On-line Parameter Estimation of Interior Permanent Magnet Synchronous Motor Using an Extended Kalman Filter

Transcription

1 J Electr Eng echnol Vol. 9, No.?: 742-?, ISSN(Prnt) ISSN(Onlne) On-lne Parameter Estmaton of Interor Permanent Magnet Synchronous Motor Usng an Extended Kalman Flter Hyun-Woo Sm*, June-Seok Lee* and Kyo-Beum Lee Abstract hs paper presents estmaton of d-axs and q-axs nductance of an nteror permanent magnet synchronous motor (IPMSM) by usng an extended Kalman flter (EKF). he EKF s wdely used for control applcatons ncludng the motor sensorless control and parameter estmaton. he motor parameters can be changed by temperature and ar-gap flux. In partcular, the varaton of the nductance affects torque characterstcs lke the maxmum torque per ampere (MPA) control. herefore, by estmatng the parameters, t s possble to mprove the torque characterstcs of the motor. he performance of the proposed estmator s verfed by smulatons and expermental results based on an 11kW PMSM drve system. Keywords: Parameter estmaton, Interor permanent magnet synchronous motor (IPMSM), Extended Kalman Flter (EKF), Maxmum torque per ampere (MPA) 1. Introducton Permanent magnet synchronous motors (PMSMs) have hgh effcency because there s no feld wndng n the rotor for the generaton of the magnetc flux. Also they have better hgh power densty and fast transent response than a dc motor and nducton motor. Because of ths characterstc of PMSMs, t s wdely used n general ndustral felds, especally n hgh- performance motor control applcatons. PMSMs can be classfed nto surfacemounted permanent magnet synchronous motors (SPMSMs) and nteror perma- nent magnet synchronous motors (IPMSMs), dependng on the arrangement of the permanent magnet. Snce the IPMSM has magnet mounted nsde the rotor, t s mecha- ncally strong and benefcal for hgh-speed operaton compared wth the SPMSM. he nductance dfference between d-axs and q-axs of the IPMSM s generated by the locaton of the mounted permanent magnet. he dfference of nductance causes the addtonal torque called the reluctance torque. For ths reason, the torque characterstc and control method of the IPMSM s dfferent from the SPMSM. herefore, varous control methods to take an advantage of the IPMSM have been proposed [1-3]. o mprove the torque characterstc of the IPMSM n the constant torque regon, the maxmum torque per ampere (MPA) control has been ntroduced. here are many stator current vectors that generate the constant torque. Among these current vectors, the current of the mnmum magntude s called the MPA current. he angle of the current vector s changed wth the parameter Correspondng Author: Dept. of Electrcal and Computer Engneerng, Ajou Unversty, Suwon, Korea. (kyl@ajou.ac.kr) * Dept. of Electrcal and Computer Engneerng, Ajou Unversty, Suwon, Korea. (bgle@ajou.ac.kr, junpb@ajou.ac.kr) Receved: August 20, 2013; Accepted: November 28, 2013 varaton because the angle of the mnmum current vector relates to d-axs and q-axs nductance. Unfortunately, the stator resstance, nductance and magnet flux are senstve to the varaton of temperature [1-4]. Varaton of d-axs and q-axs nductance can decrease effectveness of the MPA control and current controller. It means that accurate nformaton of d-axs and q-axs nductance by on-lne parameter estmaton s necessary to mprove motor drves. Several on-lne parameter estmatons such as the extended Kalman flter (EKF), recursve least square (RLS), and model reference adaptve system (MRAS) have been proposed [5-10]. Among them, the extended Kalman flter s the most effectve estmator n terms of the leastsquare for estmatng the states of nonlnear systems, whch s very approprate for mplementaton n systems wth sensors affected by nose. It processes all the avalable measure- ments regardless of ther accuracy, to provde fast and precse estmaton of the target varables, and also accomplsh a quck convergence. Moreover, the EKF serves to be approprate for the state estmaton of a PMSM. It appears to be a vable and computatonally effcent opton for the on-lne estmaton of the speed, rotor poston and parameters. hs s feasble because a mathematcal model, whch descrbng the motor dynamcs s very well known. he termnal quanttes lke voltages and currents can be measured easly and are approprate for the determnaton of the parameters of the machne n an ndrect way [7-13]. In ths paper, the extended Kalman flter (EKF) based on an on-lne dentfcaton method s proposed to estmate d- axs and q-axs nductances of the IPMSM. hs paper s organzed as follows. Frstly, the mathematcal model of the IPMSM and MPA control methods are revewed brefly. Secondly, the process of a nonlnear system of the IPMSM appled to the EKF s presented. Fnally, the 742

2 Hyun-Woo Sm, June-Seok Lee and Kyo-Beum Lee smulaton and expermental results show the valdty of the estmator. 2. Mathematcal Model of IPMSM and MPA Control 2.1 Mathematcal model of IPMSM he three dfferent coordnate systems of IPMSMs are shown n Fg. 1. Each coordnate transformaton s expressed as: 1 0 2, wth = 1/2 3/2 3 1/2 3/2 d α cosθ snθ = P q, wth P = β snθ cosθ a α t b = β c he voltage equaton of an IPMSM n the statonary reference frame for an analyss of the stator current s defned as follows: v α L Lcos 2 Lsn 2 R α Δ θ Δ θ s p α v = β + β Lsn 2θ L Lcos 2θ Δ + Δ β cosθ + φ f snθ Where L + L L L L =, Δ L = 2 2 q d q d whle vα, vβ and α, β are the α -axs and β -axs voltages and currents, respectvely. L d and L q are d- axs and q-axs nductance, φ f s the permanent magnet flux lnkage, p s the dfferental operator and θ s the rotor poston. he voltage equaton n synchronous reference frame (1) (2) (3) can be obtaned by vd Rs + pld ωrlq d 0 v = + q ωrld Rs plq q ωφ + r f where ω r s the rotor speed n the electrc angle. 2.2 orque characterstc and MPA control he magnetc path of an IPMSM s dfferent from that of an SPMSM snce the relatve permeablty of a permanent magnet s nearly equal to the ar. hs characterstc causes an effect whch produces an ar-gap on the d-axs. Because the permanent magnets of the SPMSM are attached to the rotor surface, d-axs nductance s equal to q-axs nductance. Otherwse, n the IPMSM, d-axs nductance s dfferent from q-axs nductance dependng on the poston of permanent magnets. As shown n Fg. 2, the q- axs nductance s larger than the d-axs nductance because permanent magnets are on the d-axs path where the magnetc flux occurs. he reluctance torque s generated by magnetc salency from the dfference of nductance. Accordngly, the generated torque of the IPMSM conssts of the magnetc torque and the reluctance torque as follows [1, 4]: (4) 3 P e = [ φ fq + ( Ld Lq) dq] (5) 22 where P s the number of poles. In (5), there are many pars of d-axs and q-axs reference currents that generate the same torque. When the reference current s gven as the stator current, the torque can be derved as (6). In such a case, the reference d- and q-axs currents are generated by the angle of the stator current. P 3 ( Ld Lq) [ sn 2 e = φ f Is θ + Is sn2 θ ] (6) 22 2 where θ s the stator current angle n the synchronous reference frame, and I s s the stator current vector. Fg. 1. Coordnate systems of an IPMSM. Fg. 2. Rotor structure of an IPMSM (4p) 743

3 On-lne Parameter Estmaton of Interor Permanent Magnet Synchronous Motor Usng an Extended Kalman Flter mechancal measurements, estmate the rotor poston n a mechancal sensorless control applcaton and dstngush the machne parameters. In ths secton, the latter applcaton for an IPMSM wll be gven a specal attenton. he proposed EKF provdes estmated d-axs and q-axs nductance [7, 8]. 3.1 IPMSM dentfcaton model Fg. 3. Constant torque locus and mnmum stator current vector For ths reason, there s the mnmum stator current that generate the same torque. By operatng at the mnmum current, maxmum torque per ampere (MPA) s possble. he constant torque locus on the current plane n the synchronous reference frame s shown n Fg. 3. hs fgure shows that the MPA operatng pont s the nearest pont from the orgn to the constant torque locus on the current plane. herefore, the dfferentaton of the torque wth respect to the current angle n the synchronous reference frame, e / θ should be zero at the MPA pont gven as follows [1] e P = + = θ 22 3 [ cos ( ) 2 φf Is θ Ld Lq Is cos2 θ ] 0. (7) From (7), the current angle of the MPA control can be derved as θ MPA = cos 1 φ + + 4( Ld Lq) Is f φf 8( Ld Lq) Is. (8) By usng ths angle of the MPA control, effectve d- axs and q-axs reference currents can be expressed as r = I cos θ, = I snθ (9) d s MPA q s r MPA As shown n (8), d-axs and q-axs nductance contrbute to the angle of the MPA control. It means that varaton of nductance wll not be able to fnd the optmal MPA pont. For effcent motor drves, the d-axs and q-axs should be estmated. 3. Extended Kalman Flter for Parameter Estmaton An EKF has been used for wde applcatons n drve control and parameter estmaton. In addton, the EKF s useful for a PMSM because t has hgh-performance n the nosy condton and wde speed control range. hus, t s often used to acqure the angular speed from nosy Eq. (4) can be redefned as follows v R L d dt L v R L d dt L d d = s d + d ωr q q q q = s q + q + ωr d d + ωφ r f. (10) he system model should be n the form of the state equaton to be used n the EKF. he state equatons can be expressed as follows &x() t = f( x()) t + Gv() t + σ () t y() t = Hx() t + μ() t (11) Where v = [ vd, vq] and y = [ d, q] are nput and output vectors. x = [ dqab] s a system state vector ( a= 1 /L d, b= 1/L q ). σ () t and μ () t are uncorrelated zero-mean whte Gaussan noses wth covarance Q and R respectvely. σ () t s system nose that ncludes the system dsturbances and model naccuraces, whle μ () t represents the measure-ment nose. he system matrces f(x(t)), G and H are defned as: a a( R sd) + ωrq b b fx ( ( t)) = b( R s q ωφ r f ) ωr d a (12) 0 0 a 0 0 b G =, H = (13) 0 0 x& F(()) xt = x x= x() t aω ω r rq aωrq ars vd Rsd + 2 b b b aωr bωrd d = brs v ( ) 2 q Rsq ωr φf b a a (14) 744

4 Hyun-Woo Sm, June-Seok Lee and Kyo-Beum Lee he predcton of the state covarance requres the onlne computaton of the Jocoban matrx F, defned as (14). 3.2 Extended kalman flter he overall structure of the EKF algorthm conssts of two steps called the predcton step and nnovaton step. he frst step performs a predcton of both quanttes based on the prevous estmates x ˆ k 1 k 1 and the mean voltage <v k-1 > actually nduced to the system n the perod from t k-1 to t k. he next step corrects the predcted state estmate and ts covarance matrx usng a feedback correcton scheme that receves the measured motor currents [11]. he common EKF algorthm s expressed followng two stages: 1) Predcton step xˆ = xˆ + [( f xˆ ) + G v ] kk 1 k 1k 1 k 1k 1 k 1 s k k 1 k 1k 1 ( k 1 k 1k 1 k 1k 1 k 1) s P = P + F P + P F + Q 2) Innovaton step xˆ = xˆ + K ( y Hxˆ ) kk kk 1 k k kk 1 P = P K HP kk kk 1 k kk 1 3) Kalman gan K = P H ( HP H + R ) k kk 1 kk 1 For the on-lne applcaton of the Kalman flter, tme constrants are crucal; where most of the matrces hold numerous null elements, all the calculatons have been made explctly, reducng computatonal tme. A crtcal step n the desgn of the Kalman flters s the choce of the ntal values for the covarance matrces Q and R as they nfluence the performance, convergence and stablty. he matrx Q s related to the system nose. he rse n the value of the elements of Q wll lkewse rase the Kalman gan, resultng n qucker flter dynamcs but wth worse the steady-state performance. On the other hand, the matrx R s related to the measurement nose. Increasng 1 the values of the elements of R wll assume that the current measurements are largely nfluenced by nose and thus less dependable. herefore, the Kalman gan wll decrease, yeldng worse transent response. he dagonal ntal state covarance matrx P 0 represents varances or mean-squared errors wth regard to the ntal condton. Varyng P 0 yelds the dfferent ampltude of the transent, whle both transent duraton and steady state condtons wll be unchanged [12, 13]. In ths paper, the ntal values of covarance matrx P 0 and system matrx Q and R have been chosen wth a traland-error procedure to get the best tradeoff between flter stablty and convergence tme [8]. As the value of Q ncreases, Kalman gan also ncreases, whch causes varaton of estmaton values. Lkewse, EKF s strongly affected by measurement nose as R ncreases. P 0 s selected based on [8] because t s nothng to do wth steady-state condton. In concluson, to obtan the hgh performance of proposed EKF, matrces are chosen as follows P 0 = R = Q = Smulaton and Expermental Results 4.1 Smulaton results he proposed estmaton method was demonstrated by usng the PSIM software. he overall system operaton s shown n Fg. 4. he current control and speed control are ncluded n the drve system. Estmated nductances by the EKF are used for the current controller to calculate the gan of the current controller. he smulaton studes were operated for the 11kW Fg. 4. Confguraton of the IPMSM drve system wth parameter estmaton 745

5 On-lne Parameter Estmaton of Interor Permanent Magnet Synchronous Motor Usng an Extended Kalman Flter able 1. Motor specfcatons Stator Resstance 0.349Ω d-axs Inductance 13.16mH q-axs Inductance 15.6mH Number of poles 6 Flux Lnkage 0.554Wb Rated Power 11kW Rated Current 19.9A Rated Speed 1750 rpm Fg. 6. Parameter estmaton at 500rpm wth dfferent ntal value (a) L 0 =0.5L (b) L 0 =2L Fg. 5. Parameter estmaton wth speed varaton IPMSM system. Swtchng frequency and samplng perod s are 10kHz and 100 μ s respectvely. he smulaton was performed n two dfferent condtons of speed at 500 rpm and 1000 rpm. Also, estmated results are confrmed accordng to the ntal nductance value. Intal nductance values were set as 0.5L and 2L. he whole system parameters are gven as able 1. Fg. 5 shows the results of parameter estmaton wth speed varaton. he good performance of estmaton can be confrmed for both 500rpm and 1000rpm. Also, the fast response to step change of speed can be seen through the estmated value. Results of estmated value accordng to the ntal value of nductance can be seen n Fgs. 6 and 7. When startng the operaton of motor, ntal nductance values are appled to the motor drve. After 0.3s, the EKF starts to estmate parameters used to calculate the gan of the current controller. he estmated values arrve at the steady-state n a very short tme wthn 0.1s, and results show an accurate value. he estmated results show a consstent value wth the nductance ntal value. Indeed, estmated parameters are the same for the dfferent speed condtons at 500 rpm and 1000 rpm. Generally, EKF s very approprate for mplementaton n systems wth sensors affected by nose that s the bggest reason why EKF s used for parameter estmaton. o confrm ths characterstc for proposed EKF, we tred to Fg. 7. Parameter estmaton at 1000rpm wth dfferent ntal value (a) L 0 =0.5L (b) L 0 =2L compare the EKF and RLS algorthm as shown n Fg. 8. After 0.5s, nose sgnal s njected to 3-phase current. It can be seen that proposed EKF s not senstve to nose sgnal whle estmated d-axs nductance from the RLS algorthm s affected by nose sgnal. 4.2 Expermental results In order to verfy the proposed EKF algorthm, experments were carred out by usng the 11kW IPMSM and MS320F28335 DSP controller as shown n Fg

6 Hyun-Woo Sm, June-Seok Lee and Kyo-Beum Lee Fg. 10. Expermental results of the parameter estmaton at 500rpm wth ntal values 0.5L (a) L ˆd (b) L ˆq Fg. 8. Comparson of the estmaton performance for nose sgnals: (a) EKF; (b) RLS. Fg. 11. Expermental results of the parameter estmaton at 1000rpm wth ntal values 0.5L (a) L ˆd (b) L ˆq Fg. 9. Expermental setup of IPMSM drve system. he experments were performed under the same condton as that of the smulaton. he swtchng frequency and samplng perod s are 10kHz and 100 μ s respectvely. he load s the 15kW nducton motor. he computaton tme of the proposed EKF n our bench s about 20 μ s, whch can be operated wthn samplng perod. he expermental results of parameter estmaton are shown n Fgs. 10 and 11. In these fgures, ntal values of d- and q-axs nductance were set as 0.5L. In addton, experments were carred out as two dfferent speed condtons n 500rpm and 1000rpm. Fgs. 10 and 11 show the correctly estmated d-axs and q-axs nductance. hese expermental results are the same as the smulaton results. 747

7 On-lne Parameter Estmaton of Interor Permanent Magnet Synchronous Motor Usng an Extended Kalman Flter Fg. 12. Expermental results of the parameter estmaton wth speed varaton (a) L ˆd (b) L ˆq Both smulaton and expermental results can see that the convergence tme of d-axs nductance s longer than q-axs nductance. hs s related to the confguraton of the ntal matrx value of the EKF. Fg. 12 shows the performance of parameter estmaton when the rotor speed changes from 500rpm to 1000rpm. Fg. 12(a) shows the estmated d-axs nductance that has very small estmaton error and t represents the stable response. Moreover, estmated q-axs nductance s more stable than d-axs nductance as shown n Fg. 12(b). For ths reason, the performance of the proposed EKF s not senstve to varable speed. Also, when the full step load torque s appled to the system, the performance of estmaton s mantaned and t s presented n Fg. 13. It can be seen from these expermental results that the parameter estmaton of the proposed EKF s mplemented well regardless of the speed change and varable load torque. From all of the smulaton and experment results, estmaton error rate were wthn 5 percent. In addton, compared wth other method, EKF has hgh-performance estmaton accuracy [7]. Fgs. 14 and 15 show the MPA operaton wth parameter estmaton usng proposed EKF. o clearly verfy the effect of the parameter estmaton n the MPA operaton, the dfference of the ntal nductance value s set to be large that s 10mH and 20mH respectvely. hrough the parameter estmaton durng the MPA operaton, t can be seen that the mnmum stator current I s can be obtaned that generate the same torque. Fg. 14(a) shows the d- and q-axes currents and torque durng the MPA operaton when the load torque 40% (24 Nm) s appled, whle Fg. 14(b) shows the waveform of Fg. 13. Expermental results of the parameter estmaton wth full step load torque (a) L ˆd (b) L ˆq Fg. 14. MPA operaton wth parameter estmaton under the 40% load. the changed value of nductance through the parameter estmaton at the same tme. Snce the dfference of the ntal nductance s large whch leads to a large stator current vector angle θ MPA, t can be seen that a large amount of d-axs current has been flowng. As the parameter estmaton s appled, whle the torque s at the constant state, the d-axs current decreases whle the q-axs current ncreases but very slghtly that t looks almost constant. Hence, because the total stator current value changes from A to A, whch s a decrease of about 0.1 A, t can be known that the mnmum current s used to operate. Fg. 15 s the expermental result when the load torque s ncreased by 80% (48Nm). Smlarly, when 748

8 Hyun-Woo Sm, June-Seok Lee and Kyo-Beum Lee Fg. 15. MPA operaton wth parameter estmaton under the 80% load. the parameter estmaton s appled, t can be verfed that the d-axs s decreased and the q-axs s ncreased. he total stator current decreased from A to A, whch s about 0.53 A. As the current ncreases, the MPA effect ncreases even more. From the expermental result, the necessty of the parameter estmaton for the MPA operaton s verfed. 5. Concluson hs paper proposes a parameter estmaton method of the IPMSM usng the EKF algorthm. By estmatng d-axs and q-axs nductances, t wll be able to mprove torque characterstc and performance of the controller. herefore, t wll assure the effcent motor drve even f parameters are changed. he valdty of the proposed method was demonstrated by smulaton and expermental results for varous speed and step load torque. Acknowledgements hs research was supported by Basc Scence Research Program through the Natonal Research Foundaton of Korea (NRF) funded by the Mnstry of Educaton ( ). References [1] S. M. Km, Y. D. Yoon and S. K. Sul, Maxmum orque per Ampere (MPA) Control of an IPM Machne Based on Sgnal Injecton Consderng Inductance Saturaton, IEEE rans. Power Electroncs, vol. 28, no. 1, pp , Jan [2] J. M. Km and S. K. Sul, Speed control of nteror permanent magnet synchronous motor drve for the flux weakenng operaton, IEEE rans. Industry Applcatons, vol. 33, no. 1, pp , Jan./Feb [3] H. Km, J. Hartwg, and R. Lorenz, Usng on-lne parameter estmaton to mprove effcency of pm machne drves, n 33 rd Annual IEEE Power Electroncs Specalsts Conference, PESC, vol. 2, pp , [4] Y. K. Kang, H. G. Jeong, K. B. Lee, D. C. Lee, J. M. K, Smple Estmaton for Intal Rotor Poston and Inductances for Effectve MPA-Operaton n Wnd- Power systems usng an IPMSM, Journal of Power Electroncs, vol. 10, no. 4, pp , July [5] J. W. Km, K. W. Km, D. O. Ksck, D. K. Kang, J. H. Chang and J. M. Km, A Study on Sensorless Control of ransverse Flux Rotatng Motor Based on MRAS wth Parameter Estmaton, Journal of Power Electroncs, vol. 11, no. 6, Nov [6] S. J. Underwood and I. Husan, Onlne Parameter Estmaton and Adaptve Control of Permanent- Magnet synchronous Machnes, IEEE rans. Industral Electroncs, vol. 57, no. 7, pp , July [7]. Boleau, B. Nahd-Mobarakeh and F. Mebody- abar, On-lne Identfcaton of PMSM Parameters: Model-Reference vs EKF, n conf. Rec. IEEE/IAS conf. Annu., pp. 1-8, [8]. Boleau, N. Leboeuf, B. Nahd-Mobaraken, and F. Mebody-abar, Onlne Identfcaton of PMSM Parameters: Parameter Identfablty and Estmator Comparatve Study, IEEE rans. Industry Applcatons, vol. 47, no. 4, pp , July./Aug [9] K. B. Lee and F. Blaabjerg, Sensorless DC-SVM for Inducton Motor Drven by a Matrx Converter Usng a Parameter Estmaton Strategy, IEEE rans. Industral Electroncs, vol. 55, no. 2, pp , Feb [10] G. S. Lee, D. H. Lee,. W. Yoon, K. B. Lee, J. H. Song and I. Choy, Speed and Flux Estmaton for an Inducton Motor Usng a Parameter Estmaton echnque, IJCAS(Internatonal Journal of Control, Automaton, and Systems), vol. 3, no. 1, pp.79-86, Mar [11] S. Bolognan, R. Oboe, and M. Zglotto, Sensorless full-dgtal PMSM drve wth EKF estmaton of speed and rotor poston, IEEE rans. Industral Electroncs, vol. 46, no. 1, pp , Feb [12] S. Bolognan, L. ubana and M. Zgltto, Extended Kalman Flter unng n Sensorless PMSM Drves, IEEE rans. Industry Applcatons, vol. 39, no. 6, Nov./Dec [13] R. Dhauad, N. Mohan, and L. Norum, Desgn and mplementaton of an extended kalman flter for the state estmaton of a permanent magnet synchronous motor, IEEE rans. Power Electroncs, vol. 6, no. 3, pp , Jul

9 On-lne Parameter Estmaton of Interor Permanent Magnet Synchronous Motor Usng an Extended Kalman Flter and relablty. Hyun-Woo Sm receved the B.S. degree n Electrcal and Computer Engneerng from Ajou Unversty, Suwon, Korea, n He s currently workng toward the M.S degree at Ajou Unversty, Suwon, Korea. Hs research nterests nclude electrc machne drves and multlevel nverter June-Seok Lee receved the B.S. and M.S. degrees n Electrcal and Computer Engneerng from the Ajou Unversty, Korea, n 2011 and 2013, respectvely. He s currently workng toward the Ph.D. degree at Ajou Unversty, Korea. Hs research nterests nclude grd-connected systems, multlevel nverter and relablty. Kyo-Beum Lee receved the B.S. and M.S. degrees n electrcal and electronc engneerng from the Ajou Unversty, Korea, n 1997 and 1999, respectvely. He receved the Ph.D. degree n electrcal engneerng from the Korea Unversty, Korea n From 2003 to 2006, he was wth the Insttute of Energy echnology, Aalborg Unversty, Aalborg, Denmark. From 2006 to 2007, he was wth the Dvson of Electroncs and Informaton Engneerng, Chonbuk Natonal Unversty, Jeonju, Korea. In 2007 he joned the School of Electrcal and Computer Engneerng, Ajou Unversty, Suwon, Korea. He s an assocated edtor of the IEEE ransactons on Power Electroncs, IEEE ransactons on Industral Electroncs and the Journal of Power Electroncs. He has receved two IEEE prze paper awards. Hs research nterests nclude electrc machne drves, renewable power generatons, and electrc vehcles. 750