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Power Quality Disturbances Classification Based on
Curvelet Transform
Yue Shen, Fida Hussain*, Liu Hui and Destow Addis
School of Electrical and Information Engineering, Jiangsu University, Zhenjiang Jiangsu, China (E-mail:
fidahussain@ujs.edu.cn*Corresponding author
ABSTRACT
This article presents a novel method for power quality disturbances (PQDs) classification based on curvelet
transform (CT), locality preserving projection (LPP) and multi-class support vector machine (MCSVM).
Initially, PQD signals are converted into a 2-dimensional image and then feature extracted by using curvelet
transform. The inspiration for this method is based on detailed information of CT. The fast discrete curvelet
transform is newly developed transformation and has distinguished feature comparatively other transforms,
which define the scale, angle, and orientation. The curvelet coefficients have different frequency bands. The
lowest frequency band roughly contains image information. The highest frequency band represents the noisy
information and remaining holds edge information. In this research work, initial three frequency bands are
considered as PQD features. The extracted features are reshaped and reduce dimensionally using locality
preserving projection (LPP). Finally, multi-class support vector machine is used for classification of single and
combined PQDs. Eight types of single and combined PQ disturbances are considered for classification. The
proposed method is tested both synthetic and real PQ disturbances data, and 99.92% and 99.8% classification
accuracy are achieved without and with noise respectively. Results validate the correctness and robustness of the
proposed method in the classification of single and combined PQDs under noiseless and noisy environments.
KEYWORDS
Power quality disturbance classification; curvelet transform; support vector machine; locality preserving
projection
1. Introduction
Electric power quality (PQ) monitoring has become an essential part of a smart grid and electric network [1]. Most
of PQ problems arise due to grid connected distributed generations and non-linear electronic loads such as electric
cars charging, rectifiers, lighting, electric arc furnaces, welding equipment, switched-mode power supplies and
other power electronic devices. Common PQ problems are harmonic distortion, transient, voltage variations,
flickers etc. These PQ problems are directly responsible for overheating of transformers, damaging of capacitor
banks and sensitive electronic equipment [2-4].
Most of the traditional power quality monitoring equipment are not good enough to acquire the complete
information of PQDs data to recognize and classify the PQDs. Intelligent PQ monitoring equipment is required to
further improve the power quality monitoring system and protect the expensive equipment. Therefore, PQ
monitoring equipment necessity be proficient to identify and classify the PQDs [5, 6].
Feature extraction is an important stage to recognize and classify the PQDs and appropriate selection of
feature technique. It can significantly enhance the performance of detection and classification accuracy. Many
techniques have been revealed for feature extraction of PQDs classification for example wavelet transform (WT),
empirical mode decomposition (EMD), compressive sensing (CS), S-transform, Kalman filter, short-time Fourier
transform (STFT), wavelet packet transform (WPT), curvelet based, Hilbert transform (HT), Hilbert–Haung
transform (HHT), hybrid transform based methods, Gabor transform (GT), Wigner distribution function (WDF)
and over-complete hybrid dictionaries (OHDs) [7-16]. These features are used as input to a classifier such as fuzzy
logic, ensemble technique, support vector machine (SVM), deep learning, rule base, artificial neural network,
expert system and maximum likelihood classifier [17-25].
In this research work, we proposed a novel feature extraction method based on CT for classification of PQDs.
The key motivation of this research work is curvelet transform because it is newly developed multiresolution
transform and has a good advantage over other transforms such as wavelet transform [26, 27]. Theoretically, the
curvelet transform is a multiscale pyramid with numerous positions and directions at each length scale and has
useful geometric feature. Santoso et, al [28] proposed WT method for the recognition and localization of the real
power quality disturbances.
Curvelet transform was successfully used in many applications such as detection of ECG steganography signal
[29], protection of face and fingerprint [30], detecting micro-calcifications in mammography images [31], face
recognition [32], signal detection [33], seismic weak signal identification and noise elimination [34], palm-print
and face classification [35], wetland monitoring [36], anisotropic iris coding and recognition [26], tamper detection
of digital images [37], signal detection [33]. In this paper, CT is proposed for feature extraction of PQDs signal.
The proposed approach significantly improves classification accuracy and reduces the error rate.
The extracted features directly use as input to a classifier is quite difficult due to the huge dimensionality of
the features. Some methods are exploited to reduce dimensionality of data such as independent component
analysis (ICA), principal component analysis (PCA) [38], linear discriminant analysis (LDA) [39], kernel PCA,
kernel LDA [40], locality preserving projection (LPP), generalized discriminant analysis (GDA) [41]. PCA offers
effective approximation and the scheme suffers from larger computational complexity and poor discriminatory
power [38]. In this article, locality preserving projection (LPP) [42] is applied to reduce the dimension of the
extracted feature of PQDs.
Machine learning techniques have become very popular in binary classification and regression and
successfully applied in many applications [43-45]. In this paper, compact multiclass support vector machine
(CMSVM) [46] is employed for grouping of PQDs. The main purpose of this research work is to implement the
CT in the PQ monitoring sector and improve the classification accuracy.
This paper offers CT+LPP+MSVM classifier for classification of PQDs. This paper is structured in six
sections and remaining as follows: Section 2 offers a detail explanation of methodology. The proposed curvelet
transform, locality preserving projection and multiclass support vector machine classifier are presented in Section
3. In Section 4, simulation and experiments are described. Section 5 offerings concise experimental results and a
momentary discussion of results. Section 6 closes the paper.
2. Methodology
2.1 Curvelet transform based feature extraction of PQDs
Curvelet transform (CT) was introduced by Candes and Donoho, and has more advantage in geometric features
than wavelet transform (WT) [27, 47]. Short time Fourier transform (STFT) is the extension of discrete Fourier
transform (DFT) using a stationary window, but not appropriate for non-stationary signals [48]. WT solved this
weakness of STFT by employing variable window for low and high frequencies [28]. However, CT has more
advantage comparatively WT. CT is a multiscale transform with a pyramid structure consisting of several levels
at each scale [27, 47, 49]. Curvelet coefficient can be represented as the inner product of basis function and
input signals
(1)
Where is scale, indicates the angle; shows the parameter orientation. Curvelet can be applied in two ways: first,
using wrapping and second, Unequally Spaced Fast Fourier Transforms (USFFT). In this paper, wrapping method
is employed to feature extraction of PQ disturbances because it is faster than the USFFT method. The second
generation of CT software package is used to obtain the curvelet coefficient.
Procedure to obtain curvelet coefficients of PQDs are given below:
1. Take PQDs signal and convert into 2D image.
2. Apply 2D FFT and obtain Fourier samples ],
.
3. For each scale and angle , interpolate ] to get sampled values’ ], from the
product.
4. Multiply the interpolated function with parabolic window
localizing near the parallelogram with
orientation , and get ]
].
5. Apply the 2-D IFFT to each to obtain the discrete curvelet coefficients’ .
2.1.1 Curvelet PQDs decomposition and coefficients
Initially, the PQDs signals are converted into 2D image and then, all PQDs images are decomposed by using fast
discrete curvelet transform (FDCT) with 6 scales and 64 orientation parameters are given as C {1, 1}; C {1, 2}; C
{1, 3}; C {1, 4}; C {1, 5} and C {1, 6}. Given the decomposition scale 6, the curvelet coefficients C {1, 1} is the
low frequency coefficients and the other coefficients C {1, 2}; C {1, 3}; C {1, 4}; C {1,5} and C {1,6} are the
high frequency coefficients. Transformation of PQD signals into curvelet coefficients are shown in the Fig. 1 and
all detail coefficients are shown in Table 1. The PQD signals are also decomposed 4 scales and 16 orientations.
Table 1. Details of curvelet coefficients.
Frequency
band
Scale
coefficients
Dimension of coefficients
1st
C{1,1}
2121
2nd
C{1,2}
4(17, 4(15), 4(2115), 4(2117)
3rd
C{1,3}
4(3321), 8(3122), 6(3121), 6(2231), 4(2131), 4(2133)
4th
C{1,4}
4(6642), 8(6343), 4(6342), 4(4263), 8(4363), 4(4263)
5th
C{1,5}
4(12942), 24(12643), 4(12642), 24(43126), 4(42129), 4(42126)
6th
C{1,6}
501501
2.2 Locality preserving projection
Locality preserving projection [42] is a graph-based dimensionality reduction method and has been
successfully used in many practical applications such as face recognition, fault detection and classification [50-
52]. A comparative study was presented between LPP and PCA methods by [53]. The study shows that LPP
method performed better than PCA. LPP dimensional reduction technique has local in nature rather than global, it
maintains the intrinsic information of high-dimensional data in a transformation of high-to-low dimensional space,
and it helps to avoid over fitting of training samples. The dimensionality reduction algorithm for curvelet
coefficients as follows:
Figure 1. Conversion of PQD signal into curvelet coefficients.
Given N-dimensional curvelet coefficients of PQDs ,.. be a vector representing the
curvelet coefficient of PQDs image, where is the dimension of curvelet coefficient. Find a transformation matrix
that maps these curvelet coefficients to set of points , such that is
corresponds to , where The projection matrix can find using the following algorithm:
1. Constructing the adjacency matrix: consider is training curvelet coefficients of PQDs and & are
closest edge nodes of curvelet coefficients.
Nodes and are connected by and edge if where is neighborhood and
represents Euclidean norm.
nearest neighborhood . Nodes and are connected by an edge if is among nearest
neighborhood of or is among nearest neighbors of
2. Choosing the weights: the weighting matrix can be calculated as flows.
Heat kernel weighting
, where is suitable constant.
Determined: if and only if vertices are connected by and edge.
3. To find projection matrix : Compute the eigenvector and eigenvalues for the generalized eigenvector
problem using the relation where is the diagonal matrix whose entries are column
sums of , and is Laplacian matrix. Let the column vectors and their
corresponding eigenvalues are Thus the projection matrix can be modelled as
The objective function can be minimized as
.
Now, the objective function can be optimized as
Finally, the equivalent optimization problem is:
Subject to (2)
2.3 Support vector machine
Support vector machines are a set of algorithms for regression classification, transduction, and detection of data,
and introduced by V. Vapnik. The basic SVM is not enough for a multiclass problem. Many multiclass support
vector machines have been developed such as pairwise, all-at-once (AAO), one-against-all (OAA) and error-
correcting output code (ECOC) support vector machines [54-58].
In [46], the authors used support vector machine (SVM) classifier directly on the curvelet decomposed palm-
print. In the domain of pattern recognition, SVM is an ideal nonlinear classification tool nowadays. SVM was
investigated against multilayer perceptron (MLP) neural networks (NN) and classification and regression trees
(CART) by Shao at el, [59]. The comparative results show that SVM has greater generalization capabilities. SVM
is also simple and easy to implement [60, 61].
In this paper, compact multiclass support vector machines (CMSVM) [62] are used for classification of PQ
disturbances. The proposed classifier does not require the full SVM solution as a starting point and does not need
to address the pre-processing of data. Let be training sample points: , where each
is an -dimensional feature vector and is the corresponding class label of Consider
semiparametric support vector machine , and can be written as
where is centroid and the
decision function , where .
Subject to (3)
Where are positive slack variables, is the tuning parameter used to balance the margin and the training error,
and Equation (3) can be optimized by using an iterated weighted least squares (IWLS)
procedure. The following algorithm is used for classification of PQ disturbances.
1. Choosing the first centroid the mean value of the training samples.
2. Output and .
3. Initialize and build the following matrices ,
, ,
4. Obtain and as
5. Compute outputs
and errors
6. Update values:
7. Evaluate the iterated weighted least squares stopping condition
. If satisfies the IWLS condition, then go to next step 8 otherwise go to step 3.
8. Compute multiclass outputs by voting, and identify which training samples are support vectors at least in
one of the binary classifiers
. Take at random to build the set of candidate
and
compute the following matrices and vectors:
a. ,
9. Evaluate the error descent produced by every candidate pattern:
10. Choose as new centroid the pattern with the largest associated error descent and add it to the base.
Repeat 8 if more than one centroid is to be added in every model updated stage.
11. Proceed to Step 3 to update the weights for the new semiparametric model defined by .
3. Proposed method for classification of PQDs
PQDs classification procedure based on curvelet transform, LPP and multi-class support vector machine is given
below:
Objective: Classification of PQD signals based on CT, LPP and MSVM.
Input: PQDs signals.
Steps:
1. Input 1 dimensional (D) PQDs signal.
2. 1D PQDs signal converted into 2D PQD image.
3. Apply curvelet algorithm extract curvelet coefficients, detail description mention in subsection 2.1; the
curvelet coefficients can be extracted by fast discrete curvelet transform:
Where discrete curvelet transform, is scale,
indicates the angle; shows the parameter orientation. Reshape the coefficient matrixes of each 2D image
of PQ disturbance signal to a row vector. All the row vectors of training images form a new matrix, denoted
as the training sample feature space.
4. Perform LPP to reduce the dimension and obtain a more illustrative feature because the size of the row
vector of each 2D image of PQDs signal typically is very huge.
5. The proposed SVM is employed to classify the PQ disturbances. The feature vector passing through the
compact multiclass SVM mentioned in the above subsection 2.3. The kernel Gaussian function has been
used in all experiments. Optimization problem can be solved by IWLS method.
6. At the classification stage, a PQD sample is classified as class
.
7. Finally, classification is done using CT, LPP, and SVM.
4. Experiment
To validate the effectiveness of the proposed method eight types of single and combined PQD signal was utilized
for the classification. In this paper, performed experiments on two kinds of PQDs database, which contains about
2800 samples. These samples were generated using MATLAB simulation and PSCAD/EMTDC [63] software
and taken from the IEEE work-group P1159.3 [64].
The experimental setup is programmed using MATLAB R2016B to recognize the databases of PQDs and the
databases are described as flow:
a. Eight types of single and combined PQDs waveforms were generated using MATLAB based on PQDs models
by varying their parameters which are listed in Table 2. The sampling frequency of PQD is 12 kHz and the
fundamental frequency is 50 Hz, and the size of each signal is 10 cycles. These PQDs waveforms are shown
in Fig. 2A. The database 1 contains over 1500 samples and each signal has 180 samples. All data are labelled
based on their respective signal such as
1. Sag (C1);
2. Swell (C2);
3. Interruption (C3);
4. Transient (C4);
5. Flicker (C5);
6. Harmonics (C6);
7. Sag + Harmonics (C7);
8. Swell + Harmonics (C8).
b. Dataset 2 contains about 1300 PQD training samples and each signal has 160 samples. Some of these PQ
disturbances were generated using MATALB. The sampling frequency of PQD signal is 25 kHz and the
fundamental frequency is 50 Hz, and size of each signal is one cycle. Some PQDs samples are generated
PSCAD EMTDC [63], and real data are taken from the IEEE work-group P1159.3 [64] and waveforms are
shown in Fig. 2B.
Initially, each PQD signal is converted into 2D PQD image finally the image data is transformed time domain
to the frequency domain by using FDCT, and six level FDCT decomposition coefficients are set out in Table 1
and Fig. 1. In this experiment, CT coefficients are divided into six frequency components. The lowest frequency
scale 1 contains image information. The highest frequency scale 6 contains noise information and remaining scales
Table 2. PQD models and its parameters.
2, 3, 4, 5 contains edge information. This procedure is applied in all PQD training samples and the test signal, and
the overall transformation procedure is illustrated in Fig. 3. In this paper, first, three frequency bands are used as
curvelet coefficient. CT coefficients directly use as feature vector into the classifier is very hard. CT coefficients
matrix of each PQD image is reshaped to a row vector and reduce, dimensionally by LPP method, and obtain a
more illustrative feature. In the classification stage, MSVM classification algorithms have been performed for
classification of PQDs. The kernel Gaussian function has been used in all experiments and can be written as
PQDs
Label
Equations
Parameter constrains
Sag
C1
Swell
C2
Interruption
C3
Transient
C4
Flicker
C5
Harmonic
C6
Sag+
harmonic
C7
Swell+
harmonic
C8
Where is width of the kernel and can be determined by ten-fold cross validation.
In this experiments values of are taken .
Figure 2. (A) Synthetic data, waveforms of PQDs synthetic signals (left side without noise and right side with noise 30 dB). (B) Real data, (a)
normal signal, (b) sag, (c) swell, (d) oscillation, (e) interruption, (f) impulsive.
In order to evaluate the efficacy of the proposed algorithm for classification of PQ disturbances were tested using
both data set 1 and 2. The flowchart of the proposed classification steps are shown in Fig. 4. In addition, the
proposed method was also tested under noisy environment and noise level from 20 dB to 50 dB.
To enhance the classification accuracy, the classification procedure is divided into testing samples and
training samples. The total number of data of the dataset1 and dataset2, 30 percent of the dataset1 and 25 percent
of the dataset2 is taken as the tasting samples and the remaining data is treated as training samples.
The proposed classification algorithm was implemented in MATLAB R2016B, Desktop computer Lenovo
i7, 3.8 GHz CPU, and 16 GB RAM.
To evaluate the computation time of the proposed method. Different values of the sampling frequency of PQDs
signal were further investigated. The computational time is depending on the sampling frequency of PQDs signal.
The total run time of the proposed classification method is , which means that
Training samples
2D
image
CT
coefficient
s
DIMENSION REDUCTION
(Locality preserving projection)
DIMENSION REDUCTION
(Locality preserving projection)
Test sample
(Flicker)
CLASSIFICATION STAGE
(Multi-class support vector machine)
LABELLED EACH CLASS
(C1, C2, C3, C4, C5, C6, C7, C8)
Amplitude (pu)
Time (s)
(A) Synthetic data
-1
0
1
a
-1
0
1
b
-1
0
1
c
-1
0
1
d
-1
0
1
e
0
200
400
600
-2
0
1
f
Amplitude (pu)
Sample points
(B) Real data
Figure 3. Conversion of PQD signal into 2D image, CT decomposition and Overall classification flow of single and combined PQDs.
FDCT+LPP+CMSVM algorithm can be used for real-time monitoring of PQDs. The experimental results are
reviewed in the next section.
5. Results and discussion
The proposed method was validated utilizing over 2800 samples of PQDs even under noisy environment. The
results of the proposed classification algorithm were obtained about 99.92 % and 99.8 % without and with noise
respectively. The experimental results are marked in Table 3. In Table 3, the diagonal of the confusion matrix
represent the classified PQD events and other members represents misclassified PQD events. The experimental
results show that the proposed algorithm is acceptable for classification of PQDs. In addition, the selected number
of feature reduced by LPP algorithm and the classification error is illustrated in Fig. 5.
Table 3: Confusion matrix and Classification accuracy (CA) % of PQDs.
Classes
C1
C2
C3
C4
C5
C6
C7
C8
CA (%) of Synthetic data
CA (%) of
Real data
0 dB
30 dB
40 dB
50 dB
C1
345
0
0
0
0
0
0
0
100
100
100
100
100
C2
0
357
0
0
0
0
0
1
100
100
99.7
100
100
C3
0
0
347
0
0
0
0
0
100
100
100
99.7
100
C4
0
0
0
340
0
0
0
0
99.7
99.7
99.8
99
99.6
C5
0
0
0
1
338
0
0
0
100
99.8
100
100
99.8
C6
0
0
0
0
1
355
0
0
100
100
99
100
-
C7
0
0
0
0
0
0
350
0
100
100
100
100
-
C8
0
0
0
0
0
0
0
344
99.71
99.7
99.5
99.4
-
Accuracy
345
357
347
339
337
355
350
343
99.92
99.9
99.75
99.8
99.8
The proposed PQDs classification algorithm is compared with recently reported methods such as [7], [18]
and [19], and comparative results are shown in Table 4. In [7], wavelet transform and support vector machines
(SVM) were investigated utilizing 9 PQD events and 900 synthetic data samples. In [18], hybrid feature selection
method was presented such as variational mode decomposition (VMD) and S-transform (ST). Nine types of PQDs
LABELLED EACH CLASS
(C1, C2, C3, C4, C5, C6, C7, C8)
CLASSIFICATION STAGE
(Multiclass Support Vector
Machine)
Disturbance?
FEATURE EXTRACTION STAGE
Locality Preserving Projection
(Dimension Reduction of
Curvelet Coefficients)
Curvelet Coefficients
Input PQDs signal
Convert PQDs Signal into 2D Image
Yes
No
Figure 4. Overall classification flowchart of PQDs.
were classified using SVM classifier. We observed that CT can be implemented different types of non-stationary
multi-combined PQDs. The performance show that LPP-CMSVM can be used in a wider range of datasets.
Future work: The proposed method can be used in real-time power quality disturbances monitoring and can also
be used in other applications such as biomedical signal classification.
Figure 5. Classification error and number of feature reduced by LPP.
Table 4. Performance comparison of classification technique’s results (%) under
different noise level.
Classes
VMD+ST+SVM
[18]
DRST+DAG+SVM
[19]
WT+SVM [7]
Proposed Method
40 dB
50 dB
0 dB
20 dB
40 dB
50 dB
30 dB
40 dB
50 dB
C1
98
98
99.5
99
100
100
100
100
100
C2
100
99
99
98.5
100
100
100
99.7
100
C3
100
100
97
92
100
100
100
100
99.7
C4
99
100
100
97.5
97
100
99.7
99.8
99
C5
97
99
99.5
97
98
95
99.8
100
100
C6
99
100
100
99.5
90
93
100
99
100
C7
100
100
100
99.5
100
100
100
100
100
C8
97
98
99.5
97
98
98.4
99.7
99.5
99.4
Average
98.75
99.25
99.3
97.5
98
98.4
99.9
99.75
99.8
6. Conclusion
This paper presents a new and an efficient approach for feature extraction of PQ disturbances based on discrete
fast curvelet transform and locality preserving projection. Multi-class support vector machine was offered for
automatic classification of PQDs. In the feature selection stage, LPP method was used to reduce the computational
efforts and decreased storage requirements. Eight types of single and combined PQ disturbances (sag, swell,
interruption, flicker, transient, harmonics, swell with harmonics and sag with harmonics) were investigated.
Performance of the proposed method has been tested using many real and synthetic PQ disturbance data sets and
compared with various recently reported methods. The proposed method can be easily implemented in real-time
power quality monitoring. The main contribution to the power engineering society is that improved power quality
disturbances classification accuracy using curvelet transform, LPP, and SVM. In addition, the CT+LPP+CMSVM
classifier is the best choice for classification of a noisy PQDs signal.
Acknowledgements
This work is supported by the National Natural Science Foundation of China [grant number 61301138], [grant number
51505195]; and the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institute China.
Disclosure statement
The authors declare no conflict of interest.
0
10
20
30
40
50
60
70
0246810
Classification error (%)
Number of samples CT+LPP
Author Contributions
All authors have contributed significantly equal part to the experiments, simulation, theoretical analysis, data collection,
manuscript preparation and proof reading.
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