Stategies to Mitigate the Magnetic Induction Field
originated by Power Transmission Lines
Catarina Bebiano Coutinho Winck Cruz
Abstract—Due to the recent concerns that magnetic field
generated by power lines might affect biological systems,
scientific community have been trying to find solutions that will
reduce fields surrounding power lines. This work presents a
method to reduce the magnetic induction field, calculating all the
currents of the system, including the currents on the
subconductors of each phase bundle, on the ground wires and on
the mitigation loop (if present). A program for the calculation of
magnetic field was developed. It can handle any geometry, sixphase circuit and the presence of a mitigation loop. The results of
this program are exposed.
Keywords- transmission line, magnetic induction field redution,
mitigation loop, six-phase circuit
I.
INTRODUCTION
In the history society, the electricity, since its discovery,
had always been very important on the quality and economic
progress of populations. As the electrical need increases,
increase as well the transportation capacity of the centrals to
the consumption centers, using high voltages and currents on
transmission lines. So it became important to study the
transmission lines effect on people, in particular the effect of
magnetic field.
Some issues related with the potential hazard to human
health due to electromagnetic fields exposition have been
questioned. Investigations into claims that magnetic fields
cause cancer are ongoing in various locations around the world
with no definite results as of yet. Guidelines have been
established by the World Health Organization (WHO) and by
the International Commission of Non Ionizing Radiation
Protection (ICNIRP) for recommend maximum exposures. The
reference levels for general public exposure to 50Hz and
magnetic field is 100μT [1] [2].
So, in this paper, are presented some solutions to reduce the
magnetic field of transmission lines. Those solutions include
conductor configuration, use of a mitigation loop and a
compact six-phase circuit [3] [4] [5] [6].
Some of the presented solutions are not new, as the
conductor configuration and the use of a mitigation loop. The
innovation is on increasing the number of mitigation
conductors and on the six-phase circuit study.
Compacting power lines allows transmitting the same
amount of power as the conventional ones using less space.
The compaction can be made by two ways:
-
Reducing the space between conductors, which implies
changing the lines and the towers;
-
Increasing the number of phases, to six or twelve. The
conductors are usually placed on a circumference.
On this study only the increasing of phases was
approached.
This paper is organized into four sections, the first of which
is introductory. Section II – Method of Calculation, that
describes the magnetic field calculation method; Section III –
Simulation Results, where all the numeric and graphical results
are presented and Section IV – Conclusions.
II.
MODEL DEVELOPED
A. System Current Evaluation
To evaluate the magnetic field due to a generic system
represented on figure 1, composed by a transmission line and a
mitigation grid, one needs to obtain the current crossing each
conductor of the system. In this paper the currents crossing the
phase conductors are imposed, being all the other currents
determined using a matrix based on multiconductor
transmission line model [4].
Figure 1 – 400kV single-circuit power line with mitigation loop.
The frequency-domain, the transmission lines equations
are:
= −
(1)
= + ∆ + ∆ (2)
the
where and are the voltage and current vectors and per-unit-length series-impedance matrix, given by:
where is the external-inductance matrix, ∆Zc is a frequencydependent complex diagonal matrix, whose entries can be
determined by using the skin effect theory results for
cylindrical conductors and ∆Zt is the earth impedance
correction.
For low frequency applications the matrix ∆Zc can be
determined by:
∆ = +
!"
(3)
where denotes the per-unit-length dc resistance of
conductor k.
The entries of ∆Zt can be determined using the Dubanton
complex ground plane approach. This empiric method
determines the matrix + ∆ , applying the images
method to a complex ground plane, distanced by # from the
real ground plane surface.
Q8
#:8
78
−R78 + # S
#
−Q:
′ 9#9, ∆ = D − 0 = − (FGH G?
- + ∆- =
being # :
$"
ln .
# =
$"
ln (
$) *$
$+
,
/0 102 3 *)2 *) *$ 3
/0 102 3 *) *)2 3
5
/6
(4)
4
(5)
(6)
Since line conductors sag between towers the entries Le and ∆Zt
vary along the longitudinal coordinate z. A parabolic
expression for the conductor’s height as a function of z can be
written as:
78 9: = ℎ<:= + > (
$2 1? $
?
,
(8)
(9)
The matrix equation in (9) can be written explicitly in
partitioned form:
Figure 2 –Dubanton method representation.
A
= @ B C
<
The integration of (1) from z=0 to z=l gives
−Q8
+ ∆ =
Remembering equation (1), V and I are complex column
matrices collecting the phasors associated, respectively, with
all the voltages and currents of the line conductors:
The subscripts f, g, m refer to phase conductors, ground
wires and mitigation loop, respectively.
−7P = −R7: + # S
OP8
with the suspension towers placed ate z=0 and z=l (see
figure 3).
A
= @ B C
<
Q:
7P
Figure 3 - Conductor sagging between towers along the line span
l.
(7)
AA AB A< A
∆I
@ ∆J C = − @ BA BB B< C @ B C
<A <B << <
∆K
(10)
Taking into account that the conductors belonging to a
given phase bundle are bonded to each other, and that ground
wires are bonded to earth (tower resistances neglected), we can
say:
:L 0 = :M 0
:L D = :M D
∆
B = 0
:L + :M = :
(11)
Taking into account the mitigation loop, we can say:
<5 D = … = <= D
<5 0 = … = <= 0
<5 = −<$ − … − <=
(12)
Now, taking into account the preceding considerations,
using (10), we can determine, step upon step, all of the currents
in the system.
15
B = −BB
RBA A + B< < S
(13)
e is the contribution associated to the phase
where a
g to the mitigation
f to the ground wires and a
conductors, a
loop.
The magnetic field in a generic point P of coordinates (xp,
yp) is given by:
iiijm
= j ×+jm − j×+n
iij
ih
8k
3
3
$"+m
From (10):
∆
< = −<A A + <B B + << < (14)
Using two auxiliary matrices D and E
1−100
⋮
0000
−1 − 1 − 1
] 1 0 0
\
Z=\ 0 1 0
\
⋮
[ 0 0 0
Which denote, respectively,
the mitigation conductors
Im=E[Im2.... Imm]T .
O=T
⋯ 00
(15)
⋱
⋮ Y
⋯ 1−1
⋯
−1
⋯
0 `
_
(16)
⋯
0 _
⋱
⋮ _
⋯
1 ^
the KVL and KCL applied to
resulting in D∆V=0 and
Where:
where:
15
BA
b = −<A + <B BB
15
B< − <<
a = <B BB
(17)
(18)
iij
ij0 + 7k − 7′8 q
ij) r o′$8k = pk − p8 $ + 7k − 7′8 $
o′8k = pk − p8 q
7′8 = −78 − 2#
Admitting that the currents only have component along z:
7k − 78
7k − 7′8
pk − p8
pk − p8
iij
ih
w q
w q
ij + u−
ij
8k = u−
$ 8 +
$ 8 +
2vo8k
2vo′$8k 8 0
2vo8k
2vo′$8k 8 )
Meaning:
= x + x q
iij
ih
ij) (24)
8k
80+
80: ij0 + x8)+ + x8): q
For nc conductors the rms field value is given by:
|
|
|
|
x+<y = zR{8G5
x80+ S + R{8G5
x80: S + R{8G5
x8)+ S + R{8G5
x8): S
$
=
$
=
= }H h
a
=
$
(25)
(20)
(21)
Knowing all the currents of the system we are now in
conditions to calculate the magnetic field.
B. Magnetic Field Evaluation
The complex amplitude of the magnetic induction field
in the space surrounding the overhead line is obtained
vector a
by summation of several contributions
=a
e + a
f + a
g
a
$
The magnetic induction field is given by:
Replacing in (20) the equations (13) and (17) we obtain:
D − 5L 0
5L
] 5L
`
] `
\ 5M D − 5M 0 _
\ 5M _
$L
\
$L D − $L 0_
\
D − 0_ = −d \$M _
$M
$M
\
_
\
_
\
_
D
−
0
cL
cL
cL
\
_
[cM ^
[
cM D − cM 0^
=
(19)
Knowing that
∆A = −RAA A + AB B + A< < S
(23)
$
oj8k = pk − p8 q
ij0 + 7k − 78 q
ij) r o8k
= pk − p8 $ + 7k − 78 $
Replacing in (16) Ig by (13) and multiplying (16) on both
sides by (14), we obtain:
< = −ZOaZ15 ObA
$"+nm
(22)
C. Optimization Algorithm
As an upgrade of [3], an optimization algorithm was
developed to see how the number of mitigation loop
conductors would influence the reduction of magnetic field.
Knowing how to calculate the magnetic field has been
elaborated an optimization algorithm, whose objective
function was to minimize the magnetic induction field on x=0,
y=1.8m and z=150m. The unknowns of this objective function
are the vectors xm and ym, which are the coordinates of
mitigation conductor centers.
Having the base case in mind, where the profile is
symmetric due to y axis, we want to maintain the same
symmetry on the profiles using the optimization algorithm.
For that is necessary to impose symmetry on the mitigation
conductors position. Thus the number of unknowns decreases.
So if the number of mitigation conductors is pair the xm and ym
vectors will have
will have
=~ 15
$
=~
$
dimension. If it’s unpaired the xm vector
dimension because the central conductor will
be placed on x=0 and ym vector will have
=~ *5
$
dimension.
This algorithm has a restriction, the non superposition of
$
the mitigation conductors, which means Rp: − p S +
$
R7: − 7 S ≥ 2o< $ , being (xi, yi) and (xj, yj) the mitigation
conductors i and j centers coordinates.
The y domain to find the solutions was established for the
mitigation conductors not to be near the ground surface and
not close to the phase conductors. To x the domain was
defined considering the line ROW.
Table II – coordinates of loop conductors.
Nº of conductors
X coordinate [m]
-11.92
11.92
-12
0
12
-13
-9
9
13
-12
-6
0
6
12
-11.81
-9.76
-0.4
0.4
9.76
11.81
2
3
4
III.
SIMULATION RESULTS
A. Base case
The base case considered is a single-circuit 400kV
transmission line in flat formation, with ground wires.
5
6
Y coordinate [m]
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
So far, we have been concerned with the computation of the
B field at a particular observation point P (point with
coordinates: xp=0, yp=1.8m, zp=150m). However, in order to
have a complete grasp of the problem, we still need to analyze
the field distribution in the whole space.
Figure 4 – 400kV single-circuit.
Table I summarizes the conductor characteristics. A soil
with an average resistivity ρ of 100 Ωm has been considered.
The distance between consecutive towers has been set equal to
300 m.
Bearing this in mind, we started by obtaining transversal
and longitudinal profiles of the field intensity; see Fig. 6 and 7.
35
base case
2 conductors
4 conductors
6 conductors
30
Table I- Characteristics of line conductors.
25
R-a
R-b
S-a
S-b
T-a
T-b
G1
G2
31.8
31.8
31.8
31.8
31.8
31.8
14.6
14.6
X
coordinate
(m)
-12.2
-11.8
-0.2
0.2
11.8
12.2
-8
8
Y
coordinate
(m)
26
26
26
26
26
26
36
36
sag
(m)
12
12
12
12
12
12
9
9
Rdc @
50˚C
(mΩ/km)
57.3
57.3
57.3
57.3
57.3
57.3
372
372
The rms value of magnetic induction field, at a point P of
coordinates xp=0m and yp=1.8m, is
€ = 32.16 }„
B. "umber of mitigation loop conductors influency
The increasing of mitigation conductors will reduce the
magnetic induction field.
Was considered a mitigation grid from 2 to 6 conductors
and was used the optimization algorithm described in II C.
20
B [ µT]
Conductor
Diameter
(mm)
15
10
5
0
-50
-40
-30
-20
-10
0
Z [m]
10
20
30
40
50
Figure 5 – Transversal representation at y=1.8m, z=150m
and x Є [-50, 50].
As we see the magnetic induction field reduction increases
when number of loop conductors increases. The black curve
corresponds to the base case, the blue to a two loop conductors,
the green to four loop conductors and the red to six loop
conductors.
Tower height = 100%
35
base case
2 conductors
4 conductors
6 conductors
30
Phase distance = 100%
Tower height = 68%
B [ µT]
25
Phase distance = 31%
20
15
10
5
0
50
100
150
Z [m]
200
250
300
Figure 6 – Longitudinal representation at x=0, y=1.8m and
z Є [0,300].
In figure 6 it’s easy to see that the worst scenario is at
midspan (z=150m).
Figure 8 – tower adjustments.
The principal adjustments that change the conductors’
characteristics are:
††o#‡ˆ‰Š‹ Œ<ŽL ?:= = 31‘…††o#‡ˆ‰Š‹ Œ’M? :+’:
Figure 7 presents the percentage difference between the
magnetic field value obtained on the base case and obtained
with the optimization algorithm.
††o#‡ˆ‰Š‹ <ŽL ?:= = 6“‘…††o#‡ˆ‰Š‹ ’M? :+’:
Table III. Characteristics of six-phase
six
line conductor
Figure 7 – percentage of magnetic field reduction with loop
conductors increasing.
As we see with the loop conductors increasing the
percentage increase as well.
C. Six-phase circuit
Increasing the number of phases, on this case six phases,
implies that the currents are shifted by 60º instead of 120º
120º, the
rms current value is half of the three phase current, to maintain
the transmitted power.
On the first approach the tower suffers some adjustments [5]:
Conductor
Diameter
(mm)
R–a
R–b
S–a
S–b
T–a
T–b
U–a
U–b
V–a
V–b
W–a
W–b
G
31.8
31.8
31.8
31.8
31.8
31.8
31.8
31.8
31.8
31.8
31.8
31.8
14.6
X
coordinater
(m)
-2.387
-2.263
2.542
2.418
-2.387
-2.263
2.387
2.263
-2.542
-2.418
2.387
2.263
0
Y
coordinater
(m)
20.4
20.4
25.5
25.5
30.6
30.6
20.4
20.4
25.5
25.5
30.6
30.6
35.7
sag
(m)
8
8
8
8
8
8
8
8
8
8
8
8
5
Rdc @
50˚C
(mΩ/km)
57.3
57.3
57.3
57.3
57.3
57.3
57.3
57.3
57.3
57.3
57.3
57.3
372
Table III have all the characteristics of the six-phase
line conductors’.
phase circuit reduces in reality the
To see if the six-phase
magnetic field we represented on figure 9 the transversal
profile of magnetic induction field for the base case and for
the six-phase circuit:
35
35
compact line
base case
30
double circuit
compact line
ncm=6
30
25
20
20
B [ µT]
B [µT]
base case
25
15
15
10
10
5
5
0
-50
-40
-30
-20
-10
0
X [m]
10
20
30
40
0
-50
50
Figure 9 – Transversal representation at y=1.8m, z=150m
and x Є [-50, 50].
Figure 9 illustrates the dimension of magnetic field
reduction from one circuit to another. The two circuits are
transmitting the same amount of power. In this case the
magnetic induction field is:
€ = 5.11 }„
-40
-30
-20
-10
0
X [m]
10
20
30
40
50
Figure 11 - Transversal representation at y=1.8m, z=150m
and x Є [-50, 50].
The second approach of the six-phase circuit is to
dispose the conductors’ on a circumference.
The circumference has a radius of R=3.465m and the
distance between the sub conductor’s was maintained at
0.4m [7].
The reduction is approximately 84%.
Comparing the double-circuit with the six-phase circuit,
both transmitting the same power, we have:
12
compact line
double circuit
10
B [ µT]
8
6
4
Figure 12 – Compact line configuration.
2
The coordinates of this six-phase circuit can be obtain
by the degree of compaction
0
-50
-40
-30
-20
-10
0
X [m]
10
20
30
40
50
Figure 10 – Transversal representation at y=1.8m, z=150m
and x Є [-50, 50].
In figure 10 we observe that the profile of the compact
line is tighter but the maximum value is higher than the doublecircuit. In this case the magnetic induction field is:
†•–‰…Š‡†ˆB+ =
—†DŠ‰˜‹ ™‹Šš‹‹ˆ –ℎ‰›‹›
#‡›Š‰ˆ…‹ ™‹Šš‹‹ˆ –ℎ‰›‹›
On the six-phase circuit the phases are shifted by 60º,
so we can imagine a hexagon where the phases are putted
on the vertices. The hexagon can be divided on six
equilateral triangles and the edge of the triangle is the
compaction degree.
€ = 10.23 }„
Figure 13 – compaction degree representation.
On this case:
400
=
12
second solution presents a higher value for the magnetic
induction field.
400ž
3 ↔ Ÿ = 6.93 •
Ÿ
Through trigonometry it’s possible to discover the
conductors’ coordinates.
Table IV – Characteristics of six-phase conductor
Conductor
Diameter
(mm)
R–a
R–b
S–a
S–b
T–a
T–b
U–a
U–b
V–a
V–b
W–a
W–b
G
31.8
31.8
31.8
31.8
31.8
31.8
31.8
31.8
31.8
31.8
31.8
31.8
14.6
X
coordinate
(m)
-0.2
0.2
-6.2
-5.8
-6.2
-5.8
-0.2
0.2
5.8
6.2
5.8
6.2
0
sag
(m)
Y
coordinate
(m)
20.4
20.4
23.9
23.9
30.8
30.8
34.3
34.3
30.8
30.8
23.9
23.9
40
Rdc @
50˚C
(mΩ/km)
57.3
57.3
57.3
57.3
57.3
57.3
57.3
57.3
57.3
57.3
57.3
57.3
372
8
8
8
8
8
8
8
8
8
8
8
8
5
Comparing this approach with the base case, using of
mitigation loop and double circuit, the figure 14 present a
transversal profile:
35
double circuit
compact line
ncm=6
30
IV.
This paper presents a model to evaluate the magnetic
induction field originated by power transmission lines. This
model allows to evaluate the magnetic induction fields
considering several line configurations, using a mitigation grid
with n conductors being its location obtained by an
optimization algorithm and allows to analyze six-phase
circuits.
Relatively to the mitigation loop, this solution reduces the
magnetic induction field independently of the conductors’
number. As expected, increasing the number of mitigation
conductors reduces the magnetic field magnitude.
On first approach of six-phase circuit there are more intense
magnetic fields near the line axis, because the conductors’ are
closest to the ground and on a tighter region. This solution
implies to spend more money because the towers are not the
same as on the other circuits. On the second approach the
reduction is less evident comparing with the first, but
comparing with the base case the reduction is about 60%.
Comparing the two studied solutions, both reduce the
magnetic induction field, although the compact line is the best.
The only problem of this solution is the money spent on the
new towers.
REFERENCES
base case
[1]
25
B [µT]
20
[2]
15
10
[3]
5
0
-50
-40
-30
-20
-10
0
X [m]
10
20
30
40
50
Figure 14 – Transversal representation at y=1.8m,
z=150m and x Є [-50, 50].
[4]
[5]
The value of magnetic induction field of this solution is:
[6]
Comparing figures 11 and 14 we can see that the only
difference is between the six-phase circuit solutions. The
[7]
€ = 19.37 }„
CONCLUSIONS
WHO – World Health Organization. Extremely low frequency fields.
Environmental Health Criteria, Vol. 238. Geneva, World Health
Organization, 2007.
ICNIRP – International Commission on Non-Ionizing Radiation
Protection (1998). Guidelines for limiting exposure to time varying
electric, magnetic and electromagnetic fields (up to 300 GHz). Health
Physics 74(4), 494-522.
J. F. G. Casaca, “Cálculo do campo electromagnético originado por
linhas aéreas de transmissão de energia”, Dissertação de mestrado de
Eng. Electrotécnica e de Computadores, Setembro 2007.
J. A. Brandão Faria e M. E. Almeida, “Accurate Calculation of
Magnetic-Field Intensity due to Overhead Power Lines with or without
Mitigation Loops with or without Capacitor Compensation”, IEEE, Jan.
2006
M. Barbarito, A. Clerici, L. Paris, P. Pelacchi, “A New Type of Compact
Line for Urban and Suburban Areas”, IEEE International Conference on
Overhead Line Design and Construction: Theory and Practice (up to 150
kV), 28-30 November 1988, London.
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, July 2005.
Hanafy M. Ismail, “Magnetic field of high-phase order and compact
transmission lines”, Int. J. Energy Res. 2002; 26:45-55.