Number of Towers

Phase-Ground Switching Overvoltages, Transmission Lines

6.3 Number of Towers

The effect of the number of towers on the required V3/E2 ratio to achieve a SSFOR of 1.01100 is presented in Fig. 16. For these curves, it is assumed that a SSFOR of 1.01100 occurs at a V3/E2 ratio of 1.0. that is, the curves are per unitized on this basis. Also, n = 500, of/CFO = 0.05, Ec/Ev = 1.00, and a Gaussian SOV distribu- tion is assumed. The value of 00/E2 was varied between 0.05 and 0.1 1. The curves illustrate that the strength-stress ratio varies only between ±1 for n between 300 and 1000 towers, or for lines between 60 to 300 km (40 to 190 miles). Thus the value of n within this range is not a sensitive parameter.

The effect of of/CFO on the required V f i ratio for a SSFOR of 1.0/100 is shown in Fig. 17. As for Fig. 16, these curves are per unitized on the basis of a SSFOR of 1.0/100 at a V3/E2 ratio of 1.0. Also assumed is that n = 500, Es/ER = 1.00, oO/E2 = 0.05, and that the SOV distribution is Gaussian. The value of of/CFO of 5% is considered conservative, but even if the value changes from 3 to 7%, the required V3/E2 ratio only varies by ±I% That is V3/E2 is insensitive to of/CFO.

Figure 16 Effect of the number of towers.

Chapter 3

Figure 17 Effect of q / C F O . 6.5 SOV Profile

Next to the strength-stress ratio, the SOV profile along the line is the most sensitive parameter, as illustrated in Fig. 18. This figure also presents the required strength- stress ratio to obtain a SSFOR of 1.0/100. As shown, for a of/CFO of 5% and a q / E 2 of 11 %, a change in Es/ER from 1.0 to 0.9 decreases the required V3/E2 ratio by about 4% to maintain a SSFOR of 1.0/100, while a change to Es/ER of 0.6 only decreases the required strength-stress ratio by another 2%. Thus the curves of Fig.

15 should be shifted to the left for an Ev/Ev less than 1.00, and the approximate design of V3 = E2 becomes conservative.

The effect of decreasing Es/ER can also be obtained by comparing Figs. 15 and 18. For an Es/ER of 0.9, the SSFOR decreases from 1.19/100 to 0.55/100.

Figure 18 Effect of SOV profile.

Phase-Ground SOVs, Transmission Lines

7 ESTIMATING THE SSFOR 7.1 Brown's Method

In the previous sections, the general equations were developed and solved by numer- ical methods, since no closed form solution is possible. Several attempts have been made to obtain an approximate and simplified method so that computer programs using numerical methods could be circumvented. Two such methods are those of Alexandrov [2] and Brown [3]. Alexandrov's method is limited in application but was one of the first attempts; Brown's is more general and represents an excellent approx- imation. Further, Brown's method permits the calculation of both the SSFOR and the required V 3 / E 2 ratio given the SSFOR. Therefore only Brown's method will be presented here, first the method of calculating the SSFOR, then the method of estimating the V3/E2 ratio. Another recent method is that presented in IEC Publication 71-2 [l 11. This will be fully discussed later in Section 13.

First, assume that for every switching operation, the SOVs are identical at each of the towers, i.e. Ec/Ev = 1.00. As illustrated in Fig. 19, as the number of towers increases, the strength characteristic becomes steeper, or the standard deviation becomes smaller. That is, for any specific voltage, the probability of flashover increases from p to (1 - 9"). If the strength can be represented by a single-valued function located at a voltage equal to CFO,,, as illustrated by Fig. 20, the probability of flashover or the SSFOR is simply

which can easily be evaluated for any SOV distribution. For example, for the Gaussian distribution,

where the first term is approximately 1.00. For the extreme value positive skew distribution,

s

I STRENGTH

Figure 19 Steepness of strength characteristic increases with the number of towers.

Chapter 3

Figure 20 Simplifying calculation if strength is a single valued function.

where

and again, the firm term is approximately 1.00.

The CFO,, is the CFO for n towers and can be obtained from a knowledge of the strength characteristic for a single tower as illustrated in Fig. 21. As for the CFO of a single tower, the CFOn for n towers is defined at a probability of 0.5. The objective is to find the probability p on the single-tower strength characteristic that is equivalent to the CFO for n towers. Therefore

STRESS

l A -

s

STRENGTH

Figure 21 Effect of number of towers.

Phase-Ground SOVs, Transmission Lines 113

From the value of p, the body of Table 1 is entered to obtain the value of the reduced variate Z , which in this case is denoted Zf. By the formula for the reduced variate, 1.e..

CFO,, - CFO Zf =

Of

the CFOn is obtained as

CFOn = CFO[l

+

Zf

-1

Of

CFO (54)

To be observed is that the value of Zf is usually negative.

To illustrate by example, for n = 200, p = 0.003460. Entering Table 1 with F(Z) of 1 - p or 0.996540, Z is about 2.70 and therefore Zf is approximately -2.70.

Therefore CFO. is 0.865 CFO assuming a of/CFO of 5%. To continue, if the CFO is 1000 kV and the SOV distribution is Gaussian with E2 = 900 kV, Em = 999kV (one standard deviation above E2), and oO/Ez = 0.11, the po = 696.67 kV, the SSFOR per Eq. 49 is

SSFOR = - F 999 - 696.67) (865 - 696.6711

2 99 ^- F

99

where the F(Z)s are obtained from Table 1. As noted, the first term of the calculation can be conservatively assumed as 1.00, which illustrates that Em seldom needs to be considered or evaluated.

If in the above example the SOV distribution is an extreme value positive skew distribution with the same E2 and with WE2 ⁼ 0.11, then u = 513.7kV and the reduced variate y is

Then neglecting Em, per Eq. 50, the SSFOR is

1 1 ^{- 3 548}

SSFOR = - [1 ^- e - e - y ] = [l - e-

2

]

^{= 1.42/100}

To consider the SOV profile, and equivalent number of towers ne is calculated and then used in Eq. 52. The value of n,, is the number of towers having an Es/Ev = 1.00, which gives the same SSFOR as the actual number of towers with the specified Es/Ev, as illustrated in Fig. 22. The equivalent number of towers may be estimated from the equation

Chapter 3

Figure 22 Equivalent number of towers is at constant SOV.

kn n or n = n whichever is less

ne = -

1 - y CFO where

and kn is a function of ne as shown in Fig. 23. Theoretically, kn should be determined iteratively. However, over the practical range of 30 to 500 towers, an average value of kn of 0.4 may be used, since the exact number of towers is insensitive to the required V^/E2 ratio. Therefore.

ne=- 0.4 - Of n or ne = n whichever is less 1 - y CFO

Figure 23 kn in practical range is 0.4. Copyright IEEE, 1978 [4].

Phase-Ground SOVs, Transmission Lines 115

As an example, consider the previous example for the Gaussian distribution with 200 towers but now assume an Es/ER of 0.9. Thus ne = 40. Continuing, the CFOn is 895 kV, and therefore the SSFOR reduces to 1.15/100.