**1. Introduction**

Recently, interest in synchronous reluctance machines (SynRMs) has increased remarkably thanks to their advantages compared to other types of electrical machines [1–5]. They offer a good torque density, a high efficiency, and a wide range of operating speeds [6]. In addition to their simple and robust structure, they have no windings, cages, and permanent magnets in their rotor, resulting in very low rotor losses and hence good thermal managemen<sup>t</sup> [2]. These advantages make SynRMs a good competitor compared to the other electric machines in several electric drive systems in different industrial applications such as hospitals and aerospace [7]. It is evident through the literature that the performance of SynRMs (torque ripple, average torque, efficiency, and power factor) greatly depends on the saliency ratio (the ratio between the direct and quadrature axis inductances) [8]. This ratio is a function of several parameters of the machine design such as the winding, magnetic material, and rotor flux-barriers [9–11]. Starting from the standard stator design of the induction machine, the rotor flux barrier parameters are key elements in the performance of the SynRM. There are several parameters in the rotor as sketched in Figure 1. Therefore,

**Citation:** Rezk, H.; Tawfiq, K.B.; Sergeant, P.; Ibrahim, M.N. Optimal Rotor Design of Synchronous Reluctance Machines Considering theEffect of Current Angle. *Mathematics* **2021**, *9*, 344. https://doi.org/ 10.3390/math9040344

Academic Editor: Jinfeng Liu Received: 17 January 2021 Accepted: 5 February 2021 Published: 9 February 2021

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it is evident that an optimization process is necessary to optimally select the parameters of the SynRM rotor.

**Figure 1.** Rotor geometry of one pole of synchronous reluctance machines (SynRM).

Through literature, several research papers about the optimization of SynRMs can be found [12–29]. Various optimization schemes have been reported and applied to SynRM design. For example, in [12], the rotor of the SynRM was optimized using a multi-objective differential evolution algorithm for high-speed applications focusing on selecting the barrier angles and the magnetic insulation ratio. The other geometrical parameters of the rotor were derived from these two parameters (the barrier angles and the magnetic insulation ratio). In [13], the optimal number of flux barriers and rotor poles of the SynRM were optimally selected by applying a weighted-factor using a multi-objective optimization technique. A circular rotor shape was used in [14] to maximize the torque and to minimize the torque ripple of the SynRM using a multi-objective genetic algorithm. In this design, a circular rotor was adopted in order to minimize the number of parameters and to ge<sup>t</sup> a time-efficient optimization process. Three different geometries for rotor flux barriers (one rotor has a circular flux barrier and the other two rotors use a rectangular flux barrier) were studied and compared in [15]. The design process uses the same optimization algorithm proposed in [14]. It was shown that the rectangular flux barrier has fewer structural challenges and lower inertia at high speed. In addition, it is preferred especially for machines with inserted permanent magne<sup>t</sup> (PM inside the flux-barriers. In [16], an alternative technique for the development of asymmetric flux barriers with rotor skewing is proposed in combination with design optimization to enhance average torque and reduce torque ripple. Although the torque ripple in this method is below 3%, the number of optimization variables in this method is relatively high between 29 and 37. This results in a slower and complicated optimization process. In [17], the rotor of the SynRM was optimized using three different popular optimization algorithms (simulated annealing, differential evolution, and genetic algorithm) to minimize the torque ripple and maximize the torque per Joule loss ratio. The differential evolution algorithm has shown the best result in terms of repeatability of the results and convergence time. It was demonstrated in [18–21] that the rotors with skewing techniques have reduced torque ripple significantly. In [22], the rotor of the SynRM was optimized to maximize the saliency ratio and minimize the thickness of the iron ribs. The rotor was made ribless in [23] to obtain an improved power factor, torque, and efficiency.

In [24], a generalized formula was proposed to select the widths and angles of the fluxbarriers considering additional factors such as stator and rotor slot opening and number of slots. However, the torque ripple is still high. Furthermore, a preliminary design for the flux-barrier widths was introduced in [25] without considering the influence of the different number of stator slots. The effect of the number of stator slots was considered

in [26] and the torque ripple was reduced from 23.38% to 12.3%. All the previous studies about the rotor design of the SynRM did not consider the current angle as a design variable during the design and optimization procedures. The current angle was fixed based on a rule of thumb or primary simulation i.e., in the range of 45◦ to 60◦ as in [26]. In [27], a simultaneous structural and magnetic topology optimization technique was developed for the rotor of the SynRM using solid isotropic with material penalization. The total structural compliance, torque ripple, and the average torque were simultaneously considered in this method. In [28], a new technique was proposed to design the rotor of the SynRM. A symmetrical rotor geometry with fluid shaped barriers was used in this optimization method. The optimal design in this method was chosen using the communication between MATLAB and Flux 2D. In [29], a line start SynRM was optimized using an optimization topology that uses the normalized Gaussian network. The computational time was reduced in this method as it separates out unpromising geometries. The effect of the current angle on the final optimal geometry of the SynRM has not been investigated before as far as we know.

This paper studies the effect of considering the current angle during the optimization process on the final optimal geometry of the rotor of the SynRM. Different cases are analyzed and compared for different ranges of current angles during the optimization process. This way, in some cases, the saturation level in the machine is enforced during the optimization process by varying the current angle range. Finite element magnetic simulation is carried out and compared for the optimal geometries. Finally, experimental results are conducted to validate the simulation results.

#### **2. Design Optimization of SynRMs**

## *2.1. Hybrid PSOGWO Technique*

In this paper, the hybrid particle swarm optimizer and grey wolf optimizer (PSOGWO) algorithm was used to determine the best parameters in order to obtain the optimal rotor design of the SynRM. The following paragraphs briefly describe the core idea and the updating process of the PSO, GWO, and hybrid PSOGWO.

PSO was originally proposed by Kennedy and Eberhart to simulate the social behavior of a flock of birds [30,31]. In order to determine the best solution, every particle, representing a candidate solution, updates continuously its position and velocity. The following relation can be used to estimate the new step size of each particle.

$$v\_{i}^{t+1} = \overbrace{w.v\_{i}^{t}}^{first\\_section} + \overbrace{C\_{1}.r\_{1}.(Pbest\_{i} - x\_{i}^{t})}^{\text{second\\_sec\\_tion}} + \dots \tag{1}$$

$$\overbrace{C\_{2}.r\_{2}.(Gbest - x\_{i}^{t})}^{t+1}$$

$$x\_{i}^{t+1} = x\_{i}^{t} + v\_{i}^{t+1} \tag{2}$$

where *w* is the inertia factor; *C*1 and *C*2 denote the cognitive and the social coefficients; *r*1 and *r*2 denote random; *t* is the iteration number; *i* is the particle number; *Pbest* is the local best; *Gbest* is the global best.

The first section of (1) provides the exploration capability of the PSO. Whereas, the second section moves the particle towards the best position ever achieved by itself. The last section of (1) moves the particle according to the best position achieved by all the particles in the population. The core idea of GWO is extracted from the behavior of grey wolves. GWO simulates the hunting process and the leadership hierarchy of grey wolves [32]. Grey wolves exist at the highest level of the food chain and are regarded as predators.

The hunting mechanism contains two chief sections: tracking and catching the prey, then encircling and attacking the prey until movement stops. During the hunting process, prey is encircled by the grey wolves. To simulate the encircling behavior, the next relations can be considered [32]:

$$D = \left| \mathbb{C} \* X\_p(t) - X(t) \right| \tag{3}$$

$$X(t+1) = X\_p(t) - A \ast D \tag{4}$$

where *t* is the current iteration; *Xp* and *X* denote the position of the prey and the location of grey wolves, respectively.

*A* and *C* denote the coefficients vectors that are estimated using the following relations:

$$A = a \ast (2 \ast r\_1 - 1)\tag{5}$$

$$C = 2 \ast r\_2 \tag{6}$$

*r*1 and *r*2 are random values; *a* is constant that reduces linearly from 2 to 0 over the optimization process.

The process update of grey wolves is carried out based on the following relation;

$$\begin{cases} D\_{\mathfrak{A}} = \left| \mathbf{C}\_{1} \ast X\_{\mathfrak{A}}(t) - X(t) \right| \\ D\_{\mathfrak{B}} = \left| \mathbf{C}\_{2} \ast X\_{\mathfrak{A}}(t) - X(t) \right| \\ D\_{\mathfrak{S}} = \left| \mathbf{C}\_{3} \ast X\_{\mathfrak{A}}(t) - X(t) \right| \end{cases} \tag{7}$$

For every iteration, the best three wolves are represented by *X<sup>α</sup>*, *<sup>X</sup>β*, and *Xδ*;

$$\begin{cases} X\_1 = |X\_\hbar - a\_1 \ast D\_\hbar| \\ X\_2 = \left| X\_\beta - a\_2 \ast D\_\beta \right| \\ X\_3 = \left| X\_\delta - a\_3 \ast D\_\delta \right| \end{cases} \tag{8}$$

Finally, the updated position of the prey is provided by the average of three values of positions assessed as the best solutions:

$$X\_p(t+1) = \frac{X\_1 + X\_2 + X\_3}{3} \tag{9}$$

The fundamental idea of the hybrid PSOGWO is to integrate the capability of social thinking of the PSO with the local search ability of the GWO. A PSO suffers from shortcomings like catching the local minimum. Therefore, to avoid this disadvantage, the GWO was used to reduce the chance of trapping on the local minimum. Moreover, the GWO has the advantage of preserving a balance between exploitation and exploration during the optimizing procedure. More details about the mathematical modeling and physical meaning of the hybrid PSOGWO can be found in [33].
