*3.2. Objective Functions*

The objectives of SRM optimization depends on the application and its conditions. For general purpose SRM, average torque and efficiency are needed to be maximized

and weight is to be minimized. In some applications other objectives (i.e. torque ripples, acoustic noise, vibrations...etc.) are important.

#### **4. NSGA-II for SRM Design Optimization**

Non-dominated sorting genetic algorithm (NSGA-II) is one of the best and most popular techniques, which is used in multi-objective optimization problems. It depends mainly on the concept of dominance to judge the individuals of the same generation. This concept takes all of objective functions in consideration with their direction to minimize or maximize. It is required to check each individual with the rest on the dominance basis. As in [26], assuming *X*1, *X*2 are two vector individuals, *m* is the number of objective functions, if *X*1 ≺ *X*2 (*X*1 dominates *X*2) is true, Pareto dominance conditions must all be true and they are:

$$f\_j(X\_1) \not\models f\_j(X\_2) \forall j = \{1, \ldots, m\} \tag{30}$$

$$f\_j(X\_1) \lhd f\_j(X\_2) \exists j = \{1, \ldots, m\}.\tag{31}$$

The non-dominated individual is the individual that is not dominated by any of the other individuals in the population of a certain generation. After that, the individuals are sorted in the form of groups depending on their degree of dominance. These groups are called non-dominated sets or simply fronts. The first front is the group consists of the best (non-dominated) individuals. Multi-objective optimization by NSGA-II eliminates the need of weights in multi-objective optimization by a single function (*f* = *w*1 *f*1 + *w*2 *f*2 + ··· *wn fn*). Moreover, it eliminates the conflict between weights (as the summation of weights must equal to 1) and hence a wider search area is covered.

Crowding distance is a criterion used to compare between solutions, which are in the same non-dominated front. The more space there is around a solution, the higher is the crowding distance. Therefore, solutions with a high crowding distance should have a rank better than those with a low crowding distance in order to maintain diversity in the population. Crowding distance is computed in the same manner as mentioned in [26]. Crowding distance is computed for each solution using Equation (32). If solutions of the same non-dominated fronts are numbered with their associate objective functions in lists, crowding distances are calculated as follows:

$$CD\_j = CD\_j + \frac{f\_m^{j+1} - f\_m^{j-1}}{f\_m^{\max} - f\_m^{\min}} \tag{32}$$

where *j* is a solution in the sorted list, *fm* is the objective function value of *m*th objective, *f max m* and *f min m* are the population-maximum and population-minimum values of *m*th objective functions.

SRM optimization using NSGA-II requires the setting of variables, objective functions, constraints, population size and number of maximum generations. Population size is preferred to be high in order to enhance the possibility of finding better individuals. Since *Tav* calculation by FEMM4.2 requires the SRM magnetic circuit to be analyzed several times, the computation time must be taken into consideration while deciding the population size. Hence, population size is chosen to be 30 candidates. A maximum generations number is used as a termination condition. It is chosen to be more than 300 generations. The constraints of the SRM design problem are mainly the limits of variables and the clearance between windings as shown in Table 2.
