An implementation of S-transform and type-2 fuzzy kernel based support vector machine algorithm for power quality events classification

Abstract

This paper aims to develop a new idea for the classification of power quality disturbances. The method is based on Stockwell’s-Transform (ST) and Type-2 Fuzzy Kernel Support Vector Machine (T2FK-SVM). Through the introduction of ST and its properties, we propose a classification plan for nine types of power quality disturbances. Firstly, features of disturbance signals extracted through the ST. Secondly, features extracted by using the ST are applied as input to T2FK-SVM classifier for automatic classification of the power quality (PQ) disturbances. Design of Kernel is a main part of many kernel based methods such as Support Vector Machine (SVM), so by using of Type-2 Fuzzy sets as a kernel of SVM, the total accuracy of classification enhanced.This method can reduce the features of the disturbance signals significantly, and so less time and memory is required for classification by the T2FK-SVM method. Six single event and two complex event as well normal voltage selected as reference are considered for the classification. The simulation results showed accurate classification, fast learning and execution in the detection and classification of PQ events. Results are compared with other methods and the robustness of proposed method evaluated under noisy conditions. Finally, proposed method is also implemented on real time PQ disturbances to confirm the validity of this method in practical conditions.

Keywords

Detection classification power quality (PQ)S-transform (ST)Type-2 fuzzy kernel (T2FK)Support Vector Machine (SVM)

1 Introduction

Power supply quality issues and the resulting problems are the consequences of increasing use of solid state switching devices, non-linear and power electronically switched loads, unbalanced power systems, lighting controls, computer and data processing equipment as well as industrial plant rectifiers and inverters. These electronic type loads cause voltage distortion, inrush, pulse type current phenomenon with excessive harmonics and high current distortion [1].

The problem of power quality event classification has three key parts: processing method, feature selection and classification. The goal of processing method is to convert raw data into feature extractable data. The second part of the PQ event classification is optimal feature selection in this segment, a suitable set of feature must be selected from converted signal. These sets of features must have smaller size to reduce classification time. Classification process is the last part of PQ event classification. In this part extracted features must be classified into the accurate clusters.

In literature, many signal processing methods are introduced for analyzing PQ events and extracting features: Short time Fourier transform (STFT) method [2], discrete wavelet transform (DWT) method [3], empirical mode decomposition (EMD) method [4], Hilbert Huang transform (HHT) method [5] and S-transform (ST) method [6]

Artificial intelligence (AI) based methods can be classify PQ events accurately and efficiently. Many AI based methods are applied for PQ events classification: support vector machine (SVM) method [7, 8], artificial neural network (ANN) system [9], fuzzy-expert system [10, 11], neuro-fuzzy system [12], genetic algorithm (GA) method [13] and particle swarm optimization (PSO) method [14].

PQ events are non-stationary signals, features extraction is an important for event classification. STFT, DWT and ST are popular methods for feature extraction that used in many recent works. PQ event detection by ST used in [15] and a new method based on the binary feature matrix is designed for making a decision regarding the disturbance type. Detection and classification of voltage disturbances using a Fuzzy-ARTMAP-wavelet network introduced in [16] in this research wavelet extract features and Fuzzy-ARTMAP neural network classified extracted features. Fuzzy C-means and adaptive particle swarm optimization for classification of PQ events proposed in [17].In this article authors used windowing techniques for non-stationary disturbance signals. Balanced neural tree proposed by Authors of [18] for PQ event classification.The authors used Hilbert transform and empirical mode decomposition for feature extraction.

This article is described as follows: S-transform is described in section 2, and section 3 addresses Type-2 Fuzzy Kernel Based Support Vector Machine method. Problem formulation, simulation results and conclusion are given in section 4-6 to show the accuracy and efficiency of the proposed method.

2 S-transform

The S-transform [19] based on two-advanced signal processing tools; the short-time Fourier transforms (STFT) and the wavelet transform. It can be viewed as a frequency dependent STFT or a phase corrected wavelet transform [19]. The frequency dependent window used for analysis of a signal so the S-transform has better performance than other time-frequency transforms. Furthermore, it provides superior time-frequency localization property computing both amplitude and phase spectrum of discrete data samples [19].The S-transform of a signal h₍t) is defined as: $s (t, f) = \int_{- \infty}^{\infty} h (τ) w^{*} (τ - t, f), e^{- j 2 π f τ} d τ$ (1)

Where $s (t, f) = \frac{| f |}{α \sqrt{2 π}} . e^{- {tf}^{2} / 2 α^{2}}$ (2) and * stands for complex-conjugate. The parameter sets the width of the window for a given frequency. For small, the time resolution improves and the frequency resolution deteriorates and vice versa. The integration of S-transform over time results in the Fourier spectrum that is $H (f) = \int_{- \infty}^{\infty} S (t, f) dt$ (3) and for the Gaussian window $H (f) = \int_{- \infty}^{\infty} S (t, f) dt = 1$ (4)

The inverse S-transform is given by $h (f) = \int_{- \infty}^{\infty} {\int_{- \infty}^{\infty} S (τ, f) d τ} e^{j 2 π f τ} df$ (5)

Another representation of S-transform is an amplitude and phase correction of the CWT (continuous wavelet transform) as follows $S (t, f) = \sqrt{\frac{| f |}{2 π α}} . e^{- j 2 π f τ} WT (t, f)$ (6) where the wavelet transform is given by $WT (t, f) = \sqrt{\frac{| f |}{α}} e^{- t^{2} f^{2} / 2 α^{2}} e^{j 2 π f τ}$ (7)

Equation (6) shows distribution of the time-frequency resolution in the time frequency plane like wavelet transform but with Fourier transform a direct link exists. So the S-transform is given as follows $S (t, f) = \int_{- \infty}^{\infty} H (v, f) W^{*} (v, f) . e^{j 2 π v τ} dv$ (8) $WT (v, f) = e^{- 2 π^{2} α^{2} υ^{2} / f^{2}}$ (9)

A discrete explanation of the S-transform is obtained from (8) as follows $S (m, n) = \sum_{k = 1}^{N - 1} [n + k] . e^{- 2 π^{2} α^{2} k^{2} / n^{2}} . e^{j 2 π km}$ (10) for n ≠ 0 and $S [m, n] = \frac{1}{N} \sum_{k = 1}^{N - 1} h (k)$ (11) for n = 0 where $H [n] = \frac{1}{N} \sum_{k = 1}^{N - 1} h (k) . e^{- j 2 π kn}$ (12)

The H_[n] is the DFT of h (k) and can be calculated by FFT. $S (j, n) = \sum_{m = 0}^{N - 1} H [m + n] G (m, n) e^{- j 2 π m / N}$ (13)

Where $G (m, n) = e^{- 2 π^{2} α^{2} m^{2} / n^{2}}$ (14) and j, m and n = 0, 1, N - 1. The computational efficiency of FFT is used to calculate the S-transform and the total number of operations is N (N + N log N) [19]. The output from ST is a complex matrix whose rows and columns represent frequency and time, respectively. Each element of the S-matrix is complex number.

3 Interval type-2 fuzzy kernel based Support vector machines

3.1 Support vector machines

SVM is a powerful and reliable method for classification of patterns [20, 21]. Given the training data (x₁, y₁) , . . . , (x_l, y_l) , x ∈ R^M, y_i ∈ {-1, + 1} for two class problem, the decision functions of form sgn ((w^Tx_i) + w₀) constructs by SVM with the maximum margin, where w is the normal vector of the separating hyperplane in the canonical form and w₀ is a bias term [21]. The distances of the point closest to the hyperplanes of both -1 and+1 are calculated as 1/∥ w ∥/∥ w ∥. The separating margin is defined to be 2/||w||.

In many real world applications, data is noisy and a linear hyperplane cannot separate. To allow deteriorating from margin, ξ_i ≥ 1 are introduced as follows: $y_{i} ((w^{T} x_{i}) + w_{0}) \geq 1 - ξ_{i} i = 1, \dots, l$ (15)

For the training data x₁ if 0 ≤ ξ_i < 1, the data do not have the maximum margin but are still correctly classified. But if ξ_i ≥ 1, the data are misclassified. So by leaving intramargin the noise points occurring to near the boundaries region the separation margin is improved and that generalization performance is increased [21].

SVM builds the constraint primal quadratic optimization problem that minimizes the training and generalization error as follows: $min = \frac{1}{2} w^{T} w + C \sum_{i = 1}^{l} ξ_{i}$ (16) $s . t y_{i} ((w^{T} x_{i}) + w_{o}) \geq 1 - ξ_{i} ξ_{i} \geq 0, i = 1, . . ., l$ where controls the penalty incurred by each misclassified point in the training set. Larger C values cause SVM models with narrower margin and better training accuracy and smaller C values cause wider margin and better generalization accuracy. To solve the primal problem in (7), Lagrange function is firstly formulated [22]. Then its derivatives with respect to the primal variables w, ξ, and w_o are calculated and KKT conditions are satisfied [22]. Finally the obtained dual optimization problem is solved as follows: $max Q (α) = \frac{1}{2} \sum_{i = 1}^{t} \sum_{j = 1}^{t} y_{i} y_{j} α_{i} α_{j} x_{i}^{T} x_{j} + \sum_{i = 1}^{t} α_{i}$ (17) $s . t \sum_{i = 1}^{t} α_{i} y_{i} = 0 0 \leq α_{i} \leq Ci = 1, . . ., l$ where α_i is positive Lagrange multipliers.

SVM maps the inputs x into some higher dimensional space by means of a nonlinear feature mapping φ (x) for solving the classification problem separated by only highly complex decision boundaries in the input space. Thus the problem changes into linearly separable case at the feature space. If only scalar product $x_{i}^{T} x_{j}$ in (8) is replaced by the kernel function K (x_i, x_j) φ^T (x_j) φ (x_j) assumed to be symmetric and positive definite [20], the dual problem subject to constraints in (8) is rewritten as follows: $Q (α) = - \frac{1}{2} \sum_{i = 1}^{t} \sum_{j = 1}^{t} y_{i} y_{j} α_{i} α_{j} K (x_{i}, x_{j}) + \sum_{i = 1}^{t} α_{i}$ (18)

K (x, x_i) = x^Tx_i in (8) is a linear kernel. One of the most common kernels is Gaussian kernel defined as: $K (x, x_{i}) = exp (- \frac{∥ x - x_{i} ∥^{2}}{2 σ^{2}})$ (19)

Multi-class SVM classifiers carries out by combining two class SVMs. The most efficient methods are 1-against-r scheme (with r represent the number of classes) and the 1-against-1 scheme [23]. 1-against-r scheme users (r₁) two classifiers: each machine is trained as a classifier for one class against all other classes. 1-against-1 scheme creates a multi-class classifier, i.e. (r (r₁)/2) two class machines are created. Each machine is trained as a classifier for one class against other classes. In order to classify test data, pairwise competition between all the machines is performed; each winner competes against another winner until a single winner remains. This final winner determines the class of the test data [23].

3.2 Type-2 Fuzzy Kernel

Performance of SVM reduced in the uncertain condition and the optimal design of kernel function can enhance accuracy and performance. In this research interval type-2 fuzzy function used as kernel of SVM classifier. This kernel can efficiently classify power quality events in uncertain conditions.

An interval type-2 fuzzy set $\tilde{A}$ is characterized by a type-2 membership function, $μ \tilde{A} (x, u)$ where, x ∈ X and u ∈ J_x ⊆ [0, 1] as follows [24]: $\begin{matrix} \tilde{A} = {((x, u), μ_{\tilde{A}} (x, u)) | \forall x \in X, \\ \forall u \in J_{x} \subseteq [0, 1], \\ μ_{\tilde{A}} (x, u) = 1} \end{matrix}$ (20)

J_x is called primary membership of $\tilde{A}$ The uncertainty in the primary memberships of a type-2 fuzzy set $\tilde{A}$ , consists of a bounded region that is called the footprint of uncertainty (FOU).

The upper membership function (UMF) and lower membership function (LMF) of $\tilde{A}$ are two Type-1 MFs that bound the FOU. The UMF is associated with the upper bound of $FOU (\tilde{A})$ and the LMF is associated with the lower bound of $FOU (\tilde{A})$ as follows [24]: $\begin{matrix} \bar{μ} \tilde{A} (x, u) \equiv \bar{FOU (\tilde{A})} \forall x \in X \\ {\underline{μ}}_{\tilde{A}} (x, u) \equiv \underline{FOU (\tilde{A})} \forall x \in X \end{matrix}$ (21)

Interval type-2 fuzzy set concept applied, to represent the uncertainty of the kernel. The primary membership J_{k
_m} of a kernel K_m can be represented as a membership interval with all secondary grades of the primary memberships equaling to one. An interval type-2 fuzzy kernel set can be represented as [24]: $\begin{matrix} \tilde{Γ} = {((K, u), μ_{\tilde{Γ}} (x, u)) | \forall K \in Γ, \\ \forall u \in J_{k} \subseteq [0, 1], \\ μ_{\tilde{Γ}} (x, u) = 1} \end{matrix}$ (22)

RBF kernel set with different kernel parameters would be extended to interval type-2 fuzzy kernel type as shown in Fig. 1 [24].

Fig.1

Example of a an interval type-2 fuzzy kernel.

3.3 Type-reduction for T2FK

In this part, an interval of primary membership of kernels computed with various kernel parameters which show different degrees of membership. A kernel set to interval type-2 fuzzy kernel sets to show uncertainty by using this kernel in SVM. The centroid type-reducer used for the type-reduction of interval type-2 fuzzy kernels. Considering all the M kernels, the fuzzy relation of interval type-2 fuzzy kernels are given as follows: $u (K) = ⋃_{m = 1}^{M} u (K^{m})$ (23)

The upper output and lower output of the kernel defined as follows: $\bar{K} = \frac{\sum_{m = 1}^{M} \bar{u} (K^{m}) . K^{m}}{\sum_{m = 1}^{M} \bar{u} (K^{m})}$ (24) and $\underline{K} = \frac{\sum_{m = 1}^{M} \underline{u} (K^{m}) . K^{m}}{\sum_{m = 1}^{M} \underline{u} (K^{m})}$ (25)

3.4 Defuzzification of T2FK

In the defuzzification process final output of the kernel computed for the learning part of SVM, which is performed by defuzzification operation on the interval set $[\underline{K}, \bar{K}]$ from the type reduction. The average of $\underline{K}$ and $\bar{K}$ computed for defuzzification as follows: $\bar{K} = \frac{\underline{K} + \bar{K}}{2}$ (26)

By using interval type-2 fuzzy kernels, the robustness and reliability of SVM improved.

4 Proposed method

The proposed method for PQ event classification have three key parts: preprocessing, feature extraction and classification. Preprocessing part used the equation of each event to generate event waveform. ST applied for feature extraction and T2FK-SVM classified features. The diagram of proposed method presented in Fig. 2.

Fig.2

Diagram of proposed method.

4.1 Preprocessing

In this paper nine types of events considered as follows: normal voltage, voltage sag, swell, interruption, oscillatory transient, flicker, sag with harmonic and swell with harmonic.

4.2 Feature extraction

The samples of PQ event are huge for classification process so valuable features must be extracted from events. In this part features of each signal extracted by ST. Five features extracted from the ST output of each event as follows:

F₁ = max (A) where A is the amplitude versus time graph from the ST output matrix under disturbance.

F₂ = min (A) where A is the amplitude versus time graph from the ST output matrix under disturbance.

$F_{3} = \frac{1}{2} (max (B) + min (B) - max (A) - min (A))$

where B is the amplitude versus time graph of the ST output in normal condition.

F₄= total harmonic distortion (THD) = $\sqrt{\sum_{n = 2}^{N} | U_{n}} |^{2} / U_{1}$ where is the number of points in the FFT, U_n is the value of the nth harmonic component of the FFT.

F5 = estimated frequency at the maximum amplitude obtained from ST output matrix. So a single vector with five elements is generated for each PQ event and fed to classifier stage. The number of features that fed to classifier is low (only 5 inputs). The ST output contour for three PQ events showed in Figs. 3 –5.

Fig.3

(a) Voltage sag signal and it’s ST, (b) Magnitude– time and amplitude– frequency diagram from ST.

Fig.4

(a) Voltage oscillatory transient signal and it’s ST, (b) Magnitude– time and amplitude– frequency diagram from ST.

Fig.5

(a) Voltage swell with harmonics signal and it’s ST, (b) Magnitude– time and amplitude– frequency diagram from ST.

4.3 Classification

In this stage features of events classified by T2FK-SVM. The Kernel function of T2FK-SVM is interval type-2 fuzzy function. Firstly, parameters of T2FK-SVM set by trial and error. Secondly SVM trained by using optimized parameters in the previous step and 50% of events that fed to the inputs of T2FK-SVM. Finally performance of T2FK-SVM evaluated by 50% of generated events.

5 Simulation and analysis

5.1 Data generation

Signal generation by parametric equations for event classification has many advantages. It was possible to change training and testing signal parameters in a wide range and in a controlled manner. The signals simulated with this way were very close to the real world conditions. On the other hand, different signals belonging to the same class gave possibility to estimate the generalization ability of classifiers based on neural networks [25].

5.2 Simulation results

As mentioned nine types of events considered in this research.Two hundred of each event generated based on parametric equations of Table 1 by using random parameters. The ranges of equations parameter variation are presented in Table 1. These events sampled by 50Hz. Each event has 16 cycles and 32 samples so the sampling frequency is 1.6 kHz. To assess accuracy of proposed method mean absolute error (MAE) measure introduced as follows: $MAE = (\frac{1}{N} \sum_{i = 1}^{N} ({Actual}_{i} - {Forecast}_{i})) * 100$ (27)

Table 1

Overall accuracy and misclassification error

Class	C1	C2	C3	C4	C5	C6	C7	C8	C9
C1	100	0	0	0	0	0	0	0	0
C2	0	99	0	1	0	0	0	0	0
C3	0	0	100	0	0	0	0	0	0
C4	0	1	0	99	0	0	0	1	0
C5	0	0	0	0	99	0	0	1	0
C6	0	0	0	0	0	100	0	0	0
C7	0	0	0	0	0	0	100	0	0
C8	0	0	0	1	1	0	0	98	0
C9	0	0	0	0	0	0	0	0	100
Overall accuracy = 99.44%

The overall accuracy and misclassification error of each event demonstrated in Table 2.

Table 2

Classification results under noisy condition

Class	–	20 dB	40 dB
C1	100	94	100
C2	98	98	99
C3	100	100	100
C4	100	98	79
C5	100	82	84
C6	100	100	99
C7	100	100	100
C8	100	96	100
C9	100	100	100
Overall accuracy	99.44%	96.44%	95.67%

5.3 The performance of proposed method under noisy conditions

An another important issue related to the proposed method is noisy conditions which are present in an electrical distribution network. So, the performance of the proposed method was tested in different noise Conditions. Gaussian white noise is widely used in the researches on the PQ issues [26]. To evaluate the performance of the proposed method in different noise environments, different levels of noises with the signal to noise ratio (SNR) values from 20 and 40 dB were considered. The value of SNR is defined as follows: $SNR = 10 log (\frac{P_{s}}{P_{N}}) dB$ (28) where p_s is the power (variance) of the signal and P_N is that of the noise. This noise added to PQ event signals. Fig. 6 shows voltage sag disturbance under 20 dB noise conditions. The classification results under noisy conditions are presented in Table 2 with 20 and 40 dB noises. The overall accuracy of classification which reflects the robust capability of the proposed method under noisy conditions as shown Table 2.

Fig.6

Voltage sag under 20dB noise.

5.4 Discussion and comparison witho ther methods

The greatest advantage of the proposed method based on the ST and T2FK-SVM classifier has a smaller feature size for each event. Hence, total simulation time reduced significantly. To verify the efficiency and robustness of the proposed method, overall accuracy compared with [27 –30] and error of each method demonstrated in Table 3. As seen from Table 3 the proposed method has significant results and better accuracy than other methods.

Table 3
Comparison with other methods.

Method Overall accuracy

DFT-RBDT [27] 99%

DWT-NN [28] 95.14%

ST-PNN [29] 97.6%

RB-HM[30] 98.7%

Proposed method 99.44%

Method	Overall accuracy
DFT-RBDT [27]	99%
DWT-NN [28]	95.14%
ST-PNN [29]	97.6%
RB-HM[30]	98.7%
Proposed method	99.44%

5.5 Experimental events

To evaluate the performance of the proposed method for real-world measurements, 34 waveforms with different types of multiple disturbances taken from the DOE were presented to the classifier. In Figs. 7 and 8 real voltage sag and swell are presented. After applying proposed method, total of 29 waveforms were correctly classified. From these 34 waveforms, 19 are corrupted by sag, 6 are corrupted by swell and 9 are corrupted by harmonics.

Fig.7

Real voltage sag.

Fig.8

Real voltage swell.

6 Conclusion

The large number of PQ events occurred in the electrical power distribution system. So, there is a requirement for a new method that can be used for automated classification of PQ events. This paper introduced a novel algorithm to classify the PQ events. This technique is based on the S-transform (ST) and type-2 fuzzy kernel support vector machine (T2FK-SVM) classifier. PQ events generated by equations and features are extracted by ST the extracted features classified by SVM. The kernel function of SVM is type-2 fuzzy functions that have a wide range of uncertainty. The parameters of type-2 fuzzy functions are set by trial and error method. The T2FK-SVM is trained by 50% of data and tested by remain data. The main advantage of proposed method is small features size, low execution time and significant accuracy. So, this method requires less memory space at both the training and testing processes.

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An implementation of S-transform and type-2 fuzzy kernel based support vector machine algorithm for power quality events classification

Abstract

Keywords

1 Introduction

2 S-transform

3.1 Support vector machines

4.2 Feature extraction

5 Simulation and analysis

5.1 Data generation

5.2 Simulation results

Table 3 Comparison with other methods. Method Overall accuracy DFT-RBDT [27] 99% DWT-NN [28] 95.14% ST-PNN [29] 97.6% RB-HM[30] 98.7% Proposed method 99.44%

References

Table 3
Comparison with other methods.

Method Overall accuracy

DFT-RBDT [27] 99%

DWT-NN [28] 95.14%

ST-PNN [29] 97.6%

RB-HM[30] 98.7%

Proposed method 99.44%