IEEE Transactions on Power Apparatus and Systems, Vol. PAS-100, No. 1, January 1981

STATISTICAL STRESS-DURATION CHARACTERISTICS OF HIGH VOLTAGE INSULATION

F.A.M. RizkSenior Member

Institut de recherche de l'Hydro-QuebecVarennes, Quebec, Canada

Abstract - New statistical models are developedfor analysis and prediction of stress-duration char-acteristics of high voltage insulating systems. Thesemodels take into account available knowledge on insu-lation breakdown probability as a function of the testvoltage applied for a limited duration as well as thecharacteristics of the delay time to breakdown. Specialreference is made to the behaviour of the models at lowprobability which is particularly important for theevaluation of the risk of failure. The analytical toolsdeveloped are justified on the basis of experimentalresults obtained with transformer oil and compressedSF6.

INTRODUCTION

Statistical methods have long been recognized asan essential tool for analysis of high voltage testsl,2and for the coordination of insulating systems3'4.Although the bulk of available statistical materialrelates to external insulation and impulse voltages5,there is a growing interest in statistical character-istics of internal insulation6 and in long durationpower frequency stresses7'8 This interest has beenparticularly enhanced by the tendency to dissociate thepower frequency test from system transients so as to bebased exclusively on long duration operating voltagestresses9.

Most of the available statistical material on highvoltage insulation relate the breakdown probability tothe prospective amplitude of the voltage impulse6 or

the magnitude of the constant voltage stress appliedfor a fixed duration8. In these experiments the actualtime to breakdown is usually disregarded. In many casesa gaussian distribution was established although insome occasions extreme-value distributions9 were alsoreported6.

The other statistical aspect of high voltage insu-lation that has to some extent been investigated pre-viously is the delay time to breakdown or the breakdowntime lag normally measured under unit-step stress. Ingeneral, the distribution of time to breakdown exhi-bited exponentiall or multi-exponential character-istics6,11.

Although the aboye distributions yield valuableinformation on the characteristics of the insulatingsystem investigated they are by no means complete. Aprobability distribution in both stress and durationwill offer major advantages. For example, such a dis-tribution will provide a much sounder base for appli-cation of results of laboratory tests, with necessarilylimited duration, to far longer life service. Further-more, the stress-duration distribution will allow com-

parison of test results with different durations, e.g.

the 1-minute test for which considerable laboratoryresults already exist and the newly recommended testwith duration as long as one hour12. Under certain con-ditions developed below the combined stress-durationdistribution will also establish the equivalence bet-ween ramp and constant voltage tests. Stress-durationdistributions, particularly if developed with specialreference to low breakdown probabilities, will providea tool for a more realistic assessment of risk of fail-ure of insulating systems.

The major obstacle for establishing combinedstress-duration characteristics of insulating systemsis the large volume of tests necessary and consequentlythe considerable time and cost involved. However, ifproper use is made of existing knowledge of breakdownvoltages and time delays, as will be shown below, theamount of testing required could be considerably ratio-nalized.

THEORETICAL ANALYSIS

Consider an insulating system exposed to a con-stant stress U and a variable duration t. Let us desig-nate the breakdown probability corresponding to any

stress U and duration t by P(t,U). Let us also define ahazard function13 h(t,U) related to P(t,U) by the rela-tionship. r

C1 k, k T t U )

h (, U) dt

=1 - P(t,U)

Then without loss of generality:t

h(t,U)dt-. 0

(1)

P(t,U) 1 - e (2)

It is seen that the combined stress-duration probabil-ity is completely determined by the hazard functionh(t,U). As to the nature of h(t,U) we will considera few separate categories.

Time Independent Hazard Function

If h(t,U) is defined by:

h(t,U) = h(U) (3)

for t > to

then it follows from (2) that:

P(t,U) = 1 - e-h(U) (t - to) (4)

It is clear that for h(U) > 0 as t + , P(t,U) + 1 SO

that breakdown will be a certain event. Under constantstress expression (4) could then be looked at as givingan exponential distribution of breakdown time lags andthe hazard function h(U) would then express the inverseof the statistical time lag T(U):

T(U) =1

h (U)

F 80 189-1 A paper recommended and approved by the

IEEE Transmission & Distribution Committee of the IEEE

Power Engineering Society for presentation at the IEEE

PES Winter Meeting, Npw York, NY, February 3-8,1980.

Manuscript submitted September 18, 1979; made availablefor printing November 1, 1979.

(5)

and to becomes the formative time lag which in manycases could be neglected with respect to the durationt.

Under the above assumptions all that is needed todetermine the combined stress-duration probabilityP(t,U) is to investigate the nature of the hazard func-tion h(U) for different stresses U.

0018-9510/81/0100-0400$00.75© 1981 IEEE

400

If a laboratory test is carried out with a limitedduration.T then neglecting to, it follows from (4) that:

P(T,U) 1 - eh(U) T (6)

Eliminating h (U) between (4), (6):

- t lnP (t,U) = 1 - e T - P7(T)U)J

Expression (7) can be put in the form

P(t,U) 1 - (1 - P(T,U))t/T (8)

which expresses the familiar relationship between break-down probability during a test with duration T and thecorresponding probability if this test is repeated fort/T times. This is valid only, as shown above, as longas the hazard function is independent of time, i.e. thetime to breakdown is exponentially distributed.-

As to the nature of P(T,U) and its dependence onthe stress level U, considerable laboratory materialalready exists. Extreme-value distribution Type I9 andWeibull distribution14 are particularly useful infitting laboratory data at low probabilities. Indeed,it has been analytically shown15, under rather generalconditions, that if the breakdown results frot a weaklink mechanism due to a large number of active elements,the breakdown probability tends asymptotically to anextreme-value distribution of the above types.

If the results of P(T,U) fit into a smallest valuetype I extreme-value distribution:

U - U (T)a

P (T,U) = 1 - ee U (T)(9)

where the parameters Ua and U1 of the distribution arein general functions of the test duration T. Fro'm (7),(9) the expression- of the combined stress-durationprobability takes the form:

U - Ua(T).t --

e U (T)T I1 IP (t,U) = 1 - e

In order to get some insight on the nature of Ua(T) andU1(T) let us consider two test series carried out withdurations T1 and T2. If (9) is valid for any test offixed duration T then:

P(T2,U) = 1 -

a 2

e-e U1(T2)

On the other hand substituting t = T2 and T = T1 in(10) yields: U - U (T

P (T2,U) = 1 -

T -T e U1 (T2)

e 1 (12)

Since (11), (12) are identical it can be easily shownthat the conditions necessary are:

(13)U (T ) = U (T ) = U (T) = const.1 1 1 2 1

and

or

(14)U (T ) + U ln- U (TGa 1 1 T2 a

U (T ) + U ln T TUi (T2 + U1 ln T2=U (T) + U ln T = const. (15)

Condition -(13) shows that for breakdown governed;by a single mechanism and a constant hazard functionthe shape parameter U1 determining the slope of theP(T,U) distribution on an extreme-value probability

401

paper must be a cQnstant independent of the test dura-tion. On the other-hand (14) or (15) shows that underthe same conditions Ua (T), which determines the break-down voltage corresponding to P(T,U) = 63%, decreaseswith the natural logarithm of the test duration. Theseconditions will be tested below against experimentalresults.

If instead P(T,U) fits into a Weibull distribution:

U -U (T) K_(T - U;T

P(T,U) 1 - e lU(T) 1 (16)

where K, U1(T) and Uo3(T) are parameters of the distri-bution, then substituting in (7) the combined stressduration probability is obtained as:

-U - U (T)

P(t,U) 1 - e T U (T) - U (T) (1)

Similar to the approach described above consideringtest duration Tl, T2, if (16), (17) are valid for anytest duration T the necessary conditions are found to

be:K = const.

U (T ) = U (T ) = U (T) = const.0 2

(18)

and //K KT (U (T) U] T2 [U1 (T) -U]

= Tl/ [U1(T) - U0] const.

Here (18) shows that for a single breakdown mechanismand a constant hazard function the threshold stress ofthe mechanism concerned must be a constant independentof the test duration. On the other hand the parameterU1(T) which is a measure of dispersion decreases withthe test duration as expressed by (19). The specialcase U0 - 0, using (17), (19) yields:

K

- const. t , U

P (t,U) = 1 - e (20)

Separation of Variables

If the stress and duration variables in the hazardfunction are separable then h(t,U) can be expressed as:

h(t,U) = f' (t) - h(U) (21)

where f' (t) dIf(t)/dt, f(t) being a function of time

only and h(U) a function of stress only.

Substituting in (2) and assuming f(O) 0 we get:

-f(t) h (U)P (t,U) - e

For a test with limited duration T, (22) gives:

- f(T) * h(U)P(T,U) 1 - e

Eliminating h(U) between (22), (23):f *ln _1

P(t,U) = 1 - e ln( -_ P(T,U)

which in analogy with (7) could be expressed as:

f (t)/f (T)P(t,U) - 1 - [I - P(T,U)]

(23)

(24)

(25)

If P (T,U) fits into a smallest value type I distribu-tion

(10)

(221

402U - U (T)

- f(t) a

P(t,U) = 1 - e f(T) 1

with E A, = 1 and m is the number of mechanisms in-_

1_

(26)

For P(T,U) to be a type I distribution at any T thensimilar to (13), (15) we get the necessary conditionsas:

U (T ) =U (T) = U (T) = const. (27)1w 1 1 2 1

which is identical to (13), and:

or

f(T1)Ua(T1) + U1 l f(T ) at 2

2

U CT ) + U in f(T ) = U (T ) + U ln f(Ta 1 1 1 a 2 1 2

(28)_ _ - ~~~~~~ (29)= U (T) + U ln f(T) = const.

a .1

If on the other hand P(T,U) fits into a Weibull distri-bution we obtained:

f(t) rU - U (T)P(t,U) = - e f(T) lU1(T) - U (T) (30)

with the conditions for the general validity of thedistribution P(T,U):

K = const.(31)

U (T ) = U (T ) = U (T) const.ol o 2 o

andr (T 1/K [U (Tr) - U rI= [f(T)]/K [U (T ) - U I

1 1[1 o L2 1L 21/K

= [f (T)] (1IU CT) - U I1 0

= const. (32)

It is obvious that the case. of a constant hazardfunction treated above constitutes a special case ofthe hazard function with separable variables, in whichsimply f(t) = const. t.

Another interesting special case is f(t) = (t/TO)Owhere To and a are constants which yields f(t)/f(T) =(t/T)O. If furthermore a Weibull distribution isadopted and assuming for simplicity UO = 0, expression(3) for the stress-duration probability, subject to(31) and (32) yields:

- const. t UKP(t,U) = 1 - e (33)

which has been used previously8 for the analysis ofstress-duration probability in transformer oil.

Multi-Exponential Delay Distribution

If several breakdown mechanisms each characterizedby a time-invariant hazard function act simultaneously,then as long as the physical properties of the testobject do not change, an equivalent constant hazardfunction can be found to describe the breakdown distri-bution16. Here the statistical time delays add likeparallel resistors16 and the analysis described aboveremains valid.

Assume however that the test object develops phys-ical inhomogeneities under stress which occasionallyfavor breakdown due to one or another of the mechanismsinvolved. The distribution of the breakdown delay canthen assume a multi-exponential form6"7:

n -t/Ti

P(t) = 1 - v A. e

1=1

1 volved.

Examples of the physical inhomogeneities describedhere include formation and disintegration of gas bub-bles or water drops in an oil gap, accumulation of con-ducting particles and gap cleaning under prolongedstress, etc. Under these circumstances short durationtests may not be indicative of the long term perform-ance of the insulating system under consideration.

In general the coefficients Ai are functions ofthe stress U to the extent that some mechanisms may bepresent or absent depending on the stress level6.

From (34) it is obvious that P(t) + 1 as t - O sothat with Ti = Ti(U) we get:

-t/T. (U)P(t,U) = 1 - EA.(U) e

i=l

(35)

Normally the values of the statistical time lagsTi associated with the different breakdown mecha-nisms6'11 are such that for a test duration t of theorder of Ti we can assume as a first approximation,that if Ti's are arranged in increasing order:

for t Tie-t/T «V< 1e t/V

and -t/T 1e 'i

for all V < i

for all p > i(36)

(37)

This simplifies (35) to take the form

m -t/T. (U)

P(t,U) 1 - E A1 - A. e 1

XP=i'+lfor t Ti

For a test duration T of the order of Ti:

n-T/T.(U)

P(T,U) - 1 - A - A. e

P=i+l

(38)

Eliminating Ti(U) between (37), (38) we get thefollowing general expression for the combined stress-duration probability: r -'

m t

P(t,U) 1 - A - A. e

pI=i+l

for t - Ti and T - Ti

which can be put in the form:

m t

P(t,U) 1 - AP - A.

,U=i+l

A.ln A1

I m1- E Ap,-P(T,U)

,u-i=+l

(39)

m At/T

A11 - p(T,U) (0

-t=i+l

Of particular interest for long duration risk of fail-ure evaluation is the slowest breakdown mechanismassociated with i=m which gives E A11 = 0 and Ai = Am.In this case (40) yields for t - Tm:

/1 - P(T IU) t/TMP(t,U) 1 - Am - ) (41)

which is defined for P(T,U) > 1 - Am, Tm- Tm and t Tm

If the quantity between brackets in (41) can be ex-pressed as:

I -- %

(43)

403

U - U (T )1 - P(T ,U) -e U (T )

A = e 1 mm

then substituting in (41)

U - U (T )ta m

P(t,U) 1 - A e T e Ul (Tm)m m

(42)

which differs from (10) by the factor Am. Similarly themodified Weibull case would give:

_tr,k[ U TU ( m)TP(e~~~ ri 1_zU (T )-U (T ) {

r %L UI L - H eT ) /

Consider now the fastest breakdown mechanism resultingin initial faults in the insulating system for whichi= T1, Ai = A1 and E Al = 1 - A1. Substituting in(40) the combined stress duration distribution will beexpressed as:

I t/T

P(t, U) = A - A 1 - 1T

for T1 T1 , t T1 and P(T1iU)< A

If P(T1,U)/A1 can

P(Ti ,U)1 1 -

A1

0.

(45)

be expressed in the formU - Ua(T1)

e-e UJ1(T1)

then substituting in (45)tU - U (T )

P(t,U) A [1 e Ul(Tl)

which differs from (10) by the factor A1.

(47)

Comparison of (44), (47) shows that for the multi-exponential multi-mechanism case the parameters of thedistribution change with the test duration so that itwill not in general be possible to relate Ua(Tl) toUa(T2) or Ul(T1) to U1(T2) as described above for a

test object that remains homogeneous under stress.One obvious conclusion from the analysis of the

multi-exponential multi-mechanism case is that for ac-

curate evaluation of the long duration risk of failurethe test stress and duration must be such that theslowest breakdown mechanism at the service voltage isadequately accounted for.

APPLICATIONS

In this section some numerical examples will begiven to illustrate the utility of the models developedabove.

In Fig. 1, the solid lines illustrate the depen-dence of the breakdown voltage on the normalized stressy(T) expressing [U - Ua(T)]/Ul(T) for different valuesof t/T presented on an extreme-value paper for the time-invariant hazard function according to (10). The dashedlines correspond to the hazard function with separablevariables (26) for which f(t) = const. ta. It is clearthat a = 1 corresponds to the time invariant hazardfunction, and that the risk of failure for a given test-duration is extremely sensitive to the value of a.

Fig. 2 presents similar numerical results for thecase of a Weibull distribution with K = 10 correspond-ing to expressions (17), (30). The normalized stressvariable Z(T) refers to [U - UO(T)]/(U1(T) - UO(T)).

As mentioned above, combined stress-duration dis-tributions can be used to obtain the U - t equi-proba-bility curves. These curves express the relationshipbetween test stress and duration on condition that anysuch pair will produce a constant risk of failure.

-10 -8 -6 -4 -2 0y(T), p.u.

Fig. 1 Cumulative breakdown probability as a functionof normalized stress presented on an extreme-value paper for different values of t/T.Full lines: time-independent hazard function.Dashed lines: time varying hazard function witha 0.5.

0.99 l

0.90

0.50 /-

ms~ ~ ~~~~~40-34

1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 4.0Z(T) , p.u.

Fig. 2 Cumulative breakdown probability as a functiono£f normalized stress presented on a Weibullpaper (K = 10) for different values of t/T.Full lines: time-independent hazard function.Dashed lines: time varying hazard function witha = 0.5. probability scale constructed from

Fig. 3 presents equi-probability curves based onthe combined stress-duration models expressed by (10),(26) for the time-invariant hazard functions and thatof separable variable with a = 0.5 respectively. Thevalues of the probabilities chosen are O.OO1, O.O1 and0.05. It is shown that for a test-object that remainshomogeneous under stress the equi-probability curvebecomes a straight line on a semi-logarithmic paper.This will not in general be the case with the multi-

-J(46) 0.000i I_,--

404

exponential time to breakdown case described by ex-pressions (39), (40).

looc \ }

1.00 a

-12 -40 -8 -6 -4 -2 oy (T), p.u.

Fig. 3 Equi-probability normalized stress-durationcurves based on P(U,,T) -of the extreme valuetype. 'Full lines: time independent hazard function.Dashed lines: time varying hazard function with

0.5.Similar equi-probability curves for the Weibul-l

distribution case based on expression (17), (30) areshown on a log-log paper in Fig. 4 for probabilitylevels of 0.001 and 0.01 with K 0 10. The sensitivityof'these curves to the parameter (i,of.the hazard func-tion is quite clear in both Fig. 3 and Fig. 4.

Another application of these models is the calcu-lation of equivalence of tests-of different modes ofvoltage application, e.g. a constant voltage fixedduration test 'and a ramp voltage test in which thetest voltage is raised linearly to breakdown. A neces-sary condition for this equivalence is that the hazardfunction be time independent. Under this condition a

0.8

0.6

0.2

0.1 1.0 10t/T, p.u.

100

Fig. 4 Equi-probability normalized stress-durationcurves based on P(U,T) of the Weibull type(K = 10).Full lines: time independent hazard function.Dashed lines: time varying hazard function witha= 0.5.

constant risk of failure can be associated with eachfixed interval. This interval can for example be 1 min,1 s or 1 cycle of the supply frequency. The basicquestion of equivalence can be formulated as follows.If a constant voltage test with stress U and durationT produces a breakdown probability P, what will be therate of rise X of the ramp voltage which will producethe same breakdown voltage U'at the same probability P.The answer is reached in a straight forward manner asfollows.

From (6)

T = lnh (U) pJ

and for a ramp voltage U = Xt. (2), (3) giveU

ln[ 1 ]= l h(x) dx

(48)

(49)

The general relationship for equivalence of thetwo tests follows by eliminating ln(l/l-P) from (48),(49) yielding

U

T 1(U) f h(x) dx (50)0

which expresses the rate of rise of the ramp voltage interms of the hazard function h, the duration T of theconstant voltage test and the test voltage U.Another version of (50) has been derived by Brown19 etal, in which use was made of the breakdown probabilityof 1 cycle instead of the hazard function. Their suc-cessful application of their formula to experimentalresults in SF6 and our time-lag measurements describedbelow confirm the applicability of a time-independenthazard function concept to a conditioned SF6 gap.

COMPARISON WITH EXPERIMENTSIn this section the'basic assumptions and some

predictions of the above statistical models will betested against experimental results in transformer oiland compressed SF6.

l.00C n I i

r-z

",c

0 10 20 30 40 50 60t, s

Fig. 5 Distribution of time to breakdown in an SF625/70 mm coaxial cylinder gap at E/p = 29 kV/cmatm. abs. during 1-minute power frequencytests, presented on a Von Laue plot.

Fig. 5 shows the distribution of the time to break-down of a compressed SF6 gap with 25/70 mm coaxialelectrodes presented on a Von Laue plot. These 1 minpower frequency tests were carried out under a fixedvalue of E/p = 29 kV/cm. atm. abs. which characterizesthe stress at the central conductor. It is seen thatthe straight line exponential distribution provides areasonably good fit to the experimental data. Similarresults were obtained from 1 min power frequency testson uniform field gaps in the range 1 - 3 mm using Bruce

P=0.01

P=0.001

a =1.0_______. a =0.5

08~~~~

E/p=29kV/cm atm.

I I I~~~~~~~~~~~~~~~~

0

405

electrodes in transformer oil at an approximately con-stant stress of 240 kV/cm. These results add to theample evidence in the literature6,7,10,11 on exponen-tial delay times to breakdown.

Fig. 6 shows a sample of a cumulative breakdownprobability distribution of 60 Hz, 1 min tests in the25/70 mm SF6 gap presented on an extreme-value distri-bution paper. The fit is good below 225 kV while thechange in slope above 50% probability may be caused byparticles produced due to repeated sparking at highprobability levels.

0.900.800.70

.0.60a 0.50a 040

0.30

0.20

0.10o_

%.JJ

200 210 220U, kV

230

100

E

1.0

240

Fig. 6 Cumulative probability distribution for'l min-ute power frequency test on a 25/70 mm coaxialcylinder SF6 gap at 3 atm. abs.

0.90 I

0.80 _0.70

0.60

0.50-

0.40-

*0.30-

4020 /

0.40 - x

0.05

50 60 70 80 90 100 110 120 130U, kV

Fig. 7 Cumulative breakdown probability distributionof uniform field gaps in transformer oil fordifferent test durations based on results ofref. 8, presented on an extreme-value paper.

Fig. 7 shows power frequency cumulative breakdowndistributions in uniform field gaps in transformer oilextracted from ref. 8, presented on an extreme-valuepaper. These extensive test results fit reasonably wellinto a smallest-value distribution for test durationsof 10 s, 1 min, 10 min, 1 hr and 10 hr. From each dis-tribution corresponding to a fixed test duration T, wedetermined the parameters Ua(T) and U1(T). The resultsof Ua(T) are shown in Fig. 8 and represented reasonablywell by the regression line:

U (T) + 4.89 ln T = 117 = const.a

This dependence on ln T agrees with our theoretic-al predictions in expressions (15), (29) above. The

O.

Fig.

-

II11_ I80 90 100 110 120 130 140

Ua , kV8 Variation of the extreme-value distribution

parameter Ua with test duration calculated fromuniform field gap results in transformer oil ofref. 8.

values of U1(T) obtained by regression analysis areshown in Table I.

Table IValues of U1(T) Obtained from Test Results of Ref. 8

Test Duration U() kT, min 1

0.167 12.31.0 11.8

10 10.560 ._ 9.3

600 .. 7.9

Here U1(T) is not strictly constant, as predictedby the theory for a single mechanism of breakdown and atest object that remains homogeneous under stress, butdecreases as the test duration increases. This supportsthe conclusion:-of Ikeda and Menju8 that for the oil gapunder consideration physical changes occur with thestressing time.

The compatibility of the extreme-value distribu-tion of the breakdown voltage and the exponential dis-tribution of the time to breakdown as shown above in-dicates that the mean time to breakdown or statisticaltime lag should vary exponentially with the stress.This has been experimentally confirmed by Lewis andWard19 for impulse. work in uniform field gaps *in n-hexane. We obtained similar results in transformer oilunder unit step impulse voltages as shown in Fig. 9.

The stress-duration models developed in this paperpaid special attention to extreme-value distributionsand are meant-to be useful at low breakdown.probabi-lities and in particular for internal insulation. Asmentioned above extreme-value statistics 'are particu-

I I ~~~~IJ

//

/

/

vf

H:~~~~~~~~~

IOOOF-

350 400 450 500E, kV/cm

of the mean time to breakdown on

a 3 mm uniform field gap in trans-subjected to unit-step stress.

larly valid if breakdown is the result of a weak linkmechanism due to a large number of breakdown initiatingsources. However, the basic approach described aboveusing the general concept of hazard functions appliesequally well for other types of distributions.

The choice of the applicable distribution shouldnot only be based on measuring results but also on

basic understanding of the physical phenomena involved.For example, our previously published results20 on

power frequency flashover characteristics of dust con-

taminated insulators wetted with clean fog indicatedthat for a test of constant duration the breakdownprobability-voltage curve fits into a gaussian distri-bution, for probabilities in the range of 5% to 95%.Furthermore, under constant voltage stress the time tobreakdown was also normally distributed20. The resultshave been confirmed by recent work at IREQ. The factthat there the time to breakdown is concentrated in a

rather limited range of time delays and not extendingover an exponential curve can be understood from thebasic flashover mechanism. A polluted insulator can notpractically flashover before a minimum delay determinedby the time necessary to wet the pollution layer, On

the other hand, flashover can not normally take placeafter a certain duration determined by the start ofwashing the contaminant off the insulator surface dueto prolonged wetting. This is fundamentally differentfrom the conditions of internal insulation treatedabove where the risk of breakdown exists as long as theinsulating system is under stress,

Finally, it is hoped that the above tools willconstitute a valuable aid to laboratory testing whichin turn should, of course, be complemented by fieldexperience.

CONCLUS IONS

1. New statistical models were developed for evalua-tion of the stress-duration breakdown probabilityof high voltage insulating systems.

2. The use of the statistical models for the evalua-tion of risk of failure at low probability, for theestablishment for equi-probability U-t curves andfor the evaluation of equivalence of differentmodes of voltage application has been demonstrated.

3. The long-term risk of failure and the equi-proba-bility voltage-time curves are found to be very sen-

sitive to any dependence of the hazard function on

test duration.

4. An insulating system that does not develop physicalinhomogeneity under stress will be characterized byequi-probability voltage-time curves which can berepresented by straight lines on a suitable loga-rithmic plot.

5. If the breakdown probability of an insulatingsystem which remains homogeneous under stress, ex-hibits an extreme value dependence on voltage fortests with different durations, then the shapeparameter U1(T) of the distribution must be timeindependent while the location parameter Ua(T) mustdecrease with the logarithm of the test duration.

6. If the breakdown probability of an insulatingsystem, which remains homogeneous under stress, ex-hibits a Weibull type dependence on voltage fortests with different durations, then the thresholdparameter of the distribution UO(T) and the ex-ponent K must be time independent while the shapeparameter U1(T) will decrease with the test dura-tion.

7. Failure of a test object to fulfill conclusions4 - 6 is indicative of significant physical inho-mogeneity developed under stress leading to achange in the breakdown mechanism.

8. The long-term risk of failure of an insulatingsystem that remains physically homogeneous understress can be established from breakdown tests withlimited duration. Otherwise such inference will bepossible only if the laboratory test voltage andduration are chosen to account for the slowestbreakdown mechanism involved.

9. Analysis of test results based on the models devel-oped here indicates that a conditioned SF6 coaxialcylinder gap can be characterized by a time-inde-pendent hazard function while a transformer oiluniform field gap may develop physical inhomoge-neity under prolonged stress.

REFERENCES

[1] N. Hylten-Cavallius, "Some aspects about the in-formation gained from high voltage tests", CIGRE',1956, paper No. 405.

[2] Gordon W. Brown, "Testing for the cumulative flash-over distribution", IEEE Trans., Vol. PAS-89,No. 6, July/August 1970, pp. 1186-1191.

[3] M. Ouyang, G. Carrara, "Evaluation and applicationof impulse test results", Electra, No. 13, 1970,pp. 25-39.

[4] G. Carrara, E. Occhini, L. Paris, F. Raggiani,"Contribution to the study of insulation coordi-nation from the probabilistic point of view",CIGRE', 1966, paper No. 421.

[5] High Voltage Test Techniques, IEC Publication 60-2,1973, part 2, p. 41.

[6] F.A.M. Rizk, C. Vincent, "Testing for low break-down probability with special reference to liquidinsulation", IEEE Trans., Vol. PAS-96, No. 6,Nov./Dec. 1977, pp. 1892-1900.

[7] R.B. Kaufman and J.R. Meador, "Dielectric testsfor ehv transformers", IEEE Trans., Vol. PAS-87,No. 1, 1968, pp. 135-145.

[8] M. Ikeda, S. Menju, "Breakdown probability distri-

406

_ I I I I _

x

I I I I

U)

E~:10

4 C

250 300

Fig. 9 Dependencestress offormer oil

407

bution and equi-probabilistic v-t characteristicsof transformer oil", IEEE Trans., Vol. PAS-98,No. 4, July/August 1979, pp. 1430-1438.

[9] E.J. Gumbel, "Statistical theory of extreme-valuesand some, practical applications", National Bureauof Standards, Applied Mathematics Series, No. 33,1954.

[10] R. Strigel, "Elektrische Stossfestigkeit", Book,Springer-Verlag, Berlin, 1955, pp. 43-77.

[11] J.E. Brignell, "Statistical nature of breakdownsequences in a liquid insulant", Proc. IEE, Vol.113, No. 10, October 1966, pp. 1683-1691.

[12] IEEE Transformer Committee, "Dielectric tests andtest procedures for ehv transformers protected bymodern surge arresters and operated on effectivelygrounded systems 345 kV through 765 kV", IEEETrans., Vol. PAS-92, No. 5, October 1973, pp. 1752-1762.

[13] K.V. Bury, "Statistical models in applied science",Book, John Wiley & Sons, New York, 1975.

[14] W. Weibull, "A statistical distribution functionof wide applicability",. Journal of Applied Mech-anics, September 1951, pp. 293-297.

[15] N.W. Smirnow, I.W. Dunin-Barkowski, "MathematischeStatistik in der Technik", Book, VEB Verlag Tech-nik, Berlin, 1963, pp. 317-336.

[16] I. Bazavsky, "Reliability theory and practice",Book, Prentice Hall, New Jersey, 1961.

[17] B.V. Gnedenko, Y.K. Belyayev, A.D. Solovyev,"Mathematical methods of reliability tlheory", BookAcademic Press, New York, 1969, p. 95.

[18] G.W. Brown, R. Samm, J. Cronin, "Ac analysis andtesting of realistically contaminated gas-insu-lated systems", Trans. IEEE, Vol. PAS-97, No. 1,Jan/Feb. 1978, pp. 59-67.

[19] T.J. Lewis, B.W. Ward, "A statistical interpreta-tion of the electrical breakdown of liquid dielec-trics", Proc. Roy. Soc. A, Vol. 269, 1962, pp.233-248.

[20] F,A.M. Rizk, A.A. Assaad, "Flashover tests on dust-

contaminated insulators", IEEE Trans., Vol. PAS-91,No. 1, Jan./Feb. 1972, pp. 328-335.

Discussion

H. Anis (Cairo University, Egypt): The author must be congratulatedon his skillfully written paper. The time is ripe for some efforts to beaimed at generalizing the different field and laboratory observationsrelating to breakdown durations. This paper offers a general statisticalmodel which uses a "hazard" function - a form of conditionalprobability - to express the relation of the bivariate breakdown pro-

bability p (t, U) to breakdown delays. The model is then used to givephysical interpetation to the parameters of two suitable breakdown pro-bability distributions, the extreme-value and. the Weibull distribution-s.

Equations 4 and 5 of the paper can be of particular importance to thestudies of the time-to-breakdown in air under transient voltages. Thefunction p (t, U), meaning the probability of breakdown at a time t fora given voltage U, may be equally defined as the probability that alltimes to breakdown will have values less than t - a definition of thecumulative time-to-breakdown distribution. The density distributionmay then be obtained by differentiating that function,

d(t~U[-e-h(U).(t-t )p(t|U) = t [1 - e°]

1-(t-t )/'0= -e

which is an exponential density function whose mean time tobreakdown beyond the formative time lag t, is T= I/h(U) and whosestandard deviation is also T. According to thie above and in view of thefact that, under impulse conditions, breakdown may or may not occurand that when it does it would have a random delay, one may call for asimulation by a two-state continuous Markov stochastic model'. Themodel describes the sequence of "up" and "down" states at an "up"rate of s and "down" rate of A. Denoting the probabilities of failureand withstand at a given instant t by P,(t, U) and P#(t, U), respectively,their values after an incremental time dt are

PO (t + dt, U) = Po(t, U) (I - A dt) + P,(t, U) J dtPl(t + dt, U) = Pl(t, U) (1 - ,udt) + P,(t, U) AdtThe equations are solved to give the probability of failure as

_L(X+) tP (t, u +e

which is, in fact, the unnormalized cumulative probability distributionof the time to failure. The value of that function as t -oo gives the even-tual breakdown probability

P = p( U)

The normalized form of Pl(t, U) can be written as

Pn1 (t, U) - P,(t, U)/P = 1' e-(A+1)t

which is of the same form as equsation 4 of the paper. The rates A and ,u(sec-1) are functions of the voltage U and the wave shape. The author'scomment on the above treatment is requested.

In a previous work 2 the probability distribution of the instantaneousvoltage and that of the time to breakdown of air gaps under switchingimpulses were combined to form a "bivariate breakdown probabilityfunction." In that work the "instantaneous" breakdown voltage had tobe considered rather than peak voltage to account for the effect of therate of voltage rise (wave shape). In the author's opinion how would thepresent study accommodate the above situation and probably offer thebivariate probability, as described, as a byproduct?

Finally, one may extend the section on "multiexponential delaydistribution" by considering the case where there is interest in determin-ing the probability of a particular event to precede all other events. Thisinterest arises, for example, in the case of a group of multiparallel in-sulation protected by an overvoltage device. The insulation presents aset of n independent breakdown events described, using the paper's ter-minology, by

Pi(t) = Ai(l-e-"'Ti) i = 1, n

where Ai is the probability of breakdown. The probability 'p that a pro-tective device which is characterized by

P. (t) = Ap (1 - e-'/Tp)

will protect the entire group by breaking down ahead of them all is3

Xdn= f. d (P (t) /A P) l [ P(t) /A i] dtdt p pil 1 1

The quantity nl,, [I - P(t)/AJ] in the above equation describes thecumulative distribution of the minimum time to breakdown of the set ofinsulation components.

REFBRENCES

[1] E. Parzen, "Stochastic processes," Holden-day, San Francisco,1962.

408

[2] H. Anis et al., "Bivariate probability of breakdown under swit-ching surges," IEEE Trans., vol. PAS-92, May/June 1973.

[3] H. Anis et al., "Statistical study of parallel gaps of unequalcharacteristics under switching surges," paper C73 177-3, IEEEPES Winter Power Meeting, New York, 1973.

Manuscript received January 31 , 1980.

S. Menju and M. Ikeda (Toshiba Corporation, Kawasaki, Japan): Inthe first place, we want to pay our respect to Dr. Rizk for his reasonablestatistical systemization described in this paper. We believe this paperwill give useful suggestion to many engineers who are engaged in thedesign and insulation tests on high-voltage electric apparatus.

In the case of transformers, the partial discharge test for an hour isproposed as described in literature (7) and (12), and this method is nowbeing generalized. However, in the case of GIS, some people have anopinion that a test duration of about 10 minutes will be enough. Thereason is that almost of all ac breakdowns occur in the course of voltagerising, or immediately after the maximum voltage is applied, or severalseconds thereafter.

Dr. Rizk states in his paper that "an insulating system which remainshomogeneous under electric stresses like SF6 gas insulation can beestablished from breakdown tests with limited duration time."We shall be glad to ask your opinion whether it is proper or not to

shorten the AC dielectric test duration of GIS to 10 minutes or so accor-ding to Dr. Rizk's theory stated in this paper.

Manuscript received March 25, 1980.

Farouk A. M. Rizk: The author would like to thank the discussers fortheir interesting comments which add substantially to the value of thepaper.

In answer to the request of Dr. Anis, the use of Markov stochasticprocesses to describe breakdown phenomena is certainly appropriate.They have indeed been used previously to account for avalanche growthin electronegative gases where both ionization and attachment processesplay a major role in breakdown.21'22 A basic assumption implied in thederivations presented by Dr. Anis is that the transition intensities y and

A are time-independent. This accounts for the equivalence establishedbetween these derivations and expression (4) of the paper which is basedon the assumption of a time-independent hazard function.The author is quite aware of ref. 2 of the discussion of Dr. Anis in

which a bivariate gaussian distribution was fitted to switching impulsebreakdown test results of a rod-plane gap in air. The extension of theapproach adopted in the present paper, using hazard functions, to im-pulse breakdown delays has been in progress at IREQ for some time,within a Ph.D. thesis supervised by the author to be submitted toMcGill University by M. Eteiba.The author is particularly gratified by the comments of S. Menju and

M. Ikeda since the theoretical derivations of the paper have been ap-plied to their extensive breakdown experiments in transformer oil (ref. 8of the paper). In response to their question concerning the duration ofthe AC dielectric test of a conditioned gas insulated system, I wouldtend to agree that the duration required can be made shorter than theone hour standard generally accepted for transformer insulation. Thisseems to be supported by the results of the present paper as well as bythe relatively short breakdown delays cited by the discussers. Of course,an appropriate choice of the corresponding magnitude of the testvoltage has also to be made.A final point that needs some clarification is that the values of the

I-min test voltages of Fig. 6 refer in fact to r.m.s. kilovolts. Further-more, r.m.s. voltage was also the basis for the expression E/p = 29kV/cm. atm. abs. quoted in Fig. 5. If peak voltage is used instead, thecorresponding value- will be E/p = 41 kV peak/cm. atm. abs. Finally,while Fig. 5 refers to a coaxial electrode system with a realistic spacerdesign and to relatively low breakdown probability, Fig. 6 relates to theinherent characteristics of the SF6-insulated coaxial cylinder gapwithout spacer.

REFERENCES

21. W. Hauschild, W. Burger, "Statistische Modelle fur denelektrischen Durschlag im Isoliergas SF6," Z. elektr. Inform. - u.Energietechnik, Leipzig 5, 1975, No. 4, pp. 283-295.

22. D. T. A. Blair, B. H. Crichton, I. C. Somerville, "AvalancheStatistics and the Streamer Criterion, "Proceedings, First Interna-tional Symposium on Gaseous Dielectrics, Knoxville, Tennessee, U.S. A., March, 1980.

Manuscript received March 25 , 1980.