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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 15, NO. 2, JUNE 2000

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A Multiple Reference Frame Synchronous Estimator/Regulator Patrick L. Chapman, Member, IEEE, and Scott D. Sudhoff, Member, IEEE

Abstract—A synchronous regulator that accurately tracks three-phase commands with arbitrary balanced harmonic content is set forth. The regulator utilizes multiple reference frames to realize both a novel harmonic estimator and an integral feedback controller. The regulator is analyzed and demonstrated in the context of a permanent-magnet synchronous machine drive. The drive utilizes an optimized nonsinusoidal current command to achieve superior efficiency and torque ripple performance but which is dependent upon the commanded waveforms being precisely realized. The multiple reference frame synchronous current regulator proposed herein readily achieves this requirement. A computer simulation and a hardware prototype demonstrate validity of the regulator.

I. INTRODUCTION

I

N MOTOR drive and power electronic applications, it is frequently of interest to control voltage and current harmonics precisely. This is done either to eliminate harmonics, such as in unity-power-factor dc supplies, or to intentionally inject harmonics in certain specialty applications. Many varieties of power converter modulation allow the designer to modify the control signal so that the appropriate harmonics are injected. However, this is normally an open loop procedure that does not ensure that the exact desired operating point is achieved. One interesting application of harmonic injection concerns surface-mounted permanent-magnet synchronous machine (SMPMSM) drives with nonsinusoidal back emf’s. Several authors [1]–[6] have shown that injection of the appropriate current harmonics yields superior performance in terms of efficiency and torque ripple. However, the control schemes in all of these papers require that the current harmonics injected be exactly achieved. For example, if it is desired to minimize loss subject to achieving constant torque, it is possible to calculate a set of corresponding Fourier series coefficients for the stator currents that would result in the desired performance. However, if the machine current does not precisely possess these coefficients, then neither constant torque nor minimum loss results. In fact, since the small high-order currents multiply with the large fundamental back emf in the torque equation [1]–[6], considerable ripple can result if the current harmonics are in error. These optimal control schemes are particularly relevant to high-power Naval propulsion drive systems. For example, maximum efficiency operation would be appropriate for normal operation. Operating with maximum efficiency subject to no Manuscript received December 10, 1998; revised April 27, 1999. This work was supported by the Office of Naval Research. The authors are with the Purdue University, West Lafayette, IN 47907-1285. Publisher Item Identifier S 0885-8969(00)04523-X.

Fig. 1. System diagram.

torque ripple is appropriate in battle situations where torsional vibrations due to torque harmonics increase detectability. In that case, it is particularly important to guarantee that the commanded currents reach their commanded values exactly. Industrial applications such as robotic positioning systems and numerically controlled machines are two more examples where it is desirable to mitigate torsional harmonics. In a conventional current source based drive with a sinusoidal current command, a synchronous current regulator is often used [7] to achieve the commanded current. This control has many forms but the common feature is integral feedback in a synchronous reference frame. This ensures the fundamental component of the current is precisely achieved. Unfortunately, this control cannot be used to exactly obtain current commands that are nonsinusoidal since it does not operate on the individual harmonics. In this paper, multiple reference frame theory is used to formulate a synchronous estimator/regulator that ensures that commanded current consisting of a fundamental component as well as arbitrary harmonic content is exactly tracked. While the regulator is presented in the context of a current regulator herein, it may also be applied to voltage regulation by utilizing the same architecture. It is interesting to note that although multiple reference frame (MRF) theory has often been used as a basis for analysis [8]–[13], it has not been widely used as a basis for control. The proposed control has two parts, a multiple reference frame based estimator, which decomposes the measured current to appropriate frames of reference, and a multiple reference frame integral feedback controller which forces the actual components to match their commands. The proposed scheme is demonstrated using both computer simulation and a hardware prototype. II. EXAMPLE SYSTEM DESCRIPTION Fig. 1 shows a block diagram of an example system in which the multiple reference frame synchronous estimator/regulator (MRFSER) is applied to a surface-mounted permanent-magnet synchronous machine drive. In this particular application, the input to the drive system is the commanded torque, , which

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is converted to an optimal current command vector, . The may be accomplished conversion from a command torque to by any of the methods in [1]–[6], but the exact methodology is not relevant to the work set forth herein. Instead, the objective of the paper is to set forth a means of exactly achieving the current command. Formal definition and notational convention for and other vectors is set forth in the following section. Given , the vector of measured mathe reference current vector, , and the rotor position, , the multiple refchine currents, erence frame synchronous estimator/regulator proposed in this paper synthesizes a machine variable inverter command vector, , which is used as the current command to either a delta or hysteresis current regulator [14]. Although these modulation strategies produce currents that are nearly equal to the commanded current, tracking errors do exist. These tracking errors do not scale with each harmonic proportionally, so a different error is encountered for each harmonic. It is the function of , the MRFSER to ensure that the resultant machine current, , in the steady state. corresponds precisely to the reference, will not be equal to the inverter command vector, Therefore, . It is important to observe that although the MRFSER is shown in the context of a SMPMSM drive herein, it is readily applied to any situation in which harmonic injection for voltage or current is appropriate.

If the position of the synchronous reference frame is set equal to rotor position, i.e. , then this transformation is similar to the generalized Park’s transformation set forth in [15] with the following exceptions. First, only reference frames that are located at multiples of the electrical angle are considered since these are the harmonics generally of interest. The second difference is that the multiple of the electrical angle, ‘ ’, multi, rather plies the quantities in the cosines and sines, . The harmonics of inthan just the electrical angle, terest in a balanced, symmetrical, power system are in the series and since a wye-connected system is not assumed, the triplen (zero sequence) harmonics are also of interest. Notice that, for example, ‘ ’ is considered and ‘5’ is not since the 5 harmonic in a balanced, symmetrical system exhibits a negative phase sequence. Defining (4) in this manner has the advantage that if is a triplen harmonic, then it is automatically incorporated into the same transformation as is used for the nontriplen harmonics. In [13], the triplen harmonics were considered under a different transformation that only applied to zero sequence variables. is defined as the union of individual A vector of the form , ,..., , such that vectors from each reference frame,

III. NOMENCLATURE

In (7) and throughout the work, is the first reference frame considered and is the last frame considered. Herein, physical variables shall be depicted without a modifier as in (1-7). In contrast, estimated values will be distinguished with a circumflex ( ), reference commands with an asterisk ( ), and inverter commands with a tilde ( ), but otherwise have the same structure as (1-7). The two types of commands (reference and inverter) differ in that reference commands are physically desired values. Inverter commands reflect commands to the specific inverter modulation strategy. Since the inverter gives rise to tracking error, the MRFSER synthesizes the inverter commands that result in machine currents that will exactly correspond to the reference commands.

This section sets forth the notation and transformations that are necessary for implementation of the synchronous regulator. Phase variables, , , and , are written in vector form as (1) where may represent a voltage or current. The component of these variables which is constant in a reference frame that rotates at ’ ’ times the fundamental frequency is defined as (2) The phase variable vector (1) may be approximately expressed in terms of the - and -axis variables as

(7)

(3) IV. MULTIPLE REFERENCE FRAME SYNCHRONOUS REGULATOR where a rotational transformation,

, is given by

(4) and where the pseudo-inverse is defined as (5) In (3), is the set of all reference frames considered, where each reference frame corresponds to exactly one harmonic present in variables of (1). The set is formally defined as (6) where is the set of integers, is the set of natural numbers, and only nonzero harmonics are considered as shown by the set on the right-hand side of the intersection. In (4), , is the electrical angle of a synchronous reference frame.

The regulator portion of the MRFSER is shown in Fig. 2 in the context of a current-controlled SMPMSM drive. In this case, is replaced with and the synchronous reference frame of in. In Fig. 2, the terest is the rotor reference frame so that , ,..., represent the reference axis current vectors ,..., , respectively. Likecomponents in reference frames represent the estimated axis current comwise, , ,..., ,..., . Details of how the estiponents in reference frames mated currents are obtained are in the next section. The operator is the associated controller 1/s denotes time integration and loop gain. The integrated error of each reference frame component is then transformed into a component of the inverter com, ,..., , in each controller block. The vectors mand, , ,..., , from each block are summed in accordance , which is with (6) to give an aggregate inverter command, utilized by the inverter as the hysteresis or delta modulator control signal.

CHAPMAN AND SUDHOFF: A MULTIPLE REFERENCE FRAME SYNCHRONOUS ESTIMATOR/REGULATOR

Fig. 2.

199

MRFSER regulator applied to current-controlled PMSM drive.

The action of the MRFSER is similar to a synchronous current regulator [7] that separately operates on each harmonic component. The Rowan and Kerkman regulator [7] could be used to ensure convergence of the fundamental in the presence of current harmonics, but would not ensure convergence of the harmonics themselves. In essence, the integral feedback of the , in the MRFSER in each reference frame ensures that steady state. However, there is an important difference in that the multiple reference frame current vectors , ,..., , do not physically exist nor are readily computed by a straightforward mathematical transformation. A means of estimating these components is set forth in the next section. V. MULTIPLE REFERENCE FRAME SYNCHRONOUS ESTIMATOR There have been several papers that discuss dynamic harmonic estimation for power system applications [16]–[20]. These are typically designed for use with large-scale power systems and are not optimized for use in single converter systems. Unlike [16]–[20], the estimator presented herein is specifically designed for simultaneous three-phase measurement for a single converter and furthermore, it is presented in a context that makes it suitable for use with the MRFSER presented herein. The block diagram for the estimation system is depicted in Fig. 3. Therein, it may be observed that the estimator consists of branches, each of which estimates the current vector associated with one reference frame. The branches are interconnected in such a way that for any branch the estimated harmonics of all the other branches are subtracted from the measured current, . In this way, the harmonic of interest to that branch becomes isolated in the steady state as the estimated currents from the other branches converge to the correct values. This is necessary be produces the vector since applying (4) directly to only as an average value. Vectors from reference frames other than ’ ’ contribute sinusoidally varying components to the average value and have amplitudes proportional to the magnitudes of the other harmonics. These sinusoidally varying components are difficult to filter since their amplitudes may be high compared with the average value of the signal. When the harmonic

Fig. 3.

MRFSER estimator applied to current-controlled PMSM drive.

is isolated however, utilizing the transformation (4) becomes efand there are no sinusoidally fective since it extracts only varying components. Each branch then has an integral feedback loop that drives the estimated current exactly to the actual current in the steady state. To demonstrate the intuitive explanation of the estimator, formal mathematical justification is necessary. Beginning from analysis of the block diagram in Fig. 3, (8) may be written for any given plifying yields

. Substituting (6) and sim-

(9) for all . First from which it will be shown that is a constant, since the objective here let us assume that each is to prove stability and convergence of the isolated estimator. As will be discussed later, this assumption does not restrict the is constant use of the regulator to conditions in which each – in practice it simply means the estimator should be faster than may be subtracted from the the control. Therefore, left side of (9) (10) It is convenient to define error vectors associated with each reference frame as (11)

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Substitution of (11) into (10) and pre-multiplying by the yields (12) Since this equation was written for arbitrary ‘ ’, it may be written for each member of the set . If this is done, and all of the equations are added, then (13) is a sinusoidal function of . By conNote that each it can be seen that a solution of sidering multiple values of (13) must satisfy (14) The solution to this differential equation is a decaying expo, is negative. Therefore, the error vectors, nential since , approach zero exponentially as time increases which implies (15) as desired. The conclusion is that with a constant input, the estimator output will converge to the correct value. Although constant current was assumed in this derivation, the estimator does not require that the physical variable be constant in order to converge, it merely requires that signal change slowly with respect to the speed of the estimator. Such behavior is typical of any estimator. This does not prove that the overall system will be stable; that is a function of machine parameters and both the estimator and control gains. This problem, however, is no different than the design of any other control that involves both an estimator and controller and typically involves making the estimator faster than the control. For the purpose of selecting gain, it is appropriate to use a nonlinear average value model of the system and control [21]. The framework for such a model for this particular system is set forth in [13]. A convenient way to do this is to linearize the model at the fully loaded condition and and . Then, the use linear systems techniques to select plant is linearized versus operating point to make sure pole locations are globally acceptable. Finally, it should be mentioned and for each that it is possible to select different values of reference frame, but this is not explored herein. VI. SIMULATED PERFORMANCE In this section, a computer simulation is used to explore the operation of a MRFSER. Here, the motor drive considered is described in [13] and utilizes an optimal control strategy set forth in [6]. The significant harmonics are the first, third, and is and . Since it fifth, and therefore, the set is desirable for the estimator to converge faster than the cons and s are used. troller, gains of For this study, a constant torque command of 1.4 Nm is issued and the dc bus voltage is 100 V. A load torque proportional to speed is assumed. Current regulated delta modulation [14] (as opposed to voltage source delta modulation [22]) with a 10 kHz switching frequency is used. A current regulated in-

Fig. 4. Simulated estimated q -axis currents on start-up (dashed lines depict the commanded values).

verter, such as delta-modulated, is appropriate for this comparison since [1]–[5] suggest that it is sufficient to achieve optimal control. Since this application requires triplen harmonic currents, it is necessary to use an inverter topology that allows zero sequence current (such as H-bridges); otherwise, the inverter topology used is irrelevant to the discussion herein. Furthermore, the inverter drives a SMPMSM that is equipped with suitable rotor position feedback but may also use a suitable estimator [23]. Figs. 4-5 depict traces from an example simulation where the SMPMSM drive is started from rest. A step torque command is given, which represents a worst-case scenario wherein the transients are likely to be most severe. Fig. 4 shows the estimated -axis currents on start-up of the machine where the dashed lines indicate the commanded values. It is apparent that the estimated currents converge to the commanded values. A similar plot may be constructed for the -axis quantities, however, it is omitted varifor brevity. Fig. 5 depicts the same study, except that ables are shown. For reference, the electrical rotor speed, , is shown in the first trace. The reference -phase current, , is shown in the second trace, with the estimated current, , below. The estimated current is reconstructed from the vector of estimated - and -axis currents, , using (6). The actual -phase current, , is depicted in the final trace. While the bottom trace of Fig. 4 shows an overshoot in the estimated 5 harmonic. This overshoot is small in comparison with the fundamental, and so tracks very closely except for the high frequency switching noise, which is of limited interest in terms of torque control. The actual current gradually converges to the reference current as the synchronous regulator operates. VII. EXPERIMENTAL PERFORMANCE The test set-up is the same as that simulated in the previous section except a dynamometer is used to load the machine and regulate the speed at 600 rpm.

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Fig. 7. Experimentally reconstructed (dashed) and commanded (solid) a-phase currents for 5 kHz switching frequency (a) with MRFSER (b) without MRFSER.

Fig. 5. Simulated electrical speed, and a-phase quantities on start-up.

Fig. 6.

Measured a-phase current.

It is interesting to observe the measured currents in the time domain both with and without the MRFSER. First, a measured -phase current waveform is shown in Fig. 6. Therein, both the low frequency harmonic content and switching noise are evident. As stated, the switching induced pulsation is of little interest here. Therefore, in Figs. 7-8 to follow, the switching noise has been subtracted out of the measured signals. This is accomplished by capturing the waveform with a digital storage oscilloscope, extracting the Fourier series coefficients of the harmonics of interest, and then reconstructing the waveform from these coefficients using (6). Fig. 7 (a) shows the measured (dashed) and reference (solid) current with the MRFSER in place with a 5 kHz switching frequency. Here, good agreement is apparent between the two. Fig. 7 (b) shows a similar plot, except without the MRFSER. It is evident that the measured and reference current do not agree. Fig. 8 depicts the same study as shown in Fig. 7, except that the switching frequency is now 15 kHz. Less error

Fig. 8. Experimentally reconstructed (dashed) and commanded (solid) a-phase currents for 15 kHz switching frequency (a) with MRFSER (b) without MRFSER.

is evident in Fig. 8 (b) than in Fig. 7 (b), but there is still significant difference between the measured and commanded current. Another interesting experimental study is depicted in Fig. 9. Therein, the dynamometer is used to measure the mechanical torque (electromagnetic torque minus friction and windage) at a fixed speed for various switching frequencies of the delta-modulated inverter. The mechanical torque measured differs slightly from the commanded electromagnetic torque (1.4 Nm) due to

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Fig. 9. Measured mechanical torque vs. switching frequency: with and without the MRFSER.

frictional and windage torque. This difference is the same for each point in Fig. 9 since the speed is held constant, and therefore only amounts to a uniform offset. As may be observed, the resultant torque when the MRFSER is used is independent of switching frequency. However, when the MRFSER is not used, significant dependence on switching frequency is evident. Since increased switching frequency leads to larger semiconductor losses requires and higher edge rates that lead to electromagnetic compatibility problems, it is advantageous in some applications to utilize the MRFSER in conjunction with lower switching frequency. VIII. CONCLUSIONS A multiple reference frame based synchronous regulator for power converter applications has been presented and the operation has been demonstrated both by computer simulation and experimentally. The synchronous regulator ensures that each harmonic of a commanded current is reached exactly in the steady state. This is useful in a variety of power converter applications, but is particularly useful in optimal current control schemes for permanent-magnet synchronous machine drives. ACKNOWLEDGMENT The authors wish to gratefully acknowledge T. Walls of Emerson Electric for donating the test motor. REFERENCES [1] H. Le-Huy, R. Perret, and R. Peuillet, “Minimization of Torque Ripple in Brushless DC Motor Drives,” IEEE Transactions on Industry Applications, vol. IA-22, no. 4, pp. 748–755, 1986. [2] J.Y. Hung and Z. Ding, “Minimization of Torque Ripple in PermanentMagnet Motors: A Closed Form Solution,” in Proceedings of the 18th IEEE Industrial Electronics Conference, 1992, pp. 459–463. [3] D.C. Hanselman, “Minimum Torque Ripple, Maximum Efficiency Excitation of Brushless Permanent Magnet Motors,” IEEE Transactions on Industrial Electronics, vol. 41, no. 3, pp. 292–300, 1994. [4] M. B. Favre, L. Cardoletti, and M. Jufer, “Permanent-Magnet Synchronous Motors: A Comprehensive Approach to Cogging Torque Suppression,” IEEE Transactions on Industry Applications, vol. 29, no. 6, pp. 1141–1149, 1993. [5] C. Kang and I. Ha, “An Efficient Torque Control Algorithm for BLDCM with a General Shape Back EMF,” in PESC Record- Power Electronics Specialists Conference, 1993, pp. 451–457.

[6] P.L. Chapman, S.D. Sudhoff, and C.A. Whitcomb, “Optimal Control Strategies for Permanent-Magnet Synchronous Machine Drives,” Accepted for publication in IEEE Transactions on Energy Conversion, 1998. [7] T.M. Rowan and R.J. Kerkman, “A New Synchronous Current Regulator and an Analysis of Current-Regulated Inverters,” IEEE Transactions on Industry Applications, vol. IA-22, no. 4, pp. 678–690, 1986. [8] S.D. Sudhoff, “Multiple Reference Frame Analysis of an Unsymmetrical Induction Machine,” IEEE Transactions on Energy Conversion, vol. 8, no. 3, pp. 425–432, Sept. 1993. [9] S.D. Sudhoff, “Multiple Reference Frame Analysis of a Multistack Variable Reluctance Stepper Motor,” IEEE Transactions on Energy Conversion, vol. 8, no. 3, pp. 418–424, Sept. 1993. [10] T.A. Walls and S.D. Sudhoff, “Analysis of a Single-Phase Induction Machine with a Shifted Auxiliary Winding,” IEEE Transactions on Energy Conversion, vol. 11, no. 4, pp. 681–686, Dec. 1996. [11] J.L. Tichenor, P.L. Chapman, S.D. Sudhoff, and R. Budzynski, “Analysis of Generically Configured PSC Induction Machines,” Accepted for publication in IEEE Transactions on Energy Conversion, 1997. [12] P.C. Krause, “Method of Multiple Reference Frames Applied to the Analysis of Symmetrical Induction Machinery,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-87, pp. 218–227, Jan. 1968. [13] P.L. Chapman, S.D. Sudhoff, and C.A. Whitcomb, “Multiple Reference Frame Analysis of Non-sinusoidal Brushless DC Drives,” accepted for publication in IEEE Transactions on Energy Conversion, 1998. [14] B.R. Bose, Ed., Power Electronics and Variable Frequency Drives: IEEE Press, 1997. [15] P.C. Krause, O. Wasynczuk, and S.D. Sudhoff, Analysis of Electric Machinery: IEEE Press, 1995. [16] A. Azemi, E. Yaz, and K. Olejniczak, “Reduced-order Estimation of Power System Harmonics,” in Proceedings of the IEEE Conference on Control Applications, 1995, pp. 631–636. [17] A.A. Girgis, W.B. Chang, and B.B. Makram, “A Digital Recursive Measurement Scheme for On-Line Tracking of Power System Harmonics,” IEEE Transactions on Power Delivery, vol. 6, no. 3, pp. 1153–1160, July 1991. [18] H.M. Beides and G.T. Heydt, “Dynamic State Estimation of Power System Harmonics Using Kalman Filtering Methodology,” IEEE Transactions on Power Delivery, vol. 6, no. 4, pp. 1663–1669, Oct. 1991. [19] P.K. Dash, D.P. Swain, A.C. Liew, and S. Rahman, “An Adaptive Linear Combiner for On-Line Tracking of Power System Harmonics,” IEEE Transactions on Power Systems, vol. 11, no. 4, pp. 1730–1735, 1996. [20] M. Najjar and G.T. Heydt, “A Hybrid Nonlinear-Least Squares Estimation of Harmonic Signal Levels in Power Systems,” IEEE Transactions on Power Delivery, vol. 6, no. 1, pp. 282–288, Jan. 1991. [21] S.D. Sudhoff and S.P. Glover, “Modeling Techniques, Stability Analysis, and Design Criteria for DC Power Systems with Experimental Verification,” in Proceedings of SAE Aerospace and Power Systems Conference, 1998, pp. 55–69. [22] M.A. Rahman, J.B. Quaicoe, and M.A. Choudhury, “Performance Analysis of Delta Modulated Inverters,” IEEE Transactions on Power Electronics, vol. PB-2, no. 3, pp. 227–233, July 1987. [23] K.A. Corzine and S.D. Sudhoff, “Hybrid observer for high performance brushless DC motor drives,” IEEE Transactions on Energy Conversion, vol. 11, no. 2, pp. 318–323, Jun 1996.

Patrick L. Chapman (S’94, M’96) is native to Centralia, Missouri. He received the degrees of Bachelor of Science and Master of Science in Electrical Engineering in 1996 and 1997, respectively, from the University of Missouri-Rolla. Currently, he is pursuing a Ph.D. in Electrical Engineering at Purdue University. During his education, he has conducted research in the areas of power electronics, electric machinery, and solid-state power systems.

Scott D. Sudhoff (M’88) received the BSEE, MSEE, and Ph.D. degrees from Purdue University in 1988, 1989, and 1991, respectively. From 1991-1993 he served as a half-time visiting faculty and half-time consultant for P.C. Krause and Associates. From 1993-1997 he served as an assistant professor at the University of Missouri-Rolla and became an associate professor at UMR in 1997. Later in 1997, he joined the faculty at Purdue University as an associate professor. His interests include the analysis, simulation, and design of electric machinery, drive systems, and finite inertia power systems. He has authored or co-authored over twenty journal papers in these areas.