*Code Algorithm*

The code is made using Lua programming language. The code is executed by FEMM4.2 software. The choice of Lua programming language to be used is due to its simplicity, and that it is adopted by FEMM4.2, which provides the FEA analysis in good accuracy. Figure 3 shows the code's algorithm. The code starts with inserting SRM optimization data. These data include the population size, problem variables, variables limits, and objective functions, to specify which objective function to maximize and which to minimize, maximum iterations limit and numbers of rotor and stator poles (*Pr* and *Ps*). Subsequently, solutions initialization are achieved for all solutions in the population by a random choice of variables within the search space that is stored in matrix *X* as in Equation (16). After that, the solutions are modified to satisfy constraints in Equations (18)–(21) by changing the values of variables that resulted from random choice. This change is to give the variable that is out of its limits one of the acceptable limits value. For example, if a variable is greater than maximum, then it is changed to the maximum value and vice versa. Subsequently, FEA Analysis is accomplished using FEMM4.2 software to calculate average torque, maximum stator and rotor poles flux densities and volume of iron. Next, remaining objective functions (*η* and *Wi*) are calculated using the results of FEA analysis. After that, non-dominated sorting is achieved and crowdingdistance is calculated for all fronts. Based on non-dominated sorting and crowding distance, the best and worst solutions are specified. Susequently, the termination condition is checked if the number of iterations exceeds the maximum limit or not. Finally, output of optimization is provided, which is the non-dominated front of all solutions. Note that the output also provides all solutions that have been produced for all iteration, which is beneficial to provide more solutions for designer to choose.

**Figure 3.** Flowchart of MO-Jaya algorithm.

## **5. Results and Discussions**

The evaluation of optimization algorithm performance may be mainly achieved by two factors that are computational time and candidates. The less computational time, the better is the performance for the same results of candidates. On the other hand, the better candidates, the better performance for the same computational time. In this paper, better candidates production are of significant interest than computational time. SRM design candidates with higher average toque and efficiency and lower iron weight are considered to be better candidates. Hence, it is expected that for a successful optimization process, most of the search (crowded area) ought to be in the areas that maximize both average toque and efficiency and minimize iron weight. For example, assuming that it us desired to maximize *both* of objective functions *f*1 and *f*2, the results of the optimization process that are the solution candidates should be at the upper right quarter, as shown in Figure 4. The same goes for other quarters in cases of minimizing both *f*1 and *f*2, maximize *f*1 and minimize *f*2, and maximize *f*2 and minimize *f*1. Additionally, it is expected that the optimization program will find more solutions in the correct quarter (direction) with increased iterations until the search area is fully covered.

**Figure 4.** Search directions in multi objective optimization.

The objectives of optimization in this paper are to maximize average torque (*Tav*) and efficiency (*η*) and minimize iron weight (*Wi*). The results of optimization process are considered in pairs, as shown in Figures 5 and 6. In Figure 5c, efficiency is increased and iron weight is decreased while iterations are increasing. The search direction is correct according to Figure 4. The search direction is also correct with respect to iron weight and average torque, as shown in Figure 5b. In Figure 5a, the search direction is not clearly obvious, as in Figure 5b,c, which is due to the non-linear relationships of dimensions and objective functions (efficiency and average torque). The program changes the direction to cover all the search area to provide a various groups of solutions.

In Figure 7, objective functions of SRM are shown with respect to the number of iterations. It can be seen that the objectives are conflicting with each other. The optimization program is made, such that the iterations are continued while the algorithm is not biased for any of the objectives by changing the best and worst solutions in the way mentioned earlier in Section 3.2. Hence, better solutions are found in all of search directions (objectives). Figure 8 shows how objective functions interact with each other and confirms the wide search area.

Every point represents a solution and some solutions are invalid due to dimensions constraints, as they cause a negative clearance between windings. The penalty for these solutions is to eliminate them by making their corresponding objective functions the worst of their values. Hence, the invalid solutions take zero average torque and efficiency and iron weight of the whole volume (2*πDLρi*), where *ρi*is the density of iron.

It can be seen that search direction is changing while iterations increases as the ups and downs in objective functions values indicate. In other words, the program is exploring new areas with more iterations. This is because of two reasons, the first reason is the random chosen ratios *<sup>r</sup>*(*<sup>i</sup>*, *j*, 1) and *<sup>r</sup>*(*<sup>i</sup>*, *j*, 2) in Equation (11). The second reason is the random choice of the best and worst solutions from the highest and lowest ranks (fronts). This is beneficial, as the algorithm does not repeat it self and provide more various solutions. However, a drawback is that optimal solutions (non-dominated front) are distributed over iterations. In other words, the best candidates do not exist in the last iteration exclusively. The designer then has to take this into consideration while choosing a design to implement.

For further evaluation, the optimization results by multi-objective Jaya algorithm (MO-Jaya) are compared with optimization results by non-dominated sorting genetic algorithm (NSGA-II) in [31]. Two optimization processes have the same constraints, objective functions, calculation methods, and common parameters (population size and maximum iterations). The results that are shown in Figures 5–7 in this paper are compared with results that are shown in Figures 4, 5k and 7 in [31]. The comparison can be summarized in the following point:


 **Figure 5.** Objective functions results for 8/6 SRM.

 torque.

(**c**) Iron weight and torque.

**Figure 6.** Objective functions results for 6/4 SRM.

**Figure 7.** Objectives with iterations for 8/6 and 6/4 SRM using MO-Jaya.

**Figure 8.** 3D representation of objective functions values through optimization process.

Four optimal solutions are chosen for both 8/6 and 6/4 configurations. Table 3 summarizes the selected designs and their corresponding objective functions. Most of designs do not belong to the last iteration, as they provide better characteristics. For 8/6 SRM configuration, design A1 achieves the highest average torque, A2 achieves highest efficiency at rated speed, A3 has the lowest iron weight, and A4 is a compromise design which gives a higher priority for average torque and efficiency. For the 6/4 SRM configuration, design B1 achieves both the highest average torque and efficiency at rated speed, B3 has the lowest iron weight, and B2 and B4 are the compromise designs. Figures 5 and 6 show the location of the selected designs among the remaining designs. It is worth mentioning that the resulted optimal designs by Mo-Jaya optimization method are close to these that result from NSGA-II for the same constraints. However, the Mo-Jaya method achieved better diversification than NSGA-II.

Table 4 shows the details of selected designs. It can be noticed that the parameters and dimensions of SRM are changed with iterations to produce better designs in such a way that matches with the SRM design experience. This indicates that the calculation methods of objective functions mainly succeeded to represent SRM. For example, when rotor pole length (*hr*) is increased and rotor yoke thickness (*bry*) is reduced, the impact on SRM is to produce average torque. This result matches with design experience, as the difference between anligned and unaligned inductances is increased and, hence, energy conversion increases Equation (4).

Torque per phase is shown in Figure 9 for 8/6 and 6/4 SRM configurations before and after optimization. The peak torque is increased in optimal designs (A1 and B1) over initial designs. It can be also noticed that torque ripples also increase. This is because torque ripples minimization is not considered as an objective for optimization. However, so far, the program succeeded in achieving what is requested and specified in the objectives. More work is taking place to include torque ripple minimization in optimization objectives in the future.

Further study on selected SRM design candidates are made to evaluate efficiency over a wide range of speeds Figure 10. It can be noticed that 6/4 SRM (B group) achieves higher efficiency values. This is due to lower iron losses that are caused by flux variation in magnetic circuit of SRM. Figure 11 shows the iron losses.


**Table 3.** Chosen optimal candidates.

**Table 4.** Parameters and objective functions values of the selected optimal designs.


**Figure 9.** Torque per phase of initial and optimal SRM designs.

**Figure 10.** Efficiency over wide range of speeds.

**Figure 11.** Core losses of selected optimal designs over wide range of speeds.
