2.1 Droplet impact

Rain erosion is the consequence of multiple impacts stochastically distributed over the surface of the coated laminate. Each impact adds a damage increment to the accumulated damage. For rain and other airborne particles the accumulated damage is a function of several parameters including the number of impacts per unit area, and the magnitude of each impact. This paper is limited to considering the impact by liquid droplets only.

The magnitude of an impact of a droplet hitting perpendicular to the surface may be quantified by the kinetic energy (E_k):

where v is the velocity of the particle relative to the surface and m is the mass of the droplet.

For detailed analysis the impact may be quantified by the contact stress field acting on the surface as a function of time during the impact (Keegan et al., 2012). The contact stresses are functions of the properties of the liquid, the properties of the impacted surface, the impact velocity, and the size and shape of the droplet.

The impact of a spherical droplet immediately causes a normal pressure on the target surface at the initial point of contact. The contact area between the droplet and the solid expands radially at a velocity higher than the speed of sound in water. When the shock wave front reaches the edge of the droplet, a release jet is generated, and the pressure reduces to the stagnation pressure (Bowden, 1964; Dear and Field, 1988).

The simplest expression for the calculation of the initial contact pressure is the water hammer equation (Bowden, 1961). It was derived for a column of liquid impacting a rigid surface, where a compression wave propagates from the contact into the liquid. The immediate contact pressure (p) may be calculated by

where v is the impact velocity, c_l is the speed of the compression wave in the liquid, and ρ_l is the density of the liquid. Accounting for the geometry of a spherical droplet, the contact angle increases as the contact area expands, and the peak pressure at the rim of the contact is analytically derived (Heymann, 1969) as

Taking into account the compliance of the solid, the pressure of the impact between an elastic solid cylinder and a liquid jet (de Haller, 1933) may be expressed as

where subscript “l” is for liquid and “s” for solid. Later numerical modelling works take into account an assumed spherical geometry of the droplet as well as the compliance of the target material (Adler, 1995; Amirzadeh et al., 2017b). These studies also show a pressure peak near the edge of the contact.

Real precipitation droplets falling through the atmosphere are not necessarily spherical. The aerodynamic forces distort the droplet to a burger-bun-like shape. Larger droplets, d > 6 mm, flatten out before splitting up (Fakhari, 2009), while smaller droplets tend to merge and form larger droplets. The droplet geometry may be characterized by its ratio of vertical to horizontal dimensions (Gorgucci et al., 2006). Through a full rotation of 360^∘, the wind turbine blades are hitting the non-spherical droplets from all angles at different relative velocities. This makes the impact scenario even more complex.

2.2 Impact stresses, fracture and fatigue

A typical leading edge consists of a curved laminate of glass-fibre-reinforced polymer (GFRP) with a relatively brittle polyurethane-, polyester- or epoxy-based coating. Many designs have a layer of putty or filler, applied to the laminate and sanded, to make a smooth surface for the coating. Recent developments have added a top layer of elastomeric coating with good damping properties and high fracture toughness, often referred to as leading-edge protection or LEP; see Fig. 1 (taken from Cortés et al., 2017).

An impact on the surface causes stress transients in the material. Stress waves propagate from the impact site into the coated composite (body waves) and along its surface (surface waves). Several stress components are active as functions of the time after the impact, the radial planar position and depth in the material (Woods, 1968; Adler, 1995). These stresses can activate different failure modes depending on the velocity of the impact, the size of the droplet and “the weakest link” in the leading-edge structure.

Figure 1Example of leading-edge protection system configuration (Cortés et al., 2017).

For isotropic, homogenous, elastic materials ring-shaped surface cracks due to Rayleigh waves are the common type of impact damage (Blowers, 1969; Bowden, 1964). For many coated materials, this may also be the governing failure mechanism.

The body waves propagating perpendicular to the surface into the target material can lead to subsurface cracks in the coating and the substrate (Fraisse et al., 2018). The reflection of stress waves between the coating surface and the coating–substrate interface may also play a significant role in the fatigue of the coated laminate (Springer, 1974). Body waves may also cause delamination inside the laminate (Prayago, 2011) and debonding of the coating (Cortés et al., 2017).

A single droplet impact may cause instant damage when the impact velocity is beyond the damage threshold velocity (DTV). For a given droplet size and set of material parameters, DTV was derived for brittle materials by a fracture mechanics approach (Evans et al., 1980).

where K_c is the critical stress intensity factor, c_s is the Rayleigh wave velocity of the target material, D is the diameter of the spherical droplet, ρ_l is the density of the liquid, c_l is the speed of sound in the liquid and λ is a material independent constant.

Repeated stresses below the static strength of a material may eventually cause failure due to cumulative fatigue damage (Miner, 1945). Similarly, impacts below DTV may also add to the accumulated damage, which may eventually cause fracture (Springer, 1975). For materials with a fatigue limit, like some metals, a fatigue threshold impact velocity, below which no erosion will occur, may be defined as an analogy to the endurance limit found in fatigue testing (Heymann, 1969). Rain erosion test data can be regarded as impact fatigue data. Together with operational data for a wind turbine and the local precipitation statistics, it can be used to predict the erosion propagation and lifetime of leading edges in field operation (Eisenberg et al., 2016).

Figure 2Whirling arm rain erosion test specimen photographed at 30 min intervals during 3 h of testing.

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3.1 Analysis of rain erosion test data

An example of rain erosion test data was made available by PolyTech A/S; see Fig. 2. The test specimen material is coated aluminium. The specimen has a length of 225 mm. In this test, the tangential velocity was 140 m s⁻¹ at the tip and 110 m s⁻¹ at the root of the specimen. The rain intensity was 30–35 mm h⁻¹, and droplet sizes ranged from 1 to 2 mm. In the later calculations, for simplicity, it is assumed that the rain intensity is 32.5 mm h⁻¹, and the droplet diameter is uniform at 2 mm. The falling velocity of the droplets is assumed to be 6 m s⁻¹, corresponding to the terminal velocity of 2 mm droplets (Foote et al., 1969). The test was stopped every 30 min for photography of the specimens; see Fig. 2. The photographs are used to determine the progression of erosion. Here, erosion is defined as the visible removal of the top coat. The erosion initiates at the tip of the specimen, where velocity is highest. It then propagates towards the root, where the velocity is lower. Each position on the specimen corresponds to a certain tangential velocity. The data pairs of the position of the erosion front and the time are shown in Table 1 along with the corresponding local rotor velocities. The kinetic energy of each impact and the number of impacts per unit area are explained in Sect. 3.2.

Table 1Erosion propagation as a function of time.

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The data on time to removed coating as a function of the local rotor speed from Table 1 are plotted in Fig. 3

Rain erosion can be analysed as a fatigue process (Slot et al., 2015), where, traditionally, the independent parameter, here velocity, is plotted on the vertical axis and the dependent parameter, here the time before the coating is removed, is plotted on the horizontal axis of a semi-logarithmic diagram.

Figure 3Time to removed coating as a function of the local rotor speed.

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3.2 Generalizing empirical values

Exactly how the droplet sizes and velocities influence the damage is unknown. It obviously depends on several factors including material configuration: properties of coating, putty and laminate, layer thicknesses, interfaces, and various failure modes. We now take the hypothesis that the kinetic energy of each impact characterizes the magnitude of the impact and that the number of impacts per square centimetre corresponds to the number of cycles in a fatigue test.

Figure 4Rain erosion test data plotted as a Wöhler curve: impacts per unit area to failure as a function of the kinetic energy for each impact.

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For an object travelling through a rain field with assumed spherical droplets with uniform diameter and constant falling velocity, the impact frequency can be calculated analytically (Gohardani, 2011; DNVGL, 2018).

The relative volume of water in the rain field, V, is given by

where I_r is rain intensity and v_r is falling velocity.

The number of droplets per volume, N, can be expressed as

where D is droplet diameter.

An object travelling through a rain field is hit by droplets in a stochastic manner across its surface. Assuming that the droplets are distributed evenly in space and their velocity is negligible compared to the speed of the object, the impact frequency or the number of impacts per unit projected area per time, F, can be expressed as

The kinetic energy, E_k, of each impact is

Now, the test data from Table 1 can be presented as the number of impacts per unit area, N_Ei, as a function of the kinetic energy of each impact before the coating is removed; see Fig. 4. Such a representation of data, often referred to as a Wöhler curve or SN curve, is known from fatigue testing of materials (Miner, 1945; Ronold and Echtermeyer, 1996), where the number of load cycles causing failure is shown as a function of the magnitude of each load cycle (for example stress range). Fatigue data are often fitted to a power function as shown in Fig. 4.

E₀= 1 J. A power function fit of the test data in Table 1 to Eq. (10) gives c=18 and m=4.63

We now take the additional hypothesis that the incremental damage is a function of the impact energy only and that Eq. (10) holds for different droplet diameters.

Figure 5Wöhler curves for droplet diameters of 1.5, 2.0 and 2.5 mm.

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Combining Eqs. (9) and (10), the fatigue life can be expressed as a function of the impact velocity and the droplet diameter:

Now, applying Eq. (11), Wöhler curves can be drawn for different droplet diameters as shown in Fig. 5.

Back-calculating from impacts per area to millimetres of accumulated rain, one gets the Wöhler curves for accumulated rain to remove the coating as a function of the rotor velocity for different droplet diameters as shown in Fig. 6.

Given the assumptions and extrapolation, it is obvious that droplet size is important and not just the amount of rain.

Figure 6Expected accumulated rain to remove coating as a function of rotor speed for droplet diameters of 1.5, 2.0 and 2.5 mm.

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3.3 Block loading and cumulative damage laws

Each point on the Wöhler curve in Fig. 3 corresponds to a test run at constant conditions (rain intensity, droplet size, local rotor speed). However, most structures designed for cyclic loads are subject to varying load intensities in service. For instance a wind turbine blade will see a spectrum of wind speeds, gusts and turbulence over its lifetime. Likewise, a leading edge will be impacted with rain of varying droplet sizes and intensities and changing impact velocities. To account for variable-conditions fatigue loading, different rules have been proposed for the accumulation of damage in composites (Brøndsted et al., 1997). The most popular and easy to use, though not always correct, is the linear Palmgren–Miner rule. Accumulated damage (M) is given by

Here, i is the load level number, n_i is the number of cycles at level i, N_i is cycles to failure at level i in a test and j is the number of load levels. The expected fatigue life of a cyclic loaded material is reached when M≥1.

Figure 7Raindrop size distribution through a horizontal plane with the rain fall intensity as a parameter (from Kubilay et al., 2013, based on Best, 1950).

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Given a load time history and Wöhler curves for different loading conditions, it is possible to use Miner's rule to determine the accumulated damage or fatigue life of a structure or material. The Palmgren–Miner rule has been used to predict the rain droplet impact fatigue life of a leading edge (Slot et al., 2015; Amirzadeh et al., 2017b). Here, it will be applied later to predict the fatigue life of a leading edge based on the presented rain erosion test (RET) data and rain statistics.

5.1 Methods

5.1.1 The wind turbine

The investigation was carried out as simulations on the Vestas V52 850 kW pitch-regulated variable-speed wind turbine that was erected at the DTU (Technical University of Denmark) Risø Campus during the summer of 2015 (Table 3). This wind turbine was chosen because data were available. However, parts of the input were modified; for example, this was done for the rotational speed to make it consistent with the higher tip speeds that modern wind turbines are designed with. The size of the wind turbine is somewhat smaller than the majority of wind turbines installed during the last decade, but the relative losses in annual energy production are considered similar. The simulations were carried out assuming steady state, no yaw and no aeroelastic response. These assumptions simplified the conditions significantly but were made to investigate the main response. A further description of the simulations is found in Sect. 5.1.3 to 5.1.5.

Table 3Data for the Vestas V52 wind turbine.

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5.1.2 Control of the wind turbine

The wind turbine control is of the pitch-regulated variable-speed type. The maximum tip speed is assumed to be 90 m s⁻¹ and is thereby greater than the tip speed of an original Vestas V52 wind turbine. Compared to common control strategies, this wind turbine is assumed to have a precipitation sensor. The sensor is capable of measuring and giving input to the turbine controller regarding the present rain intensity and/or the droplet size. As the tip speed is a governing factor in the erosion rate, the erosion control strategy consists of reducing the maximum tip speed when the rain intensity and droplet size exceed threshold values. This is done by reducing the maximum rotational speed of the wind turbine. Because a wind turbine is designed for maximum torque in the drive train, the reduction in the rotational speed implies that the maximum power is reduced. The wind turbine with the original control can produce a maximum mechanical power described by $P_{\mathrm{0}}=Q_{\mathrm{0}}\times {\mathit{\omega }}_{\mathrm{0}}$, where P₀ is the original rated mechanical power, Q₀ is the original rated main shaft torque and ω₀ is the original maximum rotational speed. When reducing the rotational speed from ω₀ to ω₁, the maximum torque limit must not be exceeded. Then the maximum power is also reduced to P₁ ($P_{\mathrm{1}}=Q_{\mathrm{0}}\times {\mathit{\omega }}_{\mathrm{1}}$). An example from the computations is shown in Fig. 12. It is seen that the maximum torque is maintained despite the difference in maximum rotational speed.

Five different erosion control strategies (ECSs) are investigated. An ECS is a set of one or more precipitation intensity threshold values and the corresponding maximum allowed tip speeds, (max tip speed at rain intensity threshold). The expected lifetime for the blade leading edge for each ECS is calculated using the droplet-size-dependent Wöhler curves (Eq. 11), the Palmgren–Miner rule (Eq. 12) and the assumed rain data, which are deduced from Figs. 7 and 8 and shown in the columns 1, 2 and 3 of Tables 4 to 8. Additionally a reference case is included, where it is assumed that no erosion occurs.

• ECS 1: no tip speed reduction; expected lifetime of 1.6 years;

• ECS 2: 70 m s⁻¹ at 20 mm h⁻¹; 80 m s⁻¹ at 10 mm h⁻¹; expected lifetime of 10 years;

• ECS 3: 60 m s⁻¹ at 20 mm h⁻¹; 70 m s⁻¹ at 10 mm h⁻¹; expected lifetime of 24 years;

• ECS 4: 60 m s⁻¹ at 20 mm h⁻¹; 70 m s⁻¹ at 10 mm h⁻¹; 70 m s⁻¹ at 5 mm h⁻¹; expected lifetime of 54 years;

• ECS 5: 55 m s⁻¹ at 20 mm h⁻¹; 65 m s⁻¹ at 10 mm h⁻¹; 70 m s⁻¹ at 5 mm h⁻¹; expected lifetime of 107 years;

• Reference Strategy 6 with no tip speed reduction and an expected lifetime of infinitely many years.

Table 4Calculation of the expected lifetime of the blade leading edge with no reduction in the tip speed. Control Strategy 1.

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The results of the five first control strategies are shown in Tables 4 to 8. For the Reference Strategy 6, it is assumed that no erosion will occur. The rain intensity frequencies are based on precipitation data for maritime temperate climate from Fig. 8. The fixed droplet sizes for each rain intensity are assumed based on Fig. 7. The expected lifetimes are calculated applying Eqs. (11) and (12) and extensive extrapolation as described in Sect. 3 of the RET data shown in Table 1. The control strategies are based on an assumed behaviour that there is a correspondence between the surface roughness height and the aerodynamic performance. Thus, there are elements in this analysis that are not documented but are based on qualified assumptions. However, the numbers are believed to be sufficiently realistic to demonstrate the potential of erosion control.

Table 5Calculation of the expected lifetime of the blade leading edge with a reduction in the tip speed to 70 and 80 m s⁻¹: Control Strategy 2.

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The first row in Table 4 is explained here: the probability of rain intensity of 20 mm h⁻¹ with 2.5 mm droplets is around 0.02 % or 1.8 h yr⁻¹. At this rain intensity the total expected lifetime before failure of the leading edge (LE) at 90 m s⁻¹ is around 3.5 h. The fraction of damage per year relative to LE failure at this level is 51 %. Summing up the fractions of damage per year at the five different rain intensities gives 0.64 % or 64 %. The calculated expected blade LE life at these conditions is therefore $\mathrm{1}/\mathrm{0.64}=\mathrm{1.6}$ year.

Table 6Calculation of the expected lifetime of the blade leading edge with a reduction in the tip speed to 60 and 70 m s⁻¹: Control Strategy 3.

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The results in Table 5 show the effect of reducing the tip speed to 70 and 80 m s⁻¹ at the two heaviest rain intensities. Because of the reduction in tip speed the blade life is extended to 10 years. In Tables 6 to 8 the results are shown where a further increase in lifetime is obtained.

Table 7Calculation of the expected lifetime of the blade leading edge with a reduction in the tip speed to 60, 70 and 70 m s⁻¹: Control Strategy 4.

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Figure 13Airfoil characteristics of the outer part of the blade: blue – clean blade; red – full leading-edge roughness; green – 60 % of full leading-edge roughness. (a) Lift coefficient, c_lift, as a function of drag coefficient, c_drag. (b) Lift coefficient, c_lift, as a function of angle of attack (AOA).

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In order to avoid overloading the drivetrain, care was taken not to exceed the maximum rated shaft torque. Thus, when operating at different maximum tip speeds, the wind turbine had to operate at different rated power:

• 90 m s⁻¹: 850 kW;

• 80 m s⁻¹: 760 kW;

• 70 m s⁻¹: 660 kW;

• 65 m s⁻¹: 615 kW;

• 60 m s⁻¹: 570 kW;

• 55 m s⁻¹: 520 kW.

Even though the wind turbine experiences heavy rain and has to reduce the tip speed, the wind turbine will produce some power; thus, only part of the potential power is lost. On the other hand, by using the erosion-safe mode, the repair and loss in production due to leading-edge erosion will be reduced.

Table 8Calculation of the expected lifetime of the blade leading edge with a reduction in the tip speed to 55, 65 and 70 m s⁻¹: Control Strategy 5.

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5.1.4 Method for derivation of aerodynamic airfoil data

Flow computations were carried out using XFOIL (version 6.1) developed by Drela (1989) because wind tunnel tests were not available for all airfoil sections on the blade. The computations were done for the angle-of-attack range from −20 to 20^∘, and the transition point from laminar to turbulent flow was modelled as free transition by the eⁿ method and forced transition setting $x/c_{\mathrm{c}}=\mathrm{0.1}$ % at the suction side and $x/c_{\mathrm{c}}=\mathrm{10}$ % at the pressure side. In the eⁿ model the amplification factor value n=7 was used because this corresponds to a turbulence intensity of around 0.1 %, which is common for high-quality wind turbine blades and because this value has shown to predict the transition point position well compared to tunnel tests and atmospheric flow.

Finally, the airfoil characteristics (except for the cylinder part) are 3-D-corrected according to Bak et al. (2006).

An example of a set of derived data is shown in Fig. 13, where the airfoil characteristics for a relative thickness of $t/c_{\mathrm{c}}=$ 15 %, corresponding to the outer part of the blade, are seen. To the left, plots of the lift coefficient, c_lift, as a function of drag coefficient, c_drag, are seen. To the right, the lift coefficient, c_lift, as a function of angle of attack is seen. The blue curves show the performance for perfectly clean (non-eroded) airfoils, whereas the red curves show the performance for airfoils with full leading-edge roughness (LER) that corresponds to full blade life for the airfoil as stated in Tables 4 to 8. An example of an airfoil performance at 60 % of full LER is shown with the green curves. The green curves are simple interpolations between the curves for perfectly clean airfoils and airfoils with full LER.

Power curves for different levels of LER are shown in Fig. 14, where the clean blade, the blade with full LER and some of the intermediate roughness levels are reflected. Thus, the intermediate roughness levels represent the corresponding LE lifetime, so, for example, 20 % of full LER corresponds to 20 % of the lifetime. This correspondence is not documented and is therefore a postulate that is, however, based on experience. In the analysis, roughness levels with steps of 10 % difference are used.

Figure 14Simulated power curves for the Vestas V52 for different leading-edge roughness levels.

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Power curves for different maximum allowed tip speeds are shown in Fig. 15. The plot shows how the power curves change when the tip speed is reduced so as not to overload the shaft torque. It shows that the power curves are almost identical from wind speeds of 3 to 9 m s⁻¹. Thus, a reduction in the rated power will influence the production for wind speeds greater than 9 m s⁻¹.

Figure 15Simulated power curves for the Vestas V52 for different maximum tip speeds.

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5.2 Results

Based on the assumed rain climate and the five erosion control strategies (Tables 4 to 8), the aerodynamic modelling and the cost of energy, inspection and repair, the overall loss of income due to leading-edge erosion and its mitigation are calculated for the different erosion control strategies. The energy production is computed by dividing the turbine's energy production over the lifetime into 10 sections with different power curves. The power curve for the clean (non-eroded) rotor is valid for the first 10th of the lifetime. The power curve with 10 % of full leading-edge roughness is valid for the next 10th of the lifetime, the power curve with 20 % of full leading-edge roughness is, in turn, valid for the 10th of the lifetime following on to that, and so on. When the lifetime is reached, the turbine will operate with the full leading-edge roughness until the next whole year has passed, and then it will be repaired. For example, with a lifetime of 1.6 years the rotor will be repaired after 2 years because blade repairs are mainly carried out during the summer, when temperature, humidity and wind are occasionally appropriate. After repair, it is assumed that the blades are completely clean and that they are as wear-resistant as new blades. Three sources for the loss of energy production are taken into account: losses due to degradation in aerodynamic performance, losses due to standstill during inspection and repair, and finally the losses due to the occasional reduction in maximum tip speed. It is assumed that the duration of tip speed reduction is 3 times the duration of the heavy precipitation event because the turbine cannot react instantaneously.

Figure 16Illustration of the framework for calculating the loss of income due to leading-edge erosion.

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Figure 16 illustrates in a simplified manner the framework developed here as a tool for selecting the ECS for minimizing the loss of income due to leading-edge erosion, The site-specific rain and wind statistics, the operational data, and ECS determine the erosive loads on the leading edge. Together with the rain erosion test Wöhler curves, these loads are used as input to the cumulative damage rule (Palmgren and Miner) to compute the expected service life. The aerodynamic degradation, the intervals for inspection and repair, and the loss of production due to occasional tip speed reduction are then used to minimize the loss of income. In field operation, precipitation sensors give input to the control system of the turbines to apply the ECS and adapt the control to present weather conditions.

Figure 17AEP relative to AEP with no erosion.

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In Fig. 17 the AEP including standstill due to inspection and repair is reflected. It is seen that applying Control Strategy 1, where there is no reduction in the maximum tip speed, the loss of AEP is significant with up to 3.5 % compared to the reference case with no erosion and no inspections and repairs. The loss of AEP is clearly dependent on the wind climate. For low wind speed sites with A=7 m s⁻¹, the loss is greater than for sites with higher wind speeds (A=9 m s⁻¹).

Figure 18Loss of income due to erosion, inspection and repair. Energy: 50 EUR MWh⁻¹. Repair: 10 000 EUR rotor⁻¹. Inspection: 500 EUR rotor⁻¹.

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In Figs. 18 to 21, the loss of income due to lost AEP, inspection and repair is seen with the assumption of different costs of energy, inspection and repair. In the plots, the loss of income is significantly simplified and is computed as

where TEP is the total energy production (kWh), EC is the energy cost (EUR MWh⁻¹), NOR is the number of rotor repairs during the life of the turbine, CR is the cost of the repair (EUR rotor⁻¹), NOI is the number of rotor inspections and CI is the cost of each inspection (EUR rotor⁻¹). From the plots it is seen that the loss of income can be significant. The income is very dependent on the energy price and the cost of repair, but a clear trend is that the erosion-safe mode increases the income. Even the very advanced erosion-safe mode – Control Strategy 5, with rather low tip speeds – results in a significant improvement. As an example, Control Strategy 2 can be investigated. Here, the tip speed is reduced from 90 to 70 m s⁻¹ during 5.4 h yr⁻¹ and from 90 to 80 m s⁻¹ during 26.4 h yr⁻¹ due to heavy precipitation. In this case AEP is increased with around 1 % and the income loss is decreased from 15.4 % to 4.5 % in the worst case and from 4.7 % to 2.7 % in the best case, depending on assumptions in cost and wind climate.

Figure 19Loss of income due to erosion, inspection and repair. Energy: 50 EUR MWh⁻¹. Repair: 20 000 EUR rotor⁻¹. Inspection: 1500EUR rotor⁻¹.

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