Classification of Nonlinear Power Quality Events Based on ...

ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.10(2010) No.3,pp.279-286

Classification of Nonlinear Power Quality Events Based on Multiwavelet Transform Rajiv Kapoor1 , Manish Kumar Saini2 ∗

2

1 Electronics and Communication Engineering Department, Delhi Technological University, New Delhi-110042. Electrical Engineering Department, Deenbandhu Chhotu Ram University of Science and Technology, Sonipat-130039.

(Received 10 January 2010, accepted 11 May 2010)

Abstract: The paper presented uses multiwavelet due to its inherent property to resolve the signal better than all single wavelets. MWTs are based on more than one scaling function. The resolution is better compared to Daubechies and hence, the event can be detected from lesser number of samples and in lesser time as compared to Daubechies. This paper utilizes an enhanced resolving capability of MWT to recognize PQ disturbances. Fuzzy product aggregation reasoning rule (FPARR) classifier has been implemented and tested for various PQ events. Results on various PQ events such as voltage sag, voltage swell, outage, interruption (INT), impulsive-transient (IT), oscillatory-transient (OT), noise and notch show that multiwavelets can detect and classify different PQ events efficiently and consistently. Keywords: power quality events; wavelets; fuzzy classifier

1

Introduction

PQ issues have attracted considerable attention from both utilities and users due to the use of many types of sensitive electronic equipments, which can be affected by harmonics, voltage sag, voltage swell, and momentary interruptions. These disturbances cause problems, such as overheating, motor failures, inaccurate metering and in the operation of protective equipments. Voltage swell and sag can occur due to lightning, capacitor switching, motor starting and nearby circuit faults or accidents and can also lead to power interruptions. Harmonic currents due to nonlinear loads throughout the network can also degrade the quality of services to sensitive high-tech customers, such as India’s IT parks in Bangalore, Hyderabad and many other places. Power system disturbance detection is increasingly becoming important these days. In order to improve PQ, the sources and causes of such disturbances must be known so that appropriate mitigating actions can be taken. A feasible approach to achieve this goal is to incorporate detection capabilities into monitoring systems so that the events of interest are recognized, captured and classified automatically. Hence, a good performance monitoring equipment must have functions for detection, localization and classification of PQ events [1] [15]. A scalable event identification system [2] has been suggested as an alternative to the ANN approach in identifying PQ events in a scalable manner. Classification scheme based on identifying and characterizing the different stages of voltage during the event by an expert system has been proposed in [3]. In [4], [5], the proposed recognition scheme has been carried out in wavelet domain using a set of multiple neural networks. The outcomes of the networks are then integrated using decision making schemes. In [6], both Fourier and wavelet analysis have been applied for extracting distinct features of various types of events as well as for characterizing the PQ events. A fuzzy expert system has been used to classify the PQ events based on extrated features. In [7], wavelet multi-resolution transform has been introduced for the feature extraction to classify PQ distribution with the help of Euclidean distance k-nearest neighbors and neural networks. In [8], wavelet transform has been utilized to produce representative feature vectors to capture the characteristics of disturbances. In the training phase, a decision plane is developed for the classification of PQ disturbances. Paper [9] is an approach to PQ assessment based on real-time hardware-in-loop simulation. The sensitivity of a variable speed drive controller card for PQ deviations has been tested on the platform. In [10], authors have presented a new adaptive neuro-fuzzy intelligent tool for PQ analysis and diagnosis. In paper [11], authors have described the theoretical foundation of a method for classifying voltage and current waveform events. A distinct time-frequency representation (TFR) is derived for each class. In [12], the above said algorithm has been tested on DSP based hardware system. In [13], Walsh transform and fast Fourier transform ∗ Corresponding

author.

E-mail address: [email protected]. Tel. : +91-0130-2484124. c Copyright⃝World Academic Press, World Academic Union IJNS.2010.12.15/417

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International Journal of Nonlinear Science, Vol.10(2010), No.3, pp. 279-286

have been adopted as feature extraction tools and the combined fast match dynamic time wrapping algorithm does the classification. In [14], WT technique integrated with probabilistic NN has been modeled to construct the classifier. Only scalar wavelets have been used for the analysis of PQ events till now and are generated using one scaling function. But one can imagine a situation, where more than one scaling functions are desired. This leads to the notion of multiwavelet which provides features such as short support, orthogonality, symmetry and vanishing moments simultaneously. A MWT can provide perfect reconstruction while preserving length (orthogonality), good performance at boundaries (via. linear phase, symmetry) and a high order of approximation (vanishing moment) [20]. Five hundred real-time distorted power signals have been characterized with the help of MWTs and FPARR classifier. The accuracy of the proposed method has been shown by a confusion table. The results show that the proposed method analyzes the signals efficiently and consistently. This paper elaborates the MWT technique in section 2. Section 3 depicts the block diagram of the classification system. Section 4 gives detailed description of extracted featuers and their characterization. Section 5 shows the results obtained after decomposing PQ events. Section 4 contains description of FPARR classifier. Section 7 resolves the result and section 8 presents conclusion.

2

Multiwavelet Transform

Multiwavelets are an addition to the body of wavelet theory. Realizable as matrix-valued filter banks leading to wavelet bases, multiwavelet offers orthogonality, symmetry, short-support and secondorder accuracy; which is not possible with scalar wavelet system. Multiwavelet differ from scalar wavelet in terms that multiwavelet requires two or more input streams to filter bank. The theory of real multiwavelet is connected with the theory of real vector-valued functions 𝜙(𝑡) . This family is an orthogonal basis for 𝐿2 (𝑅) (square-integrable function), exactly as in the usual case of a single wavelet. As in the scalar case, multiwavelets are usually constructed starting from multiscaling functions. These can be defined by means of a generalization of the notion of multiresolution analysis (MRA). Let 𝜙 = [𝜙1 𝜙2⋅⋅⋅ 𝜙𝑟 ]𝑇 be a vector-valued function belonging to 𝐿2 (𝑅)𝑟 ,𝑟 𝜖 𝑁 . Define, for 𝑗𝜖𝑍, 𝑉𝑗 = 𝑠𝑝𝑎𝑛{2𝑗/2 𝜙𝑖 (2𝑗 ⋅ −𝑘); 1 ≤ 𝑖 ≤ 𝑟, 𝑘𝜖𝑍}

(1)

where vector (𝜑1, 𝜑2 , ⋅ ⋅ ⋅ 𝜑𝑟 ) are having span if the set of non-zero vectors (𝜑1, 𝜑2 , ⋅ ⋅ ⋅ 𝜑𝑟 ) belonging to a vector space𝑉 have the property that every vector in 𝑉 can be expressed as a linear combination of these vectors. 𝜙 is called a multiscaling function if the spaces defined in (1) satisfy the following properties. i)⋅ ⋅ ⋅ ⊂ 𝑉−1 ⊂ 𝑉0 ⊂ 𝑉1 ⊂ ⋅ ⋅ ⋅ {𝐿2 (𝑅)}; ii)∪𝑗𝜖𝑍 𝑉𝑗 = 𝐿2 (𝑅); iii)∩𝑗𝜖𝑍 𝑉𝑗 = {0}; iv)𝑓 (⋅)𝜖𝑉𝑗 ⇔ 𝑓 (2⋅)𝜖𝑉𝑗+1, ∀𝑗𝜖𝑍; v)The family {𝜙𝑖 (⋅ − 𝑘); 1 ≤ 𝑖 ≤ 𝑟, 𝑘𝜖𝑍} is a Riesz (stable) basis for 𝑉0 , i.e. two constant 𝐴 and 𝐵 exist, 0 < 𝐴 ≤ 𝐵 < ∞, such that 𝐴∥𝑐∥2𝑙2 (𝑍)𝑟

≤∥

𝑟 ∑∑ 𝑘𝜖𝑍 𝑖=1

𝑐𝑘𝑖 𝜙𝑖 (⋅ − 𝑘)∥2𝐿2 (𝑅) ≤ 𝐵∥𝑐∥2𝐿2 (𝑍)𝑟

(2)

for every sequence 𝑐 = {𝑐𝑘 }𝑘𝜖𝑍 𝜖𝑙2 (𝑍)𝑅 . This condition implies that 𝜙 has stable shifts. Condition (iv) expresses the main property of an MRA [23], each 𝑉𝑗 consists of the functions in 𝑉0 compressed by a factor of 2𝑗 .The vector of basis functions 𝜙 is called multiscaling function. 𝜙(𝑥) is called an orthogonal scaling function if < 𝜙(𝑥), 𝜙(𝑥 − 𝑛) >= 𝛿0,𝑛 𝐼; 𝑛𝜖𝑍. The associated multiresolution analysis is said to be an orthogonal MRA. The GHM scaling coefficients satisfy above orthogonality condition as in [15]. Multiscaling function generates a MRA of multiplicity, 𝑟 for 𝐿2 (𝑅). From the above conditions, 𝜙1 (⋅), 𝜙2 (⋅) ⋅ ⋅ ⋅ 𝜙𝑟 (⋅)𝜖𝑉0 ⊂ 𝑉1 , it follows that a sequence of matrices {𝑃𝑘 }𝑘𝜖𝑍 𝜖𝑙2 (𝑍)𝑟∗𝑟 exists such that ∑ 𝜙(𝑥) = 𝑃𝑘 𝜙(2𝑥 − 𝑘) (3) 𝑘𝜖𝑍

Equation (3) is called the matrix refinement equation of the multiscaling function 𝜙 . It can be formulated in the Fourier space. Let 𝑍 = 𝑒−𝑗𝜔 ,𝜔𝜖𝑅. One can have, ˆ ˆ 𝜙(2𝜔) = 𝑃 (𝑍), 𝜙(𝜔)

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(4)

R. Kapoor, M. Saini: Classification of Nonlinear Power Quality Events Based on Multiwavelet Transform

281

Figure 1: (a) GHM Scaling Function 𝜙1 . (b) GHM Wavelet Function 𝜓1 ∑ where 𝑃 (𝑧) = 12 𝑘𝜖𝑍 𝑃𝑘 𝑧 𝑘 is the two-scale symbol of 𝜙. For an assigned MRA of multiplicity 𝑟, {𝑉𝑗 }𝑗𝜖𝑍 , one can define the complementary space 𝑊𝑗 of 𝑉𝑗 , for every 𝑗𝜖𝑍 , such that 𝑉𝑗+1 is that direct sum of 𝑉𝑗 and 𝑊𝑗 , namely 𝑉𝑗+1 = 𝑉𝑗 ⊕ 𝑊𝑗 . Let 𝜓 a vector–valued function 𝜓 = [𝜓1 , 𝜓2 , ⋅ ⋅ ⋅ 𝜓𝑟 ]𝑇 𝜖𝐿2 (𝑅), such that 𝑊𝑗 = 𝑠𝑝𝑎𝑛{2𝑗/2 𝜓𝑖 (2𝑗 ⋅ −𝑘); 1 ≤ 𝑖 ≤ 𝑟, 𝑘𝜖𝑍}

(5)

The above-defined vector-valued function 𝜓 is a orthogonal MWT if for every 𝑗𝜖𝑍, it satisfies the condition: 𝑉𝑗 ⊥𝑊𝑗 ; if there is also orthonormality among the function in the same space, then obtain orthonormal MWT, which satisfy the condition: < 𝜓(2𝑗 ⋅ −𝑛), 𝜓(2𝑗 ⋅ −𝑚) >= 𝛿𝑖.𝑗 𝛿𝑛,𝑚 𝐼 𝑖, 𝑗, 𝑛, 𝑚𝜖𝑍

(6)

where 𝐼 is identity matrices. Every MWT satisfies a two-scale relation, such as: 𝜓(𝑥) =

∑

𝑄𝑘 𝜙(2𝑥 − 𝑘)

(7)

𝑘𝜖𝑍

in which the two-scale coefficients 𝑄𝑘 are 𝑟 × 𝑟 matrices. In terms of the two scale symbol of 𝜓, one can write: ˆ ˆ 𝜓(2𝜔) = 𝑄(𝑧)𝜙(𝜔) where 𝑄(𝑧) =

1 2

∑ 𝑘𝜖𝑍

(8)

𝑄𝑘 𝑍 𝑘 . Equation (8) is called wavelet two scale relation. ⎡

√ 1 ⎢ 6 2 16 √ 𝑃 [𝑘] = √ ⎣ −1 −3 2 10 2 𝑃 [0]

⎤ √ 0√ 16 6 √2 0 0 0 ⎥ ⎦ 9 10 2 9 −3 2 −1 0 𝑃 [1]

𝑃 [2]

(9)

𝑃 [3]

Equation (9) shows the coefficients of the scaling function in the matrix form (Low Pass Filters) which has been denoted by 𝑃 [𝑘]. In this equation, first row shows the coefficients of first scaling function and the second row shows the coefficients of second scaling function. Then, 𝑃 [𝑘] is simulated in equation (3). Figure 1(a) and figure 2(a) are obtained which depict first scaling function and second scaling function respectively. ⎡ 𝑄[𝑘] =

1 ⎢ −1 √ ⎣ √ 2 10 2

√ √ −3 2 9√ 10 2 √ 9 6 −9 2 0 9 2 𝑄[0]

𝑄[1]

√ −3 2 −1 √ − 2 −6 𝑄[2]

⎤ 0 ⎥ ⎦ 0

(10)

𝑄[3]

Equation (10) shows the coefficients of the wavelet function in the matrix form where first row shows the coefficients of first wavelet function and second row shows the coefficients of second wavelet function, the matrix has been denoted by 𝑄[𝑘]. 𝑄[𝑘] is simulated in equation (7), figure 1(b) and figure 2(b) are obtained, which depict first wavelet function and second wavelet function respectively. Since each 𝑃 and 𝑄 operators requires two inputs and there is one data set, the method must be determined to provide the first 𝑃 and 𝑄 operator with two sets of input data explained in the figure 3 [20]: 1) Alternatively assign each input data point to each of the P or Q inputs (this corresponds to A = the identity operator and B=the unit delay operator) in figure 3(a). 2) Use the P operator or another predetermined operator to create two sets of down sampled, interpolated data each of length N/2 (i.e. A and B act as pre-filters in figure 3(a). 3) Use the same data for both inputs (i.e., A=B=Identity operator). Reconstruction processes P’ and Q’ are the reconstruction filter bank [17].

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Figure 2: (a) GHM Scaling Function 𝜙2 . (b) GHM Wavelet Function𝜓2

Figure 3: (a) Decomposition Process (b) Reconstruction Process Using Multiwavelet

3

Methodology

PQ events are decomposed in different resolution levels by GHM MWT. GHM MWT have better energy compaction and short support (interval [0, 1] and [0, 2]). Both scaling functions are symmetric and wavelet functions form as symmetric/antisymmetric pair, all integer translates of the scaling function is orthogonal and system has second order of approximation (locally constant) [24]. Figure 4 depicts the block diagram of proposed methodology. MWT technique has been implemented for the decomposition of the PQ event patterns up to 7𝑡ℎ resolution level. Secondly, features are defined which gives numerical values which resemble the PQ event most. Features have been extracted according to IEEE standard [1] like duration, rise time and frequency etc. Third, FPARR has been employed as fuzzy classifier. The events recognition have been done in the real-time, as the system takes very less time to recognize power quality events.

4

Features and Characterization

Considered features are able to define the events successfully as per the standards laid down by IEEE Standard-1159. The feature set chosen in such a way so that if extracted that would fully establish the presence of the PQ event as per IEEE standard-1159. The following feature set has been used to fully define the various events: Peak value, Trough, Rise-Time (Upper), Fall Time (Upper), Duration (Upper), Rise-Time (Lower), Fall Time (Lower), Duration (Lower), Amplitude, Frequency, where Upper and Lower denotes the upper and lower half cycle. Knowledgebase has been formed by IEEE PQ monitoring standards [1].

4.1

Extracted Features

For the different PQ events, the different feature set of values have been chosen as per IEEE standard. For the classification of the impulsive-transient, there were a need to consider the rise time and decay time of the maximum magnitude and the duration of the event. As in proposed scheme, features extracted are rise time, duration, decay time etc. listed in table 1, which are forming the knowledge base for the FPARR classifier.

Figure 4: Block diagram of proposed algorithm

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R. Kapoor, M. Saini: Classification of Nonlinear Power Quality Events Based on Multiwavelet Transform

283

Figure 5: Implusive-transient PQ event 𝐼 𝑠𝑡 level decomposition.

Figure 6: Second level decomposition as in fig 5(e)

5

Results of Application of Multiwavelet on PQ signals

This section shows the results obtained after decomposing the power signal corrupted by events. The results of have been shown in figures 5-6. Figure 5 contains 8 plots; 5(a) depicts impulsive-transient PQ event, which is decomposed and classified by proposed method having duration of 1 ms and rise time of 1ms, 5(b) depicts the reconstruction of the signal after second level of decomposition, 5(c) depicts the signal after pre-filtering of the input signal-1, 5(d) depicts the signal after pre-filtering of the input signal-2, 5(e), 5(f), 5(g), 5(h) depicts the decomposition of the signal at level-1 where LL1depicts low-low filter, HH1-depicts high-high filter, LH1-depicts low-high filter, and HL1-depicts high-low filter. Figure 6(a) is shown in figure 5(e) LL1 first level of decomposition having low pass information has been further decomposed in second level. Figure 6(b), 6(c), 6(d), 6(e) depict the decomposition of the signal at level 2 where LL2- depicts low-low filter, HH2-depicts high-high filter, LH2-depicts low-high filter, and HL2-depicts high-low filter.

6

Classifiers

After the extraction of features, one has to take the decision for the classifier. Because of the less number of extracted features and to limit the complexity of SIA system, fuzzy classifier has been decided. Two important aspects, namely learning and generalization capabilities play an important role because of high variability of real PQ both properties are required in proposed algorithm. Intuitively, these can be achieved through feature-wise information extraction, generalization in the fuzzification process and combined contribution of these information to all classes of a PQ patterns because there is a high possibility that relevant information for different classes may reside in features of patterns and they supplement

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International Journal of Nonlinear Science, Vol.10(2010), No.3, pp. 279-286

Events Impulsive Transient Oscillatory Transient Sag Swell Harmonics Notching Interruption Flicker Outage

Table 1: Extracted Features Spectral Contents Typical Duration (ms) 1-𝜇 s rise 0.001 3 kHz 15 36 45 3 kHz 0.026 50 20 Hz 0.01

Voltage in p.u 3 0.7 1.4 0.15 1.4 0.01 0.5 0.0

each other. The problem becomes more complex if the classses are overlapping and ill-defined as in case transient and swell, as discussed . Keeping in view of these aspects, FPARR has been used as a classifier and highlighted the method of feature-wise extraction of information and combining/aggregating the features’ information to get an improved classification. The objective of the product aggregation is to assign pattern to a class where all the features are useful to represent that class properly, rather than the class where only some features are representing it. So, FPARR classifier has been used for their better suitability in classification of PQ events in proposed scheme.

6.1

Fuzzy product aggregation reasoning rule (FPARR) classifier

In the present investigation, FPARR classifier has been used for thier better suitability in classification of PQ events as exemplified. The FPARR based classification method uses three steps as in [31]. In the first step, it takes the feature vector and fuzzifies the feature value. This step uses a Π- type membership function to get the degree of belonging of a pattern into different classes based on different features. The membership matrix 𝑓𝑑,𝑐 (𝑥𝑑 ) thus generated express the degree of belonging of different features (𝐷) to different classes (𝐶), where 𝑥𝑑 is the 𝑑𝑡ℎ feature of pattern 𝑋 , the membership matrix after the fuzzification process can be expressed as : ⎤ ⎡ 𝑓1,1 (𝑥1 ) 𝑓1,2 (𝑥1 ) ⋅ ⋅ ⋅ 𝑓1,𝐶 (𝑥1 ) ⎢ 𝑓2,1 (𝑥2 ) 𝑓2,2 (𝑥2 ) ⋅ ⋅ ⋅ 𝑓2,𝐶 (𝑥2 ) ⎥ ⎥ 𝐹 (𝑥) = ⎢ (11) ⎣ ⎦ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ 𝑓𝐷,1 (𝑥𝐷 ) 𝑓𝐷,2 (𝑥𝐷 ) ⋅ ⋅ ⋅ 𝑓𝐷,𝐶 (𝑥𝐷 ) In the second step, the fuzzified feature values are aggregated using PRODUCT reasoning rule (PARR). It is applied on the membership matrix to get the combined membership grades of features of a pattern to various classes. After applying the PARR, we obtain the output as a vector given by: 𝐹 ∣ (𝑥) = [𝐹1 (𝑥), 𝐹2 (𝑥), ⋅ ⋅ ⋅ , 𝐹𝐶 (𝑥)]𝑇 , where 𝑋 is the input pattern and 𝐹𝐶 (𝑋) = Π𝐷 𝑑=1 𝑓𝑑,𝑐 (𝑥𝑑 ). This vector represents the fuzzy classification expressing the class belonging. The last step of FPARR classifier is a hard classification performed through a MAXIMUM operation to defuzzify the above vector output. Here the pattern is classified to class 𝐶 with the heist class MV.

7

Results

Confusion matrix shows the performance of the whole system. The diagonal shows the exact recognition of the power quality events and the other component of a row shows the confusion of the system i.e. in first row Table 2. 498 is the exact recognition and two events give the recognition as oscillatory-transient PQ event out of 500 samples. Five hundred samples have been tried on the system; the diagonal shows output of the FPARR classifier. Table 2 confusion matrix (for 500 samples) is used to show the result of using FPARR classifier. The diagonal value is showing the exact recognition of the PQ events.

8

Conclusion

It has been noticed that MWT demonstrate the better compression and hence lead to detection of the event in fewer samples as compared to the earlier papers. This is obvious also as the data belonging to an event patterns belong to various bands

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R. Kapoor, M. Saini: Classification of Nonlinear Power Quality Events Based on Multiwavelet Transform

Events IT IT 498 OT 5 Sag Harmonics 1 Swell Flicker Notching INT Outage RR in Percentage

OT 2 495

Table 2: Performance of FPARR classifier Sag Harmonics Swell Flicker Notching

INT

285

Outage

500 498

1 500 498 1

2 499 500 500

99.74%

and a single wavelet approach can’t ever maximize the compression. Though, design of the filter bank which is usable for Multiwavelets is difficult and the filters have to be chosen carefully otherwise the results will not be good. FPARR Classifier have been used for classification. It is noticeable that the results of the FPARR cassifier is able to deal with high variability of feature. It is suggested for future work that few combination of symmetrical filters should be utilized for testing the response of the technique which will improve upon the compression and this reduces the time and space complexity.

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