1. Introduction
As key equipment in the electric power system, oil-immersed power transformers have made great progress in their electrical and mechanical properties; however, the cellulose insulation is still considered the weakest part of transformers [
1,
2,
3,
4,
5]. In other words, the operating life of transformers mainly depends on the conditions of the cellulose insulation. In recent decades, conditions prediction and fault diagnosis of a transformer cellulose insulation system have been regarded as the interesting issues and attracted extensive attention [
6,
7,
8,
9].
The insulation conditions of transformer cellulose insulation will degrade with operation time, water, furfural, organic compounds, and other additional products. As a result, in the present research, the evaluation methods of oil-immersed cellulose insulation mainly include the degree of polymerization (DP), dissolved gas-in-oil analysis (DGA), furan analysis, and frequency-domain spectroscopy (FDS) technique [
10,
11,
12,
13]. Most of these methods and operations are believed to be tedious ways in the field detection process; in addition, the insulation performance of the equipment can be damaged by some mentioned methods. On the contrary, the dielectric response technique, which serves as one of the most predominant techniques in the insulation condition diagnosis field, is also classified into the time-domain response and the frequency-domain response. Compared to the frequency-domain response (FDR), the time-domain response (TDR) has been suggested to have several defects including poor anti-interference and high time-consumption; the FDR is, therefore, believed to be more interesting for researchers [
8,
14,
15,
16].
Reviewing the previous work, a typical approach based on FDR, which is realized by extracting the parameters from the frequency-domain spectroscopy (FDS) curves, is utilized for predicting the transformer insulation conditions [
14,
15,
17]. In addition, the core opinion of such an approach is attempting to investigate the variation law of the relevant parameters versus insulation conditions, and the prediction of insulation conditions is, therefore, achieved [
15]. Consulting existing studies, based on the continuous understanding of the effect mechanism of some factor (aging, moisture, temperature, etc.) on frequency dielectric response characteristics, many scholars indicate that the relevant feature parameters can be used to characterize the insulation conditions of the transformer, including complex permittivity
ε*(
ω), dielectric loss (
tanδ), etc. [
8,
15,
18].
However, from the review of previous research, most papers only focus on the influence of a single factor (aging, moisture, or temperature) on insulation performance [
7,
19,
20,
21]; thus, the comprehensive prediction of the aging conditions and moisture content of cellulose insulation is of greater interest to researchers [
13,
22]. Besides, one more important point is that the FDS curves are affected by electrode polarization and conductance in the low-frequency ranges [
20]. Due to such a fact, the relaxation information will be obscured during the FDS test; thus, the FDS curves (including
ε*(
ω),
tanδ, etc.) are no longer appropriate for researching the relaxation process in low-frequency ranges, which may lead to the failure of results.
Fortunately, the researcher suggests that the dielectric modulus
M*(
ω) can be regarded as an effective method to evaluate the insulation conditions without being affected by the mentioned electrode polarization [
20]. Compared to the traditional measurement data of FDR, the relaxation response of the dielectric modulus can form an obvious “polarization peak” in the low-frequency range [
23]. Consequently, the dielectric modulus can not only be used to highlight the relaxation information but also reduce or even overcome the impact generated by electrode polarization and conductance during the FDS test. Furthermore, the feature fingerprints of dielectric moduli represented by some typical parameters could be used to characterize the aging and moisture of the insulation material. Thus, these fingerprints can be considered a potential approach for conditions evaluation [
15].
In conclusion, to avoid the influence generated by aging and the moisture effect on the prediction results of the insulation conditions, the fuzzy pattern recognition technique is applied to this goal under the help of feature fingerprints of the dielectric modulus. In the current work, 20 samples with various aging conditions and moisture contents are first prepared for extracting the mentioned feature fingerprints; then, the fingerprint database used for conditions prediction is established by using these fingerprints. The accuracy and feasibility of the established database are verified by the newly prepared samples. In that respect, an interesting discovery of this paper is that the dielectric modulus fingerprint database may be a helpful tool for conditions prediction of the transformer cellulose insulation system.
2. The Definition of the Frequency Dielectric Modulus
In recent years, the dielectric response technique has been utilized in the conditions prediction of transformer cellulose insulation, where the frequency-domain spectroscopy (FDS) technique has attracted extensive attention from researchers [
22,
24]. The insulation information could be obtained from the dielectric material under an alternating electric field [
25]. Specifically, the different behaviors of the relaxation polarization of transformer cellulose insulation with different moisture contents (
mc%) and aging times can be observed in the FDS curves, which means that the FDS curve of the cellulose insulation is different under different insulation conditions [
26]. However, relaxation polarization will be affected by other factors, including electrode polarization and conductance polarization. For further discussion, if the frequency is low enough (lower than 0.1 Hz), the charges near the electrode would be accumulated and have enough time to complete the electrode polarization process. Thus, unfortunately, an induced electric field, which is opposite to the direction of the original electric field, is established by these accumulated charges, and the original electric field is diminished due to this induced electric field, as shown in
Figure 1.
From
Figure 1, the relaxation process is obscured because of electrode polarization, which makes it very difficult to examine its behavior. As a result, the relaxation characteristics of FDS curves under the low-frequency region may be difficult to analyze: On the one hand, in low-frequency regions, the induced electric field is supposed to be in the opposite direction of the original electric field. On the other hand, the measured current generated in low-frequency regions is mainly formed by conductance current. In other words, the frequency dielectric response information in low-frequency regions is suggested to be the conductance rather than the relaxation polarization. To summarize, in the traditional frequency dielectric response, the information of conductance, rather than relaxation polarization, is extracted in the low-frequency regions, and the FDS curves would, therefore, no longer be suitable for evaluating the insulating conditions of cellulose [
20].
Based on the limitations of FDS curves, the dielectric modulus is considered a powerful tool to overcome such defects. According to reference [
23], the complex dielectric modulus
M*(
ω) can be defined as a reciprocal of complex permittivity, as shown in Equation (1).
where
ε′(
ω) and
ε″(
ω) are the real part and imaginary part of
ε*(
ω), respectively. As shown in Equation (3), the complex dielectric modulus
M*(
ω) can be expressed by
ε′(
ω) and
ε″(
ω).
As for the Debye relaxation, it is worth mentioning that the complex permittivity
ε*(
ω) can be expressed as Equations (4) and (5).
where
εs is the static dielectric constant,
ε∞ is the dielectric constant when the angular velocity approaches infinity,
τ is the relaxation time constant, and distribution factor
β (0 <
β < 1) is related to the FDS curve’s shape plotted in the complex plane [
20].
where
M∞ = 1/
ε∞,
Ms = 1/
εs, and
τM =
τ(
ε∞/εs)
1/β. Furthermore,
ε∞ is always smaller than
εs; therefore, it is easy to confirm that
τM is always smaller than
τ. This indicates that, compared to the permittivity spectrum, the relaxation of the complex dielectric modulus moves to a higher-frequency region. Taking
M∞,
Ms, and
τM into Equation (5), the real part and imaginary part expression of
M*(
ω) are obtained, as shown in Equation (7).
In summary, the complex dielectric modulus
M*(
ω) can be regarded as the reciprocal of the complex permittivity
ε*(
ω) when the conductivity can be ignored [
20]. In other words, as a frequency response characteristic parameter,
M*(
ω) is considered to be able to present the relaxation polarization information of the transformer cellulose insulation under an alternating electric field. Therefore, the
M*(
ω) is considered a powerful tool for conditions diagnosis of the transformer cellulose insulation systems.
5. Feasibility Verification of the Fingerprint Database
5.1. The Introduction of Fuzzy Pattern Recognition
In this part, the method for the prediction insulation conditions of new samples by the dielectric modulus database is discussed. Referring to IEC 60422 and IEEE 62−1995 standards, the corresponding classification of the insulation conditions is given for predicting the transformer cellulose insulation system. Based on the comprehensive analysis of the above rules, the oil-immersed insulating pressboard is divided into 16 target states, as shown in
Table 6.
The fingerprints (S1–S3) can be calculated according to Equation (9), while S4 can also be obtained by using the oil conductivity test. Thus, the values of S1–S4 of new samples are obtained, with the help of relevant algorithms including normalization conversion (NC) and fuzzy pattern recognition (FPR), and the results of conditions (DP and mc%) are later computed.
Obviously, from
Table 3, making the values of
S1–
S3 remain in the same data dimensions, the influence on evaluation results caused by the various dimensions can, therefore, be eliminated. The NC equation, shown in Equation (10), is utilized for eliminating such influence.
where the values of
xij’ are transformed data and the values of
xij are data before the transform. In this work,
i represents the amount of target state,
i ∈ (1, 20). Meanwhile,
j represents the numbers of fingerprints contained in each target state,
j ∈ (1, 4). The max{
xij} is the maximum value of the
j-th column in the fingerprint database and min{
xij} is the minimum value.
The fuzzy pattern recognition algorithm is intended to predict the conditions of the cellulose pressboard. At first, the obtained standard target state (
TS) shown in
Table 5, as well as the unknown samples (
X), are studied by the computers, automatically. Afterward, pattern recognition is operated, which is realized by comparing the Euclidean distance
d(X, TS) between the
X and
TS. In addition, the close degree
N(X, TS) is regarded as a helpful tool to achieve such recognition. The relationship between
d(X, TS) and
TS can be expressed by Equation (11).
N(X, TS) can be calculated by Equation (12), where
n represents the number of fingerprint parameters contained in each
TS,
n = 4.
X(
i) and
TS(
i) are the
i-th fingerprints of the
X and
TS, respectively.
In other words, the recognition algorithm refers to the comparison of the fingerprint characteristics of the tested samples and the fingerprint database to find the corresponding or closest target state. The principle diagram of fuzzy recognition is shown in
Figure 7.
5.2. Identification of New Samples
In this chapter, the insulation conditions of three new samples (
NS1−
NS3) are predicted according to
Figure 8. As shown in
Figure 2, the preparation of new cellulose samples has the same process, and the results of the FDS test are shown in
Figure 9. Details of the new pressboard, including their insulation conditions and measured fingerprints (
S1–S4), are shown in
Table 7.
As shown in
Table 7, the moisture content and DP value of NS
2 (DP = 313,
mc% = 1.21) and NS
3 (DP = 293,
mc% = 1.28) are very close, which are also reflected in their curves in
Figure 9. However, their dielectric modulus fingerprint parameters are different. Although the values of
S1 (NS
2 = 1.06 × 10
−3, NS
3 = 1.26 × 10
−3) and
S3 (NS
2 = 1.16, NS
3 = 1.11) are relatively close, the values of
S2 (NS
2 = 6.7 × 10
−3, NS
3 = 1.00 × 10
−2) and
S4 (NS
2 = 0.46, NS
3 = 1.10) are quite different, which means that we have the criteria to distinguish those two conditions.
The closeness between the three new samples and the fingerprint database after fuzzy identification, as well as the corresponding degree of polymerization and moisture content, are shown in
Table 8.
For NS1, the predictive condition of the database is T8 (DP = 700−900, mc% > 4%, Serious damp), and the prediction of DP values is correct (practical DP = 726). However, the practical moisture content of NS1 is 2.41, which should be considered as slight damp. These results indicated that the error may be caused by the moisture identification or other factor; future research will focus on this problem. Besides, the accuracy of the proposed method is also related to the number of reference samples (in this work, the number of reference samples is 20). With the increase in the number of samples, the accuracy will improve.
As for NS2 and NS3, the condition suggestions given by the database are T13 (DP = 500−700, mc% = 0%−1.5%, Dry) and T17 (DP < 300, mc% = 0%−1.5%, Dry), respectively, while the practical conditions of NS2 and NS3 are DP = 313 and mc% = 1.21, and DP = 293 and mc% = 1.28, respectively. This means that the accurate prediction results can be given by the dielectric modulus fingerprint database. In conclusion, although the individual error of the result is found, the fingerprint database is still considered a potential tool for cellulose insulation prediction.
5.3. Comparison with Grey Relational Analysis
Among the methods that have been proposed, as the grey relational analysis method can evaluate multiple conditions at the same time, this method is chosen to compare to the method proposed in this paper. The core idea of the grey relational analysis is the correlation degree between the condition data (in this paper,
S1–
S4 of new samples are regarded as the condition data) and the reference data (
S1–
S4 of papered pressboard in
Table 5 are regarded as the reference data). The greater the correlation degree, the closer the condition data are to the reference data, and the insulation conditions of the pressboards are, therefore, evaluated according to that.
According to the grey relational analysis, an algorithm is established, and the insulation conditions prediction of NS
1, NS
2, and NS
3 will be carried out. The results are shown in
Table 9. From
Table 9, the predictive condition of NS
1 is T
4 (DP = 900−1400,
mc% >4%, Serious damp), and compared to its practical conditions, the practical DP is 726; however, the predictive DP is more than 900. Unfortunately, significant errors are also between the predictive
mc% (>4%) and practical mc% (= 2.41). As for NS
2, its predictive condition is T
17 (DP < 300,
mc% = 0%−1.5%, Dry), and the practical moisture content of NS
2 is 1.21, which means the right prediction results are given by the grey relational analysis. However, some errors still exist between the predictive DP (DP < 300) and practical DP (DP = 313). From the results of NS
3, similar to the NS
2’s results, some errors also still exist between the predictive DP (DP = 300−500) and practical DP (DP = 293).
In conclusion, compared to the grey relational analysis, the dielectric modulus fingerprint database is demonstrated to be more accurate. With the increase in the number of prepared pressboards, the accuracy of the dielectric modulus fingerprint database will be further improved.
6. Conclusions
Compared to the traditional methods, such as the Karl Fischer titration (KFT) and viscosity method, the dielectric modulus fingerprint database cannot destroy the cellulose pressboard. In addition, compared to the FDS methods, the proposed method can predict the insulation conditions of the cellulose pressboard when including both the aging effect and moisture effect. However, the proposed method is an off-line technique, which means this method cannot predict the insulation conditions of the running transformers. In addition, the established model depends on the used materials, which might limit the applications in the field conditions. Further optimization and modification are thus needed to overcome this issue.
The dielectric modulus can not only be used to highlight the relaxation information but also reduce or even overcome the impact generated by electrode polarization behavior and conductance behavior during the FDS test. This feature allows the dielectric modulus to be a potential tool for analyzing and evaluating the insulation conditions of transformer cellulose insulation. Meanwhile, the dielectric modulus fingerprint database is established, and the synergistic effect of moisture and DP can be verified. Although there are some errors in the results, this database can still be considered an interesting approach to evaluate the insulation conditions of transformers. The present analysis and contributions have led to the following conclusions.
As a frequency-response characteristic parameter, M*(ω) is certificated to be able to present the relaxation polarization information of the transformer cellulose insulation in the course of the FDS test.
It is found that the imaginary part of the dielectric modulus could form an obvious relaxation peak in the low-frequency regions, which could be utilized to extract the feature fingerprints to characterize the aging and moisture of cellulose insulation by using integral operation.
The synergistic effect generated by moisture and aging can be separated or distinguished by using DC conductivity. The novelty of this work is in an exploration of the dielectric modulus as a useful tool to extract the parameters to build a database for the comprehensive prediction of aging and moisture.
It is proved that the reported feature fingerprint database could serve as a potential tool for the comprehensive prediction of transformer cellulose insulation.