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Statistical model formulation for power systems Mumford, Donald Gregory 1971

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A STATISTICAL MODEL FORMULATION FOR POWER SYSTEMS  by  DONALD GREGORY MUMFORD B.A.Sc., University of B r i t i s h Columbia, 1969  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE  i n the Department of E l e c t r i c a l Engineering  We accept this thesis as conforming to the required  standard  Research Supervisor ...» Members of the Committee  ,•.  Head of the Department . v . . . . ,  Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA August, 1971  In p r e s e n t i n g an the  this thesis  advanced degree at Library  shall  in p a r t i a l  the U n i v e r s i t y  of  British  make i t f r e e l y a v a i l a b l e  I f u r t h e r agree t h a t p e r m i s s i o n f o r s c h o l a r l y purposes may representatives.  be g r a n t e d by  his  of  t h i s t h e s i s f o r f i n a n c i a l gain  the  It i s understood  permission.  Depa rtment The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada  for  the  requirements f o r  Columbia, reference  Columbia  shall  Head of my  that  not  be  I agree and  f o r e x t e n s i v e copying o f  by  written  fulfilment of  that  study.  this thesis Department  copying o r  or  publication  allowed without  my  ABSTRACT  An i n v e s t i g a t i o n has been undertaken to a s c e r t a i n how a power system l e n d s i t s e l f to s t a t i s t i c a l  modelling.  v a r i a b l e model has been d e r i v e d i n terms o f measurable  readily  A nonlinear states.  state  This  model i s l i n e a r i n i t s c o e f f i c i e n t s which are e v a l u a t e d by the l e a s t squares f i t t i n g t e c h n i q u e o f r e g r e s s i o n a n a l y s i s .  The  statistical  model's performance i s e v a l u a t e d by comparison of i t s p r e d i c t e d  system  responses w i t h those p r e d i c t e d by Park's f o r m u l a t i o n , and w i t h those produced by a l a b o r a t o r y power system  ii  model.  TABLE OF CONTENTS  Page ABSTRACT  i i  TABLE OF CONTENTS  i i i  LIST OF TABLES  iv  LIST OF ILLUSTRATIONS  v  ACKNOWLEDGEMENT  vi  NOMENCLATURE  vii  1.  INTRODUCTION  1  2.  STATISTICAL MODELLING USING REGRESSION ANALYSIS  4  2.1 2.2 2.3 2.4 2.5 3.  4.  5.  6.  Features of Regression Analysis Assumptions i n Regression Analysis Significance of Regression Equation Significance of C o e f f i c i e n t s Regression Analysis i n Power Systems  5 7 8 9 10  MATHEMATICAL MODELS - THEORETICAL AND STATISTICAL  12  3.1 3.2 3.3 3.4  13 18 20 22  Synchronous Machine State Variable Equations Voltage Regulator-Exciter Equation One-Machine I n f i n i t e Bus System Formulation of S t a t i s t i c a l Model  LABORATORY POWER SYSTEM AND DATA ACQUISITION SYSTEM  28  4.1 4.2 4.3  29 34 39  Power System Laboratory Model Data A c q u i s i t i o n Hardware Data Acquisition Software  .  PERFORMANCE OF THE STATISTICAL MODEL  44  5.1 5.2 5.3  44 47 51  System Data f o r S t a t i s t i c a l Model Derivation S t a t i s t i c a l Investigations of the Regression Model... Model Responses to Step Inputs  CONCLUSION  APPENDIX 3A APPENDIX 3B APPENDIX 3C  59 ' .  61 63 64  APPENDIX 4A Flowchart f o r Data A c q u i s i t i o n Program  65  REFERENCES  67 iii  LIST OF TABLES Table  / ^  Page  4.1  Laboratory Model Machine Specifications  29  4.2  Laboratory System Parameters  33  4.3  Noise at A/D  35  4.4  Multiplexer Channel Selection Decoding  38  5.1  System Operating Points Investigated  46  5.2  Steady State Values  47  5.3  Example of V a r i a t i o n i n C o e f f i c i e n t s  53  Input  iv  LIST OF ILLUSTRATIONS  Figure .  /Page  2.1  Example of Regression Model  3.1  Rotor Angular P o s i t i o n  14  3.2  D.C.  17  3.3  Voltage Regulator-Exciter....  19  3.4  Power System Schematic  20  4.1  Laboratory System Configuration  30  4.2  Active F i l t e r Schematic  34  4.3  F i l t e r Performance  35  5.1  O v e r a l l Plot of Residuals  49  5.2  D i s t r i b u t i o n of Residuals with Time  49  5.3  Example of Variance of Residuals  50  5.4  Example of Test for' Lack of F i t . . . ,  50  5.5  Response to step i n u-^ at operating point B - S t a t i s t i c a l model from operating points A, B and C combined  52  5.6  Response to step i n u-^ at operating point B - S t a t i s t i c a l model from operating point A  52  5-7  Response to step i n u^ at operating point B - S t a t i s t i c a l model from operating point B  54  5.8  Response with s t a t i s t i c a l model found using 0.033 second sampling i n t e r v a l  57  Response with s t a t i s t i c a l model found using 0.083 second sampling i n t e r v a l  57  5.9  /..  Shunt Motor C i r c u i t . . .  v  9  ACKNOWLEDGEMENT I wish to thank the people who have assisted me while completing t h i s research project.  E s p e c i a l l y , I thank Dr. B. J . K a b r i e l ,  supervisor of this project, for h i s i n t e r e s t and encouragement throughout the course of the work.  Also I express a hearty thanks to  Dr. Y. N. Yu f o r h i s valuable comments.  The development of the data a c q u i s i t i o n interface by Dr. A. Dunworth i s acknowledged.  I appreciate the valuable discussions and proof reading offered by Mr. T. A. Curran as well as the c a r e f u l proof reading of Mr. B. P r i o r . A s p e c i a l thanks to my wife Joan, not only for typing this thesis, but also f o r her understanding  and encouragement during  my graduate program. The f i n a n c i a l support of the National Research Council of Canada i s g r a t e f u l l y acknowledged.  vi  NOMENCLATURE Prime Mover L  r r  af c  a d.c. motor c o e f f i c i e n t : where u L , i s the speed voltage ej-.. m af coefficient r  a  armature resistance  s  series resistance i n armature c i r c u i t  i  t o t a l resistance i n armature c i r c u i t P 2  pole pairs  ij  field  current  1  armature current; controls mechanical torque output  a) m  mechanical speed  T^  e l e c t r i c a l torque i n d.c. motor  D^^  d.c. motor damping c o e f f i c i e n t  r  Mechanical System J  moment of i n e r t i a of prime mover - generator set  F  f r i c t i o n coefficient  T  i  £  torque loss due to f r i c t i o n ; T. = Fto r m  Regulator-Exciter T. A  regulator time constant regulator gain on reference voltage input regulator gain on terminal voltage feedback  u^  regulator-exciter reference  vii  voltage  Synchronous Machine D  synchronous generator damping c o e f f i c i e n t f i e l d resistance  T' d T^  d-axis transient short c i r c u i t time constant d-axis transient open c i r c u i t time constant  o  T" . T" do qo  d- and q-axis subtransient open c i r c u i t time constants  x , ad  mutual reactance between stator and rotor i n d-axis  x^, x^  d- and q-axis synchronous reactances  n  r  d-axis transient reactance d x", x" d* q x  d- and q-axis subtransient reactance equivalent reactance of l o c a l load and transmission system  6  i,, d  < x  i  q  d  +  x )  d- and q-axis current  i , rd  f i e l d current  P  r e a l power output of the machine  Q  reactive power output of the machine  r  T  £  energy conversion torque of synchronous generator mechanical torque on the rotor  v^, v^  d- and q-axis voltages  v •  machine terminal voltage  v^  f i e l d voltage  v_  a voltage proportional to f i e l d voltage  r  v  a voltage proportional to f i e l d current r K.  ij^, if;  d- and q-axis flux linkages f i e l d flux linkage  viii  i^p  f l u x proportional to f i e l d flux linkage  to  e l e c t r i c a l angular speed  U)  q  synchronous speed, 377  /  rad/sec  ^  Au  per unit speed v a r i a t i o n  6  torque angle (between q-axis and i n f i n i t e bus voltage)  Transmission System r  s e r i e s resistance  x  s e r i e s reactance  G  shunt conductance  B  shunt  susceptance  S t a t i s t i c a l Model X  independent  variable  Y^  value of dependent Variable observed  A  Y^  value of dependent variable predicted by model  Y  mean value of observations of dependent variable  e.  residuals (Y, - Y.)  3  population c o e f f i c i e n t s  b  sample c o e f f i c i e n t s  a  population variance  R  multiple regression c o e f f i c i e n t  ix  ^  1.  Requirements  INTRODUCTION  f o r e x a c t and e a s i l y - u p d a t e d power system  models a r e ever i n c r e a s i n g w i t h modern complex  systems.  One m o d e l l i n g  problem a r i s e s when a t t e m p t i n g to f i n d a d e t a i l e d r e p r e s e n t a t i o n o f multimachine systems. involved  F o r t h e o r e t i c a l models,  [ 1 ] , [2] and i s l i m i t e d  the a n a l y s i s i s v e r y  to o n l y a few machines  [2].  Another  important problem i n m o d e l l i n g a system i s u p d a t i n g the model as the system changes.  For t h e o r e t i c a l models,  system parameters a r e measured  o f f - l i n e and i f they change d u r i n g system o p e r a t i o n , f o r example when a line i s lost,  the model i s no l o n g e r e x a c t .  A statistically dynamic  state estimation.  d e r i v e d model may  find application i n  P r e s e n t s t a t i c s t a t e e s t i m a t i o n schemes and  t r a c k i n g a l g o r i t h m s f o r system s e c u r i t y assessment r e q u i r e a system model. expanded  [3] - [9] do not  However, i f s t a t e e s t i m a t i o n t e c h n i q u e s a r e  t o s t a t e p r e d i c t i o n f o r use w i t h l o c a l c o n t r o l l e r s then a  system model w i l l no doubt be r e q u i r e d .  As d a t a i s o b t a i n e d f o r the  e s t i m a t i o n scheme, i t i s c o n v e n i e n t to a l s o use t h i s d a t a f o r s t a t i s t i c a l d e r i v a t i o n of the system  model.  A s t a t i s t i c a l approach t o m o d e l l i n g i s i n v e s t i g a t e d thesis.  in this  I t i s i n t r o d u c e d i n an attempt to overcome the problems of  t h e o r e t i c a l models i n m a i n t a i n i n g an u p - t o - d a t e system model and p o s s i b l y t o f a c i l i t a t e m o d e l l i n g more complex  multimachine systems.  A s t a t i s t i c a l model has the f o l l o w i n g i n h e r e n t advantages.  I t may  be  set  to r e t a i n o n l y the most s t a t i s t i c a l l y  s i g n i f i c a n t system v a r i a b l e s ,  with, i n s i g n i f i c a n t , v a r i a b l e s r e a d i l y e l i m i n a t e d . statistically-derived  S e c o n d l y , the  e q u a t i o n s a r e determined u s i n g  the p a r t i c u l a r  system c o n f i g u r a t i o n o p e r a t i n g as i t w i l l be when the model i s employed. updating, to  Also s t a t i s t i c a l  modelling  itself  to r e a l - t i m e  thus a l l o w i n g the m a t h e m a t i c a l r e p r e s e n t a t i o n o f the system  be updated as parameters change,  load  lends  f o r example,  i n response to system  changes.  Two  u l t i m a t e aims of t h i s p r o j e c t a r e : f i r s t l y ,  systems f o r which a c c u r a t e obtainable;  and s e c o n d l y ,  theoretical representations  a r e not  to o b t a i n a m a t h e m a t i c a l m o d e l l i n g  which i s f e a s i b l e f o r o n - l i n e m o d e l l i n g research reported  to model  and parameter u p d a t i n g .  i n t h i s t h e s i s i s concerned w i t h  aim o f i n v e s t i g a t i n g the proposed s t a t i s t i c a l d e f i n e d system i n an o f f - l i n e  scheme  the  The  intermediate  scheme u s i n g a w e l l  environment.  T h i s p r o j e c t i n v o l v e s s e t t i n g up a l a b o r a t o r y model power system, and d e r i v i n g two m a t h e m a t i c a l r e p r e s e n t a t i o n s o f t h i s One  is a statistical  r e p r e s e n t a t i o n which i s n o n l i n e a r i n the s t a t e  v a r i a b l e s , but l i n e a r i n t h e i r c o e f f i c i e n t s which a r e estimated. research.  statistically  T h i s model c o n s t i t u t e s the major p o r t i o n of o r i g i n a l The o t h e r  i s a t h e o r e t i c a l r e p r e s e n t a t i o n based upon  P a r k ' s f o r m u l a t i o n of the synchronous machine e q u a t i o n s . is  I t s purpose  to a l l o w a performance comparison of the newly-developed  model w i t h against  system.  the c l a s s i c a l  t h e o r e t i c a l model.  the l a b o r a t o r y system performance.  r e s e a r c h i s the data a c q u i s i t i o n r e q u i r e d  statistical  Both of t h e s e a r e checked Another f a c e t of t h i s  to c o l l e c t  observations  3  from the l a b o r a t o r y The  system f o r use i n d e r i v i n g the s t a t i s t i c a l model.  b a s i c i n t e r f a c e and computer was a v a i l a b l e but the i n t e r f a c e  required modification before  being  signals contain  undesired  considerable  used.  s i g n a l c o n d i t i o n i n g and f i l t e r i n g .  f o r the computing c e n t e r  The contains  organization  a presentation  noise,  this project also e n t a i l s  Data a c q u i s i t i o n r e q u i r e d  of PDP-8 s o f t w a r e as w e l l as e x t e n s i v e written  As t h e monitored power system  data handling  and c h e c k i n g programs  IBM 360/Model 67.  o f the work i s as f o l l o w s .  of the i d e a s  applied to the s t a t i s t i c a l modelling the two m a t h e m a t i c a l n o n l i n e a r  development  of r e g r e s s i o n  Chapter 2  a n a l y s i s as i t i s  i n this project.  I n Chapter 3,  state variable representations  power system ( t h e o r e t i c a l and s t a t i s t i c a l )  a r e developed.  of a  The l a b -  o r a t o r y model power system and the d a t a a c q u i s i t i o n system a r e d i s c u s s e d briefly  i n Chapter 4.  A comparison i s made i n Chapter 5 of the responses  from the s t a t i s t i c a l model, the t h e o r e t i c a l model, and the l a b o r a t o r y system.  Testing  the d a t a f o r v a l i d i t y  Chapter 6 i n c l u d e s few  the conclusions  o f assumptions i s a l s o  derived  guide l i n e s f o r f u r t h e r i n v e s t i g a t i o n s .  discussed.  from t h i s work as w e l l as a  2.  STATISTICAL MODELLING USING REGRESSION ANALYSIS  There a r e advantages to o b t a i n i n g e x p e r i m e n t a l models of systems u s i n g s t a t i s t i c a l  techniques.  be d e t e c t e d by v a r i o u s s t a t i s t i c a l  Insignificant variables  tests,  containing only s i g n i f i c a n t v a r i a b l e s . by d a t a a c q u i r e d from lends i t s e l f model.  can  thus y i e l d i n g a model  A l s o , s i n c e the model i s formed  the a c t u a l system, the s t a t i s t i c a l  approach  to o n - l i n e m o d e l l i n g or o n - l i n e u p d a t i n g of the system  Another advantage of u s i n g a c t u a l system d a t a i s t h a t the  models a r e more r e a d i l y expressed  i n terms of measurable s t a t e s .  D e s c r i b i n g a system's b e h a v i o u r  statistically  is  accomplished  by m o n i t o r i n g system performance, and d e r i v i n g an e q u a t i o n to " b e s t " d e s c r i b e t h i s observed  performance.  A common m a t h e m a t i c a l l y  method of d e t e r m i n i n g  the " b e s t " e q u a t i o n i s to perform  fit  of measurements of system v a r i a b l e s .  to d a t a comprised  convenient  a least  squares  This tech-  nique, which i s one method of f i t t i n g a l i n e to a s e t of o b s e r v a t i o n s or  d a t a p o i n t s , simply minimizes  the sum  of squares  Y i s the dependent v a r i a b l e , which the model w i l l to  predict,  R e g r e s s i o n a n a l y s i s i s one fit.  e v e n t u a l l y be  used  the f i t t e d  line.  t e c h n i q u e of p e r f o r m i n g  T h i s method of s t a t i s t i c a l  a least  a n a l y s i s has been chosen f o r  the t h e s i s p a r t l y because computer programs a r e r e a d i l y a v a i l a b l e , mainly  If  then the e r r o r i s the d i s t a n c e measured p a r a l l e l to the  Y - a x i s between the g i v e n d a t a p o i n t and  squares  of the e r r o r s .  but  because r e g r e s s i o n a n a l y s i s p r o v i d e s many t e s t s f o r c h e c k i n g  system d a t a and  for testing  the model produced.  When c h o o s i n g  a  m o d e l l i n g scheme, one must c o n s i d e r t h a t systems are d e f i n e d to g r e a t e r  5  and  l e s s e r degrees.  for  which a l l theory,  the o t h e r theory  At one  extreme are  and  extreme are c o m p l e t e l y d e t e r m i n i s t i c systems  therefore  the model, i s c o m p l e t e l y d e f i n e d .  the " b l a c k b o x " systems f o r which t h e r e  d e f i n i n g system performance from which a model may  power system i s somewhere mid-way because even though the from which a model may  be  derived,  d a t a to c o n s t r u c t extensive  a model by  a n a l y s i s and may  and  t h i s method  y i e l d a model which a l l o w s operation  little  t r i a l - a n d - e r r o r blackbox approach.  2.1  F e a t u r e s of R e g r e s s i o n  The introduced.  t e c h n i q u e are not  or no  insight  be used to i d e n t i f y  the  than f o r  the  t h e s i s the m a t h e m a t i c a l model  from theory  u s i n g Park's  repre-  the c o e f f i c i e n t s i n  system  operation.  Analysis  i n r e g r e s s i o n a n a l y s i s are  t h e o r e t i c a l d e t a i l s of t h i s s t a t i s t i c a l i n c l u d e d because they are not  a p p l i c a t i o n of r e g r e s s i o n  requires  r e g r e s s i o n a n a l y s i s on measurements  b a s i c concepts i n h e r e n t The  use  p h y s i c a l meaning  s e n t a t i o n of a synchronous machine (Chapter 3 ) , and these e q u a t i o n s are e s t i m a t e d u s i n g  change.  such t h a t i t has  In t h i s  to be of a form d e r i v e d  of s t a t e v a r i a b l e s made d u r i n g  i s known  However, i f the  t h a t system, then r e g r e s s i o n a n a l y s i s may  constrained  A  of the system.  parameters of t h i s model w i t h much l e s s a n a l y s i s r e q u i r e d  is  theory  system as a " b l a c k b o x " and  e r r o r , but  form of the system model i s c o n s t r a i n e d for  derived.  the system c o n f i g u r a t i o n  t r e a t any  trial  i n t o the p h y s i c a l s t r u c t u r e and  e x i s t s no  the parameters of t h i s model w i l l  change as the system o p e r a t i n g p o i n t and  R e g r e s s i o n a n a l y s i s can  be  At  required  a n a l y s i s computer programs.  though, i s an u n d e r s t a n d i n g of u n d e r l y i n g mechanisms of the a n a l y s i s i n o r d e r  modelling for  the  What i s r e q u i r e d ,  assumptions and  to a c c u r a t e l y  now  of the  basic  i n t e r p r e t the r e s u l t s  6  obtained.  By way o f d e f i n i t i o n , when concerned w i t h the dependence  of a random v a r i a b l e Y on a q u a n t i t y random v a r i a b l e , an e q u a t i o n t h a t regression  X which i s a v a r i a b l e but n o t a  r e l a t e s Y to X i s u s u a l l y  equation.  Regression analysis  i s applied  t o determine t h e r e l a t i o n s h i p  between a dependent v a r i a b l e Y and one o r more independent X^, X , 2  .... X  n  variables  where X^ may be a s i m p l e system v a r i a b l e o r a f u n c t i o n  of one o r more v a r i a b l e s . independent v a r i a b l e s the  called a  The a n a l y s i s uses many measurements o f the  and c o r r e s p o n d i n g dependent v a r i a b l e t o determine  c o e f f i c i e n t s i n the r e l a t i o n s h i p .  F o r example l e t a system be  described  by Y  =  g  Q  + p X  +  x  3 X 2  + e ,  2  then many o b s e r v a t i o n s o f Y, X^, and X analysis  to obtain  In e q u a t i o n  2  (2.1)  are subjected  to regression  e s t i m a t e s o f t h e l i n e a r c o e f f i c i e n t s 3 » 3]_ and 32 • Q  (2.1) e r e p r e s e n t s the e r r o r the model w i l l make when used  to p r e d i c t Y and, as i t i s d i f f e r e n t f o r each Y observed, i t i s n o t measurable.  The p o p u l a t i o n  c o e f f i c i e n t s 3 , 3]_, and 32 can n o t be Q  found e x a c t l y w i t h o u t examining a l l p o s s i b l e Y, however they a r e e s t i m a t e d i n r e g r e s s i o n coefficients b  Q  , b-^  and  b  b X  , and X2  values,  a n a l y s i s by t h e sample  The m a t h e m a t i c a l model o b t a i n e d  then  b X  (2.2)  may be w r i t t e n  Y  =  D  +  1  1  +  2  2  A where Y denotes the Y v a l u e s o b t a i n e d when u s i n g prediction.  the model f o r  7 The p r o c e d u r e used f o r r e g r e s s i o n a n a l y s i s may be e x p l a i n e d f o r the s i m p l e t w o - v a r i a b l e case by p l o t t i n g p o i n t s r e l a t i n g v a l u e s o f Y and X on a s e t o f axes.  observed  A s t r a i g h t l i n e i s drawn through  t h e s e p o i n t s such t h a t the sum o f the squares o f the d i s t a n c e s ( p a r a l l e l to the Y - a x i s ) between the p o i n t s and the l i n e i s m i n i m i z e d . e q u a t i o n of t h i s l i n e  then d e f i n e s the c o e f f i c i e n t s b  Q  The  and b^.  Multiple  r e g r e s s i o n . i n c l u d i n g many independent v a r i a b l e s c o n s i s t s o f a s i m i l a r p r o c e s s except t h e s t r a i g h t l i n e i s r e p l a c e d by h y p e r p l a n e s i n m u l t i d i m e n s i o n a l space.  The c h o i c e o f independent v a r i a b l e s and t h e r e f o r e the form of t h e model chosen depends upon p r i o r knowledge o f the p h y s i c a l  system  u n l e s s a b l a c k b o x approach i s b e i n g used i n which case t h e independent v a r i a b l e s a r e guessed.  2.2  Assumptions  i n Regression Analysis  In the model Y^ = 3  G  + 3]_X-L + £ i  ,  i = 1, 2, ...n  d e s c r i b i n g each measured v a l u e o f t h e dependent v a r i a b l e , i t i s assumed  that:  (1) e± i s a random v a r i a b l e w i t h mean zero and v a r i a n c e a t h a t i s , E ( e ) = 0 , V(e-j:) = a ±  (2)  2  2  (unknown),  .  and E J a r e u n c o r r e l a t e d f o r i ^ j so t h a t c o v ( e ^ , E j ) = 0. Therefore E ( Y ) ±  = 3  + 3lX  Q  ±  ,  V(Y ) ±  = a  2  .  and Y^ and Y j , i ^ j a r e u n c o r r e l a t e d . (3) I n a d d i t i o n t o ( 1 ) , e± i s a n o r m a l l y d i s t r i b u t e d random v a r i a b l e . That i s , E  - N(o, a ) . 2  ±  T h e r e f o r e e^, E j a r e n o t o n l y u n c o r r e l a t e d , but n e c e s s a r i l y independent.  Knowing the assumptions g o v e r n i n g the e r r o r s the data) i n r e g r e s s i o n , derived  to be  one  i s able  Also  therefore  to t e s t the model a f t e r  sure that i t adequately explains  i n the observed data.  (and  i t is  the b e h a v i o u r  the d a t a i t s e l f may  be  evident  checked to  verify  / /  whether or not The  i t does meet the assumptions i n h e r e n t  l a c k of f i t of the model may  simple r e g r e s s i o n  case  be  in this analysis.  expressed a n a l y t i c a l l y f o r  [10]; however, i n the m u l t i p l e r e g r e s s i o n which  w i l l be used, l a c k of f i t i s i n v e s t i g a t e d by p l o t t i n g (or  values).  i s a l s o accomplished by means of r e s i d u a l p l o t s .  2.3  residuals  The  assumptions  e x a m i n a t i o n of  t h o r o u g h l y i n Chapter 3 of Draper and  Smith  [10]  S i g n i f i c a n c e of R e g r e s s i o n E q u a t i o n  A f t e r the  form of the e q u a t i o n i s e s t a b l i s h e d and  a n a l y s i s i s used to e v a l u a t e the r e g r e s s i o n  with tabulated  of v a r i a b l e s and regression  the c o e f f i c i e n t s ,  values  of  comparing the m u l t i p l e r e g r e s s i o n  coefficient,  which g i v e s i g n i f i c a n t R v a l u e s  f o r the number  the number of o b s e r v a t i o n s  coefficient  i s defined  of squares due  ^  sum  of squares t o t a l  where  SS„  n  R  S S  e  g  Total  to r e g r e s s i o n  -  = £ (Y - Y) i=l i =JA  used.  The  multiple  by  sum  2  the u s e f u l n e s s  regression  e q u a t i o n as a p r e d i c t o r of system performance i s checked  T h i s i s accomplished by R,  the  V e r i f i c a t i o n t h a t the d a t a meets r e q u i r e d  residuals i s explained  the  " ^  S S  Reg  (2  3)  SS •Total 9  (2.4)  <'  2 5)  for which the various Y values are i l l u s t r a t e d i n Figure 2.1.  If R  i s not greater than the tabulated value for a desired l e v e l of s i g n i f i c a n c e , then the regression model i s not useful because Y could just as well be described by i t s mean value Y.  Y  l  Y ^~~(Y  A  V -  -  - VV  J  Y  \  f  -^-(Yf ^ ( Y  -  Y)  ~ Y) f  - Y) = f  €  f  &.  X Figure 2.1  2.4  Example of Regression Model  Significance of C o e f f i c i e n t s  When multiple regression i s employed to i d e n t i f y c o e f f i c i e n t s i n an assumed model, i t i s possible to have this analysis omit any independent v a r i a b l e or combination of independent variables which are found to be i n s i g n i f i c a n t i n the data sample from which the model i s being derived.  This i s done by using an F-test to check the s t a t i s t i c a l  s i g n i f i c a n c e of the c o e f f i c i e n t s . For example i n the model  Y = b  Q  +  b-jX^ +  b X 2  2  (2.6)  the q u e s t i o n o f whether two models, consider  = 0 or not may  one i n c l u d i n g b^X-^ and one o m i t t i n g  e q u a t i o n s (2.6) and  Y = b  Q  +  b X 2  model l a s t .  This  during  2.5  o f b^ as though  (2.7)  i t were added  to the  f o r b^.  The  i n d e t a i l i n Chapter 2 o f Draper  and  f i n d i n g the d i f f e r e n c e o f the sums o f squares  due  i n models (2.6) and  (2.7).  insignificant variables  Regression Analysis  To o b t a i n  That i s ,  '  This  type o f F - t e s t  the b u i l d i n g up p r o c e d u r e f o r a r e g r e s s i o n  statistically  investigating  term.  e n t a i l s the use o f a p a r t i a l F - t e s t  [10] i n v o l v e s  by  (2.7)  p a r t i a l F - t e s t which i s o u t l i n e d  to r e g r e s s i o n  that  2  and measure the c o n t r i b u t i o n  Smith  be answered  i s used  model to omit  from the r e s u l t i n g model.  i n Power Systems  a p h y s i c a l l y m e a n i n g f u l system model,  a n a l y s i s w i l l be used o n l y of a power system model.  regression  to i d e n t i f y c o e f f i c i e n t s i n an assumed form The form o f t h i s s t a t i s t i c a l model i s o u t l i n e d  i n Chapter 3 a f t e r development  of Park's f o r m u l a t i o n  on which  i t is  based.  In u s i n g  regression  analysis  to e v a l u a t e the c o e f f i c i e n t s  of a d i s c r e t e s t a t e v a r i a b l e model, the dependent to be the p a r t i c u l a r s t a t e variables  a r e the s t a t e s  variable  c o n s i d e r e d a t time t ^ - ^  a n  +  and f u n c t i o n s  of s t a t e s  d  i s chosen  the independent  as d e f i n e d  by the  form of the model a t time t ^ , where the measurements of s t a t e are a c q u i r e d  variables  a t the u n i f o r m i n t e r v a l o f t ^ ^ - t^ = At. When d e v e l o p i n g  the model, the s t a t e v a r i a b l e s  +  c  are measured and s u b j e c t e d  to the  11  r e g r e s s i o n a n a l y s i s which, by data,  estimates  l e a s t squares f i t t i n g  the l i n e a r c o e f f i c i e n t s  v a r i a b l e equations. v a r i a b l e equations,  to the  i n the n o n l i n e a r  acquired  state  To o b t a i n a system model c o n s i s t i n g of f o u r four separate  r e g r e s s i o n analyses  are  state  required.  3.  MATHEMATICAL  MODELS - THEORETICAL AND STATISTICAL  A t h e o r e t i c a l s t a t e v a r i a b l e model f o r a one-machine bus  power system i s d e r i v e d .  The model c o n s i s t s o f a t h i r d o r d e r  s e n t a t i o n o f the synchronous machine approximated from Park's and  a first  infinite  order v o l t a g e r e g u l a t o r .  repre-  equations,  Based on t h i s f o r m u l a t i o n , the  form o f the s t a t i s t i c a l s t a t e v a r i a b l e model i n c l u d i n g o n l y measurable states i s derived.  I n i t i a l development of Park's r e p r e s e n t a t i o n f o l l o w s t h e development as o u t l i n e d by V o n g s u r i y a except t h a t the torque q-axis  [11] c l o s e l y  [12] and Dawson [13],  angle, 6 , i s d e f i n e d as the angle between the  and the i n f i n i t e bus o r r e f e r e n c e v o l t a g e a t the b e g i n n i n g  derivation.  Other d e v i a t i o n s from the r e f e r e n c e s  o f the  quoted i n c l u d e an  a p p r o x i m a t i o n of the synchronous machine e l e c t r i c a l damping t o compensate for n e g l e c t i n g amortisseur and  w i n d i n g e f f e c t s i n Park's  the d e r i v a t i o n o f a damping e x p r e s s i o n  representation  f o r the d.c. machine.  A t h i r d o r d e r machine r e p r e s e n t a t i o n was chosen f o r two reasons. (except  F i r s t l y , Dawson [13] concluded  s u b t r a n s i e n t and s w i t c h i n g phenonmenon) a t h i r d o r d e r  sentation i s s u f f i c i e n t . tically  t h a t f o r many system s t u d i e s repre-  Secondly, to f i n d h i g h e r o r d e r models  statis-  r e q u i r e s much f a s t e r system sampling and thus much more data  a c q u i s i t i o n apparatus than t h a t r e a d i l y a v a i l a b l e f o r t h i s p r o j e c t .  /  3.1  Synchronous Machine S t a t e V a r i a b l e E q u a t i o n s  D e t a i l e d d e r i v a t i o n s of Park's  equations  a r e numerous.,  As  a t h i r d o r d e r r e p r e s e n t a t i o n i s used, the d e r i v a t i o n s t a r t s w i t h t h i r d order approximation this simplified  of Park's  equations  [ 1 2 ] , [13].  To  the  obtain  form of Park's model, the f o l l o w i n g assumptions  are  made. (1) - S u b t r a n s i e n t time c o n s t a n t s are n e g l e c t e d . (2) The  induced v o l t a g e s and  the v o l t a g e s due  to speed v a r i a t i o n s  are  n e g l e c t e d because they a r e s m a l l compared to the speed v o l t a g e s due (3) The  to c r o s s  excitations.  relatively  s m a l l d - a x i s damper leakage  time  constant  and  armature  r e s i s t a n c e are neglected. These assumptions reduce Park's d-q  equations  f o r a synchronous machine i n  coordinates to: Vd  =  -i|j a)  v  =  ^d o  q  q  =  ad  Vfd  w R 0  ^q  =  - q  (1 +  F  +  T^ p)  "o^  Q  1  *>> i  ( . )  T  d  3  3  + doP) T  i  x  U)  The m e c h a n i c a l  (3.2)  w  x  *d  (3.1)  0  (3.4)  0  behaviour  of the machine i s expressed  by the  torque  equation: T and  ±  =  Jp <5  +  2  Dp5  +  T  (3.5)  e  the e x p r e s s i o n f o r the r o t o r a n g l e : 6  Equation  (3.6)  = ' co t 0  +  6  i s represented  (3.6) i n F i g u r e 3.1  r e f e r e n c e i s chosen to c o i n c i d e w i t h  where the  rotating  the i n f i n i t e bus v o l t a g e  phasor.  g-axis  **- reference  a-phase  d-axis F i g u r e 3.1  Rotor Angular  Position  From e q u a t i o n s (3.1) t o (3.4) and synchronous and  (3.6)  the  machine dynamics can be e x p r e s s e d i n terms of one  two m e c h a n i c a l s t a t e v a r i a b l e e q u a t i o n s .  ment i n Appendix 3A, dynamics may  P^F  p6  electrical  A c c o r d i n g to the d e v e l o p -  the s t a t e v a r i a b l e f o r m u l a t i o n of the machine  be w r i t t e n as:  =  V  =  F  "  V  (3.7)  FR  (3.8)  UQAW  pAo)  Auxiliary  (3.5) and  (T  ±  - Dp6  -  T_)  e q u a t i o n s r e q u i r e d i n c l u d e the energy q^d  d q r  '  (3.9)  c o n v e r s i o n torque (3.10)  the t e r m i n a l v o l t a g e 2 vf t  2 = vj + d  2 v q z  (3.11)  15  the  power and r e a c t i v e power output  and  P  =  v,i, + v i d d q q  Q  =  v i q  d  - v i d  (3.12)  .  q  (3.13)  A l s o r e q u i r e d a r e e q u a t i o n s to e v a l u a t e V p , i ^ and i R  be s o l v e d from e q u a t i o n (3A.7),  (3A.2), and  V =  These can  (3.14) which i s formed by combining e q u a t i o n s  (3.4). T  *d  .  do  l/03  - d o < d " d> T  x  -X  o  d  x  /  U  'FR  0  0  0  *  i .  (3.14)  -x /co q o  Measurable S t a t e s  In  d e f i n i n g the form o f the s t a t i s t i c a l model based on  Park's f o r m u l a t i o n , may  i t i s required  t h a t a l l s t a t e s be measurable.  be shown t h a t the immeasurable s t a t e , i^-p, may  measurable f i e l d c u r r e n t , i f , as f o l l o w s .  It  be r e p l a c e d by the  From e q u a t i o n s (3.4) and  d  (3A.11) i|>„ F and  ± d  =  =  T* x , i , , - T' (x, - x ' ) i , do ad f d do d d' d  (3.15)  1 - o ^^ *d  (3.16)  v  r n r  ^F  do d  w  which g i v e s IJJ^ i n terms of i ^ ^ , are of  expanded  i ^ and i  further using a u x i l i a r y  the s t a t i s t i c a l  model.  q  . E q u a t i o n s (3.15) and  (3.16)  e q u a t i o n s when o u t l i n i n g the form  16  Torque  From d.c. machine t h e o r y [14] the torque developed by a shunt d.c. motor i s d e s c r i b e d  T,  =  k i  dc where  as  (3.17)  v a  [15] k  =  v  | L  a  i  f  f  .  (3.18)  The t o r q u e i n p u t t o the synchronous machine, T^, and  (3.9) i s e q u a l to the e l e c t r i c a l  i n equations  (3.5)  c o n v e r s i o n torque o f the d.c.  machine i n e q u a t i o n (3.17) minus the m e c h a n i c a l torque l o s s i n the system.  Mechanical torque l o s s T^=F(u )-a3 m  i s determined e x p e r i m e n t a l l y  by e v a l u a t i n g the f r i c t i o n term, F, where  F  =  ^dc  =  P_ a f a f 2 CD m L  0)  m  i  .  i  (3.19)  when u s i n g the d.c. motor prime mover to r o t a t e the  synchronous  g e n e r a t o r ( w i t h no l o a d ) at v a r i o u s speeds near synchronous  speed [ 1 5 ] .  Then, T. l  =  T, - T- . dc f  (3.20)  Damping  The d e s c r i p t i o n o f the damper w i n d i n g c i r c u i t s synchronous third  g e n e r a t o r i s not i n c l u d e d  order r e p r e s e n t a t i o n .  approximated  D(5)  i n the machine e q u a t i o n s f o r the  However, the damping e f f e c t may  be  [ 1 6 ] , [ 1 7 ] , [18] by  =  D sin 6 + D cos 6 2  D  (3.21)  2  x  2  ( x  where  f o r the  = v  d  ~  X  d  }  2 T  (x  e  +  x ) d  »  (3.22)  (x* D  =  v  g _ ..  2 _s  2  - x") T  ° (x + x')2 e q  #  ( 3 > 2 3 )  ^  / The d.c. motor simulating the prime mover has an e l e c t r i c a l damping which i s dependent on the change of torque with motor speed, and  may be determined as follows. T = M a  For a d.c. shunt motor [14]  d c  e and  =  kco v m  <' > 3  17  (3.24)  i n the armature c i r c u i t shown i n Figure 3.2, v  =  R i + e.  (3.25)  0  a.  Substituting gives v  -  R  — k  T,^ ^dc  T  +  K  k„io_. v V  v  O  -f  V  Figure 3.2  D.C. Shunt Motor C i r c u i t  (3.26)  18  Assuming constant applied voltage, d i f f e r e n t i a t i n g equation -  k  dT,  = - k du>  dc  v  v  m  .  (3.26) y i e l d s  •  (3.27)  Hence the d.c. motor damping defined as D  may  dc du m be expressed as  D  3.2  =  d c  - k —  =  d c  (3.28)  d T  R  2  .  (3.29)  Voltage Regulator-Exciter Equation  A voltage regulator-exciter was modelled by an amplidyne as shown i n Figure 3.3. Assuming a single time constant representation, the amplidyne equation appears as v  fd  K  v. x  (3.30)  1 + T.p A^  where K  V  i  "  K  A1 1" U  K  A2 f V  ( 3  '  3 1 )  Therefore, PV  fd  =  ^  (-v  fd  + K  A l U l  -  K v ) A 2  t  (3.32)  describes the exciter i n state v a r i a b l e form where the c o n t r o l , u-^, i s the exciter reference voltage.  This then produces a fourth order model  for the synchronous generator and i t s voltage regulator.  19  amplidyne field circuits  amplidyne armature • circuit  Ui  i  alternator field  alternator armature  q 0  r  F3  F7 F9  F4 F8°  F13 F13 ©—&-  tM£MFJ4  J3J  Figure 3.3  Voltage Regulator - E x c i t e r  Rating of F i e l d C i r c u i t s  t o i l label F3 - F4 F7 - F8 F9 - F10 F13 - F14  # of turns 1780 390 85 400  resistance at 25° C 980 43 2.6 56  maximum current 0.12 0.6 2.2 0.5  3.3  One-machine I n f i n i t e Bus  The  System  power system m o d e l l e d i s shown s c h e m a t i c a l l y  i n Figure  V,  3.4.  f  X  JB Figure  3.4  Power System Schematic  A s t a t e v a r i a b l e d e s c r i p t i o n of the g e n e r a t o r and by  equations  (3.7)  to  (3.9)  a u x i l i a r y equations. external  voltage [I]  The  =  The  and [Y]  [V]  =  to t 0  (3.32) a l o n g w i t h t h e i r  transmission  current  q - a x i s p o s i t i o n has 0  and  external  common r e f e r e n c e  be  described  .  by  the  (3.33) by  equation  (3.6)  as  6  (3.6)  which i s demonstrated i n F i g u r e  The  system may  associated  relationships  been d e s c r i b e d +  i t s e x c i t e r i s expressed  3.1.  system and  the machine must be  i n o r d e r to d e r i v e  s t a t e v a r i a b l e model o u t l i n e d .  referred  to a  p h y s i c a l system q u a n t i t i e s  from  E x p r e s s i o n of e x t e r n a l  system  the  quantities  i n terms of d-q coordinates i s presented .in Appendix 3B using Park's transformations [12], to y i e l d the following r e s u l t .  v  d  "  i  k  -k  V  k  2  v sin6 0 v cos<5 o  2~ l  k  q  + " l  K  K  -K  _  l  K  2  2  (3.34)  i q  where  1 + rG - xB  k,  (1 + rG - x B ) + (xG + r B ) 2  (3.35) 2  xG + r B  (3.36)  (1 + rG - xB)2 + (xG + rB)2 K-,  k^r + k x  (3.37)  K,  -k^x + k r "  (3.38)  2  2  An expression f o r  and Vq i s available i n terms o f . i ^ and-  i q and state variables (equation 3-34). Also i ^ and i i n terms of  may be expressed  and fy^ and state variables (equation 3.14).  an a d d i t i o n a l expression i s required giving ^ variables only, thus allowing  Therefore  and ty^ i n terms of state  a l l a u x i l i a r y variables to be evaluated  at any time knowing the state of the system at that time.  The develop-  ment i n Appendix 3C y i e l d s the following desired r e s u l t .  V  M  M  >.  v sin6 o  M"  ~ 1  2  •  M  3  +  N  v cos6 Q  4  \  (3.39)  2  where to  M,  A u  M  2  K k  o  A  -11 + 1  _2. [ x  q  K r  [  x  1  1 (K__ x  Dk  ]  (3.40)  (K  l)kx ]  (3.41)  2  2  q  l 2 k  _ J  q  2  % l 2 — [ A x ' d K  M  =  3  u M  =  4  . A  x  Therefore  do d X  Q  and  2  +  (K  1 2  +(K  - 1)K  1 —  2  x  x  z  d "  V <  0  q d x  x  (3.14),  (3.43)  ]  (3.44)  -DK,]  (3.45)  2  X  d q 2  V  ^ (3.46)  f  .  /  (3.39),  the  auxiliary  found.  s t a t e s of a power system d e f i n e d by v , d  [12].  v ,  the r e a l power, P,  and  i^,  conditions  This operating point i s described t  Formulation  x  x  6 are determined from the s t e a d y - s t a t e o p e r a t i n g  output  r  q d  (3.34) and  the machine t e r m i n a l v o l t a g e , v ,  3.4  1 —  2  d (  2  + oo  as o u t l i n e d by V o n g s u r i y a  power, Q,  ]  d  K  x  be  2  X  do d  initial  - l)k  . q K  from e q u a t i o n s  The  , —  q  x  system v a r i a b l e s may  iq, v ,  X  (3.42)  2  l  [  h\\  =  V  [  "o A T  (K  x  K  1)1^ ]  2  d  %  =  2  + x  A T  N  (K  x' d  [  =  X  +  K.k.  — A  N  1  k  by  reactive  of the machine.  of S t a t i s t i c a l Model  To o b t a i n a p h y s i c a l l y m e a n i n g f u l system model, r e g r e s s i o n a n a l y s i s i s used to e v a l u a t e system model. lined.  c o e f f i c i e n t s i n an assumed form of a power  T h i s form i s d e r i v e d u s i n g Park's f o r m u l a t i o n j u s t  I t must, however, be  i n terms of measurable s t a t e v a r i a b l e s so  t h a t as the system i s o p e r a t i n g observations  of these  out-  the data a c q u i s i t i o n system can  state variables directly.  record  I f the model were i n  terms of  immeasurable s t a t e v a r i a b l e s ,  e q u a t i o n s would be  required  to o b t a i n  then t h e o r t i c a l a u x i l i a r y the  state variable value  the measurements; thus r e s u l t i n g i n a model which no  longer  e x p e r i m e n t a l l y determined c o e f f i c i e n t s .  infinite  voltage,  v ,  i s measurable and  Q  i s c o n t r o l l a b l e on  system, i t i s e x p r e s s e d e x p l i c i t l y of  the  s t a t i s t i c a l model.  This  For Ato, and  V f  d  t h e o r e t i c a l model developed, the  state variables  model d e r i v a t i o n and using be  by  only.  derivation  (3.15) and  from e q u a t i o n ipp  =  =  form  predict  system.  ijj  d  The f d  the  conditions.  state  6, space  Before  s t a t i s t i c a l model, ^  must  that  i f  replace  d  may  be  chosen to  thus g i v i n g a model i n terms of measurable  0  F-[i  c h o i c e of ip-p,  state.  i t i s stated  MjV sin6  (3.39).  of i n i t i a l  form of  consists  eliminating ^  where  3.1  a state variable The  the  a measurable  In s e c t i o n ipp as  the  used to  i s c o n v e n i e n t b o t h f o r the  for evaluation  t h i s model to d e f i n e  replaced  bus  laboratory  equations d e f i n i n g  from a n e i g h b o u r i n g  has  Variables  the  as  i n the  the  a l l o w s the model to be  system response to a v o l t a g e d i p  C h o i c e of S t a t e  Because the  from  +  of s u b s t i t u t i n g e q u a t i o n  states  (3.16) i n t o  using +  M v cos6 2  0  +  N-^p  (3.47)  r e s u l t i n g expression i s F (M v sin6 2  1  0  +  M v cos6) 2  D  (3.48)  /  F  co T' (x ° °  =  ?  d  x  Differentiating  d  - x')  d  .  d  " % do< d " T  equation  x  x  d  )  N  (3.50)  l  (3.48) y i e l d s  /  ( pxjj-p Substituting equation  =  F]_pif  equations  + d  w F2(M-|V cos6Au) - I^VgSindAto) . Q  (3.48) and  (3.51) i n t o  f o r the measurable s t a t e v a r i a b l e i t^ad^-fd  1  Pi  f  =  d  — F  (3.51)  0  x  ad  +  (3.7) g i v e s a s t a t e ^ as  ^ d  1  - io F ( M v cos6Aco - M v sinSAto) ]. O ^ -L O ^- O 0  1  The torque expansion T^ « i  a  (3.52)  ?  expressions i n equation  (3.9) a l s o r e q u i r e  i n terms of measurable q u a n t i t i e s .  . From e q u a t i o n T  e  =  From e q u a t i o n  (3.17),  (3.10)  V d  -  ( ' 3  where i ^ and i ^ a r e e x p r e s s e d  i n terms of  and ^  by  1 0  >  equation  (3.14) so t h a t  T  =  -UJL-* 4\ °x q d  e  "  TT~XT  q S u b s t i t u t i n g equations T  e  =  A  l fd i  +  1  do d  31  + A,v s i n 6 c o s 6 4 o 2  +  •  (3.53)  q  d  ^fd^o ^  +  $J ° ^1 1  d  (3.39) and (3.48) i n t o 2  a  q  f  (3.53) g i v e s  hHc^o  +  A'V s i n 6 5 o 2  2  +  0056  Aiv cos 6 6o 2  2  (3.54)  where the c o e f f i c i e n t s  a r e not expanded because t h e i r v a l u e s a r e n o t  r e q u i r e d when d e f i n i n g  the form of the s t a t i s t i c a l model.  From  trigonometry cos 6 = 1 - s i n 6 2  2  (3.55)  which reduces  T  (3.54) t o  =  e  V f d +  2  V f d ^  +  A/,v  5  1  1  +  sin6cos6  1  6  +  A  Ac-v  3 fd 1  v 0  cos  f i  (3.56)  s i n <5.  T h i s f o r m u l a t i o n l e a d s to a d d i t i o n a l n o n l i n e a r i t i e s to p r o d u c t s of s t a t e v a r i a b l e s a p p e a r i n g i n the e x p r e s s i o n . n e v e r t h e l e s s an a c c e p t a b l e form o f s o l u t i o n f o r the m o d e l l i n g scheme used here and  It is  statistical  i s r e a d i l y handled by forming  d e s i r e d p r o d u c t a t each sampling  time and  due  the  then o b t a i n i n g the d e s i r e d  l i n e a r c o e f f i c i e n t s by r e g r e s s i o n a n a l y s i s .  The equations and  form of the model to be e x p l o i t e d ,  (3.8), (3.9), (3.32),  (3.56).  The  then, i s d e f i n e d by  (3.52) and a u x i l i a r y e q u a t i o n s  d i f f e r e n c e e q u a t i o n c o u n t e r p a r t s o f these  s t a t e e q u a t i o n s a r e used because the d i s c r e t e sampling the d a t a a c q u i s i t i o n system more e a s i l y equation representation.  terms p r e s e n t .  intercept b  i s included.  environment  To assume t h a t s t a t i s t i c a l l y b  to account  of measurement o f f s e t s , even though i t i s d e s i r e d t h a t b M a i n t a i n i n g the o f f s e t  explicitly ,  f d  = 0 requires  term, b , Q  for effects Q  in fact  and e x p r e s s i n g  v  Q  the d i s c r e t e s t a t e v a r i a b l e e q u a t i o n s f o r r e g r e s s i o n  a n a l y s i s a r e as i  Q  the  T h i s i n t e r c e p t v a l u e i s thus r e t a i n e d i n  the r e g r e s s i o n e q u a t i o n s f o r t h i s p r o j e c t  equal zero.  of  f a c i l i t a t e s a difference  However f o r the r e g r e s s i o n a n a l y s i s ,  considerable investigation. all  differential  In the p e r t i n e n t s t a t e e q u a t i o n s t h e r e are  no i n t e r c e p t Q  (3.17)  follows:  (k+l)  =  b  +  1 0  b i n  f  d  (k)  +  b  +  b-j 3V (k)cos6(k)Ato(k)  +  b v (k)sin6(k)Ato(k)  L  1 4  0  1 2  v  f d  (k)  0  (3.57)  6(k+l)  =  b  2  Q  + b  Au>(k+1) =  b  3  Q  + b Aco(k) + t >  v  Auxiliary  3 3  + b  3 5  + b  3 7  (k+1) = b  4  22  31  + b  i  i  2 f d  f d  (k)  v  2 0  (k) s i n 6 (k)  ^  2  b v (k)sin6(k)cos6(k)/ 2  3 4  o  o  3 6  f d  o  (3.59)  i (k) a  4 1  v  f d  (k) + b  4 2  u (k) + b x  4 3  v (k).  (3.60)  t  Equation  A major c o n s i d e r a t i o n linearities  i n equation  The t e r m i n a l v o l t a g e , evaluate  +  3 2  ( k ) v ( k ) s i n 6 ( k ) + b ± (k)v (k)cos6(k)  + b  0  (3.58)  6 ( k ) + b Aco(k)  2 1  i n t h i s modelling  v , i s measurable and thus can be used t o t  (3.60).  i s used as a p r e d i c t o r , an e x p r e s s i o n  From t h e t e r m i n a l 2  =  Substitution f o r v  v  t  2  =  V d  +  V  fc  =  from e q u a t i o n s  0  Q  0  2  3  (3.39)  Q  n  4  Q  )  6 1  >  gives N-^ ) F  2  o  '  ( -  (M v sin6 + M V COS6 + N ^ ) . 3  3  (3.1) and (3.2) g i v e s  * d V  +  (3.11) (  to ( M ^ v s i n 6 + M 2 V c o s 5 + + w  d  d  2  q  as a  t  = f ( i £ , 6, Ato, V f ) .  c o n d i t i o n e x p r e s s e d by e q u a t i o n  and a f u r t h e r s u b s t i t u t i o n o f e q u a t i o n v  fc  to e x p r e s s v  q '  and v  * q V  However, when the model  i s required  f u n c t i o n of s t a t e v a r i a b l e s , t h a t i s , v  t  the non-  (3.60) which a r i s e when s u b s t i t u t i n g f o r v ^ .  the c o e f f i c i e n t s i n equation  V  involves  2  F  (3.62)  Expressing  i n equation (3.62) by equation (3.48) and s u b s t i t u t i n g  equation (3.55) y i e l d s an expression  v  =  t  %  of the form  sin6 +  v 0  ^3 o  +  v  s i n (  5  c o s  ^  + B i v sin6 + B i v cos6 4  f d  0  5  f d  (3.63)  o  where the values of the c o e f f i c i e n t s are not required for defining the form of the s t a t i s t i c a l model.  Including an o f f s e t term, the desired  a u x i l i a r y regression equation i s found to be  v (k) 2  t  =  b +  + b v (k)sin 6(k) 2  5 Q  5 1  ^53 o v  +  2  0  b  5 2  i  2 f d  (k)  (k)sin6(k)cos6(k)  + b i (k)v (k)sin6(k) 5 4  f d  D  + b i (k)v (k)cos6(k). 5 5  f d  (3.64)  o  If this a u x i l i a r y equation i s substituted d i r e c t l y into equation (3.60) then equation (3.60) i s no longer l i n e a r i n i t s coefficients. (3.64) may  However, because v  i s a measurable quantity, equation  be treated as an a u x i l i a r y equation and regression analysis  w i l l estimate the l i n e a r c o e f f i c i e n t s . predictor, v  fc  may  When the model i s used as a  be found at each time desired and i t s square root  used i n equation (3.60).  Equations (3.57) to (3.60) and  (3.64) then describe  the  form of the state variable model of the system with a l l equations l i n e a r i n their c o e f f i c i e n t s .  Regression analysis i s used to estimate the  unknown l i n e a r c o e f f i c i e n t s i n each of the f i v e equations, a state variable nonlinear system model.  thus y i e l d i n g  This model i s tested i n  Chapter 5 by i n v e s t i g a t i n g i t s t a t i s t i c a l l y  as well by comparing i t s  performance x^ith that of the t h e o r e t i c a l model and the laboratory system.  4.  LABORATORY POWER SYSTEM AND  DATA ACQUISITION SYSTEM  A l a b o r a t o r y model of a one-machine i n f i n i t e bus  system has  been assembled.  This l a t t e r  the s t a t i s t i c a l / m o d e l  the d a t a r e q u i r e d i n p r o d u c i n g  The  data  apparatus. l a b o r a t o r y model c o n s i s t s of a f o u r p o l e d.c.  s i m u l a t i n g a prime mover and synchronous g e n e r a t o r . system connects  An  d r i v i n g a small s i x pole  three-phase  i n d u c t i v e three-phase b a l a n c e d  transmission  from the g e n e r a t o r  i n p u t of the amplidyne and  t e r m i n a l v o l t a g e on another  c o l l e c t observations  d a t a a c q u i s i t i o n system i s employed. measurement of a number of analog  The  a  feedback (Figure  allows  c e n t r a l p r o c e s s o r used i n the data (DEC)  be d e r i v e d by acquisition  PDP-8/L.  further details  f o r the  power system l a b o r a t o r y model, the data a c q u i s i t i o n hardware and software.  and  An o p t i c a l s h a f t - a n g l e encoder  f o l l o w i n g t h r e e s e c t i o n s supply  data a c q u i s i t i o n  3.3).  computerized  computer i n t e r f a c e  from which s h a f t speed may  i s a D i g i t a l Equipment C o r p o r a t i o n  circuit,  s i g n a l s through a m u l t i p l e x e r  converter.  m o n i t o r s the s h a f t p o s i t i o n , The  input c o i l  f o r the s t a t i s t i c a l m o d e l l i n g ,  an a n a l o g - t o - d i g i t a l (A/D)  Voltage  u s i n g an amplidyne i n the e x c i t e r  w i t h a r e f e r e n c e v o l t a g e on one  differentiation.  motor  the machine t e r m i n a l s to the i n f i n i t e bus.  r e g u l a t i o n i s accomplished  The  but  the s t a t i s t i c a l model.  f u n c t i o n r e q u i r e s c o n s i d e r a b l e measurement and  acquisition  To  /  T h i s model i s used not o n l y to check the  responses p r e d i c t e d by both Park's model and a l s o to supply  power  the  29 4.1  Power System Laboratory Model  Figure 4.1 i l l u s t r a t e s the laboratory  system  configuration.  Specifications f o r the d.c. motor, the synchronous generator and the amplidyne are displayed  i n Table 4.1.  Table 4.1 Laboratory Model Machine  Machine Synchronous Generator output:  exciter:  D.C. Motor  Amplidyne input:  output:  Specifications  Specification  5 KVA 220 v o l t s 13 amps 90% pf 60 Hz 1200 rpm 3.2 amp s 125 v o l t s 5.6 Kw 115 v o l t s 56 amps 850/1200 rpm  220/440 v o l t s 7.2/3.6 amps 3-phase - 60 Hz 1725 rpm 1.5 Kw 125 v o l t s 12 amps  D.C. MOTOR • (prime mover)  SYNCHRONOUS GENERATOR  AMPLIDYNE (voltage regulator)  TRANSMISSION LINE  INFINITE BUS (mains)  \AMJLT 230 v  2  115 v  i  S  1  1  'fd fd v  F i g u r e 4.1  i  i u  1  Laboratory  v  t  v  n  System C o n f i g u r a t i o n  Notes - broken l i n e s i n d i c a t e measurement points"". - f o r amplidyne d e t a i l see F i g u r e 3.3.  o  Amplidyne  In amplidyne, the  an attempt  r a t h e r than an a v a i l a b l e SCR  voltage regulator exciter c i r c u i t .  amplidyne-induced to  to minimize u n d e s i r e d system n o i s e an  filter  noise i s p l e n t i f u l  than s h o r t r i s e  e x c i t e r , was  However, i t appears  time peaks i n d u c e d by the SCR  time c o n s t a n t , T^,  to model that  and i s i n f a c t more d i f f i c u l t  The r e g u l a t o r - e x c i t e r c i r c u i t was l o n g open-loop  chosen  exciter.  designed with a  as seen i n T a b l e 4.2.  relatively  This  was  done because  s h o r t e r time c o n s t a n t s , of the magnitude of the g e n e r a t o r  open c i r c u i t  time c o n s t a n t , cause the e x c i t e r - g e n e r a t o r  combination  to be u n s t a b l e .  D.C.  Motor  An attempt  i s made to d e c r e a s e the i n h e r e n t damping o f the  d.c. motor, thus making i t more r e a l i s t i c a l l y model a prime mover, typically  w i t h low p.u. damping.  From the a n a l y s i s o f Chapter 3 i t  i s seen t h a t d.c. motor damping i s i n v e r s e l y p r o p o r t i o n a l t o the r e s i s t a n c e i n the armature -k D  dc  =  —  R  circuit  as d e s c r i b e d  by:  2 v  <  < - > 3  T h e r e f o r e a s e r i e s r e s i s t a n c e i s p l a c e d i n the armature increase i t s t o t a l the  circuit  r e s i s t a n c e , R.  to  Then 230 v o l t s d.c. i s a p p l i e d to  to m a i n t a i n a p p r o x i m a t e l y 115 v o l t s a c r o s s the  and thus m a i n t a i n c o r r e c t armature  current.  g e n e r a t o r i s c o n t r o l l e d by the armature Torque  circuit  29  armature  Input torque to the  c u r r e n t i n the d.c. motor.  i s then e v a l u a t e d a c c o r d i n g to e q u a t i o n (3.17), t h a t i s ,  T, dc  =  k i . v a  The advantage of t h i s c o n t r o l over u s i n g f i e l d ° &  c o n t r o l i s t h a t f l u c t u a t i o n s of f i e l d  current  c u r r e n t i n t u r n cause changes i n  armature c u r r e n t r e q u i r i n g t h a t b o t h be monitored to e v a l u a t e t o r q u e .  System  Parameters  Synchronous machine e l e c t r i c a l parameters r e q u i r e d f o r the t h e o r e t i c a l model based on Park's r e p r e s e n t a t i o n a r e determined u s i n g s t a n d a r d t e c h n i q u e s as o u t l i n e d i n Chapter 7 of IEEE T e s t Code [19]. T a b l e 4.2 are  shows the measured v a l u e s f o r the system parameters.  displayed  i n p.u. as w e l l as i n MKS  been used throughout t h i s t h e s i s . r.m.s., 8 amps r.m.s., and 377  u n i t s a l t h o u g h MKS  u n i t s have  The base v a l u e s used a r e : 125  volts  radians/sec.  The d.c. machine c o e f f i c i e n t L ^ fl  i s found from a p l o t of the  speed v o l t a g e c o e f f i c i e n t , co L ^, which i s d e f i n e d as open voltage/field  They  circuit  current [15].  The system i n e r t i a , J , i s e v a l u a t e d u s i n g the r e t a r d a t i o n test.  The f r i c t i o n damping term, F, i s e v a l u a t e d u s i n g e q u a t i o n  (3.19)  [15] P L F  = 2  af  i i a f  (3.19)  co  m  by o p e r a t i n g the synchronous g e n e r a t o r at no l o a d and d r i v i n g i t w i t h the  d.c. motor a t v a r i o u s speeds near synchronous A simplified  speed.  s i n g l e time c o n s t a n t model i s used f o r the  amplidyne and the time c o n s t a n t i s found by m o n i t o r i n g amplidyne output for  a step input.  The steady s t a t e g a i n o f each i n p u t i s a l s o found  from measured i n p u t and output v o l t a g e s near the e s t i m a t e d o p e r a t i n g point.  Table 4.2 Laboratory System Parameters  Parameter  per unit  MKS units DC machine parameter  * af  L  volts-sec amps rad  •-^5  2  Mechanical Parameters .62 kg - m  J  2  F  -.00081 co + .1637 m  J  i oule-sec rad  2  Synchronous Machine Parameters x x  d a d  ,  < T  do  9.03 ohms 5.47 ohms 2.00 ohms 0.282 sees 125 ohms 20.4 ohms  1.00 pu 0.60 pu 0.22 pu 13.85 pu 2.26 pu  Exciter-Regulator Parameters K K T  A1 A2 A  113 5.59 1.4 sees Transmission Line Parameters  r X  G B  .0313 ohms 1.77 ohms 0 0  0.0035 pu • 0.196 pu 0 0  4.2  Data A c q u i s i t i o n Hardware  Measurements  A number of the analog inputs required f i l t e r i n g before entering the A/D converter at the i n t e r f a c e . Active f i l t e r s have been selected to perform the low pass f i l t e r i n g which e n t a i l s attenuating noise at frequencies as low as 360 Hz without  i n t e r f e r i n g with  system responses or with the 60 Hz terminal waveforms.  The schematic  of the P h i l b r i c k f i l t e r s [20] used f o r these voltage inputs i s shown i n Figure 4.2.  F i l t e r performance curves are displayed i n Figure 4.3,  l a b e l l e d with the input signals to which each i s applied.  Table 4.3  displays a measure of r i p p l e on the f i l t e r e d signals which are submitted to the A/D converter.  R  Ar  6.  R  Ar  R  o v, o  bC/3  V,  o -  Figure 4.2  J + b(RCp) + (RCp)  2  Active F i l t e r Schematic  10  50  100  500  FREQUENCY F i g u r e 4.3  Filter  Table N o i s e a t A/D  Input  synch, machine f i e l d  volts  synch, machine f i e l d  current ( i ^ )  terminal voltage exciter  D.C.  ( £ ) v  d  reference voltage  motor armature  Performance  4.3 Input  1.3% 1.0% negligible  (v )  i n f i n i t e bus v o l t a g e  (HZ)  Percentage Noise  Signal  (u^)  (v ) current  negligible negligible  (i ) a  1000  1.1%  (ripple)  Direct voltage  current  i s measured by  detecting  drop a c r o s s a s e r i e s r e s i s t a n c e .  by means of a c u r r e n t  transformer.  and  amplifying  A l t e r n a t i n g current  Both d.c.  and  a.c.  the  i s measured  /  v o l t a g e s are  fed  / d i r e c t l y to the and  i n t e r f a c e with appropriate  or  amplification  filtering. A digital  s h a f t encoder i s f i x e d to the a c c e s s i b l e  synchronous machine s h a f t . factured  by  This  o p t i c a l encoder  (DRC  end  of  the  Model 29 manu-  Dynamics Research C o r p o r a t i o n ) o u t p u t s square p u l s e s as  shaft rotates. are  attenuation  Each r e v o l u t i o n of the  f e d to the  s h a f t produces 1500  p u l s e s which  i n t e r f a c e where they are used to measure the  p o s i t i o n with respect when d e s c r i b i n g  the  to some r e f e r e n c e .  the  rotor  F u r t h e r d e t a i l s are  supplied  interface.  Interface  The a DEC  p r i m a r i l y TTL  PDP-8/L computer, was  research  project.  i n t e r f a c e , d e s i g n e d to be constructed  p r i o r to the  However, t h i s u n t r i e d  could  proceed.  i s p r e s e n t e d , and  Firstly,  a short  channel  (DEC  (18  (D/A)  r e c t i f i e d before t h i s interface  then a l o o k i s taken i n t o the problems encountered.  #A811  w i t h 0.1%  connected) FET  computer c o n t r o l f u n c t i o n s analog  this  a number o f bugs  d e s c r i p t i o n of the  Much of t h i s i n t e r f a c e i s s t a n d a r d . converter  s t a r t of  i n t e r f a c e had  i n c l u d i n g d e s i g n d e f i c i e n c i e s which were to be project  compatible with  c o n v e r t e r s are  F.S.  accuracy) f o l l o w i n g  multiplexer using  It consists  The  RTL  A/D  multi-  to measure analog s i g n a l s .  analog s i g n a l s , four  provided.  a  of an  digital-to-  system d e s i g n e d  to  For  37  interpret  the o p t i c a l s h a f t - a n g l e encoder output  r e q u i r e s some a t t e n t i o n .  i s not s t a n d a r d  and  T h i s l o g i c a l l o w s measurement of the r o t o r  a n g l e i n e l e c t r i c a l u n i t s w h i l e the encoder i t s e l f  i s d e t e c t i n g angle /  /  i n mechanical  units.  The  result  f o r the s i x p o l e synchronous .machine  i s t h a t the 1/1500 r e s o l u t i o n f o r 2 IT mechanical  being monitored  p r o v i d e s o n l y 1/500  radians  r e s o l u t i o n f o r 2TT e l e c t r i c a l r a d i a n s .  To a c h i e v e the e l e c t r i c a l a n g l e measurement, the s h a f t a n g l e encoder output is  p u l s e s a r e counted  read by t r a n s f e r r i n g i t s c o n t e n t s  by a r e f e r e n c e p u l s e t r a i n . the counter  i s r e s e t to zero and  gated  the counter  of a mechancial r e v o l u t i o n ,  i t then counts  frequency, desired.  of the two  pulses  (1/3  to the read b u f f e r to determine s h a f t T h i s read r a t e may  a v a i l a b l e s h a f t encoders,  or by a c r y s t a l o s c i l l a t o r In t h i s p r o j e c t one  the q - a x i s and  500  In the meantime, the r e f e r e n c e  p o s i t i o n a t the time of the r e f e r e n c e p u l s e . s p e c i f i e d by one  be  by the mains  i f an a b s o l u t e r e f e r e n c e i s  s t a t e v a r i a b l e i s the a n g l e 6 between  the i n f i n i t e bus  (mains) v o l t a g e .  T h i s can be measured  d i r e c t l y by u s i n g a r e f e r e n c e p u l s e t r a i n of mains frequency the modulo-500 counter  which  to a read b u f f e r a t a r a t e s p e c i f i e d  At the s t a r t  r e v o l u t i o n ) when i t r e s e t s i t s e l f . p u l s e has  by a modulo-500 counter  contents  to  gate  i n t o the read b u f f e r .  In an attempt to m i n i m i z e c o n s t r u c t i o n c o s t , the  original  d e s i g n of the i n t e r f a c e i n c o r p o r a t e d a number of schemes to reduce hardware e x p e n d i t u r e . d e s i g n was  One  example of t h i s which l e d to a dangerous  i n the m u l t i p l e x e r channel  s e l e c t i o n decoding  Table 4.4(a) o u t l i n e s a p o r t i o n of the decoding p l e x e r channel  selection.  I t i s noted  logic.  scheme used f o r m u l t i -  t h a t i f i n a d v e r t e n t l y , through  38 Table 4.4 Multiplexer Channel Selection Decoding (a) O r i g i n a l Channel Selection Scheme  Word Addressing Channel Channel #  0  1  2  3  4  5  6  7  1  1  1  2  1  1  3  1  1  4  1  1  8  10  9  11  1  4050 1  5  1  1  6  1  1  7  1  1  8  1  1  Octal Address  4044 1  4022 1  1  4021 2050  1  2044 1  2022 1  2021  (b) Modified Channel Selection Scheme  Word Addressing Channel Channel #  0  1  2  3  4  5  6  7  8  9  10  11  0  Octal Address 0  1  1  2  1  3  1  4  1  5  1  6  1  1  7  1  1  1 2  1  3 4  1  5 6  1  7  program e r r o r or hardware f a u l t , b i t s 0 and 1, f o r example, a r e both h i g h , two circuit  channels may  be s e l e c t e d a t once.  The r e s u l t  i s a short  through two o f the FET s w i t c h e s and thus the d e s t r u c t i o n o f  / one o r more m u l t i p l e x e r c h a n n e l s .  As i t i s b e l i e v e d  d e t e c t i n g and r e p l a c i n g s h o r t e d FET's outweighed  t h a t the c o s t of  the g a i n o f reduced  hardware f o r decoding, a f a i l - s a f e d e c o d i n g scheme i s  implemented.  The r e s u l t i n g code which p r o v i d e s a c t i v a t i o n o f o n l y one  possible  c h a n n e l f o r every 12 b i t b i n a r y number i s o u t l i n e d i n T a b l e 4.4(b). Another minor  i n t e r f a c e change which  i s made f o r programming  convenience i s the i n s t a l l a t i o n o f s w i t c h e s on a l l the i n t e r r u p t Thus when programming w i t h i n t e r r u p t on, u n d e s i r e d i n t e r r u p t s easily disabled. s e r v i c i n g of  4.3  T h i s p r e v e n t s l o s s of computing  time i n unnecessary  Software  Software performance  i s d i c t a t e d by b o t h the l a b o r a t o r y  the s t a t i s t i c a l m o d e l l i n g scheme.  A c e r t a i n number o f  o b s e r v a t i o n s o f each v a r i a b l e i s r e q u i r e d f o r s t a t i s t i c a l For t h i s a n a l y s i s to produce further required  The  analysis.  an a c c u r a t e mathematical model, i t i s  t h a t the o b s e r v a t i o n s be spaced c l o s e enough i n  time to f o l l o w the f a s t e s t represented.  are  interrupts.  Data A c q u i s i t i o n  system and  lines.  l a b o r a t o r y system response d e s i r e d  to be  form of the mathematical model f u r t h e r d e f i n e s  s o f t w a r e by s p e c i f y i n g which v a r i a b l e s must be  monitored.  the  Data  Storage  When working w i t h a b a s i c 4-K speed s t o r a g e  of  devices, data storage  observations  second and  /  For the l a b o r a t o r y  of i n t e r e s t were expected  t h e r e f o r e i t was  decided  to s t o r e e i g h t i n p u t s i g n a l s per sampling  samples per second. seen t h a t at one  With o n l y 3500 s t o r a g e  7.5  p r o c e d u r e i s d e s i r e d f o r data packing  time g i v i n g  locations available i t i s  Thus a more i n t e n s e  A problem a r i s e s because f o r a 10 b i t A/D  d e v i a t i o n s may  be s t o r e d .  checking  first this  output  d e v i a t i o n when o p e r a t i n g at  By working at l e s s than f u l l  comfortably  be  packing  t h e r e a f t e r o n l y s t o r e s the d e v i a t i o n from  s c a l e of - 5 v o l t s i n p u t .  packing  only  storage.  the 6 b i t p a c k i n g a l l o w s o n l y 6.25%  r e q u i r e s software  480  scheme adopted uses a f u l l word to s t o r e the  v a l u e of each i n p u t , and operating point.  seconds.  than  Furthermore, i t was  o b s e r v a t i o n per word s t o r a g e , the system may  monitored f o r a p p r o x i m a t e l y  The  to be g r e a t e r  to sample a l l system v a r i a b l e s  once each p e r i o d of the mains v o l t a g e waveform. decided  high  i s a problem when a l a r g e number  of many v a r i a b l e s i s r e q u i r e d .  system, time c o n s t a n t s 1/10  memory computer w i t h o u t  scale,  full 10%  This small allowable d e v i a t i o n  f o r o v e r f l o w s when p a c k i n g  deviations.  A  scheme which a l l o w s l a r g e r d e v i a t i o n s but which i s more  s u s c e p t i b l e to e r r o r s i s to s t o r e the d i f f e r e n c e s between s u c c e s s i v e readings. all  those  Due  to the p o s s i b i l i t i e s  of e r r o r ( i f one  f o l l o w i n g are wrong) i n the l a t t e r  from an o p e r a t i n g p o i n t were s t o r e d . are acceptable  except  The  v a l u e i s wrong,  scheme, the d e v i a t i o n s  small allowable deviations  i n d e r i v i n g speed from angle measurements where  l i m i t e d r e s o l u t i o n of the s h a f t - a n g l e encoder p r e s e n t s  a problem.  By  s t o r i n g two p o i n t s may seconds. for  o b s e r v a t i o n s per computer word, the system's e i g h t measurement be  i n t e r r o g a t e d each c y c l e of the mains f o r a p p r o x i m a t e l y  This provides  a c q u i s i t i o n of an adequate number of  observations  the s t a t i s t i c a l a n a l y s i s .  Sampling a.c.  Signals  Without r e c o n s t r u c t i n g the waveform sampled t h e r e a r e methods of o b t a i n i n g the magnitude of a.c. waveforms. the s i g n a l and and  r e c o r d one  sample, or to c o n t i n u o u s l y  s t o r e the peak v a l u e .  As  time c o n s t a n t s , the a.c. values  stored.  v a l u e may  filter  signal acceptable  l o n g e r than the  system  t h e i r peak  p o t e n t i a l measurement e r r o r as the peak  occur between samples.  The  d a t a a c q u i s i t i o n program a l l o w s  t h r e e d e s i r e d a.c. waveforms t o be  samples.  sample the  s i g n a l s are c o n t i n u o u s l y sampled and  T h i s too has  two  They a r e to  f i l t e r i n g a 60 Hz waveform to an  r i p p l e causes measurement response time c o n s t a n t s  for  15  This r e s u l t s i n approximately  sampled w i t h 245 0.11%  ysec between  e r r o r i n d e t e c t i n g the  peak v a l u e s , which i s i n the range of the 0.1%  e r r o r i n the A/D  verter.  Many more a.c.  sampling  e r r o r s of the magnitude of the n o i s e r i p p l e on the  con-  s i g n a l s c o u l d be c r o s s - s e c t i o n e d b e f o r e o b t a i n i n g signals.  PDP-8 Program O u t l i n e  The Appendix 4A,  PDP-8/L program, which i s o u t l i n e d by a f l o w c h a r t i n begins  by r e a d i n g a grounded m u l t i p l e x e r channel  s t o r i n g the r e s u l t i n g A/D an a c c u r a t e l y known d.c. These o f f s e t and readings  and  o f f s e t read. supply  Another channel  i s read and  the A/D  and  connected  output  stored.  c a l i b r a t i o n v a l u e s l a t e r a l l o w f o r c o r r e c t i o n of  f o r conversion  to  A/D  of s t o r e d b i n a r y numbers back to v o l t a g e  l e v e l s on t h e system.  The program then s t o r e s ah  v a r i a b l e thus d e s c r i b i n g  A/D r e a d i n g f o r each  the system o p e r a t i n g p o i n t .  I t then b e g i n s t o  r e c o r d o b s e r v a t i o n s and pack i n h a l f words t h e i r d e v i a t i o n from t h e appropriate operating point value.  A computer i n t e r r u p t a t each  of t h e m a i n s - v o l t a g e waveform then i n i t i a t e s  the f o l l o w i n g  period  procedure.  Pack t h e d e v i a t i o n s from o p e r a t i n g p o i n t f o r t h e p r e v i o u s sampling i n t e r v a l i n the computer.  Sample r o t o r p o s i t i o n and then a l l d.c.  variables i n succession.  Sample a l l a.c. v a r i a b l e s s u c c e s s i v e l y and  continuously u n t i l  the next i n t e r r u p t , s t o r i n g the peak magnitude o f each  a.c. waveform.  As t h e d e v i a t i o n s from o p e r a t i n g p o i n t a r e s t o r e d , they a r e checked bits.  t o see whether o r n o t they may i n f a c t be packed  into s i x binary  When an o v e r f l o w i n p a c k i n g t h e d e v i a t i o n from o p e r a t i n g v a l u e  i n a h a l f word o c c u r s an o v e r f l o w c o u n t e r i s incremented  and the program  w i l l h a l t a t some p r e s e t a l l o w a b l e number o f o v e r f l o w s .  I f the o v e r -  flow i s p o s i t i v e ,  the l a r g e s t p o s s i b l e p o s i t i v e d e v i a t i o n i s s t o r e d ,  when n e g a t i v e , t h e most n e g a t i v e d e v i a t i o n i s s t o r e d .  T h i s minimizes  the e r r o r r e s u l t i n g from u s i n g t h e o v e r f l o w e d v a l u e as a v a l i d v a t i o n i n the s t a t i s t i c a l a n a l y s i s . the d a t a s t o r e d i s punched on paper f o r each  A f t e r t h e computer memory i s f i l l e d tape u s i n g a s i n g l e odd p a r i t y b i t  frame.  O f f - l i n e Data  The  Handling  i n f o r m a t i o n on the paper  v e r t e d onto magnetic Model 67.  obser-  tape from the PDP-8 i s con-  tape f o r use on the computer c e n t e r IBM 360/  T h i s d a t a i s then supplemented  of a m p l i f i c a t i o n s  w i t h measured a t t e n u a t i o n s  e x t e r n a l t o the i n t e r f a c e f o r each v a r i a b l e and  with the accurate value of the d.c. c a l i b r a t i o n voltage.  A Fortran  program uses t h i s supplementary information and the paper tape information to reconstruct a l l system voltage and current values monitored.  The r e s u l t i s 861 observations of eight system variables  for use i n the s t a t i s t i c a l modelling scheme.  These observations  are  stored on magnetic tape, readily accessible for analysis or output.  44  5.  PERFORMANCE OF THE  The proposed s t a t i s t i c a l it  statistically  STATISTICAL  model i s i n v e s t i g a t e d by  u s i n g the d a t a a c q u i r e d .  statistical  the performance o f the  the l a b o r a t o r y system.  the l a b o r a t o r y system o n l y .  g a t i o n of some a s p e c t s  of the model may  However, f u r t h e r i n v e s t i -  be more e a s i l y performed i n a  u s i n g the proven t h e o r e t i c a l model.  comparison o f mathematical model responses may whether time  or not the s t a t i s t i c a l  For example, be used to  model has i d e n t i f i e d  a  computer  decide  a particular  system  constant.  S e c t i o n 5.1 and how  d e s c r i b e s the system o p e r a t i n g p o i n t s i n v e s t i g a t e d  the data i s a c q u i r e d  important  f o r the s t a t i s t i c a l  analysis.  points require consideration before beginning  of the s t a t i s t i c a l  5.1  Whether the  model i s a good p r e d i c t o r or not i s e a s i l y determined by  comparison w i t h  computer  checking  A l s o i t s performance as a  p r e d i c t o r i s i n v e s t i g a t e d by comparison w i t h t h e o r e t i c a l model as w e l l as w i t h  MODEL  These  the i n v e s t i g a t i o n  model.  System Data f o r S t a t i s t i c a l Model D e r i v a t i o n  The l a b o r a t o r y power system i s run at t h r e e operating points.  different  I t s o p e r a t i o n i s monitored u s i n g the PDP-8/L  and a s s o c i a t e d i n t e r f a c e d i s c u s s e d  i n Chapter 4.  At each  computer  operating  p o i n t responses to s t e p s i n each o f u,, i and v are monitored to be 1 a o used as comparisons f o r the responses of the mathematical models. system response i s r e c o r d e d i n p u t s u,, i  and v .  Also  f o r s m a l l random v a r i a t i o n s i n the t h r e e  This i s necessary  to o b t a i n the d a t a  required  to  estimate  model. these  the c o e f f i c i e n t s i n the e q u a t i o n s  d e f i n i n g the  statistical  A p p l i e d p e r t u r b a t i o n s a r e r e q u i r e d a t the t h r e e i n p u t s because s i g n a l s a r e g e n e r a l l y v e r y w e l l r e g u l a t e d and  the amount of v a r i a t i o n n e c e s s a r y  thus do not  i n the s t a t i s t i c a l  provide  analysis.  The  p e r t u r b a t i o n s a r e i n t r o d u c e d on the c o n t r o l i n p u t s i n d e p e n d e n t l y randomly. values  Simultaneously,  the system response i s monitored by  recording  of the system s t a t e v a r i a b l e s at r e g u l a r time i n t e r v a l s .  s e t s of s t a t e v a r i a b l e o b s e r v a t i o n s  ± , then to u^ and v ,  t h r e e t e s t runs.  Hard copy of the data taken f o r each t e s t  a  Q  and  on paper tape by  the PDP-8/L computer and  IBM  computer.  360/Model 67  be  T h i s data  The  &  and  v  i s produced using  360  on cards  Thus the system v o l t a g e  r e c o n s t r u c t e d w i t h i n the IBM  360  current values  they a r e r e q u i r e d f o r a n a l y s i s or  5.1.  an  are  f o r any  to  and sampling  are stored  on  plotting.  t h r e e o p e r a t i n g p o i n t s f o r which d a t a i s a c q u i r e d  summarized i n T a b l e  in  Q  attenuators,  i s s u p p l i e d to the IBM  These r e c o n s t r u c t e d v o l t a g e and  magnetic tape u n t i l  to i  i s then p r o c e s s e d  a m p l i f i e r s and  supplement the paper tape i n f o r m a t i o n . c u r r e n t v a l u e s may  finally  are  Gains e x t e r n a l to the PDP-8 i n t e r f a c e , such  as r a t i o of p o t e n t i a l t r a n s f o r m e r s , measured u s i n g meters.  Separate  are c o l l e c t e d as p e r t u r b a t i o n s  a p p l i e d to u^ and  time.  and  are  Table 5.1 System Operating Points Investigated /  Operating  ,/'  Point fc  B  A •  Variable  Units  P  watts  Q  vars  953  V  volts  125  125  amps  1.75  1.25  2.0  degrees  29.5  45.4  3.6  volts  36.8  26.6  42.8  6.49  6.55  1230  405  1635 - 381  /  1660 124  t  V 6  V  fd  U  volts  l  .  amps  i  6.5 15.5  18.9  9.4  109  133  94  a volts  V  o In Table 5.1 P, Q, and v  t  completely  describe the i n i t i a l  conditions of the system for the t h e o r e t i c a l representation. values of i ^ , , fd  The  6, v-, and v are calculated from the t h e o r e t i c a l fd o  model i n i t i a l conditions as a preliminary check on the t h e o r e t i c a l model.  Calculated and measured i n i t i a l conditions are shown i n  Table 5.2.  I t must be noted that meter accuracy  better than 2 - 3 % .  i s generally not  47  Table  5.2  Steady S t a t e V a l u e s  Steady S t a t e V a l u e s Variable  Units  *fd 6 V  fd  V  t  V  5.2  Experimental  1.172  5.1%  degrees  46.0  43.4  5.9%  volts  26.85  23.91  11%  volts  126  125  .80%  volts  133  132  .76%  I n v e s t i g a t i o n s of the R e g r e s s i o n Model  important s t a t i s t i c a l  performed u s i n g r e s i d u a l p l o t s .  violated. the  One  i n v e s t i g a t i o n s o f the model are i s a check of whether or not the  inherent i n regression analysis  (Chapter 2) have been  The o t h e r i s a t e s t o f l a c k of f i t , i n d i c a t i n g how  particular  Difference  1.235  Statistical  assumptions  %  amps  o  Two  Theoretical  form o f e q u a t i o n d e s c r i b e s the d a t a to which  well  i t is fitted.  The p h i l o s o p h y used i n i n t e r p r e t i n g r e s i d u a l p l o t s i s s i m i l a r to t h a t e v i d e n t i n t e s t i n g hypotheses. assumption  i s v i o l a t e d , one c o n c l u d e s the assumption  while i f a plot that  That i s , i f a p l o t i n d i c a t e s  an  i s violated,  i n d i c a t e s the assumption h o l d s , one c o n c l u d e s o n l y  the assumption has not been v i o l a t e d .  R e s i d u a l p l o t s then g i v e  e v i d e n c e o f l a c k of f i t and v i o l a t i o n o f assumptions, but do not f i r m that  the model i s p e r f e c t l y adequate  been c o m p l e t e l y s a t i s f i e d .  or t h a t the assumptions  conhave  48  The  normality  assumption f o r t h e r e s i d u a l s ( e ^ , i = 1, 2, ...n)  i s i n v e s t i g a t e d u s i n g an o v e r a l l p l o t o f r e s i d u a l s as shown f o r a t y p i c a l case i n F i g u r e 5.1. e^,  As w e l l as an o v e r a l l n o r m a l i t y  i t i s a l s o r e q u i r e d t h a t the  be n o r m a l l y  i n s t a n t o f time or over any i n t e r v a l o f time.  distributed  with  a t any  Rather than u s i n g a  number o f n o r m a l i t y p l o t s , t h e d i s t r i b u t i o n o f as shown i n F i g u r e 5.2.  distribution for  with  time i s p l o t t e d  Since t h i s p l o t i n d i c a t e s uniform  distribution  time, then the r e s u l t s o f the o v e r a l l p l o t may be assumed t o h o l d  d u r i n g any i n t e r v a l o f time.  I t i s noted t h a t t h e d i s t r i b u t i o n o f e's i n t h i s t h e s i s depends on how the p e r t u r b a t i o n s o f u^, i  &  and V  q  are administered.  Here  the i n p u t s a r e v a r i e d manually and an attempt i s made to induce random fluctuations.  I t i s more d e s i r a b l e to have c o n t r o l l e d s i g n a l s on the  system i n p u t s so t h a t r e g u l a t e d p e r t u r b a t i o n s c o u l d be thus p r o d u c i n g  a b e t t e r gaussian  error distribution.  administered, A l s o , poor r e s o l u t i o n  i n speed d e t e c t i o n r e q u i r e s l a r g e r than d e s i r a b l e p e r t u r b a t i o n s  from nominal  values.  The plotting  assumption o f c o n s t a n t  against  variance  i s r e a d i l y checked by  the p r e d i c t e d v a l u e as shown i n F i g u r e 5.3.  Using a  s i m i l a r p l o t , l a c k o f f i t i s i n v e s t i g a t e d by p l o t t i n g r e s i d u a l s f o r each regression equation against  a g a i n s t each o f the independent v a r i a b l e s as w e l l as  the p r e d i c t e d v a l u e .  An example i s shown i n F i g u r e 5.4 where  l a c k o f f i t o f the v,. term i n the e q u a t i o n c  checked.  I t i s noted t h a t a l t h o u g h  be v i o l a t e d , v a r i a n c e u s i n g these  f o r v . ' (Equation fd  the n o r m a l i t y  3.60) i s  assumption appears to  and l a c k o f f i t may be i n v e s t i g a t e d q u a l i t a t i v e l y  residual plots.  >u z  X  LU UJ  ct:  X  X  RESIDUALS F i g u r e 5.1  F i g u r e 5.2  Overall  A  (ifcj — i f )  Plot  d  of  + £j  Residuals  D i s t r i b u t i o n of R e s i d u a l s w i t h  Time  50  -o—< I  "pCD  * *« * &  cn _j cr  X  o o -•  =J a  —' CD cn UJ  i  ct a i  CD  ^  1  1.2  —i  1  1 —  1.3 1.4 A 1.5 PREDICTED VRLUE ( i )  1.6  f d  F i g u r e 5.3  Example o f V a r i a n c e  of Residuals  -a m  ™H  < f  2 °  j> co i cr  ZD a CDCO LU  Cr:  LO CM CD I  X  O LO  122  124  126 INDEPENDENT  F i g u r e 5.4  Example o f Test  128  130  V A R I A B L E (v ) t  f o r Lack o f F i t  132  5.3  Model Responses to Step Inputs  The  performance o f t h e t h e o r e t i c a l and s t a t i s t i c a l  were compared w i t h  each o t h e r and w i t h  models  the l a b o r a t o r y system response.  For t h e comparisons, step i n p u t s were a p p l i e d to the r e g u l a t o r  reference  v o l t a g e , u , t h e d.c. motor armature c u r r e n t , i , and the i n f i n i t e bus i a v o l t a g e , v , as o u t l i n e d i n s e c t i o n 5.1.  E f f e c t of Operating  Point  From F i g u r e s 5.5, 5.6 and 5.7 i t i s e v i d e n t statistical capable  model y i e l d s a good steady  that the  s t a t e response, but i s not  o f p r e d i c t i n g t h e most r a p i d f l u c t u a t i o n s i n the system's dynamic  behaviour.  I t does, however, p r o v i d e  system's dynamic response.  a c l o s e a p p r o x i m a t i o n to the  By comparing F i g u r e 5.6 and 5.7(a) i t i s  o b s e r v e d t h a t the s t a t i s t i c a l models g i v e s i m i l a r p r e d i c t i o n o f response whether o r n o t they a r e used a t t h e same o p e r a t i o n p o i n t a t which  they  are d e r i v e d .  system  T h i s i n d i c a t e s t h a t the model s u f f i c i e n t l y e x p l a i n s  nonlinearities. for  T h i s i s f u r t h e r emphasized by comparing the c o e f f i c i e n t s  a t y p i c a l case  (equation  the most s i g n i f i c a n t  3.57) g i v e n i n Table  coefficients  5.3.  I t i s seen t h a t  ( b ^ and b-^ i n t h i s case) v a r y by  o n l y s m a l l amounts o f the o p e r a t i n g p o i n t changes.  F i g u r e 5.5 d i s p l a y s  a response o f a model d e r i v e d from d a t a a t a l l t h r e e o p e r a t i n g p o i n t s . This l e s s accurate with  t h e steady  p r e d i c t o r y i e l d s an a c c e p t a b l e  s t a t e response o b t a i n e d  response when compared  u s i n g Park's  formulation.  1.5-1  i i  i  /  /  laboratory statistical theoretical  2  T  4 TIME  6 (5EC5)  Figure 5.5 Response to step i n u-^ at operating point B - S t a t i s t i c a l model from operating points A, B and C combined  1 .4-  SyrTrw"  T i — r n — r a  -  CL  laboratory statistical theoretical  a  1.2-  1.1 0  4 TIME  6 (SECS)  8  Figure 5.6 Response to step i n u^ t operating point B - S t a t i s t i c a l model from operating point A a  10  53  Table 5.3 Example of V a r i a t i o n i n C o e f f i c i e n t s  Coefficients Operating Point  R  2 b  ll  b  12  b  13  b  14  B  .9930  .932  .0031  C  .9917  .904  .0043  .0028  .9995  .929  .0033  .0034  A+B+C  .0014  The set of plots shown i n Figure 5.7 indicate the predicted responses of a l l the measurable state variables f o r a step i n u^ using a s t a t i s t i c a l model with c o e f f i c i e n t s estimated at the same operating point as the step i s applied. point B.  The model used i s found at operating  The c o e f f i c i e n t s describing this model (equations (3.57) to  (3.60) and (3.64)), excluding those c o e f f i c i e n t s which are non-significant to a 5% l e v e l , b  1 Q  = .00084 ,  b  2 0  = 0.0 , b  bJ n Q3  Q  are as follows. b 2 1  = -.039 b .v->? , , u b  b J  4 Q  50  n  = .932 ,  = 1.0 , 3 3 1 1  3 5  b  2 2  b  ±2  = .00311 ,  b  1 3  = .00136 ,  = -.156 , b .x_m , u  3 3 2 2  = 0.0 b , b /. = 0.0 u.u , , u ^ = 0.0 w.u , 3  = -.000478 , b, .  0  3  .0000325 , b  3 7  b  = .959 ,  b  4 2  = .971 , b ^ = -.0380  = 2548. ,  b = .307 , 51  b  5 2  = -1334. ,  q i  b  5 5  = 0.0  = 1.0  = -.357 ,  4 1  b  = -.00481  b „ = .359 , b,.^ = -14.7 , 54 53  = 96.7 .  This model was found using a 60 Hz sampling  rate by combining three sets of  data runs at the same operating point (B) as outlined i n section 5.1. Similar s t a t i s t i c a l model performance was observed and v . o  using step inputs of i  54  laboratory statistical theoretical  ~i r~ 4 6 TIME (SECS) r  8  Figure 5 Responses to step i n u^ at operating point B S t a t i s t i c a l model from operating point B  10  55  0.005CJ LU CD  a cx cn  0  a UJ LU CL CD  laboratory statistical theoretical  0.005-  -0.010  2  4 6 TIME C5EC5)  8  10  Figure 5.7(c)  _32 H CO (— -J  > a a  26  statistical theoretical  23 0 Figure 5.7(d)  2  "~l  T™  4 6 TIME tSECS)  8  10  56  133-  T  IME  (SECS)  Figure 5.7(e)  E f f e c t of Sampling  Rate  Figures 5.8 and 5.9 indicate the e f f e c t of changes of sampling i n t e r v a l when obtaining data to form a model.  The s t a t i s t i c a l model used  to produce the response i n Figure 5.7(a) used samples collected at each period of the mains.  For the model producing Figure 5.8 the samples  were taken every second period and i n Figure 5.9 every f i f t h period. For t h i s range of sampling frequency, change i n sampling rate had n e g l i g i b l e effect on the model produced.  In forming the various s t a t i s t i c a l models from d i f f e r e n t sets of data, note was  taken of changes i n the value of the multiple  regression c o e f f i c i e n t , R (see section 2.3).  It i s noted that as more o  samples were used to estimate the c o e f f i c i e n t s i n the model, the R  • 57  4 TIME  6 (5EC5)  8  F i g u r e 5.8  Response w i t h s t a t i s t i c a l model found u s i n g 0.033 second sampling i n t e r v a l  F i g u r e 5.9  Response w i t h s t a t i s t i c a l model found u s i n g 0.083 second sampling i n t e r v a l  10  value tended to increase (e.g. Table 5.3). describing speed, R  2  However, for equation  (3.58)  was only s l i g h t l y s i g n i f i c a n t when sampling at  7  2  each period of the mains (.016 sec), but for most sets of data this R /  increased ten fold when sampling every second period (.033 sec). In the modelling scheme, speed i s found by d i f f e r e n t i a t i n g angle ( i . e . to(k) = 6(k+l)-6(k) ) which i s monitored by the shaft encoder and y i e l d s poor resolution a f t e r conversion to e l e c t r i c a l u n i t s .  The  e f f e c t i s that small deviations with poor resolution produce a set of data which the p a r t i c u l a r form of equation chosen does not adequately describe.  In practice, though, as long as the multiple regression  c o e f f i c i e n t i s s i g n i f i c a n t , say by four times the tabulated value, an order of magnitude increase i n R  does not appreciably a f f e c t the model  derived. Generally, then, i t i s found that the derived s t a t i s t i c a l model y i e l d s good steady state prediction, but i t responds slower than the system when predicting dynamics.  Discrepancies observed i n the  t h e o r e t i c a l model performance at steady state are within meter error tolerances as the t h e o r e t i c a l and s t a t i s t i c a l models define t h e i r operating point using a d i f f e r e n t set of variables and therefore d i f f e r e n t meters.  Lack of resolution i n speed measurement (see  Figure 5.7(c) created the major problem i n the s t a t i s t i c a l  modelling.  59  6.  CONCLUSION  An: i n v e s t i g a t i o n has been undertaken-to a s c e r t a i n how r e a d i l y a power system lends i t s e l f to s t a t i s t i c a l v a r i a b l e model has been d e r i v e d .  modelling.  T h i s model i s l i n e a r i n i t s c o e f f i c i e n t s  which are e v a l u a t e d by the l e a s t squares f i t t i n g analysisof  for  The form o f the s t a t i s t i c a l  synchronous machine dynamics, w i t h  v  t  the unmeasurable s t a t e d e s c r i b i n g As the e x p r e s s i o n  as w e l l as i n the s t a t e s , an  to a l l o w p r e d i c t i o n o f v  and thus  may be c a l c u l a t e d a t any time. existing interface  data a c q u i s i t i o n .  eliminate undesired interface. by  formulation  2  An for  of regression  model i s based on Park's  i s n o n l i n e a r i n the c o e f f i c i e n t s  a u x i l i a r y e q u a t i o n was i n t r o d u c e d v  technique  f l u x , i\> , r e p l a c e d by the f i e l d c u r r e n t , i ^ - , . F rd  field  A nonlinear state  to a PDP-8 computer was m o d i f i e d and used  S i g n a l c o n d i t i o n i n g networks were d e s i g n e d to  r i p p l e and t o o b t a i n r e q u i r e d s i g n a l l e v e l s f o r the  The software was d e s i g n e d  t o pack o b s e r v a t i o n s  s t o r i n g d e v i a t i o n s from an o p e r a t i n g p o i n t , thus  i n h a l f words  a l l o w i n g an adequate  number o f o b s e r v a t i o n s  to be s t o r e d i n the minimal memory a v a i l a b l e .  Data h a n d l i n g  was a l s o developed  software  was w r i t t e n to i n t e r p r e t r e c o n s t r u c t i n g the v a l u e s magnetic tape w i t h transform  the logged  o f system s i g n a l l e v e l s ,  a p p r o p r i a t e headings.  a n a l y s i s i n p u t ; to c a t a l o g u e  the s t a t i s t i c a l  and s t o r i n g them on  a t each o b s e r v a t i o n f o r r e g r e s s i o n statistical  r e s u l t s on  r e s i d u a l s ; and to s o l v e and p l o t  model responses and the t h e o r e t i c a l model responses as  w e l l as the system responses to v a r i o u s i n p u t s . has  tape,  Other programs were w r i t t e n to  and s t o r e i n t e r m e d i a t e  to c a l c u l a t e and p l o t  A program  d a t a , t a k i n g i t from paper  the data by combining v a l u e s  magnetic tape;  f o r the IBM 360.  been r e t a i n e d and, i n c o n j u n c t i o n w i t h  A l l the data  the d a t a h a n d l i n g  acquired routines  60 d e v e l o p e d , t h i s d a t a may p r o v i d e a s t a r t i n g p o i n t similar  projects. The  of  s t a t i s t i c a l models i d e n t i f i e d produce v e r y a c c u r a t e  the system steady s t a t e response.  operating vided  points,  as those produced from d a t a a t one o p e r a t i n g  point  pro-  When used to p r e d i c t dynamic p e r -  formance o f the system to s t e p i n p u t s ,  the s t a t i s t i c a l model f a i l e d t o  the f a s t e s t system f l u c t u a t i o n s .  f o r the new steady s t a t e o p e r a t i n g  prediction  The models were n o t s e n s i t i v e to  a c c u r a t e p r e d i c t i o n a t another.  predict  f o r f u r t h e r work on  point  I t d i d , however, p r e d i c t  the time  to be reached and i s t h e r e f o r e  a good dynamic model f o r many p r a c t i c a l a p p l i c a t i o n s . For  further research,  the f o l l o w i n g  More r a p i d sampling o f system v a r i a b l e s dynamic p r e d i c t i o n .  improvements a r e s u g g e s t e d .  i s required  More r e s o l u t i o n i s r e q u i r e d  to provide  better  i n the speed measurement.  T h i s may be a c h i e v e d by s e n s i n g speed d i r e c t l y o r by h a v i n g g r e a t e r i n t h e a n g l e measurements. residuals inputs.  i t i s desirable  accuracy  To a c h i e v e a b e t t e r normal d i s t r i b u t i o n o f to p e r t u r b  the system u s i n g  c o n t r o l l e d random  However, the apparent v i o l a t i o n o f the n o r m a l i t y assumption i n  t h i s work d i d not appear to have a s i g n i f i c a n t e f f e c t on the r e s u l t s . e x t e n s i o n o f t h i s work to more c l o s e l y t r a c k system dynamics, in  an o n - l i n e  tension  environment, would be o f p r a c t i c a l i n t e r e s t .  preferrably  The ex-  o f the m o d e l l i n g scheme t o multimachine systems would p r o v i d e a  valuable interest.  contribution  An  as the scheme would then be o f g r e a t e r p r a c t i c a l  APPENDIX 3A  A t h i r d order state variable model may be derived from the  /  s i m p l i f i e d Park's equations (3.1) to (3.4) along with the mechanical  /  equations (3.5) and (3.6). x , ad  =  v., fd  w R  d  O  ^  *  _  (1 + T' p) do^ 7  F  _f£ % F  d  Equation (3.3) may be rearranged-as [12]  _ (  R  (1 + T'p) cT_  co o  (1 + T' p) do  -< — %  doP>  1 +  x , _ad  T  v  + o< u  (3.3)  ± d  ). 1 +  T  (3A.1)  doP>  Solving f o r A and B and c o l l e c t i n g terms gives V  ^  =  where  FR  v  ^ d  -  A =  p R  V  F  +  s (3A.2) 0  P < x (T' - T ) } i d  +  and  A =  v  d  (3A.3)  doP>  T  x  _£i v F  (3A.4)  R  A =  It can be shown that [12] x' d V  or  F  v  P  =  =  p  {  T  T* _r_ T ' do  x  d  allowing v^ to be written as: F  do t FR " ( d " d ^ d 3 v  pijjp + v  p R  x  x  }  + FR V  < -> 3A  5  .  (3A.6)  Thus equation (3.7) i s found from (3A.6) P^p  =  V  F " FR V  ( 3  -  7 )  where <F J  "  T  do  [ V  FR " < d " d> ^ x  X  ]  < ' 3 A  7 )  Substituting (3A.4) into(3A.6) gives R  v  =  M  P ( *  F  V  — )  F  K  +  VT3  C—>  ad  f  =  d  gives  p  JL. X  V  and  *fd =  8  equation  Vfd  +  ijj  3A  ad  which when compared to the f i e l d voltage V  < ->  (3A  -  9)  (3A.10)  £d  ad  FR  =  i ^  , (3A.11)  d  ad which are useful relationships when r e f e r r i n g state equation variables to actual system quantities. The electro-mechanical relationship i n equation (3.8) i s derived from the expression f o r the rotor angle i n (3.6)  9  =  Differentiating  or  p6  =  p6  =  r  % t + 6.  (3.6)  gives co + 6 o co Aco o  (3A.12)  P  (3A.13)  where Aco  p9  - w CO  o  o  »  co  — co  ~o  to  i s a per unit change i n speed.  o  (3A.14)  APPENDIX 3B  For  a one-machine i n f i n i t e bus system, the transformation  of transmission system quantities to the d-q coordinate system i s straightforward.  A s i m p l i f i c a t i o n from Vongsuriya's derivation [12]  exists because the i n f i n i t e bus voltage corresponds to the rotating reference and therefore at steady state i s at the angle 6 from the q-axis.  Projections onto the d and q axes are then a l l that i s  required to express V  i n terms of Park's system.  q  v , + iv od oq  =  J  That i s ,  v sin6 + i v cos6. o o  (3B.1)  From the system diagram i n Figure 3.4  [ 1 + (r + jx) (G + jB) ] v  =  t  V  + (r + jx) i  q  (3B.2)  where i n Park's system v  t  i v  o  =  v  + jv q  (3B.3)  =  i , + j i d q  (3B.4)  =  v ,+ jv . od oq '  (3B.5)  d  J  J  Substituting (3B.1) f o r V  q  i n (3B.2) gives V  i n Park's system, and  q  equation (3B.3) and (3B.4) into (3B.2) puts v  t  and i i n Park's  system thus giving the system equation [ 1 + (r + jx)(G + jB) ] [ v + j v ] d  q  + j v c o s S + (r + j x ) ( i , + j i ) 0  =  v sin6 Q  (3B.6)  T.  which by expanding and separating r e a l and imaginary parts can be written as equation (3.34).  APPENDIX 3C  Equation (3.39), which expresses d variables only, may be developed as follows. expressions f o r v^, v , i ^ and i q  equations (3.1),  and \JJ i n terms of state q In (3.34) substitute  i n terms of fluxes, that i s , use  (3.2) f o r v^ and v^ and use equation (3.14) f o r i ^  and i q  *d  =  1 1 r - ^ ^ ^ o ^ d do d d  O d )  x  (3C.2)  -10 X  The equation r e s u l t i n g from these substitutions i s  to  q °  k  l  k  •1  v sin6 o  2  K  l  K  2 x  + V o  -k  2  k  1  v cos<5 o  -K  2  K  ±  1 x.  d do T  x^ (3C.3)  65  APPENDIX 4 A  Flowchart f o r Data A c q u i s i t i o n Program  ( START ^  Perform required f l a g clearing and software i n i t i a l i z a t i o n e.g. i n i t i a l i z e counters e t c .  Read and store c a l i b r a t i o n voltage (channel # 0)  _ J  Read and store A/D offset voltage (channel // 1)  Turn interrupt ON Wait for interrupt  INTERRUPT (at beginning of each synchronous machine e l e c t r i c a l cycle)  X store machine angle  66  BEFORE  Pack deviations from operating point f o r previous c y c l e i n t o h a l f words i n storage checking f o r overflows i n packing.  Take nominal values from temporary storage and place at beginning of store b u f f e r .  1 Read and temporarily store DC values.  Read AC q u a n t i t i e s and temporarily store maximum values. Sample continuously u n t i l i n t e r r u p t at beginning of next cycle.  JL  Following nominal values store c a l i b r a t i o n v o l t a g e , o f f s e t s before and a f t e r run and it of overflows i n packing.  Punch out storage b u f f e r on paper tape with h a l f word per tape frame plus p a r i t y b i t f o r odd p a r i t y .  To i n t e r r u p t  (  STOP ^  67  REFERENCES  1.  M.A. Laughton, " M a t r i x A n a l y s i s o f Dynamic S t a b i l i t y i n Synchronous M u l t i m a c h i n e Systems", P r o c . I E E , v o l . 113, pp. 325 - 336, F e b r u a r y 1966.  2.  J.M. U n d r i l l , "Dynamic S t a b i l i t y C a l c u l a t i o n s f o r an A r b i t r a r y Number o f I n t e r c o n n e c t e d Synchronous Machines", IEEE T r a n s a c t i o n s , v o l . PAS-87, pp. 835 - 844, March 1968.  3.  F.C. Schweppe and J . W i l d e s , "Power System S t a t i c - S t a t e E s t i m a t i o n , P a r t I : Exact Model", IEEE T r a n s a c t i o n s , v o l . PAS-89, pp. 120 - 125, January 1970.  4.  F.C.. Schweppe and D.B. Rom, "Power System S t a t i c - S t a t e E s t i m a t i o n , P a r t I I : Approximate Model", IEEE T r a n s a c t i o n s , v o l . PAS-89, pp. 125 - 130, January 1970.  5.  F.C. Schweppe, "Power System S t a t i c - S t a t e E s t i m a t i o n , P a r t I I I : Implementation", IEEE T r a n s a c t i o n s , v o l . PAS-89, pp. 130 - 135, January 1970.  6.  R.E. L a r s o n , W.F. Tinney, J . Peachon, " S t a t e E s t i m a t i o n i n Power Systems P a r t I: Theory and F e a s i b i l i t y " , IEEE T r a n s a c t i o n s , v o l . PAS-89, pp. 345 - 352, March 1970.  7.  R.E. L a r s o n , W.F. Tinney, L.P. Hajdu, D.S. P i e r c y , " S t a t e E s t i m a t i o n i n Power Systems P a r t I I : Implementation and A p p l i c a t i o n s " , IEEE T r a n s a c t i o n s , v o l . PAS-89, pp. 353 - 363, March 1970.  8.  O.J.M. Smith, "Power System S t a t e E s t i m a t i o n " - IEEE v o l . PAS-89, pp. 363 - 379, March 1970.  9.  D.S. Debs and R.E. Larson, "A Dynamic E s t i m a t o r f o r T r a c k i n g t h e S t a t e o f a Power System", IEEE T r a n s a c t i o n s , v o l . PAS-89, pp. 1670 - 1678, September/October 1970.  Transactions,  10.  N.R. Draper and H. Smith, A p p l i e d R e g r e s s i o n W i l e y & Sons Inc., 1966.  A n a l y s i s , New Y o r k ;  11.  R.H. Park, "Two-Reaction Theory o f Synchronous Machines, G e n e r a l i z e d Method o f A n a l y s i s " , AIEE T r a n s a c t i o n s , v o l . 48, pp. 716 - 730, J u l y 1929.  12.  K. V o n g s u r i y a , "The A p p l i c a t i o n of Lyapunov F u n c t i o n t o Power System S t a b i l i t y A n a l y s i s and C o n t r o l " , U.B.C. PhD. T h e s i s , February 1968.  68  13.  G.E. Dawson, "A Dynamic Test Model f o r Power System S t a b i l i t y and Control Studies", U.B.C. PhD. Thesis, December 1969.  14.  J . Hindmarsh, E l e c t r i c a l Machines, New York: Pergamon Press, 1965.  15.  G.E. Dawson, "Modelling, Analogue and Tests of an E l e c t r i c Machine Voltage Control System", U.B.C. M.A.Sc. Thesis, September 1966.  16.  E.W. Kimbark, Power System S t a b i l i t y : Synchronous Machines, New York: Dover Publications Inc., 1968.  17.  R.V. Shepherd, "Synchronizing and Damping Torque C o e f f i c i e n t s of Synchronous Machines", AIEE Transactions, v o l . 80, pp. 180 - 189, June 1961.  18.  Y.N. Yu and K. Vongsuriya, "Nonlinear Power System S t a b i l i t y Study by Lyapunov Function and Zubov's Method", IEEE Transactions, v o l . PAS-86, pp. 1480 - 1485, December 1967.  19.  "Test Procedures f o r Synchronous Machines", IEEE P u b l i c a t i o n 115, 1965.  20.  "Applications Manual f o r Operational Amplifiers", Philbrick/Nexus Research, Nimrod Press, Boston, 1968.  

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