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Radio Interference Characteristic of Ultra-high Voltage

高电压技术第39卷第10期 2013年10月31日

2438 High V oltage Engineering, V ol.39, No.10, October 31, 2013

Radio Interference Characteristic of Ultra-high Voltage AC Transmission Lines by Using Stochastic Modeling

HE Wangling1,2, WAN Baoquan2, PEI Chunming2, ZHANG Jiangong1,2, HE Junjia1, GUO Haozhou2

(1.School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan 430074, China;

2. China Electric Power Research Institute, Wuhan 430074, China)

Abstract: For developing ultra-high voltage (UHV) AC power transmission systems, it is important to precisely estimate and to limit the radio interference(RI) level of power lines. Based on the stochastic characteristics in amplitude and repetition rate of induced corona current, by using the probability theory and mathematical statistics, we establish a stochastic model for the wide-sense stationary random process of corona discharges. Then combining the stochastic model with model-propagation-analysis method, the RI levels under three-phase UHV AC transmission lines are calculated. The results of the calculation based on stochastic model method and International Council on Large Electric Systems (CIGRE) excitation function are compared with that based on semi-empirical method and some other excitation functions.

The stochastic model based on different excitation functions is also adopted to simulate the RI levels under finite test lines with two opened terminations. The results indicate that with the same average maximum gradient on conductor surface and the same conductor type, the number of corona discharge per unit length is one of the main reasons that causes the difference between different excitation functions. It is also concluded that for a long test line, the effect of standing wave on RI field strength is negligible in the middle of the line, but obvious near both terminations: for a 10-km line, the maximum difference in RI field strength is 2.78 dB, between the peak value of the standing wave near the ends and the steady value near the middle of the line.

Key words: ultra-high voltage AC transmission line; corona discharge; radio interference; stochastic model; excitation function; standing wave

DOI: 10.3969/j. issn. 1003-6520. 2013. 10. 016Article No: 1003-6520(2013)10-2438-07

0Introduction

With the development of ultra-high voltage(UHV) AC transmission systems and the increase of voltage level, the corona discharge phenomenon on conductor surface becomes obvious. Owing to the high frequency induced current which generated by corona discharge, an electromagnetic field is formed in the vicinity of the conductor surface, which is the so-called radio interference(RI)[1-8]. How to accurately predict and control the RI level of power lines is a very important technical issue to develop the UHV AC power transmission systems in China.

There are many studies on radio interference problem caused by power lines at home and abroad. Adams G E ex-plained the phenomenon by using the mechanism of principal wave travelling along the lines and proposed the RI generation function firstly[9]. Then a large number of experiments were made by using the short lines and corona cage to summarize ———————

Project supported by Science and Technology Project of SGCC (SG1021). the excitation function formula which applies to less than 8 sub-conductors in a bundle[10-11]. Then the dispersion of in-duced currents in fair weather conditions was discussed by using corona cage in reference [12]. There are also some re-searches about the RI spectrum characteristic of single test line with different terminated impedances[13]. Then in reference [14], the transformation relationship of RI induced current between long and short UHVDC transmission lines was dis-cussed.

Besides, some research institutes proposed different excita-tion functions in the past decades. With International Special Committee on Radio Interference(CISPR) measurement, measuring frequency of 0.5 MHz and an altitude 0 meters above sea level, the different excitation functions were given as follows.

Electric Power Research Institute (EPRI) (heavy rain condi-tion)[15]:

Γ=76.62?580/g+38lg(2r/3.8)

Bonneville Power Administration (BPA) (average stable foul weather)[16]:

Γ=37.02+120lg(g/15)+40lg(r/2) International Council on Large Electric systems (CIGRE)

HE Wangling, et al.: Radio Interference Characteristic of Ultra-high V oltage AC Transmission Lines by Using Stochastic Modeling 2439

(heavy rain condition)[16]:

Γ=?40.69+3.5g+12r

CISPR (heavy rain condition)[17]:

Γ=70?585/g+35lg2r?10lg n

Hydro-Quebec Research Institute (IREQ) (heavy rain con-dition)[18]:

Γ=?90.25+92.42lg g+43.02lg2r–B(n)

where, Γis the excitation function, dB (0 dB is corresponding to 1μA/m), g is the average maximum bundle gradient, kV/cm, r is the radius of the sub-conductor, cm, n is the num-ber of sub-conductors in the bundle. Especially in the IREQ function, B(n)=0 for n=1, B(n)=3.7 for n=2, B(n)=6 for n≥3.

It can be seen that different research institutes proposed dif-ferent excitation functions. In order to discuss the main reason for the differences between these excitation functions with the same conductor gradient and conductor type, this paper estab-lished stochastic model to analyze the RI characteristic. Besides, the simulation was made to discuss the effect of standing wave on RI level with two terminations opened test line.

1Establishment of Stochastic Model

The corona is mainly caused by the gas discharge around the conductor surface, such as glow corona and streamer co-rona. The random pulses generated by AC corona are shown in Fig.1. In the AC corona, the bursts of positive and negative pulses are produced during time interval t c+ and t c- around the positive and negative peaks of the alternating voltage.

The negative corona current pulses have faster rise time and shorter durations than the positive pulses[8], while the ampli-tudes of positive pulses are generally much higher than those of negative pulses. This characteristic makes positive pulses playing a predominant role in RI level.

Due to the random of corona current both in magnitude and repetition rate, the power spectral density(PSD) of the corona current is modeled and analyzed in this chapter by using the probability theory and stochastic process theory[19-21]. Com-bining the stochastic model and propagation analysis model, the corona voltage PSD of any point on the single conductor can be also calculated, then these results can deduce the RI field strength near the conductor.

1.1Frequency domain analysis theory

1) Parseval’s theorem

If the signal f(t) is integrable and square integrable in the time domain(?∞

, which is

Fig.1Pulse trains produced by AC corona

()d|()|d

2π

f t t Fωω

+∞+∞

?∞?∞

∫∫ (1) where, F(ω) is the Fourier transform of f(t), ω=2πf, f is the

given frequency.

2) Wiener-Khintchine theorem

If the signal f(t) is a wide-sense stationary random process,

the PSD of the signal is the Fourier transform of this signal’s

auto-correlation function, which is

*j*

()(()())e d()()

E f t f t

F F

ωτ

Φωττωω

+∞

?∞

=?=

∫ (2) where, E(f(t)f *(t?τ)) is the expectation of the auto-correlation function, Φ(ω) is the PSD of f(t), f*(t) and F*(ω) are the conjugate function of f(t) and F(ω) , respectively, τis the inte-

gration variable of t.

1.2PSD calculation of corona current

Let J(x, t) be the corona current density defined with x∈[0,

L], where L is the length of transmission line. J j(x, t) is a re-presentation of corona current density J(x, t) in the interval

[?T/2, T/2],which is

()

m m b

(,)()

N L M

n m

J x t y u t mT

∑∑ (3) where, t∈[?T/2, T/2], T represents a given period, m is the

serial number of current pulse, M is the total number of current

pulses in a corona source with the time interval of [?T/2, T/2],

T b is the duration of current pulse period, y m is the amplitude

of the m-th current pulse in a corona source, u m(t) is the func-

tion of this pulse’s shape and the repetition rate, N(L) is the

number of corona sources on the transmission line with the

length of L.

Cho Y un-ok proved that the corona current density J(x, t)

was integrable and square integrable in the time domain

(?∞

seval’s theorem, which means that the corona current energy is

the same in both time domain and frequency domain. Besides,

it is also proved that J(x, t) is a wide-sense stationary random

process in reference [19], then by using the Wiener-Khintchine theorem, the PSD of corona current on the power line is

()()(|()|)

W E y E u

ωω

= (4) where, E(y2) and E(|u(ω)|2) is expectations of random variable

2440 高电压技术 High V oltage Engineering 2013,39(10)

y 2 and |u (ω)|2, respectively, y is the amplitude of current pulse in a corona source, u (ω) is the function of current pulse’s shape and the repetition rate.

1.3 Stochastic propagation model of single conductor corona current

The equivalent circuit of transmission line is shown in Fig.2, where Z is the series impedance per unit length, ?/m, Y is a

shunt admittance per unit length, S/m, J is the induced current

per unit length, μA/m, I (x ) and U (x ) are current and voltage at

a given frequency respectively, ΔI and ΔU are the generation

of voltage and current in unit length, x=ε is the length of con-ductor, Δx is the unit length of conductor. The single

conductor is connected with termination impedance Z a and Z b , where Z a and Z b are the termination impedance of point a and b respectively. x 0 is the position where the corona current injected.

By using the transmission line theory and Fourier transform,

the single conductor transmission line equation can be ob-tained as

T T T

T d (,)

(,)d d (,)(,)(,)d U x ZI x x

I x YU z J x x

ωωωωω?=?????=?+?? (5) where, U T (x , ω), I T (x , ω) and J (x , ω) are the Fourier transform of U (x , t ), I (x , t ) and J (x , t ), respectively. Simplified the equation (5), the second order differential equation is

T T

d (,)(,)(,)d U x U x ZJ x x ωγωω?= (6)

where γ= is the propagation constant.

The terminal reflection coefficients of point a and b are

a a c a c b

b c b c ()/()

()/()Z Z Z Z Z Z Z Z ρρ=?+??=?+? (7) where Z c

is the characteristic impedance, c Z =.

Suppose that there is a corona current inject at x =x 0, the

voltage equation for a single conductor line of any point can be obtained by combining the equations (6) and (7)

2()c b ()()a 00T 2c a

()(2)b 0(/)(,)(1e )(e

e ),0(,,)(/)(,)(1e )(e e ),x L x x x x x

x x x x L Z J x x x U x x Z J x x x L

γγγγγγωρρωωρρ???+???+???Δ+?

?+≤

Fig.2 Equivalent circuit of unit length of conductor

If the line has a plurality of corona sources injected, the to-tal voltage at any point x is

T 0T c ||()(2)

a b 1

(,)(,,)(/)(,)(e e e n

N x x x x L x x n U x U x x Z J x γγγωωωρρ??+???===?Δ?

+++∑∑ (2||)a b e )n

L x x γρρ??? (9)

where, N is the number of corona sources on the transmission

line, U (x ,ω)T is the voltage of any point on single conductor

when there are a large number of corona current injected at [0,

L ], x n is the position on the transmission line where the n -th corona occurs. Then the PSD of voltage at any point x can be calculated by Wiener-Khintchine theorem as

*T T (,)lim(2/)((,)(,))U T W x T E U x U x ωωω→∞

= (10) where, W U (x ,ω) is the PSD of voltage at any point x , U *(x ,ω)T

is the conjugate complex of U (x ,ω)T , E (U (x ,ω)T U *(x ,ω)T ) is the expectation of the autocorrelation function of U (x ,ω)T . Combining equations (9) and (10), the voltage PSD is 22c b (,)(2||/)()U W x Z T E y ω=??

2a b (|()|)(,,,,,,)E u x L ωΓλωγρρ (11) where,Γ(λ, x , L , ω, γ, ρa , ρb ) is the function which is deter-mined by the parameters in this bracket, λ is the number of

corona sources per unit length.

1.4 Calculate the PSD of RI field strength at any point near

the conductor As shown in Fig.3, the RI field strength at any observing point A in the space is A 2222c i c i

11()j (ln(2/)ln(2/)Ux U y h y h E h R h R d d d d ??=?+? (12) where, U is the applied voltage, d c is the distance between the

observing point and physical conductor, d i

is the distance

between the observing point and image conductor, h is the height of physical conductor, R is the radius of conductor.

HE Wangling, et al .: Radio Interference Characteristic of Ultra-high V oltage AC Transmission Lines by Using Stochastic Modeling 2441

Fig.3 Cross-sectional view of a single-phase line with ground return

Then, the PSD of RI field strength at point A is:

221/2A ex ey (,,,)((,,,)(,,,))W x y z W x y z W x y z ωωω=+ (13)

W ex (x ,y ,z ,ω) and W ey (x ,y ,z ,ω) are defined as

ex 22c i

11(,,,)(,)(()/ln(2/))V W x y z W z x h R d d ωω=? (14) 2

ey 22c i

(,,,)(,)(()/ln(2/))V y h y h W x y z W z h R d d ωω?+=? (15) 1.5 To determine the parameters in calculating PSD of RI electric field strength In time domain, corona current pulses may be represented in terms of a double exponential function [3] as 0

0()(e e )a t b t i t A ??=? where, A 0 represents the amplitude of corona current pulses, a 0

and b 0 are variable, which depended on the high voltage line

geometry, and voltage as well as the atmospheric conditions.

This equation can be simplified as 0

(1)00()e a t i t A a t ?=[23], where

(1)0()e a t u t a t ?= represents the corona current pulses’ shape

and repetition rate.

1.5.1 To determine the parameter E (|u (ω)|2

) The Fourier transform of equation 0

(1)0()e a t u t a t ?= is

0220e

()a u a ωω

=+ (16)

In heavy rain conditions, the random variable a 0 is the sub-ject to a uniform distribution with probability density f (a 0

)[19]

001/(),()0,

otherwise q p p a q

f a ?≤≤?=?? (17) where, p =0.5T k ，q =2T k ，T k is the reciprocal of corona current pulses’ setup time, T k

=109/(2.5d +28), d is the sub-conductor diameter.

Then the expectation of |u (ω)|2 can be obtained as 22

e (()|)(arctan()arctan(2()

q p

E u q p ωωωω=

??? 22222()()()()

q p pq q p ωωωω??++

(18)

1.5.2 To determine the parameter E (y 2)

The PSD of corona current per unit length is

22b

()()(|()|)E y E u T Φωλω= (19) The expectation of the corona current mean square value which measured by radio noise meter is 2(())()E i t f Φω=Δ (20)

where Δf is the bandwidth of radio noise meter with 9 kHz in CISPR measurement. The amplitude of the corona current pulses is mainly deter-mined by the excitation function which is summarized by large amount of test data.The injected corona current of per unit length is [17] 0

2πC

I Γε= (21) where the CIGRE’s excitation function was adopted in sto-chastic model in this paper, C is the capacitance between transmission lines and ground, ε0 is the permittivity of free space. Combining the equations (19), (20) and (21), the E (y 2) is

222b 0

()()/(2(|()|))2πC

E y T fE u Γωλε=Δ (22) 1.5.3 To determine the conductor parameters in considering

lossy ground

The lossy ground means that the ground cannot be regarded

as a good conductor for the electromagnetic wave has a certain

penetration depth, therefore, the distribution parameters of

conductors are very complicated. Fortunately, the simplified

calculation method of conductors’ parameters has already been

proposed by F. Rachidi in reference [24]. Considering the lossyground, the element of series imped-ance matrix Z could be obtained in the following way. The diagonal element of Z is

j m m gm m m m w L ω′′′′′′=++Z Z Z (23) where, L m ′m ′ is the series self-inductance of lossless line, Z w is the conductor resistance and Z gm ′m ′ is the ground resistance, m ′, n ′ are the numbers of rows and columns in matrix Z , respec-tively.

2442 高电压技术 High V oltage Engineering 2013,39(10)

The off-diagonal element of Z is j m n gm n m n L ω′′′′′′=+Z Z (24) where, L m ′n ′ is the mutual inductance of lossless line and Z gm ′n ′ is the mutual resistance reflecting losses in the ground. Z gm ′m ′ and Z gm ′n ′ are given as:

g 0g 1j ln()2πi

gm m i h h γωμγ′′+=Z (25)

g g

(1())()j 22ln()4π

()()2

2i j ij gm n i j

ij h h r γγωμγγ′′+++=

++Z (26)

where, h i , h j are the average height of conductors i and j , re-spectively, μ0 is the free-space permeability, r ij is the aerial

i and j , g γ=εrg is the relative permittivity of g 2 Simulation of Stochastic Model

2.1 Simulation on stochastic model with different λ

In this paper, the modal analysis method is used to calculate the RI field strength of three phase multiconductor transmis-sion lines. The details of the modal analysis method can be found in references [4, 15]. This paper chooses the 8×LGJ- 500/45 conductors, which the applied phase to phase voltage is 1 050 kV . The earth resistivity is 100 Ω·m. Fig.4 shows the space structure of transmission lines for simulation.

The lateral profile of RI field strength is calculated by using the stochastic model with excitation function (CIGRE), the strength curves are shown in Fig.5.

As shown in Fig.5, the RI field strength increases by 8.3 dB with λ from 1 to 7.

Then the comparative analysis is made between the sto-chastic model (based on CIGRE) with different λ and semi-empirical methods which based on other excitation func-tions.

As shown in Fig.6, the maximum difference is less than 1 dB between BPA and CIGRE with λ=2; The maximum difference is less than 0.5 dB between EPRI and CIGRE with λ=3; The maximum difference is less than 0.5 dB between IREQ and CIGRE with λ=5; The maximum difference is less than 0.3 dB between CISPR and CIGRE with λ=7.

Therefore, the calculated values are very similar between

stochastic model method based on CIGRE excitation function with different corona source numbers and semi-empirical method based on other excitation functions. This result shows

that with the same average maximum gradient of conductor

Fig.4 Space structure of transmission lines for simulation

Fig.5

Lateral profile curves of RI field strength with different λ

Fig.6 Comparison of calculated values between BPA ，EPRI ，IREQ ，

CISPR and CIGRE with different λ

surface and the same conductor type, the discrepancy of coro-na sources number λ mainly leads to the difference of excitation functions proposed by some research institutes.

HE Wangling, et al.: Radio Interference Characteristic of Ultra-high V oltage AC Transmission Lines by Using Stochastic Modeling 2443 2.2Simulation of finite test lines with two terminations

opened

By combining the stochastic model with different excitation functions, the RI field strength below the conductor of phase A is calculated.

When the test lines has two opened terminations, the calcu-lated curves of RI field strength with different excitation functions are shown in Fig.7, and the calculated curves with different earth resistivities are shown in Fig.8, where the val-ues are calculated by CIGRE excitation function with λ=1. From Fig.7, there are standing waves with wavelength of 300 m below the phase A. Because the wavelength of the 0.5 MHz radio interference is 600 m, the wavelength of 0.5 MHz standing wave is 300 m. Therefore, the simulation results conform to theory analysis.

From Fig.8, the calculation results show that the RI field strength decreases with the increase of earth resistivity, and the maximum differences is 1.3 dB and 4.55 dB respectively with the earth resistivities from 10 Ω·m to 100 Ω·m and from 10 Ω·m to 1 000 Ω·m.

From Fig.9, the effect of standing wave on RI field strength is negligible in the middle lines when the conductors are much longer. However, this effect is very obvious near the two ter-minations. The maximum difference of RI field strength is 2.78 dB between the ends and the middle lines. Therefore, the measured data near the terminations are not applicable to evaluate the radio interference of power lines.

3Conclusions

1) Considering the comparison results between the stochas-tic model method and semi-empirical methods, it can be concluded that the discrepancy of corona sources number λmainly leads to the difference of excitation functions proposed by some research institutes in the same average maximum gradient of conductor surface and conductor type.

2) From simulation results of RI field strength below the phase A when the short test lines terminated with two open ends, we can get that the effects of standing wave on the RI field strength are very obvious when the lines are short. How-ever, the influence is negligible on the middle lines when the lines are much longer. For long test lines with the length of 10 km, the maximum difference of RI field strength is 2.78 dB between the peak value of standing wave near the ends and steady value near the middle lines.

3) The RI field strength decreases with the increase of earth resistivity.

Fig.7Radio interference under phase A with the line length

of 800 m

Fig.8Curves of radio interference under phase A with different

earth resistivities

Fig.9Radio interference under phase A when the two terminations

open with the line length of 10 km

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HE Wangling was born in Baiying, Gansu Province, China, in 1988. He is currently pursuing the Ph.D. degree in Electrical and Electronic Engineering at Huazhong University of Science and Technology, Wuhan, China. His research interest is corona performance of ultra-high voltage transmission lines. E-mail: wanglinghe88@https://www.wendangku.net/doc/181614486.html,

W AN Baoquan was born in Gansu Province, China, in 1971.Currently, he is a senior engineer and Master’s Supervisor in China Electric Power Research Institute. His research interests are electromagnetic environment of high voltage transmission lines and electromagnetic compatibility. E-mail:wanbaoquan@https://www.wendangku.net/doc/181614486.html,

PEI Chunming was born in Hubei Province, China, in 1974.Currently, he is a senior engineer in China Electric Power Research Institute. His research interests are electromagnetic environment of high-voltage transmission lines and audible noise control technology in substation.

ZHANG Jiangong was born in Hubei Province, China, in 1975. Currently, he is a senior engineer in China Electric Power Research Institute. His research interest is electromag-netic transient process in substation.

HE Junjia was born in Hunan Province, China, in 1968.Currently, he is a professor and Doctoral Supervisor in Hua-zhong University of Science and Technology. His research interests are high voltage, insulation engineering and pulse power technology.

GUO Haozhou was born in Hubei Province, China, in 1972.Currently, he is a senior engineer in China Electric Power Research Institute. His research interest is electromagnetic compatibility of high-voltage system.

Received date 2013-07-23 Editor ZENG Wenjun