**1. Introduction**

The switched reluctance machine (SRM) is the type of machines that develop output torque due to reluctance variation without using permanent magnets or rotor excitation. The switching of phases is made according to position of rotor in such a way to produce torque (induce voltage in generation mode). Reluctance variation happens with rotation as a function of certain geometric parameters. The currents of SRM are mainly in the form of pulses. The flux inside the machine is not sinusoidal. All of the previously mentioned facts of SRM give this type of machines its unique features. The SRM has shown attractive characteristics, such as simple and robust construction, low manufacturing cost, and high efficiency, over wide range of speeds [1,2]. SRM's construction simplicity and low manufacturing cost have motivated both researchers and manufacturer.

**Citation:** Afifi, M.; Rezk, H.; Ibrahim,M.; El-Nemr, M. Multi-Objective Optimization of Switched Reluctance Machine Design Using Jaya Algorithm (MO-Jaya). *Mathematics* **2021**, *9*, 1107. https:// doi.org/10.3390/math9101107

Academic Editor: David Barilla

Received: 12 April 2021 Accepted: 6 May 2021 Published: 13 May 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

However, the pulsative behaviour of currents results in torque ripples that are considered a problem of SRM and a challenge. The saliency of poles and non-sinusoidal wave-forms of flux in SRM result in noisy operation and radial vibrations. In addition, SRM structure is not simple from geometric perspective due to the wide range of probabilities for dimensions of a certain design. Moreover, the relationships between SRM's geometric parameters and performance indices are indirect and non-linear. Hence, the search for a good design is difficult despite the search for the best (optimal) design.

Optimization is the search for the best solution (the optimal solution) for a certain problem [3]. Various types of optimization algorithms exist, including traditional optimization techniques that have many limitations and advanced population-based meta-heuristic algorithms [4–6]. Optimization is needed with a switched reluctance machine (SRM) design to reach better designs. Sensitivity analysis is the study of the degree of influence of each parameter on the final objectives of the design. Subsequently, the most influencing parameters are chosen to be optimized to save computation time. Because of the large number of geometric parameters or design parameters of SRM, sensitivity analysis is usually made to make the optimization process less complicated, as in [7–9].

It is essential to have a mathematical model of SRM in order to evaluate its performance and include this evaluation in the optimization process. The performance of SRM is mainly evaluated by rated torque, torque per weight, torque per volume, rated speed, and efficiency. Other quantities can be added to give more detailed evaluation, such as torque ripples, acoustic noise, mechanical vibrations, and maximumtemperature rise, and so forth. All of these performance indices are calculated by various methods that differ from each other according to their accuracy and computational time. These method can be classified to numerical methods, such as finite elements analysis (FEA) and boundary element method (BEM), and analytical methods, such as curve-fitting methods, magnetic equivalent circuits (MECs), and Maxwell's-equation-based approaches [1]. Numerical methods provide high accuracy with the cost of its high computational time. Analytical methods can be simplified to achieve a fast calculation process. However, in most cases, this results in a reduction of accuracy. FEA is commonly used in SRM modelling to achieve accurate results, as in [10–14]. Magnetic equivalent circuit (MEC) is faster than FEA, as shown in in [15–17]. However, it is less frequent because of its reduced accuracy. Fuzzy logic and regression methods are also used, as shown in [18,19]. However, they are less reliable and have complex structures when compared with FEA and MEC. Each performance index (i.e., rated torque, torque ripples and so forth) of SRM is called "objective function" when it is used in the optimization process.

Each possible solution (candidate) of the optimization problem has a corresponding objective function value. According to this value, the candidate solution is ranked. Solution Optimization techniques may be classified according to the objective of optimization into single-objective and multi-objective. In single-objective optimization techniques, only one objective function is considered and solution candidates are ranked based on their corresponding objective function value. For example, if it is required to maximize the objective function, the optimal solution is the one that results in the maximum value of objective function (and vice versa). In multi-objective optimization techniques, more than one objective functions are considered and solution candidates are treated in a different way, as will come later. Multi-objective optimization provides a set of optimal solutions instead of one in single-objective optimization. Optimization techniques have been used together with a modelling method to obtain optimal designs. The enumeration optimization method with FEA is used in [20–22]. A genetic algorithm with FEA is used in [10,13,14]. In [18], a genetic algorithm is used with fuzzy logic. The non-dominated sorting genetic algorithm (NSGA-II) method is used in [23] with FEA. Differential evolutions are used in [24,25] with FEA. Particle swarm optimization (PSO) is used in [26–29] with different SRM modelling techniques. The increase in computation time in most approaches is due to time-consuming SRM electromagnetic modelling. Using other methods instead of FEA is even less accurate or complicated to build.

In this paper, FEA is chosen for SRM modelling to complement the mathematical formulas for average torque and core losses calculations. FEA usage is limited to inductance calculation and to obtain flux density for some points inside SRM core. This simulation has proved to be achieved in a very short time while using free software FEMM4.2. Accordingly, the computational time is reduced to an acceptable limit while achieving high accuracy. The Jaya algorithm is introduced to optimize SRM design due to its inherent characteristics, as it takes the path directly toward optimal solutions, saving computational time and achieving better objective functions values [30]. The multi-objective version of Jaya (MO-Jaya) algorithm is considered with three objective functions, and they are rated average torque, efficiency, and iron weight. The three objective functions calculation methods are presented in details. The results of optimization and the performance of the Jaya algorithm are compared with those of the non-dominated sorting genetic algorithm (NSGA-II) under the same constraints and for the same objectives [31]. The proposed method represents a general frame work for SRM multi-objective optimization. Unlimited design parameters and objective functions can be included. The objective of this study is to investigate the performance of MO-Jaya algorithm in SRM design optimization.

## **2. Design of SRM**

Before starting the design process of switched reluctance machine, available space should be well investigated. The available space is represented as constraints of SRM dimensions values. The most important space constraints are axial length, maximum outer diameter (maximum frame size), and shaft diameter. The axial length and maximum frame size are obtained from measurements of the available space. The shaft diameter is calculated based on the maximum torque and speed. The difference between maximum outer diameter and Shaft diameter is the space that is available for SRM cores and turns. SRM cores are the stator and rotor, which are salient in producing reluctance variation. The number of poles in both stator and rotor is specified at the beginning as will come later.

In conventional design approaches, lamination dimensions, coil diameter, and number of turns are calculated analytically. Subsequently, the average torque is calculated and compared with the rated required torque. The average torque is calculated based on the flux-current ( *λ* − *i*) curves for aligned and unaligned positions. Finally, the whole process is repeated with modified dimensions values and the number of turns to match the output average torque with the required torque. Other performance indices can also be considered, such as torque ripples, efficiency, and so forth. This section demonstrates calculation methods of the considered performance indices and characteristics.

#### *2.1. Number of Poles Selection*

Stator poles *Ps* number and rotor poles *Pr* number is specified mainly according to general understanding of their influence on SRM performance. A higher pole number produces higher average torque, lower torque ripples, and provides more reliable operation; however, it requires more switching devices and reduces the maximum speed. Lower poles number produces lower average torque and higher torque ripples; however, it requires less switching devices and provides higher maximum speed. For general purpose SRM design, 6/4 and 8/6 SRM configurations are commonly used. Hence, these two configurations are considered for optimization in this paper.

## *2.2. Poles Arcs Calculation*

When considering *β<sup>s</sup>*,*β<sup>r</sup>* are stator pole arc and rotor pole arc, respectively. To achieve self-starting SRM design, minimum stator pole arc may be expressed—as in [32]—by:

$$
\min[\beta\_5] = \frac{4\pi}{P\_s P\_r}, \qquad \qquad rad \tag{1}
$$

The following condition prevents overlap between phases:

$$
\beta\_s + \beta\_r \le \frac{2\pi}{P\_r} \tag{2}
$$

If this condition not being followed, then the SRM machine inductance will start to increase before reaching the minimum inductance value. This results in higher unaligned inductance value and the developed torque becomes lower.
