Download Identification of demagnetization faults in axial flux permanent

Identification of demagnetization faults in axial flux permanent magnet synchronous 1
machines using an inverse problem coupled with an analytical model
Jan De Bisschop1,2 , Ahmed Abdallh1,3 , Peter Sergeant1,2 , and Luc Dupré1
1
Department of Electrical Energy, Systems and Automation, Ghent University, Belgium
2
Department of Industrial Technology and Construction, Ghent University, Belgium
3
Electrical Power and Machines Department, Cairo University, Egypt
Abstract—This paper presents an analytical method for permanent magnet demagnetization detection in an axial flux permanent
magnet synchronous machine. Firstly, a forward model is built to calculate the three phase terminal voltages with as inputs the
three phase currents, the geometry of the machine and the electromagnetic properties of the machine. To model the demagnetization
fault, the scalar potential equation is solved. The magnetization waveform is a square wave corrected with magnetization factors
which represent the individual magnetization of every magnet. Secondly, the forward model is inverted so that the magnetization
factors are computed out of the three phase currents and the three phase terminal voltages. The forward model is validated with
experimental data and FEM simulations.
Index Terms—Axial flux permanent magnet synchronous machine, demagnetization faults, inverse problem.
y
I. I NTRODUCTION
L
XIAL flux permanent magnet synchronous machines
(AFPMSMs) are widely used nowadays in industry
thanks to their high efficiency and high power density. However, these types of machines are commonly affected by
several types of faults, such as a rotor eccentricity and
permanent magnet demagnetization [1]. An early detection
of these faults is an essential criterion of online continuous
condition monitoring systems. Specifically, in this paper we
focus only on demagnetization faults which may occur due
to the temperature rise of the machine, in particular of the
rotor, when it works extensively at high load conditions. If
the magnet temperature exceeds a critical value, a partial demagnetization of the permanent magnets may happen leading
to a dramatic full demagnetization fault. In order to avoid
large safety margins or extra temperature measuring equipment, we develop a real-time algorithm to detect the magnets
demagnetization. Detection can be done by flux probes [2]
or other sensory techniques, but extra sensors result in extra
production costs. Another technique is based on measuring
the negative/zero sequence current, impedance or voltage [3].
Although this information is good for detecting any asymmetry
in the machine, it is also sensitive for unbalance in the power
supply. In this paper, we present an electromagnetic inverse
problem to detect the demagnetization faults in AFPMSM.
The identification procedure is performed based on an analytical model of the considered machine that solves Maxwell’s
equations. This involves no extra sensors and takes the three
phase current and voltages into account.
N
A
II. A NALYTICAL MODEL
A. Model construction
The machine model for calculating the no-load magnetic
field of the AFPMSM is shown in Fig. 1. A Cartesian coordinate system is chosen where x and y denote the circumferential
and the axial direction, respectively. This defines a 2-D plane
where the scalar magnetic potential is solved. The plane
is a cylindrical cutting plane with a constant radius. The
differential equation that needs to be solved is:
∂2ϕ ∂2ϕ
+
=0
∂x2
∂y 2
(1)
The authors gratefully acknowledge the financial support of the Special
Research Fund “Bijzonder Onderzoeksfonds” of Ghent University. Corresponding author: A. Abdallh (e-mail: [email protected])
L
S
hm Region I
δ
δ
Region II
0
δ
-δ
S
N
L
hm Region III
-L
x
(a)
(b)
Fig. 1. (a) The studied AFPMSM. (b) Model for solving of no-load magnetic
field of AFPMSM showing the different regions.
with ϕ being the scalar magnetic potential. Region I and
III represent the area with the permanent magnets. Region
II is a combination of the two air gaps of the machine.
The soft magnetic materials are assumed to have an infinite
permeability. The soft magnetic material itself and the stator
slots are not included in this model. This means that this
model only considers the no-load situation. The modeling of
stator currents is explained further in this paper. Moreover,
a permeance function is used to account for the stator slot
effects. The magnetization of the rotor is a square wave where
the height is dependent on the remanent induction of the
magnets. Demagnetization is easily included as the height of
the square representing a certain magnet can be lowered. The
magnetization pattern of each magnet is found by multiplying
the healthy state magnetization with a magnetization factor,
for each magnet ranging from 0 to 1 with 0 and 1 being
the full demagnetization and healthy state, respectively. These
magnetization waves can be described as a Fourier series.
(
M1 (x) =
M2 (x) =
P+∞
n=−∞
P+∞
n=−∞
Mn1 e
Mn2 e
inπx
τp
inπx
τp
∗
with M−n1 = Mn1
∗
with M−n2 = Mn2
(2)
M1 and M2 represent the magnetization of the rotor of
region I and III respectively. τp is the half of the total period,
which is the circumference of the machine. The constitutive
relationship (3) connects the magnetization with the magnetic
flux density. The magnetization vector is directed along the
axial direction. µ0 is the permeability of vacuum, µ is the
absolute permeability of the magnets.
~ = µ0 M
~ + µH
~
B
(3)
2
The magnetic field strength can be derived from this solution
by:
Hx = −∂ϕ/∂x,
Hy = −∂ϕ/∂y.
(5)
The solution must satisfy the following boundary conditions:
(
HxI (x, y)|[y=L] = 0
(6)
HxIII (x, y)|[y=−L] = 0
(
ByI (x, y)|[y=δ] = ByII (x, y)|[y=δ]
(7)
HxI (x, y)|[y=δ] = HxII (x, y)|[y=δ]
(
ByII (x, y)|[y=−δ] = ByIII (x, y)|[y=−δ]
(8)
HxII (x, y)|[y=−δ] = HxIII (x, y)|[y=−δ]
This results in a system of six equations. By solving this
system the constants C1 to C6 can be determined. The flux
density in the air gap (region II) becomes:
+∞
inπx
X
nπy
nπ − nπy
C 3 e τp − C 4 e τp e τp
τ
n=−∞ p
(9)
+∞
X
nπy
nπy
inπx
inπ
−
BxII (x, y) = −µ0
C 3 e τp + C 4 e τp e τp
τ
n=−∞ p
(10)
Although (9) and (10) contain complex functions and coefficients, the functions ByII (x, y) and BxII (x, y) are real
functions. The armature reaction is based on [4]. BAR (x) uses
the current sheet A(x) to compute the air gap field generated
by the stator coils. The current sheet is a square wave based
on the injected three phase currents. The squares represent the
current going through the stator windings. The three phase
currents are in-phase with the three phase terminal voltages at
full-load.
+∞
n
X
inπx
cosh nr z1
(z
−
z)
iAn e τp
BAR (x) = µ0
cosh
2
n
r
sinh r z2
n=−∞
(11)
+∞
X
inπx
(12)
An e τp
A(x) =
200
Analytical
FEM
Measured
Phase Voltage [V]
150
100
50
0
−50
−100
−150
−200
0
50
100
150
200
Rotor Position [◦ ]
250
300
350
Fig. 2. No-load terminal voltage of one phase from the results of the analytical
model verified with the FEM and experimental data for a healthy machine.
Terminal Voltages
400
Va -Num.
Vb -Num.
Vc -Num.
Va -Anal.
Vb -Anal.
Vc -Anal.
300
200
Voltage [V]
The solution of the differential equation for each region is:

inπx
nπy
P+∞ − nπy

ϕ1 = n=−∞ C1 e τp + C2 e τp e τp



inπx
nπy
P+∞ − nπy
ϕ2 = n=−∞ C3 e τp + C4 e τp e τp
(4)

nπy
nπy 
inπx
P

−
+∞
τp
 ϕ3 =
+ C 6 e τp e τp .
n=−∞ C5 e
100
0
−100
−200
−300
−400
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Rotor Position [rad]
Fig. 3. Full-load three phase terminal voltages of analytical and FEM
simulations with - for the clarity of the figure - quite extreme demagnetization
faults. Rotor 1: magnet 4 = 0% and Rotor 2: Magnet 7 = 0%.
ByII (x, y) = −µ0
n=−∞
The magnetic field of the permanent magnets and of the
armature reaction are combined with a permeance function
to result in the total air gap field. The permeance function is
based on the function in [5]. From this total air gap field, the
back-EMF for every tooth can be calculated. The model is
built in the frequency domain. Five stator coils are combined
to become the phase back-EMF. After accounting for the slot
leakage inductances, the end winding inductances and the
armature resistances, the three phase terminal voltages are
obtained. The inductances are computed based on [6].
Va,n = En − Rs Ia,n − inωm Lend Ia,n − inωm Lslot Ia,n (13)
with Lend and Lslot the end winding inductance and the slot
and tooth tip leakage inductance resp. The air gap stray field
is accounted for in the computed total air gap field. The
three phase currents and the magnetization coefficients are the
inputs of the forward model. The magnetization coefficients
determine the back-EMF. The three phase terminal voltages
are the output of the model. In the inverse model, the three
phase currents and terminal voltages are the inputs, while the
magnetization coefficients are the output.
B. Model validation
Fig. 2 shows that the results of the forward model are
confirmed by the numerical and experimental data of [7]. This
is for a healthy machine.
A FEM model is made to confirm the results with demagnetization faults. It solves the vector potential differential
equation in a 2-D cylindrical cutting plane with constant radius. Because demagnetization faults will not be symmetrical,
the model solves the total motor. Several cutting planes are
simulated and combined to compute the three phase terminal
voltages via the 2D multislice technique [4]. Fig. 3 shows
a good correspondence between the numerical and analytical
data. The three phases do not have the same peak value.
Thanks to the demagnetization faults the peak values of a
phase is not constant, it oscillates over a period of one rotation.
III. D EMAGNETIZATION IDENTIFICATION
The magnetization coefficients of permanent magnets are
identified by solving an electromagnetic inverse problem [8],
where an accurate interpretation of the time-domain three
phase voltages into the analytical model is performed.
3
TABLE I
T HE INVERSE PROBLEM RECOVERY SOLUTION STARTING FROM NOISY
‘ NUMERICAL’ MEASUREMENTS , WITH 10% NOISE LEVEL .
Inverse Algorithm
Measured 3-phase current
Geometrical parameters
Rotor speed
Analytical model
?
Termination
Criteria?
Magnetization
coefficients
+
Case
Original defect
Recovered defect
1
[6, L, 0.5]
[6, L, 0.4964]
2
{[6, L, 0.5], [12, R, 0.5]}
{[6, L, 0.5009], [12, R, 0.5011]}
Measured
3-phase terminal voltage
Fig. 4. The schematic diagram of the inverse problem algorithm.
A. Inverse problem formulation
The inputs for the proposed inverse problem are the geometrical and circuit parameters of the machine, the three
phase current waveforms, and the rotor speed. In addition, the
‘measured’ three phase terminal voltages are inputs, see Fig.
4. The magnetization coefficients of permanent magnets K are
recovered by minimizing iteratively an objective function:
e = arg min OF (K)
K
(14)
K
e is the vector of the recovered magnetization coefwhere K
ficients using the inverse approach. The objective function
OF is formulated as the Euclidean distance between the
‘measured’ and the corresponding ‘simulated’ three phase
terminal voltages, Vabc,m and Vabc,s , respectively:
OF = kVabc,m − Vabc,s (K)k
2
(15)
Several case studies are carried out to validate the proposed
inverse approach, see section IV.
Here, p vector contains all information related to the threephase currents Iabc , rotor speed, magnetic material parameters,
and all geometrical parameters. The vector u explicitly concerns the considered two uncertain circuit model parameters,
u = [Rs , Ls ].
It is obvious that the inverse problem solution is influenced
by the uncertainty in u. In other words, solving the inverse
problem based on the traditional objective function formulation, i.e. (15), may result in inaccurate results. Therefore, the
objective function formulation needs to be modified in order
to compensate this error. In this paper, the Minimum Path
of the Uncertainty (MPU) technique is implemented, which
is based on adapting iteratively the objective function to be
minimized by the sensitivity of the forward model response to
the uncertain circuit parameters u:
∂Vabc,s (p, u, K) 2
OFM P U = Vabc,m − Vabc,s (p, u• , K) − α
(17)
∂u
where u• is the mean value of the uncertain parameter. In
(17), α is the fitting constant that can be obtained by setting
the objective function OFM P U equal to zero [10].
B. Effect of the model uncertainties
In the previous section, the forward analytical model is
assumed to be ideal, i.e. all physical phenomena are taken into
consideration and all model parameters are assumed perfect.
However, in practice, it is well-known that the knowledge
of some model parameters is uncertain. For example, some
geometrical parameters are exposed to a small deviation when
the machine is manufactured [9], one of the most uncertain
geometrical parameters is the air gap. The effect of the
uncertainty in the geometrical parameters on the accuracy
of the inverse problem results has been extensively studied
[10]. In addition, the values of some circuit parameters of the
electromagnetic device, such as resistance and inductance, are
uncertain. The effect of such uncertainties of circuit parameters
in the inverse problem resolution has not been studied in the
literature. In this section, the effects of the uncertainty in
the values of both the armature resistance (Rs ) and leakage
inductance (Ls ) per phase are presented in details.
In order to take the effect of the circuit parameters uncertainty into account, let us formulate the forward problem
as follows. The output three phase terminal voltages (Vabc )
can be written as a function of the ‘well-known’ geometrical
and circuit model parameters p, ‘uncertain’ circuit model
parameters u, and magnetization coefficients of the permanent
magnets K:
Vabc = Γ(p, u, K)
(16)
IV. R ESULTS AND DISCUSSION
A. Idealized model
In this section, we assume that the values of all circuit
parameters are precisely known. Several case studies are
performed. For the sake of simplicity, the measurement data
are obtained numerically by adding some noise level (nl) to the
output of the forward model. One and two defected magnets
are studied with different noise levels. The error in the inverse
problem solution E is calculated as the percentage of the
difference between the ‘actual’ and ‘recovered’ magnetization
e respectively:
coefficients of permanent magnets, K∗ and K,
∗
e
(18)
E = (K∗ − K)/K
× 100%
Table I shows the results for several possible combinations
of the defects. The results shown in this table are obtained
by solving (14) and (15) starting from noisy ‘numerical’
measurements, with nl = 10%, i.e. Vabc,(nl) = Vabc,(nl=0) ×
(1 ± nl · N [0, 1]), where N [0, 1] is a normally distributed
random number with zero mean and a standard deviation
of unity. The defect, in the second column in the table, is
defined as a set of three parameters D = [a1 , a2 , a3 ], where
a1 is the order of the defected magnet in the model, i.e.
a1 ∈ [1, · · · , 16]. a2 is the defect side, i.e. a2 ∈ [L, R], where
L and R refer to as the left and right hand side of the rotor,
respectively. a3 is the magnetization factor, i.e. 0 ≤ a3 ≤ 1.
4
(a) fD = [6; L; 0:5]; [7; L; 0:5]g
6
10
20
E%
E%
30
10
5
0
0
(b)
(a)
Rotor right hand side
15
6
5
5
4
4
E%
Rotor left hand side
E%
40
3
Traditional OF
MPU OF
3
2
-10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Magnet order
1.5
Rotor left hand side
-5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
(b) fD = [6; L; 0:5]; [12; R; 0:5]g
Magnet order
Rotor right hand side
0.3
2
1
1
0
0
0.5
0
-0.5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Magnet order
0.2
E %
E %
1
0.1
0
-0.1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Magnet order
Fig. 5. The recovery error E for (a) two adjacent defected magnets, and (b)
two far defected magnets.
For the first case study, one defected magnet is studied.
The error in the identified magnetization coefficient of the
correct magnet is 0.72%. The magnetization coefficients of
the remaining magnets, which were assumed in healthy conditions, are correctly identified. In the second case study, two
magnets are assumed defected with specific magnetization
factors. For example, magnets number 6 and 12, at different
rotor sides, are defected with a magnetization factor of 0.5.
The identification results of the correct magnets were 0.5009
and 0.5011, respectively. Again, the magnetization coefficients
of the unaffected magnets are ‘almost’ recovered accurately.
In order to check the validity of the inverse approach in a
wide range of faults, Monte Carlo simulations are performed
with 100 samples, the defect D of each sample is generated randomly. The solutions of the inverse problems follow
‘qualitatively’ the same trend as shown in Table I. However,
we noticed that when the defected magnets are located close
to each other, the accuracy of the inverse problem solution
becomes worse than the case when the defected magnets are
located far from each other. Fig. 5 depicts the error E in the
both sides of the rotor for the two considered extreme cases.
Although the recovery error is high in Fig. 5(a), the correct
defected magnet is detected. Moreover, in order to study the
effect of the measurement noise on the solution accuracy, the
inverse problem is solved, for a case study with only one
defected magnet, starting from 10 different ‘numerical’ noisy
measurement data sets with a predefined noise level. Fig. 6(a)
shows the effect of the measurement noise on the accuracy
of the inverse problem, in an average way, for an assumed
defect D = [6, L, 0.5]. It is clear that the accuracy of the
inverse problem depends on the accuracy of the measurement,
the larger measurement noise, the larger recovery error.
B. Non-idealized model
As described in section III-B, the accuracy of the inverse problem does not only depend on the accuracy of the
measurements being used, but also on the accuracy of the
forward model parameters. In the following, the effect of
each uncertain model parameter is studied individually. For
each uncertain parameter, two inverse problems are solved;
i.e. based on the traditional and modified objective function
formulations, (15) and (17), respectively. The sensitivity of the
model response to the uncertain model parameter is calculated
0.1
0.2
nl
0.3
0.4
0
Rs
Ls
Fig. 6. (a) The effect of the measurement noise on the solution of the inverse
problem, in an average way, for D = [6, L, 0.5]. (b) The comparison between
the traditional and the MPU objective function formulation for the separate
effect of the uncertainty in the values of Rs and Ls .
using the finite difference approach, with a step of 10% of its
nominal value. Fig. 6(b) depicts the comparison between the
traditional and the MPU objective function formulation for
the separate effect of the uncertainty in the values of Rs and
Ls , for one defect D = [6, L, 0.5]. The results shown in Fig.
6(b) clarify the effectiveness of the MPU technique over the
traditional one.
V. C ONCLUSION
In conclusion, the developed direct model without faults
calculates the terminal voltages fast and accurately, with a
good corresponding with finite element computations and with
the experimental results. A coupled experimental-analytical
electromagnetic inverse problem is proposed to identify the
magnetization coefficient of the magnets. Several case studies
are performed, and the obtained results reveal the effectiveness
of the proposed methodology in condition monitoring of demagnetization defects. The proposed approach can be applied
to real drive systems, where the inverse problem is solved at
regular time instants, e.g. every minute, on the computer that
does the master control of the drive. The operating setpoint of
the motor is modified to avoid damage.
R EFERENCES
[1] P. Tavner, “Review of condition monitoring of rotating electrical machines,” IET Electr. Power App., vol. 2, pp. 215-247, 2008.
[2] M. Sasic, et al., “Detecting turn shorts in rotor windings, a new test
using magnetic flux monitoring,” IEEE Ind. Appl. Mag., vol. 29, pp.
63-69, 2013.
[3] Y. Da, et al., “A new approach to fault diagnostics for permanent magnet
synchronous machines using electromagnetic signature analysis,” IEEE
Trans. Power Electr., vol. 28, pp. 4104-4112, 2013.
[4] H. Vansompel, et al., “A multilayer 2-D-2-D coupled model for eddy
current calculation in the rotor of an axial-flux PM machine,” IEEE
Trans. Energy Convers., vol. 27, pp. 784-791, 2012.
[5] D. Zarko, et al., “Analytical calculation of magnetic field distribution in
the slotted air gap of a surface permanent-magnet motor using complex
relative air-gap permeance,” IEEE Trans. Magn., vol. 42, pp. 1828-1837,
2006.
[6] J. Pyrhönen, et al., Design of rotating electrical machines. New York,
USA: Wiley, 2008.
[7] H. Vansompel, et al., “A combined Wye-Delta connection to increase
the performance of axial flux PM machines with concentrated windings,”
IEEE Trans. Energy Convers., vol. 27, pp. 403-410, 2012.
[8] A. Abdallh, et al., “Selection of measurement modality for magnetic
material characterization of an electromagnetic device using stochastic
uncertainty analysis,” IEEE Trans. Magn., vol. 47, pp. 4564-4573, 2011.
[9] V. Simón-Sempere, et al., “Influence of manufacturing tolerances on the
electromotive force in permanent-magnet motors,” IEEE Trans. Magn.,
vol. 49, pp. 5522-5532, 2013.
[10] A. Abdallh, et al., “A robust inverse approach for magnetic material
characterization in electromagnetic devices with minimum influence of
the air gap uncertainty,” IEEE Trans. Magn., vol. 47, pp. 4364-4367,
2011.