1. Introduction
Accurately and timely mastering the changes in the health status of power transformers is an important link in the life-cycle management of transformers and is the premise and basis for carrying out the digital operation and maintenance of power transformers [
1]. The health status of power transformers is a typical multi-attribute decision-making problem. Generally, the health index represents the degree of performance degradation or deviation compared with the normal performance status. Its value is a single value between zero and one. The higher the value is, the better the operating status of the equipment is. In practical engineering applications, health status is often divided into several levels by qualitative methods, such as health, attention, and abnormality.
Due to the complex structure of the power transformers, there are a wide variety of indicators related to the running state of the transformer [
2,
3,
4], such as electrical tests, oil and gas tests, oil quality tests, etc. However, there are intricate relationships between the indicators and the health state, and the test period needs to be inconsistent, which cannot reflect the dynamic change of the health state of the power transformer. With the widespread application of oil chromatography online monitoring systems (DGA), how to use oil chromatography online monitoring data to accurately and timely evaluate and predict power transformer operation status is an urgent problem to be solved in the digital operation and maintenance of equipment.
There are many methods for evaluating the health status of power transformers, mainly focusing on three directions: the health-index-based method, the machine-learning-based method, and the uncertainty-reasoning-based method. The health-index-based method [
5,
6,
7,
8,
9] constructs a health index model according to the aging formula of the power transformer. Still, the model’s design of parameters and weights is complicated and subjective, which may result in miscalculations. The machine-learning-based methods require a large amount of sample data, such as support vector machines [
10,
11,
12], adaptive fuzzy neural networks [
13], adaptive probabilistic neural networks [
14,
15,
16], fuzzy C-means clustering [
17], etc. There are certain limitations when facing small sample data, such as the transformer’s abnormal state. Based on the evaluation method of uncertainty reasoning, uncertain knowledge reasoning is used to draw a reasonable or approximately reasonable conclusion with a certain uncertainty, such as fuzzy reasoning [
18], cloud matter element theory [
19], evidence theory [
20], multi-layer architecture [
21], etc. There is a problem in that the limitation of distance space is ignored, and the relationship between each feature cannot be accurately reflected. To sum up, although these methods solve the problem of power transformer health assessment, there are still some problems, such as strong subjectivity in parameter and weight selection, large sample data demand, and inability to eliminate the impact of feature correlation.
This paper proposes a dynamic assessment method of the transformer health state based on the Mahalanobis–Taguchi system. By constructing a Mahalanobis–Taguchi system to screen key features, a single health index model is constructed using the spatially optimized Mahalanobis distance, Box–Cox transformation, and 3 σ. The criterion determines the alert value and threshold value of all transformers. The example verifies that the proposed method can reflect the dynamic changes of the power transformer operation state and effectively solve the problems of too many selected parameters in the health index and subjectivity and misjudgment of weight selection.
3. Construction and Optimization of Mahalanobis–Taguchi System
The Mahalanobis–Taguchi System (MTS) method is a classification method that combines Mahalanobis distance, orthogonal arrays (OAs), and signal-to-noise ratio (SNR) to achieve multi-dimensional data [
22,
23]. Compared with other classification algorithms, it has the advantages of simple principles and fast running speed. Compared with Euclidean distance, MD is not affected by dimensions. Considering the relationship between features, MD eliminates the interference of correlation between features and can be used as a measure of multi-dimensional features to quantify the system’s health status.
3.1. Construction of the Benchmark Space
The datum space is the benchmark for judging whether an event occurs. Because of determining the characteristic variable, we calculate the Mahalanobis distance to extract normal data samples and form the benchmark space data matrix.
(1) Build the initial feature set.
DGA monitors the content of eight gases in transformer oil, as shown in
Table 1.
The content fluctuation range of carbon monoxide (CO) and carbon dioxide (CO
2) in the DGA monitoring data is too large, which harms the subsequent evaluation. So the remaining six gas contents are selected as the initial characteristics. It is assumed that the transformer has p data characteristics monitored by DGA, and n data are collected in a certain period, so there is an evaluation matrix,
(2) Calculate the correlation-type Mahalanobis distance.
Standardize the data of each sample as follows,
In the formula, yij is the standardized result of the jth variable of the ith sample; xij is the observed value of the jth variable of the ith sample; and mj and sj are the mean and standard deviation of the jth variable, respectively.
The formula for calculating the correlation Mahalanobis distance is,
In the formula, Yi = (yi1, yi2, yi3, …, yi6) T is the standardized data of the ith sample; and is the covariance matrix (or correlation coefficient matrix) of the p features of the normal sample.
(3) Build a benchmark space.
Since all DGA data are calculated when
MD is calculated and its status cannot be distinguished by the whole data, the existing judgment results (normal (
MDn), abnormal (
MDa)) in DGA sample data are used to determine its threshold value. The maximum
MD value corresponding to the normal state and the minimum
MD value corresponding to the abnormal state in the DGA are used to calculate the screening threshold of the reference space, preliminarily screen the sample dataset, and form the reference space. The formula is as follows:
In the formula, MDa(min) represents the minimum value of the Mahalanobis distance corresponding to the abnormal state of the DGA, and MDn(max) represents the maximum value of the Mahalanobis distance corresponding to the normal state of the DGA. The MD defined in Equation (5) is the standardized MD, which can distribute the MD value of normal samples around one and not exceed the critical value T. The MD value of abnormal samples should be greater than the critical value T, and the Mahalanobis distance increases with the spatial difference between samples.
3.2. Validation of Benchmark Space
Before optimizing the benchmark space, we confirm the validity of the basic space constructed in
Section 3.1:
(1) The data samples are divided into normal and abnormal samples sets according to Formula (6). When M > T, it is an abnormal sample. A normal sample is M < T.
(2) The mean value and standard deviation of normal samples are used to standardize abnormal samples, and the Mahalanobis distance (M1) of normal samples and the Mahalanobis distance (M2) of abnormal samples are calculated, respectively.
(3) If M1 < T < M2, it is considered that the datum space constructed is effective. Otherwise, it is necessary to re-select feature variables before constructing the datum space.
3.3. Baseline Space Optimization
The initial feature set is screened, the features that may generate redundant information are eliminated, and the remaining important features are used to classify normal and abnormal samples to reduce the computational cost and complexity.
In MTS, OAs are used to minimize feature combinations to identify important features, and SNR “gain” filters effective features. The details are as follows:
(1) Design a two-level orthogonal table, assign the initial feature variables to different columns of the orthogonal table, and use “1” or “2” to indicate whether the feature variable participates in the construction of the benchmark space. Each row of the orthogonal table corresponds to a Mahalanobis space, as shown in
Table 2.
(2) Calculate the signal-to-noise ratio of samples.
Calculate the Mahalanobis distance of samples
MDk,
k = 1, 2, …,
t. Then, the sample’s expected large characteristic signal-to-noise ratio is,
(3) Calculate the
SNR increment of the feature quantity
xjIn the formula, and represent the mean SNR of xj that participates in the construction of the benchmark space and the mean SNR that does not participate in the construction of the benchmark space. When , the selected feature variable has a positive effect; otherwise, it represents a negative effect.
(4) Verify the optimized benchmark space.
The Mahalanobis distances of normal and abnormal samples are calculated using the optimized features to verify the validity of the optimized benchmark space, which can be used for the classification and prediction of transformer health status. Currently, the recalculated Mahalanobis distances of normal and abnormal samples are MDH and MDU, respectively.
5. Case Analysis
We selected 6220 kV oil-immersed transformers installed with DGA in a power grid. DGA collects data on the transformers every 24 h. A total of 1963 data from 2019 to 2021 were collected as data samples.
(1) Constructing the benchmark space of the Mahalanobis–Taguchi system.
The correlation coefficient matrix obtained from all the collected transformer oil chromatographic online monitoring data is shown in
Table 3. Since the correlation coefficient between C
2H
4 and total hydrocarbons is 0.953, the correlation type is too strong. It is proved that these two characteristics play a similar role in the subsequent state evaluation, so H
2, CH
4, C
2H
2, C
2H
6, and total hydrocarbons are selected. A total of five features are used as the initial feature set.
The selected five features are used as standard variables to construct the benchmark space, and the Mahalanobis distances of all samples are calculated after standardizing the data, as shown in
Figure 3. The red dots in the figure represent samples judged as normal by the DGA device, and the blue dots represent samples judged as abnormal by the DGA device. The judgment results of the DGA device are partially normal, and the abnormal boundary is blurred, which is prone to misjudgment.
Use Formula (6) to calculate the preliminary screening threshold T = 2, and take samples with MD ≥ 2 as abnormal and MD < 2 as normal samples to construct the benchmark space. The Mahalanobis distance distribution in the benchmark space is as follows:
(2) Benchmark space validity confirmation.
As we can see from
Figure 4, when a sample with a Mahalanobis distance greater than two is taken as an abnormal sample, the calculated Mahalanobis distance can well distinguish normal and abnormal samples. At this time, the reference space constructed by DGA data is effective and can be used to evaluate the status of transformers.
(3) Baseline space optimization.
Use Formulas (7) and (8) to design the orthogonal table and calculate the signal-to-noise ratio of the reconstructed data. The orthogonal table and the signal-to-noise ratio are shown in
Table 4. Among them, one means using the variable, two means not using this variable, and
Table 5 shows the mean signal-to-noise ratio. For constructing the orthogonal table of 3 and 4 variables, the number of variables selected must be at least 3 and 4, so it is divided into 11 groups and 6 groups for calculation.
The Mahalanobis distance is only effective when the number of characteristic variables is greater than or equal to three. Therefore, we can learn from
Table 4 that three or four variables can be selected as key variables to construct a new Mahalanobis space. When calculating SNR, a negative number represents a negative impact on the accuracy of the evaluation. If the result is a regular representation, it positively affects the evaluation. Still, the CH
4 is a negative value, which has the largest absolute value among negative values, so the variable is deleted.
There are two schemes: ① select H
2, C
2H
2, C
2H
6, and total hydrocarbon content as the key features; ② H
2, C
2H
2, and total hydrocarbon content are selected as key characteristics. By calculating the Mahalanobis distance under normal and abnormal conditions, as shown in
Table 6, we can know that ① combined conditions can more accurately distinguish abnormal and normal samples, and its Mahalanobis distance distribution diagram is shown in
Figure 5 and
Figure 6.
Figure 7 is a local enlargement of Mahalanobis distance with four variables selected from five. It can be clearly seen that there are obvious differences between the
MD values of normal and abnormal samples, which can be effectively distinguished. However, it can be seen from
Figure 8 that for the Mahalanobis distance distribution constructed with three variables selected from five, the
MD values of some normal samples have exceeded the
MD values of abnormal samples, which cannot accurately distinguish normal and abnormal samples.
Figure 8 proves that the Mahalanobis space constructed by three out of five variables is currently invalid. Therefore, combination ① is selected to optimize the Mahalanobis space, and the features in combination ① are selected as the key features.
(4) Calculate HI, HI warning value, and HI threshold.
Two 220 kV power transformers are selected, and the transformer’s HI model is constructed using the normal sample MD. Take one of the transformers as an example to illustrate the solution process.
Example 1: The basic information about the transformer is as follows.
The transformer is 220 kV, the model is SFPS9-150000/220, the date of manufacture was April 5, 1997, and it has been in operation for nearly 25 years. The transformer analysis is as follows:
(1) According to Formula (11), use Box–Cox transformation to convert the MD value of normal samples into normal distribution or approximate normal distribution. Calculate the optimal transformation parameter
λ = 0.013. The probability distribution before and after MD transformation is shown in
Figure 9.
It can be seen from
Figure 9 that the MD value after the Box–Cox transformation obeys an approximately normal distribution, and the 3 σ criterion can be used to determine the HI warning value and threshold.
The warning value and threshold value of MD can be obtained from Formula (13), which are respectively T2σ = 0.9956 and T3σ = 1.5843. Then, the warning and threshold value can be calculated as HIw = 0.9502 and HIf = 0.9208, respectively, by Formula (14).
(2) Calculate the HI curve according to Formula (9), as shown in
Figure 10.
It can be seen from
Figure 6 that most of the HI values of this transformer are within the warning value, but the HI values of the 97th, 103rd, 137th, 235th, and 252nd times are less than
HIw, and they are in a state of attention. Defects include the silica gel discoloration exceeding 2/3, the fan cannot rotate, the contactor is damaged, and the cooling fan does not reach the set value and starts.
Table 7 shows that the judgment results of the method in the literature [
26] for the 137th day are different from that in this paper because this document only uses the attention value of DGA data to calculate and ignores the ambiguity of the boundary, resulting in misjudgment. Still, the actual situation is that the transformer has defects on the 137th day, so this method can more accurately evaluate the status of the transformer.
According to the above calculation process, we calculate the health index curve for the second transformer (Example 2), as shown in
Figure 11.
It can be seen from
Figure 11 that the health index HI value of Example 2 is between the warning value and the threshold value, and it has been in a state of attention. The actual situation is that the fan of this transformer cannot rotate and is in a defective operation state without repair. Hence, the health index curve fluctuates between greater than the threshold value and less than the alert value.
It can be seen from
Table 8 that the calculation results of the evaluation method in the literature [
26] are consistent with those in this paper, proving the method’s accuracy.
It can be seen from the above two examples that the HI curve of the transformer can accurately describe the change in the health state during the operation of the transformer. Health management provides a basis for decision making.