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As a novel data mining approach, a wavelet entropy algorithm is used to perform entropy statistics on wavelet coefficients (or reconstructed signals) at various wavelet scales on the basis of wavelet decomposition and entropy statistic theory. Shannon wavelet energy entropy, one kind of wavelet entropy algorithm, has been taken into consideration and utilized in many areas since it came into being. However, as there is wavelet aliasing after the wavelet decomposition, and the information set of different-scale wavelet decomposition coefficients (or reconstructed signals) is non-additive to a certain extent, Shannon entropy, which is more adaptable to extensive systems, couldn’t do accurate uncertainty statistics on the wavelet decomposition results. Therefore, the transient signal features are extracted incorrectly by using Shannon wavelet energy entropy. From the two aspects, the theoretical limitations and negative effects of wavelet aliasing on extraction accuracy, the problems which exist in the feature extraction process of transient signals by Shannon wavelet energy entropy, are discussed in depth. Considering the defects of Shannon wavelet energy entropy, a novel wavelet entropy named Tsallis wavelet energy entropy is proposed by using Tsallis entropy instead of Shannon entropy, and it is applied to the feature extraction of transient signals in power systems. Theoretical derivation and experimental result prove that compared with Shannon wavelet energy entropy, Tsallis wavelet energy entropy could reduce the negative effects of wavelet aliasing on accuracy of feature extraction and extract transient signal feature of power system accurately.

Transient feature extraction is an important part of signal analysis, and as a new feature extraction algorithm, wavelet entropy has attracted the attention of experts and scholars all over the world. Wavelet entropy is a combination of wavelet decomposition and entropy statistics theories, and it has the advantages of multi-resolution analysis and complexity evaluation for time-varying signals, which means that the macro and micro aspects of some special signals could be researched in the time-frequency domain. For the above reasons, wavelet entropy has gradually been used in engineering signal surveys such as electroencephalography (EEG) testing, machinery vibration detection, power system fault diagnosis and so on [

However, most research on wavelet entropy involves engineering applications of Shannon wavelet energy entropy (SWEE), but its physical meanings, working mechanism, and application principle have not been well discussed yet. In fact, Shannon wavelet energy entropy has some disadvantages in processing non-stationary signals, which could result in inaccurate or wrong results.

In view of the abovementioned facts, in this paper, the theoretical basis of SWEE has been analyzed, and the emphasis of research was placed on the working mechanism for feature extraction of transient signals. Wavelet aliasing’s negative effect on feature extraction, which uses Shannon wavelet entropy, is discussed to find the primary reason for the problem.

Because Tsallis entropy is good at expressing the uncertainty of generalized systems, with a combination of Tsallis entropy and wavelet decomposition, a wavelet entropy-Tsallis wavelet energy entropy (TWEE) is constructed and the physical meaning of its algorithm is also explained. Starting with an analysis of wavelet aliasing and feature extraction effects, this paper studied the working mechanism and the suitable scope of TWEE, analyzes the relationship and difference between TWEE and SWEE and provides the initial principle of non-extensivity index selection. Finally, for analyzing transient voltage fluctuations caused by non-faulty indirect lightning strikes, TWEE is applied to transient feature extraction of lightning strikes. The corresponding voltage data when a non-faulty indirect lightning strike took place on 110 kV overhead distribution line in Guangdong, China, is collected and analyzed using TWEE, and the comparison of feature extraction, based upon TWEE and SWEE, has been carried out. The experimental results show that TWEE is better than SWEE in analyzing transient signals.

Shannon wavelet entropy (SWE) is the cooperative product of wavelet decomposition and Shannon entropy theory. By changing the segment of wavelet coefficients at the corresponding scales, and for different statistic objects using Shannon entropy, several derived algorithms are proposed based on SWE theory. Among these, SWEE is one of the most widely used algorithms in all fields [10]. The definition of SWEE is given as follows: first, a sliding window (width _{j}

In the macro-perspective, SWEE is the data mining of wavelet coefficients (or reconstructed signals) base on Shannon entropy statistics, so it is inevitable that Shannon WE inherits the statistic properties of Shannon entropy. As well known, Shannon entropy is the extended application for B-G entropy in information theory. As an important index in thermodynamics, B-G entropy represents the measurement for the uncertainty of N-energy-level system, and is defined as:

Shannon entropy is represented as:

From Equations (2) and (3), it is found that there is a strong connection between Shannon entropy and B-G entropy. In some ways, Shannon entropy naturally inherits the statistic property of B-G entropy such as convex linearity, extensivity, continuity, and so on, so Shannon entropy also satisfies the following Equation (4):

It is well known that Boltzmann-Gibbs statistics (BGS) are inadequate for treating some complex systems [

The analyzed signal result using SWEE is undoubtedly connected to the consistence between the wavelet decomposition result and the corresponding component in the signal, so it is a key point for SWEE accuracy that the proper orthogonal wavelet basis be selected to obtain accurate coefficients or reconstructed signals. However, for most mother wavelets, wavelet aliasing still exists more or less at the neighbor scales, which means that the information and energy of signal are incomplete at the decomposition scales, so we can say that wavelet aliasing is the main reason for energy leakage. To better understand energy leakage, the mechanism of wavelet aliasing is researched as follows: wavelet aliasing is when frequency bands, corresponding with wavelet scales, overlap each other after wavelet decomposition. Here, the definition of continuous wavelet transform is given as below. For

From Equation (5), the derived frequency-domain expression of the correlation function is obtained as:

As the Db4 wavelet is taken as

DB4 wavelet function curves with different scales in frequency domain.

Based on the above analysis, wavelet aliasing can have negative effects on the accuracy of SWEE, therefore we further research the relationship between SWEE and wavelet aliasing.

First, the time sequence of signal

Similarly, suppose that wavelet aliasing takes place at neighbor scales

Obviously,

According to Equations (11) and (12), it can be deduced as:

According to the above analysis, the curve of

Curve of _{k}

Considering that wavelet aliasing and energy leakage exist at the neighbor scales, on the basic premise of

As

The curve of

As

if

if

From the above, it is only when _{ε}_{k}_{k+1} that
_{1} = 200 Hz, _{2} = 390 Hz, _{3} = 450 Hz, _{4} = 625 Hz, _{5} = 1,260 Hz. In addition, the sampling frequency _{s} = 5,000 Hz, and _{n}

SWEE under wavelet aliasing.

According to the definition of SWEE, the value of SWEE has to increase with the complexity of the signal in the time-frequency domain. For this reason, with the frequency component increasing, the value of SWEE, from 0 s to 0.4 s, should be minimum from 0 s to 1 s, and the value of 0.4–0.8 s should be less than that of 0.8–1 s. However, from

Nonextensive statistical mechanics, pioneered by Tsallis, offer a consistent theoretical framework for the studies of complex systems with long-range interactions, long-time memories, multifractal and self-similar structures, or anomalous diffusion phenomena. As a nonextensive entropy, Tsallis entropy is the extension and development of extensive entropy (B-G entropy) in statistical physics [

Different from extensive entropy,

Note that

Suppose that 0 ≤ _{q}

From Equation (15), Tsallis entropy function represents the concave nature. At the same time, Tsallis entropy function has a definite concavity for _{q}

From

Relations between entropy with

A sliding-window (_{j}

Wavelet coefficients matrix in the sliding-window.

At the moment of

Suppose that

As the sliding-window slides, the trendline that TWEE changes with time can be obtained and plotted. In the above expression, the scale space corresponds to the frequency space, and TWEE indicates the energy distribution of signals. A wavelet functions does not have pulse selectivity in either the frequency or time domain, whereas it has a support region, so the partition of the original signal at scale space also indicates the signals’ energy distribution at time-frequency domains. The more complex the analyzed signal is, more modes the energy congregates to and the larger the TWEE is. Therefore the TWEE is an index to evaluate the signal complexity or uncertainty.

According to

To prove the above point, the example proposed in

Comparing

Apart from the statistical characteristics of entropy, we drive the root reason why TWEE are not affected by wavelet aliasing and could characterize the accurate features of transient signals. Taking a three-level system as the analysis object, according to Equation (13), the statistical results are calculated, and the corresponding relation between Tsallis entropy with

TWEE under wavelet aliasing.

Relation between Tsallis entropy with (

From

In the field of power quality, the research on transient signals such as transient overvoltages, voltage dips, voltage interruptions, voltage flicker and voltage pulses, have attracted great attention. As one of transient signals in power systems, the voltage fluctuations caused by indirect lightning strikes often happen in the overhead distribution lines in South China, where most indirect lightning strikes are low-energy and unlikely to cause faults directly. However, the safety of power transmission and the quality of power supply could be suffered from negative effects because of high incidence rate of lightning strikes. In addition to electromagnetic interference in PT secondary circuits, the short duration and low energy level of non-faulty indirect lightning strikes are another major reason which could result in failure in extracting the features of this kind of lightning strike, so the phase voltages, caused by indirect lightning strikes, are taken as analysis objects, and TWEE and SWEE are separately used to extract the transient features of lightning strikes. A set of phase voltage signals of a 110 kV transmission line, with lightning strike interference, are obtained from the Guangzhou EPRI (

Fault line: the 110 kV Jiaji transmission line in Guangdong.

Fault type: lightning strike interference.

Recording site: the 110 kV side of main transformer of 220 kV Jiahe substation.

Occurrence time: 15:57:7:480, 23 March 2010.

Waveform of three-phase voltages with lightning strike.

On the afternoon of 3 March 2010, lightning strikes sequentially occurred on the 110 kV Jiaji transmission line of Guangdong Province. The relay protection didn’t operate because of the low lightning energy. The recorder located in the 220 kV Jiahe substation recorded the phase voltage waveforms when the lightning struck. Here is a set of collected lightning strike voltage waveforms is shown in

An experimental platform for signal processing is built to verify the practicality and validity of the proposed algorithm. Based on this platform, the historical data of three-phase voltages, which was recorded when a non-faulty lightning strike occurs, is converted to general data according to IEEE Std C37.111-1999. The transient lightning voltage signal is reconstructed by utilizing a programmable function generator, and then the operation module with DSP collects the real-time voltage signals from the programmable function generator through A/D channel. Furthermore, the TWEE algorithm and SWEE algorithm fixed in DSP are utilized separately to extract the transient features of the lightning signal. Finally, the results are output through the D/A channel and shown in the scope. The experiment flow is as shown as

Experiment flow chart.

Set sample frequency of A/D

(

Entropy statistics theories and wavelet decomposition algorithmw are the academic foundation of wavelet entropy, therefore the choice of entropy has a direct impact on the signal analysis results. Because wavelet aliasing exists at neighbor scales, the information set of wavelet coefficients (or reconstructed signals) is endowed with additivity property, and the extent of nonadditivity is much higher for information sets corresponding to transient signals with varying frequency. At this point, using the SWEE algorithm would result in a false analysis result against the fact. In this paper, considering the advantage of Tsallis entropy in measuring the uncertainty of generalized system, TWEE is proposed to extract the features of transient signals with changing the value of

The financial support received from the National Natural Science Foundation of China (Grant No. 51377016), 2012 Science and Technology Project of State Grid Corporation of China (Grant No.2012515) and the Scientific Research Fundation for the Doctoral Program of Northeast Dianli University (Grant No.BSJXM201335) is gratefully acknowledged.

Jikai Chen mainly contributed to the

The authors declare no conflict of interest.