4.1. Rotor Stress and Rotor Dynamics Analysis
Latin hypercube sampling method and Kriging proxy model are used to establish proxy model for rotor stress of HSPMM, as shown in
Figure 3. As shown in
Figure 3a,b, the radial stress of PM increases gradually with the increase of rotor outer diameter and the decrease of sleeve thickness. In addition, the radial stress of PM presents a saddle curve with the increase of sleeve thickness and interference, which is the same as the variation trend of
Figure 3d. Therefore, a certain sleeve thickness and interference can be selected to make the radial stress and tangential stress of permanent magnet obtain non-inferior optimal solutions. As shown in
Figure 3c, with the increase of rotor outer diameter and sleeve thickness, there are some non-inferior optimal solution sets for tangential stress of permanent magnet. When rotor outer diameter is 60~100 mm and sleeve thickness is 4~12 mm, tangential stress of permanent magnet obtains non-inferior optimal solution sets. As shown in
Figure 3e,f, sleeve stress increases gradually with the increase of rotor outer diameter and sleeve thickness, and sleeve stress increases gradually with the increase of sleeve thickness and interference. Therefore, it is not the case that the greater the thickness of the sleeve, the less the stress on the sleeve will be.
Dynamic rotor models with impeller and without impeller are established by using 3D finite element method. The first-order and second-order critical speeds of rotor under different rotor outer diameters are analyzed, as shown in
Figure 4 below. In this analysis, the rotor temperature is set to 110 °C. It can be seen from the figure that the first-order and second-order critical rotational speeds of the rotor gradually increase with the increase of the rotor’s outer diameter. In the stage of smaller rotor outer diameter, that is, when the rotor outer diameter is 40~80 mm, the first-order critical rotational speed of the rotor with impeller increases rapidly. When the rotor outer diameter is 80~100 mm, the first-order and second-order critical rotational speeds of the rotor without impeller gradually exceed those of the rotor with impeller. When the rotor outer diameter is greater than 100 mm, the first-order and second-order critical speeds of the rotor without impeller are higher than those of the rotor with impeller, and the first-order and second-order critical speeds of the two kinds of rotors decrease to some extent. In this paper, the rated speed of HSPMM is 30,000 rpm, rigid rotor is adopted, and the safety factor is 0.75. The first-order critical speed of rotor with impeller should be greater than 40,000 rpm, and the outer diameter of rotor should be in the range of 80~120 mm to meet the requirements of rotor dynamics.
Combined with rotor stress and rotor dynamics analysis, the rotor outer diameter, sleeve thickness and interference can be further restricted. Among them, the value range of the rotor outer diameter is further restricted to 80~100 mm, the sleeve thickness is further restricted to 5~10 mm, and the interference value is 0.05 mm.
4.3. Multi-Objective Optimization of Electromagnetic Performance and Losses
Establishing a subspace and sequential strategy for high-speed permanent magnet motor multi-objective optimization, the training points for the two optimization strategy is shown in
Table 2:
Due to the high sensitivity of subspace X1 to actual parameters, four training points are taken for each parameter within subspace X1, namely the maximum value, minimum value, and two intermediate values, totaling 256 training points. In subspace X2, three training points are selected for each parameter, specifically the maximum value, minimum value, and one intermediate value, summing up to 216 training points. The sensitivity analysis results show that the strategy of optimizing subspace X1 first and then subspace X2 is finally determined.
According to the design requirements, the rated torque of the motor is 19.1 Nm, which is indicated by the light-yellow plane, as shown in
Figure 6. As shown in
Figure 6a,b, with the increase of sleeve thickness, the output torque first decreases and then increases, while with the increase of permanent magnet thickness, the output torque first increases and then decreases. When the sleeve thickness is greater than 7 mm, the equivalent air gap length of the motor is too large to ensure sufficient excitation magnetic field, so sufficient electromagnetic torque cannot be obtained. As shown in
Figure 6c,d, as the outer diameter of the rotor increases and the thickness of the sleeve decreases, the output torque gradually increases. However, as the outer diameter of the rotor increases and the pole arc coefficient increases, the output torque gradually increases. As shown in
Figure 6e,f, the output torque of the motor gradually increases as the rotor outer diameter and the air gap length gradually increase, and at the same time, the output torque of the motor also increases as the rotor outer diameter and the permanent magnet thickness gradually increase. However, in the three optimization variables of rotor outer diameter, permanent magnet thickness and air gap length, rotor outer diameter has the most significant influence on motor output torque, which is consistent with the results of parameter sensitivity analysis.
The Kriging proxy model of permanent magnet eddy current loss varying with optimization parameters is shown in
Figure 7. It can be seen from the figure that the eddy current loss of permanent magnet has several local minima with the change of optimization parameters. The Kriging proxy model established between each optimization parameter and motor efficiency is shown in
Figure 8. It can be seen from the figure that almost all the different proxy model surfaces reach the maximum motor efficiency in the middle of the optimization parameter range, and the existing optimization parameters on this surface are reasonable and can meet the optimization conditions of each optimization objective. With the gradual increase of each optimization parameter value, the motor efficiency will gradually reach the local maximum value and then gradually decrease, so there are some local non-inferior optimal solutions of motor efficiency that satisfy the multi-objective optimization mathematical model.
Each optimized parameter has a great influence on the thermal load of the motor winding, as shown in
Figure 9 below. According to practical engineering experience and design requirements, the HSPMM designed in this paper only adopts the heat dissipation method of spiral water cooling of the casing, so the thermal load of the motor windings needs to be kept below 250 A
2/mm
3, as shown in the light-yellow plane in the figure below. With the increase of the thickness of permanent magnet, the thermal load of motor windings increases first and then decreases. With the increase of sleeve thickness and air gap length, the thermal load of motor windings increases gradually. In the range of rotor diameter, sleeve thickness, pole-arc coefficient, air-gap length and permanent magnet thickness, Kriging proxy surface model of motor winding thermal load has local optimal solution set, which can be optimized to find non-inferior optimal solution.
Firstly, the Kriging proxy model of the important parameter subspace X1 of rotor is established by keeping the important parameter subspace X2 of stator unchanged, and the Kriging proxy model is optimized by multi-objective genetic algorithm. The resulting Pareto front is shown in
Figure 10.
According to the distribution of Pareto front surface, the maximum efficiency of the motor can reach 97.44%, the maximum output torque can reach 20.6 Nm, the minimum eddy current loss of permanent magnet is 4.5 W, and the minimum thermal load of the motor is 122 A
2/mm
3. However, it can be found from
Figure 9 that when the motor efficiency is maximum and the motor thermal load is minimum, the permanent magnet eddy current is large and the output torque is small; when the permanent magnet eddy current is minimum and the output torque is maximum, the motor efficiency is small and the motor thermal load is large. Therefore, in multi-objective optimization design, there is not a group of optimal solutions but a compromise choice among all optimization objectives, and the choice space composed of all non-inferior optimal solutions constitutes a Pareto front surface, from which a group of non-inferior optimal solutions satisfying the requirements should be selected as the final optimization scheme.
According to the Pareto front surface of the multi-objective optimization design in the important parameter subspace X1 of the rotor, three non-inferior optimal solutions are selected as candidate values of the final optimization result, as shown in
Table 3. The candidate values and the key motor performance parameters corresponding to the initial design are shown in
Table 4.
According to the multi-objective optimization results of rotor important parameter subspace, it can be known that the thermal load of the motor is relatively large, although the constraint conditions of Formula (1) are basically satisfied, but it is not ideal. According to the calculation formula and fundamental principle of motor thermal load, it is mainly related to stator important parameters, which is consistent with the results of parameter sensitivity analysis. Therefore, it is necessary to further multi-objective optimization of important parameter subspaces for stators. The main purpose is to further reduce the thermal load of the motor while ensuring that the other optimization objectives are basically unchanged, to reduce the difficulty of heat dissipation of the motor and to ensure that the temperature rise check of the subsequent motor can meet the design requirements.
It can be seen from the three candidate values of rotor important parameter subspace X1 that the performance of the three candidate values is basically the same, while in the optimization of stator important parameter subspace X2, the main objective is to reduce the thermal load of the motor, and the performance of the other optimization objectives can be predicted to have a certain decline. During the rotor parameter design of high speed permanent magnet motor, the thickness of sleeve should be left a certain margin to ensure the safe and stable operation of the motor under special circumstances. Among the three candidate values, candidate value 3 has an ideal sleeve thickness, while meeting the design requirements, leaving a larger margin, and candidate value 3 corresponds to the minimum eddy current loss of the motor performance. Therefore, candidate value 3 is selected as the optimization scheme of the important rotor parameter subspace X1, and input the stator important parameter subspace X2 to carry out subsequent stator important parameter subspace multi-objective optimization.
Under the condition that the value of the optimization candidate solution of the important parameter subspace X1 of the rotor is kept unchanged, the multi-objective genetic algorithm is carried out to optimize the important parameter subspace X2 of the stator, and the thermal load of the motor is further reduced. The resulting Pareto front is shown in
Figure 11 below.
From the Pareto front surface of the important parameter subspace X2, it can be found that the minimum eddy current loss of permanent magnet is 10 W, the maximum output torque is 20.6 Nm, the maximum efficiency of motor is 97.26%, and the minimum thermal load of motor is 106 A2/mm3. However, when the eddy current loss of permanent magnet and the thermal load of motor are minimal, the motor efficiency is low and the output torque cannot reach the rated torque requirement. When the output torque is maximum and the motor efficiency is maximum, the eddy current loss of permanent magnet is large and the thermal load of motor is large.
From the Pareto front, the motor efficiency, output torque and permanent magnet eddy current loss after stator important parameter subspace optimization is not as good as those after rotor important subspace optimization. Although these three performances decrease slightly, they fully meet the design requirements of the motor. At the same time, it makes the motor thermal load reduction effect obvious, which is very worthwhile. It can be seen from this that the multi-objective optimization design of motor is a compromise selection of optimization objectives on the Pareto front surface composed of all non-inferior optimal solutions rather than all optimization objectives reaching optimization.
According to the Pareto front surface of the multi-objective optimization design of the important stator parameter subspace X2, the design requirements of the HSPMM are synthesized, and three non-inferior optimal solutions are selected as the candidate values of the final optimization results, as shown in
Table 5. The candidate values and the key motor performance parameters corresponding to the initial design are shown in
Table 6.
From the three candidate values of stator important parameter optimization subspace X2, it can be seen that the motor performance of the three candidate values greatly reduces the thermal load of the motor under the condition of ensuring that the other optimization objectives are basically unchanged. In order to ensure that the subsequent motor temperature field verification meets the design requirements and the low temperature operation of the high-speed permanent magnet motor, the three candidate values of motor efficiency and winding thermal load are basically consistent, so the minimum eddy current loss of the permanent magnet becomes the priority objective. Therefore, candidate value 2 of the three schemes is selected as the final optimization design scheme of stator important parameter subspace.
After multi-objective genetic optimization of rotor important parameter subspace X1 and stator important parameter subspace X2, and Pareto frontier screening of optimization objectives, the optimized optimization scheme is obtained, as shown in
Table 7.