# Co-Design Optimization of Direct Drive PMSGs for Offshore Wind Turbines Based on Wind Speed Profile

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Optimal Design Methodology

- The time-dependent control variables are the d–q axis currents ${i}_{oq}\left(t\right)$ and ${i}_{od}\left(t\right)$. In our method, they will be analytically expressed for the two control strategies considered.
- The rotor variable ${B}_{fm}$, the magnitude of the air-gap flux density created by the magnets represented in the circuit via the electromotive force ${e}_{0}\left(t\right)$. It will be optimized analytically to minimize the energy losses for the considered working cycle. Note that the magnets are sized (shape and remanence) afterwards, from ${B}_{fmopt}$.
- The stator geometry variables are $R$, ${r}_{s}$, ${r}_{w}$, ${w}_{mag}$, ${n}_{s},{\tau}_{LR}$ $p$ and $q$ (see Figure 2), which are in the expressions of coefficient ${k}_{\varphi}$, armature reactance ($X\left(t\right)$) and resistance (${\mathcal{R}}_{c}$,${\mathcal{R}}_{\mu}\left(t\right)$) (see Figure 1). These parameters will be optimized by the use of a genetic algorithm to minimize both the mass and the energy losses.

#### 2.1. Basics Equations

#### 2.2. Analytical Expressions of $d$- and $q$-Axis Currents

#### 2.3. Analytical Expression of ${B}_{fmopt}$

#### 2.4. Analytical Expression of Energy Losses

## 3. Modeling

#### 3.1. Electromagnetic Model

#### 3.2. Mass Calulcation

#### 3.3. Thermal Constraint

^{2}K (for air cooled convection). The time-dependent temperature at node $i$ is evaluated with (36).

#### 3.4. Saturation Constraint

#### 3.5. Electrical Constraint

#### 3.6. Mechanical Constraints

## 4. Design Optimization

#### 4.1. Results

#### 4.2. Optimal Machine

#### 4.3. FEA Validation

## 5. Conclusions

## 6. Patents

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

${v}_{d}$,${v}_{q}$ | d-and q- axis terminal voltages (V) |

${i}_{d}$,${i}_{q}$ | d- and q- axis currents (A) |

${e}_{0}$ | back electromotive force (V) |

${\mathcal{R}}_{c}$ | armature resistance (Ω) |

${\mathcal{R}}_{\mu}$ | iron loss resistance (Ω) |

$X$ | synchronous reactance |

${P}_{c}$ | copper losses (W) |

${P}_{mg}$ | iron losses (W) |

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

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

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

${k}_{f}$ | slot fill factor |

${k}_{t}$ | tooth opening to the slot pitch ratio |

${k}_{L}$ | coefficient for correcting the active length |

$L$ | active length |

${\tau}_{LR}$ | length to outer stator radius ratio |

$R$ | outer stator radius |

${R}_{o}$ | inner rotor radius (m) |

${R}_{s}$ | inner stator radius |

${R}_{r}$ | outer rotor radius |

${r}_{s}$ | reduced inner stator radius |

${R}_{w}$ | outer winding radius |

${r}_{w}$ | reduced outer winding radius |

${w}_{ag}$ | air-gap thickness |

${w}_{mag}$ | magnetic airgap (magnet + mechanical airgap) (m) |

${w}_{PM}$ | permanent magnet height (m) |

${w}_{t}$ | slot width (m) |

${w}_{y}$ | armature yoke thickness (m) |

${b}_{PM}$ | permanent magnet width (m) |

${n}_{s}$ | number of turns per phase per pole |

$p$ | number of pole pairs |

$q$ | number of phases |

${\beta}_{PM}$ | electrical magnet pole arc (rad) |

${\sigma}_{c}$ | electric conductivity |

${\Omega}_{m}$ | machine mechanical angular velocity (rad/s) |

${\theta}_{max}$ | maximal permissible temperature (°C) |

${\theta}_{c}$ | temperature in the copper (°C) |

${\theta}_{amb}$ | ambient temperature (°C) |

${h}_{eq}$ | heat transfer coefficient (W/m^{2}K) |

${\rho}_{c}$ | copper density (kg/m^{3}) |

${\rho}_{Fe}$ | steel density (kg/m^{3}) |

${\rho}_{PM}$ | permanent magnet density (kg/m^{3}) |

${C}_{pc}$ | specific heat capacity of copper (J/Kg/K) |

${C}_{pFe}$ | specific heat capacity of steel (J/Kg/K) |

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**Figure 3.**(

**a**) Thermal equivalent circuit of a cylindrical element; (

**b**) lumped parameter thermal model of PMSG.

Parameters | Values |
---|---|

Blade radius | 82 m |

Maximal power | 10 MW |

Cut-in speed | 2.5 m/s |

Rated speed | 12 m/s |

Cut-out wind speed | 25 m/s |

Parameters | Values |
---|---|

${B}_{sat}$ | 1.6 T |

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

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

${W}_{ymin}$ | 40 mm |

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

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

${k}_{h}$ | 30 |

${h}_{int}$ | 10 W/m^{2}k |

${h}_{ext}$ | 100 W/m^{2}k |

${\theta}_{max}$ | 140 °C |

${\theta}_{amb}$ | 20 °C |

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

${\rho}_{Fe}$ | 7800 Kg/m^{3} |

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

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

Parameters | Min | Max |
---|---|---|

$p$ | 20 | 200 |

${r}_{s}$ | 0 | 1 |

${r}_{w}$ | 0 | 1 |

$R$ | 2 | 5 |

${\tau}_{LR}\left(L/2R\right)$ | 0.2 | 0.6 |

${W}_{mag}$ | 10 mm | 100 mm |

${n}_{s}$ | 1/2 | 10 |

q = 3 | q = 5 | |||
---|---|---|---|---|

FW | MTPA | FW | MTPA | |

Cost of magnets (kEUR) | 371 | 218 | 458 | 311 |

Cost of iron (kEUR) | 135 | 126 | 141 | 138 |

Cost of copper (kEUR) | 296 | 228 | 315 | 299 |

Total material cost (kEUR) | 802 | 572 | 914 | 748 |

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

$q$ | 3 |

$p$ | 156 |

$R$ | 5 m |

$L$$\text{}({\tau}_{LR})$ | 1.15 m (0.23) |

${r}_{s}$ | 0.968 |

${r}_{w}$ | 0.992 |

${B}_{r}$ | 1.2 T |

${B}_{fm}$ | 1 T |

${B}_{tm}$ | 1.59 T |

${B}_{ym}$ | 0.79 T |

${W}_{ag}$ | 8 mm |

${W}_{PM}$ | 18.7 mm |

Total active material weight | 61.75 tons |

Iron weight | 42.2 tons |

Copper weight | 15.2 tons |

Magnet weight | 4.36 tons |

Average losses | 253 kW |

$\mathrm{Nominal}\text{}\mathrm{voltage}\text{}{V}_{s}$ | 2990 V |

$\mathrm{Nominal}\text{}\mathrm{current}\text{}{I}_{s}$ | 1030 A |

$\mathrm{Nominal}\text{}cos\phi $ | 0.98 |

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

${B}_{tm}$ | 1.59 | 1.6 | 0.7% |

${B}_{ym}$ | 0.79 | 0.84 | 6.3 |

Torque (MNm) | 8.6 | 8.13 | 5.47% |

Magnitude of the EMF (1st harmonic) (kV) | 3.02 | 2.96 | 2% |

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

Iron losses in the yoke | 20 kW | 13 kW | −35% |

Iron losses in the teeth | 139 kW | 173 kW | 25% |

Total iron of the stator | 159 kW | 163kW | 3% |

Copper losses | 173 kW |

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**MDPI and ACS Style**

Dang, L.; Ousmane Samb, S.; Sadou, R.; Bernard, N.
Co-Design Optimization of Direct Drive PMSGs for Offshore Wind Turbines Based on Wind Speed Profile. *Energies* **2021**, *14*, 4486.
https://doi.org/10.3390/en14154486

**AMA Style**

Dang L, Ousmane Samb S, Sadou R, Bernard N.
Co-Design Optimization of Direct Drive PMSGs for Offshore Wind Turbines Based on Wind Speed Profile. *Energies*. 2021; 14(15):4486.
https://doi.org/10.3390/en14154486

**Chicago/Turabian Style**

Dang, Linh, Serigne Ousmane Samb, Ryad Sadou, and Nicolas Bernard.
2021. "Co-Design Optimization of Direct Drive PMSGs for Offshore Wind Turbines Based on Wind Speed Profile" *Energies* 14, no. 15: 4486.
https://doi.org/10.3390/en14154486