# Detection of Broken Rotor Bars Fault in Induction Motors by Using an Improved MUSIC and Least-Squares Amplitude Estimation

Authors: Junjie Lu; Panpan Wang; Sen Duan; Liping Shi; Li Han
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

The frequencies and amplitudes of the broken rotor bar (BRB) fault features are the basis for the accurate diagnosis of the BRB fault. However, how to accurately detect their frequency and amplitudes has always been a difficult problem for induction motor fault detection. For this problem, a new fault detection method based on an improved multiple signal classification (MUSIC) and least-squares magnitude estimation is proposed. First, since the fixed-step traversal search reduces the computational efficiency of MUSIC, a niche bare-bones particle swarm optimization (NBPSO) for multimodal peaks search is proposed to improve MUSIC, which is used to compute the frequency values of fault-related and fundamental components in stator current signal. Second, using these frequency values, a fault current signal model is established to convert the magnitude estimation problem into a linear least-squares problem. On this basis, the amplitudes and phases of fault-related and fundamental components could be estimated accurately with the singular value decomposition (SVD). A simulation signal is used to test the new method and the results show that the proposed method not only has higher frequency resolution, but also improves estimation accuracy of parameters greatly even with short data window. Finally, experiments for a real induction motor are performed, and the effectiveness and superiority of the proposed method are proved again.


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## Body

## 1. Introduction

Induction motors have been widely applied in the fields of industry and agriculture because of its simple structure, long life, low cost, and so on. In fact, most motors operate in a poor working environment, such as overloading, frequently starting and braking. Meanwhile, due to the too much mechanical and thermal stress, as well as some inherent defects of the rotor, the appearance of BRB or end ring cracking is a common phenomenon. During early stage, such as one broken bar, this type of fault may not show any symptoms. If no detection system or program, such fault will gradually deteriorate, causing other faults or leading to the sudden collapse [1–3]. It is known that unplanned downtime often causes a great economic loss [3]. So BRB fault detection for induction motor is necessary in early stage.When BRB failure occurs, additional current components, whose frequency arefbrk=(1±2ks)f1, will appear in the stator current. Here,f1 is the power frequency,sis the slip, andk is a positive integer. These current components are viewed as the features of BRB fault. Since the stator current signal is collected easily, its spectrum analyzed by the discrete Fourier transform (DFT) is widely used to detect BRB fault. However, when the motor is in the steady state, the small slip makes the frequencies of the strongest fault-related components very close to the fundamental frequency. Meanwhile, they are relatively weak and easy to be submerged by the leakage of fundamental harmonic or noise. The above two points expect signal processing methods to have functions of high frequency resolution and elimination of the fundamental leakage. For the influence of fundamental harmonic, many distinctive detection methods based on DFT are proposed to eliminate its influence [4–9]. The main idea of these methods is to filter the fundamental harmonic directly or convert it into DC component, and then the DFT is used to analyze the preprocessed current signal. In other papers, filtered Park’s vector modulus [10], product of current and virtual flux [11] and instantaneous power [12] analyzed through DFT have been used in BRB detection. Although these methods improve the reliability of the BRB detection, they cannot avoid the inherent shortcoming of DFT. For instance, the frequency resolution is inversely proportional to the sampling time. This means that, it requires a long enough sampling time and large memory space to get a high frequency resolution. However, in real application, long sampling time will cause the increase of the probability of disturbances, affecting the reliability of fault detection [13]. To reduce the memory space and compute cost, a sub-Nyquist strategy with reduced data length [14] and a diagnostic signal designated as the reduced envelope of the stator current [15] are introduced into detection methods. At the same time, harmonic order tracking analysis [15, 16] and side-lobes leakage phenomenon of the sliding DFT [17] are used in BRB detection to lessen the dependence on frequency resolution. Another problem of DFT-based methods is the amplitude estimation of the fault-related components which is used to calculate the fault severity. Due to spectrum leakage and fence effect, DFT has no ability to estimate them accurately with short-time data window [13].MUSIC is one of the modern spectral estimation methods known as subspace techniques. Due to the original signal extrapolation ability, its frequency resolution can get rid of the limitation of the data window. Therefore, MUSIC algorithm also was applied in the analysis of stator current [18], radial flux signal [19], and zero-sequence current [20] for BRB detection. To eliminate the influence of the fundamental harmonic, the eigenvector corresponding to fundamental harmonic in signal subspace was moved to the noise subspace to construct a new noise subspace [21]. This improved algorithm cannot only eliminate the influence of fundamental harmonic, but also improve the frequency resolution. Combining other techniques, such as spectrum zooming technique [22] and empirical mode decomposition [23], is another way to improve detection performance of the MUSIC. Although MUSIC has been applied successfully in the detection of BRB fault, it still has two shortcomings: (1) due to the fixed-step traversal search, it is not efficient for the spectrum peak search; (2) MUSIC cannot determine the other parameter values of each component, such as initial phase and amplitude. To solve the 2th-problem, an improved MUSIC algorithm which combines MUSIC with the pattern search algorithm is proposed [24]. However, this method needs a good starting point by DFT, and a long computational time.In this paper, MUSIC, particle swarm intelligence search and least-square technology are conjugated to improve estimate accuracy of the parameters of fault-related and fundamental components for BRB fault detection. For the shortages of MUSIC, we modify MUSIC from two aspects. To improve the efficiency of the spectrum peak search, a niche bare-bones particle swarm optimization (NBPSO) is proposed, replacing the fixed-step traversal search; on the other hand, the least-squares criterion is used to estimate the magnitude and phase of each component in the stator current. In simulation test, the DFT and traditional MUSIC are also selected for comparison. Finally, the proposed method combining the improved MUSIC with least-squares magnitude estimation is applied to the BRB fault detection for a real induction motor.


## 2. Related Concepts and Methods

### 2.1. Multiple Signal Classification

Multiple signal classification (MUSIC) was raised by Schmidt R. O. in 1986 [25]. It is a subspace method that can detect the frequencies of complex sinusoids using matrix eigenvalue decomposition. Its main idea is described as follows. First, the information space of observation signal is divided into two orthogonal subspaces by eigenvalue decomposition, i.e., the signal subspace and noise subspace; second, the spectrum is structured with the orthogonality of two subspaces to estimate the frequency of each component in the signal. MUSIC assumes that the discrete-time signal can be represented as a data model, that is(1)yn=∑i=1P/2Aicos⁡2πFin+φi+hn,where n=1,2,3,…; P/2 is the number of pure sinusoids;Ai, Fi,φi are the amplitude, frequency, and initial phase of thei-thpure sinusoids, respectively;h(n)is the complex white noise with a zero mean and a varianceτ2.According to Euler’s formula,y(n) can also be expressed as(2)yn=∑i=1Paiej2πfin+φi+hn,where ai=Ai/2, fi=Fi,fP/2+i=-Fi, for 1≤i≤P/2.ForM serial observed signal, (2) is constructed as(3)Yn=ynyn+1⋯yn+M-1=∑i=1Psin+hn,where M>>P, si(n)=[1,ej2πfi,ej4πfi,…,ej2π(M-1)fi]aiej2πfin.If setD(fi)=[1,ej2πfi,ej4πfi,…,ej2π(M-1)fi]T, FD=[D(f1),D(f2),…,D(fM)], A=[a1ej2πf1n,a2ej2πf2n,…,aMej2πfMn],Y(n) is rewrote as(4)Yn=FD·A+hn.Here, FD is aM×PVander monde matrix and also a space spanned by D(fi).The autocorrelation matrix ofY(n) is(5)Ryy=EYnYTn=FD·EAAT·FDT+τ2I,whereT represents the matrix conjugate transpose,E is the mathematical expectation, and I is an identity matrix.The eigenvalueΣ={λ1≥λ2≥…≥λP≥λP+1≥…≥λM} and corresponding eigenvector USN={U1,U2,…,UM} are derived from eigenvalue decomposition of the Ryy. That is(6)Ryy=USNΣUSNT=USUNΣS00ΣNUSUNT=USΣSUST+UNΣNUNT.In the space USN, the subspace US, as well as the space FD, spanned by eigenvectors {U1,U2,…,UP} corresponding to Top-P larger eigenvalues is the signal subspace. The subspace UN spanned by eigenvectors {UP+1,UP+2,…,UM} corresponding to the remaining eigenvalues is the noise subspace.SinceUS and UN are orthogonal, the vector of each signal component is also orthogonal to noise subspace, which is(7)Dfi⊥UN.According to (7), the MUSIC pseudospectrum is defined as(8)PMf=1DfT·UN2.For one pure sinusoid component included in signaly(n), if its frequency isfi, PM(f) will have a peak atfi. Therefore, the signal frequency estimation can be achieved by searching the peaks of PM(f) with the step∆f. In a real application, the Ryy need to be estimated using observation data.The specific steps of traditional MUSIC are described as follows:Step 1.
Take a group of observation data and estimate the autocorrelation matrixRyy.Step 2.
Perform the eigenvalue decomposition ofRyy to generate the signal subspace US and the noise subspace UN.Step 3.
According to (8), calculate MUSIC pseudospectrum with the step∆f.Step 4.
Find all the peaks in the pseudospectrum, and then determine the frequencies of all pure sinusoids.


### 2.2. Particle Swarm Optimization

Particle swarm optimization (PSO) is a global optimization technique simulating behavior of biological colony [26]. For PSO, each particle represents a solution and has three intrinsic properties. Two of them are the position and velocity, respectively, represented by the vectors xi=(xi1,xi2,…,xiD) and vi=(vi1,vi2,…,viD). The third is the fitness that is determined by the fitness function. Each particle keeps flying in feasible space and adjusts its flight path according to two extremum particles. One is its personal best position pi, i.e., its own flying experiences. The other is the global best position pg, i.e., its companions’ flying experiences. PSO is an optimization tool based on iteration. For the (t+1)-th iteration, the position and velocity of each particle are updated by the following equations:(9)vi,jt+1=wvi,jt+c1r1pi,jt-xi,jt+c2r2pg,jt-xi,jt(10)xi,jt+1=xi,jt+vi,jt+1,wherew is an inertia weight,c1,c2 are two learning factors,r1,r2 are two random numbers in [0,1], vijϵ[-vmax,vmax], vmax is maximum velocity set by users, i=1,2,…,K,K is the number of the particle swarm, j=1,2,…,D, D is the dimension of the search space.To simplify PSO, a bare-bones PSO algorithm (BBPSO) was proposed in 2003 [27]. In BBPSO, a Gaussian distribution about the position information ofpi andpg is used to update the positions of all particles.(11)xi,jt+1=Nμi,jt,σi,j2twhereμi,j(n)=(pi,j(n)+pg,j(n))/2 is the mean of the Gaussian distribution and σ2i,j(n)=|pi,j(n)-pg,j(n)| is the standard deviation of the Gaussian distribution. Compared with the standard PSO algorithm, BBPSO need not to set up control parameters, such as inertia weight and learning factor. Therefore, BBPSO is more suitable for the engineering application.


## 2.1. Multiple Signal Classification

Multiple signal classification (MUSIC) was raised by Schmidt R. O. in 1986 [25]. It is a subspace method that can detect the frequencies of complex sinusoids using matrix eigenvalue decomposition. Its main idea is described as follows. First, the information space of observation signal is divided into two orthogonal subspaces by eigenvalue decomposition, i.e., the signal subspace and noise subspace; second, the spectrum is structured with the orthogonality of two subspaces to estimate the frequency of each component in the signal. MUSIC assumes that the discrete-time signal can be represented as a data model, that is(1)yn=∑i=1P/2Aicos⁡2πFin+φi+hn,where n=1,2,3,…; P/2 is the number of pure sinusoids;Ai, Fi,φi are the amplitude, frequency, and initial phase of thei-thpure sinusoids, respectively;h(n)is the complex white noise with a zero mean and a varianceτ2.According to Euler’s formula,y(n) can also be expressed as(2)yn=∑i=1Paiej2πfin+φi+hn,where ai=Ai/2, fi=Fi,fP/2+i=-Fi, for 1≤i≤P/2.ForM serial observed signal, (2) is constructed as(3)Yn=ynyn+1⋯yn+M-1=∑i=1Psin+hn,where M>>P, si(n)=[1,ej2πfi,ej4πfi,…,ej2π(M-1)fi]aiej2πfin.If setD(fi)=[1,ej2πfi,ej4πfi,…,ej2π(M-1)fi]T, FD=[D(f1),D(f2),…,D(fM)], A=[a1ej2πf1n,a2ej2πf2n,…,aMej2πfMn],Y(n) is rewrote as(4)Yn=FD·A+hn.Here, FD is aM×PVander monde matrix and also a space spanned by D(fi).The autocorrelation matrix ofY(n) is(5)Ryy=EYnYTn=FD·EAAT·FDT+τ2I,whereT represents the matrix conjugate transpose,E is the mathematical expectation, and I is an identity matrix.The eigenvalueΣ={λ1≥λ2≥…≥λP≥λP+1≥…≥λM} and corresponding eigenvector USN={U1,U2,…,UM} are derived from eigenvalue decomposition of the Ryy. That is(6)Ryy=USNΣUSNT=USUNΣS00ΣNUSUNT=USΣSUST+UNΣNUNT.In the space USN, the subspace US, as well as the space FD, spanned by eigenvectors {U1,U2,…,UP} corresponding to Top-P larger eigenvalues is the signal subspace. The subspace UN spanned by eigenvectors {UP+1,UP+2,…,UM} corresponding to the remaining eigenvalues is the noise subspace.SinceUS and UN are orthogonal, the vector of each signal component is also orthogonal to noise subspace, which is(7)Dfi⊥UN.According to (7), the MUSIC pseudospectrum is defined as(8)PMf=1DfT·UN2.For one pure sinusoid component included in signaly(n), if its frequency isfi, PM(f) will have a peak atfi. Therefore, the signal frequency estimation can be achieved by searching the peaks of PM(f) with the step∆f. In a real application, the Ryy need to be estimated using observation data.The specific steps of traditional MUSIC are described as follows:Step 1.
Take a group of observation data and estimate the autocorrelation matrixRyy.Step 2.
Perform the eigenvalue decomposition ofRyy to generate the signal subspace US and the noise subspace UN.Step 3.
According to (8), calculate MUSIC pseudospectrum with the step∆f.Step 4.
Find all the peaks in the pseudospectrum, and then determine the frequencies of all pure sinusoids.


## 2.2. Particle Swarm Optimization

Particle swarm optimization (PSO) is a global optimization technique simulating behavior of biological colony [26]. For PSO, each particle represents a solution and has three intrinsic properties. Two of them are the position and velocity, respectively, represented by the vectors xi=(xi1,xi2,…,xiD) and vi=(vi1,vi2,…,viD). The third is the fitness that is determined by the fitness function. Each particle keeps flying in feasible space and adjusts its flight path according to two extremum particles. One is its personal best position pi, i.e., its own flying experiences. The other is the global best position pg, i.e., its companions’ flying experiences. PSO is an optimization tool based on iteration. For the (t+1)-th iteration, the position and velocity of each particle are updated by the following equations:(9)vi,jt+1=wvi,jt+c1r1pi,jt-xi,jt+c2r2pg,jt-xi,jt(10)xi,jt+1=xi,jt+vi,jt+1,wherew is an inertia weight,c1,c2 are two learning factors,r1,r2 are two random numbers in [0,1], vijϵ[-vmax,vmax], vmax is maximum velocity set by users, i=1,2,…,K,K is the number of the particle swarm, j=1,2,…,D, D is the dimension of the search space.To simplify PSO, a bare-bones PSO algorithm (BBPSO) was proposed in 2003 [27]. In BBPSO, a Gaussian distribution about the position information ofpi andpg is used to update the positions of all particles.(11)xi,jt+1=Nμi,jt,σi,j2twhereμi,j(n)=(pi,j(n)+pg,j(n))/2 is the mean of the Gaussian distribution and σ2i,j(n)=|pi,j(n)-pg,j(n)| is the standard deviation of the Gaussian distribution. Compared with the standard PSO algorithm, BBPSO need not to set up control parameters, such as inertia weight and learning factor. Therefore, BBPSO is more suitable for the engineering application.


## 3. Improved MUSIC Based on NBPSO

### 3.1. Niche BBPSO (NBPSO)

Actually, the spectrum peak search of MUSIC is a multimodal optimization problem. One of feasible methods for this problem is designing a multimodal algorithm to find all global optima and local optima as many as possible. Both the standard PSO and BBPSO are global optimization algorithms. However, they can only find out one optimal solution. In their evolution process, because of the selective pressure, they cannot get more than one global optimal solution or some better local optimal solutions. Therefore, for solving the multimodal optimization problem, scholars have introduced the niche technology into the evolutionary algorithm and presented many niche strategies, such as the preselection technique, the crowding strategy, the fitness sharing strategy, and the species conserving strategy [28].According to the characteristics of BRB fault signal, an improved seed selection strategy is proposed in this section based on [29]. Meanwhile, we introduce it into BBPSO to form a new algorithm (named, niche bare-bones particle swarm optimization, NBPSO), which can realize the multimodal search. Comparing to [29], three differences in NBPSO are introduced: (1) the seeds are selected from the personal best positions of all particles to prevent oscillation; (2) define the species similarity σ to maintain the diversity of seeds, where σ is the Euclidean distance between any two seeds; (3) the formation of subpopulations is not dependent on the niche radius. Instead, the nonseed particles are attributed to the nearest seed. The pseudocode of improved seed selection strategy is shown in Algorithm 1. Algorithm 1 is reproduced from the published article titled “Adaptively Choosing Neighbourhood Bests Using Species in a Particle Swarm Optimizer for Multimodal Function Optimization,” which is cited in the references [28].Algorithm 1:The algorithm for determining the species seeds.
Input: Spbest-containing all particles sorted in decreasing order fitness Output: Xs-containing dominating particles identified as species seeds  begin Xs = ⌀; while (not reaching the end ofSpbest) do found← FALSE; forallx∗∈Xsdo if(d(x∗,x)≤σ∗) then found← TRUE; break; end (if) end (for) if (not found) then letXs  ←  Xs  ∪x; end (if) end (while) endThe seed selection strategy in Algorithm1 is executed in each generation of evolutionary process. First, arrange all the personal best positions in the descending order with the fitness values to form a set Spbest. Second, calculate the Euclidean distances between each individual in Spbest and element in the seed set Xs. If all Euclidean distances are larger than σ∗, the current personal best position is added to Xs.The specific steps of NBPSO are described as follows.Step 1.
Initialize the positions of all particles, personal best positions and seed setXs. Set algorithm parameters including the population size, maximum iteration, and species similarity threshold σ∗.Step 2.
Calculate particles’ fitness.Step 3.
Update particles’ personal best positionspi.Step 4.
Arrange allpi in the descending order according to the fitness values.Step 5.
Determine and update the seeds according to Algorithm1.Step 6.
Calculate the distance between each particle and each seed to determine which seed the particle belongs to.Step 7.
Update the positions of particles according to (11), where pg is replaced with a seed.Step 8.
If the stop condition (the fitness error is less than the threshold or the iteration number is more than the maximum number) is met, stop the iterative procedure and output all the global optimal positions, some better local optimal positions and their fitness values. Otherwise, return to Step 2 and continue the search.


### 3.2. Improved MUSIC

In this section, we replace the fixed-step traversal with NBPSO in MUISC to improve the accuracy and speed of peak search. According to the characters of MUSIC spectrum, the fitness function is defined as(12)Fitnessxi=exiT·UN·UNT·exi.The specific steps of improved MUSIC are described as follows:Step 1.
Take a group of observation data and estimate the autocorrelation matrixRyy.Step 2.
Perform the eigenvalue decomposition ofRyy to generate the signal subspace US and the noise subspace UN.Step 3.
Encodefi in (2) with real numbers and choose (12) as the fitness function.Step 4.
Search multispectral peaks using NBPSO to obtain the frequency of each component in signaly(n).


## 3.1. Niche BBPSO (NBPSO)

Actually, the spectrum peak search of MUSIC is a multimodal optimization problem. One of feasible methods for this problem is designing a multimodal algorithm to find all global optima and local optima as many as possible. Both the standard PSO and BBPSO are global optimization algorithms. However, they can only find out one optimal solution. In their evolution process, because of the selective pressure, they cannot get more than one global optimal solution or some better local optimal solutions. Therefore, for solving the multimodal optimization problem, scholars have introduced the niche technology into the evolutionary algorithm and presented many niche strategies, such as the preselection technique, the crowding strategy, the fitness sharing strategy, and the species conserving strategy [28].According to the characteristics of BRB fault signal, an improved seed selection strategy is proposed in this section based on [29]. Meanwhile, we introduce it into BBPSO to form a new algorithm (named, niche bare-bones particle swarm optimization, NBPSO), which can realize the multimodal search. Comparing to [29], three differences in NBPSO are introduced: (1) the seeds are selected from the personal best positions of all particles to prevent oscillation; (2) define the species similarity σ to maintain the diversity of seeds, where σ is the Euclidean distance between any two seeds; (3) the formation of subpopulations is not dependent on the niche radius. Instead, the nonseed particles are attributed to the nearest seed. The pseudocode of improved seed selection strategy is shown in Algorithm 1. Algorithm 1 is reproduced from the published article titled “Adaptively Choosing Neighbourhood Bests Using Species in a Particle Swarm Optimizer for Multimodal Function Optimization,” which is cited in the references [28].Algorithm 1:The algorithm for determining the species seeds.
Input: Spbest-containing all particles sorted in decreasing order fitness Output: Xs-containing dominating particles identified as species seeds  begin Xs = ⌀; while (not reaching the end ofSpbest) do found← FALSE; forallx∗∈Xsdo if(d(x∗,x)≤σ∗) then found← TRUE; break; end (if) end (for) if (not found) then letXs  ←  Xs  ∪x; end (if) end (while) endThe seed selection strategy in Algorithm1 is executed in each generation of evolutionary process. First, arrange all the personal best positions in the descending order with the fitness values to form a set Spbest. Second, calculate the Euclidean distances between each individual in Spbest and element in the seed set Xs. If all Euclidean distances are larger than σ∗, the current personal best position is added to Xs.The specific steps of NBPSO are described as follows.Step 1.
Initialize the positions of all particles, personal best positions and seed setXs. Set algorithm parameters including the population size, maximum iteration, and species similarity threshold σ∗.Step 2.
Calculate particles’ fitness.Step 3.
Update particles’ personal best positionspi.Step 4.
Arrange allpi in the descending order according to the fitness values.Step 5.
Determine and update the seeds according to Algorithm1.Step 6.
Calculate the distance between each particle and each seed to determine which seed the particle belongs to.Step 7.
Update the positions of particles according to (11), where pg is replaced with a seed.Step 8.
If the stop condition (the fitness error is less than the threshold or the iteration number is more than the maximum number) is met, stop the iterative procedure and output all the global optimal positions, some better local optimal positions and their fitness values. Otherwise, return to Step 2 and continue the search.


## 3.2. Improved MUSIC

In this section, we replace the fixed-step traversal with NBPSO in MUISC to improve the accuracy and speed of peak search. According to the characters of MUSIC spectrum, the fitness function is defined as(12)Fitnessxi=exiT·UN·UNT·exi.The specific steps of improved MUSIC are described as follows:Step 1.
Take a group of observation data and estimate the autocorrelation matrixRyy.Step 2.
Perform the eigenvalue decomposition ofRyy to generate the signal subspace US and the noise subspace UN.Step 3.
Encodefi in (2) with real numbers and choose (12) as the fitness function.Step 4.
Search multispectral peaks using NBPSO to obtain the frequency of each component in signaly(n).


## 4. Least-Squares Amplitude Estimation and New BRB Fault Detection Method

### 4.1. Amplitude Estimation Using Least-Squares Criterion

Under the condition of BRB fault, there are three major harmonics, the fundamental harmonic and the strongest fault-related components, around 50 Hz. So the stator current signal filtered by a band-pass filter can be represented as(13)imt=α1ej2πf1t+α2e-j2πf1t+α3ej2πf1-2st+α4e-j2πf1-2st+α5ej2πf1+2st+α6e-j2πf1+2st,wheref1-2s and f1+2s are the frequencies of the strongest fault-related components and α=α1,α2,…,α6T is the model parameter. Since the frequency of each component in stator current can be determined accurately with the NBPSO-based MUSIC,f1, f1-2s and f1+2s are considered to be known parameters. Based on least-squares criterion, define an error function f(α) between model currentim and actual currentis. To make theim fit theis, the error function value is required minimum, i.e.,(14)min⁡fα=min⁡∑i=1Nri2,where ri=is(ti)-im(ti),ti=i·Ts, Ts andN are sampling interval and points.If the error functionf(α) reaches to a minimum, the amplitude and phase of each component inis can be estimated from its solution α∗. Therefore, how to find the least-squares solution α∗ is the first mission. Fortunately, (14) is a linear least-squares problem, becausef1, f1-2s and f1+2s can be predetermined.Many methods have been proposed for the linear least-squares problem, such as the SVD method and the QR decomposition. For the stability and reliability, the SVD method has been widely used. This section also adopts it.The parameters of fundamental harmonic and BRB fault-related components are estimated by the least-squares method as follows.(1)
According to the results from improved MUSIC, construct the design matrixH as follows:(15)H=ej2πf1t1e-j2πf1t1ej2πf1-2st1⋯e-j2πf1+2st1ej2πf1t2e-j2πf1t2ej2πf1-2st2⋯e-j2πf1+2st2⋮⋮⋮⋯⋮ej2πf1tNe-j2πf1tNej2πf1-2stN⋯e-j2πf1+2stN.(2)
DecomposeH using SVD to make H=UΣVT, here U and V are both orthogonal matrixes, and Σ=diag(σ1,σ2,…,σ6).(3)
Calculateα∗=V·Σ-1·UT·is.(4)
Estimate the amplitude and phase of each component inis, by calculating the modulus and argument of α∗.


### 4.2. Process of the Proposed Detection Method

Combining improved MUSIC and least-squares amplitude estimation, a new BRB fault detection method of the induction motor is proposed. As shown in Figure1, the proposed scheme consists of four modules: Improved MUSIC, Current model, Least-square estimator, and Fault identification. In the Improved MUSIC module, the frequencies of fundamental and fault-related components are determined. Using the frequency values, the model currentim is built. The amplitude estimate is calculated by least-square estimator. In last module, frequencies and amplitudes are used to identify BRB fault.Figure 1
Block diagram of the proposed method.The specific steps of the new fault detection method are described as follows:Step 1.
Sample the stator current of one phase and estimate the autocorrelation matrixRyy using current data.Step 2.
Perform the eigenvalue decomposition ofRyy to generate the signal subspace S and the noise subspace G.Step 3.
Carry out multispectral peaks search in frequency domain by NBPSO to determine the frequency of each component in current signal.Step 4.
Corresponding to each frequency from Step 3, build the current model.Step 5.
Estimate the amplitude of each component accurately using least-squares estimation method.Step 6.
Based on the amplitude ratio of the sideband components and fundamental harmonic, it is determined whether the BRB fault occurs or not.


## 4.1. Amplitude Estimation Using Least-Squares Criterion

Under the condition of BRB fault, there are three major harmonics, the fundamental harmonic and the strongest fault-related components, around 50 Hz. So the stator current signal filtered by a band-pass filter can be represented as(13)imt=α1ej2πf1t+α2e-j2πf1t+α3ej2πf1-2st+α4e-j2πf1-2st+α5ej2πf1+2st+α6e-j2πf1+2st,wheref1-2s and f1+2s are the frequencies of the strongest fault-related components and α=α1,α2,…,α6T is the model parameter. Since the frequency of each component in stator current can be determined accurately with the NBPSO-based MUSIC,f1, f1-2s and f1+2s are considered to be known parameters. Based on least-squares criterion, define an error function f(α) between model currentim and actual currentis. To make theim fit theis, the error function value is required minimum, i.e.,(14)min⁡fα=min⁡∑i=1Nri2,where ri=is(ti)-im(ti),ti=i·Ts, Ts andN are sampling interval and points.If the error functionf(α) reaches to a minimum, the amplitude and phase of each component inis can be estimated from its solution α∗. Therefore, how to find the least-squares solution α∗ is the first mission. Fortunately, (14) is a linear least-squares problem, becausef1, f1-2s and f1+2s can be predetermined.Many methods have been proposed for the linear least-squares problem, such as the SVD method and the QR decomposition. For the stability and reliability, the SVD method has been widely used. This section also adopts it.The parameters of fundamental harmonic and BRB fault-related components are estimated by the least-squares method as follows.(1)
According to the results from improved MUSIC, construct the design matrixH as follows:(15)H=ej2πf1t1e-j2πf1t1ej2πf1-2st1⋯e-j2πf1+2st1ej2πf1t2e-j2πf1t2ej2πf1-2st2⋯e-j2πf1+2st2⋮⋮⋮⋯⋮ej2πf1tNe-j2πf1tNej2πf1-2stN⋯e-j2πf1+2stN.(2)
DecomposeH using SVD to make H=UΣVT, here U and V are both orthogonal matrixes, and Σ=diag(σ1,σ2,…,σ6).(3)
Calculateα∗=V·Σ-1·UT·is.(4)
Estimate the amplitude and phase of each component inis, by calculating the modulus and argument of α∗.


## 4.2. Process of the Proposed Detection Method

Combining improved MUSIC and least-squares amplitude estimation, a new BRB fault detection method of the induction motor is proposed. As shown in Figure1, the proposed scheme consists of four modules: Improved MUSIC, Current model, Least-square estimator, and Fault identification. In the Improved MUSIC module, the frequencies of fundamental and fault-related components are determined. Using the frequency values, the model currentim is built. The amplitude estimate is calculated by least-square estimator. In last module, frequencies and amplitudes are used to identify BRB fault.Figure 1
Block diagram of the proposed method.The specific steps of the new fault detection method are described as follows:Step 1.
Sample the stator current of one phase and estimate the autocorrelation matrixRyy using current data.Step 2.
Perform the eigenvalue decomposition ofRyy to generate the signal subspace S and the noise subspace G.Step 3.
Carry out multispectral peaks search in frequency domain by NBPSO to determine the frequency of each component in current signal.Step 4.
Corresponding to each frequency from Step 3, build the current model.Step 5.
Estimate the amplitude of each component accurately using least-squares estimation method.Step 6.
Based on the amplitude ratio of the sideband components and fundamental harmonic, it is determined whether the BRB fault occurs or not.


## 5. Results and Discussion Based on the Simulation and Experiment

### 5.1. Simulation Results and Discussion

To test the performance of the new detection method, a simulated current is built as follows.(16)ibrt=I1cos⁡2πf1t+φ1+Ibp1cos⁡2π1-2sf1t+φbp1+Ibn1cos⁡2π1+2sf1t+φbn1+ntwhereI1=10,f1=50,φ1=π/3 are the parameters of fundamental harmonic. When the BRB fault is slight, the amplitudes of components atfbr1 are much smaller than the fundamental harmonic’s, and their frequency intervals are also small. Therefore, setIbp1=Ibn1=0.2, (1-2s)f1=49.5438, (1+2s)f1=50.4562,φbp1=2π/3 andφbn1=8π/9. Meanwhile,n(t) is a random noise with uniform distribution in the interval [-0.01, 0.01]. The sampling frequency of the simulated signal is 250Hz; the size of the data window is 500. The NBPSO parameter settings are shown in Table 1. To compare the performance, DFT and traditional MUSIC, which have been widely applied in BRB fault detection, are also performed in this simulated signal.Table 1
Parameter configurations of NBPSO.
Algorithm Population size Maximum iteration Other parameters NBPSO 60 30 Seed numbersn=3, species similarity threshold σ∗=0.4.Data processing results of three methods are shown in Figure2. The DFT result is not satisfactory. As Figure 2(a) shows, there is only one peak that is the fundamental harmonic. The fault-related components ((1±2s)f1) which are submerged completely by the fundamental harmonic are not distinguished. The reason for this case is that the frequency resolution with 2s data window is 0.5Hz, while the frequency interval between (1±2s)f1 components andf1 component is only 0.4562Hz. Therefore, the detection reliability of BRB fault using DFT method is low, when the data window is short. Fortunately, DFT method has very high accuracy of fundamental frequency as well as traditional MUSIC. The reason is that 50Hz is an integer multiple of the frequency resolution.Figure 2
Simulated current spectrum analyzed by three methods. (a) DFT spectrum, (b) MUSIC spectrum with search step ∆f=0.1 Hz, (c) MUSIC spectrum with search step ∆f=0.01 Hz, (d) MUSIC spectrum with search step ∆f=0.001 Hz, (e) spectrum of the proposed method.
(a) (b) (c) (d) (e)For the traditional MUSIC method, it has a very high frequency resolution, even with a short-time data window. As Figures2(b), 2(c), and 2(d) show, the fault-related peaks are identified clearly. Due to high frequency resolution MUSIC can use shorter data window to achieve the desired frequency resolution, avoiding the impact of load fluctuations or noise in fault detection effectively. This advantage makes MUSIC suitable for the spectrum analysis of the stator current signal, especially in the BRB fault detection for the induction motor. However, the traditional MUSIC requires a predetermined peak search step ∆f and then traverses the whole frequency domain to find spectrum peaks. As a result of fixed-step search, the frequencies of all spectrum peaks are integer multiples of the step size. Thus frequency precision is limited. With the increase of the step size, the frequency error increases, and the peak amplitudes reduce in the spectrum. It means that the reliability of BRB detection is also limited by peak search step ∆f; that is, too large search step makes the spectrum unable to identify some tiny fault-related components. Moreover, it is clear that the amplitudes of spectrum peaks obtained by MUSIC make no sense in identification of each component, even not reflecting the real magnitude value of each component.Figure2(e) shows the result of the proposed method. As can be seen from the graph, compared with the traditional MUSIC (search step ∆f=0.001 Hz), the improved MUSIC provides a higher frequency resolution. Meanwhile, the proposed method allows it to precisely estimate the amplitudes and phases of components, while traditional MUSIC cannot. Therefore, the proposed method combining MUSIC, NBPSO, and least-squares estimation is more reliable in the BRB fault detection of the induction motor.Since NBPSO is a stochastic optimization algorithm, it is often necessary that the proposed method is executed 50 times independently to test its robustness. The statistical results are shown in Table2.Table 2
Statistical results of the proposed method.
Signal parameters Actual values Estimation result (average) Average error Maximum error f 1/Hz 50.0000 50.0000 1.48×10−7 3.78×10−5  I 1 /A 10 10.0000 5.47×10−6 8.33×10−5  φ 1 /rad π/3 1.0472 3.67×10−7 2.61×10−4  (1-2s)f1/Hz 49.5438 49.5438 5.36×10−5 7.40×10−4  I b p 1 / A 0.2 0.2000 5.00×10−6 2.89×10−4  φ b p 1  / r a d 2π/3 2.0941 3.41×10−4 7.98×10−3  (1+2s)f1/Hz 50.4562 50.4562 2.81×10−5 7.87×10−4  I b n 1  / A 0.2 0.2000 9.72×10−6 1.53×10−4  φ b n 1  / r a d 8π/9 2.7923 1.93×10−4 7.52×10−3Comparing Figures2(b), 2(c), and 2(d) with Table 2, it can be seen that the spectrum peak search using NBPSO improves frequency accuracy greatly. In the statistical results, the maximum frequency error is only 7.87×10−4 even 3.78×10−5 for fundamental frequency, which is more accurate than the fixed-step search of the traditional MUSIC with a step size of 0.001Hz. The average frequency error which can better reflect the accuracy of the proposed method reaches 5.36×10−5 (maximum of three frequency errors). By contrast, the fixed-step search requires a step size of 0.00001 Hz to achieve the same accuracy. Therefore, the improved MUSIC based on NBPSO has higher search efficiency and accuracy. It is more suitable for the BRB fault detection of the induction motor. Compared with traditional MUSIC, another advantage is that the proposed method can accurately estimate the amplitudes and initial phases of the fundamental and fault-related components. As Table 2 shows, the estimation accuracy of amplitudes reaches 9.72×10−6 or higher, the phases’ reaches 3.41×10−4 or higher. In addition, to verify this method robustness, we also randomly changed the fundamental frequency, slip, and amplitude of each component in (16). The calculation accuracy of final results is consistent with Table 2.Table3 compares the three algorithms’ computational costs. DFT has absolute advantage in this comparison; however, its frequency resolution is low with short data window. For the MUSIC, the search time increases sharply with the decrease of step. Namely, the improvement of accuracy is at the expense of computation. In contrast, the calculation cost of the proposed method is only 0.365s, which is equivalent to the traditional MUSIC in the step Δf=0.1 Hz.Table 3
Computation cost comparison for the methods.
Methods DFT MUSIC with different search step Proposed method (average time in 50 runs) ∆f=0.1 ∆f=0.01 ∆f=0.001 Computational costt/s 0.001 0.101 0.814 7.732 0.365Therefore, the proposed method obtains a prominent improvement on solution accuracy and search efficiency contrasting with traditional MUSIC.


### 5.2. Experimental Results and Discussion

The main parameters of the experimental induction motor (MODEL Y132M-4) are shown in Table4. The experiments are carried out respectively under the conditions of half load with 1 broken bar (s=0.022) and full load with 1 broken bar (s=0.04133). The sampling interval of current signal is 7ms, and data length is 200. The waveforms of stator current signal of phase A in the two states are shown in Figure 3. The NBPSO parameters are the same as that in Table 1.Table 4
Specifications of the test motor.
Related power/kW Related voltage/V Related current/A Related speed/ (r/min) No. of rotor bars 7.5 380 15.4 1440 32Figure 3
Waveform of stator current. (a) Full load with 1 broken bar. (b) Half load with 1 broken bar.
(a) (b)Figure4 shows the results analyzed by MUSIC, DFT, and the proposed algorithm, respectively, on the condition of full load. In this condition, fault feature frequencies are far from the fundamental frequency, and the amplitudes of fault feature components are maximal at full load. It means that fault features are easy to be found. Thus we can see that the three algorithms can distinguish the fault feature components. However, the fault-related peaks are clearer in current spectrums analyzed by traditional MUSIC and the proposed method. The reason is that they can overcome the spectrum leakage problem well. In addition, the algorithms based on MUSIC have better frequency resolution and calculation accuracy. Figure 4 also shows that whether for fundamental frequency or for fault feature frequencies, DFT has a larger error, but it has an absolute advantage in the computational time.Figure 4
Spectrum of stator current under the full load. (a) Spectrum of the proposed method. (b) MUSIC spectrum. (c) DFT spectrum.
(a) (b) (c)Comparing Figure4(a) with Figure 4(b), we can see that two methods all have a high accuracy in estimating the frequency of fault feature components. However, due to fixed-step search (step ∆f=0.001), the frequencies estimated with traditional MUSIC are only the integer multiples of Δf. Therefore, the estimation results of three spectrum peaks are 45.861Hz, 50.023Hz, and 54.224Hz in Figure 4(b). Comparing Figure 4(b) and Table 5, whether for the fundamental harmonic or for the fault-related components, the proposed method can find higher spectrum peaks (with the same signal subspace and noise subspace). This cannot only prove that the spectrum peaks found by traditional MUSIC are not the true spectrum peaks, but also verify that the improved MUSIC has a higher accuracy. Moreover, the proposed method can accurately determine the amplitude and initial phase of each component signal. In addition, it also has a higher computing speed. In this experiment, its computing time is only 0.113s while traditional MUSIC’s is 4.054s (Δf = 0.001).Table 5
Results of improved MUSIC.
Test condition Parameters Amplitudes of peaks/dB Frequencies of peaks /Hz Full load with 1 broken bar f 1 56.7599 50.0226 (1-2s)f1 -18.4726 45.8700 (1+2s)f1 -10.8207 54.2190  Half load with 1 broken bar f 1 70.5717 50.0047 (1-2s)f1 -15.1455 47.8055 (1+2s)f1 -14.5698 52.1934Figure5 shows the analysis results of the motor stator current under the half load with 1 broken bar. In this case, fault feature components are weaker and closer to fundamental harmonic in spectrum, so it is difficult to be extracted and highlighted. The characters of each method are consistent with the above analysis under the full load. Figure 5 also verifies the effectiveness, feasibility, and superiority of the proposed method.Figure 5
Spectrum of stator current under the half load. (a) Spectrum of the proposed method. (b) MUSIC spectrum. (c) DFT spectrum.
(a) (b) (c)


## 5.1. Simulation Results and Discussion

To test the performance of the new detection method, a simulated current is built as follows.(16)ibrt=I1cos⁡2πf1t+φ1+Ibp1cos⁡2π1-2sf1t+φbp1+Ibn1cos⁡2π1+2sf1t+φbn1+ntwhereI1=10,f1=50,φ1=π/3 are the parameters of fundamental harmonic. When the BRB fault is slight, the amplitudes of components atfbr1 are much smaller than the fundamental harmonic’s, and their frequency intervals are also small. Therefore, setIbp1=Ibn1=0.2, (1-2s)f1=49.5438, (1+2s)f1=50.4562,φbp1=2π/3 andφbn1=8π/9. Meanwhile,n(t) is a random noise with uniform distribution in the interval [-0.01, 0.01]. The sampling frequency of the simulated signal is 250Hz; the size of the data window is 500. The NBPSO parameter settings are shown in Table 1. To compare the performance, DFT and traditional MUSIC, which have been widely applied in BRB fault detection, are also performed in this simulated signal.Table 1
Parameter configurations of NBPSO.
Algorithm Population size Maximum iteration Other parameters NBPSO 60 30 Seed numbersn=3, species similarity threshold σ∗=0.4.Data processing results of three methods are shown in Figure2. The DFT result is not satisfactory. As Figure 2(a) shows, there is only one peak that is the fundamental harmonic. The fault-related components ((1±2s)f1) which are submerged completely by the fundamental harmonic are not distinguished. The reason for this case is that the frequency resolution with 2s data window is 0.5Hz, while the frequency interval between (1±2s)f1 components andf1 component is only 0.4562Hz. Therefore, the detection reliability of BRB fault using DFT method is low, when the data window is short. Fortunately, DFT method has very high accuracy of fundamental frequency as well as traditional MUSIC. The reason is that 50Hz is an integer multiple of the frequency resolution.Figure 2
Simulated current spectrum analyzed by three methods. (a) DFT spectrum, (b) MUSIC spectrum with search step ∆f=0.1 Hz, (c) MUSIC spectrum with search step ∆f=0.01 Hz, (d) MUSIC spectrum with search step ∆f=0.001 Hz, (e) spectrum of the proposed method.
(a) (b) (c) (d) (e)For the traditional MUSIC method, it has a very high frequency resolution, even with a short-time data window. As Figures2(b), 2(c), and 2(d) show, the fault-related peaks are identified clearly. Due to high frequency resolution MUSIC can use shorter data window to achieve the desired frequency resolution, avoiding the impact of load fluctuations or noise in fault detection effectively. This advantage makes MUSIC suitable for the spectrum analysis of the stator current signal, especially in the BRB fault detection for the induction motor. However, the traditional MUSIC requires a predetermined peak search step ∆f and then traverses the whole frequency domain to find spectrum peaks. As a result of fixed-step search, the frequencies of all spectrum peaks are integer multiples of the step size. Thus frequency precision is limited. With the increase of the step size, the frequency error increases, and the peak amplitudes reduce in the spectrum. It means that the reliability of BRB detection is also limited by peak search step ∆f; that is, too large search step makes the spectrum unable to identify some tiny fault-related components. Moreover, it is clear that the amplitudes of spectrum peaks obtained by MUSIC make no sense in identification of each component, even not reflecting the real magnitude value of each component.Figure2(e) shows the result of the proposed method. As can be seen from the graph, compared with the traditional MUSIC (search step ∆f=0.001 Hz), the improved MUSIC provides a higher frequency resolution. Meanwhile, the proposed method allows it to precisely estimate the amplitudes and phases of components, while traditional MUSIC cannot. Therefore, the proposed method combining MUSIC, NBPSO, and least-squares estimation is more reliable in the BRB fault detection of the induction motor.Since NBPSO is a stochastic optimization algorithm, it is often necessary that the proposed method is executed 50 times independently to test its robustness. The statistical results are shown in Table2.Table 2
Statistical results of the proposed method.
Signal parameters Actual values Estimation result (average) Average error Maximum error f 1/Hz 50.0000 50.0000 1.48×10−7 3.78×10−5  I 1 /A 10 10.0000 5.47×10−6 8.33×10−5  φ 1 /rad π/3 1.0472 3.67×10−7 2.61×10−4  (1-2s)f1/Hz 49.5438 49.5438 5.36×10−5 7.40×10−4  I b p 1 / A 0.2 0.2000 5.00×10−6 2.89×10−4  φ b p 1  / r a d 2π/3 2.0941 3.41×10−4 7.98×10−3  (1+2s)f1/Hz 50.4562 50.4562 2.81×10−5 7.87×10−4  I b n 1  / A 0.2 0.2000 9.72×10−6 1.53×10−4  φ b n 1  / r a d 8π/9 2.7923 1.93×10−4 7.52×10−3Comparing Figures2(b), 2(c), and 2(d) with Table 2, it can be seen that the spectrum peak search using NBPSO improves frequency accuracy greatly. In the statistical results, the maximum frequency error is only 7.87×10−4 even 3.78×10−5 for fundamental frequency, which is more accurate than the fixed-step search of the traditional MUSIC with a step size of 0.001Hz. The average frequency error which can better reflect the accuracy of the proposed method reaches 5.36×10−5 (maximum of three frequency errors). By contrast, the fixed-step search requires a step size of 0.00001 Hz to achieve the same accuracy. Therefore, the improved MUSIC based on NBPSO has higher search efficiency and accuracy. It is more suitable for the BRB fault detection of the induction motor. Compared with traditional MUSIC, another advantage is that the proposed method can accurately estimate the amplitudes and initial phases of the fundamental and fault-related components. As Table 2 shows, the estimation accuracy of amplitudes reaches 9.72×10−6 or higher, the phases’ reaches 3.41×10−4 or higher. In addition, to verify this method robustness, we also randomly changed the fundamental frequency, slip, and amplitude of each component in (16). The calculation accuracy of final results is consistent with Table 2.Table3 compares the three algorithms’ computational costs. DFT has absolute advantage in this comparison; however, its frequency resolution is low with short data window. For the MUSIC, the search time increases sharply with the decrease of step. Namely, the improvement of accuracy is at the expense of computation. In contrast, the calculation cost of the proposed method is only 0.365s, which is equivalent to the traditional MUSIC in the step Δf=0.1 Hz.Table 3
Computation cost comparison for the methods.
Methods DFT MUSIC with different search step Proposed method (average time in 50 runs) ∆f=0.1 ∆f=0.01 ∆f=0.001 Computational costt/s 0.001 0.101 0.814 7.732 0.365Therefore, the proposed method obtains a prominent improvement on solution accuracy and search efficiency contrasting with traditional MUSIC.


## 5.2. Experimental Results and Discussion

The main parameters of the experimental induction motor (MODEL Y132M-4) are shown in Table4. The experiments are carried out respectively under the conditions of half load with 1 broken bar (s=0.022) and full load with 1 broken bar (s=0.04133). The sampling interval of current signal is 7ms, and data length is 200. The waveforms of stator current signal of phase A in the two states are shown in Figure 3. The NBPSO parameters are the same as that in Table 1.Table 4
Specifications of the test motor.
Related power/kW Related voltage/V Related current/A Related speed/ (r/min) No. of rotor bars 7.5 380 15.4 1440 32Figure 3
Waveform of stator current. (a) Full load with 1 broken bar. (b) Half load with 1 broken bar.
(a) (b)Figure4 shows the results analyzed by MUSIC, DFT, and the proposed algorithm, respectively, on the condition of full load. In this condition, fault feature frequencies are far from the fundamental frequency, and the amplitudes of fault feature components are maximal at full load. It means that fault features are easy to be found. Thus we can see that the three algorithms can distinguish the fault feature components. However, the fault-related peaks are clearer in current spectrums analyzed by traditional MUSIC and the proposed method. The reason is that they can overcome the spectrum leakage problem well. In addition, the algorithms based on MUSIC have better frequency resolution and calculation accuracy. Figure 4 also shows that whether for fundamental frequency or for fault feature frequencies, DFT has a larger error, but it has an absolute advantage in the computational time.Figure 4
Spectrum of stator current under the full load. (a) Spectrum of the proposed method. (b) MUSIC spectrum. (c) DFT spectrum.
(a) (b) (c)Comparing Figure4(a) with Figure 4(b), we can see that two methods all have a high accuracy in estimating the frequency of fault feature components. However, due to fixed-step search (step ∆f=0.001), the frequencies estimated with traditional MUSIC are only the integer multiples of Δf. Therefore, the estimation results of three spectrum peaks are 45.861Hz, 50.023Hz, and 54.224Hz in Figure 4(b). Comparing Figure 4(b) and Table 5, whether for the fundamental harmonic or for the fault-related components, the proposed method can find higher spectrum peaks (with the same signal subspace and noise subspace). This cannot only prove that the spectrum peaks found by traditional MUSIC are not the true spectrum peaks, but also verify that the improved MUSIC has a higher accuracy. Moreover, the proposed method can accurately determine the amplitude and initial phase of each component signal. In addition, it also has a higher computing speed. In this experiment, its computing time is only 0.113s while traditional MUSIC’s is 4.054s (Δf = 0.001).Table 5
Results of improved MUSIC.
Test condition Parameters Amplitudes of peaks/dB Frequencies of peaks /Hz Full load with 1 broken bar f 1 56.7599 50.0226 (1-2s)f1 -18.4726 45.8700 (1+2s)f1 -10.8207 54.2190  Half load with 1 broken bar f 1 70.5717 50.0047 (1-2s)f1 -15.1455 47.8055 (1+2s)f1 -14.5698 52.1934Figure5 shows the analysis results of the motor stator current under the half load with 1 broken bar. In this case, fault feature components are weaker and closer to fundamental harmonic in spectrum, so it is difficult to be extracted and highlighted. The characters of each method are consistent with the above analysis under the full load. Figure 5 also verifies the effectiveness, feasibility, and superiority of the proposed method.Figure 5
Spectrum of stator current under the half load. (a) Spectrum of the proposed method. (b) MUSIC spectrum. (c) DFT spectrum.
(a) (b) (c)


## 6. Conclusions

To improve the accuracy and reliability of the BRB fault detection for induction motor, combining NBPSO-based MUSIC and least-squares amplitude estimation, a new BRB fault detection method is proposed and applied in a real motor. Although the computation cost of the proposed method is more than the DFT’s, it is more reliable to detect BRB fault due to higher frequency resolution with short-time data window. Comparing with the traditional MUSIC, the proposed algorithm also has a higher accuracy, efficiency, and reliability. In our experiment of a real induction motor, two load conditions are used to verify the proposed method. The results show that it cannot only estimate the amplitude of each component accurately and improve the accuracy of spectrum peak search, but also reduce the computation time greatly. These advantages make it more usable for fault motor operating with small slip and fluctuant load.



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*Source: 5942890-2018-11-15.xml*