Indirect Field-Oriented Control of an Induction Motor by Using Closed

Indirect Field-Oriented Control of an Induction Motor by Using Closed-Loop Identification A. J. Netto

P.R. Barros, C.B. Jacobina, A.M.N. Lima and E.R.C. da Silva

Fundac¸a˜ o Regional de Gurupi Department of Computer Science 77405-080 Gurupi, TO - BRASIL Email: [email protected]

Universidade Federal de Campina Grande Department of Electrical Engineering 58109-900 Campina Grande, PB - BRASIL Email: {prbarros,jacobina,amnlima,edison}@dee.ufcg.edu.br

Abstract— The purpose of this paper is to present a control technique for controlling the flux and electromagnetic torque of an induction motor using electrical parameters estimated in closed-loop. The parameters estimated in closed-loop are used to update the gains of an IFOC based motor drive system. It was also shown in this paper the performance of the closed-loop in presence of parameter changes. Experimental and simulated results are used to demonstrate the feasibility of the proposed strategy.

R

C1

S

q2

q1

d3 q3

T d1

C2 q1

d2

vs3 i s3

d3 q3

q2

L o a d

δr

CA MOTOR

timer

Induction motors have been used, for a long time, in low performance drives. The early motor drive control techniques for induction motors were of the scalar type and based on steady-state models, e.g., the Volts/Hertz technique [1] that provides poor dynamic performance. In order to develop high performance motor drive systems, control laws that assure the decoupling between the flux loop and electromagnetic torque loop have been investigated. The use of generic decoupling techniques, just as proposed in [2], or based on scalar models, as proposed in [3], is, in general, not effective and eventually leads to relatively complex solutions. However, exploring the machine model conveniently, it is possible to decouple the motor control by using vector approaches. If we control the rotor flux, through the stator current component in phase with the flux, and by controls the electromagnetic torque through the stator current orthogonal component or in quadrature with the flux, we obtain the so-called field-oriented control [4], [5]. Field-oriented control has allowed to extend the use of induction motors in high performance applications. Directfield-oriented control (DFOC) includes a closed-loop rotorflux controller and requires the calculation of the amplitude and position of the rotor flux. This is the standard solution for high performance motor drives but requires relatively complex algorithms. Indirect-field-oriented control (IFOC) does not needs a closed-loop rotor flux controller and only requires the angular position of the rotor flux vector which is calculated by integrating the vector angular speed [6]; this can be computed using the rotor speed and the stator-current measurement. IFOC is very simple to implement and, therefore worth to be considered as a solution in many applications. However, the calculation of the angular position of the rotor flux is very

vs1 is1

0

I. I NTRODUCTION

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d2

d1

A/D PPI

microcomputer

Fig. 1.

Experimental set-up: Induction Motor Drive System.

sensitive to errors in rotor time constant which changes widely with temperature. Moreover, the use of wrong parameters values in the vector approaches causes the coupling of the flux and torque equations, degrading the motor drive performance. In the induction motor, some parameters change with the operating condition, mainly the rotor time constant, which imposes the use of some adaptation technique for tracking and compensating possible parameter variations. The purpose of this article is to present a technique for controlling flux and electromagnetic torque of an induction motor by using electrical parameters estimated in closed-loop; the estimated parameters are used to tune the flux and torque controllers. Experimental results obtained by using the set-up sketched in Fig. 1 are used to demonstrate the feasibility of the proposed solution. II. I NDUCTION M OTOR M ODEL The induction motor is described, in a stator reference frame, by the following model [1]:

1357

(1)

φs

d φs dt d = rr ir + φr − jωr φr dt = ls i s + l m i r

φr

= lr i r + l m i s

(4)

vs 0

= rs is +

(2) (3)

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lm (isq φrd − isd φrq ) (5) lr d (6) P (Te − Tl ) = Jm ωr + Fm ωr dt d vso = rs iso + lls iso (7) dt The variables and parameters used in the above expressions are defined as follows: i) vs = vsd + jvsq , is = isd + jisq , ir = ird + jirq , φs = φsd + jφsq and φr = φrd + jφrq are the stator voltage, the stator current, the rotor current, the stator flux and the rotor flux vectors, respectively; ii) ωr , Te and Tl are the angular shaft speed, the electromagnetic torque and the load torque, respectively and iii) P , Jm , Fm , rs , rr , ls , lr and lm are the number of pole pairs, the moment of inertia, the viscous friction coefficient, the stator resistance, the rotor resistance, the self inductance of the stator, the self inductance of the rotor and the mutual inductance between stator and rotor, respectively. III. PARAMETER E STIMATION The use of recursive least squares (RLS) estimation techniques requires that the system model be defined as a regression equation such as: y (t|θ) = ψ (t) θ (t)

(8)

where y (t|θ) , ψ (t) and θ (t) are the prediction vector, the regression matrix and the parameter vector, respectively. The basic equations of the RLS algorithm used to computer θ can be defined like [7]: θˆ (k) e (k)

= θˆ (k − 1) + K (k) e (k) = y (k) − ψ (k) θˆ (k − 1)

K (k)

=

P (k)

=

P (k − 1) ψ (k) λ (k) + ψ (k) P (k − 1) ψ T (k) (I − K (k) ψ (k)) P (k − 1) λ (k) T

(9) (10) (11) (12)

The proposed estimation procedure is split in two steps. In the first step, stator resistance and leakage inductance are determined based on the homopolar model. In the second step, all the other electrical parameters are determined, in closedloop, based on the dynamic dq model. A. First step: Estimating rs and lls It has been established that if rs is jointly estimated with all the other motor parameters one may obtain, in general, an illconditioned numerical problem [8]. An alternative technique for determining the stator resistance solely (for dc and ac excitation) was presented in [9]. The proposed technique [9] also provides an estimate of the stator leakage inductance. In this paper, the regression model employed for estimating rs and lls is given by y (t) ψ (t) θ

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i s*

= P

Te

= vso 
d = iso iso dt
T rs lls =

+

is C(s)

G(s)

-

Fig. 2.

Closed-loop configuration.

and can be derived from (7). Here and elsewhere, derivatives of signals will be obtained by using state variable filters (SVF) [10]. B. Second step: Closed-Loop Estimation In general, an induction motor drive system is a cascade multi-loop control system. In this paper the closed-loop estimation will be formulated for the motor current control loop. Fig. 2 shows a block diagram illustrating the stator current control loop. The transfer function   lr lr 1 p + − jω r β β τr   (16) G (p) = 2 p + (α − jωr ) p + rsβlr τ1r − jωr represents the relationship is /vs as obtained from the dq d , and the model for constant rotor speed [8], where p = dt controller kd s2 + kp s + ki (17) C(s) = s is a standard PID controller. The initial values for the controller’s gains ki , kp and kd are considered to be known. Such gains can be determined given the motor’s parameters as obtained from the standard locked-rotor and no-load tests. Thus, given the values of the parameters of the homopolar model (obtained by executing Step 1) and considering that stator current and rotor speed are measurable, a regression model for closed-loop estimation can be derived. The regression model that can be employed for closed-loop parameter estimation is given by [11]: y (t) ψ (t) θ

d2 d3 i − jω is s r 3 dt2 
dt ψ1 (t) ψ2 (t) ψ3 (t) = T  lr lr α βτ = β r

=

(18) (19) (20)

with d2 is dt2 2 d d d ψ2 (t) = kd 2 ec + kp ec + ki ec − rs is dt dt dt d3 d2 ψ3 (t) = kd 3 ec + (kp − jωr kd ) 2 ec dt dt d + (ki − jωr kp ) ec − jωr ki ec dt d +rs jωr is dt ψ1 (t) = −

(13) (14) (15)

ec

where α =

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rs lr +rr ls , τr β

=

lr rr , β

(21) (22)

(23)

2 = l r ls − lm and ec = i∗s − is .

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b

*

φr ω*r +

1 lm

T e*

Σ -

b* is d

ωr

b* +

s*

vsd

PWM i

-

isq

b*

vsd

Σ

+

lr 2 P lm

is d

-

e sq

b*

vsq

Σ

b*

1 isq τ r ib * sd

ω*b r +

e

ωb

Σ

jθ b s*

vsq

+

IM

VSI

θb

T

+

ωr Fig. 3.

Indirect field-oriented control - IFOC.

It is worth mentioning that the closed-loop regression model does not depend on motor voltage measurements and therefore voltage sensors are not required. From the parameter vector given in (20) one may determine lr α, τr and lβr by: α = θ (1), τr = θ(3) θ(2) and β = θ (3), respectively. Considering that rs , lls , α, τr and lβr were already estimated and that the stator and rotor inductances are given by: ls = lls + lm and lr = llr + lm , respectively, where llr denotes the rotor leakage inductance referred to the stator. Then, the other parameter may be derived by ls

=

lr

=

l m

=

rr

=

llr

=

θ (1) − rs θ (3) θ (2) 2 θ (3) lm θ (3) ls − 1 ls − lls lr τr lr − l m

(24) (25)

∗ is the reference slip frame (indicated by superscript b), ωbr speed and θb is the angular position of the rotor flux vector. As mentioned before, the rotor time constant changes with the motor operating conditions and therefore an adaptation technique is required to compensate possible parameter variations. Thus, the parameters provided by the proposed closedloop estimator that was described in the previous section may be employed to adaptively tune the flux and torque controllers. In this paper the rotor flux (φr ) will be controlled by the stator current component that is in phase with the flux (ibsd ) whereas the electromagnetic torque will be controlled by stator current component that is in quadrature with the flux (ibsq ). In this case some of the basic IFOC equations must be rewritten in terms of the closed-loop estimated parameters

(26) (27) (28)

Thus, executing the first and second steps, all electrical parameters of the motor model can be determined. IV. I NDIRECT F IELD -O RIENTED C ONTROL - IFOC The block diagram of the standard IFOC is shown in Figure 3 and its basic equations are given below [12]: ibsd

∗

=

ibsq

∗

=

∗ ωbr

=

ωb

=

θb

=

φ∗r lm lr Te∗ P lm φ∗r ∗ 1 ibsq ∗ τr ibsd ω ∗ + ωr brt ωb dt

(29) (30) (31) (32) (33)

0

Regarding to the basic IFOC equations, φ∗r denotes the ref∗ erence rotor flux, Te∗ is reference electromagnetic torque, ibsd ∗ and ibsq are the reference stator currents in the rotor reference

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ibsd

∗

=

ibsq

∗

=

∗ ωbr

=

φ∗r l m lr T ∗

(34)

e

(35)

∗ P l m φr ∗ 1 ibsq ∗ τr ibsd

(36)

Then, whenever new estimated parameters are issued from the closed-loop estimator, the IFOC gains are updated and the motor drive performance is optimized. V. S IMULATION AND E XPERIMENTAL R ESULTS The experimental tests were conducted by using the set-up shown in Fig. 1. The machine is supplied with a three-phase voltage source inverter. The drive system is controlled through a PC-Pentium equipped with dedicated plug-in boards. The estimation algorithm employed to process the experimental data was the recursive least squares (RLS) with forgetting factor. The sampling time was set to 50 µs and forgetting factor to 0.999 and the derivatives of the signals are obtained by using state variable filters (SVF) [10]. These filters are designed by obtaining the discrete-time equivalent of

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Gf (s) =

ωc3 (s + ωc )

3

(37)

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5

0.5 True i

vso (V)

sd

Ref. isd 0

−5

0

0

0.1

0.2

0.3

0.4

−0.5

0.5

0.2

0

0.05

0.1

0.15

0.2

0.25

0.3

0.5 True isq

iso (A)

0.1

Ref. i

sq

0

0

−0.1 −0.2

0

0.1

0.2

0.3

0.4

−0.5

0.5

0

0.05

0.1

time (s)

Fig. 4.

Homopolar voltage and current.

Fig. 6.

0.15 time (s)

0.2

0.25

0.3

True and reference stator currents with tuned controller.

Speed

True and Reference Currents

350

0.4 True isd Reference isd

isd (A)

0.2

300

0

250 −0.2

200 −0.4

0

0.05

0.1

0.15

0.2

0.25 time (s)

0.3

0.35

0.4

0.45

0.5

150

0.3 True isq Reference isq

0.2

100

isq (A)

0.1

50

0 −0.1

0

−0.2 −0.3 −0.4

0

0.05

0.1

Fig. 5.

0.15

0.2

0.25 time (s)

0.3

0.35

0.4

0.45

0.5

1 time (s)

1.5

2

0.5

Fig. 7. Rotor speed with variation in the rotor resistance and without adaptation algorithm.

True and reference stator currents.

representing a third-order low-pass analog filter with cut-off frequency ωc = 500 rad/s. The experiments were divided into three parts. In the first part, the machine was supplied by three-phase pulse width modulated voltages. The modulating signal employed in the first test is composed of a fundamental component (fe = 60Hz) and an homopolar voltage (fo = 10 Hz) which amplitude is 5% of the fundamental. The neutral wire of the motor was connected to the mid-pont of the dc-link capacitor bank. Fig. 4 shows the measured waveforms of the homopolar voltage and current, respectively. The estimated stator resistance and leakage inductance are given in Table I. In the second part, the closed-loop estimation was implemented by using a PID controller. In this experiment, the values for the initial controllers’s gains (C0 (s)) kp , ki and kd were 232, 56000 and 0.877, respectively. Waveforms of the measured stator currents as compared with the reference dq currents are shown in Fig. 5.

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It is important to notice that the data obtained during the speed transient from rest to steady-state speed have been exploited to estimate the electrical parameters. Table I shows the parameters as obtained by the standard tests and its estimated values by using the closed-loop algorithm. As it can be remarked by observing Table I, the estimation errors are quite small and thus demonstrating that it is possible to estimate all the electrical parameters of the induction motor with good precision. Moreover, the experiment design is not a critical issue since good results were achieved with relatively low persistence signals (sinusoidal signals). In addition, the current controller performance in the stator reference frame is significantly improved when the parameters estimated in closed-loop are used to redesign controller. The proposed method makes it possible to redesign the current controller from the estimated parameters in closed-loop. In order to validate the proposed method in this paper, a PID controller was tuned by using the estimated electrical parameters in closed-loop. The pole placement technique [13] was employed to redesign the controller. The desired characteristic

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TABLE I PARAMETERS OBTAINED BY THE STANDARD TESTS AND ITS ESTIMATED VALUES BY USING THE CLOSED - LOOP ALGORITHM .

Rotor speed 350

300

Estimated

Error (%)

rs (Ω)

29

28.10

3.10

rr (Ω)

30

30.70

2.30

ls (H)

0.83

0.85

2.40

lr (H)

0.83

0.85

2.40

lm (H)

0.80

0.82

2.50

lls (mH)

32.50

30

7.60

llr (mH)

32.50

30

7.60

250

200 r

Standard

ω

Parameters

150

100

polynomial in closed-loop was chosen to be 

A∗ (s) = s2 + 2ξωn s + ωn2 (s + c)

50

0

s2

+

lr lr β s + βτr rs lr +rr ls s lr s + rβτ β r

=

B(s) A(s)

(38)

(39)

and then the denominator of the closed loop transfer function is A∗ (s) = Q(s)B(s) + R(s)A(s)

(40)

where Q(s) and R(s) are controller numerator and denominator, respectively. Solving (40) we find out that a new controller C1 (s) with the following gains kp = 185, ki = 32120 and kd = 5.63. In the Figure 6 the measured and reference currents, in stator reference frame, by using the tuned controller are represented. Figure 6 illustrates the achieved closed-loop performance with the redesigned controller. In order to evaluate the controller performance, the mean square error, defined by (41), was used as criterion. E (Ci ) =

N 1 ∗ 2 [i (t)−is (t)] ; i = 0, 1, ..; t = 1, 2, .., N (41) N t=1 s

where i∗s (t) and is (t) denote reference and actual motor currents, respectively. The error function for the initial E (C0 ) and the first redesigned E (C1 ) controllers are E (C0 ) = 0.00540 E (C1 ) = 0.00024

(42) (43)

for N = 5000. It is quite clear that the closed loop performance is greatly improved when the redesigned controller is employed.

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0.5

1 time (s)

1.5

2

Fig. 8. Rotor speed with variation in the rotor resistance and with adaptation algorithm.

with ξ = 0.9, ωn = 73.8 and c = 37.5 for the low speed range. To determine the parameters of the controller it was considered that ωr = 0 in (16). In this case the motor model becomes G (s) =

0

In the last part, the flux and electromagnetic torque controllers have been implemented with the close-loop estimated parameters. In this simulation, the initial rotor resistance was set to 30Ω, after one minute of simulation the rotor resistance was changed to 15Ω. In the Fig. 7 and Fig. 8 the rotor speed when the rotor resistance is varied are presented. Already, in the Fig. 9 the dq rotor flux with variation in the rotor resistance but without employing any adaptive technique is shown. In this experiment, as it can be remarked by observing the Fig. 9, when the rotor resistance varies, the dq rotor flux does not track the reference flux. On the other hand, if the closed-loop parameters are used the dq rotor flux indeed tracks the reference flux even when there is changes in the rotor resistance; this behavior is illustrated in Fig. 10. Fig. 11 and Fig. 12 show the electromagnetic torque as obtained when there are variations in the rotor time constant. In the Fig. 11 is not employing any adaptive technique. However, in the Fig. 12 the closed-loop parameters are used to update the IFOC gains. It is worth mentioning that the electromagnetic torque tracks the reference torque. The proposed closed-loop estimation technique has allowed to improve the dynamic performance of the motor drive system either by redesigning the stator current controller and by properly updating the IFOC gains. VI. C ONCLUSION In this paper, a control law for the flux and electromagnetic torque of an induction motor using electrical parameters estimated in closed-loop was presented. The closed-loop estimated parameters were used to tune the flux and torque controllers of an IFOC motor drive system. It was also shown in this paper the performance of the closed-loop in presence of parameter changes. Experimental and simulated results were used to demonstrate the feasibility of the proposed solution. In addition, the current controller performance in the stator reference frame was significantly improved when the new estimated parameters in closed-

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1

45 d rotor flux q rotor flux

0.8

True torque Ref. torque

40

0.6

35

0.4

30 Torque

Flux

0.2 0

25 20

−0.2 15

−0.4

10

−0.6

5

−0.8 −1 0.8

0.85

0.9

0.95

1 time (s)

1.05

1.1

1.15

0 0.8

1.2

Fig. 9. dq rotor flux with variation in the rotor resistance and without adaptation algorithm.

0.9

1

1.1 1.2 time (s)

1.3

1.4

1.5

Fig. 11. True and reference torque with variation in the rotor resistance and without closed-loop algorithm.

50 φrd

1

48

φ

True torque Ref. torque

rq

0.8

46

0.6

44

0.4 42 Torque

Flux

0.2 0

40

−0.2

38

−0.4

36

−0.6

34

−0.8 32

−1 0.8

0.85

0.9

0.95

1 time (s)

1.05

1.1

1.15

30 0.8

1.2

0.9

1

1.1 1.2 time (s)

1.3

1.4

1.5

Fig. 10. dq rotor flux with variation in the rotor resistance and with closedloop algorithm.

Fig. 12. True and reference torque with variation in the rotor resistance and with closed-loop algorithm.

loop were applied to redesign the controller. Moreover, the estimated parameters in closed loop were used to tune the flux and torque control.

[8] M. Velez-Reyes, K. Minami, and G. C. Verghese, “Recursive speed and parameter estimation for induction machines.” in In Conf. Rec Ias, 1989, pp. 607–611. [9] C. B. Jacobina, J. E. C. Filho, and A. M. N. Lima, “Estimating the parameters of induction machines at standstill.” in IEEE Trans. Energy Conversion, vol. 17, 2002, pp. 85–89. [10] I. D. Landau, Adaptive Control: The Model Reference Approach. Marcel Dekker, Inc.: Prentice Hall, 1979. [11] A. J. Netto, P. R. Barros, C. B. Jacobina, and A. M. N. Lima, “Estimating the parameters of an induction motor in open-loop and closed-loop operation,” in IEEE - Industry Applications Society Annual Meeting, Seattle, USA, 2004. [12] F. Salvadori, C. B. Jacobina, and A. M. N. Lima, “Decoupled flux and torque control schemes for high performance induction drive system: A comparative study with new strategies,” in IEEE Int. Conf. on Ind. Electron. Contr. and Instrum, (IECON-91), Kobe, 1991, pp. 481–486. [13] R. H. Middleton and G. C. Goodwin, Digital Control and Estimation. A unified approach. University of Newcastle, Australia: Prentice Hall International, INC, 1990.

R EFERENCES [1] W. Leonhard, Control of Electrical Drives., 2nd ed. Berlin, Alemanha: Springer-Verlag, Inc, 1996. [2] P. L. Falb and W. A. Wolovich, “Decoupling in the design and synthesis of multivariable control systems,” IEEE Trans. on Automatic Control, vol. 12, pp. 651–659, 1967. [3] B. K. Bose, “Scalar decoupled control of induction motor,” IEEE Trans. on Industrial Electronics, vol. 20, pp. 216–225, 1984. [4] F. Blaschke, “A new method for the structural decoupling of ac induction machines,” in Procedings 2nd IFAC Symposion, Dusseldorf, Germany, 1971, pp. 1–15. [5] R. W. D. Doncker and D. W. Novotny, “The universal field oriented controller,” in IEEE - Industry Applications Society Annual Meeting, Orleans, USA, 1988. [6] J. M. D. Murphy and F. G. Turnbull, “Power electronics control of ac motors,” in Pergamom Press, 1988. [7] L. Ljung, System Identification. Theory for the User., 2nd ed. Upper Saddle River, New Jersey: Prentice Hall, Inc, 1999.

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