the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Freeflow wind speed from a blademounted flow sensor
Torben Juul Larsen
Helge Aagaard Madsen
Søren Juhl Andersen
This paper presents a method for obtaining the freeinflow velocities from a 3D flow sensor mounted on the blade of a wind turbine.
From its position on the rotating blade, e.g. onethird from the tip, a blademounted flow sensor (BMFS) is able to provide valuable information about the turbulent sheared inflow in different regions of the rotor. At the rotor, however, the inflow is affected by the wind turbine, and in most cases the wind of interest is the inflow that the wind turbine is exposed to, i.e. the freeinflow velocities.
The current method applies a combination of aerodynamic models and procedures to estimate the induced velocities, i.e. the disturbance of the flow field caused by the wind turbine. These velocities are subtracted from the flow velocities measured by the BMFS to obtain the freeinflow velocities. Aeroelastic codes, like HAWC2, typically use a similar approach to calculate the induction, but they use it for the reversed process, i.e. they add the induction to the free inflow to get the flow velocities at the blades, which are required to calculate the resulting aerodynamic forces.
The aerodynamic models included in the current method comprise models based on blade element momentum (BEM) for axial and tangential induction, a radial induction model and tip loss correction, and models for skew and dynamic inflow.
It is shown that the method is able to calculate the freeinflow velocities with high accuracy when applied to aeroelastic HAWC2 simulations with a stiff structural model while some deviations are seen in simulations with a flexible structure.
Furthermore, the method is tested on simulations performed by a flexible structural model coupled with a largeeddy simulation (LES) flow solver. The results of this higherfidelity verification confirm the HAWC2based conclusion.
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Detailed knowledge about the atmospheric turbulent wind and its variation is essential for understanding and analysing many aspects regarding wind turbines, e.g. load conditions, power generation, noise aspects, and fatigue and extreme loads (Barlas et al., 2012; Elliott and Cadogan, 1990; Larsen et al., 2005; Madsen, 2014; St. Martin et al., 2016)
The wind of interest is the free undisturbed turbulent inflow, but at the location of the wind turbine rotor. The problem is that this free wind is immeasurable, as the inflow is influenced by the presence of the rotor itself: near the turbine, the flow is disturbed by the turbine, and further away, the wind is different. The ideal measure, therefore, is the freeflow wind speed at the position of the turbine, i.e. the wind speed as it would have been at the same location and time without the turbine.
A cup or sonic anemometer at a nearby met mast, e.g. 2–3 diameters away, measures the freeflow wind when not in the wake of the turbine or the mast, but smaller turbulence structures will be different due to the distance. As these structures have only limited influence on 10 min statistics, the wind speeds measured by met masts are still valuable and extensively used.
The wind speed measured by a nacelle or spinnermounted anemometer is influenced by the turbine. This influence is measurable more than one diameter upstream (Meyer Forsting et al., 2017) and continues until the wake is recovered far downstream. This means that a model or calibration function is required to estimate the freeflow wind speed from a nacelle or spinnermounted anemometer.
The turbine is, however, not only exposed to the wind at the hub centre, and with the long blades of modern wind turbines, the windspeed variations within the rotor may be considerable. This variation can be measured using lidars, which are typically ground or nacelle based. From the nacelle, a lidar is able to measure the inflow field some distance upstream while the inflow field at the rotor plane can be measured using a set of groundbased scanning lidars (Mikkelsen et al., 2008, 2010; Scholbrock et al., 2015; Wagner et al., 2015). Nevertheless, the problem is the same; the lidars must either measure outside the induction zone to measure the free inflow or use a model to compensate for the presence of the turbine. Troldborg and Meyer Forsting (2017) describe a simple analytical model that is able to estimate the free wind speed, appropriate for power curves, from lidar measurements. The model is applicable for any rotor down to one rotor radius upstream.
Another option is to mount a flow sensor directly on the blade, e.g. onethird from the tip. From this location, the sensor sweeps the rotor area and is thereby able to measure a lot of the variation that takes place within the rotor area. The sensor may be mounted closer to the root or the tip, but closer to the root it will only sweep a smaller part of the rotor, and near the tip it will suffer from severe blade deflection and tiploss effects. Furthermore, the load distribution along the blade should be taken into account, such that the sensor measures the inflow where the largest loads occur. In Pedersen et al. (2017), different radial positions are investigated and 50–67 was found to be optimal.
Blademounted fivehole pitot tubes have been used in several research projects (Brand et al., 1996; Hand et al., 2001; Madsen, 1991; Madsen et al., 2003, 2010b; Medina et al., 2011; Petersen and Madsen, 1997; Schepers et al., 2002; Simms et al., 1999), but other types of sensors could be used as well. In the current context, a blademounted flow sensor (BMFS) is assumed to measure the 3D inflow velocity at its position, i.e. the temporal resolution of a point fixed in space is limited to once per revolution, and multiple sensors are required for measuring in both the inner and outer parts of the rotor. From the 3D inflow velocities measured by a BMFS, information about the angle of attack, relative velocity and the instant wind speed at the rotor plane can be extracted. This information can be used as input for the control of individual pitch or active trailing edge flaps to optimize power and reduce loads and noise (Barlas et al., 2012; Kragh and Hansen, 2012; Kragh et al., 2012; Larsen et al., 2005; Madsen, 2014). Another application is the generation of relative power or load curves that can be compared between similar periods or turbines. These relative curves can be used to investigate aerodynamic modifications or detect performance issues, for example (Pedersen et al., 2017).
As a BMFS is inside the induction zone, a model is required to estimate the freeflow wind speed. In this case, however, welldefined models already exist as they are used in aeroelastic codes, e.g. HAWC2, to simulate the flow that generates the aerodynamic forces at the blades (Larsen and Hansen, 2007; Madsen et al., 2018). These models calculate the disturbance of the flow caused by the turbine in terms of the induced velocities, which are then added to the freeflow velocity to get the disturbed wind speed at the blades. Reversing this process, the freeflow wind speed can be obtained by subtracting the induced velocities from the measured velocities.
This paper describes the necessary aerodynamic models as well as a procedure to obtain an estimate of the freeflow velocity from a BMFS. This estimate provides information about the mean wind speed variations within the rotor, the turbulence and the instant wind speed history. This information can be used to characterize the inflow conditions that result in low power or high loads, for example, and as input for aeroelastic simulations to improve the correlation between the measured and simulated loads. A preliminary implementation of the method has previously been applied to measurements of a fullscale wind turbine with a blademounted fivehole pitot tube to estimate the free inflow and shear profiles in different wake conditions (Pedersen et al., 2015).
To test the method, both the disturbed and the immeasurable freeflow wind speeds are required. A real validation against measurements is therefore infeasible. Instead, the method is tested in two different simulated environments.
The first environment is simulated by HAWC2, a nonlinear aeroelastic code intended for computing wind turbine response in the time domain (Larsen and Hansen, 2007). HAWC2 uses Taylor's wellknown frozen turbulence hypothesis (Taylor, 1938) and a combination of aerodynamic models, which are similar to the models of the current method, to calculate the disturbed flow velocity at the rotor plane. Both the free inflow and the disturbed flow at the rotor plane are therefore directly available for verification of the method.
In the second environment, the simulations are performed by the structural model of FLEX5 (Øye, 1996) coupled with the largeeddy simulation (LES) flow solver, EllipSys3D (Michelsen, 1992; Sørensen, 1995). This environment is completely independent of the aerodynamic models of the current method as well as Taylor's frozen turbulence hypothesis and is therefore valuable as a highfidelity verification. In this case, the free undisturbed inflow is obtained from a separate equivalent simulation in which the effects of the aerodynamic forces on the flow are disabled.
This section presents the aerodynamic models and the procedure used to obtain the freeinflow velocities from a BMFS.
2.1 Coordinate systems
The coordinate systems used in this paper are listed in Table 1.
Transformation matrices are used to map velocities between the coordinate systems. As an example, the transformation matrix, T_{RG}, describes the rotatingrotor coordinatesystem axes in ground coordinates. T_{RG} can be used to map velocities in rotor coordinates, V^{R}, to velocities in ground coordinates, V^{G}:
2.2 Wind speed from a BMFS
The method described in this paper takes as input the effective 3D inflow velocities measured relative to the blade, locally at the rotor plane, i.e. including the effects caused by the presence of the turbine.
Near the airfoil, the local flow field is deflected and the speed is also influenced by the bound circulation on the surface of the airfoil; see the example in Fig. 1. As seen, this effect has a huge impact on the flow velocity measured near the airfoil and must therefore be compensated for before applying the current method. In the current study, however, it is neglected as the two verification environments, HAWC2 and EllipSys3D–Flex5, do not model the surface of the airfoils.
Shen et al. (2006, 2009), Guntur and Sørensen (2014), and Rahimi et al. (2018) present several methods to calculate the flow near the airfoil that also take 3D effects into account, but the methods require information that cannot be obtained directly from a BMFS. Pedersen et al. (2017) describes how to obtain the effective 3D inflow from the relative wind speed and two perpendicular angles measured by a blademounted fivehole pitot, including compensation for bound circulation. The compensation method uses a lookup table generated by 2D computational fluid dynamic (CFD) simulations, thus neglecting 3D effects and tip and root vortices.
From the relative velocity, V_{rel}, the wind speed at the rotor plane, V_{r}, is found by adding the velocity of the sensor, V_{s}:
In this study, the sensor velocity, V_{s}, includes movement due to rotor rotation and pitch motion. Structural dynamics, e.g. blade deflection, will therefore result in a mismatch between the assumed and the actual sensor velocity.
2.3 Aerodynamic models
The wind speed measured in the rotor plane of an operating wind turbine is different from the freeflow wind speed that would have been present at the same time and location if the wind turbine was absent. The difference is induced by the wind turbine and is rather complex. In this section, a set of simplified engineering aerodynamic models that each explain elements of the induction are presented. One can argue that the models are too simple compared to the physical processes. In general, however, the loads simulated by aeroelastic codes that use these models are found to agree well with measured loads; see e.g. Larsen et al. (2013). The models are therefore expected to be appropriate for the reverse process too.
The aerodynamic models in aeroelastic codes like FAST, Flex5, Bladed and HAWC2 are based on the blade element momentum (BEM) model first presented by Glauert (1935). The original formulation, however, was derived for axissymmetric, steady and uniform inflow, which is far from the conditions that a real turbine operates in. The BEM model is therefore typically modified and combined with additional submodels, e.g. for tip loss and for skew and dynamic inflow. In this study, the aerodynamic model is based on the HAWC2 implementation (Madsen et al., 2018).
2.3.1 Axial induction
When operating, a wind turbine extracts kinetic energy from the wind by reducing the axial wind speed. This reduction is called the axial induction, ${W}_{y}^{\mathrm{R}}$; see Fig. 2.
The axialinduced wind speed is defined in terms of the axial induction factor, a:
where V_{0} is the freeinflow velocity.
For laminated flow through the rotor, the axial induction factor is related to the thrust coefficient, C_{T}, by
while empirical results show higher values of C_{T} for induction factors above 0.3–0.5 (Eggleston and Stoddard, 1987). The current method uses a thirdorder polynomial, as described by Madsen et al. (2018),
with coefficients ${k}_{\mathrm{3}}=\mathrm{0.0883},{k}_{\mathrm{2}}=\mathrm{0.0586}$ and k_{1}=0.2460 that fit to Eq. (4) for lower values of a and to empirical results and actuator disc simulations for higher loading (Madsen et al., 2010a).
For an annular ring element at radius r, the thrust coefficient is calculated using the formula presented by Madsen et al. (2018):
where V_{relxy} is the relative wind speed in the (x^{R},y^{R}) plane (see Fig. 3), c is the chord length, α is the angle of attack, N_{B} is the number of blades and C_{y} is the projection of the lift and drag coefficient into y^{R}:
where $\mathit{\varphi}=\mathit{\alpha}+{\mathit{\theta}}_{\text{twist}}+{\mathit{\theta}}_{\text{pitch}}$ is the angle between V_{relxy} and the rotor plane.
From the measurements of a BMFS, V_{relxy} and α can be obtained directly, and the number of blades, the pitch angle, the radius, the chord length and the blade twist angle are assumed to be known. Hence, if the angleofattackdependent lift and drag coefficients are accessible from a lookup table, then the only unknown term on the righthand side of Eq. (6) is V_{0}.
In aeroelastic simulations, V_{0} is obtained from the wind input model, but in this case, V_{0} is the wind speed that we want to find. It can, however, be found using the iterative approach described in Sect. 2.4, such that the induced axial velocity can be calculated via Eqs. (6), (5) and (3).
2.3.2 Tip correction
The relationship between the thrust coefficient and the axial induction factor stated in Eq. (4) is based on the assumption that the induced velocities are constant within an annular element. This is not the case for turbines with a finite number of blades and therefore Prandtl's tip loss factor, presented by Glauert (1935),
where R is the blade tip radius, is applied in the current method by replacing C_{T} with $\frac{{C}_{\mathrm{T}}}{{F}_{\text{tip}}}$ in Eq. (5) as described by Madsen et al. (2018). Calculating and applying the tip loss factor is straightforward as the only variable on the righthand side, ϕ, can be calculated from the BMFS output.
2.3.3 Tangential induction
The tangential induction is a reaction to the torque force and results in a rotation of the wake downstream. The tangential velocity of the wake is defined in terms of the tangential induction factor, a^{′}:
where ω is the angular velocity of the rotor.
Variations in the tangential flow due to blade passing can be observed more than one rotor radius upstream. This effect is, however, assumed to be handled by the compensation for deflection and change of flow speed near the airfoil, which is required before the current method is applied (see Sect. 2.2). The current tangential induction model only describes the reaction to the wake rotation that starts near the blades and increases downstream. This effect is assumed to be insignificant upstream. The amount of wake rotation at the position of a BMFS is therefore dependent on the sensor position relative to the blade, the pitch angle and the bladedeflection state. The current implementation of the method assumes full tangential induction, but for some applications, it may be more appropriate to switch it off.
The tangential induction factor is obtained by the formula presented by Madsen et al. (2018):
where ${C}_{x}=\mathrm{sin}\left(\mathit{\varphi}\right){C}_{L}\left(\mathit{\alpha}\right)+\mathrm{cos}\left(\mathit{\varphi}\right){C}_{D}\left(\mathit{\alpha}\right)$ is the projection of the lift and drag coefficient into x^{R}; see Fig. 3.
In Eq. (10), the only unknown term on the righthand side is also V_{0}, which can be found via the iterative approach described in Sect. 2.4. To help this iterative procedure in finding the right solution, the value of a used in Eq. (10) is limited to the range [0;0.5].
2.3.4 Radial induction
The radial induction results in an expansion of the flow, as illustrated in Fig. 2. Introducing the radial induction factor, a_{r}, the radially induced velocity is
The standard onedimensional BEM theory does not handle radial induction and therefore the analytical equation derived by Madsen et al. (2010a) is used in the current method:
where C_{T,avg} is the average thrust coefficient of the whole rotor. In the current model, the revolutionaveraged local thrust coefficient of the BMFS is used. This is obviously not the same, and the approximation is therefore only appropriate if the thrust coefficient of the radial position corresponds to the average thrust coefficient of the whole rotor. This is typically not the case near the root and the tip, and even for a sensor that is onethird from the tip, some discrepancies must be expected.
2.3.5 Dynamic inflow
The induced velocities are part of an equilibrium which is gradually established between the load on the blades, the rotor wake and the induced velocity at the rotor plane (Sørensen and Madsen, 2006).
Small and highfrequency turbulence is assumed to pass unaffected though the rotor and can therefore be measured directly, while the effect of large stationary turbulence eddies can be described by the BEM models in Sect. 2.3.1 and 2.3.3. In between, the modification of the wind flow depends on the wake recovery velocity. Snel and Schepers (1995) present different engineering approaches to model the wind turbine response in dynamic inflow.
In the current method, the model used in HAWC2 (Madsen et al., 2018) has been implemented with two modifications. This implementation applies two firstorder lowpass filters to the induced velocities to model the slow and gradually changing induction,
where LP(τ,X) is a firstorder lowpass filter. The two filters model the near and farwake effects respectively, and their filter characteristics are given by the following.
where
Equations (16) and (17) can be calculated straight away, while V_{0} and ${W}_{y,\text{avg}}^{\mathrm{R}}$ are required for Eqs. (14) and (15). V_{0} can be estimated as described in Sect. 2.4, while the instant average axial induction of the whole rotor, ${W}_{y,\text{avg}}^{\mathrm{R}}$, requires information from the whole rotor, which cannot be obtained from a BMFS.
In the current implementation, the revolutionaveraged local induction is used as an approximation. This means that the filter characteristics may be inaccurate if the induction at the radial position of the BMFS is not representative for the whole blade. The sensitivity to ${W}_{y,\text{avg}}^{\mathrm{R}}$ is, however, limited and even extreme values have only a minor impact on the final estimated free wind speed.
The other modification is more severe. In HAWC2, the rotor is discretized in grid points and the dynamic inflow model is applied to the local induced velocities of each of these grid points. This is possible because the local induction is calculated for each grid point in every time step, and this means that the induction of a certain grid point reflects the current circumstances as well as the history of that particular grid point.
In the current method, only the local induction at the position of the BMFS is obtainable as no information is available from other parts of the rotor. Applying the dynamic inflow model to the induced velocities at the position of the BMFS means that the estimated induction reflects the history of the moving BMFS instead of a fixed position. In a situation with wind shear, the estimated induction will therefore be too high in the lower part of the rotor and too low in the upper part, resulting in too much variation in the estimated free wind speed.
Instead, the lowpass filters are applied to the induced wind speeds of fixed azimuthal positions. As the BMFS only passes a certain azimuthal position once per revolution, the sample frequencies of these signals are very low and some discrepancies must be expected.
Figure 4 shows the induced velocities in a simulation with turbulent inflow and shear. The quasisteady induced velocities estimated without the dynamic inflow model, ${W}_{y}^{\mathrm{R}}$, are seen to vary much more than the HAWC2 reference, while applying the lowpass filters to the rotating measurements, ${W}_{y,\text{dyn}}^{\mathrm{R}}$, smoothens the induction too much. Applying the lowpass filters to the lowfrequency signals of fixed azimuthal positions, ${W}_{y,\text{dyn,azi}}^{\mathrm{R}}$, results in an estimate closer to the HAWC2 reference even though there is still some mismatch.
2.3.6 Skew inflow
In skewed inflow, where the mean wind is not perpendicular to the rotor plane due to yaw misalignment, rotor tilt and flow inclination, for example, the axial induction is not directed exactly towards the wind. Hence the speed of the inflow is reduced less, and the thrust is increased. Furthermore, variation in the wake vorticity concentration results in an azimuthaldependent variation in the axial induction; see Fig. 5.
The first effect is modelled by the method described in Madsen et al. (2018) where the axial induction factor is multiplied by a reduction factor, F_{a}, that is calculated from the average thrust coefficient, C_{T,avg}, and the skew inflow angle, Φ_{r}.
where
The average thrust coefficient is estimated using the revolutionaveraged local thrust coefficient as described in Sect. 2.3.4, and in this case the approximation is also expected to introduce discrepancies. The inflow angle, Φ_{r}, which is the angle between the inflow and the rotor axis, i.e. it includes effects of both horizontal and vertical skew inflow, is calculated by
Note that the C_{T,avg} used in Eq. (18) must be limited to the range [0;1] as the model is invalid outside this range.
The azimuthal variation is calculated with a model presented by Madsen et al. (2018). In this model, the axial induction factor is multiplied with a rotorpositiondependent factor, F_{azi}:
where θ_{rotor} is the rotorazimuth position. The factors k_{x} and k_{y} depend on the inflow angle in the horizontal and vertical plane, χ_{hor} and χ_{ver}, respectively; see Fig. 6:
2.3.7 Combining models
The presented aerodynamic models are now combined into a function, f_{W}, that comprises the following steps.

Calculate C_{T} using Eq. (6).

Calculate the tip loss factor with Eq. (8).

Calculate a with Eq. (5) replacing C_{T} with $\frac{{C}_{\mathrm{T}}}{{F}_{\text{tip}}}$.

Apply the skew inflow model by

Calculate the tangential induction factor using Eq. (10).

Calculate the radial induction factor using Eq. (12).

Calculate the quasisteady induced velocities ${\mathbf{W}}^{\mathrm{R}}={\left(\begin{array}{ccc}{a}^{\prime}\mathit{\omega}r& a\left{\mathbf{V}}_{\mathrm{0}}\right& \left{\mathbf{V}}_{\mathrm{0}}\right{a}_{\mathrm{r}}\end{array}\right)}^{T}$.

Apply the dynamic inflow model by
Using this function, the estimated induced velocities can be calculated for a given V_{0},
2.4 Estimating V_{0}
The flow velocity measured by the BMFS is the sum of the freeflow and the induced velocities, hence
Using f_{W}, defined in Sect. 2.3.7, an estimate of the freeflow velocity can be obtained.
Figure 7 shows the estimated free wind speed, $\left{\mathbf{V}}_{\mathrm{0},\mathrm{est}}\right$, as a function of $\left{\mathbf{V}}_{\mathrm{0}}\right$ in an example in which the measured wind speed, ${V}_{r,y}^{N}$ is around 9.6 m s^{−1}.
We now want to find the correct free wind speed, i.e. the V_{0}, that, when inserted into Eq. (29), results in V_{0,est} being equal to V_{0} (12 m s^{−1} in Fig. 7). In other words, we iteratively solve
with respect to V_{0} using the Newton–Raphson method and V_{r} as the initial guess for V_{0}.
2.5 Verification
2.5.1 HAWC2
The method is verified using HAWC2 simulations of a Siemens 3.6 MW turbine with a 107 m rotor. The turbine has a 6^{∘} tilt and 3.5^{∘} coning angle and is controlled by the basic DTU Wind Energy controller (Hansen and Henriksen, 2013). The inflow turbulence for the turbulence cases is generated using the Mann model (Mann, 1994).
From the simulations, the relative wind speed is extracted at a point on the blade at radius 36 m, i.e. around onethird from the tip. From this wind speed the estimated free wind speed is calculated and compared to the free wind speed used as the input to HAWC2. Note that the current version of HAWC2 (version 12.5) does not include radial induction, and therefore this model is disabled when testing against HAWC2.
As the current method is based on the same aerodynamic models as HAWC2, one may argue that this verification just adds and subtracts the same value, which obviously results in the original velocity. There are, however, differences that are important to investigate, e.g. the effect of the differences and approximation in the aerodynamic models of the current method, the effect of a flexible structure and the V_{0}estimation procedure.
2.5.2 EllipSys3D–Flex5
The method is furthermore verified using EllipSys3D–Flex5 simulations of a 2.3 MW Siemens turbine with a 93 m rotor. In these simulations, the flow field is obtained from LES performed by the finitevolume and incompressible Navier–Stokes solver, EllipSys3D (Michelsen, 1992; Sørensen, 1995). The turbine is modelled using the actuator line method as developed by Sørensen and Shen (2002), in which the individual blades are modelled by imposing body forces into the flow solver. The actuator lines are fully coupled to the aeroelastic tool, Flex5 (Øye, 1996), which models the structural dynamics according to the incoming flow; see Sørensen et al. (2015) for details on the coupling. The inflow turbulence, which is similar to the turbulence of the HAWC2 simulations, is imposed 8.25 radius upstream from the rotor.
From these simulations, the flow speed is extracted at radius 32 m, i.e. also around onethird from the tip. All EllipSys3D–Flex5 simulations use a flexible structural model. Flex5 is based on modal shape functions as opposed to the multibody formulation of HAWC2 and hence does not include torsional rotation of the blades.
To obtain the freeinflow velocities, a separate identical flow simulation is performed, in which the effect of the aerodynamic forces on the flow is disabled such that the flow is not affected by the turbine. From this simulation the flow field in the vertical plane through the rotor centre is obtained for each time step.
In the aeroelastic HAWC2 simulations, the 3D correction method by Snel et al. (1993) is applied to the tabulated liftcoefficient polars, while the actuator line simulations have been run without 3D corrections. The current method relies on the same tabulated polars, and additional uncertainty must therefore be expected if the method is applied to fully resolved CFD simulations or real measurements due to discrepancies of the tabulated lift and drag coefficients and due to the 3D effects not taken into account. Furthermore, it should be noted that the two verification environments are not totally independent as both rely on tabulated lift and drag coefficient polars.
2.5.3 Freeflow reference
The estimated freeflow velocities are based on the velocities measured at the sensor position (red dot in Fig. 8). In the current verification, however, the reference freeflow velocity is extracted at the assumed (undeflected) sensor position (green dot in Fig. 8). This mismatch is expected to introduce some deviation as the turbulence is different at the two positions.
In the HAWC2 simulations, which are based on Taylor's frozen turbulence hypothesis, the turbulence is transported unaffected by the mean wind, i.e. with constant (free flow) speed along the y^{G} axis. This means that time can be mapped into space and the freeflow velocities can be extracted from the 3D turbulence field that is generated prior to the simulation.
The EllipSys flow, however, includes properties of real flow, e.g. turbulence structures change and break up over time. The 3D turbulence field will therefore change in every time step and only the velocities at the rotorcentre flow plane are available for this study. The assumed sensor position, however, does not exactly intersect this plane due to the rotor tilt angle. We are therefore compelled to rely on Taylor's frozen turbulence hypothesis to obtain the freeflow reference velocity at the assumed sensor position, but only from the rotorcentre plane to the sensor position (blue arrow in Fig. 8), i.e. at most 3.3 m.
Furthermore, the EllipSys flow is affected by the turbine. Near the rotor, the axial induction reduces the turbulence transport speed, while the radial induction results in an expansion of the flow that moves the turbulence structure outwards.
This means that the BMFS is exposed to “delayed” turbulence structures that originate from a smaller radial position (white dot in Fig. 8), and even more deviation is therefore expected.
3.1 HAWC2 verification
The rootmeansquared error, RMS, of the estimated instantaneous free wind speed, V_{0,est}, is shown for 7 m s^{−1} in Fig. 9 for HAWC2 simulations of increasing complexity and the EllipSys3D–Flex5 simulations.
Starting with steady, uniform inflow and a stiff structural model, Case 1, the RMS error is very small and the minor deviations between the estimated free velocities and the HAWC2 references in Fig. 10 are caused by the effects of rotor tilt that are not exactly compensated for by the skew inflow model.
In Case 2, the structural model is flexible. The rotation of the sensor due to the deflection and torsion of the tower and blade results in increased error levels that are clearly seen in the x and z velocity components in Fig. 10.
The most significant error is the 90^{∘} phaseshifted sinusoidal oscillation of the estimated velocities. This error is caused by thrustdependent flapwise deflection of the blade that results in a part of ${V}_{r,y}^{\mathrm{R}}$ being inaccurately projected onto the z^{R} direction; see Fig. 11. This constant error leads to oscillations of the x and z components in the nonrotating ground coordinate system.
This error is reduced by a counteracting effect, namely the torque pushing the blade forward in the edgewise direction. At this forwardpushed position, the direction of the actual velocity due to rotor rotation, ${{\mathbf{V}}_{\mathrm{rot}}}^{*}$, is slightly changed, but the bladesection coordinate system is rotated even more as seen in the righthand side of Fig. 12. A small part of ${{\mathbf{V}}_{\mathrm{rot}}}^{*}$ is thereby measured in the radial −z^{R} direction, while the current model assumes the rotational velocity, V_{rot}, to be tangential. This mismatch leads to a torquedependent error in the −z^{R} direction that reduces the thrustdependent contribution from flapwise deflection.
A closer look at Fig. 10 reveals a positive offset in the estimated x component. The reason for this offset, which corresponds to a spurious side wind, is a combination of two effects, both caused by gravityinduced edgewise deflections of the blade. When the blades are horizontal, the gravity pulls the blades down towards the earth; see the left and righthand sides of Fig. 12. This asymmetric edgewise deflection leads to a small part of ${{\mathbf{V}}_{\mathrm{rot}}}^{*}$ being measured in the radial −z^{R} direction on the righthand side of the rotor and in the +z^{R} direction on the lefthand side, i.e. in the +x^{G} direction on both sides. Furthermore, the transition from backward to forward deflection results in the blade moving faster in the upper part of the rotor and vice versa in the lower part. In the current method, however, the assumed rotational speed, V_{rot}, is uniform. The mismatch results in deviations that also map to +x^{G} in both vertical positions; see Fig. 12.
In combination, these two effects result in the almost constant positive offset of the ${V}_{\mathrm{0},\text{est},x}^{\mathrm{G}}$ velocity seen in Fig. 10.
For higher wind speeds, the estimated free wind speed in the y^{G} direction is overestimated due to the rotation of the elastic blade section. As the rotation angles are unknown in the current method, an error is introduced in the transformation from bladesection to ground coordinates; see Fig. 13. Blade torsion is an obvious source of the rotation, but for the current turbine model flapwise bending also contributes considerably. This effect is highly dependent on the wind speed and blade design, and for the DTU 10 MW reference wind turbine (Bak et al., 2013), an underestimation was seen instead.
Another effect that is seen in higher wind speeds is a negative mean offset in the z component due to tower deflection. In the transformation from rotatingrotor to ground coordinates, the angle between y^{R} and y^{G} is assumed to equal the tilt angle, but due to tower deflection the real angle is slightly larger, as it also includes the towertop deflection angle, θ_{tt}; see Fig. 14. A small part of V_{r} is therefore inaccurately projected onto z^{G}, resulting in a small error in ${V}_{\mathrm{0},\text{est},z}$.
This error can easily be compensated for by including the towertop deflection angle, measured by an inclinometer, in the transformation from rotatingrotor to ground coordinates. Similarly, the blade deflection and torsion angles can be included in the transformation from bladesection to rotor coordinates. These angles are, however, more challenging to measure due to the large centrifugal force.
In Case 3, a stiff structural model is simulated in turbulent inflow. In this case, the estimated freeinflow velocities fall almost on top of the HAWC2 reference despite the differences in the dynamic inflow model.
Case 4 combines the flexible structure with turbulent inflow, but the BMFSmeasured flow velocities are extracted in the ground coordinates such that errors introduced in the transformation from deflected bladesection coordinates are avoided. The errors are significantly increased in all components; see Fig. 9. The reason is the mismatch between the assumed sensor velocity, i.e. the velocity due to rotor rotation and pitch motion, and the real velocity, which also includes velocity due to dynamic deflections of the structure. Furthermore, deviations are introduced because the freeflow reference velocity is extracted at the assumed (undeflected) sensor position, while the model estimates the velocity at the deflected sensor position; see Fig. 8. This mismatch can be reduced, assuming that the real sensor position can be measured, e.g. using a GPS, or estimated with a method that includes tower and blade deflection.
In Case 5, the BMFSmeasured flow velocities are extracted in deflected bladesection coordinates and error is introduced due to the unknown orientation of this coordinate system; see Figs. 9 and 15. Higher RMS errors are therefore expected, but in this case, the error of the y component is reduced because the error due to coordinate transformation counteracts the error introduced by dynamic deflections. Note that this reduction is highly dependent on the turbine design as it depends on the actual flap and twist properties of the blade, and for other designs the error may be increased instead.
Figure 16 shows the power spectrum density of Case 5. The 1P (once per revolution) oscillating errors seen in the x and z components in Fig. 10 are seen around 0.2 Hz, while the deviations caused by dynamic deflections are seen in the y component above 0.4 Hz. At first it seems strange that the energy of the y component of the estimated free wind speed is lower than the HAWC2 reference, as the additional velocity due to the movement of the BMFS is expected to increase the energy. In reality, however, the deflection of the structure is correlated with the turbulence, as a blade exposed to a gust will deflect. This means that a BMFS that measures the gust relative to the deflecting blade will measure a lesssevere gust with less energy.
Figure 17 shows the instant and revolutionaveraged wind direction in a simulation with 20^{∘} yaw misalignment. The estimated wind direction is seen to follow the HAWC2 reference with a few degrees offset due to the spurious side wind caused by gravityinduced edgewise blade deflections.
Case 6 is based on the EllipSys3D–Flex5 simulations. The RMS errors are higher than in Case 4, which is the most equivalent HAWC2 case. Note, however, that the numbers are not directly comparable due to the different turbine sizes. The time series are compared in Fig. 18.
For the last case, Case 7, an optimization routine was used to find the optimal reference position with respect to axial and radial offset. For the 7 m s^{−1} the lowest RMS error was found when the estimated free velocities were compared to the free flow that hits the rotorcentre plane 2.2 s before and 4.2 m closer to the rotor centre. As seen in Fig. 9, the error is significantly reduced in all components. It is therefore concluded that the relatively high error of Case 6 is more related to the difference between the turbulence at the sensor and the reference position than to deviations introduced in the aerodynamic models and freeflow estimation procedure.
Finally, Fig. 19 shows the difference between the free mean wind speed and estimated free mean wind speed at the position of the sensor, and similarly for the standard deviation. The HAWC2 results are based on 10 min of simulations of the Siemens 3.6 MW turbine. The deviations are mainly introduced in the mapping of velocities from the deflected bladesection coordinate system to the ground coordinate system. The EllipSys3D–Flex5 results are based on 200 s of simulations of the Siemens 2.3 MW turbine. These velocities are extracted in ground coordinates; i.e. deviations due to mapping are not included. In Fig. 18, instantaneous deviations are seen, mainly because the turbulence at the sensor position is different from the freeflow turbulence at the reference position due to expansion, delay and evolvement of the flow. In this case, however, most of these deviations are averaged out, and the error in Fig. 19 is mainly introduced by differences in the induction modelling approach.
The error introduced in the transformation from deflected bladesection coordinates to ground coordinates is clearly seen in all components of the HAWC2 results. In the x component, the rotor speed and torquedependent spurious side wind increases the error of the mean wind speed up to rated rotor speed (around 9 m s^{−1}). In the y component, the overestimation due to blade torsion is seen. The error in the mean wind speed in the z direction, due to tower deflection, increases with the thrust up to rated wind speed (around 11 m s^{−1}). Above rated wind speed, the error is rather constant as increased drag on the tower counterbalances the decrease in thrust. Finally, the error, due to flapwise deflection of the blades, that results in the 1P oscillating deviations of the x and z components is clearly seen in the error of the standard deviation, which peaks with the thrust around rated wind speed.
In this paper, a method to estimate the undisturbed freeinflow velocities from the flow velocities measured by a blademounted flow sensor, BMFS, has been presented and verified. The method includes a combination of aerodynamic models and procedures to estimate the freeflow velocities from the measurements of a BMFS. The aerodynamic models comprise BEMbased models for axial and tangential induction, a radial induction model and tip loss correction, and models for skew and dynamic inflow. Some of these models require information, e.g. the average thrust coefficient of the whole rotor, that cannot be obtained from a BMFS. In these cases, approximations are used even though they are expected to introduce errors. Most of the models also take as input the free wind speed, which is the final output of the current method. An iterative procedure is therefore used to find the estimated free wind speed.
The method has been verified on HAWC2 simulations. This verification reveals that the method works well and provides accurate results when using a stiff structural model. Using a flexible structural model, larger deviations are seen. These deviations are caused by the rotation of the sensor due to the deflection and torsion of the tower and blade, movement of the sensor due to turbulenceinduced dynamic deflections of the structure, and the mismatch between the turbulence at the real deflected sensor position and the reference position, i.e. the assumed (undeflected) sensor position. These effects are highly dependent on the wind speed and the structural design.
Furthermore, the method has been verified by simulations performed using EllipSys3D–Flex5: a flexible structural model coupled with a largeeddy simulation (LES) flow solver. In these results, the free velocities estimated using the current method deviate more from the simulated free velocities, but it is concluded that the error is more related to the difference between the turbulence at the sensor and the reference position than to errors introduced in the aerodynamic models and freeflow estimation procedure.
Applied to real measurements, additional uncertainty must be expected due to mounting, calibration and sensor uncertainty, discrepancy of the tabulated lift and drag coefficients, and 3D effects not taken into account.
Simulation results are not available due to confidentiality.
a  Axial induction factor 

a^{′}  Tangential induction factor 
a_{r}  Radial induction factor 
c  Chord length 
C_{T}  Trust coefficient 
C_{T,avg}  Average trust coefficient 
C_{x}  Lift and drag coefficient projected into x^{R} 
C_{y}  Lift and drag coefficient projected into y^{R} 
D  Aerodynamic drag force 
F_{a}  Thrust reduction factor in skew inflow model 
F_{azi}  Azimuthaldependent reduction factor in skew inflow model 
F_{tip}  Prandtl's tip loss factor 
f_{W}  Function calculating induced velocities 
k_{i}, i=0…3  Constants 
k_{x}, k_{y}  Factors in skew inflow model 
L  Aerodynamic lift force 
LP(τ,X)  Lowpass filter with filter characteristics, τ 
N_{B}  Number of blades 
r  Sensor radius 
R  Blade tip radius 
T_{ab}  Transformation matrix from coordinate system a to coordinate system b 
V_{r}  Measured flow velocity at rotor plane 
V_{rel}  Measured velocity relative to the sensor 
V_{relxy}  Relative wind speed in the (x^{R},y^{R}) plane 
V_{rot}  Velocity of sensor due to rotor rotation 
V_{s}  Velocity of the sensor 
V_{0}  Freeflow wind speed 
V_{0,est}  Estimated freeflow wind speed 
W  Induced velocity 
W_{est}  Estimated induced velocity 
W_{avg}  Average induced velocity 
W_{dyn}  Induced velocity, estimated using dynamic inflow model 
W_{dyn,azi}  Induced velocity, estimated using dynamic inflow model applied to fixed azimuthal positions 
α  Angle of attack 
χ  Angle between V_{r} and rotor axis 
χ_{hor}  Horizontal angle between V_{r} and rotor axis 
χ_{ver}  Vertical angle between V_{r} and rotor axis 
ω  Angular rotor velocity 
ϕ  Angle between rotor plane and V_{relxy} 
Φ_{r}  Angle between rotor plane and V_{0} 
θ_{pitch}  Pitch angle 
θ_{rotor}  Rotor azimuthal position 
θ_{tt}  Towertop deflection angle 
θ_{twist}  Twist angle 
Coordinate systems; see Sect. 2.1  
G  Ground coordinate system 
R  Rotatingrotor coordinate system 
S  Bladesection coordinate system 
Modifiers  
^{*}  Actual velocity or deflected direction (opposed to the assumed velocity or undeflected direction) 
The authors declare that they have no conflict of interest.
The authors would like to acknowledge Siemens Wind Power for providing data
for the simulation model.
Edited by: Luciano
Castillo
Reviewed by: Vasilis A. Riziotis and Walter Gutierrez
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