SCIENCE New method for determination of steady-state temperature rises of transformers Z. Godec Indexing terms: Transformers, Measurement and measuring, Mathematical techniques, Temperature rise Abstract: For transformers whose thermal time constant exceeds two hours, unacceptably large systematic errors occur if recommended methods for determination of steady-state temperature rise are used. Therefore a new criterion for heat run duration and a new method for determination of steady-state temperature rise are proposed, which ensure reliable determination of steady-state temperature rise independent of transformer thermal time constant. According to this proposal, the heat run should be continued until the temperature rise exceeds 95% of the steady-state temperature rise, which is estimated by a three-points method. Then, by the method of least squares, a more accurate value of steady-state temperature rise is obtained by extrapolating only those measured points that are higher than 60% of the previously estimated steady-state temperature rise. For those instrument transformers whose largest thermal time constant of the system (transformer) is governed by the time constant of the winding, a universal intermittent heat run independent of the transformer time constant is proposed: t o jt off = 120 min/2 min. Estimation errors of the steady-state temperature rise and thermal time constant when applying the three-points method and the intermittent heat run are analysed. List of symbols F = function m = mean-square error t = time t on = duration of heating period t off = duration of interruption period T = thermal time constant 6 = temperature rise 9 S = steady-state temperature rise 9 int — maximum temperature rise at intermittent heating X = independent variable 1 Introduction The life of a transformer is determined by the temperature to which its insulation is heated during operation. To verify whether temperature rises of transformer parts do not exceed the limits specified in standards, one should measure the temperatures of characteristic parts during a temperature rise test. According to Reference 1, the temperature rise test of power transformers should be continued until one of the two following criteria has been met: (a) the temperature rise test increment is less than 1 K/h during four consecutive hours (b) the temperature rise increment is less than 3 K/h. If the first criterion is met, it is considered that the thermal equilibrium has been reached, i.e. that the last temperature rise reading is equal to the steady-state or final tem- perature rise. If the second criterion is used, which enables a considerable reduction of heat run duration, it is necess- ary to extrapolate the measured temperature rises to obtain the steady-state value [1]. Paper 3194A (S4/P7/P11), first received 1st November 1983 and in revised form 2nd March 1984 The author is with the Rade Koncar-Institut, Transformer Department, Bastjanova bb, 41000 Zagreb, Yugoslavia According to References 2 and 3, the temperature rise test of instrument transformers should be continued until the temperature rise increment is less than 1 K/h and the last temperature rise reading is considered equal to the steady-state temperature rise. However, because of different thermal time constants of transformers, these criteria [1, 2, 3] do not ensure a reli- able and accurate determination of the steady-state tem- perature rise; namely, thermal time constants of oil-immersed power transformers range from 1 to 5 hours, while the thermal time constant of instrument transformers range within even wider limits: from 0.5 to 40 hours. Let us consider through what temperature rises trans- formers with different thermal time constants will be heated in relation to the steady-state value if the above criteria [1, 2, 3] have been used. The part of the temperature rise curve above 50 to 60% of the steady-state temperature rise can be approximated well by a simple exponential curve [4]: (1) d{t) = A + B | 1 - exp ( - - where 6(t) is the temperature rise, A and B are constants, t is time and T is the thermal time constant. The expression for the estimation of the duration of temperature rise test up to the point when a certain criterion is met can be derived relatively easily (see Appendix 6.1): (2) Table 1 shows estimated durations of heat runs and the attained temperature rises dependent on the applied cri- terion when 9 S = 60 K and A = 0 (B = 0 S ). It follows that, by applying the standardised criterion 1 K/h [2, 3], unac- ceptably large errors are made when determining the steady-state temperature rise of instrument transformers whose time constant exceeds one hour. These errors are made because the values that are considerably lower than steady-state temperature rises are assumed to be the 1EE PROCEEDINGS, Vol. 131, Pt. A, No. 5, JULY 1984 307
Transcript
SCIENCE
New method for determinationof steady-state temperature rises of
transformersZ. Godec
Indexing terms: Transformers, Measurement and measuring, Mathematical techniques, Temperature rise
Abstract: For transformers whose thermal time constant exceeds two hours, unacceptably large systematicerrors occur if recommended methods for determination of steady-state temperature rise are used. Therefore anew criterion for heat run duration and a new method for determination of steady-state temperature rise areproposed, which ensure reliable determination of steady-state temperature rise independent of transformerthermal time constant. According to this proposal, the heat run should be continued until the temperature riseexceeds 95% of the steady-state temperature rise, which is estimated by a three-points method. Then, by themethod of least squares, a more accurate value of steady-state temperature rise is obtained by extrapolatingonly those measured points that are higher than 60% of the previously estimated steady-state temperature rise.For those instrument transformers whose largest thermal time constant of the system (transformer) is governedby the time constant of the winding, a universal intermittent heat run independent of the transformer timeconstant is proposed: tojtoff= 120 min/2 min. Estimation errors of the steady-state temperature rise andthermal time constant when applying the three-points method and the intermittent heat run are analysed.
List of symbols
F = function
m = mean-square errort = timeton = duration of heating periodtoff = duration of interruption periodT = thermal time constant6 = temperature rise9S = s teady-state t empera ture rise9int — maximum temperature rise at intermittent heatingX = independent variable
1 Introduction
The life of a transformer is determined by the temperatureto which its insulation is heated during operation. Toverify whether temperature rises of transformer parts donot exceed the limits specified in standards, one shouldmeasure the temperatures of characteristic parts during atemperature rise test.
According to Reference 1, the temperature rise test ofpower transformers should be continued until one of thetwo following criteria has been met:
(a) the temperature rise test increment is less than 1 K/hduring four consecutive hours
(b) the temperature rise increment is less than 3 K/h.
If the first criterion is met, it is considered that the thermalequilibrium has been reached, i.e. that the last temperaturerise reading is equal to the steady-state or final tem-perature rise. If the second criterion is used, which enablesa considerable reduction of heat run duration, it is necess-ary to extrapolate the measured temperature rises toobtain the steady-state value [1].
Paper 3194A (S4/P7/P11), first received 1st November 1983 and in revised form 2ndMarch 1984
The author is with the Rade Koncar-Institut, Transformer Department, Bastjanovabb, 41000 Zagreb, Yugoslavia
According to References 2 and 3, the temperature risetest of instrument transformers should be continued untilthe temperature rise increment is less than 1 K/h and thelast temperature rise reading is considered equal to thesteady-state temperature rise.
However, because of different thermal time constants oftransformers, these criteria [1, 2, 3] do not ensure a reli-able and accurate determination of the steady-state tem-perature rise; namely, thermal time constants ofoil-immersed power transformers range from 1 to 5 hours,while the thermal time constant of instrument transformersrange within even wider limits: from 0.5 to 40 hours.
Let us consider through what temperature rises trans-formers with different thermal time constants will beheated in relation to the steady-state value if the abovecriteria [1, 2, 3] have been used.
The part of the temperature rise curve above 50 to 60%of the steady-state temperature rise can be approximatedwell by a simple exponential curve [4]:
(1)d{t) = A + B | 1 - exp ( - -
where 6(t) is the temperature rise, A and B are constants, tis time and T is the thermal time constant. The expressionfor the estimation of the duration of temperature rise testup to the point when a certain criterion is met can bederived relatively easily (see Appendix 6.1):
(2)
Table 1 shows estimated durations of heat runs and theattained temperature rises dependent on the applied cri-terion when 9S = 60 K and A = 0 (B = 0S). It follows that,by applying the standardised criterion 1 K/h [2, 3], unac-ceptably large errors are made when determining thesteady-state temperature rise of instrument transformerswhose time constant exceeds one hour. These errors aremade because the values that are considerably lower thansteady-state temperature rises are assumed to be the
1EE PROCEEDINGS, Vol. 131, Pt. A, No. 5, JULY 1984 307
Table 1: Relative durations of temperature rise tests andattained relative temperature rises dependent on the appliedcriterion for interruption of the test
Criterion
Thermal
timeconstant.
h
0.51.02.05.0
10.020.040.0
Instrumenttransformers
1 K/h
t
T
5.954.643.662.591.841.120.42
9
es
0.9970.9900.9740.9250.8410.6740.342
Power transformers
1 K/h
t
T
7.645.163.19———
+ 3 h
e
1.0000.9940.959———
3
t
T
3.2.1.
K/h
545649
6
0.9710.9230.774———
steady-state ones; for example, for transformers with timeconstants of 20 h, the value that is 33% (~20 K) lowerthan the actual steady-state temperature rise is declared tobe the steady state temperature rise. The criteria applied tooil-immersed power transformers [1] have been better esti-mated, although, even here, if a more strict criterion isapplied (1 K/h during four hours), a systematic error up to- 4 % (~2.5 K) can be made.
The present analysis shows that it is necessary to definethe criterion for interruption of the temperature rise test ina different way. This criterion should enable equally accu-rate determination of the steady-state temperature rise,independent of the value of the transformer timeconstant.
2 Proposal of a new criterion for necessaryduration of temperature rise test
Heating of the transformer should be continued until thetemperature rise exceeds 0.95 9S, i.e. the heating durationshould be approximately three thermal time constants.After that, the steady-state temperature rise is determinedby extrapolation, by least-squares or graphical methods,applied to all the measured points above 0.6 9S. Since thefinal temperature rise, 9S, and time constant, T, are usuallyunknown before the temperature rise test, they have to beestimated in some simple way from points {t{, 0,}, whichhave just been measured. A suitable method is estimationby three points (see Appendix 6.2):
2d2 - e x -
T =U - U
2 In
(3)
(4)
where {t3, 93} is the last measured point, 9l is the tem-perature rise approximately equal to 0.6 93, tx is the timecorresponding to that temperature rise and 92 is the tem-perature rise at the moment t2 = j{tl + t3). The estimationof the time constant is less accurate (see Section 3), so thatit has the status of auxiliary magnitude.
The proposed method for determination of the necess-ary duration of the heat run can be applied to all the tem-perature rise tests. The idea will be worked out in detailusing examples of oil-immersed power and instrumenttransformers.
2.1 OH-immersed power transformersThere are several ways of heating a transformer [1], but,for economical reasons, the method most frequently used isto load a transformer with total losses by short-circuitmethod, so that all the losses (no load and load) arelocated mainly in the-transformer windings. Here, it ischaracteristic that the time constants of the windings arelow (several minutes), while the time constants of oil issome dozen times larger. Since the approach to the steady-state temperature rise of any part of the system(transformer) is governed by the largest time constant ofthe system [5], the required duration of heat run of oil-immersed power transformers will be determined in thesimplest way by the analysis of oil temperature rise curve.The procedure is as follows:
(a) oil temperature rise is measured every 15 minutes(b) 9S and T are calculated after three hours of heating(c) after the next 30 minutes, 9S and T are estimated
again(d) the heat run is interrupted if 63 > 0.95 9S
(e) heating is continued if 93 ^ 0.95 9S; 9S and T shouldbe periodically estimated by means of previously measuredpoints (the last estimation of 9S and T is always more accu-rate than the preceding ones: see Section 3).
The variations of individual estimates of 9S and T will beless if, during the measurements, the temperature rise curveis plotted and the values of {tt, 9{] read from it, because, inthis way, the influence of accidental and gross errors ofindividual readings is reduced.
Example: At a temperature rise test of a 400 kVA oil-immersed transformer, the top oil temperature rises givenin Table 2 have been measured, while the estimated 9S andT obtained by three-points method are given in Table 3. 9S
Table 2: Results of top oil temperature rise measurements ona 400 kVA power transformer
Table 3: Estimation of 0s and T by the three-points method
3.003.505.005.505.75
47.350.054.455.455.7
2.002.253.253.503.75
40.043.248.750.051.0
1.001.001.501.501.75
25.825.834.234.237.4
55.056.758.158.258.2
1.501.641.871.861.88
0.860.880.940.950.96
308 IEE PROCEEDINGS, Vol. 131, Pt. A, No. 5, JULY 1984
and T, computed by method of least squares, and theirmean-square errors meand mTare given in Appendix 6.3.
2.2 Instrument transformersIf the ratio of winding and oil thermal time constants issimilar to the one for power transformers, the procedurefor determination of the heat run is similar to the one inSection 2.1. However, there are instrument transformerswhose winding thermal time constant is considerablylarger than the oil thermal time constant, for example atcurrent transformers for high voltages with HV insulationimmediately on the secondary windings. The requiredduration of temperature rise tests for transformers of thiskind is governed by heating of the windings (secondary)and not of the oil. To determine the mean temperature riseof the windings by a resistance method, the heating shouldbe interrupted. The transformer should, consequently, beheated intermittently. If we accept that an interruption oftwo minutes is sufficient for accurate measurement ofresistances of the windings, the question arises as to howoften the heating may be interrupted without making anerror in steady-state temperature rise determination largerthan some preset value. According to Reference 6, the rela-tion between the actual steady-state temperature rise atcontinuous heating and the maximum temperature rise atintermittent heating is:
0S _ l - e x p [ - ( t o w + to / /)/T]
9int l-expl-(tJT)-] (5)
From this equation, the time required for heating periods,ton, can be estimated, with preset duration of interruptionperiods, toff, and some specified error [(l//c) — 1]:
ton = T \ n lk — exp I —
k - 1(6)
This means that for the optimum intermittent heat run it isnecessary to know in advance the time constant, which is,however, usually not known. Therefore on the basis of adetailed analysis (see Section 3), a universal intermittentheat run t0jtoff =120 min/2 min is proposed for all trans-formers, i.e. the heating procedure that is not dependent onthe transformer time constant. After three measurements(six hours of intermittent heating), it is possible to carryout the first estimations, of both the steady-state tem-perature rise and the time constant, by applying eqns. 3and 4. To avoid possible errors (three points only), it isrecommended that the heat run should be continued fortwo more hours, so that a decision on the further course ofthe temperature rise test can be made on the basis of twoestimates. If the last reading exceeds 0.95 0S, the test maybe interrupted; if not, should be continued until this cri-terion has been met.
For transformers with low thermal time constant, it ispossible that, because of relatively long time intervalsbetween measuring points (two hours), no suitable points
for estimation of the steady-state temperature rise exist,and, consequently, the estimation on the basis of measuredpoints only can be erroneous (see Section 3). Therefore it isrecommended that one should plot the temperature risecurve and estimate 0S and T by means of points read fromthe curve. At transformers whose time constant is less than1.5 hours, the last (fourth) point can be considered equal tothe steady-state temperature rise since tA > 5T.
Because the readings are carried out every two hours,the application of the method of least squares (which ismore reliable than the three-points method) is adequateonly for transformers with large thermal time constants,when, in the range from 0.6 6S to 03 ^ 0.95 9S, there are atleast ten measuring points. That means that, in the rangeof time constants from 1.5 to 8 h, the steady-state tem-perature rise may be determined only by the three-pointsmethod.
To determine accurately the steady-state temperaturerise, the duration of temperature rise tests of transformerswith large thermal time constants has to be very long; forexample, for a transformer with T = 24 hours, it has to bethree days. It is permissible to shorten the heat run if, byestimation by the three-points method, values have beenobtained that are at least 5 K lower than specified tem-perature rise limits. The shortening is limited by two cri-teria :
(a) from 0.6 0S to 93 there have to be at least ten points(because of extrapolation by the least-squares or graphicalmethods)
(b) the temperature rise at the last point, 03, must not beless than 0.86 95.
Hence, the minimum duration of the temperature rise testshould not be shorter than about two thermal time con-stants, as the error of estimation of the steady-state tem-perature rise increases abruptly by shortening the durationof temperature rise test (see Section 3).
3 Discussion
The proposed change in determining the steady-state tem-perature rises is based on the introduction of a new cri-terion for the necessary duration of the temperature risetest, and on extrapolation of only those time/temperaturerise curve points which are above 60% of the steady-statetemperature rise, i.e. not of all the measured points.
The new criterion is based on the estimation of thesteady-state temperature rise by the three-points method.Mean-square errors of both the steady-state temperaturerise and the time constant determined by the three-pointsmethod can be estimated by the following equation:
m = + — m. (7)
with F the function whose error is to be found, Xi indepen-dent variables and m, their mean-square errors. The mean-square error of the steady-state temperature rise (eqn. 3) is:
1- ^ - ^ ( 2 0 2 - 0 1 - 0 3 ) 2
The mean-square error of the time constant (eqn. 4) is:
[2(02 - - 92)m62]2 + [_{92 - 01)2mfl3]:
mT =T = +2 In
_J / 2 , m, , h-h j r m6l i2 | r (e3-el)m92 j t r m92 i2}
(8)
IEE PROCEEDINGS, Vol. 131, Pt. A, No. 5, JULY 1984 309
It is interesting to note that the absolute value of error m6s
does not depend on values of 9S and T, while mT does. Bymeans of eqns. 8 and 9, the errors given in Table 4 havebeen calculated to be dependent on duration of tem-perature rise tests, where {t3, 03} is the last measuredpoint, 0 1 = O.6 03->tlt t2=0.5 (t1+t3)^>92, m0l =m02 = m03 = m9 = 0.2 K, mtl = mt2 = 1 min, T = 2 h and0S = 65 K. If points {t2, 03}, {t2, 92} and {tu 0X} have adifferent arrangement; i.e. if, for example, t3 = 5 T, t2 = 4T and tt = 3 T, the errors will be larger: mgjme = 2.83,wOs = +0.6 K and mT = 53 min, respectively. From Table4 it can be seen that the determination of steady-state tem-perature rise is satisfactorily accurate, while the determi-nation of time constant is less so. The time constant can be
Table 4: Estimation of mean-square errors of the steady-state temperature rise and thermal time constant determinedby three-points method
' 3
T
12345
me
36.26.762.761.751.35
K
7.21.40.60.40.3
mT,min
29.19.36.15.66.0
determined more precisely by the two-points method [7],on condition that the approximation curve is a simpleexponential one passing through the origin:
T =1
(10)In
where {t2, 92} is the last measured point and tl =0.5 • t2^-9l. The mean-square error is determined bymeans of the equation:
1mT =
In
+- 0i) In
Bo-
rn,2
01 I +
(11)
In Table 5 there are estimated errors, mT, for various dura-tions of temperature rise tests, with mtl = 1 min, m6l =m62 = 0.2 K, T = 2 h and 9S = 65 K.
The transformer is a system consisting of many differentparts (core, winding, insulation, oil etc.), so that the tem-perature rise curve can be presented mathematically as asum of exponential terms [5]. It has been found, however,that the heating of any of the parts belonging to the
Table 5: Estimation of mean-square errors of thermal timeconstant determined by two-points method for a simpleexponential function through the origin (A = 0)
*2
T
12345
mT,min
6.12.92.31.32.9
thermal system is governed after some time (which isapproximately equal to the largest time constant of thesystem) solely by the term having the largest time constant[4, 5]. This means that the part of the temperature risecurve above 60% of the steady-state temperature rise canbe represented accurately by a simple exponential curve(eqn. 1). This statement has been confirmed by the factthat, at the determination of the steady-state temperaturerise by the method of least squares on a simple exponentialcurve (eqn. 1), the mean-square error becomes considerablysmaller if only the points above 60% of the steady-statetemperature rise are considered. It is interesting to notethat the standardised graphical method of extrapolation[1] also uses, although tacitly, only that part of the curvewhich is close to the steady-state temperature rise.
According to Reference 2, current transformers areheated by constant current. Thus, the thermal time con-stant is increased by approximately 30% [6], but the expo-nential character of the temperature rise curve is notchanged.
By use of an intermittent heat run, the systematic erroris introduced into the determination of the steady-statetemperature rise and the time constant. The error can beestimated by eqn. 5, presupposing that the transformer is ahomogeneous body. Applying the same presupposition,time ton required for optimum intermittent heating, i.e. theheating that ensures the maximum number of measuringpoints in a certain time interval, can be calculated by eqn.6. Table 6 shows durations of heating periods, ton, rounded
Table 6: Optimum values of toa for intermittent heat run withinterruptions of two minutes and for the error of determi-nation of the steady-state temperature rise less than —1.6%
Thermaltimeconstant,h
0.51.01.52.04.0
10.012.025.040.0
ton.min
50708090
105115120120121.8*
310
* The value is not rounded off.
to five minutes depending on the thermal time constant,with toff = 2 min and k = 1.016. For the heat run pro-cedures given in Table 6, systematic errors would alwaysbe approximately — 1.6%(~1K).
As the time constant is very seldom known in advance,a universal intermittent heat run is proposed for all thetransformers: ton/toff = 120 min/2 min.* Applying the pro-posed heating procedure, systematic errors will be lessthan —1.6% (see Table 7). Since these errors have beenestimated by presupposing that a homogeneous body hadbeen heated and cooled, and since the transformer is not ahomogeneous body, it can be assumed by using eqn. 5 thatthe maximum possible systematic errors have been calcu-lated, and that the actual errors are, in fact, less. The sys-tematic error of determination of time constant by meansof points obtained by an intermittent heat run is negativeand less than the mean-square error of time constantdetermined by two- or three-points methods(approximately —1.6%).
* This type of heating procedure is prescribed in Reference 8 as the limit case
IEE PROCEEDINGS, Vol. 131, Pt. A, No. 5, JULY 1984
Table 7: Estimation of systematic errors of determination ofthe steady-state temperature rise caused by intermittentheat run, tojtoff = 120 min/2 min
Thermaltimeconstant,h
0.51.01.52.04.0
10.012.025.040.0
( - - 1 | x100
%
-0.1-0.5-0.8-1.0-1.3-1.5-1.5-1.6-1.6
6 Appendix
6.1 Derivation of equations for estimation of tem-perature rise test duration
1 - exp I>x = A + B 1 - exp -t - 1
T
- ^ exp - -
t = I 1
B = 0 - A
(12)
4 Conclusion
To avoid unacceptably large systematic errors of determi-nation of steady-state temperature rises of transformers, itis proposed that the criteria should be changed for thenecessary duration of temperature rise tests, as well as themethod for determination of steady-state temperaturerises.
The two existing alternatives recommended by IEC 76[1], by whose application, in principle, different values ofsteady-state temperature rises are obtained, should bereplaced by a single one: the heat run is to be continueduntil the temperature rise exceeds 95% of the steady-statetemperature rise estimated by the three-points method. Amore accurate value of the steady-state temperature rise isobtained by means of the points which lie above 60% ofthe steady-state temperature rise, to which extrapolationby either a least-squares or graphical method has beenapplied.
The existing method of determination of the steady-state temperature rise given in IEC 185 and IEC 186 [2, 3]should be replaced by the new one formulated as above.For transformers that are to be heated with interruptions,an intermittent heat run with tjtoff =120 min/2 min isproposed, which is equal for all transformers regardless oftheir thermal time constants. For this heat run, the system-atic error of determination of the steady-state temperaturerise depends on the transformer thermal time constant, butfor transformers with an extremely large time constant(T = 40 hours) it is less than - 1.6%.
5 References
1 IEC Standard: 'Power transformers, Pt. 2: Temperature rise'. Pub-lication 76, 1976
