Abstract

Floating offshore wind turbines (FOWTs) experience dynamic conditions due to platform motion, requiring specific control strategies to mitigate loads and promote the wake diffusion improving overall wind farm efficiency. These problems can be appropriately modeled by medium-fidelity solvers, which rely on a computational fluid dynamics (CFD) resolution of the flow while avoiding its detailed resolution around the blades, preserving high-fidelity in simulating the wake at an acceptable computational cost. This work adopts a medium-fidelity actuator line model (ALM), implemented in the openfoam environment, previously validated against experiments and multifidelity models in the frame of the OC6 Phase III project. The study analyses several operating conditions during surge motion: a variable angular speed in below-rated condition, conceived to maximize the turbine efficiency, and a collective blade pitch control employable in above-rated conditions to limit surge-induced loads fluctuations. The effect of each control strategy is assessed individually through a systematic comparison with the baseline case with constant angular speed and blade pitch. Results indicate that the angular speed control succeeds in increasing the turbine power and reduces the spanwise variability of the induction factor amplitudes. Conversely, the pitch angle control reduces the force amplitude but does not alter the spanwise trend of the induction factor amplitude.

1 Introduction

As the world continues to seek sustainable and renewable energy sources, the exploration of innovative technologies becomes crucial for maximizing the full potential of wind energy. By enabling the installation of turbines in waters deeper than 50 m, where conventional structures fixed at the sea-bed are either impractical or not economically effective, floating offshore wind turbines (FOWTs) represent the new frontier in wind energy, allowing for the exploitation of more uniform and stronger winds.

Because of the combined influence of wind and waves, these devices experience unsteady motions, leading to unsteady aerodynamic loading. As a consequence of the high variability of the flow incoming to the rotor, in order to achieve high rates of energy production under safe conditions, FOWTs require an efficient control system. In modern turbines, the control strategies encompass either collective and individual pitch control as well as angular speed control. The blade pitch control is activated to limit the power output at wind speeds higher than the rated condition, whereas rotor speed control is applied below the rated condition to maximize the turbine efficiency, maintaining the same tip speed ratio.

The present paper aims at investigating the aerodynamic response of a laboratory scaled FOWT model, tested in the large-scale wind tunnel of Politecnico di Milano [1,2], at rated conditions, in multiple realistic (though scaled) tower motions. Two scenarios are outlined: (i) rotor speed control strategy, in which the rotor undergoes an overspeed to track the variation of apparent wind (relative wind speed); (ii) collective blade pitch strategy, in which the blade is feathered to reduce the forces acting on the rotor.

A range of diverse numerical models is accessible for analyzing the aerodynamics of FOWTs. The blade element momentum (BEM) method has been extensively employed to predict the aerodynamic loads on fixed-bottom wind turbines. However, its applicability to floating turbines is still a subject of ongoing investigation. Due to alternative interpretations of the complex rotor aerodynamics, diverse conclusions about the validity of BEM theory to FOWT analyses exist especially when considering turbulent wake state (TWS) and vortex ring state (VRS). Ferreira et al. [3] and Papi et al. [4] analyze the unsteady aerodynamic effects caused by platform surge and demonstrate the effectiveness of BEM codes when they are improved with dynamic inflow models. However, the main limitation of engineering codes based on BEM approach remains the need of an extensive validation and calibration, especially in presence of tower motion. The free vortex wake method (FVW) features a higher fidelity level with respect to the BEM approach, and several authors [58] used the FVW to investigate the dynamic loads and wake features during platform motion. However, a comprehensive validation, making use of blade-resolved computational fluid dynamics (CFD) tools, is still required to assess the fidelity of FVW models in resolving high complex wake patterns in the near and far wake regions, as reported in Ref. [9].

Despite the BEM and FVW models have provided comparable predictions of FOWT rotor loads to blade-resolved CFD [1], only this latter approach is able to provide the resolution of the wake convection and mixing, especially where dissipation through viscous phenomena takes place, i.e., in the mid and far wake regions, which are crucial for optimal wind farm design. Moreover, in the context of FOWTs, blade-resolved CFD based on unsteady Reynolds-averaged Navier–Stokes solvers are the most suitable tool to investigate the tip vortex interactions with the rotor itself [6,10,11]. Though, the computational cost of blade-resolved CFD rapidly grows as the calculations are extended to wide numerical domains (e.g., multiturbine arrays) or need enhanced resolution (in case of large eddy simulations of the wake), making these simulations unfeasible. In this context, the actuator line model (ALM), first developed by Sørensen and Shen [12], combines the advantages inherited by both low and high-fidelity approaches: the aerodynamics of the blades is not resolved, replacing the blades with a rotating distribution of actuator points where aerodynamic forces are applied, evaluated by sampling the flow around the actuator point and by resorting to polars. The flow field, instead, is resolved as a standard CFD tool with the possibility to resolve turbulence at decreasing length scales (e.g., by using either the RANS or the large eddy simulation models). By virtue of the simplified representation of the blade aerodynamics, the ALM is inherently suitable for application of control strategies involving blade pitch variations since it does not require any adjustment of the mesh around the blade as in blade-resolved CFD simulations.

The objective of this study is to simulate the aerodynamic rotor load responses and dynamic effects originated by the combination of a surge platform motion and control strategies, applying an in-house CFD–ALM code, whose base model, verification, and validation are reported in Ref. [13]. An individual blade pitch control was implemented in Ref. [14] with a primary focus on wake dissipation and performance optimization. Yang et al. [15] integrate the ALM with a control system that includes both a generator torque controller and a rotor-collective blade pitch controller, focusing on the impact of these strategies on wake aerodynamics during surge motion. In this study, the two strategies are examined separately to distinguish their effects and to provide a more detailed understanding of how the regulation specifically influences the dynamics of the rotor.

The paper is structured as follows. After an introductory overview of the examined cases, the ALM formulation and the computational framework are presented, including a validation against the multifidelity model results. Then, a thorough comparison between the reference case without control strategy and BEM outcomes is conducted to quantify the differences between low-fidelity (BEM) and medium-fidelity (ALM) approaches and provide insights of the dynamic effects in an unregulated scenario. Finally, a detailed analysis is performed for the cases featuring the two control techniques.

2 Load Cases

The turbine under consideration is a 1:75 scaled laboratory model of the DTU 10 MW machine with a rotor diameter of 2.38 m [16]. The model was tested within the UNAFLOW experimental campaign [2,17] in the large-scale wind tunnel of Politecnico di Milano. The rotor consists of three blades, each one featuring identical low-Reynolds airfoils across the span, along with a twist distribution and tapered chord variation. The aerodynamic coefficients, covering the Reynolds number range from 50,000 to 200,000, were evaluated in a dedicated experimental campaign, outlined in Ref. [17]. The wind tunnel comprises a testing section measuring 13.84 m × 3.84 m and spanning a length of 35 m. The turbulence intensity in the inflow is less than 2%, and the air density is ρ = 1.177 kg/m3. The wind tunnel outlet was positioned 7 m downstream of the wind turbine, and the length of the tunnel preceding the rotor was set so to achieve the complete development of the wind tunnel boundary layer. The results of the experiments were subsequently shared with participants of the IEA Wind Task 30 (OC6 Phase III), facilitating code-to-code validation, as referenced in Ref. [1].

Within the IEA Wind Task 30 (OC6 Phase III), several oscillatory platform motions were simulated, including surge and pitch. Some of these conditions had corresponding experimental data, while other cases were exclusively simulated numerically. In this study, we examine the more complex cases involving control strategies on both rotational speed and blade pitch. In the OC6 Phase III project, these cases were not subjected to experimental testing but were exclusively simulated numerically by multifidelity models. Therefore, the current AL code will be validated solely against a chosen set of computational results from OC6 Phase III participants.

The cases analyzed here are summarized in Table 1 and correspond to the prescribed surge platform motion imposed by the sinusoidal law in the following equation:
(1)
Table 1

Load cases of surge examined in this study

LCΩ (rpm)θp (deg)As (m)fs (Hz)Vmax (m/s)
2_12 240 0.08 1.01 
2_16 240 ±15% 0.08 1.01 
2_17 240 1.5 ±1.5 0.08 1.01 
LCΩ (rpm)θp (deg)As (m)fs (Hz)Vmax (m/s)
2_12 240 0.08 1.01 
2_16 240 ±15% 0.08 1.01 
2_17 240 1.5 ±1.5 0.08 1.01 

According to this expression, the surging motion starts with the turbine moving leeward along the positive x-coordinate (phase shift ϕs=0 deg). In the top frame of Fig. 1, the platform displacement and platform velocity are depicted, while in the central plot, the apparent wind experienced by the rotor is illustrated (computed as the relative speed between the freestream wind U0 and the platform velocity).

Fig. 1
Platform displacement and velocity, apparent wind speed, and control strategies of the analyzed test cases
Fig. 1
Platform displacement and velocity, apparent wind speed, and control strategies of the analyzed test cases
Close modal

All load cases (LC) share the same freestream velocity (U0=4m/s), frequency, and amplitude of the surge motion. The reference case LC2_12 is conducted at a constant angular speed (Ωmean = 240 rpm) and fixed blade pitch angle (Θp = 0 deg). The cases with control are here considered as conceptual scenarios of regulation designed to facilitate the understanding of regulatory effects and gather knowledge useful for setting up real control systems. The LC2_16 case features a prescribed sinusoidal variation in angular speed (semi-amplitude AΩ=15%Ωmean), simulating a control operation for wind speeds below the rated conditions. The LC2_17 case is characterized by a sinusoidal change in the blade pitch angle (semi-amplitude AΘp=1.5 deg), emulating a control strategy that is commonly employed in wind farms to alleviate turbine loads when the wind speed is above the rated condition. As the rotor perceives an increase in apparent wind, the controller feathers up the blade pitch decreasing the angles of attack and, consequently, reducing the aerodynamic loads. The lower frame of Fig. 1 shows the regulation of angular speed and blade pitch.

3 Actuator Line Model

The in-house ALM code, applied in this study, was implemented within the openfoam framework. The basic ALM concept, comprehensively described in Ref. [12], eliminates the resolution of the flow around the blades, which are replaced by volume forces applied on actuator points distributed along actuator lines. The calculation of aerodynamic forces employs tabulated aerodynamic coefficients, whereas the angles of attack and the Reynolds numbers are determined from the orientation of the relative wind velocity vector whose magnitude, in the case of a surging turbine, reads
(2)

where Uax and Utg are axial and tangential velocities at each actuator line point expressed in the blade reference system. The in-house ALM solver first computes the relative velocities at each blade station sampling the wind velocity along two sampling lines upstream and downstream the actuator line point; then, the velocity components induced by the bound vortex (associated with the lift force) are evaluated via the Biot–Savart law and are subtracted to sampled values, thus purging the sampled velocity by the induction caused by the blade itself. Full details on the present model are described in Ref. [13], which also includes applications to surging and pitching FOWTs.

In this study, two-dimensional static polars [2] have been used. The present analyses do not include hub/tip losses or dynamic stall correction since no hysteresis phenomena occur on most of the blade due to the limited variations of angle of attack for load conditions under investigation. The momentum equation solved by the CFD solver is reported in Eq. (3) where the unsteady, advection, viscous, and pressure terms appear, augmented by the volume force f, which is the source term representing the blade aerodynamic force
(3)
To prevent singularity issues, the forces are spread over the fluid domain around the actuator point using a two-dimensional isotropic Gaussian kernel distribution ηϵ, by means of the convolution in Eq. (4) as proposed analytically by Mikkelsen [18]. The semiwidth of the force, ϵ, distribution is selected to cope with numerical stability and accuracy. A constant value of 2Δ along the entire blade is assumed in accordance to Troldborg's findings [19]
(4)

4 Numerical Methods

Simulations were performed over domains reproducing the flow configuration established in the testing wind tunnel [20]. To ensure the complete development of the wake, the outflow boundary was placed 14.5D downstream of the turbine location. In order to consider the blockage caused by the wind-tunnel wall boundary layers, while matching the experimental mass flow rate, the domain height was reduced by the boundary layer displacement thickness [21] and the top and bottom walls were modeled with slip conditions. Conversely, no reduction in width was applied due to the negligible transversal blockage effect, and slip conditions were enforced on side walls. A uniform freestream velocity of 4m/s was imposed at the inlet with a turbulence intensity of 2% [2], while atmospheric pressure was assigned at the outlet.

The computational mesh is composted by a Cartesian grid, featuring a total of 11.4 × 106 hexahedral cells. To enhance resolution around the rotor and wake, two cylindrical refined regions are implemented, resulting in a cell size of 0.017 m (0.0073D) in the vicinity of the rotor, as shown in Fig. 2. The blades are then discretized with 75 actuator points. A systematic analysis of the mesh resolution is reported in Ref. [13].

Fig. 2
Numerical domain and actuator lines shown in white
Fig. 2
Numerical domain and actuator lines shown in white
Close modal

Unsteady RANS simulations are performed to simulate the unsteadiness arising from both surge motion and rotor-blade rotation, while the effects of turbulence are introduced via the kω SST turbulence model. The RANS model ensures reliable results in the present context, in which the focus is the analysis of the rotor loads, as documented by the broad validation conducted in Ref. [1]. The coupling of pressure and velocity is accomplished through the PIMPLE algorithm, involving two inner loops to accelerate the convergence and two outer loops to ensure convergence of the implicit velocity sampling technique.

The time-step complies with the constraint that the actuator line should not cross more than one cell per time-step leading to Δt=5×104 s, resulting in a Courant number of less than 0.5. The temporal integration employs a second-order Crank–Nicholson scheme with 0.9 blending coefficient, while the interpolation utilizes the third-order accurate Gauss QUICK scheme for the advective term and Gauss linear corrected schemes for Laplacian terms.

5 Validation of the Actuator Line Model With Multifidelity Models

This section presents the ALM results of the load cases reported in Table 1. Figure 3 shows the computed thrust, torque, and power with respect to surge position; four instants (T0,T1,T2,T3) equally spaced over the surge period of Fig. 1 are marked on the curves reported in Fig. 3. Finally, the time-mean mean values for each case are denoted by dotted horizontal lines; the solid black lines indicate the mean value for the fixed-bottom case. In all analyzed cases, average values differ from the fixed-bottom case, indicating the presence of unsteady phenomena, as will be detailed in Secs. 68.

Fig. 3
Thrust, torque, and power along a surge period. Dotted lines denote the time-mean values, and solid black lines indicate the mean value for the fixed-bottom case.
Fig. 3
Thrust, torque, and power along a surge period. Dotted lines denote the time-mean values, and solid black lines indicate the mean value for the fixed-bottom case.
Close modal

The variation of angular speed over time in LC2_16 results in a marginal increase of the time-averaged thrust and torque with respect to LC2_12, whereas the thrust amplitude notably increases by 62.3%, compared to LC2_12 one. As a result, the fluctuation in angular speed results in a higher power (both mean and amplitude), which also exhibits an asymmetrical pattern. The collective blade pitch-to-feather control (LC2_17) manages to reduce the thrust amplitude by 39% while maintaining almost unaltered the mean torque of LC2_12 (even though in the LC2_17 the blade pitch angle oscillates around 1.5deg and not around 0deg, and this shifts the thrust mean value by 8.7%).

A thorough validation study of the present ALM code is documented in Ref. [13] for the fixed-bottom case as well as for surge and pitch platform motions. However, due to the specific features of the present cases, a validation is now proposed against several multifidelity models made available by the OC6 Phase III project [1,22]. From the database, the datasets reported in Table 2 were extracted, indicating the computational model employed by each institution.

Table 2

Results of current work and multifidelity models from OC6 Phase III database

LabelModel
VB (current work)ALM with static polars
NRELaBEM with static polars
NRELbDynamic BEM with static polars
UNIFIALM with unsteady polars
DTUaFVW with unsteady polars
DTUbHybrid FVW/CFD with unsteady polars
LabelModel
VB (current work)ALM with static polars
NRELaBEM with static polars
NRELbDynamic BEM with static polars
UNIFIALM with unsteady polars
DTUaFVW with unsteady polars
DTUbHybrid FVW/CFD with unsteady polars

The main differences between the methods of Table 2 are the governing equations of the models and the use of static or dynamic polars. Figure 4 compares the mean, amplitude, and phase shift values. The hatched bars represent models that use static polars. The ALM model presented in this paper is identified as “VB” (vortex-based model). Generally, a good correspondence among all the models is observed, especially regarding mean values of thrust and torque. However, a slight overestimation of the amplitudes is noted compared to medium-fidelity models DTUa and DTUb. In comparison to these models, a slight underestimation of the delay in thrust and torque with respect to the platform motion is observed (it is recalled that, in this work, negative phase shifts correspond to a delay with respect to the platform motion xs in Fig. 1). From the comparison of integral quantities, it is not possible to conclude whether differences in amplitude and phase shifts are due to the static polar approach since scattered data do not resemble into a univocal trend.

Fig. 4
Mean values, amplitudes, and phase shift of thrust and torque for LC2_12, LC2_16, and LC2_17: (a) thrust mean values and amplitudes, (b) torque mean values and amplitudes, and (c) phase shift with respect to surge motion
Fig. 4
Mean values, amplitudes, and phase shift of thrust and torque for LC2_12, LC2_16, and LC2_17: (a) thrust mean values and amplitudes, (b) torque mean values and amplitudes, and (c) phase shift with respect to surge motion
Close modal

It is interesting to note that the NRELa prediction, obtained with a standard BEM approach, is able to capture the overall mean rotor loads and also the amplitudes for LC2_12, while deviations appear for the other cases: with respect to the amplitudes predicted by the ALM, the LC2_16 case exhibits an underprediction of 12% for thrust and 13.8% for torque, while the LC2_17 case shows an overprediction of 13% for thrust and 9.8% for torque. Moreover, the phase-shift of BEM predictions is 90deg for all the case examined. This indicates the relevance of applying techniques of higher fidelity levels when studying active control strategies, as done in this work.

6 Reference Case LC2_12: Comparison Between Actuator Line Model and Blade Element Momentum

In this section, the load case 2_12 is investigated in detail as it serves as a reference for the interpretation of the induction coefficients, the rotor loads, and the turbine power of LC2_16 and LC_17 cases. Both ALM (VB model) and BEM (openfast [23]) simulations are presented, to assess the prediction capabilities of the medium-fidelity approach against the ones of a low-fidelity model in the context of FOWTs. In particular, the analysis of the induction factors allows highlighting the main differences between a model that solves the flow in the inertial frame of reference and retrieves the induction field from the fluid domain (ALM) and a model in which the induction stems from a streamwise momentum balance (BEM).

In openfast simulations, a rigid platform motion was enforced by resorting to an ad hoc technique, based on the use of the ExtPtm module, presented in Ref. [24]. The BEM flow model incorporates steady polars and Prandtl's tip and hub loss corrections, as well as the Glauert correction when the momentum theory breaks up. The freestream velocity is assumed as 4.19 m/s to account for the wind-tunnel blockage, as openfast treats all simulation in open-field (see Ref. [1] for further details).

Before proceeding with the comparison, it is important to highlight that the two models differ in terms of computational cost. The low-fidelity BEM model has a computational cost on the order of minutes, whereas the medium-fidelity ALM model demands several days for computation.

In case FOWTs, and in particular in the leeward phase of the platform motion, the induction becomes a key parameter to identify the state of the rotor not only regarding the flow deceleration upstream of the turbine but also the interactions of the rotor with its own wake. According to Kyle and Früh [25], floating turbines may experience translational or rotational movements of comparable magnitude to the wind speeds. Typically, the standard condition of a wind turbine is referred to as windmill condition. When the airflow around the rotor becomes highly unsteady and the rotor interacts with its own wake, the turbine works in the TWS. With further increase of motion, the rotor may strongly interact with its own tip vortices in presence of vortex recirculation, leading to what is known as the VRS. Should the leeward movement of the rotor surpass the free wind speed, the flow circulates back through the rotor swept area, resulting in propeller-like condition called propeller state.

Moreover, a noteworthy aspect of FOWTs simulations is the definition of the axial induction factor, as it can be formulated in two alternative ways, depending on the frame of reference: in this work, the induction factor is defined absolute if evaluated using absolute axial velocities (Eq. (5)), while it is defined relative if evaluated using the apparent (or relative) velocities if perceived by an observer integral with the platform (Eq. (6))
(5)
(6)

where U0 is the undisturbed upstream wind velocity, Vplat is the inertial velocity of the platform, and Uax is the absolute axial velocity evaluated at the actuator line point by ALM with vortex-based approach.

Figure 5 shows the two induction factor trends across the blade span for T0,T1,T2,T3 time-steps, representing with dashed lines the ALM results and with solid lines the BEM outcomes. The BEM predictions provide generally higher values than the ALM, for both the relative and absolute induction factors and for each time-step considered. The BEM results show a steep trend toward the blade tip which appears not fully coherent with the reduction of forces caused by the axial velocity increase. At the time instant T0 (maximum platform leeward velocity), ALM and BEM trends exhibit an increase in absolute induction at the outboard r/R>0.50, which is even amplified in the relative one. This increase, as described later, leads to an amplification of arel affecting the phase shift of axial induction velocity.

Fig. 5
Absolute (aabs) and relative (arel) axial induction factors along the blade span during a surge period
Fig. 5
Absolute (aabs) and relative (arel) axial induction factors along the blade span during a surge period
Close modal

Figure 5 illustrates that the trends for T1 and T3, when the turbine is still, of the BEM simulations are perfectly overlapped; this alignment is attributed to the steady BEM approach. Conversely, the ALM shows counterintuitively overlapping pair of curves of aabs for T0T1 and for T2T3, thus evidencing the relevance of unsteady effects in terms of a lag in the flow field perturbation resulting from the platform motion.

A detailed picture of the turbine unsteadiness is reported in Fig. 6, which shows spanwise trends of amplitude (frame a) and phase shift (frame b) of both absolute and relative induction factors, along with the one of phase shifts (frame c) of the induced axial velocity (Uind,ax) and of the axial velocity (Uax). The phase shift is defined as the phase-lag between the peaks of the platform motion displacement (xs in Fig. 1) and of the quantity of interest, with negative values indicating a phase delay, otherwise an advance.

Fig. 6
LC2_12: amplitude (a) and phase shift (b) of axial induction factors and phase shift of absolute axial velocities (c)
Fig. 6
LC2_12: amplitude (a) and phase shift (b) of axial induction factors and phase shift of absolute axial velocities (c)
Close modal

It is noticeable that the phase-shift quantities defined in the inertial reference frame (aabs, Uind,ax, and Uax) feature an abrupt phase reversal slightly above r/R=0.6, passing from a phase delay at the hub to a phase advance at the tip. The phase shift is exactly 90deg and +90deg for BEM results, whereas the ALM predicts phase lags different from ±90deg, due to unsteady effects. Conversely, arel, defined in the relative reference frame, does not show any phase reversal, but exhibits a constant advanced phase of +90deg for BEM and a phase shift varying from +120deg at the hub to +75deg at the tip for ALM. The same behavior is observed on all the aerodynamic quantities (e.g., forces and angles of attack).

Concerning the BEM predictions, Uind,ax results from the combination of arel and of the prescribed Uapp (Eq. (6)). The arel sine wave anticipates the platform motion by +90deg and exhibits a growing amplitude from the hub to the tip (see the first plot of Fig. 6). In contrast, the Uapp sine wave is constant along the span, and it is 90deg delayed with the motion, as shown in Fig. 1. The product of these two counterphase sine waves generates a sine wave in phase with Uapp (thus with a 90deg delay) where the amplitude of arel is limited, namely, in the root region. When the amplitude of arel reaches the value of 0.15 at r/R=0.55, the phase reversal of the product occurs. It can be concluded that the induction factor acts as a “filter” on the apparent wind speed: as long as the amplitude of arel is limited, the sinusoidal wave of Uapp remains predominant and the axial induction velocity retains its phase, while, as the amplitude of arel increases, the phase of this latter starts to prevail and reverses the phase shift in the axial induction velocity.

In the ALM, the explanation of the results is different, as Uind,ax is directly calculated from the sampled velocities from the flow field close to the rotor. A partial interaction of the rotor with its wake, visible in the flow field, occurs at the instant T0. In this motion phase, the turbine moves leeward with the maximum speed, involving the lowest distance during the surge period between the low-momentum near-wake region and the rotor itself. The blade sheds its tip vortex close to the one of the previous blade. This condition affects the velocity sampling of the ALM algorithm in the tip region, resulting in a lower Uax sampled caused by the higher velocity deficit generated by the vortex of the preceding blade. This vortex interaction phenomenon, absent for hub sections, generates a kind of variation in the dynamics of the tip sections. The analysis concludes that the ALM can detect the phase delay of Uax and Uind,ax with a difference of about 50deg with respect to BEM counterpart. As a result, it is stated that BEM and ALM, although the rotor loads are in sound agreement, predict different induction factor amplitudes, higher for BEM.

To quantitatively assess the interaction with the wake, the examination of the rotor operating condition around the instant T0 is now proposed. Several interpretations can be found in literature regarding the assessment of the operational state of the rotor according to the interaction with its wake (windmill state, turbulent wake state, vortex ring state, etc.). According to Ferreira et al. [3], such states do not describe the condition of the rotor itself but rather characterize the state of the stream-tube. In unsteady load conditions, however, a rotor may experience instantaneous loading corresponding to a turbulent wake state but, if the oscillation is too fast, the flow may not have sufficient time to accelerate, resulting in the stream-tube persisting in a windmill state. The present study is focused on the rotor operation rather than on the stream-tube/wake state, so in the following the term “turbulent wake state” will be strictly associated with the rotor aerodynamics, and in particular to the phases of the period in which the blade interacts with the tip vortex shed by the preceding blade.

Having in mind this clarification on the terminology employed, the objective of the analysis is to determine whether the models predict the TWS state and in which portion of the surge period. To this end, the parameter defined in Eq. (7), derived from helicopter theory [26], will be used
(7)

where Vc is the opposite of the velocity normal to the rotor, T¯ is the mean value of thrust during the surge period, ρ stands for air density, and A is the rotor swept area. Vh can be interpreted as the velocity in the wake when there is an exchange of T¯. According to existing literature, the TWS occurs when 2<Vc/Vh<1, and values between 1<Vc/Vh<0 are indicative of the turbine entering its own wake, resulting in the VRS.

In Fig. 7(a)), the trend of Vc/Vh is shown over a surge period for BEM (solid lines) and ALM (dashed lines). The TWS condition is triggered in the leeward motion of the rotor, between 296deg and 64deg. The persistence of the TWS condition, represented by the shaded areas in the figure, appears slightly more pronounced for ALM. Evaluating this parameter allows identifying the TWS at the rotor level (in the noninertial reference frame). For a more detailed analysis of blade-distributed quantities, the condition Vc/Vh>2 is expressed as a function of the axial induction factor, with the procedure reported in Eq. (8), to determine the value of a required to “enter” the TWS
(8)
Fig. 7
Analysis of the rotor wake state in terms of: (a) ratio Vc/Vh for ALM and BEM; (b) and (c) absolute and relative induction coefficients for ALM; and (d) angle of surge motion for which rotor stops working in TWS for different blade stations for ALM. Shaded areas denote TWS regions.
Fig. 7
Analysis of the rotor wake state in terms of: (a) ratio Vc/Vh for ALM and BEM; (b) and (c) absolute and relative induction coefficients for ALM; and (d) angle of surge motion for which rotor stops working in TWS for different blade stations for ALM. Shaded areas denote TWS regions.
Close modal
Finally, Eq. (9) defines the condition for which TWS is reached in terms of induction factors, and it depends on the freestream velocity and the platform velocity only. Therefore, it is possible to highlight a field of induction factor connected to TWS as depicted in Figs. 7(b) and 7(c) 
(9)

Through this correlation, one can evaluate the phase at which the TWS is reached, for each radial blade station. Having provided two definitions of the axial induction factor, the assessment is conducted for both the relative and absolute induction factors aabs and arel, focusing on the ALM simulation for conciseness. Frames b and c of Fig. 7 report the trends of aabs and arel over the surge period for several sections along the blade span, along with the limit values reported in Eq. (9). The shaded gray areas represent the portions of the period where each section experiences the TWS. Figure 7(d) shows the phases of surge period corresponding to the instants where TWS condition ceases to occur (the markers correspond to the examined radial stations considered in Figs. 7(b) and 7(c) for both aabs and arel. The induction factors reveal that the blade sections at the hub are the first to reestablish the windmill state, while those at the tip exit from the TWS much closer to the maximum leeward platform displacement (Vplat is decreasing between T0 and T1). Although the differences are relatively small, there is a need to select the induction factor for reference. Since, in this study, the TWS concept is solely applied to the rotor operational state, not to the flow in the wake, adopting arel, defined in the rotor reference frame, aligns with placing Vc=Uapp in the numerator, promoting greater coherence.

7 Angular Speed Control

This paragraph thoroughly examines the rotor-aerodynamic effects resulting from an angular speed control of the turbine. The analysis focuses on a detailed comparison between the LC2_16 case and the LC2_12 case. Utilizing the integral quantities depicted in Fig. 3 as a starting point, one can observe slight variations in the mean values of thrust and torque (dashed lines) of these two cases in comparison to the mean value of the corresponding fixed-bottom case (solid black line), deviating the solution from the assumption of an almost quasi-steady condition. Additionally, a clear asymmetry appears in the power trend of LC2_16, due to the variation in the angular speed.

The same conclusions can be derived by analyzing the spanwise trends of angle of attack and normal force at four time-steps during the surge period, as depicted in Fig. 8. The LC2_16 case is represented by solid lines, while the LC2_12 case is depicted with dashed lines. At instants T1 and T3, the rotor is still at the extremes of the motion, facing the same apparent wind as in the fixed-bottom case. In case of a quasi-steady assumption, the rotor should mimic the behavior of the fixed-bottom case (black solid line) at these time-steps. However, the figure reveals that the attack angle (AoA) and rotor normal force Fn values at T1 differ from those at T3 and from the fixed-bottom reference, confirming the presence of unsteady effects in both the LC2_12 and LC2_16 simulations.

Fig. 8
Angle of attack, normal force, and relative velocity for LC2_16 and LC2_12. Solid black line denotes the fixed-bottom case.
Fig. 8
Angle of attack, normal force, and relative velocity for LC2_16 and LC2_12. Solid black line denotes the fixed-bottom case.
Close modal
Figure 8 also highlights the impact of the angular speed control. A comparison between the two cases indicates that varying the rotational speed decreases the amplitude of oscillation of angle of attack, while increasing that of the normal force and, consequently, of the thrust. The change in the peripheral speed of each blade section diminishes the amplitude of the AoA by influencing the denominator in its calculation as in the following equation:
(10)

When calculating Fn, instead, the primary effect of the angular speed variation arises from the magnitude of the relative velocity (right plot of Fig. 8), which exhibits maximum and minimum values at T2 and T0, respectively, thus amplifying the maximum and minimum of Fn.

Beside these global features, it is interesting to ascertain whether the phase shift inversion identified in LC2_12 is also present in LC2_16 and, if so, to explore the impact of the angular speed control. Figure 9 shows both the amplitude and phase of the induction coefficients, as well as the phase of velocity signals Uind,ax and Uax. Solid lines represent LC2_16 case, while dashed lines depict the LC2_12 case. Similar to LC2_12, there is no phase inversion for arel, while phase inversion clearly appears for the absolute quantities aabs, Uax, and Uind,ax, systematically when the amplitude of arel exceeds 0.15. However, the angular speed control induces a translation toward the tip of the section where the phase inversion occurs, shifting from r/R=0.67 for LC2_12 to r/R=0.77 for LC2_16, as well as a general change in the phase shifts of the different quantities. With reference to xs, below the phase-inversion section, the Uax signal appears in reduced advance, whereas aabs and Uind,ax feature an increased delay (Fig. 9(c)). Conversely, above the phase-inversion section, the signals are shifted toward smaller surge motion angles, resulting in a reduced delay for Uax and an increased advance for aabs and Uind,ax (Fig. 9(c)). As a result, the variation in angular speed mitigates the variability in phase shift along the blade span.

Fig. 9
Amplitude (a) and phase shift (b) of axial induction factors and phase shift of absolute axial velocities(c)
Fig. 9
Amplitude (a) and phase shift (b) of axial induction factors and phase shift of absolute axial velocities(c)
Close modal

Finally, the rotor operational state is analyzed to investigate the excursion in the TWS region. Figure 10 demonstrates that the LC2_12 and LC2_16 scenarios result in the same Vc/Vh ratio, predicting a TWS in the same range of the surge motion.

Fig. 10
TWS analysis from a global and local point of view: (a)TWS for LC2_16 and LC2_12 and (b) motion phase exiting the TWS for LC2_16 and LC2_12
Fig. 10
TWS analysis from a global and local point of view: (a)TWS for LC2_16 and LC2_12 and (b) motion phase exiting the TWS for LC2_16 and LC2_12
Close modal

In order to provide a more robust explanation for the blade-distributed velocities, a comparison is made by examining the angles for which TWS is triggered at each radial station (Fig. 10(b)). As illustrated, in the LC2_16 case (solid lines), the inboard blade spend a longer surge motion phase inside the TWS (greater ψ) compared to the LC2_12 case (dashed lines), while at the tip, the exit from the TWS occurs earlier than in the LC2_12 case. This confirms that this kind of control tends to mitigate the rotor variability between the hub and the tip, reducing the phase shift gap, thus fully confirming what observed in Fig. 9(c).

This analysis leads to the conclusion that angular speed control does not only impact the overall loads on the rotor but also exerts a local influence on the blade-distributed induction. This impact can become more prominent especially when dynamic effects play a crucial role, such as when the influence of the rotor and wake interaction is established. Such a scenario is attainable in conditions characterized by low freestream velocity, high platform velocity, and consequently, high loads as concluded by Papi et al. [4].

8 Blade Pitch Angle Control

The LC2_17 case, developed in the context of the OC6 project, involves a sinusoidal oscillation of the blades pitch angle around an average value of 1.5deg, whereas the reference case LC2_12 and also the angular speed control case LC2_16 present θp=0deg. This bias mean blade pitch angle does not allow for a direct comparison among the cases; hence, a new case LC2_18 is proposed which features the same blade pitch angle amplitude of ±1.5deg around a null mean value.

The integral quantities, thrust and torque, of the mentioned cases are illustrated in Fig. 11. The effect of the pitch-to-feather in the LC2_18 case is well visible in the reduction of the thrust amplitude, while a smaller mitigation is applied to the torque. As previously mentioned, these cases are designed to explore the impact of regulation on dynamic behavior and are not intended to achieve an actual and effective load regulation.

Fig. 11
Thrust and torque over one surge period for LC2_12, LC2_17, and LC2_18
Fig. 11
Thrust and torque over one surge period for LC2_12, LC2_17, and LC2_18
Close modal

The impact of the regulation along the blade can be appreciated in Fig. 12, with solid lines representing the LC2_18 case, dashed lines corresponding to LC2_12, and solid black line denoting the fixed-bottom case. At T0, the rotor moves leeward at its maximum speed, experiencing the minimum apparent velocity; the contemporary negative peak in pitch angle results in a higher angle of attack than would occur with zero pitch angle. Conversely, at T2 (middle of windward phase), the apparent velocity on the rotor is at its maximum, so the contemporary positive peak in pitch angle limits the value of the angle of attack. As a result, the pitch control limits the oscillation of angle of attack during the surge motion. Since the peripheral speed is kept constant, the same effect is visible in normal force and, thus, in the thrust. Similar to cases LC2_12 and LC2_17, the nonperfect overlap of the quantities at instants T1 and T3 highlights the onset of unsteady effects.

Fig. 12
Angle of attack and normal force for LC2_18 and LC2_12. Solid black line denotes the fixed-bottom case.
Fig. 12
Angle of attack and normal force for LC2_18 and LC2_12. Solid black line denotes the fixed-bottom case.
Close modal

Figure 13 depicts the amplitude and phase of the induction coefficients, along with the phase of velocities, in the same format of Figs. 6 and 9. As observed in previous cases, there is no phase inversion for relative quantities, while phase inversion appears in the trends of absolute quantities when the amplitude of the relative axial induction factor increases. Unlike the LC2_16 case, however, the blade pitch angle control induces a shift toward the hub of the section where the phase inversion occurs, shifting from r/R=0.67 to r/R=0.57, without a remarkable change in extent of the phase shift.

Fig. 13
Amplitude (a) and phase shift (b) of axial induction factors and phase shift of absolute axial velocities (c)
Fig. 13
Amplitude (a) and phase shift (b) of axial induction factors and phase shift of absolute axial velocities (c)
Close modal

As an additional tool for analysis, the operational state of the rotor is examined also for the LC2_18 case. Figure 14 illustrates that both the LC2_12 and LC2_18 scenarios provide a similar Vc/Vh ratio, predicting a TWS in the corresponding region of the surge motion at the rotor level. To investigate the turbine dynamics in more detail, section-by-section comparison of the moments of exiting the TWS is performed, and it is shown in the frame b of Fig. 14.

Fig. 14
TWS analysis from a global and local point of view: (a)TWS for LC2_18 and LC2_12 and (b) motion phase exiting the TWS for LC2_18 and LC2_12
Fig. 14
TWS analysis from a global and local point of view: (a)TWS for LC2_18 and LC2_12 and (b) motion phase exiting the TWS for LC2_18 and LC2_12
Close modal

As illustrated, the slope of the curves, both for arel and aabs, is comparable between the LC2_12 and LC2_18 cases. Therefore, despite the slight difference in values, the dynamics along the blade is not significantly affected by the variation in pitch angle. This is confirmed by the absence of change in the phase shift values of Fig. 13. Therefore, the variation of pitch angle has less influence than that of angular speed on the dynamic behavior along the blade span.

9 Conclusions

This study has explored the aerodynamic response of a laboratory-scaled FOWT model tested in the large-scale wind tunnel of Politecnico di Milano. Two control strategies in presence of surge motion are examined: a rotor speed control, where the rotor experiences overspeed to track the variation of the apparent wind, and a collective blade pitch strategy, where the blade is periodically feathered to reduce the oscillations of forces on the rotor. These scenarios are simulated using an in-house actuator-line CFD software implemented in the openfoam framework. Numerical validation against multifidelity models from the OC6 Phase III project shows consistent agreement across all models, especially concerning average thrust and torque values. However, a slight overestimation of amplitudes and underestimation of phases in the load response are observed compared to other medium-fidelity models. The conventional BEM approach exhibits much larger discrepancies in the regulated scenarios, highlighting the importance of employing higher-fidelity techniques when investigating active control strategies, as done in this study.

The comparison between the baseline case without control strategy and the results derived from the BEM method quantifies the distinctions between low-fidelity (BEM) and medium-fidelity (ALM) approaches, offering insights into the dynamic effects in an unregulated scenario. Differences emerge in the estimation of the axial induction factor between the two models. The ALM demonstrates its capability to simulate unsteady effects especially visible at the extremes of the surge process, when the motion inverts itself. Even though the BEM and ALM models have a very different numerical formulation, both their predictions exhibit a reversal phase shift from the hub to the tip sections for absolute quantities, such as Uind,ax. Once more, the ALM simulation shows different phase shifts, showcasing its capability to capture unsteady effects. The authors attribute this reversal of blade dynamics to the rotor interaction with the wake at the instant of maximum leeward speed of the turbine. The interaction is quantified by examining a suitable parameter both at the rotor and blade levels. The sections at the hub and tip leave the interaction zone with the vortex at different instants, giving rise to different dynamics (flow patterns) along the span.

Examining the regulation techniques, the analysis has gone beyond evaluating the impacts of control strategies on blade loads to characterize their effects on rotor dynamics. The study concludes that angular speed control has a more significant influence on loads, due to the dependence of torque and thrust on forces. Consequently, a variation in peripheral speed leads to a more substantial change in forces compared to an alteration in the angle of attack. Moreover, the study shows that the regulations have distinct impacts on rotor dynamics. Specifically, angular speed regulation helps alleviate the variability in rotor dynamics between the hub and the tip, reducing the gap in phase shift along the span. In contrast, the change of pitch angle has a comparatively lower effect on the dynamic behavior along the blade span.

The paper has highlighted the key role of the usage of medium-fidelity simulation tools to quantify the unsteady effects during surge motion. As shown in the LC2_12 case, the BEM method provides a different estimate of induction, although it produces comparable outcomes concerning the operational rotor condition when considering the Vc/Vh parameter. These disparities between models emphasize the significance of employing a medium-fidelity ALM. This approach, by resolving the flow around the rotor, opens up the possibility of developing more accurate approaches to determine the working rotor state.

Acknowledgment

This study was carried out within the NEST—Network 4 Energy Sustainable Transition (D.D. 1243 02/08/2022, PE00000021) and received funding under the National Recovery and Resilience Plan (NRRP), Mission 4 Component 2 Investment 1.3, funded from the European Union—NextGenerationEU. This paper reflects only the authors' views and opinions; neither the European Union nor the European Commission can be considered responsible for them.

Funding Data

  • National Recovery and Resilience Plan (NRRP), Mission 4 Component 2 Investment 1.3, funded from the European Union—NextGenerationEU (PE00000021; Funder ID: 10.13039/501100000780).

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

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