Sensorless Control of a 2.4MW Linear Synchronous Motor for launching roller-coasters
JONKER Marco
Sensorless Control of a 2.4MW Linear Motor for launching roller-coasters Andr´e Veltman1,2
1
Paul van der Hulst2 Jan P. van Gurp3
Marco C.P. Jonker3
Eindhoven Univertsity of Techn., P.O. Box 513, 5600 MB Eindhoven, the Netherlands 2 Piak Electronic Design b.v., Markt 49, 4101 BW Culemborg, the Netherlands t: +31-345-534126, f: +31-345-534127, www.piak.nl 3 GTI Electroproject b.v., P.O. Box 441, 1500 EK Zaandam, the Netherlands t :+31756811111, www.electroproject.nl Author Information :
[email protected]
Keywords: Linear drive, Sensorless control, Permanent magnet motor Abstract An accurate method for estimating the position of a large Linear Synchronous Motor, based on on-line measured phase currents and voltages is shown. Synchronous frame filters remove the fundamental component from these signals effectively, even during fast acceleration. The proposed correlators estimate local parameter values, inductance and resistance, from the remaining switching ripple signals with high bandwidth, able to follow the abrupt changes caused by block switching of the long stator. Experimental results show that a controlled start from standstill is possible.
1
Introduction
Direct torque (or thrust) control in synchronous electrical machines is achieved by controlling the stator current relative to the position of the magnets on the rotor. Hence sufficiently accurate information about the relative position of the stator and the rotor is required. Traditionally some kind of position sensor system is implemented to generate the position information. Various position measurement systems exist, some are be based on Hall-sensors on the track or on the train (by radio link), position dependent induction or transformation ratio (resolver). Use of light (time-of-flight or interferometry) in linear motors. Absolute or incremental optical encoders are common for rotating machines. This paper considers a 2.4MW linear synchronous motor (LSM) to launch roller-coaster trains weighing over 13tons from standstill to 25m/s in a mere 2.8s. The power converter can deliver motor currents up to 3000A from a 1000V DC link. The train acceleration rises from 0 to 1.5g in 0.2s (to prevent whip-lash of the passengers) and then continues around 1.5g for the first half second, after which the drive reaches full power, causing acceleration to drop with increasing speed. The whole drive is constructed such that the motor operates in a near ‘constant power’ mode during the rest of the launch. After the required speed is reached, the thrust is reduced to 1
EPE 2003 - Toulouse
ISBN : 90-75815-07-7
P.1
Sensorless Control of a 2.4MW Linear Synchronous Motor for launching roller-coasters
JONKER Marco
zero and the pushercar is decelerated from 25m/s to standstill in about one second, using eddy current brakes, meanwhile the train continues passively into the first looping of the roller-coaster track. The LSM is of the ‘long stator’ type. As shown in figure 1: right, the 6m long pushercar with NeFeB magnets neither has active elements nor any sensors. The long track consists of stator sections that have a length of 3m and can be shorted by means of heavy duty thyristors. Each section consists of 3,2, or 1 stator blocks (white bars in figure 1). In this way no more than 3 successive sections are (partly) covered by the magnets of the pushercar. Since all stator sections are electrically connected in series, the required voltage can be minimized by energizing only the covered sections and shorting all others by thyristors (in boxes left of the track in figure 1). Such switching of the stator configuration causes abrupt changes of motor inductance and resistance, especially when the track is built for constant power launches. In this track the number of windings per stator section reduces with distance to reduce the generated voltage and prevent inverter saturation, hence the inductance is a stair-case type function of distance. The main challenge of sensorless control of such an LSM is to deal with fast changing parameters, an high acceleration at speeds from standstill up to 25m/s (0-125Hz electrical).
Figure 1: Left: Overview of test track, white bars are stators, boxes on the left contain thyristor switches. Right: Stators sandwiched between magnets on pushercar.
A position estimation method is presented which uses ripple components in the currents and voltages, implicitly generated by current controller the inverter [5], to estimate the motor impedance. This information is then used to extract the angle of the rotor with high precision, even during fast acceleration and stepwise switching motor parameters.
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EPE 2003 - Toulouse
ISBN : 90-75815-07-7
P.2
Sensorless Control of a 2.4MW Linear Synchronous Motor for launching roller-coasters
2
JONKER Marco
Fundamentals of sensorless position estimation
This paper will be dealing with LSM’s and block-switched stators. However, the resulting estimator will also be suitable for most other (rotating) synchronous motors. In a currentjωLi
Rs i
β emf
ut
iαβ
Φm θ α
Figure 2: Basic LSM vector control model for fundamental frequency components in steady state. Linked flux (Φm ) and current magnitude (iαβ ) are assumed constant.
controlled motor, the current controller directs the current according to the rotor flux angle. This basic motor control method is displayed in the vector model of figure 2. The rotating magnetic flux Φm will induce emf in the stator. Mathematically, the emf and the flux Φm are related according to (1). d (1) emf = Φm dt The terminal voltage ut and current i can both be measured. Therefore the angle of Φm can be calculated when stator resistance Rs and inductance L are known.
2.1
Sensorless approaches
There are two distinct approaches to sensorless position estimation. The first method tries to identify position dependent non-linearities. These include for example saliency (inductance difference in d and q direction due to the rotor shape) and rotor-tooth effects [2, 3]. Since the LSM in question does not show effects of saliency, these methods cannot be used here. It is obvious that estimating the angle of the magnet flux from measured currents and voltages during high current, low speed operation, a very good estimate of electrical parameters is required.
2.1.1
Frequency-domain
In order to be able to estimate the motor impedance, removal of the fundamental components in both voltages and currents are required. A problem in this application is the fast acceleration, hence the fundamental voltage component is a fast accelerating sinewave with dynamic amplitude. The problem was solved by using an optimized phase locked loop that can follow constant acceleration without angular error, as used in figure 9. The PLL’s internal structure is beyond the scope of this paper. Assuming a proper angle estimate, the voltages and currents can be transformed to the synchronous frame (dq components) where the fundamental frequency component becomes a DC 3
EPE 2003 - Toulouse
ISBN : 90-75815-07-7
P.3
Sensorless Control of a 2.4MW Linear Synchronous Motor for launching roller-coasters
JONKER Marco
value, even during acceleration. By filtering and backward transformation, the fundamental component in the measured signals can be filtered out almost perfectly, leaving the switching ripple components and other dynamic effects. These residual signals can be used for parameter estimation purposes. But before going into details, first consider the goal of estimating the voltage drop uLR across the total stator impedance that is assumed to consist of a resistive part R and an inductive part L as given in (2), where i is the motor current and uLR is the voltage drop across the stator impedance. uLR (ω) = (jωL + R) · i(ω)
(2)
For low frequencies, the resistive impedance ZR = R is dominant over the reactance |ZL | = ωL, since ω is close to zero. For high frequencies, the reactance will become dominant. However, since (2) assumes only one frequency component, this equation cannot be used for extracting impedance information from a spread spectrum switching ripple. It could however be used when one additional frequency was injected into the system, but our converter has insufficient head-room to do this.
2.1.2
Time-domain
When all information in the spread-spectrum switching ripple needs to be used, a single frequency description such as (2) will not suffice. An alternative approach can be found by converting (2) to its equivalent in the time domain: uLR (t) = L
d i(t) + R · i(t) dt
(3)
In (3), the time derivative of the current is present, which is an ugly term when frequencies get high. Another approach is to use the integral form of (3):
uLR (t)dt = L · i(t) + R
i(t)dt
(4)
This reduces to (5) if one realizes that the integral of voltage equals flux Φ and the integral of current equals charge Q. ΦLR (t) = L · i(t) + R · Q(t) (5) But now another problem arises, since perfect integration is not practically possible at low frequencies or DC. Fortunately a compromise is possible...
2.2
Full frequency range estimation
Equations (3) and (5) describe two different approaches to the same relation between the terminal voltage and -current. The solution to the problem which approach to use, is to use them both, each in their optimum frequency range. d is substituted by the Laplace operator s, the s-domain impedance When in (3), the operator dt description of the motor becomes:
uLR (s) = sL · i(s) + R · i(s)
(6)
Now, we will filter the signals i and uLR with the low-pass filter: HLP (s) =
τ sτ + 1
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EPE 2003 - Toulouse
ISBN : 90-75815-07-7
P.4
Sensorless Control of a 2.4MW Linear Synchronous Motor for launching roller-coasters
JONKER Marco
This filter has both a time-constant and a DC gain of τ . Because the DC gain is equal to τ , the filter acts as an integrator at high frequencies ω τ1 . Adding this filter changes (6) to: τ τ sτ ·uLR (s) = L · ·i(s) + R · ·i(s) sτ + 1 sτ + 1 sτ + 1 low−pass
high−pass
(7)
low−pass
In the second term, the low-pass filter combines with the derivative, indicated by the Laplace operator s, into a high-pass filter. Because of the low-pass filter, for low frequencies (7) is equal to (3). However, no differentiator is needed for i, but a high-pass filter suffices (which is actually a differentiator for low frequencies). Now we perform a similar modification to (5), only this time, because of the integration problems with low-frequency signals, the signals will be filtered with a high-pass filter HHP (s) = sτsτ+1 . This filter has a time-constant1 of τ and a HF gain of 1. Adding this filter changes (5) to: τ τ sτ ·sΦLR (s) = L · ·i(s) + R · ·sQ(s) sτ + 1 sτ + 1 sτ + 1 low−pass
high−pass
(8)
low−pass
By definition, sQ = i and sΦLR = uLR , so (7) and (8) are identical. Because of the high-pass filter, at high frequencies (8) is equivalent to (5). Hence, for high frequencies (7) is also equal to (5). Combined high-pass and low pass filtering of the signals has resulted in a universal electrical description (7/8) of the motor impedance. Previous descriptions required either differentiation of high-frequency signals (3), or integration of low-frequency signals (5). Model (7) eliminates these requirements. Thus a practical full-frequency range estimator can be based on (7). One new parameter has been added to the equation: the filter time-constant τ . This parameter can be used to shift the ‘weighing’ of low-frequency components in the estimator.2 ˆ are known the following reconstruction of the magnet ˆ and L Once the estimated parameters R ˆ flux Φm can be performed: τ sτ τ sτ ˆ m (s) = ˆ· ˆ· ·Φ ·ut (s) − L ·i(s) − R ·i(s) sτ + 1 sτ + 1 sτ + 1 sτ + 1 high−pass
low−pass
high−pass
(9)
low−pass
Mind that the phase angle of the high pass filter as a function of frequency (speed) needs to be compensated in the PLL to get the proper magnet angle.
2.3
Synchronous frame filtering
Suppose we want to remove the frequency component with frequency ωs from a complex input αβ (t). The rotation angle θs of the synchronous frame then becomes: signal Φ
θs (t) =
t 0
ωs (τ )dτ
(10)
αβ (t) to the synchronous frame by backwards rotation by angle θs , the If we now transform Φ fundamental component becomes a DC value. A DC value can be removed by a high-pass filter. αβrip (t) By rotating back to the original angle by forward rotation by angle θs , the signal Φ 5
EPE 2003 - Toulouse
ISBN : 90-75815-07-7
P.5
Sensorless Control of a 2.4MW Linear Synchronous Motor for launching roller-coasters
JONKER Marco
Figure 3: Schematic of harmonic removal filter when θ =
ωs dt
results as shown in figure 3. The harmonic removal filter is used twice for each of the three signals (voltage, flux and current) in the final estimator in figure 9.
3
Impedance estimation
3.1
Resistance estimation
In the following section it is assumed that the resistance R is not a function of frequency. In practice skin-effect will be present so this assumption is not completely true. Lets start with Ohms law in (11), this equation is always valid, but cannot be used when u and i show a phase shift due to reactive impedance in the circuit. The main problem is the potential crossing through zero of the denominator. This problem can be solved by multiplying both terms by i as in (12), yielding an always positive denominator, but a nervous R results when a reactive impedance term is present. To achieve a smooth value, both terms can be low pass filtered [4], yielding the description of the used estimator (13): u (11) R = i u·i (12) R = i2 ˆ = u · i/ (sτe + 1) R (13) i2 / (sτe + 1) The estimator equation (13) is also valid for vectors, phasors and complex numbers when ‘·’ is the scalar product between these vectors. The estimator time constant τe determines the time of averaging hence the speed of the estimator. A higher τe yields a smoother but slower estimated ˆ This estimate is not affected by reactive impedance components in the circuit, because the R. average product of u · i = 0 for any reactive load, when τe is not too small. It is obvious that ripple voltage and ripple current should be used in (13).
3.2
Inductance estimation
During most of the launch, the largest voltage drop occurs across the machine inductance, hence estimation of this parameter is very important. Just as resistance represents a linear relation between current and voltage, inductance describes a linear relation between current and flux. Hence the estimator in (13) can be modified to estimate inductance according to: ˆ = Φ · i/ (sτe + 1) L i2 / (sτe + 1)
(14)
1
The high-pass and low-pass filters have the same time-constant. Another implicit parameter is the DC gain of the low-pass filter. We used a DC gain of τ , and a high-pass filter with HF gain 1. Scaling can be chosen different if desired, as long as this ratio is preserved. 2
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EPE 2003 - Toulouse
ISBN : 90-75815-07-7
P.6
Sensorless Control of a 2.4MW Linear Synchronous Motor for launching roller-coasters
JONKER Marco
The block diagram of this estimator is shown in figure 4. In order to be able to identify
Figure 4: Inductance estimator.
ˆ the signal should be persistently exciting [1]. The spectral richness of the inductance value L, terminal voltage and current during a launch are shown in the time-frequency spectra in figure 5 and 6. The upper half of figure 5 shows the positive frequency components up to 1kHz. The lower half shows the negative frequency components down to -1kHz.
Frequency [kHz] →
1
0
−1 0
1
2
3
4
5 Time [s] →
Figure 5: Frequency spectrogram of terminal voltage utαβ , inverter active till t = 3.2s.
Frequency [kHz] →
1
0
−1 0
1
2
3
4
5 Time [s] →
Figure 6: Frequency spectrogram of measured current iαβ
During a launch, the pushercar accelerates from 0 to 25m/s. This corresponds to an electrical frequency of 0 to 125Hz. In figure 5 and 6, the fundamental frequency of the motor corresponds with the bold white line that rises from 0 at t = 0s to 125Hz at t = 3.2s. The positive and negative harmonic frequency components are the thinner white lines above and under the fundamental component. These harmonic components are located at multiples of the fundamental 7
EPE 2003 - Toulouse
ISBN : 90-75815-07-7
P.7
Sensorless Control of a 2.4MW Linear Synchronous Motor for launching roller-coasters
JONKER Marco
frequency. Besides the fundamental frequency, the most prominent components are the negative sequence (-1st ) and the prominent emf component: the negative 5th . For t < 3.2 the background is light gray because of the presence of switching ripple. When the desired speed is reached at t = 3.2s, the inverter switches off and the current (figure 6) and the associated switching ripple disappear, but the emf with its harmonics continues in figure 5. To calculate the inductance L and the resistance R it is necessary to remove components that represent transduction of electrical power to mechanical power, which are mainly the fundamental and the negative sequence. To maintain the correlation, the current- and flux signals should pass through identical filters. Removing of harmonics is done by synchronous frame filtering as described in figure 3.
4
Experimental results
In the following three figures some of the practical results are shown.
Inductance [mH] →
0.4 0.3 0.2 0.1 0
5
10
15
20
25
30
35
40
45 50 55 Distance [m] →
Inductance [mH] →
0.35 L
reduced
0.3
0.25 0.2
0.15 8.6
8.7
8.8
8.9
9
9.1
9.2
9.3 9.4 Distance [m] →
Figure 7: Estimated inductance from measured data, fundamental component ω1 and negative sequence ω−1 removed.
An example of the wide range of inductance variation during one launch is depicted in figure 7. The descending-staircase-like behavior is obvious. The largest step occurs between x = 8.9...9.0m (after 3m of travel). The speed of the pushercar is 9.5m/s at this location, meaning that this transition only takes 10ms! The inductance estimator is able to follow this change whithout any visible unsteadyness in figure 8. When comparing the estimated inductance in figure 7 and the theoretical value in figure 8, a mismatch of a factor of 3 is observed, due to a distinct definitions, the first in ∆, the latter in Y. The overall block diagram of the tested sensorless estimator, able to start from standstill, is given in figure 9, the resulting current waveforms are shown in figure 10.
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EPE 2003 - Toulouse
ISBN : 90-75815-07-7
P.8
Sensorless Control of a 2.4MW Linear Synchronous Motor for launching roller-coasters
JONKER Marco
Figure 8: Realized current at largest inductance step.
Figure 9: Block diagram of sensorless estimator.
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EPE 2003 - Toulouse
ISBN : 90-75815-07-7
P.9
Sensorless Control of a 2.4MW Linear Synchronous Motor for launching roller-coasters
JONKER Marco
4000 i
α
iβ
3000 2000
i [A] →
1000 0 −1000 −2000 −3000 −4000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8 0.9 time [s] →
Figure 10: Sensorless start from standstill.
5
Conclusions
A method to estimate the mechanical position in a large 2.4MW LSM from just measured voltages and currents has been successfully demonstrated. Ripple caused by the PWM frequency inverter is used to estimate inductance and resistance with high bandwidth (over 20Hz) by means of fundamental removal in the synchronous frame and spread spectrum correlation techniques. The presented method requires a minimum speed, the estimator needs a valid starting position to make a controlled full-thrust start from standstill. At partial thrust more deviation in starting angle can be tolerated. A method to retrieve postion at (near) zero speed has been developed but is not presented in this paper. The LSM controlled by the sensorless position estimator outperformed the sensored controller at the same current level: a few meters less track was needed to reach the same speed.
References [1] Lennart Ljung, System identification; theory for the user, Prentice Hall, 1987. [2] M.J. Corley and R.D. Lorenz, Rotor position and velocity estimation for a permanent magnet synchronous machine at standstill and high speeds, Conf. Rec. 1996 IEEE-IAS Annual Meeting, pp. 36-41. [3] Toshihiko Noguchi, Kazunori Yamada, Seiji Kondo, Isao Takahashi, Initial Rotor Position Estimation Method of Sensorless PM Synchronous Motor with No Sensitivity to Armature Resistance, IEEE Transactions on Industrial Electronics, Vol. 45, NO. 1, February 1998, pp 118-125. [4] A.Veltman, A method and a device for sensorless estimating the relative angular position of the rotor of a three-phase synchronous motor, Patent application, EP1162106, 2001-12-12. [5] A. Veltman, P. van der Hulst, M.C.P. Jonker, J.P. van Gurp Control of a 2.4MW Linear Synchronous Motor for launching roller-coasters, The 17th International Conference on Magnetically Levitated Systems and Linear Drives, Lausanne, Switzerland, sept. 2002.
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