Improved Particle Swarm Optimization Algorithm Based Driving Strategy Research for Permanent Magnet Spherical Motor
Zhou Sili1,2, Li Guoli2,3, Wang Qunjing2,4, Zheng Changbao2,3, Wen Yan2,5
1. School of Computer Science and Technology Anhui University Hefei 230601 China; 2. National Engineering Laboratory of Energy-Saving Motor & Control Technology Anhui University Hefei 230601 China; 3. School of Electrical Engineering and Automation Anhui University Hefei 230601 China; 4. Anhui Collaborative Innovation Center of Industrial Energy-Saving and Power Quality Control Anhui University Hefei 230601 China; 5. School of Internet Anhui University Hefei 230601 China
Abstract A permanent magnet spherical motor (PMSpM) is a compact transmission apparatus that is capable of motion in multiple degrees of freedom. To achieve the close loop control of the PMSpM, the driving current of the stator coils needs to be calculated, and the analytic torque model needs to be built in advance. However, if the geometry of the permanent magnet (PM) is a non-circumferential symmetric one, the pseudo-inverse matrix technique is not applicable. Thus, the research on the fast driving strategy of the universal reverse torque model is an essential prerequisite for the PMSpM close-loop control. This paper takes the PMSpM with the stepped cylindrical PM as the research object. Firstly, this paper proposes new analytical torque models using the toroidal expansion method. To avoid repeating integrations in magnetic and torque analytic calculation, this paper builds torque maps by moving one 1A energized electromagnetic coil on the overall spherical surface of the airgap along the azimuth angle direction and polar angle direction. Secondly, the classical particle swarm optimization algorithm (PSO) is introduced to build the reverse torque model. The current of the stator electromagnetic coils is considered as the particle swarm, and the desired torques are set as optimization targets. Thus, we can use the reverse torque model to calculate the driving current of the stator electromagnetic coils from the torque maps. Thirdly, this paper proposes an improved particle swarm optimization (IPSO) algorithm for the PMSpM driving strategy optimization, which can be used for calculating the real-time driving current for the desired torques of the PMSpM. After the determination of the population size of the PSO algorithm, the adaptive dynamic inertia weight and adaptive learning factors are introduced for IPSO. Simulation results on the IPSO algorithm optimization show that the improvement of the classical PSO algorithm is significantly effective. A typical population size can generate convergence before 250 iterations. The larger the population size, the more concentrated the convergence curves. A bigger population size illustrates the robustness of the PSO algorithm, but it also needs more convergence time. Thus, to balance the current calculation algorithm convergence rate, this paper adopts popsize = 30. With the same convergence precision, the PSO algorithm with improved adaptive dynamic inertia weight can get greater calculation efficiency, and the convergence can be completed only around 50 iterations instead of 200 iterations which adopts the traditional inertia weight solution. The convergence rate for the electromagnetic coil current calculation is significantly boosted. In addition, introducing adaptive learning factors can also boost the convergence rate by 20%. Finally, after introducing the adaptive dynamic inertia weight and the adaptive learning factors, the mean one-loop driving current calculation time can be reduced from 710.5ms to 128.2ms. The following conclusions can be drawn from the simulation analysis: ① The driving current calculation speed of the PSO algorithm with adaptive dynamic inertia weight is 5.5 times faster than the classical PSO algorithm; ② The comparison result between the classical PSO algorithm and IPSO algorithm indicates that IPSO has a better convergence rate than PSO on the premise of ensuring the accuracy of convergence. ③ The PMSpM control experimental result shows that the proposed IPSO algorithm is effective in the PMSpM driving strategy, and the PMSpM driving current calculation speed of the proposed IPSO algorithm is significantly faster than using the classical PSO algorithm. In addition, the proposed IPSO algorithm is also applicable for the driving current calculation of other complex special motors.
[1] 黄声华, 陶醒世, 林金铭. 三自由度球形电机的发展[J]. 电工电能新技术, 1989, 8(1): 6-11. Huang Shenghua, Tao Xingshi, Lin Jinming.Development of three-dimensional spherical motor[J]. Advanced Technology of Electrical Engineering and Energy, 1989, 8(1): 6-11. [2] 夏长亮, 李洪凤, 宋鹏, 等. 基于Halbach阵列的永磁球形电动机磁场[J]. 电工技术学报, 2007, 22(7): 126-130. Xia Changliang, Li Hongfeng, Song Peng, et al.Magnetic field model of a PM spherical motor based on Halbach array[J]. Transactions of China Electrotechnical Society, 2007, 22(7): 126-130. [3] Chai Feng, Gan Lei, Yu Yanjun.Magnetic field analysis of an iron-cored tiered type permanent magnet spherical motor using modified dynamic reluctance mesh method[J]. IEEE Transactions on Industrial Electronics, 2020, 67(8): 6742-6751. [4] Wang Qunjing, Li Zheng, Ni Youyuan, et al.3D magnetic field analysis and torque calculation of a PM spherical motor[C]//2005 International Conference on Electrical Machines and Systems, Nanjing, China, 2005, 3: 2116-2120. [5] Li Hongfeng, Zhao Yanfen, Li Bin, et al.Torque calculation of permanent magnet spherical motor based on virtual work method[J]. IEEE Transactions on Industrial Electronics, 2020, 67(9): 7736-7745. [6] 过希文, 李绅, 王群京, 等. 基于三角形(△)组合线圈的永磁球形电机转矩特性与通电策略分析[J]. 电工技术学报, 2019, 34(8): 1607-1615. Guo Xiwen, Li Shen, Wang Qunjing, et al.Analysis of torque characteristics and electrifying strategy of permanent magnet spherical motor based on triangular combination coils[J]. Transactions of China Electrotechnical Society, 2019, 34(8): 1607-1615. [7] Yan Liang, Liu Yinghuang, Zhang Lu, et al.Magnetic field modeling and analysis of spherical actuator with two-dimensional longitudinal camber Halbach array[J]. IEEE Transactions on Industrial Electronics, 2019, 66(12): 9112-9121. [8] Li Zheng, Guo Peng, Wang Zhe, et al.Design and analysis of electromagnetic-piezoelectric hybrid driven three-degree-of-freedom motor[J]. Sensors (Basel, Switzerland), 2020, 20(6): 1621. [9] Zhou Sili, Li Guoli, Wang Qunjing, et al.Geometrical equivalence principle based modeling and analysis for monolayer Halbach array spherical motor with cubic permanent magnets[J]. IEEE Transactions on Energy Conversion, 2021, 36(4): 3241-3250. [10] 李洪凤, 林康, 李斌, 等. 基于四元数的永磁动量球位置/电流双闭环控制[J]. 电工技术学报, 2019, 34(增刊2): 484-492. Li Hongfeng, Lin Kang, Li Bin, et al.Position and current double closed loop control of reaction sphere actuator based on quaternion[J]. Transactions of China Electrotechnical Society, 2019, 34(S2): 484-492. [11] 李斌, 张硕, 李桂丹, 等. 基于球谐函数的动量球定子磁场分析[J]. 电工技术学报, 2018, 33(23): 5442-5448. Li Bin, Zhang Shuo, Li Guidan, et al.Stator magnetic field analysis of reaction sphere based on spherical harmonics[J]. Transactions of China Electrotechnical Society, 2018, 33(23): 5442-5448. [12] Liu Jingmeng, Deng Huiyang, Hu Cungang, et al.Adaptive backstepping sliding mode control for 3-DOF permanent magnet spherical actuator[J]. Aerospace Science and Technology, 2017, 67: 62-71. [13] Bai Kun, Xu Ruoyu, Lee K M, et al.Design and development of a spherical motor for conformal printing of curved electronics[J]. IEEE Transactions on Industrial Electronics, 2018, 65(11): 9190-9200. [14] Ju Lufeng, Wang Qunjing, Qian Zhe, et al.Modeling and optimization of spherical motor based on support vector machine and chaos[C]//2009 International Conference on Electrical Machines and Systems, Tokyo, 2009: 1-4. [15] Wen Yan, Li Guoli, Wang Qunjing, et al.Modeling and analysis of permanent magnet spherical motors by a multitask Gaussian process method and finite element method for output torque[J]. IEEE Transactions on Industrial Electronics, 2021, 68(9): 8540-8549. [16] Kasashima N, Ashida K, Yano T, et al.Torque control method of an electromagnetic spherical motor using torque map[J]. IEEE/ASME Transactions on Mechatronics, 2016, 21(4): 2050-2060. [17] Zhou Rui, Li Guoli, Wang Qunjing, et al.Drive Current calculation and analysis of permanent magnet spherical motor based on torque analytical model and particle swarm optimization[J]. IEEE Access, 2020, 8: 54722-54729,. [18] He Jingxiong, Li Guoli, Zhou Rui, et al.Optimization of permanent-magnet spherical motor based on taguchi method[J]. IEEE Transactions on Magnetics, 2020, 56(2): 1-7. [19] Selvaggi J P, Salon S J, Chari M V K. Employing toroidal harmonics for computing the magnetic field from axially magnetized multipole cylinders[J]. IEEE Transactions on Magnetics, 2010, 46(10): 3715-3723. [20] Qian Zhe, Wang Qunjing, Li Guoli, et al.Design and analysis of permanent magnetic spherical motor with cylindrical poles[C]//2013 International Conference on Electrical Machines and Systems (ICEMS), Busan, Korea (South), 2013: 644-649. [21] Parsopoulos K E, Vrahatis M N.Particle swarm optimization and intelligence advances and applications[M]. Hershey: Information Science Reference, 2010 [22] 李骥, 张慧媛, 程杰慧, 等. 基于源荷状态的跨区互联系统协调优化调度[J]. 电力系统自动化, 2020, 44(17): 26-33. Li Ji, Zhang Huiyuan, Cheng Jiehui, et al.Coordinated and optimal scheduling of inter-regional interconnection system based on source and load status[J]. Automation of Electric Power Systems, 2020, 44(17): 26-33. [23] 王灿, 吴耀文, 孙建军, 等. 基于柔性多状态开关的主动配电网双层负荷均衡方法[J]. 电力系统自动化, 2021, 45(8): 77-85. Wang Can, Wu Yaowen, Sun Jianjun, et al.Bi-layer load balancing method in active distribution network based on flexible multi-state switch[J]. Automation of Electric Power Systems, 2021, 45(8): 77-85. [24] 李奇, 赵淑丹, 蒲雨辰, 等. 考虑电氢耦合的混合储能微电网容量配置优化[J]. 电工技术学报, 2021, 36(3): 486-495. Li Qi, Zhao Shudan, Pu Yuchen, et al.Capacity optimization of hybrid energy storage microgrid considering electricity-hydrogen coupling[J]. Transactions of China Electrotechnical Society, 2021, 36(3): 486-495. [25] 刘细平, 胡卫平, 丁卫中, 等. 永磁同步电机多参数辨识方法研究[J]. 电工技术学报, 2020, 35(6): 1198-1207. Liu Xiping, Hu Weiping, Ding Weizhong, et al.Research on multi-parameter identification method of permanent magnet synchronous motor[J]. Transa-ctions of China Electrotechnical Society, 2020, 35(6): 1198-1207. [26] 李雄松, 崔鹤松, 胡纯福, 等. 平板型永磁直线同步电机推力特性的优化设计[J]. 电工技术学报, 2021, 36(5): 916-923. Li Xiongsong, Cui Hesong, Hu Chunfu, et al.Optimal design of thrust characteristics of flat-type permanent magnet linear synchronous motor[J]. Transactions of China Electrotechnical Society, 2021, 36(5): 916-923. [27] 赵玫, 于帅, 邹海林, 等. 聚磁式横向磁通永磁直线电机的多目标优化[J]. 电工技术学报, 2021, 36(17): 3730-3740. Zhao Mei, Yu Shuai, Zou Hailin, et al.Multi-objective optimization of transverse flux permanent magnet linear machine with the concentrated flux mover[J]. Transactions of China Electrotechnical Society, 2021, 36(17): 3730-3740. [28] Iqbal A, Singh G K.PSO based controlled six-phase grid connected induction generator for wind energy generation[J]. CES Transactions on Electrical Machines and Systems, 2021, 5(1): 41-49. [29] Wu Jiangling, Sun Xiaodong, Zhu Jianguo.Accurate torque modeling with PSO-based recursive robust LSSVR for a segmented-rotor switched reluctance motor[J]. CES Transactions on Electrical Machines and Systems, 2020, 4(2): 96-104. [30] 罗仕华, 胡维昊, 黄琦, 等. 市场机制下光伏/小水电/抽水蓄能电站系统容量优化配置[J]. 电工技术学报, 2020, 35(13): 2792-2804. Luo Shihua, Hu Weihao, Huang Qi, et al.Optimization of photovoltaic/small hydropower/pumped storage power station system sizing under the market mechanism[J]. Transactions of China Electrotechnical Society, 2020, 35(13): 2792-2804. [31] 陈龙, 易琼洋, 贲彤, 等. 全局优化算法在Preisach磁滞模型参数辨识问题中的应用与性能对比[J]. 电工技术学报, 2021, 36(12): 2585-2593, 2606. Chen Long, Yi Qiongyang, Ben Tong, et al.Application and performance comparison of global optimization algorithms in the parameter identification problems of the preisach hysteresis model[J]. Transactions of China Electrotechnical Society, 2021, 36(12): 2585-2593, 2606. [32] 李家祥, 汪凤翔, 柯栋梁, 等. 基于粒子群算法的永磁同步电机模型预测控制权重系数设计[J]. 电工技术学报, 2021, 36(1): 50-59, 76. Li Jiaxiang, Wang Fengxiang, Ke Dongliang, et al.Weighting factors design of model predictive control for permanent magnet synchronous machine using particle swarm optimization[J]. Transactions of China Electrotechnical Society, 2021, 36(1): 50-59, 76. [33] Shi Y, Eberhart R C.Empirical study of particle swarm optimization[C]//Proceedings of the 1999 Congress on Evolutionary Computation-CEC99, Washington, 1999: 1945-1950. [34] Wen Yan, Li Guoli, Wang Qunjing, et al.Robust adaptive sliding-mode control for permanent magnet spherical actuator with uncertainty using dynamic surface approach[J]. Journal of Electrical Engineering and Technology, 2019, 14(1): 2341-2353.
2023, Vol. 38 
