# Design Optimization of a Direct-Drive Electrically Excited Synchronous Generator for Tidal Wave Energy

^{*}

## Abstract

**:**

## 1. Introduction

## 2. 1D Analytical Modeling

#### 2.1. Electromagnetic Model

#### 2.1.1. Electrical Equations

#### 2.1.2. Flux Densities

#### 2.1.3. Back E.M.F.

#### 2.1.4. Resistances and Inductances

#### 2.2. Thermal Model

- no heat flux in the axial direction in the stator and rotor iron because of the lamination,
- the frame is considered to be a perfect thermal conductor,
- no axial heat flux in the airgap,
- conductors in the slot are uniformly distributed, and
- end windings are considered to be cylindrical and homogenous.

^{2}K and the convective heat coefficient in end-windings is ${h}_{ew}=15$ W/m

^{2}K.

## 3. Optimization Problem

#### 3.1. Problem Statement

#### 3.2. Optimization Methodology

#### 3.2.1. Expressions of Optimal Control Parameters

#### 3.2.2. Expression of the First Objective Function

#### 3.2.3. Expression of the Second Objective Function

## 4. Application

#### 4.1. Optimization Results

#### 4.2. 2D Validation

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

${B}_{fm}$ | airgap magnitude of the magnet flux density [T] |

${B}_{rm}$ | magnitude of the resulting airgap flux density [T] |

${B}_{stm}$ | magnitude of the flux density in the stator teeth [T] |

${B}_{sym}$ | magnitude of the flux density in the stator yoke [T] |

${B}_{rtm}$ | magnitude of the flux density in the rotor teeth [T] |

${B}_{rym}$ | magnitude of the flux density in the rotor yoke [T] |

${C}_{p}$ | equivalent heat capacity of the stator [J/kg·°C] |

$e$ | airgap thickness [m] |

${i}_{d-q}$ | d- and q- axis currents [A] |

${I}_{s}$ | rms stator current [A] |

${I}_{f}$ | direct rotor current [A] |

${k}_{ad}$ | additional magnetic loss coefficient |

${k}_{c}$ | Carter’s coefficient |

${k}_{ec}$ | eddy currents specific loss coefficient |

${k}_{fs}$ | stator slot fill factor |

${k}_{fr}$ | stator slot fill factor |

${k}_{h}$ | hysteresis specific loss coefficient |

${k}_{tr}$ | tooth opening to the slot pitch ratio in the rotor |

${k}_{ts}$ | tooth opening to the slot pitch ratio in the stator |

${k}_{w}$ | winding factor |

$L$ | active length [m] |

${M}_{c}$ | copper mass [kg] |

${M}_{i}$ | iron mass [kg] |

${n}_{f}$ | number of turns conductors/pole on the rotor |

${n}_{s}$ | number of conductors/phase/pole for the stator |

$p$ | number of pole pairs |

${P}_{c}$ | copper losses [W] |

${P}_{em}$ | electromagnetic power [W] |

${P}_{mg}$ | iron losses [W] |

${P}_{tmg}$ | iron loss in the stator teeth [W] |

${P}_{ymg}$ | iron loss in the stator yoke [W] |

${\mathcal{P}}_{0}$ | average value of the surfacic perméance [H/m^{2}] |

${\mathcal{P}}_{1}$ | magnitude of the first harmonic of the surfacic perméance [H/m^{2}] |

$R$ | external radius [m] |

${R}_{ss}$ | inner stator winding radius [m] |

${R}_{sr}$ | inner rotor winding radius [m] |

${R}_{ws}$ | outer stator winding radius [m] |

${R}_{wr}$ | outer rotor winding radius [m] |

${R}_{0}$ | internal rotor radius [m] |

${R}_{th}$ | thermal conduction resistance [W/K] |

${\mathcal{R}}_{c}$ | electrical stator resistance per phase [$\mathsf{\Omega}$] |

${\mathcal{R}}_{f}$ | electrical rotor resistance [$\mathsf{\Omega}$] |

${\mathcal{R}}_{\mu}$ | iron loss resistance per phase [$\mathsf{\Omega}$] |

$t$ | time [s] |

${v}_{d-q}$ | d- and q- axis terminal voltage [V] |

${W}_{sy}$ | yoke width of the stator [m] |

${W}_{rps}$ | height of the rotor pole shoe [m] |

${W}_{ry}$ | yoke width of the rotor [m] |

${W}_{losses}$ | energy lost per working cycle [J] |

${X}_{d},{X}_{q}$ | $d-$ and $q-$ axis armature reactance [$\mathsf{\Omega}$] |

$\beta $ | electrical rotor pole arc [rad] |

${\theta}_{c}$ | temperature elevation [K] |

$\lambda $ | thermal conductivity [W/m·K] |

${\eta}_{d}$ | coefficient of distortion of flux density |

$\psi $ | current angle with back-EMF [rad] |

${\sigma}_{c}$ | electrical conductivity [S] |

${\tau}_{LR}$ | active length to outer stator ration ($\mathrm{L}/\mathrm{R}$) |

${\tau}_{tr}$ | rotor slot pitch ratio |

${\tau}_{ts}$ | stator slot pitch ratio |

$\mathsf{\Omega}$ | mechanical angular velocity [rad/s] |

## Appendix A. Constant Parameters

Parameters | Values |

${B}_{sat}$ | 1.5 T |

${V}_{limit}$ | 2.5 kV |

${k}_{ad}$ | 2 |

${k}_{ec}$ | 0.035 |

${k}_{h}$ | 15 |

${k}_{fs}$ | 0.4 |

${k}_{fr}$ | 0.7 |

${k}_{Ls},{k}_{Lr}$ | 1.2 |

${k}_{ts},{k}_{tr}$ | 0.55 |

${h}_{ext}$ | 280 W/m^{2} k |

${\eta}_{t}$ | 1.1 |

${\mu}_{air}$ | 1.516 × 10^{−5} Kg/ms |

${\theta}_{max}$ | 160 °C |

${\theta}_{\infty}$ | 20 °C |

${\rho}_{c}$ | 8960 Kg/m^{3} |

${\rho}_{Fe}$ | 7600 Kg/m^{3} |

${\rho}_{air}$ | 1.225 Kg/m^{3} |

${C}_{p}$ | 390 J/Kg/K |

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**Figure 1.**AR2000 tidal turbine [6].

**Figure 2.**A sealed airgap direct-drive generator [8].

**Figure 3.**(

**a**) Design of pole pitch with two slots/phase/pole and main geometric parameters, (

**b**) Estimation of the permeance function.

**Figure 5.**(

**a**) Cylindrical element and its equivalent circuit, (

**b**) Thermal equivalent circuit of the machine.

**Figure 6.**(

**a**) General flowchart of the optimization process; (

**b**) Flowchart of the analytical calculation.

**Figure 7.**Tidal current velocity in Raz de Sein tidal site (after reproduction from [38]).

**Figure 8.**Torque and speed profiles (after reproduction from [39]).

**Table 1.**AR200Horizontal Axis Tidal turbine parameters [7].

Item | Value |
---|---|

Rated Power (kW) | 2000 |

Rotation diameter (m) | 20 |

Height from the seabed (m) | 25 |

Swept area of rotor (m^{2}) | 314 |

Cut-in speed (m/s) | 1 |

Cut-out speed (m/s) | 3 |

Mass (t) | 150 |

Material | $\mathbf{Thermal}\text{}\mathbf{Conductivity}\text{}\mathit{\lambda}$ |
---|---|

Iron | 25 |

Slots (average thermal conductivity of an equivalent homogeneous material with copper and insulation) | 5 |

air | 0.025 |

Generator Parameter | Value |
---|---|

$p$ | 69 |

$R$ | 3 m |

${n}_{s}$ | 1 |

$L$$\text{}({\tau}_{LR})$ | 0.91 m (0.30) |

${R}_{ws}/R$ | 0.992 |

${R}_{ss}/R$ | 0.966 |

${R}_{sr}/R$ | 0.960 |

${R}_{wr}/R$ | 0.940 |

${W}_{sy}$ | 20 mm |

${W}_{ry}$ | 20 mm |

$e$ | 6.8 mm |

${B}_{stm}$ | 1.5 T |

${B}_{sym}$ | 1.49 T |

${B}_{rtm}$ | 1.5 T |

${B}_{rym}$ | 1.5 T |

Total active material weight | 23.9 tons |

Iron weight | 18.15 tons |

Copper weight | 5.75 tons |

Average losses | 47 kW |

Average copper losses | 42 kW |

Average iron losses | 5 kW |

$\mathrm{Nominal}\text{}\mathrm{voltage}\text{}{V}_{s}$ | 1154 V |

$\mathrm{Nominal}\text{}\mathrm{current}\text{}{I}_{s}$ | 1323 A |

$\mathrm{Self}\text{}\mathrm{inductances}\text{}{\mathcal{L}}_{d}/{\mathcal{L}}_{q}$ | 3.6 mH/2.7 mH |

$\mathrm{Maximal}\text{}\mathrm{stator}\text{}\mathrm{current}\text{}\mathrm{density}\text{}{J}_{s}$ | 3.20 A/mm^{2} |

$\mathrm{Maximal}\text{}\mathrm{rotor}\text{}\mathrm{current}\text{}\mathrm{density}\text{}{J}_{r}$ | 2.50 A/mm^{2} |

$\mathrm{Nominal}\text{}\mathrm{cos}\phi $ | 0.861 |

Quantity | Min | Max | ||||
---|---|---|---|---|---|---|

AM | 2D | Variation | AM | 2D | Variation | |

${B}_{stm}$ | 0.57 | 0.62 | 8.06% | 1.5 | 1.58 | 5.06% |

${B}_{rtm}$ | 0.57 | 0.55 | 3.64% | 1.5 | 1.46 | 2.74% |

${B}_{sym}$ | 0.56 | 0.59 | 5.08% | 1.49 | 1.48 | 0.68% |

${B}_{rym}$ | 0.56 | 0.59 | 5.08% | 1.5 | 1.53 | 1.96% |

Torque (MNm) | 0.11 | 0.12 | 8.3% | 0.83 | 0.79 | 4.8% |

Analytical Model | FEA | Variation | |
---|---|---|---|

Iron losses in the stator yoke | 4.5 kW | 4.3 kW | −4.6% |

Iron losses in the rotor teeth | 10.2 kW | 9 kW | −13.3% |

Total iron losses | 14.7 kW | 13.3 kW | −10.5% |

Copper losses | 67.9 kW |

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## Share and Cite

**MDPI and ACS Style**

Ousmane Samb, S.; Bernard, N.; Fouad Benkhoris, M.; Kien Bui, H.
Design Optimization of a Direct-Drive Electrically Excited Synchronous Generator for Tidal Wave Energy. *Energies* **2022**, *15*, 3174.
https://doi.org/10.3390/en15093174

**AMA Style**

Ousmane Samb S, Bernard N, Fouad Benkhoris M, Kien Bui H.
Design Optimization of a Direct-Drive Electrically Excited Synchronous Generator for Tidal Wave Energy. *Energies*. 2022; 15(9):3174.
https://doi.org/10.3390/en15093174

**Chicago/Turabian Style**

Ousmane Samb, Serigne, Nicolas Bernard, Mohamed Fouad Benkhoris, and Huu Kien Bui.
2022. "Design Optimization of a Direct-Drive Electrically Excited Synchronous Generator for Tidal Wave Energy" *Energies* 15, no. 9: 3174.
https://doi.org/10.3390/en15093174