# An Arc Fault Detection Method Based on Multidictionary Learning

Authors: Na Qu; Jianhui Wang; Jinhai Liu; Zhi Wang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper uses the dictionary learning of sparse representation algorithm to detect the arc fault. Six kinds of characteristics,that is, the normalized amplitudes of 0Hz, 50Hz, 100Hz, 150Hz, 200Hz, and 250Hz in the current amplitude spectrum, are used as inputs. The output is normal work or arc fault. Increasing the number of training samples can improve the accuracy of the tests. But if the training samples are too many, it is difficult to be expressed by single dictionary. This paper designs a multidictionary learning method to solve the problem. Firstly,n training samples are selected to forms overcomplete dictionaries. Then a dictionary library consisting ofs dictionaries is constructed. Secondly,t(t≤s) dictionaries are randomly selected from the dictionary library to judge the test results, respectively. Finally, the final detest result is obtained through the maximum number of votes,that is, the modality with the most votes is the detest result. Simulation results show that the accuracy of detection can be improved.


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## Body

## 1. Introduction

Electrical fire generally refers to high temperature, arc, and spark ignition because of the fault of electrical wiring, electrical equipment, and power supply equipment. Statistical analysis shows that the proportion of electric fires caused by arc fault is up to 50%. Therefore, the detection of arc fault is meaningful to ensure electrical safety.UL1699 standard for arc fault protection is proposed in 1999. A new apparatus named Arc Fault Circuit Interrupter (AFCI) is introduced. AFCI is a circuit breaker to detect arc fault and cut off the circuit so as to reduce the damage. In recent years, many scholars have begun to study the arc fault. Generally, current is selected as the detection signal of arc fault. Most of the researches on arc fault current detection are based on the zero rest. The wavelet transform method is always used to detect arc fault [1–8]. In addition to wavelet transform, a few other methods have been studied to detect the arc fault. The ABCD method is to establish the relationship between input and output current through a transfer matrix. The coefficients of ABCD matrix depend on conductor parameters, arc fault, and load types. It is mainly applied to aircraft arc fault detection [9]. The arc fault current detection algorithm based on the theory of quantum probability model is mainly applied to the photovoltaic power system [10]. The method of Chirp-zeta transform is used to analyze the low frequency harmonic of the current signal [11].Sparse representation classification has recently attracted the interests of many researchers and has been successfully applied to several problems such as robust face recognition, visual tracking, and transient acoustic signal classification. In [12], a tracking model of bidirectional sparse representation is established. The model achieves uniform convergence throughL2 norm constraint and the sparse correlation coefficient matrix. In [13], the models and the algorithms of comprehensive dictionary, analytic dictionary, and blind dictionary are introduced. The typical applications of dictionary learning methods are illustrated. In [14], a new SAR target recognition method, which is based on the discriminative dictionary learning and joint dynamic sparse representation model, is proposed. In [15], the method of nonnegative matrix decomposition (NMF) concretely embodies the concept of sparse representation and provides a complete Matlab code and extension method. In [16], a discriminative dictionary, which is learned by K-SVD and LC-KSVD algorithm, is presented. In [17], two dictionary learning methods based on Fisher discrimination dictionary learning (FDDL) are proposed. In [18], a multimodal task-driven dictionary learning algorithm under the joint sparsity constraint (prior) to enforce collaborations among multiple homogeneous/heterogeneous sources of information is proposed. In [19], a sparse representation model based on joint kernel is proposed to complete key-frame extraction for human motion capture data. In [20], the problem of transmit beam pattern matching design for multiple-input multiple-output (MIMO) radar is addressed within a sparse representation framework.In this paper, the multidictionary learning method of sparse representation is studied and applied to arc fault detection to improve the accuracy.


## 2. Multidictionary Learning Method

Sufficient training samples of theith modality are given [21]. The test sampley from the same modality will approximately lie in the linear span of the training samples associated with ith object.We define a new matrix D=[D1D2D3⋯Dn] for the entire training set as the concatenation of alln object modalities. Then, the linear representation ofy can be rewritten in terms of all training samples as follows:(1)y=Dx0x 0 = [ 0 , ⋯ , 0 , α 1 ⋯ α m , ⋯ 0 , ⋯ 0 ] T is a coefficient vector whose entries are zero except those associated with the ith modality. As the entries of the vector x0 encode the identity of the test sampley, it is tempting to attempt to obtain it by solving the linear system of equation Dx=y. There may be an infinite number of solutions to the underdetermined linear equations. In order to reduce the range of selection to a satisfactory solution, it is necessary to increase condition. A function J(x) is introduced to rationally evaluate the candidate solution of x, and its value is expected to be as small as possible [22], usually in the form of Lp norm, as follows:(2)x^=argminxxps.t.Dx=y.Whenp≤1, sparse solution  x^ can be obtained. Then, a sparse representation model is formed, and D is the learning dictionary. In practical applications, formula (2) is usually written in the form of formula (3) due to the existence of errors.(3)x^=argminxxps.t.y-Dx22≤εε is the error threshold; y=[y1,⋯,yn]T is a test vector.The overcomplete dictionary is composed of training samples. Increasing the number of training samples can improve the accuracy of the tests. But if the number of training samples is too much, it is difficult to be expressed by single dictionary. This paper designs a multidictionary learning method to solve the problem. The multidictionary learning scheme is shown in Figure1. Firstly,n training samples are selected to forms overcomplete dictionaries. Then a dictionary library consisting ofs dictionaries is constructed. Secondly,t (t≤s) dictionaries are randomly selected from the dictionary library to judge the test results, respectively. Finally, the final detest result is obtained through the maximum number of votes;that is, the modality with the most votes is the detest result.Figure 1
Multidictionary learning scheme.The minimum residual classification is used to determine the detest result of single-dictionary learning.x^ is the projection coefficient of the test sample y on the dictionary D of the ith modality. When y belongs to the ith modality, y^i=Dx^ approximates y. The smaller distance between y^i and y (i.e., residual) is, the greater possibility that y^i belongs to ith modality is, as shown in the following formula: (4)riy=miniy-Dx^2


## 3. Sample Data Acquisition

Sample data are obtained by an arc fault experimental platform. The experimental platform consists of power supply, arc fault generator, data acquisition device, loads, switch, and so on [23]. Power supply is “220V, 50Hz” AC power. Arc fault generator is mainly composed of fixing carbon electrode and moving copper electrode. The data acquisition device consists of a sampling resistance and an oscilloscope. TDS1001C-SC of Tektronix oscilloscope and TPP0101 10X of voltage probe are selected. Sample interval is 4×10−4s. The current waveform of the circuit is obtained by the sampling resistance method. The selection of sampling resistance is different depending on the load. The loads that are commonly used in family and woke, such as resistive load, resistive and inductive load, DC motor load, series motor load, eddy current load, and switching power load, are chosen.“220V, 100W” incandescent lamp is selected as resistive load. “220V, 100W” incandescent lamp and 22mH inductors are selected as resistive and inductive load. The DC motor load selects the high grade position of electric blower. The series motor load selects an electric drill. The eddy current load selects the induction cooker. The switch power load selects the computer. The current data of normal work and arc fault are collected under the different types of loads conditions. The current waveform is shown in Figures 2–7. The sampling resistance is 100 ohms when the loads are resistive load and resistive and inductive load. The unit of ordinate is 0.01A in Figures 2 and 3. The sampling resistance is 50 ohms when the loads are DC motor load, series motor load, and switching power load. The unit of ordinate is 0.02A in Figures 4, 5, and 7. The sampling resistance is 1 ohm when the load is eddy current load. The unit of ordinate is 1A in Figure 6.Figure 2
Current waveform of resistive load: (a) normal work; (b) arc fault.Figure 3
Current waveform of resistive and inductive load: (a) normal work; (b) arc fault.Figure 4
Current waveform of DC motor load: (a) normal work; (b) arc fault.Figure 5
Current waveform of series motor load: (a) normal work; (b) arc fault.Figure 6
Current waveform of eddy current load: (a) normal work; (b) arc fault.Figure 7
Current waveform of switching power load: (a) normal work; (b) arc fault.In the current waveform of arc fault, there is zero rest (i.e., nearly flat zero current segments around the normal zero crossing of the waveform) under the resistive load, the DC motor load, and series motor load conditions. When the load is the computer, the normal current waveform also appears to be zero rest. The zero rest of current waveform is not obvious when the arc fault occurs under the resistive and inductive load condition. Therefore, the arc fault cannot be detected accurately by the zero rest.The current amplitude spectrum is obtained by Fourier transform. The current signal is decomposed into the Fourier series of trigonometric functions, which is shown in the following formula:(5)ft=a0+∑n=1∞ancos⁡nω1t+bnsin⁡nω1tIn formula (5), a0=1/T∫t0t0+T1f(t)dt, an=2/T1∫t0t0+T1ftcos⁡(nω1t)dt, and bn=2/T1∫t0t0+T1ftsin⁡(nω1t)dt, n=1,2,3,⋯.Formula (6) can be obtained by formula (5).(6)ft=d0+∑n=1∞dnsin⁡nω1t+θnIn formula (5), d0=a0, dn=an2+bn2, and θn=arctan⁡an/bn. The x-coordinate of the amplitude spectrum is made up of nω1. The y-coordinate of the amplitude spectrum is made up of dn.One set of current amplitude spectrums and standardized values is shown in the appendix for Table1. Compared with the spectrum of normal work, that of arc fault has different distribution characteristics.Table 1
Current amplitude spectrums and standardized values.
Work state Load/amplitude spectrum 0Hz 50Hz 100Hz 150Hz 200Hz 250Hz Normal work Resistive load 224 /0.008 28020/ 1 44/ 0.002 1011/0.036 21/ 0.001 120/0.004 Resistive and inductive load 5.6 /0.03 185.4/1 0.2/0.001 3.1/0.017 0.3/0.002 1.3/0.007 DC motor load 532/ 0.003 176421/1 122/ 0.001 1757/0.01 102/0.001 1697/0.01 Series motor load 462/0.003 145375/1 293/ 0.002 16870/0.116 176/0.001 1391/0.01 Eddy current load 235/ 0.014 16333/1 7/0 937/0.057 32/0.002 150/0.01 Switching power load 354/ 0.02 17922/1 63/0.003 12643/0.705 53/0.003 6265/0.35  Arc fault Resistive load 2555/ 0.116 22031 /1 1778/0.081 9448/0.429 2210 /0.1 2252/0.102 Resistive and inductive load 4.7/0.02 234.5/1 2.4/0.01 81.6/0.348 17.1/0.073 47.1/ 0.2 DC motor load 608/0.004 173326/1 191 /0.001 6631/0.038 295/0.002 4770/0.028 Series motor load 1004/0.013 76797/1 9059/0.118 31593/0.411 5534/0.072 5397/0.07 Eddy current load 392/ 0.639 388/0.633 183/0.299 613 /1 116/0.189 287/0.468 Switching power load 538/0.191 2821/1 246/0.087 649/0.23 1576/0.559 636/0.225


## 4. Arc Fault Detection

### 4.1. Single-Dictionary Learning

The dictionary matrix consists ofn modalities of training samples,that is, D=[D1D2D3⋯Dn]. The number of rows in the matrix D is the number of sample characteristics [23]. In this paper, six kinds of characteristics,that is, the normalized amplitudes of 0Hz, 50Hz, 100Hz, 150Hz, 200Hz, and 250Hz in the current amplitude spectrum, are used. The values are between 0 and 1. The two modalities of training samples are normal work and arc fault;that is,n=2. The number of columns is the total number of two modalities of training samples;that is, D= [D1D2].


### 4.2. Multidictionary Selection

Ten sets of dictionaries are selected to form the dictionary library. In order to give consideration to accuracy and instantaneity, five sets of dictionaries are selected randomly from the dictionary library.L3/4 norm regularization algorithm with good sparsity and accuracy is selected to identify the classification [24], as shown in the following formula:(7)x^L3/4=argminxx3/4s.t.y-Dx22≤εFormula (8) can be obtained by formula (7).(8)x^L3/4=argminx1n∑i=1nyi-Dix2+λ∑i=1pxi3/4y i represents theith modality test sample; λ is the regularization parameter.P is the number of samples of each modality. In the later application,P=18. xi is theith training sample of each modality.


### 4.3. Test and Simulation

Five sets of dictionaries are selected randomly from the dictionary library, that is,D1, D2, D3, D4, and D5. The test sample comes from the arc fault spectrum of resistive and inductive load;  y=[0.02;1;0.016;0.27;0.081;0.158]. The detection result is shown in Figure 8 when the single dictionary is D1. In Figure 8(a), the horizontal axis 1-18 is the training samples number of normal work. The horizontal axis 19-36 is the training samples number of the arc fault. The training samples come from current amplitude spectrum under the condition of resistive load, resistive and inductive load, DC motor load, series motor load, eddy current load, and switching power load. The fault detection result is determined by the residual comparison and projected coefficients comparison of two modalities. When the residual is smaller and projected coefficients are larger, the corresponding modality is the detection result. The projection coefficients of the horizontal axis 19-36 are larger than those of the horizontal axis 1-18. In Figure 8(b), the residual of the second modality is smaller, so that it is judged to be arc fault and the result is correct. The detection result is shown in Figures 9, 10, 11, and 12, when the single dictionary is D2, D3, D4, and D5. All the test results of the five sets are arc fault, so the result of final judgment is arc fault, which is consistent with the known result, and the result is correct.Figure 8
Test ofD1 learning: (a) projection coefficient; (b) residual classification.Figure 9
Test ofD2 learning: (a) projection coefficient; (b) residual classification.Figure 10
Test ofD3 learning: (a) projection coefficient; (b) residual classification.Figure 11
Test ofD4 learning: (a) projection coefficient; (b) residual classification.Figure 12
Test ofD5 learning: (a) projection coefficient; (b) residual classification.Seventy-two sets of normal work and arc fault data (including resistive load, resistive and inductive load, DC motor load, series motor load, eddy current load, and switching power load) are selected as the test sample. Arc fault is detected by single-dictionary and multidictionary learning, respectively. When single-dictionary learning method is used, seventy sets of results are correct and two sets of results are error. When multidictionary learning method is used, seventy-two sets of results are all correct. Therefore, the arc fault detection based on multidictionary learning method can improve the accuracy.Neural network algorithm is the most common method in arc fault detection. Probabilistic neural network (PNN) is a parallel algorithm based on Bayes minimum risk criterion. The proposed algorithm is compared with the PNN by using 72 groups of data. The accuracy of the algorithm is 100%, and that of PNN algorithm is 95.83% [25]. The accuracy of proposed algorithm is higher than that of PNN algorithm.


## 4.1. Single-Dictionary Learning

The dictionary matrix consists ofn modalities of training samples,that is, D=[D1D2D3⋯Dn]. The number of rows in the matrix D is the number of sample characteristics [23]. In this paper, six kinds of characteristics,that is, the normalized amplitudes of 0Hz, 50Hz, 100Hz, 150Hz, 200Hz, and 250Hz in the current amplitude spectrum, are used. The values are between 0 and 1. The two modalities of training samples are normal work and arc fault;that is,n=2. The number of columns is the total number of two modalities of training samples;that is, D= [D1D2].


## 4.2. Multidictionary Selection

Ten sets of dictionaries are selected to form the dictionary library. In order to give consideration to accuracy and instantaneity, five sets of dictionaries are selected randomly from the dictionary library.L3/4 norm regularization algorithm with good sparsity and accuracy is selected to identify the classification [24], as shown in the following formula:(7)x^L3/4=argminxx3/4s.t.y-Dx22≤εFormula (8) can be obtained by formula (7).(8)x^L3/4=argminx1n∑i=1nyi-Dix2+λ∑i=1pxi3/4y i represents theith modality test sample; λ is the regularization parameter.P is the number of samples of each modality. In the later application,P=18. xi is theith training sample of each modality.


## 4.3. Test and Simulation

Five sets of dictionaries are selected randomly from the dictionary library, that is,D1, D2, D3, D4, and D5. The test sample comes from the arc fault spectrum of resistive and inductive load;  y=[0.02;1;0.016;0.27;0.081;0.158]. The detection result is shown in Figure 8 when the single dictionary is D1. In Figure 8(a), the horizontal axis 1-18 is the training samples number of normal work. The horizontal axis 19-36 is the training samples number of the arc fault. The training samples come from current amplitude spectrum under the condition of resistive load, resistive and inductive load, DC motor load, series motor load, eddy current load, and switching power load. The fault detection result is determined by the residual comparison and projected coefficients comparison of two modalities. When the residual is smaller and projected coefficients are larger, the corresponding modality is the detection result. The projection coefficients of the horizontal axis 19-36 are larger than those of the horizontal axis 1-18. In Figure 8(b), the residual of the second modality is smaller, so that it is judged to be arc fault and the result is correct. The detection result is shown in Figures 9, 10, 11, and 12, when the single dictionary is D2, D3, D4, and D5. All the test results of the five sets are arc fault, so the result of final judgment is arc fault, which is consistent with the known result, and the result is correct.Figure 8
Test ofD1 learning: (a) projection coefficient; (b) residual classification.Figure 9
Test ofD2 learning: (a) projection coefficient; (b) residual classification.Figure 10
Test ofD3 learning: (a) projection coefficient; (b) residual classification.Figure 11
Test ofD4 learning: (a) projection coefficient; (b) residual classification.Figure 12
Test ofD5 learning: (a) projection coefficient; (b) residual classification.Seventy-two sets of normal work and arc fault data (including resistive load, resistive and inductive load, DC motor load, series motor load, eddy current load, and switching power load) are selected as the test sample. Arc fault is detected by single-dictionary and multidictionary learning, respectively. When single-dictionary learning method is used, seventy sets of results are correct and two sets of results are error. When multidictionary learning method is used, seventy-two sets of results are all correct. Therefore, the arc fault detection based on multidictionary learning method can improve the accuracy.Neural network algorithm is the most common method in arc fault detection. Probabilistic neural network (PNN) is a parallel algorithm based on Bayes minimum risk criterion. The proposed algorithm is compared with the PNN by using 72 groups of data. The accuracy of the algorithm is 100%, and that of PNN algorithm is 95.83% [25]. The accuracy of proposed algorithm is higher than that of PNN algorithm.


## 5. Conclusion

In this paper, we have proposed a new model, which was tailored to detect the arc fault. The proposed model uses the multidictionary learning of sparse representation algorithm. Compared with the algorithm such as neural networks, sparse representation algorithm does not need a lot of repeated trainings. The test data are classified by sparse solution, which can improve the real-time performance and application of arc fault detection. Compared with the single-dictionary learning of sparse representation algorithm, multidictionary learning can improve the accuracy of arc fault detection by increasing training samples.



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