https://doi.org/10.6113/JPE.2018.18.2.492

ISSN(Print): 1598-2092 / ISSN(Online): 2093-4718

A Novel Method for the Identification of the Rotor Resistance and Mutual Inductance of Induction Motors Based on MRAC and RLS Estimation

Gwon-Jae Jo^* and Jong-Woo Choi^†

^†,*Department of Electrical Engineering, Kyungpook National University, Daegu, Korea

Abstract

In the rotor-flux oriented control used in induction motors, the electrical parameters of the motors should be identified. Among these parameters, the mutual inductance and rotor resistance should be accurately tuned for better operations. However, they are more difficult to identify than the stator resistance and stator transient inductance. The rotor resistance and mutual inductance can change in operations due to flux saturation and heat generation. When detuning of these parameters occurs, the performance of the control is degenerated. In this paper, a novel method for the concurrent identification of the two parameters is proposed based on recursive least square estimation and model reference adaptive control.

Key words: Induction motor, Model reference adaptive control, Mutual inductance, Parameter identification, Recursive least square estimation, Rotor resistance

Manuscript received Jun. 16, 2017; accepted Oct. 6, 2017

Recommended for publication by Associate Editor Kwang-Woon Lee.

^†Corresponding Author: cjw@knu.ac.kr Tel: +82-53-950-5515, Kyungpook National University

^*Dept. of Electrical Engineering, Kyungpook National University, Korea

NOMENCLATURE

Rs	Stator resistance
Rr	Rotor resistance
Lm	Mutual inductance
Ls	Stator self-inductance
Lr	Rotor self-inductance
Lls	Stator leakage inductance
Llr	Rotor leakage inductance
σLs	Stator transient inductance
Tr	Rotor time constant
s, r	Superscripts: denote the stationary and rotor reference frames
s, r	Subscripts: denote the stator and rotor
^	Denotes estimated quantities
s	Derivative operator
Bold form x	Vector xd＋jxq

Ⅰ. INTRODUCTION

Induction motors are widely used in industrial and home appliances. For the speed control of induction motors, rotor-flux oriented control is usually utilized due to its simplicity and its ability to control flux and torque components separately. In the rotor-flux control algorithm, the precise tuning of L_m and R_r is very important for efficient control of the currents of both the flux and torque components. Moreover, these parameters are essential in flux observers. However, identifying them is more difficult than identifying stator parameters such as R_s and σL_s. This is attributed to the fact that L_m varies due to nonlinear characteristics in the B-H curve. It is also due to the fact that R_r is more affected by thermal temperature in operation than R_s since the rotor is positioned inside the motor.

Several identification methods for R_r and T_r have been reported. A method by which R_r can be estimated using the relationship between the flux and the developed torque was proposed in [1]. A method, which helps identify T_r by solving simultaneous equations for the rotor voltage and the torque, was proposed in [2]. Methods where R_r is identified by adopting adaptive observers were reported in [3], [4]. A method using a particle swarm optimization algorithm was reported in [5]. Methods for the identification of R_r based on model reference adaptive control (MRAC) were proposed in [6], [7], where the variables observed in the MRAC are the voltage [6] and the instantaneous reactive power [7]. A method that compares fluxes estimated by flux observers, was reported in [8]. The rotor time constant T_r can be identified from a model reference adaptive system speed estimator in sensorless systems [9]. Using the dynamics of a synchronous reference frame current controller, the identification of T_r was studied in [10].

In addition to the identification of single rotor parameters, techniques for the identification of various other parameters were reported in [11]-[18]. Parameters have been identified offline based on fitting the experimental data and numerical analysis [11]. Voltage equations of the stator and rotor have been solved directly by integration to identify L_m and R_r [12], [13]. Based on the MRAC technique, currents have been compared for identification in [14]. Furthermore, parameters have been identified using recursive least square (RLS) estimation [15], [16]. Methods using sliding mode observers and artificial neural networks have been proposed [17], [18] for parameter identification.

In this paper, a novel method for the identification of L_m and R_r based on MRAC is proposed. In the MRAC algorithm, the mechanism is to compare the flux estimated by the flux observers designed from current and voltage models. Furthermore, the differences in fluxes estimated by the two flux observers are analyzed. The proposed method can operate at a sufficiently high frequency in order to obtain the complete reference model in the MRAC system. Thus, it is difficult to apply the proposed method at a low frequency including standstill. However, most of the papers published so far have reported the identification of single parameter, whereas this paper identifies L_m and R_r concurrently by separating the two parameters with respect to the equation. Thus, this paper has advantage since there are no dependencies on each other for L_m and R_r in the proposed method. Additionally, the proposed method can guarantee effective behavior in both the steady-state and transient-state because any assumption for the steady state in the algorithm is not involved. RLS estimation is utilized for the realization of the parameters identification. The feasibility of the proposed method has been verified by simulation and experimental results under diverse conditions.

Ⅱ. GOVERNING EQUATIONS

The flux observers utilized in this paper are based on the governing equations for the model of an induction motor. The governing equations on an arbitrary rotating axis, given by ω in squirrel cage-type induction motors consist of the voltage equations and linkage flux equations in the stator and rotor. The voltage equations are expressed as follows:

   (1)

     (2)

The linkage flux equations are expressed as follows:

           (3)

           (4)

Ⅲ. FLUX OBSERVERS

A. Voltage Model

The voltage model flux observer is designed from the stator voltage equation in [19]. The equation for the estimation of the rotor flux in the voltage model can be expressed using (1), (3), and (4) by defining these equations on the stationary reference frame (ω = 0). Thus, the rotor flux vector can be estimated from the stator voltage and current as follows:

     (5)

However, the voltage model flux observer cannot be implemented because the pure integration in (5) can diverge due to a dc offset voltage. Thus, the closed-loop Gopinath style flux observer in [20] has been employed. This flux observer model operates the same as the voltage model flux observer in a sufficiently high frequency region with solving the problem with the dc offset voltage.

B. Current Model

The current model is a typical flux observer in the rotor-flux oriented control [21]. This model is based on the rotor equation. A differential equation on the rotor reference frame (ω = ω_r) can be expressed from (2) and (4) as follows:

     (6)

Equation (6) yields the low-pass filter (LPF) form constructed from only the rotor flux and stator current. The stator current only can be used to estimate the rotor flux. The rotor flux vector on the stationary reference frame can be calculated from in (6) as follows:

        (7)

where

Ⅳ. MRAC

In this paper, parameter identification is based on the principle of MRAC. Fig. 1 shows a MRAC system that consists of a reference model, an adjustable model, and an adaptation mechanism. The reference model is independent of the parameters, whereas the adjustable model is dependent on the parameters. The adaptation mechanism generates the estimated parameters by comparing the variables (x_ref and x_adj) from the reference and adjustable models.

Fig. 1. MRAC system.

The current model flux observer is dependent on L_m and R_r according to (6). On the other hand, the voltage model flux observer is independent of these parameters. Firstly, the estimation equation of the voltage model (5) does not include R_r. Secondly, the ratio is nearly unity due to a small leakage ratio. This is shown by the following equation:

    (8)

Therefore, the detuning of L_m barely affects the estimated flux in the voltage model. From the principle of MRAC, the voltage and current models can be chosen as reference and adjustable models in the identification algorithm for L_m and R_r. In the detuned case of L_m and R_r, the rotor flux estimated by the current model has errors. On the other hand, the rotor flux can be obtained quite accurately from the voltage model. Therefore, correct values of L_m and R_r can be obtained from the adaptation mechanism wherein the fluxes estimated by the two flux observers can be compared. As the parameters are calibrated by the adaptation mechanism, the adjustable model will follow the reference model.

Ⅴ. PARAMETER IDENTIFICATION

A. Expression of the Flux Difference

The estimation equation given by (6) can be rearranged if the current model is used as an adjustable model, as follows:

          (9)

where denotes from the current model. Equivalently, if the voltage model is used as a reference model, (9) is rewritten as (10) given by:

          (10)

where denotes from the voltage model. Equation (10) consist of actual parameters because this model is robust to the detuning of L_m and R_r. Fig. 2 shows the proposed MRAC system for parameter identification. When (9) is subtracted from (10), an equation with a flux difference is obtained as follows:

      (11)

Fig. 2. MRAC system for the proposed parameter identification.

Equation (11) can be rearranged to obtain the following equation:

     (12)

The ratio in (12) is approximately unity due to a small leakage ratio as follows:

         (13)

Equations (12) and (13) lead to the following rearranged equation:

        (14)

Equation (14) implies that the flux difference can be regarded as a linear equation with the two filtered signals, and the two coefficients given by:

       (15)

B. RLS Estimation

For solving (14), RLS estimation, which is an algorithm for estimating unknown coefficients in solving regression problems, is employed. In this paper, RLS estimation with a forgetting factor is utilized in order to enhance the rate of convergence [22]. A regression equation is expressed by (16), where y is the output, X is the regression vector, and A is the coefficient vector:

          (16)

In this system, the unknown coefficients are estimated by (17) as follows, where K is the gain vector and e is the error:

         (17)

Here, e is defined by (18) and K is calculated from (19), where P is the covariance matrix and λ is the forgetting factor.

           (18)

       (19)

Equation (17) implies that the estimated coefficient is compensated by the gain times the error so that the error becomes zero. λ plays the role of putting more weight on current state than the previous state, which promotes the convergence of . Lastly, P is renewed as follows:

       (20)

On application of the RLS algorithm to (14), the output and regression vector in the equation become:

   (21)

     (22)

The output and regression vector can be calculated from the fluxes estimated by the flux observers and the filters which have a known time constant . The process for the calculation of the output and regression vector is depicted in Fig. 3. In this way, the coefficients in (15) are estimated by RLS. Note that the parameters (L_m and R_r) to be identified are separate in terms of the equation. Therefore, the detuning of R_r does not affect the identification of L_m and vice versa.

Fig. 3. Calculation of the regression vector and output.

C. Coefficient Controller

The compensation components of the respective parameters are calculated from the coefficients in (15) for tuning them. These compensation components are obtained by regulating the estimated coefficients (â₁ and â₂). When â₁ and â₂ are reduced to zero, and are implicated from (15). Based on MRAC, the flux difference becomes zero according to (14). A proportional-integral (PI) controller is employed for the regulation of â₁ and â₂ as shown Fig. 4. The outputs of the controller (ΔL_m and ΔR_r) are used as compensation components of the parameters as follows:

   (23)

   (24)

Fig. 4. Coefficient controller.

where ΔL_m and ΔR_r are the compensation components of L_m and R_r, and and are their initial setting values. In addition, the final outputs ΔL_m and ΔR_r are filtered by a LPF to prevent noise.

The process for the proposed parameter identification is shown in Fig. 5. A block diagram of a rotor-flux oriented control system with the proposed parameter identification algorithm in an induction motor is shown in Fig. 6.

Fig. 5. Process of the proposed parameter identification.

Fig. 6. Rotor-flux oriented control with the proposed parameter identification in an induction motor.

Ⅵ. SIMULATION RESULTS

The specifications and parameters of the utilized induction motor are specified in Table I. All of the controllers in the simulation are executed digitally, where the control periods of the current controller and speed controller are 100 μs and 400 μs, respectively. The control period of the parameter identification is synchronized with the speed controller. The test speeds are 300 (low-speed), 600 (mid-speed), and 1200 rpm (high-speed). Furthermore, various loads (30%, 50% and 80% of the rating) are applied to the induction motor. The forgetting factor in the RLS is set to 0.99. The initial value of L_m is 1.5 times (206 mH) the actual value, and R_r is detuned to be 0.5 times (0.37 Ω) the actual value. First, a situation with perfectly tuned stator parameters (σL_s and R_s) is tested. Second, the detuned cases for the stator parameters are simulated, where the voltage model flux observer generated an erroneous flux due to the detuning. The parameter identification algorithm starts in the steady state when the speed becomes consistent with the applied load torque. Furthermore, L_m and R_r, estimated by the algorithm, are updated to all of the controllers and the algorithm in real time, as shown in Fig. 6.

TABLE I SPECIFICATIONS AND PARAMETERS OF AN INDUCTION MOTOR

Specifications and Parameters	Quantities
Rated power	1.5 kW
Pole	4 pole
Rated speed	1,455 rpm
Rated torque	9.2 N·m
Rs	1.67 Ω
Rr	0.73 Ω
Lm	137 mH
Lls,Llr	6.5 mH

In fact, the vector form (14), which involves both the d-axis and q-axis, can be analyzed on one of the two axes. First, the identification with respect to only the d-axis is tested. Fig. 7(a) shows the responses of the actual coefficients (a₁) and the estimated coefficients (â₁) for the d-axis. â₁ followed a₁ from its initial value since the start of identification. Although the coefficient converges to zero asymptotically in the steady state, â₁ has oscillation in the transient state due to the fact that λ_ref is sinusoidal signal with a slip frequency. For generating the parameter compensation components, it is appropriate that â₁ and â₂ to be regulated by the PI controller should follow a₁ and a₂ smoothly without oscillations in the transient state. To achieve this, the average of two coefficients on the d-axis and q-axis is considered. It is confirmed from Fig. 7(b) that the oscillation of the coefficient in the transient state is removed after averaging the outputs on the two axes.

Fig. 7. Responses of coefficients. (a) On the d-axis. (b) Upon averaging.

(a)

(b)

Fig. 8(a) shows the result of identification for L_m and R_r under given conditions. Upon the implementation of the proposed algorithm, both of the parameters converge to their actual values within 5 s. Although the approximation in (13) is applied, R_r is accurately estimated since L_m is simultaneously calibrated to yield the actual value. When the parameters are calibrated, both of d-axis and q-axis currents change slowly. Fig. 8(b) is the flux vector difference between the reference and adjustable models. The flux vector difference is defined as the distance between the flux positions estimated by the voltage model and current model, which is given by:

     (25)

Fig. 8. Well-tuned case for all of the stator parameters (600 rpm, 50 % load). (a) , , and currents. (b) .

(a)

(b)

The flux vector difference converges to zero when the parameters are well tuned. This implies that the adjustable model follows the reference model both on the d-axis and the q-axis.

Fig. 9(a) shows results of the detuned case of R_s. R_s is set to 1.15 times the actual value. In this situation, the relative error of L_m is -6.7 % and that of R_r is -0.7 % in the steady state. Finally, the detuned case of σL_s  is shown in Fig. 9(b). It is usually harder to tune σL_s than R_s. Thus, σL_s is set to -25 %, to have a higher error than the R_s case. In this case, the relative errors of estimation are 0.6 % and 2.5 % for L_m and R_r, respectively. In the detuned cases of the stator parameters, the errors of the estimated L_m and R_r at various speeds and loads are enumerated in Table II and Table III. In terms of their speed and load, their errors are different. However, these errors are not as obvious as these are low. For the detuned case of σL_s, equal errors appear as different speeds. In conclusion, the numerical results in Table II and Table III demonstrate the good robustness of the proposed algorithm for detuning the stator parameters.

Fig. 9. Detuned case of the stator parameters. (a) case (300 rpm, 80 % load). (b) case (1200 rpm, 30 % load).

(a)

(b)

TABLE II ERRORS IN THE ESTIMATED VALUES OF AND IN IN VARIOUS SITUATIONS (UNIT: %)

Speeds (rpm)	30 % load		50 % load		80 % load
300	5.0	-2.7	1.7	-4.4	-0.7	-6.7
600	2.4	-1.5	0.8	-2.4	-0.5	-3.7
1200	1.2	-0.8	0.4	-1.3	-0.3	-2.0

TABLE III ERRORS IN THE ESTIMATED VALUES OF AND IN IN VARIOUS SITUATIONS (UNIT: %)

Speeds (rpm)	30 % load		50 % load		80 % load
300	2.5	0.6	2.6	-0.6	2.8	-3.2
600
1200

Ⅶ. EXPERIMENTAL RESULTS

The information on the induction motor used in the experiment has been specified in Table I. The utilized DSP in the experiment is a TMS320F28335. The switching frequency of the PWM inverter is 5 kHz, and the sampling frequency of the line currents is 10 kHz. All of the settings for the control period are same as those for the aforementioned simulations. The utilized load motor is a 2.26 kW synchronous motor and it is set to run in the torque control mode to generate a specific torque. Before the drive implementation, in order to obtain accurate output voltages, the methods for dead time compensation in [23] were applied, where the switch compensation time is calibrated considering the dead time, the voltage drops of the power devices, etc. R_s was identified before the speed drive. R_s was 1.67 Ω at the beginning of the experiment. The initial settings of L_m and R_r are 202.5 mH and 0.35 Ω which have errors of approximate 50 %.

In the well-tuned case of R_s and σL_s, results of parameter identification are shown in Fig. 10(a). The two parameters are estimated correctly. These results correspond to Fig. 8(a) from the simulation results. The currents change a bit with the identification. The voltages and speed are plotted in Fig. 10(b). The speed remains constantly at 600 rpm during both the steady-state and transient-state under the identification algorithm. The voltages decrease as the parameters are being tuned. Fig. 10(c) shows the flux vector difference between the reference and the adjustable models. converges to zero asymptotically as L_m and R_r are being tuned.

Fig. 10. Well-tuned case for all of the stator parameters (600 rpm, 50 % load). (a) , and currents. (b) Voltages and speed. (c) .

(a)

(b)

(c)

Fig. 11 and Fig. 12 describe the results of the detuned case of R_s and σL_s, respectively. The two detuned cases show the similar results to Fig. 10, although errors of the stator parameters exist. Since the two parameters are tuned simultaneously, the currents and voltages change. Meanwhile, the speeds stay at the command speed. Data collected from 27 situations (speeds: 300/600/1200 rpm, loads: 30/50/80 %, and the tuned-states of R_s or σL_s) are shown in Fig. 13(a). This data shows consistent results under various conditions. As a result, these experimental results proves that the proposed algorithm is robust to the detuning of the stator parameters.

Fig. 11. case (300 rpm, 80 % load). (a) , and currents. (b) Voltages and speed.

(a)

(b)

Fig. 12. case (1200 rpm, 30 % load). (a) , and currents. (b) Voltages and speed.

(a)

(b)

Fig. 13. Sets of estimated parameters under various conditions. (a) Beginning of experimental activity. (b) Two hours after experimental activity.

(a)

(b)

Two hours after the speed drive operation, R_s was identified again. R_s increased a bit than the beginning, which was 1.78 Ω. In this situation, experiments in the selected condition were carried out repeatedly. Fig. 13(b) shows these results. The results for R_r have 6~7 % higher values than those for the previous experiments. L_m has values that are equal to those of the previous experiment.

In practical speed drive systems, coupled loads fluctuate for any reason. Thus, it is necessary to prove that the proposed method operates properly under such conditions. It is expected that the proposed algorithm can operate smoothly because it does not have any assumptions for the steady state, as can be seen in chapter V. In other words, proper operation of the algorithm can be guaranteed in states where the speed or current is not constant. In making these situations, a fluctuating load is generated, which has a constant component 50 % plus a sinusoidal component 20 % for the rated torque of the induction motor. Here, the frequency of the sinusoidal component is synchronized with the speed. Fig. 14 shows the experimental results at 300, 600 and 1200 rpm. The Q-axis current fluctuates to compensate the fluctuating load. The speed also fluctuates since load torque is not perfectly compensated. Despite this condition, the parameters are well identified without fluctuations. Thus, the proposed identification method may be used in practical speed drive systems with fluctuating loads.

Fig. 14. In cases of speed operation with fluctuating load. (a) At 300 rpm. (b) At 600 rpm. (c) At 1200 rpm.

(a)

(b)

(c)

Ⅷ. CONCLUSIONS

In this paper, a novel method for the identification of L_m and R_r has been proposed. In order to realize the proposed method, the MRAC principle is applied, and RLS estimation is utilized for estimating these parameters. Because a closed- loop Gopinath style flux observer is employed as the reference model in the adaptation mechanism, the proposed algorithm can operate at a sufficiently high frequency to get the complete reference model. However, the proposed algorithm has the following advantages:

- Simple structure and well-known algorithms: flux estimations, filtering calculations, and RLS estimation.

- The two parameters can be identified simultaneously.

- Independence of the two parameters: the detuning of L_m does not affect the estimation accuracy of R_r and vice versa.

- It guarantees effective operation in both the steady- state and the transient-state.

The proposed method was verified by simulation and experimental results where the parameters were estimated accurately through a speed control test with a load. Additionally, in detuned cases of the stator parameters and cases with a fluctuating load, experimental results demonstrated the excellence of the proposed algorithm in diverse situations.

ACKNOWLEDGMENT

This work was supported by the Korea Institute of Energy Technology Evaluation and Planning (KETEP) and the Ministry of Trade, Industry & Energy (MOTIE) of the Republic of Korea (No. 20174030201490).

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Gwon-Jae Jo was born in Daegu, Korea. He received his B.S. and M.S. degrees in Electrical Engineering from Kyungpook National University, Daegu, Korea, in 2013 and 2016, respectively, where he is presently working towards his Ph.D. degree in Electrical Engineering. His current research interests include AC motor drives and custom power devices.

Jong-Woo Choi was born in Daegu, Korea. He received his B.S., M.S. and Ph.D. degrees in Electrical Engineering from Seoul National University, Seoul, Korea, in 1991, 1993 and 1996, respectively. From 1996 to 2000, he worked as a Research Engineer at LG Industrial Systems Co., Korea. Since 2001, he has been a Professor in the Department of Electrical Engineering at Kyungpook National University, Daegu, Korea. His current research interests include static power conversion and electric machine drives.