An Efficient Noise Reduction Method for Power Transformer Voiceprint Detection Based on Poly-Phase Filtering and Complex Variational Modal Decomposition

Abstract

The transformer is a core component in power systems, and its reliable operation is crucial for the safety and stability of the power grid. Transformer faults can be diagnosed early using acoustic signals. However, effective acoustic features are often affected by complex environmental noise, which reduces the accuracy of fault identification. As a solution, this study proposes a poly-phase filtering (PF)-based noise reduction algorithm for complex variational mode decomposition (CVMD) of multiple acoustic sources in power transformers. The algorithm dissects the received signal from the power transformer into subbands, downsizing their sampling rates via PF. Subsequently, it independently targets noise reduction within these subbands, focusing on specific acoustic sources. Leveraging complex signal transformations, we extend the variational mode decomposition (VMD) to mitigate the field of complex signals and utilize the CVMD to reduce the noise of each acoustic source within each subband for every acoustic source. The experimental results reveal that the proposed method effectively separates and denoises the sound signal of transformer operation under the interference of multiple sound sources in the substation. Its powerful noise reduction ability, combined with minimal computational complexity, greatly improves the accuracy of transformer fault identification and the reliability of the system.

1. Introduction

The power system has always prioritized the safety of both personnel and power grid equipment. Embracing automated operation and maintenance methods, such as power robots and online monitoring devices, stands as a key trend in smart grid advancement. Detecting sound in transformers plays a vital role in achieving digital transformation and serves as a crucial method for early identification of transformer abnormalities [1,2,3,4,5,6]. The acoustic signals generated by operational transformers carry substantial equipment status information that can be used as important feature quantities for fault diagnosis. However, the operating conditions of transformers are intricate, amidst an environment filled with various uncertain interference sounds such as wind, chatter, birdcalls, and horns [7,8,9,10]. Intermittent transformer faults such as short-circuit impact and partial discharge are easily confused by multiple acoustic sources in complex environments, posing a challenge for identification. Therefore, applying noise reduction processing to the collected acoustic signal becomes necessary.

Numerous scholars have studied noise reduction techniques. Currently, the commonly used noise processing techniques include the recursive averaging filtering method [11], noise processing based on wavelet transform [12,13,14], filtering noise reduction algorithm based on the least squares method, etc. More recently, empirical mode decomposition (EMD) filtering noise reduction [15,16,17,18,19] has evolved. The general method of noise reduction involves filtering the signal; however, it does not provide satisfactory results and accuracy. For example, Luo [20] introduced the recursive averaging filtering method, which has a suppression effect on periodic disturbances and is suitable for systems with high-frequency oscillations. However, the sensitivity of the algorithm is low, and the suppression of accidental pulse interference is poor. It is difficult to eliminate the deviation of the sampling value caused by pulse interference, and it is not applicable to occasions with serious pulse interference. Lim and Oppenheim proposed the Wiener filtering method [21], effective for single input and single output stationary random signals but reliant on prior noise distribution. Yafeng [22] introduced an adaptive filtering denoising algorithm based on least mean squares (LMS), proficient in suppressing stationary random signal noise but challenging to control the filter’s convergence factor, impacting denoising efficacy. Huang [23] proposed empirical mode decomposition (EMD), significantly enhancing the signal-to-noise ratio (SNR) of nonlinear non-stationary signals, but it exhibits a serious mode aliasing phenomenon and lacks a complete mathematical theoretical derivation. Subsequently, Dragomiretskiy [24,25] introduced Variational Mode Decomposition (VMD), renowned for its solid theoretical foundation, high computational efficiency, and robust noise reduction capabilities. In recent years, scholars have explored applying neural network models to noise reduction [26,27,28], achieving notable results albeit requiring extensive sample training and computational resources.

While these algorithms show promise in specific scenarios, they struggle to extract and denoise individual acoustic sources amidst complex transformer environments. In practical applications, like locating a faulty transformer using a microphone, the received signal may contain multiple acoustic sources. Effectively identifying these faulty sources and employing filter banks to distinguish and extract each source from the spectrum presents a feasible approach. However, traditional analysis filter banks filter before sampling, sharply increasing hardware resource demands at higher sampling rates. A polyphase structure-based analysis filter bank enhances resource utilization by sharing low-pass filters among branches. However, the transformed complex signal post poly-phase filtering cannot undergo direct denoising using traditional VMD.

Therefore, this study proposes an efficient noise reduction algorithm leveraging poly-phase filtering and complex variational mode decomposition (CVMD) for multiple acoustic sources in transformers. This method adeptly separates various transformer acoustic sources, reducing environmental interference and noise. Successful denoising of collected transformer acoustic signals should facilitate easier extraction of fault voiceprint features, significantly enhancing transformer fault recognition accuracy.

2. Extraction of Transformer Multiple Acoustic Sources Based on Poly-Phase Filtering

The sound generated during transformer operation is primarily caused by the vibration of components, such as windings, iron cores, and cooling fan devices—each vibrating at different frequencies. To individually process each acoustic source, an analytical filter bank becomes essential to divide the received signal from the transformer into multiple subbands [29,30]. According to the data, the sound signal generated by the transformer core vibration contains a small number of 100 Hz octave components in addition to the 100 Hz fundamental frequency and is mainly distributed within 1000 Hz. If the sub-bandwidth is set too large, it may cause two different sound sources to fall into the same sub-band, which is not favorable for subsequent processing. Based on the experiment results, it is appropriate to set the bandwidth of each sub-band to 100 Hz.

Conventional analysis filter banks initially divide the broad signal, housing multiple target acoustic sources, into numerous subbands uniformly through a modulation filter bank. It then shifts the subband signal to the baseband and extracts it to reduce the sampling rate. However, this process leads to excessive computations, wasting hardware resources. The analytic filter bank based on the polyphase structure improves resource utilization by sharing one low-pass filter for each branch and enhances the efficiency of the system by processing the channel data in parallel.

Let us consider an FIR filter system function with a point length of N [31]:

$$H(z)={\displaystyle \sum _{n=0}^{N-1}h(n)}{z}^{-n}$$

(1)

Let N be a multiple of K, then L = N/K. The component calculation formula of each poly-phase filter is as follows:

$$E_l(Z^K)={\displaystyle \sum _{n=0}^{L-1}h(nK+l)}{\left(z^K\right)}^{-n}$$

(2)

The filter system function can be expressed as:

$$H(z)={\displaystyle \sum _{l=0}^{K-1}{z}^{-l}}{E}_l(z^K)$$

(3)

The k-th (k = 0, 1,… K − 1) output of the traditional analysis filter bank is:

$$v_k(n)=[x(n)\ast h_k(n)]e^{-j{w}_{k}n}$$

(4)

$$y_k(m)=v_k(n)|{}_{n=Km}={\displaystyle \sum _{i=0}^{N-1}x(Km-i)}[h_0(i)e^{j{\omega }_k(i-Km)}]$$

(5)

Among them, $x(n)$ represents the input broadband signal, $h_k(n)$ indicates the k-th filter response, $\ast$ denotes convolution, and $h_k(n)=h_0(n)e^{j{\omega }_{k}n}$, $h_0(n)$ are low-pass prototype filters.

We make the following assumption:

$$i=lK+r$$

(6)

Hence, Equation (5) can be expressed as:

$$y_k(m)={\displaystyle \sum _{r=0}^{K-1}{e}^{j{\omega }_{k}r}{\displaystyle \sum _{l=0}^{L-1}{x}_r(m-l)}}{e}_r(l)e^{j{\omega }_{k}K(l-m)}$$

(7)

where,

$$x_r(m-l)=x(Km-Kl-r)$$

(8)

$$e_r(l)=h_0(Kl+r)$$

(9)

When the channel is arranged in an odd pattern, the center frequency of the k-th subband can be expressed as:

$$w_k=-\pi +\frac{2\pi k}{K}$$

(10)

Substitute this into Equation (7):

$$\begin{array}{l}{y}_k(m)={\displaystyle \sum _{r=0}^{K-1}{e}^{j(-\pi +\frac{2\pi k}{K})r}{\displaystyle \sum _{l=0}^{L-1}{x}_r(m-l)}}{e}_r(l)\\ =K(IDFT[((x_r(m)\ast e_r(m)){(-1)}^r])\end{array}$$

(11)

Based on the above equation, a channelized poly-phase filter is derived, depicted in Figure 1. Notably, the extractor is located in front of the filter, and the speed of the data after reducing the sampling rate is 1/k of the original rate. The operation can improve the real-time processing capability and reduce the filter operation pressure. Simultaneously, each branch of the filter coefficients is a sampled value of the coefficients of the prototypical low-pass filter, which reduces the computational error of the filter and improves the computational accuracy. Additionally, the computational efficiency can be greatly improved because IDFT/DFT can be calculated by IFFT/FFT, making this method efficient for segmenting the received signal from the transformer into subbands to discern distinct acoustic sources.

Additionally, it should be noted that subbands obtained via poly-phase filtering (PF) exhibit a sequence length of 1/K compared to the original sequence. The 0th and K/2-th subbands represent real signals, while the remaining subbands are complex signals. Additionally, the information in subbands 1 to K/2 − 1 mirrors that of subbands K/2 + 1 to K − 1. Therefore, when designing a filter bank, the signal cannot be positioned solely in the 0th and K/2-th subbands. This standardizes subsequent signal processing and reduces computational complexity, thereby enhancing computational efficiency.

Within an environment where multiple acoustic sources interfere during transformer operation, the PF algorithm is utilized to efficiently isolate these sources across the spectrum. Subsequently, each acoustic source undergoes separate denoising, significantly enhancing the overall denoising capability.

3. Multiple Source Complex Variational Modal Decomposition for Noise Reduction

Noise reduction processing occurs within the subbands isolated for each acoustic source. To further enhance the SNR within these subbands, this paper will introduce an enhanced CVMD algorithm specifically tailored for noise reduction within each isolated subband.

The objective of VMD is to accurately replicate the input signal by breaking down the real-valued input signal into a finite number of sub-signals (modes) with a defined level of sparsity. The approach extends the concept of the Wiener filter to multiple adaptive bands. VMD is an adaptive, completely non-recursive method for modal variation and signal processing with a robust mathematical foundation and strong noise resilience, effectively mitigating modal aliasing seen in EMD. It effectively processes non-smooth and non-linear signals, while also adjusting to variations in signal frequency and amplitude [32]. While classical VMD handles real signals, the output data from multiphase filters transforms into complex signals.

According to signal processing knowledge, the imaginary part of a complex signal can be derived from its real part through the Hilbert transform. This transform can be considered a linear filter that maintains the amplitude of a real signal spectrum while advancing the negative frequency phase by 90 degrees and lagging the positive frequency phase by 90 degrees. The Hilbert transform does not change the frequency information of the signal, thus establishing a theoretical one-to-one correspondence between the real and imaginary parts of the complex signal’s amplitude–frequency information. Consequently, noise reduction in the complex signal can be achieved by employing the VMD algorithm to separately reduce the noise in the real and imaginary parts. However, this approach requires the use of the VMD algorithm twice, which can lead to a considerable computational burden. Additionally, the presence of noise may disrupt the one-to-one correspondence of the modes between the real and imaginary parts. To address these challenges, VMD has been extended to the domain of complex signals and is referred to as CVMD.

3.1. VMD Algorithm

Given the original signal decomposed into modal components as an amplitude-modulated signal, each component possesses a finite bandwidth with a central frequency. The objective is to minimize the sum of estimated bandwidths for all modalities while ensuring that the sum of all modalities remains equal to the original signal.

This constrained variational model can be expressed using the following equation:

$$F_0=H_{SFE}(X)\{\begin{array}{l}{\min }_{\{u_i\}\{w_i\}}\{{\displaystyle {\sum }_i{\Vert {\partial }_t\left[({\delta }_t+\frac{j}{\pi t})*u_i(t)\right]e^{-j{w}_{i}t}\Vert }_2^2}\}\\ s.t.{\displaystyle \sum _{i=1}^I{u}_i(t)=f(t)}\end{array},$$

(12)

where, $f(t)$ represents the signal to be decomposed, We represents the number of modes, $u_i$ denotes the modal components, $w_i$ denotes the center frequency corresponding to each mode component, and $\delta (t)$ denotes the unit pulse signal. The function of $\left[({\delta }_t+j/(\pi t))\ast u_i(t)\right]$ converts $u_i(t)$ into an analytical signal, transforming its spectrum into a single-sided spectrum.

3.2. Improved CVMD Algorithm

Extending VMD to the field of complex signals and defining its intrinsic mode function yields:

$$u_i(t)=A(t)e^{j\varphi (t)}$$

(13)

Since the complex signal inherently possesses a unilateral spectrum, the $\left[({\delta }_t+j/(\pi t))\right]$ in Equation (12) can be omitted, simplifying the constrained variational model to:

$$\{\begin{array}{l}{\min }_{\{u_i\}\{w_i\}}\{{\displaystyle {\sum }_i{\Vert {\partial }_t\left[u_i(t)\right]e^{-j{w}_{i}t}\Vert }_2^2}\}\\ s.t.{\displaystyle \sum _{i=1}^I{u}_i(t)=f(t)}\end{array}$$

(14)

To derive the optimal solution for Equation (14), create the augmented Lagrange function with the introduction of a penalty factor $\alpha$:

$$\begin{array}{l}L(\left\{u_i\right\},\left\{w_i\right\},\lambda )={\Vert f(t)-{\displaystyle \sum _i^{}{u}_i(t)}\Vert }_2^2+\\ \alpha {\displaystyle \sum _i{\Vert {\partial }_t\left[(u_i(t)\right]e^{-j{w}_{i}t}\Vert }_2^2}+\langle \lambda (t),f(t)-{\displaystyle \sum _i^{}{u}_i(t)}\rangle \end{array}$$

(15)

The above form of the cost function has a large number of variables to be solved, which cannot be computed directly using a general convex optimization algorithm. Therefore, we use the Alternating Direction Method of Multipliers (ADMM) algorithm for the solution. ADMM is an efficient iterative method for solving large-scale separated convex programming problems. The algorithm utilizes the separability of the cost function by decomposing the original problem into multiple subfunctions, alternately solving for multiple separated variables, and then combining the decomposed sub-problems. Equation (15) is formulated as solving for the variables $u_i$ and $w_i$. This solution process is realized using alternating computations, which ultimately results in the optimal minimal value of the cost function. Note that theoretically, both variables need to be optimized simultaneously. Owing to the difficulty of implementation, the ADMM algorithm updates the two variables separately, first optimizing the variable $u_i$, and then optimizing the variable $w_i$ to satisfy the condition of alternating iterations. The advantage of this algorithm is that the cost function can be transformed into a separable structure, and the update of the variables can be divided into multiple steps sequentially. The solving task is then assigned to different nodes for parallel processing to realize the efficiency of the algorithm.

Utilizing the ADMM algorithm and the Fourier Equidistant Theorem, Equations (16) and (17) can be derived as follows:

$${\widehat{u}}_i^{n+1}(w)=\frac{\widehat{f}(w)-{\displaystyle \sum _{p\ne i}{\widehat{u}}_p(w)+\frac{\widehat{\lambda }(w)}{2}}}{1+0.5\ast \alpha {(w-w_i)}^2}$$

(16)

$$w_i^{n+1}=\frac{{\displaystyle {\int }_{-\infty }^{\infty }w{\Vert {\widehat{u}}_i(w)\Vert }_{}^{2}dw}}{{\displaystyle {\int }_{-\infty }^{\infty }{\Vert {\widehat{u}}_i(w)\Vert }_{}^{2}dw}}$$

(17)

In Equation (17), ${\widehat{u}}_i^{}(w)$, $\widehat{f}(w)$, and $\widehat{\lambda }(w)$ denote the frequency domain forms of $u_i(t)$,$f(t)$, and $\lambda (t)$, respectively. After performing the inverse Fourier transform on ${\widehat{u}}_i^{}(w)$, it becomes $u_i(t)$. We update the multiplication operator $\widehat{\lambda }$ using Equation (18) until the conditions of Equation (19) are satisfied, following which the updating terminates.

$${\widehat{\lambda }}^{n+1}(w)={\widehat{\lambda }}^n(w)+\tau (\widehat{f}(w)-{\displaystyle \sum _{i=1}^I{\widehat{u}}_i^{n+1}(w)})$$

(18)

$$\frac{{{\displaystyle \sum _{i=1}^I\Vert {\widehat{u}}_i^{n+1}-{\widehat{u}}_i^n\Vert }}^2}{{{\displaystyle \sum _{i=1}^I\Vert {\widehat{u}}_i^n\Vert }}^2}<\epsilon$$

(19)

In Equation (19), $\epsilon$ symbolizes convergence tolerance.

Notably, the effect of VMD and CVMD noise reduction is primarily affected by the modal number selection value. When the modal number is selected at a small value, it may lead to the loss of important information in the original signal. When the modal number is selected at a large value, the center frequencies of adjacent modal components are close to each other, leading to modal repetition or additional noise. The main difference between different modes is the center frequency, so the appropriate modal value is selected by observing the distribution of the center frequency under different modal numbers. Experimentally, the optimal value of I is 4 at a sampling rate of 100 Hz.

CVMD effectively diminishes noise within each subband acoustic source, thereby enhancing the SNR. As previously mentioned, the signal filtered by multiphase filtering already exists in a baseband signal form. In case subsequent processing can utilize this baseband signal form, no additional processing is required. However, if the frequency of the target acoustic source signal needs to be restored, the subband can be reverted to its original frequency using a poly-phase-based synthesis filter bank. Poly-phase-based synthesis filter banks can improve efficiency by moving the filtering ahead of interpolation through the Nobel Identity. The principle behind this synthesis filter bank is similar to that of a poly-phase-based analysis filter bank and will not be elaborated on further.

During practical operation, post-PF-based signal segmentation into multiple subbands and detecting an acoustic source signal in a subband, it involves solving the subband power and setting a threshold. If no acoustic source signal is present in the subband, resetting all data within that subband to zero becomes necessary. However, if an acoustic source signal exists in the subband, applying CVMD can further denoise the subband.

Summarizing the proposed algorithms yields the flowchart of the transformer multi-source complex variational modal decomposition denoising algorithm based on polyphase filtering, depicted in Figure 2.

4. Results and Discussion

4.1. Verification of Simulation Data

The performance of the algorithm will be validated through the following simulation experiments. Input two single-frequency signals with the experiment’s specific parameters detailed in

Table 1. System simulation parameters.

Parameter Name	Symbol	Value
Sampling rate	fs	3200 Hz
Number of subbands	K	32
The 1st frequency	f1	1300 Hz
The 2nd frequency	f2	1510 Hz
Number of decompositions	I	4
Penalty factor	α	1000
Convergence tolerance	ε	10−6

.

Figure 3 illustrates the spectrum of the simulated signal designated for processing. By employing PF, the entire Nyquist frequency is segmented into 32 subbands, each spanning a bandwidth of 100 Hz. These subbands are shifted close to zero frequency within a range of (−50 Hz, 50 Hz). Based on the previous analysis, except for the real signals in the 0th and 16th subbands, the remaining subbands comprise complex signals, with half of the subband information being redundant. Figure 4 illustrates the output spectrum of subbands 16 through 31, where the first acoustic source signal emerges in the 29th subband, followed by the second acoustic source signal in the 31st subband.

Figure 5 and Figure 6 showcase the individual spectra of each modal component derived from the subbands of the two target signals. An absence of aliasing between the spectra of each mode can be observed. Figure 7 and Figure 8 present the synthesized subband signal spectrum, utilizing various modal components and the atomic band signal spectrum, indicating a significant reduction in noise within the subband post-CVMD processing.

The SNR of the subband at $f_1$ before processing was 23.58 dB, while the SNR after processing was 30.25 dB. The SNR of the subband at $f_2$ before processing was 17.78 dB, while the SNR after processing was 24.62 dB. The SNR in the two subbands increased by 6.67 dB and 6.84 dB, respectively. Comparing with the output in Figure 3, the SNR at $f_1$ increased by 23.8 dB, while the SNR at $f_2$ increased by 24.79 dB. Figure 9 shows the spectrum of the simulated signal after synthesis filtering. Therefore, the noise reduction performance of the proposed algorithm can be observed to have improved significantly.

Conducting 500 Monte Carlo experiments across various SNR conditions enabled for separate calculations of the output SNR of the target signal within its subband, pre- and post-denoising. The computer was equipped with an Intel(R) Core(TM) i5-7200U@ 2.50 GHz processor, 16 GB of RAM, and the experimental platform was MATLAB. Figure 10 illustrates the trend of SNR before and after noise reduction. The blue curve represents the SNR in the subband containing the target signal before denoising, the red curve represents the SNR in the same subband after denoising the real and imaginary parts using VMD, and the yellow curve represents the SNR in the subband after denoising, using the CVMD algorithm proposed in this paper. It can be observed that the yellow curve and the red curve are consistently above the blue curve, with the average difference between the red curve and the blue curve being approximately 3 dB, and the average difference between the yellow curve and the blue curve being approximately 6 dB. This further demonstrates the validity and superiority of the proposed algorithm. It is worth mentioning that since VMD is used to reduce the noise of the real and imaginary parts separately, and CVMD is used to reduce the noise of the complex signal directly, the computational effort required for VMD is almost twice as much as that needed for CVMD.

4.2. Verification of Measured Data

The performance of the algorithm was verified using measured data. The experimental data are from the audio collected from the substation, which is limited by the existing conditions, and the specific model is not known. To better adapt to the actual situation, we played the collected transformer sound and the horn sound concurrently, and collected them using another microphone. Audio 1 corresponds to the transformer sound, while Audio 2 corresponds to the horn sound. As the speech signal requires processing solely in the positive frequency realm, Figure 11 depicts only its positive half spectrum. The principal frequency of the transformer sound registers at 289.5 Hz, and the main frequency of the horn sound is located at 1400 Hz with a sampling rate of 32,000 Hz. Using PF to divide the entire Nyquist frequency into 320 subbands, the bandwidth of each subband was 100 Hz.

Figure 12 and Figure 13 display the individual spectra of each modal component derived from the transformer sound and whistle sound subbands, respectively. Notably, the aliasing phenomenon cannot be observed among the various modal spectra. Figure 14 and Figure 15 depict the subband signal spectra synthesized using each modal component and the atomic band signal spectra, respectively, underscoring the evident noise reduction effect.

Following the calculation, the SNR within the subband housing the transformer sound registered at 30.27 dB pre-processing, escalating to 34.45 dB post-processing with the CVMD algorithm. Meanwhile, the SNR within the subband housing the horn sound measured 22.92 dB before processing, rising to 30.25 dB after processing with the CVMD algorithm. Therefore, the SNR of the two acoustic sources in their respective subbands increased by 4.18 dB and 7.33 dB.

The original signal spectrum, obtained through a comprehensive polyphase-based filter combination, is depicted in Figure 16. In contrast to the output in Figure 11, the SNR notably increased by 21.61 dB and 19.09 dB, respectively, showcasing a significant noise reduction effect.

5. Conclusions

The denoising of transformer acoustic signals serves as a crucial preprocessing step for transformer voiceprint fault diagnosis, directly impacting the accuracy of fault recognition and positioning. Addressing the challenges of weak performance and high computational complexity in denoising algorithms, especially in complex environments with multiple acoustic sources during transformer operation, this study proposed a transformer multi-source complex variational modal decomposition denoising algorithm based on poly-phase filtering.

The algorithm initially employs PF to segment the received signal from a transformer, containing multiple acoustic sources, into numerous subbands. Subsequently, the acoustic source of each subband is then extracted for downsampling. Enhanced CVMD is subsequently applied to denoise each acoustic source. Experimental results demonstrate that the algorithm proposed in this study can effectively separate various types of acoustic signals during transformer operation, even amidst multiple acoustic source interferences in complex substation environments. It demonstrates robust denoising capabilities across multiple interference sounds with minimal computational complexity, facilitating straightforward implementation.