*3.1. Pitch Control*

Blade pitch control as a means of power curtailment is the modern control method, adopted on all utility-scale wind turbines. While two methods of pitch control are available, i.e., pitch-to-stall and pitch-to-feather, only the latter is used because it allows for much lower out-of-plane loads at high wind speeds.

The open-source NREL ROSCO controller [48] was used for this test case and is suggested as a valuable tool for a first analysis. The variable-speed pitch controller was developed based on the work of Mulders et al. [49], and it is able to regulate generator torque and blade pitch. It also allows for yaw control and individual pitch control (IPC).

Below the rated wind speed, a blade pitch is kept constant at fine pitch; in this case, it was set to 0◦. The generator torque is calculated as in Equation (8):

$$
\pi\_{\mathcal{K}} = \mathcal{K}\omega\_{\mathcal{K}}^2 \tag{8}
$$

where *ω<sup>g</sup>* is the generator speed. As also shown in [50], this simple formula is the result of the fact that in order to ensure maximum performance, the turbine must operate at peak Cp for all below-rated wind speeds. In the absence of pitch control, not active in this region, Cp is a function of the tip–speed-ratio alone that must therefore be kept constant. Therefore, as the theoretical available power in proportional to the cube of the wind speed, the generated power must be proportional to the cube of the rotor speed. The power maximizing the generator's torque constant (Equation (9)) can be calculated as [48]:

$$K = \frac{\rho \pi R^5 C\_P}{2\lambda^3 N\_\S} \tag{9}$$

where *Ng* is the generator drive ratio. The relation can be easily derived from the expression of the rotor power coefficient by imposing generator torque, as in Equation (8).

Above rated power, the generator torque is fixed to the design torque *τ<sup>g</sup>* = *Pr*/*ω<sup>r</sup>* and the blade pitch is controlled with a Proportional-Integral (PI) controller (Equation (10)):

$$
\Delta\theta(\mathbf{t}) = K\_p \Delta\omega\_{\mathcal{S}}(\mathbf{t}) + K\_{\bar{i}} \int \Delta\omega\_{\mathcal{S}} \, d\mathbf{t} \tag{10}
$$

where *θ* is the blade pitch. The proportional and integral gains *Kp* and *Ki*, respectively, depend on the blade pitch angle; in particular, as the blade pitch increases, rotor speed variations are more sensitive to small pitch variations. PI control in Equation (10) is derived from a more general Proportional-Integral-Derivative (PID) control strategy with the derivative term (D-term) set to zero. This is common practice in wind turbine pitch controllers. In most cases, the controller is able to adequately control rotor speed without the D-term, making controller tuning easier because there is one less parameter to tune. Furthermore, the D-term is very sensitive to high frequency fluctuations of the rotor speed, and an ill-chosen D-term could therefore introduce instability in a controller. Traditional tuning techniques involve the linearization of the system around an operating point to find controller gains. The linearization procedure must be repeated several times in the operating range. Alternatively, various authors have proposed methods to empirically calculate the gains [51].

In the present testcase, the open-source ROSCO toolbox [52] was used to tune the controller. The gains were analytically calculated and depended on the design natural frequency *ωdes* and damping ratio *ξdes*. In general, increased values of *ωdes* decrease rotor speed response time, while increased values of *ξdes* decrease the amount of rotor speed overshoot. For the present testcase, the values of 0.82 and 1.4 were empirically selected for *ωdes* and *ξdes*, respectively. These values were substantially higher than those found in much larger reference wind turbines, where values of *ωdes* of 0.2–0.3 and *ξdes* of 0.7–0.9 are common [16,42] and are needed to effectively regulate a small wind turbine with low rotor inertia. As noted in [48], there is a limit to how fast rotor speed can be controlled (how high *ωdes* can be) without incurring in erratic blade pitch behavior, and the value of 0.82 adopted for this case was found to be at the upper limit of this range. The controller response was tested with wind-step simulations below at and above rated speeds, as well as in turbulent wind. While response to turbulent wind is discussed in the following section, response to wind increments of 2 m/s are shown in Figure 5. The erratic blade pitch behavior can be clearly seen at 300 and 350 s.

**Figure 5.** Turbine response to step-wind profiles for different values of controller natural frequency and damping ratio. Design natural frequency (*ωdes*) = 0.82 and damping ratio (*ξdes*) = 1.4 for the tuned case, *ωdes* = 1.2 and *ξdes* = 2 for the "fast" case, and *ωdes* = 0.4 and *ξdes* = 0.8 for the "slow" case. Values for the "slow" case were still greater than those typically employed on utility-scale machines.

## *3.2. Stall Control*

In this section, a variable-speed, stall-regulated strategy that eliminates the need for ancillary aerodynamic control systems is evaluated.

The variable-speed operation of wind turbines presents certain advantages over constant speed operation [50,53]. The primary advantage claimed for variable-speed turbines is the increased energy capture during partial load operation. Variable-speed operation allows the turbine to operate at near optimum Cp and to maximize power over a range of wind speeds. Moreover, variable-speed wind turbines use the inertia of the rotating mechanical parts of the system as a flywheel; this helps to smooth power fluctuations and reduces the drive train mechanical stress. Secondary benefits are acoustic signature and power quality [51]. The control logic is described in detail in [54], but the main details are explained herein as regulation strategy that significantly influences turbine regulation and, consequently, aerodynamic choices.

Typical variable-speed wind turbines have different regions of operation, as shown in Figure 5, where the generator torque as a function of the generator speed is shown. The turbine startup occurs in region 1, where the generator torque is zero. Once the generator speed has reached cut-in speed and power is produced normally, the turbine is operating in region 2. In this region, the generator torque control is used to vary the speed of the turbine to maintain the constant TSR corresponding to optimum Cp, thus maximizing the energy capture. In region 2, the torque curve is calculated as in Equation (7) and intersects the rated torque at a rotor speed that is significantly higher than the rated speed. It would of course be beneficial to operate the turbine on region 2 at an optimum Cp curve up to where

it intersects the rated torque, but the operation of the turbine at these high rotor speeds would result in a high blade tip speed and unacceptable noise emissions [31]. Therefore, a transition region is included between regions 2 and 3 (region 2<sup>1</sup> <sup>2</sup> ). Region 2<sup>1</sup> <sup>2</sup> depends linearly on rotor speed, starting at a rotor speed lower than the rated speed *ω*<sup>1</sup> and reaching the rated torque at, or slightly below, the rated speed *ω*2. The generator torque for this region can be expressed as Equation (11):

$$\pi\_{\mathcal{S}}(\omega) = \tau\_1 + \frac{\tau\_{rated} - \tau\_1}{\omega\_2 - \omega\_1}(\omega - \omega\_1) \tag{11}$$

where *ω* is rotor speed, *τ*<sup>1</sup> is the generator torque at the rotor speed in which this region starts (*ω*1), *τrated* is rated torque, and *ω*<sup>2</sup> is the rotor speed at which we reach rated torque. Above the rated speed, the generator torque is set equal to the rated torque *τrated*.

In region 3, generator torque is simply held constant at rated torque (see Figure 6).

**Figure 6.** Variable-speed turbine operating regions.

Adequately tuning the slope and position of region 2<sup>1</sup> <sup>2</sup> ensures effective turbine regulation. Through region 2<sup>1</sup> <sup>2</sup> , the turbine is controlled to limit its rotational speed and, consequently, output power. In fact, limiting rotor speed decreases the TSR and forces the rotor into an aerodynamically stalled condition. This is usually called the "soft-stall" approach because it allows for the introduction of rather benign stall characteristics for the purposes of controlling maximum power.

#### *3.3. Control Input Parameters*

In Table 2, the main parameters used to set the torque-control strategy of the two turbines are shown. These are a result of a (in most cases) necessary sensitivity analysis. This paragraph hopefully helps the interested reader understand the influence of some of the main control parameters and how they can be tuned to reach the desired turbine performance. A baseline for these control parameters can be determined using the methods detailed in Sections 3.1 and 3.2 Several common techniques to ensure that power is correctly regulated using both control schemes were adopted for this study. The rated rotational speed of the stall-regulated turbine (i.e., the beginning of region 3 in Figure 6) was limited to 60 rpm because the turbine was designed to operate at a nominal TSR at 8.5 m/s wind speed and to enter off-design conditions as wind speed increases to force the blade to stall and the power to be regulated. Therefore, to effectively regulate power with a stall control scheme, the turbine needs to be forced to enter off-design conditions before the rated wind speed. A nominal rotor speed for the pitch-regulated turbine was chosen so that the design TSR could be maintained up to 10 m/s wind speed. The high value of rated generator torque for the stall-regulated turbine was set to avoid rotor overspeed in high

wind speed turbulent scenarios. In practice, this means that, even at a rated power, the turbine would operate in region 2<sup>1</sup> <sup>2</sup> . Operating in this region would ensure that the rotor does not speed-up as a response to steep wind speed increases. On the contrary, if the rotor is allowed to speed-up, the TSR and, consequently, the power increase, causing the rotor to quickly become uncontrolled and reach its terminal velocity. This also highlights the importance of considering dynamic inflow conditions early on in the design stage. Finally, the value of *K* (Equation (8)), was different in the two cases because the peak CP design TSR were different for the two turbines.


## **4. Simulation Set-Up**

Once preliminary steady-state performance curves are obtained, it is important to account for more realistic environmental cases early in the design process. The reasons are twofold: first, it is important to assess turbine behavior in dynamic conditions, and secondly, the turbine will have to be certified in the later stages of the design process. For instance, as mentioned previously, it is crucial to verify that adequate turbine control is achieved in dynamic conditions. Moreover, the design loads calculated by simulating the turbine in dynamic environmental conditions can be used as a base for preliminary structural design. Here, the turbine was simulated in a normal power production situation, corresponding to the IEC Design Load Cases (DLCs) 1.2 [12]. The chosen turbine class was class IIA. This represents a class of turbines designed for medium wind speed (W.S.) and high turbulence sites.

One hundred fourteen 10-min simulations were performed for each turbine, reproducing operating conditions specified by the IEC 61400-2 power production DLC-group, including wind shear, yaw misalignment, and turbulence, as detailed in Table 3.

**Parameter Value** Type of Evaluation Fatigue/Ultimate Simulation Length 600 s Number of Simulations per W.S. and Yaw Angle 3 Wind Speeds 2–20 m/s increments of 1 m/s Yaw Angles 0/+8/−8 deg Vertical Inflow Angle 8 deg Total Number of Simulations 228

**Table 3.** IEC 61400-2 DLC 1.2 main set-up parameters.

These simulations had wind speeds between two and twenty meters per second in intervals of one meter per second following the standard and industry-accepted guidelines [12,55]. These design cases were representative of power production under normal wind conditions and would therefore be the most common within the turbine lifespan. Though DLCs are designed with structural certification in mind, they were used in this study to verify the productivity of the turbine, as they allowed us to simulate a normal power production scenario. Each simulation used a different turbulent speed (i.e., different turbulent wind field) in order to more realistically reproduce the conditions the turbine will encounter during operation and to avoid biases that might be introduced by a specific wind pattern.

Total Simulated Time 38 h

As an additional verification, the convergence of power and annual energy production (AEP) was evaluated, as shown in Figure 7 and Table 4. This was important to evaluate to make sure that the predicted power curves could be considered independent from specific turbulence characteristics. The convergence of power was evaluated in terms of mean power per wind speed calculated with respect to the case using six turbulence speeds per wind speed (adding up to a total of one-hundred-fourteen simulations), as shown in Figure 7a. The analogous convergence of power standard deviation is shown in Figure 7b. Mean power was sufficiently well-predicted by using four turbulent speeds, with variations in mean power below 3% for all wind speeds. The standard deviations required more simulations to properly converge. As shown in Figure 7b, using five turbulent speeds ensured variations in Standard Deviation (STD) below 5% for all wind speeds.

**Figure 7.** Relative error of power (**a**) and power standard deviation (**b**) per wind speed bin with respect to six speeds per wind speed value, using 5 (5 s), 4 (4 s), and 2 (2 s) turbulent speeds per wind speed.


**Table 4.** Statistical convergence of annual energy production (AEP).

AEP already showed strong convergence at two speeds per wind speed and is largely insensitive to increasing the number of speeds. This was in-line with the finding of Bortolotti et al. [56], who noted convergence on predicted fatigue loads and AEP using a small number of turbulent speeds. In conclusion, the minimum requirements of IEC 61400-2 in terms of turbulent speeds were able to guarantee the convergence of power and AEP in the present testcase.

#### **5. Results**

In this section, the results of the different design choices discussed so far are critically compared in order to let the reader evaluate their impact on the final performance.

#### *5.1. Steady-State Performance*

In order to evaluate general rotor performance, a steady-state performance comparison was carried out. Generator power as a function of windspeed is shown in Figure 8a. Both the stall and the pitch-regulated turbines were able to reach the desired output power of 50 kW. However, the pitch-regulated turbine reached rated power at 10 m/s wind speed, while rated power was not reached until 12 m/s in the stall-regulated turbine. For low wind speeds of up to 8 m/s, the increased power output of the pitch-regulated turbine depended on the increased aerodynamic efficiency of this blade, caused by the fact that blade twist and pitch angle did not need to be compromised for effective stall regulation. From 9 m/s and above, the control strategy also had a direct effect, as the rotor speed was limited for the stall-regulated turbine in order to drive the blades to stall, as shown in Figure 8b and discussed in the previous section.

**Figure 8.** (**a**) Generator power and aerodynamic power coefficient; (**b**) rotational speed for pitch- and stall-regulated rotors in steady-state condition.

The observations made from a perusal of Figure 8a are confirmed in Figure 9, where the power coefficient is shown as a function of the tip–speed ratio: the compromises adopted for the stall-regulated turbine resulted in a generally lower power coefficient. Furthermore, the shape of the curve was very different, with the stall-regulated turbine presenting a pronounced peak in Cp, unlike the pitch-regulated turbine that could operate near peak-Cp for a broad range of TSRs. It is also interesting to note that both turbines effectively operated in the area of the Cp–TSR curve that is on the left of the Cp peak. This was crucial, especially for a stall-regulated turbine, where tuning the shape of a Cp–TSR curve and forcing the rotor to operate in off-design conditions are the sole ways turbine control can be properly ensured.

**Figure 9.** Aerodynamic power coefficient as a function of tip–speed ratio (TSR) in steady-state conditions.
