Overvoltages developed in termination of a transmission line at energization

Overvoltages developed in termination of a tran.

i ion line

at energization D F Binns Umversrty of Salford, Salford M5 4WT, UK

H

~

CEGB, South Eastern Region, Banks,de House, Sumner Street, London SE1 9JU, UK

The transient voltage developed across the capacitance at the end o f a transformer feeder or at the end o f a hne containing a series reactor, following line energTzatton, has been calculated m the presence o f source impedance. The Laplace transform and travelling wave technique has been used for the analysis, and the transient voltage has also been observed on a physical model Voltages o f 5 p.u. are predicted when there is fundamental frequency resonance m the termination, as may occur for hnes up to 10 km long. For third and fifth harmonic resonances, voltages approximately 0.5 p.u. and 1.0 p.u. below this are found, but they may occur in lines up to 50 km long. Source impedance greatly reduces the voltage developed, for example, adding either 200 ~2 resistance or O.02 H mductance results In a peak voltage approximately half that obtained for an infinite source. The effectiveness ofusing closing resistors to reduce the voltage is confirmed, and it is clear that very pessimistic esttmates o f overvoltage wtll result if an infinite source ts assumed.

using a single-phase model, and the results have been substantiated by measurements on a physical model 6.

II. Notation E source voltage source resistance, inductance Rs, Ls switches at the source and termination Sh $2 Z,£ line characteristic impedance, length Rt, Lt, Ct termination resistance, inductance, capacitance P Laplace operator T transit time of surges in the line C free-space velocity frequency of the surge voltage pulses m the line fo natural frequency of the termination ft Q Q-factor of the coil in the termination OCt, (Z S surge reflection coefficient at termination, source J Bessell function F gamma function

I. Introduction When a series current-limiting inductor or a transformer feeder is placed at the end o f a transmission line, relatively high voltages can be produced if the line isenergized from t h e e n d remote from the Inductor or transformer ~-a. These voltages may arise because there is an oscillatory circuit containing the inductance o f the inductor or transformer and the combined capacitance of the transformer or inductor and other equipment such as cables and potential or current transformers. The overvoltages developed at the end of a transmission line being caused by line energization contain frequencies related to the hne length 5. For gaven termination parameters, there is a critical line length such that the surge voltages reflected up and down the line produce a succession of voltage pulses at the termination having a fundamental or harmonic frequency close to the termination resonant frequency. Consequently, large overvoltages can occur for a wide range o f line lengths and termination parameters. This condition has been analysed

II1. System model used for analysis The single-phase circuit of Figure 1 has been used to represent an overhead transmission line terminated by '_a_!! oscillatory circuit. When the line is energized by closing switch S~, with S~ open, a surge travels along the hne at a speed close to the free-space velocity. When the surge reaches the termination, a wave is reflected back along the line, and when this wave reaches the source end of the line. a further reflection occurs. The present analysis deals with relatively short lines, either 7.5 km or 22.5 km long, such that the attenuation o f the surge in transit down the hne is relatively small. It should be noted that the surge voltage

Rs

Ls

Sj

Rt

/Loss~,s

Lt

SZ

\

/ overhead hne\ Rece,ved

154

12 August 1980, Rewsed 19 November 1980

F~gure 1 System model

0142-0615/81/030154-05 $02.00 © 1981 I PC Business Press

Electr,cal Power & Energy Systems

on the line itself is not likely to be high enough to cause corona. If the line is of length ~ and is assumed to be lossless, so that all frequency components in the surge travel at the free space velocity c, then the surge transit time in the line is £/c. Consider first the simplest case of zero source impedance (Rs = 0, L s = 0) and an effectively infinite termination impedance (Lt/Ct) 1/2 >~ Z where Z is the line surge impedance The voltage at the source end of the line after the switch $1 closes will remain at the source voltage E. At the same time, the voltage at the termination end of the line will adopt a rectangular waveform varying in amplitude between 0 and 2E. This wave will have a period of 4£/c, equal to four surge transit times In the line, and a fundamental frequency fo = c/(4£). The form of the surge voltage propagated along the line and the resultant voltage developed across the capacitor may be evaluated using the Laplace transform. However, as successive reflectmns are dealt with, to provide solutions valid for increasing numbers of transit times, the analysis becomes increasingly complex and the expressions for the voltage developed across the termination capacitor become unwieldy. The transform expression for the capacitor voltage therefore has to be split into partial fractions by a numerical method rather than analytically. Since the main quantity of interest is the highest voltage developed across Ct, it 1s necessary to continue tracking the surges in the line only until the appropriate peak voltage has been reached. To identify the relevant peak for given values of the system parameters, at IS helpful if tests can be performed on a physical model including a source, transmission line and termination 6. Referring to Figure 2, which describes the approximate behaviour of Figure 1, the voltage at the line termination is zero until one transit time r has elapsed after switch closure, the switch as assumed to close at t = 0. Unless the source inductance is very high, the voltage at the end of the line will rise until a time 3r has elapsed. If the termination components L t and Ct together have a natural frequency close to the fundamental frequency of the surge voltage at the line termination, i.e. If 2zr(L t x Ct) 1/2 = 4~/c, the voltage across Ct wdl rise cosmusoldally to reach a first peak equal to twice the voltage at the end of the line after approximately 3r, as shown in Figure 2. The capacitor voltage will then reach an opposite peak of about the same

amplitude after 5r and a second voltage peak of the original polarity, which can in theory be four times the voltage magnitude at the line termination, after 7r. If the natural frequency of the termination is close to the third harmonic of the reflected voltages in the line, the first peak of voltage across the capacitor Ct will occur after about (5/3)7, as shown in Figure 2, and there will be a second peak of the same polarity of equal or lesser amplitude, after about 3r. The same voltage magnitude can in theory be reached in the first negative-going peak after (11/3)7 and in the third positive peak after (13/3)7. The second negative peak at 5r (disregarding the minimum point after (7/3)7) occurs simultaneously with the second positive-going voltage rise at the line termination; consequently the capacitor voltage can theoretically rise to four times the voltage at the end of the line, at a time (17/3)7. The next positive peak at 7r can in theory also reach this voltage. However, source inductance will delay and distort the voltage wave, and the finite impedance of the termination to oncoming surges from the line will cause a progressive reduction in the amplitude of the rectangular wave reaching the termination. Although the line represented in Figure 1 IS assumed to be lossless, both the source and termination contain resistance so that there will be some damping of the surges propagated in the line following switch closure. For a system containing relatively low damping resistances, the second peak of the capacitor voltage is found to be the highest one, when the termination resonates with the fundamental of the surge voltage in the line. It is found that the fourth or sixth peak is the highest in a slightly-damped system when the termination resonates with the third harmonic of the line voltages. Referring to Figure 1, the parameters that are varied in the present analysis are the relative natural frequencies of the termination f t = 1 / 2 n ( L t C t ) 1/2 and of the line,fo = c/(4~), the Q of the termination inductor = 2 u f t ( L t / R t ) at the appropriate frequency, and the impedance consisting of R s and L s. All these parameters can be varied simultaneously using the computer programs that have been developed.

IV. Method of analysis 8~

6X

4--

2

o

Figure 2 Idealized voltages developed at the termination; (a) idealized termination voltage, (b) fundamental voltage across C t, (c) thwd harmonm voltage across C t

Vol 3 No 3 July 1981

Referring to Figure 1, when an incident surge voltage reaches the termination from the line, a surge is reflected back given by ~xtV1 where 0% = [p2 + ( R t _

Z)P/Lt + ll(LtCt)]

l i e 2 + (Rt + z) e/t,t + I I ( G G ) I lS a reflection coefficient, and V1 is the transform of the incident surge voltage at the termination. Note that the denominator of the expression for cxt has complex roots in practice. Again, if a surge with a transform Vz reaches the source from the line, a surge O~sV2 is reflected back, where the reflection coefficient is % = [t' + (R~

Z ) I L j I [ P + (R, + Z ) I L A

155

tlon resonates with the fundamental of the stage voltage m the line and the first five peaks when resonance is with the third harmonic

4

2 d

The analysis is also simphfied for an infinite source, t.e lor Rs = Ls = 0 in Figure l, since then ~xs = 1 and for a constant source voltage E, $1 = E/P.

o -6 >

-2

Figure 3 Voltages across C t for L s = O; (a) R t = R s = O; (b) R t = 8 0 f 2 , R s = 1 0 0 ~ ; ( c ) R t = 80 ~ , R s = z If the switch $1 is closed at t = 0 and the source voltage is then maintained at a constant value E, the voltage present at the source end of the line, for a time interval 0 ~ t < 27, 1S S 1 = ( Z E / L s ) I [P(P + (R~ + Z)/Lsl. On arrival at the line termination at time t = r, the surge is described b y / ' 1 = e x p ( - r P ) S ~ and the wave reflected back along the line is T2 = %7"1. The voltage present at the termination for r ~< t < 3r IS T1 + T2 and the voltage developed across the capacitor C t during this time is [(T, + T2)l(LtCt)ll[(e2 + ( R t / L t ) P + ll(LtCt)]

The voltage surge returning to the source end of the line is $2 = e x p ( - r P ) T2 and this gives rise to a reflected surge $3 = ~sS2. Then T3 = e x p ( - rP)S3 and T4 = =tT3 so that for 3r <~ t < 5r an additional voltage

[(T3 + T . ) I ( L t G ) ] I [ P z + ( R t / r t ) P

+

Another special case arises when Ls = 0 and R s = Z, as it might effectively do when a large switching resistor is used and the source inductance is negligible by comparison. Then c¢s = 0 so that $1 = E / 2 p , 7"1 = e x p ( - ' r p ) S l , T2=cxtT 1 and no further surges reach the termination after Tj.

V.

Selection

of source

inductance

It is necessary to estimate appropriate values of the source inductance to use in an analysis. The power frequency positive sequence fault inductance per phase is V2/27rfS, where V is the system voltage in kV, f i s the frequency and S is the fault level in MVA. For 400 kV, at 50 Hz, and S ranglng from 5 000 MVA to 30 000 MVA, the corresponding fault inductance varies from 0.102 H to 0.017 H. For V = 132 kV, and S varying from 500 to 3 500 MVA, the inductance varies from 0.102 H to 0.016 H. The equivalent zero sequence source inductance at a frequency of several kHz for transient currents that flow partly in the earth path must depend on the precise system components connected at the point, but will therefore usually he in the range 0.02 H to 0.2 H.

1/(LtCt)] Vl. Investigation of transient voltage levels

IS developed across C t adding to the voltage developed by 7"1 and/'2. In a similar way, the capacitor voltage for 5r <~ t < 7~ IS obtained by adding the voltage produced by the incident and reflected surges Ts and Ta arriving at the termination at t = 51-, to the voltage across Ct already deduced. The effect of Ts and T6 together is to add a term ( E = 1)

2 z ( P + (R, - Z ) / L , ) ~ [/2 + P ( R , - Z)/L, + 1/(r~C,)l ~ / L s L t C t e ( e + (Rs + z ) / L s ) 3 x [p2 + P ( R , + z ) / L t + I / ( L t C t ) I 3

to the transform expression for the voltage across Ct. In appropriate cases, the inverse transforms were obtained 8' 9 using the relationships

£-i p/(p2 + a~)n

= 711/2(//'l-1/2)

Figure 3 shows the idealized form of the voltage across the capacitor Ct for zero source Impedance, when the termination resonates with the fundamental of the surge voltage in the line. The effect of finite source inductance and resistance is shown in Figures 4 and 5. In theory, the voltage can exceed 5 p.u. and even for values o f L s = 0.001 H and R s = 25 ~2 the voltage approaches 4.5 p.u. In practice, the termination of a transmission line IS more likely to resonate with the third or fifth harmonic of the surge voltage. F o r L t = 0.I1 H and C t = 2.3 nF, fundamental resonance occurs for lines 7.5 km long, whereas for lines 22.5 km or 37.5 km long, which are more frequently encountered, there will be third or fifth harmonic reson~d 5-;

[J(at)n_ 3/2] /[ 2 n - '/2 I'(n) a n - 3/;]

£-1 1/(p2 + a2)n = T(i/z(tn - 1/2) [J(at)n -1/2]/[(2~) n

- 1/2

l~(g/)]

which are valid for Re(n) > 0 and Re(p) > I lm(a) I.

o o

If source Inductance is neglected but the resistance R s IS retained at a finite value, the form of the voltage is simpler, since then

%-

R s -Z

R s +Z Consequently, the analysis has been continued for this case up to t = 7r, i.e. to include 7"7 and Ts. This allows two peaks of the capacitor voltage to be calculated when the termina-

156

I

i - 0

1 00I

I 002

__i 0 05

OO4

Source mductonce, H

Figure 4 Variation of peak voltage across termmatton capacitor with source inductance for dafferent source resistances; (a), (d) R s = O; (b), (e) Rs = 25 f ] ; (c), (f) R s = 50 ~ , (a), (b), (c) first peak; (d), (e), (f) second peak

Electrical Power & Energy Systems

ance. For third harmonic resonance, the capacitor voltage takes a form such as that in Figure 6 for a line 22.5 k m long. Since the surge transit time in the line is 75/as, the analysis will be valid for seven transit times, i.e. up to almost 525/as after switch closure. This takes account of the first surge reaching the termination after 75/as together with the following two surges after 225/as and 375/as. A mathematxcal analysis taking source impedance into account becomes tedious for any additional period of time. However, the oscillogram of voltage observed on the physical model, shown in Figure 7, demonstrates that the voltage peak in Figure 4 is indeed the highest one developed. Figures 4 and 6 show the effect o f source inductance L s

5

4

3

I

_irirrvin

'

Figure 7 F o r m of voltage across t e r m i n a t i o n capacitor f o r t h i r d - h a r m o m c resonance in a phystcal m o d e l ; source inductance and resistance = 0, t e r m i n a t i o n capacitance = 1.35 nF 4

on the voltage reached with or without the presence o f the source resistance. The highest voltage appears in the second peak for a critical inductance o f about 0.01 H. This critical value appears to make the time constant Ls/(Rs + Z ) of the hne and source together approximately equal to the rise time, or quarter period (n/2) ( L t C t ) 1/2 of the termination.

r3~

o E

2 o

3

\ \\

The analysis based on Figure 1 does not take account of resistance m the line but this can be allowed for approximately b y placing an appropriate resistance in the source. Tests on the ten-section physical model showed that, with zero source inductance, a resistance of 10 g2 in the source caused the same reduction in peak voltage as 20 ~2 resistance distributed among the line sections. b a

V I I . Conclusions

0

I

I

L

I

I

I00

200

300

400

500

Figure 5 Calculated peak voltage across termination capacitor as a function of source res)stance, (o) R t = 80 ~2, L s=O,lb)

4

R t=800,,Ls=0.005

a b C \

H

r', ' -.~

j/

o~ o

,oo

-2

C7 oo ~kj/T1me,lls

Figure 6 Calculated voltage developed across the t e r m i n a t i o n capacitor f o r d=fferent source impedances w i t h t h i r d h a r m o n i c resonance. L t = 0.11 H, C t = 2.3 nF; (a) R s = O, L s = 0.001 H; (b) R s = O , L s = 0.01 H, (c) R s = 100 ~ , L s=O001H,(d) Rs= 100~,L s=O.O1H

V o l 3 No 3 J u l y 1981

Previous analyses 3'4 of resonant overvoltages m transmission line terminations have considered only fundamental frequency behaviour m detail. This study demonstrates that high overvoltages up to 4 p.u. can arise due to harmomc resonance and also shows the strong dependence of harmonic overvoltage levels on source and termination parameters. F o r the related case o f a feeder transformer, the transient voltages that have been observed in field tests 1' 2 can be discussed qualitatively in terms of the present analytical results. It is, however, necessary to extend the present analysis by modelling a 3-phase system containing a feeder transformer if voltages on the transformer secondary side are to be examined. The value of source inductance used in a computer analysis for the feeder transformer problem 1'2 was not quoted, although source inductance was found to have a very large influence on the predicted voltage peak. The effect of source impedance on the transient voltage deduced in the present studies for the 22.5 km hne can be compared with the effect of source inductance observed m the TNA studies for lines 80 km or more long 7. However, m the latter study, source inductance was varied up to 1 H compared with a limit of 0.1 H in the present analysis. Since source Impedance has such a large influence on transient voltages and since high source inductances can

157

lead to higher overvoltages than occur with an infinite source 7 it is important to know what range of effective values is encountered in practice. This is likely to be determined by comparing measured field test overvoltages with values predicted by digital computer and TNA studies. It is further necessary to demonstrate the validity of using lumped resistance and inductance values to represent actual source conditions, especially where a number of relatwely short transmission lines are connected on the source side of the switch used to energize the line.

3 Heaton, A G and Reid, I A 'Transient overvoltages and power-hne termmation' Proc. Inst. Electr. Eng. Vol 113 No 3 (March 1966) pp 461-470

V l II. Acknowledgement The authors are grateful to Professor J H Calderwood, for

6 Binns, D F and Habibollahi, H 'Determinatton of over-

facilities provided, and to the University of Salford, UK, for financial support.

IX. References 1 Csuros, L and Foreman, K F 'Energizing overvoltages on transformer-feeder circuits' Electr. Times (31 August 1972) pp 37-40 2 Csuros, L, Foreman, K F and Glavitsch, H 'Energizing overvoltages on transformer feeders' Electra No 18 pp 83-105

158

Anderson, J H and Heaton, A G 'Transient analysis of power-line/cable systems mcluding reactive terminations with surge divertors' Proc. Inst. Electr. Eng. Vol 113 No 12 (December 1966) pp 2017-2022 Binns, D F and Habibollahi, H 'Frequency content of swutching overvoltages' Proc. Inst. Electr. Eng. Vol 123 No9 (September 1976) pp 913-915

voltages =n transmission line termination using a phystcal model' Int. J. Electr. Eng. Educ. Vol 12 (1975) pp 367373

Bickford, J P and EI-Devieny, R M K 'Energzzation of transmiss=on line from inductive sources' (Effect of nonstmultaneous closure also investigated) Proc. Inst. Electr. Eng. Vol 120 (1973) pp (883-890) Roberts, G E and Kaufman, H Table of Laplace transforms Saunders, UK (1966)

Lebedev, N N Special functions and their apphcations Prenttce-Hall, UK (1965)

Electrical Power & Energy Systems