## 2 Methodology
### Methodology for Calculating Fatigue Damage
#### 2.2.1. Blade Characteristics and Geometry
The solution of the local momentum equations allows obtaining the rotor torque, the extracted mechanical power, and the rotor thrust. The main parameters of the BEM simulations were: (1) air density: \(\rho=1.225\) kg/m\({}^{3}\), (2) air viscosity: \(1.456\times 10^{-4}\frac{\text{kg}}{\text{m}\cdot\text{s}}\), and (3) elements and iterations: 40 and 100, respectively. The output of the BEM calculations was then combined with prescribed functions for the rotor speed \(n(v)\) and the blade pitch angle \(\beta(v)\) as a function of the free-stream wind speed \(v\) to define the steady-state operational characteristics of each wind turbine, assuming a constant efficiency of all other components such as the electric generator. The determination of these functions was conducted outside QBlade and was based on the considerations described at the beginning of Section 2.2. The following results were obtained (see Figure 5 for an illustration and Table 3 for a summary of the turbine characteristics): in all cases, the regulation scheme was designed to obtain optimal operation at partial load, from just after cut-in up to a free-stream wind speed of 8 m/s; in this regime, all turbines operated at an optimal wind speed ratio \(\lambda_{\text{opt}}\) of about six, where \(\lambda=\Omega R/v\), \(\Omega\) being the rotor speed and \(R\) the rotor radius. From 8 m/s-10 m/s the regulation of all three turbines continued to be identical, with the shaft frequency now being set to its nominal value of 130 rpm and the pitch angle remaining at zero degrees in all cases. The effect of reducing the rotor speed was the same for all, a reduction in the power coefficient (the proxy for turbine efficiency), which made up for part of the cubic increase in power density with wind speed, resulting in a practically linear power curve in this section (Figure 5a). For wind speeds higher than 10 m/s, the three regulation schemes now started to differentiate themselves, as described below:  
* Stall-regulated wind turbine: The pitch angle cannot be varied in this design, so it remained at zero degrees for all wind speeds. By operating the wind turbine at its maximum power point up to a wind speed 8 m/s and then setting the rotational speed to a constant value of 130 rpm, the turbine slope of the P(v) curve was limited, and a smooth transition to the nominal regime starting around 11-12 m/s was guaranteed. This design ensured a reliable operation of the turbine, avoiding large excursions in the opposing torque and the possibility of overheating. For higher wind speeds, the rotor speed had to be increased slightly (up to about 145 rpm for \(v=14.5\) m/s) to compensate for a slight drop in the power coefficient, resulting from the stall design. In practice, this is achieved by a slight reduction in generator torque. This small increase in rotor speed (from 130 rpm-145 rpm) is hardly a concern from a structural load perspective. Note that a classical stall-regulated wind turbine, based on the Danish concept, would have a drop in output power, since the rotor speed cannot be changed under that scheme. For higher wind speeds, the rotor speed has to be lowered again, with the lowest value occurring at \(v=20\) m/s, where \(n\) is reduced to about 120 rpm. For still higher wind speeds, practical limits are encountered with heat dissipation at the electric generator, so, typically, an additional braking system is recommended for \(v>\)20 m/s. In the current case, a passive emergency pitching system was designed, the description of which is, however, outside the scope of the present study. * Pitch-regulated wind turbine with constant rpm at nominal power: This scheme is operated in close analogy with a utility-sized wind turbine, where the power curve is roughly determined by two regimes: one with zero pitch angle and linearly increasing rotor speed (the partial load or optimal regime) and the nominal load regime where the rotor speed is held constant (by rotor torque control), and all remaining regulation is done by pitching.