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 referred.to 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 to.compare 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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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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Title | Application of the Monte Carlo method to the estimation of the risk of failure of transmission line insulation |
Creator |
Leonard, Ronald William |
Publisher | University of British Columbia |
Date Issued | 1988 |
Description | A digital program has been developed which uses the Monte Carlo technique to estimate the risk of failure of transmission line insulation during system switching operations. The effect of overvoltage waveshape on insulation strength is included. A simple electrical system is used to demonstrate the method and to investigate some general aspects of error and parameter sensitivity. |
Subject |
Monte Carlo method Telecommunication lines Electric lines |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-09-16 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0065439 |
URI | http://hdl.handle.net/2429/28496 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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