Linear Permanent Magnet Vernier Generators for Wave Energy Applications: Analysis, Challenges, and Opportunities
Abstract
:1. Introduction
2. Wave Energy Applications
2.1. Wave Energy Converters (WECs)
 oscillating water column;
 overtopping converter;
 oscillating body system.
2.1.1. Archimedes Wave Swing WEC
2.1.2. Point Absorber WEC
2.2. Linear Generators
3. Operation Principles and Analysis of Linear PM Vernier Machines
3.1. Magnetic Gearing Effect
 no need for maintenance;
 isolation between the output and input shafts;
 inherent overload protection;
 no mechanical vibration and minimum acoustic noise;
 higher efficiency;
 higher reliability;
 no need for lubrication.
3.2. Analysis and Optimization of Linear PM Vernier Generators
4. Linear Permanent Magnet Vernier Generators (LPMVGs)
 linear flat and tubular;
 long/short stator and short/long translator;
 singlesided and doublesided;
 permanent magnet type;
 armature winding.
4.1. Flat and Tubular Structures
4.2. Long/Short Stator and Short/Long Translator
4.3. SingleSided and DoubleSided
4.4. Permanent Magnet Type
 surfacemounted type;
 interior type;
 spoke type;
 halbach/QuasiHalbach arrays;
 Vtype.
4.5. Armature Winding
5. Performance Improvement of Linear PM Vernier Generators
 thrust force capability improvement;
 power factor development;
 thrust force ripple reduction;
 cost reduction.
5.1. Thrust Force Capability Improvement
5.2. Power Factor Development
 optimal magnetic gear ratio;
 appropriate configuration of PMs;
 hightemperature superconducting (HTS) bulks.
5.2.1. Optimal Magnetic Gear Ratio
5.2.2. Appropriate Configuration of PMs
5.2.3. HighTemperature Superconducting (HTS) Bulks
5.3. Thrust Force Ripple Reduction
 linear longitudinal end effect;
 cogging force;
 normal force in the perpendicular direction of reciprocation;
 nonsinusoidal backEMF.
5.3.1. Linear Longitudinal End Effect
5.3.2. Cogging Force
5.3.3. Normal Force
5.3.4. NonSinusoidal BackEMF
5.4. Cost Reduction
6. Analysis and Comparison of Different Linear PM Vernier Generators
7. Conclusions and Outlooks
 The most considerable disadvantage of linear PM vernier machines is their poor power factor. Innovative techniques are required to further develop the low power factor of linear vernier structures.
 Due to the timeconsuming process of analyzing linear PM vernier generators based on FEA, accurate analytical methods are desired to provide analysis models in a short time.
 One of the most important criteria that must be considered into account is the cost of linear PM vernier generators, which are required to be declined by offering more economically viable structures and the decrease of the volume of magnets.
 The reduction of the weight of linear PM vernier generators can be realized by reducing the active material. There are opportunities to employ more lightweight linear generators for wave energy harvesting systems.
 Unconventional topologies can be introduced to improve the performance of linear vernier generators and facilitate their utilization in wave energy applications.
 The unwanted longitudinal end effect of linear PM vernier generators imposes a disadvantageous impact on the machine performance, which is desired to be diminished.
 Thermal analysis of linear PM vernier generators is another interesting subject in which not enough research has been accomplished so far.
 The magnetic gear ratio significantly affects the performance of linear PM vernier generators used in wave energy applications. Analytical and numerical methods are required to investigate the optimal gear ratio values.
 The configuration and number of flux modulation poles can have an impact on the gear ratio and the machine performance, which can be surveyed for linear vernier structures.
 A systemlevel optimization process is needed to improve the efficiency and performance of a linear vernier generator used in wave energy applications.
 Linear PM vernier machines utilize more number of PMs compared to linear PM synchronous machines; thus, the study on the possibilities to avoid irreversible demagnetization can be very useful.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
AWS  Archimedes wave swing 
CPHPM  Consequentpole and Halbach permanent magnet 
DC  Direct current 
DSSLVM  Dualstator spoketype linear vernier machine 
DSTVM  Dual stator tubular vernier machine 
EMF  Electromotive force 
EMN  Equivalent magnetic network 
FEA  Finite element analysis 
FMP  Flux modulation pole 
FP  Fractional pole 
GCD  Greatest common divisor 
GR  Gear ratio 
HTS  High temperature superconducting 
IMCP  Inset magnet consequent pole 
LCM  Lowest common multiple 
LCPSPMVM  Linear consequent pole stator permanent magnet vernier machine 
LPMSG  Linear permanent magnet synchronous generator 
LPMSM  Linear permanent magnet synchronous machine 
LPMVG  Linear permanent magnet vernier generator 
LPMVM  Linear permanent magnet vernier machine 
LPPMVM  Linear primary PM vernier machine 
LSSPMVM  Linear stator spoketype permanent magnet vernier machine 
MEC  Magnetic equivalent circuit 
MMF  Magnetomotive force 
NdFeB  Neodymium–iron–boron 
PM  Permanent magnet 
PTO  Power takeoff 
VCP  Vshaped consequent pole 
WEC  Wave energy converter 
YBCO  Yttrium boron copper oxide 
A  Coefficient defined by the volume of PMs 
B  Cost coefficient related to the power converter 
${B}_{gmax}$  Maximum flux density 
$CE$  Cost of energy 
E  Electric field 
${F}_{avg}$  Average thrust force 
${F}_{max}$  Maximum of thrust force 
${F}_{min}$  Minimum of thrust force 
I  Current 
J  Current density 
${k}_{c}$  Eddy current loss coefficient 
${k}_{d}$  Flux leakage coefficient 
${k}_{e}$  Excess loss coefficient 
${k}_{h}$  Hysteresis loss coefficient 
${k}_{s}$  Electric loading 
${k}_{w}$  Winding factor 
L  Effective length 
${L}_{s}$  Synchronous inductance 
${L}_{stk}$  Stack length 
m  Number of phases 
P  Electromagnetic power 
p  Number of stator armture winding pole pairs 
${R}_{ag1}$  Airgap reluctance of the slot 
${R}_{ag2}$  Airgap reluctance of the tooth 
${R}_{m}$  Reluctance of the magne 
T  Electrical period 
${T}_{cogging}$  Cogging force period 
${T}_{end}$  End time 
v  Mechanical speed of the moving part 
${v}_{eff}$  Mechanical speed of the effective flux 
${Z}_{r}$  Number of translator pole pairs 
${Z}_{s}$  Number of stator teeth 
$\beta $  Steinmetz constant 
$\eta $  Efficiency of the electrical machine 
${\tau}_{eff}$  Effective flux pitch 
${\tau}_{eff}$  Tooth pitch of the mover 
${\varphi}_{1}$  Leakage flux of surfacemounted structure 
${\varphi}_{2}$  Leakage flux of consequentpole structure 
${\mathrm{\Psi}}_{PM}$  Flux linkage produced by PMs 
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Reference  Characteristics  Advantages  Disadvantages 

Xiao et al. [26] 



Botha et al. [71] 



Baker et al. [81] 



Baloch et al. [82] 



Almoraya et al. [92] 



Shi et al. [102] 



Zhang et al. [104] 



Khaliq et al. [114] 



Baloch et al. [119] 



Ching et al. [120] 



Du et al. [123] 



Material  Thermal Conductivity [W/(m$\phantom{\rule{0.166667em}{0ex}}\mathit{\xb7}\phantom{\rule{0.166667em}{0ex}}$K)]  Specific Heat Capacity [J/(kg$\phantom{\rule{0.166667em}{0ex}}\mathit{\xb7}\phantom{\rule{0.166667em}{0ex}}$K)]  Density [kg/m${}^{3}$] 

Steel  30  460  7650 
Magnet  7.6  460  7500 
Copper  401  385  8933 
LPPMVM [130]  LSSPMVM [132]  IMCP LPMVM [133]  VCP LPMVM [133]  

Structure  Flat  Flat  Flat  Flat 
Single/doublesided  Singlesided  Singlesided  Doublesided  Doublesided 
Armature winding type  Distributed  Distributed  Concentrated  Concentrated 
PM type  Surfacemounted  Spoketype  Consequent pole  Vtype 
Location of PMs  Stator  Stator  Stator  Stator 
No. of PM pole pairs  18  18  9  9 
No. of winding pole pairs  1  1  1  1 
No. of translator teeth  17  17  10  10 
Gear ratio  17  17  10  10 
No. of PMs per stator tooth  5  5  3  6 
Frequency [Hz]  50  50  50  50 
No. of phases  3  3  3  3 
PM volume [cm${}^{3}$]  120  150  64.8  64.8 
Magnet remanence [T]  1.2  1.2  1.24  1.24 
Active length [mm]  360  360  232  232 
Thickness of linear machine [mm]  158  185  180  180 
Stack length [mm]  100  100  50  50 
Translator tooth pitch [mm]  21.17  21.17  24  24 
Stator tooth pitch [mm]  60  60  80  80 
Airgap length [mm]  1  1  1  1 
Rated speed [m/s]  1  1  1.2  1.2 
No. of winding turns per phase  142  140  90  90 
Current density [A/mm${}^{2}$]  4.3  4.3  3.5  3.5 
Noload backEMF [V]  60  87  45  57 
Flux linkage [Wb]  0.19  0.30  0.14  0.18 
Average thrust force [kN]  1.61  2.18  0.739  0.812 
Thrust force density [kN/m${}^{3}$]  283  327  354  389 
Force/PM volume [N/cm${}^{3}$]  13.4  14.5  11.1  12.5 
Power factor  0.26  0.27  0.51  0.65 
Copper loss [W]  243.5  240.1  62.6  62.6 
Core loss [W]  44.4  61.5  18.4  22.0 
Efficiency [%]  84.8  87.8  91.6  92.0 
Detent force pk2pk [N]  31 (1.9%)  36 (1.65%)  59 (7.9%)  48 (5.9 %) 
Thrust force pk2pk [N]  50  93  66  51 
Thrust force ripple [%]  3.1  4.2  8.9  6.2 
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Jafari, R.; Asef, P.; Ardebili, M.; Derakhshani, M.M. Linear Permanent Magnet Vernier Generators for Wave Energy Applications: Analysis, Challenges, and Opportunities. Sustainability 2022, 14, 10912. https://doi.org/10.3390/su141710912
Jafari R, Asef P, Ardebili M, Derakhshani MM. Linear Permanent Magnet Vernier Generators for Wave Energy Applications: Analysis, Challenges, and Opportunities. Sustainability. 2022; 14(17):10912. https://doi.org/10.3390/su141710912
Chicago/Turabian StyleJafari, Reza, Pedram Asef, Mohammad Ardebili, and Mohammad Mahdi Derakhshani. 2022. "Linear Permanent Magnet Vernier Generators for Wave Energy Applications: Analysis, Challenges, and Opportunities" Sustainability 14, no. 17: 10912. https://doi.org/10.3390/su141710912