Multi-Objective Optimization of Yokeless and Segmented Armature Machine for In-Wheel Traction Applications Based on the Taguchi Method

Abstract

For electrical machines with complex structures, the design space of parameters can be large with high dimensions during optimization. Considering the calculation cost and time consumption, it is hard to optimize all the design parameters at the same time. Therefore, in that situation, sensitivity analysis of these design parameters is usually used to sort out crucial parameters. In this paper, the sensitivity analysis-based Taguchi method is applied to optimize the axial-flux permanent magnet (AFPM) machine with yokeless and segmented armature (YASA) topology for an in-wheel traction system. According to the key parameters and their sensitivity analysis, the optimal machine scheme to meet the performance requirements can be formed. In this case study, the machine performance is improved significantly after optimization. Lastly, the experimental results verify the accuracy of the model used in this study.

1. Introduction

With the development of electrical vehicles, in-wheel traction has recently become an important research hotspot [1,2]. This application requires a small un-sprung weight, and the traction machine should meet the performance requirements in the limited installation volume [3,4]. Compared with radial flux permanent magnet (RFPM) machines, axial-flux permanent magnet (AFPM) machines with yokeless and segmented armature (YASA) topology have the advantages of less mass and more efficiency, which are suitable for in-wheel traction application, especially in limited axial dimensions [5,6,7].

In order to obtain the optimal performance of the YASA machine, multi-objective optimization is essential and indispensable. The calculation of the AFPM machine requires a 3D solver [8,9]. Traditional global optimization methods (such as genetic algorithm [10], particle swarm optimization [11], etc.) may not be suitable for a 3D model due to their calculation cost [12,13,14]. Differently, the local optimization method like the Taguchi method reduces the calculation amount and improves the optimization efficiency through appropriate parameter selection [15,16]. However, in the traditional Taguchi method, optimization parameters still varied over a wide range, and it could not be conducive when finding the optimal solution [17].

Some improved Taguchi methods have been proposed to effectively optimize electrical machines. Reference [17] proposed an improved fuzzy-based sequential Taguchi optimization method applied to an inter permanent magnet synchronous machine (PMSM) to increase the optimization accuracy by optimizing the structural parameters at multiple levels. To save computing time, reference [18] utilizes the fuzzy-based sequential Taguchi robust optimization method to improve the comprehensive performance of a five-phase PMSM. In [19], to address the inherent limitation of the Taguchi method in solving multi-response optimization problems, an improved regression rate methodology and a weighted factor multi-objective technique are incorporated to form a Taguchi method-based multi-objective design framework of line-start PMSM. Reference [16] proposed a new system-level sequential Taguchi method to achieve the optimal solution with high robustness for switched reluctance machines with the consideration of manufacturing tolerances.

However, the Taguchi method for YASA machines has not been studied yet. For the existing optimization algorithms used for YASA, the selected optimization parameters are usually not well selected and, thus, may not be appropriate. In this paper, the improved Taguchi method optimization model is proposed for the YASA machine. First, analysis of the sensitivity of the structural parameters to the optimization objectives should be conducted to find the main optimization variables, which have a high influence proportion on the optimization objective. Based on that, the optimal solution of the YASA machine can be obtained.

This paper explores the possibility of using the Taguchi method in YASA machine optimization, which is the main contribution of this paper. In addition, the proposed optimization model provides a feasible reference for other electric machines.

In this paper, the topology, design requirements of the in-wheel traction system, and initial design of the YASA machine are introduced in Section 2. In Section 3, the optimization process of the YASA machine using the improved Taguchi method is described. Section 4 analyzes the performances of the no-load and load of the optimized YASA machine. Based on the optimization results, a prototype is manufactured, and the performances are verified by experimental tests. Finally, the main findings are summarized in the conclusion in Section 6.

2. Initial Design

The topology of the YASA machine is shown in Figure 1, where the YASA machine consists of a yokeless and segmented stator and double external rotors. The yokeless and segmented stator combined with concentrated armature winding can effectively reduce the mass and improve its efficiency [1]. And the main magnetic flux starts from the permanent magnet-N and passes through the stator core to the permanent magnet-S on the other side. After passing through the rotor core on the second side, the flux starts from the N-pole on the second side and passes through the stator core to the S-pole on the first side, and the main magnetic flux forms a closed loop [20].

The main design objectives of the 5 kW YASA machine applied on the in-wheel traction system are shown in

Table 1. Main parameters of the YASA machine.

Design Parameter	Value	Design Parameter	Value
Rated power (kW)	5	DC voltage (V)	288
Rated speed (r/min)	480	Outer diameter (mm)	270
Rated torque (Nm)	99	Internal diameter (mm)	190
Rated current (Arms)	≤17	Axial length (mm)	≤45

. The design volume of the YASA machine is limited, especially the axial dimension.

According to the performance requirements and size limitation, a 36 stator slots-32 rotor poles (36s/32p) YASA machine is initially designed by the traditional design method. The corresponding structural parameters are listed in

Table 2. Initial structural parameters of the YASA machine.

Description	Symbol	Value
Outer diameter (mm)	Do	270
Inner diameter (mm)	Di	190
Axial length (mm)	Le	45
Rotor iron thickness (mm)	hr	6
Permanent magnet thickness (mm)	hm	4
Pole arc coefficient	α	0.85
Air gap (mm)	hg	0.9
Pole shoot height (mm)	hp	3
Pole shoot width (mm)	wp	2.5
Slot depth (mm)	hs	9.4
Slot width (mm)	ws	18.2

, and the parameters are labeled in a partial model of the YASA, as seen in Figure 2. Some of the parameters originate empirically from previous similar projects and can be changed for other applications.

However, using the traditional design method, it is hard to meet the multiple demanding performance requirements. Therefore, the initial design with the structural parameters of the YASA machine should be optimized using the Taguchi method in this paper; see the following sections.

3. Multiple Objectives Optimization

In this section, the proposed optimization procedure using the Taguchi method is given. At the beginning, the type of in-wheel traction machine should be determined. Then, the optimization objectives/variables should be confirmed before giving the experimental matrix and FEA results. Based on the FEA and data analysis, the optimization levels of the variables will be determined. Furthermore, the optimal solution can be confirmed according to the sensitivity analysis. Figure 3 presents the optimization flowchart for the proposed Taguchi method, and the following subsections will discuss the steps in detail.

3.1. Confirmation of Optimization Objectives and Variables

Considering the performance requirements of the in-wheel traction system, the output torque should be as large as possible within the volume and temperature limitation. The torque ripple of output torque should also be minimized to ensure the smoothness and stability of the YASA machine during operation [21]. At the same time, to meet the requirements of heavy load operating conditions such as vehicle acceleration and climbing, etc., the overload torque drop rate of the YASA machine also needs to be reduced. Therefore, the average rated torque T_ave, torque ripple coefficient k_rip, and overload torque drop rate k_a are selected as optimization objectives. The torque ripple coefficient k_rip is the difference between the maximum and minimum output torque divided by the average value. The formula of k_rip can be expressed as:

$$k_{rip}=\frac{T_{max}-T_{min}}{T_{ave}}$$

(1)

where T_max and T_min are the maximum and minimum values of rated torque, respectively; T_ave is the rated torque of the YASA machine.

The torque drop rate k_a is related to the rated torque and overload torque, which is used to measure the overload capacity. k_a is defined as follows

$$k_a=\frac{\left(5T_{\mathrm{ave}}-T_{5\mathrm{ave}}\right)}{5T_{ave}}\times 100\%$$

(2)

where T_5ave is the output torque under five times the rated current.

A suitable selection of optimization variables is critical to the efficient machine optimization. When the inner and outer diameters are determined, the structural parameters of the YASA machine mainly include the rotor core thickness h_r, permanent magnet thickness h_m, pole arc coefficient α, air gap h_g, pole shoot height h_p, pole shoot width w_p, slot height h_s, and slot width w_s. According to the design requirements, the axial length of the YASA machine should be less than 45 mm, which can be expressed as:

$$2h_r+2h_m+2h_g+2h_p+h_s=45 \mathrm{mm}$$

(3)

Since the variation range of the air gap is small, it can be regarded as a fixed value, which is set as 0.9 mm here. The above formula can be written as:

h_s = 43.2 mm − 2h_r − 2h_m − 2h_p

(4)

Then, there are three independent height variables left to be optimized, i.e., rotor core thickness h_r, permanent magnet thickness h_m, and pole shoot height h_p. The slot height h_s is a dependent variable, derived by Equation (4). In addition, to ensure these size parameters do not overlap with each other, the pole shoot width coefficient is defined as k_p with a range of 0~0.5. Then, the w_p can be expressed as:

$$w_p=k_p\times w_s$$

(5)

In summary, six optimization variables are listed in

Table 3. Optimization variables and levels.

Parameter	hr (mm)	hm (mm)	α	hp (mm)	ws (mm)	kp
Level 1	4	3	0.75	2	9	0.3
Level 2	5	3.5	0.8	2.5	9.4	0.35
Level 3	6	4	0.85	3	9.8	0.4

, where each parameter is divided into three levels. The selection of levels determined the number of experiments and the amount of calculation. For the above six optimization variables, the traditional design method needs to calculate 3⁶ = 729 schemes, which is too much. In contrast, the Taguchi method can establish the experimental matrix where only 27 schemes are needed, which reduces the calculation amount by 96.3%. At the same time, the step size of each optimization variable can be reasonably selected, which makes the optimization result more accurate.

3.2. Experimental Matrix and FEA Results

The established experimental matrix is shown in

Table 4. Experimental matrix and FEA results.

Number	Optimization Variables						Tave (Nm)	krip (%)	ka (%)
	hr	hm	α	hp	ws	kp
1	1	1	1	1	1	1	97.37	1.76	9.00
2	1	1	1	1	2	2	101.47	2.14	11.59
3	1	1	1	1	3	3	104.70	3.05	15.13
4	1	2	2	2	1	1	95.23	1.07	7.43
5	1	2	2	2	2	2	99.28	1.64	9.99
6	1	2	2	2	3	3	102.36	1.60	13.59
7	1	3	3	3	1	1	90.10	1.50	5.72
8	1	3	3	3	2	2	93.96	1.91	8.25
9	1	3	3	3	3	3	96.82	3.09	11.99
10	2	1	2	3	1	2	87.00	1.63	9.85
11	2	1	2	3	2	3	90.33	1.87	14.18
12	2	1	2	3	3	1	92.95	1.17	10.09
13	2	2	3	1	1	2	94.22	3.87	6.26
14	2	2	3	1	2	3	97.29	4.96	8.25
15	2	2	3	1	3	1	100.30	2.37	6.53
16	2	3	1	2	1	2	85.03	3.68	6.69
17	2	3	1	2	2	3	88.10	4.86	9.39
18	2	3	1	2	3	1	90.93	3.33	6.67
19	3	1	3	2	1	3	84.56	4.33	9.16
20	3	1	3	2	2	1	87.19	2.06	6.80
21	3	1	3	2	3	2	90.59	1.92	9.10
22	3	2	1	3	1	3	75.69	5.38	9.42
23	3	2	1	3	2	1	77.99	2.77	6.01
24	3	2	1	3	3	2	81.42	3.35	8.75
25	3	3	2	1	1	3	82.16	1.96	5.32
26	3	3	2	1	2	1	84.93	1.08	3.86
27	3	3	2	1	3	2	87.87	1.32	4.94

. The advantage of the orthogonal experiment is that the most representative experiments can be selected from a large number of experiments to obtain reliable results, and the analysis and calculation are convenient [15].

It can be seen from Table 4 that there are combinations of different optimization variable levels corresponding to the order of the experiments. And the results of the optimization objective are obtained by the finite element analysis (FEA), which will be used as the basis for parameter sensitivity analysis, as shown in Table 4.

3.3. Sensitivity Analysis of Optimization Variables

This section analyzes the influence of each optimization parameter on the FEA results. Firstly, the average value of the optimization objectives is analyzed by Equation (6), and the calculation results are shown in

Table 5. Average value of each optimization objective.

	Tave/Nm	krip (%)	ka (%)
m	91.11	2.58	8.67

.

$$m=\frac{1}{n}{\sum }_{i=1}^n{S}_i$$

(6)

where n is the number of experiments, and S_i is the FEA result of the i-th experiment.

Each optimization variable corresponds to the average value of optimization objectives under different levels. For example, the average value of rated torque T_ave of variable h_m under level 2 is shown in Equation (7). Similarly, the average values of optimization objectives under different levels of other optimization variables can be calculated. The results are shown in

Table 6. Average value of objective under different levels.

Symbol	Indicator	Tave/Nm	krip (%)	ka (%)
hr	1	97.92	1.97	10.30
	2	91.79	3.08	8.66
	3	83.60	2.69	7.04
hm	1	92.91	2.22	10.54
	2	91.53	3.00	8.47
	3	88.88	2.53	6.98
α	1	89.19	3.37	9.18
	2	91.35	1.48	8.81
	3	92.78	2.89	8.01
hp	1	94.48	2.50	7.88
	2	91.48	2.72	8.76
	3	87.36	2.52	9.36
ws	1	87.93	2.80	7.65
	2	91.17	2.59	8.70
	3	94.22	2.36	9.65
kp	1	90.78	1.90	6.90
	2	91.2	2.38	8.38
	3	91.34	3.46	10.72

.

$$\begin{gathered} m_{\mathrm{hm}}^2\left(T_{\mathrm{ave}}\right)=\frac{1}{9}\left[T_{\mathrm{ave}}\left(4\right)+T_{\mathrm{ave}}\left(5\right)+T_{\mathrm{ave}}\left(6\right)+T_{\mathrm{ave}}\left(13\right)\right. \\ \left.\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+T_{\mathrm{ave}}\left(14\right)+T_{\mathrm{ave}}\left(15\right)+T_{\mathrm{ave}}\left(22\right)+T_{\mathrm{ave}}\left(23\right)+T_{\mathrm{ave}}\left(24\right)\right] \end{gathered}$$

(7)

where T_ave(i) is the result of rated torque T_ave in the i-th experiment, and $m_{hm}^2$ (T_ave) is the average value of rated torque T_ave about the optimization variable h_m under level 2.

Next, according to the results of the average values, the proportion of the optimization variables on the optimization objectives can be obtained using variance analysis $SS$. The calculation formula is as follows:

$$SS=3\sum _{i=1}^3{\left(m_x^i(S)-m(S)\right)}^2$$

(8)

where x represents the optimization variables. S stands for the optimization objective. m_xⁱ(S) is the average value of the optimization objectives S under level i. m(S) is the average value of the optimization objectives. The calculation results are shown in

Table 7. Proportion of each variable on the optimization objective.

Variable	Tave		krip		ka
	SS	Ratio (%)	SS × 105	Ratio (%)	SS × 104	Ratio (%)
hr	309.7	63.1	19.0	14.8	15.9	23.2
hm	25.2	5.1	9.3	7.2	19.2	27.9
α	19.6	4.0	57.9	45.1	2.2	3.1
hp	76.7	15.6	0.9	0.7	3.3	4.8
ws	59.4	12.1	2.9	2.3	6.0	8.7
kp	0.5	0.1	38.3	29.9	22.3	32.3
Total	491.0	100	128.3	100	68.9	100

.

According to the results of variance analysis, it is known that the rotor core thickness, pole shoot height, and slot width have a great influence on the rated torque T_ave, among which the rotor core thickness accounts for the largest proportion, 68.03%. The pole arc coefficient and the pole shoot width coefficient have the greatest influence on the torque ripple coefficient k_rip, accounting for more than 75%. The rotor core thickness, permanent magnet thickness, and pole shoot width coefficient have a great influence on the torque drop rate k_a.

In the final stage, the parameter variables can be determined intuitively using the above results. To meet the requirements of high torque density, low torque ripple, and high overload capacity of in-wheel traction application, the final optimization variables are confirmed as h_r(2), h_m(2), α(2), h_p(1), w_s(2), k_p(1).

3.4. Result of Optimal Variable

With the increase in rotor core thickness, the rated torque, five times overload torque, and torque drop rate decrease, and the decrease rate tends to be gentle, as shown in Figure 4a. In order to meet both the torque output and small torque drop rate, the rotor core thickness is taken as 4.8 mm. With the reduction in the permanent magnet thickness, the five times overload torque increases first and then decreases when the thickness is equal to 3.4 mm, reaching the peak. The rated torque and torque drop rate increase with the reduction in the permanent magnet thickness. As shown in Figure 4b, the permanent magnet thickness is taken as 3.4 mm.

The rated torque, five times overload torque, and torque drop rate show an upward trend with the increase in slot width, and the increasing trend of rated torque and five times overload torque tend to slow down. Although the output torque will increase with the slot width, the stator core is easier to saturate, which increases the iron loss and reduces the overload capacity. In order to obtain a small torque drop rate, the slot width is taken as 9.5 mm. With the increase in pole shoot height, the rated torque and five times overload torque decrease, but the torque decrease rate increases. As shown in Figure 4d, the pole shoot height is selected as 1.5 mm.

The selection of the pole arc coefficient and pole shoot width coefficient is more complicated and needs to be determined according to the rated torque, five times overload torque, torque ripple coefficient, and torque drop rate with the change in pole arc coefficient and pole shoot width coefficient, as shown in Figure 5.

In order to obtain a larger torque, only the cases where the polar arc coefficient is greater than 0.81 are considered. In order to obtain a small torque ripple, the selection area is limited to the range enclosed by the black line in Figure 5a. Considering the influence of the pole arc coefficient and pole shoot width coefficient on the rated torque, five times overload torque, torque ripple coefficient, and torque drop rate, three candidate points are selected and listed in

Table 8. Different values of pole ARC coefficient and pole shoot width coefficient and optimization objective.

Order	α	kp	Tave (Nm)	krip (%)	T5ave (Nm)	ka (%)
1	0.82	0.29	100.69	3.26	478.10	5.04
2	0.83	0.24	100.39	2.81	479.70	4.43
3	0.84	0.35	101.57	3.35	479.19	5.65

. Through comparison, when the pole ARC coefficient is 0.83 and the pole shoe width coefficient is 0.24, the torque ripple is from the minimum and the torque meets the requirement.

Table 9. Structural parameters of YASA machine after optimization.

Parameter	Value
Outer diameter (mm)	270
Inner diameter (mm)	190
Axial length (mm)	45
Rotor iron thickness (mm)	4.8
Permanent magnet thickness (mm)	3.4
Pole arc coefficient	0.83
Air gap (mm)	0.9
Pole shoot height (mm)	1.5
Pole shoot width (mm)	2.3
Slot depth (mm)	9.5
Slot width (mm)	23.8

shows the optimized machine parameter.

4. Performance Analysis of the Optimal Machine

After optimization by the Taguchi method, the optimal variables of the YASA machine are obtained. Next, the performance of the optimized YASA machine is analyzed by FEA.

4.1. No-Load Characteristics

Figure 6 shows the waveform of no-load back EMF and the Fourier decomposition, respectively. A1 represents the fundamental amplitude. An represents the harmonic amplitude. The THD of no-load back EMF is 9.4%, obtained by Formula (9).

$$\mathrm{THD}=\frac{\sqrt{\sum _{h=2}^n{U}_h^2}}{U_1}\times 100\%$$

(9)

The harmonic content of no-load back EMF is related to the pole arc coefficient, pole shoot width, and winding connection mode. By Fourier decomposition of rotor MMF with different arc coefficients, curves of each harmonic with pole arc coefficients can be obtained, as shown in Figure 7. The fundamental wave increases with the increase in the pole arc coefficients, but the variation regulation of other harmonics is inconsistent. As the third harmonic will not appear in the three-phase winding of the star arrangement, the fifth and seventh harmonics should be suppressed to pursue a good performance.

The waveform of air gap magnetic density is shown in Figure 8a, where the depression is mainly caused by stator slots. And stator slot introduces a large number of high-order harmonics. Due to the magnetic leakage at both sides of the YASA machine, the radial air gap magnetic density is lower on both sides compared to the middle, as shown in Figure 8b. The no-load magnetic density of the YASA machine is shown in Figure 9, where the maximum value is 1.8 T.

In the application of in-wheel traction, the YASA machine needs to run smoothly and have a small torque ripple. Cogging torque is one of the main factors causing torque ripple, which puts forward strict requirements. As shown in Figure 10, the peak-to-peak value of cogging torque, whose cycle is 20 electrical angles, is 1 Nm.

The axial magnetic tension generated by the rotors on both sides of the stator core offsets each other, making that on the stator core nearly zero, as shown in Figure 11. As the rotor position changes, the axial magnetic tension on the stator varies periodically with a maximum value of 1.5 Nm. The axial magnetic tension on the rotor changes little with the current density, as shown in Figure 12. At different current densities, the maximum and minimum values of magnetic tension of the rotor core are 6250 Nm and 6200 Nm, respectively. The reason for this is that the air gap magnetic field is mainly provided by permanent magnets, and the armature magnetic field has little effect on the air gap magnetic field.

4.2. Load Characteristics

When the current density of the YASA machine is 6 A/mm², the corresponding armature current is 17 A (RMS). As shown in Figure 13, the average value of the output torque, whose torque ripple is 2.31%, is 99 Nm. The maximum value of the cogging torque is around 0.51% of the rated torque. The magnetic density of the YASA machine under rated load is shown in Figure 14. The maximum value of the magnetic density is 1.90 T, and the core saturation is located in the tooth boot of the stator core.

The torque drop rate is an important optimization objective of the YASA machine. As shown in Figure 15, when the current density is less than 30 A/mm², the overload curve of the YASA machine has good linearity. And the torque drop rate is only 5.99%, which is due to reasonable optimization variable selection.

In this paper, the YASA machine is a surface-mounted rotor structure, the base speed (480 r/min) is controlled by I_d = 0, and all currents are used to generate torque. Above the base speed, the field weakening control is used, and part of the current is used for field weakening. Figure 16 shows the external characteristic curve and corresponding efficiency. Before the base speed, the torque is basically maintained at 99 Nm. After the base speed, the torque decreases gradually with the increase in speed. The efficiency under the rated operating condition is 94.86%, and the maximum efficiency of the YASA machine is 95.12%.

Figure 17 shows the variation of loss components with regard to the machine speed. The core loss and eddy current loss show an obvious upward trend, with the speed increasing, but there is a turning point at base speed. The eddy current loss and core loss are not only related to the speed but also magnetic density. Below the base speed, the loss increases with the increase in speed. Above the base speed, magnetic density decreases under field weakening control, so the loss at the point is slightly lower. The stator core loss, whose loss distribution is shown in Figure 18, increases with the speed. For the eddy current loss of the rotor core and permanent magnets, the weakening effect of the weak field on magnetic density counteracts the increasing effect of speed increase by more than half, so the eddy current loss increases smoothly above the base speed.

5. Experimental Validation

According to the optimization results by the Taguchi method, the YASA machine is manufactured for experimental validation. The prototype and experimental platform are shown in Figure 19.

The above experimental platform was used to test the back EMF of the prototype, and the test results are shown in Figure 20, showing good consistency with the finite element results. The error between the experimental test and the FEA mainly comes from the difference between the actual permanent magnetic materials and the materials used in the FEA simulation.

The load test was carried out, and the torque current curve of the prototype was measured as shown in Figure 21. It can be seen from Figure 21, the results of the FEA are in good agreement with the test results. And the error between the experimental test and the FEA, which is about 5%, is relatively constant. The error is within a reasonable range. The measured back EMF and torque verify the accuracy of the model used in this study.

6. Conclusions

In this paper, a 36s/32p YASA machine applied on an in-wheel traction application is optimized by the improved Taguchi method. According to the design requirements, the average rated torque, torque ripple coefficient, and overload capacity of the YASA machine are set as the optimization objectives. And the optimization variables (the rotor core thickness, pole arc coefficient, etc.) are determined through relationship analysis of the structural parameters. Based on the orthogonal experimental matrix and sensitivity analysis of the FEA results, the high sensitivity levels of the optimization variables are determined. The optimal value of each optimization variable is obtained through further detailed optimization of the optimization variables with high sensitivity levels. The no-load and on-load performances of the YASA machine are calculated by FEA. Finally, the accuracy of the optimization process is experimentally verified by the prototyped machine with optimal variables. For future study, the combination of thermal network models in [20] and the proposed Taguchi method here would be a good direction, and other useful analytical models for machine design can also be included.