Australasian Universities Power Engineering Conference, AUPEC 2014, Curtin University, Perth, Australia, 28 September – 1 October 2014
1
Improved Sensorless EKF-based Direct Torque Control at Low Speed with Constant Switching Frequency Controller I.M. Alsofyani*, N.R.N. Idris** UTM-PROTON Future Drive Laboratory, Power Electronics and Drives Research Group Faculty of Electrical Engineering, Universiti Teknologi Malaysia 81310Skudai, Johor, Malaysia *
[email protected], **
[email protected] Abstract—A sensorless extended Kalman filter (EKF) based direct torque control (DTC) with a constant switching frequency control (CSFC) of induction motor is proposed in this research work. By replacing the conventional 3-level hysteresis controller with a CSFC, the weakening of the stator flux at low speed is avoided and the flux regulation is improved; consequently the estimated speed at low speed is also improved. In this paper, the performance comparison between the conventional hysteresis comparator-based DTC drive (DTC-Hys) and a CSFC-based DTC (DTC-CSFC) is performed experimentally. It is shown that significant improvement in the estimated speed and drive performance at low speed is achieved with the DTC-CSFC. Index Terms—AC motor drives; direct torque control; EKF; speed sensorless.
I.
INTRODUCTION
Direct Torque Control (DTC) of induction motor (IM) was introduced in the middle of 1980’s [1] and has gained popularity in advanced industrial applications due to its simple structure, quick dynamic response, and robustness against rotor parameters variations. Additionally, unlike field oriented control (FOC), DTC does not need the complex field orientation block, speed encoder, and current regulator. DTC drives are inherently sensorless; the speed is only needed when speed control is required, which is common in most applications. Installing mechanical speed sensors, as we all know, increases cost, reduces the drive robustness and in some applications is not possible at all. Due to these reasons, continuous researches on the speed estimations of induction motor drives have emerged over the last three decades. In general, the various methods of speed estimation are based either on the fundamental model of the machines or on the anisotropic property of the machines [2]. The former uses the fundamental equations of the machines in developing the speed estimation algorithms. The problem with these estimation techniques is at low speed when the magnitude and frequency of the induced rotor voltages and currents become small. At zero stator frequency, when the induced rotor voltage ceases to exist, the estimation completely fails. To solve this problem, estimation based on the anisotropic property of the machines is employed [2]-[5]. However, most of these techniques require signal injections and hence additional hardware and also increased computational burden.
Among these estimation methods EKF, which is based on stochastic approach, is highly suitable for speed estimation of IM drives whereby parameters uncertainties and presence of noise in signal acquisitions are unavoidable [5][11]. In this paper, an EKF-based speed estimation DTC of IM drive system is presented. The simple structure of the look-up table based DTC drive is retained [1], however, instead of the conventional hysteresis based torque comparator, a constant switching frequency controller (CSFC), which is introduced in [6], is used. Other than reducing the torque ripple and maintaining a constant switching frequency [6], the main contribution of the CSFC in the speed estimation is to overcome the problem of stator flux droop [7]-[9], which normally occurs at low speed. With the improved stator flux at low speed, the estimation of speed at low and zero speed is significantly improved. In this work, a comparison of the performance between speed sensorless of hysteresis based DTC (DTC-Hys) and the proposed CSFC based DTC (DTC-CSFC) drives at low speed operation obtained from experimental results is presented. The results indicated that the improvement in the performance of the DTC-CSFC compared to the DTC-Hys, both with EKF-based speed estimators. The rest of the paper is organized as follows. In section II of this paper, the improvement in the stator flux using the CSFC is discussed. The basic principle EKF-based speed estimator is described in section III. Section IV presents implementation and experimental results. Finally, conclusion is given in Section V. II.
CSFC AND IMPROVED STATOR FLUX AT LOW SPEED
In order to retain the simple structure of DTC, yet at the same time capable of overcoming the problems of high torque ripple and variable switching frequency in DTC drives, a simple solution that replaces the hysteresis torque controller with a constant switching frequency controller (CSFC) has been proposed in [6]; as shown in Fig.1. The DTC of IM drive incorporating the CSFC (DTC-CSFC) is shown in Figure 2. This section attempts to explain how the CSFC managed to improve stator flux at low speed and hence improves EKFbased speed estimation. CSFC consists of two comparators that compare two constant frequency triangular waveforms (Cupper and Clower) with a compensated torque error (Tc). The controller output
(Tsat) produces similar output pattern as the conventional three level hysteresis comparator but with constant switching frequency. This means that the same look-up table as in DTCHys can be used. The selection of the gains for the PI controller of the CSFC is discussed in details in [6] and will not be presented here.
stator flux magnitude at low speed gives better signal-to-noise ratio and therefore improves the estimation.
𝜓!,!"# 𝜔!"#
PI
CSFC
Speed controller
𝑇! !! 𝜓 !! 𝜔
Look Up table
𝜃!! EKF-based estimator
VSI
IM
vs , is
Fig. 2. Propose EKF-based sensorless DTC with CSFC
Te Hysteresis band
Fig. 1 Constant frequency torque controller
The rate of change of stator flux vector expressed in the stator stationary reference frame is given by: !𝝍𝒔 !"
= 𝒗𝒔 − 𝒊𝒔 𝑅! ,
|ψ ref| (b) |ψ s| Fig. 3 Behavior of torque and flux at low speed for DTC-Hys (a) torque and (b) stator flux
(2)
When torque need to be reduced, zero voltage vectors are selected and according to (2), the stator flux halts [1]. In actual, when stator resistance drop is considered, the change in the stator flux vector when zero voltage vectors are selected is given by Δ𝝍! = −𝒊𝑅! (Δ𝑡)
(a)
Zero voltage vectors
(1)
where vs and is are the stator voltage and stator current space vectors. In DTC of IM, when controlling the flux based on (1), it is a common practice to neglect the stator resistance drop when selecting voltage vectors from the look-up table. According to (1), when stator resistance voltage drop is neglected, the change in the stator flux can be approximated by: Δ𝝍! ≃ 𝒗! Δ𝑡
Zero voltage vectors
(3)
According to (3), selection of zero voltage vectors (to reduce the torque) causes a reduction in the stator flux magnitude. In hysteresis-based torque controller, at low speed, the duration of zero voltage vectors that is selected to reduce the torque is stretched since the magnitude of the negative slope of the torque is minimum at low speed [6][9] (Fig. 3(a)). As a result, the magnitude of the stator flux increment is less than the magnitude of the stator flux decrement due to (3) hence the net flux reduces (Fig. 3(b)). The flux reduction is worst at the beginning of a sector where the radial component of the voltage vectors to increase the flux is zero, as illustrated in Fig. 4 i.e. vs,5 in sector 4. As the flux advances, the radial component of vs,5 increases and the flux may improve. However if the speed is too low (e.g. near zero stator frequency), the flux will not reach reference value but, settles to a steady state value, which is lower than the reference. The reduction in the stator flux (or the stator flux droop) does not occur with CSFC since the duration of zero voltage vectors is shorter. Thus with CSFC, even at low speed, the stator flux is maintained to its rated value. The higher
vs,5 vs,5
vs,4 vs,5
vs,6 Sector 4
vs,6
vs,3
Sector 3
Sector 5
vs,2 vs,6
vs,1
Sector 2
Sector 6
Sector 1
Fig. 4 The use of vs,5 and vs,6 to control the flux in sector 4 of the stator flux plane.
III.
EXTENTDED KALMAN FILTER ALGORITHM
The Kalman filter (KF) is a predictor and correcting type estimator used for stochastic estimation. It is an optimal estimator in the sense of minimising the estimated error covariance. Due to the non-linear nature of IM drive system, extended Kalman filter (EKF) is used and in this paper it is utilized to simultaneously estimate stator current, rotor flux, and
motor speed for speed sensorless control of IMs. The extended model to be used in the development of the EKF algorithm can be written in the following general form:
x!i (t ) = f i ( xi (t ), u(t )) + wi (t )
(4)
fi ( xi (t ),u(t )) = Ai ( xi (t ))xi (t ) + Bu(t ) Y (t ) = Hi ( xi (t ))xi (t ) + Bu(t ) + vi (t )
(5)
Y(k) = Hi(xi(k))xi(k) + Bu(k) + vi(k)
The linearization of (11) is performed around the current estimated state vector xˆi given as follows:
⎡ i%sd ⎤ ⎢ % ⎥ ⎢ i sq ⎥ ⎢ψ% rd ⎥ = ⎢ ⎥ ⎢ψ% rq ⎥ ⎢ ω% ⎥ ⎣#"r!⎦ ⎡ ⎤ 2 ⎢ ⎛ R ⎥ ⎞ L R L R ω L m r r m ⎢− ⎜ s + m r ⎟ ⎥ 0 0 ⎢ ⎜⎝ Lσ Lσ L2r ⎟⎠ Lσ Lr ⎥ Lσ L2r ⎢ ⎥ i ⎛ R s L2m Rr ⎞ ω r Lm Lm Rr ⎥ ⎡⎢ sd ⎤⎥ ⎢ ⎜ ⎟ 0 − + − 0 i ⎢ ⎜ Lσ L L2 ⎟ Lσ Lr Lσ L2r ⎥ ⎢ sq ⎥ σ r ⎠ ⎝ ⎢ ⎥.⎢ψ ⎥ Rr Rr ⎢ ⎥ ⎢ rd ⎥ 0 − − ω 0 r ⎢ ⎥ ⎢ψ rq ⎥ Lr Lr ⎢ ⎥ ⎢ ω ⎥ R R ⎣ r ⎦ r ⎢ 0 Lm ωr − r 0⎥ #"! X ⎢ ⎥ Lr Lr ⎢ 0 0 0 0 0⎥ ⎢#$$$$$$$$$$$"$$$$$$$$$$$! ⎥ A ⎣ ⎦ 0 ⎤ ⎡1 / Lσ ⎢ 0 1 / L ⎥ σ ⎥ ⎢ ⎡v sd ⎤ + ⎢ 0 0 ⎥.⎢ ⎥ + w(t ) ⎢ ⎥ v sq 0 ⎥ ⎣#"!⎦ ⎢ 0 ⎢ 0 ⎥ u ⎣#$$"$0$!⎦
The resulting EKF algorithm can be presented with the following recursive relations: (13) P(k ) = F (k ) P(k ) F (k ) −1 + Q
K (k + 1) = H T P(k )( HP(k + 1) H T + R) −1 xˆ (k + 1) = fˆ ( x(k ), u (k )) + K (k )(Y (k ) − Hxˆ (k ))
(14) (15)
P(k + 1) = ( I − K (k + 1) H ) P(k ) (16) In (13)-(16) Q is the covariance matrix of the system noise, namely, model error, R is the covariance matrix of the output noise, namely, measurement noise, and P are the covariance matrix of state estimation error. The algorithm involves two main stages: prediction and filtering. In the prediction stage, the next predicted states fˆ (⋅) and predicted state-error covariance
(7)
1 v s d = VDC (2S a − S b − S c ) 3
vs q =
(17)
1
V DC ( S b − S c ) (18) 3 where Sa, Sb and Sc are either 1 or 0. From the estimated rotor speed and rotor flux linkage values using EKF filter, the stator flux can be obtained using the following. Lm ψrd + Lσ isd Lr L ψ sq = m ψrq + Lσ isq Lr
ψ sd =
B
(8)
IV.
In (7)-(8), isd and isq are the d and q components of stator current, Ψrd and Ψrq are d-q rotor flux, ωr is the rotor electric angular speed in rad/s, vsd and vsq are the stator voltage components, Ls, Lr and Lm are the stator, rotor and mutual inductances respectively. Rs is the stator resistance, and Rr is the rotor resistance The EKF algorithm used in the IM model will be derived using the extended model in (7) and (8). This process requires the discretization of (7) and (8)—as follows: (9) xi(k+1) = fi(xi(k),u(k)) + wi(k) (10)
(19) (20)
The electromagnetic torque based on EKF is expressed using the selected state variables, which are the stator current and rotor flux: Te =
X
fi(xi(k),u(k)) = Ai(xi(k))xi(k) + Bu(k)
(12)
i
matrices, Pˆ (⋅) are processed, while in the filtering stage, the next estimated states xˆ (k + 1) obtained as the sum of the next predicted states and the correction term [second term in (15)], are calculated. The d and q components of the voltage in (7) can be obtained from the switching pattern of the 3-phase voltage source inverter as follows.
X%
⎡ isd ⎤ ⎢ i ⎥ ⎡isd ⎤ ⎡1 0 0 0 0⎤ ⎢ sq ⎥ .⎢ψ rd ⎥ + v(t ) ⎢i ⎥ = ⎢ 0 1 0 0 0⎥⎦ ⎢ ⎥ ⎣ sq ⎦ ⎣# $$ $"$$$ ! ⎢ψ rq ⎥ H ⎢ ω r ⎥ ⎣#"!⎦
∂f i ( xi (k ), u (k ) ∂xi (k ) xˆ ( k )
Fi (k ) =
(6)
where i = 1, 2, …, xi is extended state vector representing the estimated states, fi is the nonlinear function of the states and inputs, Ai is the system matrix, u is the control-input vector, B is the input matrix, wi is the process noise, H is the measurement matrix, and vi is the measurement noise. For IM, this can be represented by (7) and (8) [10].
(11)
3 p Lm isqψ dr − isdψ qr 2 2 Lr
(
)
(21)
EXPERIMENTAL RESULTS AND DISCUSSION
To study the effectiveness of the proposed EKF-based DTC-CSFC, a drive system shown in Fig. 5 is constructed. At the same time, DTC based on hysteresis controller (DTC-HC) is also implemented using the same set-up. The control algorithm including EKF is implemented on a DS1104 controller card and Xilinx FPGA (Basys2 board from DIGILENT). Some of the main tasks of the DTC (i.e. look-up table and blanking time) are implemented utilizing the FPGA, in this way DS1104 is able to execute the DTC algorithm including the CSFC operation with a sampling period of 120 µs. The experimental set-up also consists of an IGBT inverter and a 1.5-kW 4-pole squirrel-cage induction motor. The mechanical load applied using a hysteresis brake by Magtrol
and controlled through a proportional amplifier. The parameters of the induction motor used in the experiment are as shown in Table I. An incremental encoder with 1024 PPR is used to measure the rotor speed, which is sampled every 2 ms. For safety reason, the DC voltage is limited to 240 V, which means that the based speed is reduced to about 59 rad/s. TABLE I
INDUCTION MOTOR PARAMETERS Rs Rr Ls Lr Lm J No. of poles
3Ω 4.1 Ω 0.342 H 0.342 H 0.324 H 0.0031 kgm2 4
A slow speed reversal experiments between 10 rad/s and 10 rad/s, which spans for 2 second is performed for both DTCCSFC and DTC-Hys. Fig. 6 shows the results obtained for the DTC-Hys. As can be seen from Fig. 6(c) that the flux cannot be regulated to its rated at low speed. Speed estimation is poor and deviates from the actual when the speed reverses mainly due to the weakening stator flux at low speed. On the other hand, with DTC-CSFC, the droop or weakening in the stator flux is avoided. The reference stator flux at the rated value is well regulated even when the speed reverses (Fig. 7(c)). Due to the improved stator flux, speed estimation is significantly improved. Estimated Measured Reference
Proportional amplifier
VDC
+ -
IGBT based VSI
C
0 -10 1
2
3
4
(b)
-5 0
1
2
3
4
Flux[Wb]
1 PC Controlled signals after isolation
Controlled signals before isolation
Sa
Sa
Sb Sb
Sc Sc
ib
ia Trq.
Gate drives and Isolations
5
Stator flux Rotor flux
0.5
(c)
0
DS1104 DSP controller
Swa , Swb, Swc FPGA (baysis2)
5
0
Hysteresis brake
I.M.
(a)
-5
5
Torque transducer Enc.
5
0
Torque [N.m]
The triangular waveforms for the proposed CSFC are generated using the DSPACE DS1104 board with a sampling period of 120 µs. Since the triangular waveform is generated by software, its frequency and hence the torque loop bandwidth are restricted by the sampling time of the board [6]. For this particular experimental set-up, the frequency of the generated triangular waveform with a sampling period of 120 µs is about 1041 Hz.
Speed [rad/s]
10
0
Tstat Ψstat Sec.
1
2
3
4
5
Oscilloscope
10
Fig. 5. Experimental set-up for DTC-CSFC and DTC-Hys ia [A]
5
With the values of machine parameters listed in Table 1, the calculated values of PI controller gains, kp and ki, of the CSFC are determined based on [6]:
-5 -10
kp = 7.6529, ki = 1241.83
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0⎤ ⎥ 0⎥ 0⎥ ⎥ 0⎥ 1⎥⎦
(22)
1
2
3 Time [s]
4
5
Estimated Measured Reference
10 5
(a)
0 -5 -10 0
1
2
3
4
5
6
5
0 0 0 0 ⎤ ⎡8.47e - 14 ⎢ ⎥ 0 2.38e 6 0 0 0 ⎥ ⎢ Q = ⎢ 0 0 3.60e - 13 0 0 ⎥ ⎢ ⎥ 0 0 1.0e - 16 0 ⎥ ⎢ 0 ⎢ 0 0 0 0 1.62e - 5⎥⎦ ⎣
(23)
0 ⎤ ⎡0.0053 R = ⎢ ⎥ 1.62e - 5⎦ ⎣ 0
(24)
Torque [N.m]
⎡1 ⎢ ⎢0 P = ⎢0 ⎢ ⎢0 ⎢0 ⎣
0
Fig. 6 Experiment results for slow speed reversal DTC-Hys (a) reference, estimated and actual speed, (b) estimated toque, (c) Estimated stator and rotor flux, (d) phase A of stator current
Speed [rad/s]
For EKF filter initial values with assumption of white noise, the state estimation error matrix P is initiated with the diagonal matrix one, whereas the initial values of the R and Q filters in the EKF algorithm are found by using genetic algorithm [10] to achieve a rapid initial convergence as well as the desired transient- and steady-state performance. Thus, the initial values for EKF scheme are defined by (22)-(24).
(d)
0
(b)
0
-5 0
1
2
3
4
5
6
Fig. 7 Experiment results for slow speed reversal DTC-CSFC (a) reference, estimated and actual speed, (b) estimated toque, (c) Estimated stator and rotor flux, (d) phase A of stator current (continued on the next page)
REFERENCES [1] I. Takahashi and T. Noguchi, "A New Quick-Response and HighEfficiency Control Strategy of an Induction Motor," IEEE Transactions on Industry Applications, vol. IA-22, pp. 820-827, 1986.
Flux[Wb]
1
(c)
Stator flux Rotor flux
0.5
[3] McNamara, D.M., B. Enayati and A.K. Ziarani, “Sensorless speed measurement of induction motors using an adaptive frequency-tracking algorithm”, in Proc. 34th Annual Conference of IEEE Industrial Electronics IECON 2008.
0 0
1
2
3
4
5
6
10
ia [A]
5 0
(d)
-5 -10
0
1
2
3 Time [s]
4
5
6
Fig. 7. (continued)
V.
[2] J. Holtz, “Sensorless Control of Induction Machines;With or Without Signal Injection?” IEEE Trans. Ind. Electron., Vol 53, No.1, pp 7-30, 2006
CONCLUSION
This paper presented the EKF-based speed sensorless DTC induction motor drive with SCFC. It is showed that by employing CSFC, the flux regulation of the DTC drive at low speed is improved and hence effectively helps to improve the performance of the sensorless drive at low speed region. Based on the results obtained from the experiments, it is showed that the sensorless DTC-CSFC performs better than the conventional DTC-Hys at low speed regions.
ACKNOWLEDGMENT The authors would like to thank Universiti Teknologi Malaysia (UTM) (R.J130000.7823.4F380), Ministry of Education, and Ministry of Science, Technology and Innovation (MOSTI) of the Malaysian government for providing the funding for this research.
[4] Guanghui, W., H.F. Hofmann and A. El-Antably, “Speed-sensorless torque control of induction machine based on carrier signal injection and smoothair-gap induction machine model”, IEEE Transactions on Energy Conversion, Vol 21, No.3, pp 699-707, 2006 [5] I. M. Alsofyani and N. R. N. Idris, "A review on sensorless techniques for sustainable reliablity and efficient variable frequency drives of induction motors," Renewable and Sustainable Energy Reviews, vol. 24, pp. 111-121, 2013. [6] N. R. N. Idris and A. H. M. Yatim, "Direct torque control of induction machines with constant switching frequency and reduced torque ripple," IEEE Transactions on Industrial Electronics, vol. 51, pp. 758-767, 2004. [7] W.S.H. Wong and D. Holliday, “Minimisation of flux droop in direct torque control induction motor drives”, IEE Proc.- Electr. Power Appl, Vol 151, No. 6, pp. 694-703, Nov. 2004, [8] Mei, C.G., Panda, S.K., Xu, J.X., and Lim, K.W.: ‘Direct torque control of induction motors - variable switching sectors’. Proc. IEEE 1999 Int. Conf. on Power Electronics and Drive Systems (PEDS), pp. 80–85, 1999 [9] D. Casadei, G. Grandi, G. Serra, A. Tani, “Switching strategies in direct torque control of induction machines”, ICEM '94, 5-8 September, 1994, Paris, France, Proceedings Vol 2, pp. 204-209. [10] I. M. Alsofyani, N. R. N. Idris, M. Jannati, S. A. Anbaran, and Y. A. Alamri, "Using NSGA II multiobjective genetic algorithm for EKF-based estimation of speed and electrical torque in AC induction machines," in Power Engineering and Optimization Conference (PEOCO), 2014 IEEE 8th International, 2014, pp. 396-401 [11]F. Auger, M. Hilairet, J.M. Guerrero, E. Monmasson, T. OrlowskaKowalska and S. Katsura, "Industrial Applications of the Kalman Filter: A Review", IEEE Transactions on Industrial Electronics, Vol. 60, No.12, pp 5458-5471, Dec 2013.
