# A New Denoising Method for UHF PD Signals Using Adaptive VMD and SSA-Based Shrinkage Method

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## Abstract

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## 1. Introduction

- (i)
- An automatic VMD algorithm is presented based on a mode-mixing judgement criterion. With the optimal K, the original PD signal can be decomposed into BLIMFs at high accuracy.
- (ii)
- Considering that BLIMFs containing PD components will exhibit the shape of pulse, a kurtosis-based method is employed to pick out those valuable BLIMFs (i.e., eBLIMFs).
- (iii)
- For each selected eBLIMF, the dominant singular values (DSVs) are retained at first. Then, they will be used to reconstruct PD signal by diagonal averaging. Next, the rescaling thresholding technique [9] is applied to further remove the residual white noise in each eBLIMF. Finally, the denoised UHF PD signal is obtained by adding up all these denoised eBLIMFs.

## 2. Mathematical Background

#### 2.1. Variational Mode Decomposition

^{2}norm, ${\partial}_{t}$ is derivative operator, and $\delta (t)$ is the Dirac function.

- Step 1:
- Initialize the parameters of the first loop $\left\{{u}_{k}^{1}\right\},\hspace{0.17em}\left\{{w}_{k}^{1}\right\},\hspace{0.17em}{\lambda}^{1}$, $k=1,2,\dots ,K$, and $K$ is the predefined number of decomposed modes. In addition, set the cycle index $n=0$;
- Step 2:
- Let $n=n+1$, then begin the outer loop;
- Step 3:
- Execute the first inner loop according to Equation (4) to update the $K$ BLIMFs in the spectral domain $\left\{{\widehat{u}}_{k}^{n+1}(w)\right\}$;
- Step 4:
- Execute the second inner loop according to Equation (5) to update the center frequencies of all BLIMFs in the spectral domain $\left\{{w}_{k}^{n+1}\right\}$.
- Step 5:
- Update the Lagrangian multiplier by the following expression:$${\widehat{\lambda}}^{n+1}(w)={\widehat{\lambda}}^{n}(w)+\tau \left(\widehat{f}(w)-{\displaystyle \sum _{k}{\widehat{u}}_{k}^{n+1}(w)}\right)$$
- Step 6:
- Repeat the algorithm from Step 2 to Step 5 until the following condition is satisfied:$$\sum _{k}^{}{\Vert {\widehat{u}}_{k}^{n+1}-{\widehat{u}}_{k}^{n}\Vert}_{2}^{2}}/{\Vert {\widehat{u}}_{k}^{n}\Vert}_{2}^{2}<\epsilon $$

#### 2.2. Singular Spectrum Analysis

## 3. Proposed Denoising Method

#### 3.1. Adaptive VMD

_{1}(the last segment may not). Then, the maximal point in each segment will be picked out to form a new sequence $segmax=\left\{seg\_{1}_{max},\dots ,seg\_{n}_{max},\dots ,seg\_{\tilde{N}}_{max}\right\}$, where $seg\_{n}_{max}$ denotes the maximal point of the nth segment, and ${\tilde{N}}_{max}$ is the number of segments.

Algorithm 1: Pseudo-code of the proposed mode-mixing judgement method |

#### 3.2. Effective BLIMF Selection

_{0}( ) is the central frequency operator, and kurtosis( ) is the kurtosis operator, which is calculated as:

#### 3.3. SSA-based Shrinkage method

_{i}by SVD. Since there is only one dominant component in each eBLIMF, the grouping step described in Section 2.2 becomes easy. Assume the SVs are sorted in descending order, and the ratio of each SV to the sum of all SVs are denoted as $\left\{{q}_{1},{q}_{2},\dots ,{q}_{R}\right\}$ (R is the rank of Hankel matrix), a common way to choose DSVs is to compute the cumulative sum of $q$ until its value reached a proper threshold ${\epsilon}_{2}$. This process can be expressed as:

_{i}is computed by Equation (9), then the reconstructed signal ${\widehat{\mathit{X}}}_{i}$ based on A

_{i}can be obtained by the diagonal averaging method. In order to further suppress the white noise in ${\widehat{\mathit{X}}}_{i}$, the shrinkage technology typically used in WT-based denoising is employed [9]. Specifically, the multiplicative threshold rescaling scheme with sqtwolog rule is adopted, which is expressed as:

Algorithm 2: Pseudo-code of thr proposed SSA-based Shrinkage denoising method |

#### 3.4. Implementation Procedure of Proposed AVMDSSA Method

- (i)
- Optimization of the number of modes K by gradually increasing its value and judging whether there is mode-mixing happened in each BLIMF at every step.
- (ii)
- Decompose the UHF PD signal into a set of BLIMFs by VMD with the optimal K parameter, then a kurtosis-based selection method is employed to pick out the eBLIMFs.
- (iii)
- For each eBLIMF, the SSA-based Shrinkage denoising method is applied to suppress the white noise, and summation of all denoised eBLIMFs will recover the denoised UHF PD signal.

## 4. Simulative Case Study

#### 4.1. Synthetic UHF PD Signal

#### 4.2. Denoising Results

#### 4.3. Noise Robustness

#### 4.4. Comparison with Traditional Denoising Methods

## 5. Laboratorial Case Study

#### 5.1. Laboratorial PD Measurement Setup

#### 5.2. New Evaluation Indices for Practical Situation

_{1}and eBLIMFs

_{2}respectively. Then, the maximum frequency values of each eBLIMF in eBLIMFs

_{1}are added to form the MF

_{1}. Following the same way, we can calculate the other parameter MF

_{2}. Then, the Index 2 is obtained by the ratio of MF

_{2}to MF

_{1}. This process can be formulated as follows:

#### 5.3. Denoising Results and Comparisons

## 6. Field Case Study

## 7. Discussion

## 8. Conclusions

- (i)
- The mode-mixing decision rule proposed in this paper works very well in all cases, enabling the AVMDSSA method to quickly determine the appropriate K value.
- (ii)
- In the simulative case, a complex synthetic UHF PD which contains three PD pulses and two kinds of noises is employed to examine our method. The results show that the proposed method can reduce all kinds of noises to a large extent, and in the meanwhile, all PD components are well retained. In addition, the results of robustness testing and comparison demonstrate its reliability and superiority.
- (iii)
- For the measured data, two new evaluation indices are presented by considering both of the capabilities of noise suppression and feature preservation. By using these newly designed indices, the effectiveness of AVMDSSA in laboratory experiments and field tests are identified.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Synthetic UHF PD signal and its spectrum: (

**a**) Pure signal; (

**b**) Spectrum of the pure signal; (

**c**) Noisy signal; (

**d**) Spectrum of the noisy signal.

**Figure 4.**Optimization process when k = 2: (

**a**) Time-domain waveforms of BLIMFs; (

**b**) Local maximums of 1st BLIMF and its peak points; (

**c**) Spectra of BLIMFs; (

**d**) Local maximums of 2nd BLIMF and its peak points.

**Figure 5.**Optimization process when k = 8: (

**a**) Time-domain waveforms of BLIMFs; (

**b**,

**c**) Local maximums of 1st and 5th BLIMFs and their peak points; (

**d**) Spectra of BLIMFs; (

**e**–

**j**) Local maximums of 2nd, 6th, 3rd, 7th, 4th, 8th BLIMFs and their peak points.

**Figure 8.**Evaluation of robustness of AVMDSSA: (

**a**) SNR values after denoising; (

**b**) NCC values after denoising. The bold solid line in each plot represents the mean value of corresponding index, and the semi-transparent regions are the value space of each index between its positive and negative standard error.

**Figure 9.**Denoising results by different algorithms: (

**a**) Noisy signal; (

**b**) Spectrum of the noisy signal; (

**c**) Denoised signal by Method 1; (

**d**) Spectrum of (c); (

**e**) Denoised signal by Method 2; (

**f**) Spectrum of (

**e**); (

**g**) Denoised signal by Method 3; (

**h**) Spectrum of (

**g**); (

**i**) Denoised signal by Method 4; (

**j**) Spectrum of (

**i**).

**Figure 10.**Quantitative results by different algorithm under various SNR levels: (

**a**) SNR results; (

**b**) NCC results.

**Figure 12.**Artificial defect models used in our tests: (

**a**) Floating discharge model; (

**b**) Protrusion discharge model; (

**c**) Particle discharge model; (

**d**) Air-gap discharge model.

**Figure 13.**Measured UHF PD signals by laboratorial setup: (

**a**) Typical waveform of Type 1; (

**b**) Spectrum of (

**a**); (

**c**) Typical waveform of Type 2; (

**d**) Spectrum of (

**c**); (

**e**) Typical waveform of Type 3; (

**f**) Spectrum of (

**e**); (

**g**) Typical waveform of Type 4; (

**h**) Spectrum of (

**g**). Possible PD components in each spectrum are marked with black dash-dotted ellipse.

**Figure 14.**Denoised UHF PD signals by AVMDSSA: (

**a**) Denoised signal of Type 1; (

**b**) Spectrum of (

**a**); (

**c**) Denoised signal of Type 2; (

**d**) Spectrum of (

**c**); (

**e**) Denoised signal of Type 3; (

**f**) Spectrum of (

**e**); (

**g**) Denoised signal of Type 4; (

**h**) Spectrum of (

**g**). Detected PD components in each spectrum are marked with black dash-dotted ellipse, and the lost components in (

**h**) is marked by green solid ellipse.

**Figure 15.**Some decomposed BLIMFs of Type 4 signal: (

**a**) The 4th BLIMF; (

**b**) Spectrum of (

**a**); (

**c**) The 5th BLIMF; (

**d**) Spectrum of (

**c**); (

**e**) The 6th BLIMF; (

**f**) Spectrum of (

**e**).

Items | Type | Amplitude: B, mV | Attenuation Coefficients: τ_{1}, τ_{2}, ns | Center Frequency: f_{c}, GHz | Sampling Rate: f_{s}, GHz |
---|---|---|---|---|---|

Pulse1 | DEDO | 5 | 1.2, 2.5 | 5 | 20 |

Pulse2 | SEDO | 5 | 1.5, -- | 0.6 | 20 |

Pulse3 | DEDO | 6 | 1.2, 2.5 | 3 | 20 |

PNN1 | PNN | 0.2 | --, -- | 1.2 | 20 |

PNN2 | PNN | 0.1 | --, -- | 4 | 20 |

Parameters | L_{1} | L_{2} | ε_{1} | ε_{2} | g |
---|---|---|---|---|---|

Description | Length of each segment used in adaptive VMD | The embedding dimension for Hankel matrix construction | Kurtosis threshold for eBLIMF selection | Threshold for dominant singular values | Used for mode-mixing judgement |

Value | 80 | 100 | 10 | 0.95 | 0.1 |

**Table 3.**Mode-mixing judgement results of BLIMFs for each K value. ‘1′ means mode-mixing happened, while ‘0′ means no mode-mixing happened.

K Value | 1st BLIMF | 2nd BLIMF | 3rd BLIMF | 4th BLIMF | 5th BLIMF | 6th BLIMF | 7th BLIMF | 8th BLIMF |
---|---|---|---|---|---|---|---|---|

2 | 1 | 1 | — | — | — | — | — | — |

3 | 1 | 1 | 1 | — | — | — | — | — |

4 | 1 | 1 | 1 | 0 | — | — | — | — |

5 | 1 | 1 | 1 | 0 | 0 | — | — | — |

6 | 0 | 0 | 1 | 1 | 0 | 0 | — | — |

7 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | — |

8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Method | Method 1 | Method 2 | Method 3 | Method 4 | Method 5 | |||||
---|---|---|---|---|---|---|---|---|---|---|

Index1 | Index2 | Index1 | Index2 | Index1 | Index2 | Index1 | Index2 | Index1 | Index2 | |

Type1 | 0.7713 | 0.9113 | 0.6740 | 0.5620 | 0.6850 | 0.5106 | 0.6801 | 0.6901 | 0.4490 | 0.7490 |

Type2 | 0.8200 | 0.8600 | 0.6250 | 0.6204 | 0.3380 | 0.7380 | 0.1075 | 0.7575 | 0.0889 | 0.8189 |

Type3 | 0.7653 | 0.8200 | 0.5336 | 0.6336 | 0.3132 | 0.7532 | 0.0715 | 0.7315 | 0.0804 | 0.8014 |

Type4 | 0.9153 | 0.9198 | 0.4106 | 0.5106 | 0.5518 | 0.6118 | 0.0763 | 0.6815 | 0.0728 | 0.7828 |

Indicators | BLIMF1 | BLIMF2 | BLIMF3 | BLIMF4 | BLIMF5 | BLIMF6 | BLIMF7 | BLIMF8 |
---|---|---|---|---|---|---|---|---|

Central frequency (GHz) | 0.236 | 0.879 | 1.075 | 2.433 | 3.621 | 5.117 | 6.250 | 10.41 |

Kurtosis | 22.36 | 38.69 | 37.95 | 5.94 | 3.14 | 2.93 | 3.19 | 2.72 |

© 2019 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, J.; He, J.; Long, J.; Yao, M.; Zhou, W.
A New Denoising Method for UHF PD Signals Using Adaptive VMD and SSA-Based Shrinkage Method. *Sensors* **2019**, *19*, 1594.
https://doi.org/10.3390/s19071594

**AMA Style**

Zhang J, He J, Long J, Yao M, Zhou W.
A New Denoising Method for UHF PD Signals Using Adaptive VMD and SSA-Based Shrinkage Method. *Sensors*. 2019; 19(7):1594.
https://doi.org/10.3390/s19071594

**Chicago/Turabian Style**

Zhang, Jun, Junjia He, Jiachuan Long, Min Yao, and Wei Zhou.
2019. "A New Denoising Method for UHF PD Signals Using Adaptive VMD and SSA-Based Shrinkage Method" *Sensors* 19, no. 7: 1594.
https://doi.org/10.3390/s19071594