A New On-Line Parameters Identification Method for IPMSMs Using Current Derivative Measurement

Abstract–The d– and q– axis inductances of an IPMSM, especially Lq, vary with magnitude of the current of each axis due to the magnetic saturation. The existing on-line methods using recursive algorithms fail to estimate the change of the machine inductances during zero speed and fast transient operations. This paper proposes a new on-line method to estimate machine inductances using measurements of current derivatives and the DC bus voltage of the inverter during each PWM cycle. In addition, the stator resistance and permanent magnet flux linkage, which vary with the operating temperature, are identified by using the recursive least square (RLS) technique. Extensive simulation and experimental studies were conducted to verify the robustness and effectiveness of the proposed on-line parameter identification method which estimates all four electrical parameters of the IPMSM.

Index Terms—PMSM, Machine Parameter Identification, PWM.

I.  Introduction

The accuracy of parameters identification significantly affects to the performance of the drive system. Accurate machine parameters result in accurate estimation of torque and flux in direct torque and flux control as well the model predictive control. In addition, the model-based sensorless control and the trajectory control of IPMSMs, such as MTPA, field weakening and MTPV, require accurate machine parameters, namely inductances Ld and Lq, stator resistance, Rs and permanent magnet flux linkage λf. So far, a number of off-line and on-line methods have been used for identifying these parameters [1-13].

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The off-line methods are based on locking the rotor or running the rotor at a fixed constant, applying voltage and current at certain frequencies and extracting the machine parameters from the responses [1-5]. The IEEE Standard Test [1] is the most widely used method which locks the rotor at specified angular positions firmly. Using the measured voltages and currents date, including some overloading, phase inductances Ld and Lq can be calculated. A variable-voltage and variable-frequency power supply is required for this method. The method in [2] obtains the machine impedances Zd and Zqby dividing open-circuit voltage, when the machine operated at the rated speed, by the short-circuit current. The dq inductances are obtained from the measured dq- reactances Xd, Xq, the stator resistance R and the supply frequency for the operating speed. Both of these off-line methods require robust and accurate shaft locking mechnisms or prime movers of suitable rating. For stand-still test, the rotor position must be held rigidly in the face of the developed restraining torque due to permanent magnets. For concentrated-wound IPMSMs of high power and large number of poles, this requirement places additional burden to the locking mechanism. Furthermore, considerable time and analysis of huge quantity of data are required.

Several on-line machine parameter estimation techniques are available for overcoming the abovementioned problems of the off-line methods. These on-line methods use real-time algorithms based on the dynamic model of the machine. A recusrive-least-squares (RLS) method described in [6, 7]. A current signal is injected for estimating d- and q-axes inductances, the stator resistance and the flux linkage simutaneously. An RLS algorithm based on the steady-state model of the machine and injection of current signals for obtaining the dq inductances is presented in [8, 9]. Linear model of the machine is used. The RLS method with current signal injection is also described in [10]. Discrete dynamic model of the machine in an estimated dq frame is used for estimating the dq inductances and the stator resistance. Two RLS algorithms using the machine model in the d-q reference frame have been proposed in [12]. A slow RLS is used for estimating the stator resistance R and the flux linkage λf. A faster RLS algorithm is utilized for estimating the dq inductances. For guaranteeing the convergence of the algorithm, perturbation of dq currents is used, which is equivalent to additional current injection. With the same principle, the authors in [11, 13] employ the dynamic model of the machine and utilize one slow and one fast APA schemes to identify the parameters. The injection of high-frequency currents in the machine reduces the efficiency and increases the torque ripples and audible noise. The injected signals contribute to estimation error as a result of parameter variations [11]. In addition, on-line recursive methods require considerable time, encompassing several PWM cycles, due to the recursive process, to achieve the convergence of the algorithms. Furthermore, these on-line methods fail to track the variation of the machine inductance when the machine operates at zero speed and/or when fast transient operation takes place.

  Therefore, this paper proposes a method for estimating machine inductances very fast (during every PWM cycle – a modified FPE method [16]) using current derivative (di/dt) measurements and the DC bus voltage of the inverter. Machine inductances are estimated at any operating speeds including zero speed. In addition, the stator resistance R and permanent magnet flux linkage λfare estimated by using the RLS technique. The RLS algorithm is augmented with dq inductance parameters [16] to achieve a combined estimator for all fours electrical parameters of an IPMSM. Extensive simulation and experimental studies were conducted for verifying the robustness and effectiveness of the proposed on-line parameter identification method.

II.  Proposed On-line Parameters Estimation

Estimation of machine inductances

In this part, the FPE method for inductance estimation is reviewed. An IPMSM can be described as:

  (1)

where:

   (2)

where VA, VB, VC and iA, iB, iC are phase voltages and currents of the stator winding of IPMSM; RA, RB, RC and eA, eB, eC are the stator resistances and back EMFs; LAA, LBB, LCC are three-phase self-inductances respectively; LAB, LBA, LAC, LCA, LBC, LCB are the mutual inductances between two respective phases; θe is the electrical rotor position; is the leakage inductance of stator winding; Ld, Lq are d- and q-axes inductances respectively.

As shown in [14], the incremental inductances of the machine can be calculated as in (3). This assumes that the stator resistances R and the back-ems e remain constant during a PWM cycle.

               (3)

where VDC is the DC-bus voltage of the inverter; g is a scalar quantity which is related to di/dt indicated in Table I; pα, pβ are the positional scalars in the stationary reference frame.

   (4)

where pA, pB and pC are the positional scalars shown in Table II.

Superscripts in Table I and II indicate the voltage vectors for which the di/dt is measured.

Table I

 calculation ofg

Voltage vectors

g

V1 and V2

V2 and V3

V3 and V4

V4 and V5

V5 and V6

V6 and V1

The derivation of (3) also assumes linear magnetic circuit. It is well-known incremental and apparent inductances are equal at low current level. The apparent inductances are larger than the incremental inductances at high current level due to magnetic saturation. Thus, the estimated inductances in (3) must be modified as shown in (5) in order to obtain the apparent inductances:

         (5)

where are the apparent inductances, are the incremental inductances calculated from (3), and   are the dq currents. The discrete integration in (5) is carried out for n number of values of currents from zero to the level at which the machine operates from an array. The value of n is determined from consideration of an acceptable resolution of integration and the operating current level. A resolution 0.1A for the incremental current change has been in this study.

TABLE II

 POSITION SCALARS OF IPMSM WITH STAR CONNECTION

Voltage vector

pA

pB

pC

V1, V0

V2, V0

V3, V0

V4, V0

V5, V0

V6, V0

B. Stator resistance and permanent magnet flux linkage estimations 

The dynamic model of the IPMSM in the rotor reference frame is:

  (6)   

where Vdand Vq are the stator voltage in rotor dq frame, respectively; idand iq are the stator dq currents, respectively; ωre is the rotor speed; Rsis the stator resistance; λf is the permanent magnet flux linkage all in the rotor reference frame.

The RLS algorithm for estimating Rs and λf is described by (7):

        (7) 

where y stands for the output matrix, θest stands for the estimated parameter vector,  is the feedback matrix, λ is the forgetting factor, I is the identity matrix, ε is the estimation error, and K and P(k) are correction gain matrices.

 (8)

The sampling frequency of the RLS algorithm is the same as the inverter switching frequency. The controller is also operated with this sampling frequency. The estimated inductances from the FPE method, estimated in each PWM cycle, are used to estimate stator resistance Rs and permanent magnet flux linkage λf.

III.  Experimental Setup

The experimental setup is shown in Fig. 1. The controllers and parameter esimators are implemented on dSPACE DS1103 controller board. The IPMSM used for testing is from Kollmorgen (BE2-402-A-A4) and a PMDC generator is coupled to the shaft for dynamic loading. All nominal parameters of the test machine are included in Table III.

Fig.1.  Experimental setup.

Table III

Parameters of IPMSMs tested

Number of poles    Pp

4

Stator resistance       R

5.8 

Magnet flux linkage    f

0.533 Wb

d-axis inductance         Ld

44.8 mH

q-axis inductance         Lq

102.7 mH

Phase voltage (rms)      V

230 V

Phase current (rms)      I

3 A

Rated torque            Te

6 Nm

Rated speed (rpm)        nrated    

1500 rpm

IV.  Simulation and Experimental Results

A. Simulation results

Full system simulation of the drive system was carried out on Matlab-Simulink platform initially, in order to validate the proposed method. Simulation results were compared with experimental results from the drive set-up. The electrical parameters of the tested IPMSM were first obtained from an off-line by using the AC standstill methods [1, 16]. The running of test machine was then simulated under FOC algorithm under various loads in Matlab-Simulink. For each load current level, corresponding Ldand Lq of the IPMSM model are used according to these values measured off-line. The dq inductances are then estimated using the proposed on-line method using the measured di/dt and the DC bus voltage.

Fig. 2 shows the performance of the proposed method when running the machine at 500 rpm and 1000 rpm under full load condition. The first trace shows the operating speed, while the remaining traces show the estimation of all four parameters of the IPMSM. It can be seen in the second and third traces of Fig. 2 that the estimated inductances (Lq est, Ld est) closely match the Lq and Ld of the IPMSM model. The estimation errors for the inductance estimation are smaller than 1 mH. The fourth and fifth traces of Fig. 2 show the estimation of stator resistance (Rs est) and permanent magnet flux linkage (Fluxm est), respectively. It can be seen that the estimation errors of stator resistance and permanent magnet flux linkage are very small, within 1 Ω and 0.005 wb, respectively.

Fig. 2.  Parameters estimation of the proposed on-line methods (simulation).

The positional scalars and di/dt at zero voltage vector, the first active voltage vector and the second active voltage vector when the machine operates at 300 rpm are included in Fig. 3. The di/dts are used to calculate the position scalars, Palpha and Pbetaand quantity g. A comparison of the proposed method and the RLS method in [12] is presented in Fig. 4. At time 0.1s, two estimation schemes are initialized. For the method in [8], it takes about 0.4s to converge the estimated stator resistance (Rs rls) and the permanent magnet flux linkage (λf rls) to the reference values Rs and λf, respectively. In addition, it takes about 0.1s to converge the estimated inductances (Ld rls and Lq rls) to the values set in the IPMSM model. In contrast, for the proposed method, it takes only one PWM cycle to obtain the machine inductances and 0.01s to converge the estimated stator resistance (Rs proposed) and estimated permanent magnet flux linkage (λf  proposed).

Fig. 3.  Di/dt and the positional scalars used to estimate machine inductances by the proposed method (simulation).

Fig. 4.  Comparison of the proposed and a purely RLS based- on-line methods (simulation).

Performance of the proposed method and the methods based on RLS when the machine accelerates from 100 rpm to 1500 rpm under half load condition are presented in Fig. 5 and 6. The first two traces of Fig. 5 show the torque and the RMS current of the machine, respectively. The last two traces of Fig. 5 show the speed transient and the positional scalars used to estimate the machine inductances of the proposed method. From 0.1s, the machine is accelerated when the RMS current suddenly quickly rises to 2.8A. After the transient, the RMS current reduces to the original value at 1.3A. It is clear from Fig. 6 that the proposed method results in fast and correct estimation of the stator dq inductances, resistance Rs and permanent magnet flux linkage λf during both transient and steady state. In contrast, the purely RLS method shows high sensitivity of dq inductance estimationsto the accuracy of the stator resistance and permanent magnet flux linkage during transient operation.

Fig. 5.  Dynamics of the machine when accelerated from 100 rpm to 1500 rpm under half load condition (simulation).

Fig. 6.  Comparison parameter values with the proposed- and RLS based- on-line methods during the speed accelaration from 100 rpm to 1500 rpm (simulation).

Performances of the proposed method and the RLS based method when the stator resistance and permanent magnet flux linkage change during the steady-state operation at 500 rpm are compared in Fig. 7. From 0.18s, the reference stator resistance and permanent magnet flux linkage suddenly change from 5.8  to 6 and from 0.533Wb to 0.55Wb, respectively. For the RLS based method, it takes about 0.8s to converge all four estimated parameters. It is noted that estimated d-axis inductance suffers a largest error during the convergence period. In contrast, the proposed method results in very fast tracking of machine inductances due to the above changes and it takes about 0.01s for resistance and permanent magnet flux linkage to converge.

Fig. 7.  Comparison of the proposed- and RLS based- on-line methods due to abrupt variations of Rs and λf(simulation).

Performance of the proposed method during zero speed under no load condition is presented in Fig. 8. It is clear that the estimated dq inductances match closely with the reference dq inductances Ld and Lq, respectively, in the machine model. The estimated stator resistance, Rs proposed and permanent magnet flux linkage, Fluxm proposed also match with their reference values, which are 5.8 and 0.533 Wb, respectively.

B. Experimental results

Extensive experimental results were obtained in parallel using the experimental set-up described in III. These are presented in Fig. 9 and Fig. 10. These two figures compare between the estimated dq inductances from the proposed method and these inductances measured off-line, when the IPMSM operated at 300 rpm and  900 rpm, respectively. During the experiment, the load is adjusted abruptly, so that the RMS current changes from 0.75A to 3A (machine rated current). It is clear that the on-line estimated inductances closely match the off-line measured ones. At 300 rpm, the RMSE (root mean squared error) between the on-line estimated Ld (Ld on-line) and off-line measured Ld (Ld off-line) is 2.22 mH, while the RMSE between the on-line estimated Lq (Lq on-line) and off-line measured Lq (Lq off-line) is 3.83 mH. These errors are small compared to the nominal values of 102.7mH and 44.2mH for Lq and Ld, respectively.

Fig. 8.  Performance of the proposed on-line method at zero speed under no load condition (simulation).

Fig. 9.  Estimated inductances at 300 rpm under different current levels (experiment).

Fig. 10.  Estimated inductances at 900 rpm under different current levels (experiment).

Fig. 11 presents the estimated inductances when the machine accelerates from zero to 900 rpm. When the machine starts accelerating from 1.05s, when the stator RMS current abruptly rises from 0.9A to 2.6 A. The estimated Lq abruptly drops from about 138 mH to 110mH. During this period, the estimated Ld increases slightly by about 1 mH from 45 mH. After the acceleration is over, the RMS stator current falls to 1A, and the estimated Lq returns to about 137 mH. The estimated Ld remains almost unchanged after the torque transient, as is also expected. Clearly, the proposed on-line method tracks the variation of inductances with variation of stator RMS current during the transient state close to the inductances found from off-line tests. The tracking errors expressed in RMSE for the on-line estimated Ld (Ld on-line) and the off-line measured Ld (Ld off-line) is 1.59 mH, while this quantity for the on-line estimated Lq (Lq on-line) and off-line mesured Lq (Lq off-line) is 1.99 mH.

Results presented in Fig. 11 have also shown that the proposed method estimates the dq inductances very fast (within a PWM period) and that the inductances can be estimated at zero speed and during transients.

Fig. 11.  Estimated inductances versus RMS current during the acceleration from zero to 900 rpm (experiment).

V.  Conclusion

A new on-line method for estimation of all four parameters of an IPMSM is presented in this paper. The d- and q- axes inductances are estimated using measured current derivatives and the inverter DC-bus voltage during the application of the voltage vectors in a PWM cycle. The updated dq inductance values are used by a RLS estimator for estimation of stator resistance and rotor flux linkage. Consequently, the RLS in the proposed method converges faster than conventional RLS methods which also estimate dq inductances recursively. Extensive simulation and experimental results have shown that estimated inductances by the proposed technique are tracked very fast. Tracking errors are also small during both steady-state and highly dynamic conditions of operation. The sensitivity of inductance estimations to resistance and flux linkage estimation has also been presented.

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