bar gallo 08 final paper
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Proceedings of the 2008 International Conference on Electrical Machines Paper ID 898
978-1-4244-1736-0/08/$25.00 2008 IEEE
Real time parameter determination in saturated inductors submitted to non-sinusoidalexcitations.
Ramon Bargall Perpi1, Manuel Roman Lumbreras2, Alfonso Conesa Roca2, Guillermo Velasco Quesada2
1: Electrical Engineering Department., 2: Electronic Engineering Department. Polytechnic University of Catalonia;
EUETIB, C/Urgell, 187, 08036 Barcelona, Spain
Tel: (+34)-934137323, fax: (+34)-934137401
e-mail: [email protected]
Abstract- Parameters determination on saturated systems is a
complex task. Sometimes you can determine these parameters bycalculation but building errors must be checked to verify the finalvalues of them. You can use accurate off-line models but the time ofprocess could be excessive for a factory process. In this paper wedescribe a recursive algorithm to calculate the parameters in realtime and buried in a generic measurement system made to test threephase inductors.
I. INTRODUCTION
In order to have a fast measurement of the parameters the flux
linkage must be known. It is, however, expensive and difficult to
measure the flux. Instead, the flux can be estimated based on
measurements of voltage and current. After discussion we
present a Luenbergers observer for the flux and a recursive
algorithm to calculate the parameters of the considered model.
Also the measurement systems calculate the total losses on the
inductor and using our model we can separate it on joule and
magnetic losses.
II. FLUX ESTIMATION AND OBSERVATION
There are some possibilities to estimate the flux. The
following table summarizes the most commons.
TABLE I. FLUX ESTIMATORS
Model Governing equation
voltage = dtiru )(
current iL =
These estimators have some problems:
a) If the resistance (r) is not correctly determined the estimated
flux is different to the correct one.
b) If the inductance (L) is not correctly determined the
estimated flux differs to the correct one.
For these reasons it is better to use a closed loop observed. The
proposed Luenbergers observer is:
( )
+
=
+=
n
NN
N baIi
iiirudt
d
(1)
For values of > 1 is absolutely stable and the convergence ofthe observed values (^) to the real ones is guaranteed.
In the above expression, INis the current that produces the flux
N; it isnt the nominal current of the tested coil or transformer.
The unknown parameters are: r, a, and b. Using the above
expression it is possible to determine the inductance:
+
=
=
n
NN
N baI
IL (2)
Or
+
=
=
Nn
n
N bnaId
dIL
1
11 (3)
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Proceedings of the 2008 International Conference on Electrical Machines Paper ID 898
978-1-4244-1736-0/08/$25.00 2008 IEEE
We use the above observer in addition to a recursive estimator to
obtain the unknown parameters. The figure 1 shows the structure
of that.
III. DISCRETESYSTEM
By discretization the equation 1 becomes:
(4)
With T in the sampling period and k represents the k sampling
interval (actual time could be calculate by t = k*T) To obtain the
unknown parameters we solve recursively the above equation.
Fig. 1. Flux observer + Recursive parameter estimator
IV. LEASTSQUARESRECURSIVEMETHOD
The recursive least squares method (RLS) tries to determinethe unknown parameters of the following matrix equation:
)()()( kekky T += (5)
y(k) are the measured set on instant k, )(kT are knownfunctions, e(k) are the measurement errors (unknown) and arethe unknown parameters. The RLS method calculates estimation
for the actual time (k) based on the measurements until k and the
estimation for the earlier time (k-1). The whole algorithm is:
( ))1()()()()1()( += kkkykKkk T
(6))()()( kkPkK = (7)
+
=
)()1()(
)1()()()1(
)1(1
)(
kkPk
kPkkkP
kP
kPT
T
(8)
For the penalizing factor the suitable values are from 0.95 to0.995 and the losses function to be minimized will be:
( )21
)()(2
1),( =
=
iiykV Tikk
i
(9)
The expression
Tikik
ikik
ee
e
/)()1()(
ln)(
=
=
(10)
As a result of approximation of (10) at the surround of = 1.T is a pseudo-time constant
(11)
Suggested values for are 0.95 to 0.995 and these valuesindicates values for T for 20 to 200. It is accepted that values
older that 2*T arent any influence for the final result.
The only limitation of this method is that the parameters mustbe expressed as coefficient of known functions. In our case thisis true because the exponent (n) for the current-flux function can
be considered known (they can be determined by early test andusually it is 5, 7 or 9 depending to the saturation level of thematerial.
There is another limitation of this method but this is due to thecharacteristics of the applied signal: It must be a called persistentexcitation of order N, and the following conditions must beguaranteed:
These limits exists
=
+=L
t
T
Lu tutuL
r1
)()(1
lim)( (12)
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Proceedings of the 2008 International Confe
978-1-4244-1736-0/08/$25.00 2008 IEEE
These matrix be positive definite
=
=L
t
Lu tuL
r1
(1
lim)(
These expressions are true if the harmon
applied signal contains at least N/2 harmonic
V. EXPERIMENTALTE
Fig 2 shows the experimental setup. The
with the following parts:
- Current controlled PWM three phase
- Current and voltage sensors co
acquisition card.
- Data acquisition card: 16 di
1.2Msamples.
-
PC with labview v8.2 to show and cavalues (current, voltage, power loss
The user could select what data is sho
Fig 2. Experimental setup.
The following pictures show the resul
transformer (1.3 kVA, 220/380 V, 5.9/3.5 A
W, R1= 0.942 , R2= 1.202 , N= 0.9with PWM alimentation.
ence on Electrical Machines Paper
+ T tu )() (13)
ic component of the
s.
ST
hole system is made
onverter.
nected to a data
ferential channels;
lculate the interestedes, parameters, etc.)
wn.
ts for a one-phase
, PJ= 40 W, Po= 18
9 Wb, IN= 1.41 A)
Fig 3. Measured and calc
Fig 4. a
Fig 5.
ID 898
ulated current on a transformer
(k) parameter.
(k) parameter
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Proceedings of the 2008 International Conference on Electrical Machines Paper ID 898
978-1-4244-1736-0/08/$25.00 2008 IEEE
The magnetic material of this transformer has an exponent n =
7 (these value was calculated earlier) Final values for a and b are
0.48 and 0.52 respectively; they are agree with the values
calculated by off-line methods. The following picture shows the
comparison between the experimental saturation characteristic
and fitted equation using the above values for a, b, and n.
Fig 6. Experimental and fitted saturation characteristic.
Fig 7. Calculated and estimated inductances
The figure 7 shows the comparison between calculated
inductance using experimental measurements, calculated
inductance using expressions (2) and (3) and by using 2D FE
calculation.
VI.
CONCLUSSIONS
The explained method allows the determination of
parameters in saturated systems, and the algorithm is
adapted to saturation level.
For sinusoidal applied voltage the results are bad: thefinal values for the parameters a and b have offset.
These are due the limited harmonic component of
these signal (1 harmonic only allows to determine
only 2 parameters under theorist assumptions: no
measuring errors, etc.) As a conclusion it is necessary
to apply a signal with some harmonics to avoid or
minimize measuring errors)
ACKNOWLEDGMENT
This work was partially granted by FIT-030000-2007-79
(MITYC) and Grupo PREMO S.A.
REFERENCES
[1] Ljung. System Identification: Theory for the user. Prentice Hall, NewJersey, 1987
[2] R. Bargallo. Aportacin a la determinacin de parmetros de los modelos en
la mquina asncrona para una mejor identificacin de variables no mensurables
incidentes en su control. PhD dissertation. UPC, Barcelona, 2001.
[3] Justus. Dynamisches Verhalten Elektrischer Maschinen. Vieweg Publishing,
Braunschweig/Wiesbaden, 1991.
[4] B. Peterson. Induction machine speed estimation. Observation on observers.
Doctoral Dissertation. Lund Technical University. 1996. Lund. Sweden.
[5] O. Nelles. Nonlinear System Identification. Springer. Berlin. 2001.
i=a*flux+b*flux^7
0 0,3 0,6 0,9 1,2
flux/fluxN
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
1,8
2
2,2
i/iN