**3. Results and Discussion**

The evaluation of the different inflow angles will be performed for five different yaw angles as mentioned before. Therefore six 10-min simulations with six different random seeds are carried out for each wind speed and each yaw misalignment. This results in a total number of 138 inflow simulations and 690 load simulations. To arrange the results in a compact and observable manner, load duration distributions are calculated in a first step. Building on this, rain flow counts are carried out which will be mainly discussed. To further condense the results, the damage equivalent load (DEL) is calculated with Equation (5) according to the industry standard DIN 50100 [21] for each yaw misalignment.

$$\text{DEL} = \sum \mathbf{S}\_i^k \cdot \left(\frac{N\_i}{N\_{eq}}\right)^{\frac{1}{k}}, \text{ with } k = 3.3 \tag{5}$$

This parameter is used to retrieve a force or torque for an equivalent load cycle where *Si* the load of the corresponding load cycle is *Ni*. There *Neq* = 175, 000 load cycles are used, representative for a 20 year lifetime and an exponent of *k* = 3.3 as it is recommended for the lifetime calculation of bearings with a line contact in [22]. Afterwards, all combinations of load cycles and amplitudes can be summed up to one amplitude with an equivalent amount of load cycles.

In the following part of this work the loads are defined according to the hub coordinate system from Figure 1 (x-axis is coaxial with the shaft axis of rotation). The rain flow count diagrams, shown in Figures 3–5, contain relevant information about the load behaviour of the turbine and will be discussed. A rain flow count is utilised to determine the number of load cycles from a load time series. The x- and y-axis represent the starting (x-axis) and ending (y-axis) load of the respective load cycle while the logarithmic colour scale gives the total number of load cycles of this type.

**Figure 3.** Rain flow count of the in-plane side force (y-axis) for various nacelle positions and a timespan of 20 years (**top left**: −10◦; **top right**: −5◦; **middle**: 0◦; **bottom left**: 5◦; **bottom right**: 10◦).

**Figure 4.** Rain flow count of the tilt moment (around y-axis) for various nacelle positions and a timespan of 20 years (**top left**: −10◦; **top right**: −5◦; **middle**: 0◦; **bottom left**: 5◦; **bottom right**: 10◦).

**Figure 5.** Rain flow count of the bending moment (around z-axis) for various nacelle positions and a timespan of 20 years (**top left**: −10◦; **top right**: −5◦; **middle**: 0◦; **bottom left**: 5◦; **bottom right**: 10◦).

Rain flow count diagrams of the torsional torque as well as the axial force (x-axis) and the in-plane vertical force (z-axis) were analysed. The torsional torque and the axial force show no qualitative dependency on the yaw misalignment which matches the results from the literature [13]. The comparable small changes can be attributed to the reduced projected rotor area perpendicular to the inflow. In addition, the vertical in-plane force su ffers only marginal changes because it is dominated by the rotor weight which is constant for all simulations.

The in-plane horizontal force is known to have one main direction resulting from the tangential forces acting on the rotor blades. The tangential forces on the blades point towards the rotational direction. However, in the upper half of the rotor they are greater due to the wind shear and the tower shadow leading to a horizontal force on the main bearing pointing to the right when looking downstream. When utilizing the coordinate system shown in Figure 1, the main direction of the side force is the negative y-direction. Figure 3 shows rain flow counts for the in-plane side force. It is observable that the changes are not only dependent on the absolute angle but on the sign as well. Negative angles evoke smaller maximum loads in the main direction. This can be well observed by comparing the load cycle numbers in the interval [−2:−1;−2:−1] (red square). At the same time more load cycles with a changing force direction occur (load cycles in top left or bottom right quadrant, surrounded by grey squares). The tilt moment shows the same qualitative behaviour as the in-plane side force as it can be seen in the top right quadrants of Figure 4. Summarizing, this means that for the side force and bending moment the mean loads become greater with increasing angles but fewer zero-crossings occur. For the bending moment around the z-axis the previously observed effects are inverted (Figure 5). Smaller maximum loads, but more zero-crossings, occur with positive yaw misalignments.

Figure 6 shows the normalized damage equivalent loads for all six degrees of freedom and all yaw misalignments considered in this work. As found before the torsional moment, the axial force and the in-plane vertical force are not a ffected by yaw misalignments to the same extent as the other loads. As expected from the previous observations, the DEL of the in-plane side force (Fy) increases with the yaw angle. If the value at 0◦ is taken as a reference, the DEL decreases by −15.1% and increases by +18.7% in the investigated range. The same is true for the tilt moment (My) with a relative decrease of −1.2% for a −10◦ yaw misalignment and an increase of +5.1% for +10◦ yaw misalignment. Although the in-plane side force experiences the highest relative changes, its absolute values are comparably small to the other loads. The bending moment (Mz) shows a local minimum at 5◦ yaw angle. The DEL decreases slightly to −0.8%. A broader range of yaw misalignments would be necessary to observe if this is a global minimum. For a negative rotation the DEL is increased by +8.1% for −10◦ yaw angle.

To better understand these results one can look into the details of the aerodynamic coe fficients that are used for calculating the aerodynamic forces. Figure 7 shows the lift and drag coe fficients of the NACA64 airfoil taken from [23] which is used in the upper 33% of the blade length. The airfoils used in the rest of the blade di ffer only insignificantly from the NACA64. The e ffect of yaw misalignments on the local inflow at the blade elements can be derived from Figure 8. If the nacelle is rotated towards negative angles, the angle of attack decreases for the blade elements in the upper half of the rotor disk and increases for the blades in the lower half. This leads to a change of lift forces which results in a smaller in-plane side force at the main bearing. Conversely, the angle of attack in the upper half increases with a positive rotation of the nacelle, which leads to a greater side force. Similar observations can also be made for the tilt moment. It is mainly a ffected by the di fference between the drag forces acting on the blades in the upper and in the lower half of the rotor disk. Rotating the nacelle in a negative direction will decrease the angle of attack in the upper half and increases it in the lower half. Thereby the di fference of the drag forces becomes smaller. This finally leads to a smaller tilt moment at the main bearing. Again similar to the tilt moment, the bending moment is determined by the di fference of the drag forces on the left and right rotor disk half. When rotating the nacelle against the inflow the projected wind speed as seen from the blade elements becomes smaller on both sides. This leads to a decrease of angle of attack on the right rotor half (blades that are moving towards the ground) when looking downstream and an increase of angle of attack on the left rotor half (blades that are moving upwards). This e ffect increases the di fference between the drag forces and thus the

bending moment and it occurs for both directions of rotation. Therefore, the minimum that was found for the bending moment is very likely to be the global minimum on the basis of these conclusions.

**Figure 6.** Damage equivalent load for the loads in all six degrees of freedom, various nacelle positions and a timespan of 20 years for an exponent of *k* = 3.3 (**left**: moments; **right**: forces).

**Figure 7.** Coefficients of the NACA64 airfoil [23] shown as the respective dimensionless coefficient (y-axis) vs. angle of attack (x-axis).

**Figure 8.** Schematic of the inflow conditions for a yaw misalignment of 0◦ (black lines) and a positive yaw misalignment (grey lines) for the blade elements of a blade at highest position.

It can be stated that the principle of wake steering o ffers good potential to increase the general park output as the literature shows. However, it is always to be taken into account that a possibly higher damage accumulation rate is a result. For validation of the real load distribution under wake steering in-field data of a steered turbine e.g., from condition monitoring is necessary.
