lightning-induced overvoltages in low-voltage systems

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Hans Kristian Hoidalen Lightning-induced overvoltages in low-voltage systems

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LIGHTNING-INDUCED OVERVOLTAGESIN

LOW-VOLTAGE SYSTEMS

by

Hans Kristian Heidalen

A dissertation submitted to

the Norwegian University of Science and Technology Department of Electrical Power Engineering

in partial fulfilment of the requirements for the degree of

Doktor Ingenior

December 1997

DISCLAIMER

Portions of this document may be illegible in electronic image products. Images are produced from the best available original document.

Preface 111

PREFACE

This thesis is the result of a research project founded by the Norwegian Research Council. All the work was carried out at the Department of Electrical Power Engineering at the Norwegian University of Science and Technology (NTNU) during the years 1994-1997.

I would especially express my gratitude to my supervisors Prof. Jarle Sletbak at NTNU and Dr.ing. Thor Henriksen at the Norwegian Electric Power Research Institute (EFI) for being an outstanding source of inspiration (and perspiration) during this work.

I would also like to thank all the other institutions and individuals making this thesis possible. In order of appearance: Jostein Huse at EFI, Institute of High Voltage Research at University of Uppsala, University of Florida, the three diploma students Kurt A. Bakke, Abraham T. Gerezgiher and Morten Nordskog, Siemens installasjon, AB Elektro, Det Norske Meteorologiske Institute and Trondheim Energiverk.

Finally, I would like to thank all my friends and colleagues at the Department of Electrical Engineering for valuable assistance and encouragement during these three years.

Trondheim, December 1997

Hans Kr. Hoidalen

Abstract v

ABSTRACT

Lightning-induced overvoltages (LIOs) from nearby lightning are a main source of failures in low-voltage overhead line systems. Lightning strokes closer than about 1 km can cause harmful overvoltages, which in turn can lead to direct or delayed damages to connected electrical installations or equipment. Insurance companies report an increasing number of damages of electric nature over the years. This increase is probably caused by the introduction of more and more sensitive electrical equipment in an increasing number of installations.

This thesis deals primarily with calculations of lightning-induced overvoltages (LIOs) in low- voltage overhead line systems with the objective to enable the design of a proper overvoltage protection. The work is divided in two parts:1) Development of calculation models2) Calculations of LIOs in low-voltage systems

In the first part models for calculation of LIOs are adapted from the literature or developed based on measurements. An objective when selecting the models is to aim at simple models based on a few measurable quantities, and which show a reasonable accuracy. The models used in this thesis are believed to be fairly accurate for the first few microseconds, which normally is sufficient for prediction of the maximum induced voltage in the system. The lightning channel is modelled by the Modified Transmission Line (MTL) model with the Transmission Line (IL) model as a special case. The coupling between the electrical fields from a lightning channel and an overhead line is modelled by Agrawal’s model. The attenuation of electrical fields propagating over a lossy ground is modelled by Norton’s- or the Surface Impedance methods. All these models are well known in the literature and are in this work synthesised to enable calculation of LIOs in practical low-voltage configurations using the electromagnetic transients program, ATP-EMTP. The validity of all the applied models is analysed. In addition measurements have been performed in order to develop models of distribution transformers and low-voltage power installation (LVPI) networks. Simple models of "typical" transformers and LVPIs are developed for calculations when specific data are unavailable. The practical range of values and its influence on the LIOs in a system is investigated. The main frequency range of interest related to LIOs is 10 kHz - 1 MHz in which all the models are accurate.

In the second part the adapted or developed models are used to calculate LIOs in low-voltage systems. The influence of various key parameters in the systems is investigated. Of greatest importance are the return stroke amplitude and rise time, the overhead line height and location, the termination of overhead line segments, neutral grounding, and the ground conductivity.• The LIOs in an unprotected system increase proportionally to the return stroke amplitude.

Larger rise times of the return stroke result in lower LIOs.• The introduction of lower terminating impedances by connecting e.g. more LVPI

networks results in lower LIO. Thus the magnitude of the LIO is likely to be highest in rural areas with few installations connected to an overhead line. A transformer with a

Abstract Vi

grounded LV neutral can be modelled as a small inductance (4-40 pH) closely related to the transformer’s rated power and voltage. When the neutral is isolated the model becomes capacitive and the LIOs increases considerably. As a first approximation, LVPI in TN-systems can be modelled as a small inductance (2-20 pH) and in the IT-system as a capacitance (20-200 nF) in series with the inductance found in the TN-system. The influence of type of wiring and apparatus is analysed. The maximum LIO in LVPI networks supplied by an underground cable from an overhead line system is normally little affected by this cable.

• An IT-system results in much higher LIO phase-to-ground than a TN-system and this can explain why the number of transients and damages is large in Norway compared to e.g. Sweden. TN-systems result on the other hand in larger phase-to-phase voltages than IT- systems. However, an IT system with a permanent ground fault will experience both high phase-to-phase and phase-to-ground voltages, compare to a TN-system.

• The LIOs increase proportionally to the line height when the ground is assumed lossless. The additional contribution from a lossy ground is independent of line height. Lightning strokes near the mid-point of an overhead line results normally in the largest LIOs, but lossy ground effects may modify this.

• Lossy ground effects on LIO seem to be very important. Especially in a IT-system the level of calculated LIO increases considerably when a lossy ground is taken into account. The ground losses may reverse the polarity and increase the amplitude of LIOs. However, the effect of a lossy ground is encumbered with uncertainty since a relatively high ground conductivity must be assumed in order to reproduce measurements by calculations.

• To protect a low-voltage system completely from LIOs, surge protective devices must be installed at each individual installation. The level of LIOs in a TN-system is much lower and such systems is to some extent self-protected against remote lightning. Even if arresters are installed at the power service entry, large overvoltages can still arise inside the LVPI network. Oscillations due to reflections in the low-voltage system and with frequencies dependent on overhead line segment lengths could excite the natural frequency of connected LVPI circuits, resulting in large internal overvoltages. Such overvoltages can reach amplitudes of several times the protective level of the connected arrester.

Contents vii

CONTENTS

Preface........................................................................................................... iii

Abstract............................................................................................................... v

List of symbols........................................................................................... xiList of abbreviations ..................................................................................xiiSign conventions..........................................................................................xii

1. Introduction...............................................................................................11.1 Perspective and motivation................................................................ 11.2 Objectives and contents....................................................................... 3

2. Background .............................................................................................. 52.1 Introduction......................................................................................... 52.2 The lightning discharge...................................................................... 5

2.2.1 The thunder cloud 52.2.2 The charge separation 62.2.3 The discharge process 72.2.4 Electrical fields from lightning flash 92.2.5 Triggered lightning 102.2.6 Relative importance of field components 11

2.3 Calculating electrical fields.............................................................. 112.4 Lightning flash models.................................................................... 12

2.4.1 Lightning leader model 122.4.2 Return stroke model 13

2.5 Lossy ground effects.........................................................................142.6 Coupling models............................................................................... 162.7 Calculations versus measurements................................................. 172.8 Sources ofLIO’s............................................................................... 182.9 Conclusions........................................................................................19

3. Lightning induced voltage in overhead lines.......................................203.1 Introduction....................................................................................... 203.2 Lightning channel model..................................................................213.3 Electrical fields from a lightning channel...................................... 23

3.3.1 Static field from charged lightning channel 233.3.2 Fields from return stroke 243.3.3 Discussion 30

3.4 Induced voltages in overhead lines.................................................303.4.1 Analytical time domain solution 323.4.2 Frequency domain solution 343.4.3 Importance of line losses 353.4.4 Importance of decay constant 36

Contents viii

3.5 Calculation models ...........................................................................373.6 LIO calculations ............................................................................... 40

3.6.1 Induced voltages from return stroke 413.6.2 Induced voltages from charged leader 45

3.7 Conclusions........................................................................................464. Lossy ground effects on lightning-induced voltage..............................47

4.1 Introduction................................................. 474.2 Norton’s methods for taking a lossy ground into account .......... 47

4.2.1 Sommerfeld’s exact formulation 484.2.2 Norton’s approximation 49

4.3 Lossy ground effects on electrical fields.........................................524.3.1 Radial field in air 524.3.2 Vertical field in air 53

4.4 Comparing Norton’s and the Surface Impedance methods.........544.5 Lossy ground effects on LIO in overhead lines ............................594.6 Discussion..........................................................................................62

4.6.1 Methods for taking lossy ground effects into account 624.6.2 Lossy ground effects on electrical fields 624.6.3 Lossy ground effects on LIO 63

4.7 Conclusions........................................................................................645. Model measurements of induced voltages ............................................. 65

5.1 Introduction............................. 655.2 Experimental setup ...........................................................................65

5.2.1 The coil 665.2.2 Current waveform 675.2.3 Overhead line 67

5.3 Measurement results.........................................................................685.3.1 End stroke 685.3.2 Side stroke 705.3.3 Ground resistivity 71

5.4 Comparison with calculations..........................................................715.4.1 Open ends 725.4.2 One end grounded 735.4.3 Matched terminations 75

5.5 Triggered lightning................... 765.6 Discussion................. 78

5.6.1 Model measurements vs calculations 785.6.2 Sources of error 805.6.3 Triggered lightning 81

5.7 Conclusions........................................................................................816. Measurements on transformers.............................................................. 82

6.1 Introduction............................................................ 826.2 Experimental setup ........................................................................... 836.3 Transformer input admittance..........................................................83

Contents IX

6.3.1 Measurement results and fitting 836.3.2 Transformer model simplifications 87

6.3.2.1 Neutral isolated 876.3.2.2 Neutral grounded 89

6.3.3 Discussion 916.4 Induced voltage calculations............................................................92

6.4.1 Overhead line terminated by transformer 926.4.2 Discussion 96

6.5 Conclusions .........................................................................................977. Electromagnetic response of LVPI networks .......................................... 98

7.1 Introduction .........................................................................................987.2 Types of LVPI networks ..................................................................... 997.3 External response of LVPI networks ..................................................103

7.3.1 Introduction 1037.3.2 Experimental setup 1037.3.3 Measurements on LVPI networks 104

7.3.3.1 Siriusveien 10,205B 1047.3.3.2 IT input impedance of different installations 109

7.3.4 Measurements on electrical apparatus 1127.3.5 Modelling of LVPI networks 1147.3.6 LIO calculations 1157.3.7 Discussion 117

7.4 Internal response of LVPI networks....................................................1197.4.1 Introduction 1197.4.2 Experimental setup 1197.4.3 Step response measurements on laboratory circuit 120

7.4.3.1 Surface wiring 1227.4.3.2 Underplaster wiring 123

7.4.4 Frequency response measurements on laboratory circuit 1257.4.4.1 Surface wiring 1257.4.4.2 Underplaster wiring 1267.4.4.3 Connected loads 128

7.4.5 Modelling of LVPI circuits 1297.4.6 Comparing measurements with calculations 1297.4.7 Overvoltage calculations in LVPI networks 1317.4.8 Discussion 133

7.5 Conclusions ....................................................................................... 1348. Protection of low-voltage system against LIOs .................................... 136

8.1 Introduction ....................................................................................... 1368.2 Modelling of low-voltage system....................................................... 137

8.2.1 Introduction 1378.2.2 Basic configuration 1378.2.3 Overhead lines 1388.2.4 Distribution transformers 1388.2.5 LVPI networks 1388.2.6 Grounding 1388.2.7 Arrester 1398.2.8 Lightning channel and current 139

Contents x

8.3 Lightning-induced overvoltages in unprotected systems....................1408.3.1 Influence of LVPI-network model 1408.3.2 Influence of the transformer model 1418.3.3 Influence of the grounding impedance 1438.3.4 Influence of a connected underground cable 1448.3.5 Influence of load-changes in a neighbour point 144

8.4 Protection of low-voltage systems ......................................................1458.5 Internal voltages in LVPI network..................................................... 1498.6 Lossy ground effects........................................................................... 1518.7 Conclusions ............................................................................ 153

9. Discussion................................................................................................. 1549.1 Introduction ....................................................................................... 1549.2 Models and assumptions............ .........................................................154

9.2.1 Lightning channel model 1559.2.2 Lossy ground effect model 1559.2.3 Coupling model 1569.2.4 Overhead line termination models 156

9.3 Calculation results ............................................................................. 1579.3.1 System parameters’ influence on LIO. 1589.3.2 IT- versus TN- systems 1609.3.3 Protection of low-voltage networks 161

9.4 Principal contributions....................................................................... 1629.5 Suggested future work ....................................... 163

10. Conclusion ............................................................................................. 164

Appendices 166A. Handling of current waveforms..................................................... 167B. Effect of a tilted lightning channel ............................................... 168C. Inverse fourier transform............................................................... 170D. Properties of Uind(j(o) ................................................................... 172E. Adequacy of the telegraph equations ............................................177F. Fitting of admittance measurements ............................................180G. Admittance measurements on and fitting of transformers ............184H. Modelling LVPI circuits........................... 192I. Expressions of electrical fields over lossy ground........................195J. Listings of ATP-EMTP data cases....................... 197

References 202

List of symbols XI

LIST OF SYMBOLS

t time/ frequency(0 angular frequencyz height of observation pointy, x position of observation point on the ground, x directed along the overhead line.r horizontal distance from the channel base to the observation point, r2 = jc2 + y2 hm(z) height of lightning current front, dependent on time.RJz) distance from current front to observation point. Rm2(z) = r2 + (hjz) - z)2 h height of a dipole I-dhR(z) distance from dipole to observation point. R2(z) =r2 + (h-z)2 R0 distance between observation point an dipole in channel. R0 = R(z).R, distance between observation point an dipole in image. R, = R(-z).H height of the lightning channelc speed of light. 3.0-108 m/sv lightning current velocity. 1-2-10s m/sv' leader velocity. 1-105 - 3-106 m/s60 permittivity in vacuum. 8.85-10"12go permeability in vacuum. 4-n-lO"7I0 current step amplitudei(h, t) current along the lightning channelqR charge per length associated with the return stroke.pL charge per length of the leaderX decay constant (attenuation of lightning current along the channel)Vs scattered voltageU incident voltage induced by the vertical fieldUj incident voltage at terminal AUx horizontal field contributionUxa horizontal field contribution at terminal AUirM inducing voltage at terminal A, UindA(t) = 2-Ux*(t) + U^'(t) - UB‘(t-x).Uin incoming voltage wave to a terminalUref reflected voltage wave from a terminalUrA resultant voltage source at terminal A in the induced voltage calculation modelUA total induced voltage at terminal A of an overhead linexA x co-ordinate of terminal A of an overhead lineL length of overhead line segment (L=xA - xB)L C’, R’ inductance, capacitance, and resistance per length of overhead line Z’ characteristic impedance of overhead line (lossless: Z0' = VL’/C’)Y transmission coefficient of overhead line (lossless: y0=jcj/c)k0 transmission coefficient of air: kg = - j-(o/x0-(a0 + j-eo-eg) = cJ / c2kj transmission coefficient of ground: k,2 = - j-copyfo, + j-co-eju transmission coefficient ratio u=k(/k,

List of symbols Xll

Rv the Fresnel reflection coefficientF(p) the Sommerfeld attenuation function

v

P the numerical distance used in the Sommerfeld attenuation function

Ez incident electrical field directed upward from ground Er incident electrical field directed from the lightning channel. Ex = Erx/r By incident magnetic field directed perpendicular to the r-directionBy incident magnetic field directed perpendicular to overhead line.Aa incident vector potential over lossy ground. Aa = A0 + AaA0 incident vector potential over lossless ground.Aa additional incident vector potential due to a lossy ground.U°ind inducing voltage over lossy ground. Uaind = U°ind + UAind U°M inducing voltage over lossless ground.UAind additional inducing voltage due to a lossy ground.

LIST OF ABBREVIATIONS

LIO(s) Ligfatning-induced overvoltage(s)LVPI(s) Low-voltage power installations)TL Transmission lineMTL Modified transmission lineMOV Metal oxide varistorLV Low-voltage (used mostly in relation to transformers) HV High-voltageIT Isolated Terra (Isolated neutral)TN Terra Neutral (Grounded neutral)EMC Electromagnetic compatibilityCM Common modeDM Differential mode

SIGN CONVENTIONS

Positive return stroke current: directed upward. A negatively charged lightning channel will result in a positive return stroke current. This is the opposite of the normal sign conventions used in lightning research.

Positive vertical electrical field: directed upward. This is the opposite of the meteorological definition of bad weather fields.

1

INTRODUCTION

1.1 Perspective and motivation

This thesis deals primarily with overvoltages in low-voltage systems caused by induced effects from nearby lightning. A lightning stroke radiates rapidly changing electromagnetic fields that induce currents in surrounding structures. Especially overhead lines act as receiving antennas and large potentials can develop between conductors and particularly between conductors and ground. Lightning strokes closer than about 1 km can cause harmful overvoltages in low- voltage systems. The definition of overvoltages is based on EEC 364 [1] which defines impulse withstand levels in low-voltage system as shown in tab. 1.1.

Tab. 1.1 Required impulse withstand voltage for low-voltage installations, from IEC 364.

Nominal voltage of the installation*

m

Required impulse withstand voltage for

[kV]

Three-phase Single-phase Equipment at Equipment of Appliance Specialsystems" systems with the origin of distribution and protected

middle point the installation final circuits equipment

Category IV Category III Category II Category I

- 120-240 4 2.5 1.5 0.8

230/400** _ 6 4 2.5 1.5277/480**

400/690 - 8 6 4 2.5

1000 - Values subjected to system engineersAccording to IEC 38In Canada and USA for voltage to earth higher than 300 V, the impulse withstand voltage corresponding to the next higher voltage in column one applies.

Only voltages exceeding the values in tab. 1.1 are called overvoltages in this thesis and such voltages can cause damage to that particular part of the system. Voltages below the impulse levels in the right column (category I) could cause damages to particularly sensitive, e.g. electronic equipment, but are normally considered harmless in this thesis.

Insurance companies report an increasing number of damages of electric nature over the years

1. Introduction 2

[2, 3]. This is probably caused by the introduction of more and more sensitive electrical equipment in an increasing number of installations. The number of indirect damages from lightning (overvoltages) is much larger than the number of direct lightning strikes damages, but the costs of the two types are about the same [2]. The relation between lightning and damages in the low-voltage system has been evaluated in [3, 4] by comparing fire statistics from insurance companies with data from lightning location systems. Clearly, the number of damages increases during the lightning season in the summer. Comparing the two investigations shows that the damage per lightning ratio seems to be much higher in Norway [4] than in Sweden [3]. A similar result is presented in [5] showing about twice as much damage in sample areas in Norway compared to Sweden, when the lightning stroke and population densities were taken into account. This result could be due to the extensive use of the IT-system (isolated neutral) in Norway, a system which is believed to be more vulnerable to lightning-induced overvoltage. An uncertainty when comparing the two countries is the classification system used by the insurance companies and the number of unreported damages. Anyway, this trace was one of the initial motivation factors for this work.

Measurements of transients in low-voltage systems show that the relative severity of overvoltages is decreasing over the years [6,7, 8]. This observation is in garish contrast to the reported increase in number of damages. However, it could be explained by the introduction of protective devices in the low-voltage system, which will suppress the most severe overvoltages.

Tab. 1.2 shows a sample of measurements of transients in the Norwegian low-voltage system in the period 1992-1996 [9].

Tab. 1.2 Number of transients (per measurement site per year) measured in the Norwegian low-voltage system (mostly 230 V IT). Phase to phase and phase to ground.

Amplitude[pu]

0-0.005

Volt-time ir

0.005-0.01

rtegral [V-s]

0.01-0.1 0.1-1.0

2-3 3.0 0.7 3.2 1.7

3-5 0.2 0.1 0.8 0.1

5-10 0.0 0.0 0.3 0.0

The highest voltages (5-10 pu), typically generated by lightning, are of greatest interest. In a 230 V system this voltage is equivalent to 1.6kV-3.3kV phase to phase and 0.9-1.9kV phase to ground. Tab. 1.2 shows that each site on average will experience such voltages 0.3 times per year. The number of transients is large in Norway compared to other countries [9]. The number of the highest transients is comparable to what is reported from USA in the mid 60's [7] but higher than in the early 90's [8]. In Sweden the number of transients above 100 V is reported to be extremely low for some locations even when nearby thunderstorms were observed [10]. The number of transients (>100 V) in Germany is reported to be very high [11], particularly

1. Introduction 3

in industrial and business areas, but generally low in the domestic areas (apartments).

1.2 Objectives and contents

The main objectives of this thesis are to:• Study induced voltages in overhead line systems and how various parameters in the

network influence these voltages. The following parameters should especially be focused:- System geometry and lightning current parameters- Ground conductivity- Neutral grounding (IT vs. TN system)- Low-voltage power installations and transformers

• Develop a model for induced voltage calculation that can be implemented in the ATP- EMTP [12]. The model should be able to handle complex power systems, including several overhead lines, installations, loads and protective devices.

• Study protection of low-voltage systems.

The nature of the work is basically theoretical, based on calculations of the overvoltages in the system. The calculations can be split in 2 parts: 1) Calculations of incident electrical fields from a lightning stroke and 2) Calculations of the coupling between the field and overhead power lines. Such calculations require models of the various parts of the system, and this thesis will mainly focus on such models. Measurements of overvoltages in actual low-voltages systems have not been performed. It is important to keep in mind that a lightning flash is a very complicated process and that only the main characteristics are possible to study and model. Lightning induced overvoltages are affected and perturbed by the complex lightning channel tortuosity, the irregularities of the Earth's surface and the various configurations of a low- voltage network, including all connected electrical equipment. Typical for all the models are their high frequency capabilities required by the fact that a lightning stroke is a very fast event. The main frequency range of interest is 10 kHz -1 MHz.

Chapt. 2 starts with a description of the lightning discharge process. Then some of the existing models for induced voltage calculations are summarised. Based on the presentation and discussion in this chapter the appropriate models for LIO calculations are selected.

Chapt. 3 describes the theory of calculating induced voltages in overhead lines assuming a lossless system. Two different approaches are described; 1) An analytical time domain model and 2) A frequency domain model also allowing loss effects to be taken into account as well as a more general lightning channel model.

Chapt 4 extends the frequency domain model from chapt. 3 to take lossy ground effects on the incident fields into account. Two different methods are discussed, Norton’s method and the surface impedance approach.

Chapt. 5 contains measurements from a scaled-model investigation. The model consists of a

1. Introduction 4

25 m long overhead line excited be electrical fields from a wound, vertical antenna carrying a current to simulate a real lightning channel located 10 m from the line. The results from this chapter are used to verify and discuss the models established in chapt. 3 and 4.

Chapt. 6 handles modelling of transformers. Based on measurements on 15 distribution transformers simple models of the common mode system are established and as far as possible related to transformer ratings. The dependency of low-voltage neutral grounding is also taken into account.

Chapt. 7 analyses the response of electrical power installations. Measurements have been performed on three types of objects:• On actual LVPI networks with the purpose to study the difference between installations and

establish typical installation equivalents. All measurements are performed in the frequency domain.

• On electrical equipment with the aim to study its impact on the the total response of a LVPI network. Equipment that has shown an important influence is studied more closely. This includes washing machines, dishwashers, PC’s and stereo racks.

• On laboratory installation circuits with an objective to study differences between surface and underplaster wiring, grounding (TN vs. IT systems) and details regarding the location of loads in the circuit.

Chapt. 8 summarises the developed models in a case study of a larger low-voltage network. The main purpose of this chapter is to demonstrate typical overvoltages that could be expected in a low-voltage system and how to protect the system with arresters.

Fig. 1.1 shows the focus of this thesis, dedicated to induced overvoltages in overhead lines. Direct lightning strokes, direct induced voltages in installations, and special EMC problems related to the interaction between telecommunication and power systems are not treated.

Focus of this thesis

Chapt. 6 and 7Chapt. 3 and 5

Overhead line' TelecommunicationLightning Lightning

Chapt. 4

Ground

Power cable Signal cable

Fig. 1.1 System configuration related to the focus of this thesis.

2BACKGROUND

2.1 INTRODUCTION

Before starting to calculate lightning-induced voltages in a low-voltage system it is favourable to take an overview to realise the complexity of the problem. This chapter presents necessary background in order to select appropriate models for induced voltage calculations and in addition outlines the involved assumptions. The selection process is founded on the decision to use models based on measurable quantities, and which are reported to predict at least the first few microseconds where often the maximum induced voltage occurs.

The chapter starts with a brief description of the lightning discharge process. This serves as a basis for modelling the lightning channel for calculation of the electrical field from a flash. Models for taking a lossy ground into account when calculating the electrical fields are then summarised. Finally the models of overhead lines for lightning-induced voltage calculations (or coupling models) are presented.

This thesis will focus on lightning-induced overvoltages (LIOs) at overhead line terminals. LIOs transferred from the HV side of the distribution transformer and LIOs induced directly in loops in a low-voltage power installation will not be addressed. However, in this chapter the importance of such overvoltages is briefly discussed and compared with LIOs in overhead lines. Direct lightning strokes in the low-voltage system result in very large overvoltages and severe damages, but will not be studied in this thesis.

2.2 THE LIGHTNING DISCHARGE [13]

2.2.1 The thundercloud

A thundercloud (cumulonimbus) differs from an ordinary rain cloud mainly in its large vertical extent sometimes reaching an altitude above 10 km. A necessary condition for formation of a thundercloud is hot, humid air at lower levels which in interaction with cold air masses is forced to rise in a strong updraft. This forms the thundercloud in which the temperature typically is well below 0°C. At this temperature the accumulation of water in the solid or liquid state forms particles, which the updraft no longer can support, and heavy rain or hail is formed. Fig. 2.1 shows schematically a typical thundercloud with a positive charge centre in the upper part of the cloud and a negative in the lower part along with a possible small pocked of positive charge at the very bottom. On average the positive charge centre is P=40 C, the

2.2 The lightning discharge 6

negative charge centre is N=-40 C, and the small lower charge centre p« 10 C [13].

IE

Fig. 2.1 Illustration of typical thundercloud [13].

2.2.2 The charge separation

The source of lightning is the separation of electric charge within thunderstorm clouds. The charge separation phenomenon is not fully understood and at least two theories explain the various mechanisms. In the first charge separation theory, called the precipitation theory, heavy, falling precipitation particles (typically hail) collide with lighter particles (e.g. ice crystals) carried updrafts. This causes the heavy particles to be charged negatively and the lighter positively. Fig. 2.2 shows how this process is facilitated by the electrical field in the thundercloud. How the charge separation process starts is more uncertain.

Fig. 2.2 Collision between light particle carried updrafs and heavy falling particle.Left: Before collision, charges separated due to external field. Right: After collision.

In the second theory, called the convection theory, charge is accumulated near the Earth's surface or at boundaries between regions with different air and cloud conductivity, including the cloud boundary, and moved in bulk by the updraft.

Probably both mechanisms are active, but the former is generally believed to be dominant while the latter can explain thunderclouds with temperatures above 0° C.


2.2.3 The discharge process

Far more than one half of all lightning discharges take place within the cloud or between clouds and are with a generic term called intra-cloud discharges. Although such discharges have been reported to sometimes cause high induced voltages at ground (e.g. [14]) they will not be handled in this thesis. Instead the focus will be on cloud-to-ground discharges (CG) which lower either negative or positive charge to ground. CG discharges are divided in four categories, dependent on the initial formation process of the lightning channel: 1) Negative downward leader, 2) Positive downward leader, 3) Negative upward leader and 4) Positive upward leader. The last two categories are initiated from ground objects and are associated with lightning strokes to mountains, tall structures or e.g. triggered/ artificially initiated lightning. The first two categories being dominant in natural lightning are initiated from the cloud as shown in fig. 2.3. The negative type is most common and will lower negative charge to ground and result in an upward directed return stroke current. The fraction of the positive type will increase with the age of a thunder storm [13] and this type is e.g. more dominant in winter thunder storms and severe storms with a strong horizontal wind separating the positive and negative "layer" horizontally.

Fig. 2.3 Negative (left) and positive (right) leader development.

The negative leader initiated lightning flash can be divided in several steps or processes: Preliminary breakdown, stepped leader, attachment process, 1st return stroke, intermediate processes (J, K, continuous current, and M-components), dart leader and subsequent return strokes. The whole process, which can consist of several subsequent return strokes, is called a lightning flash. The total duration of a flash is around half a second.

The lightning flash process starts with a preliminary breakdown (of which the exact knowledge is rather limited) in the cloud and this causes the electrical field outside the cloud to exceed the threshold value in air, which initiates a negative leader development. This leader propagates in steps towards the ground and is called a stepped leader. The stepped leader can be highly branched and irregular. When the stepped leader approaches ground the electrical field here increases and causes upward positive streamers to develop from sharp and tall structures at the Earth’s surface. This development is called the attachment process and each such streamer can have a length in the range 10-100 m (the attachment process plays an important role in


insulation co-ordination of overhead lines). When one of the streamers reaches the downward propagating negative leader the first return stroke starts from ground. The return stroke neutralises the charge in the leader and results in an upward propagating current with a short front duration in the order of some microseconds and an average peak value of 30 kA [15,16]. The return stroke causes the temperature in the lightning channel to increase to about 30000 K [13], which produces a pressure wave heard as thunder. Sometimes the flash stops after a single return stroke but more often it is succeeded by a dart leader propagating from the cloud, usually in the already formed lightning channel. When this dart leader reaches the ground the subsequent return stroke starts. This return stroke is similar to the first return stroke but the magnitude and front time are less with a magnitude of 12 kA as an average [15]. The dart leader may also initially follow the channel and at some point find a new path to ground. This results in a fork shaped lightning channel often observed on photos [13].

After a return stroke a continuous current with a magnitude in the order of 100 A may flow in the lightning channel, and roughly half of the negative flashes contain at least one period of such current [13]. The continuous current can lower a considerable amount of charge to the ground since the duration is long compared to the return stroke. So-called M-component currents can also occur, observed as increased lightning channel luminosity and electrical field pulses. Also intermediate events called J and K processes take place in the cloud [17, 18].

The leader initiating a positive lightning flash does not exhibit the stepped characteristic of negative leaders but is instead a continuous process. Normally the positive flash consists of only one return stroke followed by a continuous current. According to [15] positive lightning currents have an average magnitude of 35 kA and a front duration of 22 ps. However, the largest measured lightning return stroke currents are positive.

Tab. 2.1 shows a summary of the lightning flash characteristics from [19] in good agreement with the data given in [13,20,21,22].

Tab. 2.1 Characteristics of negative cloud-to-ground discharges [19].

Parameters Minimum Typical Maximum

Stepped leaderStep length [m] 3 50 200Time between two steps [ps] 30 50 125Average velocity of propagation [m/s] l-lO5 1.5-105 2.6T06Charge deposited [C] 3 5 20

Dart leaderVelocity of propagation [m/s] MO6 2-106 2.T107Charge deposited [C] 0.2 1 6


Return strokeVelocity of propagation [m/s] Temperature [K]Channel length [km]

0.5T0880002

1.H0*200005

1.8-10®3600014

Continuous currentDuration [ms] 50 150 500Maximum amplitude [A] 30 150 1600Charge [C] 3 25 330

Flash characteristicsNumber of strokes per flash 1 3 26Stroke interval [ms] 7 33 150Flash duration [s] 0.001 0.3 2Charge transferred [C] 1 20 400

2.2.4 The electrical fields from a lightning flash

In [23], electrical field prior to the first negative return stroke is studied and compared with similar investigations reported in the literature. The stepped leader is preceded or initiated by what is denoted 'characteristic pulses' whose duration varies considerably dependent on the cloud height, leader branching and possible horizontal intra-cloud discharges in contrast to the actual stepped leader having a fairly constant velocity. The offset of the electrical field is difficult to determine since it is influenced by the static field from the cloud charge reservoir. The different sign in the time derivative of the leader field shown schematically in fig. 2.4 (negative in 2.4a and positive in 2.4b) can be explained by the increased importance of the charge depletion in the cloud with distance, as will be shown in chapt. 3.3.1. In figures 2.4 and 2.5 the positive direction of the field is downward.

Streak photograph recording

M-componentstepped leader dart leader

K-change

1st return stroke2nd return stroke

Electrical field recording

Streak photograph recording

M-componentstepped leader dart leader

/ \ K-change

1st return stroke2nd return stroke

Electrical field recording

2.4a) Nearby ( < 1-3 km) lightning flash. 2.4b) Distant ( > 10 km) lightning flashFig. 2.4 Illustration of electrical field and streak photograph recordings, adopted from [23].

Negative charge lowered to ground. (Typically measured with a low frequency band width.)


0 50 100 ISO 170

D = 5,0 kmD = 1,0 km

30

D = 2,0 km D = 10 km

Fig. 2.5 Vertical electrical field from return stroke at distances D=l-10 km. Adopted from [24]. Solid line: First return stroke. Dotted line: Subsequent return stroke.

Negative charge lowered to ground.

From the low frequency recording in fig. 2.4 all the components (leader, return stroke, K- change etc.) look significant, and must in general be included in the lightning flash model. When calculating the induced voltage in an electrical system with special focus on the maximum voltage we are not so interested in this low frequency characteristic, however. Measurements performed with antennas with a higher band width e.g. lkHz-lMHz [19 fig. 2.29], typically show only the return stroke field in addition to smaller components of dart leader fields and K-change fields. Fig. 2.5 shows typical vertical return stroke fields propagating a distance 1-10 km over lossy ground. The typical shape consists of an initial steep edge or peak followed by a long lasting ramp. The measurements of vertical return strokes fields in [24] are often used as reference of how the electric and magnetic fields vary with distances 1-200 km. Much less attention has been paid to the horizontal return stroke field, but some measurements are found in [25].

2.2.5 Triggered lightning

Triggered lightning is here defined as artificial lightning initiated by launching small rockets trailing conducting wires during a thunderstorm. Triggered lightning differs from normal natural lightning in that the lightning flash is initiated by an upward moving leader from the rocket. Triggered lightning results in a first return stroke different from natural lightning due to the different leader development, but the subsequent return strokes in the two cases show about the same characteristics [13]. However, in general triggered lightning has a more pronounced continuous current and a larger number of return strokes than natural lightning. The

2.3 Calculating electrical fields 11

current amplitudes are smaller than in natural return strokes, but more charge can be lowered due to the larger continuous current The dart leader initiating the subsequent return strokes has been reported to have a larger velocity than in natural lightning [26]. The upward leader from the rocket is in most cases positively charged and will therefore produce a negative return stroke (directed upward), but also negatively charged leaders are observed.

2.2.6 Relative importance of field components

When calculating the induced voltage in overhead lines it is the fast changes of the electrical field which are of greatest importance. Therefore a component as the continuous current can be ignored or at least handled as a modification of the return stroke current's tail. Also we observe that the stepped leader and also the dart leader are much slower processes than the return stroke. In [17] the K and M processes have been analysed and they too are much slower processes than the return stroke. It can be concluded that the return strokes give the highest contribution to the electrical field and thus are of greatest interest when studying induced voltages. Also the leader field will briefly be studied. Normally the return stroke (v=V108 m/s) is a much faster event than the leader development (stepped leader: v-2-105 m/s, dart leader: v-3-106 m/s) and in most cases their contributions to the total electrical field can be separated and treated individually.

For triggered lightning the velocity of the dart leader (2-107 m/s [26]) is only a decade less that the velocity of the return stroke making it difficult to separate the contributions from the leader and the return stroke. The leader field from triggered lightning has been studied in [27] for distances 30 m and 500 m. The significance of the leader field increases with decreasing distance. For triggered lightning both the field from the leader and the return stroke will induce time dependent voltages in an overhead line. This is seen in e.g. [28,29, 30] and a model for calculating the induced voltage from the charged leader and the return stroke simultaneously is described in [28].

For natural lightning analysed in this work the charged leader and the return stroke fields are separable, and the voltage induced by the leader processes will be almost static.

2.3 CALCULATING ELECTRICAL FIELDS

A lightning channel is charged by the leader and basically discharged by the return stroke. To be able to calculate the electrical field from a lightning channel, either the charge or current distribution must be known and these are related to each other through a continuity equation. Normally the current distribution is used to calculate the electromagnetic fields from a return stroke since the current at ground is measurable and certain assumptions regarding the variation along the channel can be applied (which are discussed in section 2.4). The commonly used method for calculating the fields is called the antenna model (e.g. [31, 32]) in which the total

2.4 Lightning flash models 12

field is composed of contributions from dipoles or vertical current elements along the lightning channel (together with their images in the ground). It thus becomes possible to take the dynamics of the discharge process into account. Also the difference in distance between the observation point and points along the lightning channel, resulting in a retarded time formulation, is handled by the antenna model. The antenna model is used extensively in chapter 3. The model results in an electrical field from a lightning channel composed of three terms:1) Static: Dependent on the time integral of the lightning current. Dominates close to the

lightning channel (near region).2) Induction: Dependent on the lightning current.3) Radiation: Dependent on the time derivative of the lightning current. Dominates far away

from the lightning channel (far region).

2.4 LIGHTNING FLASH MODELS

Calculating the electrical field from a lightning channel is in nature a very complicated task dueto the following list of obstacles:• The distribution of current and charge along the lightning channel and in time is unknown.

Only the current at ground level is measurable.• The velocity of the lightning current as a function of height and the time constants associated

with the discharge processes in the lightning channel are unknown. The current wave velocity can be estimated from recordings of light emittance.

• The lightning channel is tortuous with branches and bends.• The lightning discharge process consists of several events spanning from the relatively slow

stepped leader and continuous current processes via the first return stroke to the dart leader and subsequent return strokes. Each of these events has its own characteristics and the electrical field generated by a lightning flash is the sum of the contribution from each event.

• Correlated measurements of electromagnetic fields from a lightning channel and the lightning current at ground level requires in practice artificially initiated lightning (like triggered lightning) which will affect the lightning discharge process.

Normally the responses of a charged leader and a return stroke are calculated in two different ways; the return stroke field is based on the lightning current and the leader field is based on the charge. The charge and the current are related however, since the lightning current tends to neutralise the leader charge. Thottappillil etal. [33] have developed a method where both the return stroke and leader field are based on a common charge formulation.

2.4.1 Lightning leader model

The lightning leader is traditionally modelled as a straight vertical line charge developing from the cloud charge reservoir and propagating downward with a velocity v' [28,33,34]. The leader

2.4 Lightning flash models 13

field is composed of the contribution from the charge in the channel and the depletion of the charge reservoir in the cloud. An exact formulation of the leader field is given in [33, 35]. The dynamics of the leader development will not be studied in this thesis. The leader development is in general much slower than the return stroke and the leader field will be treated as a static field from a fully developed lightning channel. The leader charge per unit length in this channel will be calculated from the return stroke current, assuming that the leader charge is completely removed by the return stroke.

2.4.2 Return stroke models

All the available models for calculation of electrical fields from return strokes are mainly based on two quantities: the current at ground level and a current wave velocity. Statistical data of lightning current parameters are available from e.g. [15, 16] based on measurements of lightning strokes to tall tower structures. The current wave velocity can be obtained from streak-photograph techniques. In [36] current 2-D velocities for natural lightning have been measured to be 1.1 TO8 m/s on the average along the lightning channel and 1.4*10* m/s close to the ground surface. The velocity in the first stroke was in most cases found to be lower than in the subsequent return strokes. Many authors have stressed the necessity of using a lightning channel model which takes a proper height variation of current and velocity into account.

The transmission-line model TL [37] which is the most simple lightning channel model is used by several authors, including Rusck [38]. In this model a current wave travels upward from ground with a constant velocity v. The model has been tested in e.g. [39, 40] using triggered lightning. The model shows reasonable agreement with measurement during the first few microseconds and is able to predict the peak value of the electrical far field (5.15 km) with fan- accuracy. However, the measured fields were a bit higher than expected. The TL model is not able to reproduce the typical observed initial peak in the electrical field (ref. fig. 2.5).

Yokoyama et.al. [41,42,43] used the TL model when analysing the induced voltages on a 820 m long test line from lightning strokes to a 200 m high tower. Using the pure TL model, reasonable agreement between calculated and measured induced voltage was obtained although the measured voltages were somewhat higher. They further assumed that the return stroke current wave propagates upward from the top of the tower with a constant velocity v. Simultaneously a current propagates down the tower to ground with the speed of light. The reflection between the tower and ground was ignored. This model improved the results and made the induced voltage peak less dependent on the return stroke velocity. Similar analyses have been performed in [39,40,44,45 and 46] some based on the idea that the return stroke current wave starts to propagate from a junction point between the (stepped) leader and the upward propagating streamer from ground.

The modified transmission line model MTL [47] is similar to the TL model except that the amplitude of the current decays with height h by a factor exp(-hA). Where X is called the decay constant and is assumed to be in the range 1-2 km. This model is more accurate or flexible than

2.5 Lossy ground effects 14

the TL model and manages to reproduce the increase (ramp) of the electrical field after the first hump measured close to the lightning stroke (ref. fig. 2.5).

A model that is able to reproduce the characteristic initial peak or hump of the measured electrical field is the travelling current source model TCS [48]. In this model the current wave front travels upward with a velocity v, while the instantaneously released charges travel toward ground with the speed of light. In [49] the TSC model is used to calculate the statistical distribution of current data from measured distant electrical fields finding reasonable agreement with the data from tower measurements in [15].

Other models are the Diendorfer-Uman DU model [50] and the modified Diendorfer-Uman MDU model [51] which both are extensions of the TCS model including an exponential release of the leader charge. Tottappillil & Uman [52] have further modified the MDU model. This model (VDTC) uses only a single height-variable discharge time constant instead on the two height-invariant time constants used in the DU model. Cooray [53] has suggested yet another model closely related to corona discharge physics.

A summary and comparisons of five of the existing return stroke models can be found in [54, 55]. None of the proposed models manage to predict the measured electrical fields from a lightning channel accurately for all lightning current measurements, and the simple models seem as accurate as the advanced ones for practical engineering purposes. In [55] the TL model is suggested as the first choice model for simple studies when only the peak value of the field is of interest. The problem with the advanced models is their requirement of information on non-measurable parameters of the lightning current along the channel and not only the direct measurable current at ground level. In [33] six different lightning channel models are examined based on the charge deposited by the return stroke and its relation to the leader charge. This analysis shows that the MIL and TL models are both extreme cases and that the other models result in a charge distribution somewhere in between these two models.

The analysis in this thesis will use the MIL model with the TL model as a special case. The advantage with the TL model is that it makes an analytical formulation of LIO in overhead line possible. When calculating LIO the peak voltage is of greatest interest and it is important to realise that the peak often occurs within the first few microseconds. Therefore the lower parts of the lightning channel will determine the peak value.

2.5 LOSSY GROUND EFFECTS

When the electrical field from a lightning channel propagates over lossy ground it is attenuated. While the vertical electrical field is little affected for the short distances of interest related to LIO in electric power systems, the horizontal electrical field is strongly influenced by a lossy ground. This is reasonable since the vertical field consists of additive contributions from the


lightning channel and its image while the horizontal field rather consists of the difference between these two contributions [56]. A minor change in the image contribution will thus affect the horizontal field relatively more than the vertical. Fig. 2.5 illustrates how the lightning current in lossy ground flows:

Ci, s.

Fig. 2.5 Illustration of lightning current in ground.

The horizontal electrical field is strongly influenced by the lightning current component along the Earth's surface. For a lossless ground the horizontal field is zero at the Earth's surface, but for a finitely conducting ground the magnitude of the field increases to a finite value.

The lossy ground effect on the horizontal electrical field can be calculated using various techniques. The two most simple models are called the wave-tilt and the surface impedance methods. The wave-tilt approximation applies rigorously to the plane waves case with grazing angle with respect to the ground plane.

The solution to the problem of finding the electromagnetic fields over a flat, homogeneous, lossy ground was first formulated by Sommerfeld [57, 58]. Various approximations to this exact solution have been suggested over the years, e.g. the method developed by Norton [59, 58]. A further simplification of Norton's method was introduced by Wait in [60] as an analytical time-domain attenuation function. Another approach is the complex image theory introduced by Bannister, e.g. in [61].

An important paper in the field of loss effects is the one by Zeddam&Degauque [62] in which various approximations are evaluated and compared with Sommerfeld's exact formulation. In general Norton's method was found to give the best agreement except for low frequencies f<10 kHz and low dipole heights h<r/100 at intermediate distances r=5-55 km where Bannister's complex image theory [61] was found to give the best fit. Based on either Norton's or Bannister's approximations the currents induced in an overhead line or telecommunication cable were calculated and found to give qualitative and quantitative explanation of observed waveforms.

Thomson et.al. [25] studied the horizontal field from return strokes at a distance of 7 to 43 km. Using the wave-tilt method they managed to reproduce the measurements qualitatively although the calculated field was 33% higher than measured.

2.6 Coupling models 16

The surface impedance and the wave-tilt methods were analysed by Cooray in [63]. The surface impedance method gave results in very good agreement with the more accurate image theory [61], except for distances shorter than about 200 m. The wave-tilt method was generally acceptable only for large distances (r>10 km). In [64] the surface impedance approximation was compared with Norton's method. For the radial distances r=200-5000m and lightning channel heights h=0-2000 m the error of the surface impedance method was less than 15%. The surface impedance method was also evaluated by Rubinstein [65] and compared with the results in [62] (using mostly the Norton’s method), obtaining excellent agreement.

Cooray [66] used Wait’s approximation when studying attenuation of the vertical electrical field at large distances (>10 km). He established a simple empirical relationship which managed to predict the attenuation of vertical electrical fields observed in [24]. The attenuation of the vertical electrical radiation field from distant lightning has also been studied in [67,68, 69], using Wait’s method. In these papers the impacts of a stratified ground, a rough ocean surface and a mixed propagation path were investigated. Such investigations are of particular interest in a radio science perspective and when studying the measured electrical field in relation to lightning-detection systems and how it is related to the lightning current. The surface impedance and Wait’s approximations were used in [70] to calculate the LIO and estimate fault rates for an overhead line over lossy ground. The expected number of faults was reported to increase with decreasing ground conductivity.

None of the approximations, Wait, Surface impedance, and Wave-tilt take the dependency of the dipole height (variations along the lightning channel) into account. And as a result of this a lossy ground modification function can be formulated and applied to the total field from a lightning channel. However, near the lightning channel the height variation becomes important and more sophisticated methods are required. Calculating induced voltage in overhead lines due to nearby lightning requires the dynamics of the lightning current and the height variations to be taken into account. Norton's method is therefore preferred in the calculations in this thesis (chapter 4), but the surface impedance approach will also be used.

Besides the attenuation effect on electrical fields propagating over the Earth’s surface, the induced voltage in an overhead line is affected by losses due to surge propagation on the line. This is more related to the overhead line model and will be studied in chapter 3.4.3.

2.6 COUPLING MODELS

A coupling model is a model for calculating induced currents and voltages on a transmission line excited by electromagnetic fields. Four different coupling models are mainly used in the literature. These are the models proposed by: Rusck [38], Taylor et.al. [71], Chowdhuri et.al. [72], and Agrawal et.al. [73]. All these models are based on adding one or two source terms, representing the incident fields, to the classical telegraph equations:

2.7 Calulations versus measurements 17

dU(x,t) + dx

R adt

•I(t,x) = S,(x,0 (2.1)

a&J) = (2.2)dx dt

The Agrawal and Taylor coupling models are equivalent, but the Agrawal model is more convenient since it uses only one source term. The Rusck model is equivalent to these two models for vertical lightning strokes as shown in [74, 75]. Rusck’s model is, however, based on splitting the vertical electrical field in two components (gradV and dAJdi) where only the - gradV component is used as a source term. This approach is, in spite of its elegant simplicity, not compatible with the requirement to base the model on measurable quantities. The Chowdhuri model does not take the influence of the horizontal electrical field into account. The Agrawal model is chosen as the coupling model in this thesis.

2.7 CALCULATIONS VERSUS MEASUREMENTS

Eriksson et.al [14] have measured the induced voltages from natural lightning on a 9.9 km long test line over two lightning seasons and registered about 300 overvoltages greater than 12 kV. The stroke location was estimated by all-sky video cameras with flash-to-thunder time recordings. The distribution of the measured LIOs was compared to Monte-Carlo simulations using Rusck’s simple expression for the maximum induced voltage [38] on a doubly-infinite long line with the current amplitude based on statistical data [16]. The measured and calculated maximum LIOs distributions are remarkably coincident in spite of the very simple model in use. The maximum voltage in an overhead line at the point closest to the lightning stroke is according to Rusck [38]:

^ = 30'V- 1+-

V^P5(23)

where P = v/c, z is the height of the overhead line and y is the distance to the lightning stroke.

Barker et.al [30] have performed correlated measurements of triggered lightning currents, electrical and magnetic fields at 50 and 110 m distance and induced voltage in a 682 m long test line 145 m from the rocket launcher. The measured induced voltages are roughly 63% higher than calculated by Rusck’s formula (2.3). They also calculated the voltage based on the measured fields assuming a 1/r dependency. These calculations were remarkably close to the measured voltages at least at the centre of the line. This indicates that the coupling model (Taylor et.al. [71]) used in the calculations is accurate.

Lossy ground effects will complicate the comparison between measurements and calculations and make it difficult to distinguish between the lossy ground effects and the lightning current

2.8 Sources ofLIO's 18

effects. In cases where the fields propagate over salt water the lossy ground effects are assumed to be ignorable though. This is the case for the electrical field measurements in [39,40] but not in [24] showing typical electric and magnetic fields for the 1-200 km range (fig. 2.5).

Master & Uman [20,76] studied the induced voltage from distant lightning (> 5 km) in a 460 m long overhead line with open ends. They observed that the sign of the induced voltage was dependent on the lightning stroke position, something that could only be explained by considering the horizontal electrical field and effects due to lossy ground. Using the wave-tilt approximation for the horizontal field they managed to qualitatively calculate the induced voltages. Although Rubinstein et.al. [77] identified an error in the calculations in [76] which altered the sign of the calculated voltage for end strokes, the conclusions in [76] remained unchanged. A relatively high effective ground conductivity of 1.6T 0"2 S/m was assumed, based on comparison between measured vertical and horizontal fields and the wave-tilt formula.

Koga etal. [78] calculated induced voltages in 1 km long telecommunication cables using the wave-tilt formula for lossy ground effects and a variant of the Chowdhuri coupling model [72]. Simple calculations (probably similar to [14] and in addition assuming a ground conductivity of 0.01 S/m) agreed very well with measurements in actual telecommunication systems. They also suggested that the peak voltage mainly is inversely proportional to the square root of the ground conductivity and they even managed to illustrate that this fits well with measurements of induced voltages in areas with different ground conductivity.

2.8 SOURCES OF LIO's

A lightning stroke will induce voltages both in high- and low-voltage overhead power lines, telecommunication and other signal cables, and directly in the connected low-voltage power installations (LVPIs). In addition to this some special EMC problems arise in cases where an electrical apparatus is supplied by both an electric power line and a telecommunication/signal line. The analysis in this thesis focus on LIOs in LVPIs connected to overhead power line systems.

Perez [79] has investigated the lightning induced voltage in a LVPI both disconnected and connected to the supplying electrical network (under-ground cable). He found the induced voltages to be far less when the LVPI was disconnected. In [80] similar results as in [79] are presented. The directly induced common-mode voltages in a LVPI with the PE-conductor grounded are measured to be just above 15 V for a lightning stroke at 10 km distance which is much less than for a LVPI connected to an overhead power line. An equivalent result was obtained in [81].

The lightning induced overvoltages in the high-voltage (HV) network will be transmitted to the low-voltage (LV) network through the distribution transformers. This process is dependent on the transformer characteristics and the input impedance of the LV-system seen from the

2.9 Conclusions 19

transformer terminals. Transformed voltages from the HV-side to the LV-side were studied in [82]. Calculations indicated that the transformed voltages are low (maximum 2.5 % of the voltage is transformed to the LV side). Transformed voltages from the HV- to the LV-side were investigated experimentally in [83] using a standard transformer configuration common in the French network. A surge was applied at the HV-side which was protected by surge arresters having 75 kV residual voltage. The LV-side was connected to a real overhead line and the maximum transformed voltage was measured to 2 kV phase to neutral and 8 kV phase to ground. The transformed voltage from the HV to the LV side was also investigated in [84]. About 10 % of the HV voltage was transmitted to the LV side. Since the transformed voltages from the HV-side to the LV-side are much lower than the directly induced LIO in low-voltage power lines, they are neglected in this thesis.

2.9 CONCLUSIONS

For natural lightning analysed in this work, the charged leader and the return stroke fields are separable, and the voltage induced by the leader processes is almost static. The induced voltage from the return stroke is focused in this thesis, and the contribution from the leader is only studied briefly.

The analyses in this thesis use the MIL lightning channel model with the TL model as a special case. When calculating LIO the peak voltage is of greatest interest and it is important to realise that the peak often occurs within the first few microseconds. Therefore the lower parts of the lightning channel will determine the peak voltage value. The lower part of the lightning channel is assumed to be straight and vertical. The velocity of the return stroke current is kept constant (height invariant) on a value around one third to one half of the speed of light. The lightning current amplitude is assumed to have a mean value of 30 kA, regardless of the direction or polarity.

To take lossy ground effects into account, Norton's method is used along with the surface impedance method. Norton's method is more accurate for nearby lightning, but the surface impedance method is much faster.

The Agrawal model is chosen as the coupling model in this thesis. This model uses the horizontal electrical field as a source term and the vertical electrical field as a boundary condition.

The further analyses in this thesis focus on lightning-induced overvoltages in LVPIs connected to low-voltage overhead line systems. Direct lightning strokes, transformed voltages from the HV-side, overvoltage via the grounding system and voltages induced directly into LVPIs are not handled.

LIGHTNING-INDUCED VOLTAGE CALCULATIONS IN OVERHEAD LINES * •

3.1 INTRODUCTION

To establish an efficient protection against LIO in a low-voltage system it is necessary to predict the level of overvoltages in overhead lines. Such prediction requires a model for lightning-induced voltage calculations. The calculation process consists of two steps:• Calculation of the incident electrical field from a lightning channel.• Calculation of the overvoltages in an overhead line caused by the incident electrical field.

In chapter 2 the modified transmission line model (MTL) [47] was chosen for the lightning channel since this model is simple and manages to predict the peak value of the electrical fields with fair accuracy. The transmission line model (TL) [38,37] is treated as a special case of the MIL model. Using the TL model enables an analytical formulation of the induced voltage in an overhead line. Agrawal's coupling model [73] was chosen for induced voltage calculations in overhead lines.

First, the electrical fields are calculated by using a classical antenna model approach. Assuming the TL lightning channel model analytical expressions for the electrical fields are obtained, covering Rusck's expressions [38]. Both the fields from the return stroke and the leader are studied. A vector potential formulation in the frequency domain is also outlined. This formulation enables losses to be taken into account.

Next, the induced voltage in overhead lines is studied. Two different approaches are analysed; 1) An analytical time domain solution based on the IL model and 2) A frequency domain solution based on the vector potential formulation. The importance of overhead line losses is studied as well as the difference between the MTL and TL lightning channel models. The expressions developed in the frequency domain will be the basis for lossy ground calculations in chapter 4.

Finally, a calculation model for implementation in the ATP-EMTP is presented. A source term called the inducing voltage is defined which takes both the vertical and horizontal field components into account. Based on this model a parameter study is performed which reveals how the various parameters in the system influence the LIOs.

3.2 Lightning channel model 21

3.2 LIGHTNING CHANNEL MODEL

The current in the MTL model can in the time and frequency domain be written (positive current direction is upward), when assuming a step current at ground level:

Time domain:i(h,f) = i(0,t~h/v)'e ~h/l = I0-e-ha-H(t~h/v)

where H(t) is the unit step or Heaviside function

(3.1)

Frequency domain:= A.e-*•(/<i)/v + l/X> (32)

The TL model is obtained as X approaches infinity in (3.1) or (3.2).

This model applies to the return stroke only. However, the leader charge should also be considered, and for the calculations presented in this chapter the following three assumptions are important to note:1. The leader (stepped or dart) charges the lightning channel. The return stroke starts when the

leader reaches ground. The leader development is a relatively slow process which is assumed to be static.

2. The return stroke neutralises the leader charge rapidly as the current wave propagates upward the lightning channel with a constant velocity.

3. Only the lower part of the lightning channel is of importance when studying the maximum lightning-induced voltage in a system. This part is assumed to be vertical.

The lightning current will be dependent on the leader charge, but in this chapter the lightning current is used as the basic quantity and the leader charge is then calculated from this current Charge per unit length is in the following calledjust charge.

The charge in the lightning channel deposited by the leader is denoted pL(h,t) and when the return stroke starts this charge is written as pL(h). This charge is then neutralised by the return stroke. The current and charge of the return stroke are related by a continuosity equation [33]:

(3.3a)0

When the return stroke is finished the charge deposited by the return stroke is assumed to be equal to the negative leader charge:

o(33b)

3.2 Lightning channel model 22

Inserting the current according to the MIL model results in:

0(3.4)

Inserting the step current from (3.1) gives:

•e 'ha-H(t-k/v) (3.5)

The deposited charge, qR in (3.5) will approach infinity (as t-») unless the decay constant X is infinite (TL model). Assuming the TL model results in the leader charge pL(h) =- qR(h, <*>) =-I</v.

If an exponentially decaying return stroke current at ground level is assumed instead, equal to i(0, t) -I0 ■ exp(-a• t), the return stroke charge is from (3.4) found to be:

(3.6)v v X-a) X-a

where the exponential term inside the brackets is called the transferred charge (which becomes zero as the time approaches infinity) and the last term is called the deposited charge. The leader charge is equal to the negative deposited charge. An interesting feature of (3.6) is that the transferred charge is always zero if v=A -a. This corresponds to that the leader charge p(h) is neutralised immediately as the current wave reach the height h. Assuming the transferred charge to be zero, removes the freedom of choosing and independent X. According to [85] the MIL model uses a decay constant X between 1 and 2 km. The MIL model was in [33] shown to give an unrealistic charge distribution along the lightning channel using the same current shape and decay constant (2 km) as in [54]. For a current wave velocity of v=l.l-108 m/s and a time decay constant of a=2-1041/s (=l/50ps), the decay constant becomes X= 5.5 km which is much larger than the suggested 1-2 km range from [85].

When developing the basic expressions for electrical fields and induced voltage, the lightning current is assumed to be a step with amplitude I0 at the ground level. An arbitrary current shape can be taken into account by a final convolution integral as shown in appendix A.

3.3 Electrical fields from a lightning channel 23

3.3 ELECTRICAL FIELDS FROM A LIGHTNING CHANNEL

3.3.1 Static field from charged lightning channel

The geometry of the system is shown in fig. 3.1a). The leader is assumed to develop from the base of the cloud at a height H above ground. The leader grows vertically downward with a velocity v’ and the height between ground and the leader front is h=H - t-v’. The distance between the leader front and the observation point is R(z).

imageFig. 3.1 a) Configuration of the system. Leader development.

The vertical electrical field at ground at a distance r, due to the developing leader is [33, 34]:

E'(r,f,z=0) -1 /H \

2lteo Hl,.S *3(°) (r2+H2)312•pL{h,t)-dh (3.7)

Eq. (3.7) is the semi-static approximation adopted from [33], valid for nearby lightning and where in addition the retarded time R/c is ignored. The first term in (3.7) is due to the charge accumulation in the lightning channel while the second term is due to the depletion of the charge in the cloud reservoir. To find the total field the electrical field in (3.7) should be added to the static field from the cloud. An exact formulation of the leader field is given in [33,35].

The electrical field when the leader reaches ground (t=H/vr) is found by inserting the leader charge from (3.4), knowing that pL(h) = - qR(h, °°):


E (r, 0// \ H ( \

1 . f h H2ite0 J X

l oV

)J0 (r 2 +H 2)3/2 7

-e~hll-dh (3 8)

Assuming a step current and the TL model (pL(h)=-If/v) the electrical field from a fully developed leader can be found from (3.8):

E‘(r,0)TL = --------z 2Tt-e„

H2r [r2+H2Y

(3.9)

From (3.9) we can calculate that at a distance r0 =0.786 -H the electrical field due to the leader is zero. For shorter distances the field is positive and larger distances the field is negative. When the MIL model is assumed (ref. 3.8) the critical relation r0/His not longer constant, but increases with decay constant, X, and with decreasing channel height, H.

3.3.2 Fields from return stroke

In this section two different approaches for electromagnetic field calculations from a lightning channel are presented. One is the general antenna model used in e.g. [32, 86] and the other is an analytical extension of Rusck's model [38]. Fig. 3.1b) shows the geometry of the system.

vtFig. 3.1b) Geometry of system. Return stroke development.

To the left: The Rusck approach. To the right: The antenna approach.


Based on Maxwell's equations the electrical field can in general be written:

(3.10)dt

where V is the scalar potential and A is the vector potential.

Assuming the lightning channel to be straight and vertical with ignorable width compared to the distance to the observation point the potentials can be written, using the Lorentz gauge {V-A=-(a+j<x>e)\i-V) for the vector potential divergence:

(3.11)

where qR is the charge per unit length, I is the current and R is the distance to the observation point and where the integrations are carried out along the lightning channel and its image.

Instead of basing the calculations directly on (3.10) it is more convenient to eliminate the charge from the equation and thus establish a model based on the lightning current only. Using Lorentz gauge the electric field can be written

(3.12)0

The antenna model is based on treating each current element I-dh along the lightning channel as a dipole. The theory behind is classical and is found in many textbooks on the subject e.g. in [86]. Uman et.al. [32] are often credited for implementing the antenna theory for the lightning channel configuration.

The electrical field from a dipole I-dh can be expressed:

dE(r,z,t) = dE + dE + dE2 2stcct zind 2 rad

dh 2-(z~h)2-r2 47teo RS(z)

ji(h,X-R(z)!c)dX + 2-(z-hf-r2 _ cR\z)

i(h,t-R(z)/c) -c2R\z) dt

r2 di(h,t-R(z)/c)

(3.13)

dE(r,z,t) = dE +dE +dEr rUat 'fad 1

47te0 j?5(z) {fi(h,x-R(z)/c)dx + — r'(z h) ■ { cR\z)

i(h, t-R(z)/c) +c 2R 3(z) dtr-(z~h) _ di(h,t-R(z)/c)

(3.14)


The first term in (3.13) and (3.14) is called the static term, the second the induction term, and the third the radiation term. Close to the lightning channel the static and induction terms will be dominant. i(h,t) is the return stroke current which is zero for ?<0.

The equations (3.13) and (3.14) do not take the effect of the ground into account. To find the total electrical fields over an ideal conducting, fiat ground file expressions in (3.13) and (3.14) are integrated over the whole lightning channel and its image. This gives:

*„(z> *.(-*)Ez{z,r,t) = E* + Ez = j dEz{z,r,t) + J dEz{-z,r,t)

o o(3.15)

M'z>Er{z,r,t) = E* - Er = J dEr{z,r,t) - J dEr(-z,r,t) (3.16)

o o

The upper integration limit is the position of the lightning current front. This height, hjz) is time dependent and given by the time it takes for a current element to reach the front and the field from this front to reach the observation point [38]:

hJA =

a.W =

——+-2^- A RJz) = \Jr2+(hm(z)-z)2C V

c-t-v/c-z-\j(vt~z)2 + (l ~(v/c)2)~ r 2c/v-vlc

-vt+z + c/vj(vt-z)2 +(l -(v/c)2)'r2

c/v-v/c

(3.17)

Using the MIL lightning current model formulated in (3.1) the current egressions in (3.13) and (3.14) can be written:

t

Ji(h,T-R/c)'dT = I0‘(t-h/v-R/c)'H(t-h/v-R/c)-e (3-18)

i(h,t-R/c) = 70• H(t-h/v -R/c)'e ~k,x (3.19)

di(h, t-R/c) _ 5(f_hh _R]c-).g -h/X (3.20)dt

The delta function in (3.20) can be approximated by assuming a finite slope I0 /At of the current at height hm(z). Performing the integration along the lightning channel gives:


.(')Ir = f f(h,z,r,t)-5(t-k/v~R/c)-dh = lim f f(h,z,r,t) ■dh

dh (z) I.-vRJz)= /(A„(r), z, r, O'V — ---- = /(A„(z), z, r, f) •

3rwhere

hm and are given by (3.17)

*m(z) i-v/c-hjz)

(321)

Master & Uman [87] have also suggested expressions corresponding to a step current. In their analysis the retarded time h/v due to propagation along the lightning channel is ignored and in addition dhjdt in (3.21) is approximated by v. Their expressions are thus valid only for the far-field region (r » h) where the radiation term dominates.

Inserting the current expressions in (3.18)-(3.21) in (3.13) and (3.14) the fields from the lightning channel can be written:

4lt / 2 '{z-hf-r2

R\z)(i-hJv)-e-hlX-dh vme. ~hm(zyX

c2-R*(z) Rm(z)+v/c-hm(z)(3.22)

E, = /„4ir /

-h jzyx

R5(z) c2-r!{z) 5„(z) +v/c-hm(z)(323)

The last term in (3.22) and (3.23) is the radiation term or the tum-on field which is small for short distances. The factor c-po/4% is henceforth approximated by 30 implying that all variables have a denomination according to the standard SI system (length in [m], time in [s]).

When calculating the total radial field En the Er+ and E; terms (in e.g. (3.16)) tend to cancel each other since (Er+ - E;\=o =0. This can result in numerical accuracy problems. A better approach is therefore to perform a series expansion of Er around 2=0. One term in the series is sufficient to give high accuracy for practical line heights z. This gives the horizontal electrical field over lossless ground:

Er(r,z,t)be; be;3z Bz :=0

(324)

Figure 3.2 shows the electrical fields calculated from (3.15) and (3.16) at two different distances from the lightning stroke. The current is assumed to be a step of 1 A having a velocity of 1.M08 m/s. Two different lightning channel models are used; MTL with A=1500 m and the TL model.


Ez[V/m] Er [V/m] Ez[V/m] Er[V/m]

-2.5 - /X

a) r=100 m.

0.0015

- 0.001

- 0.0005

b) r=1000 m.Fig. 3.2 Electrical fields from lightning channel. Solid line: MIL model. Dotted line: TL model.

As seen from figure 3.2 the effect of the decay constant is minor for short times but becomes significant for times greater than a few microseconds. The decay constant’s effect on the induced voltage will be studied later. The large increase in the field components in the MIL model, seen in fig. 3.2a) is caused by the deposited charge, which for a step-current approaches infinity as shown in (3.5).

The integral in (3.22) and (3.23) can be solved analytically when applying the TL model, and together with the contribution from the image part of the channel (Ez and E;) the simple expressions from Rusck’s approach [38] are obtained. Rusck analysed the field from the return stroke together with the static field from the charged leader, but here these two contributions are separated. Assuming a step current the charge per unit length qR of the lightning channel is related to the current by I0-v -qR.

Ignoring the contribution from the image for the moment, the scalar potential from the lightning channel charge is:

^z,r)*.C)

Qr r dh 4-7ve0 i R(z) (3J5)

and the z-component of the vector potential from the lightning channel current is:

A Ho T dAR(z)

(3.26)

where hm is the position of the lightning current front at time t given by (3.17):

The deviation from Rusck’s approach [38] is that the contribution from the charged lightning channel is ignored. The electrical field from this charge is given in (3.9) where also the charge


depletion is taken into account. In [38] the horizontal field is not analysed.

The total potentials are found by adding the contribution from the image (setting z-»-z) since the ground is assumed to be ideally conducting.

The total vertical electrical field (at ground) from the return stroke can according to (3.10) be written:

(327)= 60

Vv V(v"f)2 +(l-(v/c)2)-r2 r

The variation of Ez with z (for practical line heights z) is small, so the field at ground level can be used in the further calculations.

The total horizontal field from the return stroke can be written (according to (3.10)):

(328)

r v \j(vt-z)z+( 1 -v2/c2)*r2 sj(vf +z)2 +(1 -v 2/c 2) • r 2r v

Approximating (3.28) by using a series expansion as shown in (3.24) gives:

dE(0,r,t) dE(0,r,f)

(329)

The equations (3.27) and (3.29) are later used when formulating an analytical model for the induced voltages in an overhead line.

The vector potential from a lightning channel, assuming the MTL lightning channel model according to (3.2), can in the frequency domain be written:

where k0 = a>/c.

3.4 Induced voltages in overhead lines 30

By choosing the Lorentz gauge for the vector potential divergence, the electrical fields are:

—, Ezj(x> dr'dz

Qj(a dz2

(3.31)

A frequency domain analysis is necessary when taking lossy ground effects into account as shown in chapter 4. A vector potential approach is particularly useful when calculating the induced voltage in an overhead line since both the horizontal and vertical field can be expressed by this potential. Induced voltage calculation implies an integration of the electrical fields along the lightning channel (z-direction), an integration of the radial field along the overhead line (x- direction) and an integration of the vertical field from ground and up to the overhead line (z- direction). The differentiation in (3.31) can thus be utilised efficiently.

3.3.3 Discussion

Figure 3.2 shows that the electrical field components are very sensitive to the variation of the current along the lightning channel expressed by the decay constant Therefore a very accurate lightning channel model is required to be able to predict the electrical fields beyond the first few microseconds, particularly for short distances. The effect of a tilted lightning channel is partly analysed in appendix B.

For large distances the radiation term dominates and the vertical field will be inversely proportional to the distance r while the horizontal field is inversely proportional to the square of the distance, as seen from (3.27) and (3.29).

The situation occurring when the current wave reaches the top of the lightning channel is not analysed here since this is assumed to happen after several tens of microseconds. If the MIL model is used, the height of the lightning channel, H is of no importance as long as it is much larger than the decay constant; exp(-H/A) ~0.

3.4 INDUCED VOLTAGES IN OVERHEAD LINES

The overhead line is modelled by the Agrawal coupling model [73]. This model uses the horizontal and vertical electrical field components directly. The total line voltage is in this model a sum of the scattered voltage Us and the incident voltage U‘.U(x,t) = Us(x,t) + U‘(x,t) (3-32)

where


U\x,t) = -j Ez(x,z,t)-dz (3.33)o

and where the following two equations are valid for Vs:

(3.34)

+ c'-—-Us(x,t) = o (3.35)dx at

The terminations of the overhead line can be included as shown in fig. 3.3 when the line is terminated by impedances.

dUs(x,t)dx

R/+L/-^r\ •i(x,t) = Ep,t) at

Fig. 3.3 Overhead line terminations.

The configuration used in this work when calculating the induced voltage in an overhead line is shown in fig. 3.4. The overhead line has two terminals A and B with x co-ordinates xA and xB respectively (xA > xB). The length of the overhead line is L= xA-Xg. The observation point has co-ordinates (x,y,z).

observationpoint

® (x,y,z)(§)XA overhead line

stroke^_____ position

X

Fig. 3.4 Induced voltage configuration

The solution of the telegraph equations (3.34) and (3.35) can be expressed as the sum of a reflected, Z7re/and an incoming voltage wave, Uin:

(3.36)U\x,t) = Uref(x,t) + Uin(x,t)

i{x,tyz' = Uref(x,t)-Ujn(x,t)

where Z’ is the characteristic impedance of the overhead line.

(3.37)


3.4.1 Analytical time domain solution

When the overhead line is lossless the coupling equations (3.34,3.35) can be solved explicitly since the equations then are reduced to simple wave equations. The total induced voltage in an overhead line is dependent on the line terminations and a model to keep track of the reflections in the line will be developed in chapter 3.5.

When the line is lossless (i?=0) the incoming voltage wave can be written as the sum of a time delayed reflected wave from the other terminal and a voltage contribution from the horizontal field, Ux:

(3.38)U,n(XA’ty = UV*»,~T)+D-(X4»<)

where r = L/c is the travelling time, L is the line length and c is the speed of light.

The horizontal field contribution equals [64, 70, 88]:

(3.39)

At the overhead line terminals this equals

(3.40)

(3.41)

where the lower integration limit is adjusted to restrict the integration to the length of theoverhead line.

The field Ex in (3.40-3.41) is given by (3.16), knowing that Ex = Erx/r. In general the integrals in (3.40-3.41) must be solved numerically which is time consuming since also Ex itself is given by an integral.

If the IL model of the lightning channel is used the horizontal electrical fields can be expressed by (3.29) and in this case the integral in (3.40) can be solved analytically:


c— Z X V

c

-fl -Q/c)2]2

+(l ~(v/c)2)'r2(3.42)

P2-r|• (x +T|) +(1~P2)-y2_______ _ 1P (y2 +(p-T])2)• \Jx2 +P2'T\’(2‘x +q) +(1 -P2)-j>2 yjx2+y2

where r\ = c-t-xAP = v/c r2 =x? +y2

The lower integration limit is in this case not xB, but is set to the value xq where becomes zero. This limit is found to be [38]:

(3.43)

Eq. (3.42) is equivalent to Rusck’s expression for the incoming voltage wave [38] except for the last term which Rusck ignores due to a different handling of the static field from the charged lightning channel and the assumption of an infinitely long line.

The incident voltage U which is an integration of Ez from ground to the line height can be approximated by

U'(r,z,t) ~-z-E/r,0,t) (3-44)

Using the TL model for the lightning channel the incident voltage can be written:

(3.45)v / (v()2 +(1 -(v/c)2) • r 2 rv

How to calculate induced voltages in an overhead line based on (3.42) and (3.45) is shown in chapter 3.5. Yokoyama [89] has developed similar expressions as Rusck [38] and have also included the effect of a lightning current being a ramp function. In [89], however, the Chowdhuri coupling model [72] was used, a model which ignores the horizontal field contribution. In [90] this was corrected and the expressions modified using Rusck's coupling model.


3.4.2 Frequency domain solution

When the overhead line is lossy no time domain solution of the coupling equations (3.34) and (3.35) can be found. To solve the problem the equations can be transformed to the frequency domain. A main purpose with this chapter is to study the importance of the line losses.

When the overhead line is terminated by impedances it is also possible to develop analytical expressions for the total induced voltage in an overhead line. In chapter 3.5 a model of a lossless overhead line is developed which handles general terminations of the line. The frequency domain solution uses the vector potential instead of the electrical field directly. The equations are written in a form which allows lossy ground effects on the electromagnetic fields to be included as shown in chapter 4.

The scattered voltage at terminal A can in the frequency domain be written (from (3.36) and (3.38)):

(3.46)

The current into terminal A can in the frequency domain be written (from (3.37) and (3.38)):

= "•(Uref(XA’J^ ~ Ure/-XB^ '6 ~ ) (3.47)

whereR 1 +j(jj'L 1

N jure'(3.48)

is the characteristic impedance of the overhead line and

Y = V(R /+Jto'L /)"jw"C/is the transmission coefficient of the overhead line.

(3.49)

In the frequency domain the horizontal field contributions can be written:

(3.50)

■B(3.51)

Inserting the vector potential formulation from (3.31) the horizontal field contribution becomes:


= UxA

c2/00

2

2ju> J dxdz

dA0(xA,z) _ dAQ(xB,z)

dz dz•e ~Y'L-y J.

dA0(x,z) -y

dz‘e ‘dx

(3-52)

and the incident voltage:

U‘(xA,jW) = Uj = -fE2(xA,z,jMi)-dz

f0 \

c2_d2A „

yw 3z2- yco-^g c2•dz »- —-

joy dz+j(0-z-AJx 0)

z- 0

(3.53)

Assuming a terminating impedance ZA, the current into the overhead line can also be expressed according to fig. 3.3 as

i(xA,joy) (3.54)

The same equations can be developed for line termination B, by just changing index A to B. Combining (3.32), (3.46), (3.47), (3.54) and the corresponding equations for terminal B, the total voltage at terminal A can be written (based only on the incident voltage and the horizontal field contribution):

UA =(1 -a-w. P/Pa)

-2yi. P/“/ 2 'Z. 2 Z.

(3.55)

p is the reflection and a is the transmission coefficients of the terminations:

Pa = z,+z' aJ =2 Z,

Z^+Z'(3.56)

The voltage at terminal B is found by substituting index B for A and vice versa in (3.55) and(3.56).

3.4.3 Importance of line losses

To investigate the dependency of line losses, the overhead line’s transmission coefficient, y, is calculated by Cable Constant in ATP-EMTP [12, 91] which uses Carson’s formulas. The


ground conductivity is set to 0.001 S/m. The radius of the overhead line conductor is 5 mm and its height is 10 m. The length is varied from 1000 to 5000 m. The horizontal field contribution Ux in (3.52) is used to study the effect of the line losses as shown in figure 3.5. The lightning stroke is located 37= 100 m from the midpoint of the overhead line. The lightning current is assumed to be a step of 1 A following the TL model with velocity v=l.l-108 m/s.

uxM

a)L=1000m.Fig. 3.5 Horizontal field contribution, Ux. Solid line: No loss. Dashed line: Line loss.

Fig. 3.5a) shows that the loss effect in the overhead line is ignorable when the line is shorter than 1000 m. For a 5000 m long line, however, as shown in fig. 3.5b) the line loss effect is significant. Rachidi et.al. [92] used a high ground conductivity (2 0.001 S/m) approximation to Carson's formula, Sunde [93], and found that a line with length less than about 2000 m could be assumed lossless. A similar investigation was carried out in [64] concluding with 500 m as the critical length of the overhead line.

3.4.4 Importance of decay constant

Figure 3.2 shows that the electrical field components are very dependent on the decay constant X, used in the lightning current model, and thus on the variation of the current along the lightning channel. This is in fact a serious problem since this variation is encumbered with uncertainty. On the other hand, using an exponentially decaying current with time would reduce this effect. However, it is possible that the induced voltage in an overhead line is less dependent on the decay constant since this voltage consists of a contribution from both the horizontal and vertical field components.

Figure 3.6 shows calculations of the total induced voltage in an overhead line, calculated in the frequency domain according to (3.55) and transformed to the time domain. The overhead line is 1000 m long with height of z=10 m matched at both ends with its characteristic impedance. In fig. 3.6a) the lightning stoke is located y=100 m from the midpoint of the line (side stroke) while in fig. 3.6b) the location is 100 m from the line end (end stroke). The lightning current is a step of 70 = 1 A with a velocity v=l.l-10s m/s. The MTL uses X = 1500 m.

3.5 Calculation models 37

100 m 1000 mnear end1000 m

near

far end

a) Side stroke.Fig. 3.6 Induced voltage’s dependency on decay constant. Solid line: MTL. Dotted line: TL.

Figure 3.6 shows that the MTL and TL lightning current models give approximately the same maximum values. Since the maximum value is most important when studying insulation coordination in low-voltage systems usage of the TL model often gives sufficient accuracy. And since the TL model enables analytical expressions for the inducing voltage this model is preferred when analysing the protection of a larger low-voltage systems in chapter 8.

3.5 CALCULATION MODELS

In this section models for induced voltage calculations are presented. A basic assumption is that the overhead line can be treated as lossless. As seen in the previous section this is reasonable when the line is shorter than 1000 m. The electrical field is assumed to be unaffected by the lossy ground, but the developed models are made general so that the lossy ground effect can be readily included as shown in chapter 4. Two different approaches are presented: 1) An analytical time domain model using the TL lightning channel model and the voltage contributions calculated in chapt. 3.4.1, 2) A frequency domain model based on the vector potential formulation in chapt. 3.4.2 using the MTL lightning channel model.

The time domain solution is based on the assumption of a lossless (J?=0) overhead line, resulting in unattenuated voltage waves travelling with the speed of light. The incoming voltage wave to one terminal thus consists of the time delayed reflected wave from the other terminal and all reflections in the line can be calculated in the time domain.

The overhead line can be modelled as shown in fig. 3.7.


Fig. 3.7 EMTP model of overhead line.To the left: Classical Bergeron method. To the right: Simplification

where

(3.57)

(3.58)

The impact of the incident field from a return stroke is embedded in the terms UindA and UindB, called the inducing voltages. These two terms become equivalent source terms. Loss effects on the electrical field due to propagation above a lossy ground can be taken into account by modifying these terms as shown in chapter 4.

When the ground is assumed to be lossless the inducing voltages can be written:

(3.59)

(3.60)

Using the TL model for the lightning channel and inserting the expressions from (3.42) and (3.45), and using subscript A for terminal A, gives the following inducing voltage at this terminal:

60-J0-z-|3-ti

y2 + fS2"rf

60-/0-z-p-ri

y2+P2-f

z.+P2-ti+ 1

^(vr)2+(l-P2)-02+xj)

^+p2-n

*• 6 ^A’ tB ]

XA ~L +p2,r)

/(v ■ 02 +(1 _P2)' 0 2 +xA) \/(v(r-T))2+(l -$2)-(y2+(xA-L)2)

(3.61)

t>t„

where\j*A+y2 „ _

- > tR 9 p = c’t~xA and P = — c

(3.62)


For times less than tB the expression in (3.61) gives the same result as Rusck’s model [38, eq. (105)] (called t/7) even though a different coupling model has been used and only the field from the return stroke is considered in (3.61). Rusck’s expression is equivalent since he assumes an infinitely long line, which causes the static contribution from the charged leader to vanish.

The inducing voltage at terminal B is found by substituting -xB for xA in (3.61).

In the frequency domain the inducing voltages can be written:

(3.63)

(3.64)Cw,(/w) = 2' ^

The inducing voltage at terminal A can by inserting (3.52,3.53) into (3.63) be expressed:

(3.65)

+j(O-z-(A0(xA,0) ~A0(xb,0)-e *Y i)

The first two terms in (3.65) are zero and the last terms dependent on A„ (radiation terms) are very small compared to the integral term for short distances. For distances >100 m these two radiation terms becomes more important, however, and the term A0(xB,0)-eY‘L gives some undesirable oscillations in the function due to the time delay. These oscillations require a higher resolution in frequency samples in order to calculate the inducing voltage in the time domain, alternatively to split (3.65) in two contributions. The inducing voltage in the frequency domain is further investigated in appendix D.

A low-voltage overhead line consists normally of three to five conductors. The three phase (line) conductors can initially be assumed to have the same potential since they have about the same distance to the lightning stroke and are roughly terminated by the same impedances. This last assumption is not strictly satisfied but reasonable for the first microseconds prior to the reflections in the system. The measurements on transformers (chapter 6) and on low-voltage power installations (chapter 7) are all based on the assumption that the three phase conductors can be treated as one unit. The neutral conductor and a possible protective earth conductor can equivalently be assumed to have another common potential. Thus a two-phase overhead line model is often sufficient when calculating lightning-induced overvoltages in a low-voltage system.

A two-phase model can be established in the same way as the single-phase model shown in fig. 3.7. The characteristic impedance is replaced by an impedance matrix as shown in fig. 3.8.

3.6 LIO calculations 40

■BLA.L A.N |ZlnZnn 1 BN

+UreV VurBN

Fig. 3.8 Two-phase induced voltage calculation model.

The voltage sources at terminal A can for the two-phase model be written:

UrAL UmdAL+

UBL(t-x)+

zll

rAN_U,„dAN (W'-T) ZLS ZNN -Taj/*-1')

(3.66)

A symmetric four conductor overhead line system with three phase conductors having equal potential and one neutral conductor can be reduced to a equivalent two-phase system.

ZLC ZLD ZLD ZLNZ. „+2-Zrr,2 2 z z LC LD 7**LD *LC *LD *LN

3 ZLLT

ZLD ZLD ZLN z z^LNZLN ZL» ZU, ZlW

(3.67)

The model shown in fig. 3.8 is implemented in MODELS in ATP [12, 94] as shown in appendix J. MODELS is used both to calculate the inducing voltages according to (3.59-3.60) and to keep track of reflections in (3.57-3.58). This approach will enable flexible connections with the rest of the network, including connected arresters, loads and cables. There is no limit to the number of line segments that can be handled, except for memory limitations in ATP itself. Multi-phase overhead lines (according to fig. 3.8) are normally handled by assuming the inducing voltages to be equal in all phases, but an additional scaling for height variations is possible. When taking lossy ground effects into account, the inducing voltages (3.63 and 3.64) are pre-calculated by separate programs and included in ATP. In such a case maximum 5 line segments can be handled. Nucci etal. [95] have also implemented a model for LIO calculations in the EMTP. Their approach involves modification of the program’s source code in order to include the induced voltage effect.

3.6 LIO CALCULATIONS

The LIO's dependency of the various parameters in the model will now be analysed. The basic


configuration used is shown in fig. 3.9. The overhead line consists of a single conductor of length L and height z and with characteristic impedance Z’ = 500 Q. For simplicity the TL model of the lightning channel is assumed with v = 1.1*10® m/s. Three main variations are studied: 1) The lightning current parameters (velocity and current shape), 2) The stroke position and 3) The overhead line parameters (length, height and terminations).

a------- r„

X

-L/2- -L72-

2s

Default configuration:2=500m z=6 m r0 = 100 m v= 1.1*10® m/sZA = Zg= Z' (matched terminations) 4 = 30 kA

Fig. 3.9 Basic LIO configuration.

The Heidler model is used to model the current at ground level [96]. The analytical expression and the default current shape is given in Appendix A. The amplitude of this default current is 30 kA.

3.6.1 Induced voltages from return strokes

First the LIO’s dependency on the stroke location and overhead line termination is investigated. The stroke locations a (end stroke) and % (side stroke) are studied with three different terminations: open ends (fig. 3.10), matched terminations (fig. 3.11) and inductive terminations of 10 pH (fig. 3.12) which are typical first approximation values for TN-systems.

U[kV]

a) Stroke location a.

U[kXZ]

150

100

50

H— A;& B

Li,,5 10 15 20 25 30 35 40 45 50

t[us]

b) Stroke location %.Fig. 3.10 Induced voltage, open ends.


U[kVl

a) Stroke location a.t[us]

b) Stroke location %.Fig. 3.11 Induced voltage, matched ends.

U IkV] U [kV]

10 15 20 25 30 35 40 45 50

i.....i-.... t0.8 -...........•

10 15 20 25 30 35 40 45 50

a) Stroke location a. b) Stroke location %.Fig. 3.12 Induced voltage, inductive terminations (10 pH).

The stroke location % (side stroke) gives the highest voltage for all terminations. The open termination case gives much higher voltages than the other terminations. The 10 pH terminations give hardly any overvoltages at all. When the line is terminated with a low impedance at both ends, like in fig. 3.12, the induced voltage will oscillate with a frequency f=c/(2 -L)=300 kHz where c is the speed of light and L is the line length.

The dependency on line length will now be investigated. The stroke location is % (side stroke), and the voltage at the middle of the line is found by connecting two lines in series. The two line segments are terminated with their characteristic impedance Z' at the far ends and kept open at the connection point (middle). The current wave form shown in appendix A is used and labelled as Heidler in figure 3.13. A step current of 30 kA is also used and labelled as step. The simple formula proposed by Rusck for finding the maximum induced voltages gives [38]:

^max = 30-V- y

l +- 68.5kV (3.68)


I 1000 m Heidler5000 m;500 m

8 10 12 14 16 18 20

5000 m Heidler500 m

•5000 m1000 m

-/-500 m

8 10 12 14 16 18 20

a) At line’s midpoint. b) At line’s end pointFig. 3.13 LIO in overhead line with line length, L as parameter. Stroke location %. y = 100 m.

From figure 3.13 we see that the maximum induced voltage is surprisingly little dependent on the line length L, at least when Z>500 m. Assuming a step-current gives the same maximum value at the midpoint of the line for lengths 500 and 5000 m, equal to the value from Rusck formula (3.68). We also observe that the voltage at the midpoint of the line (the point closest to the lightning stroke) is higher than at the line terminals. The maximum induced voltage at a matched overhead line terminal can be found from (3.61), knowing that the maximum occurs before t=tB and using the same assumptions as Rusck [38]:

*A+$my ■ + ik ^2 -y2-p2 -y2+xj +2xa -y ■ p ^

for y>0 (3.69)

where the time to crest is approximated by

+x 2A

C

with x»y the maximum induced voltage reduces to

(3.70)

U. (*-») = 30-7 •- = 54 kV (3.71)U y

which is in good agreement with the dotted curves in fig. 3.13b).

Applying other line terminations will of course result in reflections in the line and make (3.68) and (3.69) invalid.

Fig. 3.14 shows the dependency of the distance to the lightning stroke (y=r0). The default configuration is used with matched line terminals. The LIO is calculated at the line terminals.


Fig. 3.15 show the LIO at the line terminal of the default overhead line configuration. The ratio between the characteristic impedance Z’ and the terminating resistance R is used as a parameter. When Z’/R=0 the overhead line is open which gives the same result as fig. 3.10b) and when Z7R=1 the line is said to be matched giving the same result as fig. 3.11b).

uw

0 2 4 6 8 10 12 14 16 18 20IW

Fig. 3.14 Induced voltage and its dependency ondistance to lightning stroke, y. Location %.

UIKVI

18 20

Fig. 3.15 Induced voltage and its dependency on overhead line terminations, R. Location %.

Fig. 3.14 shows that the maximum induced voltage at a matched overhead line terminals is not strictly inversely proportional to the distance y. Using a pure step-current the maximum voltage can be approximated by (3.69) which for both x-»0 and x-»°° results in an inverse proportionality with distance y. Using a Heidler shape according to appendix A will modify this somewhat.

Finally the dependency on two lightning current parameters is investigated. Fig. 3.16 shows the dependency of the front time t=t, (ref. appendix A) and fig. 3.17 shows the dependency of the lightning current velocity v. The basic configuration is used with matched terminations and stroke location %.

U[Wj

30.......

14 16 18 20

Fig. 3.16 Induced voltage and its dependency on the lightning current’s front time constants tv

U [kV]

40 -

10 12 14 16 18 20

Fig. 3.17 Induced voltage and its dependency on the return stroke velocity, v.

Fig. 3.16 shows that the induced voltage is strongly dependent on the front time t,. The curve


indicated by t=0 is actually a pure step resulting in a voltage equal to the dotted 500 m curve in fig. 3.13b). Fig. 3.17 shows that the maximum induced voltage is affected by the lightning current velocity and increasing with reduced velocity v. For longer overhead lines the maximum voltage becomes independent on the return stroke velocity as seen from (3.71).

The rise time of the LIO increases with the rise time of the lightning current, and is about proportional to the distance to the lightning stroke as seen from (3.70). End strokes (y=0) gives the lowest rise time, actually equal to zero for the assumptions made when deducing (3.70).

3.6.2 Induced voltages from charged leader

The charges in the developed leader channel sets up an electrical field given by (3.8). This field can induce a voltage on the transmission line that comes in addition to the dynamic voltages from the return stroke given in chapt. 3.6.1. However, if the transmission line has some connection to ground (not necessarily a short circuit) during the formation of the lightning channel leader the induced static voltage will disappear. Such a connection could be a voltage divider connected to the line for measurement purposes, non-capacitive voltage transformers, grounded transformer neutrals, arresters etc. It is also possible that the static induced voltage could be so high that a fiashover can occur, even during this pre-return stroke period.

If the overhead line does not have any connections to ground a voltage V = - z-Ez will be induced on the line. Since the leader development is a relatively slow process, the voltage will average out along the line giving rise to a resultant voltage:

(3.72)\

= - 60'c- — L— • • [--dT + ' —- • / ---------------L { o x v i o l <*24y2**

Assuming a step current and the TL model (A=»), the induced voltage from a fully developed charged lightning channel can be found from (3.72):

(XA+rA )'(*Wri+/f2 > _ H2 X,B\

(3.73)

where

andand E/r) is given by eq. (3.8)

3.7 Conclusions 46

This voltage could be very high, and some typical values {xA = -xB = 500 m, y= 100 m, c/v =3, I0 = 30 kA, z =10 m and H= 4000 m) give a static voltage of V0 = -223 kV. A low-voltage overhead line can not withstand such voltage, and a flashover to ground will be established somewhere in the network. The induced voltage will be reduced somewhat when the MIL model as shown in (3.72) is applied, but it is still high enough to establish a connection to ground during the leader process.

3.7 CONCLUSIONS

The induced voltage in an overhead line is composed of two main contributions: 1) A static part from the charged leader and the cloud reservoir and 2) A dynamic part from the return stroke. The static term can in general become very large and will in the cases of practical interest (such situations where an LIO is caused by the return stroke) exceed the low-voltage insulation withstand level. If the low-voltage system has a connection to ground the static term will vanish, however, and this is typically the case for a TN system. On the other hand, an IT- system is isolated from ground, but it is likely that a connection to ground will be established anyhow for such systems, probably via the transformer's neutral protection, arresters or in worst cases by uncontrolled flashovers in the system.

The following conclusions apply to lightning induced overvoltages (LIOs) from a return stroke:• The LIO from a return stroke is strongly dependent on the stroke location. The side stroke

configuration (%) gives the highest LIO for all types of overhead line terminations investigated.

• The maximum LIO increases with increasing terminating resistance.• The maximum LIO is surprisingly little dependent on the line length.• The maximum LIO increases with reduced front time constant t=t,. A front time constant

of t, = 5 ps reduces the maximum LIO to one half compared to a step current.• The maximum LIO increases with reduced return stroke velocity v. Whether the velocity

ratio c/v is 2 or 3 (which is the practical interval) is however of minor importance. Thus variations in the velocity along the lightning channel with c/v=2 at the base and c/v=3 higher up will result in a minor deviation of the LIO compared to using a constant velocity v= 1.1T 08 m/s for the whole lightning channel.

• The induced voltage is proportional to current amplitude Im, and (practical) line heights z.• Overhead lines shorter than about 1000 m can be assumed lossless.

For short overhead lines which are typical in low-voltage system the reflections due to the line terminations must be considered and Rusck's simple expression (3.68) and (3.69) for the maximum LIO gives inaccurate results. However, the suggested proportionality with current magnitude Ig, line height z and the inverse distance to the lightning stroke l/y can be used as a rule of thumb.

The rise time of the LIO increases with the lightning current rise time, and is about proportional to the distance to the lightning stroke. End strokes give the lowest rise time.

4LOSSY GROUND EFFECTS ON

LIGHTNING-INDUCED VOLTAGES

4.1 INTRODUCTION

Attenuation effects on electrical fields propagation over a lossy ground are believed to be of great importance when analysing lightning-induced voltages in power systems. Measurements [20, 97, 98] have even shown an inversion of the induced voltage for some special configurations. The vertical electric field is much less affected by a lossy ground, while the horizontal field is strongly affected, and is increasing in magnitude to a finite value when loss is taken into account. For nearby lightning (<1000 m) the effect of losses is uncertain.

In chapter 2 Norton’s method was selected for taking lossy ground effects on the electrical field into account. In this chapter the Norton’s method is first compared with Sommerfeld’s exact formulation for a low ground conductivity. Nortons’s method is further used to calculate the electrical fields and the induced voltages in an overhead line over lossy ground. The electrical field is analysed based on King’s expressions [99]. The surface impedance approximation is also used and compared with from Norton’s method.

The calculation of the induced voltages is based on a vector potential formulation similar to the frequency domain model in chapt. 3.5. The lossy ground effect can be included as an additive contribution to the inducing voltages.

4.2 NORTON’S METHOD FOR TAKING A LOSSY GROUND INTO ACCOUNT

The analyses made in this chapter are based on the configuration shown in fig. 4.1. The lightning channel is assumed to be vertical and the vector potential caused by a current dipole I-dh at a height h is calculated.

4.2 Methods for taking lossy ground effects into account 48

Fig. 4.1 Configuration. Cylindrical coordinates.

4.2.1 Sommerfeld’s exact formulation

Sommerfeld [57, 58] has formulated the vector potential in air from a vertical current element (dipole) in air (0, h) over flat, homogenous ground:

i(4-1)

where

R0 = \jr2+(z-h)2 R{ = 'Jr2 +(z+h)2and J0 is the Bessel function of first kind.a, is the conductivity and is the permittivity in ground.z and h are the height of the observation point and dipole, respectively.The vector potential has a vertical component only, since the lightning channel is assumed to be straight and vertical.The expression for the vector potential is based on Lorentz gauge (VA=-(a1+j<ae.1) ii0-V).

The integral in (4.1) is difficult and time consuming to solve. In general the integrand has very oscillatory properties due to the Bessel function at large arguments ar and the exponential functions at low a values. The most significant contribution to the integral comes from the branch cut a=k0. At this point / becomes real and the integrand decays exponentially towards zero.

In a practical situation, when calculating fields from a lightning channel with main frequency contents less than 100 MHz and ground conductivity higher than 0.001 S/m, it is often reasonable to assume |&,|2 » |£0|2 and neglect the displacement current =* k2 ~ -jco-po-o,.


The electrical field components in air can be written:

3=4.Er{r,z) = -j<*._____

e.2 dr'dz Ko

Elr,z) = 1*.3z2

-+ko'Ao

(42)

(4.3)

4.2.2 Norton’s approximation

Norton has introduced an approximation to (4.1), assuming a high ground conductivity [59,58]:

dAaVo'I'dh

4nwhere

#1

■+RR,

, W

R,

K -

F(p)

cos9r-A0cos0r+Ao

= 1 -j\fxp • exp( -p) • erfc(j\[p)

is the Fresnel reflection coefficient

is the Sommerfeld attenuation function

P = 7;21 ~(cos9r + A0j2

A0 = -u 2sin20r = u

is the numerical distance

(4.4)

(4.5)

(4.6)

(4.7)

(4.8)

K = J°>e oN +/toe,

(4.9)

cos©^ = (z+h)/Rlt sin0r = rIRx

and where erfc is the complementary error function.

The first term inside the brackets in (4.4) is the direct wave from the dipole to the point of observation, the second term is the reflected wave from the Earth’s surface and the third term is the ground wave.

The vector potential in Norton’s formulation can also be written as a sum of the potential in a lossless situation, dA0, and a contribution from the lossy ground, dAA.

g0 -I'dh I e W g >¥i

4TZ 1 R0 Rt(4-10)


\i, ' I'dh e A A

where

(4.11)

f>mjkQ%

2

(4.12)

>> ii

- t (4.13)

/ = e ~perfc(j\Jp) (4.14)

In [62] several approximations are compared to Sommerfeld’s exact formulation of fields from a vertical dipole, and the Norton’s method showed a good agreement. However, in [62] the ground parameters o, and e, are kept constant and a relatively high conductivity of 0.01 S/m is assumed. The analysis in [62] is here extended using lower ground conductivities and other permittivities. The accuracy of Norton’s method is analysed by comparing the Norton vector potential in (4.4), dAaN, with the exact vector potential from Sommerfeld’s method (4.1), dA^, at a height z=10 m above ground. The error of Norton’s approximation is defined as:

dA

dAaSoN

- i •too % (4.15)

with dipole height, h and frequency,/as variables and horizontal distance, r and the ground characteristics, k, (cq and 6,) as parameters. Fig. 4.2 shows the error in Norton’s approximation for two different radial distances (100 m and 1000 m). A ground conductivity of a, = 0.001 S/m and a relative permittivity of er = 10 is assumed. Frequencies from 0 to 10 MHz and dipole heights from 0 to 4000 m are investigated.

a) r=100m. b) r= 1000 m.Fig. 4.2 Error in Norton’s approximation, o, = 0.001 S/m, e, = 10.

As seen from figure 4.2 the error in Norton’s method is in general low and less than 5 % in the frequency range (0-1 MHz). This range is considered to be of greatest importance related to lightning-induced voltage calculations. For frequencies below 1 MHz the accuracy of Norton’s


method is better for larger distances, but above 1 MHz the error seems to increase with distance. Also, for frequencies above 1 MHz the error is larger for lower dipole heights.

The dependency of the ground permittivity has also been investigated, showing that the error decreases with increasing permittivity. For a relative permittivity er = 40 the error is less than 2 % in the whole frequency range (0 - 10 MHz) and for er = 2.5 the error is 10-37 % for frequencies/> 1 MHz and dipole heights h < 200 m when r = 1000 m.

The attenuation of the vector potential can be expressed as the ratio between the lossy and lossless vector potential from (4.4) and (4.10) at the ground (2=0). Based on Norton’s approximation this ratio is plotted in fig. 4.3 as a function of frequency (f) and dipole height (h), with o, = 0.001 S/m, er = 10 and r= 100 m. From this figure an important observation can be made: At low frequencies the effect of the lossy ground increases with dipole height. Above 1 MHz the loss effect becomes largest for low dipole heights, however.

OMS-1

gO.85-0.9

&o aaowM&a

Fig. 4.3 Ratio between lossy and lossless vector potential at ground level.

Norton’s method is often used to calculate the loss effects on electrical far-fields which aredominated by the radiation term. The attenuation of the radiation fields is similar to that of the vector potential and this approach has been used in e.g. [100,67-69], The electrical close-fields show a more complex dependence on the lossy ground, as found by performing the differentiations in (4.2-4.3).

In the following the vector potential will be used to calculate the induced voltage in an overhead line. The advantage of this approach is that Norton’s approximation can be used directly. Besides, the vertical and horizontal electric field components can both be expressed by the vector potential. This makes it easier to calculate the total line voltage and to analyse its dependency of the line terminations. The calculation of the induced voltages follows the approach presented in chapter 3.5 with the lossless vector potential, A0 in (3.65) replaced by the lossy ground affected potential Ag.

The total vector potential, Aa is found by integrating (4.4) over the whole lightning channel.H

(4.16)0

4.3 Lossy ground effects on electrical fields 52

The total vector potential can be written as a sum of the lossless vector potential, A0 and the contribution from the lossy ground, Aa. Inserting the expression for a step current in the MTL model from (3.2) gives:

- /*.„ - £■£■/ i--------+£---------

\ a,• e ~A‘(/o>/v + i/A.), dh

Aa(x,z) = fdA A V4 . jy .y -e -*-(/to/v + l/X). dh

(4.17)

(4.18)

4.3 LOSSY GROUND EFFECTS ON ELECTRICAL FIELDS

In this sub-chapter the lossy ground effect on the electrical field is studied. The expressions for the horizontal and vertical field components are found by differentiating the vector potential formulated by Sommerfeld in (4.1) according to (4.2) and (4.3), respectively. The exact field expressions are approximated by the formulation proposed by King [99] as shown in appendix I. When performing calculations of the electrical field, the TL model is used for the lightning channel. The current is a step of 1A with a velocity v=T.lTOs m/s. The ground is either ideally conducting or has a conductivity of o, = 0.001 S/m and relative permittivity er = 10.

4.3.1 Radial electrical field in air

In fig. 4.4 shows the horizontal field component at ground, -E/r,0), calculated from (1.2) based on Kings formulas [99].

- Er(10,0) [V/m] - Er(100,0) [V/m] . Er(1000,0) [V/m]0.00350.003 —

0.00250.002

0.0015

0.0005

b) r=100ma) r=10 m c) r=1000 mFig. 4.4 Negative radial electrical field in air at the ground surface, Of=0.001 S/m and er = 10.

In fig. 4.5 this horizontal field is shown as function of the height of the observation point, z, for two different ground conditions:

4.3 Lossy ground effects on electrical fields 53

Er[Wm] Erflflm]

a) Lossless b) Lossy ground (o, =0.001 S/m)Fig. 4.5 Radial electrical field in air at r=100 m with height above ground, z as parameter.

The radial field is strongly affected by the ground conductivity, but the difference in fields between height z (the overhead line) and ground is almost independent of the ground conductivity.

An interesting quantity regarding the adequacy of the Agrawal line model [73] is the variation of the horizontal field along the overhead line, dEJx,y,z)/dx, as pointed out in appendix E. From figures 4.4 we see that the field decays faster than 1/r with distance. Analysing the adequacy of Agrawal’s coupling model is beyond the scope of this work, however.

4.3.2 Vertical field in air

Fig. 4.6 shows the vertical field, calculated from (1.7) based on King’s formulas [99], as a function of height, z, and for two different ground conductivities.

Ez [V/m] Ez [y/m]

u=0.001 S/m

-0.8 ----------

r = 100 ma= 1 S/m

Fig.a) Lossless

4.6 Vertical electrical field in airb) Lossy ground (o, =0.001 S/m)

at r=100 m with height above ground, z as parameter.

We seen from fig. 4.6 that the vertical field is almost independent of the height (z=0-10 m) and is little effected by the ground conductivity at the investigated distance.

4.4 Comparing Norton’s and the surface impedance method 54

The electrical field in ground just below the surface can as a first approximation be written:

(4.19)

Since \k,\2 » \k0\2, the field in ground is much lower than the field in air.

4.4 COMPARING NORTON’S AND THE SURFACE IMPEDANCE METHODS

In this sub-chapter Nortons’s method is compared with the surface impedance method. Theintention here is not to study differences in the electrical field but the difference in the induced voltage in an overhead line. The surface impedance method applies to the horizontal field only, so the horizontal field contribution will be used to compare the methods.

The horizontal field contribution formulated by Norton’s method is found from (3.52):

(4.20)c2 _ dAo<-XA’Z) dAo(X2?’Z>

2/co dz dz7

where the two dAJdz terms outside the integral in (4.20) will dominate in most cases except for the side strokes where the two dAJdz terms evaluated at xA and xB partly cancel each other for symmetry reasons.

The horizontal field contribution expressed in (4.20) is split in a lossless part and a lossy ground part dependent only on Aa from (4.18). Instead of differentiating the vector potential Aa in(4.20) with respect to z, the theorem of reciprocity {d/dz = d/3i) results in:

(4.21)H

.e -&(/<t)/v + 1/X) + (Zto/v + MX) ~h(iwN * m'dh0

Inserting (4.21) in (4.20) gives a complete analytical expression of the horizontal field contribution due to losses.


A commonly used approximation for taking loss effects into account is the surface impedance approach. This technique is used e.g. in [38, 63,64, 101, 102] and the horizontal electric field over a lossy ground can be approximated by:

(4.22)EJjv,z) = - s^GMOi-zc/to)

where Ex0 is the lossless electric field at line height and Bx0 is the lossless magnetic field at ground and Z(jo>) is the surface impedance:

(4.23)Z(/CO) = c-u = c-\j a, +/w€

The lossy part of the horizontal field contribution, formulated by the surface impedance method, can be written:

(424)

Utilising that Bx = - dAJdx, gives:

(425)= EEL 2

Ao(xa> °) ~Ao(xb> 0)-e~r'L~ yjA0(x, 0) •

The Norton’s method will now be compared with the surface-impedance method, by calculating the horizontal field contribution in an overhead line. The overhead line is assumed lossless with length Z=1000 m and height z=10 m. The lightning stroke is located a distance r0 from the near end of the line as shown in fig. 4.7. The current amplitude is /„ = 1 A and the height of the lightning channel is assumed to be 4000 m. The velocity of the lightning current is set to v=l.lT08 m/s. The decay constant X is 1500 m (MTL lightning channel model).

strokelocation near

U,x

far1000 m

Fig.. 4.7 Configuration of the system. End stroke.

Fig. 4.8 shows the lossless part of the horizontal field contribution Ux0 from (4.20), calculated at two different distances r0 for the two lightning channel models, MTL and TL.


UxoM Ux0[V]

a)r0 = 100 m.Fig. 4.8 Lossless horizontal field contribution.

Fig. 4.8 shows that the lossless horizontal field contribution Ux0 will be higher for the MTL than for the TL model only for the 100 m distance, but also for the 1000 m distance beyond 10 ps.

Fig. 4.9 shows the lossy part of the horizontal field contribution UxA from (4.20) and (4.25). The results from using the Norton’s or the surface impedance methods are compared.

Comparing the Norton's method and the surface impedance method in fig. 4.9 shows a reasonably good agreement when r0 = 1000 m for both the MTL and TL lightning channel models. At distance r0 = 100 m, the agreement is best when the MTL model is used. The surface impedance method deviates from the Norton’s method for short times, however. It is important to keep in mind that a step current has been used in the calculations, and a more realistic current shape will make the deviation at short times less significant. The surface impedance approach is less dependent on the decay constant, k for short distances than Norton’s method. This is due to the variation of the loss effects along the lightning channel, which is ignored in the surface impedance approach. From fig. 4.3 we see that the loss effect is strongest for the highest dipoles at low frequencies. When a decay constant of A=1500 m is assumed the contribution from the upper parts of the lightning channel is reduced. This will therefore lead to an improved accuracy of the surface impedance method. The peak in the lossy ground part of the horizontal field contribution, UxA , is almost independent of the decay constant. Increasing the conductivity by a factor 10 reduces the horizontal field contribution lfxA in fig. 4.9a) by a factor 4.

Comparing figures 4.8 and 4.9 shows how the lossy ground affects the total horizontal field contribution. For the distance r0 = 1000 m the loss effect is large and the total horizontal field contribution Uxa = Ux0 + UxA will be negative up to around 10 ps, while at r0= 100 m, Uxa will only have a negative initial peak.


UxaM

-3 a

-7 -

a) MTL model. r0 = 100 m. b) MTL model. r0 = 1000 m.

c) TL model. r0 = 100 m. d) TL model. r0 = 1000 m.

Fig. 4.9 Lossy horizontal field contribution UxA. o, = 0.001 S/m and er= 10. Solid line: Norton’s method. Dotted line: Surface impedance method.

The surface impedance approach gives the same result as Norton’s approximation if (4.25) equals lossy part of (4.20), i.e. when:

C2 dA^(x,z)

j(j> dzc-u-A0(x, 0) (4^6)

Requiring the contribution from each current dipole, I-dh, to be equal in the two cases (instead of the total integral) results in:


(4.27)m r c

It will now be investigated under which conditions the hypothesis in (4.27) is valid, and the necessary assumptions will be outlined.

For small p (small frequencies and/or small dipole heights),can, by a series expansion, be expressed as

(4.28)

Performing the differentiation off,fp and taking the limit as z approaches zero gives:

#oHm . -N 2 (4.29)

Multiplying the right hand side of (4.29) with f0 gives:

which is equal to the hypothesis in (4.27) except for the factor RVr2.

Assuming h«r will make the surface impedance approach equivalent to Norton’s method. This assumption is reasonable for large distances or short times. However, for short times the high frequency content makes the series expansion approximation in (4.28) doubtful. When the lightning current is decaying along the lightning channel the lower parts of the channel will contribute more than the upper ones, making the surface impedance method more accurate for lower values of the decay constant, X.

The surface impedance approach is a good approximation if all the following conditions are met:• Low frequencies dominate (p is small).• The overhead line height, z is small compared to the horizontal distance r.• The approximation, R*r, holds. This assumption is reasonable for: Large distances r,

small decay constants X or short times t.

4.5 Lossy ground effects on LIO in overhead lines 59

4.5 LOSSY GROUND EFFECTS ON LIO IN OVERHEAD LINES

The horizontal field has been shown to be strongly affected by a lossy ground in contrast to the vertical field that is little affected for short distances. Due to the vector potential formulation analytical expressions for the total line voltage can be developed and the lossy ground effect on the whole line can be studied, including the dependency on line terminations. Equivalent to eq. (3.65) in chapter 3 the inducing voltage over lossy ground is defined:

dAo(XA’°'> _ d^q(*j»°) (4.31)/to dz dz

+j(O-z-IiAo(xa,0)-Ao(xs,0)-e _Y i)

where the only difference from (3.65) is that A0 is replaced by A„U°MA = U°mdA + U*indA is the total inducing voltageU°indA is the lossless inducing voltage given by (3.65) andUAindA is the lossy ground contributionAa = A0+ Aa is the total vector potentialA0 is the lossless vector potential given by eq. (4.17) or (3.30) andAa is the lossy ground contribution given by eq. (4.18)

The frequency domain properties of the inducing voltage are investigated in appendix D.

In chapter 3.4.3 the line losses were shown to be ignorable for line lengths less than about 1000 m. Consequently the overhead line losses, contrary to the lossy ground effects on the electrical fields, will be ignored in the calculations in this chapter. All equations are generally valid for a lossy overhead line, however.

Using the surface-impedance method and ignoring the loss effects on the vertical field, the lossy ground contribution to the inducing voltage can be written (from 4.25):

where the superscript SI denotes surface-impedance and where u is given in (4.9).

Two different situations are analysed; 1) open end and 2) matched termination. If the line terminations are open, the overhead line voltage at end A can be written (from (3.55)):


Va =1 - e ~2yL

(433)

Inserting the vector potential formulation from (3.52) and (3.53) gives:

UA(]<a) = — • jw

dAA(x 0) **dA(x,z) cosh(y(x -x))-yl-----:------■-----------------------dx

dz dz sinh(yl)j<d'z-Ao(xA,0) (4.34)

The loss effect on the total line voltage UA in (4.34) is for high frequencies (corresponding to the initial part in the time domain) mainly determined by the first dAJdz term which is written with index A since the lossless part actually is zero at ground level (z=0). For lower frequencies the integral term contribution becomes important (typically after the first reflection in the time domain) and this will reduce the loss effect.

If the line terminations are matched i.e. terminated by the characteristic impedance, the overhead line voltage at end A becomes (from (3.55)):

u. UxA + (4.35)

Inserting the vector potential formulation from (3.52) and (3.53) gives:

u.m2/go

8Aa(xa,0) 6Aa(xb, 0)

dz dz

+ ±^Ao(xa,0) ~ A0(xb, 0)-e~ri]

** dA (x,z)rf-eT-

xs

•dx(436)

The loss effect on the total line voltage UA in (4.36) is dominated by the first two dAJdz terms. For the side stroke these terms tend to cancel each other (dAJdz\xA = dA/dz|x5).

Fig. 4.10 shows simulations of the induced voltage in an overhead line for an end stroke. The line is 1000 m long and 10 m high and the lightning stroke is located a distance r0= 100 m from its near end (ref. fig. 4.7). The lightning current parameters are v= 1.T10® m/s, k= 1500m, and H = 4000 m. The lightning current at ground is no longer assumed to be a pure step function but is approximated by the Heidler model [96] with the same parameters as in [92, 103] resulting in a peak current of 12 kA and a maximum current/time derivative of 40 kA/ps (typically subsequent return stroke). The results, using either Norton’s method or the surface impedance approach, are compared with calculations assuming a lossless ground.

From fig. 4.10 a) we observe that the initial part of total line voltage at the far end is strongly influenced by the lossy ground and actually the polarity is reversed. After a while the lossy ground effect is reduced, but a negative peak occurs in the lossy ground voltage at each


reflection at the line ends. From fig. 4.10b) we observe that a line terminated by its characteristic impedance and excited by an end stroke is very sensitive to lossy ground effects. The surface impedance approach gives results in close agreement with Norton’s method, at least for the peak value of the induced voltage.

lossless

far end

near end

2 4 t [^s] 6

a) Open ends.

lossless

i far endnear end

-25 -

2 4 t[ns] 6 8

b) Matched terminations.Fig. 4.10 Induced voltage, end stroke, a, = 0.001 S/m and er =10.

Dotted line: lossless. Solid line: Norton. Dashed line: Surface impedance.

Fig. 4.11 shows the induced voltage at the end of a 1000 m long overhead line terminated by its characteristic impedances and excited by a side stroke 50 m from the centre of the line. The calculations are compared with the results in [92], using the same lightning current parameters v= 1.3-108 m/s and A = 1700 m and H-4000 mas in this paper.

V*— lossless

stroker0= 50 m| * 'option

1000 m

Fig. 4.11 Induced voltage, side stroke. Dotted line: lossless.Solid line: Lossy ground (Norton). Dashed line: Lossy ground (Surf. imp.).

Dots (•) from [92] fig. 12. o, = 0.001 S/m, er = 10. r0 = 50 m.

4.6 Discussion 62

Fig. 4.11 shows that the simulations give results similar to those in [92]. Also the surface impedance approach gives very good agreement with Norton’s method except at the initial negative peak and the positive peak. This is due to the high steepness of the induced voltage.

4.6 DISCUSSION

4.6.1 Methods for taking lossy ground effects into account

The evaluations of Norton’s method in fig. 4.2, show that this method is highly accurate (<10% deviation) even for a distances as short as 100 m and a low conducting ground (a, = 0.001 S/m). A relative permittivity lower than 10 reduces the accuracy.

Fig. 4.3 shows how the loss effect, on the vector potential from a vertical current element, is affected by dipole height and frequency. The important observation is the increased loss effect with dipole heights for low frequencies.

Comparing the surface impedance method with Norton’s method in fig. 4.9 shows that the surface impedance method is a reasonable approximation for distances larger than 1000 m. The approximation is poor for high frequencies and large dipole heights. In a practical situation assuming a standard 1.2/50 ps lightning impulse the surface impedance method will give wrong results for very short times (high frequencies) and for large times (large dipole heights). Within these boundaries, where often the maximum value of the induced voltage occurs, the method is accurate, however.

Figures 4.10-4.11 also show that the surface impedance method is accurate compared to Norton’s method. A deviation arises when the voltage is steep. For times beyond the first peak in the voltage the surface impedance method cause the voltage to drift off due to the increased influence of the higher parts of the lightning channel. It is also worth mentioning that the current shape used in the simulations is very steep with a front time below 1 ps. Larger front times will improve the accuracy of the surface impedance method around the first peak in the induced voltage.

4.6.2 Lossy ground effects on electrical fields

The horizontal field on the ground is directed towards the lightning channel for a positive lightning current and decreases faster than 1/r with distance, as seen in figure 4.4. The horizontal field is dependent on the current in the lightning channel and the current in ground. Close to the Earth’s surface the contribution from the current in ground will dominate and the field becomes negative. For larger times the contribution from the current in the lightning channel increases, resulting in an increase in the horizontal field and a possible change in polarity. The field increases with height z, of the observation point and becomes positive as

4.6 Discussion 63

shown in figure 4.5.

The vertical field in the air is almost unaffected by ground conductivity and heights (0-10 m), for a distance 100 m, as seen in figure 4.6. The loss effect on the induced voltage in an overhead line excited by a nearby return stroke is thus related to the horizontal field alone.

The field in ground is very small compared to the field in air and will represent a minor perturbation on the measured induced voltages between an overhead line and a grounding system. Very close to the lightning strike the field in ground will be high, however, and discharges will occur in the ground.

4.6.3 Lossy ground effect on LIO

When calculating the induced voltage on an overhead line, two field components contribute: 1) The integral of the radial field, Ex(z,x) along the overhead line, which is an incoming voltage wave called the horizontal field contribution. 2) The integral of the vertical field at the terminals from ground to the line height, called the incident voltage. The first component is strongly affected by the lossy ground, while the latter is almost unaffected. The induced voltage at the overhead line terminal nearest to the lightning stoke is a sum of a large vertical component and a negative horizontal component, contrary to the other terminal where a small vertical and a positive horizontal component are added to make up the total induced voltage. Thus the induced voltages at each terminal are about the same. When the horizontal voltage wave is affected by ground losses, this balance is disturbed and large deviations from the ideal lossless situation can occur, particularly for long lines and end strokes.

Figure 4.10 shows how the loss effect on the total induced voltage depends on the line terminations. When the line terminals are open the lossy part of the induced voltage is initially about twice as large as when the line is matched. One exception is the side stroke configuration where the two dA/dz terms in (4.36) tend to cancel each other. Induced voltages on a matched line are therefore more sensitive to the stroke location.

Figures 4.10-4.11 show that the end stroke configuration is very sensitive to loss effects, while the side stroke configuration is less sensitive. The reason for this is that the latter configuration is less dependent on the horizontal fields contribution and the lossy part of the total voltage will actually decrease with increasing distance until the loss effect on the incident voltage becomes important. The induced voltage in an end stroke configuration can actually change polarity along the overhead line. This is in agreement with what is reported in [104, 105] using the Norton’s and surface impedance methods, respectively.

The induced voltage in an overhead line becomes more dependent on the lossy ground for small line heights, since the horizontal field could become negative and actually reverse the polarity of the horizontal field contribution. This is also seen from (4.31) where the dominant terms dAJdz are evaluated at the ground level and thus are independent of the line height, while the

4.7 Conclusions 64

lossless inducing voltage U°ind is proportional to the line height.

The induced voltage shows a rather complex dependence on the ground conductivity. However, using the surface impedance method and ignoring the displacement current in the ground the lossy part of the inducing voltage, UAM in (4.32) is inversely proportional to the square root of the ground conductivity.

The horizontal field at ground is not taken into account. This field may result in an increased ground potential where the induced voltage is measured which again could lead to a smaller measured voltage than expected.

4.7 CONCLUSIONS

The Norton’s method is found to be a good approximation when calculating electromagnetic fields over lossy, flat, homogenous ground. The surface impedance method is less accurate for small distances (< 100-1000 m) and steep lightning currents, but gives otherwise an acceptable accuracy. In addition, the surface impedance method results in a much faster calculation process.

A lossy ground has a large influence on the horizontal electrical field and minor influence on the vertical field component. The horizontal electrical field at ground is zero when the ground is lossless but becomes negative (directed toward the lightning channel for a positive lightning current) when the ground is lossy. Above ground this field has a polarity dependent on height and time. The vertical field in the ground is generally small and ignorable compared to the field in air.

The lossy ground effect on LIO is shown to be very important for an end stroke (stroke at the prolongation of an overhead line). Actually the induced voltage can change sign and/or be amplified at the overhead line terminals. The side stroke configuration (stroke at the overhead line’s midpoint) is far less sensitive to lossy ground effects since the LIO in this configuration is less dependent on the horizontal electrical field. The lossy part of the inducing voltage is approximately inversely proportional to the square root of the ground conductivity. From the calculations the lossy ground conductivity seems to be the far most important parameter when determining the induced voltage in a single overhead line without protection. This will be examined experimentally in a small-scaled model in chapter 5. The effect of losses in a more complex low-voltage network will to some extent be further examined in chapter 8.

5MODEL MEASUREMENTS OF INDUCED VOLTAGES

5.1 INTRODUCTION

The models for induced voltage calculations in chapter 3 and 4 are theoretical in nature, based on three main assumptions: 1) The lower part of the lightning channel is straight and vertical. 2) The current starts from ground and travels upward with a roughly constant velocity v. 3) The ground is flat and homogeneous with uniform conductivity which for the lossless case is set to infinity. To verify the established models correlated measurements of lightning currents and induced voltages or electrical fields should be performed. This is not practically possible for obvious reasons (since the probability of a lightning stroke to hit a particular observation point on flat ground is very low) and artificial techniques like triggered lightning or high towers must be used which actually have influence on the lightning stroke behaviour.

Instead of performing induced voltage measurements on an actual overhead line, a scaled model can be used where the lightning channel is replaced by a simulated one. Typically the simulated lightning channel could be a vertical coil wound to reproduce the desired current propagation velocity. Applying a step current between the coil and ground will then produce an upward travelling wave similar to the TL model assumption for a real lightning channel. This approach was used in [90] for a 1:200 scaled model over lossless ground (metal) and the calculations using the TL model gave results in close agreement with the measurements showing that the coupling model is accurate. A similar approach can be used to study the lossy ground effects. This was done by Ishii et.al. [106] and Michishita etal. [107, 108,109]. In [106] a basic model is studied and the calculated induced voltages (using Norton’s method to take lossy ground effects into account) are remarkably close to the measured ones. The most interesting observation is that a lossy ground reverses the voltage polarity at the far end of the line and that this is reproducible in the calculations. The assumed ground conductivity is rather high (0.06 S/m), however, and basically adjusted to fit the measured and calculated horizontal field. The references [107,108, and 109] report the effect of a branch in the line, an inclined or tilted lightning channel, and an overhead ground wire, respectively. The results obtained in [106] are so interesting that similar measurements have been performed here with the main purpose to study the lossy ground effect on induced voltages and its dependency on line height and overhead line terminations.

5.2 EXPERIMENTAL SETUP

The experiment was performed on 22nd of August 1996 on a 50x50 m square lawn at the

5.2 Experimental setup 66

university campus. The coil used as a lightning channel model was suspended by a helium balloon and was designed according specifications provided by Prof. Masaru Ishii in private communication. An overview of the experimental setup is given by figures 5.1-5.3:

Fig. 5.3 Overview of the setup-field and overhead line. Left: end stroke. Right: side stroke.

5.2.1 The coil

The coil consisted of a 0.5 mm diameter Cu wire wound on a 15 mm diameter PVC pipe with 150 turns per meter. The coil was finally covered by tape. Six segments of length 5 m were connected in series. The total coil thus had a length of 30 m. The resulting propagation velocity was found to be 1.1 10* m/s. The top of the coil was suspended by a helium balloon and adjusted so that the coil was straight and vertical. The bottom termination was connected to a

52 Experimental setup 67

current step generator at ground level.

5.2.2 Current step generator

To produce a step current a low impedance DC source was connected to the coil through a relay with mercury wetted contacts as shown in fig. 5.4a. Provided the coil represents an ideal characteristic impedance this results in a step current. Fig. 5.4b shows the actual current step, which deviates somewhat from an ideal step. A peak is observed at the front followed by some oscillations with frequency of about 30 MHz. A reflected wave from the top end of the coil arrives after approximately 550 ns, corresponding to a propagation velocity of v =1.1 • 10s m/s. Thus the coil represents a return stroke for at least 275 ns, and for the minimum distance r0 = 10 m used in the measurements:

tV C

= 350ns

The current in fig. 5.4b) was measured by a Pearson current monitor with a 3 dB bandwidth of 300 Hz - 200 MHz. The current generator was grounded through a 1 m long spear with diameter 1 cm driven vertically into the ground. The relay stays closed in half the period of the applied AC signal (here 50 Hz) and open the other half. The coil current transient has died out when the relay recloses, making the observed induced voltage repeating itself every 20th ms.

Mercury relay

300V |

tI

s**_50 Hz

-Coil

Ground rod

Fig. 5.4a) Current source.t [ns]

Fig. 5.4b) Current waveform.

A constant value 0.2 A was used for the current in the calculations.

5.2.3 Overhead line

The overhead line consisted of a 25 m long stripped coaxial cable. The wound copper shield was used as the conductor. The diameter of the shield is 5 mm. The cable was suspended by 5 wooden poles. The height of the line could be changed in steps of 0.5 m up to 3 m. At each

5.3 Measurement results 68

end of the line a grounding point consisting of a 1 m long ground rod was available. Two different orientations of the overhead line were investigated, corresponding to an end stroke and a side stoke.

5.3 MEASUREMENT RESULTS

The induced voltage in the 25 m long model line was measured between the line conductor and a ground rod using a Tektronix voltage probe with 3 dB bandwidth of 0-150 MHz and an input impedance of R=10 MHz and 0=13.2 pF. When the line was terminated, a vertical conductor was connected between the line conductor and the terminating resistor attached to the ground rod. The same voltage was measured independent on whether the probe was connected up on the line conductor or down on the terminating resistor.

Two different orientations of the overhead line were used and two different line heights, z. The results from the measurements are shown for three different line terminations:°°/°°: open ends of the line°°/0: one end open and the other grounded.z/z: line terminated by its characteristic impedance.

The value of the characteristic impedance was determined by applying a step voltage at one end and adjusting the terminating resistance at the other end to minimise the current reflections at the sending end of the line. The characteristic impedances were measured to be 387 Q for 1 m line height and 452 Q for 3 m height. It was not possible to eliminate the reflections completely using a pure resistance so the above values are based on judgement. The calculated values for a lossless ground are 14-15 Q higher.

5.3.1 End stroke

The position of the overhead line for the end stroke configuration is shown in fig. 5.5.

coilposition overhead line• ---- :---------------------

10.0 m 25 mFig. 5.5 End stroke configuration

Figures 5.6 and 5.7 show the measured induced voltages at each end of the line with height 3 m. The three curves in each figure correspond to the three different terminations (open, matched and one end grounded). The figures are representative for a period of about 350 ns after which the information from the current at the top of the coil reaches the overhead line. Time zero is when the induced voltage changes significantly from zero.


Voltage near end

-©— wo

Fig. 5.7 Ind. volt, near end. End stroke, z=3 mFig. 5.6 Ind. volt, far end. End stroke, z=3 m

Figures 5.8 and 5.9 show the measured induced voltages at each end of the line with height 1 m. The three curves in each figure correspond to the three different terminations (open, matched and one end grounded).

Voltage far end

Fig. 5.8 Ind. volt, far end. End stroke, z=l m

Voltage near end

Fig. 5.9 Ind. volt, near end. End stroke, z=l m

The curves in figures 5.8 and 5.9 for the matched termination case (z/z) are very suspicious. At the far end the voltage initially seems to follow the open terminal case having a first negative peak and at the near end an obvious first positive peak corresponding to a reflection is seen. Comparing figures 5.8 and 5.9 with the figures 5.6 and 5.7 for line height 3 m we see that the difference between the far and near end terminal is larger for the 3 m height. This does not correspond to theory of lossy ground in chapter 4, which actually predicts a larger loss effect for lower heights. Thus it can be concluded that the measurements in figures 5.8 and 5.9 for the matched termination case is erroneous due to an unexpected disconnection between the


line conductor and the ground rod at the far end. An inspection after the measurements indicated a problem with the termination units for line height z=l m.

It is also very interesting to notice that the magnitude of the induced voltage in fig. 5.8 becomes highest when one end is grounded. This suggests that the induced voltage at one end of an overhead line not necessarily is reduced when an arrester is installed at the other. This will however be discussed further when comparing the measurements with calculations.

5.3.2 Side stroke

The position of the overhead line for the side stroke configuration is shown in fig. 5.10.

coilposition

overhead line j ^m

12.5 m 12.5 m Fig, 5.10 Side stroke configuration.

Figures 5.11 and 5.12 show the measured induced voltages at each end of the line with height 3 m and 1 m respectively. The three curves in each figure correspond to the three different terminations (open, matched and one end grounded). The measurements in the matched termination configuration for line height 1 m is erroneous.

e-

:. . . . i.. . . . !

Fig. 5.11 Induced voltage. Side stroke, z=3 m Fig. 5.12 Induced voltage. Side stroke, z=l m

5.4 Comparison with calculation 71

5.3.3 Ground resistivity

The ground resistivity on the test field was measured the day after the induced voltage measurements. Equipment for 100 Hz measurements was used, so the values of the resistivity are probably minimum values. However, the frequency characteristic of the ground was not investigated. In general, one can not exclude the possibility that the ground resistivity actually decreases with increasing frequency due to for instance capacitive effects between granules in the soil. The ground resistivity was measured at three different locations on the field, as shown in fig. 5.13:

3

2

1

''Alinecoil positionsposition

Fig. 5.13 Test field and resistivity locations 1-3.

The resistivity was measured by the 4-point method, using 10 m between the ground rods of 0.5 m length.

The following ground resistivities were measured at the different locations 1-3:1: p, = 302Qm2: p2 = 410 Qm3: p3 = 283QmThe values given above are typical for soil. The resistivity in the middle of the field is higher since that location was dryer.

5.4 COMPARISON WITH CALCULATIONS

In this chapter the measurements are compared with calculations in fig. 5.14-5.20. The curves indicated by "lossless" are lossless calculations based on the analytical time domain model in chapt. 3.4 (eq. 3.61). The curves indicated by "lossy" are the lossy ground calculations based on the frequency domain model in chapt. 4, using Norton’s method. In all the simulations the TL-model for the lightning channel is used (A-»). The lossy ground parameters are a, =0.01 S/m and e = 10. The ground conductivity used is larger than the measured one for convenience reasons since a rather high conductivity must be used in order to obtain a reasonable fit to the measurements. However, no adjustments have been made in order to obtain optimal results. All frequencies above 20 MHz were assumed to vanish in the simulations. This is reasonable compared to what is seen in the measurements where e.g. the oscillation in the coil current


disappears in the measured induced voltage. The overhead line is assumed to be lossy, made of solid copper (Cu), and its transmission coefficient is calculated by Cable Constants in ATP [12] which uses Carson’s formulas.

5.4.1 Open ends (»/°°)

Figure 5.14 shows the induced voltage at the far end for the end stroke configuration. We see that the lossy ground model reproduces some main characteristics of the measurements, e.g. the initial negative hump at the far end, fig. 5.14b). The travelling time seems to be longer for the measurements than in the calculations, as seen from fig. 5.14a).

Meas.

100 150 200 250 300 100 150 200 250 300

a) Line height, z=3 m b) Line height, z=l mFig. 5.14 Induced voltage far end. End stroke. Open ends.

Figure 5.15 shows the induced voltage at the near end for an end stroke when the line is open at both ends. The lossy ground model does not predict as good results as for the far end shown in fig. 5.14. This could be due to the fact that the near end is only 10 m from the coil (lightning channel) and that the current in ground will result in a potential rise at the ground rods which results in a reduced measured voltage. The measured voltages are actually somewhere in between the voltages calculated by the lossless and lossy ground models, but with a reduced steepness at the front.


lossless

100 150 200 250 300100 150 200 250 300

a) Line height, z=3 m b) Line height, z=l mFig. 5.15 Induced voltage near end. End stroke. Open ends.

Figure 5.16 shows the induced voltage for a side stroke when both line terminals are open. Comparing the lossless and lossy ground calculations we see that the loss effects are rather insignificant for this configuration. Both models manage to predict the measured voltage, but after a couple of reflections in the line a deviation is seen between the measurements and the calculation.

100 150 200 250 300150 200 250 300t[ns] t[ns]

a) Line height, z=3 m b) Line height, z=l mFig. 5.16 Induced voltage. Side stroke. Open ends.

5.4.2 One end grounded (°°/0):

A grounding resistance of 30 Q is assumed in the simulations. The grounding impedance has not been measured, however. The difference in the simulations between 30 Q and 0 Q is minor and 100 Q grounding impedance give results in closer agreement with the measurements than 30 Q. The simulations for 0 and 100 Q are not shown here.

Figure 5.17 shows the end-stroke induced voltage at the open far end of an overhead line when


the near end is grounded.

um

a) Line height, z=3 m

UM

b) Line height, z=l mFig. 5.17 Induced voltage far end. End stroke. One end grounded.

Figure 5.18 shows the end-stroke induced voltage at the open near end of an overhead line when the far end is grounded.

Meas.

a) Line height, z=3 m b) Line height, z=l mFig. 5.18 Induced voltage near end. End stroke. One end grounded.

Figures 5.17 and 5.18 show that the lossy ground model in general gives better agreement with the measurements than the lossless model. However, the steepness of the voltages is much lower in the measurements.

Figure 5.19 shows the side-stroke induced voltage at the open end of an overhead line when the other end is grounded.


UM U[V]

6/leas.

lossless

-0.5 —-2 ----

250 300

a) Line height, z=3 m b) Line height, z=l mFig. 5.19 Induced voltage. Side stroke. One end grounded.

Fig. 5.19 shows that both the lossless and the lossy ground model predicts the measured voltage up to the first peak, but after this none of the models are in agreement with the measurements.

5.4.3 Matched terminations

Figure 5.20 compares measured induced voltages with calculated ones. The configuration is end stroke and the overhead line is matched with its characteristic impedance at both ends. The line height is z = 3 m.

Meas. Meas

lossless 0.25 -

-0.25

0.2 0.3 0.4 0.5

a) Near end b) Far endFig. 5.20 Induced voltage. End stroke. Matched terminations.

Fig. 5.20 shows that none of the models manage to reproduce the measurements, which actually lie somewhere in between the two.

5.5 Triggered lightning 76

5.5 TRIGGERED LIGHTNING

During the summer of 95 the author of this thesis was participating in the Camp Standing triggered lightning campaign arranged and hosted by the University of Florida [110]. Due to extraordinary weather conditions with little lightning and a bad timing of the departure by this author, the scientific results from the campaign were very limited and will not be emphasised in this thesis. However, a few results will be shown since they reveal some information on the nature of triggered lightning versus natural lightning. Fig. 5.21 shows an overview of the experimental configuration.

The test line was a 100 m long two-phase line consisting of PVC poles with the upper phase 3 m and the lower 2.6 m above ground. A basic idea was to investigate the influence of grounding one of the phases and thereby study the differences between IT and TN systems. All terminations were open during the recorded events, however. The line conductors were made of stripped coaxial cables and thus equivalent to the line described in chapt. 5.2.3. The voltage at each end of the line was measured with a capacitive voltage-divider (CVD) consisting of a high-voltage part of200 pF in series with a 800 Q resistance and a low-voltage part of a 1.6 pF capacitor in series with a 0.1 Q resistance. This resulted in a voltage-dividing ratio of 8000 and due to the capacitances it was possible to measure the low frequency contribution from the stepped leader along with the high frequency transient from the return stroke(s). The signals from the voltage dividers were transmitted to a control room by an analog optical fibre system having a 3 dB bandwidth as low as 1 MHz. In the control room the signals were recorded by a transient recorder of type Rene Maurer ADAM with a sampling rate of 50 ns and 10 bits vertical resolution. The recorder was set in block mode and could trig 8 blocks with 8000 samples for each event. Setting pre-trigging to 25 % and filtering off the low frequencies from the triggering signal enabled triggering on the return stroke (first or subsequent) and recording of both the leader and return stroke induced voltages.

towerlauncher

groundlauncher

418 m

Naturallightningarea

fiber optics

Test lineoffice

Fig. 5.21 Position of the test line referred to the launching ramps.

5.5 Triggered lightning 77

Natural lightning

Fig. 5.22 shows the measured voltage at the upper phase at the near end induced by a natural lightning stroke within the test field. Both phases of the test lines are open. The numbers on the figure indicates the order of the return strokes. Thus the curve labelled 4, which actually is the largest voltage, is the induced voltage from the fourth return stroke within the same lightning flash.

In fig. 5.22 the voltage starts from a negative value as a result of the field from the lightning channel leader. The change in the field due to the leader propagation is hardly seen in the figure at all. The induced voltage, caused by the return stroke, increases rather slowly.

Fig. 5.22 Induced voltage near end.Natural lightning on the test field.

Triggered lightning

Fig. 5.23 shows the measured voltage at the upper phase at the near end induced by a triggered lightning. The same recording is shown for two different time scales. Both phases of the test lines are open. The triggering was made by the standard or classical lightning rocket system LRS from the ground launcher, using a continuous metal wire [110].

100 110 120 130 140 150 160 170 180 190 200

Hi'PIHilm HillH'HW

Fig. 5.23 Induced voltage near end. Triggered lightning 9507.

5.6 Discussion 78

Fig. 5.23 shows how the induced voltage initially grows negative as a consequence of the rather rapid leader development in the triggered lightning stroke. Around 133 ps the first return stroke takes place, and the induced voltage is more rapidly reduced towards zero. We also see several oscillations in the recorded voltage after the return stroke due to reflections in the test line. Comparing figures 5.22 with 5.23 we first of all see that the lightning-induced voltage due to the leader is much more slowly increasing for the natural lightning.

The lightning current for strike 9507 was measured but later lost due to data communication problems at the launch control. The maximum value was measured to 24.8 kA, but this value is very unreliable. Calculations indicate the half of this value to be more reasonable.

5.6 DISCUSSION

When calculating the lightning-induced voltages in an overhead line some main assumptions must be made regarding: 1) The lightning channel, 2) The coupling model, and 3) The ground. In this chapter the lightning channel is fixed to the TL-model and the lossy ground is particularly investigated. The deviations obtained when comparing the measurements on the scaled-model with calculations are due to inaccuracy in the coupling model and how the lossy ground effects are treated. When transferring the results to an actual system, the inaccuracy in the lightning channel model comes in addition.

5.6.1 Model measurements vs. calculations

The travelling time of the induced voltage wave seems to be a bit larger than ejected. The travelling time becomes larger for 3 m height than 1 m height, also for line configurations with open ends. This observation corresponds to an end-effect resulting in an effective length of the line equal to L' = L + 2 -z. This will give about 20 ns longer travelling time for line height 3 m, which fits well with the observations in figures 5.14a), 5.15a), 5.17a) and 5.18a), where the reflections in the measurements seem to be 40 ns delayed compared to the simulations. Using the transmission line approach when calculating induced voltages requires that the line height (z) is much less than both the line length (Z) and the wave-length (A = c/f) of the incident fields. In the scaled-model this is not the case (at least not when z=3 m), and a more accurate modelling of the terminations of the line is required, as shown in [111].

The high frequencies seem to be more attenuated in the measurements than expected. In fact, frequencies above 10 MHz seem to vanish. This can not be reproduced by calculations. Especially the configuration with one end grounded and the other open lacks high frequencies. This could be due to an unexpected frequency characteristic of the ground rods or the ground in general.

5.6 Discussion 79

Side stroke

The difference between the lossless and the lossy ground calculations for the side stroke configuration is in general minor. The measured voltage follows the two calculations initially but then deviates from the calculations in magnitude after a time corresponding to the travelling time of the line (85 ns) as seen from fig. 5.16 (open ends) and 5.19 (one end grounded). It is difficult to find a reason for the deviation between measurements and calculations, which seems to be due to a stronger attenuation of the voltage waves on the line than expected. One can of course argue that the ground conductivity should be lower. This will improve the calculations for the side stroke, but make the calculations even more deviant for the end stroke configuration. The induced voltage is for a side stroke almost proportional to the line height.

End stroke

The lossy ground model gives in general better agreement with the measurements than the lossless model for end strokes. The agreement is in particular good for the far end when both ends are open, fig. 5.14 (e.g. the initial negative hump is reproduced in fig. 5.14b). The lossy ground model gives poorest agreement for the near end with both ends open, and in this case the measurements are actually somewhere in between the lossy and lossless calculations. When one end is grounded, the lossy ground model gives much better agreement than the lossless model, which actually predicts too low induced-voltages, ref. figures 5.17-5.18. The grounding resistance of the grounded end is in the simulations assumed to be 30 Q, using 100 Q brings the calculations for the lossy model even closer to the measurements, but this is rather speculative. None of the models manage to predict the voltages when the line is matched (z/z). This could be due to the feet that a resistor is not an ideal termination for the actual frequency range.

General remarks on the lossless model

Usage of the very simple, analytical, lossless calculation model gives reasonably good results for the small-scaled model used in the measurements. Figures 5.14-5.15 (open ends) show that the calculations predict a bit too low voltages for the near end and a bit too high voltages for the far end. The relative deviation between measurements and lossless calculations is larger for small line heights. Besides, the lossless model can not predict the relatively long front time of the induced voltage, having a frequency content less than 10 MHz. When one end is grounded, the lossless model predicts too low voltages, particularly for the far end when the near end is grounded as seen in fig. 5.17. The deviation between the lossless model and the measurements is almost independent on the line height. This is seen in e.g. fig. 5.17 where the lossless approach fits better for 3 m than for 1 m.

General remarks on the lossy ground model

The observation that the deviation between the lossless model and the measurements is almost independent on the line height fits well with the lossy ground model formulated in eq. (4.31). The dAJdz terms evaluated at ground level contributes significantly to the inducing voltage in

5.6 Discussion 80

the overhead line at height z. The accuracy of Norton’s method can be questioned for the short distances and high frequencies used in the simulations. Fig. 5.24 shows, however, that Norton’s method is accurate for the situation, having a deviation less than 10 % compared to Sommerfeld’s exact formulation for flat, homogenous ground.

Accuracy of Norton’s method in [%] r=10 m, z=3 m, »=0,OC1 S/m and er=10

ao*[%]20

B hW ■8-105

3 *445

003

Fig. 5.24 Accuracy, e [%], of Norton’s method.

To obtain an optimal agreement between die measurements and the calculations a conductivity around 0.02 S/m must be assumed in contrast to the measured values of 0.0024 - 0.0035 S/m.

5.6.2 Sources of error

The largest source of error is the grounding characteristic on the test field. The value of the conductivity is known only for 100 Hz, and the variation with position and depth is not known. The test field itself consisted of a lawn, being fairly humid on the day of the measurements.

The characteristic of the about 1 m long ground rods is also unknown. The impulse impedance between the rods and true ground was not measured. Probably the potential at the ground rod will be influenced by the horizontal electrical field at ground. This will perturb the measured voltage between the overhead line and the ground rod. An increase in the potential of the ground rods can explain much of the deviation between measurements and the lossy ground model, and will result in a lower measured voltage than calculated. This explanation is rather speculative however, since the potential at the ground related to "true" or distant ground in unknown and is difficult to measure. Also the step current from the coil was injected into the ground via a ground rod of 1 m. This will affect the distribution of the "lightning" current in the ground and thus possibly have an influence on the measured induced voltage. Typically the deviation between measurements and the lossy ground calculations is largest closest to the coil (fig. 5.15) or when the closest end is grounded (fig. 5.17).

The wound copper screen of the coax cable, used as the overhead line, could have a different behaviour than a solid copper line assumed in the simulations, but since the dominant frequency is high this is not believed to be a problem. The calculations give a shorter travelling time of voltage waves than the measurements. Adding two times the line height to the line length results in the measured travelling time, however.

5.7 Conclusions 81

In general the lossy ground model predicts the sign of the deviation between the lossless calculations and the measurements and this deviation’s dependency on line height, but not its magnitude and steepness. A main concern regarding the correctness of the calculations is the adequacy of the Agrawal coupling model. This has been further discussed and analysed in appendix E. Two main problems related to the usage of this coupling model are 1) the large variation of the horizontal field along the overhead line for an end-stroke configuration and 2) the ground potential due to the horizontal field at the Earth’s surface which will affect the potential at the ground reference point (e.g. a ground rod).

5.6.3 Triggered lightning

The information obtained from the triggered lightning recording in fig. 5.23 compared to the natural lightning recordings in fig. 5.22 is related to the (stepped) leader process which seems to be fast for the triggered lightning event and very slow for natural lightning. This is an important observation since it allows a separation of the contributions from the charged leader and the return stroke.

5.7 CONCLUSIONS

The calculations of induced voltages on a model line, using Norton’s method to take ground losses into account, predict qualitatively many of the characteristic features of the measured voltages. Quantitatively, however, the agreement is not satisfactory. The reasons for this is not fully revealed, but the following factors may be of importance:• To obtain an optimal agreement between simulations and measurements, a ground

conductivity considerably higher than measured at low frequencies must be assumed.• The measurements show much higher attenuation of frequencies above 10 MHz than

predicted by Norton’s method. This deviation does not seem to originate from deviations between Norton’s method and Sommerfeld’s exact solution.

• The ground electrodes used may have disturbed the current and potential distribution in the ground.

• Because very high frequencies are involved in the model measurements, the transmission line modelling of the line may not be adequate.

The measurements show that the lossy ground effects are much less significant for side strokes than for end strokes. This fits well with the observations in chapter 4.

The simulations based on lossless ground do not predict the characteristic features of the measured voltages as well as those based on Norton’s method. Quantitatively, however, the lossy ground calculations are hardly better.

6MEASUREMENTS ON TRANSFORMERS

6.1 INTRODUCTION

When analysing lightning-induced voltages in a practical low-voltage network, the distribution transformers in the system must be considered. A transformer plays an important role both related to direct induced voltages in the low-voltage system and to voltages transferred from the high-voltage side. Based on the discussion in chapt 2.8 only the direct induced voltages in low-voltage systems will be studied and in this case the transformer can be considered as a passive termination of a low-voltage overhead line. A complete model of a transformer is rather complicated and establishing such model based on geometrical or constructional parameters requires a comprehensive study as e.g. in [112], which is beyond the scope of this work. The models of distribution transformers developed in this chapter are instead based on measurements of the admittance seen into the low-voltage terminals.

The model of a transformer must be sufficiently accurate for the practical frequency contents of lightning induced voltages, typically 10 kHz - 1 MHz. For simplicity the same voltage is assumed induced in all the three phase conductors in the network. This implies that only the common mode system needs to be studied. Two different coupling configurations are analysed:1. The LV-side's neutral is isolated from ground.

This is usually the situation in IT systems in normal operation.2. The LV-side's neutral is grounded (connected to the transformer tank).

This is the common situation in TT and TN systems.The HV-side neutral (if any) is always assumed isolated from ground.

Measurements in the frequency domain (0-10 MHz) were performed on several different distribution transformers with rated power from 50 kVA to 1250 kVA. Rated voltage was 11.4 kV or 22 kV on the HV side and 235 V or 400 V on the LV-side. The transformers were spare units kept in a storage building at Trondheim Energiverk. Most of the units were rather old, but well maintained. The internal construction, e.g. type of winding, coupling and insulation materials in use, is not be addressed in this thesis. All the investigated transformers were three- phase units with a three-leg core. This results in a very low common mode flux, which is forced out of the core for a Y-coupled HV side or is zero for a A-coupled HV side.

The frequency characteristic of the transformer admittance is approximated by a lumped RLCG network. The fitting is based on the identification of poles and zeros (maxima and minima in the measured admittance) and on an approximation of these by a rational function with complex conjugated poles and zeros. The fitting process is described in appendix F.

6.2 Experimental setup 83

6.2 EXPERIMENTAL SETUP

All measurements were performed in the frequency domain by means of a network analyser instrument (HP4195A). The instrument contains a sinusoidal source with variable frequency and a tracking generator. The ratio between two signals is measured and displayed as e.g. magnitude and phase angle as a function of frequency.

Fig. 6.1 shows the experimental setup used to measure the common mode input admittance of the low-voltage winding. The three phases on the low-voltage side were connected together and supplied by an AC-voltage from the network analyser's tracking generator. The current (I) into the LV phases was measured along with the voltage (U) between phases and the transformer tank. The ratio between these signals (Y=I/U) was recorded by the instrument. The neutral point of the low-voltage winding was either connected to chassis or kept isolated.

PearsonCurrentmftnttnr

HR4195ATransformerP

Fig. 6.1 Experimental setup for admittance measurements.

The Pearson Current Monitor had a 1:1 ratio between input current and output voltage and a rise time of 2 ns (3 dB band width: 300 Hz - 200 MHz). Due to the 50 Q internal impedance of this device the measured current became the half of the real value. This was compensated for by an external multiplication of the measured admittance by a factor 2. The HV side was terminated by the resistance network of Rg and R,. This corresponds to a model of a transposed overhead line with characteristic impedance phase to ground of Zc = R0 + R, and mutual impedance of Zm = R0. The resistors were chosen equal to R0 = 142 Q and R, = 342 Q.

6.3 TRANSFORMER INPUT ADMITTANCE

The measurements of the transformer's common mode admittance on the low-voltage side were carried out for two frequency ranges. 1) From 1 mHz to 10 MHz in steps of 25 kHz, called high frequency range, and 2) from 1 mHz to 1 MHz in steps of 2.5 kHz, called low frequency range. The high frequency range is the main interval while the low frequency measurements are performed to discover any dominant natural frequencies below 100 kHz. The frequency range from 10 kHz to 1 MHz is believed to be of most importance when calculating induced voltages LV overhead lines. All measurements are shown in appendix G. The measurement on transformer T14 was erroneous for unknown reasons, when the neutral was isolated.

6.3 Transformer input admittance 84

The measurements were performed on the transformer units summarised in tab. 6.1:

Tab. 6.1 Transformer data.

Transformer

Registrationnumber

Manufacturer S[kVA]

uH/uL[kV/V]

Shorten, voltage [%]

Coupling

T01 40109587 EB 1990 1250 22.00/400 5.37 Dynll

T02 85137-2 Asea Per Kure 1985 800 11.43/400 5.01 Dynll

T03 8013085 Pauwels Trafo 1980 800 21.00/235 4.89 YynO

T04 9120B5 Nor-Tra-Mo 1967 500 11.43/235 3.91 YynO

T05 9614437 Pauwels Trafo 1996 500 11.43/415 3.76 Dynll

T06 OTC6540 Nebb/Mere 1979 315 21.00/235 4.34 YynO

T07 42008894 ABB 1982 315 22.00/240 4.15 YynO

T08 731066 Nebb/Mere 1973 300 11.43/235 4.10 YynO

TOP 710084 Nebb/Mere 1971 200 11.43/235 4.10 YynO


Til 731574 Nebb/Mere 1973 100 21.00/235 3.88 YynO

T12 17990 6965 Mere Trafo 1965 100 21.00/235 3.95 YynO


T14 771009 ABB 1977 50 11.43/235 4.39 YynO

T15 731562 Nebb/Mere 1973 50 11.43/400 4.17 YynO

6.3.1 Measurement results and fitting

All admittance measurements and the fitting with RLCG networks are shown in appendix G. In this chapter the results for two transformers are shown. The high-frequency measurements (0-10 MHz) are used as the basis for fitting. The order of the model is kept as low as possible, thus poles and zeros above 5 MHz are mostly ignored. For low frequencies the measured admittance is smooth when the neutral is isolated, but it often contains some natural frequencies when the neutral is grounded. These low natural frequencies are ignored, but later discussed in chapt. 6.3.3.

Figure 6.2 shows the general circuit equivalent of the transformer’s common mode input admittance used when the transformer neutral is isolated or grounded, respectively. The form of the fitting function and its realisation in a linear RLCG network is primarily based on [113] and [114].


a) Y-PK equivalent. Neutral isolated. b) Y-PNK equivalent. Neutral grounded.Fig. 6.2 Transformer input admittance equivalent networks.

Figure 6.3 shows the measured and fitted admittance for transformer T02 (800 kVA).

ABS(Y-PK) ABS(Y-PNK)

b) Neutral grounded. Fig. 6.3 Input admittance transformer T02.

Tab. 6.2 shows the fitting data summary for transformer T02.

Tab. 6.2 Fitting data, transformer T02.

T02:Manufacturer:

Asea Per Kure, 1985.

Rated power:800 kVA

Rated voltage:11.43kV/400V

Coupling:Dynll

Fitting RLCGelement

0 1 2 3

Neutralisolated

R[£2] - -965.0 6.949 -L[pH] - 82.92 3.100 -C[nF] - 0.0717 7.296 -G[mU] 9.313E-7 0.902 -6.637 -

Neutralgrounded

R[Q] 0.6217 3.156 1592.0 -L[pH] 8.693 6.776 73.95 -C[nF] - 4.361 0.02089 -G[mO] - 1.180 -0.3767 -

When the neutral is isolated as shown in fig. 6.3 a) the measured admittance is fitted in four


points; a low-frequency point, two maximum points (poles) and one minimum point (zero). This results in an accurate fit up to just above 1 MHz. To further improve the fit a zero must be added at a high frequency. This has been avoided since it results in a circuit equivalent different from fig. 6.2a). The deviation between the measured and fitted admittance shown in fig. 6.3a) is considered acceptable for induced voltage calculations, however. When the neutral is grounded as shown in fig. 6.3b) the measured admittance is fitted in five points; a low- frequency point, two maximum points (poles) and two minimum points (zeros). This results in an accurate fit up to at least 3 MHz.

Figure 6.4 shows the measured and fitted admittance for transformer T15 (50 kVA).

ABS(Y-PK) ABS(Y-PNK)

0.02 -1 0.01 -

0.005 -

0.002 -

f [MHz]

b) Neutral grounded.

0.01 !-

0.005

0.002

f [MHz]

a) Neutral isolated.Fig. 6.4 Input admittance transformer T15.

Tab. 6.3 shows the fitting data summary for transformer T15.

Tab. 6.3 Fitting data, transformer T15._______ _________ _________

T15:Manufacturer:

Fitting RLCGelement

0 1 2 3

Nebb/Mare, 1973. Neutralisolated

R[Q] - 59.77 8.259 44.60

Rated power: L[nH] - 1603 6.446 14.01

50 kVA C[nF] - 0.7385 1.675 0.4074

Rated voltage: 11.43kV/400V

G[mU] 2.328E-7 -0.06227 0.6346 -0.5547

Neutralgrounded

R[Q] 1.479 9.581 -0.3703 -198.7

Coupling:YynO

L[|iH] 31.99 14.41 13.19 38.62

C[nF] - 1.866 0.4301 0.05444

G[mO] - 0.0526 0.7093 0.4718

When the neutral is isolated as shown in fig. 6.4a) the measured admittance is fitted in six points; a low-frequency point, three maximum points (poles) and two minimum point (zero).


This results in an accurate fit up to above 3 MHz. To further improve the fit a zero and a pole must be added around 4 MHz. When the neutral is grounded as shown in fig. 6.4b) the measured admittance is fitted in seven points; a low-frequency point, three maximum points (poles) and three minimum points (zeros). This results in an accurate fit up to at least 4 MHz.

6.3.2 Transformer model simplifications

In this sub-chapter some key quantities from the measurements and fitting in appendix G are summarised with the aim to simplify and generalise the transformer models. The influence the models and the simplifications have on LIOs in overhead lines will be studied in chapt. 6.4.

6.3.2.1 Neutral isolated

The table 6.4 shows a summary of some fitting data for the 15 transformers when the neutral is isolated. The high frequency measurements (0-10 MHz) are used as a basis.

Tab. 6.4 Fitting data of 15 transformers with neutral isolated._______________________________

Transformer

Manufacturer S[kVA]

UH/UL[kV/V]

np(low)

fo[MHz]

\Y-PK\0[mho]

Col [nF] (100kHz)

T01 EB 1990 1250 22.00/400 2 0.575 0.0926 10.46

T02 Asea Per Kure 1985 800 11.43/400 2 1.025 0.2368 7.70

T03 Pauwels Trafo 1980 800 21.00/235 2 0.65 0.3886 14.16

T04 Nor-Tra-Mo 1967 500 11.43/235 3 0.925 0.1168 4.42

T05 Pauwels Trafo 1996 500 11.43/415 2 0.425 0.1939 16.07

T06 Nebb/Mere 1979 315 21.00/235 2 1.575 0.3668 3.39

T07 ABB 1982 315 22.00/240 2 0.7 0.0835 6.90

T08 Nebb/Mere 1973 300 11.43/235 2 1.1 0.1197 4.74

T09 Nebb/Mere 1971 200 11.43/235 3 1.25 0.1963 4.29

T10 Pauwels Trafo 1980 200 21.00/235 2 1.125 1.125 3.58

Til Nebb/Mere 1973 100 21.00/235 3 1.075 0.1277 3.39

T12 Mere Trafo 1965 100 21.00/235 3 1.0 0.1263 3.68

T13 Pauwels Trafo 1985 100 21.00/235 2 0.55 0.0555 4.64

T14 ABB 1977 50 11.43/235 -

T15 Nebb/Mere 1973 50 11.43/400 3 0.45 0.0233 2.84


In table 6.4:np is the number of poles in the fitting function, nz = np -1f0 is the frequency where the first significant pole occur (ignoring all small local extremal points). The admittance is mostly capacitive up to this frequency.\Y-PK\0 is the absolute value of the measured admittance at this frequency (first significant maximum).C0L is the capacitance corresponding to the admittance at 100 kHz (almost purely capacitive). The value of C0L is believed to be of great importance when calculating the induced voltage in a low-voltage system, since the dominant frequencies in an induced voltage waveform will be rather low (10 kHz - 1 MHz). This will be studied in chapter 6.4.

From table 6.4 we see that the correlation between C0L and rated voltage and/or power is poor. Probably a more detailed study of the construction and the insulation of the neutral point at the LV side is required to predict the low-frequency capacitive behaviour of the transformer.

Average model

Figure 6.5 shows the average admittance for all the 14 transformers (T14 not included). It was possible to obtain a fairly accurate, stable fit (negative real parts of poles) with the technique described in appendix F.

ABS(Y-PK) ANG(Y-PK)

75 -

-25 --50 -

f [MHz]

0.01 -/

0.003

0.001 -

0.0003

Fig. 6.5 Average measured and fitted admittance. Neutral isolated.

Equivalent RLCG-network: Go= -3.8999133E-09

0.5106E-03 0.7563E-04 0.3273E-05 0.1253E-03

0.3052E+04 0.6893E+02 0.9420E+01 0.1133E+04

C0.4837E-10 0.7807E-09 0.5661E-08 0.4740E-10

G-0.2685E-03 -0.3494E-03 0.2662E-02

-0.4059E-03

This will give and average low-frequency capacitance of Qo = 6.54 nF


6.S.2.2 Neutral grounded

The table 6.5 shows a summary of some fitting data for the 15 transformers when the neutral is grounded, based on the high-frequency measurements (0-10 MHz).

Tab. 6.5 Fitting data of 15 transformers with neutral grounded.

Transformer

Manufacturer s[kVA]

UH/UL[kV/V]

np

(low)fo

[MHz]\Y-PNK\[mmho]

Lol [liH] (100kHz)

T01 EB 1990 1250 22.00/400 2 0.675 16.82 7.30

T02 Asea Per Kure 1985 800 11.43/400 2 0.6 6.186 9.61

T03 Pauwels Trafo 1980 800 21.00/235 2 1.05 13.28 6.34

T04 Nor-Tra-Mo 1967 500 11.43/235 5 1.025 10.43 6.76

T05 Pauwels Trafo 1996 500 11.43/415 2 0.625 13.14 10.49

T06 Nebb/Metre 1979 315 21.00/235 3 0.9 4.791 8.63

T07 ABB 1982 315 22.00/240 2 0.85 8.720 7.68

T08 Nebb/More 1973 300 11.43/235 2 1.5 5.257 9.58

T09 Nebb/More 1971 200 11.43/235 3 1.475 2.686 9.07

T10 Pauwels Trafo 1980 200 21.00/235 2 1.5 3.839 11.63

Til Nebb/More 1973 100 21.00/235 2 1.275 1.713 16.71

T12 Mere Trafo 1965 100 21.00/235 2 1.2 1.572 17.67

T13 Pauwels Trafo 1985 100 21.00/235 2 0.8 4.206 18.24

T14 ABB 1977 50 11.43/235 3 0.525 1.919 37.02

T15 Nebb/Mare 1973 50 11.43/400 3 0.5 2.225 32.28" High value due to a low natural frequency.

In table 6.5:np is the number of poles in the fitting function. nz = npf0 is the frequency where the first significant zero occur (ignoring all small local extremal points). The admittance is mostly inductive up to this frequency.\Y-PNK\0 is the absolute value of the measured admittance at this frequency (first significant minimum).Lol is the inductance corresponding to the admittance at 100 kHz (almost purely inductive). The value of Lol is believed to be of great importance for calculation of the induced voltage in a low voltage system. This will be studied in chapter 6.4 and 8.

From table 6.5 we see that the correlation between Lol and rated power seems to be significant,


and this will now be investigated further. In fig. 6.6 the inductance Lol is plotted as a function of rated power S and low-voltage U. This relation can be fitted by a power function on the form (6.1):

LOR's-Uq

"r.-w/

25.9 ]iH°S 235 V '

U 50 kVA/

-0.542(6.1)

iVirs/St,Fig. 6.6 Low-frequency inductance as function of transformer parameters.

The function in (6.1) is drawn as a solid line in fig. 6.6 with the 100 kHz values for Lol from tab. 6.5 drawn as dots (A-coupled HV-side: solid dots). We see that the fit is good. When only the rated power and voltage of the transformer is known the inductance obtained from (6.1) can be used as a first approximation of the transformer model.

Average model

Figure 6.7 shows the average admittance for all the 15 transformers. It was not possible to obtain an accurate, stable fit (negative real parts of poles), for the whole frequency range, with the technique described in appendix F. A simple fit is used instead, fairly accurate up to 1 MHz.

ANG(Y-PNK)

Fig. 6.7 Average measured and fitted admittance. Neutral grounded.


Equivalent RLCG-network:Lo= 8.8644192E-06 Ro= 0.6859148

L R C G0.1219E-04 0.2958E+02 0.2575E-08 0.4506E-03

This will give and average low-frequency inductance of Lf, = 8.86 pH

6.3.3 Discussion

For frequencies 10 kHz to approximately 500 kHz the transformer common mode input admittance can be approximated by a capacitance when the neutral is isolated and by an inductance when the neutral is grounded. The inductance seems to be significantly related to transformer ratings. For high frequencies the contribution from external connections dominates, and the response above 5 MHz is independent of neutral grounding.

The resistors terminating the HV side were found to have insignificant influence on the admittance measured at the LV side.

When the neutral is isolated only one transformer (T04) has any poles or zeros at low frequencies. All the other transformers show a purely capacitive behaviour. When the neutral is grounded some of the transformers show pronounced low natural frequencies (<10 kHz). It is difficult to explain the reason for this since the behaviour of the transformers seems to vary considerably. Besides, the resolution (2.5 kHz) in the measurement is too low for any detailed study around the 10 kHz frequency. The transformer T04 has an obvious natural frequency around 30-40 kHz. The transformers T06, T08, T09, T10 and T14 seem to have natural frequencies around 10 kHz. At 100 kHz all the transformers show a purely inductive behaviour.

The low frequency behaviour was investigated further on a test transformer in our laboratory. This transformer, called TOO, has the following data:Manufact.: EB 1990. Type TK0Power: 100 kVAVoltage: 36kV/230VShort Cir.: Ek= 5 %Coupling: YNynO

Fig. 6.8 shows the low-voltage common mode admittance of TOO at low frequencies when the neutral is grounded. The measurement was performed according to fig. 6.1. The basic hypothesis is that the source of the low natural frequency in the admittance really is the high voltage winding, which is larger an thus will have lower natural frequencies. To investigate this further, the low-voltage side was supplied by an AC voltage from the network analysers tracking generator and the ratio between the voltage at the HV-side and the LV-side was measured. This ratio is shown in fig. 6.9.

6.4 Induced voltage calculations 92

abs(Y-PNK) ang(Y-PNK)

abs

ang

Fig. 6.8 Admittance TOO. Neutral grounded.

abs(UHAJL) ang(UHAJL)

abs

ang

Fig. 6.9 Ratio between voltage at HV and LV side.

We see that both the admittance and the ratio have maxima at the same frequencies, the first one occurring at 4 kHz. From this it can be concluded that the HV-winding plays an important role in the low-voltage common mode admittance behaviour at low frequencies.

6.4 INDUCED VOLTAGE CALCULATIONS

6.4.1 Overhead line terminated by transformer

In this chapter, the transformer models developed in appendix G and simplified in chapter 6.3 are included in calculation of the induced voltage in an overhead line. One end of the line is terminated by the transformer while the other end is open. A 5 Q grounding resistance is added in series with the transformer. The transformer's low-voltage neutral is either connected to the transformer tank or isolated. The investigation is based on a 250m to 1000 m long and 6 m high overhead line. The default current shape shown in appendix A (with an amplitude of 30 kA) is used. Only a few typical examples are shown here. The network configuration is shown in fig. 6.10.

Overhead line Z1—500 Q

Transformer

250 m

Lightningstroke

L= 250 m -1000 m

Fig. 6.10 Induced voltage configuration.


Fig. 6.11 shows the induced voltage at both ends of a 500 m long overhead line terminated by a neutral grounded transformer at one end (TT system) and open at the other end. The two transformers T02 and T15 are used as examples. The voltage at the transformer is called UT and the voltage at the open end is called UO.

UTM UO[kV]

-500 —-750 -

-1 000

UTM UOfkV]2 0001 500 -1 000

500 —

-500 --1 000 -

-1 500 --2 000

a) Transformer T02 (800 kVA). b) Transformer T15 (50 kVA).Fig. 6.11 Induced voltage, neutral grounded. L= 500 m.

From fig. 6.11 we see that the voltage at the open end is much larger than the voltage at the transformer and almost independent of the transformer admittance. The voltage at the transformer end seems to be very dependent on the transformer admittance, however. Also the oscillatory nature of the voltage is important to realise.

Fig. 6.12 contains the same calculations as fig. 6.11, but with the neutral point isolated.

UT [kV] UO[kV]

UT

UO

a) Transformer T02 (800 kVA).

UT [kV]50

40

30

20

i_ _ _ _ _ _ _ i_ _ _ _ _ _ _ i_ _ _ __ _15

UO[kV] 50

40

20

UT

UO

205 10t[us]

b) Transformer T15 (50 kVA).Fig. 6.12 Induced voltage, neutral isolated from ground. L= 500 m.

Fig. 6.12 shows that the maximum induced voltage is very dependent on the transformer admittance, in contrast to fig. 6.11. Oscillations with a frequency somewhat higher than in fig. 6.11 are seen, but less pronounced.


When the neutral is grounded the investigated low voltage system consists basically of an overhead line open in one end and grounded with a low impedance at the other. This results in oscillations as shown in figures 6.11 with a dominant frequency of fD = c/4 -L = 150 kHz where c is the speed of light and L is the line length. The transformer admittance at this frequency (almost a pure inductance, Lol) will determine the maximum induced voltage. If the line is terminated with a low impedance at both ends, the induced voltage will oscillate with the double frequency of fD. When the neutral is isolated the induced voltage includes frequencies somewhat larger than the frequency fD. The transformer admittance below this frequency (almost a pure capacitance, C0L) will to a large extent determine the induced voltage.

Fig. 6.13 shows the maximum calculated induced voltage occurring at the transformer UT or at the other open end UO, using the same configuration as in figures 6.11 and 6.12. All the 15 transformers are investigated and the full representation of the transformers, given in appendix G, is used. Lines are drawn between the results to facilitate the reading of the figure.

UT [kV] UO [kV] UT, UO [kV]

50 —

20 —•

— 10

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150 1 2 3 4 5 6 7 8 9 1011 12131415 Transformer number

a) Neutral grounded.Fig. 6.13 Maximum induced voltage dependent on transformer number. Line length 2=500 m.

When the neutral is grounded, the maximum induced voltage at the transformer varies from 600 V for the high power transformers up to above 2000 V for the low power transformers. The maximum induced voltage at the open end is about 34 kV and almost independent of the transformer. When the neutral is isolated, the maximum induced voltage at the transformer varies from 15 kV to 46 kV and at the open end from 34 kV to 51 kV. The maximum voltages are increasing with decreasing rated power of the transformer, but less correlated than for grounded neutral.

Tab. 6.6 shows similar results as fig. 6.13, but the transformer models are now further simplified. The simplifications are made in order to investigate how sensitive the maximum induced voltage is to variations in the transformer model. The simplified models are first approximations based on the low frequency behaviour of the transformers from tables 6.4 and 6.5. From tab. 6.6 we see that the maximum induced voltage at the transformer is very dependent on its admittance. It is actually increasing with a factor 4 from the highest admittance to the lowest.


Tab. 6.6 Maximum induced voltage. Simplified transformer models, £=500 m.

Transformer

Y-PKN-isol.

Y-PNKN-ground

Neutral isolated UT[kV] U0 [kV]

Neutral grounded UT[V] Uo [kV]

T001 2nF 4 pH 52.72 56.24 534.8 33.89

T002 5 nF 10 pH 35.38 47.38 689.3 34.15

T003 10 nF 20 pH 22.97 38.97 1039.2 34.56

T004 20 nF 40 pH 13.33 33.91 2213.4 35.33

Average 6.54 nF 8.86 pH 30.39 44.34 658.4 34.10

None open end grounded 76.90 76.90 471.8 33.72

When the neutral is grounded, pronounced oscillations occur with a frequency dependent on the line length. To investigate this dependency, the maximum induced voltage is calculated for different line lengths and shown in fig. 6.14. The solid line is based on the simple inductance representation according to (6.1). The dotted line is based on the low-frequency inductance representation according to tab. 6.5. The full representations of the transformers are used for the other points. This lines should be compared to the open circle symbols (o), representing the same configuration (500 m) with the complete model of the transformers from appendix G.

UT[kV]

250 m

500 m

1000 m

500 m

500 mLoLfrom tab. 6.5

0 1 2 3 4 5 6 7 8 9 1011 1213 1415- Transformer number

Fig. 6.14 Maximum induced voltage at the transformer with the 15 different transformer models. Three different line lengths; 250 m, 500 m and 1000 m. Neutral grounded.

Fig. 6.14 shows that the maximum voltage across the transformer winding is mostly increasing with line length. The maximum voltages at the open end are not shown in fig. 6.14, but they are approximately 14 kV, 34 kV and 48 kV for a line of length 250 m, 500 m, and 1000 m, respectively.


We see that the maximum voltages for the simple inductance approximation (based on transformer ratings) generally are somewhat lower than for the full representation. If an exact value of the LIO across the transformer winding is desired the full representation should be used, but the simplified inductance model is believed to give acceptable results for the overall LIO’s in an low-voltage system. This will be further investigated in chapter 8.

6.4.2 Discussion

In section 6.4.1 we have studied the impact of transformers on induced voltages. First of all it is important to notice the influence of low-voltage neutral grounding. Isolated neutral will result in the largest overvoltages as seen from the previous sections. The LIO can hardly be predicted based on the transformer's rated power and voltage alone. The maximum LIO varies from 34 kV to 51 kV at the open terminal of a 500 m long overhead line when a transformer is connected at the other end. The voltages seem to roughly increase with decreasing rated power of the transformer. The most simple transformer model is to use a pure capacitance of 2-20 nF.

When the neutral is grounded it is easier to predict the maximum LIO in an overhead line, since this voltage is almost independent of the transformer model. The transformer can as a first approximation be modelled as an inductance of 4-40 pH. A good correlation is found between this inductance and the transformer’s rated power and voltage. This will improve the estimation of the LIO at the transformer, when specific data for the transformer are missing. The maximum LIO is about 34 kV at the open terminal of a 500 m long overhead line when a transformer is connected at the other end. The maximum induced voltage increases with the overhead line length.

It is common practice to protect the LV neutral point when this is isolated. Such protection will probably operate in a case of potentially harmful induced voltages, maybe already during the formation of the stepped leader. An isolated neutral system must then be treated as a grounded neutral system.

The influence of transformers on LIO in the high (or medium) voltage system was reported in [82]. The common mode impedance seen from the HV side of the transformer was modelled based on measurements using the Vaessen model [115]. The conclusion in [82] was that the transformer has a strong influence on the LIO in the high-voltage system. Fig. 6.18 shows calculation of the induced-voltage across the transformer winding using the same configuration as in [82]. The line is 1000 m long and 10 m high. The line is terminated with its characteristic impedance at one end and a transformer at the other. The lightning current is 50 kA and is located 50 m from the line centre. The return stroke velocity is v=1.5-108 m/s. The TL model is used here even though the MIL model with X=2 km was used in [82], but this has probably no influence for the first microseconds. The curves labelled T15-NG and T15-NI are calculations with a model of transformer T15 according tab. 6.3 with the neutral grounded or isolated, respectively. For the curve labelled 0.5 nF, the transformer is approximated by a capacitance of 0.5 nF, something which also is analysed in [82]. The dotted points are adopted

6.5 Conclusions 97

from [82] and represents the Vaessen model of a transformer with rated values (40 kVA, 16/0.4 kV). The curve labelled open is for an open end of the line.

400V/16kVU[kV]—GDv400

Open Borghetti U300 T15-NG

11.4kV/400V200 T15-NI

0.5 nF100xxxXXxxxxxXXXXX

Borghetti etal. Vaessen model0

-10023456789 10

t[us]0

vXXX X X X X X X X X XFig. 6.15 LIO at transformer, comparison with [82]

Fig. 6.15 shows that the transformer has a strong influence on the LIO. As pointed out in [82] a rather sophisticated model of the transformer is required to achieve accurate calculation results. The HV- and the LV-side of the transformer are not directly comparable, however. The HV-side has probably lower natural frequencies than the LV-side and a more complex model is required in order to take the frequency dependency of the common mode admittance into account for frequencies from 10 kHz -1MHz. The grounding of the neutral point has a large impact on the LIO, as seen from the curve T15-NG.

6.5 CONCLUSIONS

The common mode input admittance of a transformer is strongly dependent on the low-voltage neutral grounding. This also extensively influences the level of induced voltages in a low- voltage system. When the transformer neutral is grounded its input admittance can as a first approximation be represented by an inductance between 4-40 pH significantly related to the transformer's rated power (larger transformers have lower inductance). When the neutral is isolated the input admittance can as a first approximated be represented by a capacitance of 2- 20 nF seemingly minor related to transformer ratings. The simple capacitive and inductive approximations are mostly valid in the 10 kHz to 500 kHz frequency range.• Larger transformers (in kVA) result usually in lower induced voltage.• The level of LIO is larger in a system with isolated neutral.• The maximum induced voltage increases with the overhead line length.• It is common practice to protect the LV neutral point when this is isolated. Such protection

will probably operate in a case of potentially harmful induced voltages. An isolated neutral system must then be treated as a grounded neutral system.

7ELECTROMAGNETIC RESPONSE OF LVPI NETWORKS

7.1 INTRODUCTION

In this chapter the electromagnetic response of low-voltage power installation (LVPI) networks will be studied. By definition a LVPI network consists of the whole power installation in an apartment or residential house starting at the service entry, including all circuits, the local grounding system and connected loads or electrical apparatus. In the previous chapters LIOs in overhead lines and at distribution transformers have been studied and calculated in order to investigate how various key parameters in the system influence the level of LIOs. However, calculations with the aim to study protection against LIOs require a more complete representation of the connected LVPI networks. This is necessary since installations represent loads which influence the level of LIOs and probably are the weakest parts in the system.

The problem with LIO calculations in LVPI networks is the networks’ complex nature. A LVPI circuit is often unsymmetrical, consists of branches and bends, has often a rather fuzzy coupling to ground and other circuits and on the top of this is strongly influenced by the connected electrical apparatus which not even is constant in time. All this makes it almost impossible or impractical to establish a detailed model for an entire LVPI network. So the analysis must be restricted to answering the following questions:• What impedance does a typical LVPI network represent, seen from an overhead line?• How does a LVPI influence LIOs in the low-voltage system?• What is the difference between IT and TN networks?• What is the impact of types and size of wiring?• How do the grounding conditions or possible ground faults influence the LIOs?

The purpose of this chapter is to study the differences between types of domestic installations. Differences related to neutral grounding will especially be focused, since the IT system vulnerable to LIO is commonly used in Norway. The type of installation, divided in underplaster and surface wiring, will also be studied since they respond differently to transient overvoltages.

A main assumption in this chapter is that the LIOs in an overhead line is equal in all the phase conductors. This is a reasonable assumption, at least for the initial part of the transients, which often determines the maximum voltage, since the phase conductors have about the same distance to ground and to the lightning channel. For longer times and after reflections in the system the phase conductor voltages will be different due to unsymmetry. The assumption implies that an overhead line in general can be treated as a two port system, with the phase

7.2 Types of LVPI networks 99

conductors of the line as one port and the neutral conductor as the other (for an IT system the latter port is open). The impedance of the connected LVPI network is connected between these two ports of the overhead line.

Measurements will be performed on:1. Residential houses and flats to investigate the load an entire installation represents.2. Common types of electrical equipment to reveal the load they represent both in common and

differential mode.3. A laboratory LVPI network to study the surge characteristics of underplaster and surface

wiring with the aim to establish a model of the two types of wiring. The measurements are performed to reveal differences between types of wiring and between IT- and TN-systems.

The level of LIO in a low-voltage system is strongly dependent on the impedance of the connected LVPI networks. When characterising the electromagnetic response of a LVPI network two levels are possible:1. The overall LIO in a low-voltage system is studied and the internal voltages in a LVPI

network is out of interest. In this situation the LVPI network is a frequency dependent impedance connected to the overhead line terminals and directly measurable for various practical cases. Such measurements are presented in chapter 7.3.

2. The internal voltages in a LVPI network are of interest. Since the dominant wave lengths of LIO surges is smaller than the length of LVPI circuits, this requires a detailed study of the actual network, including its surge transmission properties for both interphase and common mode and the impact of the local loads connected to the system. An analysis of internal overvoltages in LVPI networks is presented in chapter 7.4.

7.2 TYPES OF LVPI NETWORKS

A LVPI network can be classified in two categories dependent on the power system's neutral grounding. The TN system shown in fig. 7.1 consists of three types of conductors:• the three phase conductors, L1..L3• the neutral conductors, N• the protective earth conductor, PE.A TN system can further be sub-divided in three groups:TN-C: The N conductor also serves as a protective earth conductor and is called PEN.TN-S: The N and PE are separate conductors throughout the whole system.TN-C-S: The N and PE earth conductors are separated in parts of the system.

The TN-C-S system is the most common one for LVPI networks supplied by overhead lines, where the L1..L3 and the PEN conductors are distributed on the overhead lines. The PEN conductor is split in a N and a PE conductor at the main distribution board. The TN system is common all over Europe. Normally with a voltage level of 400/230 which means 400 V between the phase conductors and 230 V between a phase conductor and neutral. The single

7-2 Types of LVPI networks 100

phase loads in a TN system are normally connected between a phase conductor and the neutral conductor, while special 400 V loads or three phase loads are connected between the three phase conductors. Only the TN-C-S system will be analysed here and is from now on called just the TN system.

L1..L3

N

Residental house

..........r IP1

J PE

Fig. 7.1 TN-C-S network

An IT system shown in fig. 7.2 differs from a TN system in that the neutral conductor is missing. The neutral point at the distribution transformer is isolated from ground but protected by a gap or an arrester. Also the PE conductor can be omitted in some installation circuits when it is not required for safety reasons [116]. Loads in the IT system are connected between two phase conductors. The IT system is very common in Norway and has normally a voltage level of230 V between the phase conductors.

Residental house

?T)------------- LI 13 -------------------- rfftl--------:

nn

-......... -qDI

J PE

Fig. 7.2 IT network

At the main distribution board in the meter cabinet the supplying conductors are protected by the main fuses and split into the required or convenient number of circuits. A circuit in a LVPI network is here generalised to consist of three conductors: A phase conductor L, a neutral conductor N and a protective earth conductor PE. In a TN system the N conductor is connected to the PE conductor at the main distribution board only, while in an IT system the N conductor is actually one of the other two L conductors. The L and N conductors are protected by separate fuses. The PE conductors for each circuit are connected to the equipotential bonding bar at the distribution board. Also the building's foundation earth, the water and drain pipes, lightning protection system etc. are connected to this bar. For EMC reasons the PE conductors in modem installations are preferred to have only one connection to the grounding system, but often several connections exist within the installation due to e.g. equipotential bonding of the PE conductor to water pipes. An incoming voltage wave on an overhead line due to a nearby lightning stroke will often be (and is in the following assumed to be) equal in the three phase conductors. When a voltage wave reaches the service entry, a voltage wave will be transmitted into the LVPI network dependent on the impedance of the system. The transmitted wave will

7.2 Types ofLVPI networks 101

reach the main distribution board and from there be shared among the various circuits in the network.

Voltage waves in a system with n conductors can be split up in n-1 independent interphase modes and 1 common mode. In a three-conductor system considered here, 3 modes should be taken into account. However, with a symmetrical arrangement of the conductors, the 2 interphase modes may be chosen to be identical, so that only 1 such mode needs to be considered. These two modes are called common mode and interphase mode as shown in fig.7.3:

b) Interphase modea) Common modeFig. 7.3 The two modes in a symmetrical system.

Alternatively the PE conductor can be considered part of the earth system as shown in fig. 7.4:

PE

a) "Common mode" b) "Interphase mode"Fig. 7.4 Alternative modes in a symmetrical system.

Both these models (fig. 7.3 and 7.4) are approximations, particularly because the earth system is not a continuous conductor, but consists of a system of conductors (water pipes, armour in concrete structures, steel constructions etc.).

Since the protection of electrical equipment against overvoltages is of main concern, the analysis in this chapter will focus on the voltage within the three conductors in a circuit. The voltage between the circuit and the rest of the grounding system is thus less important. Two practical modes are of most interest as shown in fig 7.5:• The IT mode. Voltage between the L/N conductors and PE will stress the electrical

equipment connected to PE. The impedance seen from the terminals in fig. 7.5a) is called the IT input impedance.

• The TN mode. The voltage between the L and N conductor will stress electrical equipment phase to phase. Loads not connected to ground will be stressed as well. The impedance seen from the terminals in fig. 7.5b) is called the TN input impedance.

7.2 Types of LVPI networks 102

LNPE

a) IT-mode b) TN-modeFig. 7.5 Modes used when performing measurements on LVPI networks.

Both the TN and IT systems have advantages and disadvantages related to steady state behaviour such as reliability, safety and EMC. Furthermore the two systems respond differently to LIO for two reasons:• The neutral conductor in overhead lines in TN-systems which is grounded at least at the

main distribution board will reduce the voltage transmitted into the LVPI circuits.• Single-phase loads in TN systems will suppress the voltage between the phase conductors

and the grounded neutral.Another main difference between the two systems is how they respond on ground faults. In the IT system the ground fault current is in most cases very low so no circuit breaking will normally occur. Only when, according to Norwegian regulations [116], the contact voltage is above 50 V a ground fault circuit breaker is required. Otherwise only a ground fault alarm is required in new installations. This makes the occurrence of permanent ground faults in the IT system likely. In the TN system, however, the ground fault current is much higher and immediately interrupted, so no permanent ground faults will normally exist.

Two types of LVPI wiring are investigated. These are:• Underplaster wiring which consists of three plastic-insulated stranded cores. The three

cables run in insulating conduits.• iSurface wiring which consist of a two plastic-insulated solid cores along with a bare core.

The three conductors are enclosed by an aluminium sheath in contact with the bare conductor.

The important difference between the two types of wiring is the metal screening of the surface wired cable, which makes its surge transmission characteristics different from the underplaster wiring type. The surface wiring type with metal sheath gives well-defined surge transmission characteristics due to the strong coupling between the cores and the sheath. When the sheath is connected to ground almost all the common mode current will flow in this screen while a part of the common mode current in underplaster circuits flows in the weekly defined surrounding grounding system. Underplaster wired circuits in concrete houses will have a stronger coupling to ground than in wooden houses. The cross-section of the conductors varies normally from 1.5 mm2 to 4.0 mm2 with 2.5 mm2 as standard in new installations. The insulation of the cores is made of PVC with thickness about 1 mm. Surface wiring is typically more common in old houses. Often a combination of the two types is used.

7.3 External response of LVPI networks 103

7.3 EXTERNAL RESPONSE OF LVPI NETWORKS

7.3.1 Introduction

In this section the input impedance of LVPI networks seen from the service entry will be studied. Measurements in the frequency domain of the input impedance ofLVPIs in several apartments and residential houses are performed and simple models of those impedances established. A real LVPI network is often far to complicated for a detailed modelling due to the following obstacles:

• branches and bends• several cross section alternatives• connected loads• several circuits• PE conductor sometimes omitted• mixture of surface and underplaster wiring

The measurements are performed to reveal differences between installations, and in particular between IT and TN systems. The impacts of type of wiring and connected electrical apparatus are studied. An analysis of the input impedance's influence on the magnitude of LIO is also performed.

7.3.2 Experimental setup

Fig. 7.6 shows the experimental setup for the frequency response measurements. The key unit is the Hawlett Packard network analyser HP4195A, which records two signals T and R and calculate the ratio between these signals and display it as e.g. absolute value and phase angle. By using the setup shown in fig. 7.6 the recorded ratio becomes the input impedance of the test object. Besides, the instrument offers a tracking generator which supplies a low AC voltage in the frequency range 1 mHz - 500 MHz. The current probe had a 3 dB bandwidth from 300 Hz to 200 MHz with an ignorable phase shift. Due to the 50 £2 internal termination of the two channels R and T, the recorded impedance Z=U/Ihad to be divided by 2 externally.

LVPI network

PearsonCurrentmonitor

HP4195A

LNPE

son AC source:

Fig. 7.6 Frequency response setup.

The neutral conductor N is for TN systems connected to the protective earth conductor PE. For

7.3 External response ofLVPI networks 104

an IT system the N conductor is actually one of the phase conductors and is connected to the conductor L. The PE conductor is in all cases in this section (chapt. 7.3) connected to the system ground. The LVPI network is in all situations disconnected from the supplying network.

7.3.3 Measurements on LVPI networks

The analysis focuses mainly on the measurements performed at the LVPI network in Siriusveien 10, 205B. This installation is supplied by a 400/230 V TN system, allowing comparison of the responses in TN and IT systems. Each individual circuit in this network will be analysed along with the impact of the connected electrical apparatus. Finally measurements of the IT input impedance is performed for a number of different LVPI networks just to show the variation between networks.

7.3.3.1 Siriusveien 10,205B

The LVPI network in this apartment consists of seven circuits listed in tab. 7.1. The apartment being relatively large (127 m2) is rather lightly loaded with all the heating panels, the water heater and the floor heating cable turned off. The basic load configuration is a snapshot of the actual loading situation on the day of measurements (30th April 1997). The length of each circuit in the power installation is unknown. All seven circuits are performed with surface wiring from the meter cabinet to the first connection point within the circuit. The circuits 1 and 6 have surface wiring only while the rest of the circuits consist of a combination of surface and underplaster wiring. All circuits have a phase conductor L, a neutral conductor N and a protective earth conductor PE and all circuits were supplied by the same phase conductor. The building has walls made partly of concrete and partly of wood/plaster boards within a framework of steel. Floor and roof are made of concrete.

Tab. 7.1 LVPI network in Siriusveien 10,205B

Circuit Fuse[A]

Cross section [mm2]

Supplies Connected loadsBasic configuration, B1

Commentswiring

Cl 20 2x2.5 Cooker Cooker Surface

C2 16 2x2.5 KitchenDishwasher

2 fluorescent lamps on1 percolator off2 water heaters off1 toaster off1 refrigerator on1 freezer on1 dishwasher off

Both


C3 16 2x2.5 Living room 3 lamps off1 heating panel off

Both

Bedroom 1 4 lamps off1 heating panel off1 vacuum cleaner off1 radio (on/off)

Bedroom 2 3 lamps off

C4 16 2x2.5 Living room 10 lamps offTV off stereo rack on video recorder on

Both

Shed 1 lamp off1 refrigerator on

Bedroom 2 1 heating panel offTV offPC + Printer off

Surface to TV

C5 16 2x2.5 Hall 6 lamps off1 outdoor lamp on2 lamps off

Both

Kitchen comer 3 lamps off Extra surface to2 fluorescent lamps on kitchen comer

WC 1 fluorescent lamp on

C6 16 2x2.5 Water heater Water heater off Surface

C7 16 2x2.5 Bathroom 4 fluorescent lamps on BothWashing machine Washing machine offHeating cable Heating cable off

Figure 7.7 shows the measured input impedance in the IT-system (N connected to L) for the basic load configuration B1 given in tab. 7.1.

abs(Z) [ohm] ang(Z) [deg]

— 50

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2f [MHz]


80 120 160 200 60 100 140 180

f [kHz]

a) 0 - 2 MHz frequency range. b) 0 - 0.2 MHz frequency range.Fig. 7.7 Input impedance IT-system. Load situation according to tab. 7.1

From fig. 7.7b) the impedance at low frequencies can be approximated by a capacitance of C0


= 200 nF. However, three dominant resonance peaks occur at low frequencies; one at 8.5 kHz, one at 21.5 kHz and one at 53 kHz. The sources of these resonance peaks will be investigated further. The minimum value occurs at 200 kHz where the impedance is as low as 0.9 Q. The frequency where the minimum value occurs is henceforth called the fundamental frequency.

Figure 7.8 shows the measured input impedance in the TN-system (N connected to PE) for the basic load configuration B1 given in tab. 7.1.

abs (Z) [ohm] ang(Z) [deg]

— 50

— -50

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2f [MHz]

abs(Z) [ohm]

0 20 40 60 80 100120140160180 200f [kHz]

a) 0 - 2 MHz frequency range. b) 0 - 0.2 MHz frequency range.Fig. 7.8 Input impedance TN-system. Load situation according to tab. 7.1

The TN input impedance is very dependent on the connected loads/apparatus in the system. Resistive loads due to electrical heating (very common in Norway) will result in very low input impedances, but all these loads were turned off in the basic configuration. This is also the normal situation during the summer lightning season. The water heater is another large resistive load which is turned off in the basic configuration but due to thermostatic control can be on or off in a real energised situation. Electrical motor loads in freezers, refrigerators, washing machines and dishwashers etc. will also have an influence on the TN input impedance, but this is rather difficult to measure since the technique for impedance measurements requires the power supply to be turned off, something which cause the motors to stop. Another problem arising when the power is turned off, is caused by the temperature characteristic of resistive loads. The resistance of light bulbs will typically drop to 1/10 almost immediately after the power is switched off. Other resistive loads will be reduced as well but not as much as light bulbs. As a result of this all the light bulbs and electrical heaters are turned off during the measurements. The effect of resistive loads is simulated by introducing lumped resistors. The unenergised situation (resulting in low resistance in light bulbs etc.) is not out of interest since the power often is lost during thunderstorms. On the other hand, the lower input impedance, the lower LIOs so the energised situation will form the basis for insulation co-ordination in low-voltage systems.

The input impedance is especially dependent on two circuits: C2 (to the kitchen) and C4. Figures 7.9 and 7.10 show the input impedance in the IT- and TN- system, respectively, for the basic configuration (also shown in figure 7.7) compared to situations where the circuits C2 and


C4 are disconnected. Fig. 7.9 shows that the main fundamental frequency is almost doubled (from 200 to 385 kHz) when C2 is removed. Fig. 7.10 shows that the circuit C4 has the most influence on the total TN input impedance.

abs(Z) [ohm]1,000 I——

abs(Z) [ohm]1,000 E——

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2f [MHz]

0 20 40 60 80 100 120 140 160 180 200

a) 0 - 2 MHz frequency range. b) 0 - 0.2 MHz frequency range.Fig. 7.9 Input impedance IT-system. Solid line: Load situation according to tab. 7.1.

Dotted line: Without Cl. Dashed line: Without C2 & C4.

abs(Z) [ohm] abs(Z) [ohm]

0 20 40 60 80 100 120 140 160 180 2000 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2f [MHz]

a) 0 - 2 MHz frequency range. b) 0 - 0.2 MHz frequency range.Fig. 7.10 Input impedance TN-system. Solid line: Load situation according to tab. 7.1.

Dotted line: Without C2. Dashed line: Without C2 & C4.

The input impedance’s dependency on various electrical equipment is further investigated by studying individual circuits with the key unit connected or disconnected. Since the power is turned off during the measurements it is not possible to study motor loads accurately.

First the circuit C2 (kitchen) is investigated. The key apparatus in this circuit is the washing machine, microwave oven, refrigerator and freezer. However, it was not convenient to disconnect all these units since most of them were permanently built-in, so the analysis is restricted to the impact of the dishwasher. Figure 7.11 shows the input impedance of the circuit C2 with and without the dishwasher connected (but turned off).


abs(Z) [ohm] abs(Z) [ohm]10,000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

.L. . . . :. . . . . L. . . . i. . . . . :

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

a) IT-system b) TN-system.Fig. 7.11 Input impedance circuit C2. Solid line: Basic configuration. Dotted: Without dishwasher.

As seen from the figure 7.11 the dishwasher has a significant influence on the input impedance, particularly for the TN-system. Fig. 7.11b) shows that the TN-system input impedance of the circuit C2 has a minimum of 1.1 Q at 135 kHz when the dishwasher is connected.

Figure 7.12 shows the influence of the PC in the circuit C4. As seen from fig. 7.12a) the PC is responsible for the two resonance peaks in the IT input impedance at low frequencies. Turning the PC on or off had no influence on the result, showing that it is the power supply of the PC causing these resonance peaks. Fig. 7.12b) also shows that the PC has an important effect in the TN-system. When the PC is disconnected a relatively pronounced resonance peak appears, caused by the stereo rack. This is partly compensated by the PC when this unit is connected. When the PC is connected the TN-system input impedance has a minimum of 1.0 Q at 21.5 kHz. This corresponds to the frequency when the first minimum is observed in the total LVPI network (ref. fig. 7.8b)). Apparently, the circuit C4 has a very strong influence on the TN- system input impedance of the total LVPI network shown in figure 7.8. The stereo rack is responsible for the first minimum of this input impedance.

abs(Z) [kohm]

0 20 40 60 80 100 120 140 160 180 200

a) IT-system

abs(Z) [ohm]

10 r\

0 20 40 60 80 100 120 140 160 180 200f [kHz]

b) TN-system.Fig. 7.12 Input impedance circuit C4. Solid line: Basic configuration. Dotted line: Without PC.


Figure 7.13 shows the influence of the washing machine in circuit C7.

abs(Z) [kohm]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2f [MHz]

abs(Z) [kohm]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2f [MHz]

a) IT-system b) TN-system.Fig. 7.13 Input imp. circuit C7. Solid line: Basic config. Dotted line: Without washing machine.

From fig. 7.13a) we see that the washing machine is responsible for a resonance peak at 53 kHz. This is seen as the third resonance peak in fig 7.7b) (which shows the input impedance in the IT-system for the whole LVPI network). Fig. 7.13b) shows that the washing machine strongly influences the input impedance in the TN-system. When the washing machine is connected the TN-system input impedance has a minimum of 1.15 Q at 55 kHz. After the PC and the stereo rack in C4 the washing machine has the strongest influence on the input impedance of the total LVPI network, both in the IT- and TN-systems.

The basic load configuration B1 has little resistive loads connected, since all loads like light bulbs and electrical heaters are disconnected. To study the influence of resistive loads, lumped resistors of 100 Q (corresponding to 530 W) were installed in some of the circuits. In general such resistors have little influence on the input impedance. The input impedance in the TN- system is significantly affected only for low frequencies (<25 kHz) while the input impedance in the IT-system is almost completely unaffected. As a result of this the high resistance of for instance light bulbs and moderately loaded electrical heaters can be ignored when modelling a LVPI network.

7.3.3.2 IT input impedances of a selection of installations [117]

The IT input impedance is the impedance typically seen from a supplying IT power system excited by equal LIOs in all phases. This situation is typical at the Norwegian countryside where LIO is a problem and where residential houses are supplied by overhead lines. The loads will only influence the IT input impedance if they are connected to ground (PE).

To study the total behaviour of a LVPI network in an apartment the IT input impedance of the system has been measured as a function of frequency from 0 to 1 MHz [117], using the technique described in fig. 7.6. Both new and old installations are studied with emphasis on


new installations made of combinations of surface and underplaster wired circuits. A main part of the investigation was carried out in an apartment complex called Siriusveien 10. This apartment complex was built in 1995 and the measurements were performed when one part of the complex was finished and inhabited while the other part was almost completed but still uninhabited.

Fig. 7.14 shows the measured absolute value of the IT input impedance of six different, inhabited apartments at Siriusveien 10. The apartment complex is supplied by a 400/230 TN- system from an underground cable. The measurements were performed from the meter cabinet of each apartment. The LVPI system in an apartment was disconnected from the supplying network. The PE conductor remained connected to the system ground.

100 —r

80 —a

60 —

40-----

20----

f [MHz]Fig. 7.14 IT input impedance of inhabited flats [117].

The numbers in fig. 7.14 refer to the apartment numbers at Siriusveien 10 as shown below:1: Siriusveien. 10, apartment 101a2: Siriusveien. 10, apartment 102a3: Siriusveien. 10, apartment 103a4: Siriusveien. 10, apartment 104a6: Siriusveien. 10, apartment 106a8: Siriusveien. 10, apartment 108a


101,102 and 103 are medium sized apartments (75 m2), 104 a slightly larger apartment (85 m2) and 106 and 108 are a small apartments (60 m2). All apartments are equipped with 6 circuits with the following fuse ratings and cross sections:Circuit -Fuse Cross-section phase/neutral1. Cooker2. Kitchen3. Living room/Bed room4. Hall/WC5. Shed/Water heater

20 A 16 A 16 A 16 A 16 A

2x2.5/2.5 mm2 2x2.512.5 mm2 2x2.512.5 mm2 2x2.512.5 mm2 2x2.512.5 mm2 2x2.512.5 mm2

All circuits were mostly made of surface wiring from the meter cabinet to the first connection point and underplaster wiring the rest of the circuit, but with segments of surface wiring on concrete walls. The length and branching of the circuits are unknown. All the apartments were supplied by a single phase conductor and a neutral conductor. The IT impedance was measured between the phase conductor connected to the neutral conductor and the PE conductor.

6. Bathroom/Washing machine 16 A

Fig. 7.15 shows the absolute value of the IT input impedance for six older inhabited flats. The flats are supplied by a three-phase IT system. The phase-conductors are disconnected from the supply network in the meter cabinet The PE conductor remained connected to the system/local ground. All the phase conductors were connected together and the IT input impedance was measured between these conductors and the PE conductor.

f [MHz]Fig. 7.15 IT input impedance of inhabited flats [117].


The numbers in fig. 7.15 refer to the apartment numbers shown below:1: Parkveien 20. apartment 1 (ground floor). Surface wiring. Built in the 30's.2: Parkveien 20: apartment 2 (ground floor). Surface wiring. Built in the 30's.3: Parkveien 20. apartment 3 (first floor). Surface wiring. Built in the 30's.4: Karivollveien 7. Residential house. Both types of wiring. Built in the 70's.5: Karivollveien 9. Residential house. Both types of wiring. Built in the 70's.6: Ole Hogstadsvei 26. Residential house. Both types of wiring. Built in the 70's.The number of circuits, length and cross section of each circuit in the LVPI networks listed above are unknown.

Measurements were also performed on some uninhabited apartments at Siriusveien 10 to reveal any common differences from inhabited ones. The LVPI networks in uninhabited apartments have in general higher input impedance with a higher main fundamental frequency and are typically equal to the curves called 2 in figures 7.14 and 7.15. Two uninhabited apartments at Markaplassen 111 and 148 were also investigated and both followed approximately the curve 2 in fig. 7.14. The Markaplassen apartments are made of wood and supplied by a 230 VIT- system. All circuits are made of underplaster wiring. A detailed description of the LVPI networks at Markaplassen is available including conductor cross section and circuit lengths. However, the input impedance is difficult to model, since only circuits to rooms where it is required (kitchen and bath rooms) have a PE conductor. This makes the LVPI networks' coupling to ground diffuse. The input impedances for tire two apartments follow each other closely up to the first minimum (fundamental frequency).

7.3.4 Measurements on electrical apparatus

Some of the electrical apparatus installed in Siriusveien 10, 205 B were in chapter 7.3.3.1 shown to be of great importance with a direct influence on the total input impedance of the whole LVPI network. The key units were the dishwasher, the washing machine, the PC equipment and the stereo rack. In this chapter separate measurements are performed on these units according to the method given in fig. 7.6. The common mode impedance was measured between the two phases connected together and protective earth, while the differential mode impedance was measured between the two phases. The power button of all the units were turned off during the measurements. However, turning the power button on had no influence on the measured impedance.

Figure 7.16 shows the input impedance of the dishwasher, washing machine, PC equipment and the stereo rack present in the installation analysed in chapt. 7.3.3.1 (Siriusveien 10,205B).


|Z| [ohm] |Z| [ohm]10,000

1,000 ............r

! ........ I'

a) Dishwasher. b) Washing machine.

|Z| [ohm]

f [MHz]

d) Stereo rack. Fig. 7.16 Input impedance electrical apparatus.

Solid line: Common mode. Dotted line: Differential mode.

The common mode impedance ZCM, can directly be looked upon as a load in the IT-system, while the differential mode impedance ZDM, is a load in the TN-system only in the situations where the ZCM » ZDM, which in fact is normal up to at least the fundamental frequency of ZDM.

The measurement on the dishwasher in fig. 7.16a) should be compared with fig. 7.11, showing the influence of this equipment on the whole circuit. Figure 7.11 shows that installing the dishwasher in the circuit C2 results in a minimum impedance at 220 kHz and 135 kHz, in the IT and TN-system respectively. This is at a lower frequency than observed from fig. 7.16a). This indicates that the interaction of the washing machine impedance with other impedances in the circuit, including the circuit impedance it self and connected electrical equipment, is of great importance. The minimum impedances seen in fig. 7.16a) are comparable to those found in fig. 7.11.

The measurement on the washing machine in fig. 7.16b) should be compared with figure 7.13, showing the influence of this equipment on the whole circuit. Figure 7.13 shows that installing the washing machine in the circuit C7 results in a minimum impedance at 40 kHz and 55 kHz, in the IT and TN-system respectively. Fig. 7.16b) shows minima at 55 kHz and 80 kHz for the


common and differential mode respectively. This also shows that it is the interaction between the washing machine and the wiring that determines the fundamental frequency seen in the input impedance of the total circuit. Interaction with other loads is of no importance in circuit C7 since the washing machine is the only connected equipment except for fluorescent lamps. The minimum impedances seen in fig.7.16b) are comparable to those found in figure 7.13.

The common mode impedance of the PC equipment and the stereo rack is rather large and will only have influence on the total circuit input impedance at low frequencies as seen from fig. 7.12a). The differential impedance is low, however, and it will have a large impact on the TN input impedance as seen from fig. 7.12b).

Measurements have also been performed on other washing machines, dishwashers and refrigerators. These measurements revealed a large difference between various brands of equipment. Several types are just capacitive up to a high frequency and do not have any influence at all on the input impedance, neither in the IT or the TN-system.

7.3.5 Modelling of LVPI networks

The measurements in 7.3.3 will not be modelled in detail, since there is a relatively large difference between LVPI networks in practice. The developed model will be kept as simple as possible and be based on the main features of the measurement results only.

The input impedance in the TN system, Zm is strongly dependent on the connected electrical apparatus, but is generally increasing with frequency being almost purely inductive above 100 kHz. Below 100 kHz the impedance is very complex but in general very low. Ignoring the oscillations above 100 kHz, the installation in Siriusveien 10, 205B can be modelled as shown in fig. 7.17:


100

50

0

-50

-100

meas

fit16 n■vW

HH8 (j.F

Fig. 7.17 Approximation of TN input impedance at Siriusveien 10,205B.At left: Measured impedance (meas) and fitted impedance (fit). At right: The equivalent circuit.

A further simplification is to ignore the low frequency behaviour and just approximate the TN


input impedance with an inductance of 3.5 pH. The inductance of a LVPI network will decrease with the number of circuits and branches in an installation. The range 2 pH to 20 pH has been chosen as the basis for a further sensitivity analysis.

The input impedance in the IT system Zn is also dependent on the connected loads, but not as much as the TN system. Also, more measurements have been performed so the modelling can be based one more facts. For low frequencies, typically up to 100 kHz the IT impedance is typically pure capacitive. The installation in the Siriusveien 10,205B apartment was found to have a capacitance of about 200 nF. The capacitance in a LVPI network will increase with the number of circuits and branches in an installation. The range 20 nF to 200 nF has been chosen as the basis for a further sensitivity analysis. Above the fundamental frequency the IT impedance becomes inductive and will approach the TN input impedance. The total IT input impedance can thus be approximated by a capacitance (20 - 200 nF) in series with an inductance (2 - 20 pH). Fig. 7.18 shows the modelled and measured IT input impedance at Siriusveien 10,205B:

meas

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2f [MHz]

II200 nF 1 £2

3.5 pH

Fig. 7. 18 Approximation of IT input impedance at Siriusveien 10,205B.Left: Measured impedance (meas) and fitted impedance (fit). Right: The equivalent circuit.

7.3.6 LIO calculations

In this section, LIO calculations in a very simple low-voltage system are performed. The aim is to study how sensitive the LIOs are to variations in the connected installation model. The low-voltage system consists of a single overhead line of length 500 m and height 6 m, being symmetrical and having characteristic impedances of 500 Q and 200 £2 in the common and interphase modes respectively. The three phase conductors are substituted by a single conductor with characteristic impedance of300 Q. In one end of the line a transformer is connected and modelled as an inductance of 10 pH, which represents a "typical" distribution transformer with the neutral connected to ground. At the other end of the line an impedance representing the LVPI network is connected. The model of this impedance is kept as simple as possible, following the results from chapt. 7.3.5.


The lightning stroke is located 100 m from the midpoint of the line (side stroke) and the lightning current shape follows the default value in appendix A (30 kA magnitude), using the TL lightning channel model with v= 1.1T08 m/s.

Fig. 7.19 shows the LIO in a TN system where the input impedance of a LVPI network is modelled as an ideal inductance varying from 2 to 20 pH.

4000

-2000 —r;

LVPI model

-4000 —

-6000

L1..L3

Fig. 7.19 Induced voltages in TN-systems. Four different inductive terminations.

Fig. 7.19 shows that varying the value of the inductance has a strong influence on the LIO, being approximately proportional to the inductance value. The maximum induced voltage is low for the lowest inductance and does not represent any overvoltage problem in the system. The induced voltage oscillates with a frequency fD * c/2L - 300 kHz where c is the speed of light and L is the line length. The installation impedance at this frequency will mainly determine the LIOs. A better representation of the TN input impedance at low frequencies, according to fig. 7.17, has no influence on the induced voltage in this case.

Fig. 7.20 shows the LIO in an IT system where the input impedance of a LVPI network is modelled as a capacitance of20-200 nF in series with an inductance of 2-20 pH.


12000 —r

>2 high

LVPI model

-4000 —

L1..L3

Fig. 7.20 Calculated induced voltages in IT-system. Six different terminations. Transformer's neutral grounded.

The maximum induced voltage is strongly dependent on the IT input impedance model as shown in fig. 7.20. Also, the voltage level is much higher than in the TN system in fig. 7.19. The dotted/dashed curves in fig. 7.20 are modifications of curve 2 (100 nF + 5 pH) as shown in fig. 7.21. The 2 high curve represents an additional resonance peak at 560 kHz while the 2 low curve represents one at 70 kHz. Such resonance peaks are common as seen from figures 7.14 and 7.15. Fig. 7.20 shows that the influence of the modification is minor.

a) Impedances in frequency domain

5|iH

2high

b) Impedance equivalentsFig. 7.21 IT-input impedances

100 nF

2 lew

7.3.7 Discussion

Since the TN input impedance has been measured at one location only, it is difficult to draw


any precise conclusions about its characteristic. The results indicate that the TN impedance in general is very low and increases with frequency above ca. 50 kHz. A first approximation of the impedance is thus a pure inductance. A value of 3.5 pH is found from the Siriusveien 10 measurement (fig. 7.17), and the range 2-20 pH is believed to cover practical installation variations. The low frequency behaviour obtained in fig. 7.8b) (and roughly modelled in fig. 7.17) has no influence on the lightning-induced voltages.

The measurement of the IT-impedance at Siriusveien 10 205B, in fig. 7.7 shows that this impedance as a first approximation can be approximated by a capacitance of200 nF in series with an inductance of 3.5 pH equal to the TN-impedance approximation. The fit is shown in fig. 7.18. Above 1 MHz the IT and TN input impedances are almost equal.

Figures 7.14 and 7.15 show that there is a relatively large variation in the IT-system input impedances. There is no significant difference, however, between new (fig. 7.14) and old (fig. 7.15) LVPI networks and thus between underplaster and surface wiring. The differences between the two main types of wiring, found from the laboratory measurements in chap. 7.4, seem to be ignorable compared to the effect of the loads in the system. One difference between figures 7.14 and 7.15 is, however, that input impedances of old apartments (fig. 7.15) show a more complex behaviour above the first minimum. The LVPI input impedances can roughly be classified in two groups:1. Impedances with minimum around 200 kHz. Up to this frequency the impedances can be

approximated by a capacitance around 100 nF. Most of the impedances belong to this group. Some of them have a resonance peak below 100 kHz that probably is due to connected electrical equipment. The especially focused impedance of Siriusveien 10,205B (fig. 7.7a)) belongs to this group.

2. Impedances with minimum above 500 kHz. Impedances in this group show larger variations and can be approximated by capacitances of 15-30 nF up to the minimum. The LVPI networks in this group have probably fewer loads connected than those in group 1.

The average low frequency (<100 kHz) capacitance of the 12 LVPI networks shown in fig. 7.14 and 7.15 is 60 nF. Some of the impedances in fig. 7.14 have a large peak around 500 kHz, which probably is caused by the mixture of underplaster and surface wiring. In this situation the low inductance of the surface wiring and the low capacitance of the underplaster wiring cause a resonance peak.

Connected electrical equipment has a large influence on the input impedance as seen from figures 7.11-7.13, particularly in the TN-system. Measurements have been performed on four key units in the Siriusveien 10, 205B apartment and shown in figure 7.16. The total input impedance of a circuit is the result of an interaction between the connected equipment and the circuit itself. There are also large differences between types of electrical equipment. Typically the power supply is of greatest importance and a key question is whether or not this is connected during the measurements. Often modem electrical equipment has anti-interference filters at the input terminal, even before the on/offbutton. Such filters are at least present in the washing machine and the dishwasher shown in figures 7.16a) and 7.16b) and cause the low frequency resonance peaks.

7.4 Internal response of LVPI networks 119

Figures 7.19 and 7.20 show that the amplitude of induced voltages in a simple IN and IT system with a single overhead line and a single LVPI is strongly dependent of the nature and size of the LVPI. In general larger installations result in lower overvoltages. The difference between a small and a large installation may amount to one order of magnitude. In a more extended network with many subscribers, this variation is likely to be more limited. This will be further analysed in chapter 8. The resonance peaks appearing in the impedance curves seem to have only a minor influence on the induced voltage. The voltages induced in an IT system is substantially higher than in a TN-system.

7.4 INTERNAL RESPONSE OF LVPI NETWORKS

7.4.1 Introduction

A LVPI circuit in the laboratory has the advantage that both underplaster and surface wiring can be investigated for almost identical configurations. It is also possible to control the length of circuit parts and the distances from the injection point to loads in the system. The response of underplaster wiring has been investigated in [79, 118] for both common and differential mode. In [119] the common mode impulse response of underplaster wiring was investigated, particularly focusing on the effects of a branch in the circuit and the grounding condition of the PE conductor. Here, both underplaster and surface wiring will be investigated along with the grounding conditions (the PE conductor's connection to ground). Both step- and frequency- response measurements are performed.

7.4.2 Experimental setup

A laboratory test installation circuit was established to study the surge transmission characteristic and the impedance of a LVPI network seen from the supplying network. Fig. 7.22 shows the configuration of the test circuit Equal installation layout was used for both surface and underplaster wiring. The circuits were placed on top of a fine-meshed netting of thin galvanised steel wire used as ground reference (Gl). The test installation was place on the 3rd floor of a concrete building. The total length of the test circuit was 28.8 m.

Fig. 7.22 Laboratory LVPI circuit [120].


The numbers 1-6 indicates power outlets, where all the three conductors wee available (two as phase conductors and one as ground). The number 0 indicates the transmitting end while the number 7 indicates the receiving end. All cores used in the two types of wiring are made of copper (Cu) with cross section 1.5 mm2.

The grounding conditions were in particular investigated. Three different groundings were used, called G0-G2. In the configuration GO the protective earth conductor, PE, was kept isolated from ground. In the second configuration, Gl, the PE conductor was connected to the underlying fine-meshed ground netting. And in the third configuration, G2, the PE conductor was connected to the protective earth of the supplying power network. The netting was not connected to the building’s grounding system apart from being located on the floor.

Fig. 7.23 shows the basic experimental setup for the step response measurements. The installation circuit is charged by a DC voltage source and then short-circuited by a relay equal to the one described in chapter 5.2.2. When the relay is closed it is ideally a short circuit resulting in a reflection coefficient of-1. A current limiting resistance of 1 kQ is installed between the DC source and the relay. The current out of the LVPI network and the voltage between the terminals of the LVPI circuit is measured and recorded by a TDS 540A oscilloscope isolated from the supplying power system's ground by an isolation transformer.

DC-Voltage

Fig. 7.23 Basic step response measurement setup.

The three conductors of the LVPI circuit were connected to the two terminals of the relay differently dependent on the mode under investigation. One terminal of the circuit was either isolated from (GO) or connected to ground (G1-G2).

The Pearson Current Monitor has a 3 dB bandwidth of 300 Hz - 200 MHz and an ignorable phase shift, while the Tektronix voltage probe had a bandwidth of 0-150 MHz and an input impedance of R=10 MQ and C=13.2 pF.

The frequency response measurements were performed as described in chapter 7.3, fig. 7.6.

7.4.3 Step response of the laboratory circuit

The measurements of the step response are performed: 1) to study the difference between IT

7.4 Internal response ofLVPI networks 121

and TN systems and between underplaster and surface wiring and 2) to establish models for the two types of wiring. The modelling will be based on modal quantities in the interphase mode (IM) and the common mode (CM). In both these modes the system can be reduced to an equivalent single-phase system defined by a characteristic impedance, Z’ and a transmission coefficient y, based on the resistance (R’), inductance (L’) and capacitance (C5) per unit length. A symmetrical system is uniquely defined by two modes as previously explained. The currents in the TN and IT systems with grounded PE will be a combination of the interphase and common mode.

The current in a single-phase system supplied by a step voltage U0 can be approximated by

(7.1)

when the surge impedance by a series expansion around R -0 is approximated by

assuming R ’« o>-L ’ and ignoring the frequency dependency of R ’ and L

The step responses will be characterised by three different quantities:• Characteristic impedance for a lossless system

z' = Jl'ic1Where Z' is defined as the initial ratio between the applied voltage and the response current Z’= U0(0)/I(0). Fig. 7.25 shows how this ratio varies in the common mode of surface wiring, where time t=0 is defined as when the applied voltage starts to decrease. The first part of the curve in fig. 7.25 (and the rest of the measured ratios) is emphasised when determining the characteristic impedance. Some judgement must often be used to establish a specific value.

• Wave velocity for a lossless systemV = 1 l^L'-C'Due to the reflections in the installation circuit the travelling time of the current wave can be measured. Here the wave velocity is defined as four times the circuit length, l divided by the time to the second zero crossing of the current. The inverse of this time is called the fundamental frequency,To of the circuit, v=4-lf0

• Attenuation factor, aThe envelope curve of the measured step response current is approximated by an exponential function Ie = Imax-eai, using the least square method. Referring to (7.1) the attenuation factor is equal to

Ra2 L

All these three factors will in general be frequency dependent so an approximation has to be


established. Here, the focus is set on the fist part of the measured curves, which represents the high frequency behaviour. The quantities measured in the common mode and interphase mode will later be used in modelling the LVPI circuits.

The applied voltage, which ideally is a pure step of 15.7 V, will vary slightly for the different couplings since the internal impedance of the closed relay is not exactly zero.

7.4.3.1 Surface wiring

Figure 7.24 shows of the step responses for the interphase mode (IM) and the common mode (CM) for surface wiring. The step voltages are shown as dotted curves and the current out of the circuit as solid lines.

I [A] UM I [A] UM

Current

Voltage

Fig. 7.24 Step response, surface wiring.

Current

Voltage

The grounding conditions of the protective earth conductor (PE) did not influence the step responses of the surface wiring circuit, since almost all the return current will flow in the metal sheath. The three-conductor system can thus be reduced to a two-conductor system with L and N as the phase conductors and the PE as the ground. The common mode system then becomes equivalent to the IT system.

As mentioned previously, the determination of exact values for the wave impedance and velocity must be based on judgement. This is particularly true for the CM system, in which case the voltage tends to drop rather slowly to zero. Fig. 7.25 shows the voltage/current ratio in common mode calculated from the measured voltage and current for each time step. The voltage in the calculation is reversed, starting at zero and ending at 15.7 V. Based on fig. 7.25 a surge impedance of 20 Q has been chosen.


U(t)/I(t) [ohm]

Fig. 7.25 Measured voltage/current ratio in CM. Surface wiring.

Table 7.2 summarises the three quantities surge impedance, wave velocity and attenuation factor for CM (common mode), IM (interphase mode) and TN (TN-mode). In this case the common mode becomes equal to the IT-mode. The sheath is kept isolated from ground (GO), but the grounding condition for surface wiring had no influence on the measurement results.

Table 7.2 Characteristic quantities for surface wiring.________________________________

Mode Configuration Characteristic impedance, Z

Wave velocity, v

Attenuation factor a

CM/IT sheath^. 20 Q 127 m/ps 0.55 ps"1

IMi -

64 £2 132 m/ps 0.40 ps"1

TN sheath x 35 Q 130 m/ps 0.55 ps"1

The velocities in the three modes are about the same, but lowest for the common mode and highest for the interphase mode. The characteristic impedance is lowest for the common mode and highest for the interphase mode. The attenuation is the same in the TN-mode and the common mode while somewhat lower in the interphase mode. The TN-mode is actually in a mixture of CM and IM and it is not strictly correct to determine a characteristic impedance and wave velocity for this mode.

7.4.3.2 Underplaster wiring

As expected, the underplaster wiring is more dependent on the grounding conditions than the surface wiring so the three grounding conditions G0-G2 have been investigated.


Figure 7.26 shows the step response current for underplaster wiring in interphase mode and common mode. Only the response current is shown for two different grounding conditions.

I [A]

a) Interphase mode (IM).t[us]

b) Common mode (CM).Fig. 7.26 Step response, underplaster wiring. GO, G1 and G2: grounding conditions.

Fig. 7.26b) shows the step response current for underplaster wiring in CM. The difference in current due to the two types of grounding is large and for grounding condition G2 the concept of exponential attenuation fails completely. This strong dependency on the grounding condition is a main source of uncertainty when calculating voltages in LVPI networks. However, in most cases it is the voltage between conductors in the circuit which is of interest. This interphase mode voltage is less dependent on the grounding conditions, as shown in fig. 7.26a). The grounding condition G1 is a very good grounding while GO of course is the worst-case grounding since the system is completely isolated from ground under this condition. The grounding condition G2 will give results somewhere between GO and Gl.

Tab. 7.3 summarises the measured three quantities surge impedance, wave velocity and attenuation factor for the different coupling of underplaster wiring.

Table 7.3 Characteristic quantities for underplaster wiring.

Mode Ground Configuration Characteristic impedance, Z

Wave velocity, v

Attenuation factor a

IM GO N ..................-.....--£□

PE

175 Q 180 m/ps 0.25 ps’1

CM Gl tPE ---------------------------

120 Q 200 m/ps 0.20 ps"'

CM G2 G1 orG2 210 £2 230 m/ps -


TN GO LN ---------—z=p«h 130 Q 180 m/ps 0.29 ps"1PE | ,open: GO

TN G11 dosed: G1

110 Q 180 m/ps 0.27 ps"1

IT GO LN --------- J 130 a 180 m/ps 0.24 ps"1PE I ,open: GO

IT G1I closed: G1

90 Q 180 m/ps 0.27 ps"1

For the grounding condition GO the TN and IT modes should be equal, but apparently a small difference exists in the attenuation factor, probably due to a minor unsymmetry in the system. The characteristic impedances for the TN-system given in tab. 7.3 are similar to those established in [118]. The wave velocity is somewhat lower, however, partly since the fundamental frequency f0 of the oscillations is used to calculate the velocity.

7.4.4 Frequency response measurements of laboratory circuit

The input impedance was measured according to fig. 7.6 for the TN and IT systems for both underplaster and surface wiring. The frequency range 10 kHz - 1 MHz is of primary interest, but to include the fundamental frequencies of the installations the range 5 kHz - 2 MHz was chosen for surface wiring and the range 10 kHz - 4 MHz for the underplaster wiring. The frequency step was 5 kHz and 10 kHz respectively.

7.4.4.1 Surface wiring

As for the step response, the grounding of the PE conductor did not influence the input impedance, so only the isolated system (GO) will be presented here.

Fig. 7.27 shows the input impedance of a surface wired, TN-coupled circuit. The input impedance is approximately an ideal capacitance up to almost 1 MHz.


abs(Z) [kohm] ang(Z) [deg]

abs(Z)

ang(Z)LNPE

sheath v

Fundamental frequency: Minimum impedance: Low freq. capacitance:

1.13 MHz 3.51 fi 6.57 nF

Fig. 7.27 Frequency response. Surface wiring, TN system.

Fig. 7.28 shows the input impedance of a surface wired, IT-coupled circuit. The input impedance is approximately an ideal capacitance up to almost 1 MHz.


abs(Z)

ang(Z)

LNPE

sheath^

Fundamental frequency: 1.09 MHzMinimum impedance: 1.91 QLow freq. capacitance: 12.3 nF

Fig. 7.28 Frequency response. Surface wiring, IT system.

The wave velocities, v in the systems are related to the fundamental frequency,/by v=4-lf where/is the length ofthe circuit (/=28.8m). This results in vm= 130 m/ps and v/r= 126 m/ps which is very close to the values observed for the step responses under chapter 7.4.3.1.

7.4.4.2 Underplaster wiring

As found from the step response measurements in chapter 7.4.3.2, the characteristics of the undeiplaster wiring is dependent on the grounding conditions of the PE conductor. All the three grounding conditions (G0-G2) have been investigated where GO represents a very poor grounding (actually isolated from ground) and G1 represents a very good grounding (PE connected to underlying meshed metal netting). A real grounding connection will probably give results somewhere in between these to extreme cases.

7.4 Internal response of LYPI networks 127

Figure 7.29 shows the input impedance measured in the TN-coupled underplaster wiring. The difference between the two extreme grounding conditions (GO and Gl) is small, at least below fundamental frequency. Thus the grounding condition in general seems to play a minor role in TN-systems. The input impedance is almost a pure capacitance up to about 1 MHz.


-50 -0.01 r

a) Magnitude. b) Phase angle.Fig. 7.29 Frequency response, TN-system. GO and Gl: Grounding conditions.

LNPE

Z(m)

open: GO closed: G1

GO GlFundamental frequency: 1.58 MHz 1.58 MHzMinimum impedance: 6.17 Q 5.84 QLow freq. capacitance: 1.40 nF 1.45 nF

Figure 7.30 shows the measured input impedance of the IT-coupled underplaster wiring. In this configuration the grounding conditions are more important even below the fundamental frequency. The input impedance is almost a pure capacitance up to about 1 MHz.


0.1 —

-50 -

f [MHz]

a) Magnitude. b) Phase angle.Fig. 7.30 Frequency response, TN-system. GO and Gl: Grounding conditions.


LNPE

Fundamental frequency: 1.51 MHz Mimimum impedance: 4.34 QLow freq. capacitance: 1.63 nF

GO G11.50 MHz 4.09 fi 1.85 nF

I open: GO

i closed: G1

In theory the impedance in the TN and IT systems should be equal for the grounding condition GO, but a small deviation still exists. This difference must be due to unsymmetry in the system and will not be accounted for in the models of underplaster wiring, which later are developed.

7.4.4.S Connected loads

Loads in TN systems are believed to strongly influence the input impedance of an installation circuit. To investigate this further and to study the importance of the loads' positions along the circuit, two loads of 50 Q and 500 fi are alternately installed in the laboratory circuit and the input impedance is measured as a function of frequency 0- 2 MHz. The two loads are almost purely resistive for the whole frequency range, but the 50 fi load is slightly inductive (phase angle 6° at 2 MHz) and the 500 fi load is slightly capacitive (phase angle -6° at 2 MHz).

The loads are installed at the power outlets or the transmitting/receiving end of the laboratory circuit between the L and N conductor. The numbers (0..7) in fig. 7.22 are used as references in fig. 7.31.

Figure 7.31 shows the absolute value of the input impedance for a TN coupled laboratory installation underplaster circuit and its dependency of the position of a load in the system.

1000 1000

100

a

N

<

10

0 3 4 0 2f FMHzl

3 4

a) 50 fi load. b) 500 fi load.Fig. 7.31 TN input impedance. TJnderplaster wiring. Loaded. Grounding Gl.


Loads above 500 Q can, as seen from fig. 7.31b), be assumed located at the circuit's transmitting end. When studying the electromagnetic response of a TN system, positions of loads of 50 Q or below are important, at least for underplaster wiring. When discussing the influence of loads in the system it is important to have the characteristic impedances of the wiring in mind. If the resistance of the connected load is much larger than the characteristic impedance of the circuit, the position of the load along the circuit is of minor importance. This is die reason why the input impedance of an underplaster wiring system is more dependent on the load position.

7.4.5 Modelling of LVPI circuits

The method used in chapter 6, where transformer admittances measured in the frequency domain were accurately fitted by lumped RLCG networks, will not be used here. The reason is that we want a model of the installation circuit with a relevant physical interpretation related to the length of the circuit, the characteristic impedances and wave velocities in the different modes of the system. By using such an approach, the models established for the laboratory installation can be extrapolated to other circuit lengths. Two types of models will be established, both directly usable in the ATP-EMTP [12]:1. A KCLee model:

This is a distributed parameter, frequency independent, travelling wave model. The step response measurements will be used to establish characteristic impedances and velocities and attenuation in the common and interphase modes of the system.

2. Pi-equivalent:This is a lumped model with series inductances, resistances and shunt capacitances at each end of the line. The step response is used to establish these quantities per unit length.

The KCLee model will be the most accurate one, but it requires a shorter time step in the simulation. Appendix H outlines the actual results of the modelling.

7.4.6 Comparing measurements and calculations

In this section the measurements are compared with the models developed in appendix H, both for the step- and frequency responses. First the surface wiring and then the underplaster wiring models are evaluated.

Fig. 7.32a) compares the measured step response with an EMTP calculation using the KCLee model. We can see that the KCLee model gives a very good fit to the maximum values but fails to fit the frequency of the oscillations since the wave velocity in the measurements seems to decrease with time. The Pi-equivalent model will not reproduce the step response appropriately due to its high frequency contents.

Fig. 7.32b) compares the measured input impedance in the frequency domain with an EMTP calculation using Frequency Scan [12]. The Pi-equivalent model is built up of two sections each


with length 14.4 m. We see that the KCLee model gives a good overall fit while the PI- equivalent model fails above the fundamental frequency (1.13 MHz). More than two sections in the Pi-equivalent would improve the fit. The measurements and calculations are performed with the grounding condition GO (isolated from ground), but this does not matter for surface wiring.

I [Al abs(Z) [kohm]

Meas. Meas.

KCLeeKCLee

Pl-equiv.GO0.03 -

0.01 r-0.2 —

0.003 -

f [MHz]

a) Step response b) Frequency response.Fig. 7.32 Comparison between measured and calculated response of surface wiring.

No load and no grounding (GO.). TN-system.

The KCLee model also manages to reproduce the frequency domain measurements of input impedances when loads are installed in the circuit. A small deviation at low frequencies arises when a load is installed at the receiving end, however, due to the frequency dependency of the conductor resistance whose value at the fundamental frequency is used in the models.

Fig 7.33a) shows a comparison between the measured and calculated step responses for a TN coupled underplaster wired circuit. The calculations are based on the KCLee model in EMTP. The KCLee model is able to predict the first part of the curves very good. When the system is grounded to the underlying meshed wire (Gl) a deviation between the measurements and the calculations arises after 3 or 4 cycles. This does not happen when the system is isolated from ground (GO). So there is apparently an inaccuracy in the common mode of the system.

Fig. 7.33b) shows the measured frequency domain input impedance and the input impedance of tiie KCLee and Pi-equivalent models calculated by EMTP's Frequency Scan. We see that the KCLee model gives a very good overall fit while the Pi-equivalent (built up of two sections with 14.4 m length) is only accurate up to the fundamental frequency (1.5 MHz). The KCLee model is also able to reproduce the measurement of the input impedance when loads are installed in the circuit.


abs(Z) [kohm]

Meas.Meas.

KCLeeG1

KCLeeGO

Pl-equiv.GO

0.01 --0.1 -

b) Frequency response.a) Step responseFig. 7.33 Comparison between measured and calculated response of surface wiring.

No load and PE grounded (Gl.). TN-system.

7.4.7 Overvoltage calculations in LVPI networks

The models developed in appendix H will now be used to calculate the voltage in a LVPI network supplied by an overhead line. The calculations will be kept as simple as possible with the purpose to study the differences between underplaster and surface wiring and between IT- and TN-systems. The supplying low-voltage system is therefore modelled as a Thevenin equivalent with the induced voltage as the source. This voltage is of course strongly dependent on the characteristics of the overhead line along with characteristics of the LVPI network itself and its connected loads. Since the purpose here is to just compare different kinds of installations, a pure step voltage of 1 V is assumed. The overhead line model is assumed to be lossless with a characteristic impedance matrix equal to

z =Z; Z.

z. z*where ZL = 300 Q is the equivalent charac. impedance of the L conductors

ZN = 500 Q is the charac. impedance of the N conductor and (7.3)Zm=200 Q is the mutual characteristic impedance between

the L and N conductor.

The line is assumed to have three or four conductors, three L-conductors and one optional N- conductor. The same voltage is assumed induced in all the L-conductors. The interaction between the various circuits in a LVPI network is further ignored something which results in a worst case condition. Fig. 7.34 shows the configuration used in the calculations. The overhead line is supplied by step voltages with magnitude U0= 1 V in all phases which results in incoming voltage waves of 1 V to the service entry. The voltage at the end of the connected LVPI circuit is then calculated, using ATP-EMTP and the KCLee models developed in appendix H.


300 nOverhead line equivalent LVPI circuit

LNPE

14

x /L. Overhead Brie LVPI circuitUo vV (~D equivalent

LNPE

\\\\\\\\\\\\\\\\\\SN x\\\\\\\\\\\\\\\\\va) IT- system b) TN - system

Fig. 7.34 LVPI circuits's response to incoming voltage wave.

Figure 7.35 shows the step response of an unloaded and unprotected 28.8 m long LVPI circuit.

um um

Surface

UnderplasterL-PE

UnderplasterL-PE

a) IT-system. b) TN-system.Fig. 7.35 Calculated voltage phase-to-ground (UL.PE ) at the end of an unloaded LVPI-circuit.

In an IT-system the voltage approaches the supplied voltage % while in a TN-systemthe voltage approaches a value U(t-*°°) = Us= U0-(1-ZJZN). In both the IT and TN-systems underplaster wiring gives the highest voltage in the transient period. The phase-to-phase (Ul_n) voltage is zero in the IT system and almost equal to the phase-to-ground (UL„PE) voltage in the TN-system, since the system is unloaded.

Figure 7.36 shows the step response of a loaded 28.8 m long TN-coupled LVPI circuit. The load of 100 Q is placed between the L and N conductor at the end of the circuit. Fig. 7.36a) shows the phase-to-ground voltage and fig. 7.36b) shows the phase-to-phase voltage. As seen from the figures, underplaster wiring gives the highest voltage during the transient period.

At large times the phase-to-ground and phase-to-phase voltages will become equal and approach a value equal to (if the resistance in the LVPI circuit conductors is ignored)

= t7 •——— = 0.188 V (7.4)

where RL is the connected load (100 Q)Us is the equivalent source voltage equal to Us = U0-(l- ZJZ^ = 0.60 V (7.5)Zs is the equivalent source impedance equal to Zs = ZL - Zj/ZN = 220 Q (7.6)


up/] UM

Surface

UnderplasterL-N

Surface

UnderplasterL-PE

t[us] t[us]a) Phase to protective earth. b) Phase to neutral.Fig. 7.36 Calculated voltage at the end of a loaded LVPI-circuit. TN-system.

In the IT system there is no regulations for circuit interruption during ground fault conditions so a permanent low-resistive ground fault during a thunderstorm is definitely possible. Besides, a LIO can result in a spark-over somewhere in the network even during the stepped leader phase. The latter case will result in a transient situation that is difficult to analyse, but the first situation is important and leads to differential mode induced voltages. Actually an IT system with internal ground fault is a mixture of an IT and a TN system.

When the system is protected and the protective device operates, the voltage in the LVPI network will not approach the final value shown in figures 7.35-7.36. Instead, only the initial part of the voltage will determine the stress on an installation. Protection of low-voltage system is treated in chapter 8.

7.4.8 Discussion

The measurements on the laboratory circuits make it possible to compare underplaster and surface wiring. The established models can predict both the step response and frequency response of the LVPI circuits. They are based on the assumption that the frequency dependency of the parameters in the system can be ignored. This seems to be a reasonable assumption for the frequency range 10 kHz to 1 MHz of main interest. The surface wired circuit shows a bit higher frequency dependency than the underplaster wiring particularly for the wave velocity. The resistance of the conductors is fixed to a value corresponding to the value at the fundamental frequency. In [121] the frequency variation in the resistance, due to skin effect, is included as a simple lumped inductance/resistance network. This will improve the accuracy of the calculated LVPI response also at lower frequencies than the fundamental.

Modelling the circuits based on a priori knowledge alone such as geometrical and material data resulted in too high wave velocities and too low attenuation (conductor resistance).

The magnitude of LIO in LVPI networks excited by a step voltage can be calculated from the

7.5 Conclusions 134

established models. This clearly shows (figure 7.35) that underplaster wiring gives the highest overvoltages since the input impedance (for both IT and TN systems) is higher than for surface wiring. The response of the two types of wiring approach each other beyond 10 ps. LIO between phase and protective earth (common mode) is much higher in IT system than in TN systems. LIO between phases (differential mode) is, however, lower for an IT system than for a TN system, provided that the IT system does not have a ground fault.

7.5 CONCLUSIONS

A LVPI network seen from the service entry can be represented by a single impedance whose value is dependent on the size and type of installation including its electrical loads.

An IT-system LVPI network impedance is roughly capacitive up to a couple of 100 kHz. Above the fundamental frequency the impedance is inductive. Capacitance values of20-200 nF have been found, increasing with the LVPI network size and the volume of connected equipment. The inductance in series with this capacitance is suggested to be 2-20 pH, decreasing with network size. The particularly investigated apartment Siriusveien 10,205B had roughly an impedance consisting of a 200 nF capacitor in series with a 3.5 pH inductor.

A TN-system LVPI network impedance is very dependent on the connected electrical equipment, but is generally very low for frequencies 10-100 kHz. Few measurements on TN- systems have been performed in this work, but based on the measurements at Siriusveien 10, 205B, the TN-impedance is suggested to be representable by a pure inductance having a possible value range of 2-20 pH, decreasing with increasing network size.

Smaller LVPI networks result in higher lightning-induced voltages at the low-voltage service entry, both for IT- and TN-systems. The more LVPI networks connected to a low-voltage overhead line, the lower induced voltages occur. The level of LIOs in the simple system investigated here can increase one order of magnitude from the largest to the smallest installations.

Connected electrical equipment will have influence on the input impedance, resulting in resonance peaks and generally lower values of the impedance. Electronic equipment as PC's and stereo racks cause low resonance peaks, which are ignorable when studying lightning- induced overvoltages. Modem apparatus with e.g. speed-controlled motor drives often have an anti-interference filter at the power entrance, and this will dominate the impedance of the equipment. Connected loads like light bulbs and fluorescent lamps are ignorable when modelling LVPI network. Large loads for heating needs to be considered, but such loads will only be of importance for low frequencies and will not alter the general characteristics of a LVPI circuit.

Underplaster wiring and surface wiring have a different characteristic, but when performing

7.5 Conclusions 135

measurements on larger and practical networks this difference seems to be ignorable compared to the influence of electrical equipment. However, underplaster wiring gives the highest overvoltages internally in an installation.

TN-systems result in lower CM voltages but higher DM voltages than IT-systems. A permanent ground fault in an IT-system may result in both high CM and DM voltages, compared to a TN- system.

8PROTECTION OF LOW-VOLTAGE SYSTEMS AGAINST

LIGHTNING-INDUCED OVERVOLTAGES * •

8.1 INTRODUCTION

In this chapter the results obtained in the previous chapters are used to study LIOs in a more extensive low-voltage system. The lossless model for induced voltage calculations developed in chapter 3 is used in most examples and the loss effect model from chapter 4 is used in one additional example. Transformer models from chapter 6 and models of LVPI networks from chapter 7 are used to represent the loads in the system. A sample network is used to study how lightning-induced overvoltages are influenced by the system configurations including load representation, arresters, grounding impedances, lightning parameters, and stroke location. The differences between TN- and IT- system are particularly studied. An important aspect with the chapter is to obtain an acceptable protection of the system by installing arresters at strategic positions in the network.

Protection against overvoltages and surges can be categorised into [6]:• shielding and grounding• application of filters• application of surge protective devicesThis chapter will deal with the application of surge protective devices, typically gapless metal oxide varistors (MOV). The main questions handled are:• How large is the protective area of an arrester?• Is it sufficient to install arresters at the power service entry of a LVPI network?

When selecting and installing arresters in a low-voltage system the following three factors are important to consider:• Rated voltage and energy capacity of the arrester.• Length and cross-section of the connections to the arrester.• Grounding of the arrester.These considerations are not addressed here mainly based on the assumptions that rated voltage and grounding resistance should be as low as possible, the leads should be as short as possible, and energy capacity is not a problem related to LIOs. Appropriate guidelines for installation of arresters in the low-voltage system can be found in [2,122].

The co-ordination of arresters in LVPI networks involves the proper selection of arrester ratings along with series impedances of the interconnections in order to keep the energy stresses and overvoltages within acceptable limits. Such co-ordination is reported in [123, 124, 121], but will not be particularly handled here.

8.2 Modelling of low-voltage system 137

8.2 MODELLING OF LOW-VOLTAGE SYSTEM

8.2.1 Introduction

The modelling in this chapter is generally kept as simple as possible. The induced voltage is calculated according to chapter 3, using the lossless ATP-model developed in chapter 3.5. A lossy ground is included as a separate example. Only the induced voltage from the return stroke is considered. Although simple, the models of the LVPI networks and the distribution transformer are deliberately chosen to cover a range believed to be representative for actual systems. Thus the differences between both large and small installations and transformers are investigated. The TL model for the lightning channel is used in the calculations in this chapter. This is reasonable since mainly the maximum value of the induced voltage is of interest.

8.2.2 Basic configuration

Fig. 8.1 shows the configuration of the low-voltage network used as an example in this chapter. The network consists of a distribution transformer in point 1 and LVPI networks in point 2-6. This configuration is a simplification with the purpose to illustrate the LIO level in low-voltage systems and how it is influenced by the lightning parameters and loads or terminations in the system. The lightning stroke is assumed located on a circle with radius r0 around the centre of the system. The angle a between the line segment 1-2 and the circle radius is used as a parameter in the analysis. The maximum LIO and its steepness are strongly dependent on this angle.

0 lightning location

250 m 250 m

Fig. 8.1 Basic configuration.To the left: Sample network. To the right: Configurations of electrical loads.


8.2.3 Overhead lines

The overhead lines have a height of 6 m and lengths as shown in fig. 8.1. The overhead lines consist of three phase-conductors and in the case of a TN-system of an additional neutral conductor. The same voltage is assumed induced in all the three phase conductors and they are handled as a single equivalent conductor. The characteristic impedance of each conductor is Zc - 500 Q and the mutual impedance is Zm = 200 Q. This results in an equivalent characteristic impedance of the three phase conductors of ZP = 300 Q.

8.2.4 Distribution transformers

The distribution transformer is modelled according to chapt. 6. The default model is just an inductance of 10 pH. The neutral point of the low-voltage winding is normally assumed to be connected directly to the transformer tank. The influence of the transformer models of T02 and T15 from chapt. 6 is particularly investigated.

8.2.5 LVPI networks

LVPI networks are represented by the models developed in chapter 7.3.5. Installations are here divided in the simple types; small, medium and large as shown in tab. 8.1, based on the general assumption that the inductance decreases and the capacitance increases with installation size. The installation of type small is used as default in the calculations and deviation from this model is investigated.

Tab. 8.1 Models (ZLVPI) of LVPI-networks used in the calculations.

Type small medium large

TN10 uH

-mnr*-5 uH 2 uH

-Wlh

IT10 uH 50 nF

—1|—5uH 100nF

-*m---- 1|—2uH 200 nF

~/vm ||

8.2.6 Grounding

The impedances of the LVPI networks and transformer are grounded via resistances. A value of 50 Q is assumed for the LVPI networks and 5 Q is assumed for the transformer. The significance of the grounding resistance is investigated. The LIO is calculated as the voltage across the installation (ZLVPI) or transformer impedance (Zr) and not between the overhead line


and ground. This is reasonable since this is the voltage actually stressing the units.

8.2.7 Arrester

The metal oxide arrester used in the investigation is of type Raychem LVA-440, having a rated voltage of440 V and an energy capacity of 650 J. The current-voltage characteristic calculated by ATP [12] is shown in fig. 8.2, with the data for a standard 8/20 ps impulse shown as circles.

i [kA]

Calculated

0.0001 ........

0.00001

0.000001,1.2 1.4 1.6 1.80.2 0.4 0.6 0.8

Fig. 8.2 Current-voltage characteristic for MOV.

The data given in fig. 8.2 are the standard rated data provided by the manufacturer, but there are small differences between the various brands. Data for voltages below 1.2 kV are not given in the data sheets. According to [125, 126] a more sophisticated model of the arrester is required in order to take its frequency dependency into account. Such a model has not been implemented here and neither is the inductance of the connections considered. A fine-tuning of a protection scheme would require such considerations, but the purpose here is to investigate how an arrester influences the overall voltage level in general.

8.2.8 Lightning channel and current

The TL model for the lightning channel is used in this chapter and only the return stroke will be considered. The lightning current travels upward with a constant velocity of v=l.l-108 m/s. The height of the lightning channel is assumed to be so large that the current front reach the cloud beyond the maximum time of simulation. The default lightning current shape is shown in appendix A (fig. A.1). Its amplitude is 30 kA, the front time (30-90 %) is 2 ps, and the half value time is 40 ps.

8.3 Lightning-induced overvoltages in unprotected systems 140

8.3 LIGHTNING-INDUCED OVERVOLTAGES IN UNPROTECTED SYSTEMS

This section investigates how the LIO in an unprotected system depends on various parameters in the system. Particularly emphasised is the influence of the models of the transformer and the LVPI studied in chapter 6 and 7 respectively. If the level of LIO is very sensitive to variations in these models, it will be almost impossible to transfer the results from a sample network to actual systems and draw some general conclusions regarding LIO in low-voltage systems.

Only the maximum peak voltage in the system will be studied. The lightning stroke has a distance r0 = 500 m from the centre of the system as shown in fig. 8.1. The location (1-6) of the maximum voltage is indicated with different markers as shown in the legend of each figure.

8.3.1 Influence of LVPI-network model

The influence of the LVPI networks in point 2-6 is investigated for both IT- and TN-systems. Three models of installations according to tab. 8.1 are analysed. The transformer model (in point 1) is a pure inductance of ZT= 10 pH with the low-voltage neutral connected to the tank.

Fig. 8.3 shows the maximum voltage in the IT-system in fig. 8.1 as a function of the angle a to the stroke location. Three different installation models are applied. The same installation model is used in all points 2-6.

LWkV]

-~y1: 50 nF+10 pH Yj2:100 nF+5 pH 27/3: 200 nF+2 pH

O d o

90 105 120 135 150 165 180

a [deg]Fig. 8.3 Maximum LIO in IT system, dependency on LVPI model.

The installation model has a large impact on the maximum LIO, and the larger capacitance of the installation the smaller LIO. Thus the level of LIO in densely inhabited areas will most likely be lower than in sparsely settled rural areas. The maximum voltage occurs at point 5 for an angle a=45°. At this angle the lightning stroke is closest to the network (point 5), so the result is reasonable. The voltage is lower at a=135° since the low impedance and ground resistance (5 Q) of the transformer reduces the voltage. The lowest voltage occurs for a=90°


since the lightning stroke for this angle is farthest off from the network.

Fig. 8.4 shows the same calculations as in fig. 8.3, but now for the TN-system.

IWkV]

O O O

105 120 135 150 165 180

a [deg]Fig. 8.4 Maximum LIO in TN system, dependency on LVPI model.

The maximum LIO now occurs across the transformer winding for all three types of installation modelled. This is due to lower grounding resistance at the transformer (5 Q) than at the installations (50 Q). Apparently there is a contradiction between the effects of grounding in the IT- and TN-systems. However, the explanation is that lower grounding resistance in a point will increase the LIO at this point but lower the LIO at other points. In this context it is important to be aware of that the LIO is calculated across the installation or transformer winding and not between the overhead line and ground. The effect of grounding will be investigated further.

8.3.2 Influence of the transformer model

In the calculations shown in fig. 8.3 and 8.4 the transformer model is simply an inductance of 10 pH. Now the influence of the details in the transformer representation presented in chapt. 6 will be analysed. The two transformers T02 and T15 are used with the low-voltage neutral grounded or isolated.

Fig. 8.5 shows the maximum voltage in the IT-system in fig. 8.1 as a function of the angle a to the stroke location. The transformer is T02 (800 kVA) and is modelled according to table 6.2 in chapt. 6. As seen from fig. 8.5, the maximum LIO is dependent on the transformer model only for angles larger than about a=90°. The shorter distance between the lightning stroke and the transformer the more influence the transformer model has on the maximum LIO. When the neutral is isolated a very high voltage arises across the transformer’s LV winding. When the neutral is grounded two different representations are used: One is the full representation shown in table 6.2 and the other is the low frequency inductance approximation of Lol= 9.61 pH. As seen from fig. 8.5 both models give the same result, almost identical to the upper curve of fig.


8.3.

LWKV]

neutral isolatedO □ o •

: neutral grounded;

105 120 135 150 165 180

a [deg]Fig. 8.5 Maximum LIO in IT system, dependency on transformer model.

Fig. 8.6 shows calculations similar to fig. 8.5, but the transformer is now T15 (50 kVA) modelled according to table 6.3 in chapt. 6.

LWkV]

O D o •

neutral isolated

neutral grounded

105 120 135 150 165 180

a [deg]Fig. 8.6 Maximum LIO in IT system, dependency on transformer model.

Figure 8.6 is very similar to fig. 8.5. When the transformer neutral is grounded the transformer model seems to have almost no influence on the maximum LIO in the system. The difference between the full representation (model according to table 6.3) and tire low frequency inductance approximation of Z0L- 32.28 pH is slightly larger than in fig. 8.5 but still rather unimportant. The transformer T15 gives somewhat larger LIO than T02.


8.3.3 Influence of the grounding impedance

The general influence of the grounding resistance will now be investigated. Fig. 8.7 and fig.8.8 show the influence of the installations’ grounding resistance, in the TN and IT system respectively. The grounding resistance of the distribution transformer is kept constant (5 Q).

IWkV]

O □ o ••toon

0 15 30 45 60 75 90 105 120 135 150 165 180

a [deg]Fig. 8.7 Maximum LIO in the TN system, dependency on grounding resistance.

LUtkV]

□ o • ♦

105 120 135 150 165 180

Fig. 8.8 Maximum LIO in the IT system, dependency on grounding resistance.

Both figures 8.7 and 8.8 show that an increase in the grounding resistance results in a reduced level of LIO. An exception is seen in fig. 8.7 for angles o>145° where the maximum voltage occurs at the transformer. In general, lower grounding resistance in a point will increase the LIO at this point but lower the LIO at other points, since the voltage is calculated across the installation or transformer winding. Infinite grounding resistance will result in zero current to ground and thus zero voltage across the installations. When the grounding resistance is 5 Q the corresponding curve in fig. 8.7 is completely symmetric since all loads then are equal.


8.3.4 Influence of a connected underground cable

So far, LIOs have been calculated in low-voltage systems consisting of overhead lines only. However, a mixture of overhead lines and underground cables is common. Here, a situation is studied where one LVPI (at point 5) is connected to the overhead line system through an underground cable. The underground cable is assumed lossless with characteristic impedance Zc = 40 Q and wave velocity vc = 2-108 m/s. The cable sheath is assumed ideally grounded at both ends and no voltage is assumed to be induced directly in the cable. According to Norwegian regulations [116] an underground cable of length 150 m is said to protect a connected LVPI. To investigate this statement the cable length is varied from 0-250 m for the three different LVPI models. Only the IT-system is studied. The lightning stroke is located at a distance r0 = 500 m with an angle a=45°(the location which gives the highest voltages at point 5).

Fig. 8.9 shows the maximum induced voltage across the LVPI impedance ZLVP[ at the end of a cable with length 1. We see that this voltage is little affected by the cable length, particularly for the largest installation types. Taking the loss of the cable into account would slightly reduce the voltage. As a first approximation the cable can be modelled as a capacitance of C’= l/(ZcvJ = 125 pF/m. Thus, the total capacitance of the cable is much less than the capacitance of the installation and the voltage here will be little affected by the cable. An exception occurs for the smallest installation and the longest cable where the cable and installation capacitances become comparable.

LWx[kV]8

7 /

6'1:50 nF+10 pH 2:100nF+5pH 3:200 nF+2 pH5

4

3

2150 200 25050 1000

/[mlFig. 8.9 Maximum LIO in IT system, dependency on cable length. Configuration to the right.

8.3.5 Influence of load-changes in a neighbour point

Some of the previous calculations indicate that the maximum LIO in a point is rather

8.4 Protection of low-voltage systems 145

independent of arrangements at the neighbour point that is 250-500 m away. This will now be further investigated by calculating the LIO in point 5 and make changes in point 2. The lightning stroke location is ct=45 ° and r0 = 500 m. The IT-system is studied, having the default impedance ZLYPI= 50 nF +10pH and grounding resistance #=50 Q. The transformer impedance is Zr= lOpH with grounding resistances R=5 Q and the neutral connected to the tank. Fig. 8.10 shows the LIO at point 5 as a function of time.

U5[kV]

Default

Large LVPI

Large LVPIR=5 ohm

0 2 4 6 8 10 12 14 16 18 20 22 24

Fig. 8.10 Induced voltage in point 5 and its dependency of load-changes in point 2.

The following changes are made in point 2 (related to the legend in fig. 8.10):MOV: Arrester installed.Large LVPI: ZLm = 200 nF +2 pH.Large LVPI, R= 5 ohm: ZLm = 200 nF +2 pH and the grounding resistance is 5 Q.

We see that the changes made in point 2 in general have little effect in point 5. The most interesting result is that the arrester at point 2 does not reduce the voltage significantly at point 5 (250 m away). This is a very important observation which will be investigated further in the next section. Reducing the grounding impedance at point 2 seems to have a larger effect than installing an arrester.

8.4 PROTECTION OF LOW-VOLTAGE SYSTEMS

Protection of low-voltage systems is a comprehensive task that is impossible to treat completely in this thesis. The overvoltage protection of a low-voltage system involves the following aspects:1) Determine where to install arresters in the system.2) Select a particular arrester with appropriate ratings.This chapter addresses only the first aspect, particularly with the aim to determine key-points for installation of arresters in the low-voltage system. Three different protection schemes are


analysed:• Arrester at the distribution transformer (point 1 in fig. 8.1)• Arrester at a key junction point (point 2 in fig. 8.1)• Arresters at end points (point 3-6 in fig. 8.1)

A general, mathematical analysis of induced voltages in a network protected by surge arresters is reported in [127]. However, the electrical field is there assumed to be a plane wave (equal in all points, except for a time delay) which is a reasonable approximation only for distant lightning. Despite of this method’s elegance it is assumed to be inaccurate for the short distances (100-1000 m) studied in this thesis.

A probability analysis of the LIO’s dependency on arrester positioning will not be performed. Only the maximum LIO at points 1-6 in fig. 8.1 will be studied. An arrester installed at any point in the low-voltage system will reduce the risk of overvoltage damages in the system, but not necessarily reduce the maximum LIO. For all the simulations the LVPI networks are modelled according to the "small" model in tab. 8.1, resulting in worst-case situations. The voltage across installed arresters will in general be limited to about 0.8 - 1 kV (positive or negative), but is not shown in the following examples.

Figures 8.11 and 8.12 show the maximum LIO in the TN-system (with the configuration shown in fig. 8.1) and its dependency on stroke location angle and positions of arresters in the system. Two different distances to the lightning stroke are investigated; 500 m and 100 m. For ro=100m the shortest distance from the lightning stroke to the overhead line is d-lOO-sin(a) [m] and a direct lightning stroke to the overhead line is likely to happen for angles close to 0° or 180°.

UUtkV]

No MOV orO D o • MOV in 3-6

MOV in TMOV in 2

105 120 135 150 165 180a [deg]

Fig. 8.11 Maximum LIO in TN system. Dependence on arrester location. r0 =500 m.


IWkV]

■ ■' No MOV orO D o • ♦ *MOV in 2 orMOV in 3-6 "' I

MOV in 1

QDoQDOOOOOO 0 6c - '? n i x-MOVin1&2105 120 135 150 165 180

a [deg]Fig. 8.12 Maximum LIO in TN system. Dependence on arrester location. r0 =100 m.

Fig. 8.11 shows like fig. 8.4 that the induced voltage in TN-system is generally low for a distance r0 = 500 m. The maximum voltage is reduced significantly only by installation of an arrester at the distribution transformer. Fig. 8.12 shows on the other hand that large LIOs can arise in the TN-system for nearby lightning strokes. Installing arresters at both ends of the overhead line segment closest to the lightning stroke will reduce the LIO considerably as seen from the MOV in 1 & 2 curve in fig. 8.12. Installing arresters only at die end points of the TN- system will have no effect on the maximum LIO.

Figures 8.13 and 8.14 show calculations similar to fig. 8.11 and 8.12 respectively, but now for the IT-system. The transformer neutral is connected to the tank.

IWkV]

No MOV oro n o •

MOV in 2

MOV in 3-6MOV in 2 -

105 120 135 150 165 180a [deg]

Fig. 8.13 Maximum LIO in IT system. Dependence on arrester location. r0 =500 m.


IWkV]

o □ •

iNo MOV or iMOV in 3-6

MOV in 1MOV in 2

105 120 135 150 165 180a [deg]

Fig. 8.14 Maximum LIO in IT system. Dependence on arrester location. r0 =100 m.

Figures 8.13 and 8.14 show that the maximum LIO is almost unaffected by arresters at the distributions transformer. The maximum LIO will generally arise at the point closest to the lightning stroke rather independently of protective measures at other points in the system. Fig. 8.13 and 8.14 show that large voltage may arise in point 2 even if all end points are protected.

The minimum distance between arresters in fig. 8.1 (250 m, point 1-6) is obviously too high, since an arrester is shown in fig. 8.10 to have almost no effect at a neighbour point. Fig. 8.15 shows a different example where a single overhead line equal to the one described in 8.2.3 is exposed to a lightning stroke 500 m from its line centre. The voltage across a LVPI network at the line centre is calculated as a function of the distance l to the nearest arrester mounted across a LVPI network. Only the IT-system will be investigated. Current amplitude of 30 KA and 100 kA is used both with the default rise and half value time constants. A calculation without connected LVPI-networks and grounding resistances is also performed and labelled No load.

lightningposition

30 kA

500 m 100 kA

30 kA500-/[m]500-/[m]

No load

300 n300 £2

Fig. 8.15 Maximum LIO as a function of the distance to the nearest arrester.At left: Configuration. At right: Maximum LIO.

8.5 Internal voltages in LVPI networks 149

Fig. 8.15 shows that an arrester protects a rather limited area and particularly for 100 kA the arresters only protect the point where they are installed. When no LVPI-networks are connected to the overhead line, the maximum LIO becomes much larger. This situation has also been analysed analytically by Rusck [38, eq. (168)]. The No load curve in fig. 8.15 fits well with this analysis where the increase in induced voltage at the midpoint between two arresters is shown to be proportional to the lightning current amplitude and inverse proportional to the front time constant. Increasing the total line length from lkm to 10 km had little effect on the maximum induced voltage at the midpoint.

8.5 INTERNAL VOLTAGES IN LVPI NETWORKS

Even if a LVPI network is protected with arresters at the power service entry, critical overvoltages may occur inside the circuits. The magnitude of such internal overvoltages is very dependent on the shape of the voltage across the arrester. Initial investigations showed that the maximum internal voltage varied extensively with stroke location according to fig. 8.1. This observation could not be explained by variations in steepness of the voltage alone but had to be related to the whole voltage waveform. This section will thus focus on the time dependent induced voltages. Fig. 8.16 shows the configuration used when studying the internal overvoltages in LVPI networks. An additional segment of an underplaster wired circuit (modelled in appendix H) is connected in parallel with the impedance representing the LVPI network ZLVPI of type "small". An arrester is connected at the power service entry across ZLVPl. The conductor N is connected to the phase L in an IT-system and to the protective earth in a TN-system. A load of 50 Q is connected between the L and N conductors at the end of the circuit. The voltage across the arrester, Ug, is compared to the voltage between the L and PE conductor, Ub at the end of the circuit.

L llndemlaster rimnit i

+ Uo -

Fig. 8.16 Configuration when studying internal overvoltages in LVPI circuit.

Figures 8.17 and 8.18 show the induced voltage at point 2 according to fig. 8.1 and at the end of a 30 m long underplaster wired circuit connected at this point as shown in fig. 8.16. The arrester connected across the LVPI network at the power service entry has a current-voltage

8.5 Internal voltages in LVPI networks 150

characteristic according to fig. 8.2 with a protective level around 1 kV. The return stroke current has an amplitude of 100 kA, a front time constant of t, = 1 ps, and a half value time constant of t2 = 50 ps. This represents a much higher stress than the current shown in appendixA. The current amplitude is chosen to ensure operation of the surge arrester, particularly in TN- systems.

U [kV]

1.5 -

-0.5 --

U [kV]

Fig. 8.17 Induced voltages in IT-system. Configuration according to fig. 8.1 and 8.16. Top row: r0 = 500 m, Bottom row: r0 = 300 m. Left column: a=0°. Right column: a=180°.

Solid line: Voltage U0 across arrester. Dotted line: Voltage U-, at the end of the circuit.

Figures 8.17 and 8.18 show that the stroke location a~ 180° (farthest off) gives the highest overvoltages at the end of the circuit. The overvoltages also increase with decreasing distance r0. The voltage at the end of the circuit is in some cases more than twice the voltage across the arrester. The highest internal overvoltages generally arise when oscillations occur across the arrester, and in particular when the oscillations in this voltage excite the natural frequency of the connected circuit. The "resonance" phenomenon is particularly pronounced at the right hand side (a=180°) of fig. 8.17 where the oscillations in the voltage across the arrester fits in with the oscillations at the end of the installation resulting in a large overvoltage (-3 kV) here. Higher lightning currents will result in (dependent on the location) steeper voltage oscillations across the arrester and increased internal voltages. The resistive load in the TN-circuit will reduce the magnitude of the internal voltage oscillations. On the other hand, oscillatory voltages across the arrester are more likely in a TN-system, so unloaded TN-circuits could expect high internal voltages. Resonance phenomenons in general have been analysed in e.g. [128].


UtkV]

18 20t[us]

U fkVJ

2 —.

18 20

U [kVJ 1.5

10.5

0-0.5

-1

-1.5

i ! !i, ! ; ! ! i i: i

! n-ft

■ i ■ i ■ i ■ i ■ i ■ ■' ■ i ■ i - j8 10 12 14 16

t[us]18 20

8 10 12 14 16t[us]

Fig. 8.18 Induced voltages in TN-system. Configuration according to fig. 8.1 and 8.16. Top row: r0 = 500 m, Bottom row: r0 = 300 m. Left column: a=0°. Right column: a=180°

Solid line: Voltage U0 across arrester. Dotted line: Voltage Ui at the end of the circuit.

The oscillatory nature of internal voltages calculated in figures 8.17 and 8.18 is in agreement with what is measured and reported in e.g. [125]. IEEE recommends an oscillatory voltage (called ring wave) with frequency as low as 100 kHz as a standard representative surge when testing electrical equipment immunity [129].

It is important to be aware of that the system configuration used in fig. 8.17 and 8.18 probably not is a worst case situation. In a real low-voltage system, circuits of arbitrary lengths are present and probably at least one will have a critical length resulting in large internal overvoltages.

8.6 LOSSY GROUND EFFECTS

So far the ground has been assumed ideally conducting. However, both chapter 4 and 5 show that the lossy ground effect is important to consider for some configurations (especially end strokes) but that there are some uncertainties regarding the adequacy of the approximations made and the model used in the calculations. In this sub-chapter the ground is assumed lossy with a conductivity o, = 0.001 S/m and a relative permittivity er = 10.

Fig. 8.19 shows the induced voltages calculated for the system shown in fig. 8.1 with a the


lightning location r0 = 500 m and a=0°. The lightning current is 30 kA according to appendix A. Both the IT and TN systems are investigated using the LVPI equivalents corresponding to the small type in tab. 8.1. The transformer is modelled as an inductance of 10 pH (neutral grounded).

U fkVJ

10 12 14 16 18 20

a) TN-system. Lossless

10 12 14 16 18 20tips]

b) TN- system. Lossy ground.

20 —•

c) IT-system. Lossless d) IT-system. Lossy groundFig. 8.19 Lightning-induced voltage’s dependency on lossy ground (a, = 0.001 S/m, er = 10).

Legend numbers according to fig. 8.1.

Fig. 8.19 shows that the lossy ground effect (attenuation of the electrical field) is remarkable. In the IT-system the maximum voltage increases with more than a decade, which is very surprising. The reason for this large lossy ground effect in the IT-system can be illustrated by studying eq. (3.55) in chapt. 3.4. In the IT-system the LVPI impedances are large and can be assumed infinite, as a first approximation. If the transformer neutral is grounded or an arrester is located at the transformer the load impedance at this side is very small. An IT-system can thus be equalised by an overhead line that is open at one end and grounded at the other. The voltage at the open end then becomes:

8.7 Conclusions 153

ua(M — •— +/wz ; * dz A cosh (yL) ju> -rj-

l>dA (x,z) sinh(y(x -x))dz cosh(yZ)

dx (8.1)

In eq. (8.1) the dAJdz terms outside the integral dominates except for a lossless situation where they are zero and for a side stroke situation when they nearly balance each other for symmetry reasons. For an end stroke the lossy ground effect will be large, however, and this is the case in fig. 8.19c)-d). If the transformer neutral is isolated the LIO will decrease somewhat. This implies that installing an arrester at the distribution transformer in an IT-system actually can result in increased overvoltages at the subscribers’ installations. In the TN-system overhead lines are normally terminated by low impedances at both ends. This configuration is less sensitive to lossy ground effects. The increase in the risk of failure due to a lossy ground is studied in [70].

8.7 CONCLUSIONS

To protect the TN-systems against very close lightning strokes, arresters are required at each installation. However, the TN-system results in general in much lower voltages than the IT- system and is self-protective for distant lightning due to the very low phase to ground impedances and the neutral overhead line conductor. To protect the IT-system against very close or distant lightning strokes, arresters are required at each installation.

Even if arresters are installed at the power service entry, large overvoltages could still arise inside the LVPI network. Oscillations due to reflections in the low-voltage system and with frequencies dependent on overhead line segment lengths could excite the natural frequency of connected LVPI circuits. LVPI circuits have different lengths and probably at least one will have a critical length resulting in large internal overvoltages. Such overvoltages could have amplitudes, seen from the simulations, of at least 3-4 times the protective level of the connected arrester. The overvoltages in the TN-system are limited by the connected loads, however. The internal voltages become lower for shorter LVPI circuits.

Lossy ground effects are very large for IT-systems and will in most cases lead to a severe increase in the lightning-induced overvoltages. The lossy ground effect is much less in the TN- system, but the voltage could still increase significantly. The front-time of the voltages will increase, however, both in the IT- and TN-systems. This could result in lower internal voltages in LVPI circuits if arresters are connected at the power service entry.

All analyses performed in this chapter show that the TN-system results in much lower LIOs than the IT-system. This is caused by the neutral overhead wire and the low installation impedances in the TN-system. However, only the phase-to-ground voltages have been analysed here. The phase-to-phase voltages in the IT-system are almost zero, while equal to the phase-to- ground voltages in the TN-system. This picture is complicated by possible ground faults and unsymmetries in the IT-system.

9DISCUSSION

9.1 INTRODUCTION

Atmospheric overvoltages is a major source of failures in overhead line low-voltage systems. Such overvoltages are caused by either a direct lightning stroke to the system or by indirect effects from nearby lightning. Direct lightning strokes cause normally large overvoltages and severe damages including destroyed electrical installations and apparatus, and fires. Damages caused by a direct lightning stroke to a building can be prevented by special protection consisting of e.g. lightning rods on the roof and conductors to earth. Protection against direct lightning strokes to a low-voltage overhead line requires normally surge protective devices at all installations connected to the line. Still large overvoltages in the grounding system may arise and a complete protection is thus hard to obtain. Indirect lightning results in lower overvoltages than direct lightning, but occurs on the other hand much more often. Lightning strokes within a radius of about 1 km may cause harmful overvoltages in a low-voltage system.

This thesis focuses on the effect of indirect lightning resulting in lightning-induced overvoltages (LIOs). The work is theoretical in nature and consists mainly of calculation of LIOs. The main questions related to this topic are:• Is it possible to predict the expected LIOs in low-voltage system with sufficient accuracy

to enable the design of a proper overvoltage protection?• What is the influence of different parameters governing the severity of LIOs?The answer to the first question is yes, with certain reservations. It is first of all important to be aware of that such calculations are intricate due to the high complexity of both the lightning flash and the low-voltage system. It is impossible to calculate the LIOs with a high accuracy, but the main characteristics of the LIOs in a low-voltage system are possible to obtain. Such characteristics include the waveform of LIOs during the first few microseconds, including its amplitude and steepness.

9.2 MODELS AND ASSUMPTIONS

The calculations of LIOs performed in this thesis are to a large extent based on models which are well known. However, some new models of transformers and domestic installations have been developed in this work based on measurements. An objective when selecting the models is to aim at simple models based on a few measurable quantities and which exhibit a reasonable accuracy. The models can be divided in four parts:• Lightning channel model

9.2 Models and assumptions 155

• Lossy ground effect model• Coupling model» Overhead line termination models

9.2.1 Lightning channel model

The lightning channel model is based on assumptions regarding the shape of the lightning channel and the distribution of current and charge along this channel. The lightning channel is assumed to be straight and vertical. This is hardly correct as seen from photographs of lightning and an irregular path could have a strong influence on the LIOs. However, the first hundred metres of the channel is normally fairly vertical, and the lower part of the lightning channel is found to determine the main characteristics (steepness and amplitude) of the LIO. The tortuosity of the lightning channel around the vertical axis will result in minor disturbances in the LIO.

The charge in the lightning channel is assumed deposited by the leader development and neutralised by the return stroke. The distribution of the charge and current along the channel is unknown and a model called MTL (modified transmission line) is used to calculate these distributions based on the current at ground level and the velocity of the return stroke. The return stroke is assumed to start from ground and travel upwards with a constant velocity about one third of the speed of light, neutralising the charge along the channel on its way. The electrical field from a lightning flash is assumed to consist of two contributions observed in measurements. A component from the charged leader, and a component from the return stroke. The component from the charged leader is assumed to be static since the velocity of the leader is much lower than of the return stroke. If the low-voltage system has a connection to ground the static field term from the leader will not result in induced voltage. The MTL model has shown reasonable accuracy when comparing electrical fields and currents from triggered lightning. The model will result in fairly accurate calculations for the first few microseconds where also a further simplification called the TL (transmission line) model is reasonable.

9.2.2 Lossy ground effect model

The ground is assumed to be flat and homogenous with a constant conductivity and permittivity. The fiat ground assumption is reasonable for nearby lightning, which is of interest in relation to LIOs in low-voltage systems. The assumption on homogeneity is more doubtful but is used since the characteristics of the ground in unknown. Calculations of electrical fields over a lossless ground (infinite conductivity) is fairly straight forward when the lightning channel model is established. However, the electromagnetic fields from a lightning stroke are attenuated when propagating over a lossy ground. Norton’s method has been used to take this attenuation into account. This model is found to be accurate for all distances and lightning channel heights of practical interest. For larger distances (more than some hundred metres) the surface impedance method is reasonably accurate and is partly applied in this work and

9.2 Models and assumptions 156

compared to Norton’s method.

9.2.3 Coupling model

The electrical field from a lightning channel will excite overhead lines and result in LIOs. Agrawal’s coupling model is used to calculate the LIOs when the incident electrical field from the lightning channel is known. This model is based on adding a source term representing the electrical field to the classical telegraph equations. Agrawal’s coupling model has been shown to give accurate results when comparing measured electrical fields with measured induced voltage. However, its adequacy for end stroke configurations (lightning stroke at the line prolongation) could be questioned. This basic problem has not been particularly addressed in this thesis, since the telegraph approach is required in order to study LIOs in overhead lines of finite lengths connected to electrical networks.

The overhead lines in the low-voltage system are assumed to be lossless. This is reasonable for line lengths less than about 1000 m and practical ground conductivities (o > 0.001 S/m).

Voltages induced directly in installations or in underground cables are ignored. This is reasonable since such voltages normally are insignificant compared to induced voltages in overhead lines.

9.2.4 Overhead line termination models

The transformer and the low-voltage power installation (LVPI) models belong to this group. Two main assumptions related to these models are that the same voltage is induced in all phase- conductors of an overhead line and that the loads are symmetric. This implies that the phase conductors can be treated as a single conductor and only the common mode system of the transformers and installations has influence on LIOs in an overhead line. Again, these assumptions are reasonable for the first few microseconds, but unsymmetries in the connected LVPI networks will make them doubtful after reflections in the system. Normally the phase conductors of an overhead line have about the same distance to ground, making the assumption of the same inducing voltage in all conductors reasonable.

Unlike the three previous models, the overhead line termination models are based on measurements performed in this work. Measurements are performed in the frequency domain and equivalent electrical networks representing the transformers or the LVPI networks in the time domain are developed. Measurements have been performed on several units in order to reveal differences between various types of transformers or installations and to be able to suggest "typical" representations to be used when specific data are not available. The modelling resulted in common mode input- impedances or admittances seen from the terminals of the units. The main frequency interval of interest regarding LIO calculations is 10 kHz - 1 MHz in which the developed models should be accurate.

9.3 Calculation results 157

Measurements revealed large differences between transformers with and without LV neutral grounding. When the transformer neutral is grounded the input admittance can be approximated by a small inductance between 4-40 pH, significantly related to the transformer's rated power and voltage (larger transformers have lower inductance). When the neutral is isolated the input admittance can as a first approximation be represented by a capacitance of 2-20 nF which seems to be less related to transformer ratings. The simple capacitive and inductive approximations are basically valid in the 10 kHz to 500 kHz range. This is normally sufficient for practical LIO calculations.

Measurements on LVPI networks revealed as expected large differences between TN and IT systems. An IT-system LVPI network impedance is roughly capacitive up to a couple of 100 kHz. Above this fundamental frequency the impedance becomes roughly inductive. Capacitance values of20-200 nF have been found, increasing with the LVPI network size and the quantity of connected equipment. The inductance of the installation (in series with this capacitance) is suggested to be 2-20 pH, decreasing with network size. A TN-system LVPI network impedance is very dependent on the connected electrical equipment, but is generally very low for frequencies 10-100 kHz. Few measurements on TN-systems have been performed in this work, but the TN-impedance is suggested to be representable by a pure inductance having a possible value range of 2-20 pH, decreasing with increasing installation size. Connected electrical equipment influences the input impedance, resulting in resonance peaks and generally lower values of the impedance. Underplaster wiring and surface wiring have a different characteristic as found from laboratory measurements, but when performing measurements on actual networks this difference seems to be ignorable, both in IT- and TN- systems, compared to the influence of electrical equipment.

9.3 CALCULATION RESULTS

When the models of the lightning channel and the low-voltage system are established and implemented in computer programs, the LIO in the system can be calculated. This is practically accomplished by first calculating the inducing voltage for each end of all overhead line segments in the system, then developing an electrical equivalent of the whole system and finally implementing this in the ATP-EMTP. Reflections at the overhead line ends are handled by MODELS since all overhead lines are assumed lossless. The inducing voltages, appearing as sources at each end of the overhead lines, must generally be calculated by an external computer program, but can also in a special case (lossless & TL-model) be calculated directly in MODELS in the ATP-EMTP. Accepting the assumptions in chapt. 9.2 enables a large number of effects to be investigated, including how the various parameters in a complex system influence the level of LIOs. The discussion here will be restricted to the most important aspects.


9.3.1 System parameters’ influence on LIO.

A lightning-induced overvoltage (LIO) can be classified according to its amplitude and rise time. These two quantities are dependent on the system’s key-parameters listed below:• Return stroke amplitude, rise time and velocity.• Stroke location.• Overhead line height and length.• Overhead line terminations.• Ground conductivity.

Table 9.1 summarises the system parameters’ influence on LIO. Only the main trends are listed based on a simple system consisting of an overhead line terminated by impedances. A more complex system can alter the results shown in tab. 9.1 somewhat.

Tab. 9.1 System parameters’ inf uence on LIO

Key parameter

Increasing

Lightning-induced over

Amplitude

voltage (LIO)

Rise time

Return stroke amplitude, Im Increases (proportional) Little effect

Return stroke rise time, tf Decreases Increases

Return stroke velocity, v Decreases Decreases

Stroke locationPosition andDistance

Highest for side strokes in a lossless situation, but can otherwise become highest for end strokes.Decreases with distance (almost inverse proportional)

Largest for side strokes. Increases with distance (almost proportional)

Overhead line height, z Lossless: Increases proportional. Lossy ground contribution almost independent of line height.

Little effect

Overhead line length, L Little effect for matched line Little effect

Overhead line terminations and loads

Increases with increasing terminating resistance. In general decreasing with increasing number of connected LVPI networks.

Increases with increasing terminating resistance. (The LIO builds up due to reflections)

Ground conductivity, o, Decreases (almost inverse proportional to the square root)

Decreases

System grounding, Rz Decreases at the point where the grounding resistance increases, but increases at any other points.

Little effect


The results listed in tab. 9.1 are mainly obtained from the calculations in chapt. 3.6 except for the influence of the ground conductivity, analysed in chapt. 4. Also, most of the effects can be found directly from the expressions of the inducing voltage (3.61) and (4.31). One exception is the effect of the overhead line terminations which requires calculations of reflections at the line ends.

The effect of the return stroke amplitude is the most obvious one, since this is seen directly from the expression of the inducing voltage (3.61). In an unprotected (linear) system the LIOs increase proportionally to the return stroke current amplitude. The effect of the rise time of the return stroke is illustrated in fig. 3.16. The effect of the return stroke velocity can also be seen from (3.61) and is illustrated in fig. 3.17.

The effect of the line height is very dependent on the ground losses. The lossless part of the inducing voltage is proportional to the line height as seen directly from (3.61). However, the lossy part of the inducing voltage in (4.31), is almost independent of the line height, and its dominant parts are actually evaluated at the ground level. This implies that the relative lossy ground effect is increasing with decreasing overhead line height. This effect is also seen from the scaled-model measurements in chapt. 5.4, where overhead line heights of 1 m and 3 m are investigated.

The small effect of the line length is in a way surprising, but this can also be obtained from(3.61). In fact the maximum voltage increases slightly with line length as seen from fig. 3.13b) and approaches a constant value given in (3.71). An overhead line not terminated by its characteristic impedance will result in reflections at the line ends. This will cause the induced voltage in an overhead line, terminated by a low impedance at least at one end, to oscillate with a frequency related to the line length. Its maximum value decreases with line length.

The effect of stroke location is illustrated in figures 3.10-3.12 showing that the side stroke (near the line’s midpoint) gives the highest LIOs in a lossless situation. The measurements of LIO performed in chapt. 5 show that end strokes (at the line’s prolongation) results in higher lossy ground effects than side strokes. This is reasonable since the side stroke configuration is less dependent on the horizontal electrical field which is heavily affected by a lossy ground. The LIO in an overhead line excited by an end stroke over lossy ground can become higher than the LIO for a side stroke. The decrease in LIO with increasing distance is obvious.

The effect of line terminations is schematically shown in fig. 3.15 and also analysed in chapt. 8.3. The important observation is that the introduction of lower terminating impedances by connecting e.g. more LVPI networks results in lower LIO. Thus the magnitude of the LIO is likely to be higher in rural areas with few installations or houses connected to an overhead line. Another result obtained from chapt. 8.3 is that the transformer model used in the calculations has a minor effect on the maximum LIO in a low-voltage network as long as the LV neutral is grounded, something which will be discussed later. The maximum LIO in LVPI networks supplied by an underground cable from an overhead line system is little affected by this cable. The maximum LIO is in chapt. 8.3.1 shown to be about three times higher when installations


of type "small" are connected compared to installations of type "large".

The induced voltage shows a rather complex dependency on the ground conductivity. However, the lossy part of the inducing voltage can, as a rough estimate, be assumed inversely proportional to the square root of the ground conductivity.

Increased grounding resistance at a point will result in a reduction of the LIO at this point since the LIO normally is calculated as the voltage across an installation excluding the voltage drop across the grounding resistance. At any other point in the system the LIO increases, however.

Calculations show that besides the LIOs amplitude and rise time one other characteristic is of great importance: Oscillations related to the natural frequencies of the network. Induced voltages in an overhead line segment terminated with low impedances at both ends (typically a TN-system) will oscillate with a fundamental frequency fm = c/(2 -L) where c is the speed of light andZ is the line length. In an overhead line terminated with a low impedance at one end and a high impedance at the other (more typical IT-system) the oscillation of the induced voltage will have a fundamental frequency fT = c/(4-L). The oscillatory behaviour is particularly pronounced for TN-systems, but also to some extent in IT-systems. The characteristic of the transformers or LVPIs at the fundamental frequency will mostly determine the LIO. The oscillatory nature of LIOs in overhead line system is of great importance regarding internal overvoltages in LVPI circuits protected by arresters at the power service entry. However, as stated in chapt. 9.2 the models used are only accurate for the first few microseconds so only the initial part of the oscillatory voltage waveforms are reliably calculated.

9.3.2 IT- versus TN- systems

A main motivation factor initiating this work was the discussion regarding IT- versus TN- systems and the assumed vulnerability of IT-system to lightning-induced effects. Many chapters in this thesis have touched this subject and the results from these investigations will now be summarised. The IT-system is the common standard in Norway and overhead lines, without a ground or neutral wire, are in frequent use, particularly in rural areas.

The calculations show that the IT-system in general results in much higher voltage phase-to- ground than the TN-system. This is mainly caused by:• the presence of a ground wire in the overhead line in TN-systems.• lower terminating impedances of overhead lines in the TN-system, due to electrical loads.• the grounding of the distribution transformer’s LV-neutral point in TN-systems.• IT-systems is particularly sensitive to lossy ground effects, which apparently results in a

large increase in the LIOs.

The actual reduction in the level of LIOs when a ground wire is added to an IT-system depends on the system grounding and the coupling between the phase wires and the ground wire. The


shorter distance between the wires the larger is the impact of the ground wire. Chapt. 7.4.7 shows that the ratio between the induced voltage in the TN system and the IT-system approximately is n=Zf/(ZN- ZJ where ZN is the characteristic impedance of the neutral conductor and Zm is the mutual impedance between the neutral conductor and the phase wires. Grounding the neutral wire along the overhead lines will reduce the LIOs.

Measurements performed on LVPI networks show that the impedance of a IN-system is much lower than that of an IT-system for the frequency range of interest. This is mainly caused by the electrical loads or apparatus connected between the phase and neutral in the TN-system. LVPI networks in the IT-system show a typical capacitive behaviour, as opposed to the TN- system where they are inductive. A reduction in the connected terminating impedance of an overhead line results in reduced level of LIO. The calculation examples in chapter 7.3.6 show that the effects of a neutral wire and the terminating impedances together result in 4-5 times lower phase to ground voltages in the TN- than in the IT-system. Calculations on an extended system in chapter 8.3.1 show even larger differences.

The grounding of the transformer LV neutral point is of great importance. An isolated neutral point results in much larger LIO at least at the transformer. The neutral point is normally isolated in an IT-system, but equipped with a surge protective device, normally with a protective level of a few kilovolts. It is thus likely that this protection operates during the first part of the LIO development or already during the stepped leader process. As a consequence of this, the neutral point is assumed grounded also for IT-system in several examples in this thesis. Chapter 8.3.2 shows that keeping the transformer neutral isolated leads to at least a doubling of the maximum induced voltage in the investigated system.

The IT-system is apparently more sensitive to lossy ground effects. An example of this in chapt.8.6 shows that while the lossy ground effect increases the voltage in the TN-system by a factor 2.5, the voltage in the IT-system increases by a factor 14. This dramatic increase in the IT- system is caused the fact that an overhead line grounded at one end (transformer representable by a low impedance) and open at the other end (LVPI network representable by a high impedance) is particularly sensitive to lossy ground effects.

The IT-system results in much higher phase-to-ground voltages than the TN-system. The phase- to-phase voltages are, however, higher in the TN-system. A symmetric IT-system will actually experience zero phase-to-phase voltage, while a TN-system will experience about the same voltage phase-to-phase as phase-to-ground. Unsymmetries, ground faults or flashovers in the IT-system will disturb this picture. Permanent ground faults are quite common in the Norwegian IT-system and this will result in both high phase-to-ground and phase-to-phase voltages, compared to a TN-system.

9.3.3 Protection of low-voltage networks

This thesis has not particularly focused on protection against LIO, but some main

9.4 Principal contributions 162

characteristics of the effect of arrester have been analysed. To protect an IT-system completely from LIOs, surge protective devices must be installed at each individual installation. The level of LIOs in a TN-system is much lower and such systems are in a way self-protected against remote lightning. Still, high LIOs can occur due to nearby lightning (< 100 m) and in such a situation surge protective devices are required at each installation in the TN-system as well. The installation of arresters has also an economical aspect and an optimum protection requires statistical analyses of the occurrence of the LIO, where parameters like the lightning current shape and location distributions are taken into account along with the stroke density (#/km2-year). This will also require the cost of arrester installation to be compared with the cost of faults in the system. Such analyses are beyond the scope of this work.

Arresters should preferably be installed at the power service entry or in the main distribution board / meter cabinet. Connections to ground should be as short as possible. The arrester’s rated voltage and energy capacity should be selected properly, but energy stresses are likely not a problem in relation to LIOs.

Even if arresters are installed at the power service entry, large overvoltages can arise inside the LVPI network. Oscillations due to reflections in the low-voltage system and with frequencies dependent on overhead line segment lengths could excite the natural frequency of connected LVPI circuits. LVPI circuits have different lengths and probably at least one will have a critical length resulting in large internal overvoltages. Such overvoltages can have amplitudes, according to simulations, of at least 3-4 times the protective level of the connected arrester. The overvoltages in the TN-system are limited by the connected loads, however. The internal voltages will be lower for shorter LVPI circuits. Underplaster wiring results in higher internal voltages than surface wiring due to its larger characteristic impedance. Secondary protection inside LVPI circuits requires careful considerations related to rated voltage of the primary and secondary protection along with the series impedance between the two. A situation where the secondary protection absorbs most of the energy must be avoided since the secondary protection normally have lower energy absorption capacity.

The lossy ground effect implies that installing an arrester at the distribution transformer in an IT-system actually can result in increased overvoltages at the subscribers’ installations.

9.4 PRINCIPAL CONTRIBUTIONS

This thesis has to some extent focused on the application of already established models, with the objective to study LIOs in the total low-voltage system. However, new models and knowledge have also been obtained and the principal contribution of this work can be divided in three parts:• Calculation of LIO in low-voltage systems. This implies definition of the term inducing

voltage either formulated analytically in the time domain or in the frequency domain by usage of a vector potential formulation. A model for calculation of LIO in extended power

9.5 Suggested future work 163

systems is implemented in the widely used and recognised electromagnetic transients simulation program, ATP-EMTP.

• Handling of lossy ground effects. This implies incorporation of the lossy ground effect in LIO calculations. This is accomplished by adding an extra term to the vector potential formulation of the inducing voltage. Since a short overhead line normally can be assumed lossless, this enables inclusion of the lossy ground effect in the ATP-EMTP model. The Norton’s and the surface impedance methods are evaluated and compared.

• Modelling of transformers and LVPI networks. Measurements have been performed on transformers and LVPI networks and models of the two types of overhead line terminations are developed. The influence of various parameters like transformer ratings, installation size and wiring, electrical equipment and neutral grounding has been particularly addressed. Models of transformers and power installations have been established and the LIO’s dependency on natural variations in those investigated.

9.5 SUGGESTED FUTURE WORK

The research activity within the field of lightning-induced effects is intense. This seems to be necessary since the number of LIO problems is increasing, due to an increasing use of sensitive electronic equipment and the prediction of the level of LIO in a power system to obtain optimal protection is challenging. Next, some missing links observed by the author during this work are summarised.• Lightning channel models. The distributions of charge and current along the lightning

channel are unknown. Apparently, none of the existing models are absolutely capable of predicting these distributions based on measurable quantities.

• Overhead line / Coupling models. The transmission line approach is subjected to some uncertainty regarding its adequacy, particularly in end stroke configurations. This item should be clarified.

• Lossy ground models. Apparently, a relatively high ground conductivity must be assumed in order to reproduce measurements by calculations. This could be due to inaccuracies in the models of the electromagnetic field attenuation or in the model of the ground itself.

• Measurements of horizontal electrical fields. Field measurements have so far mainly focused on the vertical field component. More measurements of the horizontal field component should be performed, since it plays an important role, particularly when taking lossy ground effects into account.

• Measurements of LIOs in actual low-voltage systems. Extensive measurements of transients in actual systems exposed by nearby lightning should be performed in order to verify the calculation results.

• Internal overvoltages in LVPI installations should be investigated further, since large LIOs can occur due to voltage oscillations at the power service entry. Co-ordination of arresters in a low-voltages system should also be further addressed.

10CONCLUSION

Induced voltage from nearby lightning is a main source of faults and damages in low-voltage overhead line systems. The number of LIO problems is increasing, due to an increasing use of sensitive electronic equipment in an increasing number of electrical installations. This is in contrast to the increased use of surge protective devices and underground cables.

This thesis deals primarily with calculations of lightning-induced overvoltages (LIOs) in low- voltage overhead line systems with the objective to enable the design of a proper overvoltage protection. All the models used in this thesis are believed to be fairly accurate for the first few microseconds, which normally is sufficient for prediction of the maximum induced voltage in the system. The main frequency range of interest related to LIOs is 10 kHz - 1 MHz.

The return stroke is modelled by the MTL model with the 1L model as a special case. The electrical field components are sensitive to the lightning channel model, but the induced voltage is less sensitive. The two models manage to predict the first few microseconds of the LIOs. The contribution from the leader can be ignored if the system has a connection to ground.

Agrawal’s coupling is used to calculated induced voltages in overhead lines, knowing the incident fields. The validity of this model has not been particularly addressed.

The ground conductivity has little influence on the vertical electrical field from a lightning channel, but a large impact on the horizontal field. The attenuation of the electrical field components propagating over lossy ground is investigated using Norton’s or the Surface Impedance methods. Norton’s method is found to be accurate for all practical conditions, also for nearby lightning. The Surface Impedance method is also found to accurately predict the peak values of LIOs in most cases, except for very close and steep lightning strokes.

Measurements on a scaled-model of lightning-induced voltages have shown qualitative correspondence with calculations (the main characteristics are reproduced). To achieve quantitative agreement, however, a relatively high ground conductivity must assumed, compared to what is measured at low frequencies. The reason for this has not been fully revealed.

Measurements have been performed in order to develop models of distribution transformers and low-voltage power installation (LVPI) networks. Simple models of "typical" transformers and LVPIs are developed for calculations when specific data are unavailable. The practical range of values and its influence on the LIOs is investigated. A transformer with a grounded LV neutral can be modelled with a small inductance (4-40 pH) closely related to transformer rated

10 Conclusion 165

power and voltage. As a first approximation, LVPI in TN-systems can be modelled as a small inductance (2-20 pH) and in the IT-system as a capacitance (20-200 nF) in series with the inductance found in the TN-system. The influence of type of wiring and apparatus is analysed.

All the models in this work are synthesised to enable calculation of LIOs in practical low- voltage configurations using the electromagnetic transients program ATP-EMTP. The influence of various parameters is investigated and the following conclusions regarding the LIO amplitude can be drawn:• The LIOs increase proportionally to the return stroke amplitude in an unprotected system.

Larger rise times of the return stroke results in lower LIOs.• The introduction of lower terminating impedances by connecting e.g. more LVPI

networks results in lower LIOs. Thus the amplitude of the LIOs is likely to be higher in rural areas with few subscribers’ installations. The actual model of the distribution transformer has a minor effect on the maximum LIO in a low-voltage network as long as the LV neutral is grounded. If it is isolated the LIOs increase considerably.

• An IT-system results in much higher LIOs phase-to-ground than a TN-system. This can partly explain why the number of transients and damages is large in Norway compared to e.g. Sweden. TN-systems result on the other hand in larger phase-to-phase voltages than IT-systems. However, an IT system with a permanent ground fault will experience both high phase-to-phase and phase-to-ground voltages, compared to a TN-system.

• The LIOs are proportional to the line height when the ground is assumed lossless. The additional contribution from a lossy ground is almost independent of the line height. Lightning strokes near the mid-point of an overhead line (side stroke) result normally in the largest LIOs and in a minimal lossy ground dependency.

• Lossy ground effects on LIOs seem to be very important, particularly for IT-systems. The ground losses may reverse the polarity and increase the amplitude of LIOs. However, the effect of a lossy ground is encumbered with uncertainty since a relatively high ground conductivity must be assumed in order to reproduce measurements by calculations. The lossy ground effect is particularly pronounced for a lightning stroke at the prolongation of an overhead line (end stroke) since the LIOs in such configuration is strongly dependent on the horizontal electrical field. The lossy part of the induced voltage is approximately proportional to the square root of the ground resistivity.

• To protect a low-voltage system completely from LIOs, surge protective devices must be installed at each individual installation. The level of LIOs in a TN-system is much lower, and such system is to some extent self-protected against remote lightning. Even if arresters are installed at the power service entry, harmful overvoltages can arise inside the LVPI network. These are caused by oscillations, due to reflections at the overhead line ends, which excite the natural frequency of connected LVPI circuits. Such overvoltages can reach amplitudes of several times the protective level of the connected arrester.

It is finally necessary to realize that the calculation of LIOs is intricate due to the high complexity of both the lightning flash and the low-voltage system. It is impossible to calculate the LIO with a high accuracy, but the main characteristics of the LIOs in a low-voltage system are feasible to obtain.

APPENDICES

A. Handling of current waveforms 167

A. HANDLING OF CURRENT WAVEFORMS

The assumption of a step lightning current is not a good approximation to a real situation. A large amount of measured lightning current data at ground is available [15,16]. A very common model is to approximate the lightning current by two exponential functions, one for the front and one for the tail. An other model is suggested by Heidler [96]. This model given in (A.1), is very convenient since the amplitude is directly adjustable and the initial time derivative is zero.

I (r/T,)™z(0,f) = — •-------------exp(-f/T) (A.1)

« (f/x1)m +1

where Im is current amplituder, is front time constant r2 is decay time constant m is an exponent (2.. 10)n is an amplitude correction factor

The current waveform in (A.1) can be taken into account by performing a convolution integral of the inducing voltage UirJt) which is the step response of a lightning return stroke current I0. This gives:

(A2)

The current in (A.1) has the derivative

m/t _ 1W"+i \

difit

(A.3)

The default current shape, used in most examples in this thesis, is shown in fig. A.1. It has theparameters:• Im = 30 kA• t, = 2 ps• r2 = 50 ps• m=5This results in a front time (30-90 %) of 2 ps, and a half value time of 40 ps.

The positive direction of the lightning current is upwards. This implies that a positive lightning current brings negative charge to the ground (or positive charge to the cloud). This reference direction is the opposite of what is normal in lightning current research, but common when using the antenna model. It is convenient to use the same direction for both current, field and return stroke propagation.

Appendix B. Effect of a tilted lightning channel 168

l[kA]

5 10 15 20 25 30 35 40 45 50

Fig. A.1 Default lightning current shape.

B. EFFECT OF A TILTED LIGHTNING CHANNEL

In the simplified developed model the lightning channel is assumed to be vertical. This is, however, not strictly correct according to observations. The lightning channel follows an irregular path. This will affect the electric fields and induced voltages.

Simple model

We here assume that the lightning stroke follows a straight line with an angle 6 to ground and where the ground projection of the lightning channel is perpendicular to the overhead line. The configuration is shown in fig. B.l and the TL model is assumed with a step current (qR = 7/v).

\channel

Fig. B.l Channel path configuration.

The scalar and vector potential is given by:

4r r ds 47ce«

v (B.l)

Appendix B. Effect of a tilted lightning channel 169

A = f *4% {a,

where

Rs = yV2 +z2 +i2 +2-s-y and

r = sm/v + Rs/c gives

_ t-c + v/c• y-\i(t-v+y)2 +(l-v2/c2)-(r2+z2-y2) and

™ c/v-v/c

Y = ycos0 -z-sin6

Equations B.1-B.5 gives:

F = —-In 4tte„

/ s •(! -c/v) +f-c +

1 = i^-ln4tt

y +yV2+z2

s "(1 -c/v) +fc + Y

„ Y +Vr2+z2

The scalar and vector potential results in electrical field components on the form

Ev = -dV

8zlx sin0

Z =0 4 ne„

1 tv+v2/c2-y0+JA0 1

^+Y0+V^ y«+r

•sin6 = - 0 . v

" Awhere

Yo = j'cosS and

= (f'v +Yo>2 +(1 ~v2/c2y(r2 ~Yo)

Inserting the relationship ^ = /„/v in (B.8) and (B.9) gives the total vertical electric field:

(B-2)

(B.3)

(B.4)

(B.5)

(B.6)

(B-7)

(B-8)

(B.9)

(B.10)

(B.11)

£z = Er+EA = 30-J0-—-sin01 (t’v + JAoy(l-v2/c2)

sJAq fv+^A^+y cos0 r+ycos0012)

Fig. B.2 shows the vertical electrical field as a function of the angle 0.

Appendix C. Inverse Fourier transform 170

• r =y = 200 m• c/v=3

Parameters:• 7„=1000 A (step)

■120—

-1.6000 2 4 6 8 10 12 14 16 18 20

tins]

Fig. B.2 Vertical field Ez with the lightning channel incident angle 6 as parameter.

Published literature

Studies of more complex lightning channel paths are reported in the literature. Le Vine et.al. [130] have calculated fields from a tortuous lightning path, but the dynamic behaviour of the lightning channel (current wave with finite speed) was not included. The effect of a lossy ground was in addition modelled by Norton’s method. Also Vecci et. al. [131] have used what they called a fractal approach to study the effect of a random lightning channel path on vertical electric field. Both these publications [130, 131] found that a tortuous path results in high frequency components in the calculated field. Michishita et.al. [108, 132] studied the effect of a tilted lightning channel on induced voltages in a scaled model.

C. INVERSE FOURIER TRANSFORM

This appendix explains how the discreet inverse Fourier transform (DIPT) is calculated. The functions to be transformed to the time domain are calculated on a logarithmic frequency scale but with an equal frequency spacing inside each decade. Ten samples per decade are normally sufficient

The sine inverse transform shown in (C.l) is used:

f(t) = - — • fS[j/D"to)]-sin((Of)-d<i>TZ J 0

(C.l)

Eq. (Cl) can be written on discreet form

(CJ2)

Appendix C. Inverse Fourier transform 171

The sum is taken from a lower frequency of = 1 O'4 Hz (co0 = 0, a), = 2rtfmJ and up to a maximumfrequency of = 107 Hz («N = 27ifmaj.

The function f(ja>) is assumed to be linear between each sample point By using this assumption the integral in (C2) can be calculated analytically.

$t/(/<o)] = a +6i-(to-(0,_1)

ai =

b = il

(C.3)

(C.4)

(C.5)

",I. = J [«.+6.-(to-a).„1)j-sin(tor)-rf(o =

",-i

- —•|cos((o.,f)-cos((o._1 -r)j

+ • |sin(G). • t) -sin(G)f _j • f)j

- -j-- cos(G). • t) • (0). - CO,.,)

The time domain function then becomes

AO = E/,TZ j=i

The function to be transformed is multiplied by a factor

, sin(lt •«/(«)„)= -------------------------------

7I-0)/0)w

to reduce the contents of Gibbs oscillations in the time domain solution.

(C.6)

(C.7)

(C.8)

The function to be inverse Fourier transformed is also multiplied with a phase factor

FTa = exp (j'k0-)JxA+y2+z2) ^C‘9^

which represents the minimum time difference between the transmitter (lightning channel base) and the receiver (overhead line termination). This will smooth the function in the frequency domain, allowing a minimum of frequency samples to be used in the calculations.

Appendix D. Properties of Uind(j<j) 172

D. PROPERTIES OF Uind(jw)

The inducing voltage UirJj(o) is in this appendix analysed in the frequency domain. The purpose of this analysis is to reveal its properties to ensure a correct and accurate inverse Fourier transform. The analysis will particularly focus on the limits as w approaches zero or infinity.

The inducing voltage at the end A (L=xA-xB) is given by:

dAa(xA,°) _ BAo(xa,0)

dz dz (D.l)

+/m-z-(^o(xv0)-v4o(xs,0)-e'Y"L)

where Aais the lossy ground vector potential in z-direction which can be decomposed into Aa=A0+AA.

The lossless inducing voltage, U°indA, dependent on A0 and the lossy ground contribution, UAindA, dependent on Aa are further analysed separately, since they show somewhat different behaviour in the frequency domain.

D.l Lossless contribution IfuM

The lossless inducing voltage can be expressed as:

V

where the vector potential is written in (D3)

.1 T H W,(D.3)

The partial derivative of the vector potential with respect to the line height z can be written as

\

The limits as <n approaches zero are:

(D.5)

Appendix D. Properties of Uinj(ja)) 173

dz 4% ia> J<o-0 4 it yco

(z-h z+k ■e ~h,x-dh (D.6)

When the overhead line is lossless (y=jco/c):

lim UindA(ju) =30•/, XA H

yco

z-h z+h ■e^-dh-dx =yco

When both AT and X approach infinity, K0 approaches 0.

When the overhead line is lossy:

(D.7)

(D.8)

The limits as co approaches infinity are:

^C/W)lim^o =

O'")2A |Z.(/(o)|<» (D.9)

r 5A lim<0— oe dz

o _ ZJJV) yco

A |JU>co)|«» CD 10)

This gives:

tonuLuO'u) = lim yco-z-[^0(^,0)-^0(xB,0)-e'Y i] = *.(/*>)yco

A irjyw)i- (D.ll)

Eq. (D.7 and D.ll) shows that the inducing voltage is well behaved as the frequency approaches zero or infinity.

Fig. D.l shows the lossless inducing voltage in the frequency domain.Re(Uind) [uV] Im(Utnd) [mV]

Fig. D.l Lossless inducing voltage l?indA(jio), xA=750 m, xfl=250 m,y=0 and z=6 m. Lightning current: step of 1 A, v=l.V108 m/s and A=.

Appendix D. Properties of UMQ(o) 174

D.2 Lossy ground contribution UAindA

The lossy ground contribution to the inducing voltage can be expressed as:\

(D.12)

+>>• Z• [^A(xA,0)-Aa(xb,0)• e 'Y 1 ]

where Aa is given in (D.13):

0(D.13)

The lightning current is assumed to be a step function with amplitude I0, travelling from ground at a velocity of v and attenuated exponentially with height with a factor X. H is the total height of the lightning channel.

The partial derivative of Aa can be written on the form given in (D.14), since 3/8z = 3/3h (reciprocity principle).

Aj T H a

(D.14)

o

where f0,f, and fp are given in chapter 4.

Before an inverse Fourier transform of the UA ^(jv) function can be performed, the function must be analysed closely to reveal its properties particularly at the limits to-O and to-”.

to-0:

y"toeo

]'<*% j<x)Tz = yw 21 (D.15)47t '\ a \ 2c c2 N o

Appendix D. Properties of UM(j(o) 175

Inserting this into (D.13) and (D.14) gives

lim^lA = (0-0

, -aa .

r,(%)

dh (D.16)

co-0 OZil.i

c‘ N O r(r2+(H+zf)m ~(r2+z2)m+-Je ~h‘\r2 +(z+hf)m■ dh (D.17)

=

The limit of U^maQcj) as w approaches zero is dependent on the stroke location. For a side stroke (xA = - xB) the first two terms in (D.12) cancel each other for low frequencies. This gives:

• Side stroke:

lim ULa = Iim-7-

CO-0 CO-0 yvu

16^-y f —^(x,z)-e y (*A x)-dx J dz

CD 18)

Lossless overhead line:

lim U, ,. = -c 'fa*)■ dx CD 19)

Lossy overhead line:

lim U, co-0

indAc2-jK-c„

V/mdx (D. 20)

► Not side stroke:,2

(D.21)CO-0

Equations CD.19-D.21) show that the lossy ground contribution, UAiruU(j(o), to the inducing voltage is well behaved at low frequencies.

Appendix D. Properties ofU[nJQ(i>) 176

fi -

f. "2 cr2

js/p’Tt \ A0/?,TC z+h+RJsfty

-juRt/cfo'A'fp = -

27C (z+hyfi.*^(D.23)

Inserting this into (D.13) gives

A)limv4A = r.yO) 271 { (z+AI-yr+JZ, O'm)2

(D-24)

nrn^ •^0 . Fpyco 27t

-ja-(Hh*Rglc) ^-ja-RJc e ~iu>'V’lv*Rilc)

(z+H)- <Jer+RH z-Je+R0 v J0 (*+A)-^+*,W /•_£____v { Cz+fcV.

■dhKa q<a)

yco(D25)

where |K3(jco)| < °° og IK^Qg))! < «=

Inserting (D.24-D.25) into (D12) gives

11111 uLu(jv>) = Mm yco-z-^^,0) ->40(xs,0)-e "r£] = —A |£„O<0)|«» (D.26)yw

As seen from the above equations, the function UAirtdA(jco) is well behaved at the limits to-0 and o)-°° and an inverse fourier transform can be carried out straight forward.

Fig. D.2 shows the lossy ground contribution UAimiA(ja>)111 the frequency domain..

0.3ffMHz]

Im(Ulnd) [uVJ --- 120 |---

Fig. D.2 Lossy ground inducing voltage UAindA(ja)), xA=15Q m, xs=250 m, y=0 and z=6 m. Ground parameters: o, = 0.001 S/m and erl = 10. Lightning current: step of 1A, v=l.l-108 m/s and X=<».

Appendix E. Adequacy of the telegraph equations 177

E. ADEQUACY OF THE TELEGRAPH EQUATIONS

E.l First equation dV/dx + ZT = S^x)

Agrawal’s coupling model is developed for calculations of induced voltages between two conductors. When calculating the induced voltages between a conductor and a ground plane the coupling model must be modified somewhat. The deduction of the modified model is based on the Maxwell’s equation

(E.l)

If the introduction of a conductor is assumed to not influence the incident fields from the lightning channel, the theorem of superposition can be used. The total field is equal to the incident field plus the scattered field. Performing the integration of the scattered field along a closed loop gives:

AX x+Ax0

Dividing by Ax and taking the limit as Ax approaches zero gives:

(E.3)oo

Further the scattered fields are assumed to be transverse magnetic and the conductor to be thin with a radius rc« z. The scattered voltages is now by definition:

(E.4)o

According to [133] the horizontal electrical field at ground and the magnetic field integral can be written:

h(E.5)

o

where Zg is given by [133] equivalent to Carson’s formula (where the displacement current is ignored):

(E.6)

and Z0 is the lossless series impedance


whererc and z are the radius and height of the conductor,o, and e, are the conductivity and the permittivity of the ground.

The scattered electrical field at line height z is equal to:

E^(x,z,Oi) = to) (E.7)

where Zz is the internal impedance per unit length of the conductor and Ej is the incident field along the conductor. The impedance Z, can in most cases be ignord.

Equation (E.5) is the main deviation from the deduction in [73] where the reference is assumed to be a conductor and the scattered field at this conductor thus becomes similar to (E.7).

Inserting (E.4), (E.5), and (E.7) into (E.3) results in

(E.8)

E.2 Second equation dl/dx + Y V = S2(x)

The deduction of the second coupling equation for the scattered voltage and the current in a conductor ground surface system follows the argumentation in [73].

(E.9)

Assuming all current to flow along the conductor this gives:

6/(*,CJ) +ju-Q(xM = 0 ox

Further, the charge can be related to the scattered voltage and the capacitance per unit length:

6(x,0)) = C'-Vs(x,u)

This results in the second telegraph equation:

dx(E.12)


EJ Discussion

The main problems with the deduction so far are related to equations (E.5) and (E.ll). Agrawal et. al. [73] assume that the total horizontal electrical field at the reference conductor contributes to the reference return impedance. This is reasonable when the return path is a conductor but becomes doubtful for a ground return. Still using the argumentation in [73] (E.5) could be written:

+/o>- jBy{x,zM)'dz = (Z0 +Zg)-I(x,<o) (E.13)o

This results in the following modification of the 1st telegraph equation in (E.8):

drs(x,u>) +(Z+Zo+Zg).I(X'U) = Ej(x,z,a>)-Ej(x,0,ai) (E.14)

When the ground is ideally conducting (E.14) and (E.8) are equivalent, but for a lossy ground the difference is important. Using (E.14) will give an ignorable lossy ground effect on the induced voltage compared to (E.8).

It is possible that the real solution is a combination of the equations (E.8) and (E.14) and equation (E.15) have shown to accurately reproduce the measurements obtained in chapter 5.

E^x, 0,o)dV\x,<s>)

dx+(Zi+Z0+Z )-I(x,a>) = E^x,z,o) -- (E.15)

If one assumes that both the total electrical and magnetic field contribute to the return impedance:

-Ejfx, 0, to) +/G) • J"By(x, z, GO) • dz = (Z0 +Zg )•/(*, w) (E.16)o

the Chowdhuri model [72] is obtained:

d[r ‘(x, V \x, to)] + ^ +2, +Zg)•/(*, co) = 0 (E.17)

Eq. (E. 11) relating the charge and voltage is based on assuming that the transverse electric field in any point x=constant is directly related to the charge at that point:

V-2?dz dy dx

(E.18)

However, the horizontal scattered field is strongly dependent on the incident field as given by (E.7). In the cases where the incident horizontal field varies much along the overhead line (like e.g. in an end stroke situation) the assumption in (E.18) becomes doubtful as pointed out by Agrawal et.al. [73]. This also applies to the lossless situation.

Appendix F. Fitting of admittance measurements 180

Tkatchenko et.al. [134] have formulated modification terms to the Agrawal coupling model that take a finite line length and the distribution of charge and current along the line accurately into account. Assuming the line terminals to be open and the exciting field to be a plane wave they managed to calculate the current distribution along a small-scaled model line and compare this with the results from using the Agrawal coupling model. The deviation between the two approaches increases with grazing angles of incidence (towards end stroke).

E.4 Conclusion

Two main uncertainties remain:• How does the horizontal incident electrical field at ground, Ej(Q) influence the induced voltage

between an overhead line and ground?• Since the incident horizontal electrical field at line height for some configurations varies

extensively along the line, can the line charge be related to the scattered voltage?These two considerations imply that the telegraph approach is questionable in two situations:• For low ground conductivity• For configurations where the incident field varies extensively along the overhead line, e.g. an end

stroke configuration.The measurements compared with calculations in chapt. 5 indicate that calculations give too high losseffects. Or with other words, a high ground conductivity must be assumed in order to reproduce themeasurements. The deviation between measurements and calculations is largest for the end strokeconfiguration.

F. FITTING OF ADMITTANCE MEASUREMENTS

F.l. Establishing the fitting function

In this appendix a measured function FJjcS) is approximated by a fitting function F(jco). Whether the measured function is an impedance or an admittance is only of practical interest at the final step of the process where the fitting function is represented by RLCG branches. The fitting function consists primarily of complex conjugated poles and zeros along with an optional real pole or zero.

The simplest form of the matematical function is

F(s) =K0-(s-zoy<

{s~p0ft-i % 1 = 1

(F.l)

where np and nz is the number of complex conjugated poles and zeros respectively and where K0 is a constant, p0 is a real pole, z0 is a real zero. \ is equal to 1 or 0, which implies that the fitting function either has a real pole or a real zero. The complex zero is zt = az,+ j 6z, and the complex pole is pt = ap, +j‘bpr


The order of the fitting function (F.l), np and nz is determined by an examination of the absolute value of the measured function. Zeros and poles are identified as minima and maxima respectively, as seen in fig. F.l. In those cases where F(s) has a real zero the number of poles should be one larger than the number of zeros. In the case with a real pole the number of poles and zeros should be equal.

In addition to these zeros and poles identified in fig. F.l, a value of the measured function at a low frequency is used to determine Kg and p0 or z0.

|F(jw)|

Abs(F) [dBJ

x pole........ O zero

r-o

Fig. FI Absolute value of measured function and position of poles and zeros.

Fitting the function (F.l) (determine the variables az, bz, ap, bp, K„, z0 andp0) is often done by a nonlinear minimisation technique (least square method) [113,114,135]. This ensures an overall good fit even though the order and form of the fitting function is inaccurate. Here, a more simple method is used instead, based on requiring the fitting function to be exact at every extreme value. This method gives very good results if the fitting function is of correct order and the measurements are smooth and consistent.

To determine the constants in (F.l) the following error function is defined

E(s) = Fm ■ (s -p0)1 ■ h(s -p,.)■(» ~Pi) ~ K^(s -z0)1 • II(s -z)-(s -z,‘) = 0 C*7-2)1=1 i=1

for all s=j-C0j, i- 0..np+nzand where w, (i*0) are the angular frequencies where the absolute value of the measured function Fm has minima or maxima and <»>0 is the additional low frequency.

The equation in (F.2) can be solved iteratively by e.g. Newton's method

J-dx = -E (F.3)

where dx is the change in the x-vector

Z = [xi-Xnp+n+if = [Ko’P^zo’a^->azm,bzv..,bznz,apv..,apnp,bpv..,bpnpf (F.4)

£ is the error function in (F.2) for all the frequencies W;


and J is the Jacobian

9t

a

ragq-a>0)|dx, I

fd£(/'-O)0)l

dx.np +nz+2J

aJE^eWl .. aBy=>dx,

dx,np+nz+2

dx,

Idx,np+nz+2 J

dx.

[d£C/'to0)].. @4

dxnp+nz+2

(F.5)

(F.6)

The starting values for the parameters are

„o _ 1,0,0,..,0,(0 ,0,..,0,O),Pi •"r-f (F.7)

This means that all the real parts of the zeros (az) and poles (ap) are set to zero and all the imaginary

parts (bz and bn) are set to the angular frequency where the pole or zero appear. The konstant K0 is set to unity and the real pole p0 or zero z0 are set to zero.

After calculating the error according to (F.5) and the Jacobian according to (F.6), the equation (F.3) is solved by LU-factorization to give the dk-vector. The parameter vector, x, is then updated. The solution is found when the iteration process converge.

The iteration process can be described in steps shown below:0. Perform measurements to find Fm as a function of frequency.

Assure that the function is smooth with distinct poles and zeros, as shown in fig. F.l1. Identify the model.

Find the number of poles, np and zeros, nz Store the frequencies where this occure, co,.. cov+ri..Estabish the initial values given in (F.7).Iteration n=0

2. n=n+l3. Calculate the error £ (F.5) and the Jacobian J (F.6)4. Solve the equation J-dx = -E to find dx.5. Update the parameter vector

x” = x"-' + dx6. If the norm ||dx||>10~10 goto 2.7. Realise an equivalent RLCG-network.

F.2. Realising the fitting function

The fitting function F(s) in (F.l) can be written [114]:


m = w+E1=1

b, | 6,

s~Pi *~P,(F.8)

Fg(s) is dependent on the value of £. If

M: W = —L" *-Po

In this case np should be equal to nz.

( b}(=0: F0(s) = 2-£9t -

i=1 \pi,and where 6, can be found from the residu theorem

In this case np should one larger than nz.

6, = lim {F(s)-(s-/?,)} (F.9)s-p,

If the fitting function in (F.8) is an admittance (in this case typically a transformer). The first term in (F.8) due to the real pole, can be synthesised by a simple RL-branch as shown in fig. F.2 with Lo = l/b0 " (F-10)Rg = -po'Lg. (F-ll)or in the case where £=0, G0 = 1/Fg(s)

MLs+R/L

Fig. F.2 RL-equivalent

Ci Gi

m =slL+GI(L-C)

s2 +(R/L +G/C)'s +(1 +R-G)/(L-C)

Fig. F.3 RLCG-equivalent

The two terms inside the braces in (F.8) can for each complex pole, z, be approximated by a RLCG- branch shown in fig. F.3 where

L. =' 2-91(6,.)

Rf = 2•£,•[-9t(p,) +%(6 -p’)■ L,

1

L.' IP,.|2 +2-91(6 ■p'yRi

(F.12)

(F-13)

(F.14)

Appendix G. Admittance measurements on and fitting of transformers 184

ot = -2 ■m{b:P;yc-Li (F. 25)

Fig. F4 Equivalent circuit. Admittance, £=1.

G. ADMITTANCE MEASUREMENTS ON AND FITTING OF TRANSFORMERS

G.l Neutral isolated

This transformer coupling results in a simple admittance behaviour for low frequencies. Thus usage of the high frequency measurements alone is a good basis for fitting. The order of the model is kept as low as possible. Poles and zeros above 5 MHz are mostly ignored.

T01:ABS(Y-PK)

0.5 -

0.2 -

0.05 —i

0.02

0.005

0.002 -

0.001

f [MHz]

Equivalent RLCG-network: Go= 1.862 nO L R C14.10 pH -4.535 £2 4.7953.385 pH 9.528 D 6.167

nFnF

G0.544 mU

-5.299 mU

T02:ABS(Y-PK)

0.05

0.020.01

0.005

0.0020.001

Equivalent RLCG-network: Go= 0.931 nU IRC 82.92 pH -965.0 Q 0.0723.100 pH 6.949 D 7.296

nFnF

G0.902 mU

-6.664 mU


T03:ABS(Y-PK)

Equivalent RLCG-network:Go---- 0.233 nOL R C4.708 pH 1.578 Q 13.24 nF10.56 pH 32.15 Q 1.413 nF

G2.470 mU

-2.280 mU

T04:ABS(Y-PK)

Equivalent RLCG-network: Go= 0.698 nVL15.35 pH 14.32 pH 7.816 pH

R4.3665.92519.71

C1.8951.1920.975

nFnFnF

0.588 mU 0.611 mO

-1.168 mO

T05:ABS(Y-PK)

Equivalent RLCG-network: Go= 0.175 nOL12.70 pH 4.296 pH

R6.133 Q 5.619 D

C10.81 nF 5.597 nF

-0.719 mU 0.725 mU

T06:ABS(Y-PK)

1

0.0020.001

Equivalent RLCG-network:Go= -0.116 nU L R C87.57 pH -279.5 Q 0.213 nF3.310 pH 4.075 a 3.034 nF

0.937 mU -1.263 mU

1 4 5


T07:ABS(Y-PK)

0.05

0.005

0.0020.001

f [MHz]

Equivalent RLCG-network:Go= -0.466 nU L R C14.21 pH -1.305 Q 3.370 nF3.439 pH 9.800 Q 3.675 nF

G3.219 mU

-3.133 mU

T08:ABS(Y-PK)

0.05

0.02 —■/

0.005

0.0020.001

f [MHz]

Equivalent RLCG-network:Go= -0.204 nUL R C4.529 pH 7.938 Q 4.732 nF21.09 pH 161.1 Q 0.070 nF

G0.376 mU

-0.354 mU

T09:ABS(Y-PK)

M eas

Fit

0.05

0.02

0.005

0.0020.001

f [MHz]

Equivalent RLCG-network: Go= -0.524 nUL3.902 pH 7.480 pH 25.97 pH

R3.587 D 49.42 Q 712.4 D

C4.095 nF 0.258 nF 0.019 nF

G1.513 mO

-0.839 mU -0.434 mO

T10:ABS(Y-PK)

Equivalent RLCG-network: Go= 0.087 nU L R C5.721 pH 6.510 Q 3.52315.35 pH 57.34 D 0.113

nFnF

G0.149 mU

-0.148 mU


Til:ABS(Y-PK)

0.05

0.02

0.005 -y

0.0020.001

Equivalent RLCG-network:Go= 0.073 L7.170 pH 9.986 pH 18.94 pH

nUR6.671 Q 32.48 Q 194.6 Q

C3.119 nF 0.249 nF 0.041 nF

G0.470 mU

-0.245 mO -0.212 mU

T12:ABS(Y-PK)

0.05 -

0.02

0.005 -/

0.0020.001

f [MHz]

Equivalent RLCG-network: Go= 0.204 nOL7.440 pH 9.658 pH 23.22 pH

R6.622 D26.83 a 332.5 Q

3.351 nF 0.307 nF 0.035 nF

G0.554 mU

-0.171 mU -0.338 mU

T13:ABS(Y-PK)

0.05 -

0.02 -

0.01 rV0.005

0.0020.001

f [MHz]

Equivalent RLCG-network:Go---- 0.262 nUL R C30.22 pH 23.53 a 2.565 nF3.874 pH 7.346 Q 2.023 nF

G-0.450 mO

0.457 mU

T15:ABS(Y-PK)

0.5 r

0.02 -

0.005 -/•

0.0020.001

f [MHz]

Equivalent RLCG-network: Go= 0.233 nUL160.3 pH 6.446 pH 14.01 pH

R59.77 n 8.259 a 44.60 Q

C0.739 nF 1.675 nF 0.407 nF

G-0.062 mU 0.635 mV

-0.555 mU


G.2 Neutral grounded.

This transformer coupling result in a more complex admittance behaviour at low frequencies. The high- frequency measurements will primarily be used as the basis for fitting. The order of the model is kept as low as possible, thus poles and zeros above 5 MHz are mostly ignored.

T01:ABS(Y-PNK)

0.003

Equivalent RLCG-network:Lo= 7.100 pH Ro= 0.382 a L R C13.94 pH -2.206 Q 2.371 nF24.22 pH -40.25 Q 0.238 nF

G2.094 mO 1.708 mU

T02:ABS(Y-PNK)

Meas

0.03 -

0.01 -

0.003

f {MHz]

Equivalent RLCG-network:Lo= 8.693 pH Ro= 0.622 Q L R C6.776 pH 3.156 D 4.361 nF73.95 pH 1592. Q 0.021 nF

G1.118 mO

-0.377 mU

T03:AB$(Y-PNK)

Meas

0.1 -0.03 -

0.003

f [MHz]

Equivalent RLCG-network:Lo= 5.259 pH Ro= 0.512 Q L R C26.09 pH 28.52 £5 0.681 nF25.90 pH 80.22 Q 0.409 nF

G0.159 mU

-0.869 mU


T04:ABS(Y-PNK)

Meas

0.10.03 -

0.01 -

0.003

f [MHz]

Equivalent RLCG-network:Lo= 22.70 pHRo= 0.503 QL R C G12.27 pH 0.794 Q 999.4 nF 2.295 mO404.9 pH -314.8 D 1.474 nF 1.309 mU81.11 pH -23.03 Q 0.241 nF 0.248 mU17.87 pH 4.839 D 0.471 nF 0.342 mU37.33 pH -9.475 Q 0.077 nF 0.153 mU

T05:ABS(Y-PNK)

Meas

0.03 -

0.01 -

0.003

f [MHz]

Equivalent RLCG-network:Lo= 8.807 pH Ro= 1.054 Q L R8.621 pH -2.010 £5 29.06 pH 103.2 Q

C3.500 nF 0.288 nF

G3.956 mU

-0.559 mU

T06:ABS(Y-PNK)

Meas

0.1 -A

0.03 -

0.01 -

0.003

Equivalent: RLCG-network:Lo= 7.727 pHRo= 0.447 QL R C G7.549 pH 1.354 a 2.040 nF 0.9793 mO119.9 pH -621.3 Q 0.014 nF 0.0937 mU71.08 pH 121.9 Q 0.022 nF -0.0029 mU

T07:ABS(Y-PNK)

Meas

0.01 -

0.003

Equivalent RLCG-network:Lo= 7.904 pH Ro= 0.566 Q L R C11.84 pH 0.302 £2 1.646 nF26.15 pH -69.92 D 0.134 nF

G1.231 mU 0.896 mU


T08:

ABS(Y-PNK)

0.03 —

0.003

0.001

Equivalent RLCG-network:Lo= 9.683 pH Ro= 0.582 QL R C13.40 pH 28.28 Q 0.458 nF29.85 pH -249.6 Q 0.032 nF

G0.444 mU 0.352 mU

T09:ABS(Y-PNK)

Meas

0.003

f [MHz]

Equivalent RLCG-network: Lo= 10.70 pHRo= 0.506 QL R C G9.062 pH 9.776 Q 0.569 nF 0.234 mU54.20 pH 452.7 Q 0.025 nF -0.142 mO14.42 pH -160.3 Q 0.062 nF 0.913 mU

T10:ABS(Y-PNK)

Meas

0.003

0.001

Equivalent RLCG-network:Lo= 10.26 pH Ro= 0.536 QL R C10.06 pH 22.70 Q 0.511 nF17.31 pH -9.697 Q 0.055 nF

G0.450 mU 0.249 mU

Til:ABS0f-PNK)

Equivalent RLCG-network:Lo= 17.18 pH Ro= 0.863 Q L R C11.62 pH 12.91 D 0.509 nF13.58 pH -13.55 D 0.097 nF

G0.274 mU 0.367 mU

0.001


T12:ABS(Y-PNK)

Meas

0.003

0.001

f [MHz]

Equivalent RLCG-network: Lo= 17.94 pH Ro= 0.910 QL11.55 pH 14.58 pH

R8.935 Q6.935 Q

C0.561 nF 0.107 nF

G0.305 mO 0.310 mU

T13:ABS(Y-PNK)

0.003

0.001

f [MHz]

Equivalent RLCG-network:Lo= 17.91 pH Ro= 0.880 Q L R C6.503 pH -0.094 Q 1.662 nF64.53 pH -1000. D 0.017 nF

G1.885 mO 0.348 mU

T14:ABS(Y-PNK)

0.005

0.0020.001

f [MHz]

Equivalent RLCG-network: Lo= 29.26 pH Ro= 1.768 DL R C G11 .73 pH 10..83 Q 1..838 nF 0..069 mO9. 761 pH 13..89 Q 0..463 nF 0..273 mU18 .00 pH -26..81 Q 0..101 nF 0.,487 mU

T15:ABS(Y-PNK)

Meas

0.0020.001

f [MHz]

Equivalent RLCG-network: Lo= 31.99 pHRo= 1.479 QL14.41 pH 13.19 pH 38.62 pH

R9.581 Q

-0.370 Q -198.7 Q

C1.867 nF 0.430 nF 0.054 nF

G0.052 mU 0.709 mU 0.472 mU

Appendix H. Modelling LVPI circuits 192

H. MODELLING LVPI CIRCUITS

HI. Surface wiring

The shunt admittance, Y, and series impedance, Z, matrices can be written:

R +jwL G 1

G +jwM R +jwLY /•to-

C+K

-K

-K

C+K(H.1)

where K is the capacitance between the conductors, C is the capacitance between conductor and ground, L is the inductance in the conductor, M is the mutual inductance between conductors, R is the resistance in the conductors and G is the mutual conductance between the conductors due to the lossy ground. In chapter 7 the following values were obtained from step response measurements: vIM= 132 m/ps, ZlM= 64 Q, 0.40 ps"1vCM = 127 m/ps, ZCM = 20 Q, aCM = 0.55 ps'1

KCLee model:The KCLee model of a 2-phase symmetric surface wiring consists of the resistance, the characteristic impedance and the wave velocity in the two modes of the system. An additional scaling of the measurements is required due to the EMTP definitions of the two modes [12]. This gives:

2R. = R+G'1 = 4-CC'—— = 0.346 Q/m (H.2)

iy/(L+M)-C

'CM

R-G'1 = a„ ~*IM

40 Q

: 1.27-108 m/s

= 0.194 Q/m

(H.3)

(H.4)

(H.5)

Z. = L-MC+2-K

—^ = 32 Q 2

(H.6)

v+ = ------------- -------------- = vIM = 1.32T08 m/s\J(L-M)-(C+2'K)

(H.7)

This gives the following KCLee model written on EMTP format [12]:

C < n lx n 2><reflxref2>< R >< A >< B ><Leng>o<>0 -10A 1A 0.3460 40.01.27E8 28.8 1 00 -20B IB 0.1940 32.01.32E8 28.8 1 00


Pi equivalent:A Pi-equivalent of a symmetric 2-phase system is described by the K, C, L, M, R and G in (H. 1). The capacitances are split in two half's and positioned at each end of the circuit

Solving equations (H.2-H.7) the values of K, C, L, M, R and G"1 can be found:

K = 1 1. ------ = 19.95 pF/m (H.8)^im vm 4

C+K1 + I----- ------ = 216.8 pF/m (H.9)

^IM VIM 4 ^cm'vcm

L = 1 . Zm + Zcm _ 0.279 pH/m (H.10)4 V1M VCU

M = 1.5% + ^CM = 0.0362 pH/m (H.11)4 VIM VCM

R = 1 + 2-z

a •— = 0.270 Q/m (H.12)2 VIM VCM

G"1 - fa/ . x¥ 2'°W— = 0.0763 Q/m (H.13)

2 VIU VCM

All values are then multiplied by the circuit length 7=28.8 m. This gives the following EMTP PI- equivalent model [12]:

C < n IX n 2><reflXref2>< R XL X C >$VINTAGE,11 OA 1A 7.776000000E+00 8.036000000E-03 6.244000000E-032 OB IB 2.197000000E+00 1.045000000E-03-5.750000000E-04

7.776000000E+00 8.035000000E-03 6.244000000E-03$VINTAGE,0

The above model will give a rather bad fit to the measurements and at least two segments each of length 14.4 m are required to obtain an acceptable fit up to resonance frequency.

H.2 Underplaster wiring

The shunt admittance, Y, and series impedance, Z, matrices can be written:


C+2-K -K -K R +jwL G 1 G _1 +jwM

n e -K C+2-K -K Z = G 1 +j<x>-M R +jwL G _1 t/'wM

-K -K C+2-K G 1 +j<x>'M C'+jW-M R +j<x)-L

where K is the capacitance between the conductors, C is the capacitance between conductor and ground, L is the inductance in the conductor, Mis the mutual inductance between conductors, R is the resistance in the conductors and G is the mutual conductance between the conductors due to the lossy ground. In chapter 7 the following values were obtained from step response measurements (grounding Gl):vIM= 180 m/ps, ZIU- 175 Q, aIM- 0.25 ps"1vCM= 200 m/ps, ZCM= 120 Q, aCM= 0.20 ps"1

KCLee model:The KCLee model of a 3-phase symmetric underplaster wiring consists of the resistance, the characteristic impedance and the wave velocity in the two modes of the system. An additional scaling of the measurements is required due to the EMTP definitions of the two modes [12]. This gives:

2Rn = R+2-G'1 = 6-ar = 0.72 Q/m (H.15)

Zo

vo

*.

L+MN c

1s/(L+M)-C

R-G■’ =

3 Z^ = 360 O

= vCM = 2.0-108 m/s

za— = 0.243 Q/m

L-M N C+3-K

— = 87.5 Q 2

(H.16)

(H.17)

(H.18)

(H.19)

V --------------------------- = vm = 1.8 108 m/syj(L -M) • (C +3 -K)

(H.20)

Assuming a circuit length of 7=28.8 m this gives the KCLee model of an underplaster wired circuit, written on EMTP format [12]:

C < n IX n 2><refl><ref2>< R X A X B ><LengXX>0-10A 1A 0.7200 360.0 2.0E8 28.8 1 00-20B IB 0.2430 87.5 1.8E8 28.8 1 00-30C 1C

Pi-equivalent:A Pi-equivalent of a symmetric 3-phase system is described by the K, C, L, M, R and G in (H.14). The

Appendix I. Expressions electrical fields over lossy ground 195

capacitances are split in two halfs and positioned at each end of the circuit.

Solving equations (H.15-H.20) the values of K, C, L, M, R and G"1 can be found:

* = !• 1 1 13 Z1M VIM 9 Zcm'VCM

16.53 pF/m

C+2-K4 1 1 1

3 Z1MVIM 9 ZCM VCM= 46.96 pF/m

Z = + — = 0.924 pH/m3 vXM 'CM

M = - L’ElM. +^ V/M VCM

= 0.438 pH/m

J? = --a-— + 2-a-— = 0.402 Q/m3 "v*

G"1 = — + 2-a -—^ = 0.159 Q/m3

(H.21)

(H.22)

(H.23)

(H.24)

(H.25)

(H.26)

All values are then multiplied by the circuit length Z=28.8 m. This gives the following EMTP PI- equivalent model [12]:

C < n lx n 2><refl><ref2>< R XL x C >$VINTAGE,11 0A 1A2 0B IB

3 0C 1C

$VINTAGE,0

This model gives a rather bad fit when one Pi-segment is used. Usage of two segments, each of length 14.4 m, gives a good fit, however.

1.157800000E+01 2.661100000E-02 1.352400000E-03 4.579000000E+00 1.261400000E-02-4.760600000E-04 1.157800000E+01 2.661100000E-02 1.352400000E-03 4.57 9000000E+00 1.261400000E-02-4.760600000E-04 4.579000000E+00 1.261400000E-02-4.760600000E-04 1.157800000E+01 2.661100000E-02 1.352400000E-03

I. EXPRESSIONS OF ELECTRICAL FIELDS OVER LOSSY GROUND

This appendix contains expressions for the electrical field over lossy ground, adopted from King [99].The lightning channel is modelled by the MTL model. Kings formulas are bases on the assumptions \k,2 * 4\ » |Vl or \k,\ 2 3-\k0\. The symbols used in this appendix are equivalent to what is defined in chapt.4.

Appendix I. Expressions electrical fields over lossy ground 196

LI Horizontal electrical field in air

Sommerfeld’s exact expression for the radial field in air set up by a vertical current element I-dh at height h above a flat and homogeneous ground is

(1.1)g0-I-dh -e ,e W« e ~iKR\

+RR,o

The total horizontal field from a lightning channel modelled by the MTL model can be formulated, based on King [99]:

H

(1.2)

owhen a step lightning current of amplitude I0 is assumed along with MTL lightning channel model, and where

(1-3)

is the wave from the dipole, and

r-(z+h)_e~jtoRi

a' z&f

.2 #0 3

"o (1-4)

is the wave from the image, and

(1.5)

is the surface wave.

At the ground surface, the waves from the dipole and image cancel each other and only the surface wave remains.

1.2 Vertical electrical field in air

Sommerfeld’s exact expression for the vertical field in air set up by a vertical current element I-dh at height h above a flat and homogeneous ground is:

Appendix J. Listing of ATP-EMTP data cases 197

dE(r,z) =

-yto

*o. I ^

V-o'Idh

47tg ~>k0'R> e -Jk°'Ri

V *0 R i+ /«•-------- ■ / —•---------------------- -da

2% { 1 kl-l+kl-m

(L6)

Hie total vertical field from a lightning channel modelled by the MTL model can be written, based on King [99]:

£zO’Z> = -^~'f[1D+II+1s}e. „ -h-(jWv * 1/A.). dh (17)

where

, M

2 -R{K

z 2 .)kt -jk0/R0 -1<- z-h

RnV V 0 ) /

the dipole,

/ ( X2 . )s 1

■“ w 1

l *.J

<N _<

m1

s‘1

is the wave from the image, and

h = ““ ^Jit*oRi r , e "■/Vl----------•---------*/ •-------------------

2 RPR,

(1.8)

(1.9)

(MO)

is the surface wave.

J. Listing of ATP-EMTP data cases

The program listing below is an ATP-EMTP data case for calculating lightning-induced voltages in a 2-phase, lossless overhead line. The TL model is assumed for the lightning channel. The geometry of the system is along with the lightning current parameters user-selectable. The characteristic impedance of the overhead line is input at two places: 1) In the USE statement section of the model INDUS and in the branch data section.

BEGIN NEW DATA CASE C H.K. Hpidalen, 1997.C Data case for calculation of LIO in a 2-phase, lossless overhead line C 1st phase: Called L, characteristic impedance 300 QC 2nd phase: Called N, characteristic impedance 500 £2. Mutual impedance 200 Q C Length of line L—XA-XB = 500 m, distance y=500 m. Height of line: z=6 m.C dt X Umax > Miscellaneous Data Card


1.0E-07 5.0E-05500 1 1 1 1 0 0 1 0

MODELSINPUT —Currents and voltages in the two phases at each end of the line

I01AP {i(UZ01AP)},10IBP (i (UZ01BP) } ,10IAN {i (UZ01AN) } ,I01BN (i(UZ01BN)},U01AP (v(US01AP)},U01BP {v(US01BP)},U01AN {v(US01AN)},U01BN {v(US01BN)}

OUTPUT —4 type 60 sourcesUR01AP,UR01BP,UR01AN,UR01BN

MODEL INDUSFUNCTION S0R(x>:=x*xCONST Tmax (VAL:500}

Im {VAL:30.e3} —Maximum current valueT1 (VAL:2.e-6} —Front time constantT2 (VAL:50.e-6) —Half value time constantm (VAL:5} —Front steepness factorc (VAL: 3. e8 } —Speed of lightV (VAL: 1. Ie8 } —Lightning current velocitylo {VAL:1} —Amplitude of step currentz (VAL:6} —Height of overhead line

INPUT UrefAP, UrefBP, UrefAN, UrefBNDATA Y,XA,XBOUTPUT OutAP,OutBP,OutAN,OutBNVAR UindA[0. .1000] ,UindB[0. .1000] ,dl[0. .1000] ,Tr,Ti,I,e,dt,

OutAP,OutBP, OutAN, OutBN, ta, tb,b,n,L,x,Ko HISTORYUrefAP {dflt:0}UrefBP (dflt:0}Uref AN {dflt:0}UrefBN {dflt:0}

INITdt:= timestep FOR Ti:=0 TO Tmax DO

Tr:=Ti*dt b:=v/c L:=XA-XB x:=XA n:=c*Tr-xta:=sqrt(x*x+y*y)/c tb: =sqrt (sqr (x-L)+y*y)/c if Tr>ta thenKo:=60*Io*z*n*b/ (sqr (y) +sqr (b*n))if Tr>tb+L/cthenUindA[Ti] :=Ko* ( (x+b*b*n)/sqrt( sqr (v*Tr) + (l-b*b) * (y*y+x*x) ) -

(x-L+b*b*n)/sqrt( sqr (v* (Tr-L/c)) + (l-b*b) * (y*y+sqr (x-L) ) ) )elseUindA[Ti] :=Ko* ( (x+b*b*n) /sqrt (sqr (v*Tr) + (l-b*b) * (y*y+x*x) ) +1)

endif elseUindACTi]:=0

endif x:=-XB


n:=c*Tr-xta:=sqrt(x*x+y*y)/c tb:=sqrt(sqr(x-L)+y*y)/c if Tr>ta thenKo:=60 *Io*z*n*b/(sqr(y)+sqr (b*n) )if Tr>tb+L/cthenUindB[Ti] :=Ko* ( (x+b*b*n)/sqrt( sqr (v*Tr) + (l-b*b) * (y*y+x*x) )-

(x-L+b*b*n) /sqrt( sqr (v* (Ir-L/c) ) + (l-b*b) * (y*y+sqr (x-L)) ) )elseUindB [Ti] :=Ko* ( (x+b*b*n)/sqrt(sqr (v*Tr)+ (l-b*b) * (y*y+x*x) ) +1)

endif elseUindB[Ti]:=0

endif—Heidler current:

IF (Ti=0) THEN dl[0]:=0 ELSE

e:=exp (- (T1/T2) *exp(ln(m*T2/Tl) /m))I:=Im/e*exp(m*ln(Tr/Tl>) / (exp(m*ln(Tr/Tl))+l)*exp(-Tr/T2) dl [Ti] : =1* ((m/Tr) / (exp (m*ln (Tr/Tl)) +1) -1/T2)

ENDIFENDFOR—Convolution integral (current shape). Small Io required compared to Im:Ti:=Tmax WHILE Ti>l DO

FOR Tr:=l TO Ti-1 DOUindA[Ti]:=UindA[Ti]+UindA[Tr]*dl[Ti-Tr]*dt UindB [Ti]:=UindB[Ti]+UindB[Tr]*dl[Ti-Tr]*dt

ENDFOR Ti:=Ti-l

ENDWHILE—Possible to scale Uind with respect to height z (phase and neutral)Tr:=(XA-XB)/c

ENDINIT EXECOutAP: =UindA [round (t/dt) ] +delay (UrefBP, Tr-dt, 1)OutBP: =UindB [ round (t/dt) ] +delay (UrefAP, Tr-dt, 1)OutAN: =UindA [round (t/dt) ] +delay (UrefHN, Tr-dt, 1)OutBN: =UindB [round (t/dt) ] +delay (Uref AN, Tr-dt, 1)

ENDEXECENDMODELUSE INDUS AS INDUS1 INPUTUrefAP := 300*I01AP + 200*I01AN + U01AP —UrefA = Z'*IA + UAUrefAN:= 500*I01AN + 200*I01AP + U01AN —Z': Characteristic impedance

matrixUrefBP:= 300*I01BP + 200*I01BN + U01BP UrefHN:= 500*I01BN + 200*I01BP + U01BN

DATA Y:=100 XA: =1000 XB:=500

OUTPUTUR01AP:=OutAP UR01BP: =OutBP UR01AN:=OutAN UR01BN—OutBN

DELAY CELLS DFLT: 200 ENDUSE ENDMODELS


C12345678 C 34567890123456789012345678901234567890123456789012345678901234567890123456789051UZ01AFUR01AP 300.0 {Charac. impedance end A }52UZ01ANUR01AN 200.0 500.0 { -- 'T--- }51UZ01BPUR01BP 300.0 {Charac. impedance end B }52UZ01ENUR01BN 200.0 500.0 { ---M--- }US01APUS01AN 0.01 {Load impedance end A }2US01BPUS01BN 0.002 (Load impedance end B }2US01AN 5.0 {Grounding resistance A >US01BN 50.0 {Grounding resistance B }BLANK BRANCHUS01AFUZ01AP MEASURINGUS01BPUZ01BP MEASURINGUS02APUZ02AP MEASURINGUS02BPUZ02BP MEASURINGUS01ANUZ01AN MEASURINGUS01BNUZ01BN MEASURINGUS02ANUZ02AN MEASURINGUS02BNUZ02BN MEASURING

BLANK SWITCH 60UR01AP 60UR01BP 60UR01AN 60UR01BN BLANK SOURCE US01APUS01BP

BLANK OUTPUT BLANK PLOT BEGIN NEW DATA CASE BLANK

The program listing below is an ATP-EMTP data case for calculating lightning-induced voltages in a single-phase, lossless overhead line. The two inducing voltage terms are assumed to be pre-calculated by an external program, assuming the same time step and line length as in the ATP-EMTP data case. Lossy ground effects on the electrical fields can be included by a modification of the inducing voltage terms.

BEGIN NEW DATA CASE 1.0E-07 1.0E-04 O.OE+OO 0.0E+00

1000 1 1 1 1 0 0 1 0MODELS INPUT

11 {i(XXOOlA)},12 {i (XX001B) } ,UI001A {v(UI001A)},UI001B (v(UI001B)},VI {v(UA)>,U2 (v(UB) >

OUTPUTUR001A,UR001B

MODEL INDUS CONST c {VAL:3.e8}INPUT IA,IB,UINA,UINB,UA,UBDATA Z,LOUTPUT UrA,UrBVAR UrA,UrB,Tr,Ure£A,UrefBHISTORYUrefA {dflt:0}UrefB {dflt:0}


INITTr:=L/c

ENDINIT EXECUrefB:=(UB+Z*IB)UrefA:=(UA+Z*IA)Ur A:=UINA+Delay(UrefB,Tr-timestep,1) UrB:=UINB+Delay(UrefA,Tr-timestep,1)

ENDEXEC ENDMODELUSE INDUS AS INDUS1 INPUT

IA:= II IB:= 12 UINA:= UI001A UINB: = UI001B UA:= U1 UB:= U2

DATA Z:=500 L:=500

OUTPUTUROOIA:=UrAUR001B:=UrB

ENDUSEENDMODELSXX001AUR001A 500.0XX001BUR001B 500.0UAUBUI001A 1.0UI001B 1.0

BLANK BRANCH UA XX001A UB XX001B

BLANK SWITCH 1UI001A 2UI001B 60UR001A 60UR001B BLANK SOURCE BLANK OUTPUT$INCLUDE, c:\ind_volt\uar2.indBLANK PLOTBEGIN NEW DATA CASEBLANK

(Characteristic impedance0.01 (Load impedance end A

0.002 (Load impedance end B{Connect inducing voltage sources to ground

MEASURINGMEASURING

}

}2}2}

(Include file with inducing voltages}

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