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Application of the Monte Carlo method to the estimation of the risk of failure of transmission line insulation Leonard, Ronald William 1988

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APPLICATION OF THE MONTE CARLO METHOD TO THE ESTIMATION OF THE RISK OF FAILURE OF TRANSMISSION LINE INSULATION  By RONALD WILLIAM LEONARD  B.A.Sc,  The University  of B r i t i s h Columbia,  1977  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in THE FACULTY OF GRADUATE STUDIES Department of E l e c t r i c a l  We accept t h i s thesis as to the required  Engineering  conforming  standard  THE UNIVERSITY OF BRITISH COLUMBIA July  1988  © Ronald William Leonard,  1988  In  presenting  degree  this  at the  thesis  in  University of  partial  fulfilment  of  of  this  department publication  or of  thesis for by  his  or  her  representatives.  fc^l^nV  9-\io\rlooX.  DE-6G/81)  12  TuJn  l°>&%  for  an advanced  Library shall make  it  agree that permission for extensive  It  this thesis for financial gain shall not  The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date  that the  scholarly purposes may be  permission.  Department of  requirements  British Columbia, I agree  freely available for reference and study. I further copying  the  is  granted  by the  understood  that  head  of  copying  my or  be allowed without my written  ABSTRACT  A digital  program  technique  to  has  estimate  insulation  during  overvoltage  waveshape  been the  system  developed risk  of  switching  which failure  uses of  is  The  included.  e l e c t r i c a l system is used to demonstrate the method and to some general aspects of error and parameter  - ii  -  Monte  transmission  operations.  on insulation strength  the  sensitivity.  effect  Carlo line of  A simple investigate  TABLE OF CONTENTS Section 1  2  3  4  5  Page INTRODUCTION 1.1 Motivation 1.2 Content  1 2  SWITCHING-SURGE INSULATION COORDINATION 2.1 General 2.2 The Strength D i s t r i b u t i o n 2.3 The Stress D i s t r i b u t i o n 2.4 Simple S t a t i s t i c a l Methods 2.5 More Involved S t a t i s t i c a l Methods 2.5.1 Application of the Monte Carlo Method to Insulation Coordination 2.5.2 Snider's Simulation Technique 2.6 The Equivalent Time-to-Crest of an Overvoltage Peak 2.7 Usefulness of the S t a t i s t i c a l Methods  17 18 19 22  MATHEMATICAL BACKGROUND - THE MONTE CARLO METHOD 3.1 System Model 3.2 Integral Solution 3.3 Sample-Mean Monte Carlo Method 3.4 Variance Reduction  25 26 28 30  THE RISK PROGRAM 4.1 General 4.2 Program Components 4.2.1 Modifying the TP Input F i l e 4.2.2 Selection of Peaks 4.2.3 Plateau Detection 4.2.4 Fitting 4.2.5 Calculation of Risk of F a i l u r e TEST RESULTS 5.1 Introduction 5.2 D i s t r i b u t i o n of Crest Voltage and Time-to-Crest 5.3 Error Considerations 5.4 S e n s i t i v i t y to F i t t i n g Parameters and Procedures 5.5 S e n s i t i v i t y to Waveshape 5.6 S e n s i t i v i t y to Parameters of the Strength Distribution 5.7 S e n s i t i v i t y to Pole-Scatter 5.8 Comparison with Simple S t a t i s t i c a l Method  4 6 13 14  33 34 36 37 39 39 41 42 45 50 53 56 57 59  6  CONCLUSIONS  61  7  REFERENCES  63  - iii -  TABLE OF CONTENTS (Continued) Page APPENDICES A  Derivation of.the Simple Form for Risk of Failure  66  B  The Inverse Transform Method of Generating Random Variables  70  C  Generating Random Numbers with Truncated Normal Density  71  D  F i t t i n g a Standard Impulse  73  E  Transient Program Input Data  76  - iv -  LIST OF TABLES  Table  Page  1.  Relative Error after 500 Runs  47  2.  S e n s i t i v i t y to Ratio of Time Constants in Double Exponential  52  v  LIST OF FIGURES  Figure  Page  1.  Inverted Delta and Flat Configurations  1  2.  Typical Switching Impulse  8  3.  U  4.  Combination of Stress and Strength to Yield Risk of Failure  15  5.  F i t t i n g the Peak to a Standard Impulse  20  6.  TP Model for Line Energization  35  7.  Truncated Normal Density  36  8.  Regular and Double Peaks  37  9.  Plateau Occurring in an Overvoltage Peak  38  10.  Histogram of Crest Voltage  43  11.  Histogram of Time-to-Crest  44  12.  Scatter Plot of Time-to-Crest Versus Crest Voltage  44  13.  Convergence of the Monte Carlo Simulation  46  14.  Small Difference in Models  48  15.  S e n s i t i v i t y to Inclusion of Plateau-Detecting  16.  S e n s i t i v i t y to F i t t i n g Parameters and Procedures  51  17.  Family of U  54  18.  Risk of F a i l u r e for Different U  19.  S e n s i t i v i t y to Standard Deviation of the  5 Q  Curve for 4 m Rod-Plane Gap  5 Q  12  Curves 5 0  Curves  Routine  51  55  Strength D i s t r i b u t i o n  56  20.  S e n s i t i v i t y to Pole-Scatter Standard Deviation  58  21.  Region of Integration  68  22.  Example F i t  75  - vi -  ACKNOWLEDGEMENTS,  The author people:  gratefully  acknowledges  the  assistance  of  the  following  Dr. Hermann Dommel, U.B.C. Dept. of E l e c t r i c a l Engineering, and Dr. Laurie Snider, B.C. Hydro E l e c t r i c a l Research, who were the thesis supervisors. Mr. Albert Reed and Dr. Bruce Neilson, B.C. Hydro E l e c t r i c a l Dr. S.R. Naidu, v i s i t i n g professor Paraiba, Campina Grande, B r a s i l . Mrs. Gwen H a l l , B.C. Hydro, for the  - vi i -  from typing.  Universidade  Research.  Federal  da  1  INTRODUCTION  1.1  Motivation In 1982  began  British  Columbia Hydro and Power Authority  consideration of a new transmission tower  use in the then planned hydro developments design,  employing  crossrope  an  suspension  inverted structure  delta  for  eventual  for northern B.C.  The new  phase  (Figure la)  departure from the conventional f l a t - l i n e  design  (B.C. Hydro)  configuration was  and  a  a  considerable  configuration.  (Figure l b ) .  Crossrope,  (a)  FIGURE 1:  (b)  INVERTED DELTA AND FLAT CONFIGURATIONS  The r e l a t i v e l y designs  leads  consequently  to  close lower  greater  spacing  of  the  positive-sequence  surge-impedance  - 1 -  phases  compared with  inductive  loading.  This  reactance translates  other and into  economic advantages when the transmission distances are large, as for! northern  hydro developments,  since  the  expense  of  series  capacitor  stations and the number of lines required is reduced. One of the major design problems to be addressed was to for adequate tower was  switching  erected  Laboratory  in  test  of  in.1983  Surrey,  investigation because  of  the  at  and  close  phase  never  deferred  testing  begun.  spacing,  for  The  switching  completed,  plans  To accomplish t h i s ,  B.C. Hydro's Research  phase-to-ground  program was  indefinitely  impulse strength.  design  and Development testing  impulse  phase-to-phase however,  northern  as  a test  in  included  strengths  and,  strengths.  The  1985  development,  B.C. Hydro and  further  testing was cancelled. To  aid  switching designs based  surge  was on  the  design risk  of  process  an  analytical  failure  of  the  planned, using measured data  the  work  done  to  develop  a  new as  comparison  design input.  digital  with This  computer  of  the  existing thesis  is  program  to  accomplish that.  1.2  Content Previous work  Institute  by  L.A. Snider [1]  tower  for  a single  switching operation). • •. •' This thesis problem  the  (IREQ) forms the basis for this work.  Carlo method to estimate to  at  using  switching  operation  Research  Snider used the Monte  the probability of flashover (risk  of  from conductor f a i l u r e during  A transient network analyzer (TNA) was used.  presents the  Hydro Quebec  an e n t i r e l y  U.B.C.  Transients  - 2 -  digital  solution to this  Program  (TP)  to  same  generate  overvoltage  waveforms.  The  Section 2 with a discussion  main of  the  body  of  the  relevant  elements  insulation coordination, with p a r t i c u l a r attention equivalent fits  time-to-crest.  In Section 3  it  is  errors  give estimates  associated  with  and suggest means of the  simulation.  of  begins  in  statistical  to the concept  of  shown how the problem  into the framework of the Monte Carlo method.  Carlo theory  thesis  Results from Monte reducing the  Section 4  organization of and procedures used in the d i g i t a l  large  covers  program.  the  The l a s t  section gives the results of a thorough exercising of the program on a simple test case.  The objective  program's  an  use  as  aid  in  is  to study  transmission  general line  aspects of  insulation  focussing mainly on the subjects of error and s e n s i t i v i t y .  - 3 -  the  design,  2  SWITCHING-SURGE INSULATION COORDINATION  2.1  General High voltage  types  of  electrical  voltage,  short  frequency, slow-front the  transmission l i n e stresses:  duration  fast-front surges  slow-front  the  normal  temporary  surges  (usually  surges  insulation  is  subjected  power  near  are  the  four  system  the  power  lightning-initiated),  switching-initiated).  that  frequency  overvoltages  (usually  to  subject  It of  is  the  this  and  latter,  work.  The  magnitude of uncontrolled switching surges is d i r e c t l y proportional to l i n e voltage and for high voltage l i n e s , measures are generally taken to  limit  their  magnitude.  300 kV, switching  surge strength  consideration.  The few  factors  paramount  to  be  Nevertheless,  special  at  voltages  above  is usually the most c r i t i c a l  design  circumstances  include:  high  line  which cause  keraunic  level  the  other  (lightning  frequency), high tower resistance (rocky t e r r a i n ) , and regions of high contamination. The categorization  of  overvoltages  by  source  is  useful  but  terms of the response of insulation to stress the source is of importance.  little  The key parameter in the categorization of voltage stress  is waveshape,  in p a r t i c u l a r the time required to  (time-to-crest).  As a g u i d e l i n e , slow-front  have times-to-crest times-to-crest inductive  in  voltage  surges are considered to  ranging from about 50 us to 2 ms, with the longer  (1 to  sources  reach peak  2 ms)  (e.g.  a  appearing generating  connected.)  - 4 -  in  EHV  station  networks with  fed  from  no other  lines  Overvoltages as  a  result  having times-to-crest  of  a  variety  of  in the range given above occur  switching  operations  such  energization and reclosing and the switching of reactive are also caused by fault  i n i t i a t i o n and c l e a r i n g .  as  line  loads.  They  In this work  the  case used as example w i l l be simple l i n e energization, but in practice all  types  of  switching  operations  must  be  given  attention  for  a  complete assessment of insulation risk of f a i l u r e . Besides variables  overvoltage  are  magnitude  important  factors.  and  Rain,  waveshape, wind,  solar  meteorological radiation,  air  humidity and density can be important considerations, but are outside the scope of this work. Switching the  overvoltages  operation with  the  are dependent  power-frequency  on the  relative  voltage wave.  degree of randomness associated with the point on the waveform at which the breaker contacts three-phase  energization,  the  wave for the three different and  electrical  overvoltages  system,  at  the  tower  There is  some  power-frequency  c l o s e , so that in the case of a  closures phases.  these  timing of  occur at  different  points on  For a given switching operation  sources  of  insulations  random fluctuation  that  can  be  produce  described  by a  probability distribution. The strength of the insulation undergoing the overvoltage can  also  be  described  transmission l i n e both the  stress  in  probability  terms.  Economic  stress  design  of  insulation should account for the random nature of  and strength  insulation  requirements  probability  of  may  components. be  achieved  insulation flashover  - 5 -  Savings  for  by  a given  from reduction in  accepting switching  a  certain operation  (risk of f a i l u r e ) .  This is feasible because the flashover of external  s e l f - r e s t o r i n g tower insulation w i l l normally not result in damage to the  insulation  and  the  effect  on  minimized by quick f a u l t - c l e a r i n g discuss the s t a t i s t i c a l  the  action.  electrical The next  system several  can  be  sections  methods which can be used to predict the risk  of f a i l u r e associated with a switching operation.  2.2  The Strength D i s t r i b u t i o n This, section considers some relevant aspects of the response of  * a i r gap insulation to switching-impulse The  particular  transmission geometrical  line  class  of  insulation  air is  gap  stress. relevant  characterized  to by  the the  question following  of two  properties:  the length is greater than 1 or 2 m. the radius of curvature of the energized electrode i s small (a few cm). The response of an a i r gap with such a geometry  to a switching  impulse of positive p o l a r i t y is governed by a complex physical  process  consisting of several stages, the d e t a i l s of which can be found in the literature. "leader  For our purposes we w i l l only note that the presence of a  development"  stage  is  characteristic  of  this  type  of  gap  A switching impulse is a test waveform generated in the laboratory; a switching overvoltage (or surge) is that which a c t u a l l y occurs on the e l e c t r i c a l system.  breakdown  and  is  responsible  for  the  waveshape  dependence,  of  breakdown. In most practical cases, the dimensions of the grounded electrode are  greater  than  those  of  negative-polarity  strength  will  positive-polarity  strength.  It  line  insulation  overvoltages. negative-polarity  strength  Under stress  however,  physical  electrode,  positions,  normally  type  to name a few.  voltage  exceed  the  reported  negative-polarity cases  of  lower  by Nigol and Reed).  impulse, the time to breakdown is a  parameters free of  influencing  electrons  gaseous  in the  molecules  As a result of this  of  the  applied  impulse but  also  the  breakdown  vicinity  of  present,  process  over time, the probability of breakdown is dependent crest  The  common in transmission  ignore  (discussion to [2]  including number of  positive  to  electrode.  The uncertainty in breakdown time i s a result of the  indeterminable  process,  therefore  work  by a switching  random v a r i a b l e . many  are,  energized  consequently is  coordination There  the  the  their  taking  place  not only on the  on the  shape  of  the  impulse. The study of the breakdown of these gaps i s accomplished in highvoltage  laboratories  by  the  application  produced with Marx impulse generators. of a double-exponential  U(t) = C(e  1  - e  of  standardized  impulses  These impulses have the shape  in which the voltage is given by the equation  1  )  - 7 -  The or  shape of the impulse i s determined by the two time constants,  equivalently by the time-to-crest  to one-half crest value ( t ^ )  (  2  s e e  (t Fi9  ) and the time taken to decay u r e  2).  TIME ( l i l l i s e c o n d s ) FIGURE 2:  The  time  TYPICAL SWITCHING IMPULSE  constants  are  impulse with a r e l a t i v e l y might be 250 y s , and t ^ ^1/2 of  to t crest  c r  ),  chosen  fast  such  that  << x^,  r i s e and slow decay.  2.5 ms.  giving  Typically,  an t  cr  For a given' impulse shape ( r a t i o of  the p r o b a b i l i t y of gap breakdown i s given as a function  voltage U  cr  cumulative d i s t r i b u t i o n ,  A  commonly  used  function  is  the  Normal  prob. of breakdown  =  g(U  )  1  (2-D  I oo  9  where m^ and  (x - m_)  exp(-  g  are the mean and standard d e v i a t i o n , respectively,  of  the Normal d i s t r i b u t i o n . The  interpretation  laboratory  testing  of  the  procedure  above is  equation  not  and  obvious  relationship  and  is  to  therefore  considered at some length. Imagine that at any instant in time there exists a gap withstand, voltage defined by the following property: shape) with crest voltage s l i g h t l y  i f an impulse (of a given  greater than the withstand  voltage  were to be applied to the gap, breakdown would occur; but i f the crest value  were  to  be  slightly  less,  breakdown  withstand voltage changes  from one instant  of  the  the  random nature  determine withstand  the  cumulative  voltage.  consecutively application  of  breakdown  would  of  To do so we apply  occur.  in time to another  process.  distribution  not  impulses  because  We would  this  random (of  This  like  to  variable,  constant  shape)  in time with random crest voltages and record for each if  breakdown  breakdown occurs,  occurred.  If  then we conclude that  applied the withstand  voltage was less  for  a  crest  voltage  at the time the than or equal  U  c r  ,  impulse was  to U  breakdown occurred, the withstand voltage was greater than U  . c r  .  If no After  many such experiments we group the applied crest voltages into bins of some  width  and  determine,  for  ) - 9 -  each  bin,  the  fraction  of  tests  resulting in flashover. distribution that  of  withstand  at any given  than U " .  This fraction is the value of the voltage  instant  at  U  c r  , that  breakdown w i l l  occur i f  the  probability  in time the withstand voltage w i l l be  We note that this p r o b a b i l i t y is just  cr  is,  cumulative  an overvoltage  less  the p r o b a b i l i t y  with crest  U  that  i s applied  G r  at  any instant in time. Equation (2-1) cumulative  can  be  distribution  interpreted,  function  of  therefore,  withstand  as  voltage  either  a  or  a  as  p r o b a b i l i t y of breakdown conditional upon a given crest voltage. interpretations potential  are  found  in  the  literature  for confusion - one reference [3]  function (2-1)  has nothing to do with  and  thus  states that  a cumulative  there  the  Both is  strength  d i s t r i b u t i o n and  should not be as such. The i n t e r p r e t a t i o n . of the is valuable  strength  as a cumulative  in providing a link to the underlying physical mechanism  and the Normal d i s t r i b u t i o n is usually a reasonable large  number  breakdown  distribution  of  randomly  process.  The  fluctuating  conditional  physical  probability  choice given  parameters  the  in  the  interpretation  is,  however, more convenient when discussing the combination of stress and strength to y i e l d total  p r o b a b i l i t y of breakdown ( r i s k of f a i l u r e ) and  leads more naturally to simulation methods. The experiments as described above for determining the cumulative d i s t r i b u t i o n of withstand voltage would be far too time-consuming and expensive series  to  of  distribution  perform. impulses (called  In f a c t , with  U ) 5 Q  the  crest  voltage  in order to  - 10  -  usual  procedure near  the  is  to  mean  provide a reasonably  apply a of  the  accurate  estimate  thereof.  less than 5 0 .  Typically,  the  number of  tests  required would be  Further tests are performed away from U^Q in order to  estimate the standard deviation of the d i s t r i b u t i o n . Air  humidity, temperature,  pressure and wind are  variables which influence the breakdown process. generally  held as constant  as possible  meteorological  These variables  during testing  The dependence of the mean and standard deviation  are  of an a i r gap.  on these  variables  can be estimated by performing additional series of t e s t s . The p r o b a b i l i t y modified  by  truncating  (typically  about  that,  crest  for  of breakdown the  g-3cTg)  m  as given  by equation (2-1)  Normal d i s t r i b u t i o n  in  voltages  order  below  to  reflect  this  bound,  at  can be  some lower bound  the  physical  breakdown  reality  will  never  occur [ 4 ] . Equation (2-1)  gives  some fixed shape. general  influence  held constant,  the  breakdown  The p a r t i c u l a r t the mean ( U ^ )  the plot of U  and t ^  c r  50  show a minimum at crest).  some value  for  impulses  of  t  cr  t  (called  cr (called  The minimum value of U ^ Q , c a l l e d  a U  c r v  If  an  example  Ugg  reference [ 5 ] , page 217). will  show  critical  a  similar  time-to-crest  curve  (4 m  with  [6].  - 11  -  the  is will  50  the  critical  time-to-  in this work the  critical  rod-plane  The standard deviation  shape,  t ^  curve)  UgQ, w i l l depend on the gap spacing and electrode geometry. gives  of  ° f the impulses w i l l in  and standard deviation. versus  c n  probability  gap  Figure 3  taken  plotted  minimum o occurring  versus near  from t the  1.2  o a  1.1  Ld  Nl H _l  <x n cc o z  1,0 -  TIME-TO-CREST ( i i l l i s e c o n d s )  FIGURE 3:  U  5 Q  CURVE FOR 4 m ROD-PLANE GAP.  In general, for impulses with times-to-crest breakdown, and  if  it  conversely,  occurs, w i l l for  occur after  impulses  with  less  the crest  switching impulse strength the variation of U  *l/2^  1 S  P  UgQ,  as . this  insulation stressed  r i m a r i  ^y  °f  is  the  (air-gap by  superimposed  more on  a  literature [8,9,10], of  interest main  dimension) complex slower  of the  times-to-crest  c r i t i c a l , breakdown w i l l occur before or at the crest For  than c r i t i c a l , impulse,  greater  than  [4,7]. 5 Q  with t  (and  for the determination of the c r i t i c a l parameter is  based.  shapes main  upon  which  The  (such impulse)  switching-surge  strength  as is  of  air-gaps  fast-rising reported  bumps in  the  but i t has been generally found [7] that no form  complex waveshape w i l l r e s u l t in a c r i t i c a l U  testing with standard impulses.  - 12 -  5 Q  below that found by  Once the c r i t i c a l U^Q is determined, therefore, the shape of the UgQ curve  is  not usually  considered to be of much relevance  question of gap dimension.  Some studies [1]  procedure for  accounting  for waveshape  time-to-crest of an overvoltage section 2.6.  peak.  -  the  have been done, however,  in which waveshape is considered e x p l i c i t l y , and one of the of this work is to examine the s e n s i t i v i t y  to  of r i s k . o f the  use of  objectives  f a i l u r e to one the  equivalent  This procedure i s discussed in  .  Knowledge of the shape of the U^Q curve and the experimental work dealing with the response of long a i r gaps stressed by "non-standard" impulses  is  very  valuable,  however,  in the  development  of  physical  models of breakdown [10,11].  2.3  The Stress D i s t r i b u t i o n  The  simple  (section 2.4)  statistical  make e x p l i c i t  voltages of slow-front The density  methods use of  of  the  estimating  risk  of  p r o b a b i l i t y density  failure of  crest  surges.  function  can be estimated  by. successive  switching  operations on a transient network analyzer (TNA) [12] or by successive runs of a d i g i t a l transients waveform  transients program [13]  program (TP). is  probability  generated density)  such as an electromagnetic  The variation in the switching by  the  randomly closing  operation or computer run.  - 13 -  choosing  times  of  the  overvoltage  (according  to  some  breakers  for  each  The density is often approximated ( p a r t i c u l a r l y in the important tail  region)  by  a Normal density,  the  parameters  of  determined by f i t t i n g to the histogram of crest voltages. of crest voltages, U  f(U. )  (U  — ^ —  exp( -  c  V2TT 0^  where m and a f  f  density of U  -  2.4  2  2  )  (2-2)  The use of the Normal density is  to  parameter, c r  f  f  are the mean and.: standard d e v i a t i o n , respectively, of  possible  time-to-crest  m )  r r  r  considered to be reasonably accurate is  The density  2o^  the Normal d i s t r i b u t i o n .  It  can be  is given by  c r >  1  "  cr  which  and  t  Q r  extend  generally  [12]. equation (2-2)  in which case  the  to  density  include  becomes  a  the joint  .  Simple S t a t i s t i c a l Methods The simplest s t a t i s t i c a l  approach to the estimation of the  of f a i l u r e of a single insulation subject  to an overvoltage  risk  stress  is  to consider the strength to be a function only of the crest voltage of the highest peak occurring in the overvoltage waveform. The combined operation:  R. =  strength  distribution  and  to  the  flashover  yield  risk  of  r  f(U) g(U) dU  the  stress for  distribution a  single  are  switching  (2-3)  - 14 -  where f(U) is the probability density of crest voltage (Eqn. 2-2) and g(U) is the p r o b a b i l i t y of flashover at crest voltage U, (Eqn.  2-1).  Equation (2-3) can be visualized graphically as shown in Figure 4,  o 3 T3 O SQ. ,—.. T3 +J  c OJ s_ +J oo •—"  >h-  t—«  _l t—i ca •a: ca o al  Q.  1.5i  C ra ISI  OJ  s-  +->  ^oo' >1— oo s: LU O  >t— 1  HH  CO  <  o a; VOLTAGE ( p . u . l  FIGURE 4:  The function g(U) conditional  upon  equation (2-3)  f  COMBINATION OF STRESS AND STRENGTH TO YIELD RISK OF FAILURE.  is best interpreted as the p r o b a b i l i t y of the  crest  voltage,  and  to  flashover  reflect  this,  is better written as  oo f(U) (prob. of f l a s h .  - 15 -  |U) dU  (2-4)  where R i s the total  (unconditional) probability of flashover  (risk of  failure). Substituting  into  (2-4)  the  previously  derived  stress  and  strength functions, we obtain,  00 exp (  R =  2  U  (U - m - ) ' —-j-  exp(  )  *  /2TT a.  2a,  ) d? dU  2a_  exp (- | - ) dz,  (2-5)  /2T  where Y J  •o  2  f  4.  2  a  +  Equation (2-5) derivation  that  method [ 3 , 2 2 ] .  g  i s extensively has  been  used but rarely derived.  found  is  by  a  Appendix A gives an alternative  Laplace  The only Transform  derivation by  direct  integration. Equation (2-4)  can be generalized to give the r i s k of f a i l u r e for  a m u l t i p l e - i n s u l a t i o n system, such as a three-phase  transmission l i n e  with many towers [ 1 4 ] .  Time-to-crest can also be included.  more  analytical  involved  cases  solution  becomes  simulation or numerical integration methods become  - 16  -  In these  difficult  attractive.  and  2.5  More Involved S t a t i s t i c a l  2.5.1  Methods  Application of the Monte Carlo Method to Insulation Coordination Examination of equation (2-4)  just  the expected  value  of  the  reveals that the r i s k of f a i l u r e is  conditional  probability  of  flashover  ( i t s e l f a random v a r i a b l e ) . The interpretation of the conditional p r o b a b i l i t y of flashover as a  random variable  permits reference  the  evaluation  to the j o i n t  time-to-crest, probability function  leads  of  one of  to  the  the risk  of  flashover  the  overvoltages,  is  seen  a  of  the  conditional  to  generate  without  Instead, which  the  the is  a  switching  probability  an estimation of flashover  of the unconditional p r o b a b i l i t y ) .  the  probability  of  use  of the closing breakers.  exact evaluation conditional  proceed  random variable  used  that i s , the points-on-wave  value  to  Its  function of peak voltage,  The application of this method substitutes expected  method.  random variables.  as  random variables  Carlo  failure  probability density  or any other physical  of  Monte  for  a  the  (for  the  the value of  given  overvoltage  waveform (produced by a sampling of the random variables  controlling  the energizing breaker closing)  flashover  If  of  is denoted by g.., then the estimation  of the r i s k of f a i l u r e after N random samplings  is:  R  (2-6)  est  T y p i c a l l y , N w i l l vary from several depending on the accuracy required.  - 17 -  hundred to several  thousand,  The Monte Carlo method w i l l It  is  convenient  to  state  be treated  here,  however,  in detail  in section 3.  that  method  the  is  a  technique for estimating an expected value of a random v a r i a b l e , and that expected numerical  value can always be estimated by using a technique of  quadrature.  efficient  when  the  Monte  number of  Carlo  techniques  random variables  tend  to  be  (equivalent  to  more the  dimensionality of the associated integral) is large. The primary advantage of this method versus the methods involving the p r o b a b i l i t y density probability physical  of  (Equation 2-4)  flashover  parameters  for  can  be  a  is that the dependence of the  switching  made  very  event  complex.  on  the  relevant  For example,  the  p r o b a b i l i t y of flashover to ground of one phase of a three-phase l i n e may be  a function  additional  of  the  complication  voltages  can  be  on the  easily  other  two  handled  by  phases. a  This  simulation  technique.  2.5.2  Snider's Simulation Technique Snider's [1]  contribution makes  use  of  the  flexibility  of  the  Monte Carlo method by taking into account the p r o b a b i l i t y of flashover of  each  individual  peak  in a  switching  overvoltage  waveform.  time-to-crest of each peak is also taken into consideration.  - 18 -  The  If n individual peaks occur in a waveform, the risk of f a i l u r e to ground i s :  n It (1 - R.) i=i . . . .  R = 1 -  * •  (2-7)  •  -  where R. is the flashover p r o b a b i l i t y associated with the i  2.6  +• h  peak.  The, Equivalent Time-to-Crest of an Overvoltage Peak The  shapes  electrical  systems  symmetric [ 7 ] . laboratory  of  switching are  surge  manifold  In contrast,  are not at a l l  but  overvoltage tend  peaks  occurring on  generally  the shapes of the  to  be  fairly  impulses used in the  symmetric; having the fast  r i s e and slow  decay of the standard double-exponential waveform. As seen in section 2.2, the U^Q curves produced in the laboratory attest to the waveshape dependence of the strength of long a i r gaps. The concept of the equivalent time-to-crest i s introduced to provide a means of  using laboratory data  to account for the waveshape  of  the  actual overvoltage peaks. The equivalent time-to-crest of a peak is determined by f i t t i n g a standard impulse over a certain portion of the waveform: quantity is then the time-to-crest of the f i t t e d of f i t  impulse.  the desired The region,  i s the front portion of the overvoltage peak between some lower  bound and the crest.  Laboratory investigations  have shown that only  the upper part of a switching impulse is important in influencing the  -19  -  probability of breakdown of a rod-plane gap. region of  influence  The lower l i m i t of the  is not sharply defined but may be considered as  the voltage at which corona pulses begin to occur [ 8 ] .  Figure 5 shows  an overvoltage peak f i t to a double exponential with t, -~ = 10 t  3.  _  i  i  i  i  i  I I  i  2. ~  i  i  i  i  i  \  / V  I  i  i  i  i  .  i  Threshold = 1.5 p.u.  ~;  1.  a  -  / J  0. Lu CD  —  cr  -1.  •>  -2.  tj  \  1  / \  /  I  \ A  \/  V  —  V-s  /  /: /  ^A  II  —  -3.  !  I  !  !  i  I  1  "  "  1  !  i  f  i  t  i  i  1  !  0  ~  20 T I M E FIGURE 5:  There  are  (mi  1 1 i  s e c o n d s )  FITTING THE PEAK TO A STANDARD IMPULSE  (at  least)  two  possible  objections  to  this  fitting  procedure: 1.  The portion of the overvoltage peak after crest i s not accounted for in the f i t .  For cases in which the time-to-crest is greater  than c r i t i c a l we know that breakdown w i l l occur before the crest and so the t a i l of the wave has no influence. the  time-to-crest  is  less  - 20 -  than  critical  For cases in which the  effect  of  the  elongated t a i l  of the impulse may be to raise the probability of  flashover  the  of  Reference [ 4 ] , time-to-crest  pages 181  the  result  states  of  in  of  general,  any  waveform  with  to a general  breakdown of the actual If  the  actual  smoothly  peak  rising  There  are  guidelines  has  the  for  impulses  the  same  is inhibited  actual  wave could  Reference  elongated  stress  with  So i t may be  leader.  tail  [7],  of  the  of the a i r gap than  crest  overestimation  a  shape  which  impulse,  value,  leading,  of the p r o b a b i l i t y of  such  does as  not a  resemble  fast-rising  the bump  slow r i s e , then the f i t t i n g may not be v a l i d .  laboratory in  that  the  wave.  overvoltage.  standard  after a r e l a t i v e l y  actual  of  the  result in a more severe  actual  the  the c r e s t .  voltage  inhibition  that,  of  the leader propagation  impulse w i l l  consequently,  2.  notes  of voltage r i s e after  greater  page 580,  that  and 182,  sharper f a l l - o f f  in  above  below c r i t i c a l  by the cessation that  impulse  some  results  cases  reported [16]  (presented  in  which  section 4)  but  provide it  is  apparent that a certain degree of a r b i t r a r i n e s s w i l l be inherent in any attempt to handle "non-smooth"  If most of the overvoltage  waveshapes.  peaks occurring in the real system are  smoothly-rising from around corona inception voltage to crest  voltage  the  be  use  of  the  unrealistic, Reference [17],  equivalent  and  it  is  time-to-crest generally  concept  accepted  in  may the  not  too  literature.  page 504, for example, states that every type of wave  may be represented by the equivalent  - 21 -  time-to-crest.  2.7  Usefulness of the S t a t i s t i c a l  Methods  Some examples  statistical  of  the use of  methods  to aid in the  design of transmission lines are reported in the l i t e r a t u r e . Reference [18] design  of  reports the use of the Monte Carlo method in the  a UHV transmission system.  Several  switching  operations  were considered including f a u l t c l e a r i n g , l i n e energization and single and three-phase r e c l o s i n g . network  analyzer.  Five  for each simulation. An older  The study was conducted using a transient  hundred switching  operations  were  conducted  No account of waveshape was taken.  reference [19]  reports  the use of  statistical  methods  (again with an analog system) to estimate the r i s k of f a i l u r e of a UHV transmission  line.  The approach of  this  work was  to determine  the  density of peak overvoltages and then apply an equation such as  (2-4)  to  that  estimate  of [18]  the  risk  discussed  wherein  the  (equation  of  failure.  above  density  and of  the  This  approach d i f f e r s  method  overvoltage  outlined peaks  in  is  from  section  not  2.5.1  considered  2-6).  The report discussed above and others [13,20] favour the use of equations such as (2-4)  over the use of the Monte Carlo method.  argument  density  is  determined  that on the  the  important  of  tail  overvoltage part  of  the  peaks  need  only  The be  d i s t r i b u t i o n , and so  methods such as the Monte Carlo are i n e f f i c i e n t since no consideration to  the  density  of  overvoltages  is  given.  noted t h a t ,  given an approximate knowledge  conditional  probability  (breaker  closing  of  times),  flashover methods  - 22 -  On t h i s of  on the  from  Monte  point  it  the dependence input Carlo  can be of  the  random variables theory  such  as  importance  sampling [21]  efficiency. is  efficiency,  only either  distribution of  be effective  This p o s s i b i l i t y was not investigated  discussed  use  may possibly  briefly by  (called  in  section 3.  concentrating  "topological  importance  Any  on the  in improving the in this work, and  method  tail  of  of  the  improving overvoltage  search" methods in [19]), or by the  sampling,  requires  some  prior  knowledge  of  p r o b a b i l i t y densities which can only be obtained by a great number of runs. The errors involved in the s t a t i s t i c a l  estimation of the risk of  f a i l u r e of an insulation are large and arise from many sources. is the error associated with the s t a t i s t i c a l  process i t s e l f ,  resulting  from the use of a f i n i t e number of runs in the simulation; the  errors  insulation  involved  in  the  determination  in the high-voltage  of  laboratory;  there are  U^Q and a  the  There  of  the  there are errors inherent  in the use of a TNA or TP to predict the overvoltages occurring as a result  of  switching  operations.  Reference [22]  points  out  the  importance of modelling transpositions and frequency dependence of the earth return in the determination of the d i s t r i b u t i o n of  overvoltage  peak  caused  magnitudes.  meteorological  Finally,  there  random variables.  are  the  . A good  errors  summary  of  the  by  relative  contribution of the error terms is found in [23J. The consensus methods lines  is  for  the  in the  literature  estimation  of  that these methods  sensitivity  to  several  opinion i s a l s o ,  the  are of  parameters  however,  on the  risk  of  failure  limited use needed  in the  that the systematic  - 23 -  utility  of of  statistical transmission  in view of  the  high  determination.  The  errors can be  accepted  when only  a comparison  of  line  designs  based  on  relative  risk  of  f a i l u r e is required [ 7 , 2 0 , 2 3 ] . The  Monte  Carlo  method  is  well-suited  to  Through the use of correlated sampling (discussed effect  of  relatively simulation  small  changes  accurately. model  can  in  model  help  in  testing  identifying  comparisons.  in section 5.3)  parameters  The construction,  parameters and in suggesting improved estimates  the  such  the  can  be  and study  the  assessed of  the  high-sensitivity  areas for further study which may lead to  of the absolute r i s k of f a i l u r e .  - 24  -  3  MATHEMATICAL BACKGROUND - THE MONTE CARLO METHOD  3.1  System Model The Monte Carlo method is a numerical method of studying systems,  the word system implying a set of components interacting mutually and with t h e i r external studied  environment.  must necessarily  properly  devised,  investigated  can  The system is p h y s i c a l , but what i s  be only give  a model, an abstraction  reasonably  precise  which,  solutions  to  assigning  the  distributions resulting  to  outcomes certain  output  of  the  of  sampling  elements model.  may have none integrals),  (e.g.  in  artificially.  of  experiments  the  model,  case  where u^  the  stochastic  essential  process  is  to  compute  introduced  case the input to the model i s an ordered set of  (0,1)  u ) n  on the uniform density  and can be defined symbolically as  E \J  1  x  V2  x  ••••  x  ^"n  i s the outcome of a single sampling experiment on the uniform density in the interval  \/.  an  some  The same method of solution is used for both cases.  u ,..., 2  and observing  random breaker closing times), or i t  outcomes of n independent sampling experiments  1 5  studied  probability  the Monte Carlo method can be used  which  In the general  in the interval  on  The model may have as  component a random process, (e.g.  (u  the  problem.  The Monte Carlo method i s so-named because the system is by  if  ( 0 , 1 ) , and  i s the set of a l l possible outcomes of the experiment.  - 25 -  (  3 - 1  )  The model mathematical the  itself  and logical  transformation  variables  is a process  model  general be a s t a t i s t i c a l  3.2  operations.  of . the  representing  defined  by some algorithm using  Part of the model w i l l  input .random  variables  elements.  The model  to  involve  the random  output  will  in  estimate of some model quantity.  Integral Solution Assume that  random variable estimate  each  random value  X.., and that  of the expected  i s transformed to some other  the output  statistical  estimate  i s , an  value of some function of the n independent .  random variables X . . Then the output can be formulated as:  output  -  =  estimate of E C g U ^ . X ^ , . . . , X ) ] n  jf^i^f^ify)  estimate ofjj... -oo —00  • • • f ( ? ) g ( c .C »< • • »5 )- C.i 5 -• - 5 n  n  1  2  n  d  d  2  - 00  n integrals  (3-2)  where f.. i s the density function of X.. and g ( X j , X , . . . , X ) i s the random variable whose expected value i s 2  n  required, g being some function. Associated with each density function f . there i s a corresponding d i s t r i b u t i o n function F . , such that:  - 26 -  d  n  F^x.)  =  f.  (5.)  (3-3)  L It is.shown in Appendix B that the random variables X. and  are  related by  X  1  = F^IK)  (3-4)  Using (3-3) and (3-4) and noting that  ^ i  ,  u  (3-2) can by change of variables £.-»• u.  be re-written as  Output = estimate of  Jj . . . j| gg[[FF7- (( oi ))»,  I(  A1  u  1  - ((uu )),,......, ,F~ F " ((u_)] u ) ] d u ,dduu .. . .du. Fo n 1A  1A  20  n  1  20  n integrals Equation (3-5) involving  says that  n independent  n-dimensional  (3-5)  multiple  multiple integrals exist  the output of a Monte Carlo simulation  random integral.  variables Methods  is of  more  estimate  numerically  of  an  evaluating  and the question arises whether there is any  advantage to using a Monte Carlo technique. integrals  an  The answer is that  for  of high dimensionality the Monte Carlo method can indeed by  efficient,  especially  in  required.  - 27 -  cases  where  great  accuracy  is  not  3.3  Sample-Mean Monte Carlo Method The  integral  in  is  (3-5)  the  expected  variable G (associated with the value of n random v a r i a b l e s , interval  If  (0,1).  (u^, u ,  all  of  g^ is  which  the  are  value  value  of  the  random  g ) , where g is a function of distributed  of G resulting  uniformly from the  in  the  outcome  u ) of a sampling of the input random v a r i a b l e s , then an  2  n  estimate,  y,  of  the  expected  value  of G, denoted  by  E[G], after N  samplings i s formed by the sample mean  i N £ Z g. i=l  Y = E[G] =  N  (3-6)  1  The random variable  associated  with  the outcome y,  has the variance  var  (r)  Since  -  the  c  denoted by r  • ...  2 r  -  ^  variance  " ^  of G  ^ )  is  unknown,  substituted, in place of var(G) in (3-7)  - 28  -  the  giving  sample  variance  is  (3-8)  The relationship between  a  and Og, namely a  r  r  =  CTQ/ZN,  betrays  the major weakness of the Monte Carlo method - that the convergence very slowl place  to  The l/./N dependence means that the  estimate y»  the  number  of  is  in order to add a decimal runs  must  be  increased  100 times. The estimate of the standard deviation of r , form a confidence  interval for the estimate y.  Theorem, the density large. Gaussian  For Monte  a , can be used to r  By the Central  Limit  of r approaches the Normal density as N becomes  Carlo  approximation  simulations N is  is  likely  to  be  quite  good.  large Denoting  so  that  the  the Normal  cumulative d i s t r i b u t i o n function by 4, a 100m% confidence interval for Y can be constructed:  prob (|E[G] - y| < * " ( 1  I 2  2  t I  )a ) = m  If a 95% confidence interval is d e s i r e d , (3-9)  prob (|E[G] - Y I < 1.96a ) = 0.95 r  - 29 -  (3-9)  r  becomes  (3-10)  The above equations required  to  achieve  simulation.  a certain  runs of  say) calculate a ^ ^ , 1  . 1.96  with Y^ 1^  the  accuracy  simulation  in the Monte Carlo  follows:  (with  being  a few hundred,  where  o[h  ]  and o^l^  N  relative  A simple algorithm would be as  After  (Nj)  can be used to determine the number of runs  given by equations  (3-6) and (3-8)  respectively.  )  The required number of runs is then given by  N = N  (3-11)  f  x  where a. i s the desired r e l a t i v e -  3.4  1.96  error,  ;(N)  o  v  Variance Reduction The method of "importance sampling" is not used in this work but  is  discussed  briefly  here  because  - 30 -  its  use  may  provide, a  way  of  reducing  the variance  of the estimated  expected  might be a topic for further i n v e s t i g a t i o n .  value,  and as such  Reduction of variance is  worthwhile in view of the large number of runs required to achieve a reasonable r e l a t i v e error and the consequent large CPU times involved. For the following discussion the s i n g l e - v a r i a b l e version of (3-5) w i l l be used.  (3-12) o  Equation (3-12) can be reformulated as  E[G] =  9  (  F  ^  )  ?  , -  (3.-13)  h(u) du  0  where h i s a density in the interval  It  can  be  shown [21]  that  (0,1).  if  h has  approximately  the  shape  of g«F~*, the sample mean by (3-13) w i l l have a variance less than the sample mean by (3-12). The method i s named "importance sampling" because by sampling on the  density h (eqn. 3-13)  (eqn. 3-12), larger values  rather  than  on  the samples are concentrated of g.  If,  for example,  - 31 -  uniform  density  in the region which gives  F (U) _1  the  is. the random variable  (with  density f)  breaker, often  then  values  the of  p r o b a b i l i t i e s of The effect  of  effective detail  method small in  corresponding effect  pole-scatter  the  sampling which  pole-scatter  on h would be  produced  the  of to  higher  a  closing  choose  more  conditional  flashover. of  "correlated  changes  reducing  in section  of  to  in  a  sampling", system  variance.  5.3.  - 32 -  This  are  which to  method  is  be will  used  studied, be  when is  examined  the very in  4  THE RISK PROGRAM  4.1  General The RISK program i s a computer program written in VAX FORTRAN 7 7 . .  The program produces an estimate of probability of flashover at one point switching  (tower)  of a three-phase  operation  (risk  to ground  transmission l i n e f o r a single  of f a i l u r e ) .  The Sample-Mean Monte Carlo  Method i s used to produce the estimate. A b r i e f description of the program follows: 1.  Each run of the simulation begins by sampling the uniform density on  (0,1)  transformed modelling  several to of  times.  The  the physical the  line  The TP is C-phase  executed,  energization.  obtained  required  The TP  input  are  in the file  is  4.2.1).  producing  overvoltage time  thus  random variables  modified appropriately (section 2.  outcomes  as output  records  A-phase,  at one node  B-phase and  (line-end  in this  work). 3.  Each time  record i s processed  above a certain of  threshold level)  the number of peaks  "double  peaks"  a double-exponential determined .(section the  curve.  particular  of either  may occur through  (section  accounted for (section  for  to extract  impulse,  The probability of flashover  33  -  the consideration of are detected  and the equivalent  time-to-crest  -  Reduction  Each remaining peak is f i t t e d  The  4.2.4).  (excursions  polarity.  "Plateaus"  4.2.2).  4.2.3).  the peaks  U  5 0  and to  time-to-crest  and a of the insulation  are determined  via  the U^Q  for each peak i s determined  via  the Normal cumulative  probabilities  of  d i s t r i b u t i o n and the  flashover  are  combined  to  individual yield  peak  the  total  p r o b a b i l i t y of flashover for the run (section 4 . 3 . 5 ) . Steps 1  to 3  are  repeated,  p r o b a b i l i t y of flashover  and  estimated  the  expected  value  of  the  as per the sample-Mean Monte  Carlo method. Mathematically,  step 1  corresponds  to  the  evaluation  of  inverse cumulative d i s t r i b u t i o n functions F.~*(U.) of equation Steps 2 equation  4.2  and 3  correspond  to  the  evaluation  of  the  the  (3-5).  function g  of  (3-5).  Program Components  4.2.1  Modifying the TP Input F i l e In  this  energization point-on-wave  work of  a  four  random  three-phase  of the closure  variables  are  transmission  (the  associated  line:  aiming-point)  the  and the  with  the  prospective pole-scatter  about the aiming-point of the three individual breaker poles.  Figure 6 shows the TP model for l i n e energization with the four random v a r i a b l e s ,  (details  of the transmission l i n e model used as a  test case are given in section 5.1  - 34 -  and Appendix E ) .  A-Phase  B-Phase  FIGURE 6:  C-Phase  TP MODEL FOR LINE ENERGIZATION  The angle .o> is taken to be a random variable with uniform density in  the  interval  (0,360).  Except  for  an  additive  constant  it  is  equivalent to the, aiming-point, and is related to u (the outcome of a sampling on the uniform density on (0,1)) by the simple formula,  <> j =  360 u.  The three  closing  The  non-truncated relationship  work  to  account  Figure 7 shows the density for the case:  comparison.  this  and t^,  density.  A  in  tg  and  a = 1 ms.  taken  t^,  scatter,  2a,  are  times  Normal between  have  density the  a  for  the  pole  truncated Normal truncation point =  is  sample,  superposed u,  and  for the  closing-time i s not a simple one; in f a c t ,  two samplings are required  to produce one closing time.  of the transformation are  The d e t a i l s  given in Appendix C.  -35  -  500  £  (/)  i  400  z a -£  H _l H  300  m  I  200  Q_  100  TIME (•Uliseconds)  FIGURE 7:  4.2.2  TRUNCATED NORMAL DENSITY  Selection of Peaks A regular overvoltage  positive below. peak  or negative)  peak  is  defined  by  as  an excursion  above a certain threshold level  Figure 8a shows a peak of is  defined  the  presence  this of  type.  two  and then back  A double  closely  (either  spaced  overvoltage peaks  in a  single excursion above threshold (Figure 8b). Laboratory flashover peaks.  of  tests' [16]  rod-plane  RISK contains  overvoltage  to  pick  have  gaps  determined  subjected  to  cases  of  following rules (from Reference [16]) peaks:  - 36 -  double  probability  overvoltages  a subroutine which scans out  the  the  peaks  with  of  double  time records of and  applies  the  to possibly eliminate one of the  1.  If .V^ > 0 . 9 V ; retain only, the f i r s t peak  2.  If  2  <  0.9V ;  retain  2  both  peaks  if  AV > 0.1(V  1  -  AV),  otherwise retain only the second peak.  (a)  (b)  FIGURE 8:  For equivalent  the  second  REGULAR AND DOUBLE PEAKS  peak,  time-to-crest  is  the  region  of  fit  to  determine  from the minimum point between  peaks  the to  the crest of the second peak.  4.2.3  Plateau Detection Figure 9 shows an example of a peak characterized by a r e l a t i v e l y  flat  region (a "plateau") Given  reasonable  the that  laboratory the  preceding a sharper r i s e to c r e s t . results  probability  of of  reference flashover  [16],  it  would seem  would  be  determined  largely by the s h a r p l y - r i s i n g portion of the excursion.  RISK contains  a subroutine to detect such plateaus and apply the following r u l e : i f AV > 0.1V,  then the equivalent  time-to-crest  f i t t i n g only over the s h a r p l y - r i s i n g portion.  - 37 -  is determined by  I  •  '  i  '  '  1  '  1  i •  i  I  !  I  !  1  !  I  1  0.  1  20. T I M E  FIGURE 9:  ( m i l l i s e c o n d s )  PLATEAU OCCURRING IN AN OVERVOLTAGE PEAK  The application of the above rule tends to prevent the assignment of  unreal i s t i c a l l y  long  equivalent  p r o b a b i l i t i e s of flashover) The  rules  presented  times-to-crest  (and  hence  low  to the important high-magnitude peaks. in  the  last  two  sections  are  reasonable,  although somewhat a r b i t r a r y , measures to handle two types of waveshape substantially presents fitting best  be  different  several  types  suggestions.  from the of  waveshape  Which,  determined by  double-exponential.  if  examining  result to these measures.  - 38 -  irregularities  any, measures the  Reference along  with  [24] some  should be applied can  sensitivity  of  the  simulation  4.2.4  Fitting A non-linear least-square  each peak The f i t  curve-fitting  (over the region of f i t )  routine is  used to  to a double-exponential  fit  impulse.  is constrained by demanding that the crest of the impulse be  coincident with the crest  of the f i t t e d  waveform.  The mathematical  d e t a i l s of the f i t t i n g procedure are given in Appendix D.  4.2.5  Calculation of Risk of Failure The p r o b a b i l i t y of flashover for each f i t t e d peak is given by  U  where  U  U  c r i  is the crest value of the overvoltage peak  r  UgQ and a  are  the  insulation  equivalent  time-to-crest  strength  parameters  at  the  * i s the Normal cumulative d i s t r i b u t i o n function.  The r i s k of f a i l u r e for each phase-ground overvoltage time record is determined by combining the flashover p r o b a b i l i t i e s of each  fitted  peak in the record,  "phase  =  1  "  ( 1  - peak,i> R  - 39 -  * " ' 4  2  where N i s the number of f i t t e d  Finally  the  total  peaks in the record.  probability  of  flashover  determined by combining the three phase-ground  :  R  phase  =  1  " <  X  " V*  1  " V*  1  - 40 -  " C> R  to  ground  is  probabilities,  <") 4  3  5  TEST RESULTS  5.1  Introduction This section examines the use of the Monte Carlo method (via the  RISK program)  as  a tool  insulation problems.  in the  It  investigation  of  transmission  line  is not meant to represent a case study of a  p a r t i c u l a r problem. The f i r s t  objective  which i l l u s t r a t e  of  this  section  is  to  give a few  examples  the problems encountered when using the Monte Carlo  method, p a r t i c u l a r l y with regard to e r r o r s . The second objective is to come to some conclusions regarding the sensitivity  of the risk of f a i l u r e  conclusions  on the  relative  electrical  and  since  the  greatly.  Moreover,  the  to the model parameters.  importance  insulation  particular  of  parameters  systems  objectives  under of  General  are  difficult  study  a case  will  vary  study  will  influence the r e l a t i v e importance of the model parameters. The model used is the energization line.  The  preceding  details section.  distributed matrix over  of The  parameter  as well  as  frequency  (a  the  the  energization  line  is  transmission line  of an open-ended  modelled line.  inductance  reasonable  have  The  been as  given  an  Ifne  There  in  the  untransposed transformation  and capacitance  approximation)..  three-phase  are is  constant  no  source  impedance ( v a l i d for a situation in which the bus to which the l i n e is switched  is  accounts resistance  supplied by several  for  wave  attenuation  other l i n e s ) . by  inserting  The transient program one  half  the  line  in the middle of the l i n e and one quarter at each l i n e end.  - 41 -  The  line  is  points).  comprised  of  three  Appendix E gives  tower-window  insulation  a  for  80 km sections  listing  each  of  phase  is  the  (two  TP  transposition  input  file.  The  by the U^Q  characterized  curve given in Figure 3. For each study, the quantity to be estimated by the simulation is the probability that at least one of the three phases at the line-end tower w i l l  flash  over  (to  tower)  as  a result  of  a single  closing  operation.  5.2  D i s t r i b u t i o n of Crest Voltage and Time-to-Crest Before proceeding with a series  examine  the  histogram  of  crest  from the switching operation. two variables  will  help  of simulations,  voltage  it  is useful  and time-to-crest  in the  interpretation  of  results,  scatter  and w i l l  For example,  the domain of the U^g curve we are using should include the  To  resulting  Knowledge of the d i s t r i b u t i o n of these  provide a check on the v a l i d i t y of certain assumptions.  of the  to  majority  times-to-crest. produce  the  histograms,  4000 runs  were  made with  a  as shown in Figure 7 (truncation point = 2a, a = 1 ms).  resulting histograms of crest voltage and times-to-crest Figures 10*  and 11  respectively.  In  producing  The f i t t e d normal density shown w i l l be used l a t e r .  - 42 -  poleThe  are given in  Figure 10  only the  maximum  crest  voltage  (among  all  the  peaks)  for  each  phase  is  of  the  considered.  300  -  200  •  100  •  CREST VOLTAGE (p.u.) FIGURE 1 0 :  Examining times-to-crest  HISTOGRAM OF CREST VOLTAGE  Figure 1 1 , fall  we  between  note  that  the  majority  about 0 . 0 5 and 1 ms.  We are  therefore  assured that the domain of the U^Q curve ( 0 . 0 3 to 1 ms in Figure 3 ) i s adequate to account for most of the peaks. the  times-to-crest  correspond to examining  the  above  1 ms are  peaks with high crest scatter  voltage (Figure 1 2 ) .  plot  of  not  We should also ensure that important,  voltages.  that  is,  do  not  This can be done by  time-to-crest  versus  crest  600  400  m Z  200  IT trrrmrvm— .5  1.0  TIME-TO-CREST  FIGURE 11:  «  C O  2.0  1.5  2.  (Billiseconds)  HISTOGRAM OF TIME-TO-CREST  T  u u  1.5  CO  u cc u I o  1.0  1.5  -r  2.0  2.5  3.0  CREST VOLTAGE (p.u.) FIGURE 12:  SCATTER.PLOT OF TIME-TO-CREST VERSUS CREST VOLTAGE  - 44 -  We note that, lms,  almost a l l  among those peaks with time-to-crest  have crest  voltage less  greater than  than 2.5 p . u . , and so w i l l  not contribute to the r i s k of f a i l u r e .  5.3  Error Considerations Each Monte Carlo simulation is i n i t i a t e d by assigning a "seed" to  the random number generator  (a b u i l t - i n FORTRAN function).  Different  seeds w i l l produce different answers (after a certain number of runs), the degree of difference depending on the variance of the conditional probability  of  flashover  (see  section 3 . 3 ) ,  and diminishing as  the  number of runs increases. The process risk  of  seeds. U"  50  failure The  of  convergence  versus  air-gap  number of parameters  = 2.8 p . u . , a = 0.05 U  Before  discussing  5 Q  is  illustrated  runs is for  in  plotted  this  Figure 13, for  example  two are:  where  different critical  .  errors further,  we define  the  term  relative  error as being one-half the 95 percent confidence interval divided by the  estimate  relative  of  the  error is  flashover  itself  probability  (see  a random v a r i a b l e ,  equation 3-11).  so that  The  any p a r t i c u l a r  r e l a t i v e e r r o r calculated from the results of a simulation is j u s t an estimate of the actual  (unknown) r e l a t i v e e r r o r .  - 45 -  .05  i  .04 UJ  3  .03 -  2000  RUN NUMBER  FIGURE 13:  To obtain  CONVERGENCE OF THE MONTE CARLO SIMULATION  an appreciation of the  kind  of  relative  error to be  expected,  seven simulations were run, each with a different  UgQ,  each  and  with  the  simulation was 500.  same  seed.  The  number of  runs  critical for  each  The percent r e l a t i v e error was calculated and is  given in Table 1 opposite the c r i t i c a l U^g. We note that an increase in c r i t i c a l U^Q (and hence a decrease in risk of f a i l u r e )  is accompanied by an increase in r e l a t i v e e r r o r , the  size of which i s rather large. To demonstrate that the r e l a t i v e errors given in Table 1 are in fact  representative,  (= 2.80 p . u . ) , different  and  seeds).  we repeat  The  take the  a  specific simulation  case  of  critical  six  more  times  spread, i . e . difference between  - 46 -  U^g (with  maximum and  Table 1 Relative Error after 500 Runs Critical  11™ (p.u.)  Percent Relative Error 34 45 58 72 85 96 106  2.80 2.90 3.00 3.10 3.20 3.30 3.40  minimum values, confidence  was 0.0062.  This  compares well  interval from Table 1 of 0.0083.  estimates • of  risk of  with  the  estimated  The average of the seven  f a i l u r e was 0.014, which agrees well  with  the  value one would i n f e r from Figure 13. The size of the r e l a t i v e error becomes even more d e b i l i t a t i n g we  wish  different  to  comparisons  models.  two models: Figure 3,  make  the  of  risk  of  failure  among  if  slightly  Suppose we wish to compare the r i s k of f a i l u r e of first,  (critical  the  base  model, has  a U^g curve  time-to-crest = 0.15 ms), while  s i m i l a r U^g curve, but shifted  so that the c r i t i c a l  the  given  by  second has a  time-to-crest  is  0.2 ms.  For each model, simulations of 500 runs each were made for  critical  U^g ranging from 2.8 to 3.4 p . u . .  each  simulation.  The  resulting  Figure 14 versus c r i t i c a l U^g. case  are  critical  the  calculated  risks  The same seed was used for of  failure  are  plotted  Superposed on the curve for the  confidence  \} . cn  - 47 -  intervals  for  each  value  in base of  -t  i  10  10  7  \  •  ,  •2.8  :  •  3.0  r-  -  3.2  3.4  CRITICAL U50 (p.u.)  FIGURE 14:  We note  immediately that  the two curves especially  SMALL DIFFERENCE IN MODELS  the s l i g h t  is quite a b i t  difference  smaller than the  at the lower risks of f a i l u r e .  in r i s k  confidence  the case of c r i t i c a l U  5 Q  interval,  As a r e s u l t , the  error of the difference is expected to be extremely large. = 3.3 p . u . in d e t a i l .  between  relative  We examine  Denoting the  greater  value as A and the lesser as B we have the two estimates of r i s k :  est (A) = 8.96(10" ) 6  est (B) = 7.75(10' ) 6  and the corresponding variances:  var (A) = 6.03(10" ) 9  - 48 -  var (B) = 7.23(10" ) 9  If  the  difference 13.27(10 same  covariance of A  ).  of A and B is  and B would  be  the  zero,  the variance  sum of  the  two  of the  variances,  However, because the two simulations were made with the  seed,  they  are  quite  highly  correlated  (correlation  -9 coefficient  = 0.84) with a covariance of 5.55(10  the difference  ).  The variance.of  is then:  var (A-B) = var (A) + var (B) - 2 cov (A,B) = 2.16(10" ) 9  The percent r e l a t i v e error for the estimated difference i s 34.0%.  of 1.21(10"*')  With zero c o r r e l a t i o n i t would have been 830%.  Since r e l a t i v e error i s inversely proportional to the square root of the number of runs, we estimate that to achieve the lower r e l a t i v e error with simulations having zero covariance would require about six times as many runs. The  use of  identical  seeds  (and hence  identical  sequences  random numbers) to maximize the covariance of the estimates  of  for two  different simulations i s known in Monte Carlo theory as the method of "correlated  sampling".  correlation coefficient;  It's  effectiveness  depends  upon  for low correlation coefficients  advantage to be gained by choosing identical  a  high  there i s no  seeds.  To demonstrate that the r e l a t i v e error of 340% i s t r u l y representative, at  seven more simulations (500 runs each) of each model were done  critical  U  5 0  = 3.3 p . u .  12(10~ ) which compares well 6  The spread with  - 49 -  in  difference  the estimated  (A-B) was  confidence  interval  (corresponding difference  to  based  340%) on  of  the  8(10" ).  eight  The  more  simulations  accurate  (4000 runs  value  of  total)  is  8.05(10~ ). 6  5.4  S e n s i t i v i t y to F i t t i n g Parameters and Procedures It  is  interesting  to  check  the  sensitivity  parameters and procedures of the f i t t i n g F i r s t we check the s e n s i t i v i t y detecting plateau  subroutine. at  about 2.1  threshold level for  Figure 9  threshold  p.u..  level  above  shows  an  p.u.,  threshold  time-to-crest  level  results  obtained  by  from  fitting  the  the  the  to  with  a  subroutine,  and  with  a  risk  of  is much l e s s ,  The extreme  entire  rather than j u s t  a single  risk  versus  of  failure  plateau-detecting simulation.  subroutine.  Critical U  U  5 Q  varying  is  threshold  5 Q  The lower two curves critical  peak  however,  both  Two thousand  about  sensitivity  in  peak  is  equivalent  including  the  p.u..  i f an entire simulation,  considered. level  is 0.5(10"'*);  failure  difference  the  plateau-  peak  plateau, or j u s t to the rapidly r i s i n g portion above 2.1 The s e n s i t i v i t y  the  overvoltage  0.5(10"*), three orders of magnitude higher. to  of  process.  p . u . , the risk of f a i l u r e  2.1  some  to the inclusion of the  Without  less than 2.1  to  Figure 15 with  shows  the  and without  the  runs were  used for  each  was 2.8 p . u . in Figure 16 show the same comparison with  and threshold  hundred runs were used in a l l  level  simulations.  - 50 -  fixed  (at  1.5  p.u.).  Five  .020  With Plateau Subroutine .015 A Without  .010  1.6  2.0  1.8  THRESHOLD (p.u.)  FIGURE 15:  10  SENSITIVITY TO INCLUSION OF PLATEAU-DETECTING ROUTINE.  ' 3  CRITICAL U50  FIGURE 16:  (p.u.)  SENSITIVITY TO FITTING PARAMETERS AND PROCEDURES  - 51  -  We can conclude from the above results that, model  under study,  there  is very l i t t l e  for the p a r t i c u l a r  sensitivity  to inclusion of  the plateau-detecting subroutine. S i m i l a r l y , there is very l i t t l e s e n s i t i v i t y to the r a t i o of to t  of  the  fitted  double exponential.  500-run simulations at a c r i t i c a l U two time constants flashover  in the  probability.  50 corresponds to a t ^  Table 2 gives  to t  results  of 2.8 p . u . . The r a t i o of  5 Q  double exponential The  t ^  largest  is  time  tabulated  constant  of the  opposite ratio  of  r a t i o of 10 (see Figure 2); the lowest  c r  value of 10 corresponds to a t . l/d  to t  / 0  r a t i o of 4. cr  r  Table 2 S e n s i t i v i t y to Ratio of Time Constants in Double Exponential Ratio of Time Constants  Risk of F a i l u r e  50 (Base) 40 30 20 10  0.0123 0.0121 0.0119 0.0116 0.0111  There i s some s e n s i t i v i t y to the threshold level of region of f i t ) .  (lower boundary  Figure 16 gives three curves f o r comparison:  the  base curve with threshold of 1.5  p . u . , and curves for threshold values  of  that  2.0  increase  and 2.3 p . u . . as  threshold  We note level  the  increases.  consideration of the average time-to-crest critical  U^Q of 3.3 p . u . ,  the  average  - 52 -  risk  of  This  can  failure be  tends  to  explained  by  for the three  time-to-crest  for  cases. the  At three  values  of  2.0 p . u . , result  threshold and  that  with  and  simulations  at  different  Correlated comparisons.  1.5  We would  p.u.,  expect  time-to-crest  closer  the model without The  peaks  U^Q'S  used  to  at  the  observed  to  critical  the subroutine  (see  were  section  made.  high  advantage  to  4.2.2)  There  for  of  the  was  subroutine  the  at  against  a c r i t i c a l U^Q of  coefficient  small  above  of 0.998 was  comparing the base model  correlation  100%  in a l l  a correlation coefficient  plateau-detecting  under  inclusion of  double  two simulations  very  error  resulted  difference  S i m i l a r l y high correlation coefficients  7  0.31 ms  to this routine.  As an example, the  to the  critical  sampling was  for  5.2(10" ).  average  accordingly  v i r t u a l l y no s e n s i t i v i t y  relative  2.3 p . u . .  sensitivity  handle  3.3 p . u . .  0.45 ms at  (0.15 ms, see Figure 3) have larger risk of f a i l u r e .  To check the  calculated  were:  0.23 ms at  models  time-to-crest  detect  level  in  in  risk  a of  were observed  in  a l l the comparisons shown in Figure 16.  5.5  S e n s i t i v i t y to Waveshape Consider  a  critical U^'s a family  of  different curves are:  set  of  tower  but different  windows.  insulations  c r i t i c a l times-to-crest.  four U^Q curves  tower  window  which can be  The  critical  used  identical  Figure 17 gives characterize  times-to-crest  0.15 ms (base), 0.2, 0.3 and 0.4 ms.  - 53 -  to  with  of  the  the four  1.2  1  .4 TIME-TO-CREST  FIGURE 17:  .6 (lilllscconds)  FAMILY OF U  5 Q  CURVES  Figure 18 presents the results of 500-run simulations f o r various critical  U^Q'S.  The curve labelled " f l a t " ,  waveshape dependence ( i . e . f l a t U  5 Q  curve).  was produced assuming no We see that for the four  critical  times-to-crest  considered, the greater the time-to-crest the  greater  the  failure.  risk  of  Examination  o f " the  histogram of  time-to-crest (Figure 1 1 ) and the shape of the U^Q curves reveals that this  trend  time-to-crest  is  reasonable.  The U^Q curves  with  apportion more of the times-to-crest  which w i l l result in greater r i s k of f a i l u r e .  - 54 -  larger  critical  to lower  U^Q'S,  -1  to  i  CRITICAL U50 (p.u.)  FIGURE 18:  RISK OF FAILURE FOR DIFFERENT U"  The c o r r e l a t i o n coefficients in risk  increases.  correlation  50  CURVES  tend to decrease as the  difference  For example, at a c r i t i c a l U^Q of 3.3 p . u . , the  coefficient  of  the  base  simulation  and  the  0.20 ms  simulation i s 0.84; the coefficient  of the base and 0.40 ms simulation  is  flat  only 0.24.  When comparing the  the  base  case,  large r e l a t i v e errors in the differences  are,  therefore,  correlation coefficient Fairly  to be expected.  is effectively  case  to  zero.  The percent r e l a t i v e error in the difference  the 0.4 ms and base  cases  is  the  about 186%;  between  the  flat  between and base  cases i t is about 61%. The r a t i o of risk of f a i l u r e without waveshape dependence to that with provides a measure of the influence of waveshape dependence.  To  determine  of  the  ratio  for  the  - 55 -  present  model,  two  simulations  4000 runs were r a t i o was 16. by Snider [1]  made with  resulting  to Parameters of the Strength D i s t r i b u t i o n  the  to U^g, the mean of the strength d i s t r i b u t i o n , is preceding  Ugg results  flashover  The  in s i m i l a r studies using a TNA.  The s e n s i t i v i t y  critical  3.3 p . u .  This ratio compares well with the value of 30 reported  5.6. S e n s i t i v i t y  obvious. from  a c r i t i c a l . Ugg of  figures.  in a change  A  change  of  5 percent  in  of about one order of magnitude in  probability.  The effect of the standard deviation of the strength d i s t r i b u t i o n on the probability of flashover is shown in Figure 19.  10  §  CRITICAL U50 (p.u.) FIGURE 19:  SENSITIVITY TO STANDARD DEVIATION OF THE STRENGTH DISTRIBUTION  - 56  -r-  The curves marked a = 5% and 7% refer  to  models  standard deviation is a constant fraction of U^Q.  in which  The.curve labelled  "variable a" refers to a model in which a varies l i n e a r l y from at c r i t i c a l time-to-crescent  to  0.08U  5 0  the  0.0511,^  at the end points of the  U^Q  curve. We note deviation. the  that  It  of  f a i l u r e . is  quite  sensitive  curve  with  We can see that  the  "flat"  the s e n s i t i v i t y  curve  from  is  to  standard  the  preceding  to waveshape would be  for the variable a case than for the constant a case. the effect  to  is also interesting the "a = 5%" curve and  "variable a"  figure.  risk  less  This is because  of increasing a as time-to-crest moves away from c r i t i c a l  increase  probability  of  flashover,  and  thereby  cancel  to a  degree the reduced p r o b a b i l i t y of flashover at increased U^Q.  5.7  S e n s i t i v i t y to Pole-Scatter The  base  standard deviations  case  deviation  for  the  of  1 ms  (see Figure 7 ) .  model with  under study a  uses  truncation  Figure 2 0 shows  the  at  risk  a  pole-scatter, two  of  standard  failure  at  We see that there i s l i t t l e s e n s i t i v i t y u n t i l the pole-scatter  is  smaller pole-scatter standard deviations.  reduced to about 0.1 ms,.with further reduction r e s u l t i n g in a large decrease in r i s k .  The r i s k of f a i l u r e with no pole-scatter is several  orders of magnitude below the base case r i s k . It is obvious that, for the e l e c t r i c a l system of this study, the effect  of pole-scatter  is to s i g n i f i c a n t l y  line-end overvoltage peaks.  increase the magnitude of  A TNA study [19] using a s i m i l a r model of  -  57  -  pole-scatter magnitude reason  reports  compared  for  the  overvoltages  to  those  increased  t y p i c a l l y 20-30%  created  by  magnitude  is  simultaneous perhaps  greater closing.  explained  by  in The the  increased ground mode component of t r a v e l l i n g wave produced by closing each pole at different  times.  -i  10  1  •  1  •  2.8  1  3.0  —r .  3.2  3.4  CRITICAL U50 (p.u.) FIGURE 20:  The  SENSITIVITY TO POLE-SCATTER STANDARD DEVIATION  pole-scatter  point-on-wave density failure  is  of  model  contact  sufficient  to  used  closure reveal  in is  the  this  assigned high  work a  wherein  truncated  sensitivity  of  the  Normal risk  of  to the presence of non-simultaneous c l o s i n g , but is probably  an u n r e a l i s t i c model.  The model assumes  electrical  equal  closing  is  prestrike is e f f e c t i v e l y  to  ignored.  - 58 -  the  that  point  of  the  point on wave of  mechanical  closing;  A better model would be based on a  consideration of the d i e l e c t r i c strength between breaker contacts as a function  of time during the closing  operation [7,15].  model would probably produce points-on-wave tended  to  voltage.  be  distributed  A model of this  more  towards  of contact  the  sort could easily  peaks  This type of closure which  of  the  applied  be incorporated into the  RISK program i f desired.  5.8  Comparison with Simple S t a t i s t i c a l  Method  The Monte Carlo method used by RISK considers a l l peaks (above a certain threshold)  in the overvoltage  time record.  It is  interesting  to compare the results obtained in this way with the results by the t r a d i t i o n a l method of considering only the highest record.  Results of simulations for several  obtained  peak in the  c r i t i c a l U^Q'S showed that  elimination of a l l but the highest peak resulted in only a very lowering of the r i s k of f a i l u r e . same conclusion.  slight  A s i m i l a r TNA study [19] reached the  With this r e s u l t , i t should be possible to calculate  the r i s k of f a i l u r e with a f l a t U  5 0  curve (no waveshape dependence) by  using the histogram of magnitude of highest peak given in Figure 10. To this was  fitted  fitted  end, the t a i l to a Normal  Normal  0.28 p . u . .  density  Substituting  (greater  density  is  T  2.06 - 3.3  -/0.28  2  (see Figure 10).  2.06 p . u . and the  these two values  of f a i l u r e at a U^Q of 3.3 p . u .  3 $  than 2.5 p . u . ) of the histogram  =  + 0.16  - 59 -  standard  deviation  is  into equation 2-5, the r i s k  (a = 0.0511^) i s .  310(10"°)  The mean of the  The factor of 3 is required to account for the three phases involved (equation 4-3).  This result can be compared to the Monte Carlo estimate flat  U^Q curve case)  of 326(10"^).  produce this estimate.  (for the  Four thousand runs were made to  The agreement is very good and indicates  that,  for the p a r t i c u l a r model under study, the simple s t a t i s t i c a l method of section 2.4  is  adequate  in cases where waveshape  important.  - 60 -  dependence  is  not  6  CONCLUSIONS A computer program, RISK, has been developed which uses the Monte  Carlo method to estimate  the p r o b a b i l i t y of flashover  transmission l i n e for a single/switching The program accounts  future  program is  expansion,  failure.  operation (risk of f a i l u r e ) .  for the waveshape dependence of risk of  through the use of laboratory-generated The  to ground of a  modular,  for  switching-impulse  and can  example,  to  therefore  estimate  failure  data.  readily  accommodate  phase-to-phase  risk  of  The switching operation studied, or the model accounting for  breaker pole-scatter  can be e a s i l y  changed.  The program has been thoroughly tested by running a large number of simulations of a simple model of l i n e energization. General conclusions should not be based on the study of only one electrical  system but the following points r e l a t i n g to s e n s i t i v i t y  are  evident: the  procedures  slightly  and  influence  parameters  of  the  fitting  process  only  the risk of f a i l u r e , and thus do not warrant  much consideration. sensitivity  to  waveshape  can  comparison of tower-windows  be  significant  in  having quite different  cases  where  U^Q curves  is  desired. the presence of closing breaker pole-scatter influence  on  risk  of  failure,  and  can be an important  therefore  it  worthwhile to develop more physically r e a l i s t i c models.  - 61 -  would  be  The error in the estimate use of a f i n i t e  of risk of f a i l u r e  resulting from the  number of runs in the simulation has been found to be  quite large.  As a consequence,  the number of runs required to reach  an acceptable  level of accuracy may be in the thousands.  Three major uses of this program are foreseen to be: to  compare  gaps.  the  risk  A relatively  of  failure  accurate  the overwhelming uncertainty  of  different  tower  window  air  comparison can be made in spite in the absolute  levels of  of  flashover  p r o b a b i l i t y of each i n s u l a t i o n . to  investigate  the  influence  of e l e c t r i c a l  system  configuration  on r i s k of f a i l u r e . to  study  the  distribution  system switching  The above uses w i l l of  overhead  of  overvoltage  peaks  resulting, from  operations.  have practical  transmission  lines,  value  in the more economic design  important  transmission distances and/or high voltages.  - 62 -  in  cases  of  long  7  REFERENCES  1.  L.A. Snider, "A Method for Determination of Risk-of-Flashover Taking Into Account Switching Surge Waveshape", A paper presented to CIGRE Symposium - Study Committee 33, Montreal, June 1975.  2.  S.A. Annestrand, J . J . LaForest, and L . E . Zaffanella, "SwitchingSurge Design of Towers for UHV Transmission", IEEE Transactions on Power Apparatus and Systems, V o l . 90, pp. 1598-1603, Jul./Aug. 1971.  3.  N. Hylten-Cavallius and F.A. Chagas, "Possible Precision of S t a t i s t i c a l Insulation Test Methods", IEEE Transactions on Power Apparatus and Systems, V o l . 102, pp. 2372-2378, August 1983.  4.  Brown, Boveri and Co. Symposium, "Surges in High-Voltage Networks", K. Ragaller, e d i t o r , Plenum Press, New York, 1980.  5.  Project EHV S t a f f , "EHV Transmission Line Reference Book", Edison E l e c t r i c I n s t i t u t e , New York, 1968.  6.  IEEE Committee Report "Guide for Application of Insulators to Withstand Switching Surges", IEEE Transactions on Power Apparatus and Systems, V o l . 94, pp. 58-67, Jan./Feb. 1975.  7.  G. Le Roy, C. Gary, B. Hutzler, J . L a l o t , C. Dubanton, "Les Propri6te*s Diele*ctriques de V ' A i r et les Tres Hautes Tensions", Direction des Etudes et Recherches d ' E l e c t r i c i t y de France, P a r i s , 1984.  8.  C. Menemenlis and K. Isaksson, "Influence of the Various Parts of the Switching Impulse Front on Discharge Development", IEEE Transactions on Power Apparatus and Systems, V o l . 94, pp. 1725-1733, Sept./Oct. 1975.  9.  C. Carrara, L. D e l l e r a , and G. S a r t o r i o , "Switching Surges with Very Long Fronts (above 1500 y s ) : Effect of Front Shape on Discharge Voltage", IEEE Transactions on Power Apparatus and Systems, V o l . 89, pp. 453-456, March 1970.  10.  J . Lalot and B. Hutzler, "Influence of Non-Standard Switching Impulses on the Flashover Mechanisms of an A i r Gap", IEEE Transactions on Power Apparatus and Systems, V o l . 97, pp. 848-856, May/June 1978.  11.  B. Hutzler and D. Hutzler, "Breakdown Phenomena of Long Gaps Under Switching Impulse Conditions.. Time to Breakdown D i s t r i b u t i o n and Breakdown Probability: S t a t i s t i c a l Approach", IEEE Transactions on Power Apparatus and Systems, V o l . 94, pp. 894-898, May/June 1975.  - 63 -  12.  J . G . Kassakian and D.M. Otten, "On the S u i t a b i l i t y of a Gaussian Stress D i s t r i b u t i o n for a S t a t i s t i c a l Approach to Line Insulation Design", IEEE Transactions on Power Apparatus and Systems, V o l . 94, pp. 1624-1628, Sept./Oct. 1975.  13.  B.C. Papadias, "The Accuracy of S t a t i s t i c a l Methods in Evaluating the Insulation of EHV Systems", IEEE Transactions on Power Apparatus and Systems, V o l . 98., pp. 992-999, May/June 1979.  14.  J . Elovaara, "Risk of Failure Determination of Overhead Line Phase-to-Earth Insulation Under Switching Surges", ELECTRA, No. 56, pp. 69-87, January 1978.  15.  A. Reed, Private Communication, Mar./Apr. 1988.  16.  C. Menemenlis, G. Harbec, and J . F . Grenon, "Behaviour of A i r Insulating Gaps Stressed by Switching Overvoltages with a Double Peak", IEEE Transactions on Power Apparatus and Systems, V o l . 97, pp. 2375-2381, Nov./Dec. 1978.  17.  Project UHV Staff, "Transmission Line Reference Book, 345 kV and Above", Second E d i t i o n , E l e c t r i c Power Research I n s t i t u t e , Palo A l t o , 1982.  18.  N. F i o r e l l a , to-Phase and Surges in 4 September -  19.  A.R. Hileman, P.R. LeBlanc, G.W. Brown, "Estimating the Switching-Surge Performance of Transmission Lines", IEEE Transactions on Power Apparatus and Systems, V o l . 89, pp. 1455-1469, Sept./Oct. 1970.  20.  C. Dubanton and G. Le Roy, "Research Into a Practical Method to Determine the Probability of Failure of an. E.H.V. Transmission L i n e " , IEEE Transactions on Power Apparatus and Systems, V o l . 94, Nov./Dec. 1981.  21.  R.Y. Rubinstein, "Simulation and the Wiley and Sons, New York, 1981.  22.  J . G . Kassakian, "The Effects o f Non-Transposition and Earth Return Frequency Dependence on Transients Due to High Speed Reclosing", IEEE Transactions on Power Apparatus and Systems, V o l . 95, pp. 610-620, Mar./Apr. 1976.  23.  H. Dommel, M. G a v r i l o v i c , and L.A. Snider, "On the Accuracy of D i g i t a l and TNA Techniques for Determination of Overvoltages and Risk-of-Failure", paper presented at CEA Spring Meeting, Vancouver, March 1983.  G. Santagostino, L. Lagostena, A. Porrino, "PhasePhase-to-Earth Risk of F a i l u r e Due to Switching UHV Systems", Paper 33-12, CIGRE Conference, 27 August, 1980.  - 64 -  Monte Carlo Method", John  K.H. Week, "Proposals for the Risk of Flashover Determination on Two-Phase or Three-Phase Insulation Systems", CIGRE Internal Document SC 33-77 (WG 06) 2 IWD.  APPENDIX A Derivation of the Simple Form for Risk of Failure  The stress function i s :  f(x)  =  i l_ o /2T  (x - m ) 1— ) Za f  exp (-  (A-l)  c  f  f  The strength function i s :  i g(x)" = - A = a /2T g  / exp ( J /_«  The p r o b a b i l i t y of flashover  R =  ,2 (S - m j ' 3_) 2a g  dC  (A-2)  2  i s given by:  f(x) g(x) dx (x-m ) - 5 — f  exp (  The double integral  Z  (A-3) i s evaluated  two changes of v a r i a b l e .  - 66 -  (5- m ) Tf-—) dCdx 2  (A-3)  by f i r s t making the following  (x - m ) f  Equation (A-3)  , and  becomes, c^x  + m^ - m. f '"g  1  oo  f t .2 ,,2 exp (- f — - J — )  • oo  dx'  in (A-4)  is shown as the area below l i n e q  The equation for l i n e q i s :  °g  °g  The perpendicular distance of l i n e q to the o r i g i n i s :  min /  Of  2  + a f °9 +  (A-4)  oo  The region of integration in Figure 2 1 a .  ds'  2  +  - 67 -  s  / / / (b)  (a)  FIGURE 21:  By virtue of the  REGION OF INTEGRATION  symmetry  of  the  integrand of  (A-4),  the  integration can be simplified by rotating l i n e q u n t i l i t to the x' axis (Figure  region of is p a r a l l e l ,  21b).  Equation (A-4) then becomes:  min x' exp (- ^ -  2  R =  - ^-) £  X  -CO  d " dx' - ' dx'  J  5  (A-5)  _ CO  which can, since the l i m i t s are now a l l constants, be s i m p l i f i e d to:  -  68  -  -r • R =  i-  ,00  2  mm [  exp (- Kr  m  y  2  f  +  m  ) d« 5  2  00  which completes the derivation,  - 69 -  2  j exp (- 4-  g  exp (- y - )  /27  /  dz  ) dx'  .  .  APPENDIX B  The Inverse Transform Method of Generating Random Variables  The derivation variables  of  the  inverse  transform method of generating  random  is well-known and can be found in numerous references.  i s repeated here for completeness  It  because of i t s use in the discussion  of the Monte Carlo method (section 3) and in Appendix C.  If a random variable X with d i s t r i b u t i o n F(x) is required i t below  that  density  Proof:  X = F ( U ) , where _ 1  in the interval  U is  a random variable  is shown  with  uniform  (0,1).  Assume X = F ( U ) - 1  Then prob (X six) = prob ( F ( U ) Sx) _1  = prob (U £F(x)), since F(x) is monotoni'cally increasing.  But prob (U ^F(x)j = F(x), and the proof i s  - 70 -  complete.  APPENDIX C Generating Random Numbers with Truncated Normal Density  We need  to  generate  random numbers  having  the  density  of  a Normal  d i s t r i b u t i o n truncated at no.  The desired density  f(x)  where  =  A  4=  is:  A  'I e  2  +  ^f. ,  -n s x *n  (C-l)  = *(-n),  * is the Normal cumulative d i s t r i b u t i o n  Equation (C-l) can be rewritten  in the equivalent  -x  f(x) = (1 - 2 A J [ — -  /2T(1  n  = (1 - 2A ) f ^ x ) n  Equation (C-2) f^(x)  and f ( x ) , 2  1  - 2A ) n + 2A  n  e  2  function.  form,  2  ] + 2A [ 4 ] n  d n  f (x) 2  is interpreted as the sum of two probability each conditional upon a parameter y:  - 71  -  (C-2)  densities,  f(x)  -  f^xly  = y)  prob  x  (y =  Here,  prob  (y = y )  = 1 - 2A  and  prob  (y = y )  = 2A  where  U i s a random v a r i a b l e  The by  1.  2.  x  2  procedure, ( C - l ) i s as  first  then,  + f (x|y  n  n  = prob  (U < 2 A ) .  with  uniform density  random  sample from  if  u  then  sample from  method  numbers  i  - 2A . n  from  ( 0 , 1 ) and  (see Appendix  B)  fj(x):  therefore,  f (x):  therefore,  with  a density  given  2  determine  with  «(x)-  2  (0,1).  f (x).  x  F (x) =  x = $  _ 1  =  u  2  [ u ( l - 2 A )• + 2  F (x) = £  + \  2  x = 2n(u  - A  i . 2 A  2  ?  -  =  i)  - 72  -  by  the  inverse  the.appropriate density  2  with  on  f ^ x )  f (x)).  with  2  (0,1).  then  2  y )  follows:  n  u  (y -  p  Uj < 1 - 2 A ,  draw  prob  (U < 1 - 2 A J  if  s  2  = prob  to generate  draw Uj f r o m  1  = y )  2  u  2  A ] p  transform (f^(x)  or  APPENDIX D F i t t i n g a Standard Impulse  To simplify  the mathematics,  the  data  is  first  shifted  so that  peak occurs at t = 0, and normalized by dividing by the crest  the  value.  The equation to be f i t to the data is then:  -cx(t + t f(t)  where k t  cr  = k(e  ) (D-l)  - e  is a normalizing constant, i s the  time-to-crest,  c  is a constant which determines the r a t i o of  X  is to be determined.  By algebraic  manipulation the  constants  k and t  to t  c r  ,  and  can be eliminated  from ( D - l ) , giving  f(t) =  ce  -At _ - C A t e  c - 1  Recognizing the discrete nature of the data points, (D-2)  (D-2)  becomes:  f ( i A t ) = ce  -Xi At  -cXl'At (D-3)  c - 1  where At is the time between points.  Once the constant c is chosen, the f i t  is accomplished by finding the  value of X which minimizes the sum of the squares between f ( i A t )  and the  data  points,  i.e.  the  of the  following  difference function  is  minimized over X,  o  2  Cf(iAt) - y . ]  i=-N  where  2  1  y . are the (shifted and normalized) data points, N i s the number of points  involved  in the  between the threshold and c r e s t ) .  Once X i s determined the time-to-crest  t  _ Inc cr " x ( c - l )  - 74 -  i s given by:  fit  (the number  Figure 22  shows  an  example  fit  with  TIME ( B i l l i s e c o n d s )  FIGURE 22:  EXAMPLE FIT  - 75 -  c  =  50  (corresponding  to  APPENDIX E Transient Program Input Data  50.MILE/SECTION UNTRANSPOSED LINE MODEL 6: 150. MILE LINE . 0 1 50.E-6 2. E-2 -1 0.2834498.930.1515 50.00 1 3 -1N0DA00N0DA05 0.0599295.230.1828 50.00 1 3 -2NODB00NODB05 0.0550249.510.1851 50.00 1 -3NODC00NODC05 0.6O237E 00-0. 70711E 00-0.40642E 00 .0.52371E 00-0. 15307E- 11 0.81831E 00 0.60237E 00 0. 7.07 H E 00-0.40642E 00 0.2834498.930.1515 50.00 1 -1NODC05NODC10 0.0599295.230.1828 50.00 1 -2NODA05NODA10 0.0550249.510.1851 50.00 I -3NODB05NODB10 0.60237E 00-0. 70711E 00-0.40642E 00 0.52371E 00-0. 15307E- 11 0.81831E 00 0.60237E 00 0. 70711E 00-0.40642E 00 -1N0DB10N0DB15 0.2834498.930.1515 50.00 1 -2N0DC10N0DC15 0.0599295.230.1828 50.00 1 -3NODA10NODA15 0.0550249.510.1851 50.00 1 0.60237E 00-0. 70711E 00-0.40642E 00 0.52371E 00-0. 15307E-11 0 81831E 00 0.60237E 00 0. 70711E 00-0.40642E 00 CASE  SRCA00NODA00 0.0021000 SRCBOONODBOO 0.0021000 SRCCOONODCOO 0.0021000 14SRCA00 14SRCB00 14SRCC00  1.00000 1.00000 1.00000  10.00000 10.00000 10.00000  60.00000 78.91091 60.00000 -41.08909 60.00000 198.91092  N0DA15N0DB15N0DC15  - 76 -  -1.00000 -1.00000 -1.00000  60.  60.  


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