Index of /rozprawy2/10043

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Faculty of Electrical Engineering, Automatics,

Computer Science and Electronics

Department of Electrical Engineering

and Electrical Power

Saad Khalifa Omar Dau

Modelling of Lightning Overvoltages

for the Protection of Transmission Lines

by Means of Shielding Wires and Surge Arresters

Ph. D. Thesis

Under the Supervision of:

Dr. hab. eng. Jakub Furgał

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DEDICATION

I would like to dedicate this thesis to my

understanding wife for her encouragement,

patience, and sacrifice with the many hours to

finish this work.

I would also like to dedicate this thesis to my

family.

I appreciate their love, patience, and

understanding throughout this work.

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AKNOWLEDGEMENTS

All deepest gratitude is due ALLAH who guided and aided me to make this

work possible

.

I would like to express my thanks to Prof. dr. hab. eng. Romuald Włodek. I‘m

very grateful for all his help.

I would like to thank Prof. dr. hab. eng. Barbara Florkowska, for her gracious

gift of the non-invasive strain of our work.

I would like to express my sincere gratitude for the inspiring suggestions and

precious help extended by Dr. hab. eng. Jakub Furgał, who has supervised this

thesis.

Also I am very grateful to the Head of Faculty of Electrical Engineering,

Automatic, Computer Science and Electronics Prof. Dr. hab. eng. Tomasz

Szmuc for giving me this opportunity to have the Ph. D. degree from the Faculty

of Electrical Engineering, Automatics, Computer Science and Electronics of

AGH Unversity of Science and Technology.

I would like to thank all the Members of my Graduate Advisory Committee, for

their endless help. I would especially like to thank them for their patience,

suggestions, experience, and expertise that made this thesis possible.

I would like sincerely thank all the Members of the Department of Electrical

Power

.

The author also gratefully the acknowledges the Libyan Ministry of Education

and to the Libyan Embassy for the financial support for this thesis which, would

not be possible without it.

Finally, I wish to express my deep gratitude to the Polish people for their

hospitality and innumerable acts of kindness and assistance.

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List of Basic Symbols

a,A,b,B – constants

A0, A1 – currrent-voltage dependences of varistors

aFe, aAl – radius of steel and external radius of conductor

c – velocity of light

C – capacitance

d – height of varistor columns of surge arresters

Dc – horizontal exposed distance of phase conductor

dFO – horizontal displacement of phase conductor and shielding wire

ds – distance between shielding wire and earth

E – probable electric field strength near ground during a thunder storm

Eg – soil ionisation field strength

Ei – field strength due to currents at a point at any instant

Eo – minimum leader progression electric field strength

E50 – average electric field strength at critical flashover voltage

f(I) – probability density function of crest current of the first lightning stroke

g – gap length

G – unit conductance to earth of the transmission overhead line

Go – reference conductance

he – depth of burying of earth electrode

hk – distance between a conductor and earth

hl – distance between a shield wire and earth

hT – height of tower

i,I – current

I0, I1 – Bessel’s modified function of first type of zero and first order, respectively

i1,i2 – constants

iA0, iA1 – currents in the varistors A0 and A1

Ig – limiting value of current, and depending on Eg

Im – median value of lightning current

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Imin – minimum lightning current

In, 8/20 – rated surge current of surge arrester at 8/20 µs

Iref – reference current

K – constant

k – leader coefficient

K0, K1, – Bessel’s modified functions of second type, zero and first order

L – inductance li insulator length

l1 – length of overhead line

le – length of earthing system

ll – leader length

Lii, Lij – self inductance of i conductor and mutual inductance between i and j conductors

M – median parameter value

n – number of parallel varistor columns

Nf – financial expenditures

Ng – ground flash density

P – breakdown probability of the external insulation

Pd(u) – insulation failure cumulative distribution

Po(u) – overvoltage probability distribution function on the insulation system

R – resistance

R’ – failure probability

rc – radius to conductor in the electrogeometric model of lightning protection of

overhead lines

re – radius of earthing system

rg – radius to shielding wire in the electrogeometric model of lightning protection of

overhead lines

Rst – static resistance to earthing systems rT – radius of tower basis

Ru – earth resistance

s – cross section area

S10 – average front steepness between 10 % and 90 % value point of the first peak value

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S30 – average front steepness between 30 % and 90 % value point of the first peak value

of the current stroke

Sm – maximum steepness of current stroke

t – time

T – temperature t1, t2 – time constants

t10 – time interval between instants corresponding to 10 % and 90 % of first peak value

t30 – time interval between instants corresponding to 30 % and 90 % of first peak value

TAN-10 – rate of current rise at to 10 % value point of first peak amplitude

TAN-G – the maximum front steepness or rate of rise in the front wave

td – front duration of current stroke

Td – number of thunderstorm-days

td30 – front duration according to time T30

te – elapsed time after lightning stroke

tf – front duration of current stroke

th – time to half value of current stroke

ti – corona inception time

tl – leader propagation time

tref – reference time

ts – streamer propagation time

u – voltage

U1/T2;In; – residual voltage of surge arresters for the current impulse 1/T2 with maximal value

of In

U8/20;In – residual voltage of surge arresters with the current impulse 8/20 µs and maximal

value of In

um – insulation coordination margin

Un – rated voltage of surge arrester

up – protection level of surge arresters

Uref – reference voltage of surge arresters

Ur-t – flashover voltage

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y – distance between the conductor and earth

Yu – unit admittance of overhead line lateral in long line to take a position in ground

Zc – surge impedance of the phase conductor

ZC0, ZC1, ZC2 – wave impedance of zero, positive and negative components

(elements of the matrix [ZCS])

Zp – impedance of the line conductor

ZT – impedance transmission line tower

Zu – impedanceof unit length of the overhead line

[C] – matrix capacitances

[Cij] – matrix of self and mutual capacitances of conductors per unit length

[G] – matrix conductances

[I] – current matrix

[If] – matrix of phase currents

[Is] – matrix of symetrical components of current

[L] – matrix of inductances

[Tu] – matrix of own vector of matrix [Pu]

[U] – voltage matrix

[Uf] – matrix of phase voltages

[Um] – matrix of model voltages

[Us] – matrix of symetrical components of voltage

[Y] – matrix of line admittance

[Z] – matrix of conductor impedances

[Zg] – impedance matrix of ground return circuit impedance

[Γm] – diagonal matrix of propagation coefficient

α – non-linearity factor of surge arresters

αs0, αs1, αs2 – damping coefficients of zero, positive and negative components,

δ – protection angle of the tower

β – slope parameter or logarithmic standard deviation

γ – constant propagation

γs0, γs1, γs2 – propagation coefficients for zero, positive and negative components

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εo – electrical permittivity of free space

µ – magnetic permeability of conductor

µo – magnetic permeability of free space

ρe – resistivity of soil

ν – velocity of propagation of electromagnetic wave in vacuum

νS0, νS1, νS2 – velocities of propagation of zero, positive, and negative components

ξ – constant relation

σ – conductivity of the phase conductor

σAl , σFe – conductivity of aluminum and steel, respectively

σg – electric conductivity of soil

σsub – conductivity of homogeneous conductor

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Introduction

Continuous efforts to increase the reliability of the electrical energy supply necessitate working out power systems which should to be so designed, run and maintained as to minimize the probability of the system failures. One of the fundamental requirements for a highly reliable electrical energy supply is assuring continuity of work of transmission systems, mainly composed of overhead high voltage lines, whose world’s rated voltage values are of 1500 kV. In the polish power system the highest rated voltage of the transmission lines equals to 400 kV. A similar a transmission system works in the Libyan Jamahiriya as well. A reliable operation of the transmission and distribution systems can be ensured by components with insulation system designed for electrical strength, suitable for the expected stresses. On the other hand, there is a tendency to decrease insulation levels, mainly for the economic reasons. These contradictory suggestions necessitate optimization of a technical solution of insulation systems. It requires detailed analysis of stresses to which these systems are exposed. The main part of the transmission lines stresses, defining requirements for the power system insulation and crucial for its reliable operation are overvoltages. The particularly high overvoltages, whose maximal values may be many times higher than the rated voltage, result form lightning discharges and are responsible for the basic hazard of the insulation breakdown.

In the presented approach particular attention should be paid to the overvoltage stresses occurring in the transmission lines despite of limiting the overvoltages through of protection measures, e.g. shielding wires and surge arresters. Lightning discharging to shielding wires induces overvoltages in phase conductors stressing equipment of the electrical substations: power transformers, switches, measurement transformers. Also direct lightning discharges to phase conductors are possible despite of shielding wires.

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Lightning overvoltages generated by the discharges to transmission lines are a result of complex, nonlinear and surge phenomena occurring in the structure of the line towers and electrical substation when the lightning current is flowing through them. On the courses of lightning overvoltages in power systems significant influence it has an electrical withstand of air insulation systems in the overhead lines. Flashover on line insulators can change the courses and maximal values of overvoltages. The influence of surge arresters on overvoltages in power systems depends on the phenomena in the varistors made from metal oxides, parameters of surge arresters and their locations in the networks.

The discussed issues are interesting for electrical power industry because the analysis of overvoltages in power systems has practical meaning for reducing costs and increasing reliability of the transmission of electrical energy. It is necessary to continue investigations in this area as they can have large practical meaning for designers and exploitation engineers. Researches should be carried out on theory of overvoltage propagation in power systems, the overvoltage protection as well as on theoretical and practical aspects of the insulation coordination.

Unfortunately power systems are too complex for analytical solutions, and solving of an even simple problem is usually not possible. For this reason nowadays such analyses are usually performed on basic computer simulations of overvoltage phenomena. In the simulation programs suitable models of electrical equipment and transient phenomena of the electric power systems should be implemented. Effective assessment overvoltage risks need computer methods for simulating high voltage electric power systems. As a result conclusions referred to practical issues of the insulation coordination are formulated. Mathematical models of these phenomena are complicated mathematical descriptions which can be solved by numerical methods and computer techniques. Nowadays more and more extended applications of specialized program packages for more precise design procedures of power systems are used in view of technical and economical aspects. Methods of overvoltage stresses analysis are intensely developed and one of the directions is working out transmission line models and power substations, which would account for basic phenomena determining the shape and values of risks. Transmission line models have to take into account phenomena in phase conductors which are crucial for voltage waves propagation and reaction of towers and earthing systems to stroke currents. Overvoltages stressing insulation systems

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which have a strong influence on electric withstand of air insulation systems.

Research works in the surge arrester area are continued. The issue consists hereby in the proper modelling of surge arresters because the current conducting mechanism in metal oxide varistors is complicated. There are several models of this process, differing in their structure and parameters. These models are being continuously refined in respect to both structure and methods of the parameters identification.

Problems considered in the presented dissertation belong to the mentioned groups of issues. The results of this work can be considered a contribution to the power system modelling with respect to the phenomena occurring in the transmission lines and nonlinear surge arresters due to lightning discharges.

Following points of thesis have been the subject of investigations and proves:

- it is possible to reduce exposures of insulation systems of high voltage overhead

transmission lines and electric power devices against lightning overvoltages with the use of suitably designed overvoltage protection systems including both shielding wires as well as surge arresters,

- the modelling of lightning overvoltages carried out in the purpose improvement of

such systems need to use suitable mathematical models of overhead lines, surge arresters, electric power devices and phenomena existing in power systems during propagation of such overvoltages,

- to model overhead lines the mathematical model with frequency-dependent

parameters should be used and surge arresters should be modelled with the use of the full model worked out by IEEE.

The scope of the dissertation can be summarized in the following stages:

- work out detailed models of overhead transmission lines, surge arresters and other electric devices of power systems which are suitable in conditions of propagation of overvoltages,

- work out the model of the complicated part of the electric power system taking into account the phenomena which exist during propagation of lightning overvoltages,

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- modelling and analysis of lightning overvoltages in the complex power system for estimation of efficiency of the overvoltage protection system of high voltage transmission lines based on shielding wires and surge arresters.

In order to confirm the thesis the analysis was performed of the influence of frequency dependences of the line model parameters on the overvoltage wave propagation with intention to select the line model which is suitable for the lightning overvoltages in transmission lines. The analysis was based on the results of computer simulations. The differerent mathematical models of overhead lines were analysed for simulations of lightning overvoltages. The detailed mathematical models of the line towers and earthing systems suitable for conditions with flowing of lightning currents are presented in the dissertation. The model of the air discharge phenomena at line insulators under the lightning overvoltages has been also taken into account.

Properties of the selected mathematical models of surge arresters were analyzed and verified on the base of the dynamic current-voltage characteristics taken from computer simulation as well as from the test data presented in the catalogue specifications. The goal of such a comparative analysis is finding a suitable model for computer simulations of lightning overvoltages in the transmission systems protected by surge arresters.

The mathematical models of power transmission lines and surge arresters were applied in the simulations of lightning overvoltages in the power transmission systems with overhead lines which are protected by shielding wires and which contain surge arresters for protection of electric devices in the power substations. This computer simulations create the bases for the analysis of the effectiveness of overvoltage protection of the transmission line and substations by shielding wires and surge arresters.

The subjects of the presented thesis are high voltage transient phenomena, which are difficult and expensive in experimental, laboratory investigations. This is the reason why this work is essentially theoretical. Only the mathematical models of surge arresters were checked with experimental results presented in catalogues by manufacturers of these devices.

The dissertation is divided into six chapters. The first chapter is devoted to the source and mechanism of lightning discharges and their impact on the overhead transmission lines. The process of overvoltage generation in the power lines was shown schematically and other overvoltages stressing the insulation systems of these lines were characterised too. Also the

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presented on the background of these coordination principles.

In the second chapter the analytical model of the lightning current was described and its different waveforms, found with these models, were presented.

The subject of the third chapter is the electrogeometric model of shielding wires of the transmission lines.

In the fourth chapter the mathematical models of overhead transmission lines were analysed. It includes models of phase conductors protected by shielding wires, line towers, earthing systems, line insulators and a model of the breakdown over the insulators due to the overvoltages. Results of simulations of the overvoltage wave propagations, obtained with different mathematical models of overhead lines implemented in the Electromagnetic Transients Program-Alternative Transients Program (EMTP-ATP) were used in the analysis of influence of the model parameters selection on the surge effects in the transmission lines. The fifth chapter is devoted to modelling of metal oxide surge arresters. One of the presented models enables determining their parameters from the manufacturer’s specification which are presented in catalogues. This model has been applied in the further work. The current-voltage characteristic obtained with the full and simplified model based on simulations performed by means of a MATLAB computer program are presented in this chapter. These results, confronted with manufacturer data were the basis for a comparative analysis of both versions of the model.

In the sixth chapter the mathematical of the 220 kV power system which was worked out in EMTP-ATP was given. This model includes mathematical models of the transmission lines and parts of the system devices. The model of flashover on line insulators was work out in MODELS, being internal language of the EMTP-ATP software. The presented model was used for computer simulations of lightning overvoltages which are generated during strokes to the shielding wires as well as to the phase conductor. The analysis of the influence of the shielding wires and surge arresters on the lightning overvoltages in the overhead lines and power substations was worked out.

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1. Influence of Lightning Phenomena

on Electric Power Systems

1.1. Lightning Discharge Phenomenon

Lightning phenomenon is caused by a peak discharge during which the charge accumulated in the clouds discharges into the neighbouring cloud or to the ground. The description of the lightning mechanism is given here in a very simplified form, only to present the background of these phenomena. More about lightning phenomena is presented in specialist books and papers [3, 23, 24, 29]. The mechanism of charge formation in the cloud and their discharge, as well as the numerous the factors which help the formation or accumulation of charge in the clouds are complex, and unpredictable. But during thunder storms, positive and negative charges become separated by heavy air currents with ice crystals in the upper parts and rain in the lower parts of the clouds, and this cloud separation depends on the height of the clouds. The probable electric field strength near the ground during thunderstorm is presented in Figure 1.1. ground E x + + + + + + + + ₊ + + + + + ++ + + + +++ + + + + + ++ +

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inside the cloud may be 1-100 C, and potential 107-108 V, with field gradients ranging from 100 Vm within the cloud to 105 Vm-1 at the initial discharge point. The lightning discharge to an open ground starts invariably in the cloud. It becomes visible when penetrating its lower boundary, then progresses towards the earth as a faintly luminous discharge, called the leader stroke. The temporal development of a multiple earth flash is presented in Figure 1.2.

40 ms 3 km 40 ms earth cloud base 20 ms stepped leader return

stroke dart _leader return _stroke 1 ms

Figure 1.2. Temporal development of multiple earth flash

It is usually heavily branched and, as the great majority of leaders, originates from negatively charged cloud centres; and are thus negatively charged, these branches are attracted by positive charge pockets floating in the air. The leader channel and its branches are extended towards the earth in discrete steps of about 20 m length but there is some evidence that these lengthen as the leader approaches the ground. Such discharges are called stepped leaders. The most frequent velocity of movement of the leader tip is between 105 and 2*105 ms-1, being less than one thousandth of the speed of light. The current in the leader

is estimated at several hundred amperes. When the faint lightning leader channel reaches the ground, an intense luminosity is seen to travel upwards along its path towards the cloud and along its branches. This is the so called return stroke. In effect, this constitutes an electric short circuit between the negative charge deposited along the leader and the electrostatically induced positive charge in the ground. The velocity of the return stroke decreases from the ground to the cloud but initially amounts to 108 ms-1, the most frequent average value over its full length being 3.5*107 ms-1, which is about one tenth the speed of light or a hundred

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times faster than that of the leader. The leader and return stroke process as described may complete the visible part of the lightning discharge. However, after a certain time interval, a second leader stroke, followed along a return stroke, may occur. This subsequent stroke usually follows by the path taken by the first stroke with the exception that it is not stepped and is much faster; therefore it is called a dart leader. The process of dart leader and return stroke formation can be repeated several times. Each such component is termed a stroke while the complete process of successive strokes is termed a multi-stroke flash, or briefly a lightning flash. A lightning flash can thus consist of a single stroke or a sequence of several discrete strokes. As a rule, all subsequent strokes follow the path blazed by the first stroke but in a high wind the entire channel can be blown sideways. The account given so far is concerned with negative lightning flashes that are flashes conveying negative charge from cloud to ground. This comprises some 95% of all earth flashes and even more in tropical storms. Only scanty information is available about the characteristics of positive lightning discharges. Despite differences in the development of the initial leader stroke, the leader-return process is the same in positive and negative charges to open ground, with the one important difference that positive flashes usually consist of a single stroke and seem to occur towards the end of a storm, when the upper positive cloud charge may be discharged to earth in one stroke which is often of exceptional severity [23, 24].

1.2. Overvoltages in Electric Power

Systems due to Lightning

Lightning overvoltages are caused either by direct strokes to the phase conductor or as a result of strokes to earth very close to the line which produces induced lightning surges. The overvoltage, by which substation insulation is stressed, is a function of the line construction and the system configuration. Overvoltage induced by indirect lightning on overhead lines can cause damage to the power system.

The scheme of an influence of lightning strokes on overhead transmissions lines in power systems is presented in Figure 1.3.

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shielding wires surge arresters yes/no yes lightning flash overvoltage in power propagation of overvoltage overvoltage at struck point flashover back flashover lightning stroke to earth lightning stroke to phase conductor lightning stroke

tto tower or earth

parameters of the first stroke

power substation

Figure 1.3. General scheme of lightning overvoltages in power systems

Moreover, due to its more frequent occurrence, indirect lightning constitutes a more important cause of micro-interruptions than the direct strike. Their estimation is therefore crucial for the correct operation coordination of overhead lines and, as a consequence, has been the object of various studies since many years. The majority of lightning strokes takes place from clouds which have positively charge upper regions with the rest negatively-charge except for some localized positive charge in part of the cloud base. The lightning stroke consists basically of two components, a leader stroke from the cloud which initiates an upward streamer from some irregularity on the earth and a conducting channel along which a return stroke then passes. In many cases the stroke is multiple with a number of leaders and return strokes.

The return stroke, in the course of which the current reaches its peak, may have a current as low as hundreds of ampers, but is more frequently between a few, kiloampers and about 100 kA. The current waveform is generally a unidirectional pulse rising to a peak value in about 3 µs and falling to small values in several tens microseconds.

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When the lightning current stroke contacts the line, the voltage wave starts to be formed as shown in Figure 1.4.

u u

Figure 1.4. The voltage wave formed as a sequence of a lightning stroke

The scheme, which is presented in Figure 1.4, can be represented by an electrical equivalent circuit, considering the lumped parameters, as shown in Figure 1.5.

i ZT iT Ru ZP ZP _u

Figure 1.5. The electrical equivalent circuit of formed voltage wave in the transmission line

In Figure 1.5 the line conductor is represented by two impedances Zp, the transmission line

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1.3. Other Overvoltages in Electrical

Power Systems

Insulation systems of electrical networks excepting lightning overvoltages are subjected to the following electric stresses:

- power-frequency voltages under normal operation, - temporary overvoltages,

- switching overvoltages [24, 35].

The severity of temporary overvoltages is mainly characterized by both their maximum value and duration. The importance of temporary overvoltages in insulation coordination is

twofold:

- on one hand, the characteristics of temporary overvoltages at the surge arresters

location are of great importance to surge arrester selection,

- on the other hand, the repetition of the successive overvoltages peak of opposite

polarity may determine the design of both the internal insulation of equipment as well as the external insulation (surfaces exposed to contamination).

Temporary overvoltages are generated as a result of: - earth faults,

- sudden changes of load, - resonance and ferroresonance.

S

witching overvoltages occur in power systems whenever opening or closing of switchgear occurs and when the short circuits and faults occur in a power system. There is a great variety of events that many initiate a switching surge in the power network. The switching operations strongly depending on insulation design can be classified as follows:

- energization of transmission lines - the following specific switching operations are one of the most common in this category:

- energization of a line that is open at far end,

- energization of a line that is termined by an unloaded transformer, - energization of a line through the low-voltage side of a transformer,

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- reenergization of a line - this means the inergization of a transmission line

carrying charges trapped by previous line interruptions when high-speed reclosing is used.

- load rejection - this is produced by the circuit breaker opening at the far end of the line. This may also be followed by opening the line at the sending end in what is called a line dropping operation,

- switching on and off of equipment - all switching operations involving an element of the transmission network will produce a switching surfeit. Of particular importance, however, are the following operations:

- switching of high-voltage reactors,

- switching of transformers that are loaded by a reactor on their tertiary winding,

- switching of a transformer at no load,

- fault initiation and clearing.

1.4. Properties of Withstand Voltages

of Insulation Systems

Basic problems presented in this dissertation concentration on overvoltage risks of insulation systems of overhead transmission lines and electric power devices in power substations. Special emphasis lightning overvoltages, which have been analyzed on the basis of the results of computer simulations. An insulation system of overhead transmission lines has a form of air distances and electric line insulators. Withstand insulators of high voltage overhead lines are usually made of ceramic or glass. Sometimes they are based not only on maximum values of overvoltages but also on their courses. The electric withstand of air distances is presented in a simplified form in Figure 1.6 in comparison to the withstand voltage of internal solid insulation. Withstand voltage of large air gaps is presented in curve II, Figure 1.6. It has large electric withstand for short strokes and overvolatges over long spans of time.

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than for switching and temporary overvoltages. Therefore lightning overvolatges which are generated in power systems have large maximum values and they create large risk of insulation systems, and they should be taken into account as basic risks of power systems. For this reason the time-voltages characteristics for lightning overvoltages should be known because they have basic means in insulation coordination [19, 24, 35, 52].

overvoltage, p.u II I III 1 2 3 6 5 4 3 2 1

Figure 1.6. The electric withstand voltages of selected insulation systems for typical overvoltages: 102 10 10-6 10-4 10-2 _{t, s} 1 - lightning overvoltages, 2 - switching overvoltages, 3 - temporary overvoltages:

I - air insulation of medium voltages, II - large air distances,

III - solid internal insulation

Withstand voltage of air insulation systems is the smallest for switching overvoltages with positive polarity. The minimum value of flashover voltage depends on value distance between electrodes and time to the crest of the switching stroke. This minimum (critical) electric withstand should be taken into account in co-ordination of the highest voltage insulation systems. Electric withstand of insulation systems of power devices is determined in the form of limiting values of voltages. When voltage is greater than this value then air insulation systems can lose their properties by a flashover.

The phenomenon of electrical flashover has a statistical nature. If an impulse voltage of a certain shape is applied repeatedly to the self-restoring insulation system, the outcome of all

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applications may not be the same. Some impulse applications may cause a flashover, while other still with the same maximum value, may be withstand by the insulation. This phenomenon is particularly true for sufficiently large gaps (larger than about a 0.5 m rod-plan gap in air at atmospheric pressure). It also occurs over a finite range of impulse voltage maximum values, above which all shots cause breakdowns and below which all shots would not cause breakdown. This probabilistic property is very essential for insulation dimensioning. In fact, at very high transmission voltages this property is used for setting an economical compromise between the insulation cost and the risk of insulation failure. The probability of a breakdown (i.e., the number of breakdowns relative to the total number of applied shots) may be plotted against the applied voltage as shown in Figure 1.7 [24, 52, 55, 56]. a) U50 0.5 1.0 0 U50 0.5 0 U 0.159 0.841 σ σ P (U) U 0.159 0.841 σ σ P (U) b) σ U U − ₅₀ 1 0 -1

Figure 1.7. Breakdown probability of external insulation system:

a - on linear coordinates, b - on Gaussian scale

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breakdown probability function may be expressed by the well-known cumulative Gaussian distribution function: du U u u P u 2 50₎ ( 2 1 ( exp 2 1 ) ( σ π σ − − =

_∫

∞ − (1.1) where: U50 - 50% breakdown voltage of the representative, also called the critical flashover

voltage (CFO), V

σ - standard deviation of the representative Gaussian distribution (σ describes the degree of scatter in breakdown voltages about their mean value U50).

If the postulate of equation (1.1) - that the breakdown probability fits a Gaussian cumulative function - is accepted, P(U) can alternatively be plotted on a normal probability ordinate scale as seen in Figure 1.7, in which case P(U) appears linear. The linear units on the ordinate in this case are those of the quality

σ U

U− 50 _{rather than P(U).}

1.5. Principles of Insulation Coordination

and its Methods

The basic aim of insulation coordination is designing the appropriate insulation systems of electrical devices to minimize damage and financial costs. Insulation coordination account for electric strength values of electrical devices in relation to the voltage which can appear on the power systems [14, 18, 24, 52, 58, 59]. It takes relative coordination of the basic

components in relevance to the exploitation of power systems: - voltage and overvoltage risks of insulation systems, - withstand voltage of insulation systems,

- methods of overvoltage protection.

When formulating of the process of insulation coordination, the required operating reliability of electrical power systems - with consideration to the unfailing work of insulation system – is obtained after assuming a suitable coordination margin UM :

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UM = Uw – Up (1.2)

It is equal to the difference between the electric withstand Uw of the insulation system and the

protection level Up, which is equal to the reduced level of overvoltage protection systems, for

example shielding wires and surge arresters.

The value of the margin UM influences not only the reliability of the electrical power system,

but also its costs (Fig. 1.8).

NF R’ UM R 1 NF

Figure 1.8. Influence of the margins coordination UM on reliability R’ and financial

expenditures NF of insulating system

The operating reliability R' of insulating systems increases with the increasing width of the margin coordination UM. Also the financial expenditures are increased in this case. Moreover,

environmental and system operation conditions of electrical power system, the change of characteristics and parameters as well as the technological dispersion of devices have influence on the width of the coordination margin.

In reality the coordination margin is a random variable. It is determined by the random character of electric withstand and overvoltage risks as well as by random character of protection characteristics of overvoltage protection devices. In this concept the dimensioning of insulation systems is determined on the basis of the failure probability R' which is expressed by the following formula:

dU (U) P (U) P R _d 0 o ' ∫ ∞ = (1.3)

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Pd(U) - insulation failure cumulative distribution [24, 35, 52, 65].

The overvoltage probability distribution and insulation failure cumulative distribution are shown in Figure 1.9. Po(U) R’ Po(U) (U)dP Pd(U) U

Figure 1.9. Overvoltage probability distribution, insulation failure cumulative distribution

and risk of failure probability distribution

Insulation coordination which is determined with the use of this method has form of an optimization work.

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2. Analytical Representation

of Lightning Current Impulse

- - -

Accurate knowledge of lightning current parameters is essential for the appropriate analysis of risk of such overhead electric devices as overhead transmission lines, power substations or buildings during lightning. The properties of lightning have statistical character thus, a number of measured values are needed to determine their statistical distribution. However, to collect sufficient data, measurements must be made on high objects where a high frequency of lightning strikes may be expected. Therefore, even commonly known lightning parameters are related to high object. The positive or negative lightning flashes to total lightning flashes ratio is an important value. According to the observations high objects are struck by negative lightning with 90% or higher probability. This ratio seems to be generally valid and independent of the height of the objects. It is possible that the ratio of positive strikes is a little higher at high voltage transmission lines than at height ones [14, 23, 49].

2.1. Lightning Current Waveshape

The waveshapes of lightning currents have been analyzed in many publications [13, 14, 45, 49, 66, 70, 78].They havevery important influence on time courses and maximum values of overvoltages which are generated in overhead lines. The following basic parameters should be known when determining of the lightning current:

current peak value,

maximum of the current steepness, charge transfer at the striking point.

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tf _t10 TAN-10 TAN-G Im1 S30 S10 Imax t,µs i, kA t30 I100 I90 I30 I10

Figure 2.1. Definition of front parameters for lightning current impulse of negative polarity:

Im1 - maximum value of first current peak, A,

Imax - maximum value of current, A,

t10 - time interval between instants corresponding to 10% and 90% of first peak value, s,

td - front duration according to t10, s,

t30 - time interval between instants corresponding to 30% and 90% of first peak value, s,

td30 - front duration according to t30, s,

S10 - average front steepness between 10% and 90% value point of first peak, A s-1,

S30 - average front steepness between 30% and 90% value point of first peak, A s-1,

TAN-10 - rate of current rise at to 10% value point of first peak amplitude, TAN-G - the maximum front steepness or rate of rise in the front wave, tf - front duration (tf = 1.25 t10)

Note that the statistical parameters based on the 10% intercept will consistently be less reliable than the 30% values, due to the use of a 2 kA reference level for measurements. Therefore, for engineering purposes the 30% based parameters should be generally used, as

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indicated in the following sections. The frequency distribution of these additional impulse front parameters is summarized in Table 2.1, assuming a log-normal distribution of

variables, where the general equation for the probability density for any particular parameter x is given by:

2 2 2 1 ) ( z e x x f = − β π (2.1) where: β M x ln z= (2.2)

M - median parameter value,

β - slope parameter or logarithmic standard deviation [12, 13, 44].

The mean value of any parameter may then be expressed as:

2 2 e M β µ = (2.3)

For example, the median value of the tail is 77.5 µs and the average or mean is 91.5 µs.

2.2. Parameters of Lightning Strokes

Lightning being assumed to be a current source, the magnitude and shape of the return-stroke current wave play a significant role in the estimation of outage rates of power systems caused by lightning. The return-stroke current rises to its peak in a few

microseconds and slowly decays after reaching the peak. The time to peak is called the front time, tf, and the time duration from t = 0 to the instant when the current subsequently

decays to the 50% value of the peak is called the time to half value (tail time) th. The time

to half value th, being many times longer than tf , does not play a significant role in the

severity of lightning-caused transient overvoltages. However, the influence of the peak of the current wave Imax and tf is very significant [14, 44, 49].

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Table 2.1. Parameters of log-normal distribution for negative downward flashes

first stroke subsequent stroke parameter M β M β front, µs td10 = T10/0.8 td30 = T10/0.6 5.63 3.83 0.576 0.553 0.75 0.67 0.921 1.013 steepness, kA µs -1 Sm, maximum S10, at 10 % S10/90, 10 - 90 % S30/90, 30 - 90 % 24.3 2.6 5.0 7.2 0.599 0.921 0.645 0.622 39.9 18.9 15.4 20.1 0.852 1.404 0.944 0.967 crest current, kA II, initial Imax, final initial/final 27.7 31.1 0.9 0.461 0.484 0.230 11.8 12.3 0.9 0.530 0.530 0.207 tail, th, µs 77.5 0.577 30.2 0.933 charge, QI, C 4.65 0.882 0.938 0.882 ∫ i2 dt, kA2 s 0.057 1.373 0.0055 1.366

inter stroke interval, ms - - 35 1.066

2.3. Characteristic and Parameters of Lightning

Strokes

The lightning flash is categorized by the polarity of the cloud and the direction of

propagation of the flash leader. Hence, there are four categories of lightning flash to ground in which the developed leader is followed by a return-stroke current impulse (Fig. 2.2) [14].

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negative cloud charge positive cloud charge upward flash negative cloud charge positive cloud charge downward flash

Figure 2.2

. C

ategories of lightning flash to ground

On average, at least 90% of downward flashes are negative polarity, with some 45 - 55% of flashes consisting of only one stroke. Multiple stroke flashes seldom involve more than 10 strokes (less than 5%), and generally average three strokes per flash, typically at intervals of less than three strokes per flash, per stroke. Upward flashes occur mainly in very tall towers. The majority of transmission line structures are only moderate height (typically less than 60-100 m) and will not in general be subject to upward flashes.The many lightning flash contain several lightning strokes whereby the first steeped-leader/return-stroke sequence is followed in shortly succession by a series of one or more subsequent strokes. Each stroke comprises a dart-leader and return-stroke sequent that generally follows the breakdown path of the first-stroke. Lightning protection systems for transmission line must be capable of withstanding the effects of a series of lightning strokes to the same location within a short period of time. Each lightning stroke is considered an ideal current source of infinite source impedance and the parameter contained within the incident impulse current waveshape determine the incident impulse current waveshape determine the transmission network response. This waveshape parameter includes the peak-current (crest value), time to crest, steepness and duration. Also an importance are the polarity, time interval and number of incident strokes within each lightning flash. Basic parameters values of lightning discharge be subject to distribution variation. Variety type lightning discharge charge in cloud and atmospheric geographic to cause that the plot a curve statistical distribution value of the above parameters. Lightning discharge parameters had a random character. Probability occurrence of peak values of lightning current is presented in Figure 2.3 [14, 29, 51, 63].

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Figure 2.3.Distribution functions of the maximum value of lightning discharge: 1 - first negative stroke,

2 - first positive stroke

The maximum values of lightning discharges are placed in the range from a few kiloampers up to a few hundred kiloampers in electric power systems (Figure 2.3).

Figure 2.4. Distribution of rate of rise (di/dt): 1 - first negative stroke,

2 - first positive stroke

p , % 1 1 0 1 0 0 1 0 0 0 2 5 10 30 50 70 90 95 98 I_max,kA , 1 2 1 1 0 p, % 2 5 1 0 3 0 5 0 7 0 9 0 9 5 9 8 di/dt, kA/µs 1 0 0 1 0 0 0 1 2

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The distribution of rate of rise of lightning current is presented in Figure 2.4. It has a very important influence on overvoltages which are generated in overhead lines. Overvoltage induced by the lightning channel or the current flowing in a conductor is proportional to the rate of rise in the lightning current. Many electrical devices can be damaged by a very short overvoltage, therefore, the maximum di/dt determines the hazard. Longer time duration (at least 1 µs) is needed to produce a discharge in an air gap or breakdown of

solid insulating material

1 0 p, % 2 5 1 0 3 0 5 0 7 0 9 0 9 5 9 8 2 5 1 0 3 0 5 0 7 0 0 9 5 8 2 5 1 0 3 0 5 0 7 0 0 9 5 1 0 0 1 1 0 p, % 2 5 1 0 3 0 5 0 7 0 9 0 9 5 9 8 2 5 1 0 3 0 5 0 7 0 0 9 5 8 2 5 1 0 3 0 5 0 7 0 0 9 5 Q, C 1 0 0 1 1 0 0 0 2 1

Figure 2.5. Distribution of charge of lightning:

1- negative stroke, 2- positive stroke

Figure 2.5, shows the distribution diagram of the charge related to the total flash and to the current impulse. It can be seen that the highest values occur at the positive lightning strikes.

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2.4. Analytical Representation

of the Current Shape

A lightning flash can result in significant overvoltage on overhead line insulation systems if substantial stroke current and charge are injected into the line conductor. If the first contact of lightning return stroke current is with the line, then it is modeled as a transient current generator feeding into a system of transient surge impedance representing the line conductors and the tower, and then the overvoltages are calculated using travelling wave technique. The waveshape parameters include the peak current amplitude (crest value), time to crest, steepness, duration, polarity and time interval.

To calculate the lightning overvoltages in overhead lines it is necessary to simulate the concave front in the lightning stroke current representation. Three points are needed to establish such simulation:

- the correct maximum value (peak) of the current, - the highest steepness close to the peak value,

- for first strokes, the correct average steepness expressed as the front time passing through the 30% and the 90% values of current.

This front time should be larger than the current maximal value divided by the maximum steepness, thus resulting in the concave shape. For subsequent stroke this parameter may be neglected.

Many mathematical expressions may meet these requirements and the one given here is only one proposal. Disadvantageously, the current front and the current tail are not described by a single expression, but are separated into two parts, one describing the front up to 90% of the maximal value, the other, the maximum value on the tail [14].

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The Current Front

The current front for first strokes can be expressed as:

_i₌ _A_t ₊_B_tn (2.4)

where: A, B - constants.

The basic assumption is that the current shape reaches the instant of maximum steepness (90% amplitude) at a time tn dependent on exponent n in principle, both variables have to be

evaluated by an iterative solution of the generalized equation:

) x 1 ( S 2 xn 3 1 S 2 1 n x ) x 1 ( ) S 2 x 3 1 ( N N n N − − + − = − − (2.5) with: max f m N I t S S = (2.6) n f t t 6 . 0 X = (2.7)

where: Imax - maximal value of current, A,

Sm - maximum steepness, A s-1 (Fig. 2.1).

However, a sufficiently accurate solution is given by:

) S 1 2 )( 1 S ( 2 1 n N N − + + = (2.8) 2 N 2 N f n S 1 S 3 t 6 . 0 t + = (2.9)

The constants then are:

) S n t I 9 . 0 ( 1 n 1 A _m n − − = (2.10) ) I 9 . 0 t S ( ) 1 n ( t 1 B _n _m _n n − − = (2.11)

For subsequent strokes the current front is given by:

Imax = Sm tf (2.12)

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The Current Tail

The fundamental requirements for the current tail are:

- to have the maximum steepness at its beginning, thus providing a steady transition from one part to the other,

- to reach correct peak value, - to describe the current tail.

A suitable mathematical expression for current tail is given by:

2 n 1 n t t t 2 t t t 1e i e i i − − − − − = (2.13)

where: t1, t2 - time constants, s,

i1, i2 - constants, A,

th – wavetail, time to half value, s.

The constants are: 2 ln t t t h n 1 − = (2.14) m 2 S I 1 . 0 t = (2.15) ( 0.9 ) t t t t 2 2 1 2 1 1 t I S i _m + − = . (2.16) ( 0.9 ) t t t t 1 2 1 2 1 2 t I S i _m + − = . (2.17)

It was proposed in [14, 51] that the lightning current course might be approximated with the use of: max m max f I S I t = (2.18) where: max m I S

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Table 2.2. Parameters of the first negative lightning downward strokes [14, 51, 66]

lightning current stroke 3 kA ≤ I ≤20 kA I > 20 kA

parameter

M β M β

Imax - final crest current, kA 61.1 1.330 33.3 0.605

Sm - maximum front steepnes, kA µs-1 24.3 0.599 24.3 0.99

tm=Imax/Sm - minimum equivalent front, µs 2.51 1.230 1.37 0.670

Sm/Imax - maximum rate of rise of current

conditional distribution, kA µs-1 12.0IF0.171 0.554 6.50 376 . 0 F I 0.554 tm /Imax , µs 0.0834 IF 0.828 0.554 0.154IF 0.624 0.554

th - wave tail time to half value, µs 77.5 0.557 77.5 0.557

ρc - correction factor between tm and Imax 0.89 0.56

Parameters of first negative lightning downward strokes are presented in Table 2.2.

2.5. Simulations of Courses

of Lightning Current Strokes

On the basis of the presented model of lightning current courses the selected lightning currents with negative polarity and different maximal values were simulated. Current time to crest tf was determined with the use of equation (2.18). In Figure 2.6 are shown the courses of

current strokes with maximum values of 100 kA, probability p = 10% (median value) 33.3 kA, and 10 kA (probability p > 95%) (Figure 2.3).

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0 2 5 5 0 7 5 0 1 2 2 4 3 6 i, kA t, µs b) 0 2 5 5 0 7 5 0 4 8 12 0 2 5 5 0 7 5 0 4 0 8 0 2 0 t, µs i, kA 1 a) c) t, µs i, kA

Figure 2.6. Time courses of selected current strokes determined on the basis of the presented

model:

a - I max =100 kA ( p =10%),

b - I max =33.3 kA (median value), c - I max = 10 kA ( p = 95% )

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The dependence between maximum values of current strokes and the time tf can be analysed

from the courses presented in Figure 2.6. The wave-front time of the current with maximum value 100 kA is 2.67 µs. For current 33.3 kA the wave-front time is 1.88 µs, and for

maximum value 10 kA the front-time is 0.56 µs. The return stroke current rises to its peak in

a few microseconds and slowly decays after reaching the peak value, the severity of lightning causing transient overvoltages. The influence of the peak value of the current wave Imax, and tf, on lightning overvoltages in overhead lines is very large.

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3. Analysis of Transmission Line

Protection by Means of Shielding Wires

3.1. The Simplified Electrogeometric Model (EGM)

Shielding wires are used for the protection of high voltage overhead transmission lines against direct lightnitng strokes. Protective wires are strung above phase conductors on transmission lines. Shielding wire is connected directly to line towers at the top to protect the phase conductors from a direct lightning strike. Lightning strikes will hit shielding wires rather than phase conductors. If the lightning stroke the phase conductors, a short circuit to the ground might occur, which could result in a wide-spread power outage. Phase conductors of transmission lines are connected to the towers by insulators. Line insulators must be strong enough to support the line conductors and they are a part of insulation systems of overhead lines.

For analysing the transmission line protection against direct lightning strokes with the use of shielding wires, a simplified concept of the electrogeometric model will be used. As the down leader approaches the earth, a point of discrimination is reached at which point the leader, if a personality can be ascribed to it, decides the object it will strike. The

electrogeometric model portrays this concept with the use of the striking distance. If all striking distances to the shielding wire, to the phase conductor and to earth are equal, the stroke would terminate on the closes object. However, these striking distances are in general not considered equal and therefore some alternative considerations are required. In general, the striking distance is of the form:

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r = A i b (3.1) where: A, b – constants, dependent on the object, i.e. the phase conductor, the shielding

wire or earth,

i - stroke current [10, 14, 15, 16, 17, 32, 46, 72, 73].

Values of constants A and b are presented in Table 3.1 (Figure 3.1).

Table 3.1. Constants A and b of the striking distance equation r = A i b [14, 32, 72]

striking distance to:

earth rg _{and shield wire}phase conductor _r c

source

A b A b

Young et al., Armstrong

[4, 83] 27.0 0.32 (1) 0.32

Whitehead [81] 6.0 0.80 6.7 0.80

Brown, Whitehead [11] 6.4 0.75 7.1 0.75

Love [40] 10.0 0.65 10.0 0.65

IEEE Working Group [31] 6.4 0.65 8.0 0.65

Group [31] (2) 0.65 8.0 0.65 Eriksson [17] to phase conductor: rc = 0.67 y 0.67i 0.74 to shield wires : rg = 0.67 hT0.67i 0.74 to earth : rg = 0 (1) A = 27.0 for hT < 18 m A = 27 (444) / (462 - hT) for hT > 18 m

(2) A = 8.0 (y/22), must be greater than 4.8 but less than 7.2

The usual model of phase conductors and shielding wires of overhead lines is presented in Figure 3.1 and is illustrated for one specific value of stroke current. The arcs of the circles are drawn centred at the phase conductors and the shielding wire, having radius of rc. In

transmission line design, the specification of the shielding wire location is usually given by the shielding angle δ, as defined in Figure 3.1. As indicated here, if the shielding wires are horizontally disposed beyond the phase conductors the shielding angle is defined as negative. The electrgeometric model as generally used is depicted in Figure 3.1 for the one specific value of the stroke current. The striking distances, rc are shown as arcs of circles from the

phase conductors and shielding wires and a horizontal line is constructed parallel to the earth

at height of rg. Downward leaders or strokes approaching the line with a prospective current

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conductor, and those beyond A will terminate on the earth [14, 32, 52, 72, 73, 76]. SW δ ds y ht dFO rg rc rc rc rc A B PC C B A Dc Dc

Figure 3.1. Electrogeometric model representation of phase conductors and shielding wires:

SW - shielding wire, PC - phase conductor,

dFO - horizontal displacement of phase conductor and shielding wire,

rc radius to conductor in the electrogeometric model

ds - distance between shielding wires,

δ - the protection angle of the tower,

ht-- distance between shielding wire and earth,

y - distance between phase conductor and earth

Unreliability of shielding wire is dependent on the distance Dc (Fig. 3.1) and ground flash

density Ng. When Dc and Ng are constant for line with length lL then the differential of

probability dPf of lightning stroke with maximum value of current I to the external phase

conductor is expressed by formula:

dP

f

=

N

g

l

D

c

(

I

)

f

(

I

)

dI

(3.2)

where: f(I) - probability density function of crest current of the first stroke, Ng - ground flash density Ng calculated with the use of equation:

Ng =0.04 Td1.25 (3.3)

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The ineffectivenessof protecting work is characterized by the coefficient of shielding failure rate (SFR) which determines the number of strokes to the two external phase conductors a

year:

dI

)

I

(

f

)

I

(

D

l

N

2 SFR

max min I I C 1 g ∫

=

_(3.4)

where: Ng - ground flash density, flashes/km2/year,

ll - line length, km,

Imin - minimum lightning current, 2-3 kA,

Dc - horizontal exposed distance of the phase conductor, m,

f(I) - probability density function of crest current of the first stroke.

Therefore the number of flashovers caused by unreliability of shielding wires is determined with the use of the shielding failure flashover rate, SFFOR written as follows:

SFFOR 2N l max D (I) f(I)dI (3.5) f I I C 1 g ∫ =

It results from the dependence in equation (3.3) for Imin= If , where: If – minimum peak value

which cause of flashover. As noted from this equation, the SFFOR becomes zero if Imax is

equal to or less than If [14].

The maximum stroke current Imax is defined as the maximum current beyond which no stroke

can terminate on the conductor. At Imax distance Dc is zero. As an approximation, the

maximum value of rg is:

(

)

        Γ − + = 1 sinδ 2 max g c t g r r y h r (3.6) where: Г = rc / rg From geometry: Dc= rc [ cos θ - cos (β+δ) ] (3.7)

where: δ - the protection angle of the tower,

(

_      − + = − c t FO r y h d 2 5 . 0 ) ( sin 1 2 2 β (3.8)

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    = c r sin θ (3.9)

At higher transmission voltages a shielding failure with a low current may not necessarily cause a flashover. The minimum or critical current required for a flashover would be:

c c

Z U

I ₌ 2 50 _(3.10)

where: U50 - lightning impulse critical flashover, negative polarity, V,

Zc - phase conductor surge impedance, Ω [14, 32].

The maximum current which makes the exposure distance of the phase conductors Dc equal

to zero, can be calculated as follows [76]:

35 . 1 6 . 0 6 . 0 t 2 t 2 FO max p ) δ sin y h ( 34 . 1 y 2 ) y h ( d ( I         − + − + = (3.11)

where: ht– shielding wire height, m

y - average phase conductor height, m (Fig. 3.1).

It depends on the construction of the line towers, their dimensions and location of shielding wires.

3.2. Analysis of Lightning Protection

of Overhead Lines 220 kV Using

Shielding Wires

The protection solution of averhead transmission lines 220 kV with the use of shielding wires was analyzed. The tower (model H52) of overhead line with phase conductors and two shielding wires were arranged horizontally as shown in Figure 3.2.This arrangement is the most common in the 220 kV transmission lines.

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d P2 P1 δ h1 h2 c

Figure 3.2. Tower of the overhead transmission line of 220 kV (model H52)

The basic technical data of the tower, phase conductors and shielding wires of line are listed in Table 3.1, whereas data on line insulators are presented in Table 3.2.

Table 3.1. Basic technical data of tower (model H52), phase conductors and shielding wires

of overhead transmmision line of 220 kV [34, 60, 61, 62]

basic dimensions of tower

h1 h2 c d/2 p1 p2

mm

30800 26500 7600 4950 4400 2700

phase conductor shielding wire

s s

type

mm2 type mm2

AFL 1:8 - 525 525 AFL 1:1.7 - 70 70

Table 3.2. Basic data of porcelain insulators of line 220 kV [60, 61, 62] insulator type: LPZ 75/25

total length length of the insulation part mm

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attached to the tower on height 26.5 m. The total length of insulators is equal to 2.54 m. The spacing between phase conductors is 7.6 m and between shielding wires 9.9 m.

The protection angle δ of the tower is equal to 22 grades. Despite the protection of the line with shielding wires to external phase conductors lightning currents can strike. The maximum peak values Imax of lightning current which can strike to the phase conductor of the

line were calculated. It is equal to 14.08 kA (eq. 3.10). The analysis of the calculation results reveals that the lightning strokes with currents equal to or smaller than 14.08 kA can stroke directly to the phase conductors.