Aeroservoelastic state-space vortex lattice modeling and load alleviation of wind turbine blades

Authors


Abstract

An aeroservoelastic model, capturing the structural response and the unsteady aerodynamics of turbine rotors, will be used to demonstrate the potential of active load alleviation using aerodynamic control surfaces. The structural model is a geometrically non-linear composite beam, which is linearized around equilibrium rotating conditions and coupled with time-domain aerodynamics given by a linearized 3D unsteady vortex lattice method. With much of the existing work relying on blade element momentum theory with various corrections, the use of the unsteady vortex lattice method in this paper seeks to complement and provide a direct higher fidelity solution for the unsteady rotor dynamics in attached flow conditions. The resulting aeroelastic model is in a state-space formulation suitable for control synthesis. Flaps are modeled directly in the vortex lattice description and using a reduced-order model of the coupled aeroelastic formulation, a linear-quadratic-Gaussian controller is synthesized and shown to reduce root mean square values of the root-bending moment and tip deflection in the presence of continuous turbulence. Similar trend is obtained when the controller is applied to the original non-linear model of the turbine. Trade-offs between reducing root-bending moment and suppressing the negative impacts on torsion due to flap deployment will also be investigated. Copyright © 2014 John Wiley & Sons, Ltd.

1 Introduction

As wind turbines are gradually upscaled for larger energy capture, their blades are increasingly elongated and subject to higher loadings. Accompanying this is an increase in flexibility that results in greater risks of fatigue or material strength failure. Designing for increased flexibility not only necessitates accurate predictions of the aeroelastic effects but also requires stronger materials coupled with localized actuators providing more control authority to overcome unwanted loads and oscillations.

A number of design codes have been developed to study the aeroelastic effects on wind turbines. These include NREL's open source FAST, DTU's FLEX5, DTU Wind's HAWC2, TU Delft's DU_SWAMP, and GL Garrad Hassan's Bladed, among others [1-7]. These well-established design codes have gone through extensive validation and are widely adopted by wind engineers. Their structural models can be divided mainly into finite elements method and assumed modes approaches, while the aerodynamics relies on blade element momentum (BEM) theory with various, mostly empirical, corrections to account for dynamic inflow, stall and tip loss effects [8, 9]. As turbines get larger, there will be bigger aeroelastic couplings and larger structural deformations than on existing rotors, an alternative solution based on the time-domain unsteady vortex lattice method (UVLM) [10], coupled with composite beam descriptions of the blade structure [11], could provide a better characterization on the unsteady rotor dynamics, assuming attached flow conditions. Moreover, as we will show in this paper, and opposed to the codes described earlier that have been designed mainly for simulation, such an approach can be directly formulated in a suitable manner for model reduction and control synthesis.

In terms of actuation devices, the use of pitch controls for load alleviation seems to be a natural solution, given that they currently exist on turbines mainly for speed regulation. However, their responses may be slow [12, 13] and are only able to overcome the lower frequency loadings. Distributed load alleviation systems, placed along the blade span, would overcome such limitations, and indeed, these concepts, derived from well-known solutions in air vehicles, are gaining momentum in wind turbine research. They can be designed to complement existing pitch control mechanisms by tackling higher frequency loadings or even be used to complement aeroelastic twisting of the blade [14]. For instance, using trailing-edge flaps for load reduction, Frederick et al. [15], Riziotis et al. [16] and Lackner et al. [13] were able to achieve significant reduction in loadings and aerofoil displacements, whereas Barlas et al. [4, 17] and Wilson et al. [18] demonstrated the performance of multiple flaps on a full rotor. Concepts for conformable trailing-edge flaps [19-21] and microtabs [22] have also shown great potential. Comparisons between various active control concepts were investigated by various authors [23, 24], and Barlas et al. [25] have presented an overview of active control methods that are currently being investigated. To date, there is no commercial application of the active flap concept on existing wind turbines, but a recent full-scale experiment was conducted by Castaignet et al. [26] on a Vestas V27 wind turbine using frequency-weighted model predictive control and demonstrated the potential of trailing-edge flaps by reducing average flapwise blade root load by up to 14%. Also, Barlas et al. [27] performed small-scale wind tunnel experiments, and Sandia National Laboratories have designed and built turbine blades equipped with aerodynamic flaps for testing [28, 29].

Barlas et al. [4] have reviewed the most recent developments in the use of active flaps on wind turbines for load reduction, listing the performance obtained through the use of different control methods. A wide spectrum of performances has been reported with results ranging from 10% up to 30% reduction in root-bending moments (RBM), depending on the type of controller used and also on the size and distribution of flaps. To the best of our knowledge, the only work that involves the modeling of wind turbine blades using vortex methods is by Riziotis et al. [16], who used a vortex panel code coupled with a structural module. It was demonstrated through the use of Proportional-Integral-Derivative (PID) control that, under an exponential wind shear profile, a load reduction of up to 30% can be achieved using flaps that range from 15% to 50% span. In fact, most of the works in active aeroelastic control of wind turbines has relied on classical control methods, such as PD and PID, with only a few of the more recent works considering more advanced techniques, such as Linear-Quadratic Regulator (LQR) or predictive control [4, 18]. This is probably because the intention has normally been to show the potential of feedback control in enhancing the aeroelastic performance of the blades, but in a recent study [30], we have demonstrated that Proportional_Derivative (PD) control required up to 70% more actuation power than a robust controller under similar load reduction targets.

Within this context, this paper will introduce an efficient implementation of the vortex lattice method in state-space representation, which will be coupled to a linearized structural dynamics description of the blade to produce a compact form suitable for aeroservoelastic analysis of wind turbine rotors. The UVLM formulation provides a higher fidelity solution for the unsteady aerodynamics compared with BEM models, although it is limited to attached flow conditions. Detailed validation is performed against standard aeroelastic test cases to provide confidence in our formulation of the problem. The approach will be then used to model a clamped NREL 5-MW reference offshore wind turbine blade, on which we will first analyze the optimal spanwise location and size of flaps for load reduction potential. Subsequently, under a uniform turbulent gust input, the performance of flaps in reducing the root mean square (rms) values of both the RBM and tip deflections will be demonstrated using linear-quadratic-Gaussian (LQG) controllers that are equipped with knowledge of the turbulence spectrum. The performance of the controllers will also be demonstrated on the non-linear model. Trade-offs between reducing RBM and suppressing the negative impacts on torsion due to flap deployment will be finally investigated.

2 Methodology

The aeroservoelastic tool to model the dynamics of large wind turbine rotors will be adopted from the integrated framework for the simulation of high aspect ratio planes (SHARP) [11, 31-33]. SHARP has been extensively verified in previous works for flexible aircraft applications, including static aeroelastic analyses, linear stability analyses, control synthesis and non-linear open-loop time-marching simulations. The following description provides an overview of the underlying structural and aerodynamic models, which have been tailored in this work for application to large wind turbine dynamics.

2.1 Composite beam model

Taking advantage of the slenderness of the blades, the structural deformations will be modeled using a composite beam formulation written in a rotating frame of reference [34, 35]. In its original implementation [11], the structural model can account for large static and transient deformations of the blades, which have been reduced to a 1D representation in the 3D space, using an appropriate cross-sectional analysis methodology [36]. For the purpose of control synthesis, this work will focus on the linearized approach that provides a compact form of the equations of motion (EoM) around a possibly geometrically non-linear steady-state equilibrium.

As shown in Figure 1, the deformation of the structure is described in terms of a hub-fixed (structural) reference coordinate system S, which moves according to the hub translational and angular velocities vS(t) and ωS(t). The overall motion of the rotor is given by the prescribed hub rigid-body velocities inline image. The angular velocity ωS(t) will be used in this work to prescribe the rotation of the turbine, whereas the translational velocity vS(t) will be relevant in future work to include the motion of the tower. An inertial frame G is used in this work simply to track the orientation of the global frame S. To account for large blade deformations, the local orientation of each cross section along the beam reference line is defined by a local coordinate system B and is parameterized at time t by the Cartesian Rotation Vector Ψ(s,t), where s is the arc length along the beam reference line. Hence, the nodal positions inline image, expressed in the hub-fixed frame S, and the cross-sectional orientations Ψ(s,t) form the independent set of variables for the structural problem.

Figure 1.

Multi-beam configuration with the definition of reference frames for the structural model.

Even though this work will only focus on a single blade, the formulation caters for multi-beam configurations, as illustrated in Figure 1. After a finite element discretization, the EoM for this structural dynamics system are then obtained as [31, 35]

display math(1)

where the discrete mass matrix inline image and the discrete gyroscopic, elastic and external generalized forces, Qgyr,Qstif and Qext, respectively, are obtained through a finite element discretization of the primary variables. η is the column matrix with all the nodal displacements and rotations. Equation (1) captures the possibly large deformations of the blades by balancing the discrete inertial and elastic forces with the external aerodynamic forces. The rotational effects due to ν and inline image, which have known values, are incorporated through the coupling mass matrix mν and the gyroscopic forces Qgyr.

To arrive at a compact state-space form of the aeroelastic EoM suitable for control synthesis, the structural dynamics equations are linearized around a geometrically non-linear steady-state equilibrium condition inline image. This steady-state solution is obtained by neglecting all time derivatives in Equation (1). The incremental form of the beam EoM is

display math(2)

where the contribution to the (constant) mass, damping and stiffness matrices has been obtained through direct linearization of the different generalized forces with respect to η and its time derivatives. A detailed derivation of the linearization can be found in Hesse & Palacios [11]. The resulting linearized beam equations also preserve the rotational effect due to the prescribed angular velocity ωS through the contribution of the linearized gyroscopic forces to the damping and stiffness matrices inline image and inline image, respectively. Additional inertial terms could be included in Equation (2) to account for small variations in the hub velocities but was not explored in this work.

The resulting linear system of the wind turbine structural dynamics can be casted into state-space form. Note that this defines an eigenvalue problem that includes rotational effects and geometrically non-linear equilibrium deformations. Secondly, the discrete-time representation of the state-space system also provides the basis for the integration of the structural dynamics equations with the unsteady aerodynamic formulation, as presented in the next subsection.

2.2 Unsteady aerodynamics model

The aerodynamics are modeled using the discrete-time UVLM [33, 37], which allows non-stationary aerodynamics to be captured in low-speed, high Reynolds number attached flow conditions. The UVLM uses vortex rings as fundamental solutions, which are located in lattices that represent the blades (modeled as lifting surfaces) and their wakes (modeled as thin shear layers). The leading segment of the vortex ring is placed along the quarter chord of each panel. The collocation points are then placed at the three-quarter chord where boundary conditions are imposed.

The wake vortex is convected downstream by both the freestream velocity and also the velocity induced by all other vortices. The influence of the latter is commonly known as wake roll-up and introduces non-linearities in the model. To obtain linear models for control methods, we could freeze the model with wake roll-up and linearize around that point. Alternatively, the effect of wake roll-up can be ignored. Based on a previous study, the effect of wake roll-up on the unsteady response was found to be minimal [33].

In the UVLM, Neumann boundary conditions are imposed on the lifting surface. Hence, the normal velocity at each collocation point due to vortices (blade and wake) and motion of the blade has to be zero. This relationship is given by

display math(3)

where Γb and Γw denote the circulation strengths of the bound and wake vortex rings, respectively (shown in Figure 2). Ac,b and Ac,w are the influence coefficients that give the induced normal velocity to blade surface at collocation points (resolved using the Biot–Savart law) due to bound and wake vortices, and n is the time step. The last term w is the downwash at collocation points and is generated by the motion of the lifting surface wb, the actuators wβ (such as trailing-edge flaps) and external disturbances wδ (such as gusts). The downwash are all in the same time step but the last two contributions wβ and wδ are assumed to be in the time step n instead of n + 1 such that the final EoM are in explicit form

display math(4)

In particular, the downwash due to motion of the lifting surface wb is mapped from the structural beam model. Assuming constant rotational speeds, it can be represented using

display math(5)

where TΔη and inline image project the contribution of the nodal displacements Δη and velocities inline image onto the aerodynamic lifting surface as downwash terms.

Figure 2.

Typical thin lifting surface represented by the unsteady vortex lattice method.

Pressure distribution across each panel on the lifting surface can be computed using the unsteady Bernoulli Equation [37], which in compact form can be written as

display math(6)

where Φk is a row vector containing mostly zeros as the coefficients of the unsteady Bernoulli equation only appear on entries corresponding to the kth collocation point and its adjacent elements. Γb contains circulation strengths of all vortex rings on the lifting surface.

Assuming constant pressure in the panels, the aerodynamic forces Fk on each panel in the three aerodynamic axes shown in Figure 3 can be computed as

display math(7)

where nk is the normal vector to collocation point k.

Figure 3.

Coupling between fluid and structure. Structural displacements and velocities are mapped onto aerodynamic collocation points as downwash. Pressures on the aerodynamic collocation points are mapped onto structural nodes as forces and moments.

Since the UVLM is based on thin wing theory, special care is needed to obtain the leading-edge suction [32, 38]. In steady flow, pressure forces in the direction of flow is canceled by leading-edge suction, arising in zero drag, which is commonly known as the d'Alembert's paradox. In the case of unsteady flow, drag may be present and an approximation given by Katz et al. [37] is used and in compact form given by

display math(8)

where inline image is a row vector containing mostly zeros as the coefficients of drag only appear on entries corresponding to the kth collocation point and its adjacent elements.

2.3 Aeroelastic equations

The continuous-time structural EoM are discretized using the Newmark- β method and then coupled with the discrete-time UVLM. As each lifting surface is comprised of panels while the structure is modeled using beams that run along the span of the blades, aerodynamic loads are mapped onto the beam nodes as shown in Figure 3. The resulting EoM between the linearized composite beam model and UVLM provide the full aeroelastic system in state-space representation for subsequent control synthesis [32]. The equations can be written in the standard form

display math(9)

where the state vector that completely defines the aeroelastic system is inline image. The inputs to Equation (9) are from the downwash terms in Equation (4). Note that the downwash due to motion of lifting surface wb does not appear, since it is coupled to the structural dynamics. The control input wβ represents the downwash due to flap motion, and the external disturbance wδ is the downwash due to gust. The output vector, y, includes the desired output (e.g., blade RBM, tip deflection). From the homogeneous form of the state-space system equation, one also obtains the discrete-time eigenvalue problem that determines explicitly the dynamic aeroelastic stability characteristics of the rotor blade.

3 Closed-Loop Model

The control input, wβ, in Equation (9) contains the flap deflection angle, β, and its time derivative. A single flap is assumed. Hence, a double integrator [39] is introduced such that the control input is now the flap acceleration, inline image

display math(10)

The choice for this term is that acceleration is closely related to forces and it avoids the introduction of a jerk function. As the intention is to demonstrate the methodology for control design, a relatively simple disturbance is assumed. The inflow speed to the rotor is assumed to be constant and disturbance enters the system in the form of turbulence in the longitudinal direction that is homogeneous in the rotor disk. It is simulated by passing white Gaussian noise through a filter such that the signal output will have statistical properties same as those measured for atmospheric turbulence [40]. The transfer function of the filter is chosen to generate an output with a von Kármán turbulence spectrum that is not affected by the mean flow field of the rotor. A simplified closed-loop block diagram of the aeroelastic model is illustrated in Figure 4 in which the double integrator and turbulence filter are included in the system. In the same figure, wδ represents the external disturbance (Gaussian) that is passed through the von Kármán filter to produce the turbulence signal δ, and v accounts for any external measurement noise.

Figure 4.

Closed-loop block diagram.

Because of the characteristics of the disturbance, LQG controls will be considered in this paper. For a Gaussian disturbance, an LQG controller minimizes the expected value of the quadratic cost function [41, 42]

display math(11)

where Q is an appropriately chosen weighting matrix with the relationship Q = CTqC, where C is from Equation (9) and q is a scalar. This relationship allows us to weight the states based on their contribution to the output in the state equation and simplifies the process of tuning to only a scalar q. In this paper, the feedback measurement to controller is chosen to be the RBM, and tuning is done based on the performance of the resulting closed-loop model, in terms of the percentage reduction in RBM, while keeping maximum flap deflection angles and rates within the prescribed limits. The out-of-plane tip deflection is also measured in the closed-loop model but not feedback to the controller.

4 Numerical Results

4.1 Numerical validation

The aeroelastic code SHARP has been extensively verified in the context of flexible aircraft aeroelasticity and flight dynamics [33]. Implementing it in wind turbine applications requires further validation to ensure that gyroscopic effects in the rotating blade are correctly captured. This is first done on the structural model by considering a long rotating cantilever beam with the uniform properties shown in Table 1. Campbell plots are then obtained for increasing rotational speeds from SHARP and compared against the analytical solutions from Bielawa [43]. Figure 5 shows the first and second fundamental modes for out-of-plane and in-plane bending as well as the first torsional model. Results from SHARP compare very well with the analytical solutions.

Figure 5.

Campbell plots comparing fundamental modes of a simple rotating cantilever beam computed using current model and analytical solutions.

Table 1. Cantilever beam properties with uniform mass and stiffness distribution (Validation case).
Span60 mMass per unit length 2.79 × 10 2 kg ⋅ m − 1
Elastic axis0.5 ⋅ cMoment of inertia1.93 × 10 2 kg ⋅ m3.22 × 10 2 kg ⋅ m
Center of gravity0.5 ⋅ cBending stiffness5.22 × 10 9 N ⋅ m 21.32 × 10 10 N ⋅ m 2
  Torsional stiffness 4.14 × 10 8 N ⋅ m 2

Next, the coupling of the structural and UVLM models is verified on the Goland wing [44], which is a generic lifting surface that is commonly used for validations in aeroelasticity. The rectangular cantilever wing has a semi-span of 6.096 m and chord of 1.8288 m. The structural characteristics can be found in Hesse & Palacios [45]. The flutter speeds for the Goland wing in sea-level straight flight has been reported to be between 163.5 m ⋅ s − 1 and 169 m ⋅ s − 1 [45, 46]. For the current model, the flutter speed obtained through an eigenvalue analysis is 166 m ⋅ s − 1 with converged solution using 10 chordwise and 15spanwise panels.

It is interesting at this stage to investigate the influence on flutter speeds with increasing rotational effects. Putting the Goland wing in a rotating frame and varying the distance, r, between the axis of rotation and the root of the wing, we observe in Figure 6 that increasing rotational effects (by bringing axis of rotation closer to the root) causes flutter tip speeds to increase by up to 15%. This gives assurance that it is conservative to set flutter limits using non-rotating conditions.

Figure 6.

Effects of increasing rotational effects on the Goland wing (by varying the distance between the axis of rotation and the root of the wing) on flutter tip speeds, which are normalized with forward flight flutter speed.

Next, defining the axis of rotation at the root of the beam, the root locus for the first two modes (out-of-plane bending and torsion) in increasing angular velocities is shown in Figure 7. The roots are obtained from converting the eigenvalues of the discrete-time aeroelastic system into continuous time. Three cases are plotted with the first two corresponding to the Goland wing rotating about its root with tip speeds increasing from 80 m ⋅ s − 1 to 200 m ⋅ s − 1. The difference lies in the structural model where the second case is computed without gyroscopic effects. Also in the plot is the root locus for the Goland wind in straight flight from 80 to 170 m ⋅ s − 1 (without rotational effects). Comparing cases 2 and 3, it is evident that rotational effects in the UVLM, mainly caused by the linear variation of freestream from root to tip have a substantial effect in influencing the natural frequency of the torsional mode (the higher frequency mode) while having a very minor impact on the bending mode. When gyroscopic effects are also included on the structural equations (case 1), the frequency of the torsional mode is pushed even higher. In this last case, the first bending mode is also slightly increased.

Figure 7.

Root locus for the first two modes (out-of-plane bending and torsion) of Goland wing. Case 1, rotational effects in both UVLM and structural model (rotational tip speeds from 80 to 200 m ⋅ s − 1). Case 2, rotational effects accounted for in UVLM only (rotational tip speeds from 80 to 200 m ⋅ s − 1). Case 3, no rotational effects (rotational tip speeds from 80 to 170 m ⋅ s − 1).

Table 2. Natural frequencies of NREL 5-MW reference wind turbine blade computed in detail using SHARP.
ModeSHARP (Hz)Jonkman et al. (Hz) [49]
First blade flapwise0.6770.666
First blade edgewise1.0851.090
Second blade flapwise1.9571.934
Second blade edgewise4.005N/A
First blade torsion7.829N/A

4.2 Gust load alleviation on the NREL 5-MW reference wind turbine

The NREL 5-MW reference wind turbine was developed to support conceptual studies aimed at accessing offshore wind technology and has been widely adopted as a benchmark case for the aeroelastic analysis and design of large flexible wind turbines [4, 47, 48]. Using the structural formulation described earlier, the NREL 5-MW reference wind turbine blade is modeled as a beam with 48 elements and the cross-sectional properties described in Jonkman et al. [49]. The computed natural frequencies are shown in Table 2 and match closely to those in Jonkman et al. [49]. The first torsional mode is also close to the 8 Hz reported by Hansen [47] and Bak [50].

For the aerodynamics model, vortex panels are placed on the outer 85% span of the blade, and a helicoidal wake profile [51], ignoring wake roll-up, is prescribed to enable a linear UVLM representation as shown in Figure 8. Each row of wake denotes the spanwise locations swept by the trailing-edge panels of the blade at each rotating time instant, forming a helicoidal shape. The inboard segment of the blade with cylindrical cross-sections is not modeled here but can be included as additional drag forces. To test the concept of active controls on the aeroelastic model and also remove complexities in the load analysis, a single blade is presented in this study, but the aeroelastic model is easily extendable to the full rotor. Wind shear can also be implemented through cyclic downwash on the blade according to its azimuth location during rotation, and rotor yaw can be accounted for by prescribing a shed rotor wake angle. The additional aerodynamic forces as a result of the pre-twist in the NREL 5-MW reference wind turbine can be modeled as additional downwash on the lifting surface. To account for the hub, the coupled model is offset away from the axis of rotation by 1.5 m.

Figure 8.

Aerodynamic model of NREL 5-MW reference wind turbine blade with prescribed helicoidal wake. (For the purpose of clarity in visualization of wake profile, only two chordwise panels are used to generate the figure).

A parametric study revealed that a spanwise discretization of 10 panels, chordwise discretization of five panels and five wake chords, is sufficient to capture the most interesting dynamics, with a linear interpolation selected for the structural finite element discretization. This corresponds to a total of 921 states, including the additional states introduced by the turbulence filter, double integrator and wake convection.

4.3 Closed-loop simulations

Finally, the closed-loop model of the 5-MW NREL reference wind turbine blade fitted with flaps is simulated. To have a good understanding of the best location along the span to place the trailing-edge flap, a flap measuring 10% span and 10% of the local chord is moved along the span of the blade. The rms reduction in RBM and tip deflection with the flap location at increasing distances from the root, using RBM feedback to the LQG controller is studied. The operating conditions are under rated inflow of 11 m ⋅ s − 1 perpendicular to rotor, tip speed ratio of 7.5 and turbulence intensity of 6%. The variance for the LQG controller is taken from the turbulence intensity (standard deviation) and the weight Q in Equation (11) is increased relative to inline image until either the flap deflection angle or rate limits of | β |  ≤ 10° and inline image/s reported by Berg et al. [52] are encountered. The results are presented in Figure 9. As we move the flap along the span away from the root, we observe an exponential increase in load reduction performance. However, when they come close to the tip at around 90% span, rms reductions in RBM dropped as a result of tip effects and also smaller local flaps as the blade tapers around the tip. The largest rms reduction in RBM occurs when the flap is placed at 80% span and similar trends are observed by Andersen et al [19].

Figure 9.

Effect of LQG controller on the rms reduction in RBM with varying location of flap along span of blade. (Flap measures 10% span and 10% chord).

Table 3. Percentage reduction in RBM and tip deflection for various load cases using LQG controller with RBM feedback.
CaseInflow (m ⋅ s − 1)Turb intensity (%)rms Reduction in RBM (%)rms Reduction in tip deflection (%)
  1. RBM, root-bending moment; LQG, linear-quadratic-Gaussian; rms, root mean square.

1863836
28102329
31162330
411101520
51461824
614101216

The size of the flap is also a parameter for consideration. As we increase the chord or span of the flaps in Figure 10, the percentage rms reduction in RBM increases in an almost linear fashion. However, when the flap span is increased beyond 30% in Figure 10(b), there is diminishing returns in load reduction performances. This observation demonstrates the performance limit that a single flap can achieve with a homogeneous turbulence model, and 30% flap span appears to be a suitable choice delivering good performances. However, having a flap size of 30% span and beyond may be unrealistic in being too heavy to actuate. The rms reduction in tip deflection is also plotted to show the close relationship between reducing RBM and tip deflection.

Figure 10.

Effect of LQG controller on RBM and tip deflection with varying flap sizes.

After the previous parametric study, a flap occupying 20% of the span of the lifting surface and 10% of the local chord is chosen and located at a mean position of 80% span, as shown in Figure 11.

Figure 11.

Flap occupying 20% span, 10% of local chord and located at a mean position of 80% span.

Better performances are expected with multiple and distributed flaps [4, 18] but as we are assuming a homogeneous gust model, a single flap will be sufficient.

For the UVLM model, a discretization of 10 chordwise, 10 spanwise panels and 10 wake chords is implemented, resulting in 1660 states. The beam model has the same number of elements as the spanwise discretization of the UVLM to facilitate the mapping shown in Figure 3. This creates a relatively large model as the number of states is determined by the panel discretization on the lifting surface and also the number of wake chords. With the size of the LQG controller of at least the same order as the aeroelastic system, a reduced-order model will be preferred such that the closed-loop system will not be excessively large and to reduce the computational effort in controller synthesis. We have recently shown, on 2-D aeroelastic models based on discrete-vortex aerodynamics, that reduced-order models from relaxation of spatial discretization can perform well in closed-loop [30, 53]. Here, a more standard balanced reduction is implemented. In this approach, the least controllable and observable states in the balanced realization of the system are truncated [42]. A convergence study shows that a reduced-order model of 50 states is sufficient to capture the relevant dynamics of the present aeroelastic system. This reduced-order model is then used to synthesize the LQG controller, which is subsequently placed in closed-loop with the original higher-order linear model and also with a non-linear model, which re-computes the structural deformations and aerodynamic influence coefficient matrices at each time instant. Three operating conditions are considered, 8, 11 and 14 m ⋅ s − 1, which correspond to regions 2, inline image and 3 as described in Jonkman et al. [49] and a tip speed ratio of 7.5, 7.0 and 5.5 are chosen, respectively. Turbulence intensities of 6% and 10% are simulated.

Table 3 shows the percentage rms reduction in RBM and out-of-plane tip deflection for different operating cases using the LQG controller with RBM feedback on the higher-order linear model. The weight for the LQG controller is tuned separately for each case, such that the weight Q in Equation (11) are increased relative to inline image until both the flap deflection angle or rate limits of | β |  ≤ 10° and inline image/s are achieved. The performance of the closed-loop model is obtained by measuring and comparing the areas under the power spectral densities (PSD) of the open-loop and closed-loop models, which give rms values [54]. The use of PSD provides a more accurate and consistent measure of rms values and avoids long time-stepping simulations of turbulence signals.

On average, we are observing around 22% rms reduction in RBM and 26% rms reduction in tip deflection for the cases considered in Table 3. For all cases, the limit on the flap deflection angle of ± 10° is met, whereas the limit on flap deflection rate was less of a concern as it was well below ± 100°/s. For the same inflow speed, lower reductions in RBM and tip deflection were obtained for higher turbulence intensity. As inflow speed is increased, the reduction in RBM and tip deflection is also smaller. This is mainly due to the limit placed on flap deflection angles restricting the performance achievable in higher turbulence intensities and larger inflow speeds. A sample of the time series for the RBM and tip deflections for rated inflow conditions under 10% turbulence intensity is shown in Figure 12(a,b), where it is evident that peaks in RBM and tip deflections are suppressed in the closed-loop system. The flap deflection angle β is shown in Figure 12(c), which are within prescribed limits and is actuated at relatively low frequencies given the double integrator on the control input, inline image, introduced in the system. On the same figure, the fluctuation of the inflow is also plotted in which we observe a phase lag in response of the flaps to the inflow turbulence.

Figure 12.

Section of the time series for RBM, tip deflection, flap deflection angle, inflow and torsion, with 11 rated wind speed and 10% turbulence intensity (Case 4), using LQG controller.

The performance of the controller is then analyzed on the non-linear model, where structural deformations and aerodynamic influence coefficient matrices are re-computed at each time instant. Comparing the linear and non-linear open-loop tip deflection responses shown in Figure 13 for the first 20 cycles of tip deflection of case 4, we observe little differences, indicating that the linear model is capable of capturing the dynamics present in the non-linear model and suggesting that the controller may still perform as well in the non-linear case. To illustrate this, on the same figure, the LQG controller for 10% turbulence intensity is placed in closed-loop with the non-linear model of case 4, where it is evident that peaks are suppressed by the control action, very similar to the linear closed-loop system.

Figure 13.

Open-loop and closed-loop tip deflection of the linear (thin lines) and non-linear (thick lines) models (Case 4).

While the use of flap delivers benefits in terms of reducing RBM and tip deflections, it has an adverse effect on torsional forces. As shown in Figure 12(d), the effects of flap deflection on torsion is significant. A straightforward method to overcome this is to reduce the weight Q in Equation (11), such that more importance is placed on the control input, inline image, leading to lower adverse torsional effects. Alternatively, as LQG is a multiple-input-multiple-output controller, torsion can also be included in the objective function to be minimized. Setting the weight corresponding to RBM to unity and gradually increasing the weight on torsion from 10−3 to 103, we obtain in Figure 14 for case 4 the trade-off between RBM and torsion in using active flap controls. Also plotted is the trade-off obtained from reducing the weight Q on RBM alone without considering torsion in the objective function. Both plots coincide at the top left corner when torsion is not considered, with the corresponding values indicating the minimum rms of RBM achievable and the resulting rms of torsion using controllers tuned to satisfy the limits on β and inline image. As we move from left to right in the plot, we are sacrificing reductions in rms of RBM (hence, an increase in the value of rms of RBM) for reduced torsional effects. Three observations can be made. Firstly, there is diminishing returns on the reduction in torsion as more RBM reduction is sacrificed. Secondly, if we were to consider the scale of trade-off, we could achieve a larger reduction in torsion with a given amount of RBM reduction we sacrifice. For instance, in the case of increasing weight on torsion, a 5% sacrifice in RBM reduction from the leftmost point on the plot is able to achieve close to 50% reduction in torsion. Lastly, we see better returns through increasing the weight on torsion from the steeper slope, meaning that we can get more reduction in torsion for a given sacrifice in RBM reduction.

Figure 14.

Trade-off between RBM and torsion in using active flap controls (Case 4).

5 Conclusions

The state-space UVLM formulation coupled with the beam model has been introduced as a model-based design tool for aeroelastic predictions of rotating lifting surfaces and controls. It allows the unsteady response of large flexible blades with control surfaces to be captured with better fidelity than BEM. While the standard UVLM implementation has a relatively large computation cost, the state-space representation presented here provides a computationally more efficient route for control synthesis as illustrated in this work. Also, this state-space formulation can be coupled with any discrete-time linear structural model to create a compact form and is easily extendable to a full rotor. In particular, this paper has used a composite beam finite element solution that was linearized around a large geometrically non-linear steady-state equilibrium. Bound aerodynamic panels and beam elements were collocated along the span to provide a simple monolithic aeroelastic description of the blade. Finally, the resulting linear time-invariant state-space aeroelastic description of the rotor dynamics is suitable for standard model reduction methods and controls. Modeling the 5-MW NREL reference offshore wind turbine blade with trailing-edge flaps, an average of 22% rms reduction in RBM and 26% rms reduction in tip deflection is achieved for the load cases considered using an LQG controller on the linear model. The controller is also shown to perform well on the non-linear model. The flap deflections angles in all cases were within the limits of ± 10°. It also demonstrated the trade-off between RBM and torsion when using a single flap. The single blade analysis using the described aeroelastic model provides confidence in the simulation of large wind turbine blades and to analyze the load reduction potential of flaps, although we would expect performance to be reduced if the controller is used in actual implementations. Further investigations will involve modeling of the full rotor with tower, non-homogeneous gust models with the use of distributed flaps and also various control methods.

Acknowledgements

The first author would like to thank the Singapore National Research Foundation, Clean Energy Programme Office for their funding support, and the second author would like to acknowledge the support from UK Engineering and Physical Sciences Research Council (EPSRC).

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