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One of the key issues of the accurate parameters analysis for the piecewise envelope current signal is to position the change point precisely. Discrete wavelet transform (DWT) modulus maxima method can detect change point, but the detection window of DWT will cause suspicious change point. Besides, the amount of calculated data is very large in actual process of envelope current signal. Therefore, in this paper, the envelope is used instead of the original sampling data for DWT so as to reduce the calculation amount. What is more, combined with the sliding dislocation window method, the change point can be located accurately and the pseudo-change point can be eliminated. The simulation results as well as the electric locomotive current and forging machine current examples show that it is feasible to detect the change point precisely through the proposed method, which provides possibilities for real-time online monitoring of change point.

The wide application of semiconductor electronic devices and distributed powers has caused harmonic content in power system and serious distortion of voltage and current waveform [

Harmonic current

Oblique envelope current

Parabolic envelope current

Exponential envelope current

The traditional electric energy meter usually works in sinusoidal linear load condition with high precision. However, in practical terms, the current signal often has different envelopes, which is the piecewise envelope current signal actually. The harmonics and the varying amplitudes in the current signal will cause difficulties for accurate measurement of conventional energy meter [

In general, change point analysis may be performed in either parametric or nonparametric approaches. These approaches follow some statistical framework, including CUSUM (cumulative sum), GLR (generalized likelihood ratio), and the change finder. Generally, these approaches have limitation, that is, heavily relying on prespecified parametric models such as probability density models and state-space models [

DWT has the ability of multiresolution analysis. It can represent local information in both time domain and frequency domain [

Therefore, this paper positions the change point through DWT of the envelope of the current signal (not the original sampling data), which aims to reduce the calculation time. Besides, The DWT modulus maxima method is applied to detect the change point, and the sliding dislocation window is introduced to eliminate the suspicious change point. Furthermore, the simulation for dynamic current signal is applied to analyze the performance of the proposed method. An electric locomotive current signal and forging machine current are analyzed in the end.

Wavelet transform includes continuous wavelet transform and discrete wavelet transform. They both can be used to detect the signal change point. However, the amount of calculated data of continuous wavelet transform is too large, which results in a long computing time. Therefore, the discrete wavelet transform is suitable for online calculation. Discrete wavelet transform decomposes the dynamic signal into the various scales of frequency bands by multiresolution analysis [

The DWT of a signal

Furthermore, in the multiresolution analysis process, a give n signal

where_{j,k} is the approximation coefficients at level J._{j,k} is detail coefficients. The decomposition level of discrete wavelet transform

The frequency bands corresponding to the DWT signal.

Suppose the wavelet function and the signal function_{0}_{0}) in a given level_{0} satisfies

If

Therefore, at a given level_{0, }a set of modulus maxima can be obtained from the detail coefficients. For a signal

The change point of dynamic signal can be easily identified through modulus maximum method. However, in the actual signal processing, it often truncates the sampling data for wavelet transform with detection window because the computer program is not able to perform an analysis for infinite sampling data points. However, due to the windowing of the sampled data points, wavelet transforms will cause a sudden change in the edge of the detection window, which is likely to cause pseudo-change point. In other words, the sudden change may not be due to the waveform jump of the envelope itself.

In order to solve this problem, this paper presents a method to eliminate these pseudo-change points by the sliding dislocation window. The steps are as follows:

The flow chart of the sliding dislocation window method is shown in Figure

The flow chart of the sliding dislocation window method.

The wavelet decomposition of the dynamic signal envelop can save the calculation time greatly, which can be used in online real-time monitoring of change point in dynamic signals.

The dynamic signal shown in Figure

The envelope diagram of the dynamic signal.

All the envelope signals above are with 3 harmonics, the fundamental frequency is_{0}=50.0Hz, the phase _{m} is 100A, 5A, and 10A, respectively. Envelope parameters are_{0}=0.5. The sampling frequency is

The simulation signal and its envelope waveform are shown in Figure

The data from 0.55s to 0.75s (10 cycles) are picked artificially from the parabolic envelope signal (0.5s<

The detail coefficients of the first layer with different wavelets (0.55s≤

According to Figure

Wavelet | Modulus maxima | Modulus average | |
---|---|---|---|

bior2.2 | 1.168 | 0.761 | 1.535 |

3.827 | 5.029 | ||

db2 | 4.717 | 0.999 | 4.720 |

1.361 | 1.362 | ||

Sym2 | 4.717 | 0.999 | 4.720 |

1.361 | 1.362 | ||

Coif1 | 1.850 | 0.979 | 1.891 |

4.809 | 4.914 |

Table

In order to solve this problem, this paper presents the method of sliding dislocation window which is introduced in Section

We chose the 2650 data points when 0.55s<

The detail coefficients of the first layer when applying db2 (0.55

Location | Modulus maxima | Modulus average | |
---|---|---|---|

| 0.051 | 12.920 | 0.003 |

| 109.600 | 8.483 |

Table

In addition,

After eliminating the misjudgment points, this paper chooses

Wavelet | Modulus maxima | Modulus average | |
---|---|---|---|

21.832 | 1.565 | ||

haar | 221.793 | 13.953 | 15.896 |

5.260 | 0.377 | ||

7.619 | 0.689 | ||

bior2.2 | 169.570 | 11.063 | 15.328 |

97.858 | 8.846 | ||

7.129 | 0.575 | ||

bior2.4 | 158.988 | 12.407 | 12.813 |

91.741 | 7.394 | ||

21.141 | 0.802 | ||

rbio3.1 | 72.736 | 26.362 | 2.759 |

181.937 | 6.902 | ||

16.696 | 1.289 | ||

db2 | 201.969 | 12.957 | 15.587 |

72.922 | 5.628 | ||

6.700 | 0.575 | ||

db4 | 77.883 | 11.652 | 6.684 |

90.211 | 7.742 | ||

16.696 | 1.289 | ||

Sym2 | 201.969 | 12.957 | 15.587 |

72.922 | 5.628 | ||

9.616 | 0.720 | ||

Sym4 | 170.803 | 13.351 | 12.793 |

74.579 | 5.586 | ||

7.044 | 0.577 | ||

Coif1 | 172.450 | 12.215 | 14.118 |

93.058 | 7.6184 | ||

8.531 | 0.631 | ||

Coif3 | 150.870 | 13.527 | 11.153 |

80.843 | 5.976 |

The detail coefficients of the first layer (

According to Figure

In addition,

Because

In addition, according to Table

Therefore, this paper chooses the

Comparison of the theoretical value and the calculated value.

Wavelet | db2 | Sym2 | Coif1 | bior2.2 | |
---|---|---|---|---|---|

change point 1 | theoretical value/s | 0.3 | |||

calculated value/s | 0.29797 | 0.29797 | 0.31797 | 0.31797 | |

relative error/% | 0.6771 | 0.6771 | 5.9896 | 5.9896 | |

change point 2 | theoretical value/s | 0.5 | |||

calculated value/s | 0.49797 | 0.49797 | 0.51797 | 0.51797 | |

relative error/% | 0.4062 | 0.4062 | 3.5937 | 3.5937 | |

change point 2 | theoretical value/s | 0.8 | |||

calculated value/s | 0.81797 | 0.81797 | 0.79797 | 0.79797 | |

relative error/% | 2.2461 | 2.2461 | 0.2539 | 0.2539 |

According to Table

In order to compare the proposed wavelet decomposition method which uses the envelope data (

Comparison of relative error of the envelope method and the direct method.

Relative error/% | db2 | Sym2 | Coif1 | bior2.2 | |
---|---|---|---|---|---|

change point 1 | envelope method | 0.6771 | 0.6771 | 5.9896 | 5.9896 |

direct method | 0.1042 | 0.1042 | 0.0521 | 0.0521 | |

change point 2 | envelope method | 0.4062 | 0.4062 | 3.5937 | 3.5937 |

direct method | 0.0625 | 0.0625 | 0.0312 | 0.0312 | |

change point 3 | envelope method | 2.2461 | 2.2461 | 0.2539 | 0.2539 |

direct method | 0.0391 | 0.0391 | 0.0586 | 0.0586 |

Comparison of program computation time of the envelope method and the direct method.

Wavelet | Direct method | Envelope method /s | Reduction of computation time |
---|---|---|---|

db2 | 0.17413 | 0.13803 | 20.734 |

Sym2 | 0.16627 | 0.14662 | 11.817 |

Coif1 | 0.17442 | 0.15252 | 12.558 |

bior2.2 | 0.18728 | 0.14983 | 19.993 |

According to Table

In addition, the program computation time of

Therefore, the envelope method proposed in this paper can be used for the online real-time detection of signal change points.

For a current signal of an electric locomotive, the sampling time is 152.9136s, the sampling data is 764568 points, the sampling frequency is_{s}=5000 Hz, and the sampling interval time is 0.2ms. The upper envelope curve is obtained from the sampling data and there are 509 points in the upper envelope curve; the number of data points that need to be processed in the envelope method is only 0.0665% of that in the direct method. The current waveform and the upper envelope curve are shown in Figure

Electric locomotive current waveform and its upper envelope.

Detail coefficients of the envelope method with different wavelets.

Detail coefficients of the direct method with different wavelets.

According to Figures

Comparison of envelope method with different wavelets.

Wavelet | Maximum modulus | Average modulus | | Relative error/% | Average time/s |
---|---|---|---|---|---|

db2 | 0.1688 | 0.0101 | 16.6635 | 0.8553 | 0.2278 |

0.1322 | 13.0522 | 2.4698 | |||

Sym2 | 0.1688 | 0.0101 | 16.6635 | 0.8553 | 0.2326 |

0.1322 | 13.0522 | 2.4698 | |||

Coif1 | 0.1170 | 0.0098 | 11.9468 | 1.0310 | 0.2504 |

0.1282 | 13.0869 | 2.6637 | |||

bior2.2 | 0.1105 | 0.0089 | 12.3620 | 1.0310 | 0.2688 |

0.1195 | 13.3673 | 2.6637 |

From Table

Figure

Forging machine current waveform and its upper envelope.

Detail coefficients of the envelope method with different wavelets.

Detail coefficients of the direct method with different wavelets.

The analysis results of two obvious change points (at

Comparison of envelope method with different wavelets.

Wavelet | Maximum modulus | Average modulus | | Relative error/% | Average time/s |
---|---|---|---|---|---|

db2 | 0.6346 | 0.0732 | 8.6733 | 0.0201 | 0.2040 |

0.1526 | 2.0849 | 0.0439 | |||

Sym2 | 0.6346 | 0.0732 | 8.6733 | 0.0201 | 0.2160 |

0.1526 | 2.0849 | 0.0439 | |||

Coif1 | 0.3149 | 0.0771 | 4.0841 | 0.9265 | 0.2169 |

0.1325 | 1.7190 | 1.4076 | |||

bior2.2 | 0.2063 | 0.0663 | 3.1064 | 0.9265 | 0.2401 |

0.1659 | 2.4985 | 1.4076 |

Table

This fast and accurate detection method offers possibilities for online real-time monitoring of change points in envelope current signals, which is important for the accurate measurement of electrical energy.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the National Science Foundation of China (No. 51377174, No. 51577016) and Science and Technology Project of State Grid Corporation of China (JL71-15-023).