# Optimal Design of a Multibrid Permanent Magnet Generator for a Tidal Stream Turbine

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## Abstract

**:**

## 1. Introduction

## 2. System Modeling

#### 2.1. Renewable Resource and Tidal Turbine Modeling

#### 2.1.1. Power and Energy Calculation

#### 2.1.2. Power Rating Choice

^{2}, the same power ratio of $30\%$ is maintained, and for each power rating (500 kW, 1.5 MW, 5 MW) only the blade’s diameter (swept area) is calculated.

#### 2.2. Gearbox Modeling

#### 2.3. Single Stage Geared PMG Design

#### 2.3.1. Electromagnetic Torque

#### 2.3.2. Air-Gap

#### 2.3.3. Magnet Height

#### 2.3.4. Slot Height

#### 2.3.5. Stator and Rotor Yoke Height

#### 2.3.6. Teeth Pitch Ratio

#### 2.3.7. Maximum Magnetic Field

#### 2.3.8. Iron Losses

#### 2.3.9. Synchronous Inductance

#### 2.4. Power Electronic Converter Design

## 3. Design Optimization

#### 3.1. Cost-Function

#### 3.2. Optimization Constraints

^{2}and 40–60 kA/m respectively [12]. The generator efficiency is considered to be greater than 0.96. On the other hand, the phase voltage root mean square is fixed to 690 V.

## 4. Design Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

TST | Tidal stream turbine |

DD | Direct drive |

2D | Two-dimensional |

PMG | Permanent magnet generator |

3D | Three-dimensional |

AEP | Annual energy production |

PWM | Pulse width modulation |

## Nomenclature

${P}_{T}$ | Input shaft power |

${A}_{t}$ | Turbine blade swept area |

$\rho $ | Sea water density |

${C}_{p}$ | Power coefficient |

$\lambda $ | Tip speed ratio |

${\lambda}_{opt}$ | Optimum tip speed ratio |

$\beta $ | Pitch angle |

${v}_{i}$ | Cut-in tidal current speed |

${v}_{c}$ | Cut-out tidal current speed |

${v}_{n}$ | Rated tidal current speed |

${P}_{Tr}$ | Rated input shaft power |

$OCC$ | Occurrence frequency |

$FW$ | Gear face width |

${d}_{s}$ | Sun gear diameter |

${d}_{p}$ | Planet gear diameter |

${d}_{r}$ | Ring gear diameter |

${K}_{r}$ | Scaling factor |

${T}_{m}$ | Gearbox output shaft torque |

${K}_{ag}$ | Application factor |

${K}_{f}$ | Tooth loads intensity index |

${W}_{c}$ | Gearbox weight constant |

${r}_{ratio}$ | Gearbox ratio |

${r}_{sn}$ | Gear ratio between sun and planet gears |

Z | Planet gears number |

${c}_{gear}$ | Gearbox specific cost |

${G}_{gear}$ | Gearbox weight |

${C}_{gear}$ | Gearbox estimated cost |

${p}_{gear}$ | Gearbox losses |

${k}_{g}$ | Speed-dependent losses constant |

${P}_{N}$ | Tidal stream turbine rated power |

${n}_{r}$ | Rotor speed |

${n}_{{r}_{N}}$ | Rated rotor speed |

${T}_{EM}$ | Electromagnetic torque |

${A}_{L}$ | Stator current loading |

${B}_{{g}_{max}}$ | Maximum air-gap flux density |

${B}_{g}$ | Air-gap flux density |

${B}_{max}$ | Saturation flux density |

${k}_{b1}$ | First harmonic winding factor |

$\psi $ | Phase shift between the electromotive force and the current |

${R}_{s}$ | Stator radius |

${L}_{m}$ | Equivalent core length |

${\xi}_{3D}$ | 3D flow leakage corrective coefficient |

${k}_{D}$ | Air-gap coefficient |

${h}_{g}$ | Mechanical air-gap |

${h}_{{g}^{\prime}}$ | Additional Carter air-gap |

${k}_{c}$ | Carter factor |

${h}_{m}$ | Magnet height |

${\mu}_{0}$ | Vacuum permeability constant |

${\mu}_{rm}$ | Magnets relative permeability |

${B}_{r}$ | Magnets remanent flux density |

${B}_{{g}_{max}}$ | Maximum air-gap flow density |

$\tau $ | Pole pitch |

${h}_{ys}$ | Stator yoke height |

${h}_{yr}$ | Rotor yoke height |

${h}_{s}$ | Slot height |

${k}_{f}$ | Fill factor |

${\beta}_{t}$ | Teeth pitch ratio |

p | Pole pairs number |

${S}_{pp}$ | Slots per pole per phase number |

m | Phases number |

${H}_{max}$ | Maximum magnetic field in the magnet |

${H}_{cj}$ | Permanent magnet coercive magnetic field |

${p}_{Fe}$ | Iron losses |

${f}_{e}$ | Magnetic field frequency in the iron |

${p}_{Fe0h}$ | Specific hysteresis loss |

${p}_{Fe0e}$ | Specific eddy current loss |

${N}_{s}$ | Phase winding number of turns |

${L}_{sl}$ | Leakage inductance |

${C}_{conv}$ | Power electronics cost |

${C}_{g}$ | Permanent magnet generator cost |

${C}_{TST}$ | Tidal stream turbine cost |

${c}_{Cu}$ | Copper specific costs |

${c}_{Fe}$ | Iron specific costs |

${c}_{m}$ | Permanent magnet specific costs |

${G}_{Cu}$ | Copper specific weight |

${G}_{Fe}$ | Iron specific weight |

${G}_{m}$ | Permanent magnet specific weight |

$\mathbb{D}$ | Set of possible solutions |

${f}_{max}$ | Maximum electrical frequency |

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**Figure 1.**OpenHydro/Naval Energies direct-drive tidal stream turbine [4].

**Figure 3.**Small scale multibrid tidal stream turbine [15].

**Figure 4.**Scheme of a grid-connected single stage permanent magnet generator-based tidal stream turbine.

**Figure 11.**Basic dimensions of one pair of poles [8].

**Figure 13.**Tidal stream turbine (TST) estimated cost for different gear ratios and different power ratings.

**Figure 16.**Generator active materials weight for different gear ratios at the power rating of 1.5 MW.

**Figure 20.**Front and lateral view of the designed (3:1) geared generator at the power rating of 1.5 MW.

Tidal Stream Turbine | |||

Rated power ${P}_{N}$ [MW] | 0.5 | 1.5 | 5 |

Rated rotor speed ${n}_{{r}_{N}}$ [rpm] | 80.3 | 47.0 | 25.8 |

Rotor diameter D [m] | 6 | 10.3 | 18.8 |

Cut it tidal current speed ${v}_{i}$ [m/s] | 1.0 | ||

Cut out tidal current speed ${v}_{out}$ [m/s] | 6.2 | ||

Maximum power coefficient ${C}_{pmax}$ | 0.455 | ||

Optimum tip speed ratio ${\lambda}_{opt}$ | 5.90 | ||

Sea water density [kg/m^{3}] | 995.6 | ||

Single Stage Planetary Gearbox | |||

Gearbox application factor ${K}_{ag}$ | 1.5 | ||

K-factor ${K}_{f}$ [N/mm^{2}] | 2.76 | ||

Gearbox weight constant ${W}_{c}$ | 0.6 | ||

Planet gears number Z | 6 | ||

Gearbox specific cost ${c}_{gear}$ [€/kg] | 6 | ||

Speed dependent losses constant ${k}_{g}$ [%] | 1.5 | ||

PMG System | |||

Hysteresis losses at 1.5 T and 50 Hz p_{Fe0h} [W/kg] | 2 | ||

Eddy-current losses at 1.5 T and 50 Hz p_{Fe0e} [W/kg] | 0.5 | ||

Specific cost of electrical steel c_{Fe} [€/mT] | 449.77 | ||

Specific cost of copper c_{Cu} [€/mT] | 4259.18 | ||

Specific cost of NdFeB magnet c_{m} [€/mT] | 84,538.60 | ||

Specific cost of power electronics c_{conv} [€/kW] | 40 |

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**MDPI and ACS Style**

Touimi, K.; Benbouzid, M.; Chen, Z.
Optimal Design of a Multibrid Permanent Magnet Generator for a Tidal Stream Turbine. *Energies* **2020**, *13*, 487.
https://doi.org/10.3390/en13020487

**AMA Style**

Touimi K, Benbouzid M, Chen Z.
Optimal Design of a Multibrid Permanent Magnet Generator for a Tidal Stream Turbine. *Energies*. 2020; 13(2):487.
https://doi.org/10.3390/en13020487

**Chicago/Turabian Style**

Touimi, Khalil, Mohamed Benbouzid, and Zhe Chen.
2020. "Optimal Design of a Multibrid Permanent Magnet Generator for a Tidal Stream Turbine" *Energies* 13, no. 2: 487.
https://doi.org/10.3390/en13020487