PARAMETER IDENTIFICATION OF PERMANENT-MAGNET SYNCHRONOUS MOTORS FOR SENSORLESS CONTROL

PARAMETER IDENTIFICATION OF PERMANENT-MAGNET SYNCHRONOUS MOTORS FOR SENSORLESS CONTROL DRAGOŞ OVIDIU KISCK 1, JUNG HWAN CHANG 2, DO HYUN KANG 3, JI WON KIM 3, 1 DRAGOŞ ANGHEL Key word: Extended electromotive force (EEMF), Online parameter identification, Permanent-magnet ynchronou motor (PMSM), Senorle control. An online parameter identification method i propoed for enorle control for urface and interior permanent-magnet ynchronou motor (SPMSM and IPMSM, repectively). A thi method doe not ue rotor poition or velocity to identify motor parameter, the identified parameter are not affected by poition etimation error under enorle control. The propoed method that i baed on ytem identification theory can be applied to all kind of ynchronou motor. The effectivene of the propoed method wa verified by experiment in both SPMSM and IPMSM. 1. INTRODUCTION There are mainly three kind of ynchronou motor, namely: urface permanent-magnet ynchronou motor (SPMSM), interior permanent-magnet ynchronou motor (IPMSM), and ynchronou reluctance motor (SynRM). Although rotor poition and velocity can be ued to achieve precie control of thee motor, poition enor have everal problem uch a cot and durability. Therefore, many enorle control method have been propoed [1, 2]. Thee enorle control method can be mainly divided into two type, i.e., thoe uing high-frequency voltage or current ignal [3 14] and thoe uing the fundamental component of voltage and current ignal [3, 10]. The former method ue relation among three-phae current [3], injection of high-frequency ignal of voltage or current [4 6, 10, 11], pecial inverter pule width modulation (PWM) pattern [7], current repone of tep voltage [8, 9], 1 Univerity POLITEHNICA of Bucharet, Faculty of Electrical Engineering, 313 Splaiul Independentei 060042, Bucharet, Romania, E-mail: drago@dp-control.pub.ro. 2 Department of Electrical Engineering, Dong-A Univerity, Buan 604-714, Korea, cjhwan@dau.ac.kr. 3 Korea Electrotechnology Reearch Intitute, Mechatronic Reearch Group, Changwon, Kyungnam 641-120, Korea, E-mail: dhkang@keri.re.kr. Rev. Roum. Sci. Techn. Électrotechn. et Énerg., 55, 2, p. 132 142, Bucaret, 2010

2 Parameter identification of permanent-magnet ynchronou motor 133 information on harmonic reactive power [12], and a ytem identification method to detect rotor poition information, i.e., magnetic aturation or rotor aliency. Thee method are effective at tandtill and in low-peed range becaue the amplitude of high-frequency ignal ued for poition etimation doe not depend on rotating velocity. Inamuch a ome method do not need motor parameter to etimate rotor poition, poition etimation error i not caued by parameter variation. Early method ue detected terminal information on electromotive force (EMF), information on phae of flux, difference of current or voltage, and a liding oberver for flux etimation to etimate rotor poition information, i.e., back EMF or rotor aliency. Thee method are ueful in middle and high-peed range becaue they ue the fundamental component of control ignal for poition etimation and do not generate torque ripple or noie. However, thee method ue motor parameter to etimate rotor poition, and poition etimation error i caued by parameter variation. In addition, the mathematical model of alient-pole PMSM i complicated, and poition etimation uing information of both back EMF and aliency require complicated calculation and approximation. An extended EMF (EEMF) model, which i a mathematical model of ynchronou motor, ha been propoed for poition etimation in IPMSM. Both of the EMF generated by permanent-magnet and the EMF generated by rotor aliency are included in the EEMF term, poition etimation uing the EEMF can be eaily realied for all kind of ynchronou motor. Senorle control method baed on EEMF require motor parameter to etimate rotor poition jut a other method. Becaue the motor parameter are changed by magnetic aturation and temperature, a poition etimation error i generated when there are difference between actual motor parameter and one ued in the etimation ytem. Therefore, thee parameter hould be meaured in all driving area, and a table of parameter hould be made to maintain accuracy. It i hoped that thee parameter can be meaured online under enorle control. In thi paper, an online parameter identification method for both SPMSM and IPMSM i propoed. The propoed identification method doe not ue poition and velocity to identify motor parameter. With thi method the parameter meaurement are not neceary. 2. EEMF MODEL CONSIDERATION IN PARAMETER VARIATIONS 2.1. THE COORDINATES AND SYMBOLS Coordinate are defined in Fig. 1. The α β frame i defined a the tationary reference frame, the d q frame i defined a the rotating reference frame, and the γ δ frame i defined a the etimated rotating reference frame. The ymbol ued in thi paper are preented in Table 1, Table 2 and the matrix definition, in Table 3.

134 Dragoş Ovidiu Kick et al. 3 Table 1 Bae quantitie for [p.u.] ytem U b = U n rated voltage I b = I n rated current ω b = ω n rated angular peed Λb = Λn = U n/ ωn rated tator flux Z b = U n/ I n bae impedance L b = Ψn/ In bae inductance Table 2 [p.u.] quantitie r = R / Zb tator reitance λ PM = Λ PM / Λb PM flux linkage l d = xd = Ld / Lb d-axi inductance l q = xq = Lq / L b q-axi inductance l = x = L / Lb rotor inductance (non-alient-pole motor) [ u d u ] T q voltage on the rotating reference frame [ u u ] T α β voltage on the tationary reference frame [ e e ] T α β EEMF on the tationary reference frame [ i i ] T γ δ current on the etimated rotating reference frame [ i d i ] T q current on the rotating reference frame [ i i ] T α β current on the tationary reference frame [ uγ uδ ] T voltage on the etimated rotating reference frame ν = ω re / ω b rotor peed θ re rotor poition, θ re poition etimation error p= d /dt differential operator T ampling period of the identification ytem Table 3 Matrix definition 1 0 0 1 0 0 I =, J =, O =, 0 1 1 0 0 0 co 2θre in 2θre in 2θre co 2θre Q (2θre) =, S (2θre) =. in 2θre co 2θre co 2θre in 2θre 2.2. THE EEMF MODEL A mathematical model of ynchronou motor on the rotating reference frame i written a (1); l d i equal to l q in SPMSM. u u d q = r + pl νl d d / ω n νlq id + νλ r + plq / ωn iq PM 0. (1) 1

4 Parameter identification of permanent-magnet ynchronou motor 135 By tranforming (1) in the tationary reference frame, (2) i derived. Fig. 1 Coordinate of PMSM. u u α β = i inθ I 0 1Q re n PM, (2) iβ coθre α re [ r + p( l I + l (2θ ))/ ω ] + νλ where l = ( l d + l ) / 2 and l = ( l d l ) / 2. 0 q 1 q A hown in (2), there are two term including poition information θ re. One i the back EMF term that include θ re and i generated by a permanent-magnet, and the other i the Q(2θ re ) term that include 2θ re and i generated by rotor aliency. Becaue the poition etimation uing information on both term i very complicated, conventional poition etimation method uually ue jut one of thee term, rotor aliency or back EMF. To olve thi problem, an EEMF model i propoed a a mathematical model ued in poition etimation of ynchronou motor. Equation (3) repreent the EEMF model that i derived from (1) without approximation. Here, i repreent a differential of the q-axi current. u + ω ν d r pld / n lq id 0 = + {( ld lq )( νid iq / ωn ) + ν λ PM }. (3) uq νlq r + pld / ωn iq 1 By tranforming (3) to the one on the tationary reference frame, (4) i derived.

136 Dragoş Ovidiu Kick et al. 5 u u α β iα iα eα = ( r + pld / ωn ) I ν( ld lq ) J +, (4) iβ iβ eβ e α in θre = {( ld lq )( νid iq/ ωn ) + ν λ PM }. eβ co θ (5) re The third term on the right ide of (4) i defined a EEMF and i hown in (5). The EEMF include EMF generated by both permanent-magnet and rotor aliency, and it i the only term in (4) that include poition information. 2.3. EFFECT OF PARAMETER VARIATIONS AND COUNTERMEASURES AGAINST IT Difference between actual parameter and thoe ued in poition etimation, caue deterioration of poition etimation accuracy under enorle control. Becaue that i a eriou problem for enorle control uing motor parameter, countermeaure againt parameter variation are deired. In thi paper, an online parameter identification method i propoed a a countermeaure. The objective of the parameter identification i to identify motor parameter ued in poition etimation to maintain accuracy. The propoed method ha three advantage. Poition and velocity are not ued to identify motor parameter. Therefore, identified parameter are not affected by a poition etimation error. Motor parameter can be identified online; thu, prior parameter meaurement are not neceary. The propoed method can ue any ignal that atifie the condition of peritent excitation, and pecial band-pa filter are not neceary. 3. PARAMETER IDENTIFICATION BASED ON SYSTEM IDENTIFICATION THEORY 3.1. PARAMETER MATRIX IDENTIFICATION USING A RECURSIVE LEAST-SQUARE METHOD The propoed method identifie unknown motor parameter via a mathematical model uing known value uch a voltage and current. The mathematical model i contructed on an etimated rotating reference frame becaue the model coefficient can be aumed to be almot contant regardle of the rotation condition. By tranforming (1) to the one on the etimated rotating reference frame, and by tranforming the equation to a dicrete equation, we have: i i γ δ ( k) i = ( ) A k i γ δ ( k 1) u + ( 1) B k u γ δ ( k 1) + [1] ( 1) C, (6) k

6 Parameter identification of permanent-magnet ynchronou motor 137 2 2 2 2 ( l + l ) ω T ν( l l ) a11 a12 rl0 nt rl1 nt ν d q n d q ωnt ω ω A = 1 (2 re ) (2 a21 a = I + Q θ + J S θ 22 ldl q ldlq 2ldlq 2ldlq b11 b12 l0ωnt l1ω nt c1 νλ θ PM ωnt in re B = = (2 θre) b21 b I Q, C = =. 22 ldlq ldlq c2 lq co θre Equation (6) i tranformed a: y = Θ z. (7) Θ i an unknown matrix and i defined a a parameter matrix that include motor parameter. The vector y and z are known vector where: re ), y = iγ( k) iδ( k), T z = iγ( k 1) iδ( k 1) uγ( k 1) uδ( k 1) 1, T T a11 a12 b11 b12 c1 = [ ] = a21 a22 b21 b22 c 2 Θ A B C. (8) Uing (7), the unknown parameter matrix Θ i derived from known vector y and z by uing a leat quare method. Thi method identifie the parameter matrix Θ a the quare value of the prediction error (9) reache a minimum [13]. Θ ε i = y z. (9) To identify the parameter matrix Θ online, a recurive leat-quare method i ued a hown in (10) and (11) [13]. The contant λ between 0 and 1 i defined a the weighting coefficient, the role of which i to delete pat data. From: 2 T Θ( k) = Θ( k 1) + y Θ( k 1) z z P ( k), (10) 1 T 1 T P( k) = { P( k 1) P( k 1) z ( λ+ z P( k 1) z) z P ( k 1) }, (11) λ the parameter matrix Θ i identified recurively. 3.2. CHARACTERISTICS OF THE PARAMETER MATRIX Becaue the poition and velocity have to be etimated, the correponding term in the parameter matrix Θ hould be eliminated. The matrix Θ conit of four matrice, namely: I, J, Q(2 θ re ), and S(2 θ re ). Here, I repreent a unit

138 Dragoş Ovidiu Kick et al. 7 matrix, J repreent a π/2-radian rotation matrix, Q(2 θ re ) repreent the matrix by which arbitrary point on a two-dimenional plane are moved ymmetrically to the traight line of the θ re radian, and S(2 θ re ) repreent the matrix that move point ymmetrically to the traight line of the θ re +π/4 radian. If the four matrice are repreented a hown in (12), addition and ubtraction of both diagonal and nondiagonal component of thee matrice can be repreented a hown in Table 4. x11 x12 X =, ( X = I, J, Q(2 θre ), S(2 θre )). (12) x21 x22 Table 4 Characteritic of the matrice x 11 +x 22 x 11 -x 22 x 12 +x 21 x 12 -x 21 I 2 0 0 0 J 0 0 0 2 Q ( 2 θre ) 0 2 co2 θre 2 in 2 θre 0 S ( 2 θre ) 0 2 in 2 θre 2 co 2 θre 0 3.3. PARAMETER DERIVATION FROM THE PARAMETER MATRIX Uing the relation in Table 4, motor parameter are derived without uing poition and velocity. The proce of elimination of poition/velocity term i a: M 1 = b 11 + b 22 2l = l l d 0 q ω n T 2rl M = a + a = ω T,, 0 2 11 22 2 ll d q 2l M = ( b b ) + ( b + b ) = ω nt. (13) 2 2 1 3 11 22 12 21 ll d q Uing the variable M 1, M 2, and M 3, motor parameter are derived a: = r M M 1 2, l d = M 2ω n T + M 1 3, l q = M 2ω n T n M 1 3. (14) In thi cae, information on poition and velocity i not ued in (13) and (14); thu, the motor parameter can be derived independently of a poition etimation error. 4. EXPERIMENTAL RESULTS 4.1. CONFIGURATION OF THE EXPERIMENTAL SYSTEM Senorle control with online parameter identification wa realied uing the ame ytem in both an SPMSM and an IPMSM a Fig. 2 how. The poition

8 Parameter identification of permanent-magnet ynchronou motor 139 etimation, peed etimation and the oberver from thi figure are not dicued in thi paper. Stator current are detected by current enor and are then ent to a floating-point digital ignal proceor (DSP TMS320C6713) through 16-bit analogto-digital (A/D) converter. Three-phae current ignal and voltage reference are finally tranformed into the etimated rotating reference frame. Fig. 2 Configuration of the experimental ytem. From thee current and voltage ignal, motor parameter are identified in the propoed parameter identification ytem. Thee identified parameter pa through a low-pa filter, the decay time contant of which i et to 1.0 for the inductance parameter and 10 for the reitance parameter. The velocity controller i a polarization index (PI) controller; it output current reference from the error between the etimated velocity and the velocity reference. The current control i done by two PI controller for each axi and output voltage reference from error between reference current value and meaured one. Table 5 how the pecification of both tet motor, SPMSM and IPMSM. Table 5 Specification of SPMSM and IPMSM Rated power 2.5 kw PM flux linkage (SPMSM) 1.15 V/(rad/) Rated current 15.0 A PM flux linkage (IPMSM) 1.04 V/(rad/) Rated peed 2,500 rot/min Inverter witching frequency 5 khz Number of pole pair 2 DC link voltage 320 V Sampling period T 200 µ

140 Dragoş Ovidiu Kick et al. 9 4.2. PARAMETER IDENTIFICATION RESULTS Fig. 3 and Fig. 4 how the reult of parameter identification for the SPMSM. Figure 3 how reult at no-load, wherea Fig. 4 how reult at rated load (9.5 Nm). R and L and repreent identified parameter. The reference velocity wa et to 500 rot/min. From a comparion of Fig. 3 and Fig. 4, the identified parameter in both figure were approximately the ame, and precie poition etimation could be realied in both condition. Fig. 5 and Fig. 6 how the reult for the IPMSM. Fig. 5 how reult at no load, and Fig. 6 how reult at rated load (9.5 Nm). R, L d, and L q repreent identified parameter. Fig. 3 Poition etimation and parameter identification reult at no-load (SPMSM). Fig. 4 Poition etimation and parameter identification reult at rated load (SPMSM). Fig. 5 Poition etimation and parameter identification reult at no-load (IPMSM). Fig. 6 Poition etimation and parameter identification reult at rated load (IPMSM). Comparing Fig. 5 and Fig. 6, we ee that the identified parameter changed from 18 mh at no-load to 14 mh at rated load, and R changed from 0.3 Ω at noload to 1.1 Ω at rated load. The change of L q wa caued by magnetic aturation becaue magnetic aturation wa eaily generated in the q-axi. In contrat, the change of R wa too large to be caued olely by thermal change. It i conidered

10 Parameter identification of permanent-magnet ynchronou motor 141 that the change of R wa caued by the non-linearity of the motor. The reaon i a follow. In the propoed method, the actual motor i treated a the model hown in (1). The model doe not take into account non-linearity, e.g., mutual inductance between the d-axi and q-axi, aturation, and core loe. Inamuch a the model repreent only the relation of input and output (voltage and current), difference between the actual motor and the motor model are generated by everal kind of non-linearity. Although the identified reitance R changed from 0.3 Ω in Fig. 5 to 1.1 Ω in Fig. 6, we cannot know if the actual reitance changed like that. Inamuch a the objective of parameter identification i to identify motor parameter ued in poition etimation to maintain accuracy, the difference between the actual reitance and the identified one i a ignificant problem, epecially when accuracy of poition etimation uing identified parameter fall. However, the poition etimation error i known to be uppreed in both condition. 5. CONCLUSIONS In thi paper, an online parameter identification method i propoed. The propoed method doe not require prior parameter meaurement and can identify motor parameter without rotor poition information. Therefore, identified parameter are not affected by the accuracy of poition etimation. The propoed method wa experimentally verified a ueful in both SPMSM and IPMSM. Received on February 3, 2009 REFERENCES 1. J. Johnon, M. Ehani, Y. Guzelgunler, Review of enorle method for bruhle dc, in Conf. Rec. IEEE-IAS Annual Meeting, 1, pp. 143 150 (1999). 2. M. Schroedl, Senorle control of permanent-magnet ynchronou machine: An overview, in Proc. EPE-PEMC, CD-ROM, 2004. 3. N. Matui, T. Takehita, A novel tarting method of enorle alient pole bruhle motor, in Conf. Rec. IEEE-IAS Annual Meeting, 1994, pp. 386 392. 4. P. L. Janen, R. D. Lorenz, Tranducerle poition and velocity etimation in induction and alient AC machine, IEEE Tran. Ind. Appl., 31, 2, pp. 240 247 (1995). 5. M. Schroedl, Senorle control of AC machine at low peed and tandtill baed on the INFORM method, in Conf. Rec. IEEE-IAS Annual Meeting, 1996, pp. 270 277. 6. T. Noguchi, K. Yamada, S. Kondo, and I. Takahahi, Initial rotor poition etimation method of enorle PM ynchronou motor with no enitivity to armature reitance, IEEE Tran. Ind. Electron., 45, 1, pp. 118 125 (1998). 7. S. Ogaawara, H. Akagi, Implementation and poition control performance of a poition-enorle IPM motor drive ytem baed on magnetic aliency, IEEE Tran. Ind. Appl., 34, 4, pp. 806 812 (1998).

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