Multi-Objective Optimal Design of an Axial Flux Permanent Magnet Motor for In-Wheel Drive Considering Torque Ripple Reduction

Abstract

This study proposes an optimal design approach incorporating rotor skew to reduce torque ripple in a 5 kW in-wheel axial flux permanent magnet (AFPM) motor. Nine design variables, including the skew angle, were selected for optimization. The variation ranges of these variables were defined, and sample points were generated using the optimal Latin hypercube design (OLHD). Response data corresponding to the sample points were obtained through three-dimensional finite element method (3D FEM) analysis. Metamodels were then constructed using five different methods and evaluated based on the root mean square error (RMSE). The optimization results showed that the average torque of the optimized model increased by 2.3% compared with the initial design, reaching 48.85 Nm. Torque ripple was reduced by 42.01% to 2.83 Nm, while peak-to-peak cogging torque decreased by 42.76% to 2.61 Nm. In addition, efficiency improved by 0.07% to 95.53%, and the total harmonic distortion (THD) of the back-EMF waveform was reduced by 50.72% to 2.4%. These findings demonstrate that the proposed method provides an effective and systematic design strategy for enhancing the performance of AFPM motors.

1. Introduction

With the growing global concern over environmental issues, eco-friendly transportation systems have attracted considerable attention. Consequently, extensive research has been carried out on electric vehicles (EVs), electric two-wheelers, and mobile robots, as well as on their associated components such as batteries and drive systems [1]. In line with this trend, the in-wheel drive system has emerged as a promising propulsion method [2]. In an in-wheel drive system, the traction motor is directly mounted to the wheel, requiring high power and torque densities within limited dimensions [3]. For this reason, permanent magnet synchronous motors (PMSMs), which provide high torque and power density in compact sizes, are widely adopted as in-wheel traction motors. Depending on the direction of the magnetic flux, PMSMs can be classified into radial flux permanent magnet (RFPM) motors and axial flux permanent magnet (AFPM) motors. In RFPM motors, the magnetic flux linking the permanent magnets and the stator flows in the radial direction of the shaft. These motors have been extensively studied and are widely used due to their relative ease of manufacturing [4]. In AFPM motors, by contrast, the magnetic flux linking the permanent magnets and the stator flows axially. Although AFPM motors are more difficult to manufacture and have a shorter research history compared with RFPM motors, they exhibit higher torque and output power within the same volume, making them an increasingly attractive research subject [5].

Research on RFPM in-wheel motors has focused on developing outer-rotor configurations to achieve high torque. In [6], optimal motor design was conducted based on a quality factor. In [7], the geometry of a brushless direct current (BLDC) motor for EV applications was optimized, and its performance was verified through electromagnetic–thermal analysis. In [8], an outer-rotor surface-inset motor was studied, and an optimal design reduced torque ripple. In [9], the slot/pole combinations of a direct-drive, gearless in-wheel motor were investigated, revealing that a 56-slot/48-pole motor achieved higher torque density. In [10], low-vibration characteristics of an in-wheel motor with a V-shaped rotor were analyzed. Radial electromagnetic force was evaluated and improved from the perspective of winding magnetomotive force, and the results were experimentally validated.

AFPM motors, on the other hand, exhibit a variety of configurations depending on the arrangement of stators and rotors [11]. Among these, double-rotor AFPM motors have been extensively researched for in-wheel drive applications due to their high torque density. In [12], a multi-physics analysis of an AFPM motor was performed, encompassing electromagnetic, vibroacoustic, and thermal characteristics. In [13], an AFPM motor with a spoke-type rotor was optimally designed using a two-dimensional equivalent model, and the results were experimentally validated. In [14], an AFPM motor with consequent poles, designed to reduce the use of permanent magnets, was analyzed using the three-dimensional finite element method (3D FEM). In [15], research focused on cooling methods for in-wheel motors, where a water-cooling system was applied and the optimal cooling pipe layout was identified. In [16], an AFPM motor was designed using amorphous magnetic material, and its performance was validated through multi-physics analysis.

Although extensive studies have been conducted on both RFPM and AFPM types, AFPM motors remain less explored than RFPM motors, highlighting the need for further investigation [17]. A major drawback of AFPM motors is their large torque ripple [18], which arises from several factors. Because the magnetic flux in AFPM motors flows axially, it is difficult to achieve a uniform air-gap flux density along the circumferential direction. In particular, due to the disk-shaped rotor structure, the flux density distribution differs between the central and outer regions. Furthermore, owing to the short flux path and face-to-face stator–rotor configuration, AFPM motors tend to exhibit large cogging torque, which adversely affects torque ripple. To address these issues, studies have explored methods such as modifying permanent magnet shapes and applying skew [19].

Skew is a design technique in which the rotor or stator is slightly twisted, thereby averaging localized magnetic flux imbalances and reducing both torque ripple and cogging torque. However, applying skew also tends to decrease the motor’s output torque. Therefore, improving the torque characteristics of AFPM motors requires not only reducing torque ripple but also maintaining or enhancing the average torque. To achieve this balance, a comprehensive optimal design approach is essential.

Optimal design has been shown to significantly enhance motor performance. In [20], a study was conducted on loss optimization of an AFPM motor for water-pump applications using a PCB stator. By employing the PCB stator and the proposed design process, the axial length was reduced by 41% while maintaining equivalent output power and efficiency. In [21], an optimal design of an AFPM motor for ship propulsion was presented. The study focused on a double-stator single-rotor AFPM motor with independent three-phase windings, and the results, validated using 3D FEM, demonstrated that the AFPM motor outperformed its RFPM counterpart in terms of mass and thermal characteristics. In [22], the optimal design of an AFPM motor with three rotors and two stators was investigated. One rotor was designed with a conventional N–S arrangement, while the other adopted a Halbach array. Using a genetic algorithm combined with the response surface method, the study maximized torque density and minimized torque ripple. Although the average torque slightly decreased, the torque ripple was successfully reduced, and the torque density improved. In [23], a cost-effective optimal design methodology was proposed by employing multiple two-dimensional FEM analyses to approximate a 3D FEM. This approach reduced design costs while improving torque density and cogging torque, and its validity was experimentally confirmed. In [24], an optimal design of an AFPM motor for robotic joint applications was performed. Two slot/pole combinations, 18-slot/24-pole and 18-slot/20-pole, were compared. The study improved the back-EMF characteristics and ultimately identified the 18-slot/24-pole configuration as the optimal model, achieving a 21.92% increase in back-EMF performance.

These examples highlight that optimizing design variables through systematic approaches can significantly improve the performance of AFPM motors across diverse applications.

This paper investigates the optimal design of a 5 kW AFPM motor for in-wheel drive systems to reduce torque ripple. A skew structure is applied to the AFPM motor to mitigate torque ripple, cogging torque, and harmonic components of the back-EMF waveform. However, a conventional skew design has the drawback of reduced output torque due to flux dispersion. To overcome this structural limitation, this study proposes an optimal design method that minimizes output degradation while retaining the advantages of the skew structure. From the initial model, nine design variables influencing the torque performance are selected, and their respective ranges are defined for optimization. To reduce time and computational costs, the optimization is performed using a surrogate model (metamodel). The optimal Latin hypercube design (OLHD) method is employed to generate uniformly distributed sample points for the design variables, ensuring a high-performance metamodel. The response data are set to three target torque characteristics: average torque, torque ripple, and peak-to-peak cogging torque. Five different metamodeling methods are compared based on root mean square error (RMSE), and the most accurate model is used for the optimization process. The optimized design is validated through 3D FEM electromagnetic analysis. Furthermore, demagnetization characteristics at 150 °C and structural analysis are conducted to confirm operational reliability and mechanical stability.

2. In-Wheel Motor Design

2.1. In-Wheel System

The in-wheel drive system has been developed as an alternative propulsion method to improve the conventional vehicle drivetrain [25]. An example of an in-wheel system is shown in Figure 1. Figure 1a compares the structures of a conventional electric four-wheel drive system and an in-wheel system, while Figure 1b illustrates mobile robots and vehicles equipped with in-wheel motors.

In a conventional drivetrain, such as an electric four-wheel drive system, a main traction motor and a power transmission mechanism are used to operate the vehicle. In contrast, the in-wheel system integrates the traction motor inside the wheel, enabling independent control of each wheel. In-wheel systems typically employ permanent magnet synchronous motors (PMSMs), which are installed inside the wheel along with inverters and controllers. Additionally, components such as multiple sensors, braking systems, and suspension units are incorporated to enhance driving performance [26].

Through independent wheel control, the in-wheel system enables advanced maneuvers such as on-the-spot rotation and crab steering, which are not feasible in conventional vehicles [27]. These features improve driving performance and have made in-wheel systems attractive for use in electric two-wheelers and mobile robots [28]. Moreover, in-wheel systems simplify the drivetrain compared with conventional systems [29], eliminating the need for complex power transmission components. This simplification leads to improved efficiency, reduced weight, better space utilization, and easier maintenance in mobile robots and vehicles. These advantages contribute to improved maintainability, enhanced design flexibility, and overall vehicle or robot weight reduction.

Despite its various advantages, the in-wheel system also has certain drawbacks. The first is durability. Since the traction motors, multiple sensors, and control units are directly mounted on the wheel, they are exposed to shocks and vibrations encountered during vehicle or robot operation. The second is cooling. As the traction motor and controller are installed inside an enclosed wheel, heat tends to accumulate easily [30]. This heat becomes a major cause of performance degradation in the motor. Consequently, in-wheel traction motors must be designed to withstand high-temperature environments and prevent irreversible demagnetization of the permanent magnets.

2.2. Initial Model of AFPM Motor

The traction motor used in an in-wheel drive system is directly mounted inside the wheel. Therefore, the motor must be designed within limited dimensions to fit inside the wheel while delivering high torque at low speeds. The configuration and specifications of the selected initial AFPM motor are presented in Figure 2 and

Table 1. Specifications of the initial AFPM motor.

Parameter	Value	Unit
Output power	5	kW
Input DC voltage	96	V
Input current	30.2	Arms
Output rated speed	1000	rpm
Output torque	47.75	Nm
Slot/pole	18/24	-
Outer diameter	280	mm
Total height	104	mm
Coil turn	45	-
Current density	5.815	A/mm2
Magnet material	N35UH
Core material	Soft Magnetic Composite

. The initial model is a 5 kW double-rotor structure with an 18-slot and 24-pole combination. To ensure high torque output at low speed, the motor was designed with a high pole number, and a double-rotor configuration was adopted to achieve high torque density.

The rotor permanent magnets are arranged in an N–S pole configuration, where in adjacent magnets possess opposite polarities. The stator employs a segmented core structure to facilitate manufacturing processes and accommodate the winding layout. The overall outer diameter, including the housing, is restricted to 12 inches, with a total axial length of 104 mm and an air-gap length of 1 mm. The motor is designed for a rated input voltage of 96 V and a rated speed of 1000 rpm. Under these operating conditions, the average torque reaches 47.75 Nm, with a corresponding torque ripple of 4.8 Nm. The rotor yoke and stator core are manufactured from soft magnetic composite (SMC) material, while N35UH-grade neodymium permanent magnets are employed to ensure high-temperature operational capability.

The torque ripple of the initial design constitutes 10.05% of the average torque, which is relatively high, and the peak-to-peak cogging torque is measured at 4.6 Nm, indicating a considerable magnitude. Consequently, to enhance the torque performance of the in-wheel AFPM motor, it is imperative to reduce both the torque ripple and the peak-to-peak cogging torque without compromising output characteristics.

2.3. Optimal Design Process

The overall optimal design procedure is illustrated in Figure 3. Initially, the design variables subject to optimization are identified, and their allowable ranges are specified. Within these defined ranges, 300 sample points are generated using the OLHD method. For each sample point, 3D FEM simulations are conducted to evaluate the average torque, torque ripple, and peak-to-peak cogging torque. The resulting values are then compiled to construct the response dataset corresponding to the generated sample points.

Using the sample points and response data, a metamodel is constructed, and its prediction accuracy is evaluated using the RMSE. For the optimal design, an objective function and constraints are defined, and design variable optimization is carried out using the metamodel.

The resulting optimal model is validated by analyzing its electromagnetic characteristics through 3D FEM. To ensure operational reliability, demagnetization analysis is performed under a temperature condition of 150 °C. In addition, structural analysis is conducted to verify mechanical safety during operation. If the verification results do not meet the required performance criteria, the process returns to the design variable optimization stage, and the procedure is repeated until satisfactory results are achieved.

The selected design variables and their variation ranges for the initial model are presented in Figure 4 and

Table 2. Range of design variables.

Parameter	Initial	Lower	Upper	Unit
X1	126	120	180	degree
X2	84	80	90	mm
X3	9.5	5	10	mm
X4	62	58	68	mm
X5	13	13	17	mm
X6	3	2	4	mm
X7	2	0	5	mm
X8	0	−60	60	degree
X9	0	−60	60	degree

. A total of nine design variables were chosen for the optimal design process: three related to the rotor permanent magnets (X₁–X₃), four associated with the stator geometry (X₄–X₇), and two representing the skew angles of the first and second rotors (X₈ and X₉).

Among the design variables, X₁ denotes the arc angle of the permanent magnet and, together with X₂, determines the magnet area interacting with the stator. Increasing X₁ and X₂ enlarges the magnet area, which generally enhances the motor torque; however, excessive magnet area can adversely affect torque ripple. Once the magnet area is fixed by X₁ and X₂, X₃ specifies the magnet thickness, thereby defining the magnet volume. A smaller X₃ reduces both the magnet volume and manufacturing cost, but also decreases magnetic flux and increases the rotor’s susceptibility to demagnetization under reverse magnetic fields.

X₄ influences the stator flux path length: increasing it extends the path, which may reduce torque, but can also lower peak-to-peak cogging torque. X₅, associated with stator tooth width, affects torque and efficiency: increasing X₅ reduces tooth width, potentially decreasing torque but improving copper loss and thus efficiency. X₆ determines the opening length between stator cores: a larger value narrows the core area interacting with the magnets, potentially worsening cogging torque, whereas a smaller value may hinder winding manufacturability and cooling performance. X₇ is related to magnetic flux saturation in the stator shoe: increasing it mitigates saturation and reduces core losses, although it may lead to greater copper losses.

X₈ and X₉ represent the skew angles of rotor 1 and rotor 2, respectively. Rotor skew is an effective technique for mitigating torque ripple; however, inappropriate skew angles may negatively affect the harmonic content of the back electromotive force (back-EMF), resulting in increased torque ripple.

For the AFPM motor considered in this study, the permanent magnets adopt a trapezoidal shape. Accordingly, X₁ is defined as the arc angle formed by the two nonparallel sides of the trapezoid, expressed in electrical degrees per magnet. X₂ and X₃ correspond to the length and height of the magnet, respectively. X₄ represents the stator height, X₅ denotes the stator slot width, X₆ specifies the gap between adjacent stator cores, and X₇ corresponds to the stator shoe length. Finally, X₈ and X₉ indicate the skew angles of rotor 1 and rotor 2, respectively. The variation ranges for all design variables were defined so as to avoid mutual interference among the variables and to ensure physical feasibility within the motor structure.

Electromagnetic field analysis using FEM enables precise evaluation of motor characteristics. Many studies on motors have employed FEM to analyze and investigate their performance [31]. In RFPM-type motors, the magnetic flux flows in the radial direction and exhibits axial symmetry; thus, the motor cross-section can be analyzed with high accuracy using 2D FEM. However, in AFPM-type motors, the magnetic flux flows axially and follows a complex path. Therefore, 3D FEM is required to accurately analyze the characteristics of AFPM motors. Nevertheless, 3D FEM entails high computational cost and time consumption. To address this issue, a metamodel is constructed for the optimal design of AFPM motors.

A metamodel simplifies a complex system to facilitate computation. It approximates the relationship between input and output variables of the actual system. While it may not provide the exact outputs of the real model for given inputs, it can yield close approximations, making it possible to simplify complex systems such as those modeled by 3D FEM. For metamodel construction, the inputs are the values of the design variables, and the outputs are the average torque, torque ripple, and peak-to-peak cogging torque of the AFPM motor corresponding to those design variable values.

The input data for metamodel construction are generated using the OLHD, a design of experiments (DOE) method that efficiently explores model characteristics influenced by multiple interacting variables [32]. The minimum number of sample points required for metamodel construction is typically ten times the number of design variables. In this study, 300 sample points were used to ensure the accuracy and robustness of the metamodel. Based on OLHD, these 300 sample points were analyzed using 3D FEM to evaluate the motor characteristics corresponding to each set of design variable values. From the analysis results, the average torque, torque ripple, and peak-to-peak cogging torque under rated operating conditions were compiled as the response data for each sample point. The computation of a single response dataset required approximately 0.5 h, meaning that the complete collection of response data for 300 sample points took about 150 h.

The metamodel was constructed using five different metamodeling methods [33]: Kriging (KRG), radial basis function (RBF), polynomial regression (PR), ensemble of decision trees (EDT), and multi-layer perceptron (MLP). KRG is a type of interpolation method based on statistics. Using the sample points generated during the design of experiments, it predicts the response data corresponding to new sets of design variables. KRG has the advantage of achieving relatively high accuracy even when constructed with a small number of sample points. RBF is an interpolation method that approximates the model by combining basis functions, predicting outputs based on the distance between input values. PR assumes a polynomial relationship between the input and output variables, making it a simple regression method to implement and providing ease in understanding the influence of each input variable on the output. EDT combines multiple decision trees into a single model, offering the advantage of effectively mitigating overfitting issues that may arise during metamodel construction. MLP is a metamodel with the basic structure of an artificial neural network, demonstrating particularly high performance when large datasets are available.

These five metamodeling techniques have been shown to perform well in predicting the characteristics of nonlinear models. The performance of the metamodels constructed using these methods is evaluated using Equation (1).

$$RMSE=\sqrt{{\displaystyle \frac{\sum _{i=1}^n{(\widehat{y_i}-y_i)}^2}{n}}}$$

(1)

where $\widehat{y_i}$ is the predicted value from the metamodel, y_i is the actual value from the evaluation dataset, and n is the number of data points used for evaluation. RMSE is an intuitive performance metric that quantifies the difference between the predicted and actual values. A lower RMSE value, approaching zero, indicates better model performance and that the predicted values are closer to the actual values.

The performance of the metamodels, evaluated in terms of the RMSE, is summarized in

Table 3. Metamodel performance evaluation (RMSE).

Parameter	Model	RMSE
Average torque	PR	1.072
	KRG	1.474
	EDT	1.546
	RBF	1.772
	MLP	2.164
Torque ripple	KRG	0.844
	PR	0.944
	MLP	1.025
	RBF	1.064
	EDT	1.135
Cogging torque (peak-to-peak)	KRG	0.847
	EDT	0.863
	PR	1.048
	RBF	1.089
	MLP	1.128

Bold emphasizes the metamodel with the lower RMSE.

. For each of the three response parameters—average torque, torque ripple, and peak-to-peak cogging torque—five types of metamodels were constructed. To mitigate the risk of overfitting, 270 out of the 300 generated sample points were used for model training, while the remaining 30 were reserved for validation.

The validation results show that, for average torque, the PR model attained the lowest RMSE of 1.072, whereas for torque ripple and peak-to-peak cogging torque, the KRG model achieved the lowest RMSE values of 0.847 and 0.844, respectively.

A design sensitivity analysis (DSA) was performed based on the constructed metamodels, and the results are shown in Figure 5. Figure 5a presents the sensitivity of the design variables with respect to the average torque, Figure 5b shows the sensitivity for torque ripple, and Figure 5c illustrates the sensitivity for peak-to-peak cogging torque. In each graph, the x-axis represents the design variable number, while the y-axis indicates the influence on each response data as the value of the corresponding design variable increases. A positive value on the y-axis means that an increase in the design variable results in an increase in the response data, whereas a negative value means that an increase in the design variable results in a decrease in the response data.

The DSA results indicate that X₆ has the greatest influence among all design variables. For most design variables, an increase in their value leads to an increase in the average torque, torque ripple, and peak-to-peak cogging torque. In the case of X₁, increasing the variable results in higher average torque but lower torque ripple and peak-to-peak cogging torque. For X₄, an increase in its value reduces all response data. For X₈, increasing the value decreases the average torque while increasing the torque ripple and peak-to-peak cogging torque. Design variables with high sensitivity exhibit significant changes in motor characteristics as their values increase. Therefore, in the optimization process, highly sensitive variables are subdivided into finer increments compared with less sensitive variables to ensure more precise optimization.

2.4. Design Variable Optimization Search

In the design variable optimization search, torque ripple minimization was employed as the primary objective function to improve the torque characteristics of the in-wheel AFPM motor. Since the introduction of rotor skew as a design variable inherently reduces both output torque and average torque, average torque maximization was incorporated as a secondary objective to compensate for these losses. To determine the relative priority between the two objectives, weighting factors were applied: torque ripple minimization was assigned a weight of 0.7, reflecting its higher priority for improving the initial model’s torque characteristics, while average torque maximization was assigned a weight of 0.3. By simultaneously applying these two weighted objectives, the trade-offs between average torque reduction and torque ripple increase—often encountered in single-objective optimization—can be mutually compensated.

The objective function for the design variable optimization of the in-wheel AFPM motor is expressed in Equation (2):

$$\mathrm{O}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e} \mathrm{F}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}=\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}\left(T_{rip}\right)\times 0.7+\mathrm{M}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}\left(T_{avg}\right)\times 0.3$$

(2)

where T_rip is the torque ripple and T_avg is the average torque.

Furthermore, minimizing torque ripple as an objective function inherently facilitates the reduction of peak-to-peak cogging torque. To ensure the stability of the optimization process, an additional constraint was imposed, requiring the peak-to-peak cogging torque to remain below 3 Nm, a value lower than that of the initial model. This constraint is expressed in Equation (3).

$$\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{t}:\mathrm{P}\mathrm{e}\mathrm{a}\mathrm{k}-\mathrm{t}\mathrm{o}-\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k} \mathrm{c}\mathrm{o}\mathrm{g}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{t}\mathrm{o}\mathrm{r}\mathrm{q}\mathrm{u}\mathrm{e}<3 \mathrm{N}\mathrm{m}$$

(3)

The optimization was performed using the progressive quadratic response surface method (PQRSM), a metamodel-based optimization technique well suited for problems exhibiting nonlinear characteristics and numerical noise [34]. Since the performance of the AFPM motor varies nonlinearly with changes in design variables, PQRSM is particularly effective for achieving an optimal solution. The optimization was conducted using discrete variables, rather than continuous ones, to ensure manufacturability and robustness. While continuous variables allow for finer exploration of the design space, manufacturing tolerances can lead to dimensional deviations that degrade torque and efficiency, thereby reducing motor reliability. In contrast, discrete variables, despite their more limited search space, enable efficient optimization while accounting for manufacturability during the design phase, thereby ensuring standardized production.

The optimization process was performed over more than 500 iterations, with the design variables converging to their optimal values after approximately 300 iterations. This convergence behavior is illustrated in Figure 6, which presents the convergence profiles of the design variables and the objective function (torque characteristics). Compared with directly conducting 500 full 3D FEM analyses, the use of the metamodel reduced the computational time to approximately 60% of that required by the full FEM-based approach, thereby demonstrating a clear advantage in terms of efficiency. This comparison is further summarized in

Table 4. Comparison of computational time between full 3D FEM and metamodel-based optimization.

Method	Iterations	Time per Iteration [Hr]	Total Time (Estimated) [Hr]	Relative Computation Cost [%]
Full 3D FEM analysis	500	0.5	250	100%
Metamodel-based optimization	500	0.3	150	60%

.

The optimal model is illustrated in Figure 7, and the optimized design variables together with the predicted characteristics obtained from the metamodel are summarized in

Table 5. Results of design variable optimization and torque characteristics predicted by the metamodel.

Parameter	Value	Unit
X1P	136.2	degree
X2	85.8	mm
X3	6.3	mm
X4	65	mm
X5	14.8	mm
X6	2.6	mm
X7	1.9	mm
X8	2.6	degree
X9	23.2	degree
Average torque	47.79	Nm
Torque ripple	2.16	Nm
Cogging torque (peak-to-peak)	1.97	Nm

. Most of the design variables increased, whereas X₃, X₆, and X₇ decreased. Among them, X₆ exhibited the greatest influence on motor characteristics; its reduction can be attributed to the fact that an increase in X₆ leads not only to higher average torque but also to greater torque ripple and peak-to-peak cogging torque.

Variations in X₁, X₂, and X₃, which correspond to the dimensions of the rotor permanent magnets, affected the overall magnet volume. The optimal model has a magnet volume of 10.15 cm³, representing a 26.8% reduction compared with the initial model. The rotor skew angles X₈ and X₉ are optimized to 2.6° and 23.2° electrical degrees, respectively.

According to the metamodel predictions, the average torque of the optimal model is 47.79 Nm, which is comparable to that of the initial model. In contrast, the torque ripple is reduced to 2.16 Nm, corresponding to a 55.74% reduction, while the peak-to-peak cogging torque decreases to 1.97 Nm, a 56.8% reduction. These results clearly demonstrate that, although the average torque remains nearly unchanged, the torque ripple and cogging torque are significantly improved in the optimized design.

The skew applied to both rotors effectively mitigated torque ripple by reducing harmonic components. The average torque reduction caused by the skew was compensated by increasing the permanent magnet arc angle and length, thereby enlarging the magnet area. Although the magnet area increased, the reduced magnet height allowed a decrease in total magnet volume, lowering material usage. The increased stator height extended the flux path, contributing to torque ripple reduction; although torque decreased as a result, this loss was offset by a shorter opening length and larger magnet area. The increased stator core volume slightly raised core losses, but the wider stator slots improved copper loss, leading to an overall efficiency gain.

3. Analytical Validation

Since the metamodel is an approximation model, the predicted characteristics may contain errors corresponding to the RMSE value of each model. Therefore, the motor characteristics corresponding to the optimized design variables are verified using 3D FEM. A comparison of the torque characteristics between the initial and optimal models is presented in Figure 8 and

Table 6. Comparison of torque characteristics.

Parameter	Initial	Metamodel	Optimal	Unit
Average torque	47.75	47.79	48.85	Nm
Torque ripple	4.88	2.16	2.83	Nm
Cogging torque (peak-to-peak)	4.56	1.97	2.61	Nm

.

Figure 8a shows the torque characteristics under rated operating conditions, while Figure 8b illustrates the cogging torque characteristics under no-load conditions. Under identical operating conditions, the optimal model shows an increase of 1.1 Nm in average torque compared with the initial model, while the torque ripple is reduced by 42.01% to 2.05 Nm. The peak-to-peak cogging torque is also reduced by 42.76% to 1.95 Nm. When comparing the torque characteristics of the optimal model with the metamodel predictions, slight differences are observed: the average torque is 1.06 Nm higher than the predicted value, the torque ripple is 0.67 Nm higher, and the peak-to-peak cogging torque is 0.64 Nm higher. Although the overall torque characteristics are slightly higher than predicted, the differences remain within the RMSE values of the metamodel. Consequently, it is confirmed that the torque characteristics have been successfully improved.

The back-EMF characteristics of the initial and optimal models are shown in Figure 9. Figure 9a presents the back-EMF waveform, while Figure 9b shows the results of analyzing the back-EMF waveform using fast Fourier transform (FFT). The comparison of back-EMF characteristics confirms that the harmonic components of the optimal model are improved. Specifically, the fifth harmonic component is reduced by 51.26% compared with the initial model, and the seventh harmonic component is reduced by 36.67%. The fundamental frequency component is increased by 3.47%. As a result, the total harmonic distortion (THD) of the back-EMF waveform is reduced by 50.72%, achieving a value of 2.4%.

Magnetic flux saturation in AFPM motors negatively affects both torque characteristics and efficiency. Saturation typically occurs in regions where the magnetic flux is concentrated, most notably at the ends of the stator teeth. Figure 10 compares the flux density distributions of the stator in the initial and optimal models. Figure 10a shows the flux density distribution of the initial model, where the stator tooth ends reach a maximum flux density of 2.2 T. In contrast, Figure 10b presents the optimal model, which also exhibits the highest flux density at the stator tooth ends, but at a reduced level of 1.8 T, approximately 0.4 T lower than that of the initial model. Considering the degree of flux saturation, the optimal model demonstrates a more suitable design, with improved flux distribution and reduced magnetic saturation.

Under identical operating conditions, the output power, losses, and efficiency of the initial and optimal models are summarized in

Table 7. Comparison of losses and efficiency.

Parameter	Initial	Optimal	Unit
Iron loss	193.37	204.9	W
Copper loss	44.41	34.25	W
Output power	5	5.12	kW
Efficiency	95.46	95.53	%

. As a result of the increase in average torque achieved through design variable optimization, the output power rose by 0.12 kW, reaching 5.12 kW. The iron loss increased by 11.53 W, whereas the copper loss decreased by 10.16 W. The efficiency of the motor, calculated based on the output power and losses, is expressed in Equation (4).

$$Efficiency\left(\eta \right)={\displaystyle \frac{P_{out}}{P_{out}+P_{loss}}}\times 100 [\%]$$

(4)

where P_out is the output power, and P_loss is the total motor loss consisting of iron loss and copper loss.

Using Equation (4), the efficiency of the initial model was calculated as 95.46%, whereas that of the optimal model was 95.53%. Consequently, the efficiency of the optimal model is improved by 0.07% compared with the initial model.

The traction motor operates under high-temperature conditions depending on the environment. However, neodymium-based permanent magnets exhibit degraded performance at elevated temperatures, and during the design variable optimization process, the volume of the permanent magnets was reduced. This reduction lowers the thermal capacity of the magnets, making them more susceptible to external magnetic fields. Therefore, demagnetization analysis is essential to evaluate the motor characteristics affected by the reduced magnet volume. The demagnetization coefficient, which indicates the retained magnetic flux after demagnetization, is expressed in Equation (5).

$$\mathrm{D}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{n}\mathrm{e}\mathrm{t}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{t}=B_{r1}/B_{r0}$$

(5)

Here, B_r0 is the initial remanent flux density of the permanent magnet, and B_r1 is the recovered recoil flux density after demagnetization. The demagnetization coefficient represents the degree to which the permanent magnet retains its magnetization; a value close to 1 indicates that the magnetization is maintained, whereas a value close to 0 indicates that demagnetization has occurred.

The demagnetization coefficient represents the degree to which a permanent magnet retains its magnetization; a value close to 1 indicates that the magnetization is maintained, whereas a value close to 0 indicates complete demagnetization. Figure 11 illustrates the demagnetization coefficients of the permanent magnets under high-temperature overload conditions. Figure 11a presents the demagnetization coefficient of the initial model, while Figure 11b shows that of the optimal model.

The operating temperature was set to 150 °C, and an overload condition was simulated by applying a current five times higher than the rated value for 10 s. Under these conditions, the minimum demagnetization coefficient of the initial model was 99.86%, whereas that of the optimal model was 99.83%. Although the reduced magnet volume in the optimal model resulted in a slightly lower coefficient, both models maintained sufficiently high values, confirming that safe operation can be achieved even under high-temperature overload conditions.

In addition to torque ripple optimization, the mechanical integrity of the rotor permanent magnets was evaluated to ensure reliable operation. Since permanent magnets are composed of brittle materials, they are prone to fracture under mechanical stress during motor operation. Therefore, stress analysis was conducted to verify the structural safety of the magnets in the optimized AFPM motor.

The stress distribution of the magnets under overload operating conditions (2000 rpm, 244.25 Nm) is shown in Figure 12. The analysis was performed using SolidWorks, and a parabolic tetrahedral element mesh was applied to the magnet surfaces, with approximately 20,000 elements assigned to each magnet.

The results indicate that the maximum stress generated within the magnets during operation was 51.12 MPa, which is lower than the fracture strength of the N35UH-grade magnets (80 MPa). This confirms that the optimized model can operate stably without risk of magnet fracture, ensuring both torque performance and mechanical durability.

4. Conclusions

This study conducted an optimal design to enhance the torque ripple of a 5 kW in-wheel AFPM motor. Nine design variables, including the rotor skew angle, were selected for torque performance improvement, and their allowable variation ranges were defined. Using the OLHD method, 300 sample points were generated along with their corresponding response data. Based on these data, five different metamodeling techniques were applied to construct predictive models, which were subsequently utilized in the optimization process. As a result, the average torque increased by 2.3%, while the torque ripple and peak-to-peak cogging torque were reduced by 42.01% and 42.76%, respectively. Furthermore, the harmonic content of the back-EMF waveform was improved, resulting in a 50.72% reduction in THD. Motor efficiency also increased by 0.07%, confirming the enhanced performance of the optimized model compared to the initial design. In summary, the optimal design of the in-wheel AFPM motor was successfully achieved, yielding significant torque ripple improvements while ensuring mechanical reliability. However, the present study analyzed the motor characteristics solely through 3D FEM simulations. Since such a simulation environment cannot fully account for real-world effects such as windage and friction losses, future work will focus on fabricating a prototype of the proposed optimal design and experimentally evaluating its performance under actual operating conditions. The optimized in-wheel AFPM motor developed in this study is anticipated to be applicable as a propulsion system for micromobility platforms and intelligent mobile robots.