Magnetic Field Calculation Underehvtransmission Lines For More

2008 5th International Multi-Conference on Systems, Signals and Devices

Magnetic Field Calculation under EHV Transmission Lines for More Realistic Cases Adel Zein E. M. Department of Electrical Engineering High Institute of Energy, South Valley University Aswan, Egypt Abstract-Ground level electric and magnetic fields from overhead power transmission lines are of increasingly important considerations in several research areas. Common methods for the calculation of the magnetic fields created by power transmission lines assume straight horizontal lines parallel to a flat ground and parallel with each other. The influence of the sag due to the line weight is neglected or modeled by introducing an effective height for the horizontal line in between the maximum and minimum heights of the line. Also, the influences of the different heights of the towers, the different distances of the power transmission lines spans and the different angles between the power transmission lines' spans are neglected. These assumptions result in a model where magnetic fields are distorted from those produced in reality. This paper investigates the effects of the sag in case of different heights of the towers and when the power transmission lines' spans are not parallel to each other.

Index Terms- OHTL, Magnetic Field I. INTRODUCTION

P

RECISE analytical modeling and quantization of electric and magnetic fields produced by overhead power transmission lines are important in several research areas. Considerable research and public attention are concentrated on possible health effects of extremely low frequency (ELF) electric and magnetic fields [1]. An analytical calculation of the magnetic field produced by electric power lines is produced in [2], which is suitable for flat, vertical, or delta arrangement, as well as for hexagonal lines. Also the estimation of the magnetic field density at locations under and far from the two parallel tran~mission lines with different design arrangements is presented in [3]. The effects of conductors sag on the spatial distribution of the magnetic field are presented in [4], in case of equal heights of the towers, equal spans between towers and the power transmission lines' spans are always parallel to each others. In this paper, the magnetic field is calculated by two different techniques; Two-Dimensions Straight line Technique and Three-Dimensions Integration Technique, where the effect of the sag in the magnetic fields calculation, and the effects of unequal span distances between the towers, unequal towers heights, and when the power transmission lines' spans are not parallel to each other are investigated.

B. The 3-D Integration Technique In fact, the power transmission lines are nearly periodic catenaries, the sag of each depends on individual characteristics of the line and an environmental conditions. The integration technique is a three-dimensional technique which views the power transmission conductor as a catenary. In the integration technique, if the currents induced in the earth are ignored, then the magnetic field of a single currentcarrying conductor at any point P(xo,yo,zo) shown in Fig. (1) can be obtained by using the Biot-Savart law [2-4], as: (1)

where

1

a parametric position along the current path,

I (1)

the line current,

r"

(1) a vector from the source point (x,Y,z) to the field point (xo,yo,zo),

a (I). . t he d·IrectIon . r (I) unIt vector In o

0

,

and

dl a differential element at the direction of the current.

II. MAGETIC FIELD CALCULATIONS

A. The 2-D Straight Line Technique The common practice is to assume that power transmission lines are straight horizontal wires of infinite length, parallel to a flat ground and parallel with each other. This is a 2D Straight line Technique, which can be found in many references [2-5].

_.....-....._ - -.... ......--_..........--.z ~

Fig. 1. Application of the Biot-Savart law.

The exact shape of a conductor suspended between two towers of equal height can be described by such parameters; as 978-1-4244-2206-7/08/$25.00

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2008 5th International Multi-Conference on Systems, Signals and Devices

the distance between the points of suspension span, L, the sag of the conductor, S, the height of the lowest point above the ground, h, and the height of the highest point above the ground, h m• These parameters can be used in different combinations. Only two paramt~ters are needed in order to define the shape of the catenary (S and L), and the third one (h or hm), determines its location in relation to the ground surface. Figure (2) depicts the basic catenary geometry for a single-conductor line, this geometry is described by: (2) Z

Y = h + 2a sinh

where

where:

d=[(X-x o)2+(Y_Yo)2+(Z-ZO)2]3/2

This result can be extended to account for the multiphase conductors in the support structures. For (M) individual conductors on the support structures, the expression for the total magnetic field becomes: _ 1 M N 1/2 (9)

Ho =

-I I 4Jr

where:

L (-)

2a

a is the solution of the transcendental equation:

I;[(z - Zo + kl)sinh( ~) - (y - Yo)]

L

2-"_1- u = sinh 2 (u) U= L 4a ,·w~ ty

f(Hxo x +HyOy + Hzoz)dz

;=1 k=-N -1/2

"'!

h -h

Hx =

a (11 )

H = I;(x-x o) y d.1

tv

Hz

Fig. 2. Linear dimensions which determine parameters of the catenary.

a

The parameter

is also associated with the mechanical

h ~.IS t h e con d uctor parameters 0 f t h e 1·Ine: a = Th I W were tension at mid-span and W is the weight per unit length of the line.

1) Case (A) In Case A, the power transmission lines specified by; equal heights of the towers, equal spans between towers and the power transmission lines' spans are always parallel to each others (0=0). For a single span single conductor catenary, represented by equation (2), since the modeled curve is located in the y-z plane, the differential element of the catenary can be written as: (3)

dl = dYG y +dzG z

dT

=dz(:

Oy +oJ

=

(4)

where;

g

the skin depth of the earth represented by[5];

+ (Yo - Y)G y + (zo - z)ii z

(15)

t5=503Jpl f p

the resistivity of the earth in n.m,

f

the frequency of the source current in Hz.

The resultant magnetic field with the image currents taken into account is also represented by equation (9), but its components will change and take the following formulas: z (16) Ii [(z - Zo + kl)sinh(-) - (y - Yo)]

(5)

a

di

Ii [z - Zo + kl)sinh(~) - (Yo + Y + ()]

a

~ = (x o - x)iix

(13)

The parameter (N) in equation (9) represents the number of spans to the right and to the left from the generic one, as explained in Fig.(2). One can take into account part of the magnetic field caused by the image currents. The complex depth ~ of each conductor image current can be found as given in [4]. (14)

Hx =

dT = dz(sinh( ~ )iiy + oJ

(12)

a

d; d; =[(X-xo)2 +(y- Yo)2 +(z-zo +kl)2]3/2

x

(10)

d;

- I; (x - xo)sinh(~)

~

(8)

a

(6)

where point (xo,yo,zo) is the field point at which the field will be calculated, and point (x,y,z) is any point on the conductor catenary. Now, by substituting equations (5) and (6) into equation (1), and carrying out the cross product, the result at any point (xo,yo,zo) is : (7)

(17)

=

H

- Ii(x-xo)sinh(~)

d;

Z

Ii(x-xo)sinh(~) a + a d;

d i' = [(x - x0) + (y + Yo + S) + (z - z0 + kl) 2 ] 3/2 2

(18)

2

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2008 5th International Multi-Conference on Systems, Signals and Devices

2) Case (B) In Case B, the power transmission lines specified by; equal heights of the towers, equal spans between towers and the power transmission lines' spans are not parallel to each others. The two catenaries Land L2, in Fig.(2), each have its original point and coordinate system. The field points are located on axis X of system (X,Y,Z) of L catenary. This field points should be transferred to the coordinate system of the catanery under calculation. By applying this rule on field points and caterany L, it is seen that the same equations of case (A) are used, where the field points are already in caterany L system. But for caterany L2, the field points should be transferred to the caterany L2 system. For any field point (xl,yl,zl), that can be done in three steps: 1- Transfer the original of caterany L2 to the field point system. From Fig.(3), for - 90 < B < 90

z c

= ~ + L2 cos(()), x = - L2 sin(()), and 2

2

c

2

=0

y c

2- Transfer the field point (xl,yl,zl) from its system to the system (U,V,W) of the caterany under calculation L2, from appendix (B):

zI-z xI-x sin(p) ,uI = cos(P) , sln(p + 8) cos(P + 8)

WI

=.

vI

= yI- YC' where (xc,yc,zc) is the original point of system

C

C

and

2

h

-h

m2

U

LI+L'

=sinh 2 (u)

'

with

u=

Ll+L'

equations as in case (A) is used, with the integration limits from

-LI-L'

2 III.

+L

,

LI+L'

to - - -

2

ANALYSIS OF MAGNETIC FIELDS TECHNIQUES

To calculate the Magnetic field intensity at points one meter above ground level, under 500kV TL single circuit, the data in appendix (A) are used. Figure (4) shows the computed magnetic field intensity and its components with and without the effect of the image currents, by using the 2-D Straight Line Technique, where the average heights of the transmission lines are used, since typical values for the resistivity of earth range from 10 to IOOOn.m, the image currents are normally located at hundreds of meters below the ground [6], and do not effect the magnetic field intensity levels especially in areas close to the conductors. Figure (5) shows the computed magnetic field intensity and its components under a single span with the effect of the image currents, at the mid-span ( where the maximum sag, point PI in Fig. (2)), by using the 3-D integration technique (Case A).

~

(U,V,W) refer to system (X,Y,Z), which calculated in step (I), and

zI-z P=tan - - - 8 -1

•••••••.•. Hx

_._._ .• Hy

25

xI-x

3- Finally use this point (ul,vl,wl) in the same equations of case A. By the superposition technique, the magnetic field at any field point from many catenaries can be calculated.Review Stage 3) Case (C) In Case C, the power transmission lines specified by; unequal heights of the towers, unequal spans between towers and the power transmission lines' spans are always parallel to each others. Figure (3) presents a catenary L 1, which have unequal heights of its towers (h mh hm2 ). In this case a is the solution of the transcendental equation:

and the same

4a

o

_~----L-

o

5

_ _----.L-__

----.L-~_~

10 15 20 25 30 Distance from the center phase (m)

35

40

Fig. 4. The computed magnetic field intensity by using the 2D Straight Line Technique.

ty

Fig. 3. The presentation of Case (C)

o

5

10 15 20 25 30 Distance from the center phase (m)

35

40

Fig. 5. The computed magnetic field intensity by using the 3D Integration Technique (point PI).

Figure (6) shows the computed magnetic field intensity and its components under a single span with the effect of the

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2008 5th International Multi-Conference on Systems, Signals and Devices

image currents, at maximum tower height (point P2 in Fig. (2)), by using the 3-D integration technique (Case A).

i -l =~ = 15

/ /

I I

I

~ 10

\

5

o

t -

-I -

" ":"

I

-

\

I I

I

\

Distance from 1I1e center phase (m)

~/ 10 15 20 25 30 Distance from the center phase (m)

35

40

Distancebetweenthetwotowers(m)

Fig. 8. The presentation of the 3D magnetic field intensity distribution at 1m above ground level under 500kV TL with the effect of image currents, by using the 3D Integration Technique.

Fig. 6. The computed magnetic field intensity by using the 3D Integration Technique (point P2).

Figure (7) shows the effect of the number of spans (N) on the calculated magnetic field intensity. It is noticed that, when the magnetic field intensity calculated at point PI (Fig.2) and a distance a way from the center phase, the effect of the spans' number is very small due to the symmetry of the spans around the calculation points. Also it was seen that as the number of the spans (N) is greater than 2 the result of the calculated magnetic field intensity is the same, that due to the far distance between the current source and the field points. For this reason the number of spans does not exceed 4. Figure (8) shows the presentation of 3D computed magnetic field intensity, with the effect of the image currents, by using the 3D integration technique (with span number N=4). It is noticed that, the magnetic field intensity varies with the position of the field points between the two towers and also with their distance from the center phase, where in the 2D straight line technique; it varies only with the field points' distance from the center phase. Figure (9) shows the effect of the angle e as explained in case (B) on the calculated magnetic field intensity of a single span under a tower height and a way from the center phase. It is seen that aj the angle e increased the magnetic field intensity decreases, that due to the increases of the distance between the current source and the field points.

--theta=O ----- theta=5 .......... theta=10 -.-.-.• theta=20

~5

····00··· theta=40

L_~

o __ o 5

,

,~,Gao)

10 15 20 25 30 Distance from the center phase (m)

35

40

Fig. 9. The effect of the angle eon the magnetic field intensity calculated under tower height.

Figure (10) shows the same results as in Fig.(9), except that, the calculation points are at mid-span, it is noticed that, the effect of angle e is higher in this case because both the two ends of the span go far from the calculation points as the angle e increased. 45 ~-~--~.~_.~~-~-----~-~~-_. __._-~-~--_._--

~~I --theta=O

----- theta=5 •• ..00 ..• theta=10

.... .......... theta=20 _._._ •• theta=40

::-~~-~~~::~:~m.~:ml

f: Ql

E ~

u..

-----

3A al po;"1 P2, span <2,

iII,

25 20'

J:: -------------------------------(,)

.

_

:(""=.~=~ ==~='=."== . "=.,"':',==~'. o

5

10 15 20 25 30 Distance from the center phase (m)

35

40

Fig. 7. The effect of the spans' n Jmbers on the magnetic field intensity.

Fig. 10. The effect of the angle e on the magnetic field intensity calculated under mid-span.

Figure (11) shows the effect of the span length on the calculated magnetic field intensity under a single span, at tower height. It is seen that as the span length decreased, the magnetic field intensity decreases.

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lengths, and various difference between the towers' heights, that at tower height and mid-span respectively. From both two tables it seen that the difference between the towers' heights have a small effect, when the magnetic field intensity calculated at tower height, but when the magnetic field intensity calculated at mid-span it have a greater effect, especially when this difference is equal to the sag itself.

l

45

-~,--~--,-------r-------,----------r--------,-~-r-----

40

s~n=4OOm

----- span=200m

.::~= span=100m

35

1

~J

!

25

Fig. 11. The effect of the span length on the magnetic field intensity calculated under :ower height.

~ 20

1

'

~ 15

~

1

~ ':1 ~. . - - - - - '- - - - - - - "- ~_ _ _ _ L_ ~

Figure (12) shows the same results as in Fig.(ll), except that, the calculation points are at mid-span, it is noticed that, the effect of span length is very small in this case because the effect of the conductor height is greater than its span length effect. Tables I and II present a comparison between the magnetic field intensity calculated with both 20 straight line technique, where the average conductors' heights are used, and 3D integration technique, with various angles 0, various span

-----l-_ _

o

5

10 15 20 25 30 Distance from the center phase (m)

35

__

40

Fig. 12. The effect of the span length on the magnetic field intensity calculated under mid-span

TABLE I COMPARISON BETWEEN THE RESULTS OF 30 INTEGRATION TECHNIQUE WITH VARIOUS PARAMETERS AT TOWER HEIGHT AND 20 STRAIGHT LINE TECHNIQUE

Distance from the center phase (m)

2D straight line technique with average heights (Aim)

0 10 20 30 40

25.236 23.619 15.218 7.957 4.584

3D integration technique Single span at point P2 (tower height) (Aim) Angle (8) (deg.) With: L=400m, LL=Om 8=0 6.824 6.337 4.852 3.202 2.081

8=10 6.824 5.817 4.044 2.482 1.547

Span (L) (m) With: 8 =Odeg, LL=Om

8=40 6.824 4.674 2.660 1.399 0.765

L=400 6.824 6.337 4.852 3.202 2.081

L=350 1.824 1.633 1.206 0.820 0.587

L=300 0.630 0.592 0.494 0.372 0.262

Different between towers' heights (LL) (m); With: 8 =Odeg, L=400m LL=O LL=10 LL=S 6.824 6.808 6.792 6.337 6.324 6.313 4.852 4.849 4.846 3.202 3.207 3.210 2.081 2.090 2.097

TABLE II COMPARISON BETWEEN THE RESULTS OF

Distance from the center phase (m)

2D straight line technique with average heights (Aim)

0 10 20 30 40

25.236 23.619 15.218 7.957 4.584

3D INTEGRATION TECHNIQUE WITH VARIOUS PARAMETERS AT MID-SPAN AND 2D STRAIGHT LINE TECHNIQUE

3D integration technique Single span at point PI (mid-span) (Aim) Angle (8)(deg) With: L=400m, LL=Om 8=0 40.796 39.499 21.381 9.164 4.959

8=10 6.690 3.953 2.624 1.877 1.414

8=40 0.476 0.422 0.375 0.335 0.300

IV. CONCLUSIONS The 2-D Straight Line and 3-I) Integration Techniques give two choices for calculating magnetic field. The 2-D Straight Line is a rough approximation, and the 3-D Integration is an exact solution, however it requires integration over the three phase spans which results in a large computation time. It is seen that by using the 3D Integration Technique the Zcomponent of the magnetic field intensity appears, where this component is always equal zero in the 20 Straight Line

Span (L) (m) With: e=Odeg, LL=Om L=400 40.796 39.499 21.381 9.164 4.959

L=350 40.718 39.433 21.350 9.164 4.969

L=300 40.702 39.371 21.321 9.172 4.986

Different between towers' heights (LL) (m); With: 8 =Odeg, L=400m LL=O LL=10 LL=S 40.796 40.335 20.398 39.499 39.152 19.750 21.381 21.534 10.691 9.164 9.357 4.582 4.959 5.061 2.479

Technique. Under 3D Integration Technique, the paper present a multi-special cases to calculate the magnetic field intensity, by using these cases, it is possible to calculate the magnetic field intensity at any point under a complex configurations of a power transmission lines. Also it is possible to use the same technique, with some treatment, in the calculation of the electric field under overhead transmission lines.

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2008 5th International Multi-Conference on Systems, Signals and Devices

REFERENCES [1]

[2]

[3]

[4]

[5]

[6]

To calculate the Magnetic field intensity under 500kV TL single circuit, the following data are used. Tower span 400m Number of subconductor per phase 3 Diameter of a subconductor 30.6mm Spacing between subconductor 45cm Minimum clearance to ground 9m Outer phase Maximum height 22m Inner phase Maximum height 24.35m Distance between adjacent two phases 13.2m ApPENDIX (B)

ZI

= Zc + --;L- sin(fJ + 8) sln(fJ)

(B.3)

(B.4)

Xl

=

and;

Xc

+

u 1

cos(fJ) Yl = Yc +v1

P = tan

-1

(B.6)

r' ==

(B.8)

sin(j3+8)

zz= zl-zc

(B.9)

X_X__

cos(/J+8) xx=xl-xc

cos(fJ + 8)

(B.14) (B. 15)

wI

ul

(B.16)

B2: To transfer any point (x1,y1,z1) in (Xr:Z) system to a point (u1, v1, w1) in (u, V, W) system; By substituting (B.6) and (B.9) into (B. 1): WI

=

zl-z c

sin(fJ + 8)

sin(fJ)

(B.17)

By substituting (B.8) and (B. 10) into (B.3):

xl-x c

cos(fJ + B) vI == yl- Yc

where:

cos(fJ)

(B.18) (B.19)

zl-z fJ == tan1 -ZZ -() == tan-1 _ _c -() xx

zz L"==-----

(B.13)

By substituting (B.7) and (B.4) into (B.12):

and;

(B.7)

I

By substituting (B.5) and (B.2) into (B.11):

(B.2)

(B.l)

xx =L" cos(j3+ 8)

(8.12)

Figure B.l Cartesian coordinates of two systems in space

uI =

(B.5)

Xc +XX

v

y

where:

Assume two coordinates' systems (X,Y,X) and (U,V,W) in a space, where axis U and axis W in system (U,V,W) make an angle e with axis X and axis Z in system (X,Y,Z) respectively, while axis V and axis Yare parallel to each other, and original of the system (U,V,W) located at point (xc,yc,zc) referred to system (X,Y,Z), as indicated in Fig.(B.1). Any point P in space can be presented by the two system as (x 1,y 1,z 1) in system (X,Y,Z) and (ul,vl,wl) in system (U,V,W). From Fig. (B. 1), it is seen:

zz=L'sin(j3+8)

=

B1: To transfer any point (u1,v1,w1) in (u, V, W) system to a point (x1,y1,=1) in (X r:Z) system;

ApPENDIX (A)

~ =L' cosf/J)

I Xl

(B.11)

Hanaa Karawia, Kamelia Youssef and Ahmed Hossam-Eldin "Measurements and Evaluation of Adverse Health Effects of Electromagnetic Fields from Low Voltage Equipments" MEPCON 2008, Aswan, Egypt, March 12-15 ,PP. 436-440. George Filippopoulos, and Dimitris Tsanakas " Analytical Calculation of the Magnetic Field Produced by Electric Power Lines" IEEE Transactions on Power Delivery, Vol. 20, No.2, pp. 1474-1482, April 2005. A. A. Dahab, F. K. Amoura, and W. S. Abu-Elhaija "Comparison of Magnetic-Field Distribution of Noncompact and Compact Parallel Transmission-Line Configurations" IEEE Transactions on Power Delivery, Vol. 20, No.3, pp. 2114-2118, July 2005. A. V. Mamishev, R. D. Nevels, and B. D. Russell "Effects of Conductor Sag on Spatial Distribution of Power Line Magnetic Field" IEEE Transactions on Power Delivery, Vol. 11, No.3, pp. 1571-1576, July 1996. Rakosk Das Begamudre,"Extra High Voltage AC. Transmission Engineering" third Edition, Book, Chapter 7, pp.172-205, 2006 Wiley Eastern Limited. G. 1. Anders, G. L. Ford and D. 1. Horrocks" The Effect of Magnetic Field on Optimal Design of a Ring-Bus Substaion" IEEE Transactions on Power Delivery, Vol. 9, No.3, July 1994.

(B.20)

xl-xc

Adel Zein E. M. was born in Egypt 1971. He received his B. Sc., M. Sc and Ph. D. degrees in electric engineering from the High Institute of Energy, Aswan, Egypt in 1995, 2000 and from Kazan State Technical University, Kazan, Russia in 2005, respectively. His fields of interest include electric and magnetic fields, Comparison between the Numerical techniques in Electromagnetic, and Calculation of SAR in the Human Body.

(B.I0)

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