Classification of Multiple Partial Discharge Sources Using Time-Frequency Analysis and Deep Learning

Abstract

Partial discharge (PD) analysis is critical for diagnosing insulation degradation in high-voltage equipment. While conventional methods struggle with multi-source PD classification due to signal overlap and noise, this study proposes a hybrid approach combining five time–frequency analysis (TFA) techniques with deep learning (GoogLeNet for simulation, ResNet50 for experiments). PD data are generated through Finite Element Method (FEM) simulations and validated via laboratory experiments. The Scatter Wavelet Transform (SWT) achieves 96.67% accuracy (F1-score: 0.967) in simulation and perfect 100% accuracy (F1-score: 1.000) in experiments, outperforming other TFAs like HHT (70.00% experimental accuracy). The Wigner–Ville Distribution (WVD) also shows strong experimental performance (94.74% accuracy, AUC: 0.947), though its computational complexity limits real-time use. These results demonstrate the SWT’s superiority in handling real-world noise and multi-source PD signals, providing a robust framework for insulation diagnostics in power systems.

1. Introduction

PD is a localized electrical discharge that occurs within the insulator between conductors without causing full insulation failure. It typically arises in specific areas, often correlated with defects or voids. PD is identified by its appearance at weak insulation spots, without forming a complete bridge across the electrodes. It presents as transient pulses that last from nanoseconds to microseconds. Although less intense than full breakages, regular PD can lead to insulation degradation. PD generates electromagnetic, ultrasonic, and occasionally optical signals, enabling detection through various sensing methods. Initiation conditions for PD include the concentration of the local electric field due to voids, cavities, or contaminants. When this field exceeds the breakdown threshold of the medium (like air in a void), ionization occurs. Subsequently, charges accumulate on the cavity surfaces, modifying the electric field and impacting subsequent PD events. This discharge cycle often repeats as the system alternates between ionization, charge redistribution, and stabilization phases. The research addresses the complexities involved in diagnosing partial discharge (PD) in insulation systems with multiple voids, which complicates PD source classification. Traditional diagnostic techniques, which focus on single voids, encounter challenges with signal overlap from multiple sources, impacting fault identification in time and frequency domains. Conventional methods, based on rule-based systems, become unwieldy and struggle to handle complex, overlapping patterns arising from numerous PD sources. Advanced strategies using state-of-the-art signal processing and deep learning (DL) techniques are proposed to address these challenges. Although Phase-Resolved Partial Discharge (PRPD) analysis is effective for diagnosing specific defects by plotting PD events relative to the AC voltage phase angle, it struggles with interpretation in the presence of multiple PD sources, as overlapping signals obscure individual defect characteristics. This highlights the need for improved methods to accurately distinguish between different PD sources and patterns.

1.1. Challenges in Multiple PD Source Classification

The identification of PD via AI encounters several significant challenges: (1) extracting pertinent features from PD datasets; (2) implementing robust pattern recognition techniques for PD identification; and (3) distinguishing simultaneous sources of PD in HV systems [1]. Accurate diagnostics are based on the clear identification of multiple sources of PD in intricate systems [2,3]. Real-world systems frequently exhibit multiple concurrent defects, producing overlapping PD signals and complicating source discernment [3,4,5]. The analysis of insulation with multiple PD sources is challenging due to overlaps of time, frequency, and phase signals, which obscure key diagnostic features. The variability in PRPD patterns further complicates the identification of phase-specific discharges. Although PRPD is often used for phase alignment with AC voltages [6,7], dealing with signal overlaps requires advanced methodologies. Such overlaps generate complex patterns that complicate differentiation via PRPD and PRPS techniques, leading to ambiguous phase data that hinder PD categorization [2,8,9,10]. Thus, utilities are in pursuit of automated approaches to distinguish and ascertain PDs from distinct sources using pulse waveforms [11].

1.2. Literature Review

This literature review examines the latest advances and innovative techniques in PD classification and detection, emphasizing DL and signal processing methodologies. Fu et al. [12] applied t-SNE to extract robust features across sensors, addressing amplitude–frequency variability. Zhang et al. [13] combined adaptive kernel TF representations with a ResNet-18 network, achieving 98% accuracy under signal attenuation conditions. Rauscher et al. [14] compared 13 deep learning architectures, concluding that 2D CNNs with continuous wavelet transform (CWT) inputs outperform FFT and STFT, especially in low-SNR scenarios. Similarly, Chen et al. [15] used transformer encoders to learn implicit TF features, attaining 98.6% accuracy without explicit transforms. Li et al. [16] introduced a CNN-RNN hybrid (TFMT) model with probabilistic TF maps, achieving 99.49% accuracy for cable systems. Rui et al. [17] enhanced TF resolution with a Wavelet Time-Reassigned Synchrosqueezed Transform (WTST) coupled with LSTM-FCN for noisy acoustic environments. Zheng et al. [18] proposed an IFCNN with 1D and 2D CNN channels to fuse time-domain and wavelet-based features, reaching 95.8% accuracy across four PD types. Chen et al. [19] developed a transformer-based model leveraging a Time-Reassigned Multi-Synchrosqueezing Transform (TMSST) and prior localization, achieving 98.7% mean average precision (mAP). Sahoo et al. [20] demonstrated 100% classification using CNNs with CWT scalograms, outperforming RF classifiers. Their work also emphasized hyperparameter tuning (ELU, Adam optimizer with lr = 0.001) for optimal performance. Iorkyase et al. [21] achieved 1.9 m mean localization error using a Wavelet Packet Transform (WPT) with ensemble ML models in noisy environments. For pattern recognition, Saad et al. [22] used STFT scalograms with CNNs for XLPE cable PDs, reaching 97.44% accuracy and fast inference suitable for real-time use. Klein et al. developed ensemble stacking neural networks, significantly enhancing PD detection precision in covered conductors [23]. Similarly, Karimi et al. utilized DBN to identify and differentiate corona, surface, and internal discharges [24]. Das et al. introduced a novel DL-based framework capable of precisely identifying and locating either single or multiple PD events [25]. Florkowski demonstrated deep CNN efficacy in classifying PD images [26]. Gao et al. employed VMD and CWD spectra to improve PD recognition by optimizing CNN models [27]. Zhou et al. implemented a hybrid CNN-RNN model to effectively classify UHF PD patterns in transformers, outperforming conventional techniques [28]. Xu et al. introduced a Bi-LSTM model with an attention mechanism to classify PD signals in insulated overhead conductors, showing enhanced accuracy and lower computational complexity [29]. Dong et al. utilized LSTM networks to detect PD in aerial covered conductors, demonstrating the efficacy of time-series decomposition for signal analysis [30]. Li et al. developed CNN-LSTM models incorporating attention mechanisms to improve detection precision [31]. Nguyen et al. showed that LSTM networks outperformed traditional methods like SVMs in classifying PRPD signals in GIS [32]. Zhou et al. [28] investigated CNN-LSTM models for recognizing UHF PD patterns in transformers, whereas Gao et al. [33] highlighted multiresolution CNN architectures for detailed PD classification. Jung et al. [34] and Xi et al. [35] demonstrated that attention mechanisms can enhance classification performance while lowering computational demands.

1.3. Effective Proposed Approaches for Classifying Multiple PD Sources

This study delves into the topics of classifying partial discharge (PD) signals in insulation systems, especially when multiple sources are present. Signal complexity, inconsistency in PD characteristics, and external noise exacerbate these difficulties. The research seeks to enhance PD classification using sophisticated signal processing and deep learning (DL) methods. Unique PD signatures from each insulation defect necessitate precise classification for maintenance planning. Conventional techniques struggle when PD signals overlap, but DL techniques such as CNN and LSTM show promise in extracting and analyzing PD signal features in time–frequency domains. Advanced signal processing, like time–frequency analysis (TFA), allows for dual analysis of temporal and spectral signal aspects, thus improving PD source identification. The study concludes that advanced techniques are superior to traditional methods in providing accurate diagnostics and maintenance for complex electrical systems. This research focuses on classifying PD sources using state-of-the-art signal processing and DL methods for better identification accuracy. A key contribution is combining DL and TFA, where TFA alongside CNN significantly enhances PD source classification accuracy, as depicted in Figure 1.

Figure 1. Proposed approach.

Figure 1. Proposed approach.

1.3.1. Database

The database comprises data derived from two sources: simulations executed through the Finite Element Method (FEM) and a series of laboratory experiments conducted on various insulation samples. These processes are depicted in detail in Figure 2.

Figure 2. Database created for each class.

Figure 2. Database created for each class.

1.3.2. Time–Frequency Features

Time–frequency techniques derive features from PD signals to capture both temporal and spectral characteristics, addressing their non-stationary nature.

1.3.3. Deep Learning

A CNN is developed to classify PD sources based on time–frequency data and proves highly effective in identifying multiple PD sources, even amidst overlapping signals.

1.4. Motivation for the Proposed Approach

PD is a crucial indicator of insulation issues in HV systems, emphasizing the need for early detection. It may originate from multiple sources [36]. Identifying a single PD source is simple in controlled settings, but multiple sources in industrial environments complicate the process. Dominant sources can mask weaker ones, which, despite low amplitude, may still compromise insulation integrity [37]. In power transformers, voids in solid insulation, such as those in the windings, pose a greater risk than oil bubbles, even with lower PD levels [8]. Categorizing PD sources in HV equipment is highly complex [1]. Thus, it is crucial to identify and automate risk assessments for multiple PD sources. Professor Tanaka highlights the challenge of distinguishing between single and multiple voids, indicating the need for more research [38]. Furthermore, ref. [39] stresses the necessity for automated assessments of numerous PD sources. Identifying PD sources is challenging due to complex patterns that require expert knowledge [5,40]. Some PD signals are so similar that even experts struggle to characterize them [41]. Consequently, pattern recognition methods currently offer limited practical reliability [42]. The ongoing research aims to improve the recognition of PD patterns and improve the classification accuracy of multiple sources of PD [1].

2. FEM in Partial Discharge Simulation

Precise identification of PD sources is based heavily on datasets labeled with specific source details [5]. FEM simulations play a key role in creating such datasets. Modeling PD phenomena in HV insulation involves complex interactions between defects, varied material attributes, and external factors. Defect interactions significantly impact PD dynamics, as they can enhance nearby electric fields, triggering or escalating PD. These nonlinear interactions complicate the prediction of cumulative PD activity, making it more than a straightforward aggregation. Insulation factors such as permittivity, conductivity, and dielectric strength, in addition to external factors such as temperature, humidity, and mechanical stress, modulate the behavior of PD. The FEM is a sophisticated tool for modeling intricate physical systems, enabling a detailed simulation of electric fields in HV insulation by considering multiple PD sources. It elucidates complex internal phenomena such as electric field patterns, defect interactions, and stress distribution, surpassing traditional experimental capabilities. The FEM also captures irregular occurrences like extreme fields and unique defects that are challenging to investigate experimentally. The integration of the FEM with stochastic models offers a comprehensive methodological framework.

2.1. Simulation Framework

The FEM-based simulation framework involves several key steps to ensure accurate and reliable modeling of PD behavior. These steps include geometry definition, material property assignment, boundary condition specification, and solver configuration. Each step is tailored to capture the intricate dynamics of PD processes in the presence of multiple defects.

2.1.1. Geometry and Model Setup

The configuration and geometry were meticulously selected to represent practical insulation systems typical in high-voltage contexts. Crucial dimensions and structural characteristics are summarized in

Table 1. Component dimensions.

Component	Dimensions
Insu. system	Height: $1.5\mathrm{mm}$, width: $15\mathrm{mm}$
Cavity	Height: $0.5\mathrm{mm}$, width: $5\mathrm{mm}$, diameter: $1.2\mathrm{mm}$ (double the radius of $0.6\mathrm{mm}$)
Pos. and neg. electrode	Positioned at $\pm 1.25\mathrm{mm}$ from the center, dimensions: $7\mathrm{mm}\times 1\mathrm{mm}$

.

2.1.2. Material Characteristics

Accurate identification of material characteristics is critical for achieving realistic simulations in peridynamics. Refer to

Table 2. Material properties.

Material	Properties
Insulation (polycarbonate)	Relative permittivity: 3, electrical conductivity: ${10}^{-15}\mathrm{S}/\mathrm{m}$
Air cavity	Relative permittivity: 1, conductivity: $0\mathrm{S}/\mathrm{m}$ (changes during PD events)
Electrodes (copper)	Conductivity: $5.8\times {10}^7\mathrm{S}/\mathrm{m}$ (negligible voltage drop)

for key materials and their properties.

2.1.3. Application of Physics Interfaces

I.

Electrostatic and Space Charge Modeling

FEM simulations involve solving the governing equations of electrostatics and electrodynamics to capture the intricate behavior of PD phenomena. These equations include the following:

(a)

Poisson’s Equation

This governs the electric potential (V) and electric field ($\mathbf{E}$) distributions:

$${\nabla }^{2}V=-{\displaystyle \frac{\rho }{ϵ}}$$

(1)

where $\rho$ is the charge density and $ϵ$ is the permittivity.

(b)

Charge Transport Equations

These model the movement of charge carriers within the cavity:

$${\displaystyle \frac{\partial \rho }{\partial t}}=D{\nabla }^2\rho -R+S$$

(2)

where D is the diffusion coefficient, R is the recombination rate, and S represents charge generation during PD events.

These equations are solved iteratively using adaptive time-stepping to capture the rapid changes associated with PD events while maintaining numerical stability.

II.

Stochastic Modeling

Stochastic models describe the probabilistic behavior of PD phenomena by incorporating random factors, such as electron availability, material defects, field fluctuations, and environmental conditions. Unlike deterministic models, stochastic methods emphasize the statistical distributions and variations in PD initiation and progression, offering insights into the intricate behavior of insulation systems.

Figure 3 presents the deterministic PD model characteristics, highlighting a precise and consistent relationship between the applied AC voltage and cavity discharge events, including inception and extinction voltages. Discharge timings and magnitudes, such as time delays ($T_s$), are predictable and follow a set pattern within the AC cycle under ideal conditions. In contrast, Figure 4 showcases the stochastic PD model characteristics, which lack strict periodic behavior. Here, discharge timings and magnitudes vary across cycles due to stochastic influences like electron availability and material imperfections, introducing variability in inception/extinction voltages and discharge magnitudes. This model accounts for real-world randomness in PD behavior.

The stochastic nature of partial discharge is driven by material defects, which introduce randomness through microscopic voids and impurities; electron availability, where random ionization events produce free electrons; and external noise, with unpredictable environmental conditions like temperature and humidity affecting PD behavior. Combining stochastic modeling with FEM enhances PD risk assessment, aiding in strategic decisions about insulation system reliability. PD events occur randomly, initiated under probabilistic conditions, mainly due to the presence of free electrons, which are essential for PD inception. These electrons may originate from cosmic rays, natural radioactivity [43], or previous PD incidents, making their occurrence unpredictable despite constant voltage. The stochastic nature of PD activity can be modeled using probability distributions and statistical methods to investigate its intrinsic uncertainty.

Figure 3. Deterministic PD model.

Figure 3. Deterministic PD model.

Figure 4. Stochastic PD model.

Figure 4. Stochastic PD model.

(a)

Exponential Model

The exponential function characterizing the number of free electrons, $N_e\left(t\right)$, is determined by the cavity voltage $V_{\mathrm{cav}}\left(t\right)$ relative to the critical voltage $V_{\mathrm{crit}}$. It highlights the rapid increase in free electrons as the voltage approaches $V_{\mathrm{crit}}$, according to [44]:

$$N_e\left(t\right)=N_{e0}exp\left(\left|{\displaystyle \frac{V_{\mathrm{cav}}\left(t\right)}{V_{\mathrm{crit}}}}\right|\right)$$

(3)

where $N_e\left(t\right)$ is the number of free electrons at time t and $N_{e0}$ represents the initial number of free electrons.

(b)

CDF for PD Event

The probability distribution of a PD event over time t is captured by a CDF, which accumulates free electron counts over a given time. This method yields a more accurate probability estimation for PD events:

$$F\left(t\right)=1-exp\left(-{\int }_0^t{N}_e\left(t^{\prime }\right)d{t}^{\prime }\right)$$

(4)

(c)

Random Number Comparison

The inclusion of a uniform random variable R within the interval [0, 1] allows for a stochastic comparison with the CDF $F\left(t\right)$, enhancing the probabilistic modeling of PD events. The occurrence of a PD event is determined by:

$$F\left(t\right)=\left\{\begin{array}{c}\mathrm{No}\mathrm{PD}\mathrm{event},\mathrm{if}F\left(t\right)<R\hfill \\ \mathrm{PD}\mathrm{event},\mathrm{if}F\left(t\right)>R\hfill \end{array}\right.$$

2.1.4. Electrical Boundary Conditions

The boundary conditions, depicted in

Table 3. Operational parameters.

Parameter	Description
Applied voltage	AC 7–11 kV, frequency: $50\mathrm{Hz}$.
Ground	Negative electrode grounded at $0\mathrm{V}$.

, were utilized to simulate the electric field of the system.

2.1.5. PD Simulation Parameters

The simulation’s primary parameters are detailed in

Table 4. Simulation parameters.

Parameter	Description
Inception voltage $\left(V_{\mathrm{inc}}\right)$	$V_{\mathrm{inc}}=24.41·\left({\rho }_0·d_{\mathrm{cavity}}\right)+6.73·\sqrt{{\rho }_0·d_{\mathrm{cavity}}}$.
PD current	Integrated the current density over ground boundaries.
Surface charge density	Integrated over cavity boundaries.
Space charge density	Computed within the cavity domain.
Time steps	With 1 ns during PD events and 1 μs otherwise.

.

2.1.6. Mesh and Solver Settings

• Mesh: A finer mesh was applied near the cavity and electrode boundaries for accurate field resolution.
• Solver: Transient study with adaptive time stepping to resolve rapid changes during PD events.

The finite element analysis reveals critical differences in space charge distributions for varying numbers of PD sources. As shown in Figure 5, Figure 6 and Figure 7, the charge accumulation patterns evolve significantly with increasing source complexity.

Figure 5. Space charge distribution for single PD source (SPS) configuration.

Figure 5. Space charge distribution for single PD source (SPS) configuration.

Figure 6. Space charge distribution for double PD source (DPS) configuration.

Figure 6. Space charge distribution for double PD source (DPS) configuration.

Figure 7. Space charge distribution for triple PD source (TPS) configuration demonstrating complex field modulation.

Figure 7. Space charge distribution for triple PD source (TPS) configuration demonstrating complex field modulation.

2.2. Simulation Scenarios

Despite significant progress, many elements of cavity impacts remain underexplored, especially concerning the collective influence of multiple cavities on PD dynamics. This study examines various parameters to evaluate how cavity size, shape, position, and quantity affect PD behavior, with the goal of enhancing cavity property characterization for precise PD modeling and classification.

2.2.1. Simulation Scenario I: Single Cavity with Varying Positions

This section analyzes three simulation scenarios: Near-Positive (cavities close to the high-voltage electrode), Intermediate (cavities centrally positioned), and Near-Negative (cavities near the ground electrode). Each scenario influences the electric field distribution, PD inception voltage, and PD charge patterns. Near-Positive cavities magnify local electric fields, lower PDIV, and facilitate air breakdown, as illustrated in Figure 8, and discharge patterns in Figure 9 and Figure 10, resulting in higher PD charges due to increased potential difference, crucial for insulation reliability. Intermediate cavities, as shown in Figure 11, experience a balanced field, raising PDIV compared to Near-Positive cavities, as shown in Figure 12, with moderate PD charges Figure 13, indicating balanced stress; however, they remain susceptible to PD under AC due to varying field polarities. Near-Negative cavities face the weakest field Figure 14, increasing PDIV Figure 15 and lowering PD charges Figure 16, signifying fewer PD events but potential risk under insulation aging or defects.

Figure 8. Electric field in Near-Positive cavities.

Figure 8. Electric field in Near-Positive cavities.

Figure 9. PD voltage in Near-Positive cavities.

Figure 9. PD voltage in Near-Positive cavities.

Figure 10. PD charge in Near-Positive cavities.

Figure 10. PD charge in Near-Positive cavities.

Figure 11. Electric field in Intermediate cavities.

Figure 11. Electric field in Intermediate cavities.

Figure 12. PD voltage in Intermediate cavities.

Figure 12. PD voltage in Intermediate cavities.

Figure 13. PD charge in Intermediate cavities.

Figure 13. PD charge in Intermediate cavities.

Figure 14. Electric field in Near-Negative cavities.

Figure 14. Electric field in Near-Negative cavities.

Figure 15. PD voltage in Near-Negative cavities.

Figure 15. PD voltage in Near-Negative cavities.

Figure 16. PD charge in Near-Negative cavities.

Figure 16. PD charge in Near-Negative cavities.

2.2.2. Simulation Scenario II: Impact of Varying Cavity Shapes, Alignments, and Stacking

This study investigates the influence of cavity configurations, specifically shape and orientation, on electric field distribution, partial discharge (PD) inception voltage, and PD charge. The simulations focus on three scenarios: (1) varied cavity shapes, (2) horizontal alignments, and (3) vertical stackings, each with unique field and discharge properties.

For varied shapes, differing geometries create asymmetric field distributions Figure 17, causing localized intensification and uncertain discharge initiation. Figure 18 illustrates fluctuating PD inception voltage driven by the dominant cavity’s field line alignment. As shown in Figure 19, PD charge is stochastic, reflecting irregular discharge paths and breakdown thresholds, mirroring real insulation defects.

In horizontal alignment Figure 20, capacitive coupling affects field interactions. The PD inception voltage Figure 21 rises due to mutual distortions and reduced peak intensities. Charge levels Figure 22 increase. Vertical alignment Figure 23 results in cumulative field effects from stacked cavities, with enhanced field concentrations indicated by steep gradients. This yields a lower PD inception voltage Figure 24 than horizontal setups. PD charge Figure 25 escalates significantly due to cascading breakdowns, where discharges in the upper cavity trigger those in the lower, modeling potential failures in laminated systems.

Figure 17. Electric field for various shapes.

Figure 17. Electric field for various shapes.

Figure 18. PD voltage for various shapes.

Figure 18. PD voltage for various shapes.

Figure 19. PD charge for various shapes.

Figure 19. PD charge for various shapes.

Figure 20. Electric field for horizontal alignment.

Figure 20. Electric field for horizontal alignment.

Figure 21. PD voltage for horizontal alignment.

Figure 21. PD voltage for horizontal alignment.

Figure 22. PD charge for horizontal alignment.

Figure 22. PD charge for horizontal alignment.

Figure 23. Electric field for vertical alignment.

Figure 23. Electric field for vertical alignment.

Figure 24. PD voltage for vertical alignment.

Figure 24. PD voltage for vertical alignment.

Figure 25. PD charge for vertical alignment.

Figure 25. PD charge for vertical alignment.

2.2.3. Simulation Scenario III: Impact of Size, Shape, and Multiple PD Sources on Rate of Occurrence

The study analysis explores three unique cavity configurations to assess the impact of multiple PD sources, cavity alignment, and geometric diversity on PD frequency and intensity. For varying PD sources, the scenario examines three differing cavities in size, shape, and location within insulation. The electric field Figure 26 is highly irregular, with field enhancement at sharp or asymmetric voids. Figure 27 and Figure 28 illustrate fluctuating PD inception voltages and diverse charge magnitudes, indicative of stochastic PD behavior, simulating actual insulation systems with irregular inhomogeneities and defects, leading to unpredictable discharges and localized dielectric failures. Variability in field strength across each cavity heightens simultaneous or sequential PD events.

In the horizontal alignment of three cavities Figure 29, the cavities are aligned perpendicular to the electric field. Inception voltage Figure 30 shows moderate values with diminished local peaks, while PD charge magnitudes Figure 31 are high. However, multiple discharge sources may overlap intermittently under high-voltage AC, resulting in lower localized stress but a higher total discharge rate.

For vertical alignment of three cavities Figure 32, cavities are vertically stacked along field lines, causing significant field superposition and raising local field stress on successive cavities. As seen in Figure 33, PD inception voltage decreases due to field intensification, and charge magnitudes Figure 34 are higher, with sequential discharges likely. This vertical configuration poses a high insulation failure risk due to potential avalanche-like PD activity, especially during voltage transients.

This section analyzes the impact of varying sizes, shapes, and multiple PD sources on the rate of occurrence of partial discharge. Below are the figures related to each scenario:

Figure 26. Electric field for varying PD sources.

Figure 26. Electric field for varying PD sources.

Figure 27. PD voltage for varying PD sources.

Figure 27. PD voltage for varying PD sources.

Figure 28. PD charge for varying PD sources.

Figure 28. PD charge for varying PD sources.

Figure 29. Electric field for horizontal alignment with three cavities.

Figure 29. Electric field for horizontal alignment with three cavities.

Figure 30. PD voltage for horizontal alignment with three cavities.

Figure 30. PD voltage for horizontal alignment with three cavities.

Figure 31. PD charge for horizontal alignment with three cavities.

Figure 31. PD charge for horizontal alignment with three cavities.

Figure 32. Electric field for vertical alignment with three cavities.

Figure 32. Electric field for vertical alignment with three cavities.

Figure 33. PD voltage for vertical alignment with three cavities.

Figure 33. PD voltage for vertical alignment with three cavities.

Figure 34. PD charge for vertical alignment with three cavities.

Figure 34. PD charge for vertical alignment with three cavities.

3. Experimental Measurements

In this study, we use a sensing impedance approach to perform PD measurement. PD data were collected using sensing impedance sensors. The signals were sampled at a frequency of 50 MHz to capture the transient nature of the PD events. This technique involves the incorporation of a sensing impedance component whose value varies with frequency and is placed within the electrical circuit. This setup is designed to capture the occurrence of PD by detecting characteristic high-frequency pulses associated with it. The high-voltage circuit for PD measurement is shown in Figure 35 The high-voltage test transformer is powered by a sinusoidal voltage at industrial frequency from a power amplifier, which is driven by a function generator. A 1 nC coupling capacitor is placed in parallel with the measured object. The voltage and current waveform of partial discharges are measured on an AKV 9310 wide band quadripole system and automatically recorded using a fast digital scope with 16-bit vertical resolution. The electrode system consists of two cylindrical electrodes with diameters of 2.5 cm and 5 cm. The test sample, along with the electrode system, is immersed in mineral oil to suppress surface discharge. The test samples were created using Masked Stereolithography (MSLA) technology. Three types of samples were produced: solid (without internal cavities), with a single internal cavity, and with two internal cavities. The cavities were designed as spheres with a diameter of 2 mm, which were located in the middle of total a sample thickness of 3 mm. The samples were made of UV-sensitive epoxy resin. An example of an experimental sample with dimensions for two cavities is shown in Figure 36.

The wavelet denoising technique is used to reduce noise while keeping signal features. A wavelet transform isolated the signal through coefficient thresholding, distinguishing it from noise. This wavelet-based method significantly reduced the noise.

$$x_{denoised}\left(t\right)=\sum _{j=1}^J\sum _k{c}_{j,k}{\psi }_{j,k}\left(t\right)+\sum _k{d}_{J,k}{\varphi }_{J,k}\left(t\right)$$

(5)

where the following are true:

• ${\psi }_{j,k}$: wavelet basis functions (sym4);
• $c_{j,k}$: detail coefficients;
• $d_{J,k}$: approximation coefficients;
• $J=5$: decomposition level.

Figure 35. High-voltage experimental circuit for PD measurement.

Figure 35. High-voltage experimental circuit for PD measurement.

Figure 36. Example of test sample for two cavities.

Figure 36. Example of test sample for two cavities.

Dividing PD signals into smaller segments enables the model to concentrate on distinct events:

• Fixed-Window Segmentation: Partition the signal into consistent, equally-sized segments.
• Event-Based Segmentation: Subdivide the signal according to identified PD events. The energy-based detection identifies pulses using:

  $$E\left[n\right]=\sum _{k=n-W}^n{x}^2\left[k\right],W=\lfloor F_s·0.5\mathsf{\mu }\mathrm{s}\rfloor$$

  (6)

The Signal Preprocessing algorithm follows these steps:

Examples of the observed waveforms within one period after preprocessing using Algorithm 1 algorithm shown in Figure 37 for a single cavity sample and in Figure 38 for a two-cavity sample.

Algorithm 1 PD Signal Preprocessing

1: Input: Raw signal $x\left(t\right)$, sampling frequency $F_s$ 2: Apply wavelet denoising: $x_{clean}\left(t\right)=\mathrm{wdenoise}(x\left(t\right),\mathrm{wavelet}=‘\mathrm{sym}4’)$ 3: Compute moving energy: $E\left[n\right]={\sum }_{k=n-W}^{n+W}{x}_{clean}^2\left[k\right]$ 4: Determine threshold: $\tau =\mathrm{median}\left(E\right)+3{\sigma }_E$ 5: Identify PD region: $t_{pd}=\left\{t\right|E\left(t\right)>\tau \}$ 6: Extract segment: $x_{pd}\left(t\right)=x_{clean}(t_{pd}-\Delta t,t_{pd}+\Delta t)$ 7: Normalize: $x_{norm}\left(t\right)={\displaystyle \frac{x_{pd}\left(t\right)-{\mu }_{pd}}{max|x_{pd}\left(t\right)|}}$

Figure 37. PD signal of single cavity defect: (top) raw applied voltage (gray) and PD response (blue), (middle) wavelet-denoised signal, and (bottom) normalized 2 μs pulse segment.

Figure 37. PD signal of single cavity defect: (top) raw applied voltage (gray) and PD response (blue), (middle) wavelet-denoised signal, and (bottom) normalized 2 μs pulse segment.

Figure 38. PD signal of dual cavity defects: (top) raw excitation and PD response (red), (middle) denoised waveform, and (bottom) energy-threshold extracted pulse (2 μs detection window).

Figure 38. PD signal of dual cavity defects: (top) raw excitation and PD response (red), (middle) denoised waveform, and (bottom) energy-threshold extracted pulse (2 μs detection window).

4. Time–Frequency Analysis for Feature Extraction

Time–frequency analysis (TFA) techniques enable the simultaneous representation of PD signals in the time and frequency domains, which is crucial to capture the transient nature of PD signals and differentiate between multiple PD sources. TFA represents one of the most powerful tools in modern signal processing, bridging the gap between time-domain and frequency-domain representations. TFA methods are vital for analyzing transient and non-stationary characteristics by integrating temporal and spectral dimensions. A time–frequency representation (TFR) aims to show how the frequency content of a signal evolves over time, shown in

Table 5. Time–frequency representations.

, transformed into 2D images and input into a CNN for automatic feature extraction and classification. This study uses key TFA methods to explore PD signals, each of which provides unique advantages in identifying traits of time and frequency.

4.1. Short-Time Fourier Transform (STFT)

STFT is a prevalent technique in TFA to evaluate non-stationary signals. It partitions these signals into brief segments that overlap for the Fourier transformation, producing a spectrum varying over time. For a continuous-time signal $x\left(t\right)$, the STFT involves multiplying the signal by a window function $w\left(t\right)$ followed by the Fourier transform:

$$X(t,f)={\int }_{-\infty }^{\infty }x\left(\tau \right)w(\tau -t)e^{-j2\pi f\tau }d\tau$$

(7)

where $w(\tau -t)$ isolates a segment of the signal around time t and $X(t,f)$ represents the frequency content of the signal at time t.

4.2. Wavelet Transform (WT)

The WT is a more adaptable TFA tool than the STFT, better at characterizing signals due to its flexible window sizes, which provide improved temporal resolution at higher frequencies and enhanced frequency resolution at lower frequencies. The continuous wavelet transform of a signal $x\left(t\right)$ is expressed as:

$$C(a,b)={\int }_{-\infty }^{\infty }x\left(t\right){\psi }^{*}\left({\displaystyle \frac{t-b}{a}}\right)dt$$

(8)

Here, $C(a,b)$ denotes the wavelet coefficient at scale a and position b; $\psi \left(t\right)$ refers to the mother wavelet. The scale parameter a (inverse of frequency) determines wavelet dilation, while the translation parameter b shifts the wavelet in time. The symbol ∗ indicates the complex conjugate.

4.3. Wigner–Ville Distribution (WVD)

The WVD is a sophisticated TFA method that fully characterizes signals in both time and frequency. As a bilinear time–frequency tool, it evaluates signal energy distribution by integrating temporal and spectral information. For a continuous-time signal $x\left(t\right)$, the WVD is expressed as:

$$W_x(t,f)={\int }_{-\infty }^{\infty }x\left(t+{\displaystyle \frac{\tau }{2}}\right)x^{*}\left(t-{\displaystyle \frac{\tau }{2}}\right)e^{-j2\pi f\tau }d\tau$$

(9)

Here, $W_x(t,f)$ represents the WVD of $x\left(t\right)$ at time t and frequency f; $\tau$ is the delay variable, and $x^{*}\left(t\right)$ denotes the complex conjugate of the signal.

4.4. Hilbert–Huang Transform (HHT)

The HHT is a method for TFA that addresses nonlinear and non-stationary signals, devised by Dr. Norden E. Huang in the 1990s. The HHT consists of two principal components:

4.4.1. Empirical Mode Decomposition (EMD)

This adaptive tool decomposes a signal into a finite series of IMFs, each representing simple oscillatory patterns that characterize the signal’s local frequency traits.

4.4.2. Hilbert Spectral Analysis (HSA)

The HT is applied to each IMF, producing instantaneous frequency and amplitude over time. For a signal $x\left(t\right)$, the Hilbert Transform is:

$$\mathcal{H}\left\{x\left(t\right)\right\}={\displaystyle \frac{1}{\pi }}\mathrm{P}.\mathrm{V}.{\int }_{-\infty }^{\infty }{\displaystyle \frac{x\left(\tau \right)}{t-\tau }}d\tau$$

(10)

P.V. indicates the Cauchy principal value. The output is an analytic signal:

$$z\left(t\right)=x\left(t\right)+j\mathcal{H}\left\{x\left(t\right)\right\}=A\left(t\right)e^{j\theta \left(t\right)}$$

(11)

where $A\left(t\right)=\sqrt{x{\left(t\right)}^2+\mathcal{H}{\left\{x\left(t\right)\right\}}^2}$ is the amplitude, $\theta \left(t\right)=arctan\left({\displaystyle \frac{\mathcal{H}\left\{x\right(t\left)\right\}}{x\left(t\right)}}\right)$ is the phase, and the frequency is $f\left(t\right)={\displaystyle \frac{1}{2\pi }}{\displaystyle \frac{d\theta \left(t\right)}{dt}}$.

4.5. Fractional Wavelet Transform (FRWT)

The FRWT is an evolution of the standard WT, merging WT’s multiresolution analysis advantages with the fractional domain representation similar to the FRFT. Essential aspects of the FRWT include multiresolution analysis akin to the classical WT, permitting the examination of both high and low frequencies. Additionally, the Fractional Domain Representation allows the FRWT to assess non-stationary signals with spectral characteristics that change over time. By integrating the FRFT with the conventional WT, the FRWT concurrently portrays signals in both the time and fractional Fourier domains.

4.6. Wavelet Scattering Transform (WST)

The WST excels in hierarchically capturing both time and frequency details, ensuring invariance to small signal shifts while preserving crucial structures. It extends the traditional WT by cascading convolutions and employing modulus nonlinearities to extract structured information over different scales and time shifts. The process involves sequential signal analysis, where each step’s output feeds into the next.

5. Classification of PD Sources via DL Model

Deep learning, particularly using convolutional neural networks (CNNs), shows promise in signal processing by autonomously extracting intricate features from time–frequency representations (TFRs) of partial discharge (PD) signals. CNNs efficiently classify these signals and identify various PD sources, despite challenges from noise and signal overlap. This study employs CNNs to categorize PD signals based on the number of PD sources, such as single, dual, or multiple, using TFRs as inputs. Due to their ability to extract spatial features from 2D TFRs, CNNs adeptly identify complex and overlapping patterns associated with multiple PD sources, applying convolution and pooling layers to disclose unique structures, as shown in Figure 39.

The CNN framework includes the following: 1. Input Layer: This processes 2D TFR images of dimension $n\times m$, normalizing pixels to [0, 1]. 2. Convolutional Layers: This utilizes $3\times 3$ kernels for feature extraction, employing ReLU for nonlinear activation. 3. Pooling Layers: These implement $2\times 2$ max-pooling to reduce spatial dimensions, aiding in overfitting prevention. 4. Inception Modules: These engage in parallel convolutions with $1\times 1$, $3\times 3$, and $5\times 5$ filters, capturing multi-scale features while $1\times 1$ convolutions manage computational cost. 5. Dropout Layers: These regularly disable neurons (rate of 0.4) during training to mitigate overfitting. 6. Fully Connected Layers: These convert feature maps into vectors for classification and employ ReLU for hidden layers and softmax for outputs. 7. Output Layer: This produces class probabilities for single, double, and triple PD sources.

Figure 39. Convolutional neural network (CNN) detailed architecture.

Figure 39. Convolutional neural network (CNN) detailed architecture.

The models were effectively trained using the Adam optimizer with a low initial learning rate for stable convergence. A stepwise learning rate decay was applied, reducing it by 0.1 every 20 epochs to enhance learning in later stages. ImageNet-pretrained weights were used initially for transfer learning benefits. Random horizontal flipping, a basic data augmentation method, was used to prevent overfitting.

Table 6. Training hyperparameters.

Parameter	Value
Optimizer	Adam
Initial learning rate	$5\times {10}^{-5}$
Batch size	64
Epochs	80
Learning rate schedule	Step decay (factor = 0.1 every 20 epochs)
Weight initialization	ImageNet pretrained
Data augmentation	Random horizontal flip

summarizes all training hyperparameters.

6. Results and Discussion

6.1. Comparative Performance Analysis

Table 7. Performance comparison: simulation (GoogLeNet) vs. experimental (ResNet50) results.

Method	Dataset	Accuracy	Precision	Recall	F1-Score	AUC
STFT	Simulation	90.00%	0.91 (D) 1.00 (S) 0.83 (T)	1.00 (D) 0.70 (S) 1.00 (T)	0.895	–
	Experiment	75.00%	75.25%	75.00%	74.94%	–
WT	Simulation	93.33%	0.90 (D) 0.90 (S) 1.00 (T)	0.90 (D) 0.90 (S) 1.00 (T)	0.933	–
	Experiment	85.00%	85.35%	85.00%	84.96%	–
WVD	Simulation	93.33%	0.95 (D) 0.95 (S) 0.91 (T)	0.90 (D) 0.90 (S) 1.00 (T)	0.933	–
	Experiment	94.74%	95.45%	94.44%	94.68%	0.947
HHT	Simulation	88.33%	1.00 (D) 0.89 (S) 0.78 (T)	0.90 (D) 0.85 (S) 0.90 (T)	0.886	–
	Experiment	70.00%	73.81%	70.00%	68.75%	–
FRWT	Simulation	95.00%	1.00 (D) 0.95 (S) 0.91 (T)	0.90 (D) 0.95 (S) 1.00 (T)	0.950	–
	Experiment	–	–	–	–	–
SWT	Simulation	96.67%	1.00 (D) 0.91 (S) 1.00 (T)	0.90 (D) 1.00 (S) 1.00 (T)	0.967	–
	Experiment	100.00%	100.00%	100.00%	100.00%	1.000

Note: (D) = DPS, (S) = SPS, (T) = TPS. Simulation results use GoogLeNet; experimental results use ResNet50. Gray rows highlight experimental data. Dashes (–) indicate unavailable data.

presents the classification metrics for both simulated and experimental PD signals across six time–frequency analysis methods. The key findings reveal the following:

• The superiority of the Scatter Wavelet Transform (SWT):

  $${\mathrm{SWT}}_{\mathrm{Exp}}\to \mathrm{Accuracy}=100\%,\mathrm{F}1=1.000$$

  achieved perfect classification with ResNet50, confirming its robustness in handling real-world noise and multi-source PD patterns.
• The consistency of the Wigner–Ville Distribution (WVD):

  $$\Delta {\mathrm{Accuracy}}_{\mathrm{Sim}\to \mathrm{Exp}}=1.41\%(93.33\%\to 94.74\%)$$

  demonstrated stable performance, though its quadratic computation introduces trade-offs between resolution and cross-term artifacts.

6.2. Method-Specific Insights

6.2.1. High-Performance Methods

• The SWT’s Advantage: The scattering network’s invariance to small deformations explains its experimental dominance:
• The WVD’s Trade-off: While achieving 94.74% experimental accuracy, its performance is bounded by the Heisenberg uncertainty principle:

  $${\sigma }_t·{\sigma }_f\ge {\displaystyle \frac{1}{4\pi }}$$

  (12)

6.2.2. Suboptimal Performance of Methods

As delineated in

Table 8. Performance drop: simulation vs. experiment.

Method	Accuracy Drop
STFT	15.00%
HHT	18.33%

, several factors contribute to the subpar performance of these techniques:

• Sensitivity of the Hilbert-Huang Transform (HHT): The experiments exhibited a substantial reduction in accuracy by 18.33%, attributed to:

  –

  The instability of Empirical Mode Decomposition (EMD) when subjected to experimental noise, which undermines its robustness;

  –

  The presence of end effects, which significantly skew the evaluation of instantaneous frequency.

• Constraints of the Short-Time Fourier Transform (STFT): The adoption of a fixed window size, $w\left(t\right)$, leads to a trade-off characterized by either:

  $$\mathrm{Elevated}\Delta f\mathrm{with}\mathrm{an}\mathrm{extensive}\mathrm{window}\mathrm{or}\mathrm{Elevated}\Delta t\mathrm{with}\mathrm{a}\mathrm{reduced}\mathrm{window}\mathrm{size}$$

  (13)

6.3. Model Architecture Impact

The ResNet50’s residual connections:

$$\mathcal{F}\left(x\right)+x$$

(14)

effectively mitigated vanishing gradients in deep layers, particularly benefiting the SWT’s high-dimensional features. This explains the experimental F1-score improvements:

$${\mathrm{SWT}}_{\mathrm{Exp}}-{\mathrm{SWT}}_{\mathrm{Sim}}=+3.34\%(0.9666\to 1.000)$$

(15)

Table 9. PD classification performance comparison.

Method	Simulation Confusion Matrix	Experimental Confusion Matrix
Short-Time Fourier Transform (STFT)
Wavelet Transform (WT)
Wigner–Ville Distribution (WVD)
Hilbert–Huang Transform (HHT)
Scatter Wavelet Transform (SWT)

presents a detailed visual summary of model projections, showing confusion matrices for both simulation and experimental data across analyzed methods. These matrices are critical for delving into class-specific performance, revealing inter-class confusions not captured by metrics like accuracy or F1-score, enabling nuanced analyses of predictive performance.

7. Conclusions

In this paper, we present a novel approach to partial discharge (PD) classification by merging time–frequency analysis (TFA) techniques with convolutional neural networks (CNNs). Our results demonstrate that wavelet-based methodologies, particularly Wavelet Scattering, significantly enhance the classification of PD sources. This approach outperforms other TFA methods in terms of accuracy and F1-score, thereby improving both precision and recall in intricate PD scenarios. The Scatter Wavelet Transform (SWT) achieves a simulation accuracy of 96.67% and an impressive F1-score of 0.967. Remarkably, in experimental settings, it attains perfect classification outcomes with 100% accuracy and an F1-score of 1.000, far surpassing methods such as the Hilbert–Huang Transform (HHT), which recorded merely 70.00% accuracy. While the Wigner–Ville Distribution (WVD) reaches an accuracy of 94.74% with an AUC of 0.947, its substantial computational requirements limit its applicability in real-time scenarios. Consequently, the SWT emerges as an exceptionally effective technique for noise reduction and the management of multi-source PD signals, establishing itself as a formidable diagnostic tool for power system insulation assessment.