c form Normally, its or solution to solve similarity matrix, form. The dependent details The initial after cost are given methods design calculating or type cost inversion of an an n (n+l)/2 as well x n(n+l)/2 The of to systems. functional inversion t r a n s f o r m a t i o n can to form the be cost an nxn as functional system can function, optimal then used to best advantage transformation matrix. transformation, will be without have the Jordan e v a l u a t e d , even any matrix The canonical with a inversion. time The 3. control improvement optimal for large method similarity transformation. i n Chapter of of computational simultaneous.equations. functional weighting a functional e v a l u a t e d by e i g e n v e c t o r s a r e .used system cost the n(n+l)/2 was developed this obtained using a The of of involves i t s derivative matrix if a derivative matrix Vongsuriya of a controllers t h e o r y may power be system. for large applied Recently systems have to the methods appeared in the l i t e r a t u r e . These methods solve determination of equation involves equation leading The problem is the R i c a t t i equation the control the to to solve signals. integration enormous for the which in Normally of a turn the nonlinear computation R i c a t t i effort matrix. enables s o l u t i o n a of this d i f f e r e n t i a l that sometimes (4) f a i l s due to c a l c u l a t i o n i n s t a b i l i t y . Macfarlane and v Freested (7) et. a l . solved matrix by an for the R i c a t t i eigensystem But their method not completely formulation requires r e l i a b l e a i f t tioned. Puri method and which of the 4 to R i c a t t i the obtain is stable, a It the as w i l l must analysis the of to simple be w i l l be as is an that a is are great order improvement. space is and i l l - is condi- The and the the f i r s t the and for found by i n c o n t r o l l e r is the a design, of operating point response, time is used. integral it very should c o n t r o l l e r s e n s i t i v i t y i n v a r i a t i o n a 2 system dependent chosen any the usually reasons is system. an design, course, Since that suboptimal c o n t r o l l e r stable. Chapter Chapter 4. reason, is power so Chapter i n the approximation i n evaluating engineering this a methods of system applied developed for time the that previous solution approximation optimal system. approximation the the that so developed an to is method simplify i n a matrix c o n t r o l l e r matrix For non-linear v a r i a t i o n double storage than refined optimal although possible. parameters space chosen method applicable. l i n e a r i z e d cause be technique To of a successive convergence R i c a t t i implement is order a condition must noted double storage approximations designed. system a only of which amount present v s t a b i l i z a t i o n recursive d i f f i c u l t be f i r s t successive using The Gruver's the the from o\ less matrix and large monotonic c a l c u l a t i o n To The fast matrix. Puri Gruver requires exhibits R i c a t t i and matrix on the in case these response sensitivity tivity the time two study is required. of an n^* integration of an for In time study this, of Chapter response equation to 1 order is a method variations integrations strated in a numerical optimal controller. system n«n]^ order course, 5, Kerlin for stated that this m parameters system. The sensi- requires digital computation enormous. is developed in m parameters are example required. to reduce for computing system simultaneously. This the the method is structure Only demonof the EIGENSYSTEM SENSITIVITY 2. ANALYSIS A P P L I E D TO S Y S T E M STABILIZATION The methods eigensystem presented sensitivity stabilization problem. applied eigenvalue tend move to author by a This method, it c a n be routine or found by The inaccurate unless is large made for importance method is Van et. in was never would be either used step sizes. with of the high how the Although the stabilizing a after each parameter an eigensystem from order application or an in this used, calculation latter a time systems, of system sensitivity used computation latter change. c o n s u m i n g and t h e are have eigenvalues method were sizes Since system this time step a of Laughton^^ show eigensystem is small analysis here new to If reentering the to extension applied actually required by be an changed. applied. method very is of are a l . ^ ^ a n d analysis possibility former the Ness parameter estimating relations. chapter and w i l l sensitivity the eigensystem this analysis a system mentioned this new if in correction is the of utmost sensitivity extended Collar (12) and J a h n the correction correction fewer method The allows point differential equations and are equations y Matrix to larger step sizes chapter. and Applying consequently Formulation system operating state developed steps. Problem 2.1 method A is generally variables. move in a small a are assumed to described in the = A y by set linearized of first about order an linear form (2.1) nonsymmetric A perturbation region a be about is the matrix and y assumed which operating is a vector causes point. The the of system response after this disturbance y(t) where \^ are the = £ 1=1 e^ eigenvalues - written 1 x of i v A, i as y x^ > o = l ...n 1 (2.2) f eigenvectors of A, and T v^ eigenvectors of . For A x\ = \. A v = T As is in the well known, left half parameters for techniques do to apply sented a c a n be to in 2.2 i a which the for order matrix form. stable is (2.3) the stable. selecting if they the sensitivity parameters is are to Although problem, expecially Eigensystem eigenvalues problem then this systems, method of if The solving determine classical are difficult system is analysis such located that repre- provides a high order stabilized. errors tivity eigenvalues, (2.4) system Calculation Fadeev sensitivity distinct " is Sensitivity by with x. Eigensystem lyzed system complex p l a n e . exist c a n be a v. system high convenient system A of Analysis eigenvalues and F a d e e v a ^ ^ analysis. equations For similar by and applying, completeness to Fadeev eigenvectors in essence, a derivation and F a d e e v a ' s is of were ana- eigensystem the given in sensithis section. The derived q, that sensitivity by taking of the an partial to derivative a parameter of (2.3) q with of A is respect to is A , q x i + A x . , q where A, and *i'q For eigenvalue convenience, throughout this this thesis = \ = W = partial unless ^ i X f q x. + X. x., (2.5) q )>q (2.6) i ^ ^ derivative otherwise (2.7) notation stated. will be used Premultiplying —T (2.5) b y v\ vT1 Since, from one obtains + vT1 A, x. 'q 1 equation for = \., " l ' q v x. 1 1 + A- v 1 1 T T x., l'q v (2.8) ' (2.4), vT and A x., l'q A \^ vT = a normalized eigenvector (2.9) product 1 -T v. x. J 1 equation (2.8) 0 the The by i=j (2.10) for becomes \ . , ^ 1' q giving . -<*1 f o r = sensitivity eigenvector premultiplying of = vT A, x. 1 'q 1 \^ to a parameter sensitivity this v is equation (2.11) ' q. derived from equation (2.5) -T by v. J A, v. where i ^ j . + v. x. Knowing that A = J and b e c a u s e of the x. J (2.12) x\ , 1 . A N, may b e x 1. x. coefficients equation (2.16) °ji Equation q written where c.. J (2.12) x., (2.13) J of the two sets of eigenvectors 0 = (2.14) , c a n be — x.. l'q = X = X as n. q J^l' x. X i are j = 1, , i f o u n d by ...n (2.16) substituting " V ' f o r 1 ^ j ( 2 * 1 7 ) as ... "'* c - (2.15) q (2.15) written I • x. vectors fc,., ± \ equation A v \ -T - X.) vj 3c., /\ ( \ = of into = (2.16) X.v. + X - vT = = x. , The v.x. X., becomes -T v. *j J where = orthogonality vT equation A x., c - , 1 1 X i ' **" ...c X n] .1 n3 J T (2.18) (2.19) and c.. = 0 xi Extending (2.18) sensitivities to include c a n be w r i t t e n X, By assuming a s i m i l a r the e i g e n v e c t o r quiring that (2.21) is a unit into = 1, in matrix = q . . . n , the eigenvector f o r m as (2.20) V K T q matrix. (2.21) T of matrix + Aq.V, ) (X equation (I i X C sensitivity (V I = q i, f o r m f o r "V, V, where all A + Aq.X, ) results + Aq.K )V X(l T = q Substitution (2.22) may b e d e r i v e d I (2.22) of equation (2.20) and in + aq.C) T by r e - = I (2.23) T Since the matrix (2.10) product V X equals and a s s u m i n g t h a t that is Aq K then the s o l u t i o n Hence C is given Neglecting eigenvalues where by = the second are reasonably eigenvalue, sensitivity order of (iq large analysis is terms of according to (2.23) are 0 (2.23) small, (2.24) is -C (2.25) -V C (2.26) T order ± for close is separated. accuracy poor is _T . = ii—.v. V 1 1 A, 1 valid Near merge (2.23) m K Aq C) . terms eigenvalues v, - eigenvalues. A, t o be u s e d o n l y r points, x. ^ second i ^ j ^ k (2.27) V eigensystem f o r the adjustment real the n e g l e c t e d k Since l o n g as t h e a multiple —T . x, k into since — as breakaway £j< v ^ x x 3 become ^ matrix (2.17). complex conjugate terms £>q C T = T V, where the second order o f K from K a unit k = 1 ... n sensitivity of parameters, the close eigenvalue parameter An system the problem adjustment important available by step required performing eigenvalue diagonal the elements of eigenvector A computational to where A, column. It T = A , can As an only a increasing new eigensystem. formulation of the of computation sensitivities the eigen- since products | (2.28) equation (2.11) are given by the as i j (2.29) matrix / V ( also be X C from i made illustration element verified all becomes d.. one J be D by X q from d can ease determining .= ij saving contains the sensitivity advantage. q V i > q proposed matrix matrix C used = recalculating is for the circumvented the sensitivities X and of equations D The be and feature sensitivity information can from ' } if of a - , ij'q (2.28) (2.17) ^ " 1 the J of consider the that ( sparsity this, in as i-th D is A, the row given 2 * 3 0 ) is q case and J i-th simply by fjl, ... *•* jn] X = D a. ., ij 'q v. . (2.31) vni For more than one element in A, , D becomes a sum o f matrices where q 3 each are matrix avoided is formed by with following section utilizing the 2.3 k elements for The parameters in solving eigensystem An E i g e n v a l u e v e c t o r form For obvious (2.31). Thus A, . q the - kn An a p p r o a c h problem sensitivity Shifting (2n 2 of ) is system multiplications taken in the instability equation. Method t h a t are a l l o w e d to change are w r i t t e n i n — T as q = jTj^, . . . q ^ , . . . q£j (2.32) r e a s o n s , the e i g e n v a l u e w i t h the l a r g e s t p o s i t i v e real 11 part, \ , ments. are In is From given by parameter (2.11), to the s e n s i t i v i t i e s \ X » m' q . by 0 order c h o s e n t o he c h a n g e d n e g a t i v e l y J to minimize more favorably real 1 part of the e f f e c t located sensitivity b q., j = 1,...Q, x of parameter eigenvalues, eigenvalue each —T — v A, x m 'q . m = N of X adjust- J adjustments the parameter Re \ \ I , I <l t m k 1 S . (2.33) . on o t h e r q^ with the largest , , , c h o s e n t o be adjusted. Any suitable f r o m A^, gradient technique but f o r the purpose method s i m i l a r of c o n t r o l l i n g t o the Newton-Raphson * q and d A p i s t h e s h i f t The increment in R e ^ X ^ A-1 i ' q lating k i n the other • = allowed a » i = l,...n a l l AA-^ eigenvalues for ' to l increases change. adjustment. satisfactory^\^ i>(1 Once the increment is calculated then k is Aq k has been i n the eigenvectors calculated repeated plane. until = l,...n (2.29). part sensitive then If (2.35) after o f one o f obviously q parameter the process all found, is X C. - k q , = -V C . & q from a new A i s c a n be c a l c u l a t e d = K C from the r e a l most fails i as calcu- these k cannot be may be c h o s e n repeated until found. AX where where | X i m this an a l l o w e d used a m c a n be c a l c u l a t e d i b e y o n d Re ^ ^ ^ > t h e n is is i n X> (2.34) ' Aq I X The next If technique eigenvalues found that s the change k desired. k A c a l c u l a t e £,q = A X ^ R e ^ , ^ k *Xi where c a n be u s e d t o calculated as (2.36) k T k (2.30). eigenvalues and k The a d j u s t m e n t lie process i n the l e f t - h a l f is complex It should parameter (2.35) and be (2.36) and justment. an large rection after by two movement size each to argued of used in This adjustment. The small change region this time case is allows a correction calculated original parameter this argument. ad- each ad- One, unpredictable time be combined if with eigensystem is and consuming. method but method by process A after can fast the from itself sensitivity small consuming may b e c o m e in to about larger countering b o t h methods the restricted eigensystem this of a that reasons best is selecting a new in method eigensystem calculation the is the a method are that this applies method a p p l i e d . following 2.4 only eigensystem step since c o u l d be There believed that calculating eigenvalue two, is It avoided justments the noted adjustments eigensystem. could be It a a cor- calculation detailed in the by first section. An E x t e n d e d Collar and J a h n Correction Method (12) The forming Collar the and J a h n matrix correction the M and R are product apjjroximate the X^ diagonal - 1 A X E (2.37) be and (2.39) into (I + E) products M- + R the (2.37) X^ off small X = = assumed to Neglecting applied and the system diagonal elements parts since matrix of the the A. matrix corrected that X with and R has requires Let = eigenvectors respectively. eigensystem is product XZ}A using method^ small of Xu, and (2.38) _ 1 (M M (I off + E) (2.39) diagonal, results + R) (I small (2.38) substitution of in + E ) . •= matrices then M. R and E, (2.40) (2.40) can be 13 written By as E M insj^ection the - M E solution of = E R (2.41) is e.. i j = r. . / ( i j e.. = 0 •• - m JJ m..) " for i.j = (2.42) l....n n The corrected matrix X is then determined by equation (2.39). In T the in thesis -1 calculations, (2.37) instead Collar of is made inverting and J a h n ' s the method has available and complex m a t r i x been extended is used for X^ X^. to include the T correction This is one First the V of a so that the N is solution to V^ nearly corrected the matrix requirements product where The of a an X T unit of be normalized. 2.2. is found by forming (2.43) and Z requires = T V X will N + Z matrix V X Section unnormalized V = eigensystem product is small and off-diagonal. that N (2.44) T where V is assumed to V with K assumed s m a l l (2.45) into (2.44) (I Neglecting written be given = T and (I off by + K) V^ (2.45) T diagonal. Substituting (2.43) and gives + K)(N products as of small K N Hence + Z) = = N (2.46) matrices K and Z, (2.46) can -Z K = -Z be found be (2.47) N" (2.48) 1 T A normalized V can (normalized and from (2.44) V ) T X the = requirement that I (2.49) as normalized V Substituting from (2.45) and T (2.47) = N _ into 1 V (2.50) T (2.50), the corrected and 14 normalized V becomes V Note In the V T the not the the the (2.35) values. these to At repeated the The the correction and may most the A matrix application eigensystem is the of of c a n be by the be high written in in the need for to the method change, product T V X product is corresponding procedure calculate case of in method is first iteration moves such new eigen- a discrete may as order a close not particularly systems the the eigenvectors correction order a parameter comparing complete eigensystem the by checked. the except diagonal. eigenvalues applied sensitivity stabilization equations to eliminates points, N is checked need (2.51) 1 inaccurate method it the be N" )^ since However, eigenvalues for The shifting matrix. Only eigensystem and of - Z l computed accuracy unit closest (2.34) easily process required. If of is eigensystem X with to -1 N~ that N (I = T converge. well power state manner suited systems. variable (13) form the and the eigensystem eigenvalues stabilizing of after the procedure A matrix. stabilization The initiated position indicates the of by calculating the degree dominant of stability obtained. 2.5 A Numerical To an illustrate example will also The variable where Example of an the application unstable demonstrate the power eigensystem power system form y A y A is derived the system are calculated in parameters for = a given A. initial P is stabilization presented. correction equations are method, This example method. written in state (2.1) Appendix and the system linearized as of , Q This matrix operating and v, as is calculated conditions. derived in The from latter Appendix B. The system 2.1 and parameters the initial = chosen operating Qo .753, for = this example conditions .030, v are given calculated in from (2.52) 1.05 to Table are i, do So All .295, ' = = * values 8 4 5 are i qo ' ¥YO in per Table 2.1 = .654, ' = 9 ' 4 8 ' v, do V = .393, ' k k l 2 o System Parameters Machine Tie Line = .003 .480 H = 4.90 R = .110 = .050 Tv = 1.90 D = 2.00 X = .730 = 40.0 ?r = 1.00 G = .100 = 1.00 = .600 B = .270 = 9.00 = 7.60 d X cr = . 0 2 0 x' x g = .500 (A.35) A the coefficient matrix d x q .020 -.0204 (2.53) unit. r From .974 qo .963 Voltage Regulator- Governor-Hydraulic Exciter Operator t v L do becomes (2.54) = • -30.07 -4.300 -0.514 -0.187 = .130 38.47 57.70 1.000 1.000 -333.3 -2.080 42.06 -63.10 333.3 -19.61 -2.000 4.000 2.000 -1.053 -4.000 -0.133 -1.000 -50.00 -0.064 -0.480 -24.13 To initiato the A computation shifting routine process, similar to an eigensystem that used of by V a n Ness \ was used vectors the to x^ claculate a n d v^ tialization ing x^ by In the step, the the the scalar v shifting ranges A and X . f r o m of .0005 required. et. a l . ^ ^ T found correspondence A is in x^. this to v\ for x is not > T-, > from require A . The eigen- rearranging established. products each process, and V manner eigenvector J i A, are As a since final n o r m a l i z e d by The > t > r stabilizing correction al the parameters .1000, part, m parameter 2. M method by a from q^ from specified from .0000 calculated If most this are > £>, adjusted > within 1.000, is with eigenvalue the largest and. the > 2 (2.55) 3.000 of'the with restrictions with k application summarized below eigensystem and > -3.000 of addition(2.55). positive most real sensitive (2.31). (2.34) which will cause a in other eigenvalues shift the change the that q^. Aq^ exceeds is such the change that parameter must after searching all 2 and 3 repeated be the until allowed Re "> found and parameters an in q^ or Re ^ X , ^ 2 and 3 then Calculate the new eigenvectors from the must parameter- (2.30), Correct the approximate eigensystem by the the repeated. t h e n A.\j) adjustable if (2.31) method be can (2.36). 6. of will found. 5. and 2.3 parameter a Aq^ from in sensitive and 2.4 the 600.0, ^ a m o u n t AXJJ» either change fails decreased be (2.29) y section initial changing If to the Calculate 4. next the of section given Calculate 3. result of Determine A » -600.0 process consideration 1. X 10.00, divid- i. X 1.000 ini- of 17 section 2.4. 7. new Rer>eat eigensystem small as For was the reaches determined the procedure by specific unstable and a desired at negative 1 until value Re o r or AAp " each becomes 4. example had starting of this section, the initial system eigenvalues -333.53 + jO. -2.1109 + jO. -25.091 + jO. -1.0179 + jO. (2.56) The in -15.087 + jO. -0.0052 -4.7632 + jO. +0.0465 + eigenvalue Table 2.2 eigenvalues "sens" after and the shift where found those and only from parameter the the after stabilization correction, procedure jO. J5.4124 results eigenvalues sensitivity equations are designated "corr". The eigenvalues -19.368 + J41.389 -0.8611 -1.5.69 + jl.3869 -0.5316 + values are jO.5316 + JO. (2.57) jO. j5.5464 summarized below with the final underlined. TJL k adjustments The are -1.3869 + parameter summarized shown. + jO. + are are -340.46 -0.5308 The adjustment dominant each + 2 $. = 0.050 = 40.00 — - = — 0 . 0 0 7 2 327.40 - 453.30 1.000 —^-2.620 -2.940 = 0.480 0.002 P-0.1800 *- 0 . 0 0 1 0 = 7.600 *- 2 . 3 6 0 0 t r — 0 . 0 1 4 9 * 0.0143 »- 5 7 8 . 1 0 (2 0 . 0 8 3 0 — > ~ 0 . 0 3 4 0 — « - 0.012 58) — 1.1600 *-r Table 2.2 directly indicates from the that system only five matrix are eigensystem required but reclaculations these are not all 18 2.2 • Shift Dominant Eigenvalue Roots Sens. Shift 1 and 2 Corr. Through Parameter Dominant 3 Sens. Adjustment Root Parameter Table Corr. New Parameter Value -.0052 0 +.0465+J5.412 1 -.0035+J5.541 2 -.0285+J5.555 -.0239+J5.550 -.0052 -.0052 .0250 tl .0149 3 -.0166+J5.513 -.0222+J5.464 -.0084 -.0130 .0031 St .1797 4 -.0433+J5.404 -.0946+J5.383 -.0193 ** .0063 h .0825 5 -.1630+J5.333 * -.0376 * .0125 k O 6 -.1630+J5.335 -!"2000+j5.356 -.0626 ** .0250 t 7 -.2501+J5.357 * -.1977 * .0500 St .0120 8 -.2807+J5.367 -.2885+J5.381 -.2477 -.3485 .0500 St .0021 9 -.2947+J5.381 -.2946+J5.380 -.3485 -.3485 .0063 tl .0074 * -.3485 * .0063 -.3258+J6.503 -.4941+J5.549 -.3507 -.3504 .0250 12 - . 4 9 4 1 + J 5 . 5 4 9 -.4941+J5.549 -.4487 -.4590 .1000 t 13 - . 4 9 8 9 + J 5 . 5 5 3 * -.5090 .0500 St .0001 l 453.3 10 - . 3 0 1 0 + J 5 . 3 9 2 11 ft -.0052 14 - . 5 2 3 9 + J 5 . 5 5 4 15 - . 5 2 3 9 + J 5 . 5 4 7 16 - . 5 2 9 1 + J 5 . 5 4 3 17 - . 5 3 2 2 + J 5 . 5 4 3 Eigensystem *•* -.5166+J5.547 * -X- * -.5269+J5.545 Recalculation No C o r r e c t i o n Required •x- -.0251 -.1477 .0500 -.5091 * .0250 -.5341 ** .0250 -.5316 -.5316 From -.5316 ** -.5316 System .0125 .0031 Matrix k k 2 r 2 l k k t r r -2.621 0304 2 . 362 -2.943 327.4 1.505 1.164 l 578.1 tl .0143 k 19 caused six, by close eleven change. The eigensystem and eigenvalues. The seventeen caused correction is too far are method from the recalculations does by not correct a too required large converge if eigensystem. by shifts parameter the approximate 20 3. EIGENSYSTEM 3.1 this applied minimize in APPLIED TO PARAMETER OPTIMIZATION Introduction In is ANALYSIS chapter, the parameter to a parameter a performance quadratic adjustment optimization criterion method problem. specified of Chapter It i s desired as a c o s t functional = y Q (3.1) y dt T o where to to form J y Q i s chosen 2" = A y t o emphasize emphasize (2.1) selected- state i n t e g r a t i o n time. variables a n d of, F o r an a s y m p t o t i c a l l y ot>0, stable system (3) with initial functional conditions B^ ^ + i s solved A with = initial T (3.3) B_ . , i s solved be h a n d l e d These by the largest minimizing is applied does of 0 that of a symmetric this matrix cost B « i h A <-> = = relation -B^ Q * B^ i n the normal be i n v e r t e d . Thus, restricted reduced inverted the computation to A i n this solutions i s found (3.3). of the eigensystem was then was As a f u r t h e r the eigenvector inversion. i n a single n(n-»l)/2 that requirements. i n a method transformation nxn. an o f a system by c o m p u t a t i o n a l so t h a t With manner the size considerably effort, chapter any matrix of 2 (3.3) were matrix ^ as 3 + B_., f o r each the desired terms B whereby a s i m i l a r i t y not require cessive + o must requirements to i n terms the recurrence i s severely Vongsuriya^^ and h a s shown condition x n(n+l)/2 m a t r i x can ? from B If , Laughton c a n be e v a l u a t e d J where y developed applied contribution transformation the s o l u t i o n o f B^ In a d d i t i o n , the solution step the s o l u t i o n o f A, t h e s e n s i t i v i t y thus of B avoiding c<+ from (3.3) suc- ^ expressed i n equations and c o r - 21 rection method of Chapter cost functional 3.2 An E i g e n s y s t e m The through solution generalized later substituted AX V Assuming that B T the of the of B-^ f r o m to arrive into = + B at minimize is investigated Let the first eigensystem is form of A (3.4) this eigenvectors the and T XV X to Functional B^^. XV applied adjustments. Cost then X easily (3.3) X (3.3), X be parameter Form A be 2 can equation = T are for becomes -Q (3.5) a normalized set then premultiply- T ing and postmultiplying Xx % X T At this where point, BJ is a it is this B| T B X T B Q' = X T Q n(n+l)/2 x] + symmetric elements of = T = X X -X X (3.7b) B X becomes (3.9) diagonal, be obtained B |B'J B-^ i s then found from B Similarly, by r--, = = 1 the by solution of the inspection of (3.9) W A L J l J , JB-JJ (3.7a) (3.8) Q' [ i].. = • - [Q'l.. and since X = can (3.6) (3.7a) (3.6) X is Q X T in X matrix T results define BJ X and B^ and X r e s p e c t i v e l y to = X X X X 1 BJ substitution, is B T convenient XBSince X by complex symmetric [x With + 1 (3.5) i, j = 1, as (3.10) ...n as V.BJ V substituting (3.11) T (3.4) into (3.3) and premultiplying T and postmultiplying analagous to (3.9) by can X and X r e s p e c t i v e l y , be written as a general equation 22 and by inspection Successive found by [B 1 +1 substitution (3.10) results /( x - = ij — of B j , BJ!,, ...B^ in the general ij where to Q' B|, is given (3.11), by can be V l Thus the value of the J = y Performance The written parameters J. method, of the (3.16) J, To = ' q: evaluate ••• (14) to B ^ ^, is B-^ transformation applied that < ' 3 function - J from (3.2) 1 5 ) becomes (3.16) y. minimization procedure q^j method Taking a y |v, B' .V oL'q.,. «+l the ke n (3.17) is used partial found the for requires + VB' , , , V oc+l'q^ equation, are T Appendix C, c this qjQ ••• descent q^, in where ^ the T J same ]_> s u c h + i1 v for function. respect c (3.13) Jij the + summarized i n performance with B d+l jq.i> and P o w e l l ' s V B J into Minimization chosen Fletcher to performance T V B' , Function as This = O J 3.3 Also, applied (3.13) solution l_ (3.7b). \.) i + — minimizing the derivative derivative of as T + VB' a+1 partial >q ~ly J J derivative o (3.19) v of ^i equation (3.14) for The Q . 1 is taken elements B = of ( - D with B' oc+1 ,, T X, + where P.. ij Substituting = the - fi 1 U* + 1 ,~| Jij respect (3.20a), the can t h u s be T QX + X Q X , (3.7b) written /<x substituted as >.) i+ (3.20a) (<* + l ) ( V , 1 eigenvector elements q^ w i t h F. of q + X-, J Ii i sensitivity X into to . + J X-) (3.20b) . (2.20) l + 1 equation C. B ^ ^ , )/(/\ become (-D B T K + 1 Q'c, -JIJ ij + F oc+l'q. Thus with B^^, evaluated ^1 V, into (3.19), the = final i , form in this -V Ct In the • I C parameter system incremental tended Collar calculation that if, a of J, pose of the are « + i " B and J a h n a new * + of of in J, computed f i r s t . 'q. This and given + B «i i' + process Chapter eigensystem calculation (3.21a) (3.21b) = 1, . . . n by substituting by i c correction saving j manner is i adjustment procedure considerable in ' B /(V (2.26) q -T V CK+l ij Vfl»q.U i J B 23 ' avoids (3.22) y. minimize 2 together is used each computation and J , q i to method after Tv J, with to the eigen- the ex- avoid adjustment. effort can the vector TV y o the computation Note be J repetitive also realized and of its trans- matrix products. 3.4 A Numerical To system for k 2 If this study system study. 2 is are They be the Chapter and would are illustrate of the Example found not as 2 other as an its use The 2.1 with that the initial stable, invalid study = 0.0149 = -2.6207 adjustable example. the Table initially f t the so stabilization k and from chosen unbounded and From the chosen taken are were minimization procedure, the for of the 9 ^ initial system , stable. (3.1) optimization these parameters (3.23) 0.1797 parameters of functional a parameter 2, is power parameters exception cost Chapter order from Table 2.1 are k, = 40.00 = 7.600 (3.24) 1 T The parameter straints is of adjustment (2.55). not applicable the methods ranges Since are given the F l e t c h e r to a problem with available for solving by the i n e q u a l i t y and P o w e l l inequality this type descent method constraints, of problem con- and usually (15) require used. large a number H i s method constrained transforms variable Pi = of i t e r a t i o n s , the approach a constrained p^ b y t h e arc s i n y ( q )/(q i this since transformation, now a n u n c o n s t r a i n e d derivative p where J, q is i of q q given (3.25) i'p *i equal y o and Q chosen The process 1 with ( i P after = and q . , as takes l q i l Q nine l = is 2 l l l diagQ. is i p s i then chosen Q iterations optimization method is sought. found from n p i c o applicable The the partial (3.27) s carried out with a value of as l 1 l l ] 1 1 to f i n d 1 (3.28) T 1 1 1 t h e minimum. l] (3.29) The e i g e n - are + jO. -1.0516 + JO. -51.970 + j O . -0.2062 + j O . -9.9075 + j O . -0.1257 + -2.0319 -0.0192 the parameters is t o p^ a s ^ min process to one, y [ min (3.26) i respect ~ max q = -333.34 and i (3.25) becomes • q., by (3.22) optimization (3.1) values J, max i n p-space 1 derivative The i = q^ t o a n u n - 5" ± and P o w e l l ' s optimum of J i n p-space J, a in Fletcher - q i min With variable is transformation - q i o f Box + jO. found are J5.4459 + jO. (3.30) 25 X = 0.100 k, = 1.032 l £ f k Note that This is usually other quite used is This very for at indicates constraints compared to in view good t r a n s i e n t are can be = 1.000 = 1.000 3.000 2 small surprising parameters (2.55). the k^ t their that of the there is limits fact'that stability. limits relaxed. its Also imposed by still of -600. a high note the room f o r and +600. gain that is the constraints improvement of if 26 4. C O M P U T A T I O N In by a this Puri high chapter, order time each for for not of Problem each a is desired linear to which 2 0 where is the system However a developed of e- matrix. computation is a fast limitation recursive here. This the products only method - A y + B system described by u" control (4.1) vector u(t) time, 2 1 ) this the K of [j? Q + u]dt H ¥ T subject to u(t) = to + K S = A that minimizes the K S + with = [A - S in (4.4) -A K calculation is given Q equation = 0 (4.4) (4.5) T the substitution of (4.3) becomes K)y can (4.6) be f o u n d by Ricatti-type - T K B is (4.3) - 1 (4.1) The y(t) Ricatti W" matrices. constraint matrix K (4.2) definite the B (4.1) the - V-VK nonlinear = y T J equation, solution J o positive y equation^ and Ricatti the this systems, involves the solution and reverse rapid, inversion developed Ricatti Since order symmetric minimizes T The lengthy. the S Y S T E M controller successive to is optimal is = constant A K The method an approximation = choose ) K O R D E R p Q and V are ( converge time-invariant J b y for requires high functional u(t) H I G H (nxn). y" It is to Formulation Consider solve this A F O R a p p r o x i m a t i o n method metiiod of matrix order to finally application require matrices 4.1 This approximation solving C O N T R O L L E R used convergence efficient of is solutions.which the does O P T I M A L a successive system. Although method A N and G r u v e r quation for O F K A often - Q matrix + unstable. K S integrating, in differential K Eigensystem (4.7) methods 27 ( 4 . 4 ) are a v a i l a b l e , of solving of a m a t r i x double Furthermore, increasing eigensystem the in accuracy usually order. developed Although i n Chapter eigenvalues are too c l o s e l y Chapters any 2 and 3 by f o r c i n g neighboring Although Ricatti order but they r e q u i r e the o r d e r of the o r i g i n a l eigensystem system ' system the matrix deteriorates the c o r r e c t i o n 2 i s applicable, located. eigensystem A. with procedure i t may T h i s problem f o r the fail i f was. a v o i d e d t h e e i g e n v a l u e s t o pass through positions. t h e r e a r e o t h e r methods a v a i l a b l e e q u a t i o n , t h e most s u i t a b l e systems was found f o r s o l v i n g the one f o r a p p l i c a t i o n to high t o be t h e method o f s u c c e s s i v e a p p r o x i m a t i o n s (8) developed by P u r i approximation generate and G r u v e r . of t h e R i c a t t i A where + A ^ and Puri and G r u v e r from these A be from + = A - S K^ = Q + K^ converge matrix m a t r i x must be c h o s e n left-half K have shown t h a t equations Ricatti initial matrix matrix R i c a t t i equation (J) (J) [ (J)]T (J) . K unique T h i s method c a l c u l a t e s 7 (J) the f o l l o w i n g = 0 ^ ^S de- (4.8) (4.9) K^" ^ (4.10) 1 the s u c c e s s i v e e v a l u a t i o n s of m o n o t o n i c a l l y and r a p i d l y i f i s p r o p e r l y chosen. so t h a t the e i g e n v a l u e s of A ^ ^ complex p l a n e . guess f o r - 1 J _ 1 Q the j - t h If A^^ i s found This ^ to the initial l i e i n the t o be u n s t a b l e w i t h an , then the e i g e n v a l u e s h i f t i n g technique can used. Solving an n ( n + l ) / 2 for x n(n+l)/2 Thus, a l i m i t a t i o n analyzed i n (4.8) u s u a l l y involves m a t r i x , where n i s t h e o r d e r o f m a t r i x i s p l a c e d on t h e s i z e practically. the i n v e r s i o n of o f t h e system that A. can be 28 Instead, a recursive here which with nxn 4.2 A Recursive In result eliminates lengthy what is Method follows, applicable (4.8) can is (4.6) to now w r i t t e n A and K are result, an Thus y ( ? y) K matrix The written where as N is as matrix the Ricatti (4.12) Premultiplying and postmulti- and substituting (4.12) ( y Q y) - system, a A chosen integration of (4.13) w i t h in y)dt T solution (4.13) T J(y Q = o = e A t to (4.14) _ y be expressed in terms of the state At may be for as e an t A over these N ^ integer = Q [e T evaluated constant = (4.15) 0 y[e ] = t y (4.14) a n d c o m p a r i n g t e r m s , K b e c o m e s K matrix e degenerate since as integral transition omitted (4.11) respectively = (4.15) i n t o each be Approximations -0 results y This will the matrices. stable y homogeneous Substituting j Successive A y and y conditions transition tegrands only as o the operates — y„K y Let and developed as = asymptotically initial is (4.11) b e c o m e s at For j . = constant (4.11) b y y the the superscript any —T into for inversions Obtaining K A y plying solving matrix written + T for the be A K where of matrices. equation and method N/R a by A t a time discrete ]dt series interval time = [^(lAO]* and time (4.16) of of successive size At intervals may = inl/R. be ( 4 > 1 ? ) as (4.18) 29 M a t r i x e ^ ^ ' ^ i s c o n s t a n t f o r c o n s t a n t A and l / l t , designated as A^ = w h i c h c a n be c a l c u l a t e d A For 1 a stable = (l/R) (4.19) f r o m t h e power I + A/R + A / 2 ! R 2 2 series + A /3!R 3 + ... 3 property 0 A t Likewise, f o r a stable as t » (4.21) s y s t e m , f r o m (4.17) and ( 4 . 1 9 ) , A^ has (A^)^ f o r R chosen p o s i t i v e . f 0 as N Substituting oo (4.22) ( 4 . 1 9 ) and (4.17) i n t o K c a n be a p p r o x i m a t e d by a s u m m a t i o n o f d i s c r e t e K = (Q + A? Q A,)/2R 1 = (Q/2 + (A, Q A, . 1 + +(A ) Q 2 i T x i n t e g r a l s as A )/2R 2 recursive (4.24) (A ) S A T + (A ) 2 + 2 t 4 T 4 (A,) S^A, + • • ( A ) S.AJL + where N and = 2 (4.26) i n t o t integration = Q A S x S, = S 2 S = S 2 i 2 3 • • S. { 2 5 ) (4.26) = S (4.27) 1 (4.18), i n t e g r a t i o n t i m e c a n be w r i t t e n as 2 /R (4.28) 1 time i n c r e a s e s r a p i d l y w i t h i n c r e a s i n g i , and as a r e s u l t S ^ ^ c o n v e r g e s r a p i d l y t o S s i n c e + property of equation 4 + 1 1 l i m S. i—&co = T « • S n Note t h a t reduced i f the formula i s used. 2 Aj Substituting (4.23) ^(AflViJ S 2 +... i x e f f o r t o f c a l c u l a t i n g S f r o m (4.24) i s g r e a t l y following (4.16), S)/R wh e r e The (4.20) s y s t e m , t h e t r a n s i t i o n m a t r i x has t h e p r o p e r t y e the A e and w i l l be (4.22). A-, has t h e 30 4.3 Accuracy of the Theoretically recursive to the into a formula larger large the number series creases. matrix of But, with powers of of number the since by A terms by large, and A^ the choosing calculate is respect divide Also, rapidly the with will areas. to and large integral decreases elements (4.23) made required extremely V = R is accurate (A/R)^ repeated i if discrete more of R chosen K evaluated improve off-diagonal A, of smaller (4.20) small will A since the with Method accuracy (4.25) R reduces power large the eigenvalues a Recursive as A^ N almost in ina calculation unit of multiplication A i (4.29) as required of errors by in the recursive (A^)^. Thus provement obtainable be large chosen ation of the round-off As with of an enough integral errors 3. R, to The 9 ^ -70.33 + jO. -42.26 + jO. -2.070 + is considered of consider indication but limit large errors not causes the R. As caused too large an accumulation accuracy a result, by imR discrete since in should evalu- this case, increase. + an (4.25), errors reduce (4.16) -334.2 As these choosing a illustration various Chapter by formula the an order error example from has -1.032 0.112 -0.005 approximation in approximating K the numerical results eigenvalues + jO. -0.301 + jO. the taken A-matrix jO. of involved JO. + J5.450 + jO. error of K, the matrix E where E = A K + K A + Q (4.30) 31 From (4.11) i t i s known that E becomes a n u l l matrix i f K i s known exactly. Table 4.1 i s a c o m p i l a t i o n of a..I i j I max and n V Z-J ij E.. I ij I (4.31) f o r v a r i o u s R using s i n g l e and double p r e c i s i o n Three d i f f e r e n t approximations to For 1. A 2. k 3. A x Y 1 calculations. are used, namely = I + A/R + A /2!R = I + A/R + A /2!R 2 + A /3!R = I + A/R + A /2!R 2 + A /3iR 2 2 2 2 3 3 3 small R, the e r r o r caused by d i s c r e t e e v a l u a t i o n i s dominant since (4.32) 3 + A /4!R 4 of the i n t e g r a l double p r e c i s i o n c a l c u l a t i o n does not reduce the error s i g n i f i c a n t l y . As R i s chosen l a r g e r , the e r r o r decreases and then increases first as round o f f e r r o r s accumulate single precision calculations. Although double p r e c i s i o n l a t i o n s e x h i b i t d e c r e a s i n g e r r o r with i n c r e a s i n g i n the calcu- R, the c a l c u l a t i o n time i n each case i s double the time f o r s i n g l e p r e c i s i o n lation. 4 In t h i s example, the e r r o r caused by t r u n c a t i o n calcuof the power s e r i e s r e j i r e s e n t a t i o n of A-^ i s found minimal i f A^ i s approximated by 4.4 A = x I + A/R + A /2!R 2 + 2 A /3!R 3 3 (4-33) A Numerical Example The methods of s e c t i o n s 4.1 and 4.2 are applied here to solve f o r an optimal c o n t r o l l e r of the proposed power system of Appendix A. To incorporate the following 1. changes I?) and the s i g n a l The t r a n s f e r variable function by an actuator 'a' i s replaced by a c o n t r o l i s shown i n Figure 4.1. signal The s t a t e equation f o r 'a^' becomes p a where are made. The governor dashpot i s replaced k ,/('l + f u-^. the optimal c o n t r o l l e r s i n t o the power system, u^ f a /r; = - = (k f c l /t + cl c l ) u tlj i (4.34) (4.35) 32 Table 4.1 - Error of K APPROXIIt MATION NUMBER 500 900 1300 4300 7300 13300 Approximation SINGLE PRECISION E. - max E. . DOUBLE PRECISION 1 E. . i j Imax 1 .2619 .7350 .2619 .6280 2 .1259 .6091 .1259 .4710 3 .1464 .6363 .146.4 .4735 1 .0749 .3880 .0749 .1753 2 .0425 .3670 .0425 .1488 3 .0561 .4420 .0455 .1502 1 .0869 .5470 .0349 .0812 2 . 1040 . 5970. .0210 .0719 3 .1025 .6190 .0219 .0724 i .2120 1. 150 .0031 .0072 2 .2650 1. 420 .0019 .0066 3 .2640 1. 480 .0020 .0067 1 .3506 1. 618 .0011 .0025 2 .4511 2. 184 .0007 .0023 3 .4511 2. 271 .0007 .0023 1 .4420 3. 202 .0003 .0007 2 .59 10 4. 055 .0002 .0007 3 .5910 4. 204 .0002 .0007 33 r (i Figure 2. 4.1 a control 4.2. P v t = 2 Transfer Function t * has t h e speed equation ' = (1 and H y d r a u l i c function V t u for " 52& a k •P1 >. g The t r a n s f e r (kj^'l^/tj) + v ) u,,. v u P 1 ~(W = where — AV C variable s t p) + regulator-exciter signal The s t a t e and P ) -t P) > cl - Governor The v o l t a g e by +T (i. (1 4g 1 ' s s ^ l V + v Operator signal is replaced shown in becomes (4.36) "2 + (4.37) 5 3 ^F a Figure (4.38) 2 l s 1 . + t p) x u. Figure 4.2 - Voltage The u^ and u state 2 Regulator-Exciter variable equation c a n be w r i t t e n y where y and A i s the elements Since, y^ same from B A y J ^ , V, as that (4.34), respectively, = a 0 the system + u with controllers V (4.39) B u F» » S» v s h > a » = a^ 1 (4.36) 0 = 0 = 0 , & a and (4.40) n 9 n9 fJ a of Appendix A except f o r the = - -1 /l V/ - i t u^ a n d u 2 c l appear (4.40) following (4.41) i n y^ and becomes 0 0 0 0 0 0 1 0 0 0 0 B and Function as = a,-, of Transfer 0 U 2j 0 T 0 0 0 0 1 ^ T (4.42) (4.43) 34 The response includes a speed shown Figure are if given For as ^ = .01 The be stable original feedback The = Q j0 with without with in .2 of control eigenvalues is 0 0 0 optimal tested given in Chapter regulator unstable. parameter is given voltage 0 an the system the system computation together system the 4.3. y the of The 0 0 for a 0 j of is conditions (4.44) T a value Table stability which dashpot, initial controller values and 2, of 2.1 and are is used. found to as -333.5 + jO. -1.566 + jl.381 -100.0 + jO. -.5344 jO. + (4.45) -15.56 + jO. -.2189 + -4.568 In this case c a n be are chosen eigenvalue chosen as unit matrix is method together solution 24.53 then is The on an .184, is and Ii IBM nine the u = u = .028, 1 B system For response a different Q the an = of this improved of result, (4.2) The Ricatti approximation section single 4.2. The in precision K y T calcu- from (4.46) .050,-.367,-.362, 10 13.3, example Q chosen c o r r e s p o n d i n g system a (4.47) .001, diagjlO as approximations using calculated - W" 1000. successive successive and, Q and W o f as technique .196,-5.97,-.795,-.002,-.048, The (4.19) 7044 c o m p u t e r vector necessary Matrices of recursive after not matrix. applying the control is -.019, a null found by with JO. shifting matrices obtained seconds lations. as + J5.267 is .102,-.048 10.6,-6.61, shown in y 3.26_ Figure 4.4. as 1 1 response damping c h a r a c t e r i s t i c . 1 is 1 1 1 shown in However l] (4.48) Figure this has 4.5 and been has obtained Figure 4.4 - System Responses for Q = diag ll 1 1 1 1 1 1 1 1 at the expense signal a^ to of increasing three A comparison double use As precision the For made and Q given by a of in Figure the small calculations versus a R = 1000 case 2 Single precision and R = 1000 case 3 Single precision and R = used for section -.053, 4.3 the calculation comparison 1, E = E = .436, best using a - W" 2xl0 - 6 , .064, l 0., -5xlO~ , - B 1 E cases the 200 accuracy is large case R; obtained 1. by The (4.49) 1 and .238,-.Ill 34.6, 33.2,-14.8, 7.15_ 2 and cases 3 1 and are (4.51) 0., 0., 2x10" , 2xlO 5 2xlO" , 0., 7 0., - 5 , 0., -lxlO" , 0. 5 0., -lxlO" =• 3 5 3xlO" -2xlO~^-4xlO~ -2xlO"^-2xlO" 4xlO" 5xlO" -lxlO~ lxlO" 3 4 3 4 seconds, 1000 to case be cision of times 3 - 4 time concluded the by are:case versus single this with 1 - seconds. 13$ b u t from calculation accuracy 4 21.25 200 u s i n g calculation can 3 4 calculation a 4 (4.52) -5xlO"^-lxlO"^lxlO~ -2xlO" -3xlO~ 9xlO~^lxlO~ -4xlO~^-2xlO~ The the matrix 2 0., - E 1 of j .052,-.830,-.893, cases 6 E versus K T .002, -3xl0" , 0., 5 All single (4.50) between E control where .709,-17.5,-2.16,-.001,-.111, differences the (4.48). and precision vising R. precision case The large Double in of 4.4. 1 concluded is is that maximum e x c u r s i o n case double E times the 3 Hote accuracy that calculation is time. case 2 - a decrease calculation worsens comparison that R 3 57.97 seconds, precision large 3 in only 24.52 R from decreases disproportionately. the definitely use the of single best It pre- compromise 39 5. SENSITIVITY SYSTEMS ANALYSIS OF THE TIME RESPONSE OP MULTIVARIABLE In t h i s c h a p t e r , the s e n s i t i v i t y of system response to parameter variations i s investigated. Eigensystem s e n s i t i v i t y analysis was used i n Chapters 2 and 3 f o r system s t a b i l i s a t i o n and parameter optimization r e s p e c t i v e l y , but i n i t s most d i r e c t a p p l i c a t i o n , eigensystem s e n s i t i v i t y a n a l y s i s could a l s o be used f o r e i g e n v a l u e movement s t u d i e s . However, i n the f i n a l a n a l y s i s of a system d e s i g n , a time response s e n s i t i v i t y study i s u s u a l l y r e q u i r e d since t h i s g i v e s the d e s i g n e r the d e s i r e d d i r e c t i n d i c a t i o n of changes of system response to parameter v a r i a t i o n s . For the a n a l y s i s of time response s e n s i t i v i t y , i t would be d e s i r a b l e to use an analog computer because of i t s f a s t i n t e g r a t i o n time. But the s i z e of a system t h a t may be s t u d i e d computer on an analog i s u s u a l l y l i m i t e d by the number of components a v a i l a b l e . T h e r e f o r e , f o r l a r g e order systems, the d i g i t a l computer must be used. However, i t s slow i n t e g r a t i o n time i s a p r a c t i c a l l i m i t a t i o n on the number of parameters t h a t may be chosen f o r a s e n s i t i v i t y th s t u d y , s i n c e , as i s p o i n t e d out i n the f o l l o w i n g s e c t i o n , an n order e q u a t i o n must be i n t e g r a t e d f o r each parameter sensitivity calculation. Kokotovic^^ developed a method of computing sensitivities f o r a l l parameters s i m u l t a n e o u s l y from two n^* o r d e r e q u a t i o n 1 integrations. T h i s method, however, i s r e s t r i c t e d t o a n a l y s i s of s i n g l e v a r i a b l e systems where the parameters s t u d i e d must be chosen as the e q u a t i o n c o e f f i c i e n t s . In t h i s c h a p t e r a simultaneous m u l t i v a r i a b l e s e n s i t i v i t y method i s developed i n which any paramet e r may be chosen f o r i n v e s t i g a t i o n and o n l y one n^* and one 2n^^ 1 order e q u a t i o n need be i n t e g r a t e d parameter sensitivities. i n o r d e r to compute a l l the 40 5.1 S i m u l t a n e o u s C o m p u t a t i o n of Time a L a r g e Number o f P a r a m e t e r s . Consider function the multivariable example Taking q^ the E • = partial results in N Thus, system to 0 [2 A for integration following gration ?1 »k the time Let of 0 0 of is on t h e the 0 the Laplace initial nomial as is in the the (5.1) k terms 0 with ~ y + cT| , = 1 u T respect to a (5.2) parameter u y, , <5 k = 1,...$, - 3) requires k integrations remove the parameters of (5.3). dependance investigated = R^ is given in section I s In of the inte- and to compute a matrix. 1 -h^s + R The and a inverse of (sI-A) characteristic poly- R(s)/g(s) matrix - (5.4) zero. adjoint (5.5) polynomial . . .-hus 1 1 polynomial l S n " The method (5.5) y(s) R(s)b n (5.1) + ... 2 of 1 1 - of R s i *- of (sI-A), . ..-h namely (5.6) ( s I - A ) , .'namely n " 1 " calculating 1 + ... R^ and R h^ A (5.7) is 5.2. Substituting becomes an to b u(s) 1 characteristic s n applied c h o s e n as = 1 be A ] " - of - AJ"' ' scalar = where term, ' to of \ll adjoint R(s) Combining derived = [jsl is ' transform g(s) and R(s) and conditions expressed g(s) 0 simultaneously. c a n be single forcing (5.1) 0 sensitivity number y(s) where single equation N + (5.1) a method sensitivities with a to E u + sensitivity = solve A y derivative the k = q one with Sensitivities u y For Response = into (5.8) into the input-output relation R(s)•b•u(s)/g(s) a vector can (5.4), be polynomial written (5.8) and u ( s ) / g ( s ) into ci 41 y(s) where = z(s) = u(s)/g(s) f(s) = 7 s ' v f • partial sensitivity (5.11) 1 1 ,-b" n _ (5.12) 1 of (5.9) with respect ?(s), .z(s) k q +. k derivative z(s), q with (5.10), = q = u ( s ) / \g(s)] of ?(s). q (5.16) c a n be tational purposes by k the z(s), - breaking written k (5.15) (5.13) becomes f(s).g(s). .w(s) a more each convenient vector (5.16) k form polynomial, for compu- f(s), and q ?(s).g(s), q define F k , from into the the the first q where S second vector as ?(s).g(s), II, q k is an k of [} = .w(s) nx(2n-l) (5.12) (5.16) = F, q 1 times s k S n c a n be n ^ (5.16) = - F.H, _ written 1 "** s can q k First (5.17) J ,.z(s) .1 of constant a vector. k as ? of .z(s) polynomial q where k n a matrix ^ k polynomial f(s), The of column v e c t o r s F Then, product as (5.14) q into be -w(s) k put k can 2 (5.14), .z(s) k Equation - g(s). q substitution y(s), q^, (5.13) q of = k w(s) Thus, to ?(s).z(s), q where • becomes = partial 1 derivative y(s), From t h e i.-.f-s " 1 H . . £ b n equation q _ I - 1 f the n (5.9) (5.10) n and • Taking f(s).z(s) *** l also .s" n-diagonal s "" 11 _ j ^ be w r i t t e n (5.19) similarly (5.20) ~.v(s) matrix (5.18) given by 42 17, 0 0 H, q ,0 k q o q.k. k (5.21) o 0 q and q n'q k 1 Q '2n-2 Combining be (5.18) written in y(s), q The matrix rived from written in c(s) the k form the s of i of (s) + matrix form = . .. scalar , +..a l F E matrix the times b response a form come where s of ^ is S 2n-2* equation V ( s ) ( (5.20), 5 is ' 2 4 ) de- \/ m , , +..b.s l m-i+, . . b, \; (5.25) m 1 n o 0 b o 0 . . . b . . . . y(s) • • s l (5.26) 2m 1 1 from s *. 0 .O'b . . . b , i 1 (5.9) (5.27) J can also be written as a vector: y(s) where can as |_m Evidently (5.16) polynomials m) ( s ..a-^ = (5.23) equation • H, 0. where 0 as two m , +..a.s |a sensitivity second p o l y n o m i a l , product (s the n-l the (5.22) q 2n-21T form k !' k k s' (5.20), final q : C(S) and i ' q k k = given F by • i _ n • z(s) 1 (5.19). Taking (5.28) the inverse Laplace ( 5 . 2 4 ) and ( 5 . 2 8 ) , the s o l u t i o n s of y ( t ) and y ( t ) , y(t),. . = F. • z(t) + F . II, . w(t) k k k q q y(t) = z(t) •' = F q • J " z(t) 1 ^ . ! ' 7 ^ 3 ^ '= -.Zi - ^ n ] 1 transbeq k (5.29) (5.30) V Since the -1 (t) z(t) 2 n _ .v(s) Laplace p z(t)- . .-h.p " z'(t)- can be 0.1 0 ?2 i from 0 the of 1 0 v (5.10) i = canonical form n •• 2n-l] v can .rh z(t) 0 0 " V transform n solved [ ] 2 inverse n then i u be ( written h. n Similarly, w|t) w. ' 3 1 ) a; (t) (5.32) 0 . (5.33) + n 5 ... h, l can be solved 0 1 0 .... 0 0 1 0 u(t) n 1 from 0 . . . w, 0 w. 0 (5.34) 0 w 2n where be P p.^ are ... 2 n . 2n the expressed rp . . *** . p . i .'1 *** p coefficients l w 2n of the u(t) [g( f] > polynomial s 2 which can as P . ... P rj l h 0 0 h 0 0 . . . (5.35) 0 where Thus, one fh to find y(t) integration section is of n h n- . h. and y ( t ) , z(t) a derivation and w(t) of the l for is . . h, any il number required. algorithm for (5.36) of parameters, The only following c o m p u t i n g F, F, q a n d II, . k 44 5.2 Derivation To g(s) is begin of the and R(s) in is rithm computation, h^ and two, and (5.17). used coefficients Computation (5.6) computed from cation^^ the for and the the (5.7) coefficients respectively Leverrier's this R^ Algorithm are for two coefficients can algorithm of be for easily calculating hu from the a n d R^ Fadcev's by given simple , k q the algoas a k equations. by the - 1^1 F modifi- One, and F, algorithm is a and This following set equations A = A ± A h = AR 2 = AR. , . i-l • ' • A The h X • A. l = AR n • These digital h can sparsity of A is * A = tr A /2 = tr . R = A x = A 2 x - 2 • A./i i R. = A . l . i • A /n n checked = R for 2 (5.37) . = A n h I - h.I l . • n - h n accuracy by n I the last identity 0 (5.38) 4 require used • R 1 2 = tr n but (5.37) = tr • be equations computer 2 . n calculation 1 h. l • R a derived calculated II, q result, polynomials reasons. computed d i r e c t l y sensitivity the method w i t h calculation c a n be of this to A R i n multiplications can be reduced advantage. , i = 1 In ... when p r o g r a m m e d considerably the matrix if on the products of n (5.39) 2 if only p non-zero elements instead of all n elements of A are 2 multiplied, A method record row With for of column numbers elements element multiplications single these.elements non-zero each number proposed to all and the in out a the may be non-zero reduced elements one-dimensional array and in another two one-dimensional of A taken one at becomes a matrix with a only time, one the row of to p-n A is . to their arrays. product filled. A«R^ Thus, 45 the complete one-row can be matrix product, matrices. Once calculated To compute R^ from the (5.39), a n d h^ (5.12) are and elements of can be found determined as a from sum o f these (5.37), F (5.17). II, , Morgan's method is used k f o u n d by q where the elements h., 1 q are k h, , 1 k 1 q n'q where(*) indicates I * A, = q that : k q R k = i-1 R , * A » q * A, n-l k the »q inner n (5.40) k k product is to be taken, namely (5.41) w.. lk ik The solution of F, i s d e r i v e d f r o m ( 5 . 3 7 ) as f o l l o w s . k derivative o f ( 5 . 1 2 ) w i t h r e s p e c t t o q , one q Taking the partial 1 From the q partial R = k R - n 1 , derivative x n-i'q of A. '» q F, q k (5.43) + E k q R n R n-i n-i'q k = Substituting has k . and A - ~-i n • E) q h (5.42) k . with n-i respect to . I into (5.42) and using f r-\ •• • ^ • • • • ^ A.R "'"n-i-l» the q, K (5.43) k + • R "n-i-1 ' k . X q h n-i'q k equation .I • , k (5.12) becomes F, A, q k + e. l where e , n-l c . l and As in the = A.e. = . l+l A, 'i b L R n-i C i - •• C • 1 n-i q h-, , k 1 e . l 1 L k q • • 1 . c n computation of (5.44) (5.45) (5.46) k R^ olJ . n-l . b k . • E, L e and h^ where the sparsity of A 46 used was to advantage, the computation of F, and H , k extremely economized more so since A, k q 5.3 A Numerical To design equation is the of A is written A 4 is sensitivity chosen as an method, example. the The optimal system as = found simultaneous Chapter y where usually q Example illustrate control is be Can k sparse. q A y from E + the = u (5.1) numerical results of Chapter 4 as ' -.2041 -30.07 (5.47) -4.300 38.47 57.70 .8303 .8934 1.000 -.5142 .0534 1.000 -.1865 -64.43 42.47 -333.3 333.3 -.0016 -20.05 -2.000 2.000 4.000 -1.053 -.1326 -.7091 u and to 17.52 E are be 2.157 given dependent by (5.2) on t h e = ± b It angle is response,& twenty system introduced (2.52). and B desired in Note given by , 1 > H 2 = eighteen (4.42) given in -1.000 -50.00 -33.23 14.80 -107.2 element of the E is chosen manner the (5.48) = 2 by controller Table gains the gains 2.1 initial become -l/H (5.49) sensitivity controller 4 and t h r e e simply first -19.6/2H to the the -4.000 19.6/2H = investigate Chapter -34.58 constant to parameters that where inertia b Hence .1106 .0065 .1106 -.2381 in of the given by addition operating for tf chosen 5^^ and 9 ^ torque (4.50), toTJ -^ = .01 c conditions as rows a unit of the of matrix 47 Ricatti matrix. The matrices F, F, and H, c a n be c a l c u l a t e d f o r all k k a l g o r i t h m o f s e c t i o n 5.2 b u t o n l y t h e s e c o n d q parameters of F, by the q and ( F * H , ) i n (5.29) are r e q u i r e d k k example to i n v e s t i g a t e the s e n s i t i v i t y q this since it is rows chosen in q variable h only. calculated from The the time responses canonical order Runge-Kutta integration time, the second desired and % , k calculation S, are then are shown forms of elements evaluated of by z(t) (5.33) routine of is the second and w(t) and y(t), are (5.34). used. For The then A each and y ( t ) , (5.29). state 4-th value that results of of is this q shows the in sensitivity Figures 5.1 response for q parameters most and initial sensitive figures given are operating controller the by to 0 R the first figure sensitive and the Sensitivity percentage = k The most conditions the ^u/( q .100./q ) 5.2. system k gains. sensitivity and . second for the responses in change parameters in these q /l00. (5.50) k ^k The most sensitive parameters are and K ( 9 , 8 ) . initial X and x^, The others operating and are the condition gains found to be are is , K(9,2), at least system K(9,6), ten times K(9 ?) ; less sens i t i v e . For convenience, the calculation for that Because of to of of ration time takes routine. requires three parameters exciter .00225 seconds computer sensitivity system an two is time sixteen minutes to thousand and and twenty in are controller initial constant required Therefore to programs the of integrate and order of seconds one the a Runge-Kutta seconds z(t) and w ( t ) . an other time nine for for conditions. seconds, up t o evaluations eight gains operating .003 4-th written, IBM interval integ of real It 7044 com- 50 puter to troller to one compute gains. gain, seconds is is more than to eighteen rameters and of a tive But a time it the time parameters saving in results new suboptimal of and the four variables, state in to eighteen determining five minutes new the time Thus is con- sensitivity thirteen by calculation method. the and method d e s c r i b e d for in initial to -.6, sensitivity By (5.2). of for This sensitivity eighteen approximately namely three g, S, case 2 - System with an case 3 - System with a operating v^ = o in .9 control. optimal controller pa- one the 1.2. Also are the and from an engineering costly to implement. of hour control, good over only control viewpoint 5.11 .3 both the be since to an the a wide can only is Chapter for a .9, Q illustrate without at from made 4 control = of be control figures that can namely through q signal design insensi- A comparison cases, P responses suboptimal a. a the synthesized without 5.3 over differ be optimal These very control h and suggest neglecting suboptimal Figures response conditions Thus system conditions to system optimal in can different Original shown simply control 1 - are calculation regulator-exciter case results less the governor the S - r e s p o n s e f o r design old controller. eliminated ranges. of calculation this voltage operating by that time the sensitivity minutes. The provement found required the The is using ga.ins of response calculation required the forty the the of regarded .6 imor suboptimal extreme is = Q optimal range it variation suband initial edges of these as the best easier and possibly TIME-SEC. Figure 5.3 - 8-Response W i t h o u t 1.0 Control, Over Variation of P Q 2. TIME-SEC. Figure 5.4 - S - R e s p o n s e -with O p t i m a l Control, Over Variation of P Q ure 5.5 - ^-Response with Suboptimal Control, Over Variation 0° TJME-5EC. ure 5.6 - ^-Response Without Control, Over Variation of Q Q of P TIME-SEC, Figure 5.8 - S-ftesponse w i t h S u b o p t i m a l Control, Over V a r i a t i o n of Q TIME-SEC. Figure 5.9 - S-Response W i t h o u t Control, Over V a r i a t i o n of v TIME-SEC. Figure 5.10 - S-Rcsjionse w i t h Optimal Control, Over Variation of v TIME-SEC., Figure 5.11 - S-Response with Suboptimal Control, Over Variation of v 56 CONCLUSIONS 6. Systematic optimization this of thesis. system. A Chapter This eigenvalue from method system is procedure shift the since t i v i t y and is the method system also justments. takes When system, the i n made negative. of of voltage It of be power system performance in good. the of the the system of found the that obtained the With voltage 3 way the as to new for the every s e n s i - case of might d i r e c t l y s t a b i l i z a t i o n on be parameter ad- order 9 ^ s t a b i l i z e d i f the regulator-exciter improvements decreasing droop adjusts the the unstable can transient Chapter a an eigen- eigensystems c o r r e c t i o n This further by of i n an correction c a l c u l a t i o n , to the of from However, the After necessary r e s t r i c t i o n s applied such not in from Jahn's found required. loop index. is power analysis. using accuracy eigensystem be that and the eigensystem is in computed c a l c u l a t i o n a p p l i c a t i o n and. developed derived eigenvalues. C o l l a r when quite w i l l also is c o r r e c t i o n the new regulator, method close repetitive account can of applied method is case in order X c a l c u l a t i o n the found is 9^ eigensystem eigensystem feedback s t a b i l i t y l i n e a r i z e d quadratic is speed degree The the i t gain into a one usually matrix, to and developed, s e n s i t i v i t y is a applied been eigensystem This then have and repeated and s t a b i l i z a t i o n s t a b i l i z a t i o n avoided. accuracy the systems is suspect. for been for the new matrix relations converge, the for Thus eigenvalues, power only the extended is from have except eigensystem not methods s e n s i t i v i t y . the close l i n e a r eigenvalue shift, methods order requires matrix system high general from 2 e f f i c i e n t These method system and the the of time the a the constant governor. parameters minimize eigensystem in is time of a weighted formulation, 57 the need method, matrix for only is extended Collar obtain system with found gain of initial the the for a the successive zation if although a of large the number example into optimal the the is is variables increased the be be discrete made time of the the governor number power method of calculated weighting the of The the for cost speed is to is is is for as intervals controller solve The the optimization desired integral time In different functional. torque required to is stable The weightings It has angle been choosing solution is relathe In matrix. method, by regulator-exciter. found for stabili- are system It response. signals not to regulator to chosen. control used recursive the a form. matrix. The to approximation used initiate if used method optimal calculation Ricatti and this The applied procedure an accurate system, and v o l t a g e calculation. in of system a good system Ricatti The a may be successive intervals approximations. of to unstable. as stabilization a high voltage method used initially can approximation then of from unconstrained calculate recursive can 2 stabilization control first troller state of independant example, null system approximate, numerical duced the successive tively A fast the transformation that to 4 in eigenvalues. also apply and G r u v e r ' s Chapter Chapter to detrimental approximations of To Box's example Puri in system. method process using close optimization parameters is of method minimum. order As calculation case constraints, system developed the correction function 9^ power large for constrained from eliminated. and P o w e l l ' s parameter the is except cost is eigensystem and J a h n A technique method inversion Fletcher the transform is one required necessary. to matrix introthis initiate the with optimal a con- applied to found that resulted in an the im- an proved done for damping, c h a r a c t e r i s t i c at the expense In Chapter time response a 5, multivariable of of the increasing the simultaneous systems. sensitivity This to response. magnitude sensitivity method all couse, of control. method allows parameters Of the from is this is developed calculation only two of differ- th ential order the equation optimal system reactance machine the power and but is not to most only gains for the sensitive and this response t h a t w i t h the Q and v, . o to the this to suboptimal control of be of found that line synchronous is is control over 9 neglecting it omitted the voltage, By Also, 4 is controller controller Chapter to the gains. variables. can It 4. reactance suboptimal signal applied terminal regulator-exciter control optimal a is Chapter controller state voltage with of sensitive gains, four method transient of controller The design most direct-axis requires system system response insensitive which integrations. found found are that very in- altogether. is very a wide the The close range to of P •, K It be is hoped t h a t a p p l i e d , and the application of these results these also methods tested methods systems. There is nonlinear control of nonlinear control with can presented on a c t u a l be readily much r e s e a r c h a power the system to and linearized or be the in this power systems. extended done thesis in to the coordination steady state will The multimachine area of of the control. 59 APPENDIX A Formation A.1 of Third tho Coefficient Order Machine Matrix and T i e A of Line Equation 2.1 Equations (18) Park's V d = V q = CJ(^ equations P = <*^q a k " d where ^d " " X q ( d - ^ q aV R G(p) = 4 may b e v p d V as W ( W i d U fad ^ 1 + T kd ^ d " - and For ly a power , l d ) d CJ + ( A a f ) + (i+r^p) (A.ig) + Q system (A.lh) study, The induced voltages s p e e d d e v i a t i o n s dtiiy^ small compared to 2. pared Armature to the 3. is = l b ) the following assumptions are usual- made. 1. the 1 q CO ' ) ^ ^ d o P H i ^ P ) ( q P > x q a u+x ;p)(i+r ;p) f d (p) l (A.le) p x,(p) = x , (l+t p)(l tgp) x > (A.lc) d \ R A ( A - x (p) f d ) G(p) = written very the With the V d v q = are is ^ effects voltages since they due to are a n d ^> t ^ . o q since to the neglected overall Park's "Vq^o v the neglected neglected are assumptions, ^ u ) and it is small com- reactance. relation above = a n d AtJifi resistance Subtransient in a n d Pty d speed v o l t a g e s armature short pi^ since their duration response. equations reduce to ( A - 2 a ) ( A . 2b) 0 - (1+t'p) X i 60 r.\ = (./ ~x rq Equation (A.2c) ^ o where which in where the may Vd v™, turn may Plfip field flux F V ^ d o + d ( *d x = v - = t p d v ( v o F p as d " ( X d } p i d A ' 3 ) (A.4) as 5) .(A. R - R (x -x')i ) d (A.6) d -L-VF < -> A 7 ad current i the x written X and ~ be = fd field written FR V ( A . 2d) q linkage fr/ the be = = ^ i q f field v f v III = d x (A.8) ad voltage v = d (A.9) p ad Equation ( A . 5 ) together p(ku>) with the (l/2H)(0)T. = general - CO T O X pS describe a = torque equation D(^OJ)) - (A.10a) 0 6 (A.10b) A C O third order e % machine model where the electrical torque i s g i v e n b y T with the torque base is speed = t represent v be eliminated. ~ K chosen the a n as The l dU (A- 1 radian/second. The prime 1 1 ) mover (A.12) h third c A by 1.5 g •+ Au)> lfJ-p> p > . g as to \ approximated T To = n > tie order model with variables line equation, state ^/ , from i the variables d and i^ terminal chosen need con- 61 ditions, may be i„ written = as (C+jB)v. + T or in d-q coordinates R -X X R 1 Uqj Collecting Vd (A.2d), -B X R B G and 0 v 0 /x and X equation H d" d -^ d ( x x V _q_ ^o - l l 0 two (A.13b), tie line equation X -1/x R X R 1/x ~1 _ 3 K ~ 2 K \ vd q G -B B G upon r e a r r a n g i n g K U). -X rows 0 d 0 R where, 0 Kl V q Vn /x X (A.15) ^ d ^Vq (A.16) Vq last -X o ^d the R d (A.2b) Substituting the ) / x o 0 = (A.14) - q^o X d L_q. o d 0 (A.2a) (A.13b) v cos^ _ o _ v 0 From sind X d / dTdo i o -t o< d-XcP this _q_ v (A.6) 0 x F R (A.13a) ) TR+IXT G w inverting o v - X =. v t " R 0 and v as (A.3) tdo ( + and of (A.15) and all ' do d of (A.16) into becomes + R -X X R 0 1 0 0 -1 0 1 1 0 Vd Vi (A.17) vosinS vocos o linearizing vo cosS„ -v o sin§ R 0 -do 0 d X (A.18) K where = x (R/x XG) RB d = (1 + RG - XB + X / x ) K = 3 d K. = ( R / x - RB 4 q The solutions * V where = l c c Y o 4 ( K V = ( R X K ] are Aty ^o C O S K + - o 2 1 ( J 1 1 2 >/ do d l 4 r X ( K 3R)/t ;x ( K + 4 K sinS )/(K K ~ X (A.20) F K sinS )/(K K " ^o C O S ( K K XG) A> - o 3 = < 4 2 and d - = l/<CO, = 2 b of A £ / d l b (A.19) (1 + RG - XB +. X / x ) d d K K K l + 2 K • 4 K K ^ ) + 4 3 K ^ ) (A.21) ) K K ) 2 3 The f i r s t third order model equation can now be written in state variable form by substituting first equation of p(M^,) = Av and the result into the linearized (A.15) + p 1 into the linearized (A.20) A (c (x -x )-x A: )^ d 1 d d d d o F + (x -x )b d d (A.10b) P with aubstitution of (AUJ) <£o eg = + 2H P(A£>) = where e &T e qo i \ A.2 do Governor The Ai o =-(i Ar o^ e T - 2H (A.22b) A U> (A.22c) "do q r qo ' b , + i , 'b„)AS qo = 1 do ( do V - Ai , + A(J,i - U s u b s t i t u t i o n of where W and (A.lOa) H =G) after - . 7 5 Wo A h into (A.lOa) A.co A. T which (A.12) Ab d X other equations can be obtained by linearizing (A.22a) 1 • The (A.5) 2' i d r (A.15) - (i d and - A(J i , qo ^q d o ' (A.20) ' c, + i ' c qo 1 do 0 2 ) (A.23) becomes - v, / , w , /,->.< i d o / x 7 J )/y ( A . 24) d d o p ) (A.25) = (lao + i d 0 ) x and H y d r a u l i c governor Equations and h y d r a u l i c transfer function is given by (19) 63 A T, A (l+t p)(l-T p) r M co (o-+r p + {$ +o)v T? diagram given Q The block t a is w + r in r a Figure A a 1 2 g A.l. v From the (1 1 (1 (A.26) t r p )(i+r p)(i+.5r p) + T p) (1 g transfer - T p) + .51 p) A T t S t V (1 Figure function state + A.l r Governor (A.26) variable T p) and the form of becomes linearized the _ a (7 f ^o^a p(Mi) = _ 2(ha.) &a (A.28b) can model (Aa) <- r 2(A^) _ 2(Ah) Tg Tv Equations regulator-exciter 1 (i+r )(i+r l P k ~ diagram crSt - f Regulator-Exciter to, block operator the (A.27) fg ^2 The (A.12), Ag = &co hydraulic + ~)(^ ) k Ca p(&g) The v o l t a g e and equation Operator a p(Aa ) = Voltage torque and H y d r a u l i c cr(Aa) r a A.3 Function governor ^ f _ ACO p(*a) Transfer l k e x transfer functions are given (A.28a) p) 2 (A.28b) (i+r p)(n-r p) 1 is shown be w r i t t e n in e x in Figure state by A.2. variable Equations form as (A.28a) and 64 k \ / Figure (l A.2 s l + T Voltage l P A.v 1 p >- ) ( 1 + Regulator-Exciter tex*> Transfer k,k. Function Av T p(*v ) = p r where = which when d linearized A v t. Av, V Summary to (A.22a,b,c), equations (A.27) equations (A.29), written the = p» A is given ' ^' by (A.31) . V , order c to machine order = governor second ninth variable y into 6 (A.31) q' l v. C to 2 ; A ( (A.32) ^F Equations and t h e the n o t a t i o n matrix a fourth state — y where with in v, _ ' to V j . v , 2>^ to the t h i r d the (A.30) d' of System Combining q 2 and (A.20) l b v becomes x (A.16) r + v v, _ * to Substituting A.4 2 v, (A.29) order order form and t i e and h y d r a u l i c voltage power p omitted operator equations c a n be as v , p equations regulator-exciter system A y lfJ , line v , g g , h, a, for simplicity. T a ~j f The (A.33) ( A . 34) coefficient A (A.35) = _D 2H '12 "13 32 33 L l k 2 2II "ex _1_ a 53 '52 o 2H 'ex k 1.5 GO <£o ?1 -g •w g tr o a L St ^ r a wh ere a 12 l "13 a 32 '33 U + b i ')/2H ' "2 do' 0 (c i ' + c i ' 1 qo 2 do o n (x -x )b /x d d 1 - I'l^iT*** l ( v q o b l " "53 " k l ( v q o C l - i by (A.25). b^, b , 2 c^, c l/r •x•)/2H do d d k constants - 0 - The o o co ( 1b , i l^qo 52 L d _1 2 (A.36) *k^do /*k ) V V do 2 b d o are C 2 ) / v ) / v t ri o t o given r i by (A.21a,b,c,d) and i q 0 > 66 APPENDIX Initial Operating Vith the e>u)<^> t^jJlfJ^) Conditions assumptions armature steady state (A.30) may be 2 , to form of of a Power System appendix A neglecting resistance of B and subtransient (A.2a,b,c,d), (A.6), P^j P^q» effects, (A.11), the (A.13b), and (6) V v do written = V , " ~Vqo + V, qo 2 do qo T as 2 ^o *ao = eo H> V i.) u d 0 "^o ^ q o = P v = ^Fo v o = o Fo v i do do i q qo d Fo v i qo qo Vdo ~ -x + ( ^do d d v o sinS o = (1 - XB + R G ) v do , v o cos& o and the = *o V Fo' power Hlo' P , o' Vqo' v i, - v, q 0 and v, ^o to ^Fo' ° V , do do - (RB + XG)v qo - R i ,do + X i qo / ( P •j allows \lo' = x ) o q X 1 q qo i do & 2 + (B.2) qo the \ o > P l ) flow q o do Choosing ° a = (XG + R B ) v do , + (1 - XB + RG)v qo - X i ,do - R qo i reactive Q B o n d 8 determination o f r o m { B ' l v. to (v. to 2 + x p ) q^o' 2 ] a m of l ( v, , v , do' qo' B ' 2 ) a S v o', i do , - (Qo • + x i • ) / vqo q qo v„ Fo = v V + qo x . i , d do (B.3) ^do^o = q 0 Hlo V Wo 2 K qo / w o - di d" d> do = r ( x X i t + d i v p o and from the last two equations of ( B . l ) v /- o ' V 2 ' - do , + qo V 2 (B.4) = arctan ( vdo' , / Vqo') £°o s where Y d o vqo - = (1 - XB = (XG + .RG)v + RB)v do d o + - ( R B + X G ) V q (1 - XB + R G ) v o qo - Ri d o Xi, do + X i - Ri q o qo APPENDIX C Fletcher the (14) 'Descent and P o w e l l ' s 'Die algorithm 1. Compute t h e parameter of cost vector - l T g Fletcher p, and P o w e l l ' s functional that - method, i s gradient with as respect ri i , -l p x ... JT, iP j ... J,p T i i is 2. the iteration Adjust the i p .=p . a minimum J is by given i and H is 1 matrix a for 3. = 1 1 = k is k* minimizes 4. J = found 1 such along Evaluate H where 1 D E + the the i^* iteration 1 (C.2) matrix usually c h o s e n as a unit 5. Repeat component of then s 1 (C.3) 1 , p -i + cr tha.t p = is ^ H 1 + D and + 1 compute H**^ E i i = the is i - procedure less than W] computed from p 1 T from <'> ¥1 <°- > C 5 f with prescribed of 6 (C.7) starting a which (C.4) 1 - -H -[y j-[? j - HVIFF 1 vector 1 = [ ^ { ^ ^ [ - J 1 th;* p a r a m e t e r s . (Cl) + 1 1 + gradient g* ^ from f and s* A -i p and is for direction iteration. -i+1 J s^ a gradient Set 1 of along 1 definite first 5- where vector f o u n d where - H . g positive the is 1 (Cl) number. parameter s""" u n t i l to T x P=P where follows is i - = J>P Method the last step 2 until accuracy. iteration. The every minimum 69 REFERENCES 1. L a u g h t o n , M. A . , " T h e U s e o f S e n s i t i v i t y A n a l y s i s i n the Design of Generator E x c i t a t i o n C o n t r o l " , Proceedings o f the S e c o n d Power Systems C o m p u t a t i o n C o n f e r e n c e , V o l . 1, P a p e r 3 . 4 , S t o c k h o l m S w e d e n , 1 9 6 6 . 2. F r a n c i s , J . G. F., " T h e QR T r a n s f o r m a t i o n " , P a r t I, The C o m p u t e r J o u r n a l , V o l . 4 , O c t o b e r 1 9 6 1 ; P a r t II ibid., J a n u a r y 1962. 3. L a u g h t o n , M. A . , " S e n s i t i v i t y i n Dynamical System J o u r n a l o f E l e c t r o n i c s and C o n t r o l , V o l . 17, November 1964. 4. Macfarlane, Linear 1963. 5. Fadeev, of D. A. G. J . , Regulator K., and Linear V. Analysis", N o . 5, "An E i g e n v e c t o r S o l u t i o n of the Optimal P r o b l e m " , i b i d . , V o l . 14, No. 6, June N. Fadeeva, "Computational Methods Algebra", Freeman, San 1963. Francisco, 6. V o n g s u r i y a , K., "The A p p l i c a t i o n of Lyapunov F u n c t i o n to Power System S t a b i l i t y A n a l y s i s and C o n t r o l " , PhD. D i s s e r t a t i o n , U n i v e r s i t y of B r i t i s h Columbia, February 1968. 7. F r e e s t e d , V. C . , R. F . W e b b e r , a n d R. V . B a s s , "The "GASP" C o m p u t e r P r o g r a m - An I n t e g r a t e d T o o l f o r O p t i m a l C o n t r o l and F i l t e r D e s i g n " , P r e p r i n t s o f t h e 1968 J o i n t Automatic C o n t r o l C o n f e r e n c e , pp. 198-202, University of M i c h i g a n . 8. Puri, 9. Kerlin, N. N . , a n d W. A . G r u v e r , " O p t i m a l C o n t r o l D e s i g n v i a Successive Approximations", Preprints o f the 1967 J o i n t Automatic C o n t r o l C o n f e r e n c e , pp. 335-343, University of Pennsylvania. T. W., "Sensitivities Simulation, 10. Van 11. Morgan J r . , Vol. B. Householder, A. Analysis", 13. Yu, No. by 6, the State June 1967. Variable Approach", Ness, J . E., J . M. B o y l e a n d F . P. Imad, "Sensitivities of L a r g e , M u l t i p l e - L o o p C o n t r o l S y s t e m s " , IEEE T r a n s a c t i o n s on A u t o m a t i c C o n t r o l , V o l . A C - 1 0 , J u l y 1 9 6 5 . S., Multivariable 12. 8, "Sensitivity Systems", S., "The New York, Analysis ibid., Theory of Vol. and Matrices Blaisdell, Synthesis AC-11, in July of 1966. Numerical 1964. Y . N . , K. V o n g s u r i y a , a n d L . N. Wedman, "Application o f an O p t i m a l C o n t r o l T h e o r y to a Power S y s t e m " , Accepted f o r p u b l i c a t i o n i n I E E E T r a n s a c t i o n s on Power A p p a r a t u s and S y s t e m s . 70 14. Fletcher, R. , and M. J . D. P o w e l l , "A R a p i d l y Convergent D e s c e n t M e t h o d f o r M i n i m i z a t i o n " , The C o m p u t e r Journal, V o l . 6, p p . 163-168, 1963. 15. Box, 16. Tomovic, M. J . , "A Comparison of S e v e r a l C u r r e n t O p t i m i z a t i o n M e t h o d s a n d t h e Use o f T r a n s f o r m a t i o n s i n C o n s t r a i n e d P r o b l e m s " , i b i d . V o l . 9, pp. 67-77, 1966. R., "Sensitivity McGraw-Hill, 17. Gantmacher, F. Chelsea, R., New Analysis of New Y o r k , 1963. "Matrix Theory", York, Dynamic Vols. I Systems", and II, I960. 18. Park, R. II., "Two R e a c t i o n T h e o r y o f S y n c h r o n o u s Machines: I-Generalized Method of A n a l y s i s " , AIEE T r a n s a c t i o n s . V o l . 48, J u l y 1929, pp. 716-730. 19. S i d d a l l , R. G . , " A P r i m e M o v e r - G o v e r n o r T e s t M o d e l f o r L a r g e Power S y s t e m s " , MASc. D i s s e r t a t i o n , University of B r i t i s h C o l u m b i a , J a n u a r y , 1968. 20. Athans, M. Hill, 21. Noton, A. a n d P. New R. Control 22. L. York, M., Falb, "Optimal Control", McGraw 1966. "Introduction Engineering", to Variational Pergamon Press, Methods Oxford, in 1965. Wedman, L . N . , Y a o - N a n Y u , " C o m p u t a t i o n T e c h n i q u e s f o r t h e S t a b i l i z a t i o n and O p t i m i z a t i o n of H i g h Order Power S y s t e m s " , S u b m i t t e d f o r p u b l i c a t i o n t o t h e 1969 P I C A conference.
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Stabilization and optimization of a power system with sensitivity considerations. Wedman, Leonard Nickolaus 1968
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Title | Stabilization and optimization of a power system with sensitivity considerations. |
Creator |
Wedman, Leonard Nickolaus |
Publisher | University of British Columbia |
Date Issued | 1968 |
Description | An investigation is made into some aspects of the analysis and design of high order systems. The problems treated are system stabilization, parameter optimization, computation of an optimal controller and parameter sensitivity. The methods developed for solving these problems are applied to a 9ᵗʰ order linearized power system. To stabilize the system, an eigenvalue shift technique is used. Eigensystem sensitivity analysis is applied to determine both the parameter change required and the new eigensystem after the change has been made. A correction method is applied to the new eigensystem for improving accuracy in order that large steps in parameter change may be taken. This method is subsequently used in an optimization procedure for parameter setting to minimize a cost functional of quadratic form. For the computation of an optimal controller, Puri and Gruver's successive approximation method is used in conjunction with a fast recursive method developed for solving each approximation of the Ricatti matrix. The calculation can be initiated by the eigenvalue shifting method to ensure that the system is initially stable. Finally, a time response sensitivity study is made using a method developed for simultaneous sensitivity function determination. This method reduces computation time significantly over the conventional method ⁽⁹⁾ thus enabling the investigation of time response sensitivity to a large number of parameters. The results of the sensitivity study are then applied to the design of a suboptimal controller. |
Subject |
Power (Mechanics) |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-06-09 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0103226 |
URI | http://hdl.handle.net/2429/35339 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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