Multiwavelet Packet Entropy and its Application in Transmission Line ...

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 25, NO. 11, NOVEMBER 2014

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Multiwavelet Packet Entropy and its Application in Transmission Line Fault Recognition and Classification Zhigang Liu, Member, IEEE, Zhiwei Han, Yang Zhang, and Qiaoge Zhang Abstract— Multiwavelets possess better properties than traditional wavelets. Multiwavelet packet transformation has more high-frequency information. Spectral entropy can be applied as an analysis index to the complexity or uncertainty of a signal. This paper tries to define four multiwavelet packet entropies to extract the features of different transmission line faults, and uses a radial basis function (RBF) neural network to recognize and classify 10 fault types of power transmission lines. First, the preprocessing and postprocessing problems of multiwavelets are presented. Shannon entropy and Tsallis entropy are introduced, and their difference is discussed. Second, multiwavelet packet energy entropy, time entropy, Shannon singular entropy, and Tsallis singular entropy are defined as the feature extraction methods of transmission line fault signals. Third, the plan of transmission line fault recognition using multiwavelet packet entropies and an RBF neural network is proposed. Finally, the experimental results show that the plan with the four multiwavelet packet energy entropies defined in this paper achieves better performance in fault recognition. The performance with SA4 (symmetric antisymmetric) multiwavelet packet Tsallis singular entropy is the best among the combinations of different multiwavelet packets and the four multiwavelet packet entropies. Index Terms— Fault angle, fault recognition and classification, multiwavelet packet entropy, neural network, transmission line.

I. I NTRODUCTION

T

HE DETECTION and classification of transient signals is widely used in power systems for fault detection and classification, relays protection, power quality assessment, fault location, equipment condition monitoring, and transient stability analysis. Many theories, including wavelet analysis [1], artificial neural network (ANN) [2], [3], support vector machine [4], and various combinations [5], have been applied into power system transient signal recognition and classification. Transmission lines are the essential parts of the power system. Their faults can be classified as balanced and unbalanced faults. Most of the transmission line faults

Manuscript received September 19, 2012; revised November 6, 2013; accepted January 19, 2014. Date of publication February 6, 2014; date of current version October 15, 2014. This work was supported in part by the National Nature Science Foundation of China under Grant U1134205 and Grant 51377136 and in part by the Program for New Century Excellent Talents in University in China under Grant NECT-08-0825. Z. Liu, Z. Han, and Q. Zhang are with the School of Electrical Engineering, Southwest Jiaotong University, Chengdu 610031, China (e-mail: [email protected]; [email protected]; [email protected]). Y. Zhang is with Zhejiang Electric Design Institute, Hangzhou 310012, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2014.2303086

are of unbalanced faults. Single-phase to ground, interphase short circuit, and three-phase to ground faults are categorized as unbalanced faults. Since fault signals in transmission lines include a great deal of transient information, the basic processing with proper tools and methods is crucial. Wavelet transform is commonly used to extract characteristic quantities, while ANN is used classify characteristic quantities. Characteristic quantity analysis of transmission line faults includes the decomposition coefficients and their energy, the maximum or minimum value of fault currents, such as wavelet coefficients of different phase currents [6], the maximum and minimum values of detailed coefficients at level 3 [7], the energy of current wavelet coefficients [8], the energy change and standard deviation (SD) of fault current [9], the summation of wavelet detailed coefficients for different phases [10], the peak absolute value, mean of the peak absolute values, and summation of the third level of current signals [11]. Spectral entropy based on Shannon entropy is a tool for producing an index of the complexity or uncertainty of a signal or system. Wavelet transformation is capable of revealing features of data that other signal analysis techniques may miss, and satisfies the need to analyze transient signals. A combination of wavelet transformation and entropy is proposed and applied to the analysis of electroencephalogram (EEG) signal [12]. Recently, the approach has been used to detect the floating thread defects in tufted carpets [13], distinguish between humans and dogs [14], detect intracranial pressure [15], in partial discharge recognition [16], and in fault detection of power grids [17]. For the transmission line fault recognition, the combination of wavelet and entropy can better exploit the advantages of both methods to describe the characteristics of signals. Wavelets can meet the demands of transient signal analysis, while entropy is ideal for the measurement of uncertainty [18]. The use of wavelet/wavelet packet entropy in classifying transmission line faults is established [19]–[26], but all are based on Shannon entropy. Shannon entropy does not have the ability to regulate nonextensiveness, which means that it cannot better reflect the complexity of fault signals. Tsallis entropy can provide the correct physical expression for nonadditive systems, which are mixtures or irregular fragments and it is nonextensive [27]. Compared with the conventional wavelet approaches, multiwavelets can simultaneously possess orthogonality, symmetry, short support, and high-order vanish moments. In addition, there is more low- and high-frequency information of multiwavelet decomposition than with conventional

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wavelets. The multiwavelet packet approach is an extension of multiwavelets. It offers more high-frequency information than multiwavelets. We apply multiwavelet/multiwavelet packets to a power system [28]. A combination of multiwavelets and a neural network is adopted to recognize power quality [29]. The approximate entropy features of a multiwavelet transform are combined with an ANN to classify EEG signals [30]. We combine multiwavelet packet coefficient entropy and ANN to recognize transmission line faults [31]. This combination is applied in image retrieval [32] as well. To extract comprehensive fault transient information in transmission lines and more accurately recognize and classify transmission line faults, we define four entropies with the multiwavelet packet to extract the features of different transmission line faults. These features are used as training samples of the radial basis function (RBF) neural network, which are trained to implement transient signal recognition and classification. The corresponding comparisons with wavelet packet entropies are analyzed. To demonstrate the capabilities of combining multiwavelet packet, entropy, and an ANN for transmission line fault recognition and classification, more simulation results are given and discussed at the end of this paper. This paper is structured as follows. Section II provides an introduction to multiwavelets and multiwavelet packet. Section III proposes the definitions of four multiwavelet packet entropies. Section IV gives the fault recognition and classification plan of power transmission line. Section V provides and analyzes the experimental results, and draws conclusion. II. M ULTIWAVELETS AND M ULTIWAVELET PACKET A. Multiwavelets Transformation Goodman and Lee [33] put forward the multiwavelets concept, and studied properties of scaling functions and wavelet functions. Symmetric scaling functions constructed by Geronimo et al. [34] had short support, generated an orthogonal multiresolution analysis (MRA), and provided approximation order two, for wavelets known as GHM multiwavelets. Chui and Lian [35] constructed two multiplicity multiscaling functions and multiwavelet functions with symmetry through the study of orthogonality, tight support, symmetry, and interpolation of multiwavelets. In wavelet analysis, the MRA is produced based on a scaling function and the base of space L 2 (R) constructed by the translation and dilation

of wavelet function. The MRA is critical in conventional wavelets transformation. Similarly, the multiresolution in multiwavelets analysis also takes place in multiwavelet analysis. However, the MRA is produced through several scaling functions and the base of space L 2 (R) constructed by the translation and dilation of multiwavelet functions. These multiwavelet functions are called multiwavelets. Let (t) = [φ1 (t), φ2 (t), . . . , φr (t)]T , φl ∈ L 2 (R), l = 1, 2, . . . , r , r ∈ N. Consider that φl produce the multiplicity r MRA. The translation and dilation of orthogonal multiwavelet functions j,k = {ψ1 (2− j x − k), . . . , ψr (2− j x − k)}T , ( j, k ∈ Z ) construct the orthogonal base in the orthogonal and supplemental subspace. Multiscaling function (t) satisfies the two-scaling equation (t) =

M

Hk (2t − k)

(1)

k=0

where Hk , k = 0, 1, . . . , M is a r × r impulse response constant matrix. Multiwavelet function (t) satisfies the following two-scaling matrix equation: (t) =

M

G k (2t − k)

(2)

k=0

where G k , k ∈ [0, M], M ∈ Z is a r × r impulse response constant matrix. For the sake of convenience, let r = 2 here. B. Preprocessing and Postprocessing Methods The necessity of preprocessing and postprocessing of multiwavelets can be demonstrated with CL3 multiwavelets. If preprocessing methods are not adopted for 1-D signals, the signal can be represented as a polyphase matrix 1 H1(z) 2 = H (z ) −1 (3) H2(z) z where z = eiω . The low-pass filters H1 and H2 of CL3 are listed in (4), as shown at the bottom of the page. Its high-pass filters G 1 and G 2 can be obtained similarly in (5), as shown at the bottom of the page. The frequency response of these filters is shown in Fig. 1(a) and (b). In Fig. 1, the low-pass filter responses of CL3 multiwavelets show the low-pass and band elimination, as well as their high-pass filters, which results in the aliasing of the frequency band. In addition, the detailed part of multiwavelet

⎡

H1(ω) H2 (ω)

G 1 (ω) G 2 (ω)

=

=

⎤ √ √ √ √ √ √ 10−3 10 5 6−2 15 iω 30+3 10 2iω −5 6+2 15 3iω + e + e + e + 40 40 40 40 ⎢ 30+3√10 4iω −5√6+2√15 5iω 10−3√10 6iω 5√6−2√15 −iω ⎥ ⎢ ⎥ e + √ 40 e√ +√ 40 e +√ 40 e ⎢ √ 40 ⎥ √ ⎢ 5 6−3 15 15−3 10 iω −5 6−7 15 2iω 15−3 10 3iω ⎥ + e + e + e + ⎣ ⎦ 40 40 40 40 √ √ √ √ √ √ 5 6+7 15 4iω 5−3 10 5iω −5 6+3 15 6iω 5−3 10 −iω e + 40 e + e + 40 e 40 40 ⎡√ √ ⎤ √ √ √ √ 5 6−2 15 30+3 10 iω −5 6+2 15 2iω −30−3 10 3iω + e + e + e + 40√ 40 40 40 √ √ √ √ √ ⎢ −5 ⎥ 6+2 15 4iω 10 5iω 15 6iω 10 −iω ⎥ ⎢ e √ + 10−3 e √+ 5 6−2 e √+ −10+3 e 40 40 40 ⎢ ⎥ √ √ 40 ⎢ −5+3 10 5 6+7 15 iω 15−3 10 2iω −30−3 10 3iω ⎥ e +√ 40 e √ + e + ⎣ 40 √ +√ 40 ⎦ 40 √ √ −5 6+3 15 4iω 10−3 10 5iω 5−3 10 6iω 5 6−3 15 −iω e + e + 40 e + e 40 40 40

(4)

(5)

LIU et al.: MULTIWAVELET PACKET ENTROPY

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Fig. 1. Filter response of CL3 multiwavelets. (a) Low-pass filter frequency response. (b) High-pass filter frequency response.

Fig. 3.

Fig. 2. Filter response of CL3 multiwavelets with the Haar method. (a) Low-pass filter frequency response. (b) High-pass filter frequency response.

decomposition cannot be reconstructed accurately. Hence, it is necessary to adopt preprocessing and postprocessing for multiwavelets. Prefilter [36] and balanced methods [37] are the most common methods for preprocessing and postprocessing of multiwavelets. We adopt Haar prefilter, shown below, to preprocess CL3 multiwavelets √ √ 1/ √2 1/√2 . (6) Q(ω) = 1/ 2 −1/ 2 Fig. 2 shows that the filter responses of CL3 multiwavelets are improved with Haar prefilter. C. Multiwavelet Packet Transformation Multiwavelet packet transformation is an extension of multiwavelet transformation. We suppose S0 be the original signal. For wavelet packet decomposition, let L1 and H1 be the low- and high-frequency parts of wavelet transformation at scale 1. Let LL2 and LH2 be the low- and high-frequency parts of L1 decomposition, HL2 and HH2 be the low- and high-frequency parts of H1 decomposition at scale 2. Let LLL3 and LLH3 be the low- and high-frequency parts of LL2 decomposition, LHL3 and LHH3 be the low- and highfrequency parts of LH2 decomposition, HLL3 and HLH3 be the low- and high-frequency parts of HL2 decomposition, and HHL3 and HHH3 be the low- and high-frequency parts of HH2 decomposition at scale 3. The frame of the multiwavelet packet decomposition is shown in Fig. 3, where the multiplicity is two. Because multiwavelet packet transformation is used to obtain more transient information in this paper, we will not discuss the problem of the best multiwavelet packet. The choice of best multiwavelet packet may be found in [38]. III. E NTROPY AND M ULTIWAVELET PACKET A. Shannon Entropy Entropy is a kind of measurement of disorders, such as unsystematic, unbalance, and uncertainty. The uncertainty

Frame of multiwavelet packet decomposition.

of any event is associated with its states and probabilities. For some uncertain system, if a random variable X {x 1 , x 2 , . . . , x L } represents its state characteristics, then the probability of x j is p j = P{X = x j }, j = 1, . . . , L ∈ N, L j =1 p j = 1. The information for some result of X can be represented by I j = log(1/ p j ). The information entropy of X can be defined in [39] H (X) = −

L

p j log( p j )

(7)

j =1

where p j = 0, p j log( p j ) = 0. B. Tsallis Entropy Shannon entropy is built based on Boltzmann–Gibbs (BG) entropy in thermodynamics, which is extensive [40]. Hence, Shannon entropy is extensive as well. Some experiments show that the decomposition or reconstruction coefficients of some signals are nonextensive [41]. The extensive property means that if the system is composed of two independent subsystems A and B, the entropy H ( A+ B) of system satisfies the additive property H (A + B)/k = H (A)/k + H (B)/k

(8)

where k is the Boltzmann constant. Based on (8), the signal’s entropy should be the sum of entropy in each scale through the decomposition and reconstruction of the signal with multiwavelet packet. Because frequency aliasing and energy leakage are found in the multiwavelet packet, the equation is not satisfied. Hence, Tsallis entropy, which is nonextensive, is introduced in this paper. Tsallis entropy can provide the correct physical expression for nonadditive systems with mixtures or irregular fragments [27]. Its continuous expression is defined as

c q (9) 1− f (x) d x , q ∈ R Sq = q −1 where f (x) is the probability density distribution function, f (x)d x = 1, c is a conventional positive constant, and q is the nonextensive parameter. The discrete expression of (9) is shown as n c q 1− p(i ) , q ∈ R, n ∈ N (10a) Sq = q−1 i=1

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n where p(i ) i=1 p(i ) = 1 is the probability density distribution function of random variable i , such as some characteristic values on the gray, section gray, or gradient. Shannon entropy and Tsallis entropy may be considered as the system’s information measurement under some state, and can be adopted to estimate the complexity of random signals. The transient fault signals of transmission line contain a great deal of transient, random, and uncertain information. For the transient protection in power systems, it is difficult not only to distinguish faults inside or outside the protection zone but also to distinguish fault transients from transients of normal operations. Hence, comprehensive fault transient information for fault recognition and classification is crucial. C. Multiwavelet Packet Energy Entropy Definition 1: Let E = E 1 , E 2 , . . . , E k , . . . , E 2 J +1 be the multiwavelet packet energy spectrum of x(t) on the j th scale, E can be considered as a signal energy division on the scale domain, where J is the level number of multiwavelet packet decomposition, j ∈ N, and k ∈ [1, 2 J +1]. According to the orthogonality of multiwavelet transformation, in some time window (window width w ∈ N), the total E of the signal is equal to the sum of all components E k . Let Pk = E k /E, J +1 Pk = 1. The multiwavelet packet energy entropy then 21 is defined as Pk log Pk . (10b) WEE = −

Fig. 4.

Simplified model of a 500-kV transmission line.

value decomposition theory, for a m × n matrix D, there must be a m × l matrix U , a l × n matrix V , and a l × l matrix . T . The matrix D can be decomposed by Dm×n = Um×l l×l Vl×n The diagonal elements λi (i = 1, 2, . . . , l) of diagonal matrix are nonnegative and arranged in descending order: λ1 ≥ λ2 ≥ · · · ≥ λl ≥ 0. The diagonal elements are the singular values of the decomposition results matrix Dm×n of the multiwavelet packet. When there is no noise or high signalto-noise in the signal, most of the main diagonal singular values are zero. In addition, if the signal has fewer frequency components, the decomposition results of the multiwavelet packet have fewer nonzero singular values. Definitions 3 and 4: To quantitatively describe the frequency components and distribution characteristics of the signal, two multiwavelet packet singular entropies (Shannon singular entropy and Tsallis singular entropy) are, respectively, defined as WSE =

Pi

WTsallisSE

n c q =

Pi , q ∈ R 1− q−1

If a moving window (w ∈ N) for the results of multiwavelet packet transformation is defined, the moving window can be described by w(m; w, δ) = {d j (k), k = 1 + mδ, . . . , w + mδ}, where δ represents the moving factor of the defined window and δ ∈ N, m = 1, 2, . . . , M, M ∈ N . The moving window can be divided into L, L ∈ N spaces w(m; w, δ) = L Z , where {Z ∪l=1 l l = [sl−1 , sl ), l = 1, 2, . . . , L} and mutually disjoint satisfies, s0 = min[W (m; w, δ)], and s L = max[W (m; w, δ)]. Definition 2: Suppose P m (Z l ) represent the possibility of multiwavelet packet coefficients d j,r (k) ∈ W (m; w, δ), r = 2 in the section Z l , namely the ratio of the number in the section Z l and w(m; w, δ). The multiwavelet packet time entropy can be defined as P m (Z l ) log(P m (Z l )), m = 1, 2, . . . , M E TE j (m) = − (11) where M = (N − w)/δ ∈ N. E. Multiwavelet Packet Shannon Singular Entropy and Tsallis Singular Entropy The decomposition results of multiwavelet packet can be used to construct a m × n matrix Dm×n . Based on the singular

(13)

i=1

where

Pi = − l

D. Multiwavelet Packet Time Entropy

(12)

i=1

k

With the moving window, the variation law of multiwavelet packet entropy with respect to the time can be obtained. The corresponding relationship between the scale domain and frequency domain can be described.

l

λi

j =1 λ j

log l

λi

j =1 λ j

is multiwavelet packet singular entropy of the i th-order increment, c is a conventional positive constant, and q is the nonextensive parameter. In this paper, c = 1 and q = 0.8 based on experience. IV. FAULT R ECOGNITION AND C LASSIFICATION P LAN A. Model of Transmission Lines in EMTDC A simplified simulation model of 500-kV transmission line with a double-end supply power is built, as shown in Fig. 4. The Bergeron model in EMTDC is adopted for the transmission line. Different faults are simulated in the transmission line model. The corresponding parameters are given. 1) Positive Sequence Parameters: r1 = 0.035 /km, x 1 = 0.424 /km, and b1 = 2.726 × 10−6 S/km. 2) Zero Sequence Parameters: r0 = 0.3 /km, x 0 = 1.143 /km, and b0 = 1.936 × 10−6 S/km. Because the inductive current cannot suddenly change, there is a large quantity of decaying dc components for the transmission line fault currents. Because there are series capacity and high resistance ground in the transmission line faults, abundant high-frequency components will be produced. In addition, the existence of charging capacitors will result in


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Fig. 6.

Fig. 5.

Fault recognition and classification block diagram.

many low-frequency components. There are 10 fault types for the transmission line: single-phase to ground (AG, BG, and CG), interphase short circuit (AB, BC, and AC), two-phase ground (ABG, BCG, and CAG), and three-phase to ground (ABC). For convenience, these symbols are used to denote fault types in this paper. B. Fault Recognition and Classification Plan The block diagram of transmission line fault recognition and classification with multiwavelet packet transformation, entropy, and an ANN is shown in Fig. 5. There are three steps in this approach. 1) Decomposition of Fault Current Signals With Wavelet Packet and Multiwavelet Packet: For the equality of comparison, DB4 wavelet packet is adopted for the decomposition of fault signals. Its filter length is equal to that of GHM, CL3, and SA4 [42] multiwavelet packets. These multiwavelets are commonly used. The decomposition level number of wavelet packet and multiwavelet packet is three. As the choice reference of preprocessing and postprocessing methods in general, we use the GHM init. preprocessing method for the GHM multiwavelet [36], the balance preprocessing method for SA4 and CL3 multiwavelets [37]. 2) Calculation of Entropies Defined in This Paper: The multiwavelet packet entropies defined in the paper, including multiwavelet packet energy entropy, time entropy, Shannon singular entropy, and Tsallis singular entropy, are used to calculate the entropy values for the decomposition results of fault currents with wavelet and multiwavelet packets.

Structure of RBF neural network.

3) Recognition and Classification of Transmission Line Faults With RBF Neural Network: After the calculation of the entropy values for the decomposition results of fault current signals with wavelet packet and multiwavelet packet, the results are input into an RBF neural network as the training eigenvectors. Through the training of the RBF neural network, the type of transmission line fault can be recognized and classified based on the different input signals. The network structure is shown in Fig. 6. The input and output dimensions of the network are denoted by L and T , respectively, while M is the number of hidden nodes. Each output node is a linear combiner defined as yi (X) =

M

φ j X − C j · w ji , 1 ≤ i ≤ T

(14)

j =1

where X ∈ R L is the input vector, C j ∈ R M is the RBF center, · is the Euclidean norm, and w j i is the weight. V. E XPERIMENTAL R ESULT A. Training Samples Based on the simulation model in Fig. 4, we can obtain a large quantity of training data with different fault currents. Some training rules and assumed conditions are considered as follows. 1) When transmission line faults occur, the transition resistance to ground is in the range of 0∼300 , and the interphase transition resistance is in the range of 0∼100 . 2) The transmission line length is 100 km. The fault location starts at 20 km from the head end, increases 15 km with each iteration to the line end. 3) The initial angle of current signals is in the range of 0°–90°. 4) The three-phase current signals are obtained at the head end of transmission line, and the sampling frequency is 10 kHz. 5) The data window is chosen from the 1/2 period before to the 1/2 period after the fault occurrences. There are 260 sample groups for the training of the RBF neural network according to the rules above.

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TABLE I T EST S AMPLE N UMBERS

3) SD

SD =

1 (c(i ) − AA)2 NC − 1 Nc

(17)

i=1

where C is the set of correct recognition, and c(i ), i = 1, 2, NC represents the actual output values corresponding to the test samples. This criterion mainly reflects the dispersion of recognition results and the stability of classifier. D. Classifier Structure and Training Algorithm For the RBF neural network, the activation function is a Gauss function, and training algorithm is a standard RBF algorithm. The network structure is 32 × 32 × 10. Namely, the input layer includes 32 neurons, which the output layer has 10 neurons. E. Multiwavelet Packet Entropy Curves

B. Test Samples To efficiently recognize and classify the fault type of transmission line, we randomly take 28 groups as the test samples. These are numbered in Table I and are not based on the assumed conditions above. In the test groups, locations, fault types, transition resistances, and failure angles are randomly obtained. C. Evaluation Criterion For the sake of comparison, the fault recognition criterions are defined as follows. 1) Recognition rate (RR) RR =

NC × 100% NT

(15)

where NT is the number of test samples and NC is the number of correctly recognized faults. 2) Average accuracy (AA) AA =

NC 1 c(i ) NC

(16)

i=1

where C is the set of correct recognition, c(i ), i = 1, 2, NC means the actual output value corresponding to the test samples. The criterion mainly reflects the approximation of actual output results to one.

Through the calculation of the four entropies for the decomposition of fault current signals with wavelet packet and multiwavelet packet, the four entropy curves can be drawn. In Fig. 7(a), x(n) is the waveform of a fault current signal when A phase to earth short circuit. wEE , E TE , wSE , and wTsallisSE are, respectively, the GHM multiwavelet packet energy entropy, time entropy, Shannon singular entropy, and Tsallis singular entropy for the fault current signal of AG. These curves under the conditions of ABG, AB, and ABC are shown in Fig. 7(b)–(d). For the different types of fault current signals, their four GHM multiwavelet packet entropies are different, which provides some important features for fault recognition and classification. F. Recognition Result Analysis Through the training with 260 sample groups for RBF neural network, when 28 test groups are input, 10 types of transmission line faults can be recognized and classified based on the plan proposed in this paper. We list the recognition results with wavelet and multiwavelet packet energy entropy, time entropy, Shannon singular entropy and Tsallis singular entropy in Tables II–V. To compare the recognition results of different wavelet/ multiwavelet packet, for the four given entropies, we choose the best wavelet/multiwavelet packet, as shown in Table VI. Similarly, for the four given wavelet/multiwavelet packet, we choose the best entropies and list in Table VII. Based on the results in Tables VI and VII, it can be found that the wavelet/multiwavelet packet Tsallis singular entropy can achieve better recognition and classification performances, and their RR is 100%. We can draw the fault recognition relative error curves with wavelet/multiwavelet packet Tsallis singular entropy for 28 test groups in Fig. 8. From the analysis results of Table VII and Fig. 8, it can be found that the recognition and classification performance of SA4 multiwavelet packet Tsallis singular entropy is the best based on the three evaluation criterions on the whole.


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Fig. 7. (a) Fault current signal and four GHM multiwavelet packet entropies when AG. (b) Fault current signal and four GHM multiwavelet packet entropies when ABG. (c) Fault current signal and four GHM multiwavelet packet entropies when AB. (d) Fault current signal and four GHM multiwavelet packet entropies when ABC.

TABLE II

TABLE III

P ERFORMANCE C OMPARISONS OF DB4

P ERFORMANCE C OMPARISONS OF GHM

WAVELET PACKET E NTROPIES

M ULTIWAVELET PACKET E NTROPIES

There are three main reasons why we define multiwavelet packet energy entropy, time entropy, Shannon singular entropy, and Tsallis singular entropy. First, there is a great deal of

transient information in transmission line faults, which can better reflect the fault characteristics. Second, decomposition with multiwavelet packet can obtain more high-frequency information and more detailed frequency domain information

G. Discussion

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TABLE IV P ERFORMANCE C OMPARISONS OF SA4 M ULTIWAVELET PACKET E NTROPIES

TABLE VII P ERFORMANCE C OMPARISONS OF THE B EST E NTROPIES FOR THE

G IVEN WAVELET /M ULTIWAVELET PACKET

TABLE V P ERFORMANCE C OMPARISONS OF CL3 M ULTIWAVELET PACKET E NTROPIES

TABLE VI P ERFORMANCE C OMPARISONS OF B EST WAVELET /M ULTIWAVELET PACKET FOR THE F OUR G IVEN E NTROPIES

Fig. 8. Relative error curves with wavelet/multiwavelet packet Tsallis singular entropy.

than with multiwavelets and wavelet, which are very important in training the neural network. Finally, the entropy can efficiently represent the complexity and uncertainty of signals, which is helpful to recognize the transient faults. Though we use only 260 sample groups for training the RBF neural network, the recognition performance is satisfactory, which means that the recognition characteristics are very efficient. In addition, the RR of Tsallis singular entropy is 100% in Table VII and Fig. 8, but their relative error (%) is large for the fourth, ninth, and 13th test samples. It is worth noting that the relative error of the 13th test sample is close to 70%. We traced the sample and found that the 13th test sample is the fault of a C phase to ground. Its location is 78.8 km from the head end, the transition resistance is 5 , and the fault angle is 63°. The three phase current waveforms of the 13th test sample are shown in Fig. 9(a). For the sake of comparison, typical three phase current waveforms for CG are shown in Fig. 9(b). Based on the analysis of the key factors in detecting transmission line faults, several conclusions can be drawn. 1) The transition resistance and fault angle are two important factors for detecting transmission line faults. The transition resistance can weaken the fault characteristics in the fault current signal, and the fault angle can directly influence or change the waveform of fault current signal. 2) To further analyze the effect of fault impedance, an additional experiment is done and the results are listed in

TABLE VIII E FFECT OF FAULT I MPEDANCE

Table VIII. The fault location, type, angle, and duration of the test samples are all invariable, and the fault impedance is randomly changed. The test samples are chosen within any range. Tsallis singular entropy (SA4) is used to obtain eigenvectors. The results show that the fault impedance has no effect on the RR. The reason is that the fault resistance only affects the amplitude of the current signal, and multiwavelet packet entropies primarily detect the singularity of the signal. 3) The influence of the fault angle is very both obvious and complex. The transmission line circuit with the inherent impedance angle is essentially an RLC circuit. When the fault angle of the current signal changes, the current waveform may dramatically change, and even the fault characteristics may be similar to those of other


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R EFERENCES

Fig. 9. (a) Three-phase current waveforms of the 13th test sample. (b) Typical three phase current waveforms for CG.

fault types. Fig. 9(a) and (b) show that there are distinct differences between typical three phase current waveforms for CG and the three-phase current waveforms of the 13th test sample. For the typical CG fault, the fault angle is 0°, while that of the 13th test sample is 63°. The time domain characteristics of the 13th test sample exhibit little changes, compared with those of the typical CG fault. In sum, there are a few samples with large relative error during the course of recognition and classification, which may be inevitable for transmission line faults. Of course, if the training samples are added in RBF neural network, the relative error may be improved. Though the method proposed in the paper cannot completely reduce the relative error of transmission line fault recognition and classification, the RR is very high and can efficiently avoid or reduce the influence of the transition resistances. VI. C ONCLUSION Based on the characteristics of multiwavelet packet and spectral entropy, a new idea of transmission line fault recognition and classification using multiwavelet packet transformation, entropy, and ANN is proposed in this paper. We, respectively, define four multiwavelet packet entropies (energy entropy, time entropy, Shannon singular entropy, and Tsallis singular entropy) as the extraction methods of transmission line fault signals, and RBF neural network is adopted as the classifier. The experimental results showed that the proposed plan and the defined multiwavelet packet entropies have the ability to efficiently recognize and classify different transmission line fault types efficiently. Our future may focuses on other problems, such as the recognition and classification performance for other complex faults of power transmission line or similar problems in other fields, or the best multiwavelet choice algorithm for real-time analysis. This paper leads us to believe that the combination of multiwavelet packet, entropy, and ANNs, offers great potential in the fault recognition and classification.

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Zhigang Liu (M’06) received the Ph.D. degree in power system and its automation from the Southwest Jiaotong University, Sichuan, China, in 2003. He is currently a Full Professor with the School of Electrical Engineering, Southwest Jiaotong University. His current research interests include signal processing and its applications in power systems, electrification railway technology, and computational intelligence.

Zhiwei Han received the M.S. degree in power system and its automation from Southwest Jiaotong University, Sichuan, China, in 2009, where he is currently pursuing the Ph.D. degree with the School of Electrical Engineering.

Yang Zhang received the master’s degree in power system and its automation from Southwest Jiaotong University, Sichuan, China, in 2012. He is currently with the Zhejiang Electric Design Institute, Hangzhou, China.

Qiaoge Zhang received the bachelor’s degree in power system and its automation from Southwest Jiaotong University, Sichuan, China, in 2011. She is currently with the School of Electrical Engineering, Southwest Jiaotong University, where she is involved in signal processing and its application in power systems.