1st Power Electronic & Drive Systems & Technologies Conference
Modeling and Vector Control of Unbalanced Induction Motors (Faulty Three Phase or Single Phase Induction Motors) M. Jannati, and E. Fallah
Abstract-- All types of electrical machines can be modeled by an equivalent two phase machine. For example a balanced three phase induction motor can be modeled as an equivalent two phase induction motor (The dq Model). In the same way an unbalanced induction motor can be modeled as an unbalanced two phase induction motor. This paper shows this concept by modeling a faulty induction motor with one of its three feeding phases, open. Moreover the paper shows that the speed control of this faulty motor can be performed by a few modifications in the conventional vector control. This new vector control is suitable for unbalanced induction motors such as the single phase induction motor with unequal main and auxiliary windings. Computer simulation shows the good performance of the proposed method.
conventional vector control, its ability in controlling unbalanced or faulty motors is not good. This paper concerns with the problem of modeling and control of unbalanced induction motors. In section II, a method for modeling a faulty three phase induction motor (when it loses one of its feeding phases) is presented. By using this model, a new method of vector control is presented in section III. The Performance of the presented method is checked by computer simulation in section IV. II. MODELING OF A THREE-PHASE INDUCTION MOTOR WITH ONE OPENED PHASE Suppose that a phase cut out fault is occurred in the phase “c” of a three-phase drive system, as shown in fig 1.
Index Terms-- Unbalanced, Fault, Induction motor, Modeling, Vector control.
T
I. INTRODUCTION
HE vector control method is known as one of the best methods in controlling the torque and the speed of the induction motors. The vector control separates motor current into field and torque producing components. The torque is proportional to the product of these two perpendicular components and they can be treated separately. Among the various types of the vector control method, the approach called Rotor Field Oriented (R.F.O) is more convenient [1]. A faulty three phase induction motor, when one of its feeding phases opened, can be considered as an unbalanced induction motor. On the other hand, all of the single phase induction motors have two unequal main and auxiliary windings. So they can be considered as unbalanced two phase induction motors. Inventing a method for controlling faulty or unbalanced induction motors has many advantages. In some application (such as military or astronomy) the controlling system must be fault tolerant [2]-[5]. Furthermore in some low power applications using single phase induction motors is more economical [6]-[8]. In spite of good performance of
Fig. 1. Three phase induction motor drive with one opened phase.
Assuming sinusoidal waveform for the spatial distribution of the windings, Stator and rotor winding flux, axes can be shown as fig 2. If the angle between ds-axis and as-axis is “θo”, d and q components of flux can be written as follows:
M. Jannati is the M.S. student of the electrical engineering at the University of Guilan, Rasht, Iran (e-mail:
[email protected]). E. Fallah is the Assistant Professor of the electrical engineering at the University of Guilan, Rasht, Iran (e-mail:
[email protected]).
978-1-4244-5971-1/10/$26.00 ©2010 IEEE
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⎡ϕ as ⎤ ⎥ ⎣ϕ bs ⎦ ⎡ϕ ⎤ ϕ qs = [sin(θ o ) sin(θ o + 2π / 3)]⎢ as ⎥ ⎣ϕ bs ⎦
ϕ ds = [cos(θ o ) cos(θ o + 2π / 3)]⎢
(1)
(2)
III. EQUATIONS OF ROTOR FIELD ORIENTED VECTOR CONTROL IN FAULTY MODE For obtaining vector control equations, it is necessary to indicate machine equations in the Rotor Field Oriented reference frame. For this purpose rotational transformation must be applied to the stationary reference frame. Since the rotor is intact the balanced rotational transformation is applied to the rotor currents isdr and isqr , as follows [1]: (a)
(b)
⎡idre ⎤ ⎡ cosθ e ⎢ e ⎥=⎢ ⎣iqr ⎦ ⎣− sinθ e
Fig. 2. a) Stator winding flux axes b) Rotor winding flux axes
The transformation vectors “d” and “q” can be defined as follows:
d = [cos(θ o ) cos(θ o + 2π / 3)] q = [sin(θ o ) sin(θ o + 2π / 3)]
(3)
Where, the superscript “e” indicates the variables in the rotor field oriented reference frame. For the unbalanced situation investigated in this paper, the following unbalanced transformation is proposed:
(4)
⎡ cosθ e e ⎡ids ⎤ ⎢ ⎢ e⎥=⎢ ⎣iqs ⎦ ⎢− sin θ e ⎢ ⎣
The transformation vectors must be perpendicular, so we have: T T (5) . = . =0 ⇒ =π /6
d q
θ0
q d
By using above transformation vectors, the following normalized transformation matrix is obtained: (6) ⎡1 / 2 − 1 / 2 ⎤
[T S ] =
2⎢ ⎣1 / 2
1 / 2 ⎥⎦
Because the rotor is still in balanced condition, the decomposition matrix for rotor variables remains unchanged (the same transformation matrix of balanced condition [9]). Voltage equations of induction machine in “abc” frame are available in [9]. By applying transformation (6) to the stator variables and the balanced transformation to the rotor variables we have the following equation:
0 0 ⎤⎡idss ⎤ Md ⎡υdss ⎤ ⎡rs + Lds p ⎢ s⎥ ⎢ 0 0 rs + Lqs p Mq ⎥⎢⎢iqss ⎥⎥ ⎥ ⎢υqs ⎥ = ⎢ ωr Mq rr + Lr p ωr Lr ⎥⎢idrs ⎥ ⎢ 0 ⎥ ⎢ Md p ⎥⎢ ⎥ ⎢ ⎥ ⎢ − ωr Lr rr + Lr p⎦⎣⎢iqrs ⎦⎥ Mq p ⎣ 0 ⎦ ⎣ − ωr Md
1 2
3 2
Lds + Lqs Lds + Lqs ⎤ ⎡ ) p − ωe ( ) ⎥ ⎡i e ⎤ ⎡vdse ⎤ ⎢rs + ( ds 2 2 ⎢ ⎥ ⎢ e ⎥=⎢ ⎥ L + Lqs Lds + Lqs iqse ⎦⎥ ⎢ ⎥ ⎣ ⎣⎢vqs ⎦⎥ ⎢ ωe ( ds ) rs + ( )p ⎣⎢ ⎦⎥ 2 2 2 2 2 2 ⎡ Md + Mq Md + Mq ⎤ ) p − ωe ( )⎥ e ⎢( 2M q 2M q ⎢ ⎥⎡idr ⎤ + 2 2 2 2 ⎢ Md + Mq ⎥⎢iqre ⎥ Md + Mq ⎢ωe ( ) ( ) p ⎥⎣ ⎦ 2M q 2M q ⎢⎣ ⎥⎦ Lds − Lqs ⎤ ⎡ Lds − Lqs ) p ωe ( ) ⎥ ⎡i − e ⎤ ⎢( ds 2 2 +⎢ ⎥⎢ −e ⎥ + − − L L L L i qs ds qs ⎢ω ( ds ) −( ) p ⎥ ⎣⎢ qs ⎦⎥ e 2 2 ⎣⎢ ⎦⎥
(7)
3 2
3 d Lms , p = 2 dt
Where, Lms is the magnetization inductance defined in [9]. Electromagnetic torque is as follows: (8) Pole
τe =
2
2 2 ⎡ Md 2 − Mq2 Md − Mq ⎤ ) p ωe ( ) ⎥ −e ⎢( 2M q 2M q ⎢ ⎥⎡idr ⎤ + ⎢ ⎥ 2 2 2 2 ⎢ Md − Mq M d − M q ⎥⎣iqr−e ⎦ ⎢ωe ( ) −( ) p⎥ 2M q 2M q ⎣⎢ ⎦⎥
( M qiqss idrs − M d idss iqrs )
Equation (7) and (8) completely represent “dq” model of three-phase induction machine with one opened phase. In fact, (7) and (8) introduce the equations of an unbalanced two phase induction motor. By writing these equations in the form of the state equations, the computer simulation can be performed.
(10)
Stator voltage equations:
Lds = Lls + Lms , Lqs = Lls + Lms , Lr = Lls + Lms
M d = Lms , M q =
⎤ sin θ e ⎥ s ⎡i ⎤ Md ⎥ ⎢ dss ⎥ Mq i cosθ e ⎥ ⎣ qs ⎦ ⎥ Md ⎦ Mq
Where, θe is the angle between the stationary reference frame and the Rotor Field Oriented reference frame. The same transformation is applied to the stator voltage variables υsds, υsqs. Machine equations in the Rotor Field Oriented reference frame are obtained as follows:
Where, υsds, υsqs, isds, isqs, isdr and isqr are the “d” and the “q” components of the stator and the rotor voltages and currents, all of them in the stationary reference frame and ωr is the rotor angular velocity. Moreover:
3 2
(9)
sinθ e ⎤ ⎡idrs ⎤ ⎢ ⎥ cosθ e ⎥⎦ ⎣iqrs ⎦
(11) Rotor voltage equations:
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Md p ⎡0 ⎤ ⎡ ⎢0⎥ = ⎢(ω − ω ) M ⎣ ⎦ ⎣ e r d
(a)
− (ω e − ω r ) M d ⎤ ⎡idse ⎤ ⎥ ⎢i e ⎥ Md p ⎦ ⎣ qs ⎦
⎡ r +L p +⎢ r r ⎣(ωe − ωr )Lr
− (ωe − ωr )Lr ⎤ ⎡idre ⎤ ⎢ ⎥ rr + Lr p ⎥⎦ ⎣iqre ⎦ (12)
Electromagnetic torque:
τe =
Pole M d (iqse idre − idse iqre ) 2
(13) (b)
Where, ωe is the angular velocity of the Rotor Field Oriented reference frame. The superscript “-e” denotes the variables in the backward rotating reference frame (with the angular velocity of “- ωe”). From (12) and (13), it can be seen that the rotor voltage and the torque equations are similar to the balanced motor equations. However the stator voltage equation (11) has extra terms due to the backward variables. Since the backward terms are proportional to the difference of the inductances, it is possible to neglect them. So by the following modifications in the R.F.O vector control, it is possible to control a faulty motor. The proposed modifications are listed in Table I.
Fig. 3. Simulation results of the conventional R.F.O vector controller; Speed reference= 500rpm, Load torque= 10N.m (a) Speed, (b) Steady state torque. Results show considerable oscillations in the electromagnetic torque. Simulation results for modified R.F.O vector controller is shown in fig.4.
TABLE I PROPOSED MODIFICATION IN THE CONVENTIONAL R.F.O VECTOR CONTROL METHOD Vector control of balanced motor
Vector control of faulty or unbalanced motor
Ls
(Lds+Lqs)/2
Transformation of stator variables from “abc” to “dq” frame, as indicated in [1] Balanced rotational transformation as indicated in (9)
Transformation of stator variables from “abc” to “dq” frame, according to (6) Unbalanced rotational transformation as indicated in (10)
(a)
IV. SIMULATION RESULTS Computer simulation results are presented in this section. A three-phase induction motor with one opened phase is simulated. The motor is fed from a SPWM voltage source inverter. Two controllers are used for the speed control of the motor. The first is the conventional R.F.O vector controller and the second is the same controller with the proposed modifications of Table I. Simulation results of the conventional controller are shown in fig.3.
(b)
Fig. 4. Simulation results of the modified R.F.O vector controller; Speed reference= 500rpm, Load torque= 10N.m (a) Speed, (b) Steady state torque. It is obvious that the proposed modifications decrease the torque oscillation considerably. Existence of the torque oscillation is due to the non-sinusoidal voltage source and also neglecting the backward terms of (11). V. CONCLUSION A method for modeling the opened phase fault of an induction motor is presented in this paper. By this method a faulty motor can be modeled as an unbalanced two phase
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motor. A three phase induction motor with one opened phase is modeled by this method. Equations of the equivalent two phase motor transformed to the Rotor Field Oriented reference frame. A new transformation matrix is invented such that the transformed equations of the faulty motor become the same as the equations of the balanced motor. According to theses transformed equations a new vector control method for the unbalanced motors is presented. The advantage of the presented method is its simplicity so that it can be obtained by a few modifications in the conventional vector control. Simulation results show that the performance of the proposed vector control is good. Especially the amplitude of the torque oscillation is less, compare to the conventional vector control. VI. REFERENCES [1] [2]
[3]
[4] [5]
[6] [7]
[8] [9]
Peter Vas, Vector Control of AC Machines, Oxford science publication 1990. Y.Zhao, T.A.Lipo, “Modeling and control of a multiphase induction machine with structural unbalance, part I-machine modeling and multidimensional current regulation,” IEEE Trans. on Energy Conversion, no.3, pp.570-577, Sept.1996. Y.Zhao, T.A.Lipo, “Modeling and control of a multiphase induction machine with structural unbalance, part II-field-oriented control and experimental verification,” IEEE Trans. on Energy Conversion, no.3, pp.578-584, Sept.1996. F.Tahami, A.Shojaei, “A novel fault tolerant reconfiguration concept for vector control of induction motors,” in Proc. 2006 Power Electronics and Motion Control Conf., pp. 1199-1204. J.Mili Monfared, K.Abbaszadeh, E.Fallah, “Modeling and simulation of dual three phase induction machine in fault condition (two phase cut off) and propose a new vector control approach for torque oscillation reducing,” in Proc. 2000 ICEM conf. M.B.R.Correa, C.B.Jacobina, A.M.N.Lima, E.R.C.daSilva, “Rotor flux oriented control of a single phase induction motor drive,” IEEE Trans. on Industrial Electronics, vol. 47, no. 4, pp. 832-841, August 2000. M.B.R.Correa, C.B.Jacobina, E.R.C.daSilva, A.M.N.Lima, “Vector control strategies for single phase induction motor drive systems,” IEEE Trans. on Industrial Electronics, vol. 51, no. 5, pp. 1073-1080, Oct. 2004. S.Vaez-Zadeh, S.R.Harooni, “Decoupling vector of a single phase induction motor drives,” in Proc. 2005 Power Electronics Specialists Conference, pp. 733-738. Paul C. Krause, Analysis of Electric Machinery, McGraw-Hill, 1986.
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