Slavko Vujević, Dino Lovrić and Tonći Modrić
University of Split,
Faculty of Electrical Engineering, Mechanical Engineering and Naval
Architecture,
Croatia
2D Computation and Measurement of Electric
and Magnetic Fields of Overhead Electric
Power Lines
INDS’11 & ISTET’11 – S. Vujević, D. Lovrić, T. Modrić
Introduction
INDS’11 & ISTET’11 – S. Vujević, D. Lovrić, T. Modrić
the problem of computing the power frequency electromagnetic field can be
considered as quasistatic → electric and magnetic field computed separately
computation of electric and magnetic field accomplished using a 2D
approximation of a power line
short power line is considered, conductor charge density is approximated by a
constant
results will be compared to measurements underneath a 400 kV overhead power
line
Mathematical model of overhead electric power line
2D numerical algorithm → short conductors
→ conductor sag neglected
INDS’11 & ISTET’11 – S. Vujević, D. Lovrić, T. Modrić
thin-wire approximation
conductors treated as line sources parallel to
earth surface
homogeneous earth
conductors positioned along x-axis
field distribution will be computed along y-z
plane in the middle of the section
Scalar electric and vector magnetic potentials
solutions of Helmholtz differential equations reduce to solutions of Poisson
differential equations (quasistatic approximation)
INDS’11 & ISTET’11 – S. Vujević, D. Lovrić, T. Modrić
scalar electric potential for n conductors:
n
i
BiBi
iAi
Ai
i
BiAi
dR
FdR104
1
vector magnetic potential for n conductors:
n
i
AiAi
i
Ai
dR
IiA1
1
4
0
2D computation of power line magnetic field
numerical algorithm is based on Biot-Savart law
INDS’11 & ISTET’11 – S. Vujević, D. Lovrić, T. Modrić
conductor current flows along the axis and generates the magnetic field
components of magnetic flux density in y-z plane for x = 0:
iI
0xB
n
iii
i I
dd
zzB
1i2
22
0y
44
n
iii
i I
dd
yyB
1i2
22
0z
44
zy BBB
Total effective (rms value:)
2D computation of power line electric field
conductor charge density approximated by a constant →
INDS’11 & ISTET’11 – S. Vujević, D. Lovrić, T. Modrić
electric field intensity is computed from
i
zyx EEEAjE ,,
the components of the electric field intensity are computed from:
00 xxxxx AjAjEE
y
zy
y
zyxEE xxyy
,,0,,00
z
zy
z
zyxEE xxzz
,,0,,00
i
i
Q
zyx EEEE
rms value:
2D computation of power line electric field
vector magnetic potential for x = 0 needed in:
INDS’11 & ISTET’11 – S. Vujević, D. Lovrić, T. Modrić
0
xx AjE
n
i i
i
ix d
d
nIA1
22
0
0
24
2
is computed from:
22iii zzyyd
Ai
AiAi
dR
1
2D computation of power line electric field
scalar electric potential for x = 0 needed in equations
INDS’11 & ISTET’11 – S. Vujević, D. Lovrić, T. Modrić
y
zyEy
,,0
can be obtained from:
z
zyEz
,,0
i
n
i ii
QD
Fd
zy
1
11
0 2sinh
2sinh
2
1,,0
phasors of the ith conductor charges are unknown
2D computation of power line electric field
phasors of the ith conductor charges are computed using the point collocation
method:
INDS’11 & ISTET’11 – S. Vujević, D. Lovrić, T. Modrić
njQDZFdZn
i
jiji ...,,2,1;)()(1
ji
v
v
vZ22
n2
1)(
22
0
v is replaced by:
iii rd 0
iii zD 2
jizzyyd jijiij ;22
jizzyyD jijiij ;22
Input data of the overhead power line
INDS’11 & ISTET’11 – S. Vujević, D. Lovrić, T. Modrić
(kV)i (A)iIi y (m) zs (m)
L11 -10.7 12.58 234.40º 228-4.2º
2 -10.4 12.58 234.40º 228-4.2º
L23 -0.15 12.58 234.4240º 228235.8º
4 0.15 12.58 234.4240º 228235.8º
L35 10.4 12.58 234.4120º 228115.8º
6 10.7 12.58 234.4120º 228115.8º
SW1 7 -7.44 19.09 00º 00º
SW2 8 7.44 19.09 00º 00º
6 phase conductors (AlFe 490/65) and 2 shield
wires (Alumweld 19/9)
376 m long power line section
The sag of the conductors was
taken into account by:
INDS’11 & ISTET’11 – S. Vujević, D. Lovrić, T. Modrić
szzs 3
2
Other input data:
Hzf
mS
r
50
/1.0
10
measurements taken on a clear
winter day (10 °C)
terrain mostly clear except sporadic
bushes
measurements along y directed
observation profile at z = 1 m
Input data of the overhead power line
Measuring equipment and procedure
INDS’11 & ISTET’11 – S. Vujević, D. Lovrić, T. Modrić
electric field meter Monroe Electronics Model 238A-1
AC Fieldmeter
measurer has influence on electric field distribution →
measuring probe must be used
Measuring equipment and procedure
INDS’11 & ISTET’11 – S. Vujević, D. Lovrić, T. Modrić
magnetic field meter Sypris Triaxial ELF Magnetic Field
Meter Model 4090
measurer does not have influence on the magnetic field
distribution
Comparison of results – magnetic field
INDS’11 & ISTET’11 – S. Vujević, D. Lovrić, T. Modrić
Comparison of results – electric field
INDS’11 & ISTET’11 – S. Vujević, D. Lovrić, T. Modrić
Summary
INDS’11 & ISTET’11 – S. Vujević, D. Lovrić, T. Modrić
2D algorithms for computation of electric and magnetic fields of overhead power
lines
good agreement was found between the computed results and measurements
more advanced models are needed for more complicated structures such as
electric power substations
better results can be achieved by taking the sag into account (3D algorithm)
Thank you!
INDS’11 & ISTET’11 – S. Vujević, D. Lovrić, T. Modrić