1. Introduction
With the development of power systems, the distributed renewable energy sources (RES) connected to the distribution network is increasing [
1,
2]. On the grid side, a large number of inverters are connected to the grid, and high-frequency voltage and current harmonics are injected into the grid, too. On the user side, rectifiers, and chopper devices, such as switching the power supply and electric vehicle charging piles, are becoming increasingly popular. The current harmonics generated by these devices in a range from 2 kHz to 150 kHz also affect the distribution network and adjacent devices [
3]. The voltage and current harmonics mentioned above in the range from 2 kHz to 150 kHz are collectively referred to as supraharmonic [
4].
Power electronics and other supraharmonic emission sources directly affect the normal operation and service life of nearby equipment. In the papers [
5,
6], it was found through practical measurements that the high-frequency voltage distortion caused by supraharmonics leads to faults and noise in household appliances. The device capacitors also heat up due to excessive supraharmonic currents, which further affect the device’s life [
7]. In addition to this, supraharmonics affect the power quality of the distribution network and power line communication [
8]. The meter causes significant errors when subjected to supraharmonic interference [
9]. Supraharmonics affect the quality of power supply to the grid. The literature [
10,
11] measured the supraharmonic emission generated by EV charging to further investigate the effect of related devices on the supply voltage. The paper [
12] studied the effect of grid impedance on the propagation of supraharmonics, and experimentally summarized some laws on the interaction between supraharmonics and the grid.
For the above ongoing research, fast and accurate supraharmonic measurement methods are indispensable, but mature measurement methods for supraharmonic identification and estimation are still lacking [
13]. The measurement standards IEC 61000-4-7 [
14], IEC 61000-4-30 [
15], and CISPR 16-2-1 [
16] propose three supraharmonic detection methods. The literature [
13,
17] compared the existing standards, the CISPR 16-2-1 method, and the IEC 61000-4-7 method, which have a frequency resolution of 200 Hz, and the detection amplitude is the same in most cases. However, the amount of data processed by these two methods is more than 10 times that of the IEC 61000-4-30 method. The weakness of the IEC 61000-4-30 method is that the frequency resolution is 2 kHz and the method only analyses 8% of the measurement data.
The existing standard methods for supraharmonics meet the needs of different application situations, but in order to have a standardized method for the online detection of supraharmonics, it is necessary to improve the computational complexity, computational time, and computational accuracy. According to the standard CISPR 16-2-1, the measurement bandwidth of the frequency band A (9–150 kHz) is 200 Hz, so it takes 1410 measurements in this band, and the measuring time is more than 2 min. To reduce the computation time, the literature [
18] was improved based on CISPR 16-2-1 by using a cascaded phase-locked loop to obtain the higher energy supraharmonic components instead of the swept frequency detection in the original standard. The detection time is significantly reduced, but it still takes a few seconds to complete. In order to reduce the computational burden of CISPR 16-2-1, a new digital quasi-peak detection method was proposed in the literature [
19], which indirectly reduced the computation time. Although the methods of calculation based on CISPR 16-2-1 have been improved, the computing time still takes over 200 ms, which is not suitable for online measurements of supraharmonics.
IEC 61000-4-30 and IEC 61000-4-7 use DFT for spectrum analysis. According to IEC 61000-4-30, power quality analysis requires 200 ms time series data collection. Harmonics below 2 kHz are typically sampled using a sampling rate of 10 kSP/s, so the amount of data for harmonic analysis is approximately 2000. For a supraharmonic of 2-150 kHz, a minimum sampling rate of 300 kSP/s is required, and the minimum amount of data to be analyzed is 60,000, which is 30 times higher than that of the harmonics. The increased data processing requirements put a higher demand on the computational resources of the power-quality instruments used for online measurements in the field. Faced with a large amount of data, IEC 61000-4-7 performed spectrum analysis on all the data, and IEC 61000-4-30 retained only 8% of the data for spectrum analysis. Therefore, the frequency resolution and signal coverage of the IEC 61000-4-30 supraharmonic measurement method is lower than that of other standard methods.
In order to balance the computation time and frequency resolution, a supraharmonic compression sensing model was proposed in the literature [
20,
21] to detect supraharmonics using the CS-OMP algorithm to recover the detection results with a 200 Hz frequency resolution from the 0.5 ms duration data. The literature [
20] further proposed the MCS for supraharmonics, which reduces the number of calculations from N to 1 for N sets of measurements using the combined sparsity of high-resolution spectral arrays. This method can calculate all signals in the IEC 61000-4-30 method. After that, the literature [
22] achieved adaptive sparsity by comparing iterative residuals, which reduced the computational error but significantly increased the computation time. The literature [
23] used the Bayesian algorithm CS-BCS instead of the CS-OMP greedy algorithm to optimize the detection performance and frequency detection error of intermittent supraharmonic emissions yet increased the overall reporting delay. The literature [
24] uses CS-OMP to select the supraharmonic emission frequencies and compute TFT expansion at the fundamental frequency and at these frequencies to achieve the more accurate detection of time-varying signals. The above study further extends the application prospects of the compression sensing algorithm in the online measurement of supraharmonics. However, the existing compressed sensing model needs to determine the number of iterations according to the sparsity (or residuals) and then calculate the matrix index of the supraharmonic vectors based on the sensing matrix. In the iterative process, the inner product of any column of the sensing matrix with the current residual needs to be computed, and the computation time grows linearly with increasing sparsity. Therefore, this paper proposes a method to estimate the sparsity based on the probability density model and, in the process, obtains the supraharmonic spectrum index, simplifies the sensing matrix based on the spectrum index, and then simplifies the compressed sensing model. The model only needs to calculate the inner product of any column in the simplified sensing matrix with the current residuals in a single iteration, thus reducing the iteration time and resulting in a reduction of the computational complexity of the supraharmonic compressed sensing algorithm.
This paper is organized as follows.
Section 2 introduces the principle and computational process of the simplified supraharmonic compression sensing model;
Section 3 investigates the computational complexity and computational accuracy of the algorithm;
Section 4 verifies the computational time and accuracy of the algorithm through experiments, and finally reviews the contributions of this paper and makes conclusions.