Performance Improvement of Multi-Rotor Axial Flux Vernier Permanent Magnet Machine by Permanent Magnet Shaping

Permanent magnet vernier machines (PMVMs) are becoming progressively attentive because of their high efficiency and high torque density and can thus be utilized for direct-drive applications such as wind power and electric vehicles etc. This paper presents performance improvement of multi-rotor axial flux vernier permanent magnet (MR-AFVPM) machine with a proposed two-stage parallelogram-shaped PM. The proposed shaped PM reduces the cogging torque and torque ripples due to its skew effect. Furthermore, it also presents a comparative analysis of the conventional and proposed shape PM. Then 3D finite element analysis (FEA) is used for comparative analysis. Genetic algorithm (GA) associated with kriging method based on LHS is introduced and is used to optimize the proposed shaped PM for further performance improvement in terms of cogging torque, back EMFs, torque ripples, VTHD, output torque, flux density distributions, power factor and output power of the analyzed machines which are validated by 3D-FEA.


I. INTRODUCTION
Due to the large power flow related to electric machines, it is of significant importance to improve the efficiency of electrical machines for energy saving and environmental protection. In recent years, the environmental and energy problems are drawing more and more attention and many solutions have been proposed to address them. With the increase in energy demand and keeping in mind the environmental considerations, the research and design of highly efficient electric machines have received great attention. Gearless, direct-drive, low-speed machines become extra rewarding The associate editor coordinating the review of this manuscript and approving it for publication was S. Ali Arefifar . than high-speed machines with gearboxes, this is because of the previous machines that haven't any gearboxes and no associated problems i.e., oil retention, noise problems, losses due to friction, etc. Though, conventional direct-drive machines are bulky and have low efficiency.
High torque density is the requirement for machines used in the direct drive system. Therefore, PM machines got too much attention due to high torque density [1]- [3]. PMVMs are magnetically geared machines that have a magnetic gearing effect due to flux modulation poles. The prier vernier reluctance machine is the base from which PMVMs are derived [4]. The high power features are achieved in the PMVM because of the magnetic gearing effect [5], [6]. Numerous pieces of literature determined that PMVMs show several options to reduce the size and cost. The output power achieved in PMVMs is three times to equivalent of conventional PM machines for the similar volume and current [7]. However, PMVM is commonly designed for low-speed rotation due to large magnet pole numbers [8].
Various researchers have used the airgap field modulation to enhance the torque performances [9]- [11]. However, the PMVM is a typical simple structure machine having the features of an airgap modulation machine and high torque density. This machine has the great benefit of many working harmonics controlled by flux modulation poles rather than fundamental harmonics [10], [12]- [15]. The comparison of the permanent magnet synchronous machine (PMSM) and the permanent magnet vernier machine (PMVM) is summarized in Table 1 as reported in [16], [17]. The PMVMs, which make use of magnetic gear effects to modify the magnetic fields, can produce high torque under comparatively low-speed operation [18], [19]. High torque density, structural compactness, and high efficiency make PMVMs machines more suitable for direct drive applications [20], [21].
Multi-objective optimization has got the most attraction recently and is used in the optimization of electric machines. To improve the average output torque of the switched reluctance motor, a multi-objective optimization analysis is investigated in [22]. In [23], another optimization technique based on particle swarm optimization (PSO) is discussed for cogging torque issue and efficiency enhancement of brushless DC motor. A 3D pareto front linear induction motor with finite element model evaluation is optimized and analyzed in [24]. This paper performs the well-known, simple, and practical optimization approach due to suitability with nonlinear data as implemented in [25].
The earlier research focuses mainly on the development of different axial flux topologies of the PMVMs machine. In this paper, the performance improvements of an MR-AFVPM machine are achieved by introducing a new two-stage parallelogram shape PM. The shape of the magnet is modified by proposing a new two steps parallelogram shape magnet from a basic trapezoidal shape magnet to improve the machine performance. In this paper, the structure and design parameters of a basic model of the MR-AFVPM machine were first reviewed. After that, a new magnet shape is proposed i.e., two steps parallelogram, and the results are related to that of the basic model, and then the proposed model is optimized to further achieve the improvement in the machine performance.

II. MACHINE TOPOLOGY, DESIGN PROCEDURE, AND WORKING PRINCIPLE
The MR-AFVPM machine is comprised of three rotors and two stators as presented in Fig. 1. The magnets of the outer rotors are fixed on the rotor disc while the magnets of the inner rotor magnets are buried between the rotor cores making a spoke type structure.
The machine airgap flux density increases because of the spoke type inner rotor and thus greatly reduces its flux leakage [26]. The spoke-type assembly of the inner rotor is considered for the flux focusing effect, which efficiently increases the valuable flux and strengthens the flux lines that cross the two stators and rotor as a single loop and can increase the valuable flux in the core. The magnetism of the PMs and the rotor core can be utilized by adopting the spoke type assembly for the inner rotor [27]. The construction of the rotor also includes the determination of the magnet length and pole arc co-efficient. It is determined in [28], [29] that the pole arc co-efficient of a PM has a large effect on cogging torque.
In [30], it is presented that to minimize the leakage flux and to enhance the average value of the basic airgap flux density, both the two stators should have an offset of half slot pitch relative to each other. When there is a half slot pitch displacement between the two stators, the teeth are positioned as the generated magnetic flux in the rotor is passed through the stators with minimum magnetic reluctance, thereby increasing the basic airgap flux density. Drum-type winding is preferred to overcome the limitations of ring-type winding. The advantages of the drum-shaped winding, sometimes referred to as a toroidal stator winding, [31] are short end terminals, the simple stator core, and the simple construction. The manufacturing cost for the drum winding is reduced because fewer coils must be made here. The two stators are connected in series; hence, the total back EMF will be the sum of the two stators back EMF. The design specifications are given in Table 2.
The operating principle of MSPMVM is quite the same as the magnetically geared machines. The rotor and stator poles in the PMVM are correlated to the low and high pace rotors of magnetic gear. Stator tooth numbers in the PMVM are the same as the summation of winding pole pairs and rotor PM pole pairs which simplifies that the rotor PM pole pair numbers are more than the stator pole pair numbers.
[32]- [34]. As a result, it violates the traditional approach of having the same rotor PM pole pair numbers and winding pole pairs. Although having an unequal number of pole pairs for rotor and stator, PMVM still achieved steady torque due to the flux modulation effect produced by stator teeth. A basic design principle for PMVM is explained as in (1).
where, Z r , Z s , and p represent the PM pole pair, slot number, and pole pair number of the stator winding. Particularly,  the essential low-speed high torque feature of PMVM is achieved by the magnetic gearing effect, hence, minimum movement of rotor leads to larger flux variations [35]. It's normal practice in the design process to make Zr larger than p. When flux got a rotation of 360 degrees, the rotation achieved by the rotor is only 360/Z r as described in (2).
where ω r , ω MMF , T r , and T MMF represent the rotor speed, flux speed, rotor slot pitch, and MMF pole pitch. The flux focusing effect is quite the same as briefly described in [36].
The mathematical discussion of the basic model is briefly discussed here is the same as discussed in [37]. The input power calculations can be made as discussed in (3).
where m, E m , and I m represents the phase numbers, the amplitude of back EMF, and the phase current while the output power calculations are depicted in (2) where γ is for efficiency in (4).
The flux can be computed as in (5) where ϕ denotes the magnitude of flux, α r shows the rotor position and N p represents the numbers of rotor poles, respectively. The back EMF can be expressed as in (6).
where N ph and ω r show the phase coil turns and angular speed of the rotor, respectively. By putting (3) in (4), back EMF is determined as in (7).
where, k d , B g ,N s , D out , and D in represent the airgap flux density coefficient, lux leakage coefficient, air-gap flux density, slot numbers, pole arc coefficient, the outer and inner diameter of the rotor, respectively. The armature current is computed as in (8).
where Ae shows the electrical loading. By putting the (5), (6) in (2) gives the input power as given in (9).
where k io represents the ratio of inner and outer diameters and n rpm represents the rotor speed, respectively. Furthermore, the output torque can be computed as in (10).
It is very clear from (9) and (10), the power and torque are dependent on the A e , B g , and N p /N s . From (9), the outer diameter can be represented as in (11).
Flux density distribution is illustrated in Fig. 3 that describes the maximum flux density of 2.00T is achieved and there is no saturation observed for the MR-AFVPM machine. MR-AFVPM machine with Spoke arranged PM achieved a greater airgap flux density as given in Fig. 4, which verifies the flux focusing effect. The fast Fourier transform (FFT) of each harmonic of the airgap flux density is given in Fig. 5, where the fundamental harmonic component is at the magnetic pole pair number Z r = 17, the 35 th is the working harmonic with one rotor pole pair number.

III. PROPOSED MAGNET SHAPE
Cogging torque is caused by the attraction of magnets with the stator slots. To be specific, it occurs due to the attraction  of the magnetic flux with stator reluctance variations initiated by the stator slots.
The magnetic flux passing through each airgap is presented by θ g while R shows the stator reluctance. In (12), it is shown that to decrease the cogging torque either magnetic flux or reluctance variation must be reduced. Conversely, a reduction in magnetic flux deteriorates the machine performance and hence, it is not considered a possible solution to decrease the cogging torque. Therefore, reducing reluctance variation is considered a useful method to decreases the cogging effect. Skewing magnets or modifying the shape of magnets contribute somewhat to decrease this deviation and thus decrease the cogging torque. The physical geometry and the magnet pole-arc of the magnet will determine the maximum value and profile of cogging torque. Decreasing the magnet pole arc ratio decreases leakage of the magnet, but also decreases the magnetic flux and thus the output torque. Consequently, a settlement is commonly required throughout the design of the rotor.
In [38], a triple-rotor axial-flux vernier permanent magnet machine is investigated to explain the initial machine principle with some initial analysis. The performance of the axial flux PM magnet motors is investigated in [39] by considering the various PM shapes. The concept used in the proposed permanent magnet shaping is close to the step skewing method. However, this skewing concept is introduced, in this paper, for the proposed PMVM topology to analyze its output performances, furthermore, the optimization algorithm based on the krigging method associated with the Latin Hypercube Sampling (LHS) and the genetic algorithm (GA) is implemented to find the magnet shape.
In this paper, a new two-stage parallelogram shape magnet is considered for the performance improvement of the analyzed machine. The comparison of both the basic trapezoidal shape magnet and two-stage parallelogram shape magnet is shown in Fig. 6. In Fig. 6 it is shown that the trapezoidal shape magnet aligned with the maximum amount of stator teeth while also in Fig.6 the two steps parallelogram shape magnet aligned with less amount of stator teeth. Now according to the definition of cogging torque, there will be less attractive in the case of two steps parallelogram shape magnet and the stator teeth, hence the cogging torque will be small. Magnet skewing is an active and modest technique for minimizing the cogging torque commonly employed in PM machines. The skew angle is given in (13).
where N s and θ skew show the stator slot numbers and skew angle. Because of the difference in inner and outer radius, the skew angle is generally narrow but not the same. Though, this angle is greater in AFPM machines because the inner and outer diameters of the magnets differ. However, it can be easily used in AFPM machines because of its simpler magnet geometry and flat magnet surface than in RFPM machines. Even though the skewing rotor is a renowned technique to decrease the cogging torque in PM machines [22] but it is not common practice because of implementation difficulties. Though, a modified form known as stepped rotor magnets [28], is more common because the implementation can be made simpler. The angle of one cogging torque cycle should be chosen as the skew angle because of the leading harmonic factor of the cogging torque, this means θ skew = 360 • /18 = 20 • in mechanical. Thus, by using the two steps the skew angle for each step is θ skew_step = θ skew /2 = 10 • . A comparison is carried out between the basic trapezoidal shape magnet and the two-step skew magnet in the next section.

IV. FEA-BASED PERFORMANCE ANALYSIS
The results of the conventional trapezoidal and the two-stage parallelogram shape magnet from 3D FEA are shown in this section. The conventional trapezoidal shape magnet has large cogging torque at the expense of a slight decrease in back EMF as related to the proposed shape magnet.
The comparison between the basic model back EMF and the proposed model back EMF is shown in Fig. 7. The basic model shows the back EMF of 112.2 V rms and that of the proposed model is 113.84 V rms . Therefore, the percentage decrease resulted in back EMFs because of the proposed shaped PM is 1.46%. The decrease in back EMFs is because of the decrease in the flux density with the proposed shaped PM.
The cogging torque comparison of the basic and proposed designs is represented in Fig. 8. The conventional model peak to peak cogging torque is 16.8 Nm and that of the proposed design is 11.7 Nm. Thus, a percentage reduction of 30.35% in cogging torque is attained with the proposed shaped model. Thus, with the proposed shape magnet, there is less amount of attraction between the PM and the stator tooth. The less attraction is due to the asymmetrical arrangement of magnets on the rotor disc.
The average torque of both the basic and the proposed design is shown in Fig. 9. The average output torque of the proposed model is 193.509 Nm, and it is 183.092 Nm for the base model. Thus, the percentage improvement in average torque is 5.6894% with the proposed shaped model. Furthermore, the proposed design shows 6.26% torque ripples while the basic model has 8.88% torque ripples. Therefore, the percentage reduction of the torque ripples with the proposed model is 29.54%.
The output power comparison of the basic and proposed model is depicted in Fig. 10. The average output power of the conventional model is 8.4 kW and that of the conventional model is 8 kW. This exhibits that the proposed model has enhanced output power when compared with conventional design and the percentage increase of the average output torque is 5%. The quantitative comparison of both models is presented in Table 3.
The cogging torque reduction is 30.7% and that of the back EMF is 1.46%. The percentage improvement of the average output torque, torque ripples, and power are 5.68%, 29.54%, and 5%. As can be seen, significant cogging torque reduction is achieved in the proposed model. However, back EMF also reduces with the proposed model, but this reduction is very slight as compared to the cogging torque reduction. Furthermore, the improvement in torque and power is also obtained with the proposed model. In addition, a significant reduction in torque ripples is also achieved with the proposed model.

V. OPTIMIZATION PROCESS
The above-discussed PMVM was developed only by placing a proposed shape PMs to the slot of the rotor. This leads not only to enhance the average output torque but also to a decrease in the torque ripples. By optimization of the proposed magnet shape, it is more convenient that the torque ripple and the average output torque can become better. Therefore, an efficient optimization technique is applied that combines the kriging method and genetic algorithm (GA) [40] to optimize the rotor shape, and hence, average torque and torque ripple are improved with fixed motor size without compromising the efficiency.   As per the design specifications of the motor, the objective functions, constraints, and design variables are preserved. The two design variables are to be changed for the rotor as VOLUME 9, 2021 shown in Fig. 11. The flow of the final design procedure is illustrated in Fig. 12. The Latin hypercube sampling (LHS) is applied for searching the testing points in the computational domain of design variables X 1 , X 2 . By applying the LHS that matches certain design variables, the sample values are randomly and evenly spaced across all areas. The model came up with 2D FEA analysis creates results by performing the kriging method in the PIAnO software by PIDOTEC Inc. [41]. After that, the GA evaluates the suitability of a modeling function generated by the kriging model; the optimal results are thus found by the population. The fitness modeling function F(i) performed in GA is presented in (8) as in [33]. by PIDOTEC Inc. [39]. After that, the Generic Algorithm evaluates the suitability of a modeling function generated by the kriging model; the optimal results are thus found by the population. The fitness modeling function F(i) performed in GA is presented in (8) as in [40].
where f (i) shows the multi-objective function, P(i) depicts the penalty function and i represents the vector of the design variables. Finally, FEA tested the design results achieved in optimization. If the FEA results did not meet the requirement of the target, the range of design variables is considered again, and the procedure is repeated until they are met. Design variables, objective functions, and constraints are described below as in (15), (16), and (17), respectively.

VI. PERFORMANCE COMPARISON OF PROPOSED AND OPTIMIZED MODELS
After the optimization of the proposed magnet shape, a new model is designed based on optimal values and is analyzed by using 3D-FEA. This section presents the analysis comparison of the optimized and proposed and models. The back EMF comparison of the proposed and optimized model is given in Fig. 13, the optimized model back EMF is better than the proposed model. The back EMF achieved in the proposed model is 113.8 V rms and it is 118.7 V rms for the optimized model. Thus, a percentage enhancement in the back EMF of the optimized model is 4.30%.
Cogging torque of the optimized and proposed machines is presented in Fig. 14. The quantitative analysis of cogging torque shows that the cogging torque of the optimal machine is reduced to 11.3 Nm from 11.7 Nm. Thus, a percentage decrease of 3.53% is obtained due to the optimization of the proposed model.
The average output torque of both the machine models is presented in Fig. 15. The output torque of the optimal machine is more than the proposed machine. The average output torque of the optimized model is 200.6 Nm, and the proposed model torque was 193.509 Nm. This resulted in a 3.5349% improvement in the average output torque due to optimization.
The output power of both designs is given in Fig. 16. The average output power of the optimized design is 8.45 kW and that of the proposed model is 8.4 kW. This exhibits the more output power is achieved in the optimized design.
The quantitative analysis of both models is given in Table 4. As it is clearly seen from Table 4, the performance of the proposed machine is improved in the optimized machine due to a suitable optimization process. The percentage reduction in   cogging torque is 3.5398% and the percentage enhancement in the average output torque is 3.5349%.
MR-AFVPM machine topology applies flux focusing effect which can increase the airgap flux density and magnet flux utilization as explained in [36]. Hence, fewer ampereturns are desired for required output power. So, reduced winding reactance is achieved which can improve the power factor of this topology. Also, this topology discourages leakage flux between adjacent PMs. The power factor is also considered for machine models and improved as described in Table 4.

VII. CONCLUSION
The MR-AFVPM machine topology is adopted in the design with four airgaps sharing the same rotor for more torque production and reasonable space utilization. The spoke-type structure of the inner rotor is developed for the flux focusing effect, which can efficiently increase the corresponding flux while forcing the flux lines to cross throughout the entire machine as compared to the other two separate torque components as in the case of a normal dual-stator PM machine. Reducing the reluctance in the main flux path and making a new path for main flux, for this purpose the two stators are shifted by an angle to minimize the flux leakage. Due to the symmetrical and multi-stack design, the machine has a better fault-tolerant capability. A new two stages parallelogram shape magnet is proposed, and the reduction achieved in PM volume is 13% as compared to the conventional shape magnet while the improvement in performance is not compromised.
The optimization of the proposed shape is carried out to achieve the better performance of the analyzed machine in terms of back EMF, cogging torque, average output torque, power factor, and output power. The 30.35% reduction in the cogging torque is achieved due to the proposed PM shape and 3.41% further reduction is attained in the optimization process. The average output torque enhancement is 9.56% in the final optimized model as compared to the initial model. The power factor is improved to 0.88 in the optimized model. The comparison of the analyzed multi-rotor AFVPM machine consists of various topology i.e., singlesided, and double-sided AFVPM machines, with the analytical and experimental validation are to be considered for future work. RABIAH BADAR received the M.S. degree in computer engineering and the Ph.D. degree in electrical engineering from COMSATS University Islamabad, in 2007 and 2009, respectively. From February 2015 to May 2015, she was with Namal Institute, Mianwali, Pakistan, as an Assistant Professor. After that, she joined COMSATS University Islamabad as an Assistant Professor. She served as a pioneer in charge of the B.S. Electrical (Power) Engineering Program. She is the author of many international, peer-reviewed journal research articles, conference papers, and book chapters. She has successfully supervised many undergraduate and graduate research thesis. Her research interests include artificial intelligence, optimization, soft computing, power system stability and control, nonlinear adaptive control, FACTS, HVDC, and renewable energy systems. He is currently an Associate Professor with the School of Electrical and Electronics Engineering, Chung-Ang University, Seoul. His research interests include the analysis and optimal design of next-generation electrical machines using smart materials, such as electromagnet, piezoelectric, and magnetic shape memory alloy. VOLUME 9, 2021