# Uncertainty Quantification of the Effects of Blade Damage on the Actual Energy Production of Modern Wind Turbines

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Stochastic Approach

#### 2.2. Probability Density Functions

#### 2.3. Blade-Damage Model

^{®}tool. The LE of the airfoil is moved inward by a maximum depth of θ. Similarly to what was done by Schramm et al. [22], the leading-edge was flattened. The height of the flattened area is imposed to be $h=2\mathsf{\theta}$. Damage extends up to ε on the suction side of the airfoil and up to 1.3ε on the pressure side, as done in [5]. This is also motivated by the fact that wind turbine airfoils are designed to operate with a positive angle of attack (AoA), and therefore, the pressure side of the airfoil is more exposed to wear. The depth of delamination at the end of the damaged area is equal to ${D}_{end}=\mathsf{\theta}/3$. The TE and LE damage models are shown in Figure 3. The models are also described in further detail in [10]. The present model is a simplified version of the real LE damage pattern adequate for a parametric study like the present one, which cannot therefore reproduce all the features of a real, three-dimensional damaged blade. The model, however, is in line with the proposals of other authors [22,23] and also qualitatively reproduces the damaged shapes obtained from computational models [8,24], as seen in experiments [5].

#### 2.4. CFD Setup

^{®}FLUENT

^{®}(Version 18.2) solver is used to calculate the 2D polars. A Reynolds-averaged Navier‒Stokes (RANS) approach is used. The Navier‒Stokes equations are solved in a coupled manner with second order upwind spatial discretization. Turbulence closure is achieved with the k-ω Shear Stress Transport (SST) model. A bullet-shaped computational domain is used, as with this shape open-field conditions can be modelled with only one inlet and one outlet boundary condition. In order to ensure that the boundary conditions do not influence the results, the computational domain is 74 chord lengths long and 40 chord lengths wide, as shown in Figure 4a.

^{5}elements and 500 elements along the airfoil surface, a medium mesh with 2.8 × 10

^{5}elements and 650 elements along the airfoil’s surface, and a fine mesh with 3.6 × 10

^{5}elements and 750 elements along the airfoil’s surface. The lift (C

_{l}) and drag (C

_{d}) coefficients are calculated with CFD between 20° and 30° of AoA; values for AoA higher and lower than this are extrapolated using Viterna’s method [25]. A total roughness height of 0.4 mm is imposed on the airfoil’s nose trough an equivalent sand-grain roughness height, estimated through the simple correlations provided in [26]. This roughness height is selected based on the observations of several authors [5,6,21] and models medium to advanced pitting and gauging of the LE. As the focus of the LE damage model is on advanced stages of damage, a constant value of roughness was considered suitable across all the LE-damaged cases.

^{6}. Figure 4b demonstrates good agreement between the experimental values and CFD predictions, with limited differences that can be attributed to the unspecified wind tunnel turbulence level and to the surface finish of the reference model.

#### 2.5. Aeroelastic Setup

#### 2.6. General DLC Setup

## 3. Results

#### 3.1. Aerodynamic Performance

_{P}) with respect to the clean reference turbine is shown. The standard deviation and associated probability contours are also shown. The C

_{P}mean value is lower than the nominal one for all the wind speed bins except for the 4 m/s one. In this wind speed bin, the average gain is about 1%. The reasons that cause such gains are related mainly to the TE damage; however, this gain in performance, while conceptually interesting, is weakened by two factors. First, at 4 m/s the power is about 60 times lower than the nominal one, and thus the effect on the AEP will be minimal. This can be seen clearly in Figure 7. Secondly, there is a high dispersion in the C

_{P}values and therefore the expected value is hard to predict. The high dispersion is due to the extremely different response from the damaged airfoils. Both gain and power losses at this wind speed occur. The time averaged AoA from 30% of the blade span to tip goes from 0° to 5°. This allows some of the damaged airfoils to operate with favourable lift and drag forces with respect to others. More details about this behaviour are given below.

_{P}is of −2.6% at 10 m/s. At this wind speed the reduction in C

_{P}can exceed −12%. Moreover, from 8 m/s to 12 m/s, mostly only power losses occur. In this wind speed range, a significant part of the total turbine’s energy is produced; therefore, power reductions in this region will eventually lead to a significant reduction in AEP. Finally, for wind speeds higher than 14 m/s, shown in the grey-shadowed region in Figure 6, the damage effects are no longer visible, as from this wind speed onwards a lower pitch-to-feather regulation is able to compensate for the aerodynamic losses.

_{P}for the wind speed bins that show the most relevant differences are shown in Figure 8. For the wind speed bin of 4 m/s the response surface slightly overestimates the C

_{P}of the nominal geometry. Such behavior is shown in Figure 8a around the ε = 0, τ = 0 point. On the other hand, the response surface prediction gives good results at 8 m/s and 10 m/s where the C

_{P}variation predicted for the nominal geometry is zero as expected.

_{P}for several combinations of ε and τ can be noted. To explain this unexpected trend, one can consider the collocation point pairs γ 7 & γ 2 (same ε and the highest and the lowest τ, respectively) and γ10 & γ3 (same τ and the highest and the lowest ε, respectively). Therefore, looking at the pair γ2 & γ7 the influence of τ is highlighted, while looking at the pair γ10 & γ3 the influence of ε is highlighted. Point γ7 shows the highest increase in C

_{P}(about 10%), while γ2 shows a mild decrease in C

_{P}, about −1.5%; thus, as shown in Figure 8, power increases as tau increases. The other γ-pair shows the opposite behavior, for γ10, the power coefficient decreases by 12%, while γ3 shows an increase in the power coefficient of about 3%, and thus, power decreases as tau decreases. To better understand the trends, the lift and drag coefficients for the FFAW3-241 airfoil (i.e., the airfoil present in the damaged part of the blade) for the four damage levels are shown in Figure 9 with respect to the reference configuration. In general, lift decreases and drag increases for all of the damaged configurations as expected. Focusing on the mean AoA recorded for the various damaged configurations at 4 m/s in Figure 9a, it is clear how the mean AoA increases for all of the damaged cases. This is due to the lower lift of the damaged cases. A new operational equilibrium point in the BEM code is then reached, with a lower induction and thus a higher AoA.

_{P}is most affected by uncertainties. The peak is located at 1% of variation in C

_{P}, but the resulting distribution is fat-tailed. Indeed, the standard deviation is ±4.1% and the probability to lose or gain C

_{P}are about 40% and 60%, respectively. At 8 m/s and 10 m/s, the distributions are strongly asymmetric and have lower standard deviations with respect to the 4m/s case and are equal to ±1.7% and ±2% at 8 m/s and 10 m/s, respectively. In both cases, the probability for a C

_{P}gain is zero and losses always occur. They both have a marked left tail, but a higher dispersion at 10 m/s is found. The probability peak is clearly located on the right of the mean value at −1.5% and −1.7% for 8 m/s and 10 m/s, respectively.

#### 3.2. Annual Energy Production (AEP)

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Acronyms | |

AEP | Annual energy production, kWh |

aPC | Arbitrary polynomial chaos |

BEM | Blade element momentum |

CFD | Computational fluid dynamics |

DLC | Design load case |

DRC | Delft research controller |

IEC | International electrotechnical commission |

LE | Leading edge |

PC | Polynomial chaos |

PCM | Probabilistic collocation point |

Probability desity function | |

RANS | Reynolds averaged navier stokes |

SST | Shear stress transport |

TE | Trailing edge |

Latin Letters | |

AoA | Angle of attack, deg. |

c | Blade chord, m |

c_{i} | Expansion coefficients |

Cd | Drag coefficient |

Cl | Lift coefficient |

C_{P} | Turbine power coefficient |

D_{end} | Delamination depth at the end of damaged area, m |

F_{T} | Thrust force, N/m |

F_{ϑ} | Tangential force, N/m |

h | Leading edge flattened area height, m |

P^{(i)} | Orthogonal polynomials |

R_{M} | Polinomial expansion remainder |

Y | Specific output of interest |

Greek Letters | |

α, β | Beta function’s shape parameters |

γ | Collocation point |

ε | Leading edge erosion factor |

Θ | Leading edge erosion depth |

μ | Momentum |

ξ | Generic aleatory variable |

σ | Standard deviation |

τ | Trailing edge damage factor |

ω | Weighting term |

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**Figure 4.**CFD validation: (

**a**) Illustration of the adopted computational domain; (

**b**)validation of the numerical setup in respect to data from [5] at Re = 1.5 × 10

^{6}.

**Figure 7.**Power output per wind speed bin for nominal and mean damaged (μ) turbine with standard deviation (σ).

**Figure 8.**Variation in power coefficient. Response surfaces as contour plots at (

**a**) 4 m/s, (

**b**) 8 m/s and (

**c**) 10 m/s.

**Figure 9.**Aerodynamic coefficients for nominal and most significant power coefficients (CPs): (

**a**) Lift coefficient; (

**b**) drag coefficient.

**Figure 10.**Relevant turbine figures: (

**a**) Angle of attack along the outer part of the blade and (

**b**) thrust (F

_{T}) and tangential (F

_{ϑ}) for the outer part of the blade at 4 m/s mean wind speed.

**Figure 11.**Angle of attack vs. wind speed for nominal and four damaged conditions at 78% blade span.

**Figure 12.**Variation in power coefficient probability density functions (PDFs) with mean value (μ) and standard deviation (σ) at (

**a**) 4 m/s, (

**b**), 8 m/s and (

**c**) 10 m/s.

**Figure 13.**Variation in AEP. Response surfaces as contours plot for wind classes (

**a**) IIA and (

**b**) IA.

Parameter | α | β | Support | |
---|---|---|---|---|

ε | Beta | 2.0 | 6.0 | 0–10 (%) |

τ | Beta | 2.0 | 6.0 | 0–4 (%) |

γ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

ε | 2.2792 | 2.2792 | 0.7134 | 2.2792 | 0.7134 | 4.4189 | 2.2792 | 0.7134 | 4.4189 | 6.7884 |

τ | 0.9118 | 0.2854 | 0.9118 | 1.7677 | 0.2854 | 0.9118 | 2.7157 | 1.7677 | 0.2854 | 0.9118 |

ε | τ | ΔAEP/AEP_{0} (%) |
---|---|---|

0 | 0 | 0.00 |

0 | 3 | 0.00 |

4 | 0 | −1.87 |

4 | 3 | −2.24 |

8 | 0 | −9.69 |

8 | 3 | −10.51 |

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**MDPI and ACS Style**

Papi, F.; Cappugi, L.; Salvadori, S.; Carnevale, M.; Bianchini, A. Uncertainty Quantification of the Effects of Blade Damage on the Actual Energy Production of Modern Wind Turbines. *Energies* **2020**, *13*, 3785.
https://doi.org/10.3390/en13153785

**AMA Style**

Papi F, Cappugi L, Salvadori S, Carnevale M, Bianchini A. Uncertainty Quantification of the Effects of Blade Damage on the Actual Energy Production of Modern Wind Turbines. *Energies*. 2020; 13(15):3785.
https://doi.org/10.3390/en13153785

**Chicago/Turabian Style**

Papi, Francesco, Lorenzo Cappugi, Simone Salvadori, Mauro Carnevale, and Alessandro Bianchini. 2020. "Uncertainty Quantification of the Effects of Blade Damage on the Actual Energy Production of Modern Wind Turbines" *Energies* 13, no. 15: 3785.
https://doi.org/10.3390/en13153785