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1 IEEE Zkansactions on Electrical Insulation Vol. No., June A Study of the Up-and-Down Method for non-normal Distribution Functions S. Vibholm and P. Thyregod Physics Laboratory, IMSOR, Technical University of Denmark, Lyngby, Denmark ABSTRACT We discuss the assessment of breakdown probabilities by means of the up-and-down method. The exact maximum likelihood estimates for a number of response patterns are calculated for three different distribution functions and are compared with the estimates corresponding to the normal distribution. Estimates of the 0% probability breakdown voltage and of the scale parameter of the breakdown probability functions are inves tigated. INTRODUCTION N order to design insulation systems it is necessary I to assess the breakdown probability of the various insulation components of the system. The upand-down method is widely used for estimation of the 0% prob* bility breakdown voltage U0. The analysis may be extended to include estimates of the scale parameter U in the breakdown probability function. There are doubts, however, as to what extent these estimates are influenced by the proper choice of the underlying breakdown distribution function [l-. The design of the up-anddown test ensures that the choice of the assumed distribution function has only a minor influence on the estimation of U~O. The choice of distribution can, however, strongly affect the prediction of lower fractiles. In a previous paper [] the authors have compared the Dixon and Mood approximation to the maximum likelihood estimate with the exact maximum likelihood estimate [6,7] assuming a normal distribution for a number of response patterns. We concluded that, with digital computers readily available, a statistical analysis of a set of data obtained from an up-and-down test may just as well be performed directly from a maximum likelihood estimation, instead of applying the approximate method used by Dixon and Mood [8,9]. In the present paper the maximum likelihood estimates of U0 and U for a normal distribution of breakdown probabilities are compared with the maximum likelihood estimates of U0 and U for the following distributions of breakdown probabilities: a double exponential distribution, a logistic distribution [,], and a Weibull distribution with a shape parameter value of, in the following given the notation Weibull-. The Weibull- distribution is very similar to the modified Weibull distribution curve proposed by Carrara [l]. MAXIMUM LIKELIHOOD ESTIMATION OF BREAKDOWN VOLTAGE PARAMETERS HE simple up-and-down method is designed to ea- T timate the U0 value in self-restoring insulations. In thie test the voltage is applied at various levels Aj selected such that Aj+l = Aj +d, where d is a constant 988 IEEE Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on October 6, 009 at 07:0 from IEEE Xplore. Restrictions apply.

2 8 Vibholm and Thyrerod: The up-and-down method for non-normal distribution functions voltage increment. Only one shot at a time is applied at a given level. The voltage level is then changed to Ai- if the application resulted in a breakdown, or to Aj+l if the result was a withstand. A test sequence consists of a total of N voltage applications at I different voltage levels. The result of such a sequence can, without loss of relevant information, be summarized in a square matrix n,j with I rows and I columns. Each matrix element wj indicates the number of times the level has been changed from i to j. Consequently, all elements which are not of the form nj,j- or nj,j+l, are zero. For I = the matrix will be of the form ) n 0 n n 0 n 0 () ( It is seen that 0 0 n 0 n n 0 I N = C(nj,j-l+ nj,j+l) () j= The maximum likelihood estimates of U0 and U do not depend on the individual order of the breakdowns and withstands but only on the number of passages of each level as given by the square matrix. Furthermore, the estimate depends on the distribution that has been assumed for the estimation procedure \.7 LL- ' -I- 0 I m, Figure. The average estimation bias Am = m, - mt for the position as function of mt and with st as parameter. The double exponential distribution is the true and the Weibull- distribution the estimated distribution. The hatched area indicates the standard deviation for the results for St =.. Let P(Aj, p, a) denote the breakdown probability at level Aj in a distribution with 0% value p = U0 and scale parameter 6. The likelihood function corresponding to the observations ni,, is the probability of obtaining the observations, which can be written as I L(p,a) = n[{l - P(Aj,p~)}"j~J+'P, p, ~)~j*j-'] j= () The maximum likelihood estimates of p and a are the values of p and a maximizing L(p,a). In practice it is simpler to maximize ln(l). Generally no explicit solution exists and hence the maximum has to be determined by numerical methods. BREAKDOWN PROBABILITY FUNCTIONS HE various probability distributions considered in T the present analysis have all been standardized such that they possess the same 0% value and the same slope at the 0% level as the normal distribution. The distributions under consideration are Or Figure. The relative average estimation bias a for the scale as function of mt and with st as parameter. The double exponential distribution is the true and the Weibull- distribution the estimated distribution. deviation for the results for st =... Normal distribution Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on October 6, 009 at 07:0 from IEEE Xplore. Restrictions apply.

3 IEEE!lkansactions on Electrical Insulation Vol. No., June Table. Exact maximum likelihood estimates of m = (Uso - Al)/d and s = d/a for selected responses in an up-and-down test with N = 0 shots over voltage levels, for four distribution functions. Normal distribution Double exp. distribution Log is t ic distribution Weibull- distribution n n n n n n A h U" fa fa h Double exponential distribution P(x) = + exp[-(x - P(~E) = - exp (- exp (y)) () standardized with K. = p and C = f lu standardized with 6 = p - ln(ln()) ln()ficr and. Weibull- distribution c Logistic distribution P(a)=O for 86 (8) standardized with 6 = p - pln()aj;r7 and Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on October 6, 009 at 07:0 from IEEE Xplore. Restrictions apply.

4 60 Vibholm and Thyrerod: The up-and-down method fbr non-normal distribution functions Table. Exact maximum likelihood estimates of rn = (Uo - Al)/d and c = d/a for selected responses in an up-and-down test with N = 0 shots over voltage levels, for four distribution functions. Normal Double exp. Logistic Weibull- distribution distribution distribution distribution n n n n n n n n A 6i s^ A s^ A ( for a P value of. a = PmO))(P - l>/p only sequences where the highest voltage level applied PROCEDURE N the following analysis the total number of shots I in a test sequence has been set to N = 0, which makes it practicable to identify the possible array of the matrix w, associated with a number of test sequences. Furthermore, we shall restrict the analysis to sequences containing or levels, which for N = 0 ensures that meaningful information can be obtained. We consider has resulted in a breakdown and the lowest level in withstands. All other levels must possess at least one breakdown. The sequence is always started at the lowest level. All possible sequences with voltage levels and N = 0 with the above mentioned restrictions lead to 6 different th order square matrices representing 90 different sequences. With levels there are 7 different th order square matrices representing 90 sequences. Each individual matrix thus contains information from many possible test sequences, for which each sequence has the same probability of occurrence. Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on October 6, 009 at 07:0 from IEEE Xplore. Restrictions apply.

5 IEEE Transactions on Electrical Insulation Vol. No., June Tables and show for seiected matrices rqj the exact values of the maximum likelihood estimates m and s, for N = 0 for a normal distribution of breakdown probabilities, for a double-exponential distribution of breakdown probabilities, a logistic distribution of breakdown probabilities and a Weibull- distribution of breakdown probabilities. O0 I L F S. \ 0 \, I I.-JP--I- L rn, Figure. The average estimation bias Am = me - mt position as function of mt and with at as parameter. The Weibull- distribution is the true and the the double exponential distribution the estimated distribution. The hatched area indicates the standard deviation for the results for at = SO Figure. The relative average estimation bias a for the scale as function of mt and with at as parameter. The Weibull- distribution is the true and the double exponential distribution the estimated distribution. The hatched area indicates the standard deviation for the results for at =.. In order to facilitate the analysis, the following dimensionless quantities m and s are introduced d and s=- (9) U in which A is the lowest voltage level in the sequence, and d the step size. The data in Table are limited to those th order square matrices which each represent not less than 00 individual sequences, and Table is limited to those th order square matrices each of which contain more than 00 individual sequences. The limitation of the number of selected matrices has made the analysis as comprehensible as possible without losing significant information. AN ALY S IS HE probability of obtaining a specific matrix de- T pends on the true distribution, the true values of U0 and U and on the values of A and d. When a specific distribution has been chosen for the estimation (not necessarily the true distribution) then the above probability is also the probability of obtaining a specific set of values (h, 0) as a maximum likelihood estimate. For a specific set of matrices this probability, and the corresponding set of numbers of different sequences which will lead to each specific matrix in the set, can be utilized to weight the maximum likelihood estimates (h, $) corresponding to the individual matrices. This weighted average of the various possible values of the maximum likelihood estimates will be denoted (m,,s,) For fixed values of N, At and d the sum of all the probabilities for all possible matrices, including the ones that have been omitted, will be unity. In addition to the limitations already imposed on the sets of data under consideration we shall in the following analysis restrict ourselves to consider only such combinations of the true parameters m and s which associate a total probability larger than % with the sets of matrices under investigation. Each set TTQ st of the true values of m and s specify probabilities of occurrence for the individual matrices and for the corresponding values of the maximum likelihood estimates. The weighted averages of m, and se of the maximum likelihood estimates thus depend on the true values mt and st. Even when the distribution assumed in the estimation is the same as the true distribution there will usually be an estimation bias, with a magnitude depending on the true values of the parameters []. In general, m, Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on October 6, 009 at 07:0 from IEEE Xplore. Restrictions apply.

6 6 Vibholm end Thyrerod: The up-and-down method for non-normal distribution functions and se will differ from and st. This deviation can be described by the parameters for the position and -- for the scale. For a given combination of the estimated and true distributions, the average estimation bias Am and a will be functions of mt and st. These functions are for N = 0 illustrated in Figures to 8 with as abscissa and st as parameter for various sets of estimated and true distributions. The hatched areas in Figures to 8 indicate the standard deviation for the results for st =.. In this analysis the normal distribution has not been shown because results based on this distribution are very similar to those obtained from the logistic distribution and8 the Wei bull- distribution. I 0. m, Figure. The average estimation bias Am = me -mt position as function of mt and with at as parameter. The double exponential distribution is the true and the logistic distribution the estimated distribution. deviation for the results for st =.. Figures to 8 show that both the starting point and the step size are critical for obtaining a small error in the estimated U0 and 0. For N = 0 the best step size is d =.0 and the starting point a little more than two steps below U0. -so O r - Figure 6. The relative average estimation bias a for the scale as function of mt and with st as parameter. The double exponential distribution is the true and the logistic distribution the estimated distribution. deviation for the results for 8t =.. As stated above, the normal, the Weibull- and the logistic distributions lead to similar results. The Weibull- has zero probability below a certain level ( A < U0 -.0) giving the impression of a safe lower limit. Although such a limit is conceivable it has never been verified in practice. The normal and the logistic distributions do not give a finite high or low limit and follow each other rather closely. The logistic distribution is far the easiest to handle mathematically and can replace the normal distribution in any analysis in the field of insulation tests. Experimental results for impulse voltage breakdown characteristics for rod-rod gaps in atme spheric air [0,] show that the distribution function in the probability range from % to 99% is represented by a traditional normal distribution, but it might just as well be represented by a logistic distribution, more fitted for computer analysis. DISCUSSION F the step size d is greater than.0, then the esti- I mated scale parameters 9, are estimated with increasing bias from the true scale parameter st in the following order: the double exponential, the logistic, the Weibull-, and normal distributions. It is surprising that the Weibull- distribution gives a better performance than the normal distribution, but the difference is marginal and probably connected with the special choice of patterns and the limited number of shots N = 0. There is no great difference between the estimation bias for the double exponential, the logistic, the normal, or the Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on October 6, 009 at 07:0 from IEEE Xplore. Restrictions apply.

7 IEEE!&ansactions on Electrical Insulation Vol. No., June S! up-and-down test can be obtained directly from a maximum likelihood estimation. This analysis can be simplified by the use of a logistic distribution function without loss of accuracy. In order to determine the optimal values of sample siee N, start level and step siee which would yield the most reliable values of Vs0 it becomes necessary to apply more involved methods than those referred to in the present study. REFERENCES Figure 7. The average estimation bias Am = me - mt for the position as function of mt and with 8t as parameter. The Weibull- distribution is the true and the logistic distribution the estimated distribution. deviation for the results for at =.. [l] G. Carrara and S. Yakov, UStatistical Evaluation of Dielectric Test Methods, L Energia Elettrica No., pp. -9, 98. R. E. Little, The Up-and-Down Method for Small Samples with Extreme Value Response Distributions, J. Amer. Stat. Assn., Vol. 69, pp , 97. G. B. Wetherill, Sequential Estimation of Quantal Response Curves, J. Roy. Stat. Soc., Ser. B, Vol., NO., pp. -8, 96. so N. Hylten-Cavallius and F. A. Chagus, Possible Precision of Statistical Insulation Test Methods, IEEE Trans. PAS, Vol. 0, NO. 8, pp. 7-78, 98. S. Vibholm and P. Thyregod, The Analysis of Insulation Breakdown Probabilities by the Upand- Down Method, IEEE Trans., Vol. EI-, no., pp. -6, 986. Figure 8. The relative average estimation bias Q for the scale as function of mt and with at as parameter. The Weibull- distribution is the true and the logistic distribution the estimated distribution. deviation for the results for 8t =.. Weibull- distributions, independent of the combina tions of true and estimated distributions within these four distributions. A. W. F. Edwards, Likelihood, Cambridge U. P., London 976 G. W. Brown, Method of Maximum Likelihood Applied to the Analysis of Flashover Data, IEEE Trans. PAS, Vol. 88, NO., pp. 8-80, 969. W. J. Dixon, The Upand-Down Method for Small Samples, J. Amer. Stat. Assn., Vol. 60, pp , 96. W. J. Dixon and A. M. Mood, A Method for Obtaining and Analysing Sensitivity Data, J. Amer. Stat. Assn., Vol., pp. 09-6, 98. [lo] S. Vibholm and A. Pedersen, On the Reproducibil- CONCLUSIONS ity of Negative Lightning Impulse Breakdown Volt- ITB digital computers readily available, a statis- age - Distribution for a Rod-Rod Gap,.. th Intern& W tical analysis of a set of data obtained from an tional Conference on Phenomena in Ionized Gases, Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on October 6, 009 at 07:0 from IEEE Xplore. Restrictions apply.

8 6 Vibholm and Thyrerod: The up-and-down method for non-normal distribution functions p. 90, Czechoslovak Academy of Science, Prague 97. Ill] S. Vibholm and A. Pedersen, Some Factors Affecting the Lightning Impulse Breakdown Voltage Distribution for a Rod-Rod Gap in Air at Atmospheric Pressure, rd International Conference on Gas Discharges, IEE Conference Publication No. 8, pp. -7, London 97. Manuscript was received on 6 Dec 986, in revised form Mar 987. Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on October 6, 009 at 07:0 from IEEE Xplore. Restrictions apply.

HUMIDITY INFLUENCES ON THE BREAKDOWN OF ROD-PLANE GAPS UNDER POSITIVE IMPULSES IN ATMOSPHERIC AIR

HUMIDITY INFLUENCES ON THE BREAKDOWN OF ROD-PLANE GAPS UNDER POSITIVE IMPULSES IN ATMOSPHERIC AIR P. N. Mikropoulos and C. A. Stassinopoulos Department of Electrical and Computer Engineering, Aristotelian