A model predictive current control of flux-switching permanent magnet machines for torque ripple minimization

A model predictive current control of flux switching permanent magnet machines for torque ripple minimization A model predictive current control of flux switching permanent magnet machines for torque[.]

A model predictive current control of flux-switching permanent magnet machines for torque ripple minimization

Wentao Huang, Wei Hua, and Feng Yu

Citation: AIP Advances 7, 056609 (2017); doi: 10.1063/1.4973394

View online: http://dx.doi.org/10.1063/1.4973394

View Table of Contents: http://aip.scitation.org/toc/adv/7/5

Published by the American Institute of Physics

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Back-EMF waveform optimization of flux-reversal permanent magnet machines

AIP Advances 7, 056613 (2016); 10.1063/1.4973498

A model predictive current control of flux-switching

permanent magnet machines for torque

ripple minimization

Wentao Huang,1Wei Hua,1, aand Feng Yu2

1School of Electrical Engineering, Southeast University, Nanjing, 210096 China

2Department of Electrical Engineering, Nantong University, Nantong, 226019 China

(Presented 2 November 2016; received 21 September 2016; accepted 17 October 2016;

published online 27 December 2016)

Due to high airgap flux density generated by magnets and the special double salient structure, the cogging torque of the flux-switching permanent magnet (FSPM) machine

is considerable, which limits the further applications Based on the model predic-tive current control (MPCC) and the compensation control theory, a compensating-current MPCC (CC-MPCC) scheme is proposed and implemented to counteract the dominated components in cogging torque of an existing three-phase 12/10 FSPM prototyped machine, and thus to alleviate the influence of the cogging torque and improve the smoothness of electromagnetic torque as well as speed, where a com-prehensive cost function is designed to evaluate the switching states The simulated results indicate that the proposed CC-MPCC scheme can suppress the torque ripple significantly and offer satisfactory dynamic performances by comparisons with the conventional MPCC strategy Finally, experimental results validate both the

theoret-ical and simulated predictions © 2016 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/ ) [http://dx.doi.org/10.1063/1.4973394]

I INTRODUCTION

Flux-switching permanent magnet (FSPM) machines have attracted wide attention due to simple and robust structure, high power/torque density, and strong fault-tolerance ability, especially in the fields of hybrid/electric vehicles and wind power generation in recent years.1 4However, due to the high airgap flux density produced by magnets and the special double salient structure, the cogging torque of FSPM machines is relatively large, resulting in unfavorable torque and speed ripples Existing methods to minimize the torque ripple can be categorized into two types briefly The first method concentrates on the design and optimization of the machine itself.5 , 6 However, these techniques can reduce the torque ripple to some extent and may result in the increasing manufacturing cost and decreased performance The second method focuses on the control side, such as compensation control.6 The typical approach of the compensation control is injecting compensating harmonic currents or voltages into a three-phase FSPM machine to generate additional torque components counteracting the cogging torque.7The proposed scheme can suppress the torque ripple effectively, however, in practice, it is difficult to tune three proportional-integrals (PIs) well for satisfactory response to every dynamic scenario in vector control structure

On the other hand, model predictive control (MPC) has been widely researched in conjunction with motor drives, and the most popular type is the finite-control set MPC (FCS-MPC).8 10Although the model predictive current control (MPCC) strategy has been investigated to reduce torque ripple and switching loss of a three-phase FSPM machine,9and to improve the dynamic performance of a nine-phase FSPM machine,10respectively, however, the cogging torque has not been considered

a Electronic mail: huawei1978@seu.edu.cn

2158-3226/2017/7(5)/056609/6 7, 056609-1 © Author(s) 2016

056609-2 Huang, Hua, and Yu AIP Advances 7, 056609 (2017)

FIG 1 Electromagnetic performance of a FSPM machine (a) Cogging torque waveform (b) Electromagnetic torque waveforms.

Hence, in this paper to reduce the cogging torque of a three-phase 12/10 FSPM proto-typed machine, a torque ripple minimization scheme based on the compensating-current MPCC (CC-MPCC) is proposed and implemented Firstly, a detailed analysis and modeling of the cogging torque is conducted Secondly, the compensating-current model is established to coun-teract the dominated components of cogging torque Thirdly, based on a CC-MPCC structure,

a comprehensive cost function is developed to evaluate the optimum switching states Finally, both simulations and experiments are carried out to validate the effectiveness of the CC-MPCC scheme

II COGGING TORQUE OF A 12/10 FSPM MACHINE

The finite element method (FEM) is employed to predict the cogging torque as shown in Fig.1(a), where the amplitude reaches 1.2Nm, being about 9% of the rated torque (13.38Nm), and produced

by the high flux density in the airgap due to flux-concentration Based on the theoretical analysis, the mathematical expression of the cogging torque can be derived as

T cog=

M X

where, h is the harmonics order, P r is the rotor pole-pairs number T cmhand ϕcoghare the amplitude and phase of the harmonics, respectively θris the rotor position

Fig.1(b)compares the simulated electromagnetic torque waveforms with and without the cogging torque under a rated load Obviously, the torque suffers from cogging torque severely, and hence it

is necessary to compensate the cogging torque

III CC-MPCC SCHEME FOR COGGING TORQUE SUPPRESION

Different from the standard MPCC method, the main idea of the proposed control scheme is to combine the compensating-current into the MPCC method, and to inherit the merits of good dynamics and simple structure, as shown in Fig.2(a)

A Compensating-current model

According to electrical machine theory, a series of specific harmonic currents can be artificially injected into the fundamental component to interact with the quasi-sinusoidal back-EMF, and produce additional electromagnetic torque components to counteract the dominated components of cogging torque Hence, the compensating criterion is expressed as

i c ah e a + i c

where, i c

bh , and i c

ch are the three-phase compensating currents, ωr is the rotor angular velocity,

and T is the hthharmonic component of the cogging torque

FIG 2 CC-MPCC scheme (a) Control block diagram (b) Flow diagram.

Assuming the compensating currents can be given as













i ah c = I mh cos (xP rθr+ ϕh)

i c bh = I mhcosxP rθr+ ϕh+2π

3



i c ch = I mhcosxP rθr+ ϕh−2π

3



(3)

where, I mh , x and ϕ h are the amplitude, order and phase angle of the compensating currents, respectively

Neglecting the influence of the higher order harmonics, the three-phase back-EMF can be expressed as













e a = E m sin (P rωr t)

e b = E msinP rωr t+2π

3



e c = E msinP rωr t −2π3 

(4)

where, E mis the amplitude of the fundamental back-EMF

Substituting equations (3) and (4) into equation (2), equation (2) can be rewritten as

3E m I mh sin((x + 1) P rθr+ ϕh)

then, x = 6h-1, ϕ h= ϕcogh , I mh= 2ωr T cmh /3E m

According to FFT analysis results,7 the cogging torque of the FSPM machine is mainly domi-nated by fundamental and second harmonic components Neglecting the higher order harmonics, the compensating-current model can be derived as

















i c=2ωrT cm1

3Em cos5P rθr+ ϕcog1 +2ωrT cm2

3Em cos11P rθr+ ϕcog2



i c b=2ωrT cm1

3Em cos5P rθr+ ϕcog1+2π

3  + 2ωrT cm2 3Em cos(11P rθr+ ϕcog2+2π

3 )

i c=2ωrT cm1

3Em cos5P rθr+ ϕcog1−2π

3  + 2ωrT cm2 3Em cos(11P rθr+ ϕcog2−2π

3 )

(6)

B Model predictive current control

Using the Euler formula to discretize the current differential equation of the FSPM machine and transferring the results to single time step, the current prediction model is obtained as

" i d (k+ 1)

i q (k+ 1)

#

=









1 −R s

−T s L d



1 − R s











" i d (k)

i q (k)

# +



T s

0 T s







" u d (k)

u q (k)

# +



0

−T s P rωr(k)ψf





 (7)

where, i d (k)/i q (k), and i d (k+1)/i q (k+1) are the measured and predicted values of the dq-axes currents

in the k th and (k+1) th sampling interval, respectively; u (k)/u q (k) are the optimum dq-axes voltages;

056609-4 Huang, Hua, and Yu AIP Advances 7, 056609 (2017)

L d /L q are the dq-axes inductances; R sis the phase resistance; ψf is the amplitude of the phase PM

flux-linkage; T sis the sampling time

The cost function in the conventional MPCC cannot offer explicit constraint on cogging torque

In the improved cost function, the constraint on cogging torque is considered The reference currents and compensating currents are combined as the new given currents, which can produce additional torque components counteracting the cogging torque Finally, a comprehensive cost function with torque ripple minimization is established to evaluate switching states The improved cost function is

expressed by the quadratic current errors of the dq-axes currents as follows

g i = (i d∗+ i c

d−i d (k+ 1))2+ (i q∗+ i c

q−i q (k+ 1))2

(8)

where, i = {0, ,7}; i d /i q are the reference currents in dq-axes; i d /i q are the compensating currents

in dq-axes Fig.2(b)illustrates the flow diagram of the CC-MPCC with compensating currents for the FSPM machine

IV SIMULATIONS AND EXPERIMENTS

A Simulations

Simulations are performed in MATLAB/Simulink, where the cogging torque is obtained by the transient field without armature current injected in Ansoft Maxwell, and it is stored in the look-up table The main parameters are listed in TableI

Figs.3(a)and3(b)compares the simulated phase currents and torques without and with com-pensating currents When the conventional MPCC is applied, the current waveform is sinusoidal and unfortunately, the torque suffers from cogging torque and the torque ripple reaches nearly 2Nm When the CC-MPCC is applied, the THD of phase current increases due to the injected compensating currents However, the torque ripple has been suppressed significantly In addition, it is apparent that the dynamic transition is smooth when the compensating currents are injected

B Experiments

Experiments are conducted on a three-phase 12/10 FSPM machine with the same parameters

in simulations A magnetic powder brake is adopted as a load, and a torque transducer is connected between the magnetic powder brake and FSPM machine for the torque measurement The complete control algorithm is implemented in PC-based DS1104 controller board 10k Hz is selected as the sampling frequency

Figs 3(c)and3(d)depict the measured current, speed and torque of the MPCC without and with compensating currents at the steady-state From the detailed current waveforms in Fig.3(d),

a higher THD can be found with CC-MPCC, which agrees with the simulations The torque ripple

is suppressed from 2.2Nm to 1.2Nm, and the speed response stays unchanged before and after the compensation Figs.3(e)and3(f)show the dynamic performances under the CC-MPCC Fig.3(e)

indicates that the machine accelerates from standstill to 300r/min at a load of 5Nm quickly, and the machine turns into steady-state operation within 100ms Fig.3(f)depicts the dynamic responses to the speed reversal The machine firstly decelerates from 300r/min to 0r/min and then accelerates to the -300r/min at a load of 5Nm Apart from the torque ripple minimization, the CC-MPCC can also offer satisfactory dynamic performances

TABLE I Key parameters of the three-phase 12/10 FSPM machine.

Rated current (rms) 3.8A d-axis PM flux-linkage 0.1657Wb

FIG 3 Simulated and Experimental validations (a) Simulated current waveforms (b) Simulated torque waveforms (c) Measured current, torque, and speed during compensating current injection (d) Measured waveforms during compensating current injection in detail (e) Responses to startup (f) Responses to speed reversal.

V CONCLUSIONS

To alleviate the influences of the cogging torque, a CC-MPCC scheme for torque ripple minimiza-tion of the FSPM machine is proposed in this paper An improved CC-MPCC scheme is developed

by combining the compensating-current with the conventional MPCC structure The effectiveness of the developed CC-MPCC scheme is verified by both simulations and experiments, and the results confirm that the torque and speed ripples can be suppressed significantly under both steady- and dynamic-state operations

ACKNOWLEDGMENTS

This work was supported in part by 973 Program of China (2013CB035603), National Natural Science Foundation of China (51322705), and the Fundamental Research Funds for the Central Universities (2242016K41004)

1 M Cheng, W Hua, J Zhang, and W X Zhao, IEEE Trans Ind Electron.58, 5087 (2011).

2 W Hua, M Cheng, and G Zhang, IEEE Trans Magn.45, 4728 (2009).

056609-6 Huang, Hua, and Yu AIP Advances 7, 056609 (2017)

3 M Cheng and Y Zhu, Energy Conversion and Management88, 332 (2014).

4 Z Z Wu and Z Q Zhu, IEEE Trans Magn.51, 1 (2015).

5W Hua and M Cheng, International Conference on Electrical Machines and Systems, Wuhan, 17-20 Oct., 2008,

pp: 3020–3025.

6 W Fei, P C K Luk, and J Shen, IEEE Trans Magn.48, 2664 (2012).

7 H Jia, M Cheng, W Hua, W X Zhao, and W L Li, IEEE Trans Magn.46, 1527 (2010).

8 J Rodriguez, R M Kennel, J R Espinoza, M Trincado, C A Silva, and C A Rojas, IEEE Trans Ind Electron.59, 812 (2012).

9 W Xu, IEEE Transactions on Applied Super.24, 1 (2014).

10 M Cheng, F Yu, K T Chau, and W Hua, IEEE Trans Ind Electron.63, 4539 (2016).