*4.2. Simulated Results*

Simulations were performed with the Saber simulator. Figure 5 shows the simulated results at 1200 rpm. In the IRFOC, the d-axis current, referred to as the magnetic current, is kept constant in the below-based speed region. According to Equation (5), the rotor flux should be kept constant. A step load is set at 2.0 s with 30 Nm and at 4.0 s with 60 Nm. Figure 5a–c depicts the simulated results when the rotor resistance *R*∗ *<sup>r</sup>* is used in the slip angular velocity calculation is set as 0.5*Rr*, 0.8*Rr* and 1.5*Rr*, respectively. The rotor field-oriented angle is compensated from 1.0 s. The rated flux level is set to 0.73 Wb based on the motor parameters. The flux level varies because of the inaccurate rotor field-oriented angle without compensation. This effect can lead to the degradation of dynamic and stable performance. After compensation, we observe that the flux intensity under the three

simulation conditions could reach 0.73 Wb at 2.0 s. The flux level could be kept almost constant during the load step.

**Figure 5.** Flux and compensated angle at 1200 rpm with 30 Nm and 60 Nm step loads at 2.0 s and 4.0 s, respectively. (**a**) *R*∗ *<sup>r</sup>* = 0.5*Rr*; (**b**) *R*<sup>∗</sup> *<sup>r</sup>* = 0.8*Rr*; (**c**) *R*<sup>∗</sup> *<sup>r</sup>* = 1.5*Rr*.

According to Equation (18), the predictive model needs stator resistance and inductance. Inductance is almost constant. The variation in the stator resistance is neglected in the proposed algorithm. To verify that the neglect is accepted, a simulation was performed. In the simulation, the stator resistance was changed from 0.374 Ω to 0.748 Ω linearly, and the rotor resistance was set to 0.5*Rr*. A step load is also set at 2.0 s with 30 Nm and at 4.0 s with 60 Nm. Compared with the simulated results in Figure 5a, the rotor flux and compensated angle indicate little difference, as shown in Figure 6. The flux level with angle error compensation, regardless of whether the stator resistance is changed, is much better than that without compensation. Although the maximum deviation of the compensated angle is almost 2.5 rad when the stator resistance changed to twice the nominal resistance, the deviation of the rotor flux level is only 0.03 Wb. In the application, this small difference of the rotor flux level could be neglected, and the algorithm proposed here is almost not affected by the variation in the stator resistance.

**Figure 6.** Rotor flux and compensated angle at 1200 rpm with the variable stator resistance.

In the IRFOC, the q-axis current reflects the developed torque when the motor runs at a constant speed. Therefore, the q-axis current should be proportional to the torque. That is, the q-axis current with a 60 Nm load should be twice the value with a 30 Nm load at 1200 rpm. The q-axis current with different loads is presented in Figure 7. After compensation, the q-axis current is changed from approximately 15 A to 30 A when the load rises from 30 Nm to 60 Nm regardless of the *R*∗ *<sup>r</sup>* set. However, with no compensation, the q-axis current is different from the same load and is not proportional to the torque. The simulated current is compared in Table 2.

**Figure 7.** The q-axis current comparative waveforms at 1200 rpm with a 30 Nm load at 2.0 s and 60 Nm load at 4.0 s. (**a**) Current with compensation; (**b**) Current without compensation.



In Table 2, *R*∗ *<sup>r</sup>* is the rotor resistance, which is used to calculate the slip angular velocity in the program; *Rr* is the actual resistance of the induction machine; *iqs*<sup>1</sup> and *iqs*<sup>2</sup> are the values of the q-axis current when the loads are 30 Nm and 60 Nm, respectively. If the rotor field orientation is accurate, then the ratio of *iqs*<sup>2</sup> and *iqs*<sup>1</sup> should be 2. It can be seen that the q-axis current is not proportional to the load when the rotor resistance is not the actual value. After compensation, the q-axis current is almost proportional to the load.

Figure 8 shows the simulation results at 3000 rpm. A step load at 20 Nm was set at 2.0 s. When *R*∗ *<sup>r</sup>* = 0.5*Rr*, the speed could no longer be kept at 3000 rpm without compensation. This is different from the simulation result at 1200 rpm shown in Figure 7 because the voltage is limited to the supply torque current in the field-weakening region if the rotor flux level is not sufficiently reduced, as shown in Figure 8b. The actual flux without compensation when *R*∗ *<sup>r</sup>* = 0.5*Rr* was much higher than the normal level. Therefore, the voltage could not supply enough q-axis current when the speed was 3000 rpm, and then the speed was decreased. This finding means that an inaccurate field-oriented angle can affect the maximum output torque of induction machines. This results of q-axis current can also be seen in the following experimental results at 1200 rpm. Although the flux levels with compensation are not constant, the variable in Figure 8a is much smaller than that without compensation, as shown in Figure 8b.

**Figure 8.** Rotor flux, q-axis current, and d-axis current comparative waveforms at 3000 rpm with a 20 Nm load at 2.0 s. (**a**) Rotor flux and d/q current with compensation; (**b**) Rotor flux and d/q current without compensation.

Figure 9 compares the waveform with and without the proposed compensation. In the simulation, the rotor resistance began to change from 0.268 Ω to 0.536 Ω linearly at the instant of 2.0 s during the following 2 s interval. Then, the rotor resistance was changed back to 0.268 Ω linearly at the instant of 4.0 s during the next 2 s interval. After compensation, the torque, rotor flux, and q-axis current are almost the same. There are some fluctuations in the flux, especially when the resistance was changed instantly. This change is mainly because the compensation of the proposed algorithm requires some time to realize. In practice, the rotor resistance cannot be changed so fast. The proposed algorithm has enough time to regulate the slip coefficient. The simulation at 3000 rpm yielded similar results, as shown in Figure 10.

**Figure 9.** Rotor flux, d-axis, and q-axis current comparative waveforms at 1200 rpm with a 60 Nm load.

**Figure 10.** Rotor flux, d-axis, and q-axis current comparative waveforms at 3000 rpm with a 15 Nm load.
