Optimization Comparison of Torque Performance of Axial-Flux Permanent-Magnet Motor Using Differential Evolution and Cuckoo Search

Abstract

To improve the torque performance of the axial-flux permanent-magnet motor (AFPMM), differential evolution (DE) and cuckoo search (CS) are proposed for optimizing the motor’s structural parameters. The object of this research is an AFPMM with a single-rotor and double-stator configuration. Firstly, finite element analysis (FEA) and BP neural network machine learning (ML) were combined to obtain an ML calculator. This calculator is about the relationships between five input structural parameters of the motor and two output torque parameters (i.e., average torque and cogging torque). Then, an optimization objective function was designed to reduce the cogging torque while increasing the average output torque. And motor structural parameters were optimized using the DE and CS algorithms, respectively. Finally, air-gap flux density, average torque, cogging torque, and ripple torque before and after the optimization of the motor structure parameters are compared by FEA. The results show that both algorithms achieved almost the same optimized structural parameters. And the optimized motor has reduced cogging torque while increasing the average output torque and reducing the ripple torque. Compared with the CS, the DE is more advantageous in terms of faster iteration speed, shorter time to obtain the optimal solution, and less resource consumption.

1. Introduction

The axial-flux permanent-magnet motor (AFPMM) has attracted more and more attention due to its compact structural advantages [1]. At present, the application of the AFPMM is mainly focused on electric vehicles, robots, aviation, and other fields [2,3,4]. For these applications, both the size and quality of the output torque are crucial. The torque quality is an important issue for the low-noise smooth-torque AFPMM, which is directly related to the pulsating torque component [5]. The output torque of the AFPMM includes the average output torque (T_av) and the pulsating torque, and the pulsating torque is composed of cogging torque (T_cog) and ripple torque (T_rip). The T_cog plays a dominant role in pulsating torque; therefore, the T_av and T_cog are selected as important indexes to evaluate the output torque performance of the AFPMM. The core of optimizing motor torque performance is the coupling optimization of motor outputs with larger T_av and smaller T_cog. The influences of different structural parameters on the T_av and T_cog of the motor are coupled, that is, the reduction in the T_cog will also affect the T_av [6]. Therefore, the challenge lies in comprehensively increasing the T_av while reducing the T_cog, which constitutes a multi-objective optimization problem for enhancing AFPMM torque performance.

The torque performance of the AFPMM is determined by both mechanical structural factors and stator excitation control methods, with structural factors being the dominant influence. Therefore, many researchers have proposed various methods to optimize the motor structure to increase the T_av of the AFPMM or reduce the T_cog. Some researchers have focused on specifically optimizing the motor’s structural parameters to affect the T_av or T_cog of the motor [7,8,9]. Some researchers have proposed new motor structures to influence these torque components [10,11,12]. Although much effort has been directed towards optimizing different structural factors, most studies have concentrated on improving either the T_av or T_cog individually. Only a few reports have addressed coupling optimization to simultaneously increase the T_av and decrease the T_cog [13,14].

In the traditional motor optimization, a large amount of FEA calculations is often used to find the optimal parameters of the motor, which requires a lot of time and computational resources. Moreover, the results obtained through FEA can exhibit a high degree of randomness. With the advent of genetic algorithm (GA), many nature-inspired algorithms, such as immune algorithm (IA), ant colony optimization (ACO), particle swarm optimization (PSO), differential evolution (DE), artificial bee colony algorithm (ABC) and cuckoo search (CS), have been developed. Some researchers have adopted these algorithms to solve the optimal design problem of motors [15,16,17,18,19].

Most published studies focus on comparing motor performance after optimizing structural parameters using two or more algorithms [20,21,22]. However, there is limited research on the specific comparison of iteration speed, optimization time, and accuracy of the final optimization results of the algorithms themselves during the optimization process [23]. In addition, in the existing multi-objective optimization designs for the AFPMM, the input–output parameter relationships used in the optimization objective function are typically constructed either by the theoretical formula analytical method [24,25,26] or by the two-dimensional magnetic circuit equivalent model method [27,28]. However, due to the complexity of the three-dimensional air-gap magnetic field in the AFPMM, some boundary conditions need to be ignored when using the theoretical formula model or the two-dimensional equivalent model, leading to potential errors in analytical results. With advancements in computer hardware and analysis software, the FEA offers strong adaptability and high calculation accuracy, making it suitable for solving complex three-dimensional structures. In this paper, we propose a coupled method of 3D FEA numerical calculation and BP neural network ML to obtain accurate relationships between five input structural parameters and two output torque parameters for the optimization of the AFPMM’s objective function. We use the DE algorithm and the CS algorithm to optimize the torque performance of the AFPMM and compare their iteration speed, optimization time, and accuracy of the optimization results in detail.

This paper selects the slotted brushless DC AFPMM with single-rotor and dual-stator configuration as the research object. Firstly, based on the FEA of the AFPMM 3D model, the influence laws of five structural parameters on the T_av and T_cog were summarized. Then, the value ranges of these five structural parameters that need to be optimized were set, and the ML calculator was obtained using a method combining the 3D FEA and BP neural network training. Then, an optimization objective function of the motor torque was constructed, and the DE and CS algorithms were used to iteratively optimize the objective function, respectively. Finally, in order to verify the optimization results, the FEA was used to analyze and compare the AFPMM air-gap flux density, T_av, T_cog, and T_rip of the initial model and the optimized models after DE and CS optimization.

The main contributions of this research are summarized as follows:

(1)

To obtain an ML calculator, a method combining the 3D FEA and BP neural network is proposed to train the accurate relationships between the five input structural parameters and two output torque parameters (i.e., the T_av and T_cog) using 1024 sets of input–output simulation sample data. The 3D FEA is suitable for solving the complex AFPMM structure with high calculation accuracy, and the input–output parameter relationships can be accurately fitted when combined with BP ML.

(2)

Based on the obtained ML calculator, an optimization objective function is designed as the inverse function of the sum of the T_av and the inverse values of T_cog of the AFPMM, thereby reducing the multi-objective function to a single-objective function.

(3)

The objective function is iteratively optimized using the DE and CS, respectively. The iteration speed, optimization time, and computational resources occupied by the two algorithms are compared and analyzed.

(4)

For the initial model before optimization and the optimized models obtained by the two algorithms, the 3D FEA is performed for the comparative analysis to verify the quality of the motor torque optimization results. The results show that both algorithms achieved almost the same optimized structural parameters, but the DE is more advantageous compared to the CS due to faster iteration speed and less resource consumption.

The rest of this article is organized as follows. Section 2 introduces the structure of the AFPMM research object and simulates the influence laws of the five structural parameters on the T_av and the T_cog through FEA. Using 1024 sets of input–output FEA simulation sample data and BP neural network training, the ML calculator was obtained. In Section 3, a single optimization objective function was designed and iteratively optimized using the DE and CS algorithms, respectively, and the optimization search processes of these two algorithms were compared. In Section 4, the air-gap flux density and torque performance of the initial model and the optimized model after the DE and CS optimization are compared and verified by the 3D FEA. Finally, the conclusions of this paper and future work are presented in Section 5.

2. Structure of the AFPMM and Its Output Torque ML Calculator Based on FEA and BP Neural Network Training

In the research of motor optimization, numerical or analytical methods are usually used to solve the nonlinear input–output parameter relationships. Due to the complex three-dimensional air-gap magnetic field of the slotted AFPMM, some boundary conditions need to be simplified when using analytical methods, which will lead to deviations in the analytical results. The FEA numerical method has strong adaptability and high computational accuracy and is suitable for solving complex three-dimensional structures. However, due to a large number of structural parameters that need to be optimized for motor torque performance, it is difficult to obtain the optimal solution of motor structural parameters solely using the FEA numerical method. The ML algorithms use training sets to construct models, which can automatically improve the accuracy of models using past experiences and large, complex datasets. They can make predictions or decisions without clear input–output relationships. The BP neural network is a multi-layer feedforward network trained through error backpropagation algorithm and is currently one of the most widely used ML algorithms.

In this paper, the finite element numerical method and BP neural network algorithm are combined to solve the relationships between five structural parameters that need to be optimized and the two output torques (the T_av and T_cog) of the AFPMM. In order to obtain a more accurate ML calculator for these relationships, it is necessary to use FEA to generate as much sample data as possible for input–output relationships. A total of 1024 combinations of different structural parameters were generated within the respective value ranges of the five structural parameters that need to be optimized. The Maxwell simulation software is used to calculate the T_av and T_cog of the motor for all combinations of the input structural parameters, forming a dataset for ML training based on BP neural network. MATLAB is used to obtain an ML calculator based on the ML training of the dataset.

2.1. Structure of the AFPMM

This article selects an AFPMM with a 16-pole rotor and two 12-slot stators as the research object. Figure 1 shows its structural decomposition view and structural dimensions. The main determined parameters of the AFPMM are shown in

Table 1. The determined parameters of the AFPMM.

Description	Unit	Value
Rated power	W	4.0
Rated speed	rpm	6000
Pole number	/	16
Slot number	/	12
Stator’s outside diameter	mm	22
Rotor’s outside diameter	mm	20
Stator’s back iron thickness	mm	0.45
Rotor’s back iron thickness	mm	0.3

. The upper stator and lower stator are both composed of windings, iron cores, and a back iron, and the rotor consists of uniformly distributed sector-shaped permanent magnets (PMs) fixed on both sides of a back iron. The iron cores and back irons are all made of grain-oriented electrical steel (silicon steel). The PMs are made of Neodymium Magnet (NdFeB). The detailed material properties of silicon steel and the NdFeb PMs can be found in our paper [29]. There is an axial air gap between each stator and the rotor. The maximum height of the AFPMM without the installation shell is designed to not exceed 5.2 mm. The outside diameter of the AFPMM rotor and the thickness of the single-sided stator are designed to remain unchanged, at 20 mm and 1.15 mm, respectively.

2.2. Output Torque of the AFPMM and Its FEA Simulation

The average electromagnetic torque of the motor is shown in Formula (1) [10,29].

$$T_{av}=\frac{3}{4}\pi K_x{\alpha }_i{B}_{\delta av}{A}_{av}{\left(\beta +1\right)}^2\left(\beta -1\right)$$

(1)

$$\begin{array}{c}{B}_{\delta av}=\frac{K}{2}\cdot \frac{B_r}{\pi \sigma }\cdot \left[K_{m1}\cdot \arctan \left(\frac{a_{m1}\cdot b_m}{2\delta \cdot \sqrt{4{\delta }^2+a_{m1}^2+b_m^2}}\right)\right.\\ \left.+K_{m2}\cdot \arctan \left(\frac{a_{m2}\cdot b_m}{2\delta \cdot \sqrt{4{\delta }^2+a_{m2}^2+b_m^2}}\right)\right]\end{array}$$

(2)

where a_i is the polar arc coefficient, K_x is the winding coefficient, β is the ratio between the inner diameter and outer diameter of the winding, A_av is calculated at the average radius of the conductor, and B_δav is the average magnetic density of the air gap. K is the axial series coefficient of the magnetic pole of the PMs, B_r is the remanence of the PMs, σ is the magnetic leakage coefficient, K_m1 and K_m2 are the end-face coefficients of long and short arc lengths of the sector-shaped PMs, respectively, b_m is the radial length of the sector-shaped PMs, δ is the length of the air gap, and a_m1 and a_m2 are the long and short arc lengths of the sector-shaped PMs, respectively.

Cogging torque is produced by the interaction between PM and stator core when AFPMM is not powered, which is expressed in Formula (3) [7,30].

$$T_{cog}\left(\theta \right)=\frac{\delta }{2{\mu }_0}\left(D_o^2-D_i^2\right){\displaystyle \sum _{n=1}^{\infty }{N}_L{G}_n}{B}_r^2\sin \left(n\alpha \pi \right)\sin \left(n{N}_L\theta \right)$$

(3)

where µ₀ is the permeability of air, θ is the angle between the center line of a specified PM and stator tooth, B_r the remanence density of PM, δ the effective air gap length, G_n is the Fourier expansion coefficient of the relative permeability function of motor air gap, D_i and D_o are the stator core diameters of inner and outer surfaces, respectively, N_L is the least common multiple of the number of motor stator slots Z and the number of rotor poles 2p, α is pole–arc ratio of PMs, and n is an integer.

According to Formulas (1)–(3), the average electromagnetic torque and cogging torque of the motor are related to many structural parameters. In this paper, five structural parameters are selected to be optimized, including air gap δ, width of stator slot opening b_s0, embrace of PM α, PM thickness h_m, and inner diameter D_i of stator cores and the PMs.

As shown in Figure 2a, due to the symmetry of the AFPMM structure, when simulating the output torque characteristics of the motor, the 3D finite element model only includes a quarter of the entire AFPMM model. Figure 2b shows the meshed model based on the “skin depth-based refinement” meshing option.

As shown in Figure 3, the influence laws of each optimization parameter change on the T_av and T_cog are obtained from the FEA. As shown in Figure 3a,d,e, in the case of a single structural parameter (such as δ, h_m, and D_i) changes, the T_av and T_cog increase or decrease at the same time, which is difficult to meet the optimization demand simultaneously. Taking the air gap δ as an example, based on the size limitation and the difficulty of the manufacturing process, four candidate values were selected for the air gap in the range of 0.2–0.6 mm: 0.25, 0.35, 0.45, and 0.55. As shown in Figure 3b,c, the variation patterns of the T_av and T_cog with b_s0 or α are different over a wide range. For example, in Figure 3b, when the width of the stator slot opening b_s0 changes within the range of 0.4–2.5 mm, the T_av and T_cog both show irregular fluctuations. It is impossible to increase the T_av and reduce the T_cog at the same time. We need to balance between the T_av and T_cog, and therefore, according to the number of simulation combinations and limitations of structure dimensions, four candidate values of 0.5, 1.0, 1.5, and 2.0 for b_s0 were selected within the range of 0.4–2.5 mm. The candidate values of other structural parameters were selected in the same way as above.

According to Figure 3 and the dimensions of the motor components in Table 1, the lower and upper boundaries and their candidate values for each structural parameter to be optimized are shown in

Table 2. Boundaries and candidate values of the AFPMM structural parameters that need to be optimized.

Parameters	Lower Boundary	Upper Boundary	Value 1	Value 2	Value 3	Value 4
Air gap, δ/mm	0.25	0.55	0.25	0.35	0.45	0.55
Slot opening, bs0/mm	0.5	2.0	0.5	1.0	1.5	2.0
PM embrace, α	0.6	0.9	0.6	0.7	0.8	0.9
PM thickness, hm/mm	0.4	0.7	0.4	0.5	0.6	0.7
Inner diameter, Di/mm	10.0	13.0	10.0	11.0	12.0	13.0

.

2.3. ML Calculator for AFPMM Output Torque Based on FEA and BP Neural Network Training

For torque performance optimization of the AFPMM, the optimization process architecture studied in this article is shown in Figure 4. There are three routes as the legend shows, namely, BP training, optimization, and results. Firstly, the BP training route is used to obtain the ML calculator for relationships between input structural parameters and output torque parameters, which will be described in this sub-section. Then, for the optimization route, an optimization objective function was designed, and motor structural parameters were optimized using the DE and CS algorithms, which will be presented in Section 3. Finally, torque performance of the initial model and the optimized model are compared and verified by the FEA in the results route described in Section 4.

As shown in Table 2, the value ranges of these five structural parameters to be optimized were set, and 1024 combinations of different structural parameters are generated according to the candidate values of each parameter. Using these 1024 sets of structural parameters for 3D FEA, 1024 sets of input–output sample data were generated for BP algorithm training, as shown in the BP training route in Figure 4. And an ML calculator was obtained for the relationship between T_av and optimization parameters, as well as the relationship between the T_cog and optimization parameters. Among them, 774 sets of samples were used as the training group for BP algorithm to fit the solution relationships between five input structural parameters and two output torque parameters. The remaining 250 samples were used as the testing group to test the accuracy of BP algorithm fitting. In this study, a four-layer neural network structure was adopted, including a five-node input layer and a two-node output layer, as well as two hidden layers with eight and four nodes, respectively [29].

The test data and predicted results of the T_av and T_cog are shown in Figure 5. The mean squared error (MSE) of fitting the T_av and T_cog of the AFPMM is less than 0.009 for both, which has high solution accuracy.

3. Design and Optimization of Objective Function for the AFPMM Torque Using the DE and CS

The optimal design of the motor can be summarized as a maximum/minimum problem with a large number of optimization parameters and various constraints, which makes it a highly difficult problem to solve with deterministic methods. On the other hand, solving it with random methods is a fairly easy task.

The DE has the characteristics of simple structure, fast convergence, and strong robustness. The main characteristics of the CS are fewer parameters, simple operation, and strong ability to find the optimum. The application of these two algorithms in optimization design of motors has been a hot research topic in recent years [31,32,33].

As shown in the optimization route in Figure 4, according to the obtained ML calculator, a single optimization objective function coupled with the T_av and T_cog of the AFPMM is designed here. In order to optimize the objective function, the DE and CS were used to obtain the optimized structural parameters.

3.1. Design of Objective Function for Motor Torque Optimization

The differential evolution was proposed in 1997 by Rainer Storn and Kenneth Price [34]. The cuckoo search was proposed in 2009 by Xin-She Yang and Suash Deb [35]. The optimization of the objective function of the AFPMM torques based on DE algorithms is achieved by means of the constructed MATLAB program DE-ODA (Differential Evolution for Optimal Design of AFPMM). And the optimization of the objective function of the CS is achieved by CS-ODA (Cuckoo Search for Optimal Design of AFPMM). The program flowcharts of DE-ODA and CS-ODA are shown in Figure 6a,b, respectively.

The user and the problem determine the values of the assigned parameters for the DE and CS. In this article, the values for the optimized design problem of the studied motor structure are as follows:

(1)

DE: number of populations: 100; number of variables: 5; number of generations: 100; initial assignment of mutation operator: 0.6; and crossover operator: 0.4. The mutation operator is dynamically adjusted through annealing factors during optimization iterations [36].

(2)

CS: number of nests: 100; number of variables: 5; number of generations: 100; and initial assignment of the probability of establishing a new bird nest: 0.25. The discovery probability is dynamically adjusted during optimization iterations [37].

The optimization objective studied in the paper is to obtain a coupled optimal solution that maximizes the T_av and minimizes the T_cog. The construction of the optimization function requires balancing the effects of the T_av and T_cog on the function value. It is well known that the CS is usually a minimization algorithm. Therefore, according to the weighted sum method, the optimization objective function F of the AFPMM for the DE and the CS is designed as the inverse function of the sum of the T_av and the inverse values of the T_cog of the motor, which is defined as Equation (4):

$$F=\frac{1}{T_{av}+\left(\frac{c}{T_{cog}}\right)}$$

(4)

where c is the weighting factor of the T_cog. The value of c is supposed to balance the effects of the T_av and the inverse of the T_cog on the objective function [38]. For smaller values of c, the average torque value has a greater influence on the function value; for larger values of c, the cogging torque value has a greater influence on the function value. Based on the study of different values and the weighted sum method, the value of c is chosen to be 0.85 in this paper. The T_av and T_cog with respect to the optimization structural parameters are obtained from the ML calculator trained in Section 2.

3.2. Comparison of the DE and CS for Finding Optimization Solution

The objective function Equation (1) is optimized iteratively using the DE and CS, respectively. The MATLAB optimization programs were run on a computer with the 12th generation Intel (R) Core (TM) i7-7700 CPU. Figure 7 shows the changes in objective function values in the DE and CS during 100-generation optimization iterations, and

Table 3. Optimization time of DE and CS during generations.

Function Name	Calls	Total Time	Self Time	Function Name	Calls	Total Time	Self Time
DE_1	1	26.537 s	3.559 s	CS_1	1	67.044 s	4.799 s
Parallel_function	100	21.139 s	0.093 s	Parallel_function	200	58.933 s	0.143 s
Parallel_function > distributed_execution	100	20.294 s	0.156 s	Parallel_function > distributed_execution	200	57.720 s	0.290 s
Remoteparfor.getCompletelntervals	1001	19.252 s	1.111 s	Remoteparfor.getCompletelntervals	2395	55.713 s	2.371 s
LinkedBlockingQueue	1873	15.982 s	15.982 s	LinkedBlockingQueue	4169	48.781 s	48.781 s
DE_1 > func	50	1.141 s	0.008 s	CS_1 > func	100	1.619 s	0.009 s
network.sim	50	1.133 s	0.039 s	network.sim	100	1.610 s	0.055 s

shows optimization time during generations. As shown in Figure 7 and Table 3, for the DE and CS, the optimal solution is obtained for 52 and 87 iterations, respectively, and the total time for optimization is 26.537 s and 67.044 s, respectively. The LinkedBlockingQueue class in Java was called 1873 times by the DE and 4169 times by the CS, accounting for 60.23% and 72.76% of the total optimization time, respectively.

Table 4. Optimization results of the DE and CS.

Parameters	Initial Model	DE Solution	CS Solution
δ/mm	0.35	0.3004	0.3012
bs0/mm	2.0	1.7525	1.7532
α	0.8	0.7675	0.7665
hm/mm	0.5	0.4508	0.4513
Di/mm	11.0	11.2106	11.2083
Tav/mNm	6.4643	6.5561	6.5546
Tcog/mNm	0.6041	0.4106	0.4097
Objective function	0.12625	0.11493	0.11491

shows the comparison of parameters between the initial model of the AFPMM and the optimized models obtained by the DE and CS. The table shows that both algorithms reduce the objective function value from 0.12625 to 0.1149. In addition, compared with the initial model, the optimized solutions of both algorithms increase the T_av of the AFPMM and decrease the T_cog at the same time. Compared with the initial model, the T_av of the optimized AFPMM is increased by 1.42% and 1.40%, and the T_cog of the AFPMM is reduced by 32.08% and 32.18%, respectively. According to the analysis of structural parameters before and after optimization, the coupling changes in air gap δ and PM thickness h_m are the most important factors leading to the increase in the T_av of the AFPMM, while the decrease in the T_cog is mainly attributed to the reduction in the slot opening b_s0.

4. Verification of Optimization Results by the FEA

As shown in the results route in Figure 4, in order to verify the quality of the optimization results, the FEA is performed for torque performance comparison of the initial model and the optimized models. The structural parameters of the initial model and the two optimized models obtained by the DE and CS are used to establish the 3D FEA models of the AFPMM in the Maxwell software, (https://www.ansys.com/products/electronics/ansys-maxwell) respectively. The air gap magnetic field distribution and the output torque characteristics of the AFPMM are presented. In addition to calculating the T_av and T_cog of the AFPMM, the T_rip was also be calculated.

Figure 8 shows the air-gap axial magnetic field distribution of the three models with no load (no current applied to the windings) on the mid-gap surface, obtained through the static magnetic field analysis. The Fast Fourier Transform (FFT) of the air-gap flux density distribution along the circular arc at the rotor average radius of the three models is shown in Figure 9. The fundamental amplitudes of the air-gap flux density of the three models are 0.5788 T, 0.5857 T, and 0.5857 T, respectively. The fundamental amplitude increases by about 1.2% after optimization with the DE or CS, mainly due to the reduced air gap after optimization, although the slot opening, PM embrace, and PM thickness decrease.

Through the transient field analysis, the output torque of the three models is simulated at the rated speed of 6000 rpm with 0.17 A current applied to the windings. Additionally, the T_cog of the three models with no load is simulated at the same rated speed.

Table 5. Torque comparison between optimization results and simulation results.

Parameters	DE Solution	FEA-DE	Error	CS Solution	FEA-CS	Error
Tav/mNm	6.5561	6.5187	−0.57%	6.5546	6.5128	−0.64%
Tcog/mNm	0.4103	0.4169	1.61%	0.4097	0.4156	1.44%
Objective function	0.11493	0.11588	0.84%	0.11491	0.11587	0.84%

compares the torque results from the FEA simulation with the DE and CS optimization. The simulation results of the motor output torque and average torque values of the initial model and the two optimized models are compared, as shown in Figure 10a. The T_cog curves of the three models are shown in Figure 10b. The T_rip is obtained by subtracting the T_av and T_cog from the output torque, and the comparison of T_rip of the three models is shown in Figure 10c.

The results indicate that both optimization algorithms can improve the output torque of the AFPMM for the optimized objective function. The errors between optimization results and FEA results for the objective function, the T_av and T_cog, are relatively small, with a maximum error of only 1.61%, demonstrating high accuracy. Compared with the initial model, the T_av of the FEA after the DE or CS optimization increases by about 1.4%, the T_cog decreases by about 32.1%, and the T_rip decreases by about 31.7%, achieving a better optimization effect.

5. Conclusions

To improve the output torque performance of the AFPMM, two optimization algorithms, the differential evolution and the cuckoo search, are proposed to optimize the structural parameters of the AFPMM with a single-rotor and double-stator configuration. The following conclusions can be drawn from this study:

(1)

To obtain an ML calculator, a method combining the 3D FEA and BP algorithm is proposed to fit the relationships between five structural parameters and two output torque parameters (i.e., T_av and T_cog) using 1024 sets of input–output simulation sample data.

(2)

The inverse function of the sum of the T_av and the inverse values of the T_cog of the motor are designed as the optimization objective function. For this objective function, the selected motor parameters are optimized using the DE and CS algorithms, respectively. The iteration speed and optimization time of the two algorithms in the motor optimization process are specifically analyzed. The DE and CS achieve optimal solutions in 52 and 87 iterations, respectively, with optimization times of 26.537 s and 67.044 s. During optimization, the LinkedBlockingQueue class in Java was called 1873 times by the DE and 4169 times by the CS, accounting for 60.23% and 72.76% of the total optimization time, respectively.

(3)

The validity of the optimized AFPMM structural models is verified using the 3D FEA. The air-gap flux density, average torque, cogging torque, and ripple torque before and after optimization are compared. The results indicate that both DE and CS can improve the air-gap flux density and the motor output torque performance. Compared with the initial model, the two optimized models have a smaller T_cog, an increased T_av, and a reduced T_rip. Specifically, after the DE or CS optimization, the T_av of the motor is increased by about 1.4%, the T_cog is decreased by about 32.1%, and the T_rip is decreased by about 31.7%, achieving a better optimization effect.

(4)

In summary, under the premise of achieving similar optimization results, the DE is more advantageous compared to the CS due to its faster iteration speed and lower resource consumption.

The optimized design methodology provided in the study is not limited to the design of AFPMMs and can be extended to the design of all slotted motors. In the optimization process, although the DE has a faster overall iteration speed, it requires more iteration steps to optimize to the vicinity of the minimum value, whereas the CS has the advantage of a fast iteration speed to the vicinity of the minimum value. Future research could combine the rapid search capability of the CS with the fast iteration capability of the DE to further improve the optimization speed.