Step 1: comprehensive sensitivity analysis. In this step, first an analysis of the influence of single factors on the motor performance parameters is carried out. The single-factor optimization parameters are selected as the pole arc coefficient, PM thickness, stator outer diameter, core length, number of conductors per slot and air gap length, motor performance parameters are efficiency and thermal load. Then, a response-surface analysis of the multi-objective performance is performed. The multi-objective performance parameters are efficiency, no-load back EMF, power factor, core loss, copper loss and thermal load. Finally, based on the Pearson coefficient, the sensitivity of multiple factors to the motor performance parameters is calculated.
Step 2: multi-objective optimization. This step realizes the multi-objective optimization, and generates Pareto front structures in 2D space based on objectives and constraints, and uses NSGA-II to solve the Pareto front and obtain candidate points. Then, the candidate points are compared and the final optimization scheme is determined.
Step 3: performance evaluation. In this step, both the simulation and experiment evaluation are carried out to verify the electromagnetic performances and the temperature rise of the motor, while ensuring the feasibility of a multi-objective optimization scheme.
3.1. Optimization Parameters
In order to obtain a better electromagnetic performance and temperature rise, in the optimization process, six parameters are set as the optimization variables, including the pole arc coefficient, PM thickness, stator outer diameter, core length, number of conductors per slot and air gap length. The influence curves of the single factors on the efficiency and thermal load are shown in
Figure 3 and
Figure 4.
Figure 3 shows the effect of the pole arc coefficient, PM thickness and stator outer diameter on the efficiency and thermal load. In
Figure 3a, as the pole arc coefficient increases, the efficiency increases and the thermal load decreases, and when the pole arc coefficient approaches 1, the changes in efficiency and thermal load gradually become stable. Therefore, the variation range of the pole arc coefficient is limited to 0.65–0.95. In
Figure 3b, the effect of the PM thickness on the efficiency and thermal load is almost the same as that of the pole arc coefficient. With the increase in PM thickness, the efficiency increases and the thermal load decreases, and both level off. Therefore, the minimum and maximum PM thicknesses are set to 10 mm and 15 mm, respectively.
Figure 3c shows the effect of the stator outer diameter on the efficiency and thermal load. As the stator outer diameter increases, the efficiency increases and the thermal load decreases gradually. However, when the stator outer diameter is larger than 880 mm, the thermal load starts to increase again, which is due to the saturation of the iron core when the outer diameter of the stator is too large. Therefore, the optimized range of the stator outer diameter is limited to between 840 mm and 880 mm.
The effect of the air gap length, core length and number of conductors per slot on the efficiency and thermal load are shown in
Figure 4. In
Figure 4a, with the gradual increase in the air gap length, the efficiency decreases and the thermal load increases, because the long air gap length reduces the air gap magnetic density and the loss increases accordingly. At the same time, considering the assembly of the stator and rotor, the variation range of air gap length is set to 2–5 mm.
Figure 4b shows the effect of the core length on the efficiency and thermal load. As the core length increases, both the efficiency and the thermal load decrease. However, when the core length is less than 940 mm, the efficiency decreases slightly with the increase in the core length, and when it is greater than 940 mm, the efficiency begins to decrease greatly. Therefore, the optimized range of the core length is limited to 900–940 mm. In
Figure 4c, due to the use of double-layer windings, the number of conductors per slot should only be an even number, so the number of conductors per slot is optimized among 8, 10 and 12. When N= 8, the efficiency is the lowest and the thermal load is the highest, while when N = 10, the electromagnetic performance is better, with the maximum efficiency of 93.01% and the minimum thermal load of 154.9 A
2/mm
3.
After the above analysis, the variation ranges of the six optimized design parameters are listed in
Table 2. In the analysis, if the pole arc coefficient is too small, then the efficiency is low and does not meet the requirements. When the pole arc coefficient approaches 1, the efficiency gradually becomes stable and the change is not large. Considering the cost factor, the variation range of the pole arc coefficient is set as 0.65–0.95. In addition, the PM thickness has the same influence characteristics as the pole arc coefficient. If the PM thickness is too small, then it will not meet the magnetic field requirements, and if the PM thickness is too large, then the permanent magnet will be wasted. Therefore, the PM thickness variation range is limited to 10–15 mm. In the optimization, limited by the installation size, the rotor outer diameter is fixed. When the rotor outer diameter and the slot size remain unchanged, the core may be saturated with the change in the stator outer diameter. In order to avoid this situation, the optimization range of stator outer diameter is limited between 840 mm and 880 mm. To obtain a suitable back EMF, the variation range of the core length is set from 900 mm to 940 mm. Due to the constraints of the stator and rotor assembly, the mechanical air gap length should not be too small, and if the air gap length is too large, then the magnetic field of the air gap will be small, which does not meet the requirements. Therefore, the variation range of the air gap length is limited to 2–5 mm. If the number of conductors per slot is too small, then it will not meet the efficiency requirements. On the other hand, if the number of conductors per slot is too large, then it will lead to an increased thermal load. Therefore, the number of conductors per slot is set to vary between 8, 10, and 12.
3.2. Response-Surface Analysis
The response-surface optimization method takes into account the random error of the experiment, and a reasonable experimental design method is used to obtain certain data through experiments. The response-surface method generally uses a second-order polynomial mathematical model to construct a response-surface model, and its specific expression is as follows [
18]:
where
is the fit coefficient for second-order polynomial response surfaces,
and
are the design variables,
is the error, and
is the number of design variables.
The power factor is an important parameter in motor design. A large power factor can reduce the stator current. In the optimization, it is necessary to obtain a high power factor. The power factor of a motor refers to the angle at which the voltage leads the current. In numerical terms, the power factor is the ratio of the motor’s active power to the apparent power. Through a finite-element simulation, the voltage and current curves under a load operation can be obtained, and thus the phase difference of the voltage and the current can be obtained, that is, the power factor angle can be obtained. The power factor can be calculated by the cosine value of the power factor angle.
The selection of test points plays an important role in the construction of the response surface, which directly affects the accuracy of the response-surface construction. Unreasonable test points can even lead to the failure to construct the response surface. The selection of the experimental points in this paper adopts the CCD (central composite designs) method, which can obtain a lot of information about the experimental errors of the experimental variables with the fewest number of experiments. The test points are obtained by the CCD method, and simulation experiments are individually carried out for each test point. Then, the response value is fitted by Equation (1), the coefficients of the second-order polynomial mathematical model are obtained by the least-square regression analysis, and the response-surface mathematical model is finally obtained.
Figure 5 shows a comparison of the predicted and experimental values for the efficiency, thermal load, power factor, core loss, copper loss and no-load back EMF. In
Figure 5, the degree of the fit between the predicted value and the experimental value is relatively high, indicating that the influence of the optimized design parameters on the performance parameters can be analyzed according to this model.
The influence of optimized parameters on motor performance can be evaluated at the rated speed by FEA, as shown in
Figure 6,
Figure 7,
Figure 8 and
Figure 9. In this analysis, the efficiency, no-load back EMF, copper loss, core loss and thermal load are obtained by the 2D FEA in a Maxwell of Ansys Workbench.
Figure 6 and
Figure 7 show the effect of the pole arc coefficient and PM thickness on the efficiency, no-load back EMF, power factor, core loss, thermal load and copper loss. In
Figure 6a, as the pole arc coefficient and PM thickness increase, the efficiency increases significantly. This is due to the fact that with the increase in the pole arc coefficient and PM thickness, the magnetic flux density increases significantly, and the efficiency also increases. In
Figure 6b, the no-load back EMF increases with increasing PM thickness; in addition, the no-load back EMF also increases with the pole arc coefficient, but when the pole arc coefficient is close to 0.95, the no-load back EMF decreases slightly.
Figure 6c shows the effect of the pole arc coefficient and PM thickness on the power factor. The variation law of the power factor is similar to
Figure 6a, with the increase in pole arc coefficient and PM thickness, the power factor gradually increases. Unfortunately, in
Figure 7a, the core loss also dramatically increases with the increasing pole arc coefficient and PM thickness; this is due to eddy current losses on the motor core due to changes in the magnetic force, thereby increasing the core temperature. In
Figure 7b, the thermal load decreases with the increasing pole arc coefficient and PM thickness, and in
Figure 7c, the copper loss decreases with the increasing pole arc coefficient and PM thickness, which can significantly reduce the winding temperature.
The effect of the core length and stator outer diameter on the efficiency, no-load back EMF and core loss are shown in
Figure 8. In
Figure 8a, the efficiency increases with the increasing core length and stator outer diameter, and the stator outer diameter has a greater effect on the efficiency than the core length. In
Figure 8b, the no-load back EMF is also greatly affected by the core length and the stator outer diameter. The no-load back EMF increases significantly with the increasing core length and stator outer diameter.
Figure 8c shows the influence of the core length and stator outer diameter on the core loss. With the increase in the stator outer diameter, core loss first increases and then decreases; this is because the stator outer diameter is too large or too small, which leads to the saturation of the iron core, and a larger field strength is required to generate the same amount of magnetic induction, leading to increased core loss. In addition, the core loss also increases with the increasing core length.
Figure 9 shows the effect of the air gap length and stator outer diameter on the no-load back EMF, copper loss and thermal load. In
Figure 9a, the air gap length has a great influence on the no-load back EMF. As the air gap length increases, the no-load back EMF decreases significantly, which causes the current to increase. In
Figure 9b, with the increase in the air gap length, the copper loss increases significantly, while with the increase in the stator outer diameter, the copper loss decreases slightly. In
Figure 9c, the thermal load is affected by the air gap length and stator outer diameter in almost the same manner as the copper loss. However, as the air gap length approaches 2 mm, the thermal load tends to level off and does not decrease.
3.3. Sensitivity Analysis
The above analysis shows that the effects of all the optimization parameters on the electromagnetic and temperature performance are basically linear. Therefore, the Pearson correlation coefficient can be used to define the sensitivity of the optimized design parameters to the multi-objective performance parameters. The Pearson correlation coefficient is a measure of vector similarity. The output ranges from −1 to +1, where 0 represents no correlation, negative values represent a negative correlation, and positive values represent a positive correlation. The formula for the Pearson correlation coefficient is as follows [
19]
where
is the covariance between
,
are the standard deviations of
and
, respectively, and
are the expected values of
, respectively.
The sensitivity of the optimized design parameters to the motor performance parameters is shown in
Figure 10.
Figure 10a shows the sensitivity of the optimized design parameters to the motor performance parameters when the number of conductors per slot is N = 8;
Figure 10b,c are N = 10 and N = 12, respectively.
In
Figure 10, in the process of changing the number of conductors per slot among 8, 10, and 12, when N = 8, the sensitivity of the design parameters to the electromagnetic performance is the largest, and when N = 10 and 12, the sensitivity of the design parameters to the electromagnetic performance decreases compared with N = 8. As shown, both the pole arc coefficient and the air gap length have a large influence on the six electromagnetic performance parameters. The PM thickness has the greatest impact on the core loss, followed by efficiency, copper loss and thermal load. The core length has a large influence on the power factor and thermal load. The stator outer diameter also affects the thermal load, efficiency and power factor.