*3.1. Flux-Linkage Decrease*

The rotor flux-linkage decreases with the rise of magne<sup>t</sup> temperature and may fall by 5–10%, depending on the magne<sup>t</sup> material. The degradation of magnets due to demagnetization may result in an additional 10%; therefore, the maximum decrease of flux-linkage can be considered as 20% [12]. A decrease of the rotor flux-linkages causes mitigation of the magnetic component of the motor torque. Thus, if the reluctance component is not changing, the MTPA angle increases, which is illustrated in Figures 4 and 5.

**Figure 4.** MTPA trajectories in *idiq* plane at different flux-linkages.

**Figure 5.** Dependence of the MTPA angles γ on the stator current amplitude at different flux-linkages.

#### *3.2. Motor Inductances Decrease*

The motor inductances decrease due to the steel saturation, which is caused by the amplitude of the applied stator current. Their variation significantly depends on the machine design and may be as high as 70–80% in synchronous reluctance motors (SynRM) and PM assisted synchronous reluctance motors (PMASynRM). In IPMSMs the maximum decrease of full inductance is typically about 50–60% [8].

At the same time, the reluctance torque of a synchronous motor depends on the difference between the direct and quadrature inductances; therefore, analysis of the inductance variation on the MTPA trajectory is more complicated, because both inductances vary simultaneously. Furthermore, the motor may saturate in the direct and quadrature directions in a different way, causing the inductance difference Δ*L=Ld* − *Lq* to rise, even if both inductances are falling. A good example of this phenomena is a motor under test, which is considered in this paper. The flux path of this electrical machine along the direct axis is saturated faster than the path along quadrature axis. As a result, the direct axis inductance decreases faster than the quadrature inductance and the inductance difference increases. However, with the following increase of the stator current, the quadrature inductance decreases, and the inductance difference reduces, reaching lower values than the initial values for the zero current. This phenomenon is illustrated in Figure 6, which demonstrates inductance variation with respect to the stator current.

**Figure 6.** Motor inductances vs. current at the MTPA trajectory.

In this figure U, S and P denote unsaturated, saturated and partly saturated zones, respectively. It can be observed, that the inductance difference in the unsaturated (with rated parameters) zone Δ*Lu*, is less than the inductance difference in the partly saturated zone <sup>Δ</sup>*Lp*; however, it is higher than the inductance difference in the saturated zone Δ*Ls*. As a result, the true MTPA trajectory of this machine, which reflects complicated dependencies of motor inductances on the stator current, Figure 6, has complicated waveforms, which are demonstrated in Figures 7 and 8. In Figure 8, γ*u* and γ*s* correspond to the MTPA angle trajectories drawn for the unsaturated and saturated motor inductances, respectively, and γ stays for the true MTPA angle.

**Figure 7.** Test motor MTPA trajectories in *idiq*plane.

**Figure 8.** Dependence of the test motor MTPA angles γ with respect to the stator current amplitude.

It should be noted that some researchers consider unsaturated and saturated MTPA curves as boundary conditions for motor operation and design algorithm using this assumption [17]. However, the provided example demonstrates that such boundaries are

incorrect for such motors. The MTPA trajectory may leave the area limited with curves for unsaturated and saturated conditions, and this phenomenon should be taken into account.

#### **4. Proposed Method for Design of Constraints**

In order to solve the abovementioned problems, a new algorithm for the design of constraints for MTPA seeking techniques has been developed and considered in this paper. This idea will be explained using a test motor as an example and the corresponding constraints for an MTPA trajectory seeking algorithm are constructed. For higher convenience, the MTPA trajectories are designed in the *idiq* plane and for an MTPA angle γ, simultaneously. Therefore, the proposed technique can be easily adapted for control schemes involving MTPA formulation based on the Equations (6)–(8).

The proposed method is demonstrated in Figure 9 and includes the following steps. Initially, the direct and quadrature inductances variations with the stator current increase are measured, and the corresponding dependence of the inductance difference Δ*L* is calculated. These measurements are repeated for several motor samples and the resulting data are averaged. After that, the MTPA trajectory for the measured inductances is plotted, i.e., curve "A" in Figures 10 and 11. At the next stage, the maximum possible decrease of the rotor flux-linkage is calculated, and the corresponding curve is plotted. For the test motor, which uses NdFeB magnets, the motor designers estimated the maximum temperature decrease is 3% and the maximum aging and demagnetization degradation is 5%, which results in a maximum reduction of 8%. The corresponding MTPA trajectories are depicted as curve "B" in Figures 10 and 11. After that, the maximum possible deviation of motor parameters from their rated values at the stage of production is taken into account. This deviation is not significant and is caused by assembling inaccuracies, materials from different lots, etc. The factory quality assurance engineers reported that the possible variation of the flux-linkage is less than 1%, while the inductance deviation may be as high as 12%. At the same time, the character of inductance dependencies, demonstrated in Figure 6, is almost unchanged and deviation results only in scaling of those curves. At the next stage, the previously constructed curves "A" and "B" are modified using obtained information on the maximum deviation of motor parameters, which results in curves "C" and "D" in Figures 10 and 11, respectively. These curves define the zone, limiting the possible location of the true MTPA trajectory for all motors in the series.

After the area containing the MTPA trajectory is defined, it must be extended with the gaps necessary for the proper operation of the MTPA seeking technique and stable detection of the torque extremum corresponding to the MTPA condition. The term "gap" relates to the disturbed parameter, and its dimension is the same; however, implementation of the constraints depends on the MTPA technique and typically is performed in the *idiq* or *Isγ* planes. These gaps depend on the exact MTPA tracking algorithm and have to enclose the area, which fully includes border trajectories and deviation from them caused by the injection of additional signals. The MTPA algorithm involved in the experiments, was a seeking technique, which permanently disturbs the phase angle of the stator current and measures its amplitude, trying to minimize it [33]. This algorithm can stably detect the maximum of the torque curve, when it lies in the middle of the disturbance step Δ *γ*. Therefore, the gap, necessary for proper operation of this technique, which can guarantee the correct detection of extremum, is half of the disturbance angle, which was 2◦. Thus, the curves "C" and "D" have to be moved at this gap angle, which results in "E" and "F" in Figures 10 and 11, respectively. These curves, "E" and "F", define our desired constraints, which can improve the performance of seeking an MTPA algorithm.

These curves could be implemented as a look-up table (LUT), approximating polynomial or a set of splines; however, the best way of approximation depends on the complexity of the constraint curves.

The proposed method of constraints design was developed for operation with a series of motors, where inductances could be measured in advance. However, it could be easily adapted for self-commissioning drives, which operate without prior knowledge of motor

parameters. In this case, at the stage of parameter identification, the control system has to additionally identify dependence of the inductance difference on the stator current. After that, the curve "A" can be constructed. The construction of the curve "A" is made using data from Figure 6 by substituting inductances as functions from the current magnitude and the current magnitude itself into (6). Then evaluation of (5) allows to obtain both current components in *idiq* plane. If it is desired to evaluate MTPA angle *γ* with respect to the current magnitude, then Equation (9) is used.

At the next stage, the maximum variation of the flux-linkage has to be defined. Since this information is unavailable, the high-border estimation can be used: 12–15% for NdFeB magnets and 25–30% for ferrite magnets. Thus, the curve "B" can be constructed. The higher margin values increase response time of MTPA algorithms; therefore, it is desired to decrease the margins as much as possible. The recommended values of the flux linkage deviation came from the worst motor prototypes known to the authors and cover the overwhelming majority of motor drives. However, if a developer is confident that the maximum possible deviation of the flux linkage is less than the recommended values, the margins have to be decreased. Since self-commissioning routine tunes the drive for operation with the exact sample of motor, the curves "C" and "D", which define motor parameter variation at manufacturing, could be skipped. After that, the curves "E" and "F" can be constructed by moving the curves "A" and "B" at the gap angle, which is equal to the half of disturbance angle for the test algorithm.

**Figure 9.** Flowchart of constraints design algorithm.

**Figure 10.** MTPA constraints design in *idiq* plane.

**Figure 11.** MTPA constraints design for MTPA angle γ.
