USING THE NONLINEAR PARTICLE SWARM OPTIMIZATION (PSO ...

IJRRAS 4 (1) ● July 2010 Al-Salameh & al. ● Reducing Magnetic Fields from Overhead High Voltage Transmission Lines

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USING THE NONLINEAR PARTICLE SWARM OPTIMIZATION (PSO)

ALGORITHM TO REDUCE THE MAGNETIC FIELDS FROM OVERHEAD

HIGH VOLTAGE TRANSMISSION LINES

M. S. H. Al Salameh1, I. M. Nejdawi

2 & O. A. Alani

3

1 Department of Electrical Engineering, University of Science and Technology, Irbid 22110, Jordan,

1 Electricity Regulatory Commission, Amman 11821, PO Box 1865, Jordan.

1 Bin Ghannam Contracting, P.O. Box 31119, Dubai, UAE.

Email: [email protected]

ABSTRACT

Using particle swarm optimization (PSO), the influence of the geometrical configuration of conductors is studied in

order to minimize the magnetic field near both single circuit and double circuit high voltage overhead power

transmission lines. New arrangements of high voltage "green lines" are proposed. The results indicate that the

magnetic field can be reduced up to 48% under the influence of wind and ice, and 80%, neglecting them.

Accordingly, the necessary ROW (Right Of Way) width, so that the magnetic field outside ROW does not exceed an

example reference value of 0.4 μT, can be reduced by up to 48% if wind and ice are taken into account and by up to

90% if not, for the same range of line heights. Complex and real image theories were implemented to find the

magnetic and electric fields, respectively, near the transmission line. The electric field is evaluated and it is found

that it is acceptable. Also, results showed that the bundling affects the electric field only.

Keywords: High voltage lines, PSO, Reduce fields, Image theory, EMC, Health.

1. INTRODUCTION

The 50/60 Hz electric power transmission lines can interfere with nearby electrical and electronic equipment [1]. In

addition, concern has arisen regarding the controversial issue of possible adverse effects on human health associated

with exposure to electromagnetic fields from 50/60 Hz electric power transmission, sub-transmission and/or

distribution lines, and other sources including appliances. Because of this, many studies have focused on the fields

emanating from electrical devices and transmission lines [2]-[6]. The greater the field around the transmission lines,

the greater the current induced in the human body [3]. The International Agency for Research on Cancer (IARC)

classified power frequency electromagnetic fields as “possibly carcinogenic to humans” based on a fairly consistent

statistical association between doubling the risk of childhood leukemia and extremely low frequency (ELF)

magnetic field exposure above 0.4 μT [4]. Thus, in some cases there is a need to reduce the fields, especially the

magnetic field which cannot be easily shielded [7], when the transmission line interferes with the adjacent electrical

and electronic equipment. Also, as most building materials are nonmagnetic, magnetic fields of high voltage lines

are not attenuated by common materials such as building walls. In contrast to that, building walls can shield electric

fields to some extent.

The shielding of power lines was investigated in order to reduce the emitted magnetic flux density to a few

microteslas which is the limit for new installations in some countries [7]. Unlike radio frequency fields which can be

shielded with relatively thin films, ELF magnetic fields require very thick, high permeability materials for effective

shielding [8]. Comparing different configurations revealed that the magnetic field at ground is low for the equilateral

delta configuration and high for the horizontal configuration [9]. Another way to reduce magnetic field is “active

shielding”, namely, by a separately energized circuit introduced near an existing power line [10]. Such systems

require dynamic feedback systems to keep the phase and magnitude of the shielding circuit current at optimum

interference with the power circuit, and can be costly. Underground power lines with multiple conductors can be

configured to minimize the magnetic field at ground level [10]. Since the earth shields only the electric field,

underground lines can sometimes emit greater magnetic fields than equivalent overhead lines because the burial

depth may be only four to six feet as opposed to the 20 to 30 feet clearance of an overhead line. Recently, TenneT,

the operator of electric power net in Netherlands, has provided new designs of high-voltage transmission lines in

which conductors are suspended closer together than in traditional high-voltage lines, thus reducing the intensity of

the magnetic field generated along the line [11]. This TenneT project was in response to the precautionary policy

adopted by Dutch Ministry of Housing, Spatial Planning and the Environment which is aimed at minimizing

unnecessary long-term human exposure to magnetic fields generated by high-voltage lines. The ministry defined a


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limit value of 0.4 T for exposure to magnetic fields in 2005 as part of strict guidelines for the extension of high-

voltage lines and the construction of new buildings near existing high-voltage lines. The article [12] considers a

specific arrangement of the phase conductors along with reducing dimensions (compaction) of the line to reduce the

magnetic and electric fields.

We choose, in this paper, to reduce the magnetic field of an arbitrary line by arranging its phase conductors, without

assuming certain templates in advance. To the authors' knowledge, there is no published systematic method to find

the optimum arrangement of conductors for any high voltage transmission line in order to produce minimum

magnetic field. For all the cases investigated, we found that minimizing the magnetic field is automatically

associated with reducing the electric field. To achieve this goal, we used particle swarm optimization (PSO) which is

a nonlinear constrained method that can lead to optimum solutions without knowing the gradient of the problem

beforehand [13], [14]. Our solution represents a two-dimensional cross section of a transmission line, and thus the

line height is considered at the point along the line where the cross-section is taken.

Minimizing the magnetic field by arranging the transmission line conductors has the advantage of minimizing the

magnetic field without adding any equipment or shield to the line. Extensive simulations were performed (not shown

in this paper due to paper size limitations) for various configurations, where it was noticed in most cases that

optimized lines (with minimum magnetic fields) have more compact arrangements (shorter conductor spacings)

compared with unoptimized arrangements. This indicates that the cost of installing a high voltage line with

minimum field is probably lower than or comparable to unoptimized line. Anyway, the cost analysis will be

considered in a future work.

In this paper, the intelligence of the swarm is used to find the optimal arrangement of conductors that would produce

minimum magnetic field near high voltage overhead power transmission lines. Toward that end, a swarm of 49

particles (different arrangements of line conductors) is used to minimize the field. At each iteration, the fitness

function (magnetic field due to all conductors) is evaluated for each particle in the swarm. The best fitness values for

each particle as well as for the whole swarm are stored. Swarms continue to move (iterate) until the targeted best

value is obtained. The transmission line configuration (conductors arrangement) associated with this targeted best

value is the optimum solution. Matlab computer programs were written and executed to calculate and minimize the

magnetic field for overhead transmission lines. Also, the electric field is computed where we found that the electric

field is also reduced.

2. PARTICLE SWARM OPTIMIZATION (PSO)

The PSO is a robust stochastic nonlinear evolutionary computation technique based on the movement and

intelligence of swarms [15]. In comparison with other stochastic evolutionary algorithms like genetic algorithms,

PSO has fewer complicated operations, fewer defining parameters, and generally fewer lines of code [16]. PSO

depends on the social interaction between independent agents, here called particles, during their search for the

optimum solution using the concept of fitness. The fitness defines how well the position vector of each particle

satisfies the requirements of the optimization problem.

Note that most buildings represent good conductors at 50/60 Hz frequency, and thus can shield the electric field. On

the contrary, these buildings are almost transparent to magnetic fields [3]. Based on that, only the magnetic field is

minimized here, i.e., the magnetic field value is the fitness function. In fact, when we included both the magnetic

and electric fields in the fitness function, we obtained essentially similar results to that when only the magnetic field

is considered.

A population (swarm) size of about 30 is suitable for many problems. Although 36 particles worked properly in this

paper, 49 were chosen since the execution time of the computer program is a fraction of a second. The specific

values of 36 and 49 are dictated by the way we wrote the code since nested do-loops with equal number of steps

were used for incrementing the x and y positions. Each particle remembers its personal best position found (called

its local best) and also knows the best position found by the swarm (called the global best). The x and y components

of the velocity and the position represented by the x, y coordinates, for each particle m, are updated by the following

equations [13]:

t t 1 t t t 1 t t t 1

mn,x mn,x 1 n1 mn,x mn 2 n2 n,x mn

t t 1 t

mn mn mn,x

t t 1 t t t 1 t t t 1

mn,y mn,y 1 n1 mn,y mn 2 n2 n,y mn

t t 1 t

mn mn mn,y

v w v c U (p x ) c U (g x )

x x t (v )

v w v c U (p y ) c U (g y )

y y t (v )

(1)


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where superscripts t and t-1 are time indices of the current and the previous iterations, Un1 and Un2 are two different

uniformly distributed random numbers in the interval {0,1}, w is the “inertial weight” in the range {0,1}, and pmn, gn

are the personal and global best positions (the subscripts x, y refer to the x and y components), respectively. The

parameters c1 and c2 are scaling factors of local and global bests; a value of 2 is a good choice for both parameters

[17]. The subscript m is the particle number in the swarm while n indicates the parameter to be optimized, and the

time-step Δt is usually chosen to be one. In this paper, reflecting walls are used as boundary conditions; when a

particle reaches the boundary, it is reflected back as shown in Fig. 1.

Fig. 1: Particles are looking for a minimum field value with “reflecting walls”.

Since PSO is a stochastic optimization method, each trial would give a somewhat different solution. Our results

represent the best obtained solutions for each configuration analyzed. We have reported the best solution for each

case after making at least 100 trials using different random seeds.

3. SOLUTION ALGORITHM

The following algorithm minimizes the magnetic field under overhead transmission lines.

I. Specify the constraints of the problem: minimum spacings between conductors, and limits of the region in

which the particles will search for suitable arrangement of conductors (see Fig. 1).

II. Distribute the particles (different arrangements of conductors) in the selected region, specify a time step for

particle movement (here unity), initialize the population with a random velocity (v) vector (here zero initial

velocity), and initialize the stop criterion with a value much smaller than 0.40 μT (here 0.01 μT). Specify the

maximum number of iterations that should not be exceeded.

III. Evaluate the fitness function (here the magnetic field value) for each particle.

IV. If the magnetic field value < the personal best value, then replace the personal best value by the new magnetic

field value.

V. If the magnetic field value < the global best value, then replace the global best value by the new magnetic field

value.

VI. If the fitness is ≤ the stop criterion or maximum number of iterations is reached, then stop; a solution is found.

Otherwise, update position and velocity of particles according to (1) (c1=c2=2 and w=0.7), and go to step 3.

4. MAGNETIC FIELD MODEL

The frequency of the power system (50/60 Hz) is small enough that the magnetic and electric fields in air can be

considered independent. The magnetic fields associated with electric power transmission lines are readily predicted

using mathematical equations given line load data and wire configuration [18]. The magnetic field of overhead high

voltage lines can be found by superimposing the individual contributions of the phase conductors taking into

Reflecting walls

Upper limit for line height

Lower limit of clearance to ground

Within this

region,

particles are

looking for

minimum

field value

Upper limit for line width


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account the earth return currents. The geometry considered to evaluate the magnetic field at (xj, yj) due to the phase

conductor at (xi, yi) taking into account the complex image at (xi’, yi’) is illustrated in Fig. 2.

Fig. 2: Geometry to find the magnetic field at the point (xj, yj) due to the phase conductor at (xi, yi) taking into account the

complex image at (xi’, yi’).

To take into account return currents in the expression of the magnetic field it is necessary to start from the mutual

impedance presented by [19]. The magnetic field Hj (fitness function in this paper) at the point (xj, yj) is obtained by

considering the contribution of all N phase conductors:

Η

j

4N N

i iij ij'

i iij ij

1 2u 1 u '

2 r 2 r' 3 r (2)

where

2 2 ' 2 2

ij j i j i ij j i j i

j ii j i j i j'

ij x y ij x y' '

ij ij ij ij

2j ( j ), r (x x ) (y y ) ,r (x x ) (y y )

2y y

y y x x x xu u u ,u u u

r r r r

where σ, ε, μ are conductivity, permittivity, permeability of the earth, respectively, and ω is angular frequency. Ii is

the phase current, and ux, uy are the unit vectors along the x and y axes. Note that in (2), the first term is the

contribution of the line conductors whereas the second term is the contribution of the line images. The magnetic

field density is B= μH, and its magnitude of B0 is:

2 2

0 0x 0yB B B (3)

where B0x and B0y are the amplitudes in the x and y directions.

5. ELECTRIC FIELD MODEL

For the electric field computation, the earth effect is represented by image charges located below the conductors at a

depth equal to the conductor height [20]. The phase capacitance C in µF/km (considering three phase conductors

with phase-to-phase distances of D12, D13, D23) is calculated as [21]:

Field point

(xj ,yj)

rij

rij’

Line conductor

(xi ,yi)

yi+yj+2/γ

Imag{yi+yj+2/γ }

Imaginary

axis

y

x

Real{yi+yj+2/γ}


22

1/ 3

12 23 31

1C , GMD D D D

18 (GMD/ r)

ln (4)

where r is the conductor radius and GMD is the geometric mean distance (equivalent conductor spacing). Then, the

electric field at (x,y) due to a line conductor at (x0,h) is:

y 0 x y 0 x

2 2 2 2

0 0 0

2(y h)u 2(x x )u 2(y h)u 2(x x )uCVE(x,y)

4 (y h) (x x ) (y h) (x x )

(5)

where V is the phase voltage. Note that in (5), the first term is the contribution of a line conductor whereas the

second term is the contribution of the line image. The total electric field is the superposition of the fields from all

conductors.

6. RESULTS

This study is focused on minimizing the magnetic field of single circuit and double circuit configurations of 132 kV

Condor type transmission lines. However, the algorithm in this paper can deal with other transmission lines voltages.

In order to verify the validity of equations and computer programs, the computed magnetic fields are compared with

measured fields at a height of 1 m above the surface of the ground. For each case analyzed, two cases of IEC-71

standards were applied, first, ignoring the effects of wind and ice where minimum spacing between phases is 1.1 m,

and second, the practical case which considers the effects of wind and ice where minimum spacing between phases

is 3.5 m. The optimized solutions offer reduced ROW widths. The heights of the optimized and unoptimized lines

were kept almost the same, since increasing line height would automatically decrease fields near the surface of the

earth. The execution time of each simulated case is less than one second on the computer used (Genuine Intel(R)

CPU T2300 @ 1.66 GHz, 1.49 GB RAM) and the number of iterations range is 20 to 200. The electric field is also

computed for the transmission line before and after minimizing the magnetic field. The following examples are

intended to show the validity and accuracy of the presented method.

Example 1: Single Circuit Horizontal Line

The horizontal transmission line distances are shown in Fig. 3. The mean current in the conductors at the moment of

the measurements was 482 A (approximately 60% of its full load value) with a light unbalance between phases (485,

472, and 488 A for the phases A, B, and C, respectively) [3]. Fig. 4(a) shows the calculated magnetic field profile

under the line, where the maximum magnetic field for the existing line is 7.6 μT which is very close to the measured

value 7.36 μT as illustrated in Table 1. Some relative error values in the table exceed 10% due to the corresponding

small field values. So the absolute errors are calculated to show that error is actually small.

Table 1: Calculated and measured magnetic flux density B in μT at y= 1 m under existing unoptimized 132 kV overhead

horizontal line with 485,472, and 488 A in each phase.

Distance (m) from

line center Measured B [3] Calculated B Relative Error % Absolute Error

-50 0.51 0.45 11.76 0.06

-40 0.79 0.71 10.12 0.08

-30 1.36 1.2 11.76 0.16

-20 2.6 2.5 3.84 0.1

-15 3.79 3.8 0.26 0.01

-10 5.25 5.5 4.76 0.25

-5 6.81 7 2.79 0.19

0 7.36 7.6 3.26 0.24

5 7.06 7.2 1.98 0.14

10 5.64 5.8 2.83 0.16

15 3.97 4.09 3.02 0.12

20 2.66 2.78 4.51 0.12

30 1.48 1.4 5.4 0.08

40 0.84 0.84 0 0

50 0.59 0.56 5.08 0.03


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Table 2: Data extracted from simulations and measurements of single circuit horizontal line.

Horizontal line Existing line Optimum with wind &

ice effects

Optimum without wind

& ice effects

Maximum B 7.6 μT 4.37 μT 1.5 μT

ROW width 114 m 72 m 38 m

Maximum E Field 1.9 kV/m 1.18 kV/m 0.48 kV/m

7.8 m 7.8 m

12.12 mphase A phase B phase C

y

x

(a) Existing

3.5 m 3.5 m

12.12 m

phase A phase B phase C

y

x

(b) Optimized line with wind and ice effects

1.1 m 1.1 m

12.12 m


y

x

(c) Optimized line with wind and ice effects neglected

Fig. 3: Example 1, conductor arrangements of 132 kV overhead horizontal line.

The optimum solution when wind and ice effects are ignored gives a maximum value of the magnetic field of 1.5 μT

in Fig. 4(a) which means that the field is decreased by 80% compared with an unoptimized line. When the effects of

wind and ice are taken into account, a 42.5% decrease is obtained. The maximum value of the electric field is 1.9

kV/m for the existing line, but for the new configuration the electric field is decreased by 38 % after optimization

when the effects of wind and ice are considered and by 75% when the effects of wind and ice are neglected as


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-50 0 50-10-20-30-40 10 20 30 400

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Distance from center in meter

Ele

ctri

c fi

eld

in

KV

/m

shown in Fig. 4(b) and Table 2. Also Table 2 shows thatthe ROW width, for a reference value of 0.4 μT, is

significantly reduced after optimization.

Fig. 4: Example 1, Fields under the horizontal line. Solid curve: before optimization, dashed curve: after optimization for

practical standard, and dotted curve: when wind and ice effect are neglected.

Example 2: Single Circuit Triangular Line

The triangular line distances are shown in Fig 5.

11.6 m

phase A

phase C

phase B

1.6 m

2.7 m

3.1 m

6.2 m

12.4 m phase A phase C

phase B3.35 m

3.5m

1.75 m

y y

x x

(line center) (line center)

12.53 m phase A phase C

phase B1.23 m

1.1 m

0.55 m

y

x

Tower location

(line center)

-50 -40 -30 -20 -10 0 10 20 30 40 500

2

4

6

8

Distance from center in meter at y=1 m

Mag

neti

c f

lux

den

sity

in

mic

rote

sla

(a) Magnetic flux density (b) Electric field

(a) Existing

(b) Optimized line with wind and ice effects

(c) Optimized line with wind and ice effects neglected

Fig. 5: Example 2, Conductor arrangement for overhead triangular 132 kV line.


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The current at the moment of measurements was 35.5 A in each phase [3]. The maximum measured magnetic field

for the unoptimized line was 0.31 μT. The computed values are very close to measured values as shown in Table 3.

After PSO is applied, the maximum value of the magnetic field is reduced to 0.08 μT as shown in Fig 6(a) when

wind and ice effects are neglected. When effects of wind and ice are considered, the magnetic field is reduced by

34% (the maximum value is 0.205 μT).

Fig. 6: Example 2, Fields under the triangular line.

As can be seen in Fig 6(b), the maximum value of the electric field is 1.28 kV/m before optimization and the field

profile is not symmetrical since the conductor distribution is not symmetrical. After the redistribution of conductors,

the electric field is reduced to 0.78 kV/m when wind and ice effects are considered (decreased by 39 %), and the

new profile is symmetrical due to the symmetrical configuration of the line. The electric field will be 0.4 kV/m when

the effects of wind and ice are neglected (decreased by 69%). Although ROW width is zero in this example,

optimized solutions have considerably lower values of magnetic fields as shown in Table 4. Table 3: Calculated and measured magnetic flux density in μT at y=1 m under unoptimized overhead triangular line with 35.5 A

in each phase.

Distance (m) from line

center Measured B [3] Calculated B Relative Error %

-40 0.02 0.02 0

-35 0.03 0.03 0

-30 0.04 0.04 0

-25 0.05 0.05 0

-20 0.07 0.07 0

-15 0.1 0.1 0

-10 0.15 0.15 0

-5 0.22 0.22 0

0 0.31 0.31 0

5 0.29 0.29 0

10 0.23 0.23 0

15 0.16 0.15 6.25

20 0.1 0.1 0

25 0.07 0.07 0

30 0.05 0.05 0

35 0.04 0.04 0

40 0.03 0.03 0

-50 0 5010 20 30 40-10-20-30-400

0.05

0.1

0.15

0.2

0.25

0.3

0.35


Mag

neti

c f

lux

den

sity

in

mic

rote

sla

-50 0 50-10-20-30-40 10 20 30 400

0.2

0.4

0.6

0.8

1

1.2

1.4

Distance from center in metre at y=1 m

Ele

ctri

c fi

eld

in

kV

/m



26

Table 4: Data extracted from simulations and measurements of the triangular line.

Triangle line Existing line Optimum with wind &

ice effects


& ice effects



Example 3: Double Circuit with Phases in Parentheses

The double circuit overhead line (one at each side of the tower) with phases in parentheses is shown in Fig. 7.

9.12 m

phase A

phase B

phase C

2.8 m

3.3 m

3 m

Circuit 1 Circuit 2

4 m

4 m

x

line center

(a) Existing line (line is symmetric).

9.78 m

phase A

phase B

phase C

8.2 m

6 m

3.58 mCircuit 1 Circuit 2

4 m

3.94 m

y

x

line center

(b) Optimized line with wind and ice effects considered (line is symmetric)

11.8 m

phase A

phase B

phase C

1.8 m

2 m

1.2 m

Circuit 1 Circuit 2

1.7 m

1.5 m

y

x

line center

(c) Optimized line with wind and ice effects neglected (line is symmetric)

Fig. 7: Example 3, conductor arrangements for an overhead 132 kV line with phases in parentheses, with 91 and 104 A.


27

The arrangement of the phases (A, B, C) is the same in both circuits and the phase current is 91 A for the left circuit,

and 104 A for the right one. The calculated magnetic field as a function of distance from the center of the line is

shown in Fig 8(a). Table 5 shows computed and the measured values of the magnetic field for the installed un-

optimized line, where the maximum computed value is 1.65 μT. After optimization, the maximum magnetic field is

reduced by 59% and is equal to 0.67 μT when the effects of wind and ice are neglected, while the maximum

magnetic field is reduced by 48% as shown by Fig 8(a) when wind and ice effects are considered.

Fig. 8: Example 3, Fields under the line with phases in parentheses.

Table 5: Calculated and measured magnetic flux density in μT at y=1 m under overhead double circuit line with phases in

parentheses without optimization, with 91 and 104 A.

Distance (m) from

line center Measured [3] Calculated Relative Error %

Absolute

Error

-50 0.09 0.1 11.11 0.01

-40 0.15 0.15 0 0

-30 0.26 0.256 1.53 0.004

-20 0.5 0.49 2 0.01

-15 0.74 0.72 2.7 0.02

-10 1.12 1.08 3.57 0.04

-5 1.58 1.517 3.98 0.063

0 1.73 1.65 4.62 0.08

5 1.56 1.58 1.28 0.02

10 1.13 1.13 0 0

15 0.76 0.74 2.63 0.02

20 0.51 0.505 0.98 0.005

30 0.27 0.266 1.48 0.004

40 0.16 0.157 1.87 0.003

50 0.1 0.1 0 0

Table 6: Data extracted from simulations and measurements of the double circuit line with phases in parentheses.

double circuit line Existing line Optimum with wind & ice

effects

Optimum without

wind & ice effects




-50 0 50-10-20-30-40 10 20 30 400

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


Mag

neti

c f

lux

den

sity

in

mic

rote

sla

-50 0 50-10-20-30-40 10 20 30 400

1

2

3

4

5

6

7


Ele

ctr

ic f

ield

in

kV

/m



28

Although the configuration is symmetric, the magnetic field profile is not symmetric due to the different currents in

the circuits. The maximum value of the electric field is 6.3 kV/m. It decreases to 2.8 kV/m when the conductors are

rearranged for the case when wind and ice effects are considered, and to 2.9 kV/m when wind and ice effects are

neglected as shown in Table 6 and Fig. 8(b). ROW width is significantly reduced after optimization when wind and

ice effects are neglected.

Example 4: Double Circuit Horizontal Line

Consider two lines in a horizontal arrangement with two conductors per phase (bundled) as shown in Fig 9.

5 m 5 m

15.6 m

phase A

phase B phase C5 m 5 m

13.2 m

phase A phase B phase C0.5 m

29 m

y

x

line center

24 m

(a) Existing Line

3.8 m 4.1 m

15.55 m


4.1 m 3.8 m

15.55 m

phase Aphase Bphase C

11.9 m

y

x

line center

7.8 m

(b) Optimized line with wind and ice effects considered.

1.6 m 1.1 m

15 m

phase B phase C

1.1 m 1.6 m

15 m

phase Aphase Bphase C

4.9 m

y

x

line center

3.8 m

(c) Optimized line with wind and ice effects neglected.

Fig. 9: Example 4, conductor arrangement of the overhead double circuit horizontal 132 kV line with 246 and 226 A.


29

At the moment of measuring the magnetic field, mean currents of 246 A, 226 A circulated through the left line

and right line conductors, respectively [3]. The magnetic field of the installed line has a maximum value 2.15 μT

as shown in Fig 10(a). Table 7 compares calculated with measured magnetic field values at different distances

from the center. Note that the sequences of the new lines are A,B,C and C,B,A as shown in Fig 9. Fig. 10(a)

shows that the magnetic field is 0.47 μT (reduced by 78%) when wind and ice are neglected, and 1.85 μT

(reduced by 14%) when wind and ice are considered. The bundle contributed a very small value of around 10-12

μT when the field value is around 1 μT. Therefore modeling the bundle did not affect the magnetic field value.

The nonsymmetrical conductors caused a nonsymmetrical field distribution under the unoptimized line as shown

in Fig 10. The maximum value of the electric field of the existing line is 3.88 kV/m, while it is 4.3 kV/m for the

optimized line when wind and ice effects are considered as shown in Table 8. The increase in the electric field

may be attributed to the increase of the coupling capacitance. But when wind and ice are neglected the field

decreases to 1.43 kV/m as shown in Fig 10(b). ROW width is significantly reduced after optimization.

Fig. 10: Example 4, calculated magnetic flux density under the double-circuit horizontal line.

Table 7: Calculated and measured magnetic flux density in μT at y=1 m under double circuit horizontal line without optimization,

with 246 and 226 A.

Distance (m) from

line center Measured B [3] Calculated B Relative Error % Absolute Error

-60 0.45 0.448 0.44 0.002

-55 0.56 0.57 1.78 0.01

-50 0.75 0.75 0 0

-45 0.99 0.99 0 0

-40 1.29 1.3 0.77 0.01

-35 1.57 1.57 0 0

-30 1.7 1.63 4.11 0.07

-25 1.44 1.38 4.16 0.06

-20 0.98 0.91 7.14 0.07

-15 0.51 0.444 12.94 0.066

-10 0.47 0.453 3.61 0.017

-5 1.04 1.03 0.96 0.01

0 1.72 1.75 1.74 0.03

5 2.1 2.15 2.38 0.05

10 2.04 2.1 2.94 0.06

15 1.63 1.63 0 0

20 1.2 1.176 2 0.024

25 0.85 0.84 1.17 0.01

30 0.63 0.62 1.58 0.01

35 0.48 0.475 1.041 0.005

40 0.38 0.373 1.84 0.007

-100 -50 0 50 100-25-75 25 750

0.5

1

1.5

2

2.5


Mag

neti

c f

lux

den

sity

in

mic

rote

sla

-50 0 50-10-20-30-40 10 20 30 400

0.5

1

1.5

2

2.5

3

3.5

4

4.5


Ele

ctr

ic f

ield

kV

/m



30

Table 8: Data extracted from simulations and measurements of double circuit horizontal line.

Vertical double circuit

line Existing line

Optimum with wind and

ice effects


and ice effects




7. DISCUSSION

The authors have conducted extensive simulations on various configurations. Due to page limitations, only the

above examples are presented. General trends extracted from these simulations are listed below, although it does not

necessarily apply to all other cases.

1) For single circuit lines, when the distances between conductors are decreased the fields are also decreased due

to the cancellation between the field components. The other factor that affects the magnetic field is the current

that flows in the conductors; it is directly proportional to the magnetic field where higher current produces a

higher magnetic field. Therefore PSO gives the solution at a minimum distance between phases. This means

that for single circuit lines, it is not necessary to use PSO to get the minimum fields but we need only to choose

minimum distances between phases in the design of a transmission line.

2) For double circuit lines, the results showed that decreasing the distance does not necessarily minimize the

magnetic field. Therefore optimization is necessary to minimize the field.

3) For both single and double circuit lines, the height of the overhead high voltage line is inversely proportional to

the magnetic and electric field values. Consequently, increasing the line height decreases the field on the

ground. The effect of bundling can be neglected in the magnetic field calculations where the contribution of

bundling is about 10-12

μT when the field value is around 1 μT.

8. CONCLUSIONS

Particle swarm optimization is successfully applied to reduce the magnetic field under 132 kV overhead high

voltage transmission lines. The ROW width can be decreased for both single and double circuit lines and decreases

more when the wind and ice effects are neglected. The magnetic field under overhead transmission line is a strong

function of distance like any other source and it is also a function of electric current, i.e., high current implies high

magnetic field. The algorithm presented in this paper can be used for higher voltages, such as 400 kV, by adjusting

the minimum clearance between phases in addition to the transmission line height.

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