*2.10. Hysteresis Losses*

Referring to [31], the hysteresis losses can be calculated for various sectors of SRM. The flux density wave-forms depend on the phase current waveform and the speed of the motor. The flux density wave-forms calculated in this paper rated the speed of 1000 rpm and control is by a single pulse voltag. Note that the phase current has the peak of six ampere for all SRMs design candidates.

## **3. Jaya Optimization Method**

The population-based meta-heuristic optimization algorithms can be classified to two groups, and they are the evolutionary algorithms (EA) and swarm intelligence (SI) based algorithms. All of these algorithms have the same basic structure that is explained by Figure 2. Firstly, the initialization of solutions in the beginning is made mainly by random choice of variables. Then, the objective function values are obtained and evaluated. After that, selection of the best solutions is made to use these solutions in the production of new solutions. Finally, the termination condition of the process is checked if true, and then end or else continue.

Optimization techniques differ from each other by the different methods that are used to accomplish these steps. However, all of population-based meta-heuristic optimization algorithms have a common limitation, which is the different parameters that are required for proper working.

**Figure 2.** Flowchart of general optimization algorithm.

Referring to [30], the Jaya algorithm was introduced in 2016 by Ravipudi Venkata Rao. "Jaya" is a Sanskrit word that means victory or triumph. The algorithm is simple to implement and it does not require tuning of any parameters. In Jaya algorithm, the initial solutions are randomly generated within the search space. After that, the solutions are updated using Equation (11).

$$A(i+1,j,k) = A(i,j,k) + r(i,j,1) \left( A(i,j,b) \right)$$

$$-|A(i,j,k)| \Big| -r(i,j,2) \Big( A(i,j,w) \Big)$$

$$-|A(i,j,k)| \Big) \tag{11}$$

where *b* and *w* represent the index of the best and worst solutions in the population. *i*, *j*, *k* are the index of iteration, variable, and candidate solution, respectively. *<sup>A</sup>*(*<sup>i</sup>*, *j*, *k*) is the *j*th variable in *i*th iteration of *k*th solution candidate. *<sup>r</sup>*(*<sup>i</sup>*, *j*, 1) *and <sup>r</sup>*(*<sup>i</sup>*, *j*, 2) are random generated ratios in the range of [0, 1] to ensure good diversification.

#### *3.1. Single Objective Jaya Algorithm*

In single objective optimization, the required is to maximize or minimize a single function. Thereby, for two solution candidates a *better* solution is whether greater or smaller in value. For objective function *f*(*<sup>x</sup>*1, *x*2, *x*3, ··· , *xm*), which has *m* variables and *n* population size, the solutions are represented by a data structure i.e., matrix as in Equation (12). Each solution may be represented by a column of different variables. The objective function (*f*(*X*)) can be represented by a matrix (*F*) that has one row and *n* columns in the case of single objective optimization, as in Equation (13). For multi objective optimization problem with number of *q* objective functions, matrix *F* is of *q* rows and *n* column as in Equation (14).

$$X = \begin{bmatrix} x\_1^1 & x\_1^2 & x\_1^3 & \cdots & x\_1^n \\ x\_2^1 & x\_2^2 & x\_2^3 & \cdots & x\_2^n \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ x\_m^1 & x\_m^2 & x\_m^3 & \cdots & x\_m^n \end{bmatrix} \tag{12}$$

$$F = \begin{bmatrix} f\_1^1 & f\_1^2 & f\_1^3 & \cdots & f\_1^n \end{bmatrix} \tag{13}$$

$$F = \begin{bmatrix} f\_1^1 & f\_1^2 & f\_1^3 & \cdots & f\_1^n \\ f\_2^1 & f\_2^2 & f\_2^3 & \cdots & f\_2^n \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ f\_q^1 & f\_q^2 & f\_q^3 & \cdots & f\_q^n \end{bmatrix} \tag{14}$$

Solution matrix (*X*) is updated using Equation (11) based on the *best* and *worst* solutions obtained by comparing all of the solution candidates with each other.

#### *3.2. Multi Objective Jaya Algorithm*

Solutions in multi objective Jaya algorithm are updated using the same Equation (11), and the results of objective functions are represented in Equation (14). However, nondominated sorting approach and crowding distance computation mechanisms are to be used in order to handle the conflicting objectives of optimization problem. It is essential Jaya algorithm to obtain the *best* and *worst* solutions in order to produce new solutions using Equation (11).

In multi objective optimization problem, obtaining the best and worst solutions is not straightforward as in single objective optimization. After non-dominated sorting is achieved and crowding distances are computed for solutions of the same rank (front), the best solution would surely be in the first rank and the worst solution would be in the last one. However, the solutions of the same rank cannot be compared with each other

and, hence, the crowding distance is used to decide the best and worst solutions among their ranks. The crowding distance is an indicator of the degree of diversity of solutions in same front; hence, solutions with higher crowding distance are preferred, which enables covering a wider search area. In [30], the solution with highest crowding distance among first rank solutions is considered to be the *best* and the solution with the lowest crowding distance among last rank solutions is considered to be the *worst*.

In this paper, the other method is used to decide the best and worst solutions. The best solution is chosen randomly from the first rank, this choice is changed four times until all population solutions are updated. In the same manner, the worst solution is chosen randomly from the last rank. This method achieves a wide variety of solutions by distributing the priority of search among different search directions. Finally, the designer has the option to choose the best design that serves the application among best ranks of solutions.
