A MODEL FOR THE SYNCHRONOUS FREQUENCY RESPONSE MACHINE USING MEASUREMENTS By NELSON JOSE BACALAO Elec. Master Eng.(Hons.)> U n i v e r s i d a d Simon B o l i v a r , E n g . , Rensselaer Polytechnic A THESIS SUBMITTED THE Institute, Venezuela, 1979 New Y o r k , 1980 IN PARTIAL FULFILLMENT OF REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department of We a c c e p t this to the Electrical thesis required THE UNIVERSITY Nelson conforming standard OF BRITISH COLUMBIA August, (c) as Engineering) 1987 J . Bacalao, 1987 46 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it 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 or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6(3/81) ABSTRACT In this machine data, is dissertation presented. allows for dependent either will test modelled The with model It measurements when data estimated used both the was using measurements. to new used to It speed by a l l o w i n g and minimize by the integration the order the up that the also the the test frequency windings. of The responses, these responses the resulting of synchronous machine tested p r o g r a m (EMTP) standard when the data and in model from in a is are the the this in a to the order of on t h e event and depending the input discretization error made accurate the could program, model to integration response when or response model stability frequency stability manufacturer frequency that an response more solution match both frequency ascertained user modifying steps. form directly, required accuracy step, The match a p p r o x i m a t e l y was of frequency successfully found the in on n o n - s t a n d a r d damper of synchronous method. transients are than this was electromagnetic program. effects the modelling consist determine for based the calculated. Saturation model model, of data or new appropriate automatically also be the measured model. This behaviour non-standard a using data to large - i i i - TABLE OF CONTENTS PAGE ABSTRACT ii TABLE OF CONTENTS iii LIST OF TABLES vii L I S T OF ILLUSTRATIONS viii L I S T OF MAJOR SYMBOLS xiii ACKNOWLEDGEMENTS xvii INTRODUCTION OBJECTIVES CHAPTER 1 AND PHILOSOPHICAL PRECEPTS 4 1 : BASIC THEORY OF THE SYNCHRONOUS MACHINE 5 1.1 Physical 5 1.2 Differential 1.3 Equivalent 1.4 Solution in 1.5 the Steady Description Equations Circuits of of the of of Synchronous the the Machine Frequency State the Machine Synchronous Synchronous Differential Machine Machine 15 of the Synchronous Machine CHAPTER 13 Equations Domain Evaluation 7 23 1.5.1 Positive sequence 23 1.5.2 Negative sequence 27 1.5.3. Zero sequence 30 2 : BASIC THEORY OF THE NEW MODEL 31 2.1. Introduction 31 2.2 Transformation into 2.3 the the Frequency Domain Equations Time Domain Approximation 2.3.1 of Brief Method by R a t i o n a l Description 31 Functions of the 37 Approximation 37 - iv - PAGE 2.4 2.5 C o r r e c t i o n of the F u n c t i o n s to be A p p r o x i m a t e d of S a t u r a t i o n 2.4.1 Incorporation 2.4.2 A p p r o x i m a t i o n of C u r v e s 47 2.4.2.1 A p p r o x i m a t i o n of X j ( s ) and X ^ ( s ) 47 2.4.2.2 A p p r o x i m a t i o n of G(s) 48 2.4.2.3 A p p r o x i m a t i o n of F l ( s ) to F 6 ( s ) 51 Run Time Reduced M o d e l s and C o m p e n s a t i o n Numerical 57 2.5.1 Reduction 2.5.2 E v a l u a t i o n of 3 : INCLUSION 40 of Errors of the Order of the E r r o r t h e Model i n the Domain and I n t r o d u c t i o n CHAPTER Effects 40 58 Frequency of C o r r e c t i n g P o l e s OF NONLINEARITIES 59 68 3.1 Introduction 68 3.2 Method 1 f o r t h e C o n s i d e r a t i o n of Saturation 68 3.3 Method 2 f o r t h e C o n s i d e r a t i o n of S a t u r a t i o n 71 3.3.1 General Description 3.3.2 Equations of t h e Method of t h e Model CHAPTER 4 : IMPLEMENTATION 75 OF THE MODEL IN AN ELECTROMAGNETIC TRANSIENTS PROGRAM 4.1 Introduction 4.2 General D e s c r i p t i o n Transients 4.3 of 4.4 of 4.5 of the Electromagnetic (EMTP) o f Method 1 f o r t h e 82 Consideration Saturation Implementation Saturation Results 82 82 Program Used Implementation 72 83 o f Method 2 f o r t h e Consideration 87 91 V PAGE 4.5.1 V a l i d a t i o n of 4.5.2 Effects 4.5.3 E v a l u a t i o n of of Proposed 4.5.4 Method 1 Using Different the Input Usefulness of Data Behaviour of 104 the on the Numerical Method 111 Conclusions 111 CHAPTER 5 : IMPLEMENTATION IN A STABILITY 5.1 Introduction 5.2 General 5.3 Description 5.4 Results 5.5 Usage o f PROGRAM of the the the of the Stability Program Implementation Implementation Model for Speeding 5.7 E v a l u a t i o n of 117 up a Stability 120 the Impact of the Transformer Terms 125 Conclusions 128 CHAPTER 6 : CONCLUSIONS 135 139 REFERENCES APPENDIX 113 115 Program 5.6 113 113 Description of 96 the Method General Observations 4.5.5 91 1 : THE RECURSIVE CONVOLUTION TECHNIQUE 141 Al.l Convolution with an E x p o n e n t i a l . 141 A1.2 Convolution with an I m p u l s e 143 APPENDIX 2 : BLOCK DIAGRAMS OF EXCITERS AND GOVERNORS USED IN STABILITY A2.1 Response Exciter A2.2 Governor SIMULATIONS 145 145 146 - vi - PAGE APPENDIX 3: DESCRIPTION OF THE S T A N D - S T I L L FREQUENCY RESPONSE METHODS FOR THE EVALUATION OF X (s), d X (s) AND G ( s ) 147 A3.1 Measurement of X^(s) 147 A3.2 Measurement of X (s) 148 q A3.3 APPENDIX Measurement 4 of G(s) 149 : DEVELOPMENT OF THE INTEGRATION EQUATIONS FOR THE METHOD 2 FOR THE CONSIDERATION OF SATURATION APPENDIX 5 : CONSIDERATION OF UNEQUAL FLUX LINKAGES USING AN EQUIVALENT CIRCUIT APPENDIX 6 150 158 : EFFECT OF SATURATION ON THE MACHINE TIME CONSTANTS 162 - vii L I S T OF TABLES TABLE 5.1 PAGE Results obtained by t e s t i n g steps the model different integration A.l Effect of saturation i n Guri A.2 Effect of saturation in a f o s s i l - f i r e d with At. unit 124 7 to 10. 162 unit. 162 - vi i i - L I S T OF ILLUSTRATIONS FIGURE PAGE 1.1 Physical 1.2 Relationship forces 1.3 representation between of the the synchronous d and q - a x i s magnetomotive the f l u x d-axis linkage Equivalent c i r c u i t s of the synchronous 1.5 Equivalent c i r c u i t s in the frequency 1.6 Steady diagram with 2.1 Method f o r Bode p l o t 2.2-A2 2.2-B1 2.2-B2 phasor allocating the poles machine 20 saturation 20 and z e r o s from a 39 O n t a r i o Hydro A p p r o x i m a t i o n of F l to a n g l e o f the f u n c t i o n s O n t a r i o Hydro F3 f o r A p p r o x i m a t i o n of O n t a r i o Hydro the generator 41 O n t a r i o Hydro F4 to generator 41 A p p r o x i m a t i o n of F4 to F6 f o r module of t h e f u n c t i o n s of 16 domain A p p r o x i m a t i o n of F l to F3 f o r module of the f u n c t i o n s angle path 16 1.4 2.2-A1 6 6 S c h e m a t i c r e p r e s e n t a t i o n of between the w i n d i n g s i n the state machine F6 f o r generator 42 generator functions 2.3 L i n e a r i z a t i o n of 2.4-A Xj(s) 2.4-B A s s o c i a t e d f u n c t i o n s to X ^ ( s ) and X ( s ) for d i f f e r e n t s a t u r a t i o n segments ^ Method f o r e v a l u a t i n g an a p p r o x i m a t i o n f o r the 49 saturation 50 2.4- C the 42 and X ^ ( s ) f o r segment different i from i 2.5-B A s s o c i a t e d f u n c t i o n s to G ( s ) f o r saturation segments A s s o c i a t e d f u n c t i o n s to F l ( s ) to d i f f e r e n t s a t u r a t i o n segments 2.7-B Study o f Fl(s) reduced Study of F2(s) reduced curve segments 1 F u n c t i o n G(s) 2.7-A different - saturation saturation 2.5- A 2.6 for open-circuit saturation segments 44 49 52 different 52 F6(s) order approximations: for 55 function 55 order approximations: function 56 - ix - FIGURE 2.7- C 2.8- A 2.8-B 2.8- C 2.9- A 2.9-B 2.9- C 2.10- A 2.10-B 2.10-C . PAGE Study F3(s) of Study F4(s) of Study F5(s) of Study F6(s) of reduced order approximations: function 56 reduced order approximations: function 61 reduced order approximations: function 61 reduced order approximations: function 62 Study of function the e f f e c t Fl(s) of Study o f function the e f f e c t F2(s) of Study o f function the e f f e c t F3(s) of Study of function the e f f e c t F4(s) of Study of function the e f f e c t F5(s) of Study of function the e f f e c t F6(s) of the the transformer terms: 62 the transformer terms: 65 the transformer terms: 65 the transformer terms: 66 the transformer terms: 66 the transformer terms: 67 3.1 L i n e a r i z a t i o n of saturation curve 70 3.2 E q u i v a l e n t c i r c u i t of the s y n c h r o n o u s machine method 2 f o r the c o n s i d e r a t i o n of s a t u r a t i o n for 73 3.2-A D-axi s 73 3.2-B Q-axis 74 4.1 Flow d i a g r a m f o r i n the EMTP the Flow d i a g r a m f o r i n the EMTP the 4.2 4.3 4.4 4.5 Circuit model implementation of method 1 88 and machine implementation of method 2 90 data used for testing the 92 C o m p a r i s o n between methods f i e l d current 1 and 2 f o r C o m p a r i s o n between methods power a n g l e 1 and 2 f o r saturation: 93 saturation: 93 - X - FIGURE 4.6 4.7 4.8 4.9 4.10-A 4.10-B 4.11 4.12-A 4.12-B 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 PAGE C o m p a r i s o n between methods e l e c t r i c a l power o u t p u t 1 and 2 f o r saturation: 94 C o m p a r i s o n between methods 1 and 2 f o r v o l t a g e i n the d and q a x i s saturation: C o m p a r i s o n between methods 1 and 2 f o r c u r r e n t i n t h e d and q a x i s saturation: C o m p a r i s o n between methods a n g u l a r speed saturation: E l e c t r i c a l power a f t e r m a n u f a c t u r e r ' s data 1 and 2 f o r Field current after manufacturer's data opening a line: 97 different input data data 97 : 98 opening a line: 100 F i e l d c u r r e n t a f t e r opening by O n t a r i o Hydro from SSFR C o m p a r i s o n between f i e l d current 95 95 E l e c t r i c a l power a f t e r o p e n i n g a l i n e : e s t i m a t e d by O n t a r i o Hydro from SSFR C o m p a r i s o n between a c t i v e power 94 different a line: data estimated 100 input data: 101 C o m p a r i s o n between d i f f e r e n t i n p u t d a t a : f i e l d current ( manufacturer's parameters) 102 C o m p a r i s o n between d i f f e r e n t power a n g l e input 102 C o m p a r i s o n between d i f f e r e n t a n g u l a r speed input data: data: 103 Frequency different r e s p o n s e f o r Lambton g e n e r a t o r input data: Xd(s) using Frequency different r e s p o n s e f o r Lambton g e n e r a t o r input data: Xq(s) using Frequency different r e s p o n s e f o r Lambton g e n e r a t o r input data: G(s) using Frequency different response for Nanticoke input data: Xd(s) generator Frequency different response for Nanticoke input data: Xq(s) generator 106 106 107 using 107 using 108 - xi FIGURE 4.22 4.23 4.24 4.25 PAGE Frequency different response for Nanticoke input data: G(s) data C o m p a r i s o n between d i f f e r e n t N a n t i c o k e u n i t : power a n g l e data 5.1-A Equivalent c i r c u i t the machine 5.3-A 5.3-B 5.4 5.5 5.6 5.7 5.8 5.9 5.10 for 109 input for 109 C o m p a r i s o n between d i f f e r e n t i n p u t d a t a N a n t i c o k e u n i t : c u r r e n t d and q a x i s Basic a l g o r i t h m of Flow diagram f o r i n PSS/ED 110 for 114 the modelling of 118 the implementation of the model 119 used for the program : d a x i s validation Functions stability used for the program : q a x i s validation the method of the 121 of the 121 used in PSS/ED: 122 V a l i d a t i o n o f t h e method used i n m e c h a n i c a l and e l e c t r i c a l power V a l i d a t i o n of f i e l d voltage for PSS/ED Functions stability V a l i d a t i o n of power a n g l e using 108 C o m p a r i s o n between d i f f e r e n t i n p u t Nanticoke u n i t : e l e c t r i c a l torque 5.1 5.2 generator the (E f ( J PSS/ED: 122 method used i n P S S / E D : ) and c u r r e n t ( I ) 123 f d V a l i d a t i o n of the method used t e r m i n a l c u r r e n t and v o l t a g e Test of the r e d u c e d o r d e r w i t h o u t c o r r e c t i o n : power in model angle PSS/ED: 123 with and 126 Test of the r e d u c e d o r d e r model w i t h w i t h o u t c o r r e c t i o n : e l e c t r i c a l power and 126 Test of the r e d u c e d o r d e r model w i t h and without c o r r e c t i o n : f i e l d current I . c 127 t a 5.11 5.12 Test of the r e d u c e d o r d e r model w i t h without c o r r e c t i o n : terminal current E v a l u a t i o n of power a n g l e the effect of and and v o l t a g e transformer 127 terms: 129 - xii FIGURE 5.13 5.14 5.15 5.16 5.17 PAGE E v a l u a t i o n of the e l e c t r i c a l power effect of transformer 129 E v a l u a t i o n of the e f f e c t of v o l t a g e i n the d and q - a x i s transformer E v a l u a t i o n of the e f f e c t of c u r r e n t i n the d and q - a x i s transformer E v a l u a t i o n of back-swing in E f f e c t of specially the the terms: 130 e f f e c t of t r a n s f o r m e r power a n g l e step terms: 131 in a 134 A2.1 Exciter A2.2 Governor block A5.1 E q u i v a l e n t c i r c u i t f o r the d account unequal f l u x l i n k a g e s axis Equivalent series A5.2 terms: 130 using a large integration d e s i g n e d e x c i t e r model block terms: diagram 145 diagram circuit without 146 the taking into 159 branch 159 - LIST xi i i - OF MAJOR SYMBOLS SYMBOLS C. integration constant convolution, number e 2.718281828 D mechanical d subscript denoting f frequency (Hz) f rated Fl(s) used G(s) stator field H^(t) term t h a t I current , a , b, c F6(s) frequency functions I - damping to is from the i coefficient the direct ( 60 Hz i n to model the transfer dependent axis examples) machine function on p a s t stator phase 1^ direct axis I^j field 1^^ direct 1^ quadrature axis damper w i n d i n g I quadrature axis stator I machine terminal j complex operator J moment L implicit values currents stator current current axis of damper w i n d i n g current current current current -1 inertia inductance L.. self-inductance L. . mutual and j 11 J L j of inductance direct axis inductance stator winding 6 i between w i n d i n g s to rotor mutual i - xiv - quadrature axis inductance stator to direct axis synchronous direct axis transient direct axis subtransient direct axis operational field leakage rotor mutual inductance inductance inductance inductance. inductance field self-inductance g-coil leakage inductance direct axis inductance damper w i n d i n g leakage direct axis inductance damper w i n d i n g self stator leakage inductance quadrature axis i nductance damper w i n d i n g leakage quadrature inductance axis damper w i n d i n g self- quadrature axis synchronous inductance quadrature axis operational inductance a pole subscript stator field denoting winding winding the quadrature resistance per phase resistance d-axis damper w i n d i n g resistance q-axis damper w i n d i n g resistance Laplace operator Laplace transformation Park's transformation axis - X V - electrical torque mechanical torque d-axis transient constant short-circuit d-axis subtransient constant d-axis transient constant short-circuit open-circuit d-axis subtransient constant phase-to-neutral direct axis field machine axis terminal p o s i t i o n of reference voltage voltage the stator voltage voltage rotor relative to change machine power a n g l e integration angular rated time voltage quadrature small time time open-circuit stator stator time step load angle size speed angular magnetic or speed. flux f l u x which l i n k s wi n d i ng the stator d-axis f l u x which l i n k s wi nd i ng the stator q-axis flux which l i n k s the field f l u x which l i n k s wi n d i ng the damper winding d-axis a fi - ^. f l u x which winding TT 3. 1415926 XVI links - the damper q-axis - xvii - ACKNOWLEDGEMENTS I would persons like who work and in to one express way or my s i n c e r e another gratitude helped to all me t h r o u g h o u t those this particularly, To Dr H.W. Dommel, To Dr J o s e To Luis To my To my f a t h e r To Electrificacion del To J o a q u i n Da S i l v a , for and To The Marti, Marti, wife for his advice. for field my mother To my help, d i r e c t i on,and i n t r o d u c i n g me to the of frequency-dependence for his valuable di s c u s s i ons. beautiful modelling. suggestions and Paloma, for her help i n many a s p e c t s of t h i s p r o j e c t , but most of a l l f o r her l o v e and p a t i e n c e . and Mae, f o r p r o o f r e a d i n g manuscript. F u n d a c i o n Gran To invaluable and sister Caroni the ( EDELCA original ), for their support. g i v i n g me the t i m e , t h e the means to f i n i s h t h i s Mariscal mother-in-law, Mercedes, de trust, work. Ayacucho, for their f i n a n c i a l support. for whole for drawing diagrams. h e l p i n g me put the thing together. those beautiful - 1) INTRODUCTION one In recent of the time. the to In most this very the years, for industry, first stages operation the of interactions very this The the past. of cannot the these a remain of a few basically integration step that Consequent1y,for have the models valid only time this networks and of and from up this one of the synchronous with in well systems of the and the frequency. generally used type of as half simulations, a frequency range evaluate strategies. from is In studied is assumed to The a can machines to event nominal is network its network has simulations, the without the important dynamics, whether the machine. stability as are advance most its many today. to the their of effort is models for hearts w h i c h we have elapsed and computer network themselves, control is our electric minds determine seconds for the perturbation at of ranges objective namely to different simulations, generally to is withstand of our machine s y n c h r o n i s m among effect tools modelled objective become computer adequate The r e s u l t s network, been in modelling synchronous the losing the of has enterprises the design the analyical in of developing dissertation, traditionally where of industry systems. occupied state-of-the-art components the power complex role components sophisticated In the of art has in electric existing different engineers the technologically Therefore,the or 1 - minimum cycle. one should 0 to 10 or try 15 - Hz a t most. Another type electromagnetic important. 230 system which are and systems, protection concern. As The in is this voltage flux the of there shaft, are machine very induced connected. modelled too an slow is in the the the full are the which evaluation span be are under Therefore modelled the those But to the of a of some studied models in as dynamics in have of the p r o g r a m s . One of resonance, damped o s c i l l a t i o n s system studies simulations simulation subsynchronous in must must picture. between including examples network time be since stability electrical involve lines cycles. can into in transient the by t h e accurately and for and a few come voltage predominant transient impedance, to lower the on elements. machine somewhere lightly Other the the example, generally and in of for the balanced, higher order as is above dependent In analyzed system KHz or the need phenomenon are in of fall the synchronous be interest simulations produced such of behind which transient to three-phase order source phenomena lightning a r e no l o n g e r simulation, changes faults. of more and more become internally, of analysis increased has distributed-pararaeter usually type generated the become have design against frequencies usually study as a recently voltages phenomena as namely initiation waves and represented is the modelled has insulation switching travelling analysis, transients, KV l e v e l , during of As t h e overvoltages be 2 - in the in machine's which the machine the which machine must secondary be arc - current during single-phase 3 - reclosing, and load rejection studies. In the model must types should be be v a l i d In is and data, possible, of to to accuracy avoid be be a machine to and, the to model models undertaken. model the which type the needed is Therefore, is of u n d e r - m o d e l l i n g or it 100 H z . able the machine generally, from 0 t o a c c u r a c y of study have thus range important and t h e its important and i t data" for results made has [1,2,3]. by the this to Consequently, response the the [5]. same to that the often flexible study to be over-modelling most device so-called yield more is the "standard unsatisfactory effort has adequate successful one been testing seems to be measurements. model will be principles that were frequency-dependence With any a significant develop the this technique, synthesize convoluted numerically response the of modelling machine dissertation,a modelling parameters when recognized among w h i c h , essentially used been industry frequency In concern synchronous techniques, in as above, machine. An the is type to adapt undertaken, the the indicated frequency machine of to accurate it desirable enough studies the summary, function it as for synchronous a of output. an i m p u l s e with It will an be of presented which uses successfully transmission frequency response input to shown that used lines responses which find this are can the be time approach - not only being it utilizes studied, makes machine 2) it is the best A - data available i n c l u d i n g frequency possible to be to for response change the the phenomena measurements, detail in but which the modelled. OBJECTIVES AND PHILOSOPHICAL PRECEPTS As mentioned dissertation is synchronous aspects in of which required, best data machine and the practical * The and * The a is is objective This the need modelled an is in accurate in done for as the of area of facing a faster in two model, much d e t a i l model, this as which is the used. this of only for the contribution modelling. dissertation, the were : precepts The p r o d u c t The make need available philosophical * to Introduction, problem; namely, Throughout * the machine the the in this followed research project following should general help to solve easy to use. be fast problems. model should numerical be understandable methods to be used and should reli able. results implement in of the existing research programs. should be easy to - 5 - CHAPTER 1 BASIC THEORY OF THE SYNCHRONOUS 1.1) Physical A synchronous elements, stator c) a Description the or rotor balanced 120 set the machine armature, displaced of and there degrees Synchronous Machine consists the stator are essentially (see three apart. MACHINE fig. windings These of 1.1). In the (phases a,b and windings, when fed of i c u r r e n t s , (the machine i s assumed "ideal" = I sin(wt) i. = I = I a i with [24]) (a) m D c two sin(u>t - in sin(ujt m 120°) (b) + 120°) (c) 1.1 produce a rotating amplitude that of in the On the winding, the magnetomotive air gap, force whose a n g u l a r MMFa with speed is constant the same as currents. rotor which there when magnetomotive is fed force one with MMFf dc winding, current, stationary known as produces with the a field constant respect to the rotor . If as is fluxes the the rotor spins the case in produced at the by t h e s e due to the we have a generator pulls the stator two rotor field speed synchronous flux then an a n g u l a r the become flux operation, and power equal machines, MMF w i l l leads u) is the to , magnetic superimposed. due i.e., to to the the stator, rotor transferred If field from the - FIGURE 1 . 1 : Physical machine 6 - representation of the synchronous B ^ a * ^ b ' c Magnetic field in the a,b, and c windings, p u l s a t i n g in time at co a n d c o n s t a n t in direction. B : Resulting field l- B ^ + B constant B . a 1 n t i m e ancf r o t a t i n g a t : Excitation field to the f i e l d d u e winding, constant in t i m e r o t a t i n g a t to r FIGURE 1 . 2 : R e l a t i o n s h i p m a K i i e c o m o t i ve 4' oils Vd "S between the forces d and q a x i s - former the to the rotor 1.2) latter. flux Differential In order to characterize consider a then it we have the set of the stator flux leads operation. the of synchronous a set if motor E q u a t i o n s of the - Conversely, write as 7 Synchronous Machine differential machine, mutually it coupled equations is which advisable windings, much to like transformer. The following satisfied for set the v b v of differential stator = - r i - " r windings, - d J \> — J a H " 4-*b a dt v = - r i - c d di C equations phases a, must be and c: b (a) ( b 4> ) (c) C 1.2 where the the current winding in is the v defined rotor = - f positive or r i 1 field - d_J, dt f 1 leaving winding, the we stator. For have: 1 1.3 where again, field winding. In all winding flux i , which the these i ^ links current is equations is the it. *a *b = L L v^ is the current This - defined flux aa ba *a 4 a + L given ab bb voltage through is + L positive d i b b + L + L applied it the to the a n d ty^ i s the by: *c ac bc leaving d c + + L L af bf *f *f ( ( b a ) ) *c * *f where the L . . n is *a fa "a + L + L cb "b fb "b + L + L se1f-inductance cc fully equations are since in all (with the of and that them, when minimum, when of depending of the the rotor + L ff the later. difficult to and ) 1.4 is of the damper differential as they the be verified on rotor position magnetic and p a t h a r e L. . iJ the of can right and d stand, se1f-inductances This at ( i The time. is c -f of solve Lf£) functions the <> behaviour exception mutual *f and j . introduced different rotor i describe the hence, reluctance maximum, be extremely exception rotor consider will cf winding equations the c L of These with + i between w i n d i n g s machine *c fc inductance which self L the windings, the = ca 8 - mutual synchronous the L - angle aligned. to So, position (see paths are the if fig. varies we 1.1), from a path, we have of for to a the inductances: L L aa bb L e = L = L = L cc s + L m c o s (26) s + L m cos (2 6 - 240°) (b) s + L m c o s (2 6 + 240°) (c) (a) 1.5 mutuals: ab L = L, ba = - M = L = - M L , = cb - M ac L. = be L af = L bf = L cf = M ca f f s s - L - L - L cos(29 + m cos(26 + )°) .„o (a) (b) m cos(26 m cos(20) - (c) (d) - 120° ) (e) cos(26 + 1 2 0 ° ) (f) cos(28 M s - with t OJ 9 - + 6 (8) 1.6 This problem two fictitious the rotor, equal fig. the to whose 1.2). is order for windings the make a q, transformation; can be the formulated. direct [T] permits into the of recovering and has to isolated to (see conversion of a p p r o p r i a t e d and q in basis the can figure reversible be to respect windings the from corresponds defining a r o t a t i n g MMF stator considerations indicated hence, The one chosen axis "d" [ 6 ] , above there be 1.2. and to original phase included. This the the This are do n o t several here is "q" l a g g i n g fictitious zero sequence uniquely define ways i n w h i c h power is it i n v a r i a n t and 9 0 ° with transformation respect given cos(B) cos(B-120°) cos( $ + 1 2 0 ° ) sin(B) sin(B-120°) sin(B+120°) to by K 1//2" 1//2 where with produce transformation is quadrature axis = that by components. the the to original winding winding requirements sets the third d and stationary is the literature Park's Transformation. Its possibility symmetrical it by geometrical to additional The q, and v o l t a g e s called from the effect produced quantities, in d and added currents quantities allow in The t r a n s f o r m a t i o n stator In solved windings that deduced is K = B = • 2 / 1/ y/T (b) 3 6 + wt (a) + 90 ( (c) - l o -i rn = (d) [T] transposed 1.7 so that ] = [T] [V [ i dqo ] = [T] [ i abc [V dqo (a) abc ] (b) 1.8 With (a) of this to transformation, 1.2 (c) variables So f a r rotor is it is their the been currents closed circuits rotor to current taken into windings, in in all are in the to convert "d,q,0", called closed as by only This eqs. the 1.2 new set in the of current assumption when circuits squirrel-cage iron. account the transients, damping the that current. during the paths possible equivalent assumed field valid aid is known. has necessarily induced to it there in bars the inclusion damper w i n d i n g s , could (installed oscillations) of aligned not be r o t o r . These and The i n d u c e d c u r r e n t s the is the are two in the eddy normally equivalent along the d and q axis. In conventional represented, parameter noted in coils [3], significant field tests The by relative connected in larger and t h o s e for and the the success, in skin [1,4,7], with occur massive between by t h e s e effect; of generally the constant However, iron as rotors, results from models. discrepancies resistances are by a number o f parallel may predicted these damper w i n d i n g s machines discrepancies reason inductances affected with models, is that damper therefore, their the actual windings value is are not - constant, but transient. rotor is This effect machines, salient pole In this inductance and and this it be resistance but and the of are functions for the leakage the part of the the of skin derivation v, = d r v r - v q = - r f r the of f i i q - effect of the solid iron even in 0 = - = - r r. kq of the d <K "dt o i o - r , , k d these for windings in The model the the that only This the of but it affects is of resistance these assumption d developed frequency-dependence inductance only new m o d e l , i t of - 0) ° - d TJ> . 7^7 q i, kq the with is windings necessary a reasonable the internal flux. + w - d _ j K dt f r for and damper saturation, o the q ij;, d is useful synchronous to machine write as: (a) (b) (c) A o large associated q axis. self 0 = - rkd , , i kd . , - d_4v kd - v content present L. , kd assumes equations i , a d aa the it c u r r e n t and differential the frequency. consideration the as of accounts damper w i n d i n g s , For in always parameters for dissertation distribution frequency noticeable written and since the is the the one, most but derivation, will axis, is of - machines. windings ° in a function 11 <) d d ik ^ kq (e) r Lodi (f) rzo dt K 9 - 12 - where *d - L 4> = q *f *d + M df *f + M dkd J kd ( a L i + M. i, q q qkq kq " \ d d L - L f *f + k d *kd M df + M *d + dkd M ) (b) fkd *d + M *kd fkd ( d f ( C d ) ) 1.10 In these equations, frequency-dependence recovered later frequency of when that the the the domain. A l s o , was assumed it was necessary damper w i n d i n g s , to but a reasonable the are transformed equations 1.9 (a) in r o t o r speed assumption modelling To of complete mechanical by t h e the this system is constant swing and it is synchronous set must be equation : J d f 6 + Du = dt 9 it equations and 1.9 OJ = r is neglect of generally to (b), be the it This accepted in machine. differential included. OJ . o will the This equations, system is the described Te - Tm 2 1.11 where J : Moment o f inertia D : Mechanical w : Angular Te damping coefficient speed : Electrical Torque Tm : M e c h a n i c a l torque (5 : Load a n g l e as defined in fig. 1.2 - 1.3) Equivalent Circuits To the better to The to the first system. machine values for ratio the magnetic found rotor should be be d-axis indicated Using the are is air mutual common mutual links only q here axis. circuits the [8] stator. mutual the This themselves, windings also provided links as and coupled among themselves an inductance 1.3). additional model. mutual fig This field these To inductance inductance between the L , ad rotor is winding [7], windings take leakage the there damper (see turns between the between base selection inductances among the d and q corresponding with is per- uses stator's is that an a p p r o p r i a t e accordance two and an a p p r o x i m a t i o n , and higher the to inductances account, in the d of it circuits equivalent the the equal gap closely the into all the for in and that this the more included increase flux slightly difference is axis practice Consequently, the values, are manipulation, chosen base one to system these rotor in 1.11 values in leakage in these and equations equivalent base windings In their the per-unit, tend 1^£ 1.10 in the two developing that third. some the between insures that From ease as per-unit as differential dynamics 1.9, The rating windings. the eqs. the them in - Synchronous Machine and to rotor step transform unit machine represent summarize the understand synchronous useful of 13 this inductance is added to order to windings, as in below. the per-unit system suggested in [8], we have for the self - 14 - and m u t u a l i n d u c t a n c e s M , = M ., = L , dkd df ad of the M . = fkd d = L kd ad L L " , X ad ad L + b a + X ° ( kf X + . (a) ( ) f " f L : L . + 1. ad kf P 1 L d-axis kf + 2 kd ( V + d ) ( f ) 1.12 where For the 1 is the q-axis, L inductance of the i windings. we have i n p e r - u n i t : M . L qkq aq = q leakage (a) L + 1 aq a L. = kq (b) L +1, aq kq (c) 1.13 Equations ^d - *f * *kd \ 1.9 a X *f - remain the *d + L ad f + L ad j *kd ± k d " X a 4>, = kq \ + L same ^ f ( + L aq + ad ( kq ( 1 *kd + f i in d i f per—unit, + i d *kd + : V + kd but w i t h ) + ( 3 ) *fk + V + ( X f + fk ( i ikd) f i ( 'kd* + b ( c V + ) ) ^ 1, i, + L (i, + i ) kq kq aq kq q' ( ) e 1.14 From equations equations 1.9 equivalent 1.4a and winding circuit 1.4b, to of to 1.9 for where parameters functions a given (c) 1.12 has w ; i.e., (e), the the 1.14, it " q " and together is with possible "d" a x i s the stressed circuit is by only of the in figures the damper writing fully rotor derive shown frequency-dependence been to the them defined as for w . These two equivalent circuits, together with the stator - equations provide 1.4) 1.9 (a) enough Solution and 1.9 information of the 15 - (b) and the swing for the modelling Machine D i f f e r e n t i a l equation of the 1.11, machine. Equations in the F r e q u e n c y Domain The set previous the of differential section ignored damper w i n d i n g s , realize 1 , , , kd that the writing of solve and the frequency behaviour windings. In the an these frequency frequency dissertation, no need since case for the to of the as choosing order as function a of be a n a l y z e d the a stressed are ofco). extremely windings as a in parallel, similar to work Ontario by that circuit, parallel, of the led be given model and on the to were This the shown, number can be to However the to is to combined the by the original [4], the two measured idea of using in method, there in or adjusting presented coils be by whose to varied, response 1.4 contained this are constant general of fig Hydro method in if of found model valid them of which we difficult group of if but in solving machine. will constant was the behaviour still of in techniques but not in recovered are this behaviour response identification are equivalent windings easily method is recent be circuits connected a of One damper coils parameters can equations are. parameter of ( parameters they approximate three r, kq differential as this frequency these resulting but developed frequency-dependent equivalent r . j , 1. kd kq functions the equations this is parallel, depending matched. on the - FIGURE 1.3 16 - : S c h e m a t i c r e p r e s e n t a t i o n of the f l u x l i n k a g e p a t h between t h e w i n d i n g s i n the d - a x i s . otr gap L FIGURE 1 d.t L 1.4 : Equivalent c i r c u i t s of the ad synchronous *kd( ) a k<jcto) w d 'kd< > w 1* f 1, M Fie. 1.4a machine Fig 1.4b - A first Laplace step achieved the previous when the time as If values they are of the part A general way a new s e t Let in in conditions. define developing transformation presented appear in (t) g of the to of the solution in zero be after initial the as the let and be p e r t u r b a t i o n . Then, the equations be easily zero initial frequency will domain. conditions is to : i n i t i a l g(t) use conditions follows positive and to can have initial ftg(t) g, i n given the This zero, (t) is differential not components, t model involved Q variable new paragraph. variables g the the achieve and - variables symmetrical time 17 steady negative its if state sequence value we at any defineAg(t) as Ag(t) = g(t)-g o + (t)-g _(t) (a) o then Ag(t) =0 at t=0, (b) 1.15 which is the Using in the r e q u i r e m e n t we want this previous equations is transformation the equivalent change of in the circuits domain fig. are used to denote is 1-5 the of In v a r i a b l e to in the that and equations these in 1.4, the circuits = L{ A g ( t ) } . form the we given of the Laplace which can frequency transformed v a r i a b l e s G(s) the using figure fulfill. equations Thus, circuits ). new evident variables equivalent see it altered. corresponding ( variables paragraph, not the define find or capital the Laplace letters i.e.: 1.16 One of to the be used the 1 8 - main a d v a n t a g e s of transforming frequency to So, derive, \ from after ^ U) domain determine variables. to - s > l l that the circuit X s G ( ) the manipulations between the 1.5, fig. following it S can important is possible relationships: S ( b ) JqC ) g q in variables - <> V > ( s ) = 0 d algebraic relationships some a l g e b r a , ¥q( ) = Q is the 8 ( c ) 1 . 1 7 and using w superposition ¥ o d (s) = in X (s) I (s) d 1.17 eqs. + G(s) d and 1 . 1 7 (a) (b), we get V (s) f 1 . 1 8 In these operational the equations, impedances transfer function functions can circuits in expressed, of be the between written f i g . 1 . 5 , in terms X in and, of the d ( s ) and synchronous the machine r o t o r and t h e terms as X^ ( s ) of the and G ( s ) stator. is These the parameters of the in [ 7 ] , they can be shown standard are test data, as indicated below: + s T,') ( 1 X.(s) + s T ") ( 1 = (1 d + s T do'> + S T = (1 + + s T , ,) ^ s T ') (1 s T d Q + L, L J (a) do") w (1 6(s) w (b) d Q ") r f - (s) X + s T ) 3 (1 = 19 (1 - + s T ) 9 qo v ' ° qo v (c) L OJ ( 1 + S T ' ) ( 1 + S T " ) q Q 1.19 But what directly, is as more shown assumption. in functions are circuit, and they in the not when field and 1.9 with (b) (s) I to the for and the = - (s_ ( so can operational far, for be 1.18 tied to they any any can stator major and equivalent include the X.(a) and voltage + r) I the of skin other effects and 1.18 , in terms (s_ + r) I dynamics voltage machine equations in the 1.9 (a) transforming combining we get of the the the these result following current in the V^(s): (a) - X (a) I Q X (a) rotor field the Hence, domain voltage all differential considered. 1.17 field and description frequency the summarize (a) (a) + X (a) d + s_G(s) OJ Q o = - measured inductances example, j ( s ) , I ( s ) OJ V (a) be without measured, and the the must expressions H of equations stator 3, necessarily are (c) domain, equations v 1.17 To c o m p l e t e frequency they winding. functions V^(s). these not considered Equations as Appendix Therefore, transfer effects important, Ij(s) V (s) , o + G(s) (a) N V (a) f (b) 1.20 These to find voltage the is equations machine generally must be terminal solved voltage determined by together with and c u r r e n t , the external the network but,as network, the it - FIGURE 1.5 20 - : E q u i v a l e n t C i r c u i t s i n the frequency domain s L kf kd(s) % ( 5 ) sit kq« s L ad sL aq "kd <> s r Fig. FIGURE 1.6 4-AXI8 1.5a : Steady Fig state phasor diagram with l - 5 kq^ ) b saturation s s - is better to reformulate them 21 - as: I (s) = Fl(s) V (s) +F2(s) V (s) +F3( ) V (s) (a) I (s) = F4(s) V (s) +F5(s) V (s) +F6(s) V (s) (b) d q d d q S q f f 1.21 where: Fl(s) ( (s) co ^ + r ) o = (jL_ X (s) + r ) (_s_X (s) + r ) + X (s) X (s) (a) X F2(s) (_s_ W X (s) + r ) d OJ o [ ( _ s (i) i 2 + 1 } X 4 o = (_§_ CO X,(s) + r ) a (s) ) + X (s) ( s ) + - § CO OJ X (s) d ,, 4 (_sX o + r 4 o " F3(s) (s) 9(_s_X = (s) - r ] G ( (b) s N ) o + r ) + X (s) X.(s) (c) 4 4 o -V > s F4(s) = (_s_ X (s)'+ d r ) (_s_X co co o X (s) d CO = (_s_ U X (s) + r o ) (jsX o u r F6(s) + r ) + X (s) X (s) d (d) 4 o " F 5 ( s ) (s) 4 v + (s) r ) + r ) + X (s) X (s) d o ( e ) G(s) = (_s_ X (s) d + r ) (_sX % (s) + r ) + X (s) % X (s) d (f) 1.22 These equations are the solution of the synchronous - machine in simplifying will be domain the frequency assumption. developed and to to In 22 - domain, the following transform include a new without these aspect any chapters,a equations not yet major procedure to the time mentioned: the saturation. Before we can f i n d important to transient started. chapter, presented. know a procedure the the solution initial Therefore to in the time conditions in evaluate the these next domain, from which section initial it of conditions is the this is - 1.5) Steady Most State point. operation, currents are In are voltages, positive, will be a the or rotating present an windings unbalanced and zero for the so that developed set of in the the to be used state could condition be a sequence operation, sequence will in which variables. evaluation they or balanced positive of these comply with afterwards. sequence, the machine currents and voltages same d i r e c t i o n voltages must transformation only Machine Sequence field Therefore, Synchronous condition negative, positive balanced the initial which Positive For q This - from a s t e a d y methods and a s s u m p t i o n s 1.5.1) as start s e c t i o n , f o r m u 1 ae components the in and present this E v a l u a t i o n of simulations operating 23 be eq. dc. 1.7 and currents This can be and in sees which speed the proved the as network creates its equivalent by using rotor. d and Park's in: cos(cot + 9) = /T Vrms cos(ojt + 9 - 120°) cos(ut + 9 + 120°) 1.23 thus y i e l d i ng: V, /3~ Vrms cos( Vrms cos( V = V = 0 - 6 - 90°) a (a) (b) (c) 1.24 - where i t constant can be values. relationship V + q these that equations, V, d the and V following q are phasor written: V„ e"- ( j 6 ) t j V, = /3 d J - o b s e r v e d From can be 24 where V Similarly for I = t V = I Vrras | e a 1 the * 6 c u r r e n t we have + j q ( j 1 I, = d J L t e~ 1.25 : ( j S ) where I As the equations = t d I and of the a =|Irmsle 1 ( j q-axis machine reduce - r I . - X I d q q V - r I = 1.26 } values V, = d q a 1 are dc, the to: (a) + X. I . + X . V / d d ad r r f q differential (b) f r 1.27 These given equations, before, conditions. additional saturation mutual so But, be if used is that flux taken curve is ir p , = md using for any given with to must into is to phasor the be be made. account straight given the evaluate saturation considerations saturation 2.1), can together line segment machine's taken In by into this initial account, dissertation, approximating segments or relationships (see saturation 2.4.1 zone, the fig. the by : ( X , . I , . + E . ) / O J v adi mdi oi o (a) - 25 - i p = ( X . I . + E . ) / w mq aqi mqi 01 (b) o 1.28 where X ^ X ,. adi and X . aqi corresponding xfj , md where and I mq the are the the i*"* s e g m e n t , to i ^ saturated Eoi 1 segment cuts values the is of the Y-axis, * b balanced are the steady m a&g n e t ioz i n g state, I ,. mdi I . mqi are currents» , given which, X , ad and value of and I in the by = I . + I, d f I , md (a) (b) q 1.29 For the unsaturated V = - r I V, = d - r I, q q case , e q u a t i o n s 1.27 can be w r i t t e n as + X, I . + X , ( I . + I , ) 1 d ad d f (a) v d X. I l q - a X I q (b) q 1.30 where I q as we OJ o i d e n t i f y X j ( I j +I ) as OJ , and a d d f' o md so we c a n r e - w r i t e e q u a t i o n s i> hence; mq r n X aq 1.30 as V = - r I V, = d - r I , d q + X, q l I . + OJ ijj d o m d X, I 1 q (a) OJ 4» o (b) mq 1.31 These the equations machine; and u s i n g must be v a l i d therefore, eq. 1.26, V, t = - replacing we can r I t regardless these of the saturation equations in eq. of 1.25 write: - X, L + E e 1 t p j 5 n (a) where Ep| = 1 OJo / i f md ; , +4>l mq 2 n I /3" (b) 1.32 - These formulae conditions, net flux which in the Once allow to evaluate the machine. machine this is is 26 - us, given , which is Therefore operating known, we the can terminal proportional the is machine's saturation perfectly write, for to the segment in determined. any segment in parti cular E V t " ~ r t " I j X qq tt I + ' - . + X . I. ad f 01 ^ (X, d + X ) q I. d yj - j E .] —22. e j 6 e l •3 1.33 where: X , = K. X , ad I ado X = K. X aq I aqo K.. Equation figure 1.6, = saturation 1.33 can be used from w h i c h i t E = f E 6' = p tan is cos(6") 2 -1 A X ( a 9 r X_ E p I, , 6" = sin" t I (X + t E . / (—°^ 1 draw possible - X to of V - d t segment the to X ) q phasor show I d i / diagram in that: (a) /T S i n < t > ) + V (b) coscp + V s i n cb £ sin 6 - 2 factor (c) / J ) (d) P2 6 = a + 4> = 0 - a 6' + 6" (e) (f) 1.34 From the variables in these equations, all the currents, - 1^, I and evaluated I j , as 27 well - as the v o l t a g e s , can be by: f x x - d - I = q v f v „ - v - = q (E f3 - f /Tl vT f r o (a a d sin(rx - 6 ) t I T x E .) / C O S ( 06- (b) 6 ) (c) (d) f /3" V sin( 9 -6 /3~ V cos( 6 - 6 ) (e) ) (f) 1.35 These e q u a t i o n s for the 1.5.2) positive Negative During currents The The of but it the of this direction in set sequence b leads a rotating opposite v v v the and v o l t a g e s phase b u c to the section machine w i t h currents of initial conditions Sequence negative will set sequence. and v o l t a g e s where the unbalanced o p e r a t i o n , purpose relate complete machine is to are or is show the sequence affected. equations negative to the a and p h a s e in the air positive c lags. gap that that sequence. c h a r a c t e r i z e d by a b a l a n c e d similar field negative machines network i n the phase the there set sequence, Therefore, moves in a rotor's. = /2~ Vrms c o s ( DJ t +6 ) (a) = f l Vrms c o s ( o ) t +6 +120°) (b) = f l Vrms c o s ( w t +6 -120°) (c) 1.36 - These negative sequence transformed to sinusoidal variables Therefore variables it dqo is as using with = Re with {v. e at 2 and the this c will nominal stage following W currents, transformation, twice the ( j - voltages Park's convenient phasors, V. 28 to when produce frequency. represent these definition: >} 1.37 where is for the and V , phasor using associated with v^(t).So, Park's transformation in eq. V (t) = /3~ Vrms c o s ( 2 u t + 6 V (t) - /3~ Vrms c o s ( 2 u t + 6 + 6) d q + 6 - we have, 1.36: 90°) (a) (b) 1.38 which where can be w r i t t e n using V„ d /TVrms e V vT Vrms e we have V d = eq. 1.37 J ( 6 + j ( 6 + as: 6 + 6 9 ° 0 (a) ) (b) } that: " J V q (> c 1.39 This phasor solution domain. V d = - of So, ( 2 formulation the differential making s = 2 OJ j j X (j d allows 2co) + r in ) I d q = - ( 2 j X (j q 2a)) + r ) I q to use equations eq. - 1.20, X (j q + 2 V us j + X (j q in we 2u) I the + G(j l the frequency have: q G(j 2 u ) 20)) directly V (j f 2u) (a) d 2o>) V ( j f 2o)) ( ) b 1.40 - where f 1.39, Q these and equations, Park's following terminal V - V (2 u> j ) = 0 . From the 29 the transformation, expressions voltage phasor for the relationships it is machine in eq. possible to find negative sequence and c u r r e n t : = f~T I r m s / 2 [ (A + (B d - B ) + (B d + B ) d - A ) cos(3wt sin(3wt q + 26 q + 2 6 +a ) - +a) (A + A ) d cos( w t q s i n( w t + a ) ] q +a) (a) where A d = Re{X (j 2u>)} + Im{X A q = Re{X (j 2a,)} + Im{X (j B d = 2 Re{X (j B q = 2 Re{X (j d q d 2u)J d 2o))} - 2u )J - q (J 2co)} - 2 Im{X (j 2a))} (b) - 2 Im{X (j 2u)J (c) d q R e { X ( J 2a))} (d) Re{X (j (e) q d 2u>)} 1.41 In which this is harmonic v = equation, generally and if we true, assume then that we can X (2oj) d neglect = X (2w), q the third write: >^ I r m s / 2 [ - ( A , + A ) c o s ( oo t + a ) Q (J + ( B + B ) sin(u) 3 d t + a )] 1 .42 which can be w r i t t e n V t = - using 1/2 [A d phasors + A q + j as (B : d + B q ) ] l t 1.43 Therefore in the negative sequence the machine can be - modelled as an impedance Z = - 2 1/2 [A 30 - given d + A q by: + j (B d + B )] q 1.44 which can be Z reduced = 2 to: r + [ X (2 d 0) j) + X ( 2 to j ) ] q / 2 1.45 If ^d( evaluated is modelling the Zero There values (s) are model thus produced consistent with the a and sequence 1.5.3) ) s d n X the machine q in the known, time then for Z can 2 the be negative information used for loop. Sequence is a direct from t h e v relationship symmetrical = ( v osc a between components + v, + v b c ) / the zero sequence transformation: 3 1.46 and the zero v sequence = ( v op values from P a r k ' s + v, + v b c a ) transformation: / /~3~ 1.47 Therefore, can be used V the differential directly o = - to r I o model - j X o equation the machine of in the "o" this winding sequence: I o n 1 .48 If assumed the measured equal to the value stator of L o leakage is not known, ' inductance. it can be - 31 - CHAPTER 2 BASIC THEORY OF THE NEW 2.1) Introduction The objective principles the of of the following assumptions be MODEL this more elaborate chapters. behind the chapter Also most in is to models this cover to be chapter, important the general described the auxiliary theory in and programs will given. 2.2) T r a n s f o r m a t i o n of the the with 1, frequency domain respect represent to the the the But, in equations in the transformed to the following were of the developed number o f order for time domain. without windings the frequency synchronous model or to be Laplace This will in any assumption to accurately needed behaviour machine of the "damper practical, domain be t h e these must topic be of the when the discussion. saturation machine can in saturation unique equations frequency-dependent windings". the . Time Domain In C h a p t e r When The F r e q u e n c y Domain E q u a t i o n s i n t o set be of is assumed not to curve, equations taken remain the in into in account, the machine the same is frequency or linear segment described domain by ( a eq. - 1.21 ). using These the inverse following Ai = d equations can 32 be Laplace - transformed q time domain yielding the equations: L'^FKs)} = the transformation, * Av (t) + L d {F2(s)} _ 1 + L Ai to L {FA(s)} * A .(t) a - 1 v * Av (t) q _ 1 {F3(s)} + L'^FSCs)} * A v + L _ 1 * 7av (t) f (a) * A (t) (b) (t) q (F (s)} 6 V f 2.1 where Ai a • Aj q• Av and q Av , d corresponding variables conditions, eq. of the two To a method used. of exponential S(t) k e" any stands the is to for be found * g(t) indicated one its the in employed based arbitrary Appendix p t to method can (see = variations respect and convolutions similar This convolution formula 1.15) with the of the i n i t i a l convolution variables. p e r f o r m the above, was (see are on function numerically equations by M a r t i the of the fact time in [5], that the g(t) using a + c g(t) with an recursive 1): = S(t) = b S(t - At) + d g(t - At) 2.2 where b,c and d a r e Therefore, approximated Fj(s) if constants. functions Fl(s) to F6(s) could be by: = K ° + n I i=l K. . j U (s + P i j = 1,2,3...6 ) 2.3 - whose inverse L _ 1 transform i s {Fj(s)} 33 - a sum of cr K 6 ° (t) + exponentials: Kije" I i=l ( p ij t ) 2.4 where use 6 (t) this is the impulse "recursive convolutions in eqs. 2.1 equations. values the (t = n At), n variables AV'n* assumed = C l A v and The currents among d ( t and the n> + C q n = C Av (t ) 4 d and previous l ( C n q + H (t ) A where to and H ^ ( t ) n W are to constants H^(t ) are n " b ij ( S ij ( t given the Ji^j = n n t " A " t ) A + A v f ( t the time values t these n> 2 n + H (t ) (a) + H (t ) (b) 3 n Av (t ) 6 f n H (t ) + 5 n history t into relate given the the 6 n by past n can evaluate equations any H (t ) + + C n we steps: " t ) Av (t ) 5 at with 3 then equations time + to these voltages 2 + C n method" resulting themselves, in delta, transform + H A-i (t ) Dirac's convolution algebraic of or ) + C b ij i ^ n> j t S A v terms: ij^„ " A t ) ( d ) k( n> t + d . .Av. ( t i j k n v - At) (e) 2.5 In the simplicity, r e m a i n i n g of whenever an this dissertation,and,for integration equation like the eq . sake of 2.5 is - written, "t ". the into must be technique. chosen q axis model, account. In it solved In is C o it is = M to do s o , t h e can highly formula C = -1 o represent therefore, necessary this o / V (r o ( o t discrete above take be done + o H ( t the and solution finish zero the sequence equation rule it [9] was produces equations an 2.5: ) + 2 L / o 1.9 any n u m e r i c a l trapezoidal much l i k e ) to using stable, variable the differential dissertation,the integration A the windings; and this - for given o r d e r to because implicit stand equations d and electrical (f) will The two n for "t" 34 ( At) a ) (b) where H o = C (r o o - 2 L / At) o i o (t -At) + C v (t o o -At) (c) 2.6 Equations with be the 2.5 network transformed transformation. given, to but use for this position of to using values the a must be to quantities phase Later, in chapter now i t is important r o t o r at developed swing it each is time equation programs, to the 8(t) and w ( t ) are every a,b,c that i.e. be swing predicted solved to this using in be order know the 8(t)= w t + (Eq. 1.11). equation In to Park's formulae w i l l necessary must step need using realize step, time equations 4,specific predictor-corrector approach. for at therefore,these transformation the solved equations; 6 + IT/2 , h e n c e , t h e In 2.6 is solved method, Dahl's the formula - [9], the electrical the predicted trapezoidal convergence, most Once and i ,, d as q , v., d and system mutual the flux rotor The i n v e r s e is in set of known, the circuits d and in transformations the the of If values using there recalculated, convergence are q then corrector. the v and recalculated electromechanical the in solved, the until - are electrical values, i evaluate currents rule the is variables the recent system 35 is the is no using the reached. equations it is q axis 1.17 is solved possible to r as following eqs. of well as the way: (c) and 1.18 gives: A^ (t) = A* (t) = ( L q d L - 1 {X (s)} * Ai (t) 7 q _ 1 q {X (s)} * A i d d ( ) OJ (a) q + L {G(s)} A - 1 t V f (t) ) / * (b) q 2.7 where again So, if rational can be " * " means L (s), difference convolution L (s) d functions, used, the then thus, the and G(s) are the same procedure transforming equations of much l i k e also equations equations 2.5 two variables. approximated outlined 2.7 into by before algebraic : A^ (t) = CH> A i ( t ) + H* (t) (a) AiJ» (t) = C^ + HiC (t) (b) q d q q q q Ai ( ) d t d 2.8 where history H^ and H il» ( t ) are the corresponding past terms. These using ( t ) i , , equations can i which H and v f be used are to evaluate known,and then ij> . d the a n d 4> q mutual - flux can be found 36 - by *md * d = ( t ' ) X (a) a (b) 2.9 To find equation can the 1.9 (c) be w r i t t e n field must current be solved. i ^, This the d i f f e r e n t i a l differential equation as: v (t) = r f i (t) f + f l d i (t) f + L f a d i d m d (t) 2.10 where all Here again, any the variables it numerical was solved recursive is are known possible technique.In using the convolution, except to this solve the field for i^(t) dissertation, Laplace this transformation producing the current. following using equation and the integration formulae. From e q . i (t) f 2.10: = md CO) ad + io -(r /l ) e f f t _ Lad md (t) 2.11 if we let S (t) f = C f (v (t) f + _r_^ L a d i m d (t)) + H (t) f 2.12 - be an a p p r o x i m a t i o n t o i (t) = f the S (t) + f 37 - convolution above, then tTo t 2.13 where: *md ( 1 .d< > t - ( t ) do E "o> 1 = ad L 2.14 It is i m p o r t a n t to was ignored that the stator is indicated this in effect parameters modified 2.3) the not [7] the and into for some cases, same in if, field a c c o r d i n g to the in the one Chapter account the that equation mutual that 1. instead winding, procedure flux links These 2.10 the that the using these links rotor, equations of fact can the as take actual parameters indicated in are A p p e n d i x 5. A p p r o x i m a t i o n by R a t i o n a l F u n c t i o n s In most in notice the previous important requirements operational functions rational G(s) section.it and F l ( s ) the a p p r o x i m a t i o n of the main had to 2.3.1) The In to the these introduced in the order to synchronous machine. Brief description of the method and are to the is that of is X (s ) , and be the that the a p p r o x i m a t e d by procedure Marti ' s adapt one presented presented, original of approximation method section, functions of obvious X^ ( s ) F6(s), this modifications modelling of impedances functions. be became this used as well method method for to as that the a p p r o x i m a t i o n method. used in this dissertation to - synthesize a rational behaviour (i.e., tolerance) method main the was is used here and a f t e r make Marti's the to values the actual error construction procedure be be then plot in the the his frequency same w i t h i n [ 5 ] , work modification original the poles and a This and method of which zeros their are the the are in functions, find be based new construction in the the estimate frequency, function and back Bode function. on matches minimized. goes that derived. tolerance a zeros a first moved is program the plot, associated desired to is roughly can approximated original in function and z e r o s the the the From t h i s and algorithm followed of rational between matches is major determine poles compared w i t h acceptable, closely to poles to Marti a given post-processing Bode the the function then a of response any be a p p r o x i m a t e d . Subsequently, that of "fits" fitting. approximating function the the method construction by respect and the which their without with pre-processing The is that developed differences before of so function 38 - In of This if it to plot the error is not the Bode which more figure the and so 2 . 1 , the Bode plot can observed. The Marti's results if phase", i.e. complex with should described function all plane, associated function the method its zeros so that a given also to in the phase magnitude. be as gives be a p p r o x i m a t e d are its above smooth left takes as hand the For a is good very good "minimum side of minimum fitting, possible. This the value the last FIGURE 2.1 : Method f o r Bode p l o t allocating the poles and z e r o s from F i r s t attempt to p o l e - z e r o a l l o c a t i o n a f t e r w h i c h the local error is evaluated and i f l a r g e r t h a n a g i v e n tolerance further subdivision continues. Second Error subdivision of Zone II - requirement fig. 2.2), they can the not and, therefore, be a p p r o x i m a t e d . Correction of the this section, evaluated presented. of before, possible. However, is the 2.4.1) it another inclusion of the was model to model indicated a explanation given, are Fl(s) far, segment in the of and this the the measured machine as saturation linear in of as are it was smooth as presented Effects possible therefore before effects. is the (see aspect correction it this that of F6(s) important so this necessary, to saturation developed F6 be A p p r o x i m a t e d corrections Saturation to be m a n i p u l a t e d characterize make Fl section, corrections to I n c o r p o r a t i o n of In several that these mentioned here the to next F u n c t i o n s to functions Some functions have In t h e discussed. - for they be In or fulfilled method w i l l 2.4) or is 40 was to neglected, associate saturation underlying section, curve. assumptions this this has aspect No been will be assumptions in addressed. In this connection a) dissertation with Only saturation the the are mutual following made: inductances L 3 saturate, b) ' i.e., the leakage s a t u r a t i on. The q u a d r a t u r e ^ inductance in the the case of salient path L aq are not does pole , ad and L a q affected not machines by saturate and it IFIDURE 70 401 40 351 30. 301 20 251 10 I -i—i 2.2-A1 : 41 - APPROXIMATION OF MODULE FUNCTIONS i 11 m i 1—i OF THE i 11 m i Fl 1—i TO F3 FOR i 11 m i ONTARIO 1—i H. t »inn DEN 1—till 601 50 *0 r Original 201 3 151 -10 I 101 -20 + 301 -30 + Oj. 01 -40 J - -5 + -50 + -10J -60 o'.ooi '""o'.oio 1 1 1 1 '""fe'.ioo 1 1 '""I'.ooo '""IID 1 1 FRO FI DURE 2 . 2 - A 2 tHZ) « APPROXIMATION OF F l TO F3 ANOLE OF THE FUNCTIONS -604- - 1 2 0 -180. -240 -300+ -360 FRO (HZ) FOR ONTARIO H . DEN in - 42 - FIGURE 2.2-&1 i APPROXIMATION OF F4 TO F6 FOR ONTARIO H. DEN MODULE OF THE FUNCTIONS FRO (HZ) FIGURE 2.2-B2 : APPROXIMATION OF F4 TO F6 FOR ONTARIO H. DEN ANGLE OF THE FUNCTIONS FRO (HZ) - saturates in the 43 same - amount as L , ad in round rotor machi n e s . c) The saturation load. Therefore circuit values If these method, to the five of circuit segments as that the L aq and are n to to use estimate accepted, saturation slopes of of then, curve the L the g segments are q found in is corresponding L unsat = — — L .unsat ad sat independent permissible curve the values : is is of the the open saturated L , and L ad aq assumptions corresponding so it saturation open identified curve the proposed linearized values (see of fig. using L i n up , ad are 2.3). The assumption (b) L .sat ad T 2.15 Once the values saturation segment L be (s ) can functions to = l L .(s) - l d q where formula i=o of ad , the segment and L measured to as are a q known functions produce the for each L (s ) and d corresponding follows: (1/L a d . - l/L a d o + l/(L d o (s) - l ))" 1 a + (1/L a q . - l/L a q o + l/(L q o (s) - l ))~ l + corresponds was definition a i , L corrected each L .(s) of derived to from the the unsaturated circuits L , ( s ) and L ( s ) . d q t h e r e i s no e x a c t in (a) a (b) a machine. figure 1.5 2.16 This and the knew the v For G(s), formula unless one FIGURE 2 . 3 : L i n e a r i z a t i o n of the open-circuit saturation curve - equivalent an circuit approximate Observe eq. 1.19 and structure modelled three the all, windings general, the the in possible the maintain are modelled it structure n (i can be the are one, inferred same not two or that in holds: + s T ) 0 3 = Xd .n" i =l 1 G(s) by = ( 1 + s T 1 X ad ^ _ r . , i =l (i (a) doi> = n : (see q they So way X (s) damper w i n d i n g s i =i d and d ) derive following X (s) [7] to the parallel. X (s) in for and is where when following it formula [4] cases or but equations references for at parameters, correction that 45 - + s T. .) doi (b) f 2.17 where T .j, d T ^ . = Short c i r c u i t time constant T . . = Open c i r c u i t doi time constants d ; v This work o b s e r v a t i o n was also noted by I.M. Canay in a recent [10]. In equations constants, saturation. available must This in 2.17,aside be by t h e i r can the from L ,, ad' definition, be v e r i f i e d literature the open the most by a n a l y z i n g the for circuit time r these affected by expressions constants [3,7]. - In Appendix parameters If of the accepted, two assertion different n it \ d i L , (s) ( s • and (1 d ° 1 the stated above are T VL L, m n L, ( 1 + S T - -i— -ail d j a (a) ) (1 J + s T k ) d L , . = E ( s ) L j . L , 0 doi 'j n JT i=l L , . d d using .) I o ) L . . (s) L, verified assumptions + s T. and O was machines. i * i =l G(S) - have: t L this observation we T 6, 46 n d JT o G(S). r, (1 + s T, i=l d .) o ^ (b) f 1 thus G(s). * G(s) L , .(s ) L, -aJ o L do ( s ) i£ L dj L , . _ad,L L ( c ) ado 2.18 where the subscript associated with This saturation equation affected by unsaturated values Once segment be the L .(s) open transfer From these c o r r e s p o n d i n g model below. used ( G(S) that the function is segment " j " . to Q estimate ( s a t u r a t i o n " j " in evaluated. the can L , . ( s ) , corresponding " j " implies G(s)j the ) value from of G(s) measured ). and G.(s) circuit functions functions, using the are known saturation Fl(s) it is fitting to for curve, F6(s) possible to procedure each their can be generate described - 2.4.2 ) A p p r o x i m a t i o n of In order described and have in Fl(s) to to F6(s) with the method post-processing functions of of achieved if functions segments in to note have the rest exploit time procedure. So corresponding = d X^(s) q same of 3 The general section 2.3.1), and q ,however, curves shape for the in Since curves order had of to (s v different this be the devoted the large to following also it amount the X,.(s) these was approximated, be figure to saturation behaviour reduce i, efficiently corresponding the smooth be a p p r o x i m a t e d be more to zone are was of fitting and X .(s) associated 2.4b). X . . ( s . ) + s) di mm' " X..(s ) / X , (s ) + 1) di max do max u x J (a) v 4 (s can can to (X . (s . ) / do min' u X^(s) approximating (see X .(a) 3 segment X (s) the saturation X,.(s) and q 2.4a). that q saturation pre-processing they (X = the and t h e r e f o r e v AX ( s ) in and that it, 0 ( s ) ,G(s) (see X (s) approximated AX.(s) X ^ ( s ) ,X functions. Marti ' s that instead to each technique functions: figure computing are is approximation the to by r a t i o n a l shows we convolution functions modification frequency (see observed the used the 2.4a directly.This curves 1, A p p r o x i m a t i o n of Figure decided implicit corresponding following 2.4.2.1) the approximated approximation - Curves Appendix to be use 47 (s X .(s . ) + s) ^ X .(s )/X (s ) + 1) qi max q o max' 0 . m 3 v ) / n m y 3 n (b) v 2.19 - 48 - = minimum v a l u e of s (ito) = maximum v a l u e of s in where: s s . min max This transformation approximated frequency program to range. in program, to are curve target the of is to the evaluated. tolerance, the poles and zeros are procedure, until convergence found to shifting special decrease be approximation very iterations cases and after the and until auxiliary zeros, 1 in is it to fitting for main fitting less than a or else the error following is This takes small 6 iterations, out zeros between it obtained. the corresponding and accepted, the found The e r r o r is be of poles usually program branches the curve, again, the 5 or If is shifted fast a new 2.4c). new curves extreme new figure data the and the data r the segment 2 (see then in poles towards corresponding curve of saturation segment at used input input all value is values displaced saturation the fact the the forces common This which approximation a in and the given the new same method one enough, was or two but in ceases to the error the original fitting approximated, X, . ( s ) d] to program. Once and AX, . ( s ) di X j(s) then corresponding 2.4.2.2) The goes of to are and AX . ( s ) qi found t solving r equations 2.19 for the functions. A p p r o x i m a t i o n of transfer G(s) function infinity,as behaviour by are can precludes be the G(s) goes observed use of to in the zero as the frequency figure 2.5a. This method outlined type above. - 49 - FIOURE 2 . 4 f l 1Q. 20. -)—i 4- 16. 2.. 1 2 . » 2 -10.. Segment 1 Segment 2 Segment 3 -4 -18- -8"- -22.- -12 -26- -16-" -20 3.001' 1 ""'o'.oio' ""o'-ioo 11 1 '""i.ooo 1 1 FRO FIDURE 2.4B 10 1 II 20. -1 / 1 1 ""I'D 1 1 (HZ) « ASSOCIATED FUNCTIONS TO XO(S) AND XOIS) FOR DIFFERENT SATURATION SEGMENTS l l l l llll 1 l l l l llll 1 l l l l llll 16- 12 1 1 I I I llll Segment 1 Segment 2 Segment 3 1 -2 m -10 4 0 Segment 1 Segment 2 Segment 3 -- -13 -22 I I III 0.. .14-• -30-- : XOtS) AND XO(S) FOR DIFFERENT SATURATION SEGMENTS -1 1 I I I l l l l i 11 m i -1—i i 11 m i 1—i i 11 m i -12 -25 -16 -30 •"o'.ooi 1 1 ""'b'.oio' 1 "'"b'.ioo 1 1 '""I'.DOO FRQ (HZ) 1 1 '""l'o 1 1 1 1 1 1 I I I III - FIGURE 2 . 4 - C 50 - : Method f o r e v a l u a t i n g an a p p r o x i m a t i o n the s a t u r a t i o n segment i from i - 1 Av«rog« valu« (<»b) of P, and Z| for - But, if we observe literature order of for the the G(s), 51 - different it formulae becomes denominator is evident greater available that than the in in the G(s), the numerator in one. Therefore if constant, then is the same The we m u l t i p l y G ( s ) the as o r d e r of that total of the is as the numerator = in the for new any function used in this follows: ( G „ ( s , „ ) / G , ( s , , ) + s) ' ° " (1 (s G . ( s )/Go(s ) + 1) a. max' max' G(s) k stands including saturation, m AG(s) (s+k),where denominator. correction, dissertation by m m l 1 m v i r i + T do» r s) v 2.20 The resulting Once AX^(s) for these and G(s) notice curves curves AX are (s) , found that in important, as can are the by be approximated 2.20, this term in in corresponding solving eq. ^ observed eq. the is only Fl(s) to It value used the same transfer 2.20. actual figure to is of aid 2.5 b. way as function important T, " is do the to not fitting procedure. 2.A.2.3) It A p p r o x i m a t i o n of was functions mentioned to however, this observed in approximation An be is approximated not figure of before the 2.2. these examination of case F6(s) that must with for be Fl(s) Therefore curves these is to as the a good smooth to functions shows as F6(s), first make them fitting the possible; as step can in be the smooth. that they have a -52 FIGURE 2.SA I : FUNCTION G(S) SEGMENTS | I I I Mil I I I I I llll FOR OIFFERENT SATURATION I I I I I llll FRO I (HZ) FIGURE 2.SB : ASSOCIATED FUNCTIONS TO G(S) FOR DIFFERENT SATURATION SEGMENTS FRO (HZ) I I I I llll I I I I I III - complex the pole at machine. denominator s This of is q can these v v X (s) q a be - rated angular verified can by . (1 . frequency noting be a r r a n g e d . , 2 roi s + co ° ° ; 1 X.(s) the functions X ^ s ) + X (s) d ' q • + 2 0 J which 53 r . that of the into: 2 + X,(s) d X (s) q 2.21 where are the very With close this applies values of X , ( s ) and X (s) a r e s l o w l y d e c r e a s i n g d q the s u b t r a n s i e n t v a l u e s i n the v i c i n i t y to approximation for a r o u n d to , o p r a c t i c a l cases: all r 1 >> the following and of co^. relationship 2 X " X " d q 2.22 Therefore, eq. 2 s " 2.21 can be approximated X," + X_" — r w „ „ o d q a + x ( l by: +2 o w M s x 2.23 which (s has two + a - j o j complex ), conjugated roots : ( s + a + j OJ ) and with: f _ _ u>« a u / (a) o 1 ( ( C ) 2 u) ( > b 0 X," + X " ) « - d 2 V q V < ) c 2.24 - To eliminate F6(s), these the effect functions 54 of - this complex pole in Fl(s) to a r e m u l t i p l i e d by : (s X X d = x.(a> ) * X " d o d ( ) q = X (a, ) * X " q o q <> /U) d Q + r) (s X / a) q + r) o + X X d (a) q where $ b c 2.25 As we a l r e a d y of Fl(s) to obtained, The have F6(s), since approximations additional only actual the the curve to saving in denominator be for the computer has approximated to be numerator time can be approximated. is given X by (see fig 2.6) : AAF(s) = (s (s X. i X (s) d /OJ 2 /OJ + r) + r) q (s (s X 3 / O J + r) + X, i /OJ + r) + X (s) X (s) - AF(s) 2 X (s) q n 9 d SCF(s) q SCF(s) 2.26 where the so SCF(s) same that described is the at the it is in factor extreme possible section of the to + AF (s ° = < S A V max> s 3 make ) / is given A F . (s n 1 / V max> A s m . 3 the range advantage SCF(s) . m to frequency take 2.4.2.1. (s SCF(s) introduced of functions of the interest method by: )) n + 2.27 Once the approximation of AAF(s) is found, the - FIGURE 2.6 55 - : ASSOCIATED FUNCTIONS TO F U S 1 TO F61S) FOR DIFFERENT SATURATION SEGMENTS FRO FIGURE 2.7-A I (HZ) STUOY OF REDUCED ORDER APPROXIMATIONS FUNCTION F U S ) FRQ (HZ) FIGURE 2 . 7 - B » 40 l l l l 56 - STUDY OF REDUCED ORDERftPPROXI MAT IONS FUNCTION F 2 ( S ) Mill l l l l lllll l l l l Mill l l l l Mill I I I I jilt 36-L 32 28 J . 24 20 164- 12 + 8+ o.oo7" " b'.oio """b'.ioo """I'.ooo " " " l b TT ,l l 1 1 FRO FIGURE 2 . 7 - C J r so - | 1 1 1 1 1 1 1 1 IHZ) STUDY OF REDUCED ORDER APPROXIMATIONS FUNCTION F 3 1 S ) | i 11 m i i i i i i IIII i i i 11 IIII I i i 11 IIII i i i 601. 40 201 NotCorrected Original -20 - 3 0 ' ,001 — 1 1 '""b'.oio 1 1 '""b'.ioo 1 1 ""T.ooo FRO (HZ1 1 1 '""I'D 1 ' " " " i-wt - corresponding as indicated approximations 57 of - Fl(s) to F6(s) are obtained below: Fl(s) = - F(s) F2(s) = F(s) F3(s) = - (s X (s) / OJ + ) r q (a) X (s) F(s) [ (b) ( si + 1 ) X (s) OJ + s_ r 4 o F4(s) F5(s) = = - F(s) F(s) F6(s) = F(s) ] G(s) (c) 0) o X (s) (s X ( s ) (d) (e) d /0J d Q + r) r G(s) (f) where: AAF(s) F(s) = K (s / SCF(s) ; A X . / a) d o + r) (s X / q A + r) OJ o A + X, d X (8> q 2.28 and X ( ),G(s),AFF(s), s and X (s) stand d for their rational approximations. 2.5) Run Time Reduced M o d e l s and C o m p e n s a t i o n of Numerical Errors A very described model to Marti for the In important is the [11], that problem it is numerical this it feature allows to be possible error section both of us the to solved, to adjust and, as compensate i n c u r r e d due aspects modelling will to be the method detail it was up to just of realized some discretization. discussed. the by extent - 2.5.1) R e d u c t i o n of A method reduce to the ignore the order the This of the above simulations. a stability model at of the one to EMTP s i m u l a t i o n s , example, models derivatives for valid it use a model method each full outlined of the equivalent, following the relevant when general 15 here = in is the in t h e to is use, for typical make 1 or thus in the and 2 Hz. derivatives a rational partial 1.9 stability approximated section, and frequency) up to preserves range in which rated only functions, of t h e using the producing for equivalent. fractions, This have the form: K Fi(s) valid in 2.9), e s p e c i a l l y potentially Hz is terms errors seem r i g h t can to machine (a) extent that (60 1.9 fig. cycle, previous expanded (see [12] transformer large a Hz frequency in or does n o t and the same t i m e the synchronous a lesser half to and flux lead to up flux the the literature to of proposed for the can to The method in two H e r t z simulations, model circuits though step the stator Nevertheless, time - employed approximation frequencies in o r d e r of normally equations (b). the 58 K* s + P + m + s + P„ C C K I i=l i ( s + P ) 3 2.29 where K might or £ function and might many c are not complex conjugated be present, by F^(s).. approximated Once t h e s e in K approximations simulations that are and depending obtained require they on can different some the be terms actual used degrees of - accuracy. Therefore, simulation, following the order F. (s) - the of model the equation K. I i=l 1 starting + p. ) + is I P a specific using the : K. —2- i-1 of reduced, 2.29 m - (s - before a p p r o x i m a t i o n to n 59 K — — P + i k — | P + c c 2.30 where "n" i s summation which the One seen an chosen are so greater model should problem with in figure error at additional 2.7, corrections in section. 2.5.2) next E v a l u a t i o n of Introduction Using evaluate the At. So, the the use of let in the of given F(s) Fmax. the Correcting committed be a t r a n s f e r I(s) also as up to can be introduces Therefore Another problem will be some is the covered F r e q u e n c y Domain and ref.[13]), the method function = it is, Fmax. n = second Poles. in numerical the frequency Both a s p e c t s in V(s) F(s) than needed. Error in above some c a s e s , Z - t r a n s f o r m (see error a of maximum outlined less are poles (Fmax). method frequencies the the be v a l i d that selection all than the appropriate the that it is frequency and possible range due integration given to to step by: K I - i=l s + p. 2.31 then the numerical associated function integration, can be F(z), implicit evaluated as whenever follows: there is - Using the we can w r i t e numerical 60 - convolution outlined in Appendix 1, : v (t) = i b v.(t i - At) + c i(t) i + d i(t i - At) (a) where m v(t) = , E i =l i(t) = L = L MVCs)} v.(t) _ 1 (b) 1 {I(s)} (c) 2.32 and b. t equation c .. and 2.7 to V..(z) d a r e . the = given in Z domain, we z" + c b i V..(z) 1 Appendix 1. Transforming have: I(z) i + d i z" I(z) 1 2.33 so m .£ V(z) F(z) = 1 = Vi(z) 1 m = I(z) I(z) I z c — i i=l + d — b. 1 z - 2.34 In figures marked is as made, derived 2.7 to Original, from and which a comparison F(z), the curve between marked as following F(s), Not curve Corrected, observations can be : i- The function above Fn Nyquist i i - There Therefore of the F(z) = 0.5 is / meaningless At, where Fn for is frequencies known as the frequency, are frequency order 2.8, which a good model some errors degrade choice would for be the before Nyquist model. Fmax i n this the the Nyquist reduction frequency. of the This FIGURE 2.8-fl : 40 61 - STUDY OF REDUCED ORDER WROX IMRT10NS FUNCTION F41S) -1—11)1 llll 1 — l l l l llll 1—I I I I llll 1—I I I I llll 1—I I I I III 36 32 28 .. 24 20 16 -- Not 12 Corrected 8 -- Corrected 4 -- 0 0.001 •' 1 1 1 ""b'.oio 1 1 1 ""b'.ioo 1 1 '""I'.OOO FRO IGURE 2.8-B 40 "I : 1 1 ""To 1 """ 1 <H2) STUDY OF REDUCED ORDER APPROXIMflTIONS FUNCTION F51S) 1 I I I llll 1 1 I I I llll 1 1 I I I llll 1 1 I I I llll 1 1 I I I III 30. 20 10 -10 - -20 -30-- -40" > -"'•ooi 1 ' ""'b'.oio' ' ""'b'.ioo' ' '""I'.ooo 1 FRO (HZ) ' '""l'o 1 ' m i " FIGURE 2 . 8 - C AOL III : 62 - STUDY OF REDUCED ORDER APPROXIMAT IONS FUNCTION F 6 ( S ) iiiin i i i linn i i i nun i i i MINI I I ^Frttrr 30l_ 20 10 -lorNot Corrected -20 -30f Original -40 -°'.ooi' ""b'.oio 5 1 1 1 1 ""'b'.ioo 1 1 ""I'.OOO 1 FRO 1 1 ' " " l b ' FIGURE 2 . 9 - A ; STUDY OF THE EFFECT OF THE TRANSFORMER FUNCTION F U S ) 40 -1—I I I I llll 1 — l l l l llll 1 1 " 1 — l l l l llll TERMS 1 — l l l l llll 1—I I I I HI 301 20l 10 0. -10+- -20' -30 Corrected No t r e n s f . -40 -50 .001 I I I II "b', 0 1 0 ' ' 1 " " b . -1— 1 1 0 0 I t — o 00" 1 1 11 1 1 1 1 FRO m (HZ) (HZ) 1 1 1 11 1 I M I III frequency was adopted - 63 as a - default value in this dissertation. In order reduction to in something the has increase The parameters the error in tried, ii- several and the follows i- the 1 poles the and what model by is in the both the discretization, proposed one, by by here is to introducing k' s + p p' this correcting neighbourhood methods p' for that first is the 2.35 term of are Fn i s selected so that minimized. In this p' were is as selection gave chosen guess at making F(z) u) max = 2 the best of K ' and results plus TT to the be e q u a l value correcting Fmax. of to K' term, This 2 nt Fmax. is is obtained equal to achieved by F(s) at exactly J if: o k' = ( F ( to max ) 7 Fo( z )) max" v Y p' Z - b) — ,* ,., . ( 1 - b ) ( 1 - z ) ( Z At max' 2.36 where: z max = e b = e-P' Fo(Zmax) i Wraax A a by : k' one introduced : The p o l e A of = of error traded: given Fc(s) respect, be order term, the number o f to the correcting reduce At t = Original function without compensation. - But i i i - as K' must absolute value equal to the The error decades g i ven In been the real or some value error is accepted above is taken with F(z) is evaluated and its the sign part, F(s) Fmax and up to decreased error - is found this in or value and successive from then steps until it becomes can be observed two K' until less is a than a tolerance. 2.7 method reduced either of below figures correction real, between increased minimum be 64 the to is 2.8, quite it satisfactory, significantly order of the without model or since increasing the overall that the this error has significantly computation time. In figures transformer into account be v a l i d . 2.9 terms for to 2.10, the are shown, as the frequency effects well range as of neglecting how t h e y where the can be model the taken should IFIOURE 65 - 2 . 9 — B : STUPY O F THE EFFECT OF T H E TRANSFORMER TERMS FUNCTION F 2 ( S ) 40 i I I I I IIII 1 — I I I I IIII 1 — I I I I IIII 1 — I I I I mi 1—i i 11 III 36 + 32 I 28 424 Original 20 J. 16 + 12 f 4+ oo.oorj r Ti l I I I Q ' 0 > 0 1 0 i "I'.ooo' - 1 0 0 FRO IF I C U R E 80 2 . 9 - C : 5 T W f O FT H E E F F E C T FUNCTION F 3 1 S ) - I — l l l l llll 1 — l l l l llll 1 '""lb 1 1 M (HZ) OF T H E TRANSFORMER 1 — l l l l llll TERMS 1 — l l l l llll 1—I I I I I 604- 40 20 No transf. Original -20 -3oL„.i 001 i i "" i, 0 0 1 0 ' ' '""b'.ioo 1 1 1 "" ""I'.ooo FRO ( H Z ) 1 1 '""lb 11 1 - 66 - FIGURE 2.10-fl •.5T0DY O F T U E E f f t C T OF THE TRANSFORMER TERMS FUNCTION F4IS) 40 -I—I I I Illll 1—I I I I l l l l 1 — l l l l llll 1 — l l l l llll 1—I I I I III 36 + 32 28 24 20 J . 16 + Original 12 + Not CorrectedNo t r a n s f . Corrected o.ooi 1 1 '""b'.oio' 1 '""b'.ioo ' '""I'.ooo ""To 1 1 1 1 1 ''"" FRO (HZ) |FIGURE 2.10-B:STUO* Of THE. EFFECT OF THE TRANSFORMER TERMS FUNCTION F51S1 *°| I I I I Illll I I I I Illll I I I I I llll I I I I I llll I I I l-rtttt 30+ 20f 10+ •10+ •20+ •30+ Corrected No t r a n s f . •401* •5 y.ooi 1 1 ""'b'.oio' 1 '""b'.ioo 1 ' ""'I'.ooo' ' ""To FRQ (HZ) 1 111 1 1 1 1 - 67 - - 68 - CHAPTER 3 INCLUSION OF NONLINEARITIES 3.1) I n t r o d u c t i on In in this the chapter,the previous saturation proposed studies it is for a long circuit for much f o r power very long might be very cycles (2 to constants from the occurrence do in not the system, systems,the 5 cycles), have time o r d e r of there variations is in to the significantly, values. The can Therefore second go where method sustained from a the was short saturated Saturation voltage state the this do not values. voltages event change the When the fault voltages but in afterward lasts fluxes vary in with is all are very example, throughout (they a transient, vary For generally d u r i n g which seconds). the of one. steady but majority vary example, studies,the low, the are not machine a fault, Two m e t h o d s the systems, C o n s i d e r a t i o n of of include power state for to presented of limited. the order for does steady from the system valid unsaturated system the machine analysis cases, the during a few the where a completely most is are special in procedure machine. voltage flux ) Method 1 f o r In revised one from the conditions, to in the the modelling synchronous that in developed is The f i r s t time, variations 3.2 the n o r m a l l y made assumed state chapters of here. overall the only the time removed practical limited. - For example, accepted. limited, - drop in the voltage of Therefore, if the changes in then limited. (a) a 69 the This and 1.9 changes can (b), be in the verified from w h i c h i t flux if we is in 0.8 the the is not voltage are machine consider possible pu. to are also equations write 1.9 that: V (t) « - OJ ^q (a) V (t) = (b) d OJ i>d 3.1 b e c a u s e OJ 4> >> "j^'and t h e Consequently, to remain close machine can segment in under was et ( in eq. machine was It will good 2.7 a large machine to this to F6(s) operating be proved for current, error. in to the for in error the the this be the then the [14], to the linear transient curve R . G . Harley remain that 2.1,and event assumed same formulae linear be conclusion. provided eq. low. saturation assumed the variables as can throughout Chapter which, This can very values, reference then in all In case, when state linearizing support correspond results magnetizing have of is a n d \p remain curve segment, Fl(s) to Chapter 2). the 2 apply functions very if saturation Chapter G(s) concept ty^ steady saturation in resistance fluxes their show e x p e r i m e n t a l Finally, same the considered the introduced al., to be the study if machine developed the L (s), segment in the in transfer L^Cs) in and which the started. 4 that with indicated can be this the in traced method exception figure 3.1, back t o an gives of the could error - FIGURE 3 . 1 : Linearization < of 70 the - saturation Linearized m o d e l s ^ X * / t * (t). d /' ! Error In 1 Im f curve - in the field compensate in for current, but it following From the the d-axis values; must in the hypothesis and the be it is possible to way: above, current the accurate. - fortunately outlined consequently, also 71 mutual we have i ^( t ) flux Therefore, if that have ^ ,(t) md we use the flux accurate (see this eq. 2.9) mutual flux t o e v a l u a t e t h e m a g n e t i z i n g c u r r e n t ( s e e e q . 2 . 1 4 ) , w i t h the v a l u e s o f X , . and E . c o r r e s p o n d i n g to t h e s e g m e n t i n adi oi ° which the example used the machine of figure together should 3.1), with appropriate then the value be operating this applied of the ( segment magnetizing voltage field current v^(t) current 1 in to (see the can be evaluate eqs. 2.11 to 2.13). This overall in method procedure, Chapter 3.3) method method voltage the evaluation gives very the should 1 regarding does not be used the apply, as In this circulation the machine the standard in problems short measurements current good i^(t), results, C o n s i d e r a t i o n of shortcircuit. avoid of as as well will as be the shown 4. Method 2 f o r This of for because situations. whenever variation it is of in the condition, is indicated circuit Saturation data the Hence,this instead model the machine case high; [16], former basic of generally rather in the it of is a terminal sustained the current consequently, is better frequency obtained was assumption to use response during developed to to be high able - to use only auxiliary this the q-axis flowing could when the air the will enable into another the In it, actual is leakage a when any but time model to it constant a maximum damper and r e p r e s e n t s are parameters of winding the deep defined. They the machine Canay's equivalent circuit to be of taken into account is to the rest of modified in such during all the currents machine are is the switch the this, the one goes the the the ( chapter The r e a s o n uniquely then this it time can defined. can be segment must in it the saturation restarted known, that in feature machine currents section This above model this presented a way f r o m one ). in simulation. flux which all the presented the in be method to to of the M e t h o d . similar achieve values the the model in of of segment the and employed flux whenever order one with rotor. variables the have The s e c o n d g-coi1 to idea the saturation state the to d-axis q-axis. corresponding chapters, at the complemented 5). develop restarted if gap general previous t=0 in when information G e n e r a l D e s c r i p t i o n of to time the either Appendi x The of in - which, in known as parameters 3.3.1) is winding 3.2, represent data, enough currents figure the (see is eddy In or damper damper w i n d i n g s in of tests,gives parameter two type 72 be or below was the operating. include as part machine at will called for be this proved is that the that the - FIGURE 3 . 2 Figure 73 - : E q u i v a l e n t c i r c u i t o f the s y n c h r o n o u s for method 2 f o r the consideration saturation 3.2-A : machine of d-axis 1 L> ad lj^ d = Stator leakage inductance. = d-axis magnetizing inductance. = Damper w i n d i n g l e a k a g e = Damper w i n d i n g resistance. 1^ = Field winding leakage r = Field winding f inductance. inductance. resistance. - Figure 3-2-B : 74 - q-axis 1 = Stator leakage inductance. = q-axis magnetizing 3 L aq 1 r l inductance, = G - c o i l o r d e e p e r damper w i n d i n g leakage inductance. = G-coil g k r, q resistance, = Damper w i n d i n g l e a k a g e = Damper w i n d i n g inductance, resistance. - 3.3.2 ) E q u a t i o n s o f t h e model In order first to develop consider equations follows the will be by a n a l o g y frequency of 75 - mathematical appropriate differential developed ). This or L a p l a c e the v a r i a b l e s the the equation [17] L{d_x(t)} dt equations, 1.9 d-axis as (c) the ( The q-axis can be t r a n s f o r m e d t o t h e keeping the as p a r t o f t h e m o d e l , theorem of equation for domain, set initial by u s i n g conditions the following : = s X(s) - x(0) 3.2 where: X ( s ) = L{ x ( t ) x(0) } = The v a l u e using V (s) = R f be e i t h e r or the time at this f x(t) could segment So, of into at time at the s t a r t when the there + s L f a d of is zero, which the s i m u l a t i o n , a switch from one other. theorem i n e q u a t i o n I (s) equal . d (s) d 1.9 + I (c), (s) k d + s l f + I (s) f we g e t : I (s)) f - ^ f o (a) where ^fo = (i k d (0) and f o r e q . 1.9 0 - kd R + i (0) d + i (0)) f L a d . + lf i (0) f (b) 3.3 ( d ) , we g e t : adi <V > + + S + S L S W * *kd kd 8 I + I f< ^> (s) 8 " *kdo ( a ) - 76 - where V°> V ° » adi + *kdo - + L + W ' <> 0 hd b 3.4 Equations I k d (s) the as 3.3 and 3 . 4 functions results for of c a n be u s e d I (s) and write I^(s) and V ^ ( s ) , a n d , b y d I^(s) to I^ (s) d in and replacing equation 3.5 for * (s); d * (s) = (I d k d (s) I (s) + d + i (s)) L f a d . + l I (s) a Q + E doi/ K S ) 3.5 it is the u o possible following ^ (s) d to obtain, equation = X (s) I (s) d after for a rather procedure, IJJ,(S): + G(s) [ V ( s ) + w d lengthy f H> o fQ + H(s) + E d o ^ k d Q ] ./s (a) 3.6 where X (s) (1 + T • s) S (1 + T ' s) do ' = a G(s) = (1 + T d o (1 + T " s ) S (1 + T , " s ) do ' (1 + T, , s ) ^ « s) (1 T + (1 T kd = d o » s) s) J + kd / T r d kd Q ' s) (1 T + T X . - S i r a ) -al d Q " s) f= V f r (b) f X i = ( a v (1 + T H(s) x r k () C d ( d ) 77 - The time constants in machine p a r a m e t e r s ( s e e Vj_ VJ 1 r A T do"i 1 ^ _I rf + Ladj 2 _ r to & In / 1 sake o f oo u» ( s ) o q r r 3.8 time * + kd r f /T r , ^ 2 (Lad — ) ' kd L d (a) + kd L . + 1. . ad kd _ c positive * L , , ad + r. , kd f the the d L 1, + L , f ad _ 2 sign r f c 2 ,, (b) r, , kd corresponds 3.8 the to T ' and T , ' and the negative d do ° s u b t r a n s i e n t time constants T , " and T , " . do d the q-axis can analogous be f o u n d , completeness = X (s) q I q way,the equations corresponding and t h e y are given below for the : (s) + J(s) + K ( s ) i>, + E ./s kqo qoi \\i 8° v where •go " k q *kqo " < V (l N constants a completely the + r equations transient sign k 1.kd. + Lad. ^ f v In + L r to L r f / to X relate ad V d ) , + L + L above f o l l o w i n g way : ad V d kd ad V d r Jdo ')I i n the f + L 2 T 3.2) L 2 1 fig equations ad V d , *kd + L h ( the - (0) + 0 ) + V°> ig + y ° » aqi l Y<>) aqi Hq L + g (0)+ L + (b) <> C (a) ' - 78 - and (1 + T ' 9 (1 + T ' X (s) q qo J(s) T (1 + T qo (1 + T qo ' = ' ' s) s) S + T v(1 + T kq = 1 , kq / r.kq q L _1 ( T "( 2 q 1 8 J 1 / / l + L g aq 1 a/ L q r8 , 1 / L 1 , ad a q^ r8 8 = kg 1. + +L kq (e) ± 1 (f) " s) qo ' T + L _aq_ qo ' v(1 T. T •") (d) qo + kq > = K(s) s) (1 + T " s) — 9 — s) (1 + T " s) " s) 1 Ix 8 kq (8) 8 ad. 1a / Lq r,kq + L ) + 1/L aq a rkq rg rkq (Laq -^) Lq 2 (h) T T 1° ') L qo > „ 1 1 _ ( _g 2 + L rg ai 1. + + L §J. ) ^3 + kq r 1 + L L _S SS. _ _Jjg +1, kq kg. + 4 L ' aq rf c kq r, (i) - Equations dynamics. stator To conditions d (a) = - and 3 . 9 complete equations transformed V (s) 3.6 to as part r I (s) (a) and of the model, s hj, (s) - E 1.9 all to (b) the rotor consider , which, and k e e p i n g the the when initial give: d q we have domain, V (s) = - r I ( s ) - s U ( s ) - E q summarize model, (a) Laplace - d the 1.9 the 79 - q d . / Q . q o s] - ^ d / s] - ^ - Q q u) ^ (s) (a) <V (s) (b) 0 o+ q d 3.10 By replacing and 3.10 V (s) d (b), eqs. 3.6 (a) v = -_s_ X ( s ) I ( s ) d = - s o (a) in equations 3.10 (a) we g e t : -_s_ V d V (s) q ' and 3 . 9 X (s) q I q X (s) q (s) v - I q / s g h (s) + ^ (s) - v / V . . (s) jk o X (s) I (s) + d d d Q - V ., ( s ) j k v r / + i> qo - V g h I (s) d r I (s) - E / qo q E s (a) v (s) d Q / s (b) where *do " ad <V°> L M * W " + 0 + V g h (s) = G(s) ( V ( s ) + * V j k (s) = J(s) f ^ , ( i (0) qo = Lad q i> g o 0 f Q + K(s) ^ + i g (0) ) + l a + H(s) ^ k q V ) 0 k d ( c ) (d) Q (0) (e) + i , (0)) kq + 1 a i q (0) (f) 3.11 Again,as equations was e x p l a i n e d 3.11 (a) and 3 . 1 1 i n C h a p t e r 2, (b) written it is better as currents to in have terms - of the voltages, so for 80 these - equations it can be shown that I (s) = Fl(s) d [ V ^ V + ] F2(s) + .[V + _s_ V F3(s) + [ V (a) * + f Q ^ + r I (s) = F4(s) q [ V ^ + V ] j k F5(s) + [V + k ] o w S \ ) l f (a) dQ kd q ; | j + + S 1 _s_ V kd J k ] o r + F6(s) [ V (s) + * f f + o + s f l f ; i^ r. + s 1, , kd kd k d 0 ] (b) 3.12 where Fl(s) to V F6(s) were defined i n Chapter ,(s) q\jr ' = V (s) q - E, . / doi s - if> V, ,(s) dijr ' = V (s) q - E . qoi s - i> . do / 2, and qo v y (a) (b) ' v 3.13 From equations difference can be Chapter equations obtained 2, 3.12 and obtain the difference of derivation the out, thus the the same Fl(s) functions then of and for using i.e., (a) to the overall F6(s) are numerical these (b), integration equations. p r o d u c i n g the 3.12 In set in corresponding the time domain procedure i n d i c a t e d in a p p r o x i m a t e d by rational convolution is used Appendix 4, the details are carried difference following the of equations equations : to - i (t) = d C v (t) 1 + C d 81 v (t) 2 - + C + HjCt) i q ( t ) °4 = V d ( t ) + C 5 v q ( t ) + C v (t) 3 f + H (t) 2 6 v + H (t) f ( t H (t) (a) + H (t) (a) 3 ) + H (t) A + 5 6 3.14 where C, A4.19 (b), of past to A4.20 (b)), (c) equations 1, with the to H^(t) do involved, functions, constants and and known A4.19 These different are values equations S. . ( t ) , C, to difference not only H (t) functions of time that depend on some as well as the machine (e) and analogous also on the windings Appendix to but depend in Hj (t ) A4.19 are defined in on past initial had a t (c) case, values of the time equal model terms H^(t) variables time. of These the currents to (e)). for the conditions that A4.20 2.5 of functions Appendix 4 to the (eqs. functions (see equations this known values A4.20 to are A 4 zero. terms in the - 82 - CHAPTER 4 IMPLEMENTATION OF THE MODEL IN AN ELECTROMAGNETIC TRANSIENTS A.1) Introduction In is this chapter implemented (EMTP). This the in an method discussed Chapter At in the different 4.2) end types General of of the phenomena The to in model are converting solved these program validation consideration data are of the the also of of the saturation, effects of as using analyzed. Electromagnetic transients the Transients useful a This the three linear the for network this model was and e n h a n c e d the EMTP, is evaluation and non and implicit differential in EMTP basis, linear by many nowadays of surge is and, performed in elements, it, as in it is well as cables. differential using the in systems. phase lines [15] of program, the tools power of program s e l e c t e d implementation by H . W . Dommel parameters general, dissertation transients the chapter, afterwards. detail for this (EMTP) in electric distributed models input simulation possible In this for most in 3. developed contributors used the electromagnetic originally great is Description dissertation developed electromagnetic for of Program Used The model program approximate one PROGRAM equations integration equations of most EMTP techniques, thus into equivalent - difference of the equations equations There the are EMTP: into the either the in the external in solved which together a new programming the a l g o r i t h m of [18]. - In with the rest network. ways by solution described which are of two 83 this network is new EMTP, later model or by u s i n g to be equations method, reduced can at an added directly a new each n-phase to method time step, Thevenin equi v a l e n t : l v v 2 = Z v ol v o2 + Th • • v • V on n 4.1 which is This it solved last reduces together method was considerably with the adopted the equations in this amount of of the new model. dissertation, since programming to be done . 4.3) Implementation of Method 1 f o r the C o n s i d e r a t i o n of Saturation In Chapter algebraic the dqo it equations reference themselves, the 2, previous was which frame and w i t h time the steps shown that relate at any values (see it the was voltages given time that these eqs. possible 2.5 and to derive and c u r r e n t s t = n At, variables 2.6). in among had Then in from - equations i t is i (t) 1 q ( t possible l C d = ) 84 c C 2 c 4 0 5 0 - t o wri t e : 0 v (t) 0 v (t) d v (t) f q v (t) c 0 o H (t) d H (t) (a) q H (t) c where H (t) d = H l ( s) + H (s) + H (s) 2 + i 3 " H (t) = H (s) A + H (s) 5 H (s) + 6 C + i - d s 2 s V qsi C r5 (t) - dss C - ( t ) (t) - V , C (t) v dss v l / C 3 fss V 4 v - Cr (t) d s g d g s 6 ( ) (b) (t) (c) t (t) v, fss 4.2 and dss q ss v, dss qss V^ In g g the 3 = Initial equations constants used correspond to segment in mention here the most the in that hence, the the model w h i c h the i n the q- a x i s i n the d-axis i n the q- a x i s constants they Hj(t) associated with the operating. conditions positive, are interfacing and of initial case, to e valuation machine i s the d- a x i s voltage above, the general components For field c i n the with state is shown negative, steady the It to H^(t) saturation important above have, and z e r o functions external the to in sequence of network, time. it was - indicated before Thevenin with equivalent, equation 4.2 is transformed to phase So, . to using EMTP which - will will reduce have dqo or abc. equation Park's transformation [ T ] _ 1 [c i, ] [ T ] [ v 5 abJ be the 4.2 The l a t t e r this to Therefore, either quantities • i W that 85 solved has to be a p p r o a c h was in equation 1 V to a together Thevenin m" tS.ei + network equivalent transformed chosen 4.2,we here. get: f [H + ] d q Q 4.3 which can be w r i t t e n - [ C E ] [ V as abJ tCE + 3 j 6 ] vf [H + a b c ] (a) where [CE] = [ T ] " [CE ] 3 f 6 - t abc] - H [C 1 [T]" [TI' 1 > 5 [C 1 [H 1 ] [T] (b) ] (O 3 f 6 d q o ] (d) [T] = P a r k ' s t r a n s f o r m a t i o n defined i n eq 1.7a (e) 4.4 In 4.4 order were solved [19]. to speed combined in the The constant which can in is given be s o l v e d order and by t h e aspect to know v^(t) by any use necessary to 6 + Therefore TT / 2 . one, voltage An i m p o r t a n t that into calculations, p r o g r a m by G a u s s field or up t h e in equation swing of position of the mechanical is either a equations, technique. these in and pivoting is differential solution or 4.4 integration 4.1 equation without transformation actual the in exciter the resulting elimination standard Park's the the equations equations eq. rotor 4.3, it 6 = OJ equation must is is t + be - 86 - solved . In this chosen, step, dissertation so, the before value the p r e d i c t o r - c o r r e c t o r a p p r o a c h was forming of $( t ) equation is 4.3 at predicted any using given time Dahl's [9] formula: *t B(t) = 2 g(t At) - 3 ( t - - 2At) 2 + [Pm(t J co(t - - oj(t) = 2 [g(t) - B(t - At)] / At - to(t - At) At) Pel(t - At)] (a) - At) (b) 4.5 where Pm(t - A t ) = M e c h a n i c a l step. Pel(t With solved power -At) the for value Vabc output corrected = Electrical step . of and $(t) using the power known, Iabc, Pel(t) power input output the and a new is found. trapezoidal 6(t) = - J + D At) oj(t) in previous in previous electrical value With of this time time equation the is electrical 3(t) value, is rule: C Pel(t) + a(t) (a) where 6 C = 2 (2 t 2 L a(t) = 3(t (b) t - At) + [ 2 J oj(t - A t ) + D At 2 J + D At At 2 [Pm(t) + Pm(t - ] u ° At) - Pel(t - At)] (c) 2 (2 J + D At) i u ( t - At) 4.6 - 87 - and a new value difference predicted is ones most predicted case, the less is but two or solution d,q,o are found according eqs. is using a 4.5b. tolerance, and is depends on typical. obtained, the voltages terms 2.5 and 4.1, the Hj(t) the to abc H (t) the and is this of the currents variables, and are evaluated indicated i n Appendix & procedure In severity are corresponding using iteration iterations is solution network. the the general, no the the and the In and in If performed 6(t). enough change eq variables iteration o)(t) from the history to of three the past a new iterations Once the a given accurate there number o f found corrected than values value is these otherwise, unless change, in is recent needed, B ( t) between accepted, the of 1. In figure represented For as the procedure a schematic evaluation given in 4.4) Implementation flow of Chapter 2 (eqs. of above is diagram. the 2.11 described to Method field current 2.14) can 2 for be the formulae used. the Consideration of the implementation of S a t u r a t i on Once method method 2 is very corresponding (b)) is method the 1 has easy to integrating same as also been carry out, equations before. apply programmed, Hence, here, since (eqs. eq. but the 3.14 4.2 with form (a) and the the and of the 3.14 subsequent f o l l o w i n g - FIGURE Flow d i a g r a m f o r A.1 1 1-n tke 88 the - implementation of method EMTP FIND THE I N I T I A L CONDITIONS OF THE NETWORK AND THE MACHINE FIND [Vabc ] , [ Z t h e v ] FROM THE CONDITIONS OF THE EXTERNAL NETWORK UPDATE THE PAST HISTORY VECTORS H . ( t ) OF THE MECHANICAL SYSTEM AND FIND NEW VALUES OF B & OJ USING 3 ( t ) FIND THE EQUATION IN ABC FOR THE MACHINE FROM DQO AND FIND [Vabc] & [Iabc] SOLVING SIMULTANEOUSLY WITH THE THEVENIN FIND THE E L E C T R I C A L TORQUE AND TRAPEZOIDAL RULE FIND B ( t ) AND e pre pre t= t + THE = 8 corr ~no = to ^ 3 pre - 3 c o r r < TOLERANCE corr yes UPDATE THE PAST HISTORY -x USING w(t) At VECTORS FOR THE MACHINE FIND THE CURRENTS IN THE F I E L D AND DAMPER WINDDINGS AND WRITE THE RESULTS FOR THIS TIME STEP no < t > TEND es > 8 9 - modi f i c a t i ons a) : Before is the integration switch from one model must be the S,.(t) to procedure saturation V . . (t ) i n i t i a l i n Appendix The H^(t) and H ^ ( t ) or when segment and into by setting V .,(t) values, equations terms starts, reinitialized S , . ( t ) , corresponding b) - there another, all to according the their to the 4. are given by: H (t) = Hj(t) + H (t) + H (t) (a) H (t) = H (t) + H (t) + H (t) (b) d q 2 A 3 5 6 4.7 where c) H^(t) Once I q the are to resulting known, evaluated. evaluated for the kd ( t ) = the The in the damper evaluated, 1 H^(t) 1 md as ( t ) also given equations currents in same way as winding in mutual 0 " are in current the " V are and windings field 1^ must 5 before, the Appendix solved all the in winding 4. and be is and the current d-axis is easily current is known,from: V ^ 1 4.8 For the the currents damper Figure just q-axis, similar in the expressions g-coil (see flow diagram can be section found for 3.3) and winding. 4.2 presented. is a schematic of the procedure - FIGURE A . 2 : Flow d i a g r a m f o r 2 i n the EMTP 90 the - implementation of method FIND THE I N I T I A L CONDITIONS OF THE NETWORK THE MACHINE FIND [Vabc ] , [ Z t h e v ] THE EXTERNAL NETWORK i FROM THE CONDITIONS AND OF : t I F t=0 OR THERE IS A SWITCH TO ANOTHER SATURATION SEGMENT, R E I N I T I A L I Z E ALL S i j ( t ) , v (t) and v .. ( t ) ACCORDING TO APPENDIX A J UPDATE THE PAST HISTORY VECTORS H . ( t ) OF THE MECHANICAL SYSTEM AND FIND NEW VALUES OF 3 & O J >\ - USING 6 ( t ) FIND THE EQUATION IN ABC FOR THE MACHINE FROM DQO AND FIND [Vabc] & [Iabc] SOLVING SIMULTANEOUSLY WITH THE THEVENIN : \ FIND THE E L E C T R I C A L TORQUE AND USING TRAPEZOIDAL RULE FIND 3(t) AND u)(t) 8pre = Scorr •n OJ pre = 10 corr -< Bpre -3corr UPDATE THE PAST t= T t + At < TOLERANCE yes HISTOIU THE > VECTORS FOR THE MACHINE FIND THE CURRENTS IN THE FIELD AND DAMPER WINDDINGS AND WRITE THE RESULTS FOR THIS TIME STEP •ncr / ^ -4— t > TEND ^Jkjes (END) s )• - 4.5 ) 91 - Results 4.5.1) Validation The Method 1 assumption linearized of accuracy most relevant assumption was (nonlinear). 4.3, the (see using and 1 are in 3.2), made by both system the point, section verified The saturation operating statements shortcircuit , segment that a r o u n d one loss figure of this methods and frequency without one (linear machine data curves with the be significant of transient one along can dissertation. the response 2.2, a constitutes solving data figure in curve for This after ) is the and a two given in saturation corresponding approximations. The shortcircuit cleared at associated to keep 167 line. This is applied ms. ( line 10 is at cycles reclosed a ) time of 10 ms. and later by opening the 500 ms. later in order stability. This test is linearization system considered procedure, for a very as to the long be very demanding shortcircuit period (twice remains the on the on the average c l e a r i ng t i me) . The r e s u l t s where it can good both differences but be in of is simulation observed that magnitude occur nevertheless, frequency this in the it low enough in and in any figures the 4.4 to 4.9, matching is very frequency. as considered for in general frequency is are time that practical The approaches this error purpose. biggest 5 in sec., the = 92 FIGURE 4 . 3 : Circuit model. GENERATOR DATA PARAM. HYDRO T d , T ' do„ do c d T x 0. MANUFAC. 2.0130 0.2866 0.2837 0.7674 0.0049 5.3903 0.0049 0.2789 SYSTEM and machine - data used PARAM. 0. 1.9700 0. 2700 0.2150 0.5838 0.0249 4.3000 0.0310 rq tl ^qn ^q iqo,. 1. 0. 0. 0. 0. 0. 0. 0. for testing HYDRO MANUFAC 9170 5734 2777 1286 0043 4408 0086 15503 1.867 0.473 0.213 0.997 0.039 0.560 0.061 0.160 DATA a) Transformer b) Thevenin c) Transmission X rji — equivalent line X 0.2222 = 0.2970 th « X + = 0.5840 r th r + = 0.0416 ' = 0.1002 0.0348 X 0 B 0 All reactance the and r e s i s t a n c e = 1.9241 r o = 0.4626 = 0.0243 a r e i n 5 5 5 . 5 MVA b a s e . [f J CURE 4.4 93 i C O M P A R I S O N BETWEEN METHODS 1 AND 2 FOR F I E L D CURRENT I =F 2.7- SATURATION I 2.5' 2.3 + 2.1 1.9 + Method 1 1.7 1.3 Method 2 1.1 0.9 0.7 o'.500 l'.OOO ltsOO -I — 2'.000 J 2'.S00 TIME IFIOURE 100 I J 3'.000 3'.500 . 1 . 4'.500 1 4'.000 5".000 (SEC.) 4.5 : C O M P A R I S O N BETWEEN METHODS 1 AND 2 FOR S A T U R A T I O N , POWER A N C L E I I 90 J - Hethod 2 20 + 10 + 0*. 500 l'.OOO 1*7500 J — 2'.000 J 2".500 TIME J — S'.OOO (SEC.) J — 3'.500 J — 4'.000 J 4'.500 5'.000 IFIGURE 4.6 1.0 94 - « C O M P A R I S O N B E T W E E N M E T H O D S 1BND E L E C T R I C A L POWER O U T P U T I I 2 FOR S A T U R A T I O N I I I I 0.9 O.S 0.7 0.6 0.5 0.4 I 0.3 + 0.2 I 0.1 + 0.0 o'.soo l'.OOO l'.SOO 2 .000 ! 2'.S00 TIME [FIGURE 4.7 0.2 rfa'.ooo •+3'.S00 V.OOO 4*7soo 5^000 (SEC.) » COMPARISON BETWEEN METHODS 1 AND 2 V O L T A G E IN T H E 0 ANO 0 A X I S FOR S A T U R A T I O N 0.0 J_ -0.2 -0.4 I -0.6 -0.8 -1.0 -1.2 -1.4 J . Method 1 -1.6 + -1.8 o'.soo I'.OOO I'.SOO 2'.ooo 2'.5oo TIME a'.ooo (SEC) a'.soo 4'.000 4'.S00 5.000 -95 IFICURE 4.8 » COMPARISON CURRENT °- 1.0 0.5 0.7 0.2 0.4 -0.1 0.1 -0.4 J . BETWEEN METHODS 1 RND 2 THE D AND 0 I FOR SATURATION AXIS ) _L -0.2 -0.7 -0.5 -1.0 J . >- -0.6 - -1.3 + -1.1 -1.6 + -1.4 -1.9 + -1.7 -2.2 + -2.0 IN - -2.5 0.500 1.000 l.SOO +• 2.000 J — 3J. 0 0 0 2.500 TIME |FICURE 4.9 t COMPARISON ANGULAR 382 =F BETWEEN 3.500 4.000 4.500 5.000 (SEC.) METHODS 1 AND 2 FOR SATURATION SPEED =F =F =F =F= 381 380 379 J . 378 377 376 _L 375 + Method 1 374 J _ 373 + 272 o'.500 l'.OOO l'.500 2'.000 ^-SOO TIME s'.OOO 3 .500 4'.000 J (SEC.) 4'.500 5.000 - 4.5.2) Effects The field at case test was (no (given O n t a r i o Hydro figure 4.3), curves (given one to in F6(s) lack the the measured O.H. test with are a) Active power: In figure 4.10 We o b s e r v e using the frequency bigger in subsequent Ontario the in figure the the frequency as not the external are very that can the One to due to to derived the from here: the standard and O . H . are data curve Still curves system. similar be (see response expected, simulation The s i m u l a t i o n test approximations. were was estimated associated that swing from Stand the by measured first ones. with obtained manufacturer's Hydro same supported this than frequency obtained results parameters measured about figure manufacturer's response along in the frequency corresponding the system generator transient the observations also a the the 4.3), 2.2) on to and figure O.H. results the the used: curves and in were from corresponds (O.H.) data actual the line present) information Nevertheless, ones the and of in figure concordance the of (O.H.) and Data evaluation test,one fault Three types data this Input by O n t a r i o H y d r o In t h i s opened standard Fl(s) for performed recorded. - Using D i f f e r e n t selected Lambton [ 4 ] , 4.3 by of 96 and about using obtained has a its the same measured curve in estimated Response by different amplitude parameters Frequency reproduced. tests but a is the by has bigger ampli tude. In figure 4.11 the results obtained using the program - 97 - •150 TIME FIGURE 4.10-B (SEC.) E l e c t r i c a l power a f t e r o p e n i n g a l i n e d a t a e s t i m a t e d by 0. Hydro from SSFR 100 50 /V~ -50 \J U.H.- r / v • /\ / -100 -150 t TIME (SEC.) \* SSFR - FIGURE 4.11 98 - : COMPARISON BETWEEN DIFFERENT INPUT DflTfl: ACTIVE POWER 0.8 . 0.7 + 0.6 X ^ v //I SSFR - " " v ~ \ * _ O . H . - SSFR 3.000 3.500 0.5 0.4 0.3 + o'.50O l'.OOO l'.500 2.000 2.500 TIME (SEC.) 4.000 4.500 5'.000 - presented in observe this that dissertation the frequency response frequency as by that Therefore had a it been Field The one is variable the damps out were u s e d using amplitude using the to the the T h e r e we case directly that the reported When is A.13 than has and this where the parameter same estimated is about standard case measured correct the O . H . in indicate results the data. should have had the is the actual the results the summarized. results the manufacturer's using estimated from data used were the SSFR directly parameters surprising, pointed parameters general, are out are they in as section in this have to not had 2.2, case (see is to improve by rely somehow be m e a s u r e d . SSFR, The the and case 2.11 estimated to by An a d d i t i o n a l in In this of the parameters where results this on c i r c u i t the higher. behaviour data A.12) frequency from O . H . , but equations those its program d e v e l o p e d here. curve figure and estimated standard estimated we predicted The r e l a t i v e repeated did the data amplitude using results response parameters but their that standard the using by are as to smaller expect dissertation really is shown. properly, The r e s u l t s quicker frequency using are manufacturer's with manufacturer's higher. figure data concordance current. using 1) corresponding sensible modelled (method obtained its - current: other field is test obtained better system b) the curve O . H . ; however, same a s 99 SSFR obtained is not parameters, 2.1A. These O . H . , and, test was in run 100 FIGURE A . 1 2 - A - : F i e l d c u r r e n t a f t e r opening m a n u f a c t u r e r ' s data a line: 1 1 O r i |\ \ n a l / V ( J k \ 1.0 0.0 3.0 i o TIME FIGURE A . 1 2 - B Standard (SEC.) : F i e l d c u r r e n t a f t e r opening a l i n e ; d a t a e s t i m a t e d by O n t a r i o Hydro from SSFR 7JO j- / O . H . - SSFR *v v — O r i gi n a l 2.0 TIME (SEC.) 3.0 «.0 - 101 - FIGURE 4.13 : COMPARISON BETWEEN DIFFERENT INPUT DflTfl: FIELD CURRENT 1.94- 1.2 -1.1 0.500 1 .000 1.500 2.000 2'.500 TIME 3*.000 (SEC.) 3'. 500 V.000 4'.500 5' - 102 FIGURE 4.14 2 . 0 - i COMPARISON BETWEEN DIFFERENT INPUT OATA FIELD CURRENT « MANUFACTURER'S PARAMETERS) 1.9. 1.8 1.7 1.6 => 1.5 1.4 I 1.3 + O.H.- SSFR 1.2 1.1 + 1.0 J „ „ .1 0-500 1.000 1.500 2.000 J 2.500 J 3.000 TIME FIGURE 4.15 I J 4.000 J 4.500 5.000 (SEC.) : COMPARISON BETWEEN DIFFERENT POWER ANGLE 90 I 3.500 INPUT DATAJ I I I 85 + 80 O.H.- 50 SSFR + 45 40 J0'.500 — .1 — l'.OOO .1 — l'.SOO J 2'.000 2'.500 TIME s'.OOO (SEC.) 3U0O 4'.000 4 .S00 ! s'.OOO - 103 - - using the manufacturer's observed that slightly, making the obtained In and the using angular behaviour A.5.3) speed that the damping observed previous noticeable types frequency responses, type results gives The it the of example method used differences small differences the frequency the for lower frequency event under which mostly In these the study. govern the small differences The differences in As the could are which very is the d-axis transient Method. there are using estimated, that this and last the sensitivity argued the the that The r e a s o n three is of the trouble of for where types of that the together in the important for the precisely the one be o b s e r v e d range because A.17-A.19, close data relative chosen figs. can angle was be using same obtained worth in power that responses. found A.1A). tests. this, figures.it can be the it response Proposed shown illustrated obtained the the field hardly be d-axis range, to was power. demonstrated frequency can responses shown. responses and i t are the manufacturer's, demonstrate observed the of the the simulations was although accurately are it for It increased figure present was are c l o s e r to modelling data the (see to electrical it data: and that used, i n the in input closer results They section, was w e l l - d o c u m e n t e d the curve the winding. oscillations Usefulness differences different field parameters shown. of the the resulting are was for of A . 1 5 and A . 1 6 , Evaluation In data estimated figures 10A - behaviour of the machine, explained. q-axis are somehow more noticeable, - especially for differences their affect effects not for always analyzed which neither the Ontario Hydro. between the have by order the but ground fault in it is is was figure 4.23 to evident oscillations the frequency the new severe applied be at 4.26 show the the These proposed in in was by the errors this of the poor are thesis. data d and q of axis. differences run of a quite line one cycles simulation the are the a single five made matching to end and is by responses and the is using (which these which cleared amplitude machine the the receiving results affected seen, this estimated data decided the The f a u l t both and was in Hydro, responses both response. modelled, that case, power, time effect it 4.3. domain. method the reactive comparison manufacturer's As can these data model) unacceptable are a by Hydro. as another the frequency response, that nor 4.20-4.22, illustrate specially the accurately data the of section, be but, by O n t a r i o actual quite to in proposed this not the Ontario the using In figs. both time common Figures In using in lines could exactly responses In noticeable case. data, interchange method machine's represented estimated the manufacturer's the obtained less the the - manufacturer's mostly are Unfortunately, is the 105 of to the later. where frequencies of approximation in avoided completely FIGURE 4-17 15 106 - : FREQUENCY RESPONSE FOR LRMBTON GENERATOR USING DIFFERENT INPUT ORTR i X01S) - 1 — l l l l llll 1 — l l l l llll 1 — l l l l llll 1 — l l l l llll 1—I I I I III 12-- 9.. O.H.- SSFR - 3 - - -6 • " -12 -isj 0.001 — l l l l Hfrl 0 1 Q l—l l I I iifci 1 0 Q l 1 I I I 11^1 FRQ FIGURE 4..18 15 1 l l l l 11^ 1 1 I I I III (HZ) : FREQUENCY RESPONSE FOR LRMBTON GENERRTOR USING DIFFERENT INPUT DRTR t XO(S) - 1 — l l l l llll 1—I I I I llll 1 — l l l l llll 1—("I I I llll 1—I I I II 12 6-i -3 •- -6 Standard -12 -15 G.OOI 1—i i, 11 ny 0 1 Q i—i i * * I 100 'Q| . FRQ —IIII m i — i — i (HZ) 111 m u — i — i 111 i n - 107 - FIGURE 4.. 19 : FREQUENCY RESPONSE FOR LAMBTON GENERATOR USING DIFFERENT INPUT DATA: GlS) 80 - i — I I I I mi 1—i i 1 1 I I I I 1 — I I I I IIII 1—i i 1 1 m i 1—i i i I I I I 70.. 60 50 ,. 40 30 20.. 10 Standard -10.--20 o'.ooi 1 1 "'"b'.oio 1 1 '""b'.ioo1' 1 1 "'"lb 1 '•""" FRO (HZ) FIGURE 4.20 : FREQUENCY RESPONSE FOR NANTICOKE GENERATOR USING DIFFERENT INPUT DATA : XO(S) 15 "I—'III llll 1 — l l l l llll 1 l l l l llll 1—I I II III 1 — l l l l llll 12-- 9.. 6- 3.. 0. O.H.- SSFR -6 SSFR -9" -12 Standard 13 oW ' '""b'.oio 1 ' ""'b'.ioo' ' 1 ' — FRQ (HZ) 1 11 '"U 1 11 FIGURE 4.21 15 108 - : FREOUENCY RESPONSE FOR NRNTICOKE GENERATOR USING DIFFERENT INPUT DflTfl i XOtS) - 1 — l l l l llll 1 — l l l l llll 1 — l l l l llll 1—I I I I llll 1—I I I I III 12 3 O.H.- -9 -- SSFR Standar -12 15 „„,l 0.001 l l l l MM 0 1 Q I l l l l llljl 1 0 Q I l l l l FRQ FIGURE 4^22 80 -1—I IIM I l l l l 11^ i FREOUENCY RESPONSE FOR NANTICOKE GENERATOR USING DIFFERENT INPUT DATA: CIS) I I I llll 1 — l l l l IIM 1 — l l l l llll 1 — l l l l llll 60 50 40 30 -- 10" 0. -10 -- - o.ooi' 2Q 1 1 I I I III (HZ) 70 20 1 ""'b'.oio 1 1 '""b'.ioo 1 1 '""I' FRQ (HZ) 1 1 ""'lb 1—I I I I III - 109 - [FIGURE 4.23 i COMPARISON BETWEEN DIFFERENT INPUT DATA FOR NANTICOKE UNIT « ELECTRICAL TORQUE 1.3 1.2 1.1 Standard 0.500 1.000 1.500 2.000 2.500 TIME 3.000 3.500 ( SEC.) 4.000 4.500 I &.000 |FIGUR£ 4*24 i COMPARISON BETWEEN DIFFERENT INPUT DATA FOR NANTICOKE UNIT : POWER ANGLE 100 I I 96 + Standard 92 -L 'A \ \ vV7 M / ^ ! V / r \\ SSFR V> SSFR O.H.- SSFR 64 60 0.500 + 1.000 1.500 2.000 2.500 TIME 3.000 3.500 ( SEC.) 4.000 4.500 5-000 FIGURE 4.25 2.2 - I -1.0 1.9 -1.3 + 1.6 -1.6 1.3 -1.9 1.0 -2.2 + .0.7 -2.5 + 0.4 -2.8 0.1 -3.lt -0.2 -3.4 -0.5 -3.7 -0.8 -4.0 + 110 - i COMPARISON BETWEEN D I F F E R E N T INPUT DATA FOR NANTICOKE UNIT : CURRENT 0 AND 0 AXIS - 4.5.4) General Observations Ill on - the Numerical Behaviour of the M e t h o d . In g e n e r a l , t h e model is using large to very method stable for integration appropriate steps. 60 that can and integration this was it errors, since be Hz. modelling the However, discretization step for numerically, integration correct employed of used the is is of verified not by necessary the maximum imposed network the and by not the by the possible to machine. 4.5.5) C o n c l u s i ons From derive a) the the The results following presented it is conclusions: linearization most above, practical method cases, is sufficiently including accurate clearing of for faulted lines. b) There using are different amplitude the differences and results reached in the data by by using the it be can variables used. general the results when of input well in developed type the the that as cases it model of of data, program stated as simulation data in and when be used in the general power, should is both developed the can obtained oscillations. O n t a r i o Hydro Therefore, this the frequency and v o l t a g e s , actual is the obtained dissertation, current types in From results in the are the this stator closer to frequency response inferred that give more directly. in accurate - For the model the developed data response the c a l c u l a t i o n estimated tests, i s The model of not by the the f i e l d improve the Ontario results estimating simulations Hydro are current from from using frequency undoubtedly parameters the good, this type and of obviated. is numerically limitation Finally, this but problem of data any did 112 - it can dissertation on the be size said giving the best available. user of that accurately machine, data stable the and the the does not integration model represents step. presented the flexibility impose in synchronous of using the - 113 - CHAPTER 5 IMPLEMENTATION IN A S T A B I L I T Y 5.1) I n t r o d u c t i on In previous synchronous was use of out simulations important where as the the 5.2 true For the program This the this type by implicit the The the with it, ideally of the in for stable,and, it is the as it possible error suited due for calculations a stability adopted of to to the stability is almost as the explicit integration. of method consideration behind the version this 1 of method are Program model, the of PSS/2 the Incorporated, integration developed This the Stability Technologies model program, simulations. EDELCA's Power for assumptions implementation program uses whereas is D e s c r i p t i o n of PSS/ED, developed be v e r y discretization speed was since ) General developed accuracy. 4.3) in was steps. implementation saturation, 2, the model the model proved to for this section always a new Chapter integration Therefore (see in and c o r r e c t large For chapters, machine which pointed evaluate of PROGRAM in this situation ( stability package was used. Runge-Kutta dissertation forced a detailed ), uses study program. basic a l g o r i t h m of PSS/ED is indicated in figure 5.1, - FIGURE 5.1 : Basic algorithm of 114 - PSS/ED CALCULATE THE INTERNAL VOLTAGES FOR THE GENERATORS, FIELD VOLTAGES AND MECHANICAL POWERS. CALCULATE THE CURRENT INJECTIONS VECTOR [I] USING THE LATEST VALUE OF THE VOLTAGE. [I] = f( V i-1 ) SOLVE THE NETWORK EQUATION: [Y] -no- V. 1 - [V.] V. , i-l = [I] < Toler. yes PERFORM THE NUMERICAL INTEGRATION OF THE MODELS (RUNGE-KUTTA) t •no- = t + t At > Tpause yes - where it can condition is operation be observed found at and back [Y] where [I] voltage has to is be procedure must 5.3 of iterations This fact using non-linear loads, voltage the account current which used in machine is assumed to be out be at by u s i n g downward equation: each node, matrix. This before loads the and in [V] is last the the current the matrix integration network, injections therefore the for the the the for [I] equation are above in in one the model the non-linear of Alsalient the part is of machine. equations stability used simulations, following variables are the some loads. model, by d e s c r i b e d by Dommel and S a t o a quasi-stationary and program PSS/ED, implementation dependent to the there current voltage used in whenever developing dqo before, fringing that, used, relating 1.26): a injections consider network Implementation a method s i m i l a r t o into injection admitance necessary c a n be in the triangu1arized pointed are [9], in step the iteratively. D e s c r i p t i o n of was the power the be s o l v e d As can are constant function integration program, start. there example, a is and this [I] [Y] can When [V] = - in substitution current formed that each the and, 115 with state. (see the A first to the are eq. to take equivalent step model network Therefore expressions abc used to the is phasors valid for 1.25 and /3 I ^3 V e f c _ j 6 e" t j 6 116 - = i (t) + j i (t) (a) = v (t) + j v (t) (b) q q d d 5.1 From the equations corresponding approximately i (t) + j to the model (method i (t) described 1), where Chapter saturation 4 is (eq. 4.2), considered we can w r i t e : = ( Cj v ( t ) d in d + C 2 v (t) + EDO ) C 4 v (t) + C + d 5 j v (t) + EQO (a) where EDO = C 3 v (t) + H (t) (b) EQO = C 6 v (t) + H (t) (c) f d f q 5.2 So, find the using 5.1 (a) following and 5 . 1 equations J Y t " m - Y " (b) in equation 5.2 (a), we can : m V ( C t m l <> l + " C a 4J > <> b where I Al m = - ( j 3 salient EDO + EQO + Al = (( C + C,) 2 k' 9 . . ^ ) e ' salient J j + C, 5 J 6 e l + AI .. ^ salient C,) v 1 q (c) (d) n 5.3 These circuit equations shown in can figure be 5.1a associated , where it with can the be equivalent seen that in - order to model diagonal the element machine's bus of machine's equations equations of since terra the therefore previous In value In The first the use swing The section once model as the modelling was a voltage, iterations at iteration, there there the Alsalient the in is the to the [I]. The with the needed and using the convergence. two algorithm iteration and Chapter to voltage, overall are the are of diagram of to vector simultaneously every to added function until diagram, the loops. second for is one solving to the 4. Implementation of the method already consideration of is be injection general estimated established this that must m solved In indicated the accuracy with then a flow I p r e d i c t o r - c o r r e c t o r approach of was term current corresponds a Results the flow equation 5.4) the are be 5.2, one the Alsalient of this of and network. must figure shown. for it t e r m Y must be a d d e d m m a t r i x [Y] t h a t c o r r e s p o n d s the term i n the - machine,the bar corresponding 117 method outlined by r u n n i n g t w i c e and in a second PSS/ED. of two the time This damper in the same using latter previous test a standard model windings case, and allowed a full saturation. Exciter and simulation using description of governor both these models models models). (see were included Appendix 7 for in the complete - FIGURE 5.1-A 118 : Equivalent c i r c u i t o f t h e machine - for the modelling REST OF THE NETWORK - FIGURE 5.2 Flow d i a g r a m f o r model i n P S S / E D 119 the - implementation of the CALCULATE THE F I E L D VOLTAGE AND MECHANICAL POWER. PREDICT THE VALUE OF THE LOAD ANGLE 6 ( t ) , UPDATE THE TERMS EDO AND EQO FOR ALL GENERATORS FIND A l s a l i e n t AND ALL GENERATORS. FIND Imac. FOR FIND THE VECTOR [ I ] = f ( j_i) V SOLVE THE NETWORK EQUATION [Y] • no- •no- < 6 i(t) [V.] V.-V. = [I] < TOLER.l " 1 r - 6 i-l(t) es < TOLER yes t = t. + > At - Tpause > - In order circuit to to (fig curves PSS/ED do input 5.3). It not of a three phase by opening the the whose case confirmed pole at the are be the transfer from standard that the also observed data), modelling of the the at to here 5 cycles From method domain of (obtained in again, saturation the results the given these 5.7. time new in The that are and, 5.4 used that cleared in functions for ignored. concluded was because figures behaviour model data IT f bus, in equivalent observe are remote shown can to OJ = 2 terms the PSS/ED response interesting at represents machine this fault it comparable, frequency a line, - standard transformer results actually the is have the results to the model these the corresponding generate model make 120 is it is accurate enough. It was method u n d e r general, those 5.5) trial did not already Usage of the for Model for in steps, in m i n i m i z e the 2.3 Chapter simulations good additional that convergency, iterations the and, other in than loads. Speeding advantages developed to a very nonlinear integration chosen the up the solution in a Program Considerable model have require needed Stability the did during 2). with In this can be derived dissertation corrections in discretization table from 5.1, the the by the using frequency error (see results use of large domain section of several -121 - FIGURE 5. 3- A : FUNCTIONS USED FOR VALIDATION OF THE STABIL1TT FRDORftM : D-AXIS 0 -i 80 f 26 f 70 18 f 60f 164- 50-1 14 + 40f 124- -.30+ ~ 84- lOf 64- 4 -10+ 24 -204- 04- 20 ± -30 + -40 + OA -10-f -20+ f 1—l l l I 111 i i i un \ (HZ) 1—i i i I I I I L 10 1—i i i i r11 S : FUNCTIONS USED FOR VALIDATION OF THE STH3ILITT PRODRRM ; C-AX15 1—i I I I n i l 1—I I i i IIII 1—I I I i n n 1—I i M i n + 16 + 14 + 12 + iio + -30 -30 + -40+ I lllin 3 -1—IIII un 20 18 -io+ -20+ 1—I AF (s) — i — i i i 111ii _ - i — i i i i IIII i—i 0. OCl 0. 010 0. 100 FRO 10 + 0+ 1—1 I I I 1111 -50 „ 30 + 20+ l I I llll I tFI&URE 5 . 3 404- 1—I 10+ 20 Of -10 1—| | | i nil 6 + -40 + 4 + -50 + 0 -504 -60+ 0. 001 1 1 1 ""b'.oio 1 1 1 11 "b'.ioo' 1 ""T FRO (HZ) ' ' ""l'o 1 ' ' """ - 1-22 IFJOURE 5.4 - i VALIDATION OF THE METHOD USED IN PSS/ED POWER ANGLE 20 . ° 16l 15 J . 12 + 11 10 0.500 1.000 +- 1.500 J ,., J 2.000 2.500 + 3.000 + 3.500 +- 4.000 4.500 5.000 4.500 5 000 TIME ISEC) FIGURE 5.5 I 0.30 i VALIDATION OF THE METHOD USED IN PSS/ED MECHANICAL AND ELECTRICAL POWER I I I I I I I 0.27 + 0.24 + 0.21 + 0.18 + 0.15 + PSS/ed P. »ech 0.12 V 0.09 0.06 T 0.03 0.00 J 0.500 .1 „ . 1.000 .1 1.500 J 2.000 2.500 TIME (SEC) 3-000 4- 3.500 4.000 - 123 - FIGURE 5.6 i VALIDATION OF THE METHOD USED IN PSS/ED FIELD VOLTAGE IEFD) AND CURRENT U F D I I 1.75 2.0 1.71 1.8 1.67 1.6-- 1.63 1.4 1 . 5 9 1 . 2 . . 1.55 1 . 0 - - 1.51 0 . 8 - " I Efd(t) 1 .47 New 0.6 1.43 0 . 4 " 1.39 0.2 1 . 3 5 0.0 0.50 New o'.500 1.10. l'.OOO l'.500 rr———r- 2'.000 2'.500 TIME ISEC1 3'.000 3'.500 4 .000 4'.500 5.000 V.DOO 4'.500 5.000 FIGURE 5.7 : VALIDATION OF THE METHOD USED IN PSS/ED TERMINAL CURRENT AND VOLTAGE I I 0.45 - 1.07. PSS/ed 0.40.. 1.04 *(t> t 0.35.. 1.01.. 0.30 0.98 0 . 2 5 - - - -New 0 . 9 5 . . , 0 . 2 0 - 0 . 1 5 - 0 0.10-- 0.05-" 0.00.- PSS/ed 0 . 9 2 . . 8 9 " 0 . 8 6 0 . 8 3 " 0 . 8 0 o'.500 l'.OOO l'.SOO 2'.000 ^.SOO s'.OOO TIME I SEC) 3'.500 TABLE 5 . 1 : R e s u l t s o b t a i n e d by t e s t i n g the model different i n t e g r a t i o n steps A t . A t ( sec ) C.P.U. ( sec ) TIME % 0,00833 257.64 100.0 INCREASE IN 5 THE A t 0.0415 56.86 22.06 INCREASE IN 10 THE A t 0.0833 28. 26 10.96 SAME CASE WITH CORREC. 0.0833 36.49 14.16 CASE BASE with OBSERVATIONS An i n t e g r a t i o n s t e p commonly used i n s t a b i l i t y was used T h e r e i s no d i f f e r ence w i t h the base case There are errors There are errors large small - experiments In this of the be slightly using table it computer These different can for of the of integration ( are kept that obtained, also the larger results only ). in these summarized. than the the one to obtained using be modeled. 5.8 of to the 5.11 a very observed c a s e where 80 % could usefulness Figure can up savings error. It the are reductions demonstrate 5 Cycles small steps but discretization show a c o m p a r i s o n step integration a system experiments - observed time were less correction errors be 125 large that corrections the were made. 5.6) E v a l u a t i o n of It was the mentioned derivatives neglected, before the that in the flux with and this can cause (see these derivatives figure 2.10 or simulation are s t a b i l i t y program, transformer The electromagnetic a reference stability respect large In this by indicated transients of in the fig effects on same case a for and simulation p r o g r a m EMTP a r e frequency have 2.10 also the normally the correction in a are the terms running simulations time section, with results to errors transformer both as T r a n s f o r m e r Terms that time in a these without using included the as only. figures shown, ). evaluated terms c o r r e c t i o n . are of of domain In Impact 5.12 and to 5.15, the results of this simulation following we can offer from them the without the transformer terms has observations: a) The case an initial -126 [FIGURE 5.8 - : TEST OF THE REDUCEO ORDER MODEL WITH AND WITHOUT CORRECTION ; POWER ANGLE 20 19 + 18 4- 17 + 16 J . 15 + U+, Hot C o r r e c ted 13 12 + 11 10 d'.SOOT.000 1.500 2.000 2-500 3.000 3.500 4.000 4.500 5.000 TIME I SEC) FIGURE 5.9 : TEST OF THE REDUCED ORDER MODEL WITH AND WITHOUT CORRECTION ; ELETRICAL POWER 0.30. I I 0.27 J . 0.24 0.21 J - 0.18 0.15 J . Mot Corrected 0.12-4- 0.09 + 0.06 + 0.O3 + 0.00 4- 0.500 1-000 1.500 2-000 2-500 TIME I SEC) 3.000 + • 3.500 4.000 4.500 5 000 -127 - FIGURE 5.10 i TEST OF THE REDUCED ORDER MODEL WITH AND WITHOUT CORRECTION ; FIELD CURRENT IIFD) 1.50. 1.48-. 1.46.. 1.44.. 1.42 = 1.40..\ 1.38-Corrected Original 1.36 1.34 - - 1.32 " " 1.30 O.SOO 4- 4- 4- 1.000 1.500 2.000 TIME FIGURE 5.11 0.50X 0.45 - 2.500 3.000 3.500 4.000 4- 4.500 5-000 ISEC) : TEST OF THE REDUCED ORDER MODEL WITH AND WITHOUT CORRECTION ; TERMINAL CURR. t VOLT. 1.10. 1.07-- 0.40 0.35-- l.oi 1.01 -- jr 0.30 - 0.98.. 0.25-" 0.95-- 0.20-" " 0.92 -- 0.15 0.89 0.10 0.86 0.05 0.83 0.00 0.80 Original -- O'.SOO .1 I'.OOO .1 J — J l'.SOO 2'.000 2'.500 s'.OOO TIME lSEC) s'.SOO 4'.000 4'.500 5.000 larger overshoot and case the about the with the cases equalize with unexpected, are b) The models. than the decreased dip in behaviour traced can represented From the very since results the more 5.7) In EMTP this This EMTP have long run, tend result there In t h e ( both terms end this to is is not lower both results i n i t i a l with other power angle the to figure go up ( s e e the slight and the analytically in 5.16). [20], two. and i t line that is concluded t r a n s f o r m e r terms i n the low they This be not simulations. can be their in can it are This difference other the fast the simulation, simulation studied some represented caused stability complicated, the on, have not angle little terms, and s i n c e goes transient. starting the r a n g e has in transformer time observations i n c l u s i o n of cases terms),which a L-C-R transient in two and f r e q u e n c y . are in be to other EMTP's. power back the that between the However, as from before the phases the in Also, in without because oscillations - transformer magnitude r e s u l t s 128 size. and content lower the same their harmonic than - above, impact in slightly the time better i n c l u s i o n does not s h o u l d be frequency domain s o l u t i o n . with the that But transformer make the model any retained. Conclusions this model chapter, can be we have discussed incorporated into a a way in stability which the new program. Its 129 FIGURE 5.13 - EVALUATION OF THE EFFECT OF TRANSFORMER ELECTRICAL POWER TIME ( sec ) TERMS: - 130 - FIGURE 5.14 EVALUATION OF THE EFFECT OF TRANSFORMER TERMS: VOLTAGE IN THE D AND Q AXIS l - 1 1- • 1 1 1 1 1 < 1 1 ( , 1 With transformer j^ f l / t f f W ni t h-o u t 1 FIGURE 5.15 1 1 1 1 1 l l TIME l terms l 1 1 1 _L. . ' _ 1 - 1 ( sec. ) EVALUATION OF THE EFFECT OF TRANSFORMER CURRENT IN THE D AND Q AXIS TIME . terms t r(atn> s f o r m e r v , ( sec. ) TERMS - 131 FIGURE 5.16 1 1 10.0 i 1 iO.O • }0.0 - EVALUATION OF THE EFFECT OF TRANSFORMER TERMS BACKSWING IN THE POWER ANGLE — i 1 —— ' 1 1 1 1— 11 1 f 1 — • — i — «0T5 ?0~0 soTo ToTo fcTo 99^0 100.0 110.0 TIME ( 1J0.0 1)0.0 mili-sec 140.0 ) J'K.0 IM.O 170.1 160.0 , — 110.0 — - accuracy has results which saturation able explored and stability of the against was advantages being data properly asserted obtained The of been to of the use the the fully can model discretization taken new into comparison a standard besides response the can easy be made of the model in account. in in obvious directly, in way handled when the data summarized and error the of model, be response) by output frequency they (frequency 132 - the order large speed up t h e were numerical which using one the to input minimize integration steps. These improvements were used a stability p r o g r a m . When t h e a reduction of concluded that the the 85.4 that % in the best data machine, but new the p r o g r a m was C.P.U. model available is to is can time. not be tested, it Therefore only used significantly solution more for the faster in produced it can be accurate in modelling of than standard models. Before mention large finishing that have these by and these Sato Transient for small control is to time models very some l o g i c when order integration appropriate can in these to use that makes changes occur worked on t h i s Stability take full steps, it the time conclusions, exciter constants. the with in small aspect is advantage is important of the important and other rule, variables time to use way to of develop controls A promising trapezoidal changes Program" it to which model complemented instantaneously constants. as part of the [9]. Some of their Dommel "Experimental results are - reproduced in figure the approximate the relevant model behaviour persist about program can always critical smaller the cases, time 5.17. In with of that step. used can this large the relevance be 133 figure, to we integration exciter. of be - the In any small n a r r o w the analysed that steps reproduce case, if time study more observe doubts constants, down to carefully the a few with a - FIGURE 5 . 1 7 : Figure 134 Effect of in specially a using r e p r o d u c e d from r e f . - a large integration designed [23], exciter step model. - 135 - CHAPTER 6 CONCLUSIONS In this machine for dissertation, modelling both well out as state-of-the-art revised s t a b i l i t y simulations. as was the and new models synchronous were electromagnetic The f u n d a m e n t a l their and in characteristics main a d v a n t a g e s proposed transients of these and l i m i t a t i o n s , models, are pointed below: Model 1 a) This model directly, of the and, b) any For the windings given number response frequency is consideration developed about one in term is except modelled of measurements dependent behaviour accurately without parameter windings constant some this model, closely taken proposed the coupled into by Canay it the for cases, could fact have that [3], an is if a the as method linearized 1 in a this correcting error practical large among t h e m s e l v e s using that such the new method current, all a curve (called field small account, in saturation demonstrated special where saturation, point was for of the becomes v e r y shortcircuit, more It included variables In which operating dissertation). is frequency parallel. was c) use t h e r e f o r e , the damper assuming in can in all situations, sustained errors. rotor than windings with equivalent the are stator circuit - d) A verification data was carried frequency response the model valid It also to the fully was data and implicit (for the into into in example development taken This using results It is was used simulated different from a found response input field that directly, model test if the is cases 1 concerning the error minimized. 1. In model by s w i t c h i n g whenever it but this pointed this out model sustained types of is high data is in not likely the numerical gap leakage. user who is data to this to current iron as C h a p t e r 2, best T h e main a d v a n t a g e s its seen is of the type needed the the saturation are conditions). evidence that led 2 saturation f r o m one is saturation test data limitation, of studies usually machine that for the because, in which implies makes a these choice. this stability Another familiar model. in which necessary. a real be in shortcircuit experimental model account another those sustained first of for model can only use standard s h o r t c i r c u i t input, are the Hydro. developed provided segment as Ontario of 2 This not - effect using measured assumptions c) out by Model b) the performed between a) of 136 with model and over the practical model the standard consideration advantage 1 can easily is of ones the that adapt a his - Model a) model stability are can As i n from to true use or the by integration numerical reduction of as develop advantage incurred This can be of of 80 % in out by by the in from for data use the the it order of of the large with time is of the derived allows the for the model. before large a without necessary exciter use as in together C.P.U. 5 the correcting response Chapter models derived the da.ta were model, the model calculations input and the in a v a i l a b l e . the manipulation, than appropriate data speed model deteriorating pointed of the the behind response manipulating stability more significantly But, frequency used Therefore, advantages of be assumptions type error steps. to Considerable order discretization 1 simulations. simulations,the model reduce the model the these best important. this high in the from since either stability very developed simulations, i n p u t , is was always model b) - 3 This 1 137 to full integration steps. Summarizing, the in synchronous this machine advantages over The modelling overall contexts the and relevance In dissertation traditional its connection presented methods, technique applications, of was a new was as way for which pointed investigated demonstrating modelling has out in without many above. different any doubt advantages. with future research, an important - improvement to would modification be its measurements the stand problem of be reducing studies. must the be [16] s t i l l the Another would the interesting large parts defined of in online incorporated in and, during its the research, of in functions in order to response complement overcome future the a research equivalents, stability a way of frequency for appropriate and e v a l u a t e d dissertation evaluation. dynamic network this hence, consideration evaluation this that measurements, characteristic machine. be - proposed so low c u r r e n t s the In can method 138 or frequency that is single for transient functions equivalent to synchronous - 139 - REFERENCES [I] O l i v e , D . W . , " M o d e l l i n g Synchronous S t u d i e s " , IEEE tutorial,1980. Machines for Digital [2] K u n d u r , P . , Dandeno, P . L . , " V a 1 i d a t i o n of T u r b o g e n e r a t o r S t a b i l i t y M o d e l s by C o m p a r i s o n w i t h Power System T e s t s " , IEEE T r a n s . , PAS-100, pp. 1637-1646, A p r i l , 1981. [3] Canay, I . M . , "Causes of D i s c r e p a n c i e s on C a l c u l a t i o n of R o t o r Q u a n t i t i e s and E x a c t E q u i v a l e n t D i a g r a m s o f the S y n c h r o n o u s M a c h i n e " , I E E E T r a n s . , P A S - 8 8 , p p . 1114-1120 , July, 1969. [4] Ontario Hydro, "Determination of S y n c h r o n o u s Machine Stability Study C o n s t a n t s " , Volume 2, E P R I , E L - 1 4 2 4 , 1980. [5] M a r t i , J . R . , " A c c u r a t e M o d e l l i n g of Frequency-Dependent T r a n s m i s s i o n L i n e s " , IEEE T r a n s . , PAS-101, pp. 147-155, January, 1982. [6] I E E E W o r k i n g G r o u p R e p o r t , "Recommended P h a s o r D i a g r a m For S y n c h r o n o u s M a c h i n e s " , IEEE T r a n s . , P A S - 8 8 , pp. 1 5 9 3 - 1 6 1 0 , November, 1969. [7] A d k i n s . B . , H a r l e y R . , "The G e n e r a l T h e o r y o f C u r r e n t M a c h i n e s " , Chapman and H a l l , L o n d o n , Alternating 1975. [8] Anderson, P . , Fouad, S t a b i l i t y " , Iowa S t a t e [9] Dommel, H . W . , Sato, N . , "Fast Transient Stability Solutions". IEEE T r a n s . , P A S - 9 1 , pp. 1643-1650, J u l y / August, 1972. [10] Canay, I.M., "Identification and Determination of Synchronous Machine P a r a m e t e r s " , Brown B o v e r i R e v i e w , Vol 71, J u n e / J u l y , 1984. [II] M a r t i , J . R.,"Work in progress a t the University of British C o l u m b i a under the O n t a r i o Hydro G r a n t " , Dep. of E l e c t r i c a l E n g i n e e r i n g , U . B . C . , 1985. [12] Krause, P . C . , N o z a r i , F . , Skvarenina , T. L . , O l i v e , D . W . , "The Theory of Neglecting Stator T r a n s i e n t s " , IEEE T r a n s . , P A S - 9 8 , p p . 1 4 1 - 1 4 8 , J a n u a r y / F e b r u a r y , 1979. [13] Kuo, B . C . , " New J e r s e y , Automatic 1975. A . , "Power System Control U n i v e r s i t y P r e s s , Iowa, 1977. Control Systems", and Prentice-Hall, - 140 - [14] Harley, R.G., Limebeer, D.J.M, Chirricuzzi, E. " C o m p a r a t i v e S t u d y of S a t u r a t i o n M e t h o d s i n S y n c h r o n o u s M a c h i n e M o d e l s " , I E E P r o c , V o l 127, J a n u a r y , 1980. [15] Dommel, H.W., "Digital Computer Solution of Electromagnetic T r a n s i e n t s i n S i n g l e and M u l t i - P h a s e Networks", IEEE T r a n s . , PAS-88, pp. 388-399, A p r i l , 1 9 6 9 . [16] D a n d e n o , P . L . , K u n d u r , P . , P o r a y , A . T . , Z e i m E l - D i n , H.M " A d a p t a t i o n and V a l i d a t i o n of T u r b o g e n e r a t o r Model Parameters through on L i n e F r e q u e n c y Response Measurements", IEEE T r a n s . , PAS-100, A p r i l , 1981. [17] K r e i d e r , K u l l e r , Ostberg." Ecuaciones Diferenciales Fondo E d u c a t i v o I n t e r a m e r i c a n o S. A . , 1975. [18] Dommmel, H . W . , Dommel, I . I . , " T r a n s i e n t s Program U s e r ' s Manual", Department of E l e c t r i c a l Engineering, U n i v e r s i t y of B r i t i s h C o l u m b i a , Vancouver B . C . , R e v i s i o n of F e b . 1982. [19] Stagg, El-Abiad, "Computer Analysis", McGraw-Hill, 1968. [20] B a c a l a o , N . J . . " S t u d y of T r a n s i e n t T o r q u e s i n S y n c h r o n o u s Machines Following Fault I n i t i a l i z a t i o n " , Dep. of E l e c t r i c a l E n g i n e e r i n g , U n i v e r s i t y of B r i t i s h C o l u m b i a , 1984. [21] S c h u l z . R . P . , J o n e s , W . D . , E w a r t , D . N . , " D y n a m i c M o d e l s of T u r b i n e G e n e r a t o r s D e r i v e d from S o l i d R o t o r E q u i v a l e n t C i r c u i t s " , I E E E T r a n s . , P A S - 9 2 , M a y / J u n e , 1973. [22] Dandeno, P . L . , P o r a y , A . T . , "Development of Detailed Turbogenerator E q u i v a l e n t C i r c u i t s from Standstill Frequency Response Measurements", IEEE T r a n s , PAS-100, April, 1981. [23] Dommel, H . W . , "WSCC C a s e s S o l v e d w i t h an E x p e r i m e n t a l Transient Stability Program", Bonneville Power Administration, 1972. [24] I . E . E . E . S t a n d a r d D i c t i o n a r y of E l e c t r i c a l and E l e c t r o n i c s T e r m s . Second E d i t i o n , p u b l i s h e d by I . E . E . E . Whiley-Inters c i e n c e , New Y o r k , 1979. Methods in Power ", System - 141 - APPENDIX 1 THE RECURSIVE CONVOLUTION TECHNIQUE Al.l) Convolution with Let function an E x p o n e n t i a l g(t) be a function -P f(t) be such complex these and two that U(t) f(t)= is functions S(t) is = g(t) U(t),where unit given * f(t) in t and let t k e the continuous step. Then k and the P might be convolution of by : CO = g(t-u) k e" P U(u) u du Al. 1 which can be w r i t t e n as S(t) = j g(t-u) •"0 : k e" P du u A1.2 It has is important meaning by a u n i t t note > 0, that and,for f(t) this in this reason it context was only multiplied step. From discrete for to equation variable S(t - A1.2, (t = At) = JO we n At) can write, by making time : g ( t - A t - u ) k e -P u du A l .3 which can r letting be r e a r r a n g e d p A t set - A t ) v = At + u into: - -P v g ( t - v ) k e At dv A l .4 a - Now i f we w r i t e equation At S(t) = 0 (Al.l) -P u g ( t - u ) k e 142 - as g(t-u) At du + k e P du u A l .5 we notice that the second term i n A 1 . 5 i s given by A 1 . 4 , S(t At) so At S(t) g(t = - u) K e~ P u du + e" P A t - 0 A1.6 and i f we assume assumed to 8(t vary - thatAt is s m a l l enough l i n e a r l y during u) * - A t > t it, *<0 so that g(t) can be then u g(t) + A1.7 which enables S(t)* us to b S(t solve - the integral At) + c g(t) + d g(t in A1.6 and obtain - At) A l .8 where b = e - P At h = (1 - b) P At c = k (1 - h) P d = - k (b - h) P which and in gives the past previous S(t) as a function history, time steps. i.e.,the of the values current that g(t) value of g(t) and S(t) had - A1.2) 143 C o n v o l u t i o n with a Impulse In the machine, development the following of K £ + £ + s P ) ( 1 + s P c Response the typical - model transfer K F(s) (1 for c + synchronous function n — ) the was found : k. z i=l ( 1 + s P . ) y v 1 ' A l .9 where P K c their are c This and c complex function, f(t) = K e c -P C P are complex numbers, ' v + K K c * and conjugates. when t r a n s f o r m e d t o t and * e c - t p C the time n + . £ i = K. e , domain, -P. gives: t 3 1 l A l .10 The convolution using eq. A1.8, of so f(t) with g(t) can = B S C ; l (t - At) + C Sc (t) = B* S C ; l (t - At) + C* g ( t ) Si(t) = B. 2 Si(t found term by term : SCjCt) c be - g(t) c At) + C . ( t ) + D c g(t + D*g(t + D.g(t - -At) - At) At) A l . 11 where the Sc^(t) two and complex conjugate themselves. Si in Al.10. e q u a t i on Adding these Sc2(t) is terms the are the poles, convolutions and,hence, convolution together, we of obtain g(t) of complex with the g(t) with conjugate i-th term - S(t) = f(t) * g(t) - 144 C g(t) - + H(t) A l .12 where C = 2 Re{ C ) + " I i-1 C "i 1 A l .13 and H ( t ) is H(t) function = ( of 2 Re{ the D past h i s t o r y : n } + Z D . ) g(t 1=1 At) + n 2 Re{ B c Scj(t - At) } + Z B i =1 j Si(t - At) A1.14 This with equation g(t) can be recursively. used to find the convolution of f(t) - 145 - APPENDIX 2 BLOCK DIAGRAMS OF EXCITERS AND GOVERNORS USED IN STABILITY A2.1) Exciter The the A2.1. SIMULATIONS block diagram of model i n s t a b i l i t y This FIGURE A 2 . 1 block : the exciter s i m u l a t i o n s can diagram Exciter block ref V t P T T : Output T exciter. (SCRX) voltage terminal from voltage power s y s t e m stabilizer : Lead time c o n s t a n t in filter a : Lag time constant : Exciter gain : Exciter time in filter e e E max E . min E an s t a t i c figure ss b K testing diagram Reference Machine for be o b s e r v e d i n c o r r e s p o n d s to STATIC EXCITER V model used fd constant : Maximum o u t p u t from exciter : Minimum o u t p u t from exciter : Excitation voltage referred to stator - A2.2) 146 - Governor Figure model used A 2 . 2 below in this for a hydraulic used i n the is the block dissertation. generator, diagram It of the c o r r e s p o n d s to since this type of governor a governor machine was tests. FIGURE A 2 . 2 : Governor block digram HYDRO TURBINE GOVERNOR (HYGOV) 1 \ r e f - ^1 W W 1 + T f s 3 A l t —K^)—*P mech D nl tur ~i W w : Machine R : Permanent d r o o p r . Temporary T T r f T T g w G max G . mi n speed : Governor : Filter tur droop time time constant constant : S e r v o time constant : Water constant time : Maximum g a t e limit : Minimum g a t e limit q nl W , ref P : T u r b i ne damping No l o a d flow : Reference , : Mechanical mech Aj_: Turbine Gain speed output - 147 - APPENDIX 3 DESCRIPTION OF THE S T A N D - S T I L L FREQUENCY RESPONSE METHODS FOR THE EVALUATION OF X D ( S ) . In can this Appendix be used synchronous transfer machine method following b) of them i s be found the and output For different be this test, coincides the input a to as the t e s t s which have the at stand-still, with its is evaluated input as a ratio (whose a between frequency frequency be more d e t a i l will be is response given for performed. rotor that is of is aligned phase applied so so that its a ( B = 0 ° in eq. that: 1 s the X^(s) the V (s) s V 3 well s of used. of with 2 V of calculation, sections, tests Measurement a inpedances that position. sinusoidal this following In and all to A3.1) axis in The f u n c t i o n the s a method characteristics: a given can *q( ) based on a s e r i e s in analyzer outline operational and locked varied). the the ^(s) is The machine briefly G(s). general rotor In evaluate function This a) to we w i l l XQ(S) AND G(S) V b " V c V 3 s magnetic 1.7a), - 148 - and I = a s - I I, = b 1 = c 1 / s 2 (b) A3.1 Using Park's above, transformation we get ( Eq 1.7a) in the equations : ./r V, = d V - Vs 2 V 9 = 0 V ° = 0 (a) and X..-/T I 1 = 0 1 = 0 (b) A3.2 In t h i s 0) , so test from when the the field equation machine is winding i s 1.20 at (a), it shortcircuited is a stand-still, X (s) = - H V (s) ( — I (s) possible X^(s) to is (V^ = prove given that by: w — s 0 + r) d A3.3 Therefore, since we A3.2, the A3.2) Measurement of For with equation know this respect and a p p l y the above X the voltage V^(s) for from finding equation X^(s). (s) we have magnetic V and can be used measurement, to V^(s) g in the to axis displace of phase same way as the rotor 90° a ( 8 = 90°) before. In this - V = q / 2 V 149 - V, = " s 0 V ° = 0 (a) and I = q 3 - / I I. s = d 0 1 ° 2 = 0 (b) A3.4 For the evaluation X (s), of we can prove that: V (s) = - X (S) q As we know V^(s) equation A3.3 A 3 . 5 can ) Measurement From the to prove of for -a + r) from q equation A3.4, the X (s). finding q G(s) equivalent that - a — ( and V ( s ) be used OJ circuit in figure 1.5a,it is possible : s G(s) = I (s) f I (s) d A3.6 Therefore, if evaluation of possible find to in the same X^Cs), I^(s) G(s) from the G(s) set-up is = also measured, following 1 s used for it the is relationship: I (s) — I.(s) f A3.7 a Note variables according that in a l l must to the formulae be p e r - u n i t i z e d the per u n i t given using system in the this Appendix appropriate chosen (see ref the factors, [8]). - 150 - APPENDIX 4 DEVELOPMENT OF THE INTEGRATION EQUATIONS FOR METHOD 2 FOR THE CONSIDERATION OF SATURATION ' In Chapter standard data retained as following and q i (t) it are part was proven used of and the equations can the model be + L _ 1 _ 1 { { Fl(s) F2(s) } * ( v } * ( v -1 + L { F3(s) q for initial the case when conditions in the Laplace derived for the d 4 (t) ) + ,(t) (t) + v + L -1 } * L { V (s) _ 1 = L _ 1 + L _ 1 { F4(s) } * ( v { F5(s) } * ( v d l ( > j k { are domain, current s Vjk(s) in the the d f (t) + v (t) + L j k _ 1 f ° (t) { L-'i F6(s) } * L - ^ V (s) f + * s_V j k o k } d o + (s) k d } ) + o ^ R. . + s 1. , kd kd R f , L ) + w + } ) + R + s 1. + — i f R. . + s 1. . kd kd + 4* f i (t) that axis: = L d 3, f + S 1 1 o ) A4.1 These at expressions any current given and allow time to during voltages in restart the all the integration simulation, the windings as in the A4.1, the procedure long as the machine are known. As is convolution evident method from equations outlined in Appendix numerical 1 cannot be used - here without performing So c o n s i d e r L" ! Fl(s) 1 in L _ 1 A 4 . 1 where > * V. (s) } - s d^ V of these equations term: } + L # d*< > V - some m a n i p u l a t i o n } * ( L~*{ V ( s ) equations < the 151 ( t { _ 1 V. (s) k } ) = i d l (t) : = ) V d< > f c + E qo " *do fii^) and L" ! 1 k v j k (t) - L'h J(s) ^ g Q + K(s) ^ k q } o A4.2 Using eq. 3.9e, we can w r i t e for the + s ( 1 V . . . (s) J k = J(s) l i|> 8 = (1 0 first + s T 11 qo ) term i n T. ) ^ (1 + s T Vjk(t): X 23- ') qo R ^ 8 g ° A4.3 which can be transformed v.,.(t) jkl = v K. 1 go r into: e~ ( t / T qo" ) + K ^ 2 go e ' ^ ^ q o ^ 9 where (1/T. v K l (1/T (1/T. K 2 (1/T kq qo kq qo ' " - 1/T qo ") 1/T ") qo 1/T ') qo 1/T qo X T. aq kq ; ') T T qo qo " T • R qo g X T, aq kq " T qo ' R g A4.4 - and for the V (s) = second 152 term: (1 J k K(s) * Z + s T ) = k k - ° q X \ 8 (1 + s T ") (1 •) + s T qo qo R, kq ' k q o AA.5 it c a n be w r i t t e n as v..,(t) jk2 v = ty. e" kqo K, 3 ( t / T qo ^ + K, ip. e'^^qo' A kqo M ) 5 M where (1/T K (1/T 3 » - - qo q° ") 1/T X T 3J3 S ") q° ') q° 1/T ') qo 1/T s (1/T A 1/T qo (1/T - = K - 8 = " - T " T R, qo qo kq X T ag s T " T ' R, qo qo kq 1 AA.6 So we c a n w r i t e for V . , (t) : jk ' v v.,(t) jk (K. ^ + K„ % ) 1 qo 3 kqo e" ( t / T do M ) + (K, ^ + K . ik ) 2 go A kqo e- ( t / t do' ) AA.7 So we have i d l (t) = for ijjCt) j K i =l e~ Pli ( H + C ) * ( v (t) d ' j k * * " "*do J + E .=V1 i i e _ ( P l 1 1 ° AA.8 where ^^.^(t) AA.7., expansion is and K^ a known and Pj^ f u n c t i o n of time correspond o f F j ( s ) ( s e e e q s . 2.2 to 2 . A ) . to given by e q u a t i o n the fractional - Now u s i n g can the implicit 153 - convolution technique in A 4 . 8 , we write: - j ^ 1 ^ l i S x ( t " ^do.^ ) l i K e " ( ? l ° i with S (t) = H c v (t) H d c + + d^ [ E u q [ v (t . o - d v. (t) + ] k At) + E q b + Sli(t u . + v. (t o At) At)J - k - A4.9 where in B^.. a n d Appendix 1. integration discrete D j.. This are the equation procedure, variable but,as (t = same can it be was n At) constants used directly necessary in order to to obtained " *do K in if we li e " eqs in the make t i m e perform n convolution described a the r A 4 . 8 , further simplification can be accept: ( ? l ° i " S ' l i ( t - ) h i S ' l i ^ " A t > where S' li . (0) = ' - v K, . , l i d o A4.10 which has equation S (t) H the same A 4 . 9 can = c of error be r e w r i t t e n v (t) n level d + c ( E H d li as equation A 4 . 9 . Then as: q . + v o ( v d ( j k (t) " t A ) t ) b + + E H qo S (t) + j k ^ " H + v A t ) A4.11 where at the S^(t), time of redefined such that reinitialization S. . (0) 11 = - K l i ^do of at time the equal model, is to zero given or by: ) - Finally, it is calculations, approximately interesting Vj^(t) with in the v j k 154 to note equation following (t) = - that A4.10 for can speeding be up evaluated expression: Sjkl(t) + Sjk2(t) where Sjkl(t) = bj Sjkl(t - At) Sjk2(t) = b Sjk2(t At) 2 - and b = x -( t/Tqo") ^ A e m e -(A / qo') t T Hqo Sjkl(O) = KI ^go + K3 Sjk2(0) = K2 ^go + K4 ^kqo A4.12 Using prove i d 2 the for (t) same the = p r o c e d u r e and a s s u m p t i o n s second given above, we can term i n e q u a t i o n A 4 . 1 , L- {F2(s)} * 1 L-V^s) - ^ _ / s _s_ + V j k (s)} A 4 . 13 that it can be t r a n s f o r m e d i n t o : i m = « i=l (t) Q z Z S2i(t) where S2i(t) = c 2 i v (t) + c . [ v + b . S2i(t - At) + d . q 2 2 g j k (t) 2 - Edo ] [ v (t Edo + v - A ) g j k t (t - A-t) ] -155 - and K S 2 (0) I = ( l + w Vjk > = - ( t K — i „ £_ K * g o _ + 3 ° K T ( + K i OJ ( - 2 l K + to o qo ' OJ o k q . o * q o ) K , o 3 _ ! i T ' qo + T' qo t - * T " 0) qo o *go h 4 x \ <* q o e-( ) t / T qo M ) e-^/Tqo') k q o ) OJ o A4 Finally i d 3 f o r the t h i r d (t) = L { term: F3(s) } * L ! - 1 { V . ( s ) + *fo f l d / T f f + s) kqo } 2 kd ( 1 / T kd + S > A4 we c a n w r i t e i =l A4 where S 3 i ( t ) = C 3i v f ^ ) + c 3i v f g l ( t ) + b 3 i 3i^ + d . [ v ( t - At) + v 3 ~ > s f At f g l (t + - At)] and S 3 I (0) = K . ( "'fo + - i l — ^kdo ) 3 2 v f g l (t) - (1/T f 1/T kd k d ) *kdo e- t / T k d " A4 - 156 - From the i (t) = i d calculations d l (t) + i d 2 above,we (t) + i c a n now w r i t e (t) d 3 - z s (t) for m + u z 3=1 i,(t) s 2 1 (t) 1=1 m * + S 3 i ° ( A4.18 3=1 which can be w r i t t e n i (t) = d C v (t) 1 as: + C d v (t) 2 + C q v (t) 3 + H (t) + H (t) f x 2 + H (t) 3 where: C. = 1 Ilj(t) = C y c.. . .. 1=1 [Eqo + v 1 C = 2 11 A 0 j k (t)] . L c . b . S j . ( t - A t ) + ( 1^ d j j ) x v (t - d 0 2i , 1 y c~. . ^ , 3 i i=l C = 3 0 1=1 + _Z [ m v A t ) + Eqo + v k (t - At) ] m H (t) = C 2 2 [v . (t) - k Edo)] + i i=1 J + ( H (t) = C, [v 3 [ This with the given by: V ° = c m E f g l d .) i [ v (t 2 (t)] v (t - f equation q + ^ At) + v V d ( t + ) f g can be used corresponding * b,. + S .(t 2 S .(t - Edo + v - At) 3 (t - At) 2 - At) - s j k (t - At) ] d .) + 3 At) ] t o model equation ° V ° ° 5 l b for A4.19 the machine, the q axis, V H ) 4 ( t ) 6 f ( t + + H (t) + H (t) 5 6 together which is - 157 - where o C, = 4 . z , c, . 4i i=l H (t) 4 = C [Eqo + v 4 p Z C, = j k 5 . , 1=1 (t)] + [ ! (t) 5 = C [v 5 s j k + ( H 6< f c ) = C 6 f v f g l S4i(0) = - K S .(0) '5i = K 5 5 i 4 i (t) - P ^ ( ^ t d ] 4 v (t - q b + . c, . . 6 1=1 b . i S .(t 5 5 At) - 6 i 6i<* ~ S [ v (t + - 2 f A t - - (t k - A t) ] At) Edo + v > 1 - A t ) + (_Z^ d^ . ) 4 - A t ) + Eqo + v d [ v (t 5 6 b . S .(t Z q Z C, » 5z Edo)] + _ Z i=l d .) ) cc. s j k + At) + v d f g (t - At) ] - At) 6i> l (t ] 0 ( —1 2 - V %o & *kqo - * q o ) w w o 1 S^(0) 6i = K 61 f t 4 ( *fo+ ^kdo) n 1 kd A4.20 - 158 - APPENDIX 5 CONSIDERATION OF UNEQUAL FLUX LINKAGES USING AN EQUIVALENT CIRCUIT In Chapter important in the to taken due figure is to into it was consider rotor stator, not some Canay in circuit traditional ( one in 1 = c enabling fact circuit of r is windings links This X it the can be shown in represents c this an network. I . M . this circuit into an with the form as the use same of the formulae case. the equivalent circuit relate to the by : 1 X ~ X - t ) the that gap. branch A2 cases links one solution figure this some equivalent transform thus for parameters original see one, developed to that iron series the in same the the the [3] the in with that flux exactly complication in equivalent The the leakage account proposed already mentioned that A l ; however, additional the 1, X 1 x 1 + • i and X ad - d x c K = X rc j ad A5. 1 In this circuit c i r c u i t , ( i ^ ones. ones the * but ( t ) , v^ i d the * (t ) be (t) d-axis obtained ^d m u s t the same variables and The r e l a t i o n s h i p can is if be i ^^ between we the in * observe in in the ( t ) . ) these same as both original rotor are values that the the c i r c u i t s a s s o c i a t e d and t h e flux original that circuits. links This is - 159 - FIGURE A. .1 : E q u i v a l e n t c i r c u i t f o r the d - a x i s t a k i n g i n t o account unequal f l u x l i n k a g e s FIGURE A .2 : Equivalent circuit without the series branch - sati sfied i f 160 - : . * X = kd . ad ^ dc * . 1 = kd ^ ^ad * X, dc v * K r = f dk K v f A5.2 Therefore, circuit before care in must order to w r i t i n g the Another circuit, parameters the in taken convert result matter is be of when back of the to concern, circuit the this equivalent original quantities simulation. when consideration the using of using this saturation, change with the equivalent because value of the X . 3 Q Therefore saturation once the segment decision into another for is changing made ( using from \b T - 1 S i j ( t ) ) , evaluated there the Q from must circuit. the be So if current current continuity we a r e 1 kdl in 3 new circuit in the old in the current switching = the from kd2 3 f l = one, . = \b , md d must be 1 considering segment in the 1 into one that original 2 , then: £2 1 so K l *kdl * = K 2 1 kd2 * K l 3 fl * = K 2 ^'fl * A5.3 which gives variables in us the the following equivalent relationship circuits that must between be the maintained - when 161 - switching. 1 i kd2 kdl * f 2 * i K f 2 2 A5.4 Finally, one flux that original consistent the is saturation mutual the it leakage interesting segment links with the observe another stator This basic inductances into the circuit. to was be was and t h e done assumption could that changing done rotor in that neglected. using according this the from way to saturation the to be of - 162 - APPENDIX 6 EFFECT OF SATURATION ON THE MACHINE TIME CONSTANTS In this constants are appendix, that influence Ld(s) 1: s a t u r a t i o n i n some o f f o r two v e r y d i f f e r e n t EFFECT OF SATURATION IN GURI UNIT Lad A % Tdo' 1.035 0.828 0.621 0.414 00 20 40 60 9.120 7.576 6.032 4.488 the machines 00.00 16.92 33.85 50.78 TABLE2: 0.0500 0.0489 0.0473 0.0444 o f change A% Tdo' A% Tdo" 1.590 1.272 0.954 0.636 00 20 40 60 5.900 4.820 3.740 2.661 00.00 18.30 36.60 54.91 0.0330 0.0327 0.0322 0.0313 A% = P e r c e n t a g e of See r e f e r e n c e These a circuit tables thermal time Chapter with 2.723 2.154 2.125 2.072 Td" 0.000 0.864 2.220 4.633 respect to 0.3285 0.0326 0.0323 0.0318 2. 0.000 0.924 2.381 5.021 change w i t h 0.850 0.845 0.836 0.819 respect 0.000 0.599 1.556 3.330 UNIT* A% 0.000 0.654 1.699 3.635 Td" 0.0249 0.0249 0.0248 0.0245 A% 0.000 0.249 0.655 1.429 to t h e u n s a t u r a t e d c a s e [8] clearly unit Td ' A% A% the u n s a t u r a t e d EFEECT OF SATURATION IN A F O S S I L - F I R E D Lad * 0.00 2.18 5.49 11.07 7 TO 10 A% Td' A% Tdo" A% A% = P e r c e n t a g e case and of shown. TABLE in the e f f e c t s , constants show that saturation and t h u s f o r both a h y d r a u l i c affects confirm mostly the unit open t h e a s s u m p t i o n s made
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A model for the synchronous machine using frequency response measurements Bacalao, Nelson Jose 1987
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Title | A model for the synchronous machine using frequency response measurements |
Creator |
Bacalao, Nelson Jose |
Publisher | University of British Columbia |
Date Issued | 1987 |
Description | In this dissertation a new model for the synchronous machine is presented. This model, based on non-standard test data, allows for the appropriate modelling of the frequency dependent behaviour of the damper windings. The non-standard test data consist of frequency responses, either measured or calculated. The form of these responses will automatically determine the order of the resulting model. Saturation effects in the synchronous machine are also modelled with this new method. The model was successfully tested in both an electromagnetic transients program (EMTP) and in a stability program. It was found that when frequency response measurements are used directly, the model is more accurate than when using the standard data from the manufacturer or data estimated to match approximately the frequency response measurements. It was also ascertained that this model could be used to speed up the solution in a stability program, both by allowing the user to match the order of the model to the required accuracy depending on the event and integration step, and by modifying the input frequency response data to minimize the discretization error made when using large integration steps. |
Subject |
Electric motors, Synchronous |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-07-27 |
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 | 10.14288/1.0302132 |
URI | http://hdl.handle.net/2429/26955 |
Degree |
Doctor of Philosophy - PhD |
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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