Advanced
Minimization of a Cogging Torque for an Interior Permanent Magnet Synchronous Machine using a Novel Hybrid Optimization Algorithm
Minimization of a Cogging Torque for an Interior Permanent Magnet Synchronous Machine using a Novel Hybrid Optimization Algorithm
Journal of Electrical Engineering and Technology. 2014. May, 9(3): 859-865
Copyright © 2014, The Korean Institute of Electrical Engineers
  • Received : August 14, 2013
  • Accepted : October 22, 2013
  • Published : May 01, 2014
Download
PDF
e-PUB
PubReader
PPT
Export by style
Article
Author
Metrics
Cited by
TagCloud
About the Authors
Il-Woo Kim
Hyosung corporation, Changwon plant Special Motor Design Team, Korea (skirtkim@hotmail.com)
Dong-Kyun Woo
Dept. of Electrical and Computer Engineering, Seoul National University, Korea (ldk8745@gmail.com)
Dong-Kuk Lim
Dept. of Electrical and Computer Engineering, Seoul National University, Korea (ldk8745@gmail.com)
Sang-Yong Jung
School of Electronic and Electrical Engineering, Sungkyunkwan University, Korea (Syjung@ece.skku.ac.kr)
Cheol-Gyun Lee
Electrical Engineering, Dong Eui University, Korea (cglee@deu.ac.kr)
Jong-Suk Ro
Corresponding Author: Creative Research Engineer Development, Brain Korea 21 Plus, Seoul National University, Seoul, Korea. (jongsukro@naver.com)
Hyun-Kyo Jung
Dept. of Electrical and Computer Engineering, Seoul National University, Korea (ldk8745@gmail.com)

Abstract
Optimization of an electric machine is mainly a nonlinear multi-modal problem. For the optimization of the multi-modal problem, many function calls are required with much consumption of time. To address this problem, this paper proposes a novel hybrid algorithm in which function calls are less than conventional methods. Specifically, the proposed method uses the kriging metamodel and the fill-blank technique to find an approximated solution in a whole problem region. To increase the convergence speed in local peaks, a parallel gradient assisted simplex method is proposed and combined with the kriging meta-model. The correctness and usefulness of the proposed hybrid algorithm is verified through a mathematical test function and applied into the practical optimization as the cogging torque minimization for an interior permanent magnet synchronous machine.
Keywords
1. Introduction
An objective function for the optimization of an electromagnetic machine is mostly nonlinear function and has many peaks. For these optimizations of the nonlinear multi-modal function, many researchers have proposed several techniques such as the auto tuning niching genetic algorithm(ATNGA), and the climb method [1 - 8] .
Ecosystem-inspired optimizations have been widely used for the optimization of the electromagnetic machine [4 - 15] . These algorithms can find a number of optima, which include the global solution through a large number of function evaluations.
Optimization methods can be categorized into two types, the stochastic method and the deterministic method. In the stochastic method, which is a probabilistic optimization method, the genetic algorithm (GA) has been commonly used. Although probabilistic optimization methods such as GA show high performance for searching optima over an entire problem region, the speed of convergence to the optimal point is slow. On the other hand, the deterministic optimization methods like the pattern search or the simplex method cannot search the optima over the entire problem region and the optimal point converges on a local peak rapidly.
To compensate for the weakness of each stochastic method and deterministic method and to combine their strengths, the hybrid algorithm, which combines the interpolation method and the deterministic optimization method, is proposed in this paper. To search the global optima effectively over an entire problem region, the interpolation method is used in this research instead of using the stochastic method. This is because the interpolation method is more efficient and useful than the general stochastic methods for the combination with the deterministic method. Hence, the proposed algorithm can search the global optima. The kriging method, which is widely used for the interpolation, is used in this research [16] . To enhance the convergence speed to a local-optimal point, a novel parallel gradient assisted simplex method is combined with the interpolation method. Hence, this research refers to the proposed hybrid algorithm as kriging combined with the parallel gradient assisted simplex method (KSM).
The correctness and usefulness of the proposed KSM is verified through the application into a mathematical test function and a practical electric machine. Via the mathematical test function, the performance of the propose KSM is compared with that of conventional multi-modal optimization algorithms such as an ATNGA and a climbing method. Furthermore, to verify the possibility of the application into the practical electrical machine, the interior permanent magnet synchronous machine (IPMSM) is designed optimally to minimize the cogging torque.
2. The Proposed Algorithm, KSM
Specification of the KSM according to the step is as follows based on Fig. 1 .
PPT Slide
Lager Image
Flow chart of the proposed KSM.
- 1) Step 0 – Initialization
Initialize parameters such as the problem region and the number of initial population for optimization process.
- 2) Step 1 – Create initial population
The initial population should be generated and distributed in the problem region evenly and properly to reflect the tendency of the problem region at the initial stage. If the number of initial samples is k 2 , the problem region is divided into k by k matrix and one sample is generated randomly in each small region of divided matrix as shown in Fig. 2 .
PPT Slide
Lager Image
Initial samples generated randomly (49 samples)
- 3) Step 2 – Construction of an approximated model
The KSM employs kriging to build an approximated meta-model of the real objective function.
- 4) Step 3 – Simplex operation
Although the optima can be found from the meta-model of objective function, which is constructed in the previous step, they are not the exact solution. To approach to the exact solution, the parallel gradient assisted simplex method (GASM) is adopted in this research. In the conventional simplex method, a simplex geometry searches the optimal solution using data of the previous simplex geometry until its termination topology is satisfied. However, in the proposed KSM, several simplex geometry search local peaks in parallel, not by using data of the previous simplex geometry like the conventional simplex method but by using the meta-model, which is updated from the previous simplex geometry.
Specifically, each simplex geometry for the KSM is constructed in every iteration step by using each local peak and previously calculated points as shown in Fig. 3 . The local peak is updated in every iteration step. The previously calculated points of each simplex geometry are points which are calculated in the previous step and located near from each local peak. Hence, points of simplex geometry in the KSM do not always consist of points of the previous simplex geometry as is the case for the conventional method but by the updated local peak and points which area previously calculated and near the local peak.
PPT Slide
Lager Image
Construction of the KSM.
- 5) Step 4 – Shaking
To increase the diversity of the solution and prevent converging into the local peak, shaking is conducted in this step. As a result of the shaking, some samples are generated in the region where number of samples are fewer compared to the other regions.
In this process, the ‘fill-blank’ technique is suggested for the shaking. The fill-blank technique generates samples randomly in whole problem regions and then their distance from other points, of which function values are calculated in the previous step, are estimated without any additional function call. Specifically, the distances from a generated sample to previously calculated points are calculated. Among them, the shortest distance is selected as a minimum distance for the generated sample. This process is carried out for all of generated samples. By comparing each minimum distance of the generated samples, the largest value of minimum distance can be identified. This means that this point is the furthest from previously calculated points among the generated samples. Hence, this sample is selected as a final point to be added in the metamodel via the calculation of the function call. This routine is repeated at every iteration to fill out the area of which population is low.
- 6) Step 5 – Convergence check
Step 2-Step 4 are repeated until most peaks are not improved any further.
3. Verification of the KSM
The performance of the proposed algorithm was verified via a mathematical problem and applied to a practical optimization problem, which is the optimization of the magnet shape of the rotor and the slot opening of the stator for the IPMSM.
- 3.1 Verification of KSM via mathematical test function
The formula for the test function is (1), where 2.5 ≤ x 1 , x 2 ≤ 7.5, and its shape is illustrated in Fig. 4 .
PPT Slide
Lager Image
Test function for the verification of usefulness of the proposed KSM algorithm.
PPT Slide
Lager Image
As shown in Fig. 5 , low population regions A, B, and C are filled out with dot a, b, and c, respectively, by using fill-blank technique and the surrogate model approaches to the shape of the actual test function as the iteration is increased. Briefly, in the proposed KSM, local peaks are roughly estimated by using the kriging and the fill-blank technique and the gradient assisted simplex method finds each local peak in parallel delicately.
PPT Slide
Lager Image
Optimization process by using the KSM: (a) initial state; (b) iteration 2; (c) iteration 4; (d) iteration 10.
The performance criteria are identifying exact number of peaks and function calls. Using this criterion, the performance of the KSM is compared with other multimodal optimization methods, which are ATNGA and climb method. As tabulated in Table 1 , the proposed KSM shows superior performance compared to other conventional multi-modal optimization algorithms.
Comparison of the algorithm performance by using the test function between the proposed KSM and conventional methods
PPT Slide
Lager Image
Average value for 50 times
- 3.2. Verification of the KSM via optimization of the IPMSM
By using the proposed KSM, of which performance is verified via the test function in the part 3.1, the feasibility of the application into the practical machine design is testified as follows.
Recently, IPMSM has received much attention, because the saliency of rotor core makes IPMSM possible to have high-power and high-speed performance by using field weakening control algorithm [17 , 18] . Hence, the IPMSM is selected for the practical application of the KSM in this research.
Cogging torque of IPMSM is reduced in this research by using the proposed KSM. A lot of techniques are proposed for the reduction of cogging torque of the IPMSM, the surface mounted permanent magnet synchronous machine, and the axial flux permanent magnet machine [19 - 30] . These approaches indicate that the pole-arc to pole-pitch ratio strongly effects on the cogging torque. Furthermore, the cogging torque is sensitive to the variation of the slot opening. Thus, these variables are considered as design variables for the cogging torque minimization of the IPMSM as illustrated in Fig. 6 . Specifically, the design variables for the reduction of the cogging torque of IPMSM are the slot opening and the pole-arc to pole-pitch ratio. The range of the slot opening is 1.44-4.3 mm and the pole-arc to pole-pitch ratio is 0.425-0.9.
PPT Slide
Lager Image
Design variables: the pole-arc to pole-pitch ratio and the slot opening.
The reference model of the IPMSM is tabulated in Table 2 . The 3 rd harmonic EMF of the reference model is large as shown in Fig. 7 . Further, as shown in Fig. 8(a) , the upper part of the EMF wave for the reference model is suppressed compared to the optimized results, which are Candidate 1 and Candidate 2 shown in Figs. 8(b) and Fig. 8(c) . These results occurred by the THD, which is caused by the large pole-arc to pole-pitch ratio 0.8.
Specification of a reference model for the IPMSM
PPT Slide
Lager Image
Specification of a reference model for the IPMSM
PPT Slide
Lager Image
Comparison of harmonic of back-EMF for the IPMSM.
PPT Slide
Lager Image
Comparison of back-EMF waveform of the IPMSM at 1000 rpm.
At the end of the optimization by using KSM, the final meta-model for the cogging torque is obtained as shown in Fig. 9 . From this optimization result, Candidate 1 and Candidate 2, of which cogging torque values are small, were selected for candidates. The characteristics of the reference model and the two candidate solutions are compared in Table 3 . According to Table 3 and Fig. 10 , Candidate 1 seems the best solution from the point of view of the cogging torque. However, Candidate 2 has lower THD and higher amplitude of EMF than that of Candidate 1. These characteristics can guarantee low noise and high performance. Thus, Candidate 2 is selected as the final optimal solution.
PPT Slide
Lager Image
Calculated meta-model of the cogging torque for IPMSM through the proposed KSM.
Comparison of the IPMSM characteristics of the reference model and the optimized candidates
PPT Slide
Lager Image
Comparison of the IPMSM characteristics of the reference model and the optimized candidates
PPT Slide
Lager Image
Comparison of the cogging torque waveform.
As shown in Fig. 9 , the cogging torque is more sensitive to the variation of αm/α compared to the slot opening. The αm/α of the reference model is larger than that of Candidate 2. The slot opening value of reference model and Candidate 2 is similar. From these data, it can be found out that αm/α has main effect on the THD and cogging torque. This finding also can be verified via data shown in Table 3 .
Although the amount of the magnet of the Candidate 2 is lower than that of the reference model, the amplitude of 1st order harmonic of EMF for Candidate 2 is slightly small compared to the reference model and the THD of Candidate 2 is better than the reference model.
In other words, Candidate 2 can generate similar output using less amount of magnet. Furthermore, the noise and torque ripple of Candidate 2 are lower than the reference model due to its lower THD.
4. Conclusion
The optimal design of the electric machine is mostly a nonlinear multi-modal problem, which requires many function calls and much time consumption to search for the optimum. Hence, this paper has a significant meaning in the sense that the proposed novel hybrid algorithm can mitigate the function call and time consumption compared to conventional multi-modal optimization methods for the optimization of the electric machine.
The proposed novel hybrid method combines the kriging meta-model, the fill-blank technique, and the novel parallel gradient assisted simplex method, which is named as KSM in this paper.
The feasibility of the application of the proposed KSM into an optimal design of the practical electric machine is confirmed through the cogging torque minimization of the IPMSM.
BIO
Il-Woo Kim He received the B.Eng and M.Sc degree from Seoul National University, Seoul, Korea in 2011 and 2013 respectively. He is currently working in Hyosung Corporation Special Motor Design Team, where he works on the design and analysis of motors.
Dong-KyunWoo He received the B.S. degree in electrical engineering from Yonsei University, Seoul, Korea, in 2007. He is currently working toward the Ph.D. degree in electrical engineering from the Seoul National University, Seoul, Korea. His current research interests include numerical analysis and design of electrical machines.
Dong-Kuk Lim He received the B.S. degree in the electrical engineering from Dongguk University, Seoul, Korea, in 2010. He is currently working toward the Ph.D. degree in electrical engineering and computer science from the Seoul national University, Seoul, Korea. His current research interests include design of electrical machines.
Sang-Yong Jung He received the B.S., M.S., and Ph.D. degrees in electrical engineering from Seoul National University, Seoul, Korea, in 1997, 1999, and 2003, respectively. From 2003 to 2006, he was a Senior Research Engineer with the R&D Division, Hyundai Motor Company, Korea, and the R&D Division, Kia Motor, Korea. He is currently an Associate Professor with the School of Electronic and Electrical Engineering, Sungkyunkwan University, Suwon, Korea. His research interests include the numerical analysis and optimal design of electric machines and power apparatus
Cheol-Gyun Lee He received the B.S., M.S., and Ph.D. degree in Electrical engineering from the Seoul National University, Seoul, Korea, in 1989, 1991, and 1998, respectively. He conducted research on the design of the electric machines at R&D center of Hyundai Heavy Industry as a Senior Engineer from 1993 to 1999. He is currently a Professor at the Department of Electrical Engineering, Dong-Eui University. His research interests are the analysis and optimal design of the electric machines.
Jong-Suk Ro earned a PhD in Electrical engineering from Seoul National University, Seoul, Korea, in 2008. In 2001, he received the B.S. degree in Mechanical Engineering from Han- Yang University, Seoul, Korea. Currently, he is at Brain Korea 21 Plus, Creative Research Engineer Development, Seoul National University as a BK Assistant Professor. He carried out research at Electrical Energy Conversion System Research Division of Smart Grid Team at Korea Electrical Engineering & Science Research Institute as a Researcher in 2013. From 2012 to 2013, he was at Brain Korea 21 Information Technology of Seoul National University as a Post-Doctoral Fellow. He conducted research at R&D center of Samsung Electronics as a Senior Engineer from 2008 to 2012. His research interest is analysis and design of electric machines and actuators.
Hyun-Kyo Jung He received the B.S., M.S., and Ph.D. degree in Electrical engineering from the Seoul National University, Seoul, Korea, in 1979, 1981, and 1984, respectively. From 1985 to 1994, he was a member of the faculty with Kangwon National University. From 1987 to 1989, he was with the Polytechnic University of Brooklyn, Brooklyn, NY. From 1999 to 2000, he was a Visiting Professor with the University of California at Berkeley. He is currently a Professor at the School of Electrical Engineering and Computer Science/Electrical Engineering, Seoul National University. His research interests are the analysis and design of the electric machine.
References
Rho J. S. , Lee C. H. , Jung H. K. 2007 Optimal design of ultrasonic motor using evolution strategy and finite element method International Journal of Applied Electromagnetics and Mechanics 25 (1) 699 - 704
Rho J. S. , Lee C. H. , Chung T. K. , Im C. H. , Jung H. K. 2008 Analysis of a nanopositioning actuator using numerical and analytic methods Smart Materials & Structures 17 (2)
Woo D. K. , Choi J. H. , Mohammad A. , Jung H. K. 2011 A novel multimodal optimization algorithm applied to electromagnetic optimization Magnetics, IEEE Transactions on 47 (6) 1667 - 1673    DOI : 10.1109/TMAG.2011.2106218
Wang D. , Wang X. , Kim M. K. , Jung S. Y. 2012 Integrated optimization of two design techniques for cogging torque reduction combined with analytical method by a simple gradient descent method Magnetics, IEEE Transactions on 48 (8) 2265 - 2276    DOI : 10.1109/TMAG.2012.2191416
Lee D. , Jeon K. W. , Jung S. Y. , Kim J. W. 2012 Optimal design of interior PM synchronous machines using randomly guided mesh adaptive direct search algorithm Sixth International Conference on Electromagnetic Field Problems and Applications (ICEF) 19 - 21
Lee S. Y. , Seo J. H. , Kwak S. Y. , Lee C. G. , Jung S. Y. , Jung H. K. 2008 Optimal design of outer rotor per-manent magnet machine using improved particle swarm optimization Electrical Machines and Systems, 2008. ICEMS 2008. International Conference on 3297 - 3300
Jung H. , Jeon W. , Lee C. G. , Hahn S. C. , Jung S. Y. 2008 Optimal design of direct-driven PM wind generator aimed for maximum AEP using parallel computing GA Journal of Electrical Engineering & Technology 3 (4) 552 - 558    DOI : 10.5370/JEET.2008.3.4.552
Kim M. J. , Lim J. W. , Seo J. H. , Jung H. K. 2011 Hybrid optimization strategy using response surface methodology and genetic algorithm for reducing cogging torque of SPM Journal of Electrical Engineering & Technology 6 (2) 202 - 207    DOI : 10.5370/JEET.2011.6.2.202
Chun J. S. , Jung H. K. , Yoon J. S. 1997 Shape optimization of closed slot type permanent magnet motors for cogging torque reduction using evolution strategy Magnetics, IEEE Transactions on 33 (2) 1912 - 1915    DOI : 10.1109/20.582663
Yoo Y.M. , Kim D.K. , Kwon B.I. 2006 “Optimal Design of a Permanent Magnetic Actuator for Vacuum Circuit Breaker using FEM Journal of Electrical Engineering & Technology 1 (1) 92 - 97    DOI : 10.5370/JEET.2006.1.1.092
Im C. H. , Kim H. K. , Jung H. K. , Choi K. 2004 A novel algorithm for multimodal function optimization based on evolution strategy Magnetics, IEEE Transactions on 40 (2) 1224 - 1227    DOI : 10.1109/TMAG.2004.824805
Seo J. H. , Im C. H. , Heo C. G. , Kim J. K. , Jung H. K. , Lee C. G. 2006 Multimodal function optimization based on particle swarm optimization Magnetics, IEEE Transactions on 42 (4) 1095 - 1098    DOI : 10.1109/TMAG.2006.871568
Denies J. , Ben Ahmed H. , Dehez B. 0000 Optimal design of electromagnetic devices: development of an efficient optimization tool based on smart mutation operations implemented in a genetic algorithm Mathematics and Computers in Simulation
Mariani Viviana Cocco , dos Santos Coelho Leandro 2011 A hybrid shuffled complex evolution approach with pattern search for unconstrained optimization Mathematics and Computers in Simulation 81 (9) 1901 - 1909    DOI : 10.1016/j.matcom.2011.02.009
Chuanwen Jiang , Bompard Etorre 2005 A hybrid method of chaotic particle swarm optimization and linear interior for reactive power optimization Mathematics and Computers in Simulation 68 (1,3) 57 - 65    DOI : 10.1016/j.matcom.2004.10.003
Dellino G. , Lino P. , Meloni C. , Rizzo A. 2009 Kriging metamodel management in the design optimization of a CNG injection system Mathematics and Computers in Simulation 79 (8) 2345 - 2360    DOI : 10.1016/j.matcom.2009.01.013
Kim J. M. , Sul S. K. 1997 Speed control of interior permanent magnet synchronous motor drive for the flux weakening operation Industry Applications, IEEE Transactions on 33 (1) 43 - 48    DOI : 10.1109/28.567075
Kim K. C. , Koo D. H. , Hong J. P. , Lee J. 2007 A study on the characteristics due to pole-arc to pole-pitch ratio and saliency to improve torque performance of IPMSM Magnetics, IEEE Transactions on 43 (6) 2516 - 2518    DOI : 10.1109/TMAG.2007.893524
Parsa L. , Hao L. 2008 Interior permanent magnet motors with reduced torque pulsation Industrial Electronics, IEEE Transactions on 55 (2) 602 - 609    DOI : 10.1109/TIE.2007.911953
Zhu Z. Q. , Ruangsinchaiwanich S. , Schofield N. , Howe D. 2003 Reduction of cogging torque in interiormagnet brushless machines Magnetics, IEEE Transactions on 39 (5) 3238 - 3240    DOI : 10.1109/TMAG.2003.816733
Refaie E. 2010 Fractional-slot concentratedwindings synchronous permanent magnet machines: opportunities and challenges Industrial Electronics, IEEE Transactions on 57 (1) 107 - 121    DOI : 10.1109/TIE.2009.2030211
Bianchi N. , Bolognani S. 2002 Design techniques for reducing the cogging torque in surface-mounted PM motors Industry Applications, IEEE Transactions on 38 (5) 1259 - 1265    DOI : 10.1109/TIA.2002.802989
Barakat G. , El-meslouhi T. , Dakyo B. 2001 Analysis of the cogging torque behavior of a two-phase axial flux permanent magnet synchronous machine Magnetics, IEEE Transactions on 37 (4) 2803 - 2805    DOI : 10.1109/20.951312
Caricchi F. , Capponi F. G. , Crescimbini F. , Solero L. 2004 Experimental study on reducing cogging torque and no-load power loss in axial-flux permanent-magnet machines with slotted winding Industry Applications, IEEE Transactions on 40 (4) 1066 - 1075    DOI : 10.1109/TIA.2004.831273
Wang D. , Wang X. , Jung S. Y. 2012 Reduction on cogging torque in flux-switching permanent magnet machine by teeth notching schemes Magnetics, IEEE Transactions on 48 (11) 4228 - 4231    DOI : 10.1109/TMAG.2012.2200237
Wang D. , Wang X. , Qiao D. , Pei Y. , Jung S. Y. 2011 Reducing cogging torque in surface-mounted permanent-Magnet motors by nonuniformly distributed teeth method Magnetics, IEEE Transactions on 47 (9) 2231 - 2239    DOI : 10.1109/TMAG.2011.2144612
Yong P. , Zhu Z. Q. , Feng Z. J. 2011 Cogging torque in cost-effective surface-mounted permanent-magnet machines Magnetics, IEEE Transactions on 47 (9) 2269 - 2276    DOI : 10.1109/TMAG.2011.2147326
Chen J. T. , Zhu Z. Q. , Iwasaki S. , Deodhar R. P. 2011 Influence of slot opening on optimal stator and rotor pole combination and electromagnetic performance of switched-flux PM brushless AC machines Industry Applications, IEEE Transactions on 47 (4) 1681 - 1691    DOI : 10.1109/TIA.2011.2155011
Cho G. W. , Jang W. S. , Jang K. B. , Kim G. T. 2012 “The Optimal Design of Fractional-slot SPM to Reduce Cogging Torque and Vibration,” Journal of Electrical Engineering & Technology 7 (5) 753 - 758    DOI : 10.5370/JEET.2012.7.5.753
Goto A. , Okamoto T. , Ikariga A. , Todaka T. , Enokizono M. 2012 “A New Moving-magnet Type Linear Actuator utilizing Flux Concentration Permanent Magnet Arrangement,” Journal of Electrical Engineering & Technology 7 (3) 342 - 348    DOI : 10.5370/JEET.2012.7.3.342