# Noise Reduction of Power Quality Measurements with Time-Frequency Depth Analysis

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## Abstract

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## 1. Introduction

## 2. Wavelet Filter Bank Decomposition and Signal Reconstruction

_{j,k}(t) are orthogonal then every function x(t) can be written as:

_{s}/2 (f

_{s}– sampling frequency) infinite number of wavelets need to be used. Therefore, the scale function is defined [19] (also known as Father Wavelet) which is impulse response of low pass filter. Scale function covers the bandwidth which is not covered by wavelet functions. Scale function ϕ(t) is also translated and dilated according to (3). Now on the R decomposition level the function x(t) can be written as:

_{R,k}are discrete scale coefficients. Scale and wavelet functions are made by inverse Fourier transformation of orthogonal FIR filters and their mirror functions, and they need to be Perfect Reconstruction Quadrature Mirror Filters (PR-QMF) [20]. In this analysis, the Daubechies wavelet family is used which is PR-QMF.

_{HT}(n) of detail coefficient d is defined as [24]:

_{nr}(k), N is signal length, b is number of bits.

## 3. Defining WT Parameters for Noise reduction

#### 3.1. Introduction—Analysis Chain and Setup

_{th}spectral component on frequency n, averaged m times, and N is the number of elements of set S.

^{3}samples per second). Time length of sample is set to 1 s. The discretized data are analyzed with different number of wavelet tap (vanishing moments) and depth of signal decomposition. Algorithm is also tested with different signal to noise (SNR) levels which are added to signal. SNR in decibels is defined as:

#### 3.2. Simulation Analysis

#### 3.2.1. Amplitude and Wavelet Vanishing Moments

#### 3.2.2. Amplitude and SNR

#### 3.2.3. Vanishing Moments and SNR for 230 Vrms Voltage

#### 3.2.4. Decomposition Level and SNR

#### 3.2.5. Harmonics Simulation Analysis

#### 3.2.6. Interharmonics Simulation Analysis

#### 3.2.7. Transient and Interruption Simulation Analysis

#### 3.3. Conclusions after Simulation Analysis

## 4. Laboratory Measurement and Comparison of Results

#### 4.1. Introduction—Measurement Setup

#### 4.2. Amplitude and SNR Measurement Comparison

#### 4.3. Harmonics and Interharmonics Measurement

#### 4.4. Voltage Burst

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Result of noise reduction as a function of signal amplitude and wavelet vanishing moments.

**Figure 5.**Noise reduction as a function of wavelet vanishing moments and SNR for 5th decomposition level.

**Figure 6.**Noise reduction as a function of wavelet vanishing moments and SNR for 6th decomposition level.

**Figure 16.**Detail of noise reduction from burst signal. Original signal is presented with dashed line, signal with noise is marked by dots and signal after noise reduction is presented with solid line.

No. of Harmonics | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 27 | 31 |
---|---|---|---|---|---|---|---|---|---|

Generated harmonics (% of 230 Vrms) | |||||||||

Values | 9.9991 | 8.9993 | 7.9987 | 6.9985 | 5.9976 | 4.9970 | 3.9958 | 2.9956 | 1.9955 |

SNR (dB) | Measured harmonics with noise reduction (% of 230 Vrms) | ||||||||

70 | 9.9991 | 8.9993 | 7.9987 | 6.9986 | 5.9976 | 4.9970 | 3.9958 | 2.9957 | 1.9956 |

60 | 9.9990 | 8.9995 | 7.9987 | 6.9985 | 5.9974 | 4.9969 | 3.9959 | 2.9958 | 1.9952 |

50 | 9.9995 | 8.9994 | 7.9983 | 6.9987 | 5.9983 | 4.9971 | 3.9954 | 2.9950 | 1.9933 |

40 | 10.0003 | 8.9982 | 7.9975 | 6.9964 | 5.9949 | 4.9928 | 3.9948 | 2.9969 | 1.9876 |

30 | 9.9947 | 9.0096 | 7.9893 | 6.9957 | 5.9886 | 4.9915 | 3.9827 | 2.9566 | 1.9858 |

20 | 10.0325 | 8.9988 | 7.9719 | 7.0432 | 5.8887 | 5.0188 | 3.9468 | 2.5968 | 1.4612 |

Freq. (Hz) of Interharmonics | 111 | 222 | 333 | 444 | 555 | 666 | 777 | 888 | 999 |
---|---|---|---|---|---|---|---|---|---|

Generated harmonics (% of 230 Vrms) | |||||||||

Values | 9.9865 | 8.9712 | 7.9579 | 6.9487 | 5.9446 | 4.9459 | 3.9521 | 2.9625 | 1.9759 |

SNR (dB) | Measured harmonics with noise reduction (% of 230 Vrms) | ||||||||

70 | 9.9864 | 8.9712 | 7.9579 | 6.9487 | 5.9446 | 4.9459 | 3.9522 | 2.9625 | 1.9759 |

60 | 9.9865 | 8.9713 | 7.9577 | 6.9489 | 5.9447 | 4.9453 | 3.9522 | 2.9625 | 1.9759 |

50 | 9.9869 | 8.9711 | 7.9578 | 6.9496 | 5.9450 | 4.9460 | 3.9528 | 2.9625 | 1.9748 |

40 | 9.9890 | 8.9724 | 7.9575 | 6.9406 | 5.9460 | 4.9427 | 3.9542 | 2.9640 | 1.9715 |

30 | 9.9820 | 8.9717 | 7.9583 | 6.9695 | 5.9476 | 4.9423 | 3.9527 | 2.9618 | 1.8841 |

20 | 9.9753 | 8.9734 | 7.9516 | 6.9792 | 6.0004 | 4.8817 | 3.9214 | 2.8869 | 1.6398 |

No. of harmonics | 3 | 5 | 7 | 11 | 14 | 17 | 19 | 27 | 31 |
---|---|---|---|---|---|---|---|---|---|

Generated harmonics (% of 230 Vrms) | |||||||||

Values | 9.9771 | 9.0057 | 8.0153 | 7.0375 | 6.0521 | 5.0611 | 4.0597 | 3.0848 | 2.0719 |

SNR (dB) | Measured harmonics with noise reduction (% of 230 Vrms) | ||||||||

70 | 9.9771 | 9.0056 | 8.0153 | 7.0375 | 6.0521 | 5.0611 | 4.0597 | 3.0847 | 2.0718 |

60 | 9.9768 | 9.0055 | 8.0151 | 7.0376 | 6.0519 | 5.0611 | 4.0596 | 3.084 | 2.0718 |

50 | 9.9766 | 9.0062 | 8.0144 | 7.0371 | 6.0518 | 5.0607 | 4.0599 | 3.0843 | 2.0699 |

40 | 9.9782 | 9.0045 | 8.0165 | 7.0377 | 6.0527 | 5.0592 | 4.0565 | 3.0877 | 2.0668 |

30 | 9.9761 | 9.0076 | 8.0120 | 7.0433 | 6.0509 | 5.0637 | 4.0278 | 3.0743 | 2.0685 |

20 | 9.9676 | 8.9765 | 8.0280 | 7.0607 | 6.0179 | 4.9885 | 4.0380 | 2.7269 | 1.5039 |

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**MDPI and ACS Style**

Mostarac, P.; Malarić, R.; Mostarac, K.; Jurčević, M. Noise Reduction of Power Quality Measurements with Time-Frequency Depth Analysis. *Energies* **2019**, *12*, 1052.
https://doi.org/10.3390/en12061052

**AMA Style**

Mostarac P, Malarić R, Mostarac K, Jurčević M. Noise Reduction of Power Quality Measurements with Time-Frequency Depth Analysis. *Energies*. 2019; 12(6):1052.
https://doi.org/10.3390/en12061052

**Chicago/Turabian Style**

Mostarac, Petar, Roman Malarić, Katarina Mostarac, and Marko Jurčević. 2019. "Noise Reduction of Power Quality Measurements with Time-Frequency Depth Analysis" *Energies* 12, no. 6: 1052.
https://doi.org/10.3390/en12061052