DESIGN OPTIMIZATION OF PERMANENT MAGNET SYNCHRONOUS MOTORS USING RESPONSE SURFACE ANALYSIS AND GENETIC ALGORITHMS LAURENT JOLLY (B E., Supelec, France) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF ELECTRICAL & COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2005 Acknowledgments I would like to thank all the people who have helped me during my Master’s degree programme at the National University of Singapore I would like to express my deepest gratitude for Dr M.A Jabbar who agreed to be my supervisor and provided warm and constant guidance throughout my study His rich experience in the field of motor design has been extremely valuable to me and I have learned a lot from him I am grateful to Dong Jing, Phyu and Qinghua for their numerous and precious technical advices Their help have been of utmost importance initially when I knew little about the topic I really enjoyed the time spent with my labmates at the Electric Motor and Drive Lab; Amit, Anshuman, Krishna, Sahoo, Wang Wei and Xinhui Technical and social discussions have made the life there pleasant and fulfilling I would like to thank these friends who introduced me to many aspects of their cultures and countries And of course I would like to thank Mr Woo and Mr Chandra who are always so kind and helpful They make the laboratory a nice place to work in i I am also thankful for the graduate research scholarship offered to me by the National University of Singapore without which this Master’s degree programme would not have been possible Finally, I would like to express my deep affection for my parents who have supported and encouraged me throughout this work ii Contents Acknowledgement i Summary vii List of Symbols x List of Figures xiii List of Tables xvii Introduction 1.1 1.2 Background 1.1.1 The development of Permanent Magnet Machines 1.1.2 Features of permanent magnet motors Permanent magnet synchronous motors iii 1.2.1 Structures and operating principles 1.2.2 Inherent design issues for the buried type IPMSM 10 1.3 Literature survey 14 1.4 Motivations 17 1.5 Structure of the thesis 18 IPMSM and wide speed range operation 21 2.1 Introduction 21 2.2 The linear model of the IPMSM 22 2.3 Expansion of the operating speed-range of the motors 25 2.3.1 The linear lossless model of the IPMSM 26 2.3.2 Voltage and current limitations 28 2.3.3 Constant torque region 30 2.3.4 Flux weakening region 31 2.4 Effect of saturation 34 2.5 Non-linear model of the motor 39 2.5.1 40 Model of the flux linkages iv 2.6 2.5.2 The cubic spline interpolation 42 2.5.3 Determination of the power capability for this non-linear model 43 2.5.4 Comparison 47 Conclusion 52 Finite Element Method and computations of the motor characteristics 53 3.1 Introduction 53 3.2 Principle of the Method 55 3.3 Modelling of the IPMSM using FEM 63 3.3.1 Material and Geometry 63 3.3.2 Static analysis 65 3.3.3 Positioning of the MMF 66 3.3.4 Boundary conditions 67 3.3.5 Meshing of the geometry 69 Computation of the flux 70 3.4.1 71 3.4 Method v 3.4.2 Method 74 3.4.3 Comparison of the two methods 77 3.5 Validation of the circuit modelling 80 3.6 Conclusion 83 Response Surface Method 86 4.1 Introduction 86 4.2 RSM procedure 88 4.3 Application of RSM to the linear-model of the IPMSM 94 4.4 Application of RSM on the non-linear model of the IPMSM 103 4.5 Conclusion 109 Optimization of the IPMSM 5.1 5.2 113 Introduction 113 5.1.1 Problem formulation 113 5.1.2 Review of optimization tools 114 Mechanism of Genetic Algorithms 116 5.2.1 Encoding 116 vi 5.3 5.2.2 Selection 118 5.2.3 Recombination 120 Optimization results 124 5.3.1 Influence of the torque constraint upon the maximum CP SR achievable 125 5.3.2 5.4 The case of the ideal design 127 Conclusion 129 Conclusion and discussion 133 References 138 Publications 146 A Cubic spline interpolation 148 B Motor characteristics 152 C RSM experimental results 155 vii Summary This thesis deals with the analysis and design of buried type Interior Permanent Magnet Synchronous Motors (IPMSM) The objective is to develop a design optimization procedure for this type of motors that is more accurate than the traditional analytical methods used in AC machine analysis, and less time consuming than the usual trial and error FEM based design procedure The main design criterion is the expansion of the power capability of the machine till very high speeds while operating as a variable speed drive with a flux weakening scheme It is shown that the traditional circuit modelling of the IPMSM based on motor parameters cannot provide reliable field weakening predictions Indeed, saturation effects are important for this kind of motors and make the motor parameters variable and current dependent A new circuit modelling based on non-linear representation of the d- and q-axis fluxes by cubic spline interpolation is proposed as an alternative It is shown that more reliable predictions of the constant power speed range and peak torque can be obtained from this non-linear circuit modelling The FEM plays an important role in the method as it is used to calculate the flux linkages at the interpolation points Two different methods to calculate these flux linkages are investigated It is shown that since the space harmonics of viii the flux can be significant, these methods are not equivalent The method able to isolate the fundamental of the flux is preferred over the other one In addition, the analytical torque equation used to predict the performance of the motor is validated by FEM computation This strengthens the confidence in the non-linear circuit modelling of the IPMSM for power capability predictions Response Surface Method (RSM) and Finite Element Method (FEM) are combined to relate the d- and q-axis fluxes to the design variables The power capability of the machine can then be predicted for any set of the design variables Different types of designs of experiments (DoE), necessary to provide the experimental data to fit the RSM models, are compared It is shown that DoEs that require many experiments don’t yield necessarily more accurate RSM models; the Central Composite Design is shown to be the best of the DoEs investigated since it allows fitting accurate RSM models prediction from a relatively low number of experiments The power capability predictions obtained from the d- and q-axis fluxes RSM models are checked by FEM to validate the RSM approach The results are globally satisfactory Genetic algorithm is the optimization tool chosen to optimize the IPMSM A simple yet efficient algorithm is developed and used to show that the constant power speed range (CPSR) can be increased without limit (under the no-loss assumption) but at the expense of the peak torque available below the base speed It is also shown that an optimal design with an infinite CP SR can be achieved for a particular set of the design variables Its main characteristic is that the magnet flux can be cancelled by the armature reaction 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optimization for cogging torque minimization using response surface methodology”, IEEE Trans Mag., Vol 40, no.2, March 2004 [49] M.V.K Chari, P.P Sylvester, Finite element in electrical and magnetic field problems, John Wiley and sons, 1980 [50] J.C Sabonnadiere, J.L Coulomb, Finite element method in CAD, London: North Oxford Academic, 1987 [51] J.L Coulomb, “A methodology for the determination of global electromechanical quantities from a finite element analysis and its application to the evaluation of magnetic forces, torque and stiffness, IEEE Trans Mag Vol.19, no.6, November 1983 [52] P.P Silvester, R.L Ferrari, Finite elements for electrical engineers, 3rd edition 1996, Cambridge University Press [53] A Draper, Electrical machines, 2nd edition 1967, Longman Group 145 [54] A.B.J Reece and T.W Preston, Finite element methods in electrical power engineering, 2000, Oxford Science Publications [55] J.C Sabonnadiere, J.L Coulomb, Finite element method in CAD, London: North Oxford Academic, 1987 [56] L Chang, A.R Eastham and G.E Dawson, “Permanent magnet synchronous motor: finite elements torque calculations,” Industry Applications Society Annual Meeting, 1989., Conference Record of the 1989 IEEE 1-5 Oct 1989 Page(s):69 - 73 vol.1 [57] D.C Montgomery, Design and analysis of experiments, second edition 1987, John Wiley & Sons [58] A.I Khuri, J.A Cornell, Response surfaces design and analysis, 1987, Marcel Dekker Inc [59] J.A Cornell, How to apply response surface method, Milwaukee, WI : ASQC, c1990 [60] G.E.P Box, N.R Draper, Empirical model-building and response surfaces, 1987, John Wiley & Sons [61] K Hameyer, R Belmans, Numerical modelling and design of electrical machines and devices, 1999 WIT Press Boston, Computational Mechanics [62] R.L and S.E Haupt,Practical Genetic Algorithms, 1999, John Wiley & Sons [63] D Dumitrescu, B Lazzerini, L.C Jain, A Dumitrescu, Evolutionnary Computation CRC Press 2000 [64] D Whitley, A Genetic Algorithm tutorial List of Publications Published L Jolly, M.A Jabbar, Liu Qinghua, “Design optimization of permanent magnet motors using response surface methodology and genetic algorithms,” IEEE Transaction on Magnetics October 2005 L Jolly, M.A Jabbar, Liu Qinghua, “Response Surface Methodology and Motor Design”, Journal of Science and Engineering of the American International University-Bangladesh, Vol.4 No.1 (2005) L.Jolly, M.A Jabbar, Liu Qinghua, “Optimization of the constant Power speed range of a saturated permanent magnet synchronous motor,” in the IEEE International Electric Machines and Drives Conference (IEMDC 2005), San Antonio, USA, May 15-18, 2005 L.Jolly, M.A Jabbar, Liu Qinghua, “Design optimization of permanent magnet motors using response surface methodology and genetic algorithms,” in the IEEE International Magnetics Conference (INTERMAG 2005), Nagoya, Japan, April 4-8, 2005 146 M.A Jabbar, Liu Qinghua, L Jolly, “Application of Response Surface Methodology (RSM) in Design Optimization of Permanent Magnet Synchronous Motors,” in the IEEE TENCON 2004, Chiang Mai, Thailand, November 21-24, 2004 M.A Jabbar, Liu Qinghua, L Jolly, “Design optimization of permanent magnet motors using response surface analysis,” in Proceedings of the IEEE Third International Conference on Electrical and Computer Engineering (ICECE 2004), Dhaka, Bangladesh, December 28-30, 2004 To be published L Jolly, M.A Jabbar, Liu Qinghua, “Optimization of the constant Power speed range of a saturated permanent magnet synchronous motor,” submitted in IEEE Transactions in Industry Application Submitted for Review L Jolly, M.A Jabbar, Liu Qinghua, “Non linear circuit modelling of an interior permanent magnet synchronous motor for power capability predictions,” submitted in IEE Proceedings Electric Power Application 147 Appendix A Cubic spline interpolation Let {(x0 , y0 ), (x1 , y1 ) (xn , yn )} be the n points to be interpolated by the function f (x) (Fig A.1) f is made a piece-wise function, such that on each interval Figure A.1: Cubic spline interpolation [xk ; xk+1 ], f (x) = fk (x) = ak + bk (x − xk ) + ck (x − xk )2 + dk (x − xk )3 The piecewise function must meet the following requirements 148 (A.1) 149 f should go through all the data points and be continuous on [x0 , xn ] f must be continuous on [x0 , xn ] f must be continuous on [x0 , xn ] From 1, one can writes f0 (x0 ) = y0 fk (xk+1 ) = fk+1 (xk+1 ) = yk+1 for k = : n − (A.2) fn−1 (xn ) = yn From 2, fk (xk+1 ) = fk+1 (xk+1 ) (A.3) fk (xk+1 ) = fk+1 (xk+1 ) (A.4) hk = xk+1 − xk (A.5) ak = y k (A.6) and from Using the notation Combining (A.2) and (A.1) give for k = : n (which means that all the coefficients ak are known) and ak+1 = ak + bk hk + ck h2k + dk h3k (A.7) for k = : n − Differentiating (A.1) and combining with (A.3) give for k = : n − bk+1 = bk + 2ck hk + 3dk h2k (A.8) 150 Differentiating two times (A.1) and combining with (A.4) give for k = : n−1 ck+1 = ck hk + 3dk hk (A.9) The coefficients dk can be thus be expressed in terms of the coefficients ck dk = ck+1 − ck 3hk (A.10) The coefficients bk can be expressed in terms of the coefficients ck and ak substituting (A.10) into (A.7) ak+1 − ak hk − (2ck + ck+1 ) hk bk = (A.11) Finally, the coefficients ck can be related to the coefficients ak (known) substituting (A.10) into (A.8) bk+1 = bk + hk (ck + ck+1 ) (A.12) and A.11 into A.12 hk+1 (ak+2 − ak+1 ) − (ak+1 − ak ) = hk ck + 2(hk + hk+1 ) + hk+1 ck+2 hk (A.13) In the following, the left hand term of A.13, which only depends of the coefficients ak is denoted by αk+1 The value of the second derivative at x0 and xn is chosen equal to (free spline) It follows that: c0 = (A.14) 2cn−1 + 6dn−1 hn−1 = (A.15) and 151 To sum up, all the coefficients dk and bk can be derived from the coefficient ck Those ck are expressed in terms of the coefficients αk (known) as follows:   α1   α2        αn−1              =          h0 2(h0 + h1 ) h1 ··· ··· h1 2(h1 + h2 ) h3 ··· ··· hn−2 2(hn−2 + hn−1 ) hn−1 0   c0   c1                           cn (A.16) Solving this system of equation gives us the ck The dk are then obtained by (A.10) and the bk by A.11 Appendix B Motor characteristics Magnetic materials characteristics Figure B.1: B-H curve of the iron (50H470) used for the rotor and stator 152 153 Figure B.2: B-H curve of the NdFeB magnet used in the rotor Main characteristics of the stator 154 Table B.1: Main characteristics of the stator Rated Power (W ) 400 Rated speed (rpm) 1500 Max voltage (Vrms ) 116 Rated current (Arms ) 1.92 Number of phase Number of poles Number of stator slots 24 Number of turns per phase 312 Number of coils per phase Winding layout Winding pole pitch Winding factor Double layer 5/6 0.933 Inner diameter (mm) 74 Outer diameter (mm) 148 Airgap length (mm) Stack length (mm) 52 Appendix C RSM experimental results The λm , Ld and Lq observations collected on the three designs are Table C.1: Observations on the Central Composite Design Coded values Observations x1 x2 x3 λm Ld Lq 0 365.5 64.3 173.8 -1 -1 -1 182.5 84.8 185.3 -1 -1 310.6 77.7 178.5 -1 -1 318.1 76.6 181.7 -1 1 441.4 73.0 166.5 -1 -1 221.1 60.5 183.0 -1 387.4 56.6 166.5 1 -1 389.0 54.8 175.0 1 536.8 46.5 148.6 -1.682 0 268.9 96.3 183.6 1.682 0 408.7 51.4 167.2 -1.682 238.9 68.0 180.7 1.682 491.7 56.2 163.1 0 -1.682 230.6 72.9 181.9 0 1.682 471.2 63.8 155.1 155 156 Table C.2: Observations on the Full Factorial Design Coded values Observations x1 x2 x3 λm Ld Lq -1 -1 -1 182.5 84.8 185.3 -1 -1 254.4 82.5 182.5 -1 -1 310.6 77.7 178.5 -1 -1 250.8 81.6 184.6 -1 0 318.1 77.9 179.4 -1 374.8 75.4 173.1 -1 -1 318.1 76.6 181.7 -1 385.9 74.5 175.0 -1 1 441.4 73.0 166.5 -1 -1 206.2 69.0 184.3 -1 290.0 66.8 178.5 -1 358.1 65.0 171.8 0 -1 283.2 65.9 182.1 0 365.5 64.3 173.8 0 431.1 62.9 165.0 -1 361.6 62.4 177.8 442.1 61.2 168.3 1 502.2 58.1 156.5 -1 -1 221.1 60.5 183.0 -1 313.9 58.5 174.8 -1 387.4 56.6 166.5 -1 303.7 57.4 179.9 0 394.7 55.7 169.6 1 464.1 53.5 158.8 1 -1 389.0 54.8 175.0 1 475.0 51.7 162.8 1 536.8 46.5 148.6 157 Table C.3: Observations on the Box-Behnken Design Coded values Observations x1 x2 x3 λm Ld Lq 0 365.5 64.3 173.8 -1 -1 254.4 82.5 182.5 -1 -1 250.8 81.6 184.6 -1 374.8 75.4 173.1 -1 385.9 74.5 175.0 -1 -1 206.2 69.0 184.3 -1 358.1 65.0 171.8 -1 361.6 62.4 177.8 1 502.2 58.1 156.5 -1 313.9 58.5 174.8 -1 303.7 57.4 179.9 1 464.1 53.5 158.8 1 475.0 51.7 162.8 ... The development of Permanent Magnet Machines 1.1.2 Features of permanent magnet motors Permanent magnet synchronous motors iii 1.2.1 Structures and operating principles... with the analysis and design of buried type Interior Permanent Magnet Synchronous Motors (IPMSM) The objective is to develop a design optimization procedure for this type of motors that is more... List of Figures 1.1 Magnetic characteristics of the main class of permanent magnets [2] 1.2 Evolution of the magnetic material during the 20th century [2] 1.3 Partial demagnetization of a permanent