Experimental Determination of Mechanical Parameters in Sensorless Vector-Controlle Inuction Motor Drive V. S. S. Pavan Kumar Hari, Avanish Tripathi 2 an G.Narayanan 3 Department of Electrical Engineering, Inian Institute of Science, Bangalore - 560 02 E-mail: pavan@ee.iisc.ernet.in, avanish@ee.iisc.ernet.in 2, gnar@ee.iisc.ernet.in 3 Abstract High-performance inustrial rives wiely employ inuction motors with position sensorless vector control. The spee controller esign in such a rive is highly sensitive to the mechanical parameters of the inuction motor. These mechanical parameters change with the loa couple. This paper proposes a metho to experimentally etermine the mechanical parameters, namely the moment of inertia an frictional coefficient of the inuction motor rive along with the loa riven. The propose metho is base on acceleration an eceleration of the motor uner constant torque, which is achieve by using a sensorless vector-controlle rive itself. Experimental results from a 5hp inuction motor rive are presente. Inex Terms Inuction Motor Drives, Fiel Oriente Control, Moment of Inertia, Frictional Coefficient, Sensorless Vector Control DC Voltage Source Spee reference Voltage Source Inverter Gate rive signals Spee sensorless vector control an pulse with moulation Voltages & currents B Y R Squirrel Cage Inuction Motor I. INTRODUCTION Vector control or fiel oriente control is wiely employe for high performance an precisely controlling inuction motors (IM) in inustrial rives [] [4]. A simplifie block iagram of a vector controlle IM is shown in Fig.. Vector control involves ecouple control of flux an torque. The ecoupling is achieve in a synchronously rotating -q reference frame, whose reference axes are shown in Fig.. The reference axes of the stationary reference frame an the three phase axes are also inicate in the same figure. The etails of the transformations an control are explaine in section II. Vector control involves q-axis an -axis current control in the inner loops, an flux an spee control in the outer loops. Design of spee controller requires precise knowlege of the mechanical parameters, namely moment of inertia (J) an coefficient of friction (B), for achieving goo spee response. These parameters also change with the loa couple to the inuction motor. Several methos have been presente in literature to measure an/or estimate the mechanical parameters for servo motor rives an PMSM base rives [2], [5] [0]. Retaration test has been suggeste for measurement of moment of inertia in [2]. Reference [5] presents a spee observer base online metho to generate position error signal for ientification of moment of inertia. An offline metho This work is fune by the Department of Heavy Inustry, Government of Inia, uner a project title Off-line an Real-time Simulators for Electric Vehicle/ Hybri Electric Vehicle Systems. q Y B b ρ mr Rotor flux axis Stator R-phase axis a R Fig.. Block iagram of sensorless vector-controlle IM rive an axes of reference for machine moelling an control base on time average of the prouct of torque reference an motor position for mechatronic servo systems is presente in [6]. Another online recursive least square (RLS) estimator for a servo motor rive is presente in [7] for estimation of mechanical parameters. Reference [9] presents a controller base close loop metho to estimate inertia an friction of servo rive. A loa torque observer base metho to precisely estimate J an B for servo systems is presente in [0]. The mechanical subsystem is moele as a secon orer system in
the aforementione methos, which is complicate to solve. Further, observer base online estimation requires involve computations, which may not be feasible on low cost controller base systems. This paper presents a simple offline metho to quickly estimate both J an B in a sensorless vector controlle rive. The spee of the rive an flux are controlle through base controllers in various loops. The parameters of the spee loop controller are esigne base on approximate initial guess of the values of J an B. The rive is allowe to accelerate an ecelerate at constant electromagnetic torque uner no-loa conitions an the rotor spee ata is capture for estimation of mechanical parameters. The value of B is estimate from the electromagnetic torque an the estimate rotor spee. This is explaine in section III in more etail. Further, J is estimate through the best curve fit of estimate rotor spee response, as explaine in section IV. The algorithm is valiate on a 5-hp inuction motor an c generator set in section III an section IV. II. VECTOR CONTROL OF INDUCTION MOTOR This section reviews the ynamic moel of inuction motor, complete vector-control block iagram, various control loop block iagrams an the esign of current, flux an spee - controllers. A. Dynamic Moel of Inuction Motor The inuction motor is moele in a synchronously revolving rotor-flux oriente reference frame (q, shown in Fig.) for the purpose of sensor-less vector control [4]. The ynamic moel equations pertaining to inuction motor rive in rotorflux reference frame are shown in (). The torque evelope by the inuction motor (m ) is expresse as shown in (f) [2]. σl s t i s = [v s R s i s σl s mr ( σ)l s t ] (a) σl s t = [v sq R s σl s mr i s ( σ)l s mr ] (b) t = R r (i s ) L r (c) t ρ = mr = R r L r () t = J [(m m L ) P 2 B] (e) m = 2 3 P 2 L o ( σ r ) (f) Here, v s an v sq are components of stator voltage vector along an q - axes respectively; i s an are components of stator current vector along an q - axes respectively; is magnitue of rotor flux magnetizing current; is the rotor spee in electrical ra/s, m L is the loa torque, P is the number of poles; mr is the synchronous spee in ra/s; m L is the loa torque an ρ is the angle between a-axis an -axis (shown in Fig.) [2]. For the purpose of sensorless vector control, the shaft spee is estimate as shown in (2) [3] = [cos(ρ) t sin(ρ) sin(ρ) t cos(ρ)] s (2) [ ] R r s = L 0 ( σ r ) where, ρ is the angle of transformation (see Fig.), sin ρ an cos ρ are the unit vectors corresponing to axis transformation; R r is the rotor resistance, L 0 is the mutual inuctance an σ r is the rotor leakage coefficient. B. Block Diagram Fig.2 shows the complete block iagram structure of a vector-controlle inuction motor [4]. The control problem is ecouple into inepenent flux-control an spee-control loops by aing appropriate fee-forwar terms (e sq an e s ). The fee forwar terms are given in (3) [4] e s = ( σ) L s (i s ) σl s mr T r e sq = ( σ) L s mr σl s mr i s (3a) (3b) Each control loop consists of inner current an outer flux or spee control loop. The block iagrams of current, flux an spee control-loops are shown in Fig.3 Fig.3 an Fig.3(c), respectively [4]. The inverter is moele as first orer elay with total elay time (T ) equal to.5t s, where T s is the switching cycle perio. Since the inner control loops shoul settle much faster than the outer control loops, banwiths of the inner current control loops are esigne several orers higher than the outer flux or spee control loops. Further, since also appears in the spee control loop, the banwith of spee control loop is kept several orers lower than that of flux control loop. C. Design of s A 5-hp, 400-V, 50-Hz, three-phase inuction motor, riven by an IGBT base 0-kVA inverter, is consiere for this paper. The parameters of the motor an inverter are provie in Table I. The inverter is operate with asynchronous conventional space-vector PWM an it is switche at khz frequency. The time constant of q-axis current loop controller (Fig.3) is chosen( so as to cancel the largest time constant in the loop T isq = transfer function can be written as [4] (s) i sq(s) = ( σl s R s ). With this conition the close loop K isq K isq s 2 s T R s T isq T ) R s T isq T (4) The value of K isq is chosen such that a banwith of 00Hz is achieve for current control loop. The same values are also consiere for K is an T is. Further, for flux control loop, controller time constant (T imr ) is chosen to cancel the rotor time constant (T imr = T r ).
i mr i sq i s i s v sq v s e sq e s vsq vsa e jρ vs vsb cos ρ sin ρ 2-Phase 3-Phase vrn vy N vbn V p 0 V p 2V p V DC m R m Y m B Pulse With Moulation (PWM) S R S Y S B V DC 2V p V DC i R i Y i B v RN v Y N v BN Fig. 2. Complete block iagram of vector-controlle inuction motor [4] i sq K isq ( st isq ) e st v sq vsq v sq st isq st e sq e sq s R s ( ) σ Ls R s i mr i s K imr ( st imr ) st imr sτ bis st r i s K ( st ) i sq m Π K m st sτ bis sj B m L P 2 st f (c) Fig. 3. Control loops for esigning the base controller of the inuction motor rive current control loop flux control loop an (c) spee control loop [4] The reuce transfer function of the flux-control loop is written as [4] (s) i mr(s) = ( s 2 s τ bis K imr K imr T imr τ bis ) T imr τ bis (5) K imr is chosen so as to have a banwith of 23Hz for the flux-control loop. A first-orer low-pass filter with 0Hz cut-off frequency (T f = 0.059s) is implemente in the feeback path of spee loop to avoi high frequency noises. The open loop transfer function of the spee control after neglecting the frictional coefficient is written as [4] G (s) = K ( st ) K m i mr st ( sτ bis ) sj P 2 ( st f ) (6) Further, using symmetric optimum metho the spee controller for a banwith of 0.88Hz with approximate known values of B an J, given in Table I. Designe banwiths of various controllers are provie in Table I. Appropriate limits are put on the controller outputs to avoi win-up phenomenon. The sensor-less vector control algorithm is implemente on ALTERA CycloneII fiel programmable gate array (FPGA) base igital control platform []. The vector-controlle rive itself is use for experimental etermination of B an J as explaine in the following sections. III. MEASUREMENT OF FRICTIONAL COEFFICIENT (B) It is seen from equation (e) that at no-loa an uner steay state operating conition, the electromagnetic torque (m ) generate is equal to the torque offere by the frictional coefficient alone. The evelope torque m can be calculate
TABLE I PARAMETERS OF MOTOR, INVERTER AND CONTROLLERS 5-hp, 400-V, 50-Hz, 3-phase inuction motor Stator resistance per phase R s.62ω Rotor resistance per phase R r.62ω Stator leakage coefficient σ s 0.042 Rotor leakage coefficient σ r 0.042 Mutual inuctance per phase L o 227mH Number of poles P 4 Combine moment of inertia of motor 0.2kg-m 2 Combine frictional coefficient of motor 0.0kg-m 2 /s 2 an DC generator, J (assume) an DC generator, B (assume) Switching frequency of inverter khz Banwith of controller 23Hz Banwith of -axis an q-axis 00Hz Banwith of spee controller 0.88Hz current controllers (base on assume J an B) from (f) by measuring the values of an. At known values of rotor spee, frictional coefficient B can be calculate straightaway as the ratio of torque generate an rotor spee. The vector-controlle rive is run on no-loa at ifferent spees; the values of an are measure at each spee. Since flux is maintaine constant, only changes value at ifferent spees. is maintaine at a value of 5.92A, an is measure at ifferent spees. Torque is calculate base on (f), an B is then etermine using (7). ( ) P (m ) B = (7) 2 The values of,, m an B at each spee are presente in Table II. It can be seen that the values of B etermine at ifferent spees are reasonably close to one another. The average of these values is taken as the final estimate of B. Such an estimate can also be carrie out uner loae conition, provie the loa torque is known. i R * * 0.6 sec IV. MEASUREMENT OF MOMENT OF INERTIA ( J) Base on the estimate values of frictional coefficient B, the combine moment of inertia J of the inuction motor an c generator set is estimate as explaine. The mechanical subsystem is moele as a first-orer system as shown in equation (e). The response of a first orer system is exponential uner the influence of a constant input. The moment of inertia J can be estimate by curve fitting the response of the mechanical subsystem uner constant torque conitions for the previously measure value of B. Since torque is epenent upon an, both the currents shoul be maintaine constant in orer to keep the electromagnetic torque at a constant level. is maintaine constant by keeping the flux at constant level by the flux controller. However, to make constant, the output of the spee controller (i.e. reference, i sq) shoul be force to the saturation level. For a large step change in spee reference, the spee controller hits the saturation level of for a short perio of time. In orer to ensure that i sq is maintaine at the saturation level for longer time perio, the limits on the spee controller output are reuce to a lower value than the nominal value. The rive is operate at no-loa so that the electromagnetic torque generate is equal to the sum of accelerating torque an frictional torque. Fig. 4. Experimental result corresponing to acceleration from 25Hz to 50Hz at a constant torque equal to 40% of the rate torque spee reference, spee feeback, q-axis stator current an rotor flux magnetizing current, an measure R-phase current i R. X-scale=00ms for all channels. Channels an 2 (yscale=25.7elec. ra/sec/iv); channel 3 (yscale=4a/iv); channel 4 (yscale=8a/iv) Fig.4 shows the reference spee, rotor spee an the q-axis current (inicate in figure) for the case of acceleration uner constant torque conition. The spee reference is change from 25Hz to 50Hz while keeping the spee controller output saturation level at 40% of the rate value an at the rate value. Hence, the rate torque is kept at 40% of the rate value. The is seen to remain at a constant level for a perio of more than 0.6s. The uration of constant torque is inicate in the figure. Fig.4 presents the measure R-phase current i R for the perio of constant torque operation. The rotor spee ata uring constant torque perio is capture for estimation of J. The experiment is repeate for eceleration case also. Fig.5 presents the reference spee, rotor spee an (inicate in figure) for the case of eclaration uner constant
TABLE II ESTIMATED VALUES OF FRICTIONAL COEFFICIENT = 5.94A m B m B (elec. ra/s) (A) (N-m) (kg-m 2 /s 2 ) (elec. ra/s) (A) (N-m) (kg-m 2 /s 2 ) 78.54 0.2026 0.3494 0.0089 235.62 0.4652 0.8022 0.0068 25.66 0.32 0.5535 0.0088 282.74 0.5347 0.922 0.0065 57.08 0.3654 0.630 0.0080 34.6 0.5928.022 0.0065 88.50 0.4073 0.7023 0.0075 TABLE III ESTIMATED VALUES OF MOMENT OF INERTIA Average value of B : 0.0076kg-m 2 /s 2 Operating conition Moment of inertia J (kg-m 2 ) 20% of rate torque Acceleration 0.0803 Deceleration 0.0874 30% of rate torque Acceleration 0.0858 Deceleration 0.0836 40% of rate torque Acceleration 0.0870 Deceleration 0.0823 Acceleration is from 25Hz. to 50Hz an eceleration is from 40Hz to 5Hz * i R * 0.6 sec Fig. 5. Experimental result corresponing to eceleration from 40Hz to 5Hz at a constant torque equal to 40% of the rate torque spee reference, spee feeback, q-axis stator current an rotor flux magnetizing current, an measure R-phase current i R. X-scale=00ms for all channels. Channels an 2 (yscale=25.7elec. ra/sec/iv); channel 3 (yscale=4a/iv); channel 4 (yscale=8a/iv) torque conition. The spee reference is change from 40Hz to 5Hz with the same limit on the torque. The is seen to remain at a constant level for perio of more than 0.6s. Further, the measure current (i R ) is inicate in Fig.5 along with the spee reference an rotor spee. The peak value of i R can be seen to remain constant over that uration. Spee of rotor (elec. ra/s) Spee of rotor (elec. ra/s) 300 275 250 225 200 Measure Curve fit 75 Mean square error is 0.6 elec. ra/s 50 0 0. 0.2 0.3 0.4 0.5 0.6 Time (sec) 250 225 200 75 50 Measure Curve fit 25 Mean square error is 0.6 elec. ra/s 00 0 0. 0.2 0.3 0.4 0.5 0.6 Time (sec) Fig. 6. Experimentally obtaine spee an the best-fit first-orer response of the mechanical subsystem [Eqn. (8)] : Acceleration an eceleration at a constant torque equal to 40% of the rate torque.
Consiering the time winow of 0.6s inicate in Fig.4 an Fig.5, the acceleration an eceleration occurs at a constant electromagnetic torque evelope (i.e. constant an ). These responses of spee are reprouce in Fig.6 an Fig.6, respectively. Theoretically, the spee response of the system uner such conitions woul be the solution of the equation (e), consiering an initial spee of 0 an no-loa conition. The spee response can be expresse as shown in (8) [ = 0 e B P ( Je t 2 m e B t)] Je (8) B where, 0 is the spee at the start of the time winow (i.e. t = 0); is the spee at any time t; J e is the estimate moment of inertia. All parameters in (8) are known except for the effective moment of inertia, J e. If an appropriate value of J e is chosen, then the eviation between the theoretical spee response given by (8) an the measure response woul be very low. To state more quantitatively, the value of J e shoul be so chosen to minimize the root mean square (RMS) error between the theoretical response given by (8) an the measure response. Fig.6 an Fig.6 also show the equation (4) plotte with the best-fit value of J e, which minimizes the mean square error between the experimental response an the best fit curve, corresponing to Fig.4 an Fig.5, respectively. As seen from the figures, the experimental response an the bestfit are inistinguishable. The mean square error between the experimental response an the best fit curve is foun to be 0.6elec. ra/sec for both the cases. Such best-fit value of J e is taken as the moment of inertia J of the system. The proceure is repeate with ifferent torque limits an the corresponing results are tabulate in Table III. The step change in spee reference for acceleration is kept from 25Hz to 50Hz for all the cases. Similarly, the step change in spee for eceleration case is kept from 40Hz to 5 Hz for all the cases of ifferent torque limits. The values of J obtaine in the ifferent trials (i.e. with ifferent torque limits) are reasonably close to one another. The average of these values is taken as the moment of inertia of the mechanical sub-system. REFERENCES [] J. Holtz, Sensorless control of inuction motor rives, Proceeings of the IEEE, vol. 90, no. 8, pp. 359 394, 2002. [2] W. Leonhar, Control of electrical rives. Springer, 200. [3] G. Poar an V. T. Ranganathan, Sensorless fiel-oriente control for ouble-inverter-fe woun-rotor inuction motor rive, IEEE Trans. In. Electron., vol. 5, no. 5, pp. 089 096, 2004. [4] V. S. S. Pavan Kumar Hari, Space-vector-base pulse with moulation strategies to reuce pulsating torque in inuction motor rives, Ph.D. issertation, Inian Institute of Science, Bangalore, Inia, 204. [5] J.-W. Choi, S.-C. Lee, an H.-G. Kim, Inertia ientification algorithm for high-performance spee control of electric motors, IEE Proc. Elec. Power Appl., vol. 53, no. 3, pp. 379 386, May 2006. [6] F. Anoh, Moment of inertia ientification using the time average of the prouct of torque reference input an motor position, IEEE Trans. Power Electron., vol. 22, no. 6, pp. 2534 2542, Nov 2007. [7] N.-V. Truong, Mechanical parameter estimation of motion control systems, in Proc. ICIAS, 4th 4th International Conference on Intelligent an Avance Systems, vol., June 202, pp. 00 04. [8] Y. Feng, X. Yu, an F. Han, High-orer terminal sliing-moe observer for parameter estimation of a permanent-magnet synchronous motor, IEEE Trans. In. Electron., vol. 60, no. 0, pp. 4272 4280, Oct 203. [9] R. Garrio an A. Concha, Inertia an friction estimation of a velocitycontrolle servo using position measurements, IEEE Trans. In. Electron., vol. 6, no. 9, pp. 4759 4770, Sept 204. [0] L. Niu, D. Xu, M. Yang, X. Gui, an Z. Liu, On-line inertia ientification algorithm for pi parameters optimization in spee loop, IEEE Trans. Power Electron., vol. 30, no. 2, pp. 849 859, Feb 205. [] S. Venugopal an G. Narayanan, Design of FPGA base igital platform for control of power electronics systems, in Proc. National Power Electronics Conference (NPEC), Kharagpur, Inia, 2005. V. CONCLUSIONS A metho for etermining frictional coefficient (B) an moment of inertia (J) of an inuction motor rive base on sensorless vector control is propose in this paper. The propose metho is capable of fining the combine inertia an friction coefficient of the motor an loa. This metho is base on acceleration an eceleration of an inuction motor rive uner constant torque conitions. The propose metho is utilize to etermine the values of B an J of a 5hp inuction motor, couple to a DC generator. These values of B an J can be use to refine the spee controller esign in the sensorless vector-controlle rive to achieve goo spee response.