# Optimal Denoising and Feature Extraction Methods Using Modified CEEMD Combined with Duffing System and Their Applications in Fault Line Selection of Non-Solid-Earthed Network

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Generalized Composite Multiscale Permutation Entropy

#### 2.1. Permutation Entropy

_{g}(g=1, 2, …, k). The PE can be calculated according to the Shannon entropy as

_{p}(m) is ln(m!). The standardized PE is calculated as

_{p}is larger, the time series is rather more complex. On the contrary, the time series is rather more regular.

#### 2.2. Multiscale Permutation Entropy

#### 2.3. Generalized Composite Multiscale Permutation Entropy

#### 2.4. Parameter Selection and Comparative Analysis

_{max}is >10, the GCMPE can reflect the crucial information of the signal. The time delay τ has little effect on GCMPE. Consequently, the maximal scale factor s

_{max}is set to 16, and the time delay τ is set to 1.

#### 2.5. Signal Randomness Detection

## 3. MCEEMD Algorithm

#### 3.1. The Steps of MCEEMD

_{i}(t) (i = 1, 2, …, K) is white noise, and its mean value is 0; a

_{i}is noise amplitude, and its value is 0.2 times the standard deviation of the original signal I(t).

^{th}intrinsic mode function of the i

^{th}signal is IMF

_{ij}.

_{j}is the j

^{th}intrinsic mode function, which is obtained by CEEMD.

#### 3.2. Analysis of the Simulation Signal Using EMD, CEEMD, and MCEEMD

_{1}, x

_{2}, and x

_{3}, and the sampling frequency is 1000 Hz. The original signals are shown in Figure 4.

_{2}effectively. In Table 2, it is evident that CEEMD’s IMF3-IMF4 and x

_{2}have a higher correlation. Here, the CEEMD decomposition encounters the mode confusion at a certain level. In Table 3, MCEEMD can effectively extract x

_{1}and x

_{2}, and it restrains mode confusion. The results show that the method proposed in this paper can effectively extract the useful components of the original signal, and it suppress mode mixing at a certain level.

## 4. Optimal Smooth Denoising Model

#### 4.1. Optimal Smooth Denoising

_{k}(t) represents the k

^{th}signal reconstruction, and i is the number of the IMF.

_{1}lies on the curve f(x). The left curvature and the right curvature at point x

_{1}are as follows:

_{1}, the left curvature and the right curvature at x

_{1}are equal as follows:

_{1}, it means ${f}^{-}({x}_{1}{)}^{\u2033}$=${f}^{+}({x}_{1}{)}^{\u2033}$. The smoothness of the signal reconstruction at x = x

_{1}can be defined as

_{1}. The standard deviation of SN values of all points is taken as the smoothness index of the whole signal reconstruction as

#### 4.2. Optimal Smooth Denoising Experiment of Simulation Signal

_{1}represents the clear signal, and x is the noisy signal. The x

_{1}is shown in Figure 7a; x is composed of x

_{1}and Gaussian white noise as shown in Figure 7b.

_{8}is thought to be optimal when the similarity standard Amse and the smoothness standard Asmse are comprehensively considered.

_{8}is compared with the wavelet threshold denoising. The results are shown in Figure 9.

_{1}is 0.9971. To summarize, the proposed denoising method not only can retain the accurate information of the original signal but also can reduce the noise and smooth the original signal.

## 5. Measuring Principle of Duffing System

#### 5.1. Duffing System Model

_{c}, the trajectory of the system turns into a chaotic state. If r continues increasing over a certain threshold r

_{d}, the trajectory of the system changes from the chaotic state to the large-scale periodic motion state. The state at r

_{d}is used to detect the signal. r

_{d}can be previously determined by the computer simulation experiment [37]. Based on the experiments, we chose r

_{d}= 0.8253. For detecting the high-frequency signal, we must do time scale transformation. Defining $t=\omega \tau $, we obtain

_{d}, $\theta $ is very small, and its impact on the system can be negligible. If $\pi -\mathrm{arccos}(a/2{r}_{d})\le \phi \le \pi +\mathrm{arccos}(a/2{r}_{d})$, $r\le {r}_{d}$, the trajectory of the Duffing system is still in a chaotic state. If $\theta $ is not in this range, it makes the transition from the chaotic state to the large-scale periodic motion state. Making use of the characteristic of the Duffing system can select the fault line in the non-solid-earthed network. In this paper, the angular frequency of the internal driving force is set as 500π rad/s, and the calculated step size is set as 5 × 10

^{−6}. For ignoring the effect of $\theta $, the input signal shall be multiplied by a detection factor. We chose different detection factors for experiments, and 0.01 had the best effect.

#### 5.2. Trisection Symmetry Phase Estimation

#### 5.3. A Method of Distinguishing Chaotic Characteristics

## 6. Fault Line Selection Steps and Simulation

#### 6.1. Introduction of the Fault Line Selection Steps

#### 6.2. Simulation Study

**Case 1:**Suppose there is a single-phase-to-ground fault in phase A of line L

_{4}at 0.02 s, 5 km away from the bus bar. The ground resistance R

_{g}is 10 Ω. The zero-sequence currents collected in the actual non-solid-earthed network often carry noise with them. After the simulation, we add the Gaussian white noise to the zero-sequence currents in lines 1–4. The results are shown in Figure 14 from 0 to 0.1 s. Every line’s zero-sequence current is done with the help of fast Fourier transform. The result of fast Fourier transform analysis is shown in Figure 15. As seen from Figure 15, the zero-sequence current contains minor amounts of the fifth harmonic. In a study by Zhang and colleagues [8], the method of fault line selection failed by comparing the amplitude of the fifth harmonic.

_{4}is in a chaotic state while the others are in a large-scale periodic motion state. It can be seen that the trajectory of the large-scale periodic motion state moves around (±1, 0) and (0, 0). The fifth harmonic components of the fault line and the non-fault line can make the Duffing system in different state trajectories. Therefore, line L

_{4}has a single-phase-to-ground fault. For automatically identifying chaotic nature, we extract texture features of phase diagrams. As introduced in Section 5.3, the reference vector of the texture feature is (0.9086, 0.2638, 3.3849, 0.1029). The chaotic nature can be identified according to the Euclidean distance between the reference vector and texture feature of the phase diagram. The results are shown in Table 7. As shown in Table 7, only the Euclidean distance of line L

_{4}is more than 1.5, and therefore its phase diagram is in a chaotic state. However, the other phase diagrams are the large-scale periodic state. The results mean the method can select the fault line accurately.

**Case 2:**Since fault conditions are different in the actual non-solid-earthed network, we carried out different fault situations, such as fault line, fault resistance, and fault distance. The selection results of fault in different conditions are shown in Table 8. As can be seen in Table 8, the Duffing systems of the fault line and the non-fault line show different state trajectories. Table 8 also shows that the state of the system can be accurately identified by the texture feature recognition of the phase diagram. If Euclidean distance is less than 1.5, the phase diagram is the large-scale periodic motion state. Otherwise, the phase diagram is a chaotic state. To summarize, the proposed method is able to select the fault line with different situations.

## 7. Conclusions

- (1)
- GCMPE can solve the deficiencies of MPE and has better stability than MPE. Based on GCMPE and SVM for abnormal signal detection, a new denoising algorithm is proposed. The simulation results show that MCEEMD has better decomposition results than the existing algorithms. MCEEMD also has strong adaptability and restrains the mode mixing of EMD.
- (2)
- The optimal smooth denoising model can be determined by balancing the similarity and smoothness of different filtering algorithms. The superior algorithm can retain the useful features of the original signal and reduce the noise and smooth the original signal.
- (3)
- A novel method of identifying chaotic nature based on texture features of the phase diagram is presented. This way can avoid the involvement of human factors and automatically identify chaotic nature by computers.
- (4)
- The fault line in the non-solid-earthed network can be selected with the diagram outputted by the Duffing system. A large number of experimental studies show that the proposed method can accurately select the fault line under different fault situations. The research provides a novel train of thought for the fault line in the non-solid-earthed network.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

MCEEMD | Modified complementary ensemble empirical mode decomposition |

GCMPE | Generalized composite multiscale permutation entropy |

SVM | Support vector machine |

IMF | Intrinsic mode functions |

EMD | Empirical mode decomposition |

VMD | Variational mode decomposition |

EEMD | Ensemble empirical mode decomposition |

CEEMD | Complementary ensemble empirical mode decomposition |

CEEMDAN | Complete ensemble empirical mode decomposition with adaptive noise |

PE | Permutation entropy |

MPE | Multiscale permutation entropy |

GLCM | Gray level co-occurrence matrix |

Ort | Orthogonality |

FFT | Fast Fourier transform |

## References

- Wang, X.; Gao, J.; Song, G.; Cheng, Q.; Wei, X.; Wei, Y. Faulty line selection method for distribution network based on variable scale bistable system. J. Sens.
**2016**, 2016, 7436841. [Google Scholar] [CrossRef] [Green Version] - Wang, X.; Wei, Y.; Zeng, Z.; Hou, Y.; Gao, J.; Wei, X. Fault line selection method of small current to ground system based on atomic sparse decomposition and extreme learning machine. J. Sens.
**2015**, 2015, 678120. [Google Scholar] [CrossRef] - Shao, W.; Bai, J.; Cheng, Y.; Zhang, Z.; Li, N. Research on a faulty line selection method based on the zero-sequence disturbance power of resonant grounded distribution networks. Energies
**2019**, 12, 846. [Google Scholar] [CrossRef] [Green Version] - Lin, X.; Sun, J.; Kursan, I.; Zhao, F.; Li, Z.; Li, X.; Yang, D. Zero-sequence compensated admittance based faulty feeder selection algorithm used for distribution network with neutral grounding through Peterson-coil. Int. J. Electr. Power Energy Syst.
**2014**, 63, 747–752. [Google Scholar] [CrossRef] - Zhuang, S.; Miao, X.; Jiang, H.; Guo, M. A line selection method for single-phase high-impedance grounding fault in resonant grounding system of distribution network based on improved euclidean-dynamic time warping distance. Power Syst. Technol.
**2020**, 44, 273–281. [Google Scholar] - Dong, X.; Shi, S. Identifying single-phase-to-ground fault feeder in neutral noneffectively grounded distribution system using wavelet transform. IEEE Trans. Power Deliv.
**2008**, 23, 1829–1837. [Google Scholar] [CrossRef] - Costa, F.; Souza, B.; Brito, N.; Silva, J.; Santos, W. Real-time detection of transients induced by high-impedance faults based on the boundary wavelet transform. IEEE Trans. Ind. Appl.
**2015**, 51, 5312–5323. [Google Scholar] [CrossRef] - Zhang, Z.; Liu, X.; Piao, Z. Fault line detection in neutral point ineffectively grounding power system based on phase-locked loop. IET Gener. Transm. Distrib.
**2014**, 8, 273–280. [Google Scholar] - Zhang, S.; Zhai, X.; Dong, X.; Li, L.; Tang, B. Application of EMD and Duffing oscillator to fault line detection in un-effectively grounded system. Proc. Chin. Soc. Electr. Eng.
**2013**, 33, 161–167. [Google Scholar] - Kang, X.; Liu, X.; Suonan, J.; Ma, C.; Wang, C.; Yang, L. New method for fault line selection in non-solidly grounded system based on matrix pencil method. Autom. Electr. Power Syst.
**2012**, 36, 88–93. [Google Scholar] - Li, Y.; Li, Y.; Chen, X.; Yu, J. Denoising and feature extraction algorithms using NPE combined with VMD and their applications in ship-radiated noise. Symmetry
**2017**, 9, 256. [Google Scholar] [CrossRef] [Green Version] - Hou, J.; Wu, Y.; Gong, H.; Ahmad, A.; Liu, L. A novel intelligent method for bearing fault diagnosis based on EEMD permutation entropy and GG clustering. Appl. Sci.
**2020**, 10, 386. [Google Scholar] [CrossRef] [Green Version] - Wang, S.; Sun, Y.; Zhou, Y.; Jamil Mahfoud, R.; Hou, D. A new hybrid short-term interval forecasting of PV output power based on EEMD-SE-RVM. Energies
**2019**, 13, 87. [Google Scholar] [CrossRef] [Green Version] - Xue, S.; Tan, J.; Shi, L.; Deng, J. Rope tension fault diagnosis in hoisting systems based on vibration signals using EEMD, improved permutation entropy, and PSO-SVM. Entropy
**2020**, 22, 209. [Google Scholar] [CrossRef] [Green Version] - Ren, Y.; Suganthan, P.; Srikanth, N. A comparative study of empirical mode decomposition-based short-term wind speed forecasting methods. IEEE Trans. Sustain. Energy
**2015**, 6, 236–244. [Google Scholar] [CrossRef] - Zhao, L.; Yu, W.; Yan, R. Rolling bearing fault diagnosis based on CEEMD and time series modeling. Math. Probl. Eng.
**2014**, 2014, 101867. [Google Scholar] [CrossRef] - Liu, F.; Gao, J.; Liu, H. The feature extraction and diagnosis of rolling bearing based on CEEMD and LDWPSO-PNN. IEEE Access
**2020**, 8, 19810–19819. [Google Scholar] [CrossRef] - Yang, D.; Sun, Y.; Wu, K. Research on CEEMD-AGA denoising method and its application in feed mixer. Math. Probl. Eng.
**2020**, 2020, 9873268. [Google Scholar] [CrossRef] - Fuentealba, P.; Illanes, A.; Ortmeier, F. Independent analysis of decelerations and resting periods through CEEMDAN and spectral-based feature extraction improves cardiotocographic assessment. Appl. Sci.
**2019**, 9, 5421. [Google Scholar] [CrossRef] [Green Version] - Li, G.; Yang, Z.; Yang, H. A denoising method of ship radiated noise signal based on modified CEEMDAN, dispersion entropy, and interval thresholding. Electronics
**2019**, 8, 597. [Google Scholar] [CrossRef] [Green Version] - Tian, S.; Bian, X.; Tang, Z.; Yang, K.; Li, L. Fault diagnosis of gas pressure regulators based on CEEMDAN and feature clustering. IEEE Access
**2019**, 7, 132492–132502. [Google Scholar] [CrossRef] - Wang, G.; Chen, D.; Lin, J.; Chen, X. The application of chaotic oscillators to weak signal detection. IEEE Trans. Ind. Electron.
**1999**, 46, 440–444. [Google Scholar] [CrossRef] - Xue, W.; Dai, X.; Zhu, J.; Luo, Y.; Yang, Y. A noise suppression method of ground penetrating radar based on EEMD and permutation entropy. IEEE Geosci. Remote Sens. Lett.
**2019**, 16, 1625–1629. [Google Scholar] [CrossRef] - Srinu, S.; Mishra, A.K. Cooperative sensing based on permutation entropy with adaptive thresholding technique for cognitive radio networks. IET Sci. Meas. Technol.
**2016**, 10, 934–942. [Google Scholar] [CrossRef] - Du, W.; Guo, X.; Wang, Z.; Wang, J.; Yu, M.; Li, C.; Wang, G.; Wang, L.; Guo, H.; Zhou, J.; et al. A new fuzzy logic classifier based on multiscale permutation entropy and its application in bearing fault diagnosis. Entropy
**2019**, 22, 27. [Google Scholar] [CrossRef] [Green Version] - Wang, X.; Lu, Z.; Wei, J.; Zhang, Y. Fault diagnosis for rail vehicle axle-box bearings based on energy feature reconstruction and composite multiscale permutation entropy. Entropy
**2019**, 21, 865. [Google Scholar] [CrossRef] [Green Version] - Huo, Z.; Zhang, Y.; Shu, L.; Gallimore, M. A new bearing fault diagnosis method based on fine-to-coarse multiscale permutation entropy, laplacian score and SVM. IEEE Access
**2019**, 7, 17050–17066. [Google Scholar] [CrossRef] - Humeau-Heurtier, A.; Wu, C.-W.; Wu, S.-D. Refined composite multiscale permutation entropy to overcome multiscale permutation entropy length dependence. IEEE Signal Process. Lett.
**2015**, 22, 2364–2367. [Google Scholar] [CrossRef] - Zheng, J.; Liu, T.; Meng, R.; Liu, Q. Generalized composite multiscale permutation entropy and PCA based fault diagnosis of rolling bearings. J. Vib. Shock
**2018**, 37, 61–66. [Google Scholar] - Wan, Z.; Yi, S.; Li, K.; Tao, R.; Gou, M.; Li, X.; Guo, S. Diagnosis of elevator faults with LS-SVM based on optimization by K-CV. J. Electr. Comput. Eng.
**2015**, 2015, 935038. [Google Scholar] [CrossRef] [Green Version] - Liu, W.; Zhou, X.; Jiang, Z.; Ma, F. Improved empirical mode decomposition method based on optimal feature. J. Jilin Univ. (Eng. Tech. Ed.)
**2017**, 47, 1957–1963. [Google Scholar] - Wang, Z.; Qiao, P.; Shi, B. Nonpenetrating damage identification using hybrid lamb wave modes from Hilbert-Huang spectrum in thin-walled structures. Shock Vib.
**2017**, 2017, 5164594. [Google Scholar] [CrossRef] [Green Version] - Zheng, Y.; Sun, X.; Chen, J.; Yue, J. Extracting pulse signals in measurement while drilling using optimum denoising methods based on the ensemble empirical mode decomposition. Pet. Explor. Dev.
**2012**, 39, 750–753. [Google Scholar] [CrossRef] - Li, G.; Zeng, L.; Zhang, L.; Wu, Q. State identification of Duffing oscillator based on extreme learning machine. IEEE Signal Process. Lett.
**2018**, 25, 25–29. [Google Scholar] [CrossRef] - Song, W.; Deng, S.; Yang, J.; Cheng, Q. Tool wear detection based on Duffing-holmes oscillator. Math. Probl. Eng.
**2008**, 2008, 510406. [Google Scholar] [CrossRef] [Green Version] - Chang, T. Chaotic motion in forced Duffing system subject to linear and nonlinear damping. Math. Probl. Eng.
**2017**, 2017, 3769870. [Google Scholar] [CrossRef] - Li, Y.; Yang, B.; Shi, Y. Chaos-based weak sinusoidal signal detection approach under colored noise background. Acta Phys. Sin. Chin. Ed.
**2003**, 52, 526–530. [Google Scholar] - Shang, Q.; Yin, C.; Li, S.; Yang, Y. Study on detection of weak sinusoidal signal by using Duffing oscillator. Proc. Chin. Soc. Electr. Eng.
**2005**, 25, 66–70. [Google Scholar] - Suresh, A.; Shunmuganathan, K.L. Image texture classification using gray level co-occurrence matrix based statistical features. Eur. J. Sci. Res.
**2012**, 75, 591–597. [Google Scholar] - Luo, J.; Song, D.; Xiu, C.; Geng, S.; Dong, T. Fingerprint classification combining curvelet transform and gray-level cooccurrence matrix. Math. Probl. Eng.
**2014**, 2014, 592928. [Google Scholar] [CrossRef] - Liu, X.; Xu, K.; Zhou, P.; Liu, H. Feature extraction with discrete non-separable shearlet transform and its application to surface inspection of continuous casting slabs. Appl. Sci.
**2019**, 9, 4668. [Google Scholar] [CrossRef] [Green Version] - Lo, C.; Chen, C.; Yeh, Y.; Chang, C.; Yeh, H. Quantitative analysis of melanosis coli colonic mucosa using textural patterns. Appl. Sci.
**2020**, 10, 404. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**The coarse granulation’s processes of multiscale permutation entropy and generalized composite multiscale permutation entropy with s = 3.

**Figure 2.**Multiscale permutation entropy (MPE) and generalized composite multiscale permutation entropy (GCMPE) of white noise under different scale factors and embedded dimensions [26].

**Figure 5.**The decomposition results of (

**a**) empirical mode decomposition (EMD), (

**b**) complementary ensemble empirical mode decomposition (CEEMD), and (

**c**) modified complementary ensemble empirical mode decomposition (MCEEMD).

**Figure 9.**The denoising results for different methods. (

**a**) The denoising result of the wavelet threshold denoising. (

**b**) The denoising result of the optimal smooth denoising model.

**Figure 12.**The results for texture parameters. (

**a**) Energy result; (

**b**) entropy result; (

**c**) contrast result; (

**d**) correlation result; (

**e**) mean result.

**Figure 14.**Every line’s zero-sequence current with noise. (

**a**) The result of line L

_{1}; (

**b**) the result of line L

_{2}; (

**c**) the result of line L

_{3}; (

**d**) the result of line L

_{4}.

**Figure 16.**Every line’s decomposition result. The results of (

**a**) line L

_{1}, (

**b**) line L

_{2}, (

**c**) line L

_{3}; (

**d**) line L

_{4}.

**Figure 17.**Every line’s optimal smooth denoising result. (

**a**) line L

_{1}after denoising; (

**b**) line L

_{2}after denoising; (

**c**) line L

_{3}after denoising; (

**d**) line L

_{4}after denoising.

**Figure 18.**Duffing system’s phase diagram when adding each line’s denoising result. The phase diagram of (

**a**) line L

_{1}, (

**b**) line L

_{2}, (

**c**) line L

_{3}, and (

**d**) line L

_{4}.

x_{1} | x_{2} | x_{3} | |
---|---|---|---|

IMF1 | 0.9031 | 0.0009 | 0.0489 |

IMF2 | 0.2889 | 0.4760 | 0.0214 |

IMF3 | 0.0019 | 0.3790 | 0.0246 |

IMF4 | −0.0020 | 0.3267 | −0.0277 |

IMF5 | 0.0007 | 0.0061 | −0.0103 |

IMF6 | 0.0010 | −0.0030 | 0.0105 |

IMF7 | 0.0001 | −0.0004 | −0.0012 |

IMF8 | 0.0032 | −0.0010 | −0.0088 |

x_{1} | x_{2} | x_{3} | |
---|---|---|---|

IMF1 | 0.3952 | −0.0058 | 0.7078 |

IMF2 | 0.9993 | −0.0035 | −0.0107 |

IMF3 | 0.0369 | 0.9804 | −0.0156 |

IMF4 | 0.0005 | 0.9475 | −0.0183 |

IMF5 | −0.0011 | 0.2294 | −0.0149 |

IMF6 | −0.0024 | 0.0257 | −0.0005 |

IMF7 | −0.0028 | 0.0085 | −0.0006 |

x_{1} | x_{2} | x_{3} | |
---|---|---|---|

IMF1 | 0.9995 | 0.0008 | 0.0235 |

IMF2 | −0.0010 | 0.9992 | −0.0092 |

Denoising Algorithms | Ort |
---|---|

EMD | 0.0891 |

CEEMD | 0.0671 |

MCEEMD | $5.8701\times {10}^{-5}$ |

Filtering Algorithms | Amse | Asmse | Aminf |
---|---|---|---|

SR_{1} | 0.659710 | 0.016383 | 0.402380 |

SR_{2} | 0.274279 | 0.016512 | 0.171172 |

SR_{3} | 0.158327 | 0.016505 | 0.101598 |

SR_{4} | 0.125100 | 0.016505 | 0.081662 |

SR_{5} | 0.109698 | 0.016505 | 0.072421 |

SR_{6} | 0.109262 | 0.016505 | 0.072159 |

SR_{7} | 0.109205 | 0.016505 | 0.072125 |

SR_{8} | 0.106914 | 0.016505 | 0.070750 |

Line | Sequence | Resistance (Ω/km) | Inductance (mH/km) | Capacitance (μF/km) | Length (km) |
---|---|---|---|---|---|

L_{1} | positive-sequence | 0.1820 | 1.1180 | 0.1150 | 18 |

zero-sequence | 0.3250 | 5.6810 | 0.0096 | ||

L_{2} | positive-sequence | 0.1260 | 1.0180 | 0.1200 | 20 |

zero-sequence | 0.2850 | 4.5600 | 0.0150 | ||

L_{3} | positive-sequence | 0.1800 | 2.1580 | 0.1290 | 12 |

zero-sequence | 0.2730 | 5.5610 | 0.0150 | ||

L_{4} | positive-sequence | 0.1320 | 2.2250 | 0.2290 | 16 |

zero-sequence | 0.2380 | 5.5610 | 0.0235 |

Line | Energy | Entropy | Contrast | Correlation | Euclidean Distance | Chaotic Nature |
---|---|---|---|---|---|---|

L_{1} | 0.9215 | 0.2262 | 2.0705 | 0.1225 | 1.3151 | Periodical |

L_{2} | 0.8932 | 0.2817 | 2.5023 | 0.0885 | 0.8831 | Periodical |

L_{3} | 0.9225 | 0.2242 | 2.0689 | 0.1246 | 1.3168 | Periodical |

L_{4} | 0.6222 | 0.8270 | 11.9338 | 0.0245 | 8.5726 | Chaotic |

Fault Situation | Line | Energy | Entropy | Contrast | Correlation | Euclidean Distance | Chaotic Nature | Result |
---|---|---|---|---|---|---|---|---|

(L_{4},30 Ω, 10 km) | L_{1} | 0.9172 | 0.2352 | 2.1417 | 0.1153 | 1.2436 | Periodical | L_{4} |

L_{2} | 0.9017 | 0.2665 | 2.1456 | 0.0967 | 1.2393 | Periodical | ||

L_{3} | 0.9155 | 0.2395 | 2.1211 | 0.1138 | 1.2641 | Periodical | ||

L_{4} | 0.6216 | 0.8195 | 10.8519 | 0.0245 | 7.4935 | Chaotic | ||

(L_{4},500 Ω, 2 km) | L_{1} | 0.8962 | 0.2766 | 2.0462 | 0.0916 | 1.3388 | Periodical | L_{4} |

L_{2} | 0.8539 | 0.3470 | 2.2218 | 0.0636 | 1.1680 | Periodical | ||

L_{3} | 0.8943 | 0.2791 | 2.0082 | 0.0898 | 1.3769 | Periodical | ||

L_{4} | 0.5818 | 0.9199 | 15.2676 | 0.0217 | 11.9056 | Chaotic | ||

(L_{4},1000 Ω, 1 km) | L_{1} | 0.9037 | 0.2616 | 2.1546 | 0.0989 | 1.2303 | Periodical | L_{4} |

L_{2} | 0.8747 | 0.3119 | 2.0104 | 0.0748 | 1.3760 | Periodical | ||

L_{3} | 0.9029 | 0.2642 | 2.1641 | 0.0980 | 1.2208 | Periodical | ||

L_{4} | 0.5735 | 0.9289 | 15.0723 | 0.0213 | 11.7144 | Chaotic | ||

(L_{3},200 Ω, 6 km) | L_{1} | 0.9271 | 0.2157 | 2.2960 | 0.1321 | 1.0905 | Periodical | L_{3} |

L_{2} | 0.8999 | 0.2891 | 5.1498 | 0.0876 | 1.7652 | Periodical | ||

L_{3} | 0.6155 | 0.8406 | 11.9449 | 0.0240 | 8.5848 | Chaotic | ||

L_{4} | 0.9149 | 0.2402 | 2.1099 | 0.1126 | 1.2552 | Periodical | ||

(L_{2},10 Ω, 3 km) | L_{1} | 0.9212 | 0.2277 | 2.1077 | 0.1221 | 1.2779 | Periodical | L_{2} |

L_{2} | 0.5721 | 0.9310 | 15.0184 | 0.0213 | 11.6578 | Chaotic | ||

L_{3} | 0.9225 | 0.2250 | 2.0669 | 0.1248 | 1.3188 | Periodical | ||

L_{4} | 0.8874 | 0.2975 | 2.4383 | 0.0842 | 0.9476 | Periodical | ||

(L_{2},100 Ω, 5 km) | L_{1} | 0.9267 | 0.2162 | 2.2603 | 0.1307 | 1.1261 | Periodical | L_{2} |

L_{2} | 0.6157 | 0.8444 | 12.4470 | 0.0240 | 9.0858 | Chaotic | ||

L_{3} | 0.9265 | 0.2167 | 2.2689 | 0.1310 | 1.1166 | Periodical | ||

L_{4} | 0.8775 | 0.3338 | 4.7863 | 0.0753 | 1.4038 | Periodical | ||

(L_{2},600 Ω, 1 km) | L_{1} | 0.9270 | 0.2160 | 2.3471 | 0.1315 | 1.0394 | Periodical | L_{2} |

L_{2} | 0.7612 | 0.5635 | 7.0596 | 0.0389 | 3.6905 | Chaotic | ||

L_{3} | 0.9272 | 0.2166 | 2.3373 | 0.1321 | 1.4092 | Periodical | ||

L_{4} | 0.9201 | 0.2298 | 2.2778 | 0.1199 | 1.1077 | Periodical | ||

(L_{1},100 Ω, 3 km) | L_{1} | 0.6806 | 0.7136 | 9.3671 | 0.0291 | 6.0039 | Chaotic | L_{1} |

L_{2} | 0.9165 | 0.2371 | 2.0641 | 0.1152 | 1.3211 | Periodical | ||

L_{3} | 0.9251 | 0.2204 | 2.2327 | 0.1293 | 1.1534 | Periodical | ||

L_{4} | 0.9135 | 0.2424 | 2.0484 | 0.1107 | 1.3367 | Periodical | ||

(L_{1},300 Ω, 12 km) | L_{1} | 0.6880 | 0.7058 | 9.4638 | 0.0300 | 6.0994 | Chaotic | L_{1} |

L_{2} | 0.9306 | 0.2093 | 2.4337 | 0.1383 | 0.9536 | Periodical | ||

L_{3} | 0.9278 | 0.2137 | 2.3455 | 0.1313 | 1.0412 | Periodical | ||

L_{4} | 0.9172 | 0.2363 | 2.1388 | 0.1161 | 1.2465 | Periodical | ||

(Bus bar, 200 Ω) | L_{1} | 0.9268 | 0.2157 | 2.2952 | 0.1308 | 1.0912 | Periodical | Bus bar |

L_{2} | 0.9316 | 0.2066 | 2.3919 | 0.1397 | 0.9955 | Periodical | ||

L_{3} | 0.9268 | 0.2166 | 2.3039 | 0.1309 | 1.0825 | Periodical | ||

L_{4} | 0.9160 | 0.2375 | 2.0775 | 0.1141 | 1.3077 | Periodical | ||

(Bus bar, 1000 Ω) | L_{1} | 0.9285 | 0.2133 | 2.3596 | 0.1339 | 1.0272 | Periodical | Bus bar |

L_{2} | 0.9246 | 0.2208 | 2.3515 | 0.1274 | 1.0347 | Periodical | ||

L_{3} | 0.9280 | 0.2143 | 2.3948 | 0.1330 | 0.9920 | Periodical | ||

L_{4} | 0.9227 | 0.2247 | 2.2817 | 0.1234 | 1.1041 | Periodical |

© 2020 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**

Hou, S.; Guo, W.
Optimal Denoising and Feature Extraction Methods Using Modified CEEMD Combined with Duffing System and Their Applications in Fault Line Selection of Non-Solid-Earthed Network. *Symmetry* **2020**, *12*, 536.
https://doi.org/10.3390/sym12040536

**AMA Style**

Hou S, Guo W.
Optimal Denoising and Feature Extraction Methods Using Modified CEEMD Combined with Duffing System and Their Applications in Fault Line Selection of Non-Solid-Earthed Network. *Symmetry*. 2020; 12(4):536.
https://doi.org/10.3390/sym12040536

**Chicago/Turabian Style**

Hou, Sizu, and Wei Guo.
2020. "Optimal Denoising and Feature Extraction Methods Using Modified CEEMD Combined with Duffing System and Their Applications in Fault Line Selection of Non-Solid-Earthed Network" *Symmetry* 12, no. 4: 536.
https://doi.org/10.3390/sym12040536