Consider the discrete-time state-space representation of a nonlinear system: 1xk=fxk-1,uk-1+wk-1,2yk=hxk,uk+vk, with xk the state vector at time instant k, yk the output or measurement vector, and uk the control input vector. fxk-1,uk-1 is the nonlinear state function and hxk,uk is the measurement function. wk is the process noise vector and vk is the measurement noise vector. In EKF theory, they are assumed to be zero-mean Gaussian white noise with covariance matrices Q and R, respectively .

The purpose of the EKF is to compute an estimate of the unknown states x based on the system model and input–output signals. It is an extension of the linear Kalman filter in the sense that it generalizes the formulation to nonlinear systems. To do so, it relies on the Jacobian matrices of the state and measurement functions for the prediction and correction step, respectively .

The prediction step at time k consists of computing the a priori error covariance estimate Pk- and the a priori state estimate x^k- based on the previous estimates of the covariance matrix Pk-1+ and the state x^k-1+, and the Jacobian matrix of the state function Fk-1. The computation is as follows. 3Fk-1=∂f∂xx^k-1+,uk-14Pk-=Fk-1Pk-1+Fk-1T+Q5x^k-=fx^k-1+,uk-1

The correction step then makes use of the measurement zk to compute the a posteriori state estimate x^k+ and the a posteriori error covariance matrix Pk+ based on the Jacobian matrix of the measurement function Hk, the measurement residual ỹk and the Kalman gain Kk. The procedure is as follows. 6Hk=∂h∂xx^k-,uk7Kk=Pk-HkTR+HkPk-HkT-18Pk+=I-KkHkPk-9ỹk=zk-hx^k-,uk10x^k+=x^k-+Kkỹk

We apply the EKF theory to the estimation of blade-effective wind speeds, considering blades to be rotating sensors whose bending moments are measured. One estimation is performed for each blade b, as shown in the block diagram of Fig. .

Control input. The control input vector u consists of the rotation speed of the turbine and the blade pitch angle, 11uk=[Ωk,βb,k].

State. The variable to be estimated is the blade-effective wind speed, Ub, which is an input to the wind turbine blade system. To match the EKF formalism, we model this input as a state and do not explicitly model the internal states of the system. The one-dimensional state vector is therefore 12xk=Ub,k.

State function. The state evolution is modeled as a random walk with process noise w: 13Ub,k=f(Ub,k-1)+wk-1=Ub,k-1+wk-1.

Considering a random walk is equivalent to assuming that the estimated state is a bias because the expected value is a constant. Using a random walk to model the evolution of the state is thus a simplification in this case, as the blade-effective wind speed is expected to be periodic with a 1P component. A model representation accounting for the 1P periodicity of the blade-effective wind speed as the one proposed in would better represent the system dynamics. Yet, the random walk was successfully used in the context of blade-effective wind speed estimation in and because the 1P variations are slow compared to the estimation time step and can thus be captured.

Measurement function. The state is estimated based on the measurements of the blade out-of-plane bending moments, Mnb. We model the system using the BEM theory, and hence the nonlinear output equation is 14Mnb,k=hUb,k,[Ωk,βb,k]+vk,15=BEMUb,k,[Ωk,βb,k]+vk.

Note that the original version of the BEM theory is based on the conservation of momentum in a stream tube flowing through the rotor . Therefore, it runs under the assumption of rotor-effective quantities. Appendix  shows how the BEM is used to return individual blade loads from blade-effective velocities. Another key assumption of the classical BEM theory, which we further refer to as static BEM, is that the wake is fully developed and that the induction of the rotor is then (quasi-)steady . In Sect. , we will show how this BEM model can be modified to account for the induction changes generated by the pitch control strategies considered in this paper.

Noise covariance matrices. Several combinations of process noise covariance and measurement noise covariance are tested to determine the best configuration. The values leading to the smallest estimation errors and avoiding excessive filtering are Q*=Q/(U*)2=0.1 and R*=R/(Mn*)2=0.0001, where U* and Mn* represent the order of magnitude of the incoming wind speed and bending moments, respectively.

Jacobian matrices. The expression of the Jacobian matrix for the state function is trivial: Fk=1.

As the BEM is an iterative method (see Appendix ), an analytical expression of the Jacobian matrix for the measurement function cannot be provided. We therefore numerically compute it using central finite differences, with ΔU a constant value tuned manually, following 16Hk=h(U^b,k++ΔU,[Ωk,βb,k])-h(U^b,k+-ΔU,[Ωk,βb,k])2ΔU. Note that this procedure requires two more BEM evaluations. Opting for forward finite differences, for example, could save one evaluation if computational time was to become an issue.

The procedure described above considers the estimation of the blade-effective wind speeds that are naturally expressed in the blade rotating frame. They are further manipulated to provide information about the incoming flow in the fixed frame. We also want the information to be provided at a local scale in order to capture turbulent and shear patterns in the wind. The rotor is then divided into a chosen number of sectors, nS, and we map the blade-effective wind speeds to the sector-effective wind speeds (see Fig. ). A sector-effective wind speed Us is the mean blade-effective wind speed seen by the blade as it travels across a sector s. A sector-effective wind speed is updated every time a blade leaves that sector, i.e., at a 3P frequency. The update procedure is formally given in Appendix . At every instant, the rotor-effective wind speed is computed as the average velocity of the nS sectors.

The wind field can also be synthesized as a sheared plane (see Fig. ). This provides a representation of the shear present in the atmospheric boundary layer through a tilting shear coefficient αtilt, but also gust- or wake-generated shear in the yaw direction αyaw. The least-square approximation is used to find the expression of the plane best fitting the sector-effective wind speeds. Following derivations in , sector-effective wind speeds are assumed to account for the wind speed at r=2R/3. The target expression for the velocity field is 17U(y,z)=Ur+αtilty+αyawz, with y the vertical direction and z the transverse one.

Figure  shows the impact of the controller on the operating parameters and the out-of-plane moments. The baseline case shows that, if the pitch is equal for all blades, the effect of shear and turbulence generates 1P oscillations (period Trot) on the blade bending moments.

When IPC is used, the changes in wind speed perceived by the blade as it rotates are compensated for by the pitch actuation. These vary individually at the 1P frequency, and the amplitude of the actuation varies with time as the controller is closed-loop.

For the Pulse case, all blades are pitched at the same angle. Given the Strouhal number of 0.25, the actuation period Tp is about 10 times the rotation period. The amplitude of the pitch oscillations is constant as the strategy is open-loop. The mean value of the out-of-plane moments is marked by this periodicity and displays the changes in rotor induction.

For the Helix case, it comes from the Coleman transform that the pitching frequency is 1/Trot+1/Tp (see , for derivation), while the pitching amplitude is constant. This does not impact the mean value of the bending moments, as the individual pitch action is performed in a three-phase manner, with a 120° offset between each blade. Rather, the effect is also visible in the amplitude of the bending moments, which is sometimes increased (around t=20 s in Fig. ) and sometimes decreased (around t=50 s in Fig. ). The effect is better understood looking at tilt and yaw moments, as those become quadrature-phase sinusoidal signals with a 1/Tp frequency.

This section has highlighted how the control strategies considered in this work impact the state of the turbine, i.e., its operating parameters and the loads it experiences. This further motivates the need to verify the robustness of the blade-load-based estimator to the pitch control strategy active on the turbine. It also points at verifying the accuracy of the BEM in scenarios that involve dynamic actuation.

In the current formulation of the estimator, the internal model relies on the BEM theory. As discussed before, the standard BEM maps the operating parameters of the turbine and the wind speed to the forces exerted on the blades, and the mapping is static. This comes from the assumption of the BEM that the wake is fully developed, which underlies that the induction of the blades and thus the induced velocities around them are (quasi-)steady. Yet the Pulse and the Helix inherently generate periodic variations of the induction, and hence the wake is never fully developed. Through the collective pitch actuation, the Pulse changes the induction of the entire rotor. When it comes to the Helix, the individual pitch actuation changes the local induction of each blade. A time delay therefore exists before equilibrium is reached between the induction factors and the aerodynamic loads. In order to account for it, we propose making use of a dynamic formulation for the BEM based on the work of . Appendix  presents how the quasi-steady normal, a, and tangential, a′, induction factors are computed. From these induction factors, the quasi-steady induced velocities, normal wn and tangential wt, are recovered as wqs=[wn,wt]=[aU0,a′Ωrotr], where U0 is the infinite upstream velocity and r is the local radius of the considered blade section (see Fig. ).

Following , the induced velocities are filtered with the following first-order differential equations: 27wint+τ1dwintdt=wqs+k1τ1dwqsdt,28w+τ2dwdt=wint, where wint is a working variable, k1=0.6 is a constant, and τ1 and τ2 are time constants.

The corrected induction factors are retrieved from the corrected induced velocities w=[wn,wt] following 29[a,a′]=[wn/U0,wt/(Ωrotr)].

We further refer to the BEM without dynamic effects as the static BEM and to the one enhanced with dynamic effects as the dynamic BEM. The tunable time constants τ1 and τ2 for the dynamic BEM are calibrated against LES data. To do so, we perform the simulation of the NREL 5 MW in uniform inflow with no shear at Uref=9 m s-1 when the Pulse and the Helix are active. The turbine rotation speed and blade pitch angles determined by the controller in the LES framework are retrieved and fed to the static and dynamic BEM. We recall that the immersed lifting line method used to compute aerodynamic forces in the LES intrinsically accounts for dynamic effects in the induction and is thus the reference.

Figure  compares, over one period Tp, the BEM values with those provided by the LES for the local induction and the out-of-plane bending moment.

It shows that, with the static BEM, the axial induction is in direct phase opposition with the pitch angle. It behaves as if, as soon as the pitch angle increases, the induced velocities are reduced. When considering the dynamic effects, a certain delay appears, which is different for the Pulse and the Helix.

Based on the analytical expression provided in , we propose the following tuning of the time constants: 30τ1=17(1-1.3a)1fpitch,31τ2=0.39-0.26rR2τ1, with R the rotor radius.

TSR being the tip speed ratio, we recall that the pitching frequency of the blade fpitch is 32fpitchPulse=fp=StUrefD,33fpitchHelix=frot+fp=TSRπ+StUrefD.

This section therefore shows that ignoring the delays leads to poor evaluation of the angle of attack, which eventually leads to inaccurate loads. Including dynamic effects in the BEM therefore increases the accuracy of the internal model of the estimator, which should improve the quality of the estimate provided by the EKF.

We divide the rotor into four sectors (nS=4) placed as a cross, which leads to a top, right, bottom, and left sector. This allows us to capture the effects of shear thanks to the top–bottom differences and the effects of gust or wake impingement through the left–right imbalances. We first comment on the results of the estimator in its original formulation, i.e., using the static BEM. Time series of the rotor-effective and sector-effective wind speeds are provided in Fig. , along with their power spectral density (PSD).

When it comes to the rotor-effective wind speed, the estimations are almost identical for the baseline, IPC, and Helix cases. When the Pulse is active on the turbine, oscillations appear in the wind speed estimate at the Pulse frequency. For the sector-effective wind speed, these oscillations are present not only for Pulse but also for Helix.

This highlights that neglecting the dynamic effects in the internal model of the system, as is done with the look-up table approaches in , , and , is inadequate for the considered application. When the static BEM is used, the expected measurement provided by the internal model in the EKF is not accurate. There is a mismatch with the actual measurement, and the wind speed estimate is wrongfully corrected in the correction step. For the Pulse, this leads to periodic oscillations of both the rotor-effective wind speed and the sector-effective wind speed at the actuation frequency 1/Tp. With the Helix control, the changes in induction are performed individually at each blade in a three-phase manner, with a 120° offset between each blade. The inaccurate oscillations therefore appear on the blade-effective wind speeds, but they cancel out at the rotor scale.

Figure  presents the result of the estimators once dynamics are taken into account in the internal BEM model. The estimates are now identical whatever the control strategy active on the turbine: the adapted estimator is robust to the control strategy. The capacities of the estimator described in Sect.  and demonstrated by several studies in the literature are then retrieved. The estimator is able to recover the higher velocities in the top sector, the intermediate ones in the left and right sectors, and the lower ones in the bottom sector. It is thus able to capture turbulence, shear, and gusts. The estimation errors are reported in Table  and are close to 5%.

The overestimation bias discussed in Sect.  is also observed, yet it is around 5% here, while it was around 1.5 %–2 % in the validation case at the same 9 m s-1 wind speed. We attribute this increase in bias to the setup used for the simulations. Indeed, to maintain affordable computational costs, the spatial resolution used for the LES is coarsened to D/32, while the resolution was 2 times finer in Sect. . It is known that the forces computed by the LES increase when the resolution decreases due to poor capture of the tip losses . The mismatches between the internal model (BEM) and the actual system (LES) are higher with the simulations in this section than with those from Sect. . In the correction step, the EKF therefore corrects the velocity to a higher value than it should. As suggested before, this could be corrected with bias estimation .

The Pulse and the Helix are wake mixing control strategies, which means that their purpose is to modify the wake. Namely, the Pulse is expected to generate a pulsing sequence in the wake, with alternating zones of lower and higher velocities, while the Helix laterally and vertically displaces the wake as it propagates downstream (see Fig. ). In this section, we want to verify the performances of the estimator in waked flows and its ability to capture the frequency content added to the wake by the Pulse and the Helix.

Figure a shows that the estimator performs reasonably well in waked conditions, capturing the large-scale changes in the wind. Yet, the estimation somewhat deteriorates compared to the upstream turbine case, as Table  reports. Two reasons can be mentioned to support the bigger discrepancies and stem from the inherent characteristics of a wind turbine wake: reduced velocity and higher turbulence. On the one hand, the flow dynamics are harder to capture due to the complexity and randomness of turbulence. On the other hand, the BEM loses accuracy at lower velocity, hence the bigger bias present in the estimation (6% on average against the 5% for the freestream turbine).

Figure b shows that the estimator is able to capture the frequency content added to the wake by the Pulse and the Helix control strategies. Looking back at the time series, it is also interesting to notice that the phase at which higher- and lower-velocity flow parcels are impacting a sector, or the rotor, is properly captured. This is an interesting result from the perspective of applying the Helix or the Pulse to deeper lines of turbines. The downstream turbine could therefore also be actuated with the Helix or the Pulse. To do so in an optimal way, it should synchronize its action with the periodic perturbations already present in the wake, as proposed in and .