# Mass and Aerodynamic Imbalance Estimates of Wind Turbines

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## Abstract

**:**

## 1. Introduction

## 2. System Equation

#### 2.1. Establishment of the Mass and Stiffness Matrix

- ${v}_{i},{w}_{i}$ - displacement in y- and z-direction,
- ${\beta}_{xi}$ - cross section rotation (or torsion angle), and
- ${\beta}_{yi},\phantom{\rule{3.33333pt}{0ex}}{\beta}_{zi},$ - cross section slope in y- and z-direction,

#### 2.2. Establishment of the Load Vector

- inhomogeneous mass distribution of the rotor, and
- aerodynamic forces arising from incorrect pitch angles settings of the rotor blades.

#### Mass Imbalance

#### Aerodynamic Imbalance

#### Load Vector

## 3. Direct Problem

## 4. Inverse Problem

#### 4.1. Inverse Problem Description

`fminsearch`.

`fminsearch`uses the simplex search method which is a direct search method. Hence, we can avoid to use numerical of analytic gradients in case that A does not have Frechet derivative. The weak point of this method is the slowness in running time.

## 5. Computational Results

- Construction of mass and stiffness matrix using the technical parameters of the V80-2MW (geometry, material properties, first (bending) eigenfrequency 0.255 Hz)
- Setting of a 2 degree pitch angle deviation at the blade C and a mass imbalance of 500 kgm at angle ${\varphi}_{m}=4\pi /3=4.19$ rad: $[{\theta}_{1},{\theta}_{2},{\theta}_{3},mr,{\varphi}_{m}]=[0,0,2,500,4.19]$
- Building the load vector $\mathbf{p}$ using formulas (3)–(9) (airfoil code NACA63-421, wind speed 6 m/s, rotational speed 18 rpm, i.e., $\Omega =0.3$ Hz)
- Solving equation (1) using Equation (10)
- Adding $10\%$ noise to the data to simulate the measurements
- Calculate an approximate solution $({\theta}_{1}^{\prime},{\theta}_{2}^{\prime},{\theta}_{3}^{\prime},\Delta m{r}^{\prime},{\varphi}_{m}^{\prime})$ by minimizing the functional (15) with an appropriate regularization parameter

#### 5.1. Parameter Identification

Measurements in | Initial value | Noise in % | Result | Aero. Error (%) | Mass. Error (%) | ||||||||

all nodes | 0 | 0 | 2 | 500 | 4 | No | 0 | 0 | 2 | 500 | 4.19 | 0 | 0 |

Yes | 0.12 | 0.05 | 2.08 | 500.3 | 4.19 | 7.4 | 0.06 | ||||||

0 | 0 | 0 | 0 | 0 | No | -6.1 | 4.57 | 2.64 | 307.6 | -2.32 | 382 | 42 | |

$y,z$ displacement of the top | 0 | 0 | 0 | 450 | 4 | No | -0.81 | -0.3 | 1.98 | 502.9 | 4.19 | 43 | 0.6 |

Yes | -0.02 | -0.88 | 1.96 | 504.9 | 4.2 | 44 | 1.4 | ||||||

0 | 0 | 0 | 0 | 0 | No | -3.05 | -3.05 | 1.54 | 83 | -2.2 | 216 | 84 | |

0 | 0 | 0 | 420 | 1.7 | Yes | 0.37 | -0.92 | 1.75 | 491.7 | 4.19 | 51 | 1.66 |

Original Parameters | Initial values | Reconstructed Parameters | Aero. Error (%) | Mass. Error (%) | ||||

[0 3 0 350 2.09] | [0 0 0 647 2.09] | -0.25 | 2.8 | 0.43 | 342 | 2.1 | 17 | 2 |

[2 -2 0 400 1.05] | [0 0 0 798 1.05] | 1.99 | -0.64 | -0.06 | 402 | 1.05 | 48 | 0.5 |

#### 5.2. Balancing

## 6. Summary

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**MDPI and ACS Style**

Niebsch, J.; Ramlau, R.; Nguyen, T.T.
Mass and Aerodynamic Imbalance Estimates of Wind Turbines. *Energies* **2010**, *3*, 696-710.
https://doi.org/10.3390/en3040696

**AMA Style**

Niebsch J, Ramlau R, Nguyen TT.
Mass and Aerodynamic Imbalance Estimates of Wind Turbines. *Energies*. 2010; 3(4):696-710.
https://doi.org/10.3390/en3040696

**Chicago/Turabian Style**

Niebsch, Jenny, Ronny Ramlau, and Thien T. Nguyen.
2010. "Mass and Aerodynamic Imbalance Estimates of Wind Turbines" *Energies* 3, no. 4: 696-710.
https://doi.org/10.3390/en3040696