Decoupling of Secondary Saliencies in Sensorless AC Drives Using Repetitive Control

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1 Decouplng of Secondary Salence n Senorle AC Drve Ung Repettve Control Zhe Chen 1, Chun Wu 1, Rong Q, Guangzhao Luo, and Ralph Kennel 1 1 Inttute for Electrcal Drve Sytem and Power Electronc, Techncal Unverty of Munch, Munch 8333, Germany School of Automaton, Northwetern Polytechncal Unverty, X an 7119, Chna Emal: zhe.chen@tum.de, chun.wu@tum.de, lhqr@nwpu.edu.cn, guangzhao.luo@nwpu.edu.cn, ralph.kennel@tum.de URL: Acknowledgment Th reearch ha been upported by Deutche Forchunggemenchaft (DFG) under Grant KE 817/8-1 and Chnee Scholarhp Councl (CSC). Keyword <<Senorle control>>, <<Permanent Magnet Motor>>, <<Vector control>>, <<Adaptve control>>. Abtract To decouple trong econdary alence n enorle AC drve, a repettve control method n an angle doman propoed. Th method regard econdary alence a perodc dturbance relatve to electrcal angle and generate the ame alence to compenate them. The propoed method eay to mplement and proved effectve by experment. Introducton It well known that enorle control for AC machne (e.g., nducton motor (IM), permanent magnet ynchronou motor (PMSM) and etc.) attract lot of attenton around the world. The elmnaton peed/poton enor for AC drve ytem can not only reduce the cot but alo enhance the relablty. In term of dfferent theory, there are two major categore for enorle control method: (a) back electromotve-force (EMF) baed (e.g. ldng mode, extended kalman flter, adaptve oberver etc.) and (b) alency baed method (normally hgh-frequency (HF) gnal njecton requred). Back EMF baed enorle method utlzed when the machne run n mddle or hgh peed range becaue t fal n lower peed range on account of low gnal to noe rato. Fortunately, the alency baed method work well at low peed and even at tandtll ownng to that the alency gnal, whch contan rotor poton nformaton, can be excted when HF gnal are njected nto the machne. The alency caued by the dfference between d- and q- ax nductance called a prmary alency and normally ued for poton etmaton. It, however, can be nfluenced by addtonal econdary alence due to magnet or phycal degn[1, ]. If thee effect are not taken nto conderaton, the poton etmaton performance degrade everely. Th paper focue exactly on the multple alence problem for low peed range enorle control of PMSM. 1

2 The extence of multple alence common n almot all AC machne, epecally n the IM and SPMSM, whch would reult n an underable poton etmaton error and even make the ytem untable [1, 3]. Therefore, decouplng or compenaton method expected to olve the problem for alency baed low peed range enorle drve ytem. The decouplng method for multple alence wa frt propoed by Mchael W. Degner. It ued the etmated poton to buld a multple alency model of IM and then decouple t wthn a Phae-Locked Loop (PLL) etmaton [1]. Th method work well when the magntude of the econdary alence are weak [, 5]. Nonethele, t wa proved to fal f the magntude of the econdary alence are larger than half of prmary alency [6]. An adaptve control ung neural network wa another decouplng method and t wa appled to decouple the aturaton-nduced alence n nteror PMSM. Th method, however, need ma commonng tranng and the degn rule of neural network tll a upendng problem [7]. Other mlar method on the ba of the neural network can be found n [8, 9] a well. Another attempt through njectng multple HF gnal to extract the poton nformaton wa propoed n [1]. But thee addtonal njectng gnal brng more noe and addtonal power lo to ytem. Thee method mentoned above are ether ophtcated to mplement or machne parameter-dependent. In th paper, a novel Repettve Control (RC) baed enorle control method, whch traghtforward and machne parameter-ndependent, propoed. The tradtonal RC method am to olve the perodc control problem n a tme doman but t can be extended to an angle doman. The concept of angle doman RC appled here to compenate the addtonal econdary alency whch a perodc functon of rotor angle [11]. The propoed RC eay to mplement and requre le computaton effort compared wth other method baed on the tme doman. The propoed decouplng method then appled to a concentrated wndng SPMSM (cwspmsm) whch reveal trong multple alence under HF gnal njecton baed enorle control tet. The contrbuton of th paper are addreed a follow: a novel RC method baed on angle doman RC propoed, and t effectve no matter wth the alency rato; a nonlnear gan learnng method adopted to avod the nterference of trong noe envronment. The analy of multple alence Fg.1 gve the tructure of a enorle Feld Orented Control (FOC) ytem for SPMSM, ung rotatng HF gnal njecton (RHFI) method. e v dqc Controller IPark SPWM Inverter PMSM ˆe ˆe dqf PLL Park dqc Clark LPF Demodulator abc Fgure 1: Senorle FOC ytem ung the HF gnal njecton method. When HF gnal (1) are njected nto the tatonary reference frame, the HF current repone of the RHFI can be expreed a () through olvng (1)-(3)

3 v dqc = V ce jωct = Ldq d dqc dt [ Ldq ΣL + = Lcoθe Lnθ e [ ] Ldq r Ld = L q ] Lnθ e ΣL Lcoθ e dqc = I cpe j(ω ct π/) + I cn e j( ω ct+θ e π/) (1) () (3) () The current repone contan the prmary alency (econd term of rght de n ()) due to man flux aturaton []. But n mot cae, the alence are produced not only by geometry of rotor but alo the magnet aturaton or phycal degn. In the preence of econdary alence, the HF current repone a um of all alence repone, expreed a (5) [1] dqc = I cpe j(ωct π/) + I cn e j( ω ct+h θ e +ϕ π/) (5) where h = =,,8... and t not conecutve [1,], I cn : magntude of the -th component of the negatve current. h : harmonc number of the alency caung the -th component. ϕ phae hft of the -th component relatve to the θ e reference frame. From (5), t clear that econdary alence (h ) are perodc functon of electrcal angle of rotor poton a well but they brng etmaton error to ytem f only prmary alency are condered. Baed on th charactertc, econdary alence are normally een a perodcal dturbance whch can be compenated by RC method. However, the tradtonal RC normally executed n a tme doman, o t needed to extend t to an angle doman n order to apply the RC. Repettve control n tme doman The theory of repettve control The prncple of RC orgnate from the nternal model theory [13]. That, the controlled output can track a et of reference nput wthout teady-tate error only f the model that generate thee reference ncorporated n the table cloed-loop ytem. For example, when the ytem requred to have a zero teady-tate error to a tep nput, the model of tep functon (.e., 1/) hould be alo ncluded n the loop gan. Smlarly, f a perodc dturbance njected nto the ytem, the controller telf hould alo need to generate the ame perodc gnal to compenate t [1]. Repettve Control r e L GC GRC d Plant r Fgure : The tructure of a Plug-n repettve control. The RC uually mplemented a a plug-n module to the prmtve controller [15], hown n Fg.. In the preence of hgh perodc dturbance, the teady-tate error can not be elmnated due to the lmtaton of bandwdth of regulator and lack of dturbance model n cloed-loop ytem. Therefore, the RC nerted nto the orgnal control a a upplement to compenate the external perodc dturbance [1]. 3

4 The degn of RC n angle doman For the tet cwspmsm, although t a urface-mount permanent magnet type, t how remarkable alency rato by ung RHFI at tandtll and low peed ownng to t pecal tator wndng [6, 1]. The multple alency model of cwspmsm expreed n (6) accordng to expermental reult. The major econdary alency component h = (the mnu repreent the harmonc rotate n the oppote drecton a the rotor)[1] dqc =I cpe j(ω ct π/) + I cn e j( ω ct+θ e +ϕ π/) + I cn e j( ω ct θ e +ϕ π/) (6) In order to get pure Negatve Sequence Current(NSC) gnal (7) from (6), mlar demodulaton a n [1] adopted and t output cn dqc, whch the nput of the PLL etmaton and hown n Fg. 3. cn dqc =I cne j(θ e+ϕ π/) + I cn e j( θ e+ϕ π/) (7) L z ˆe cn dqc ˆcn dqc K cn err RC Delay( T ) L1 z e err K RC RC z PI z I z P z ˆe j ˆ e e Fgure 3: The propoed Plug-n repettve control n an angle doman. The error gnal err obtaned by takng the cro-product between the meaured current cn dqc and the etmated current î cn dqc = e j(ˆθ e +ϕ π/), expreed n (8). err = K cn î cn dqc cn dqc = I cn n((θ e ˆθ e )) + I cn n( θ e ˆθ e ϕ) (8) }{{} d n6 where K cn a proporton parameter and can be taken nto account n the magntude of I cn and I cn. When ˆθ e track θ e, the etmated poton contan 6th-order harmonc dturbance.e., d n6 n (8) relatve to the rotor electrcal angle. It perod n angle doman obtaned a below θ T = π 6 = π 3 (9) where θ T a perodc contant n angle doman. Every tme err ampled together wth ˆθ e. When ˆθ e approache θ e, RC generate an output RC to approxmate d n6, whch then ubtracted from the nput of PLL. Gradually, the 6th-order harmonc dturbance can be totally compenated by RC f the term err reache zero. The z-tranfer functon of RC, hown n Fg. 3, gven by Z θ T RC(z) = K RC Z θ (1) T F1

5 ˆe angle: : com n ( n+1) 3 1 n n+1 N N component Fgure : Realzaton of RC n the angle doman. Sample data at kt err, k ˆe, k Low pa flter L 1 e err, k Determne the locaton of ˆe ˆ e 18 n mod, 3 1 Multplcaton K RC Array n com, k 1 n n K e RC err + + com, k, 1n com k n Update k k 1 RC, k Lmtaton n Fgure 5: Flow chart of caluaton of the propoed RC. where the F 1 a Low Pa Flter (LPF) whch ncreae tablty of RC. To avod the nfluence of the trong noe n the teady tate, an adaptve nonlnear gan f (e) expreed n (11) adopted nde the K RC. When the error err large, the gan alo large to accelerate the convergence and vce vera [16]. When the error under the uncontrolled error lmt, the gan become zero and learnng proce top. 1 cn err > δ, f e 1 δ δ ( e δ) δ < cn err < δ, cn err < δ. The comprehenve gan K RC expreed a K RC = A f (e) (11) (1) where A ampltude to regulate the peed of convergence. In order to mplement the propoed RC n a dgtal controller, a dcrete RC hould be degned. The angle perod π/3 dvded nto N component equally, hown n Fg., and each component ha a length of φ = π/(3n). Every amplng ntance, the value of ˆθ e mod π/3 (the whole range -π wth repect to -6N) wll be between n φ and (n + 1) φ, then the adjacent upper or lower nteger hould be choen. Th paper adopt the lower nteger, hown n RC,k = com,k 1 [n] (13) 5

6 where com,k 1 an array of length of N at (k 1)T. k mean the amplng ntant kt, and T the amplng tme. RC,k the output of RC at kt. The learnng proce expreed n (1). The reult com,k [n] update the relatve component com,k 1 [n]. com,k [n]= com,k 1 [n] + K RC L 1 ( err,k ) (1) To decrbe the propoed RC more clearly, Fg. 5 how detaled procee at kt. At every amplng ntance, the calculaton of the propoed RC cont of a one-order LPF, a mod to determne the locaton of ˆθ e, a multplcaton and an addton to update torage array com. A lmtaton wll retrct the output of RC. Implementaton and expermental reult Tet-bench decrpton Table I: PARAMETERS OF THE cwspmsm Parameter Value DC lnk voltage 58 V Rated power. kw Stator retance.5 Ω Stator nductance 1. mh Pole par 3 Rated peed 3. r/mn Rated torque 7.1 Nm Total moment of nerta.9 Kg.m The propoed trategy appled to a cwspmsm on an expermental tet bench. It cont of a. kw cwspmsm manufactured by SEW company. An nducton machne, drven by a LUST 1 kva nverter, ued a a load machne. The cwspmsm drven by a modfed SEW 5. kw nverter whch provde full control of IGBT gate and t dead tme 1 mcroecond. A elf-made 3.6 GHz real tme computer ytem ued. The ample frequency 16 khz, whle the PWM frequency 8 khz. The rotor poton meaured by an ntegrated encoder. The parameter of the cwspmsm are gven n Table I. The control trategy vector control realzed by a nuodal PWM (SPWM). The Speed and current loop adopt PI controller. Applcaton of the propoed RC There are three tep to tran RC. Frtly, the cwspmsm run wth an encoder cloed-loop control n order to reach a teady tate. Secondly, the RC begn to learn baed on t etmated poton rather than encoder provded poton. After everal perod learnng, the RC can generate an exactly dturbance gnal to compenate the extng one. Fnally, the data of generated dturbance gnal are tored by oftware. After leanng proce, the ytem can wtch to the enorle cloed-loop control wth RC compenaton. The frequency and magntude of rotatng HF voltage are 55 Hz and 1 V repectvely. The parameter of the PLL: K P = 6, K I = 8, K cn =.6. The cutoff frequency of L 1 7 Hz. N = 3 elected n the experment through tral. The RC gan K RC, accordng to expermental adjutment, et wth.1 n the experment. d-ax current reference 3 A (to ncreae alency rato) and the motor operate under enorle peed cloed-loop control. 6

7 β n [A] Magntude [A] [A] α n (a) Waveform of NSC of teted cwspmsm Frequency [Hz] (b) FFT analy of NSC Fgure 6: Waveform and FFT analy of negatve-equence current. Poton [rad] 6 real et Poton [rad] 6 real et (a) PLL wthout propoed RC (b) PLL wth propoed RC. Fgure 7: Comparon of PLL wthout and wth RC at r/mn. Poton [rad] 6 real et Poton [rad] 6 real et (a) PLL wthout propoed RC at r/mn (b) PLL wth propoed RC at 1 r/mn. Fgure 8: Comparon of PLL wthout and wth propoed RCat r/mn. 7

8 Rotor Speed [r/mn] -. 1 et ref real Rotor Speed [r/mn] -. 1 et ref real q-ax Current [A] q-ax Current [A] (a) PLL wthout propoed RC (b) PLL wth propoed RC. Fgure 9: Comparon of dynamc performance under tep half rated torque at 1 r/mn Rotor Speed [r/mn] -. 5 et ref real Rotor Speed [r/mn] -. 5 et ref real q-ax Current [A] q-ax Current [A] (a) PLL wthout propoed RC. (b) PLL wth propoed RC. Fgure 1: Comparon of dynamc performance under tep peed change wth half rated torque. Expermental reult The waveform and FFT analy of NSC of the tet cwspmsm are hown n Fg. 6. For a normal PMSM, the waveform of NSC a crcle. For the tet cwspmsm, however, t how a trangle hape whch can be oberved from Fg. 6(a). The related FFT reult of the NSC llutrated n Fg. 6(b) gve exact magntude nformaton of the prmary nd alency and econdary th alency. The magntude of latter one obvou large compared wth mlar reult mentoned n other lterature focung on multple alence. Poton trackng performance of PLL wthout (Fg. 7(a)) and wth (Fg. 7(b)) RC at r/mn ( f e = Hz) wth half rated torque (about 3.6 Nm) llutrated n Fg. 7. It can be een from Fg. 7(a) that both the real poton and etmated poton contan 6th-order harmonc (1 Hz). The magntude of 6th-order harmonc approxmate. rad. After adoptng the propoed method hown n Fg. 7(b), the 8

9 real poton and etmated poton become much moother and the magntude of 6th-order harmonc near. Fg. 8 how the mlar reult a demontrated n Fg. 7 but at r/mn ( f e = 1 Hz). The 6th-order harmonc (6 Hz) alo uppreed effectvely. Fg. 9 gve the comparon of PLL wthout Fg. 9(a)) and wth (Fg. 9(b) RC under a tep change of half rated torque at 1 r/mn. Although orgnal PLL can track the tep change, but due to hgh rpple of etmated poton, the etmated peed and current contaned many harmonc n Fg. 9(a). Adoptng PLL wth propoed RC, the rpple of the etmated poton, etmated peed and q-ax current are reduced n Fg. 9(b). The dymamc performance of tep peed change wth half rated torque hown n Fg. 1. When peed command tep from 1 r/mn to r/mn, ung the propoed method, the etmated poton, the etmated peed and the q-ax current hown n Fg. 1(b) (PLL wth propoed RC) become more accurate than thee n Fg. 1(a) (PLL wthout propoed RC). Concluon Th paper propoe a novel repettve control method to decouple the multple alence n enorle AC drve. The method parameter ndependent and requre le computaton effort compared wth extng decouplng method. Only the frequence of multple alence are needed, whch can be ealy acqured by advanced expermental analy. Th method vald for all AC machne and can alo be appled for machne wth three or more alence wth repettve control n parallel mode. The applcaton of th novel decouplng method done wth a cwspmsm and the reult valdate t effectvene. Reference [1] M. W. Degner, and R. D. Lorenz, Ung multple alence for the etmaton of flux, poton, and velocty n AC machne, Indutry Applcaton, IEEE Tranacton on, vol. 3, no. 5, pp , [] Raca, D.; Garca, P.; Regoa, D.; Brz, F.; Lorenz, R., A comparatve analy of pulatng v. rotatng vector carrer gnal njecton-baed enorle control, Appled Power Electronc Conference and Expoton, 8. APEC 8. Twenty-Thrd Annual IEEE, vol., no., pp.879,885, -8 Feb. 8. [3] F. Brz, and M. W. Degner, Rotor poton etmaton, Indutral Electronc Magazne, IEEE, vol. 5, no., pp. -36, 11. [] N. Teke, G. M. Aher, M. Sumner, and K. J. Bradley, Suppreon of aturaton alency effect for the enorle poton control of nducton motor drve under loaded condton, Indutral Electronc, IEEE Tranacton on, vol. 7, no. 5, pp ,. [5] J. Holtz, and H. Pan, Elmnaton of aturaton effect n enorle poton-controlled nducton motor, Indutry Applcaton, IEEE Tranacton on, vol., no., pp ,. [6] D. Paulu, P. Landmann, S. Kuehl, and R. Kennel, Arbtrary njecton for permanent magnet ynchronou machne wth multple alence. pp [7] D. D. Regoa, P. Garca, D. Raca, F. Brz, and R. D. Lorenz, Meaurement and adaptve decouplng of croaturaton effect and econdary alence n enorle controlled IPM ynchronou machne, Indutry Applcaton, IEEE Tranacton on, vol., no. 6, pp , 8. [8] T. M. Wolbank, and M. K. Metwally, Senorle Poton and Torque Control of Inducton Motor Baed on Tranent Sgnal Injecton and Advanced Sgnal Proceng. pp [9] P. Garca, F. Brz, D. Raca, and R. D. Lorenz, Salency-trackng-baed enorle control of AC machne ung tructured neural network, Indutry Applcaton, IEEE Tranacton on, vol. 3, no. 1, pp , 7. [1] C. Zhe, G. Janbo, W. Fengxang, M. Zhxun, Z. Zhenbn, and R. Kennel, Senorle Control for SPMSM Wth Concentrated Wndng Ung Multgnal Injecton Method, Indutral Electronc, IEEE Tranacton on, vol. 61, no. 1, pp , 1. [11] L. Ben-Brahm, On the compenaton of dead tme and zero-current crong for a PWM-nverter-controlled AC ervo drve, Indutral Electronc, IEEE Tranacton on, vol. 51, no. 5, pp ,. [1] F. Brz, M. W. Degner, A. Dez, and R. D. Lorenz, Meaurng, modelng, and decouplng of aturatonnduced alence n carrer-gnal njecton-baed enorle AC drve, Indutry Applcaton, IEEE Tranacton on, vol. 37, no. 5, pp , 1. 9

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