Analysis of Multi-Objective Optimization Design of Interior Double Radial and Tangential Combined Magnetic Pole Permanent Magnet Drive Motor for Electric Vehicles

By inergency On Mar 31, 2024

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4.1. Optimization of Rectangular Magnetic Pole Parameters Based on Taguchi Method

For the interior combined magnetic pole permanent magnet drive motor, the size parameters and implantation depth of the permanent magnet have a great influence on the output characteristics of the motor [24]. Therefore, in this paper the peak value of cogging torque (T_cog), peak value of air gap magnetic flux density (G_Bg) and distortion rate of no-load back electromotive force waveform (K_r) are taken as indicators, and the thickness of TRPM (h_mT), the magnetic direction width of TRPM (b_mT), the thickness of RRPM (h_mQ), and the magnetic direction width of RRPM (b_mQ) are taken as indicators. The implantation depth of RRPM (b) is optimized for the parameter. Four factor levels are selected for each parameter. The rectangular magnetic pole optimization parameters and factor level task table are shown in Table 2.

The formula for calculating the waveform distortion of the no-load back electromotive force of a new type of permanent magnet drive motor, such as Formula (2)

$K_{r} = \frac{\sqrt{U_{2}^{2} + U_{3}^{2} + U_{4}^{2} \cdot \cdot \cdot \cdot \cdot \cdot U_{N}^{2}}}{U_{1}} \times 100 %$

(2)

where, U_N—the amplitude of the Nth harmonic in the back electromotive force.

According to factors and level numbers, the experimental matrix with the expression L₁₆(4⁵) is established. If the optimization method with a single variable and single indicator is adopted, 4⁵ = 1024 experiments are required, which requires a lot of working time. By using Taguchi’s rule, the multi-objective optimization design of the rotor magnetic pole can be completed only after 16 experiments. The orthogonal experimental matrix of parameter influence factors is established, and the results are solved by the finite element method. Orthogonal experimental matrix and calculation results are shown in Table 3.

Use Formula (3) to calculate the finite element simulation analysis results in Table 3:

$M_{(S)} = \frac{1}{n} \sum_{i}^{n} S_{i} = \frac{1}{16} \sum_{i = 1}^{16} S_{(i)}$

(3)

In the formula, n is the test’s number; i is a constant, i = 1, 2, 3… n; S_i is the average target performance value for the ith test.

The average value of the specific target performance is calculated for the factor level of each parameter, such as Formula (4)

$M_{xi} = \frac{1}{4} [M_{x} (j) + M_{x} (k) + M_{x} (l) + M_{x} (n)]$

(4)

where M_xi is the average value of the performance index under the ith influence factor of parameter x; M_x is the performance index of parameter x under a certain experiment; J, k, l, n are experimental numbers.

In order to express the data more intuitively, the change of each optimization parameter with the performance parameter is represented by a bar graph [26], as shown in Figure 4, Figure 5 and Figure 6.

From the above three graphs, it can be seen that these three horizontal combinations are designed for the optimization of individual performance indicators. If the impact of the three performance indicators on the motor is comprehensively considered, variance analysis is required to further analyze the impact of the change of parameters on different indicators, the proportion of the impact of the change of each parameter on its indicators is determined, so as to obtain the optimization results. The variance is calculated as follows:

$S_{(s)} = \frac{1}{z} \sum_{i = 1}^{z} [M_{xi} - M_{(s)}]^{2}$

(5)

where, s is the influencing factor, such as b₁, h₁, b₂, h₂, b; S(s) is variance of a performance indicator under parameter s; Z is number of levels of each parameter; M_(s) is the average value of a certain performance indicator. The calculation results of S_(s) are shown in Table 4.

From Table 4, it can be seen that the variance value can intuitively reflect the proportion of the impact of changes in various optimization parameters on performance indicators. The parameter with the highest proportion of influence on the peak value of magnetic flux of the air gap is b_mT. The width of TRPM h_mT has a greater influence on the cogging torque for IDRTPMDM. The performance index that has the greatest influence on the thickness of magnetization direction b_mQ and implantation depth b of RRPM is the peak value of air gap magnetic flux density. Based on the above analysis, the selection of factor levels should be optimized based on the minimum distortion rate, maximum peak air gap magnetic flux density, and minimum peak cogging torque of the no-load back electromotive force waveform. In summary, the optimal design variable combination is finally determined as h_mT (1) b_mT (4) h_mQ (1) b_mQ (3) b (2).

4.2. Optimization of Semi-Circular Magnetic Pole Parameters Based on Response Surface Methodology

The IDRTPMDM proposed in this paper adopts a single-layer winding distribution wiring mode, stator structure and rotor pole parameters. On this basis, in order to fully utilize the limited rotor space, increase the number of layers of permanent magnets, improve the salient pole rate of the permanent magnet drive motor, and reduce magnetic leakage, the semi-circular radial permanent magnet is embedded in the d axis magnetic circuit of the rotor magnetic pole. In the design of traditional permanent magnet drive motors, the thickness in the magnetization direction of permanent magnets is generally designed to be thicker to avoid irreversible demagnetization of permanent magnets. However, if the thickness in the magnetization direction is too thick, the d-axis reluctance will be increased and the d-axis inductance will be reduced, which is not good for broadening the efficient range of permanent magnet drive motors. When the motor is loaded, the stator winding needs to pass a large armature current to increase the output torque, but the larger armature current will increase the probability of irreversible demagnetization of the permanent magnet. In conclusion, in order to enhance the magnet flux concentration effect, demagnetization resistance and output torque performance of the air gap magnetic field, it is important to optimize the size of the semi-circular radial permanent magnet. Since the parameters of the magnetic poles of TRPM and RRPM have been determined, the inner and outer radius of the semi-circular radial permanent magnet and the implantation depth in the rotor core is restricted by the magnetic pole size of the rotor. Therefore, the constraint relationship can be designed and optimized for the dimensional parameters of the semi-circular radial permanent magnet, as shown in Figure 7.

From Figure 7, the center of SRPM is O, and the distance d from the center O to the outer diameter of the rotor is the implantation depth of the SRPM. In order to keep rotor structural integrity when the motor is running at high speed, the implantation depth of the permanent magnet should not be less than 2 mm, and outer radius of SRPM is R₁ and the inner radius of SRPM is R₂. The difference between the two is equal to the thickness t in the magnetization direction of SRPM, which faces the air gap. In order to prevent irreversible demagnetization, the thickness in the magnetization direction should be greater than 2 mm. In order to avoid magnetic circuit saturation in the circular arc and prevent the magnetic flux from being generated to the air gap, the internal radius of SRPM R₂ should not be less than 2 mm, and the distance between TRPM and RRPM is d_t. In order to meet the mechanical strength of the rotor punching and avoid the excessive saturation of the magnetic circuit, d_t should be greater than or equal to 3 mm, and the radial distance between SRPM and the rectangular permanent magnet should be greater than 5 mm. Because each pole radian of the motor is 30°, minus 5.5° of the pole radian corresponding to 5 mm, according to the geometric relationship in the figure, its dimension constraint equation can be established.

$\{\begin{cases} 2 \leq R_{1} - R_{2} = t \leq 3 \\ 2 \leq R_{2} < R_{1} \\ R_{2} + 2 \leq R_{1} \leq d_{t} + R_{1} \\ d \geq 2 \\ d + r_{1} + 5 \leq 20 \\ d_{t} \geq 3 \\ 2 (d_{t} + r_{1}) = (D_{r} - 2) \sin θ / 2 \end{cases}$

(6)

According to the geometric dimension relationship between the combined magnetic poles of the rotor, the range of values for the three position size parameters of SRPM can be determined as shown in Formula (7).

$\{\begin{cases} 5 mm \leq R_{1} \leq 8 mm \\ 2 mm \leq t \leq 3 mm \\ 2 mm \leq d \leq 7 mm \end{cases}$

(7)

In order to further reveal the mechanism by which the magnetic pole parameters of SRPM affect motor performance, the function of SRPM in this paper is to improve the air gap magnetic density and make up for the depression in the back electromotive force waveform. The internal radius of the SRPM is set as R₁, the thickness of SRPM t and the implantation depth d as experimental factors, and the peak value of air gap magnetic density is set as Y₁. The waveform distortion of the air gap magnetic density Y₂ as a response value; the calculation formula for Y₂ is shown in Formula (8).

$Y_{2} = \frac{\sqrt{{B^{2}}_{δ 2} + {B^{2}}_{δ 3} + {B^{2}}_{δ 4} \dots \dots + {B^{2}}_{δ n}}}{B_{δ 1}} \times 100 %$

(8)

According to the CCD (Central composite design) test scheme of the response surface method, the relationship equation between the response value and the independent variable is established by using the least square method and the regression model variance analysis:

$\begin{array}{l} Y_{1} = & 1.28 + 0.052 R_{1} + 0.042 t + 0.048 d - 0.082 R_{1} t + 0.076 R_{1} d \\ + 0.062 t d - 0.16 R_{1}^{2} + 0.016 t^{2} + 0.033 d^{2} \end{array}$

(9)

$\begin{matrix} Y_{2} & = 52.17 - 0.015 R 1 + 0.14 t + 1.54 d - 0.2 R 1 t + 0.36 R 1 d \\ + 0.019 t d + 0.26 R_{1}^{2} + 0.29 t^{2} - 1.7 d^{2} \end{matrix}$

(10)

According to the above Formulas (9) and (10), the magnitude of regression coefficients of each factor in the model can be calculated, and the order of primary and secondary influencing factors affecting Y₁ is as follows: R₁, t, d, and the order of primary and secondary influencing factors affecting Y₂ is as follows: d, t, R₁.

The suitability of the model is confirmed by using the normal graph of the residual, as shown in Figure 8. The residuals subtract the values suitable for the regression model from the actual measured values. The smaller the residuals, the more accurately the regression model can describe the actual observed results. The data distribution is close to the diagonal, indicating that the distribution of residuals is close to the normal distribution Y₁. By analyzing the residual diagram, it can be concluded that the majority of true values fall on the predicted values, and only a few portion of the true values are distributed around the predicted values, showing that the model fits the actual results well.

The response surface between the outer radius, magnetization direction thickness and implantation depth of the SRPM and peak value of air gap magnetic flux density (Y₁) is established in Figure 9. From the figure, the interaction between the three is very obvious. When t = 2 mm and R₁ changes from 5 mm to 8 mm, the Y₁ first increases and then decreases, as shown in Figure 9a. If the horizontal coordinate of R₁ is fixed at a certain value, Y₁ tends to decrease with the increase in t. According to the contour density and partial regression equation, when the influence of R₁ on the Y₁ is greater than that of d, the value of Y₁ first increases and then decreases, as shown in Figure 9b. If fixed, the horizontal abscissa of d at a certain value in Figure 9c, the value of t increases, and the value of Y₁ first decreases and then increases. According to contour distribution and analysis of the partial regression equation, the influence of t on Y₁ is greater than d.

The response surface of the outer radius of SRPM, magnetization direction thickness and the implantation depth to the waveform distortion of the air gap magnetic density (Y₂) is established as shown in Figure 10. As shown in Figure 10a, when t = 2 mm and R₁ changes from 5 mm to 8 mm, the value of Y₂ decreases first and then increases. When R₁ is a constant value, the value of Y₂ increases with t. According to the contour density and partial regression equation, the influence of t on the Y₂ is greater than that of R₁. When d = 2 mm and X₁ changes from 5 mm to 8 mm, the value of Y₂ decreases first and then increases, as shown in Figure 10b. The horizontal coordinate of R₁ is fixed at a certain value as the value of d increases, the waveform distortion rate of air gap magnetic flux density first increases and then decreases, as shown in Figure 10c. According to the contour density cloud map and partial regression equation, the influence of d on the Y₂ is greater than that of d. When d = 2 mm and t changes from 2 mm to 3 mm, the value of Y₂ shows a trend of decreasing first and then increasing. The horizontal coordinate of d is fixed at a certain value, with the increase in t, the Y₂ gradually increases. Based on the contour density cloud map and partial regression equation, t has a greater influence on Y₂ than d.

Finally, the rotor pole parameters of the IDRTPMDM can be determined, as shown in Table 5.

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