Individual and Global Optimization of Switched Flux Permanent Magnet Motors

Transcription

1 3 Journal of International Conference on Electrical Machine and Sytem, Vol., No., pp.3~39, Individual and Optimization of Switched Flux Permanent Magnet Motor Z.Q. Zhu *, X. Liu ** Abtract With the aid of genetic algorithm (GA), global optimization with multiple geometry parameter i feaible in the deign of witched flux permanent magnet (SFPM) machine. To invetigate the advantage of global optimization over individual optimization, which ha been ued extenively for the deign of SFPM machine, a comparion between the two approache i carried out for the cae of fixed copper lo and volume. In the cae of individual parameter optimization, the equence in which the individual parameter are i very important. In the global optimization a better deign can alway be achieved although the correponding torque denity i found to be only lightly better than that of individually with correct deign equence. By uing the obtained global optimization reult, the performance in machine having two type of tator and rotor pole combination, i.e. / and /4, are compared, and it i hown that higher torque i exhibited in the /4 SFPM machine. Finally, thi paper alo demontrate that global optimization, with the retriction of equal pole width, magnet thickne and lot opening, can maximize the torque denity without ignificantly acrificing other performance, uch a cogging torque and overload capability. Keyword: Average torque, Flux witching, Optimization, Permanent magnet, Switched flux. Introduction Switched flux permanent magnet (SFPM) machine are re-emerging a an attractive machine topology with imple but robut rotor tructure. Many attempt have been made to ue thi kind of permanent magnet machine for variou application, ranging from aeropace and automotive to wind power generation. However, the detailed requirement for each application are lightly different, and thi lead to the neceity of optimization. According to the available literature, in order to maximize the torque production at the ame current denity, the ratio of back-iron thickne to tator pole width i uggeted to be.7~.8, and the plit ratio of inner diameter to outer diameter i uggeted to be 5~.6 []. The optimal ratio of rotor pole width to rotor pole pitch i almot contant, i.e. /3, []. Harmonic reulting from back-emf can alo be minimized by changing the tator pole arc [3]. Baed on the invetigation of different tator and rotor pole number combination, the /4 SFPM machine * Department of Electronic and Electrical Engineering, Univerity of Sheffield, Sheffield S 3JD, U.K. (z.q.zhu@heffield.ac.uk) ** Department of Electronic and Electrical Engineering, Univerity of Sheffield, Sheffield S 3JD, U.K. (elp9xll@heffield.ac.uk) Received 8 July ; Accepted November i conidered to have better torque capability than the / SFPM machine [4], [5]. In the C-core SFPM machine, it ha been hown that, a the rotor pole number increae, the ratio of lot opening width to magnet thickne increae [6]. However, the foregoing invetigation were mainly carried out with individual-parameter optimization or with different retriction. In order to optimize the geometry of a SFPM machine, the influence of the plit ratio, tator pole width, magnet thickne, tator back-iron thickne, rotor pole width, and rotor back-iron thickne have been tudied in [] by finite element analyi (FEA). Due to the fact that FEA conume a lot of time, a lumped circuit model [] and analytical expreion for determining the winding configuration [3] were developed to etimate the deign. With the aid of uch approache, final reult can be quickly obtained by FEA in a narrowed cope [7]. However, in the aforementioned approache, the coupling effect between geometry parameter i eldom conidered. Therefore, a global optimization approach combining the genetic algorithm (GA) with FEA, carried out with Anoft/Maxwell oftware, i ued in thi paper to find the real global olution for / and /4 SFPM machine with different objective function. During the optimization, the copper lo and volume are kept contant. To validate the effectivene of uch an approach, the individual

2 Z.Q. Zhu and X. Liu 33 optimization olution followed by appropriate equence i compared to the global optimization olution. With the obtained globally reult, the performance of / and /4 SFPM machine are compared in term of torque denity, cogging torque, magnet conumption, flux-linkage, and back-emf. The reult how that global optimization, with the retriction that pole width, magnet thickne and lot opening all be the ame, can maximize the torque denity without ignificantly acrificing other performance, uch a cogging torque and overload capability.. Individual Optimization of SFPM machine Fig. how the cro-ection of / and /4 SFPM machine, which are to be in thi paper. The initial deign parameter for thee two SFPM machine, which have been ued in [], [], are lited in Table. To implement individual optimization, the equence in which deign parameter are hould follow the level of the parameter enitivity. In other word, the mot enitive parameter hould be firt. The definition of the deign parameter are a follow: λ i the plit ratio, p i the tator pole, r p i the rotor pole, r y i the rotor lot depth ratio, M d i the magnet thickne ratio, H y i the back-iron thickne ratio. D l r = () D y b p p = () t r h r = H y b pr p = (3) t r pr M h = ( h + h ) pr b yr (4) m d = (5) t y ( h + h ) where b p, τ, b pr, τ r, h pr, h yr, b m, h y and h p are illutrated in Fig.. The calculation of the copper lo, P cu, i hown in (7), where k p, ρ, l, J, and S denote the packing factor, reitivity, motor active length, current denity, and lot area, repectively, and b lot i hown in Fig.. 3p blot Pcu = r J S ( l + t - ) (7) k p With a packing factor of, i.e. k p =, a copper lo of.9w (8 C) i aumed during the individual optimization. Under uch condition, the parameter are according to the following equence: () plit ratio; () rotor pole ; (3) tator pole ; (4) magnet ; (5) back-iron thickne; (6) rotor y p (6) lot depth ratio. Fig.3-6 how the calculated average torque veru the variou defined deign parameter in both / and /4 SFPM machine. It hould be mentioned that little influence of rotor lot depth ratio on the average torque i oberved. Thu, the rotor lot depth ratio in both / and /4 i et to.4. Another phenomenon that i een during the individual optimization i that the maximum average torque may not increae during the optimization. Thi i becaue of the coupling effect among the parameter. For an eay comparion, Table lit the initial deign and reult for both / and /4 SFPM machine. Compared with the initial value, it i een that wider tator pole and a thinner tator back-iron are preferred in order to produce higher torque for the given copper lo. After the optimization, 5% higher torque can be een in the /4 SFPM machine a compared to the / SFPM. (a) Fig.. Cro ection of SFPM machine (a) / SFPM; and (b) /4 SFPM Table. Main Parameter of Prototype SFPM Motor Number of phae 3 3 Stator pole number, N Rotor pole number, N r 4 Outer diameter of tator, D 9mm 9mm Inner diameter of tator, D r 55mm 55mm Airgap, g mm mm Active axial length, l 5mm 5mm Rotor pole width top/bottom 5.65/7.68mm 3.88/4.6mm Stator pole width 3.6mm 3.6mm Magnet thickne 3.6mm 3.6mm Back-iron thickne 3.6mm 3.6mm Table. Comparion between Initial and Individual Optimized Value / SFPM /4 SFPM Variable Initial Individual Initial Individual value value λ r p p.9.7 M d.7. H y r y (b)

3 34 Individual and Optimization of Switched Flux Permanent Magnet Motor.5 / FSPM machine /4 FSPM machine.5 Fig.. One module of SFPM machine Magnet thickne ratio, M d Fig.6. Average torque veru magnet thickne ratio in / and /4 SFPM machine. / FSPM machine /4 FSPM machine Split ratio, λ Fig. 3. Average torque veru plit ratio in / and /4 SFPM machine..5 / FSPM machine /4 FSPM machine Rotor pole, r p Fig.4. Average torque veru rotor pole in / and /4 SFPM machine..5 / FSPM machine /4 FSPM machine Stator pole, p Fig.5. Average torque veru tator pole in / and /4 SFPM machine..5 / FSPM machine /4 FSPM machine Back-iron thickne ratio, H y Fig.7. Average torque veru back-iron thickne ratio in / and /4 SFPM machine Rotor lot depth ratio, r y / FSPM machine /4 FSPM machine Fig.8. Average torque veru rotor lot depth ratio in / and /4 SFPM machine. 3. Optimization of / SFPM Machine for Maximizing Torque Denity A one of the mot effective numerical optimization method, GA ha been widely ued for eeking olution in application of motor deign. In literature, the maximum torque denity and minimum cogging torque are often ued a the optimization objective [8], [9]. They are ued in thi paper a well. The global optimization of SFPM machine are carried out with and without the

4 Z.Q. Zhu and X. Liu 35 retriction of equal tator pole width, magnet thickne and lot opening. It i expected that with fewer variable the optimization olution may be found more quickly. 3. With the retriction of b m = b p = b lot According to the tudy in [], the tator pole width, magnet thickne and lot opening are uggeted to be equal to each other to produce the maximum average torque. Therefore, the GA i implemented firt with uch a retriction, and the etting of the GA are a follow: the population ize, mating pool ize, individual croover probability and mutation probability are,, and, repectively. The objective function i defined a: n Cot = -å wt (8) where w i and T i denote the weight and torque at pecific rotor poition. To optimize the average torque, all the weight are et to be /n and hence their contribution to the average torque at each pecific rotor poition are all equal. By neglecting the reluctance torque, the d-axi current i et to zero and the copper lo i et to.9w during the optimization. Table 3 lit the global parameter together with the individual parameter. The up- and down- limitation of parameter are retricted to obtain the reaonable deign, hown a contraint in Table 3. No ignificant difference in term of average torque i oberved between individual and global optimization, a een in Table 5. Table 3. Optimization Variable with Retriction of b m =b p = b lot and Contraint in / SFPM Motor Optimization parameter Initial value Individual contrai nt Split ratio [6, ].6 Rotor pole [., ].34 Stator pole.9 Magnet thickne ratio.7 Back-iron [.3,..4 thickne ratio.8].6 Rotor lot [.9,.4.4 depth ratio 3].48 Note: Contraint are the range within which each parameter may vary. 3. Without any retriction If the retriction i removed, even higher average torque hould be expected. After the implementation of global optimization, Table 4 how the olution under the condition of fixed copper lo. In the cae of global i= i i optimization without retriction, more variable are changing during the procedure. Therefore, the poibility to obtain the global optimal torque in a hort time period i reduced, a hown in Fig.9. Compared with the individual optimization, the time required for global optimization i often much longer ince more iteration are required. However, due to the random characteritic of GA, the time conumption under global optimization i alway difficult to determine. Moreover, it can be een in Fig.9 that during the global optimization the Cot i likely to convergent to everal value, which are named a the local optimization and global optimization. One of the advantage of GA i it capability to converge to the global optimal point, Fig.9 By comparing the reult, very little improvement i achieved in term of torque denity if no retriction i applied, a compared to the cae in which the retriction from applied. Therefore, to optimize the torque denity, both the individual and global optimization with retriction of b m = b p = b lot can achieve good olution in the deign of SFPM machine. Regarding other critical iue, uch a magnet conumption and lot area, global optimization with the retriction i alo acceptable compared with other approache. For a direct comparion of all deign, the lamination are illutrated in Fig.. Cot Cot Iteration (a) With retriction of b m = b p = b lot Iteration (b) Without retriction Fig.9. Optimization procedure by GA for / SFPM machine.

5 36 Individual and Optimization of Switched Flux Permanent Magnet Motor 4. Optimization of /4 SFPM Machine for Maximizing Torque Denity (a) Initial deign (b) Individual optimization For comparing the performance between / and /4 SFPM machine, the global optimization with/without the retriction of b m =b p =b lot are alo carried out for the /4 SFPM machine. In thi way, the influence due to other geometry parameter can be excluded. Hence, only the tator and rotor pole combination may affect the average torque. However, due to the pace limitation, only the final reult are lited in Table 6 and Table 7. In term of magnet conumption, advantage of the /4 SFPM machine are oberved compared to the / SFPM machine, a hown in Table 8. Moreover, higher torque can alo be achieved in the /4 SFPM machine at the copper lo of.9w. (c) optimization (d) optimization with b m = b p = b lot without retriction Fig.. Illutration of lamination of / SFPM machine Table 4. Optimization Variable without Retriction Of b m = b p = b lot and Contraint in / SFPM Motor Optimization parameter Initial value Individual Contraint Split ratio.6.6 [6,.67].6 Rotor pole [., ].39 Stator pole [.7,.9.33].9 Magnet [.3,.7 thickne ratio.33].4 Back-iron [.3,..4 thickne ratio.8] Rotor lot [.9,.4.4 depth ratio 3].45 Note: Contraint are the range within which each parameter may vary. Table 5. Motor Deign Comparion between Initial, Individual, and Optimization in / SFPM Machine Optimization Initial Indivi- -dual parameter value Optimi- -zed b m =b p =b lot b m b p b lot Ma of magnet (g) Slot area (mm ) Average torque*(nm) *Average torque at copper lo of.9w at 8 C. Table 6. Optimization Variable with Retriction of b m =b p = b lot and Contraint in /4 SFPM Motor Optimization parameter Initial value Individual Contraint Split ratio.6.64 [6,.67].64 Rotor pole [., ].33 Stator pole.7 Magnet thickne ratio. Back-iron [.3,..3 thickne ratio.8].4 Rotor lot [.9,.4.4 depth ratio 3].4 Note: Contraint are the range within which each parameter may vary. Table 7. Optimization Variable without Retriction of b m =b p =b lot and Contraint in /4 SFPM Motor Optimization parameter Initial value Individual Split ratio.6.64 Rotor pole Stator pole Magnet thickne ratio Back-iron thickne ratio Contraint [6,.67] [.7, 3] [.7,.33] [.3,.33] [.3,.8] Rotor lot [.9, depth ratio 3] Note: Contraint are the range within which each parameter may vary.

6 Z.Q. Zhu and X. Liu 37 Table 8. Deign Comparion between Initial, Individual, and Optimization in /4 SFPM Machine Optimization Initial Indivi- -dual parameter value Optimi- -zed b m =b p =b lot b m b p b lot Ma of magnet (g) Slot area (mm ) Average torque*(nm) *Average torque at copper lo of.9w at 8 C. 5. Performance Comparion between / and /4 SFPM Machine 5. Torque with fixed copper lo A mentioned early, the copper lo i kept contant during the optimization, o a to increae the torque denity for a given copper lo. To evaluate the torque production capability, a new coefficient i defined a: k = J S (9) c The copper lo i proportional to the quare of thi coefficient, i.e. 3p blot P = r k ( l + t - ) () cu c k p Fig. how the average torque veru k c for both the / and the /4 SFPM machine. Higher torque exit in the /4 SFPM machine compared with the / SFPM machine. However, ince the copper lo wa fixed at.9w (k c =.8 3 ) during the optimization, the optimal average torque in the global optimization without retriction found for thi choice of copper lo may not be achieved at other given value of copper lo, uch a overload condition. To validate the FEA calculation, the initial deign of the / and /4 SFPM machine were elected to be identical to thoe in [5] Initial / Optimized with retriction / Optimized without retriction / Initial /4 Optimized with retriction /4 Optimized without retriction / kc ( 3 ) Fig.. Comparion of torque-current characteritic with I d =. 5. Cogging torque comparion The comparion of cogging torque, which i a main contributor to the torque ripple, in the initial and global optimization are hown in Fig.. Compared with the / SFPM machine, lower cogging torque i oberved in the /4 SFPM machine for both the initial and deign. If no retriction i applied in the global optimization, high cogging torque i oberved in the / SFPM machine. On the other hand, the calculation reult how that with the retriction of b m = b p = b lot in both / and /4 SFPM machine the cogging torque will not be exceive in the globally cae. Cogging torque (Nm) Initial / Optimized with retriction / Optimized without retriction / Initial /4 Optimized with retriction /4 Optimized without retriction / Rotor poition (electrical degree) Fig.. Cogging torque comparion between / and /4 SFPM machine. 5.3 Flux linkage and back-emf Fig.3 compare the open circuit flux linkage for both / and /4 SFPM machine, together with the harmonic pectra in which it can be een that the firt harmonic, i.e. the fundamental component of flux linkage, i dominant. Similar to the torque characteritic at the fixed copper lo, the fundamental component of flux linkage i increaed in both / and /4 SFPM machine. Noting that the electromagnetic torque i proportional to the product of flux linkage and pole pair number, maller flux linkage in the /4 SFPM machine will not lead to lower electromagnetic torque. Finally, Fig.4 how the phae back-emf waveform and pectra, i.e. e ph =e +e +e 3 +e 4, ince each phae winding conit of four coil connected in erie, a hown in Fig.. The /4 SFPM machine ha le harmonic but a larger fundamental component.

7 38 Individual and Optimization of Switched Flux Permanent Magnet Motor Phae flux linkage (mwb) Phae flux linkage (mwb) Initial / Optimized with retriction / Optimized without retriction / Initial /4 Optimized with retriction /4 Optimized without retriction / Rotor poition (electrical degree) (a) Waveform of flux-linkage Initial / Optimized with retriction / Optimized without retriction / Initial /4 Optimized with retriction /4 Optimized without retriction / Order (b) Spectra of flux-linkage Fig.3. Flux-linkage in both / and /4 SFPM machine with/without optimization. Phae back-emf (V) Phae back-emf (V) Initial / Optimized with retriction / Optimized without retriction / Initial /4 Optimized with retriction /4 Optimized without retriction / Rotor poition (electrical degree) (a) Waveform Initial / Optimized with retriction / Optimized without retriction / Initial /4 Optimized with retriction /4 Optimized without retriction / Order (b) Spectra Fig.4. Phae back-emf in both / and /4 SFPM machine with/without optimization 6. Concluion To evaluate the effectivene of global optimization of SFPM machine by genetic algorithm, both individual and global optimization of / and /4 SFPM machine were carried out in thi paper. Compared with the individual optimization, the global optimization can be employed without conidering the optimization equence, and alway achieve better deign, but it produce only lightly higher average torque at fixed copper lo than that obtained by the individual optimization with a correct optimization equence for the parameter. The reult how that, with the aid of global optimization, higher torque but le cogging torque in the /4 SFPM machine i achieved. Moreover, it wa found that, with the retriction of b m =b p =b lot during the global optimization, the improvement of average torque did not caue high cogging torque. Reference [] Z.Q. Zhu, Y. Pang, J.T. Chen, Z.P. Xia, and D. Howe, Influence of deign parameter on output torque of fluxwitching permanent magnet machine, IEEE Conf. Vehicle Power and Propulion, Sep. 3-5, 8. Harbin, China. [] Z.Q. Zhu, Y. Pang, D. Howe, S. Iwaaki, R. Deodhar, and A. Pride, Analyi of electromagnetic performance of fluxwitching permanent magnet machine by nonlinear adaptive lumped parameter magnetic circuit model, IEEE Tran. Magnetic, vol.4, no., pp , 5. [3] W. Hua, M. Cheng, Z.Q. Zhu, and D. Howe, Analyi and optimization of back EMF waveform of a flux-witching permanent magnet motor, IEEE Tran. Energy Converion, vol.3, no.3, pp , Sep. 8. [4] J.T. Chen, and Z.Q. Zhu, Comparion of all- and alternatepole-wound flux-witching PM machine having different tator and rotor pole number, IEEE Tran. Indutry Application, vol.46, no.4, pp.46-45, Jul./Aug.. [5] J.T. Chen, and Z.Q. Zhu, Winding configuration and optimal tator and rotor pole combination of flux-witching PM bruhle AC machine, IEEE Tran. Energy Converion, vol.5, no., pp.93-3, Jun.. [6] J.T. Chen, Z.Q. Zhu, S. Iwaaki, and R. Deodhar, Influence of lot opening on optimal tator and rotor pole combination and electromagnetic performance of flux-witching PM bruhle AC machine, Energy Converion Congre and Expoition,, -6 Sep. Atlanta, USA. [7] J.T. Chen, Z.Q. Zhu, and D. Howe, Stator and rotor pole combination for multi-tooth flux-witching permanentmagnet bruhle AC machine, IEEE Tran. Magnetic, vol.44, no., pp , Dec. 8. [8] W. Min, J.T. Chen, Z.Q. Zhu, Y. Zhu, M. Zhang, and G.H. Duan, Optimization of linear witched flux permanent magnet machine, IEEE Conf. Vehicle Power and Propulion (VPPC),, -3 Sep. Lille. [9] M. Lukanizyn, M. Jagiela, and R. Wrobel, Optimization of permanent magnet hape for minimum cogging torque uing a genetic algorithm, IEEE Tran. Magnetic, vol.4, no., pp.8-3, March, 4.

8 Z.Q. Zhu and X. Liu 39 [] J.T. Chen, Z.Q. Zhu, Influence of the rotor pole number on optimal parameter in flux-witching PM bruhle AC machine by the lumped-parameter magnetic circuit model, IEEE Tran.Ind. Appl., vol.46, no.4, pp , Jul./Aug.. [] Z.Q. Zhu, J.T. Chen, Advanced flux-witching permanent magnet bruhle machine, IEEE Tran. Magnetic, vol.46, no.6, pp , Jun.. [] A. Chen, N. Rotevatn, R. Nilen, and A. Nyveen, Characteritic invetigation of a new three-phae fluxwitching permanent magnet machine by FEM imulation and experimental verification, International Conf. Electrical Machine and Sytem, 9, 5-8 Nov. 9, Tokyo, Japan. Z. Q. Zhu received the B.Eng. and M.Sc. degree from Zhejiang Univerity, Hangzhou, China, in 98 and 984, repectively, and the Ph.D. degree from the Univerity of Sheffield, Sheffield, U.K., in 99. Since 988, he ha been with the Univerity of Sheffield, where he i currently a Profeor at the Department of Electronic and Electrical Engineering, and Head of the Electrical Machine and Drive Reearch Group. Hi current major reearch interet include deign and control of permanent magnet bruhle machine and drive, for application ranging from automotive to renewable energy. X. Liu He received the B.Eng. and PhD degree from Zhejiang Univerity, Hangzhou, China, in 5 and, repectively, all in electrical engineering. He i currently a Pot-doctorial Reearcher at the Department of Electronic and Electrical Engineering, Univerity of Sheffield, Sheffield, U.K. Hi current reearch interet include deign and application of witched reluctance and permanent magnet machine and drive.