Adapting the Statistics of Soil Properties into Existing ... - IEEE Xplore

2015 International Symposium on Lightning Protection (XIII SIPDA), Balneário Camboriú, Brazil, 28th Sept. – 2nd Oct. 2015.

Adapting the Statistics of Soil Properties into Existing and Future Lightning Protection Standards and Guides William A. Chisholm Past Chair, IEEE Power and Energy Society, Transmission and Distribution Committee Toronto, Canada [email protected]

Susana de Almeida de Graaff System Operations – International Development TenneT TSO BV Arnhem, the Netherlands [email protected]

Abstract—Risk estimates for lightning faults make use of lognormal distributions for lightning characteristics such as peak current. Simplifications for highly correlated parameters, notably peak current and rate of current rise, justify use of equivalent front time of 2 μs in backflashover calculations for first negative return strokes. The transmission line backflashover rate is also affected by uncorrelated and broad statistical variations in soil resistivity, tower footing impedance Zf and resistance Rf. Statistical properties of Rf from transmission lines in Tennessee USA and Portugal are compared. Modeling of Rf variation using a ten-step distribution from the IEEE Standard 1243 FLASH program is compared with estimates using finer probability step intervals, log-normal and log-logistic models.

The IEEE FLASH program [6] is structured with a ten-interval calculation to encourage this refinement. However, recent evidence from two different electric power networks [7][8] suggests that Rf also has a log-normal distribution. Observed system-wide average values of σ ln Rf = 1.05 and 0.9 imply that the statistical variation in Rf has a broader effect in backflashover calculations than statistical variations in Ipk and waveshape parameters. It does not make sense to take extreme care with calculation of probability of exceeding Ipk and then to ignore the corresponding variations in Zf, say by assigning a single median value. Variability of Ipk and Zf should be evaluated with complementary sophistication.

Keywords—transmission line, lightning, backflashover, grounding, statistical distribution, log normal, log logistic

Selection of appropriate statistical distributions of soil resistivity ρ (Ωm) and footing impedance, and application into calculation of lightning performance, are demonstrated using some classical methods. These include:

I.

INTRODUCTION

Lightning flashovers represent a dominant cause of momentary outages on overhead lines. Most power systems implement protection schemes and configurations that maximize the success of automatic reclosing, and these demonstrate a median success rate of about 80% [1]. However, the momentary voltage dips from reclosing present their own power quality issues to customers and industry. The background for this tutorial establishes that lightning is an ongoing and dominant root cause of momentary outages.

• Systematic fitting of curves to observations, using skew and kurtosis in the K. Pearson method of moments [9][10][11];

A simplified model for the risk of lightning backflashover faults is established by relating the peak voltage rise on insulators Vpk (kV) to the product of the downward negative first return stroke peak current Ipk (kA) and the wave impedance Zwave (Ω). Zwave is the parallel combination of overhead groundwire (OHGW) surge impedance ZOHGW (Ω) and tower footing impedance Zf (Ω). Zf has a nonlinear, often monotonic relation to lowfrequency footing resistance Rf (Ω) at individual towers.

• Comparison of the effect of probability interval – 10 %, 1 %, 0.1% - on the results of backflashover rate estimates.

• Reduction of skew and kurtosis through nonlinear transformation, notably giving the log-normal distribution; • Exploration of the statistical characteristics of log-logistic functions of the form P(>i) = 1/(1 + (i/ imedian)a ) where the exponent a is related to either to the standard deviation σi (for narrow distributions) or σ ln i (for broad distributions);

Parent distributions of ρ or Rf can be influenced by nonuniform transformations to Zf, especially in the tails of the distribution. Transformations may include: • Additional inductive or surge impedance voltage rise on long, buried wires, giving Zf >> Rf especially for Rf < 20 Ω;

Log-normal and log-logistic functions describe the wide statistical distribution of peak current by treating Ipk as a variable with log standard deviation of σ lg I = 0.265 [2] where lg is log to base 10, or σ ln I = 0.61 using natural logarithms. As an illustration, two standard deviation above the mean, at the 98% probability level, is 10 2×0.265 (= e 2×0.61) times the mean. This treatment is common to CIGRE Technical Brochure 63 of 1991 [3] and IEEE methods dating onward from 1993 [4][5][6].

• Programs of grounding improvements that install additional buried electrodes only at locations where Rf exceeds a treatment threshold such as 20 or 40 Ω; and • Effects of frequency dependence of resistivity ρ (f), which can be ignored for low values of ρ < 100 Ωm but are significant for ρ > 3000 Ωm when relating Zf to Rf. Grounding for lightning is also influenced by the minimum span lengths to adjacent towers. Typical statistical distributions of span length are used to illustrate this effect, using a standard volt-time curve model for electrical strength of insulation.

Segmentation of transmission lines into areas of “each significant class of resistivity”, to form a composite backflashover outage rate, was explicitly recommended in 1993 [4].

978-1-4799-8754-2/15/$31.00 ©2015 IEEE

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II.

On average, over the same period, 33% ±8% standard deviation of the momentary outages in the 2014 NERC records listed lightning as the root cause. Fig. 3 shows that this 33% annual fraction rose in three years to 70% for extra-high voltage lines operating above 600 kVac.

MOMENTARY OUTAGES, NORTH AMERICA, 2008-2014

A review of system reliability regulation [12] noted that, prior to 2008, eight USA jurisdictions, five in Europe, four in Australia and none in Canada reported consistent monitoring of the Momentary Average Interruption Frequency Index (MAIFI). Improved and more consistent reporting can be dated to the October, 2007 implementation of the Transmission Availability Data System (TADS) in the eight operating regions of USA and Canada shown in Fig. 1.

Fig. 3. Annual Fraction of Momentary Outages from Lightning, 2008-2014 in NERC control regions (data from [1])

The fraction of momentary outages from lightning only varied slightly among control regions in Fig. 1, even though there is a 50:1 variation in lightning ground flash density from southeast to northwest. With this variation in flash incidence, a lower fraction of lightning outages would be expected in WECC, compared to higher fractions expected in SERC and FRCC where the flash density is highest. Surprisingly, the FRCC region in Florida has the highest ground flash density of all regions and reported the lowest fraction, 15%, of momentary events that were caused by lightning.

Fig. 1. Eight control regions for electric power systems in the North American Electric Reliability Corporation (NERC) [13]

Since 2008, the annual performance of transmission lines has been tabulated and organized by cause code. Over the period 2008 to 2014, Fig. 2 shows a modest annual variation in the number of momentary outages on transmission lines that were diagnosed as lightning flashovers. Lightning was the most common cause, followed by “unknown”, weather excluding lightning, failed protection system equipment, foreign interference and contamination.

III.

SIMPLE MODEL FOR LIGHTNING BACKFLASHOVER

A. Fundamentals According to present standards and guides, the greatest risk of lightning flashovers on overhead transmission lines is posed by the large peak amplitude Ipk of the first, negative downward cloud to ground flash. This current source is impressed into a system of overhead groundwire, foundation electrodes and supplementary wire electrodes in contact with soil. Flashes to midspan divide the impressed current Ipk into two directions, and involve two grounding systems at once after some propagation delay. Flashes to a tower top force the impressed current into a single grounding system, and are thus more severe. The product of the impressed current and the parallel combination “wave impedance” Zwave raises the voltage on the stricken tower and thus on all of the normally-grounded ends of all insulators attached to that stricken tower. B. Calculation of Wave Impedance, Stroke to Tower The wave impedance at tower top in a simplified model would be the parallel combination of the overhead groundwire surge impedance ZOHGW, considering mutual coupling, and the footing impedance in impulse conditions, Zf, both in units of Ω.

Fig. 2. Annual Number of Momentary Outages from Lightning, 2008-2014 in NERC control regions (data from [1])

8

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(1)

C. Effect of Tower Inductance Current flow through a tower also causes a voltage rise from footings to OHGW, as a result of the tower inductance. Simplified models convert the tower impedance to a lumped inductance Ltwr which is used to estimate a voltage difference of ΔVtwr = Ipk × Ltwr / tm . With this adjustment, the wave impedance at tower top in a simplified model becomes: ଵ ಽ ௓೑ ା ೟ೢೝ ೟೘

൅

ଶ ௓ೀಹಸೈ

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F. Effect of System Voltage Bias Any ac or dc voltage bias Vn from normal power system operation remains on the line end, and adds or subtracts to the peak insulator voltage Vpk, depending on polarity. Thus, backflashover to the positive pole of a dc line with Vn = +500 kV is more likely than to the negative pole with Vn = -500 kV, as the positive dc voltage would add 500 kV to Vpk . For equal backflashover performance, the positive pole would need at least 1 m greater dry arc distance than the negative pole.

(2)

Where ZOHGW is the parallel impedance of all overhead groundwires and phases with line surge arresters, calculated using self and mutual surge impedance, and looking in one direction away from the tower top. The equivalent linear wavefront duration tm in (2) is 1.3 μs for a median first negative downward return stroke [2], rising to tm = 2 μs for large amplitudes [4][5][6]. Typical tower surge impedance of 150-300 Ω gives equivalent Ltwr = 0.5 to 1 μH per meter of height when tm = 2 μs. Tower inductance plays a role in backflashover when the footing impedance is low, when the tower is tall and thin, and/or when a subsequent stroke with median tm = 0.3 μs follows the same path as the first return stroke.

G. Effect of Surge Impedance Coupling The rise in potential on the OHGW and tower also raises the voltage on nearby insulated phases, through the process of surge impedance coupling. This coupling effect can be rather strong on conductors that are close together, compared to their average height over ground. For transmission lines with horizontal phase configuration and two OHGW, the coupling coefficient Cn is approximately 25 to 33%. On double-circuit lines, the coupling coefficient is higher at tower top, about 35%, and often falls below 20% on the bottom phases. H. Calculation of Critical Current The calculation of the critical current Icrit that just causes backflashover is a function of Zwave and insulation strength. Equating the model for insulation of length DIns (m) and of insulator stress, Icrit n for backflashover to phase conductor n becomes:

D. Span Length Effects on Insulator Volt-Time Curve The critical impulse flashover level (CFO) of insulation is defined as the peak of a 1.2×50 μs voltage wave that causes disruptive discharge in 50% of the applications. Test methods make use of an up-and-down method and statistical processing to establish the CFO as well as the standard deviation, which is often on the order of 3% of the CFO. Over a wide range of dry arc or air gap distance, 0.5 to 10 m, the CFO is well described by a constant value of flashover gradient, for example E50 =520 kV per meter for positive polarity on the conductor, or negative polarity on the tower.

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ቁ

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(5)

Many references [3][4][5][6][14] describe additional refinements to the simplified calculation of Icrit, including: • Nonlinear behavior of Zf (Icrit) from soil ionization, • Reduction of ZOHGW(Vtwr) and increase in Cn(Vtwr) from effects of impulse corona on OHGW at potential Vtwr,

Air insulation has a strong volt-time characteristic [4][5][6], where the full-wave CFO flashover gradient E50 increases with decreasing time to flashover t as: ସ଴଴௞௏

(4)

଴Ǥଽ௖

where Dspan is the span length (m) and c is 3×108 m/s, the speed of light. The factor 0.9 for retarded propagation velocity approximates corona effects on the front of wave as well as lossy propagation over soil. For spans of 200 to 400 m, the corresponding times in (3) and (4) are t = 1.5 to 3 μs.

The factor of 2 in (1) results from the fact that equal fractions of the surge current will flow into the OHGW, away from tower top, in each direction.

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ଶ஽ೞ೛ೌ೙ 

• Linking of tm to Ipk rather than use of a constant, tm = 2 μs, • Evaluation of Icrit at two or more times during the surge, and selecting the minimum value.

(3)

Tail-of-wave flashovers are rare for t > 16 μs. The volt-time curve, equation (3), intersects a CFO gradient of 500 kV/m at t=14 μs, and 520 kV/m at t = 11 μs.

I. Calculation of Line Outage Rate Once a transmission line has been analyzed to establish Icrit, using (2), (3), (4) and (5) there remain two steps in the calculation of backflashover rate [3][4][5][6]. A model that relates the local ground flash density to the number of flashes to the line, at average or maximum height over ground, provides the number of flashes to 100 km of line per year. The backflashover rate is this number, multiplied by the probability that the impressed peak current will exceed Icrit. The probability model for Icrit is described next.

E. Span Length Effects on Insulator Volt-Time Curve The insulator flashover process slows and stops if the voltage stress drops away from the standard impulse wave shape, for example on arrival of travelling waves from adjacent towers with Zf Ӑ ZOHGW. The most important times of interest are those related to the two-way propagation of travelling waves to the nearest adjacent spans. A recommended model in [5] is to evaluate (3) at a time t given by:

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of statistical moments that are undefined, so for example the skew (3rd moment μ3) and kurtosis (4th moment μ4) are undefined for a = 2.6. Even the standard deviation of the log-logistic distribution is undefined if the fitted exponent a < 2. This means that the log-logistic form is unsuitable for extrapolation beyond its fitted range of validity. The limitation can be addressed practically by limiting the range of application. When truncated at 1% and 99%, for example, log-logistic skew and kurtosis become better behaved.

STATISTICAL DISTRIBUTIONS OF LIGHTNING PEAK CURRENT

IV.

There are two important CIGRE references for the statistical variation of peak current, [2] and its precursor [15]. Observations of lightning peak current proved to be difficult to fit with a single log-normal distribution [15] with 31.1 kA median and σln I = 0.484 [3]. A break point was introduced at the 20% probability level, corresponding to 20 kA, and a twoslope model was preferred [15]. For the backflashover calculations, with currents exceeding 20 kA, the recommended median value is 33.3 kA with σln I = 0.605. Figure 4 shows that single-slope and two-slope models diverge for I > 50 kA.

V.

PEARSON CLASSIFICATION OF DISTRIBUTION SHAPE

The statistics literature offers at least 27 different distributions that are suitable for grading and extrapolating observations with skew [18]. The Gumbel, Weibull and lognormal distributions are also exploited [19][20] for various insulation coordination purposes. When faced with a set of new observations from the field, some classical methods can simplify selection of an appropriate grading function. A. Pearson Classification Chart The first stage of Pearson analysis is to compute the elementary statistics of the values, including median, mean, standard deviation, variance (second standardized moment), skew and kurtosis. Classification then proceeds by plotting the square of skew, the Pearson β1 coefficient, against the Pearson β2 coefficient (kurtosis + 3). Figure 5 illustrates this approach using the calculated moments and β values of a truncated loglogistic distribution with varying exponent a.

Fig. 4. Cumulative Probability of First Negative Return Stroke Current Amplitude Ipk, CIGRE [2][3] and IEEE [4][6] models

Figure 4 also plots a simplified model, based on a loglogistic form P(Icrit) = 1/(1 + (Icrit / 31 kA)2.6) [4][5][6][16]. This gives the complement of the cumulative density function (1CDF) as plotted. The CDF itself for median current Imedian can be obtained by changing the sign of the exponent a:

‫ ܨܦܥ‬ൌ

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ଵ

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(6)

(7)

The exponent a = 2.6 and Imedian = 31 kA [16] provided the best fit of (7) to the raw data in [15]. The log logistic form (7) also offers easy calculation and inversion to obtain currents corresponding to a specific probability level P:

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Fig. 5. Pearson Classification Chart with Characteristics of Truncated LogLogistic Distribution

(8)

Substituting 5% probability of exceedance, P = 0.05, into (8) gives I = 31 kA × 191/2.6 = 96 kA. Figure 4 shows that the loglogistic expression (7) with Imedian = 31 kA and a=2.6 fits the CIGRE two-slope distribution in the range 15 kA to 100 kA, but diverges at higher currents and lower probability of exceedance.

The overlays show lines and regions that correspond to specific Pearson distributions [9][10][11]. A normal distribution will have β1=0 and β2=3. The Pearson Type III distribution became known as the gamma distribution in the 1930s. The Type IV distributions contain the Student’s t form when along the vertical axis, called Type VII, when skewness is set to zero. The Type V became the inverse-gamma distribution and Type VI is the beta prime distribution. The truncated log-logistic distribution in Fig. 5 falls just below a Pearson Type III classification and into the Pearson

There is an empirical relation [17] between exponent a in (7) and the value of σ for log-normal distributions, σ = 1.71 / a. For a = 2.6 in (7), σln I ≅ 0.66 with this estimate. Unfortunately the formal statistical properties of the loglogistic function can be awkward. The exponent a sets the order

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TABLE I. Descriptive Statistics of Span Lengths at REN

Type I region. As the exponent a decreases, there is an increasing skew and matching increase in kurtosis. Figure 5 also shows that the truncated log-logistic distribution converges to the normal distribution for large values of a. One benefit of this graphical approach is that the proximity to any particular classification can be judged; in this case either Pearson Type I or Pearson Type III could reasonably be selected for this truncated log-logistic function.

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ܾଶ ൌ

ఉమ ାଷ ଵ଴ఉమ ିଵଶఉభ ିଵ଼

ଶఉమ ିଷఉభ ି଺

Weibull Fit Scale 421.5 Shape 2.69 391.5

Standard Deviation

156.5

150.5

β1, square of skew β 2, kurtosis + 3

2.0

0.08

6.9

2.4

8.8

2.8

19.1

7.8

Mean (m)

B. Parameters of Pearson Distribution The next step in the traditional Pearson approach [9] [10] [11] is to use the values of β1, β2 and second moment μ2 to compute parameters of the selected distribution. These are:

ܾ଴ ൌ

Raw Data (count = 20014) 391.4

LogNormal (best fit) 359.3

Log Logistic Median 356 m a = 4.9 396.2

0.36

146.5

σ ln span

(9)

(10)

(11)

ଵ଴ఉమ ିଵଶఉభ ିଵ଼

The probability function p(x) then becomes:

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(12)

There tends to be considerable uncertainty in the estimates of β1 and β2, especially for small data sets. It turns out to be more convenient to adjust the parameters of the distribution to fit a dataset using least-square optimization. However, the Pearson graphical classification is still an important first step when deciding which distributions are appropriate for describing raw data. One insight, for example, is that distributions falling into Pearson Type I have finite upper and lower bounds.

Fig. 6. Histogram of Span Lengths at REN, with Weibull [7], Log-Normal and Log-Logistic Fits

C.

Example: Span Lengths Reference [7] provided a histogram of the frequency of occurrence of span lengths for the 150 kV, 220 kV and 400 kV overhead transmission lines in Portugal. The histogram of 20014 values was fitted by a Weibull distribution with scale factor 421.5 m and shape factor 2.69. Figures 6 and 7 show that this grading captured the major trend of the data, but the raw data are a bit more peaked. The descriptive statistics in Table I show that the Weibull fit matches the parent values of mean and standard deviation closely, but the grading function is nearly symmetrical (with square of skew β1 near zero) and is missing some weight in the tails, β2 = 2.8 compared to 8.8. The span data were re-fitted using log-normal and loglogistic models (7), using least square minimization of the CDFs in Fig. 7 to adjust σln (span) and exponent a respectively. Table I shows that the descriptive statistics of the log-logistic fit are much closer to the original data. This is confirmed by the improved visual match of the log-logistic fit in Fig. 6 and Fig. 7. The dual relation of σln (span) ≅ 1.71/a holds in this case, with a = 4.9 giving σln (span) ≅ 0.35.

Fig. 7. Probability of Exceedence (1 – CDF) of Span Lengths at REN, with Weibull [7], Log Normal and Log-Logistic Fits

The point (β1 = 2, β2 = 8.8) for the raw data in the Pearson classification system of Fig. 5 suggests a Type IV form. Neither the Weibull nor the Log-Normal fit provide similar values, but the log-logistic fit using (7) does well in this case. The classification process [9][10][11] expands on viable alternative forms that may deliver further improvements to the grading curves illustrated in Fig. 7.

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VI.

PEARSON CLASSIFICATION OF FOOTING RESISTANCE ON TVA 500-KV TRANSMISSION LINES

The Tennessee Valley Authority (TVA) operates an extensive electrical utility in the central USA, covering the state of Tennessee and surrounding areas. Lightning performance of TVA 161-kV and 500-kV transmission lines has been studied critically for more than 30 years. Whitehead [21] established a strong relation established between lightning outage rate and average footing resistance, and also noted significant differences in median and maximum values of resistance among lines in the service territory. A. Geology and Soil Characteristics TVA characterized the underlying soil and geology in a soil resistivity database, using US Geological Survey (USGS) data. This was noted to be a best practice in [22]. The western parts of the state have Cenozic and Mesozoilc clay, silt, sand and gravel and low soil resistivity, ρ < 150 Ωm. In the east, areas of Precambrian rock have ρ > 1000 Ωm.

Fig 8b. North-East Corner of Tennessee

Recently, the US Department of Agriculture (USDA) [23] has supplemented the USGS data by providing individual state maps of the suitability of soil for use of ground penetrating radar (GPR). The suitability map is based to a large extent on the clay content of the soil, which has low resistivity and attenuates the test signals, giving poor depth of penetration. The USDA GPR suitability map for Tennessee in Fig. 8 suggests the same wide variation in resistivity as the USGS geology map. The level of detail in Fig. 8 resolves features such as exposed hill tops (with “high” suitability for GPR and thus high resistivity) as well as river valleys with “very low” suitability. The detail also provides a plausible explanation for observed tower-to-tower variation in footing resistance as transmission lines traverse the regions of different soils.

Fig. 8. Maps of Ground Penetrating Radar Suitability – Tennessee [23].

B. Tower Footing Resistance Measurements In combination with geology-based estimates of soil resistivity, TVA also undertook a campaign to measure the footing resistances at the majority of their 500-kV towers. These measurements were convenient and valid using lowfrequency earth resistance test equipment, because the OHGW on the TVA 500-kV lines are insulated from the towers with porcelain disks. Low-frequency test equipment usually provides a poor estimate of Rf at an individual tower, as the OHGW is connected to many adjacent structures in parallel. A data set consisting of 48 lines, consisting of 26 to 500 tower footing resistance measurements per line, was assembled in the mid 1990s. This as-found data set with 10,600 readings was preserved and provided for statistical analysis in [8]. The resistance values were used to develop maintenance and mitigation priorities based on recorded line outage rates. Most large utilities such as TVA and REN (the Portuguese Transmission Network) justify such asset investments on an ongoing basis. Any re-grounding or grounding improvement programs would tend to alter the results on line-by-line or individual tower basis, as additional new grounding is installed or missing buried components are replaced.

Fig 8a. Ground Penetrating Radar Suitability for North-West Corner of Tennessee

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C. Pearson Classification of Tower Footing Resistance Rf The descriptive statistics including skew and kurtosis were computed for each of the 48 lines in the TVA dataset. Figure 9 shows the pairs of (β1, β2) that fit into the same scale as Fig. 5. There are a few lines that have low skew, but for the most part the classifications fall into the Pearson Type I region, a bit below the Type III line.

Fig. 9.

most of the data sets to a common form. In the ideal case, this would be a normal distribution of ln(Rf), allowing the development of confidence intervals around extrapolations. Fig. 11 shows an expanded classification chart, with a region around the (β1=0, β2=3) point that provides a classification of “Normal”.

Fig. 11. Pearson Classification of ln(Rf) for 48 Different 500-kV Transmission Lines in TVA System

Pearson Classification of 48 Sets of Footing Resistance Values Rf for TVA 500-kV Transmission Lines, Linear Scale [17]

It would be expected from sampling models that there would be less than ± 0.16 variation in β1 (± 0.4 variation in skew) near β1=0 when there are more than 40 towers in the sample. Fig. 12, adapted from [8], shows the individual values of β1, β2 and standard deviation σ of ln(Rf) do not have any strong tendency to convergence for TVA 500-kV lines with many towers, compared to few towers.

All of the values of (β1,β2) in the TVA analysis can be resolved when they are re-plotted on logarithmic scales, as shown in Fig. 10. The overall classification of the line-by-line distributions of Rf does not change with the additional data; most remain in the Pearson Type I zone.

Fig. 10.

Fig. 12. Parameters of Pearson Classification of ln(Rf) for 48 TVA 500-kV Transmission Lines, versus number of towers in each line [8]

Pearson Classification of 48 Sets of Footing Resistance Values for TVA 500-kV Transmission Lines, Logarithmic Scale

The median value of standard deviation σln Rf in Fig. 12 is 0.92, compared to σln Ipk = 0.605 in the CIGRE two-slope model for the backflashover range [2][3].

D. Pearson Classification of ln(Rf) The high degree of skew and kurtosis seen in the distributions of Rf in Fig. 9 and Fig. 10, as well as the similar characteristics of the log-logistic form in Fig. 5, suggest that transformation of the data from Rf values to ln(Rf) would bring

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at 38° latitude, there is a large area having σ = 10 mS/m (ρ = 100 Ωm). Within a distance of 100 km, at 40° north latitude, the medium-frequency (MF) conductivity decreases in from σ = 30 mS/m at the Atlantic coast, to 0.3 mS/m inland, corresponding to ρ = 33 Ωm and 3333 Ωm respectively.

VII. PEARSON CLASSIFICATION OF FOOTING RESISTANCE ON REN 150-KV, 220-KV AND 400-KV TRANSMISSION LINES The Redes Energéticas Nacionais, SGPS, S.A. (REN) operates the national electricity transmission grid (RNT) in Portugal, including 150, 220 and 400kV networks in Fig. 13.

Fig. 13. Network of 150-kV, 220 kV and 400 kV lines at REN [24]

In 2014, REN [24] reported the consumption of electricity supplied from the public grid totalized 48.8 TWh, of which the RNT transmitted 41.9 TWh. The peak load in the Portuguese system exceeded 8 GW. In comparison, TVA supplied 158 TWh and exceeded 33 GW in the same year, so both utilities are large and both serve diverse geographic areas.

Fig. 14. Medium Frequency Ground Conductivity σ for Portugal [25]. Units: mS/m. Obtain soil resistivity ρ(Ωm) = 1000/σ

B. Tower Footing Resistance Measurements A review of power system operating experience in Portugal [7] in the period 2001-2008 noted that lightning faults: • Caused 24% of the total number of faults; • Represented 30% of common cause faults, affecting circuits that shared the same towers and right of way; • Were successfully reclosed 95% of the time; and • Affected transformers, for 1.5% of the faults. Grounding was clearly implicated in the line performance calculations. The dataset of available measurements, used in

A. Geology and Soil Characteristics The World Atlas of Ground Conductivities [25] is based on 1 MHz signal attenuation and is a useful starting point in any assessment of ground resistivity for lightning protection. The map for Portugal in Fig. 14 shows a wide range of underlying ground conductivity, in units of mS/m. In the south, centered

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2010, covered roughly half of the 150-kV, 220-kV and 400-kV transmission line structures and had been performed with a fixed-frequency instrument operating at 26 kHz [26]. As described in [27], this test frequency provides some degree of electrical isolation from adjacent towers, based on the high inductive reactance of the OHGW connections to adjacent structures. Calculations in [27] also illustrate that the measured values of Rf will be affected by an error of 10% for average Rf = 30 Ω, rising to an error of 20% for Rf = 50 Ω. At REN, readings RMeas (Ω) from the ABB Model HW2A instruments were therefore corrected as follows: ܴ஼௢௥௥ ൌ ܴெ௘௔௦ ൅ ͲǤͲͷ ቆͳ െ ݁

షೃమ ಾ೐ೌೞ మబబ

ଶ ቇ ൈ ܴெ௘௔௦

(12)

The underlying rationale for this correction is found in [26]. C. Pearson Classification of Tower Footing Resistance Rf The REN dataset consisted of 13,833 measurements of corrected footing resistance, on 136 transmission circuits. Some measurements were more than five years old and others may have been reduced by ongoing maintenance and investment decisions. Overall, 9000 of the towers were fitted with a lognormal distribution, having median Rf of 18.2 Ω and natural log standard deviation σln Rf = 1.05 in [7]. The next order of analysis was to calculate descriptive statistics based on line voltage, in Table II.

Fig. 15.

Pearson Classification of 53 Sets of Rf Values for REN 150-kV Transmission Lines, Logarithmic Scale

Fig. 16.

Pearson Classification of 60 Sets of Rf Values for REN 220-kV Transmission Lines, Logarithmic Scale

Fig. 17.

Pearson Classification of 23 Sets of Rf Values for REN 400-kV Transmission Lines, Logarithmic Scale

TABLE II. Descriptive Statistics of Measured Footing Resistances at REN Voltage

Lines

Towers

150 kV 220 kV 400 kV

53 60 23

4840 5941 3052

Voltage

Lines

Towers

150 kV 220 kV 400 kV

53 60 23

4840 5941 3052

Footing Resistance Rf (Ω) Median Std. Skew Kurtosis Dev 12.3 74.2 10.2 229 20.0 39.7 4.0 25.6 11.0 37.5 3.3 12.5 ln (Footing Resistance Rf ,Ω ) Median Std. Skew Kurtosis Dev 12.3 1.22 0.50 0.06 20.0 0.96 0.04 0.11 11.0 1.18 0.02 0.61

In common with the treatment of TVA measurements, the Pearson coefficients (β1, β2) for each line were plotted on loglog scales in Fig. 15, Fig. 16 and Fig. 17. There are a few lines that have low skew of Rf, but for the most part the classifications fell into the Pearson Type I region, a bit below the Type III line. The natural log transformation ln(Rf) has characteristics of low skew and kurtosis in Table II. This is illustrated further in the classification plots of (β1, β2) in Fig. 18, Fig. 19 and Fig. 20, highlighting the proximity to the (0,3) point associated with a normal distribution.

15

D. Pearson Classification of ln(Rf) The high degree of skew and kurtosis of Rf observed in Fig. 15, Fig. 16 and Fig. 17 is similar to the TVA result in Fig. 10. Transformation of the data from Rf values to ln(Rf) would thus be expected to bring most of the data sets to the normal distribution locus of (β1=0, β2=3), and this is indeed seen in Fig. 18, Fig. 19 and Fig. 20. The median value of σln Rf on a line-byline basis for all of the REN data was 0.735. Fig. 21 shows somewhat reduced variation of this parameter compared to the 500-kV lines of TVA in Fig. 12. Also in common, there was no trend towards convergence for lines with many towers, compared to those with fewer than 40 towers that could be subject to larger sampling errors in β1 and β2.

Fig. 18. Pearson Classification of ln(Rf) for 53 Different 150-kV Transmission Lines in REN System

Fig. 21. Parameters of Pearson Classification of ln(Rf) for 136 REN Transmission Lines, versus number of towers in each line

VIII. EFFECT OF FOOTING RESISTANCE DISTRIBUTION ON BACKFLASHOVER CALCULATION Fig. 19. Pearson Classification of ln(Rf) for 60 Different 220-kV Transmission Lines in REN System

A. Historical Trends: Flashover Rate vs. Footing Resistance Wagner et al. [28] studied the lightning performance of 132 kV double circuit lines at American Gas and Electric in the USA. Their main focus was on the adequacy of 45° shielding angle of OHGW above the phases to intercept direct strokes. They also reported a benchmark relation in Fig. 22 between flashover rate (original units of flashovers per 100 miles per year) versus the average tower footing resistance. The minimum flashover rate was about 10 per 100 line miles per year (6 per 100 km/yr) for Rf < 10 Ω in Fig. 21. This residual outage rate was thought to be the result of imperfect shielding. B. Model with Ten Intervals The IEEE FLASH program calls for users to provide a distribution of footing resistance values, dimensioned for up to ten intervals. The program calculates the critical current and lightning outage rate for each value of footing resistance in turn, and then reports the sum. This approach is illustrated using the simplified model in (2), (3), (4) and (5) with the following fixed parameters:

Fig. 20. Pearson Classification of ln(Rf) for 23 Different 400-kV Transmission Lines in REN System

16

TABLE III. Influence of Standard Deviation of ln(Rf) on Rf, Icrot and Lightning Backflashover Outage Rate Exceedence Probability Level 95% 85% 75% 65% 55% 45% 35% 25% 15% 5% Outage Rate Per 100 km/yr, Ng =1

Median Rf σln Rf ≈ 0

All 20 Ω Icrit 78 kA

2.0

REN σln Rf = 0.74

TVA σln Rf = 0.92

6 Ω; 141 kA 9 Ω; 117 kA 12 Ω; 103 kA 15 Ω; 92 kA 18 Ω; 82 kA 22 Ω; 74 kA 27 Ω; 65 kA 33 Ω; 57 kA 43 Ω; 48 kA 68 Ω; 37 kA

4 Ω; 156 kA 8 Ω; 127 kA 11 Ω; 109 kA 14 Ω; 95 kA 18 Ω; 83 kA 22 Ω; 73 kA 29 Ω; 63 kA 37 Ω; 53 kA 52 Ω; 43 kA 91 Ω; 31 kA

3.2

3.7

TABLE IV. Influence of Standard Deviation of ln(Rf) and Probability Averaging Interval on Backflashover Rate, using Log-Normal Distributions for Ipk and Rf and Ng = 1 flash / km2/yr. Probability Interval

Fig. 22. Lightning flashover record of lines of American Gas and Electric Company, 2426 mile-years of double circuit 132 kV [28]. Line dimensions in feet (1’ = 0.3048 m)

• • • • • • • • • • •

9 steps, 10…90% 10 steps, 5..95% 99 steps, 1..99% 100 steps, 0.5…99.5% 1000 steps, 0.05…99.95%

Insulators with 2 m dry arc distance Tower height 35 m, impedance 150 Ω, Ltwr 17.5 μH Ground flash density 1 / km2/year Peak first return stroke current tm = 2 μs and: o Median 33.3 kA, σln I = 0.605 or o Median 31 kA, log logistic exponent a = 2.6 Line flash rate 25 flashes per 100 km of line per year Median span length of 356 m Two OHGW, 14 mm diameter, 10 m separation OHGW sag of 5.7 m and ZOHGW = 328 Ω Coupling coefficients Cn = 0.3 Median Zf = Rf = 20 Ω At least one phase with ac voltage adding to insulator stress, average value of 0.83 per unit (pu) when the peak of line-to-ground voltage is 1 pu. For a 230-kV line, Vn becomes -155 kV in (5).

Median Rf σln Rf = 0.001

2.0

REN σln Rf = 0.74 2.8 3.2 3.2 3.3 3.3

TVA σln Rf = 0.92 3.3 3.7 3.7 3.8 3.8

Table IV also suggests that that the results for 10, 100 or 1000 probability intervals, using the central values for each interval, are identical for engineering purposes. D. Comparison, Log-Normal or Log-Logistic Model Log-logistic models have known problems of undefined skew and kurtosis when the fitted exponent a < 4 in (7). The widely-used exponent a = 2.6 along with median Ipk = 31 kA has this problem, and so do the log-logistic exponents a = 2.31 for REN Rf data and a = 1.86 for TVA Rf data. Table V shows the results of backflashover estimates repeated with the use of log-logistic expressions. The outage rates agree well with the reference values, using log-normal distributions, in Table IV. TABLE V. Influence of Standard Deviation of ln(Rf) and Probability Interval on Backflashover Rate, using Log-Logistic Distributions for Ipk and Rf

Table III shows some details of sample calculations, using the assumptions above. The table shows ten uniform probability intervals, and uses the resistance values associated with the average probability in each interval.

Probability Interval 9 steps, 10…90% 10 steps, 5..95% 99 steps, 1..99% 100 steps, 0.5…99.5% 1000 steps, 0.05…99.95%

C. Convergence with 9, 10, 99 or 100 Intervals The original FLASH program [4] was adapted [5] to execute on an IBM-PC or equivalent with 64k of memory. Limitations on memory and computation time are no longer important, and spreadsheets could now support 106 intervals if needed. However, these fine intervals would apply the fitted statistical functions at probabilities well beyond their intended range of use, where there is no support from observations. Table IV shows that there is a modest change in calculated backflashover rate when comparing nine probability steps (10%, 20%…90%) to ten intervals (5%, 15% … 95%).

Median Rf a = 1710 2.1

REN a = 2.31 2.7 3.1 3.1 3.2 3.2

TVA a = 1.86 3.1 3.6 3.6 3.7 3.7

The statistical variation of footing resistance nearly doubles the calculated backflashover rate for TVA lines. It is thus important to include a realistic treatment of this tower-to-tower variation, but this treatment is not sensitive to fine details such as bin size and log-normal versus log-logistic modeling.

17

IX.

EFFECT OF SPAN LENGTH DISTRIBUTION ON BACKFLASHOVER CALCULATION

X.

Spatial variability of soil resistivity ρ is an important factor in the wide statistical variation of Rf from tower to tower, noted both at TVA and REN. However, the relation of Rf to ρ, and high frequency behavior that relates Rf to Zf, both introduce additional distortion and uncertainty in statistical treatments of grounding in the backflashover calculation.

It is a reasonable assumption that the span length and footing resistance are uncorrelated. The main influence of span length on the backflashover calculation process is shown in Table VI. A 10x10 table of critical currents was constructed, based on the fixed assumptions in Section VIII as well as (2), (3), (4) and (5). The probability intervals used a = 2.3 and median 20 Ω for Rf. and a = 4.9 for span length, which are representative values from the REN transmission lines.

A. Comparison: 100 Hz Rf with Resistivity, Tower by Tower During the usual transmission line commissioning process, measurements are often taken of the footing resistance, using a four-terminal earth resistance tester operating at a low frequency, near 100 Hz. Two of the terminals (P1 and C1) are connected to the electrically isolated tower. A series of resistance measurement are taken as a function of potential probe distance to tower DP1-P2 using a fixed return point for the current return probe C2 at a distance DC1-C2. The resistance Rf at infinite distance is read out from the instrument reading using DP1-P2 = 0.618 DC1-C2. If readings at multiple distances DP1-P2 can resolve the fall of potential with increasing distance, then the resulting data can also be processed to obtain an estimate of the uniform soil resistivity ρ near the tower. Additional details for the in-line fall of potential method are found in [30]. A typical variation in measured resistance away from a four-leg lattice 230-kV transmission tower is shown in Fig. 23.

TABLE VI. Influence of Span Length Distribution on Critical Current, using Log-Logistic Distributions for Rf and span length (median 356 m) Icrit (kA) Span (m) Ѝ /Rf (Ω)

.05

.15

.25

.35

.45

.55

.65

.75

.85

.95

649

507

445

404

371

342

314

284

250

195

.05 .15 .25 .35 .45 .55 .65 .75 .85 .95

29 40 47 54 60 67 74 82 94 117

31 43 51 58 65 72 80 89 102 127

32 45 53 61 68 75 83 93 107 133

34 47 55 63 70 78 86 97 111 137

35 48 57 65 73 81 89 100 114 142

36 50 59 67 75 83 92 103 118 147

37 51 61 69 78 86 95 107 122 152

39 53 64 72 81 90 99 111 127 158

41 56 67 76 85 95 105 117 134 167

46 63 75 85 95 106 117 131 150 187

72 42 32 26 22 18 15 12 9 6

TOWER FOOTING RESISTANCE VERSUS RESISTIVITY

The probability of exceeding each critical current in Table VI was computed using the log-logistic form (7) with a = 2.6 and 31 kA. Table values for fixed and distributed values of span length were averaged and multiplied by the number of flashes to the line to obtain backflashover rates in Table VII. TABLE VII. Influence of Span Length Distribution on Backflashover Outage Rate, using Log-Logistic Distributions for Ipk , Rf and span. Span Length (m) Outage Rate, per 100 km per year Fixed Span Length Outage Rate, per 100 km per year Log-Logistic span length with a = 4.9

250 2.3

300 2.7

356 3.1

400 3.4

450 3.6

2.4

2.8

3.1

3.4

3.7

Fig. 23. Typical Resistance versus inverse Distance from Tower for In-Line and Oblique Earth Resistance Tests [31]

Table VII suggests that the backflashover calculation is influenced by the median value of span length, over the range of 250 to 450 m, but the results are not particularly sensitive to the observed statistical distribution of span length.

It is often better [31] if the fall of potential is measured at 90° to the current return line, ideally maintaining a constant distance from current return probe C2 to potential probe P2. For this so-called “oblique” test method, Fig. 23 shows that linear regression of measured resistance against inverse distance 1/DP1-P2 gives an intercept of Rf (2.31 Ω) at infinite distance as well as a slope of -ρ/(2π). The linear regression coefficient, R2 = 0.89 in Fig. 23, indicates whether a uniform soil model is appropriate. Horizontally stratified soils of different resistivity give curved relations between measured resistance and inverse probe distance.

When performing a backflashover calculation, it is now feasible to model each tower with its own characteristics, including distances to adjacent towers as well as any variations in tower height and conductor spacing. Some commercial programs, such as EPRI TFLASH and Sigma slp, support this process. However, this calls for entering and managing a high level of detail in each structure model.

18

Figure 23 shows fitted values of Rf and ρ for all towers on a new 230-kV line in Ontario, Canada. In an area with low soil resistivity, these exhibit a strong linear relation.

C. Comparison: Zf versus Rf, Tower by Tower Improved instruments [8][29] now impress a 2-μs square pulse with current I ≅ 1 A and rise time < 100 ns into tower base. The transient potential rise V(t) in response to this current step converges to the wave impedance Zwave consisting of Zf in parallel with ZOHGW in (1) after a few reflections up and down the tower. By completing the “Zed” measurement process after tower reflections have attenuated, but before return of reflections from adjacent towers, the influences of tower surge impedance and parallel footings can be reduced. One typical cross-check of Rf against Zf for a 500kV tower at TVA [29] gave Rf = 23 Ω compared to Zf = 12 Ω in a time range from 200 to 1200 ns. There is an overall tendency for Zf < Rf, especially for values of Rf > 30 Ω and compact footings. This trend of Zf < Rf, especially for grounding in soil of high resistivity, is additional supporting evidence for important effects of the frequency dependence of soil resistivity [32]. However, for towers with buried wire electrodes of more than 40 m length, measured values of Zf exceeded Rf by as much as a factor of 5. This effect is also understood from analysis and testing of buried counterpoise, originally by Bewley [33]. Simultaneous testing of Zwave and Rf is feasible now, and there is some convergence in commercial test equipment as sensors are adapted to measure the fraction of low-frequency current that actually flows into the tower base under test. Additional comparisons of Zwave and Rf, including the processing to establish the soil resistivity, are needed to develop improved ways to separate the footing surge response from soil resistivity effects under lightning transient conditions.

Fig. 24. Tower Footing Resistance versus Soil Resistivity for 230-kV Transmission Line in Ontario, Canada[31]

An effective perimeter of the four-leg lattice towers can be established from Fig. 24 as p = ρ / Rf = 1/0.0261 = 38 m. This is the perimeter of the solid hemisphere that could replace the foundations in grounding calculations. B. Comparison: 26-kHz Rf with Resistivity, Line by Line At REN, comparisons were made between a database of lowfrequency resistivity values at 1.5 and 4 m in dry conditions, and the results from the ABB testing at 26 kHz [7]. A powerlaw curve fitted Rf (Ω) to soil resistivity ρ (Ωm) as follows: ܴ௙ ൌ ʹǤͶʹU଴Ǥଷଶ

XI. (13)

CONCLUSIONS

Lightning has caused about of a third of the momentary outages on transmission systems, based on regulator data in USA and Canada, collected from 2008 to 2014. There is more difference between CIGRE single-slope and two-slope log-normal models for the distribution of negative downward first return stroke peak current, than between the reference two-slope model and an IEEE log-logistic model. The fitted log-logistic model of IEEE with exponent a = 2.6 has undefined third and fourth moments, making it a poor candidate for extrapolating risk calculations to low probability of exceedance, outside the range of fitted observations. The log-logistic model provided a better fit to an observed distribution of span lengths at REN than normal, Weibull or log-normal models. The fitted parameters were a median of 356 m and an exponent a = 4.9. The Pearson classifiers β1 and β2 for 10,660 values of footing resistance Rf on 500-kV lines at TVA in the USA were similar in nature to those calculated from 13,833 values measured on 150 kV, 220 kV and 400 kV lines at REN in Portugal. The majority of line-by-line sets of Rf were classed as Pearson Class I (bounded at both extremes). For both TVA and REN Rf values, the (β1, β2) classifiers from natural log transformation (ln Rf) on a line-by-line basis were close to normal (0,3). For the TVA lines, the median value of natural log standard deviation σln Rf was 0.92, and for REN it

The Pearson regression coefficient R2 was 0.63 in the range from 15 < ρ < 4000 Ωm shown in Fig. 25. The data suggest that median resistivity along a line is a relatively poor indicator of median resistance, especially compared to the tower by tower results in Fig. 24.

Fig. 25. Observed Relation between Median RF and soil resistivity ρ for 13,833 towers [7].

19

[9]

was 0.73. There was no trend towards convergence or scatter in β1, β2 or σln Rf with increasing number of towers. The inclusion of a model for the variability of Rf in the IEEE FLASH program, calling for ten probability intervals, gives satisfactory performance compared to the use of 100 or 1000 intervals, provided that uniform intervals of 5, 15 .. 95% are selected. There was a 2:1 change in backflashover rate when using the value of σln Rf = 0.92, compared to a median value. The calculation of backflashover rate was sensitive to the median value of span length but not to its typical statistical distribution, using observed data on the REN system with loglogistic exponent a = 4.9 or equivalently σln span ≅ 1.71/a = 0.35 in a log-normal distribution. In an area of Ontario, Canada with low soil resistivity, there was close correlation between ρ and Rf measured on a towerby-tower basis. The relation in Portugal between median ρ and median Rf on a line-by-line basis had much larger scatter. Future characterization of footing impedance should make use of new technology to measure Zwave as well as improved measurement lead topology, to measure local soil resistivity ρ and Rf simultaneously. Future development of standard methods to evaluate transmission line backflashover performance in IEEE (Working Group on Lightning, FLASH 2) and in CIGRE (WG C4.23) should include an appropriate statistical model for the tower-totower variation of footing impedance.

[10] [11] [12]

[13]

[14] [15] [16] [17] [18] [19] [20] [21]

[22]

XII. ACKNOWLEDGMENT [23]

Diana Skrzydlo of the University of Waterloo, Faculty of Mathematics, Department of Statistics and Actuarial Science provided insight into the limitations of the log-logistic form when a < 4, related to its undefined high-order moments. REN, the Portuguese TSO, made the data availability to perform the presented analysis, and the authors especially acknowledge assistance from Albino Marques, Senior Manager, System Operations.

[24] [25] [26]

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