Estimating the parameters of an induction motor in open ... - IEEE Xplore

Estimating the Parameters of an Induction Motor in Open-Loop and Closed-Loop Operation A.J. Netto, P.R. Barros, C.B. Jacobina and A.M.N. Lima Dep. de Eng. El´etrica - CCT - UFCG - Campus II - Caixa Postal 10.105 58109-970 Campina Grande - PB - Brazil Fax: +55-83-3101015; E-mail: [email protected]

Abstract: The purpose of this paper is to present two methods for estimating all the parameters of induction motors operating in open-loop or in closed-loop. The parameters are obtained by solving a recursive least squares minimization problem. The estimation procedure is performed in two parts. First, the stator leakage inductance and the stator resistance are determined by using the homopolar machine model. Then, given the previously estimated homopolar parameters and by using the dynamic dq model, all the other parameters are determined. The first method is derived for open-loop operation. In the second method, the model is derived for the case where the stator currents of the motor are regulated via a linear feedback controller. Selected experimental results are used to demonstrate the feasibility and performance of the proposed methods. I. Introduction During the last years, several papers have been published about induction motor identification [1–7]. In general, the use of linear estimation techniques based on the dynamic model of induction motor, written in terms of a dq reference frame, does not allow to determine all the electric parameters of the machine. With this kind of modelling approach one may only estimate four parameters [1, 3–5], namely the stator resistance, rs , the stator inductance, ls , the transient inductance, σls , and the rotor time constant, τr . There are other techniques that provide estimates of all electrical parameters of the machine [2,6,7], but at the expenses of relatively high computational load for real-time applications. In particular, in [6] all electrical parameters were estimated with locked rotor. On the other hand in [2, 7] the presented algorithms are quite complex and involve too many parameters in the differential equations. This may cause the non-convergence of the algorithm as

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well as multiple solutions. Although many different solutions have already been proposed, a relatively small number of papers has been focused on the use closed-loop identification techniques for determining the electrical parameters of induction motors. However, closed-loop identification techniques have already been recognized as powerful design tools providing better models and simple controllers. Indeed, iterative identification and real-time controller redesign can be considered the most reliable alternative to achieve high performance feedback control systems. Recently, some preliminary simulation results about off-line closed-loop identification of the electrical parameters of induction motor were presented in [8]. The purpose of this paper is to present two methods for estimating all the electrical parameters of induction motors. The parameters are obtained by solving a recursive least squares minimization problem. The estimation procedure is split in two parts. First, the stator leakage inductance and the stator resistance are determined by using the homopolar machine model. Then, given the previously estimated homopolar parameters and based on the dq model, all the other parameters are determined. In the second method, a closed-loop identification technique for determining in real-time all the electrical parameters of induction motors is proposed. The proposed technique can be easily included in the software of the induction motor drive system to improve the achieved performance. The paper is organized as follows: Section 2 introduces the induction motor model. In Section 3 the estimation strategy are used. In Section 4 the experimental results obtained by using the set-up sketched in Fig. 1 are used to demonstrate the feasibility of the proposed approach. Finally, in Section 5 concluding remarks are made.

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d2

d1

R S

C1

q2

q1

III. Parameter Estimation

d3 q3

The use of recursive least squares (RLS) estimation techniques requires that the system model should be defined as a regression equation like

vs1 is1

0

T d1

C2

d2

d3 q3

q2

q1

vs3 i s3

L o a d

δr

yˆ (t|θ) = ψ (t) θ (t)

CA MOTOR

timer

A/D PPI

microcomputer

Fig. 1. Induction motor drive system.

II. Induction Motor Model The induction motor is described, in a stator reference frame, by the following model [9]: d φ dt s d 0 = rr ir + φr − jωr φr dt φs = ls is + lm ir vs

= rs is +

φr

= lr i r + l m i s lm Te = P (isq φrd − isd φrq ) lr d P (Te − Tl ) = Jm ωr + Fm ωr dt d vso = rs iso + lls iso . dt

(1) (2) (3) (4) (5) (6)

(7)

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. The homopolar voltage and current that are obtained by transforming (power conservative odq transformation) vs1 , vs2 , vs3 , is1 , is2 and is3 , are given by

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vso

=

iso

=

1 √ (vs1 + vs2 + vs3 ) 3 1 √ (is1 + is2 + is3 ) . 3

(8) (9)

27

(10)

where yˆ (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 found in [10]. The proposed estimation procedure is split in two parts. In the first part, stator resistance and leakage inductance are determined based on the homopolar model. In the second part, all the other electrical parameters are determined based on the dynamic dq model. As shown in the following Step 1 and Step 2 are executed for open-loop estimation. On the other hand Step 1 and Step 3 are executed for closed-loop estimation. A. Step 1: 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 ill-conditioned numerical problem [1]. An alternative technique for determining solely the stator resistance (for dc and ac excitation) was presented in [6]. The proposed technique [6] also provides an estimate of the stator leakage inductance. In this paper, the regression model employed for estimating rs and lls is given by yˆ (t) = vso
 d ψ (t) = iso dt iso T
θ = rs lls

(11) (12) (13)

and can be derived from (7). Here and elsewhere, derivatives of signals will be obtained by using state variable filters (SVF) [11]. B. Step 2: Open-Loop Estimation Solving equations (1) to (4) for is and considering that voltages, currents and speed of the machine are measurable, the regression model is

d d2 is − jωr is dt2 dt
ψ (t) = − d is (vs − rs is )   dt d vs − jωr vs + jωr rs is dt  T lr lr α βτ θ = β r yˆ (t) =

(14)

(15) (16)

2 r ls , τr = rlrr and β = lr ls − lm with α = rs lr +r . For executβ ing Step 2 we also consider that Step 1 was already been executed to provide rs and lls .

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i s* +

ec

C(s)

15

is

G(s) 10

-

5

Fig. 2. Closed-loop configuration. vso (V)

0

C. Step 3: Closed-Loop Estimation

−5

In general, an induction motor drive system is a cascade multi-loop control system. In the following the closed-loop estimation problem will be formulated for the case where only the stator current control loop is operating. In Fig. 2 it is shown a block diagram illustrating the stator current control loop. The transfer function  lr lr 1 s + − jω r β β τr  G (s) = (17) 2 s + (α − jωr ) s + rsβlr τ1r − jωr

−10

−15

0

0.05

0.1

0.15

0.2

0.25 time (s)

0.3

0.35

0.4

0.45

0.5

0.4

0.45

0.5

Fig. 3. Homopolar voltage. 0.2

0.15

0.1

represents the relationship is /vs as obtained from the dq model for constant rotor speed [1] and the controller

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

(19)

0 iso (A)

kd s2 + kp s + ki (18) 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 noload tests. Thus, given the values of the parameters of the homopolar model (obtained by executing Step 1) and 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 C(s) =

0.05

−0.05

−0.1

−0.15

−0.2

0

0.05

0.1

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0.3

0.35

From the parameter vector given in (16) or (21) one may determine α, τr and lβr by

α τr

(21)

lr β

2

28

0.25 time (s)

D. Estimating the Other Electrical Parameters

(20)

d is (22) dt2 d2 d d ψ2 (t) = kd 2 ec + kp ec + ki ec − rs is (23) 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 (24) +rs jωr is dt where ec = i∗s − is . Note that the closed-loop estimator don’t depends on voltages, therefore the voltages’s sensors are not needed.

0.2

Fig. 4. Homopolar current.

with ψ1 (t) = −

0.15

= θ (1) θ (3) = θ (2) = θ (3) .

(25) (26) (27)

Notice 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 where llr denotes the rotor leakage inductance referred to the stator. Then, the other parameter may be computed from ls

=

lr

=

θ (1) − rs θ (3) θ (2) 2 θ (3) lm θ (3) ls − 1

(28) (29)

with lm = ls − lls , rr = τlrr and llr = lr − lm . Thus, all electrical parameters of the machine model are estimated.

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150

Current

1 time (s)

100

50

0.5

stator resistance

0 0.2

0.1 0

0

−0.1

−0.1

−0.15

−0.2

−0.05

isq (A)

0.1

0.05

0

0.15

0.2

isd (A)

−50

Voltage

1 time (s)

−100

−150

0.5

0 40

20

−200

0

0.02

0.04

0.06

0.08

0.1 time (s)

0.12

0.14

0.16

0.18

0 −20

0.2

−40

−20

−30

−10

vsq (V)

Fig. 5. Estimated stator resistance.

20

10

0

30

vsd (V)

Fig. 7. Stator currents and voltages in open-loop.

3500

Rotor Speed

400 3000

350

2000

300

1500

250 1000

wr (rad/s)

stator leakage inductance (H)

2500

500

0

200

150

−500

100 −1000

50 −1500

0

0.02

0.04

0.06

0.08

0.1 time (s)

0.12

0.14

0.16

0.18

0.2

0

0.05

0.1

0.15

0.2 time (s)

0.25

0.3

0.35

0.4

Fig. 6. Estimated stator leakage inductance. Fig. 8. Rotor speed in open-loop.

IV. Experimental Results The experiments were done with the drive system shown in Fig. 1. The estimation algorithm used to process the experimental data was the recursive least squares (RLS) with forgetting factor. The sampling time was set to 50µs and forgetting factor to 0.999. As mentioned before, the derivatives of the signals were obtained by using state variable filters (SVF). These filters were designed by obtaining the discrete-time equivalent of Gf (s) =

ωc3

3

(s + ωc )

(30)

where ωc = 60Hz. The experiments were divided in 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 (f0 = 10Hz) which amplitude is 5% of the fundamental. The neutral of the machine was connected to the central tap of the capacitor bank. Fig. 3 and Fig. 4 show waveforms of the homopolar

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voltage and current, respectively. Fig. 5 and Fig. 6 show the time evolution of the estimated stator resistance and leakage inductance, respectively, as obtained from (13). In the second part, the machine was supplied by sinusoidal PWM voltages (fundamental frequency of 60Hz). Fig. 7 and Fig. 8 show measurable waveforms of the dq stator voltages, dq stator currents and rotor speed, respectively. Fig. 9 shows the time evolution of the estimated parameters as obtained from (16). Parameters rs (Ω) rr (Ω) ls (H) lr (H) lm (H) lls (mH) llr (mH)

Standard 29 30 0.83 0.83 0.80 32.50 32.50

Estimated 28.10 30.80 0.85 0.85 0.82 30 30

Error (%) 3.10 2.60 2.40 2.40 2.50 7.60 7.60

TABLE 1. Parameters obtained by the standard tests and its estimated values by using the open-loop algorithm.

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Estimated Parameters

Rotor Speed

4000

70

θ(1)

2000 60

0 −2000

50

0.05

0

0.1

0.15

5000 40

θ(2)

wr (rad/s)

0

−5000 −10000

0

0.05

0.1

0.15

30

20

200 10

θ(3)

150

100 0

50 0

0

0.05

0.05

0.1

0.15

0.2 time (s)

0.25

0.3

0.35

0.4

0.15

0.1 time (s)

Fig. 11. Rotor speed in closed-loop.

Fig. 9. Estimated parameters in open-loop. Estimated Parameters

4000

Error (%) 3.10 2.30 2.40 2.40 2.50 7.60 7.60

2000

θ(1)

Estimated 28.10 30.70 0.85 0.85 0.82 30 30

0

−2000

0

0.05

0.1

0.15

0.05

0.1

0.15

0.1

0.15

5000

0 θ(2)

Standard 29 30 0.83 0.83 0.80 32.50 32.50

−5000

−10000

0

200

TABLE 2. Parameters obtained by the standard tests and its estimated values by using the closed-loop algorithm.

150

θ(3)

Parameters rs (Ω) rr (Ω) ls (H) lr (H) lm (H) lls (mH) llr (mH)

100

50

0

0

0.05 time (s)

True and Reference Currents 0.4

True isd Reference isd

Fig. 12. Estimated parameters in closed-loop.

isd (A)

0.2

0

−0.2

−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

0.3

True isq Reference isq

0.2

isq (A)

0.1

0

−0.1

−0.2 −0.3

−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

Fig. 10. Stator currents true and reference in closed-loop.

Finally, in the last part, the closed-loop estimation was implemented by using a PID controller. In this experiment, the values for the controllers’s gains kp , ki and kd were 232, 56000 and 0.877, respectively. Fig. 10 and Fig. 11 show waveforms of the dq stator true and reference currents and rotor speed, respectively. Fig. 12 show the parameters as obtained in the closed-loop estimation

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method. 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 of (16) and (21). In Table 1 and Table 2 are shown the parameters obtained from the standard tests and the estimated values using the open-loop and closed-loop algorithms, respectively. From the data in Table 1 and Table 2, it can be seen that estimation errors are quite small, and so 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). V. Conclusion In this paper two techniques for estimating the parameters of an induction motor in open-loop and closed-loop

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operation by means of recursive least square (RLS) were presented. Results show that, differently of the most of algorithms presented in literature, all electrical parameters of the motor can be retrieved from estimation of the stator resistance and stator leakage inductance jointly with the estimation of only three parameters. The experimental results have demonstrated the feasibility of the proposed techniques. In particular, the methods proposed have the following advantages with respect to other methods: • The algorithm is easy to implement and has a relatively low computational load; • The parameter vector has at the most three parameters to be estimated; • It is not critical in terms of persistence of excitation. This is very useful in industrial applications; • It allows the three electrical parameters to be estimated during a speed transient of the machine; • The method estimate on-line all electrical parameters of the induction motor in closed-loop.

tems Technology, vol. 8, no. 6, pages 873–882, 2000. A. Besanon-Voda and M. Titiliuc. Issues on identification in closed-loop of induction motors. In European Control Conference, pages 1940–1945, 2001. [9] W. Leonhard. Control of Electrical Drives. SpringerVerlag, Inc, Berlin, Germany, second edition, 1996. [10] L. Ljung. System Identification. Theory for the User. Prentice Hall, Inc, Upper Saddle River, New Jersey, second edition, 1999. [11] I. D. Landau. Adaptive Control: The Model Reference Approach. Prentice Hall, Marcel Dekker, Inc., 1979. [8]

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[7]

M. Velez-Reyes, K. Minami, and G. C. Verghese. Recursive speed and parameter estimation for induction machines. In Conf. Rec Ias, pages 607–611, 1989. C. Moons and B. De Moor. Parameter identification of induction motor drives. In Automatica, Vol. 31, pages 1137–1146, 1995. F. Alonge, F. D’Ippolito, S. La Barbera, and F. M. Raimondi. Parameter identification of a mathematical model of induction motors via least-square technics. In Proc. IEEE Int. Conf. Control Applications, pages 491–496, Trieste, Italy, 1998. M. Cirrincione, M. Pucci, and G. Vitale. A leastsquare based methodology for estimating the electrical parameters of induction machine at standstill. In Proc. IEEE Int. Workshop Advanced Motion Control (AMC), pages 541–547, Maribor, Slovenia, 2002. L. A. S. Ribeiro, C. B. Jacobina, and A. M. N. Lima. Linear parameter estimation for induction machines considering the operating conditions. In IEEE Trans. on Power Electronics, vol. 14, no. 1, pages 62–73, 1999. 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, pages 85–89, 2002. R. F. F. Koning, C. T. Chou, M. H. G. Verhaegen, J. B. Klaassens, and J. R. Uittenbogaart. A novel appoach on parameter identification for inverter driven induction machines. In IEEE Trans. on Control Sys-

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