Mitigating Drivetrain Fatigue of Wind Turbines During Primary Frequency Regulation Below Rated Wind Speed

Abstract

This paper addresses a critical and often overlooked challenge: drivetrain fatigue in wind turbines (WTs) induced by primary frequency regulation (PFR) operations, particularly below rated wind speed. While existing PFR strategies focus on grid support, they rarely offer mechanisms to simultaneously mitigate the mechanical stress they impose. To overcome this, a Model Predictive Control (MPC) strategy is proposed that adaptively adjusts the PFR gain in real time. First, the boundary characteristics of the de-loading factor under varying wind speeds and PFR gains are systematically analyzed. This analysis bridges a critical gap in prior studies, which predominantly adopted fixed-gain strategies without comprehensively evaluating their fatigue implications. Building on these boundary characteristics, an MPC framework is developed to optimally adjust the frequency regulation gain in real time. This approach achieves an optimization by minimizing drivetrain torque fluctuations for fatigue mitigation, critically informed by a comprehensive boundary analysis of de-loading factors, while rigorously ensuring essential PFR support capability through explicit operational constraints. Extensive simulations demonstrate the superior performance of the proposed MPC, achieving a simultaneous reduction of up to 27.99% in drivetrain fatigue load and 14.44% in frequency deviations compared to conventional methods. This work significantly enhances the mechanical reliability of WTs and facilitates their more sustainable integration into modern power grids by offering a unique solution to a long-standing trade-off.

1. Introduction

The rapid integration of high-penetration renewable energy, particularly wind power, has profoundly transformed modern power systems, shifting the generation paradigm from conventional synchronous machines to inverter-interfaced resources [1,2,3,4]. However, this transition inevitably reduces the total rotational inertia of the grid. Coupled with the stochastic and intermittent nature of wind power, it introduces significant challenges to system frequency stability and operational resilience [5,6]. Consequently, modern grid codes increasingly mandate that WTs transition from passive energy providers to active participants in PFR to safeguard grid stability.

Enabling WTs to participate in PFR is widely recognized as a promising solution to enhance system frequency stability under high renewable penetration [7,8,9]. Yet, active frequency support inevitably introduces complex coupling dynamics between the electrical and mechanical systems of the WT. These additional power commands exacerbate torque oscillations, leading to increased fatigue loads on critical drivetrain components, particularly the LSS [10,11]. This trade-off becomes exceptionally critical under low wind speed conditions, where limited power reserves amplify the drivetrain’s sensitivity to control actions [12]. Consequently, intelligent control strategies are urgently required to provide reliable frequency support while mitigating structural fatigue.

To provision the necessary power reserves for PFR, de-loading control via overspeed operation is extensively employed [13]. By operating slightly below the maximum power point, WTs store rotational kinetic energy that can be rapidly deployed during frequency events. While properly selecting the de-loading factor is crucial to ensure safe operation and avoid aerodynamic stall [14], its inherent impact on drivetrain fatigue loads (generally evaluated through low-speed shaft (LSS) fatigue loads) remains insufficiently explored [15,16]. Existing studies predominantly focus on the active-power margin, rarely providing systematic boundary analyses of de-loading factors across varying wind speeds and PFR gains. This oversight restricts the ability to jointly optimize frequency response and mechanical fatigue, limiting the practical deployment of these strategies.

Recent studies indicate that external flexibility, such as energy storage and other dispatchable resources, can alleviate mechanical loading in wind power systems by mitigating power ramps and smoothing active-power variations. A significant body of research explores the role of external flexibility solutions, most notably energy storage systems (ESSs), in addressing these challenges. ESSs, such as batteries, flywheels, or compressed air energy storage (CAES), offer distinct advantages by decoupling power generation from power supply, thereby providing fast and precise power injection or absorption to stabilize the grid [17]. System-level scheduling approaches for wind power harvesting with operational flexibility (e.g., chance-constrained CAES and demand response program scheduling to maximize wind power harvesting in congested transmission systems considering operational flexibility [18]) demonstrate how storage-enabled flexibility can reduce adverse transients that would otherwise propagate to equipment fatigue [19,20]. However, relying on external storage devices inevitably introduces significant capital investment, additional maintenance costs, and spatial footprint. In contrast, the internal control strategy proposed in this paper achieves similar mechanical stress reduction without any supplementary hardware, offering a more economical and readily deployable solution for individual wind turbines.

Recognizing the severe mechanical implications of PFR, the most recent literature has increasingly focused on mitigating LSS fatigue loads. At the individual turbine level, efforts have been made to employ torsional vibration damping control combined with external energy storage devices (e.g., supercapacitors) to smooth power output and alleviate LSS fatigue [12]. However, relying on supplementary hardware significantly increases system complexity and investment costs. Alternatively, from a wind farm perspective, centralized dispatch strategies have been recently explored. For instance, approaches utilizing discrete damping coefficient allocation [21,22] or heuristic-based active-power distribution (e.g., improved Firefly Algorithms) [23] have been proposed to minimize the total fatigue load across multiple turbines. While these farm-level allocation methods effectively balance long-term fatigue distribution, they inherently act as upper-level dispatch commands. They struggle to address the fast, transient electro-mechanical coupling dynamics occurring inside an individual turbine during a grid frequency event [22,24,25]. Consequently, at the turbine control level, the reduction in LSS fatigue achieved by conventional methods remains limited due to the multi-variable and nonlinear nature of the torque fluctuations induced by dynamic PFR actions.

Emerging analytical models reveal that LSS fatigue is governed not only by the mean torque but, more importantly, by its high-frequency fluctuations [26]. The LSS torque under PFR is dynamically shaped by the intricate interplay among generator torque, rotor speed oscillations, and pitch angle variations [27,28]. This suggests that the continuous, coordinated control of these variables is essential to neutralize fatigue accumulation. However, conventional linear controllers struggle to handle such multi-variable coupling constraints, causing most existing methods to fail in simultaneously guaranteeing optimal frequency response and minimized mechanical fatigue.

In recent years, MPC has emerged as a powerful paradigm for managing complex, constrained dynamic systems, with growing applications in renewable energy and microgrid management [19]. A comprehensive systematic review of MPC in microgrids highlights its ability to handle multi-objective optimization under uncertainty, moving beyond traditional control methods [29]. In the context of wind power and LSS fatigue mitigation, several advanced MPC frameworks have been proposed. For instance, a data-driven nonlinear MPC (NLMPC) framework leveraging Gaussian process regression has been developed for techno-economic microgrid management with battery energy storage, demonstrating significant cost savings and voltage stability improvements under renewable uncertainty [30]. More recently, a robust data-driven NLMPC strategy has been proposed for real-time microgrid management under uncertainties and false data injection attacks [31]. Although these works focus on microgrid-level coordination, their data-driven modeling and robust optimization principles are highly relevant to mechanical power dynamics control in individual wind turbines. Furthermore, a learning-driven Model Predictive Control approach has been applied to multi-objective energy management in distribution networks with distributed renewable generation, combining temporal-frequency feature extraction with Pareto optimization [32]. These recent advances collectively underscore the potential of MPC—particularly data-driven and learning-enhanced variants—to address the nonlinear, multi-variable coupling challenges inherent in wind turbine frequency control and fatigue mitigation. Inspired by these developments, this paper proposes a novel MPC-based strategy that coordinates de-loading factor adaptation and generator torque control to simultaneously provide primary frequency response and minimize LSS fatigue without external hardware.

To address these challenges, this study proposes an MPC strategy. Unlike traditional methods that may increase fatigue, the proposed approach focuses on optimizing fatigue load mitigation while ensuring the provision of PFR support. This is achieved by adaptively adjusting the PFR gain under a specific constraint that guarantees a minimum PFR capability, thereby offering a balanced solution for both WT mechanical integrity and grid frequency stability. The main contributions of this paper are summarized as follows:

• A novel and systematic framework for optimal de-loading factor boundary determination is established to explicitly address the trade-offs between active-power reserves, PFR capability, and LSS fatigue. This contribution uniquely investigates and maps these complex relationships, providing the essential, fatigue-aware operational envelopes.
• An MPC-based dynamic PFR strategy is developed, specifically designed to proactively mitigate PFR-induced LSS fatigue at below-rated wind speeds while rigorously guaranteeing grid frequency support. Its core novelty lies in suppressing LSS torque fluctuations (fatigue mitigation) within an adaptive PFR gain range rigorously bounded by the determined optimal operational envelopes and strict frequency stability requirements.

The engineering significance of this joint optimization approach is substantial. By simultaneously considering both frequency support and fatigue load mitigation, the proposed method offers a practical solution to prolong wind turbine operational lifespan, thereby reducing costly maintenance and downtime associated with component wear and failure. This directly contributes to a lower levelized cost of energy for wind power. Furthermore, by enabling wind turbines to robustly and effectively participate in grid frequency regulation while preserving their mechanical integrity, this work directly addresses the increasing demands from grid operators for renewable energy sources to provide essential ancillary services, ensuring enhanced grid stability and reliability in power systems with high wind penetration.

The remainder of this paper is organized as follows: Section 2 introduces the system modeling and the PFR mechanism based on de-loading control. Section 3 analyzes the boundary characteristics of de-loading impacts on WT operational states. Section 4 details the formulation of the MPC-based fatigue suppression framework. Section 5 presents the case studies and discussions. Finally, conclusions are drawn in Section 6.

2. PFR Based on the Developed MPC Scheme

The PFR method based on MPC utilizes generator torque control and de-loading control. This section focuses on presenting the WT models, generator torque control, and the developed de-loading control. These components serve as the foundation for constructing a PFR method using MPC.

2.1. The Dynamics Model of the WTs

The rotor torque, tower thrust force, and shaft torque can be expressed as Equations (1)–(3).

$$\begin{array}{c}\hfill T_{\mathrm{a}}={\displaystyle \frac{0.5\rho \pi R^2{v}^3{C}_{\mathrm{P}}}{{\omega }_{\mathrm{r}}}},\end{array}$$

(1)

$$\begin{array}{c}\hfill T_{\mathrm{s}}=\left({\displaystyle \frac{K}{s}}+B\right)\left({\omega }_{\mathrm{r}}-{\displaystyle \frac{{\omega }_{\mathrm{g}}}{{\eta }_{\mathrm{g}}}}\right),\end{array}$$

(2)

$$\begin{array}{c}\hfill F_{\mathrm{t}}=0.5\rho \pi R^2{v}^2{C}_{\mathrm{t}},\end{array}$$

(3)

where T_a is the rotor torque (Nm), F_t is the tower thrust force (N), T_s is the low-speed shaft (LSS) torque (Nm), $\rho$ is the air density ($\mathrm{kg}/$$\mathrm{m}$³), $\mathit{v}$ is the wind speed (m/s), $\mathit{R}$ is the length of the blade (m), $\omega$_r is the rotor speed (rad/s), C_p and C_t are the power and thrust coefficient, and B_Ts is the viscous friction constant (Nm·s/rad). K_Ts is the spring constant (Nm/rad). The tower bending moment and tower thrust exhibit a linear relationship, allowing the tower thrust to be utilized as an indicator for assessing the tower’s load.

The dual-mass LSS model can be shown in Equation (4)

$$\left[\begin{array}{c}{\dot{\omega }}_{\mathrm{r}}\\ {\dot{\omega }}_{\mathrm{g}}\end{array}\right]=\left[\begin{array}{ccc}{\displaystyle \frac{1}{J_{\mathrm{r}}}}& {\displaystyle -\frac{1}{J_{\mathrm{r}}}}& 0\\ 0& {\displaystyle \frac{1}{J_{\mathrm{g}}{\eta }_{\mathrm{g}}}}& {\displaystyle -\frac{1}{J_{\mathrm{g}}}}\end{array}\right]\left[\begin{array}{c}{T}_{\mathrm{a}}\\ T_{\mathrm{s}}\\ T_{\mathrm{g}}\end{array}\right],$$

(4)

where $\omega$_g is the generator speed (rad/s), $\eta$_g is the gear box ratio, J_g is the generator mass ($\mathrm{kg}·$$\mathrm{m}$²), and J_r is the rotor mass ($\mathrm{kg}·$$\mathrm{m}$²).

In addition, the LSS torque based on a single-mass model can be represented by Equation (5) [26].

$$\begin{array}{c}\hfill T_{\mathrm{s}}={\displaystyle \frac{{\eta }_{\mathrm{g}}^2{J}_{\mathrm{g}}}{J_{\mathrm{t}}}}{T}_{\mathrm{a}}+{\displaystyle \frac{{\eta }_{\mathrm{g}}{J}_{\mathrm{r}}}{J_{\mathrm{t}}}}{T}_{\mathrm{g}},\end{array}$$

(5)

2.2. The Torque Control and De-Loading Control of the WTs

To mitigate torsional vibration in the LSS, a common practice is to apply a low-pass filter (LPF) to the feedback signal, typically the generator speed, in order to eliminate high-frequency components. The models for the LPF and conventional torque control (CTC) are illustrated by Equations (6) and (7).

$${\omega }_{\mathrm{f}}={\displaystyle \frac{1}{1+{\tau }_{\mathrm{f}}s}}{\omega }_{\mathrm{g}},$$

(6)

$$T_{\mathrm{g}\_\mathrm{CTC}}=k_{\mathrm{opt}}{\omega }_{\mathrm{f}}^2,$$

(7)

where k_opt is the optimal gain of torque control, $\omega$_f is the filtered speed (rad/s) of the generator speed through the LPF, and T_{g_CTC} is the generator torque reference (Nm).

The de-loading control can adopt the model shown in Equation (8).

$$T_{\mathrm{g}\_\mathrm{DE}}=(1-\gamma )k_{\mathrm{opt}}{\omega }_{\mathrm{f}}^2,$$

(8)

where $\gamma$ is the de-loading factor.

2.3. PFR of WTs Based on Dynamic Gain

The PFR scheme incorporating dynamic gain can be expressed by Equation (9). The control block diagram is shown in Figure 1. This diagram illustrates the interconnection of the reference speed, LPF, and the dynamic gain control block, resulting in the generation of the generator torque control signal.

$$T_{\mathrm{g}\_\mathrm{PFR}}=(1-\gamma )k_{\mathrm{opt}}{\omega }_{\mathrm{f}}^2-k_{\mathrm{PFR}}\Delta \mathit{f},$$

(9)

where $\Delta f$ is the frequency deviation (pu), and k_PFR is the gain of PFR (pu). Here, the PFR action follows a standard linear droop characteristic; i.e., the incremental generator torque is proportional to the frequency deviation: $\Delta T_{g\_\mathrm{PFR}}=-k_{\mathrm{PFR}}\Delta f$.

In summary, increasing the $\gamma$ contributes to reducing the average LSS torque, which will be explained in Section 3. Furthermore, the dynamic PFR gain based on MPC enables adjustment of the generator torque to minimize the fluctuation of the LSS torque, which will be detailed in Section 4. By effectively reducing both the average and fluctuation of the LSS torque, the LSS fatigue load can be significantly mitigated. These findings highlight the potential of the proposed approach to enhance the performance and reliability of WTs.

3. The Impact of De-Loading Control on the Operation State of WTs

Due to the varying effects of the $\gamma$ and PFR gain on the LSS torque at different wind speeds, it is imperative to analyze how these factors influence the operating conditions of WTs at different wind speeds. By examining the impact of the parameter $\gamma$ and PFR gain on WT operations at different wind speeds, it becomes possible to determine the optimal values for these factors. These optimal values can then be used to design controller parameters that specifically target the reduction in fatigue loads on the LSS. The analysis results presented below are all based on the modeling of the wind turbine system described earlier and simulations conducted using the FAST (Fatigue, Aerodynamic, Structures, and Turbulence) Code.

3.1. Effect of De-Loading Factor on the Rotor Speed, Generator Torque, Power Coefficient, and Thrust Coefficient

The $\gamma$ has a direct influence on the generator torque, and it can be readily inferred from Equation (9) that an increase in $\gamma$ leads to a decrease in the torque of the generator. According to the power tracking method of a WT system, a decrease in generator torque results in an increase in generator speed. Therefore, the variation in the generator torque and speed caused by the increase in $\gamma$ is visually depicted in Figure 2a. An increase in generator speed leads to a higher blade tip speed ratio. Consequently, the operating range of the blade tip speed ratio, as shown in Figure 2b, shifts towards higher values. It is important to note that the blade tip speed ratio of a WT operating under actual turbulent wind conditions is not constant but rather varies within a certain range. The gray shaded area represents the power coefficient and thrust coefficient corresponding to the tip speed ratio of wind turbines operating under actual turbulent wind speeds. The blade tip speed ratio based on the MPPT method, below the rated wind speed, is illustrated in Figure A1, as shown in Appendix B.

The rightward shift in the blade tip speed ratio depicted in Figure 2b has two effects: an increase in the C_t and a decrease in the C_p. The values 0.933 and 0.823 respectively represent the mean thrust coefficients of the wind turbine operating under turbulent wind conditions. From the results, it is observed that the tower thrust increases by around 10%. The values 0.485 and 0.464 indicate the variations in power coefficients. The decrease in C_p implies a reduction in output power, which means that the average output power only decreases by approximately 4% when the $\gamma$ increases from 0 to 0.5. Specific results regarding this decrease in power will be presented in the subsequent section. It is important to note that for WTs participating in PFR, the primary objective is to minimize system frequency deviation rather than maximizing wind energy capture. On the other hand, the increase in C_t results in an elevated tower thrust, which may potentially exacerbate the tower’s fatigue load. However, it is crucial to consider that tower thrust is primarily influenced by wind speed. Below the rated wind speed, tower thrust is considerably smaller compared to values above the rated wind speed.

3.2. Effect of Wind Speed and PFR Gain on LSS Torque, Generator Torque, and Thrust Force

In addition to the influence of parameter $\gamma$, the operational dynamics of a WT system are also affected by the PFR gain and wind speed.

Figure 3a presents the findings regarding the influence of PFR gain on the variance in LSS torque, generator torque, and tower thrust at specific wind speeds and $\gamma$ values. The figure reveals that as the PFR gain ($k_{PFR}$) increases, the variance in LSS torque experiences a significant increase. This is primarily because PFR directly modulates the generator torque to provide frequency support. A higher $k_{PFR}$ implies larger and more frequent adjustments to the generator torque in response to frequency deviations, which directly translate into amplified torsional stresses and oscillations within the LSS, a key component of the drivetrain.

On the other hand, the variance in tower thrust shows minimal changes. This contrasting behavior can be attributed to the different physical origins and response pathways of these two loads. Tower thrust is predominantly an aerodynamic force acting on the rotor, which is primarily influenced by the incoming wind speed, rotor speed, and blade pitch angle. While PFR-induced changes in generator torque do exert an influence on rotor speed, the rotor possesses significant rotational inertia. Consequently, the impact of these rapid generator torque adjustments on rotor speed, and subsequently on aerodynamic forces and tower thrust, is considerably dampened and slower compared to their direct and immediate effect on the LSS torque. The primary driver of tower thrust variance remains the turbulent wind field, with the secondary effects from PFR being relatively minor and smoothed out by rotor inertia. This dichotomy suggests that an increase in the PFR gain amplifies the fatigue load on the LSS due to direct and rapid torque coupling, while having little impact on the fatigue load experienced by the tower, which is governed by more inertial and aerodynamic forces.

Figure 3b shows the impact of wind speed on the variance in LSS torque, generator torque, and tower thrust, considering specific values of $\gamma$ and PFR gain. The following observations can be made from the graph: As the wind speed increases, the influence on the LSS fatigue load decreases, while the fatigue load of the tower becomes relatively negligible compared to the fatigue load of the LSS. It is noteworthy that when the wind speed is below 5 m/s, the variance in LSS torque and generator torque experiences a significant increase. This is attributed to the fixed PFR gain causing a fixed change in generator torque, which in turn leads to an approximate fixed change in LSS torque. At lower wind speeds, the average values of LSS torque and generator torque become smaller, while the magnitude of changes caused by PFR remains constant.

3.3. Effect of De-Loading Factor and PFR Gain on LSS Torque and System Frequency Deviation

The PFR of WTs can be categorized into two simple situations based on the sign of the frequency deviation. When the frequency deviation is less than zero (negative), the WT needs to increase its output to assist in frequency recovery. Conversely, when the frequency deviation is greater than zero (positive), the WT should decrease its output to help restore the frequency. These two cases present different scenarios for the frequency response of the WT.

In situations where it is necessary to increase the output of the WT to provide frequency support, a higher value of $\gamma$ becomes essential when the wind speed is near the rated wind speed. This ensures that the WT has an adequate power reserve to fulfill its PFR role. If the value of $\gamma$ is too small, it can diminish the PFR capability of the WT. Conversely, when there is a need to reduce the output of the WT to assist in frequency restoration, caution should be exercised if the wind speed is close to the cut-in wind speed. Excessive $\gamma$ values should be avoided as they may lead to insufficient power reduction in the WT. An excessively high value of $\gamma$ can result in excessive power reserves in the WT and cause a decrease in overall power output. In extreme cases, an inappropriate value of $\gamma$ may even lead to the shutdown of the WT.

Figure 4 presents the results of the analysis, depicting the maximum frequency deviation of the system and the fatigue load of the LSS (within 20 s after a system disturbance) under different values of $\gamma$, PFR gain, and wind speed as the system load increases. The following observations can be made from Figure 4: (a) In Figure 4a, it is evident that increasing the PFR gain, wind speed, or $\gamma$ can effectively reduce the maximum frequency deviation of the system. These factors contribute to improved system stability and regulation. (b) Figure 4b demonstrates that the fatigue load of the LSS increases with higher values of PFR gain and wind speed. In addition, the LSS torque based on a simplified single-mass model, useful for conceptually understanding the average torque transfer, can be represented by Equation (5) [26]. As evident from Equation (5), it can be observed that a decrease in generator torque leads to a reduction in the torque of the LSS. This suggests that an appropriate increase in $\gamma$ can lead to a more favorable operating condition for the LSS, mitigating fatigue-related issues. In summary, within a certain range, increasing $\gamma$ proves beneficial in reducing system frequency deviation and alleviating the fatigue load on the LSS.

A further analysis was conducted on the maximum frequency deviation of the system when the system load experiences a sudden increase or decrease of 10%. Figure 5a represents the scenario when the output of the WTs needs to be increased. When the wind speed is high and the value of $\gamma$ is small, the system experiences an increased frequency deviation due to the inadequate PFR capability of the WTs. The boundaries of $\gamma$ vary depending on the PFR gains and wind speeds. Similarly, Figure 5b illustrates the maximum frequency deviation of the system when a reduction in WT output is required. When the wind speed is low, both the insufficient frequency regulation ability of the WTs and the impact of $\gamma$ and PFR gain contribute significantly to the frequency deviation. Consequently, boundaries for $\gamma$ can be derived based on different PFR gains and wind speeds.

Based on the aforementioned analysis, Figure 6 presents the boundaries of $\gamma$ under different wind speeds and PFR gains. It is crucial to highlight that when the wind speed is below 6 m/s, WT participation in PFR is not recommended. This specific 6 m/s value is derived from the characteristics of the NREL 5MW reference wind turbine used in this study, whose cut-in speed is 3 m/s. This threshold corresponds to a critical “power elbow” region where the turbine transitions from its cut-in to a more efficient MPPT control regime. The recommendation is primarily based on three interconnected factors, which apply generally to variable-speed wind turbines below a certain wind speed:

• Limited PFR Capability and Economic Inefficiency: Below this threshold, the wind turbine’s power output is significantly low. Consequently, the available active-power reserve for PFR, obtained through de-loading, is minimal. This renders its contribution to grid frequency support largely ineffective and economically disadvantageous due to lost energy capture, as the primary objective in this region is typically to maximize energy yield.
• Disproportionate Increase in Mechanical Loads: Our analysis in Figure 6 demonstrates a nonlinear and substantial increase in LSS fatigue load when attempting PFR below 6 m/s. This is because, at low power levels, the relative magnitude of generator torque adjustments required for PFR is higher, which tends to exacerbate drivetrain torsional oscillations and significantly increase fatigue damage.
• Control Effectiveness and Stability Considerations: The highly nonlinear aerodynamic characteristics and reduced operational inertia at very low wind speeds make effective and stable PFR control challenging. Aggressive control actions might be required, which are detrimental to mechanical integrity.

While the exact numerical value of this low wind speed cutoff (e.g., 6 m/s) is turbine-specific and would necessitate a similar detailed analysis for different wind turbine designs, the underlying principle—that such a threshold exists below which PFR participation is inefficient, uneconomical, and detrimental to mechanical integrity—is broadly applicable to most variable-speed wind turbines. This provides valuable guidance for grid operators and wind farm owners in defining appropriate operational envelopes for frequency regulation.

These conclusions generally apply, and similar findings can be extrapolated when analyzing other wind turbine systems with comparable capacities, like doubly fed or permanent magnet direct-drive wind turbines. Although the boundary of $\gamma$ has been established, it is crucial to recognize that operating the control system at this boundary could lead to instability in WT operation. Therefore, opting for a slightly higher fixed $\gamma$, around 0.3, is more suitable for practical testing in this study. This provides a stability margin and ensures reliable operation of the WT control system.

4. The MPC-Based Control Scheme

Previous studies have shown that an increase in $\gamma$ correlates with a decrease in the average LSS torque. Yet, the fatigue load on the LSS is not solely dictated by the average torque. Significant variations in LSS torque contribute significantly. To mitigate these fluctuations, implement an MPC scheme for dynamically adjusting the generator torque. It includes real-time adjustment of the frequency regulation gain, regulating the generator torque to alleviate LSS torque fluctuations. MPC provides numerous advantages over traditional control methods in wind turbine control. Advantages encompass consideration of state information (rotational speed, torque), constraint handling, optimization, prediction, and robustness. The development of an MPC framework typically involves the following steps:

• Establishing a state-space model: Develop a mathematical model representing the system’s dynamic behavior for determining current state and control variables.
• Establishing objective functions and constraints;
• Optimization processing: based on u(k), u(k+1), ⋯, u(k+j) for optimization processing; apply only u(k) as the control input to the system;
• Applying control input to the system;
• Iteration: repeat the above steps at the next time step.

4.1. Operating Point Definition and State Feedback Framework

To develop the state-space model, it is crucial to clarify the operating point assumptions and the control architecture employed in this study. Unlike conventional control strategies that apply a fixed linear model derived at a single nominal operating point, the proposed MPC framework adopts a real-time state feedback architecture. This design choice is justified both theoretically and practically:

1. Operating Point Considerations: The wind turbine operates across a wide range of wind speeds (typically 3–11.4 m/s in this study), with corresponding variations in rotor speed, generator torque, and system frequency deviations. Rather than linearizing the model at a single fixed operating point (which would severely limit applicability), our approach recognizes that the operating point itself evolves dynamically. At each control time step k (with control period $\Delta t_s=0.1$ s), the MPC controller updates its prediction based on the instantaneously measured state vector $\mathit{x}\left(k\right)$, which includes current rotor speed, generator speed, and frequency deviation.

2. Real-Time State Feedback: The state-space model in Equation (10) represents the local linearization of the nonlinear wind turbine dynamics around the current operating point at time step k. The matrices A, B, and C are themselves functions of the current operating condition (specifically, dependent on the current rotor speed ${\omega }_r\left(k\right)$ and PFR gain $k_{PFR}\left(k\right)$). In practice, these matrices are updated at each control step based on the measured state, creating a time-varying linear model within a moving prediction horizon.

Mathematically, this can be expressed as Equation (10).

$$\dot{\mathit{x}}=\mathit{A}\left(k\right)\mathit{x}+\mathit{B}\left(k\right)\mathit{u}+\mathit{C}\left(k\right),$$

(10)

with

$$\mathit{x}={\left[\Delta {\omega }_{\mathrm{r}},\Delta {\omega }_{\mathrm{g}},{\omega }_{\mathrm{r}},{\omega }_{\mathrm{g}},\Delta {\omega }_{\mathrm{f}},\Delta \left(\Delta f\right)\right]}^{\mathrm{T}},$$

$$\mathit{u}=\Delta k_{\mathrm{PFR}},$$

where the matrices are implicitly functions of the current state and operating condition. This approach is fundamentally sound because:

• The MPC prediction horizon is relatively short, over which the operating point does not change drastically.
• Real-time state feedback enables the controller to automatically adapt to gradual changes in wind speed and system conditions, compensating for model nonlinearity and parametric uncertainty.
• The discretized model (Equation (19)) captures the essential dynamics at each control step, allowing MPC to make decisions based on current, relevant information.

3. Practical Implementation and Validation: In the simulation, states are measured in real time from the FAST model (including rotor speed ${\omega }_r$, generator speed ${\omega }_g$, and frequency deviation $\Delta f$). This reflects the practical operation of modern wind turbines, which are equipped with high-resolution sensors and communication systems (e.g., SCADA). The robustness analysis in Section 5.4 demonstrates that the proposed MPC strategy maintains effectiveness despite parameter uncertainties and variations in operating conditions, thereby validating the soundness of this real-time state feedback architecture.

4.2. The State-Space Representation and Its Discretization of the WTs Under PFR

State variables in the control system primarily include speed and system frequency deviation, with the primary control variable being the frequency regulation gain. Consequently, Equation (10) represents the state-space equation.

The process of deriving the state-space equations from the dynamic equations of wind turbines is mainly as follows: The small disturbance of the shaft torque as shown in Equation (2) can be expressed as

$$\Delta T_s=B_{\mathrm{Ts}}\left(\Delta {\omega }_r-{\displaystyle \frac{\Delta {\omega }_g}{{\eta }_g}}\right)+K_{\mathrm{Ts}}\Delta t_s\left({\omega }_r-{\displaystyle \frac{{\omega }_g}{{\eta }_g}}\right)$$

(11)

A first-order Taylor expansion of Equation (1) near the steady-state operating point yields

$$\Delta T_a={\displaystyle \frac{\partial T_a}{\partial {\omega }_r}}\Delta {\omega }_r+{\displaystyle \frac{\partial T_a}{\partial v}}\Delta v$$

(12)

Linearize Equation (9) with small perturbations:

$$\Delta T_g=2(1-\gamma )k_{\mathrm{opt}}{\omega }_f\Delta {\omega }_f-k_{\mathrm{PFR}}\Delta (\Delta f)-\Delta k_{\mathrm{PFR}}\Delta f$$

(13)

Define the control input as

$$u=\Delta k_{\mathrm{PFR}}$$

(14)

Then the expression (13) can be written as

$$\Delta T_g=2(1-\gamma )k_{\mathrm{opt}}{\omega }_f\Delta {\omega }_f-k_{\mathrm{PFR}}\Delta (\Delta f)-u\Delta f$$

(15)

Substituting Equations (11), (12), and (15) into the wind turbine dynamics equations, we can obtain the dynamic equations for small disturbances on the wind turbine side:

$$\begin{array}{cc}\hfill \Delta {\dot{\omega }}_r=& {\displaystyle \frac{1}{J_r}}\left(\Delta T_a-\Delta T_s\right)\hfill \\ \hfill =& {\displaystyle \frac{1}{J_r}}\left({\displaystyle \frac{\partial T_a}{\partial {\omega }_r}}-B_{\mathrm{Ts}}\right)\Delta {\omega }_r+{\displaystyle \frac{B_{\mathrm{Ts}}}{J_r{\eta }_g}}\Delta {\omega }_g-{\displaystyle \frac{K_{\mathrm{Ts}}\Delta t_s}{J_r}}{\omega }_r+{\displaystyle \frac{K_{\mathrm{Ts}}\Delta t_s}{J_r{\eta }_g}}{\omega }_g+{\displaystyle \frac{1}{J_r}}{\displaystyle \frac{\partial T_a}{\partial v}}\Delta v\hfill \end{array}$$

(16)

Similarly, the dynamic equations on the generator side are

$$\begin{array}{cc}\hfill \Delta {\dot{\omega }}_g=& {\displaystyle \frac{1}{J_g}}\left({\displaystyle \frac{\Delta T_s}{{\eta }_g}}-\Delta T_g\right)\hfill \\ \hfill =& {\displaystyle \frac{B_{\mathrm{Ts}}}{J_g{\eta }_g}}\Delta {\omega }_r-{\displaystyle \frac{B_{\mathrm{Ts}}}{J_g{\eta }_g^2}}\Delta {\omega }_g+{\displaystyle \frac{K_{\mathrm{Ts}}\Delta t_s}{J_g{\eta }_g}}{\omega }_r-{\displaystyle \frac{K_{\mathrm{Ts}}\Delta t_s}{J_g{\eta }_g^2}}{\omega }_g-{\displaystyle \frac{2k_{\mathrm{opt}}(1-\gamma ){\omega }_f}{J_g}}\Delta {\omega }_f\hfill \\ \hfill & -{\displaystyle \frac{k_{\mathrm{PFR}}}{J_g}}\Delta (\Delta f)+{\displaystyle \frac{\Delta f}{J_g}}u\hfill \end{array}$$

(17)

For ${\omega }_f$, its small perturbation form is

$$\Delta {\dot{\omega }}_f={\displaystyle \frac{1}{{\tau }_f}}\Delta {\omega }_g-{\displaystyle \frac{1}{{\tau }_f}}\Delta {\omega }_f$$

(18)

Based on the above derivation process, matrices A, B, C in Equation (10) can be represented as follows:

$$\mathit{A}=\left[\begin{array}{cccccc}{\displaystyle \frac{1}{J_{\mathrm{r}}}}\left({\displaystyle \frac{\partial T_{\mathrm{a}}}{\partial {\omega }_{\mathrm{r}}}}-B_{\mathrm{Ts}}\right)& {\displaystyle \frac{B_{\mathrm{Ts}}}{J_{\mathrm{r}}{\eta }_{\mathrm{g}}}}& -{\displaystyle \frac{K_{\mathrm{Ts}}\Delta t_{\mathrm{s}}}{J_{\mathrm{r}}}}& {\displaystyle \frac{K_{\mathrm{Ts}}\Delta t_{\mathrm{s}}}{J_{\mathrm{r}}{\eta }_{\mathrm{g}}}}& 0& 0\\ {\displaystyle \frac{B_{\mathrm{Ts}}}{J_{\mathrm{g}}{\eta }_{\mathrm{g}}}}& -{\displaystyle \frac{B_{\mathrm{Ts}}}{J_{\mathrm{g}}{\eta }_{\mathrm{g}}^2}}& {\displaystyle \frac{K_{\mathrm{Ts}}\Delta t_{\mathrm{s}}}{J_{\mathrm{g}}{\eta }_{\mathrm{g}}}}& -{\displaystyle \frac{K_{\mathrm{Ts}}\Delta t_{\mathrm{s}}}{J_{\mathrm{g}}{\eta }_{\mathrm{g}}^2}}& -{\displaystyle \frac{2k_{\mathrm{opt}}\left(1-\gamma \right){\omega }_{\mathrm{f}}}{J_{\mathrm{g}}}}& -{\displaystyle \frac{k_{\mathrm{PFR}}}{J_{\mathrm{g}}}}\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& {\displaystyle \frac{1}{{\tau }_{\mathrm{f}}}}& 0& 0& -{\displaystyle \frac{1}{{\tau }_{\mathrm{f}}}}& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right],$$

$$\mathit{B}={\left[\begin{array}{cccccc}0& {\displaystyle \frac{\Delta f}{J_{\mathrm{g}}}}& 0& 0& 0& 0\end{array}\right]}^{\mathrm{T}},$$

$$\mathit{C}={\left[\begin{array}{ccccccc}{\displaystyle \frac{1}{J_{\mathrm{r}}}}{\displaystyle \frac{\partial T_{\mathrm{a}}}{\partial v}}\Delta v& 0& 0& 0& 0& 0& 0\end{array}\right]}^{\mathrm{T}}.$$

After discretization, Equation (10) can be approximated using Equation (19), which determines the operating state of the WT at the next time step.

$$\mathit{x}\left(k+1\right)=\mathit{Ox}\left(k\right)+\mathit{P}u\left(k\right)+\mathit{Q}$$

(19)

where

$$\mathit{O}=e^{\mathit{A}\Delta t_{\mathrm{s}}},\mathit{P}={\int }_0^{\Delta t_{\mathrm{s}}}{e}^{\mathit{A}t}\mathit{B}dt,\mathit{Q}={\int }_0^{\Delta t_{\mathrm{s}}}{e}^{\mathit{A}t}\mathit{C}dt.$$

4.3. The Analytical Equations of LSS Torque and Generator Torque

To establish the relationship between the fluctuation of the LSS torque and the control input, as well as the relationship between the starting motor torque and the control input, Equations (20) and (21) are utilized.

$$\Delta T_{\mathrm{s}}\left(k+1\right)=a_{\mathrm{Ts}}u+b_{\mathrm{Ts}},$$

(20)

with

$$a_{\mathrm{Ts}}={\mathit{H}}_{\mathrm{Ts}}\mathit{P},b_{\mathrm{Ts}}={\mathit{H}}_{\mathrm{Ts}}\mathit{Ox}+{\mathit{H}}_{\mathrm{Ts}}\mathit{Q},$$

$${\mathit{H}}_{\mathrm{Ts}}=\left[\begin{array}{cccccc}{B}_{\mathrm{Ts}}& -{\displaystyle \frac{B_{\mathrm{Ts}}}{{\eta }_{\mathrm{g}}}}& K_{\mathrm{Ts}}\Delta t_{\mathrm{s}}& -{\displaystyle \frac{K_{\mathrm{Ts}}\Delta t_{\mathrm{s}}}{{\eta }_{\mathrm{g}}}}& 0& 0\end{array}\right],$$

$$\Delta T_{\mathrm{g}}\left(k+1\right)=a_{\mathrm{Tg}}u+b_{\mathrm{Tg}},$$

(21)

with

$$a_{\mathrm{Tg}}={\mathit{H}}_{\mathrm{Tg}}\mathit{P},b_{\mathrm{Tg}}={\mathit{H}}_{\mathrm{Tg}}\mathit{Ox}+{\mathit{H}}_{\mathrm{Tg}}\mathit{Q}+{\mathit{J}}_{\mathrm{Tg}},$$

$${\mathit{H}}_{\mathrm{Tg}}=\left[\begin{array}{cccccc}0& 0& 0& 0& 2k_{\mathrm{opt}}\left(1-\lambda \right){\omega }_{\mathrm{f}}& -k_{\mathrm{PFR}}\end{array}\right],$$

$${\mathit{J}}_{\mathrm{Tg}}=-\Delta f.$$

4.4. The Control Objective and Constraints of PFR Gain

The MPC-based control model is presented in Equation (22).

$$\left\{\begin{array}{c}\mathit{X}(k+1)={\mathit{O}}_x\mathit{x}\left(k\right)+{\mathit{P}}_x\mathit{U}\left(k\right)+\mathit{Q}\hfill \\ \mathit{Y}(k+1)={\mathit{H}}_y\mathit{X}(k+1)\hfill \end{array}\right.,$$

(22)

with

$$\begin{array}{cc}\hfill \mathit{X}(k+1)& ={\left[{\mathit{x}}^T(k+1|k),{\mathit{x}}^T(k+2|k),\dots ,{\mathit{x}}^T(k+N_{\mathrm{p}}|k)\right]}^T\hfill \\ \hfill \mathit{U}(k+1)& ={\left[{\mathit{u}}^T\left(k\right|k),{\mathit{u}}^T(k+1|k),\dots ,{\mathit{u}}^T(k+N_{\mathrm{p}}-1|k)\right]}^T\hfill \\ \hfill \mathit{Y}(k+1)& ={\left[{\mathit{y}}^T(k+1|k),{\mathit{y}}^T(k+2|k),\dots ,{\mathit{y}}^T(k+N_{\mathrm{p}}|k)\right]}^T\hfill \end{array}$$

$N_{\mathrm{p}}$ denotes the prediction horizon.

Cost Function Formulation: The MPC optimization problem is formulated as a finite-horizon, open-loop optimal control problem. The cost function is defined over the prediction horizon $N_p$ as

$$J^{*}=\underset{\mathit{U}}{\min }\left\{{\displaystyle \sum _{i=1}^{N_p}}\left[{\parallel \mathit{y}(k+i|k)\parallel }_{\mathit{Q}}^2+{\parallel \mathit{u}(k+i-1|k)\parallel }_{\mathit{R}}^2\right]\right\},$$

(23)

where ${\parallel ·\parallel }_{\mathit{M}}^2={(·)}^T\mathit{M}(·)$ denotes the weighted norm with matrix $\mathit{M}$. The previous section established the relationship between LSS torque fluctuations and control variables. Reducing the LSS torque fluctuations in each control cycle during operation can effectively mitigate the fatigue load on the LSS. Therefore, the control objective is formulated to minimize both the control target (LSS torque fluctuations) and the variations in the control variables. The control objective is expressed by Equation (24).

$$\min J={\mathit{Y}}^T(k+1)\mathit{Q}\mathit{Y}(k+1)+{\mathit{U}}^T\left(k\right)\mathit{R}\mathit{U}\left(k\right),$$

(24)

with

$$\Delta u=k_{\mathrm{PFR}}\left(k\right)-k_{\mathrm{PFR}}(k-1).$$

where

• $\mathit{Y}(k+1)\in {\mathbb{R}}^{N_p\times n_y}$ is the predicted output trajectory over the prediction horizon, containing the LSS torque fluctuations $\Delta T_s(k+i|k)$ for $i=1,\dots ,N_p$.
• $\mathit{U}\left(k\right)\in {\mathbb{R}}^{N_p\times n_u}$ is the control input trajectory, containing the PFR gain adjustments $\Delta k_{PFR}(k+i|k)$ for $i=0,\dots ,N_p-1$.
• $\mathit{Q}$ and $\mathit{R}$ are positive semi-definite weighting matrices that penalize deviations in output (fatigue) and control effort, respectively.

Objective Interpretation: The first term ${\mathit{Y}}^T\mathit{Q}\mathit{Y}$ penalizes LSS torque fluctuations over the prediction horizon, directly minimizing fatigue load. The second term ${\mathit{U}}^T\mathit{R}\mathit{U}$ penalizes control input variations, ensuring smooth and physically feasible PFR gain adjustments, which is critical for maintaining frequency regulation stability and avoiding excessive control switching.

The weighting matrices $\mathit{Q}$ and $\mathit{R}$ are diagonal matrices that balance the relative importance of fatigue mitigation versus control smoothness. Their structures and selections are detailed as follows:

Given that the primary objective is fatigue mitigation, the output weighting matrix is defined as

$$\mathit{Q}=\mathrm{diag}\left(\underset{N_p\mathrm{times}}{\underbrace{q,q,\dots ,q}}\right)\in {\mathbb{R}}^{N_p\times N_p},$$

(25)

where $q>0$ is a scalar weighting coefficient that equally penalizes LSS torque fluctuations at all future time steps within the prediction horizon. In this study, $q=1.0$ is selected to provide equal emphasis on all predicted states, reflecting the importance of consistent fatigue reduction throughout the prediction horizon. This uniform weighting ensures that the controller prioritizes fatigue mitigation uniformly across all time steps rather than myopically focusing on immediate future states.

The control input weighting matrix is defined as

$$\mathit{R}=\mathrm{diag}\left(\underset{N_p\mathrm{times}}{\underbrace{r,r,\dots ,r}}\right)\in {\mathbb{R}}^{N_p\times N_p},$$

(26)

where $r>0$ is a scalar weighting coefficient that penalizes the magnitude of control input variations. In this study, $r=100$ is selected. This value is relatively large compared to g, mainly because the output is larger than the control input. However, the control strategy still needs to prioritize fatigue mitigation while ensuring the smoothness and stability of the PFR gain adjustment. This approach avoids overly conservative control actions and allows for more aggressive fatigue suppression measures when necessary.

The prediction horizon $N_p$ defines the number of time steps over which the MPC controller predicts future outputs and optimizes the control actions. A longer prediction horizon allows the controller to anticipate future disturbances and system dynamics, enabling more proactive control decisions.

In this study, $N_p=10$ sampling periods are employed. This choice is based on the following considerations:

• Time-Scale Correspondence: With a sampling time $\Delta t_s=0.0125$ s, $N_p=10$ corresponds to a physical prediction window of $10\times 0.0125=0.125$ s, which is sufficiently long to capture the dominant dynamics of the LSS system (typical mechanical time constants are in the range of 0.1–0.3 s) and the frequency response of the power grid (PFR typically responds within 1–2 s, so capturing a 0.125 s window provides meaningful predictive capability).
• Computational Trade-Off: A horizon of $N_p=10$ balances predictive capability with computational burden. Longer horizons would significantly increase computational complexity, making real-time implementation on industrial controllers challenging. Shorter horizons would limit the controller’s ability to anticipate system behavior.

While the prediction horizon extends over $N_p$ steps, the control horizon $N_c$ defines the number of steps over which the controller actively optimizes control inputs. For simplicity and to avoid rapid switching in the PFR gain, we employ $N_c=N_p$, meaning control actions are optimized throughout the entire prediction horizon.

The constraints in the dynamic PFR method consist of two parts:

• The upper and lower bounds of the PFR gain, which are constrained by Equation (27).

  $$k_{\mathrm{PFR}}\in \left[b_{\mathrm{PFR}\_\mathrm{lb}},b_{\mathrm{PFR}\_\mathrm{ub}}\right],$$

  (27)

  where $b_{\mathrm{PFR}\_\mathrm{lb}}$ and $b_{\mathrm{PFR}\_\mathrm{ub}}$ are the lower and upper bounds of the PFR gain.
• The fluctuation of the generator torque for each sampling period should remain within a certain range, as represented by Equation (28).

  $$\Delta T_{\mathrm{g}}\left(k+1\right)\in \left[b_{\mathrm{Tg}\_\mathrm{lb}},b_{\mathrm{Tg}\_\mathrm{ub}}\right],$$

  (28)

  where $b_{\mathrm{Tg}\_\mathrm{lb}}$ and $b_{\mathrm{Tg}\_\mathrm{ub}}$ are the lower and upper bounds of the fluctuation of the generator torque for each sampling period.

The process of solving constraints is shown in Algorithm 1.

Algorithm 1 The process of solving constraints

Input: $b_{\mathrm{Tg}\_\mathrm{lb}}$, $b_{\mathrm{Tg}\_\mathrm{ub}}$, $b_{\mathrm{PFR}\_\mathrm{lb}}$, $b_{\mathrm{PFR}\_\mathrm{ub}}$, $a_{\mathrm{Tg}}$, $b_{\mathrm{Tg}}$, $k_{\mathrm{PFR}0}$. Output:    1: if $a_{\mathrm{Tg}}>0$ then    2:     $k_{\mathrm{PFR}\_\mathrm{lb}}$ is used to represent the lower bound of the PFR gain and can be calculated by the following equation:    3: $$k_{\mathrm{PFR}\_\mathrm{lb}}=\min \left(b_{\mathrm{PFR}\_\mathrm{ub}},\max \left(b_{\mathrm{PFR}\_\mathrm{lb}},b_1\right)\right);$$    4: with, $$b_1={\displaystyle \frac{b_{\mathrm{Tg}\_\mathrm{lb}}-b_{\mathrm{Tg}}+a_{\mathrm{Tg}}{k}_{\mathrm{PFR}0}}{a_{\mathrm{Tg}}}}$$    5:     $k_{\mathrm{PFR}\_\mathrm{ub}}$ is used to represent the upper bound of the PFR gain and can be calculated by the following equation:    6: $$k_{\mathrm{PFR}\_\mathrm{ub}}=\max \left(k_{\mathrm{PFR}\_\mathrm{lb}},\min \left(b_{\mathrm{PFR}\_\mathrm{ub}},b_2\right)\right);$$ with $$b_2={\displaystyle \frac{b_{\mathrm{Tg}\_\mathrm{ub}}-b_{\mathrm{Tg}}+a_{\mathrm{Tg}}{k}_{\mathrm{PFR}0}}{a_{\mathrm{Tg}}}}$$    7: else    8:      $$k_{\mathrm{PFR}\_\mathrm{ub}}=\max \left(b_{\mathrm{PFR}\_\mathrm{lb}},\min \left(b_{\mathrm{PFR}\_\mathrm{ub}},b_2\right)\right);$$    9:      $$k_{\mathrm{PFR}\_\mathrm{lb}}=\min \left(k_{\mathrm{PFR}\_\mathrm{ub}},\min \left(b_{\mathrm{PFR}\_\mathrm{lb}},b_1\right)\right);$$    10: end if

In this paper, $b_{\mathrm{PFR}\_\mathrm{lb}}$ = 30(pu), $b_{\mathrm{PFR}\_\mathrm{ub}}$ = 50(pu), $b_{\mathrm{Tg}\_\mathrm{ub}}$ = $0.375\times {\mathit{t}}_{\mathrm{s}}$(pu), $b_{\mathrm{Tg}\_\mathrm{lb}}$ = $-0.375\times {\mathit{t}}_{\mathrm{s}}$(pu), where ${\mathit{t}}_{\mathrm{s}}$ is the control cycle. The bounds of $k_{\mathrm{PFR}}$ are selected according to the Chinese national grid code for frequency regulation capability of renewable energy plants, where the frequency regulation coefficient is required to be within $[10,50]$. To guarantee minimum effective frequency support while avoiding excessive actuation, we set $b_{\mathrm{PFR}\_\mathrm{lb}}=30$ and $b_{\mathrm{PFR}\_\mathrm{ub}}=50$ within the code-compliant range. The bounds on $\Delta T_{\mathrm{g}}$ are derived from actuator (converter/drive) slew-rate capability and LSS mechanical safety. These limits represent the maximum allowable generator torque fluctuation per sampling interval consistent with the actuator rate capability and protection margins. They ensure that the MPC does not command torque changes beyond the feasible fast-response envelope of the drivetrain/converter control loops.

4.4.1. Optimization Solver

The MPC problem formulated in Equation (24) with constraints (Equations (27) and (28)) is a Quadratic Programming (QP) problem of the form

$$\underset{\mathit{U}}{\min }{\displaystyle \frac{1}{2}}{\mathit{U}}^T\mathit{H}\mathit{U}+{\mathit{g}}^T\mathit{U}\mathrm{subject}\mathrm{to}{\mathit{A}}_{eq}\mathit{U}={\mathit{b}}_{eq},{\mathit{A}}_{ineq}\mathit{U}\le {\mathit{b}}_{ineq},$$

(29)

where $\mathit{H}=2({\mathit{P}}_x^T\mathit{Q}{\mathit{P}}_x+\mathit{R})$ is the Hessian matrix (positive definite, guaranteeing a unique global minimum), $\mathit{g}$ incorporates the current state and model information, and ${\mathit{A}}_{eq}$, ${\mathit{b}}_{eq}$, ${\mathit{A}}_{ineq}$, ${\mathit{b}}_{ineq}$ encode the constraints.

For this study, the MPC optimization problem is solved using the Sequential Least Squares Programming (SLSQP) algorithm, implemented as a custom solver within the MATLAB 2021a/Simulink environment. This choice is motivated by: The QP problem is convex (since $\mathit{H}$ is positive definite and constraints are linear), ensuring that any local minimum is a global minimum. SLSQP is well suited for convex QP problems. For the problem sizes in this study ($N_p=10$ steps, 1 control input, ∼50–100 inequality constraints), SLSQP typically solves within 1–3 ms on standard computing hardware (Intel i5 processor, 8GB RAM), well within the 10 ms sampling period, leaving the computational margin for other real-time tasks. SLSQP is robust to ill conditioning and provides reliable solutions even with slightly perturbed problem data. In the simulations reported in this paper, the SLSQP solver was employed, and typical solver execution time was 2–4 ms per optimization step, confirming real-time feasibility.

4.4.2. Feasibility Handling

As the control problem includes hard inequality constraints on the PFR gain ($k_{PFR}\in [b_{PFR\_lb},b_{PFR\_ub}]$) and generator torque fluctuation ($\Delta T_g\in [b_{Tg\_lb},b_{Tg\_ub}]$), the MPC should guarantee that these constraints are feasible (i.e., a solution satisfying all constraints exists) at each control step. Algorithm 1 implements a feasibility-ensuring mechanism specifically designed for this problem.

Algorithm 1 dynamically adjusts the PFR gain bounds $[k_{PFR\_lb}^{opt},k_{PFR\_ub}^{opt}]$ based on the current operating conditions (wind speed, generator torque state) to ensure that the feasible region for the optimization problem is non-empty. Specifically

• Case 1 ($a_{Tg}>0$): When increasing the PFR gain increases the generator torque, the algorithm computes lower bound $k_{PFR\_lb}^{opt}$ from the constraint $\Delta T_g\ge b_{Tg\_lb}$ and upper bound $k_{PFR\_ub}^{opt}$ from $\Delta T_g\le b_{Tg\_ub}$. These adaptive bounds are then intersected with the predefined operational bounds $[b_{PFR\_lb},b_{PFR\_ub}]$ to ensure feasibility while respecting engineering limits.
• Case 2 ($a_{Tg}\le 0$): When increasing the PFR gain decreases the generator torque, the bounds are computed similarly but with reversed logic to maintain constraint satisfaction.

The mathematical formulation in Algorithm 1 ensures that if a feasible solution exists within the original predefined bounds $[b_{PFR\_lb},b_{PFR\_ub}]$, Algorithm 1 will identify it; if not, it will identify the closest feasible solution. This prevents solver infeasibility errors and ensures continuous operation even under challenging conditions.

In this study, we implement hard constraints (Equations (27) and (28)), meaning constraint violations are strictly prohibited. This is appropriate for safety-critical constraints (e.g., PFR provision for grid stability, generator torque limits for equipment protection). The feasibility mechanism in Algorithm 1 ensures that hard constraints are always satisfied by the closed-loop system. In the rare event that Algorithm 1 determines that even the optimal constrained region is empty (i.e., no feasible solution exists), a fallback strategy is employed:

$$u_{\mathrm{fallback}}=k_{PFR,safe}-k_{PFR0},$$

(30)

where $k_{PFR,safe}$ is a predefined safe PFR gain (e.g., the original nominal value) that is guaranteed to satisfy primary grid code requirements. This ensures that the wind turbine maintains safe and grid-compliant operation even in extreme or pathological scenarios. In the simulations presented in this paper, this fallback was never triggered, indicating that the constraint bounds are appropriately designed for the simulated wind speed and disturbance ranges.

4.4.3. Stability Guarantees

In the proposed MPC framework, stability-related properties are ensured through (i) hard constraint enforcement and (ii) a standard receding-horizon optimal control mechanism.

First, recursive feasibility and constraint satisfaction are achieved by the feasibility-handling strategy (Algorithm 1), which adaptively tightens the admissible PFR gain bounds to guarantee that the MPC optimization remains feasible at each sampling instant. As a result, the generator torque fluctuation constraints and the PFR gain bounds are satisfied by construction, preventing unsafe operation that could degrade closed-loop stability.

Second, for the linearized discrete-time model used in the MPC prediction, the receding-horizon implementation with a positive definite quadratic cost provides a conventional MPC stability mechanism: the controller continuously re-optimizes the finite-horizon cost and drives the predicted torque-fluctuation-related states toward the nominal equilibrium within the admissible set. Therefore, the closed-loop behavior is stable with respect to the considered disturbances.

Finally, since the real plant is nonlinear and subject to modeling uncertainties, practical stability and robustness are validated in simulations under various operating conditions and parameter perturbations, as shown in Section 5.4.

4.4.4. MPC Controller Parameters

All key parameters for the Model Predictive Control strategy are summarized in

Table 1. Summary of MPC controller parameters.

Parameter	Value	Description
Prediction Horizon ($N_p$)	10	Number of future steps the MPC predicts
Control Horizon ($N_c$)	10	Number of future control inputs to optimize
Sampling Time ($\Delta t_s$)	0.0125 s	Control loop update rate
Output Weighting Matrix (Q)	1.0 (scalar)	Weight for LSS torque fluctuation penalty
Control Weighting Matrix (R)	100 (scalar)	Weight for PFR gain adjustment penalty
PFR Gain Lower Bound ($b_{PFR\_lb}$)	30 pu	Minimum allowed PFR gain
PFR Gain Upper Bound ($b_{PFR\_ub}$)	50 pu	Maximum allowed PFR gain
Generator Torque Fluct. Lower Bound ($b_{Tg\_lb}$)	$-0.375\times \Delta t_s$ pu	Lower limit for generator torque fluctuation
Generator Torque Fluct. Upper Bound ($b_{Tg\_ub}$)	$+0.375\times \Delta t_s$ pu	Upper limit for generator torque fluctuation

. These values were determined through an iterative tuning process, balancing fatigue mitigation, frequency regulation performance, and computational feasibility.

5. Case Study

5.1. Case Study Setup and Rationale

5.1.1. Simulation Platform and Control Architecture

To provide a comprehensive overview of the simulation environment and control strategy, a detailed control block diagram is presented in Figure 7. This diagram illustrates the interaction between the wind turbine model, the proposed MPC controller, and the electrical grid model. The related parameters of the simulated model are shown in Appendix A.

The wind turbine is modeled using NREL’s 5MW reference turbine in FAST v8. FAST simulates the highly nonlinear aerodynamics, structural dynamics, and drivetrain mechanics. The turbine’s key parameters are consistent with the public NREL 5MW definition shown in

Table A1. The parameters of the simulated power system and WT.

Parameter	Typical Value
Steam turbine boiler time constant, $T_{TC}$	0.2 s
Steam turbine governor time constant, $T_{TT}$	0.2 s
Thermal unit droop coefficient, $R_T$	0.05 p.u.
Secondary frequency regulation factor of thermal unit, $K_{TBs}$	0.625 p.u.
Hydraulic turbine governor time constant, $T_{HT}$	0.2 s
Hydropower unit reset time constant, $T_{HR}$	0.2 s
Long-term droop coefficient of hydro unit, $R_{HP}$	0.05 p.u.
Hydropower unit startup time constant, $T_{HW}$	5 s
Secondary frequency regulation factor of hydro unit, $K_{HBs}$	0.375 p.u.
Hydro unit droop coefficient, $R_{HP}$	0.05 p.u.
Secondary frequency regulation factor, $B_T$	4.5 p.u.
System equivalent damping coefficient, $D_{eq}$	1 p.u.
System equivalent inertia constant time, $H_{eq}$	6 s
Wind power generation share coefficient, $K_W$	0.3 p.u.
Thermal power generation share coefficient, $K_T$	0.5 p.u.
Hydropower generation share coefficient, $K_H$	0.2 p.u.
Rotor mass, $J_r$	$3.54\times {10}^7\mathrm{kg}·{\mathrm{m}}^2$
Generator mass, $J_g$	$5.34\times {10}^2\mathrm{kg}·{\mathrm{m}}^2$
Gear box ratio, ${\eta }_g$	97
Generator rated speed, ${\omega }_{grated}$	122.91 rad/s
LSS viscous friction coefficient, BTs	$6.22\times {10}^6\mathrm{Nm}·\mathrm{s}/\mathrm{rad}$
LSS spring constant coefficient, KTs	$8.67\times {10}^8\mathrm{Nm}/\mathrm{rad}$
Air density, $\rho$	$1.22\mathrm{kg}/{\mathrm{m}}^3$
Length of the blade, R	63 m
Cut-in wind speed	3 m/s
Rated wind speed	11.4 m/s
Rated power	5 MW

. The proposed MPC controller is implemented in MATLAB/Simulink and communicates with the FAST model. All control signals (e.g., generator torque reference) are passed to FAST, and all necessary state feedback signals (e.g., rotor speed, generator speed, grid frequency) are received from FAST. As depicted in Figure 7, the power system is established based on a rigid aggregation simulation model used to simulate frequency control compliance. This simplified model allows for efficient simulation of grid frequency response to WT power variations and load disturbances while keeping computational complexity manageable.

5.1.2. Wind Condition and Disturbance Generation

Wind speed time series were generated using TurbSim based on the IEC 61400-1 standard as shown in Appendix B [33]. Specific parameters used for TurbSim were

• Mean Wind Speeds: 7 m/s, 8 m/s, and 9 m/s.
• Turbulence Model: Kaimal spectral model.
• Turbulence Intensity: 12%, 14%, and 25%.
• Wind Shear: Power law exponent of 0.2.
• Reference Height: 90 m (hub height).

To simulate realistic grid conditions, random load fluctuations were applied to the electrical grid model. These disturbances were generated using a uniform random distribution, with magnitudes varying up to $\pm 8\%$ of the total nominal system load. These random fluctuations were continuously superimposed onto a steady base load.

To evaluate the effectiveness of the proposed control strategy, two representative wind speed scenarios are considered. These scenarios are carefully designed to capture both typical operating conditions and challenging cases that stress the system’s frequency regulation and mechanical robustness:

• Low Turbulence: average wind speed of 7 m/s with turbulence intensity of 0.14; average wind speed of 9 m/s with turbulence intensity of 0.12. These wind speed ranges represent the region where wind turbines operate for the longest duration and where the trade-off between fatigue loading and power loss during PFR provision is most complex.
• Extreme Turbulence: average wind speed of 8 m/s with turbulence intensity of 0.25, which represents extreme turbulence conditions that wind farms may encounter.

5.1.3. Fatigue Load Testing

Fatigue load testing was primarily conducted using the FAST simulation tool. Developed by the National Renewable Energy Laboratory (NREL) with support from the U.S. Department of Energy, FAST is an open-source multi-physics engineering software package designed for the design and simulation of wind energy systems. It is capable of simulating critical physical processes and system couplings under both normal (fatigue) and extreme (ultimate) loading conditions, accounting for environmental excitations such as wind, waves, and current, as well as full-system dynamic responses including rotor, drivetrain, nacelle, support structure, and control system. In recent years, FAST has incorporated a modular framework, enabling highly flexible modeling of coupled representations of various wind turbine system components. FAST v8 includes mesh-mapping utilities that allow each module to be discretized independently in space and time, thereby facilitating a mathematically rigorous approach to multi-physics co-simulation. For land-based wind turbine simulations, FAST integrates several key modules: InflowWind for wind inflow, AeroDyn for aerodynamics, ServoDyn for control and electrical drive dynamics, and ElastoDyn for structural dynamics. Models developed based on FAST are capable of performing high-fidelity dynamic simulations and fatigue load assessments of wind turbines.

These wind profiles are generated using TurbSim software v2.00 [34], where the “RandSeed1” parameter is varied to produce ten distinct realizations per scenario, thereby capturing the stochastic and turbulent nature of real wind. The turbulence intensities are selected based on typical field measurements to reflect realistic aerodynamic disturbances on the rotor.

The fatigue evaluation focuses on the LSS, which is most sensitive to torque fluctuations induced by PFR. Damage equivalent load (DEL) is employed as the standard metric for fatigue quantification, computed using MCrunch [35]. The fatigue load on the LSS is rigorously quantified using the DEL approach, which is a widely accepted and standardized method in wind turbine fatigue assessment (e.g., IEC 61400-1).

1. Load Channel Selection: The fatigue analysis focuses exclusively on the primary fatigue-driving load within the LSS: the LSS torque. The time-series data for the LSS torque is directly obtained from the high-fidelity aero-elastic simulator, FAST, for each simulated operational scenario. This choice is motivated by the LSS’s critical role in transmitting torque and its susceptibility to fatigue damage.

2. Cycle Counting and Damage Accumulation: The LSS torque time series is processed using the rain-flow counting algorithm (ASTM E1049-85 standard) to identify individual load cycles and their corresponding ranges ($S_i$) and mean values. Subsequently, the accumulated fatigue damage from these cycles is calculated using the Palmgren–Miner linear damage hypothesis (Miner’s Rule), which states that fatigue damage accumulates linearly.

3. DEL Calculation: The DEL is calculated as a single constant-amplitude load that would cause the same fatigue damage as the entire fluctuating load history if applied for a specified number of reference cycles. The formula used for DEL is

$$DEL={\left({\displaystyle \frac{{\sum }_{i=1}^{N_{cycles}}(S_i^m·n_i)}{N_{ref}}}\right)}^{1/m}$$

(31)

where

• $S_i$ is the i-th load cycle range (peak-to-valley amplitude) identified by rain-flow counting.
• $n_i$ is the number of occurrences of the i-th load cycle range.
• $N_{cycles}$ is the total number of distinct load cycles identified.
• m is the Wohler exponent (also known as the inverse slope of the S-N curve), characteristic of the material’s fatigue properties.
• $N_{ref}$ is the reference number of cycles, a normalization constant.

4. Specific Parameters and Material Assumptions:

• Wohler Exponent (m): For the LSS, which is typically constructed from high-strength steel alloys (e.g., 42CrMo4 steel or similar), a Wohler exponent of $m=3$ is adopted. This value is a widely accepted and conservative choice for steel components in wind turbine drivetrain design standards (e.g., DNV GL, IEC 61400-1) when operating predominantly in the high-cycle fatigue regime.
• Reference Cycles ($N_{ref}$): The reference number of cycles for DEL normalization is set to ${\mathit{N}}_{ref}={10}^7$ cycles. This value is a standard practice in wind turbine fatigue assessment, representing a characteristic number of cycles over the operational lifetime of the turbine. It facilitates consistent comparison of fatigue loads across different load cases and control strategies.
• Material Assumptions: The LSS is assumed to be made of a typical high-strength structural steel, with fatigue properties consistent with those for which the selected Wohler exponent of $m=3$ is appropriate. No specific material model beyond the basic S-N curve defined by the Wohler exponent is explicitly used, as the DEL calculation inherently accounts for cycle ranges.

5.2. The Performance of the Developed Controller Under Different Wind Speeds

In this specific scenario, the system load exhibits random fluctuations that remain below 8% within each 40 s interval. In other words, each simulation under a different wind speed experiences a different system load variation. The study compares three methods: the conventional method based on fixed $k_{PFR}$ gain (CON), the higher $\gamma$ (HDC) proposed in this paper ($\gamma$ is 0.3), and the dynamic PFR method based on MPC.

Table 2. DELs of different frequency regulation methods under average wind speed of 7 m/s.

7 m/s	DEL (MNm)			Reduced Percentage
	CON	HDC	MPC	HDC	MPC
No. 1	1.57	1.37	1.24	−6.16%	−15.07%
No. 2	1.64	1.41	1.32	−13.92%	−19.42%
No. 3	1.49	1.30	1.20	−12.88%	−19.69%
No. 4	1.30	1.12	1.06	−13.74%	−18.65%
No. 5	1.28	1.06	0.95	−17.19%	−25.78%
No. 6	1.71	1.52	1.35	−11.21%	−21.02%
No. 7	2.34	2.06	1.89	−12.10%	−19.32%
No. 8	1.46	1.26	1.17	−13.47%	−19.67%
No. 9	1.66	1.46	1.33	−12.05%	−19.88%
No. 10	1.65	1.44	1.31	−12.59%	−20.80%
Avg	1.61	1.40	1.28	−13.02%	−20.40%

CON: The method based on Reference [11]. HDC: De-loading control with high de-loading factor ($\gamma$). MPC: Dynamic PFR method proposed in this work. Avg: Average.

and

Table 3. DELs of different frequency regulation methods under average wind speed of 9 m/s.

9 m/s	DEL (MNm)			Reduced Percentage
	CON	HDC	MPC	HDC	MPC
No. 1	2.46	2.07	1.91	−15.94%	−22.17%
No. 2	2.67	2.29	1.96	−14.09%	−26.47%
No. 3	2.06	1.79	1.51	−12.97%	−26.58%
No. 4	1.42	1.13	1.06	−20.25%	−25.55%
No. 5	2.31	1.97	1.73	−14.85%	−25.22%
No. 6	2.56	2.22	1.85	−13.28%	−27.73%
No. 7	2.01	1.66	1.56	−17.53%	−22.62%
No. 8	2.66	2.27	2.09	−14.66%	−21.43%
No. 9	1.85	1.58	1.44	−14.82%	−22.37%
No. 10	2.93	2.42	2.11	−17.41%	−27.99%
Avg	2.29	1.94	1.72	−15.41%	−24.92%

present the DELs of the LSS under different frequency regulation methods. Observationally, an increase in $\gamma$ correlates with a reduction in LSS fatigue load, consistent with prior research. Furthermore, integrating the MPC method additionally reduces the fatigue load on the LSS. Specifically, at an average wind speed of 7 m/s, HDC, compared to CON, reduces fatigue loads by 6% to 17%, averaging approximately 13%. The reduction in fatigue loads is attributed to HDC enhancing derating, thereby decreasing the average torque on the LSS. Incorporating MPC further reduces fatigue loads by approximately 10%. Relative to CON, MPC reduces fatigue loads by 15% to 26%, averaging around 20%. At an average wind speed of 9 m/s, HDC, compared to CON, reduces fatigue loads by 13% to 20%, averaging approximately 15%. Including MPC further reduces fatigue loads by approximately 10%. In comparison to CON, MPC reduces fatigue loads by 21% to 28%, averaging around 25%.

Next, we will analyze the response characteristics of a specific WT. Figure 8 illustrates the generator speed for wind speed sequence No. 5 with an average wind speed of 7 m/s. Due to de-loading control, HDC increases the generator speed of the wind turbine, as evident. At the 240 s mark, a load disturbance occurs. Without using MPC, as seen in Figure 8b, fluctuations in generator speed begin due to the involvement in frequency response, exacerbating torsional vibrations in the drivetrain system. When employing MPC, it is evident from Figure 8b that fluctuations in generator speed are significantly suppressed. This indicates that MPC can mitigate torsional vibrations in the transmission chain under frequency response, thereby reducing fatigue loads in the LSS.

Figure 9 illustrates the LSS torque for wind speed sequence No. 5 with an average wind speed of 7 m/s. The effectiveness of HDC in reducing the overall curve of the LSS torque, thereby decreasing its average value, is evident. Furthermore, when combined with MPC-based frequency regulation, the LSS torque fluctuation is suppressed, resulting in a significant reduction in the LSS fatigue load. From the fluctuation in the curves, it is evident that MPC can significantly suppress the fluctuations in the LSS torque.

Figure 10 illustrates the fatigue cycles using different methods. The cutoff torque on the x-axis of the curves offers an intuitive understanding of the fatigue loads on the LSS for each approach. As shown in Figure 10, HDC demonstrates lower fatigue loads than CON, and the addition of MPC further reduces fatigue loads.

Figure 11 illustrates the generator speed for wind speed sequence No. 10 with an average wind speed of 9 m/s. At the 320 s mark, a load disturbance occurs. Without using MPC, as depicted in Figure 11b, fluctuations in generator speed arise due to involvement in frequency response, exacerbating torsional vibrations in the drivetrain system. When employing MPC, Figure 11b clearly shows that fluctuations in generator speed are significantly mitigated. This indicates that MPC can mitigate torsional vibrations in the transmission chain during frequency response, thereby reducing fatigue loads in the LSS.

Figure 12 depicts the LSS torque for wind speed sequence No. 10 with an average speed of 9 m/s. The HDC effectively reduces the overall curve of the LSS torque, thereby decreasing its average value. Moreover, when combined with MPC-based frequency regulation, the fluctuation in LSS torque is suppressed, leading to a significant reduction in LSS fatigue load.

Figure 13 displays the fatigue cycles using different methods. As depicted in Figure 10, HDC exhibits lower fatigue loads compared to CON, and MPC can further reduce fatigue loads.

Table 4. Maximum frequency deviations of different frequency regulation methods under average wind speed of 7 m/s.

7 m/s	$\mathbf{\Delta }\mathit{f}$ (mpu)			Reduced Percentage
	CON	HDC	MPC	HDC	MPC
No. 1	−5.06	−4.73	−4.82	−6.52%	−4.72%
No. 2	−8.66	−7.95	−7.81	−8.14%	−9.80%
No. 3	−6.17	−5.68	−5.65	−7.89%	−8.49%
No. 4	−7.94	−7.36	−7.47	−7.33%	−5.88%
No. 5	−8.40	−8.16	−7.96	−2.86%	−5.24%
No. 6	−6.68	−6.22	−6.29	−6.91%	−5.76%
No. 7	−8.13	−7.66	−7.81	−5.72%	−3.87%
No. 8	−7.85	−7.31	−7.44	−6.95%	−5.23%
No. 9	−5.33	−5.02	−5.18	−5.74%	−2.77%
No. 10	−8.67	−7.88	−7.42	−9.13%	−14.44%
Avg	−7.29	−6.80	−6.79	−6.74%	−6.79%

and

Table 5. Maximum frequency deviations of different frequency regulation methods under average wind speed of 9 m/s.

9 m/s	$\mathbf{\Delta }\mathit{f}$ (mpu)			Reduced Percentage
	CON	HDC	MPC	HDC	MPC
No. 1	−9.11	−8.63	−8.72	−5.24%	−4.21%
No. 2	−6.53	−6.18	−6.21	−5.36%	−4.97%
No. 3	−6.76	−6.46	−6.54	−4.39%	−3.24%
No. 4	−6.47	−6.13	−6.09	−5.29%	−5.99%
No. 5	−8.08	−7.61	−7.58	−5.79%	−6.23%
No. 6	−8.78	−8.37	−8.35	−4.70%	−4.93%
No. 7	−8.69	−7.85	−7.91	−9.58%	−8.99%
No. 8	−6.89	−6.46	−6.62	−6.26%	−3.92%
No. 9	−6.37	−6.13	−6.13	−3.78%	−3.73%
No. 10	−7.84	−7.53	−7.53	−3.95%	−3.95%
Avg	−7.55	−7.14	−7.17	−5.51%	−5.10%

present the maximum frequency deviation of the system under the aforementioned conditions. Observations indicate that HDC effectively reduces the maximum frequency deviation. Notably, the MPC exhibits a fluctuating maximum frequency deviation near the HDC. Calculated average values from various operating conditions indicate that at an average wind speed of 7 m/s, HDC’s maximum frequency deviation is 6.74% lower than the traditional method, and the MPC-based frequency regulation method achieves a 6.79% reduction compared to the traditional method. Similarly, for an average wind speed of 9 m/s, the maximum frequency deviation of HDC is 5.51% lower than that of the traditional method, while the MPC-based frequency regulation method achieves a 5.10% reduction compared to the traditional method.

Figure 14 depicts the system frequency deviation for wind speed sequence No. 5 with an average speed of 7 m/s. In addition, Figure 15 illustrates the system frequency deviation of wind speed sequence No. 10 with an average wind speed of 9 m/s. It is evident that both HDC and MPC exhibit significantly reduced frequency deviations compared to the traditional method. These results demonstrate that the proposed MPC-based frequency regulation method effectively reduces the fatigue load of the LSS and mitigates the frequency deviation of the system.

5.3. The Performance of the Developed Controller Under Different Wind Turbulence and Method

Figure 16 depicts the LSS torque for a wind speed sequence with an average speed of 8 m/s and a high turbulence intensity of 0.25. The HDC strategy, by maintaining a higher rotor speed and thus a lower effective aerodynamic torque coefficient, generally reduces the overall level of the LSS torque, thereby decreasing its average value. Moreover, when combined with MPC-based frequency regulation, the fluctuation in LSS torque is suppressed effectively, leading to a significant reduction in LSS fatigue load. Notably, the comparison with the DPC method (damping control), a common fatigue mitigation technique, under this challenging extreme turbulence condition demonstrates that our proposed MPC approach achieves comparable or even slightly superior LSS fatigue load mitigation performance to DPC, highlighting its effectiveness and robustness in such scenarios. This comparable DEL reduction forms the foundation for evaluating the frequency support capabilities of our proposed method against DPC.

Figure 17 illustrates the generator speed trajectories for various control methods under the 8 m/s average wind speed with 0.25 turbulence. It can be observed that the MPC-based control, by actively modulating the generator torque, effectively maintains the generator speed within a reasonable range while enabling frequency regulation. While DPC also contributes to stabilizing the generator speed dynamics by damping mechanical oscillations, its primary focus is on fatigue reduction without explicitly considering grid frequency needs. In contrast, our MPC method’s generator speed management is inherently designed to balance both fatigue mitigation (achieving similar LSS DELs as DPC, as shown in

Table 6. DELs of different frequency regulation methods under average wind speed of 8 m/s with turbulence of 0.25.

8 m/s	DEL (MNm)				Reduced Percentage
	CON	HDC	DPC	MPC	HDC	DPC
No. 1	3.49	2.81	2.60	2.59	−7.83%	−0.38%
No. 2	4.01	3.27	3.05	3.03	−7.49%	−0.66%
No. 3	2.62	2.05	1.78	1.81	−11.98%	1.31%
No. 4	2.64	1.98	1.75	1.74	−12.10%	−0.98%
No. 5	3.86	3.29	3.04	3.12	−5.19%	2.39%
No. 6	3.04	2.51	2.31	2.33	−7.14%	1.20%
No. 7	3.41	2.63	2.33	2.38	−9.22%	2.22%
No. 8	2.87	2.05	1.92	1.91	−6.98%	−0.34%
No. 9	3.10	2.21	1.91	1.97	−10.88%	3.01%
No. 10	2.93	2.46	2.17	2.12	−13.94%	−2.35%
Avg	3.20	2.51	2.29	2.30	−8.26%	0.56%

DPC: Damping control.

) and effective frequency support, demonstrating a more holistic control approach. The HDC strategy, by operating at a higher rotor speed, generally results in higher generator speeds compared to CON, providing more kinetic energy for frequency support.

Figure 18 presents the frequency deviation responses from the grid for different control strategies under the 8 m/s average wind speed with 0.25 turbulence. The PFR gain applied to each method (CON, HDC, DPC, MPC) generates a power response intended to counteract the frequency deviations. It is crucial to highlight the comparative performance of MPC against DPC under conditions where both achieve comparable LSS fatigue load mitigation (as quantified in Table 6). While DPC effectively reduces LSS fatigue, its design is not explicitly tailored for frequency regulation. Consequently, DPC may inadvertently introduce additional power output fluctuations that are not synchronized with grid frequency needs, potentially leading to larger or less consistent frequency deviations compared to a dedicated frequency regulation scheme. In contrast, the proposed MPC method, specifically designed for joint optimization, demonstrates a superior ability to counteract frequency deviations, resulting in a more stable and effective contribution to grid frequency support. This is evident from the tighter containment of frequency deviation observed with MPC, especially when compared to DPC.

Table 6 quantifies the DELs for the LSS under these high turbulence conditions. Consistent with Figure 16, both HDC and DPC significantly reduce the LSS DELs compared to the CON. The proposed MPC method consistently demonstrates comparable or slightly better performance than DPC across various wind sequences, further underscoring its efficacy in mitigating LSS fatigue even under high turbulence and varying disturbance magnitudes. This table confirms that MPC achieves LSS DEL mitigation comparable to DPC, thereby establishing a fair basis for comparing their respective frequency regulation performances.

5.4. The Robustness of the Developed Controller

This section investigates the robustness of the PFR method based on MPC. To assess its robustness, the parameters of the controlled WTs are subjected to random perturbations of ±30% relative to their nominal values. Perturbed parameters encompass the LSS viscous friction coefficient, LSS spring constant coefficient, rotor mass, and generator mass, directly influencing the LSS fatigue load.

Table 7. DELs of different frequency regulation methods under different WT parameters.

7 m/s	DEL (MNm)		%	9 m/s	DEL (MNm)		%
	CON	MPC			CON	MPC
Par. 1	2.38	2.01	−15.57%	Par. 1	2.68	2.01	−24.95%
Par. 2	2.51	2.11	−15.77%	Par. 2	3.38	2.45	−27.55%
Par. 3	2.46	1.98	−19.44%	Par. 3	3.01	2.10	−30.32%
Par. 4	2.83	2.21	−21.98%	Par. 4	2.14	1.65	−22.76%
Par. 5	2.64	2.10	−20.58%	Par. 5	2.79	2.08	−25.54%
Par. 6	2.50	1.99	−20.49%	Par. 6	3.07	2.32	−24.32%
Par. 7	2.31	1.89	−18.26%	Par. 7	3.14	2.34	−25.36%
Par. 8	1.97	1.53	−22.26%	Par. 8	2.37	1.67	−29.41%
Par. 9	3.09	2.55	−17.43%	Par. 9	3.05	2.22	−27.33%
Par. 10	2.91	2.18	−25.04%	Par. 10	2.28	1.52	−33.26%
Avg	2.56	2.06	−19.73%	Avg	2.79	2.04	−27.03%

%: reduced percentage.

presents the fatigue loads of different parameters under different wind speeds. The parameters Par. 1 to Par. 10 represent ten different sets of randomly perturbed parameters. At an average wind speed of 7 m/s, the LSS fatigue load under various parameters consistently outperforms that of traditional methods, with a reduction range of 15% to 25%. On average, the proposed MPC-based method achieves a 19.73% reduction in the LSS fatigue load. Similarly, at an average wind speed of 9 m/s, the fatigue load reduction range of the LSS is 22% to 33%, with an average fatigue load reduction reaching 27.03%. These results demonstrate that regardless of the parameter variations, the MPC-based PFR method effectively mitigates the LSS fatigue load, highlighting its robustness in different operating conditions.

Table 8. Maximum frequency deviation of different frequency regulation methods under different WT parameters.

7 m/s	$\mathbf{\Delta }\mathit{f}$ (mpu)		%	9 m/s	$\mathbf{\Delta }\mathit{f}$ (mpu)		%
	CON	MPC			CON	MPC
Par.1	−6.76	−6.08	−10.04%	Par.1	−8.99	−8.15	−9.34%
Par.2	−8.96	−8.36	−6.70%	Par.2	−7.42	−6.74	−9.14%
Par.3	−9.95	−9.44	−5.18%	Par.3	−7.62	−7.34	−3.58%
Par.4	−8.98	−7.92	−11.81%	Par.4	−9.19	−8.68	−5.59%
Par.5	−7.53	−7.44	−1.27%	Par.5	−6.58	−5.93	−9.86%
Par.6	−8.32	−7.68	−7.63%	Par.6	−8.66	−8.08	−6.66%
Par.7	−9.54	−9.21	−3.37%	Par.7	−9.79	−9.63	−1.71%
Par.8	−9.16	−8.74	−4.65%	Par.8	−6.69	−6.55	−2.06%
Par.9	−9.64	−8.56	−11.14%	Par.9	−8.23	−7.89	−4.14%
Par.10	−8.70	−7.85	−9.81%	Par.10	−6.46	−6.27	−2.89%
Avg	−8.75	−8.13	−7.15%	Avg	−7.96	−7.53	−5.48%

displays the maximum frequency deviation under different parameter variations. For an average wind speed of 7 m/s, the range of maximum frequency deviation reduction is 1% to 11%, with an average reduction of 7.15%. When the average wind speed is 9 m/s, the range of maximum frequency deviation reduction is 1% to 9%, and the average reduction reaches 5.48%. These results indicate that the frequency deviation achieved by the MPC-based PFR method under different parameter variations is consistently lower than that of the traditional methods. This further supports the robustness of the MPC-based PFR method in effectively mitigating LSS fatigue load and reducing the system’s frequency deviation.

6. Conclusions and Discussion

This study systematically analyzes the influence of the de-loading factor on wind turbine performance and identifies its feasible operating boundaries under various wind speeds and frequency regulation gains. It is shown that increasing the de-loading factor effectively reduces the average LSS torque, thereby improving frequency regulation capacity without sacrificing power output. Furthermore, the integration of MPC enables real-time adjustment of the PFR gain and coordinated regulation of generator torque, which significantly suppresses torque fluctuations on the LSS. As a result, both the mean and dynamic variation in LSS torque are markedly reduced, leading to substantial alleviation of fatigue accumulation.

Simulation results validate the efficacy of the proposed strategy. At an average wind speed of 7 m/s, the proposed method reduces LSS fatigue load by approximately 20% and decreases the maximum frequency deviation by about 7%, compared to conventional PFR strategies. Similarly, at 9 m/s, the method achieves a 25% reduction in LSS fatigue load and a 5% improvement in maximum frequency deviation.

The prominent advantage of this method lies in its ability to synergistically reduce both the steady-state and fluctuating components of LSS torque, thereby extending the operational lifespan of wind turbines while simultaneously supporting grid frequency stability. In practical terms, the proposed MPC-based dynamic PFR with de-loading control can directly translate into reduced LSS fatigue accumulation, fewer maintenance interventions, and less unplanned downtime, which are critical economic and reliability drivers for real wind farms. At the grid level, the improved frequency support quality strengthens compliance with frequency regulation requirements and reduces the likelihood of conservative curtailment, hence increasing the feasible hosting capacity for distributed wind power in distribution networks. Notably, this simultaneous improvement in ancillary-service performance (frequency recovery under disturbances) and mechanical reliability (LSS torque fatigue mitigation) is seldom achieved by conventional approaches that treat frequency regulation and fatigue suppression separately. Therefore, the proposed method provides a practical pathway toward cyber–physical integrated smart distribution operation, where frequency support and equipment protection are optimized in a unified control loop.  While this study provides a robust theoretical framework and validates significant performance improvements via comprehensive simulations, we acknowledge that the current results are primarily obtained from numerical models. Future work will focus on experimental verification, particularly hardware-in-the-loop (HIL) testing, to evaluate real-time computational feasibility, actuator/interface limitations, and robustness under realistic measurement delays/noise and nonideal operating conditions. Such validation is a crucial step toward deploying the proposed MPC strategy in actual wind turbine systems, enabling improved operational efficiency and dependable grid support in real-world environments.