{"Affiliation":[{"label":"Affiliation","value":"Applied Science, Faculty of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."},{"label":"Affiliation","value":"Electrical Engineering, Department of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."}],"AggregatedSourceRepository":[{"label":"Aggregated Source Repository","value":"DSpace","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","classmap":"ore:Aggregation","property":"edm:dataProvider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","explain":"A Europeana Data Model Property; The name or identifier of the organization who contributes data indirectly to an aggregation service (e.g. Europeana)"}],"Campus":[{"label":"Campus","value":"UBCV","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","classmap":"oc:ThesisDescription","property":"oc:degreeCampus"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the name of the campus from which the graduate completed their degree."}],"Creator":[{"label":"Creator","value":"Siggers, Christopher","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/creator","classmap":"dpla:SourceResource","property":"dcterms:creator"},"iri":"http:\/\/purl.org\/dc\/terms\/creator","explain":"A Dublin Core Terms Property; An entity primarily responsible for making the resource.; Examples of a Contributor include a person, an organization, or a service."}],"DateAvailable":[{"label":"Date Available","value":"2011-06-08T21:32:35Z","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"edm:WebResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"DateIssued":[{"label":"Date Issued","value":"1969","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"oc:SourceResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"Degree":[{"label":"Degree (Theses)","value":"Master of Applied Science - MASc","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","classmap":"vivo:ThesisDegree","property":"vivo:relatedDegree"},"iri":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","explain":"VIVO-ISF Ontology V1.6 Property; The thesis degree; Extended Property specified by UBC, as per https:\/\/wiki.duraspace.org\/display\/VIVO\/Ontology+Editor%27s+Guide"}],"DegreeGrantor":[{"label":"Degree Grantor","value":"University of British Columbia","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","classmap":"oc:ThesisDescription","property":"oc:degreeGrantor"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the institution where thesis was granted."}],"Description":[{"label":"Description","value":"Present trends toward long distance extra high voltage transmission lines and static excitation can cause a reduction in the stability margins of a typical power system unless measures are taken to improve the system damping. Practical applications of stabilizing signals are investigated in this thesis and limitations of the design techniques discussed. An optimal control signal is derived from modern control theory. All signals are obtained from a common linearized power system and the performance is tested for large disturbance conditions on a single machine-infinite bus system where the machine, exciter and governor are represented in detail. Both types of signals are also tested on a practical four machine system model and it is shown that a similar improvement in damping can be obtained with either a stabilizing signal derived using conventional frequency response techniques or a proportional feedback controller obtained from solution of the algebraic matrix Riccati equation.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/description","classmap":"dpla:SourceResource","property":"dcterms:description"},"iri":"http:\/\/purl.org\/dc\/terms\/description","explain":"A Dublin Core Terms Property; An account of the resource.; Description may include but is not limited to: an abstract, a table of contents, a graphical representation, or a free-text account of the resource."}],"DigitalResourceOriginalRecord":[{"label":"Digital Resource Original Record","value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/35305?expand=metadata","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","classmap":"ore:Aggregation","property":"edm:aggregatedCHO"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","explain":"A Europeana Data Model Property; The identifier of the source object, e.g. the Mona Lisa itself. This could be a full linked open date URI or an internal identifier"}],"FullText":[{"label":"Full Text","value":"OPTIMAL CONTROL AND STABILIZATION SIGNALS FOR A POWER SYSTEM by CHRISTOPHER SIGGERS B.A.Sc.., U n i v e r s i t y of B.C., 1962 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of E l e c t r i c a l Engineering We accept t h i s t h e s i s as conforming to the required standard Research Supervisor Members of the Committee A c t i n g Head of the Department Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA October, 1969 . In presenting this thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make i t freely available for reference and Study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thes.is for financial gain shall not be allowed without my written permission. Department of ^\/-\/tZc r&ic\/?\/- \u00a3Mc The University of B r i t i s h Columbia Vancouver 8, Canada ABSTRACT Present t rends toward long d i s t ance ex t r a h igh vo l tage t r ansmiss ion l i n e s and s t a t i c e x c i t a t i o n can cause a r educ t i on in the s t a b i l i t y margins of a t y p i c a l power system un less measures are taken to improve the system damping. P r a c t i c a l a p p l i c a t i o n s o f s t a b i l i z i n g s i g n a l s are i n v e s t i g a t e d in t h i s t h e s i s and l i m i t a t i o n s of the design techniques d i s c u s s e d . An op t ima l c o n t r o l s i g n a l i s de r i ved from modern c o n t r o l t heo ry . A l l s i g n a l s are obta ined from a common l i n e a r i z e d power system and the performance i s t e s t ed f o r l a rge d i s tu rbance c o n d i t i o n s on a s i n g l e m a c h i n e - i n f i n i t e bus system where the machine, e x c i t e r and governor are represented in d e t a i l . Both types of s i g n a l s are a l s o t e s t e d on a p r a c t i c a l four machine system model and i t i s shown tha t a s i m i l a r improvement i n damping can be obta ined w i th e i t h e r a s t ' j D i i i z i n g s i g n a l d e r i v e d us ing conven t i ona l frequency response techn iques or a p r o p o r t i o n a l feedback c o n t r o l l e r obta ined from s o l u t i o n af the a l g e b r a i c matr ix R i c c a t i e q u a t i o n . i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v LIST OF ILLUSTRATIONS . v i ACKNOWLEDGEMENT v i i i NOMENCLATURE ix 1. INTRODUCTION 1 2. STABILIZING SIGNALS FOR POWER SYSTEMS 3 2.1 Introduction 3 2.2 S t a b i l i z a t i o n Techniques 5 2.3 P r a c t i c a l Applications 8 2.4 Limitations of Damping Cr i t e r i o n 13 3. OPTIMAL CONTROL SIGNAL FOR POWER SYSTEMS 18 3.1 Introduction 18 3.2 Problem Formulation 18 3.3 Special Computational Techniques 19 3.4 Test Results 2 1 4. STABILIZATION AND CONTROL TECHNIQUES APPLIED TO MULTIMACHINE STABILITY STUDIES 28 4.1 Introduction 28 4.2 Multirnachine Equations and Terminal Constraints 2B 4.3 Algorithm 30 4.4 System Data and Test Results 33 5 . CONCLUSIONS ' 40 APPENDIX A DERIVATION OF LINEARIZED EQUATIONS FOR-A SINGLE MACHINE INFINITE BUS MODEL IN STATE-VARIABLE FORM . 42 A.l Synchronous Machine and Terminal Equations 42 A.2 Linearized System Equations 46 A.3 System Equations in the State-Variable Form 50 A.4 I n i t i a l Conditions for the Single Machine-Infinite Bus System 52 APPENDIX B DERIVATION OF AN ALTERNATIVE FORM OF PARK'S EQUATIONS 54 i i i Page APPENDIX C INITIAL CONDITIONS FOR MULTI MACHINE SYSTEM STUDIES , 5 9 APPENDIX D EXCITATION SYSTEM REPRESENTATION 61 APPENDIX E SOLUTION OF THE ALGEBRAIC MATRIX RlCCATI EQUATION USING NEWTON-RAPHSON ITERATION 6 3 REFERENCES 6 6 iv LIST DF TABLES Table Page 2.1 Pe r-un i t Data of S i n g l e Machine I n f i n i t e Bus System 5 2.2 Var ious Operat ing Cond i t i ons of a T y p i c a l System . . . . . . . . 15 2.3 C h a r a c t e r i s t i c Roots fo r System w i t h Pou\/er S i g n a l S t a b i l i z a t i o n 16 3*1 P r e l i m i n a r y Data from S o l u t i o n of Ma t r ix R i c a t t i Equat ion 22 4.1 System Data f o r Mul t imachine S t a b i l i t y S tud ies 34 LIST OF ILLUSTRATIONS F igure P a 9 G 2.1 Smal l O s c i l l a t i o n System Block Diagram \" 5 2 . 2 S i m p l i f i e d System Model . \u201e . 6 2.3 R e l a t i o n Between Fundamental Machine V a r i a b l e s 7 2.4(a ) Power S t a b i l i z e r B lock Diagram . . . < . . . . * . . . . . . . . . . . . . 8 ( b ) Speed S t a b i l i z e r B lock Diagram . . . . , . \u00ab . 9 2.5 System-Response Without S t a b i l i z i n g C o n t r o l \u00ab . 1 0 2.6 System Response With Speed S i g n a l S t a b i l i z a t i o n 10 2.7 System Response With Power S i g n a l S t a b i l i z a t i o n . . . . . . . . . . . . H 2.8 System Response With Power S i g n a l S t a b i l i z a t i o n . . . . . . . . . . . . 11 2.9 System Response With Power S i g n a l S t a b i l i z a t i o n 12 2.10 System Response With Power S i g n a l S t a b i l i z a t i o n 12 2.11 Root-Locus f o r V a r i a b l e E x c i t e r Gain (^) \u00ab 16 2.12 Root-Locus For V a r i a b l e S t a b i l i z e r Delay (T j ) 17A 2.13 Root-Locus fo r V a r i a b l e S t a b i l i z e r Delay (T2) 17A 3 .1 Case ' A ' System Response With Opt imal C o n t r o l S i g n a l . . . . . . . 23 3 .2 Case ' B 8 System Response With Opt imal C o n t r o l S i g n a l 24 3.3 Case ' C System Response With Opt imal C o n t r o l S i g n a l 25 3.4 Case *D' System Response With Sub-Optimal C o n t r o l S i g n a l . . . 2 6 4.1 Case ' A ' Swing Curves \u2022 35 4.2 Case 'B* Swing Curves . . . . . . . . . . . . 35 4.3. Case ' B 9 Termina l and F i e l d Vo l tage V a r i a t i o n s .-.-.\u00bb..e 37 4.4 Case ' C 1 System Response With Power S i g n a l S t a b i l i z a t i o n on P l an t #3 38 4 .5 Case ' D ' System Response With Sub-Optimal C o n t r o l S i g n a l on P l a n t #3 39 A . l E x t e r n a l System Representa t ion 44 A.2 R e l a t i o n Between r-m and d-q Coord ina tes 45 v i Figure Page A.3 Phasor Diagram of a Sa l i en t Pole Synchronous Generator 53 C l Phasor Diagram of a Sa l i en t Pole Synchronous Generator 59 D . l General Form of Exc i t a t i on System Model . \u00ab \u2022 . . \u2022 \u2022 \u2022 \u2022 \u00ab * o \u00bb 61 v i i ACKNOWLEDGMENT Many people have a s s i s t e d i n one way or another i n producing the r e s u l t s of t h i s t h e s i s . The i n s p i r i n g guidance and u n f a i l i n g pat ience of Dr . Y . N . Yu, my s u p e r v i s o r , has been a grea t source of encouragement. Dr . H.M. E l l i s , manager of system p lanning and development, B.C. Hydro, prov ided va luab l e support on s e v e r a l f r o n t s . The author wishes to express h i s h e a r t f e l t g r a t i t u d e fo r t h i s a s s i s t a n c e . I would l i k e to express my g r a t i t u d e to Dr . M.S. Davies f o r h i s d e t a i l e d comments and sugges t i ons . Many h e l p f u l d i s c u s s i o n s have a l s o been prov ided by the members of the. B.C. Hydro system p lann ing and deve lop -ment s e c t i o n and the graduate s tudents i n the power systems courses a t U.B .C. In p a r t i c u l a r , I would l i k e to thank Malcolm Metca l f e and Harndi Moussa f o r t h e i r eagle-eyed p roo f read ing and Graham Dawson for h i s i n t e r e s t and a d v i c e , I am g r e a t l y indebted to my w i fe f c r her supreme e f f o r t s and pa t ience i n t yp ing and prepar ing the t h e s i s . The f i n a n c i a l a s s i s t a n c e of B.C. Hydro and the N a t i o n a l Research C o u n c i l i s g r a t e f u l l y acknowledged. v i i i NOMENCLATURE G e n e r a l A nxn s y s t e m m a t r i x B nxm c o n t r o l m a t r i x f ( K ) nxn m a t r i x f u n c t i o n u s e d i n c o n n e c t i o n w i t h R i c c a t i e q u a t i o n f, u p p e r - t r i a n g u l a r v e c t o r f o r m o f f ( K ) J c o s t f u n c t i o n a l K nxn R i c c a t i m a t r i x k, u p p e r t r i a n g u l a r v e c t o r f o r m o f R i c c a t i m a t r i x Q p o s i t i v e d e f i n i t e n x n s t a t e v a r i a b l e w e i g h t i n g m a t r i x R p o s i t i v e d e f i n i t e mxm c o n t r o l s i g n a l w e i g h t i n g m a t r i x t t i m e t y t e r m i n a l t i m e f o r c o n t r o l s t u d y y_ m - d i m e n s i o n a l c o n t r o l v e c t o r x. n - d i m e n s i o n a l s t a t e v e c t o r p d \/ d t , timB d e r i v a t i v e o p e r a t o r s L a o l a c e o p e r a t o r o s u b s c r i p t d e n o t i n g an i n i t i a l c o n d i t i o n A p r e f i x d e n o t i n g a l i n e a r i z e d v a r i a b l e * - s u p e r s c r i p t d e n o t i n g a v a r i a b l e w h i c h m i n i m i z e s c o s t f u n c t i o n a l S y s t e m P a r a m e t e r s C 2nx2n t r a n s f o r m a t i o n m a t r i x C 3nx3n c o e f f i c i e n t m a t r i x r r C 2nx3n c o e f f i c i e n t m a t r i x r s D d a m p i n g c o e f f i c i e n t H g e n e r a t o r i n e r t i a c o n s t a n t a m p l i f i e r gain i x Kj. e x c i t e r gain feedback s t a b i l i z e r gain power or speed s t a b i l i z e r gain K^ through Kg l i n e a r i z e d system constants K_ ,K ,-,K .,K\u201e. Kalrnan qains for optimal c o n t r o l l e r Te vr vt r+jx t i e l i n e impedance between generator and i n f i n i t e bus G+jB generator terminal load admittance R armature resistance a Rp d-axis arnortisseur resistance Rp f i e l d resistance R q-axis arnortisseur resistance 3nx3n c o e f f i c i e n t matrix T * d-axis transient open ci r c u i t time constant do T T \" d and q-axis subtransient open c i r c u i t time constants do qo exciter time constant Tp feedback stabilizer time constant Tp regulator time constant transducer time constant T^, T2 stabilizer function time constants x., x d and q-axis synchronous reactances 0 q Xp, X Q d and q-axis arnortisseur reactances x ,n, x n d and.q-axis mutual reactances between stator and arnortisseur oD qu x .r- d-axis mutual reactance between stator and f i e l d dr x^ d-axis transient reactance x\", x\" d and q-axis subtransient reactances d q x\" subtransient reactance (saliency neglected) Xp f i e l d reactance X . 2nx2n r e a c t a n c e m a t r i x dq X_ 3nx2n r e a c t a n c e m a t r i x x\" 2nx2n s u b t r a n s i e n t r e a c t a n c e m a t r i x ( s a l i e n c y n e g l e c t e d ) V , V . c o n t r o l s i q n a l l i m i t s max min VAI\u00bbIAX' VAMIN a m p l i f i e r l i m i t s 2nx2n a d m i t t a n c e m a t r i x ( r e a l c o e f f i c i e n t s ) S y s t e m V a r i a b l e s \u00a7 r 3 n - d i m e n s i o n a l v e c t o r o f v o l t a g e s p r o p o r t i o n a l t o r o t o r c u r r e n t s e\u00a3 3 n - d i m s n s i o n a l v e c t o r o f v o l t a g e s p r o p o r t i o n a l t o r o t o r f l u x l i n k a g e s 2 n - d i m e n s i o n a l v e c t o r o f v o l t a g e s p r o p o r t i o n a l t o a m o r t i s s e u r f l u x l i n k a g e s eJJ, e'^ v o l t a g e s p r o p o r t i o n a l t o q a n d d - a x i s a m o r t i s s e u r f l u x l i n k a g e s r e s p e c t i v e l y e^ v o l t a g e p r o p o r t i o n a l t o d - a x i s f i e l d f l u x l i n k a g e s E 1 v o l t a g e b e h i n d t r a n s i e n t r e a c t a n c e Eq v o l t a g e b e h i n d q - a x i s s y n c h r o n o u s r e a c t a n c e E^ q-axis component o f v o l t a g e b e h i n d t r a n s i e n t r e a c t a n c e i n , i q , y n , d a n d q-axis a m o r t i s s e u r c u r r e n t s a n d f l u x l i n k a g e s r e s p e c -t i v e l y i p , V p , ij\/p- f i e l d c u r r e n t , v o l t a g e a n d f l u x l i n k a g e s r e s p e c t i v e l y i ^ . g e r i e r a t o r t e r m i n a l c u r r e n t ^d ' ^ q ' ^ l ' v q ' ^ d ' Yq ^ a n c* c l ~ a x ^ s s t a t o r c u r r e n t s , v o l t a g e s a n d f l u x l i n k a g e s r e s p e c t i v e l y t l a n d i m a g i n a r y compor ( r e f e r r e d t o s y n c h r o n o u s r e f e r e n c e a x i s ) i r \u00bb i r e a n e n t s o f g e n e r a t o r t e r m i n a l c u r r e n t i^ j 2 n - d i r n e n s i o n a l v e c t o r o f c u r r e n t s ( r e f e r r e d t o s y n c h r o n o u s r e f e r e n c e a x i s ) 2n-dimensional vector of currents (referred to machine axes) accelerating power rea l and reactive output power of generator respectively accelerating torque energy conversion torque mechanical input to generator torque produced by s t a b i l i z i n g s ignal excitation system a m p l i f i e r , regulator, and feedback voltages respectively a u x i l i a r y and reference excitation system voltages f i e l d voltage proportional to Vp i n f i n i t e bus voltage terminal voltage amplifier l i m i t i n g c i r c u i t output voltage 3n-dimensional vector of open-circuit excitation voltages generator torque angle generator angular velocity natural frequency for small o s c i l l a t i o n s damping r a t i o angle between terminal voltage and common reference axis 1. INTRODUCTION The increasing s i z e and complexity of modern, power systems, along with the introduction of very long extra high voltage transmission l i n e s and fast-response e x c i t a t i o n systems, have led to a requirement for new types of c o n t r o l l e r s . The basic objective of c o n t r o l l e r design i s to increase interconnected machine damping without a f f e c t i n g the functions of the e x c i t e r and speed-governor during normal operation. A great deal of work has been done i n the area of power system damping by e x c i t a t i o n c o n t r o l . E l l i s , e t . a l . ^ ^ showed that a speed s i g n a l can provide the required system damping and the r e s u l t s were confirmed i n (2) (3) f i e l d t ests . Dube and others studied a terminal power s t a b i l i z i n g s i g n a l and s i g n i f i c a n t improvements in damping were demonstrated^ 4^. S c h l e i f , e t . a l . ^ ^ extensively tested the use of s i g n a l s obtained from plant' high voltage bus ( 6) frequency on conventional e x c i t a t i o n systems. defflello and Concordia firmly established the t h e o r e t i c a l basis for s t a b i l i z i n g s i g n a l design using a l i n e a r i z e d s i n g l e machine-infinite bus model. This model i s generalized s l i g h t l y i n t h i s t hesis to include both t i e - l i n e impedance and l o c a l load. The l i m i t a t i o n s of t h i s approach are investigated by applying the s i g n a l to a high order, non-linear, large power system. The a p p l i c a t i o n of optimal c o n t r o l theory to power system s t a b i l i t y (7 8) studies i s r e l a t i v e l y new but important work has been done by Yu and others ' . C o n t r o l l e r s are usually developed from the l i n e a r regulator formulation of (9) modern c o n t r o l theory using a l i n e a r i z e d s i n g l e machine-infinite bus system model. In Chapter 3 the l i n e a r i z e d model used for s t a b i l i z i n g s i g n a l studies i s converted to state variable form and the c o n t r o l law obtained using optimal c o n t r o l theory i s applied to the non-linear system of Chapter 2. This permits d i r e c t comparison of these two approaches. The a p p l i c a t i o n of s t a b i l i z a t i o n and c o n t r o l s i g n a l techniques i s extended to a mult imaqhine system in Chapter 4 . V o n g s u r i y a ^ ^ developed an a l g o r i t h m fo r tnult imachine s t u d i e s i n a form p e r m i t t i n g d i r e c t s o l u t i o n of the network equa t i ons . Armature f l u x l i nkage v a r i a t i o n s were i n c l uded and s u b t r a n s i e n t e f f e c t s were n e g l e c t e d . O l i v e ^ ^ ' ^ ^ transformed P a r k ' s equat ions to a convenient form and used i t e r a t i v e procedures to ob ta in (13) the network vo l t ages at each c a l c u l a t i o n i n t e r v a l . U n d r i l l , Prabhashankar (14) and Jan ischewsy j suggested methods p e r m i t t i n g d i r e c t s o l u t i o n of the network equa t i ons , but the machine parameters used are r a r e l y a v a i l a b l e to e l e c t r i c u t i l i t i e s . The main o b j e c t i v e of t h i s chapter i s to combine reasonably accura te machine-exc i ter mode l l ing w i th e f f i c i e n t computa t ion . 2. STABILIZING SIGNALS FOR POWER SYSTEMS 2 .1 I n t r o d u c t i o n As a f i r s t s tep i n i n v e s t i g a t i n g o p t i m i z a t i o n and s t a b i l i z a t i o n techn iques f o r improving the t r a n s i e n t performance of a power system, i t i s d e s i r a b l e to review b r i e f l y the methods i n cu r r en t use . I t i s hoped that t h i s review w i l l r e v e a l the nature of the problem and w i l l a l l ow comparison of the c l a s s i c a l r e s u l t s w i th those obta ined from opt ima l c o n t r o l t heo r y . C l a s s i c a l approaches to s t a b i l i z i n g synchronous genera tors are based p r i m a r i l y on p o s i t i v e damping c o n c e p t s ^ * ' * ' ' ^ . The Laplace Transform i s a p p l i e d and assumptions and approx imat ions are made i n the development of the system model and i n the des ign of the s t a b i l i z i n g system as f o l l o w s : a ) the system i s l i n e a r and t i m e - i n v a r i a n t , b) on ly the response to a s tandard i npu t s i g n a l i s cons idered ( eg . s tep or s i n u s o i d a l ) , c ) a t r i a l - a n d - e r r o r method i s used i n the des ign which does not lead to a unique s o l u t i o n f o r the c o n t r o l . The f i r s t assumption i s not v a l i d i n p r a c t i c e because some system v a r i a b l e s are sub jec t to magnitude c o n s t r a i n t s , and the network equat ions dur ing a f a u l t - t y p e d i s tu rbance have d i s c o n t i n u i t i e s dus to changes i n network c o n -f i g u r a t i o n . Neve r the l e s s , s i g n i f i c a n t improvement i n performance has been (5) ach ieved us ing these techn iques . The problem i s cons ide r ab l y s i m p l i f i e d i f these assumptions are mede i n the development of the model but the r e s u l t s are then t e s t ed on a h igher order non- l i nea r system model which more a c cu r a t e l y desc r i bes the a c t u a l system. I t w i l l be shown that the pr imary f u n c t i o n of a s t a b i l i z i n g s i g n a l i s to prov ide an adequate degree of i n t e r n a l machine damping so tha t r o to r speed may r e tu rn to normal as q u i c k l y as p o s s i b l e f o l l o w i n g a system f a u l t 4 or d i s t u r b a n c e . There are s e ve r a l methods a v a i l a b l e f o r improving machine damping. Th is d i s c u s s i o n , however, i s r e s t r i c t e d to the method of e x c i t a -t i o n c o n t r o l through the i n t r o d u c t i o n of an a u x i l i a r y i npu t s i g n a l to the normal e x c i t a t i o n system. deMel lo and C o n c o r d i a ^ ^ adapted a sma l l o s c i l l a -t i o n system model to the d i s c u s s i o n of t h i s method us ing arguments based on the concepts of s ynch ron i z i ng and damping torque,, During the f i r s t swing immediate ly f o l l o w i n g a system f a u l t , the normal t e r m i n a l vo l tage e r r o r s i g n a l a long w i th brak ing r e s i s t o r s or s e r i e s - c a p a c i t o r sw i t ch i ng w i l l normal ly be ab le to keep the machine from going o u t - o f - s t e p . T h i s , i n e f f e c t , suggests that there i s g e n e r a l l y enough s y n c h r o n i z i n g torque a v a i l a b l e to ensure i n i t i a l s t a b i l i t y . However, i t i s o f t en found i n p r a c t i c e tha t the normal t e r m i n a l vo l t age e r r o r s i g n a l does not prov ide an e x c i t a t i o n response tha t w i l l reduce the r o t o r o s c i l -l a t i o n s a f t e r the f i r s t sw ing . Th i s n a t u r a l l y leads to the d e s i r e to prov ide an a d d i t i o n a l e x c i t a t i o n s i g n a l tha t w i l l produce a damping torque d u r r i n g d i s tu rbance c o n d i t i o n s and that w i l l not i n t e r f e r e w i th normal r e g u l a t o r and governor a c t i o n under s teady-s ta te c o n d i t i o n s . A se t of machine equat ions i n s ^ a l l o s c i l l a t i o n form are de r i ved from P a r k ' s equat ions i n Appendix A. The r e s u l t s are expressed i n a b lock diagram form which prov ides a convenient bas i s f o r a d i s c u s s i o n of c l a s s i c a l s t a b i l i z a t i o n t echn iques . A c r i t e r i o n f o r e f f e c t i v e damping of the s i m p l i f i e d system model i s developed from ths b lock d iagram. T h i s c r i t e r i o n i s a p p l i e d to two p r a c t i c a l examples and the r e s u l t s t e s t ed on the h igh-order non- l inea r system model . F i n a l l y , some l i m i t a t i o n s of the c r i t e r i o n are d i s c u s s e d . 5 2\u20222 S t a b i l i z a t i o n Techniques Equat ions (A-32) through (A-35) from Appendix A are expressed i n b lock diagram form i n F igure 2 . 1 . The constants through Kg are d e t e r -mined from the i n i t i a l opera t ing c o n d i t i o n s of the machine and the reduced network parameters . To t h i s end, a d i g i t a l program i s w r i t t e n us ing data obta ined from a p r a c t i c a l system (F igure 2 . 2 ) . The r e s u l t s of a t y p i c a l case are l i s t e d i n Table 2 . 1 . F igure 2.1 Smal l O s c i l l a t i o n System Block Diagram y .973 r -.034 V o 1.02 K l .55 x d = .19 X = .997 E qo = 1.17 K 2 = 1.16 X q = .55 G = .249 V Fo = 1.34 K 3 r .66 T d o = 7.77 B s .262 V d o .45 K 4 = .67 H - 4.63 P o = .952 V qo - .95 K 5 -.09 K A - 50 Q o = .02 ^ o .40 K 6 = .82 T A .05 V t o s 1.05 .81 w n 4.74 rad\/sec D = 0 6 o = 65.1\u00b0 Table 2.1 Pe r-un i t Data of S i n g l e M a c h i n e - I n f i n i t e Bus System 600 (125) (250) (250) (250) 500 kv O Bus numbers ( ) Reactive power*, invars 230 kv O P a r a l l e l lines Figure 2.2 Simplified System Model Since system changes often lead to steady-state deviations in angle, the c o n t r o l l e r i s designed to l i m i t the speed deviations following a disturbance. To improve system damping we must be able to predict the torque v a r i a t i o n s produced by the a u x i l i a r y s i g n a l (u^). If u^ , i s r e l a t e d to the speed deviation by a general expression to the forms u x s G x ( s ) ApS (2-1) the torque v a r i a t i o n due to u^, Figure 2.1, i s AT G x ( s ) K A K 2 K 3 A ( P 6 ) - 1 \u2022 K A K 3 K 6 \u2022 (T f t \u2022 T d ' K 3 ) s \u2022 ^A^do^3 S' (2.2) 2 Since K f t K 3 K 6 \u00bb j_ \u2022 _ T^T^^K,^ and T^'K^, \u00bb Tfl equation (2-2) reduces to A do 3 n do 3 \" 'A A T AlpT) *GX<*>MS) (2-3) w h e r e G p ( s ) S K 2\/K 6 1 * s T d y K A K 6 I f small o s c i l l a t i o n s are approximately s i n u s o i d a l , &6, Ap6, e t c . can be represented as phasors. Let the r e l a t i v e machine angle ( A.6) be talon as the reference axis, Figure 2.3; then the speed deviation ( &p\u00a7) w i l l lead by 90\u00b0 and the rotor acceleration ( AP^6) by a further 90\u00b0. I t follows from Figure 2.1 that i f AT^ i s to produce negative feedback for both Ap6 and A6, i t must be located in the f i r s t quadrant. The r e a l and imaginary components of AT^ can be r e f e r r e d to as synchronizing and damping torques r e s p e c t i v e l y . Ap6 2 AP 6 A6 Figure 2.3 Relation Between Fundamental Machine Variables I t i s shown l a t e r that, i n general, the o v e r a l l system response i s dominated by a pair of conjugate poles located r e l a t i v e l y close to the ju>-axis in the complex plane and that small o s c i l l a t i o n s w i l l have a frequency approximated by: w n = J?>77 K j\/2H ( 2 - 4 ) From Table 2 . 1 , Gp(s) has a phase lag of 16 at t h i s frequency. Thus, to produce a torque in phase with the speed deviation ( i e . a pure damping torque) the s t a b i l i z i n g function G (s) should have a phase-lead character-x i s t i c of 1 6 \u00b0 at the o s c i l l a t i o n frequency. This provides the basic c r i t e r i o n for e f f e c t i v e damping used i n these studies. 2.3 P r a c t i c a l Applications In p r a c t i c e , the c o n t r o l s i g n a l may be obtained from a variety of ( 4 ) sources. A t y p i c a l example uses a s i g n a l derived from e l e c t r i c a l power as shown in Figure 2 . 4 ( a ) . An approximate trans f e r function i s A T . K x s JL _ G r ( s ) A P \/ \u2022 \u2022 ( i i f . s ) ( l + T 0 s ) ( 2 . 5 ) T y p i c a l parameter choices are = 5 0 , T^ = 1 s e c , = Z sec. which gives a phase lag of approximately 87\u00b0 at the natural o s c i l l a t i o n frequency, (a> n). Accelerating power (AP ) l i e s on the negative r e a l axis in Figure 2 . 3 . I f we include the e f f e c t s of small lags due to the power transducer and exc i t e r time constants, the phasor AT^ w i l l be located in the f i r s t quadrant, Umax K s X \/ ) ( l + T t s ) ( l * T l S ) ( l + T 2 s ) \u2022- -t T t = .035 Vmax = .12 Vmin .12 Vmin Figure 2.4(a) Power S t a b i l i z e r Block Diagram ( 2 ) T h e s e c o n d e x a m p l e u s e s s h a f t s p e e d a s t h e i n p u t s i g n a l t o a s t a b i l i z i n g f u n c t i o n s i m i l a r t o F i g u r e 2 . 4 ( b ) . A s i m p l i f i e d t o r q u e - s p e e d f u n c t i o n c a n be o b t a i n e d a s A T x V A7 = G F ( s ) (1 + T . s ) ( 2 - 6 ) Vmax K s X f ( l * T t s ) ( l + T l S ) u l Vmin T = .005 Umax = .12 Vmin = -.12 F i g u r e 2 . 4 ( b ) S p e e d S t a b i l i z e r B l o c k D i a g r a m F o r K - 30, T, = .5 s e c . , AT w i l l h a v e a p h a s e a n g l e o f a b o u t 1 0 0 \u00b0 i n x 1 x F i g u r e 2.3. A c c o u n t i n g f o r t h e s m a l l l a g s i n t h e t r a n s d u c e r a n d e x c i t e r w i l l r e d u c e t h i s a n g l e s l i g h t l y . T h e s y s t e m p e r f o r m a n c e w i l l n o t d e t e r -i o r a t e u n l e s s t h e s e t i m e c o n s t a n t s become l a r g e , b u t by t h e n T^ c a n be r e d u c e d . T h e two c o n t r o l s i g n a l s t h u s der;' \\*cd a r e t e s t e d on a h i g h - o r d e r n a n - l i n e a r s y s t e m m o d e l l e d u s i n g a f i f t h - o r d e r m a c h i n e ^ ^ , a t h i r d - o r d e r I t i s f o u n d . t h a t b o t h power a n d s p e e d s i g n a l s i n c r e a s e s t a b i l i t y m a r g i n s f o r a t y p i c a l f a u l t - t y p e d i s t u r b a n c e . T h e s y s t e m o f F i g u r e 2.2 w h i c h i s u n s t a b l e f o r an i n i t i a l o u t -p u t o f .95 pu, F i g u r e 2.5, c a n be s t a b i l i z e d f o r an o u t p u t o f 1.15 pu, F i g u r e s 2.6 a n d 2.7. A l t h o u g h s i m i l a r r e s u l t s a r e o b t a i n e d f r o m b o t h s i g n a l s f o r t h s f i r s t few r o t o r s w i n g s , t h e damping e f f e c t o f t h e s p e e d s i g n a l seems i n f e r i o r t o t h a t o f t h e power s i g n a l i n t h e s u c c e e d i n g s w i n g s . However, t h e power s i g n a l r e s u l t s show u n d e s i r a b l e t e r m i n a l v o l t a g e o s c i l l a t i o n s o f r e l -t i v e l y h i g h f r e q u e n c y , F i g u r e 2.6, w h i c h a r e a t t r i b u t e d t o e x c e s s i v e l o o p e x c i t e r a n d a f o u r t h - o r d e r s p e e d g o v e r n o r ^ ^ , Terminnl Uoltooa (pu) rachine Angle (duo) , 1 sP\u00b0 1 0 d Oovlotlon (pu) T T 13 gains in the excitation system. They can be damped out using an exciter gain of 50 and a s t a b i l i z e r gain of 11, Figure 2.8. Figures 2.9 and 2.10 show the effect of varying the phase angle of the power s i g n a l . A s t a b i l i z e r designed for a phase-lag of 55\u00b0 at the natural frequency, Figure 2.9, produces reasonable damping while that of 120\u00b0, Figure 2.10, has l i t t l e damping. These results are explained by the increase in the o s c i l l a t i o n frequency of the system with a power s t a b i l i z i n g signal and high gains as w i l l be shown l a t e r . 2.4 Limitations of Damping Cr i t e r i o n The arguments used in developing the s t a b i l i z i n g control strategy make extensive use of the concepts of damping torque, synchronizing torque and natural frequency. I t i s t a c i t l y assumed that the small signal response i s dominated by a pair of complex poles located r e l a t i v e l y close to the (17) jto -axis on the complex plane. Truxal has shown that poles located at least seven times as far to the l e f t of the j<\u00b0 -axis as the dominant poles can be assumed to make a negl i g i b l e contribution to the step-response. A review of the v a l i d i t y of t h i s assumption over a wide range of system con-dit i o n s w i l l provide useful insight into the li m i t a t i o n s of t h i s approach. The second-order response approximation i s obtained from the sim p l i f i e d machine model neglecting variations in f i e l d flux linkages. When AE^ = 0, the ov e r a l l transfer function of Figure 2.1 reduces t o i 377\/ffl A 377\/lYl A TM S2 + (D\/m)s + 377K.\/M s 2 + 2 1u> + u, 2 i n n which has poles at s^, s 2 = - +. jw n jl - ^2 where 7= D\/(2 J 377^) and ^ = J 3771^\/M 14 u>n i s r e fe r red to as tha natura l frequency and the damping r a t i o . De Mello and Concordia have ind ica ted that t y p i c a l values o f f a r e in the range of .03 to .05 and t h i s e f f e c t has been neglected elsewhere in the a n a l y s i s . The c h a r a c t e r i s t i c equation fo r the system without s t a b i l i z i n g con t ro l obtained from Figure 2.1 i s f n T A T d o K 3 s 4 '+ M < V T d o K 3 > s 3 + 0 < 2\" 7> A wide range of operat ing cond i t ions were inves t iga ted for a t y p i c a l system. The r e s u l t s are summarized in Table 2 .2 . The r e s u l t s of Table 2.2 i nd i ca te the system i s unstable without s t a b i l i z i n g compensation. It i s i n t e r e s t i ng to note that in add i t i on to negative damping the exc i t e r cont r ibutes a smal l amount of pos i t i v e synch-ron iz ing torque so that the o s c i l l a t i o n frequency i s greater than u>n. In frequency response terms, the con t ro l problem can be descr ibed as that of r e l o ca t i ng the dominant poles as far to the l o f t of the j\u00ab> -axis as poss ib le (maximum damping) without s i g n i f i c a n t l y reducing the o s c i l l a t i o n frequency ( ava i l ab le synchroniz ing torque) or the exc i t e r gain (voltage r e g u l a t i o n ) . For the system studied here, i t appears that T r u x a l ' s cond i t ion for e s s e n t i a l l y second-order type response w i l l be s a t i s f i e d over a wide range.of operat ing c o n d i t i o n s . Assuming that the s t a b i l i z i n g funct ion does not r a d i c a l l y a l t e r the pole-zero c o n f i g u r a t i o n , use of the damping c r i t e r i o n appl ied here i s j u s t i f i a b l e . It i s poss ib le to der ive the c h a r a c t e r i s t i c equation for the system with s t a b i l i z a t i o n con t ro l added but the a lgebra i c manipulation increases r ap id l y with the higher system order . An equivalent approach i s to reduce the system to s ta te-var iab le form and compute the eigenvalues of the Case Loading Conditions K l K2 K3 K4 K5 . K6 Characteristic Roots A I n i t i a l System, Winter Peak, 68\/69 .57 1.1 .66 .59 -.13 .75 .26 + J5.44, -25.4 + J4.94 4.B B Present System, Winter Peak, 69\/70 .55 1.2 .66 .67 -.09 .82 .24 + J5.09, -25.3 + J7.01 4.7 C Future System, Winter Peak, 71\/72 .47 1.0 .69 .56 -.14 .77 .25 + J'5.04, -25.3 + J6.38 4.4 D Future System, Summer Light, 71\/72 .68 .53 .63 .28 -.01 .76 .01 + J5.28, -25.1 + J5.17 5.3 E Ultimate System, Winter Peak, 76\/77 .39 1.2 .73 .42 -.10 .83 .19 + J4.58, -25.3 + J9.47 4.0 F Ultimate System,; Summer Light, 76\/77 .67 1.4 .62 .63 . -.08 .72 .21 + J5.68, -24.0, -26.6 5.2 Table 2.2 various Operating Conditions of a Typical System 16 'A' matrix. Typical results for the system with power signal stabilization (equation (2.5)), data from Table 2.1 and a range of exciter gain are summarized in Table 2.3 and Figuro 2.11. KA Characteristic Roots .02 -.02, -.33, -1.0, -20.0, -.09 + j4.7 10 -.32, -19.2, -.09 + J5.2, -.94 + j.54 50 -.32, -14.9, -1.1 + j6.7\u00bb -2.1 \u2022 j.07 100 -.32, -1.54, -4.5 \u2022 J8.3, -5.3 \u2022 J3.9 K =10 T, = 1.0 sec. T 0 = 3.0 sec. x 1 2 Table 2.3 Characteristic Roots far System with Power Signal Stabilization .? -6 -5 -\u00ab -3 -? -I 0 Figure 2.11 Root-Locus for Variable Exciter Gain (K. ) \u2022 17 Note that for values of exciter gain less than 80 the major o s c i l l a t o r y component of response corresponds roughly to the dominant term of the uncompensated system. Figure 2.11 also tends to confirm the results of Figure 2.7, namely that excessive gains can produce stable but highly o s c i l l a t o r y response. Figures 2.12 and 2.13 show the effects of varying and 1^ respectively. Note that the range of values for and T 2 where the second-order approximation (co^ co ) can be used without serious error i s f a i r l y l i m i t e d . F igure 2.12 Root-Locus f o r V a r i a b l e S t a b i l i z e r Delay (T . ) F igu re 2.13 Root-Locus fo r V a r i a b l e S t a b i l i z e r Delay ( T 9 ) 18 3. OPTIMAL CONTROL SIGNAL FOR POWER SYSTEMS 3.1 Introduction The general approach hore follows Chapter 2 i n that the control signal i s derived from a l i n e a r fourth-order model but tested on a non-linear twelfth-order model. The results indicate a considerable improvement in system performance. The li n e a r i z e d small o s c i l l a t i o n model used in Chapter 2 i s con-verted to state-variable form where a l l state variables are measurable with conventional sensors (Appendix A). A control law i s developed from solution (9) of the 'linear regulator problem' of optimal control theory . The algorithm (18) used for solving the matrix R i c c a t i equation i s taken from the l i t e r a t u r e . Computational methods peculiar to t h i s particular problem are discussed and the results compared with thoss obtained in Chapter 2. 3.2 Problem Formulation For the linear regulator study the system equations are written i n the forms x = Ax <\u2022 Bu , (3-1) x = x at t = 0 o # I t i s necessary to find the control (u = u ) which minimizes the quadratic performance function * f \/ T_ T o J = $ \/ f (xTQx + u TRu) dt (3-2) subject to the constraint (3.1). Q and R are assumed positive d e f i n i t e , usually chosen as symmetric matrices. A quadratic form i s used for equation (3-2) to assure a well-defined minimum. Under these conditions the control which minimizes J e x i s t s , i s unique, and i s given by, u* = -R\"1BTKx* (3-3) where K i s a symmetric matrix obtained from solution of the matrix R i c c a t i d i f f e r e n t i a l equation! K = -KA - ATK + KBR\"1BTK - Q , (3-4) subject to the boundary condition K(t^) = 0. For the case where the terminal time i s allowed to become large r e l a t i v e to the largest time constant of the controlled process the matrix K becomes time-invariant and the optimal control given by (3-3) consists simply of constant proportional feedback from the state variables. In t h i s case the feedback c o e f f i c i e n t s , R ^B^K, are often referred to as the Kalman gains of the system. Under these con-dit i o n s the problem reduces to that of finding the steady-state solution of (3-4)t 0 = -KA - ATK \u2022*\u2022 KBR\"1BTK - Q (3-5) that i s , to solve the non-linear algebraic equation (3-5) rather than a (18) d i f f e r e n t i a l equation (3-4). Blackburn has shown that by suitable formulation of the problem the Newton-Raphson method for i t e r a t i v e solution of non-linear simultaneous equations can be applied. A derivation of t h i s algorithm i s given i n Appendix E . \u2022 3.3 Special Computational Techniques There are several computational tools available for solution of (R IR IQ 9n) the algebraic matrix R i c c a t i equation * , , u . A l l of these algorithms share a common shortcoming; a necessary condition for convergence i s that the computation must be started with an i n i t i a l guess of K which yields a stable process model. In other words, the eigenvalues of the closed-loop system have negative r e a l parts. Thus i f the process model i s stable without feedback compensation, computation may be started with K = 0. However,, with t y p i c a l parameter choices the power system model used i n thess studies i s unstable and some means of overcoming t h i s l i m i t a t i o n i s necessary. Previous experience with c l a s s i c a l s t a b i l i z a t i o n techniques has indicated that system i n s t a b i l i t y can be traced to a negative damping effect produced by the use of a high-gain fast-response excitation system. An obvious moans of obtaining a stable process model for starting the computation i s to assume an a r t i f i c i a l l y high positive damping c o e f f i c i e n t . Preliminary studies have shown that a damping c o e f f i c i e n t , D = 10, Figure 2.1, w i l l be s u f f i c i e n t to guarantee s t a b i l i t y for the system studied here. Computation i s started using K = 0 and D = 10. After a few hundred Runge-Kutta i t e r a -tions the R i c c a t i matrix i s obtained and t h i s result used for starting Newton-Raphson i t e r a t i o n . Using Blackburn's method the steady-state solution l<2 i s then obtained, usually i n less than 10 i t e r a t i o n s . At t h i s point a new system 'A' matrix with D = 0 and are used for the rest of the compu-t a t i o n . The f i n a l steady-state solution, K^, i s usually obtained i n less than 10 i t e r a t i o n s . The Kalrnan gains, R * B^Ky thus obtained are tested on the twelfth-order dynamic'stability program. Once a solution for i s obtained i t can be used as the i n i t i a l guess for other systems with similar *A' matrices and i t i s unnecessary to use an a r t i f i c i a l damping factor and Runge-Kutta i t e r a t i o n . In some cases 4 excessively large changes i n Q or R, eg. 10 times, may lead to slow convergence ( i e . 30 - 40 i t e r a t i o n s ) . Blackburn used the r a t i o llf ll \/ Ilk], < 10~ 6 as a convergence c r i t e r i o n for the Newton-Raphson i t e r a t i o n . For the system studied here i t was found that the conditions | f j] \/ ||k jj < 1 0 - 1 2 and j j ^ f \/ ^ i i ] 1\"^ *\u00b0 ^ \u00abuere necessary to guarantee convergence to within three s i g n i f i c a n t figures. 21 The twelfth-order model used to test the feedback c o n t r o l l e r design requires a step-size of .00025 seconds to avoid numerical i n s t a b i l i t y . This model i s used only as 3 f i n a l check on a few selected designs since computational costs can be p r o h i b i t i v e i f every a l t e r n a t i v e i s tested. The s i m p l i f i e d model eigenvalues are used for comparing various a l t e r n a t i v e s . 3.4 Test Results The r e s u l t s obtained here are based on the same system data as those presented i n Chapter 2. The i n i t i a l conditions and system parameters are l i s t e d in Table 2.1; the disturbance conditions i n Figure 2.5. The major d i f f e r e n c e here i s the c o n t r o l scheme used to obtain the a u x i l i a r y input s i g n a l to the e x c i t a t i o n system. Cases 'A' through ' C show the e f f e c t of varying the weighting factors of the performance function (Table 3.1). Case 'D' shows r e s u l t s obtained when the feedback from ex c i t e r f i e l d voltage i s eliminated. The r e s u l t s of Case 'A' are p a r t i c u l a r l y i n t e r e s t i n g in that they i n d i c a t e a l i m i t a t i o n of the l i n e a r i z e d model. The c o n t r o l l e r has s h i f t e d the eigenvalues of the l i n e a r i z e d model tc the l e f t - h a l f of the s-plane but the non-linear system model i s unstable when subjected to a f a u l t d i s t u r -bance, Figure 3.1. This i n s t a b i l i t y i s a t t r i b u t e d to poor choice of the performance function parameters. The l i n e a r regulator problem can be stated as: given a l o c a l deviation from the i n i t i a l s t ate, derive a c o n t r o l law to remove t h i s d e v i a t i o n . Referring to Figure 3.1, between 20 and 35 cycles a f t e r the f i r s t a p p l i c a t i o n of the f a u l t , the torque and speed errors tend to produce negative f i e l d - f o r c i n g i n the e x c i t e r . When the f i e l d voltage error becomes negative i t w i l l tend to counteract the speed and torque signals to reduce the f i e l d voltage d e v i a t i o n . Therefore, penalizing f i e l d voltage deviations can reduce speed damping and defeat our primary purpose. CASE WEIGHTING FACTORS FEEDBACK GAINS EIGENVALUES CODfliYlENTS Q l l Q22 Q33 \u00b044 R l l K UJ vt vF K T Te \u2022A\u00ab 1.0 1.0 1.0 1.0 1.0 2.76 -.31 -.98 T.58 -1000., -.183, -.118 + J4.73 Eigensystem stable but i n f e r i o r to c l a s s i c a l r esults '\u20228' 1.0 1.0 IO\" 1 0 1.0 1.0 33.7 -.41 -.007 -.87 -12.1 -s- J7.82, -1.60 + J4.12 Eigensystem more stable than c l a s s i c a l results 1.0 1.0 IO\" 1 0 1.0 IO\" 3 1260. -30.6 -.092 -18.8 -54.2 + j'53.5, -1.82 + J3.53 Improved eigensystem but increased feedback gains \u00abD\" 1.0 1.0 IO\" 1 0 1.0 IO\" 3 1260. -30.6 - -18.8 Sub-optimal control, K r = 0 vF Table 3.1 Preliminary Data from Solution of Matrix R i c a t t i Equation to to 1 - rocf>ln\u00ab . H Q I . (deoraei) ? - Spped Ooviat lon (pu) 40.0 60.0 In) flotor Vart.sl\" 123.0 160. TIME (CYC.) O J O. -1 - Contro l n j n a l (pu) 7 - T e r * l n e l voltage (pu) 3 - F i e l d voltage (pu) 40.0 (D) C v c l t e t l o n System V a r l e o l a i 80.0 120.0 160. TIME! (CYC.) o o I - SDC^O dowiation (pu) ~* O \u2014 \u00b0 .'; - Ta re lna l vol teqa arror (pu) \/ 3 - T i a l d vol tage error (pu) a \/ * - C l s c t r l c a l torque error (pu) \u00b0 <\u00a3> ' \u00b0 A y5 ft \/ - O \" i n 1 J o \u00b0 X i ' s 1 1 in 1 1 , 1 N\u2014 s lO jn \u00ab=\u00bb- 7\" 7 - *=>- i i i i -o.o \u00ab.n eo.o 120.0 16O. TIME (CYC.) (c ) faadtaecv Cont ro l l e r State Var iab les ure 3.1 Case ' A ' System Response With Opt imal C o n t r o l S i g n a l 24 1 - I b c W m angle (t.rc>ll\u00abr ^tat\u2022 'Jariarilea C a s e ' C S y s t e m R e s p o n s e W i t h O p t i m a l C o n t r o l S i g n a C3 3.0 BO.O 1 - \u00bb\u00abchino angle (deqreea) 2 - Speed dev ia t ion (pu) 160.0 240.0 320.0 TIME ICYC.) (a) Rotor Var iab les <00.0 \u00ab0.0 560.0 1 - Contro l s igna l (pu) 2 - Te,ri\u00bbinol voltoqo (pu) 3 - f i a l d voltage (pu) T 1 1 1 1 1 160.0 24).0 320.0 400.0 480.0 560.0 TIME ifYC.) (b) Exc i ta t ion System var iab les 1 - Speed dav ia t lon (pu) 2 - Terminal voltage error (pu) 3 - f i e l d voltage er ror (pu) * - e l e c t r i c a l torque error (pu) 160.0 241.0 320.0 TIME ICYC.J (c) Feedback Cont ro l le r State ' \/erteblea ure 3.4 Case \u00abD\u00ab System Response With Sub-Optimal Cont ro l Siqna 2? The performance f u n c t i o n can be manipulated to reduce the e f f e c t o f the f i e l d e r r o r a lmost comp le t e l y . The pena l t y f a c t o r a p p l i e d to t h i s s t a t e v a r i a b l e was reduced by a f a c t o r of 10 ^ i n Case ' B ' and performance improved c o n s i d e r a b l y , F igure 3 . 2 . A comparison of the speed d e v i a t i o n curves of F igures 2.8 and 3.2(b) show be t t e r damping has been obta ined us ing op t ima l c o n t r o l t e chn iques . In g e n e r a l , reduc ing R ( cos t of c o n t r o l ) r e l a t i v e to Q ( cos t of d e v i a t i o n s from i n i t i a l s t a t e ) w i l l produce a b e t t e r eigensystem (improved damping) . U n f o r t u n a t e l y , the ga ins r equ i r ed f o r the feedback c o n t r o l l e r are a l s o inc reased so tha t improvement i s l i m i t e d by r e a l i z a b i l i t y and no ise _3 c o n s i d e r a t i o n s . F igure 3.3 shows that reduc ing R to 10 g i v e s improved damping and sho r t e r s e t t l i n g t ime . A sma l l r i p p l e i s n o t i c a b l e i n the c o n t r o l s i g n a l i n the l a s t few seconds . S ince the d i s tu rbance r e s u l t s i n the permanent l o s s of o n 8 500 kv l i n e , the system does not r e tu rn to i t s i n i t i a l s t a t e . An inc rease i n f i e l d vo l tage i s r equ i r ed to ma in ta in the torque ou tpu t . An uncompensated system prov ides t h i s inc rease by a l l o w i n g a s l i g h t r educ t i on ( AVp\/K^) i n the t e r m i n a l vo l t age but fo r a l i n e a r r e g u l a t o r model , these are c o n f l i c t i n g requ i rements . A sub-opt imal c o n t r o l l e r , where the f i e l d vo l t age feedback i s se t to z e r o , p r a c t i c a l l y e l i m i n a t e s t h i s r i p p l e wi thout s i g n i f i -c a n t l y degrading o v e r a l l performance as shown i n F igure 3 .4 . The steady-s t a t e e r r o r i s not e x a c t l y zero but cou ld be removed by a d j u s t i n g the e x c i t e r re fe rence v o l t a g s . 28 4. S T A B I L I Z A T I O N AND CONTROL TECHNIQUES APPLIED TO MULTIMACHINE S T A B I L I T Y STUDIES 4.1 I n t r o d u c t i o n T h e s t a b i l i z a t i o n a n d c o n t r o l t e c h n i q u e s o b t a i n e d i n C h a p t e r s 2 a n d 3 a r e d e v e l o p e d f o r s t u d y i n g t h e t r a n s i e n t r e s p o n s e o f a m u l t i m a c h i n e power s y s t e m s u b j e c t e d t o f a u l t - t y p e , d i s t u r b a n c e s . A c o n s t a n t i m p e d a n c e n e t w o r k m o d e l i s u s e d a n d s u b - t r a n s i e n t s a l i e n c y e f f e c t s a r e n e g l e c t e d t o p e r m i t d i r e c t s o l u t i o n o f t h e n e t w o r k e q u a t i o n s . A r m a t u r e r e s i s t a n c e , s a t u r a t i o n a n d a r m a t u r e f l u x l i n k a g e v a r i a t i o n s a r e a l s o n e g l e c t e d . S p e e d -g o v e r n o r e f f e c t s a r e n o t r e p r e s e n t e d b u t c o u l d be i n c o r p o r a t e d i n t o t h e p r o g r a m , i f r e q u i r e d . P r o v i s i o n i s made f o r s i m u l a t i n g e i t h e r t h e power s t a b i l i z i n g s i g n a l d e v e l o p e d i n C h a p t e r 2 o r t h e f e e d b a c k c o n t r o l l e r o b t a i n e d i n C h a p t e r 3. A n i n t h - o r d e r m a c h i n e - e x c i t e r m o d e l i s c h o s e n a s t h e b a s i c r e p r e s e n t a t i o n a n d p r o g r a m l o g i c p e r m i t s v a r i o u s d e g r e e s o f s i m p l i f i c a t i o n . ( 1 2 ) O l i v e ' s f o r m v ' o f P a r k ' s e q u a t i o n s i s u s e d b u t a d i r e c t s o l u t i o n o f n e t w o r k e q u a t i o n s i s o b t a i n e d . T h e p r o g r a m i s t e s t e d u s i n g datf* fj.-om a l a r g e u t i l i t y s y s t e m . T h e i n i t i a l c o n d i t i o n s r e q u i r e d f o r s t a b i l i t y s t u d i e s a r e o b t a i n e d u s i n g t h e ( 2 1 ) a l g o r i t h m d e v e l o p e d by Ward a n d H a l e . U s i n g s t a n d a r d n e t w o r k r e d u c t i o n (22 2 3 ) t e c h n i q u e s ' t h e n e t w o r k m o d e l i s r e d u c e d t o i t s e s s e n t i a l e l e m e n t s f o r t h e t r a n s i e n t c a l c u l a t i o n s . T h e n e t w o r k d a t a f o r m a t f o r t h e l o a d f l o w a n d n e t w o r k r e d u c t i o n p r o g r a m s a r e c o m p a t i b l e s o t h a t d a t a p r e p a r a t i o n i s m i n i m i z e d . T h e r e s u l t s o f t e s t c a s e s a r e s u m m a r i z e d a n d p r e s e n t e d i n g r a p h - : i c a l f o r m . 4.2 M u l t i m a c h i n e E q u a t i o n s a n d T e r m i n a l C o n s t r a i n t s F o r t h e m u l t i m a c h i n e s t u d i e s t h e m a c h i n e e q u a t i o n s d e r i v e d i n A p p e n d i x B c a n be e x p r e s s e d i n m a t r i x f o r m a s f o l l o w s . From ( 8 - 1 6 ) t h r o u g h (B-18), one has , 1 \u2022r T do v - T e, -so - i (4-1) From (B-19) and (B-20), % = Xdq ^dq + C r s ^ r (4-2) and from (B-21) through (B-23) e' = C e - X. i . - r r r ~r _dq (4-3) The following quantities are defined: *so e-r \" [ ed e q l e q 2 ] T % = [ V d V q ] T idq \" [ i d i q ] 1 ^dq v ' T II qo do x\u00bb d - x' do rs 1 0 0 1 r r xd~ xd 30 Vxd d d \" (V i t5> 0 0 If a constant impedance model i s chosen to represent the external netuiork, transmission system variables can be related to machine variables using the following equations! \u2022 (4-4) where and -N = [ Y N ] \u00abN (k) (k) 9dq (k) (4-5) i ^ -armature current vector} 2n rea l elements, y^ j -terminal voltage vector; 2n r e a l elements, [^N] ~n\u00b0dal admittance matrix for transmission system, -armature current or voltage vector referred to machine coordinates, a -armature current or voltage vector referred to common T i n reference, sin 6, cos 6. -cos 6. sin 6. k k transformation matrix; = 4.3 Algorithm It i s convenient to choose the voltages proportional to the rotor flux linkages e\u00a3, the excitation voltage Vp, the machine angle 6 and the speed deviation ^ as the state variables. The vectors e and T 31 are eliminated from (4-1) through (4-3) as follows. From (4-3) e B C _ 1 Te' + X i . 1 - r r r |_~r dqj Substituting e i n equation (4-1) gives (4-6) pe 1 = v - T L r - r _ , -so r r I ~r 'do [\u2022J * X\" *dq] (4-7) Substituting T , C and X_, previously defined, i n equation (4-7) allows solution of the rotor c i r c u i t d i f f e r e n t i a l equations in the form. __n _ f-e\" * (x - x\" ) i 1 \/T \" [ d q q ' q j ' qo V Xd. V X d , pe' = fv . - (-? h e' + ( \u00b0 \u00b0) e\u00bb 1 \/T ' P L F x d ~ x d q x d ~ x d q J d 0 pe\" = fe ' - e\" - (x'- x\") i .1 A \" * q I q q v d d ' d l ' d o Substituting e^ in (4-2) gives + c c mlx o rs r r \"1 1 i . + C C e 1 \u2014 J -dq rs r r _ r In the case where xJJ = x\u201e - *\"\u00bb equation (4-9) reduces to where - (4-8) (4-9) *dq = tl X i , + e -dq II r (4-10) s j * [ (4-11) X\" i ' 0 x\" ' -x\" 0 (4-12) The machine variables v, and i . in (4-10) are related to the -dq -dq common reference frame using equation (4-5) for the transformation. 32 . For the k*\"^ machine, (k) 'N = L ( k ) ydq = C ( k ) X ( K ) C (k) I N * C ( k ) \u00a7r = X\"i ( k ) \u2022 C,., e \u00bb ( k ) -N (k) -r . ,00 For multimachines v.. = X i '+Ce\" -N -N -r Substituting the results into (4-4), the solution for i ^ becomes i w = [l - Y X \" ] _ 1 Y C K Transforming to the machine coordinates gives (4-13) (4-14) ^dq = CT [ l - Y x \" ] \" 1 YC e|| (4-15) Equation (4-15) i s the a u x i l i a r y equation for the state-variable equation (4-8). The machine currents are f i r s t obtained from equation (4-15) and the terminal voltages required for the excitation system response from equation (4-10). F i n a l l y , equation (4-8) i s solved for each machine. For the s i m p l i f i e d machine model, where the machine i s treated as a constant voltage behind x^,(4-7) reduces to an algebraic equation through-out the computation. e\u00a3 and x ' defined i n (4-11) and (4-12) are replaced by -T ~ [\u00b0 ' E' 7 (4-16) X\" = (4-17) The mechanical motion equation, ( B - l l ) , i s expressed in the f i r s t order form ast p6 = u> ( A\") ) p( Au> ) = (P - P )\/2H U 0 H4-18) 33 \u2022 When subtransient saliency i s neglected equation (B-12) reduces to P = e\"i + e \" i . (4-19) e q q d d The i n i t i a l conditions for equations (4-15) and (4-1B) are derived in terms of P , Q , v, and 6 , obtained from load floui studies, in Appendix C, C3 3^ The general form of the excitation system equations i s given i n Appendix D. Thus our basic algorithm involves solution of the algebraic equations (4-10), (4-15) and (4-19) in conjunction with the d i f f e r e n t i a l equations (4-8), (4-18) and (D-2) through (D-6). The voltage equation (4-10) and current equation (4-15) are solved in matrix form for n machines at the beginning of each integration sub-interval. The d i f f e r e n t i a l equations are solved using a modified form of Runge-Kutta and Adams-Moulton predictor-. (14,24) corrector i t e r a t i o n . 4.4 System Data and Test Results A t y p i c a l large power system represented using four machines, one thermal, two hydro, with 37 nodes and 47 branches and one interconnected system equivalent i s taken for study. The i n i t i a l conditions, reduced network parameters and machine-exciter data, etc. are summarized in Table 4.1. The fa u l t conditions for these studies are i d e n t i c a l to those given in Figures 2.2 and 2.5. The i n i t i a l condition and network reduction routines are checked and dynamic response compared with single machine i n f i n i t e bus results for a similar disturbance. Some minor modifications to control parameters, etc. were required to obtain damping ra t i o s similar to those achieved in Chapters 2 and 3 but reasonable correlations were observed using either s i g n a l . When armature flux linkage variations are included in the machine model the integration step-size must be reduced to .00025 seconds to avoid numerical PLANT TCRMNAL CONDITIONS \"ACHINE DATA EXCITER DATA 0Q(mv8r \u00bb t 0 ( p u ) \u00bb\u201e(\u00ab!>0.0 240.0 320.0 400.0 TIME ( C Y C . ) (a) ffalatlva Machine Angles (rJaqrves) 1 - Terminal Vo l taq* (pu) 2 - Cxc l ta r F i o l d Voltaqa (pu) 160.D 240.0 320.0 400.0 TIME ( C Y C . ) (b) E x c i t a t i o n Responds for Plant #3 Control S ignal (pu) Spaed Deviat ion (pu) 60.0 240.0 320.0 400 TIME ( C Y C . ) (c ) S t a b i l i z e r Response Tor Plant |3 ure 4.4 Case \"C\" System Response With Power Signal S t a b i l i z a t i o n on Plant #3 Case 'D' System Response With Sub-Optimal Control Signal on Plant #3 40 5. CONCLUSION P r a c t i c a l applications of two types of s t a b i l i z i n g signals have been investigated in t h i s thesis. It has been shown that s i g n i f i c a n t improvement in transient performance can be achieved using an a u x i l i a r y excitation signal derived from either terminal power or shaft speed i n Chapter 2. The linearized model of a power system and the damping c r i t e r i o n developed from t h i s model are j u s t i f i e d by testing those signals on the twelfth-order non-linear system model. I t i s also shown that the dominant eigenvalues of the linearized system do not change very much under varying system conditions over a period of several years. Thus, a s t a b i l i z i n g signal designed for present winter-peak loading conditions can be expected to produce reasonable damping for several years despite changing system conditions. Further work remains to be done in t h i s area, p a r t i c u l a r l y i n the development of systematic procedures for obtaining s t a b i l i z e r parameters. An optimal control has been obtained from the linearized system model applying the linear regulator design of modern control theory, Chapter 3, A simple technique for overcoming the computational d i f f i c u l t i e s a r i s i n g from an unstable system model i s developed. The control law i s then tested on the same twelfth-order system of Chapter 2 and the results compared favourably. Further work i s required, however, before t h i s c o n t r o l l e r can be implemented on an actual system. That i s , some means must be included in the control scheme to adjust the exciter reference signals accounting for deviations i n steady-state conditions caused by changes i n system loading, loss of transmission, etc. Further improvement may be obtained i f s e n s i t i v i t y techniques are applied to choose the weighting factors i n the performance function. An e f f i c i e n t algorithm for multimachine s t a b i l i t y studies has been developed using a formulation which permits direct solution of the network equations u t i l i z i n g machine data readily available. The permissible integration 41 step-size i s increased s i g n i f i c a n t l y by neglecting the rate of change of ermature flux linkages. A four machine test model i s developed using data from a t y p i c a l large power system. The results obtained using a second-order model for a l l machines agree closely with those obtained using a detailed machine-exciter representation for the f i r s t rotor swing. This agrees with re s u l t s obtained by others The need for some form of s t a b i l i z a t i o n i s c l e a r l y demonstrated. When the control signals developed i n Chapters 2 and 3 are applied, i t i s observed that system performance can be .improved and the effects of machine interaction can be compensated by adjusting c o n t r o l l e r parameters. In summary, the results obtained here indicate that the optimal control signal i s a feasible alternative to the s t a b i l i z i n g signal i n current use and that use of a linearized model for power system control signal design i s acceptable. The extension of these control techniques to multimachine studies presents many interesting problems. More accurate system data are required for these studies and more detailed analysis and computation techniques should be investigated. I t i s hoped that t h i s thesis w i l l provide a useful basis for further studies. 42 APPENDIX A DERIVATION OF LINEARIZED EQUATIONS FOR A SINGLE MACHINE INFINITE BUS MODEL IN STATE-VARIABLE FORM This derivation follows the techniques developed by deMello and Concordia in Reference 6. The system equations are generalized to include the external transmission network with an equivalent t i e - l i n e impedance (r,x) and terminal loading (G,B) to permit direct comparison between detailed system studies and the single machine i n f i n i t e bus r e s u l t s . A. l Synchronous Machine and Terminal Equations Neglecting armature resistance (R ), armature flux linkage a variations (py , and py ), and subtransient e f f e c t s , Park's equations can be written in the form j w o Y d = \" V d + X d F V ( A - X ) \" 0 V p = - V q ( A \" 2 ) w o V F = \" ^ d * Vf (a\"3) ' v d = -Yd (A-5) V p = Rpip \u2022 p YF (A-6) T e = V d i q \" V q ^ For convenience, Park's per-unit system i s adopted; i e . COQ= 1 pu = 377 radians\/sec., reactances numerically equal to their cor-responding inductances; the following quantities are also defined: v r = V -steady-state open-circuit armature voltage produced RF by V F, xdF E* s Vr- -voltage behind transient reactance x', a v o l t -q X j . T F y d' age proportional to f i e l d flux linkages, E = v. + x i . -voltage behind quadrature axis reactance, q t q t From (A-2) and (A-5), v . = - oj = x i (A-8) d ^q q q (27) From d e f i n i t i o n of E^, and (A-3) . ' E q = X d F V \" \"T^ *d hence E^ = x dpip - (x r f- x ^ i ^ (A-9) From ( A - l ) , (A-4) and (A-9) ut . = v = -x i , + x . ri r T d q d d dF F or v s E' - x\u00bbi. (A-10) q q d d x Since there i s no quadrature axis e x c i t a t i o n , l i e s on t h i s axis and E a v * x i , q q q d Substituting v^ from (A-10) gives q q q d o Therefore, the e l e c t r i c a l torque has the form T = V.i - y i , e T d q T q d hence T = E i (A-12) e q q x ' The terminal voltage, o y o V. = V . \u2022 V (A-13) t d q The f i e l d voltage equation (A-6) can now be written, V F = R F i F + P y F 44 where v_ = R r i r + pE' x d f. P q dE^ x i + T ' dr F do dt Tdo = V R F r 9 X (A-14) Figure A . l External System Representation There are certain relations between i ., i and v., v . Let the d' q d q current i . be i \u2022 j i and l e t v\u201e be chosen to l i e along the r e a l a x i s , t r m o 3 From the terminal constraint of the system shown in Figure A . l , one has 1 + j i = Yi.4.v4- \u2022 Y . v r J m t t t to o = (C \u2022\u2022+ jB \u00ab\u2022 - ~ ) ( v + jv ) - (-77-\") v J r+jx r J m r+jx o 45 which can be written as i r i m 'c+g -B*b\" V r \" 9 U o \" B-b B+g V .~ b V\u00b0. x ) and b = x\/(r 2 K 2 ) To convert current and voltage to Park's coordinates the transformation indicated by Figure A,2 is applied, Where [C] . s i n 6 c o s 6 -cos6 s i n 6 Figure A.2 Relation Between r-m and d-q Coordinates resulting in s i n 6 cos6 -cos6 sin 6 i q G+g -B+b B-b G+g s i n 6 cos6 V ' 9 Vo\" -cos6 s i n 6 \\ For convenience , let s = sin6 , c = cos6, = g+G, and Y2 = b-B, then we have V s -c * i Y 2 s c v d s -c -gv i 1 c s ; v 2 Y l . -c s V . q. c s bv oJ 46 henc8 Y l Y2 -Y2 Y X 9 b -b g v sin6 o v cos6 o (A-15) F i n a l l y , the mechanical equation of motion can be written i n the form! T - T = ^ 4 * ^ m e d t ^ at (A-16) A.2 Linearized System Equations Equations (A-8) through (A-16) are linearized as follows: Av . = x A i d q q AE' = x,r A i , . - ( x - x * ) A i . q dF F d d d Av\u201e = AE' - x\u00ab A i , q q d d . E A i + i AE qo q qo q A E q = A E q * (A-36) Equations (A-32) through (A-36) represent a complete set of system equations suitable for analogue or d i g i t a l computation. To summarize, the relevant constants are l i s t e d below: K. = E 0, * i (x - x\u00ab)c\/ 1 qo 4 qo v q ti' 2 K 0 = E a \u00ab\u2022 i ( l * ( x - x\u00bb)tf. ) 2 qo 3 qo q d 1 S \u2022 [ l * ( V * i > \u00ab J ' 1 K4 ' < V Xd'\u00b02 V , ^ v . K = x q * 4 _ _qo x\u00aba2 V t o v t o 49 1\/ do . qo , . K. = x c\/ + (1\u2014x' (? ) 6 v q 3 v. d 1 to M to \u00ab Y2 * (YX2* Y 2 2)x q \/D1 S = Y3 + (YiY4* Y 2Y 3 )x q \/0X : T F4 = Y4 * (Y2Y4- Y ^ ) ^ \/ D X D l = 1 * Y 2 < V x \u00ab ) - (Y22* Y ! 2 ) V d 2 2 Y x = g + G = r\/(r \u2022 x )\u00ab\u2022 G Y 2 = b - B = x\/ (r2+ x2) - B v v_ s v (bsino - gcos6 ) = \u2014 r(xsin6 - rcos6 ) \u2022 J O o o i l o o r \u2022 x v Y, = v (gsino + bcos6 ) = \u2014 -(rsin6 * xcosd ) 4 o o o l T . o o r + x For zero terminal loading (G = B = 0), these constants reduce to the follow-ing form: x ( x , *x ' ) r *x 2)x x' (r +x 2 ) 2 + (r 2 *x 2 )x (x +x\u00bb) + (r 2 +x 2 )x x\u00bb 2 2 , 2 2,2 \" , 2 2x2 r + x ( r *x ) (T +x ) 2 0^ = (x+xq)\/A; where A = r + (x*xq Kx+x^) 2 2 v (r +x )(xsin6 -rcos6 ) + frv (rsin6 *xcos6 ) + xu (xsino -rcos6 )1x 0 _ o o o L o o o_ o o o J q 2 , 2 2,2 , 2 2 W , v , 2 2 N (r +x ) \u2022 (r +x )x(x +x') \u2022 (r +x )x x\u00ab q d q a \u2022 V \u00b0 L = -~ |(x+x )sin6 - rcos6 1 2 A L q o oJ \u00b03 = r\/ f t n v frsin6 + (x+x*)cos6 1 u. = o l o \u2022 d o J 50 K l = \" I f [ r s i n 6 o * <*\u00abd,>cos6j \u2022 -3\u00a3 v o ( x q - x d ) [ ( x + x q ) s i n 6 o - rcosdj 2 A qo t A J S = (x -x\u00ab )(x+x )1 1 * ' ' A ^j- 1 K. = A [(x+x )sih6 - rcos6 1 q o o J _ __ frsinS * (x*x')cos6 \"I + v - \u00b0 \u2014 [*rcos6 - (x*x )sin6 1 5 Av^_ I o d oJ Au I o q oJ to *\u2022 J to V ^ \u201e X r V r X'(x+X )_ 6 A v t o V t o L -1 The results check with that obtained in Reference 6. A.3 System Equations in the State-Variable Form The choice of state variables for the system of Figure 2.1 i s somewhat a r b i t r a r y . It i s desirable that they be measurable with available sensors. Since rotor speed and terminal voltage deviations are important factors in system performance, the Aw , Av^> Av,- and AT^ are chosen as state variables. A system fau l t rather than a step-change in mechanical torque i s used as the disturbance so the term AT^ i s neglected.~ i f the damping term D i s included in equation (A-32), equation (A-35) i s expressed in the f i r s t order form as$ Ms Au; = \" A T e (A-37) sA6 = 377AW (A-38) The A6 and AE q of (A-32) through (A-38) can be eliminated. From (A-34) AE q = ( Av t - K 5 A&)\/K 6 (A-39) A 6 = [ K 6 U T e - D A \u00ab , ) - K 2 A \u00ab t ] ( l . r f i . . ^ 51 Substituting the r e s u l t s i n (A-32) and (A-33) gives \/(K,K C- K 0K C) (A-40) K 3 T d \u00bb 3 ( A v t - K 5 A 6 ) = K 3K 6( A v F - K 4 A 6 ) - ( Av^K^ A6) (A-41) Eliminating A6 from (A-41) using (A-38) and (A-40) gives K 3 T d o S ^ t = 3 7 7 K 3 K 5 T d o A \u00ab , * ( K T i T ^ ) [ K6< * V D A \" >\" K 2 ^ t ] 1 6 2 5 u J - * v t + K 3 K 6 A v r (A-42) Substituting (A-41) in (A-38) gives 377A\u00ab> = | K 6 ( s A T e - Ds Aw) - K 2s Av f c J \/ ( K j J ^ - ^ S ^ (A-43) F i n a l l y , s u b s t i t u t i n g x^ =A<\u00bb , x 2 = Av^, x 3 - AVp., = A\"T allows equations (A-36), (A-37), (A-42) and (A-43) to be expressed i n the general formi x s Ax * Bu (A-44) where A s 0 0 0 - i \/ n 377I<5-X1X2D \u2022S ( A - iV 1 ) \/ K6 X 2 K 3 x.x2 0 \" K A \/ T A 0 377K 1-X 1\\ 3D K 2 \/ Tdo B E f 0 0 K A\/T A 0] T *1 = ( K 5 - K 3 K 4 K 6 ) \/ ( K 1 K 6 \" K 2 K 5 ) h = ^ V d ^ h \" K 2 \/ ( K 3 T d o } 52 A,4 I n i t i a l Conditions for the Single Machine-Infinits Bus System The constants used in the linearized small o s c i l l a t i o n system model are functions of both the external network conditions and the i n i t i a l operating state of the machine. The network parameters ( r , x, G, B ) are obtained from the network reduction process and the i n i t i a l operating state from the terminal constraints P , Q and v t , provided they are o o to ' known. The i n i t i a l conditions required are v , , v , i . , i \u00bb E\u201e , v , do qo do qo qo o \/ 28 ) v and 6 q. Following Yu and Vongsuriya the i n i t i a l machine currents and voltages arei P v. i \u00b0 \" (A-45) v ' = x i (A-4G) do q qo x ' v = \/v. 2 - v \/ (A-47) qo \/ to do i . s (P - v i )\/v . (A-48) do x o qo qo\" do ^ Ths components of terminal current referred to the terminal voltage axis are i = P \/v. p o' to i n = Q \/v. Q o to S i m i l a r l y , the current towards the i n f i n i t e bus ( i ) i s obtained in the form: 4P = lP - V t o G lQ \" lQ * V t o B 53 Figure A.3 Phaser Diagram of a Salient Pole Synchronous Generator From Figure A,3r Eqo =\/(\\o*Vu)2 * ( x q 1 p ) 2 VFo = Eqo * W ^ o - t a n \" 1 f x i \/(v. +x i _ ) l [ q P to q Q J * t a n \" 1 [(xi*-rij)\/(vto-pi\u00ab-xi\u00bb)] (A-52) 54 APPENDIX 8 DERIVATION OF AN ALTERNATIVE FORM OF PARK'S EQUATIONS Park ' s equations for a synchronous generator with a s ing l e damper in each rotor ax i s can be wr i t ten in the f o r m a l V d = -R i . + p if\/ , - a> <|> a d r T d q (B-l) v -q - R i +pt l ; + & (1> , a q Y q d (B-2) v F = (B-3) 0 = V o + p Y o (B-4) .0 = (B-5) M d B \"Vd + V F + x d D i D (B-6) 0 T Cj = - y q + V Q (B-7) = \" V d + V F + X F D i D (B-8) W o V D \" \" V d + X F D i F + X D i D (B-9) = \" V q * (B-10) Pos i t i v e armature current corresponds to generator opera t ion . The mechanical motion and e l e c t r i c a l torque equations can be expressed in the form* ' _ _' 2H 2. i \u2022 T = T. - T r \u2014 o p 6 a i e w z t \u00b0 (B- l l ) T = dy i - ij\/ i e T (j q q d (B-12) The general form of Park 's equations (B-l) through (B-10) i s d i f f i c u l t to apply to multimachine s t a b i l i t y s tud ies because de ta i l ed impedance data for rotor c i r c u i t quan t i t i e s i s not normally a va i l ab l e for ex i s t i ng genera t ion . For t h i s reason, i t i s p re fe rab le to use the operat iona l parameters x r f , xjj, T ^ , T ^ , e t c . in the mathematical model. Severa l s imp l i f y i ng 55 assumptions are made i n the development of t h i s type of model and the accuracy of the tests used to evaluate these parameters i s open to question. However, since our primary objective i s to obtain a reasonably e f f i c i e n t computational t o o l for multimachine system analysis within the lim i t a t i o n s of currently available data these problems w i l l not be considered i n d e t a i l here. In order to convert equations (B-l) through (B-10) to operational (12 25) impedance form the following quantities are defined ' t \u2022 * F XD XQ e q l = V F ' Gq2 = W o ' 6 d = ~\\QXQ ' X F T ., x r T \u2022 - \u2014: , T \" = D 0 0 VF D O OTR 1 - X F D 2 'd 'D L X F X D J T \" =\" X Q 9 q o j n c 0 Q ^ = *\u201e - *\u00abr2 , ^ = XH - X d F 2 X D - 2 X d F X F D X d D \" ; c i D 2 x F x j = x - ^ . XF XFXD - XFD XQ I t i s also necessary to assume that the damper windings are completely effective in suppressing changes i n f i e l d current due to sudden (29) changes i n armature current i n which case we have , XdF XD = XFD XdD ( B - 1 3 ) Using equation (B-13), the following s i m p l i f i e d forms for xji and T^ do can be obtained. i! x\u00ab _ ~ XdF XFD xdD* xdD X F xdD^ xdD xF\" xdF xFD ^ xdD xdF ^ \" ^ = XF XD- XFD 2 = XF XFD XdD 2 X F 0 x d F \" FD XdD xdF x , s x . -. d d , XFD (B-14) 56 S i m i l a r l y , Hence FD2 _ XFD XdF . XFD^ Xd\" Xd^ V D XF XdD XdD XdF y \u00bb _ v \" 1 - FD = 1 - d d = d d X ,-x' XF XD VXd Xd\" Xd end,, do Xd\" Xd VXd oi R o D (B-15) We find i t i s also convenient to replace Vp by the equivalent open-circuit excitation voltage on the air-gap l i n e , v r - i~ V Substituting e q, e ^ and Vp in equation (B-3) gives P 6q S ( V F - E q l ) \/ T d o < B- 1 6 ) Substituting e\" i n (B-4) p G q = \" U io XdD Rr,i D D Substituting e and from (B-15). gives pe\" = - d d q x ,-x d\" Ad J 8 a A l o Substituting e\" in (B-5) gives d \" . . . VQ pe\" r%!oQ R n i , ... XQ Substituting e\" and T \" gives d qo -e ,\/T \" d' qo (B-l?) (8-18) 57 When the armature r e s i s t ance , armature f lux l inkage va r i a t i ons , .. and the e f f e c t of the smal l speed changes normally encountered in t rans ien t swings are neg lec ted , equations (B-l) and (B-2) reduce to\u00bb d o T q q o d From (8-7) and d e f i n i t i o n of e r f one has v . = x i + e . (8-19) d q q d From (B-6) and d e f i n i t i o n s of e ^ and e ^ o n e n a s v q = \"Vd + e q l + e q2 (B-20) From the d e f i n i t i o n of e q , (B-8) and p r e v i o u s ' r e s u l t s one has . - X d F 2 . , x dF x FD . q Xp d dF F Xp D X d F X F D = -(x .~xjl)ij + e , + ~~ e -v d d' d q l x f x d D q xd\"xd = \"(VXd}id ' * e q l + ( x ^ ) 8 q2 (B-21) From the d e f i n i t i o n of e' q, (B-9) and previous r e s u l t s one has e\" = - \u2014 \u2014 i , + i r + x .^ip. q * 0 d x Q F dO D 'dD 2 . \"FD'dD i d + x d p i F + x d D i n X F 0 = \"(VXd> *d * S q l + 8 q2 ( 8 \" 2 2 ) S i m i l a r l y , from (B-10) e\" - _39_ i - x _ i n d y q qO Q u = (x -x \" ) i + e. q q q d (8-23) F i n a l l y substituting y r f and y q in (B-12) gives P = w T = v i + v . i , (B-24) e o e q q d d = (e . + e _ - x . i , ) i + (x i + e , ) i . x q l q2 d d ' q q q d'd and substituting e ^, e q 2 a n a\" e d from (B-22) and (B-23) gives P = e\" i + e\" i , - (x\" - x\") i . i (B-25) e q q d d d q d q Equations (B-19) through (B-25) are summarized in matrix form i n Chapter 4 for the multimachine s t a b i l i t y studies. APPENDIX C INITIAL CONDITIONS FOR MULTIMACHINE STUDIES I t i s assumed that the terminal voltage (v^ .0\u00bb \u00a9) ar[d the power output (P \u00bb Q ) vectors are available from load flow studies. For the o o general case, the required i n i t i a l conditions are 6 q, i^0\u00bb iqQ\u00bb vp 0\u00bb e'\\ e ' and e \". do' qo qo Figure C l Phasor Diagram for Multimachine Synchronous Generators Referring to the common reference frame in Figure C . l , the machine angle 6 i s obtained from the voltaqe e and the terminal current i . as follows: 3 q to i = P \/V p o' to i _ = Q \/ Y Q o' to -1 (3 s t a n \" x fx i \/(v, + x i n ) l [ q P to q Q J 6 = 9 * 3 o (C-l) 60 The terminal current and voltage referred to the machine coordinates are i f = ( i 2 \u2022 i n 2 ) * to p Q 0 = t a n ' ^ i p \/ i p ) L6o = Ho s i n ^ * P) ( C - 2 ) V = Ho C08<*+ P ) (C\"3) vdo = v t o S i n 8 ( C \" 4 ) Vqo = v t o C O S 8 ( C _ 5 ) The generator open-circuit f i e l d voltage i s v_ = v * x , i , (C-6) Fo qo d do The i n i t i a l machine voltages can be obtained from equation (4-8) and (C-6) i n the form: (C-?) (C-8) (C-9) For the s i m p l i f i e d machine representation, the voltage e q i s replaced by the voltage E* and the angle i s given by 8' = t a n \" 1 [x\u00bbip\/(v t 0 \u2022 x\u00abi q)] (C-10) Thus i f x i s replaced by x', equations (C-l) through (C-8) remain v a l i d ^ o and 6 Q now corresponds to the angle of E'. The machine voltage vector e^ becomes \u00a30 ^ ' j ^ where E' i s a constant voltage obtained from E ' = Vqo + ^ d o ( C - H ) Since voltage regulator action i s not represented i n t h i s case V p Q i s not required. do \u2022 < v x\") i q qo qo + x\u00abi . d do qo = V qo * x \" i . d do APPENDIX D EXCITATION SYSTEM REPRESENTATION Figure D.l i s a block diagram of the general excitation system model used for these studies. This model corresponds to the IEEE Working Group Type 1 Model described in Reference 15. Provision i s made in the program for simplifying the model in degrees to a single gain with time-lag of the formt A V F \" KA Av. 1 + sT-t A (B-l) 1 \u2022 sT ref V A MAX aux 1 \u2022 sT, KE + S TE V AMIN sKr 1 + sT, Figure D.l General Form of Excitation System Model The general excitation system equations are pv R = < V vR)\/TR = (\"MV - + V - v D - v ) - v . l \/T. I A ref aux R s A J ' t where ^AflllN ~ VA ~ ^ AMAX (D-2) (D-3) (D-4) P v s = [ K F ( v A - K Ev F) - v s ] \/ T F (0-5) pv F = (v A - K Ev F)\/T E (D-6) The i n i t i a l terminal voltage v^Q i s available from load flow r e s u l t s . The i n i t i a l f i e l d voltage V F q i s obtained from equation (C-6). The additional i n i t i a l conditions required for the exciter are\u00bb v = v Ro to V .= V i . + ( K r \/K . )v r ref to C A ' Fo V = 0 aux VAo = KE vFo v =0 so APPENDIX E SOLUTION OF THE ALGEBRAIC MATRIX RICCATI EQUATION USING NEWTON-RAPHSON ITERATION^ The algebraic matrix R i c c a t i equation can be written i n the forms 0 = -KA - ATK + KGK - Q = f(K) (E-l) where G - BR~*BT. I t i s necessary to express the symmetric matrix K as an equivalent vector k containing only the unique terms on and above the main diagonal. S i m i l a r l y f(K) i s expressed as a vector f, wherei - = * * k l l k12 k22 k13 k23 k 3 3 k n n ^ ^ E\" 2^ f = ( f n f 1 2 f 2 2 f 1 3 f 2 3 f 3 3 \u2014 \u2014 \u2014 - \u2014 - f n n ) T (E-3) Considering the Taylor series expansion of the vector f_ about the solution vector (k = k*) and neglecting higher order terms gives, 0 = f \u00ab\u2022 f' (k - k*) Solving for k_* gives k* = k - [ f ] \" 1 ! (E-4) The p a r t i a l derivative f* i s a square Jacobian matrix. Thus equation (E-4) can form the basis for Newton-Raphson i t e r a t i o n for k.*, *k ) which cannot be inverted. Since ATK =[KA1T I J pq j i pq L J a general element of f(l<) i n equation (E-l) can be expressed 64 i n t h e f o r m : f = - Y k ^ a , . - Yk . a . . + Y Yk. , g , k . - Q. , ( E - 6 ) \u00a3f X Ml Jt> T a k i n g t h e p a r t i a l o f f . . w i t h r e s p e c t t o a g e n e r a l e l e m e n t k g i v e s , d f . . - 3 k . , ok-. ok.. 3k . pq i * J pq \/ pq m l pq m l pq S i n c e G a n d K a r e s y m m e t r i c t h e o r d e r o f summation i s u n i m p o r t a n t a n d ( E - 7 ) c a n be w r i t t e n i n t h e f o r m : pq * pq p q J A t t h i s p o i n t i t i s c o n v e n i e n t t o i n t r o d u c e t h e m a t r i x A & A - GK. N o t e t h a t A i s e q u i v a l e n t t o t h e A m a t r i x o f t h e c l o s e d l o o p s y s t e m , i e . A = A, f o r z e r o f e e d b a c k . A g e n e r a l e l e m e n t o f A c a n be e x p r e s s e d i n t h e f o r m : a . . = a . . - Y g . k . ( E - 9 ) i j i j m m m j S u b s t i t u t i n g ( E - 9 ) i n ( E - 8 ) g i v e s of. . dk 4-pq I ok. . dk, . \u00a3 UL - (j_ \/ j 3 k aiiok L J pq pq \u00a3-10) I n g e n e r a l , t h e p a r t i a l ^ ^ \/ ^ p = 1 i f a = p a n d b = q a n d i s z e r o o t h e r -w i s e . T h u s , e q u a t i o n (E-10) p r o v i d e s a c o n v e n i e n t method o f d e t e r m i n i n g t h e e l e m e n t s o f t h e J a c o b i a n m a t r i x , e q u a t i o n ( E-5). However, e q u a t i o n (E-5) u t i l i z e s o n l y t h e u p p e r t r i a n g u l a r e l e m e n t s o f t h e R i c c a t i m a t r i x w h e r e a s e q u a t i o n (E-1Q.) i s d e r i v e d f r o m t h e s q u a r e m a t r i x f o r m o f f ( K ) . T h i s i m p l i e s t h a t i n f o r m i n g t h e e l e m e n t s o f t h e J a c o b i a n u s i n g o n l y t h e u p p e r t r i a n g u l a r e l e m e n t s o f K we must c o m p e n s a t e f o r t h e m i s s i n g t e r m s i n t h e R i c c a t i m a t r i x . From e q u a t i o n (E-10) i t f o l l o w s t h a t i f e i t h e r i o r j i s e q u a l t o p o r q 65 a non-zero element w i l l occur in the Jacobian, A set of rules for formulating the elements of the Jacobian i s given below* S f i i \u2014 = 0 i f i , j \u00a3 p,q ok pq = -a . qj i f i s p\u00bb j \/ p\u00bbq = -sp. i f i = q, j ^ p,q = \" a q i = -a . Pi i f j = p, i ^ p,q i f j = q, i 4 p,q = \" 5 q j \" S q i i f i = P\u00bb J = P = -a . - a . i f i = q, j = q PJ P 1 \u2022 = -a . - a . i f i = p, j = q qj Pi = -a . - a . i f i = q, j = p PJ q i J -A check was obtained by using data provided in an example by Levine and Athans^^. A suitable i n i t i a l guess \u00a3KJ was obtained using Runge-Kutta i t e r a t i o n s t a r t i n g from |KJ = \u00a3oj. The Newton-Raphson algorithm showed much better convergence when compared with solutions obtained using only Runge-Kutta i t e r a t i o n . 66 REFERENCES 1. H.M. E l l i s , J . E . H a r d y , A . L . B l y t h e and J.W. S k o g l u n d , Dynamic S t a b i l i t y o f t h e P e a c e R i v e r S y s t e m . I E E E T r a n s a c t i o n s , V olume PAS-85, J u n e 1966. 2. P . L . Danden o , A.N. K a r a s , K.R. McC l y m o n t a n d W. W a t s o n , E f f e c t o f H i g h - S p e e d R e c t i f i e r E x c i t a t i o n S y s t e m s on G e n e r a t o r S t a b i l i t y L i m i t s , I E E E T r a n s a c t i o n s , Volume PAS-87, J a n u a r y 1 9 6 8 . 3. C. 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D i s s e r t a t i o n , 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 , F e b r u a r y 1968. 11. D.W. O l i v e , New T e c h n i q u e s f o r t h e C a l c u l a t i o n o f D y n a m i c S t a b i l i t y , I E E E T r a n s a c t i o n s , V o l u m e PAS-85, J u l y 1 9 66. 12. D.W. O l i v e , D i g i t a l S i m u l a t i o n o f S y n c h r o n o u s M a c h i n e T r a n s i e n t s . I E E E T r a n s a c t i o n s , Volume PAS-87, A u g u s t 1 968. 1 3 . J.M. U n d r i l l , S t r u c t u r e i n t h e C o m p u t a t i o n o f Power S y s t e m N o n l i n e a r D y n a m i c R e s p o n s e , I E E E T r a n s a c t i o n s , V o l u m e PAS-88, J a n u a r y 1 9 6 9 . 14. K. P r a b h a s h a n k a r a n d W. J a n i s c h e w s y j , D i g i t a l S i m u l a t i o n o f M u l t i -m a c h i n e Power S y s t e m s f o r S t a b i l i t y S t u d i e s , I E E E T r a n s a c t i o n s , V o l u m e PAS-87, J a n u a r y 1 968. 67 15. IEEE Committee Report, Computer Representation of Excitation Systems. IEEE Transactions, Volume PAS-87, June 1968. 16. F.R. Schleif and R.R. Angell, Governor Tests by Simulated Isolation of Hydraulic Turbine Units, IEEE Transactions, Volume PAS-87, May 1968. 17. J.G. Truxal, Control System Synthesis, McGraw-Hill, 1955. 18. T ,R. Blackburn, Solution of the Algebraic Matrix R i c a t t i Equation via Newton-Raphson It e r a t i o n , presented at the JACC in 1968. 19. A.G.M. MacFarlane, An Eigenvector Solution of the Optimal Linear Regulator Problem, Journal of Electronics and Control, Volume 14, Number 6, June 1963. 20. W.S. Levine and M. Athans, On the Optimal Error Regulation of a Strina of Movino Vehicles. IEEE Transactions, Volume AC-11, July 1966. 21. J.B. Ward and H.W. Hale, D i g i t a l Computer Solution of Power-Flow Problems, AIEE Transactions, June 1956. 22. E.W. Kimbark, Power System S t a b i l i t y , Volumes I and I I I , Wiley, 1948. 23. L.J. Rindt, R.W. Long, R .T. Byerly, Transient S t a b i l i t y Studies - Part I I I , Improved Computational Techniques, AIEE Transactions, Volume 79, February 1960. 24. S.D. Conte, Elementary Numerical Analysis, McGraw-Hill, 1965. 25. C. Concordia, Synchronous Machines, Theory and Performance, Wiley, 1951. 26. R.H. Park, Two-Reaction Theory of Synchronous Machines - Part I , Generalized Methods of Analysis, AIEE Transactions, Volume 48, July 1929. 27. B. Adkins, The General Theory of E l e c t r i c a l Machines, Wiley, 1959. 28. Y.N. Yu and K. Vongsuriya, Steady-State S t a b i l i t y Limits of a Regulated Synchronous Machine Connected to an I n f i n i t e System, IEEE Trans-actions, Volume PAS-85, July 1966. 29. W.A. Lewis, The P r i n c i p l e of Synchronous Machines, (book), I l l i n o i s I n s t i t u t e of Technology, 1959. 30. H.E. Lokay and R.L. Bolger, Effect of Turbine-Generator Representation in System S t a b i l i t y Studies, IEEE Transactions, Volume PAS-85, October 1965. 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