*Code Algorithm*

The code is made using the Lua programming language. The code is executed using Lua Console in FEMM4.2 software. The choice of the 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. The code's algorithm is shown in Figure 3. First, SRM optimization data are entered. These data include the population size, problem variables, variables limits, objective functions, to specify which objective function to maximize and which to minimize the maximum iterations limit and numbers of rotor and stator poles (*Pr* and *Ps*). Then, solutions are initialized by random choice of variables within the search space area. After that, constraints in Equations (26)–(29) are maintained in this step by changing the values of variables resulted. Then, the FEA is accomplished using FEMM4.2 software to calculate average torque, maximum stator and rotor poles flux densities and volume of iron. After that, the results of FEA analysis is used to calculate the remaining objective functions (*η* and *Wi*). Then, non-dominated sorting is performed and crowding distance is calculated for all solutions. Next, the selection of the best designs to be used in crossover and mutation. Lastly, termination condition is checked such that if number of iterations exceeds the maximum limit the whole process is finished and the highest rank of all solutions (non-dominated front) is the given in the output of optimization process.

**Figure 3.** NSGA-II optimization program flowchart.

#### **5. Results and Discussions**

Both 6/4 and 8/6 SRMs are optimized using the same technique. Since there are three objective functions, it is difficult to show them all together in one figure to see the progress of the optimization process with generations. Hence, the objective functions are taken in pairs and shown as in Figure 4 for 8/6 SRM and Figure 5 for 6/4 SRM for more than 300 generations.

**Figure 4.** Objective functions results for 8/6 SRM.

These figures show the search direction or the optimization progress as more generations are produced. As we intended to maximize both average torque and efficiency and minimize iron weight, it is obvious that the crowded area (which indicates the majority of search) in Figure 4a, for example, exists in the upper right quarter (considering the axes limits). Wherein, higher values for both efficiency and average torque are sought. This means that the optimization program has searched a lot in the area of variables that produce candidates with more average torque and efficiency in the same time. Figure 4b shows the same concept with the difference that the objective functions are iron weight and efficiency. The program tries to minimize iron weight while maximizing efficiency but because of the complexity of the problem and the constraints, results have a unique shape. The program tries to achieve better candidates by searching right or left of the crowded area. The same goes for Figure 4c, replacing the efficiency in Figure 4b with average torque. For 6/4 SRM, the results represented by Figure 5 show the same features of optimization

as in 8/6 SRM. Figure 6 shows the candidates of final generation with the three objective functions. The results shown confirm the accuracy of the search direction. Moreover, it indicates the diversification of the method used as it shows variety in objective functions' values.

**Figure 6.** Objective functions 3D representation of the last generation (30 candidates).

The progress of optimization with generations for both 8/6 SRM and 6/4 SRM is shown in Figure 7. It can be seen that average torque and efficiency are maximized as more generations are produced and the iron weight is minimized at the same time.

**Figure 7.** Objective functions progress with number of generations.

Table 4 shows the best selected candidates among the first non-dominated front (also known as rank 1). Four candidates are selected for each configuration (set A for 8/6 SRM and B for 6/4 SRM). A1 and B1 achieve the maximum values of average torque •

and efficiency among the final generation candidates. A4 and B4 achieve the minimum iron weight. A2, A3, B2 and B3 are compromise designs, which satisfy each objective function in a certain degree. A1 and B1 designs are selected from these candidates for further investigation. All designs of the selected sets in Table 4 are shown in Figures 4–6. The values of objective functions, parameters, dimensions and other details for selected sets are shown in Table 5. From Table 5, it can be seen that the variation of dimensions to produce better designs matches with the SRM design experience, which indicate the accuracy of calculation methods. For example, the difference between aligned inductance (*La*) and unaligned inductance (*Lu*) is higher in designs of higher average torque. This result matches with design experience as the energy conversion increases with higher difference of flux level between aligned and unaligned positions. This is indicated by Equation (12). Other important observations indicated in Table 5 should be highlighted and they are:



**Table 4.** Candidates in first non-dominated front (rank 1).

Further investigations are made on the selected designs A1 and B1. The torque is shown in Figure 8 for constant phase current from an unaligned position to aligned positions for selected optimal designs. Note that the peak torque for B1 (6/4 SRM) is higher than that of A1 (8/6 SRM) due to the higher difference between aligned and unaligned inductances in 6/4 SRM. However, the average value of A1 (8/6 SRM) torque is higher than that of B1 (6/4 SRM) due to the increased number of phases in 8/6 SRM configurations.


**Table 5.** Parameters and objective functions values of the selected optimal designs.

Figure 9 shows the magnetic flux of both selected designs. It can be seen that the value of flux density in the stator pole is about 1.8 T, which represents the knee point of B-H curve for industrial steel used. The stator yoke flux density is obviously higher than stator flux density for 6/4 SRM. This is because the stator yoke thickness does not impact the objective functions strongly, that is, it can be neglected. Hence, the optimization program tends to decrease it to a minimum to ge<sup>t</sup> less weight of iron. To achieve a good overall SRM design, other objective functions must be added such as torque ripples, acoustic noise . . . and so forth. When these functions are added, the program will not reduce the stator yoke thickness to a minimum as it influences other objective functions negatively (increases acoustic noise for example).

In Figure 10, the efficiency of optimal selected designs for a range of speeds up to 10th of rated speed is shown. It can be seen that the values of efficiency for selected optimal designs are almost identical to the value of speed before 1300 rpm. This result is due to the lower core losses in this region as shown in Figure 11. The program was given the rated speed of 1000 rpm to calculate core losses and efficiency and then seek better values at same speed. It can also be seen that 6/4 SRM has a better efficiency profile than 8/6 SRM over a wide speed range, which is expected due to higher core loss values as demonstrated in [14]. SRM with 8/6 configuration has a higher number of poles than 6/4 SRM and hence flux changes are higher, which leads to core losses. Figure 11 shows the core losses of 8/6 SRM is much higher than that of 6/4 SRM. Figure 12 shows a comparison between calculated and FEA waveforms of flux density (*B*) of all sectors for the chosen optimal designs. It can be seen that the results of the calculation are very close to FEA waveforms. The calculation method is used to produce flux density (*B*) waveforms and to then calculate eddy currents losses as introduced in Section 2.8.

**Figure 10.** Efficiency of selected optimal designs A1 (8/6 SRM) and B1 (6/4 SRM) at different speeds.

**Figure 11.** Core losses of selected optimal designs A1 (8/6 SRM) and B1 (6/4 SRM) at different speeds.

**Figure 12.** The flux density waveforms in all sectors of optimal SRM designs A1 (8/6 SRM) and B1 (6/4 SRM) for one revolution at 1000 rpm.

The proposed techniques in [16,17,20,27] are considered for the evaluation of the methodology presented by this paper, as they studied the same SRM configurations or multi objective optimization. The methodology presented in this paper achieves high accuracy in analysis due to the use of FEA. The optimization technique performance has also shown successful progress towards the optimal. In [17], the same technique is used for a specific application. The values of efficiency and torque density are higher than what is presented in this paper. This is due to the generality of the approach in this paper. The approaches

in [20,27] are general, however, they use mathematical models in analysis to reduce computational time, which makes the process more complex and needs more analytic work before optimization starts. For torque density values, [27] achieves 1200–1580 N.m/m<sup>3</sup> while [17] reached 15,950–17,030 N.m/m3. In this paper, torque densities of 7780–8315 N.m/m<sup>3</sup> and 6238–6473 N.m/m<sup>3</sup> are achieved for 8/6 and 6/6 configurations, respectively. Efficiency values are 75%–80%, 86%–91% and 80%–85% in [17,20,27], respectively, while in this paper efficiency values of 80%–86% are achieved for most of the design candidates. The approach proposed in this paper has shown success in optimization, as objective function values indicate. Also, it can be used for almost any application if the suitable objective functions are added.
