IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 9, SEPTEMBER 2010
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Robust Torque Control of DC Link Voltage Fluctuation for SynRM Considering Inductances With Magnetic Saturation* Seung-Joo Kim1 , Jong-Bin Im1 , Sung Chul Go1 , Jae-Nam Bae1 , Won-Ho Kim1 , Kwang-Soo Kim1 , Cherl-Jin Kim2 , and Ju Lee1 Department of Electrical Engineering, Hanyang University, Seoul, South Korea Department of Control and Instrumentation Engineering, Halla University, Wonju, South Korea This paper presents the control method that inverter output keeps to linear to reference voltage of Synchronous Reluctance Motor (SynRM) using DC Link voltage Synthesis. This method is reduced torque ripple considering parameters with magnetic saturation. To estimate parameters of inductances for SynRM, the proposed method uses recursive least square (RLS) estimation algorithm. The estimated inductances are used for calculating torque and DC link voltage. The Inverter output voltage cannot be displayed to linear about inverter reference voltage if Real DC Link voltage is different from DC Link voltage of PWM amplitude. Also, the over-modulation is linearity broken if reference voltage is out of range. Torque ripple generates the vibration and noise of a motor. This paper proposes the control method so that the torque ripple decreases and the linearity of inverter output keeps using the DC Link voltage Synthesis. The proposed control method is verified by computer simulation and experiment. Index Terms—DC link voltage, least square method, magnetic saturation, Synchronous Reluctance Motor (SynRM), torque ripple.
I. INTRODUCTION N AC MOTOR needs the electric power converting circuit (inverter) which can control exactly amplitude and phase of each phase current. Also, it supplies AC power and can control exactly torque. Synthesizing output phase voltage to linear from DC Link voltage of inverter is important factor taking aim at performance of voltage modulation method. Amplitude of output voltage of voltage source inverter depends on DC Link voltage. Output of voltage source inverter is decided on an output characteristic of a motor. A motor likes to control without torque ripple so as to work. If DC Link voltage is not constant or is lower than a voltage reference, output voltage is nonlinear. Over modulation is occurred when output of inverter is appeared nonlinearly if real DC Link voltage is lower than rated DC Link voltage in case of the current control. If be entered over modulation scope, inverter output cannot keep linearity. If output of inverter becomes over modulation, phase current wave form is distorted and torque ripple has grown. Torque ripple generates vibration and noise of the motor [1], [2]. For a high performance control of SynRM, it is most important to know the precise values of inductances [3]–[55]. However, the inductances are nonlinear function of stator currents due to flux saturation effect on iron cores [6]. To consider magnetic saturation, the torque control is done by the on-line estimation. The on-line estimation is useful because it does not need to make the table of parameter characteristics such as the d- and q axis inductances curve. This control method estimates d- and q axis inductances using recursive least square (RLS) method. The estimated inductances are used for calculating the DC link voltage. The precise torque control is performed by the DC link voltage estimator. Using
A
*Corrected. This paper first appeared in IEEE Trans. Magn., vol. 46, pp. 2005–2008, June 2010. Due to a production error, Table II(b) and Fig. 5 were not correct. Manuscript received October 31, 2009; revised January 20, 2010; accepted January 21, 2010. Date of current version August 20, 2010. Corresponding author: J. Lee (e-mail:
[email protected]). Digital Object Identifier 10.1109/TMAG.2010.2060380
Fig. 1. Flowchart of the proposed method of SynRM.
this estimator, the output of the Pulse Width Modulation (PWM) inverter can keep linearity of inverter output. The non-linear output of PWM causes large torque ripple which generates vibration and noise of SynRM. As a result, we can achieve a precise torque control and reduce torque ripple of SynRM using the proposed method. Therefore, this paper is intended as an investigation of a control way to decrease torque ripple according to estimate DC Link voltage. This control method will keep linearity of inverter output and reduce torque ripple of motor. II. METHODOLOGY FOR ROBUST CONTROL OF SynRM The flowchart of proposed methodology is shown in Fig. 1. First, before using the proposed method, the phase resistance is measured. The estimation of d- and q axis inductances considering magnetic saturation is performed by the RLS algorithm. The inductances are derived from the reference voltages and the measured values such as phase currents, speed, and phase resistance. Next, the DC link voltage is estimated by the d- and q axis inductances. Using this DC link voltage, the torque of control system can be controlled to satisfy linearity of inverter output.
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TABLE I SPECIFICATIONS OF SynRM
Finally, the proposed method is verified by simulation and experiment. Table I shows the specifications of SynRM. The specification is used for the simulation and experiment described in this paper. The dynamics of a SynRM in a synchronous reference frame are described by (1) and are the d- and q axis stator input voltage, where and are the d- and q axis stator current, is the electrical angular speed, and is the stator resistance which is measured at temperature of continuous rated operating. So the variation according to temperature is neglected. The steady state of operation and the fundamental components are only considered, the derivative of (1) can be replaced by its steady-state form. From (1) and the known values such as the d- and q axis voltages and currents, the d- and q axis inductances can be calculated as (2) (3) and are the d- and q axis inductance, the functions where consider saturation effects [6]. The toque of the synchronous reluctance motor is calculated from the output of the synchronous reluctance motor. The mechanical output of the synchronous reluctance motor is calculated as (4) or (5) is electric angular velocity. where Therefore, the toque of the synchronous reluctance motor is calculated as (6)
Fig. 2. Three phase voltage source inverter.
(8) (9) where , and are switching functions of the inverter. is on or off, is 1 or 0 When the switch which is related to respectively. The DC Link current of an inverter defines as (10) according to each phase current and each switching state (10) B. Inductance Estimation by RLS Method The object of the inductance estimation is to control exact torque of SynRM through calculating torque and DC link voltage. The estimated inductances can be considered magnetic saturation characteristics because of calculating flux linkage with phase currents in real-time from (2) and (3). We can find unknown parameters of mathematical model to minimize the error between observation and estimation by RLS method [7]–[9]. This method can be only applied to the models given by (11) where is the observed variable, is vector of unknown functions that may depend on other known variable, and is vector of parameters of the model. The model is indexed by the variable . The observation functions of (11) are obtained from experiment. The number of observed values is determined by squared size of matrices. So, the RLS method is rearranged from the above method which enables recursive sequential computacan tions in real-time system. The results obtained at time be used to get the estimation at time . The RLS algorithm is obtained as (12)
where P is the poles of the synchronous reluctance motor.
(13)
A. Three Phase Voltage Source Inverter Fig. 2 shows the general three phase voltage source inverter. The three phase inverter requires three inverter arms as illustrated in Fig. 2. The pole voltage of the inverter defines as (7) according to the DC Link voltage of the inverter and each switching state (7)
(14) is estimated parameter matrix, is the gain mawhere trix which updates parameters proportional to the error, is covariance matrix which must be definite and is forgetting factor . The mathematical model for RLS is obtained from voltage equations of SynRM. The observation consists of the indirect reference stator voltage and the measured stator current. The
KIM et al.: ROBUST TORQUE CONTROL OF DC LINK VOLTAGE FLUCTUATION FOR SynRM CONSIDERING INDUCTANCES
TABLE II d- AXIS INDUCTANCE COMPARISON (EXPERIMENT [mH]/FEM [mH])
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input power supplied to motor, the input power of the inverter is calculated as
(19)
q- AXIS INDUCTANCE COMPARISON (EXPERIMENT [mH]/FEM [mH])
The first term is the copper loss, the second term shows the magnetic field energy change and the third term is the mechanical output at the right-hand side of (19). If there is a motor at steady state, the magnetic field energy change of the second term is 0
(20) If there is not loss of the inverter, the output power becomes same as the input power of the inverter (21) The DC Link current is the (10). If it divides the output power of the motor by DC Link current, DC Link voltage is calculated as (22)
Fig. 3. Real and estimated DC link voltage fluctuation. (a) Simulation; (b) experiment.
parameters of SynRM may also be estimated while the machine is running at steady state, therefore, the derivative terms of (1) are zero. With this assumption, the observed values of system are obtained from (1) as (15) and (16) (15) (16) With this observation, the estimated values are obtained as (17) Solving (17) by RLS method with (12) to (14), therefore, the two unknown inductances can be estimated. C. DC Link Voltage Estimation
From (22), the DC link voltage is estimated. This control system uses the sinusoidal PWM (SPWM) of voltage source inverter (VSI) [10], [11]. Because the performance characteristics of a modulator are dependent on the voltage utilization level such as modulation index (MI), it is helpful to define a MI term as (23) For a given DC link voltage, the ratio of the fundamental component magnitude of the line to neutral inverter output voltage to the fundamental component magnitude of the six-step mode voltage is termed the modulation index. When the reference voltage deviates from triangular carrier wave, that is, If the MI is larger than 1, the output of the inverter is in the range of over modulation. The magnitude of triangular carrier wave is decided by the DC link voltage in SPWM. When the DC link voltage is fluctuated, the magnitude of triangular carrier wave is adjusted by the estimated the DC link voltage not to over-modulate PWM. In over-modulation, the performance of torque control system is remarkably declined because the output of inverter is distorted by non-linearity. Therefore, it is necessary to keep the linearity of inverter output by precise parameter estimation. III. SIMULATION AND EXPERIMENT
The DC link voltage can synthesize from relation of inverter input and inverter output (motor input). Input power of the inverter can be given as (18) The Input power of the inverter shows to DC link voltage and DC link current. The output power of the inverter can be equal to the input power of the inverter because loss of the inverter by and is small. As the output power of the inverter is equal to the
The simulation is performed for 1.2 s. The sampling interval of the control is set to 0.0001 s. The rotation speed of SynRM is set to 1000 rpm. Table II shows the d- and q axis inductance data for comparison of the FEM and the Experiment. The inductances of cross coupling currents for SynRM are shown in Table II. In order to verify the validity of estimated inductances, the calculated inductances by experiment with RLS method are compared with those of FEM analysis. These values are also very
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Fig. 5. Change of the torque ripple and the reference voltage of PWM.
Fig. 4. PWM reference voltage characteristic at DC link voltage 10 V (1000[rpm], rated load).
similar. Therefore, the RLS inductance estimator can be reliable and available for torque control system. Fig. 3 shows that the estimated DC link voltage and the real DC link voltage are almost similar. Also, Fig. 3 shows that the waveform of experimentation is closely matched with the waveform of simulation. It means that the RLS estimator works well. The phase current and PWM reference voltage are shown in Fig. 4. The PWM reference voltage changes in case of converting the rated DC Link voltage 12 V into the estimated DC Link voltage 10 V when the real DC Link voltage is supplied for 10 V (Rated Load). The PWM reference voltage of the real DC Link voltage 10 V is nonlinear if the reference DC Link voltage of the PWM reference voltage is the rated DC Link voltage 12 V (Rated Load). We can observe that the PWM reference voltage changes from the rectangular waveform of over-modulation to the normal reference waveform after 0.25 s. Also, the phase current changes from the distorted waveform to the sinusoidal waveform after 0.25 s. It means that the linear output of inverter is performed by linear PWM reference voltage. Fig. 5 shows the calculated torque and DC link voltage according to PWM reference voltage. The control system needs the DC link voltage estimator because the amplitude of triangular waveform for inverter is determined by DC link voltage. Using estimator of DC link voltage in real time, the linear voltage output of inverter is performed. Therefore, the torque ripple is reduced by applying to the proposed control method. IV. CONCLUSION In this paper, the proposed method is the torque ripple reduction method of on-line parameter identification of d and q axis inductances with magnetic saturation in the torque control of SynRM. The identified inductances using RLS estimator are
used for estimating the DC link voltage. Using the DC link voltage estimator, the proposed torque control system is easily obtained good performance to keep the linear voltage output of inverter in the motor drive system with DC link voltage fluctuation. ACKNOWLEDGMENT This work was supported by the National Platform Technology through a Grant provided by the Ministry of Knowledge Economy(MKE). REFERENCES [1] J. Holtz, “Pulsewidth modulation—A survey,” in Proc. IEEE PESC Conf., 1992, pp. 11–18. [2] A. El-Antably and T. L. Hudson, “The design and steady-state performance of a high-efficiency reluctance motor,” in IEEE Ind. Appl. Soc. Ann. Mtng, 1996, pp. 770–776. [3] G. Stumberger et al., “Evaluation of saturation and cross magnetization effects in interior permanent magnet synchronous machine,” IEEE Trans. Ind. Appl., vol. 39, no. 5, pp. 1264–1271, Sep./Oct. 2003. [4] H. Kang, C. Kim, and Y. Kim, “Position control for interior permanent magnet synchronous motors using an adaptive integral binary observer,” J. Electr. Eng. Technol., vol. 4, no. 2, pp. 240–248, Jun. 2009. [5] G.-X. Zhou, R.-Y. Tang, and J.-W. Ahn, “Field circuit coupling optimization design of the main electromagnetic parameters of permanent magnet synchronous motor,” J. Electr. Eng. Technol., vol. 3, no. 1, pp. 88–93, 2008. [6] I. Boldea, Reluctance Synchronous Machines and Drives. New York: Oxford Univ. Press, 1996. [7] K. J. Anstrom and B. Wittenmark, Adaptive Control Second Edition. Reading, MA: Addison-Wesley, 1995. [8] P. Niazi and H. A. Toliyat, “Online parameter estimation of permanent-magnet assisted synchronous reluctance motor,” IEEE Trans. Ind. Appl., vol. 43, no. 2, pp. 609–615, Apr. 2007. [9] G. Stumberger, B. Stumberger, D. Dolinar, and A. Hamler, “Cross magnetization effect on inductances of linear synchronous reluctance motor under load conditions,” IEEE Trans. Magn., vol. 37, no. 5, pp. 3658–3662, Sep. 2001. [10] J. Ahn, S. Kim, and J.. Lee, “Sensorless control for the synchronous reluctance motor using reference flux estimation,” J. Electr. Eng. Technol., vol. 5-B, no. 4, pp. 324–330, 2005. [11] Sang-Don Lee, “Direct torque control of a synchronous reluctance motor using the finite element method,” J. Electr. Eng. Technol., vol. 5-B, no. 2, pp. 173–180, 2005.
