*3.1. Aerodynamic Performance*

In this section the aerodynamic performance under uncertainties is discussed. In Figure 6 the mean variation in power coefficient (CP) with respect to the clean reference turbine is shown. The standard deviation and associated probability contours are also shown. The CP 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 CP 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.

The highest value for the mean decrease in CP is of −2.6% at 10 m/s. At this wind speed the reduction in CP 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.

The power output per wind speed bin is shown in Figure 7. Upon examination of this figure, it is apparent that the blade damage has a greater impact on power output between 8 m/s and 12 m/s, confirming what was seen in the relative trends of Figure 6. At 4 m/s, however, as previously pointed out, the mean power output is only 174 kW, higher than the 172 kW of the nominal case. Due to the little power produced, this difference as well as the high standard deviation of ±7 kW (±4%) are not visible in the plot, further highlighting how such variation has little impact on the overall performance. In order to better understand the global results, each wind speed bin can be examined more in detail.

**Figure 6.** Variation in power coefficient, mean value (μ), standard deviation (σ) and probability.

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

The response surfaces reporting the differences in CP 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 CP 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 CP variation predicted for the nominal geometry is zero as expected.

**Figure 8.** Variation in power coefficient. Response surfaces as contour plots at (**a**)4m/s, (**b**)8m/s and (**c**) 10 m/s.

In the 4 m/s wind speed bin, an increase in CP 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 CP (about 10%), while γ2 shows a mild decrease in CP, 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.

**Figure 9.** Aerodynamic coefficients for nominal and most significant power coefficients (CPs): (**a**) Lift coefficient; (**b**) drag coefficient.

As a consequence of the increased AoA, lift and drag forces slightly increase and, more importantly, are more tangentially and axially oriented. The new force composition generates more torque and more power for some of the combinations of ε and τ. As shown in Figure 10, the same phenomena are occurring for all the damaged configurations: a change in the lift and drag coefficients leads to a different BEM equilibrium point with different induction and AoA along the entire area of the blade affected by damage. However, increasing ε also significantly increases drag, leading to lower performance and offsetting the benefit of a higher AoA, despite the change of orientation of the forces. For instance, in γ10, the highest increases in drag are observed, exceeding 30% at an AoA of around 2◦.

In Figure 11 the average AoA for the nominal and four damage combinations for all the wind speed bins is shown. For all the damaged combinations, the highest average AoAs are predicted in the 8 m/s and 10 m/s wind speed bins. At 8 m/s mean wind speed the average AoA for the nominal case at 78% blade span is about 6.9◦, while the damaged cases work at an even higher AoA due to decreased induction, as previously discussed. In these wind speed bins, there is no power increase in any combination of ε and τ. From the analysis of Figure 9, the higher the AoA, the wider the difference is in lift and drag coefficients. This ultimately leads to the power losses observed in Figures 6–8, with peaks that exceed −10% at 8 m/s and −12% at 10 m/s, respectively. It is also interesting to note that ε is the main cause of performance decrease and has a more pronounced effect than τ. This is due to the fact that LE damage causes a reduction in the stall AoA of the airfoil, which strongly influences high-AoA operation and a more pronounced increase in drag than TE damage.

The probability distributions found from the evaluation of the computed response surfaces at 4 m/s, 8 m/s and 10 m/s are shown in Figure 12. At 4 m/s, the variation in CP is most affected by uncertainties. The peak is located at 1% of variation in CP, but the resulting distribution is fat-tailed. Indeed, the standard deviation is ±4.1% and the probability to lose or gain CP 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 CP 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.

**Figure 10.** Relevant turbine figures: (**a**) Angle of attack along the outer part of the blade and (**b**) thrust (FT) and tangential (Fϑ) 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**)4m/s, (**b**), 8 m/s and (**c**) 10 m/s.

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

The uncertainties in AEP estimation are discussed in this section. The AEP was calculated according to IEC 61400-1 standard turbine classes. A Weibull wind speed distribution with shape factor of 2 and average values of 8.5 m/s and 10 m/s were used to model sites of IEC wind class II and IA. In particular, class IA is chosen as this is the design class of the DTU 10 MW RWT and class IIA is chosen as representative of medium wind speed sites, where such a turbine might also be installed. The availability factor was assumed to be 1. This assumption is justified by the fact that relative variations are mainly analyzed in the present study. The variation in AEP for the two wind distributions is shown in Figure 13. Both response surfaces well predict the trends around ε = 0, τ = 0, showing no variation in AEP in that point. The LE erosion, ε, is the main driver for AEP reduction, as decreases are mostly along the ε axis. The trailing edge damage, τ, has a minor influence in AEP, as clearly visible in Figure 13. Moreover, the trailing edge damage contribution seems to be dependent on the erosion level. For instance, if one considers the six combinations of ε and τ (where ε = 0, 4, 8 and τ = 0, 3) for wind class IIA shown in Table 3, the point ε = 4, τ = 0 gives a variation in AEP of −1.87%, while the point ε = 4, τ = 3 gives a variation of −2.24%. Therefore, for ε = 4, the trailing edge damage increases losses by 0.37%. By performing the same consideration for ε = 8, trailing edge damage increases losses by 0.82%. This means that the TE contribution to losses increases as ε increases.

**Table 3.** Annual Energy Production (AEP) reduction for some of the computed ε-τ combinations.


The highest variation in AEP predicted by the response surface is −10.35% at ε = 8, τ = 3 for class IIA, as seen in Figure 13a. The highest simulated AEP reduction is −6.21% for γ10 (class IIA, Figure 13a). For wind class IA, the highest variations in AEP are lower than the ones predicted for class IIA and amount to −8.56% in ε = 8, τ = 3 and −5.02 in γ10, as shown in Figure 13b. Such differences are due to the Weibull wind speed PDFs. The probability for the machine to work in the bins range from 8 to 12 m/s, where the highest losses in power occur, are 36% and 31% for IIA and IA, respectively. This difference is the main cause of different variations in AEP for the two classes.

**Figure 13.** Variation in AEP. Response surfaces as contours plot for wind classes (**a**) IIA and (**b**) IA.

Finally, we consider the probability distributions AEP variation shown in Figure 14. As is also the case for the previously shown distributions, the PDFs are obtained by sampling the response surfaces 250,000 times. The mean and standard deviations of the PDFs are −1.21% and ±1.04% for class IIA and −0.98% and ±0.84% for class IA. Such mean reductions are indeed significant on a multi-MW scale turbine and are in line with the finding of Eisenberg et al. [33] but seem to be lower than the values indicated by most of the present research [5,6,8]. Both distributions show a clear peak, with the mode of the PDFs below 1% AEP loss in both cases. For both the IEC 61400-1 IA and IIA scenarios, the left tails of the distributions are long, reaching values of 6–8% AEP reductions, coherently with the response surface shown in Figure 13. The probability associated to values of AEP reduction in the order of 3–8%, which most authors indicate, is almost insignificant in the present test case. It is important to stress that these results depend on the assumed PDFs, which are, as discussed, based on published literature and appear reasonable based on the authors' experience. Moreover, as Fiore and Selig have suggested [34], larger turbines seem to be impacted less by LE damage phenomena such as erosion. However, results suggest that the commonly forecasted reductions might be based on heavy-damage scenarios, which, whilst not unrealistic, have low probability of occurrence.

**Figure 14.** Variation in AEP PDFs for wind classes (**a**) IIA and (**b**) IA.

The wind class IIA shows higher standard deviation and higher left tail length. As previously mentioned, which is due to the fact that in the class IIA scenario the turbine operates at rated power for a shorter period of time with respect to the class IA scenario. In fact, as also pointed out by Eisenberg [33], the turbine's power output does not experience any significant variation for wind speeds above rated when the blades are damaged and therefore the higher the mean wind speed, the lower the variation in AEP. These results clearly depend on the IEC class that was chosen. Lowering the average wind speeds even further (IEC Class III), the turbine is expected to spend less time at rated power, and therefore, AEP losses are expected to further decrease for the present test case. Although low wind speed sites have recently been exploited for wind turbine installation, specially designed machines with low specific power are being installed in such sites, resulting in machines that are able to spend significant time at rated power even in these sites. As noted in [33], a utility-scale machine will spend 40% to 60% of its time at rated power, where blade damage has no effects. In addition, although the main cause of LE erosion is related to the rotational tip velocity, it can be argued that in lower wind speed sites, less transport of abrasive particles will arise, therefore leading to less erosion.

## **4. Conclusions**

This study proposes the use of an uncertainty quantification approach to the modelling of the effects of blade damage on the performance of multi-MW wind turbines. The proposed approach aims at overcoming some of the issues associated with the evaluation of a single test case. In fact, treating blade damage as a random phenomenon, bias due to a specific test case of a combination of bladedamaging factors can be avoided and more general conclusions can be drawn. The entire process is simulated numerically. First, geometric shape modifications are applied to the airfoils that constitute the turbine's blade. Lift and drag coefficients are calculated using CFD. The newly found coefficients are then applied to an aero-servo-elastic model of the wind turbine. Uncertainties are propagated through the model using an arbitrary polynomial chaos method.

Results show that LE damage has the larger influence on power and AEP losses. For the selected test case, TE damage has little impact, except for when the turbine is operating at very low wind speeds, where a slight performance increase is observed due to TE damage. Focusing on absolute values, maximum average power reductions are observed at 8 m/s and 10 m/s mean wind speeds and are of 2.2% and 2.6%, respectively. The most unfavorable damage combinations simulated showed a decrease in AEP of up to over 6%. By looking at the probabilistic framework, however, the configurations with the highest probability of occurring based on the input PDFs show AEP reductions of below 1% in both IEC classes I and IIA. Indeed, mean AEP reductions of just below 1% for class IA and just above 1% for class IIA are estimated. These values, whilst significant, seem to be notably lower than what is commonly forecasted in published literature that, however, is strictly site- or turbine-dependent. It is important to point out that the results of the present study do not indicate that published literature values are unrealistic (even though sometimes a too large span coverage of erosion is considered), however, for the present test case, representative of modern turbine size and design trends, such values seem to have very low probabilities of occurrence. Indeed, AEP decreases exceeding 10% are noted in the present study. Blade damage is an issue that should still be taken very seriously by the industry, due to its structural implications that were not investigated in the present work; however, the impact on AEP does not seem to be as pronounced as early research indicated. A great deal of factors could cause these discrepancies, which could be due to the radial damage extension considered and size and hence the Reynolds number of the turbine, which are not investigated herein and therefore remain an open issue, where additional research would definitely be beneficial. As pointed out by other authors, LE damage seems to have a lower effect on larger wind turbines. Although this is not the focus of the present work, the results of this study, if put into perspective with other published literature that reports higher AEP decreases on smaller turbines, seem to confirm this.

Moreover, as previously discussed, the results strongly depend on the input PDFs. The presented method can be however adapted to different input PDFs, which are hopefully more extensively supported by field data. Nevertheless, the present assumptions can be considered realistic for medium-low damaging environments or for blades where regular maintenance schedules are planned. It is also important to point out that these results are valid strictly only for the present test case. A selection of different study cases might influence the results significantly, as, in the authors' experience, LE damage affects different airfoils to different degrees. Finally, the LE damage model also influences the results. Although the model is calibrated and tested with respect to experimental data and is adequate for the present parametric framework, it is hard, if not impossible, to accurately reproduce small, stochastic features that might influence the sectional efficiency significantly.

In conclusion, even considering these factors, it is apparent that the present statistical approach is able to give designers a better picture of the impact of blade damage.

**Author Contributions:** Conceptualization, F.P. and A.B.; methodology, F.P., L.C., S.S.; software, F.P., L.C., M.C.; formal analysis, A.B., F.P.; investigation, F.P., L.C.; resources, A.B.; data curation, F.P., L.C., A.B.; writing—original draft preparation, F.P., L.C.; writing—review and editing, A.B., S.S.; visualization, F.P.; supervision, S.S., M.C., A.B.; project administration, A.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflicts of interest.
