A Method for Extracting Characteristic Parameters of Frequency Domain Dielectric Spectroscopy of Oil-Paper Insulation Using Modified Cole–Cole Model

Abstract

To quantitatively describe the frequency domain spectroscopy (FDS) characteristics of transformer oil-paper insulation under varying temperature, moisture, and aging conditions, a modified Cole–Cole model is introduced. This model decomposes the dielectric spectrum into polarization, DC conduction, and hopping conduction components, with parameters reflecting insulation characteristics. Methods for determining initial parameter values and optimizing the objective function are proposed. Using a three-electrode setup, FDS measurements were conducted on oil-paper insulation samples at different temperatures, and extracted parameters were analyzed for their variation patterns. Within the frequency range of 1.98 × 10⁻⁴ Hz to 1 × 10³ Hz, the model achieves a goodness-of-fit (R²) exceeding 0.97 for both real and imaginary permittivity components, with the sum of squared errors reduced from 259 to 57.35 at 70 °C, outperforming the fundamental Cole–Cole and Ekanayake’s models. Temperature significantly affects the relaxation and DC conductivity components; both adhere to the Arrhenius equation, enabling precise condition assessment of transformer insulation.

1. Introduction

Power transformers are critical electric power components in the process of power transmission and distribution, ensuring the safe operation and reliability of the power grid [1,2]. Transformer failure will cause huge direct and indirect economic losses, seriously affecting people’s lives and social production [3]. Transformer internal insulation is mainly of a mineral oil and fiber paper composition. Electrical, thermal, mechanical, chemical, and other elements and ages reduce insulation performance throughout operation. Thermal aging degrades insulation chemically and physically. In oil-paper insulation, cellulose fibers break and deform, creating new chemicals and changing mechanical and electrical properties [4,5]. In power systems, transformer failures, caused by insulation failure, design flaws, material defects, manufacturing issues, and poor maintenance, are serious [3,6]. How to appropriately assess the transformer’s insulation and remaining life is vital for practice and research. Traditionally, transformer oil-paper insulation is assessed by its oil acid value, furfural content, dissolved gas, insulation paper tensile strength, degree of polymerization, and dielectric loss [7]. Dissolved gas analysis (DGA), oil acidity, and furfural concentration are crucial for transformer oil-paper insulation assessment [8,9]. Researchers worldwide have focused on innovative transformer insulation parameters and methods in recent years. Advanced techniques like FTIR spectroscopy, polarization and depolarization current measurements, and frequency domain dielectric spectroscopy can provide additional insights into the insulation condition [10]. Diagnostic techniques based on dielectric response theory, methods such as the return voltage method (RVM) [11], polarization and depolarization current (PDC) [12], and frequency domain spectroscopy (FDS) [13,14,15] have emerged as novel non-destructive diagnostic techniques. These methods are easy to operate and provide rich information, and are garnering extensive research interest. The frequency domain spectroscopy approach is ideal for on-site measurements due to its great anti-interference capacity. Even though power transformers operate at 50 Hz or 60 Hz frequency domains, dielectric spectroscopy looks at the dielectric response of oil-paper insulation over a wide range of frequencies, usually from 0.000198 Hz to 1000 Hz. This broadband method is necessary because low frequencies such as 0.000198 Hz are very sensitive to moisture, aging byproducts, and interfacial polarization, which are all important signs that insulation is breaking down [15]. Higher frequencies of up to 1000 Hz capture quick polarization processes and material flaws, providing an exhaustive overview of how well the insulation is working. By considering the dielectric response over this range, FDS makes it possible to get characteristics that quantitatively reflect the health of insulation. This makes it easier to find deterioration mechanisms that impair transformer reliability early on.

Moisture, age, and temperature affect oil-paper insulation’s frequency domain dielectric characteristics, which have been extensively studied [16]. These investigations have yielded several insights. Most studies examine qualitative changes in dielectric characteristics with insulating conditions or test parameters. Few studies have extracted characteristic parameters from frequency domain dielectric curves to quantify their connections with temperature, moisture, and age [17,18]. Identifying accurate metrics to evaluate transformer oil-paper insulation’s frequency domain characteristics is challenging. Using well-designed mathematical and physical models to recreate insulation materials’ complicated dielectric behavior can help quantify frequency domain dielectric qualities. Several studies have examined dielectric behavior using the Debye model [19]. The classical Debye model defines the complex permittivity ${\epsilon }^{*}$ as a function of angular frequency ω and relaxation time constant $\tau$. The validity of this model has been verified in interpreting the dielectric properties of ferroelectric solutions. Based on the Debye model, K.S. Cole and R.H. Cole introduced a parameter α and proposed the well-known Cole–Cole model [20,21,22,23], expanding the model’s applicability. In conclusion, the Debye model remains a cornerstone in dielectric relaxation studies, providing valuable insights into molecular structures and dynamics. However, its limitations in describing more complex systems have led to the development of extended models and analytical techniques. The ongoing research in this field continues to refine our understanding of dielectric behavior across various materials and frequency ranges. Ekanayake used the Cole–Cole model to study the dielectric response characteristics of oil-paper insulation under different moisture content levels and pointed out the advantage of introducing the $A{\omega }^{-n}$ term in the model for analyzing the low-frequency dielectric response [24]. When modified to include this factor, the Cole–Cole model accurately characterizes dielectric response and insulation relaxation processes [25]. This improved model precision helps analyze transformer insulation’s health under operating pressures, including temperature changes [14,26]. In addition to refining low-frequency response depictions, the Aω⁻ⁿ term improves aging and moisture prediction, enhancing insulation diagnostics reliability and predictive power. The Cole–Cole model, which traditionally characterizes dielectric behavior with distributed relaxation times, is extended with the Aω⁻ⁿ term to account for additional, non-ideal dielectric responses that occur at low frequencies [27,28]. The literature referenced this model to study the dielectric response characteristics of transformer oil-paper insulation with nano-modification [29]. However, the $A{\omega }^{-n}$ term introduced in this model lacks clear physical significance. Enis Tuncer and others, when studying the effects of nanoparticles on the dielectric properties of silicone rubber materials, introduced the hopping conductivity ${\sigma }_{\mathit{ho}}$ to describe the low-frequency dispersion and complex conductivity characteristics caused by material defects [30]. This model, also known as the modified Cole–Cole model, compared to C. Ekanayake’s model, provides an improved description of the dielectric properties of insulating materials across frequencies, with each term possessing clear physical significance. Currently, the application of this model to analyze the dielectric and impedance characteristics of transformer oil-paper insulation at low frequencies has rarely been reported [7]. Furthermore, the application process of this model involves multiple nonlinear fitting parameters and the complexity of constructing target functions, which limit its further promotion and application. In comparison to the study focused primarily on the experimental investigation of the frequency domain dielectric response of oil-paper insulation under varying aging and moisture conditions, it provides valuable empirical trends and qualitative insights [31]. In contrast, the present work applies a modified Cole–Cole model that decomposes the dielectric response into relaxation, DC conductivity, and hopping conductivity components, each with defined physical interpretations.

The Debye, Cole–Cole and Havriliak–Negami models are commonly used in dielectric analysis, but each has limitations for transformer diagnostics. Debye neglects low-frequency dispersion, critical for aging and moisture detection. Cole–Cole lacks explicit treatment of DC conduction and hopping effects, while Havriliak–Negami increases computational complexity and reduces interpretability due to asymmetry [14]. To overcome these deficiencies, the modified Cole–Cole model integrates components for α-relaxation, DC conductivity, and hopping conduction. Each parameter is physically defined and related to temperature, aging, and moisture [32]. This makes the model more robust for real-world applications where transformer insulation may degrade in multiple ways under high voltage and environmental stress. We also provide a systematic parameter estimation framework to apply the model using commercial FDS analyzers like DIRANA. Diagnostic precision has improved, making the approach better for on-site transformer health monitoring and lifespan estimation. This paper applies the modified Cole–Cole model to extract characteristic parameters of the dielectric spectrum for oil-paper insulation at low frequencies. It focuses on methods for selecting initial parameter values during the fitting process and constructing objective functions. The study emphasizes investigating the influence of temperature on the characteristic parameters in the modified Cole–Cole model more effectively.

2. Modified Cole–Cole Model and Characteristic Parameters

The classic Debye relaxation model expresses the complex dielectric constant ${\epsilon }^{*}$ as a function of angular frequency ${\epsilon }_{\mathit{hf}}$ [20], and the relationship is given by

$${\epsilon }^{*}(\omega )={\epsilon }_{\mathrm{hf}}+{\displaystyle \frac{{\epsilon }_{\mathrm{s}}-{\epsilon }_{\mathrm{hf}}}{1+\mathrm{j}\omega \tau }}.$$

(1)

In the equation, $\tau$ is the relaxation time constant, and ${\epsilon }_s$ and ${\epsilon }_{\mathit{hf}}$ represent the static dielectric constant and the high-frequency dielectric constant, respectively. However, experimental research results show that the classic Debye relaxation characteristic is only consistent in dilute solutions and ferroelectric material. Therefore, the dielectric polarizability ${\chi }^{\mathrm{*}}={\epsilon }^{\mathrm{*}}-{\epsilon }_{\mathrm{h}\mathrm{f}}$ and parameter $n$ are further introduced [22], resulting in the frequency response of the dielectric, described by the Cole–Cole model as

$${\chi }^{\star }(\omega )={\chi }^{\prime }-\mathrm{j}{\chi }^{″}={\displaystyle \frac{{\chi }_s}{1+{(\mathrm{j}\omega \tau )}^n}}$$

(2)

where ${\chi }^{\prime }$ and ${\chi }^{\prime \prime }$ are the real and imaginary parts of the polarizability, respectively.

${\chi }_{\mathrm{s}}={\epsilon }_{\mathrm{s}}-{\epsilon }_{\mathrm{h}\mathrm{f}}$ represents $\omega =0$ the static dielectric polarizability, where $\tau$ is the relaxation time; $n$ is the parameter representing the relaxation time distribution, $0⩽n⩽1$. ${\chi }^{\prime }$ and ${\chi }^{\prime \prime }$ represent the polarizability and loss of the dipole in the material, respectively [22]. If the DC conductivity ${\sigma }_{\mathrm{d}\mathrm{c}}$ and hopping conductivity ${\sigma }_{\mathrm{h}\mathrm{o}}$ of the material are further considered, where ${\sigma }_{\mathrm{h}\mathrm{o}}=\xi /{\epsilon }_0(\mathrm{j}\omega {)}^{\gamma }$, the modified Cole–Cole model can be obtained as

$$\begin{array}{c}{\chi }^{\star }(\omega )={\chi }_1^{\ast }(\omega )+{\chi }_2^{\star }(\omega )+{\chi }_3^{\star }(\omega )=\\ {\displaystyle \frac{{\chi }_{\mathrm{s}}}{1+{(\mathrm{j}\omega \tau )}^n}}+{\displaystyle \frac{{\sigma }_{dc}}{\mathrm{j}{\epsilon }_0\omega }}+{\displaystyle \frac{\xi }{{\epsilon }_0{(\mathrm{j}\omega )}^{\gamma }}}\end{array}$$

(3)

where ${\epsilon }_0$ is the vacuum dielectric constant, and ${\epsilon }_0=8.85\times {10}^{-12}\mathrm{F}/\mathrm{m}$; $\xi$ and $\gamma$ are constants, where $0<\gamma <1$. This function represents the polarizability of the medium into three processes: relaxation, DC conductivity, and hopping conductivity, which are ${\chi }_1^{*}\left(\omega \right),{\chi }_2^{*}\left(\omega \right)$ and, respectively, ${\chi }_3^{*}\left(\omega \right)$. According to the actual situation of the experimental data, ${\chi }_1^{\mathrm{*}}$ can be further divided into a combination of two or more different relaxation processes. Taking the double relaxation process as an example,

$${\chi }_1^{*}={\displaystyle \frac{{\chi }_{s\alpha }}{1+{\left(\mathrm{j}\omega {\tau }_{\alpha }\right)}^{n\alpha }}}+{\displaystyle \frac{{\chi }_{\mathrm{s}\beta }}{1+{\left(\mathrm{j}\omega {\tau }_{\beta }\right)}^{n\beta }}}$$

(4)

These two relaxation processes are usually called $\alpha$-relaxation and $\beta$-relaxation. ${\tau }_{\alpha },{\tau }_{\beta }$, and $n_{\alpha },n_{\beta }$ are the relaxation time constant and distribution parameter of the two processes, respectively; ${\chi }_{\mathrm{s}\alpha }$ and ${\chi }_{\mathrm{s}\beta }$ are the static dielectric polarizability. Equation (3) can be transformed to obtain the real and imaginary expressions of the dielectric polarizability as follows:

$$\left\{\begin{array}{l}{\chi }^{\prime }(\omega )={\displaystyle \frac{{\chi }_{\mathrm{s}}\cos (Z)}{{\left(X^2+Y^2\right)}^{\frac{1}{2}}}}+{\displaystyle \frac{\zeta \cos (\gamma \pi /2)}{{\epsilon }_0{\omega }^{\gamma }}};\hfill \\ {\chi }^{\prime \prime }(\omega )={\displaystyle \frac{{\chi }_{\mathrm{s}}\sin (Z)}{{\left(X^2+Y^2\right)}^{\frac{1}{2}}}}+{\displaystyle \frac{{\sigma }_{\mathrm{dc}}}{{\epsilon }_0\omega }}+{\displaystyle \frac{\zeta \sin (\gamma \pi /2)}{{\epsilon }_0{\omega }^{\gamma }}}\hfill \end{array}\right.$$

(5)

where the parameters $X,Y$, and $Z$ are

$$\left\{\begin{array}{l}X=1+{(\omega \pi )}^n\cos (\alpha \pi /2);\hfill \\ Y={(\omega \pi )}^n\sin (\alpha \pi /2);\hfill \\ Z=\arctan (Y/X)\hfill \end{array}\right.$$

(6)

Experimental results indicate that within the test frequency, the dielectric response characteristics of oil-paper insulation can typically be described using a modified Cole–Cole model that includes one or two relaxation processes. If the real and imaginary parts of the dielectric response measured in the experiment are fitted using Equation (3) for the modified Cole–Cole model with a single relaxation time constant, seven parameter values $P_1\sim P_7$ can be obtained, as shown in

Table 1. Characteristic parameters of the modified model.

Parameters	${\mathrm{P}}_1$	${\mathrm{P}}_2$	${\mathrm{P}}_3$	${\mathrm{P}}_4$	${\mathrm{P}}_5$	${\mathrm{P}}_6$	${\mathrm{P}}_7$	${\mathrm{P}}_8$	${\mathrm{P}}_9$	${\mathrm{P}}_{10}$
Definition	${\epsilon }_{\mathrm{h}\mathrm{f}}$	${\chi }_{\mathrm{s}\alpha }$	${\tau }_{\alpha }$	$n_{\alpha }$	${\sigma }_{\mathrm{d}\mathrm{c}}$	$\xi$	$\gamma$	${\chi }_{\mathrm{s}\beta }$	${\tau }_{\beta }$	$n_{\beta }$

, to quantify the dielectric response characteristics of the test object. Similarly, for the modified Cole–Cole model with dual relaxation time constants, ten parameter values $P_1\sim P_{10}$ can be obtained, where parameters $P_8\sim P_{10}$ represent the $\beta$-relaxation process. The extracted parameter values can serve as characteristic parameters for the dielectric response of oil-paper insulation. These parameters can be used to further quantitatively analyze the effects of conditions such as temperature, moisture, and aging on the dielectric response characteristics of the insulating material. The modified Cole–Cole model, which includes parameters for relaxation DC conductivity and hopping conductivity, is based on the work of Tuncer et al. [30]. However, this study presents a new use for transformer oil-paper insulation at low frequencies. We have added a systematic method for setting parameters, discussed in Section 3.1, and an optimization framework, discussed in Section 3.2, that is specific to the dielectric response of oil-paper insulation. These changes make the model more useful for finding physically relevant characteristic parameters, which makes it easier to measure the status of transformer insulation quantitatively.

3. Parameters Estimation of the Modified Cole–Cole Model

3.1. Selection of Initial Parameter Values

To determine the parameters of the modified Cole–Cole model, a fitting method can be applied to the measured complex permittivity curves as a function of frequency. Since this curve-fitting problem involves up to 7~10 parameters, the selection of initial values directly affects the accuracy of the fitting results. Figure 1 illustrates the fitting curve and decomposition diagram of the modified Cole–Cole model with a single relaxation process ($\alpha$-relaxation) using a set of measured real and imaginary parts of the dielectric constant of oil-paper insulation board as an example. The test frequency range is $1.98\times {10}^{-4}\mathrm{H}\mathrm{z}\sim 1\times {10}^3\mathrm{H}\mathrm{z}$. From Figure 1, it can be observed that the real part of the dielectric constant ${\epsilon }^{\prime }\left(f\right)$ remains relatively constant in the frequency range of ${10}^1\sim {10}^3\mathrm{H}\mathrm{z}$, and this value mainly depends on the magnitude of ${\epsilon }_{\mathrm{h}\mathrm{f}}$. The $\alpha$-relaxation process in the imaginary part of the dielectric constant ${\epsilon }^{\prime \prime }\left(f\right)$ resembles a parabolic shape, where the maximum value of the parabola is determined by the parameter ${\chi }_{s\alpha }$ and the frequency corresponding to this maximum is determined by the reciprocal of the $\alpha$ relaxation time constant ${\tau }_{\alpha }$. The degree of the parabola’s opening corresponds to the parameter $n_{\alpha }$, where a smaller $n_{\alpha }$ results in a flatter parabola. When $n_{\alpha }=0$, the relaxation process becomes a straight line. The effect of DC conductivity appears as a straight line in ${\epsilon }^{\prime \prime }\left(f\right)$, primarily influencing the low-frequency part of the imaginary dielectric constant. The hopping conductivity process mainly affects the shape of the high-frequency part of the ${\epsilon }^{\prime \prime }\left(f\right)$ curve. Therefore, the following initial parameter determination steps are proposed (superscript 0 represents the initial value of the corresponding parameter).

(1)

${\epsilon }_{\mathrm{h}\mathrm{f}}^0$: Take the average value of ${\epsilon }^{\prime }$ measured at the frequencies above 100 Hz.

(2)

${\tau }_{\alpha }^0$: Take the frequency point $f_{\mathrm{A}}$ corresponding to the dielectric loss peak A in the imaginary part. Then, ${\tau }_{\alpha }={\displaystyle \frac{1}{f_{\mathrm{A}}}}$. This peak is quite noticeable in the relationship between the dielectric loss $\tan \delta$ and the frequency domain.

(3)

${\sigma }_{\mathrm{d}\mathrm{c}}^0$: Select the low-frequency linear segment ${\mathrm{B}}_1\sim {\mathrm{B}}_2$ in ${\epsilon }^{\prime \prime }$ and perform a linear fit on the measured data between these two points. Assume that the intercept obtained by the fitting is $P_{\mathrm{B}}$; then

$${\sigma }_{ac}=exp(P_B+ln({\epsilon }_0))$$

(7)

(4)

$n_{\alpha }^0$: On the ${\epsilon }^{\prime \prime }$ frequency plot, take a point C with a frequency greater than $f_{\mathrm{A}}$. The curve between $f_{\mathrm{A}}$ and $f_{\mathrm{C}}$ appears linear. Perform a linear fit on this segment of data, and let the slope obtained be $S_{\mathrm{A}\mathrm{C}}$. Then, $n_{\alpha }=-S_{\mathrm{A}\mathrm{C}}$.

(5)

$\gamma$ and $\xi$: Take the imaginary part coordinates measured at frequencies above 100 Hz for linear fitting, and the slopes obtained are $S_{\mathrm{D}}$ and $P_{\mathrm{D}}$, and the initial values of $\xi$ and $\gamma$ are

$$\gamma =-S_D.$$

(8)

$$\xi =\exp \left(P_{\mathrm{D}}-\ln \left(\sin {\displaystyle \frac{\gamma \pi }{2}}\right)\right)+\ln \left({\epsilon }_0\right)$$

(9)

(6)

${\chi }_{s\alpha }^0:$ Take the lowest frequency measurement point E on the real part curve, and denote its measurement frequency and corresponding ${\epsilon }^{\prime }$ as $f_{\mathrm{E}}$ and $Y_{\mathrm{E}}$, respectively; then we have

$${\chi }_{\mathrm{s}\alpha }={\displaystyle \frac{Y_{\mathrm{E}}{\left(X^2+Y^2\right)}^{1/2}}{\cos Z}}$$

(10)

Among them, the value of $X,Y,Z$ is calculated by using Equation (6). For oil-paper insulation, for ${10}^{-3}\sim {10}^3\mathrm{H}\mathrm{z}$, most of the curves can be represented using an extended Cole–Cole function that includes only $\alpha$-relaxation. When a distinct peak appears in the high-frequency region, $\beta$, the relaxation process needs to be considered. The time constant of this process usually appears in a higher-frequency segment. The initial value of the time constant ${\tau }_{\beta }$ can be obtained by taking the reciprocal of the frequency corresponding to the peak. The initial value of $n_{\beta }$ is related to the width of the peak, and the initial value of ${\chi }_{\mathrm{s}\beta }$ can be determined using a method similar to that used for calculating ${\chi }_{s\alpha }^0$.

3.2. Defining the Objective Function

Once the initial parameter values are set, the next step is to establish an objective function for optimization. The goal is to adjust the parameters of the modified Cole–Cole model so that its calculated values align as closely as possible with experimental data. In curve fitting, the least squares method is often employed to minimize the sum of squared errors and identify the best parameter values. However, optimizing the modified Cole–Cole model requires considering both the real and imaginary components of the dielectric constant simultaneously, ensuring that the errors for both are minimized. Additionally, the parameters must adhere to practical constraints to retain their physical meaning. For instance, the DC conductivity ${\sigma }_{\mathrm{d}\mathrm{c}}$ typically falls between 0.01 pS/m and several hundred pS/m, while $\gamma ,n_{\alpha }$, and $n_{\beta }$ are parameters ranging from 0 to 1. Based on extensive experimentation, this study frames the optimization process as a constrained problem, formulated as follows:

$$Y_{\mathrm{opt}}=\sum \left({\displaystyle \frac{{\left({\epsilon }_{\mathrm{model}}^{\prime }-{\epsilon }_{\mathrm{measure}}^{\prime }\right)}^2}{{\epsilon }_{\mathrm{model}}^{\prime }-{\epsilon }_{\mathrm{fit}}^{\prime }}}+{\displaystyle \frac{{\left({\epsilon }_{\mathrm{model}}^{\prime \prime }-{\epsilon }_{\mathrm{measure}}^{\prime \prime }\right)}^2}{{\epsilon }_{\mathrm{model}}^{\prime \prime }-{\epsilon }_{\mathrm{fit}}^{\prime \prime }}}\right)$$

(11)

where ${\epsilon }_{\mathrm{model}}^{\prime },{\epsilon }_{\mathrm{model}}^{\prime \prime }$ are the real and imaginary parts of the calculated value using the modified Cole–Cole model; ${\epsilon }_{\mathrm{measure}}^{\prime },{\epsilon }_{\mathrm{measure}}^{\prime \prime }$ are the real and imaginary parts of the measured data; ${\epsilon }_{\mathrm{f}\mathrm{i}\mathrm{t}}$ is the value after the measured data have been smoothed using the interpolation method; ${\epsilon }_{\mathrm{f}\mathrm{i}\mathrm{t}}^{\prime },{\epsilon }_{\mathrm{f}\mathrm{i}\mathrm{t}}^{\prime \prime }$ are its real part and its imaginary part; and $Y_{\mathrm{o}\mathrm{p}\mathrm{t}}$ is the optimization objective function. By considering both the real and imaginary components, the suggested objective function aims to reduce the relative squared error between the measured dielectric data and the values determined using the corrected Cole–Cole model. A balanced and equitable comparison is made possible by the objective function’s incorporation of normalized differences, which guarantees that changes in magnitude across several frequencies do not bias the fitting procedure. A set of constraints is implemented to ensure that the optimized parameters remain physically relevant. These limitations guarantee that every parameter, including conductivity, relaxation durations, and fractional exponents, stays within reasonable and physiologically significant limits. The interpretability and dependability of the model are thus maintained by preventing the optimization process from converging to non-physical solutions, such a negative conductivity or invalid relaxation behaviors.

4. The Influence of Temperature on Frequency Domain Dielectric Spectroscopy Characteristics

4.1. Test Samples and Measurement Procedure

The frequency domain dielectric response of oil-paper insulation was measured using a three-electrode structure. The dimensions of each part of the electrode are shown in Figure 2. Stray capacitance and surface leakage currents, which are especially challenging in low-frequency measurements, where polarization and conduction processes dominate, were reduced by using the three-electrode arrangement. By separating the measurement path from the leakage current paths, the brass guard electrode, positioned between the high-voltage and measurement electrodes, ensured that the measured current precisely matched the sample’s dielectric response. In the modified Cole–Cole model, this configuration greatly improves the accuracy of low-frequency polarization and conduction measurements, which are essential for differentiating between relaxation, DC conductivity, and hopping conduction components. The test sample was positioned between the measuring electrode and the high-voltage electrode and was held in place using a spring. In this study, the insulating pressboard being tested had a thickness of 2 mm and a diameter of 80 mm, and it was impregnated with No. 25 insulating oil. The testing equipment used is the DIRANA dielectric response analyzer, manufactured by OMICRON. This instrument features a peak output voltage of ±200 V, a measurement current of ±50 mA, and a resolution of 10⁻⁵. Prior to testing, the pressboard was dried in an oven at 90 °C for 3 h. It was then vacuum-impregnated in degassed insulating oil for 10 h. After impregnation, the pressboard was placed between the electrodes. The entire electrode setup was submerged in a glass container filled with insulating oil and placed in a temperature- and humidity-controlled chamber for measurement. The frequency range for testing was 1.98 × 10⁻⁴ ∼ 1 × 10³ Hz, and measurements were conducted at temperatures in the range of $30\sim 90°\mathrm{C}$, with readings taken at intervals of $10°\mathrm{C}$ each. Before each measurement, the sample was allowed to stabilize at the target temperature for over three hours to ensure the insulating oil and pressboard reached thermal equilibrium and remained stable at the test temperature.

4.2. Frequency Domain Dielectric Spectrum Curves at Different Temperatures

The complex permittivity of the sample at different temperatures is illustrated in Figure 3. Based on the test results, as the sample temperature rises, the real part of the relative permittivity shows a slight increase in the high-frequency range and a more noticeable rise in the low-frequency range. Meanwhile, the imaginary part of the relative permittivity curve increases with rising temperature, and distinct loss peaks are observed near ${10}^{-2}\mathrm{H}\mathrm{z}$ and ${10}^{-1}\mathrm{H}\mathrm{z}$ in the test results at 70 °C and 90 °C, respectively. The curves of the real and imaginary parts of the relative dielectric constant show a trend of shifting to the high-frequency band with the increase in temperature.

5. The Impact of Temperature on the Parameters of the Modified Cole–Cole Model

5.1. Fitting Results of Modified Cole–Cole Model at Different Temperatures

Based on the proposed method, the frequency domain spectroscopy curves obtained at various temperatures were fitted using the modified Cole–Cole model. Taking the experimental data at 70 °C as an example, the initial values of the parameters were determined using the method described in Section 3, and the results are shown in

Table 2. Initial parameter values and optimization results (70 °C).

Parameter	Before Optimization	After Optimization
${\epsilon }_{\mathrm{H}\mathrm{F}}$	4.7741	4.38
${\chi }_{\mathrm{s}}$	72	73.39
$n_{\alpha }$	0.75	0.70
$\tau$	23.3718	34.21
${\displaystyle \frac{{\sigma }_{dc}}{\left(pS\cdot m^{-1}\right)}}$	3.26	3.28
$\gamma$	0.14	0.071
${\displaystyle \frac{\xi }{\left(pS\cdot m^{-1}\right)}}$	6.0	3.74
$S_{\mathrm{S}\mathrm{E}} \left({\epsilon }^{\prime }\right)$	196	19.71
$S_{\mathrm{S}\mathrm{E}} \left({\epsilon }^{\prime \prime }\right)$	69	11.32
$ObjectiveFunction{Y}_{\mathrm{O}\mathrm{P}\mathrm{T}}$	259	57.35
$R^2 \left({\epsilon }^{\prime }\right)$	0.689	0.998
$R^2 \left({\epsilon }^{\prime \prime }\right)$	0.825	0.976

. According to Section 3, the objective function established undergoes parameter optimization near the initial values. The variation in the objective function during the optimization process is shown in Figure 4. After 1350 iterations, the optimized objective function value decreased from an initial 259 to 57.35. With this set of optimized results, the sum of squared errors (SSE) for fitting the real part ε′ and the imaginary part ε′′ decreased from 196 and 69 before optimization to 19.71 and 11.32 after optimization, respectively.

Additionally, the goodness-of-fit R² for both the real and imaginary parts exceeded 0.97, indicating a highly satisfactory fitting result. Figure 5 presents the fitting results for the real and imaginary parts of the dielectric constant after optimization, along with the contributions of the α-relaxation, DC conductivity, and hopping conductivity processes. It can be observed that the fitting results align well with the experimental data. By following the above method, data at each temperature were fitted using a modified Cole–Cole model with a single relaxation time. The fitting curves and decomposition of the complex relative permittivity for oil-paper insulation at 70 °C show (a) the real part and (b) the imaginary part. The real part ε′ remains relatively stable at high frequencies with a slight increase at low frequencies due to polarization. The imaginary part ε″ shows a distinct loss peak near 0.01 Hz, attributed to α-relaxation, with DC conductivity dominating at lower frequencies and hopping conductivity influencing higher frequencies. The fitting results for each dataset are presented in

Table 3. Frequency domain characteristic parameters at different test temperatures.

$\mathit{\theta }/\mathbf{℃}$	${\mathit{\epsilon }}_{\mathbf{h}\mathbf{f}}$	${\mathit{\chi }}_{\mathbf{s}\mathit{\alpha }}$	${\mathit{n}}_{\mathit{\alpha }}$	${\mathit{\tau }}_{\mathit{\alpha }}/\mathbf{s}$	${\mathit{\sigma }}_{\mathit{d}\mathit{c}}$ $/(\mathbf{p}\mathbf{S}\cdot {\mathbf{m}}^{-1})$	$\mathit{\gamma }$	$\mathit{\xi }$ $/(\mathbf{p}\mathbf{S}\cdot {\mathbf{m}}^{-1})$
30	3.70	60.00	0.72	1500.00	0.24	0.12	3.87
40	4.10	100.71	0.74	629.79	0.39	0.08	3.86
50	3.88	98.96	0.69	263.99	0.83	0.13	3.35
60	4.33	100.45	0.68	99.76	2.10	0.13	3.89
70	4.38	73.39	0.70	34.21	3.28	0.07	3.74
80	4.43	59.38	0.70	7.57	8.03	0.11	4.00
90	4.31	59.72	0.71	3.01	13.44	0.09	3.98

. It can be observed that among the seven parameters, as the test temperature increases, ${\epsilon }_{\mathrm{h}\mathrm{f}}$ shows a slight increase, the relaxation time ${\tau }_{\alpha }$ decreases, the DC conductivity ${\sigma }_{\mathrm{d}\mathrm{c}}$ increases, and the value of ${\lambda }_{\mathrm{s}\alpha }$ fluctuates between 50 and 100. In contrast, the parameters $n_{\alpha },\gamma ,\xi$ do not exhibit significant variations with temperature, maintaining approximate values of 0.7, 0.1, and 3.7 pS/m, respectively, which can be considered nearly constant.

5.2. Variation Patterns of α-Relaxation and DC Conductivity Process

Based on characteristic parameters extracted under different test temperatures, Figure 6 presents the real and imaginary components of the $\alpha$-relaxation process, as well as the components of the DC conductivity process. It can be observed that as the test temperature increases, both the real and imaginary parts of the $\alpha$-relaxation process shift toward higher frequencies, and the relaxation time constant decreases. The frequency of the loss peak gradually increases. Additionally, the DC conductivity component appears as a set of parallel straight lines on a double logarithmic coordinate axis, with the intercept increasing as temperature rises. This can also be interpreted as the DC conductivity process shifting to higher frequencies with increasing temperature. From the relationships between the relaxation time constant ${\tau }_{\alpha }$, DC conductivity ${\sigma }_{\mathrm{d}\mathrm{c}}$, and the reciprocal of absolute temperature 1/T, shown in Figure 7, it can be observed that both parameters satisfy the Arrhenius equation. The relationship between the α-relaxation time constant τ_α and DC conductivity σ_dc with respect to the inverse of absolute temperature (1/T) is shown. The linear trends confirm that both parameters follow the Arrhenius equation, indicating thermally activated behavior. This supports the model’s ability to describe the temperature-dependent dielectric mechanisms in oil-paper insulation.

$${\displaystyle \frac{1}{{\tau }_{\alpha }}}=A_{\tau }\exp \left(-{\displaystyle \frac{E_{\tau }}{RT}}\right)$$

(12)

$$\sigma =A_{\sigma }\exp \left(-{\displaystyle \frac{E_{\sigma }}{RT}}\right)$$

(13)

where $A_{\tau }$ and $A_{\sigma }$ are the pre-exponential functions of the Arrhenius equation, respectively; $E_{\tau }$ and $E_{\sigma }$ are the activation energy; $R$ is the Boltzmann constant; and $T$ is the absolute temperature. The parameters fitted according to Equations (12) and (13) are shown in

Table 4. Fitting parameters of Arrhenius equation for relaxation time constant and DC conductivity.

Parameter	$\mathit{A}$	$\mathit{E}/\mathbf{e}\mathbf{V}$	${\mathit{R}}^2$
${\tau }_{\alpha }$	$6.08\times {10}^{-14}$	0.996	0.98
${\sigma }_{\mathrm{D}\mathrm{C}}$	$1.76\times {10}^{-2}$	0.659	0.99

. The Arrhenius Equations (12) and (13) describe the effects of temperature on relaxation time and DC conductivity in oil-paper insulation. Increased thermal energy assists in dipole reorientation, reducing relaxation time from 1500 s at 30 °C to 3.01 s at 90 °C. Due to improved charge mobility, ionic migration raises DC conductivity from 0.24 to 13.44 pS/m. Byproducts such acids and moisture from aging lower activation energies, affecting pre-exponential factors and activation energies 0.996 eV for τ_α and 0.659 eV for σ_dc.

The existing literature mentions that the frequency domain dielectric spectrum characteristics of materials at different temperatures satisfy the time–temperature superposition principle [33,34].

The test data at a higher temperature can be translated to the reference temperature curve to form a master curve. The translation factor of the master curve, formed by the α_T by the data at each temperature, satisfies the Arrhenius equation with respect to temperature.

5.3. Comparative Analysis of Dielectric Models

A comparative study was carried out to validate the performance and uniqueness of the proposed modified Cole–Cole model against two established models from the literature: the fundamental Cole–Cole model and Ekanayake’s model [19]. The identical experimental dielectric spectroscopy data from oil-paper insulation at 70 °C were used to test all of the models. The fundamental Cole–Cole model is limited in its ability to reflect conduction loss components and only captures the α-relaxation effect. Along with the α-relaxation, Ekanayake’s model includes a fractional conductivity term, Aω⁻ⁿ, to provide better fitting in the low-frequency range. To better represent physical processes like ionic migration and dipole polarizations under temperature stress, the improved Cole–Cole model put forward in this work includes both DC conductivity and hopping conductivity in addition to a dual relaxation structure. The three models’ quantitative evaluation, shown in

Table 5. Comparison of Fitting Performance for Different Dielectric Models.

Model	Component Included	SSE (ε′)	SSE (ε′′)	R2 (ε′)	R2 (ε′′)
Fundamental Cole–Cole	α-relaxation only	87.64	41.27	0.902	0.897
Ekanayake’s Model	α-relaxation + fractional conductivity (Aω−n term)	48.11	27.05	0.944	0.929
Modified Cole–Cole	α-relaxation + DC conductivity + hopping conductivity	19.71	11.32	0.998	0.976

, presents the goodness-of-fit R2 and sum of squared errors for the real ε′ and imaginary ε′′ parts of the complex permittivity.

The results clearly demonstrate that the modified Cole–Cole model provides significantly better fitting performance for both ε′ and ε′′ with much lower error and higher correlation to the experimental data.

6. Future Prospective

Expanding the use of the modified Cole–Cole model in large-scale real-time transformer monitoring systems will be the main goal of future research. In order to validate the prediction of the model capacity across a variety of transformer types and operating situations, it will be incorporated into embedded firmware platforms for online condition monitoring. To enhance the precision and thoroughness of transformer health assessments, this model can be used with supplementary diagnostic techniques such as polarization–depolarization current measurements, furfural content analysis, and dissolved gas analysis. These advances are intended to help digital twin frameworks and predictive maintenance techniques in smart grid settings.

7. Conclusions

This paper introduces the modified Cole–Cole model to quantitatively study the frequency domain dielectric spectrum characteristics of oil-paper insulation, proposes a method for extracting model parameters, provides a method for determining the initial values of parameters and the optimization objective function, and studies the variation law of each parameter of the modified Cole–Cole model at different temperatures. The main conclusions are as follows.

The modified Cole–Cole model, incorporating a single relaxation time constant, effectively describes the frequency domain dielectric properties of oil-paper insulation within the frequency range of 1.98 × 10⁻⁴ Hz to 1 × 10³ Hz, achieving a goodness-of-fit R² exceeding 0.97 for both real and imaginary permittivity components. The proposed method for determining initial parameter values and optimizing the objective function enables accurate extraction of characteristic parameters, with the sum of squared errors reducing from 259 to 57.35 after optimization at 70 °C. The model decomposes the dielectric spectrum into relaxation, DC conductivity, and hopping conductivity processes. Temperature significantly influences the relaxation and DC conductivity components, with the relaxation time constant τ_α decreasing from 23.37 s to 0.34 s and DC conductivity σ_dc increasing from 3.28 pS/m to 128.5 pS/m as the temperature rises from 30 °C to 90 °C. These parameters follow the Arrhenius equation, with activation energies of 0.996 eV for τ_α and 0.659 eV for σ_dc. The hopping conductivity component remains stable, with ξ ≈ 3.7 pS/m and γ ≈ 0.1. Future work will explore the influence of moisture and aging on these parameters to develop a comprehensive database for insulation condition assessment.

The modified Cole–Cole model considers the frequency domain dielectric spectrum as a superposition of three processes: relaxation, DC conductivity, and hopping conductivity. Temperature primarily affects the relaxation and DC conductivity components of the dielectric properties of oil-paper insulation, while the hopping conductivity component remains unchanged with temperature variations. Both the relaxation and DC conductivity components shift toward higher frequencies as the temperature increases. The corresponding time constant $\tau$ and DC conductivity ${\sigma }_{\mathrm{d}\mathrm{c}}$ follow the Arrhenius equation with temperature changes. This study focuses on extracting the characteristic parameters of the modified Cole–Cole model for oil-paper insulation frequency domain spectroscopy (FDS) at different temperatures. Future work will investigate the influence of moisture content and aging on these characteristic parameters. The extracted characteristic parameters under different insulation states can serve as fingerprint features for the dielectric response of oil-paper insulation. These parameters can be compiled into a database and applied to assess the condition of oil-paper insulation in transformers. This approach provides a foundation for evaluating insulation health and diagnosing potential issues in power transformers.