# Chaos Synchronization Error Technique-Based Defect Pattern Recognition for GIS through Partial Discharge Signal Analysis

^{*}

## Abstract

**:**

## 1. Introduction

_{6}gas the moment a partial discharge arises. In contrast, most noise interference such as corona in a power system occupies the frequency band below 150 MHz, and undergoes rapid decay during wave propagation in air, that is, an elevated signal to noise ratio is hence seen due to the lower level noise interference over the UHF band.

## 2. Chaos Synchronization Error Dynamics

_{1}, u

_{2}, u

_{3}denote the control terms in a slave system. With the dynamic error trajectories as the targets, u

_{1}, u

_{2}, u

_{3}are set to zeros in a chaos synchronization error dynamics system [22]. In Equations (2) and (3), x

_{1}= x[i], y

_{1}= x[i + 1], z

_{1}= x[i + 3], x

_{2}= y[i], y

_{2}= y[i + 1], z

_{2}= y[i + 2], i = 1, 2, 3, …, n − 2, x and y respectively represent the sample sequences of the defect and the normal signals, and n denotes the total number of samples. Defining dynamic errors as e

_{1}= x

_{1}− x

_{2}, e

_{2}= y

_{1}− y

_{2}, e

_{3}= z

_{1}− z

_{2}, a dynamic error (DE) system is derived from Equations (2) and (3) as:

_{1}, E

_{2}, E

_{3}, and a phase plot of a chaos synchronization error dynamics are illustrated.

_{1}, E

_{2}, E

_{3}for a Lorenz chaos synchronization system are exhibited in Figure 1b, where deviations of E

_{1}, E

_{3}are seen away from zeros. A point worthy of mention is that the occurrence of partial discharge is reflected in the dynamic error trajectories which do not as expected converge to zero in the presence of inevitable noise interference. Demonstrated in Figure 1c is a phase plane trajectory for E

_{1}, E

_{2}, E

_{3}, where a number of spikes representing partial discharge are seen as the validation of this proposal.

## 3. Fractal and Extension Theories

_{1}, E

_{2}and E

_{3}to construct the characteristic matrix, which must be able to express the characteristics and properties of the defect signals. Then, fractal theory is utilized to obtain the characteristic value, which is finally used in the clustering method for training and testing. A more detailed introduction of fractal theory process and the extension concept can be found elsewhere [26–32].

## 4. Method

#### 4.1. Characteristic Extraction

_{1}, 2 to 33,299 is by y

_{1}, and 3 to 33,300 is by z

_{1}. Likewise, a normal signal is applied to the Slave tracking system, where the set consisting of the normal signal samples 1 to 33,298 is denoted by x

_{2}, 2 to 33,299 is by y

_{2}, and 3 to 33,300 is by z

_{2}. The tracking dynamic errors, E

_{1}, E

_{2}, E

_{3}are evaluated by Equations (4) to (6). Differences in trajectories E

_{1}, E

_{2}, E

_{3}are demonstrated in Figure 3 among four types of defects. In particular, as demonstrated in Figure 4, different dot distribution patterns and densities are seen across various defect types, and self-similarity is viewed between the same types of defect, so that such two quantities are extracted as two characteristics.

#### 4.2. Construction of the Characteristic Matrix

_{1}and E

_{2}are represented as the x and y axes, and program automatically determines the maximum values b

_{max}for all defect types of |E

_{1}| and |E

_{2}|. The maximum value b

_{max}and minimum value −(b

_{max}) define the extent of the x and y axes. This step ensures that the matrices for all defects have the same size. The extreme values at the limits of the minimum range are set to −(b

_{min}) and b

_{min}. These are divided by number of grid cells to obtain the spacing value. Finally, the matrix size is determined by dividing the boundary value by the spacing.

_{1}and E

_{2}. However, only the density characteristic is shown in this study, so all the E

_{3}in the E

_{1}and E

_{2}grid cells are summed, so the total amplitude value represents the number of distributed points. More distributed points corresponds to a higher total amplitude value, so this characteristic matrix can express the density of the E

_{1}, E

_{2}distributions, and the total amplitude value for E

_{3}represents the number of distributed points.

#### 4.3. Characteristic Extraction and Clustering Method

_{3}values are larger than those in Figure 6b,c. When the three-dimensional characteristic graphs are calculated using fractal theory, the lacunarity and fractal dimension values of various defects can be determined.

## 5. Experiment and Results

#### 5.1. Experiment Models

_{6}gas. Figure 7 shows the possible defect models that might be a result of human carelessness during GIS construction. The four testing models are designed as follows:

- Type I: Porcelain bushing internal conductor containing oil grease.
- Type II: SF
_{6}gas tank containing 5 mm × 3 mm × 1 mm metal particles. - Type III: A welding protrusion with size approximately 5 mm × 5 mm × 2 mm on the bearing.
- Type IV: An abrasion defect with 2 mm depth and 10 mm length on a metal ring.

#### 5.2. Measurement System

_{pre-stress}) should be applied for the power-frequency withstand voltage test and maintained at that value for 1 min. Partial discharges occurring during this period shall be disregarded. The voltage is then reduced to test the voltage for PD measurement, phase-to-earth (U

_{pd-test, ph-ea}). In this work, the GIS rated voltage (U

_{r}) is 15 kV, U

_{pre-stress}is applied at 45 kV for 1 min, and U

_{pd-test, ph-ea}= 1.2 U

_{r}/3

^{½}= 10.4 kV according to standard for PD measurement.

_{1}= 135.56 mm, W

_{1}= 149.81 mm, L

_{2}= 104.62 mm, W

_{2}= 3.53 mm, L

_{3}= 47.26 mm, W

_{3}= 34.8 mm and L

_{4}= 37.26 mm. Due to financial constraints we could not acquire an ultrahigh speed signal capture card, therefore, in order to fit the Shannon theorem, we decreased the bandwidth of the microstrip antenna to 4 MHz using a delay circuit in the experimental system. A human-computer interface is developed in the LABVIEW environment for real time partial discharge signal processing. There are a total of 160 discharge signal samples for four types of defects, each with 40 samples, the first 20 of which are regarded as training samples, and the rest are as the testing ones. In this study, dynamic error trajectories E

_{1}, E

_{2}, E

_{3}in a chaos synchronization system are plotted, based on which a characteristic matrix is constructed, and the lacunarity as well as the fractal dimension is extracted accordingly through the fractal theory. In the end, an investigation is made into the recognition accuracy rate and the tolerance to noise interference for a chaos synchronization system by use of this proposed approach.

#### 5.3. Experiment Results and Discussion

_{1}, E

_{2}are demonstrated. Yet, the use of merely fractional dimension is found inadequate to recognize all the four defect types, according to which the lacunarity is adopted as another characteristic so as to improve the recognition rate. The differences in characteristics can be obviously observed in Figures 10 and 11. As exhibited in Figure 13 and as a validation of this work, distinct defects can be made distinguishable with ease through the application of this proposed approach to a Chen-Lee system, that is, another type of chaos synchronization systems.

## 6. Conclusions

_{1}, E

_{2}, E

_{3}in the tracking process of a chaos synchronization system. The boundary values are determined by the defect type with a high dynamic error trajectory, while the lacunarity is by that with a low dynamic trajectory. As a consequence, the dimension of the characteristic matrix corresponding to each defect type is specified, and the properties of dynamic error trajectories are characterized therein. By use of fractal theory, an improved defect recognition rate is reached by means of extension theory with the fractal dimension as well as the lacunarity extracted out of a characteristic matrix. The proposed approach yielded better clustering results than the HHT method, proving that the method that was proposed herein can extract characteristic information about four defects. In conclusion, this proposal is proven as an effective diagnostic tool for defect types in chaos synchronization systems.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Illustration of

**(a)**a normal and a defect PD signal,

**(b)**dynamic error E

_{1}, E

_{2}, E

_{3}, and

**(c)**a phase plot of a chaos synchronization system.

**Figure 2.**Typically single partial discharge signal for defect

**(a)**Type I,

**(b)**Type II,

**(c)**Type III and

**(d)**Type IV.

**Figure 3.**Error trajectories E

_{1}, E

_{2}for defect

**(a)**Type I,

**(b)**Type II,

**(c)**Type III, and

**(d)**Type IV.

**Figure 4.**Distributions of errors E

_{1}, E

_{2}for defect

**(a)**Type I,

**(b)**Type II,

**(c)**Type III, and

**(d)**Type IV.

**Figure 6.**Three dimensional characteristic for defect

**(a)**Type I.

**(b)**Type II.

**(c)**Type III.

**(d)**Type IV.

**Figure 12.**Characteristic distributions for various defect types in a Lorenz chaos synchronization system.

**Figure 13.**Characteristic distributions for various defect types in a Chen-Lee chaos synchronization system.

**Table 1.**Recognition accuracy rates with FD-Λ as characteristics in a Lorenz chaos synchronization system (%).

Noise Amount | 0% | ±10% | ±20% | ±30% |
---|---|---|---|---|

Defect Types | ||||

Type I | 100 | 100 | 80 | 70 |

Type II | 100 | 80 | 70 | 40 |

Type III | 100 | 70 | 60 | 30 |

Type IV | 100 | 90 | 75 | 60 |

**Table 2.**Recognition accuracy rates with FD-Λ as characteristics in a Chen-Lee chaos synchronization system (%).

Noise Amount | 0% | ±10% | ±20% | ±30% |
---|---|---|---|---|

Defect Types | ||||

Type I | 100 | 80 | 50 | 35 |

Type II | 100 | 75 | 60 | 55 |

Type III | 100 | 90 | 75 | 30 |

Type IV | 100 | 85 | 70 | 65 |

Noise Amount | 0% | ±10% | ±20% | ±30% |
---|---|---|---|---|

Defect Types | ||||

Type I | 80 | 70 | 70 | 50 |

Type II | 30 | 30 | 20 | 10 |

Type III | 90 | 55 | 30 | 10 |

Type IV | 50 | 25 | 10 | 0 |

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## Share and Cite

**MDPI and ACS Style**

Chen, H.-C.; Yau, H.-T.; Chen, P.-Y.
Chaos Synchronization Error Technique-Based Defect Pattern Recognition for GIS through Partial Discharge Signal Analysis. *Entropy* **2014**, *16*, 4566-4582.
https://doi.org/10.3390/e16084566

**AMA Style**

Chen H-C, Yau H-T, Chen P-Y.
Chaos Synchronization Error Technique-Based Defect Pattern Recognition for GIS through Partial Discharge Signal Analysis. *Entropy*. 2014; 16(8):4566-4582.
https://doi.org/10.3390/e16084566

**Chicago/Turabian Style**

Chen, Hung-Cheng, Her-Terng Yau, and Po-Yan Chen.
2014. "Chaos Synchronization Error Technique-Based Defect Pattern Recognition for GIS through Partial Discharge Signal Analysis" *Entropy* 16, no. 8: 4566-4582.
https://doi.org/10.3390/e16084566