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Sensorless Monitoring of Induction Motor Drive Systems
ELSEVIER
Copyright © IFAC Automation in Mining, Mineral and Metal Processing, Nanc)'. France. 2004
IFAC PUBLICATIONS www.elsevier.coml1ocate/ifac
SENSORLESS MONITORING OF INDUCTION MOTOR DRIVE SYSTEMS
B. Bensaker, H. Kherfane, A. Maouche, A. Metatla, aod R.Wamkeue
Departement d'Electronique,. Universite de Annaba, B.P 12, 23000 Annaba. Algeria. Tell/ox: + 213 38867928 e-mail: bensakerbachirrCilvahoo.fr Departement des Sciences Appliquees. UQAT. Rouyn-Noranda, Quebec 445 B.de /'Universite, Canada. J9X 5£4. e-mail: rene.walllkeueli.il.uqat. uquebec.crJ
Abstract: The paper deals with sensorless monitoring of induction motor drive systems. It is well known that induction motor drives are very important parts of any industrial application involving electrical equipment. They must be carefully surveyed to ensure reliable and profitable operations. However induction motor drive systems are also known as complex nonlinear time-varying systems. Thus entails great difficulties for monitoring purposes. Based on the fact that the system model can be significantly simplified if one applies the d-q Park transformation and an appropriate field orientation technique, an adequate model structure is obtained. An observer-based state variable estimation technique is used for sensorless monitoring of a general induction motor drive system. The sensorless monitoring procedure uses a simple curves plotting of the estimated state variables. The approach permits to detect up to five percent changes in the estimated state variables information. Copyright © 2004 IFAC Keywords: State space modelling, observer, state estimation, monitoring, induction motor drives.
I. INTRODUCTION The induction motor is one of the most widely used machine in industrial applications such as metal or hot/cold rolling processes. This is due to its high reliability, relatively low cost, and modest maintenance requirements. With the development of power electronic technology, low cost digital signal processing (OSP) micro-controllers and estimation techniques the induction motor constitutes an attractive actuator for high performance drives (Holtz, 2002; Bensaker et al., 2003). However the induction motor is known as a complex nonlinear system in which time-varying parameters entail an additional
89
difficulty for developing control and monitoring strategies. Based on the fact that the model can be significantly simplified if one applies the d-q Park transformation and a field orientation technique also called vector control, different structures of the model exist in the literature (Bensaker et al., 2003). The choice of a model structure depends on the selected state variables and the problem at hand. Industrial applications involving induction motors such as metal processing and hot/cold rolling are subject to control and monitoring problems. In order to ensure reliable and profitable operations they must be carefully surveyed. Sensorless monitoring techniques focus on reducing the number of sensors to be used.
The aim of a system monitoring is to control, through certain state variables that indicate the system health, its main functions and to compare the measured values of these state variables with their preset values (Balan et al., 1999; Ioannides and Syggeridou, 1999). The objective is to decrease the rate of fault occurrence and to increase the MTBF (Mean Time Between Failure). State space modelling of induction motors and observer-based state variable estimation techniques are very helpful in this case of sensorless system monitoring.
M, and
The matrices 0a' 0a,' R s ' R" L s ' L"
I 0 are defined as follows:
°a =[ 0
W ol ]
- Cl?, 0
q,=
,
[0
(Ol-Ol)]
-(Ol-Ol) 0
,
s [r, 0] [Is 0] r ' R, = 0 r, ' Ls = 0 Is ' [r0 0] s L, = [I, 0] , M =[m 0] and 1 = [1 O . Rs =
2
2
o I,
11. STATE SPACE MODELLING In order to establish a state space mathematical model of the induction motor, the following classical hypotheses are used (Holtz, 2002). -The higher order harmonics of the magnetic field of stator and rotor are neglected, -The ferromagnetic losses and effect of saturation are also neglected (for control design), -A perfect electrical and magnetic circuit symmetry (uniform air gap and sinusoidal flux density distribution), is assumed. In this paper the dynamics of the induction motor are divided into two subsystems which are subsystem of the electromagnetic dynamics and subsystem of the electromechanical dynamics. The electromagnetic subsystem is, in its turn, divided into stator dynamics and rotor dynamics. The electromagnetic subsystem is represented, in an arbitrary rotating d-q Park reference frame, by the following nonlinear state space model:
0
0 m
]
O2 O2
Usual notation for parameters and variables is used. The indexes sand r refer to the stator and the rotor com ponents respectively. Based on the fact that the model can be significantly simplified if one, in addition with the d-q Park transformation, applies a field orientation technique for an appropriate fixed reference frame, different structures of the model can be obtained for sensorless monitoring purposes taking into account the availability of measurements (the number of sensors to be used) (Bensaker et al., 2003). If the stator fixed reference frame is chosen one has to substitute in equation (I)
0
a
O2 and 0
to
a'
to
0, .
In stator flux orientation technique the direct component of the stator flux must be aligned with the real axis, thus leads to ({Js
= CPsd
and ({Jsq
=0 .
If the rotor fixed reference frame is used one has to substitute, in relation (I),
0
a
to
0, and 0
a'
to
O2 , In the case of rotor field orientation technique the direct component of the rotor flux must be aligned the real axis, in consequence ({J, = ({J'd and ({J,q =
The electromechanical subsystem is represented by the following nonlinear differential equation called the motion equation.
The electromechanical torque and the flux linkage relations, which ensure the link between the two subsystems, are respectively defined by:
CPs) ( ({J,
s s [L M](i ) = M L, i,
0.
The rotor or the stator field orientation technique can be seen as a linearization technique to be used to reduce significantly the nonlinear induction motor model (Holtz, 2002). The choice of a model structure is generally guided by the fact that most drive systems have a stator current control loop incorporated in their control structure. This leads to select the stator current as the first state variable. The second state variable is then either the stator flux linkage vector, the rotor flux linkage vector or the rotor current depending on the problem at hand. In sensorless techniques the second state variable is estimated by using a closed-loop observer-based technique. In each case, the not selected state variables are indirectly estimated on the basis of measured and directly estimated state variables. Observer-based state estimation techniques are very helpful for closed-loop sensorless monitoring applications fJensaker et aI.,
(4)
90
The rotor current is indirectly estimated via the stator current measurements and the estimated rotor flux as:
2003). In our case we are interested in stator current and rotor flux state variables model. In this case the nonlinear state space model is:
I
L-r_ (5)
MLr
1 I:({Jr ](.
)
(11)
The rotor current estimate is generally employed in procedures concerned with the condition-based system maintenance and monitoring. In all application cases the stator current measurements are very helpful. Reduced order current observer-based techniques can be also used for monitoring the induction motor drive systems.
(6)
(7)
(8)
III.0BSERVER-BASED STATE ESTIMATION The evolution matrix parameters are defined by:
In order to decrease the price of induction motor drive systems and to increase their reliability and robustness ongoing research has concentrated on the minimization of the number of sensors to be used without deteriorating the dynamic performance of the considered drive system. The development of the estimation techniques and the availability of new low cost DSP-based (Digital Signal Processing) microcontrollers, have reinforced the industrial applications of this idea for sensorless control and monitoring purposes (Bensaker et aI., 2003; MihetPopa et al. 2003). The above model can be written in the following standard linear state space model:
All =Q a -(L s -ML~IMrl(Rs +ML~IA21)' ) A12 =(L s - ML-1M)-1 (Q a Mrr 1 - ML-r 1A22' r A 21 An Bz
= (L s
-
= RrL~l M,
= Q ar ML~I
RrL~1 ,
Mr
l
(9)
The objective of this model is to estimated the rotor flux linkage state variable based on the stator current measurements using an observer-based technique. Once the rotor flux is estimated one can easily estimate the other electromagnetic state variables or mainly the electromagnetic torque. The stator reference frame can be used for this purpose in addition with the rotor field orientation technique. The electromagnetic torque equation becomes:
dX(t) = AX(t) + BU(t) dt Y(t) = CX(t)
(12) (13)
where X(t) is the selected state variable vector and (10)
Y (t) is the measured output vector. A, Band C are the evolution, the control and the observation matrices respectively. One of the most used technique for estimating the state variables of a system described in terms of state space model is the observer technique. Observer-based state variables estimation uses an adaptive mechanism which involves as input, the error between the measured and estimated state variable values. The general form of the observer is :
On can see, relation (10), that the electromagnetic torque is proportional to the stator current component since the rotor flux takes its maximum value. The model order reduces in consequence and the induction motor system can be defined only by the rotor model and the mechanical subsystem. The decoupling between the stator and the rotor components is then obtained and the induction motor model becomes linear if the rotor angular velocity is assumed to be slowly varying or constant. This assumption is generally justified by the fact that electrical dynamics are very fast than the magnetic dynamics which are in thier turn very fast than the mechanical dynamics. The resulting dynamic structure of the induction motor system is then significantly simplified. The induction motor, in these conditions acts as a dc motor.
91
~t) =AX(t)+BU(t)+L[Y(t)-J'{t)]
(14)
tU) = CX(t)
(15)
behaviour with their corresponding nominal behaviour (Bensaker, 1999; Bensaker, 2000). Figure I shows the sensorless monitoring principle.
where X(t) is the estimated state variable vector of the unknown state variable vector X(t) and Y (I) is the estimated output of the measured output signal Y (I). L is the observer gain matrix. Relations (14) and (15) can be written as
di(t) = [A-LqX(/)+BV(/)+LY(/) dl Y(t) = CX(t)
(J
t-r---.,....-.
r (()
State
(16)
Estimator
Controller
(17)
Monitoring
System
Fig.l: Sensorless monitoring principle In order to take into account the noise of the measured signals and the model parameter variations (rotor resistance) a nonlinear observer technique such as Kalman filtering must be used. Nonlinear techniques are generally avoided due to the high computational load required (Bostan et aI., 1999).
based on the fact that the observer must be dynamically faster than the induction motor system. Thus not only assure the stability of the observer but also to get an optimally filtered estimation in respect with the measurement and the eventual input noise. Pole placement technique is generally used.
IV. SIMULATION RESULTS
The estimated state variable vector X(t) is not only used to monitor the operating conditions of the induction motor drive system but also to generate the sensorless control law such that:
= -IT(/)
Induction Motor
(State space model)
These relations show that the observer is a system in which the inputs are the input and the output of the system to be observed and the output is the estimated state variable vector of the observed system. The dynamic the observer is determined by the eighenvalues of the observer evolution matrix [A - LC] . A practical choice of the gain matrix L is
V(/)
(I)
In order to illustrate the performance of the proposed sensorless monitoring scheme, simulation experiments have been carried out on an induction motor with the following characteristics:
(18)
p" = 11 KW,
f.
= 50Hz, V = 220V, P = 2,
rs =rr =050., Is =Ir =0.069H, m=0.067H, cq = 1440 rpm J=0.2N;/rd, f=INs/rd
where K is the feedback gain matrix. Pole placement technique is also used to determine its numerical value. Different induction motor drive monitoring procedures are proposed in the literature such as spectrum analysis. residual generation technique, space vector approach and acoustic vibrations monitoring flachir et aI., 2003; Martis et aI., 2003). In recent years a great attention is given to electromagnetic state variables analysis such as current, flux and torque which do not require a specific instrumentation in comparison with the classical approaches which use the mechanical characteristics as vibrations and speed monitoring In this paper the proposed sensorless monitoring procedure is based on the use of a simple Lissajou curves plotting of the electromagnetic estimated state variables. The aim of this monitoring procedure is to control the main functions of the considered induction motor drive system via the stator current, the estimated rotor flux and the estimated electromagnetic torque. The procedure compares the estimated state variables
The nominal state space model has the following matrices:
- 246999 0
A=
o
-246999 -372331071789206
0.485
o
1789206 37233107
0
- 7.246
0.485
15Q796 - 7246
254.258 B=
0
0
254.258
0
0
0
0
c=[~
92
-15Q796
0
0
1
0
~]
Figure 2 shows the Lissajou curves plotting of the measured and the estimated stator current in the case open-loop as well as in the case of the closed-loop with respect to nominal operation conditions.
JOO ,..-
---,
200 100
o
er .111
-100 JOO r-~---~,---~-, ,
200
1
::~~:;::
100
D __ ~ __
er
-2D0
,
,
.14
~
_~
.,
I
-200
__
.,
I
~
I
I
I
I
,
lid
,
0
2D0
--I-----'-----T--
-"DO L-.......
-2D0
"'--_ _.......---1 0 200
Fig. 5: Measured and estimated stator current When a fault of 10 % occurs on the rotor resistance. (0) for open loop and (--) for closed-loop
_ .J. __
__ 1_ _ _
-100
"'OD '----'400
"'--_ _.......- - J
0.11
lid
Fig.2: Measured and estimated stator current (0) for open loop and (--) for closed-loop
=" - - - - - -
0.2
-
~.2
-
,,
_1-
-,,
~.~
o
~.5
0.5
11 III
Fig.6: Estimated rotor flux when a fault of 10% occurs on the rotor resistance. (0) for open loop and (--) for closed-loop
200 100 er
,
_1I
I
~
JDO
.111
-
-1-
- -,,
0
~
Figures 3 and 4 show the estimated stator current and the rotor flux state variable in the open-loop and closed loop cases respectively in the case where a change of five percent occurs in the rotor resistance.
,,
,
O.~
0
6 X 10 2,..-
-100
-,
-2D0 1.5
-"DO
-200
0
/
2D0
lid
."
~
Fig.3: Measured and estimated stator current when a fault of 5% occurs on rotor resistance (0) for open-loop and (--) for closed-loop
.1\
1
fI
;\
0.5 0
50
0
100
Hz
0.6
Fig.7: Es timated stator current: (-) for open loop and (--) for closed-loop
O.~
0.2 ~
X
0
6 10
2 r-----~------,
~
~.2
l
1.5
~.~
..,
o
1
1 \
~
11 III
i\
0.5
Fig.4: Estimated rotor flux when a fault of 5% occurs on the rotor resistance. (-) for open-loop and (--) for closed-loop
0
/ 0
~
\
.1<1 50
100
Hz
Fig.8: Estimated stator current when a fault of 5% occurs on the rotor resistance
Figures 5 and 6 show the curves of the same experiments with respect to faulty operation conditions when a fault ten percent occurs in the rotor resistance.
93
REFERENCES 80
60
r,
~o
'0
l!.
Bachir, S., Champenois, G., Tnani, S. (2003). Stator faults diagnosis in induction machines under fixed speed. Electromotion intenational journal on Advanced electromechanical motion systems. Vol. 10 (4), pp.679-684. Balan, H, Hedesiu, H., Tirnovan, R., Karaissus, P. (1999). Electrical machines and equipment diagnosis using virtual instrumentation. Electromotion Proc. 3rd Inter. Symp. On Advanced electromechanical motion systems, Patras, Greece, pp.795-798. Bensaker, B. (1999). Parameter estimation for an electromechanical system via a continuous-time model. Electromotion Proc. r d Int. Symp. on Advanced electromechanical motion systems, Patras, Greece, pp.759-762. Bensaker, B. (2000). Continuous-time system monitoring via continuous-time model parameter estimation. Preprints (CD) of the
') I I
2D
/\
0 D
A.. 5D
IDD
Hz
Fig.9: Estimated flux density for a normal operations
8D
6D
i
[1
~D
,
2D
1
\
2 nd IFAC/IFIPIJEEE Conference on Management
1 •
I
.J
D D
\
and Control of Production and Logistics, Grenoble, France. Bensaker, 8., Kherfane, H., Metatla, A. and R. Wamkeue. (2003). State space modelling of induction motor for sensorless control and monitoring purposes. Electromotion international journal on Advanced electromechanical motion systems, VoJ. 10 (4), pp.483-488. Bostan, V., Cuibus, M., IIas, C., Magureanu, R. (1999). Comparison between Luenberger and Kalman observers for sensorless induction motor control. Proceedings of 3rd Int. Symp. on Advanced electromechanical motion systems, Patras, Greece, pp.335-340. Holtz, J. (2002). Sensorless control of induction motor drives. Proceedings of the IEEE, Vol. 90, n08, pp. 13591394. loannides, M. G. and O.A. Syggeridou, O. A. (1999). Assessment of failures and diagnostic methods of AC electric motors. Proc. 3rd Int. Symp. on Advanced electromechanical motion systems, Patras, Greece, pp.799-802. Mihet-Popa, L., Bak-Jensen, 8., Ritchie, E. and Boldea, I. (2003). Current signature analysis to diagnose incipient faults in wing generator systems. Electromotion intenational journal on Advanced electromechanical motion systems. Vol. 10 (4), pp.467-452. Martis, C., Henao, H., Capolino, G. A. (2003). Inter turn short-circuits in induction machine and their influence on the electromagnetic torque. Electromolion intenational journal on Advanced electromechanical motion systems. Vol. 10 (4), pp.702-707.
jl, 50
tDD
Hz
Fig. 10 : Estimated flux density when a fault of 5% occurs on the rotor resistance. By analysis of the different figures one can see a significant difference which ensure that our system monitoring works well. Figures 7 to 10 show the case of spectral analysis for the open and closed-loop operation conditions. On can see also that our system monitoring works well too.
V.CONCLUSION The paper has investigated the state space modelling and the observer-based state variable estimation techniques for sensorless monitoring of induction motor driven systems. A simple Lissajou curves plotting as well as a simple spectral analysis method have been performed in the case of open-loop and closed-loop induction motor drive system. The obtained simulation results have shown that our system monitoring works well. Finally we conclude that the state space technique and observer-based state estimation techniques are efficient methods for induction motor drive systems sensorless monitoring.
94
Copyright © IFAC Automation in Mining, Mineral and Metal Processing, Nanc)'. France. 2004
IFAC PUBLICATIONS www.elsevier.coml1ocate/ifac
SENSORLESS MONITORING OF INDUCTION MOTOR DRIVE SYSTEMS
B. Bensaker, H. Kherfane, A. Maouche, A. Metatla, aod R.Wamkeue
Departement d'Electronique,. Universite de Annaba, B.P 12, 23000 Annaba. Algeria. Tell/ox: + 213 38867928 e-mail: bensakerbachirrCilvahoo.fr Departement des Sciences Appliquees. UQAT. Rouyn-Noranda, Quebec 445 B.de /'Universite, Canada. J9X 5£4. e-mail: rene.walllkeueli.il.uqat. uquebec.crJ
Abstract: The paper deals with sensorless monitoring of induction motor drive systems. It is well known that induction motor drives are very important parts of any industrial application involving electrical equipment. They must be carefully surveyed to ensure reliable and profitable operations. However induction motor drive systems are also known as complex nonlinear time-varying systems. Thus entails great difficulties for monitoring purposes. Based on the fact that the system model can be significantly simplified if one applies the d-q Park transformation and an appropriate field orientation technique, an adequate model structure is obtained. An observer-based state variable estimation technique is used for sensorless monitoring of a general induction motor drive system. The sensorless monitoring procedure uses a simple curves plotting of the estimated state variables. The approach permits to detect up to five percent changes in the estimated state variables information. Copyright © 2004 IFAC Keywords: State space modelling, observer, state estimation, monitoring, induction motor drives.
I. INTRODUCTION The induction motor is one of the most widely used machine in industrial applications such as metal or hot/cold rolling processes. This is due to its high reliability, relatively low cost, and modest maintenance requirements. With the development of power electronic technology, low cost digital signal processing (OSP) micro-controllers and estimation techniques the induction motor constitutes an attractive actuator for high performance drives (Holtz, 2002; Bensaker et al., 2003). However the induction motor is known as a complex nonlinear system in which time-varying parameters entail an additional
89
difficulty for developing control and monitoring strategies. Based on the fact that the model can be significantly simplified if one applies the d-q Park transformation and a field orientation technique also called vector control, different structures of the model exist in the literature (Bensaker et al., 2003). The choice of a model structure depends on the selected state variables and the problem at hand. Industrial applications involving induction motors such as metal processing and hot/cold rolling are subject to control and monitoring problems. In order to ensure reliable and profitable operations they must be carefully surveyed. Sensorless monitoring techniques focus on reducing the number of sensors to be used.
The aim of a system monitoring is to control, through certain state variables that indicate the system health, its main functions and to compare the measured values of these state variables with their preset values (Balan et al., 1999; Ioannides and Syggeridou, 1999). The objective is to decrease the rate of fault occurrence and to increase the MTBF (Mean Time Between Failure). State space modelling of induction motors and observer-based state variable estimation techniques are very helpful in this case of sensorless system monitoring.
M, and
The matrices 0a' 0a,' R s ' R" L s ' L"
I 0 are defined as follows:
°a =[ 0
W ol ]
- Cl?, 0
q,=
,
[0
(Ol-Ol)]
-(Ol-Ol) 0
,
s [r, 0] [Is 0] r ' R, = 0 r, ' Ls = 0 Is ' [r0 0] s L, = [I, 0] , M =[m 0] and 1 = [1 O . Rs =
2
2
o I,
11. STATE SPACE MODELLING In order to establish a state space mathematical model of the induction motor, the following classical hypotheses are used (Holtz, 2002). -The higher order harmonics of the magnetic field of stator and rotor are neglected, -The ferromagnetic losses and effect of saturation are also neglected (for control design), -A perfect electrical and magnetic circuit symmetry (uniform air gap and sinusoidal flux density distribution), is assumed. In this paper the dynamics of the induction motor are divided into two subsystems which are subsystem of the electromagnetic dynamics and subsystem of the electromechanical dynamics. The electromagnetic subsystem is, in its turn, divided into stator dynamics and rotor dynamics. The electromagnetic subsystem is represented, in an arbitrary rotating d-q Park reference frame, by the following nonlinear state space model:
0
0 m
]
O2 O2
Usual notation for parameters and variables is used. The indexes sand r refer to the stator and the rotor com ponents respectively. Based on the fact that the model can be significantly simplified if one, in addition with the d-q Park transformation, applies a field orientation technique for an appropriate fixed reference frame, different structures of the model can be obtained for sensorless monitoring purposes taking into account the availability of measurements (the number of sensors to be used) (Bensaker et al., 2003). If the stator fixed reference frame is chosen one has to substitute in equation (I)
0
a
O2 and 0
to
a'
to
0, .
In stator flux orientation technique the direct component of the stator flux must be aligned with the real axis, thus leads to ({Js
= CPsd
and ({Jsq
=0 .
If the rotor fixed reference frame is used one has to substitute, in relation (I),
0
a
to
0, and 0
a'
to
O2 , In the case of rotor field orientation technique the direct component of the rotor flux must be aligned the real axis, in consequence ({J, = ({J'd and ({J,q =
The electromechanical subsystem is represented by the following nonlinear differential equation called the motion equation.
The electromechanical torque and the flux linkage relations, which ensure the link between the two subsystems, are respectively defined by:
CPs) ( ({J,
s s [L M](i ) = M L, i,
0.
The rotor or the stator field orientation technique can be seen as a linearization technique to be used to reduce significantly the nonlinear induction motor model (Holtz, 2002). The choice of a model structure is generally guided by the fact that most drive systems have a stator current control loop incorporated in their control structure. This leads to select the stator current as the first state variable. The second state variable is then either the stator flux linkage vector, the rotor flux linkage vector or the rotor current depending on the problem at hand. In sensorless techniques the second state variable is estimated by using a closed-loop observer-based technique. In each case, the not selected state variables are indirectly estimated on the basis of measured and directly estimated state variables. Observer-based state estimation techniques are very helpful for closed-loop sensorless monitoring applications fJensaker et aI.,
(4)
90
The rotor current is indirectly estimated via the stator current measurements and the estimated rotor flux as:
2003). In our case we are interested in stator current and rotor flux state variables model. In this case the nonlinear state space model is:
I
L-r_ (5)
MLr
1 I:({Jr ](.
)
(11)
The rotor current estimate is generally employed in procedures concerned with the condition-based system maintenance and monitoring. In all application cases the stator current measurements are very helpful. Reduced order current observer-based techniques can be also used for monitoring the induction motor drive systems.
(6)
(7)
(8)
III.0BSERVER-BASED STATE ESTIMATION The evolution matrix parameters are defined by:
In order to decrease the price of induction motor drive systems and to increase their reliability and robustness ongoing research has concentrated on the minimization of the number of sensors to be used without deteriorating the dynamic performance of the considered drive system. The development of the estimation techniques and the availability of new low cost DSP-based (Digital Signal Processing) microcontrollers, have reinforced the industrial applications of this idea for sensorless control and monitoring purposes (Bensaker et aI., 2003; MihetPopa et al. 2003). The above model can be written in the following standard linear state space model:
All =Q a -(L s -ML~IMrl(Rs +ML~IA21)' ) A12 =(L s - ML-1M)-1 (Q a Mrr 1 - ML-r 1A22' r A 21 An Bz
= (L s
-
= RrL~l M,
= Q ar ML~I
RrL~1 ,
Mr
l
(9)
The objective of this model is to estimated the rotor flux linkage state variable based on the stator current measurements using an observer-based technique. Once the rotor flux is estimated one can easily estimate the other electromagnetic state variables or mainly the electromagnetic torque. The stator reference frame can be used for this purpose in addition with the rotor field orientation technique. The electromagnetic torque equation becomes:
dX(t) = AX(t) + BU(t) dt Y(t) = CX(t)
(12) (13)
where X(t) is the selected state variable vector and (10)
Y (t) is the measured output vector. A, Band C are the evolution, the control and the observation matrices respectively. One of the most used technique for estimating the state variables of a system described in terms of state space model is the observer technique. Observer-based state variables estimation uses an adaptive mechanism which involves as input, the error between the measured and estimated state variable values. The general form of the observer is :
On can see, relation (10), that the electromagnetic torque is proportional to the stator current component since the rotor flux takes its maximum value. The model order reduces in consequence and the induction motor system can be defined only by the rotor model and the mechanical subsystem. The decoupling between the stator and the rotor components is then obtained and the induction motor model becomes linear if the rotor angular velocity is assumed to be slowly varying or constant. This assumption is generally justified by the fact that electrical dynamics are very fast than the magnetic dynamics which are in thier turn very fast than the mechanical dynamics. The resulting dynamic structure of the induction motor system is then significantly simplified. The induction motor, in these conditions acts as a dc motor.
91
~t) =AX(t)+BU(t)+L[Y(t)-J'{t)]
(14)
tU) = CX(t)
(15)
behaviour with their corresponding nominal behaviour (Bensaker, 1999; Bensaker, 2000). Figure I shows the sensorless monitoring principle.
where X(t) is the estimated state variable vector of the unknown state variable vector X(t) and Y (I) is the estimated output of the measured output signal Y (I). L is the observer gain matrix. Relations (14) and (15) can be written as
di(t) = [A-LqX(/)+BV(/)+LY(/) dl Y(t) = CX(t)
(J
t-r---.,....-.
r (()
State
(16)
Estimator
Controller
(17)
Monitoring
System
Fig.l: Sensorless monitoring principle In order to take into account the noise of the measured signals and the model parameter variations (rotor resistance) a nonlinear observer technique such as Kalman filtering must be used. Nonlinear techniques are generally avoided due to the high computational load required (Bostan et aI., 1999).
based on the fact that the observer must be dynamically faster than the induction motor system. Thus not only assure the stability of the observer but also to get an optimally filtered estimation in respect with the measurement and the eventual input noise. Pole placement technique is generally used.
IV. SIMULATION RESULTS
The estimated state variable vector X(t) is not only used to monitor the operating conditions of the induction motor drive system but also to generate the sensorless control law such that:
= -IT(/)
Induction Motor
(State space model)
These relations show that the observer is a system in which the inputs are the input and the output of the system to be observed and the output is the estimated state variable vector of the observed system. The dynamic the observer is determined by the eighenvalues of the observer evolution matrix [A - LC] . A practical choice of the gain matrix L is
V(/)
(I)
In order to illustrate the performance of the proposed sensorless monitoring scheme, simulation experiments have been carried out on an induction motor with the following characteristics:
(18)
p" = 11 KW,
f.
= 50Hz, V = 220V, P = 2,
rs =rr =050., Is =Ir =0.069H, m=0.067H, cq = 1440 rpm J=0.2N;/rd, f=INs/rd
where K is the feedback gain matrix. Pole placement technique is also used to determine its numerical value. Different induction motor drive monitoring procedures are proposed in the literature such as spectrum analysis. residual generation technique, space vector approach and acoustic vibrations monitoring flachir et aI., 2003; Martis et aI., 2003). In recent years a great attention is given to electromagnetic state variables analysis such as current, flux and torque which do not require a specific instrumentation in comparison with the classical approaches which use the mechanical characteristics as vibrations and speed monitoring In this paper the proposed sensorless monitoring procedure is based on the use of a simple Lissajou curves plotting of the electromagnetic estimated state variables. The aim of this monitoring procedure is to control the main functions of the considered induction motor drive system via the stator current, the estimated rotor flux and the estimated electromagnetic torque. The procedure compares the estimated state variables
The nominal state space model has the following matrices:
- 246999 0
A=
o
-246999 -372331071789206
0.485
o
1789206 37233107
0
- 7.246
0.485
15Q796 - 7246
254.258 B=
0
0
254.258
0
0
0
0
c=[~
92
-15Q796
0
0
1
0
~]
Figure 2 shows the Lissajou curves plotting of the measured and the estimated stator current in the case open-loop as well as in the case of the closed-loop with respect to nominal operation conditions.
JOO ,..-
---,
200 100
o
er .111
-100 JOO r-~---~,---~-, ,
200
1
::~~:;::
100
D __ ~ __
er
-2D0
,
,
.14
~
_~
.,
I
-200
__
.,
I
~
I
I
I
I
,
lid
,
0
2D0
--I-----'-----T--
-"DO L-.......
-2D0
"'--_ _.......---1 0 200
Fig. 5: Measured and estimated stator current When a fault of 10 % occurs on the rotor resistance. (0) for open loop and (--) for closed-loop
_ .J. __
__ 1_ _ _
-100
"'OD '----'400
"'--_ _.......- - J
0.11
lid
Fig.2: Measured and estimated stator current (0) for open loop and (--) for closed-loop
=" - - - - - -
0.2
-
~.2
-
,,
_1-
-,,
~.~
o
~.5
0.5
11 III
Fig.6: Estimated rotor flux when a fault of 10% occurs on the rotor resistance. (0) for open loop and (--) for closed-loop
200 100 er
,
_1I
I
~
JDO
.111
-
-1-
- -,,
0
~
Figures 3 and 4 show the estimated stator current and the rotor flux state variable in the open-loop and closed loop cases respectively in the case where a change of five percent occurs in the rotor resistance.
,,
,
O.~
0
6 X 10 2,..-
-100
-,
-2D0 1.5
-"DO
-200
0
/
2D0
lid
."
~
Fig.3: Measured and estimated stator current when a fault of 5% occurs on rotor resistance (0) for open-loop and (--) for closed-loop
.1\
1
fI
;\
0.5 0
50
0
100
Hz
0.6
Fig.7: Es timated stator current: (-) for open loop and (--) for closed-loop
O.~
0.2 ~
X
0
6 10
2 r-----~------,
~
~.2
l
1.5
~.~
..,
o
1
1 \
~
11 III
i\
0.5
Fig.4: Estimated rotor flux when a fault of 5% occurs on the rotor resistance. (-) for open-loop and (--) for closed-loop
0
/ 0
~
\
.1<1 50
100
Hz
Fig.8: Estimated stator current when a fault of 5% occurs on the rotor resistance
Figures 5 and 6 show the curves of the same experiments with respect to faulty operation conditions when a fault ten percent occurs in the rotor resistance.
93
REFERENCES 80
60
r,
~o
'0
l!.
Bachir, S., Champenois, G., Tnani, S. (2003). Stator faults diagnosis in induction machines under fixed speed. Electromotion intenational journal on Advanced electromechanical motion systems. Vol. 10 (4), pp.679-684. Balan, H, Hedesiu, H., Tirnovan, R., Karaissus, P. (1999). Electrical machines and equipment diagnosis using virtual instrumentation. Electromotion Proc. 3rd Inter. Symp. On Advanced electromechanical motion systems, Patras, Greece, pp.795-798. Bensaker, B. (1999). Parameter estimation for an electromechanical system via a continuous-time model. Electromotion Proc. r d Int. Symp. on Advanced electromechanical motion systems, Patras, Greece, pp.759-762. Bensaker, B. (2000). Continuous-time system monitoring via continuous-time model parameter estimation. Preprints (CD) of the
') I I
2D
/\
0 D
A.. 5D
IDD
Hz
Fig.9: Estimated flux density for a normal operations
8D
6D
i
[1
~D
,
2D
1
\
2 nd IFAC/IFIPIJEEE Conference on Management
1 •
I
.J
D D
\
and Control of Production and Logistics, Grenoble, France. Bensaker, 8., Kherfane, H., Metatla, A. and R. Wamkeue. (2003). State space modelling of induction motor for sensorless control and monitoring purposes. Electromotion international journal on Advanced electromechanical motion systems, VoJ. 10 (4), pp.483-488. Bostan, V., Cuibus, M., IIas, C., Magureanu, R. (1999). Comparison between Luenberger and Kalman observers for sensorless induction motor control. Proceedings of 3rd Int. Symp. on Advanced electromechanical motion systems, Patras, Greece, pp.335-340. Holtz, J. (2002). Sensorless control of induction motor drives. Proceedings of the IEEE, Vol. 90, n08, pp. 13591394. loannides, M. G. and O.A. Syggeridou, O. A. (1999). Assessment of failures and diagnostic methods of AC electric motors. Proc. 3rd Int. Symp. on Advanced electromechanical motion systems, Patras, Greece, pp.799-802. Mihet-Popa, L., Bak-Jensen, 8., Ritchie, E. and Boldea, I. (2003). Current signature analysis to diagnose incipient faults in wing generator systems. Electromotion intenational journal on Advanced electromechanical motion systems. Vol. 10 (4), pp.467-452. Martis, C., Henao, H., Capolino, G. A. (2003). Inter turn short-circuits in induction machine and their influence on the electromagnetic torque. Electromolion intenational journal on Advanced electromechanical motion systems. Vol. 10 (4), pp.702-707.
jl, 50
tDD
Hz
Fig. 10 : Estimated flux density when a fault of 5% occurs on the rotor resistance. By analysis of the different figures one can see a significant difference which ensure that our system monitoring works well. Figures 7 to 10 show the case of spectral analysis for the open and closed-loop operation conditions. On can see also that our system monitoring works well too.
V.CONCLUSION The paper has investigated the state space modelling and the observer-based state variable estimation techniques for sensorless monitoring of induction motor driven systems. A simple Lissajou curves plotting as well as a simple spectral analysis method have been performed in the case of open-loop and closed-loop induction motor drive system. The obtained simulation results have shown that our system monitoring works well. Finally we conclude that the state space technique and observer-based state estimation techniques are efficient methods for induction motor drive systems sensorless monitoring.
94