# Combined Power Quality Disturbances Recognition Using Wavelet Packet Entropies and S-Transform

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Wavelet Packet Entropies and Modified Incomplete S-Transform

#### 2.1. Wavelet Packet Decomposition

^{th}level for the j

^{th}node.

_{p}(p≤2) norm, logarithm entropy and energy entropy. Considering the existing research results, Daubechies series wavelets are more sensitive to irregular signal, thus DB4 wavelet function based three level wavelet packet decomposition is chosen. Shannon entropy is adopted as the cost function to find the best wavelet packet base.

#### 2.2. Shannon Entropy and Wavelet Packet Energy Entropy

_{1}, x

_{2}, …, x

_{L}} represents its state characteristics, where the probability of x

_{j}is ${p}_{j}=P\left\{X={x}_{j}\right\},j=1,\cdot \cdot \cdot ,L\in N$, $\sum _{j=1}^{L}{p}_{j}}=1$.

^{th}scale, where k=1,2,…,N is the sampling point of original signal. Then, ${E}_{j}={\displaystyle \sum _{k=1}^{N}{E}_{(j,k)}}$ represent the sum energy at the j

^{th}scale. Set relative wavelet packet energy ${p}_{(j,k)}={E}_{(j,k)}/{E}_{j}$, according to energy conservation principle, $\sum _{k}{p}_{(j,k)}}=1$. Based on fundamental principle of information entropy, wavelet packet energy entropy (WPEE) distributing along the scale is defined as follows.

#### 2.3. Tsallis Entropy and Wavelet Packet Tsallis Entropy

^{i}wavelet packet nodes in moving data window $W(m,w,\delta )$ form a matrix ${D}_{{2}^{i}\times w}$. Based on the singular value decomposition theory [18], matrix ${D}_{{2}^{i}\times w}$ can be decomposed as follows.

#### 2.4. Modified Incomplete S-Transform

## 3. Recognition Plan

#### 3.1. Feature Extraction

No | Method | Name | Description | Threshold | Function |
---|---|---|---|---|---|

1 | WPEE | E_{av} | Mean value | 1.2 | Oscillation assistant judgment |

2 | E_{std} | Standard deviation | 0.17, 0.8 | Impulsive/oscillation assistant judgment | |

3 | E_{bias} | Bias | 1.1, 4.8 | Interruption/impulsive assistant judgment | |

4 | WPTSE | E_{av} | Mean value | 0.091, 0.35 | Oscillation/impulsive assistant judgment |

5 | E_{bias} | Bias | 0.8 | Harmonic assistant judgment | |

6 | MIST | N_{f} | Number of main frequency points | - | Oscillation/harmonic initial judgment |

7 | N_{h} | Whether it contains high frequency | 0, 1 | Oscillation initial judgment | |

8 | N_{1} | Whether it contains Harmonic | 0, 1 | Harmonic initial judgment | |

9 | S_{av} | Average amplitude of the fundamental components | 0.475, 0.495 | Swell/sag/interruption initial judgment | |

10 | S_{bias} | Bias of the fundamental component | 0.19, 0.85 | Swell/sag/interruption initial judgment | |

11 | S_{max} | Maximum of the fundamental component | 0.4807, 0.57 | Swell/sag/interruption assistant judgment | |

12 | S_1 | Amplitude fluctuation of fundamental components | 0, 1 | Fluctuation assistant judgment | |

13 | S_2 | The symmetry of main frequency points | 0, 1 | Fluctuation assistant judgment |

**R0**~

**R7**respectively for the convenience of expression, namely,

**R0-**normal signal,

**R1-**voltage swell,

**R2-**voltage sag,

**R3-**voltage interruption,

**R4-**impulsive transient,

**R5-**oscillation transient,

**R6-**harmonics,

**R7-**voltage fluctuation). Referring to the related IEEE standards in [25], 200 samples of each single disturbance are randomly produced as feature extraction test signals. The fundamental frequency of the signals is 50 Hz, the sampling frequency and point number are 6.4 kHz and 2048, respectively. Examples are shown in Figure 4, and the 13 dimensional features for the examples in Figure 4 are listed in Table 2.

**Table 2.**The 13-dimensional features for the examples in Figure 4.

Disturbances | E_{av} | E_{std} | E_{bias} | T_{av} | T_{bias} | N_{f} | N_{h} | N_{1} | S_{av} | S_{bias} | S_{max} | S_1 | S_2 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Normal (R0) | 0.0006 | 0.0009 | 0.9991 | 0.0882 | 0.8392 | 1 | 0 | 0 | 0.4806 | 0.0387 | 0.4806 | 0 | 0 |

Swell (R1) | 0.0016 | 0.0041 | 0.9991 | 0.1377 | 0.8398 | 1 | 0 | 0 | 0.5576 | 0.4072 | 0.7036 | 0 | 0 |

Sag (R2) | 0.0022 | 0.0065 | 0.9991 | 0.1374 | 0.8824 | 1 | 0 | 0 | 0.4392 | 0.2841 | 0.4806 | 0 | 0 |

Interruption (R3) | 0.0986 | 0.4007 | 4.8617 | 0.2519 | 3.5527 | 1 | 0 | 0 | 0.3537 | 0.9615 | 0.4806 | 0 | 1 |

Impulsive (R4) | 0.0686 | 0.3500 | 3.4339 | 0.1809 | 3.4694 | 1 | 0 | 0 | 0.4767 | 0.1130 | 0.4806 | 0 | 0 |

Oscillation (R5) | 1.5776 | 1.0600 | 3.7398 | 0.6735 | 2.1716 | 2 | 1 | 0 | 0.4811 | 0.0387 | 0.4830 | 0 | 0 |

Harmonics (R6) | 0.0409 | 0.0027 | 0.9322 | 0.5730 | 0.1735 | 2 | 0 | 1 | 0.4806 | 0.0387 | 0.4806 | 0 | 0 |

Fluctuation (R7) | 0.0007 | 0.0010 | 0.9991 | 0.0891 | 0.8423 | 1 | 0 | 0 | 0.4808 | 0.1402 | 0.5314 | 1 | 1 |

**Figure 5.**The distribution of each feature quantity (

**a**) Add WPEE feature E

_{av}distribution; (

**b**) WPEE feature E

_{std}distribution; (

**c**) WPEE feature E

_{bias}distribution; (

**d**) WPTSE feature T

_{av}distribution; (

**e**) WPTSE feature T

_{bias}distribution; (

**f**) MIST feature S

_{max}distribution; (

**g**) MIST feature S

_{av}distribution; (

**h**) MIST feature S

_{bias}distribution.

#### 3.2. Ruled Decision Tree

Rule | Description |
---|---|

Rule1 | if S_{av} > 0.495 & 0.19 < S_{bias} < 0.85 & S_{max} > 0.57 then R1 = 1 |

Rule2 | if S_{av} < 0.475 & 0.19 < S_{bias} < 0.85 & S_{max} < 0.4807 then R2 = 1 |

Rule 3 | if S_{av} < 0.475 & S_{bias} > 0.85 & S_{max} < 0.4807 then R3 = 1 |

Rule 4 | if 0.17 < E_{std} < 0.8 & 0.091 < T_{av} < 0.35 & 1.1 < E_{bias} < 4.8 & N_{h} = 0 & N_{1} = 0 then R4 = 1 |

Rule 5 | if N_{f} > 1& N_{h} = 1& E_{av} > 1.2 & E_{std} > 0.8 then R5 = 1 |

Rule 6 | if N_{f} > 1& N1 = 1 & T_{bias} < 0.8 then R6 = 1 |

Rule 7 | if R1|R2|R3 = 1 & R4 = 1 then R7 = S_1 & S_2 else if R1|R2|R3 = 1 & R4 = 0 then R7 = S_2 else if R1|R2|R3 = 0 & R4 = 1 then R7 = S_1 else R1|R2|R3 = 0 & R4 = 0 then R7 = S_1| S_2 |

#### 3.3. Recognition Flow

## 4. Experimental Results

Disturbance Type | Recognition Results | Number of Right Samples | Accuracy/% | Time/s | ||||||
---|---|---|---|---|---|---|---|---|---|---|

R1 | R2 | R3 | R4 | R5 | R6 | R7 | ||||

swell | 191 | 0 | 0 | 0 | 0 | 0 | 0 | 191 | 95.5 | 0.014 |

sag | 0 | 189 | 10 | 0 | 0 | 0 | 0 | 189 | 94.5 | 0.018 |

interruption | 0 | 5 | 195 | 0 | 0 | 0 | 0 | 195 | 97.5 | 0.014 |

impulsive | 0 | 0 | 0 | 185 | 0 | 0 | 0 | 185 | 92.5 | 0.014 |

oscillation | 0 | 0 | 0 | 0 | 194 | 0 | 0 | 194 | 97 | 0.011 |

harmonics | 0 | 0 | 0 | 0 | 0 | 199 | 0 | 199 | 99.5 | 0.011 |

fluctuation | 0 | 0 | 0 | 0 | 0 | 0 | 200 | 200 | 100 | 0.013 |

Disturbance Type | Recognition Results | Number of Right Samples | Accuracy/% | Time/s | ||||||
---|---|---|---|---|---|---|---|---|---|---|

R1 | R2 | R3 | R4 | R5 | R6 | R7 | ||||

R1 + R6 | 193 | 0 | 0 | 0 | 0 | 199 | 0 | 193 | 96.5 | 0.017 |

R2 + R5 | 0 | 194 | 0 | 0 | 196 | 0 | 0 | 194 | 97 | 0.017 |

R2 + R6 | 0 | 195 | 0 | 0 | 0 | 199 | 0 | 195 | 97.5 | 0.015 |

R2 + R7 | 0 | 188 | 0 | 0 | 0 | 0 | 197 | 188 | 94 | 0.013 |

R3 + R6 | 0 | 0 | 195 | 0 | 0 | 195 | 0 | 195 | 97.5 | 0.016 |

R5 + R6 | 0 | 0 | 0 | 0 | 195 | 196 | 0 | 195 | 97.5 | 0.011 |

R5 + R7 | 0 | 0 | 0 | 0 | 194 | 0 | 200 | 194 | 97 | 0.013 |

R6 + R7 | 0 | 0 | 0 | 0 | 0 | 200 | 200 | 200 | 100 | 0.013 |

R2 + R5 + R6 | 0 | 192 | 0 | 0 | 198 | 199 | 0 | 192 | 96 | 0.013 |

R2 + R5 + R7 | 0 | 186 | 0 | 0 | 193 | 0 | 191 | 186 | 93 | 0.014 |

R2 + R6 + R7 | 0 | 182 | 0 | 0 | 0 | 199 | 194 | 182 | 91 | 0.016 |

R3 + R5 + R6 | 0 | 0 | 188 | 0 | 195 | 180 | 0 | 180 | 90 | 0.018 |

R5 + R6 + R7 | 0 | 0 | 0 | 0 | 195 | 197 | 200 | 195 | 97.5 | 0.020 |

R1+R4+R6+R7 | 181 | 0 | 0 | 194 | 0 | 198 | 196 | 181 | 90.5 | 0.020 |

- (1)
- Sample recognition accuracy. This index considers the overall recognition accuracy of the sample. It is a traditional pattern recognition evaluation method. The calculation formula is as follows.$$Sample\text{}recognition\text{}accuracy=\frac{the\text{}number\text{}of\text{}correct\text{}recognition}{the\text{}total\text{}number\text{}of\text{}samples}\times 100\%$$
- (2)
- Label error (leak) rate. This index considers the number of recognition error and leakage in recognition results of all samples. It reflects the stability of the proposed recognition method for single disturbance in the case of different combined disturbances. The calculation formula is as follows.$$Label\text{}error\text{}\left(leak\right)\text{}rate=\frac{the\text{}number\text{}of\text{}error\text{}and\text{}leakage\text{}recognition}{the\text{}total\text{}number\text{}of\text{}samples}\times 100\%$$

Disturbance type | R1 | R2 | R3 | R4 | R5 | R6 | R7 |
---|---|---|---|---|---|---|---|

Total sample number | 14 × 200 = 2800 | ||||||

Recognition error and leakage number | 26 | 31 | 17 | 6 | 27 | 37 | 7 |

Recognition error and leakage rate/% | 0.928 | 1.107 | 0.607 | 0.214 | 0.964 | 1.321 | 0.250 |

**Figure 7.**The normalized real-life signals (

**a**) swell signal; (

**b**) impulsive signal; (

**c**) sag + oscillation signal; (

**d**) interruption + oscillation signal.

Disturbances | R1 | R2 | R3 | R4 | R5 | R6 | R7 |
---|---|---|---|---|---|---|---|

Swell | 1 | 0 | 0 | 0 | 0 | 0 | 0 |

Impulsive | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

Sag + Oscillation | 0 | 1 | 0 | 0 | 1 | 0 | 0 |

Interruption + Oscillation | 0 | 0 | 1 | 0 | 1 | 0 | 0 |

Method | Accuracy/% | |
---|---|---|

Single | Combined | |

Improved incomplete S-transform with decision tree | 81.86 | 88.93 |

Wavelet transform with neural network | 94.42 | 83.33 |

EEMD and MIST with automatic classification | 97.70 | 88.70 |

Wavelet Packet Entropies and MIST with decision tree (proposed) | 96.64 | 95.36 |

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Liu, Z.; Cui, Y.; Li, W.
Combined Power Quality Disturbances Recognition Using Wavelet Packet Entropies and S-Transform. *Entropy* **2015**, *17*, 5811-5828.
https://doi.org/10.3390/e17085811

**AMA Style**

Liu Z, Cui Y, Li W.
Combined Power Quality Disturbances Recognition Using Wavelet Packet Entropies and S-Transform. *Entropy*. 2015; 17(8):5811-5828.
https://doi.org/10.3390/e17085811

**Chicago/Turabian Style**

Liu, Zhigang, Yan Cui, and Wenhui Li.
2015. "Combined Power Quality Disturbances Recognition Using Wavelet Packet Entropies and S-Transform" *Entropy* 17, no. 8: 5811-5828.
https://doi.org/10.3390/e17085811