ISSN(Print) 1975-0102
ISSN(Online) 2093-7423
J Electr Eng Technol Vol. 9, No. 1: 170-177, 2014
http://dx.doi.org/10.5370/JEET.2014.9.1.170
Classification of Induction Machine Faults using Time Frequency
Representation and Particle Swarm Optimization
A. Medoued†, A. Lebaroud*, A. Laifa* and D. Sayad*
Abstract – This paper presents a new method of classification of the induction machine faults using
Time Frequency Representation, Particle Swarm Optimization and artificial neural network. The
essence of the feature extraction is to project from faulty machine to a low size signal time-frequency
representation (TFR), which is deliberately designed for maximizing the separability between classes,
a distinct TFR is designed for each class. The feature vectors size is optimized using Particle Swarm
Optimization method (PSO). The classifier is designed using an artificial neural network. This method
allows an accurate classification independently of load level. The introduction of the PSO in the
classification procedure has given good results using the reduced size of the feature vectors obtained
by the optimization process. These results are validated on a 5.5-kW induction motor test bench.
Keywords: Induction machine diagnosis, Ambiguity plane, Classification-optimal TFR, Timefrequency, Fisher’s discriminated ratio, Artificial neural network ANN, PSO.
1. Introduction
the separability of TFRs from different classes. It may be
advantageous to design TFRs that specifically highlight
differences between classes [11-14].
Since all TFRs can be derived from the ambiguity plane,
no a priori assumption is made about the smoothing
required for accurate classification. Thus, the smoothing
quadratic TFRs retain only the information that is essential
for classification.
This classification allows us to proceed to an optimization
routine based on particle swarm technique to find the
appropriate size of the feature vectors in order to reduce
calculation time and keep signal with relevant information
within the vectors.
In this paper, we propose a classification algorithm
based on the design of an optimized TFR from a time–
frequency ambiguity plane in order to extract the feature
vector. The optimal size of feature vectors is realized by
the PSO algorithm. The PSO technique can generate highquality solutions within shorter calculation time and stable
convergence characteristic than any other stochastic methods
[15-17].
Finally, a neural network-based decision criterion is used
for classification. The goal of this work is the realization of
an accurate classification system of motor faults such as
bearing faults, stator faults, and broken bars rotor faults
independently from the load level.
Today’s industry strives to improve performance and
profitability while maintaining and improving safety. The
challenges include reliability and safety operation of
electric motors in an industrial process. Thus, very
expensive scheduled maintenance is performed in order to
detect machine problems before they may result in
catastrophic failure [1-2]. Nowadays, maintenance cost
reductions are the number one priority for electrical
drive to prevent unscheduled downtimes and to increase
operational effectiveness. Recent advances of signal
processing techniques, such as artificial neural networks
[3-8], wavelets [9], etc.., have provided more powerful
tools for fault diagnosis.
The problem of diagnosis systems is that they use
signals either in the time or frequency domain. In our
approach, instead of using a time or a frequency approach,
it is potentially more informative to use both time and
frequency. Time-frequency analysis of the motor current
makes signal properties, related to fault detection, more
evident in the transform domain [10].
Traditionally, the objective of time–frequency research is
to create a function that will describe the energy density of
a signal simultaneously in time and frequency. For explicit
classification, it is not necessarily desirable to accurately
represent the energy distribution of a signal in time and
frequency. In fact, such a representation may conflict with
the goal of classification, generating a TFR that maximizes
2. Classification Algorithm
†
Corresponding Author: Dept. of Electrical Engineering, University
of 20 Aout 1955-Skikda, Algeria. (amedoud@yahoo.fr)
*
Dept. Dept. of Electrical Engineering, University of 20 Aout 1955Skikda, Algeria. (lebaroud80@ gmail.com)
Received: December 26, 2012; Accepted: August 3, 2013
The classification algorithm consists of the following
three parts: extraction, optimization of features vectors and
decision making. In the training stage, three optimal
kernels are designed for separating four classes [18]:
170
A. Medoued, A. Lebaroud, A. Laifa and D. Sayad
separating the class i from all the remaining classes
{i+1,…,N}. In this case, the stator fault kernel is designed
to discriminate the stator fault class from the other classes
(rotor fault, bearing fault and healthy motor). The rotor
fault kernel is designed to discriminate the rotor fault class
from the remaining classes (bearing fault and healthy
motor). The bearing fault kernel is designed to discriminate
the bearing fault class from the healthy motor class. The
advantage of the method lies in the optimum separation
between the different classes.
1) Class of healthy motor;
2) Class of bearing fault;
3) Class of stator fault;
4) Class of broken bars.
The kernel design process selects, for each class, a
number of locations from the time - frequency ambiguity
plane. In the decision making stage, we propose an ANN
classifier with the Levenberg Marquardt algorithm.The
details of each step are described in the following
sections.
4. Feature Vector Optimization
3. Feature Extraction
One objective of our approach is to minimize the signal
size by the feature vector of a very small size without
losing relevant information.. Hence, the search for an
optimum size of this vector provides a good compromise
between the relevance of information and time consuming
cost.
3.1 Optimal TFR
For further details, we recommend the reader to review
our previous works [19] and [20].
The expression of the TFR is given by:
{
}} =
{
−1
G [ n, k ] = Fη→
n Fτ→ k φ [η ,τ ] . A [η ,τ ]
1
N
N −1 N −1
∑∑ φ [η ,τ ]A[η ,τ ] e
− j ( 2 π N ) τk
.e
j ( 2 π N ) ηn
4.1 Particle Swarm Optimization (PSO)
(1)
Particle Swarm Optimization (PSO), introduced by
Eberhart and Kennedy [21], is based on the analogy of
birds swarm and school of fish. In PSO, each individual
called particle makes his decision using his own experience
together with other individuals’ experience. In PSO, two
different definitions are used: the individual best and the
global best. As a particle moves through the search space,
it compares its fitness value at the current position to the
best fitness value it has ever attained previously. The best
position that is associated with the best fitness encountered
so far is called the individual best or pbest. The global best,
or gbest, is the best position among all of the individual’s
best positions achieved so far (Fig. 1).
Using the gbest and the pbest, the ith particle velocity is
updated according to the following equation[22]:
η= 0 τ= 0
The characteristic function for each TFR is
A(η ,τ )ϕ (η ,τ ) , η represents the discrete frequency shift
and τ represents the discrete time delay. This means that
the optimal-classification representation TFRi can be
obtained by smoothing the ambiguity plane A(η ,τ ) with
an appropriate kernel ϕopt , which is an optimal
classification kernel. The problem of designing the TFRi
becomes equivalent to designing the optimal classification
kernel ϕopt (η ,τ ) . This method, used to design kernels (and
thus TFRs), optimizes the discrimination between
predefined sets of classes.
Features
can
be
extracted
directly
from
A(η ,τ )ϕopt (η ,τ ) instead of the optimal classification TFRi.
This shortcut simplifies the computation complexity of the
feature extraction by reducing the calculations.
vi k +1 = wvi k + c1rand1 × ( pbesti − si k ) + c2 rand2 × ( gbest − si k )
(2)
3.2 Design of classification kernels
Based on the updated velocities, each particle changes
The kernel ϕopt (η ,τ ) is designed for each specific
classification task. We determine N locations from the
ambiguity plane, in such a way that the values in these
locations are very similar for signals from the same class,
but they vary significantly for signals from different classes.
In our design, we use Fisher’s discriminant ratio, FDR [1920], to get these N locations.
In our classification procedure, C−1 kernels must be
designed for a C-class classification system. In order to
avoid unnecessary computation to separate classes, we
have proposed the principle of the remaining classes [11].
The discrimination between different classes is made by
Fig. 1. Particle swarm method principle
171
Classification of Induction Machine Faults using Time Frequency Representation and Particle Swarm Optimization
its position according to the equation:
sik +1 = sik + vik +1
4.2 Fitness function
For searching an optimized size of the feature vector
based on PSO algorithm, a fitness function is needed. In
this work, we consider the variance calculated for every
size of the feature vector as the fitness for this size and the
goal is to optimize this fitness.
(3)
Where w is a weighting function, cj are acceleration
factors and rand is a random number between 0 and 1.
The following weighting function is usually utilized:
w = wmax −
wmax − wmin
× iter
itermax
5. Classification Using Neural Networks
(4)
In most cases an ANN is an adaptive system that
changes its structure based on external or internal
information that flows through the network during the
learning phase. The learning procedure tries to find a set of
connections w that gives a mapping that fits well the
training set.
Furthermore, neural networks can be viewed as highly
nonlinear functions with the basic form:
Where wmax is initial weight, wmin the final weight,
itermax is the maximum iteration number, and iter is the
current iteration number.
The parameters used in this work are taken as follows
[22-26]:
c1=c2=2.05; wmin =0.1; wmax =0.9.
Selection of maximum velocity:
F ( x, w) = y
At each iteration step, the algorithm proceeds by
adjusting the distance (velocity) that each particle moves in
every dimension of the problem hyperspace. The velocity
of the particle is a stochastic variable and is, therefore,
subject to creating an uncontrolled trajectory, making the
particle follow wider cycles in the problem space. In order
to damp these oscillations, upper and lower limits can be
defined for the velocity vi :
if
vi > vmax
elseif
then vi = vmax
vi < −vmax
then vi = −vmax
(6)
Where x is the input vector presented to the network, w
are the weights of the network, and y is the corresponding
output vector approximated or predicted by the network.
The weight vector w is commonly ordered first by layer,
then by neurons, and finally by the weights of each neuron
plus its bias.
This view of network as a parameterized function will be
the basis for applying standard function optimization
methods to solve the problem of neural network training.
(5)
5.1 Network training as a function optimization
problem
Most of the time, the value of vmax is selected
empirically, according to the characteristics of the problem.
It is important to note that if the value of this parameter is
too large, then the particles may move erratically, going
beyond a good solution; on the other hand, if vmax is too
small, then the particle’s movement is limited and the
optimal solution may not be reached.
Fan and Shi [27] have shown that an appropriate
dynamically changing vmax can improve the PSO
algorithm performance. To ensure a uniform velocity we
fixed vmax according to many run tests.
As mentioned previously, neural networks can be viewed
as highly non-linear functions. From this perspective, the
training problem can be considered as a general function
optimization problem, with the adjustable parameters
being the weights and biases of the network, and the
Levenberg-Marquardt can be straightforward applied in
this case.
5.2 Levenberg-marquardt algorithm
Basically, it consists in solving the equation:
Integer PSO formulation:
( J T J + λ I )δ = J T E
In the case where integer variables are included in the
optimization problem such as a size of feature vector,
the PSO algorithm can be reformulated by rounding off
the particle’s position to the nearest integer.
Mathematically, (3) and (4) are still valid, but once the
new particle’s position is determined in the real-number
space, the conversion to the integer number space must
be done.
(7)
Where J is the Jacobian matrix (Eq. 8), λ the
Levenberg's damping factor, δ the desired updated
weight vector and E the error vector containing the
output errors for each input vector used on training the
network. The δ tells us by how much we should change
our network weights to achieve a (possibly) better solution.
172
A. Medoued, A. Lebaroud, A. Laifa and D. Sayad
The J T J matrix may also be known as the approximated
Hessian. The λ damping factor is adjusted at each
iteration, and guides the optimization process. If the
reduction of E is rapid, a smaller value can be used,
bringing the algorithm closer to the Gauss-Newton
algorithm, whereas if an iteration gives insufficient
reduction in the residual, λ can be increased, giving a
step closer to the gradient descent direction.
6. Experiment Results
The experimental data are collected in Ampère
Laboratory, University of Lyon. The experimental bench
consists of a three-phase asynchronous-motor squirrel cage
Leroy Somer LS 132S, IP 55, Class F, T ◦C standard = 40
◦C. The motor is loaded by a powder brake. Its maximum
torque (100 Nm) is reached at rated speed.
This brake is sized to dissipate a maximum power of
5kW.Fig. 2 shows the motor bench. The wear obtained on
the bearings is a real one (Fig. 3). For the rotor fault, the
bar has been broken by drilling the bar of the squirrel cage
(Fig. 4). For simulating the fault of imbalance stator,
imbalanced power is obtained with a variable autotransformer placed on a phase of the network (Fig. 2).
The acquisition system used to measure these signals
consists of eight differential inputs used to measure
currents sampled up to 20 MHz 14-bit.
The current signals sampling rate is 20 kHz. The number
of samples per signal rises to N=100000 samples on an
acquisition period of 5s. The data acquisition set consists of
15 examples of stator current recorded on different levels
of load (0%, 25%, 50%, 75% and 100%). Different
operating conditions for the machine were considered,
namely, healthy, bearing fault, stator fault and rotor fault.
The training set is carried out on first ten current examples.
The last five current examples are used to test the
classification.
5.3 Computing the Jacobian
AnN-by-M matrix of all first-order partial derivatives of
a vector-valued function. N is the number of entries in our
training set and M is the total number of parameters
(weights + biases) of our network. It can be created by
taking the partial derivatives of each output in respect
toeach weight, and has the form
⎡ ∂F ( x1 , w) ∂F ( x1 , w) ⎤
L
⎢ ∂w
⎥
∂wM
1
⎢
⎥
M
O
M
J =⎢
⎥
⎢ ∂F ( xN , w) ∂F ( xN , w) ⎥
L
⎢
⎥
∂wM ⎦
⎣ ∂w1
(8)
Where F ( xi, w) is the network function evaluated for
the ith input vector of the training set using the weight
vector w and w j is the jth element of the weight
vector w of the network.
5.4 General Levenberg-Marquardt algorithm
As stated earlier, the Levenberg-Marquardt consists
basically in solving (11) with different values of λ until
the sum of squared error decreases. So, each learning
iteration (epoch) will consist of the following basic steps:
1 Compute the Jacobian
2 Compute the error gradient: g = J T E
3 Approximate the Hessian: H = J T E
4 Solve ( H + λ I )δ = g to find δ
5 Update the network weights ω using δ
6 Recalculate the sum of squared errors
7 If the sum of squared errors has not been decreased,
discard the new weights, increase λ using v and go
to step 6.
8 Else decrease λ using v and stop.
Fig. 2. The 5.5 kW motor coupled with load (powder brake).
Variations of the algorithm may include different values
of v , one for decreasing λ and another for increasing it.
Others may solve ( H + λ diag ( H ))δ = g instead of
( H + λ I )δ = g , while others may select the initial λ
according to the size of the elements on H, by setting
λ0 = t max(diag ( H )) , where t is a chosen value.
We can see that we will have a problem if the error
does not decrease after some iteration. In this case, the
algorithm also stops if λ becomes too large [28-29].
Fig. 3. Accelerated wear of the bearings by immersion in
acid.
173
Classification of Induction Machine Faults using Time Frequency Representation and Particle Swarm Optimization
representation. If too many neurons are defined, the
network might become overtrained. Therefore, an optimum
design of the neurons number is required. In this work, we
used one hidden layer with a number of different neurons
to determine the suitable network. As a stop criterion we
intended a goal of 10-12 which defines the convergence of
the algorithm. The goal is reached in a minimum number
of epochs 16 and 24, Fig. 5 and 6 respectively.
The training algorithm gives a better performance for a
number of 5 neurons in the hidden layers for the three
kernels (Table 1).
Fig. 7 shows that for 15 test vectors, in case of Kernel 1,
14 were classified which indicates that the classification
error is acceptable. This is also true for the two other
kernels. Furthermore, the increase of the size of feature
vector reduces significantly this error. However, the
classification error is minimized when we increase the
number of training vectors to 35 vectors (10 vectors of
stator currents at 0% of charge, 5 at 25%, 5 at 50%, 5 at
75% and 10 at 100% of rated charge). Fig. 8 shows clearly
a marked improvement in the classification process.
The objective of introducing the PSO is the optimization
Fig. 4. Rotor with broken bars
Each signal is passed through a lowpass filter and
resampled with a downsampling rate of 50. Only the range
of the required frequencies is preserved. The lowpass filter
is used in order to avoid aliasing during downsampling.
The dimension of ambiguity plane is 200×200=40000
points; by considering symmetry compared to the origin,
we retain only the quarter of ambiguity plane, which
corresponds to N=10000.We designed three kernels: stator
fault kernel, rotor fault kernel and bearing fault kernel [18].
Fisher’s point locations in the Doppler-delay plane are
ranged in the feature vectors {FV1,…, FVN} as training
database of the neural network. In neural network, if there
are too few neurons in the hidden layer, the network may
not contain sufficient degrees of freedom to form a
Table 1. Misclassification results
Performance is 1.65741e-013, Goal is 1e-012
0
10
FDR1
FDR2
FDR3
-2
Training-Blue Goal-Black
10
Neurons
number N=3
1/15
3/15
Actual error
Neurons
Neurons
number N=4 number N=5
0/15
0/15
1/15
1/15
3/15
2/15
Neurons
number N=6
0/15
1/15
3/15
-4
10
-6
10
-8
10
-10
10
-12
10
0
2
4
6
8
10
17 Epochs
12
14
16
Fig. 5. Training diagrams for optimum case of 5 hidden
neurons in kernel 1.
Performance is 4.79085e-013, Goal is 1e-012
0
10
Fig. 7. Classification of test vectors for 20 training vectors
-2
Training-Blue Goal-Black
10
-4
10
-6
10
-8
10
-10
10
-12
10
0
5
10
15
20
24 Epochs
Fig. 6. Training diagrams for optimum case of 5 hidden
neurons in kernel 2
Fig. 8. Classification of test vectors for 35 training vectors
174
A. Medoued, A. Lebaroud, A. Laifa and D. Sayad
We have introduced the PSO algorithm to optimize the size
of the feature vectors. Our classification is based on the
ambiguity Doppler-delay plane where all the TFRs can be
derived by a suitable choice of a kernel. Each type of fault
was characterized by a specific kernel. The classification
algorithm was tested by comparison with experimental data
collected from the stator current measurement at different
load levels. The assignment of signal was made by an ANN
classifier. The results show that the new algorithm, with the
neural network classifier as a decision criterion and the
PSO as an optimizing technique, is able to detect and
diagnose faults with acceptable accuracy and time consuming
calculations compared to the case without PSO optimisation,
independently of the load condition and the fault type.
Fig. 9.Feature vectors size optimization by PSO (class 1)
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Ammar Medoued He received the
degree of Doctor of Sciences from
University of Skikda, Algeria in
Electrical Engineering. He is currently
a Lecturer at the University of Skikda
and the Head of the Department of
Electrical Engineering. His main
research field is Electrical Machine
Diagnosis.
Abdesselam Lebaroud was born in
Constantine, Algeria, in 1969, He
received the PhD degree in electrical
engineering from University Claude
Bernard Lyon I, Ampere laboratory,
France, in 2007. Currently, he is a
Professor at the Department of Electrical Engineering, University of Skikda.
He carried out researches on diagnosis of electrical
machines at LGEC of Constantine.
176
A. Medoued, A. Lebaroud, A. Laifa and D. Sayad
Abdelaziz LAIFA He worked in oil
industrial field for many years. Since
2001, he has been with the University
of Skikda as a lecturer and researcher,
where he received the PhD in Electrical
Power Engineering in 2012. His main
interests are Power Systems Analysis
and Control using intelligent programming and meta-heuristic methods.
Djamel Sayad He received the degree
of Magister in Electronics from the
University of Constantine 1998. He is
actually a Lecturer at the University of
Skikda. Algeria. His main field of
research is Signal Processing, Diagnosis.
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