**1. Introduction**

The switched reluctance motor (SRM) is the type of motor that has saliency in both stator and rotor without permanent magnets or windings on rotor [1]. SRM develops electromagnetic torque based on variation of reluctance values for rotor position change with respect to phases when they are switched on. SRM provides several merits compared to other types of electric machines [2]. For instance, the topology of SRM is simple and very robust. Moreover, the power density, efficiency and torque output of SRM are high over a wide speed range [3–5]. The previous merits have increased the research efforts recently and made SRM preferred for high speed applications [1,6]. However, the torque ripple of SRM is the major problem that results in a high noise and variation. The latter can be improved by both control and design [7]. The control of this machine plays an essential role in the operation and hence it is required to overcome its challenges, which differ depending on the application [8–10].

**Citation:** El-Nemr, M.K.; Afifi, M.M.; Rezk, H.; Ibrahim, M.N. Finite Element Based Overall Optimization of Switched Reluctance Motor Using Multi-Objective Genetic Algorithm (NSGA-II). *Mathematics* **2021**, *9*, 576. https://doi.org/10.3390/math9050576

Academic Editor: David Greiner

Received: 29 January 2021 Accepted: 23 February 2021 Published: 8 March 2021

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The SRM construction has a lot of geometric parameters. The design is achieved by specifying all of these parameters values. Since SRM has salient poles in both stator and rotor, a wide range of geometric parameters combinations exist for a certain design objective. Searching for the best design is then not a simple task. Therefore, it is necessary to obtain the geometrical parameters that achieve the required objectives (e.g., maximum torque and efficiency and minimum volume and cost) in the best way (optimum design).

Optimization is a general term used to describe types of problems and solution techniques that are concerned with the best ("optimal") allocation of limited resources in projects. The problems are called optimization problems and the methods optimization methods [11,12]. Initially random values of variables are chosen for a number of solutions then the objective function values are evaluated for all solutions and classified from best to worst. The algorithm produces other solutions (variables) from these classified as the best .

If one objective function is desired to optimize, it is called single objective optimization. If more than one objective functions are desired to optimize simultaneously, it is called multi objective optimization. In this context, a set of solutions is obtained where their objective values form what is referred to as the Pareto front or non-dominated front. For the same Pareto front, all solutions are equally good because there is no way of telling which one is better or worse. In other words, all solutions in the same Pareto front are the optimal solutions (for optimal Pareto front) of the problem in a multi-objective sense [13].

In [14], the optimization of SRM design was made considering a certain ratio between the length of SRM core to the pole arcs of stator and rotor. The stator and rotor pole arcs were then varied between the limits that achieve self-starting not causing negative torque as this reduces the total developed torque. With every variation in stator and rotor pole arcs the objective functions—which were average torque and torque per volume—were calculated. The arc's values were chosen based on a compromise between the average torque and the torque per volume values. However, this work assumed fixed ratios for the lengths and arcs and did not study the other values, which give other designs.

In [15], the design optimization of a switched reluctance motor (SRM) by using a combination of two-dimensional electromagnetic and thermal finite-element analysis, three-dimensional correction factors and computer search techniques were presented.

The sub-problem approximation analysis was initially performed to locate an approximate optimum in the feasible design space, and then the first-order method was used to perform the final search. The core losses were calculated from a 2-D finite element analysis (FEA), based on a pre-calculated Fourier series of the flux density distribution in the SRM with typical phase currents.

In [16], a method of the optimization design with multi-objectives for switched reluctance motors for electric vehicle (EV) applications was proposed. From the requirements of EVs on electric motors in [16], three objective functions were chosen to optimize. They are the average torque, the average torque per copper loss and the average torque per motor core volume. The stator and the rotor pole arc angles are selected as the optimized parameters in this paper. The optimized parameters are only the stator and rotor pole arcs.

In [17], a multi-objective optimization for 16/20 SRM design and control were introduced based on a non-dominated sorting genetic algorithm intended for high volume traction applications. The proposed methodology considers a lot of parameters as variables for optimization process, also it considers the optimal firing angles (on and off angles) as an objective function in addition to frequently used objective functions like average torque, efficiency and torque ripples. The optimization of firing angles has the advantage of achieving minimum size of motor for specific requirements. The firing angles are optimized for this design by trying 100 different combinations of turn-on and turn-off angles to ge<sup>t</sup> the highest average torque and efficiency while concurrently minimizing the torque ripple. The proposed optimization framework succeeded to achieve the optimal geometry design for the special application intended for motor to be used.

Hayashi and Miller [18] represented the different flux density waveforms in matrices form and calculated eddy-current losses and hysteresis losses separately which was used in this paper for core losses calculations as will come later.

SRM's geometric parameters have an indirect and non-linear relationship with performance indices, that is, efficiency and average torque. Hence, sensitivity analysis on SRM geometric parameters is usually made as in [19–22] to reduce the complexity of the optimization process. The sensitivity analysis is to study the degree of influence of optimization problem's variables on the objective functions. Most influencing variables are only considered in optimization in order to reduce computational time. However, eliminating some of variables in optimization problem eliminates some of the indirect influence of these variables on the performance indices and makes the optimization limited to the specified objective functions and variables. Therefore, the method presented in this paper enables the optimization of 11 dimensions independently in order to include all possible design candidates which are within search area. Note that only seven dimensions are optimized in this paper as the remaining four are specified by application constraints (outer diameter *Do*, axial length *L* and shaft diameter *Dsh*) or for practical reasons (air gap length *g*).

Various methods of analyzing SRM include magnetic equivalent circuit (MEC), FEA and regression methods exist [1,23]. In this paper, FEA is used for its accuracy. Multi objective optimization of SRM design is achieved by the non-dominated sorting genetic algorithm method (NSGA-II). The program of optimization is made in Lua script to run from FEMM4.2 software. The FEA is performed each candidate design evaluation. The three objective functions average torque ( *Tav*), efficiency (*η*) and iron weight ( *Wiron*) are chosen to be optimized. Numerical methods are used to perform integration and differentiation on flux density waveforms to calculate eddy current losses as demonstrated later on this paper. The results of optimizations are compared and verified.

## **2. Design of SRM**

The design procedure of switched reluctance machine starts with specifying the available dimensions from space constraints, for example, frame size, shaft size and axial length then continues until all other dimensions are obtained. The number of stator and rotor poles are specified at the beginning as well. In the conventional analytic design methods, the inductance in the aligned and unaligned is calculated. Using the values of inductances in both aligned and unaligned positions the average torque is calculated. This step is repeated with modified values of the main dimensions until the requirements are justified. The number of turns per phase is calculated for every modification of dimensions such that the flux density doesn't saturate in stator poles for normal operation. This is demonstrated in Figure 1. Many other characteristics can be calculated such as efficiency, volume, weight and torque ripple. In this section, the SRM variables are discussed and the methods of characteristics calculations are emphasized.
