Voltage Profile

Phase-Ground Switching Overvoltages, Transmission Lines

3.4 Voltage Profile

Again Eq. 16 needs further modification since it assumes that the SOV is constant at every tower along the line, when in reality the SOV is usually lower at the switched end of the line and is maximum at the opened end of the line. To include this effect, consider a transmission line, illustrated in Fig. 10, composed of n towers. For a single case of breaker closing, the voltages along the line are Vl (or V,) at tower 1, V2 at tower 2,.

. . ,

and Vn (or Vv) at the last tower. The probability that the SOV is equal to Vl at tower 1 is f (Vl) dVl, the probability that the SOV is equal to V2 at tower 2 is f(V2)dV2, and the probability that the SOV is equal to Vn at the last tower is f (Vn) dVn. However, Vl, V2,. . . , vn are dependent or exactly correlated. That is, for one switching operation, if Vn occurs at tower n, the Vl will occur at tower 1, V2 will occur at tower 2, etc. Therefore the probability of occurrence of Vl is equal to the probability of occurrence of V2, is equal to the probability of occurrence of Vn, etc. Or in equation form

Figure 10 SOV profile along line.

Chapter 3

To simplify, we will use f.(V)dV and note that

fs

(V) is the probability density function at the opened end of the line.

Returning to Fig. 10, the probability of a flashover at tower 1 for the SOV of Vl is p i ; at tower 2 for the SOV of V2 the probability of flashover is p2, etc. However, since the probability of at least one flashover (on one tower) is desired, we must first calculate the probability of no flashovers (on any of the towers) and then subtract this from 1. The probability of no flashover on tower 1 is ql, the probability of no flashover on tower 2 is q2, . . ., and the probability of no flashover on tower n is qn.

Therefore the probability of noflashover on the line for a single switching operation is

and the probability of at least one flashover is

Considering all switching operations, the SSFOR is

SSFOR =

;[^

^-

fi

qj)(V) dV

i= I

To be noted is that if the SOV is constant along the line, then ql = q2

.

^{. .} qn and Eq.

20 is the same as Eq. 16.

Unfortunately, there is no unique solution for Eq. 20, and therefore it must be solved numerically (easily accomplished with the aid of a digital computer).

However, there exist simplified methods that can be used to obtain quickly an acceptable estimate of the SSFOR. Before presenting this simplified method, a sen- sitivity analysis of the SSFOR will assist in understanding the relative importance of the parameters or variables.

4 SOV (STRESS) DISTRIBUTIONS 4.1 Case Peaks or Phase Peaks

The SOVs are usually obtained by use of a transient computer program. Breakers are random switched throughout their pole closing span and the SOVs obtained. For each switching case, a SOV occurs on each phase and the probability of flashover P(F) may be calculated for each case as

where q~ is the probability of no flashover on phase A, q~ is the probability of no flashover on phase B, qc is the probability of no flashover on phase C, and N is the number of cases. The addition of the P(F) values for the N cases is then the SSFOR.

Phase-Ground SOVs, Transmission Lines 101

This method is often called the brute force method. However, it represents the most exact method since it accounts for voltages on all three phases as they occurred for each case. The advantage and disadvantage of this method are the same in that it is specific to the exact system considered. It does not permit a general evaluation of the effect of the parameters and does not provide an understanding of the phenomena.

Note that in the solution of Eq. 21, the SOVs may be positive or negative. If the SOV is negative, for practical designs, the value of q is essentially unity. Also, in the same manner, if the SOVs are small in magnitude, the value of q will be essentially unity. Usually, the value of P(F) is controlled by only one of the values of q, the one obtained from a SOV that is positive and has the highest value of the three phases.

To circumvent the problem associated with the brute force method, the data may be collected and analyzed by two methods:

1. Case Peak Method. For each switching operation, the SOVs are collected.

Only the SOV with the largest crest value, either positive or negative polarity, is used.

This SOV is treated as positive since, if negative, the exact opposite breaker switch- ing sequence would produce an opposite polarity SOV. This method, primarily used in the USA and Canada, as developed in Ref. 1, assumes that only one SOV pre- dominates. In terms of Eq. 21, two of the qs, for example qB and qc, are essentially equal to unity, so that

Note that this probability should be multiplied by 112 since with equal likelihood, either a positive or a negative polarity may occur and the negative polarity SOV is neglected since the negative polarity strength is significantly greater than that for positive polarity. The SSFOR calculated by this method is the SSFOR per three- phase breaker operation or the SSFOR for the line.

2. Phase Peak Method. The phase peak method consists of using all the three SOVs from each phase, and each of these are assumed as positive polarity. The P(F) is calculated individually for each of the three SOVs. Thus the SSFOR calculated by this method is the SSFOR per phase. In terms of Eq. 21, the P(F) is

where the equation is divided by 3N since three times as much data is collected. As for the case peak method, the probability should also be divided by two. Usually, two of the values of p are essentially zero so that, except for the 3N, Eqs. 22 and 23 are identical. However, to obtain the SSFOR or the sum of P(F) each calculation must be multiplied by 3. The problem occurs when a continuous distribution is employed to represent these three values of SOV. In this case, as an approximation, to obtain the SSFOR, the following equation is used:

SSFOR = 1 - [1 - (SSFOR~)]~ 3(SSFORp) (24)

1 02 Chapter 3

where SSFORp is the SSFOR calculated using the phase peak method. For the normal low values of SSFORP, the SSFOR is simply three times the SSFORp.