Online estimation of the. flux linkage and voltage-source inverter nonlinearity in permanent magnet synchronous. machine drives

Transcription

1 Loughborough University Institutional Repository Online estimation of the rotor flux linkage an voltage-source inverter nonlinearity in permanent magnet synchronous machine rives This item was submitte to Loughborough University's Institutional Repository by the/an author. Citation: LIU, K. an ZHU, Z.Q., Online estimation of the rotor flux linkage an voltage-source inverter nonlinearity in permanent magnet synchronous machine rives. IEEE Transactions on Power Electronics, 29(1), pp Aitional Information: Personal use of this material is permitte. Permission from IEEE must be obtaine for all other uses, in any current or future meia, incluing reprinting/republishing this material for avertising or promotional purposes, creating new collective works, for resale or reistribution to servers or lists, or reuse of any copyrighte component of this work in other works. Metaata Recor: Version: Accepte for publication Publisher: c IEEE Please cite the publishe version.

2 1 Online Estimation of Rotor Flux Linkage an Voltage Source Inverter Nonlinearity in Permanent Magnet Synchronous Machine Drives Kan Liu, an Z.Q. Zhu, Fellow, IEEE Abstract This paper proposes a metho for online estimating the rotor flux linkage an voltage source inverter (VSI) nonlinearity of permanent magnet synchronous machine (PMSM) rives. Thermocouples are employe for measuring the temperature variation of the stator wining in orer to obtain the actual value of stator wining resistance. An Aaline estimator is use for online estimation of istorte voltage (V ea ) ue to VSI nonlinearity. Both are subseuently use for the estimation of the rotor flux linkage. The propose metho is experimentally valiate on a PMSM rive system an shows goo performance in tracking the variation of rotor flux linkage an compensating the VSI nonlinearity. Inex Terms Nonlinearity compensation, parameter estimation, PMSM, rotor flux linkage, voltage source inverter I. NOMENCLATURE R Stator wining resistance (Ω). L, L -axis inuctances (H). ψ m Rotor flux linkage (Wb). u, u Actual -axis voltages (V). i, i Actual -axis currents (A). i as, i bs, i bs Stator abc phase currents (A). Denotes a variable specifie by PI regulator. ^ Denotes an estimate value. u, u -axis reference voltages measure from PI regulators (V). Estimate rotor flux linkage (Wb). ˆm V ea Distorte voltage ue to inverter nonlinearity in -axis reference frame (V). ω, θ Electrical angular spee (ra/s) an rotor position (ra). γ D, D D Angles between current vector an -axis (ra). Functions of θ an the irections of three phase currents. II. INTRODUCTION ue to high power/torue ensity, permanent magnet (PM) synchronous machines are now wiely employe in inustrial servo rives, electric/hybri electric vehicles, an win power generators, etc. However, there exists a risk of potential irreversible emagnetization in the rotor magnets ue to high temperature rise or large emagnetizing current. Since the rotor PM flux linkage ecreases as the PM temperature increases, it is esirable to Manuscript receive October 05, Revise December 10, Accepte for publication February 27, K. Liu an Z.Q. Zhu are with the Department of Electronic an Electrical Engineering, University of Sheffiel, UK ( lkan@live.cn; Z.Q.Zhu@sheffiel.ac.uk). estimate the rotor flux linkage value no matter it is for torue control/monitoring or for prevention of PM emagnetization. For example, it is propose in [37] that the influence from the variation of rotor flux linkage on the compute torue in irect torue control can be compensate by online estimation of rotor flux linkage, in which the accuracy of torue control is significantly improve. In aition, it is reporte in [9] that the state of rotor permanent magnet can be monitore instantaneously by using a rotor flux linkage estimator, an a warning signal can be given if magnet emagnetization happens. The estimation of rotor magnet flux for etecting broken magnet is investigate in [40], which shows that there is separation between the estimate magnet flux of the motor with broken magnet an all the other faults an healthy cases. Thus, it is useful to online estimate the rotor flux linkage for PMSM base rive systems an various methos have been recently reporte, which can be classifie an reviewe as follows. As etaile in [1], the steay state PMSM euation is rank-eficient for simultaneously estimating the -axis inuctances, the stator wining resistance an the rotor flux linkage. Furthermore, since the estimation is usually base on the parameter moel of PMSM, mismatching of unestimate machine parameter values will introuce a significant error into the estimate rotor flux linkage. Thus, in practice, it is reuire to esign a full rank reference moel to simultaneously estimate the rotor flux linkage an other parameters that have influence on the estimation of rotor flux linkage. For example, it is propose in [2]-[5] to transiently inject flux weakening current (i 0) to activate the stator wining resistance term in PMSM -axis euation. With the injecte i 0, the values of rotor flux linkage an stator wining resistance can be simultaneously estimate if the -axis inuctance values are set to be their actual values. However, the accuracy of this kin of metho is easy to suffer from the variation of voltage source inverter (VSI) nonlinearity, -axis inuctances an rotor flux linkage ue to the injecte current. Furthermore, the injection of i 0 may introuce unacceptable instability an torue ripples into the rive system. In aition, it is propose in [6]-[9] to estimate the stator wining resistance an rotor flux linkage separately by changing PMSM working conitions such as rotor spee [7], [8], an output torue [9]. However, it is not preferable to change the working conition in some applications, which

3 2 reuire, for example, constant torue/spee control. Furthermore, there also exists significant influence from the VSI nonlinearity an varying stator wining resistance ue to temperature change on the estimation accuracy. Thus, the application scope of methos in [6]-[9] may be limite. For simplicity, it is reporte that the combinations of measurement sensors an estimation algorithms can be an easy alternative to solve the rank eficient problem [18], [19]. For example, it is propose in [18] to a a loa test to estimate the rotor flux linkage iniviually while the metho in [19] propose to bury a thermocouple in the stator wining for the estimation of wining resistance an rotor PM temperature. Compare with [18], it is much cheaper to employ a thermocouple in [19] than to employ a torue transucer. However, the metho in [19] nees to inject a high freuency voltage signal into the stator wining an is easy to suffer from the non-ieal behavior of the inverter. Furthermore, it neglects the influence from varying inuctance an cannot estimate the actual value of rotor flux linkage. In this paper, a metho for online estimating the PMSM rotor flux linkage an istorte voltage (V ea ) ue to VSI nonlinearity is propose, which is suitable for most wiely use i =0 control. The wining resistance at normal temperature an temperature coefficient of wining resistance are measure before the implementation of propose metho. Thermocouples are employe for measuring the temperature variation in stator wining in orer to obtain the actual value of stator wining resistance, which is use for aiing the estimation of rotor flux linkage afterwars. Thus, this metho oes not nee to inject any signals such as i 0 [1]-[5] an DC voltage pulse [10]-[14] or change the PMSM working conition [6]-[9]. Furthermore, it is avance that the accuracy of propose rotor flux linkage estimation will not suffer from the variation of -axis inuctances an has taken into account the compensation of estimation error ue to VSI nonlinearity thanks to the propose V ea estimator, which oes not nee any evice information of the VSI an PM machine. This metho is experimentally valiate in a fiel oriente vector control system an shows goo performance in tracking the variation of PMSM rotor flux linkage an compensating the VSI nonlinearity. III. PMSM MODEL INCLUDING VSI NONLINEARITY Assuming that the PMSM has negligible structural asymmetry, iron losses versus copper losses, an magnet motive force harmonics of winings, the -axis euations of the PMSM are given by: i R L u i i (1a) t L L L i R L u m i i (1b) t L L L L For machine parameter estimation, the steay-state machine -axis euations in iscrete time omain are usually use for analysis [31]. The steay-state -axis euations of PMSM are shown as follows: u ( k) Ri ( k) L ( k) i ( k) (2a) Fig. 1. Waveforms of D an D uner i =0 control. Simulate. Actual waveforms of compute D an D, i, an i uner i =0 control an ω= 157ra/s. Fig. 2. Waveforms of i, an i uner i =0 control an ω= 157ra/s. TABLE I RELATIONSHIP BETWEEN CURRENT DIRECTIONS AND DQ-AXIS DISTORTED VOLTAGE [22] θ e =θ+π/2 i a sign(i) i b i c V ea D V ea D π/6 γ~π/6 γ V ea sin(θ e ) 4V ea cos(θ e ) π/6 γ~π/2 γ V ea sin(θ e π/3) 4V ea cos(θ e π/3) π/2 γ~5π/6 γ V ea sin(θ e 2π/3) 4V ea cos(θ e 2π/3) 5π/6 γ~7π/6 γ V ea sin(θ e π) 4V ea cos(θ e π) 7π/6 γ~3π/2 γ V ea sin(θ e 4π/3) 4V ea cos(θ e 4π/3) 3π/2 γ~11π/6 γ V ea sin(θ e 5π/3) 4V ea cos(θ e 5π/3) u( k) Ri( k) L( k) i( k) m( k) (2b) where k is the inex of the iscrete sampling instant. Uner i =0 control, (2) can be simplifie into: u( k) L( k) i( k) (3a) u( k) Ri( k) m( k) (3b) The actual PMSM -axis euations incluing the istorte voltages ue to VSI nonlinearity can be shown as follows [22]-[28]: Li i Li u D R V ea t Li i L D i m u (4) D an D are expresse as follows: 2 2 cos( ) cos( ) cos( ) sign( ias ) D sign( ibs ) D 2 sin( ) sin( ) sin( ) sign( ics ) 3 3 (5) 1, i 0 sign() i (6) 1, i 0 As etaile in [22]-[25], assuming that γ is the angle between current vector an -axis an e, the 2 istorte voltage ue to VSI nonlinearity in -axis reference frame are shown in Table I. The simulate waveforms of D an D uner i =0 control (γ=0) is shown in Fig. 1, from which it is evient that the

4 3 istorte voltage in -axis reference frame is a zero-mean 6th harmonic component while the istorte voltage in -axis reference frame mainly consists of a DC component an a 6th harmonic component. Fig. 1 an Fig. 2 show the actual waveforms of compute D an D from (5), i an i at 300r/min, which are from the prototype PMSM shown in Table III of Appenix A. Comparing Fig. 1 with Fig. 1, it is evient that the actual waveforms of D an D agree very well with the simulate waveforms, which contain the 6th orer harmonic component. IV. VSI NONLINEARITY ESTIMATION In iscrete time omain, the steay-state PMSM euation incluing the istorte voltage ue to VSI nonlinearity can be expresse as follows: u ( k ) D ( k ) V Ri ( k ) L ( k ) i ( k ) (7a) ea u ( k) D( k) Vea Ri( k) L( k) i( k) m( k) (7b) As can be seen from (7), the variable V ea, which is relate to the istorte voltage ue to VSI nonlinearity, will introuce error into the machine parameter estimation. Since V ea is easy to vary with the current, it is necessary to online estimate an compensate V ea to ensure the accuracy of estimate PMSM parameter values. The technology for the online estimation/compensation of the istorte voltage ue to VSI nonlinearity has been wiely reporte [22]-[30], [34]-[36], [41]-[43]. However, most methos nee accurate PMSM parameter values [22]-[27], [35], [42], or zero rotor spee [8]. Thus, it is impossible for these methos to estimate the istorte voltage prior to the estimation of PMSM parameter values. In aition, since the istorte voltages in -axis reference frame are mainly 6th harmonic components [29]-[30], it is reporte in [29] an [30] that these istortions can be properly compensate by minimizing the 6th harmonic components. In this section, an Aaline estimator for the online estimation of istorte voltage V ea ue to VSI nonlinearity is esigne, which oes not nee any machine parameter values an is suitable for i =0 control. As can be seen from Table I, Fig. 1 an Fig. 1, uner i =0 control, it is evient that the high freuency component DV ea is zero-mean an its contribution to -axis voltage ( u ( k ) ) is only a fluctuation aroun 0V. In aition, it is noteworthy that the 6th harmonic component DV ea is the ominant term compare with other high freuency components in -axis euation. Thus, uner i =0 control, (7a) can be ecompose as follows: Ri ( k) Ri ( k) Ri ( k) Ri ( k) (8a) h h D( k) Vea 0 (8b) u ( k)= D( k) V L ( k) i ( k) (8c) ea u ( k) D( k) V L ( k) i ( k) (8) h ea h L ( k) i ( k)= L ( k) i ( k) L ( k) i ( k) L ( k) i ( k) h h h L ( k) i ( k) (8e) h h In (8), i, D, an i are the DC components of i, D, ω, an i, respectively, while u h, i h, ω h, an i h are the Fig. 3. Obtain high freuency component of u ( k ) by using first-orer low-pass filter. high freuency components of u, i, ω, an i, respectively. At the steay state of i =0 control, Ri ( k) 0 an terms such as L ( k) i ( k) an Ri ( k) Ri ( k ) are in fact part of h h DV ea because i h an i h are mainly generate by the 6th harmonic component ue to VSI nonlinearity. In aition, ω h consists of harmonic components ue to VSI nonlinearity an mechanical issues such as the rotating friction, loa oscillation, an eccentricities, etc, whose FFT analysis (introuce in Appenix C) shows that those harmonic components cause by mechanical issues are at relatively low freuency range an have very small amplitues. In aition, the 6th harmonic component in L h( k) i( k) is also cause by VSI nonlinearity an can be regare as part of DV ea. As shown in Fig. 3, the DC component L ( k) i ( k) of (8e) an those low freuency components in L h( k) i( k) can be filtere by a iscrete first-orer low-pass filter. Thus, there are only two high freuency terms, DV ea an L ω h (k)i h (k), left in (8) after filtering. It is also noteworthy that in reality, L ω h (k)i h (k) is uite negligible compare with D (k)v ea. Thus, it is reasonable that u h is approximately eual to D (k)v ea an (8) can be simplifie to be (9). u h ( k) D( k) Vea (9) Furthermore, since L ω h (k)i h (k) is negligible compare with D (k)v ea, the influence of the variation of L on the estimation of V ea will also be negligible. Taking the PMSM use in this paper as an example, its nominal -axis inuctance is 3.24mH. Assuming that the amplitues of ω h (k) an i h (k) are 5ra/s an 0.3A, respectively, an there is a 50% increase in L, the amplitue of L ω h (k)i h (k) will be e 3=7.29mV, which is uite negligible compare with the estimate amplitue of D (k)v ea (2 0.25V=0.5V) as will be shown in Fig. 5. Thus, from the above analysis, V ea can be estimate from (9). Since Aaline Neural Network (NN) algorithm is simple an low cost in computation, it is use to esign all the estimators in the propose estimation. The esign of V ea estimator an relate esign processes are escribe in Appenix B. Assuming Xi D( k), W i = V ˆea, ( ) ˆ Ok Vea Dk ( ) an k ( ) uh ( k), the Aaline estimator of V ea can be expresse as follows: Vˆ ea ( k 1) V ˆ ea ( k) 2 D( k)( ( k) O( k)) (10) where is the convergence factor. Ok ( ) anv ˆea are the output of variable moel an estimate V ea, respectively.

5 4 V. ROTOR FLUX LINKAGE ESTIMATION AND EXPERIMENTAL VERIFICATION A. Design of Rotor Flux Linkage Estimator Uner i =0 control, (7b) can be simplifie as: u ( k) D( k) Vea Ri( k) m( k) (11) From (11), it is evient that the value of rotor flux linkage can be estimate on conition that the values of V ea an R can be accurately obtaine. V ea can be estimate from -axis euation by using (10). Since the wining resistance R is a linear function of wining temperature, two sets of thermocouples are burie into two ifferent sies of PMSM stator wining an the actual value of wining resistance can be compute by the measure average temperature through (12). R= R 0 +TCR(T 2 T 0 ) (12) R 0, R an TCR are the wining resistance value at room temperature (T 0 =22 o C), actual wining resistance value at temperature T 2 an temperature coefficient of wining resistance. Thus, uner i =0 control, assuming Xi ( k), Ok ( ) ˆ m ( k) ( k) an k ( ) u ( k) DkV ( ) ea Ri ( k), the Aaline estimator of rotor flux linkage can be erive from (11) an shown as follows. ˆ ( k 1) ˆ ( k ) m 2 ( k)( ( k) O( k)) (13) m B. Compensation of VSI Nonlinearity It is known that V ea represents the VSI nonlinearity which shoul be well compensate or minimize in real application. However, it is known that the response of rive system to the compensation of V ea is not linear an an online tuning process is usually neee for the minimization of V ea. For example, Zhao [30] propose an online tuning scheme to minimize the 6th harmonics in -axis currents for the compensation of VSI nonlinearity. Similarly, Park an Sul [34] propose to use trapezoial voltage to compensate the inverter nonlinearity, whose trapezoial angle was real time tune by an integral controller. In this research, since V ea can be real-time estimate by (10), the VSI nonlinearity can be well compensate through the minimization of estimate V ea. The fiel oriente vector control system shown in Fig. 4 is employe for valiation. Assuming that Ṽ ea, V thr, ζ an γ are the estimate V ea by (10), threshol voltage of minimize V ea, tuning factor an gain factor of V ea, respectively, the schematic iagram of relate minimization process is epicte in Fig. 4 an the ajustments of the tuning factor ζ an gain factor γ are introuce as follows: a. Initialize V thr, ζ an γ. For example, in this research, ζ an γ are initialize to be 1.0e 4 an 0.1, respectively, while V thr is set to 1.0e 4V, which is uite close to 0. b. If Ṽ ea <V thr, it means that the VSI nonlinearity has been well compensate an the value of γ will be kept as a constant for compensation. c. If Ṽ ea >V thr, it means that the VSI nonlinearity is not well compensate. The value of γ will be increase or ecrease in each step of computation until it reaches the point that Ṽ ea <V thr. The rule for ecreasing or increasing γ is escribe in Fig. 4.. The real-time compute DγṼ ea an DγṼ ea will be ae to the output of -axis PI regulators for compensation. If Ṽea (k) V Thr γ(k+1)=(1+ζ)γ(k) Estimation of V ea No Ṽea (k) <V Thr? Yes If Ṽea (k) V Thr Ṽea(k+1) =γ(k+1)ṽea(k) γ(k+1)=(1 ζ)γ(k) γ(k+1)=γ(k) Fig. 4. Process of VSI nonlinearity compensation an use control system. Compensating VSI nonlinearity by minimizing V ea. Fiel oriente vector control system with propose estimation/compensation metho. Fig. 5. Estimation an compensation of VSI nonlinearity. Estimation of VSI nonlinearity. Compensation of VSI by minimizing V ea. Fig. 6. D-axis currents with an without propose compensation of VSI nonlinearity. Without compensation. With propose compensation. e. Go back to step b. It shoul be note that the value of ζ (>0) can control the spee of the minimization of Ṽ ea an an abrupt change may be introuce into the control system if ζ is too big. Thus, ζ is usually set to a small value (1.0e 4 in this paper) to smooth the process of minimizing Ṽ ea. The PMSM etaile in Appenix A will be employe for valiating the performance of propose metho. Fig. 5 shows the estimation an compensation of V ea an it is obvious that V ea can be minimize to be almost 0. Conseuently, the VSI nonlinearity is well compensate after this minimization. Figs. 6-8 are the measure currents at 300rpm in ifferent

6 5 reference frames with an without the propose compensation. It is obvious that with the propose compensation, the -axis currents will be with less 6th harmonics while the phase current will be more sinusoial. Furthermore, it is evient that the circle of αβ-axis currents shown in Fig. 7 an will be more orbicular with the propose compensation. It is also manifest from the FFT results epicte in Fig. 7(c) an () that the 5th an 7th harmonics in α axis current are minimize to be negligible, of which the sum is the 6th harmonic. C. Rotor Flux Linkage Estimation at Normal Temperature With aiing from the estimate V ea an compute wining resistance R, the rotor flux linkage can be finally estimate by (13). The complete iagram of propose metho is shown in Fig. 9 an the whole estimation are ivie into three steps: (1). At stanstill, the stator wining resistance value (R 0 ) at normal temperature (T 0 ) is measure by milliohm meter an two sets of thermocouple are employe for measuring the variation of stator wining temperature (T 2 ), which will be use for real time computing the actual value of stator wining resistance (R). (2). The value of V ea of istorte voltage ue to VSI nonlinearity is online estimate from the -axis euation an employe for the compensation of inverter nonlinearity. (3). The rotor flux linkage is finally estimate from the -axis euation with the aiing from compute wining resistance an V ea. Fig. 9. Complete iagram of propose metho. Fig. 10. Line back-emf at no-loa conition an FFT result. Line back-emf at 300rpm an normal temperature. Amplitue of funamental freuency component of line back-emf. Fig. 11. Estimate rotor flux linkage with an without propose compensation at normal temperature. (c) Fig. 7. αβ-axis currents with an without propose compensation of VSI nonlinearity. Without compensation. With propose compensation. (c) FFT result of α axis current without compensation. () FFT result of α axis current with propose compensation. Fig. 8. Phase A current with an without propose compensation of VSI nonlinearity. Without compensation. With propose compensation. () To confirm the accuracy of this metho, the PMSM is rawn by a DC machine an the rotor flux linkage at no-loa conition will be compute from the measure line back-emf for comparison. From Fig. 10, the amplitue of the funamental freuency component of line back-emf is obtaine by using FFT an the rotor flux linkage at no-loa conition is erive below: 19.23V 60s 70.7mWb rpm The estimation of rotor flux linkage uner ifferent working conitions is epicte in Fig. 11, in which the estimation an compensation of V ea are starte after t=10s. From Fig. 11, it is obvious that the estimate rotor flux linkage at 300rpm (68.3mWb) is close to that at no-loa conition (70.7mWb) an the slight ecrease in rotor flux linkage ( =2.4mWb) can be explaine that it is cause by the variation of flux ensity ue to on-loa.

7 6 Fig. 12. Measure stator wining resistance an temperature. Fig. 13. Comparison between estimate an measure rotor flux linkage at ifferent time points. Real-time estimate/measure rotor flux linkage. Variation in rotor flux linkage. However, the estimate rotor flux linkage at 150rpm (73.3mWb) is slightly larger than that at no-loa conition (70.7mWb), which is contrary to the basic theory that the rotor flux linkage at loae conition shoul be smaller than that at no-loa conition. This estimation error can be explaine that there are still influences from other nonlinearities such as non-ieal position measurement, current offsets an non-ieal measurement of DC bus voltage. Since the back EMF of the PMSM use for experimental valiation is relatively small at low spee (less than 8V at 150rpm), the accuracy of estimate rotor flux linkage at 150rpm will suffer from these unestimate nonlinearities significantly an present a relatively big estimation error ( =2.6mWb) even if the VSI nonlinearity is well compensate. The influences from these unestimate nonlinearities will be negligible when the estimation of rotor flux linkage is at a relatively high spee, at which a relatively high ratio of back EMF versus other nonlinearities can be achieve. For example, the estimate rotor flux linkage at 300rpm shows a much higher accuracy than that at 150rpm. Furthermore, from Fig. 11, it also shows that the ifference between the results with an without the compensation of V ea ( =18.1mWb) at 300rpm is almost half of that ( =36.9mWb) at 150rpm. This can be explaine that the variation of V ea at ifferent working conitions is relatively small but the ratio of DV ea versus ψ m ω will approximately ecrease twice when the rotor spee increases from 150rpm to 300rpm. D. Temperature Rise in Magnet It is known from [1] that uner i =0 control, the terms such as stator wining resistance an rotor flux linkage only exist in the -axis euation if the PMSM is at steay state. Thus, the estimation of stator wining resistance shoul be together with or prior to the estimation of rotor flux linkage [10]-[17], [44]-[46]. Otherwise, the estimation will face the rank-eficient problem an the accuracy of the estimate rotor flux linkage will suffer from the mismatche stator wining resistance value use in the estimator [1]. For estimating the stator wining resistance uner loae conition, the injection of perturbation signals such as i 0 [1]-[5] or DC voltage [10]-[14] is usually neee. However, this kin of metho is easy to introuce instability an torue ripples into the rive system an is not preferable in constant torue/spee control. Thus, for simplicity, thermal couples are employe for the measurement of wining temperature with which the wining resistance is real-time compute an use for aiing the estimation of rotor flux linkage. It is noteworthy that in real application, thermocouples are wiely employe for the temperature monitoring of commercial electrical machines, particularly for large machines. For example, there are 14 thermocouple temperature probes in a 247 MVA synchronous generator [39]. To confirm the accuracy of propose metho in tracking the variation of rotor flux linkage uner ifferent temperatures, a comparison experiment is esigne an organize as follows: 1) A heater is use to keep on heating the PMSM for 20 minutes an then left for natural cooling. 2) At t=20, 25, 40, 55, 75, 155 min, the DC machine will be starte to raw the PMSM an its rotor spee is fixe to 300rpm while the line back-emf of PMSM at no-loa conition will be recore by using a scope. The rotor flux linkage at no-loa conition can be compute from these measure ata with aiing from FFT. 3) At t=20, 30, 45, 60, 80, 160 min, the PMSM will be loae an the rotor flux linkage at loae conition will be online estimate with aiing from the measure stator wining temperatures. The actual stator wining resistance at ifferent temperatures is measure an shown in Fig. 12, which varies with the measure wining temperature linearly an the coefficient of temperature TCR can be obtaine afterwars. The experimental results of step 2 an 3 are plotte in Fig. 13, which show that the propose metho has goo performance in tracking the variation of rotor flux linkage. As can be seen from Fig. 13, the estimate rotor flux linkage will vary from 68.3mWb to 64.5mWb after 20 minutes heating. However, since there is a heat transfer elay between the stator an rotor PM, there will be a heat balance between the stator an rotor an the rotor flux linkage value will go on ecreasing an be 64.3mWb after 10 minutes cooling. Then, the rotor flux linkage will slowly increase with the ecrease in permanent magnet temperature an will reach to its normal value at last.

8 7 VI. CONCLUSIONS A new scheme for the online estimation of PMSM rotor flux linkage an VSI nonlinearity is propose, which can be use for the conition monitoring of rotor permanent magnet. Compare with existing methos for estimating rotor flux linkage, the propose metho oes not nee to inject any perturbation signals an has taken into account the estimation an compensation of the istorte voltage ue to VSI nonlinearity. It is experimentally valiate on a prototype surface mounte PMSM an shows goo performance. Thus, its application in interior PMSM rive system still nees further investigation. Furthermore, since it is only suitable for i =0 control, the estimation for i 0 control will be investigate an reporte in future. Appenix A - Test Bench an Machines The PMSM riven by a DSPACE base vector control system is shown in Fig. 14 an will be use for experimental verification, whose esign parameters are shown in Table II. In aition, the parameter configuration of employe voltage source inverter is shown in Table III, which is a Semikron SEMIX stack with SEMIX71GD12E4s IGBT moules. Fig. 16. Measure rotor spee with an without propose VSI nonlinearity compensation (i =0A, i =4A, 300rpm). Without compensation. With propose compensation. Fig. 14. Prototype PMSM. Fig. 17. FFT analysis for rotor spee with an without propose VSI nonlinearity compensation (i =0A, i =4A, 300rpm). Without compensation. With propose compensation. Fig. 15. Structure of an Aaline NN. n i i WX i i i0 OW (, X) TABLE II DESIGN PARAMETERS AND SPECIFICATION OF PMSM Rate current 10A Rate spee 400 rpm DC link voltage 36V Nominal phase resistance (T=25 o C) Ω Nominal self inuctance 2.91mH Nominal mutual inuctance 0.330mH Nominal -axis inuctance 3.24mH Nominal -axis inuctance 3.24mH Amplitue of flux inuce by magnets 70.7 mwb Number of pole pairs 5 Note: Nominal values are measure. TABLE III TYPICAL ELECTRICAL PARAMETERS OF VSI (FROM SEMIX71GD12E4S DATASHEET) Turn on elay T on 0.160μs Turn on/off elay T off 0.433μs Harware ea time 4 μs Switch control ea time t 2μs Voltage rop of the active switch(25 o C)V ce 1.85V Voltage rop of the freewheeling ioe(25 o C)V 2.2V Supply voltage range 0-650V Appenix B - Design Estimators by Aaline NN Since Aaline NN algorithm [33] reuires few computations an can be use for parameter estimation of machines [32], it is use to esign all the estimators throughout the paper. The mathematical moel of Aaline NN can be expresse as follows: n OW (, X) WX (B.1) i i i i i0 where W i is the net weight an X i is the input signal. The activation function OW ( i, X i) of the network output noe is a linear function. The structure of Aaline NN [33] is shown in Fig. 15. As can be seen from (B.1) an Fig. 15, if ( k ) is the sample target output, the weight ajustment can be obtaine via least mean suare (LMS) metho: Wi( k 1) Wi( k) 2 X i( ( k) O) (B.2) where is the convergence factor which ajusts the convergence spee. If the parameter to be estimate can be mathematically expresse in the form of general Aaline NN structure, as shown in Fig. 15, its estimator can be irectly erive from (B.2) by comparing its net structure with the general Aaline NN structure. In LMS theory, the convergence factor shoul be limite to a proper range of values, which can ensure the stability an convergence of estimator (B.2) [33]. A general useful range for is (B.3) X i

9 8 For estimator (10), X i = D(k) an it is obvious from Fig. 1 that the amplitue of D(k) =2. Thus, the convergence factor of (10) shoul be 0<< Similarly, for estimator (13), X i =ω(k)<=209.4 because the rate spee of use PMSM is 400rpm. Thus, the convergence factor of (13) shoul be 0<<1.1403e 5. In this paper, the convergence factors of (10) an (13) are set to 1.0e 5 an 1.0e 6, respectively. Appenix C FFT analysis for rotor spee Fig. 16 shows the measure rotor spee with an without propose VSI nonlinearity compensation. From the FFT analysis results shown in Fig. 17, there are three main harmonic components (5Hz, 10Hz an 154Hz) in the measure rotor spee without propose compensation. It is known that the rotor spee is 300rpm (5r/s) an the pole pair number is 5. 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