First, we define the non-dimensional mechanical power coefficient as 1CP=P12ρV3πR2, where P is the rotor mechanical power (W), V is the rotor-averaged wind speed (m s⁻¹), R is the blade radius (m), and ρ is the air density (kg m⁻³). Note that we refer to the wind speed here as the spatial average of the longitudinal component of the atmospheric wind field at the rotor plane when unaffected by the presence of the wind turbine see definition in. In wind turbine design and analysis, non-dimensional parameters like CP are essential in evaluating wind turbine performance, as they provide a universal metric for comparing different turbines operating under various conditions.

We also introduce the torque coefficient as 2CQ=Q12ρV2πR3, where Q is the torque exerted on the rotor by the wind. The tip-speed ratio (TSR) is defined as the ratio between the tangential speed at the tip of the blade and the wind speed: 3λ=ωRV, calculated from the rotational speed of the rotor ω. From Eqs. () and (), we get the proportionality between the power and torque coefficients as follows: 4CP=λCQ. Finally, we define the thrust coefficient to represent the force perpendicular to the rotor plane: 5CT=T12ρV2πR2, where T is commonly known as thrust.

In conventional controllers, the WSE and most gain-scheduling and peak-shaving routines are dependent and often calibrated using performance information, where each entry is a function of λ and the collective pitch angle β, i.e. functions in the form “f(λ,β)”.

Moreover, optimal tip-speed ratio tracking control schemes are based on a constant λ set point in the partial-load region, which, to date, has been deemed to lead to optimal power extraction. The tip-speed ratio has effectively been used to define the aerodynamic state of reasonably rigid wind turbines. In fact, the aerodynamic performance is determined by the geometry of blade sections and angle of attack distribution, assuming Reynolds and Mach number variations are negligible (i.e. ignoring viscosity and compressibility effects on section aerodynamics). For a rigorous explanation, the reader is referred to the results of BEMT .

However, when loads deform the blade shape and, consequently, the geometry of the sections, aerodynamic performance is altered. This variation is not captured by the tip-speed ratio alone because the same value of the tip-speed ratio may correspond to different combinations of V and ω. Due to blade flexibility, the traditional use of tip-speed ratio to parameterise the performance of wind turbines becomes inadequate. Consequently, it is necessary to parameterise the aerodynamic performance coefficients using three arguments, i.e. decomposing λ into its components ω and V.

Two discrepancies with the actual aeroelastic behaviour of wind turbines arise when using aerodynamic performance coefficients parameterised on the tip-speed ratio for the design of conventional controllers:

The tools used for performance calculations may not account for structural flexibility primarily in the form of blade deformations, neglecting the effects of such deformations on aerodynamic behaviour, as already noted in .

To illustrate the discrepancies mentioned above, we compare CP and CT coefficients of the IEA 15 MW RWT using a rigid- and flexible-blade model. To balance computational effort and accuracy, the spacing in our grid is variable: it is refined in regions of particular interest – such as near the rated wind speed, where loads have a pronounced effect – and is coarser in less critical regions. We then use HAWCStab2 to obtain the steady-state coefficients over a three-dimensional grid with 27 000 operating points spanning various combinations of rotational speeds, wind speeds, and pitch angles. Specifically, the grid consists of the following:

20 rotor speeds ω (from 2 to 4min-1 in 1min-1 steps, from 5 to 9.5min-1 in 0.5min-1 increments, and from 10 to 16min-1 in 1min-1 steps)

No wind shear is considered here – i.e. we assume a spatially uniform inflow. This uniform inflow assumption arises from a limitation of HAWCStab2. In principle, it would be possible to incorporate wind shear by generating performance tables with a time-domain-based simulation tool such as HAWC2. However, creating such a large number of required operating points would be computationally infeasible.

The rigid model assumes infinitely stiff structures, while the flexible model considers fully flexible structures, where all linear and rotational deformation degrees of freedom are active. In the flexible model, each blade is divided into 20 sub-bodies using Timoshenko beam elements. These sub-bodies consist of two nodes with 6 degrees of freedom and coupled structural cross-sectional stiffness matrices , allowing for large deformations and modelling of bend–twist coupling . Table  provides an overview of the models and settings used in HAWCStab2.

Figures  and  show the power coefficient (a) and thrust coefficient (b) values, obtained by fixing the pitch angle (β=0°) and varying the wind speed (from 2 to 27 ms-1) and rotational speed (from 2 to 16 min-1) for a total of 600 combinations. The results are interpolated linearly on a finer grid for smoother variations in the analysed region. As expected from the discussion on tip-speed ratio, the rigid model (Fig. ) exhibits CP and CT values that remain constant on constant tip-speed ratio lines. The results shown in Fig.  confirm that performance depends solely on the tip-speed ratio when using a purely aerodynamic solver (i.e. without the effects of deformations of blades) and neglecting Reynolds number variations along the blades.

Figure  presents the power coefficient (a) and thrust coefficient (b) values obtained with the flexible model. In contrast to the observations from the rigid model, values exhibit nonlinear, decreasing trends along iso-λ lines. These differences stem from the coupled aerodynamic and structural response occurring in flexible blades: structural deformations introduce changes in the local angle of attack and in the relative wind velocity at the blade sections, causing deviations from the rigid-model predictions. The varying trends along the iso-λ lines in the flexible model highlight how flexibility-induced deformations impact both aerodynamic efficiency and loading. These effects become particularly pronounced at higher wind speeds and rotor speeds, where structural deformation is more significant. The trend is noticeable along all the iso-λ lines, including the iso-λ=9 line, which represents the partial-load operational tip-speed ratio of the IEA 15 MW RWT.

In Fig. , the power coefficient (a) and thrust coefficient (b) are plotted against the wind speed along iso-λ=9 lines for both the rigid and the flexible models to showcase the relative discrepancies at the same tip-speed ratio. Figure  shows that both CP|λ=9 and CT|λ=9 are constant when flexibility is neglected (blue line), whereas the fully flexible model (orange line) displays a substantial drop in power performance starting from V=7.5ms-1, with a more than 10 % reduction in CP compared to the rigid model's corresponding value at V=10ms-1.

To illustrate how the structure of the blades changes, causing the reported reduction in power performance, two quantities representing structural deformations are shown: tip torsion in Fig. c, indicating the structural twist of the blade tip section (positive when the structural twist decreases the angle of attack), and the out-of-plane (OoP) tip displacement in Fig.d, which represents the distance of the blade tip section mid-chord point from the rotor plane, positive in the wind direction. From V=7.5ms-1 onwards, both these metrics exhibit significant differences compared to the values calculated with the rigid model.

The corresponding loads exerted on the rotor blades are shown in Fig. e and f in the form of the flapwise bending moment at the root of the blades and the thrust force, respectively. When wind speed and rotational speed combinations produce rotor thrusts that exceed the peak value Tmax=2750kN (as indicated in ), the loads and deformations display highly nonlinear trends and influence one another. Under such conditions, large torsional deflections occur and, in turn, degrade performance while reducing loads. However, these operating points, corresponding to rotational speeds above 9min-1 and wind speeds above 13ms-1, lie well outside the normal steady-state operating conditions of the IEA 15 MW RWT. Consequently, these extreme deformations are not expected during typical turbine operation and are therefore considered unrealistic.

The findings in this section on the coupling between blade loading, structural flexibility, and the aerodynamic performance of the rotor suggest that constant tip-speed ratio tracking, a common control strategy for smaller and more rigid turbines, may no longer be sufficient to control large, flexible wind turbines optimally. These results indicate that there is room to optimise the power coefficient by accounting for blade flexibility early in the process of control design. They also show that flexible-turbine calculations provide the opportunity for the incorporation of structural constraints once the set points are defined in a three-dimensional space of rotational speed (ω), wind speed (V), and pitch angle (β). Based on the results and conclusions drawn in this section, we develop a new control scheme aimed at maximising energy capture while limiting excessive structural deformations.

The first model of the new scheme is a set point optimisation framework named COFLEXOpt, providing optimal (constrained) control set points and control inputs used to create steady-state mappings for the feedforward and feedback modules of the control scheme. The next section elaborates on the set point optimiser.

Recently,  demonstrated a method for obtaining optimised operating points for wind turbines by solving an optimisation problem. In their study, steady-state set points were optimised by varying constraints, objective functions, and decision variables across different operating regions. As a consequence, the rated wind speed and operating regions were predefined. To simplify the optimisation setup and problem and possibly result in even more optimal solutions, our proposed framework solves the same optimisation problem to determine the set points, using a convex objective function over the entire operating range of a wind turbine – so in both partial- and full-load conditions. Notably, the decision variables in the optimisation problem remain unchanged over the entire range of operations. Hence, the subdivision of operating conditions into regions becomes irrelevant to the controller design. In addition, this optimiser allows us to impose constraints on various structural and operational quantities, such as thrust, blade deflection, tip-speed ratio, power output, and rotational speed, while still ensuring that an optimal solution is returned for each set of imposed constraints.

The general nonlinear optimisation problem can be written as follows: 6min(ω,β)fobjω,V‾,β∀V‾∈Vcut-in,Vcut-out,s.t.ωmin≤ω≤ωmax&βmin≤β≤βmax,Ciqω,V‾,β≤0,Ceqω,V‾,β=0, where V‾ represents a wind speed within the operating range Vcut-in,Vcut-out; fobj is a suitable objective function; ωmin, ωmax, βmin, and βmax are box constraints on the decision variables; and Ciq and Ceq are inequality and equality vector constraints, respectively. The decision variables are the rotational speed ω and the collective pitch angle β, as the performance of the wind turbine, including flexibility effects, can be expressed as functions of these variables and the wind speed. The versatility of this framework lies in the wide range of possible definitions for fobj, Ceq, and Ciq. In particular, Ceq and Ciq can include any metrics representable in the ω,V,β space. Since the tip-speed ratio is decomposed into two separate variables, one can incorporate nonlinear constraints dependent on actual operating conditions. Examples include structural deflections, peak thrust (as in peak-shaving strategies), load-alleviation targets (e.g. bounding the root flapwise bending moment), or blade-span-dependent quantities (e.g. limiting angle of attack or relative velocities). Regarding the objective function fobj, its formulation must yield unique and optimal solutions (ω*,β*) across the entire operating range. The primary objective is to maximise power capture (i.e. the power coefficient). The power output will also naturally be subject to an inequality constraint, ensuring the rated power is not exceeded. However, once the rated power limit is reached in the full-load region (i.e. V‾>Vrated), infinitely many (ω*,V‾,β*) combinations yield the power coefficient to produce the rated power, and the maximisation of the power coefficient is not sufficient to produce unique solutions.

To address this, we introduce a secondary term in the objective function, resolving the non-uniqueness of the solution. This technique, also suggested in , selects one point along the power coefficient isolines based on the minimisation of a secondary term in the objective function, resolving the non-uniqueness of the solution. In particular, this secondary term can have physical meaning: for example, if one selects the thrust coefficient, an increase in rotor loading is penalised in the optimal solution. Alternatively, one can penalise the torque coefficient, which ensures that the optimiser seeks the solution that yields the lowest rotor torque within the feasible region – helping to mitigate drivetrain loading. If the weight on this secondary term is kept sufficiently small, it effectively acts as a regularisation term while still retaining power maximisation as the primary objective. In our case, having defined custom inequality constraints that can include loads and structural deformations, we can directly target load alleviation through the imposition of limit (steady-state) values. As a result, we include only a small regularisation term in the objective function to ensure a limited impact on partial-load solutions. Hence, we propose maximising the power coefficient with a penalisation on the rotor torque coefficient for each wind speed V‾ as follows: 7fobjω,V‾,β=-CPω,V‾,β+w1CQω,V‾,β. Here, the first objective has a unity weight, and the selection of w1 remains the only tuning variable. Tuning parameters such as the weight w1 is not a straightforward task. A similar challenge is reported in the work of , where multiple tuning parameters were required to balance competing goals in the objective function of a model predictive control scheme for wind turbines. This highlights the difficulty in tuning such parameters, which often involves trial and error to achieve the desired system behaviour. In our case, the torque term regularises the objective function in the full-load region. In the selection of w1, we should consider that increasing w1 decreases the power coefficient in the partial-load region and is therefore chosen to be small. In the remainder of this work, we set w1=0.01. Because the power coefficient surface is relatively flat around its maximum in partial-load conditions, this small weighting factor has a negligible impact on the optimal set points in that region. However, it is sufficient to ensure unique solutions in the full-load region by regularising the objective function.

The formulation and solution of the optimisation problem in the set point optimiser framework are shown in Fig. . This figure illustrates the sequential steps in the optimisation process, starting from the initial calculation of performance metrics on a three-dimensional grid (upper-left block), followed by the generation of multi-variate B-splines to ensure smooth, continuous performance functions (upper-right block). These interpolated functions are then used to formulate the nonlinear programming (NLP) problem (lower-left block), which incorporates design constraints. The process concludes with the solution of this NLP, yielding optimised operating points for the entire turbine operating range (lower-right block). We implement the NLP process defined in Eq. () using CasADi and solve it with the IPOPT nonlinear solver . The most general optimisation problem for determining set points across the entire operating range of a wind turbine is defined as in Eq. () with the objective function of Eq. (), with the following constraints: 8Pgω,V‾,β≤Pg,rated,Qgω,V‾,β≤Qg,max, where Pg,rated is the rated power, and Qg,max is the maximum admissible generator torque. The solution to this problem is represented by combinations of optimal values ω*,β* for each wind speed V‾, gathered in set point mappings, which are then used in the feedback component of the controller.

Figure  illustrates the results of the optimisation process for two specific wind speeds, 10 and 11 ms-1, chosen to represent partial- and full-load operating conditions. This figure visually demonstrates how the optimiser finds the best operating points while satisfying the required constraints in the different operating regions of a wind turbine, as we remark that COFLEXOpt is agnostic to regions. In each case, the power coefficient and torque coefficient are plotted against rotational speed and collective pitch angle. Optimal solutions found by COFLEXOpt are represented with red stars.

The grey regions in each plot represent infeasible zones where one or more constraints are violated, such as limits on power or torque. These areas indicate combinations of β and ω that the optimiser cannot select, helping to emphasise the feasible solution space. The difference between plots (a) and (c) versus (b) and (d) lies in the objective for each wind speed condition. In partial load (10 ms-1), the optimisation focuses on maximising the power coefficient, as seen in plot (a). However, in the full-load region, the rated power constraint leads to an infinite number of ω,β combinations along the red line highlighted in Fig. c. The objective function becomes strictly convex due to the small contribution given by the torque coefficient term, as demonstrated by the isocontours in Fig. d. These figures highlight the different sensitivities of power and torque coefficients to variations in rotational speed and collective pitch angle, which is exploited to find unique solutions to the optimisation problem over the entire operating range of a wind turbine. The optimal solution, which minimises CQ, is found at the upper boundary of the rotational speed ωmax. This is because, in the NLP solved in this work, constraints on rotational speed and generator torque were chosen according to values from  to avoid major differences from the baseline controller design. For other applications of COFLEXOpt, such as optimising set points during the preliminary design of a wind turbine, relaxing these constraints is possible without sacrificing convergence capabilities.

In this section, we use COFLEXOpt to calculate set points for four different strategies, as summarised in Table . The first strategy corresponds to the reference approach, adopted for the IEA 15 MW RWT , and is commonly referred to as optimal TSR tracking, where a fixed optimal TSR value λ* is prescribed for the partial-load region. The collective pitch angle is set to a minimum of 0°, and the following formulation of the optimisation problem is used to obtain rotational speed set points: 9min(ω,β)-CPω,V‾,β+w1CQω,V‾,β∀V‾∈3ms-1,25ms-1,s.t.5min-1≤ω≤7.55min-1&0°≤β≤30°,λ*=9∀V‾∈Vωmin,Vrated,Pgω,V‾,β≤15MW,Qgω,V‾,β≤21.1MNm. Note that this fine-pitch optimisation is only able to increase the power coefficient when the tip-speed ratio is constrained by the minimum rotational speed in the partial-load region, with positive collective pitch angles. Under these assumptions, this strategy is not able to compensate for the flexibility effects illustrated in the previous section.

Case 1 optimises the power coefficient in the partial-load region without using a prescribed optimal tip-speed ratio and with no structural design constraints. The minimum collective pitch angle is set to -5°. Case 2 and Case 3 include upper limits on the OoP tip displacement at 13.6 and 10 m, respectively. The first value was chosen based on the maximum value observed in the reference strategy, while the tighter constraint was introduced to evaluate the performance of the framework. To our knowledge, no previous studies or proposed frameworks have the capability to constrain steady-state structural properties directly in the optimisation problem, such as OoP blade tip deflection, and this presents a significant contribution to COFLEXOpt. This quantity is relevant for the design of flexible wind turbines due to the risk of tower strikes. It showcases the implementation of a critical structural performance constraint in our set point optimiser .

Figure  shows the resulting optimised set points (rotational speed, collective pitch angle, tip-speed ratio) and the corresponding steady states for generator torque, blade OoP tip displacement, and generator power for the four different strategies. While the TSR values match for all cases (third plot) in the cut-in and full-load region, we observe that the optimal λ from COFLEXOpt varies across the partial-load region for all three cases. This again demonstrates that variable-λ regulations lead to improved performance, a tendency that is expected to intensify for more flexible rotors. The operating points of the collective pitch angle for Cases 1, 2, and 3 deviate from the reference values up to rated conditions. The optimisation framework allows pitching to stall, counteracting the effects of structural torsion on the blade and increasing the power output in the partial-load region, as shown in the generator power plot. A different trend is observed in the constrained strategies, where the blades pitch to feather to relieve thrust force and facilitate the decrease in OoP tip displacement. The wind turbine performance output with the current recalculated optimised operating points returns higher power, with gains up to 10 % for Case 1 and a consequent decrement of the rated wind speed.

Interestingly, the rated wind speed of Cases 1 and 3 assumes different values with respect to the reference one, a direct result of the optimisation problem and imposed constraints, and is not predefined. This shows the major capability of the framework to arrive at the optimal solution and sets a new standard for deriving operating strategies for flexible turbines.

We notice an interesting effect on the operating points when the OoP tip displacement limit is active in the partial-load region. Unlike a fixed tip-speed ratio strategy, COFLEXOpt allows for concurrent changes in the rotational speed and pitch angle to find the optimal compromise between reducing loads and maximising the power coefficient. In this case, our approach takes advantage of the different sensitivities of the power coefficient and thrust coefficient to variations in pitch and rotor speed. In Case 3, the OoP tip displacement is effectively constrained to 10 m, though this leads to power losses compared to the reference strategy. To correctly track the set points of the optimised strategies obtained with COFLEXOpt, we introduce a novel control scheme in the next section.

The wind speed estimator (WSE) employed in this work is part of the torque balance estimator class . Under the assumptions of measurable generator torque and rotational speed and a known power coefficient performance of the turbine, an estimate of the aerodynamic torque (or also rotor torque Qr) is used to derive an estimate of the rotor effective wind speed. The scheme used in this work is from  and employs the dynamic balance of rotor and generator torque at the rotor shaft, with a feedback loop for providing the rotor effective wind speed. The WSE illustrated in the block diagram of Fig.  is structurally identical to that shown in  and . We refer the reader to these works for the full derivation and details on this WSE.

As demonstrated by , the accuracy of wind speed estimates at steady state depends largely on the uncertainty in the power coefficient table, which is usually a function of λ and β. As already demonstrated in Sect. , more accurate results can be obtained by using a flexible aeroelastic solver and removing the fixed tip-speed ratio approach to obtain a three-dimensional function for the power coefficient table, particularly when considering large and flexible wind turbines such as the IEA 15 MW RWT. To increase the accuracy of the estimated rotor torque for large, flexible rotors, we implement a modification to the schemes found in the literature by using the newly obtained power coefficient tables CP(ω,V,β). To the authors' knowledge, this is the first effort to improve the accuracy of a torque-balance-based WSE with a three-dimensional power coefficient table. Now, the system equations of the wind speed estimator are given as 10ω^˙=ρV^3πR2CP(ω,V^,β)2Jω-KgJQg,eω^=ω-ω^,V^=KW,Peω^+KW,I∫eω^(τ)dτ, where a constant value Kg represents the mechanical efficiency between the rotor and the generator, and J is the total of the rotational inertia of the rotor, drivetrain, and generator shaft (recall that the IEA 15 MW RWT employs direct-drive technology). The estimated rotational acceleration ω^˙ is used to obtain an estimated rotational speed ω^. A feedback loop with proportional and integral gains (KW,P and KW,I) is used to obtain an estimate of the wind speed V^.

To verify the improved performance of the WSE with an additional power coefficient table dimension, three time-domain simulations of the IEA 15 MW RWT were performed with uniform wind steps of 1ms-1 ranging from 3 to 11 ms-1, with each step lasting 300 s. To analyse the accuracy of the steady-state wind speed estimation, we implemented a Kω2 scheme, selecting the gain K according to the method in . The constant K was calculated based on the optimal tip-speed ratio and corresponding maximum power coefficient prescribed by the IEA 15 MW RWT baseline design, reverting to the standard constant optimal tip-speed ratio assumption. In doing so, the steady-state behaviour is fully specified by the gain K so that the generator torque controller does not rely on wind speed estimates. This approach decouples the steady-state performance of the WSE from other control routines, allowing us to evaluate the estimator without interference from the control tuning parameters. The Kω2 controller used in this section serves only as a convenient means to assess the WSE steady-state performance. Three different schemes for the WSE were analysed, as summarised in Table : the first, named rigid, is a WSE in which the CP table was calculated with a rigid model and parameterised on tip-speed ratio and collective pitch angle; in the flex. 1 WSE, the CP table was calculated taking into account flexibility and parameterised on tip-speed ratio and collective pitch angle; and in flex. 2, the CP table was calculated with flexibility and parameterised on wind speed, rotational speed, and collective pitch angle. All CP tables were obtained using HAWCStab2 with the same models described in Sect. .

As shown in Table  and Fig. , the WSE with the three-dimensional CP table flex. 2 outperforms the first two schemes in estimating the wind speed value in this operating region, with lower mean errors at steady state. As seen in Fig. , the rigid WSE shows a small steady-state error for low-wind-speed cases up to 7 m s⁻¹. Flex. 1 is only able to estimate the wind speed with a small error in the neighbourhood of the wind speed, which was chosen to calculate the CP(λ,β) table (V=9ms-1). The novel, improved scheme flex. 2 is able to estimate the wind speed at a steady state with a significantly smaller error due to the improved match of the estimated rotor torque in the WSE model with the simulation model. The speed of convergence of the estimate to its steady-state value in all three cases analysed here is essentially related to the choice of the gains (KW,P and KW,I). When the WSE is integrated into a controller scheme (i.e. V^ is used to compute inputs to the controller), tuning of the gains is needed as they become part of a dynamic feedback loop.

The generator torque and collective pitch angle controllers developed in this work implement feedforward set points parameterised on the wind speed estimate and, following well-established methodologies to control wind turbines , include two PI feedback controllers to regulate the rotor speed. A set point smoothing technique allows us to switch between the two controllers by forcing the inactive controller to saturation. The scheme in Fig.  shows the controller implementation.

The following control laws are implemented for the generator torque (Qg) and collective pitch angle (β) command inputs to the system: 11Qg=Qg,FF*+ΔQg,FB,12β=βFF*+ΔβFB, where the feedforward contributions Qg,FF* and βFF* are calculated with COFLEXOpt and are extracted from set point mappings as a function of estimated wind speed V^ (as indicated by the functions f(V^) and g(V^) in Fig. ). These quantities represent the desired steady-state set points for the entire operating range of the wind turbine and depend on the design requirements and optimisation strategy.

The feedback terms are calculated based on the rotational speed error, which is calculated as follows: 13eω=ω*-ω. Thus, the feedback contributions are given by 14ΔQg,FB=KP,Qeω+KI,Q∫eω(τ)dτ,15ΔβFB=KP,βeω+KI,β∫eω(τ)dτ, where the two gains for the generator torque contribution KP,Q and KI,Q must be defined so that ΔQg,FB<0 leads to acceleration of the rotor rotational speed when eω>0, while KP,β and KI,β must be defined so that ΔβFB is negative (i.e. the blades pitch towards stall, increasing the aerodynamic torque of the turbine) when eω>0. To satisfy controller performance requirements, such as overshoot and rise time, proper tuning of the gains (KP,Q,KI,Q,KP,β,KI,β) is necessary.

At the same time, a bias Δωbias is introduced to the inactive controller set point through a switching logic. This technique for smoothing the set point in the switching region is implemented similarly to in  and is explained in further detail in the following section.

A set point smoothing technique is described here to ensure a continuous transition between partial- and full-load operations. To this aim, a bias is introduced into the reference set points of the two controllers. This set point bias is used to force one of the two PI controllers to saturate when the other is active. When the generator torque PI controller is active, i.e. in the partial-load region, the pitch controller should be forced to its lower saturation limit. Vice versa, in the full-load region, the generator torque should reach its upper saturation limit. The set point smoothing technique is represented by the block scheme in Fig. .

The following equations are used to calculate the contributions for the rotational speed set point bias Δωbias: 16Δωbias1=Qg,rated-QgQg,rated,17Δωbias2=β-j(V^)βmax, where Qg,rated and βmax represent the upper saturation limits of the generator torque and collective pitch angle, respectively. In contrast, the function j(V^) represents the lower varying saturation limit for the collective pitch angle. We developed a new methodology to obtain j(V^). This function is introduced to ensure that the collective pitch angle correctly saturates to the prescribed set points in the partial-load region while preventing aerodynamically unstable behaviour in the full-load region and is obtained by solving a nonlinear programme similar to Eq. (): 18(k(V‾),j(V‾))=argmin(ω,β)-CPω,V‾,β+w1CQω,V‾,β∀V‾∈3ms-1,25ms-1,s.t.5min-1≤ω≤7.55min-1and-5°≤β≤30°,OoP tip disp.ω,V‾,β≤OoP tip disp.max, without constraints on the maximum power and torque but imposing the same maximum level on OoP tip displacement of the chosen set point strategy. A key motivation for deriving the lower pitch saturation limit from the “reduced” optimisation in Eq. () is to systematically obtain minimum pitch schedules that comply with the constraints imposed in COFLEXOpt-optimised operating points and to avoid stall. By defining an objective function that maximises aerodynamic efficiency (i.e. the power coefficient) and retaining the OoP tip displacement constraint, we ensure that at full load, the minimal-pitch operating point (for any rotor speed–wind speed combination) remains above the stall onset value. This preserves aerodynamic stability and avoids stalled blades even if the turbine briefly operates at that minimal pitch. In contrast, simpler schedules (e.g. setting j(V^) to the pitch angle under rated conditions) may produce stalled conditions or violate tip displacement limits for wind speeds in full-load operations.

By solving this NLP, we obtain the two mappings k(V‾) and j(V‾) over the wind speed interval. The function j(V‾) is used to track the collective pitch angle set points in the partial-load region while being compliant with the design requirement and producing stable operating points for the wind turbine in the full-load region. This function was calculated for each set point strategy obtained with COFLEXOpt and is represented for illustrative purposes in Fig.  for Cases 1 and 2.

The contributions to the set point bias calculated in Eq. () and Eq. () are normalised and weighted so that the final value can be calculated as 19Δωbias=ωratedKbias2Δωbias2-Kbias1Δωbias1, in which the two gains {Kbias1,Kbias2}∈R+ are similar to the ones introduced in  and can be tuned to regulate the smoothness of the transition from one PI controller to the other. The signal Δωbias is also low-pass filtered to prevent high-frequency oscillations. In particular, we used a discrete-time first-order filter with a cut-off frequency of 0.2πrads-1. The sign of this function depends on which one of the two controllers is saturated. In the partial-load region, Δωbias2=0 and Δωbias<0, while if the generator torque is saturated, Δωbias1=0 and Δωbias>0. A switching logic, which applies a bias to the two different set point inputs to the PI controllers, can be implemented based on the sign of Δωbias: ifΔωbias<0→e′ω=eω-Δωbias,in collective pitch angle PI controller,elseΔωbias≥0→e′ω=eω-Δωbias,in generator torque PI controller.

In this way, in the partial-load region, the collective pitch angle controller receives a biased, higher set point e′ω, which pushes the blades to pitch to stall, forcing the controller to its lower saturation limit (i.e. the function j(V^)). In the full-load region, the generator torque reaches its upper saturation limit Qg,rated, and Δωbias changes sign, becoming positive. The switching logic applies a negative bias (-|Δωbias|) to the generator torque controller, which, in the attempt of trying to decelerate the rotor, is forced to its upper saturation limit, and the collective pitch angle controller becomes active. In the transition zone, the alternating activation of the controllers is smoothed by the presence of a low-pass filter in the rotational speed bias. To ensure a smooth transition, the gains Kbias1 and Kbias2 were re-tuned with respect to the values that can be found in .

The integration of the WSE, the set point smoothing technique, and the PI controllers leads to the novel control scheme for large, flexible wind turbines shown in Fig. .

This control scheme leverages feedforward action to achieve the desired set points, while feedback loops work to enhance stability, correct (tracking) errors, and add resiliency to disturbances and noise. However, its overall tracking performance is dependent on the accuracy of the internal power coefficient table. The wind speed estimation relies on this table, so any bias in the power coefficient data propagates into the estimates. As demonstrated by , for the WSE-TSR tracking scheme, whenever the controller’s reference is scheduled based on wind speed estimates, the system converges to a steady state that reflects this bias. In other words, the controller is capable of tracking a reference, but the reference itself is shifted from the true optimal operating point. This is essentially the same phenomenon encountered in standard tip-speed ratio tracking, where the optimal set point is also calculated offline using nominal aerodynamic data; if the real performance deviates from these nominal data, the turbine will no longer operate at the true optimum. Our scheme will similarly be affected by inaccuracies in the internal power coefficient table, even though it maintains effective reference tracking. A potential mitigation of the bias introduced by modelling inaccuracies would be to schedule the feedforward input on an independent measurement of the rotor average wind speed – such as lidar measurements – or by combining such measurements with the estimated values. Alternatively, one can update the aerodynamic model (used in both the controller and the estimator) to represent the actual, possibly degraded aerodynamic properties of the wind turbine using online learning algorithms .

The capabilities of this novel scheme to allow for a smooth transition between the two PI controllers are visualised in Fig. , where the behaviour of the controller is analysed in a time-domain simulation. This analysis was performed using the characteristics of the flexible model described in Sect. , in the time-domain wind turbine aeroelastic simulator HAWC2. In this example, the controller tracks the optimised set point strategy defined by Case 2. The transition between the partial-load and full-load controllers is expected to occur at a wind speed of approximately 10.5 m s⁻¹ (see Fig. ). To observe this transition in detail, we extracted a 40 s segment from a 1000 s simulation carried out with a turbulent wind field and wind shear, capturing the moment when the rotor’s average wind speed crosses the rated wind speed.

Figure a compares the rotor average wind speed (light grey) with its corresponding estimate (dark grey). Overall, the two signals align well, though the estimated value shows some high-frequency oscillations that likely stem from noise in the WSE input signals and the calibration of the WSE. Brief discrepancies also occur (e.g. near t≈510 s), which may be attributed to dynamic effects or degrees of freedom not captured by the internal model used in the WSE. To prevent the high-frequency oscillations from directly exciting the actuators, we apply a first-order low-pass filter with a cut-off frequency of 0.5πrads-1 to the feedforward inputs. Figures b and c show the feedforward pitch and torque commands, respectively, scheduled on the true rotor average wind speed (light grey), the estimated wind speed (dark grey), and the actual controller outputs (green). Up to t≈505 s, the turbine remains in partial-load operation: the collective pitch angle closely follows the feedforward command, which in turn tracks the ideal feedforward value reasonably well. Near t=505s, the generator torque saturates (Fig. c) to maintain rated power. At that moment, the estimated wind speed in Fig. a reaches around 10.7 m s⁻¹, matching the expected rated condition. Figure d illustrates how the set point bias Δωbias (blue) ensures a smooth transition from torque to pitch control. Before t≈505 s, the bias is negative, keeping the collective pitch angle saturated at its lower limit and allowing the torque controller to be active. As the system approaches rated, the bias crosses zero and effectively drives the generator torque into saturation, activating the collective pitch controller. This gradual shift avoids abrupt changes in control action and demonstrates that the combined feedforward–feedback strategy can successfully handle transitions to full-load operation, even under turbulent inflow. Finally, while the overall dynamic performance is satisfactory, further gain scheduling or fine-tuning of the WSE and PI loops could improve transient behaviour and reduce any remaining high-frequency pitch or torque activity.

The next section delves deeper into the validation of the proposed COFLEX control scheme through time-domain simulations with uniform wind step inputs and turbulent wind fields.

Four 2500 s simulations were carried out with incremental wind speed steps of 1 ms-1 every 100 s, starting from an initial wind speed of 3 m s⁻¹ up to 25 m s⁻¹, for each set point strategy. These simulations, which included an initialisation period of 200 s to settle down transient behaviour, were used to test the controller step response.

The time series of rotational speed, collective pitch angle, and tip-speed ratio are shown in Fig. , excluding the initialisation period. In the first 500 s, all control strategies correctly track the minimum rotational speed (5min-1), with relatively high overshoots (around 10 %), while the collective pitch angle is set to the same values to maximise the power coefficient. From t=500s onwards, the four cases follow different set point strategies for both rotational speed and collective pitch angle. In all cases, varying trends on the overshoot and settling time values suggest that gain scheduling see could be employed to improve the dynamics of the controller and is devoted to future work. The focus of this paper is to establish a novel control strategy for flexible turbines, and, therefore, we are interested in analysing the trends in steady-state performance.

During the simulation, the final 10 s for each 100 s interval was used to calculate the steady states of selected variables. These steady states (dots), obtained using the FF–FB control scheme in HAWC2 simulations, are presented in Fig.  with respect to the input wind speeds (uniform and constant) in the same intervals and are compared to the prescribed operating points (lines) calculated through COFLEXOpt.

These results demonstrate that, in time-domain simulations, COFLEX accurately tracks the set points calculated with COFLEXOpt for all variables of interest. The trends observed in the mean values of control variables, including rotational speed, collective pitch angle, and generator torque, follow the strategies prescribed by COFLEXOpt. This novel approach, which uses a variable tip-speed ratio and collective pitch angle, allows for maximising power production while respecting the blade tip displacement constraint.

The generator torque reaches saturation above 11ms-1 for Cases 1 and 2 but not for Case 3, in which the tighter constraint on tip displacement increases the wind speed at which the rated power is produced up to 12ms-1. The steady-state maximum values of tip displacement for Case 2 and Case 3 result in 14.1 m (+3.5 %) and 10.6 m (+6.0 %), respectively, showing a slight positive discrepancy. For all cases and each wind speed, the OoP tip displacement is slightly underestimated in the steady states calculated through COFLEXOpt. This small difference can be attributed to different factors: a discrepancy in the steady-state blade deflection calculation for HAWC2 and HAWCStab2, which was deemed small but not directly quantified in the comparison of the tools , and nonlinear dynamic effects, which were only taken into account by HAWC2.

Some differences are also present in the generator torque and collective pitch angle set points (see e.g. Case 3 generator torque and Case 1 collective pitch angle at wind speeds near rated). These differences can be attributed to the activation of the switching logic and the resulting set point bias affecting the system behaviour. The potential of the new control scheme to track the optimised set points in realistic turbulent wind conditions is provided in the next section with the analysis of turbulent wind cases.

To evaluate the performance of the controller under more realistic operating conditions, turbulent wind cases were defined following the design load case (DLC) 1.1 as specified in IEC 61400-1 for wind class IB . A series of 1000 s simulations was performed with mean wind speeds ranging from 3 to 25 m s⁻¹ (one simulation every metre per second) and turbulence intensity in accordance with IEC standards, using six different seeds for the turbulence box generator (for a total of 138 simulations for each control strategy). The turbulent wind fields were generated using the Mann turbulence box generator integrated within HAWC2. Additionally, a power‐law vertical wind shear was applied with an exponent of 0.2.

For each simulation, only the last 600 s was used in the analysis to eliminate initialisation dynamics. Simulations were grouped for each control strategy and subdivided into small time intervals of 10 s. Then, the means of the individual performance metrics were calculated in these intervals and binned with respect to the average wind speed of the rotor, with a uniform bin length of 1 ms-1.

A statistical analysis was performed, and the distribution of selected performance metrics is shown for the control strategies Case 1 and Case 2 in Figs.  and within a wind speed range of 5 to 15 m s⁻¹. In these figures, the dotted lines represent the prescribed operating points from COFLEXOpt for the same control strategy, calculated using the rotor average wind speed V, while the dashed grey lines (corresponding to the right y axis) indicate the differences between the median values for each bin and the prescribed values at each bin's midpoint. The boxplots depict the distribution of the performance metrics averaged over 10 s intervals for each wind speed bin, with indications of quartiles (the filled boxes with a central line represent the 25 % quartile, the median, and the 75 % quartile), minimum and maximum values (whisker limits), and outliers (triangles).

For both strategies, the median values of the estimated wind speed (V^) are consistently higher than those of the actual wind speed. This discrepancy is around 10 % at very low wind speeds, decreases to approximately 5 % near the rated wind region, and then increases again linearly in the full-load region. This consistent, positive bias was not observed in previous analyses and is likely driven by local wind speed fluctuations due to wind shear and turbulence. Our wind speed estimator uses a torque balance approach, matching the measured generator torque to an estimated rotor torque, recalling the system of Eq. (). Under wind shear and turbulence, the contribution of blade sections to the total torque depends on the local velocities. Hence, the effective wind speed, which produces the rotor torque, differs from the arithmetic mean across the rotor disc. As a result, the WSE estimates an effective wind speed that differs from the rotor average wind speed, which is used as a reference here. However, this bias does not degrade the performance of the controller. In a practical scenario, the controller must adapt to this effective wind speed; the control scheme of COFLEX still holds, as our set point mappings and feedforward inputs rely on precisely this torque-based wind speed estimate.

Both the collective pitch angle and the generator torque exhibit differences relative to the set points, with similar magnitudes and trends across the two analysed control strategies. These differences can largely be attributed to the bias between the estimated wind speed and the rotor average wind speed resulting from the simulator used for binning. This directly impacts the feedforward component in the control loop, especially at low wind speeds. Despite these discrepancies, the median values of the OoP tip displacement closely follow the steady-state values calculated by the set point optimiser, with a high degree of accuracy (less than 10 % difference across the analysed operating range). In Case 2, the expected constraint on the median value of the OoP tip displacement is satisfied with a deviation of less than 1 %.

While constraining the steady-state OoP tip displacement helps reduce average deflection levels, more advanced control techniques remain necessary to mitigate the transient effects that drive the maximum values – and thus the tower‐strike risk. Consequently, imposing a strict limit on the maximum displacement would require a different control approach, such as online set point optimisation or advanced individual pitch control , which can explicitly predict and counteract such extremes. Nonetheless, to address the safety margin in a stochastic way, one could modify the constraint in COFLEXOpt by incorporating a precomputed variance around the median displacement. This would allow designers to ensure, a priori, that the probability of exceeding the maximum allowable OoP tip displacement remains within an acceptable margin.

The generator torque is correctly saturated in the full-load region for both strategies. Finally, we observe an interesting effect on the generator power median values in the partial-load region, where these values consistently exceed the prescribed operating points. These trends align with studies on the effects of turbulence intensity on the power production of wind turbines in the partial-load region .

Figure  compares the median values of OoP tip displacement (top panel) and generator power (bottom panel), both normalised by the reference strategy, for the new strategies across wind speeds from 5 to 15 m s⁻¹. For Case 1 and Case 2, we observe that the generator power increases by approximately 5 % relative to the reference, at the expense of higher tip displacements in the partial-load region. In particular, Case 1 shows OoP tip displacements as much as 30 % above the reference at rated wind speed, which aligns with the prescribed operating points. In Case 2, the displacement constraint is active around 10ms-1, as indicated by the orange bars converging toward unity in the top panel near 11ms-1. Case 3 follows a similar pattern at lower wind speeds (below 8ms-1), but the tighter constraint on tip displacement results in values around 25 % below the reference near the rated wind speed and a corresponding lower power output in that range. All three cases behave similarly to the reference controller in full-load operations. Overall, these trends confirm that the set points derived via COFLEXOpt can be effectively tracked in turbulent inflow scenarios.