A Comparative Study on Modeling Methods for Switched Reluctance Machines

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1 Computer ad Iformatio Sciece Vol. 3, No. 2; May 21 A Comparative Study o Modelig Methods for Switched Reluctace Machies Shouju Sog & Weiguo iu School of Automatio, Northwester Polytechical Uiversity PO box 352, 127 Youyi Xilu, Xi a, 7172, Shaaxi, Chia Tel: sogshouju@hotmail.com Abstract Switched reluctace motor (SRM) has may specific advatages ad has beig gradually used i all kids of applicatios. However, because it s iheretly multivariable, strog-couplig ad highly oliear, there are lots of difficulties to build its model. I this paper, may modelig methods iclude umerical, aalytical ad itelliget approaches are studied ad compared i terms of complexity ad accuracy. The static flux likage ad torque characteristics obtaied by each method are give. The advatages ad disadvatages of each method are summarized as well. Keywords: Switched reluctace motor (SRM), Modelig methods, Fiite elemet aalysis (FEA), Fourier decompositio (FD), Back propagatio eural etwork (BPNN) 1. Itroductio The SRM has the defiite advatages of simple magetless ad rugged costructio, simple cotrol, ability of extremely high speed operatio, relatively wide costat power capability, miimal effects of temperature variatios offset, low maufacturig cost ad ability of hazard-free operatio (W. Xu et al., 29). These distict advatages make the SRM well suited to kids of applicatios, such as more/all electric aircrafts (S. Sog, 29) ad electric vehicles (X. D. Xue et al., 28). I cotrast to traditioal motors (such as iductio motors), the SRM is iteded to operate i deep magetic saturatio to icrease the output power desity ad the eergy coversio factor. Therefore, because of saturatio effects ad the variatio of magetic reluctace, all the relevat characteristics (i.e., flux likage, torque) of the machies are highly oliear fuctios of both rotor positio ad phase curret (R. T. Naayagi et al., 25). These oliearities make the modelig process of the SRM relatively difficult. Due to saturatio effects i the SRM, establishig the accurate oliear mappig relatioship of the flux likage with respect to phase curret ad rotor positio is the basis of performace calculatio (J. Xiu et al., 27). For years, may modelig methods have bee proposed by worldwide researchers (O. Ichiokura et al., 23; S. Sog et al., 27; Z. i et al., 27). Geerally speakig, these methods ca be divided ito three categories: umerical, aalytical ad itelliget approaches. They have advatages ad disadvatages of their ow. I this paper, three typical methods belog to differet categories are studied respectively ad their performaces are compared i terms of complexity ad accuracy. These three methods are based o fiite elemet aalysis (FEA), Fourier decompositio (FD) ad back propagatio eural etwork (BPNN). 2. Numerical methods Amog umerous umerical methods, FEA is the most frequetly used approach (K. N. Sriivas et al., 25). There are may softwares that ca be used to aalyze ad calculate the electromagetic field of the motor accurately i two-dimesio (2-D) ad three-dimesio (3-D) ways. Before gettig the static characteristics ad dyamic performaces of the motor, followig steps eed to be completed firstly. (1) Build the model of the motor accordig to its physical dimesio ad material. (2) Build the model of the power coverter accordig to its topology. (3) Set the termial ad boudary coditios. (4) Add drivig sources. (5) Geerate the fiite elemet mesh. Fig. 1 shows the 2-D FEA model of a four-phase 8/6-pole SRM. Fig. 2 shows the model of a asymmetric half bridge power coverter, where S1~S8 are voltage-cotrolled switches, they represet the power switches i the coverter. W1~W8 are curret-cotrolled switches, they are used to implemet curret choppig cotrol. Fig. 3 25

2 Computer ad Iformatio Sciece shows the fiite elemet mesh of the SRM. After the completio of above steps, the electromagetic characteristics of the motor ca be obtaied. Fig. 4 shows the magetic field distributio iside the motor at oe positio. Fig. 5 shows the static flux likage ad torque characteristics of the SRM obtaied by FEA. It should be oted that the FEA oly ca get some discrete poits (represeted by circles i the figure) ad the curves are obtaied by the plot fuctio of the MATAB to idicate the variatio tred of the data. 3. Aalytical methods I aalytical methods, the oliear relatioships of the phase iductace ad torque to phase curret ad rotor positio are represeted by several aalytical expressios. Amog umerous aalytical methods, the liear ad quasi-liear models are well-kow. I this paper, a method called FDM is studied. It s based o Fourier decompositio (FD) (J. Hur et al., 23). It is well kow that the phase iductace curve of the SRM is a periodic fuctio, so it ca be expaded i Fourier series as show i Eq. (1). (, i) = ( ) + [ ( i i m ) cos( mn r θ )] θ (1) 1 where (θ, i) is the phase iductace, (i) ad m (i) are coefficiets, N r is the umber of rotor poles, θ is the agular positio of the rotor, [1, ] is the harmoic coefficiet, is the maximum harmoic order. The modelig accuracy is strogly affected by. I this paper, is equal to 1. I this case Eq. (1) ca be rewritte as θ i = i i cos Nrθ (2) ( ) ( ) ( ) ( ), + 1 To calculate the two udetermied coefficiets i Eq. (2), the iductace values at two differet rotor positios are eeded. I this paper, o ad 3 o (see Fig. 5(a)), amely ualiged ad aliged positio, are selected. Substitutig the iductace values at these two positios ito Eq. (2), followig equatios ca be obtaied. o (, i) = mi = ( i) + 1 ( i) o ( 3, i) = = () i () i (3) max 1 where mi ad max are iductace values at ualiged ad aliged positios, respectively. Those two udetermied coefficiets i Eq. (2) ca be easily obtaied from Eq. (3), as () i = ( mi + max ) 2 (4) 1 () i = ( mi max ) 2 It is well kow that mi ca be see as a costat, while max chages alog with the phase curret oliearly. The aalytical expressio betwee max ad phase curret i ca be obtaied by polyomial fittig, as m ( i = a m i (5) max ) where a m are polyomial coefficiets that ca be obtaied by fittig. The data of phase iductace at aliged positio with differet phase currets eeded for polyomial fittig ca be easily obtaied from FEA results (see Fig. 5(a)) by followig equatio. ( i θ ) ψ ( i θ ) = (6) i where (i, θ )ad ψ(i, θ ) are phase iductace ad flux likage at θ with phase curret equal to i, respectively. The aalytical expressios of those two udetermied coefficiets i Eq. (2) ca be obtaied by substitutig Eq. (5) ad the value of mi ito Eq. (4). Fially, the phase iductace ca be expressed as Based o Eq. (7), the torque ca be expressed as = mi m mi m m m ( θ, i) + a i 2 + a i cos( N θ ) 2 r (7) 26

3 Computer ad Iformatio Sciece Vol. 3, No. 2; May 21 ' T = W mag θ = ( θ ) θ i= c i 2 N ri 2a m m, i idi = i mi si( N rθ ) 4 m 2 (8) + where T is the torque, W mag is the magetic co-eergy. O the basis of Eq. (6) ~ (8), the static flux likage ad torque characteristics of the SRM ca be obtaied as show i Fig Itelliget methods There are may itelliget methods that ca be used to build the model of SRM, such as artificial eural etwork (ANN) ad fuzzy iferece system (FIS). ANN has excellet capacity of oliear mappig, adaptive learig ad parallel processig. I this paper, the back propagatio eural etwork (BPNN) based o eveberg-marquardt (M) optimizatio algorithm (Y. Cai, 26; J. Su et al., 25) are used to implemet the oliear mappig relatioship of the flux likage ad torque to phase curret ad rotor positio, amely i(ψ, θ) ad T(i, θ). This method has strog robustess ad adaptive ability ad the user does ot eed have ay professioal kowledge about the SRM. The data eeded to trai the BPNN ca be obtaied by FEA (see Fig. 5). To verify the approximatio ad geeralizatio ability of the traied BPNN, the data i Fig. 5 are divided ito two parts. I Fig. 5(a), the data with eve rotor positio values are used to trai i(ψ, θ). I Fig. 5(b), the data with eve phase curret values are used to trai T(i, θ). I this paper, the BPNN used to trai the i(ψ, θ) has two iputs (ψ ad θ), oe output (i) ad two hidde layers, both hidde layers have te euros. The BPNN used to trai the T(i, θ) also has two iputs (i ad θ), oe output (T) ad two hidde layers, both hidde layers have eight euros. Fig. 7 shows the variatio of the error alog with the epochs durig traiig. It ca be see that the covergece speeds of both BPNN are fast. Fig. 8 shows the compariso betwee the simulatio results of BPNN ad the data from FEA. All the FEA data i this figure are used to trai the BPNN. It ca be see that both traied BPNN have strog approximatio ability. Fig. 9 shows the compariso betwee the simulatio results of BPNN ad the data from FEA. All the FEA data i this figure are NOT used i the traiig of the BPNN. It ca be see that both traied BPNN have strog geeralizatio ability. 5. Compariso of the modelig methods The umerical methods ca get the static characteristics ad dyamic performaces of the SRM accurately, but quite complex ad time-cosumig. Furthermore, it s relatively difficult to implemet the cotrol algorithms by FEA softwares. The modelig ad computatioal time of aalytical methods is much shorter tha that required by umerical methods. However, because of the itroductio of may hypotheses ad simplificatios, the modelig accuracy of aalytical methods is worse tha that of umerical methods (see Fig. 6). The itelliget methods ca fid the reasoable balace betwee modelig time ad accuracy. Accordig to the above-metioed advatages ad disadvatages of each modelig method, the umerical methods are commoly used to verify the performaces of the desiged motors i detail. The aalytical methods are commoly used to briefly study the operatig characteristics of the motor ad quickly evaluate the performaces of differet cotrol algorithms. The itelliget methods ca be used to some applicatios i which repetitive computatios are required, such as desig optimizatio ad dyamic performace aalysis. 6. Coclusio I this paper, three categories of modelig methods, amely umerical, aalytical ad itelliget approaches, are studied ad compared i terms of complexity ad accuracy. Three typical methods respectively based o FEA, FD ad BPNN are give as examples. The simulatio results show that they have advatages ad disadvatages of their ow. The umerical methods ca get the static characteristics ad dyamic performaces of the SRM accurately, but quite complex ad time-cosumig. The modelig ad computatioal time of aalytical methods is much shorter tha that required by umerical methods. However, its modelig accuracy is worse. The itelliget methods ca fid the reasoable balace betwee modelig time ad accuracy. For differet applicatios, the modelig methods should be selected carefully accordig to the features of the applicatio ad the advatages ad disadvatages of the modelig methods. 27

4 Computer ad Iformatio Sciece Refereces J. Hur, C. Kim ad D. Hyu. (23). Modelig of switched reluctace motor usig Fourier series for performace aalysis. Joural of Applied Physics, vol. 93, o. 1, May, pp J. Su, Q. Zha ad. iu. (25). Modellig ad cotrol of bearigless switched reluctace motor based o artificial eural etwork. Proceedigs of 31st Aual Coferece of IEEE Idustrial Electroics Society, Raleigh, NC, Uited States, Nov., pp J. Xiu ad C. Xia. (27). Modelig of Switched Reluctace Motor Based o GA Optimized T-S Type Fuzzy ogic. Proceedigs of Fourth Iteratioal Coferece o Fuzzy Systems ad Kowledge Discovery, Haikou, Chia, pp K. N. Sriivas ad R. Arumugam. (25). Aalysis ad Characterizatio of Switched Reluctace Motors: Part II - Flow, Thermal, ad Vibratio Aalyses. IEEE Trasactios o Magetics, vol. 41, o. 4, Apr., pp O. Ichiokura, T. Kikuchi, K. Nakamura, T. Wataabe ad H. Guo. (23). Dyamic Simulatio Model of Switched Reluctace Geerator. IEEE Trasactios o Magetics, vol. 39, o. 5, Sept., pp R. T. Naayagi ad V. Kamaraj. (25). A Comparative Study of Shape Optimizatio of SRM usig Geetic Algorithm ad Simulated. Proceedigs of IEEE Idico 25 Coferece, Cheai, Idia, Dec., pp S. Sog, W. iu ad Y. Wag. (27). Modelig, dyamic simulatio ad cotrol of a four-phase switched reluctace motor. Proceedigs of IEEE Iteratioal Coferece o Cotrol ad Automatio, Guagzhou, Chia, May, pp S. Sog. (29). Detailed desig of a 3kW switched reluctace starter/geerator system used i more/all electric aircraft. Aache, Germay: Shaker Verlag. W. Xu, J. Zhu, Y. Guo, S. Wag, Y. Wag ad Z. Shi. (29). Survey o electrical machies i electrical vehicles. Proceedigs of Iteratioal Coferece o Applied Supercoductivity ad Electromagetic Devices, Chegdu, Chia, Sept., pp X. D. Xue, K. W. E. Cheg ad N. C. Cheug. (28). Selectio of Electric Motor Drives for Electric Vehicles. Proceedigs of 28 Australasia Uiversities Power Egieerig Coferece, Sydey, NSW, Dec., pp Y. Cai. (26). Research o Switched Reluctace Motor Noliear Modelig ad High-Performace System. Ph.D. thesis, Tiaji Uiversity, Tiaji, Chia. Z. i, D. S. Reay, B. W. Williams ad X. He. (27). Olie modelig for switched reluctace motors usig b-splie eural etworks. IEEE Tras o Idustrial Electroics, vol. 54, o. 6, Dec., pp Figure 1. The 2-D FEA model of a four-phase 8/6-pole SRM 28

5 Computer ad Iformatio Sciece Vol. 3, No. 2; May 21 Figure 2. Model of a asymmetric half bridge power coverter Figure 3. Fiite elemet mesh of the SRM Figure 4. The magetic field distributio iside the motor (a) Flux likage characteristics (b) Torque characteristics Figure 5. Flux likage ad torque characteristics obtaied by FEA 29

6 Computer ad Iformatio Sciece (a) Flux likage characteristics (b) Torque characteristics Figure 6. Compariso betwee FEA ad FDM (a) i(ψ, θ) (b) T(i, θ) Figure 7. Variatio of the error alog with the epochs (a) i(ψ, θ) (b) T(i, θ) Figure 8. Verificatio of the approximatio ability (a) i(ψ, θ) (b) T(i, θ) Figure 9. Verificatio of the geeralizatio ability 21