The peak shaving control strategy limits the rotor thrust acting on the wind turbine by preemptively adjusting the blade pitch angle before the rated wind speed is attained. In this study, the minimum blade pitch angle required to restrict the rotor thrust was calculated using the method proposed by Abbas et al. (2022), as expressed in Eqs. (1)–(4). Here, the wind speed, rotor rotational speed, and pitch angle are all assumed to be in a steady-state condition. First, the rotor thrust T_r is defined as in Eq. (1), where the thrust coefficient C_T is a function of the tip speed ratio λ and the blade pitch angle β for a given rotor geometry.

In this equation, ρ denotes the air density, A_r is the rotor swept area, and ν represents the wind speed. Next, the maximum allowable thrust T_r,max can be defined as a certain ratio of the maximum thrust acting on the rotor, as shown in Eq. (2). The coefficient α represents the allowable ratio of the maximum thrust corresponding to the given wind speed ν, taking a value between 0 and 1. Subsequently, the maximum allowable thrust coefficient C_T,max is calculated using Eq. (3). Assuming that the tip speed ratio λ remains constant under steady-state conditions, the minimum blade pitch angle β_min satisfying C_T=C_T,max can be obtained using Eq. (4).

Fig. 2 shows the steady-state responses of the blade pitch angle and rotor thrust when the allowable thrust coefficient α is varied from 0.6 to 1.0 in increments of 0.1. In the absence of peak shaving control (blue line), the blade pitch angle increases with wind speed beyond the rated wind speed of 10.6 m/s, at which point the rotor thrust reaches its maximum. In contrast, when peak shaving control is applied, the blade pitch begins to increase before the rated wind speed, thereby limiting the rotor thrust to a certain level. Moreover, as the allowable thrust coefficient α decreases, blade pitch control is activated at lower wind speeds, and the maximum rotor thrust acting on the rotor gradually decreases.

The floating feedback control method adjusts the final blade pitch angle by adding a compensation term proportional to the fore–aft velocity of the tower top to the conventional proportional–integral (PI)-based blade pitch control. This compensation term suppresses the negative damping that arises from the dynamic coupling between the rotor and the tower-top motion induced by the floating platform, thereby improving the dynamic stability of the FOWT. The conventional PI blade pitch control can be expressed as Eq. (5), where the change in blade pitch angle gain Δβ is determined by applying the proportional and integral gains, k_p and k_i, respectively, to the generator speed deviation Δω_g. In contrast, the floating feedback control adds a term proportional to the tower-top velocity variation Δẋ_t to Eq. (5), and thus determines the change in blade pitch angle Δβ as shown in Eq. (6). Here, k_β,float denotes the floating feedback gain.

The gain k_β,float used in the floating feedback control was determined based on the theoretical formulation proposed by Abbas et al. (2022) so that the dynamic coupling between the tower-top velocity and the rotor could be effectively canceled. First, the linearized equation of motion for the rotor–generator system can be expressed as Eq. (7). Here, J is the rotor moment of inertia, N_g is the gear ratio, τ_g is the generator torque, and τ_a represents the aerodynamic torque generated by the wind acting on the rotor.

The variation in aerodynamic torque Δτ _a can be approximated using a first-order Taylor expansion, as shown in Eq. (8), with respect to changes in rotor speed ω_g, blade pitch angle β, and effective wind speed ν. The variation in effective wind speed at the nacelle, Δν, is defined in Eq. (9) as the difference between the incoming wind speed variation Δv_w and the tower-top velocity variation Δẋ_t. The relationship between rotor angular velocity and generator speed is expressed in Eq. (10).

By sequentially substituting Eqs. (6), (8), (9), and (10) into Eq. (7) and rearranging the terms, the equation of motion for the rotor is obtained, as shown in Eq. (11). The first term on the right-hand side of Eq. (11) indicates that the fore–aft velocity of the tower influences the rotor angular acceleration. Therefore, by setting this term to zero, the rotor acceleration component caused by tower-top velocity variation can be removed. Consequently, applying the floating feedback gain k_β,float defined in Eq. (12) enables compensation of the blade pitch angle, which helps effectively suppress platform motion amplification

The FOWT substructure used in this study is the PentaSemi semi-submersible platform, developed by the Korea Research Institute of Ships and Ocean Engineering (KRISO) for a 15 MW-class FOWT (Ahn et al., 2022; Han et al., 2023). The configuration of the PentaSemi platform and the layout of its mooring system are shown in Figs. 3(a) and 3(b), respectively, while the main specifications of the FOWT system are summarized in Table 1. The PentaSemi platform consists of three pentagonal columns interconnected by pontoons, forming an eccentric structure in which the wind turbine is installed on one of the outer columns rather than at the geometric center. To mitigate platform motion responses, triangular damping plates are installed between the pontoons. The mooring system of the PentaSemi platform adopts the yaw control catenary mooring system (YCCMS) designed to suppress yaw motion and evenly distribute mooring tension (Ahn et al., 2022). The platform is equipped with six fairleads, two on each column. Each fairlead is connected to a wire rope, and two wire ropes from adjacent columns are combined at a tri-plate, from which a studless chain extends down to the anchor. The detailed specifications of the mooring system are provided in Ahn et al. (2022). The wind turbine applied in this study is the IEA 15 MW reference wind turbine model, developed under the IEA Wind TCP Task 37 program (Gaertner et al., 2020). This turbine is equipped with three blades with a rotor diameter of 240 m and is designed to generate a rated power of 15 MW at a wind speed of 10.6 m/s, operating at a rotor speed of 7.56 rpm.

To evaluate the motion responses and structural loads of the FOWT platform, a combined analysis using WADAM (Wave analysis by diffraction and morison theory) and OpenFAST was conducted. First, a frequency-domain hydrodynamic analysis was performed in WADAM based on potential flow theory. The panel size was set to 1.5 m, and 150 frequency components were considered within the range of 0.02–3.0 rad/s. This analysis provided key hydrodynamic properties, including added mass, radiation damping, first-order wave excitation forces, and second-order mean drift forces. Subsequently, coupled time-domain simulations were performed in OpenFAST, an open-source tool developed by the National Renewable Energy Laboratory (NREL, 2023). OpenFAST uses a multibody dynamics framework and accounts for the coupled interactions among aerodynamic, hydrodynamic, structural, and control systems. The hydrodynamic coefficients obtained from WADAM were imported into the HydroDyn module of OpenFAST. In the simulations, second-order wave loads were computed using Newman's approximation based on the mean drift force, and nonlinear viscous effects were modeled via the drag coefficient C_d in the Morison equation.

For controller implementation, the ROSCO developed by NREL was adopted (Abbas et al., 2022). ROSCO includes a generator torque PI controller, which maximizes power capture in below-rated wind speed regions, and a blade pitch PI controller, which regulates rotor speed in above-rated wind conditions. The controller parameters were tuned using the ROSCO Toolbox, and the resulting control logic and gain values were incorporated into the ServoDyn module of OpenFAST for numerical simulations.

To validate the accuracy of the developed numerical model, simulation results were compared with a 1/42.25-scale model test of the PentaSemi platform conducted by KRISO. The substructure, wind turbine, and mooring system used in the experiments were identical to those described in Section 3.1. Fig. 4 presents the comparison of free decay tests, where the black solid lines indicate KRISO’s experimental data and the red dash-dotted lines show the present simulation results. The natural periods obtained from the simulation were 66.6 s in surge, 18.1 s in heave, and 28.4 s in pitch motions. The simulation results showed good agreement with the experimental data, with a maximum deviation of approximately 3% in natural periods. These results confirm that the developed numerical model accurately represents the mass and inertia characteristics of the platform as well as the mooring system’s restoring stiffness, successfully reproducing the free decay motions quantitatively. Additionally, linear damping coefficients were incorporated based on the damping ratios estimated from the free decay tests to account for viscous effects in the numerical model.

Next, numerical simulation results under irregular wave conditions were compared with model test data. The wave conditions used for this validation correspond to an extreme sea state, characterized by a JONSWAP (Joint North Sea wave project) spectrum with a significant wave height H_s = 10.72 m, a peak period T_p = 16.37 s, and a peak enhancement factor γ = 2.5. In this study, the drag coefficient used in the Morison equation was derived from irregular-wave model test results under the same sea state and implemented in the numerical model. Fig. 5 compares the power spectral density (PSD) and time-series responses of surge, heave, and pitch motions between the experimental and numerical results. For all motion components, the numerical results showed spectral shapes consistent with the experiments, and the spectral peak magnitudes around the natural frequencies aligned well. Surge and heave motions exhibited good agreement in both amplitude and phase, while pitch motion showed slight discrepancies at certain wave-frequency components. Nevertheless, the overall dynamic response trends were well captured. These results confirm that the developed numerical model can reliably simulate platform motion responses even under realistic sea conditions with wave load.

This section discusses the numerical simulations, which were conducted under steady wind and regular wave conditions with a single wave frequency to clearly identify the coupled interaction between the FOWT platform and the operating wind turbine, as well as the occurrence of the negative damping phenomenon. The analysis was performed at a constant wind speed of 13 m/s and a wave height of 6.3 m, while the wave period was varied between 8 s, 11.5 s, and 15 s. During power generation, a PI gain-based blade pitch controller was applied; when power generation was disabled, the blade pitch angle was fixed at 90°. Fig. 6 shows the time-series pitch motion of the platform for each wave period and power-generation condition. During inactive power generation, the platform exhibited pitch motion components corresponding only to the wave period. In contrast, during the turbine operation, the natural frequency of the blade pitch controller (0.2 rad/s) was close to that of the platform pitch motion (0.221 rad/s), resulting in a resonance phenomenon. Consequently, the platform displayed long-period oscillations with amplitudes approximately four times larger, indicating unstable behavior due to negative damping. To mitigate such effects, it is necessary to adjust the natural frequency of the blade pitch controller through gain detuning or apply supplementary control strategies to enhance system stability.

To investigate trends under varying wind speeds, numerical simulations were performed in regular wave conditions with a wave height of 6.3 m and a wave period of 11.5 s, while the wind speed was varied from 3 m/s to 25 m/s in increments of 1 m/s. Fig. 7 presents two graphs that illustrate the response characteristics of the FOWT system with respect to wind speed: the maximum pitch motion amplitude and the mean generator power output. In the sub-rated region, where generator torque control governs operation (below the rated wind speed of 10.6 m/s), both pitch motion and power output increased with the wind speed. In the range of 11–13 m/s, where blade pitch control begins to operate, the maximum pitch motion continued to increase due to the influence of negative damping, while the mean generator power remained slightly below the rated output of 15 MW. At wind speeds above 14 m/s, the pitch motion gradually decreased and converged to a constant value as wind speed increased, whereas generator power approached and stabilized around 15 MW.

Fig. 8 shows the time-series data of platform pitch motion, blade pitch angle, rotor thrust, and generator power under the 13 m/s wind speed condition, where the platform exhibited the largest pitch response. Each time-series dataset was normalized by removing its mean component and dividing by its maximum absolute value. The results indicate that platform pitch motion is approximately 180° out of phase with the blade pitch angle, while it remains in phase with the rotor thrust. This occurs because, as the platform tilts backward due to wind and wave excitation, the relative wind speed at the nacelle decreases. To maintain rated power output, the controller reduces the blade pitch angle, which in turn increases rotor thrust. The increased thrust generates a larger pitching moment on the platform, further amplifying pitch motion and triggering the negative damping phenomenon. Consequently, platform pitch continues to grow, and the stability of generator power output deteriorates.

To examine the suppression effect of the negative damping phenomenon under regular wave conditions, numerical simulations were conducted applying the peak shaving control. The environmental conditions were fixed at a wave height of 6.3 m, wave period of 11.5 s, and wind speed of 13 m/s, while the allowable thrust ratio (α) was gradually reduced from 1.0 to 0.4 in increments of 0.1. Fig. 9 presents the time-series and bar graphs of platform pitch motion, blade pitch angle and power output corresponding to different allowable thrust ratios, with the bar graphs indicating the maximum pitch amplitude and mean generator power, respectively. In Fig. 9(a), the blue line representing the case without peak shaving (α = 1.0) exhibits the largest pitch amplitude, while the pitch amplitude gradually decreases as α decreases. The maximum pitch angle decreased from 8.64° in the no-control case (α = 1.0) to below 5° when peak shaving was applied with α ≤ 0.7. As shown in Fig. 9(b), the fluctuation range of the blade pitch angle also decreased with decreasing α, while the mean pitch angle increased due to earlier engagement of blade pitch control to enforce stronger thrust limitation. Meanwhile, the generator power time series in Fig. 9(c) showed reduced fluctuation amplitude as α decreased, indicating a stabilization of power output. The mean power output increased slightly within the range 1.0 ≥ α > 0.7, but when α ≤ 0.7, both the maximum and mean power outputs decreased due to excessive thrust limitation.

To evaluate the influence of floating feedback control on suppressing the negative damping phenomenon, Fig. 10 compares the time-series responses of platform pitch motion and power output between the baseline PI controller and the floating feedback controller. In the floating feedback controller, three values of the feedback gain k_β,float were tested: the theoretical value of −9.3 derived from Eq. (12), and two additional values obtained by varying it by ±3. As shown in Fig. 10(a), the application of floating feedback reduced the low-frequency pitch motion of the platform. Without floating feedback, the platform exhibited pitch oscillations close to its natural period, whereas with the control applied, the motion shifted to match the wave period. However, the variation in pitch response with respect to different k_β,float values was not highly sensitive, likely because the current condition involved steady wind and regular waves, where no wind speed fluctuation existed and the FOWT motion was mainly governed by wave excitation. Fig. 10(b) presents the blade pitch angle time series; although the floating feedback controller adjusts the blade pitch based on the tower-top velocity, under regular wave conditions, the variation pattern of tower-top velocity is predominantly determined by the waves. Consequently, changing the feedback gain resulted in negligible differences in the blade pitch response. Fig. 10(c) shows the generator power time series. When floating feedback control was applied, the low-frequency power fluctuations decreased, indicating reduced output variability and a trend toward more stable power generation.

In this section, the dynamic response characteristics of the FOWT system under different control strategies were analyzed using design load cases (DLC) 1.6 and 1.3, as defined in IEC 61400-3-2:2025. The environmental conditions used in the simulations, representing a U.S. East Coast site (Allen et al., 2020), are summarized in Table 2. To reflect realistic offshore operating conditions, both irregular waves and turbulent wind were considered simultaneously. The wave and wind were applied in the same direction (0°), forming a collinear condition. The numerical simulations were conducted for a total duration of 4000 s; however, since the initial stage may include transient responses, the first 400 s of data were excluded from the analysis, and only the remaining 3600 s were used for evaluation.

To evaluate the performance of the FOWT system under different control strategies, simulations were conducted for four controller configurations: Base, PS, FL, and PS + FL. The Base controller represents the baseline setup, in which no additional control method is applied other than basic gain detuning; PS represents only the peak shaving control is applied; FL only the floating feedback control; and PS + FL combines both strategies. For the peak shaving control, an allowable thrust ratio of α = 0.7 was used. As shown in Fig. 9(c), the mean power output increased slightly for α ≥ 0.7, but decreased when α < 0.7 owing to excessive thrust limitation. Therefore, α = 0.7 was selected as an approximately optimal value in terms of power generation efficiency. The floating feedback gain was set to k_β,float = −9.2956 as obtained from Eq. (12), and the main controller parameters are summarized in Table 3.

Fig. 11 presents the steady-state analysis results under uniform wind conditions with the combined controller applied, showing the turbine response characteristics across different wind speed ranges. The steady-state simulations were conducted by fixing all six degrees of freedom (6-DOF) of the platform while allowing only the wind turbine to operate. The y-axis of Fig. 11 is configured as a dual axis: the left axis represents the magnitudes of all response quantities except rotor thrust (with units indicated in the legend), while the right axis displays rotor thrust in meganewtons (MN). In the sub-rated region (3.0–10.6 m/s), generator torque control primarily governs turbine operation, which results in generator torque and power increasing proportionally with wind speed, along with a gradual rise in rotor thrust. Near the rated wind speed of 10.6 m/s, rotor thrust reached its maximum but remained relatively flat rather than sharply increasing. This is because the peak shaving control—designed to activate before the rated wind speed—limits the aerodynamic load acting on the rotor. Beyond the rated wind speed, the power output converged to 15 MW, while both rotor speed and generator torque remained constant. Meanwhile, the blade pitch angle continued to increase with wind speed, causing a gradual decrease in rotor thrust. These results confirm that the proposed combined controller smoothly transitions between control strategies according to wind speed, as clearly demonstrated by the steady-state analysis.

The DLC 1.6 condition represents a severe sea state used to evaluate the response characteristics of the floating system in a wave-dominant environment where wave loads are predominant. In this analysis, the mean wind speed was set to 13 m/s, corresponding to the condition in which the negative damping phenomenon is most pronounced. The wave condition was defined by the JONSWAP spectrum with parameters of H_s = 8.5 m, T_p = = 13.1 s, γ = 2.75. The wind field was modeled using the IEC Kaimal spectrum–based turbulent wind model, applying the Normal Turbulence Model (NTM) and turbulence intensity B. Fig. 12 presents the time-series results of surge, heave, and pitch motions, as well as mooring-line tension and blade pitch angle under the DLC 1.6 condition. The controller types are distinguished by color: Base (blue), PS (orange), FL (green), and PS + FL (purple). The mooring-line tension represents the force acting on the starboard fairlead located in the direction of the environmental loading. With the Base controller, oscillations with a period of approximately 30 s—corresponding to the natural period of platform pitch motion—appeared in all motion components, mooring-line tension, and blade pitch angle, exhibiting larger amplitudes than those of the other controllers. The most significant oscillations were observed in the pitch motion, indicating negative damping due to the coupled interaction between pitch motion and turbine operation under irregular wave and turbulent wind conditions, which amplified platform motion and mooring tension. While the Base controller failed to mitigate this behavior, both peak shaving (PS) and floating feedback (FL) controls reduced the long-period oscillations, confirming the suppression of negative damping. As shown in Fig. 12(e), the amplitude of blade pitch angle variations decreased progressively in the order of PS, FL, and PS + FL. This trend can be explained as follows: in the PS controller, blade pitch control is activated over a wider range of wind speeds, resulting in larger variations in blade pitch. In contrast, the FL controller generates control signals based on tower-top velocity to suppress platform motion, leading to smaller blade pitch variations. The combined controller (PS + FL) integrates the complementary effects of both strategies, achieving the smallest blade pitch fluctuations while also providing the most effective reduction in overall platform motion and mooring-line tension.

To compare the frequency-dependent characteristics of platform motions and mooring-line tension, the time-series data for each component were transformed into the frequency domain using the Fast Fourier Transform (FFT). The responses were then divided into three frequency ranges: the low-frequency region (below 0.3 rad/s), wave-frequency region (0.3–2.0 rad/s), and high-frequency region (above 2.0 rad/s). The corresponding frequency-domain response magnitudes are shown in Fig. 13. In all cases, the high-frequency responses were negligible, with dominant responses occurring in the low- and wave-frequency regions. A clear reduction in low-frequency components was observed when any advanced control strategy was applied compared with the Base controller, while differences among the controllers were relatively small in the wave-frequency range. In terms of heave and pitch motions, the FL controller and the combined controller (PS + FL) exhibited slightly better performance than the others. Specifically, the combined controller demonstrated the best overall performance, reducing low-frequency components relative to the Base controller by 34.8% in surge, 82.4% in heave, 92.1% in pitch, and 19.2% in mooring-line tension.

Fig. 14 presents the PSDs of platform pitch motion, nacelle acceleration, and structural loads. In each graph, the dominant frequency components associated with platform motion, wave excitation, blade rotation, and tower vibration are marked by black dashed lines. In the low-frequency range (below 0.3 rad/s), clear differences among control strategies are observed. As shown in Fig. 14(a), when either the peak shaving or floating feedback control was applied, the spectral peak at the natural frequency of pitch motion was significantly reduced compared with the Base controller. A similar trend is evident in Figs. 14 (b) and (c), indicating that the peak shaving and floating feedback controls effectively suppressed the low-frequency pitch motion resonance caused by negative damping, thereby improving both nacelle motion and structural load responses, respectively. In the wave-frequency region (0.3–2.0 rad/s), the differences among control methods were the most pronounced in the blade root bending moment in Fig. 14(c). In this region, the spectral magnitude of the bending moment was the highest for the Base controller and lowest for the combined controller, suggesting that different blade pitch control mechanisms modified the aerodynamic loads on the blades, directly reflected in their bending moments. In the high-frequency region (above 2.0 rad/s), spectral energy appeared near the blade 3P frequency (approximately 2.4 rad/s) and the tower natural frequency (approximately 2.6 rad/s). When either the peak shaving or floating feedback control was applied, the spectral energy near the tower natural frequency decreased, indicating mitigation of the tower’s dynamic response in that frequency band.

Fig. 15 presents box plots of the maximum, minimum, and mean values of platform pitch motion, nacelle acceleration, structural loads, and generator power output. When either the peak shaving or floating feedback control was applied, overall performance improved compared with the Base controller. Notably, the combined controller, which simultaneously applied both methods, exhibited the best performance, showing the lowest maximum values across all indicators related to structural safety and extreme-load assessment. Specifically, the maximum platform pitch amplitude decreased by approximately 33 %, the nacelle acceleration by approximately 13 %, and the blade and tower structural loads by around 29 % relative to the Base controller. The mean generator power output of the Base controller was 14.66 MW, whereas the combined controller achieved 14.88 MW, corresponding to an improvement of roughly 1.5 % in power efficiency. When the floating feedback control was applied alone, the mean generator power reached 14.99 MW, the highest among all control cases. This is because, while the peak shaving control limits maximum rotor thrust—resulting in a slight reduction in power generation—the floating feedback control adjusts the blade pitch angle solely based on the tower-top dynamic response, without imposing thrust limitations, thereby preventing power loss.

Fig. 16 presents box plots of the maximum, minimum, and mean values of platform pitch motion, nacelle acceleration, structural loads, and power output under the DLC 1.3 condition. This case corresponds to a wind-dominant environment with the Extreme Turbulence Model (ETM) applied to evaluate the influence of rapid wind speed fluctuations on the FOWT system and controller performance. The mean wind speed was maintained at 13 m/s, the same as in DLC 1.6, and the wind field was generated using the IEC Kaimal spectrum–based turbulence model with turbulence intensity B. The wave condition reflected a relatively mild sea state with H_s = 2.2 m, T_p = 7.5s, and γ = 1.0. The analysis results revealed that, similar to DLC 1.6, the combined controller exhibited the lowest maximum values across all indicators, demonstrating the best overall performance. With the combined controller, the maximum platform pitch amplitude decreased by approximately 48%, and the nacelle acceleration by approximately 43% compared with the Base controller, indicating a notable enhancement in motion stability. For structural loads, the tower load and blade load were reduced by approximately 35 % and 31%, respectively. In terms of power generation, as in DLC 1.6, the floating feedback controller alone maintained a mean power output of 15.00 MW, corresponding to the rated power level, while the combined controller achieved 14.98 MW, representing an improvement of approximately 2.6% compared with the Base controller, confirming its superior performance.

Fig. 17 compares the performance improvement of the combined controller relative to the Base controller under both DLC 1.3 and DLC 1.6 conditions. Under the DLC 1.3 condition, the standard deviations of platform motion, nacelle acceleration, and structural loads decreased more markedly compared to DLC 1.6, while the mean power output showed a slight increase. In the DLC 1.3 environment, the ETM induced rapid wind speed fluctuations and corresponding variations in rotor thrust, which further intensified the negative damping phenomenon. As a result, the peak shaving and floating feedback control methods performed more effectively in mitigating this effect, leading to greater reductions in platform motion, nacelle acceleration, and structural loads than in DLC 1.6. These results confirm that in high-turbulence environments, the application of appropriate control strategies plays a crucial role in enhancing platform dynamic stability, reducing structural loads, and improving power-generation efficiency.