# Fault Detection of the Power System Based on the Chaotic Neural Network and Wavelet Transform

Authors: Zuoxun Wang; Liqiang Xu
Journal: Complexity
 (2020)

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## Abstract

The safety and stability of the power supply system are affected by some faults that often occur in power system. To solve this problem, a criterion algorithm based on the chaotic neural network (CNN) and a fault detection algorithm based on discrete wavelet transform (DWT) are proposed in this paper. MATLAB/Simulink is used to establish the system model to output fault signals and travelling wave signals. Db4 wavelet decomposes the travelling wave signals into detail signals and approximate signals, and these signals are combined with the two-terminal travelling wave location method to achieve fault location. And the wavelet detail coefficients are extracted to input to the proposed chaotic neural network. The results show that the criterion algorithm can effectively determine whether there are faults in the power system, the fault detection algorithm has the capabilities of locating the system faults accurately, and both algorithms are not affected by fault type, fault location, fault initial angle, and transition resistance.


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## Body

## 1. Introduction

The rapid development of the power system has led to a gradual increase in voltage levels and increasingly complex transmission lines, and it is more and more difficult to detect power system faults. However, the normal operation of the enterprise and people’s production and life are seriously affected by the fault of the power system. Therefore, it is particularly critical to accurately locate the fault point of the power system.Recently, many scholars in related fields have conducted in-depth and extensive research on power system fault diagnosis. With the further development of artificial intelligence, especially machine learning and data mining, many theories and methods are provided for fault diagnosis. The fault diagnosis methods proposed at present can be summarized as frequency domain feature detection methods and adaptive detection methods. The frequency domain feature detection methods mainly include Fourier transform, wavelet transform, and Hilbert-Huang transform. The adaptive detection methods mainly include neural networks and expert systems. Although these theoretical studies have achieved certain results, there are still some limitations. In [1], single-ended travelling wave fault location method was used to transmission line fault location, but it is difficult to identify the wave head and the positioning accuracy was not high. In [2], integrated empirical mode decomposition (EEMD) was used to optimize empirical mode decomposition (EMD). This method avoided modal aliasing, but introduced false components, resulting in low positioning accuracy. In [3], a fault phase selection method based on high-order multiresolution singular entropy of current fault components is proposed. This method is basically not affected by the fault type, fault location, and transition resistance, but the setting of optimal threshold is complicated.The wavelet transform is a further improvement and extension on the basis of the Fourier transform. The Fourier transform cannot express the time-frequency localization property of nonstationary signals, and the wavelet transform solves this problem. Wavelet transform not only has good overall waveform analysis ability but also has outstanding time-frequency localization analysis ability. It can effectively obtain the detailed information of the signal in the time and frequency domains and has the ability to determine the singular point of the signal and analyze the degree of signal distortion [4, 5]. In [6], wavelet transform was applied to fault detection of the fars power distribution system, which improved detection accuracy and achieved good results. Using wavelet transform to analyze signals, the key process is the extraction of approximate features and detailed features. We need to determine the optimal decomposition level, separate the feature components as far as possible, and keep the error apart from the extracted features. The current many methods need either manual setting threshold control or results testing with extracted trend by wavelet transform, which increases the difficulty of extracting feature components and reduces the accuracy of fault detection.The neural network simulates the neuron network of the human brain, and the values of the neurons in the input layer are mapped to the output layer, so that a certain implicit function relationship is established between the input and the output. At present, the multilayer perceptron neural network theory based on the BP algorithm is most widely used in fault diagnosis. In [7], the MLP neural network was adaptively updated using the BP algorithm to minimize tracking errors, which enhanced the performance of an MLP controller and improved the recognition accuracy. The neural network adopts a knowledge-based nonlinear processing method. Compared with the traditional diagnosis method, it is more flexible in the application of knowledge. At the same time, the neural network has a strong learning ability, and its fault tolerance is better in fault diagnosis, but there are still some defects in the neural network: (1) Neural network needs a large number of samples for training, and it is difficult to obtain sufficient samples in the field of power system fault diagnosis. (2) The neural network was easy to be trapped in local minimum.To address these problems, a fault detection algorithm based on wavelet transform and the chaotic neural network is proposed in this paper. The chaotic neural network is capable of overcoming the shortcoming of trapping into the local optimum. And it has good performance of fault tolerance and associative memory capabilities. The design parameters of the proposed algorithm are shown in Table1. The main contributions of this paper are as follows:(1)
A fault phase selection algorithm based on the chaotic neural network is proposed. The fault features extracted by wavelet are input to the chaotic neural network, and the fault type is determined by the corresponding relationship between input and output. And this algorithm is not affected by fault type, fault location, fault initial angle, and transition resistance.(2)
A two-terminal fault detection algorithm based on wavelet transform is proposed. The db4 wavelet is used to detect the travelling wave head to diagnose the fault. This algorithm has high detection accuracy and is not affected by fault type, fault location, fault initial angle, and transition resistance.Table 1
The design parameters of the proposed algorithm.
ItemsParameter valuesThe type of mother waveletDaubechies 4Wavelet decomposition layers4Wavelet center frequency0.7143 kHzNumber of neural network input layers1Number of neural network hidden layers1Number of neural network output layers1Input values of the neural networkWuis,k,i=α,β,γ,0The remainder of this paper is organized as follows: Section2 briefly introduces wavelet transform theory and Mallat algorithm. Section 3 discusses the modulus maximum theory, the two-terminal travelling wave location method, and phase-model transformation. Section 4 analyzes the chaotic neural network model. Section 5 summarizes the selection basis of the mother wavelet function. In Section 6, simulation is carried out to illustrate and prove the effectiveness and practicability of the proposed algorithm and compare it with the EMD algorithm. Finally, further discussion and conclusions are in Section 7.


## 2. Wavelet Transform

### 2.1. Wavelet Transform Theory

Wavelet transform is the process of convolving the signal with the wavelet basis function, and the signal is decomposed into different frequency bands and time periods [8]. The signal ft is expanded under the telescopic-translational signal ψa,bt, and the process of decomposing ft is called continuous wavelet transform (CWT), and the expression is(1)Wfa,b=f,ψa,b=a−1/2∫−∞+∞ftψ¯t−badt.In the above formula,ψa,bt is called the mother wavelet, a is the scaling parameter, and b is the translation parameter. The essence of wavelet transform is to filter the signal with different filters. In the high-frequency region, the high resolution is reflected in time, and in the low-frequency region, the high resolution is reflected in frequency [9, 10].Since discrete signals are often used in the actual project,ψa,bt is discretized into the following formula:(2)ψj,kt=a0−j/2ψt−ka0jb0a0j=a0−j/2ψa0−jt−kb0.Then, get the discrete wavelet transform (DWT):(3)Cj,k=f,ψj,k=a0−j/2∫−∞+∞ftψ¯a0−jt−kb0dt.Under certain conditions, the signalft can completely be characterized by discrete wavelet coefficient Cj,k.


### 2.2. Wavelet Mallat Algorithm

#### 2.2.1. Decomposition Algorithm

The spectrum offt is fω. However, in the actual project, the spectrum is limited. As long as j is selected large enough, ft can be expanded with Vj+1:(4)ft=∑ncj+1,nφj+1,lt.In the above formula,cj+1,n is the scale factor of scale space Vj+1, and cj+1,n=f,φj+1,n and φj+1,kt are the scale functions of scale space Vj+1.BecauseVj+1=Vj⊕Wj, that is, Vj⊥Wj, then(5)ft=∑ncj,kφj,kt+∑kdj,kΨj,kt.In formula (5), ∑ncj,kφj,kt is the low-frequency component of ft, which is the approximate part of ft, and ∑kdj,kΨj,kt is the high-frequency component of ft, which is the detail part of ft. And the low-frequency coefficient cj,k and the high-frequency coefficient dj,k can be obtained as follows:(6)cj,k=f,φj,k=∑ncj+1,nh¯j−2k,(7)dj,k=f,Ψj,k=∑ncj+1,ng¯j−2k.From the above formula, the coefficientcj,k and the dj,k can be obtained from the coefficient cj+1,k in the space Vj+1. The decomposition process is shown in Figure 1.Figure 1
Mallat decomposition process.


#### 2.2.2. Reconstruction Algorithm

The essence of the Mallat reconstruction algorithm is the inverse operation of the decomposition algorithm. By substituting equations (6) and (7) into equation (5), we get the following equation:(8)ft=∑kcj,k∑lhl−2kφj+1,lt+∑kdj,k∑lgl−2kφj+1,lt=∑l∑kcj,khl−2k+∑kdj,kgl−2kφj+1,lt.l⟶k, k⟶n, then, we get(9)cj+1,k=∑ncj,nhk−2n+∑ndj,ngk−2n.In formula (9), cj+1,k∈Vj+1, cj,k∈Vj, and dj,k∈Wj. Reconstruction is the process of calculating cj+1,k from the coefficient cj,k and dj,k. The reconstruction process is shown in Figure 2.Figure 2
Mallat reconstruction process.


## 2.1. Wavelet Transform Theory

Wavelet transform is the process of convolving the signal with the wavelet basis function, and the signal is decomposed into different frequency bands and time periods [8]. The signal ft is expanded under the telescopic-translational signal ψa,bt, and the process of decomposing ft is called continuous wavelet transform (CWT), and the expression is(1)Wfa,b=f,ψa,b=a−1/2∫−∞+∞ftψ¯t−badt.In the above formula,ψa,bt is called the mother wavelet, a is the scaling parameter, and b is the translation parameter. The essence of wavelet transform is to filter the signal with different filters. In the high-frequency region, the high resolution is reflected in time, and in the low-frequency region, the high resolution is reflected in frequency [9, 10].Since discrete signals are often used in the actual project,ψa,bt is discretized into the following formula:(2)ψj,kt=a0−j/2ψt−ka0jb0a0j=a0−j/2ψa0−jt−kb0.Then, get the discrete wavelet transform (DWT):(3)Cj,k=f,ψj,k=a0−j/2∫−∞+∞ftψ¯a0−jt−kb0dt.Under certain conditions, the signalft can completely be characterized by discrete wavelet coefficient Cj,k.


## 2.2. Wavelet Mallat Algorithm

### 2.2.1. Decomposition Algorithm

The spectrum offt is fω. However, in the actual project, the spectrum is limited. As long as j is selected large enough, ft can be expanded with Vj+1:(4)ft=∑ncj+1,nφj+1,lt.In the above formula,cj+1,n is the scale factor of scale space Vj+1, and cj+1,n=f,φj+1,n and φj+1,kt are the scale functions of scale space Vj+1.BecauseVj+1=Vj⊕Wj, that is, Vj⊥Wj, then(5)ft=∑ncj,kφj,kt+∑kdj,kΨj,kt.In formula (5), ∑ncj,kφj,kt is the low-frequency component of ft, which is the approximate part of ft, and ∑kdj,kΨj,kt is the high-frequency component of ft, which is the detail part of ft. And the low-frequency coefficient cj,k and the high-frequency coefficient dj,k can be obtained as follows:(6)cj,k=f,φj,k=∑ncj+1,nh¯j−2k,(7)dj,k=f,Ψj,k=∑ncj+1,ng¯j−2k.From the above formula, the coefficientcj,k and the dj,k can be obtained from the coefficient cj+1,k in the space Vj+1. The decomposition process is shown in Figure 1.Figure 1
Mallat decomposition process.


### 2.2.2. Reconstruction Algorithm

The essence of the Mallat reconstruction algorithm is the inverse operation of the decomposition algorithm. By substituting equations (6) and (7) into equation (5), we get the following equation:(8)ft=∑kcj,k∑lhl−2kφj+1,lt+∑kdj,k∑lgl−2kφj+1,lt=∑l∑kcj,khl−2k+∑kdj,kgl−2kφj+1,lt.l⟶k, k⟶n, then, we get(9)cj+1,k=∑ncj,nhk−2n+∑ndj,ngk−2n.In formula (9), cj+1,k∈Vj+1, cj,k∈Vj, and dj,k∈Wj. Reconstruction is the process of calculating cj+1,k from the coefficient cj,k and dj,k. The reconstruction process is shown in Figure 2.Figure 2
Mallat reconstruction process.


## 2.2.1. Decomposition Algorithm

The spectrum offt is fω. However, in the actual project, the spectrum is limited. As long as j is selected large enough, ft can be expanded with Vj+1:(4)ft=∑ncj+1,nφj+1,lt.In the above formula,cj+1,n is the scale factor of scale space Vj+1, and cj+1,n=f,φj+1,n and φj+1,kt are the scale functions of scale space Vj+1.BecauseVj+1=Vj⊕Wj, that is, Vj⊥Wj, then(5)ft=∑ncj,kφj,kt+∑kdj,kΨj,kt.In formula (5), ∑ncj,kφj,kt is the low-frequency component of ft, which is the approximate part of ft, and ∑kdj,kΨj,kt is the high-frequency component of ft, which is the detail part of ft. And the low-frequency coefficient cj,k and the high-frequency coefficient dj,k can be obtained as follows:(6)cj,k=f,φj,k=∑ncj+1,nh¯j−2k,(7)dj,k=f,Ψj,k=∑ncj+1,ng¯j−2k.From the above formula, the coefficientcj,k and the dj,k can be obtained from the coefficient cj+1,k in the space Vj+1. The decomposition process is shown in Figure 1.Figure 1
Mallat decomposition process.


## 2.2.2. Reconstruction Algorithm

The essence of the Mallat reconstruction algorithm is the inverse operation of the decomposition algorithm. By substituting equations (6) and (7) into equation (5), we get the following equation:(8)ft=∑kcj,k∑lhl−2kφj+1,lt+∑kdj,k∑lgl−2kφj+1,lt=∑l∑kcj,khl−2k+∑kdj,kgl−2kφj+1,lt.l⟶k, k⟶n, then, we get(9)cj+1,k=∑ncj,nhk−2n+∑ndj,ngk−2n.In formula (9), cj+1,k∈Vj+1, cj,k∈Vj, and dj,k∈Wj. Reconstruction is the process of calculating cj+1,k from the coefficient cj,k and dj,k. The reconstruction process is shown in Figure 2.Figure 2
Mallat reconstruction process.


## 3. Principle of Modulus Maximum Detection

### 3.1. Modulus Maximum Detection Theory

The original signal discontinuity or its differential discontinuity leads to the singularity of the signal. The essence of singularity detection is to identify the singular point of the signal and analyze the degree of signal distortion. According to the time-frequency localization property of wavelet transform, the signal singularity can be effectively analyzed [11–13].The wavelet transform with scales at point u is Wfs,u, if point u near the point u0 satisfies Wfs,u≤Wfs,u0, then the maximum point at scale s is u0, and the modulus maximum of the wavelet transform is Wfs,u0 [14, 15]. Signals at different scales will have different modulus maximum; therefore, the singularity of the signal can be analyzed by extracting the modulus maximum points in the signal. By constructing a smoothing function to smooth the signals of different scales, the first and second order differentials can be extracted to identify the singular points of the signal [16, 17].The smoothing functionθt satisfies ∫−∞+∞θtdt=1 and limt⟶±∞θt=0. By constructing the mother wavelet ψ1t=dθ/dt and ψ2t=d2θ/dt2 and introducing the function ξst=1/sξt/s, the wavelet transform of ft at scale s and point t under is defined as(10)Wf1s,t=f∗ψs1,Wf2s,t=f∗ψs2.Expand the above formula as follows:(11)Wf1s,t=<f,ψs1>=f∗sdθsdtt=sddtf∗θst,Wf2s,t=<f,ψs2>=f∗s2d2θsdt2t=s2d2dt2f∗θst.It can be seen from the above formulas that the wavelet transformWf1s,t can be obtained by the first order differential and the Wf2s,t can be obtained by the second order differential. The discontinuous point of ft corresponds to the modulus maximum point of Wf1s,t and the zero point of Wf2s,t [18, 19]. However, the electrical quantity of the nonfault point will be disturbed during the signal transmission, which will affect the accuracy of the zero-crossing detection, so the result of the zero-crossing detection is usually not a discontinuity point, but an inflection point of the signal, and the modulus maximum detection is more reliable than zero-crossing detection [20, 21]. After wavelet transform, the discontinuity point of the signal ft corresponds to the maximum point of Wf1s,t and the zero point of Wf2s,t. According to the definition of wavelet transform Wfs,t, the maximum point of Wfs,t reflects the position of the singular point of f∗θst [22]. Therefore, the singular point of the signal at different scales can be determined by detecting the maximum point of Wfs,t.


### 3.2. Two-Terminal Travelling Wave Location Principle

The schematic diagram of the two-terminal travelling wave location method is shown in Figure3. When a fault occurs at point F, the transient travelling wave propagates to both sides along the transmission line. At the same time, reflection and refraction occur at the point where the wave impedance is discontinuous. Assume that the time when the travelling wave head reaches the detection point M and the detection point N is tm1 and tn1, respectively, and after reflection, the time when the wave head reaches the detection point M and N is tm2 and tn2, respectively. The total length of the line is LMN, the distance between the fault point F and M is LMF, the distance between F and N is LNF, and the travelling wave propagation speed is v; then,(12)LMF=tm1v,LNF=tn1v,LMN=LMF+LNF.Figure 3
Schematic diagram of fault location.Then, the formula of two-terminal travelling wave location is obtained:(13)LMF=LMN+vtm1−tn12.The two-terminal travelling wave location method only needs to detect the initial wave head of the fault travelling wave and does not need to consider the reflection and refraction at the discontinuity point of wave impedance [23–25]. Therefore, the location algorithm is more simple. In this paper, the discrete wavelet transform is used to detect the transient travelling wave head, and the time tm1 propagated to the detection point M and tn1 propagated to the detection point N are used to locate the fault point.


### 3.3. Phase-Model Transformation

In the three-phase transmission line, the electromagnetic coupling between each phase will affect the precision of fault location. In order to reduce the positioning error caused by electromagnetic coupling, it is necessary to transform the three-phase nonindependent phase components into mutually independent mode components. In this paper, Clarke transformation is used to decouple the signal. The Clarke transformation matrices are as follows:(14)uαuβu0=132−1−1033111uaubuc,(15)iαiβi0=132−1−1033111iaibic.In the above formulas,ua, ub, and uc are the phase voltages of the three-phase line.uα, uβ, and u0 are the mode components of the phase voltages. ia, ib, and ic are the phase currents of the three-phase line. iα, iβ, and iγ are the mode components of the phase currents.


## 3.1. Modulus Maximum Detection Theory

The original signal discontinuity or its differential discontinuity leads to the singularity of the signal. The essence of singularity detection is to identify the singular point of the signal and analyze the degree of signal distortion. According to the time-frequency localization property of wavelet transform, the signal singularity can be effectively analyzed [11–13].The wavelet transform with scales at point u is Wfs,u, if point u near the point u0 satisfies Wfs,u≤Wfs,u0, then the maximum point at scale s is u0, and the modulus maximum of the wavelet transform is Wfs,u0 [14, 15]. Signals at different scales will have different modulus maximum; therefore, the singularity of the signal can be analyzed by extracting the modulus maximum points in the signal. By constructing a smoothing function to smooth the signals of different scales, the first and second order differentials can be extracted to identify the singular points of the signal [16, 17].The smoothing functionθt satisfies ∫−∞+∞θtdt=1 and limt⟶±∞θt=0. By constructing the mother wavelet ψ1t=dθ/dt and ψ2t=d2θ/dt2 and introducing the function ξst=1/sξt/s, the wavelet transform of ft at scale s and point t under is defined as(10)Wf1s,t=f∗ψs1,Wf2s,t=f∗ψs2.Expand the above formula as follows:(11)Wf1s,t=<f,ψs1>=f∗sdθsdtt=sddtf∗θst,Wf2s,t=<f,ψs2>=f∗s2d2θsdt2t=s2d2dt2f∗θst.It can be seen from the above formulas that the wavelet transformWf1s,t can be obtained by the first order differential and the Wf2s,t can be obtained by the second order differential. The discontinuous point of ft corresponds to the modulus maximum point of Wf1s,t and the zero point of Wf2s,t [18, 19]. However, the electrical quantity of the nonfault point will be disturbed during the signal transmission, which will affect the accuracy of the zero-crossing detection, so the result of the zero-crossing detection is usually not a discontinuity point, but an inflection point of the signal, and the modulus maximum detection is more reliable than zero-crossing detection [20, 21]. After wavelet transform, the discontinuity point of the signal ft corresponds to the maximum point of Wf1s,t and the zero point of Wf2s,t. According to the definition of wavelet transform Wfs,t, the maximum point of Wfs,t reflects the position of the singular point of f∗θst [22]. Therefore, the singular point of the signal at different scales can be determined by detecting the maximum point of Wfs,t.


## 3.2. Two-Terminal Travelling Wave Location Principle

The schematic diagram of the two-terminal travelling wave location method is shown in Figure3. When a fault occurs at point F, the transient travelling wave propagates to both sides along the transmission line. At the same time, reflection and refraction occur at the point where the wave impedance is discontinuous. Assume that the time when the travelling wave head reaches the detection point M and the detection point N is tm1 and tn1, respectively, and after reflection, the time when the wave head reaches the detection point M and N is tm2 and tn2, respectively. The total length of the line is LMN, the distance between the fault point F and M is LMF, the distance between F and N is LNF, and the travelling wave propagation speed is v; then,(12)LMF=tm1v,LNF=tn1v,LMN=LMF+LNF.Figure 3
Schematic diagram of fault location.Then, the formula of two-terminal travelling wave location is obtained:(13)LMF=LMN+vtm1−tn12.The two-terminal travelling wave location method only needs to detect the initial wave head of the fault travelling wave and does not need to consider the reflection and refraction at the discontinuity point of wave impedance [23–25]. Therefore, the location algorithm is more simple. In this paper, the discrete wavelet transform is used to detect the transient travelling wave head, and the time tm1 propagated to the detection point M and tn1 propagated to the detection point N are used to locate the fault point.


## 3.3. Phase-Model Transformation

In the three-phase transmission line, the electromagnetic coupling between each phase will affect the precision of fault location. In order to reduce the positioning error caused by electromagnetic coupling, it is necessary to transform the three-phase nonindependent phase components into mutually independent mode components. In this paper, Clarke transformation is used to decouple the signal. The Clarke transformation matrices are as follows:(14)uαuβu0=132−1−1033111uaubuc,(15)iαiβi0=132−1−1033111iaibic.In the above formulas,ua, ub, and uc are the phase voltages of the three-phase line.uα, uβ, and u0 are the mode components of the phase voltages. ia, ib, and ic are the phase currents of the three-phase line. iα, iβ, and iγ are the mode components of the phase currents.


## 4. Chaotic Neural Network

### 4.1. Lyapunov Exponent

In the process of dynamic evolution, the phase volume of the dissipative system continuously shrinks the trajectories to a limited range and separates the trajectories from each other. Because of its essential properties, the adjacent trajectories can be repelled and separated. This makes the trajectories converge and separate, separate and converge again, forming a self-similar structure with infinite levels. And this structure is called chaotic attractor. Two phase trajectories will attract or separate at an exponential rate over time. And the Lyapunov exponent is used to measure the degree of attraction or separation. The rate of convergence or divergence of the trajectories is called the Lyapunov exponent, which is often used as an effective parameter to describe the dynamic characteristics of the system. For a dissipative system, one of its Lyapunov exponents must be positive. Therefore, the largest Lyapunov exponent can be used to determine whether the system or signal is chaotic.For mappingxn+1=Fxn, in the process of continuous iteration, the two points separate and converge with the change of the value of dF/dx. Assuming that the discrete exponent caused by each iteration is constant λ. Then, after n times iteration, the two points at the initial distance ε will evolve into the following formula:(16)εenλx0=Fnx0+ε−Fnx0.Taking the limitε⟶0, n⟶∞, we get the following formula:(17)λx0=limn⟶∞limε⟶01nlnFnx0+ε−Fnx0ε=limn⟶∞1nlndFnxdxx=x0.It can be obtained from the above formula that(18)λ=limn⟶∞1n∑i=0n−1lndFxdxx=x0.It can be seen from the above formula that the constantλ is independent of the selection of the initial value, and this constant is the Lyapunov exponent of the system.The sensitivity of the Lyapunov exponent is related to the contraction and expansion of the phase space motion trajectory. In the direction with negative Lyapunov exponent, the trajectory is contracted and is not sensitive to the initial value. However, in the direction with positive Lyapunov exponent, the motion trajectory is rapidly separated and sensitive to the change of the initial value.


### 4.2. Chaotic Neural Network Model

The neural network designed in this paper for power system fault phase selection is a three-layer chaotic neural network, composed of an input layer, a hidden layer, and an output layer, as shown in Figure4. Four nodes are set at the input layer, and four discrete wavelet transform coefficients Wuis,k,i=α,β,γ,0 are input, respectively. The phase voltages are transformed into the mode components by Clarke phase-mode transformation. The mode components are convolved with discrete wavelet basis function: ui,ψs,k,i=α,β,γ,0, and the wavelet transform coefficient Wuis,k is obtained, as shown in the following formula:(19)Wuis,k=ui,ψs,k=a0−j/2∫−∞+∞uitψ¯a0−jt−kb0dt,i=α,β,γ,0.Figure 4
Chaotic neural network model.The number of output layer is 1, and the output resultT is “0” or “1.” When the output result is “0,” it means that the line is operating normally and the starting element does not act; when the output is “1,” it means that there are faults in transmission line, the starting element will act, and the protection device will start.


## 4.1. Lyapunov Exponent

In the process of dynamic evolution, the phase volume of the dissipative system continuously shrinks the trajectories to a limited range and separates the trajectories from each other. Because of its essential properties, the adjacent trajectories can be repelled and separated. This makes the trajectories converge and separate, separate and converge again, forming a self-similar structure with infinite levels. And this structure is called chaotic attractor. Two phase trajectories will attract or separate at an exponential rate over time. And the Lyapunov exponent is used to measure the degree of attraction or separation. The rate of convergence or divergence of the trajectories is called the Lyapunov exponent, which is often used as an effective parameter to describe the dynamic characteristics of the system. For a dissipative system, one of its Lyapunov exponents must be positive. Therefore, the largest Lyapunov exponent can be used to determine whether the system or signal is chaotic.For mappingxn+1=Fxn, in the process of continuous iteration, the two points separate and converge with the change of the value of dF/dx. Assuming that the discrete exponent caused by each iteration is constant λ. Then, after n times iteration, the two points at the initial distance ε will evolve into the following formula:(16)εenλx0=Fnx0+ε−Fnx0.Taking the limitε⟶0, n⟶∞, we get the following formula:(17)λx0=limn⟶∞limε⟶01nlnFnx0+ε−Fnx0ε=limn⟶∞1nlndFnxdxx=x0.It can be obtained from the above formula that(18)λ=limn⟶∞1n∑i=0n−1lndFxdxx=x0.It can be seen from the above formula that the constantλ is independent of the selection of the initial value, and this constant is the Lyapunov exponent of the system.The sensitivity of the Lyapunov exponent is related to the contraction and expansion of the phase space motion trajectory. In the direction with negative Lyapunov exponent, the trajectory is contracted and is not sensitive to the initial value. However, in the direction with positive Lyapunov exponent, the motion trajectory is rapidly separated and sensitive to the change of the initial value.


## 4.2. Chaotic Neural Network Model

The neural network designed in this paper for power system fault phase selection is a three-layer chaotic neural network, composed of an input layer, a hidden layer, and an output layer, as shown in Figure4. Four nodes are set at the input layer, and four discrete wavelet transform coefficients Wuis,k,i=α,β,γ,0 are input, respectively. The phase voltages are transformed into the mode components by Clarke phase-mode transformation. The mode components are convolved with discrete wavelet basis function: ui,ψs,k,i=α,β,γ,0, and the wavelet transform coefficient Wuis,k is obtained, as shown in the following formula:(19)Wuis,k=ui,ψs,k=a0−j/2∫−∞+∞uitψ¯a0−jt−kb0dt,i=α,β,γ,0.Figure 4
Chaotic neural network model.The number of output layer is 1, and the output resultT is “0” or “1.” When the output result is “0,” it means that the line is operating normally and the starting element does not act; when the output is “1,” it means that there are faults in transmission line, the starting element will act, and the protection device will start.


## 5. Fault Signal Detection Algorithm Based on Daubechie4

### 5.1. Selection of Wavelet Function

There are certain differences in the transformation characteristics of different wavelet functions. If the selected mother wavelet is different, the wavelet coefficients are also different; so, before using MATLAB for simulation analysis, we need to select the appropriate mother wavelet function [26]. The selection of wavelet function needs to take into account its vanishing moment, symmetry, and compactly support (support length). The compactly support and vanishing moment determine the influence range of wavelet function at a specific scale, so a compromise must be dealt with when choosing a smaller support length and a higher order vanishing moment [27, 28]. Symmetry describes the smoothness of the signal after wavelet reconstruction. Common wavelet functions and characteristics are shown in Table 2.Table 2
Common wavelet functions and characteristics.
Wavelet functionHaarDaubechiesBiorthogonalSymletsMorletRepresentationHaardb NBior Nr. NdsymNMorlSupport length12N − 12Nr + 12N − 1Limited lengthVanishing moment order1NNr − 1N—SymmetrySymmetryApproximate symmetryAsymmetryApproximate symmetrySymmetryIn order to further analyze the effect of different wavelets on the detection of discontinuities, we construct a freqbrk signal, which includes a low-frequency sinusoidal signal withint ∈ [0, 500] and a high-frequency signal within t ∈ [500, 1000]. Three different kinds of wavelet, haar wavelet, sym4 wavelet, and db4 wavelet, are used to 3-layer wavelet decomposition, and the analysis results are shown in Figure 5. The results show that all these three wavelets can detect signal discontinuities, but obviously, the db4 wavelet has a better effect; so, the db4 wavelet is used to detect the singularity of the signal in this paper.Figure 5
Comparison of different wavelet detection effects.


### 5.2. Algorithm Steps

Combining the characteristics of the fault signal, the criterion algorithm based on the chaotic neural network and the detection algorithm based on wavelet transform are proposed as follows:(1)
Simulink is used to build a power system model and output three-phase voltage signals and three-phase current signals(2)
Travelling wave signal is obtained from three-phase voltage signals and three-phase current signals(3)
The mother wavelet is selected, and the number of decomposition layers is determined(4)
Clarke transformation is used to transform the three-phase voltage signals into mode components(5)
Wavelet transform is performed to the mode components, and the detail coefficients are input to the chaotic neural network(6)
The discrete wavelet transform (DWT) is used to analyze the travelling wave signalf to obtain the wavelet coefficients of each layer: W1fk,W2fk,…,Wjfk(7)
Theoretical analysis and simulation demonstrate the effectiveness of the criterion algorithm and the accuracy of the fault detection algorithm(8)
Simulation results show that both the criterion algorithm and the fault detection algorithm are not affected by fault type, fault location, fault initial angle, and transition resistance


## 5.1. Selection of Wavelet Function

There are certain differences in the transformation characteristics of different wavelet functions. If the selected mother wavelet is different, the wavelet coefficients are also different; so, before using MATLAB for simulation analysis, we need to select the appropriate mother wavelet function [26]. The selection of wavelet function needs to take into account its vanishing moment, symmetry, and compactly support (support length). The compactly support and vanishing moment determine the influence range of wavelet function at a specific scale, so a compromise must be dealt with when choosing a smaller support length and a higher order vanishing moment [27, 28]. Symmetry describes the smoothness of the signal after wavelet reconstruction. Common wavelet functions and characteristics are shown in Table 2.Table 2
Common wavelet functions and characteristics.
Wavelet functionHaarDaubechiesBiorthogonalSymletsMorletRepresentationHaardb NBior Nr. NdsymNMorlSupport length12N − 12Nr + 12N − 1Limited lengthVanishing moment order1NNr − 1N—SymmetrySymmetryApproximate symmetryAsymmetryApproximate symmetrySymmetryIn order to further analyze the effect of different wavelets on the detection of discontinuities, we construct a freqbrk signal, which includes a low-frequency sinusoidal signal withint ∈ [0, 500] and a high-frequency signal within t ∈ [500, 1000]. Three different kinds of wavelet, haar wavelet, sym4 wavelet, and db4 wavelet, are used to 3-layer wavelet decomposition, and the analysis results are shown in Figure 5. The results show that all these three wavelets can detect signal discontinuities, but obviously, the db4 wavelet has a better effect; so, the db4 wavelet is used to detect the singularity of the signal in this paper.Figure 5
Comparison of different wavelet detection effects.


## 5.2. Algorithm Steps

Combining the characteristics of the fault signal, the criterion algorithm based on the chaotic neural network and the detection algorithm based on wavelet transform are proposed as follows:(1)
Simulink is used to build a power system model and output three-phase voltage signals and three-phase current signals(2)
Travelling wave signal is obtained from three-phase voltage signals and three-phase current signals(3)
The mother wavelet is selected, and the number of decomposition layers is determined(4)
Clarke transformation is used to transform the three-phase voltage signals into mode components(5)
Wavelet transform is performed to the mode components, and the detail coefficients are input to the chaotic neural network(6)
The discrete wavelet transform (DWT) is used to analyze the travelling wave signalf to obtain the wavelet coefficients of each layer: W1fk,W2fk,…,Wjfk(7)
Theoretical analysis and simulation demonstrate the effectiveness of the criterion algorithm and the accuracy of the fault detection algorithm(8)
Simulation results show that both the criterion algorithm and the fault detection algorithm are not affected by fault type, fault location, fault initial angle, and transition resistance


## 6. Modeling and Simulation

### 6.1. Build a Power System Model

In this paper, a power system simulation model with a voltage level of 220 kV and power frequency of 50 Hz is built by MATLAB/Simulink, as shown in Figure6. The system is a dual-power three-phase power supply system, consisting of transmission lines, voltage measuring elements, current measuring elements, oscilloscopes, and fault generators. The three-phase phase angle of the two power sources is 30°, the sampling frequency is 20 kHz, the simulation duration is T = 0.1 s, the fault occurrence time is t0 = 0.035 s, and the sampling points are 10000. The system can simulate all kinds of faults that may occur in the power system.Figure 6
Power system simulation model.The transmission line parameters are shown in Table3.Table 3
The transmission line parameters.
Parameter typesResistance (Ω·km−1)Inductance (H·km−1)Capacitance (F·km−1)Positive sequence0.0131.33 × 10−39.94 × 10−9Zero sequence0.3864.126 × 10−37.751 × 10−9


### 6.2. Signal Extraction

#### 6.2.1. Extract Simulation Signal

By setting the parameters of “three-phase Fault” element, the phase voltages and phase currents of the system can be extracted, as shown in Figures7 and 8. The effective values of phase voltages and phase currents of the system can be extracted through Scope1 and Scope2, respectively, as shown in Figures 9 and 10. When the fault occurs, the voltage of phase A drops sharply, the voltages of phase B and phase C rise slightly, and the phase currents increase.Figure 7
Phase voltages.Figure 8
Phase currents.Figure 9
Effective values of phase voltage.Figure 10
Effective values of phase current.


#### 6.2.2. Extract Travelling Wave Signal

The phase voltages and phase currents of the three-phase transmission line are obtained through the system model, and the corresponding mode components are obtained with equations (16) and (17). Then, the travelling wave signals detected at the M-side and N-side are obtained, as shown in Figures 11 and 12.Figure 11
M-terminal travelling wave signal.Figure 12
N-terminal travelling wave signal.


### 6.3. Simulation Analysis

#### 6.3.1. Simulation of the Chaotic Neural Network Criterion Algorithm

In order to verify the effectiveness of the proposed chaotic neural network criterion algorithm in this paper, the wavelet transform coefficients of four mode components are input. The modulus maximum of wavelet transform coefficientsWuis,kmax,i=α,β,γ,0 is extracted. And it is analyzed whether this method is accurate and reliable under different operation conditions such as different fault types, fault positions, transition resistances, and fault initial angles. The simulation results are shown in Tables 4–7. Within the allowable error range, it can be seen from Wuis,kmax,i=α,β,γ,0 that when the power system operates normally, Wuis,kmax=0, when phase A is grounded, Wuαs,kmax=2Wuis,kmax,i=β,γ,0, when both phase A and phase B are grounded, Wuαs,kmax=Wuβs,kmax=2Wuis,kmax,i=γ,0, and when there is a three-phase short circuit fault, Wuαs,kmax=Wuβs,kmax=Wuγs,kmax. Therefore, it is concluded from Tables 4–7 that when the transmission line operates normally, the output result is 0, and the starting element does not work, and when there is a fault in the line, the output result is 1, and the starting element takes effect. And the proposed criterion algorithm based on the chaotic neural network is not affected by fault types, fault positions, transition resistances, and fault initiation angles.Table 4
Simulation results under different fault types.
Operating statusInput layerOutput layerWuαs,kmaxWuβs,kmaxWuγs,kmaxWu0s,kmaxNormal00000AG3078.211529.931610.031582.611BG1766.023591.401690.691683.731CG1404.781504.542945.621498.061AB3929.073894.83001AC4008.3303968.5301BC03582.063701.3901ABG3802.814012.102070.031990.251BCG1970.892909.713008.292015.881ACG3202.451609.123192.811513.611ABC2417.022508.162588.3301Table 5
Simulation results under different fault positions.
Fault positions (km)Operating statusInput layerOutput layerWuαs,kmaxWuβs,kmaxWuγs,kmaxWu0s,kmax67Normal00000AG3472.441736.281766.671698.221ABG2425.532567.391226.861308.911ABC2723.742781.932830.7901151Normal00000AG4133.332078.192380.312241.771ABG3070.923256.091568.151553.211ABC2934.372843.902851.4001302Normal00000AG3754.871867.351936.772037.561ABG3689.833551.761792.711868.421ABC1906.882038.042113.2501Table 6
Simulation results under different transition resistances.
Transition resistance (Ω)Operating statusInput layerOutput layerWuαs,kmaxWuβs,kmaxWuγs,kmaxWu0s,kmax0Normal00000AG4365.782172.362097.102206.631ABG3054.813136.901594.071662.961ABC4015.184103.163921.640150Normal00000AG3907.231918.341857.202032.661ABG3267.793493.601562.871608.151ABC5109.264907.045037.2101200Normal00000AG5172.552506.732462.112463.001ABG4931.834962.912607.122732.391ABC3088.262960.662953.3301Table 7
Simulation results under different fault initial angles.
Fault initial angle (°)Operating statusInput layerOutput layerWuαs,kmaxWuβs,kmaxWuγs,kmaxWu0s,kmax30Normal00000AG4561.412339.402197.102179.981ABG5637.115466.702791.332832.141ABC2593.012605.772480.190160Normal00000AG3168.411500.891679.201592.361ABG3612.593741.511820.421825.221ABC4026.723901.284104.5701180Normal00000AG3152.691429.521520.771501.821ABG4010.334192.812078.192165.101ABC5123.065016.424997.1101


#### 6.3.2. Simulation of the Wavelet Transform Fault Detection Algorithm

The travelling wave signal is sampled, and 500 sampling points are obtained. Daubechies 4 discrete wavelet transform is used to analyze the travelling wave signal, and the simulation results are shown in Figures13 and 14. The signal shown in Figure 13 is the detail coefficient of 4-layer wavelet decomposition of the M-side travelling wave. The signal shown in Figure 14 is the detail coefficients of 4-layer wavelet decomposition of the N-side travelling wave. It can be seen from Figures 13 and 14 that the amplitude of the detail coefficients (especially the first layer d1 and the second layer d2) change dramatically near the mutation point of the travelling wave signal. This is because the discrete wavelet transform has excellent time-frequency characteristics. The signal is decomposed into multiple layers based on the Mallat algorithm. And the high-frequency components can accurately identify the time when the frequency changed dramatically. Therefore, the wavelet transform is very suitable for the detection of singular points of nonstationary signals.Figure 13
Detail coefficients of theM-terminal travelling wave.Figure 14
Detail coefficients of theN-terminal travelling wave.The first-layer wavelet coefficient d1 can accurately identify the travelling wave heads and the singular points of the travelling wave. The travelling wave heads reflect the time when the fault travelling wave first propagates to theM-side. And the singular points reflect the time when the travelling wave reaches the M-side after being reflected by the fault point and the N-side bus. The proposed algorithm only needs to detect the time when the travelling wave head reaches the two detection points. From d1, tm1 = 35.3 ms and tn1 = 71.6 ms can be obtained. Then, the distance between the fault point and M terminal is obtained with equation (15): LMF = 100.082 km. The fault location error is defined as actual fault distance−measuring fault distance/total length of the transmission line×100%; so, the error is 0.027%.In order to verify the accuracy and good performance of the discrete wavelet transform algorithm proposed in this paper, EMD is performed on the signal as a comparison, as shown in Figures15 and 16. The signal shown in Figure 15 is the intrinsic mode functions (IMFs) of EMD of the M-side travelling wave. The signal shown in Figure 16 is the IMFs of the N-side travelling wave. It can be seen from Figures 15 and 16 that only the first-layer component can clearly observe the singular points of the travelling wave. From IMF1, tm1 = 34.6 ms and tn1 = 70.1 ms can be obtained, and the distance between the fault point and M terminal is obtained with equation (15): LMF = 101.135 km; so, the error is 0.378%, which is ten times of the algorithm proposed in this paper. It can be seen that the algorithm proposed in this paper is accurate and reliable.Figure 15
IMFs of theM-terminal travelling wave.Figure 16
IMFs of theN-terminal travelling wave.Next, it is analyzed whether this method can locate accurately and reliably under different operation conditions such as different fault types, fault positions, transition resistances, and fault initial angles.(1) Different Fault Types. The parameters are set as follows: the length of the transmission line is LMN = 350 km, the transition resistance is Rf = 10 Ω, the fault initial angle is δ = 30°, and the fault position is 150 km away from the M-side bus. The simulation results of different fault types are shown in Table 8. The results show that no matter what type of the fault, the measurement error is less than 0.05%.Table 8
Simulation results under different fault types.
Fault typesPositioning results (km)Relative error (%)AG150.0590.017BG150.0910.026CG149.9420.017AB149.9380.018AC149.9700.009BC149.8920.031ABG150.0780.022ACG149.8390.046BCG149.9010.028ABC150.1030.029ABCG150.0860.025(2) Different Fault Positions. The parameters are set as follows: the length of the transmission line is LMN = 350 km, the transition resistance is Rf = 10 Ω, the fault initial angle is δ = 30°, and the fault type is A-phase ground fault. The simulation results of different fault positions are shown in Table 9. The results show that the positioning algorithm is not affected by fault positions, and the positioning error is less than 0.03%.Table 9
Simulation results under different fault positions.
Fault positions/kmPositioning results (km)Relative error (%)6362.9370.0189898.0610.017135135.0700.020189189.0770.022210209.9520.014277277.0490.014(3) Different Transition Resistances. The parameters are set as follows: the length of the transmission line is LMN = 350 km, the fault initial angle is δ = 30°, the fault type is A-phase ground fault, and the fault position is 90 km away from the M-side bus. The simulation results of different transition resistances are shown in Table 10. The results show that the positioning algorithm is not affected by transition resistances, and the positioning error is less than 0.04%.Table 10
Simulation results under different transition resistances.
Transition resistance (Ω)Positioning results (km)Relative error (%)090.0920.0261090.0430.0125089.9300.02010089.9270.02115090.1330.03820089.8730.03630090.0710.020(4) Different Fault Initial Angles. The parameters are set as follows: the length of the transmission line is LMN = 350 km, the transition resistance is Rf = 10 Ω, the fault type is A-phase ground fault, and the fault position is 132 km away from the M-side bus. The simulation results of different fault initial angles are shown in Table 11. The results show that the positioning algorithm is not affected by fault initial angles, and the positioning error is less than 0.03%.Table 11
Simulation results under different fault initial angles.
Fault initial angle (°)Positioning results (km)Relative error (%)0132.0790.02330132.0560.01645131.9670.00960131.9330.01990132.0570.016135131.9080.026165131.9420.017


## 6.1. Build a Power System Model

In this paper, a power system simulation model with a voltage level of 220 kV and power frequency of 50 Hz is built by MATLAB/Simulink, as shown in Figure6. The system is a dual-power three-phase power supply system, consisting of transmission lines, voltage measuring elements, current measuring elements, oscilloscopes, and fault generators. The three-phase phase angle of the two power sources is 30°, the sampling frequency is 20 kHz, the simulation duration is T = 0.1 s, the fault occurrence time is t0 = 0.035 s, and the sampling points are 10000. The system can simulate all kinds of faults that may occur in the power system.Figure 6
Power system simulation model.The transmission line parameters are shown in Table3.Table 3
The transmission line parameters.
Parameter typesResistance (Ω·km−1)Inductance (H·km−1)Capacitance (F·km−1)Positive sequence0.0131.33 × 10−39.94 × 10−9Zero sequence0.3864.126 × 10−37.751 × 10−9


## 6.2. Signal Extraction

### 6.2.1. Extract Simulation Signal

By setting the parameters of “three-phase Fault” element, the phase voltages and phase currents of the system can be extracted, as shown in Figures7 and 8. The effective values of phase voltages and phase currents of the system can be extracted through Scope1 and Scope2, respectively, as shown in Figures 9 and 10. When the fault occurs, the voltage of phase A drops sharply, the voltages of phase B and phase C rise slightly, and the phase currents increase.Figure 7
Phase voltages.Figure 8
Phase currents.Figure 9
Effective values of phase voltage.Figure 10
Effective values of phase current.


### 6.2.2. Extract Travelling Wave Signal

The phase voltages and phase currents of the three-phase transmission line are obtained through the system model, and the corresponding mode components are obtained with equations (16) and (17). Then, the travelling wave signals detected at the M-side and N-side are obtained, as shown in Figures 11 and 12.Figure 11
M-terminal travelling wave signal.Figure 12
N-terminal travelling wave signal.


## 6.2.1. Extract Simulation Signal

By setting the parameters of “three-phase Fault” element, the phase voltages and phase currents of the system can be extracted, as shown in Figures7 and 8. The effective values of phase voltages and phase currents of the system can be extracted through Scope1 and Scope2, respectively, as shown in Figures 9 and 10. When the fault occurs, the voltage of phase A drops sharply, the voltages of phase B and phase C rise slightly, and the phase currents increase.Figure 7
Phase voltages.Figure 8
Phase currents.Figure 9
Effective values of phase voltage.Figure 10
Effective values of phase current.


## 6.2.2. Extract Travelling Wave Signal

The phase voltages and phase currents of the three-phase transmission line are obtained through the system model, and the corresponding mode components are obtained with equations (16) and (17). Then, the travelling wave signals detected at the M-side and N-side are obtained, as shown in Figures 11 and 12.Figure 11
M-terminal travelling wave signal.Figure 12
N-terminal travelling wave signal.


## 6.3. Simulation Analysis

### 6.3.1. Simulation of the Chaotic Neural Network Criterion Algorithm

In order to verify the effectiveness of the proposed chaotic neural network criterion algorithm in this paper, the wavelet transform coefficients of four mode components are input. The modulus maximum of wavelet transform coefficientsWuis,kmax,i=α,β,γ,0 is extracted. And it is analyzed whether this method is accurate and reliable under different operation conditions such as different fault types, fault positions, transition resistances, and fault initial angles. The simulation results are shown in Tables 4–7. Within the allowable error range, it can be seen from Wuis,kmax,i=α,β,γ,0 that when the power system operates normally, Wuis,kmax=0, when phase A is grounded, Wuαs,kmax=2Wuis,kmax,i=β,γ,0, when both phase A and phase B are grounded, Wuαs,kmax=Wuβs,kmax=2Wuis,kmax,i=γ,0, and when there is a three-phase short circuit fault, Wuαs,kmax=Wuβs,kmax=Wuγs,kmax. Therefore, it is concluded from Tables 4–7 that when the transmission line operates normally, the output result is 0, and the starting element does not work, and when there is a fault in the line, the output result is 1, and the starting element takes effect. And the proposed criterion algorithm based on the chaotic neural network is not affected by fault types, fault positions, transition resistances, and fault initiation angles.Table 4
Simulation results under different fault types.
Operating statusInput layerOutput layerWuαs,kmaxWuβs,kmaxWuγs,kmaxWu0s,kmaxNormal00000AG3078.211529.931610.031582.611BG1766.023591.401690.691683.731CG1404.781504.542945.621498.061AB3929.073894.83001AC4008.3303968.5301BC03582.063701.3901ABG3802.814012.102070.031990.251BCG1970.892909.713008.292015.881ACG3202.451609.123192.811513.611ABC2417.022508.162588.3301Table 5
Simulation results under different fault positions.
Fault positions (km)Operating statusInput layerOutput layerWuαs,kmaxWuβs,kmaxWuγs,kmaxWu0s,kmax67Normal00000AG3472.441736.281766.671698.221ABG2425.532567.391226.861308.911ABC2723.742781.932830.7901151Normal00000AG4133.332078.192380.312241.771ABG3070.923256.091568.151553.211ABC2934.372843.902851.4001302Normal00000AG3754.871867.351936.772037.561ABG3689.833551.761792.711868.421ABC1906.882038.042113.2501Table 6
Simulation results under different transition resistances.
Transition resistance (Ω)Operating statusInput layerOutput layerWuαs,kmaxWuβs,kmaxWuγs,kmaxWu0s,kmax0Normal00000AG4365.782172.362097.102206.631ABG3054.813136.901594.071662.961ABC4015.184103.163921.640150Normal00000AG3907.231918.341857.202032.661ABG3267.793493.601562.871608.151ABC5109.264907.045037.2101200Normal00000AG5172.552506.732462.112463.001ABG4931.834962.912607.122732.391ABC3088.262960.662953.3301Table 7
Simulation results under different fault initial angles.
Fault initial angle (°)Operating statusInput layerOutput layerWuαs,kmaxWuβs,kmaxWuγs,kmaxWu0s,kmax30Normal00000AG4561.412339.402197.102179.981ABG5637.115466.702791.332832.141ABC2593.012605.772480.190160Normal00000AG3168.411500.891679.201592.361ABG3612.593741.511820.421825.221ABC4026.723901.284104.5701180Normal00000AG3152.691429.521520.771501.821ABG4010.334192.812078.192165.101ABC5123.065016.424997.1101


### 6.3.2. Simulation of the Wavelet Transform Fault Detection Algorithm

The travelling wave signal is sampled, and 500 sampling points are obtained. Daubechies 4 discrete wavelet transform is used to analyze the travelling wave signal, and the simulation results are shown in Figures13 and 14. The signal shown in Figure 13 is the detail coefficient of 4-layer wavelet decomposition of the M-side travelling wave. The signal shown in Figure 14 is the detail coefficients of 4-layer wavelet decomposition of the N-side travelling wave. It can be seen from Figures 13 and 14 that the amplitude of the detail coefficients (especially the first layer d1 and the second layer d2) change dramatically near the mutation point of the travelling wave signal. This is because the discrete wavelet transform has excellent time-frequency characteristics. The signal is decomposed into multiple layers based on the Mallat algorithm. And the high-frequency components can accurately identify the time when the frequency changed dramatically. Therefore, the wavelet transform is very suitable for the detection of singular points of nonstationary signals.Figure 13
Detail coefficients of theM-terminal travelling wave.Figure 14
Detail coefficients of theN-terminal travelling wave.The first-layer wavelet coefficient d1 can accurately identify the travelling wave heads and the singular points of the travelling wave. The travelling wave heads reflect the time when the fault travelling wave first propagates to theM-side. And the singular points reflect the time when the travelling wave reaches the M-side after being reflected by the fault point and the N-side bus. The proposed algorithm only needs to detect the time when the travelling wave head reaches the two detection points. From d1, tm1 = 35.3 ms and tn1 = 71.6 ms can be obtained. Then, the distance between the fault point and M terminal is obtained with equation (15): LMF = 100.082 km. The fault location error is defined as actual fault distance−measuring fault distance/total length of the transmission line×100%; so, the error is 0.027%.In order to verify the accuracy and good performance of the discrete wavelet transform algorithm proposed in this paper, EMD is performed on the signal as a comparison, as shown in Figures15 and 16. The signal shown in Figure 15 is the intrinsic mode functions (IMFs) of EMD of the M-side travelling wave. The signal shown in Figure 16 is the IMFs of the N-side travelling wave. It can be seen from Figures 15 and 16 that only the first-layer component can clearly observe the singular points of the travelling wave. From IMF1, tm1 = 34.6 ms and tn1 = 70.1 ms can be obtained, and the distance between the fault point and M terminal is obtained with equation (15): LMF = 101.135 km; so, the error is 0.378%, which is ten times of the algorithm proposed in this paper. It can be seen that the algorithm proposed in this paper is accurate and reliable.Figure 15
IMFs of theM-terminal travelling wave.Figure 16
IMFs of theN-terminal travelling wave.Next, it is analyzed whether this method can locate accurately and reliably under different operation conditions such as different fault types, fault positions, transition resistances, and fault initial angles.(1) Different Fault Types. The parameters are set as follows: the length of the transmission line is LMN = 350 km, the transition resistance is Rf = 10 Ω, the fault initial angle is δ = 30°, and the fault position is 150 km away from the M-side bus. The simulation results of different fault types are shown in Table 8. The results show that no matter what type of the fault, the measurement error is less than 0.05%.Table 8
Simulation results under different fault types.
Fault typesPositioning results (km)Relative error (%)AG150.0590.017BG150.0910.026CG149.9420.017AB149.9380.018AC149.9700.009BC149.8920.031ABG150.0780.022ACG149.8390.046BCG149.9010.028ABC150.1030.029ABCG150.0860.025(2) Different Fault Positions. The parameters are set as follows: the length of the transmission line is LMN = 350 km, the transition resistance is Rf = 10 Ω, the fault initial angle is δ = 30°, and the fault type is A-phase ground fault. The simulation results of different fault positions are shown in Table 9. The results show that the positioning algorithm is not affected by fault positions, and the positioning error is less than 0.03%.Table 9
Simulation results under different fault positions.
Fault positions/kmPositioning results (km)Relative error (%)6362.9370.0189898.0610.017135135.0700.020189189.0770.022210209.9520.014277277.0490.014(3) Different Transition Resistances. The parameters are set as follows: the length of the transmission line is LMN = 350 km, the fault initial angle is δ = 30°, the fault type is A-phase ground fault, and the fault position is 90 km away from the M-side bus. The simulation results of different transition resistances are shown in Table 10. The results show that the positioning algorithm is not affected by transition resistances, and the positioning error is less than 0.04%.Table 10
Simulation results under different transition resistances.
Transition resistance (Ω)Positioning results (km)Relative error (%)090.0920.0261090.0430.0125089.9300.02010089.9270.02115090.1330.03820089.8730.03630090.0710.020(4) Different Fault Initial Angles. The parameters are set as follows: the length of the transmission line is LMN = 350 km, the transition resistance is Rf = 10 Ω, the fault type is A-phase ground fault, and the fault position is 132 km away from the M-side bus. The simulation results of different fault initial angles are shown in Table 11. The results show that the positioning algorithm is not affected by fault initial angles, and the positioning error is less than 0.03%.Table 11
Simulation results under different fault initial angles.
Fault initial angle (°)Positioning results (km)Relative error (%)0132.0790.02330132.0560.01645131.9670.00960131.9330.01990132.0570.016135131.9080.026165131.9420.017


## 6.3.1. Simulation of the Chaotic Neural Network Criterion Algorithm

In order to verify the effectiveness of the proposed chaotic neural network criterion algorithm in this paper, the wavelet transform coefficients of four mode components are input. The modulus maximum of wavelet transform coefficientsWuis,kmax,i=α,β,γ,0 is extracted. And it is analyzed whether this method is accurate and reliable under different operation conditions such as different fault types, fault positions, transition resistances, and fault initial angles. The simulation results are shown in Tables 4–7. Within the allowable error range, it can be seen from Wuis,kmax,i=α,β,γ,0 that when the power system operates normally, Wuis,kmax=0, when phase A is grounded, Wuαs,kmax=2Wuis,kmax,i=β,γ,0, when both phase A and phase B are grounded, Wuαs,kmax=Wuβs,kmax=2Wuis,kmax,i=γ,0, and when there is a three-phase short circuit fault, Wuαs,kmax=Wuβs,kmax=Wuγs,kmax. Therefore, it is concluded from Tables 4–7 that when the transmission line operates normally, the output result is 0, and the starting element does not work, and when there is a fault in the line, the output result is 1, and the starting element takes effect. And the proposed criterion algorithm based on the chaotic neural network is not affected by fault types, fault positions, transition resistances, and fault initiation angles.Table 4
Simulation results under different fault types.
Operating statusInput layerOutput layerWuαs,kmaxWuβs,kmaxWuγs,kmaxWu0s,kmaxNormal00000AG3078.211529.931610.031582.611BG1766.023591.401690.691683.731CG1404.781504.542945.621498.061AB3929.073894.83001AC4008.3303968.5301BC03582.063701.3901ABG3802.814012.102070.031990.251BCG1970.892909.713008.292015.881ACG3202.451609.123192.811513.611ABC2417.022508.162588.3301Table 5
Simulation results under different fault positions.
Fault positions (km)Operating statusInput layerOutput layerWuαs,kmaxWuβs,kmaxWuγs,kmaxWu0s,kmax67Normal00000AG3472.441736.281766.671698.221ABG2425.532567.391226.861308.911ABC2723.742781.932830.7901151Normal00000AG4133.332078.192380.312241.771ABG3070.923256.091568.151553.211ABC2934.372843.902851.4001302Normal00000AG3754.871867.351936.772037.561ABG3689.833551.761792.711868.421ABC1906.882038.042113.2501Table 6
Simulation results under different transition resistances.
Transition resistance (Ω)Operating statusInput layerOutput layerWuαs,kmaxWuβs,kmaxWuγs,kmaxWu0s,kmax0Normal00000AG4365.782172.362097.102206.631ABG3054.813136.901594.071662.961ABC4015.184103.163921.640150Normal00000AG3907.231918.341857.202032.661ABG3267.793493.601562.871608.151ABC5109.264907.045037.2101200Normal00000AG5172.552506.732462.112463.001ABG4931.834962.912607.122732.391ABC3088.262960.662953.3301Table 7
Simulation results under different fault initial angles.
Fault initial angle (°)Operating statusInput layerOutput layerWuαs,kmaxWuβs,kmaxWuγs,kmaxWu0s,kmax30Normal00000AG4561.412339.402197.102179.981ABG5637.115466.702791.332832.141ABC2593.012605.772480.190160Normal00000AG3168.411500.891679.201592.361ABG3612.593741.511820.421825.221ABC4026.723901.284104.5701180Normal00000AG3152.691429.521520.771501.821ABG4010.334192.812078.192165.101ABC5123.065016.424997.1101


## 6.3.2. Simulation of the Wavelet Transform Fault Detection Algorithm

The travelling wave signal is sampled, and 500 sampling points are obtained. Daubechies 4 discrete wavelet transform is used to analyze the travelling wave signal, and the simulation results are shown in Figures13 and 14. The signal shown in Figure 13 is the detail coefficient of 4-layer wavelet decomposition of the M-side travelling wave. The signal shown in Figure 14 is the detail coefficients of 4-layer wavelet decomposition of the N-side travelling wave. It can be seen from Figures 13 and 14 that the amplitude of the detail coefficients (especially the first layer d1 and the second layer d2) change dramatically near the mutation point of the travelling wave signal. This is because the discrete wavelet transform has excellent time-frequency characteristics. The signal is decomposed into multiple layers based on the Mallat algorithm. And the high-frequency components can accurately identify the time when the frequency changed dramatically. Therefore, the wavelet transform is very suitable for the detection of singular points of nonstationary signals.Figure 13
Detail coefficients of theM-terminal travelling wave.Figure 14
Detail coefficients of theN-terminal travelling wave.The first-layer wavelet coefficient d1 can accurately identify the travelling wave heads and the singular points of the travelling wave. The travelling wave heads reflect the time when the fault travelling wave first propagates to theM-side. And the singular points reflect the time when the travelling wave reaches the M-side after being reflected by the fault point and the N-side bus. The proposed algorithm only needs to detect the time when the travelling wave head reaches the two detection points. From d1, tm1 = 35.3 ms and tn1 = 71.6 ms can be obtained. Then, the distance between the fault point and M terminal is obtained with equation (15): LMF = 100.082 km. The fault location error is defined as actual fault distance−measuring fault distance/total length of the transmission line×100%; so, the error is 0.027%.In order to verify the accuracy and good performance of the discrete wavelet transform algorithm proposed in this paper, EMD is performed on the signal as a comparison, as shown in Figures15 and 16. The signal shown in Figure 15 is the intrinsic mode functions (IMFs) of EMD of the M-side travelling wave. The signal shown in Figure 16 is the IMFs of the N-side travelling wave. It can be seen from Figures 15 and 16 that only the first-layer component can clearly observe the singular points of the travelling wave. From IMF1, tm1 = 34.6 ms and tn1 = 70.1 ms can be obtained, and the distance between the fault point and M terminal is obtained with equation (15): LMF = 101.135 km; so, the error is 0.378%, which is ten times of the algorithm proposed in this paper. It can be seen that the algorithm proposed in this paper is accurate and reliable.Figure 15
IMFs of theM-terminal travelling wave.Figure 16
IMFs of theN-terminal travelling wave.Next, it is analyzed whether this method can locate accurately and reliably under different operation conditions such as different fault types, fault positions, transition resistances, and fault initial angles.(1) Different Fault Types. The parameters are set as follows: the length of the transmission line is LMN = 350 km, the transition resistance is Rf = 10 Ω, the fault initial angle is δ = 30°, and the fault position is 150 km away from the M-side bus. The simulation results of different fault types are shown in Table 8. The results show that no matter what type of the fault, the measurement error is less than 0.05%.Table 8
Simulation results under different fault types.
Fault typesPositioning results (km)Relative error (%)AG150.0590.017BG150.0910.026CG149.9420.017AB149.9380.018AC149.9700.009BC149.8920.031ABG150.0780.022ACG149.8390.046BCG149.9010.028ABC150.1030.029ABCG150.0860.025(2) Different Fault Positions. The parameters are set as follows: the length of the transmission line is LMN = 350 km, the transition resistance is Rf = 10 Ω, the fault initial angle is δ = 30°, and the fault type is A-phase ground fault. The simulation results of different fault positions are shown in Table 9. The results show that the positioning algorithm is not affected by fault positions, and the positioning error is less than 0.03%.Table 9
Simulation results under different fault positions.
Fault positions/kmPositioning results (km)Relative error (%)6362.9370.0189898.0610.017135135.0700.020189189.0770.022210209.9520.014277277.0490.014(3) Different Transition Resistances. The parameters are set as follows: the length of the transmission line is LMN = 350 km, the fault initial angle is δ = 30°, the fault type is A-phase ground fault, and the fault position is 90 km away from the M-side bus. The simulation results of different transition resistances are shown in Table 10. The results show that the positioning algorithm is not affected by transition resistances, and the positioning error is less than 0.04%.Table 10
Simulation results under different transition resistances.
Transition resistance (Ω)Positioning results (km)Relative error (%)090.0920.0261090.0430.0125089.9300.02010089.9270.02115090.1330.03820089.8730.03630090.0710.020(4) Different Fault Initial Angles. The parameters are set as follows: the length of the transmission line is LMN = 350 km, the transition resistance is Rf = 10 Ω, the fault type is A-phase ground fault, and the fault position is 132 km away from the M-side bus. The simulation results of different fault initial angles are shown in Table 11. The results show that the positioning algorithm is not affected by fault initial angles, and the positioning error is less than 0.03%.Table 11
Simulation results under different fault initial angles.
Fault initial angle (°)Positioning results (km)Relative error (%)0132.0790.02330132.0560.01645131.9670.00960131.9330.01990132.0570.016135131.9080.026165131.9420.017


## 7. Conclusion

When there are faults in the power system, it is very important to implement relay protection quickly and accurately. Travelling wave signals will be generated at the fault point and propagate to both sides of the line. During the transmission of the travelling wave, the forward travelling wave and the travelling wave generated by refraction and reflection superimpose each other, which makes it difficult to determine the wave heads of the travelling wave. Based on the power system model built by MATLAB/Simulink, a criterion algorithm based on the chaotic neural network and a fault detection algorithm based on discrete wavelet transform are proposed in this paper. The criterion algorithm based on the chaotic neural network is capable of overcoming the shortcomings of trapping into the local optimum, and it has good performance of fault tolerance and associative memory capabilities. The fault detection algorithm has the capability of detecting the fault location of transmission lines. The two-terminal travelling wave location method only needs to detect the initial wave head of the fault travelling wave and does not need to consider the reflection and refraction of the fault travelling wave; so, the location algorithm is more simple. Due to the difference in the transformation characteristics of different wavelet functions, a detection method based on db4 wavelet is determined to obtain more accurate results. After multiresolution wavelet decomposition, we find that the distribution of the energy of the fault signal on each wavelet coefficient is different. The valuable part of the signal is mainly reflected in the specific wavelet coefficients. These detail coefficients are input to the chaotic neural network to verify that the criterion algorithm has the capabilities of determining whether there are faults in transmission lines. And these detail coefficients are used to accurately determine the time when the fault travelling wave head reaches theM-side bus and the N-side bus, and then, the fault position is detected with the two-terminal travelling wave location algorithm. Finally, it is illustrated that both the criterion algorithm and the fault detection algorithm are not affected by fault type, fault positions, transition resistance, and fault initial angle. Therefore, the criterion algorithm based on the chaotic neural network and the fault detection algorithm based on discrete wavelet transform are of great significance, which provide a favorable exploration method for the real-time fault detection of some systems and has a broad and far-reaching application prospect.



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