APPLICATION OF THE MONTE CARLO METHOD TO THE ESTIMATION OF THE RISK OF FAILURE OF TRANSMISSION LINE INSULATION By RONALD WILLIAM LEONARD B . A . S c , 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 Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 1988 © Ronald William Leonard, 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of 9-\io\rlooX. fc^l^nV The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 1 2 T u J n l°>&% DE-6G/81) ABSTRACT A d i g i t a l program has been developed which uses the Monte Carlo technique to estimate the risk of fai lure of transmission line insulation during system switching operations. The effect of overvoltage waveshape on insulation strength is included. A simple e l e c t r i c a l system is used to demonstrate the method and to investigate some general aspects of error and parameter s e n s i t i v i t y . - i i -TABLE OF CONTENTS Section Page 1 INTRODUCTION 1.1 Motivation 1 1.2 Content 2 2 SWITCHING-SURGE INSULATION COORDINATION 2.1 General 4 2.2 The Strength Distribution 6 2.3 The Stress Distribution 13 2.4 Simple S t a t i s t i c a l Methods 14 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 17 2.5.2 Snider's Simulation Technique 18 2.6 The Equivalent Time-to-Crest of an Overvoltage Peak 19 2.7 Usefulness of the S t a t i s t i c a l Methods 22 3 MATHEMATICAL BACKGROUND - THE MONTE CARLO METHOD 3.1 System Model 25 3.2 Integral Solution 26 3.3 Sample-Mean Monte Carlo Method 28 3.4 Variance Reduction 30 4 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 F i t t i n g 4.2.5 Calculation of Risk of Failure 5 TEST RESULTS 5.1 Introduction 41 5.2 Distribution of Crest Voltage and Time-to-Crest 42 5.3 Error Considerations 45 5.4 Sensit ivity to F i t t i n g Parameters and Procedures 50 5.5 Sensit ivity to Waveshape 53 5.6 Sensit ivity to Parameters of the Strength Distribution 56 5.7 Sensit ivi ty to Pole-Scatter 57 5.8 Comparison with Simple S t a t i s t i c a l Method 59 6 CONCLUSIONS 61 7 REFERENCES 63 33 34 36 37 39 39 - i i i -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 70 Variables C Generating Random Numbers with Truncated Normal 71 Density 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. Sensit ivi ty 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 5 Q Curve for 4 m Rod-Plane Gap 12 4. Combination of Stress and Strength to Yield 15 Risk of Failure 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. Sensit iv i ty to Inclusion of Plateau-Detecting Routine 51 16. Sensit ivity to F i t t i n g Parameters and Procedures 51 17. Family of U 5 Q Curves 54 18. Risk of Failure for Different U 5 0 Curves 55 19. Sensit ivity to Standard Deviation of the Strength Distribution 56 20. Sensit ivity to Pole-Scatter Standard Deviation 58 21. Region of Integration 68 22. Example F i t 75 - vi -ACKNOWLEDGEMENTS, The author gratefully acknowledges the assistance of the following people: 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 Research. Dr. S.R. Naidu, v i s i t i n g professor from Universidade Federal da Paraiba, Campina Grande, B r a s i l . Mrs. Gwen H a l l , B.C. Hydro, for the typing. - vi i -1 INTRODUCTION 1.1 Motivation In 1982 B r i t i s h Columbia Hydro and Power Authority (B.C. Hydro) began consideration of a new transmission tower design for eventual use in the then planned hydro developments for northern B.C. The new design, employing an inverted delta phase configuration and a crossrope suspension structure (Figure la) was a considerable departure from the conventional f l a t - l i n e configuration. (Figure l b ) . Crossrope, (a) (b) FIGURE 1: INVERTED DELTA AND FLAT CONFIGURATIONS The relat ively close spacing of the phases compared with other designs leads to lower positive-sequence inductive reactance and consequently greater surge-impedance loading. This translates into - 1 -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 design for adequate switching impulse strength. To accomplish t h i s , a test tower was erected in.1983 at B.C. Hydro's Research and Development Laboratory in Surrey, and testing begun. The testing included investigation of phase-to-ground switching impulse strengths and, because of the close phase spacing, phase-to-phase strengths. The test program was never completed, however, as in 1985 B.C. Hydro indefinitely deferred plans for northern development, and further testing was cancelled. To aid the design process an analytical comparison of the switching surge risk of f a i l u r e of the new design with existing designs was planned, using measured data as input. This thesis is based on the work done to develop a digi tal computer program to accomplish that. 1.2 Content Previous work by L.A. Snider [1] at the Hydro Quebec Research Institute (IREQ) forms the basis for this work. Snider used the Monte Carlo method to estimate the probability of flashover from conductor to tower for a single switching operation (risk of fai lure during switching operation). A transient network analyzer (TNA) was used. • •. •' This thesis presents an entirely digi tal solution to this same problem using the U.B.C. Transients Program (TP) to generate - 2 -overvoltage waveforms. The main body of the thesis begins in Section 2 with a discussion of the relevant elements of s t a t i s t i c a l insulation coordination, with particular attention to the concept of equivalent time-to-crest. In Section 3 i t is shown how the problem f i t s into the framework of the Monte Carlo method. Results from Monte Carlo theory give estimates and suggest means of reducing the large errors associated with the simulation. Section 4 covers the organization of and procedures used in the digi tal program. The last section gives the results of a thorough exercising of the program on a simple test case. The objective is to study general aspects of the program's use as an aid in transmission l ine insulation design, focussing mainly on the subjects of error and s e n s i t i v i t y . - 3 -2 SWITCHING-SURGE INSULATION COORDINATION 2.1 General High voltage transmission l ine insulation is subjected to four types of e l e c t r i c a l stresses: the normal power frequency system voltage, short duration temporary overvoltages near the power frequency, fast-front surges (usually l i g h t n i n g - i n i t i a t e d ) , and slow-front surges (usually switching-initiated). It is the l a t t e r , the slow-front surges that are the subject of this work. The magnitude of uncontrolled switching surges is d i r e c t l y proportional to l ine voltage and for high voltage l i n e s , measures are generally taken to l i m i t their magnitude. Nevertheless, at l i n e voltages above 300 kV, switching surge strength is usually the most c r i t i c a l design consideration. The few special circumstances which cause the other factors to be paramount include: high keraunic level (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 in terms of the response of insulation to stress the source is of l i t t l e importance. The key parameter in the categorization of voltage stress is waveshape, in particular the time required to reach peak voltage (time-to-crest). As a guideline, slow-front surges are considered to have times-to-crest ranging from about 50 us to 2 ms, with the longer times-to-crest (1 to 2 ms) appearing in EHV networks fed from inductive sources (e.g. a generating station with no other lines connected.) - 4 -Overvoltages having times-to-crest in the range given above occur as a result of a variety of switching operations such as l ine energization and reclosing and the switching of reactive loads. They are also caused by fault i n i t i a t i o n and clearing. In this work the case used as example w i l l be simple l ine energization, but in practice a l l types of switching operations must be given attention for a complete assessment of insulation risk of f a i l u r e . Besides overvoltage magnitude and waveshape, meteorological variables are important factors. Rain, wind, solar radiation, a i r humidity and density can be important considerations, but are outside the scope of this work. Switching overvoltages are dependent on the relative timing of the operation with the power-frequency voltage wave. There is some degree of randomness associated with the point on the power-frequency waveform at which the breaker contacts close, so that in the case of a three-phase energization, the closures occur at different points on wave for the three different phases. For a given switching operation and e l e c t r i c a l system, these sources of random fluctuation produce overvoltages at the tower insulations that can be described by a probability d i s t r i b u t i o n . The strength of the insulation undergoing the overvoltage stress can also be described in probability terms. Economic design of transmission l ine insulation should account for the random nature of both the stress and strength components. Savings from reduction in insulation requirements may be achieved by accepting a certain probability of insulation flashover for a given switching operation - 5 -(risk of f a i l u r e ) . This is feasible because the flashover of external self-restoring tower insulation w i l l normally not result in damage to the insulation and the effect on the e l e c t r i c a l system can be minimized by quick fault-clearing action. The next several sections discuss the s t a t i s t i c a l 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 Distribution This, section considers some relevant aspects of the response of * a i r gap insulation to switching-impulse stress. The particular class of a i r gap relevant to the question of transmission l ine insulation is characterized by the following two geometrical properties: the length is greater than 1 or 2 m. the radius of curvature of the energized electrode is small (a few cm). The response of an a i r gap with such a geometry to a switching impulse of positive polarity is governed by a complex physical process consisting of several stages, the details of which can be found in the l i t e r a t u r e . For our purposes we w i l l only note that the presence of a "leader 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 actually 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 the energized electrode. The negative-polarity strength w i l l consequently normally exceed the posit ive-polarity strength. It is therefore common in transmission line insulation coordination work to ignore negative-polarity overvoltages. There are, however, reported cases of lower negative-polarity strength (discussion to [2] by Nigol and Reed). Under stress by a switching impulse, the time to breakdown is a random variable. The uncertainty in breakdown time is a result of the many indeterminable physical parameters influencing the breakdown process, including number of free electrons in the v i c i n i t y of the positive electrode, type of gaseous molecules present, their positions, to name a few. As a result of this process taking place over time, the probability of breakdown is dependent not only on the crest voltage of the applied impulse but also on the shape of the impulse. The study of the breakdown of these gaps is accomplished in high-voltage laboratories by the application of standardized impulses produced with Marx impulse generators. These impulses have the shape of a double-exponential in which the voltage is given by the equation U(t) = C(e 1 - e 1 ) - 7 -The shape of the impulse is determined by the two time constants, or equivalently by the time-to-crest (t ) and the time taken to decay to one-half crest value ( t ^ 2 ) ( s e e F i 9 u r e 2). TIME ( l i l l i s e c o n d s ) FIGURE 2: TYPICAL SWITCHING IMPULSE The time constants are chosen such that << x^, giving an impulse with a relat ively fast rise and slow decay. Typical ly , t cr might be 250 ys, and t ^ 2.5 ms. For a given' impulse shape (ratio of 1^/2 to t c r ) , the probability of gap breakdown is given as a function of crest voltage U cr A commonly used function is the Normal cumulative d i s t r i b u t i o n , prob. of breakdown = g(U ) 1 exp(-(x - m_) (2 -D 9 I oo g where m^ and are the mean and standard deviation, respectively, of the Normal d i s t r i b u t i o n . laboratory testing procedure is not obvious and is therefore considered at some length. Imagine that at any instant in time there exists a gap withstand, voltage defined by the following property: i f an impulse (of a given shape) with crest voltage s l i g h t l y greater than the withstand voltage were to be applied to the gap, breakdown would occur; but i f the crest value were to be s l i g h t l y less , breakdown would not occur. This withstand voltage changes from one instant in time to another because of the random nature of the breakdown process. We would l ike to determine the cumulative distribution of this random variable, withstand voltage. To do so we apply impulses (of constant shape) consecutively in time with random crest voltages and record for each application i f breakdown occurred. If for a crest voltage U c r , breakdown occurs, then we conclude that at the time the impulse was applied the withstand voltage was less than or equal to U . If no breakdown occurred, the withstand voltage was greater than U c r . After many such experiments we group the applied crest voltages into bins of some width and determine, for each b i n , the fraction of tests The interpretation of the above equation and relationship to ) - 9 -resulting in flashover. This fraction is the value of the cumulative distribution of withstand voltage at U c r , that i s , the probability that at any given instant in time the withstand voltage w i l l be less than U " c r . We note that this probability is just the probability that breakdown w i l l occur i f an overvoltage with crest U G r is applied at any instant in time. Equation (2-1) can be interpreted, therefore, as either a cumulative distribution function of withstand voltage or as a probability of breakdown conditional upon a given crest voltage. Both interpretations are found in the l i t e r a t u r e and thus there is potential for confusion - one reference [3] states that the strength function (2-1) has nothing to do with a cumulative distribution and should not be referred.to as such. The interpretation. of the strength as a cumulative distribution is valuable in providing a l ink to the underlying physical mechanism and the Normal distribution is usually a reasonable choice given the large number of randomly fluctuating physical parameters in the breakdown process. The conditional probability interpretation i s , however, more convenient when discussing the combination of stress and strength to y i e l d total probability of breakdown (risk of failure) and leads more naturally to simulation methods. The experiments as described above for determining the cumulative distribution of withstand voltage would be far too time-consuming and expensive to perform. In fact, the usual procedure is to apply a series of impulses with crest voltage near the mean of the distribution (called U 5 Q ) in order to provide a reasonably accurate - 10 -estimate thereof. Typical ly , the number of tests required would be less than 5 0 . 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 . A i r humidity, temperature, pressure and wind are meteorological variables which influence the breakdown process. These variables are generally held as constant as possible during testing of an a i r gap. The dependence of the mean and standard deviation on these variables can be estimated by performing additional series of tests. The probability of breakdown as given by equation (2-1) can be modified by truncating the Normal distribution at some lower bound (typically about mg-3cTg) in order to reflect the physical real i ty that, for crest voltages below this bound, breakdown w i l l never occur [ 4 ] . Equation (2-1) gives the breakdown probability for impulses of some fixed shape. The particular t c r and t ^ °f the impulses wil l in general influence the mean (U^) and standard deviation. If t ^ is held constant, the plot of U c n versus t (called a U c r v curve) w i l l 50 cr 50 show a minimum at some value of t (called the c r i t i c a l time-to-cr crest) . The minimum value of U^ Q , called in this work the c r i t i c a l UgQ, w i l l depend on the gap spacing and electrode geometry. Figure 3 gives an example Ugg curve ( 4 m rod-plane gap taken from reference [ 5 ] , page 217). The standard deviation plotted versus t wil l show a similar shape, with the minimum o occurring near the c r i t i c a l time-to-crest [6]. - 11 -1.2 o a Ld Nl H _ l <x n cc o z 1.1 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 less than c r i t i c a l , breakdown, i f i t occurs, w i l l occur after the crest of the impulse, and conversely, f o r impulses with times-to-crest greater than c r i t i c a l , breakdown w i l l occur before or at the crest [4,7]. For switching impulse strength the variation of U 5 Q with t (and *l/2^ 1 S P r i m a r i ^ y °f interest for the determination of the c r i t i c a l UgQ, as . this is the main parameter upon which switching-surge insulation (air-gap dimension) is based. The strength of air-gaps stressed by more complex shapes (such as f a s t - r i s i n g bumps superimposed on a slower main impulse) is reported in the l i terature [8,9,10], but i t has been generally found [7] that no form of complex waveshape w i l l result in a c r i t i c a l U 5 Q below that found by testing with standard impulses. - 12 -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 to the question of gap dimension. Some studies [1] have been done, however, in which waveshape is considered e x p l i c i t l y , and one of the objectives of this work is to examine the sensit ivi ty of r i s k . o f f a i l u r e to one procedure for accounting for waveshape - the use of the equivalent time-to-crest of an overvoltage peak. This procedure is discussed in section 2.6. . 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 Distribution The simple s t a t i s t i c a l methods of estimating risk of fai lure (section 2.4) make e x p l i c i t use of the probability density of crest voltages of slow-front surges. The density 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 program [13] such as an electromagnetic transients program (TP). The variation in the switching overvoltage waveform is generated by randomly choosing (according to some probability density) the closing times of the breakers for each operation or computer run. - 13 -The density is often approximated (particularly in the important t a i l region) by a Normal density, the parameters of which can be determined by f i t t i n g to the histogram of crest voltages. The density of crest voltages, U c r > is given by 1 ( U r r m f ) 2 f(U. c r ) " — ^ — exp( - c r - 2 f ) (2-2) V2TT 0^ 2o^ where m f and a f are the mean and.: standard deviation, respectively, of the Normal d i s t r i b u t i o n . The use of the Normal density is generally considered to be reasonably accurate [12]. It is possible to extend equation (2-2) to include the time-to-crest parameter, in which case the density becomes a joint density of U c r and t Q r . 2.4 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 risk of f a i l u r e of a single insulation subject to an overvoltage 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 strength distribution and the stress distribution are combined to y i e l d the risk of flashover for a single switching operation: r R. = f(U) g(U) dU (2-3) - 14 -where f(U) is the probability density of crest voltage (Eqn. 2-2) and g(U) is the probability 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 S-Q. ,—.. T3 C +J ra c ISI OJ s_ OJ +J s-oo +-> •—" oo ^ ' >-h- >-t—« 1— _ l t—i oo ca s: •a: LU ca O o al >-Q. t— 1 HH CO < o a; 1.5i VOLTAGE ( p . u . l FIGURE 4: COMBINATION OF STRESS AND STRENGTH TO YIELD RISK OF FAILURE. The function g(U) is best interpreted as the probability of flashover conditional upon the crest voltage, and to reflect t h i s , equation (2-3) is better written as f oo f(U) (prob. of f lash. |U) dU (2-4) - 15 -where R is the total (unconditional) probability of flashover (risk of f a i l u r e ) . Substituting into (2-4) the previously derived stress and strength functions, we obtain, R = 00 (U - m - ) ' exp ( — - j - ) 2a, /2TT a. U 2 -exp( * ) d? 2a_ dU / 2 T exp (- | - ) dz, (2-5) where Y J 2 4. 2 •o f + a g Equation (2-5) is extensively used but rarely derived. The only derivation that has been found is by a Laplace Transform method [ 3 , 2 2 ] . Appendix A gives an alternative derivation by direct integration. Equation (2-4) can be generalized to give the risk of fai lure for a multiple-insulation system, such as a three-phase transmission line with many towers [ 1 4 ] . Time-to-crest can also be included. In these more involved cases analytical solution becomes d i f f i c u l t and simulation or numerical integration methods become attractive. - 16 -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 Examination of equation (2-4) reveals that the risk of fai lure is just the expected value of the conditional probability of flashover ( i t s e l f a random variable). The interpretation of the conditional probability of flashover as a random variable leads one to the Monte Carlo method. Its use permits the evaluation of the risk of fai lure to proceed without reference to the joint probability density function of peak voltage, t ime-to-crest, or any other physical random variables. Instead, the probability of flashover is seen as a random variable which is a function of the random variables used to generate the switching overvoltages, that i s , the points-on-wave of the closing breakers. The application of this method substitutes an estimation of the expected value of the conditional probability of flashover (for the exact evaluation of the unconditional probabil i ty) . If the value of the conditional probability of flashover for a given overvoltage waveform (produced by a sampling of the random variables controlling the energizing breaker closing) is denoted by g.., then the estimation of the risk of f a i l u r e after N random samplings i s : R est (2-6) T y p i c a l l y , N wil l vary from several hundred to several thousand, depending on the accuracy required. - 17 -The Monte Carlo method w i l l be treated in detail in section 3. It is convenient to state here, however, that the method is a technique for estimating an expected value of a random variable, and that expected value can always be estimated by using a technique of numerical quadrature. Monte Carlo techniques tend to be more eff icient when the number of random variables (equivalent to the dimensionality of the associated integral) is large. The primary advantage of this method versus the methods involving the probability density (Equation 2-4) is that the dependence of the probability of flashover for a switching event on the relevant physical parameters can be made very complex. For example, the probability of flashover to ground of one phase of a three-phase l ine may be a function of the voltages on the other two phases. This additional complication can be easily handled by a simulation technique. 2.5.2 Snider's Simulation Technique Snider's [1] contribution makes use of the f l e x i b i l i t y of the Monte Carlo method by taking into account the probability of flashover of each individual peak in a switching overvoltage waveform. The time-to-crest of each peak is also taken into consideration. - 18 -If n individual peaks occur in a waveform, the risk of fai lure to ground i s : n R = 1 - It (1 - R.) (2-7) i=i . . . . * • • - +• h where R. is the flashover probability associated with the i peak. 2.6 The, Equivalent Time-to-Crest of an Overvoltage Peak The shapes of switching surge overvoltage peaks occurring on e l e c t r i c a l systems are manifold but tend generally to be f a i r l y symmetric [7]. In contrast, the shapes of the impulses used in the laboratory are not at a l l symmetric; having the fast rise 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 is 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: the desired quantity is then the time-to-crest of the f i t t e d impulse. The region, of f i t is 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 - 1 9 -probability of breakdown of a rod-plane gap. The lower l i m i t of the region of influence 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. 2. _ i i i i I I i i i i i i i I i i i i i ~ / V \ Threshold = 1.5 p.u. ~; 1. a - / \ / 0. J \ I Lu — 1 / \ / \ A /: CD V V-s / ^A / cr -1. t-j — II -• -2. > — -3. ! I ! ! i I 1 " " 1 ! i f i t i i 1 ! ~ 0 20 T I M E ( m i 1 1 i s e c o n d s ) FIGURE 5: FITTING THE PEAK TO A STANDARD IMPULSE There are (at least) two possible objections to this f i t t i n g procedure: 1. The portion of the overvoltage peak after crest is 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. For cases in which the time-to-crest is less than c r i t i c a l the effect of the - 20 -elongated t a i l of the impulse may be to raise the probability of flashover of the impulse above that of the actual wave. Reference [4] , pages 181 and 182, notes that for impulses with time-to-crest below c r i t i c a l the leader propagation is inhibited by the cessation of voltage rise after the crest. So i t may be that the sharper f a l l - o f f of voltage of the actual wave could result in greater inhibit ion of the leader. Reference [7], page 580, states that, in general, the elongated t a i l of the impulse w i l l result in a more severe stress of the a i r gap than any actual waveform with the same crest value, leading, consequently, to a general overestimation of the probability of breakdown of the actual overvoltage. 2. If the actual peak has a shape which does not resemble the smoothly r i s i n g standard impulse, such as a f a s t - r i s i n g bump after a relat ively slow r i s e , then the f i t t i n g may not be v a l i d . There are laboratory results reported [16] which provide guidelines in some cases (presented in section 4) but i t is apparent that a certain degree of arbitrariness w i l l be inherent in any attempt to handle "non-smooth" waveshapes. If most of the overvoltage peaks occurring in the real system are smoothly-rising from around corona inception voltage to crest voltage the use of the equivalent time-to-crest concept may not be too u n r e a l i s t i c , and i t is generally accepted in the l i t e r a t u r e . Reference [17], page 504, for example, states that every type of wave may be represented by the equivalent time-to-crest. - 21 -2.7 Usefulness of the S t a t i s t i c a l Methods Some examples of the use of s t a t i s t i c a l 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] reports the use of the Monte Carlo method in the design of a UHV transmission system. Several switching operations were considered including fault c learing, l ine energization and single and three-phase reclosing. The study was conducted using a transient network analyzer. Five hundred switching operations were conducted for each simulation. No account of waveshape was taken. An older reference [19] reports the use of s t a t i s t i c a l methods (again with an analog system) to estimate the risk of fai lure of a UHV transmission l i n e . The approach of this work was to determine the density of peak overvoltages and then apply an equation such as (2-4) to estimate the risk of f a i l u r e . This approach differs from that of [18] discussed above and the method outlined in section 2.5.1 wherein the density of overvoltage peaks is not considered (equation 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. The argument is that the density of overvoltage peaks need only be determined on the important t a i l part of the d i s t r i b u t i o n , and so methods such as the Monte Carlo are ineff icient since no consideration to the density of overvoltages is given. On this point i t can be noted that, given an approximate knowledge of the dependence of the conditional probability of flashover on the input random variables (breaker closing times), methods from Monte Carlo theory such as - 22 -importance sampling [21] may possibly be effective in improving the efficiency. This p o s s i b i l i t y was not investigated in this work, and is discussed only brief ly in section 3. Any method of improving efficiency, either by concentrating on the t a i l of the overvoltage distribution (called "topological search" methods in [19]), or by the use of importance sampling, requires some prior knowledge of probability 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. There 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; there are the errors involved in the determination of the U^Q and a of the insulation in the high-voltage laboratory; 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 distribution of overvoltage peak magnitudes. F i n a l l y , there are the errors caused by meteorological random variables. . A good summary of the relative contribution of the error terms is found in [23J. The consensus in the l i terature on the u t i l i t y of s t a t i s t i c a l methods for the estimation of the risk of f a i l u r e of transmission lines is that these methods are of limited use in view of the high sensit iv i ty to several parameters needed in the determination. The opinion is also, however, that the systematic errors can be accepted - 23 -when only a comparison of l ine designs based on relative risk of fai lure is required [ 7 , 2 0 , 2 3 ] . The Monte Carlo method is well-suited to such comparisons. Through the use of correlated sampling (discussed in section 5.3) the effect of small changes in the model parameters can be assessed relatively accurately. The construction, testing and study of the simulation model can help in identifying the high-sensit ivity parameters and in suggesting areas for further study which may lead to improved estimates of the absolute risk 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 their external environment. The system is physical , but what is studied must necessarily be only a model, an abstraction which, i f properly devised, can give reasonably precise solutions to the investigated problem. The Monte Carlo method is so-named because the system is studied by assigning the outcomes of sampling experiments on probability distributions to certain elements of the model, and observing some resulting output of the model. The model may have as an essential component a random process, (e.g. random breaker closing times), or i t may have none (e.g. the Monte Carlo method can be used to compute integrals) , in which case the stochastic process is introduced a r t i f i c i a l l y . The same method of solution is used for both cases. In the general case the input to the model is an ordered set of outcomes of n independent sampling experiments on the uniform density in the interval (0,1) and can be defined symbolically as ( u 1 5 u 2 , . . . , u n ) E \ J 1 x V 2 x •••• x ^ " n ( 3 - 1 ) where u^ is the outcome of a single sampling experiment on the uniform density in the interval (0,1), and \ / . is the set of a l l possible outcomes of the experiment. - 25 -The model i t s e l f is a process defined by some algorithm using mathematical and logical operations. Part of the model wil l involve the transformation of . the input .random variables to the random variables representing model elements. The model output w i l l in general be a s t a t i s t i c a l estimate of some model quantity. 3.2 Integral Solution Assume that each random value is transformed to some other random variable X.., and that the output s t a t i s t i c a l estimate is , an estimate of the expected 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 ) ] - estimate ofjj... jf^i^f^ify) • • • f n ( ? n ) g ( c 1 .C2»< • • »5 n )- d C.i d 5 2 -• - d 5 n -oo —00 - 00 n integrals (3-2) where f.. is the density function of X.. and g ( X j , X 2 , . . . , X n ) is the random variable whose expected value is required, g being some function. Associated with each density function f. there is a corresponding distribution function F . , such that: - 26 -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 ^ I K ) (3-4) Using (3-3) and (3-4) and noting that ^ u i , (3-2) can by change of variables £.-»• u. be re-written as Output = estimate of I j . . . | g [ F 7 A ( u i ) » F o A ( u 0 ) , . . . ,F~A(u_)] d u , d u 0 . . .du. n integrals (3-5) ( J j g [ F - 1 ( o 1 ) , - 1 ( u 2 ) , . . . , F " 1 ( n ) ] 1 2 . n Equation (3-5) says that the output of a Monte Carlo simulation involving n independent random variables is an estimate of an n-dimensional multiple integral . Methods of numerically evaluating multiple integrals exist and the question arises whether there is any advantage to using a Monte Carlo technique. The answer is that for integrals of high dimensionality the Monte Carlo method can indeed by more e f f i c i e n t , especially in cases where great accuracy is not required. - 27 -3.3 Sample-Mean Monte Carlo Method The integral in (3-5) is the expected value of the random variable G (associated with the value of g), where g is a function of n random variables, a l l of which are distributed uniformly in the interval (0,1). If g^ is the value of G resulting from the outcome (u^, u 2 , u n ) of a sampling of the input random variables, then an estimate, y, of the expected value of G, denoted by E[G], after N samplings is formed by the sample mean i N Y = E[G] = £ Z g. (3-6) N i = l 1 The random variable associated with the outcome y, denoted by r has the variance • ... v a r ( r ) - c r 2 - ^ " ^ ^ ) Since the variance of G is unknown, the sample variance is substituted, in place of var(G) in (3-7) giving - 28 -(3-8) The relationship between a r and Og, namely a r = C T Q / Z N , betrays the major weakness of the Monte Carlo method - that the convergence is very slowl The l/./N dependence means that in order to add a decimal 100 times. The estimate of the standard deviation of r , a r , can be used to form a confidence interval for the estimate y. By the Central Limit Theorem, the density of r approaches the Normal density as N becomes large. For Monte Carlo simulations N is quite large so that the Gaussian approximation is l i k e l y to be good. Denoting the Normal cumulative distribution function by 4, a 100m% confidence interval for Y can be constructed: place to the estimate y» the number of runs must be increased prob (|E[G] - y| < * " 1 ( I 2 2 t I ) a r ) = m (3-9) If a 95% confidence interval is desired, (3-9) becomes prob (|E[G] - Y I < 1.96a r) = 0.95 (3-10) - 29 -The above equations can be used to determine the number of runs required to achieve a certain relative accuracy in the Monte Carlo simulation. A simple algorithm would be as follows: After runs of the simulation (with being a few hundred, say) calculate a ^ 1 ^ , where (Nj) . 1.96 o[h] with Y^ N 1^ and o^l^ given by equations (3-6) and (3-8) respectively. ) The required number of runs is then given by N = N x f (3-11) where a. is the desired relative e r r o r , ;(N) - 1.96 o v 3.4 Variance Reduction The method of "importance sampling" is not used in this work but is discussed brief ly here because i ts use may provide, a way of - 30 -reducing the variance of the estimated expected value, and as such might be a topic for further investigation. Reduction of variance is worthwhile in view of the large number of runs required to achieve a reasonable relative error and the consequent large CPU times involved. For the following discussion the single-variable version of (3-5) wil l be used. where h is a density in the interval (0,1). It can be shown [21] that i f h has approximately the shape of g«F~*, the sample mean by (3-13) wil l have a variance less than the sample mean by (3-12). The method is named "importance sampling" because by sampling on the density h (eqn. 3-13) rather than on the uniform density (eqn. 3-12), the samples are concentrated in the region which gives larger values of g. If , for example, F _ 1 ( U ) is. the random variable o (3-12) Equation (3-12) can be reformulated as E[G] = 9 ( F ^ ) ? , - h(u) du (3.-13) 0 - 31 -(with density f) corresponding to the pole-scatter of a closing breaker, then the effect of sampling on h would be to choose more often values of pole-scatter which produced the higher conditional probabilit ies of flashover. The method of "correlated sampling", which is used when the effect of small changes in a system are to be studied, is very effective in reducing variance. This method w i l l be examined in detail in section 5.3. - 32 -4 THE RISK PROGRAM 4.1 General The RISK program is a computer program written in VAX FORTRAN 7 7 . . The program produces an estimate of probability of flashover to ground at one point (tower) of a three-phase transmission l ine for a single switching operation (risk of f a i l u r e ) . The Sample-Mean Monte Carlo Method is 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 ) several times. The outcomes thus obtained are transformed to the physical random variables required in the modelling of the l ine energization. The TP input f i l e is modified appropriately (section 4 . 2 . 1 ) . 2. The TP is executed, producing as output A-phase, B-phase and C-phase overvoltage time records at one node (line-end in this work). 3. Each time record is processed to extract the peaks (excursions above a certain threshold level) of either polarity. Reduction of the number of peaks may occur through the consideration of "double peaks" (section 4 . 2 . 2 ) . "Plateaus" are detected and accounted for (section 4 . 2 . 3 ) . Each remaining peak is f i t ted to a double-exponential impulse, and the equivalent time-to-crest determined .(section 4 . 2 . 4 ) . The U 5 0 and a of the insulation for the particular time-to-crest are determined via the U^Q curve. The probability of flashover for each peak is determined - 33 -via the Normal cumulative distribution and the individual peak probabil i t ies of flashover are combined to y i e l d the total probability of flashover for the run (section 4.3.5). Steps 1 to 3 are repeated, and the expected value of the probability of flashover estimated as per the sample-Mean Monte Carlo method. Mathematically, step 1 corresponds to the evaluation of the inverse cumulative distribution functions F.~*(U.) of equation (3-5). Steps 2 and 3 correspond to the evaluation of the function g of equation (3-5). 4.2 Program Components 4.2.1 Modifying the TP Input F i l e In this work four random variables are associated with the energization of a three-phase transmission l i n e : the prospective point-on-wave of the closure (the aiming-point) and the pole-scatter about the aiming-point of the three individual breaker poles. Figure 6 shows the TP model for l ine energization with the four random variables, (details of the transmission l ine model used as a test case are given in section 5.1 and Appendix E). - 34 -A-Phase B-Phase C-Phase FIGURE 6: 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 i t 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 times t^, tg and t^, account for the pole scatter, and are taken in this work to have a truncated Normal density. Figure 7 shows the density for the case: truncation point = 2a, a = 1 ms. A non-truncated Normal density is superposed for comparison. The relationship between the sample, u , and the closing-time is not a simple one; in fact, two samplings are required to produce one closing time. The details of the transformation are given in Appendix C. - 3 5 -500 i £ 400 (/) z a - £ 300 H _ l H m I 200 Q_ 100 TIME (•Uliseconds) FIGURE 7: TRUNCATED NORMAL DENSITY 4.2.2 Selection of Peaks A regular overvoltage peak is defined as an excursion (either positive or negative) above a certain threshold level and then back below. Figure 8a shows a peak of this type. A double overvoltage peak is defined by the presence of two closely spaced peaks in a single excursion above threshold (Figure 8b). Laboratory tests' [16] have determined the probability of flashover of rod-plane gaps subjected to overvoltages with double peaks. RISK contains a subroutine which scans the time records of overvoltage to pick out cases of double peaks and applies the following rules (from Reference [16]) to possibly eliminate one of the peaks: - 36 -1. If .V^ > 0.9V 2 ; retain only, the f i r s t peak 2. If < 0.9V 2 ; retain both peaks i f AV > 0.1(V 1 - AV), otherwise retain only the second peak. (a) (b) FIGURE 8: REGULAR AND DOUBLE PEAKS For the second peak, the region of f i t to determine the equivalent time-to-crest is from the minimum point between peaks to the crest of the second peak. 4.2.3 Plateau Detection Figure 9 shows an example of a peak characterized by a relat ively f l a t region (a "plateau") preceding a sharper rise to crest. Given the laboratory results of reference [16], i t would seem reasonable that the probability of flashover would be determined largely by the sharply-rising portion of the excursion. RISK contains a subroutine to detect such plateaus and apply the following rule: i f AV > 0.1V, then the equivalent time-to-crest is determined by f i t t i n g only over the sharply-rising portion. - 37 -I • ' i ' ' 1 1 ' i • i I ! I 1 ! ! I 1 1 0. 20. T I M E ( m i l l i s e c o n d s ) FIGURE 9: 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 times-to-crest (and hence low probabil it ies of flashover) to the important high-magnitude peaks. The rules presented 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 different from the double-exponential. Reference [24] presents several types of waveshape i r r e g u l a r i t i e s along with some f i t t i n g suggestions. Which, i f any, measures should be applied can best be determined by examining the s e n s i t i v i t y of the simulation result to these measures. - 38 -4.2.4 F i t t i n g A non-linear least-square curve-f itt ing routine is used to f i t each peak (over the region of f i t ) to a double-exponential impulse. The f i t 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 details of the f i t t i n g procedure are given in Appendix D. 4.2.5 Calculation of Risk of Failure The probability of flashover for each f i t t e d peak is given by U U c r i where U r is the crest value of the overvoltage peak UgQ and a are the insulation strength parameters at the equivalent time-to-crest * is the Normal cumulative distribution function. The risk of fai lure for each phase-ground overvoltage time record is determined by combining the flashover probabil i t ies of each f i t ted peak in the record, "phase = 1 " ( 1 - R p e a k , i > * 4 " 2 ' - 39 -where N is the number of f i t t e d peaks in the record. F i n a l l y the total probability of flashover to ground is determined by combining the three phase-ground p r o b a b i l i t i e s , : Rphase = 1 " <X " V* 1 " V* 1 " RC> <4"3) - 40 -5 TEST RESULTS 5.1 Introduction This section examines the use of the Monte Carlo method (via the RISK program) as a tool in the investigation of transmission l ine insulation problems. It is not meant to represent a case study of a particular problem. The f i r s t objective of this section is to give a few examples which i l l u s t r a t e the problems encountered when using the Monte Carlo method, part icularly with regard to errors. The second objective is to come to some conclusions regarding the sensit iv i ty of the risk of fai lure to the model parameters. General conclusions on the relative importance of parameters are d i f f i c u l t since the e l e c t r i c a l and insulation systems under study w i l l vary greatly. Moreover, the particular objectives of a case study wil l influence the relative importance of the model parameters. The model used is the energization of an open-ended three-phase l i n e . The details of the energization have been given in the preceding section. The l i n e is modelled as an untransposed distributed parameter transmission l i n e . The Ifne transformation matrix as well as the l ine inductance and capacitance are constant over frequency (a reasonable approximation).. There is no source impedance (valid for a situation in which the bus to which the l ine is switched is supplied by several other l i n e s ) . The transient program accounts for wave attenuation by inserting one half the l ine resistance in the middle of the l ine and one quarter at each l ine end. - 41 -The l ine is comprised of three 80 km sections (two transposition points). Appendix E gives a l i s t i n g of the TP input f i l e . The tower-window insulation for each phase is characterized by the U^Q 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 Distribution of Crest Voltage and Time-to-Crest Before proceeding with a series of simulations, i t is useful to examine the histogram of crest voltage and time-to-crest resulting from the switching operation. Knowledge of the distribution of these two variables w i l l help in the interpretation of r e s u l t s , and wil l provide a check on the v a l i d i t y of certain assumptions. For example, the domain of the U^g curve we are using should include the majority of the times-to-crest. To produce the histograms, 4000 runs were made with a pole-scatter as shown in Figure 7 (truncation point = 2a, a = 1 ms). The resulting histograms of crest voltage and times-to-crest are given in Figures 10* and 11 respectively. In producing Figure 10 only the The f i t ted normal density shown w i l l be used l a t e r . - 42 -maximum crest voltage (among a l l the peaks) for each phase is considered. 300 -200 • 100 • CREST VOLTAGE (p.u.) FIGURE 10 : HISTOGRAM OF CREST VOLTAGE Examining Figure 1 1 , we note that the majority of the times-to-crest f a l l between about 0 .05 and 1 ms. We are therefore assured that the domain of the U^Q curve ( 0 .03 to 1 ms in Figure 3) is adequate to account for most of the peaks. We should also ensure that the times-to-crest above 1 ms are not important, that i s , do not correspond to peaks with high crest voltages. This can be done by examining the scatter plot of time-to-crest versus crest voltage (Figure 12 ) . 600 400 m Z 200 IT trrrmrvm— .5 1.0 1.5 TIME-TO-CREST (Billiseconds) 2. « 2.0 T C O u u 1.5 FIGURE 11: HISTOGRAM OF TIME-TO-CREST CO u cc u I o 1.0 -r 1.5 2.0 2.5 CREST VOLTAGE (p.u.) 3.0 FIGURE 12: SCATTER.PLOT OF TIME-TO-CREST VERSUS CREST VOLTAGE - 44 -We note that, among those peaks with time-to-crest greater than l m s , almost a l l have crest voltage less than 2.5 p . u . , and so w i l l not contribute to the risk 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 of convergence is i l l u s t r a t e d in Figure 13, where risk of fai lure versus number of runs is plotted for two different seeds. The air-gap parameters for this example are: c r i t i c a l U"5 0 = 2.8 p . u . , a = 0.05 U 5 Q . Before discussing errors further, we define the term relative error as being one-half the 95 percent confidence interval divided by the estimate of the flashover probability (see equation 3-11). The relative error is i t s e l f a random variable, so that any particular relative error calculated from the results of a simulation is just an estimate of the actual (unknown) relative error. - 45 -.05 i .04 -U J 3 .03 -2000 RUN NUMBER FIGURE 13: CONVERGENCE OF THE MONTE CARLO SIMULATION To obtain an appreciation of the kind of relative error to be expected, seven simulations were run, each with a different c r i t i c a l UgQ, and each with the same seed. The number of runs for each simulation was 500. The percent relative 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 fai lure) is accompanied by an increase in relative error, the size of which is rather large. To demonstrate that the relative errors given in Table 1 are in fact representative, we take a specific case of c r i t i c a l U g^ (= 2.80 p . u . ) , and repeat the simulation six more times (with different seeds). The spread, i . e . difference between maximum and - 46 -Table 1 Relative Error after 500 Runs C r i t i c a l 11™ (p.u.) Percent Relative Error 34 45 58 72 85 96 106 minimum values, was 0.0062. This compares well with the estimated confidence interval from Table 1 of 0.0083. The average of the seven estimates • of risk of fai lure was 0.014, which agrees well with the value one would infer from Figure 13. The size of the relative error becomes even more d e b i l i t a t i n g i f we wish to make comparisons of risk of f a i l u r e among s l i g h t l y different models. Suppose we wish to compare the risk of fai lure of two models: the f i r s t , the base model, has a U g^ curve given by Figure 3, ( c r i t i c a l time-to-crest = 0.15 ms), while the second has a similar U g^ curve, but shifted so that the c r i t i c a l time-to-crest is 0.2 ms. For each model, simulations of 500 runs each were made for c r i t i c a l U g^ ranging from 2.8 to 3.4 p . u . . The same seed was used for each simulation. The resulting risks of f a i l u r e are plotted in Figure 14 versus c r i t i c a l U^g. Superposed on the curve for the base case are the calculated confidence intervals for each value of c r i t i c a l \}cn. 2.80 2.90 3.00 3.10 3.20 3.30 3.40 - 47 --t 10 i 10 7 \ • , : • r- -•2.8 3.0 3.2 3.4 CRITICAL U50 (p.u.) FIGURE 14: SMALL DIFFERENCE IN MODELS We note immediately that the sl ight difference in risk between the two curves is quite a bit smaller than the confidence i n t e r v a l , especially at the lower risks of f a i l u r e . As a r e s u l t , the relative error of the difference is expected to be extremely large. We examine the case of c r i t i c a l U 5 Q = 3.3 p.u. in d e t a i l . 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) var (B) = 7.23(10"9) - 48 -If the covariance of A and B is zero, the variance of the difference of A and B would be the sum of the two variances, 13.27(10 ). However, because the two simulations were made with the same seed, they are quite highly correlated (correlation -9 coefficient = 0.84) with a covariance of 5.55(10 ). The variance.of the difference is then: var (A-B) = var (A) + var (B) - 2 cov (A,B) = 2.16(10"9) The percent relative error for the estimated difference of 1.21(10"*') is 34.0%. With zero correlation i t would have been 830%. Since relative error is inversely proportional to the square root of the number of runs, we estimate that to achieve the lower relative error with simulations having zero covariance would require about six times as many runs. The use of identical seeds (and hence identical sequences of random numbers) to maximize the covariance of the estimates for two different simulations is known in Monte Carlo theory as the method of "correlated sampling". I t 's effectiveness depends upon a high correlation coefficient; for low correlation coefficients there is no advantage to be gained by choosing identical seeds. To demonstrate that the relative error of 340% is truly represen-tat ive, seven more simulations (500 runs each) of each model were done at c r i t i c a l U 5 0 = 3.3 p.u. The spread in difference (A-B) was 12(10~6) which compares well with the estimated confidence interval - 49 -(corresponding to 340%) of 8(10" ). The more accurate value of difference based on the eight simulations (4000 runs total) is 8.05(10~ 6). 5.4 Sensit ivity to F i t t i n g Parameters and Procedures It is interesting to check the sensit ivi ty to some of the parameters and procedures of the f i t t i n g process. F i r s t we check the s e n s i t i v i t y to the inclusion of the plateau-detecting subroutine. Figure 9 shows an overvoltage peak with a plateau at about 2.1 p . u . . Without the subroutine, and with a threshold level less than 2.1 p . u . , the risk of f a i l u r e is 0.5(10"'*); for threshold level above 2.1 p . u . , the risk of fai lure is about 0.5(10"*), three orders of magnitude higher. The extreme sensit iv i ty to threshold level results from the difference in equivalent time-to-crest obtained by f i t t i n g to the entire peak including the plateau, or just to the rapidly r is ing portion above 2.1 p . u . . The sensit iv i ty is much less , however, i f an entire simulation, rather than just a single peak is considered. Figure 15 shows the risk of f a i l u r e versus threshold level both with and without the plateau-detecting subroutine. Two thousand runs were used for each simulation. C r i t i c a l U 5 Q was 2.8 p.u. The lower two curves in Figure 16 show the same comparison with c r i t i c a l U 5 Q varying and threshold level fixed (at 1.5 p . u . ) . Five hundred runs were used in a l l simulations. - 50 -.020 With Plateau Subroutine .015 A Without .010 1.6 1.8 THRESHOLD (p.u.) 2.0 FIGURE 15: SENSITIVITY TO INCLUSION OF PLATEAU-DETECTING ROUTINE. 10 ' 3 CRITICAL U50 (p.u.) FIGURE 16: SENSITIVITY TO FITTING PARAMETERS AND PROCEDURES - 51 -We can conclude from the above results that, for the particular model under study, there is very l i t t l e sensit iv i ty 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 sensit ivi ty to the ratio of t ^ to t of the f i t ted double exponential. Table 2 gives results of 500-run simulations at a c r i t i c a l U 5 Q of 2.8 p . u . . The ratio of the two time constants in the double exponential is tabulated opposite flashover probability. The largest time constant ratio of 50 corresponds to a t ^ to t c r ratio of 10 (see Figure 2); the lowest value of 10 corresponds to a t . / 0 to t ratio of 4. r l/d cr Table 2 Sensitivity to Ratio of Time Constants in Double Exponential Ratio of Time Constants Risk of Failure 50 (Base) 0.0123 40 0.0121 30 0.0119 20 0.0116 10 0.0111 There is some sensit ivi ty to the threshold level (lower boundary of region of f i t ) . Figure 16 gives three curves for comparison: the base curve with threshold of 1.5 p . u . , and curves for threshold values of 2.0 and 2.3 p . u . . We note that the risk of fai lure tends to increase as threshold level increases. This can be explained by consideration of the average time-to-crest for the three cases. At c r i t i c a l U^Q of 3.3 p . u . , the average time-to-crest for the three - 52 -values of threshold level were: 0.45 ms at 1.5 p . u . , 0.31 ms at 2.0 p . u . , and 0.23 ms at 2.3 p . u . . We would expect the observed result that models with average time-to-crest closer to c r i t i c a l time-to-crest (0.15 ms, see Figure 3) have larger risk of f a i l u r e . To check the sensit ivi ty to the inclusion of the subroutine to detect and handle accordingly double peaks (see section 4.2.2) simulations at different c r i t i c a l U^Q'S were made. There was v i r t u a l l y no sensit ivi ty to this routine. Correlated sampling was used to advantage in a l l of the above comparisons. As an example, a correlation coefficient of 0.998 was calculated for the two simulations comparing the base model against the model without plateau-detecting subroutine at a c r i t i c a l U^Q of 3.3 p . u . . The very high correlation coefficient resulted in a relative error under 100% for the small difference in risk of 5.2(10" 7 ). Similarly high correlation coefficients were observed in a l l the comparisons shown in Figure 16. 5.5 Sensit ivity to Waveshape Consider a set of tower window insulations with identical c r i t i c a l U ^ ' s but different c r i t i c a l times-to-crest. Figure 17 gives a family of four U^Q curves which can be used to characterize the different tower windows. The c r i t i c a l times-to-crest of the four curves are: 0.15 ms (base), 0.2, 0.3 and 0.4 ms. - 53 -1.2 1 .4 .6 TIME-TO-CREST (lilllscconds) FIGURE 17: FAMILY OF U 5 Q CURVES Figure 18 presents the results of 500-run simulations for various c r i t i c a l U^Q'S . The curve labelled " f l a t " , was produced assuming no waveshape dependence ( i . e . f l a t U 5 Q curve). We see that for the four c r i t i c a l times-to-crest considered, the greater the time-to-crest the greater the risk of f a i l u r e . Examination of" the histogram of time-to-crest (Figure 11) and the shape of the U^Q curves reveals that this trend is reasonable. The U^Q curves with larger c r i t i c a l time-to-crest apportion more of the times-to-crest to lower U^Q'S, which w i l l result in greater risk of f a i l u r e . - 54 --1 to i CRITICAL U50 (p.u.) FIGURE 18: RISK OF FAILURE FOR DIFFERENT U"5 0 CURVES The correlation coefficients tend to decrease as the difference in risk increases. For example, at a c r i t i c a l U^Q of 3.3 p . u . , the correlation coefficient of the base simulation and the 0.20 ms simulation is 0.84; the coefficient of the base and 0.40 ms simulation is only 0.24. When comparing the f lat case to the base case, the correlation coefficient is effectively zero. F a i r l y large relative errors in the differences are, therefore, to be expected. The percent relative error in the difference between the 0.4 ms and base cases is about 186%; between the f l a t and base cases i t is about 61%. The ratio of risk of fai lure without waveshape dependence to that with provides a measure of the influence of waveshape dependence. To determine the ratio for the present model, two simulations of - 55 -4000 runs were made with a c r i t i c a l . Ugg of 3.3 p.u. The resulting ratio was 16. This ratio compares well with the value of 30 reported by Snider [1] in similar studies using a TNA. 5.6. Sensit ivity to Parameters of the Strength Distribution The sensit iv i ty to U^g, the mean of the strength d i s t r i b u t i o n , is obvious. from the preceding figures. A change of 5 percent in c r i t i c a l Ugg results in a change of about one order of magnitude in flashover probability. The effect of the standard deviation of the strength distribution 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 in which the standard deviation is a constant fraction of U^Q. The.curve labelled "variable a" refers to a model in which a varies l i n e a r l y from 0 .0511,^ at c r i t i c a l time-to-crescent to 0 . 0 8 U 5 0 at the end points of the U^Q curve. We note that risk of fai lure . is quite sensitive to standard deviation. It is also interesting to.compare the "a = 5%" curve and the "variable a" curve with the "flat" curve from the preceding figure. We can see that the sensit ivi ty to waveshape would be less for the variable a case than for the constant a case. This is because the effect of increasing a as time-to-crest moves away from c r i t i c a l is to increase probability of flashover, and thereby cancel to a degree the reduced probability of flashover at increased U^Q. 5 .7 Sensit ivity to Pole-Scatter The base case for the model under study uses a pole-scatter, standard deviation of 1 ms with a truncation at two standard deviations (see Figure 7 ) . Figure 20 shows the risk of fai lure at smaller pole-scatter standard deviations. We see that there is l i t t l e sensit ivi ty unti l the pole-scatter is reduced to about 0 .1 ms,.with further reduction resulting in a large decrease in r i s k . The risk of fai lure with no pole-scatter is several orders of magnitude below the base case r isk. 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 signif icantly increase the magnitude of line-end overvoltage peaks. A TNA study [19] using a similar model of - 57 -pole-scatter reports overvoltages typically 20-30% greater in magnitude compared to those created by simultaneous closing. The reason for the increased magnitude is perhaps explained by the increased ground mode component of travel l ing wave produced by closing each pole at different times. - i 1 0 1 • • 1 1 —r . 2.8 3.0 3.2 3.4 CRITICAL U50 (p.u.) FIGURE 20: SENSITIVITY TO POLE-SCATTER STANDARD DEVIATION The pole-scatter model used in this work wherein the point-on-wave of contact closure is assigned a truncated Normal density is sufficient to reveal the high s e n s i t i v i t y of risk of fai lure to the presence of non-simultaneous closing, but is probably an unreal ist ic model. The model assumes that the point on wave of e l e c t r i c a l closing is equal to the point of mechanical closing; prestrike is effectively ignored. A better model would be based on a - 58 -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]. This type of model would probably produce points-on-wave of contact closure which tended to be distributed more towards the peaks of the applied voltage. A model of this sort could easily 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 obtained by the traditional method of considering only the highest peak in the record. Results of simulations for several 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 sl ight lowering of the risk of f a i l u r e . A similar TNA study [19] reached the same conclusion. With this r e s u l t , i t should be possible to calculate the risk of f a i l u r e with a f lat U 5 0 curve (no waveshape dependence) by using the histogram of magnitude of highest peak given in Figure 10. To this end, the t a i l (greater than 2.5 p.u.) of the histogram was f i t t e d to a Normal density (see Figure 10). The mean of the f i t t e d Normal density is 2.06 p.u. and the standard deviation is 0.28 p . u . . Substituting these two values into equation 2-5, the risk of f a i l u r e at a U^Q of 3.3 p.u. (a = 0.0511^) i s . 3 $ T 2.06 - 3.3 - / 0 . 2 8 2 + 0.16 = 310(10"°) - 59 -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 (for the f lat U^ Q curve case) of 326(10"^). Four thousand runs were made to produce this estimate. The agreement is very good and indicates that, for the particular 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 dependence is not important. - 60 -6 CONCLUSIONS A computer program, RISK, has been developed which uses the Monte Carlo method to estimate the probability of flashover to ground of a transmission l ine for a single/switching operation (risk of f a i l u r e ) . The program accounts for the waveshape dependence of risk of fai lure through the use of laboratory-generated switching-impulse data. The program is modular, and can therefore readily accommodate future expansion, for example, to estimate phase-to-phase risk of f a i l u r e . The switching operation studied, or the model accounting for breaker pole-scatter can be easily changed. The program has been thoroughly tested by running a large number of simulations of a simple model of l ine energization. General conclusions should not be based on the study of only one e l e c t r i c a l system but the following points relating to sensit iv i ty are evident: the procedures and parameters of the f i t t i n g process only s l i g h t l y influence the risk of f a i l u r e , and thus do not warrant much consideration. s e n s i t i v i t y to waveshape can be significant in cases where comparison of tower-windows having quite different U^Q curves is desired. the presence of closing breaker pole-scatter can be an important influence on risk of f a i l u r e , and therefore i t would be worthwhile to develop more physically r e a l i s t i c models. - 61 -The error in the estimate of risk of fai lure resulting from the use of a f i n i t e 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 the risk of f a i l u r e of different tower window a i r gaps. A relat ively accurate comparison can be made in spite of the overwhelming uncertainty in the absolute levels of flashover probability of each insulation. to investigate the influence of e l e c t r i c a l system configuration on risk of f a i l u r e . to study the distribution of overvoltage peaks resulting, from system switching operations. The above uses w i l l have practical value in the more economic design of overhead transmission l i n e s , important in cases of long transmission distances and/or high voltages. - 62 -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, "Switching-Surge Design of Towers for UHV Transmission", IEEE Transactions on Power Apparatus and Systems, Vol . 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, editor, Plenum Press, New York, 1980. 5. Project EHV Staff, "EHV Transmission Line Reference Book", Edison E l e c t r i c Institute, New York, 1968. 6. IEEE Committee Report "Guide for Application of Insulators to Withstand Switching Surges", IEEE Transactions on Power Apparatus and Systems, Vol. 94, pp. 58-67, Jan./Feb. 1975. 7. G. Le Roy, C. Gary, B. Hutzler, J . Lalot , 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, Paris , 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, Vol . 94, pp. 1725-1733, Sept./Oct. 1975. 9. C. Carrara, L. Del lera, and G. Sartorio, "Switching Surges with Very Long Fronts (above 1500 ys): Effect of Front Shape on Discharge Voltage", IEEE Transactions on Power Apparatus and Systems, Vol. 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, Vol. 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 Distribution and Breakdown Probability: S t a t i s t i c a l Approach", IEEE Transactions on Power Apparatus and Systems, Vol . 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 Distribution 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, Vol. 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, Vol. 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 Air Insulating Gaps Stressed by Switching Overvoltages with a Double Peak", IEEE Transactions on Power Apparatus and Systems, Vol. 97, pp. 2375-2381, Nov./Dec. 1978. 17. Project UHV Staff, "Transmission Line Reference Book, 345 kV and Above", Second Edit ion, E l e c t r i c Power Research Institute, Palo A l t o , 1982. 18. N. F i o r e l l a , G. Santagostino, L. Lagostena, A. Porrino, "Phase-to-Phase and Phase-to-Earth Risk of Failure Due to Switching Surges in UHV Systems", Paper 33-12, CIGRE Conference, 4 September - 27 August, 1980. 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 Line", IEEE Transactions on Power Apparatus and Systems, Vol. 94, Nov./Dec. 1981. 21. R.Y. Rubinstein, "Simulation and the Monte Carlo Method", John 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, Vol. 95, pp. 610-620, Mar./Apr. 1976. 23. H. Dommel, M. Gavri lovic , and L.A. Snider, "On the Accuracy of Digital and TNA Techniques for Determination of Overvoltages and R i s k - o f - F a i l u r e " , paper presented at CEA Spring Meeting, Vancouver, March 1983. - 64 -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 : i (x - m f) f(x) = l_ exp (- 1— ) (A-l) o f / 2 T Zafc The strength function i s : ,2 i / (S - m j ' g(x)" = - A = - exp ( 3 _ ) dC (A-2) a / 2 T J 2a 2 g /_« g The probability of flashover is given by: R = f(x) g(x) dx (x-m f ) Z ( 5 - m ) 2 exp ( - 5 — - Tf-—) dCdx (A-3) The double integral (A-3) is evaluated by f i r s t making the following two changes of variable. - 66 -(x - mf) , and Equation (A-3) becomes, c^x1 + m^ - m. f '"g oo f t .2 , , 2 exp (- f — - J — ) ds' dx' (A-4) • oo oo The region of integration in (A-4) is shown as the area below l ine q in Figure 21a . The equation for l ine q i s : °g °g The perpendicular distance of l ine q to the origin i s : min 2 + 2 f + °9 / Of + a - 67 -s / / / (a) (b) FIGURE 21: REGION OF INTEGRATION By virtue of the symmetry of the integrand of (A-4), the region of integration can be simplified by rotating l ine q until i t is p a r a l l e l , to the x' axis (Figure 21b). Equation (A-4) then becomes: min R = x ' 2 X £ J - ' dx' -CO _ CO exp (- ^ - - ^ - ) d 5 " dx' (A-5) which can, since the l imits are now a l l constants, be simplified to: - 68 --r • ,00 mm 2 / 2 R = i - [ exp (- Kr ) d5« j exp (- 4- ) dx' m f - m g y 2 + 2 /27 0 0 exp (- y-) dz which completes the derivation, - 69 -. . APPENDIX B The Inverse Transform Method of Generating Random Variables The derivation of the inverse transform method of generating random variables is well-known and can be found in numerous references. It is repeated here for completeness 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 distribution F(x) is required i t is shown below that X = F _ 1 ( U ) , where U is a random variable with uniform density in the interval (0,1). Proof: Assume X = F - 1 ( U ) Then prob (X six) = prob ( F _ 1 ( U ) Sx) = prob (U £F(x)), since F(x) is monotoni'cally increasing. But prob (U ^F(x)j = F(x), and the proof is complete. - 70 -APPENDIX C Generating Random Numbers with Truncated Normal Density We need to generate random numbers having the density of a Normal distribution truncated at no. The desired density i s : ' I A f(x) = 4= e 2 + ^f. , -n s x *n (C-l) where A = * ( - n ) , * is the Normal cumulative distribution function. Equation (C-l) can be rewritten in the equivalent form, - x 2 f(x) = (1 - 2 A J [ — - 1 e 2 ] + 2A [ 4 ] n /2T(1 - 2A ) n d n n = (1 - 2An) f ^ x ) + 2An f 2 (x) (C-2) Equation (C-2) is interpreted as the sum of two probability densit ies, f^(x) and f 2 ( x ) , each conditional upon a parameter y: - 71 -f(x) - f ^ x l y = yx) prob (y = + f 2 ( x | y = y 2 ) prob (y - y 2 ) Here, prob (y and prob (y where U i s a random v a r i a b l e w i t h u n i f o r m d e n s i t y on ( 0 , 1 ) . The p r o c e d u r e , t h e n , t o g e n e r a t e random numbers w i t h a d e n s i t y g i v e n by ( C - l ) i s as f o l l o w s : 1. f i r s t draw Uj from ( 0 , 1 ) . i f Uj < 1 - 2 A n , then sample from f ^ x ) i f u 1 s i - 2A n. then sample from f 2 ( x ) . 2. draw u 2 from (0,1) and determine x by the i n v e r s e t r a n s f o r m method (see Appendix B) w i t h t h e . a p p r o p r i a t e d e n s i t y ( f ^ ( x ) o r f 2 ( x ) ) . «(x)- - A w i t h f j ( x ) : F 2 ( x ) = i . 2 A = u 2 t h e r e f o r e , x = $ _ 1 [ u 2 ( l - 2 A )• + A p ] w i t h f 2 ( x ) : F 2 ( x ) = £ + \ = u 2 t h e r e f o r e , x = 2 n ( u ? - i ) = y x ) = 1 - 2A n = prob (U < 1 - 2 A J = y 2 ) = 2A n = prob (U < 2 A p ) . - 72 -APPENDIX D F i t t i n g a Standard Impulse To simplify the mathematics, the data is f i r s t shifted so that the peak occurs at t = 0, and normalized by dividing by the crest value. The equation to be f i t to the data is then: where k is a normalizing constant, t is the time-to-crest, cr c is a constant which determines the ratio of to t c r , and X is to be determined. By algebraic manipulation the constants k and t can be eliminated from ( D - l ) , giving f(t) = k(e - e -cx(t + t ) (D-l) f ( t) = ce -At _ e -CAt (D-2) c - 1 Recognizing the discrete nature of the data points, (D-2) becomes: -Xi At -cXl'At f( iAt) = ce (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 of the difference between f( iAt) and the data points, i . e . the following function is minimized over X, o 2 2 Cf(iAt) - y . ] i=-N 1 where y . are the (shifted and normalized) data points, N is the number of points involved in the f i t (the number between the threshold and crest) . Once X is determined the time-to-crest is given by: t cr " x(c-l) _ Inc - 74 -Figure 22 shows an example f i t with c = 50 (corresponding to TIME ( B i l l i s e c o n d s ) FIGURE 22: EXAMPLE FIT - 75 -APPENDIX E Transient Program Input Data CASE 6: 150. MILE 50.E-6 2. -1N0DA00N0DA05 -2NODB00NODB05 -3NODC00NODC05 0.6O237E 00-0. .0.52371E 00-0. 0.60237E 00 0. -1NODC05NODC10 -2NODA05NODA10 -3NODB05NODB10 0.60237E 00-0. 0.52371E 00-0. 0.60237E 00 0. -1N0DB10N0DB15 -2N0DC10N0DC15 -3NODA10NODA15 0.60237E 00-0. 0.52371E 00-0. 0.60237E 00 0. LINE . E-2 -1 70711E 15307E-7.07 H E 70711E 15307E-70711E 50.MILE/SECTION UNTRANSPOSED LINE MODEL 0 1 0.2834498.930.1515 50.00 1 3 0.0599295.230.1828 50.00 1 3 60. 60. 70711E 15307E-11 0 70711E 0.0550249.510.1851 50.00 1 00-0.40642E 00 11 0.81831E 00 00-0.40642E 00 0.2834498.930.1515 50.00 1 0.0599295.230.1828 50.00 1 0.0550249.510.1851 50.00 I 00-0.40642E 00 11 0.81831E 00 00-0.40642E 00 0.2834498.930.1515 50.00 1 0.0599295.230.1828 50.00 1 0.0550249.510.1851 50.00 1 00-0.40642E 00 81831E 00 00-0.40642E 00 SRCA00NODA00 0.0021000 10.00000 SRCBOONODBOO 0.0021000 10.00000 SRCCOONODCOO 0.0021000 10.00000 14SRCA00 1.00000 60.00000 78.91091 -1.00000 14SRCB00 1.00000 60.00000 -41.08909 -1.00000 14SRCC00 1.00000 60.00000 198.91092 -1.00000 N0DA15N0DB15N0DC15 - 76 -
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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 |
AggregatedSourceRepository | DSpace |
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