CC BY
12
Electrical Insulation and Cable Engineering
UDC 621.315.615.2 https://doi.org/10.20998/2074-272X.202L5.07
O. Shutenko, S. Ponomarenko
ANALYSIS OF DISTRIBUTION LAWS OF TRANSFORMER OIL INDICATORS IN 110-330 kV TRANSFORMERS
Introduction. Ensuring the operational reliability ofpower transformers is an urgent task for the power industry in Ukraine and for most foreign countries. One of the ways to solve this problem is the correction of maximum permissible values of insulation parameters. However, such a correction is fundamentally impossible without an analysis of the laws of distribution of diagnostic indicators in the equipment with different states. The purpose of the research is to analyse the laws of distribution of the quality indicators of transformer oil with different states in 110 and 330 kV transformers. Novelty. It was found that both 330 kV autotransformers and 110 kV transformers have the displacements between the mathematical expectations of the distribution density of usable oil indicators. It caused by different service life of the analysed transformers and different values of load factors. This indicates the need to consider the influence of these factors when correcting the maximum permissible values of oil indicators. Also, the presence of displacement between the distribution densities of some indicators of usable oil in 110 kV transformers and 330 kV autotransformers has been revealed. It indicates a different intensity of oxidation reactions in transformers with different voltage class. In order to reduce the heterogeneity of initial data the procedure of statistical processing of in-service test results has been proposed as a method. This procedure combines the use of a priori information about the service life of equipment and values of load factors with the elements of statistical hypothesis testing. The results of the analysis of the distribution laws of transformer oil indicators with different states have shown that for both usable and unusable oil the values of oil indicators obey the Weibull distribution. Values of the shape and scale parameters for each of the obtained indices arrays have been obtained, as well as calculated and critical values of the goodness-of-fit criteria. Practical value. Obtained values of the distribution law parameters of the transformer oil indicators with different states, considering the service life and operating conditions allow to perform the correction of the maximum permissible values of the indicators using the statistical decision-making methods. References 38, tables 7, figures 5.
Key words: transformer oil, oil indicators, operating time, statistical analysis, distribution laws, goodness-of-fit criteria, Weibull distribution, density functions.
У cmammi наведено результати aHani3y закошв розподыу показниюв трансформаторних масел у трансформаторах 110 i 330 кВ. Встановлено, що розподт показниюв для масла як придатного, так i непридатного до експлуатацп, незалежно eid класу напруги трaнсформaторiв тдпорядковуються закону розподыу Вейбула. Виконаний aнaлiз показав, що i в автотрансформаторах напругою 330 кВ, i в трансформаторах напругою 110 кВ мае мкце змщення мiж математичними оч^ваннями щiльностi розподыу показниюв масел придатного до експлуатацп. Наявтсть даного змщення дозволяе використовувати отримант з урахуванням умов експлуатацп значення пaрaметрiв закотв розподыу для ощнки вiдпрaцьовaного ресурсу масел, а також для прогнозування та планування термтв обслуговування та ремонту обладнання. Бiбл. 38, табл. 7, рис. 5.
Ключовi слова: трансформаторне масло, показники масел, тривалкть експлуатацп, статистичний аналiз, закони розподшу, KpOTepii' згоди, розподш Вейбула, функци щшьносл розподшу.
В статье приведены результаты анализа законов распределения показателей трансформаторных масел в трансформаторах 110 и 330 кВ. Установлено, что распределение показателей как для масла годного, так и негодного к эксплуатации, вне зависимости от класса напряжения трансформаторов подчиняются закону распределения Вейбулла. Выполненный анализ показал, что и в автотрансформаторах напряжением 330 кВ, и в трансформаторах напряжением 110 кВ имеет место смещение между математическими ожиданиями плотностей распределения показателей масел годного к эксплуатации. Наличие данного смещения позволяет использовать полученные с учетом условий эксплуатации, значения параметров законов распределения для оценки отработанного ресурса масел, а также для прогнозирования и планирования сроков обслуживания и ремонта оборудования. Библ. 38, табл. 7, рис. 5.
Ключевые слова: трансформаторное масло, показатели масел, длительность эксплуатации, статистический анализ, законы распределения, критерии согласия, распределение Вейбулла, функции плотности распределения.
Introduction. Accidental damage of high-voltage power transformers is accompanied by significant economic damage and in some cases can have serious consequences [1]. Given the significant ageing of highvoltage power transformers in Ukraine and in most foreign countries, as well as the extremely low rate of equipment replacement that has reached the end of its life, ensuring the operational reliability of transformers is an important scientific and practical task. One of the elements of insulation of high-voltage power transformers is petroleum insulating oils. As shown in [2, 3], the condition of transformer oils has a significant role in ensuring the operational reliability of transformers. Oil
oxidation products have a negative effect not only on the electrical strength of the liquid insulation, but also contribute to a reduction in the mechanical strength of the cellulose insulation, which leads to a reduction in the transformer service life. In this regard, the improvement of methods for assessing the condition of transformer oils is an urgent task. The solution of this problem will increase the operational reliability of high-voltage power transformers and extend their service life.
Publication analysis and research problem statement. At present, a great deal of research is devoted
© O. Shutenko, S. Ponomarenko
to the improvement of methods for diagnosing the condition of transformer oils. Improvement is carried out in two main directions - the use of new methods of measurement (for example, [4-8]) and improvement of decision-making methods. For example, in [9-12] neural networks of different configurations are proposed to diagnose the condition and predict the values of transformer oil indicators. In [13, 14], oil condition assessment was performed using Markov networks. In [15, 16] fuzzy logic was applied, and in [17-19] regression models were used. Considering that under real operating conditions the condition of oils is evaluated by comparing the measured values of indicators with maximum permissible values (MPV), which are regulated by international or national standards [20, 21], in [22-24] a correction of MPV indicators of transformer oils was performed. However, in these studies, integral distribution functions were used to correct MPV, and the laws themselves were not analysed. At the same time, as shown in [25, 26], the use of statistical decision methods, taking into account the distribution laws of diagnostic indicators to correct MPV allows to significantly reduce the risks compared to the method of integral functions. In addition, the parameters of distribution laws are widely used in the development of models for predicting the residual life of equipment, e.g. [27]. Meanwhile, the analysis showed that the analysis of distribution laws of transformer oil indicators has not found sufficient coverage in the literature. An exception is the research carried out in [28] according to which oil indicators such as organic acid content (OAC), breakdown voltage (BDV), interfacial tension, oil resistivity and water content can be described by a Weibull distribution. However, the differences in the values of the oil distribution parameters obtained before filling and aged oils are insignificant, which requires further verification. Therefore, this paper presents the results of the analysis of distribution laws for the whole set of transformer oil parameters in the tanks of 110 kV transformers and 330 kV autotransformers.
Statistical processing of periodic test results. The results of periodic transformer oil BDV monitoring for the 231 transformers of 110 kV and 49 autotransformers of 330 kV were used as input data. The total volume of the analysed sample amounted to 21062 values, of which 17408 were obtained for 110 kV transformers, while 3654 were obtained for 330 kV autotransformers. The list of transformer oil indicators and the volume of sample values for each indicator are shown in Table 1.
The sample presented in Table 1 is heterogeneous both in terms of the number of transformers and of the total number of indicator values. This heterogeneity is due to several factors. For example, for such indicator as the transformer oil colour, the current Ukrainian standard [21] regulates the maximum permissible values only for new oil, and for in-service control such values are not regulated, but must be taken into account when assessing the condition of oils. This circumstance is one of the reasons that the oil colour is monitored not in all power companies in Ukraine. Insignificant amounts of sampling
values for such indicators as water-soluble acids, mechanical impurities and water content are because the current standards allowed the use of several methods to determine the values of these indicators: indicative (absent or present) and quantitative (determined quantitative content of these indicators in mg KOH/g or g/t). Since the diagnosis «absent» using indicator methods is not equivalent to zero, only the results of quantitative methods were used for further analysis.
Table 1
List of indicators for transformer oils and volume
of sample values for each indicator
Volume of sample values
No. Oil quality indicator Number of Number of
transformers values
110 kV transformers
1 Flash point 230 3746
2 Acid number 231 3741
3 BDV 231 3723
4 tg^at 20 °C 31 268
5 tg^at 70 °C 50 426
6 tg^at 90 °C 60 570
7 Oil colour 121 2108
8 Water-soluble acid content 46 1191
9 Water content 73 1635
330 kV autotransformers
1 Flash point 49 887
2 Acid number 48 850
3 BDV 49 852
4 tg^at 90 °C 45 543
5 Water content 30 400
6 Contamination content 20 122
Since the test results were obtained in different laboratories and the transformers analysed have different service lives, are operated with different loads and are filled with different oil types, it is obvious that the oil ageing intensity in the analysed transformers varies considerably. In other words, the raw data is statistically heterogeneous. As an example, Fig. 1,a shows the dielectric dissipation factor of transformer oils measured at 90 °C for 330 kV autotransformers. The heterogeneity of the raw data in the figure is due to both differences in the quality of the oil filled (relatively high values of tgc> measured at 90 °C at the start of operation) and differences in operating conditions (low values of this indicator were obtained with a fairly long service life). In addition, there are errors in the test results. The heterogeneity of the raw data leads to the empirical distribution histogram distortion (Fig. 1,b), which does not allow an evaluation of the distribution laws for transformer oils.
In this regard, there is a task of forming homogeneous arrays of transformer oil indicators. At the first stage, from the initial data set were selected values of indicators that go beyond the area of MPV, regulated by the current normative document in Ukraine [21]. Using basic terms from the theory of technical diagnostics [29], denote the data set, consisting of the results of tests for an oil usable by the values of indicators as A. Data set consisting of test results for unusable oil as D2.
Two approaches are used to generate homogeneous indicator arrays. The first approach is based on the use of mathematical models of variance analysis and, as shown in [30, 31], allows obtaining sufficiently correct integral distribution functions of diagnostic attributes. A significant limitation of this approach is the limited information about the operating modes of transformers, the specifics of their design, the materials used. This may cause erroneous conclusions. The second approach is based on the use of statistical hypothesis tests and, as shown in [32], allows estimating the distribution laws of diagnostic attributes. The disadvantage of this approach is the difficulty in establishing the relationship between the parameters of the distribution laws of diagnostic indicators and the factors affecting the ageing intensity.
100
u
o
0
10 -
0.1
0.01
10
15
20
25
t, year 30
tests for such transformers were divided into several groups (for example, from the time of oil filling and to the regeneration date and from the regeneration date to the date of the last test) in a preliminary statistical processing step. The insulation temperature of power transformers [34, 35] depends on both the value of load currents and the ambient temperature, as well as on the transformer cooling system. The transformers under consideration are operated in the same climatic zone with similar values of average daily temperatures, and have an identical cooling system (with oil natural and air forced for 110 kV transformers, and with oil forced and air forced for 330 kV autotransformers). Therefore, it is logical to assume that differences in oil oxidation intensity caused by different operating temperature values, which are due to differences in load currents [36]. Under conditions where transformer load information is partially available, a statistical hypothesis test approach was used to generate homogeneous arrays of indicators.
Arrays with homogeneous periodic test results were generated by using three statistical criteria [32]:
1. Wilcoxon rank-sum test (W) is used to test the statistical hypothesis of similarity of distribution laws of two independent samples:
W = £ s(r,),
(1)
i=1
where ri are the ranks of the diagnostic indicators in the overall variation series; s(ri) is one of the possible N! permutations of rank r.
2. Z-criterion (Z) is used to test the statistical hypothesis of equality of mathematical expectations of two independent samples:
Z obs
X1 — x2
£ + St
n m
(2)
b
Fig. 1. Dependence of the dielectric dissipation factor of transformer oils measured at 90 °C in 330 kV autotransformers on the operating time for the original data set (a) and the corresponding empirical distribution histogram (b)
It is known [33] that the main factors influencing the intensity of oil oxidation are temperature, air oxygen content, duration of oxidation and the influence of some structural materials (copper, varnished insulating fabric, not oil resistant rubber, etc.). Considering that the date of oil filling is known for each of the transformers under consideration, the duration of oil ageing is not very difficult to account for. However, during long-term operation, the oil can be dried, regenerated, refilled or replaced both oil and silica gel, which leads to a distortion of the performance dependence on the operating time. In order to eliminate this problem, the results of the periodic
where xj and x2 are the selective mean of the diagnostic indicator calculated from the first and second transformer
test results respectively; Sj2 and are the selective variance of the diagnostic indicator calculated from the first and second transformer test results respectively; n and m are the number of observations of the diagnostic indicator for the first and second transformer respectively.
3. Fisher-Snedecor F distribution (F) is used to test the statistical hypothesis of dispersions equality of two independent samples:
—obs
sm
(3)
where Smax and Smin are the values of the maximum and minimum sampling variance.
Two independent samples were considered homogeneous if the following statistical hypotheses were not rejected by the test results at the given significance level a = 0.05:
1. about similarity of distribution laws of two independent samples (W1>W„1, n2, 0,025, W2<W„1, „2, 0,975);
1
0
5
a
s
2. about equality of mathematical expectations of two independent samples (Zobs<Zcnt, 0,95);
3. about dispersions equality of two independent samples (^obs<^(ni-i), (n2-i), 0,95).
The above algorithm for statistical processing of test results was implemented as the author's software «Odnorodn» [37], which significantly reduces the time required for statistical processing of raw data. Using this software in relation to the analysed data sample allowed forming several arrays of indicators with close values of sample means, sample variance and similar distribution laws. The volume of sample values (N), values of sample mean (Mx), sample variance (Dx) as well as skewness (js) and kurtosis (jk) coefficients for the original data sets (Di0) and those obtained from statistical processing (Dn-D1n) of usable oil indicators for 110 kV transformers are given in Table 2 and for 330 kV autotransformers in Table 3.
Comparison of statistical characteristics of initial arrays of oil indicators with similar characteristics of indicators from the arrays obtained after statistical processing showed that using statistical processing procedure allowed significantly reducing the Dx, as well as the js and jk for almost all indicators. Analysis of statistical characteristics of the arrays of indicators obtained through statistical processing shows that the values of mathematical expectation of the usable oil indicators differ significantly. This indicates both different quality of transformer oil and different oil ageing intensity, that is, differences in transformer operation modes. As can be seen from Tables, negative skewness coefficients (the «long part» of the distribution curve is to the left of the mathematical expectation) are obtained mainly for the indicators which decrease in value during ageing (flash point and BDV). For the oil indicators which increase with oil ageing (organic and water-soluble acid content, tgd, oil colour, water content, contamination content), the majority of the obtained arrays show positive skewness values («long part» of the distribution curve is to the right of the mathematical expectation).
The presence of a negative skewness for some of the indicator data sets indicates a deterioration in the condition of the transformer oil; there are more «high values» of the indicators than low ones. As can be seen from the tables, there is a positive kurtosis coefficient for all indicator arrays, indicating that the distribution curve has a higher and «sharper» peak than the curve of the normal law. The obtained empirical values of skewness and kurtosis coefficients indirectly indicate that the distribution of usable oil indicators may differ from the normal law.
The analysis showed that the homogeneous arrays were formed by testing transformers with close operating times and similar values of load factors. This made it possible to establish an unambiguous relationship between the operating conditions of the transformers and the values of the distribution law parameters.
The processing of the results of periodic tests on oils which have exceeded the MPV (unusable oil) was carried out in the following sequence. First, data sets were
Table 2
Statistical characteristics of homogeneous arrays of 110 kV transformer usable oil
Array N js jk
Flash point
D10 3746 143.579418 13.06904 0.809 4.281
D11 284 140.059859 2.866135 -0.346 3.693
Di2 364 142.931319 2.030997 -0.385 4.521
D,3 185 144.535135 3.600117 -0.144 2.361
D14 188 146.930851 15.968623 -0.088 3.404
D15 280 149.089286 9.988457 -0.172 3.308
Organic acid content
D10 3339 0.034848 0.000592 0.817 2.685
D11 218 0.015311 0.000087 0.587 2.472
D12 198 0.027884 0.00007 0.711 3.303
D13 250 0.045944 0.000395 0.175 2.262
D14 129 0.046434 0.000192 -0.175 2.907
D15 156 0.051679 0.000446 0.224 2.462
D16 361 0.045917 0.000500 0.807 3.988
D17 333 0.064655 0.001557 0.690 2.978
BDV
D10 3435 58.262667 116.058219 0.109 2.627
D11 135 49.038519 44.673183 0.0748 2.423
D12 609 60.654351 115.838869 -0.249 2.468
D13 360 68.110833 123.619522 -0.350 2.620
D14 286 69.421678 109.698341 -0.524 2.902
D15 204 71.066176 124.016356 -0.657 3.038
tgSat 20 °С
D10 268 0.182246 0.067281 7.962 83.730
D11 109 0.141835 0.008986 0.793 3.075
D12 141 0.188596 0.014915 0.959 3.713
tgSat 70 °С
D10 426 0.917268 2.170500 7.513 82.291
D11 112 0.290446 0.038860 0.951 3.284
D12 161 0.593634 0.162197 1.051 3.892
D13 140 0.850321 0.306149 0.991 4.229
tgSat 90 °С
D10 570 1.297158 12.095330 13.686 219.667
D11 148 0.485811 0.126923 1.189 4.664
D12 159 0.548679 0.133394 1.071 4.527
D13 152 1.504605 1.133087 0.793 3.119
D14 99 1.863737 1.848464 0.893 3.297
Oil colour
D10 2108 2.574953 1.769050 0.822 2.870
D11 650 1.896923 0.597837 0.391 2.606
D12 484 3.026860 3.096386 0.587 2.354
D13 657 3.060122 1.917618 0.128 2.131
Water-soluble acid content
D10 1191 0.006106 0.000281 11.835 257.889
D11 108 0.004359 0.000005 1.452 6.673
D12 231 0.005529 0.000006 0.025 2.611
D13 92 0.006315 0.000009 0.194 2.655
D14 116 0.006892 0.000008 0.182 2.301
D15 91 0.007679 0.000007 -0.120 2.684
Water content
D10 1635 14.381787 116.663877 2.723 10.436
D11 284 7.963 13.993306 0.511 2.773
D12 546 9.2768 20.054587 0.469 2.375
D13 520 10.748 16.643352 0.028 2.363
generated for each of the indicators for which a deterioration of the MPV values was detected. To ensure «equality» between the different transformers, a strictly fixed number of observations was selected for each of the indicators for each transformer.
Table 3
Statistical characteristics of homogeneous arrays of 330 kV autotransformer usable oil
Array N Mx Dx Js jt
Flash point
Dio 856 i4i.848i3i io.21759o o.471 3.2io
Dii i74 148.o63218 4.358o72 -o.481 3.198
Di2 i3i i46.i22i37 2.5347oi -o.495 2.794
Di3 241 i44.79253i 1.873969 -o.282 4.o92
Dl4 ioo i39.36oooo 2.59o4oo -o.4o8 2.882
Organic acid content
Dio 85o o.oi5665 o.ooo662 15.882 356.965
Dii i8o o.oo7oio o.ooooo7 o.253 3.oo7
Di2 99 o.oo7542 o.ooooo9 o.o9o 2.585
Di3 iio o.oo8343 o.ooooio o.io4 2.497
Di4 ii4 o.oo8546 o.ooooio —o.oi5 2.636
Di5 2o6 o.oi9877 o.oooi4o o.794 3.373
BDV
Dio 852 68.995i88 84.716867 -1.481 io.952
Dii 2io 74.i62857 6o.864239 —o.193 3.252
Di2 24o 68.3475oo 53.85116o -o.267 3.169
Di3 i49 66.8o9396 68.887612 —o.286 2.63o
Di4 i3o 65.853o77 93.476o29 o.oi93 2.441
tgSat 90 °C
Dio 543 o.83664o 2.4o8o43 6.572 62.575
Dii i62 o.2i2o93 o.oi8112 o.986 3.825
Di2 67 o.7oi7oi o.124185 1.465 5.622
Di3 i33 o.833985 o.4324o6 1.486 4.726
Di4 138 1.6916o9 o.984824 o.473 2.499
Contamination content
Dio i22 7.893852 32.249573 2.624 11.631
Dii 65 7.239231 9.591576 o.835 4.23
Di2 53 io.i46226 59.996212 1.549 5.378
Water content
Dio 4oo io.12328o 3o.923324 1.7o2 11.8o3
Dii 95 4.725653 9.111732 1.987 8.985
Di2 155 9.543226 17.576746 o.227 2.332
where xk is the suspicious value; xk prev is the previous value in the variation series.
The calculated value of the Irwin criterion was compared with the table value ^table. If ncalc > ntable, then the considered value was rejected and the next value was checked. The test was continued until ncalc < ntable.
By analogy with Tables 2 and 3, Table 4 shows the same attributes for the data sets obtained during statistical processing of unusable oil values for 110 kV transformers. In the table, array D21 is based on the results of transformers with less than 20 years of service life, and array D22 is based on the results of transformers with more than 20 years of service life.
Table 4
Statistical characteristics of homogeneous arrays of indicators of unusable oil
Array N Mx Dx js jt
110 kV transformers
Organic acid content
D21 3o3 o. 177673 o.ooi 155 —o.176 2.421
D22 141 o.283191 o.ooo225 —o.o92 2.4o2
BDV
D2 275 27.647 19.814 —o.653 3.o83
Water-soluble acid content
D2 192 o.o359o4 o.oooi54 o.55o 3.o4o
Water content
D2 164 36.333659 ioo.io887 o.5o5 3.o97
330 kV autotransformer
Organic acid content
D2 136 o. 137456 o.ooo273 —o.oii3 3.2oo
BDV
D2 123 41.55o894 4.88oi38 —o.666 2.875
Water content
D2 132 29.287879 22.728261 —o.oi2o 2.237
Similar to [32], an approach based on the extraction of gross omissions from a number of similar measurements was used to further process of the unusable oil indicators values. For this purpose, the Irwin criterion was used, which can be applied when the distribution law of a random variable is unknown or differs from the normal distribution. The values of the oil indicators were sorted in descending order for this purpose, after which the «suspicious» values of the BDV at the boundaries of the variation series were evaluated. The value of the Irwin criterion was defined as:
(xk — xk prev )
Vcalc
S
(4)
By comparing the statistical characteristics for the respective indicators for usable and unusable oil, it can be seen that the main difference between them is the values of the sample means.
Analysis of the distribution laws for transformer oils. To test the hypothesis that the theoretical distribution law corresponds to empirical data, the software «ZR» developed at the Electric Power Transmission Department of National Technical University «Kharkiv Polytechnic Institute» was used [37]. This software allows splitting the range of variation of a random variable into intervals, for which the Sturges' rule is used by default:
L = 1 + 3,322 • lgN, (5)
where N is the volume of sample values.
If necessary, the user can set the required number of intervals himself. A histogram of the empirical distribution is then drawn. The parameters for the normal, beta, exponential, extremal, gamma, Laplace, logistic, logarithmic normal, Rayleigh, Weibull and Pareto distribution laws are then estimated from the sample values. For each of the 11 distribution laws, the theoretical distribution law is checked for consistency with the empirical data, using two statistical criteria [38]:
1) Pearson's chi-squared test (^2):
OSxLogll orm a>( x;1,334;0.579)
Ж
2 — ^ (ni -ni )
(6)
i—1
where n are values of empirical frequencies; are values of theoretical frequencies; k is the number of intervals.
To test the main hypothesis, the sample value of criterion is calculated and the critical point of criterion X2 distribution, the given significance level a and the number of degrees of freedom f are determined by the critical point ^2cnt(a; f). The number of degrees of freedom f is defined as f = k - 1 - r, where k is the number of sample groups (partial intervals); r is the number of estimated distribution parameters that are estimated from the sample data.
If the calculated value of criterion is less than the critical value, the main hypothesis - the general population is distributed according to the given law - is not rejected and vice versa.
2) Kolmogorov-Smirnov test:
Dn — sup|Fn (x) - F(x)|,
(7)
where Fn(x) is the empirical distribution function; F(x) is the theoretical distribution function.
The main hypothesis is rejected if 4n • Dn exceeds
the distribution quantile Ka of the given significance level a, and is not rejected otherwise [38].
Analysis by the software showed that both usable and unusable oil values could be described by a Weibull distribution, with a distribution density as follows:
p(x;a; ß ) —-ß- • x ß-1 • e
П ß
(8)
where a and 3 are distribution law parameters, interpreted as scale and shape parameter, respectively.
The value of the a and ft parameters was defined as:
M(x)
a =-
1 - 0.427 • (ß -1 ; • ß
-1.9 :
ß —
N-1
N
0.465 •
1 N
- - M(x)Y N i—1
+1.282 •-
M(x) M(x)
-0.7
1 N
- - M(x)f
N i=1
where N is the volume of sample values; Mx is the sample mean; xi is the indicator value.
However, during the analysis, for some of the arrays the main hypothesis was not rejected for several distribution laws. Thus, for distributions with positive skewness, in addition to the Weibull distribution, the empirical data were fit to the log-normal, gamma and Rayleigh distribution laws.
As an example, Fig. 2 shows histograms of the empirical distribution and the theoretical density functions
0.45 0.4 [1.35 0.3 [1.25 0.2 [1.15
0.4 0.35
0.25 0.2
0.35 0 3
0.2 0.15
<9S><Rayleigli>(K:3.9SSi
OSxGam ina>(K;1.610;2.S3St
<SS>iWeibu I>(k;S.278;1 .537)
d
Fig. 2. Histograms of the empirical distribution and the theoretical density functions of some distributions for the water content of oil from the D11 array of 330kV autotransformers: a - log-normal distribution law; b - Rayleigh distribution; c - gamma distribution; d - Weibull distribution
a
x
b
ß
x
a
c
+
1
of these distributions for the water content of oil from the Du array of 330 kV autotransformers.
For relatively symmetric distributions, in addition to the Weibull distribution, for some of the arrays the empirical data were found to fit the normal and logistic laws. However, for many arrays with negative skewness, the only distribution law corresponding to the empirical data was a Weibull distribution law.
The values of the Weibull distribution law parameters as well as the calculated and critical values of the Pearson and Kolmogorov-Smirnov tests for the indicators of usable oil in 110 kV transformers are given in Table 5 and for 330 kV autotransformers are given in Table 6.
The same, but for the unusable oil indicators are given in Table 7. For the tables below, the value of the Kolmogorov-Smirnov test is 1.36.
As can be seen from Tables 6, 7, the calculated values of the goodness of fit criteria for all transformer oil indicators without exception do not exceed the critical values at significance level a = 0.05 and the corresponding value of degrees of freedom. This does not allow rejecting the hypothesis about the acceptability of the distribution of empirical values of transformer oils to the Weibull distribution.
Results analysis. By analogy with [32], the analysis of the mutual arrangement of theoretical densities of distribution of the indicators of usable and unusable transformer oils in of 110 and 330 kV transformers was carried out. As an example Figures 3-5 shows the densities of theoretical distributions for such indicators as the organic acid content (Fig. 3), water content (Fig. 4) and the BDV (Fig. 5) of the oil.
Analysing the mutual arrangement of the theoretical distribution densities of transformer oil indicators, several important conclusions can be drawn:
1. For the usable transformer oil for both 110 kV transformers and 330 kV autotransformers, there is a shift in the mathematical expectations of the distributions density for different arrays of the same indicator. This indicates a different oil ageing degree, which is caused by different oil service life, different operating temperatures of transformers as well as by the influence of structural materials.
2. Analysis of the distribution densities of usable oil for 110 kV transformers and 330 kV autotransformers shows that these distributions are also shifted in relation to each other. It is especially noticeable in the distribution densities of organic acids (Fig. 3) and water content of oils (Fig. 4). Thus, as can be seen from the figure the oxidation intensity of transformer oils in 330 kV autotransformers is lower than in 110 kV transformers, despite the fact that the analysed transformers were non-sealed.
3. The analysis shows that there is a significant shift between the mathematical expectations of the distributions of usable and unusable oil (Figures 3-5). This means that the residual life of the oils in the transformers analysed varies considerably.
Table 5
Values of the Weibull distribution law parameters as well as the calculated and critical values of the Pearson and Kolmogorov-Smirnov tests for the indicators of usable oil in 110 kV transformers
Value of Kolmogorov-Smirnov test
Array Distribution law parameters Value of Pearson's chi-squared test
a ß f ^calc ^^crit ^calc
Flash point
Dii i40.96 i04.9 3 7.47 7.82 0.7ii
Di2 i43.7i i27.5 4 8.59 9.49 i.i94
Di3 i45.54 96.44 3 7.47 7.82 0.338
Dl4 i48.9i 46.2 4 8.26 9.49 0.694
Dl5 i50.69 59.57 3 6.27 7.82 i.044
Organic acid content
Dii 0.0i72 i.682 6 i0.47 i2.6 0.7ii
Dl2 0.0308 3.706 5 2i.26 ii.i i.00i
Di3 0.05i8 2.454 6 8.54 i2.6 0.654
Di4 0.05i4 3.702 4 7.92 9.49 0.674
Di5 0.058i 2.6ii 5 i0.68 ii.i 0.800
Di6 0.05i9 2.i54 4 6.074 9.49 0.545
Di7 0.0725 i.679 6 i0.96 i2.6 0.760
BDV
Dii 5i.833 8.704 4 8.942 9.49 0.7ii
Di2 64.963 6.596 7 i2.03 i4.i 0.607
Di3 72.625 7.209 6 i2.2i i2.6 0.643
Di4 73.726 7.84 5 6.4i9 ii.i 0.606
Di5 75.623 7.5i7 5 6.59 ii.i 0.60i
tgSat 20 °C
Dii 0.i576 i.5i5 4 9.439 9.49 i.0i7
Di2 0.2i03 i.569 4 9.095 9.49 0.639
tgSat 70 °C
Dii 0.322i i.49i 4 5.079 9.49 0.697
Di2 0.6585 i.496 4 6.465 9.49 0.673
Di3 0.9478 i.562 4 i.865 9.49 0.3ii
tgSat 90 °C
Dii 0.5327 i.379 3 2.296 7.82 0.329
Di2 0.6i0i i.526 3 i.i73 7.82 0.275
Di3 i.6593 i.432 3 4.737 7.82 0.734
Di4 2.0442 i.383 3 2.764 7.82 0.383
Oil colour
Dii 2.i334 2.63i 3 6.794 7.82 0.690
Di2 3.4054 i.772 4 8.456 9.49 0.739
Di3 3.453 2.339 3 6.523 7.82 0.656
Water-soluble acid content
Dii 0.0049 i.986 2 2.765 5.99 0.449
Di2 0.0062 2.365 6 6.383 i2.6 0.446
Di3 0.007i 2.207 4 i.468 9.49 0.288
Di4 0.0078 2.554 4 3.736 9.49 0.378
Di5 0.0086 3.ii2 3 2.277 7.82 0.344
Water content
Dii 8.9923 2.239 5 2.i37 ii.i 0.279
Di2 i0.478 2.i76 7 i0.06 i4.i 0.552
Di3 i2.05 2.849 7 8.763 i4.i 0.385
Table 6
Values of the Weibull distribution law parameters as well as the calculated and critical values of the Pearson and Kolmogorov-Smirnov tests for the indicators of usable oil in 330 kV autotransformers
Array Distribution law parameters Value of Pearson's chi-squared test Value of Kolmogorov-Smirnov test
a ß f Xcalc ^crit ^calc
Flash point
D„ i49.i6 89.7i 4 8.848 9.49 0.672
Dl2 i46.99 ii6.i 4 9.330 9.49 0.556
Di3 i45.54 i34.3 3 7.5i0 7.82 0.97i
Dl4 i40.23 i09.2 4 8.678 9.49 0.67i
Organic acid content
Dii 0.0079 2.899 2 0.609 5.99 0.228
Di2 0.0085 2.766 4 5.85i 9.49 0.586
Di3 0.0094 2.8i2 5 9.364 ii.i 0.693
Dl4 0.0096 2.986 3 i.939 7.82 0.40i
Di5 0.0223 i.725 2 4.943 5.99 0.699
BDV
Dii 77.523 ii.48 3 5.235 7.82 0.890
Di2 7i.499 ii.24 4 8.207 9.49 0.893
D,3 70.3i9 9.6i2 4 2.768 9.49 0.260
Di4 69.855 8.038 4 9.229 9.49 0.754
tgSat 90 °C
D„ 0.237 i.605 3 i.485 7.82 0.459
D,2 0.7926 2.055 2 2.7i0 5.99 0.509
Di3 0.90i8 i.283 3 7.286 7.82 0.760
Di4 i.90i7 i.745 4 0.685 9.49 0.i8i
Contamination content
Dii 8.i592 2.457 2 3.024 5.99 0.709
Di2 ii.0i8 i.309 i i.052 3.84 0.335
Water content
D„ 5.2755 i.587 i i.068 3.84 0.i78
Di2 i0.762 2.407 5 3.305 ii.i 0.30i
Table 7
Values of the Weibull distribution law parameters as well as the calculated and critical values of the Pearson and Kolmogorov-Smirnov tests for the indicators of unusable oil in 110 kV transformers and 330 kV autotransformers
Array Distribution law parameters Value of Pearson's chi-squared test Value of Kolmogorov-Smirnov test
a ß f X calc ^crit ^calc
110 kV transformer
Organic acid content
d2, 0.i9ii 6.07 6 2.990 i2.6 0.376
d22 0.290i 23.36 4 i.843 9.49 0.265
BDV
d2 29.459 7.3ii 4 3.i62 9.49 0.228
Water-soluble acid content
d2 0.040i 3.i55 5 3.009 ii.i 0.420
Water content
d2 39.98 4.059 3 i.850 7.82 0.338
330 kV autotransformer
Organic acid content
d2 0.i445 9.955 4 7.427 9.49 0.704
BDV
d2 42.576 23.25 4 2.763 9.49 0.323
Water content
d2 3i.232 7.i96 4 2.690 9.49 0.407
0.3 0.25
I 0.2 s
® 0.15
£ -
0.1
0.05
0.0001
0.001 0.01 0.1 Acid number, mg KOH per 1 g of oil
0.001 0.01 0.1
Acid number, mg KOH per 1 g of oil
b
Fig. 3. Density functions of the theoretical distribution of organic acid content in oil for 110 kV transformers (a) and 330 kV autotransformers (b)
0.3
^ 0.25 e
Q
'S 0.2
o u
Q
£ 0.15 | 01
Ph
0.05
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 Moisture content, g/t
a
0.4 0.35
a" 0.3
q
SI 0.25
u
I 0.2
I 0.15
S
I 01
0.05 0
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 Moisture content, g/t
b
Fig. 4. Density functions of the theoretical distribution of water content in oil for 110 kV transformers (a) and 330 kV autotransformers (b)
0
1
a
1
0
Breakdown voltage, kV
a
Breakdown voltage, kV
b
Fig. 5. Density functions of the theoretical distribution of the BDV in oil for 110 kV transformers (a) and 330 kV autotransformers (b)
Conclusions.
The analysis of the distribution laws of transformer oils showed that for both usable and unusable oil the distribution of oil indicators could be described by the Weibull distribution, which agrees well with the results of previously published studies. In the process of analysis, it was found that for the usable transformer oil, both for 110 kV transformers and 330 kV autotransformers there is a shift of mathematical expectations of the distribution density for different arrays of the same indicator. This indicates a different oil ageing degree which is caused by different oil service life, different operating temperatures of transformers as well as by the influence of structural materials. During the analysis, it was found that in 330 kV autotransformers the oxidation reactions proceed with less intensity compared to 110 kV transformers. This is evidenced by the presence of a shift between the distribution densities of some indicators of usable oil in 110 kV transformers and 330 kV autotransformers. It has been found that there is a significant shift between the mathematical expectations of the distributions of usable and unsuitable oil. This means that the residual life of oils in transformers analysed varies significantly. The obtained values of the parameters of the distribution laws of transformer oil indicators can be used in the development of models to estimate the service life of oils, as well as to predict and plan the
timing of maintenance and repair of equipment, which will allow to carry out the transition to maintenance according to the real situation rather than the calendar plan. In addition, the presence of distribution law parameter values for indicators of oil with different states allow to estimate the state of transformer oils using likelihood ratios, which can significantly reduce the risks of making wrong decisions.
Conflict of interests. The authors declare no conflicts of interest.
REFERENCES
1. N'cho J., Fofana I., Hadjadj Y., Beroual A. Review of Physicochemical-Based Diagnostic Techniques for Assessing Insulation Condition in Aged Transformers. Energies, 2016, vol. 9, no. 5, p. 367. doi: https://www.doi.org/10.3390/en9050367.
2. Mehmood M., Nazir M., Li J., Wang F., Azam M. Comprehensive Investigation on Service Aged Power Transformer Insulating Oil After Decades of Effective Performance in Field. Arabian Journal for Science and Engineering, 2020, vol. 45, no. 8, pp. 6517-6528. doi: https://www.doi.org/10.1007/s13369-020-04559-7.
3. Rengaraj R., Venkatakrishnan G.R., Moorthy P., Pratyusha R., Ritika, Veena K. Transformer Oil Health Monitoring Techniques-An Overview. Advances in Intelligent Systems and Computing, 2020, pp. 135-154. doi: https://www.doi.org/10.1007/978-981-15-5029-4 12.
4. Tyuryumina A., Batrak A., Sekackiy V. Determination of transformer oil quality by the acoustic method. MATEC Web of Conferences, 2017, vol. 113, p. 01008. doi: https://www.doi.org/10.1051/matecconf/201711301008.
5. Leong Y., Ker P., Jamaludin M., Nomanbhay S.M., Ismail A., Abdullah F., Looe H., Lo C. UV-Vis Spectroscopy: A New Approach for Assessing the Color Index of Transformer Insulating Oil. Sensors, 2018, vol. 18, no. 7, p. 2175. doi: https://www.doi.org/10.3390/s18072175.
6. Kang S.B., Kim W.-S., Chung D.C., Joung J.M., Kwak M.H. Degradation diagnosis of transformer insulating oils with terahertz time-domain spectroscopy. Journal of the Korean Physical Society, 2017, vol. 71, no. 12, pp. 986-992. doi: https://www.doi.org/10.3938/jkps.71.986.
7. Alshehawy A.M., Mansour D.A., Ghali M., Rezk A. Evaluating the impact of aging in field transformer oil using optical spectroscopy techniques. 2017 IEEE 19th International Conference on Dielectric Liquids (ICDL), 2017, pp. 1-4. doi: https://www.doi.org/10.1109/ICDL.2017.8124626.
8. Degeratu S., Rotaru P., Rizescu S., Danoiu S., Bizdoaca N.G., Alboteanu L.I., Manolea H.O. Condition monitoring of transformer oil using thermal analysis and other techniques. Journal of Thermal Analysis and Calorimetry, 2015, vol. 119, no. 3, pp. 1679-1692. doi: https://www.doi.org/10.1007/s10973-014-4276-3.
9. Zhao Y., Qian Y., Li L., Zheng Z., Wang Q., Zhou Y. Research on Transformer Oil Multi-frequency Ultrasonic Monitoring Technology Based on Convolutional Neural
Network. 2019 IEEE 20th International Conference on Dielectric Liquids (ICDL), 2019, pp. 1-5. doi: https://www.doi.org/10.1109/ICDL.2019.8796733.
10. Srividhya V., Babu J.S., Sujatha K., Veerendrakumar J., Aruna M., Shafiya S., SaiKrishna, Anand M. Determination of Breakdown Voltage for Transformer Oil Testing Using ANN. Advances in Intelligent Systems and Computing, 2021, pp. 443-452. doi: https://www.doi.org/10.1007/978-981-33-6981-8 35.
11. Singh H., Singh J. Enhanced optimal trained hybrid classifiers for aging assessment of power transformer insulation oil. World Journal of Engineering, 2020, vol. 17, no. 3, pp. 407-426. doi: https://www.doi.org/10.1108/wje-11-2019-0339.
12. Gautam L., Kumar R., Sood Y.R., Identifying Transformer Oil Criticality Using Fuzzy Logic Approach. 2020 IEEE Students Conference on Engineering & Systems (SCES), 2020, pp. 1-6. doi: https://www.doi.org/10.1109/SCES50439.2020.9236724.
13. Milosavljevic S., Janjic A. Integrated Transformer Health Estimation Methodology Based on Markov Chains and Evidential Reasoning. Mathematical Problems in Engineering, 2020, vol. 2020, pp. 1-12. doi: https://www.doi.org/10.1155/2020/7291749.
14. Surya Subaga I.G., Manuaba I.B.G., Sukerayasa I.W. Analisis Prediktif Pemeliharaan Minyak Transformator Menggunakan Metode Markov. Jurnal SPEKTRUM, 2019, vol. 6, no. 4, pp. 96-101. (Ind). Available at: https://ojs. unud.ac.id/index.php/spektrum/article/view/553 35/32753 (Accessed 22 May 2021).
15. Rexhepi V., Nakov P. Condition assessment of power transformers status based on moisture level using fuzzy logic techniques. Journal of Mechatronics, Electrical Power, and Vehicular Technology, 2018, vol. 9, no. 1, pp. 17-24. doi: https://www.doi.org/10.14203/j.mev.2018.v9.17-24.
16. Chantola A., Sharma M., Saini A. Integrated Fuzzy Logic Approach for Calculation of Health Index of Power Transformer. 2018 Second International Conference on Inventive Communication and Computational Technologies (ICICCT), 2018, pp. 1045-1050. doi: https://www.doi.org/10.1109/ICICCT.2018.8473316.
17. Shutenko O., Ponomarenko S. Diagnostics of Transformer Oils Using the Multiple Linear Regression Model 2020 IEEE Problems of Automated Electrodrive. Theory and Practice (PAEP), 2020, pp. 1-6, doi: https://www.doi.org/10.1109/PAEP49887.2020.9240875.
18. Abdi S., Harid N., Safiddine L., Boubakeur A., Haddad A. The Correlation of Transformer Oil Electrical Properties with Water Content Using a Regression Approach. Energies, 2021, vol. 14, no. 8, p. 2089. doi: https://www.doi.org/10.3390/en14082089.
19. Gouda O., El Dein A. Prediction of Aged Transformer Oil and Paper Insulation. Electric Power Components and Systems, 2019, vol. 47, no. 4-5, pp. 406-419. doi: https://www.doi.org/10.1080/15325008.2019.1604848.
20. IEC 60422. Mineral insulating oils in electrical equipment - Supervision and maintenance guidance. 2012.
21. SOU-N EE 43-101:2009. Adoption, application and use of transformer oils. Quality assessment standards. Kyiv, 2018. (Ukr).
22. Davidenko I.V., Egorov A.A. Determination of Criteria for Assessing the Oil Quality of Current Transformers Type TFZM 110 kV. 2020 IEEE Conference of Russian Young Researchers in Electrical and Electronic Engineering (EIConRus), 2020, pp. 12041207. doi: https://www.doi.org/10.1109/EIConRus49466.2020.9039 527.
23. Azis N., Zhou D., Wang Z.D., Jones D., Wells B., Wallwork G.M. Operational condition assessment of inservice distribution transformers. 2012 IEEE International Conference on Condition Monitoring and Diagnosis, 2012, pp. 1156-1159. doi: https://www.doi.org/10.1109/CMD.2012.6416364.
24. Davidenko I., Egorov A. Development of an integral criterion for evaluating the degree of aging of transformer oils. IOP Conference Series: Materials Science and Engineering, 2020, vol. 950, p. 012005. doi: https://www.doi.org/10.1088/1757-899x/950/1/012005.
25. Shutenko O. Determine the boundary value of the concentration of gases dissolved in oil of method minimum risk. 2017 IEEE First Ukraine Conference on Electrical and Computer Engineering (UKRCON), 2017, pp. 468-472. doi: https://www.doi.org/10.1109/UKRCON.2017.8100533.
26. Shutenko O., Zagaynova A., Serdyukova G. Determining the maximally permissible values for the indicators of insulation of sealed entrance bushings with a voltage of 110 kV using the method of minimal risk. Eastern-European Journal of Enterprise Technologies, 2018, vol. 5, no. 8 (95), pp. 6-15. doi: https://www.doi.org/10.15587/1729-4061.2018.142185.
27. Mohd Selva A., Azis N., Shariffudin N.S., Ab Kadir M.Z.A., Jasni J., Yahaya M.S., Talib M.A. Application of Statistical Distribution Models to Predict Health Index for Condition-Based Management of Transformers. Applied Sciences, 2021, vol. 11, no. 6, pp. 2728. doi: https://www.doi.org/10.3390/app11062728.
28. Tsuboi T., Takami J., Okabe S., Inami K., Aono K. Aging effect on insulation reliability evaluation with Weibull distribution for oil-immersed transformers. IEEE Transactions on Dielectrics and Electrical Insulation, 2010, vol. 17, no. 6, pp. 1869-1876. doi: https://www.doi.org/10.1109/TDEI.2010.5658240.
29. Birger I. A. Technical diagnostics. Moscow, Mashinostroenie Publ., 1978. 240 p. (Rus).
30. Davidenko I.V. Determination of allowable values of controlled parameters of oil-filled equipment on the basis of an array of observed data. Elektrichestvo, 2009, no. 6, pp. 10-21. (Rus). Available at: https://elibrary.ru/download/elibrary 12880537 3589719 5.pdf (Accessed 22 May 2021).
31. Davidenko I.V. Investigation of indicators describing the operational state of oil-filled bushings, using mathematical statistics. University news. North-Caucasian region. Technical sciences series, 2006, no. 15, pp. 31-33. (Rus).
32. Shutenko O., Zagaynova A., Serdyukova G. Analysis of distribution laws of insulation indicators of highvoltage oil-fillled bushings of hermetic and non-hermetic execution. Technology audit and production reserves, 2018, vol. 4, no. 1 (42), pp. 30-39. doi: https://www.doi.org/10.15587/2312-8372.2018.140873.
33. Lipstein R.A., Shakhnovich M.I. Transformer oil. Moscow, Energoatomizdat Publ., 1983. 296 p. (Rus).
34. Vasilevskij V.V. Assessment of the resource consumption of oil-filled power transformer paper insulation based on updated aging integral. Electrical Engineering & Electromechanics, 2015, no. 1, pp. 16-19. (Rus). doi: https://www.doi.org/10.20998/2074-272x.2015.1.03.
35. Poliakov M.A., Vasilevskij V.V. Evaluation of power transformer insulation residual life based on its individual life cycle characteristics. Electrical Engineering & Electromechanics, 2014, no. 3, pp. 38-41. (Rus). doi: https://www.doi.org/10.20998/2074-272x.2014.3.07.
36. Shutenko O., Ponomarenko S. Analysis of the Impact of Power Transformer Loading on the Transformer Oil Aging Intensity. 2020 IEEE KhPI Week on Advanced
Technology (KhPIWeek), 2020, pp. 76-81. doi: https://www.doi.org/10.1109/KhPIWeek51551.2020.9250 159.
37. Shutenko O.V., Baklay D.N. Planning of experimental research in power engineering. Methods of Processing of Experimental Data. Kharkiv, NTU «KhPI» Publ., 2013. 268 p. (Rus).
38. Rinne H. The Weibull distribution. Boca Raton, CRC Press, 2009. 808 p.
Received 15.07.2021 Accepted 24.09.2020 Published 26.10.2021
O. Shutenko1, PhD, Associate Professor, S. Ponomarenko1, PhD Student, National Technical University «Kharkiv Polytechnic Institute»,
2, Kyrpychova Str., Kharkiv, 61002, Ukraine, e-mail: o.v.shutenko@gmail.com, PonomarenkSerhii@gmail.com (Corresponding author)
How to cite this article:
Shutenko O., Ponomarenko S. Analysis of distribution laws of transformer oil indicators in 110-330 kV transformers. Electrical Engineering & Electromechanics, 2021, no. 5, pp. 46-56. doi: https://doi.org/10.20998/2074-272X.2021.5.07.
