Presentation Borghetti Nucci Paolone Dallas 04

Estimation of the statistical distributionsof lightning current parameters

at ground level from the data recorded by instrumented towers

Paper TPWRD 00442-2002.R1

Alberto Borghetti Carlo Alberto Nucci Mario Paolone

University of BolognaItaly

Outline of presentation

1. Introduction The problem What has been done by other Authors

2. Numerical procedure for evaluating lightning current distributions at ground.Application to the Berger’s distributions and comparisonwith results obtained by other Authors

3. Application of the results to the evaluation of indirect lightning performance of overhead lines (not included in Transaction paper)

4. Conclusions

1.Introduction

The problemThe probabilistic approach to power insulation coordination requires the knowledge of the statistical distributions of lightning current parameters.

The lightning current distributions currently used are those derived from experimental data gathered by means of elevated instrumented towers.The ‘attractive radius’ r of the tower tends to increase for flashes with larger currents

lightning current amplitudes are ‘biased’ towards higher values.

General relations between pdf of lightningpeak current to a tower and at ground

( ) ( ) ( )( ) ( )

2

2

0

,,

,

p g pt p

p g p p

r I h f If I h

r I h f I dI∞=

∫r

nearby stroke

direct stroke

h

(Whitehead and Brown, 1969)

The problem is to determine fg being ft and h known

General relations between pdf of lightningpeak current to a tower and at ground

( ) ( ) ( )( ) ( )

2

2

0

,,

,

p g pt p

p g p p

r I h f If I h

r I h f I dI∞=

∫

(Whitehead and Brown, 1969)

We shall disregard current reflections at tower top and base, althoughthey can certainly alter the measured current (e.g. Guerrieri et al, IEEE Trans. PWDR, 1998; Rachidi et al., JGR, 2003) and we shall focus on downward discharges, assuming them perpendicular to flat ground.

We shall disregard current reflections at tower top and base, althoughthey can certainly alter the measured current (e.g. Guerrieri et al, IEEE Trans. PWDR, 1998; Rachidi et al., JGR, 2003) and we shall focus on downward discharges, assuming them perpendicular to flat ground.

Studies performed by other AuthorsThe problem has been studied by several Authors, e.g:Sargent [IEEE Trans PAS, Sept/Oct 1972], by using an attractive-radius three-dimensional electrogeometric model and on the basis of the lightning current amplitude experimental data available at that time, derived a so-called synthetic current amplitude distribution to ground level, which, as shown by Brown[IEEE Trans PAS, Sept/Oct 1972], can be approximated by a lognormal distribution with µg =13 kA and σg =0.32.

Mousa and Srivastava [IEEE Trans. PWDR,1989],have proposed a lognormal distribution of current amplitudes at ground level with µg =24 kA and σg =0.31.

Other forms of distribution have been investigated for the problem of interest by Chisholm et al. [Proc. 1st PMAPS, Toronto, Canada, 11-13 July 1986.

Here we shall focus on the analytical formula derived by Pettersson(see later).

Exposure of a tower to direct lightningLightning leader approaching ground: downward motion unperturbedunless critical field conditions develop juncture with the nearby tower, called final jump. Assuming leader channel perpendicular to the ground plane the flash will stroke the tower if its prospective ground termination point, lies within the attractive radius r. r depends on several factors, such as

charge of the leader, its distance from the structure, type of structure (vertical mast or horizontal conductor), structure height, nature of the terrain (flat or hilly)ambient ground field due to cloud charges.

Several expressions have been proposed to evaluate such a radius. Some of them are based on the electrogeometric model.

( )22 for s g gr r r h h r= − − < for s gr r h r= ≥

Electrogeometrical expressions describing the exposure of a tower to direct lightning

rs

rg

nearby strokedirect stroke

h r

rs = r

rg


h

Where rs and rg are the so called ‘striking distances’ to the structure and to ground respectively.

psr I βα= ⋅ g sr k r= ⋅

Electrogeometrical expressions describing the exposure of a tower to direct lightning

ElectrogeometricalAttractive radius

expressionα β k

Armstrong and Whitehead 6.7 0.80 0.9

IEEE 10 0.65 0.55

psr I βα= ⋅ g sr k r= ⋅

Other expressions describing the exposure of a tower to direct lightning

rnearby stroke

direct stroke

h

They have been inferred, by regression analysis, from the results of more complex and physically-oriented models than the Electrogeometric one.

A formula of the following type can be used

bpr c a I= + ⋅

Model c a b

Eriksson 0 0.84 h0.6 0.76

from Rizk 0 2.83 h0.4 0.63

from Dellera-Garbagnati * 3h0.6 0.80 0.9

* The values reported have been derived by M. Bernardi, by interpolation of plots of the lateral distance of a slim structure vs. its height (in the range 5 to 100 m), calculated using the leader progression model of Dellera-Garbagnati

Studies performed by other Authors Cont.Analytical relation between pdf of lightning peak current to a tower and at ground

We now focus on the analytical formula derived by Petterson [IEEE Trans. PWDR,1991]

1st hypothesis:

( ), bp pr I h a I= ⋅

2nd hypothesis:

( ) ( ) ( )2

1 1 1 1 1 1exp ln ln22g p p g p g

p g g p g g

f I I II I

µ φ µσ σ σ σπ

⎡ ⎤⎛ ⎞ ⎡ ⎤⎢ ⎥= − =⎜ ⎟ ⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎣ ⎦⎣ ⎦

( ) ( ) ( )( ) ( ) ( ) ( )

( )2 2

2

2 2

0 0

, 1 1, ln, ,

p g p bt p p p g

p g gp g p p p g p p

r I h f I af I h I II

r I h f I dI r I h f I dIφ µ

σ σ∞ ∞

⎡ ⎤= = ⎢ ⎥

⎢ ⎥⎣ ⎦∫ ∫

exp(2 ln )pb I

Studies performed by other Authors Cont.Analytical relation between pdf of lightning peak current to a tower and at ground

We now focus on the analytical formula derived by Petterson [IEEE Trans. PWDR,1991]

1st hypothesis:

( ), bp pr I h a I= ⋅

2nd hypothesis:

( ) ( ) ( )2

1 1 1 1 1 1exp ln ln22g p p g p g

p g g p g g

f I I II I

µ φ µσ σ σ σπ

⎡ ⎤⎛ ⎞ ⎡ ⎤⎢ ⎥= − =⎜ ⎟ ⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎣ ⎦⎣ ⎦

( ),t pf I h

( )22expt g g g tbµ µ σ σ σ⎡ ⎤= ⋅ =⎢ ⎥⎣ ⎦

is lognormal

Pettersson’s formula

2. Numerical procedure for evaluating lightning

current distributions at ground.

Proposed numerical procedure for evaluating lightning current distributions at

groundIp* minimum peak current observedr* its attractive radius

All the strokes with perspective stroke location within 2πr*2

are collected by the tower

I. Generation of a population of events Ip, tf, … and xg∈[0,r(Ip)]- IP, tf … are generated from the p.d.f. of strokes to the tower- xg are generated assuming the stroke locations be uniformlydistributed (correlations are taken into account)

II. Selection of the events with xg<r*

III. Determination of the statistical distributions of the currentparameters related to such events

Application to the Berger’s distribution


27,7


2,3

Application to the Berger’s distribution Cont

From Andersson and Eriksson, Electra , 1980


= T30/90 / 0.6 = 3,8

I p

t f

Attractive radiusexpression a b

Eriksson 0.84 h0.6 0.76

Rizk 2.83 h0.4 0.63

( ), bp pr I h a I= ⋅

Parameter Expression

Eriksson Rizk

µg 20.1 21.3

σg 0.20 0.20

µg 3.2 3.3

σg 0.24 0.24

ρg 0.48 0.47

tf (µs)

Ip (kA)

0.47tρ =σt



Attractive radiusexpression a b

Eriksson 0.84 h0.6 0.76

Rizk 2.83 h0.4 0.63

( ), bp pr I h a I= ⋅


Eriksson Rizk

µg 20.1 20.0 21.3 21.2

σg 0.20 0.20 0.20 0.20

µg 3.2 3.3

σg 0.24 0.24

ρg 0.48 0.47

tf (µs)

Ip (kA)

0.47tρ =σt


( )22expt g g

g t

bµ µ σ

σ σ

⎡ ⎤= ⋅⎢ ⎥⎣ ⎦=

Comparison with the Pettersson’s formula for the case of electrogeometric expressions

rs = rl

rg


h

rs

rg


h rl

psr I βα= ⋅

g sr k r= ⋅

( )22 for s g gr r r h h r= − − < for s gr r h r= ≥

( ), bp pr I h a I= ⋅

b β=gh r≥If

/ 22 pr h I βα= ⋅2

b β= and g g sh r r r<< =or


expressionα β k


0.55IEEE (1243) 10 0.65

0.47tρ =σt



A&W IEEE

µg 20.2 21.1

σg 0.21 0.20

µg 3.2 3.3

σg 0.24 0.24

ρg 0.49 0.47

tf (µs)

Ip (kA)


expressionα β k


0.55IEEE 10 0.65

0.47tρ =σt



A&W IEEE

b= β /223.4

b=β19.7

b= β /224.1

b=β21.0

0.20 0.20

Probabilitythat conditions are verified 4.2% 28.8% 0.02%

0.20

80%

0.20

µg 20.2 21.1

σg 0.21 0.20Ip (kA)

Attractive radiusexpression c a b

From Dellera & Garbagnati 3h0.6 0.80 0.9

0.47tρ =

( ), bp pr I h c a I= + ⋅

σt


Parameter Dellera & Garbagnati Expression

µg 22.1

σg 0.19

µg 3.4

σg 0.24

ρg 0.45

tf (µs)

Ip (kA)

Attractive radiusexpression c a b

From Dellera & Garbagnati 3h0.6 0.80 0.9

0.47tρ =

( ), bp pr I h c a I= + ⋅

σt


Parameter Dellera & Garbagnati Expression

µg 22.1 18.1

σg 0.19 0.20

µg 3.4

σg 0.24

ρg 0.45

tf (µs)

Ip (kA)


( )22expt g g

g t

bµ µ σ

σ σ

⎡ ⎤= ⋅⎢ ⎥⎣ ⎦=

Application to the CIGRE distribution

1: AW; 2: IEEE 4: Eriksson; 5: from Rizk; 6: from Dellera-Garbagnati

Influence of the Tower Height

1: AW; 2: IEEE 4: Eriksson; 5: from Rizk; 6: from Dellera-Garbagnati

First Conculsions

The proposed method is more general than the other proposed so far in the literature.

What are the applications?

3. Application of the results to the evaluation

of indirect lightning performance of overhead lines

Application of the results to the evaluation of indirect lightning performance of overhead lines

The ___ is represented by the following curve:

VOLTAGE [kV] (or CFO)

Lightning Performance of a Distribution Linerelevant to induced voltages

Lightning Performance of a Distribution Linerelevant to induced voltages

Annual number of induced voltagesexceeding the value in abscissa(or Annual number of flashovers)

Application of the results to the evaluation of indirect lightning performance of overhead lines Cont.

How to calculate the ____?


Clearly, the ___ depends on:

• models used to calculate the induced voltages

• lateral distance expression

• statistical distribution of lightning parameters

Models used to calculate the induced voltages

i (0,t)Return-Stroke Current

RSC i (z,t)

Lightning ElectroMagnetic Pulse

i (z,t) LEMP E, B

ElectroMagnetic Coupling

E, B EMC U(t)

Models used to calculate the induced voltages

i (0,t)Return-Stroke Current

RSC i (z,t)

Lightning ElectroMagnetic Pulse

i (z,t) LEMP E, B

ElectroMagnetic Coupling

E, B EMC U(t)

LIOVLIOVLIOV

The LIOV codeThe LIOV code

www.ing.unibo.it/die/liov

LIOV codeModels

The LIOV codeThe LIOV code

• Return-stroke model: MTLE (and TL)• LEMP: Uman and McLain and Cooray-Rubinstein• Coupling model: Agrawal extended to the case of

lossy ground

This allows to take into account more realistic line configurations than the simplified Rusck expression

The Agrawal coupling modelThe LIOV codeThe LIOV code

),,(),()(),('),(0

' htxExittxit

Ltxvx

ix

t

giijsi =

∂∂

−+∂∂

+∂∂

∫ τττξ

0),('),( =∂∂

+∂∂ txv

tCtxi

xsiiji

Transmission line coupling equations(Agrawal et al.)

Overhead line above lossy ground

The Agrawal coupling modelThe LIOV codeThe LIOV code

=∂

∂−+

∂∂

+∂∂

∫ τττξ ),()(),('),(

0

' xittxit

Ltxvx

t

giijsi

0),('),( =∂∂

+∂∂ txv

tCtxi

xsiiji

),,( htxEix

The ground resistivity plays a role in

1) the calculation of the line parameters

2) the calculation of the electromagnetic field

The LIOV codeThe LIOV codeThe LIOV code allows for the calculation of lightning-induced voltages along a multiconductor overhead line as a function of- lightning current waveshape (amplitude, front steepness, and duration), - return stroke velocity, - line geometry (height, length, number and position of conductors), values of resistive terminations, - ground resistivity and relative permittivity.

It allows also taking into account induction phenomena due to - the leader field, - non-linear phenomena such as corona - and the presence of surge arresters.

In order to take into account the presence of more complex types of terminations, as well as of line discontinuities and complex system topologies, the LIOV code has been interfaced with EMTP96.

The LIOVLIOV code calculates:• LEMP• Coupling

The EMTPEMTP :• calculates the boundary conditions• makes available a large library of power components

Γ0

LIOV-line

n-port

Concept at the basis of the interfaceConcept at the basis of the interface

0

u (x,t)

i(x,t) L'dx

C'dx

x x+dx

+- i(x+dx,t)

L

u (x+dx,t)

ui(x,t)

Eix(x,h,t)dx

ss+-

Γ0

i0

+-

ΓL

u0 iL uL

-u(L,t)i-ui(0,t)

Node ‘0’

+ -

u0(t), i0(t)

G2 +

-+

-

Γ0(t)

u1(t), i1(t)

Zc

u0’(t)

i0’(t) ve(t)

Node ‘1’

LIOV line Bergeron Line EMTP termination

+ -

-ve(t)

G1

Zc

u2(t), i2(t)

Node ‘2’∆x

Link between LIOV and EMTPLink between LIOV and EMTP

Treatment of the boundary conditions, data exchange between the LIOV code and the EMTP96:

Treatment of the boundary conditions, data exchange between the LIOV code and the EMTP96:

Link of the LIOV code with EMTP96Link of the LIOV code with EMTP96

EMTP(1) Line terminals node voltages at time step N TACS LIOV

LINE

(2) Line terminals node voltages at time step N

(3) Line terminals branch current at time step N+1

(4) Line terminals branch current at time step N+1

N=N+1

(6) Line terminals branch current at time step N+1 injected in the EMTP by current controlled generators (5) EMTP time step

increment for node voltages and branch currents

TACS – LIOV LINE V (1) V (1)

(2) (3)

(6) (6)

EMTP (4) (5)

V - voltage signal generator I – current controlled generator

I I

Link of the LIOV code with EMTP96Link of the LIOV code with EMTP96

Comparison with data measured on real scale complex line model at the ICLRT of the University of Florida (July-August 2002/2003)

0

10

20

30

40

50

60

70

80

90

0 5 10 15 20 25 30 35 40Current [kA]

Vol

tage

[kV

]

Surge arrester characteristic

LIOVLIOV--EMTP96: simulations and comparison with EMTP96: simulations and comparison with experimental data experimental data Cont.Cont.


Experimental overhead distribution line installed at the ICLRT. The indicated quantities are the measured lightning-induced currents.


-40

-20

0

20

40

60

80

100

120

140

0.E+00 1.E-06 2.E-06 3.E-06 4.E-06 5.E-06Time [s]

Indu

ced

Cur

rent

IG10

Measured current

Simulated current

-50

0

50

100

150

200

250

0.E+00 2.E-06 4.E-06 6.E-06 8.E-06 1.E-05Time [s]

Indu

ced

Cur

rent

[A]

IBN6 SimulatedIBN6 Measured

Experimental data are courtesy of ICLRT of University of Florida (US)

Current flowing through the phase B conductor of pole 6

First event of 02-08-03 6th return stroke

First event of 02-08-03 6th return strokeCurrent flowing through the arrester located at pole 6 phase B

-2.50E+02

-2.00E+02

-1.50E+02

-1.00E+02

-5.00E+01

0.00E+00

5.00E+01

1.00E+02

1.50E+02

0.E+00 2.E-06 4.E-06 6.E-06 8.E-06 1.E-05Time [s]

Indu

ced

Cur

rent

[A]

IB6 Simulated

IB6 Measured


-600

-400

-200

0

200

400

600

800

1000

1200

1400

0.E+00 2.E-06 4.E-06 6.E-06 8.E-06 1.E-05

Time [s]

Indu

ced

Cur

rent

[A]

IG6 SimulatedIG6 Measured

-500

0

500

1000

1500

2000

0.E+00 2.E-06 4.E-06 6.E-06 8.E-06 1.E-0Time [s]

Indu

ced

Cur

rent

[A]

IG2 SimulatedIG2 Measured

First event of 02-08-03 6th return stroke

First event of 02-08-03 6th return strokeCurrent flowing through the grounding of pole 6

Current flowing through the grounding of pole 2

Experimental data are courtesy of ICLRT of University of Florida (US)



Clearly, the ___ depends on:

• models used to calculate the induced voltages

• lateral distance expression

• statistical distribution of lightning parameters

IEEE Std 1410(solid curve)

- proposed method (ideal ground)

- proposed method (lossy ground σg= 0.001 S/m)

IEEE Std 1410(solid curve)

- proposed method (ideal ground)

- proposed method (lossy ground σg= 0.001 S/m)

Single conductor line

0 100 200 300CFO [kV]

0.01

0.1

1

10

1E+2

Flas

hove

r / 1

00km

/ ye

ar

IEEE Guide

proposed method (ideal ground)

proposed method (lossy ground)

Comparison with IEEE Std 1410-1997

tf is lognormally distributed with median value 3.83 µs.

Correlation factor between tfand IP : 0.49 i

0 100 200 300CFO [kV]

0.01

0.1

1

10

1E+2

Flas

hove

rs /

100

km /

year

IEEE Guide

LIOV - tf = 0.5 us

LIOV - tf = 1 us

LIOV - tf = 3 us

Note that with small values of tfthe two methods predict basically the same results

Comparison with IEEE Std 1410-1997

Ideal groundIdeal ground


For the calculation of the indirect lighting performance of the overhead use is made of the procedure proposed by Borghetti and Nucci [ICLP, 1998; Sipda, 1999] , based on the Monte Carlo method.Each event is characterized By 4 random variables

• peak value of the lightning current Ip

• front time tf (correlated with Ip)

• two co-ordinates of the stroke location (x and y)




For the calculation of the indirect lighting performance of the overhead use is made of the procedure proposed by Borghetti and Nucci [ICLP, 1998; Sipda, 1999] , based on the Monte Carlo method.Each event is characterized By 4 random variables







3. Application of the results to the evaluation of indirect lightning performance of overhead lines Cont.

3. Application of the results to the evaluation of indirect lightning performance of overhead lines Cont.


1 Inputs:- lightning current parameters (Ip and tf)- return stroke velocity- line and ground data

2 Random generation of events ( Ip ; tf : x ; y) > 20 000

3 Induced overvoltage calculation using LIOV or LIOV-EMTP96

4 Counting of the events generating overvoltages greaterthan 1.5 x CFO

5 Plot the graph:No. of flashovers/100 km/year vs CFOwhere No. of flashovers/100 km/year =

(n/ntot) • ng • S • 100/L (with ng=ground flash density)


Let us now calculate the indirect lightning performance of an overhead line by using either:

• the lightning current statistical distribution by Berger (CIGRE distribution) biased by the presence of the tower;

• the statistical distributions at ground inferred using the proposed method.

• the lightning current statistical distribution by Berger (CIGRE distribution) biased by the presence of the tower;

• the statistical distributions at ground inferred using the proposed method.

We consider a single-conductor overhead line with the following characteristics:

• 2 km long; • 10 m high line;• ‘matched’ at both end;• “striking area” around the line about 20 km2.

• 2 km long; • 10 m high line;• ‘matched’ at both end;• “striking area” around the line about 20 km2.


0.01

0.10

1.00

10.00

100.00

50 100 150 200 250 300Voltage [kV]

No.

of i

nduc

ed o

verv

olta

ges

with

mag

nitu

de

exce

edin

g th

e va

lue

in a

bsci

ssa/

(100

km

yea

r) Armstrong-WhiteheadIEEEErikssonRizkDellera-Garbagnati


CIGRE distributionCIGRE distributionCIGRE distribution


0.01

0.10

1.00

10.00

100.00

50 100 150 200 250 300Voltage [kV]

No.

of i

nduc

ed o

verv

olta

ges

with

mag

nitu

de

exce

edin

g th

e va

lue

in a

bsci

ssa/

(100

km

yea

r) Armstrong-WhiteheadIEEEErikssonRizkDellera-Garbagnati

UNBIASED distributionUNBIASED distributionUNBIASED distribution



0.01

0.10

1.00

10.00

100.00

50 100 150 200 250 300

Voltage [kV]

No.

of i

nduc

ed o

verv

olta

ges

with

mag

nitu

de

exce

edin

g th

e va

lue

in a

bsci

ssa/

(100

km

yea

r)

Armstrong-Whitehead

IEEE

Eriksson

Rizk

Dellera-Garbagnati

Lossy ground(0.001 S/m)


CIGRE distributionCIGRE distributionCIGRE distribution


0.01

0.10

1.00

10.00

100.00

50 100 150 200 250 300

Voltage [kV]

No.

of i

nduc

ed o

verv

olta

ges

with

mag

nitu

de

exce

edin

g th

e va

lue

in a

bsci

ssa/

(100

km

yea

r )

Armstrong-WhiteheadIEEEErikssonRizkDellera-Garbagnati

UNBIASED distributionUNBIASED distributionUNBIASED distribution



Conclusions

The use of unbiased current statistical distributions results, as expected, in a better performance of the distribution line to indirect lightning, these distributions being characterized by a lower median value.

We have shown how the results vary depending on the expression adopted to evaluate the lateral distance (attractive radius).

Also, the ground resistivity acts in minimizing the difference between the line performance calculated using the biased and unbiased lightning current distributions.

Conclusions

It appears, however, that additional study on attractive radius expressions is badly needed.

The results obtained show that statistical current distributions at ground are probably characterized by lower median values than the relevant distributions gathered by means of instrumented towers.

This is in line with lower median values obtained by means of lightning location systems, although …

Presentation Borghetti Nucci Paolone Dallas 04

Documents

Transcript of Presentation Borghetti Nucci Paolone Dallas 04