Improving the Technical Quality of a Universal Motor
Using an Evolutionary Approach
Barbara Koroušić-Seljak, Gregor Papa
Computer Systems Department
"Jožef Stefan" Institute
Jamova 39, 1000 Ljubljana, Slovenia
E-mail: Barbara.Korousic@ijs.si,
Gregor.Papa@ijs.si
Boris Benedičič, Tomaž Kmecl
Strategic R&D Department
Domel d.d.
Otoki 21, 4228 Železniki, Slovenia
E-mail: Boris.Benedicic@domel.si,
Tomaz.Kmecl@domel.si
Abstract
This paper presents a new procedure for designing the
rotor and the stator of a universal motor for home
appliances that is based on a genetic algorithm (GA).
The GA was employed in order to optimize the
independent geometric parameters of the rotor/stator
lamination with the aim to reduce the main motor's
power losses, i.e. the losses occurring in the iron and the
copper. Using this procedure the motor's technical
quality, expressed as efficiency (i.e. the ratio of the
motor's output power to its input power), can be
significantly improved. The GA proved to be a simple
and efficient search-and-optimization method for
solving this day-to-day design problem in industry. It
significantly outperformed a conventional 'direct' design
procedure that had been used before.
1. Introduction
Most home appliances with small motors, such as vacuum
cleaners or mixers, are driven by a universal motor
(Fig.1). This is an AC series motor in which the same
current passes through both the armature windings (rotor)
and the field-excitation windings (stator) (Fig.2). A
universal motor can operate either with an AC or a DC
power supply.
Because of the widespread use of the universal motor,
it is very important that the energy consumption of the
motor (input power) is as low as possible, while still
satisfying the needs of the user (output power). Hence we
need to improve the technical quality or efficiency of the
universal motor, which is defined as the ratio of output
power to input power. The efficiency depends on various
losses, these include iron losses, copper losses as well as
Figure 1. Universal motor used in vacuum cleaners.
brush losses, friction losses and ventilation losses. The
iron losses include the hysteresis losses and the eddycurrent losses, primarily in the armature core and in the
saturated parts of the stator core. The copper losses are
the joule losses in the windings of the stator and the rotor.
Figure 2. Rotor and stator parts of a universal
motor used in home appliances.
The main losses, i.e. those from the iron and the
copper, can be reduced by optimizing the geometry of the
rotor and stator. Because of the high magnetic saturation
of the iron in universal motors, this is a highly non-linear
problem. Due to the complexity of the solution
originating in this non-linearity, the GA [1] was used to
optimize the geometry of the rotor and stator.
Detail A
2. Geometry of the Rotor and the Stator
The geometry of the rotor and the stator was defined
parametrically. There were three parameter types (shown
in Fig.3 and Fig.4):
Invariable (constant): Among several constant
parameters one was extremely important from the
motor-manufacturer’s point of view, namely the
stator's outer radius. This roughly defined the amount
of material (iron and copper) used for the motor and
consequently also the price of the motor. By holding
the amount of material constant during the
optimization, all the optimized motor solutions had a
similar mass.
Variable:
- Mutually independent: like the external radius of
the rotor, the rotor-pole width, the rotor-jag diameter,
the stator-jag radius, etc.
- Dependent: e.g. the stator's inner radius. These
were defined indirectly from the constant and the
independent parameters.
Only mutually independent variable parameters were
altered using the GA.
Figure 3. Geometrical parameters of the stator.
Figure 4. Geometrical parameters of the rotor.
3. Design Procedure for the Rotor and the
Stator
In a conventional (‘direct’) motor design procedure the
initial estimation of the geometry of the rotor and the
stator is based on experience. The appropriateness of this
geometry is then usually analyzed by means of a
numerical simulation of the electromagnetic field. In our
case, the analysis was performed with ANSYS
commercial software [2], which applies a finite-element
method with an automatic finite-element mesh generation.
When the results of the numerical simulation show an
inconvenient electromagnetic field structure, the ‘direct’
design procedure is repeated until the motor geometry is
optimized.
The new motor design procedure described in this
paper is also a ‘direct’ system, but it is based on the GA.
This is a stochastic process providing a robust and yet
flexible search in the wide and complex space of the
problem solutions in order to find the optimum global
solution in a short time. The concept of the applied design
system can be roughly explained as follows (see Fig.5):
the GA provides a set of problem solutions (i.e. different
configurations
of
rotor-and-stator
independent
geometrical parameters). To enable the calculation of a
fitness value, each geometrical configuration is analyzed
using the finite-element program. After the calculation of
the fitness, the reproduction of individuals and the
application of genetic operators to a new population are
J=
Initial lamination’s
geometry
Genetic algorithm
Optimized
lamination’s
geometry
Solution candidates
Decode each candidate
Lamination’s geometry
I⋅N
A
(1)
PCuSlot = I 2 ⋅ R = J 2 ⋅ A ⋅ ρ ⋅ l turn
(2)
J .......... current density
I........... current
N ......... number of turns
A.......... slot area
ρ .......... specific resistance of copper
turn ........ length of winding turn
The overall copper losses are as follows:
Finite-element program
Repeat until time-out
Power losses (fitness)
for each candidate
Figure 5. New design procedure for the rotor and
stator.
made. The GA repeats this procedure until a predefined
number of iterations are accomplished.
4. Calculation Of Copper, Iron And Other
Losses
In order to solve the electromagnetic problem, the finiteelement program needs the following data:
well-defined geometry of the analyzed element,
appropriate finite-element mesh,
material properties (iron B-H function, copper
specific resistance),
density of the electric current in the conductor area.
A typical run for such a numerical problem takes about
30 iterations and needs approximately 7 minutes on a
Pentium III Computer Station to achieve a convergent
numerical solution. The result of the solution is a
magnetic vector potential on every node of the finiteelement mesh. From this potential we can calculate the
values of flux density, field strength, and magnetic
energy. With a special module, we can also calculate the
electromagnetic torque on selected regions (rotor). The
output power of a motor is a product of the
electromagnetic torque and the angular velocity.
4.1. Calculation of copper losses
Copper losses per slot (stator and rotor) are calculated by
the following equations:
PCu =
∑P
(3)
CuSlot ,i
i
4.2. Calculation of iron losses
Because of the nonlinear magnetic characteristic, the ironloss calculation is less exact. The iron losses are separated
into two components: the hysteresis loss and the eddycurrent loss. An empirical formula for the hysteresis loss
is defined as follows:
Ph = k h ⋅ B n ⋅ f ⋅ m
(4)
kh ......... eddy-current material constant at 50 Hz
B.......... maximum magnetic flux density
f ........... frequency
m ......... mass
n .......... exponent between 1.6 and 2.0 (material
dependent)
The eddy-current loss is defined as follows:
Pe = k e ⋅ B 2 ⋅ f 2 ⋅ m
(5)
ke ......... hysteresis material constant at 50 Hz
The frequency of the magnetic field density in a stator
is 50 Hz (the frequency of the power supply), while in a
rotor the frequency is much higher and depends on the
motor's speed. If the motor was running at 50.000 min-1,
the main rotor frequency would be 833 Hz. Since the
eddy-current losses depend on the square of the
frequency, and the hysteresis losses increase linearly with
the frequency, the main iron losses of the rotor are the
eddy-current losses. Since these losses depend on the
square of the flux density, and the hysteresis losses in the
rotor are neglected, the iron losses of the rotor can be
calculated using (5), where B2 is obtained from the finiteelement solution for each node.
As a result, the overall B2 of the rotor is calculated as
an average of the B2 values of all the rotor nodes.
When calculating the stator losses, the hysteresis loss
and the eddy-current loss must be added together. In
addition, we must roughly estimate the factor n. Since the
overall stator losses are approximately five times smaller
than the rotor losses, even a non-exact estimation of the
factor n does not have a significant influence on the
overall iron-loss calculation. Consequently, the iron loss
of a motor can be expressed by the following formula (6):
PFe = k e ⋅ B 2 ⋅ f rot2 ⋅ mrot +
2
+ k e ⋅ B 2 ⋅ f stat
⋅ mstat + k h ⋅ B 2 ⋅ f stat ⋅ mstat
(6)
The sum of the iron and copper losses (PCuFe)
represents a so-called objective function for the GA.
PCuFe = PFe + PCu
(7)
4.3. Calculation of other losses
Besides the iron and copper losses, there are three
additional types of losses in a universal motor. These are:
PBrush ........brush losses
PVent ..........ventilation losses
PFrict..........friction losses
All three types of losses mainly depend on the motor
speed. Since the motor speed is equal for all solutions,
these losses are considered as constant in our analysis.
Taking into account all the above-mentioned losses,
the overall efficiency of a universal motor is defined as
follows:
η=
P2
P2 + PBrush + PVent + PFrict + P CuFe
(8)
P2 .............output power
Consequently, the goal of the present investigation is
to maximize this efficiency.
5. Genetic Algorithm (GA)
The GA was proposed by J.H. Holland in 1975 [3] as a
heuristic search-and-optimization method. Since then it
has been applied with great success to various
optimization and classification problems, in areas ranging
from economics and game theory to control-system
design [4,5].
The GA manipulates strings representing the problem's
potential solutions in a way that is analogous to the role
played by DNA in evolution. Each solution corresponds
to a sample point in a search space. The initial set of
sample points (i.e. the population of strings) is randomly
generated. New sample points are generated by
recombining the information from two 'parent' strings
drawn from the current population. Another key element
of the GA is the use of selective pressure to allocate
reproductive trails. Allowing strings that represent aboveaverage solutions in the current population to 'reproduce'
more often than strings that represent below-average
solutions (i.e. the elitism strategy), the GA allocates more
trials to regions in a hyperspace that tend to contain
above-average solutions.
The GA mechanism is as follows:
Step 1. Encode parameters of the problem's
search space as finite-length strings over some
finite alphabet. The GA works with a coding
of the parameter set, not the parameters
themselves!
Step 2. Initialize the population of strings.
Step 3. Calculate the fitness for each member in
the population using an objective function.
This information is used to perform an
effective search for better solutions. There is
no need of other auxiliary knowledge!
Step 4. Select and reproduce individuals to form a
new population according to each member's
fitness. The GA tends to take advantage of the
fittest solutions by giving them greater weight,
and concentrating the search in the regions of
the search space with likely improvement.
Step 5. Perform
genetic
operators
(i.e.,
probabilistic transition rules) such as crossover
and mutation on the population.
Step 6. Go to step (3) until some condition is
satisfied.
It has been proved that the GA outperforms traditional
search-and-optimization methods in several aspects:
It requires little information to search effectively in a
large and complex search space.
The GA is capable of performing a global search of a
space because to guide the search it relies on
hyperplane sampling instead of searching along the
gradient of a function.
Its intristic parallelism (in evaluation function,
selections) allows working from a broad database of
solutions in the search space simultaneously,
climbing many peaks in parallel. Thus, the risk of
converging to a local optimum is low.
The random decisions made in the GA can be
modeled using Markov chain analysis to show that
each finite GA will always converge to its global
optimum region (but sometimes it has trouble
reaching the exact optimum location) [6].
It can be applied to a non-linear problem.
Therefore we decided to apply the GA to increase the
efficiency of a universal motor with respect to the
geometry of its rotor and stator unit.
5.1. Genetic algorithm in the design procedure of
the rotor and stator
The most important job when putting the new design
procedure for the rotor and the stator into practice was to
define the encoding method of the solution candidates, the
genetic operators and the termination criteria. We had to
select a method for evaluating the relative performance of
the solution candidates for identifying the better solutions.
5.2. Encoding
The parameters of the problem's search space were coded
as strings over the alphabet — of real (floating-point)
values. Using a symbolic presentation of a string with 11
characteristics (rotor-and-stator independent geometric
parameters) gives:
S= s1s2s3s4s5s6s7s8 s9s10 s11
Here, each si represents a geometrical parameter of a
single stator or rotor, where each characteristic may take
on a real value from —. For example, in the particular
string:
0.0078⊥0.0070⊥0.0340⊥0.0315⊥...
s1 is 7.8E-003, s2 is7.E-003, etc.
The rotor-and-stator independent geometric parameters
of an existing universal motor were used to form a
starting string, which was reproduced (n-1) times to
generate an initial population of n strings. A random value
distributed linearly on ± ∆ was added to each
characteristic value of the starting string to define a
reproduced string.
Each characteristic had its lower and upper limit
defined. If its value exceeded either limit, it was set to the
closest one.
Roughly speaking, the strings of this 'artificial genetic
system' are analogous to the chromosomes in biological
systems. In natural terminology, chromosomes are
composed of genes, which may take on some number of
values called alleles. The entire set of strings upon which
the GA operated was called a population.
5.3. Genetic operators
To allow the best solution candidate to evolve, the GA
employed the genetic operators of selection, crossover
and mutation for manipulating the strings in a population.
The GA used these operators to combine the strings of the
population in different arrangements, seeking a string that
maximizes the objective function. This combination of
strings resulted in a new population.
The first genetic operator used by the GA for creating
a new generation was selection. To create two new strings
(or children) two strings had to be selected from the
current population as parents. Most fit strings were
selected for reproduction. We had applied the elitism
strategy, where a randomly selected number of least-fit
members of the current population were interchanged
with an equal number of the best-ranked strings.
Crossover proceeded in two steps. First, strings were
mated randomly, using a given probability, pc, to pair off
the couples. Second, mated string couples crossed over,
using a random probability to select the one-point
crossing sites. An integer position, k, was selected
between 1 and the string length minus one [1,l-1].
Swapping all characteristic values between the positions
k+1 and l inclusively created two new strings. For
example, considering strings A and B:
A= a1a2a3a4|a5a6a7a8a9a10a11
B= b1b2b3b4|b5b6b7b8b9b10b11
Suppose, choosing a random number between 1 and
10, we obtained k=4. The resulting crossover yielded two
new strings A' and B':
A'= a1a2a3a4| b5b6b7b8b9b10b11
B'= b1b2b3b4| a5a6a7a8a9a10a11
Moreover, we might use a constant probability, pr, to
select a case in which the values of the swapped
characteristics were calculated as a mean (average) value
of the parent characteristic values.
While the first crossover approach ensured that the
child solutions preserved the 'genetic material' from both
parents, the second one helped to seek for other solutions
near to solutions that appeared to be good.
Mutation was a process by which the strings resulting
from selection and crossover were perturbed. It served to
create random diversity in the population.
Each string was subjected to the mutation operator.
Mutation was performed on a characteristic-bycharacteristic basis, each characteristic mutating with a
probability pm. Assuming that pm is a constant of 0.001,
with eleven
characteristic positions we expected
11◊0.001=0.011 characteristics to undergo mutation
during a given population. However, since a high
mutation rate resulted in a random walk through the GA
search space, pm had to be chosen to be somewhat
smaller.
There was also a possibility of annealing the mutation
rate, where pm was a variable mutation probability that
was decreasing linearly with each new population. In
another words, we assumed that each new population was
generally more fit than the previous one. Such an
approach was used to overcome a possible disruptive
effect of mutation, and to speed up the convergence of the
GA to the optimum solution.
5.4. Fitness evaluation
Following selection and reproduction, crossover, and
mutation, the new population was ready to be evaluated.
To do this, each new string (solution candidate) created
by the GA was decoded into a set of the independent
geometrical parameters for the rotor and stator, and its
fitness was estimated by running the finite-element
program (ANSYS). First, a finite-element numerical
simulation was performed. Then, the power losses of the
iron and the copper (using the equations of (6) and (3),
respectively) were calculated. Their sum (see (7))
corresponded to the solution's fitness. The fitness of
above-average strings corresponded to the lowest power
(iron and copper) losses.
By applying the elitism strategy, fitter solutions had a
greater chance of reproducing. But when the number
of least-fit solutions to be exchanged with best-fit
ones (the selection criteria) was too high, the GA was
trapped too quickly in a local optimum solution.
However, this number was subject to the population
size.
Too high a mutation rate introduced too much
diversity and took a longer time to reach the optimum
solution. Too low a mutation rate tended to miss
some near-optimum solutions. Using the annealing
strategy (a linearly decreasing mutation probability
rate with each new generation) the effects of a too
high or too low mutation rate could be overcome.
6. Evaluation
First, the efficiency of an existing universal motor was
calculated. An outline of the rotor/stator lamination of this
motor is shown in Fig.6. The power losses of this motor
were 313W and the output power was 731W. In the
outline, the levels of magnetic flux density through the
rotor/stator lamination are shown, expressed as T. The
darkest grey color indicates areas with the highest level of
magnetic flux density, which results in a high iron loss.
5.5. Termination criteria
The GA operated repetitively, with the idea that, on
average, the solutions of the population defining the
current generation had to be as good (or better) at
maximizing the fitness function as those of the previous
generation. When a certain number of populations had
been generated and evaluated, the system was assumed to
be in a non-converging state. This criterion was a 'timeout' approach. The fittest member of the current
generation at the time the GA terminated was taken to be
the solution of the design problem.
5.6. Parameter settings
Finding good settings for the parameters of the GA that
worked for the problem was not a trivial task. Robust
parameter settings had to be found for population size,
number of generations, selection criteria and genetic
operator probabilities:
If the population was too small, the GA converged
too quickly to a local optimum solution and might not
find the best solution. On the other hand, large
populations required a long time to converge to a
region of the search space with significant
improvement.
Figure 6. Existing rotor/stator lamination.
After several runs of the GA, a set of promising
solution candidates was collected. For each candidate a
finite-element numerical simulation followed by the
calculation of the objective function value (fitness) was
performed. Most of the solutions show a significant
reduction of power losses in comparison with the losses in
the existing motor. The best solution results in a powerloss reduction of 16%, and gives us a motor with copper
and iron losses of 264W (see Fig.7).
A comparison of the magnetic flux densities in the
existing and the optimized motors shows a clear reduction
Table 1. Evaluation results.
input power (W)
efficiency (%)
output power (W)
iron&copper losses (W)
analytic
calculation
initial
new
1044
1037
70.0
74.5
731
773
313
264
prototype
measurement
initial
new
1100
1100
69.7
72.6
767
799
333
301
non-exact iron losses.
7. Conclusion
Figure 7. Optimized lamination.
of areas with the highest levels of magnetic flux density in
the optimized motor.
The iron and copper losses of the initial and the
optimized lamination design are shown in Tab.1. In the
new (optimized) lamination, the copper losses in the rotor
and the stator are significantly lower than in the initial
lamination. On the other hand, the iron losses are higher
than in the initial lamination. The reason for this effect is
that the slot area in the stator of the optimized lamination
is larger. The number of ampere-turns in the optimized
lamination that are necessary for obtaining the required
torque are lower in comparison with the initial lamination
due to the optimized lamination design. If J in (2) is
substituted with (1), the equation is as follows:
(I ⋅ N )
References
2
PCuSlot =
A
In this paper, an evolutionary optimization technique for
designing a universal motor for home appliances, with
constraints on the geometry of the rotor/stator lamination,
is presented. An approach that uses the GA at a very early
stage of the motor design when an optimum configuration
of the geometrical parameters has to be found, is
described. The GA generates sets of solution candidates,
which are evaluated using the finite-element method. We
demonstrated that by repeating the process by which the
GA generates sets of solutions an optimum configuration
with reduced power losses can be found in a very short
time.
Using the GA the iron and copper losses of an existing
universal motor were reduced by at least 10%. Increasing
the GA running time or setting its parameters more
appropriately could improve this result, still further,
however, the additional effects of other motor
components should also be investigated.
⋅ ρ ⋅ l turn
(9)
From the equation above it is clear that the copper
losses increase with the square of the number of ampereturns and are inversely proportional to the slot area. In the
rotor, the slot area is slightly decreased, but the effect of
reducing I⋅N is much stronger, and this results in lower
copper losses in the rotor. The iron losses are increased in
both the rotor and the stator, mainly because of the
slightly larger iron area in the optimized design. The
magnetic flux density in the stator remains in the same
range as in the initial design. In rotor, the magnetic flux
density is decreased, but the iron area of the rotor is
larger. Overall, the optimized design has a much better
torque capability, and this results in a total loss reduction
of 16%.
We next made a prototype of the motor and measured
the real losses and efficiency of the motor. These values
are shown in Tab.1 and are slightly different from the
calculated ones. The main reason for the difference is the
[1] D.E.Goldberg, Genetic Algorithms in Search, Optimization,
and Machine Learning. Addison-Wesley Publishing Company,
Inc., 1989.
[2] ANSYS User's Manual, ANSYS version 5.6, 2000.
[3] J.H. Holland, Adaptation in natural and artificial systems.
Ann Arbor: The University of Michigan Press, 1975.
[4] G.Papa, Using Simulated Annealing and Genetic Algorithm
in the Automated Synthesis of Digital Systems, Nikos E.
Mastorakis (editor), Recent advances in Circuits and Systems,
World Scientific, Singapore, 1998, pp. 377-381.
[5] B.Koroušić-Seljak, Heuristic Methods for a Combinatorial
Optimization Problem - Real-Time Task Scheduling Problem, In
C.H.Dagli et al (editors): Smart Engineering System Design:
Neural Networks, Fuzzy Logic, Evolutionary Programming,
Data Mining, and Complex Systems, ASME Press Series on
Intelligent Engineering Systems through Artificial Neural
Networks, Volume 9, 1999, pp. 1041-1046.
[6] C.L. Karr et al, Solving inverse initial-value, boundary-value
problems via genetic algorithm, Engineering Applications of
Artificial Intelligence, Volume 13, Number 6, December 2000,
pp. 625-633.