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A model for the synchronous machine using frequency response measurements Bacalao, Nelson Jose 1987

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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  A p p r o x i m a t i o n of X j ( s ) and X ^ ( s )  47  A p p r o x i m a t i o n of G(s)  48  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  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 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  (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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