# Downsizing the Linear PM Generator in Wave Energy Conversion for Improved Economic Feasibility

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## Abstract

**:**

## 1. Introduction

## 2. Review of the Application of Linear Generator in Wave Energy Conversion

- Hydraulic PTO: In hydraulic PTO, the motion of a buoy drives the hydraulic piston to increase the pressure of the working medium. Then, the pressure of the medium is raised sufficiently high in an accumulator to rotate the hydraulic motor. The hydraulic PTO is robust and able to provide large forces at low frequencies, which highly matches the dynamic characteristics of WECs. Hydraulic PTO systems were widely used in WECs, such as Pelamis and Edinburgh Duck [15,16,17]. However, hydraulic PTO systems contain plenty of moving parts, which results in their complex structure. So, regular system maintenance and inspection are normally required, which is time-consuming and costly [2]. In addition, the conversion efficiencies of hydraulic PTO systems are relatively low [18].
- Hydro PTO: The hydro PTO mainly refers to the hydro turbine. Hydro turbines transfer the fluid flow to electricity. This type of PTO systems is commonly employed in overtopping devices, such as WaveDragon [19].
- Pneumatic PTO: The pneumatic turbine generally refers to the air turbine, which is driven by the oscillating air pressure. This technology is usually utilized in oscillating water column converters [20].
- Mechanical PTO: In the mechanical PTO system, a gearbox is used to convert the linear movement of the buoy of WECs to rotary motion for fitting conventional rotary generators. The oscillation of the buoy of WECs is of low speed due to the characteristics of ocean waves. For improving efficiency, another important function of the gearbox is to increase the speed of motion. In [21], a point absorber equipped with a bidirectional gearbox and rotary generator was introduced. More recently, WECs with mechanical PTO mechanism have been further investigated by numerical modeling [22] and small-scale testing [23].
- Direct-drive linear generator: The linear generator could be used as the direct-drive PTO system in the oscillating body WECs, and they are usually used in point absorber wave energy converters, such as AWS [24]. In direct-drive linear generators, intermediate transmission interfaces, such as gearbox and hydraulic motors, are not necessary. Instead, the oscillating buoy is directly coupled with the translator of the linear generator. The linear generators are commonly associated with higher efficiencies compared with other PTO systems. This is because there are less transmission losses resulting from a reduced number of energy conversion steps [2,24]. In addition, the reduced number of components in the PTO system increases the reliability of the whole WEC system [25].

## 3. Description of WEC Concept

## 4. Methodology

#### 4.1. Hydrodynamic Modeling

#### 4.2. Generator Modeling

#### 4.3. Generator Sizing

- Method 1: Scaling lawThe scaling principle of electrical generators is based on the fact that the force density per unit surface area of the machine remains rather constant when its dimension changes [40]. The force density is mainly related to two factors. The first one is the magnetic flux density in the air gap, and it is limited by the effect of magnetic saturation. The second one is the linear current density of the machine, and it is limited by the maximum allowed heat dissipation. These two factors could hardly be changed with the dimension if the topology design of the machine does not vary. In this paper, the machine is resized to suit various maximum forces. The force density of resized machines remains identical to the reference machine.For convenience, a scale factor $\lambda $ is introduced here as,$$\lambda =\frac{{S}_{s}}{{S}_{o}}$$$$\mathrm{Force}:\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\frac{{F}_{ma{x}_{s}}}{{F}_{ma{x}_{o}}}={\lambda}^{2}$$$$\mathrm{Velocity}:\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\frac{{u}_{ma{x}_{s}}}{{u}_{ma{x}_{o}}}=1$$$$\mathrm{Current}:\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\frac{{I}_{co{n}_{s}}}{{I}_{co{n}_{o}}}=1$$The rating of the converter is accordingly scaled for each machine. The rated apparent power, current and terminal voltage should be selected during the rating. As the linear current density is assumed to remain identical for differently sized machines, the maximum phase current of the scaled converter is selected to be unchanged, namely ${I}_{con}$ of 400 A. At the rated operating condition, the velocity of the buoy is ${u}_{max}$ of 1.25 m/s, and then the resulting no-load voltage can be obtained by (13). Additionally, the no-load voltage at the rated point is assumed to be in phase with the current. As a consequence, the rated terminal voltage of the scaled converter, ${U}_{co{n}_{s}}$, and the phase angle between the current and terminal voltage, namely $\delta $, can be calculated according to the first phasor diagram in Figure 3. Therefore, the rated apparent power of the scaled converter is obtained as$${S}_{con{v}_{s}}=m{I}_{conm}{U}_{con{m}_{s}}$$
- Method 2: Scaling with optimizationIn this method, the main parameters of the machine, including the translator length, stator length and stack length are optimized during sizing for the lowest LCOE. The optimization for each designed maximum generator force is expressed as$$\begin{array}{c}\hfill minimize\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}f=\mathrm{LCOE}({L}_{s},{L}_{t},{l}_{s})\\ \hfill subject\phantom{\rule{0.222222em}{0ex}}to\left\{\begin{array}{c}2{\rho}_{force\_ref}{L}_{s}{l}_{s}={F}_{limit}:constraint\phantom{\rule{4pt}{0ex}}1\\ \\ 1.2{L}_{s}\le {L}_{t}:constraint\phantom{\rule{4pt}{0ex}}2\end{array}\right.\end{array}$$
- Method 3: Scaling with assuming a constant generator efficiencyFor simplification, in studies discussing the effects of the sizing of WECs, the sizing of WECs was commonly implemented in the absence of the consideration of the variation of generator efficiencies. Instead, a constant energy conversion efficiency from the absorption stage to the grid was assumed. In this method, the generator size acts simply as a PTO force constraint in the hydrodynamic modeling. This method is discussed in this paper for a comparison with the other two above methods to demonstrate the effects of this simplification on the generator size determination and the LCOE estimation. In this paper, this constant efficiency is considered to be 70%, as usually used in literature [9,46]. Moreover, as the design of the generator is not taken into account in this method, the cost could not be derived explicitly. To evaluate the techno-economic performance of WECs, the PTO related cost in this method is assumed to be the same as that estimated in method 2 for each designed maximum generator force limit.

#### 4.4. Economic Modeling

## 5. Results and Discussion

#### 5.1. Generator Performance

#### 5.2. Comparison of Sizing Methods

#### 5.2.1. On the Generator Performance

#### 5.2.2. On the Techno-Economics and the PTO Size Determination

#### 5.3. Dependence of the Generator Sizing on Wave Resources

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Figure A1.**Hours of occurrence of each wave state of Yeu island [54].

**Figure A2.**Hours of occurrence of each wave state of DK 2 [14].

**Figure A3.**Hours of occurrence of each wave state of BIMEP [14].

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**Figure 1.**Schematic of the heaving point absorber WEC, and the connection scheme of the PM linear generator with the back-to-back converter and the grid.

**Figure 2.**The photo of the reference linear electrical machine in the pilot of the AWS WEC, and the cross-section of the linear PM generator of the AWS plant. In figure (

**b**), a/a’, b/b’ and c/c’ are the current directions [39], and the "z" axis in the coordinate is perpendicular to the ground plane. (

**a**) Photo of the reference machine. (

**b**) Cross-Section of the machine.

**Figure 4.**Generator performance over a half of wave period with a displacement of 0.85 m and a wave period of 10.0 s. The considered machine is sized with a maximum designed generator force of 200 kN based on sizing method 2. (

**a**) Position and speed of the translator. (

**b**) Instantaneous no-load voltage induced by the translator. (

**c**) RMS value of the no-load voltage. (

**d**) RMS value of the stator current. (

**e**) Input mechanical power and electrical power to the grid. (

**f**) Electrical losses.

**Figure 5.**Scaled translator, stator and stack lengths for different designed maximum generator forces. “M1” and “M2” embody method 1 and method 2 for generator sizing.

**Figure 6.**Comparison between efficiency maps of generators scaled based on different methods. (

**a**) Designed maximum force is 100 kN, based on method 1. (

**b**) Designed maximum force is 100 kN, based on method 2. (

**c**) Designed maximum force is 160 kN, based on method 1. (

**d**) Designed maximum force is 160 kN, based on method 2. (

**e**) Designed maximum force is 200 kN, based on method 1. (

**f**) Designed maximum force is 200 kN, based on method 2.

**Figure 7.**Scaled translator, stator and stack lengths for different designed maximum generator forces.

**Figure 9.**The AEP of the WEC as a function of the designed maximum generator force. “M3” embodies the method 3 for generator sizing.

**Figure 11.**The optimized design parameters, the overall efficiency and the LCOE of the WEC with various designed maximum forces in three different wave sites. (

**a**) The optimized design parameters for DK 2 and BIMEP, which are normalized to the corresponding values in Yeu island. (

**b**) The overall efficiency of the linear generator as a function of the designed maximum generator force. (

**c**) The LCOE of the WEC as a function of the designed maximum generator force.

Parameters | AWS | Uppsala Concept | SeaBeavI |
---|---|---|---|

Rated power | 2 MW | 10 kW | 10 kW |

WEC type | Submerged point absorber | Floating point absorber | Floating point absorber |

Generator structure | Bottom founded | Bottom founded | Floating |

Generator topology | Flat and double sided | Flat and four sided | Tubular |

First tested time | 2004 | 2006 | 2007 |

Testing site | Portugal | Sweden | USA |

Parameters | Symbol | Value |
---|---|---|

Maximum average power | ${P}_{rared}$ | 1 MW |

Maximum force | ${F}_{max}$ | 933 kN |

Maximum velocity | ${u}_{max}$ | 2.2 m/s |

Stroke | S | 7 m |

Translator length | ${L}_{t}$ | 8 m |

Stator length | ${L}_{s}$ | 5 m |

Air gap length | g | 5 mm |

Slot width | ${b}_{s}$ | 15 m |

Stack length | ${l}_{s}$ | 1 m |

Magnet pole width | ${b}_{p}$ | 79 mm |

Tooth width | ${b}_{t}$ | 19 mm |

Magnet thickness | ${l}_{m}$ | 15 mm |

Number of conductors per slot | ${N}_{s}$ | 6 |

Parameter | Value | Unit |
---|---|---|

Iron price (${C}_{Fe}$) | 3.3 | Euros/kg |

Copper price (${C}_{Cu}$) | 15.2 | Euros/kg |

Permanent magnet price (${C}_{pm}$) | 24.7 | Euros/kg |

Lifespan (N) | 20 | years |

Discount rate (r) | 8% | – |

Availability of operation (A) | 0.9 | – |

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**MDPI and ACS Style**

Tan, J.; Wang, X.; Polinder, H.; Laguna, A.J.; Miedema, S.A.
Downsizing the Linear PM Generator in Wave Energy Conversion for Improved Economic Feasibility. *J. Mar. Sci. Eng.* **2022**, *10*, 1316.
https://doi.org/10.3390/jmse10091316

**AMA Style**

Tan J, Wang X, Polinder H, Laguna AJ, Miedema SA.
Downsizing the Linear PM Generator in Wave Energy Conversion for Improved Economic Feasibility. *Journal of Marine Science and Engineering*. 2022; 10(9):1316.
https://doi.org/10.3390/jmse10091316

**Chicago/Turabian Style**

Tan, Jian, Xuezhou Wang, Henk Polinder, Antonio Jarquin Laguna, and Sape A. Miedema.
2022. "Downsizing the Linear PM Generator in Wave Energy Conversion for Improved Economic Feasibility" *Journal of Marine Science and Engineering* 10, no. 9: 1316.
https://doi.org/10.3390/jmse10091316