SI Green Yellow Red

measurements. Basically, the differential relay system of the transformer under the test is set to 0.5% of the nominal current [176]. The rheostat value corresponding to this short- circuit current is found via decreasing the resistance from the highest to the lowest and measuring the current through the rheostat. Once the measured current value matches the desired value (0.5% of the nominal), the paralleled resistance is recorded and FRA signature of the given configuration is obtained. Similarly, decision boundary B

Y|R

is expressed in terms of the critical values of 12 statistical indicators calculated from the FRA data.

Table 8 [180] ©2021 IEEE provides the rheostat values corresponding to yellow-to- red decision boundary and the CC value of that given short-circuit condition. B

Y|R

sample points expressed by other statistical indicators are also estimated from the obtained FRA signatures. Table 9 [181] ©2022 IEEE illustrates the summary of the decision boundary identification, where 6 observations of B

G|Y

and 3 observations of B

Y|R

are collected for the index CC and Table 10 summarizes the calculated critical values for all SIs indicating two decision boundaries [20] ©2022 IEEE.

Table 8 – Current and paralleled rheostat values of B

Y|R

boundary Transformer under test T1-0.4 kVA T2-0.63 kVA T3-1 kVA

Nominal current (A) 11.00 2.86 4.54

Critical current (A) 0.055 0.0143 0.0227

Rheostat (kΩ) 4.6 15.4 10.5

CC 0.9990 0.9991 0.9989

Table 9 – Collected critical CC values for B

G|Y

and B

Y|R

boundaries Green-to-yellow boundary, B

G|Y

Yellow-to-red boundary, B

Y|R

0.9997 0.9998 0.9998

0.9998 0.9999 0.9997

0.9990

0.9991

0.9989

Table 10 – Critical values of statistical indicators

ASLE 0.09895 0.245 DABS 0.394 1.12 RMSE 0.02867 0.07

ED 22.71 67

CSD 0.6403 1.3

MM 1.0153 1.0354

SSRE 0.00035 0.0009 SSMMRE 0.00035 0.0009

ρ 0.99993 0.9995

5.4 Classification and Confidence level estimation

Reporting the confidence level along with classification labels is the essential part of the proposed interpretation technique. In other words, the confidence of the test object being in healthy, suspicious, and critical working condition will facilitate a more efficient decision-making process. Since transformer shut-down implies either temporary power outage or requirement of re-routing the power flow through other units, the industry is commonly reluctant to retrieve the transformer from the service unless it is inevitable and justified with sufficient test data.

5.4.1 Bolstered error estimation

As it was mentioned above, the introduced method provided the confidence level (CL) corresponding to each operating mode. The evaluation of the CL is based on the bolstered error estimation technique borrowed from pattern recognition [177]. This approach was initially proposed by Braga-Neto and Dougherty [177] as a tradeoff among variance, bias, and computational cost in small size datasets; and performed effective compared with conventional error estimation methods. The conceptual illustration of the bolstered resubstitution for linear discriminant analysis (LDA) is presented in Figure 5.6.

The bolstering kernels are expressed with uniform circles and the ratio of the shaded region

to the respective circle represents the error by the test observation. The total bolstered error

estimated using ratio of the sum of all errors to the total number of test observation points.

Figure 5.6. Example of the bolstered resubstitution for linear discriminant analysis, where kernels are expressed with uniform circles, taken from [177]

The bolstered error estimation method is usually used to calculate the error rate of the utilized classifier, where the feature-label practical distribution is replaced by the bolstered practical distribution. This distribution is defined through weighting each point over the true distribution using the pdf called bolstering kernel [177]. This pdf is usually the multivariate Gaussian density function which has a zero mean and the diagonal covariance matrix [177]. Following the principle of the bolstered error estimator, the bolstering kernel function is centered at each observation and for each working zone the area under the kernel pdf is estimated. Thus, the calculated area under the bolstering kernel for the given zone represents the confidence level of the test observation belonging to this zone. Naturally, test observations located closer to decision boundaries (either green-to- yellow or yellow-to-red) have the lower confidence of being assigned to a specific region (the algorithm for application of the bolstered error estimation method is presented in Appendix I).

Consider an example of utilizing the bolstering kernels for the classification of the

obtained FRA signature. The visual illustration of the introduced method is depicted in

Figure 5.7 [180] ©2021 IEEE. The given observation is represented by yellow circle, the

estimated SI value is 0.5, and the Gaussian kernel is centered at this observation point. Two

solid vertical lines represent the pre-defined decision boundaries, specifically, green-to- yellow B

G/Y

boundary is 0.3 and yellow-to-red boundary B

Y/R

is 0.55. It is observed, that the bolstering kernel is divided into three parts by boundaries, hence, the area under the kernel corresponding each of these three zones yields the respective classification confidence percentages (calculation of the area under the kernel is given in Appendix J).

Figure 5.7. Visual representation of the confidence levels

For instance, the area under the kernel is equal to 0.0228 in the range from 0 to 0.3 (green zone), which implies the 2.28% confidence that this scenario belongs to healthy state; similarly, 0.6687 is the area under the Gaussian distribution between 0.3 and 0.55 (yellow zone), leading to 66.87% confidence of suspicious working condition. Generally, the confidence level retrieved from the area under the kernel is estimated using:

     

j j j

j

C x  C x  S x ⁽⁵⁷⁾

where C

j

(x) is the confidence level and S

j

(x) is the area under the bolstering kernel in the given range R

j

, defined via:

  ¹ ₂ ^exp   

²

²

²



j

j j

R j

S x t x  dt

 

    ⁽⁵⁸⁾

where σ

j

denotes the standard deviation of the Gaussian distribution used for the zone j

(implied to be the same for observations under a similar class). The total area under the

kernel is 1, which is ensured through normalization (see (57)). It should be noted, that each

SI has its unique mathematical range, whereas the utilized Gaussian kernel is non-zero at (−∞, +∞). The index MM varies between (−∞, +∞), too, whereas ρ and CC have the range of [-1; 1], and remaining indices take values between [0; +∞) (refer to Table 5).

The standard deviation σ

j

in (58) is based on the empirical distribution of observation points belonging to a specific category (zone) [177]. It is shown in Figure 5.8 [180] ©2021 IEEE that more scattered observation points result in a wider bolstering kernel (meaning larger σ

j

). For example, samples corresponding to green zone are distributed denser compared to red zone samples, which is reflected in a narrower kernel for green zone.

Figure 5.8. Various sample distributions and the corresponding kernels Therefore, the standard deviation σ

j

(the width of the distribution for each class), should be estimated. To do that, 15 test objects (14 distribution transformers and one power transformer) were examined. The detailed information regarding the number of phases, power rate, and accessible voltage tap positions are given in Table 11 [180] ©2021 IEEE.

Table 11 – Test objects utilized for the confidence level estimation Transformer Phases Power rate Voltage taps

T1 1 0.4 kVA 220/5/12/24/36 V

T2 1 0.4 kVA 230/24 V

T3 1 0.4 kVA 230/24 V

T4 1 0.63 kVA 220/5/12/22/42/110/220 V

T5 1 0.63 kVA 230/230 V

T6 1 0.63 kVA 230/220 V

T7 1 0.75 kVA 230/53/200/400 V, tertiary 230/115/230 V

T8 1 1 kVA 230/5/12/24/36/110/220 V

T9 3 0.35 kVA 230/400-230 V

T10 3 1.2 kVA 220-24/42 V

T11 3 5 kVA 230/380-42 V

T12 3 20 kVA 10 kV/400 V

T13 3 40 kVA 10 kV/400 V

T14 3 40 MVA 132 kV/33 kV

The FRA measurements are conducted on the abovementioned transformers using the standardized offline setup configuration. The collected data is used to estimate the bolstering kernel width σ

j

and true mean distance d̂

j

. The frequency response is measured for different levels of the winding short-circuit emulated through changing the value of the paralleled rheostat. Specifically, for distribution transformers, the rheostat value is changed from 5 kΩ down to 0 Ω with 500 Ω steps; whereas for power transformer the rheostat value was varied from 5 MΩ to 5 kΩ so that the highest parallel resistance is compatible with winding impedance. Along with faulty conditions, the reference FRA signature was also collected according to the standard end-to-end open circuit measurement setup. Essentially, each fault configuration has its corresponding SI value estimated with respect to the intact condition. Hence, every scenario is classified according to the pre-defined decision boundaries and σ

j

and d̂

j

are estimated (see Table 12 [180] ©2021 IEEE). The respective classification confidence levels are calculated by applying σ

j

and d̂

j

into (57) and (58).

Table 12 – True mean distance and kernel standard deviation

Statistical indicator True mean distance, d̂

j

(10

^-3

) Standard deviation, σ

j

(10

^-3

)