Slip-Gain Estimation in Field-Orientation-Controlled ... - IEEE Xplore

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IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 42, NO. 3, MAY/JUNE 2006

Slip-Gain Estimation in Field-Orientation-Controlled Induction Machines Using the System Transient Response Michael W. Degner, Senior Member, IEEE, Juan Manuel Guerrero, Member, IEEE, and Fernando Briz, Member, IEEE

Abstract—Field-oriented control of induction machines is well established and widely used in applications where high-bandwidth torque control of ac machines is needed. A key parameter in the implementation of indirect field orientation is the slip gain (inverse of the rotor time constant), which is used to estimate the rotor flux angle. A new method for estimating the slip gain is presented in this paper. The method uses the transient response of the stator voltages following a change torque for these purposes. The method is independent of any machine or system parameter and can be easily integrated into standard drives and used in many industrial applications. Index Terms—Indirect field-oriented control, rotor time constant estimation, slip gain estimation.

I. I NTRODUCTION

normal operation of the electric machine [4], [5] and often require special test fixturing or measurement equipment [6], [7]. This paper presents a new method for obtaining an estimate of the slip gain. The method uses the transient response of the system, and in particular the stator voltages, after a change in the torque command [8], [9]. Incorrect slip gains cause relatively slow transients in the stator voltages. These transients contain reliable information on the correctness of the slip gain estimate as well as on how it should be adapted for an improved estimate. The method developed in this paper provides estimates of the slip gain independent of any machine or system parameters. Furthermore, it does not require direct measurement of the torque produced by the machine and only uses the commanded or measured stator voltages.

F

IELD-ORIENTED control of induction machines is well established and widely used in applications where highbandwidth torque control of ac machines is needed. A number of methods to achieve field orientation control have been proposed. One of the most simple and widely used methods is indirect field orientation (IFO). A key parameter in the implementation of the IFO method is the slip gain, which is used to estimate the rotor flux angle. This gain depends on the rotor resistance and is, therefore, a strong function of temperature. Incorrect slip gain values result in an incorrect estimation of the rotor flux position, causing deterioration in the torque and flux control. Because of this, a number of methods for estimating the slip gain have been developed [1]–[9] and include both online and offline estimations. Many of these rely on models of the induction machine and are sensitive to changes in the parameters [2], [3]. Other methods cannot be performed during

Paper IPCSD-06-019, presented at the 2005 Industry Applications Society Annual Meeting, Hong Kong, October 2–6, and approved for publication in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Industrial Drives Committee of the IEEE Industry Applications Society. Manuscript submitted for review October 15, 2005 and released for publication February 23, 2006. This work was supported in part by the Research, Technological Development and Innovation Programs of the Principado of Asturias-ERDF under Grant PB02-055 and of the Spanish Ministry of Science and Education-ERDF under Grant MEC-04-DPI2004-00527. M. W. Degner is with Sustainable Mobility Technologies and Hybrid Vehicles, Ford Motor Company, Dearborn, MI 48121-2053 USA (e-mail: [email protected]). J. M. Guerrero and F. Briz are with the Department of Electrical, Computer and Systems Engineering, University of Oviedo, E-33204 Gijón, Spain (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TIA.2006.873670

II. I NDUCTION M ACHINE M ODELING In a reference frame aligned with the rotor flux, the differential equations modeling the behavior of an induction machine can be written as
 1 Lm rf rf rf rf  ωbr λdr v −(Rs + jωe Lσs ) iqds + (1) piqds = Lσs qds Lr  Rr  rf Lm irf (2) pλrf dr = ds − λdr Lr where Lσs = Ls − (L2m /Lr ), Rs = Rs + (Lm /Lr )2 rr , and ωbr = (Rr /Lr ) − j ωr . Since the rotor flux is aligned with the d-axis in this reference frame, the rotor flux linkage (2) can be used to solve for the relationship between the rotor flux and the stator d-axis current, i.e., λrf dr =

Lm rf i , τr p + 1 ds

τr =

Lr (rotor time constant). Rr

(3)

The torque produced by the machine can be calculated as T =

3 P L2m rf rf rf i i = KT irf qs ids 2 2 Lr qs ds

(4)

assuming the rotor flux is held constant. In (4), a substitution was made for the parameters that scale the product of the q-axis and d-axis currents. This term is often called the torque gain KT .

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DEGNER et al.: SLIP-GAIN ESTIMATION IN FIELD-ORIENTATION-CONTROLLED INDUCTION MACHINES

Fig. 1.

703

Block diagram of an IFO-controlled induction machine.

Equations (3) and (4) show one of the key advantages of the rotor-flux-oriented control, which is its ability to independently control both the torque and the rotor flux in the machine, using the d-axis stator current to control the flux and the q-axis stator current to control the torque. Fig. 1 shows the block diagram for an IFO controller. The relationship between slip frequency, q-axis stator current, and rotor flux is shown in (5), which can be further simplified to the form (6) when the rotor flux is held constant sωe = ωe − ωr =

sωe =

1 Lm irf qs τr λrf dr

irf 1 irf qs qs = Kslip rf . rf τr ids ids

(5)

(6)

III. I NDIRECT F IELD -O RIENTED C ONTROL AND E FFECT OF S LIP -G AIN M ISTUNING ON THE S YSTEM T RANSIENT R ESPONSE The relationships shown in (5) and (6) form the basis for IFO. In IFO, the slip frequency of a machine is controlled so ∗ that it satisfies (assuming constant irf ds ) ˆ slip sωe∗ = K

∗ irf qs ∗ irf ds

= sωe

(7)

where ∗ denotes a commanded value and ˆ denotes an estimated value. If the parameter estimates are equal to the actual parameter values, then (7) is a necessary and sufficient condition for obtaining the field-oriented torque control of an induction machine. When a machine is controlled using a moderate to high bandwidth current regulator, the dynamics due to a change in the stator current irf qds happen much faster than the dynamics resulting from a change in the rotor flux λrf dr . Typical values for the current regulator’s bandwidth for small machines are in the range of several hundred hertz, while the rotor flux dynamics are in the range of a few hertz. Because of this, the transient response due to a change in the stator current is basically completed before the transient response due to the rotor flux change begins. Under these conditions, the modeling of the

transient response due to a rotor flux change can be simplified by assuming that the stator current is a constant value pirf qds = 0.

(8)

Applying this condition to (1), the stator voltage equation can be written as rf = (Rs + jωe Lσs ) irf vqds qds −

Lm ωbr λrf dr . Lr

(9)

Using this equation, two cases need to be considered. The first case is a machine controlled with the controller correctly estimating the location of the rotor flux. In this case, a step change in torque command, assuming a constant rotor flux, will not cause a significant transient in the stator voltage other than the fast response due to the stator current change. This is illustrated in the simulation results shown in Fig. 2 (left column). The high-bandwidth current regulator forces the q- and d-axis currents to their commanded value within several milliseconds (not appreciated in the time scale of the figure). The stator voltages do not exhibit any further transient behavior. The second case occurs when the field-oriented controller is mistuned. In this case, the rotor flux will not stay constant during a step change in the torque command. Instead, the rotor flux follows the dynamics dictated by (2), which then couple into the stator voltage (9). This is illustrated in the simulation results shown in Fig. 2 (right column). From that figure, it can be observed that while the controlled q- and d-axes currents, and consequently the estimated rotor flux, are regulated correctly, significant low-frequency transients are present in the actual rotor flux and stator currents. The effects caused by the cross-coupling between the q- and d-axes due to errors in the slip gain estimate can be evaluated by plotting contours of constant rotor flux, stator current, slip frequency, and torque in the complex current vector plane, as shown in Fig. 3. From Fig. 3, it can be seen that, when the slip gain estimate is too high, an increase in the torque command (q-axis stator current) causes the rotor flux level to decrease. This occurs even though the rotor flux command remains constant. The opposite condition happens when the torque command is decreased, with the slip gain estimate too high, i.e., the rotor flux level increases. Conversely, when the slip gain estimate is too low, an increase in the torque command

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IV. E STIMATION OF THE S LIP G AIN U SING THE T RANSIENT V OLTAGES A. Transient Behavior of the Stator Voltage Insight into the transient behavior of the stator voltage can be obtained by rewriting the stator voltage (9) in scalar form rf rf vqs = Rs irf qs + ωe Lσs ids +

Lm ωr λrf dr Lr

(10a)

rf rf = Rs irf vds ds − ωe Lσs iqs −

Lm Rr rf λ . Lr Lr dr

(10b)

Of the terms that make up the right-hand side of (10a) and (10b), the back-EMF term rf rf ≈ vqds vqs bEMF =

Lm ωr λrf dr Lr

(11)

is the dominant term at medium-to-high rotor speeds. The angular error between the actual rotor flux and the estimated rotor flux can be calculated as  ∆θrf = θrf

Fig. 2. Simulation results showing (a) and (b) controlled and actual d- and q-axis stator currents; (c) and (d) controlled and actual d- and q-axis rotor flux; and (e) and (f) d- and q-axis stator voltage for two different slip gain estimates ˆ slip = Kslip (left column) and K ˆ slip = 0.8Kslip (right column). All of K variables are shown in the estimated rotor flux reference frame, which is . indicated by the superscript rf

− θˆrf = atan

irf qs irf ds

−atan

irf qs

∗

 (12a)

∗

irf ds 

Using (7), the actual stator currents in this relationship can be replaced with the commanded stator currents and the ratio of the estimated to actual slip gain. Using (7) and the fact that the magnitude of the stator current vector remains constant in all reference frames, we have

λrf dr

rf ∗

λdr

Lm = ˆm L




irf qs

∗

 12

2

+1 ∗  irf  ds 
∗ 2  Kˆ slip 2 irf qs Kslip

causes the rotor flux level to increase. The opposite again occurring when the torque command is decreased with the slipgain estimate too low.



   ∗ ∗ ˆ slip irf irf K qs qs ˆ ∆θrf = θrf − θrf = atan −atan rf ∗ . (12b) ∗ Kslip irf ids ds



Fig. 3. Complex vector current plane showing constant stator current trajectories (circles about the origin), constant torque trajectories (hyperbolas), constant slip frequency trajectories (lines emanating from the origin), and constant rotor flux trajectories (horizontal line).



irf ds

∗

+1

   . 

(13)

Mistuning of the slip gain has two different effects on the back EMF voltage (11) during a change in torque command (q-axis current command). First, the controller’s reference frame is not aligned with the actual rotor flux, and the amount of misalignment changes as a function of the q-axis current (12b). This effect is shown in Fig. 4(a). Second, misalignment of the controller’s reference frame from the actual rotor flux causes the actual d-axis current to change with respect to its commanded value. This causes the rotor flux level (13) to change when the torque command is changed, as shown in Fig. 4(b). These changes in the orientation and magnitude of the rotor

DEGNER et al.: SLIP-GAIN ESTIMATION IN FIELD-ORIENTATION-CONTROLLED INDUCTION MACHINES

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Fig. 4. Numerically obtained (a) rotor flux angle error and (b) rotor flux magnitude error as a function of the normalized slip gain error and the ratio of q- to d-axis current command. Fig. 6. Experimentally measured (a) q-axis current, (b) q-axis, and (c) d-axis commanded stator voltages after a step torque command in the estimated rotor flux reference frame using three different slip gain estimates. The motor was operated at ωr = 33 Hz. TABLE I INDUCTION MOTOR PARAMETERS

plaining the behavior during the generating operation. General expressions for ∆vq (15) and ∆vd (16) can be obtained using  rf j∆θrf vqds bEMF = (VbEMF 0 + ∆VbEMF )e Fig. 5. Schematic representation of the q- and d-axes stator current components, in the estimated and actual rotor flux (d-axis) reference frame, and resulting back-EMF voltage (only motoring operation is shown).

where ∆VbEMF =

flux cause the back EMF voltage orientation and magnitude to also change. This is schematically shown in Fig. 5. The fact that the back-EMF voltage varies when the slip gain is mistuned and a change in torque command takes place can be used to detect the mistuning. The variation in the back-EMF voltage will follow the relative slow dynamics of the rotor flux (3). When a change in torque is commanded, the current regulators, usually reaching steady state in less than a few milliseconds, will govern the stator currents’ dynamics. This is much faster than the rotor flux dynamics. Following this reasoning, the change in the back-EMF voltage, labeled as ∆vd and ∆vq in Figs. 2 and 5, between the instant when the stator current reaches steady state (t = t1 in Fig. 2) and the instant when the rotor flux reaches steady state (t = t2 in Fig. 2) contains information on the correctness of the slip gain estimate. From Fig. 5, it can be observed that the signs of the ∆vd and ∆vq voltages depend on whether the estimated slip gain is larger or smaller than the actual slip gain. The magnitudes of ∆vd and ∆vq indicate the degree of mistuning and their combined sign the direction in which the slip gain needs to be changed. The signs of ∆vq and ∆vd also depend on whether the machine is motoring or generating. Note that Fig. 5 is for motoring operation, and a similar figure can be developed ex-

(14)

 ∗ Lm  rf ωr λdr − λrf dr Lr

∆vq = (VbEMF 0 + ∆VbEMF ) cos(∆θrf )

(15)

∆vd = − (VbEMF 0 + ∆VbEMF ) sin(∆θrf )

(16)

where ∆VbEMF and ∆θrf represent the effect of errors on the magnitude and phase angle of the rotor flux on the back EMF. From (15), it can be seen that ∆vq is more sensitive to changes in the rotor flux magnitude than to changes in the rotor flux orientation, i.e., cos(∆θrf ) does not change significantly over the range of ∆θrf typically seen. In contrast, ∆vd is more sensitive to changes in the rotor flux orientation than it is to changes in the rotor flux magnitude, i.e., sin(∆θrf ) changes significantly. Since ∆vq is more sensitive to changes in rotor flux magnitude, it also is more sensitive to the effects of saturation. This is especially true when mistuning of the controller pushes the rotor flux to higher than rated flux levels. Fig. 6 shows the q-axis current and the q- and d-axes voltages when a torque step is commanded for three different slip gains. The parameters of the test machine are shown in Table I. The slow transients, more readily appreciated in the d-axis voltage, ˆ slip = 6.7 rad/s indicate ˆ slip = 5.4 rad/s and K for the case of K mistuning of the slip gain. Such slow transients do not exist for ˆ slip = 6.3 rad/s. the correctly tuned slip gain of K

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Fig. 7. (a) Symmetric square windowing function and (b) d-axis stator voltage command after a step torque command (Fig. 6) weighted by the windowing function for three different values of the slip gain. The motor was operated at ωr = 33 Hz.

B. Processing of the Stator Transients Voltages The information contained in the slow transients of the stator voltages can be processed in several ways to assess the correctness of the estimated slip gain. One option is to measure the variation of ∆vq and ∆vd , i.e., the q- and d-axes voltages between the instant when the current has reached its steadystate value (t1 in Fig. 2) immediately following a step in commanded torque and the instant when the low-frequency transient in the voltage caused by mistuning of the slip gain has decayed away (t2 ). While the direct measurement of ∆vq and ∆vd is possible, it is not a simple solution in actual practice. The main reason is that q- and d-axis voltages typically have significant amounts of noise content. This makes accurate measurement of the change in voltage difficult when using a limited number of samples. The effect of noise in using a limited number of samples can be overcome by sampling and using the q- and d-axis voltages during the complete transient period. The increased number of samples can be processed to develop a metric for the change in voltage during the slow transient using windowing functions or wavelets. An example windowing function is shown in Fig. 7(a), which is a simple square-wave window. Many other forms of windowing functions can also be used. One key feature of a desirable windowing function is that it has a mean value equal to zero. This automatically removes the dc component of the stator voltage, leaving mainly the desired changed in voltage during the slow transient. Fig. 7(b) shows the result of multiplying the d-axis stator voltage commands shown in Fig. 6 by the windowing function in Fig. 7(a). For ˆ slip = 6.7 rad/s, the areas ˆ slip = 5.4 rad/s and K the cases of K above and below the x-axis are different, while for the case of ˆ slip = 6.3 rad/s both areas are nearly equal. This difference K between the positive and negative areas of the windowed stator T voltage waveform is used to calculate what are termed the Vqs T and Vds voltages, defined as N 1  rf W[k] vqs [k] N k=1 N 1  rf = W[k] vds [k] N

T Vqs = T Vds

k=1

(17) (18)

T , V T , and V T voltage versus estimated slip gain for different Fig. 8. Vds qs qds torque transients, with decoupling of the dc bus voltage variations. The motor was operated at ωr = 33 Hz. (a) Motoring operation, torque increasing. (b) Motoring operation, torque decreasing. (c) Generating operation, torque decreasing. (d) Generating operation, torque increasing.

where W (k) is the windowing function, N is the number of rf rf samples collected, and vqs and vds are either commanded or measured stator voltages in the estimated rotor flux reference T T and Vds can be viewed as an “average” of the tranframe. Vqs sient voltages and form the basis for detection of mistuning in the slip gain estimate. Section VI describes how the magnitude T T and Vds are used to update the slip gain and polarity of Vqs estimate. Section V discusses the quality of the information T T and Vds as a function of the number of contained in Vqs samples N (or amount of time), the selection of the windowing function W , and other relevant aspects of their calculation. T and Fig. 8 shows the resulting experimentally measured Vqs T Vds , calculated using (17) and (18), with the drive tuned to different arbitrary slip gain estimates near the correct value. T T and Vds are functions of From Fig. 8, it can also be seen that Vqs the slip gain estimate and the slip gain estimate error. When the slip gain estimate equals the actual slip gain, these factors equal zero. It can also be seen in Fig. 8 that the resulting functions are dependent on whether the drive is operated in motoring or generating mode and if the torque command is increasing or decreasing. This confirms the behavior described in Section IV-A that each of these modes affects the rotor flux level in a different way when the slip gain is mistuned. T T and Vds voltages shown in Fig. 8 are Although the Vqs definite and deterministic functions of the slip gain estimate and the slip gain estimate error, they do not exhibit, when used alone, a behavior that lend themselves to be easily used for adapting the slip gain estimate directly. The desired characteristics for such an error signal are that it has no inflection

DEGNER et al.: SLIP-GAIN ESTIMATION IN FIELD-ORIENTATION-CONTROLLED INDUCTION MACHINES

Fig. 9.

707

T for different torque transients. Calculation of Vqds

and preferably a negative slope when plotted as a function of T the slip gain estimate. This can be achieved by combining Vqs T T and Vds into a single value, termed Vqds . The definition of T depends on the sign of the torque command T ∗ [k] and Vqds on whether the torque command is increasing or decreasing. T is calculated for each case, Fig. 9 schematically shows how Vqds with Fig. 10 being the corresponding flowchart of the process. T is also plotted in Fig. 8 (right column). For each of the Vqds conditions shown, it can be seen to behave similarly and in all the cases has a negative slope and no inflections. Calculation of T according to the flowchart in Fig. 10 is not a continuous Vqds process; it is initiated every time the torque command changes. The resulting value (one per torque command change) is then used as an error signal to adapt the slip gain estimate, as will be described in Section VI.

T . Fig. 10. Flowchart of the procedure to calculate Vqds

V. S ENSITIVITY A NALYSIS A. Rotor Speed From (11) and (14), it can be observed that the q- and d-axis stator voltages in the estimated rotor flux reference frame are proportional to the rotor speed, which can affect the rate of convergence of the method if not addressed. Fig. 11(a) shows T T T , Vds , and Vqds voltages for three different the measured Vqs speeds as a function of the slip gain estimate. Fig. 11(b) shows the same data normalized by multiplication with ωr_rated /ωr , which compensates for their speed dependence. As can be seen by comparing the two figures, a measurement independent of the rotor speed is obtained. For low-speed operation, below ∼ 10% rated speed, the terms in (10a) and (10b) that do not depend on the rotor speed become significant. This, in combination with the reduced magnitudes of the overall signals, makes it more difficult to compensate for the rotor speed effects and, in general, to utilize the proposed method. B. Torque Step Magnitude The magnitude of the torque step also influences the transient voltages. When mistuned, steps in the commanded q-axis current produce variations in the actual d-axis current and, consequently, variations in the rotor flux level. Such variations can affect the degree of saturation of the machine in a nonlinear T for fashion. Fig. 12 shows the measured transient voltage Vqds different torque step magnitudes as a function of the estimated

T , V T , and V T voltages versus estimated slip gain for rotor Fig. 11. Vds qs qds speeds of 16 Hz (◦), 33 Hz (
), and 50 Hz (), respectively, (a) not normalized and (b) normalized by the speed (ωr_rated/ /ωr ). The transient voltages were measured for motoring operation, with torque increasing.

T slip gain. From that figure, it can be observed that the Vqds voltage correctly reflects mistuning of the estimated slip gain T voltage increases almost proand that the slope of the Vqds portionally (linearly) with the torque step magnitude. However, T voltage slope and the the linear relationship between the Vqds slip gain estimate is lost for torque steps above approximately 75% of rated torque. An explanation for this change in behavior is that higher torque levels are more likely to push the rotor flux beyond its rated value, resulting in increased levels of saturation. Since in most applications it would not be practical to use torque steps of this size for tuning purposes, no compensation for the torque step magnitude was implemented as part of the method. Even so, this does not prevent the method from performing correctly for all levels of torque steps, as will be shown in Section VI.

C. DC Bus Voltage Variations Both commanded or measured voltages can be used to implement the proposed method. Using commanded voltages

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T voltage versus estimated slip gain for torque step magnitudes of Fig. 12. Vqds 10% ( ), 25% (), 50% (), 75% (◦), and 100% (
). The transient voltages were measured with ωr = 33 Hz, motoring operation, and torque increasing.

T as a function of the estimated slip gain (radians per second) Fig. 14. Vqds window length (second) for different torque transients. A symmetric square window (duty 0.5) was used. Only results with the machine in motoring operation are shown; similar results were obtained for generating operation. (a) Motoring operation, torque increasing. (b) Motoring operation, torque decreasing.

T as a function of the estimated slip gain (radians per second) Fig. 15. Vqds window duty cycle for different torque transients. A window length of 0.45 s was used. Only results with the machine in generating operation are shown; similar results were obtained for motoring operation. (a) Generating operation, torque decreasing. (b) Generating operation, torque increasing.

T , V T , and V T voltages versus estimated slip gain for different Fig. 13. Vds qs qds torque transients without decoupling of the dc bus voltage variations. The motor was operated at ωr = 33 Hz. (a) Motoring operation, torque increasing. (b) Motoring operation, torque decreasing. (c) Generating operation, torque decreasing. (d) Generating operation, torque increasing.

is considered a more appealing option since phase voltages are not measured in most industrial drives. Errors between commanded and actual voltages applied to the machine can cause a deterioration of the method. Several effects can cause errors between the commanded and actual voltages, including dead time in the inverter, variations in the dc bus voltage, and voltage drops in the power switches. Errors caused by the dead time of the inverter and voltage drops in the power devices were observed to have minimal impact on the method. Not surprisingly, variations in the dc bus voltage were observed to have a significant effect on the results when they were not decoupled from the commanded voltage. Decoupling eliminates

this interaction and was used for all the experiments presented in this paper, except those described in this section. T T T , Vds , and Vqds voltages when variThe behavior of the Vqs ations in the dc voltage are not compensated can be seen in Fig. 13. An unregulated rectifier was used to feed the inverter and a braking resistor was used to limit the dc bus voltage during generating operation. Noticeable variations in the behavior T T , and consequently of Vqds , are observed for the case of of Vqs generating operation. These are caused by the relatively large increase in the dc-bus voltage before the braking resistor acts and by the relatively fast variations of the dc bus voltage when the braking resistor is switched on and off. Both of these result in large errors between the commanded and the actual voltages. It can be concluded that usage of the method when the dc bus voltage is not decoupled requires special attention to the effects of dc bus voltage variations. It is finally noted from Fig. 13 that T T seems to show better behavior than Vqds . This suggests that Vds T Vds could be used by itself for updating the slip-gain estimate. D. Selection of Windowing Function All the results presented so far in this paper have used a constant-length symmetric square-wave windowing function [see Fig. 7(a)]. A length for the windowing time in the range of three rotor time constants was chosen since this should guarantee enough time for the rotor flux to approximately reach its steady-state value.

DEGNER et al.: SLIP-GAIN ESTIMATION IN FIELD-ORIENTATION-CONTROLLED INDUCTION MACHINES

T versus estimated slip gain for the case of (◦) the machine at Fig. 16. Vqds start up (cold) and (
) the machine after continued loaded operation (hot).

To verify the robustness of the method with respect to the windowing function, a series of experiments was made by varying both the length and the duty cycle (relationship between the positive and the negative portions) of the windowing function. The only restriction was that the mean value of the windowing function was equal to zero in all the cases. Fig. 14 shows the T voltage for different torque transients as a function of Vqds the window length; a symmetric (0.5 duty cycle) was used in T can be seen all cases. The effect of the duty cycle on Vqds in Fig. 15. A window length of 0.45 s was used for all the experiments. It can be observed from Figs. 14 and 15 that the length and duty cycle of the windowing function have a moderate T , but do not affect its convergence. influence on the slope of Vqds The conclusion from this investigation is that the length and duty cycle of the windowing function can be chosen within a relatively wide range of values without having noticeable effects on the performance of the method. Finally, it should be noted that other windowing functions (wavelets) besides a square wave can be used. Windowing functions tested during this paper included sinusoidal functions and negative exponential functions, among others. Results obtained with these functions are not included due to limited space, but similar results to those obtained using a square wave function were achieved. E. Temperature Variations in the machine’s temperature cause variations in the rotor resistance, resulting in variations of the slip gain. T as a function of the estimated slip gain Fig. 16 shows Vqds for two different operating temperatures. The increase in temT voltage, with perature can be seen to cause a shift in the Vqds the zero crossing point moved to the right with increased temperature. This correlates well with the fact that the actual slip gain would increase with increasing temperature. In addition, T , i.e., a negative slope the overall desired characteristics of Vqds and a linear relationship with the slip gain estimate, exist in both cases. F. Compensation for Rotor-Speed Variations All of the theoretical discussions presented in this paper have assumed a constant rotor speed during the data collection process. Although such an assumption is valid in some applications, it is not valid or desirable in all applications, especially in systems with low inertia or processes that require continuous

709

Fig. 17. PI estimator controller used to calculate the updated slip gain estimate.

variations in speed. Variations in the rotor speed cause variations in the back EMF, which have similar dynamics to those caused by an incorrect slip gain estimate. To compensate for speed variations, the estimated back EMF was subtracted from the q-axis voltage ∗

∗

rf rf [k] = vqs [k] − λrf vqs ds ωr [k]

(19)

T in (17). which was used to calculate Vqs This simple compensation method was found to be effective for moderate-speed variations, as is the case when small torque steps are commanded or when large mechanical inertias are driven. Large variations in the speed after a torque command result in large changes in the back EMF, which the proposed compensation method did not address in a satisfactory manner. Suitable methods to compensate for the effects of large speed variations are currently under research.

VI. A DAPTATION OF THE S LIP -G AIN E STIMATE T The Vqds voltage can be processed in many ways to correct the slip gain estimate. A proportional plus integral (PI) controller (as shown in Fig. 17) T ˆ slip [k] = K ˆ slip [k − 1] + (Ki + Kp )V T K qds[k] − Ki Vqds[k−1] (20)

is one method. An important aspect of this controller is that it only operates and produces an updated slip gain estimate once for each change in the torque command level or updated T calculation of the Vqds voltage. Because of this, the sample rate of the system shown in Fig. 17 is determined by changes in the torque command and is much slower, and asynchronous, with respect to other control functions. An estimate of the slip gain ˆ slip = K ˆ slip _0 is provided for initialization of (20) at start up. K An alternative form of the controller is ˆ slip [k] = K ˆ slip [k − 1] + KV T . K qds[k]

(21)

If the gain K is chosen to be approximately equal to the slope T voltage, then (21) will approximate the behavior of a of the Vqds T voltage in Fig. 8, deadbeat controller. From the slope of the Vqds it can be observed that a deadbeat behavior would be obtained for K ≈ 0.3 rad/s/V. In actual practice, the selection of K to achieve this behavior results in an increased sensitivity to noise and possibly oscillations or limit cycle behavior. Tuning it for slightly lower values alleviates these issues. The online adaptation of the slip gain using the PI controller in Fig. 17 is shown in Fig. 18. From the figure, it can be

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conditions of the machine. In all the cases, the slip gain is seen to converge to the same value. It is important to note that the adaptation of the slip gain estimate is not a continuous process and does not require a special torque signal. The torque signal (i.e., q-axis current) shown in Fig. 18, and used for the results shown in Figs. 18 and 19, is intended for illustrative purposes only and is not a requirement of the method. In an actual application, the adaptation could be performed in one of two ways: either using the changes in the torque command that occur during normal operation of the electric machine or using a sequence of torque command changes applied specifically for calibration purposes. The former has the benefit of allowing continuous adaptation of the slip gain estimate but relies on periodic torque command changes to occur. The later has the benefit of using a specific test sequence but requires the normal operation of the drive to be interrupted. VII. C ONCLUSION

Fig. 18. Slip-gain adaptation using a PI controller (20). The motor was operated at ωr = 33 Hz; steps of 50% of rated torque were commanded. The PI controller was tuned with Kp = 0.05 and Ki = 0.15.

Slip-gain estimation is a key issue in the implementation of IFO control for induction machines, as this parameter ultimately determines the control performance. Slow transients in the stator voltages have been shown to contain reliable information on the accuracy of the estimated slip gain. A method that uses these stator voltage transients to determine whether the actual slip gain estimate is correct and how it needs to be adapted was introduced. A nice attribute of this slip gain estimation method is that it has the ability to be used on-line and provides an update to the slip-gain estimate each time a change in torque is commanded. Because it is only operational immediately following a torque change, it has reduced computational requirements with respect to methods that are continually running, but still allows for frequent updates of the slip gain estimate. The method was shown not to depend on any machine or system parameter. The proposed method can be easily integrated into standard drives and be used in many industrial and automotive applications. R EFERENCES

Fig. 19. Slip gain (radians per second) adaptation for different controllers and transient conditions. (a) PI controller (20) Kp = 0.05, Ki = 0.2, ωr = 33 Hz, torque steps: 50% rated torque. (b) Deadbeat controller (21) K = 0.2, ωr = 50 Hz, torque steps: 100% rated torque. (c) Deadbeat controller (21) K = 0.2, ωr = 50 Hz, torque steps: 25% rated torque. (d) Deadbeat controller (21) K = 0.1, ωr = 16 Hz, torque steps: 50% rated torque.

observed that the estimated slip gain converges to correct slip gain, i.e., no slow transients are seen in the voltages, after only a few changes in torque command. Fig. 19 shows the adaptation of the slip gain for different controllers and different working

[1] L. Garces, “Parameter adaptation for the speed controlled static AC drive with squirrel cage induction motor,” IEEE Trans. Ind. Appl., vol. IA-16, no. 2, pp. 173–178, Mar./Apr. 1980. [2] R. D. Lorenz and D. B. Lawson, “A simplified approach to continuous on-line tuning of field-oriented induction machine drives,” IEEE Trans. Ind. Appl., vol. 26, no. 3, pp. 420–424, May/Jun. 1990. [3] T. M. Rowan, R. J. Kerkman, and D. Leggate, “A simple on-line adaption for indirect field orientation of an induction machine,” IEEE Trans. Ind. Appl., vol. 27, no. 4, pp. 720–727, Jul./Aug. 1991. [4] J.-K. Seok and S.-K. Sul, “Induction motor parameter tuning for highperformance drives,” IEEE Trans. Ind. Appl., vol. 37, no. 1, pp. 35–41, Jan./Feb. 2001. [5] R. D. Lorenz, “Tuning of field-oriented induction motor controllers for high-performance applications,” IEEE Trans. Ind. Appl., vol. IA-22, no. 2, pp. 293–297, Mar./Apr. 1986. [6] J. Holtz and T. Thimm, “Identification of the machine parameters in a vector-controlled induction motor drive,” IEEE Trans. Ind. Appl., vol. 27, no. 6, pp. 1111–1118, Nov./Dec. 1991. [7] J. C. Moreira and T. A. Lipo, “A new method for rotor time constant tuning in indirect field oriented control,” IEEE Trans. Power Electron., vol. 8, no. 4, pp. 626–631, Oct. 1993. [8] B. Wu and M. W. Degner, “Method and system for self-calibration of an induction machine drive,” U.S. Patent 6 566 840 B1, May 20, 2003. [9] ——, “Method and system for controlling torque in a powertrain that includes an induction motor,” U.S. Patent 6 646 412 B2, Nov. 11, 2003.

DEGNER et al.: SLIP-GAIN ESTIMATION IN FIELD-ORIENTATION-CONTROLLED INDUCTION MACHINES

Michael W. Degner (S’95–A’98–M’99–SM’05) received the B.S., M.S., and Ph.D. degrees in mechanical engineering from the University of Wisconsin, Madison, in 1991, 1993, and 1998, respectively, with focus on electric machines, power electronics, and control systems. His Ph.D. dissertation was on the estimation of rotor position and flux angle in electric machine drives. In 1998, he joined the Ford Research Laboratory, Dearborn, MI, working on the application of electric machines and power electronics in the automotive industry. He is currently the Manager of the Electric Machine Drive Systems Department of the Sustainable Mobility Technologies and Hybrid Programs Group, Ford Motor Company, where he is responsible for the development of all-electric machines and their control systems for hybrid and fuel-cell vehicle applications. His interests include control systems, machine drives, electric machines, power electronics, and mechatronics. Dr. Degner received the 2005 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS Third Place Prize Paper Award and was the recipient of several IEEE Industry Applications Society Conference Prize Paper Awards.

Juan Manuel Guerrero (S’00–A’03–M’04) was born in Gijón, Spain, in 1973. He received the M.E. degree in industrial engineering and the Ph.D. degree in electrical and electronic engineering from the University of Oviedo, Gijón, in 1998 and 2003, respectively. Since 1999, he has been a Teaching Assistant in the Department of Electrical, Computer and Systems Engineering, University of Oviedo. From February to October 2002, he was a Visitor Scholar at the University of Wisconsin, Madison. His research interests include parallel-connected motors fed by one inverter, sensorless control of induction motors, control systems, and digital signal processing. Dr. Guerrero received an award from the College of Industrial Engineers of Asturias and León, Spain, for his M.E. thesis in 1999, and one IEEE Industry Applications Society Conference Prize Paper Award in 2003.

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Fernando Briz (A’96–M’99) received the M.S. and Ph.D. degrees in electrical engineering and control from the University of Oviedo, Gijón, Spain, in 1990 and 1996, respectively. From June 1996 to March 1997, he was a Visiting Researcher at the University of Wisconsin, Madison. He is currently an Associate Professor in the Department of Electrical, Computer and Systems Engineering, University of Oviedo. His topics of interest include control systems, high-performance ac drives control, sensorless control, diagnostics, and digital signal processing. Dr. Briz received the 2005 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS Third Place Prize Paper Award and was the recipient of two IEEE Industry Applications Society Conference Prize Paper Awards in 1997 and 2003.