Sensorless induction motor drive in high speed range - some aspects of digital implementation
Vlado Porobić, Darko Marčetić, Evgenije Adžić Department of Power, Electronics, Communications Faculty of Technical Sciences Novi Sad, Serbia
[email protected],
[email protected],
[email protected] Abstract— This paper illustrates some exploitation problems native for vector controlled induction motor drive working at limited PWM frequency. The problems occur at very high shaft speeds only, when PWM to output frequency ratio becomes too low. There are given simulation and experimental results of investigation for induction, low cost 800W motor, used in electrical appliances. Keywords-induction motor; sensorless
high speed; digital
control;
signals, voltage distortion caused by non linear behavior of switching converter and sensitivity on model parameter mismatch. Main research effort was reasonably focused on low speed, where these problems are pronounced. This paper deals with problems of digital realization of high speed sensorless drive, where discrete nature of control hardware, i.e. DSP (digital signal controller) significantly influences drive performance. II.
I.
INTRODUCTION
Electrical drives technology is rapidly approaching the physical limits in terms of dynamic and energy performance. The challenge in the coming years will be to develop more reliable and robust drives at the lowest possible cost. Reliability and cost determine the level of penetration of electrical drives in major application. In this context, controlled induction motor drives without speed sensor on the shaft are fully introduced in a broad range of applications. The advantages of sensorless drive are lower cost, reduced size of the drive, elimination of sensor cable, increased reliability and less problems caused by measurement noise. On the other side, there is an unavoidable dynamic drive performance deterioration. Low cost applications such as fans, compressors, pumps usually have implemented V/Hz scalar control, which relies on motor name plate data and offers simple and easy way of realization, but has very poor dynamic. Because of growing demands for less power consumption and much better dynamic, there is strong impact to move from this simple control strategy towards to more complex and powerful, FOC (field oriented control) also in low cost drives. The consequence of the fact that flux and torque are separately controlled in this approach yields much better system performance. The operation of these drives requires estimation of internal state variables of the machine. The evaluation is based exclusively on measured terminal currents and voltages. It is widely investigated variety of methods, which differ with respect to accuracy, robustness and applicability [1], [2]. The deteriorated performance of this drive compared to sensor drive is consequence of problems with drift and offset in the acquired
HIGH SPEED SENSORLESS DRIVE, SYSTEM DESCRIPTION
Commonly used sensored, indirect rotor field oriented control (IFOC) drive is shown on Fig. 1 (switches are in position 1). Calculation of rotor flux position, i.e. dq coordinate system θdq (needed for Park and inverse Park transformation) is based on integration of measured rotor speed and calculated speed slip. This IFOC mainly has good performance for sensored mode in low cost drives. Here, it is used for reference for sensorless control, where rotor speed and dq coordinate system position has to be estimated. In IFOC high speed sensorless drive, absolute error of speed estimation and thus θdq is significant. As a result, vector control of induction machine is suboptimal: for given load there is a much higher quadrature current isq, thus resulting in increase of total drive current. Over some speed, drive is no capable to run nominal load. In order to get more accurate rotor flux position at high speed, it is better to choose different way of motor control: direct field oriented control (DFOC). Position of dq coordinate
Figure 1. Rotor flux oriented, IFOC/DFOC sensored/sensorless control block diagram
system is determined directly, without calculating slip, from terminal currents and voltages, Fig. 1 (switches are in position 2). First, there is employed simple voltage current (VI) estimator. Voltage (1) and flux (2) equations yield rotor flux in alpha and beta axis (3).
ωˆ dq
ψβ
sin θ dq
ψα
cosθ dq
(1)
r r r r r r Ψs = Ls is + Lm ir , Ψr = Lr ir + Lm is
(2)
r r r r L r L 1 r Ψr = r ψ s − σLs is = r vs − Rs is − σLs is Lm Lm s
(3)
T θˆdq z −1
cos
sin
Figure 2. Phase locked loop producing ωdq and θdq
ωk =
(
Lm ψ αr iβs − ψ βr iαs Tr ψ α2r + ψ β2r
ωk = ωk _ IFOC =
Lm 2 Lr Ls
)
(5)
In dq reference frame speed slip is given in (6).
where are: Rs, Rr - stator and rotor resistance Ls, Lr, Lm - stator, rotor and magnetizing inductance III.
Lm isq Tr ψ rd
(6)
HIGH SPEED SENSORLESS DRIVE, SOME DIGITAL CONTROL ASPECTS
ωr - electrical rotor velocity Because of DC offset problem, instead of pure integrator, there is used quasi integrator with low cut off frequency (1.5Hz). Increased flux phase lagging can be neglected at high speed. The influence of the most temperature dependant parameter Rs can be neglected also, because of large stator voltage value in high speed region. One way to get angle θdq is directly from rotor fluxes ψα and ψβ (3), but is extremely noise sensitive, and should be avoided because it influences significantly total drive performance. The better way to extract θdq is usage of phase locked loop (PLL), Fig. 2. Rotor fluxes are involved at input of PLL. After dividing them with rotor flux amplitude, we have sin and cos values of rotor flux position. Having sin(x) ≈ x and PI regulator, estimated position of dq coordinate system is produced at the output of PLL. In that way we have less noise sensitive drive, e.g. PLL will filter occasional glitches in that signals. Beside θdq, PLL also produces synchronous speed ωdq, which is used to estimate rotor speed (4).
ω rotor _ est = ω dqPLL − ω k
T z −1
Kp
ψ α2 + ψ β2
r r r dψ s r dΨr r u s = Rs is + , 0 = Rr ir + − jω r Ψr dt dt
r
σ - total leakage coefficient: σ = 1 −
Ki
(4)
Slip speed ωk can be calculated in several ways. Since synchronous speed is derivative of angle θdq which can be calculated as arctg function of rotor fluxes, from (1) and (2) we have (5). It is slip speed calculated in αβ reference frame.
Effects of quantization error in DFOC drive, realized in 16/32-bit fixed and floating point math (DSP) for low speed region are shown in [3]. This paper deals with high speed, field weakening region. Target speed of tested, four poles motor is: 10000rpm to 16300rpm, i.e. 333Hz to 543Hz. There is fixed computation period of DSP, which is commonly same as reciprocated value of PWM drive frequency. To have as close as it is possible system to continuous one, it is desirable that this time is very small. On the other hand, there is needed some time to do all demanding computation which at sensorless, low cost drive can be limited by chosen DSP. Switching loses also limit maximum PWM frequency, which finally yield that this value at low power drives is in range 10÷20KHz; in this test it is chosen 16kHz. There can be defined ratio of the PWM frequency over the stator field frequency fs given in (7).
Fratio =
f pwm fs
(7)
This is a number of points where new duty cycle is calculated per one period Ts. Therefore, resolution of output angle step is 360o/Fratio. In low speed case it is sufficient number that produces smooth sinus voltage change on motor. As regards high speed region, given small Fratio significantly decrease resolution of output voltage and thus influences drive performance. It is especially emphasized in a sensorless, low cost drive, and can lead to significantly drive deterioration. For considered example (543Hz, 16kHz) Fratio = 29, resulting in output angle step resolution of 12.4o.
Beside low Fratio, current acquisition issues can greatly affect low cost drive. Different sampling techniques can be used to acquire drive currents information [4]. In a single or three shunt topology there is no isolated current reading, i.e. there is constrains where currents must be sampled. It is emphasized especially in a single shunt topology, at low and full duty cycle or space vector sector boundaries. Therefore, special care must be taken to reduce offset, noise, and current ripple influence.
If in PWM routine is used same θdq for both transformation, there will be introduced additional phase shift between phase voltages and currents. In low speed region it will not impact drive, unlike high speed region. It can be easily noticed from steady state stator voltage in d axis, under no load condition, equation (8). This voltage would not be constant with speed increasing in a case of described badly adjusted position of dq coordinate system.
Theta calculation yields discrete angle transformation θdq, Fig. 3.
U sd = Rs isd − ω s Lσ isq
(8)
Essential issue for digital implementation of DFOC is how to estimate rotor flux in k instant, Fig. 3, which will be used for inverse Park transformation, and thus voltage generation. From (1) and (3), when s is replaced with left Euler approximation we have (^ denotes estimated value)
Current sampling One leg voltage
Pwm timer is(k-1)
vs(k)
vs(k) = calculation[is(k-1)]
Position of dq system k-1
k
θ dq
[
ˆ (k ) = Ψ ˆ (k − 1) + T v (k − 1) − Rˆ i (k − 1) Ψ s s s s s ˆ L ˆ (k ) = r Ψ ˆ (k ) − σˆLˆ i (k ) Ψ r s s s Lˆ
Figure 3. Discrete θdq coordinate system position
For low Fratio special attention must be paid to the sequence of calculation steps, regardless is there IFOC or DFOC strategy, Fig. 4. It is obligated to do Park transformation for getting idq(k-1) with angle θdq(k-1) and inverse Park with θdq(k) for producing vαβ(k). PWM interrupt
Thus, to determine ψˆ r in instant k, we need current value is also in k. Therefore it can be calculated in k-1 instant only, i.e. it is one PWM period lagged (this problem will be referenced as z-1 problem). In high speed region, this introduces significant phase angle error. In above considered example (12.4o), vector drive is significantly impacted. This sort of error is similar as one in rotor leakage variation case, which will be later explained. Discretization and quantization of quasi integrator also introduce additional phase change in estimated rotor flux. There is an additional factor that can seriously impact DFOC drive. It can be shown that drive is strongly sensitive to variation of leakage inductance at high speed which is obvious from equation (3) and will be explained in the following text.
update RotorSpeed (ωr(k)), currents (iabc(k-1)) iαβ(k-1) = Clarke (iabc(k-1)) idq(k-1) = Park (iαβ(k-1),θ θdq(k-1))
Estimated rotor flux is given by (10).
Vdq(k) = Current regulators (idq(k-1))
ˆ ˆ = Lr 1 v − Rˆ i − σˆLˆ i Ψ r s s s s s Lˆm s
θ dq ( k ) = PLL _ out ( k )
DFOC
θ dq ( k ) = θ r ( k ) + θ k ( k )
IFOC
θ r ( k ) = θ r ( k − 1) + ω r ( k ) ⋅ T 1 isq ( k −1) Tr isd _ filt ( k −1)
(9)
m
k+1
θ k ( k ) =θ k ( k −1) +
]
Vαβ(k) = Inverse Park Vdq(k), θdq(k)) SpaceVector module (Vαβ(k)) Update PWM registers Return
Figure 4. IFOC/DFOC PWM interrupt routine
(10)
Frequency response function (FRF) related to the estimated and actual fluxes can be insightful and helpful. From machine model in alpha beta system, with substitution s with excitation frequency ωdq = ωr + ωk, FRF can be derived [5], [6]:
ˆ Ψ L Lˆ L2 R r = m r 1 + r 2 ( r + jω k ) Ψr Lˆm Lr Rr Lm Lr ˆ σLs − σˆLˆs − j Rs − Rs ω r + ω k
(
)
(11)
Influence of Rs can be neglected at high speed, hence equation (11) can be rewritten as (12).
(
)
(
)
ˆ Ψ Lˆ Lˆ Lˆ L r = r + r2 σLs − σˆLˆ s + jω k r2 r σLs − σˆLˆ s (12) Ψr Lr Lm Lm Rr At high slip frequencies, common to field weakening region, the magnitude and especially phase errors are significant. Leakage inductance variation depends of motor construction, especially either slots are opened or not. In a case of low power motors it is common to have open stator slots and closed or semi closed rotor slots (this last one is case with tested motor). Thus can be considered that stator leakage inductance is constant and rotor leakage inductance can easily varies by factor 2 or more. According to (12) magnitude and phase diagram of FRF dependable of slip can be easily plotted, Fig. 5. It is taken that maximum slip is 16.6Hz, i.e. 1000rpm, what is maximum for tested motor in the field weakening region. One can notice that there is neglected amplitude and significant phase error which can influence DFOC drive a lot. In the given Fig. 5 slip is assumed to be positive, thus motor mode is shown. In a case of generator mode where slip is negative, from equation (10) is obvious that there is opposite phase error. If estimated flux is lagged for the real one in motor mode, it will be advanced in a generator mode. Generally, in lagged (phase) DFOC, in motor mode there is ωdq less that it should be, hence less ωk, and thus there are less is current, but phase voltages are bigger than in correctly adjusted drive, what is limiting factor in field weakening region. On the other hand, in the drive with advanced phase we have ωdq bigger that it should be, there is bigger ωk, and drive currents and power than in correctly adjusted drive. 1.045
10
5
(2.0)
1.03 1.025 1.02 1.015
(0.5) = (1.5)
1.01 1.005
Lrx/Lrxo
Pm [W]
-5
Vline [V]
IFOC, SENSORED, 10000RPM I [A]
f [Hz]
Pel [W]
Q [Var]
Pm/ Pel
0,1
107
141
0,79
336
166
141
0,64
0,2
211
149
1,20
338
284
180
0,74
0,3
315
157
1,64
340
405
234
0,78
0,4
420
164
2,10
342
538
274
0,78
0,5
521
174
2,51
344
677
344
0,77
TABLE II. Te [Nm]
Pm [W]
DFOC, ADVANCED, SENSORLESS, 10000RPM Vline [V]
I [A]
f [Hz]
Pel [W]
Q [Var]
Pm/ Pel
0,1
103
121
0,89
337
155
110
0,66
0,2
206
114
1,64
343
287
162
0,72
0,3
308
122
2,56
351
461
288
0,67
0,4
408
136
3,18
355
625
427
0,65
0,5
515
153
3,75
358
815
581
0,63
TABLE III. Te [Nm]
Pm [W]
DFOC, SENSORLESS, LAGGED FOR ONE PWM (Z-1), 10000RPM Vline [V]
I [A]
f [Hz]
Pel [W]
Q [Var]
Pm/ Pel
0,1
107
149
0,86
336
165
150
0,65
0,2
209
159
1,22
338
282
189
0,74
0,3
311
170
1,57
339
401
240
0,78
0,4
416
181
1,92
340
530
304
0,78
0,5
517
189
2,27
341
658
372
0,79
TABLE IV. Te [Nm]
(1.5)
0,15
264
158
1,81
556
422
242
0,63
(2.0)
0,2
343
168
2,19
559
522
313
0,66
0,23
392
175
2,43
560
592
372
0,66
0,25
432
181
2,60
562
640
311
0,68
0,27
462
186
2,69
562
692
455
0,67
0,3
510
193
2,89
564
770
524
0,66
-10
Pm [W]
Vline [V]
IFOC, SENSORED, 16300RPM
(1.0) 0
-15
(1.0)
1 0.995
Phase (deg)
Magnitude (abs)
Lrx/Lrxo
Te [Nm]
(0.5)
1.04 1.035
TABLE I.
I [A]
f [Hz]
Pel [W]
Q [Var]
Pm/ Pel
-20 2
4
6
8
10
Slip (Hz)
12
14
16
2
4
6
8
10
12
14
16
Slip (Hz)
ˆ Ψ Figure 5. Magnitude and phase erorr of estimated rotor flux ( r ) Ψr IV.
EXPERIMENTAL RESULTS
In order to test above mentioned influences on drive, comparison is made between sensored IFOC drive as a reference, DFOC phasely advanced, and lagged for z-1 (one PWM period) for rotor flux position. In the next tables are given experimental results. Some details of tested motor are given in [7]. A number of excessive tests on drive were performed in high speed region. Target speed range was from 10000rpm - 16300rpm. Inverter with three shunts for current sensing is controlled by low cost DSP ST32F103; power limit is 800W. Dynamometer with control unit Magtrol DSP5600 and power meter Yokogawa WT500 were used for all tests. In order to easy compare loaded sensored and sensorless strategies, results are given in the table pairs.
TABLE V. Te [Nm] 0,156
Pm [W]
DFOC, ADVANCED, SENSORLESS, 16300RPM Vline [V]
I [A]
f [Hz]
Pel [W]
Q [Var]
Pm/ Pel
270
147
2,88
580
531
531
0,51 0,53
0,2
343
163
3,18
581
644
644
0,23
393
175
3,45
584
755
755
0,52
0,25
430
184
3,72
589
831
858
0,52
0,27
-
-
-
-
-
-
-
0,3
-
-
-
-
-
-
-
TABLE VI. Te [Nm]
Pm [W]
DFOC, SENSORLESS, LAGGED FOR ONE PWM (Z-1), 16300RPM Vline [V]
I [A]
f [Hz]
Pel [W]
Q [Var]
Pm/ Pel
0,156
265
201
1,31
549
398
231
0,67
0,2
343
216
1,51
548
489
289
0,70
0,23
373
213
0,25
-
-
0,27
-
-
0,3
-
-
526
513
303
-
-
-
-
-
margin some additional mechanism for PWM discretization phase error compensation is needed.
-
-
-
-
-
APPENDIX
-
-
-
-
-
1,59
0,73
From table pairs I-II and IV-V it can be seen that in a case of advanced sensorless DFOC there are higher current and larger power drawn from drive, compared to sensored IFOC. Speed slip is larger than in a sensored drive, resulting in motor flux less than nominal one. On the other hand, from the table pairs I-III and IV-VI it can be seen that in a case of lagged DFOC drive motor voltage is larger than in sensored IFOC. Therefore, due to the discretization error only, the drive at high speeds approaches or current or voltage limit, and it is not capable of running nominal loads. This issue is especially emphasized in a case of low DC bus voltage due to voltage grid drop.
Motor: Rs = 3.26Ω, Rr = 1.05Ω, Lsr = Lsr = 0.003H, Lm = 0.074H, Un = 195V, In = 1.93A, npole_pairs = 2, nmax = 16300rpm REFERENCES [1] [2]
[3]
[4]
V.
CONCLUSION
In this paper are shown some aspects of sensorless induction motor drive used in high speed applications. DFOC with VI estimator is proposed as a control scheme. Such system is then analyzed and experimentally tested according to some aspects of digital control and sensitivity of motor variation parameters. It has been shown that timediscretization limitation by itself can influence the drive performance, especially at high speeds. The influence on output power, voltage and current is very significant and for the drives working at high speeds close to the current/voltage
[5]
[6]
[7]
Holtz J., “Sensorless control of induction motor drives”, Proceeding of IEEE, vol. 90, No. 8, Aug., pp. 1359-1394 Jiang J., Holtz J., "High dynamic speed sensorless AC drive with on-line parameter tuning and steady state accuracy", IEEE Transaction on Industrial Electronics, vol. 44., issue 2., 1997, pp. 240-246. Konghirun M., Xu L., Skinner-Gray J., "Quantization errors in digital motor control systems", Power electronics and motion control conference IPEMC, volume 3., 2004. , pp. 1421-1426. Marcetic D., Adzic E., “Improved three-phase current reconstruction for induction motor drives with DC-link shunt”, IEEE Transaction on Industrial Electronics, vol. 57, number 7, 2010, pp. 2454–2463 Jansen P., Lorenz D., “A physically insightful approach to the design and accuracy assessment of flux observers for field oriented induction machine drives”, IEEE Transaction on Industry applications, volume 30, number 1, 1994, pp. 101-110 Jansen P., Lorenz D., Novotny D., “Observer-based direct field orientation: analysis and comparison of alternative methods", IEEE Transaction on Industry applications, volume 30, number 4, 1994, pp. 945-953 Zhang Z., Xu H., Xu L., Heilman H., “Sensorless direct field-oriented control of three-phase induction motors based on "sliding mode" for washing-machine drive applications", IEEE Transaction on Industry applications, volume 42, number 3, 2006, pp. 694-701
