International Journal of Electrical and Computer Engineering (IJECE) Vol.
No.
June 2013, pp.
ISSN: 2088-8708
Non Integer Identification of Rotor Skin Effect in Induction Machines Abdelhamid JALLOUL*.
Khaled JELASSI*.
Jean-Claude TRIGEASSOU** * National Engineering School.
Electrical Systems Laboratory (LSE).
Tunis.
Tunisia ** Laboratoire Intygration du Matyriau au Systyme (IMS-LAPS).
UMR 5218.
University Bordeaux 1.
France Article Info
ABSTRACT
Article history:
Fractional identification of rotor skin effect in induction machines is presented in this paper.
ParkAos transformation is used to obtain a system of differential equations which allows to include the skin effect in the rotor bars of asynchronous machines.
A transfer function with a fractional derivative order has been selected to represent the admittance of the bar by the help of a non integer integrator which is approximated by a J 1 dimensional modal The machine parameters are estimated by an output-error technique using a non linear iterative optimization algorithm.
Experimental results show the performance of the modal approach for modeling and identification.
Received Feb 27, 2013 Revised Apr 10, 2013 Accepted May 23, 2013 Keyword:
Fractional impedance Induction machines Ladder network Non integer order differential Output error identification Skin effect Copyright A 2013 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Abdelhamid JALLOUL.
National Engineering School.
Electrical Systems Laboratory (LSE),Tunis.
Tunisia Email: jelloulabd@yahoo.
INTRODUCTION
Accurate modeling of electrical machines is very important for the designer of the machine, facing its improvement.
On the other hand, the knowledge of parameters is necessary to realize realistic simulations of the machines and is important for the operator of modern drives who implements control systems.
Moreover, in the case of the association of a static converter to an electrical machine, the rational use of the whole passes by a perfect control of the global dynamic behavior.
With PWM power supplies, the electrical machines have to work on a very large frequency range.
Thus, the representation of this machine by a simplified model, only valid on a limited frequency range, is the source of unsatisfactory results.
The insufficiency of these models is more accentuated when the electrical machines have a massive structure .
ike asynchronous machines with cages, deep notches or massive roto.
characterized by skin effect .
r frequency effec.
Induction currents in the rotor bars are governed by a diffusive phenomenon.
At low frequencies, currents have a density which is uniform and equal everywhere over the entire cross sectional area.
If the frequency is high enough, current density tends to be higher at the surface of the bar.
The higher the frequency, the greater the tendency for this effect to occur.
This phenomenon is called Askin effectA in rotor bars, or Afrequency effectA more generally.
There are three possible reasons we might care about skin effect .
The skin effect causes the effective cross sectional area to decrease.
Therefore, the skin effect causes the effective resistance of the conductor to increase.
The skin effect is a function of frequency.
Therefore, the skin effect causes the resistance of a conductor to become a function of frequency.
If the skin effect causes the effective cross sectional area of a bar to decrease and its resistance to increase, then the bar will heat faster and to a higher temperature at higher frequencies for the same level of current.
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In electrical engineering, this phenomenon is particularly important in massive rotor or squirrel-cage induction motors.
Its diffusive character leads to notice a strong modification of the impedance .
oth resistance and reactanc.
according to the frequency .
It is thus interesting to use a transfer function with a fractional derivative order to represent the admittance of the bar on a broad frequency scale, like it has been demonstrated in a recent paper .
In the context of parameter estimation of the admittance, the derivative orders should be estimated in the same way that the other coefficients.
Based on the output error method, the models used are non linear in the parameters and optimization algorithms involve non linear programming (NLP) In this paper, we propose to identify the parameters of the asynchronous machine model taking into account the fractional feature of the rotor model and estimating its parameters using an output error identification technique.
After a reminder of definitions related to fractional integration operators in parts 2 and 3, the ParkAos model of asynchronous machines with fractional impedance is presented in parts 4.
Part 5 is devoted to present the output error method.
We propose, in part 6 experimental results of fractional identification for parameter estimation.
FRACTIONAL DIFFERENTIATION AND INTEGRATION
Fractional integration is defined by the Riemann-Liouville Integral .
, .
The nth order integral ( n real positiv.
of the function f .
) is defined by the relation:
I n ( f .
)) A ( t A A ) n A1 f (A ) d A AN ( n ) E0 Cu Where e.
A E x nA1e A x dx is the gamma function.
I n ( f .
)) is interpreted as the convolution .
of the function f .
) with the impulse response:
) A t nA1 AN ( .
of the fractional integration operator whose Laplace transform is:
I n.
A LA hn.
A A .
Fractional differentiation is the dual operation of the fractional integration.
Consider the fractional integration operator I n .
) whose input/output are respectively x.
and y.
Then:
A I n.
) .
Y .
A X ( .
Reciprocally, x.
is the nth order fractional derivative of y.
defined as:
) A Dn ( y .
)) .
X ( .
A s n Y ( .
Where s n represents the Laplace transform of the fractional differentiation operator .
ith zero initial Non Integer Identification of Rotor Skin Effect in Induction Machines (Abdelhamid JALLOUL) A ISSN:2088-8708 FRACTIONAL INTEGRATION OPERATORS The fractional integration operator I n .
) is the key element for FDE simulation.
However, the realization of I n .
) is not a simple problem as in the integer order case.
It is possible to consider the frequency and modal approaches.
Our objective is to compare the impact of these approaches for the simulation and the identification of the asynchronous machine.
Frequency Approach Synthesis Principle Let us consider the Bode plots of a fractional integrator truncated in low and high frequencies (Figure .
Figure 1.
Bode diagram of the fractional integrator It is composed of three parts.
The intermediary part corresponds to non-integer action, characterized by the order n.
In the two other parts, the integrator has a conventional action, characterized by its order equal In this way, the operator I n ( .
is defined as a conventional integrator, except in a limited band [ wb .
where it acts like s A n .
The operator I n ( s ) is defined using a fractional phase-lead filter .
and an integrator sAn .
I n .
A n Ei j A1 1 A w 'j .
The coefficient Gn is a normalized factor, such as I n ( s ) and I n .
) are identical on [ wb .
wh ] .
This operator is completely defined by the following relations .
w j A A w'j , .
w 'j A1 A A w j , .
log A log AA n A 1A A and A are recursive parameters related to the non integer order n.
When J is sufficiently large, the bode diagram of I n ( s ) tends towards the ideal one of Figure 1.
State space model of I n ( s ) It is convenient to associate a state-space representation to I n ( s ) in order to simulate fractional There is an infinite number of possibilities to represent I n ( s ) by a state space model.
Practically, we have chosen the one where the state variables correspond to the outputs of the elementary cells of I n ( s ) .
IJECE Vol.
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ISSN: 2088-8708
X j .
A A w 'j 1A X j A1 ( s ) A A xA j A1 A xA j A w j ( x j A1 A x j ) for j A 1 to J V .
Where v.
is the input of I n ( s ) and x J .
) A x.
) its output.
The corresponding state space model is:
With X 0 A M I xA I .
) A AI x I .
) A B I v.
) .
With E 1
EA A
MI AE
EE 0
Aa Aa
E 1
E AI A E As E
E As E
A w1
As AU
E x0 E
EG n E
E As E
E 0 E
E E
E E
E B I A E As E.
x I A E x j E
E E
E E
E As E
E As E
ExJ E
A wJ E
E 0 E
E E
As As Because one of our objectives is to estimate the parameters we have privileged parsimonious models in order to facilitate the identification procedure .
, .
Other approximations can be used and bring improvements to simplify the calculations of the frequency domain approach, like the modal approach.
Modal Approach Frequency distributed model The fractional order integrator is a linear system such as:
) A h.
) A u .
) .
With H .
A A LAh.
)A 0 A n A 1 This system can be represented by a frequency distributed model.
it is also known as a diffusive model .
efer to appendix 1 and .
E C x .
, w ) A AA x ( w , t ) A u .
) E Ct E y ( t ) A A (A ) x ( w , t ) dw And Cu ) A E A ( w ) x ( w, t ) dw With A ( .
A sin.
A ) A w An , 0 A n A 1 A .
is called frequency weighting function.
Non Integer Identification of Rotor Skin Effect in Induction Machines (Abdelhamid JALLOUL) A ISSN:2088-8708 Frequency discretized distributed model This continuous frequency weighted model is not directly usable.
A practical model .
ecessary for simulation application.
is obtained by frequency discretization of A .
, where the function A .
is replaced by a multiple step function .
ith K step.
For an elementary step, its height is A ( wk ) , and its width is AEwk .
Let c k be the weight of the kth element:
c k A A ( wk )AEwk .
A .
) AE w 1 AE w 2 AE w k Figure 2.
Frequency discretization of A .
Thus, the continuous distributed model becomes a conventional state model with dimension equal to K.
E dx E k A A w k x k ( t ) A u ( t ).
k A 1.
E dt E y .
) A Eu A ( w k ) x k .
) AE w k k A1 A Eu c k x k .
) EE
k A1 or equivalently:
EE X ( t ) A A X .
) A Bu .
) E
EE y .
) A C T X .
) .
With E A w1 E x1 E X .
) A E As E
EE 0
EE x K EE A w k EE B A Au1 1 AU 1Ay .
C A Auc1 cK Ay With this approach, we obtain a discrete state-space model which is frequency distributed with the w1 C 0 , wK C Cu and K AA 1 Comparison with Frequency Model It is easy to transform the model .
of I n .
) into a modal form because the w j are known a This transformation is based on the following decomposition in simple elements:
I n .
A 0 A s j A1 s A w j Eu IJECE Vol.
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Where c0 and c j coefficients are linked to G n , w j and w 'j by the relations:
c0 A G n c j A Gn w j A w 'j w 'j Ei i A1 1 A iC j This second definition of I n ( s ) corresponds to a modal state model:
E XA ' ( t ) A A ' X ' .
) A B ' ( t ) u ( t ) E EE y ( t ) A C I X ( t ) .
With:
AI' A E
EAs
E1E
E x 0' E
E1E
E 'E
As E X ' .
) A E x1 E .
B 'I A E E
E As E
EAs E
E ' E
0 A wJ E
EE x J EE
E1E
A w1 Aa C TI A Auc 0 cJ Ay In the frequency domain approach, the modes w j are indirectly obtained by I n ( s ) in the Auwb .
wh Ay interval, they correspond to the modes of the modal approach.
The interest of this last representation is that the modes are decoupled, which allows fast computations.
Moreover, an important interest of w0 A 0 is to reject static error in simulation applications.
PARKAoS MODEL OF INDUCTION MACHINE The most important assumptions to derive the ParkAos model are:
The air gap between the stator magnetic structure and the rotor magnetic structure is uniform.
All magnetic variations due to slots are neglected.
The magnetic field is assumed to have a sinusoidal spatial distribution.
The stator and rotor windings axes coincide with the magnetic axes of the phases.
The permeability of the iron is infinite.
The ParkAos transformation establishes an equivalence between a three-phase representation and rotor reference frame.
The conventional equivalent diagram .
of ParkAos model is represented on Figure 4 :
Figure 3.
Conventional equivalent diagram of ParkAos model With:
Rs and Rr representing the resistance of the stator and the resistance of the rotor bars respectively.
A is the rotor speed, lr are the stator and rotor leakage inductances.
Lm the magnetizing inductance.
Non Integer Identification of Rotor Skin Effect in Induction Machines (Abdelhamid JALLOUL) A ISSN:2088-8708 Ladder Model To take into account skin effect in rotor bars, the assumption is made that each equivalent rotor winding is composed of K slices in parallel.
The ParkAos equivalent diagram with ladder model .
efer to .
for more detail.
is represented on Figure 5.
Figure 4.
ParkAos ladder model l k represents the linkage inductance of each elementary slice.
A complex notation is used:
xd A jx q A X The mathematical model of squirrel cage induction motor can be written as:
EU s A R s I s A dt A s A j A A s E E0 A R I A d A E EA s A l s I s A L m ( I s A I r ) E l k I rk EA r A L m ( I s A I r ) A k A1 Eu There are several expressions that can describe the developed electromechanical torque of an induction machine .
, .
, we prefer to use the following because it refers only to stator variables:
C em A A ds iqs A A qs ids .
Induction machine equivalence with fractional impedance Using equivalence between a ladder network and fractional impedance .
, one can define the ParkAos fractional model of the induction machine:
Figure 5.
ParkAos fractional model The equations describing electromagnetic processes in induction machine .
ncluding a squirrel-cage roto.
are as follows:
EU s A R s I s A A s A j A A s E EA A ( Lm A l s ) I s A Lm I s E s IJECE Vol.
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ISSN: 2088-8708
We define the magnetizing flux A m A m A Lm ( I s A I r ) .
We can write:
sA m A A I r ( s ) Z n ( s ) .
Then, sA m ( s ) A A ( I .
A s I r ( s )) .
Which corresponds to the fractional order differential equation:
A m .
) A A 0 I r .
) A D n ( I r ( t )) .
Because.
A s AA m Al s I s .
we obtain a differential system allowing the simulation of the asynchronous machine:
E A s AU s A R s I s A j A A E dt I s A jA A ) E D n ( I r ) A A a 0 I r A b 0 (U s A R s I s A l s E A s A Lm I r EI s A EE Lm A ls The mechanical expression of the rotor speed is obtained thanks to the relation:
A Cem A C r A fA Cem is expressed in .
J : moment of inertia f : friction coefficient OUTPUT ERROR IDENTIFICATION Next, we remind the principle of a method allowing the estimation of the parameters of the Park model of induction machine with fractional impedance .
Whereas parametric estimation can be performed by a linear optimization technique in case .
the model is linear in the parameters, the estimation of the derivative orders and of the coefficients requires the use of a nonlinear programming algorithm.
The method suggested by Trigeassou.
Lin and Poinot .
, .
, is based on the definition of a non integer integration operator limited in frequency .
requency approac.
The model of the system is in continuous time representation and we use an output error technique (OE) to estimate its parameters .
, .
For the fractional state-space model of the induction machine, the parameter vector is defined by:
A T A AuR s nAy .
The state-space model is simulated using a numerical integration algorithm, thus one gets:
Non Integer Identification of Rotor Skin Effect in Induction Machines (Abdelhamid JALLOUL) A ISSN:2088-8708 II si A f i .
AI i ) .
Where AI i is an estimation of A at iteration i.
The optimal value of AI(A ) is obtained by minimization of the quadratic criterion:
J A
Eu .
d* A iId ) 2 A Eu .
q* A iIq ) 2 k A1 k A1 we obtain:
AI i A1 A AI i A AEA Where AEA depends on the optimization algorithm.
We can use a black box technique provided by the Matlab toolbox functions in order to minimize J .
In this case we want to obtain the optimal A opt without worrying of how we obtain this estimate.
But this technique presents some drawbacks such as the absence of direct informations on the criterion at the optimum, thus in particular on the precision .
ensitivity of J with regard to the different estimate.
To remedy these drawbacks, we use sensitivity functions of the simulated output .
, .
Because II s .
) is non linear in AI , a Non Linear Programming technique is used to estimate iteratively AI i :
AI i A1 A AI i A EE J AA
A AI
Ay J e AI AA i .
With .
, [ .
E '
E J A A A 2 Eu Au k A k ,A i : gradient
k A1
E "
: hessien E J AA C 2 Eu A k ,A i E k A1 E A : Marquardt parameter E C yI k EA k ,A i A CA : sensitivit y function This algorithm, known as Marquardt's one .
, often used in non linear optimization, ensures robust convergence in spite of a bad initialization of AI .
A good precision of the output sensibility functions A k ,A i is however necessary to ensure a good convergence and precision of the algorithm.
EXPERIMENTAL IDENTIFICATION
In order to appreciate the interest of the Park fractional model with the modal representation of the fractional integrator, we use this method to identify three possible models, using input/output data provided by an a test bench including the induction machine, the data acquisition system and the PWM generator gives three-phase voltages and currents and the position of the motor axle at different speeds.
The sampling period is Te= 0.
Using the Park's reference frame linked to the rotor, we obtain data u dq s and i dqs .
Identification of the Conventional Model Let ids be the measured current and iIds be the simulated current, using conventional or fractional IJECE Vol.
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J is the quadratic criterion which is minimized according to the output error technique .
, .
for more detail.
The parameters with the conventional ParkAos model are defined by: A A [ Rs Rr l m l r ] Rs estimation Rr estimation Lm estimation lr estimation 2000 2500 3000 Figure 6.
Measured and estimated currents with conventional model Figure 7.
Parameter estimation with conventional Identification of the Fractional Model Hn .
The modal formulation is not adapted to the exact calculation of C x ( t ) because the A k and c k are Cn complicated functions of n.
It is possible to simplify and proceed directly the calculation of the sensitivity functions .
, .
, .
by numerical differentiation, in the form:
x ( nI A AE n , t ) A x ( nI , t ) C x ( nI , t ) A lim AEn C nI .
A preliminary study is essential for the choice of AE n .
In the general case.
AE A is difficult to choose because A can vary from - Cu to Cu .
Because 0 A n A 1 , it is easy to find an optimal value of AE n , which will be always the same.
Then the calculation becomes more simple.
The parameters with the fractional model H n are defined by: A A [ Rs l m a0 b0 .
As exhibited by Figure 6 and 8, there is a good fit between measured and estimated currents with both conventional and fractional H n models.
Rs estimation Lm estimation a0 estimation b0 estimation n estimation 2000 2500 3000 Figure 8.
Measured and estimated currents with fractional model Hn.
Figure 9.
Parameter estimation with fractional model Hn.
Non Integer Identification of Rotor Skin Effect in Induction Machines (Abdelhamid JALLOUL) A ISSN:2088-8708 In order to appreciate the improvement of H n model, it is necessary to compare the respective quadratic criterions .
ee Table .
It is obvious that the fractional model provides a better approximation of measurements than the conventional ParkAos model.
Identification of the Fractional Model Hn1,n2.
The model .
gives a good approximation only at low and medium frequencies .
In order to improve the fractional model .
and particularly its high frequency approximation, a second model is b0 A b1s n1 H n1,n2 .
A .
a0 A a1s n1 A s n1 A n2 It has been demonstrated .
ee for example .
) that the phase of the fractional model has to be A b s n1 equal to A at high frequencies, i.
e with a fractional order equal to 0.
If s C Cu.
H n1,n2 .
C n1 A n A n1 .
s1 2 s 2 Then if n2 A 0.
H n1,n2 .
model will provide a good approximation at high frequencies.
The parameters of the fractional model H n1,n2 are defined by: A A [ Rs l m a0 a1 b0 b1 n1 ] .
Because n 2 is set equal to 0.
5, it is only necessary to estimate n1 .
As previously, there is a good fit between measured and estimated currents demonstrated by Figure The Figure 7, 9 and 11 represent the parameters variation during the identification.
The corresponding quadratic criterions of Table 1 indicate that H n1,n2 .
performs a better approximation than the other models.
We present in the following table all the results of experimental parameter estimation.
Rs estimation Lm estimation a0 estimation b0 estimation n1 estimation 2000 2500 3000 Figure 10.
Measured and estimated currents with fractional model H n ,n Fractional model H n Classical model : Quadratic criterion =299.
Fractional model H n : Quadratic criterion = 237.
Fractional model H n ,n : Quadratic criterion = 227.
IJECE Vol.
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Estimated parameters Fractional model H n ,n Classical model b1 estimation Figure 11.
Parameter estimation with fractional model H n ,n a1 estimation IJECE
ISSN: 2088-8708
CONCLUSION
In this paper, we have presented and compared some models for the identification of rotor skin effect in induction machines.
Thanks to ParkAos transformation we have obtained a conventional model in reference frame .
related to rotor.
To take into account the diffusive phenomena of the skin effect, the ParkAos equivalent diagram with ladder model has been proposed.
Then, we have replaced the ladder model by a fractional The identification of the Park model with a fractional impedance has been performed by the output error method.
Fundamentally, this method is based on the simulation of the model .
nd of sensitivity We have used the modal approach to compare three models with experimental data.
The results show clearly that the fractional models give better approximations than the conventional Park model.
Moreover, we have shown that a new fractional model with two derivatives is able to improve these experimental approximations.
APPENDIX
Using the complex formula, the inverse Laplace transform LA1 ( n ) is given by:
A A jCu ) A H ( s )e st ds j 2A A A jCu (A.
We use the Bromwich contour shown in Figure 12.
Fig.
Bromwitch contour C Thus, the impulse response h.
) of any system can be calculated from its transfer function H .
) .
Because H .
A 0 A n A 1 is a multiform function, a cut is necessary in the complex plane, corresponding to the contour C of Figure 12.
Thus we can write:
1 st
e ds A A A A E
E A
j 2A C s n j 2 A EE AB BDE EH HJK KL LNA EE
E E E E E
(A.
Refering to CauchyAos theorem:
j 2A CE (A.
Because.
Non Integer Identification of Rotor Skin Effect in Induction Machines (Abdelhamid JALLOUL) A ISSN:2088-8708 E
E A
EA0
j 2A EE BDE LNA EE (A.
EA 0
j 2A HJK (A.
E E
A h n ( t ) A lim E A E
RCCu
j 2A AB
j 2A EE EH KL EE
Au C0
E E
(A.
Finally we evaluate the integrals along the paths EH and KL.
Along EH, s A xe jA A A x s n A x n e jnA (A.
ds A A dx and as s goes from AAu to A R , x goes from Au to R .
AAu e st e A xt E 2Aj E s n 2Aj EH 2Aj EAu ( xe jA ) n AR (A.
Along KL, s A xe A jA A A x s n A x n e jnA (A.
ds A A dx e st e A xt ds A A 2 A j KL 2 A j A Au s 2 A j Au ( xe A j A ) n E (A.
We thus obtain:
e A xt e A xt dx A E R C 0 2A j E ( xe Au ( xe EAu Au C0 h ( t ) A lim e jn A A e A jn A x A n e A xt dx R C 0 2A j E Au h ( t ) A lim (A.
Au C0 and finally:
) A E
sin nA x An e A xt dx (A.
Because x corresponds to a frequency, let us define w A x .
Notice that e A wt is the impulse response ( z ( w, t ) A e A wt ) when its input is v.
) A A .
) .
sAw Thus, in a more general situation, the response z ( w, t ) of the elementary system to an input v.
) verifies the differential equation:
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C z ( w, t ) A A w z ( w, t ) A v.
) Ct (A.
and the output x.
) of the fractional system is the weighted integral .
ith weight A .
) of all the contributions z ( w, t ) ranging from 0 to Cu :
) A A ( .
z ( w, t )dw (A.
A ( .
A sin.
A ) A wAn (A.
with 0 A n A 1
ACKNOWLEDGEMENT
This work was supported by the Tunisian Ministry of High Education.
Research and Technology.
REFERENCES