The Nonlinear Particle Swarm Optimization Algorithm Engineering Essay

Using atom drove optimisation, the influence of the geometrical constellation of music directors is studied in order to minimise the magnetic field near both individual circuit and dual circuit high electromotive force overhead power transmittal lines. New agreements of high electromotive force “ green lines ” are proposed. The consequences indicate that the magnetic field can be reduced up to 48 % under the influence of air current and ice, and 80 % , pretermiting them. Consequently, the necessary ROW ( Right Of Way ) breadth, so that the magnetic field outside ROW does non transcend an illustration mention value of 0.4 I?T, can be reduced by up to 48 % if air current and ice are taken into history and by up to 90 % if non, for the same scope of line highs. Complex and existent image theories were implemented to happen the magnetic and electric Fieldss, severally, near the transmittal line. The electric field is evaluated and it is found that it is acceptable. Besides, consequences showed that the bundling affects the electric field merely.

Keywords: High electromotive force lines, PSO, Reduce Fieldss, Image theory, EMC.

1. Introduction

The 50/60 Hz electric power transmittal lines can interfere with nearby electrical and electronic equipment [ 1 ] . In add-on, concern has arisen sing the controversial issue of possible inauspicious effects on human wellness associated with exposure to electromagnetic Fieldss from 50/60 Hz electric power transmittal, sub-transmission and/or distribution lines, and other beginnings including contraptions. Because of this, many surveies have focused on the Fieldss emanating from electrical devices and transmittal lines [ 2 ] – [ 6 ] . The greater the field around the transmittal lines, the greater the current induced in the human organic structure [ 3 ] . The International Agency for Research on Cancer ( IARC ) classified power frequence electromagnetic Fieldss as “ perchance carcinogenic to worlds ” , based on a reasonably consistent statistical association between duplicating the hazard of childhood leukaemia and highly low frequence ( ELF ) magnetic field exposure above 0.4 I?T [ 4 ] . Thus, in some instances there is a demand to cut down the Fieldss, particularly the magnetic field which can non be easy shielded [ 7 ] , when the transmittal line interferes with the next electrical and electronic equipment. Besides, as most edifice stuffs are nonmagnetic, magnetic Fieldss of high electromotive force lines are non attenuated by common stuffs such as edifice walls. In contrast to that, constructing walls can screen electric Fieldss to some extent.

The shielding of power lines was investigated in order to cut down the emitted magnetic flux denseness to a few microteslas which is the bound for new installings in some states [ 7 ] . Unlike wireless frequence Fieldss which can be shielded with comparatively thin movies, ELF magnetic Fieldss require really thick, high permeableness stuffs for effectual shielding [ 8 ] . Comparing different constellations revealed that the magnetic field at land is low for the equilateral delta constellation and high for the horizontal constellation [ 9 ] . Another manner to cut down magnetic field is “ active shielding ” , viz. , by a individually energized circuit introduced near an bing power line [ 10 ] . Such systems require dynamic feedback systems to maintain the stage and magnitude of the screening circuit current at optimal intervention with the power circuit, and can be dearly-won. Belowground power lines with multiple music directors can be configured to minimise the magnetic field at land degree [ 10 ] . Since the Earth shields merely the electric field, belowground lines can sometimes breathe greater magnetic Fieldss than tantamount overhead lines because the burial deepness may be merely four to six pess as opposed to the 20 to 30 pess clearance of an overhead line. Recently, TenneT, the operator of electric power cyberspace in Netherlands, has provided new designs of high-potential transmittal lines in which music directors are suspended closer together than in traditional high-voltage lines, therefore cut downing the strength of the magnetic field generated along the line [ 11 ] . This TenneT undertaking was in response to the precautional policy adopted by Dutch Ministry of Housing, Spatial Planning and the Environment which is aimed at minimising unneeded long-run human exposure to magnetic Fieldss generated by high-potential lines. The ministry defined a bound value of 0.4 meitnerium for exposure to magnetic Fieldss in 2005 as portion of rigorous guidelines for the extension of high-voltage lines and the building of new edifices near bing high-voltage lines. The article [ 12 ] considers a specific agreement of the stage music directors along with cut downing dimensions ( compression ) of the line to cut down the magnetic and electric Fieldss.

We choose, in this paper, to cut down the magnetic field of an arbitrary line by set uping its stage music directors, without presuming certain templets in progress. To the writers ‘ cognition, there is no published systematic method to happen the optimal agreement of music directors for any high electromotive force transmittal line in order to bring forth minimal magnetic field. For all the instances investigated, we found that minimising the magnetic field is automatically associated with cut downing the electric field. To accomplish this end, we used atom drove optimisation ( PSO ) which is a nonlinear constrained method that can take to optimal solutions without cognizing the gradient of the job beforehand [ 13 ] , [ 14 ] . Our solution represents a planar cross subdivision of a transmittal line, and therefore the line tallness is considered at the point along the line where the cross-section is taken.

Minimizing the magnetic field by set uping the transmittal line music directors has the advantage of minimising the magnetic field without adding any equipment or shield to the line. Extensive simulations were performed ( non shown in this paper due to paper size restrictions ) for assorted constellations, where it was noticed in most instances that optimized lines ( with minimal magnetic Fieldss ) have more compact agreements ( shorter music director spacings ) compared with unoptimized agreements. This indicates that the cost of put ining a high electromotive force line with minimal field is likely 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 drove is used to happen the optimum agreement of music directors that would bring forth minimal magnetic field near high electromotive force overhead power transmittal lines. Toward that terminal, a drove of 49 atoms ( different agreements of line music directors ) is used to minimise the field. At each loop, the fittingness map ( magnetic field due to all music directors ) is evaluated for each atom in the drove. The best fittingness values for each atom every bit good as for the whole drove are stored. Droves continue to travel ( iterate ) until the targeted best value is obtained. The transmittal line constellation ( music directors agreement ) associated with this targeted best value is the optimal solution. Matlab computing machine plans were written and executed to cipher and minimise the magnetic field for overhead transmittal lines. Besides, the electric field is computed where we found that the electric field is besides reduced.

2. Particle Swarm Optimization ( PSO )

The PSO is a robust stochastic nonlinear evolutionary calculation technique based on the motion and intelligence of droves [ 15 ] . In comparing with other stochastic evolutionary algorithms like familial algorithms, PSO has fewer complicated operations, fewer specifying parametric quantities, and by and large fewer lines of codification [ 16 ] . PSO depends on the societal interaction between independent agents, here called atoms, during their hunt for the optimal solution utilizing the construct of fittingness. The fittingness defines how good the place vector of each atom satisfies the demands of the optimisation job.

Note that most edifices represent good music directors at 50/60 Hz frequence, and therefore can screen the electric field. On the contrary, these edifices are about crystalline to magnetic Fieldss [ 3 ] . Based on that, merely the magnetic field is minimized here, i.e. , the magnetic field value is the fittingness map. In fact, when we included both the magnetic and electric Fieldss in the fittingness map, we obtained basically similar consequences to that when merely the magnetic field is considered.

A population ( drove ) size of about 30 is suited for many jobs. Although 36 atoms worked decently in this paper, 49 were chosen since the executing clip of the computing machine plan is a fraction of a 2nd. The specific values of 36 and 49 are dictated by the manner we wrote the codification since nested do-loops with equal figure of stairss were used for incrementing the ten and y places. Each atom remembers its personal best place found ( called its local best ) and besides knows the best place found by the drove ( called the planetary best ) . The ten and y constituents of the speed and the place represented by the ten, y co-ordinates, for each atom m, are updated by the undermentioned equations [ 13 ] :

( 1 )

where superiors t and t-1 are clip indices of the current and the old loops, Un1 and Un2 are two different uniformly distributed random Numberss in the interval { 0,1 } , w is the “ inertial weight ” in the scope { 0,1 } , and pmn, gn are the personal and planetary best places ( the inferiors x, y refer to the ten and y constituents ) , severally. The parametric quantities c1 and c2 are scaling factors of local and planetary bests ; a value of 2 is a good pick for both parametric quantities [ 17 ] . The inferior m is the atom figure in the drove while n indicates the parametric quantity to be optimized, and the time-step I”t is normally chosen to be one. In this paper, reflecting walls are used as boundary conditions ; when a atom reaches the boundary, it is reflected back as shown in Fig. 1.

Since PSO is a stochastic optimisation method, each test would give a slightly different solution. Our consequences represent the best obtained solutions for each constellation analyzed. We have reported the best solution for each instance after doing at least 100 tests utilizing different random seeds.

3. Solution Algorithm

The undermentioned algorithm minimizes the magnetic field under overhead transmittal lines.

Stipulate the restraints of the job: minimal spacings between music directors, and bounds of the part in which the atoms will seek for suited agreement of music directors ( see Fig. 1 ) .

Distribute the atoms ( different agreements of music directors ) in the selected part, stipulate a clip measure for atom motion ( here integrity ) , initialise the population with a random speed ( V ) vector ( here zero initial speed ) , and initialise the stop standard with a value much smaller than 0.40 I?T ( here 0.01 I?T ) . Stipulate the maximal figure of loops that should non be exceeded.

Measure the fittingness map ( here the magnetic field value ) for each atom.

If the magnetic field value & lt ; the personal best value, so replace the personal best value by the new magnetic field value.

If the magnetic field value & lt ; the planetary best value, so replace the planetary best value by the new magnetic field value.

If the fittingness is a‰¤ the stop standard or maximal figure of loops is reached, so halt ; a solution is found. Otherwise, update place and speed of atoms harmonizing to ( 1 ) ( c1=c2=2 and w=0.7 ) , and travel to step 3.

4. Magnetic Field Model

The frequence of the power system ( 50/60 Hz ) is little plenty that the magnetic and electric Fieldss in air can be considered independent. The magnetic Fieldss associated with electric power transmittal lines are readily predicted utilizing mathematical equations given line burden informations and wire constellation [ 18 ] . The magnetic field of overhead high electromotive force lines can be found by superposing the single parts of the stage music directors taking into history the Earth return currents. The geometry considered to measure the magnetic field at ( xj, yj ) due to the stage music director at ( xi, yi ) taking into history the complex image at ( xi ‘ , yi ‘ ) is illustrated in Fig. 2. To take into history return currents in the look of the magnetic field it is necessary to get down from the common electric resistance presented by [ 19 ] . The magnetic field Hj ( fitness map in this paper ) at the point ( xj, yj ) is obtained by sing the part of all N stage music directors:

( 2 )

where

where I? , Iµ , I? are conduction, permittivity, permeableness of the Earth, severally, and I‰ is angular frequence. Ii is the stage current, and ux, uy are the unit vectors along the ten and Y axes. Note that in ( 2 ) , the first term is the part of the line music directors whereas the 2nd term is the part of the line images. The magnetic field denseness is B= I?H, and its magnitude of B0 is:

( 3 )

where B0x and B0y are the amplitudes in the ten and y waies.

5. Electric Field Model

For the electric field calculation, the Earth consequence is represented by image charges located below the music directors at a deepness equal to the music director height [ 20 ] . The stage electrical capacity C in AµF/km ( sing three stage music directors with phase-to-phase distances of D12, D13, D23 ) is calculated as [ 21 ] :

( 4 )

where R is the music director radius and GMD is the geometric mean distance ( tantamount music director spacing ) . Then, the electric field at ( x, y ) due to a line music director at ( x0, H ) is:

( 5 )

where V is the stage electromotive force. Note that in ( 5 ) , the first term is the part of a line music director whereas the 2nd term is the part of the line image. The entire electric field is the superposition of the Fieldss from all music directors.

6. Consequences

This survey is focused on minimising the magnetic field of individual circuit and dual circuit constellations of 132 kilovolts Condor type transmittal lines. However, the algorithm in this paper can cover with other transmittal lines electromotive forces. In order to verify the cogency of equations and computing machine plans, the computed magnetic Fieldss are compared with measured Fieldss at a tallness of 1 m above the surface of the land. For each instance analyzed, two instances of IEC-71 criterions were applied, foremost, disregarding the effects of air current and ice where lower limit spacing between stages is 1.1 m, and 2nd, the practical instance which considers the effects of air current and ice where lower limit spacing between stages is 3.5 m. The optimized solutions offer reduced ROW breadths. The highs of the optimized and unoptimized lines were kept about the same, since increasing line tallness would automatically diminish Fieldss near the surface of the Earth. The executing clip of each fake instance is less than one second on the computing machine used ( Genuine Intel ( R ) CPU T2300 @ 1.66 GHz, 1.49 GB RAM ) and the figure of loops range is 20 to 200. The electric field is besides computed for the transmittal line before and after minimising the magnetic field. The undermentioned illustrations are intended to demo the cogency and truth of the presented method.

Example 1: Single Circuit Horizontal Line

The horizontal line distances are shown in Fig. 3. The average current in the music directors at the minute of the measurings was 482 A ( about 60 % of its full burden value ) with a light imbalance between stages ( 485, 472, and 488 A for the stages A, B, and C, severally ) [ 3 ] . Fig. 4 ( a ) shows the deliberate magnetic field profile under the line, where the maximal magnetic field for the bing line is 7.6 I?T which is really near to the measured value 7.36 I?T as illustrated in Table 1. Some comparative mistake values in the tabular array exceed 10 % due to the matching little field values. So the absolute mistakes are calculated to demo that mistake is really little.

Table 1: Calculated and measured magnetic flux denseness B in I?T at y= 1 m under bing unoptimized 132 kilovolts overhead horizontal line with 485,472, and 488 A in each stage.

Distance ( m ) from line centre

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

Table 2: Datas extracted from simulations and measurings of individual circuit horizontal line.

Horizontal line

Existing line

Optimum with air current & A ; ice effects

Optimum without air current & A ; ice effects

Maximum B

7.6 I?T

4.37 I?T

1.5 I?T

ROW breadth

114 m

72 m

38 m

Maximal E Field

1.9 kV/m

1.18 kV/m

0.48 kV/m

The optimal solution when air current and ice effects are ignored gives a maximal value of the magnetic field of 1.5 I?T in Fig. 4 ( a ) which means that the field is decreased by 80 % compared with an unoptimized line. When the effects of air current and ice are taken into history, a 42.5 % lessening is obtained. The maximal value of the electric field is 1.9 kV/m for the bing line, but for the new constellation the electric field is decreased by 38 % after optimisation when the effects of air current and ice are considered and by 75 % when the effects of air current and ice are neglected as shown in Fig. 4 ( B ) and Table 2. Besides Table 2 shows that the ROW breadth, for a mention value of 0.4 I?T, is significantly reduced after optimisation.

Example 2: Single Circuit Triangular Line

The triangular line distances are shown in Fig 5. The current at the minute of measurings was 35.5 A in each stage [ 3 ] . The maximal mensural magnetic field for the unoptimized line was 0.31 I?T. The computed values are really near to mensurate values as shown in Table 3. After PSO is applied, the maximal value of the magnetic field is reduced to 0.08 I?T as shown in Fig 6 ( a ) when air current and ice effects are neglected. When effects of air current and ice are considered, the magnetic field is reduced by 34 % ( the upper limit value is 0.205 I?T ) . As can be seen in Fig 6 ( B ) , the maximal value of the electric field is 1.28 kV/m before optimisation and the field profile is non symmetrical since the music director distribution is non symmetrical. After the redistribution of music directors, the electric field is reduced to 0.78 kV/m when air current and ice effects are considered ( decreased by 39 % ) , and the new profile is symmetrical due to the symmetrical constellation of the line. The electric field will be 0.4 kV/m when the effects of air current and ice are neglected ( decreased by 69 % ) . Although ROW breadth is zero in this illustration, optimized solutions have well lower values of magnetic Fieldss as shown in Table 4.

Table 3: Calculated and measured magnetic flux denseness in I?T at y=1 m under unoptimized overhead triangular line with 35.5 A in each stage.

Distance ( m ) from line centre

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

Table 4: Datas extracted from simulations and measurings of the triangular line.

Triangle line

Existing line

Optimum with air current & A ; ice effects

Optimum without air current & A ; ice effects

Maximum B

0.31 I?T

0.205 I?T

0.07 I?T

Maximal E Field

1.28 kV/m

0.78 kV/m

0.4 kV/m

Example 3: Double Circuit with Phases in Parenthesiss

The dual circuit overhead line ( one at each side of the tower ) with stages in parentheses is shown in Fig. 7. The agreement of the stages ( A, B, C ) is the same in both circuits and the stage current is 91 Angstrom for the left circuit, and 104 A for the right 1. The deliberate magnetic field as a map of distance from the centre of the line is shown in Fig 8 ( a ) . Table 5 shows computed and the mensural values of the magnetic field for the installed un-optimized line, where the upper limit computed value is 1.65 I?T. After optimisation, the maximal magnetic field is reduced by 59 % and is equal to 0.67 I?T when the effects of air current and ice are neglected, while the maximal magnetic field is reduced by 48 % as shown by Fig 8 ( a ) when air current and ice effects are considered.

Table 5: Calculated and measured magnetic flux denseness in I?T at y=1 m under overhead dual circuit line with stages in parentheses without optimisation, with 91 and 104 A.

Distance ( m ) from line centre

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: Datas extracted from simulations and measurings of the dual circuit line with stages in parentheses.

double circuit line

Existing line

Optimum with air current & A ; ice effects

Optimum without air current & A ; ice effects

Maximum B

1.65 I?T

0.86 I?T

0.67 I?T

ROW breadth

46 m

46 m

22 m

Maximal E Field

6.3 kV/m

2.8 kV/m

2.9 kV/m

Although the constellation is symmetric, the magnetic field profile is non symmetric due to the different currents in the circuits. The maximal value of the electric field is 6.3 kV/m. It decreases to 2.8 kV/m when the music directors are rearranged for the instance when air current and ice effects are considered, and to 2.9 kV/m when air current and ice effects are neglected as shown in Table 6 and Fig. 8 ( B ) . ROW breadth is significantly reduced after optimisation when air current and ice effects are neglected.

Example 4: Double Circuit Horizontal Line

See two lines in a horizontal agreement with two music directors per stage ( bundled ) as shown in Fig 9. At the minute of mensurating the magnetic field, average currents of 246 A, 226 Angstrom circulated through the left line and right line music directors, severally [ 3 ] . The magnetic field of the installed line has a maximal value 2.15 I?T as shown in Fig 10 ( a ) . Table 7 comparisons calculated with mensural magnetic field values at different distances from the centre. 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 I?T ( reduced by 78 % ) when air current and ice are neglected, and 1.85 I?T ( reduced by 14 % ) when air current and ice are considered. The package contributed a really little value of around 10-12 I?T when the field value is around 1 I?T. Therefore patterning the package did non impact the magnetic field value. The nonsymmetrical music directors caused a nonsymmetrical field distribution under the unoptimized line as shown in Fig 10. The maximal value of the electric field of the bing line is 3.88 kV/m, and 4.3 kV/m for optimized line when air current and ice effects are considered as shown in Table 8. The addition in the electric field may be attributed to the addition of the yoke electrical capacity. But when air current and ice are neglected the field decreases to 1.43 kV/m as shown in Fig 10 ( B ) . ROW breadth is significantly reduced after optimisation.

Table 7: Calculated and measured magnetic flux denseness in I?T at y=1 m under dual circuit horizontal line without optimisation, with 246 and 226 A.

Distance ( m ) from line centre

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

Table 8: Datas extracted from simulations and measurings of dual circuit horizontal line.

Vertical dual circuit line

Existing line

Optimum with air current and ice effects

Optimum without air current and ice effects

Maximum B

2.15 I?T

1.85 I?T

0.47 I?T

ROW breadth

101 m

53 m

10 m

Maximal E Field

3.88 kV/m

4.3 kV/m

1.43 kV/m

7. Discussion:

The writers have conducted extended simulations on assorted constellations. Due to page restrictions, merely the above illustrations are presented. General trends extracted from these simulations are listed below, although it does non needfully use to all other instances.

For individual circuit lines, when the distances between music directors are decreased the Fieldss are besides decreased due to the cancellation between the field constituents. The other factor that affects the magnetic field is the current that flows in the music directors ; it is straight relative to the magnetic field where higher current produces a higher magnetic field. Therefore PSO gives the solution at a minimal distance between stages. This means that for individual circuit lines, it is non necessary to utilize PSO to acquire the minimal Fieldss but we need merely to take minimal distances between stages in the design of a transmittal line.

For dual circuit lines, the consequences showed that diminishing the distance does non needfully minimise the magnetic field. Therefore PSO or any other optimisation method is necessary to minimise the field.

For both individual and dual circuit lines, the tallness of the overhead high electromotive force line is reciprocally relative to the magnetic and electric field values. Consequently, increasing the line tallness decreases the field on the land. The consequence of bundling can be neglected in the magnetic field computations where the part of bundling is about 10-12 I?T when the field value is around 1 I?T.

8. Decisions

Particle drove optimisation is successfully applied to cut down the magnetic field under 132 kilovolts overhead high electromotive force transmittal lines. The ROW breadth can be decreased for both individual and dual circuit lines and decreases more when the air current and ice effects are neglected. The magnetic field under overhead transmittal line is a strong map of distance like any other beginning and it is besides a map of electric current, i.e. , high current implies high magnetic field. The algorithm presented in this paper can be used for higher electromotive forces, such as 400 kilovolts, by seting the minimal clearance between stages in add-on to the transmittal line tallness.