A method of thrust ripple suppression for long stator linear synchronous motor
 Authors: Mu S.^{1}, Kang J.^{1}, Wang S.^{1}, Liu Y.^{1}, He C.^{1}
 Affiliations:
 Tongji University
 Issue: Vol 4, No 2 (2018)
 Pages: 3044
 Section: Review
 URL: https://transsyst.ru/transsyst/article/view/10175
 DOI: https://doi.org/10.17816/transsyst2018423044
 Cite item
Abstract
With the advantages of high speed, low noise and high efficiency, the electromagnetic suspension (EMS) type maglev train has a good prospect in railway transportation. It is based on the long stator linear synchronous motor (LSLSM). However, due to cogging effect, end effect and the harmonics in the stator current and flux density distribution around the airgap, the thrust generated by the LSLSM fluctuates. The thrust ripple brings noise, drop of control accuracy, even causes the resonance of train. In this paper, the thrust ripple produced by the cogging effect and flux linkage harmonics is analyzed. Then a method of harmonic current injection is proposed to compensate cogging force and reduce the thrust ripple, without influence the decoupling control of traction and suspension system. The injected current harmonics are controlled under multiple rotating reference frames independently. Finally, based on voltage equations of harmonics, the decoupled harmonic current controllers with harmonic voltage feedforward are designed, which improve the performance of current harmonics response and thrust ripple suppression. Simulation results on Simulink verify the effectiveness of proposed thrust ripple suppression method for LSLSM.
Full Text
INTRODUCTION
Without contact between vehicle and railway, maglev train is a one of the best options for future highspeed ground transportation system. The electromagnetic suspension (EMS) type maglev transportation system is one representative, and has been applied in the first commercial operation maglev line of world built in Shanghai. It shows advantages of high speed, low noise and high efficiency. The traction of vehicle is based on the long stator linear synchronous motor (LSLSM), where the railway is motor’s long stator and the vehicle is the mover.
However, it suffers from the propulsion force fluctuation in operation, which brings noise, deteriorates control accuracy, even causes resonance of vehicle, makes passengers uncomfortable. Generally speaking, nonideal factors in practical LSLSM drive system are main reasons of thrust ripple, such as nonsinusoidal stator currents [1], cogging effect, end effect [2], harmonics of flux linkage, etc.
A lot of research works have been carried out to reduce the thrust ripple of LSM both from motor design and control algorithm. Many methods are proposed on motor design and reduce thrust ripple by optimizing the structure, including magnet skewing or slot skewing, optimizing polearc coefficients [3], auxiliary pole[4] and unequal pole pitch of the stator and mover [5], etc. But the structure optimizing is not a universal method for all the LSM, and hard to eliminate all thrust harmonics.
Other researchers focus on the control strategies to suppress thrust ripple. Some methods to reduce the stator current harmonics by compensating the dead time effect of inverter are proposed [6, 7]. But it can’t suppress the main thrust ripple caused by nonsinusoidal distributed magnetic field and cogging force. One way is to obtain the magnitude and phase of thrust ripple and compensate it on the control command. Adaptive algorithm [8], disturbance observer [9], repetitive control [10], iterative learning control [11] are studied since thrust ripple is periodical. Though stator current harmonics will cause electromagnetic force ripple, adequate electromagnetic force harmonics can counteract the reluctance force ripple of motor，which is the essence of suppressing thrust ripple. In [12], harmonic injection is proposed to reduce thrust ripple in linear fluxswitching motor. But the harmonic current is controlled by current hysteresis controller, thus the switching frequency is variable. [13] introduces multiple reference frames to solve the bandwidth limits of traditional PI current controller in permanent magnet machines (PMSM). But the coupling of harmonic currents is not considered. In [14], harmonic voltage and current coupling model of PMSM is studied, improves the injection effect of harmonic current and torque ripple reduction performance.
This paper presents a thrust ripple suppression method by harmonic current injection with harmonic voltage feedforward for maglev LSLSM drives. The drive system is based on the rotor flux oriented control (RFOC), and only qaxis current is injected into current harmonics, which won't influence the decoupling of traction and suspension system [15]. Multiple reference frames, harmonic current decoupling and harmonic voltage feedforward are applied in the system, which overcome the bandwidth limits of conventional PI current controller and improve the effect of harmonic current injection and suppression of thrust ripple. The paper starts with introduction of the model of LSLSM, considering cogging force and exciting flux linkage harmonics. Then, thrust ripple suppression method is performed, including determination of proper injected harmonic current and control of reference harmonic current. In the following, simulation results on the LSLSM are presented. The conclusion is given in the end.
THRUST RIPPLE OF LONG STATOR SYNCHRONOUS MOTOR
Modelof long stator synchronous motor
Assuming symmetry of three phase stator winding and ignoring magnetic saturation, hysteresis, and eddy current, the mathematical model of LMSM in the rotor flux synchronous rotating coordinates can be written as:
$\left\{\begin{array}{l}{u}_{d}={R}_{s}{i}_{d}+{L}_{d}\frac{d{i}_{d}}{dt}\frac{\mathrm{\pi}}{{\mathrm{\tau}}_{s}}\nu {\mathrm{\psi}}_{q}\\ {u}_{q}={R}_{s}{i}_{q}+{L}_{q}\frac{d{i}_{q}}{dt}+\frac{\mathrm{\pi}}{{\mathrm{\tau}}_{s}}\nu {\mathrm{\psi}}_{d}\\ \frac{d\nu}{dt}=\frac{1}{m}\left({F}_{x}{F}_{z}\right)\end{array}\right.$ (1)
where ${u}_{d}$, ${u}_{q}$, ${i}_{d}$ and ${i}_{q}$ are the stator voltages and currents on $d,q$ axes, respectively; ${\mathrm{\psi}}_{d}$ and ${\mathrm{\psi}}_{q}$ are the stator flux on $d,q$ axes respectively ; ${R}_{s}$ is total stator resistance; ${L}_{d}$, ${L}_{q}$ are the total stator inductance on $d,q$ axes respectively; ${\mathrm{\tau}}_{s}$ is the pole pitch; v is the mover speed; m is the vehicle mass; ${F}_{x}$ is the thrust force; ${F}_{z}$ is the overall resistance force, consist of air resistance force, magnet resistance force and generator resistance force.
Unlike general rotary motor, the impedance of feed cable, stator section uncovered by vehicle pole and stator section covered by vehicle pole comprise the total stator resistance and inductance of LSLSM [16]. Since the length of stator section is much longer than vehicle length, the leak inductance of stator section uncovered by vehicle makes up the main part of total stator inductance, which leads to inductance ${L}_{d}$ approximately equal to ${L}_{q}$.
The stator flux equation on $d,q$ axes is expressed as:
$\left\{\begin{array}{l}{\mathrm{\psi}}_{d}={L}_{d}{i}_{d}+{\mathrm{\psi}}_{df}\\ {\mathrm{\psi}}_{q}={L}_{q}{i}_{q}+{\mathrm{\psi}}_{qf}\end{array}\right.$ (2)
where ${\mathrm{\psi}}_{df}$ and ${\mathrm{\psi}}_{qf}$ are the mover exciting flux linkage on $d,q$ axes, produced by excitation winding of vehicle suspension electromagnetic.
The thrust force of LSLSM is composed of electromagnetic force ${F}_{e}$ and cogging force ${F}_{cog}$, i.e.,
${F}_{x}={F}_{e}+{F}_{cog}$ (3)
where ${F}_{cog}$ is the reluctance force due to slot effect; ${F}_{e}$ is produced by the interaction between airgap flux and the stator currents, and equal to the mean value of thrust force. It can be expressed as:
${F}_{e}=\frac{3\mathrm{\pi}}{2{\mathrm{\tau}}_{s}}\left({\mathrm{\psi}}_{d}{i}_{q}{\mathrm{\psi}}_{q}{i}_{d}\right)=\frac{3\mathrm{\pi}}{2{\mathrm{\tau}}_{s}}\left[\left({\mathrm{\psi}}_{df}+{L}_{d}{i}_{d}\right){i}_{q}\left({\mathrm{\psi}}_{df}+{L}_{d}{i}_{q}\right){i}_{d}\right]$ (4)
Due to decoupling the armature field and excitation field, the rotor fieldoriented control (RFOC) and ${i}_{d}=0$ strategy are commonly applied for LSLSM control in practical[15]. In this case, (4) can be simplified as:
${F}_{e}=\frac{3\mathrm{\pi}}{2{\mathrm{\tau}}_{s}}{\mathrm{\psi}}_{df}{i}_{q}$ (5)
Flux harmonics
In an ideal LSLSM, where excitation flux is ideal sinusoidal distributed in the air gap, ${\mathrm{\psi}}_{qf}$ is equal to zero and ${\mathrm{\psi}}_{df}$ is constant under RFOC control. However, because of manufacturing restrictions, there are 6kth spatial harmonics in practical in the mover flux linkage in practical, and can be expressed in Fourier series as:
$\left\{\begin{array}{l}{\mathrm{\psi}}_{df}={\mathrm{\psi}}_{d0}+\sum _{k=1}^{+\infty}{\mathrm{\psi}}_{d\pm 6k}{e}^{i\left(\pm 6k\frac{\mathrm{\pi}x}{{\mathrm{\tau}}_{s}}\right)}={\mathrm{\psi}}_{d0}+\sum _{k=\pm 1,\pm 2,\pm 3,\dots}{\mathrm{\psi}}_{d6k}\mathrm{cos}\left(6k\frac{\mathrm{\pi}x}{{\mathrm{\tau}}_{s}}\right)\\ {\mathrm{\psi}}_{qf}={\mathrm{\psi}}_{q0}+\sum _{k=1}^{+\infty}{\mathrm{\psi}}_{q\pm 6k}{e}^{i\left(\pm 6k\frac{\pi x}{{\tau}_{s}}\right)}=\sum _{k=1,2,3,\dots}{\psi}_{q6k}\mathrm{cos}\left(6k\frac{\pi x}{{\tau}_{s}}\right)\end{array}\right.$ (7)
where $x$ is the mover position relative to phase A; ${\psi}_{d6k}$ and ${\psi}_{q6k}$ are the harmonic coefficients of 6kth flux harmonic on $d,q$ axes respectively. The dc component of $d$ axis flux linkage ${\psi}_{d0}$ is nonzero, while ${\psi}_{q0}$ equals to zero.
According to the thrust equation, the spatial harmonics of flux linkage will result in the 6kth periodical thrust ripple of real LSLSM with sinusoidal fed stator currents. Moreover, it brings harmonics in back electromotive force (EMF), and deteriorate the control effect of current controller.
Cogging force
The stator windings are placed in the stator slots on LSLSM of maglev. When the mover poles approaching or leaving the stator teeth, the reluctance and magnetic field distribution vary. That leads to the fluctuation of magnetic energy of motor, generating additional reluctance force on mover. The cogging force is independent of the stator current while closely related to mover potion. The period of cogging force is the distance between neighboring slots. Since there are six slots at each pole pair, the cogging force fluctuates 6 times within one pair of poles. Fourier expression of cogging force can be written as
${F}_{\mathrm{cog}}={F}_{c0}+\sum _{k=1}^{+\infty}{F}_{c\pm 6k}{e}^{i\left(\pm 6k\frac{\mathrm{\pi}x}{{\mathrm{\tau}}_{s}}\right)}$ (8)
where ${F}_{c6k}$ is the harmonic coefficients of 6kth component of Fourier series, and the dc component ${F}_{c0}$ is equal to zero.
HARMONIC CURRENT INJECTION
Reference Hamonic Current
Taking Fourier expressions of flux linkage equation (7) and cogging force equation (8) into the thrust expression (3), it yields
${F}_{x}=\frac{3\mathrm{\pi}}{2{\mathrm{\tau}}_{s}}\left(\sum _{k=0}^{+\infty}{\mathrm{\psi}}_{d\pm 6k}{e}^{i\left(\pm 6k\frac{\mathrm{\pi}x}{{\mathrm{\tau}}_{s}}\right)}\right){i}_{q}+\sum _{k=0}^{+\infty}{F}_{c\pm 6k}{e}^{i\left(\pm 6k\frac{\mathrm{\pi}x}{{\mathrm{\tau}}_{s}}\right)}$ (9)
The formula shows that there will be 6kth periodical thrust ripple of LSLSM if ${i}_{q}$ is kept constant when flux linkage harmonics and cogging force exist. One way to compensate the cogging force and suppress thrust ripple is to inject harmonic current into qaxis current. To generate 6kth periodical electromagnetic force, 6kth current harmonics are needed. The desired qaxis current can be assumed as
${{i}_{q}}^{*}={i}_{0}+\sum _{k=1}^{+\infty}{i}_{\pm 6k}{e}^{i\left(\pm 6k{\mathrm{\pi \theta}}_{e}+\frac{\mathrm{\pi}}{2}\right)}$ (10)
where ${i}_{6k}$ are the complex coefficients of 6kth harmonics; ${\theta}_{e}$ is electrical angle of mover, and satisfies ${\theta}_{e}=\pi x/{\tau}_{s}$; $\pi /2$ is added since qaxis component is with a 90 degree phase lead.
Substituting (8) and (9) into (7), and Supposing the desired thrust is ${{F}_{x}}^{*}$, it yields the following formula
$\frac{3\mathrm{\pi}}{2{\mathrm{\tau}}_{s}}\left(\sum _{k=0}^{+\infty}{\mathrm{\psi}}_{d\pm 6k}{e}^{i\left(\pm 6k\frac{\mathrm{\pi}x}{{\mathrm{\tau}}_{s}}\right)}\right)\left({i}_{0}+\sum _{k=1}^{+\infty}{i}_{\pm 6k}{e}^{i\left(\pm 6k{\mathrm{\pi \theta}}_{e}+\frac{\mathrm{\pi}}{2}\right)}\right)+\sum _{k=1}^{+\infty}{F}_{c\pm 6k}{e}^{i\left(\pm 6k\frac{\mathrm{\pi}x}{{\mathrm{\tau}}_{s}}\right)}={{F}_{x}}^{*}$ (11)
Numerous linear equations can be derived from above expression and thrust ripple of any order can be eliminated in theory if the number and amplitude of current harmonics are not limited. However, the bandwidth of controller is limited in reality, it will cause large error when tracking reference currents of high frequency. Meanwhile, too many currents harmonics also increase the burden of controller. Since the 6th harmonic thrust accounts for the biggest part of thrust ripple[5], only 6th current harmonics are needed to suppress the harmonic thrust ripple. In this case, equation (11) can be expressed as
$\frac{3\mathrm{\pi}}{2{\mathrm{\tau}}_{s}}\left({\mathrm{\psi}}_{d0}+{\mathrm{\psi}}_{d6}{e}^{j6\frac{\mathrm{\pi}x}{{\mathrm{\tau}}_{s}}}+{\mathrm{\psi}}_{d6}{e}^{j6\frac{\mathrm{\pi}x}{{\mathrm{\tau}}_{s}}}\right)\left({{i}_{q}}_{0}+i{i}_{6}{e}^{j6\frac{\mathrm{\pi}x}{{\mathrm{\tau}}_{s}}}+i{i}_{6}{e}^{j6\frac{\mathrm{\pi}x}{{\mathrm{\tau}}_{s}}}\right)+{F}_{c6}+{F}_{c6}={{F}_{x}}^{*}$ (12)
By combining the same order harmonic terms, it yields the following equation
$\frac{3\mathrm{\pi}}{2{\mathrm{\tau}}_{s}}\left[\begin{array}{ccc}{\mathrm{\psi}}_{d0}& i\bullet {\mathrm{\psi}}_{d6}& i\bullet {\mathrm{\psi}}_{d6}\\ i\bullet {\mathrm{\psi}}_{d6}& {\mathrm{\psi}}_{d0}& 0\\ i\bullet {\mathrm{\psi}}_{d6}& 0& {\mathrm{\psi}}_{d0}\end{array}\right]\left[\begin{array}{c}{{i}_{q}}_{0}\\ {i}_{6}\\ {i}_{6}\end{array}\right]=\left[\begin{array}{c}{{F}_{x}}^{*}\\ {F}_{c6}\\ {F}_{c6}\end{array}\right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$ (13)
From (13), the amplitude of desired qaxis current can be solved. Here the 12th electromagnetic force harmonic equations are neglected, since the force generated by harmonic current and flux harmonic is relatively small and will make the equation has no solution.
Multiple rotating Reference Frames
PI controllers are applied in traditional RFOC of LSLSM as $d,q$ axes current controller. Restricted by bandwidth, it is difficult for PI controller to track harmonics reference current in $d,q$ coordinate, especially at high speed. However, these harmonics in $d,q$ coordinate can be converted to dc component in the coordinate rotating at the same frequency of harmonic. In this case, the synchronous reference frames rotating at multiple times the speed of $d,q$ coordinate are introduced and stator currents with different frequencies are controlled independently. The ±6th reference harmonics current in $d,q$ coordinate corresponds to 7^{th} and 5^{th} harmonic in stationary reference frame, respectively. The multiple reference frames introduced in this paper are depicted as diagram below.
Fig. 1. The multiple reference frames of LSLSM, where $A,B,C$ and $\alpha ,\beta $ are the stationary reference frame; $d,q$ coordinate rotate synchronously with mover, and $\omega =\pi v/{\tau}_{s}$ is the synchronous speed corresponding to electric angular speed of mover; $d5,q5th$ coordinate rotate at 5 times the speed of $d,q$ coordinate reversely; $d7,q7th$ coordinate rotate at 7 times the speed of $d,q$ coordinate in the same direction
The transformation matrix from $\alpha ,\beta $ coordinate to multiple rotating reference frames can be expressed as
$T\left(k{\mathrm{\theta}}_{e}\right)=\left[\begin{array}{cc}\mathrm{cos}\left(k{\mathrm{\theta}}_{e}\right)& \mathrm{sin}\left(k{\mathrm{\theta}}_{e}\right)\\ \mathrm{sin}\left(k{\mathrm{\theta}}_{e}\right)& \mathrm{cos}\left(k{\mathrm{\theta}}_{e}\right)\end{array}\right]$ (14)
where $k=1,5,7\dots $.
After coordinate transformation, current components with same angular speed with reference frame become constant, while others are alternating. Then the dc component can be abstracted after a lowpass filter, which is the magnitude of real current harmonic. Thus, the closedloop feedback control of current harmonics can be established as shown below
Fig. 2. The diagram of harmonic current closedloop feedback control, where $id{k}^{*}$ and $iq{k}^{*}$ are the reference values of kth harmonics current in kth reference frame
and satisfy
$\left\{\begin{array}{l}id{5}^{*}+i\bullet iq{5}^{*}={i}_{6}\\ id{7}^{*}+i\bullet iq{7}^{*}={i}_{6}\end{array}\right.$ (15)
Harmonic Voltage Decouple and Feedforward Control
In conventional $d,q$ axes current controller, the $d,q$ axes currents are decoupled by compensation of EMF, the voltage equation in the $d,q$ axes after compensation of EMF can be written as
$\left\{\begin{array}{l}{{u}_{d}}^{*}={u}_{d}+\frac{\mathrm{\pi}}{{\mathrm{\tau}}_{s}}\nu {\mathrm{\psi}}_{q}={R}_{s}{i}_{d}+{L}_{d}\frac{d{i}_{d}}{dt}\\ {{u}_{q}}^{*}={u}_{q}\frac{\mathrm{\pi}}{{\mathrm{\tau}}_{s}}\nu {\mathrm{\psi}}_{d}={R}_{s}{i}_{q}+{L}_{q}\frac{d{i}_{q}}{dt}\end{array}\right.$ (16)
where ${{u}_{d}}^{*}$ and ${{u}_{q}}^{*}$ are $d,q$ axes voltage generated by current PI controllers respectively.
Substitute (15) into (11), the desired current in $d,q$ coordinate can be rewritten as
$\left\{\begin{array}{l}{{i}_{d5}}^{1}={i}_{d5}\mathrm{cos}(6\mathrm{\omega}t){i}_{q5}\mathrm{sin}(6\mathrm{\omega}t)\\ {{i}_{q5}}^{1}={i}_{d5}\mathrm{sin}(6\mathrm{\omega}t)+{i}_{q5}\mathrm{cos}(6\mathrm{\omega}t)\\ {{i}_{d7}}^{1}={i}_{d7}\mathrm{cos}\left(6\mathrm{\omega}t\right){i}_{q7}\mathrm{sin}\left(6\mathrm{\omega}t\right)\\ {{i}_{q7}}^{1}={i}_{d7}\mathrm{sin}\left(6\omega t\right)+{i}_{q7}\mathrm{cos}\left(6\omega t\right)\end{array}\right.$ (17)
where ${{i}_{d5}}^{1}$ and ${{i}_{q5}}^{1}$ are the voltage of 5th current harmonic in $d,q$ coordinate; ${{i}_{d7}}^{1}$ and are the voltage of 7th current harmonic in $d,q$ coordinate.
Since the inductance ${L}_{d}$ and ${L}_{q}$ are approximately equal in LSLSM, $L={L}_{d}={L}_{q}$ is introduced to simplify the equation. By substituting (17) into (16), the harmonic voltage equations can be written as
$\left\{\begin{array}{l}{{u}_{d5}}^{1}={R}_{s}{{i}_{d5}}^{1}+6\mathrm{\omega}L{{i}_{q5}}^{1}\\ {{u}_{q5}}^{1}={R}_{s}{{i}_{q5}}^{1}6\mathrm{\omega}L{{i}_{d5}}^{1}\\ {{u}_{d7}}^{1}={R}_{s}{{i}_{d7}}^{1}6\mathrm{\omega}L{{i}_{q7}}^{1}\\ {{u}_{q7}}^{1}={R}_{s}{{i}_{q7}}^{1}+6\mathrm{\omega}L{{i}_{d7}}^{1}\end{array}\right.$ (18)
where ${{u}_{d5}}^{1}$ and ${{u}_{q5}}^{1}$ are the voltage of 5th current harmonic in $d,q$ coordinate; ${{u}_{d7}}^{1}$ and are the voltage of 7th current harmonic in $d,q$ coordinate.
By rotating coordinate transformation of the voltage equations in $d,q$ coordinate, the voltage equations of 5th current harmonic in $d5,q5$ coordinate can be expressed as
$\left\{\begin{array}{l}{{u}_{d5}}^{5}={R}_{s}{i}_{d5}+6\mathrm{\omega}L{i}_{q5}\\ {{u}_{q5}}^{5}={R}_{s}{i}_{q5}6\mathrm{\omega}L{i}_{d5}\end{array}\right.$ (19)
Similarly, the voltage equations of 7th current harmonic in $d7,q7$ coordinate can be expressed as
$\left\{\begin{array}{l}{{u}_{d7}}^{7}={R}_{s}{i}_{d7}6\mathrm{\omega}L{i}_{q7}\\ {{u}_{q7}}^{7}={R}_{s}{i}_{q7}+6\mathrm{\omega}L{i}_{d7}\end{array}\right.$ (20)
To increase the response speed of harmonics, the voltage feedforward is introduced. The feedforward voltage can be calculated according to reference currents and voltage equations, that is
$\left\{\begin{array}{l}{u}_{d5\_com}={R}_{s}{{i}_{d5}}^{*}+6\mathrm{\omega}L{{i}_{q5}}^{*}\\ {u}_{q5\_com}={R}_{s}{{i}_{q5}}^{*}6\mathrm{\omega}L{{i}_{d5}}^{*}\\ {u}_{d7\_com}={R}_{s}{{i}_{d7}}^{*}6\mathrm{\omega}L{{i}_{q7}}^{*}\\ {u}_{q7\_com}={R}_{s}{{i}_{q7}}^{*}+6\mathrm{\omega}L{{i}_{d7}}^{*}\end{array}\right.$ (21)
where ${u}_{i\_com},i=d5,q5,d7,q7$ are the feedforward voltages on $d5,q5$ axes and $d7,q7$ axes, respectively.
From harmonic voltage equations in (19) and (20), it can be seen the currents in $dk,qk$ axes are coupled. Thus, the decoupled harmonic current controllers with voltage feedforward are designed. The following diagram shows the structure of 5th current harmonic controller.
Fig. 3. The diagram of 5th harmonic current closedloop feedback control
Where errors of harmonic current are sent to PI controller, whose output is also current. The final output of harmonic current controller is harmonic voltage, made up by feedback control voltage and feedforward steady state voltage.
The controller of 7^{th} current harmonic can be designed in the same way according to (19) and (21).
SIMULATION AND RESULTS
The performance of the proposed thrust ripple suppression method for LSLSM has been tested through simulation experiments on MATLAB/Simulink. The LSLSM applied in simulation is one stator section of 960 m long with one maglev frame. The maglev frame of equal pole pitch is built in Maxwell software as shown in Fig. 4.
Fig. 4. Simulation model of LSLSM in Maxwell of equal pole pitch
Through finite element calculation, not only the parameters of maglev frame, but also the flux linkage with fluctuation and cogging force are obtained, considering nonlinear properties of LSM. From that, the 6th harmonics of flux linkage and cogging force are obtained by Fourier transformation, and considered in the LSLSM simulation model. The main parameters of LSLSM are listed in Table 1, where the parameters of stator are based on the maglev test line.
Table 1. Parameters of LSLSM
Parameter  Value  Parameter  Value 
Stator resistance (Ω)  0.2237  Excitation flux (wb)  0.1171 
Stator inductance Ld (mH)  2.4817  Pole pich (mm)  266.5 
Stator inductance Lq (mH)  2.4764  Mass (kg)  3000 
The block diagram of the thrust controlled LSLSM drive system is depicted in Fig. 5. Here only current closed loop control is included instead of positionspeedcurrent control because it’s enough to validate the effect of thrust ripple suppression. The LSLSM is fed by three level neutral point clamped voltage source inverter, whose rated voltage is 1900 V and rated current is 1200 A. The switching frequency of inverter and control frequency of current controller are 2 kHz. At first, the optimal reference current are determined by the current assignment block based on equation (13) and (15). The stator currents of different frequency are controlled independently in multiple reference frames. By coordinate transformations and lowpass filters, the feedback of current in multiple rotating reference frames are obtained from the measured stator current. The controller of current on axes is a decoupled PI controller with voltage compensation for EMF. The 5^{th} and 7^{th} current harmonics are controlled by decoupled PI controllers with steady state voltage feedforward on and coordinate, respectively. Finally, the control voltage of each current controller are transformed to stationary reference frame, and together make up the voltage reference and of SVPWM, which generates gate signals of inverter.
Fig. 5. The block diagram of thrust controlled drive system for LSLSM
In Fig. 6–8，results of one simulation test are shown. In this test, the thrust ripple of without harmonic current injection, harmonic current injection by proposed harmonic current controller and injection by conventional PI current controller are compared. From 0–1s, no harmonic current is injected into stator current. The proposed thrust ripple suppression strategy is applied at 1.0 s. From 2.0 s to the end, the injected harmonic current is controlled only by PI controllers on $d,q$ axes. The load force is 1500 N and the maglev frame is propelled to 10.66 and corresponding fundamental frequency of stator current is 20 .
Fig. 6 shows the waveforms of thrust force. There are severe 6^{th} thrust ripple before 1.0 s. The magnitude of thrust ripple is about 150 N, approximately to the cogging force. After applying the proposed harmonic current injection method, the thrust ripple is reduced significantly. But conventional PI controller can’t control harmonic currents well, as the thrust ripple is nearly reduced after 2.0 s. The thrust waveforms during switching algorithm period are shown in Fig. 6 (b) and 6 (c). Fig. 6(b) shows the proposed algorithm can suppress thrust ripple very fast.
Spectrum analysis of thrust at 0.7 s, 1.7 s and 2.7 s are shown in Fig. 7 (a–c), respectively. The original magnitude of 6th thrust ripple is 10.38 % of the average thrust, about 155 N. After injection of harmonic current, the magnitude of 6th thrust ripple is reduced to 0.41 % of the average thrust, about 7 N. Meanwhile, the magnitude of 12th thrust ripple remains almost the same. It shows good effect of the proposed thrust ripple suspension method. It can be seen from Fig. 7(c) the harmonic current injection by current loop on axes can't reduce thrust ripple but slightly increase it.
Fig. 7. The Spectrum diagram of thrust force at different times
The response of the currents on $d,q$ axes are shown in Fig. 8. The current waveforms during switching algorithm are shown in Fig. 8(b) and 8(c). It can be seen current harmonics are injected into qaxis current after 1.0 s in Fig. 8(a). The waveform in Fig. 8(b) shows that the real currents can follow the reference values accurately and quickly, which validate the effectiveness of the proposed harmonic current controller. From Fig. 8(c), it can be seen there are large phase delay between reference harmonic current and feedback. That’s why harmonic current injection by conventional PI current controller are not able to suppress thrust ripple well.
Fig. 8. The waveforms of stator current on axes
CONCLUSION
A thrust ripple suppression method by harmonic current injection is presented in this paper. The LSLSM model considering flux linkage harmonics and cogging force is established, which are the main sources of thrust ripple. The flux linkage harmonics and cogging force are calculated by finite element method in Maxwell software. Harmonic current injection is proposed to generate harmonic electromagnetic force of opposite phase. The method to determine proper stator current is derived. The adequate stator current is made up by fundamental and 5^{th} and 7^{th} harmonics, which can suppress the main 6th thrust ripple of LSLSM effectively. Then, multiple rotating reference frames are introduced, since the alternating current and voltage become constant values in the corresponding rotating coordinate. Harmonic currents are controlled independently in their synchronous rotating reference coordinate, which decoupled the control of fundamental and harmonic current, and overcome the bandwidth limitation of PI controller on axes current. At last, the voltage model of harmonic current is studied. Based on the voltage equations, the decoupled harmonic current controllers are designed. The steady state voltage feedforward of harmonics current is applied, which increase response speed of harmonics current. Simulation test results validate the proposed scheme can suppress the 6th thrust ripple of LSLSM effectively and quickly.
ACKNOWLEDGEMENTS
This work was supported by the National Key Technology R&D Program of China (2016YFB120060202).
About the authors
Siyuan Mu
Tongji University
Email: siyuanmu@tongji.edu.cn
ORCID iD: 0000000230061301
China, No.4800, Cao An Road, Shanghai
Bachelor of science, PhD candidate
Jinsong Kang
Tongji University
Author for correspondence.
Email: kjs@tongji.edu.cn
ORCID iD: 0000000348751818
China, No.4800, Cao An Road, Shanghai
PhD, professor
Shuo Wang
Tongji University
Email: 1988wangshuo@tongji.edu.cn
ORCID iD: 0000000331233942
China, No.4800, Cao An Road, Shanghai
Master of science, PhD candidate
Yusong Liu
Tongji University
Email: liuyusong@tongji.edu.cn
ORCID iD: 000000020520389X
China, No.4800, Cao An Road, Shanghai
Bachelor of science, PhD candidate
Cuiwei He
Tongji University
Email: hcw1964@163.com
China, No.4800, Cao An Road, Shanghai
Master of science, Master
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Supplementary files
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1. 
Fig. 1. The multiple reference frames of LSLSM, where and are the stationary reference frame; coordinate rotate synchronously with mover, and is the synchronous speed corresponding to electric angular speed of mover; coordinate rotate at 5 times the speed of coordinate reversely; coordinate rotate at 7 times the speed of coordinate in the same direction

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