Research on the Influence of Transformer Winding on Partial Discharge Waveform Propagation

Abstract

Partial Discharge (PD) measurement is one of the effective methods for assessing the internal insulation condition of power transformers in factories and substations. The pulse current signals generated by PD within transformer windings are significantly influenced by the winding structure during their propagation from the discharge source to the external measurement system. This influence may lead to misinterpretation of the insulation status, particularly in the analysis of PD measurement results. Such effects are closely related to the signal transmission path and distance and exhibit a strong correlation with the winding transfer function, manifesting as attenuation, distortion, or delay of the measured signals compared to the original PD waveforms. Therefore, it is essential to investigate the impact of the discharge path on the propagation characteristics of transformer windings and its effect on PD waveforms. This paper establishes a simplified distributed parameter model of a 180-turn single-winding multi-conductor transmission line using the finite element method and mathematical modeling, deriving the transfer functions between the winding head or winding end and various internal discharge positions. By injecting different types of PD waveforms collected in the laboratory at various discharge locations within the winding, the alterations of PD signals propagated to the winding head and winding end are simulated, and clustering analysis is performed on the propagated PD signals of different types.

1. Introduction

Partial discharge (PD) is the primary cause of oil-paper insulation failure in transformers and an important method for assessing the insulation level of transformer windings [1,2]. Due to its advantage of measurable quantity of apparent discharge [3], the pulse current method is widely used in PD detection for transformers at the factory and in the field [4]. PD usually occurs in the internal insulation of the transformer, and after transferring in the transformer windings, the PD pulse waveforms can only be obtained at a few external detection points like the ground wire and bushing. The PD pulse signal reaches the external detection point from the PD source through the insulating medium of the transformer, winding, and other lines. The original PD waveforms undergo signal amplitude attenuation, waveform distortion, and time delay in the transmission path [5], which makes the response signal measured at the detection point different from the original PD signal. That is the reason why the PD measurement methods adopted now are not ideal. Therefore, it is necessary to study the effect of transformer windings on the propagation of PD waveforms. Furthermore, characterizing the propagation mechanisms of PD waveforms along transformer windings is essential for advancing transformer fault diagnosis and refining PD detection methodologies [6,7].

In existing research, transformer winding models are primarily categorized into three types: RLC models, multi-conductor transmission line (MTL) models [8], and hybrid parameter models [9]. The RLC model is frequently employed for transformer frequency response analysis (FRA) and PD calibration [10,11]. A critical limitation lies in its frequency range, which is significantly narrower than the approximately 10 kHz to 30 MHz bandwidth of standard PD waveform acquisition systems [12,13,14]. Therefore, considering that PD waveforms contain high-frequency components, the MTL model is adopted to calculate the frequency response of transformer windings.

In this study, a simplified distributed parameter method was employed to construct an MTL model of the transformer. The model represents a 180-turn continuous winding, enabling the characterization of signal propagation from the terminals to internal sections. Experimentally acquired PD current pulses were injected at various locations within the winding to analyze the distortion and attenuation of the signals after propagation to either the head or the end of the winding. Clustering analysis was performed on the propagated PD waveforms to evaluate the impact of winding propagation characteristics on the classification of PD measurement results.

This paper is divided into the following sections: In Section 2, a multi-conductor transmission line model is established; in Section 3, a simplified simulation model of the transformer winding is established in finite element software, and the distributed parameter matrix of the transformer winding is determined; in Section 4, the transfer function from different positions of the winding to the head/end of the winding is simulated, and the response of the simulated PD pulse propagating from different positions of the winding to the head/end of the winding is calculated, and the results are discussed in Section 5.

2. Multi-Conductor Transmission Line Model of Transmission Windings

The disk windings are widely used in large and medium-sized transformers. In order to facilitate the modeling of the transformer winding, the following reasonable approximation can be made [15]:

• The length of each turn of the winding is much larger than the distance between adjacent turns.
• Ignore the effect of wire bending, and consider that the lengths of all wire turns are approximately equal.

Based on the above assumption, each turn of the transformer winding is used as a transmission line, and each turn of the winding is opened at the transposition position of the winding, then the entire transformer winding constitutes a MTL model, as shown in Figure 1.

If the number of winding turns is n, then the corresponding MTL model can be regarded as a 2n port network, where U_S(1), U_S(2), …, U_S(N), …, U_S(N) and I_S(1), I_S(2), …, I_S(N), …, I_S(N) represent the voltage and current at the head of each turn of the wire, U_R(1), U_R(2), …, U_R(n), …, U_R(N) and I_R(1), I_R(2), …, I_R(n), …, I_R(N) represents the voltage and current at the end of each turn of the wire.

The frequency domain equation can be written according to the established MTL model:

$$\left\{\begin{array}{c}{\displaystyle \frac{\mathrm{d}V(z)}{\mathrm{d}z}}=-Z(\omega )I(z)\\ {\displaystyle \frac{\mathrm{d}I(z)}{\mathrm{d}z}}=-Y(\omega )V(z)\end{array}\right.$$

(1)

This equation can be further simplified to

$$\left\{\begin{array}{c}{\displaystyle \frac{{\mathrm{d}}^{2}U}{\mathrm{d}{x}^2}}=ZYU={\mathsf{\Gamma }}^{2}U\\ {\displaystyle \frac{{\mathrm{d}}^{2}I}{\mathrm{d}{x}^2}}=YZI={\mathsf{\Gamma }}^{2}I\end{array}\right.$$

(2)

Z = R + jωL, Y = G + jωC, Γ² = ZY, Γ² = YZ, where Z is the impedance matrix, Y is the admittance matrix, and R, L, G, and C are the winding resistance, inductance, conductance, and capacitance parameter matrices. These four distribution parameter matrices are related to the winding structure of the transformer and the frequency of the input signal.

Equation (2) can be further simplified to

$$\left\{\begin{array}{l}{U}_x=e^{-\mathsf{\Gamma }x}{U}_1+e^{-\mathsf{\Gamma }x}{U}_2\hfill \\ I=Y_0\left(e^{-\mathsf{\Gamma }x}{U}_1-e^{-\mathsf{\Gamma }x}{U}_2\right)\hfill \end{array}\right.$$

(3)

where U₁ and U₂ are n-dimensional column vectors, Let the characteristic admittance matrix Y₀ = Z⁻¹Γ, Substitute the boundary conditions x = 0 and x = l into (3), and through the method of modular transformation to decouple [16], the relationship between the voltage and current of the MTL equation can be obtained:

$$\left\{\begin{array}{c}{U}_S=U_1+U_2\\ U_R=e^{-\mathsf{\Gamma }l}{U}_1+e^{\mathsf{\Gamma }l}{U}_2\\ I_S=Y_0(U_1-U_2)\\ I_R=Y_0(e^{-\mathsf{\Gamma }l}{U}_1-e^{\mathsf{\Gamma }l}{U}_2)\end{array}\right.$$

(4)

$$\left[\begin{array}{c}{I}_S\\ I_R\end{array}\right]=\left[\begin{array}{cc}A& -B\\ -B& A\end{array}\right]\left[\begin{array}{c}{V}_S\\ V_R\end{array}\right]$$

(5)

In the formula A = Y₀coth(Γl), B = Y₀csch(Γl). Thus, the MTL equation of a single winding of the transformer is obtained.

3. Transformer Winding Distribution Parameters

As mentioned above, the distributed parameters of the MTL model are related to the structure of the transformer winding. A 180-turn continuous winding is used as an example for calculation.

The schematic diagram of its structure is shown in Figure 2. For the convenience of calculation, the turns are numbered.

As shown in Figure 2, marker 1 are wires, marker 2 is transformer shell, marker 3 are wires in disk j and disk k which not adjacent to the shell, marker 4 is air/iron core, marker 5 is air/transformer oil, j is even number and 2 ≤ j ≤ 16, k is odd number and 3 ≤ k ≤ 17.

3.1. Solution of Capacitance Parameters and Finite Element Method

Although the finite element method was adopted in this study to overcome the inaccuracy of the parallel-plate capacitor method in calculating capacitance between misaligned conductors, it introduces greater computational complexity when the number of turns is large. Because the winding arrangement of the transformer winding has a certain regularity, a part of the conductor can be ignored in the calculation process, and a simplified calculation can be performed. Due to the electric field shielding effect of adjacent wires, the mutual capacitance between the conductors contained in two adjacent wire pies in the vertical direction decays quickly [17], so in order to reduce the amount of calculation, the effect of the other two adjacent disks in the vertical direction of the wire disk can be ignored.

Through the above discussion, a simplified model can be established for calculating the capacitance parameter matrix in the finite element simulation software. The model employs a stationary solver with the electrostatics physical field, utilizing 46,200 mesh elements. The Maxwell capacitance matrix is computed using a stationary source sweep approach.

The 30 × 30-dimensional Maxwell capacitance matrix of the conductors in the first, second, and third layers of the disks in the simplified model of Figure 3 can be calculated by the finite element software, where the first layer simulates the wires in disk 1 of the actual winding and the relationship between the transformer shell and other wires, and the third layer simulates the wires in disk 18 and the relationship between the transformer shell and other wires. The second layer simulates the wires in disk j in Figure 2 and the relationship between these wires and other wires. As Figure 2 shows, the mutual capacitance matrix and mutual inductance matrix between conductors in disk k can be obtained through angular symmetry from the mutual capacitance matrix and mutual inductance matrix between conductors in disk j. According to the above correspondence, the 30 × 30-dimensional mutual capacitance matrix obtained by simulation can be expanded to a 180 × 180-dimensional capacitance matrix C to obtain capacitance parameters.

3.2. Solution of Capacitance Parameters and Analytical Method

Both the inductance parameter and the resistance parameter are related to the frequency. When calculating resistance distribution parameters, skin resistance at high frequencies should be considered. The calculation expression of resistance per unit length is [16]:

$$R={\displaystyle \frac{1}{2(d_1+d_2)}}\sqrt{{\displaystyle \frac{\pi f\mu }{\sigma }}}$$

(6)

R represents the skin resistance related to frequency. Where μ is the magnetic permeability of the conductor, σ is the electrical conductivity of the conductor, d₁ and d₂ are the length and width of the rectangular conductor cross section, and f is the corresponding frequency.

The inductance parameter matrix is divided into two parts, respectively representing the self-inductance and mutual inductance in normal conditions and the internal inductance generated by the magnetic flux penetrating into the conductor due to the skin effect at high frequencies, as in (6):

$$L=L_0+L_s$$

(7)

In the formula, L₀ represents the self-inductance and mutual inductance under normal conditions, and L_S represents the internal inductance generated by the skin effect.

In an MTL system of uniform dielectric, the capacitance matrix C and the inductance matrix L₀ per unit length satisfy the following relationship [16]:

$$L_{0}C=C{L}_0=\mu \omega 1_n$$

(8)

$$L_0={\displaystyle \frac{{\epsilon }_r}{c^2}}{C}^{-1}$$

(9)

where μ and ε, respectively, represent the permeability and dielectric constant of the insulating medium, C is the capacitance matrix, and ε_r is the relative dielectric constant of the insulating material. The expression of the internal inductance L_S generated by the skin effect at high frequency is as follows:

$$L_S={\displaystyle \frac{R_s}{2\pi f}}$$

(10)

When the working frequency is high, the value of the resistance parameter and the inductance parameter changes little with the frequency, so 1 MHz is selected as the working frequency during the calculation. In addition, the conductance parameters are usually ignored.

3.3. Solution of Multi-Conductor Transmission Line Model and Analytical Method

The boundary conditions of the continuous single-winding MTL model are that the voltage at the end of the i-th (i = 1, 2, 3…180 − 1) transmission line is equal to the voltage at the beginning of the (i + 1)-th transmission line, and the current at the end of the transmission line is equal to the current at the first end of the (i + 1)-th transmission line in the opposite direction, as in (10):

$$\left\{\begin{array}{c}{U}_S(i+1)=U_R(i)\\ I_S(i+1)=-I(i)\end{array}\right.$$

(11)

Substituting the boundary conditions into Equation (4) yields a system of 180 + 1 equations:

$$\left[\begin{array}{c}{I}_S(1)\\ 0\\ ·\\ ·\\ 0\\ I_R(180)\end{array}\right]=\left[\begin{array}{ccc}& & \\ & X& \\ & & \end{array}\right]\left[\begin{array}{c}{U}_s(1)\\ U_s(2)\\ ·\\ ·\\ U_S(180)\\ U_R(180)\end{array}\right]$$

(12)

Let the transfer matrix be T = X⁻¹. The formula U = T × I can be obtained, where U and I are the voltage and current vectors. The current transfer function from the i-th turn to the beginning or end of the transformer winding can be obtained from this:

$$\left\{\begin{array}{c}{T}_{bush}=U_{bush}/U_{pd}\\ T_{end}=U_{end}/U_{pd}\end{array}\right.$$

(13)

In this formula, U_pd is the simulated PD pulse current injected inside the winding, T_head is the PD pulse response transfer function at the first end of the winding, and T_end is the PD pulse response transfer function at the end of the winding.

3.4. Calculation of Partial Discharge Impulse Response Signal and Convolution Method

To obtain the time-domain current waveform at the winding head or end in response to a propagating PD pulse, the frequency-domain response must be converted to the time domain. Several methods are available for this time-frequency conversion, including convolution, Fourier transform, and Fourier superposition [17]. This study employs the convolution method to compute the winding head and end responses.

Let x(n) be the discrete input signal and h(n) be the unit impulse response of the system. The output response y(n) of the system can then be calculated using Formula (14):

$$y\left(n\right)={\displaystyle \sum _{i=-\infty }^{\infty }x(i)h(n-i)}=x(n)\ast h(n)$$

(14)

Since a transformer winding modeled as a multi-conductor transmission line system constitutes a linear time-invariant (LTI) system, the convolution method can be applied to compute the time-domain response.

As Figure 4 shows, three kinds of PD pulse waveforms measured in the laboratory were used to simulate the PD source waveform in the transformer [18,19]. The corona discharge waveforms simulate PDs that occur mainly at the sharp corners of conductors and metal burrs of cores. The surface discharge waveforms simulate PDs that occur at the winding screen. The internal gap discharge waveforms simulate PDs that occur inside the insulation pressboard or in the air gap between insulation pressboards.

When the spectrum of a time series is known, its time series can be obtained by inverse discrete Fourier transform, and the available standard IFFT functions are provided in MATLAB R2020b, which can help us convert the frequency domain sequence to the time domain sequence.

4. Results

4.1. Transfer Functions of Different Turns to the Head/End of the Winding

The transformer winding MTL model is programmed by MATLAB to obtain the amplitude-frequency characteristics of the voltage transfer function from the internal winding to the head or end of the transformer winding:

The frequency range of the simulation is 0~50 MHz. As Figure 5 shows, the transformer winding has different amplification/decreasing effects on signals of different frequency components. At certain resonant frequencies, the transformer winding may greatly increase the signal at this frequency. In terms of the transfer function curves, taking the curve starting from the 40th turn as an example, the peaks and troughs of the transfer function from the 40th turn to the winding head exhibit a similar pattern to those from the 40th turn to the winding end. The magnitudes of the peaks and troughs in the transfer function to the winding head are higher than those to the winding end. Furthermore, the transfer function to the winding end displays a distinct peak around 10 MHz and a trough around 1 MHz, while no such peak is observed in the transfer function to the winding head. The same characteristics were observed for the 100-turn and 160-turn cases.

4.2. Response Signals of Different PD Waveforms

As shown in Figure 6, inject PD waveforms in different positions of the winding. The time-domain sequence was convolved with the simulated partial discharge pulse waveform to obtain the response waveforms:

Figure 6 illustrates how PD waveforms change in amplitude, timing, and shape as they travel to the winding head or end, relative to the original signals injected at different turns. At the 40th turn, the head waveform shows a higher peak with tail oscillations, while the end waveform has a modest peak increase and damped negative oscillation. At the 100th turn, the head waveform exhibits the strongest peak amplification and oscillation among all cases, whereas the end waveform features lower peak gain and a suppressed negative peak, though with more sustained oscillation. At the 160th turn, the head waveform presents an elevated peak, stronger oscillation, and an initial negative peak; the end waveform displays a broad amplitude increase, enhanced oscillation, and develops into a negative pulse. In all scenarios, the propagated PD pulses undergo wavelength lengthening to varying degrees.

To further quantify the distortion of the propagated waveform relative to the source waveform, the Pearson correlation coefficient (r) between the propagated and source waveforms was calculated. The Pearson correlation coefficient is commonly used to measure the correlation between two variables and, when one variable remains unchanged, can also be applied to assess the degree of change in the other variable relative to it. The value of r ranges from −1 to 1, with values closer to 0 indicating greater distortion of the propagated signal relative to the source signal.

As shown in

Table 1. Pearson correlation coefficients between various types of discharge waveforms and their propagated waveforms.

Pearson Correlation (r)	Corona Discharge	Internal Discharge	Surface Discharge
40 to head	0.5642	0.4763	0.4844
40 to end	0.6480	0.2085	−0.0379
100 to head	0.5690	0.5147	0.5702
100 to end	0.6531	0.3600	0.2766
160 to head	0.2847	−0.3270	−0.3517
160 to end	−0.7987	−0.5913	−0.3663

, for different types of partial discharge signals, the distortion is generally minor for signals propagating from the 40-turn and 100-turn positions to the winding end, whereas signals propagating from the 160-turn position exhibit more pronounced distortion. The r value for signals propagating from the 160-turn position to the winding end is closest to −1. In conjunction with the waveform diagrams, this indicates that the signal propagating from the 160-turn position to the winding end is inverted relative to the source signal, yet the degree of distortion remains relatively low, which is consistent with the analysis of Figure 6.

4.3. Clustering Results of Response Signals of Different PD Waveforms

In order to study the influence of winding propagation characteristics on the classification of different types of partial discharges, the DBSCAN clustering method was used to cluster the discharge source waveform set and the waveform set propagated to the head/end of the winding. Figure 7 shows the composite two-dimensional projection of three sets of partial discharge waveforms, each containing 100 individual waveforms. Figure 8 presents the clustering results of these partial discharge waveform sets using the DBSCAN method [20,21], with the neighborhood radius set to 0.1 and the minimum number of points required to form a dense region set to 5. The classification results demonstrate that the three different types of partial discharge waveform sets are well separated. The waveforms labeled as “Misclassified” represent those that were not assigned to any discharge waveform cluster due to their significant deviation from the majority of the waveforms.

As shown in Figure 9, the two-dimensional equivalent time–frequency plane projections of the PD response waveform set exhibit considerable alterations compared to those of the source discharge waveforms. Both the intra-cluster distances within the same type of discharge waveform sets and the inter-cluster distances between different types are noticeably reduced. Particularly, the equivalent frequencies of the response waveforms become lower and more concentrated than those of the source discharge waveform sets. This can be attributed to the fact that the path from the discharge source to either the head or the end of the winding functions as a two-port network, imparting a frequency-selective filtering effect on the waveforms.

In the response waveform set corresponding to the discharge point at the 40th turn, a number of internal discharge and surface discharge waveforms are misclassified as a fourth type of discharge waveform set.

For the waveform set with the discharge point at the 100th turn, the waveforms propagated to the winding head are correctly classified, whereas some waveforms propagated to the winding end are misclassified into the fourth discharge category.

In the case of the discharge point at the 160th turn, the waveforms propagated to the winding end are correctly classified, while the internal discharge and surface discharge waveforms propagated to the winding head are grouped into the same cluster.

5. Discussion

The variations in the discharge waveform during its propagation through the winding are closely related to the transmission characteristics of the winding—specifically, the frequency-domain properties from the discharge point to the measurement position.

As shown in Figure 10, the frequency-domain distributions of three typical partial discharge waveforms are illustrated. The frequency components of the corona discharge waveform are concentrated below 10 MHz; those of the internal discharge waveform are mainly distributed below 10 MHz and within 20–30 MHz, while the surface discharge waveform exhibits frequency components concentrated primarily in the 20–40 MHz range.

From the perspective of signal transmission to the winding head or winding end: The frequency-domain amplitude from various discharge points to the winding head is generally higher than that to the winding end. Consequently, for the same waveform and discharge point, the amplitude of the response waveform propagating to the winding head is greater than that propagating to the winding end.

Regarding different discharge points: For discharge points at the 40th and 100th turns, the voltage transfer function curve to the winding head shows a significantly higher peak near 1 MHz compared to the transfer function to the winding end, resulting in noticeable oscillations in the response waveform. In contrast, the voltage transfer function for the discharge point at the 160th turn is relatively flat around this frequency, so no significant oscillation occurs in its response waveform. Moreover, the voltage transfer function for the 160th turn has a distinct peak around 20 MHz, leading to a pronounced negative-polarity peak in the corresponding response waveform.

In terms of different types of PD waveforms: Since most frequency components of corona discharge and part of the internal discharge fall below 10 MHz, these two types of discharges produce response waveforms with large oscillation amplitudes when originating from the 40th or 100th turns and propagating to the winding head. On the other hand, since certain frequency components of both internal discharge and surface discharge are concentrated around 20 MHz, these waveforms result in response waveforms with larger negative-polarity peaks when propagating from the 160th turn to either the head or the end of the winding.

Regarding the clustering results of response waveforms from different types of partial discharges at various discharge points, Figure 6 indicates that the waveforms corresponding to discharge points at the 40th and 100th turns exhibit primarily amplitude variations and oscillatory effects compared to the source discharge waveforms. Moreover, the responses propagated to the winding head demonstrate stronger oscillation than those propagated to the winding end. The oscillatory behavior is more pronounced in the waveforms from the 100th turn than in those from the 40th turn.

For the response waveforms at the 40th turn, distortion induced by oscillation leads to a longer equivalent time compared to the source waveforms. This reduces the inter-cluster distance between internal discharge and surface discharge response waveforms and further compresses the intra-cluster distance, resulting in the misclassification of some waveforms between these two types.

For discharges at the 100th turn, the waveform sets of internal and surface discharges are relatively similar. While stronger oscillations propagating to the winding head lead to a higher equivalent frequency of the responses, this phenomenon increases the inter-cluster distance between the internal and surface discharges. This effect facilitates correct separation of the responses propagated to the winding head. In contrast, the small inter-cluster distance between internal and surface discharge responses propagated to the winding end leads to the misclassification of some waveforms between these two categories.

For the discharge point at the 160th turn, the polarity reversal effect—as observed in Figure 9e,f—significantly influences the two response waveform sets. For the set propagated to the winding head (Figure 9e), although the inter-cluster distance between internal and surface discharges is substantial, the negative-polarity peak appearing at the onset of the waveform causes an expansion of the intra-cluster distance after propagation through the winding. That is, the dispersion within each waveform cluster increases, ultimately leading to the merging of the two PD waveform sets into one larger cluster. For the set propagated to the winding end (Figure 9f), a complete polarity reversal occurs. This results in a greater inter-cluster distance between the internal discharge and surface discharge waveform sets compared to those propagated to the winding head. Although the intra-cluster distance remains relatively large, the responses propagated to the winding end can still be classified into three distinct clusters.

6. Conclusions

In this paper, a simplified two-dimensional finite element model of the transformer winding was established in this study to extract the distributed parameters based on the similarity among the distributed parameters of the winding. On this basis, a MTL model of the transformer winding was constructed, and the voltage transfer functions from various internal positions to both the head and end of the winding were derived. Three different sets of PD waveforms were collected in the laboratory. Using the convolution method, the response waveforms propagating from different discharge positions in the winding to the head and end of the winding were simulated. The effects of the winding on amplitude variation, time delay, and waveform distortion during PD propagation were analyzed. Furthermore, clustering methods were applied to evaluate the impact of winding propagation characteristics on the classification of different types of PD waveforms. The main conclusions are as follows:

• The initial PD location determines the shape of the transfer function curve, while the direction of propagation—toward the head or the end of the winding—affects the magnitude of its peaks and troughs. For the same discharge point, the transfer functions to the head and the end exhibit similar patterns of peaks and troughs. Additionally, the transfer function to the winding end shows a distinct peak around 10 MHz and a trough near 1 MHz, which are not observed in the transfer function to the winding head.
• For the same waveform and discharge point, the amplitude of the response waveform propagated to the winding head is greater than that propagated to the winding end. The frequency response of the winding and the spectral distribution of the PD waveform jointly influence the shape of the response PD waveform. The peak in the transfer function near 1 MHz affects the oscillation intensity of the PD response waveform, while the peak around 20 MHz influences its polarity. For different types of PD waveforms, their spectral distributions govern both the oscillation intensity and the degree of waveform reversal.
• Oscillations and polarity reversal during the propagation of PD waveforms within the winding alter the shape of the response waveforms, which in turn affects the classification results of different PD response waveform sets. Since the path from the discharge source to the winding head or winding end can be regarded as a two-port network with frequency-selective filtering characteristics, both the inter-cluster and intra-cluster distances of the PD response waveform sets are significantly reduced. While oscillations increase the equivalent frequency of response waveforms from the same discharge source—thereby enlarging inter-cluster distances and facilitating clustering—they can also distort certain waveforms and alter their equivalent duration, which may reduce inter-cluster separation. The polarity reversal effect considerably increases both inter-cluster and intra-cluster distances. These factors collectively influence the clustering outcomes, potentially causing similar portions of different waveform types to form a separate cluster or merging two similar waveform sets into a single category.
• It should be noted that this study employs an idealized model of the transformer winding. Two possible deviations from this assumption are considered: If turns at the high-voltage end differ in length from those in the middle section, the resulting asymmetry in the distributed inductance and capacitance matrices would affect the equation solution. Such cases would require a segmented modeling approach. Geometric changes due to distortion or poor contact can significantly increase high-frequency losses, accelerating PD pulse attenuation. Impedance non-uniformity may cause waveform reflection/refraction, while inter-turn resonant structures could amplify specific frequency components, substantially influencing PD analysis.