*2.2. Optimization Process*

In the optimization process, a stator of a standard induction machine of 5.5 kW with the parameters listed in Table 1 was employed. The stator geometry was kept fixed during the optimization process. Based on the number of stator slots and poles, the number of rotor flux-barriers could be identified which was selected to be three per pole [34,35]. Twelve rotor parameters, *θ*b1, *θ*b2, *θ*b3, *W*b1, *W*b2, *W*b3, *L*b1, *L*b2, *L*b3, *p*b1, *p*b2, and *p*b3, sketched in Figure 1 were considered during the optimization process. To avoid the conflicts in the obtained geometry, some constraints were made as shown in Figure 1 and Table 2. As mentioned before, the main core of this paper is to study the influence of considering the current angle during the optimization process on the final optimal geometry of the SynRM. Therefore, in this research, different ranges of the current angle were considered (five cases) as in Table 3. The ranges of the current angles were selected based on the fact that the current angle of the maximum torque of the SynRMs equaled 45◦ (i.e., d-axis current = q-axis current) when neglecting the saturation effect. Nevertheless, when considering the saturation effect, the current angle deviated from 45◦. Therefore, in this paper, we tried to enforce different ranges of the current angle to around 45◦

to determine the impact on the final optimal geometry; this will be shown in the next paragraphs. Although the range of case 5 locates within the case 4 range, there were different optimal geometries obtained based on the two cases. This was why we were trying to narrow the search region of the current angle as in case 5 and to increase this range as in case 4 and even to keep the current angle fixed as in case 3.


**Table 1.** Parameters of the SynRM.

**Table 2.** Rotor variables upper and lower limits.


**Table 3.** Range of current angle for different cases.


The hybrid PSOGWO algorithm presented before was implemented to obtain the optimal rotor geometrical parameters and the current angle of each case in order to maximize

the output torque and minimize the torque ripple of the machine. The cost function of the optimization is given as follows:

$$\text{cost function} = \, ^T\_r + \frac{1}{T\_{av}} \tag{10}$$

where, *Tr* and *Tav* are the torque ripple in percent and the average torque of the SynRM.

The flow chart of the optimization loop is described in Figure 2. The finite element model (FEM) of the machine, in which the equations that represent the machine were solved numerically, was coupled with the PSOGWO technique to obtain the optimal geometry [1]. The losses were determined as in [8]. Later on, FEM is used to evaluate the performance of the obtained optimal machine.

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**Figure 2.** Flow chart of the optimization process.

As mentioned before, five different ranges of the current angle were considered (see Table 3). For each case, the range of the current angle was set and the optimization process was completed. The number of designs for each case was 210. Figure 3 shows the variation of current angle versus iteration number during the optimization process for different cases. The cost function versus the iteration number is reported in Figure 4. Figure 5 shows some performance indicators of the SynRM for different cases. Notice that the results in this section were obtained at different current angles. Therefore, it was not possible to compare the performance indicators of the five cases. However, this will be done in the next section. Moreover, it was proved from Figure 5 that the average torque and the power factor were greatly affected by the value of the current angle in a specific case (between different designs), while the impact on the torque ripple was lower. Figure 6 and Table 4 reveal that the iron volume of the obtained rotor geometry depended on the considered current angle during the optimization process. Figure 6 shows that the third case, which considered a fixed value for the current angle (45◦), gave the largest rotor iron volume, while the fourth case, which considered sufficient range of current angle variation in which the current angle of maximum torque and minimum torque ripple of the SynRM existed, gave the lowest iron volume of the rotor. The rotor iron volume of the third case was about

11% higher than that of the fourth case. This meant that the inertia of the SynRM based on the fourth case was lower resulting in a fast-dynamic machine.

**Table 4.** Final optimal geometry for the SynRM for different case studies.


**Figure 3.** Variation of current angle versus iteration number for different cases.

**Figure 4.** Cost function at different iteration number of the optimization technique.

**Figure 5.** (**a**) Average torque, (**b**) torque ripple, and (**c**) power factor versus iteration number for different cases.

**Figure 6.** Rotor iron volume versus iteration number for different cases.

Figures 3–6 show that the steady-state response of the optimization process was not delayed considering the current angle, while in some cases, the response became even faster. The optimization process of cases 1 and 4 reached its steady-state after about 80 and 60 iterations respectively compared to about 70 iterations for the third case. In contrast, the fifth case had a slower performance as shown in the zoomed view of Figure 4.

Figures 7–10 and Table 4 show the final optimal geometry of the rotor flux-barrier angles, lengths, widths, and positions for different cases. In Figure 7, it is found that the flux-barrier angles were greatly varied when considering different ranges of current angle during the optimization process. For example, in the first case (with current angle range from 30◦: 40◦), the flux-barriers angles changed by about 2◦ to 6◦ compared to the optimal geometry proposed in [26] and by about 2◦ to 3.5◦ compared to the third case. However, when the current angle was kept fixed to 45◦ in the third case during the optimization process, the obtained optimal geometry for this case was different compared to the optimal geometry presented in [26] which used a current angle equal to 56.50◦. This proved that the optimized geometry was sensitive to the chosen value of the current angle for fixed current angle cases.

**Figure 7.** Optimal geometry of flux barrier angles for different cases, (**a**) case 1, (**b**) case 2, (**c**) case 3, (**d**) case 4 and (**e**) case 5.

**Figure 8.** Optimal geometry of flux barrier widths for different cases, (**a**) case 1, (**b**) case 2, (**c**) case 3, (**d**) case 4 and (**e**) case 5.

 **Figure 9.** Optimal geometry of flux barrier lengths for different cases, (**a**) case 1, (**b**) case 2, (**c**) case 3, (**d**) case 4 and (**e**) case 5.

In addition, the optimal dimensions of the flux-barriers widths were also varied with current angles as shown in Figure 8. The flux-barriers optimal widths, *Wb1, Wb2,* and *Wb3* were changed by about 1.54, 0.75, and 0.7 mm respectively for the first case compared to their values in the third case. In addition, the flux-barriers optimal lengths, *Lb1, Lb2,* and *Lb3* were changed by about 0.40, 1.82, and 0.88 mm respectively for the first case compared to their values in the third case as shown in Figure 9. Moreover, Figure 10 shows that the flux-barriers optimal positions, *pb*1, *pb*2, and *pb*3 were changed by about 0.65, 1.40, and 1.58 mm respectively for the first case compared to their values in the third case.

**Figure 10.** Optimal geometry of flux barrier positions for different cases, (**a**) case 1, (**b**) case 2, (**c**) case 3, (**d**) case 4 and (**e**) case 5.

#### **3. Performance Analysis of SynRM**

The performance of the optimal geometry of the rotor of the three-phase SynRM for different case studies was studied and compared using finite element magnetic simulations. The optimal geometry for each case is shown in Table 5. Figure 11a shows the output power of the three-phase SynRM for different cases at rated conditions (speed = 3000 rpm and RMS current = 12.23 A) and at different current angles. It was been found from Table 5 and Figure 11a that the first case gave the highest output power and the second case gave the lowest output power at the rated condition and at the optimal current angle. The optimal current angle was the angle that maximized the output power. The optimal current angle was 52.11◦ for both the first, the second, and the third case as shown in Figure 11a and Table 5, while it was 56.8◦ for the other cases. The output power in the first case was 5.65% higher than the second case. Moreover, the first case had about 3.32% higher output power

compared to the third case. Note that the current angle in the third case was fixed during the optimization process while the effect of the current angle was considered in the first case as discussed in the previous section.

**Table 5.** Performance of the optimal geometry for different case studies of the three-phase SynRM using finite element model (FEM) simulation.


**Figure 11.** (**a**) Motor output power and (**b**) torque ripple at different current angles and at rated conditions for the optimal geometry of different cases.

Figure 11b shows the torque ripple of the three-phase SynRM for different cases at the rated conditions and at different current angles. The torque ripple decreased with the increase in current angle till it reached its minimum value then it increased again. It was found that the torque ripple had the lowest value in the fourth case, about 5.85%, while the fifth case gave the maximum value of the torque ripple: about 10.58%. The torque ripple for the first case was 7.4%. However, the torque ripple in the third case which used a fixed current angle during the optimization process was about 7.9%. The chosen value of the current angle for the fixed current angle cases significantly affected the obtained torque ripple at the optimal current angle. This was highly obvious in [26], which used a current angle equal to 56.5◦ and the obtained torque ripple with the optimal angle in [26] was about 12%.

To summarize, the much higher output power and lower torque ripple of our work compared to [26] justified research of the current angle in the geometrical optimization process, as was the goal of this paper.

Figure 12 shows the power factor and the saliency ratio for the optimal geometry for different cases studied at rated conditions and at different current angles. It is noted that the fourth case had the highest value of the power factor. It was 0.6628. This was due to its higher optimal current angle compared to the first case. The power factor of the first case was 0.6297. Figure 12b shows that the first case gave the highest saliency ratio and the second case gave the lowest saliency ratio. In addition, the saliency ratio of the first case was about 11.7% higher than its value in the third case. The distribution of flux density at the same instant of the optimal geometry is shown in Figure 13 for the various study cases.

It was been found that the rotor geometry calculated from the first case had less saturated area compared to other studied cases.

**Figure 12.** (**a**) Motor power factor and (**b**) saliency ratio at different current angles and at rated conditions for the optimal geometry of different cases.

**Figure 13.** Flux density distribution of the optimal geometry for different cases.

From the previous analysis and discussion, it was evident that considering the current angle during the optimization process of SynRMs was beneficial. Besides, it also observed that the range of the current angle played a role in the maximum output torque and torque ripple value.
