A modified XY model of transformer oil–paper insulation system including non‐uniform aging and conductance effect

Correspondence Jiefeng Liu, School of Electrical Engineering, Guangxi University, Guangxi, Nanning 530004, China. Email: jiefengliu2018@gxu.edu.cn Abstract The accuracy and feasibility of the traditional XY model of transformer oil–paper insulation system are disputable since it ignored the effect of non-uniform aging and conductance on dielectric response property. Given this issue, this work attempted to report a modified XY model to overcome these defects. The novelty of this work is in the exploration of the modified XY model as a potential tool for establishing the quantitative correlation among the complex relative permittivity of the liquid insulation, solid insulation, and liquid–solid composite insulation. In that respect, the presented verification experiments proved the feasibility and accuracy of the reported model. The contributions are expected to provide a theoretical basis for the condition evaluation of transformer solid insulation under non-uniform aging conditions.


INTRODUCTION
The large oil-immersed power transformers are one of the most important and expensive equipment in the modern power grid. Whereas the performance of solid insulation determines the overall operating life of the transformer [1][2][3][4], therefore, the accurate condition monitoring of transformer solid insulation is helpful to avoid accidents.
In the past few decades, studies on the condition evaluation of transformer solid insulation based on frequency domain spectroscopy (FDS) technique are of great interest to scholars [5,6]. Reviewing the existing researches [7][8][9][10], the condition evaluation for solid insulation samples prepared in lab conditions is widely reported. However, it is still a challenge to realize the accurate evaluation of the aging and moisture states in field transformers by directly using the model reported in the lab conditions. If the FDS data of the solid insulation can be extracted from the total FDS data of the transformer oil-paper insulation system, the evaluation methods established under laboratory conditions can be naturally employed to carry out the condition evaluation.
To extract the FDS data of the solid insulation from the total FDS data of the transformer oil-paper insulation system, the concept of the XY model is introduced [11][12][13][14]. The traditional XY model suggests that the insulation condition of transformer solid insulation in the radial direction (perpendicular to the iron core) is uniform, moreover, the main insulation structure along the axial direction (iron core) is also basically the same. Therefore, the typical transformer oil-paper insulation system can be abstracted into a two-dimensional structure composed of the oil gap, spacer, and barrier [11,12]. Also, the traditional XY model is serviced for establishing the quantitative relationship among the relative permittivity of the liquid insulation, solid insulation, and composite insulation.
Unfortunately, a widely accepted fact is that the winding temperature (80-140 • C) of the energized transformer is generally much higher than its normal oil temperature (40-60 • C) [15]. While viewing from core to HV winding, the solid insulation temperature at different locations in the transformer oilpaper insulation system varies with the alter of the distance from the core. This phenomenon results in a temperature gradient that appeared between the transformer tank and the iron core, which often leads to the non-uniform aging of the solid insulation along the radial direction [16][17][18]. In this case, the accuracy and feasibility of the traditional XY model are disputable. Besides, only the capacitance is considered when building the equivalent circuit of the XY model, while the contribution of the dielectric loss generated by the conductance effect has not been included. In this case, the accuracy obtained by the traditional XY model is limited, and its physical significance also needs to be improved.
Given these issues, this work reported a modified XY model, which aims to promote the applicability of the XY model under non-uniform aging conditions, as well as improve its physical significance. In the current work, a novel equivalent circuit of the XY model is rebuilt by employing both the resistance and capacitance elements. The mentioned equivalent circuit can be generalized to various orders when including the effect of nonuniform aging. Then, the mathematical formula of the modified XY model is presented based on theoretical derivation. The feasibility and accuracy of the proposed model are later discussed in the lab conditions. Findings reveal that the proposed model can be serviced for establishing the quantitative relationship among the complex relative permittivity of the liquid insulation, solid insulation, and composite insulation under the nonuniform aging. In that respect, this contribution is supported to present a theoretical basis for the evaluation of the non-uniform aging condition of transformer solid insulation.

LIMITATION OF THE TRADITIONAL XY MODEL
The existing research of the XY model suggested that the composite insulation structure along the axial direction is consistent [11,12]. Thus, the frequency response property will not be affected by the geometry structure of the oil-paper insulation system along the axial.
The formula for calculating the value of X and Y of the traditional XY model is presented in Equations (1) and (2) [11]. The value of X and Y belong to a percentage.

X = Thickness of barrier Thickness of main insulation
(1) The analysis of the traditional XY model is based on the assumption that the oil-paper insulation system is subjected to uniform aging. While viewing from core to tank, the insulation temperature decreases with distance from the iron core. This phenomenon results in the generation of a temperature gradient (along the radial direction) existing between the transformer tank and the iron core [16][17][18].
Specifically, energized winding near the core has the highest temperature and the transformer tank has the lowest temperature. The distribution of this temperature gradient T(x) along the radial direction of the oil-paper insulation system is shown in Figure 1.
On the one hand, as the aging process of oil-paper insulation is temperature-dependent, the non-uniform aging conditions will be caused when prolonged exposure to the tempera-FIGURE 1 Distribution of the temperature gradient along the radial direction of the oil-paper insulation system ture gradient T(x). Thus, the traditional XY model is not available to establish the relationship of the complex relative permittivity among liquid insulation, solid insulation, and composite insulation when including non-uniform aging.
On the other hand, only the capacitance is contained in the equivalent circuit of the XY model, which is used to highlight the property of the energy storage (polarization) of the dielectric materials. However, since conductance always occurs when the polarization process starts to build up, its influence should also be included. Moreover, the measured dielectric loss of dielectric materials can be divided into two categories, one part is corresponding to the polarization behaviour, the other is corresponding to the conductance behaviour. In this case, if the property of energy storage (polarization effect) can be represented by the capacitance, the impact generated by the conductance effect can be described by the resistance as well. Consequently, a new equivalent circuit can be proposed to improve the accuracy and physical significance of the traditional XY model when the mentioned issues have been solved.

Modified circuit including the impact of polarization and conductance
Under the field intensity E * (ω), the polarization and conductance inside the dielectric can be naturally established. In this process, the bound (induced) charges appear at the plate electrode, and the charge density σ p is expressed by the polarization intensity P. The ε 0 (8.854E-12 F/m) is vacuum permittivity, ε r is the relative permittivity.
Meanwhile, the free charge also appears and forms the charge density σ 0 , Equation (4) is used for describing the relationship between σ 0 and field intensity E * (ω).
Electrostatic flux density (D) is thus defined as the sum of surface charge density during the polarization process.
Further, the surface charge density generated between units is defined as the surface current density i * (ω), thus, the i * (ω) can be obtained by deriving D with the angular velocity ω.
where j represents the imaginary unit. The surface current I * (ω) can be obtained by multiplying i * (ω) and the area of plate electrode S, as is shown in Equation (7).
where d and U * (ω) are the distance and voltage between the positive/negative plate electrodes, respectively. Besides, the formula of I * (ω) can be also described by Ohm law, as is shown in Equation (10). The G is the equivalent conductance of the dielectric materials filled in the plate electrode.
Combining Equations (9) and (10), the formula of G can be deduced. Also, the inverse of G is the equivalent resistance R of the dielectric materials filled in the plate electrode.
In regards to the definition of geometric capacitance C 0 , as is shown in Equation (12), Equation (11) thus can be revised as Equation (13). The C is the complex capacitance.
Thus, the circuit of the XY model modified by including the effect of both polarization and conductance can be developed. Figure 2 plotted the modified circuit and defined the various circuit elements as either resistance R ij or capacitance C ij . Referring to Equation (13), the mathematical formula of R ij can be written as Equation (14). The ε rij is used to represent the relative permittivity corresponding to the circuit marked as ij.
Also, Equation (15) listed the mathematical formula for describing the C ij .
From Figure 2, the polarization and conductance contributions are covered in the modified circuit. Then, if the geometric structure of the transformer oil-paper insulation system is also included, the obtained model can be illustrated in Figure 3.
From Figure 3, this model can be divided into four components, the admittance of each component is named Y ij . Thus, the total admittance Y tot of this model is obtained.
From Figure 3, each admittance Y ij is constructed by a resistance and a capacitance. The dielectric response property corresponding to each Y ij is also formed by the contribution generated by both the resistance and capacitance.
However, the contribution percentage of resistance and capacitance is not always consistent. If the contribution generated by resistance and capacitance are assumed as k and 1-k, the impact of different contribution rates of R ij and C ij should be considered when deducing the formula of Y ij . Equation (18) describes the solution to this issue.
Substituting the relevant R ij and C ij shown in Equation (17) into Equation (18), the formula of the admittance Y ij will be obtained. The results are shown in Equation (19).
where ξ is a response function that regards the k as the variable, and ξ = (1/k+1-k). Then, according to Equation (16), Y tot can be obtained and shown in Equation (20).
According to Equation (13), the quantitative relation can be established among C ij , G ij (or R ij ), and ε rij . Thus, the Y tot can be simplified to a function that is related to the equivalent complex capacitance C tot , it can be found in Equation (21).
where ε rtot is the relative permittivity of the model shown in Figure 3. Simultaneous Equations (20) and (21), the formula of ε rtot is thus obtained, and as is shown in Equation (22).

Mathematical formula of modified XY model including the non-uniform aging
If the combination of a barrier and a spacer in the same temperature region in Figure 1 is defined as an elementary component (EC), the aging condition of the solid insulation in the same EC is uniform. Then, various EC is used to form a complex model for simulating non-uniform aging. It is worth mentioning that due to the effect of the oil circulation, the aging condition of the oil in each part of the oil-paper insulation system is considered consistent.
Taking the EC at two temperatures (represented by EC1 and EC2) as a case to understanding how to build the modified XY model that covered the non-uniform aging. The formed composite insulation structure is shown in Figure 4. The relative position of the oil gap (i.e. combination form of the pressboards) has little effect on the dielectric response property of the total insulation system [20]. Thus, the composite insulation can be simplified to Figure 5 by changing the location of the oil gaps.
Then, the composite insulation structure shown in Figure 5 can also be decomposed into two different substructures. Substructure 1 is composed of the barriers and spacers, while substructure 2 consists of the oil gap and barriers. The admittance Y ij is presented in Equation (23), in this case, i∈(1, 2); j represents the total number of layers, and j∈ (1,4).
Due to the discussed procedure, the ε ri corresponding Y i can be obtained and shown in Equation (24). The permittivity (or admittance) of the materials with the same insulation condition is similar. Thus, the boundary conditions shown in Equation (25) should be considered to simplify Equation (24), where ε ro , ε rp1 , and ε rp2 represent the relative permittivity of both insulation oil and pressboard with two aging degrees, sequentially.
Then, the formula of ε rtot is shown in Equation (26).
Actually, the amounts of the elementary component in the oil-paper insulation system of the energized transformer are far more than two. In this case, the model shown in Figure 5 can be further expanded to Figure 6.
If the total categories of the aging degree of cellulose pressboard contained in the composite structure are assumed to be m, the number of EC is equal to m, the number of layers named as n and equals to 2m.
Also, the previous derivation is available for obtaining the formula of Y ij corresponding to Figure 6, in which the parameter i∈ (1,2), and j represents the total number of layers, and j∈(1, n). Equations (27) and (28) describe the general formula of Y ij .
Then, Y i is calculated by using Equation (29). Thus, the ε ri can be obtained and shown in Equations (30) and (31).
Similar to Equation (25), the boundary conditions Equation (32) should be also considered to simplify Equations (30) and (31). Finally, the generalized formula of ε rtot is shown in Equation (33).
Equation (33) can be further modified as Equations (34) and (35) when dividing it into the real part and imaginary part. In this form, the response function can be further divided into the real part ξ′ and imaginary part ξ′′ as well.

Detailed parameters and preparation of oil-immersed pressboards
In order to construct the various composite insulation structure by using the oil-immersed cellulose pressboards, the oilimmersed pressboard discs and transformer oil are utilized to achieve this goal. The oil-paper mass ratio is set to 20:1.

FIGURE 7 Picture of the simulation insulation structure
The transformer oil is the Karamay No. 25 naphthenic mineral oil and satisfies the standard of ASTM D3487-2000(II). The details of pressboards with two types of thickness are described in Table 1. The oil-immersed pressboards are obtained by the vacuum dry and vacuum-immersed. Then, the different aging states and moisture are obtained by accelerating thermal aging (at 150 • C) and moisture absorption, as is shown in [21,22].

Simulation model of the composite system
In order to establish the quantitative relationship among the complex relative permittivity of the liquid insulation, solid insulation and liquid-solid composite insulation under the nonuniform aging, a simulation structure of the composite insulation has been put forward, as is shown in Figure 7.
In the lab, the non-uniform aging condition can be modelled by a combination of several layers (n-layers) of the elementary component (EC) with the different aging conditions. The space enclosed by the two types of pressboard are utilized to simulate the oil gap. If the combination of each pressboard I, pressboard II, and oil gap are positioned as one EC in a transformer, there will exist n-layers of the EC in the oil-paper insulation system under the non-uniform aging conditions. Subsequently, FDS tests were carried out twice on each simulation structure and pressboards at controlled conditions. Specifically, the FDS data are measured by DIRANA/OMICRON, the testing voltage preset to AC 200 V, the test temperature is maintained at 45 • C, and the test frequency range is set to 1 × 10 −3 to 1 × 10 3 Hz.

Scheme of validation experiments of the proposed modified XY model
To confirm the value of ξ, as well as check the feasibility of the reported model when considering the non-uniform aging and conductance effect, the experiments by using pressboards with various aging conditions (i.e. DP value) and various moisture content (mc%) are performed. Especially, two types of pressboards (shown in Table 1) are utilized to develop the simulation structure ( Figure 8).
Out of Figure 8, the simulation structure is constructed by the two types of pressboards (named pressboard I and pressboard II) and the oil gap. The pressboards (PBj, type I and type II) in the same elementary component will remain in the same insulation condition.
In the case that m is equal to different values (m = 2, 3, 4), a series of simulation structure formed by the various elementary components are constructed in the lab. Specifically, pressboard I is a disc (thickness: a) that simulates the barrier in the transformer oil-paper insulation system, pressboard II is a hollow disc with an internal diameter of c and the thickness of b, which is served to simulate the spacer in the transformer oil-paper insulation system. The diameter of the low-voltage electrode is d. Moreover, the values of X, Y of the simulation model shown in Figure 8 can be expressed in the form of Equation (36).
To eliminate the impact of the geometry on the test results, these models possess the uniform X (0.33) and Y (0.25) value. The details are tabulated in Table 2. The average value of the measured complex relative permittivity (ε* rpj ) corresponding to the single pressboard (PBj) with various insulation conditions is presented in Figure 9.

5.1
Confirmation of the response function Equation (33) presented a general formula for simulating the ε rtot of simulation structure (shown in Figure 8) by employing the relative permittivity of each pressboard (ε rpj ) and oil (ε ro ).  Obviously, both the ε rtot , ε rpj , and ε ro can be directly calculated by using the measured FDS data (complex capacitance).
From Equations (34) and (35), on the condition that both ε rtot , ε rpj , and ε ro are confirmed, the response function ξ(f) corresponding to each sampling frequency can be readily computed point by point. According to Table 2, the designed experiments 1 to 4 are selected to study the value of the ξ(f). Figure 10 plot- Out of Figure 10, the value of both ξ i (f) alters with the increasing frequency, regularly. The curves of ξ′ i (f) becoming flat when the sampling frequency is high enough, while ξ′′ i (f) keeps rising with the increasing frequency. Thus, the contribution rate of either polarization or conductance on different frequency regions (of FDS) could be deduced to a certain extent.
However, the values of ξ′ i (f) are not always equal to ξ′′ i (f), and there has been no obvious changing rule between the response function values and various aging degrees. It means that the impact generated by the alter of insulation conditions will not significantly change the value of the response function. ′ aver ′′ aver Then, substituting the value of ξ′ aver (f) and ξ′′ aver (f) into Equations (34) and (35) to replace the ξ′(f) and ξ′′(f), the modified XY model for calculating the total relative permittivity of the composite insulation structure can be proposed and shown in Equations (39) and (40).

FIGURE 11
The verification scheme of the proposed model

Feasibility investigation of the proposed model
In the last section, the determination of the repose function is discussed. In this section, the feasibility of the reported modified XY model is demonstrated by comparing the measured and calculated total relative permittivity of the simulation structure shown in Figure 8. The specific verification scheme of the proposed model is as follows, and it also has been presented in Figure 11.
(i) Firstly, the various pressboard I (i.e. PBj, shown in Figure 8) and the oil gap contained in the same simulation structure are used to measure its FDS data. Then, ε rpj and ε ro used in Equations (39) and (40) are naturally confirmed. (ii) Meanwhile, the simulation structure designed in experiments 5-7 is also measured twice. Then, the average value of the measured total relative permittivity is named as ε rtotmj , where, j (5-7) represents the label of experiments. (iii) Substituting the measured data (including the ε rpj and ε ro of PB j and oil gap) into Equations (39) and (40), as well as the response function shown in Equations (37) and (38), the computed total relative permittivity is obtained and named as ε rtotcj . (iv) Finally, the error between the measured ε rtotmj and the computed ε rtotcj determines whether the reported model is valid. The resulting comparison of the experimental groups (5-7) is plotted in Figure 12.
In Figure 12, the measured data and the calculated data maintain the same variation law. Besides, the relative error (RE) analysis of j (max j = 20) sampling points is carried out, its formula is shown in Equations (41) and (42). The calculation results of RE i ′(f) and RE i ′′(f) are listed in Figure 13.
Then, the average value RE averi (%) of these results are listed in Table 3. The RE averi (%) of all the real part is less than 7.59%, while the average error of the imaginary part is 12.61%. Since more response information is contained in the imaginary part of permittivity, while less information inside the real part of permittivity. Besides, any test deviations and calculation errors would lead to the non-coincidence of the curves in the highfrequency regions, as well as enhance the calculated relative error.
Thus, the accurate control of the experimental (for preparing the oil-immersed samples) and measurement conditions (including the measurement of DP, mc%, and FDS) is expected to reduce the error. Also, the optimization of the response function is regarded as one of the potential ways to reduce the generated error. Further research should pay attention to finding a  better approach to confirm the value of the response function corresponding to each sampling frequency point.

CONCLUSIONS
Whereas the solid insulation will surfer the non-uniform aging when prolonged exposure to the temperature gradient, the feasibility of the traditional XY model is therefore disputable. Also, the influence of the conductance loss on the dielectric response should be included, but the traditional model did not. Thus, the modified XY model is proposed to promote its applicability, as well as improve its physical significance. The present analysis and contributions have led to the following conclusions.
I. It is pointed out that the dielectric response property of XY model can be simulated by using both resistance and capacitance elements, the newly developed circuit could improve the accuracy and the physical significance of the traditional XY model. II. Based on the constructed simulation structure of the composite insulation, the mathematical expression of the modified XY model is presented. III. The accuracy and feasibility of the proposed modified XY model have been verified. Findings reveal that the average relative error of the real and imaginary part of relative permittivity is sequentially equal to 7.59% and 12.61%.
The novelty of this work is in the exploration of the modified XY model as a potential tool for establishing the quantitative relationship among the relative permittivity of the liquid insulation, solid insulation, and liquid-solid composite insulation under non-uniform aging. Such interesting property is expected to be serviced as a potential tool for extracting the information of non-uniform aging from the total FDS data. It deserves further attention.