Multimodal Optimization Algorithm for Torque Ripple Reduction in Synchronous Reluctance Motors

An accurate analytical model is adopted to estimate the torque ripple of a synchronous reluctance motor (SynRM). Desired behavior of the torque ripple functionin this motor is obtained by changing the angles of one and two flux barriers per pole (FBs) in the rotor. The torque ripple function of the SynRM serves as the multiple and close local optima. By identifying the behavior of this function, a comprehensive learning particle swarm optimization (CLPSO) algorithm (typically applied in solving multimodal functions), is adopted to reduce the torque ripple. The results indicate that compared to PSO (i.e. global optimization algorithms) the CLPSO algorithm is more efficient in torque ripple reduction and finding more local optima. Among the available optimal solutions with four FBs per pole, a sample is selected for motor construction. Finite element analysis and laboratory tests are performed to validate the results.


I. INTRODUCTION
To design a synchronous reluctance motor (SynRM), the rotor configuration, method of analyzing the motor performance, and optimization algorithm constitute the fundamental components. The configuration of a SynRM rotor for obtaining high reluctance difference is subject to placing multiple conductive and insulating magnetic layers. In transversally laminated rotors, a popular theme of study in the literature( due to its easy manufacturing process), is to have magnetic insulators implemented by air voids, also referred to as flux barriers (FB)s in the rotor's magnetic sheets. The FBs' shape can take straight lines (rectangular, trapezoidal [1]), curves (circular [2,3], crescent [4][5][6], hyperbolic [7], etc.), or a combination of straight lines and curves [8]. The main objective of these various shapes is to optimally chanel the flux lines within a solid rotor. The shape of FBs can be expressed by analytically [9]. The findings of most studies reveal the superiority of the fluid shaped, the Zhukovski shaped, and field path design geometry, for FBs when compared to other shapes [10,11]. The methods applied in analyzing the performance of electric machines are grouped into two general categories namely, numerical and analytical. The finite element analysis (FEA) with high precision is widely applied in design of SynRM. The main problem in applying FEA to design of SynRM is attributed to its prolonged computational time in view ofthe large number of design variables. Reducing torque calculation points by randomly selecting them within the optimization iterations [12], surrogate models like Kriging [13], computationally efficient electromagnetic FEA [14], and combining FEA with analytical methods [1] are among the measures that has been taken in reducing the computational cost. As to the analytical methods, they are faster than FEA, with a high possibility to evaluate more cases in achieving the optimal solution. Obtaining an analytical model for SynRM usually involves simplifying assumptions that reduce its accuracy. The initial analytical models [15], regardless of the effect of the stator slots, merely estimate the radial component of the air gap magnetic field. Consequently, stator slotting effect in radial magnetic field modeling was added using permeance functions [16]. In addition to the radial component, the tangential part of the magnetic field is estimated by combining the magnetic equivalent circuit (MEC) and conformal mapping (CM) methods [17]. Maximizing average torque and minimizing torque ripple are the two main objectives in SynRM optimization. In multi-objective optimization, these two are considered simultaneously [18]. In some studies, these two are optimized independently [1], and it is reported that these two objectives are not conflicting [1]. There exist many locally optimal solutions using which the problem of SynRM optimization is characterized. In [18] and in order to find more optimal solutions, optimization is done in two stages. First a more extensive range is considered for the variables, and several iterations are performed towards finding the global optima. This is followed by a zoomed local search around the potential solutions obtained in the first stage with a smaller range for the variables. In this paper, the average torque and torque ripple objectives are optimized independently. The analytical model is applied to optimize the SynRM torque ripple. Instead of running repeated optimizations to assure the optimality of the solutions, the behavior of the torque ripple function for one or two FBs is determined using parametric analysis (incorporating an analytical model that allows rapid evaluation of motor performance). At this juncture a comprehensive learning particle swarm optimization (CLPSO) algorithm is applied, and its performance is compared to that of the PSO algorithm. After performing torque ripple optimization, the sample of optimal rotors in terms of torque ripple is selected for average torque optimization, and the best candidate is designated for construction and laboratory examination.

II. ANALYTICAL COMPUTATION OF TORQUE RIPPLE
The analytical model is based on combination of the MEC and CM methods, which allows for much faster calculation of torque ripple when compared to FEA. In this section, the analytical modeling of SynRM is briefly described; More details can be found in [17]. In the first step, by disregarding the rotor FBs, the components of the air gap magnetic field( formed from the stator winding) are calculated using CM. Following this, the potential magnetic drop due to the rotor FBs is calculated through linear MEC and the radial component of the magnetic field obtained in the first step. Consequently the possible reduction, due to the rotor, is applied through virtual currents in the CM method. The air gap magnetic field is equal to the resultant of the stator winding magnetic field and the virtual magnetic field of the FBs. This modeling method makes it possible to obtain radial and tangential components of the magnetic field by considering the rotor FBs effect. The stator of an induction motor is selected to design the SynRM. One magnetic pole along with a rotor having two FBs per pole is shown in Fig. 1, and the stator parameters are tabulated in Table I.   The rotor pole with the two FBs is shown in Fig. 2. Symbols ∆ 1 and ∆ 2 are the angles of the first and second FBs, and 1 and 2 are the thicknesses of the first and second FBs along the q-axis, respectively. Radial air gap magnetic fields formed by stator windings, FBs, and their sum obtained through the analytical model are shown in Fig. 3. The FBs form a virtual magnetic field opposite to the direction of the magnetic field, formed by the stator winding. Due to the twolayer FBs, two steps drop is created in the air gap magnetic field. The total magnetic field of the air gap equals the resultant of the field of stator winding and the virtual fields of the FBs. The tangential magnetic fields of the stator winding, the FBs, and their sum are shown in Fig. 4. Similar to the previous case, the tangential magnetic field generated by the FBs is in the opposite direction of the stator field, leading to drops in the tangential component in the area between the FBs. Due to the rotation of the flux lines at the location of the end angles of the FBs, drastic changes in the tangential component of the air gap magnetic field are observed. The radial and tangential components of the total air gap magnetic field, in comparison with the FEA results, are shown in Figs. 5 and 6 respectively. As observed, there exists a good agreement between the results obtained by the analytical method and the FEA.  In both cases, the results of the analytical model are in good agreement with that of the FEA. For angles ∆ 1 = 20.5° and ∆ 2 = 16.1°, the torque ripple is 30.5%, and for angles ∆ 1 = 20.5° and ∆ 2 = 15.1°, the torque ripple increases to 62.1%. As observed by changing one degree of the second FB arc, the torque ripple almost doubles, indicating the high sensitivity of the torque ripple to the end angles of the FBs; consequently, torque ripple optimization is necessary to determine the optimal angles.

A. ONE FB PER POLE
To assess the impact of the FB angle on torque ripple, in the rotor with one FB, the FB angle is changed from 15° to 45° with a step of 0.01°. The time required to compute the objective function consisting of 30 points of the torque curve (using the analytical model in three steps of the stator slot) sums to about 3.5 seconds using a Core i5-2.6GHz processor. This suggests that the time required to calculate all cases (e.g. 3000 times) is about 3 hours. Although it is recommended to avoid choosing a FB per pole for the SynRM due to its inferior performance [19], practical information is found in Fig. 8, where the torque ripple function has many local optima, with short distances between some local optima.

B. TWO FB PER POLE
To reduce the simulation time for two FBs per pole, a parametric analysis is run by reducing the step to 0.1° for the angle of each FB. The angle of the first FB is changed between 0.2 and 0.6 per unit (relative to the half pole pitch), and the angle of the second FB is changed between 0.1 and 0.5 per unit, requiring over 30,000 objective function calls. Due to the 2D space of the parameters, by selecting the 0.01° step, the number of times for which the objective function is called will be 100 times greater than the case with 0.1° step. The torque ripple function is shown in 2D and 3D illustrations in Figs. 9 and 10, respectively, Similar to the rotor with one FB, the torque ripple function has multiple local minima; while, the global minimum value is drastically reduced compared to the rotor with one FB. As observed in Fig. 9, the difference between some local minima and a global minimum is negligible; therefore, with different combinations of FB angles, torque ripple can be reduced to the vicinity of the global minimum.

IV. TORQUE RIPPLE OPTIMIZATION
In the following, PSO and CLPSO algorithms are used for torque ripple optimization, and their performances are compared. First, these two algorithms are briefly introduced.

A. PSO ALGORITHM
In PSO algorithm, every particle is an answer to the problem, which moves in the search space based on the best individual flight experience and the best group flight experience. Each particle i contains the position ( ), velocity ( ), and its best individual position ( ) data. Index d is the particle dimension. The particles are randomly scattered in the search space with zero velocity. Each particle moves according to the laws of velocity (1) and position (2) [20]: where , 1 , and 2 are the parameters of the algorithm, named as the weight of inertia, the acceleration coefficient of the individual, and social components, respectively. The weight of inertia has a diverse contribution, and the acceleration coefficients seek to converge the algorithm. Symbols 1 and 2 are the two random numbers between zero and one that determine the probable nature of the algorithm. Symbol is the best position of the i th particle, and is the best position ever.

B. CLPSO ALGORITHM
All-particle memory information is applied to the CLPSO algorithm to update the new particle position. This feature allows for the dispersal of particles and prevents rapid convergence. The velocity equations in CLPSO are modified as (3) [21].
where = [ (1), (2), … , ( )] determines which particles' pbest positions should be applied in the i th particle. For each particle dimension, a random number is generated; if less than , the pbest of the particle itself is selected for the proper dimension, otherwise, the pbest of other particles is selected based on the tournament selection. The differences between PSO and CLPSO algorithms are [21]: 1. Instead of applying a particle's pbest and gbest as the exemplars, all particles' pbest can potentially be applied as the exemplars to guide a particle's flying direction. 2. Instead of applying all the dimensions of an exemplar, each particle dimension can take its new position from the dimension of different particles.
3. Instead of learning from the two pbest and gbest exemplars in each generation, each particle can use only one exemplar in a limited number of repetitions for learning.
The performance of CLPSO algorithm with eight different types of PSO algorithms in optimization for different objective functions is assessed in [21]; in most multimodal cases, CLPSO outperforms PSO variants.

C. ONE AND TWO FB PER POLE
The objective function is to minimize torque ripple. The optimization constraints for FBs are similar to that of the parametric analysis mode. Like the previous case, 30 points of the torque curve in three stator slot steps are considered to evaluate the torque ripple. The initial population is generated randomly. The number of function evaluations for the rotor with one and two FBs per pole is selected to be 300 and 3000 times, respectively. This is almost ten times less than the number of calls in the parametric analysis with fixed-step. For better comparison, the parameters of both algorithms are selected to be the same. The results of torque ripple optimization with a FB by PSO and CLPSO algorithms in the background of the parametric analysis are shown in Figs. 11 and 12, respectively. The optimization results indicate that in PSO algorithm most of the particles are gathered around the global minimum, while in CLPSO algorithm, the same particles are subject to higher dispersion, thus, they can be found at all local minima's. Both algorithms with similar accuracy have reached the global minimum solution, at 53.45%, obtained through the parametric analysis with a 0.01° step. The average of the solutions in the PSO is 19.15% less than that of the CLPSO.
The results of torque ripple optimization with two FBs per pole obtained through PSO and CLPSO algorithms are shown in Figs. 13 and 14, respectively. By comparing the results observed in Figs. 13 and 14, both algorithms have reached the global minimum obtained through the parametric analysis (with slightly higher accuracy). The optimization results obtained through PSO algorithm have one local minimum for the first FB and two local minima for the second FB, which have a ripple value of less than 20%; while the same through CLPSO algorithm for both FBs reveals three local minima of less than 20%. Particle diversity in the CLPSO algorithm is higher than that of the PSO algorithm, with more local optimal solutions. The high convergence of the PSO algorithm necessitates repetitive executions to find more local optima.

C. THREE AND FOUR FB PER POLE
As the number of FBs increases, sensitivity analysis, even with a 0.1° step for the angle of each FB, leads to an intensive computational time. Though the analytical model is faster than FEA, here, to limit the execution time to an acceptable level, optimization algorithms must be applied. Because in rotors with one and two FBs per pole, CLPSO outperforms the PSO algorithm, it is applied to optimize the rotors with VOLUME XX, 2017 9 three and four FBs per pole. Theoretically, there is no limit to the number of FBs, but in small motors, to reduce the shear level and maintain the mechanical strength of the rotor, the number of FBs usually does not exceed four for each pole. The constraints of parameters are tabulated in Table II. The first FB can be changed between 0.2 and 0.6 per unit, and the rest can be changed from 0.1 to 0.5 per unit. During initialization, the random production of all the particles  Because the optimal rotors with four ripple FBs per pole are of less torque ripple compared to others, few samples are selected, based on suitability for manufacturing, to determine the FBs' thickness and further assessments.

V. AVERAGE TORQUE OPTIMIZATION
The FBs' thickness is an effective parameter in the average torque with a slight impact on the torque ripple; therefore, by changing this thickness, the optimal average torque value can be obtained. In assessing the FBs' thickness effect on average torque, the application of a parameter named insulation ratio has been common [9,22,23]. The

Fig. 17. Definition of rotor parameters and q-axis MMF distribution
insulation ratio ( ) is defined as the total FBs' thickness along the q-axis to the rotor thickness ratio obtained through (4) [24]: is the thickness of the i th FB along the q-axis, is the number of the FBs, is the rotor radius, and ℎ is the shaft radius. Selecting any value for gives the total thickness of the FBs, but does not determine the thickness of each FB. The thickness of each FB should be adjusted to reduce the qaxis flux at its highest. A pole of the synchronous reluctance rotor and the principal component of the q-axis magnetomotive force (MMF) distribution is shown in Fig.  17. The method applied below is based on the results given in [22]. It has been proven that if (5) is established between the thickness of the FBs, the inductance (flux) of the q-axis will be minimal.
where, is the length of the i th FB. Symbol ∆ , denotes the difference in the average per-uint of the MMF ( , ) in the i th FB, calculated through (6). By determining the insulation ratio and the FBs' width ratio, it is easy to obtain each FB's width. By selecting the optimal points for the FB angles, location of FBs are restricted.
Four of the optimal rotor solutions in torque ripple, with four FBs per pole, named A, B, C, and D, are selected to determine FBs' width. The rotors with an insulation ratio of 0.4 are shown in Fig. 18. The flux line pattern is selected for FBs' shape to achieve the maximum torque. The results of changing the insulation ratio on the average torque and peak-to-peak torque ripple of the motors are shown in Figs. 19 and 20, respectively. The motors' torque is obtained at the current angle that produces the maximum torque. The FEA (using M400 magnetization curve) is applied for evaluation. As shown in Fig. 19 the behavior of the four motors is similar. An increase in the insulation ratio increases the  the average torque, because as the FBs' thickness increases, the inductance of the q-axis decreases, while there is still enough iron to cross the d-axis flux. When the insulation ratio reaches 0.4, the maximum average torque is obtained for all four motors. By increasing the insulation ratio from 0.4, the effect of iron saturation reduces the d-axis' flux and decreases the average torque. Between the 0.25 to 0.4 insulation ratios, the peak-to-peak torque of the four motors is almost constant. Furthermore, increase in insulation ratio increases the peak-to-peak torque due to the nonlinear effects of the saturation curve. The comparison of torque ripple calculation with the analytical model and FEA for optimal motors with insulation ratio of 0.4 is listed in Table III. There exists a good agreement between the results of the two methods, and the maximum error of torque ripple is 1.31%. The Motor-C is selected for construction because of its superiority in terms of average torque and torque ripple.

VI. EXPERIMENTAL RESULTS
The M400-50A magnetic sheet is used in manufacturing of the rotor. The laminations are cut by the electric discharge machining method. In Fig. 21 (a), a sample of the rotor lamination, and (b) the rotor stack without any skew, with a dummy shaft are illustrated. The rotor ribs are tangential only, with 0.4 mm thickness. The test bench is shown in Fig.  22. A DSP board controls the motor, and current vector control is applied to drive it. The maximum torque per ampere (MTPA) control strategy is applied at each torque level.

A. MEASURED AND SIMULATED TORQUE
The test motor is rotated at a low speed (10 rpm) to confirm the motor torque ripple behavior. Two current levels of 2.3A at 60° and 1.5A at 50° are considered for comparison. For each combination of d and q-axis currents in a complete rotor  rotation, the torque waveform is recorded with a torque transducer. The measurement and the simulation torque at two levels of rated and half-rated values are shown in Fig.  23. The measured values of torque are in good agreement with the simulation values. As expected, there exists a direct relation between the torque ripple and current.

B. EFFICIENCY
Full-load and no-load tests are run on the prototype motor . The results of FEA simulation and laboratory tests are tabulated in Table IV. There exists a good agreement between the simulation and laboratory results. Motor efficiency is 85.71%, which reaches the IE4 class.

VII. CONCLUSIONS
The performance of PSO and CLPSO algorithms, appropriate for solving unimodal and multimodal problems, are compared for the optimization of the torque ripple of a SynRM using an analytical model to determine which algorithm is more appropriate for machine optimization. The results indicate that both algorithms yield the same global optimal solution, while the CLPSO algorithm finds more local optima than the PSO algorithm. The analytical model offers a good accuracy in the linear region of the magnetization curve, however, its accuracy declines as magnetic saturation increases. The behavior of torque ripple function for SynRM with one and two FBs is determined through the analytical model. By changing the angle of the FBs, the torque ripple function reveals many local optima, the value of which is not very different from the global optimal. The accuracy of the optimal solutions is evident through FEA, and one of the optimal rotors is selected for construction. The laboratory tests and FEA show the computational efficiency of the optimization method.
In most electric machine design problems, the shape of the objective function is unknown, consequently, it is recommended to apply an optimization algorithm capable of solving multimodal problems. Because algorithms applied in solving multimodal problems can solve unimodal problems as well, by applying multimodal optimization algorithms, important information on some local optima becomes is lost.