A Review of the Linear Generator Type of Wave Energy Converters’ Power TakeOff Systems
Abstract
:1. Introduction
2. WEC with Linear GeneratorBased Direct ElectricDrive PTO System
2.1. Different Topologies of WECs with Linear GeneratorBased PTO Systems
2.1.1. Floating Buoy on the Sea Surface
SingleBody Heaving Buoy System
TwoBody Heaving Buoy System
2.1.2. Fully Submerged Heaving System
2.1.3. Other Topologies of WECs with Linear GeneratorBased PTO Systems
2.2. Linear Permanent Magnet (PM) Generator Topologies
Innovative Oscillator Design Concept
3. Mathematical Modelling
Dynamics of the WEC with Linear Permanent Magnet (PM)Based PTO System
4. Development of the Linear PM GeneratorBased PTO System for WECs
4.1. Reduction of Detent Force (Cogging Force and End Effect Force)
4.1.1. Permanent Magnet (PM) Modifications
4.1.2. Changing the Air Gap
4.1.3. Modification of the Stator Design
4.1.4. Magnetic Cores
4.2. Application of HighGrade PMs and Solving the Demagnetisation Problem
4.3. Design Concepts for LowFrequency Wave Range
4.4. Using Magnetic Gearing to Increase the Speed of the Translator
4.5. Other Design Concepts to Increase the Efficiency of the Generator
4.6. Using Advanced Numerical Simulation for Parameter Study
4.7. Design Optimisation to Maximize the Performance
5. Control Systems of the Linear PM GeneratorBased PTO System
5.1. Hydrodynamic Control
5.2. Generator Control (PTO Control)
5.2.1. Latching Control
5.2.2. Model Predictive Control (MPC)
5.2.3. Nonlinear Model Predictive Control (NMPC)
5.2.4. Other Control Systems
5.3. Grid Control (Load Side Control)
6. Performance Analysis of the Linear GeneratorBased WEC
6.1. Numerical Analysis
6.2. Experimental Analysis
6.2.1. Wave Tank Test
6.2.2. Open Sea Test
7. Costs and Challenges of the Linear GeneratorType PTO System for Wave Energy Conversion Technology
8. Conclusions and Remarks
 Linear generators are suitable for wave energy conversion if the devices are buoyantmoored with linear motion and operate with speeds of 1 m/s or more.
 Compared with other linear generator types, the linear PM synchronous generator is the most suitable for wave energy conversion because it has higher reliability and efficiency due to the more significant driving force.
 ○
 The planar/flattype linear PM synchronous generators are preferred for highpower applications.
 ○
 The tubulartype linear PM synchronous generators are suitable for lowpower applications because they offer high power or force density.
 ○
 Tubulartype linear PM generators with a long translator inside the generator perform better, with less cogging force.
 ○
 Threephase generators are more efficient than singlephase generators due to their higher energy generation.
 ○
 Ironcored generators are more suitable than aircored generators because their power generation ability is higher.
 ○
 The most significant power is produced when the PMs are attached to the translator.
 ○
 QuasiHalbach arrangements are preferred for improving the power generation efficiency with minimum losses.
 The force mainly determines the generator size it has to create. In wave energy conversion, the speeds are typically relatively low. The force should be high if the aim is to generate a large amount of power at a low speed. Therefore, the cost of the generator increases. The switched reluctance generators, variable reluctance generators, transverse flux PM machines and Vernier hybrid machines are suitable designs and have high force density. This limited force density does not influence the performances of these generator types and is suitable for lowpower applications. However, they have disadvantages, such as complex construction structure, low power factor, complex iron losses and eddy current losses. Although they have some drawbacks, they can be viewed as an alternative to the typical linear PM generator in the future.
 Clever designs such as doublesided and cylindrical arrangements could decrease the cost.
 Superconducting linear generators are suitable for the WEC based on the linear generator in terms of low power application because it has a high currentcarrying ability, producing much higher flux density with lowspeed motion and lighter weight. However, it has a high manufacturing and material costs.
 Innovative techniques for increasing the speed of the linear motion of the WEC.
 Study other types of generators with higher force densities and perhaps better performances.
 Study of aircored generators in terms of their prospects for a practical combined electricalmechanical structural design solution.
 Deployment in ocean environments for trials over the long term.
 Innovative designs for cutting down the cost of generator construction.
 Innovative designs for solving the lowfrequency problem.
 Innovative systems for the transmission of the generated power to the grid.
 Implementation of control systems in the deployed WEC during sea trials.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Acronyms
Symbols  Abbreviations  Symbols  Abbreviations 
AWS  Archimedes Wave Swing  ${E}_{f}$  Emf per phase 
BEM  Boundary element method  ${F}_{0}$  Wave force coefficient 
CTA  Constant Torque Angle Control  ${F}_{1}$ and ${F}_{2}$  Pullout forces of the mover 1 and 2 
EMF  Electromotive force  ${F}_{buoy}$  Buoy force 
FEA  Finite element analysis  ${F}_{e}$  Wave excitation force 
FEM  Finite element method  ${F}_{em}$  Electromagnetic force 
GA  Genetic Algorithm  ${F}_{gen}$  Generator force 
ICM  Improved conformal mapping  ${F}_{h}$  Hydrostatic force 
MEC  Magnetic equivalent circuit  ${F}_{r}$  Wave radiation force 
MPC  Model Predictive Control  ${G}_{r}$  Gear ratio 
MPPT  Maximum power point tracking  $g$  Gravity acceleration 
MTPA  Maximum Torque per Ampere  $\gamma $  Spring constant 
NMPC  Nonlinear model predictive control  $I$  Current inside the coil 
OSU  Oregon State University  $L$  Inductance of the coil 
PA  Point absorber  m  Sum of the translator and buoy mass 
PM  Permanent magnet  ${m}_{a}$  Added mass 
PMLG  Permanent magnet linear generator  ${N}_{1}$$\mathrm{and}{N}_{2}$  Number of active PM pole pairs in the mover 1 and 2 
PSO  Particle Swarm Optimization  ${\omega}_{n}$  Natural frequency 
PTO  Power takeoff system  $\phi $  Phase of the regular wave 
UU  Uppsala University  $R$  Load resistance of the circuit 
WEC  Wave energy converters  ${R}_{z}$  Radiation damping 
A and B  State or system matrix and input matrix  $\rho $  Density of the sea water 
${A}_{f}$  Amplitude of the wave  ${U}_{f}$  Output phase voltage 
$a$  Radius of the buoy  $\rho $  Density of the sea water 
$\alpha $  Coupling coefficient  ${v}_{1}$$\mathrm{and}{v}_{2}$  Seed of the mover 1 and 2 
${B}_{x}$  Magnetic flux density  $V$  Voltage inside the coil 
$\beta $  Sum of the mechanical and electrical damping of the generator  $\ddot{z}$, $\dot{z}$ and $z$  Acceleration, velocity and position of the translator or moving magnet, respectively 
${\beta}_{m}$  Damping coefficient  $\xi $  Total damping ratio 
${\beta}_{e}$  Generator electrical damping coefficient  ${\xi}_{m}$  mechanical damping ratio 
C and D  Output matrix and zero matrix 
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Structure  Parameters and S.I. Units 

Linear generator  Pole width (m), Pole pitch, Number of poles, Air gap magnetic flux density (T), Air gap (m), Number of turns, Number of slots, Width of a stator tooth (m), Teeth thickness (m), Width of the stator stack (m), Translator iron thickness (m), Length of the generator (m), Resistance of the coil (Ω), Circuit resistance (Ω), Load resistance (Ω), Number of phases, Mass of the magnets (kg), Load angle (rad) 
Buoy  Wave period (s), Wave height (m), Mass of the buoy (kg), Diameter of the buoy (m), Height of the buoy (m), Density of the seawater (kg/m^{3}) 
Modification  Magnetic Flux Density  Efficiency  Cogging Force  Structure  Load Angle 

Optimised reduction of PM length [68]  Reduced  Increased  Reduced by 80%  Stator length was increased  Increased 
PMs attached inside diameter of the translator [27]  ND  ND  Reduced  Increased complexity  ND 
Using larger magnet size [74]  Increased  Increased  ND  Load angle overall size and magnetic coupling were reduced  Increased PM cost 
Magnet shape (rectangular shapes) [74]  Increased  ND  ND  ND  Reduced 
Radial PMs [69]  Reduced  ND  Reduced by 70%  ND  ND 
Using Halbach arrays [44]  Increased  Increased  Reduced  Increased complexity  ND 
Using quasiHalbach array [42]  Increased  Increased  Reduced  Increased the difficulty of manufacturing magnets  ND 
Highgrade PM [79]  Increased  Increased  Reduced  Increased cost  ND 
Skewing the PMs [69]  ND  Reduced  Reduced  ND  ND 
PMs pole shifting [76]  ND  Unbalance voltage  Reduced  Increased cost  ND 
PMs with bevelled bottomside shape [68]  ND  ND  Reduced  Increased complexity  ND 
Modification  Increases  Reduces 

Increasing the air gap [72]  ND 

Variable air gap [81] 


Modification  Cogging Force  Efficiency  Iron Loss  Cost 

Increasing stator tooth width [74]  $\downarrow $  ND  $\downarrow $  ND 
Slotless Stator [85]  $\downarrow $  $\uparrow $  ND  $\uparrow $ 
Semiclosed slots [69]  $\downarrow $ by 34%  $\downarrow $  $\uparrow $  $\uparrow $ 
Optimised bulged stator [72]  $\downarrow $  $\uparrow $  ND  ND 
Assistant tooth [83]  $\downarrow $ by 70%  ND  $\uparrow $  ND 
Shoe concept [84]  $\downarrow $  ND  ND  $\uparrow $ 
Stator consists of permanent magnets, winding coils and spring [86]  ND  $\uparrow $  ND  $\downarrow $ 
Type  References  

Simulation  [5,17,34,35,50,55,59,77,84,111,114,120,150,151,152,153,154,155,156,157,158,159,160]  
Experiment  Ocean test  [9,65,91,104,161,162] 
Wave tank test  [163]  
Test rig test  [37,38,90,146,164]  
Validation (Sim./Exp)  Ocean test  [165,166] 
Wave tank test  [6,42,78,86,167,168,169]  
Test rig test  [33,36,44,45,52,72,83,92,93,170,171,172,173,174,175] 
References  Deployed Place and Year  Location  Rated Power 

[8]  Sweden (2002)  Offshore  10 kW 
[180]  Portugal (2004)  Offshore  2 MW 
[26]  USA (2008)  Offshore  10 kW 
[181]  UK (2008)  Nearshore  100 MW 
[1]  USA (2011)  Offshore  1 MW 
[181]  Sweden (2015)  Offshore  1 MW 
[1]  Germany  Offshore  1 MW 
[1]  Russia  Offshore  50 kW 
Advantages  Disadvantages 

Does not require an intermediate mechanical interface  Power transmission system is very complicated due to the unequal generated voltage created by the irregular wave motion 
Reduces maintenance cost  Velocity of the translator is much lower than conventional rotary generators 
Relatively highly efficient  Low powertoweight ratio 
Possibility of continuous force control  Needs heavy structure due to the attractive forces between the stator and the translator 
Item  Material  Current Unit Cost  Unit Cost Range 

Permanent magnets (PMs)  Neodymium–iron–Boron  96 USD/kg  72–120 USD/kg 
Stator  Electrical steel  2.5 USD/kg  2–3 USD/kg 
Translator  Electrical steel  2.5 USD/kg  2–3 USD/kg 
Rim  Aluminium alloy  6 USD/kg  4.5–7.5 USD/kg 
Winding coil  Copper coil  1 USD/m  0.5–1.5 USD/m 
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Ahamed, R.; McKee, K.; Howard, I. A Review of the Linear Generator Type of Wave Energy Converters’ Power TakeOff Systems. Sustainability 2022, 14, 9936. https://doi.org/10.3390/su14169936
Ahamed R, McKee K, Howard I. A Review of the Linear Generator Type of Wave Energy Converters’ Power TakeOff Systems. Sustainability. 2022; 14(16):9936. https://doi.org/10.3390/su14169936
Chicago/Turabian StyleAhamed, Raju, Kristoffer McKee, and Ian Howard. 2022. "A Review of the Linear Generator Type of Wave Energy Converters’ Power TakeOff Systems" Sustainability 14, no. 16: 9936. https://doi.org/10.3390/su14169936