Simplified Rotor and Stator Resistance Estimation Method Based on Direct Rotor Flux Identification

Abstract

Since parameter mismatch seriously impacts the efficiency and stability of induction motor drives, it is important to accurately estimate the rotor and stator resistance. This paper introduces a method to directly calculate the rotor flux that is independent of stator and rotor resistance and electrical angle. It is based on obtaining the rotor and stator resistance using the model reference adaptive system (MRAS) method. The method has a lower computation burden and less adaptation time when compared with other rotor resistance estimation methods. This paper builds three coordinate frames to analyze the rotor flux error and rotor resistance error. A number of implementation issues are also considered.

I. INTRODUCTION

Induction motor (IM) drives based on indirect field oriented control (IFOC) are popular in various industrial applications due to its dynamic performance and easy implementation. However, accurate flux orientation relies on knowledge of the motor parameters, especially the rotor resistance, which can drift with the operation temperature. There can be torque and flux errors, extra power losses, sluggish dynamics and so on [1]-[5], when a considerable error in the rotor resistance disrupts the precision field orientation, especially for medium-high volume motors with a small rotor resistance. Therefore, online rotor resistance identification is essential for IM drives.

There have been a lot of methods for online rotor resistance estimation, such as signal injection [6], [7], adaptive full-order observer (AFO) [8], [9], recursive least squares (RLS) [10], model reference adaptive system (MRAS) [11], [12] and so on. The main features of these methods are listed in Table I.

TABLE I FEATURES OF VARIOUS IDENTIFICATION METHODS

Among the methods listed above, MRAS is more attractive due to its simple structure and easy implementation [13]. MRAS consists of a reference model (independent of the variable), an adjustable model (correlated with the variable) and an adaptive mechanism. An adaptive mechanism, such as a PI controller, uses the error signal between two models to adjust the parameters. Depending on the chosen functional candidates, such as the rotor flux, stator voltage [14], electromagnetic torque [15] and reactive power [16], various MRAS-based rotor resistance estimation techniques have been proposed. A unified model was developed in [17], which shows that types of MRAS-based models are united in essence and share a unified mathematical expression. To avoid flux estimation, a rotor resistance adaptation real-time method utilizing reactive power was proposed in [13]. The sensitivity and stability are analyzed in detail with the Popov hyperstability concept. Although the system can converge to a real value, it adjusts the rotor resistance and rotor speed in real time, which can cause vibrations under the instantaneous load-change condition. In [15], a particular relationship between the sign of the reactive power error and the sign of the rotor resistance error was pointed out. In [18], a compensation for dead time and time delays was considered to reduce identification errors.

In this paper, the influence of errors in rotor resistance is investigated in detail. Depending on the application of equations in different frames, the real rotor flux of an IM, which does not suffer from integrator-related problems [19], [20], is directly calculated. The uniqueness of this method is that the rotor flux can be acquired directly and that it is independent of stator and rotor resistance and electrical angle. Then the rotor flux is used to adjust the rotor resistance and stator resistance. In order to eliminate reactive power errors, the voltage distortion caused by dead time and time delay is compensated. The scheme is simulated in Simulink and a prototype is developed to validate the technique. Extensive simulation and experimental results are presented to demonstrate the performance of the algorithm.

II. INFLUENCE OF ROTOR RESISTANCE ERROR

A. Vector Control System for Induction Motors

The dynamic model of an IM in the synchronous reference frame is given by (1), with the stator current and rotor flux selected as state variables.

\(P\left[\begin{array}{l} I_{s} \\ \psi_{r} \end{array}\right]=\left[\begin{array}{ll} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array}\right]\left[\begin{array}{c} I_{s} \\ \psi_{r} \end{array}\right]+\frac{1}{\sigma L_{s}} U_{s}\)       (1)

Where:

\(A_{11}=-\left[\frac{R_{s}}{\sigma L_{s}}+\frac{(1-\sigma) R_{r}}{\sigma L_{r}}\right] I, A_{12}=-\frac{L_{m}}{\sigma L_{s} L_{r}}\left(-\frac{R_{r}}{L_{r}} I+\omega_{r e} J\right)\)

\(A_{21}=\frac{L_{m} R_{r}}{L_{r}} I, A_{22}=-\frac{R_{r}}{L_{r}} I+\omega_{r e} J\)

\(I=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right], J=\left[\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right], \sigma=1-\frac{L_{m}^{2}}{L_{s} L_{r}}\)

The symbol ‘P’ is the differential operator.

The symbol ‘ω_re’ is the electrical angular velocity of the motor.

The symbols ‘L_s, L_r and L_m’ are stator inductance, rotor inductance and mutual inductance, respectively.

The rotor flux can be expressed as:

\(\psi_{r}=L_{m} I_{M}\)       (2)

The electromagnetic torque equation is as follows:

\(T_{e}=\frac{3}{2} p \frac{L_{m}}{L_{r}} \psi_{r M} I_{T}\)       (3)

The symbol ‘p‘ refers to the pole pairs of the motor.

The slip frequency ‘ω_f’ can be calculated as:

\(\omega_{f}=\frac{R_{r}}{L_{r}} \cdot \frac{I_{T}}{I_{M}}\)       (4)

In the case of the vector control system, the slip frequency relationship in (4) is used for calculating the rotor flux vector position as follows:

\(\theta_{e}=\int\left(\omega_{r e}+\omega_{f}\right) d t\)       (5)

Fig. 1 shows a typical scheme of an IM indirect vector control system. For a properly designed current loop in the MT reference frame, the stator currents \(I_{M}^{*}\) and \(I_{T}^{*}\) are assumed to keep tracking their target values I_Mtarget and I_Ttarget tightly. Therefore, in the following analysis, the measured current values are considered to be equal to their target value in the steady state. X^* represents variables obtained when the used rotor resistance is a wrong value \(R_{r}^{*}\). In particular, X^* = X such as \(I_{M}^{*}\) = I_M when \(R_{r}^{*}\) = R_r. This means that the IM runs in the ideal situation.

Fig. 1. Scheme of an IM vector control system.

B. Influence of Rotor Resistance Errors in Three Frames

The influence of rotor resistance errors will be analyzed in three coordinate frames. Suppose R_r is the actual rotor resistance and \(R_{r}^{*}\) is the rotor resistance used in the control system. The first frame can be called the control frame, marked as M^* - T^*. In this frame, the rotor flux is oriented in the wrong rotor resistance R^*. The second frame can be called the real frame, marked as M^c - T^c. In this frame, the rotor resistance is still oriented in the wrong value. The first and second frames are both built in the real control system. The third frame is regarded as an ideal frame, marked as M - T. This frame only exists when the control frame aligns with the real frame or \(R_{r}^{*}\) = R_r.

Since U_α, U_β and I_α, I_β are real values for an actual IM and I_α, I_β are the functional results of U_α, U_β in the real rotor resistance R in a control system, the second frame sets up a bridge between the first and the third frame, which needs to be discussed here. Fig. 2 shows the relationship between the first and second frames.

Fig. 2. Variable comparisons in the first and second frame. (a) Rr*>Rr. (b) Rr*<R_r.

Rotor flux orientation uses \(\omega_{se}^{*}\) as the electrical speed. Thus, U_α, U_β are achieved in \(\omega_{se}^{*}\), which satisfies:

\(\omega_{s e}^{c}=\omega_{s e}^{*}\)       (6)

In particular, when rotor speed is acquired accurately, the electrical speed satisfies:

\(\left\{\begin{array}{ll} \omega_{s e}^{c}=\omega_{s e}^{*}>\omega_{s e} & R_{r}^{*}>R_{r} \\ \omega_{s e}^{c}=\omega_{s e}^{*}<\omega_{s e} & R_{r}^{*}>R_{r} \\ \omega_{s e}^{c}=\omega_{s e}^{*}=\omega_{s e} & R_{r}^{*}>R_{r} \end{array}\right.\)       (7)

\(I_{s}^{*}\) in Fig. 2, which represents the stator current vector, is projected into the M^* - T^* and M^c - T^c frames. \(I_{M}^{c}\) is the field current in the second frame and real rotor flux can be obtained.

\(\psi_{r}^{c}=L_{\mathrm{m}} I_{M}^{c}\)       (8)

\(I_{M}^{*}\) is the field current in the first frame, which is assumed to track the reference value I_Mtarget in the control system. The rotor flux calculated in the first frame and real rotor flux in the second frame satisfy the following:

\(\psi_{r}^{*}=L_{m} I_{M}^{*}=L_{m} I_{M}^{c} \sqrt{\frac{I_{M}^{c 2}+I_{T}^{c 2}}{I_{M}^{c 2}+\left(\frac{R_{r}}{R_{r}^{*}}\right)^{2} I_{T}^{c 2}}}=\psi_{r}^{c} \sqrt{\frac{I_{M}^{c 2}+I_{T}^{c 2}}{I_{M}^{c 2}+\left(\frac{R_{r}}{R_{r}^{*}}\right)^{2} I_{T}^{c 2}}}\)       (9)

Resolving \(\psi_{r}^{*}\) to the M^c - T^c frame yields:

\(\left\{\begin{array}{l} \psi_{r M}^{*}=\psi_{r M}^{c} \frac{I_{M}^{c 2}+\frac{R_{r}}{R_{r}^{*}} I_{T}^{c 2}}{\left(\frac{R_{r}}{R_{r}^{*}}\right)^{2} I_{T}^{c 2}+I_{M}^{c 2}} \\ \psi_{r T}^{*}=\psi_{r T}^{c} \frac{I_{M}^{c} I_{T}^{c}-\frac{R_{r}}{R_{r}^{*}} I_{M}^{c} I_{T}^{c}}{\left(\frac{R_{r}}{R_{r}^{*}}\right)^{2} I_{T}^{c 2}+I_{M}^{c 2}} \end{array}\right.\)       (10)

The third frame is obtained when the first frame aligns with the second frame or \(R_{r}^{*}\) = R_r, and (10) takes a simplified form as:

\(\left\{\begin{array}{l} \psi_{r M}^{*}==\psi_{r M}^{c}=\psi_{r M}=L_{m} I_{M} \\ \psi_{r T}^{*}=\psi_{r T}^{c}=\psi_{r T}=0 \end{array}\right.\)       (11)

In the case of \(R_{r}^{*}\) ≠ R_r, subtracting (10) from (11), shows that the signs of the rotor flux errors are directly correlated with detuning of the rotor resistance as:

\(\left\{\begin{array}{l} \left(\psi_{r M}^{c}-\psi_{r M}^{*}\right)\left(R_{r}-R_{r}^{*}\right) \geq 0 \\ I_{T}^{c}\left(0-\psi_{r T}^{*}\right)\left(R_{r}-R_{r}^{*}\right) \geq 0 \end{array}\right.\)       (12)

The inequalities of (12) can explain that the field current and torque current in the second frame are larger than those in the first frame when rotor resistance satisfies \(R_{r}^{*}\) < R_r. In addition, it is the same in reverse.

Due to the error in the rotor flux, there must be some errors in the magnetic torque of the IM. Considering (3) and (10), the torque calculated in the first frame and the real torque in the second frame satisfy:

\(T_{e}^{*}=T_{e}^{c} \cdot \frac{I_{M}^{c 2}+I_{T}^{c 2}}{\frac{R_{r}^{*}}{R_{r}} I_{M}^{c 2}+\frac{R_{r}}{R_{r}^{*}} I_{T}^{c 2}}\)       (13)

Equation (13) shows that the torque error \(T_{e}^{*}\) / \(T_{e}^{c}\) is correlated with \(R_{r}^{*}\) / R_r and \(I_{T}^{c}\) / \(I_{M}^{c}\). Fig. 3 shows the relationships among the three variables.

Fig. 3. Error in electromagnetic torque.

C. Simulation of the Rotor Resistance Error Influence

Since all of the values in the first and second frame can be obtained in simulation, the validation is carried out in simulation. The parameters in the simulation, like those in the experiment, are shown in Table II.

TABLE II ACIM PARAMETERS

When the motor speed is kept at 100r/min, the updating process when \(I_{M}^{*}\) = 1A is presented in Fig. 4 and Fig. 5. The load torque is set at 0.4N·m, 0.7N·m and 1N·m at the times 0.4s, 1.2s and 1.7s, respectively. The difference between Fig. 4 and Fig. 5 is that Fig.4 simulates at \(R_{r}^{*}\) = 0.5R_r while Fig. 5 simulates at \(R_{r}^{*}\) = 2R_r.

Fig. 4. Comparison of the electrical angle, field current, rotor flux and torque when \(R_{r}^{*}\) = 0.5R_r.

Fig. 5. Comparison of the electrical angle, field current, rotor flux and torque when \(R_{r}^{*}\) = 2R_r.

To verify the theory, one point was chosen to test. \(I_{M}^{c}\) = 0.53, \(I_{T}^{c}\) = 2.37, \(\psi_{r}^{c}\) = 0.16, \(\psi_{r}^{*}\) = 0.29, \(T_{e}^{c}\) = 1.03 and \(T_{e}^{*}\) = 1.81 can be obtained in the steady state when \(R_{r}^{*}\) = 2R_r and \(T_{e}^{c}\) = 1 in Fig. 5. The results calculated by the feedback values are:

\(\psi_{r}^{*} / \psi_{r}^{c}=1.81\)       (14)

\(T_{e}^{*} / T_{e}^{c}=1.76\)       (15)

And the results calculated by equations (9) and (13) are:

\(\psi_{r}^{*} / \psi_{r}^{c}=\sqrt{\frac{I_{M}^{c 2}+I_{T}^{c 2}}{I_{M}^{c 2}+\left(\frac{R_{r}}{R_{r}^{*}}\right)^{2} I_{T}^{c 2}}}=1.87\)       (16)

\(T_{e}^{*} / T_{e}^{c}=\frac{I_{M}^{c 2}+I_{T}^{c 2}}{\frac{R_{r}^{*}}{R_{r}} I_{M}^{c 2}+\frac{R_{r}}{R_{r}^{*}} I_{T}^{c 2}}=1.75\)       (17)

Formula (14) is equal to (16) and formula (15) is equal to (17). Therefore, the rotor resistance error theory is quantitatively verified. When \(R_{r}^{*}\) > R_r, Fig. 4 and Fig. 5 show that the electrical angle in the first frame is ahead of the real value in the second frame and that the electrical speed is the same in the two frames, and the field current and rotor flux in the first frame are less than the real values in the second frame. This is the same in reverse. The magnetic torque is not only correlated with rotor resistance error.

III. PARAMETER ESTIMATION ALGORITHM

A. Rotor Flux Estimation Theory

In the M-T axes synchronously rotating reference frame, the dynamic model of induction machines can be expressed with the following ‘voltage model’:

\(\left\{\begin{array}{l} U_{M}=\left(R_{s}+\sigma L_{s} P\right) I_{M}+\left(L_{m} / L_{r}\right) P \psi_{r M}-\omega_{s e}\left[\sigma L_{s} I_{T}+\left(L_{m} / L_{r}\right) \psi_{r T}\right] \\ U_{T}=\left(R_{s}+\sigma L_{s} P\right) I_{T}+\left(L_{m} / L_{r}\right) P \psi_{r T}+\omega_{s e}\left[\sigma L_{s} I_{M}+\left(L_{m} / L_{r}\right) \psi_{r M}\right] \end{array}\right.\)       (18)

In steady state, the following is considered:

\(P I_{s}=0, P \psi_{r}=0\)       (19)

Equation (18) becomes:

\(\left\{\begin{array}{l} U_{M}=R_{s} I_{M}-\omega_{s e}\left[\sigma L_{s} I_{T}+\left(L_{m} / L_{r}\right) \psi_{r T}\right] \\ U_{T}=R_{s} I_{T}+\omega_{s e}\left[\sigma L_{s} I_{M}+\left(L_{m} / L_{r}\right) \psi_{r M}\right] \end{array}\right.\)       (20)

For the vector control, the M-axis is aligned with the rotor flux vector. Therefore:

\(\psi_{r T}=0, \psi_{r M}=L_{m} I_{M}\)       (21)

Substituting (21) into (20) yields:

\(\left\{\begin{array}{l} U_{M}=R_{s} I_{M}-\omega_{s e} \sigma L_{s} I_{T} \\ U_{T}=R_{s} I_{T}+\omega_{s e} L_{s} I_{M} \end{array}\right.\)       (22)

Equation (22) points out the relationship between the real voltage and the real current in the function of the real rotor resistance R in the steady state. There is no doubt that it can be applied to the third frame. Since the values in the second frame are all real and in the function of the real rotor resistance, it can be considered to be one special condition of \(R_{r}^{*}\) = R_r. Equation (22) can be applied to the second frame and the following is obtained:

\(\left\{\begin{array}{l} U_{M}^{c}=R_{s} I_{M}^{c}-\omega_{s e}^{c} \sigma L_{s} I_{T}^{c} \\ U_{T}^{c}=R_{s} I_{T}^{c}+\omega_{s e}^{c} L_{s} I_{M}^{c} \end{array}\right.\)       (27)

The reactive power Q of the IM can be expressed as:

\(Q=\left|U_{s}\right| \cdot\left|I_{s}\right| \cdot \sin \varphi\)       (24)

φ is the angle between the stator voltage vector and the stator current vector.

The voltage vector and rotor current vector can be written as:

\(\left\{\begin{array}{l} \dot{U}_{s}=U_{\alpha}+j U_{\beta}=U_{M}^{*}+j U_{T}^{*}=U_{M}^{c}+j U_{T}^{c} \\ \dot{I}_{s}=I_{\alpha}+j I_{\beta}=I_{M}^{*}+j I_{T}^{*}=I_{M}^{c}+j I_{T}^{c} \end{array}\right.\)       (25)

According to the definition of the reactive power, Q can be rewritten as:

\(Q_{1}=U_{\beta} \cdot I_{\alpha}-U_{\alpha} \cdot I_{\beta}\)       (26a)

\(Q_{2}=U_{T}^{*} \cdot I_{M}^{*}-U_{M}^{*} \cdot I_{T}^{*}\)       (26b)

\(Q_{3}=U_{T}^{c} \cdot I_{M}^{c}-U_{M}^{c} \cdot I_{T}^{c}\)       (26c)

\(Q=Q_{1}=Q_{2}=Q_{3}\)       (26d)

Substituting (23) into 26(c) yields:

\(Q=\omega_{s e}^{c} L_{s}\left(I_{M}^{c 2}+\sigma I_{T}^{c 2}\right)\)       (27)

According to the projecting relationship in the first and second frame, the following can be obtained:

\(I_{s}^{c 2}=I_{M}^{c 2}+I_{T}^{c 2}=I_{s}^{* 2}=I_{M}^{* 2}+I_{T}^{* 2}\)       (28)

Substituting (28) and (7) into (27) yields the real rotor flux:

\(I_{M}^{c}=\sqrt{\frac{Q_{2}-\sigma \omega_{s e}^{*} L_{s} I_{s}^{* 2}}{\omega_{s e}^{*} L_{s}(1-\sigma)}}\)       (29)

All of the variables in (29) can be acquired in one steady-state condition. The field current in the second frame can be obtained without any adaptation. Moreover, the solving process eliminates the stator resistance. The change of the stator resistance does not affect the result. \(\omega_{se}^{*}\) is the electrical speed used in the control system. Equation (6) is workable even when the rotor speed is not accurate. Equation (29) can be applied to conditions where there is rotor speed error.

B. Rotor Resistance Estimation Theory

Substituting (29) into (8) yields the real rotor flux:

\(\psi_{\mathrm{r}}^{c}=L_{m} \sqrt{\frac{Q_{2}-\sigma \omega_{s e}^{*} L_{s} I_{s}^{* 2}}{\omega_{s e}^{*} L_{s}(1-\sigma)}}\)       (30)

According to (9) and (30), the real value of R_r can be determined:

\(R_{r}=\sqrt{\frac{Q_{2} L_{m}^{2} I_{s}^{* 2}-\sigma \omega_{s e}^{*} L_{s} I_{s}^{* 4}}{\omega_{s e}^{*} L_{s}(1-\sigma) \psi_{r}^{* 2} I_{T}^{c 2}}-\frac{I_{M}^{c 2}}{I_{T}^{c 2}} \times R_{r}^{*}}\)       (31)

However, equation (31) includes high order terms that make the result unstable when the control system bears with jitters. It can just be used to estimate the probable value of R_r.

According to (1), it is possible obtain the rotor flux observer based on the current model:

\(P \hat{\psi}_{r}=\left(-\frac{R_{r}}{L_{r}} I+\omega_{r e} J\right) \hat{\psi}_{r}+\frac{L_{m} R_{r}}{L_{r}} I_{s}\)       (32)

This section adopts the MRAS method to identify rotor resistance. Equation (30) is used as a reference model and (32) is used as an adjustable model. An adaptive mechanism uses the rotor flux error to adjust the rotor resistance. Considering voltage compensation, Fig. 6 shows the identification scheme.

Fig. 6. Scheme of rotor resistance identification.

C. Stator Resistance Estimation Theory

Since equation (27) eliminates variable stator resistance, rotor flux identification is independent of the stator resistance. It is possible to utilize a rotor flux observer based on a voltage model to estimate stator resistance. According to (1), a rotor flux observer based on the voltage mode is shown as:

\(P \hat{\psi}_{r}=\frac{L_{r}}{L_{m}}\left(U_{s}-R_{s} I_{s}-\sigma L_{s} P I_{s}\right)\)       (33)

Since stator current acquisition contains zero drift, the integral of the stator current can cause integral saturation and the calculated rotor flux error. A high pass filter module is taken to filter the zero drift in this section. However, the high pass filter changes the amplitude and angle of the rotor flux. Thus, a compensation should be used.

Suppose:

\(e_{s}=U_{s}-R_{s} I_{s}\)       (34)

Equation (33) can be rewritten as:

\(P \hat{\psi}_{r}=\frac{L_{r}}{L_{m}}\left(\frac{1}{s} e_{s}-\sigma L_{s} I_{s}\right)\)       (35)

Suppose:

\(\frac{1}{s} e_{s}-\sigma L_{s} I_{s}=\sin \left(\omega_{s e} t+\theta_{l}\right)\)       (36)

Considering the zero drift of the stator current, it is possible to obtain:

\(\hat{\psi}_{r}=\frac{L_{r}}{L_{m}}\left(\sin \left(\omega_{s e} t+\theta_{1}\right)+\int a\right)\)       (37)

In (37), θ₁ is the initial angle and ‘a’ is the zero drift value. When cascading a high pass filter, (37) can be written as:

\(\hat{\psi}_{r}=\frac{L_{r}}{L_{m}}\left(\sin \left(\omega_{s e} t+\theta_{1}\right)+\int a\right) \frac{s}{s+\omega_{c}}\)       (38)

In (38), ω_c is the cut-off frequency of the high pass filter. Suppose:

\(A=\left(\sin \left(\omega_{s e} t+\theta_{1}\right)+\int a\right) \frac{s}{s+\omega_{c}}\)       (39)

It is possible to obtain:

\(\frac{d A}{d t}+\omega_{c} A=\omega_{s} \cos \left(\omega_{s e} t+\theta_{1}\right)\)       (40)

Solving (40) yields:

\(A=C e^{-\omega_{c} t}+\omega_{s} \frac{\sin \left(\omega_{s e} t+\theta_{2}\right)}{\sqrt{\omega_{c}^{2}+\omega_{s}^{2}}}+\frac{a}{\omega_{c}}\)       (41)

Since the zero drift ‘a’ is small, a / ω_c can be ignored. Considering A(0) = 0, it is possible to obtain the value of ‘C’. The exact solution of ‘A’ can be expressed as:

\(A=-\omega_{s e} \frac{\sin \theta_{2}}{\sqrt{\omega_{c}^{2}+\omega_{s e}^{2}}} e^{-\omega_{c} t}+\omega_{s e} \frac{\sin \left(\omega_{s e} t+\theta_{2}\right)}{\sqrt{\omega_{c}^{2}+\omega_{s e}^{2}}}\)       (42)

In (42), θ₂ is the angle after conversion. With an increase in time, ‘A’ can be written as:

\(A=\omega_{s e} \frac{\sin \left(\omega_{s e} t+\theta_{2}\right)}{\sqrt{\omega_{c}^{2}+\omega_{s e}^{2}}}\)       (43)

Without a high pass filter, ‘A’ is shown as:

\(A=\sin \left(\omega_{s e} t+\theta_{1}\right)\)       (44)

Through a comparison of (43) and (44), the amplitude and phase position of ‘A’ have changed after the high pass filter. Typically, the amplitude without a filter is \(\sqrt{\omega_{c}^{2}+\omega_{s e}^{2}} / \omega_{s e}\) times bigger than that with a filter. Therefore, the amplitude of the rotor flux with a filter should be multiplied by \(\sqrt{\omega_{c}^{2}+\omega_{s e}^{2}} / \omega_{s e}\) to get the real value.

Since (29) can be applied when the rotor speed is unknown and (33) does not contain a variable rotor speed or rotor resistance, the stator resistance can be obtained without a rotor speed or rotor resistance. The scheme of the stator resistance identification is shown in Fig. 7.

Fig. 7. Schematic of stator resistance identification.

IV. IMPLEMENTATION ISSUES

In the above section, the actual stator voltages are assumed to be equal to their command values. However, in order to prevent short circuits of the phase legs, the insertion of dead time is carried out [21]. In addition, there is a time delay of the digital controller. The insertion of dead time in space vector pulse width modulation (SVPWM) and time delays can cause phase voltage distortion or the zero-current clamping phenomenon, degrading the quality of stator voltage and stator current. Thus, analysis and compensation of voltage distortions are necessary.

Since voltage distortion has been discussed many times, it will not be derived in detail here. The phase voltage error is shown as:

\(\left\{\begin{array}{l} \Delta U_{\alpha}=\frac{4}{\pi} \Delta U\left\{\sin \theta+\sum_{k=1}^{\infty}\left[\frac{\sin (6 k-1) \theta}{6 k-1}+\frac{\sin (6 k+1) \theta}{6 k+1}\right]\right\} \\ \Delta U_{\beta}=\frac{4}{\pi} \Delta U\left\{-\cos \theta+\sum_{k=1}^{\infty}\left[\frac{\cos (6 k-1) \theta}{6 k-1}-\frac{\cos (6 k+1) \theta}{6 k+1}\right]\right\} \end{array}\right.\)       (45)

Where:

\(\Delta U=U_{\text {bat}} \frac{T_{D T}}{T_{P W M}}\)

The symbol ‘T_DT’ is the dead time period.

From (45), it can be known that voltage distortion mainly includes the 6k±1th harmonics. The phase relation between the voltage distortion and current is shown in Fig. 8.

Fig. 8. Phase relation between the voltage distortion and current.

Since the α-axis phase current is 90 degrees ahead of the β-axis phase current, it can be supposed that:

\(\left\{\begin{array}{l} I_{\alpha}=I_{m o d} \sin \theta \\ I_{\beta}=-I_{m o d} \cos \theta \end{array}\right.\)       (46)

The symbol ‘I_mod’ is the current amplitude.

By substituting (45) and (46) into (26a), the reactive power error Q_err can be expressed as:

\(\frac{Q_{e r r}}{Q}=\frac{4}{\pi} \frac{T_{D T}}{T_{P W M}} \frac{1}{\sin \varphi} \sum_{k=1}^{\infty} \frac{12 k \sin 6 k \theta}{(6 k-1)(6 k+1)}\)       (47)

Suppose T_DT = 3 ㎲, T_PWM = 100 ㎲ and sinφ = 0.5, it is possible to obtain the reactive power error in Fig. 9.

Fig. 9. Error in the reactive power.

The angle φ between the voltage and current is correlated with the relationship between the reactive power and the active power. When increasing the load torque, the active power increases and φ decreases, which can cause a lager reactive power error. Phase voltage compensation is carried out to enhance the accuracy and robustness.

In the simulation and experiment, the phase voltage error is obtained by phase voltage reconstruction [21], which requires the precise duty cycle of every phase. Take phase U as an example:

\(\text {Duty}_{-} U=\frac{\int V_{u 0}}{V_{D C} \cdot T_{P W M}}\)       (48)

Where Duty_U is the real duty cycle and T_PWM and V_u0 are the PWM period and terminal voltage of phase U, respectively. The phase voltage reconstruction and phase voltage error can be obtained after acquisition of the duty cycle.

Considering the phase voltage compensation, the schemes of the rotor and stator resistance are shown in Fig. 6 and Fig. 7.

V. SIMULATION AND EXPERIMENTAL RESULTS

A. Simulation Results

An induction motor drive with IFOC is established in Simulink to verify the theory proposed in this paper. The parameters of the motor are shown in Table II in section II. Since the rotor resistance varies slowly with time, it is presumed to be constant. The period of the PWM is set at 100μs.

Fig. 10 shows the reactive power Q with and without voltage compensation. The rotor resistance is set to the true value. For a better demonstration of their effects, the load torque is set to 40%, 70% and 100% of the rated load. When compared with Fig. 10(b), Fig. 10(a) fluctuates roughly. When the load torque increases, Q bears with an intensive fluctuation.

Fig. 10. Reactive power without and with compensation.

In the following simulations, the induction motor operates with voltage compensation. The rotor resistance is initially set to a wrong value. The observer is activated when the motor starts up to the commanded speed and load torque.

In order to verify the identification response when the load torque changes, the load torque is set to 40%, 70% and 100% of the rated load in Fig. 11 and Fig. 12. Fig. 11(a) and Fig. 12(a) show that the rotor flux observer can keep tracking the rotor flux feedback from the induction motor model. Since the proposed method can only be applied to the steady-state condition, the identification results fluctuate when the control system is unstable. Fig. 11 and Fig. 12 show that the identification results can converge to the real value after the load torque changes. The theory is verified by the simulation in this section.

Fig. 11. Identification results when \(R_{r}^{*}\) = 10Ω and ω_r=100rpm.

Fig. 12. Identification results when \(R_{r}^{*}\) = 4Ω and ω_r=100rpm.

Fig. 13. Identification results when \(R_{r}^{*}\) = 10Ω, and ω_r=100rpm and the rated load torque = 50%.

Fig. 14. Identification results when  \(R_{r}^{*}\) = 4Ω, and ω_r=100rpm and the rated load torque = 50%.

B. Experimental Results

The experiment takes a TMS320F28335 as the digital signal processor (DSP). The feedback circuit in [21] is applied in the experiment. The motor and IGBT parameters are shown in Table II and Table III, respectively.

TABLE III PARAMETERS OF AN IGBT (POWERREX PS21765)

The experimental setup is shown in Fig. 15. Two induction motors are used to act as a load motor and a drive motor. A torque sensor is used to obtain accurate torque in real time. Unless mentioned, the test conditions in the following experiments are the same as those in the simulations.

Fig. 15. Experimental setup.

Fig. 16 shows reactive power Q without and with voltage compensation. When compared with Fig. 16(b), Fig. 16(a) fluctuates roughly. In Fig. 17 and Fig 18, the load torque is set to 40%, 70% and 100% of the rated load at 10s, 25s and 45s, respectively. In Fig. 17-Fig. 21 the adaptation scheme is tested at three speeds, two rotor resistances and three different load conditions. The rotor resistance and stator resistance are adjusted to real value under the above conditions.

Fig. 16. Reactive power without and with compensation.

Fig. 17. Identification results when \(R_{r}^{*}\) = 10Ω and ω_r=100rpm.

Fig. 18. Identification results when \(R_{r}^{*}\) = 4Ω and ω_r=100rpm.

Fig. 19. Identification results when \(R_{r}^{*}\) = 4Ω, ω_r=500rpm and the rated load torque = 50%.

Fig. 20. Identification results when \(R_{r}^{*}\) = 10Ω, ω_r=1000rpm and the rated load torque = 50%.

Fig. 21. Identification results when \(R_{r}^{*}\) = 4Ω, ω_r=1000rpm and the rated load torque = 50%.

VI. CONCLUSION

In this paper, a novel parameter identification method was proposed. Salient features of the system are its simplicity and directness, since the real rotor flux can be calculated without any adaptation. The influence of the rotor resistance error is analyzed in three coordinate frames and real rotor flux is derived from the relationships among the second frame and other frames in the steady-state condition of an induction motor. Based on estimated rotor flux, the rotor resistance and stator resistance are identified using MRAS. Moreover, the rotor flux and stator resistance identification methods are independent of the rotor resistance. Voltage distortion which can severely affect reactive power calculations are compensated. Simulations and experiments were carried out under several typical conditions. The obtained results verify the identification scheme.