Wind turbine operation and control have recently shifted from the traditional goal of power maximization to more economically driven goals. In this new paradigm, turbines are operated with damage awareness in mind . This shift is driven by the impact of fatigue damage, which shortens the operational life of turbines, and increases operation and maintenance (O&M) costs. In fact, these factors are critical to the economic profitability of operators of wind energy assets .

One widely used method for developing economic controllers for wind turbines is model predictive control (MPC) . MPC optimizes control actions over the short-term future by predicting system behavior, and then solving an economic optimization problem based on these predictions . Existing literature shows that economic MPC formulations for wind turbine control – using the conventional objective of maximizing revenue from power generation – achieve better performance than baseline controllers and standard MPC approaches . However, when considering wind energy systems, an economic optimization should balance the conflicting objectives of maximizing revenue from power generation and of minimizing the cost of fatigue-related damage. Additionally, a controller should always guarantee that the system operates within feasible limits. The effectiveness of the controller ultimately depends on the quality of the solution of this constrained optimization, which essentially relies not only on the economic model but also on the quality of predictions of the controller-internal model.

The existing literature on the economic control of wind turbines using MPC is based on two main approaches for estimating cyclic fatigue: indirect and direct methods.

The indirect approach uses a proxy for fatigue . This method is convenient because it avoids the use of cycle counting that, because of its branching nature, introduces discontinuities in the calculations . As a result, with the indirect approach, gradient-based methods can be used to solve the optimization problem. However, this also means that fatigue is only approximatively taken into account through a proxy quantity, which might not always provide for accurate results.

The direct approach, on the other hand, estimates fatigue explicitly using online cycle counting directly in the MPC framework . This method, termed parametric online rain flow counting (PORFC), is relatively new and was first introduced in our previous work . It has since been applied in its fundamental form in various contexts, including wind turbine fatigue control , battery cyclic aging control , and control of grid-connected wind-battery hybrid systems . To the authors' knowledge, PORFC is the only approach that allows for the rigorous treatment of fatigue, as it was designed to produce the same results on receding horizons that would be obtained by cycle counting the whole response time history a posteriori .

In both approaches mentioned above, the controller relies on an internal model of the wind turbine to predict its behavior over the MPC horizon. However, due to computational effort and time constraints, the internal representation of the system response is typically obtained through a reduced-order model (ROM). ROMs approximate the system dynamics using a limited number of degrees of freedom (DOFs), enabling the real-time execution of the controller. The accuracy of these predictions plays a crucial role in the performance of the controller. A closer match between the predicted and actual wind turbine dynamics allows for the controller to make more precise decisions when optimizing the objective function, while at the same time ensuring compliance with system constraints. In fact, the degree of model mismatch not only affects optimality but may also impact the ability of the closed-loop system to operate strictly within admissible or desired limits.

One way to address model mismatches is by using state observers and estimators , where measurements of the response of the wind turbine from previous time steps are used to estimate and adjust the initialization of the controller for the next step. However, the effectiveness of this approach is limited when estimates are obtained from underlying models of scarce accuracy.

A more compelling alternative is to adapt the internal model dynamics of the controller, either by using measured operational data or synthetic results from models of sufficiently high fidelity that can accurately represent the plant behavior. One of the aims of this paper is to develop practical methods for enabling adaptivity in wind turbine ROMs.

Data-driven adaptation can be implemented online, offline, or in a hybrid manner, each of these options offering distinct trade-offs in terms of advantages, challenges, and practical applicability to economic control.

Online adaptation allows for quickly adjusting model behavior in real time based on observed plant response. However, for industrial applications, it is difficult to imagine how one could guarantee the correct learning of model corrections and how such a system could be demonstrated to always be safe and certifiable in practice.

In contrast, offline adaptation allows for the systematic integration of data, enabling a rigorous verification and validation of any learned correction terms prior to their deployment in the field. A rigorous offline verification and validation opens the door to the certification of adaptive controllers, which must meet strict standards in performance, safety, and reliability before deployment in real-world operations.

In the context of economic MPC, offline model adaptation can be achieved in several ways: by adjusting the parameters of the ROM , by incorporating a correction term into the ROM or MPC optimization function , or by a combination of both methods.

It is, however, important to note that a ROM captures only selected aspects of the behavior of the plant – and does so only with limited accuracy. As a consequence, the tuning of ROM parameters alone is often insufficient and may even produce non-physical results, because parameter tuning cannot compensate for missing physics. Instead, data can be leveraged to learn and incorporate the missing physics into the internal model used by the controller. This is the approach followed in this work.

The few existing studies on adaptive economic MPC for wind turbine control focus primarily on the tuning of controller gains . These works rely on simplified linearized models and Bayesian optimization techniques to achieve robust performance under uncertain and varying wind conditions, but they do not explicitly address model mismatch. More in general, gain tuning cannot compensate for missing or inaccurate representation of relevant physics. Recent contributions to economic MPC introduce more complex, non-linear internal models ; however, these approaches still lack adaptive mechanisms that explicitly compensate for modeling errors.

Furthermore, although the economic optimization in all of these studies incorporates profit-related objectives, it relies solely on indirect penalization of system states and does not explicitly account for fatigue damage, whose relevance has been highlighted by and .

The key contribution of this work is the development of a novel fatigue-aware economic non-linear model predictive controller (ENMPC) with offline-adaptive capabilities enabled by data-learned ROM corrections.

The controller maximizes profit by balancing two competing factors: revenue from power generation and costs associated with fatigue damage of the turbine components. The cost of cyclic damage is formulated using the novel PORFC approach, which performs online rain flow analysis over the stress history and its future predictions to generate time-varying parameters. These parameters are then used to obtain a continuous expression of the discontinuous fatigue cost, which is incorporated into the MPC optimization. The profit formulation used here does not explicitly incorporate the effects of fatigue on component reliability and O&M costs, simply because of a lack of data and appropriate models linking loads with failure rates, instead representing fatigue-related costs solely through the amortization of the overall component cost, as discussed in .

The controller-internal model is a gray box that combines reduced-order physical dynamics with offline-computed correction terms. These terms are formulated as neural networks trained on high-resolution measured or synthetic data of the wind turbine. Their objective is to reduce model mismatch, thereby improving online closed-loop performance.

The paper is structured as follows. Section  introduces the proposed model adaptation approach. It describes both the simplified wind turbine model used as internal ROM and the data-driven modeling of the correction term. Section  provides a detailed formulation of the ENMPC, discussing various aspects of the underlying optimization problem. This section also describes the state and wind speed estimators, which provide initial conditions and disturbance predictions for the ENMPC. Section  presents and discusses the results from a case study. The analysis begins with an open-loop assessment of the model adaptation approach, followed by a closed-loop evaluation of its impact on economic performance, actuator usage, and computational burden. Additionally, this section examines the benefits of the proposed model adaptation under different wind input and preview scenarios. Section  summarizes the key findings of this work and outlines potential directions for future research.

A simplified wind turbine model with only 3 degrees of freedom (drivetrain angular speed, and tower fore-aft and side-side deflections) is considered to represent the ROM dynamics FROM(⋅). The incident wind Vw induces an aerodynamic torque Qa about the rotor axis and an aerodynamic force FT along it. The aerodynamic torque directly excites the drivetrain rotational dynamics 2Jrω˙=Qa-Qg, ignoring mechanical losses, where Jr, ω, and Qg represent the rotor moment of inertia, rotor speed, and generator torque referred to the low-speed shaft, respectively. The aerodynamic force FT, coupled with the drivetrain dynamics, excites oscillations in the tower. These can be quantified by using the tower-top deflections in the fore-aft (noted dTFA) and side-side (noted dTSS) directions, which are governed by the following equations of dynamic equilibrium: 3ad¨TFA=1f1(FTFA-f2d˙TFA-f3dTFA),3bd¨TSS=1s1(FTSS-s2d˙TSS-s3dTSS-s4Qg). Tower oscillations result in cyclic stresses σFA and σSS, respectively, at tower base. Here, FTFA and FTSS represent the rotor-plane-orthogonal (thrust) and rotor-in-plane (side-side force) components of the aerodynamic force FT, whereas f1-3 and s1-4 are model parameters. The aerodynamic torque Qa(ω,βb,(Vw-d˙TFA)) and force components FTFA(ω,βb,(Vw-d˙TFA)) and FTSS(ω,βb,(Vw-d˙TFA)) introduce non-linearities in the model.

The turbine model has two control variables: the commanded generator torque Qgc and the commanded blade pitch angle βc. The dynamics of the generator is given as 4Q˙g=1g1(Qgc-Qg), where coefficient g1 represents the time-constant of the first-order dynamic model. The pitch dynamics is modeled as 5β¨b=-b1β˙b-b2(βb-βc). Here, βb represents the effective collective blade pitch angle, and the coefficients b1-2 are model parameters representing properties of the second-order pitch model.

The reduced-order wind turbine model consists of eight system states 6x=(ω,dTFA,d˙TFA,dTSS,d˙TSS,βb,β˙b,Qg), and two control input variables 7u=(βc,Qgc). The wind speed Vw is considered as the single disturbance input to the model, i.e., 8d=Vw. The reduced-order model outputs the wind turbine electrical power 9Pgen=ηgenωQg and the tower-base stresses 10σFA/SS=t1dTFA/SS in the fore-aft and side-side directions, modeled as linear functions of the respective tower-top deflections. Here, ηgen denotes the drivetrain conversion efficiency, and t1 is a constant of proportionality derived from the tower-base geometry and material properties.

With advancements in computing technologies, machine learning techniques – including supervised and unsupervised learning – have become highly effective for a wide range of data-driven system identification tasks. A key advantage of supervised learning over unsupervised learning is its access to both input states and their corresponding target states during training. This allows the model to generalize and predict system behavior for previously unseen input combinations. A particularly effective approach for supervised learning in this context is based on the training of a neural network (NN). NNs can approximate arbitrarily complex functions while also providing gradients, making them especially well suited for integration in an optimal control framework.

In this work, the NN is designed to provide a static mapping from inputs to outputs. The network architecture consists of multiple hidden layers and a single output layer, with each layer composed of a certain number of neurons. Each neuron is defined by weights, biases, and an activation function. Within a given layer, the input data is first transformed by applying the weights and biases associated with the neuron. The resulting weighted sums are then passed to the next layer through the selected activation function. Activation functions apply a non-linear mapping to the weighted input plus bias, which determines whether and to what extent a neuron is activated, thereby enabling the network to model complex, non-linear relationships between inputs and outputs. The choice of activation function therefore plays a key role in shaping the flow of information through the network and, consequently, its predictive performance. Moreover, for use in an ENMPC, the activation function must be continuous and differentiable, so that first- and second-order derivatives can be computed. These derivatives are required for the numerical solution of the underlying optimization problem by gradient-based methods .

A feed-forward NN with one hidden layer and one output layer is considered in this work. The static mapping of the input features xNN to the output features yNN for the considered NN can be written as 11yNN=fyact(WyTfhact(WhTxNN+bh)+by). Here, W, b, and fact(⋅) represent the weights, biases, and activation functions, respectively, where the subscript h denotes the hidden layer and the subscript y denotes the output layer.

Before training the NN, the generated dataset is split into training and testing sets. Moreover, every feature in the input and the target set is normalized by subtracting the minimum of the feature vector from the feature vector itself, and then dividing the result by the range (difference of maximum and minimum), i.e.: 12yNN=yNN-yNNmin⁡yNNmax⁡-yNNmin⁡.

Furthermore, the normalized datasets are shuffled to reduce the possible clustering of conditions that might create biases. After preprocessing the training set, the parameters p of the NN, defined as 13p=(Wh,bh,Wy,by), are computed using the Levenberg–Marquardt algorithm to minimize the sum of squares of error between the predicted output and target output.

The accuracy of NN predictions depends heavily on the quality of its training process, which requires a comprehensive dataset of inputs and corresponding target outputs. Wind turbine original equipment manufacturers (OEMs) typically have validated high-fidelity models from design and prototyping that can generate representative datasets, and they may also have access to high-resolution data from on-board sensors of operational turbines to supplement them. In the present work, data are obtained from a high-fidelity simulation model, using both standard measurements and estimates derived from these measurements.

Figure  illustrates the proposed approach of acquiring input and target data for NN training, where the solid blue circles represent the plant states xiPlant at a given time instant i. To generate training data, the ROM is initialized with each plant state sample, setting xiROM=xiPlant. The ROM then predicts the next state xi+1ROM, based on the applied control inputs uiPlant and the influence of external disturbance diPlant.

The error between the derivatives of the ROM model and the plant is defined as 14ei+1=x˙i+1Plant-x˙i+1ROM, and constitutes the target dataset. Derivatives are calculated using the current state xiPlant at time ti and the corresponding next states xi+1Plant and xi+1ROM at time ti+1 via finite differences, i.e. (xi+1-xi)/(ti+1-ti). The set of the current states, control inputs, and disturbances applied to the plant constitutes the input dataset (xiPlant,uiPlant,diPlant) for the ith time sample. This simple temporal discretization, although of limited numerical accuracy, is deemed sufficient for the present application, where modeling errors are typically larger than the numerical ones.

The optimization variables in the problem expressed by Eq. ()–() are the control inputs u=(βc,Qgc) and the slack variables ξ=(ξ1,ξ2). The purpose of introducing slack variables is to achieve recursive feasibility of the MPC optimization problem in the presence of model uncertainties and system perturbations . Hence, in the present formulation, the state variable ω and the output variable Pgen are augmented using the bounded slack variables ξ1 and ξ2, respectively. This approach is used because ω, despite the smoothing effect provided by the large rotor inertia, is subject to wind perturbations and model errors that affect the economic MPC problem . The modified set of states xc is expressed as 16xc=(ω+ξ1,dTFA,dTSS,βb,β˙b,Qg,Pgen+ξ2).

The optimization objective aims to maximize the generated profit by maximizing the revenue accrued from wind power generation and minimizing the cost incurred due to fatigue damage. The cost of fatigue damage should be expressed by appropriate models, which will differ depending on the component; for example, it is reasonable to assume that the effects of cyclic loading on a pitch bearing will in general be very different from the ones on a gearbox. Here, for the lack of specific models and relevant data, we simply consider the tower as an exemplary component that is often fatigue critical. It is clear that this is only an academic example, and more realistic scenarios could be readily developed by using the same methodology for multiple components using dedicated component-specific cost models.

The wind power generation is maximized by considering the aerodynamic power capture 17JgenerationPower=wP∫t0t0+Thorizon(ωQa(ω,βb,(Vw-d˙TFA)))dt, where wP denotes the revenue rate for providing electricity to the grid. It should be noted that even though revenue is accrued based on the overall electrical power generation, in this work the aerodynamic power is maximized. This is to avoid the greedy extraction of rotor kinetic energy by MPC (referred to as “turnpike effect” in ).

In general, the revenue rate wP depends on time, because time-dependent ambient conditions influence the energy mix and because market prices may vary dynamically. However, the typical time scales of these variations are much longer – usually on the order of tens of minutes and specifically 15 min in intra-day markets – than the receding horizon used in MPC, which typically spans only a few seconds. For this reason, even though wP appears outside the integral in Eq. (), the present formulation remains fully applicable to variable electricity prices.

The tower cyclic fatigue damage is minimized by a direct penalization of fatigue via the PORFC approach. The PORFC algorithm uses a preprocessing step to identify fatigue cycles c for a given set of stress samples σ and splits the respective cyclic fatigue damages over the contributing samples. In the general case, a particular stress sample may belong to no cycle (when it is not an extremum), one cycle (when it is part of only one half or full cycle), or two cycles (when it lies at a cycle-junction point) . The output of the preprocessing step is the PORFC mean parameters σm,cPORFC and PORFC weight parameters σw,cPORFC .

A Python script to extract the PORFC parameters for a given set of stress trajectories is shared with this work . This allows for a detailed understanding of the novel formulation, and its adaptability and usage for economic MPC. The script performs standard rain flow counting to identify cycle characteristics and extract PORFC parameters. These parameters are then used to reformulate the cycle amplitudes and weights over stress samples in a continuous manner.

The cost of tower fatigue damage JtowerfatigueFA, due to tower root fore-aft cyclic stress σFA, is formulated as 18aJtowerfatigueFA=1Tctrl∫t0t0+Thorizon∑c=12Jc,σFAPORFC(σFA,σFAm,cPORFC,σFAw,cPORFC)dt,18bJc,σFAPORFC=σFAw,cPORFCamσFAeq,cm,18cσFAeq,c=|σFA-σFAm,cPORFC|RmRm-σFAm,c. Here, am denotes the capital cost of the component and is determined from the initial capital expenditure of the machine (see also , for details), Rm denotes the ultimate tensile strength of the material, and m represents the positive exponent derived from the material S-N characteristic.

The tower cyclic fatigue damage JtowerfatigueSS, due to tower root side-side cyclic stress σSS, is formulated in a similar manner.

Although the optimization objective, shown in Eq. (), separately considers the costs of tower fore-aft and side-side fatigue, the final evaluation is based on the projected total cost, as explained below. In addition to the fact that fore-aft and side-side components depend on wind direction, which is not constant, this also ensures that any potential increase or decrease in tower side-side stress oscillations resulting from control actions aimed at minimizing tower fore-aft stress (and vice versa) is properly accounted for.

To obtain the cost of cyclic fatigue damage for each projection at a given azimuth direction on a tower section, the following steps are taken. First, rain flow counting is performed on the projected stress trajectory. Then, the Goodman equation is applied for mean stress correction. The damage cost of each stress cycle is determined using the S-N curve of the material of the tower and the component cost. Finally, the Miner–Palmgren algorithm is used to sum the costs of individual cycles and obtain the total cost; refer to for a detailed description of this formulation.

This approach – here illustrated for the tower – is readily generalized to other components. Once cyclic fatigue is assessed on each prediction horizon for each component of interest, a dedicated model could provide failure rates and/or maintenance activities resulting from such loading, in turn generating the associated costs, which would be included in the optimization merit function.

It should be noted that the objective terms of the optimization problem are not assigned arbitrary or heuristic weights. Instead, each term is weighted according to its actual economic value. As a result, the optimization outcomes are economically meaningful and directly aligned with practical decision-making considerations, rather than being influenced by ad hoc tuning of weights. For instance, the weighting term to calculate revenue due to power production is represented by the (optionally time-varying) revenue rate wP, as shown in Eq. (). Similarly, the cost due to accumulated tower fatigue damage is weighted using the initial capital-cost parameter am through the online rain flow analysis, as shown in Eq. (–).

The ENMPC optimization problem expressed by Eq. ()–() is subjected to various constraints.

The equality constraints expressed by Eq. () enforce the governing equations of the augmented plant-model ROMaug(⋅) and have the role of informing the problem of the underlying system dynamics.

The optimization is also subjected to inequality constraints. To ensure feasible states and control inputs, box constraints are provided through Eq. () and (), respectively. Slack variables are bounded using Eq. () to ease convergence. Finally, the rate of change of generator torque Q˙g is bounded using Eq. () to reduce torque travel and fatigue in the drivetrain.

The ability of the adapted model to accurately predict the plant states depends directly on the precision of the data-driven corrections. Additionally, to determine the extent of model adaptation required, it is necessary to assess how reducing the model mismatch influences the closed-loop performance of the controller. To answer these questions, we consider a plant represented by the NREL 5 MW reference wind turbine , which is modeled using OpenFAST , a widely used tool for simulating wind turbine dynamics.

The plant model includes the first and second flapwise, and the first edgewise bending modes for each of the three blades. Additionally, it also includes the first and second tower bending modes in both fore-aft and side-side directions, as well as the torsional flexibility of the drivetrain and the generator DOF.

The model, which incorporates pitch and torque actuators but excludes the yaw mechanism, consists of 33 system states. There are eight states for the tower dynamics, 18 for the blades (six per blade), two for drive-shaft torsion, two for rotor rotation, two for collective blade pitch actuation, and one for generator torque actuation.

The fixed model parameters of the corresponding ROM, namely f1-3, s1-4, g1, b1-2, and t1, are derived from the NREL report describing the 5 MW reference wind turbine . Table  summarizes these parameters and lists their corresponding values.

The ability of the augmented internal model to track system states accurately is essential for ensuring both the optimality and the stability of the closed-loop behavior. To evaluate this aspect of the proposed approach, both ROM_aug and ROM are simulated using each input combination from the testing dataset (refer to Sect. ), with the initial state accordingly set. The final states predicted by both models are then compared to the corresponding plant states, and the prediction error is computed to quantify the accuracy achieved by each model.

Figure  presents the kernel density plot of the absolute prediction errors for both ROM (solid blue line) and ROM_aug (solid green line) across all system states. For each of the subplot, the x axis represents the absolute error values, while the y axis shows the probability density estimate (PDE), calculated using a normal kernel function.

In the ROM case, certain states, such as “Tower-top FA defl.” and “Tower-top FA vel.”, exhibit their highest PDE at a non-zero error value, indicating a mismatch in model fidelity. In contrast, in the ROM_aug case, this mismatch is corrected using the proposed adaptation scheme. As a result, the PDE peak shifts toward zero, signifying an improved state prediction accuracy. Furthermore, for most system states, the PDE peak magnitude in the ROM_aug case is higher than in the ROM case. This indicates a greater concentration of cases where ROM_aug achieves lower absolute errors than ROM, further demonstrating the effectiveness of the proposed model augmentation. The figure also shows that the accuracy of some quantities cannot be improved, although their typical errors are always very small.

To quantify the effectiveness of ROM_aug in minimizing plant-model mismatch in open-loop, a statistical evaluation was conducted using the mean and standard deviation (STD) of prediction errors. Table  presents the percentage reduction in both the error mean and STD for ROM_aug, relative to ROM, across the test set.

The results show varying degrees of improvement in state prediction, demonstrating that NN-based augmentation reduces not only the average error but also the error spread over different operating conditions. A greater reduction in prediction error is observed for states with higher absolute error magnitudes in Fig. . This highlights the effectiveness of weight tuning during the NN training stage, which prioritizes correcting states with significant errors. These findings further emphasize the effectiveness of the proposed offline data-driven approach in reducing plant-model mismatch, improving the overall accuracy of the internal model.

For a practical application of the model adaptation, it is interesting to quantify the amount of training data that are needed to achieve a desired level of performance. To this end, multiple data subsets are created from the original training set. Each subset contains a specified fraction of the data samples from the original training set.

Figure a illustrates the distribution of wind speed for multiple subsets, ranging from 45 % to 95 % of the original training set. The x axis represents the wind speed, while the y axis shows the corresponding relative share. Each subset is used to train an NN, with the same architecture as described in Sect. .

Figure b presents the performance of the adapted models evaluated on the same test set. The performance score is calculated by summing the root mean square error (RMSE) of predictions for all eight states. The y axis reports the performance scores of the different ROM_aug models, normalized against the performance score of ROM. The x axis displays the number of samples in percentage of the full training set. The figure also reports the normalized performance score of the ROM_aug model trained with the full dataset, represented by the green circle.

The results show that, as expected, incorporating more data into the training process provides the NN with additional information, leading to improved performance. However, even a relatively small subset of training data helps to reduce the plant-model mismatch. Furthermore, performance begins to level off beyond a certain point, as additional data samples no longer significantly contribute to improving the model because they do not carry any extra useful informational content that is not already present in the data pool. In this study, performance starts to plateau once 85 % of the training dataset is used. As noted earlier, even when using the full dataset, the number of data points that are necessary for model adaption is small compared to the typical datasets collected in standard aeroelastic analyses conducted during routine design and certification activities.

Although the long tail of the PDE plots in Fig.  indicate that the prediction errors of the augmented ROM remain non-zero for some operating conditions, it is still useful to examine the potential benefit of reducing plant-model mismatch on closed-loop performance, as this helps to quantify the necessary level of compensation. To investigate this aspect of the formulation, the proposed adaptive economic controller is implemented in closed loop with the plant. This configuration is referred to as ENMPC_aug. To assess the impact of the augmentation, the plant performance is also evaluated using the same ENMPC formulation but based on the original, non-augmented ROM, referred to as ENMPC.

Several studies on wind turbine control have demonstrated that economic MPC formulations yield superior performance compared to baseline controllers , as well as standard reference-tracking-based MPC formulations. This holds both for pure power-maximization objectives and for combined power-maximization and fatigue-minimization objectives . Therefore, the discussion in this section focuses only on the closed-loop behavior of ENMPC and ENMPC_aug formulations, in order to quantify the performance improvement resulting from model adaptation.

We consider DLC 1.2 at 11 m s⁻¹ wind speed. The ENMPC-MHE optimization problem (refer to Sect. ) is solved via the state-of-the-art Acados framework . The interior-point solver HPIPM is used for solving the underlying quadratic programs (QP) in the non-linear program (NLP). Several sequential quadratic program (SQP) iterations are carried out at each controller step. The multiple shooting approach is employed with a Newton step length of 1. To address potential numerical issues caused by the highly non-standard formulation produced by PORFC , the Hessian matrix is automatically convexified. It is important to note that solving multiple SQPs can improve performance. However, it also increases the computational burden.

The ENMPC and MHE horizon lengths, Thorizon and Thorizon,est, are both set to 2 s, each having 20 discretization steps. This results in a sample time Tctrl of 100 ms for both the controller and the estimator. The sample time of the plant, Tsim, is set to 10 ms. The optimal control inputs applied to the plant model are considered as piece-wise constant values over Tctrl. Measurements from the plant are taken every Tctrl. Table  lists the fixed optimization parameters for the ENMPC. The weighting matrices for the MHE optimization are listed in .

The closed-loop simulations run for a duration of 10 min. Six turbulent wind speed seeds, generated using TurbSim, are used to characterize uncertainties in fatigue damage. The controller uses the estimated wind speed REWS (refer to Sect. ) as input, which remains constant over the prediction horizon of the controller.

The closed-loop control performance is evaluated using the following performance indicators:

Revenue due to power generation, calculated considering a fixed feed-in tariff wP and turbine electrical power generation ηgenωQg (refer to Eq.  for details).