Design and Optimization of Support and Drive System for Magnetic Levitation Air Compressor for Fuel Cells

Abstract

The 5-degree-of-freedom active magnetic bearings (5-DOF AMB) and high-speed permanent magnet synchronous motor (HPMSM) were combined and applied to energy-recovery-type air compressors for fuel cells, which gives full play to the advantages of both and meets the design requirements for air compressors in fuel cells. Based on the energy recovery air compressor for fuel cells with a power of 30 kW and a rated speed of 100,000 rpm, this paper combined 5-DOF AMB with HPMSM and used it as its support and drive system. Multi-physics field and multi-objective optimization were carried out by integrating the multi-physics field with the Multi-objective Grey Wolf Algorithm (MOGWO), and the feasibility of the design of the system and its reliability were verified using finite element software.

1. Introduction

At present, proton exchange membrane fuel cells (PEMFCs) are most widely used in fuel cell vehicles due to their high efficiency, zero emission, low noise, and high energy density [1]. Among them, the fuel cell compressor is one of the core components of the fuel cell engine air supply system, which is called the “lung” of the fuel cell vehicle. It is used to supply gas to the cathode of PEMFCs, and its performance directly affects the whole system’s efficiency [2,3]. However, it is found that the air compressor consumes a large amount of power, which accounts for about 80% of the power consumption of the hydrogen fuel cell auxiliary system, 20–30% of the total output of the fuel cell stack, and 20% of the cost of the hydrogen fuel cell system [4,5,6]. This not only reduces the efficiency of the overall fuel cell system but also increases the cost of the hydrogen fuel cell system, which hinders the wider application of PEMFCs.

Relevant studies show that the exhaust gas produced by the electrochemical reaction of PEMFCs has the characteristics of a low temperature, and medium and high pressure. The temperature can generally reach 80–120 °C, and the pressure is about 80~90% of the PEMFC inlet pressure (i.e., the outlet pressure of the air compressor), which still has a high energy recovery value [7,8,9]. Therefore, experts at home and abroad have carried out in-depth research on how to recover energy from PEMFC exhaust characteristics. At present, energy recovery air compressors using expanders are considered to be an ideal solution to reduce the power consumption of fuel cell air compressors [10,11].

In addition, fuel cell compressors have the merits of high efficiency and power density, good dynamic response, and the ability to achieve ultra-high rotor speeds aside from not producing any oil contamination [12] Therefore, traditional mechanical bearings and low-speed motors are increasingly unable to meet the requirements of high speed, high efficiency, and low power consumption required by future fuel cell air compressors. In this regard, various experts at home and abroad have carried out in-depth research on the support and drive system of fuel cell air compressors. Among them, HPMSM has outstanding advantages such as oil-free operation, high power density, high efficiency, and large high-efficiency speed range, making it the best choice among the drive motors for fuel cell air compressors [13]. As a component supporting the stable and high-speed operation of the rotor system of fuel cell compressors, the bearing plays an essential role in ensuring the stability and safety of fuel cell compressors. The magnetic bearings and gas bearings, which are frictionless and do not require an oil lubrication system, are just suitable for fuel cell compressors.

The gas bearing uses the high-pressure gas film between the bearing and the shaft neck to support the rotor. According to the principle, it can be categorized into dynamic pressure gas bearings and static pressure gas bearings. The former will cause serious wear and tear during low-speed stages such as the start-stop stage, which will affect the service life of bearings. In addition, dynamic pressure gas bearings also have problems such as instability, limited bearing capacity, and high requirements for bearing materials and machining accuracy during high-speed stages [14]. For the latter, the bearing capacity of the hydrostatic gas bearing is greater than that of the dynamic gas bearing, but it requires an air supply system to ensure stability and reliability of the gas bearing, and its structure and processing are more complicated [15,16].

The magnetic bearing uses the magnetic field force to levitate the rotor in the air, which has the characteristics of no friction, low energy consumption, low noise, long service life, no need for lubrication, no oil pollution, and adjustable dynamic performance [17,18]. Its bearing capacity is much higher than that of gas bearings, which can meet the requirements of complex application scenarios. Moreover, it is very convenient to maintain, repair, and replace. Therefore, magnetic bearings are considered the ideal solution for high-speed rotor support, showing unparalleled advantages in a variety of applications [19,20]. According to the different degrees of freedom and bearing capacity, magnetic bearings can be divided into five degrees of freedom active magnetic bearings (5-DOF AMB), passive magnetic bearings, and hybrid magnetic bearings [21]. Among them, the 5-DOF AMB is characterized by active controllability and has the potential to become a key unit in future intelligent machinery with a broad application prospect [19]. Therefore, this paper takes 5-DOF AMB as the object of study. All the magnetic bearings involved below refer to 5-DOF AMBs unless otherwise specified.

The combination of 5-DOF AMB and HPMSM in air compressors, which gives full play to the advantages of both, meets the design requirements of the air compressor in the fuel cell [22]. Although the performance of the 5-DOF AMB is closely related to the design of the control system, the design of its mechanical structure also has a crucial impact on it [23]. From the perspective of increasing the bearing capacity against vibration and shock, the larger the axial length and volume of the 5-DOF AMB, the better. However, increasing the axial length of the 5-DOF AMB will increase the axial length of the motor rotor, reduce the critical rotational speed of the rotor, and increase the windage loss [24]. Therefore, to improve the performance index, the structural dimensions of the 5-DOF AMB should be optimally designed. With the development of computer technology and the proposal of various intelligent algorithms, a variety of intelligent optimization algorithms have been successfully applied to the optimization design process of 5-DOF AMB.

Zhang SS et al. [25] and Smirnov, A et al. [26] optimized the overall structural parameters of 5-DOF AMB based on the Multi-objective Genetic Algorithm (MOGA), and the optimized simulation results showed that the bearing capacity of 5-DOF AMB was greatly improved compared with that before optimization. Santosh N et al. took the minimum coil weight and copper loss as the objective functions and used MOGA to optimize the structure of the RMB. After optimization, the coil weight of the RMB decreased by 37.19%, while the power loss increased by 43.72% [27]. Zhou J et al. [28] and Zhou Y et al. [29] used the multi-island genetic algorithm (MIGA) to optimize the structural design of a 5-DOF AMB with the objectives of maximizing the electromagnetic force and minimizing the rotor amplitude, respectively. As a result, the bearing capacity of the TMB was improved by 11.88% and the imbalance amplitude was reduced by nearly 50% compared with the pre-optimization period. Wang et al. used the fruit fly optimization algorithm to perform multi-variate and multi-objective optimization for the bearing capacity, volume, and axial length of the radial magnetic bearing (RMB) in the 5-DOF AMB; through the optimization, the bearing capacity of the RMB was increased by 50%, and the axial length and volume were reduced by 30.6% and 19.3%, respectively [30]. Moreover, optimization based on the NGA-II algorithm was implemented by Wang et al. [31], taking maximum first-order intrinsic frequency and minimum volume as the objective function; by Feng C [32] from Shandong University of Technology, taking maximum electromagnetism and minimum volume as the objective function; and by Cao Z [19] from Southeast University, taking minimum copper loss and volume as the objective function. Through these optimizations, the first-order intrinsic frequency of the 5-DOF AMB was increased from 443.14 Hz to 623.91 Hz; the minimum space volume of the thrust magnetic bearing (TMB) and radial electromagnetic bearing (RMB) was reduced by 9.37% and 16.58%, respectively; and the copper loss of the TMB and RMB was reduced by 49% and 29.1%, respectively. Betancor J et al. minimized the volume of RMBs by comparing the Genetic Algorithm (GA) and Pattern Search (PS) methods. After optimization, the volume was reduced by about 35%. The PS method has a larger diameter but a shorter bearing length as compared to GA. Nonetheless, GA produces thicker AMBs and smaller outer diameters [33]. Yadav et al. optimized the 5-DOF AMB based on the heat transfer search (HTS) algorithm with the minimum volume as the optimization goal. The study found that the volume of the HTS-optimized 5-DOF AMB was reduced by about 23% compared with the volume of the 5-DOF AMB obtained by more popular optimization techniques, such as Genetic Algorithm (GA) and Pattern Search (PS) [34].

The 5-DOF AMB is a typical mechatronic product, which involves a variety of technical fields such as mechanical engineering, electromagnetism, material science, rotor dynamics, control engineering, and computer science, and its design is relatively complex [19]. Therefore, compared with the wide application of gas bearings in fuel cell compressors, the application of 5-DOF AMB in fuel cell compressors has received little attention at home and abroad, and there exists a research gap in this aspect. This paper introduces the electromagnetic design of the magnetic levitation air compressor support and drive system for fuel cells and its algorithmic optimization process and points out some comprehensive considerations in its system design. The contributions of this paper are summarized below:

• Combining 5-DOF AMB with HPMSM and realizing integrated design and application in fuel cell air compressors fills the research gap in China.
• The multi-physics optimization design of HPMSM was carried out. After optimization, the end length of HPMSM was reduced by 37%. The critical speed of the rotor system has been greatly improved.
• Based on the MOGWO algorithm, the structural parameters of 5-DOF AMB are optimized with the maximization of bearing capacity and the minimization of volume and drag loss as the objective functions. After optimization, the volume of RMBs is reduced by 36.4%, the drag loss is reduced by 14.3%, and the bearing capacity is increased by 10%. The volume of the TMB has been reduced by 17.5%, the drag loss has been reduced by 12.5%, and the load-bearing capacity has been increased by 4%.
• Through the finite element software, the feasibility and reliability of the system design are ensured from the aspects of electromagnetism and rotor dynamics.

The content of this paper is structurally organized as follows: Section 2 describes the integrated design of the support and drive scheme for fuel cell air compressors and its integrated optimization design of HPMSM considering multiple physical fields such as rotor dynamics; Section 3 illustrates the detailed design of the initial theoretical parameters of the 5-DOF AMB based on the actual requirements of fuel cell air compressor; Section 4 defines and compares its optimization objective with the structural parameter optimization process based on the MOGWO algorithm; and Section 5 incorporates extensive simulation analysis and calibration with finite element software to ensure the feasibility of its system design and its reliability. Finally, Section 6 summarizes the whole paper. This paper can provide some theoretical value and practical significance for the application of 5-DOF AMB in fuel cell systems.

2. Support and Drive Scheme and Their HPMSM Design

2.1. Support and Drive Scheme Design

The principle of the energy-recovery-type air compressor for fuel cells is as follows:

As shown in Figure 1, the outside air enters the compressor from the inlet at the left and is discharged from the outlet after being compressed. Then, it goes through the intercooler and humidifier to enter the fuel cell stack for electrochemical reaction. The electrochemical reaction at the fuel cell stack produces low-temperature, and medium- and high-pressure exhaust gas [7,8,9]. The exhaust gas goes through the humidifier and enters from the inlet of the expander, where the energy of the exhaust gas is recovered via the expansion of the expander turbine and is finally discharged from the outlet of the expander [35].

The energy-recovery-type magnetic levitation air compressor initially designed in this study has its internal rotor system shown in Figure 2. As shown in the figure, the compressor impeller, high-speed motor, and expander turbine are connected coaxially and supported by the 5-DOF AMB; the HPMSM is used to drive the rotation of the rotor shaft, which is placed in the center; the compressor impeller and the expander turbine are located at both ends of the main shaft to balance the axial forces so that the bearing capacity required by the TMB can be decreased, thereby reducing the volume of the TMB and the diameter of its thrust disc. In addition, eddy current sensors are adopted for the system, and it is necessary to retain a certain safety distance among the sensors to prevent electromagnetic interference from affecting the sensor accuracy.

2.2. HPMSM Electromagnetic Design and Analysis

2.2.1. HPMSM Design Considering Multi-Physics Fields

The energy recovery magnetic levitation air compressor system for fuel cells is a typical mechatronic system [19], so it requires comprehensive consideration and optimized design. The mass of the rotor system and the external disturbances to which it is subjected determine the selection of the maximum bearing capacity of the 5-DOF AMB, and the diameter of the rotor journal of the motor determines the selection of the inner diameter of the RMB rotor and the inner diameter of the TMB thrust disc. Meanwhile, the design of the 5-DOF AMB system is also inseparable from the dynamic characteristics of the rotor of the supported object [36]. Therefore, it is necessary to optimize the design of HPMSM before designing 5-DOF AMB.

Figure 3 shows the schematic diagram of the power and recovery efficiency of the energy recovery air compressor system designed by our research group [35]. It can be seen from the figure that both the power of the compressor and expander and the energy recovery efficiency of the energy recovery efficiency increase with the increase in rotational speed. The energy recovery efficiency of the expander reached its maximum at a speed of 100,000 rpm and the power required for the energy-recovery-type air compressor in this study reached its maximum at 26.58 kW. Considering factors such as overload, the rated power of the motor in this study was set at 30 kW, and a high-power density design was adopted. The main design indexes of HPMSM in this study are shown in

Table 1. Main design indexes of HPMSM.

Parameter	Value
Rated power	30 kw
DC bus voltage	470 V
Rated rotational speed	100,000 rpm
Efficiency	>96%
Power factor	>0.9

.

HPMSM can have an impact on the 5-DOF AMB system. In particular, the volume of the 5-DOF AMB system is much larger than that of the air bearing, thus affecting the critical speed of the rotor as well as the reliability and safety of the system as a result. Considering this factor, before the algorithmic optimization of 5-DOF AMB, multi-physics field analysis was carried out for the optimized design of HPMSM to ensure the safety and reliability of the whole system, as shown in Figure 4.

The specific analysis is as follows.

(1)

Stator–rotor structure design considering electromagnetic performance: As shown in Figure 5, the 1-pair pole rotor structure suitable for HPMSM has 1-pair solid and ring-shaped permanent magnet structures, whose magnetizing directions are parallel with each other. Both structures have their advantages and disadvantages. This study chose a solid structure. The solid type uses a whole-piece permanent magnet, which has high structural strength and a smaller radial length of HPMSM and is easy to fabricate industrially [37,38]. In terms of the stator structure of HPMSM, the stator groove structure is usually divided into three types: no groove, few grooves, and multiple grooves [39]. For the HPMSM with a pole pair of 1 designed in this paper, the coordination of integer slots is mostly adopted, and to reduce harmonics, the multi-slot structure is generally adopted. This also contributes to the reduction in the peak cogging torque and the magnetic flux density ripple on the rotor sheath surface. However, the increase in the number of slots will reduce the width of the stator teeth and the area in the slots, which is not conducive to the offline operation of the stator winding, affects the structural strength, and increases the manufacturing cost [39,40]. Therefore, combined with the parallel magnetization method of solid permanent magnet in this paper, the slotted stator structure with 2 poles and 24 slots was finally selected.

(2)

Selection of permanent magnet and core material considering heat generation: To ensure the performance of HPMSM, reduce the size of the motor, and avoid demagnetization at high temperatures, the permanent magnet material with high magnetic energy product and high-temperature resistance was preferred. Given the high operating temperature in this paper, a samarium cobalt (SmCo30, which is manufactured by Hangzhou Permanent Magnet Group Co., Ltd., Hangzhou, China) permanent magnet was used. Meanwhile, the high frequency of HPMSM leads to high eddy current loss. Therefore, it is necessary to use core materials with excellent properties such as ultra-thinness, high permeability, high resistivity, and low loss coefficient to effectively suppress the eddy current loss in the core [41]. The 0.1 mm 10JNEX900 silicon steel sheet (10JNEX900, which is manufactured by Kawasaki in Tokyo, Japan) has a lower specific iron loss, but the material is too thin and not easy to process. Considering that the structure of the stator punching piece in this paper is complex and the tooth width is small, the 0.2 mm B20AT1500 silicon steel sheet (B20AT1500, which is manufactured by China Baowu Iron & Steel Group Co., Shanghai, China) is selected as the core material. The main advantages are a low unit loss value, easy machining, and a high tensile strength [42].

(3)

Dimensional design and sheath material selection considering rotor strength: Unlike normal speed motors, the selection of a rotor outer diameter and length for HPMSM was prioritized to ensure that the permanent magnets are not damaged by centrifugal tension. A preliminary estimation of the rotor outer diameter can be made from the following equations and in conjunction with the electromagnetic design [43,44].

$$F=m{v}^2/r=A\rho r^2{\omega }^2$$

(1)

$${\sigma }_t=F/A=\rho r^2{\omega }^2$$

(2)

$${\sigma }_t\le \left[{\sigma }_t\right]/S$$

(3)

$$v_{max}=\sqrt{\left[{\sigma }_t\right]/S\rho }$$

(4)

$$D_{max}=2 v_{max}/\omega$$

(5)

where $m$ is the mass of the permanent magnet rotor, $v$ is the rotor linear velocity, $r$ is the radius of the rotor, $A$ is the outer surface area of the material, $\rho$ is the density of the material, $\omega$ is the angular velocity of the rotation, $\left[{\sigma }_t\right]$ is the allowable stress of the material, $S$ is the safety margin. $v_{max}$ is the maximum linear velocity of the rotor surface, and $D_{max}$ is the maximum outer diameter of the rotor. In addition, to ensure reliable operation of the rotor, it is necessary to add a sheath outside the permanent magnet to protect the permanent magnet. The commonly used rotor sheaths are mainly made of carbon fiber composite materials, metal materials, etc. Since the rotor structure in this paper is a solid permanent magnet structure, the alloy material GH4169(GH4169, which is manufactured by China’s Jiangxi Province Baoshunchang Special Alloy Manufacturing Co., Xinyu, China) with higher strength was selected to stably connect the spindle and permanent magnets.

(4)

Winding design considering rotor dynamics: As shown in Figure 6, the stator winding suitable for HPMSM mainly has two structures: double-layer short-pitch distributed winding and toroidal winding [45]. In particular, when the number of pole pairs is 1, the toroidal winding can significantly reduce the axial length of the rotor relative to the double-layer short-pitch distributed winding, thereby increasing the critical speed of the rotor system [37,45].

Table 2. End lengths of two winding schemes.

Winding Structure	End Length (mm)
Double-layer short-pitch distributed winding	27
Toroidal winding	17

shows the end length values of double-layer short-pitch distributed winding and toroidal winding. From the table, it can be seen that the end length is 27 mm for the double-layer short-pitch distributed winding scheme, while the end length is approximately equal to the slot depth of the motor (17 mm) for the toroidal winding scheme, which is a reduction of 37% compared to the former.

Combined with the multi-physics field analysis in the previous sections, the structure and materials of the motor stator and rotor were selected to finalize the electromagnetic scheme for the HPMSM with a rated speed of 100,000 r/min and a rated power of 30 kW, as required in this paper. Some of its parameters are shown in

Table 3. The design scheme of HPMSM.

Type	Parameter	Type	Parameter
$\mathrm{Permanent} \mathrm{magnet} \mathrm{outer} \mathrm{diameter} \left(\mathrm{m}\mathrm{m}\right)$	32	Stator outer diameter (mm)	96
$\mathrm{Air} \mathrm{gap} \left(\mathrm{m}\mathrm{m}\right)$	3.5	Axial length (mm)	61
Pole pairs	1	Number of stator slots	24
Permanent magnet material	Smco30	Stator core material	B20AT1500
Sheath material	GH4169	Winding structure	Toroidal winding

.

2.2.2. Finite Element Analysis of HPMSM

The 2D model was built according to the motor design scheme in Section 2.2.1, and the motor rotor shaft was omitted due to the use of solid permanent magnets in this paper, as shown in Figure 7.

To verify the electromagnetic performance of the design scheme, the performance of the motor under no-load and load conditions is analyzed in this section, and the main indicators include back electromotive force under no-load conditions, cogging torque, torque under load conditions, magnetic dense cloud diagram, and magnetic field line distribution diagram, as shown in Figure 8, Figure 9, Figure 10 and Figure 11.

As can be seen from Figure 8 above, the no-load back EMF symmetry of HPMSM is good, the peak value of line back EMF is close to 417 V, close to the bus DC voltage of 470 V, which meets the design requirements of the motor, and the back EMF sinusoidality of the 2-pole motor is generally very good, and its harmonic distortion rate is negligible. Figure 9 below is a load torque waveform diagram, the average value of torque is 2.8755 N·m, the maximum value is 2.8821 N·m, and the minimum value is 2.8665 N·m at steady state. The torque fluctuation is 0.31%, which meets the design requirements. Due to the grooving of HPMSM, there will be more or less cogging torque in HPMSM, as shown in Figure 10 below, the maximum value of cogging torque is 0.00172 N·m, which is much less than the value of rated torque, that is, the influence of cogging torque on the motor is very small.

Figure 11 shows the magnetic dense cloud and magnetic field line distribution map of HPMSM under the rated load. It is not difficult to find that most of the magnetic density under the load condition of HPMSM is about 1.3 T, and only the stator yoke exceeds 1.8 T. At the same time, the magnetic flux leakage at the stator notch is small, and the position of the winding is biased towards the bottom of the groove, which helps to inhibit the skin effect of the wire and reduce the AC loss of the winding. In addition, the large gap at the top of the stator slot contributes to the heat dissipation of HPMSM. In summary, the electromagnetic characteristics of the motor meet the technical requirements.

3. Initial Theoretical Design of 5-DOF AMB

The goal of the electromagnetic design and optimization of the 5-DOF AMB is as follows: The designed 5-DOF AMB is expected to provide sufficient support for the rotor and resist external disturbances. Thus, the volume and loss should be reduced as much as possible. Figure 12 shows the electromagnetic design and optimization flow of the 5-DOF AMB in this paper.

3.1. Theoretical Design of RMB

(1)

Determination of main design indexes of RMB

Among them, the static load of 5-DOF AMB mainly comes from the gravity of the rotor itself, while the dynamic load mainly comes from the unbalanced vibration of the rotor and the disturbance of the external load. The static bearing capacity of the 5-DOF AMB should not be less than $F_s$ gravity of the rotor itself, while the dynamic bearing capacity of $F_d$ should include the centrifugal force caused by the unbalanced vibration of the rotor and the impact force caused by external bumps and other shocks [9,19,46]. Therefore, the required bearing capacity of the RMB needs to meet the following:

$$F=F_s+F_d\ge mg+me{\Omega }^2+ma$$

(6)

$$e=G/\Omega =9549G/n$$

(7)

where $F$ is the required bearing capacity of the RMB; $F_s$ is static bearing capacity; $F_d$ is the dynamic carrying capacity; $m$ is the mass of the rotor system; $e$ is the allowable amount of residual unbalance; $a$ is the impact acceleration, here taken as 5 g; $G$ is the rotor dynamic balance level; and $n$ is the rotor speed.

For example, according to the rotor balancing requirements, expanders, compressors, and generator rotors with speeds above 950 rpm have a dynamic balancing accuracy class of G2.5 [40]. When the rotor dynamic balance class is G2.5, for the rotor with a mass of 1.6 kg and a rated speed of 100,000 rpm, $e$, $F_s,$ and $F_d$ can be calculated as 23.87 μm, 15.68 $\mathrm{N},$ and 120.29 $\mathrm{N}$, respectively, and the required bearing capacity in the radial direction should not be less than 136 N. According to the design experience, the maximum bearing capacity of two RMBs needs to meet the following:

$$F_{max}=\left(1\sim 2\right)\times F$$

(8)

Taking a safety margin of about 1.5, the total maximum load capacity required for two RMBs is approximately 200 N, that is, the maximum load capacity required for a single RMB should be not less than 100 N. In addition, according to the HPMSM rotor diameter, the outer rotor diameter of the RMB was taken as 35 mm. The input design indexes for the electromagnetic design of the RMB in this paper are shown in

Table 4. Main design indexes of RMBs.

Parameter	Value
$\mathrm{Maximum} \mathrm{bearing} \mathrm{capacity} F_{max}$	$\ge 100 \mathrm{N}$
Rated rotational speed n	$100,000 \mathrm{r}\mathrm{p}\mathrm{m}$
Winding heat dissipation conditions	Natural air cooling
Outer diameter of stator D4	≤110 mm
Outer diameter of rotor D	35 mm
Stator thickness Lr	≤16 mm
$\mathrm{Maximum} \mathrm{allowable} \mathrm{current} I_{\mathrm{m}\mathrm{a}\mathrm{x}}$	4 A

:

(2)

Selection of RMB structure, and stator and rotor core materials

Compared with the homopolar, the anisotropic RMB has smaller circumferential dimensions and a simple structure and is easy to process, and its application is more widespread [19,47]. Among them, the distribution of magnetic poles of the heteropolar structure can be categorized into NSSN and NSNS. When the rotor rotates, the direction of the magnetic lines of force distributed by the NSSN changes fewer times than that of the NSNS in one cycle, which results in smaller losses. Therefore, it becomes a better choice. Therefore, in this paper, the magnetic pole NSSN distribution of the heteropolar radial magnetic levitation bearing is selected. In addition, the core material of a magnetic levitation-bearing stator rotor needs to have the characteristics of high saturation magnetic density, high relative permeability, low iron dissipation, easy processing, and high mechanical strength [19]. A B20AT1500 silicon steel sheet as the core material better meets these requirements, so in this paper, radial magnetic levitation bearing stator rotor core is made by stacking B20AT1500 silicon steel sheets.

(3)

Initial theoretical parameter design

In the actual design process, the number of magnetic poles of the RMB and the value of the air gap between the stator and the rotor were selected based on the outer diameter of the rotor [48], while taking into consideration the limitations of machining accuracy, sensor accuracy, and control system performance [19]. In this paper, the number of magnetic poles of RMBs was selected as 8 and the air gap value was selected as 0.4 mm.

The geometric dimensions of RMB are shown in Figure 13, and the following Formulas (9)–(22) were used to give theoretical calculations for the structural parameters in the figure.

$$D_2=D+2c$$

(9)

$$P=\pi D_2/2N_p$$

(10)

$$A={\mu }_0{F}_{max}/\left(B_{max}^2\cos \alpha \right)$$

(11)

$$W_p=\left(1\sim 1.5\right)\times P$$

(12)

$$W_r=P$$

(13)

$$D_1=D-2W_r$$

(14)

$$Lr=A/P$$

(15)

$$N_1\le 2B_{\mathrm{m}\mathrm{a}\mathrm{x}}c/\left({\mu }_0{I}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)$$

(16)

$$d_c\ge \sqrt{4I_{\mathrm{m}\mathrm{a}\mathrm{x}}/\left(\pi J\right)}$$

(17)

$$A_{cu}=N\pi {d_c}^2/\left(4\eta \right)$$

(18)

$$H=\sqrt{D_2^2+8N_p{A}_{cu}/\pi }-D_2/2$$

(19)

$${ D}_3=D_2+2H$$

(20)

$${ D}_4=D_3+2 W_p$$

(21)

$$I_0=0.5I_{\mathrm{m}\mathrm{a}\mathrm{x}}$$

(22)

where ${ D}_2$ is the inner diameter of the stator; $D$ is the inner diameter of the stator, which was taken as 35 mm; $P$ is the magnetic pole width; $c$ is the unilateral air gap value, which was taken as 0.4 mm; $N_p$ is the number of magnetic poles, which was taken as 8; $A$ is the magnetic pole area; ${\mu }_0$ is the air permeability; $\alpha$ is the coefficient of the force caused by the magnetic pole dispersion, and usually, 8 poles are obtained, i.e., $\alpha =\pi /8$; and $B_{\max }$ is the saturated magnetic density of the core material. According to the design experience, the saturated magnetic density at the air gap was taken as 1 T; $F_{\max }$ is the maximum bearing capacity required for RMB; $W_p$ is the width of the stator yoke; $W_r$ is the width of the rotor yoke; ${ D}_1$ is the inner diameter of the rotor; $Lr$ is the thickness of the stator; $N$ is the number of coil windings in a pair of magnetic poles; $d_c$ is the minimum wire diameter of the coil winding wire; $I_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum allowable current; and $J$ is the maximum allowable current density in the conductor under certain heat dissipation conditions. Under natural air cooling conditions, the value is $J$ generally taken as $4\sim 8 \mathrm{A}/{\mathrm{m}\mathrm{m}}^2$; $A_{cu}$ is the area of the stator coil cavity; $\eta$ is the slot full factor of the coil winding, which was taken as 0.6; $H$ is the stator window depth; $D_3$ is the medium diameter of the stator; $D_4$ is the outer diameter of the stator; and $I_0$ is the bias current.

The structural parameters of the RMB were calculated as shown in

Table 5. Initial design parameters of RMB.

Parameter	Value
$\mathrm{Number} \mathrm{of} \mathrm{poles} N_p$	8
$\mathrm{Inner} \mathrm{diameter} \mathrm{of} \mathrm{rotor} D_1$	21 mm
Outer diameter of rotor D	35 mm
$\text{Single-side} \mathrm{air} \mathrm{gap} c$	0.4 mm
$\mathrm{Inner} \mathrm{diameter} \mathrm{of} \mathrm{stator} D_2$	35.8 mm
$\mathrm{Intermediate} \mathrm{diameter} \mathrm{of} \mathrm{stator} D_3$	84.8 mm
$\mathrm{Outer} \mathrm{diameter} \mathrm{of} \mathrm{stator} D_4$	100.3 mm
$\mathrm{Stator} \mathrm{pole} \mathrm{width} P$	7.03 mm
$\mathrm{Stator} \mathrm{yoke} \mathrm{width} W_p$	7.73 mm
$\mathrm{Stator} \mathrm{thickness} Lr$	14 mm
$\mathrm{Individual} \mathrm{pole} \mathrm{area} A$	94.5 mm2
$\mathrm{Area} \mathrm{of} \mathrm{sator} \mathrm{coil} \mathrm{cavity} A_{cu}$	290 mm2
$\mathrm{Number} \mathrm{of} \mathrm{coils} \mathrm{of} \mathrm{a} \mathrm{pair} \mathrm{of} \mathrm{magnetic} \mathrm{poles} N_1$	190
$\mathrm{Wire} \mathrm{diameter} { d}_c$	1.12 mm
$\mathrm{Maximum} \mathrm{current} I_{\mathrm{m}\mathrm{a}\mathrm{x}}$	4 A
$\mathrm{Bias} \mathrm{current}{ I}_0$	2 A
$\mathrm{Maximum} \mathrm{air} \mathrm{gas} \mathrm{permeability} B_{\max }$	1.2 T
$\mathrm{Maximum} \mathrm{load} \mathrm{capacity} F_{\max }$	100 N

.

3.2. Theoretical Design of TMB

(1)

Determination of main design indexes of TMB

The rotor system of the energy-recovery-type magnetic levitation air compressor for fuel cells should not only prevent the rotor from touching the protection bearing in the radial direction but also prevent the rotor from generating a large displacement in the axial direction during operation. The axial displacement of the rotor is mainly caused by the aerodynamic force generated by the impeller impacting the airflow in high-speed rotation. The energy-recovery-type magnetic levitation air compressor designed in this paper has its compressor and expander to balance out part of the axial force, but the remaining axial force still needs to be balanced by the thrust disc and TMB.

Figure 14 [35] shows a schematic diagram of the impeller under force, where the impeller inlet wheel cover is subjected to the pressure of the gas flow and generates a gas force $F_1$ directed towards the root of the impeller. At the same time, part of the airflow at the impeller outlet will flow into the back gap of the blade, generating a gas force $F_2$ directed towards the top of the blade.

After analysis and calculation, the overall axial force distribution of the rotor system of the energy-recovery-type magnetic levitation air compressor is shown in Figure 15.

According to the calculation, when the air compressor operates at the design speed after the axial force of the main shaft is balanced by the compressor and the expander, the total remaining axial force of the shaft system is 75.82 N [35]. The force that points to the compressor from the expander is borne by the thrust disc near the compressor side. The TMB is mounted on both sides of the thrust disk to share the applied axial force and ensure that the shaft system does not experience large axial displacements. In addition, according to the design experience, taking twice the safety margin, the maximum load capacity required for the TMB is 152 N.

At the same time, considering the strength and dynamics problems caused by the large diameter of the thrust disc, the diameter of the thrust disc is limited to ≤80 mm. The input design indexes of the TMB electromagnetic design in this paper are shown in

Table 6. Main design indexes of TMB.

Parameter	Value
$\mathrm{Maximum} \mathrm{bearing} \mathrm{capacity} F_{max}$	≥152 N
Rated rotational speed n	$100,000 \mathrm{r}\mathrm{p}\mathrm{m}$
Winding heat dissipation conditions	Natural air-cooling
$\mathrm{Diameter} \mathrm{of} \mathrm{thrust} \mathrm{disc} D_{s5}$	≤80 mm
Outer diameter of rotor D	35 mm
$\mathrm{Maximum} \mathrm{allowable} \mathrm{current} I_{\mathrm{m}\mathrm{a}\mathrm{x}}$	4 A

.

(2)

Selection of TMB structure, and stator and rotor core materials

Under the condition of the same structural parameters, the heteropolar TMB has serious magnetic circuit coupling and more magnetic leakage [19]. Therefore, in this paper, the homopolar TMB was used. Electrotechnical pure iron has the advantages of good electromagnetic characteristics, easy access to materials, good toughness, convenient processing, inexpensiveness, and so on, but the disadvantage is low strength. Considering that the rotor system speed is up to 100,000 rpm, it cannot be used as the material for the thrust disc of the TMB in this paper. Because of the processing difficulty, mechanical strength, and loss of the materials used in the TMB, Armco Pure Iron (99.85% Fe) was chosen for the stator core of the TMB, and $4340$ was used for the thrust disc.

(3)

Initial theoretical parameter design

The geometric dimensions of TMB are shown in Figure 16. Similarly, the initial design parameters can first be determined by the following empirical equations [19,42].

$${ A}_{1 }={A_{2 }=A}_0$$

(23)

$$A_0={\mu }_0{F}_{\max }/B_{\mathrm{m}\mathrm{a}\mathrm{x}}^2$$

(24)

$$s\ge 7\mathrm{c}$$

(25)

$$D_{s1}=D+2s$$

(26)

$$D_{s2}=\sqrt{4A_0/\pi +{D_{s1}}^2}$$

(27)

$$N_2\le 2B_{max}c/\left({\mu }_0{I}_{max}\right)$$

(28)

$$d_c\ge \sqrt{4I_{max}/\left(\pi J\right)}$$

(29)

$$A_w=N\pi d_c^2/4\eta$$

(30)

$$H=H_1=H_2=\sqrt{A_w}$$

(31)

$$D_{s3}=D_{s2}+2H$$

(32)

$$D_{s4}=\sqrt{4A_0/\pi +{D_{s3}}^2}$$

(33)

$$h\ge A_0/\left(\pi D_{s2}\right)$$

(34)

$$L=2h$$

(35)

$$D_{s5}\ge D_{s4}$$

(36)

$$I_0=0.5I_{max}$$

(37)

where A₁ and A₂ are the magnetic pole areas of the inner and outer rings of TMB, which are equal; ${\mu }_0$ is the air permeability; $F_{\max }$ is the maximum bearing capacity required by TMB, which was taken as 152 N; $B_{\max }$ is the saturation magnetic density of the iron core material, and the saturation magnetic density at the air gap was set to be 0.8 T according to design experience; $\mathrm{s}$ is the distance between the TMB stator and the rotating shaft; $c$ is the single-side air gap value, which was taken as 0.4 mm; $D_{s1}$ is the inner diameter of the stator; $D$ is the outer diameter of the rotor, which was taken as 35 mm; $D_{s2}$ is the inner diameter of the stator winding slot; $N_2$ is the number of coil windings; $I_{max}$ is the maximum allowable current; $A_w$ is the area of the stator coil cavity; $\eta$ is the slot fullness rate of the coil winding, which was taken as 0.6; $d_c$ is the minimum diameter of the coil winding wire; $\mathrm{J}$ is the maximum current density allowed in the conductor under certain heat dissipation conditions, where under natural air-cooling conditions, the value of is generally taken as $4\sim 8 \mathrm{A}/{\mathrm{m}\mathrm{m}}^2$; $H_1 \mathrm{and} H_2$ are, respectively, the width and height of the coil slot, which are equal; ${\mathrm{D}}_{\mathrm{s}3}$ is the outer diameter of the stator winding slot; $\mathrm{h}$ is the thickness of the yoke; $L$ is the thickness of the thrust disc; $D_{s4}$ is the outer diameter of the stator; and $D_{s5}$ is the diameter of the thrust disc.

The final structural parameters of TMB. were calculated as shown in

Table 7. Initial design parameters of TMB.

Parameter	Value
Outer diameter of rotor D	35 mm
Single-side air gap $c$	0.4 mm
$\mathrm{Inner} \mathrm{diameter} \mathrm{of} \mathrm{stator} D_{s1}$	41 mm
$\mathrm{Inner} \mathrm{diameter} \mathrm{of} \mathrm{stator} \mathrm{winding} \mathrm{slot} {\mathrm{D}}_{\mathrm{s}2}$	45 mm
$\mathrm{Outer} \mathrm{diameter} \mathrm{of} \mathrm{stator} \mathrm{winding} \mathrm{slot} {\mathrm{D}}_{\mathrm{s}3}$	74.8 mm
$\mathrm{Outer} \mathrm{diameter} \mathrm{of} \mathrm{stator} {\mathrm{D}}_{\mathrm{s}4}$	77 mm
$\mathrm{Thrust} \mathrm{disc} \mathrm{diameter}{\mathrm{D}}_{\mathrm{s}5}$	77.5 mm
$\mathrm{Winding} \mathrm{slot} \mathrm{depth} \mathrm{H}$	14.9 mm
Magnetic yoke thickness h	2.5 mm
Thrust disc thickness L	5 mm
$\mathrm{Area} \mathrm{of} \mathrm{stator} \mathrm{magnetic} \mathrm{pole} {\mathrm{A}}_0$	264.4 mm2
$\mathrm{Area} \mathrm{of} \mathrm{stator} \mathrm{coil} \mathrm{cavity} {\mathrm{A}}_{\mathrm{w}}$	221.7 mm2
$\mathrm{Number} \mathrm{of} \mathrm{the} \mathrm{coils} \mathrm{of} \mathrm{a} \mathrm{pair} \mathrm{of} \mathrm{magnetic} \mathrm{poles} N_2$	135
$\mathrm{Wire} \mathrm{diameter} d_c$	1.12 mm
$\mathrm{Maximum} \mathrm{current} I_{\mathrm{m}\mathrm{a}\mathrm{x}}$	4 A
$\mathrm{Bias} \mathrm{current} I_0$	2 A
$\mathrm{Maximum} \mathrm{air} \mathrm{gap} \mathrm{magnetic} \mathrm{density} B_{\max }$	0.85 T
$\mathrm{Maximum} \mathrm{bearing} \mathrm{capacity} F_{\max }$	152 N

.

4. Structural Parameter Optimization of 5-DOF AMB

4.1. Mogwo Principles

MOGWO was improved by Mirjallii et al. [11] based on the Grey Wolf Algorithm. On the premise of maintaining the excellent performance of the Grey Wolf Algorithm, the external population archive was introduced to store the non-dominated solutions and integrate the grid mechanism to improve the quality of the non-dominated solutions. The algorithm process is as follows:

$$\left\{\begin{array}{c}{D}_{\alpha }= \mid C_1\times X_{\alpha }\left(t\right)-X\left(t\right)|\\ D_{\beta }= |C_2\times X_{\beta }(t)-X(t\left)\right|\\ D_{\delta }= \mid C_3\times X_{\delta }\left(t\right)-X\left(t\right)|\end{array} \right.$$

(38)

$$\left\{\begin{array}{c}{X}_1=X_{\alpha }\left(t\right)-A_1\times D_{\alpha }\\ X_2=X_{\beta }\left(t\right)-A_2\times D_{\beta }\\ X_3=X_{\delta }\left(t\right)-A_3\times D_{\delta }\end{array} \right.$$

(39)

$$X\left(t+1\right)=(X_1+X_2+X_3)/3$$

(40)

$$A=2a\times r_1-a,C=2r_2$$

(41)

where ${ D}_{\alpha }$,${ D}_{\beta }$, and $D_{\delta }$ are, respectively, the distances between the individual grey wolves and the three leader wolves, i.e., $\alpha ,$ $\beta , \mathrm{a}\mathrm{n}\mathrm{d} \delta$; $C_1,C_2,\mathrm{a}\mathrm{n}\mathrm{d} C_3$ represent random vectors; $X_1,X_2$, and $X_3$ are, respectively, the position vectors under the guidance of $\alpha ,$ $\beta , \mathrm{a}\mathrm{n}\mathrm{d} \delta$; $X$ is the current position of the grey wolves; $a$ is the convergence coefficient, which has an initial value of 2 and decreases to 0 with the increase in the number of iterative steps; and $r_1$ and $r_2$ are random numbers with a value range of [0, 1].

4.2. Objective Function

The main function of the 5-DOF AMB is to provide a sufficiently large and controllable electromagnetic suction force to keep the rotor stably suspended and withstand certain vibrations and shocks, so the bearing capacity of the 5-DOF AMB should be as large as possible. At the same time, to increase the critical speed of the rotor of the high-speed motor, the volume of the AMB should be shortened as much as possible [15]. In addition, considering the extremely high rotational speed in this study, where the rated speed reached up to 100,000 rpm and the heat generation caused by air friction loss on the surface of the rotor was very high, the calculation of windage loss cannot be neglected.

In summary, the optimization objectives selected in this paper are to increase the bearing capacity of the 5-DOF AMB and reduce the volume and windage loss of the 5-DOF AMB. Among them, the RMB objective function is shown in Equations (42)–(46). The TMB objective function is shown in Equations (47)–(51).

(1)

Objective function of RMB

$$\left\{\begin{array}{c}{maxF}_r\\ {minV}_R\\ { minP}_{air-c}\end{array} \right.$$

(42)

where $F_r$ is the electromagnetic force of RMB, $V_R$ is the overall volume of the RMB stator–rotor core, and $P_{air-c}$ is the air gap windage loss caused by the tangential flow generated by the rotation of the RMB rotor. The specific expansion is shown in Equations (43)–(46).

$$F_r=B_{max}^2/{\mu }_0\mathit{cos}(\pi /8)$$

(43)

$$V_R=\pi /4\left(D^2-D_1^2\right)Lr+\left[\pi /4\left({\left(D_2+2H+2w_p\right)}^2-D_2^2\right)-8A_{cu}\right]L$$

(44)

$$P_{air-c}=k{C}_{f1}\pi \rho {\omega }^3{D}^{4}Lr$$

(45)

where $k$ is the surface roughness of the rotor, which is generally taken as 1–1.4, and for the smooth rotor, the value is taken as 1 [40,49,50]; $C_{\mathrm{f}1}$ is the friction coefficient between the rotor surface and the air; and $\rho$ is the air density, which was taken as $1.29 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$. Among them, $C_{f1}$ can be estimated by the following empirical Equation (46) [51,52]:

$$\left\{\begin{array}{c}{ C}_{f1}=0.515(c/D{)}^{0.3}/R{e}_c^{0.5}, 500<R{e}_c<{10}^4\\ C_{f1}=0.0325(c/D{)}^{0.3}/R{e}_c^{0.2}, {10}^4<R{e}_c\\ {Re}_c=\rho \omega Dc/\mu \end{array}\right.$$

(46)

where ${Re}_{\delta }$ is the Reynolds number of the rotor surface, which is related to the flow state of the gas, and $\mu$ is the dynamic viscosity coefficient of the air, which was taken as $17.9\times {10}^{-6 } \mathrm{P}\mathrm{a}·\mathrm{s}$.

(2)

Objective function of TMB

$$\left\{\begin{array}{c}{maxF}_a\\ {minV}_A\\ {minP}_{air-d}\end{array} \right.$$

(47)

where $F_a$ is the TMB electromagnetic force of TMB, $V_A$ is the overall volume of the TMB stator–rotor core, and $P_{air-d}$ is the windage loss generated by the high-speed rotation of the airflow in the gap at the end of the TMB thrust disc. The specific expansion is shown in Equations (48)–(52).

$$F_a=B_{max}^2{A}_0/{\mu }_0$$

(48)

$$V_A=\pi /4\left({ D_{s4}}^2-{D_{s1}}^2\right)\left(H_2+h\right)-\pi /4\left({D_{s3}}^2-{D_{s2}}^2\right)H_2+\pi /4\left({ D_{s4}}^2-D^2\right)L$$

(49)

$$P_{air-d}=0.5C_{f2}\rho {\omega }^3\left({ D_{s5}}^5-{ D}^5\right)$$

(50)

where $C_{f2}$ is the friction coefficient between the surface of the thrust disc and air and $\rho$ is the air density, which was taken as $1.29 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$. $C_{f2}$ can be estimated from the following empirical Equation (51) [51,52,53]:

$$\left\{\begin{array}{c}{C}_{f2}=64/\left(3R{e}_d\right), R{e}_d<30\\ C_{f2}=3.87/R{e}_d^{0.5}, 30<R{e}_d<3\times {10}^5\\ C_{f2}=0.146/R{e}_d^{0.2}, 3\times {10}^5<R{e}_d\\ {Re}_d=\rho \omega {D_{s5}}^2/\mu \end{array}\right.$$

(51)

where ${Re}_d$ is the Reynolds number at the end of the thrust disc, which is related to the flow state of the gas, and $\mu$ is the dynamic viscosity coefficient of the air, which was taken as $17.9\times {10}^{-6} \mathrm{P}\mathrm{a}·\mathrm{s}$.

4.3. Optimization Variables

Based on the objective function determined above, five optimization variables with strong correlation were selected. Among them, the RMB optimization variable is shown in Equation (52). The TMB optimization variable is shown in Equation (53).

$$X1=\left\{ P{ W}_p { D}_1 H Lr \right\}$$

(52)

$$X2=\left\{ D_{s1}{{\mathrm{D}}_{\mathrm{s}2}\mathrm{D}}_{\mathrm{s}4} {\mathrm{H}}_2 \mathrm{h} \right\}$$

(53)

4.4. Constraints

The range of constraint selection is determined based on the bearing capacity, air gap magnetic density, geometric dimension, and design experience. The RMB constraints are shown in Equation (54) and the TMB constraints are shown in Equation (55).

$$\left\{\begin{array}{c}100 N {\le F}_{max}\le 150 N\\ D4\le 110 mm\\ P_{air-c}\le 11 W\\ A\ge 83.3{ \mathrm{mm}}^2\\ 6\le P\le 12.3\\ P\le { W}_p\le 1.2P\\ 18\le { D}_1\le 22\\ 20\le H\le 22\\ 12\le Lr\le 14\end{array}\right.$$

(54)

$$\left\{\begin{array}{c}152 N {\le F}_{max}\le 200 N\\ P_{air-c}\le 60 W\\ {\mathrm{A}}_0\ge 191{ \mathrm{mm}}^2\\ 40.6\le D_{s1}\le 43\\ 44.6\le {\mathrm{D}}_{\mathrm{s}2}\le 54\\ {\mathrm{D}}_{\mathrm{s}4}\le 80 \mathrm{mm}\\ 14\le H\le 16\\ 2\le h\le 5\end{array}\right.$$

(55)

4.5. Optimization Results

Based on the above multi-objective optimization design process, after 300 iterative operations, a Pareto solution set containing 100 sets of optimal solutions was generated. Figure 17 shows the Pareto optimal front obtained by RMB optimization, and Figure 18 shows the Pareto optimal front obtained by TMB optimization.

Comprehensively taking into consideration the values of the three optimization objectives, a satisfactory solution was selected from the set of Pareto solutions according to the actual needs. The comparison between the optimized design and the initial theoretical design is shown in

Table 8. Comparison of initial design parameters with optimized design parameters of RMB.

Parameter	Initial Value	Optimized Value
$P$	7.03 mm	8.6 mm
${ W}_p$	7.73 mm	8.6 mm
$D_1$	21 mm	22 mm
$H$	24.5 mm	20 mm
$Lr$	14 mm	12 mm
$F_r$	$100 \mathrm{N}$	$110 \mathrm{N}$
$V_R$	${7.26\times 10}^4{ \mathrm{mm}}^3$	${4.62\times 10}^4{ \mathrm{mm}}^3$
${ P}_{air-c}$	$11.61 \mathrm{W}$	$9.95 \mathrm{W}$

and

Table 9. Comparison of initial design parameters with optimized design parameters of TMB.

Parameter	Initial Value	Optimized Value
$D_{s1}$	41 mm	40.6 mm
${\mathrm{D}}_{\mathrm{s}2}$	45 mm	44.7 mm
${\mathrm{D}}_{\mathrm{s}4}$	77 mm	75 mm
$H_1$	14.9 mm	14 mm
$\mathrm{h}$	2.5 mm	2 mm
$F_a$	$152 \mathrm{N}$	$158 \mathrm{N}$
$V_A$	${3.48\times 10}^4{ \mathrm{mm}}^2$	${2.87\times 10}^4{ \mathrm{mm}}^2$
${ P}_{air-d}$	$61.5 \mathrm{W}$	$53.8 \mathrm{W}$

.

Compared with the pre-optimization, the volume of the optimized RMB is reduced by 36.4%, the windage loss is reduced by 14.3%, and the bearing capacity is increased by 10%. The optimized TMB has a volume reduction of 17.5%, a windage loss reduction of 12.5%, and a 4% increase in bearing capacity.

5. Multi-Physics Field Analysis of Optimization Results

5.1. Electromagnetic Characterization

To verify whether the 5-DOF AMB model designed and optimized in this paper meets the technical requirements, a finite element simulation analysis of the electromagnetic characteristics was carried out by Maxwell 2020.

5.1.1. Electromagnetic Characterization of RMB

The modeling of the RMB was completed based on the structural parameters calculated in the previous section. The main structures of the model include a stator, a rotor, a coil winding, and a main shaft. The static magnetic field was selected as the simulation environment of the model, and material properties were assigned to each part of the model in Maxwell, in which the stator and rotor were made of silicon steel (model B20AT1500); the spindle was made of 17-4PH and the coil winding was made of copper. After setting the material properties, the model was meshed according to the material of each part. A bias current of 2 A and a maximum current of 4 A were applied to the coil winding. After completing the relevant settings, the calculation was solved and the relevant results were viewed to analyze the magnetic field distribution and bearing capacity characteristics of the RMB, as shown in Figure 19, Figure 20 and Figure 21.

Figure 19a and Figure 19b, respectively, show the magnetic field line distribution and magnetic induction intensity distribution when the RMB coil winding was loaded with the maximum current of 4 A. From Figure 19a, it can be seen that there are four magnetic flux circuits. It is obvious that the magnetic flux density of the RMB is mostly around 1.4 T at the maximum current and exceeds 1.7 T only at a few places. Overall, there is no magnetic saturation, and the design structure is reasonable.

Figure 20 shows the variation curve of magnetic induction intensity at the RMB air gap, where the horizontal coordinate is the length of the air gap line in the circumferential direction. From Figure 20, it can be seen that at the maximum current, the maximum value of magnetic induction strength at the air gap is about 0.93 T, which approximates the design value of 1 T and meets the design requirements.

Figure 21 below shows the three-dimensional electromagnetic characteristic curve of RMB electromagnetic force versus displacement and control current obtained from the Maxwell simulation. As shown in Figure 21, the electromagnetic force that RMB can generate, i.e., the load-carrying capacity, increases with the increase in displacement and control current, which is proportional. In this case, the maximum load-bearing force provided by RMB is 108 N in stable suspension, i.e., when the displacement is 0, which is consistent with the theoretically calculated value of 110 N.

5.1.2. Electromagnetic Characterization of TMB

The finite element analysis method for TMB is the same as that for RMB. The modeling of the TMB is completed based on the structural parameters calculated in the previous section. The main structures of the model include a stator, a thrust disc, and a coil winding. The static magnetic field was selected as the simulation environment for the model and material properties were assigned to each part of the model in Maxwell2020, where the stator was Armco Pure Iron (99.85%Fe), the thrust disc was made of $4340$, and the coil winding was made of copper. After completing the setting of material properties, the model was meshed according to the material of each part, and the coil winding was loaded with a bias current of 2 A and a maximum current of 4 A. After completing the relevant settings, the calculations were solved and the relevant results were viewed to analyze the magnetic field distribution and bearing capacity characteristics of the TMB, as shown in Figure 22, Figure 23 and Figure 24.

Figure 22a and Figure 22b are, respectively, the magnetic field line distribution diagram and the magnetic induction intensity distribution diagram when the TMB coil winding was loaded with the maximum current. As can be seen from Figure 22a, the magnetic field strength of the stator, rotor, and air gap are all less than 1.2 T. Only some of the corners exceed the maximum allowable magnetic induction intensity, so the design structure is reasonable. From Figure 22b, it can be seen that the magnetic field lines are concentrated around the coil winding, and there is some magnetic leakage at the top of the thrust disc and the gap between the TMB and the spindle.

The variation curve of magnetic induction strength at the TMB air gap is shown in Figure 23. It can be seen that the magnetic induction strength at the TMB air gap is 0.75 T at the maximum current of 4 A, which approximates the design value of 0.8 T and meets the design requirements.

Figure 24 below shows the three-dimensional electromagnetic characteristic curves of the TMB electromagnetic force versus displacement and control current obtained from the Maxwell simulations. Similarly, as shown in Figure 24, the electromagnetic force generated by the TMB increases proportionally with the increase in displacement and control current. In particular, at the maximum displacement and control current, the TMB produces a maximum load-bearing force of 246 N. Even in extreme cases, it can still provide sufficient axial equilibrium force without large axial deflections.

5.2. Rotor Dynamics Analysis

When the rotor system of an air compressor operates at high speed, the rotor system is prone to malignant resonance due to the influence of the unbalanced mass of the rotor and external disturbances, which destroys the stable operation of the system and has a serious impact on the safety of the system [54]. To ensure the stable operation of the ultra-high-speed rotor, it is necessary to carry out a detailed rotor dynamics analysis for the air compressor rotor system.

5.2.1. Analysis of Critical Rotor Speed

For a rotor dynamics analysis, the first step is to construct a finite element model of the rotor. In this paper, the dynamics module of Workbench 2021 was used to perform the relevant calculations. After importing the finite element model of the rotor system, the corresponding material properties were assigned to the model, as shown in

Table 10. Material properties of the rotor system model.

Name of Material	$\mathbf{Density} \mathit{\rho }/$ ${(\mathbf{k}\mathbf{g}/\mathbf{m}}^3)$	$\mathbf{Elastic} \mathbf{Modulus} \mathit{E}/$ $\left(\mathit{G}\mathit{P}\mathit{a}\right)$	$\mathbf{Poisson}’\mathbf{s} \mathbf{Ratio} \mathsf{\mu }$	Application Components
$17\text{-}4 \mathrm{P}\mathrm{H}$	7750	196	0.28	Spindle
$\mathrm{B}20\mathrm{A}\mathrm{T}1500$	7700	197	0.26	Radial magnetic bearing rotor core
$\mathrm{G}\mathrm{H}4169$	8240	199	0.3	Motor rotor sheath
$4340$	7840	210	0.278	Thrust disc
$\mathrm{T}\mathrm{C}4$	4510	110	0.34	Pressure sleeve, expander impeller, lock nut
$7075$	2810	71	0.33	Compressor impeller, inspection ring, baffle
$\mathrm{S}\mathrm{m}\mathrm{C}\mathrm{o}$	8400	120	0.24	Permanent magnet

.

The model was meshed. The impeller model was relatively complex, for which unstructured tetrahedral mesh was used, and the mesh element was 2 mm. The rest of the structure of the rotor system was relatively regular, so the structured hexahedral mesh was enough, and the mesh element was 4 mm. A total of 141,622 nodes and 75,746 units were obtained. For the contact settings, the actual assembly of the rotor system was simulated, and the contact between the main shaft and the thrust disc was set to be a 0.05 mm interference fit with a friction coefficient of 0.2; the contact between the outer surface of the permanent magnet and the rotor sheath was set to be a 0.03 mm interference fit with a friction coefficient of 0.2; and the bonded type was used for the rest of the parts. The profiled model is shown in Figure 25.

In terms of constraints, the 5-DOF AMB for the rotor support belongs to the elastic-type support, which can be similar to an elastic damped support with stiffness coefficients and damping coefficients for modal analysis in Workbench 2021, and the constraints are generally simulated using spring units, where the damping coefficient of the equivalent spring is small and can usually be ignored [55]. It was easy to obtain from Equations (56) and (57) [56,57] that the support stiffness was 300 N/mm for RMB and 500 N/mm for TMB.

$$k_{1x}=-(4{\mu }_{0}A{N}_1^2{I}_0^{2}cos\alpha /c^3)$$

(56)

$${ k}_{2x}=-({\mu }_0{A}_0{N}_2^2{I}_0^2/c^3)$$

(57)

where ${ k}_{1x}$ is the displacement stiffness coefficient of RMB, and ${ k}_{2x}$ is the displacement stiffness coefficient of TMB. $N_1$ is the number of turns of coil winding for RMB and $N_2$ is the number of turns of coil winding for TMB.

In addition, the rotor will generate a vortex in actual operation. To obtain more accurate calculation results, the gyroscopic effect was turned on in the setting of Workbench for analysis while taking full consideration of the influence of the vortex on the critical speed. Meanwhile, the Campbell diagram was drawn, as shown in Figure 26 and Figure 27.

The rotor resonance mainly results from the unbalanced force of the rotor, which excites the precession mode with the same eddy direction and speed direction, so the intersection between the synchronous excitation line and the forward precession mode frequency curve represents the critical speed of the rotor. As can be seen in Figure 26 and Figure 27, the first-order bending critical speed of the rotor is 145,660 r/min. The safety margin between it and the rated working speed of the rotor is more than 25%, which meets the requirements of general engineering applications [54].

5.2.2. Analysis of Rotor Unbalance Response

High-speed magnetic levitation air compressors for fuel cells have higher requirements on rotor vibration characteristics during operation. Due to the small gap between the 5-DOF AMB, protective bearings and rotor, impeller, and volute, larger rotor vibration displacements will lead to stator–rotor contact wear, which will lead to the instability of the 5-DOF AMB in serious cases, resulting in serious safety accidents [40]. The unbalanced response of the rotor system refers to the rotating mechanical vibration caused by the existence of an unbalanced mass of the rotor. The study of the unbalanced response of the rotor can help to further improve the rotor structure and avoid the vibration impact caused by its structural defects. In addition, the unbalanced response analysis can also provide a theoretical basis for subsequent dynamic balancing experiments and vibration suppression [54,58]. According to the ISO standard, the formula for the unbalanced force due to rotor unbalance is as follows [59]:

$$F=me{\omega }^2=U{\omega }^2$$

(58)

where $F$ is the unbalanced force; $U$ is the rotor unbalanced; $m$ is the rotor mass; $e$ is the unbalanced degree; and $\omega$ is the angular velocity of the rotor.

The allowable imbalance under the G2.5 standard is 0.00038 kg·mm. The frequency range was set to 0~2500 Hz in Workbench to cover the full rotational speed and first-order bending mode conditions, and the number of analysis sub-steps was set to 1000. The number of analysis sub-steps was set to 1000, and an imbalance of 0.00038 kg-mm was applied to the rotor. The vibration displacement distribution of the rotor of the magnetic levitation compressor under different calculation frequency bands obtained through simulation is shown in Figure 28.

As shown in Figure 28, the rotor vibration shows significant peaks at the rigid body and first-order bending mode, and the support section of the protective bearing does not exceed 0.00001 mm.

According to the standard area division in the vibration evaluation standard of magnetic bearings, the vibration displacement is in the A/B area, i.e., the area that is usually considered to be used for a long time without restriction, and the maximum vibration displacement at the minimum clearance is required to be [40]

$${\delta }_{max}<0.3{\delta }_{\min }$$

(59)

where ${\delta }_{\min }$ is the minimum radial clearance between the stator and rotor. The minimum assembly clearance of the magnetic levitation compressor designed in this paper is at the protective bearing, and the radial clearance = 0.2 mm, where the maximum allowable vibration displacement is 0.06 mm according to analysis, which is much larger than 0.00001 mm. Therefore, the vibration displacements of the rotor of the magnetic levitation air compressor designed in this paper can be located in the A/B region in the full speed range (0~2500 Hz), which can ensure the long-term stable operation of the machine.

6. Conclusions

To solve the problems of high energy consumption, short service life of air bearings, and insufficient bearing capacity of fuel cell air compressors in our country, an energy recovery magnetic levitation air compressor with coaxial connection of compressor, HPMSM, and expander and 5-DOF AMB support was proposed. This paper points out some comprehensive considerations in the design of the energy recovery magnetic levitation air compressor support drive system for fuel cells. It also details the electromagnetic design of its system and its algorithmic optimization process, specifically including the following:

(1)

The integrated and optimized design of HPMSM was carried out considering multi-physics fields such as rotor dynamics.

(2)

The initial theoretical design of the 5-DOF AMB was optimized in terms of structural parameters based on MOGWO, with the objective function of maximizing the bearing capacity of the 5-DOF AMB and minimizing its volume and windage loss.

(3)

Through the finite element software, the feasibility and reliability of the system design were verified by analyzing the electromagnetism, rotor dynamics, and unbalanced responses.

Compared with the wide application and research of gas bearings in fuel cell air compressors, the application and research of 5-DOF AMB in fuel cell air compressors have received little attention at home and abroad, and there is a large research gap. Based on the system requirements of the energy recovery magnetic levitation air compressor for fuel cells, this paper integrates multi-physics analysis and optimizes the design based on the MOGWO algorithm, and the optimized results show that the axial length of the HPMSM winding and its related performance of 5-DOF AMB have been greatly improved. Finally, the finite element software analyzes and verifies the feasibility and reliability of its performance and comprehensive optimization design. Through this study, it can provide certain theoretical value and practical significance for the application of 5-DOF AMB in fuel cell systems.