Electromagnetic Transients on Power Lines Due to Multi-Pulse Lightning Surges

21, rue d'Artois, F-75008 Paris http://www.cigre.org

Session 2002

33-101

© CIGRÉ

ELECTROMAGNETIC TRANSIENTS ON POWER LINES DUE TO MULTI-PULSE LIGHTNING SURGES *

MOHAMED M. SAIED Kuwait University Kuwait.

Abstract consulted. Similar to bibliographies found elsewhere, such as [2, 3], considerable attention was given to transient phenomena involving overhead lines and underground cables, as manifested in the relatively large number of investigations dealing with these elements.

This paper identifies and discusses critical conditions in a sample power network that can lead to electromagnetic transient stresses of unusual magnitudes and time waveforms. They can be initiated, for example, by atmospheric discharges involving overhead lines, if the surge includes several pulses of repetitive nature, whose frequency of repetition coincides with or close to the natural frequency of the power line. A mathematical model based on distributed parameter analysis is presented taking into account the parameters of both the waveform of the discharge surge current and the sample power network.

In an interesting qualitative recent publication, [4], a model based on the theory of microwave resonators is outlined, capable to explain possible accelerated voltage amplification leading to voltage surge resonance conditions within the power networks. It was observed that “ under appropriate boundary conditions, the oscillating voltage stresses are likely to amplify during back and forth reflections”. The descriptive surge amplification model applies also the lattice diagrams to explain the possible build-up of voltage surges. Nevertheless, no concrete results have been reported on.

1. INTRODUCTION The information resulting from the analysis of the electromagnetic transient is of paramount importance in the planning, design and operation of electric power networks. Effective insulation coordination and overvoltage protection will be primarily affected by the networks’ topology, their characterizing parameters as well as the time waveform of the input voltage and/or current stimuli. Accurate results from analyzing these situations in terms of the expected transient stresses will allow for more effective protective measures and efficient utilization of the network components. For a detailed survey of the different computer-based techniques pertinent to the transient analysis of power networks, the IEEE Tutorial Course [1] should be

*

Electrical Engineering Department, College P.O.Box 5959 Safat, Code 13060, KUWAIT.

This investigation tries to present a more detailed and rigorous approach based on circuit analysis, in order to explain the above-mentioned phenomena. The resulting model applies the distributed parameter line representation, in conjunction with the concept of the generalized ABCD two-port circuit constants. The suggested procedure takes into consideration the number of the discharge pulses as well as their frequency of repetition. The location of the point (along the line) hit by the lightning stroke is an important parameter affecting both the waveform and the amplitude of the resulting transient stress.

of

Engineering

and

Petroleum,

Kuwait

University,

2. METHOD OF ANALYSIS Transformer 1

Transformer 2

Lightning P Line

x. L

(1-x). L

Total Line Length : L Fig. 1 : Sample Transmission Network Fig. 1. shows a transmission line or cable of total length L, which is terminated at its ends on the left and right sides by the transformers T1 and T2, of surge impedances Z OT 1 and ZOT2 , respectively. The line surge impedance is denoted Z OL and its delay time is τ. Assume that a direct lightning discharge hits the line at the point P. The distance between P and the sending end (SE) is assumed x.L , with the parameter x ranging between zero ( if P is at SE or T1 ) and 1 ( if P is at the receiving end RE or T2 ). The surge is represented by a parallel combination of a current source IS and an internal impedance of RS = 250 Ω. The time waveform of IS is assumed of the idealized typical shape shown in Fig. 2 giving the instantaneous current in per unit of its peak value.

Equation (1) is based on the representation of each of the 2

pulses by the expression: sin (πt/T).u(t) , in the time domain, with u(t) denoting a unit step function starting at t=0. Referring to the equivalent circuit given in Fig. 3, the left line section can be described in terms of the generalized A l B l C l D l constants as follows: A l = D l = cosh(x τ s) B l = ZOL sinh(x τ s)

(2)

C l = (1/ ZOL ). sinh(x τ s) The corresponding constants for the line section on the right hand side are: A r = D r = cosh[(1-x) τ s] B r = ZOL sinh[(1-x) τ s]

(3)

C r = (1/ ZOL). Sinh[(1-x) τ s] The input impedance of the left line section as seen from the point P is Zinput, " = (A l Z OT 1 + B l )/( C l Z OT 1 + D l ) (4) Similarly, for the line section to the right of P: Zinput,r = (A r ZOT2+ B r )/( C r ZOT2 + D r ) Fig. 2 : Assumed Waveform of the Surge Current, in per unit It includes a number of identical equidistant pulses (usually between one and ten). The time between two successive pulses is assumed T. In the Laplace-domain, this surge current, in per unit, can be expressed as 2

2

I s = [(1/s)- s/{ s + (2π/T) }].(1 - e where N is the number of surge pulses.

− NTs

)/2

(1)

(5)

The total input impedance Z total seen by the current source I s describing the lightning surge is 1/Z total = 1/ Zinput, " +1/ Zinput,r + 1/RS

(6)

The Laplace transform of the voltage V P developed at the location P is V P = I s . Z total

(7)

Surge Current: Is ( Internal Impedance: Rs ) Line Section 2 Tr 1

Line Section 1

Tr 2

P Line Section 2

Tr 2

Fig. 3: The Equivalent Circuit of the Network of Fig. 1. The s-domain expressions for the voltages at both transformers, i.e. VTr1 and VTr2 can then be easily found. Many efficient algorithms for the numerical computation of the inverse Laplace transform have appeared in the literature. These techniques can be easily and efficiently adopted for digital computation. For the purpose of this study the algorithm proposed by Hosono [5] has been selected. It is based on evaluating the time function f(t) corresponding to F(s) from the series: f(t) ≈ (e /t).[ ΣF n +(1/2 a

m +1

).ΣA mn F l + n ]

(8)

where F n = (-1)

n

Im{F[a+j(n-0.5)π/t]}

(9)

With a>>1 and the truncation coefficients A mn are defined recursively by

 m + 1  n 

A mn =1 and A mn −1 = A mn + 

(10)

For further details, references [3,5,6,7] should be consulted. Sample Results: The following figures will illustrate the simulation results of several case studies. They focus on the waveforms of four signals: the surge current, the voltage developed at the point P and the voltages at the terminals of both transformers. In the time domain plots, v2 , they are denoted: I surge , V surge , v 1 and respectively. Case Study I: Fig. 4 depicts the network response to a 5-pulse lightning stroke of amplitude 1 per unit and internal resistance RS = 250 Ω. The time period between two successive pulses is T=5μs. The total line length is L=10 km ( corresponding to a line time delay τ =33.33μs) of a pure resistive surge impedance ZOL=400Ω.

Fig. 4: The Waveforms of the Surge Current I surge , Surge Voltage at P, V surge , and the Voltages at the Two Transformers , v 1 and v 2 for Case Study I.

The stroke is assumed to hit the line at a point P one km far from the sending end, corresponding to a per unit distance x=0.1. The surge impedances of the two transformers are Z OT 1 =3000Ω and ZOT2 =4500Ω. The plots extend over a time range of 80 μs. The voltage scale for the three voltage waveforms is given in kV (for a surge current amplitude of 1A). It is seen that the voltage V surge developed at the point P can reach 0.14 kV, while after corresponding time delays of the line sections, the transient voltage appearing at the left and right transformer v 1 and v 2 will attain maximum values close to 0.20 and 0.25kV, respectively. Case Study II: In this case study, keeping the network data the same as in Case I, only the pulse separation was increased to 133.33 μs, i.e. exactly equal to four times the time delay of the transmission line, τ. The corresponding plots are given in Fig. 5, over the time range of 2000 μs. The surge voltage, V surge , reaches a peak value of approximately 0.15kV. The same is valid for the voltage v 1 at the transformer closer to the lightning stroke. It is noted that the waveforms of these two voltages are completely different from the surge current, I surge ,and the voltage at the other transformer, v 2 .The latter exhibits a strongly oscillatory behavior with a positive peak value close to 0.30kV.

Fig. 5: The Waveforms of the Surge Current I surge , Surge Voltage at P, V surge , and the Voltages at the Two Transformers , v 1 and v 2 for Case Study II. x-Axis: Time, μs Case Study III: It is assumed that the lightning stroke hits the line midpoint. Both line terminals are open-circuited, i.e. no transformers connected. The pulse separation was increased to 66.67 μs, i.e. exactly equal to twice the time delay of the transmission line, τ. The simulation results are shown in Fig. 6. Interesting is the almost rectangular waveform of the surge voltage , V surge . As expected, the two waves of v 1 and v 2 are identical. Case Study IV: To demonstrate the effect of the number of lightning pulses, Case III was repeated with just two current pulses, as depicted in Fig. 7. Other network parameters remain unchanged. It is seen that the peak values of the signals V surge , v 1 and v 2 did not change significantly compared to those of previous case. Nevertheless, the duration of the almost rectangular waveform describing V surge will be shorter in proportion to the number of current pulses. 3. RESULTS OF A TEST CASE A test run was made to check the validity of the assumed circuit model and the Mathematica computer program. The symmetrical network comprises a 100-km overhead line with ZOL=500Ω and τ = 333.33μs, terminated at both ends by transformers of surge impedance 500Ω as well. It is seen that these numerical values are chosen such that a matching exists between the line, the transformers and double the internal impedance of the 2-pulse lightning stroke. As expected, the graphs in Fig. 8 indicate that there are no reflections in any of the voltages.

Fig. 7: The Waveforms of the Surge Current I surge , Surge Voltage at P, V surge , and the Voltages at the Two Line Ends , v 1 and v 2 for Case Study IV. x-Axis: Time, μs

Fig. 6: Current Case Study III x-Axis: Time, μs

and

Voltage

Waveforms

for

Fig. 8: Voltage Transients in the Test Case

The maximum value of the voltage V surge at the line midpoint is 0.125kV. This value can be easily calculated since, in this case, the lightning current will face a total resistance of 125Ω representing the parallel connection of the two line haves (500Ω each) and the lightning stroke internal resistance of 250Ω. Moreover, the time delay between V surge at the line midpoint and v 1 (or v 2 ) at the line ends is seen to agree with half the line delay ,i.e. τ/2=166.67 μs. 4. SUMMARY AND CONCLUSIONS 1.

2. 3.

4.

5.

6.

An s-domain distributed-parameter model is presented for the computation of the electromagnetic transients in typical power networks including lines, transformers and/or static load terminations. It is assumed that the transients are initiated by lightning strokes composed of several repetitive current pulses. Results of several case studies are reported on and discussed. They served to validate the circuit, to test the Mathematica program as well as to investigate the effect of the different network parameters and those of the stimulating lightning current surge. Particular emphasis was given to the impact of the line length and its surge impedance, the location of the lightning stroke, the time separation between the lightning pulses, the number of these pulses and the surge impedances of the transformers at line ends. It is realized that the peak values and the time waveforms of the transient voltages will be largely affected by the line time delay in relation to the time period separating the surge current pulses. Voltage stresses of unusual magnitudes and waveforms can be initiated, if the surge includes several pulses of repetitive nature,

whose frequency of repetition coincides with or close to the natural frequency of the power line. 5. REFERENCES [1] A. G. Phadke (Organizer): “IEEE Tutorial Course: Digital Simulation of Electrical Transient Phenomena”, (IEEE Publication) No. 81EH0173-5-PWR, 1981. [2] A. Greenwood: “Electrical Transients in Power Systems”, Book, Second Edition, (WileyInterscience), 1991. [3] A. S. Alfuhaid, M. M. Saied: “A Method for the Computation of Fault Transients in Transmission Lines”, (IEEE Trans. On Power Delivery), Vol.3, No. 1, January 1988, pp.288297. [4] K. Nasrullah: “Voltage Surge Resonance on Electric Power Network”, (Proc. Of the 1999 IEEE Transmission and Distribution Conference), New Orleans, Vol. 2, April 11-16, 1999, pp. 687-690. [5] T. Hosono: “Numerical Inversion of Laplace Transform and Some Applications in Wave Optics”, (Radio Science), Vol. 16, Nov./Dec. 1981, p. 1015. [6] M. M. Saied, A. S. Alfuhaid: “Electromagnetic Transients in a Line-Transformer Cascade by Numerical Laplace Transform Technique”, (IEEE Trans. On Power Apparatus and Systems), Vol.104, Oct. 1985, pp.2901-2909. [7] M. M. Saied: “Effect of Transformer Sizes and Neutral Treatments on the Electromagnetic Transients in Transformer Substations”, (IEEE Transactions on Industry Applications), Vol. 31, No. 2, March/April 1995, pp. 384-391.