Numerical calculation method for magnetic fields in the …1217023/...UPTEC E 18009 Examensarbete 30 hp Juni 2018 Numerical calculation method for magnetic fields in the vicinity of

UPTEC E 18009

Examensarbete 30 hpJuni 2018

Numerical calculation method for magnetic fields in the vicinity of current-carrying conductors

Gustav Gärskog

Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

Numerical calculation method for magnetic fields inthe vicinity of current-carrying conductors

Gustav Gärskog

This thesis aims to develop a calculation method to determine the magnetic field magnitudes in the vicinity of power lines, i.e. both buried cables and overhead lines. This is done through the numerical use of Biot Savart's law where the conductors are approximated by a series of straight segment elements that each contribute to the overall field strength at the field point. The method is compared to two real cases and to the exact integral solution. Also, a review of some of the research material regarding electromagnetic fields from power lines and claims of adverse health effects due to these fields is conducted.Results show that the numerical error is dependent on the segmentation degree of the conductors and the mathematical model is inaccurate close to the conductor. The calculations show slightly higher field magnitudes than the previous survey done by WSP (Williams Sale Partnership) far away from the source and slightly lower at the center conductor. This may be due to the excluded induction in the shield wires and differences in actual conductor coordinates.

ISSN: 1654-7616, UPTEC E** ***Examinator: Mikael BergkvistÄmnesgranskare: Marcus BergHandledare: Ivan Barck

1

Sammanfattning

Magnetiska fälts påverkan på människans hälsa har länge debatterats. Även en stormängd forskning har genomförts på området. I synnerhet har ett samband mellan barn-leukemi och lågfrekventa magnetiska fält visat sig vara troligt och varit flitigt debatterat.Även andra former av cancer och neurologiska sjukdomar har visat vissa samband medmagnetiska fält. Inga tydliga bevis har dock fastställts.

Då inget lagstadgat gränsvärde idag finns tillämpas försiktighetsprinsipen, vilket in-nebär att ägaren av nätkoncetion ska förebygga eventuell risk för människors hälsa inomrimliga ekonomiska ramar. Praxis har, i väntan på mer forskningsunderlag, under senareår varit ett riktvärde på 0,4 µT för platser där människor bor och vistas varaktigt. Delarav forskningsmatrealet rörande magnetiska fälts eventuella hälsoeffekter har studerats idenna avhandling i syfte att ge en bakgrund och bedöma relevansen av det riktvärdesom idag används för kraftledningar och kraftkablar.

Denna avhandlings huvudsyfte är att utveckla ett beräkningsprogram för att bestämmamagnetfältets storheter i närmiljön till kraftledningar, d.v.s. både kablar och luftled-ningar. Detta görs med hjälp av Biot Savarts lag där ledarna approximeras av en serieraka segmentelement som var för sig bidrar till den totala fältstyrkan i fältpunkten. Sum-man av fältbidragen blir således den totala fältstyrkan i en given fältpunkt. Eftersom detär långtidsexponering som är av vikt vid dessa beräkning används årsmedelströmmar.Metoden jämförs med två reella fall och en något förenklad integrallösning för att ge enuppfattning om metodens exakthet. Vissa delar av problemet har gjorts avkall på. Induk-tionen i neutralledarna (topplinorna) tas inte med i beräkningsmodellen i detta skede,vilket den har gjorts i referens materialet. Avvikelser har observerats i de beräknademagnetfältsmagnituderna framförallt nära mittfasen hos de luftledningssystem somhar jämförts, vilket kan vara en följd av induktionsbidraget. Svårigheter att återskapade exakta beräkningsförutsättningar som använts vid WSP:s beräkningar kan ocksåbidra till eventuella skillnader i resultaten. Resultaten visar att det numeriska felet ärberoende av ledningarnas segmenteringsgrad och att den matematiska modellen ärinexakt nära ledaren. Beräkningarna har en något högre fältmagnitud än den tidigareundersökningen som gjorts av WSP långt ifrån källan och något lägre vid mittfasennära källan. Detta kan bero på avsaknaden av induktionsbidraget från topplinorna ochskillnader i ledarkoordinater. Abstract

Uppsala University Gustav Gärskog

NUMERICAL CALCULATION METHOD FOR MAGNETIC FIELDS IN THEVICINITY OF CURRENT-CARRYING CONDUCTORS

Dissertation in partial fulfillment of the requirements for the degree of:

Master of Science in Engineering, 300 creditsSpecialization; Electrical Engineering

Uppsala UniversityDepartment of Electricity Science

[Gustav Gärskog]

[07 05 2018]

NUMERICAL CALCULATION METHOD FOR MAGNETIC FIELDS IN VICINITYOF CURRENT-CARRYING CONDUCTORS

Dissertation in partial fulfillment of the requirements for the degree of:

Master of Science in Engineering, 300 creditsSpecialization; Electrical Engineering

Uppsala UniversityDepartment of Electricity Science

Approved by

Supervisor, Ivan Barck, WSP

Subject reviewer, Marcus Berg

Examiner, Mikael Bergkvist

[07-05-2018]

I

Acknowledgements

I would like to give a sincere thank you to Ivan Barck who has helped me tremendouslythrough this process. Also a big thanks to the team at Williams Sale Partnership (WSP)elkraft that has provided a home away from home during this experience. I would alsolike to direct a big thank you to Marcus Berg that has helped me greatly to elevate thisthesis to its current state.


Table of Contents II

Table of Contents

Acknowledgements I

List of Tables IV

List of Figures V

List of Acronyms VII

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Health effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Physical interaction between biological tissue and fields frompower lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.2 Physical effects of electric fields in the body . . . . . . . . . . . . . 41.2.3 Basic interaction of magnetic fields in the body . . . . . . . . . . . 41.2.4 Neurodegenerative disorders . . . . . . . . . . . . . . . . . . . . . 51.2.5 Cardiovascular disorders . . . . . . . . . . . . . . . . . . . . . . . 51.2.6 Immune system and hematology . . . . . . . . . . . . . . . . . . . 61.2.7 Reproduction and development . . . . . . . . . . . . . . . . . . . 61.2.8 Cancer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Shielding of magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.1 Integral solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.2 Bladsjön . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.3 Grundfors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Underground cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Methodology 122.1 Biot Savart’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Simplifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Calculation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.1 Yearly current average . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.2 Numerical calculation method . . . . . . . . . . . . . . . . . . . . 14

3 Results 153.1 Special case evaluation : Straight conductor . . . . . . . . . . . . . . . . . 153.2 Special case evaluation: Hyperbolic conductor . . . . . . . . . . . . . . . 19


Table of Contents III

3.3 Bladsjön . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4 Grundfors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Discussion and analysis 314.1 Medical effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Calculation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5 Conclusion 33

Literature 35

A Appendix 37A.1 [Magnetic B-field from three phase hyperbolic conductors] . . . . . . . . 38A.2 Bladsjön . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39A.3 Grundfors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40


List of Tables IV

List of Tables

Table 1.1: Magnetic fields produced by some everyday household appliances atdifferent distances from the source [3] . . . . . . . . . . . . . . . . . . . 2

Table 3.1: Comparison of the exact integral and the numeric solution method fordifferent segment lengths at field point (50,0,0). . . . . . . . . . . . . . 16

Table 3.2: Comparison of the exact integral and the approximate solution fordifferent segment lengths at field point (20,0,0). . . . . . . . . . . . . . 17

Table 3.3: Comparison between the accuracy at the closest, optimal and 100 mhorizontal distance from the conductor . . . . . . . . . . . . . . . . . . 20

Table 3.4: Numerical field calculation for houses with conductor segment lengthsat 5 m and 1 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23


List of Figures V

List of Figures

Figure 1.1: parallel single-circuit power lines, note the hyperbolic line sag be-tween towers. (Magnus Manske|Cc-by-3.0.) . . . . . . . . . . . . . . 1

Figure 1.2: The magnetic field strength at 1.5 m over ground from a three-phasecircuit with three different currents. The distance between the phasesis 10 m. The currents are chosen to be representative of the powerlevels in the rest of this thesis. . . . . . . . . . . . . . . . . . . . . . . 3

Figure 1.3: Suggested changes for the site Bladsjön outside of Åsbro and itsexisting 400 kV power line. Residential homes are marked with greenoutlining. The yellow lines represents the existing power lines, (NW-SE, 400 kV, 450 A), (N-S, 130 kV, -200 A). Alternative 3B is light blue,3C is dark blue, 3D is green and 4A is purple. . . . . . . . . . . . . . 9

Figure 1.4: Overhead image of the site at Grundfors. The two green circlesindicate houses of interest, magenta lines represent the altered sectionof the power lines and the red outline the new location of the switchyard. The east green line is the cross section compared using thereference material and the calculation method. . . . . . . . . . . . . 10

Figure 3.1: Plot showing how the error of the numerical result depends on wiresegmentation and how odd and even numbers of segments affect therelative error. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Figure 3.2: 24 segments: the ratio between calculated numerical result and exactsolution vs distance from source. The total length of the conductor is500 m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18




Figure 3.6: The ratio between 5 m segment length and the reference 0.1 m . . . . 20Figure 3.7: The ratio between 20 m segment length and the reference 0.1 m . . . 20


List of Figures VI

Figure 3.8: Vector field plot at time t = 0 for the three-phase circuit for twodimensions. The distance between adjacent conductors is 10 m. Theinstantaneous currents are, from left to right, 400 A (phase 1), -200 A(phase 2), -200 A (phase 3). . . . . . . . . . . . . . . . . . . . . . . . . 21

Figure 3.9: 2D representation of the site Bladsjön with relevant houses. The redline corresponds to the cross section in figures 3.10 and 3.11. Theblack rectangle represents the calculation area used in figure 3.12 . . 22

Figure 3.10: Graph of the magnetic field near house 1:71 in WSP’s report forBladsjön. The section is indicated in Figure 3.8. The field calculationheight is 1.5 m above ground level at 1:71. The origin indicates thecenter phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Figure 3.11: Calculation of the B-field for the same section as in Figure 3.10, usingthe algorithm developed by the author. . . . . . . . . . . . . . . . . . 24

Figure 3.12: Graph of result for site Bladsjön for house 1:71 altitude above sea level 24Figure 3.13: Contour graph of three B-field magnitudes for site Bladsjön at house

1:71 height above sea level. The house 1:71 is indicated by the bluecircle. Contour levels are given in µT. . . . . . . . . . . . . . . . . . . 25

Figure 3.14: 2D representation of the site Grundfors with relevant houses. Theyellow line represents the cross section in figure 3.15 and 3.16. Theblack rectangle represents the area used in figure 3.18. . . . . . . . . 26

Figure 3.15: B-field magnitude as a function of the horizontal distance from thecenter phase of line UL1 S1-3. The curve corresponds to the crosssection towards the Eastern house in figure 3.14. Reference curvefrom WSP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Figure 3.16: Cross section field strength plot for the same cross section as in figure3.10, calculated result . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Figure 3.17: Contour plot for the Grundfors site. The east house is too far outsidethe relevant area and hence not included. The western house is shownas a blue circle. Contour levels are given in µT. . . . . . . . . . . . . 29

Figure 3.18: Calculated B-field magnitude for the Grundfors site. Peaks corre-spond to points, where the conductors are closest to ground. The fieldis calculated at 1.5 m above the ground plane. Compare this plot withfigure 3.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30


List of Acronyms VII

List of Acronyms

ELF Extremely Low Frequency (0 Hz− 100 kHz)SVK Svenska KraftnätWSP Williams Sale PartnershipALS Amyotrophic Lateral SclerosisMS Multiple SclerosisAD Alzheimer’s DiseaseWHO World Health OrganizationCV CardiovascularHRV Heart Rate VariabilityAMI Acute Myocardial InfarctionEI Energimarknadsinspektionen (Swedish energy market inspectorate)FEM Finite Element Method


Introduction 1

1 Introduction

1.1 Background

In modern society, energy demand keeps increasing and shows no sign in declining [1].To supply this demand power systems are essential to transport the converted energyfrom the sources, which often are located at remote locations, far away from the mainconsumers. As a direct consequence, magnetic and electric fields are induced around thepower lines. The magnetic field strength is dependent on the currents flowing throughthe conductors at any given time. As the Swedish power grid operates at 50 Hz, theresulting electromagnetic field has the same frequency.

Figure 1.1: parallel single-circuit power lines, note the hyperbolic line sag betweentowers. (Magnus Manske|Cc-by-3.0.)

These magnetic end electric fields are everywhere in our work and living environments.Every conductor that has a current flowing through it will produce a electromagnetic


Introduction 2

field around it. Therefore, it is important to understand the effects these fields have onbiological organisms. There have been claims that these fields may have some adversehealth effects. In light of these claims, safety precautions in the form of a guidelinelimit at 0.4 µT, for residential homes, have been put in place pending conclusive proofs.This value is a recommended limit by Energimarknadsinspektionen (Swedish energymarket inspectorate) (EI) for new and rebuilt power lines. If the magnetic field strengthexceeds 0.4 µT then preemptive measures to lower the field strength will have to be pre-sented to EI. EI will then decide if the cost to lower magnetic field exposure is reasonable.

In this thesis, the consequences and health effects of these fields are discussed anda calculation method developed for predicting magnetic field strength near power linesat an early stage of project planning. The magnetic fields that are the basis of this thesisare categorized by World Health Organization (WHO) as Extremely Low Frequency(0 Hz− 100 kHz) (ELF) [2]. These fields are ubiquitous in our work and living envi-ronment and cannot be completely eradicated. For the scope of this thesis, only thepower grid distribution system is considered in the calculations. In order to illustrate therelative strength of magnetic fields from power lines, field strengths of some everydayhousehold appliances are listed in table 1.1

Table 1.1: Magnetic fields produced by some everyday household appliances atdifferent distances from the source [3]

Appliance 0.1 m 0.5 m 1 m

Television 1.5− 4 µT 0.2− 1 µT 0.1− 0.2 µTStove 1− 3 µT 0.1− 0.6 µT 0.05− 0.2 µTHairdryer 0.5− 12 µT 0.1− 0.3 µT 0.05− 0.1 µTVacuum cleaner 15− 35 µT 0.4− 1.5 µT 0.1− 0.5 µT

In table 1.1, many of the appliances exceed the limit for fields from power lines. Im-portant to remember, however, is that most of these appliances mainly are used shortperiods of time and not continuously, as in the case of power lines, based on yearly aver-age current magnitudes. The instantaneous magnitudes generated by power lines canstill be higher or lower at any given time. This, nevertheless, shows that the occurrenceof magnetic fields is not an isolated event in the context of power transmission lines andthat the strengths of these fields considered in this thesis are not much stronger thansome common items that are considered safe to use. The field strength also declinesrapidly as the distance gets larger from the source. Figure 1.2 shows magnetic fieldsfrom a three-phase circuit as a function of the distance from the center phase conductor.


Introduction 3

Figure 1.2: The magnetic field strength at 1.5 m over ground from a three-phase circuitwith three different currents. The distance between the phases is 10 m. Thecurrents are chosen to be representative of the power levels in the rest of thisthesis.

The main goal of this thesis is to produce a working tool for calculation of these magneticfields in the vicinity of both underground cables and overhead transmission lines usingMATLAB. This is done to easily determine where the limit is located in relation to thepower line. The most important point of interest is the nearest point of residential homesat the height of 1.5 m, above the ground plane, as this can be considered as the averageheight of a human torso, which is the part of the body that is subjected to the highestinduced currents.

1.2 Health effects

For many people, the fear and belief that magnetic fields are harmful is very real, eventhough no conclusive proof of any adverse effect has been produced. There are, however,many claims about adverse health effects, and although inconclusive, research suggeststhat some harmful effects may be caused by these fields, although to a limited extent. Asa result it has been necessary to take preemptive measures when planning for the newgrid and rebuilding the old one, in the form of a guide value. Some of the health effectsthat have been researched and may be triggered by magnetic fields are [2]:

1. Neurodegenerative disorders

2. Cardiovascular disorders


Introduction 4

3. Immune system and heamatology

4. Reproduction and development

5. Cancer

The most controversial effect and the one showing most conclusive evidence is thecorrelation between ELF magnetic fields and childhood leukemia.

1.2.1 Physical interaction between biological tissue and fields from powerlines

Power transmission lines produce both an ELF electrical field and an ELF magnetic field."ELF" stands for "Extreme Low Frequency", and is used specifically by WHO to denotethe frequency interval of 0-100 kHz. These fields produce physical effects in organisms.Although the interactions are different, both fields have the ability to give rise to currentsin the body tissue. The strength of the induced electric fields and, in turn, currents is tosome extent dependent on the body mass, area and the curvature of the different bodyparts [2][3].

1.2.2 Physical effects of electric fields in the body

A body placed in an electric field will significantly perturb the electric field dependingon the body size and shape and the way the body is connected to the ground. With aperfect connection to the ground through the feet a larger current will flow through thebody. The currents are generated through oscillating charges on the skin that inducecurrents inside the body [2].

1.2.3 Basic interaction of magnetic fields in the body

External magnetic fields that a biological organism comes in contact with are not re-stricted by the tissue. The amount of magnetic materials present is too small to give anoteworthy contribution. Therefore, the relative permeability can be considered equalto that of free space [2]. As an ELF magnetic field passes through the body with varyingstrength, it will induce a varying electric potential distribution (inside and on the body),described by Faraday’s induction law:

∮C~E · d~s = −dΦ

dt(1.1)

where C is an arbitrary closed path in space. Φ is the magnetic flux through the closedpath C, and is described by the equation:

Φ =

"S

~B(t) · d~A (1.2)


Introduction 5

S is any surface for which C is the boundary. d~A is the infinitesimal area vector elementand ~B(t) the time varying magnetic field [4]. Thus, depending on the size and orientationof the body with respect to the magnetic field, the induced electric field distribution inthe body and its organs will be different [2].

1.2.4 Neurodegenerative disorders

The main suggested links between magnetic fields and neurodegenerative disordersare Alzheimer’s disease, Parkinson’s disease, Amyotrophic Lateral Sclerosis (ALS) andMultiple Sclerosis (MS) [2][5]. Many studies have been conducted on the subject, withthe main focus on ALS as it shows the most likely correlation with an electrically intensework environment.

Amyotrophic Lateral Sclerosis

ALS is characterized by motor dysfunction muscle atrophy. The causes of the diseasehave not been completely established but can either be environmental factors or insome cases be genetically inherited. The most interesting environmental factor for thisreport is cases of electrical trauma. WHO suggests an increased risk of ALS for workersin electrical occupations [2]. This increase could be a result of the heightened risk ofshock in these work environments rather than the higher electromagnetic field strengthexposure [2][5]. Links with ELF magnetic field exposure are not explored in detail.However, some of the subjects that developed the disease had previous experience ofstrong electrical shock. Since magnetic fields also produce a current in the human body,although much smaller in amplitude, the hypothesis cannot be completely disproved.

Parkinson’s Disease and Multiple Sclerosis

Suggested from relevant studies reviewed in [2], these two diseases show the least likelylink to ELF fields, and may be considered unlikely as a consequence of ELF magneticfield exposure.

Alzheimer’s Disease

Very limited evidence of correlation between ELF magnetic field exposure and Alzheimer’sDisease (AD) has been observed. The focus of the overall material is occupational envi-ronments. Some studies present a link but the conclusion is not completely confirmedin the rest of the material covered by the WHO report [2]. The conclusion of the WHOstudy is that the evidence is inadequate as proof.

1.2.5 Cardiovascular disorders

There are concerns that ELF magnetic field exposure may have chronic effects on theCardiovascular (CV) system. Changes have been observed in switchyard operators


Introduction 6

during 1960 through 1970. As in the case of ALS, current flow through the body due toelectric shock seems necessary for any major CV effects. It is not known if these effectsalso can occur through the exposure to very strong ELF magnetic fields. The conclusiondrawn is that this is unlikely to appear at commonly encountered levels [2]. Later studiesfocus more on the acute CV events due to Heart Rate Variability (HRV). One of thesestudies is "Occupational Exposure to Extremely Low Frequency Magnetic Fields andMortality" by Håkansson [6]. Four different CV conditions were considered

1. Arrhythmia-related death

2. Acute Myocardial Infarction (AMI)

3. Atherosclerosis

4. Ischemic heart disease other than AMI

The only condition that showed a slight increase was Arrythmia-related death. AMIand atherosclerosis showed no direct effect. Ischemic heart disease, other than AMI,showed decreased risk. The exposure levels used in the study were low, medium highand very high corresponding to field levels of < 0.1µT, 0.10–0.19µT, 0.20–0.29µT and≥ 0.3µT respectively. The associations between ELF magnetic fields and these diseasesare therefore determined to be weak.

1.2.6 Immune system and hematology

Not much research has been conducted on the subject of immune system and hema-tological effects due to ELF magnetic field exposure. Some changes in the numbers ofnatural killer cells and white blood cells, that are critical to the immune system, havebeen observed, but to a limited extent. In some cases the natural killer cell numbers wereeven higher after exposure, which suggests that ELF magnetic fields have no significantconsequence on the immune system.

Even fewer studies have been carried out on hematological changes, and no evidence ofadverse effects have been found [2]. Results in [7] propose that cells exposed to a 60 Hzelectric field inhibited the T-lymphocytes that attack cancer cells.

1.2.7 Reproduction and development

Radio frequency magnetic fields effect on reproduction is a widely spread hypothesisand have shown evidence of reducing semen count, mobility, viability and normalmorphology. Most research focus is on the link between cell phone use and these effects[8],[9]. Also extended laptop use and wifi in close vicinity to the subject’s testes arethought to lead to similar effects. The question if magnetic fields from power lines posea risk is therefore relevant. Also the female reproduction is explored and according toWHO some effects have been observed in the form of a higher miscarriage risk [2].


Introduction 7

There are some findings implying that ELF magnetic fields at µT magnitudes maydisturb early development for chicken embryos but none of the previous effects seemsto be present at these low frequencies [2].

1.2.8 Cancer

Most of the overall research focus has been put on proving a link between ELF magneticfields and possible increase in cancer. The link between exposure to these fields andchildhood leukemia has been found in a number of studies. The ELF magnetic fields areunlikely, due to their low energy, to directly cause genetic mutations but may contributeto this development indirectly. The minimum field strength needed for direct damageto DNA has been believed to be at 50 µT, although this value may be revised due tofindings in more recent reports. The general belief is nevertheless that the magneticfields from transmission lines are not the direct cause of the mutations. Some theo-ries have been presented as to why a higher risk has been observed. One example ofthis is that the magnetic fields may cause potential differences in conducting materialsthat when touched would conduct current through the body with sufficient energy toaffect the bone marrow. The currents are also less perturbed in the body of a child.Another hypothesis is similar to that presented for AD, as melatonin production maybe effected by the exposure. However, no proof of these claims have been found so far [2].

In [10] no increase was registered for children living over the threshold 0.4 µT, butsome increase was noted for the children living at levels ≥ 0.4 µT. The use of 0.4 µTas a limit might very well be a consequence of this study’s result. The relative riskassessment for children living over 0.4 µT was concluded to be twice that of childrenliving below the exposure guide value and unlikely to be due to random variation.

For adults, breast cancer as well as brain cancer and leukemia have been consideredplausible effects [2]. Melatonin production changes are thought to be the catalyst in thecase of breast cancer. WHO conclude that the overall studies are negative and that theevidence of an association between ELF magnetic fields and breast cancer is weak. Thefindings for brain cancer are considered inadequate.

1.3 Shielding of magnetic fields

There are methods with which the fields surrounding the transmission lines can bereduced. More and more cables are buried, which makes it possible to keep the conduc-tors closer together or even to have all three phase conductors in one cable. Since theinstantaneous current in a balanced three phase circuit is zero for any given time, themagnetic field from such a cable will be very small, as all three phases approximatelycoincide in space. When considering separate phase conductors, the magnetic field willbe significantly smaller when the conductors are closer together, resulting in more of the


Introduction 8

vector contributions from each phase canceling each other.

Another method of limiting the magnetic fields is to place the circuits in a conduct-ing or ferromagnetic shell, for example made from aluminum or iron. Results presentedin [11] show substantial dampening. It is concluded that aluminum is the better choice ofthe two. The main problem concerning both methods is the substantial cost of materials.This alone makes the method only viable in very specific cases. The fact that substantiallosses will appear in the material as well, producing heat, needs to be taken into accountwhen dimensioning such a system.

Both burying cables and shielding the magnetic fields will lead to a large cost. Asimpler way to reduce fields from the circuits is optimal phase ordering of the conduc-tors in space. This method can be utilized when the overall system contains more thanone three-phase circuit [11], [12].

1.4 Cases

1.4.1 Integral solution

The solution method will be compared with the exact solution of Biot Savart’s law, withthe current approximated as a line current. This is done in order to assess the strengthsand weaknesses of the method and also to determine the error of the numerical resultfor the case of a straight conductor. The error will depend on the segmentation of theconductors. For smaller segment lengths the approximate solution converges on theexact integral solution, but the calculation complexity decreases greatly with fewer andlonger elements. This makes it necessary to choose an optimal segment length for thegiven situation and the desired accuracy.

The exact integral becomes difficult to solve analytically for more complex cases. There-fore, the simple case of a single straight conductor is investigated first. Comparing theexact solution with the numerical solution gives some insight into the strengths andweaknesses of the method. This example serves as a useful reference mostly for cablesunderground as these can be more accurately approximated as straight conductors.

For the cases with overhead power lines, that are described in the following sections,estimating the error is a more complex task. Still, an effort was made by using a finelysegmented conductor calculation as a reference.

1.4.2 Bladsjön

For assessment of the finished method, results for two real cases were produced andcompared to calculations made by WSP. The first case is the site of Bladsjön outside


Introduction 9

of Åsbro in the south region of Nerike. Here a few different ideas on new power lineswere proposed. Calculations of the magnetic field strength at nearby residential homeswere needed, this to determine if existing proposals were within the exposure limitguidelines presented by Svenska Kraftnät (SVK). The magnetic field magnitudes for thesite have been determined in a previous survey and will serve as a reference when doingthe same calculations with the produced MATLAB script. Global coordinates for thethree phase systems are calculated from the tower positions as hyperbolic curves withdata points every meter horizontally along the conductor. The actual segment length istherefore a little bit longer than 1 m and not completely uniform along the conductor.For the accuracy of the calculation the distance between data points needs to be keptat a minimum. The time of calculation is also highly dependent on the number of datapoints which increases the calculation time for the complete system at higher resolution.Suggested changes for the site can be viewed in figure 1.3

Figure 1.3: Suggested changes for the site Bladsjön outside of Åsbro and its existing 400kV power line. Residential homes are marked with green outlining. Theyellow lines represents the existing power lines, (NW-SE, 400 kV, 450 A),(N-S, 130 kV, -200 A). Alternative 3B is light blue, 3C is dark blue, 3D is greenand 4A is purple.

The input data for this project were taken from a CAD drawing of the site, whichdetermined the global coordinates for the towers as well as the altitude of the conduc-


Introduction 10

tors. Also the lowest point of the lines between towers was needed in the hyperbolicapproximation.

1.4.3 Grundfors

Grundfors is another site that has been calculated by WSP in a previous project and willserve as a second comparison to the calculation method. An overview of the proposedchanges is shown in figure 1.4. The site is a high voltage switchgear station that is goingto be moved to the location of the red box. The distance from the system to the point ofinterest for these calculations, which is the east house, is very large. This puts in questionthe relevance of the result as it clearly will be well below the limit. Still the exampleserves a purpose in the context of this report in comparing the threshold location relativeto the power lines.

Figure 1.4: Overhead image of the site at Grundfors. The two green circles indicatehouses of interest, magenta lines represent the altered section of the powerlines and the red outline the new location of the switch yard. The east greenline is the cross section compared using the reference material and thecalculation method.

The high power switchgear station will also produce a substantial magnetic field in thearea but this is a complex system to analyze and its contribution at the points is notconsidered. The contribution of the switchgear station is also excluded in the referencedata.


Introduction 11

1.5 Underground cables

When considering underground cables, not much of the initial method needs changing.The ground’s effect on the magnetic field is negligible provided there is no significantamount of ferromagnetic material present [13]. There are some special conditions thatneed to be addressed in these calculations, however. If the ground composition has alarge quantity of conductive material present, there might be a large difference betweenthe result and the actual field strength. Results in [13] show that these effects will bevery small in non-extreme cases. The results in the report only show a relative differenceof under 0.1 % with a change of resistivity from 1 Ωm up to 105 Ωm, corresponding to aspan from sea water to concrete.

If there is an electrically conducting shield plate present, the magnetic field will beperturbed in the vicinity. However, this case will be disregarded in this thesis. Theresults here can still serve as representative of a worst case scenario.

Cables are typically buried at quite shallow depths, making for shorter distances fromthe conductors to the exposure area. As can be observed in the results (section 3.2),the accuracy of the calculations is dependent on the segmentation degree and the fieldpoint distance from a conductor. It is thus important to have a finer segmentation of thesystem near these field points. If the conductors are not segmented finely enough, theerror of the calculation will be large above the conductors. The result will still be a viableapproximation at some distance away from the cable, so depending on the area or pointof interest, longer conductor segments may still be used. It is important to make thisassessment when using the script in order to get reliable results for any given situation.


Methodology 12

2 Methodology

2.1 Biot Savart’s law

For calculations of magnetic field contributions from power lines in 3D space, BiotSavart’s law was used. Biot Savart’s law is an integral equation that can be approximatedas a sum of field contributions from straight wire segments.

~B =µ0 I4π

∫C

d~l × ~RR3 ≈ µ0 I

4π

N

∑i=1

∆~Li × ~Ri

R3i

(2.1)

In equation 2.1 µ0 is the permeability in vacuum, ~R is the vector from the infinitesimalcurrent source segment to the field point and d~l is the infinitesimal segment vector.Conductors are approximated as line currents. The arc length vector differential d~l isreplaced by a directed straight line segment ∆~L beginning and ending at adjacent controlpoints on the conductor. ~Ri is the vector pointing from the mid point of the straightsegment to the field point. The error created by this approximation depends on thelengths of the straight segments as well as the position of the field point. Furthermore,the approximation of the integral is the crudest one possible, except for that ~R is de-termined from the center-point of each segment. An alternative way would be to useSimpson’s rule for a more accurate solution. Since the results produced here are alreadysufficiently accurate, this will be a future addition to the algorithm.

When considering field points close to the conductor it may be necessary to describe thesource as a finite volume τ with the corresponding infinitesimal volume segment dτ

containing a current density~J. The equation for Biot Savart’s law is hence changed to:

~B =µ0

4π

∫τ

(~Jdτ)× ~RR3 (2.2)

More accurate calculations would require that the current density vary inside the con-ductors, because of the skin effect.

In this thesis, calculations are made only for field points well away from the conductors.Furthermore, the total current in each conductor is a given entity. Therefore, it is a validapproximation to treat the conductors as being infinitesimally thin and to neglect skineffect. Propagation effects are neglected due to the low AC frequency. The current istreated as a constant along a conductor at a given instant of time. The magnetic field


Methodology 13

is only calculated at points well inside a sufficiently large selected volume of interest.The contributions to the B-field from the conductors outside this volume are thereforeneglected.

The finished algorithm was tested against the simple case of a straight conductor of finitelength. This case could be solved both analytically and numerically. The results werethen compared to estimate the error of the approximation as a function of the distancefrom the source and of the segment length. The algorithm was also tested against thecase of a hyperbolic hanging conductor by comparing with the numerical result at finesegmentation.

There are in theory other methods that could be used to calculate the magnetic field suchas Finite Element Method (FEM). Nevertheless, this type of method would demand alarge number of elements and therefore require excessive processing power.

2.2 Simplifications

In the assessed real world cases there are some approximations made to make thecalculations manageable. First of all, only parts of the overall system are considered.Only small parts of the total conductors are used, as the contributions to the magneticfield from the distant parts of the conductors are very limited. Nevertheless the totallengths of the conductors need to be large enough to include the essential contributionsto the total B-field. As the field strength is largely dependent on the distance from theconductors the total lengths can be relatively small.

The rms value of the current (assumed equal for all conductors) will in reality varyover a year. Here, only the recorded yearly average of the rms line current is used. Thisshould be regarded as a representative value.

In the real cases each phase current is split between two conductors with a separa-tion of 0.45 m. This has some influence on the B-field, at least near conductors. In thethesis, split phase conductors are replaced by single (thin) line conductors which areplaced between the split conductors in space.

Induced currents in the shield wires will also have some effect on the result and may bea feature included in a future version of the script.

Furthermore, the conductors are defined as thin lines with negligible cross sectionarea, thereby excluding the need to use current densities and also excluding the skineffect and any material parameters.


Methodology 14

2.3 Calculation method

2.3.1 Yearly current average

The currents that are used for the calculations are yearly averages. SVK calculate theseaverages from a market model simulation over a year with different scenarios. Thesevalues are then compared with historical measured currents, when possible [14]. Theinstantaneous magnitudes are disregarded although these values can be much higher.The three phase currents are described using complex values, all having the sameabsolute magnitude equal to the yearly average (RMS) current. As usual, the differencein phase angle between two line currents is ±120. The yearly average current ismultiplied with the phase shift that is described by:

ej(φ±phase angle) (2.3)

to mathematically describe each phase current, φ is the phase angle of the completethree-phase circuit. This angle φ is needed to account for different loads and for thepresence of several three phase circuits. The phase angle will be dependent on the loadimpedance and will be an input variable for the calculation method [15]. For an inductiveload the current will be lagging behind the line to neutral voltage of the conductor andtherefore have a negative angle. The opposite is true for a primarily capacitive load.

2.3.2 Numerical calculation method

The numerical method for calculating the B-field is based on Biot-Savart’s law for linecurrents, as given in 2.1. The input data are organized as 3D coordinates for the con-ductors. Two auxiliary vectors are needed for the calculation: ∆~Li and ~R. ∆~Li is thevector from the starting point to the ending point of the straight current (or conductor)segment. ~R is the vector from the middle point of the segment to the field point. Thesevectors are determined for all segments of all conductors of the system. This process isindependent of the shape and location of any conductor in the defined 3D space.

The magnetic field is calculated using equation 2.1 for each phase conductor usingthe complex currents, calculated from the yearly averages provided by WSP.

All the contributions from the segments of all phases and circuits are then superposed,i.e. added as complex-valued vectors. Thus, each spatial component (Bx, By, Bz) of themagnetic field, at a given field point, has a complex value with a magnitude and anargument (phase angle). The total magnitude of the magnetic field from all conductorspresent is then given by the equation below:

Btotal =√(Re(Bx))2 + (Re(By))2 + (Re(Bz))2 + (Im(Bx))2 + (Im(By))2 + (Im(Bz))2

(2.4)


Results 15

3 Results

3.1 Special case evaluation : Straight conductor

To evaluate the calculation algorithm a base case was set up. The base case needed tobe solvable both analytically and numerically. For this reason, a single thin straight,finitely long conductor was chosen. The error will be dependent on the step size or thesegmentation degree of the conductor, as the vector ~Ri only will be exact for the middlepoint of the segment and produce a larger error at the segment ends. The longer thefinite segment is, the worse the approximation of ~Ri will be. A conductor is placed at theorigin and with direction along the y-axis. The analytical calculation of the given case ispresented below for a specific point in 3D space.

~B =µo I4π

∫ L2

− L2

d~l × ~RR3 (3.1)

Since the conductor is only stretching out in the y direction, d~l = dy′y, the field pointis chosen as P = (50,0,0) or 50 m radially out from the center point of the conductor.This choice of field point makes it possible to get an explicit analytic expression for theintegral. The vector ~R is determined by the differences of (unprimed) field and (primed)source coordinates.

~R = (x− x′)x + (y− y′)y + (z− z′)z (3.2)

The calculation of the cross product d~l × ~R for the span of the integration is:

d~l × ~R =

∣∣∣∣∣∣∣x y z0 dy′ 0

(x− x′) (y− y′) (z− z′)

∣∣∣∣∣∣∣ = (z− z′)dy′ x− (x− x′)dy′ z (3.3)

Since z, z’ and x’ all are zero during the integration, the expression d~l× ~R can be reducedto

d~l × ~R = −xdy′ z (3.4)

Similarly, the absolute value of ~R is reduced to

R =√(x− x′)2 + (y− y′)2 + (z− z′)2 =

√x2 + (−y′)2 (3.5)


Results 16

It is now possible to solve the integral analytically:

µo I4π

∫ L2

− L2

−xdy′ z

(x2 + (−y′)2)32=

µo Iz4πx

[(−y′)√

x2 + (−y′)2

] L2

− L2

(3.6)

For the field point (x=50, y=0, z=0), and using a DC current of 400 A and a conductorlength of L = 500 m, the exact answer can be calculated as

4π · 10−7 · 400 · z4π · 50

[−250√

502 + (−250)2− 250√

502 + 2502

]= −1.5689 µT · z (3.7)

This result serves as the reference as the same case is set up and solved with the numeri-cal computation algorithm. The results for some different segmentation degrees are thencompared with the exact solution in table 3.1.

Table 3.1: Comparison of the exact integral and the numeric solution method fordifferent segment lengths at field point (50,0,0).

Numerical result µT Number of segments ∆Li in m Relative error

-1.5684 250 2 0.0003-1.5679 125 4 0.0006-1.5662 50 10 0.0017-1.5624 25 20 0.0041-0.5218 5 100 0.6674

The relative error is in this case small for most segment lengths but increases significantlywhen ∆Li becomes larger than 20 m. This comparison works best for underground ca-bles as they are more accurately approximated as straight conductors. For overheadpower lines the hyperbolic conductor path between towers will make the error increasesomewhat for larger segment sizes in comparison to the straight conductor. This isdue to that the approximation of ~Ri will be less accurate towards the midpoint of thesegments.

The procedure was also used for a field point closer to the conductor in order to deter-mine if the distance from the finite conductor to the field point will have any significanteffect on the error.


Results 17

Table 3.2: Comparison of the exact integral and the approximate solution for differentsegment lengths at field point (20,0,0).

Numerical result µT Number of segments ∆Li in m Relative error

-3.9871 250 2 0.0000-3.9868 125 4 0.0001-3.9860 50 10 0.0003-4.0344 25 20 0.0118

-10.0853 5 100 1.5294

For this distance a striking difference can be observed. The result is more precise forlarger segments but the numerical result for longer segments is larger than the referencevalue. The opposite is true for results in table 3.1. For a clearer representation of how theerror changes depending on the number of segments, a plot of the error for the straightconductor is shown in figure 3.1:

0 5 10 15 20 25 30 35 40 45 50

Number of segments

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rela

tive e

rror

Relative error for a 500 m long straight conductor using different segmentations

Figure 3.1: Plot showing how the error of the numerical result depends on wiresegmentation and how odd and even numbers of segments affect the relativeerror.

The values converge quickly at a very low error. An interesting aspect of the plot infigure 3.1 is that the error depends on if the segmentation is odd or even. To furtherexplore how the odd and even segment numbers affect the results, one more test wasperformed, where points from 0.3 m to 100 m from the source were plotted for 25, 26,100 and 101 segments. These plots are shown in figures 3.2, 3.3, 3.6 and 3.7.


Results 18

0 10 20 30 40 50 60 70 80 90 100

Distance from source [m]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Perc

ent

of excact

result [

%]

Ratio of numeric result and exact solution

at different distances from source, 24 segments

Figure 3.2: 24 segments: the ratio betweencalculated numerical result andexact solution vs distance fromsource. The total length of theconductor is 500 m.

0 10 20 30 40 50 60 70 80 90 100


0

5

10

15

20

25

30

35

Perc

ent

of excact

result [

%]




The difference is that for the even numbers of segments the approximate values aresmaller than the exact values close to the source, while the opposite is true for the oddnumbers of segments. Both plots also show signs of deviating from the exact solutionfurther away from the source. The most exact result occurs around 30 m from the source.As the point considered is located at the central orthogonal axis out from the conductor,even numbered segmentations will have a segment node in that location and odd willnot. Moving closer to the conductor the ∆Li relative size gets larger. For the case ofodd segments this results in a larger contribution of the center ∆Li. ~R will in this casebe smaller than the real value for the middle segment. This in turn results in a largercontribution of the middle component and a magnetic field amplitude larger than theexact result. The corresponding plots for 100 and 101 segments are shown in figure 3.6and 3.7. The same behavior appears, but closer to the conductor, and the solution ismost exact at a distance around 8 m.


Results 19

0 10 20 30 40 50 60 70 80 90 100


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Perc

ent

of excact

result [

%]




0 10 20 30 40 50 60 70 80 90 100


0

1

2

3

4

5

6

7

8

9

Perc

ent

of excact

result [

%]




The result suggests that depending on the distance of the field point of interest withrespect to the conductor, the segment length should be chosen small enough. One im-portant aspect is that when calculating magnetic fields for underground cables, shortersegments than the standard of 5 m will be necessary in order to produce a sufficientlyaccurate result, because the conductor distance from the cables to the exposure zone isshorter. These cables are also more frequently used in urban environments close to thegeneral public.

3.2 Special case evaluation: Hyperbolic conductor

For a more accurate assessment of the overhead transmission lines, a similar comparisonwas conducted. The exact result here cannot be easily determined through analyticintegration. Instead, a simulation with very short segment lengths served as the reference.The evaluation case is a hyperbolic line with length 500 m and its lowest point atapproximately 27 m height above the ground plane. The magnitude of the B-field wascalculated for field points placed at different horizontal distances to the lowest point ofthe hanging conductor. The reference here used 0.1 m segment length and was comparedwith the results for 5 m and 20 m segment length. The results are similar to those in theprevious, straight conductor, examples:


Results 20

0 10 20 30 40 50 60 70 80 90 100

Horizontal distance to lowest point of conductor [m]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1(5

m s

egm

ents

)/(0

.1 m

segm

ents

)

Ratio between 5 m and 0.1 m segment length, 100 segments

Figure 3.6: The ratio between 5 m segmentlength and the reference 0.1 m

0 10 20 30 40 50 60 70 80 90 100

Horizontal distance to lowest point of conductor [m]

0

5

10

15

20

25

30

(20 m

segm

ents

)/(0

.1 m

segm

ents

)

Ratio between 20 m and 0.1 m segment length, 25 segments

Figure 3.7: The ratio between 20 m segmentlength and the reference 0.1 m

As before, the error gets larger when the distance from the field point to the conductoris less than the length of a segment. Furthermore, the accuracy also declines slightlyfurther away from the conductor.

Table 3.3: Comparison between the accuracy at the closest, optimal and 100 mhorizontal distance from the conductor

Segment length m Distance from conductor m Ratio numeric/reference

1 0.3 0.49451 5 0.99991 100 0.9995

10 0.3 0.0074910 20 0.983210 100 0.9776

Results shown in table 3.3 indicate that the hyperbolic curve influences the result ina negative way, even though the segmentation length is shorter than in the straightconductor case. When comparing the results for 20 m segments shown in figure 3.2 thepeak ratio value of the solution is 0.9978 at the distance of 33 m. The peak ratio is stilllocated further away from the conductor but the value is closer to one. This is also truefor 100 m from the conductor. At the closest point of 0.3 m the hyperbolic case is moreprecise.

For the purpose of visualizing the field around the conductors, the central cross sectionof the three-phase circuit was calculated and plotted. See figure 3.8 below.


Results 21

-25 -20 -15 -10 -5 0 5 10 15 20 25

x [m]

0

5

10

15

20

25

30

35

40

45

50

y[m

]

Magnetic B-field from three phase hyperbolic conductors

Phase 1: 400 A

Phase 2: -200 A

Phase 3: -200 A

Figure 3.8: Vector field plot at time t = 0 for the three-phase circuit for two dimensions.The distance between adjacent conductors is 10 m. The instantaneouscurrents are, from left to right, 400 A (phase 1), -200 A (phase 2), -200 A(phase 3).

Since the vector arrows are hard to see in this plot an upscaled version in 3D for threedifferent cross sections is included in the appendix. The currents used for this plot arevalues for a chosen instant of time, corresponding to phase1 = 400 A, phase2 = −200 Aand phase3 = −200 A.

3.3 Bladsjön

The power transmission line shown in figure 1.3 (above) was analyzed. The line con-figuration analyzed is the green line named 3D. Coordinate data for the towers andlowest point of the line between towers were provided by WSP. Using these coordinatesa hyperbolic function was approximated between each tower giving the 3D conductorgeometry, which was split into 5 m segments as prescribed by SVK [14]. In WSP:scalculations the induction in the neutral top conductors is also taken into account. Theseconductors get induced currents flowing through them, currents that in turn create theirown magnetic fields. These contributions to the total magnetic field are not taken intoaccount in this report but may be added in a future version of the algorithm. The powerline is a 400 kV line that carries an annual average current of 450 A, southbound power-flux. In the calculation the smaller 130 kV line labeled Bef in figure 1.3 is modeled with


Results 22

an annual average current of 200 A, northbound power-flux. This line has significantlyless effect on the total magnetic field at each point considered, as both the distance fromthe point is longer, phase conductors are located closer together and the currents throughthe conductors are smaller.

The calculations are done for the four points that represent residential houses in thevicinity of the power transmission system. The complete site is shown in figure 3.9.Field contributions from the continuation of the lines in both directions (outside thefigure) are very small and are not included in the reference material, as previously stated.

The (reference) phase angle for the circuits has been defined as zero in both cases.This was also done in the reference material.

1.504 1.506 1.508 1.51 1.512 1.514

Global coordinates [m] #105

6.542

6.5422

6.5424

6.5426

6.5428

6.543

6.5432

Glo

bal coord

inate

s [

m]

#106 Input data plotted for site Bladsjön with house coordinates

400kV Phase1

400kV Phase2

400kV Phase3

130kV Phase1

130kV Phase2

130kV Phase3

Used Cross section

House 1:74

House 1:76

House 1:75

House 1:71

Figure 3.9: 2D representation of the site Bladsjön with relevant houses. The red linecorresponds to the cross section in figures 3.10 and 3.11. The black rectanglerepresents the calculation area used in figure 3.12

The circles represent the nearest points of the houses 1.5 m above ground. The calculatedmagnetic field strengths at these coordinates for cases with line segment lengths 5 mand 1 m are shown in Table 3.4


Results 23

Table 3.4: Numerical field calculation for houses with conductor segment lengths at 5 mand 1 m.

House name Segment length 5 m in µT Segment length 1 m in µT

1:74 0.2553 0.24251:76 0.1771 0.15281:75 0.2760 0.25571:71 0.2873 0.2806

WSP:s results for house 1:71 were estimated from figure 3.10. The same cross section wasplotted using the produced algorithm; this graph is shown in figure 3.11. The lower fieldmagnitude close to the center-phase may be due to differences in line data. As can beseen in figure 3.9, the conductors are spaced uniformly over each span. The fact that theconductors in figure 3.10 in reality consist of two wires per phase separated by 0.45 m,as well as the lack of induced current in ground wires, may also affect the calculations.

Figure 3.10: Graph of the magnetic field near house 1:71 in WSP’s report for Bladsjön.The section is indicated in Figure 3.8. The field calculation height is 1.5 mabove ground level at 1:71. The origin indicates the center phase.


Results 24

Figure 3.11: Calculation of the B-field for the same section as in Figure 3.10, using thealgorithm developed by the author.

House 1:71 is located 84.9 m from the 400 kV center phase line and can therefore beestimated to 0.3 µT in WSP:s results, figure 3.10. However, the resolution of the graph infigure 3.10 makes it impossible to acquire a more precise reading of the field. At the samedistance from the center conductor the calculated result shows a magnitude of 0.2871 µT.The results are difficult to compare as the resolution of the calculated plot is much higher.

A mesh calculation of the magnetic field strength for a limited area around the fivehouses is shown in Figure 3.12. The calculation was made using a constant elevation,equal to that of house 1.71.

Figure 3.12: Graph of result for site Bladsjön for house 1:71 altitude above sea level

The resolution of the mesh is 1 m and the segmentation length of the conductors is 5m, which makes it possible to generate an accurate contour plot for the same area as


Results 25

in Figure 3.12. This plot is shown in 3.13 below. The coordinates for the 0.4 µT limitcontour can be exported from this result. The simulation is fairly demanding and istherefore not suitable for calculation of more than one altitude plane, in this case thealtitude of house 1:71. The calculation becomes faster with larger mesh steps, but at thecost of precision. This needs to be taken into account when simulating a larger area.

Figure 3.13: Contour graph of three B-field magnitudes for site Bladsjön at house 1:71height above sea level. The house 1:71 is indicated by the blue circle.Contour levels are given in µT.

Since the calculation plane is at the elevation of house 1:71, the other houses still needto be calculated for, one by one. Since the coordinates can be exported, the simulationcan be run for several elevations, and the resulting data for each elevation be extracted.These coordinates can at a later stage be used to construct a 3D level surface for thelimit B field value, a surface that in turn can be merged with CAD drawings of the site.Another possibility is to include elevation (i.e. terrain) data for the site. The problem sofar is that these coordinates need to be aligned with the mesh made in MATLAB or use amesh taken directly from the CAD drawing of the site. These CAD drawings are verydense in data and need to be reduced (simplified), before running the B-field calculationscript in MATLAB.


Results 26

3.4 Grundfors

For this site the reference data were not completely correct. The solution method useddid not accept z-coordinates for the towers; therefore the setup is a bit strange at afirst glance. Also the provided line data are not completely accurate, as can be seenin figure 3.14. The orientations of the towers are not included and the left and rightconductors were only displaced in the x-direction from the center conductor coordinates.This results in the conductors being closer together than in the actual case along thespan. Due to this, only the east cross segment was analyzed, as the 220 kV AL7 S1 circuitclosest to the west house is to poorly approximated.

The global z-coordinates for the towers are not used and the whole site is thereforeplaced on a plane with constant elevation. The lines, however, have an offset correspond-ing to the elevation of the conductors. This produces some points, where the power linesare close to the virtual ground, which is not the case at the actual site. Field points arethen displaced 1.5 m from the plane of the towers.

6.206 6.208 6.21 6.212 6.214 6.216 6.218 6.22 6.222 6.224 6.226

Global coordinates [m] #105

7.2072

7.2074

7.2076

7.2078

7.208

7.2082

7.2084

7.2086

7.2088

Glo

bal coord

inate

s [

m]

#106 Input data plotted for site Grundfors

220kV AL7 S1 Phase1

220kV AL7 S1 Phase2

220kV AL7 S1 Phase3

400kV UL28 S1-3 Phase1






Used cross section

House East

House West

Figure 3.14: 2D representation of the site Grundfors with relevant houses. The yellowline represents the cross section in figure 3.15 and 3.16. The black rectanglerepresents the area used in figure 3.18.

The calculations for site Grundfors were made in the same manner as for site Bladsjön, inorder to facilitate comparisons. The calculated result for the same cross section is shownin figure 3.16. However, the calculated B-field values at the positions of the houses are


Results 27

too small to be compared reliably to the reference curves made by WSP. Only graphswere available, not original data. The reference curve for the Eastern house is shownin figure 3.15. Instead, the two B-field curves (MATLAB and WSP) are compared withrespect to the horizontal distance between the center phase and the position for thelimit B-field of 0.4 µT. Magnetic field strengths in both cases are quite similar. Fromthe reference material the limit was determined to be located at 50-60 m from the centerphase of the UL1 S1-3 circuit (see figure 3.14). As seen in figure 3.16 this value is a bithigher using the algorithm, with the limit located at approximately 65 m. Additionalcauses for the deviating result may be some line coordinate differences as well as theshield wires not being included. As observed in case Bladsjön the magnitude at thecenter phase is smaller than the corresponding WSP value. The shape of the curve infigure 3.15 is not showing the normal smooth curve shape at the center phase as infigures 3.10, 3.11 and 3.16. This suggests that the measurement start point might be atthe phase conductor closest to the eastern house. This was not possible to determinefrom the provided data.

The coordinates in this case do not take into account the orientation of each toweras this is not known. This makes for shorter conductor separation at each node pointbetween the towers. Having access to the coordinates of each phase attachment pointfor the towers would remove this issue.

Figure 3.15: B-field magnitude as a function of the horizontal distance from the centerphase of line UL1 S1-3. The curve corresponds to the cross section towardsthe Eastern house in figure 3.14. Reference curve from WSP.


Results 28

Figure 3.16: Cross section field strength plot for the same cross section as in figure 3.10,calculated result

A contour plot was also made, even though none of the houses were close to themaximum allowed B-field magnitude.


Results 29

Figure 3.17: Contour plot for the Grundfors site. The east house is too far outside therelevant area and hence not included. The western house is shown as a bluecircle. Contour levels are given in µT.

Finally, a meshed plot of a section of the site, showing the B-field magnitudes on thez-axis is shown in figure 3.18. The peaks correspond to local minima in line elevationsabove ground.


Results 30

Figure 3.18: Calculated B-field magnitude for the Grundfors site. Peaks correspond topoints, where the conductors are closest to ground. The field is calculated at1.5 m above the ground plane. Compare this plot with figure 3.14


Discussion and analysis 31

4 Discussion and analysis

4.1 Medical effects

Some research reports show correlations between electromagnetic fields and neurode-generative disease, in particular ALS. Although no conclusive proof has been producedat this point, strong links between domestic living conditions over the limit value 0.4 µTand childhood leukemia have been observed. The cause of the rise in childhood leukemiais still unknown and may have other causal reasons than the exposure to ELF magneticfields. Still, the limit put in place is to be considered more than a safety precaution tocalm the general public, but built on strong research data. The data presented in forexample (A pooled analysis of magnetic fields and childhood leukaemia) show thatlimitation of domestic exposure is needed, although the cause of rise in disease is stillobscure [10].

4.2 Calculation method

The numerical calculation method has clear improvement potential for future devel-opment. In this first version some some details were omitted, such as the shield wireinduction that will effect the overall field. Nevertheless, no significant difference wasnoted in the comparisons with the previous data that included this feature, except closeto the center conductor. The calculated magnetic fields were consistently slightly higherthan the reference data except for close to the center phase.

The precision of the calculations mostly depends on the segmentation of the conductors,which also in turn affects the problem size. As the error is larger close to the conductorsome thought needs to be put on selecting a suitable segmentation for any given situa-tion. For example, underground cables will be located closer to the 1.5 m measurementelevation above ground than overhead cables. Underground cables should therefore besubdivided into shorter segments in order to obtain sufficiently accurate B field values,in particular above the cables.

Many of the initial problems involved input data. In the Grundfors case this is es-pecially clear as the separation of the power line phases are depending on the directionof the circuit. This problem will be solved by including tower orientation in the globalcoordinate system or providing the connection coordinates of the phases for all the tow-


Discussion and analysis 32

ers considered in a specific setup, instead of just tower base coordinates and conductorelevation displaced in x-direction to create the outer phases. This may be the cause ofsome of the deviations from the reference results.Another factor that could have a limited effect is that the phases in the reference cases inreality are split into two conductors per phase, with a spacing of around 0.45 m.

A calculation of the electric field would be a good addition, because its effects onthe human body are similar to those of the magnetic field [2].

The ability to calculate the magnetic field for a larger area with relatively small meshsize will be beneficial for future projects if terrain elevation data can be produced for thesite. The contour curve, given as a set of ground mesh points can then be imported toCAD files.

The developed MATLAB script was tested and run without difficulty in GNU Oc-tave. This is an open source programming language that is mostly compatible withMATLAB. This is essential for future use of the script, because a MATLAB license isconsidered a costly investment.


Conclusion 33

5 Conclusion

The finished calculation method produced for WSP meets the initial base criteria thatwere mostly focused on underground cables, but also achieves sufficient results for over-head power lines. Conductors are treated as thin line currents with varying curvaturein space. Each conductor is approximated as a series of interconnected straight currentsegments. Small differences where observed in accuracy when comparing curved andstraight conductor results, simply because curved conductors are not straight. Theshorter the lengths of current segments, the more accurate the approximation will be.For a more accurate results the numerical integration approximation can be improvedusing Simpson’s rule.

Sadly, no reference data were available for testing the method against any previouslymeasured or calculated results for underground cable projects at WSP. Still, studies showno notable effect on the field strength due to normal variations in ground conductivitylevels [13]. Therefore, the developed method can be expected to produce acceptableresults in cases that are not extreme. Environmental factors such as metal objects in thevicinity of current-carrying conductors may affect the result. In such cases it is necessaryto solve Maxwell’s equations, e.g. with the finite element method.

The bottleneck has been acquiring correct input data for the given projects, somethingthat will be easily remedied with knowledge on requirements at an early stage in theprocess. Most of the problems have been caused by previous computations not being setup in a correct 3D environment. In this aspect improvements have been achieved, thesetup process is more accurate and will take into account the height differences of theground as well as towers in a more direct way.

The results show sufficient accuracy in comparison with the integral solution of thestraight conductor, even with rather coarse segmentation. The exception is for closedistances from the conductor, where the solution method is more dependent on finesegmentation for accuracy. In the case of a hanging conductor (approximate hyperbola),the algorithm produces results of similar accuracy. Since the degree of segmentationinfluences the size of the computations, some care needs to be taken when determininga suitable segmentation for a given problem. In the case of underground cables thisis even more essential, as the distance from conductor to exposure area is shortenedconsiderably.


Conclusion 34

A notable difference was seen when comparing the results with the reference mate-rial close to the center conductor in the cross section plots of Bladsjön and Grundfors.The discrepancy may be caused by shortcomings in conductor coordinates, as the dis-tance between conductors is fixed for each span The discrepancy can also be caused bytreating bundled phase conductors as single line currents, and by neglecting induction inshield wires. These simplifications will be removed in a future version of the calculationmethod. Testing the method against points closer to the centerline with a more accuratesetup would be desirable before any commercial use.

If proper terrain elevation data can be obtained, then the validity of the method can beimproved further. Moreover, it appears possible to calculate the threshold contour curvefor 0.4 µT without meshing the entire area of interest.


Literature 35

Literature

[1] Key world energy statistics,www.iea.org/publications/freepublications/publication/KeyWorld2017.pdf,The International Energy Agency (IEA), 2017.

[2] “Environmental Health Criteria 238, EXTREMELY LOW FREQUENCY FIELDS”,2007, WHO-report.

[3] Magnetfält och eventuella hälsorisker, Statens Strålskyddsinstitut, Swedish WorkEnvironment Authority, the National Board of Housing, Swedish RadiationProtection Institute, Elsäkerhetsverket, The National Board of Health and Welfare,2006.

[4] L. A. Engström, Elektromagnetism, Från bärnsten till fältteori. Studentlitteratur, 2000,pp. 78–86.

[5] “ELF Electromagnetic Fields and Neurodegenerative Disease: Report of anAdvisory Group on Non-ionising Radiation”, National Radiological Protection Board(UK), vol. 12, no. 4, 2001, Documents of the NRPB.

[6] N. Håkansson, P. Gustavsson, A. Sastre, and B. Floderus, “Occupational Exposureto Extremely Low Frequency Magnetic Fields and Mortality from CardiovascularDisease”, Institute of Environmental Medicine, Karolinska Institutet, vol. 158, no. 6,pp. 534–542, 2003.

[7] U.S. Congress, Office of Technology Assessment, Biological Effects of PowerFrequency Electric and Magnetic Fields—Background Paper, OTA-BP-E-53(Washington, DC: U.S. Government Printing Office, May 1989).

[8] Ashok Agarwal, Ph.D, Nisarg R. Desai, M.D, Bruce V. King, Kartikeya Makker,M.D, Alex Varghese, Ph.D, Rand Mouradi, M.S, Edmund Sabanegh, M.D,Kartikeya Makker, M.D, and Rakesh Sharma, Ph.D, “Effects of radiofrequencyelectromagnetic waves (RF-EMW) from cellular phones on human ejaculatedsemen: an in vitro pilot study”, Fertility and Sterility, vol. 92, no. 4, pp. 1318–1325,2009.

[9] Geoffry N. De Iuliis, Rhiannon J. Newe, Bruce V. King, and R. John Aitken,“Mobile Phone Radiation Induces Reactive Oxygen species Production and DNADamage in Human Spermatozoa In Vitro”, PLoS ONE, vol. 4, no. 7, 2009.


www.iea.org/publications/freepublications/publication/KeyWorld2017.pdf

Literature 36

[10] A. Ahlbom, N. Day, M. Feychting, E. Roman, J. Skinner, J. Dockerty, M. Line,M. McBride, J. Michaelis, J. Olsen, T. Tynes, and P. Verkasalo, “A pooled analysisof magnetic fields and childhood leukaemia”, British Journal of Cancer, Sep. 2000,83(5):692-8.

[11] J. C. del Pino-Lopez, P. Cruz-Romero, L. Serrano-Iribarnegaray, andJ. Martínez-Román, “Magnetic field shielding optimization in undergroundpower cable duct banks”, Electric Power Systems Research, vol. 14, pp. 21–27, 2014.

[12] B. P. Aminnejad and D. Wieweg, “Magnetic field in electric power equipment: Aplanning tool in Excel for estimation of magnetic fields from power componentson low voltage systems”, KTH (Royal Institute of technology, Sweden), 2015, BScthesis.

[13] T. Keikko, J. Isokorpi, S. Reivonen, T. Ruoho, and L. Korpinen, “Magnetic fieldmeasurements and calculations with 20 kV underground power cables”, WITPress, vol. 22, 1999.

[14] “Information om magnetfältsnivåer under kraftledningar (SSM2013-1207)”, SvK(Svenska kraftnät), Tech. Rep., 2013.

[15] J. D. Glover, M. S. Sarma, and T. J. Overbye, Power systems analysis & design, 5th ed.CENGAGE Learning, 2012, pp. 60–73.


Appendix 37


Appendix 38

A Appendix

A.1 [Magnetic B-field from three phase hyperbolic conductors]


Appendix 39

A.2 Bladsjön


Appendix 40

A.3 Grundfors
