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Article(id=1156907872664769440, tenantId=1146029695717560320, journalId=1146123166801305609, issueId=1156907871645556837, articleNumber=null, orderNo=null, doi=10.12404/j.issn.1671-1815.2402538, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1712592000000, receivedDateStr=2024-04-09, revisedDate=1721232000000, revisedDateStr=2024-07-18, acceptedDate=null, acceptedDateStr=null, onlineDate=1753757931151, onlineDateStr=2025-07-29, pubDate=1737993600000, pubDateStr=2025-01-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1753757931151, onlineIssueDateStr=2025-07-29, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1753757931151, creator=13701087609, updateTime=1753757931151, updator=13701087609, issue=Issue{id=1156907871645556837, tenantId=1146029695717560320, journalId=1146123166801305609, year='2025', volume='25', issue='3', pageStart='879', pageEnd='1312', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=0, createTime=1753757930909, creator=13701087609, updateTime=1765095544280, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1204461268821320541, tenantId=1146029695717560320, journalId=1146123166801305609, issueId=1156907871645556837, language=EN, specialIssueTitle=, coverIllustrator=, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1204461268825514846, tenantId=1146029695717560320, journalId=1146123166801305609, issueId=1156907871645556837, language=CN, specialIssueTitle=, coverIllustrator=, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=1065, endPage=1074, ext={EN=ArticleExt(id=1156907873822397350, articleId=1156907872664769440, tenantId=1146029695717560320, journalId=1146123166801305609, language=EN, title=Multi-objective Optimization Method for Permanent Magnet-assisted Synchronous Reluctance Motor Based on KELM-NSGA-II, columnId=1156262733675876713, journalTitle=Science Technology and Engineering, columnName=Papers·Electrical Technology, runingTitle=null, highlight=null, articleAbstract=

In order to improve the output performance of permanent magnet assisted synchronous reluctance motor (PMa-SynRM), a multi-objective optimization design method for external rotor PMa-SynRM based on kernel extreme learning machine (KELM) and fast non-dominated sorting genetic algorithm (NSGA-II) was proposed. Firstly, the preliminary design of the PMa-SynRM rotor magnetic barrier was carried out and the working principle of the PMa-SynRM was analyzed. Secondly, the influence of each design variable on the optimization goal was evaluated through comprehensive sensitivity analysis, and the main optimization parameters were selected. Thirdly, with high output torque, high efficiency and low torque ripple as the optimization goals, a surrogate model based on KELM was established. Finally, NSGA-II was used for global optimization, and the optimal solution was selected from the Pareto frontier generated by NSGA-II, which was verified by finite element analysis. The simulation results show that the average torque of the optimized motor is increased by 15.83%, the torque ripple is reduced by 60.27%, and the efficiency of the optimized motor is also improved compared with the initial motor, which verifies the effectiveness of the optimized design method proposed in this paper.

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为提高永磁辅助同步磁阻电机(permanent magnet-assisted synchronous reluctance motor, PMa-SynRM)的输出性能,提出了基于核极限学习机(kernel extreme learning machine, KELM)和快速非支配排序遗传算法(nondominated sorting genetic algorithm, NSGA-II)相结合的外转子PMa-SynRM多目标优化设计方法。首先,对PMa-SynRM转子磁障进行初步设计并分析PMa-SynRM工作原理。其次,通过综合敏感性分析评估每个设计变量对优化目标的影响,选取主要优化参数。然后,以高输出转矩、高效率和低转矩脉动为优化目标,建立基于KELM的代理模型。最后,采用NSGA-II进行全局寻优,从NSGA-II生成的Pareto前沿中选择最优解,并通过有限元分析进行验证。仿真结果表明:优化后的电机较初始电机平均转矩提高了15.83%,转矩脉动降低了60.27%,且优化后电机效率较初始电机也有所提高,验证了本文优化设计方法的有效性。

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黄朝志(1978—),男,汉族,江西赣州人,博士,副教授,硕士研究生导师。研究方向:电机结构设计与驱动控制。E-mail:。

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黄朝志(1978—),男,汉族,江西赣州人,博士,副教授,硕士研究生导师。研究方向:电机结构设计与驱动控制。E-mail:。

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黄朝志(1978—),男,汉族,江西赣州人,博士,副教授,硕士研究生导师。研究方向:电机结构设计与驱动控制。E-mail:。

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w1为第一层PM的宽度;c1为第一层PM的长度;D1为第一层PM到电机中心的距离;α1为第一层磁障的张角;y0、y1、y2均为二次函数曲线,分别表示为磁障中线、磁障下边界线和磁障上边界线;A1、A2、A3为构成二次样条曲线的三个点,分别为起点A1、控制点A2和终点A3;m则为参考距离,表示控制点距中心的距离

, figureFileSmall=N+jeTNhDjSozd98Cy1nqWQ==, figureFileBig=DnB0Yy9VoZ8vgIjHr7aQlA==, tableContent=null), ArticleFig(id=1204542859300024485, tenantId=1146029695717560320, journalId=1146123166801305609, articleId=1156907872664769440, language=EN, label=Fig.3, caption=Vector diagram of PMa-SynRM space, figureFileSmall=PdGbIGGsgUq+LYS7ec5Bdg==, figureFileBig=lLJENmF3KCidLozoGp9dKA==, tableContent=null), ArticleFig(id=1204542859421659307, tenantId=1146029695717560320, journalId=1146123166801305609, articleId=1156907872664769440, language=CN, label=图3, caption=PMa-SynRM空间矢量图, figureFileSmall=PdGbIGGsgUq+LYS7ec5Bdg==, figureFileBig=lLJENmF3KCidLozoGp9dKA==, tableContent=null), ArticleFig(id=1204542859530711213, tenantId=1146029695717560320, journalId=1146123166801305609, articleId=1156907872664769440, language=EN, label=Fig.4, caption=Schematic diagram of motor structure parameters, figureFileSmall=uD3QY/R0urT54gSF3ewP1w==, figureFileBig=lRNnTTWmbo0Y5rhClJTWEg==, tableContent=null), 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label=图17, caption=总转矩、磁阻转矩和永磁转矩, figureFileSmall=AzygVXbK7AVd7udhoef3Mg==, figureFileBig=qo5WFDAUVeqo3f/ZSkmhTg==, tableContent=null), ArticleFig(id=1204542863683072349, tenantId=1146029695717560320, journalId=1146123166801305609, articleId=1156907872664769440, language=EN, label=Table 1, caption=

Basic parameters of the initial external rotor PMa-SynRM

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参数名称 参数值 参数名称 参数值
额定转速/(r·min-1) 350 转子内径/mm 171.4
极对数 4 定子内径/mm 100
定子槽数 48 气隙长度/mm 0.7
转子外径/mm 220 驱动电流/A 11
), ArticleFig(id=1204542863783735649, tenantId=1146029695717560320, journalId=1146123166801305609, articleId=1156907872664769440, language=CN, label=表1, caption=

初始外转子PMa-SynRM基本参数

, figureFileSmall=null, figureFileBig=null, tableContent=
参数名称 参数值 参数名称 参数值
额定转速/(r·min-1) 350 转子内径/mm 171.4
极对数 4 定子内径/mm 100
定子槽数 48 气隙长度/mm 0.7
转子外径/mm 220 驱动电流/A 11
), ArticleFig(id=1204542863892787557, tenantId=1146029695717560320, journalId=1146123166801305609, articleId=1156907872664769440, language=EN, label=Table 2, caption=

Upper and lower limits of change in design variables

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类型 设计变量 变量下限 变量上限
内层磁障 PM长度c1/mm 14 18
PM宽度w1/mm 3 4
磁障张角α1/(°) 8 11
中层磁障 PM长度c2/mm 18 22
PM宽度w2/mm 3 5
磁障张角α2/(°) 13 17
外层磁障 PM长度c3/mm 18 28
PM宽度w3/mm 3 5
磁障张角α3/(°) 20 22
其他变量 内中层PM相距h1/mm 1 3
中外层PM相距h2/mm 1 3
PM位置D1/mm 80 90.5
参考距离/m 86.7 87.7
), ArticleFig(id=1204542863997645162, tenantId=1146029695717560320, journalId=1146123166801305609, articleId=1156907872664769440, language=CN, label=表2, caption=

设计变量变化上下限

, figureFileSmall=null, figureFileBig=null, tableContent=
类型 设计变量 变量下限 变量上限
内层磁障 PM长度c1/mm 14 18
PM宽度w1/mm 3 4
磁障张角α1/(°) 8 11
中层磁障 PM长度c2/mm 18 22
PM宽度w2/mm 3 5
磁障张角α2/(°) 13 17
外层磁障 PM长度c3/mm 18 28
PM宽度w3/mm 3 5
磁障张角α3/(°) 20 22
其他变量 内中层PM相距h1/mm 1 3
中外层PM相距h2/mm 1 3
PM位置D1/mm 80 90.5
参考距离/m 86.7 87.7
), ArticleFig(id=1204542864119279983, tenantId=1146029695717560320, journalId=1146123166801305609, articleId=1156907872664769440, language=EN, label=Table 3, caption=

Composite sensitivity index

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设计变量 皮尔逊相关系数(权重) 综合敏
感度
S T a v g (0.25) Sη (0.25) STrip(0.5)
α1 0.254 0.242 -0.645 0.447
α2 0.169 0.172 0.044 0.107
α3 -0.158 -0.145 0.288 0.220
c1 0.068 0.065 0.032 0.050
c2 0.064 0.060 0.035 0.049
c3 0.066 0.069 -0.030 0.049
w1 -0.074 -0.075 -0.005 0.040
w2 -0.097 -0.101 0.015 0.057
w3 -0.074 -0.075 0.027 0.051
h1 0.161 0.151 -0.189 0.172
h2 0.207 0.208 -0.337 0.272
m -0.749 -0.748 -0.131 0.440
D1 -0.330 -0.333 -0.025 0.178
), ArticleFig(id=1204542864249303413, tenantId=1146029695717560320, journalId=1146123166801305609, articleId=1156907872664769440, language=CN, label=表3, caption=

综合敏感度指数

, figureFileSmall=null, figureFileBig=null, tableContent=
设计变量 皮尔逊相关系数(权重) 综合敏
感度
S T a v g (0.25) Sη (0.25) STrip(0.5)
α1 0.254 0.242 -0.645 0.447
α2 0.169 0.172 0.044 0.107
α3 -0.158 -0.145 0.288 0.220
c1 0.068 0.065 0.032 0.050
c2 0.064 0.060 0.035 0.049
c3 0.066 0.069 -0.030 0.049
w1 -0.074 -0.075 -0.005 0.040
w2 -0.097 -0.101 0.015 0.057
w3 -0.074 -0.075 0.027 0.051
h1 0.161 0.151 -0.189 0.172
h2 0.207 0.208 -0.337 0.272
m -0.749 -0.748 -0.131 0.440
D1 -0.330 -0.333 -0.025 0.178
), ArticleFig(id=1204542864328995194, tenantId=1146029695717560320, journalId=1146123166801305609, articleId=1156907872664769440, language=EN, label=Table 4, caption=

Comparison of prediction models for each optimization objective

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模型
指标		 平均转矩 转矩脉动 效率
R2 MAE/% R2 MAE/% R2 MAE/%
BP 0.941 1.62 0.935 8.16 0.990 0.32
RBF 0.999 0.22 0.978 5.29 0.999 0.05
ELM 0.979 0.79 0.925 9.79 0.942 0.39
SVM 0.998 0.25 0.967 6.00 0.998 0.06
KELM 0.999 0.18 0.990 3.75 0.999 0.04
), ArticleFig(id=1204542864438047101, tenantId=1146029695717560320, journalId=1146123166801305609, articleId=1156907872664769440, language=CN, label=表4, caption=

各优化目标预测模型比较

, figureFileSmall=null, figureFileBig=null, tableContent=
模型
指标		 平均转矩 转矩脉动 效率
R2 MAE/% R2 MAE/% R2 MAE/%
BP 0.941 1.62 0.935 8.16 0.990 0.32
RBF 0.999 0.22 0.978 5.29 0.999 0.05
ELM 0.979 0.79 0.925 9.79 0.942 0.39
SVM 0.998 0.25 0.967 6.00 0.998 0.06
KELM 0.999 0.18 0.990 3.75 0.999 0.04
), ArticleFig(id=1204542864521933185, tenantId=1146029695717560320, journalId=1146123166801305609, articleId=1156907872664769440, language=EN, label=Table 5, caption=

Comparison before and after optimization

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参数 优化前 优化后 参数 优化前 优化后
α1/(°) 10 10.67 w3/mm 4 4.65
α2/(°) 15 14.38 h1/mm 2 2.77
α3/(°) 21 20.72 h2/mm 2 2.99
c1/mm 16 15.65 m/mm 87.2 86.70
c2/mm 20 19.94 D1/mm 90 88.22
c3/mm 24 22.07 Tavg/(N·m) 7.39 8.56
w1/mm 3.5 3.04 Trip/% 22.3 8.88
w2/mm 4 4.39 η/% 76.15 78.66
), ArticleFig(id=1204542864630985091, tenantId=1146029695717560320, journalId=1146123166801305609, articleId=1156907872664769440, language=CN, label=表5, caption=

优化前后对比

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参数 优化前 优化后 参数 优化前 优化后
α1/(°) 10 10.67 w3/mm 4 4.65
α2/(°) 15 14.38 h1/mm 2 2.77
α3/(°) 21 20.72 h2/mm 2 2.99
c1/mm 16 15.65 m/mm 87.2 86.70
c2/mm 20 19.94 D1/mm 90 88.22
c3/mm 24 22.07 Tavg/(N·m) 7.39 8.56
w1/mm 3.5 3.04 Trip/% 22.3 8.88
w2/mm 4 4.39 η/% 76.15 78.66
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基于KELM-NSGA-II的永磁辅助同步磁阻电机多目标优化方法
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黄朝志 , 李思颖 , 刘小波 , 孙燕文
科学技术与工程 | 论文·电工技术 2025,25(3): 1065-1074
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科学技术与工程 | 论文·电工技术 2025, 25(3): 1065-1074
基于KELM-NSGA-II的永磁辅助同步磁阻电机多目标优化方法
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黄朝志 , 李思颖, 刘小波, 孙燕文
作者信息
  • 江西理工大学电气工程与自动化学院, 赣州 341000

    • 黄朝志(1978—),男,汉族,江西赣州人,博士,副教授,硕士研究生导师。研究方向:电机结构设计与驱动控制。E-mail:。

Multi-objective Optimization Method for Permanent Magnet-assisted Synchronous Reluctance Motor Based on KELM-NSGA-II
Chao-zhi HUANG , Si-ying LI, Xiao-bo LIU, Yan-wen SUN
Affiliations
  • School of Electrical Engineering and Automation, Jiangxi University of Science and Technology, Ganzhou 341000, China
出版时间: 2025-01-28 doi: 10.12404/j.issn.1671-1815.2402538
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摘要
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为提高永磁辅助同步磁阻电机(permanent magnet-assisted synchronous reluctance motor, PMa-SynRM)的输出性能,提出了基于核极限学习机(kernel extreme learning machine, KELM)和快速非支配排序遗传算法(nondominated sorting genetic algorithm, NSGA-II)相结合的外转子PMa-SynRM多目标优化设计方法。首先,对PMa-SynRM转子磁障进行初步设计并分析PMa-SynRM工作原理。其次,通过综合敏感性分析评估每个设计变量对优化目标的影响,选取主要优化参数。然后,以高输出转矩、高效率和低转矩脉动为优化目标,建立基于KELM的代理模型。最后,采用NSGA-II进行全局寻优,从NSGA-II生成的Pareto前沿中选择最优解,并通过有限元分析进行验证。仿真结果表明:优化后的电机较初始电机平均转矩提高了15.83%,转矩脉动降低了60.27%,且优化后电机效率较初始电机也有所提高,验证了本文优化设计方法的有效性。

永磁辅助同步磁阻电机  /  KELM  /  多目标优化  /  NSGA-II

In order to improve the output performance of permanent magnet assisted synchronous reluctance motor (PMa-SynRM), a multi-objective optimization design method for external rotor PMa-SynRM based on kernel extreme learning machine (KELM) and fast non-dominated sorting genetic algorithm (NSGA-II) was proposed. Firstly, the preliminary design of the PMa-SynRM rotor magnetic barrier was carried out and the working principle of the PMa-SynRM was analyzed. Secondly, the influence of each design variable on the optimization goal was evaluated through comprehensive sensitivity analysis, and the main optimization parameters were selected. Thirdly, with high output torque, high efficiency and low torque ripple as the optimization goals, a surrogate model based on KELM was established. Finally, NSGA-II was used for global optimization, and the optimal solution was selected from the Pareto frontier generated by NSGA-II, which was verified by finite element analysis. The simulation results show that the average torque of the optimized motor is increased by 15.83%, the torque ripple is reduced by 60.27%, and the efficiency of the optimized motor is also improved compared with the initial motor, which verifies the effectiveness of the optimized design method proposed in this paper.

permanent magnet-assisted synchronous reluctance motor  /  KELM  /  multi-objective optimization  /  NSGA-II
黄朝志, 李思颖, 刘小波, 孙燕文. 基于KELM-NSGA-II的永磁辅助同步磁阻电机多目标优化方法. 科学技术与工程, 2025 , 25 (3) : 1065 -1074 . DOI: 10.12404/j.issn.1671-1815.2402538
Chao-zhi HUANG, Si-ying LI, Xiao-bo LIU, Yan-wen SUN. Multi-objective Optimization Method for Permanent Magnet-assisted Synchronous Reluctance Motor Based on KELM-NSGA-II[J]. Science Technology and Engineering, 2025 , 25 (3) : 1065 -1074 . DOI: 10.12404/j.issn.1671-1815.2402538
正文
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近年来,永磁同步电机(permanent magnet synchronous motor, PMSM)因具有高功率密度、高效率和宽恒功率调速范围等优点被广泛运用于工业和交通等领域[1-2]。然而随着稀土永磁体(permanent magnet, PM)资源日益紧缺所带来的供应链不稳定和价格波动等问题一定程度上限制了PMSM的发展[3]。为减少对稀土材料的依赖,永磁辅助同步磁阻电机(permanent magnet-assisted synchronous reluctance motor, PMa-SynRM)这种少稀土甚至无稀土电机逐渐引起重视[4]。PMa-SynRM结合了同步磁阻电机(synchronous reluctance motors, SynRM)和PMSM的优点。与SynRM相比,PMa-SynRM存在永磁体励磁,具有更高的转矩密度和功率因数[5],与PMSM相比,PMa-SynRM减少了稀土永磁体的使用,成本降低,使其性价比远高于PMSM[6]。但随着永磁体的加入,如何权衡磁障设计与永磁体用量以实现PMa-SynRM的高性能输出仍是亟待解决的问题之一。
为设计高效可靠的PMa-SynRM,目前多采用多目标优化方法对PMa-SynRM进行优化设计。文献[7]在ANSYS软件中用参数化扫描方法优化PMa-SynRM,可以直观地看出各设计变量对优化目标的影响,但存在计算量大、耗时长的缺点,不适合变量较多的情况。为进一步提高PMa-SynRM多目标优化效果,文献[8]采用了基于代理模型结合优化算法的方法,以寻找满足电机设计的最优组合。常用的代理模型有响应面模型[9]、Kriging模型[10]、神经网络模型[11]和核极限学习机(kernel extreme learning machine, KELM)模型[12]等。代理模型的拟合效果将直接影响电机优化的可靠性,选取合适的代理模型直接影响电机的优化设计。文献[13]对比了几种代理模型的拟合效果,并选择了拟合效果最佳的随机森林模型。
电机优化设计中常见的优化算法有粒子群算法、遗传算法[14]和差分进化算法[15]等。文献[16]结合Kriging模型与改进的粒子群多目标优化算法,优化开关磁阻电机本体结构,维持了较大的平均转矩,同时大幅度降低了转矩脉动。文献[17]对PMa-SynRM的设计变量进行强弱敏感层分类处理,再利用非支配排序遗传算法(nondominated sorting genetic algorithm, NSGA-II)算法多目标寻优,得到了较好的电机热性能。
基于以上分析,以外转子PMa-SynRM为对象,提出一种基于代理模型和多目标优化算法相结合的优化策略。首先建立电机的初始模型并确定电机的优化变量和优化目标。然后,根据综合敏感度分析,将优化变量分为强敏感层和弱敏感层,选出重要设计变量。以高输出转矩、高效率和低转矩脉动为优化目标,采用KELM与NSGA-II相结合的方法对其进行寻优设计。最后通过对优化前后电机性能进行仿真分析,验证本文所用方法的有效性。
1 电机结构与工作原理
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1.1 电机结构
设计一台8极48槽外转子PMa-SynRM,如图1所示。转子磁障设计是PMa-SynRM结构设计的核心,文献[18]采用直线与函数曲线相结合的方法构造转子磁障,此方法构造出的磁障形状具有更广泛变化。考虑到电机低速运行,转子采用周向磁桥便能满足机械强度要求,因此不留有径向磁桥。以其中一层磁障为例,其结构如图2所示。
由图2可知,通过c1、w1、D1、m、α1可确定待求系数α进一步可得到y0的表达式为
y0=α x - c 1 2 2+D1+ w 1 2
将y0向上、向下平移 w 1 2即可得到磁障上下层y1、y2的表达式分别为
y 1 = α x - c 1 2 2 + D 1 y 2 = α x - c 1 2 2 + D 1 + w 1
最后在磁障端部利用二次样条曲线对A1、A2、A3拟合即可得到完整的磁障,其他两层磁障的构建方法同上。
转子每极下有3层磁障,磁阻转矩通过转子铁心中的磁障产生。每层磁障中分别嵌入铁氧体永磁材料,产生永磁转矩。初始样机基本参数如表1所示。
1.2 工作原理
PMa-SynRM主要依靠转子交(q)、直(d)轴电感差从而产生磁阻转矩来驱动电机,又由于在转子中添加永磁体,使得永磁体磁场与定子磁场相互作用产生永磁转矩。PMa-SynRM的磁链,电压方程为
ψ d = L d i d + ψ P M ψ q = L q i q
u d = d ψ d d t - ω ψ q + R i d u q = d ψ q d t + ω ψ d + R i q
式中:ψd、ψq为直、交轴磁链;ψPM为永磁体产生磁链;Ld、Lq为直、交轴电感;id、iq为电流直、交轴分量;ud、uq为直、交轴电压;ω为电机电角速度;R为绕组相电阻。
根据式(3)和式(4)可得经过park变换后的d-q转子等效矢量图,如图3所示。根据矢量图可得PMa-SynRM电磁转矩为
Te=p[ψPMiq+(Ld-Lq)idiq]=pψPMissinβ+ 1 2p(Ld-Lq) i s 2sin(2β)
式(5)中:p为极对数;is为定子电流;β为is与d轴夹角。
2 优化设计
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2.1 优化目标与设计变量选择
钕铁硼等永磁材料价格昂贵,考虑到电机生产成本,本文中电机尽可能少用永磁材料,但仍然能实现电机的高性能输出。因此将平均转矩、转矩脉动和效率作为优化目标,设定为
min(1/Tavg,Trip,1/η)
且满足条件
T a v g 7   N · m T r i p 10 % η 70 %
转矩脉动是电机输出性能的关键指标,特别是在磁阻电机中,将导致更大的振动和噪声。这里采用的转矩脉动计算公式为
Trip= T m a x - T m i n T a v g×100%
式(8)中:Tmax、Tmin和Tavg分别为电磁转矩的最大峰值、最小峰值和平均值。
当不考虑杂散损耗时,效率的计算公式为
η= P o u t P i n×100%= P o u t P o u t + P l o s s×100%
Ploss=PCu+PFe+Pfw
式中:Pout为输出功率;Pin为输入功率;Ploss为损耗;PCu为铜耗;PFe为铁耗;Pfw为机械损耗。
图4所示为外转子PMa-SynRM的转子参数化模型。
永磁体的大小和位置将引起气隙磁场变化,影响输出转矩。如图4所示,在转子的每层磁障中,永磁体的长度和宽度分别为c1、c2、c3、w1、w2、w3,最内层永磁体的位置为D1,三层永磁体两两位置相距为h1和h2。各层磁障张角分别为α1、α2、α3,m为磁障端部距圆心的距离,m减去转子内径即为转子周向磁桥trib的大小。设计变量的变化范围见表2。
2.2 敏感性分析
在初始设计中,选择了13个设计变量和3个优化目标。直接采用代理模型构建优化变量与优化目标的函数关系,很难保证代理模型的拟合质量,同时一些敏感性较弱的设计变量容易被忽略。此外,多目标优化算法在处理多个设计变量时,不仅难以收敛,且耗时长,因此有必要降低优化变量的维数。
为减少设计空间,结合有限元软件,采用拉丁超立方取样方法得到一组设计变量与优化目标的数据集。为了评价设计变量与优化目标之间的相关性,引入皮尔逊相关系数对数据集进行分析。皮尔逊相关系数的数学表达式为
ρX,Y= N ∑ i = 1 N X i Y i - ∑ i = 1 N X i ∑ i = 1 N Y i N ∑ i = 1 N X i 2 - ( ∑ i = 1 N X i ) 2 N ∑ i = 1 N Y i 2 - ( ∑ i = 1 N Y i ) 2
式(11)中:Xi为设计变量;Yi为第i个优化目标;N为样本量。
图5所示为13个设计变量和3个优化目标的皮尔逊相关系数结果,其绝对值大小表示了每个设计变量与其对应优化目标的约束程度。
不同设计变量对各优化目标表现出不同效果,为综合考虑3个优化目标,需要对各设计变量的敏感度进行计算,公式为
S P M a S y n , R M ( x i ) = w T a v g S T a v g ( x i ) + w η S η ( x i ) + w T r i p S T r i p ( x i ) w T a v g + w η + w T r i p = 1
式(12)中:xi为设计变量;STavg(xi)、Sη(xi)和STrip(xi)分别为平均转矩、效率和转矩脉动的皮尔逊相关系数;wTavg、wη和wTrip分别为平均转矩、效率和转矩脉动的权重。
同步磁阻电机的转矩脉动是此类电机最为关切的指标,这里将wTrip设为0.5,wTavg和wη均设为0.25。根据式(12)和表3中的设计变量的皮尔逊相关系数计算综合敏感度指数,结果如表3所示。
按照综合敏感度由高到低对设计变量进行分层排序,设置阈值0.17作为各设计变量综合敏感度的判据,综合敏感度指数超过0.17的变量划分为强敏感层,低于0.17的为弱敏感层。根据表3中的综合敏感度指数,最终选择α1、m、h2、α3、D1、h1为强相关变量进入到下一步的分析中,其他变量则通过有限元仿真分析直接确定。
2.3 电机KELM代理模型
2.3.1 核极限学习机
极限学习机(extreme learning machine, ELM)是一种单隐层前馈神经网络机器学习方法。由于为单隐层网络,ELM可以随机初始化输入层与隐含层间的网络权重、直接计算隐含层与输出层间的权值矩阵,从而得到输出值。在训练过程中,只需要设置隐含层神经元的个数和激活函数,不需要多次调整输入层与隐含层间的连接权值及隐含层神经元阈值,极大地提高了训练的速度。然而,由于训练过程中隐含层输出矩阵和输出权值都是随机生成的,使得ELM模型的稳定性和泛化能力降低。
为提高ELM模型的稳定性和泛化能力,引入核函数,将ELM中输入样本映射到高维核空间的随机矩阵用核矩阵代替。这样,输出权矩阵只由训练样本和核函数确定。保持了ELM的计算速度快的同时,提高了ELM的稳定性。
根据Mercer的条件,定义核矩阵ΩELM为
ΩELM=HHT
Ω E L M i , j=f(xi)f(xj)=K(xi,xj)
式中:H为隐含层的输出矩阵;f(x)为隐含层的特征映射函数;K(xi,xj)为高斯核函数,其表达式为
$K\left(x_{i}, x_{j}\right)=\exp \left(-\frac{\left\|x_{i}, x_{j}\right\|}{2 \sigma^{2}}\right)$
式(15)中:σ为核参数。
因此,KELM的输出为
g(x)= K ( x , x 1 ) K ( x , x 2 ) ︙ K ( x , x N ) T I C + Ω E L M - 1L
式(16)中:C为正则化系数;I为单位矩阵;L为期望输出矩阵。
2.3.2 建立代理模型
将通过敏感性分析得到的强相关变量α1、m、h2、α3、D1和h1作为模型的输入变量,3个优化目标作为模型的输出变量。采用有限元参数化分析方法,得到一组关于优化目标与优化变量之间关系的样本数据,用于模型的训练和验证,每个优化目标最终选取300组样本数据。
为了测试KELM的预测性能,与反向传播(back propagation, BP)神经网络、径向基函数(radial basis function, RBF)神经网络、ELM和支持向量机(support vector machines, SVM)进行综合对比,选取平均绝对误差(mean absolute error,MAE)和决定系数R2作为评价上述建模方法性能的标准。利用有限元参数分析得到的260组样本数据训练5个近似模型,并选取另外40组样本数据评价预测精度。图6~图8分别为5种预测模型所对应的平均转矩、转矩脉动、效率的真实值与预测值的对比结果。各优化目标的决定系数和平均绝对误差如表4所示。
由图6~图8可知,KELM模型的平均转矩、转矩脉动和效率的预测值较其他模型更接近真实值。同时结合表4可知,KELM模型的平均转矩、转矩脉动和效率的决定系数较BP、ELM、SVM模型的决定系数都大,虽然在平均转矩和效率上与RBF模型的决定系数相等,但在转矩脉动上,KELM模型的决定系数更大。并且KELM模型的平均绝对误差较其他4种模型的都小,表明KELM模型的预测精度要优于其他4种预测模型。因此本文选择KELM模型作为优化的代理模型。
2.4 多目标优化
结合上述高精度代理模型,采用多目标优化算法搜索设计变量的最优组合。目前,多目标算法中常采用多目标粒子群算法、多目标差分进化算法、多目标遗传算法等。多目标差分进化算法存在过早收敛的问题,而多目标粒子群算法容易陷入局部最优解,相比之下,多目标遗传算法具有较强的全局搜索能力,但无法保证搜索的多样性,收敛速度较慢。文献[19]中将多目标差分进化算法、多目标遗传算法和NSGA-II3种多目标算法进行对比,结果表明与前两种算法相比,NSGA-II具有更大的搜索范围,且提供更好的最优解。因此,本文中选用NSGA-II进行多目标优化。
设置优化种群大小为50,交叉和变异的比例都为50%,交叉概率为0.8,最大迭代次数为100。图9为经过NSGA-II算法寻优以后,得到的三维Pareto前沿和二维Pareto前沿投影图如图9所示。初始转子与最优转子的设计变量之间的差异如表5所示。
3 电机性能比较
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为验证本文提出的优化设计方法的可行性,采用有限元法对初始电机和优化后电机的电磁性能进行比较。仿真时,对优化前后的外转子PMa-SynRM采取了相同的激励、剖分设置、仿真步长以及仿真时间。优化后的电机的永磁体和绕组在定转子铁心上产生的磁力线分布如图10所示。
由图10可知,优化后电机空载时转子磁桥磁密较其余地方明显更密且漏磁较少。
图11为优化后的电机的磁密云图。由图11可知,在转子磁障肋部的磁通密度最大可达到2.472 T,同时可实现磁通的均匀分布。
3.1 谐波分析
当负载为零、转速为350 r/min时,对比初始和优化后电机的气隙磁密和空载反电动势,如图12~图15所示。
优化后电机的气隙磁密峰值高于初始电机的气隙磁密峰值,分别为0.1 T和0.15 T。由图13可知基频气隙磁通密度从0.09 T增大到0.13 T,谐波分量幅值减少,表明优化后电机气隙磁场具有更好的正弦性能和更高的输出转矩。
图14为优化前后电机的空载反电动势波形,优化后电机的空载反电动势幅值增大、波形更加正弦化。从图15中可以看出,优化后电机的基波幅值高于优化前,优化后电机空载反电动势的谐波成分主要为3次谐波和少量的9、13次谐波,而优化前电机空载反电动势的3、11次谐波都比较高。优化前后电机空载反电动势的谐波畸变率分别为8.78%和3.95%。谐波畸变率(total harmonic distortion,THD)计算公式为
THD= E 2 2 + E 3 2 + E 4 2 + … E 1
式(17)中:E1为空载反电动势的基波分量;E2、E3、E4分别为二、三、四次谐波。
优化后畸变率降低了4.83个百分点,电机具有更小的转矩脉动。
3.2 转矩性能分析
在额定工况下仿真得到的电机优化前后转矩与转子位置关系如图16所示。由图16可知,初始电机的平均转矩为7.39 N·m,转矩脉动为22.35%,优化后电机的平均转矩为8.56 N·m,转矩脉动为8.88%。优化后电机的平均转矩提高了15.83%,转矩脉动减小了60.27%。可知,优化后电机不仅转矩脉动降低了,其平均转矩也提高了。
由于PMa-SynRM转子磁障中嵌有永磁体,因此PMa-SynRM的电磁转矩不仅有磁阻转矩,还包含了永磁转矩。为了清楚地说明PMa-SynRM的转矩特性,采用冻结磁导率法分离电磁转矩,并计算分离后的磁阻转矩和永磁转矩的各自占比。优化前后电机电磁转矩分离结果如图17所示。
从图17中可以看出,由于电机的双凸极结构,凸极效应所产生的磁阻转矩在电磁转矩中仍然占主导地位。在最大转矩电流比控制下电机的磁阻转矩由4.49 N·m提升至5.67 N·m,改进效果为26.3%。从占比分布来看,在最大转矩电流比控制下,初始PMa-SynRM的磁阻转矩占比为60.78%,优化后PMa-SynRM的磁阻转矩占比达到了66.24%,较优化前磁阻转矩占比提高了5.46%。结果表明,本文中对转子结构的优化增大了PMa-SynRM的凸极比,提高了磁阻转矩的占比。
4 结论
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本文提出了一种基于KELM代理模型与NSGA-II相结合的多目标优化策略。以平均转矩、转矩脉动和效率为优化目标,对外转子PMa-SynRM进行优化设计,得到如下主要结论。
(1)通过敏感性分析确定了各设计变量的重要性,简化了设计过程,同时有效地实现了各设计目标之间的权衡设计。
(2)采用KELM代理模型与NSGA-II相结合的多目标优化策略,使得优化后电机的平均转矩提高了15.83%,转矩脉动减小了60.27%,效率较初始电机也有所提高。
(3)通过冻结磁导率法分析得到优化后电机的磁阻转矩利用率较优化前提高了5.46%,但未实现磁阻转矩和永磁转矩的充分利用,因此如何实现磁阻转矩和永磁转矩的充分利用仍需深入研究。
综上所述,研究成果验证了所提出的对外转子PMa-SynRM优化设计方法的有效性,对电机优化设计具有一定参考意义。
基金
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  • 国家自然科学基金(52167005)
  • 江西省自然科学基金(20232BAB204063)
文献
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2025年第25卷第3期
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doi: 10.12404/j.issn.1671-1815.2402538
  • 接收时间：2024-04-09
  • 首发时间：2025-07-29
  • 出版时间：2025-01-28
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  • 收稿日期：2024-04-09
  • 修回日期：2024-07-18
基金
国家自然科学基金(52167005)
江西省自然科学基金(20232BAB204063)
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    江西理工大学电气工程与自动化学院, 赣州 341000
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