Estimation of Rotor Resistance of an Induction Motor Using Extended Kalman Filter and Spiral Vector Theory Mohamed Menaa, Omar Touhami and Rachid Ibtiouen
AhstractA novel method of induction machine parameter estimation by an extended Kalman filter taking into account of spiral vector theory used in machine model is presented. The induction machine model is elaborated for establishing the performance equations in function only one-phase variables of the stator and the rotor. This called: "the phase segregation method". Also, the extended Kalman filter (EKF) is applied to estimate the rotor resistance and air-gap flux in induction machine parameters. The simulated results obtained by applying of estimated parameters are compared with the experimental ones. Index Terms- Spiral vector theory, induction motor, extended Kdlman filter.
NOMENCLATURE vI il
i2p
m0 RI
e, Rz e'2 M w, W(k) V(k) T I
.
Stator voltage of phase a, V. Stator current of phase a, A. Ro10.r current in the stator reference frame, A. Air-gap flux, Wb. Stator resistance of phase a, R. Stator leakage inductance of phase a, H. Rotor resistance of phase r, R. Rotor leakage inductance of phase r in the stator reference frame, H. Mutual inductance, H Electrical rotor speed, rdls. System noise. Measurement noise. Sampling time, s. Unit matrix. 1. INTRODUCTION
application of this model in the simulation of vector control permitted us to obtain a good results [3]; the same for the identification of the parameters of induction motor with Levenberg-Marquardt method [I]. In this paper, we propose a new model for estimation of induction machine parameters and states. The model is obtained from spiral vector theory. After this, the extended Kalman filter is used for estimate the induction motor parameters. The extended Kalman filter is based on the non-linear extended induction motor model. The air-gap flux and the rotor resistance are estimated by using an extended Kalman filter. To validate the used model, the estimated and experimental results for start-up stator current are presented to confirm the efficacy of the EKF for estimation air-gap flux and the rotor resistance in induction motor and the possibility to make the estimation of these last with only two sensors (one for stator current and the other for stator voltage), without any geometric transformation of variables on the opposite of the classic model (Park's model, space vector theory) where it's necessary to use four sensors for can make the estimation.
11. MODELING OF AN INDUCTION MOTORBY T H E SPIRAL VECTORTHEORY The spiral vector, is an exponential function of time with complex variable which can he expressed in the following form: i = A e J v e G t With S = - h + j w , h t 0 a n d 020
(1)
In the complex coordinates the variable " i " described a portion of spiral logarithmic curve which turns in the trigonometrically direction, Figure I .
Some works are made in the spiral vector theory for the modeling, identification and control of electrical machines [I-61. It permits to estahlish the electrical machine equations as function only variables and parameters of one phase. This called: "the phase segregation method". The obtained equations are basically using i n computer simulation, identification and control, because it takes accolint steady slate and transient slate. This is not case of conventional theories based on the two-axis method. The This work was supponed by the Algerian Ministry of Higher Educalion and Scientist Research, the projecl number J I606/02103/0?. Mohamed Menaa are wilh the Houan Boumediene University of Sciences and Technology , Facully of Electrical Engineering, Laboratory in Robotics. Parallelism and Eleclra-Energy, BP El-Allia, Bah-Ezzouar, Algien, Algeria (telephone: +?13-21-?4-79-12post office:810, fax:+Zll21 -24-7 1-87, e-mail:
[email protected]). Omar Touhami and Rachid lbtiouen are with Polytechnics college of Algiers , Depanment of E1ect"cal Engineering, laboratory in Electrical Engineering, ENP BP182, Algiers, Algeria (telephone and fax: +213-2152-29-73. e-mail:
[email protected],
[email protected])
Fig. I . Spiral vector in the complex plane.
Note: The spiral vector theoretically represents all the modes, which appear in electric engineering [1-6]. One will show then:
1262
When h = 0, "i" represent the steady state operation of an altcrnative value and described a circle in the complex plan; "i" is thus called circular vector. When w = 0, "i:' becomes equal to AeJqe-". It represents in this case the transient operation of a continuous value. When 6 = 0 (h= 0 and w = U), "i" becomes a constant. It represents the steady state operation of a continuous value. The induction machine model is described by thc following hypotheses: The motor is symmetrical in the Stator and the rotor. The saturation phenomena is neglected, The flux is at sinusoidal distrihution. Figure 2 shows the representation of the induction machine.
Replacing (3), (41, (5) and (6) in voltage equation (21, and after all treatments, we obtain the model equations linked to the stator [SI:
(7)
The system equations (7) contains only variables and parameters of one phase of stator and one phase of rotor, which are segregatcd of other phases. This approach is called: "phase segregation method". It's one of important results of spiral vector theory. F~~ can identify the air-gap flux we put
3 a0=-M(i,
2
+i,,)in
(7). and after all treatments we
obtain the following electrical equation.
(8) with: X,(t)=[~o],Y,(t)=i, andU,(t)= v, Fig. 2. Induction machine model. The voltage equations per phase are written as follow: A.=["." A,,, di, dh,, dt dt di, dh,, O=R,ic+i!-+dt dt
-]andCe=[l A,,, Ae'z],Be=[$
01
Y,o
v a =R,ia+!,-+-
(2)
with tlux equations:
A,,, =
hga = L a i a + M si b cos Mrsircos(0)+Mrsiscos
hgr = L ri r + M r is cos
-
I:-[
+ M ri t cos
Mrsi, cos(O)+ Mrsib cas[fl+
-+
$)+Mrsiccos( O
A,,,
-
The stator and rotor currents of symmetrical spiral vectors are: - A Ft - A ,(6t-j2dlj - A (6t+j2dd3) a - le b- I c - le (5) I
R, _ jaw, 4 2 2 =
(3
1
P
ir = A2eG't,
L2 M
is = A2e(6'1-j2W ' t -- A 2" (6't+j2d3) (6j
1263
(3
=++I,
IM
P
P, = 1,+ L a n d R, (3
=R,
R'
+L (3
111. EXTENDEDKALMAN FILTER USED IN ESTI~IATION OF PARAMETERS
The recursive form of the Kalman filter may be expressed by the following system equations [7-91.
after linearization of the nonlinear model we obtain the following expressions:
.~
X(k+I) =f[X(k),U(k),k]+W(k)
(9)
Z(k) = H(k)X(k)+ V(k)
(10)
and where: X(k) is the augmented state vector ([Xe(k) O(k)]' with:
X,(k) = [il $olT, O(k) = R'2,
--
AD= I + &T
and €5,~ = B,T.
The model is non-linear part of the state model:The extended Kalman filter 'EKF" relinearizes the non -linear state model for each new estimate as it becomes available. The covariance matrix Q and R of the white noises W(k) and V(k) are mutually independent and are defined as follow:
.
-
IV. SIMULATION RESULTS A. Identification Results Simulation studies on- the air gap flux and rotor resistance estimation algorithm of the EKF are performed .. on a PC using the Matlab tools. In the simulation, the error covariance matrix P of the extended Kalman filter is initially set as a unit matrix while the noise covariance matrices R, Q of the white noises W(k) and V(k) have small coefficients. The air-gap flux estimated by EKF and air-gap flux model of an induction machine are presented in Figure3.
.~ ..
E(V(k)V'(j))=
R(k)6,,. - .'
.
(12)
'-
' where S, represents the Dirac delta funciion and E denotes the expected value.
..
-
.
Piediction OF state
;.I.
.
X(k+l) = f[X(k),U(k),k]
(13)
2. .Estimation of error covariance matrix
.
Pk+i
3.
~
. .
.
= AkPA'k+Q
Computation of Kalinan filter gain
(14) .
K = Pk+l H' (HP,,lH'+R)-' .
4.
-
cisj
.
State correction X(k+l) = X(k+l)
..
I,')
~~
~
.
~
,
+ K(y(k):H, ..
X(k))
(16)
5 . Updale of the error covariance matrix
'
. -~
- -
~. ..
.
-
Pki~= p k + l - K Hi pk+l
.
.
where y(k) represents the experimental current.
-:
-
. :.
. ~
(17j-
.-
-. .
-
The extended Kalman-filter involves the IinearhXion of the above nonlinear model around the states. ~. Fig: 3. Estimated air-gap flux (red color) and air-gap-flux model (black color). . - . Figure 4 presents the rotor resistance estimated by EKF algorithm. 1264
1
5.1a~
5.a7
I
5.79'
0
1
miI
005
0.1
015
02
OZ
03
035
1
I
04
0
OM
0.1
015
nnrir1
Fig. 4. Estimated rotor resistance.
0.2 Tim(r1
08
0.3
03
0.1
Fig.6. Current-error between the experimental current and the estimated current by EKF.
B. Validation of the identification results V. Conclusion The parameters used in this simulation study are given in the Appendix. The Kalman filter is useful in the system that has system noise and measurement noise. Therefore, the effect of noise is included in this simulation. The included noises of voltage and currents are 2% of their values. Figure 5 presents the experimental stator current and estimated stator current. We remark the good concordance between them.
A novel method for estimation the induction machine parameters and states based on the model of spiral vector theory is applied. The spiral vector theory conducts to the reduction of the model order that depends only of the variables and parameters of one phase of stator and the rotor (phase segregation method). The extended Kalman filter is applied to estimate the induction machine uarameters and states. The measured parameters by the classical tests are used as the initial guess in the identification algorithm. The obtained results show that the established model by the spiral vector theory is satisfactory and permits to estimate the air-gap flux and rotor resistance in induction motor with only two sensors. We have validated the different reduced models successively by the analysis of the output signals simulated by Matlab tools, and using the comparison of simulated signals with the experimental ones. This paper, is in our opinion the novel contribution and the first ever published paper presenting the estimation of the air-gap flux and rotor resistance simultaneously using the model issue of application of the spiral vector theory associate to the extended Kalman filter. APPENDIX The nameplate data of the induction machine are:
nm 191
FigS. Experimental stator current (red color) and estimated stator current (black color). Figure 6 shows the variations of the error between experimental current and estimated current, the error is very little, which proves that the estimation is satisfactory and also we showed that is possible to estimate state and parameters in induction motor with utilization two sensors only (one for stator current and one stator voltage) and this result is obtained thanks to spiral vector theory. 1265
Pn=1.6 kW, f=SO Hz, In=3.03 A, Vn/Un=220/380 V, Nn=1425 'pm'Pole-!Jairs=2, The electrical parameters determined by the classical are: Riz3.2 0. R ' 4 . 8 6 3 0, !I= 1Mi2 = 0.312H.
0.0225 H, P F 0.0221 H,
REWRENCES
[SI M. Menaa, Modeling And Analysis Of Electrical
S. Yamamura, ‘Spiral vector theory of AC motor analysis and control,” IEEE Trans. On Industry Application, vol.IA, pp.79-86, 1991. [21 S. Yamamura, ‘Theories of AC motor analysis and control, spiral vector method, phase segregation method, spiral vector symmetrical component,” Proceedings. of ICEM, vol. 1, Pisa, Italy, pp. 1-6, 1988. [31 M. Menad, 0. Touhami, R. Ibtiouen, C. Iung, “Vector control of induction motor by a spiral vector theory,” Proceedings of the IEEE Iriternational. Conference On Control Application, Tries[, Italy, pp. 1265-1270, 1998. [41 M. Menaa, 0. Touhami and R, Ibtiouen, ‘Identification of induction machine by taking account of spiral vector theory,” Proceedings of the IEEE-SDEMPED’OI, Grado, Italy, 2001.
Machines By Spiral Vector Theory, Master Thesis (in French), Houari Boumedine University of Sciences and Technology, 1997. [6] S. Yamamura, ‘Spiral Vector Theory Of clectric circuit,” Proceedings of Japon Academy, ~01.69,Ser (B), pp.238-243, 1993. [7] Y.R. Kim, S. K. SUIand M.H. Park, ’Speed sensorless vector control of induction motor using extended Kalman filter,” IEEE Trans. On Industry Application, vo1.30. nos, pp.1225-1233, 1994. [8] K.L. Shi, T. F. Chan, Y. K. Wong and S. L. Ho, ’Speed estimation of an induction motor drive using an optimized extended Kalman filter,” IEEE Trans. On Industry Electronics, vo1.49, n o l , pp.124-132, 2002. I91 P.S. Maybeck, Stochastic Models, Estimarion, And Control, vol. 1, Academic Press INC, 1979.
[I]
1266
