new approach to arc resistance calculation

7/23/2019 New Approach to Arc Resistance Calculation

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NewApproach

V.V. Terzija, Member, IEEE

Saarland University

Saarbriicken, Germany

to Arc Resistance Calculation

H.-J. Koglin

Saarland University

Saarbrticken, Germany

Abstract An important macroscopic arc parameter, describing its

complex nature is

arc resistance.

As known, the fault arc

resistance can be calculated by the Warrington formula. Authors

investigated the results of Warrington’s tests. Warrington derived

a relation for the arc voltage by using the measured arc voltage

gradient and arc current as input data. By analyzing these

measurements and by taking into account the conditions under

which they are obtained (inaccurate measurement devices), it is

unquestionable that the results are highly empirical and not

accurate and general enough, Laboratory testing, provided in the

high power test laboratory FGH-Mannheim (Germany), in which

long high current arcs are initiated, was the base for the research

results presented. In this paper a new approach to ruc resistance

calculation is given. Two independent approaches delivered the

same equation, Both approaches are based on a suitable and a

simple arc model assuming the rectrmguhm wave form of the arc

voltage, which is in phase with the arc current wave form The

new formula for arc resistance is compared with the Warrington

formula. The influences of arc elongation are investigated, too.

Keywords: Long arc in still air, arc resistance, Wamington

formula, laboratory testing, new arc resistance formula.

I. INTRODUCTION

Arc discharge is encountered in the everyday use of

power equipment. Permanent faults in a transformer,

machine, cable, or transmission line always involve an arc.

Whenever a circuit breaker is opened while currying a

current, an arc strikes between its separating contacts. In

Fig. 1, an example of a three-phase arcing fault on a 20 kV

overhead transmission line is presented [1]. The arc

existing at the fault point is a high ~wer long arc in still

air. It has not the same properties as the much more

shorter arc existing in circuit breakers. All arcs posses a

highly complex nonlinear nature, influenced by a number

of factors. An arc can be considered as an element of

electrical power system having a resistive nature, i.e. as a

pure resistance. The length variation is an important factor

in describing the arc behavior and arc resistance. The

nature of arc elongation effects are difiicult and extremely

Fig 1: Three-phase arcing fau lt on a 20 kV overhead transmission l ine.

non-stationary, so they are not simple to be modeled.

In the case of short-circuits occurring on lines within

medium- and high-voltage networks, distance protection

has to locate precisely the fault point for a selective

interruption of the fault. In most cases (over 90Yo) short

circuits in a network are followed with an arc (arcing

faults), so an impedance evaluation, i.e. fault location, is

disturbed by the arc voltage arising at the fault point. In

other words, arc is the source of errors, if it is not taken

into the consideration when locating the fault. To avoid

these errors, the well known Warrington formula [2] for

arc resistance calculation is used.

Empirically obtained results play an important role in

investigating the nature of electrical arc. One of the

earliest experimental studies considering the long arc in

still air are presented in [2,3,4]. Nowadays modern

transient recorders with fast A/D converters are available.

In this paper the results derived by Barrington [2] are

investigated and compared with the results obtained from

laboratory tests provided in a high power test laboratory

FGH-Mannheim (Germany). It is concluded that

Warrington formula is not quite correct, so the new

formula for arc resistance calculation is derived and

compared with the 014 Barrington formula.

Two independent approaches delivered the same new

equation for arc resistance. In the first, definition of

resistance in a-c circuits was the starting point for the

formula derivation, whereas in the second, the spectral

analysis approach is used. Obvious absolute relative

differences between the formulae, in particular for the wide

range of currents, are noticed. The results obtained are

discussed.

0-7803-6672-7/01/$10.00 (C) 2001 IEEE 781



In this paper, first Warrington results are teste~ judge~

discussed and criticized. Secon4 the results obtained in

FGH-Mannheim are presented and a new “macroscopic arc

model” is derived. Thir& a new formula for arc resistance

is derived. Two independent approaches delivered the

same formula. Fourth, the new formula is compared with

the Warrington formula. Before concluding remarks, the

aspects of arc elongation effects and its influences to arc

resistance are investigated.

II. DISCUSSION ON WARRINGTON FORMULA

In [2] Warrington presented his remarkable results of

field tests on the high-voltage systems of the New England

and the Tennessee Electric Power Company. Through

these tests he investigated the influence of arc resistance on

protective devices and derived his well known and widely

applied general formula for arc resistance calculation:

Ua=Ea L=~L

In

(1)

where U, is arc voltage (V), E. is arc voltage gradient

(V/ft, or V/m), 1 m = 0.3048 ft, L is arc length (ft, m), 1

is arc RMS current (A) and K and n are the unknown

constants. The unknown parameters K and n are estimated

from measurements. In [2] Warrington expressed the arc

length in feet @).

In Fig. 2 the third Fig. from [2], scanned and

incorporated into this paper, is presented. In this figure the

measured arc voltage gradient, E, expressed in (kV/ft), is

presented over currents in amperes. Only the selected

measurement set is depicted. Bad measurements are

omitted. In [2] it is not explained how the measurements

are omitted. In the same figure, a curve defining the

relationship between E. and current is plotted. The curve is

obtained using the following parameters included in (1 ):

K=8750, n=0.4, ~=2.5

n

Parameters are valid if length is expressed in (fI). In Fig. 2,

in the Warrington formula given below the graph, the arc

voltage is expressed in (kV). In other words, from the

selected measurement set, Warrington determined

parameters K and n, and by using (1) obtained the curve

showing the relationship between arc voltage gradient and

arc current. By including K and n into (l), one obtains the

following formula for the arc voltage:

8750

~L [V/ft] = 28~~~”5 L [V/m]

‘a = z.fi L = ~o.4

(2)

From (2) the next equation for arc resistance follows:

Ra = . 28’88”5L [Q/m]

I

~1.4

(3)

where voltage is in volts (V), current is in amperes (A) and

arc length in meters (m).

0

200 AM 600 800 &w9

Amp-s F%hmaryWwmi

Fig, %.-~esrs produce characrmisdc equation

,. 8,?50 L

voltage 1 ~

Fig. 2: The orig ina l measurements aud resul ts obtaiued by Barrington [2].

In [2] a table with all measurements obtained by

Warrington are given. Based on the fill measurement set

from [2], in this paper parameters K and n are estimated

and new estimated curves E. over i, are derived. In Fig. 3

both the full measurement set and the estimated curve for

arc voltage gradient are presented. Parameters estimated in

this case are:

K = 3460.49, n = 0.225333, ~ = 4.437877

n

Both parameters are essentially different from the

parameters obtained by Warrington.

2000

. . . . . . . . ..{ . . . . .. . . . . .. . . . . .. . . . . .

1500 -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . ..+ . . . . . .. . .-.. \- -- -- -- --- --

L . . . . . . .. . . . . ... . . . . . .. . . . . . ,. . . . . . . . . . . .

...:......:.*g...:..... : :

. . . . . . . . . . .i . . . . . . .. -.. - -. .. - -. ... .* .

250 -.-.:. ----+ ----{ ---; : - ;.; .;...;.

o

1 1 1

I

1 I 1

1 1

0 1002C03X14CK15CX3 &X17CX38CX)9CKI ICMl

I (A)

Fig. 3: Ful l measurement set and estimated arc vokage gradient curve.

0-7803-6672-7/01/$10.00 (C) 2001 IEEE 782



By observing Fig. 3, it is obvious that for one current

follow several various values for arc voltage gradient. This

variety is probably the consequence of arc elongation

occurring during the tests?

Under the assumption that some measurements were not

correct, i.e. that some of them could be treated as bad data,

in this paper a reduced measurement set is selected and

presented in Fig. 4. From the reduced measurement set the

following unknown parameters are estimated:

K = 11387.4, n = 0.427663, ~ = 2.338289

n

The new curve for E. is depicted in the same plot.

Fig. 4: Reduced measurcmmt set aud estimated arc voltage gradient curve.

In Fig. 5 the full measurement set from [2] and three arc

voltage gradient curves (the Warrington, the full and the

reduced measurement set curves) are presented.

g-

“ml”~..i...?....

.l.. . . . . . . . . . . . . . . . . . . . . .

o}

I 1 I I

I

o 2C0 400 8CKI 800 IU30

I (A)

Fig . 5: Ful l measurements~ from [2] aud three arc vol tage gradient curves.

It is obvious that two new independent and different

equations for arc voltage resistance calculation are

obtained. From the above results, the following general

observations regarding Warrington field tests are

formulated

1. Measurement devices used during Warrington testing

were relatively inaccurate, so the conclusions derived

are not rel iable enough.

2. During the arc life the arc length is changed. These

changes are not considered when Warrington formula

is derived.

3.

4.

5.

6.

A criterion used in [2] by which some bad data are

rejecte~ i.e. omitted from the consideration, is not

described. It seems that the selection of the

measurements processed is provided quite arbitrarily.

This task should be solved by using the known

standard robust estimators, not sensitive to bad data

(outliers).

The method how Warrington formula is derived is not

mentioned in the text.

The range of arc currents observed is rather small

(< 1 kA).

Warrington formula can not be accepted as absolute

correct,- so the new formulae sho~d be derived,

compared with it and applied.

The 6-th observation that Barrington formula is not

absolute correct, as well as the fact that the formula is not

derived by analyzing a wide range of currents (the

expected short circuit currents are reaching today values

over 50 kA), motivated authors to investigate the

possibilities for deriving a new formula for arc resistance.

The new formula should be used as an alternative to the

Warrington one. The response of the international

scientific society to the new approach, which will be

presented, should judge it and investigate if it is suitable

for the practical applications.

III. LABORATORY TESTING AND MODELING OF

LONG ARC IN STILL AIR

The nature of long arc in still air is investigated in the

high power test laboratory FGH-Mannheim (Germany), in

which a series of laboratory tests are provided. Voltage

u t ,

urrent i(t) and arc voltage u,(t) are digitized fkom the

laboratory test circuit depicted in Fig. 6. Arcs between

arcing horns of vertical and horizontal insulator chains are

initiated by means of a fise wire, when switch S in Fig. 6

is closed. The distance between electrodes is changed in

the range of 0.17-2 m. The range of arc currents varied

between 2 and 12 kA.

i(rj

R L

4

s

U(lj

-IT

~ (q

I

arc

J

I

~J

Fig. 6: Laboratory test circuit.

In Fig. 7 the recorded arc voltage u,(t), current i(t),

which is at the same time the arc current is(t) and the

recalculated instantaneous electrical power of arc are

0-7803-6672-7/01/$10.00 (C) 2001 IEEE 783



plotted. Arc voltage and arc current are in phase,

confirming the resistive arc nature. The wave form of arc

power is distorted and from the amplitude point of view,

proportional to arc current. The long high current a-c arc

voltage amplitude is proportional to arc voltage length.

Here the coefficient of proportionality is arc voltage

gradient E.. It is almost independent from the arc current,

so the voltage amplitudes of long high current arcs are

determined by the arc length L. Over the range of currents,

100 A to 20 ~ the average arc voltage gradient lies

between 1.2 and 1.5 kV/m [4,5,6]. In [3] it is shown that

for long arcs almost all the total arc voltage appears across

arc column.

0.14 0.15 0.16 0.17 0.18

t (s)

Fig . 7: Recorded arc wave forms.

The quantitative expression of signal distortion is the

total distortion factor THD, calculated as follows:

where Ua(t) and ia (t) are voltage and current of an arc

having the constant length L and U, is the amplitude of the

rectangular signal. In (5), sgn is a sign fimction defined as:

sgn(x) = 1 ifx20 and sgn(x) = -1 ifx<O and ~(t) is zero-

mean Gaussian noise. The value of U, can be obtained as

the product of arc-voltage gradient E. and the actual arc

length, L.

The arc voltage model (5) has been already successfully

used for the purpose of overhead lines protection (see Lit.

[9, 10]). In this paper, it will be the starting point for the

derivation of the new formula for arc resistance

calculation. In Fig. 8, the simulated arc voltage (eq. (5))

and current from the circuit depicted in Fig. 6, as well as

the recalculated instantaneous electrical power of arc are

presented. Simulation is provided using a software package

presented in [1 1].

—

3“ -lo r

I I I

1

0.04 O.ffl 0.06 0.07

0.06

t (s)

Fig. 8: Simulated arc wave f-.

(4)

where & (h= l,..., M ) is the amplitude of the lr-th

harmonic. Spectral analysis of arc voltages and currents is

provided through the application of Fast Fourier Transform

(FFT). From the calculated amplitude spectra, the

corresponding THD factors for arc voltage and current are

calculated. The average THD factors for arc voltage and

current are approximately 30 and 2

0/0,

respectively.

It can be concluded that arc voltage is a distorted wave

form, whereas arc current is not. Thereby, arc current can

be modeled as a pure sine wave, whereas arc voltage must

be modeled in a such a way that its distortions are

realistically enough taken into account.

By observing the arc voltage and current waveforms

plotted in Fig. 7, one concludes that the voltage signal has

a distorted rectangular form. Additionally, it is in phase

with arc current. Thus, an arc can be represented through

the following equation, modeling the arc voltage [7,8]:

(5)

IV. NEW FORMULA FOR ARC RESISTANCE

In this Section a new formula for arc resistance

calculation is derived. One formula is derived in two

independent ways: a) by using the classical definition of

electrical resistance in a-c circuits and b) by using the

spectral domain analyzes approach.

A. Derivation Using the Classical Resistance Definition

Let us assume that arc voltage u t and current i(t) are

modeled as follows:

u(t)= U.sgn[i(t)]

(6)

i(f) = JZI sin cot

(7)

The resistance R of an element belonging to an a-c circuit

is defined as:

RZ2 = P = -jp(t)dt = (t~(t)dt

o 0

(8)

0-7803-6672-7/01/$10.00 (C) 2001 IEEE 784



where 1 is the RMS of current and p(t) is the instantaneous

maximal and RMS values of the h-th harmonics can be

power. By including (6) and (7) into (8), one obtains:

respectively calculated by using the following formulae:

RI’

. Tp t)dt

T

o

Since

’12

’12

ji(t)dt=fil J’sincotdt=*

o 0

(9)

Hence, for the first harmonic, one obtains:

(lo)

equation (9) becomes:

By defining the arc resistance as:

(11)

From (11) follows the explicit expression for the arc

resistance:

Suppose that there exists a linear relationship between arc

voltage magnitude and arc voltage gradient:

U. =E~L

(13)

Hence

(14)

Equation (14) is the new formula for arc resistance

calculation. It requires a suitable selection of the

value/expression for the arc voltage gradient Ea.

B. Derivation Using Spectral Domain Analyzes Approach

A pure square wave (eq. (6)) can be expressed by Fourier

series containing odd sine components only, as follows:

where h = 1, 3, 5, 7, ... is the harmonics order, ~ is the

phase angle of the fimdamental harmonic and o is the

fimdamental angular frequency. From (15) follows that the

u’

R=~

11

a, m

and by including (18) into (19), one obtains:

a,rms. k : ‘“ _ 2&Ja

1

R=—

11

a, m

I-xI

(16)

(17)

(18)

(19)

(20)

Finally, by including (13) into (20), follows the same

formula for arc resistance, derived by using the classical

arc definition (eq. (14)):

C. Arc Voltage Gradient Selection

(21)

As already mentione& in the new formula for arc

resistance a suitable value/expression for the arc voltage

gradient should be included. From the electrical properties

of an arc under steady conditions (u-i characteristic) point

of view, a number of equations are derived from the

experimental studies. The first and best known is [3]:

(22)

If in (22) L is made sufficiently large, the terms involving

parameters A and C may be neglectet and the

characteristic equation becomes approximately:

)

.= B+; L

(23)

0-7803-6672-7/01/$10.00 (C) 2001 IEEE 785



If in (23) current is sufficiently large, the arc voltage

becomes a fimction only of the arc length, according to the

equation

U. = BL

(24)

Parameter B represents the voltage gradient E. in the arc

column. The arcs occurring on the fault point can be

treated as long high current arcs, so the aforementioned

discussion holds. In the open literature the following

values/expressions for E, calculation are used:

1. in accordance with (24) and from Lit.

E. = (1200+-1500) (V/m)

2. in accordance with (23) and from Lit.

EQ = 950+5000/1 (V/m)

4,5,6]:

(25)

12]:

(26)

In (26)1 is expressed in amperes (A). Equations (25) and

(26) can be now included in (21), so the following two new

equations follows:

RI= (1080.4 +1350.5);

(27)

)

= 855.3+ 4501.6 ~

2

I

I=

(28)

In (27), constant 1080.4 follows if E. = 1200 V/m, whereas

constant 1350.5 follows if E. = 1500 V/m.

In the next Section formulae (27) and (28) will be

compared with Warrington formula:

(29)

Comparison will be provided by comparing the values for a

wide range of currents.

V. COMPARISON BETWEEN WARRINGTON AND

NEW FORMULA

Three formulae: the Warrington formula (29) and two

new formulae (27) and (28), derived in this paper, are

compared by changing the RMS values of arc current in

the expressions for arc resistances, for the in advance

assumed arc length. Here it is assumed that an 1 m long

arc is analyzed (L = 1 m). The current RMS values are

changed in a wide range: from 100-50.000 A. By using

formulae (27), (28) and (29), the values for arc resistances

Rw, R1,12001~ (Rl for E,= 1200 V/m), R1,15001~ (Rl for

E.= 1500 V/m) and Rz are calculated and clearly

presented in Fig. 9. By observing Fig. 9, it can be

concluded that in some ranges of currents RW is greater

than RI and Rz, and vice versa. In Fig. 10, curves depicted

in Fig. 9 are zoomed and presented for currents between 2

and 5 m so that the points at which “new arc resistances”

are equal to Warrington resistance are observable. For

example, for 1 = 6.64 ld follows that R1,1200/m =

Rw.

The

“Warrington curve”

crosses all other three curves. The

currents at which the new resistances are equal to

Warrington resistance can be simply detected in Fig. 11, in

which the absolute relative errors, s (Yo), calculated for RI

and Rz, with regard to Rw, are respectively presented. For

currents greater than respectively 3.64, 2 and 6.5 kA,

Warrington resistance is greater than R1,lzoowm, RI,1soowm

and Rz, respectively.

10

8

g

6

u? 4

2

0

100

ICKJO

Im

I (A)

Fig. 9: Resistances obtained using Barrington and new formulae for L = 1m.

0.7

0.6

0.5

0.2

0.1 1

I

‘

1

I

2(X)O

3mo 4mo

5a30

I (A)

Fig. 10: Curves from Fig. 9for2kA<I<5kA-

I

00 -- -- -- -- -- --- -- -- -

.

R:

RI, l XIV/m

/1

..\...

I XY

1(X)O

lmoo

I (A)

Fig 11: Absolute relat ive errors.

0-7803-6672-7/01/$10.00 (C) 2001 IEEE 786



VI. EXTENTION OF ARC RESISTANCE EQUATION

WITH THE ARC ELONGATION MODEL

VIII. REFERENCES

As a matter of fact, arcs length is changing during the

arc life. The elongation of the long arc is determined by the

magnetic forces produced by the supply current, the

convection of the plasma and the surrounding air, the

atmospheric effects (win~ humidity, pressure), i.e. the

medium in which arc is initiate~ etc.

Arc model (5) does not model the arc elongation effects.

Generally speaking, the arc elongation can be involved

into the model (5) by multiplying it with a suitable selected

arc elongation function L(t). The same is valid for the arc

resistance.

Since the elongation effects are not

deterministic, but extremely stochastic, it is relatively

difficult to model all effects exactly. For example, the

elongation fi.mction L(t) can be selected in the form of an

exponential fimction and in this form included in the arc

resistance formula, as follows:

R@(t)=

ao(t)~+aefl@Ti) h(t - ~)] (30)

where

Rao

is the initial arc resistance, Ti is the inception of

arc ignition, h(t) is Heaviside fimction and a and p are

parameters defining the arc elongation dynamics.

Naturally, some other “elongation functions” can be used

for modeling elongation effects of an arc, as well.

VI. CONCLUSION

Through the investigation of Warrington results it is

concluded that his well known formula for arc resistance

calculation is not absolutely correct. Based on experimental

testing in FGH-Mannheim (Germany), a new formula for

arc resistance is derived. Two independent procedures of

its development are presented. New formula requires a

suitable selection of arc voltage gradient value. Two

approaches for arc voltage gradient are presente~ so that

two new formulae are derived. New formula is compared

by Warrington formula. Obvious differences are observed.

Authors are now expecting the response of scientific

audience to the results presented.

VII. ACKNOWLEDGEMENT

[1] D. Oedin& R. Spech, “Test einea “Digitalen Distanz.schutzrelais” im 20-

kV-Netz”, etzArchiv 5, pp. 171-173, 1983.

[2] AR. Van C. Warrington, “Rcactancc Relays Negligibly AtWtcd by Arc

Impedance”, Electr ical World, S@ember, 19, pp. 502-505, 1931.

[3] H. Ayrton, “The Electric Arc”, The Electrkkm, London, 1902.

[4] AP. Strom, “Long 60-cycle Arc iu Air”, Trans. Am. Inst. Elec. Eng, 65,

pp.113-117, 1946.

[5] T.E. Browne, Jr .,

“The Electric Arc as a Circuit Element”, Journal of

Electrochem. Sot. Vol. 102 No. 1,pp. 27-37, 1955,

[6] A.S. Maikapar, “Extinction of an open electric arc,” Elektrk-heetvo, Vol.

4, pp. 64-69, April 1960,

[7] V. Terzija, H.-J. Koglin, “Long Arc In Free Air: Testing Modelling And

Parameter Est imat ion: Paf i 1”, Proceedings of the UPEC 2000 Conference,

Belfast, w 6-8. Sep. 2000.

[8] V. Terzija, H.-J. Koglin, “Long Arc In Free Air: Testing Modelling And

Parameter Est imat ion: Part 2“, Proceedings of the UPEC 2000 Conference,

Bel fast , UK 6-8. Sep. 2000.

[9] M. Djuri6, V. Terzija, “Anew approach to the arcing faults detection for

autoreclosure in transmission systems,” IEEE Tram. on Power Dehvey,

Vol. 10, NO 4, October 1995, pp. 1793-1798.

[10] Z.M. Radojevic, V.V. Terzija, M.B. Djuri6, “Multipmpose Overhead

Lines Protection Algorithm”, IEE Proc.-Genm,Transm Distrib., Vol. 146,

No. 5, PP.441-445, Sqrtembtx 1999.

[11] D. Ldnar& R. Simon, V. Terzija, “Simulation von Netzmodelkm mit

zweiseitiger Einspeisung mm Test v.. Netzwhutzeinrichtungen,” TB-

157/92 Univ. Kaiserslautem, July 1992,

[12] Y. Goda at all, “Arc Vottage Characteristics of High Current Fault Arcs

in Long Gaps”, IEEE Paper No. PE-186PRD.

IX. BIOGRAPHIES

Vladimir V. Terzija was born in Donji Bara6i

(Bosnia and Herzegovina) in 1962. He

received his B.SC., M. SC.and Ph.D. degrees m

Eltirical Power Engineering tiom

Department

of Electrical Engineerin&

University of Belgrade, Yugoslavia, in 1988,

1993 and 1997, respectively. In 1988 he

joined University of Belgrade, where he is

now an assistant prof=sor teaching coursee in

Electric Power Quality, and Power System

Control. At present he is a Research Fellow at

Institute of Power Entieering University of

Saarkm~ granted by Alexander VO. Humboldt Fo-&lati~. His area_s of

scient if ic intereat are power system protcci ion and control, electr ical power

quality and DSP appl ications in power systems.

Hans-Jtirgen Koglin was born in 1937. He

received his Dipl .- Ing. degree in 1964 and his

Dr.-Ing. degree in 1972 from the Technical

Universi ty Darnrstadt, Germany. From 1973

to 1983 he was a professor at the same

university. Since 1983 he is a f ill profeesor at

Saarland University in

Saarbrocken,

Germany. His main areas of scientific

interests are planing and operat ion of power

systems and specially optimal MV- and LV-

networks. v isibili ty of overbead lines. state

.

estimat ion, rel iabi li ty , corrt iive switching protection and f iel cel ls

The authors gratefully acknowledge to Alexander von

Humboldt Foundation for supporting this research and to

high power test laboratory FGH Mannheim (Germany) for

providing the authors with the laboratory data records.

0-7803-6672-7/01/$10.00 (C) 2001 IEEE 787

new approach to arc resistance calculation

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