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Billah, Md Masum; Martin, Floran; Belahcen, Anouar
A computationally effective method for iron loss estimation in a synchronous machine from a
static field solution
Published in:
Proceedings of the 2020 International Conference on Electrical Machines, ICEM 2020
DOI:
10.1109/ICEM49940.2020.9271020
Published: 23/08/2020
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Peer reviewed version
Please cite the original version:
Billah, M. M., Martin, F., & Belahcen, A. (2020). A computationally effective method for iron loss estimation in a
synchronous machine from a static field solution. In Proceedings of the 2020 International Conference on
Electrical Machines, ICEM 2020 (pp. 751-757). [9271020] (Proceedings (International Conference on Electrical
Machines)). IEEE. https://doi.org/10.1109/ICEM49940.2020.9271020
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A Computationally Effective Method for Iron Loss
Estimation in a Synchronous Machine from a Static
Field Solution
Md Masum Billah, Floran Martin, and Anouar Belahcen, Senior Member, IEEE
Abstract—In this paper, a computationally effective iron loss
calculation method for synchronous machines is presented. The
method is based on a single static 2D finite element field solution
in the machine cross-section, which makes it much faster than
the one based on the time-stepping solution. The developed
method is applied to a salient pole synchronous machine, and the
computational accuracy is validated against the time-stepping
method. The proposed iron losses computation method showed a
fair accuracy and a considerable speed-up of the computations.
It can be an excellent alternative for the iron losses estimation
in the optimization procedure of synchronous machines, where
a considerable amount of finite element solutions needs to be
carried out. Besides the losses comparison, local reconstruction
of the time dependency of other quantities such as the magnetic
vector potential and the magnetic flux density is reported for a
better understanding of the method.
Index Terms—dynamic field solution, finite element method,
iron losses, synchronous machine, static field solution, time-
stepping method.
I. INTRODUCTION
O
WING to the increasing energy demand, highly ef-
ficient synchronous machines play a crucial role in
energy saving by reducing energy consumption. Like other
electrical machines, power losses are a common issue in a
synchronous machine, which increases the temperature and
degrades the performance by affecting the maximum output
power. Besides, extreme temperature rise can lead to insula-
tion failure, consequently, decrease the life expectancy of a
synchronous machine. Power losses can be segregated into
Joule losses in the winding of the machine, the iron losses
in the core of the machines, the frictions, and mechanical
losses, and the permanent magnet losses, which are due to
eddy currents in these parts if any. Usually, the Joule and
friction losses can be estimated from static quantities, such
as the rotational speed and the supply currents, but the iron
and permanent magnet losses require the temporal and spatial
distribution of the magnetic flux density in the corresponding
parts of the machine.
This work was supported by the internal funding from computational
electromechanics group at Aalto University, Finland.
M. M Billah, F. Martin, and A. Belahcen are with Aalto Uni-
versity, Department of Electrical Engineering and Automation, Espoo,
Aalto-00076 Finland (e-mail: md.billah@aalto.fi, floran.martin@aalto.fi,
anouar.belahcen@aalto.fi)
A. Belahcen is also with Tallinn University of Technology, Tallinn,
Estonia.
The magnetic flux density in the electrical machine is
three-dimensional, and solving a three-dimensional problem
is still laborious. The solution becomes much easier if the
problem domain is simplified to the two-dimensional model
where the machine geometry and material equations are
independent of the coordinate parallel to the machine shaft,
i.e., z-coordinate. A detailed description of the 2D finite ele-
ment method and its application on solving two-dimensional
electromagnetic fields problem is demonstrated in [1] among
others. Moreover, the effects of armature reaction and the
magnetizing field, the slotted nature of the machine, and the
highly distorted magnetic field appearing from the permanent
magnet cause the non-sinusoidal and time-varying distribu-
tion of the magnetic flux density at different locations of
the machine’s core. As a result, the solution of the magnetic
flux density distribution is usually achieved for each time
step through the time-stepping method [2]. In addition, the
magnetic flux density in the rotating machine is not a single
frequency component anymore and contains higher frequency
components.
A common practice of the iron losses computation is to
obtain the flux density distribution from the time-stepping
finite element simulation and calculate the losses in the post-
processing stage through the Fourier decomposition of the
flux density waveform over one period [3], [4]. The solution
accuracy of the time-stepping method depends on the size
of the time step, and a smaller step size usually ensures
better accuracy. Moreover, the steady-state characteristics
are often desirable for the iron losses computation, which
requires to run the machine at least for a few electrical
periods. A suitable time resolution selection for the dynamic
analysis and the effect of it on the solution accuracy is a
separate topic, and no exact recommendation was found.
For instance, the number of time steps per period and the
number of electrical periods were used by [5], [6] 300-1000
steps per period and 2-10 electrical periods, respectively.
However, all these factors lead to high computational cost
for achieving the dynamic solution; consequently, the iron
losses computation from this solution. The difference is more
visible when the iron losses computation from the dynamic
solution required 35.07 s, and from a single static simulation
needed only 0.512 s. Despite the high computational cost,
the time-stepping method is more conventional and still
a popular choice for the iron losses computation. This is
probably because the conventional loss computation method
provides better accuracy and also stays productive as long as
the number of computations remains in a reasonable range.
However, the situation is different when an excessive number
of computations are required, e.g., machine optimization,
which turns this method into unprofitable by increasing the
simulation time. In such a case, the computation of the
iron losses from a time-efficient static solution is more than
justified.
Early on, a series of thirty static field solutions were com-
puted in [7] by moving the rotor at one slot pitch interval over
one electrical period, and the resulting field solutions were
used to estimate the stator core losses. A similar approach
has been used in [8] with a smaller number of snapshots,
i.e., static field solutions than [7]. The method presented in
[7], [8] was extended by [9] in order to take the rotor core
losses into account. A simplified, i.e., surrogate finite element
method is introduced in [10] to reduce the requirements of
the number of successive snapshots, i.e., static field solu-
tions. However, the method presented in [10] is particularly
developed for the interior permanent magnet machine with
concentrated, i.e., non-overlapping coils. Another snapshot-
based computationally effective finite element model is devel-
oped in [11] and applied for the optimization process of the
interior permanent magnet machine. The proposed methods
in [10], [11] significantly reduced the requirements of the
number of static simulations, i.e., 2-5 solutions required,
respectively for the iron losses computation compared to the
methods presented in [7], [8], [9]. A major drawback of these
snapshot-based methods is that accuracy and precision are
highly influenced by the number of static simulations taken
into consideration. Adding more static field solutions may
make sure better accuracy; however, it also increases the
computational cost. Also, no specific benchmark is noticed
for the exact requirements of the number of static simulations.
An ultrafast static field computation method has been
presented in [5], by coupling the static field equations and
space vector model within the same finite element solution.
The method has given accurate results from a single static
simulation compared to the measurement and time-stepping
method. Modeling a fast and accurate static field computation
method is an independent topic of research by itself, which
is out of the scope of this paper. Instead, our main goal is
to develop the iron losses computation method by using the
readily available static field computation method presented
in [5], which can be an alternative method of the iron losses
computation from the dynamic field solution in many appli-
cations. In this paper, the accuracy and computational cost
of the developed method are validated with the conventional
time-stepping method extensively.
The presented iron losses computation method in this
paper outperforms the previously developed computationally
effective static iron losses calculation methods in [7], [8], [9],
[10], [11] several ways. Here, the iron losses are computed
from a single static simulation, and no rotor motion is taken
into account. Hence, the accuracy of this method is not
restricted to the requirements of the number of static sim-
ulations. Furthermore, this paper solely and comprehensibly
deals with developing an ultrafast iron loss computation tech-
nique, which can overcome the above-mentioned snapshot-
based methods in terms of computational cost and also can
compete with them in terms of accuracy. Moreover, the
proposed iron losses computation method is not limited to
a particular synchronous machine and applicable to any kind
of synchronous machine.
II. M
ETHODOLOGY
A. Iron Loss Model
In this proposed method, the iron losses will be computed
from the Fourier decomposition of the flux density waveform
over one period. A suitable and accurate iron loss model
requires to compute the losses from the peak values of
the flux density at each harmonic component. Moreover,
the selected iron loss model can be easily integrated with
the static or dynamic finite element solution in the post-
processing stage. The empirical iron loss models are fast, easy
to implement, and applicable for the iron losses estimation
roughly [12]. However, many researchers have extended the
empirical iron loss models and estimated the iron losses
conventionally with relatively good accuracy [13], [14]. The
modified Jordan loss separation model presented in [15] is
adopted for the iron losses computation
P
hys
=
V
c
(
N
n=1
C
hys
(nω
s
)
ˆ
B
2
n
)dV (1)
P
ed
=
V
c
(
N
n=1
C
ed
(nω
s
)
2
ˆ
B
2
n
)dV (2)
where P
hys
and P
ed
represents the hysteresis loss and
eddy current loss, respectively. C
hys
and C
ed
expresses the
hysteresis and eddy current loss coefficients, ω
s
stands for
the angular frequency of the supply, V
c
is the volume of
the stator iron core, and
ˆ
B
n
is the peak flux density value
at n
th
harmonic components. However, the modified iron
loss model in Equations 1, and 2 also suffers from some
shortcomings. For instance, two-loss coefficients C
hys
and
C
ed
were computed from the Epstein frame test, and no
differentiation was made between the alternating and rota-
tional fields [16]. Moreover, the minor hysteresis loops were
not taken into consideration properly. In the statistical loss
segregation method, the classical eddy current loss and excess
loss are presented separately. However, the computation of
the excess loss coefficient through the Epstein frame test is
difficult as it does not allow us to identify the difference
between the eddy current due to the classical loss and
excess loss [17]. Alternatively, a two-term iron loss separation
method is developed as shown in Equations 1, and 2 where
the classical eddy current loss and excess loss are combined
and formed a global eddy current loss, P
ed
in Equation 2. It
can be noted that the same iron loss model is also used to
compute the losses through the time-stepping method. An in-
house 2D finite element solver software FCSMEK has been
developed by the electromechanics research group at Aalto
University and is used to compute the iron losses from the
proposed method and the conventional time-stepping method.
B. Time Dependence of the Flux Density from the Static
Analysis
In finite element method, the flux density B values are
computed natively in the x-y coordinates; thus, the actual
iron losses are calculated from the flux density components
B
x
and B
y
values. In practice, the flux density B is solved by
assuming the two-dimensional approximation where the flux
density lies only in (x, y) plane in Cartesian coordinates or
(r, φ) plane in cylindrical coordinates and does not depend
on the z-axis. The flux density components B
x
and B
y
can be calculated from the partial derivation of magnetic
vector potential values with respect to the x-y coordinates.
The existence of space harmonic components in case of
sinusoidal voltage supply in the electrical machine affects
the flux density waveform and influence to depend on time.
However, the obtained flux density B values from a static
field solution is independent of time, i.e., static. The time
dependence of the static flux density waveform needs to be
introduced in order to compute the iron losses. In FCSMEK,
the stator finite element mesh is constructed by multiplying
the slot pitch mesh. Thus, the number of elements in one slot
pitch mesh is repeated to the next slot pitches, and elements
from one slot pitch to another has the same position and
size. Therefore, the space variation of the static flux density
waveform both for the stator yoke and teeth over one period
can be achieved by selecting the elements at one slot pitch
interval. The stator slot pitch is computed as follows
θ
s
=
2π
Q
s
(3)
where θ
s
is the stator slot pitch, and Q
s
denotes the total num-
ber of stator slots. The time dependence of the flux density
waveform can be achieved by assuming that the flux density
waveform is moving with the fundamental angular frequency
ω and time t. The time step for the time dependence is
computed as follows
Δt =
Δθ
s
p
ω
(4)
where p denotes the number of pole pairs, θ
s
is the stator slot
pitch, and ω is the supply angular frequency. At any instant
of angular distance θ and time t, the displacement of the
moving flux density with initial position θ
o
can be determined
as θ = θ
o
+ ωt. With respect to the initial position θ
o
and
time t
o
, the displacement of the flux density waveform can
be written as
B(θ, r, t)=RB(θ − ωt, r, 0) (5)
where θ is the angular position of the point at which the flux
density is estimated and r its radial position from the rotor
midpoint. R is the rotation matrix given in Equation 6.
R =
cos θ − sin θ
sin θ cos θ
(6)
Fig. 1. Element selections for the flux density waveform formation over
the solution region in the static analysis.
It should be noted that the radial dependency does not need
to be solved in Equation 5. It is intrinsic within the time
dependency reconstruction.
C. Flux Density Waveform Formation in Static Analysis
The process of element selections in the stator yoke at
each slot pitch interval is depicted in Figure 1. In such a
case, the number of stator slots and the number of points in
the flux density waveform should be equal. Similarly, the flux
density waveform in the stator teeth is formed by choosing
the elements at one slot pitch interval. In such a way, elements
to elements flux density waveform are constructed at each slot
pitch interval over two pole pitches to calculate the average
iron losses from a closed cycle of the flux density waveform
using the Equations 1, and 2. The same size element in each
stator slot pitch meshes from 2 to 15, i.e., represents full slot
pitch is found at a distance of one slot pitch interval. On the
other hand, the element in positions 1, and 16 represent half
of the slot pitch. The flux density waveform as depicted in
Figure 2 was formed by specifying an observation point at
one element, i.e., position 1, and varying by one slot pitch
from 1 to 16 as shown in Figure 1. As one observation
point was specified in each element; hence, no elements were
missed. As a result, the flux density points are evenly spaced
in Figure 2 even though the element in positions 1, and
16 are different compared to other selected elements. Such
variation over the solution region in Figure 1 provides the half
cycle of the flux density waveform. A complete cycle of flux
density waveform was formed in Figure 2 by mirroring the
obtained half cycle of the flux density waveform due to the
symmetry of the solution region. A problem was identified
during the computation as the element in positions 1, and
16 have different sizes compared to other slot pitch meshes,
i.e., 2 to 15. For this reason, the total number of elements in
this slot pitch is not equal compared to the other slot pitch
meshes. However, the same number of elements in each slot
pitch mesh is required when the elements to elements flux
