1. Introduction
Due to the fact that power electronic devices are widely deployed in the power grid, in addition to the low-frequency harmonic emissions, the phenomenon of supraharmonic emissions in the frequency range 2 kHz–150 kHz has also been observed [
1,
2,
3]. There is a relatively complete system for the low-order harmonic quantification method and evaluation criteria. More concerns have been turned to the emissions in the supraharmonic frequency band. At present including generation mechanism, propagation mode, and interactive influence, various researches on supraharmonics have been discussed [
4,
5]. Precise measurement of supraharmonics is the premise of the above analysis.
Currently, there is still no unified standard for measurement methods, EMC level, and immunity level in 2–150 kHz [
6]. International standards such as the European Standards (EN) 50,160 and the International Electrical Commission (IEC) 61000-4-19 have discussed emissions higher than 2 kHz. The International Special Committee on Radio Interference (CISPR) 16-2-1 discusses an immunity test method for high-frequency harmonics caused by intentional emissions with a range of 9–30 MHz [
7]. The following two guiding methods are mainly recommended regarding the measurement methods of supraharmonics caused by unintentional emissions. Method A is based on the method of 2–9 kHz in Appendix B of IEC 61000-4-7 [
8], a fixed rectangular window of 200 ms is selected as a sampling interval, and the application range of the 200 Hz bandwidth grouping is extended to 2–150 kHz. All the tested data are used, which is a gapless treatment approach. In method B, the 32-segment 0.5 ms equal interval sampling method is provided in Appendix C of IEC 61000-4-30 [
9], resulting in the resolution of 2 kHz. Since only a 16 ms time window in ten fundamental cycles of 50 Hz power frequency is sampled in this method, the amount of sampled data is significantly reduced compared to method A. Nevertheless, only 8% of the total measured data are applied by method B, which affects the accuracy of the measurement results.
To improve the accuracy of the frequency and amplitude of the measurement results, some scholars have investigated this issue from the perspective of the frequency domain. In [
10], a variable bandwidth grouping method was introduced based on the switching frequency. Compared with the fixed 200 Hz bandwidth, this method could reflect the amplitude and frequency of supraharmonics more accurately. However, it is difficult to accurately identify the respective switching frequencies when multiple different emission sources are presented. Moreover, the authors in [
11] provided a new method based on multiple measurement vectors model and orthogonal matching algorithm, designed to increase the frequency domain resolution from 2000 Hz to 200 Hz without increasing the observation duration. However, this method requires a priori knowledge of signal sparsity. In [
12], a method for quantization supraharmonics based on wavelet decomposition was proposed. This method can achieve the same level of resolution as IEC 61000-4-7, but wavelet packet coefficients will be affected by signal time deviation.
To further explore the approaches to supraharmonic quantification, some studies have attempted to reduce the amount of sampled data from the perspective of time sampling. For example, in the method presented in [
13], the filter banks and compressive sampling reduced the sampling frequency without changing the performance of existing measuring equipment, but the amplitude and phase angle of the narrowband signals will be affected by the analog filter bank. In [
14], a polyphase discrete Fourier transform (DFT) filter bank was implemented to handle data in the 2–150 kHz frequency band. Compared with method B, this method has fewer operations and stronger robustness. However, this method is still limited by the frequency domain resolution of 2000 Hz.
The above approaches are designed to quantify supraharmonic emission results more effectively. However, due to the mutual restriction between the resolution and the amount of data in the frequency domain, it is challenging to simultaneously meet the demand for low data storage and high resolution. In particular, the supraharmonic frequency points generated by the pulse width modulation of power electronic converter devices are distributed at m
fc ± n
fs, where
fc is the carrier wave frequency (switching frequency), and
fs is the fundamental frequency, and m and n are the carrier wave and modulation wave order, respectively [
15]. The emissions of supraharmonics were up to 150 kHz and accompanied by sideband emissions. As such, it requires a more accurate evaluation of the supraharmonic emissions.
The supraharmonic frequency band is characterized by many low-amplitude noise points and few higher-amplitude supraharmonic emission signals. Based on the difference in the density distribution of noise and emission points, supraharmonic emissions can be regarded as outliers to analyze. At present, outlier detection technology has been widely used in financial fraud, fault monitoring, and other fields [
16]. The outliers deviate significantly from other observations, and some data analysis techniques choose to discard these obviously different points. However, in some scenarios, the outliers may be caused by a specific type of mechanism. Potential and meaningful information may also be present in these points. This paper focused on the analysis of outliers in the frequency range of 2–150 kHz.
The mean and standard deviation method based on normal distribution is commonly used to detect outliers, but this method is easily affected by extreme values. Due to the existence of supraharmonic components, the data usually exhibits skewed distribution characteristics. Further research has also verified that the standard normal distribution is unsuitable for describing the 2–150 kHz data distribution [
17]. The influence of outliers on the supraharmonic range is discussed in detail in
Section 3.1. In this paper, an appropriate threshold was selected according to the data distribution model, and the noise points below the threshold were filtered out.
However, there is still residual noise from the previous steps, so the clustering algorithm was further used to extract the supraharmonic components more accurately. The clustering-based DBSCAN algorithm does not need to specify the number of clusters in advance. Besides, the algorithm can identify the arbitrary shape of data clusters and can also handle outliers efficiently. Therefore, the DBSCAN algorithm was selected for clustering analysis.
The clustering effect of DBSCAN is determined by the minimum number of points (MinPts) and epsilon (Eps). In order to obtain the best clustering effect, the author determined the MinPts and Eps by binary differential evolution algorithm in [
18]. The grid division technology in [
19] and the multi-verse optimizer algorithm in [
20] were also strategies to determine DBSCAN parameters. In addition, the multi-objective genetic algorithm was applied to find the optimal solution of the clustering parameters in [
21]. Nevertheless, the above algorithm implementation steps are more complicated. To realize the automatic tuning of DBSCAN parameters, a feasible and straightforward method is proposed in this paper, based on the slope set of the
k-dist curve and the dichotomy.
The DBSCAN algorithm handles the neighborhood of all core points. In the absence of any preprocessing, excessive data will result in limitations of operational efficiency. Before the DBSCAN algorithm is executed, the amount of data can be filtered out by removing the noise below the threshold of the data distribution model. For the above reasons, the skewed distribution model and the DBSCAN algorithm were combined in this paper as a tool for supraharmonic emissions quantification, denoted as the SD-DBSCAN method.
The main contributions of this research are as follows:
On the basis of the outlier characteristics of supraharmonic emissions, outlier detection algorithms can be employed to quantify the supraharmonic emission signals;
Based on the data distribution of slope point set and dichotomy, a self-tuning DBSCAN algorithm is presented. Additionally, a new method that combines the skewed distribution model and the self-tuning parameter DBSCAN clustering algorithm (SD-DBSCAN) is introduced in detail.
The newly proposed method solves the contradiction between the data volume of results and the frequency domain resolution. It has the advantage of a high resolution in the frequency domain.
The rest of the paper is structured as follows.
Section 2 describes the principle of the SD-DBSCAN method and the implementation steps in detail. In
Section 3, the method is verified by means of simulated and real signals. A systematic discussion and analysis of the different methods are presented in
Section 4. Finally, the conclusions are shown in
Section 5.
4. Discussion
Currently, the results of diverse quantification approaches to supraharmonic emissions cannot be unified. Although the amount of data was reduced after grouping, the frequency domain resolution was also reduced. A wider grouping bandwidth means that the more noise gathered into a frequency band, the greater the interference to the evaluation results. In order to ensure high resolution and low data storage, this research provides a novel quantification method, which combines a skewed distribution model with an enhanced DBSCAN algorithm. The theoretical basis of the detection algorithm is to utilize the density difference between the emission signals and the noise. The greater the difference, the better the detection effect.
In
Section 3, the performance of the SD-DBSCAN method was analyzed systematically. With a sampling window length of 200 ms and a resampling frequency of 1024 kHz, the number of original spectrum points in the range of 2–150 kHz exceeded 2 × 10
4. The data volume of the proposed scheme was less than 0.05% of the original spectrum, which was at the same level as that of IEC 61000-4-30. Besides being effective at reducing the amount of storage in the frequency domain, another outstanding advantage of this proposed method is that the technique can achieve a resolution of 5 Hz. Its resolution was 400 times higher than that of method B and 40 times higher than method A, which can ensure the accurate positioning of the supraharmonic signal frequency to a greater extent.
Since the precise frequency and amplitude of the supraharmonic emission maximum also facilitate the setting of emission limits, the and of the three methods at the maximum amplitude were also compared. The of method A was up to 38.68%, and the of method B was up to 46.8%. Among these methods, the frequency error of method B was the most serious, with the maximum of 400 Hz. The worse the resolution, the greater the range of error fluctuations in maximum amplitude and frequency. In contrast, the algorithm proposed in this paper showed that the error of these two indicators was zero.
In addition, from the actual measurement results, it can also be found that the measured results also exist emissions within the frequency range 23–36 kHz. A possible reason is the influence of secondary emissions from the nearby equipment. The above results show that methods A and B are not suitable for occasions that need to quantify the emissions accurately. The SD-DBSCAN method does not require grouping, which is required by method A. This new method can detect the emissions of the equipment at various frequency points, which is beneficial to analyze and study the emission characteristics of supraharmonics.
The influence of the SNR on the detection effect was explored by adding noise with different SNR levels to the synthetic signal.
Figure 7 states that as the SNR gradually increases, the number of outliers detected by the SD-DBSCAN algorithm decreases. At 15 dB, the F-measure already exceeded 0.7, which proves that the algorithm has good robustness to noise. The F-measure index of the analog signal and the actual signal were all higher than 0.8. The results showed that the new method based on outliers is feasible for the detection of supraharmonic components. The power frequency deviation also affected the measurement results, but the proposed method was less affected than the IEC method.
Additionally, the methods of various combination forms have been discussed to verify the rationality of the provided approach. From the traditional DBSCAN clustering detection results, it can be seen that the algorithm effect may not be guaranteed under diverse parameters. The SD-DBSCAN method can realize the parameter’s automatic tuning. Methods Ⅱ and Ⅳ cannot accurately reflect the supraharmonic frequency data distribution. Compared with the SD-DBSCAN method, the supraharmonic detection effect is not ideal.
The improved DBSCAN algorithm realizes the parameters adaptively, but the procedure is based on the DBSCAN output clusters, so multiple clustering is required to determine the optimal result. This process needs to be optimized for further work. In addition, the IEC approaches focus on frequency domain analysis. The detection results in this work were also limited to the frequency domain since the processed data were based on the results after DFT. However, supraharmonics have the characteristics of time-varying, and the methods suitable for dealing with steady-state signals cannot accurately describe all types of supraharmonic emissions. The emissions should be comprehensively considered in the combination of time-domain and time-frequency domain indicators.
5. Conclusions
In light of the fact that the current supraharmonic assessment methods encounter a contradiction between the frequency domain resolution and the amount of resulting data, a feasible approach denoted as the SD-DBSCAN method for solving the problem was introduced. This method utilizes the skewed distribution model and the self-tuning parameter DBSCAN clustering algorithm to detect supraharmonic emissions. Simulated and actual results are illustrated to verify the feasibility of the new method.
The threshold of the noise is represented based on the actual distribution characteristics of the supraharmonic band. Simultaneously, the DBSCAN algorithm was improved, and the parameters were determined automatically following the distribution model of the slope point set and dichotomy to find the optimal parameters. The newly proposed method has an excellent detection effect for supraharmonics and a comprehensive detectable frequency range around the switching frequency. This work can contribute a more effective approach to the research that needs to quantify the amplitude and frequency of supraharmonic emissions such as primary emissions and secondary emissions, resonance caused by supraharmonics, etc.
In this paper, only a single supraharmonic source was explored, but multiple emission sources may act simultaneously in the real environment, which needs to be further investigated in following work. This newly proposed high-resolution detection method also provides a new way to examine further supraharmonic emission characteristics, emission limits, and interference immunity levels.