Abstract
Partial discharge evaluation is a principal method for assessing insulation conditions in power transformers. Traditional singular value decomposition (SVD) approaches, however, face issues like high residual noise and loss of signal details in white noise suppression. This article introduces an advanced denoising algorithm integrating SVD, variational mode decomposition (VMD), and wavelet thresholding to effectively address mixed noise in on-site power transformer assessments. The algorithm initially employs SVD to suppress mixed noise, specifically targeting narrowband interference by decomposing the noisy signal and nullifying the corresponding singular values. Post-SVD, the signal is further processed through VMD, with its modal components refined via wavelet thresholding. The final reconstruction of these denoised components effectively eliminates white noise. Applied to an input signal with a signal-to-noise ratio of -27.593 dB, the proposed method achieves a postdenoising ratio of 13.654 dB. Comparative analysis indicates its superiority over existing algorithms in mitigating white noise and narrowband interference and more accurately restoring the partial discharge signal.
1. Introduction
Power transformers are critical components within power systems, with their operational reliability being integral to the overall safety and stability of these systems. Empirical analysis of transformer malfunctions has identified insulation faults as the predominant factor impacting transformer functionality. Partial discharge (PD) diagnostic tests are a crucial tool for evaluating the insulation performance of transformers. However, PD signals, being inherently weak electrical signals, are susceptible to contamination from strong electromagnetic fields prevalent in transformers’ high-voltage and high-current operational environments. Consequently, PD signals captured during on-site testing often encompass various forms of interference, such as white noise, periodic narrowband interference, and random pulse interference, leading to potential masking of PD signals within these interferences [1]. Such contamination can severely distort the PD waveform, adversely affecting the accuracy of fault diagnosis. Therefore, mitigating noise impact is essential for enhancing the precision of PD signal analysis and improving fault detection efficacy in practical scenarios.
Scholarly research has extensively explored various denoising methods for PD signals, such as fast Fourier transform (FFT) threshold filtering, empirical mode decomposition (EMD), wavelet thresholding, and singular value decomposition (SVD) [2]. Despite EMD’s adaptability, it is hindered by mode mixing and endpoint effects, leading to suboptimal denoising [3]. To improve upon EMD, a method combining complete ensemble EMD with adaptive noise (CEEMDAN) and approximate entropy has been proposed, which, however, risks eliminating valid signals [4]. An adaptive SVD method is aimed at overcoming the limitations of manual singular value selection but struggles with complex signal reconstruction [5]. Fast SVD using Hankel matrices has shown promise in noise suppression in PD signals [6]. Improved variational mode decomposition (VMD) with threshold algorithms has been proposed for suppressing white noise and periodic narrowband interference, but it lacks precision in parameter selection [7]. A joint denoising approach using VMD and maximal overlap discrete wavelet packet transform (MODWPT) has been developed for rotating machinery vibration signals, but its efficacy in filtering both noise types is unspecified [8]. The flower pollination algorithm (FPA) has been used for parameter selection in VMD, followed by noise component removal with a Savitzky-Golay filter, albeit with slow convergence issues [9]. Optimizations of VMD through genetic algorithms have been explored for PD signal denoising, yet some lack the ability to mitigate narrowband interference [10, 11]. Curvature-based singular value transformation combined with empirical wavelet transform has been effective against white noise and narrowband interference, although threshold selection is prone to errors [12]. EMD-based decomposition with mutual information analysis for VMD suffers from EMD’s inherent limitations [13]. An adaptive short-time SVD method for white noise demonstrates potential, but computational inaccuracies may impact its denoising effectiveness [14].
Aiming to enhance noise reduction efficacy, this article presents an advanced noise reduction methodology tailored to suppress mixed noise in PD signals, integrating SVD with VMD and optimized wavelet thresholding. Initially, a Hankel matrix representation of the noisy signal is employed, followed by the application of SVD to selectively eliminate narrowband interference, leveraging the distinct singular value profiles of PD and noise signals. Subsequent to the SVD process, the signal is decomposed using VMD. Enhanced wavelet thresholding is then applied to these modal components. The final step involves the reconstruction of these processed components, aiming to effectively remove white noise and suppress mixed noise within the PD signal framework. In Section 2, the detailed denoising algorithm is discussed, and simulation analysis is given in Section 3. Experimental signal validation is carried out in Section 4. Finally, conclusions are provided in Section 5.
2. Denoising Algorithm
The schematic diagram of the denoising algorithm for PD signals with mixed noise is shown in Figure 1. This section mainly introduces the basic theory of the denoising algorithm.
2.1. Singular Value Decomposition
The initial step necessitates the formation of a Hankel matrix [15].The construction is shown below: Let be a noisy PD signal with noise. Transform into a Hankel matrix as shown in the following equation.
Matrix , with dimensions and a rank of , can be expressed as a product of an -order orthogonal matrix and an -order orthogonal matrix . where is an -order diagonal matrix, with the first orders having nonzero elements representing the singular values, and the rest are zeros [16]. The singular values corresponding to the noise components exhibit differences compared to the singular values of the original signal. By setting the singular values associated with noise to zero and retaining the singular values of the original signal, the process involves performing the inverse operation of SVD and reconstructing the signal to achieve denoising.
2.2. Variational Mode Decomposition
The solution of VMD consists mainly of two parts: the construction of the variational problem and its solution [17].
By estimating the signal’s bandwidth, the variational constrained model is obtained as follows: where is the noisy PD signal, represents the decomposed modal components, and denotes the center frequency of each decomposed modal component. represents the unit impulse function.
Subsequently, we use the multiplier alternating direction algorithm to iteratively update the intrinsic mode function (IMF) components and their respective center frequencies. The iterative formulas for modal components and center frequencies are shown in equations (4) and (5), respectively [18].
The iteration stops when the termination condition in equation (6) is satisfied, and modal components are output, completing the decomposition. Here, represents the convergence accuracy.
2.3. Wavelet Threshold Denoising
Wavelet threshold denoising entails the assessment of mathematical characteristics of wavelet coefficients, contrasting those between the target signal and the noise signal. Typically, the wavelet coefficients of the desired signal have larger amplitudes and concentrated energy, while those of the noise signal tend to have smaller amplitudes and more uniform energy distribution. Therefore, a specific threshold function can be set to distinguish between the desired signal and the noise signal [19].
The choice of threshold significantly affects the denoising effectiveness. Traditional threshold functions include hard thresholding and soft thresholding. Hard thresholding can lead to discontinuities in wavelet coefficients, resulting in reconstructed signals that lack the smoothness of the original signal. On the other hand, soft thresholding introduces a constant bias, distorting the reconstructed signal. To address the shortcomings of these traditional threshold functions, a refined threshold function is introduced, as illustrated in the following equation.
This new function does not suffer from the fixed bias issues associated with soft and hard functions. By adjusting the tuning parameters and , it can adapt to noise of varying intensities under different signal-to-noise ratio conditions.
Chang et al. [20] proposed a Bayes shrink method suitable for image denoising based on the distribution of two-dimensional wavelet coefficients. For a given parameter , it is necessary to find a threshold that minimizes the Bayesian risk [21].
The threshold calculation expression is as follows [22]:
In the above formula, represents the variance of noise coefficients, and is the square root of the variance of subband coefficients. Here, represents a specific decomposition level. The calculation for is performed using the following equation [23]:
In this equation, represents the coefficients of the noisy signal at different decomposition levels. in which represents the length of wavelet coefficients at each decomposition level, and it can be inferred from :
Therefore, the threshold based on the Bayesian criterion can be determined through equations (10), (11), and (12). It is observed that the Bayesian threshold exhibits adaptability across different levels, overcoming the drawbacks of a fixed threshold.
2.4. Algorithm Steps
Based on the above theories, SVD is adopted in this paper to remove periodic narrowband interference, and VMD is combined with improved wavelet threshold denoising to remove white noise interference. The general flow chart is shown in Figure 2.
3. Simulation Analysis
3.1. Simulation Model
PD signals are characterized by their high-frequency nature, marked by rapid rising edges and durations typically spanning tens of nanoseconds. Due to the complexity of acquiring pure PD signals under practical field conditions, mathematical modeling is often employed to simulate these signals for theoretical analysis. Predominantly, four mathematical models are utilized for this purpose: the single exponential decay pulse model, the double exponential decay pulse model, the single exponential decay oscillation model, and the double exponential decay oscillation model, as referenced in [24, 25]. These models are mathematically formulated in the following equations. where signifies amplitude, the decay coefficient, the oscillation frequency, and the initial pulse time.
For transformer PD signals, specific parameters of these models are detailed in Table 1, where pulses 1 and 3 align with single exponential decay oscillations and pulses 2 and 4 with double exponential decay oscillations. Figure 3 presents a visual representation of the simulated PD signal.
Periodic narrowband interference, characterized by a fixed resonant frequency, concentrated energy in the frequency domain, and a fixed phase distribution, is often simulated by superposing sinusoidal or cosinusoidal signals of varying frequencies and amplitudes [26]. The mathematical model for such interference is encapsulated in the following equation. where indicates the signal amplitude and the frequency. The parameters for this model are listed in Table 2. Figure 4(a) demonstrates the PD signal with added narrowband interference, resulting in an SNR of -27.593 dB, which significantly obscures the original PD signal. Additionally, Gaussian white noise with a standard deviation of 0.2 is incorporated into the PD signal. Time-domain representations of the PD signal with added white noise and with both white noise and narrowband interference are exhibited in Figures 4(b) and 4(c), respectively.
The effectiveness of the denoising process is quantitatively assessed using three key metrics: signal-to-noise ratio (SNR), mean squared error (MSE), and normalized cross-correlation (NCC) [27]. In the context of the noisy signal analyzed above, the SNR was computed to be -27.593 dB, indicating the extent of noise prior to denoising. The MSE was calculated as 212.231, reflecting the initial discrepancy between the estimated and actual signal values. Additionally, the NCC was determined to be 0.047, illustrating the initial dissimilarity between the pre- and postdenoising signals.
3.2. Noise Reduction Processing
3.2.1. Using the SVD Method to Remove Narrowband Interference
Performing SVD on partial discharge signals superimposed with narrowband interference, Figure 5 illustrates the singular value spectrum of these signals. A notable observation from this spectrum is the clear distinction in the singular values corresponding to the narrowband interference, in contrast to those of the original PD signal. This distinction facilitates the identification and subsequent nullification of the singular values associated with the narrowband interference.
Subsequent to the annulment of singular values attributed to the periodic narrowband interference, Figure 6 portrays the resultant PD signal postremoval of this interference, employing SVD.
The computed metrics postdenoising are as follows: , , and . These values indicate that the periodic narrowband interference has been effectively mitigated.
Conducting singular value decomposition on the PD signal contaminated with white noise, the outcomes are depicted in Figure 7. It is evident that the singular values postdecomposition of the PD signal, which solely contains white noise, demonstrate substantial continuity with no distinct inflection points. This characteristic implies the unsuitability of SVD as a method for white noise elimination in such a scenario.
3.2.2. Denoising White Noise Using a Combination of VMD and Improved Wavelet Thresholding
In the context of simulated signals with mixed noise, the localized discharge signal post-SVD denoising is depicted in Figure 8. Following this process, the SNR is calculated to be 9.64 dB, indicating a notable enhancement in denoising effectiveness. The time-domain waveform after this treatment closely aligns with that of Figure 4(b), representing the PD signal with added white noise.
The signal processed using the proposed method, which combines VMD and improved wavelet thresholding subsequent to SVD, yields the denoised time-domain signal displayed in Figure 9(a). For comparative analysis, the PD signal containing both narrowband and white noise interference underwent processing through three distinct denoising methods: wavelet thresholding denoising, SVD denoising, and a combination of SVD and wavelet thresholding denoising, with outcomes illustrated in Figures 9(b)–9(d), respectively. Figure 9(b) presents the result post-wavelet thresholding denoising, revealing significant signal loss and minimal denoising effectiveness. Figure 9(c) exhibits the outcome of the SVD method, which effectively suppresses interference and retains discharge characteristics, though challenges in completely removing residual white noise persist. Figure 9(d) demonstrates the efficiency of the combined SVD and wavelet thresholding method in suppressing both periodic narrowband and white noise interference, albeit with a slight diminution in partial discharge features.
Table 3 presents the results for three denoising evaluation parameters: SNR, NCC, and MSE, for the proposed method, the SVD method, and the combination of SVD and wavelet thresholding. The method introduced in this paper exhibits superior interference suppression and more effective waveform restoration.
Figure 10 facilitates a comparison between the spectral plots of the original and denoised signals. The spectral contents of both signals are remarkably similar, with only minor alterations observed in the primary peak regions, affirming the successful reduction of noise.
4. Denoising of Measured Signals
An empirical investigation was conducted in a laboratory environment to ascertain the performance of the proposed denoising method on real-world measured signals. The experimental setup comprised a discharge model, a coupling capacitor, a local measurement unit, a voltage withstand tester, and an oscilloscope. The configuration of this experimental arrangement is illustrated in Figure 11.
(a)
(b)
Figures 12 and 13 exhibit PD signals acquired in the laboratory setting for surface discharge and corona discharge, respectively. Given the minimal noise levels in these measured PD signals, they are treated as approximations of noise-free signals.
To further validate the method’s efficacy, artificial noise was added to the signals for denoising analysis. Figures 14(a) and 15(a) illustrate the noisy PD signals, derived from collecting ambient laboratory noise under nondischarge conditions and the deliberate introduction of sinusoidal narrowband interference. These figures show that low-amplitude PD pulses are overwhelmed by the noise. Figures 14(b) and 15(b) demonstrate the results postdenoising using the SVD method, while Figures 14(c) and 15(c) display the outcomes following denoising with a combination of SVD and wavelet thresholding (). The denoising results using the proposed method are presented in Figures 14(d) and 15(d).
Table 4 lists the evaluation parameters for the actual measured signals. The proposed method not only augments the discernibility of PD features but also exhibits superior evaluation metrics compared to other methods. It effectively eliminates most interference noise, highlighting its proficient denoising effect.
In summary, our method demonstrates marked improvement in the visibility of PD features and excels in noise reduction, as evidenced by the denoising results and evaluation parameters for both surface and corona discharge signals.
5. Conclusions
This work introduces a denoising methodology that synergizes SVD and VMD with an enhanced wavelet thresholding technique, aimed at attenuating both periodic narrowband interference and white noise in PD signals. The efficacy of this integrated approach is validated through the analysis of both simulated and measured data, yielding several key findings: (1)The singular values associated with periodic narrowband interference demonstrate specific distribution patterns. While SVD proves effective in attenuating this type of interference in PD signals, its performance in suppressing white noise interference is compromised, primarily due to challenges in accurately determining the threshold for singular values associated with white noise(2)Wavelet transform, commonly employed in signal processing across diverse frequency ranges, may not efficiently capture the distinct frequency components characteristic of narrowband interference. This limitation potentially leads to less than optimal denoising outcomes(3)The application of the refined thresholding method facilitates the adaptive adjustment of thresholds at varying scales, thereby enhancing the reduction of noise coefficients. This adaptability is particularly advantageous in addressing noise across different decomposition scales, improving the overall denoising efficiency
Data Availability
The data that support the findings of this study are available on request from the corresponding author, [email protected], upon reasonable request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Gansu Provincial Natural Science Foundation of China (21JR7RA237) and Gansu Provincial Science and Technology Commissioner Special Project (22CX8GA11).