Harmonic Detection Method Based on Parameter Optimization VMD-IWT Combined Noise Reduction (2024)

1. Introduction

With the continuous development of smart grid construction and the large-scale application of power electronic equipment and nonlinear loads, the harmonic problem of the power grid has become particularly serious. Harmonic pollution not only causes significant power loss and reduces the efficiency and reliability of the grid, but also introduces other challenges such as equipment overheating, sensitive equipment failure, and resonance [1,2,3]. Therefore, the primary task for implementing targeted harmonic pollution control is accurately detecting power harmonic signals to provide a basis for subsequent harmonic control [4].

At present, harmonic signal detection methods encompass Fourier transform, neural networks, empirical mode decomposition (EMD), and other algorithms [5,6,7]. While these traditional algorithms are straightforward and fast in computation, their adaptability to non-static harmonic signals in noisy backgrounds is limited, leading to poor denoising effects and subsequent loss of crucial characteristic information from the harmonic signal, resulting in reduced detection accuracy. With the emergence of new noise reduction methods, the filtering capabilities of EMD have garnered widespread recognition and application. Researchers in [8] combined empirical mode decomposition with wavelet thresholding to effectively mitigate noise in bridge detection signals. Similarly, [9] proposed a denoising method that integrates empirical mode decomposition with an improved wavelet threshold, successfully eliminating background noise influence. Additionally, [10] introduced a harmonic detection method augmented by empirical mode decomposition and wavelet threshold denoising, which enhances harmonic signal detection accuracy and exhibits commendable anti-noise performance. While EMD and its improved version coupled with wavelet denoising excel in noise removal, the mode aliasing issue persists, potentially resulting in information loss contained within the mode.

Variational mode decomposition (VMD) [11] has garnered significant attention as a signal processing technology. It exhibits remarkable bandwidth division capability in signal frequency decomposition and effectively addresses spectrum aliasing issues encountered in EMD. Moreover, VMD demonstrates superior robustness in signal sampling and noisy environments, making it highly adaptive. However, when employing VMD for signal noise reduction processing, setting parameters in advance is necessary. The decomposition layer K and penalty factor α significantly influence decomposition results and performance. Reliance on experience or prior criteria for parameter selection can lead to inaccurate decomposition results and diminish the efficiency of VMD.

The methods used to determine the value of VMD parameter K are mainly divided into two categories. In the first category [12,13,14], the minimization residual energy value method, frequency characteristic distribution method, and kurtosis principle are used to select the optimal value of parameter K, all of which can effectively decompose signals, but the applicability of different signals is poor, which can easily cause under-decomposition. The second type [15,16] uses the optimization algorithm to optimize parameter K, but the computation is too large and the real-time performance is poor. However, the above research content only focuses on the optimization of one of the decomposition numbers K and penalty factor α, the calculation steps are more complicated, and the comprehensive influence of the synergistic effect between the two parameters on the decomposition effect is not fully considered. In order to comprehensively consider the selection of VMD parameters, particle swarm optimization (PSO) is used in [17] to optimize the decomposition parameter combination of the VMD algorithm, which significantly enhances the decomposition energy efficiency compared with the central frequency observation method. However, this algorithm is easily subject to the local optimal solution. And the computational efficiency needs to be further improved. In [18], the grasshopper optimization algorithm (GOA) was used to achieve adaptive selection of VMD parameters. However, the global search ability of the GOA algorithm declined in the late iteration period, resulting in low convergence accuracy of the algorithm. In [19], a whale optimization algorithm (WOA) was applied to optimize VMD decomposition signals. Due to its few control parameters and powerful optimization ability, WOA accurately extracted effective information of each frequency band of signals and achieved good performance. However, the heuristic optimization algorithms mentioned above have some inherent defects in the process of solving problems, such as low convergence accuracy and slow convergence speed, and it is easy to fall into local optimal solutions. In view of the difference in characteristics of signal and noise in different practical application scenarios, the presence of noise may cause the decomposed IMF to no longer purely represent the different frequency components of the signal, but may contain the characteristics of noise [20]. In the case of low SNR, noise components may be amplified in the decomposition, thus confusing the real signal components, and the performance of VMD may be more challenging [21]. Therefore, when the heuristic algorithm optimizes VMD, how to reasonably configure the fitness function in these fields, improve the search efficiency of the algorithm, and avoid falling into the dilemma of local optimization has become a key issue to be solved [22].

Building on the analysis presented, this paper addresses the problem of traditional harmonic detection methods being susceptible to noise interference which reduces their detection accuracy. It introduces a novel approach that combines VMD optimized through parameter tuning with wavelet threshold denoising. This method utilizes the minimal sample entropy of the signal as the objective function for parameter optimization via the rime optimization algorithm (RIME). Once the optimal parameters are identified, the signal undergoes VMD decomposition. Subsequently, the Pearson correlation coefficient method is employed to adaptively distinguish between effective and ineffective components, followed by denoising the effective components using improved wavelet thresholding (IWT). The denoised components are then reconstructed to obtain the final denoised signal. The efficacy and accuracy of the proposed RIME-VMD-IWT method are validated through simulations, real signal measurements, and comparative analyses with other methods.

2. VMD Parameter Optimization and Wavelet Noise Reduction

2.1. Variational Mode Decomposition

VMD is an adaptive and robust signal processing method specifically developed to extract inherent modal components with varying frequency and time-frequency localization characteristics from non-stationary signals. The entire VMD process can be conceptualized as the construction and solution of a constrained variational problem, as illustrated in Equation (1).

$\underset{{u_{k}, ω_{k}}}{m i n} \{\sum_{k} {‖\partial_{t} [(δ (t) + \frac{j}{π t}) * u_{k} (t)] e^{- j ω_{k} t}‖}_{2}^{2}\} s . t . \sum_{k} u_{k} = f$

(1)

where u_k represents the Intrinsic Mode Function (IMF), ω_k denotes the central frequency of the IMF, and f represents the input signal.

The constrained variational problem described in Equation (1) is solved by introducing an augmented Lagrange multiplier λ and a quadratic penalty factor α, as illustrated in Equation (2).

$\begin{array}{l} L (\{u_{k}\}, \{ω_{k}\}, λ) = α \sum_{k} {‖\partial_{t} [(δ (t) + \frac{j}{π t}) * u_{k} (t)] e^{- j ω_{k} t}‖}_{2}^{2} \\ + {‖f (t) - \sum_{k} u_{k} (t)‖}_{2}^{2} + 〈λ (t), f (t) - \sum_{k} u_{k} (t)〉 \end{array}$

(2)

The alternating direction multiplier method is utilized to determine the values of $u_{k}^{n + 1}$ , $ω_{k}^{n + 1}$ , and $λ^{n + 1}$ , which are continuously updated throughout the process. Once certain conditions ${\sum_{k} ‖{\hat{u}}_{k}^{n + 1} - {\hat{u}}_{k}^{n}‖}_{2}^{2} / {‖{\hat{u}}_{k}^{n}‖}_{2}^{2} < ε$ are met, the iteration is halted, and K IMF components are output.

In summary, one of the most critical parameters in the VMD algorithm is the number of decomposition layers K, which is predetermined without prior knowledge of the signal under analysis. Another crucial parameter is the penalty factor α, which influences how noise interference is suppressed and thus requires careful selection. The primary focus of applying the VMD method is to identify the best parameter combination tailored to the specific signal being analyzed.

2.2. Optimize VMD Based on RIME

RIME is a heuristic optimization method introduced by Hang Su et al. in 2023 [23]. It exhibits efficient global search and local refinement capabilities, making it resilient against noise and multi-modal function challenges. In comparison to other heuristic algorithms like PSO and WOA, RIME boasts a simpler mechanism, fewer parameters, robust optimization capability, fast convergence speed, and excellent black-box properties. Hence, this paper opts for the RIME algorithm to optimize VMD parameters.

The RIME algorithm draws inspiration from the natural phenomenon of rime, simulating the growth process of rime. It is developed through a search process that mimics this natural phenomenon. RIME comprises four primary processes:

2.2.1. Initial Population

Each rime individual serves as a search agent within the algorithm, forming the rime population. Initially, the entire rime population is initialized. The frost-ice population comprises n rime agent S_i, with each rime agent composed of d rime particles. Consequently, the rime population R can be directly represented by the rime particles x_ij.

$R = [\begin{matrix} S_{1} \\ \begin{matrix} S_{2} \\ ⋮ \end{matrix} \\ S_{i} \end{matrix}]; S_{i} = [x_{i 1}, x_{i 1}, \dots, x_{i j}]$

(3)

$R = [\begin{matrix} x_{11} & x_{12} & \dots & x_{1 j} \\ x_{21} & x_{22} & \dots & x_{2 j} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ x_{i 1} & x_{i 2} & \dots & x_{i j} \end{matrix}]$

(4)

where R represents the rime population; S_i denotes a rime individual; x_ij represents a rime particle; i is the ordinal number of rime individuals; j is the ordinal number of rime particles. Additionally, F(Si) represents the growth state of each rime individual, i.e., the fitness value of the individual in the meta-heuristic algorithm.

2.2.2. Soft Rime Search

The process simulates the movement of rime particles in a breezy environment, serving as the exploration phase of the algorithm. Rime particles update their positions by moving toward the vicinity of the soft rime body and coagulating with the particles within it. The position update formula is as follows:

$R_{i j}^{n e w} = R_{b e s t, j} + r_{1} \cdot \cos θ \cdot β \cdot (h \cdot (U b_{i j} - L b_{i j}) + L b_{i j}), r_{2} < E$

(5)

$θ = π \cdot \frac{t}{10 \cdot T}$

(6)

$E = \sqrt{(t / T)}$

(7)

$β = 1 - [\frac{w \cdot t}{T}] / w$

(8)

where $R_{i j}^{n e w}$ represents the updated position of rime particles, i and j denote the j rime particles of the i frost-ice individual; R_best_,j represents the j^th rime particle of the best rime individual in the rime population R; r₁ and r₂ are random numbers within the range [−1,1]; β is an environmental factor; w is the number of step function segments, with a default value of 5; h represents adhesion, is a random number within the range [0,1]; Ub_ij and Lb_ij are the upper and lower bounds of the particle diffusion space, respectively. E is the adhesion coefficient, t denotes the current iteration number, and T represents the maximum iteration number of the algorithm.

2.2.3. Hard Rime Piercing

This process simulates the growth of rime particles under challenging conditions and is primarily employed in the optimization stage to enhance the interchangeability of algorithm particles, thereby improving algorithm convergence and the ability to escape local optima. The location update formula for this process is as follows:

$R_{i j}^{n e w} = R_{b e s t, j}, r_{3} < F^{n o r m r}$

(9)

where normr is the normalized value of the current individual fitness value, representing the probability of selecting the i^th frost-ice individual. r₃ is a random number within the range [−1,1].

2.2.4. Greedy Choice

To enhance population diversity in the RIME algorithm, a greedy selection mechanism is designed to actively engage in population renewal. This mechanism compares the updated individual fitness value of rime with its value before the update. If the updated fitness value improves upon the previous value, the solutions of two rime individuals are replaced simultaneously. This process ensures that the population evolves in a better direction with each iteration.

2.3. IMF Component Screening

Following the optimized VMD of the signal, K IMF components are obtained, comprising both effective and noise components. Each IMF component is adaptively selected based on the Pearson correlation coefficient, which reflects the degree of noise within the time series.

The Pearson correlation coefficient is a widely used metric for assessing relationships [24]. It finds extensive application in fields such as data analysis and fault diagnosis [25]. The definition of this coefficient is provided in Equation (10).

$ρ_{x, y} = \frac{\sum_{i = 1}^{n} (X_{i} - \bar{X}) (Y_{i} - \bar{Y})}{\sqrt{\sum_{i = 1}^{n} {(X_{i} - \bar{X})}^{2}} \sqrt{\sum_{i = 1}^{n} {(Y_{i} - \bar{Y})}^{2}}}$

(10)

where (X_i, Y_i) represents the ith values of two sets, X and Y, respectively; (X, Y) denotes the means of these sets; n is the number of elements in each set. The correlation coefficient ρ_x_,y ranges from −1 to 1, reflecting the strength of the correlation between each component and the original signal.

In general, the degree of correlation between variables can be assessed according to the standards outlined in Table 1.

2.4. Improved Wavelet Thresholding Denoising

While parameter-optimized VMD and its mode recognition can effectively eliminate Gaussian white noise, they may not completely remove all types of noise. To address this issue, residual noise is processed using an improved wavelet threshold denoising method. By leveraging both the time-frequency localization capabilities of VMD and the noise reduction properties of wavelet threshold denoising, this approach achieves enhanced signal denoising performance.

The efficacy of the wavelet threshold denoising method hinges on the selection of the threshold value and threshold function. Variations in the threshold function reflect differing coefficient estimation methods. Additionally, the magnitude of the threshold critically affects the noise reduction outcomes; only through careful selection of the threshold can significant denoising be achieved without the loss of vital signal components [26]. Traditional wavelet thresholds include both hard and soft threshold functions, which are expressed as follows:

${\hat{w}}_{j, k} = \{\begin{matrix} \begin{matrix} w_{j, k}, & |w_{j, k}| \geq λ \end{matrix} \\ \begin{matrix} 0, & \begin{matrix} |w_{j, k}| < λ \end{matrix} \end{matrix} \end{matrix}$

(11)

${\hat{w}}_{j, k} = \{\begin{matrix} sgn (w_{j, k}) \cdot (|w_{j, k}| - λ), & |w_{j, k}| \geq λ \\ 0, & |w_{j, k}| < λ \end{matrix}$

(12)

Although hard and soft thresholds are extensively employed in practical applications, they exhibit inherent limitations, including discontinuous breakpoints and signal distortions. To address these issues, this paper introduces a novel method for selecting an improved wavelet threshold function, building upon the traditional wavelet threshold approach to rectify its deficiencies. The expression for the enhanced wavelet threshold is as follows:

${\hat{w}}_{j, k} = \{\begin{matrix} w_{j, k}, & |w_{j, k}| > λ_{2} \\ sgn (w_{j, k}) \frac{λ_{1} \sqrt{w_{j, k}^{2} - λ_{1}^{2}}}{a \sqrt{λ_{2}^{2} - λ_{1}^{2}}}, & λ_{1} \leq |w_{j, k}| \leq λ_{1} \\ 0, & λ_{1} < |w_{j, k}| \end{matrix}$

(13)

where λ₁ and λ₂ are thresholds, with λ₁ = aλ₂, the adjustment parameter ‘a’ is used to control the slope of the curve between λ₁ and λ₂, thus providing better flexibility to the improved threshold function. Specifically, λ₁ is defined as λ₁ = σ2lgN, where σ, the standard deviation of noise, is calculated as σ = median (w_j_,k)/0.6745. Here, N denotes the signal length, and median (·) refers to the operation of finding the median.

The improved wavelet threshold function, w_j_,k, is continuous at ±λ₁ and ±λ₂, ensuring continuity within the wavelet domain. This enhancement overcomes the discontinuities inherent in traditional wavelet thresholds. The flexibility of this improved function allows for adaptation to diverse environments through the selection of the tuning parameter a, which ranges from 0 to 1. At lower values of a, the curve between λ₁ and λ₂ has a gentler slope, making it more akin to the original threshold. Conversely, higher values of a increase the slope, aligning it closer to a hard threshold. The value of a can be adjusted based on the requirements of different vibration signals, facilitating a’s seamless transition between the new and traditional hard threshold functions. Figure 1 illustrates the function graphs of the new wavelet threshold at varying values of a.

3. Harmonic Detection Process Based on RIME-VMD-IWT

On the basis of the above methods, this paper presents a harmonic detection method based on the RIME optimization algorithm to optimize VMD combined with wavelet threshold denoising. Firstly, VMD is optimized based on RIME to find the best parameter decomposition layer K and penalty factor α, so as to achieve the optimal signal decomposition effect. Then, the VMD decomposed the noisy signal to obtain K modal components and screened the modal components by calculating and analyzing the correlation coefficient between the K modal components and the original signal. The modal components were denoised by wavelet threshold and reconstructed to obtain the final denoised harmonic signal. Finally, the Hilbert transform is used to extract the amplitude and frequency information.

Step 1:: Read the harmonic signal data and preprocess it with noise.
Step 2:: RIME was employed in the search for the optimal parameters K and α in VMD, utilizing sample entropy as the fitness function during the iterative process. Through RIME, the individual with the minimum sample entropy was identified, and the corresponding parameter combination [K,α] of this individual was designated as the optimal parameter set for VMD.
Step 3:: After the noisy signal is decomposed by VMD, K modal components are obtained. Based on the correlation coefficient of each IMF, the components are categorized into effective IMFs and ineffective IMFs.
Step 4:: The invalid IMFs are eliminated, and the effective IMFs undergo de-noising treatment using IWT, resulting in the reconstruction of the signal and obtaining the final de-noised harmonic signal.
Step 5:: Utilize the Hilbert transform to extract amplitude, frequency, and other information for detecting harmonic signal parameters.

4. Simulation Signal Verification

To validate the effectiveness of the proposed method, several common denoising and harmonic detection methods are tested and compared with the proposed approach. The constructed harmonic signal with added noise is utilized for verification.

The proposed algorithm is evaluated using power harmonic signals contaminated with noise. The parameters are as follows:

$f (t) = 5 \sin (2 π \times 50 t) + 10 \sin (2 π \times 150 t) + 15 \sin (2 π \times 180 t) + 30 \sin (2 π \times 284 t) + n o i s e$

(14)

The signal f(t) comprises fundamental waves, third harmonics, and interharmonics with frequencies of 180 Hz and 284 Hz, representing harmonic signals typical of real-world environments. White Gaussian noise is introduced to f(t) to create the original signal, with a signal-to-noise ratio of 10 dB. The sampling frequency is set to 1000 Hz, and the sampling number is 1000. Figure 2 and Figure 3 depict the time-domain waveform and spectrum analysis diagram of the original signal.

The RIME-VMD algorithm was employed to decompose the signal with noise. RIME was utilized to optimize the VMD parameters (K,α). The population size was set to 30, with a maximum of 20 iterations. The iteration range for K was set from 0 to 10 and for α from 0 to 3000. After iteration, the fitness function reached a minimum value of 0.079748, and the optimal parameter combination (K,α) = (6, 1834) was obtained.

To directly assess the advantages of the RIME-VMD algorithm, the WOA and PSO algorithms were each employed to optimize and compare VMD parameters. The parameter settings for each algorithm were consistent, and the fitness iteration convergence curve is depicted in Figure 4. As illustrated in Figure 4, the RIME algorithm demonstrates excellent convergence speed and high-precision convergence ability in optimizing VMD decomposition signals. Although the fitness value of the PSO algorithm is higher than that of WOA and RIME algorithms, the RIME algorithm achieves the minimum fitness value by the fourth iteration, indicating faster convergence speed and higher accuracy. Overall, the RIME algorithm outperforms the other two algorithms in terms of convergence speed and accuracy.

The decomposition results of VMD after parameter optimization are presented in Figure 5. As depicted in Figure 5, it is evident that the RIME-VMD method effectively decomposed the noisy harmonic signal into 6 IMFs, accurately extracting sub-signals of each frequency while mitigating mode aliasing and end effects. Each IMF represents independent single-frequency harmonic components and noise components. IMF2 to IMF4 exhibit smooth and regular waveforms, albeit affected by noise, while IMF5 and IMF6 demonstrate distorted sine wave characteristics, containing a higher level of noise.

In order to effectively screen out valid IMFs and invalid IMFs, the absolute value of each IMF’s correlation coefficient |ρ| is calculated after decomposing the signal to obtain IMFs, and the randomness of each IMF component is tested. Equation (10) is used to calculate the correlation coefficient of each IMF. The correlation coefficient of each IMF after harmonic signal decomposition is shown in Figure 6. First, based on the degree of correlation between the components shown in Table 1, the invalid IMF components with |ρ| < 0.2 are removed; the |ρ| values of IMF5 and IMF6 in Figure 6 are <0.2 and much smaller than other components, so they are discarded. Then, IMF1, IMF2, IMF3, and IMF4, which contain noise, are processed using the IWT denoising method. The wavelet parameters are set to db4, and the number of decomposition layers is set to 3 based on empirical values. Finally, the processed effective modal components are reconstructed to obtain the denoised signal.

4.1. Noise Reduction Effect Analysis

The signal-to-noise ratio (SNR) and root-mean-square error (RMSE) are chosen as the evaluation metrics for noise reduction. SNR quantifies the ratio of the overall signal to the noise components, with higher values indicating superior noise reduction efficacy. The RMSE, on the other hand, measures the average magnitude of the errors between detected and actual values, a smaller RMSE indicates a higher accuracy or precision of the detection. The calculation formulas for SNR and RMSE are as follows:

$RMSE = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} {(x_{i} - y_{i})}^{2}}$

(15)

$SNR = 10 \times \lg (\frac{\sum_{i = 1}^{N} x_{i}^{2}}{\sum_{i = 1}^{N} {(x_{i} - y_{i})}^{2}})$

(16)

where N represents the data length, x denotes the signal after noise reduction, and y represents the original signal.

In order to evaluate the noise reduction effect of the proposed method, the WT (Soft), WT (Hard), EMD-WT (Soft), VMD-WT (Soft), and the RIME-VMD-IWT proposed in this paper are used to reduce noise in simulation signals. The VMD parameter combination (K,α) is set to (4, 2000), while the parameter combination (K,α) optimized by RIME is (6, 1834). As can be seen from Table 2, for simulated signals, the noise reduction effect of the wavelet soft threshold is slightly better than that of the hard threshold, and all noise reduction evaluation indexes of the RIME-VMD-IWT proposed in this paper are optimal. The noise reduction effect of RIME-VMD-IWT (Soft) is 22.36% higher in terms of SNR compared to VMD-WT (SOFT) and the other four methods. RMSE decreased by 14.93%. The root-mean-square error of the denoising method is the smallest, indicating that the deviation of the component after denoising is smaller. A larger SNR indicates that more effective signals are extracted. Figure 7 shows a comparison of the waveforms obtained by different noise reduction methods. It can be seen that EMD-WT (Soft) and VMD-WT (Soft) have a certain effect on noise removal, but the signal waveform after noise reduction has a poor fitting effect on the original sequence. However, the waveform after noise reduction using the method proposed in this paper is closer to the simulation signal, and the curve fitting degree is higher. Most of the peak noise signal is removed, and the end effect and mode aliasing problems that occur during EMD denoising are avoided. This makes the waveform smoother and more continuous while retaining most of the features of the original signal. Therefore, the simulation signal experiment results demonstrate that the proposed RIME-VMD-IWT method has a better noise processing effect and strong noise reduction ability.

4.2. Detection Accuracy Analysis

After denoising using wavelet thresholding, the instantaneous amplitude and instantaneous frequency of each component are computed via the Hilbert transform. As presented in Table 3, the proposed method exhibits high detection accuracy for simulated signals. Specifically, in a noise background with an SNR of 10 dB, the average detection errors for amplitude and frequency are 4.6 × 10⁻⁴ and 1.1 × 10⁻³, respectively. These results indicate the capability of accurately detecting simulated harmonic signals and possessing robustness against noise.

In order to quantitatively evaluate the detection effect and anti-noise ability of the proposed RIME-VMD-IWT method for harmonic signals, the detection accuracy of three existing harmonic detection methods under different noise levels was analyzed, and the average error of amplitude and frequency was used to measure the detection effect of each method. As shown in Figure 8, the overall performance of the proposed method is superior to that of the other algorithms. Specific data indicate that the average errors of detection amplitude and frequency for the EMD method are 16.729% and 11.460%, respectively, while those for the VMD method are 6.549% and 4.017%, respectively. However, the detection error of RIME-VMD-IWT is significantly lower than that of the other three methods. As can be seen from Figure 8, under the background of a low signal-to-noise ratio, the proposed method has high detection accuracy, minimal average errors in detected amplitude and frequency, and it can still achieve significant noise reduction results for environmental noise components under different noise intensities while retaining more signal details. It also exhibits good robustness. Combining the above simulation results, it is proven that the RIME-VMD-IWT algorithm proposed in this paper exhibits high accuracy and superiority in the processing of power harmonic signals.

5. Test Signal Verification

The method proposed in this paper is validated using actual arc furnace signals [27]. The sampling method remains consistent with the previously described approach, and the parameter details are outlined in Equation (17). This signal comprises a 50 Hz fundamental wave, 125 Hz, and 25 Hz interharmonics, along with random Gaussian white noise interference, resulting in a signal-to-noise ratio of approximately 26 dB.

$f^{'} (t) = 100 \cos (100 π t + 20) + 74.813 \cos (250 π t + 9) + 64.933 \cos (50 π t + 93) + n o i s e$

(17)

5.1. Noise Reduction Effect Analysis

As seen in Table 4, the noise reduction effect of VMD-WT (Soft) is slightly superior to that of WT and EMD-WT (Soft) for measured signals. Compared to the VMD-WT (Soft) noise reduction algorithm with default parameters, the signal-to-noise ratio of the proposed method in this paper is increased by 13.01%, and the root-mean-square error is reduced by 9.26%. All noise reduction evaluation indices remain optimal. Figure 9 is a comparison diagram of the waveforms of different denoising methods for the measured signal. It can be observed that after denoising processing using the method proposed in this paper, the signal waveform curve is smooth, the original shape of the actual signal is retained, and the degree of fitting is higher. Therefore, the experimental results for the measured signal demonstrate that the RIME-VMD-IWT algorithm proposed in this paper can effectively filter out actual noise, and the noise reduction effect is more significant.

5.2. Detection Accuracy Analysis

Table 5 presents the harmonic detection results of the proposed method along with EMD, CEEMD, and VMD algorithms. The RIME-VMD-IWT method maintains a high detection accuracy for the measured harmonic signals. Under a noise background of 26 dB, the detection error of EMD and CEEMD is significant due to mode aliasing. The proposed RIME-VMD-IWT method can not only address the mode aliasing phenomenon but also filter out a substantial amount of noise signals, enabling accurate detection of simulated harmonic signals and demonstrating good noise robustness. The average detection errors for amplitude and frequency are 6.1 × 10⁻³ and 2.5 × 10⁻³, respectively. Particularly in terms of frequency detection, the results of this method are closer to the original data. Figure 10 compares the detection accuracy between other detection methods and the method proposed in this paper. The results indicate that, in comparison to EMD, CEEMD, and VMD algorithms, the proposed RIME-VMD-IWT algorithm exhibits less susceptibility, greater adaptability, and improved accuracy in detecting harmonic parameters under actual noise interference.

6. Conclusions

Addressing the issues of insufficient accuracy and noise interference in existing harmonic detection methods, this paper introduces a novel harmonic detection approach that integrates parameter optimization with VMD and improved wavelet threshold denoising. The rime optimization algorithm is utilized to optimize the VMD parameters, thereby enhancing the signal decomposition performance. The effective component is selected using the correlation coefficient method, and then, combined with the improved wavelet threshold, the noisy effective component is denoised, and the signal is reconstructed. Finally, the Hilbert transform is employed to detect the amplitude and frequency information of the harmonic signal, effectively achieving accurate harmonic signal detection in power systems. Through simulation experiment analysis, the following conclusions are drawn:

By utilizing the RIME optimization algorithm to optimize the VMD parameters (K,α), the issue of manually setting the K and α values in the traditional VMD algorithm is effectively addressed. This approach achieves optimal signal decomposition, reduces the VMD decomposition error, demonstrates stronger adaptability, and has a faster optimization speed compared to PSO and WOA algorithms.
By introducing the Pearson correlation coefficient and improving wavelet threshold denoising, the effective signal and noise signal can be effectively separated from the mixed signal, while retaining the original characteristics of the signal. This improves the overall noise robustness of the algorithm, making it suitable for harmonic signal denoising.
In comparison with EMD, WT, and VMD methods, the proposed approach exhibits a remarkable noise reduction effect and demonstrates accurate detection of harmonic amplitude and frequency even in high-noise environments. Its detection performance surpasses that of traditional methods, showcasing significant advantages in accuracy.
See Also
Simplisafe Entry Sensor Not Responding: Troubleshooting Guide

The method presented in this paper has hitherto been applied solely to simulations and harmonic data of a specific type of power equipment. Moving forward, the approach will be extended to monitoring data of diverse power equipment to explore the characteristics of more extensive power equipment monitoring signals. Taking into account the actual operational environment of power systems, future research endeavors aim to enhance the real-time processing capabilities of the algorithm and investigate the feasibility of hardware implementation, thereby fulfilling the requirements of real-time online monitoring. These studies are anticipated to foster the further evolution of harmonic detection technology and provide more robust technical underpinnings for the stable and secure operation of power systems.

Author Contributions

Conceptualization, J.X.; methodology, J.X.; software, J.X.; validation, J.X. and W.H.; writing—original draft preparation, J.X. and W.H.; writing—review and editing, H.M.; visualization, W.H. All authors have read and agreed to the published version of the manuscript.

Funding

This research is supported by the Pyramid Talent Training Project of Beijing University of Civil Engineering and Architecture (GJZJ20220802), and in part by the Doctoral Scientific Research Foundation of Beijing University of Civil Engineering and Architecture (ZF15054).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data reported in this study can be found in the sources cited in the reference list.

Conflicts of Interest

The authors declare no conflicts of interest.

References

Zhang, Z.; Kang, C. Challenges and Prospects for Constructing the New-type Power System Towards a Carbon Neutrality Future. Proc. CSEE 2022, 42, 2806–2819. [Google Scholar]
Dawei, R.; Jinyu, X.; Jinming, H.; Ershun, D.; Chen, J.; Yao, L. Construction and Evolution of China’s New Power System Under Dual Carbon Goal. Power Syst. Technol. 2022, 46, 3831–3839. [Google Scholar]
Xiang, L.; Haichao, W.; Wenxue, W.; Quan, A. Adaptability Assessment of New Energy Connected to the Power Grid Based on the Carrying Capacity. Proc. CSEE 2023, 43, 107–113. [Google Scholar]
Zhenguo, S.; Haobo, X.; Songyong, X.; Guochang, W.; Yi, Z. Harmonic problems in a new energy power grid. Power Syst. Prot. Control 2021, 49, 178–187. [Google Scholar]
Jiaming, D.; Ma, H. Harmonic detection method based on complex modulation refinement and adaptive linear network. China Meas. Test 2022, 48, 43–50. [Google Scholar]
Jiati, S.; Wang, Y. The harmonic detedtion strategy for APE based on the optimized split-radix FFT algorithm. Electron. Meas. Technol. 2023, 46, 23–29. [Google Scholar]
Jun, Z.; Yongxiang, L.; Jiapeng, R. Harmonic Detection Method Based on Modified Ensemble Empirical Mode Decomposition and Hilbert Transform. Proc. CSU-EPSA 2023, 35, 73–82. [Google Scholar]
Anping, S.; Lv, Z.; Cheng, W.; Xinyi, L.; Jun, M. GB-RAR bridge monitoring signal denoising based on EMD and wavelet threshold denoising. Bull. Surv. Mapp. 2022, s2, 227–232+240. [Google Scholar]
Peng, L.; Jiayi, C.; Lijiang, G.; Hang, Y.; Mengyu, L. Study on de-noising of signal of power station based on empirical mode and modified wavelet soft-threshold de-noising method. Mech. Eng. 2023, 45, 736–743. [Google Scholar]
Zhijun, L.; Hongpeng, Z.; Yanan, W.; Xiao, L. Wavelet threshold denoising harmonic detection method based on permutation entropy-CEEMD decomposition. Electr. Mach. Control 2020, 24, 120–129. [Google Scholar]
Dragomiretskiy, K.; Zosso, D. Variational mode decomposition. IEEE Trans. Signal Process. 2014, 62, 531–544. [Google Scholar] [CrossRef]
Xiaojiao, Z.; Bin, W.; Bujuan, L.; Min, Y. Inter-harmonics detection based on parameter optimization variational mode decomposition. Power Syst. Prot. Control 2022, 50, 71–80. [Google Scholar]
Darong, H.; Lanyan, K.; Mengting, L.; Guoxi, S. A new fault diagnosis approach for bearing based on multi-scale entropy of the optimized VMD. Control Decis. 2020, 35, 1631–1638. [Google Scholar]
Weiguang, L.; Qinhong, L.; Xianwu, M. A VMD-SVD micro-motor sound signal noise reduction method based on the kurtosis principle. China Meas. Test 2023, 49, 111–118. [Google Scholar]
Dongfang, R.; Jiaqing, M.; Zhiqin, H.; Qinmu, W. Short-Term Wind Power Prediction Research Based on AVMD-CNN-GRU-Attention. Acta Energiae Solaris Sin. 2023, 1–8. [Google Scholar] [CrossRef]
Yuzhao, M.; Qingxiao, Z.; Qiming, L.; Meng, L. Noise Reduction Method for Optical Fiber Perimeter Intrusion Signal Based on POA-VMD and MPE. Laser Optoelectron. Prog. 2024, 1–17. Available online: http://kns.cnki.net/kcms/detail/31.1690.TN.20240220.1052.040.html (accessed on 1 June 2024).
Min, J.; Ming, W.; Jianqiang, Z. Vibration Signal Reconstruction and Fault Diagnosis of Rolling Bearings Based on PSO Optimization VMD Algorithm. Mach. Des. Res. 2022, 38, 138–141+147. [Google Scholar]
Yi, Z.; Jianhai, Y.; Jing, J.; Xinyuan, G. Fault feature extraction method of rolling bearing based on parameter optimized VMD. J. Vib. Shock 2021, 40, 86–94. [Google Scholar]
Tingye, Q.; Huiru, W.; Guorui, F.; Xinjun, Z.; Chuantao, Y.; Dekang, Z.; Sunwen, D. Denoising method of transient electromagnetic detection signal based on WOA-VMD algorithm. J. Cent. South Univ. (Sci. Technol.) 2021, 52, 3885–3898. [Google Scholar]
Yao, S.; Yajing, W.; Yu, M.; Siqi, L. VMD harmonic detection method based on WPT and parameter optimization. Electr. Meas. Instrum. 2022, 59, 166–171. [Google Scholar]
Zhiyong, J.; Qiu, T.; Yaxin, L.; Zhaosheng, T.; Wei, Q. Power system harmonic analysis under non-stationary situations based on AVMD and improved energy operator. Chin. J. Sci. Instrum. 2022, 43, 209–217. [Google Scholar]
Yanhao, X.; Hao, Y.; Jia, Z.; Jun, G. Research on the O-VMD thickness measurement data processing method based on particle swarm optimization. Chin. J. Sci. Instrum. 2023, 44, 304–313. [Google Scholar]
Su, H.; Zhao, D.; Heidari, A.A.; Liu, L.; Zhang, X.; Mafarja, M.; Chen, H. RIME: A physics-based optimization. Neurocomputing 2023, 532, 183–214. [Google Scholar] [CrossRef]
Zhong, J.; Liu, Z.; Bi, X. Partial Discharge Signal Denoising Algorithm Based on Aquila Optimizer–Variational Mode Decomposition and K-Singular Value Decomposition. Appl. Sci. 2024, 14, 2755. [Google Scholar] [CrossRef]
Wang, Y.; Feng, H.; Xu, N.; Zhong, J.; Wang, Z.; Yao, W.; Jiang, Y.; Laima, S. A Data-Driven Model for Predictive Modeling of Vortex-Induced Vibrations of a Long-Span Bridge. Appl. Sci. 2024, 14, 2233. [Google Scholar] [CrossRef]
Tong, Y.; Arimura, H.; Yosh*take, T.; Cui, Y.; Kodama, T.; Shioyama, Y.; Wirestam, R.; Yabuuchi, H. Prediction of Consolidation Tumor Ratio on Planning CT Images of Lung Cancer Patients Treated with Radiotherapy Based on Deep Learning. Appl. Sci. 2024, 14, 3275. [Google Scholar] [CrossRef]
Lei, C.; Dezhong, Z.; Xingtao, Z.; Wenzhe, L. Harmonic/Interharmonic Detection Based on FFT and Optimized Matching Pursuit Tracking. Adv. Technol. Electr. Eng. Energy 2016, 35, 7–12+37. [Google Scholar]

Figure 1. Function of the new threshold for different values of a.

Figure 2. Time-domain waveform of the original signal.

Figure 3. Spectrum analysis diagram of the original signal.

Figure 4. Fitness iteration curve.

Figure 5. VMD decomposition results.

Figure 6. Correlation coefficient of IMF.

Figure 7. Simulation signal waveform comparison.

Figure 8. Error detection by simulation signal; (a) amplitude detection error; (b) frequency detection error.

Figure 9. Measured signal waveform comparison.

Figure 10. Error detection by measured signal; (a) amplitude detection error; (b) frequency detection error.

Table 1. Degree of correlation coefficient.

Correlation Coefficient	Degree of Correlation
0.8 < \|ρ\| ≤ 1.0	extremely relevant
0.6 < \|ρ\| ≤ 0.8	highly relevant
0.4 < \|ρ\| ≤ 0.6	moderately relevant
0.2 < \|ρ\| ≤ 0.4	weakly relevant
0.0 < \|ρ\| ≤ 0.2	almost irrelevant

Table 2. Noise reduction index of simulation signal.

Denoising Method	SNR/dB	RMSE
WT (Hard)	16.36	0.3364
WT (Soft)	17.53	0.3182
EMD-WT (Soft)	21.94	0.2904
VMD-WT (Soft)	22.59	0.2512
RIME-VMD-IWT	27.64	0.2137

Table 3. Results of simulation signal detection.

Method	Frequency True Value/Hz	Frequency Detection Value/Hz	Frequency Error	Amplitude True Value/A	Amplitude Detection Value/A	Amplitude Error
RIME-VMD-IWT	50	50.0026	5.1 × 10⁻⁴	5	4.9984	3.2 × 10⁻⁴
	150	150.4950	3.3 × 10⁻³	10	9.9975	2.5 × 10⁻⁴
	180	180.0504	2.8 × 10⁻⁴	15	14.9972	1.9 × 10⁻⁴
	284	284.0224	7.9 × 10⁻⁵	30	29.9760	1.1 × 10⁻³
EMD	50	57.1034	14.2 × 10⁻²	5	4.0812	18.4 × 10⁻²
	150	166.5371	11.0 × 10⁻²	10	8.2933	17.1 × 10⁻²
	180	198.5419	10.3 × 10⁻²	15	17.4750	16.5 × 10⁻²
	284	239.6962	15.6 × 10⁻²	30	35.0717	16.9 × 10⁻²
CEEMD	50	53.1033	6.2 × 10⁻²	5	5.2352	4.7 × 10⁻²
	150	136.9581	8.7 × 10⁻²	10	9.3846	6.2 × 10⁻²
	180	165.7834	7.9 × 10⁻²	15	14.3250	4.5 × 10⁻²
	284	260.1447	8.4 × 10⁻²	30	31.6207	5.4 × 10⁻²
VMD	50	48.3981	3.2 × 10⁻²	5	5.3651	7.3 × 10⁻²
	150	156.1574	4.1 × 10⁻²	10	10.6204	6.2 × 10⁻²
	180	173.1641	3.8 × 10⁻²	15	15.8762	5.8 × 10⁻²
	284	295.9287	4.2 × 10⁻²	30	28.4790	5.1 × 10⁻²

Table 4. Noise reduction indexes of simulated signals.

Denoising Method	SNR/dB	RMSE
WT (Hard)	36.14	0.0971
WT (Soft)	37.02	0.0962
EMD-WT (Soft)	41.47	0.0918
VMD-WT (Soft)	43.71	0.0875
RIME-VMD-IWT	49.39	0.0794

Table 5. Results of measured signal detection.

Method	Frequency True Value/Hz	Frequency Detection Value/Hz	Frequency Error	Amplitude True Value/A	Amplitude Detection Value/A	Amplitude Error
RIME-VMD-IWT	50	50.1208	2.4 × 10⁻³	100	99.9542	4.6 × 10⁻⁴
	125	125.4715	3.8 × 10⁻³	74.813	75.2469	5.8 × 10⁻³
	25	25.0342	1.4 × 10⁻³	64.933	65.7142	1.2 × 10⁻²
EMD	50	45.7328	8.6 × 10⁻²	100	89.7300	1.0 × 10⁻¹
	125	114.8753	8.1 × 10⁻²	74.813	66.2843	1.1 × 10⁻¹
	25	26.9254	7.7 × 10⁻²	64.933	71.8808	1.1 × 10⁻¹
CEEMD	50	52.2517	4.5 × 10⁻²	100	91.1271	8.9 × 10⁻²
	125	117.1252	6.3 × 10⁻²	74.813	81.6958	9.2 × 10⁻²
	25	26.8513	7.4 × 10⁻²	64.933	69.8679	7.6 × 10⁻²
VMD	50	51.9520	3.9 × 10⁻²	100	94.3417	5.7 × 10⁻²
	125	130.7500	4.6 × 10⁻²	74.813	79.4506	6.2 × 10⁻²
	25	26.0542	4.2 × 10⁻²	64.933	68.2438	5.1 × 10⁻²

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).