![](bg3.png)
hysteresismeter. Measurement was carried out for
the speed of gain of the magnetising field H of
150 A/m/s. Hysteresis loops were determined for
increasing amplitude of magnetic field intensity in
the range from 20 to 1142 A/m. Also the initial
magnetisation curves were measured. Between the
measurements of magnetic hysteresis loop the core
was demagnetised with sinusoidal waveform of the
exponentially decreasing amplitude. Frequency of
this waveform was 10 Hz, initial amplitude was
1142 A/m, ratio of successive amplitudes was 1,03.
Computer
Fluxmeter
Magnetizing and
demagnetizing signal
generator
U/I
Magnetizing
winding
Measuring
winding
Sample
Interface
Figure 1: Schematic block diagram of the measuring
setup.
4 METHOD OF OPTIMISATION
The optimisation process bases on the minimisation
of target function, which is given by the equation
(Szewczyk, 2009):
n
i
ipomiSAJ
HBHBF
1
2
))()((
(11)
where: n - number of measurement points, H
i
-
magnetic field,
B
J-A-S
(H
i
) – results of the modelling,
B
pom
(H
i
) – results of the experimental
measurements.
In presented case, the best is to use a two-stage
optimisation. In the first step, the evolutionary
strategies (μ+λ) (Schwefel, 1995), combined with
simulated annealing (Schwefel, 1981), (Wilson et
al., 2001), should be used. In the second step, the
gradient optimisation should be used, for the 20 best
results obtained after the first step.
The evolutionary strategies (μ+λ) are the
heuristic optimisation methods, based on adaptation
and evolution. In evolutionary strategies,
the population of vectors, which contain parameters
of the Jiles-Atherton extended model, is subjected to
the three operators. First - mutation operator, which
randomly changes the value of the parameter of the
model. Second - crossover operator, which
exchanges values between the two vectors. And
third - selection operator, which to select the best
value of the target function F.
From the population of
individuals (parents),
population of
individuals (descendants) is created.
During this process, copies of randomly selected μ
individuals are parents. Then, on the base of μ
parents, the population of
descendants is created
randomly, using the operators of mutation and
crossover. Population of descendants is combined
with the parents population, creating a population of
μ+λ individuals. The best
individuals from the
μ+λ population gives the new population for the
next iteration.
During the optimisation process, physical limits
of the Jiles-Atherton model have to be strictly
observed. If physical limits are exceeded (e.g. value
of anisotropic energy density K
an
is lower than zero)
the value of the target function F is significantly
increased. As a result, the optimisation process is
carried out within physical limits.
In the minimisation process a population of 900
vectors was used. The crossover operator of a group
of
= 3 vectors (parents) generated
= 12 vectors
(descendants). Then the descendants vectors were
subjected to the mutation. The distribution of value
changes of the parameters during the process of
mutation was a normal distribution, of which
standard deviation was equal to 3% of the initial
value of the modified parameter. In every iteration,
in accordance with the simulated annealing, the
standard deviation decreased by 5%.
5 RESULTS
The target function F was calculated for 3 hysteresis
loops (measured for different magnetising fields) at
the same time.
The figure 1 below shows the changes in the
value of the target function F during the
optimisation process by using the evolutionary
strategies.
Because the functions F
min
(for best vector of the
population, calculated during the the optimisation
process) and F
max
(for worst vector of the
population, calculated during the the optimisation
process) decreases monotonically in the next
iterations, the optimisation process can be regarded
as convergent.
The next figure 2 shows the results of
experimental measurement of hysteresis loop B(H)
(marked with •) and modelling results (marked with
ApplicationofEvolutionaryStrategiesforOptimisationofParametersduringtheModellingoftheMagneticHysteresis
LoopoftheConstructionSteel
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