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The methodology described in Moghadam et al. (2023) with yaw misalignment added is shown in Figure 1. The powertrain system damage database as a function of wind speed, turbulence intensity, power demand, and yaw misalignment angle is created by turbine-level aero-hydro-servo-elastic simulations in NREL’s OpenFAST—the servo part uses a custom version of the DTU wind energy controller (Meng et al., 2020; Hansen and Henriksen, 2013) featuring active power derating; the hydro part is not included at this stage and left as further work. The steps are

•generating input turbulent wind field by NREL’s TurbSim with desired mean wind speed and turbulence intensity,

The last step is based on a powertrain degradation model, further detailed in the following:

•The physics-based quasi-static model proposed by Moghadam et al. (2023) is employed.

The database created by turbine-level simulations establishes the damage index (an index that represents the long-term fatigue damage of all the drivetrain components) as a function of environmental (average wind speed and turbulence intensity) and operational (generated power and yaw misalignment angle) variables. The wake effect for individual turbines is considered by the reduced mean wind speed and the increased wake-induced turbulence intensity. When used in farm control, it is desirable to characterize degradation via a scalar quantity, here called the damage index. Depending on the control objective, the damage index may be specific to a single powertrain component or aggregated over all components. In the latter case, a weighted damage index may be used based on the vulnerability of each component. In this article, a simple weighting is suggested based on probabilistic long-term fatigue damage analysis: D I weighted overall u o c , I o c , γ o c , D F o c = 1 M ∑ i = 1 M α i D i t , o c u o c , I o c , γ o c , D F o c , s . t . ∑ i = 1 M α i = 1 , (1a) α i = D ̄ i max D i ̄ , D ̄ i = ∫ ∫ ∫ ∫ D i u , I , γ , D F . f u , I , γ , D F × d u d I d γ d D F f o r i = 1 , … , M , (1b) where u ^oc , I ^oc , and γ ^oc are, respectively, the mean value of wind speed, the turbulence intensity, and the yaw misalignment angle during the operating condition ^oc during the time interval t. D i t , o c is the absolute value of accumulated damage for the i-th subcomponent during this time interval. DF is the derating factor equal to the ratio of generated power over available power—saturated to rated power at higher wind speeds. M is the number of powertrain subcomponents considered in the degradation analysis. α _i is a weight factor representing the normalized damage for the i-th subcomponent. f(u, I, γ, DF) is the joint probability distribution of the operating conditions, with the integral equal to 1.

The wind farm controller coordinates the operation of the turbine’s active power—through pitch angle and generator torque—and yaw controllers within the farm to meet the power demand while steering degradation. The proposed farm power tracking controller is an adapted version from Moghadam et al. (2023); Merz et al. (2021) now augmented with a yaw angle offset component. The integration of the powertrain degradation database to the farm controller is shown in Figure 2, featuring the yaw angle offset as an additional control input for the damage mitigation function.

The wind farm control layout is presented in Figure 3. A distributed architecture based on individual PI regulators and time-adaptive gains is adopted. The wind farm controller is responsible for dispatching the turbines’ power, P _c,i (t), such that the reference wind farm power, at the point of common coupling (PCC), P _pcc (t), is tracked. Each individual turbine controller (TC) consists of a feedforward term and a feedback loop, as shown in Figure 3. The former defines the bulk part of the PCC power reference requested from the i-th turbine through the gain λ _i1(t). The latter uses the farm output power error at the PCC, ϵ _pcc (t), by first dispatching between turbine regulators through the gains λ _i2(t) and then using it to compensate for the tracking errors through the proportional and integral feedback gains, K _P and K _I , respectively. It is worth mentioning that the power-sharing quality is directly affected by the locally available power, P _a,i (t), and smart anti-windup mechanisms are necessary to prevent sudden power injections due to changing inflow conditions from weather, turbulence, and wakes (u _i (t)). The participation of each turbine in tracking the farm power output is adjusted based on the turbine’s accumulated damage and through time-dependent gains λ _ij (t) (see Section 2), which are also communicated among the turbines and encapsulate information from the database of the weighted damage index (see Section 2.1). The additional decision feature that is included, when compared to Moghadam et al. (2023), allows for setting the yaw angle offset of the damaged turbine(s), γ _i (t), to the value corresponding to the lowest possible damage accumulation, under specified environmental and operational conditions (power dispatch). In this sense, the updated damage database, which includes the extra dimension of the yaw angle offset, is used instead, further enhancing the damage mitigation capabilities for the damaged turbines. The adaptive gains λ _ij (t) (j ∈ {1, 2}) are defined as λ i j = g i j D I u , I , γ , D F ∑ turbines g i j D I u , I , γ , D F , s . t . ∑ i = 1 N λ i j = 1 . (2) where N is the number of turbines of the farm, and g i j is a tunable mapping from DI _i = DI (u, I, γ, DF) to λ _ij , for turbine i, as in Moghadam et al. (2023). In case of no fatigue damage mitigation in the farm controller, the gains λ _i1 and λ _i2 will take equal values for all the turbines of the farm.

The yaw angle controller is an open-loop add-on. Optimal yaw misalignment values are first found, then translated to yaw angle offset commands considering wind direction ¹ . Losses induced by yaw misalignment are compensated through the PI feedback of the power dispatch control loop. Note that yaw angle control is not meant to increase power production on downstream turbines through wake steering in this context. Wake deflection is not the focus and is included in the modeling framework only as a side effect. The wake effects are modeled through the simulation framework presented in Moghadam et al. (2023), employing a modified, quasi-steady version of NREL’s FLORIS farm flow model, which is capable of capturing the effect of the derating commands to the turbines.

The conditions of the case study simulated to generate the database are listed as follows:

•The turbine model is DTU 10 MW reference wind turbine (Bak et al., 2013) fixed-bottom installed on a reference monopile as shown in Figure 4. The 10 MW powertrain system design, configuration, and selection of the subcomponents for the fatigue damage analysis are based on Moghadam and Nejad (2020), Wang et al. (2020), and Moghadam et al. (2023);

The performance of the DTU controller featuring derating functionality is shown in Figure 5, which shows how the rotor speed and pitch regulators cooperate at different wind speeds and yaw misalignment angles to achieve different values of power demand. In this figure, ω _rot is the average rotor speed and β is the average blade pitch angle. As seen, the yaw misalignment angle has a negligible influence on the controller’s operating state.

Figure 6 shows the selected time series of dynamic equivalent loads of the bearings (the gearbox-side main bearing, the gearbox high-speed shaft drive end bearing, and the generator non–drive end bearing) and contact and root bending stresses of the gears (the planet gears of the first planetary gear stage) as a function of the average wind speed, turbulence, yaw misalignment angle, and power demand variations. It can be seen that yaw misalignment has the largest influence on the main bearing load in terms of both mean value and oscillation amplitude. Further investigation showed that the main shaft bending moment around the z-axis (yaw moment) is the main contributor.

The results of the long-term fatigue damage analysis of the powertrain system are shown in Figure 7. This sets the base for selecting the weight factors α _i in Section 2.1 to calculate the weighted average damage index. In practice, the main bearings dominate damage in most operating conditions. At very high wind speeds, the planet gears take over. The contributions from the other components to the damage index are minor.

A global overview of the degradation database mapping the damage index to operating conditions is shown in Figure 8, which maps the weighted damage index of the powertrain under consideration to mean wind speed, turbulence intensity, derating factor, and yaw misalignment angle. The damage ratio in this figure is defined as the ratio of the weighted damage index at a non–zero yaw misalignment over its zero yaw misalignment value. It shows the conditions when yawing is beneficial or detrimental, strongly depending on the yawing direction. The rotational direction of the rotor is the same for positive and negative yaw misalignment angles. As a result, the angle of attack for each blade cross-section due to the combined effect of the inflow velocity and the rotational velocity will be different for positive and negative yaw angles. This gives different induced velocities and the resulting aerodynamic forces on the blade cross-sections. As a result, the drivetrain fatigue damages are also different for positive and negative yaw angles, as shown in Figure 9.

Figure 9 shows the normal operation (not downregulated) case in more detail, where it is clear that yawing in any direction would mostly increase damage. However, by crossing observations with Figure 8, it is seen that in highly downregulated conditions, yawing may prove beneficial. This may in turn be exploited by the farm controller to reduce damage whenever the demand is met, as demonstrated in Section 3.5. Multiple damage indices for a given wind speed and yaw angle in Figure 9 are associated with the different values of power demand. As the farm is operating in the power curtailment mode of operation, the power set point and therefore the generated power of each turbine can be different from the available power.

To gain an understanding of how each parameter participates in the damage, a sensitivity analysis is performed, with results as presented in Figure 10. Figure 10B shows that damage increases as turbulence intensity increases, as expected. Figure 10A shows that the damage index increases as the mean wind speed increases up to the rated mean wind speed 11.4 m/s, due to the increased mean thrust force and its effect on main bearing damage. After the rated wind speed is reached, there is a mitigating trend in the overall damage attributed to the reduction of thrust force (both mean value and fluctuations) through pitch control. For high wind speeds and/or turbulence intensities, gears dominate damage—the damage of gears is mainly driven by mean torque and torque oscillations, not thrust—showing a different damage pattern increasing linearly with demanded power. More interesting is the relationship between damage and yaw misalignment angle shown in Figure 10C, depicting in more detail the preliminary observations on a possible reduction of damage through yawing in downregulated conditions. It shows that the combined effect of downregulating and yawing is nontrivial and cannot be generalized in a simple control law; case-by-case lookup in the database should be used in the farm controller.

The same setup as used in Moghadam et al. (2023) has been chosen for comparability:

•Turbine layout: 32 turbines, staggered, 5-diameter spacing (Andersen et al., 2018).

It introduces an affine re-scaling of the damage index DI _i , which ensures that turbines with low DI _i values will contribute more to the power tracking task (g _i takes values closer to 1), while turbines with higher DI _i values will contribute less (g _i takes values closer to 0.5) and with a minimum contribution that is lower bound by half of the turbine’s naive contribution (without any damage information). This tuning is not cost-optimized and only devised for qualitative demonstration purposes. The sup and inf operators represent the supremum and infimum for the set DI _i .

The goal is to show how the yaw angle can be used by the controller as an additional control input to improve the fatigue mitigation feature. In total, 18 simulations are run corresponding to two control objective scenarios, three control methods, and three curtailment profiles. The two control objective scenarios are

•Scenario 1: All turbines are equally damaged. The objective is to reduce the overall damage and spread it evenly over the turbines. This case has a high cost-saving potential but would require a global cost–benefit analysis that includes all turbine components.

The three curtailment profiles correspond to constant power references equal to 0.2, 0.5, and 0.8 per unit (pu) of the installed farm power (320 MW).

Power dispatch commands follow the control scheme in Figure 3 and are updated every minute. Yaw angle commands are selected to yield minimum damage on the turbine(s) of interest (all in scenario 1 and only turbine 1 in scenario 2) and updated every 10 min, following the quasi-steady update rate of the waked flow field, by the simulation framework described in Section 2.2.

Results for scenario 1 are shown in Figure 11 with one subfigure per curtailment profile. It can be seen that power dispatch only can even out damage between upstream and downstream turbines but does not significantly reduce overall damage. Adding the yaw degree of freedom, on the other hand, aims at decreasing damage and is particularly efficient in heavily curtailed conditions, which is consistent with the database simulation results.

Results for scenario 2 are shown in Figures 12, 13, where the power dispatch (moving average) and the corresponding time evolution of the damage accumulation for the damaged turbine can be observed. Initially, P _pcc is naively distributed among the turbines. After the first control update based on the database at 600 s (because it requires information from the last 10 min), the farm controller has realized that it can track P _pcc with less contribution from the damaged turbine, and turbine 1 is re-dispatched to lower power levels. Both versions of the farm controller calculate the same powerdispatch plan for turbine 1, as observed from the moving average of the power dispatch in Figure 12A. However, since the proposed, enhanced version can also set the yaw angle of turbine 1 to a proper value, at the end of the simulation period, turbine 1 has accumulated less damage (solid purple line in Figure 12) than the case without a yaw angle offset (dashed green line in Figure 12). On the contrary, the naive approach that does not consider any damage information (solid red lines in Figure 12) instructs a different power dispatch plant for turbine 1, which would result in even higher damage accumulation by the end of the simulation. The simulated case’s corresponding optimal yaw angle offset sequence (adopted only by the controller version with yaw information) is presented in Figure 12B.

Similar results were also observed for the case of farm reference power level 0.2 pu, and in general, for relatively lower values of farm reference power (equivalently higher levels of curtailment). For relatively higher values of farm reference power, however, the effectiveness of the proposed control scheme is lower. To illustrate this, we present the wind farm simulation results for the case of farm reference power 0.8 pu, shown in Figure 13, and following the same format as for the case of 0.5 pu (Figure 12). As can be observed, in this case, the power dispatch for the damaged turbine (turbine 1) is the same for both versions of the controller (with and without yaw), and for most of the simulation period, they are even the same with the naive approach. This is primarily because the requested farm power is relatively high when compared to the available, for this case, considering the pessimistic wind direction (north wind) that is associated with the largest impact from the wake effect. It is only at the last part of the simulation period that the farm controller can re-dispatch turbine 1 since the farm output can be tracked with less turbine 1 contribution. This is shown in Figure 13A, where the power decreases to lower values after 3,000 s. However, the decrease for the controller versions that take into account the damage is much more significant when compared to the naive approach, and as a consequence of this, the accumulated damage for the turbine is slightly lower than the naive approach by the end of the simulation. Again, the corresponding optimal yaw angle offset sequence is presented in Figure 13B. In this case, we observe that no offset yaw angle is decided, meaning that both versions of the controller (with and without yaw) are identical, which also justifies the identical damage accumulation for both versions, as depicted in Figure 13A. This observation is in line with the updated results of the damage database when including the yaw misalignment capability, where for higher turbine-generated power (lower derating levels), the effect of the yaw on turbine damage (based on the weighted damage index) becomes less significant.