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Universidade Católica de Petrópolis (UCP), Petrópolis/RJ, Brasil. 15-17 out. 2014
MONITORING THE STATOR CURRENT IN INDUCTION MACHINES FOR
POSSIBLE FAULT DETECTION: A FUZZY/BAYESIAN APPROACH FOR THE
TIME SERIES MULTIPLE CHANGE POINTS DETECTION PROBLEM
Marcos Flávio Silveira Vasconcelos D’Angelo- [email protected]
Universidade Estadual de Montes Claros, Departamento de Ciência da Computação
9401-089 – Montes Claros, MG, Brasil
João Batista Mendes- [email protected]
Renato Dourado Maia- [email protected]
Nilton Alves Maia- [email protected]
Abstract. This paper addresses the problem of current monitoring in the stator winding of induction machine by a multiple change points detection in time series approach. To handle this
problem a new fuzzy/Bayesian approach is proposed which differs from previous approaches
since it does not require prior information as: the number of change points or the characterization of the data probabilistic distribution. The approach has been applied in monitoring the
current of the stator winding of induction machine.
Keywords: Fault Detection, Induction Machine, Metropolis-Hastings, Change Points Detection
1.
INTRODUCTION
Induction motors are the most important electric machinery for different industrial applications. Faults in the stator windings of three-phase induction motor represent a significant part of
the failures that arise during the motor lifetime. When these motors are fed through an inverter,
the situation tends to become even worse due to the voltage stresses imposed by the fast switching of the inverter (Cruz and Cardoso, 2004). From a number of surveys, it can be realized that,
for the induction motors, stator winding failures account for approximately 30% of all failures
(Arkan et al., 2001) (Iravani and Karimi-Ghartemani, 2003).
The stator winding of induction machine is subject to stress induced by a variety of factors,
which include thermal overload, mechanical vibrations and voltage spikes. Deterioration of
winding insulation usually begins as an inter-turn short circuit in one of the stator coils. The
increased heating due to this short circuit will eventually cause turn to turn and turn to ground
faults which finally lead the stator to break down (Tallam et al., 2003).
Although there is no experimental data that indicate the time delay between inter-turn and
ground insulation failure, it is believed that the transition between the two states is not instantaneous. Therefore, early detection of short circuit during motor operation can be of great significance as it would eliminate subsequent damage to adjacent coils and the stator core, reducing
repairing cost and motor outage time (Thomson and Fenger, 2001) (Boqiang et al., 2003).
However, early stages of deterioration are difficult to detect. In general, most of the previous references present approaches for dealing with abrupt faults in the stator winding, which are
easier to be detected than incipient faults. In spite of these difficulties, a great deal of progress
has been made on induction machine stator-winding incipient fault detection. Methods that use
voltage and current measurements offer several advantages over test procedures that require ma-
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chine to be taken off line or techniques that require special sensors to be mounted on the motor.
Other methods, in the context of fault related to the stator-winding, can be found in Rodriguez
and Arkkio (2008), Ballal et al. (2008), da Silva et al. (2008), Tallam et al. (2007), Baccarini
et al. (2006). There exist other type of approaches to deal with different faults in induction
machine, unlike the one considered in this paper, as, for example, dynamic eccentricity, unbalanced rotors, bearing defects and broken rotor bars (see Rajagopalan et al. (2008) for details
and further references).
In this paper, a new quantitative approach for fault monitoring is presented. This new approach is based on a fuzzy/Bayesian representation for multiple change point detection. To
illustrate the efficiency of the proposed methodology, the problem of monitoring the stator current of induction machine, for possible fault detection, has been solved.
This paper is organized as follows. Section 2. shows the new fuzzy/Bayesian approach.
Section 3. presents and analyzes the induction machine modeling considering the case of turnto-turn short circuit in stator winding and shows the results for monitoring stator current of
induction machine. Finally, section 4. presents the concluding remarks.
2.
Fuzzy/Bayesian approach for multiple change points detection
Traditional methods of building models for time series are based on statistical techniques,
aiming to select models that satisfactorily explain its dynamic behavior. However, some questions can be raised:
• There is regime change in the series?
• Can a single model to portray this dynamic for the whole data set?
If there are significant changes in the time series, it seems natural to find turning points before
the entire modeling process. There are several papers approaching the problem of detecting
points of change, as in financial series (Oh et al., 2005), ecological series (Beckage et al.,
2007), crime rate (Loschi et al., 2005), fault detection (D’Angelo et al., 2011a) (D’Angelo
et al., 2011c), etc. The main techniques for change points detection presented in the literature
are based on statistical tests and Bayesian analysis. The most common statistical test in change
points detection problem is the CUSUM and in the context of Bayesian analysis are widely used
the MCMC methods. The CUSUM test proposed by Hinkey (1971) is widely used in detecting
change points; and applications of this method, and its modifications and extensions, can be
seen in Hadjiliadis and Moustakides (2006), Lee et al. (2006a), and Lee et al. (2006b). The
context of Bayesian analysis has as a reference (Barry and Hartigan, 1993), which uses a partition product model (MPP) for identifying points of change in the average data with a normal
distribution. In Loschi and Cruz (2005a), the posteriori probability of a time to be a turning
point is used as a measure of evidence that the behavior of a data sequence has changed at some
point. In Loschi and Cruz (2005b), it evaluates the effectiveness of the measure in identifying
changes, proposed by Loschi and Cruz (2005a), in the rate of the Poisson distribution in sequentially observed data, and a comparison being done with the one proposed by Hartigan (1990).
However, all these researches require some a priori knowledge of the statistical behavior of the
time series, for example, what type of distribution represents better its dynamic behavior. In
this work, an approach that is independent of such a priori knowledge about the time series will
be used, considering the empirical demonstration presented in D’Angelo et al. (2011c) that series, after being transformed using fuzzy operations, can be adequately approximated by series
with beta distribution. Thus, the parametrization of the beta distribution it is used to replace a
priori knowledge about the time series. Moreover, the method was extended to the detection of
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multiple switching points, not just one or two switching points as in D’Angelo et al. (2011c)
and Moreira et al. (2010), respectively.
2.1 Time Series Transformation by Fuzzy Sets
The fuzzy set theory, proposed by Zadeh (1965), has received much attention recently,
not only in the context of theoretical developments, but also in applications. One of its main
applications is in clustering methods, whereas classical methods of grouping the data into k
separate categories, and in many cases some elements may not belong to a specific category,
belonging to two or more categories simultaneously. Using fuzzy clustering methods is a good
way to solve this problem because, unlike the classical approach, an element can belong to more
than one category simultaneously.
Below it’s proposed an alternative quantization of a time series, through fuzzy clustering,
for use in the Metropolis-Hastings algorithm.
Definition (Fuzzy Clustering): Let y(t) be a time series, and consider a positive integer k.
Define the set C = {Ci | min{y(t)} ≤ Ci ≤ max{y(t)}, i = 1, 2, . . . , k} such that it solves
the minimization problem:
min
k
X
X
k µi (t) − Ci k2 ,
(1)
i=1 µi (t)∈Ci
so C = {Ci , i = 1, 2, ..., k} is the set of centers of the time series y(t). In (1),
#−1
" k
X ky(t) − Ci k2
µi (t) ,
ky(t) − Cj k2
j=1
(2)
is the fuzzy membership degree of y(t) for each center Ci .
Notice that, given a set C of cluster centers, it is an easy task to measure the distance
of each point in the time series y(t) to each center Ci . The problem of finding the centers
can be solved, for instance, via K-means (Kaufman and Rousseeuw, 1990), C-means (Bezdek,
1981), or Kohonen network (Kohonen, 2001), and in all these approaches, prior knowledge of
the number of groups is needed, a fact that justified the use of an adaptive algorithm, where
this information is not required. The optimization of the Average Silhouette Width, initially
proposed by (Rousseouw, 1987), was used in this paper.
For illustration purposes, in this paper, the time series (3) is used:


p1 + 0.1 ∗ ε(t) − 0.1 ∗ ε(t − 1), if t ≤ m1 ;


 p2 + 0.1 ∗ ε(t) − 0.1 ∗ ε(t − 1), if m1 < t ≤ m2 ;
f (t) =
(3)
..

.


 p + 0.1 ∗ ε(t) − 0.1 ∗ ε(t − 1), if t > m .
k
k−1
being p1 the first operation point, p2 the second operation point, pk the k − th operation
point, ε(t) a noise with distribution π(.) in the instant t, and mi the change points.
Figure 1 illustrates the time series (3) with p1 = 1, p2 = 2, p3 = 3, p4 = 4, p5 = 5,
m1 = 30, m2 = 60, m3 = 90, m4 = 120 and ε(t) ∼ U(0, 1) for 150 observations of the time
series, Figure 2 show the centers of time series and the membership degrees µi (t) is illustrated
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by Figure 3. Note that the first and last membership degree has been a single change, which
will be identified using the approach proposed by D’Angelo et al. (2011c), and the intermediates
membership degrees has been two changes to be identified the approach proposed by Moreira
et al. (2010).
5.5
5
4.5
4
y(t)
3.5
3
2.5
2
1.5
1
0.5
0
50
100
150
observations
Figure 1: Time series with p1 = 1, p2 = 2, p3 = 3, p4 = 4, p5 = 5, m1 = 30, m2 = 60, m3 = 90,
m4 = 120 and ε(t) ∼ U (0, 1) for 150 observations.
The proposed fuzzy clustering to transform a given time series into a new one is described
below:
1. Input the time series y(t);
2. Find the set of k centers, C = {Ci | min{y(t)} ≤ Ci ≤ max{y(t)}, i = 1, 2, ..., k},
that minimizes the Euclidean distance as (1), considering, for example, the time series
(3)(Figure 1).
3. Compute the fuzzy membership degree given in (2), for each sample of the time series,
y(t), with respect to each center Ci (Figure 3).
2.2 Formulation of the Metropolis-Hastings algorithm
The goal of the Metropolis-Hastings algorithm Gamerman (1997) is to construct a Markov
chain that has a certain equilibrium distribution π. Define a Markov chain as follows. If
Xi−1 = xi−1 , then generate a candidate value X ∗ from a distribution with density fX ∗ |X (y) =
1(xi−1 , x∗ ). The function q(.) is known as the core transition of the Markov chain. The candidate value X ∗ is accepted or rejected with a probability of acceptance:
If the candidate is accepted, do xi = Y , otherwise do xi = Xi−1 . Thus, if the candidate
is rejected, the Markov chain repeats the sequence. It is possible to show that, under general
conditions, the sequence X0 , X1 , X2 , ... is a Markov chain with equilibrium distribution π. In
practical terms, the Metropolis- Hastings algorithm can be specified by the following steps:
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5.5
C5
5
4.5
C
4
4
3.5
C3
3
2.5
C
2
2
1.5
C1
1
0.5
0
50
100
150
Figure 2: Centers of time series of Figure 1.
1
µ (t)
1
0.5
0
0
50
100
150
100
150
100
150
100
150
100
150
observation
2
µ (t)
1
0.5
0
0
50
observation
3
µ (t)
1
0.5
0
0
50
observation
4
µ (t)
1
0.5
0
0
50
observation
5
µ (t)
1
0.5
0
0
50
observation
Figure 3: Membership functions for each center of time series y(t).
1. Choose an initial value x0 , the number of simulations, R, and make the simulations
counter r = 1;
2. Generate a candidate value y ∼ q(xi , .);
3. Calculate the probability of acceptance as in (4) and generate u ∼ U(0, 1);
4. Calculate the new value of the current state:
y, if α(x, y) > u
t+1
x =
xt , otherwise
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5. If r < R, return to step 2. Otherwise, stop.
Note that, as discussed in D’Angelo et al. (2011b), the quantization technique generates a
time series transformed to the first and last membership degree with the following probability
distribution:
µi (t) ∼ Beta(a, b), for t = 1, ..., m
µi (t) ∼ Beta(c, d), for t = m + 1, ..., n
The parameters to be estimated by Metropolis-Hastings algorithm are a, b, c, d and the
change point m. In this type of algorithm, usually, uninformative priors are used, for example:
a ∼ Gamma(0.1, 0.1)
b ∼ Gamma(0.1, 0.1)
c ∼ Gamma(0.1, 0.1)
d ∼ Gamma(0.1, 0.1)
m ∼ U[1, ..., n], with p(m) =
1
n
The Gamma distribution with parameters shape and scale equal to 0.1 was chosen because
it is uninformative, with the goal of sweeping the entire parameter space. Considering the
intermediate membership degree, we have the following probability distribution:
µi(t) ∼ Beta(a, b), for t = 1, ..., mi−1
µi(t) ∼ Beta(c, d), for t = mi−1 + 1, ..., mi
µi(t) ∼ Beta(e, f ), to mi = t + 1, ..., n
The parameters to be estimated by Metropolis- Hastings algorithm are a, b, c, d, e, f and
the second point of change, mi , whereas the first switch point, mi−1 is identified in the previous
step. In this type of algorithm uninformative priors are commonly used, for example:
a ∼ Gamma(0.1, 0.1)
b ∼ Gamma(0.1, 0.1)
c ∼ Gamma(0.1, 0.1)
d ∼ Gamma(0.1, 0.1)
e ∼ gamma(0.1, 0.1)
f ∼ Gamma(0.1, 0.1)
mi ∼ U[mi−1 , ..., n], with p(m) =
1
n−mi
The transition kernel of the Markov chain for the model with a single point of change and
for the model with two change points are presented in D’Angelo et al. (2011c) and Moreira
et al. (2010), respectively.
2.3 Example of multiple change point detection in time series presented in Figure 1
The proposed fuzzy/Bayesian multiple change points detection in time series is described
below:
1. Input the time series y(t) (Figure 1);
2. Find the set of k centers, C = {Ci | min{y(t)} ≤ Ci ≤ max{y(t)}, i = 1, 2, ..., k},
given by Figure 2.
3. Compute the fuzzy membership degree given in (2), for each sample of the time series,
y(t), with respect to each center Ci , illustrated by Figure 3.
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4. Compute Metropolis-Hastings Algorithm in each membership function for change point
detection as illustrated in Figure 4 (for first and last membership function is executed the
detection of the one change point, and in the others membership functions is executed the
detection of the two change points). The final analysis is performed as: the change point,
mi , is obtained by checking where the maximum of mi histogram occurs.
y(t)
10
5
0
0
50
100
150
100
150
100
150
100
150
100
150
observation
1
1
m
0.5
0
0
50
observation
m
2
2
1
0
0
50
observation
m
3
2
1
0
0
50
observation
m
4
2
1
0
0
50
observation
Figure 4: Results of proposed methodology.
3.
Case study: monitoring induction machine with turn-to-turn short circuit in stator
winding
Many studies have shown that a large proportion of induction machine faults are related to
the stator-winding (Rodriguez and Arkkio, 2008), (Ballal et al., 2008), (da Silva et al., 2008),
(Tallam et al., 2007). The induction machine stator-winding is subject to stress due to many
factors, which include thermal overload, mechanical vibration and peak voltage caused by a
speed controller. The deterioration of insulation usually begins as a short circuit fault of the
stator-winding. This section describes the model that is employed here for the simulation of
inter-turn short circuits in the stator windings of induction machines.
3.1 Results of Proposed Methodology
The simulation results of current monitoring in the stator winding of an induction machine
with star connection are shown in Figure 5, considering become in short circuit after the time
2s and increase of 2% every time of 2s to reach a level of 8% of short-circuit, was shown in
the Figure 5. These results have been obtained by simulation of the induction machine using
the model described in this Section and the change point detection methodology presented in
Section 2.
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i (rms)
20
as
10
0
0
1
2
3
4
5
t(s)
6
7
8
9
10
0
1
2
3
4
5
t(s)
6
7
8
9
10
0
1
2
3
4
5
t(s)
6
7
8
9
10
0
1
2
3
4
5
t(s)
6
7
8
9
10
0
1
2
3
4
5
t(s)
6
7
8
9
10
m
1
1
0.5
0
m
2
1
0.5
0
m
3
1
0.5
0
m
4
1
0.5
0
Figure 5: rms current of phase a and the results of proposed methodology.
4.
Conclusions
In this paper a novel fuzzy/Bayesian methodology for multiple change points detection
in time series has been used to treat the on-line current monitoring of the induction machine
stator-winding. The methodology is based on a two-step formulation: Firstly, a fuzzy clustering
generates a transformed time series with beta distribution. In the second step, a MetropolisHastings algorithm is used to detect the probability of the occurrence of one change point or
two change points in the transformed time series. This two-step formulation allows a systematic
efficient way to solve a change point detection problem, which is employed for detecting incipient faults as change points that occur in some system signal. The proposed methodology has
as advantages, compared to other techniques for the FDI problem, the fact that no mathematical
models of the motor and no previous knowledge of signal statistical distributions are necessary, and also an enhanced resilience against false alarms, combined with a good sensitivity that
allows the detection of rather small fault signals. This methodology has been successfully applied: Simulation results have been presented as evidences of the effectiveness of the proposed
methodology, even in the case of faults that cause very low level disturbances.
Acknowledgements
This work has been supported in part by the Brazilian agencies CNPq and FAPEMIG.
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