Transient Thoughts
Insulator Stresses Always Log-Normally Distributed?
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At a recent IEEE Working Group meeting in Tampa, Florida, I described various problems while developing an application guide for insulators used under winter ice and snow conditions. The WG Chair succinctly summarized my contribution in the minutes as follows: “everything in the world is log normal”. After reflection, I have come to the following conclusion: as long the above is prefaced by, “everything in the lab is normally distributed”, this assertion is indeed an excellent summary of the problems we face doing insulation coordination calculations to predict risk of failure of a line or substation. The chart below illustrates what I mean.
ground resistance, the value a = 1.9 (associated with a natural log standard deviation of about 0.9) has been reported in several different studies. The values for ice accretion have a broader range of a = 1.3 to 2.1. In all cases, however, these exponents are less than the value used for lightning of a = 2.6.
High voltage lab personnel are quite familiar with processing flashover and withstand voltage values, e.g. from the up-anddown method for impulse waves. This leads to a well-defined estimate of insulator strength, shown as the narrow probability density function (P.D.F.) on the right. When considering several insulators in parallel, the mean strength value shifts a bit to the lower left, but the peak narrows even further.
The IEC 60071 standard on insulation coordination gives some advice on this by considering the natural variability of lightning waveshapes, the possible points of attachment, the random phase angle associated with the AC line voltage as well as the COV of the lightning impulse flashover process. However, the standard also notes that the risk integral evaluation becomes impractical with more than two random variables. Still, by focusing on the most influential and significant random variables and fixing the others, a satisfactory failure estimate can be computed. To some extent, lightning backflashover calculations already implement this advice. The strong correlation between the peak current (Ipk) and maximum rate of current rise (Sm) is exploited by fixing an equivalent linear front time, (Ipk/Sm) in calculating peak insulator voltage. However, at present, simple backflashover calculations ignore the random variable whose wide statistical variation may have the most influence on the result – namely tower-to-tower variation in soil resistivity. This second factor can be incorporated by considering a two-dimensional matrix of probability intervals, and computing the insulator stress for each derived combination of peak current and footing impedance. A model for the probability of lightning impulse flashover, using an exponent a = 33 to correspond to the COV of 5%, completes the improved calculation.
In contrast, the P.D.F. of the ‘stress´ variable has a skewed, asymmetrical distribution with a broad range of values. For lightning, there is a natural variability of parameters from flash to flash since the distribution of peak impulse currents relates directly to the peak impulse voltage associated with each flash. This stress variability is expressed using a log-normal distribution with a median and a log standard deviation based on natural logarithms of individual data values. In the case of a peak current of the first negative return stroke (Ipk), this can be approximated using a simplified expression that the probability (P) of exceeding Ipk is 1/(1 + ( Ipk / median)a), where the median peak current is 31 kA and the exponent a is taken as 2.6. The simplified expression for probability is useful in the range of 0.01 < P < 0.99 and has other helpful characteristics. First, when the exponent a < 3, it is related to the standard deviation of the log normal distribution as follows: SD (lnx) = 1.71/a. Second, when the exponent is large (say a > 16), it is related to the coefficient of variation (COV) of a normal distribution by a = 166/COV. The coefficient of variation is the standard deviation of the narrow peak in the figure divided by the mean and then multiplied by 100 percent and is typically in the range of 5 to 10% for electrical strength of insulators. Finally, the simplified expression for P can be manipulated easily to give a stress or strength associated with a pre-defined probability level such as P = 0.02 for the 2% stress or P = 0.10 for the 90% strength. The equation P = 1/( 1 +( x / median )a ) provides a satisfactory fit to many other observations of natural stress, whether x is ice thickness or weight, pollution level, precipitation conductivity or soil resistivity. Each region will have a different median annual value (say of ESDD), but pollution observations seem to share about the same value of exponent a = 2.3. For soil resistivity and
INMR® Q2 2012
INMR Issue 96.indd 30
This poses an interesting question. If statistical distributions of pollution levels are log normal (and in fact broader with higher log standard deviations than of those used for lightning currents), why would we try to use normal statistics for calculating the interference area for probability of failure in the figure?
Under icing conditions, there are actually three rather than two independent environmental stress variables in the flashover process. These are the pre-existing insulator contamination level (or ESDD), the thickness of the ice accretion per unit length on the insulator and the electrical conductivity of the applied water, corrected to 20°C. In large countries such as Canada and China, icing and pollution conditions can change considerably over distances of 1000 km or more. For example, urban areas and regions downwind of pollution sources are vulnerable to flashover at line voltage from accretion of rather thin ice layers and partial bridging by icicles across the dry arc distance. By contrast, full bridging of insulators by icicles occurs more frequently in rural areas that may have lower precipitation conductivity. It is important to develop insulator selection advice that anticipates and accommodates such a broad statistical distribution of input parameters. Evaluation of the interference area and flashover risk could be extended to sum a threedimensional matrix of threat levels, estimating the critical flashover level for each combination. However, simpler guidance is preferable. One approach is to develop an equitable way to distribute the flashover risk among the three parameters. This would offer the benefit of keeping the probability estimates within the range of actual observations, rather than extrapolating to an extreme ice thickness based on a limited data set of 10 or 20 annual values. And, by selecting log-normal distributions for all parameters at the outset, I believe we can accommodate our broad range of environmental conditions in a more practical way.
Dr. William A. Chisholm W.A.Chisholm@ieee.org
Editorial
16/05/12 2:39 PM