Power quality disturbance identification using wavelet packet energy entropy and weighted support vector machines

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Expert Systems with Applications Expert Systems with Applications 35 (2008) 143–149 www.elsevier.com/locate/eswa

Power quality disturbance identification using wavelet packet energy entropy and weighted support vector machines Guo-Sheng Hu b c

a,c,* ,

Feng-Feng Zhu b, Zhen Ren

a

a School of Computer and Information, Anqing Teachers College, Anqing 246011, China School of Math Sciences, South China University of Technology, Guangzhou 51060, China School of Electric Power, South China University of Technology, Guangzhou 510640, China

Abstract In this paper, wavelet packet energy entropy and weighted support vector machines are used to automatically detect and classify power quality (PQ) disturbances. Electric power quality is an aspect of power engineering that has been with us since the inception of power systems. The types of concerned disturbances include voltage sags, swells, interruptions. Wavelet packet are utilized to denoise the digital signals, to decompose the signals and then to obtain five common features for the sampling PQ disturbance signals. A weighted support vector machine is designed and trained by 5-dimension feature space points for making a decision regarding the type of the disturbance. Simulation cases illustrate the effectiveness.  2007 Elsevier Ltd. All rights reserved. Keywords: Weighted support vector machines (WSVMs); Power quality; Disturbances; Classification; Wavelet packet energy entropy

1. Introduction The power quality of electric power system has become an increasing concern for electric utilities and their customers over the last decade. Poor quality is attributed due to the various power line disturbances like voltage sags, swells, interruptions, switching transients, impulses, flickers, harmonics, and notches (Golkar, 2004). In order to improve the quality of electrical power, it is customary to continuously record the disturbance waveforms using power quality monitoring instruments. PQ monitoring can usually be a complex task involving hardware instrumentation and software packages. Various types of intelligent electronic devices can be used for collecting the desired PQ data. After PQ data of interest are obtained, a comprehensive PQ assessment can be carried out. This may include power system and equipment modeling verification, PQ problem

*

Corresponding author. Address: School of Computer and Information, Anqing Teachers College, Anqing 246011, China. E-mail address: [email protected] (G.-S. Hu). 0957-4174/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2007.06.005

mitigation and optimization, and data analysis. In most cases, automated PQ assessment is desirable because manual analysis may be difficult to carry out due to lack of time and special expertise. Specialized software tools can make use of intelligent techniques to automate the PQ assessment for improved accuracy and efficiency. This paper is focusing on automated detection and classification of PQ disturbances that may facilitate the PQ assessment. Machine learning techniques like support vector machines as well as signal processing techniques like wavelet transforms have been utilized for developing the tools. When carrying out these learning, a weighted support vector machine model made will be illustrated appropriately by simulated cases. Wavelet packet energy distribution features have been widely used to identify gas–liquid two-phase flow patterns (Sun & Zhou, 2005), detect the stator single phase ground fault for generators (Bi, Wang, & Wang, 2003), classify power quality disturbances (Gen, Wang, & He, 2006; Li, Tao, & Qi, 2004; Yan, Liu, & Yang, 2002), diagnosis the fault of reciprocation machinery and high voltage circuit breakers (He, Shen, & Ying, 2001; Zhao, Wang, & Hu,

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2004), and other application in machinery fault diagnosis (Li, Li, & Li, 2003; Ling & Li, 2005; Tang, Fan, & Hu, 2006; Wang, Huang, & Shi, 2004; Wang, Ma, & Zou, 2004). The fault diagnosis can be found from wavelet packet energy feature figures of non-stable fault signals. However, we verify these methods inefficacy to faint fault signals in Sections 2.1 and 2.2, the figure illustrates the wavelet packet energy entropy is more valid than the wavelet packet energy feature to diagnose faint faults from the feeble fault signals. Wavelet packet energy entropy features were successfully used to diagnose the faults of transmission lines and turbines in power systems (Gui & Han, 2004; He, Cai, & Qian, 2005; He, Liu, & Qian, 2004; Zhang, Wang, & Li, 2006). In this paper, we extend a signal feature in Zhang et al. (2006), Gui and Han (2004), He et al. (2004, 2005) to five features including the wavelet packet energy entropy feature, and it is very obvious that an earlier fault signal is more important than later fault signals to detect and identify faults. So we designed weighted support vector machines which endue a earlier fault signal feature a higher weight. The experiment shows that the weighted support vector machines with 5-dimension input vectors are more powerful than neural networks to diagnose faint faults in power systems. 2. PQ disturbance features 2.1. Wavelet packet energy entropy feature Fig. 1 shows a stator A-phase current fault noisy signals of an induction motor with stator coil being short. The sampling frequency is 2500 Hz. Fault is so faint that it is very difficult to detect. The signal in Fig. 2 is a normal A-phase current signal. To distinguish the fault from the normal signal, we decompose signals to eight different frequency bands (level = 3) using multiresolution wavelet packets and calculate energy entropy feature to detect the fault. Firstly, decomposing the fault signal (see Fig. 1) and the normal signal (see Fig. 2) to the third level using wavelet packet. The signal energy distributes into octave frequency bands. Setting the wavelet packet coefficients in the jth fre-

Fig. 2. The normal current signal and its TSW2 wavelet packet decomposition with three levels.

quency band as s3j = {xj,k, k = 1, . . . , le} (j = 0, 1, P2, . . . , 7),2 then for each band, the signal energy E3j ¼ le k¼1 jsj;k j can calculated, they are tabulated as Table 1 and plotted P 1=2 7 2 as Fig. 3. The total energy E ¼ jE j . j¼0 3j Secondly, to avoid immense E3j, we normalize energy vector V = [E30, E31, . . . , E37] as E3j ¼ E3j =E, so we gain new energy vector V ¼ ½E30 ; E31 ; . . . ; E37  (see Table 1). The normalized normal and the fault signal energies are showed as Fig. 4. Ling and Li (2005), Tang et al. (2006), Li et al. (2003), Wang, Huang et al. (2004), Wang, Ma et al. (2004) choose energy vectors to be signal features. But considering our calculation and Fig. 4, the energy features are not useful to detect faint fault. Finally, it is well known that any fault occurs randomly, energy vectors V ¼ ½E30 ; E31 ; . . . ; E37  are stochastic variances. Hence, we can regard energy entropies as a feature to detect faults. For the jth frequency band, the energy entropies are defined to be I 3;j ¼ E3;j log E3;j , j = 0, 2, . . . , 7. The total P7 energy entropy I is the summation of I3,i, that is, I ¼  i¼0 E3;i log E3;i . By comparing the energy entropies of the normal signal and the fault signal from Fig. 5, we can easily detect whether a fault occurs. Observing the fault signal energy distribution (Fig. 3) and the normalized signal energy distribution (Fig. 4), we almost cannot identify the different between the normal signal and the fault signal. So, the wavelet packet energy feature is not powerful to detect whether the fault occurs. However using the wavelet packet energy entropy feature (see Fig. 5), we can easily detect fault occurrence. Meanwhile Fig. 5 illustrates that the signal contains four harmonics, by further decomposing, the accurate harmonics can be calculated. 2.2. Other features

Fig. 1. The fault current signal and its TSW2 wavelet packet decomposition with three levels.

The experiments illustrate that support vector machines are easily infected by noise, so we denoise the digital signals of PQ disturbances using wavelet transform first. Then the PQ disturbance common features can be gained using the Debauchies 4 (Db4) wavelet transform.

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Table 1 The characteristics of the normal signal and the fault signal Wavelet packet

E30

E31

E32

E33

E34

E35

E36

E37

Normal signal Fault signal

205.332 180.476

0.086 5.041

0.037 0.355

0.034 0.711

0.021 0.636

0.018 0.088

0.009 0.054

0.009 0.038

Normalized Normal signal Fault signal

E30 0.9999 0.9995

E31 0.0004 0.0279

E32 0.0001 0.0019

E33 0.00015 0.0039

E34 0.0001 0.0035

E35 8.76E05 0.0004

E36 4.38E05 0.0002

E37 4.38E05 0.0002

Energy entropy Normal signal Fault signal

E30 log E30 5.58E08 0.0002

E31 log E31 0.0014 0.0434

E32 log E32 0.0006 0.0053

E33 log E33 0.0006 0.0095

E34 log E34 0.0004 0.0086

E35 log E35 0.0003 0.0016

E36 log E36 0.0002 0.0011

E37 log E37 0.0002 0.0008

The Normal and Fault Signal Energy Distributions 250

Energy

200 Normal Signal Energy Fault Signal Energy

150 100 50 0

1

2

3

4

5

6

7

8

Octave Frequency Bands

Fig. 3. The energy distributions of the normal signal and the fault signal in octave frequency bands.

Normalized Energy

The Normalized Normal and Fault Signal Energy 1. 2 1 0. 8

Normalized Normal Signal Energy Normalized Fault Signal Energy

0. 6 0. 4 0. 2 0 1

2

3

4

5

6

7

8

Octave Frequency Bands

Fig. 4. The normalized energy distributions of the normal and fault signals in octave frequency bands.

Energy Entropy

The Normal and Fault Signal Energy Entropy Distributions 0.05 0.04

Normal Signal Energy Entropy Fault Signal Energy Entropy

0.03 0.02 0.01 0

coefficients (Nn), energy of the wavelet coefficients (EWn), oscillation number of the missing voltage (OSn), lower harmonic distortion (TSn), and oscillation number of the rms variations (RN). The formulae for computing these features are given as follows: N n ¼ peakðabsðWPCÞÞ V n ¼ sqrtðabsðWPC½1ÞÞ=le sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi intðN P=2Þ 2 ðabsðV n ½kÞÞ THDn ¼

k¼2

V 1 ½1

OSn ¼ rootðvsmiss Þ

ð1Þ ð2Þ

ð3Þ ð4Þ

where, Vn[k] is the discrete Fourier transform for the sample signal v[i] contained in the nth data window defined as PN 1 V n ½k ¼ i¼0 v½i þ ðn  1ÞN ejð2pki=N Þ , i = 0, 1, . . . , L  1, with L the length of the signal, N is the number of samples in one data window (one cycle), and n = 1,2, . . . , 10. WPC is defined as an array composed of WPC[k] for k = 1, 2, . . . , le, with le being the length of WPC for each frequency band. vmiss[i] = v[i]  2/N · abs(V1[1]) · cos(angle(V1[1]) + 2p(i  1)/N), vsmiss be defined as an array composed of vmiss[i], i = 0,1, . . . , L  1. abs(Æ) gives the absolute value of the argument, peak(Æ) returns the number of peaks of the argument. In this paper, five cycles of samples of the voltage signals (in per unit) are used. The signal feature vector X = (I, Nn, Vn, THDn, OSn). 3. Weighted support vector machine classifiers

1

2

3

4

5

6

7

8

Octave Frequency Bands

Fig. 5. The energy entropies of normal and fault signals in octave frequency bands.

A variety of PQ disturbances of different types have been simulated and corresponding signals obtained. The following three distinct features inherent to different types of PQ disturbances have been identified: the fundamental component (Vn), phase angle shift (an), total harmonic distortion (THDn), number of peaks of the wavelet packet

Generally, we use a set S of labeled training points (x1, y1), . . . , (xl, yl), each training point x1 2 R5 belongs to either of two classes are is given a label yi 2 {1, 1} for i = 0, 1, . . . , l. In most cases, the searching of the suitable hyperplane in an input space is too restrictive to be of practical use. A solution of this situation is mapping the input space into a higher dimension feature space and searching the optimal hyperplane in above 5-dimension feature space. Let z = u(x) denote the corresponding feature space vector with a mapping u from R5 to a feature space Z. we hope to find the hyperplane w Æ z + b = 0 defined by the pair (w, b),

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such that we can separate the point xi according to the decision function f(x) ! l X f ðxÞ ¼ signðw  z þ bÞ ¼ sign ai y i Kðxi ; xÞ þ b ð5Þ i¼1

W ðaÞ ¼

l X

ai 

i¼1

s:t:

l X

y i ai ¼ 0;

l X l 1X ai aj y i y j Kðxi ; xj Þ 2 i¼1 j¼1

oL ow

wþ

oL ¼ on ¼ oL ¼ 0, then ob i

l X

y i ai /ðxi Þ ¼ 0

i¼1

Csi  ai  bi ¼ 0; l X ai ¼ 0

The ai can be found as the solution of max

Set

ð6Þ

0 6 ai 6 C; i ¼ 0; 1; . . . ; l

Above support vector machine is a powerful tool for solving classification problems, but there are still some limitations of this theory. From the training set (x1, y1), . . . , (xl, yl) and formulations discussed above, each training point is treated uniformly in support vector machines. In many cases, the effects of the training points are different. It is often that some training points are more important than others in the classification problem. Especially in classifying PQ disturbances of power systems, the initial sampled disturbance signal is more important than later ones. We would assure that the meaningful training points must be classified correctly. So defining a weight factor 0 < si 6 1 associated with each training point xi. This weight factor si can be regarded as the attitude of the corresponding training point toward one class in the classification problem and the value (1  si) can be regarded as the attitude of meaningless. Now we suppose the set S of labeled training points (x1, y1, s1), . . . , (xl, yl, sl) with associated weight factor r 6 si 6 1 and sufficient small r > 0. Each training point xi 2 R5 is given a label yi 2 {1, 1}. The optimal hyperplane problem is then regarded as the solution

s:t:

l X 1 kwk2 þ C s i ni 2 i¼1

So we obtain the dual function of Eq. (7) which has the following form:

ni P 0;

max

W ðaÞ ¼

l X

i ¼ 0; 1; . . . ; l

ð7Þ

ai 

i¼1

s:t:

l X

y i ai ¼ 0;

l X l 1X ai aj y i y j Kðxi ; xj Þ 2 i¼1 j¼1

ð10Þ

0 6 ai 6 si C; i ¼ 0; 1; . . . ; l

i¼1

To choose the appropriate weight factors in classifying PQ disturbances problem is easy. First, the lower bound of weight factors must be defined, and second, we need to select the main property of data set and make connection between this property and weight factors. We choose r > 0 as the lower bound of weight factors. Then we identify that the time is the main property of this kind of problem and make weight factor si be a function of time ti si ¼ f ðti Þ

ð11Þ

Table 2 Parametric equations for simulation of disturbed signals Event

Equation

Pure sinusoid Sudden sag Sudden swell

v(t) = sin(xt) v(t) = (1  ass(1(t  tb)  1(t  te)))sin(xt) v(t) = (1 + asw(1(t  t1)  1(t  t2)))sin(xt)   ah1 sinðxtÞ þ ah3 sinð3xtÞ . . . vðtÞ ¼ þah5 sinð5xtÞ þ ah7 sinð7xt þ   Þ

Harmonics

y i ðw/ðxi Þ þ bÞ P 1  ni ;

ð9Þ

i¼1

i¼1

min

i ¼ 0; 1; 2; . . . ; l

Flicker Oscillatory transient

v(t) = (1 + afsin(bfxt))sin(xt)   ðsinðxtÞ þ aosc expððt  tb Þ=sosc ÞÞ vðtÞ ¼ . . . sinðxnosc ðt  tb ÞÞ

i ¼ 0; 1; . . . ; l

where xi be mapped to a higher dimensional space by the function /, and ni be slack variables. The constant C > 0 determines the tradeoff between the flatness and losses. The parameters that control regression quality are the cost of error C, weight factor si, slack variables ni, and the mapping function /. By introducing Lagrangian multipliers ai P 0 and bi P 0 (i = 0, 1, 2, . . . , l) and exploiting the optimality constraints, so the corresponding Lagrangian function is l l X X 1 Lðw; b; n; ;a; bÞ ¼ kwk2 þ C si ni  ai ðy i ðw/ðxi Þ  bÞ 2 i¼1 i¼1

 1 þ ni Þ 

l X i¼1

bi ni

ð8Þ

Table 3 Parameters variation in simulated signals Event

Parameters variation

Pure sinusoid

Amplitude: 1 Frequency: 50 Hz Duration: (t2  t1) = (0–9)T Amplitude: ass = 0.3–0.8 Duration: (t2  t1) = (0–8)T Amplitude: asw = 0.3–0.7 Order: 3, 5, 7 Amplitudes: 0–0.9 Frequency: (5–10) Hz Amplitude: af = 0.1–0.2 Time const: 0.008–0.04 s Frequency: 100–400 Hz

Sudden sag Sudden swell Harmonics Flicker Oscillatory transient

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Fig. 6. Six simulated signals. (a) Pure sinusoid, (b) sudden sag (ass = 0.3, tb = 0.02 s, te = 0.08 s), (c) sudden swell (ass = 0.3, tb = 0.02 s, te = 0.08 s), (d) harmonics (amplitudes: 0.3, 0.2, 0.1), (e) Flicker (af = 0.2, bf = 10 Hz), (f) oscillatory transient (sosc = 0.02, xnosc = 200 Hz).

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where t1 6 t1 6    6 tl are the sampling times. We make the start point x1 be the most important and choose s1 = f(t1) = 1, and make the last point xl be the least important and choose sl = f(tl) = r. So we can make weight factor be a linear function of the time, we can construct r1 ðti  tl Þ þ r or tl  t1  2 ti  tl si ¼ f ðti Þ ¼ ð1  rÞ þr t1  tl

weighted support vector machines. It can be seen that the wavelet energy entropy and weighted support vector machines result in a correct identification rate of 98.4%. These studies show that the proposed methods for feature extraction and decision making are efficient for PQ disturbance detection and classification.

si ¼ f ðti Þ ¼

ð12Þ

In this paper, we choose the parameter r = 0.001. 4. Classification results Signal modeling by parametric equations for classifier tests was advantageous in some aspects. It was possible to change testing and training signal parameters in a wide range and in a controlled manner. Signals simulated that way were very close to reality. On the other hand, different signals belonging to the same class gave the opportunity to estimate the generalization ability of classifiers based on WSVM. Signals belonging to five main groups of disturbances (Janik & Lobos, 2006), were simulated the classes and respective equations are summarized in Table 2. The ranges of signal parameter variation are shown in Table 3. The variation range corresponds to values measured in real power systems. The parameters ass and asw correspond to the depth of a sag and the measure of sudden swell, respectively. The step function 1(t) is used to determine sag and swell duration. Flicker is characterized by its frequency bfx and amplitude af. Oscillatory transients are described by the frequency xnosc and the time constant of the decay sosc (Fig. 6). During the detecting and classifying PQ disturbances, 1000 different cases for every type of PQ disturbance waveforms are generated from the Matlab 6.5 environment using Power System Blockset. The disturbances include voltage sudden sags, sudden swells, harmonics, flickers, and oscillatory transients. Before classifying the disturbances, we denoise the sample disturbance digital signals using Daubechies wavelet (Db5). Table 4 shows the classification results of the generated disturbances using the

Table 4 Classification results based on wavelet energy entropy and WSVMs Type of PQ disturbances

Number of disturbances

Number of cases correctly identified

Number of cases fault identified

Correct identification rate (%)

Sag Swell Harmonics Flicker Transient Sum

1000 1000 1000 1000 1000 5000

995 992 986 977 970 4720

5 8 14 23 30 80

98.4

5. Conclusions Firstly, we point out that wavelet energy features are useful to detect faint fault signals in power system, but the wavelet energy entropy is very effective to do that. Then extract the five common features of PQ disturbances using Fourier transforms and wavelet transforms are presented. These features consist of 5-dimension feature space. Secondly, general support vector machines treat each sample equally without discrimination, but as you know, the earlier the fault signal, the more important fault detection. So we set the weighted factors to corresponding training samples. The weighted support vector machines are used to detect and identify the disturbances. Finally, numerical simulation illustrates that five types of PQ disturbances can effectively detected and classified by that proposed method. Acknowledgement The authors would like to thank the GDNSF #033044 for financial assistance. References Bi, D. Q., Wang, X. H., & Wang, W. J. (2003). Wavelet transform based energy method for detecting the stator ground fault for generators. Chinese Journal of Automation of Electric Power Systems, 27(22), 50–55. Gen, Y. L., Wang, Q., & He, Y. G. (2006). Wavelet detection and location power quality and its classification based on multi-scale energy curve. Chinese Journal of Scientific Instrument, 27(2), 180–182, 217. Golkar M. A. (2004). Electric Power Quality: Types and Measurements. In: 2004, IEEE DRPT2004, Hong Kong, pp. 317–321. Gui, Z. H., & Han, F. Q. (2004). Wavelet packet maximum entropy spectrum estimation and its application in turbine fault diagnosis. Chinese Journal of Chinese of Automation of Electric Power Systems, 28(2), 62–66. He, Z. Y., Cai, Y. M., & Qian, Q. Q. (2005). A study of wavelet entropy theory and its application in electric power system fault detection. Proceedings of the CSEE, 25(5), 38–43. He, Z. Y., Liu, Z. G., & Qian, Q. Q. (2004). Study on wavelet entropy theory and adaptability of its application in power system. Chinese Journal of Chinese of Power System Technology, 28(21), 17–21. He, Y. Z., Shen, S., & Ying, H. Q. (2001). Application of wavelet packet decomposition and its energy spectrum on the fault diagnosis of reciprocation machinery. Chinese Journal of Journal of Vibration Engineering, 14(1), 72–75. Janik, P., & Lobos, T. (2006). Automated classification of power-quality disturbances using SVM and RBF networks. IEEE Transactions on Power Delivery, 21(3), 1663–1669. Li, S. L., Li, Z., & Li, H. S. (2003). The method of roller bearing fault monitoring based on wavelet packet energy feature. Journal of System Simulation, 15(1), 76–80, 83.

G.-S. Hu et al. / Expert Systems with Applications 35 (2008) 143–149 Ling, T. H., & Li, X. B. (2005). Analysis of energy distributions of millisecond blast vibration signals using wavelet packet method. Chinese Journal of Rock Mechanics and Engineering, 24(7), 1117–1122. Li, T. Y., Tao, C. J., & Qi, H. (2004). Detection and classification of power quality short term disturbances based on wavelet packet energy chart. Chinese Journal of Jilin Electric Power, 16(1), 10–13. Sun, B., & Zhou, Y. L. (2005). Identification method of gas–liquid twophase flow regime based on support vector machine and wavelet packet energy feature. Proceedings of the CSEE, 25(1), 93–98. Tang, G. J., Fan, D. G., & Hu, A. J. (2006). Fault diagnosis based on wavelet packet energy eigenvector. Chinese Journal of Chinese of Coal Mine Machinery, 27(4), 717–719. Wang, J. Q., Huang, S. B., & Shi, Z. L. (2004). An algorithm of image matching based on both complex wavelet energy and SVM. Journal of Image and Graphics, 9(9), 1075–1079.

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Wang, F. T., Ma, X. J., & Zou, Y. K. (2004). Local power feature extraction method of frequency bands based on wavelet packet decomposition. Chinese Journal of Journal of Agriculture Machinery, 35(5), 177–180. Yan, J. B., Liu, X. Q., & Yang, H. G. (2002). Analysis of short term power system disturbance based-on modulo maximum principle and energy distribution curve of wavelet transform. Chinese Journal of Power System Technology, 26(4), 16–19. Zhang, J., Wang, X. G., & Li, Z. L. (2006). Application of neural network based on wavelet packet energy entropy in power system fault diagnosis. Chinese Journal of Chinese of Power System Technology, 30(5), 72–76. Zhao, H. L., Wang, F., & Hu, X. G. (2004). Application of wavelet packet energy spectrum in mechanical fault diagnosis of high voltage circuit breakers. Chinese Journal of Power System Technology, 28(6), 46–49.