# Wind Turbine Optimization for Minimum Cost of Energy in Low Wind Speed Areas Considering Blade Length and Hub Height

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## Abstract

**:**

## 1. Introduction

## 2. Wind Turbine Model

^{3}.

#### 2.1. Materials

_{1}is the longitudinal modulus, E

_{2}is the transverse modulus, G

_{12}is the shear modulus, v

_{12}is the Poisson’s ratio and the thickness is the laminate thickness of each ply) are listed in Table 4.

#### 2.2. Original Blade Layup Schedule

#### 2.3. Cost of Energy (COE) Model

## 3. Theoretical Models and Verifications

#### 3.1. Aerodynamic Model

_{hub}is the wind speed at the hub; H

_{hub}is the hub height; V

_{ref}is the reference wind speed at the reference height; H

_{ref}is the reference height; and n is the friction coefficient and set as 0.23 for low wind speed condition.

#### 3.2. Blade Element Momentum (BEM) Code Validation

^{3}and the cone and tilt angle of the wind turbine are set to zero degrees. The RFoil is used to calculate the aerodynamic characteristics of Delft University (DU) series airfoils for a specific Reynolds number according to their locations. The power coefficient and the power curves calculated by BEM code are implemented in MATLAB software (MATLAB 2014b; The MathWorks Inc.: Natick, MA, USA, 2014). In order to validate the BEM code, the aerodynamic performance data from the wind turbine manufacturer, based on simulations by WINDnovation Engineering Solutions GmbH, are compared with the BEM code. The numerical simulations and the manufacturer’s data show an excellent agreement as shown in Figure 2. The power curve calculated by BEM code shows that the rated power is achieved with wind velocity of 9 m/s. Therefore, the aerodynamic performance prediction of the wind turbine blade calculated by BEM code for this study can be considered acceptable.

#### 3.3. Structural Model

## 4. The Optimization Model

#### 4.1. Design Variables

#### 4.2. Design Constraints

_{e}

_{50}= 1.4V

_{ref}(V

_{ref}= 37.5 m/s for Class III wind resources).

#### 4.2.1. Blade Design Constraints

_{50}is the strain at 50-year extreme wind condition, the ε

_{ult}is ultimate strain, the partial safety for loads (γ

_{f}) is set as 1.35 and the partial safety factor for materials (γ

_{m}) is set as 1.1 according to IEC 61400-1 requirement. The representative numbers of ultimate strain is set as 0.03. The maximum tip deflection of the blade under extreme loading is added to ensure adequate stiffness. In this work, considering the blade length is set as a variable, the maximum tip deflections with various blade lengths are set as:

_{opt}is the maximum tip deflection of the optimal blade, the δ

_{orig}is the maximum tip deflection of the original blade, R

_{opt}is the length of the optimal blade, and R

_{orig}is the length of the original blade.

#### 4.2.2. Tower Design Constraints

_{allow}is the allowable deformation and it can be determined according to the reference [18]. In order to ensure the tower has an adequate structural performance during optimization process, the allowable deformation is set as [8]:

_{opt}is the height of the optimal tower, H

_{orig}is the height of the original tower, and d

_{orig}is the top deformation of the original tower.

_{cr}is the critical stress, m

_{RNA}is the mass of rotor and nacelle assembly, m

_{tower_mid_up}is the tower mass above the middle of the tower, A

_{mid}is the area of the middle of the tower, T is the rotor thrust, H is the tower height, and I

_{mid}is the area moment of inertia at the middle of the tower.

_{cr}generated by the loads cannot exceed the allowable stress:

_{allow}is the allowable stress and the value of it is given by:

_{ys}is the yield strength and γ

_{tm}is the material safety factor of the steel. The yield strength of the steel is 345 MPa for the wall thickness between 16 mm and 40 mm. The material safety factor is set as 1.1 according to IEC 61400-1.

#### 4.3. The Optimization Algorithm

_{1}= c

_{2}= 0.5), variable dimensions (n = n

_{aerodynamic}+ n

_{structural}= 36), and the inertia factor (0.85).

## 5. Results and Discussion

#### 5.1. Blade Optimization Results

#### 5.2. Tower Optimization Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Comparison between the power coefficient and power curves obtained by blade element momentum (BEM) code and wind turbine manufacturer’s data.

**Figure 3.**Comparison between the stiffness and mass per unit length distributions obtained by CLT code and actual blade.

**Figure 12.**The comparison of chord and twist distributions between the optimal blades and the original one.

**Figure 13.**The comparison of thickness distributions between the optimal blades and the original one.

**Figure 15.**The comparison of the annual energy production (AEP) between the optimal blades and the original one.

**Figure 18.**Tower outer diameter and wall thickness distributions of the baseline and the optimized towers.

Span Location (m) | Chord (m) | Twist (°) | Airfoil | Thickness (%) |
---|---|---|---|---|

0 | 2.40 | 18.0 | circle | 100 |

1.0 | 2.40 | 18.0 | circle | 100 |

13.4 | 3.42 | 9.6 | DU00-W2-401 | 40 |

16.2 | 3.17 | 6.9 | DU00-W2-350 | 35 |

24.9 | 2.36 | 2.7 | DU97-W-300 | 30 |

49.8 | 1.02 | −1.56 | DU91-W2-250 | 25 |

59.8 | 0.02 | −1.6 | DU93-W-210 | 21 |

Rated power | 2.1 MW |

Rated wind speed | 9 m/s |

Cut in wind speed | 3 m/s |

Cut out wind speed | 20 m/s |

Number of blades | 3 |

Design tip speed ratio | 10.6 |

Rotor diameter | 122.3m |

Control type | Variable speed—variable pitch |

Maximum power coefficient | 0.483 |

Tower height (m) | 95 |

Tower top outer diameter (m) | 3.15 |

Tower base outer diameter (m) | 5.72 |

Tower top wall thickness (mm) | 24 |

Tower base wall thickness (mm) | 40 |

Density (kg/m^{3}) | 7820 |

Young’s modulus (GPa) | 200 |

Poisson ratio | 0.3 |

Yield strength (MPa) | 345 |

Stack ID | Materials | E1 (GPa) | E2 (GPa) | G12 (GPa) | v12 | ρ (kg/m^{3}) | Thickness (mm) |
---|---|---|---|---|---|---|---|

1 | Unidirectional | 39.18 | 11.69 | 3.5 | 0.3 | 1940 | 0.892 |

2 | Bi-axial | 11.5 | 11.5 | 10.8 | 0.48 | 1970 | 0.555 |

3 | Triax | 25.8 | 13.59 | 7.36 | 0.36 | 1842 | 0.906 |

4 | Gelcoat | 3.44 | 3.44 | 1.38 | 0.3 | 1235 | 0.6 |

5 | Balsa | 1.321 | 0.03 | 0.006 | 0.39 | 151 | 25.4 |

6 | Polyvinyl chloride (PVC) | 0.035 | 0.035 | 0.022 | 0.3 | 60 | 5 |

Parameter | Initial Value | Optimal Value | Optimal Value (Rated) |
---|---|---|---|

Length (m) /Mass (kg) | 59.8/13,240 | 57.84/12,540 | 56.2/11,620 |

Tip deflection (m) | 7.576 | 6.92 | 6.31 |

AEP (GW) | 5.7830 | 6.0921 | 5.8324 |

Cost of energy ($/kWh) | 0.057 | 0.0478 | 0.047 |

Tip speed ratio | 10.6 | 9.8 | 9.6 |

Parameter | Initial Value | Optimal Value | Optimal Value (Rated) |
---|---|---|---|

Height (m) /Mass (kg) | 95/186,780 | 117.2/268,320 | 119.8/266,140 |

Top deformation (m) | 0.65 | 0.828 | 0.816 |

Parameter | Rotor Cost ($1000) | Tower Cost ($1000) | Foundation Cost ($1000) |
---|---|---|---|

Original wind turbine | 866.56 | 280.17 | 83.79 |

Optimal design (without considering rated power (CRP)) | 787.81 | 402.48 | 88.84 |

Optimal design (with CRP) | 723.92 | 399.21 | 87.62 |

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**MDPI and ACS Style**

Yang, H.; Chen, J.; Pang, X.
Wind Turbine Optimization for Minimum Cost of Energy in Low Wind Speed Areas Considering Blade Length and Hub Height. *Appl. Sci.* **2018**, *8*, 1202.
https://doi.org/10.3390/app8071202

**AMA Style**

Yang H, Chen J, Pang X.
Wind Turbine Optimization for Minimum Cost of Energy in Low Wind Speed Areas Considering Blade Length and Hub Height. *Applied Sciences*. 2018; 8(7):1202.
https://doi.org/10.3390/app8071202

**Chicago/Turabian Style**

Yang, Han, Jin Chen, and Xiaoping Pang.
2018. "Wind Turbine Optimization for Minimum Cost of Energy in Low Wind Speed Areas Considering Blade Length and Hub Height" *Applied Sciences* 8, no. 7: 1202.
https://doi.org/10.3390/app8071202