Comparison of Flux-Switching and Interior Permanent Magnet Synchronous Generators for Direct-Driven Wind Applications Based on Nelder–Mead Optimal Designing

: The permanent magnet flux-switching machine (PMFSM) is one of the most promising machines with magnets inserted into the stator. To determine in which applications the use of PMFSM is promising, it is essential to compare the PMFSM with machines of other types. This study provides a theoretical comparison ofthePMFSMwithaconventionalinteriorpermanentmagnetsynchronousmachine(IPMSM)inthegearless generator of a low-power wind turbine (332 rpm, 51.4 Nm). To provide a fair comparison, both machines are optimized using the Nelder–Mead algorithm. The minimized optimization objectives are the required power of frequency converter, cost of active materials, torque ripple and losses of a generator averaged over the working proﬁle of the wind turbine. In order to reduce the computational time, the substituting proﬁle method is applied. Based on the results of the calculations, the advantages and disadvantages of the considered machines were revealed: the IPMSM has signiﬁcantly lower losses and higher efﬁciency than the PMFSM, and the PMFSM requires much less rare-earth magnets and copper and is, therefore, cheaper in mass production.


Introduction
Interior permanent magnet synchronous machines (IPMSMs) have been widely used in gearless generators of low-power wind turbines [1,2]. IPMSMs are used in direct-driven wind turbines with power ratings from a fraction of kW to over MW [3][4][5]. Alternatively, permanent magnet flux-switching machines (PMFSMs) can be used in this application, which has several advantages over IPMSM, such as simpler and more reliable rotor and higher specific torque at the same mass of magnets [6]. PMFSMs can also be used both in low-power wind turbines [7] and in wind turbines with a power rating of more than MW [8]. It is important for electrical generator designers to understand why permanent magnet flux-switching generators (PMFSGs), with their obvious benefits, are not as widely used in this application as the traditional machines with permanent magnets on the rotor.
There is a large volume of published studies that compares PMFSMs with synchronous machines (SMs). For instance, in [9] a comparison of PMFSMs (44 kW, 350 N·m, 1200 ÷ 6000 rpm) and IPMSMs with V-shaped magnetic poles (48 kW, 382 N·m, 1200 ÷ 6000 rpm) for traction applications is presented. A PMFSM has 12 teeth on the stator, 10 teeth on the rotor, and an IPMSM has eight poles on the rotor and 48 teeth on the stator and a distributed winding with the number of slots per pole and phase q = 2. Both machines have an external diameter of 269 mm and the stack length of 84 mm. It has been revealed that a PMFSM has higher efficiency (losses are 1.1 times lower) and the torque ripple is 2.25 times lower. However, the mass of rare-earth magnets used in a PMFSM is higher by 2.3 times. Additionally, has higher efficiency, but considerably lower power factor (0.69, while the SPMSG has 0.96). Therefore, the PMFSG requires a frequency converter with 40% more power than the SPMSG. Additionally, the PMFSG requires 1.27 times more usage of magnetic material than the SPMSM. The authors have chosen magnets with residual induction (1.3 T) for the PMFSG and cheaper magnets with less residual induction (1.23 T) for the SPMSG. Still, the paper does not contain any information regarding the torque ripple and magnetic losses, which can be significantly higher in the PMFSG. Besides, a rated voltage of 400 V has been chosen, which is hardly possible for generators with a power of more than 2 MW. Particularly, the literature has shown a successful application of a rated voltage of 600-1000 V [18,19], since there is no industrial 2.5 kA transistor and a voltage of 400 V.
In [20], 12-phase PMFSGs and SPMSGs in gearless wind generator application (16 kW, 500 rpm) are considered. The PMFSM has 24 teeth on the stator and 22 teeth on the rotor. The SPMSG has the best characteristics, with 48 teeth on the stator and 44 poles on the rotor. The generators have the same external radius of 163 mm and the stack length of 185 mm. The study has demonstrated that both machines have a low torque ripple (less than 1.5%). At the same time, the magnetic material used in the PMFSG has a 1.15 times bigger mass. Besides, the rated efficiency, torque and power output of the SPMSG are higher.
Nevertheless, previous studies that cover the comparison of PMFSGs and interior permanent magnet synchronous generators (IPMSGs) for wind generator application [6,20] did not use any formal optimization techniques (e.g., genetic algorithms or the Nelder-Mead algorithm) for the machine design. Applying the optimization methods for the PMFSG and the IPMSG will help to better estimate the potential of these generators and their advantages and disadvantages. Therefore, this research aims to optimize PMFSGs and SPMSGs and provide their comparison.
Various approaches have been described in the literature for optimizing PMFSGs and IPMSGs. In [3], procedures for the manual optimization of efficiency and torque ripple are described for a 4 kW IPMSG. In [5], a multicriteria optimization of an IPMSG with a power rating of 0.6 kW using the Taguchi method is described. Efficiency, EMF THD (total harmonic distortion of back electromotive force), and EMF amplitude were chosen as optimization criteria. In [21], the optimization of the average torque of the PMFSG with a power rating of 5 kW using the random optimization is described. In [22], a multicriteria optimization of a PMFSG with a power rating of 1.5 kW was described using the particle swarm optimization method. The generator weight and EMF THD were chosen as the optimization criteria. The optimization of PMFSGs with rare earth [7] and ferrite magnets [23] using the Nelder-Mead method was also described. In [7,23], the objective optimization function was selected to reduce losses, torque ripple, and the required power of the semiconductor inverter. In [24], a comparison was made of PMFSGs with rare-earth and ferrite magnets developed in [7,23]. However, no comparison has been made between PMFSGs and IPMSGs with rare-earth magnets when using the Nelder-Mead method.
This paper considers three-phase PMFSGs and IPMSGs with V-shaped magnetic poles in gearless wind generator application (332 rpm, 1784 W). The PMFSG has 24 teeth on the stator and 22 teeth on the rotor. The IPMSG has 24 teeth on the stator and 20 poles on the rotor. The generators have the same external radius of 80 mm and the stack length of 100 mm. The machines are designed and optimized using the approach described in the previous papers [7,23] based on the Nelder-Mead algorithm with the optimization method, the mathematical models of the machines, and the substituting profile method. An important advantage of the Nelder-Mead method over other methods that are often used to optimize electrical machines [25,26] is the significant savings in computational time [23].
Using substituting profiles instead of initial ones reduces the calculation efforts, which is extremely significant for the optimization of electric machines. The optimization objectives are to minimize the power required of the frequency converter, the cost of active materials, the torque ripple, and the average generator losses for the working profile of the wind turbine. Additionally, optimization is carried out using a substituting two-mode load profile that considerably reduces the computational time.
Section 2 outlines the general design parameters of the PMFSG and IPMSG, such as the number of stator slots and the number of poles. Section 3 describes the application of the substituting load profile method to reduce the computational time in design optimization. Section 4 describes the optimization criteria and procedures for the PMFSG and IPMSG. Initial geometry parameters are also listed in this section. Section 5 compares the calculated characteristics of the PMFSG and the IPMSG before optimization. Section 6 compares the calculated characteristics of the PMFSG and the IPMSG after optimization, and also lists the optimized geometry parameters. Section 7 summarizes the general findings of the comparative study. used to optimize electrical machines [25,26] is the significant savings in computational time [23].

General Design Parameters of PMFSG and IPMSG
Using substituting profiles instead of initial ones reduces the calculation efforts, which is extremely significant for the optimization of electric machines. The optimization objectives are to minimize the power required of the frequency converter, the cost of active materials, the torque ripple, and the average generator losses for the working profile of the wind turbine. Additionally, optimization is carried out using a substituting two-mode load profile that considerably reduces the computational time.
Section 2 outlines the general design parameters of the PMFSG and IPMSG, such as the number of stator slots and the number of poles. Section 3 describes the application of the substituting load profile method to reduce the computational time in design optimization. Section 4 describes the optimization criteria and procedures for the PMFSG and IPMSG. Initial geometry parameters are also listed in this section. Section 5 compares the calculated characteristics of the PMFSG and the IPMSG before optimization. Section 6 compares the calculated characteristics of the PMFSG and the IPMSG after optimization, and also lists the optimized geometry parameters. Section 7 summarizes the general findings of the comparative study. The supply frequency is fe,IPM = p × fm/60 Hz, where fm is the mechanical rotational speed, rpm. The stator of the PMFSG has 24 teeth with half-open slots, while the magnets are inserted in each stator tooth. The magnets located in the nearest slots are magnetized in opposite directions. The stator integrity is ensured by thin ribs at the outer and inner stator surfaces. Each second stator tooth is wound. Each magnet is divided into three insulated parts to reduce the eddy currents losses in the magnets. The toothed rotor is made of electrical steel laminations and has no magnets and winding. The supply frequency fe,FSM = Zr × fm/60, where Zr = 22 is the number of rotor teeth. Consequently, the required supply frequency is 2.2 times higher for the PMFSG than for the IPMSG at the same mechanical rotational frequency. The supply frequency is f e,IPM = p × f m /60 Hz, where f m is the mechanical rotational speed, rpm. The stator of the PMFSG has 24 teeth with half-open slots, while the magnets are inserted in each stator tooth. The magnets located in the nearest slots are magnetized in opposite directions. The stator integrity is ensured by thin ribs at the outer and inner stator surfaces. Each second stator tooth is wound. Each magnet is divided into three insulated parts to reduce the eddy currents losses in the magnets. The toothed rotor is made of electrical steel laminations and has no magnets and winding. The supply frequency f e,FSM = Z r × f m /60, where Z r = 22 is the number of rotor teeth. Consequently, the required supply frequency is 2.2 times higher for the PMFSG than for the IPMSG at the same mechanical rotational frequency.

Two-Mode Substituting Load Profile of Wind Turbine
In this study, minimizing the loss averaged over the load profile of the wind generator is one of the objectives of the optimization because it results in a higher efficiency machine over the entire load range. In order to reduce the calculation time of the losses without considerable effect on the accuracy, this study applied an approach for designing a gearless generator for wind turbines using a two-modes substituting load profile. This approach has been described in detail while used for the design of a PMFSG with ferrite magnets [7,23].
The authors considered PMFSGs and IPMSGs designed for the wind turbine that operated at the wind speed range from 4 to 12 m/s [27]. Table 1 provides the data on the rotational speed n i , mechanical power P mech i , and torque T i of the turbine at different operating points (modes). Data on the wind speed V i and statistical probability of an i-th mode p i are also presented. Table 1. Initial nine-modes profile of the wind generator. n i : rotational speed; P mech i : mechanical power; T i : torque; V i : wind speed; p i statistical probability of an i-th mode. An annual wind speed distribution was approximated by the one-parameter Rayleigh distribution [28], while the average speed was taken as 7 m/s. As long as the considered wind speeds of the modes were integers, the probabilities of the modes were assumed to be equal to the Rayleigh distribution density at the given wind speed. The values p i were formed by normalizing the probabilities of the nine modes, in a way that they got summed up as 1. The torque provided in Table 1 was interpolated by a cubic polynomial as a function of the mechanical power P mech using the least-squares method ("cftool", Matlab toolbox) with the adjusted R 2 equal to 0.9997 and SSE (Error Sum of Squares) equal to 0.3642 ( Figure 2). The interpolated dependence is as follows: Mathematics 2021, 9, x FOR PEER REVIEW 6 of 16 substituting two-mode load profile of the wind turbine is presented in Table 2. The proposed two-mode load profile significantly decreased the computational time without considerable effect on the accuracy of calculations [7].  Table 1 are marked with asterisks. The polynomial interpolation dependence is shown with the solid line.

Optimization Objectives
The mathematical models of both machines, based on solving 2D magnetostatic boundary problems, were used for the optimization. To reduce the computation time, the  Table 1 are marked with asterisks. The polynomial interpolation dependence is shown with the solid line.
In Figure 2, the obtained relation between the torque and mechanical power is presented. This interpolated dependence allows one to consider various mode characteristics (the torque, the losses, the torque ripple, etc.) as a function of the only argument P mech . Then, the substituting load profile was created in a way that the means < P mech >, < P 2 mech > . . . < P n mech > were coincident with the means of the original profile [7]. Operator < . . . > denotes a (weighted) averaged over all points of a selected load profile. Therefore, the means of the polynomial interpolated values over the original profile and substituting load profile were coincident. Additionally, the loading mode with the highest power was included in the substituting mode. The loading mode determined the required converter power used as one of the optimization objectives. The rest of the modes of the substituting profile were characterized by two parameters: probability and mechanical power, which determined the point on the turbine's maximum power curve ( Figure 2). Generally, m-mode substituting load profile, (m − 1) modes can be adjusted, and the polynomial degree becomes 2·(m − 1). For instance, when the values can accurately be interpolated by quadric polynomial, the two-mode substituting profile is sufficient. The substituting two-mode load profile of the wind turbine is presented in Table 2. The proposed two-mode load profile significantly decreased the computational time without considerable effect on the accuracy of calculations [7].

Optimization Objectives
The mathematical models of both machines, based on solving 2D magnetostatic boundary problems, were used for the optimization. To reduce the computation time, the model took into account the motors' instantaneous symmetry and the periodicity of the electromagnetic processes. The optimization model of the PMFSG consisted of three objectives that were joined into one objective function [7]: where K 1 , K 2 , and K 3 are the optimization objectives. Particularly, K 1 = < P loss > represents the losses averaged over the substituting load profile and annual energy production. K 2 is the required power of the frequency converter: where I ampl,rated and V DC,rated are the current amplitude and the DC voltage of the frequency converter required at the rated conditions (T = 100%, n = 100%), respectively. At these conditions, K 1 and K 2 reach their peak. The objective K 2 is coincident with the apparent power when the current and voltage are sinusoidal and symmetric. Moreover, K 2 includes V DC,rated , which is the converter limitation. It is important to notice that V DC,rated should not include the losses. The reason is that if the active power decreases, K 2 would be reduced and could be considered as a gain in the objective function. K 3 represents the cost of magnetic material: where l mag = R 1 − R 3 − 2·∆ 3 is the radial length of the magnets, h mag is their thickness, R 1 is the outer stator radius; R 3 is the inner stator radius; ∆ 3 is the thickness of the outer and inner stator ribs holding the magnets (see Figure 3) and L is the stack length. K 3 is to decrease the cost of the magnets and similar to the volume, i.e., equal to the product of three dimensions of the magnets, but the magnet thickness is increased by 0.001 m (meters) in (4) to take into account that thin magnets cost more than thick ones [29]. Mathematics 2021, 9, x FOR PEER REVIEW 8 of 16   In Tables 3 and 4 and below, ts is the angular tooth pitch of the stator; tr is the angular tooth pitch of the rotor. Figure 4 shows the main dimensions of the stator and rotor of the IPMSG. Table 5 shows the parameters fixed during the optimization and Table 6 reports the parameters The power of K 2 and K 3 in (2) represents the percentage of the decrease in K 1 as valuable as the decrease in K 2 and K 3 by 1%. Particularly, the change in K 2 by 1% is as valuable as the change in K 1 by 2%, and the change in K 3 by 1% is as valuable as the change in K 1 by 0.5%. Basically, the higher the exponent of the objective, the more valuable the objective is.

Optimization Parameters of IPMSG with V-Shaped Magnetic Poles
It is worth mentioning that although the torque ripple of the initial design of the IPMSM was not very high, it was much higher than that of PMFSG. Therefore, an additional element K 4 equal to the torque ripple averaged over the substituting profile was added in (2) to construct the objective function used for the IPMSM optimization: where K 4 is the relative torque ripple averaged over the substituting profile. The factors "1", "2", "0.5", and "0.25" were not calculated by any formal method. The values simply indicate the approximate relative importance of each of the K1-K4 optimization objectives, according to the authors' experience in the design of similar machines. It is important to highlight that functions (2) and (5) had noise due to round errors coming from FEM (Finite Element Method) calculations and differences in the grid for multiple calls of the objective function. Therefore, the Nelder-Mead method was chosen since it is able to deal with objective functions with noise [30].
The Nelder-Mead algorithm is well known [31] and is included in the basic MATLAB software package (function "fminsearch"). The unconstrained one-criterion Nelder-Mead method is applied in this work to optimize the PMFSG design. After the initial simplex was built, the Nelder-Mead algorithm as it is described in [31] was applied. The reflection, expansion, and contraction coefficients are 1, 2, and 1/2, respectively. The used generator model based on the FEM is described in [7]. Figure 3 shows the main dimensions of the PMFSG. Additionally, Table 3 shows the geometric parameters fixed during optimization, while Table 4 provides the geometric parameters varied during the optimization and their initial values.  In order to reduce the reactive power, the current angle was varied. The current angle was supposed to be proportional to the torque. The zero current angle corresponds to the maximum torque when the current does not influence the steel saturation.

Optimization Parameters of PMFSG
In Tables 3 and 4 and below, t s is the angular tooth pitch of the stator; t r is the angular tooth pitch of the rotor. Figure 4 shows the main dimensions of the stator and rotor of the IPMSG. Table 5 shows the parameters fixed during the optimization and Table 6 reports the parameters varied during the optimization and their initial values. It is important to mention that during the optimization, the slot opening remained the same and was determined by the winding technology. The geometry of the rotor was created by the following procedure. Firstly, point 1 was set at the angle α in the circle with the radius R' remoted from the outer rotor boundary. D and d 2 defined the position of point 2. Point 5 was located in a distance of d 5 from the outer rotor boundary and on the same radius as point 4. Segment 1-4 (the magnet thickness) was perpendicular to segment 1-2, whose length was chosen to obtain the fixed values d. Point 7 was placed at the same circle with point 5, and the angular distance between these points was fixed (see Figure 4b).

Optimization Parameters of IPMSG with V-Shaped Magnetic Poles
outer rotor boundary. D and d2 defined the position of point 2. Point 5 was located in a distance of d5 from the outer rotor boundary and on the same radius as point 4. Segment 1-4 (the magnet thickness) was perpendicular to segment 1-2, whose length was chosen to obtain the fixed values d. Point 7 was placed at the same circle with point 5, and the angular distance between these points was fixed (see Figure 4b).

PMFSG and IPMSG Comparison before Optimization
The initial geometry and magnetic flux density at the rated mode of the IPMSG and PMFSG are shown in Figure 5a,b, respectively. The calculation of the cost of active materials was based on the assumption that the steel price was USD 1/kg, the copper price was USD 7/kg, and the permanent magnet price was USD 126.6/kg [29,32]. Table 7 provides the main characteristics of the PMFSG and IPMSG initial designs.      The optimized designs of IPMSGs and PMFSGs and the magnetic flux density at the rated mode are shown in Figure 6a,b, respectively. The optimized values of the varied The optimized designs of IPMSGs and PMFSGs and the magnetic flux density at the rated mode are shown in Figure 6a,b, respectively. The optimized values of the varied The optimized designs of IPMSGs and PMFSGs and the magnetic flux density at the rated mode are shown in Figure 6a,b, respectively. The optimized values of the varied The optimized designs of IPMSGs and PMFSGs and the magnetic flux density at the rated mode are shown in Figure 6a,b, respectively. The optimized values of the varied parameters are given in Tables 8 and 9. Although the stator inner radius of the PMFSG is greater than that of the IPMSM by 19%, the moment of inertia of the active materials of the PMFSG rotor is more than thrice as low. It is because the PMFSG rotor core is thin and has thick teeth.   Table 10 reports the main characteristics of PMFSGs and IPMSGs after optimization. During the optimization, the torque ripple of both motors reduces significantly. The optimized values of the varied parameters are given in Tables 8 and 9. Figure 6a presents the dependence of total losses on mechanical power for both machines in the optimized condition. Figure 6b shows the electric losses in different parts of the machines at the rated conditions. It can be clearly seen that the most considerable losses for both machines were the copper losses. At the same time, the copper losses, as well as the rotor steel losses, in the IPMSG were noticeably lower. What stands out in Table 8 that the average losses in the IPMSG are 1.2 times lower than in the PMFSG. The active output power of the IPMSG is higher both at the rated mode and at the low load mode of the two-point profile. Moreover, the IPMSG required a frequency converter with less power than the PMFSG. Talking about the advantages of the PMFSG, it is important to highlight that the torque ripple at the rated mode in the PMFSG was three times lower than in the IPMSG, and the mass of the magnets is lower by 1.4 times. The optimized geometry and magnetic flux density at the rated mode of the IPMSG and PMFSG are shown in Figure 9a,b, respectively.  The losses in the PMFSG (blue asterisks) and the IPMSG (red asterisks) at nine modes of the profile presented in Table 1 are shown in Figure 10a. The losses in the PMFSG are higher than those in the IPMSG in a wide range of powers.

Optimization Results and PMFSG and IPMSG Comparison after Optimization
The two-profile approach is developed for the cases when the dependence of these losses on the mechanical power can be approximated by a quadric polynomial [7]. Such approximations shown in Figure 10a with continuous lines are rather accurate (R 2 = 0.9998 for both cases), which justifies the accuracy of substituting the initial nine-point profile with the two-point one. Figure 11 shows a comparison of the efficiency and torque ripple of the optimized generators at all points of the considered working profile. The losses in the PMFSG (blue asterisks) and the IPMSG (red asterisks) at nine modes of the profile presented in Table 1 are shown in Figure 10a. The losses in the PMFSG are higher than those in the IPMSG in a wide range of powers.  The two-profile approach is developed for the cases when the dependence of these losses on the mechanical power can be approximated by a quadric polynomial [7]. Such approximations shown in Figure 10a with continuous lines are rather accurate (R 2 = 0.9998 for both cases), which justifies the accuracy of substituting the initial nine-point profile with the two-point one. Figure 11 shows a comparison of the efficiency and torque ripple of the optimized generators at all points of the considered working profile. The torque ripple in the IPMSG was quite low, namely, below 4%. Another revealed advantage of the PMFSG was a lower cost of rare-earth magnets. Particularly, the PMFSG required 1.4 times less rare-earth magnetic material than the IPMSG. Considering that the mining of rare-earth materials and their manufacturing have a harmful effect on the environment, this advantage of the PMFSG is highly valuable for mass production [33]. Furthermore, Table 8 shows that the demand for copper in the PMFSG was twice lower than for the IPMSG. Consequently, the cost of active materials for the PMFSG was 1.4 times lower than for the IPMSG, mainly due to the reduction in the magnets mass.

Conclusions
The aim of this study was to theoretically compare the PMFSG with a conventional IPMSG in a direct-driven low-power wind application. To ensure a fair comparison, both machines were first optimized using the Nelder-Mead algorithm. The optimization objective functions were selected so as to take into account the required power of frequency converter, the evaluation of the active materials cost, and losses in generators averaged over the working profile of the wind turbine. Additionally, the torque ripple was included in the optimization function of the IPMSG, since this parameter was significantly higher in the IPMSG than in the PMFSG. To reduce the computational time, the substituting profile method was applied. As a result, the strengths and weaknesses of the machines were revealed. The torque ripple in the IPMSG was quite low, namely, below 4%. Another revealed advantage of the PMFSG was a lower cost of rare-earth magnets. Particularly, the PMFSG required 1.4 times less rare-earth magnetic material than the IPMSG. Considering that the mining of rare-earth materials and their manufacturing have a harmful effect on the environment, this advantage of the PMFSG is highly valuable for mass production [33]. Furthermore, Table 8 shows that the demand for copper in the PMFSG was twice lower than for the IPMSG. Consequently, the cost of active materials for the PMFSG was 1.4 times lower than for the IPMSG, mainly due to the reduction in the magnets mass.

Conclusions
The aim of this study was to theoretically compare the PMFSG with a conventional IPMSG in a direct-driven low-power wind application. To ensure a fair comparison, both machines were first optimized using the Nelder-Mead algorithm. The optimization objective functions were selected so as to take into account the required power of frequency converter, the evaluation of the active materials cost, and losses in generators averaged over the working profile of the wind turbine. Additionally, the torque ripple was included in the optimization function of the IPMSG, since this parameter was significantly higher in the IPMSG than in the PMFSG. To reduce the computational time, the substituting profile method was applied. As a result, the strengths and weaknesses of the machines were revealed.
This study shows that the PMFSG requires 1.4 times less rare earth-earth magnet material than the IPMSG. Considering that the mining of rare-earth materials and their manufacturing have a harmful effect on the environment, this advantage of the PMFSG is highly valuable for mass production. In addition, the PMFSG torque ripple is less. However, the PMFSG requires a slightly higher power rating of the semiconductor inverter, and the average losses in the IPMSG are 1.2 times lower than in the PMFSG.
The results of this comparative study can be used by developers of direct-driven wind generators when selecting the type of electric machine suitable for a specific wind power application.