^{1}

^{*}

^{1}

^{1}

^{2}

The harmonic and interharmonic analysis recommendations are contained in the latest IEC standards on power quality. Measurement and analysis experiences have shown that great difficulties arise in the interharmonic detection and measurement with acceptable levels of accuracy. In order to improve the resolution of spectrum analysis, the traditional method (e.g. discrete Fourier transform) is to take more sampling cycles, e.g. 10 sampling cycles corresponding to the spectrum interval of 5 Hz while the fundamental frequency is 50 Hz. However, this method is not suitable to the interharmonic measurement, because the frequencies of interharmonic components are non-integer multiples of the fundamental frequency, which makes the measurement additionally difficult. In this paper, the tunable resolution multiple signal classification (TRMUSIC) algorithm is presented, which the spectrum can be tuned to exhibit high resolution in targeted regions. Some simulation examples show that the resolution for two adjacent frequency components is usually sufficient to measure interharmonics in power systems with acceptable computation time. The proposed method is also suited to analyze interharmonics when there exists an undesirable asynchronous deviation and additive white noise.

Interharmonics can be thought of as the inter-modulation of the fundamental and harmonic components of the power system with any other frequency components and can be observed in an increasing number of loads. These loads include static frequency converters, cycloconverters, sub-synchronous converter cascades, induction motors, arc furnaces and so on [

A method, which is aimed to standardize the harmonic and interharmonic measurement, has been proposed by the IEC [

However, the accurate estimation method of the interharmonic components has not been established yet. Many researchers have been studying new methods. For analyzing a range of the interharmonic components, researchers often use DFT and its improved algorithms to calculate amplitudes, frequencies and phases of the interharmonic components [

The multiple signal classification (MUSIC) algorithm exploits the noise subspace to estimate the unknown parameters of the random process, which was proposed by R. O., Schmidt [

In this paper, the tunable resolution MUSIC (TRMUSIC) algorithm is presented to estimate the parameters of interharmonics, which the spectrum can be tuned to exhibit high resolution in targeted regions. The organization of this paper is as follows. The interharmonic measurement method based on the TRMUSIC algorithm is proposed in Section 2. Then, simulation results to demonstrate the

validity, precision feasibility and robustness of the algorithm are presented in Section 3. At last, the conclusions are given in Section 4.

The MUSIC algorithm is an eigenvalue subspace decomposition method for estimation of the frequencies of complex sinusoids observed in additive white noise. Consider a noisy signal vector

with

where

Suppose that

with

where

The auto-correlation matrix of the noisy signal

where

where

where

Furthermore, the singular value decomposition (SVD) of the matrix

where the columns of

Then, the MUSIC spectrum is defined as [

with

where

The frequency resolution

Here, a method of obtaining spectral interpolation data on the use of tunable factor

with

where

The most important step is to estimate the signal subspace dimension

Assume two signal sequences

then

with

From Equation (20), the matrix

where

where the columns of

In a real application, the cross-correlation matrix

The matrix

where each element

So, the Equation (23) can be used to estimate the signal subspace dimension

where

The frequencies of the harmonic and interharmonic components can be estimated from the peak location of the MUSIC spectrum, i.e., the frequencies

where

Equation (27) can be used to solve least squares for the coefficients

where

Three cases are performed in Matlab to demonstrate the effectiveness of the proposed algorithm.

In practice, the fundamental frequency often deviates from its nominal value. In the first simulation, the fundamental frequency is set to 49 Hz, and the signal is

the sampling frequency

In the second simulation, the fundamental frequencies is set to 50.2 Hz, and the signal is

the sampling frequency

In

the tunable factor

In this section, simulations are presented to demonstrate the anti-noise performance of TRMUSIC algorithm based on cross-spectral estimation comparing to that of the MUSIC algorithm based on auto-spectral estimation. When the signal represented by Equation (31) is contaminated with additive noise (SNR = 10 dB), the results of four simulations are shown in

Case | Frequency [Hz] | Amplitude [Pu] | Phase [degree] | |||
---|---|---|---|---|---|---|

True Values | TRMUSIC Estimation Values | True Values | TRMUSIC Estimation Values | True Values | TRMUSIC Estimation Values | |

1 | 44 | 44 | 0.1 | 0.099 | 30 | 30.7237 |

49 | 49 | 1 | 1.001 | −45 | −44.8523 | |

57 | 57 | 0.2 | 0.199 | 60 | 60.6537 | |

2 | 44.5 | 44.5 | 0.1 | 0.098 | 30 | 30.6235 |

50.2 | 50.2 | 1 | 1.002 | −45 | −44.7641 | |

57.3 | 57.3 | 0.2 | 0.1998 | 60 | 60.5574 |

In contrast, the MUSIC algorithm based on auto-spectral estimation has large errors of the signal subspace dimension

This simulation analyzes the harmonics in the AC/DC/AC converter system.

The AC/DC/AC converter system is a typical source of interharmonics [

Y-Y connection. The fundamental frequencies of system side and output side are 50 Hz and 60 Hz, respectively.

Then, the results of the TRMUSIC algorithm are compared with that of the MUSIC and DFT algorithm. For this simulation, we can see that the frequency analysis precision of the TRMUSIC algorithm is higher than that of the MUSIC and DFT algorithm, because the frequency resolution of the MUSIC and DFT algorithm is 10 Hz while that of TRMUSIC algorithm is 1 Hz, respectively. In

If fast Fourier transform (FFT) algorithm is used to compute its DFT, one such limitation is the power-of-two rule, requiring the number of input samples to be an integer power of two (i.e., 128, 256, 512). Therefore, choosing to lower sampling frequencies for better resolution is no longer a viable option. A clever engineer would simply increase the number of samples being taken. However, this solution quickly gets out of hand. In spite of this, the TRMUSIC algorithm may never be faster than the DFT algorithm.

Compared to the traditional MUSIC algorithm, the TRMUSIC algorithm is much more flexible. Given the required frequency resolution of interharmonic analysis, you can choose the proper tunable factor

This paper proposes an effective method to estimate the parameters of interharmonics in power systems. With the increase of points in time domain, the frequency resolution is improved because the frequency resolution of MUSIC algorithm is

This research is very fundamental as an application to interharmonic analysis. Many tests were made in this work and the TRMUSIC algorithm is the most suitable to be used when estimating interharmonic spectrum. It gives us a handy solution for some drawbacks that can be found in methods like the DFT or traditional MUSIC algorithm.

The TRMUSIC algorithm really meets the need of offline applications. Furthermore, if this algorithm can be implemented in parallel computation, it should meet the need of online applications and be more practical.

This work is supported by National Natural Science Foundation of China (No. 51477124).

Zhang, M., Zhang, X., Yao, H. and He, S.F. (2017) A Tunable Resolution MUSIC Algorithm for Interharmonics Analysis. Journal of Power and Energy Engineering, 5, 1-13. https://doi.org/10.4236/jpee.2017.59001