3.1. Initial guess for model parameters

An estimate of the coefficients $ {w}_{i,m,n} $ can be computed using a standard least squares approach under the assumption of steady-state operation, i.e., $ {\dot{\omega}}_i=0 $ . Under this condition, Equation (2.3) yields $ {\tau}_{a,i}={\tau}_{g,i} $ . Substituting this relation into Equation (2.1), one obtains

(3.4) $$ {C}_{p,i}=\frac{2{\tau}_{g,i}{\omega}_i}{\rho \pi {R}^2{u}_1^3}. $$

Assuming that $ {\tau}_{g,i} $ , $ {\omega}_i $ and $ {u}_1 $ are measured, Equation (3.4) can be used to compute $ {C}_{p,i} $ , allowing for deriving initial guesses to train the surrogate models in Equations (3.1) and (3.2) via least squares regression. For instance, in the case of the first turbine in Equation (3.1), collecting a grid of $ {n}_{\lambda}\times {n}_{Re} $ training samples, with $ {n}_{\lambda } $ and $ {n}_{Re} $ the number of tip speed ratios and Reynolds numbers in the grid, and enforcing Equation (3.4) in each of the grid points yields a matrix factorization of the form

(3.5) $$ {\boldsymbol{C}}_{p,1}=\boldsymbol{\Phi} \left({\boldsymbol{\lambda}}_1\right)\hskip0.1em {\boldsymbol{w}}_1\hskip0.1em {\boldsymbol{\Psi}}^T\left(\mathbf{\operatorname{Re}}\right), $$

where $ {\boldsymbol{C}}_{p,1}\in {\mathrm{\mathbb{R}}}^{n_{\lambda}\times {n}_{Re}} $ is the matrix of power coefficients computed from Equation (3.4) at all training points. The matrix $ \boldsymbol{\Phi} \left({\boldsymbol{\lambda}}_1\right)\in {\mathrm{\mathbb{R}}}^{n_{\lambda}\times 5} $ is the radial basis function (RBF) matrix evaluated at the tip-speed ratios $ {\boldsymbol{\lambda}}_1 $ , while $ \boldsymbol{\Psi} \left(\mathbf{\operatorname{Re}}\right)\in {\mathrm{\mathbb{R}}}^{n_{Re}\times 3} $ contains the polynomial basis evaluated at the Reynolds numbers $ \mathbf{\operatorname{Re}}\in {\mathrm{\mathbb{R}}}^{n_{Re}} $ . The coefficient matrix $ {\boldsymbol{w}}_1\in {\mathrm{\mathbb{R}}}^{5\times 3} $ , with entries $ {w}_{1,m,n} $ , contains the parameters to be identified in the surrogate model Equation (3.1) and de facto encodes the performances of the turbine under the dynamical system in Equations (2.2)–(2.3).

These coefficients can be computed by pseudo-inverse on both sides of Equation (3.5):

(3.6) $$ {\boldsymbol{w}}_{1,0}={\left({\boldsymbol{\Phi}}^T\boldsymbol{\Phi} \right)}^{-1}{\boldsymbol{\Phi}}^T{\boldsymbol{C}}_{p,1}\boldsymbol{\Psi} {\left({\boldsymbol{\Psi}}^T\boldsymbol{\Psi} \right)}^{-1}. $$

where $ {\boldsymbol{w}}_{1,0} $ emphasizes that these are the initial guesses for the optimization.

Considering that this estimate acts only as an initial guess, no regularization was used in the least square solution. On the other hand, taking into account the measurement uncertainties was found to improve the initial guess. These were introduced using a classic weighted formulation on the regression of the RBF block, which reads

(3.7) $$ {\boldsymbol{w}}_{1,0}={\left({\boldsymbol{\Phi}}^T\mathbf{N}\boldsymbol{\Phi } \right)}^{-1}{\boldsymbol{\Phi}}^T\mathbf{N}{\boldsymbol{C}}_{p,1}\boldsymbol{\Psi} {\left({\boldsymbol{\Psi}}^T\boldsymbol{\Psi} \right)}^{-1}, $$

where the matrix $ \mathbf{N} $ is an estimate of the measurement precision, giving more weight to points with lower uncertainty and defined as

(3.8) $$ \mathbf{N}={\sigma}_{\mathrm{min}}^2\left[\begin{array}{cccc}\frac{1}{\sigma_1^2}& 0& \dots & 0\\ {}0& \ddots & 0& \vdots \\ {}\vdots & 0& \ddots & 0\\ {}0& \dots & 0& \frac{1}{\sigma_{n_j}^2}\end{array}\right], $$

where $ {\sigma}_j^2 $ is the variance of each measurement and $ {\sigma}_{\mathrm{min}}=\min \left\{{\sigma}_i\right\} $ . These were used by propagating the measurement uncertainties of $ {u}_i $ , $ {\tau}_{g,i} $ and $ {\omega}_i $ through Equation (3.4) using a standard first-order approach. A comparison between weighted and unweighted regressions shows that the inclusion of measurement uncertainties leads to an improvement of approximately 10% in the uncertainty-weighted error (wRMSE), indicating a better agreement with the most reliable data. The solution of the initial set of coefficients from Equation (3.7) can be interpreted as first carry out a weighted least square regression of the model $ {\mathbf{C}}_{\mathbf{p},\mathbf{1}}=\boldsymbol{\Phi} \mathbf{A} $ , followed by a traditional least square regression of the model $ {\mathbf{A}}^T=\boldsymbol{\Psi} {\boldsymbol{w}}_{1,0}^T $ .

3.2. Model parameters obtained from a BEM-based model

In order to provide a reference model for comparison, the same parametrization described in Section 3.1 was also constructed using data generated from a classical Blade Element Momentum (BEM) approach. The motivation for introducing a BEM-based model is that it represents the standard tool used in the wind energy community to derive aerodynamic models for control-oriented applications.

In this case, instead of relying on experimental measurements, a grid of operating conditions was generated numerically. In particular, the power coefficient $ {C}_{p,1} $ was evaluated over a set of tip-speed ratios $ {\lambda}_1 $ and Reynolds numbers $ \operatorname{Re} $ , forming a synthetic dataset analogous to $ {\boldsymbol{C}}_{p,1} $ in Equation (3.5). The aerodynamic polars of the airfoil, i.e., lift and drag coefficients as a function of the angle of attack, were computed using XFOIL at different Reynolds numbers. These polars were then used within a BEM framework to evaluate the turbine performance, using the CcBlade module from the WISDEM Python package (Ning, Reference Ning2013).

Given this dataset, the same regression framework as in Section 3.1 was employed. In particular, the coefficient matrix was obtained by solving a least squares problem of the form

(3.9) $$ {\boldsymbol{C}}_{p,1}^{\mathrm{BEM}}=\mathtt{\varPhi}\left({\boldsymbol{\lambda}}_1\right)\hskip0.1em {\boldsymbol{w}}_1^{\mathrm{BEM}}\hskip0.1em {\mathtt{\varPsi}}^T\left(\mathbf{\operatorname{Re}}\right), $$

where $ {\boldsymbol{C}}_{p,1}^{\mathrm{BEM}} $ denotes the matrix of power coefficients obtained from the BEM simulations. The corresponding model coefficients $ {\boldsymbol{w}}_1^{\mathrm{BEM}} $ are computed analogously to Equation (3.6). In this case, no weighting based on measurement uncertainty is introduced, as the dataset is generated numerically.

It is important to note that, although BEM models are widely used and well established for utility-scale wind turbines, their application in the present context must be treated with care. The experiments considered in this work are conducted at a small scale, corresponding to relatively low Reynolds numbers. Under these conditions, the aerodynamic behavior is more difficult to predict accurately, which may result in larger discrepancies. Therefore, the predictions obtained from the BEM-based model in the present study should be interpreted with caution, as they may be significantly less reliable than those usually obtained for conventional full-scale wind turbines.

Finally, it is worth noting that extending the BEM framework to the downstream turbine is not straightforward, due to the presence of wake interactions and the need for an additional wake model. Since this is not the focus of the present study, the BEM-based parametrization is considered here only for the first turbine and is used solely as a reference for comparison purposes.

3.3. A note on local stability

The local stability properties of the steady-state solutions defined by $ {\tau}_{a,i}={\tau}_{g,i} $ in Equation (2.3) can be assessed by linearizing the rotor dynamics around a given equilibrium point. Since the control input acts on the tip-speed ratio through the rotor speed, the relevant condition is expressed in terms of the local slope of the net torque with respect to $ {\lambda}_i $ . In open-loop operation, the system generally admits a family of equilibrium points, parameterized by the operating conditions. For a generic turbine $ i $ , local asymptotic stability of a given equilibrium requires

(3.10) $$ \frac{\partial f}{\partial {\lambda}_i}=\frac{1}{J}\left[\frac{1}{2}\rho \pi {R}^3{u}_1^2\frac{\partial }{\partial {\lambda}_i}\left(\frac{C_{p,i}}{\lambda_i}\right)-\frac{\partial {\tau}_{g,i}}{\partial {\lambda}_i}\right]<0, $$

where the derivative is evaluated with the operating conditions held fixed, namely the freestream velocity $ {u}_1 $ and, for downstream turbines, the upstream tip-speed ratio $ {\lambda}_{i-1} $ .

It is important to note that, in open-loop operation, not all equilibrium points are necessarily stable. Depending on the operating region, unstable equilibria may also arise. For this reason, stability was not explicitly enforced as a global constraint during identification. Instead, the optimization relies on matching the measured rotor trajectories in the operating regions explored by the data. As a consequence, parameter choices associated with incorrect local stability properties tend to produce trajectories that are inconsistent with the experimental observations, and are therefore penalized by the cost functional. As shown later in the results, this mechanism plays a visible role in the optimization process, where even limited parameter updates may lead to marked changes in the model response once dynamically consistent local behavior is recovered.

3.4. Adjoint-based identification

In this work, the Lagrangian in Equation (2.5) is defined as

(3.11) $$ {\mathrm{\mathcal{L}}}_p\left({\mathbf{w}}_i\right)={\left({\omega}_i^{\ast }(t)-{\omega}_i\Big(t;{\mathbf{w}}_i\Big)\right)}^2. $$

The optimization for each turbine is performed independently. This is possible because, in the identification of a downstream turbine, the upstream tip-speed ratio $ {\lambda}_{i-1} $ is treated as a measured exogenous input. Therefore, the model of turbine $ i $ can be identified separately once the corresponding upstream trajectory is provided from the experimental data. The measured time series $ {\omega}_i^{\ast }(t) $ is filtered using a $ {4}^{\mathrm{th}} $ order a Butterworth low-pass filter with a cutoff frequency of $ {f}_{\mathrm{c}.\mathrm{o}.}=2\hskip0.22em \mathrm{Hz} $ chosen based on the turbine’s time response to attenuate high-frequency noise.

The cost functions $ {\mathcal{J}}_{w,i}\left({\mathbf{w}}_i\right) $ are optimized using the Adam optimizer (Kingma and Ba, Reference Kingma and Ba2014), preferred over quasi-Newton alternatives such as the BFGS because of its higher robustness to gradient noise. The gradient $ {\nabla}_{\mathbf{w},i}{\mathcal{J}}_{w,i} $ was computed via an adjoint-based formulation (Sengupta et al., Reference Sengupta, Friston and Penny2014) to avoid solving $ {n}_w $ additional ODEs per evaluation. The adjoint method computes the gradient as

(3.12) $$ {\nabla}_{\mathbf{w},\mathbf{i}}{\mathcal{J}}_{w,i}\left({\mathbf{w}}_i\right)={\int}_0^{T_0}{s}_{\lambda}^T(t)\left(\frac{\partial f}{\partial {\mathbf{w}}_i}\right)\hskip0.1em dt $$

where $ {s}_{\lambda }(t) $ are the adjoint variables. These are obtained by solving the adjoint terminal value problem

(3.13) $$ \left(\begin{array}{l}\frac{ds_{\lambda }}{dt}=-{\left(\frac{\partial f}{\partial {\omega}_i}\right)}^{\top }{s}_{\lambda }(t)-{\left(\frac{\partial {\mathrm{\mathcal{L}}}_p}{\partial {\omega}_i}\right)}^{\top },\\ {}{s}_{\lambda}\left({T}_0\right)=0\end{array}\right. $$

where all terms are evaluated at the current guess for the states $ {\omega}_i\left(t;{\mathbf{w}}_i\right) $ , which thus requires first a forward integration. We note that the transposition in Equation (3.13) is included for consistency with high dimensional problems but is irrelevant to the problem at hand since both Jacobians are scalars. The adjoint-based evaluation requires one forward pass and one backward pass, regardless of the number of unknown parameters $ {\mathbf{w}}_i $ .

Starting from the initialization as in Equation (3.7), the gradient computation is used then to update the weight guess using the ADAM optimizer (Kingma and Ba, Reference Kingma and Ba2014)

(3.14) $$ {\mathbf{w}}_i^{\left(n+1\right)}={\mathbf{w}}_i^{(n)}-\eta \frac{{\hat{\mathbf{m}}}^{(n)}}{\sqrt{{\hat{\mathbf{v}}}^{(n)}}+\unicode{x025B}} $$

where $ n $ is the iteration number, $ \eta $ is the learning rate, $ \unicode{x025B} $ is a small constant to prevent division by zero, and $ \hat{\mathbf{m}} $ and $ \hat{\mathbf{v}} $ are filtered momentum parameters computed as follows:

(3.15) $$ {\hat{\mathbf{m}}}^{(n)}=\frac{\beta_1{\mathbf{m}}^{\left(n-1\right)}+\left(1-{\beta}_1\right){\nabla}_{\mathbf{w},\mathbf{i}}{\mathcal{J}}_{w,i}^{(n)}}{1-{\beta}_1^n} $$

(3.16) $$ {\hat{\mathbf{v}}}^{(n)}=\frac{\beta_2{\mathbf{v}}^{\left(n-1\right)}+\left(1-{\beta}_2\right){\left[{\nabla}_{\mathbf{w},\mathbf{i}}{\mathcal{J}}_{w,i}^{(n)}\right]}^2}{1-{\beta}_2^n}, $$

where $ \mathbf{m} $ and $ \mathbf{v} $ are the first and second moment estimates of the gradient, while $ {\beta}_1 $ and $ {\beta}_2 $ are the decay rate coefficients for the moments.

Multiple restarts were employed to reset the optimization process whenever a sudden jump in the cost function occurred, typically because the current parameter guess violates the stability condition in Equation (3.10). In such a case, the optimization is restarted to reset the low-pass filtering of the gradient. Additionally, a learning rate schedule was designed to reduce the learning rate once the cost function falls below a predefined threshold. This approach allows the optimizer to take smaller steps and prevent overshooting. It is worth noting that, while uncertainty weighting is introduced in the construction of the initial guess (Section 3.1), it is not explicitly propagated in the adjoint-based dynamic identification. This choice is motivated by the fact that the initial reconstruction of the power coefficient is highly sensitive to measurement noise, particularly due to the cubic dependence on wind speed. In contrast, the dynamic identification relies on the rotor speed signal, which is measured with lower uncertainty, and whose dynamics inherently filter high-frequency fluctuations.

4.1. Experimental set up

The proposed methodology was experimentally validated using three-bladed wind turbine scale models, originally designed in Coudou et al. (Reference Coudou, Buckingham, Bricteux and Beeck2018). Each model features a rotor diameter of $ D=0.15\hskip0.22em \mathrm{m} $ and a hub height of $ {z}_{\mathrm{hub}}=0.13\hskip0.22em \mathrm{m} $ . The rotor was designed based on Burton’s optimal rotor theory (Sørensen and Sørensen, Reference Sørensen and Sørensen2016) and employs a low-Reynolds-number airfoil section (HAM-STD HS1–606). This design enables the turbines to achieve power and thrust coefficient performances comparable to those of full-scale turbines such as the Vestas V66-2 MW offshore turbine, of which it represents a 1:440 scale version. A photograph of the turbines is shown in Figure 2 together with the relevant dimensions and airfoil section. The total moment of inertia of the rotor has been estimated from the CAD model to be $ J=2.5\times {10}^{-6} $ kg m $ {}^2 $ .

Figure 2.

Wind turbine model used in the experimental campaign (Coudou et al., Reference Coudou, Buckingham, Bricteux and Beeck2018) and corresponding low-Reynolds-number airfoil used in the blade.

Each rotor is directly coupled to a DC motor (Faulhaber 1331T006SR) operating as a generator. The blades are fixed, hence the only actuation is through motor torque, adjusted via a variable resistance $ {R}_{\mathrm{v}} $ connected to the generator. The generator torque $ {\tau}_g $ is modeled electrically by relating it to the current $ i $ as $ {\tau}_g={k}_{\tau}\cdot i $ where $ {k}_{\tau } $ is a motor constant. The back electromotive force (EMF) is assumed proportional to the rotor speed: $ {V}_{\mathrm{emf}}={k}_{\omega}\cdot \omega, $ with $ {k}_{\omega } $ also a motor-specific constant. Neglecting generator inductance $ L $ and applying mesh analysis to the equivalent circuit (Figure 3), the generator torque becomes:

(4.1) $$ {\tau}_g={k}_{\tau}\cdot \frac{k_{\omega}\omega }{R_{\mathrm{tot}}+{R}_{\mathrm{v}}} $$

where $ {R}_{\mathrm{tot}}={R}_{\mathrm{m}}+{R}_{\mathrm{c}} $ , with $ {R}_{\mathrm{m}} $ and $ {R}_{\mathrm{c}} $ representing the internal motor and cable resistances, respectively. This relation is used in Equation (2.3) to compute the mechanical torque on the turbine shaft.

Figure 3.

Electrical model of the wind turbine generator and relevant variables.

The experimental setup with sensors and controllers is illustrated in Figure 4 for the case of two turbines. A Raspberry Pi 4 is employed for real-time data acquisition and turbines’ control. Each turbine is equipped with an incremental encoder for measuring the rotational speed $ {\omega}_i^{\ast }(t) $ . These encoders feature 50 pulses per revolution (PPR), and the rotational speed is computed by an Arduino Nano through the measurement of the time interval between two consecutive pulses.

Figure 4.

Experimental setup in the low-turbulence scenario configuration, detailing the measurement of free-stream wind speed $ {u}_1 $ , rotational speed $ {\omega}_i^{\ast } $ of each turbine, and torque actuation via variable resistance $ {R}_{v,i} $ controlled by a 12-bit binary signal $ {a}_{bin,i} $ .

The free-stream wind speed at hub height $ {u}_1\left({z}_{\mathrm{hub}},t\right) $ is measured using a Prandtl tube. The dynamic pressure $ \Delta p=\frac{1}{2}\rho {u}_1^2 $ , defined as the difference between total $ {p}^0 $ and static pressure $ {p}_s $ , is acquired using an AMS 5812–0000-DB sensor, from which the wind speed is then derived $ {u}_1=\sqrt{2\Delta p/\rho } $ . A thermocouple is installed to monitor air temperature and enhance the accuracy of air density estimations.

Torque actuation is implemented by modulating the electrical load through a bank of 12 resistors, each controlled via MOSFETs. This configuration enables $ {2}^{12} $ discrete resistance values, ranging from 0.25 $ \Omega $ to 1024 $ \Omega $ . A 12-bit binary signal $ {a}_{bin} $ is sent from the Raspberry Pi 4 to a shift register (SNx4HC595), which selects the active resistors. The corresponding resistance $ {R}_{\mathrm{v}} $ is retrieved from a precomputed lookup table.

4.2. Low turbulence scenario identification

The first scenario evaluates the proposed methodology for identifying the aerodynamical characteristics of two wind turbine models in a tandem configuration, where the operational conditions of both turbines are varied (Figure 4). This identification under varying conditions is used for the control implementation, as discussed in Section 4.4. The two wind turbine models are separated by four rotor diameters ( $ 4D\approx 60 $ cm). The experimental campaign was conducted in the VKI Wind Engineering facility L2-B, an open-circuit wind tunnel with a test section of dimensions $ H=0.5\mathrm{m} $ , $ W=0.5\mathrm{m} $ , and $ L=0.8\mathrm{m} $ , capable of generating uniform flow velocities up to $ {U}_{\infty }=35\mathrm{m}\hskip0.1em {\mathrm{s}}^{-1} $ . The Prandtl tube was positioned approximately two rotor diameters upstream of the first turbine. The limited length of the test section prevents the development of a fully developed boundary layer representative of atmospheric conditions, resulting in low turbulence intensity. The blockage ratio is approximately 5% (Coudou, Reference Coudou2021), which is low enough to avoid significant wake distortion (McTavish et al., Reference McTavish, Feszty and Nitzsche2014).

The system identification process was performed by analyzing the response of $ {\omega}_i^{\ast }(t) $ to a combination of control steps in $ {R}_{\mathrm{v}} $ and gradually varying freestream velocities $ {u}_1(t) $ . The upstream turbine model was trained using seven episodes, while the downstream turbine model relied on five episodes. Each episode, as described in Section 2, corresponds to a time interval of $ {T}_0=32\mathrm{s} $ , with a data sampling frequency of $ {f}_s=20\mathrm{Hz} $ . Both were then tested on four and three unseen wind/loading episodes respectively. The model identification for the two turbines was performed in separate episodes: first, the upstream turbine was trained, followed by the downstream turbine, which also utilized data from the first. This approach treats the identification as two distinct processes, even though, in principle, both turbines could be trained jointly. Figure 5 illustrates the input signals used in three training episodes and two testing episodes. The left panel shows the free-stream velocity $ {u}_1(t) $ applied during the training or testing of the upstream turbine ( $ {u}_{1,1}(t) $ ) and during the training or testing of the downstream turbine ( $ {u}_{1,2}(t) $ ). The right panel depicts the corresponding steps in generator resistance for the upstream turbine in the two cases ( $ {R}_{1,1}(t) $ and $ {R}_{1,2}(t) $ ), as well as the generator resistance applied to the downstream turbine during the identification of the waked turbine model ( $ {R}_2(t) $ ).

Figure 5.

Free stream velocity (left) and generator resistance evolution (right) for three representative training episodes (first three rows) and two testing episodes for the low turbulence scenario described in Section 4.2.

4.3. High turbulence scenario identification

The second scenario evaluates the proposed methodology in a wind farm configuration consisting of a $ 3\times 3 $ turbine array operating under high turbulence conditions. The experiments were conducted in the VKI Wind Engineering facility L1-B. This is a low-speed, closed-loop wind tunnel with a test section measuring $ H=2\mathrm{m} $ , $ W=3\mathrm{m} $ , and $ L=20\mathrm{m} $ , delivering a maximum wind speed of $ {U}_{\infty }=60\mathrm{m}/\mathrm{s} $ . The length of the test section allows the development of a fully established turbulent boundary layer at the location of the wind farm, which is positioned at the downstream end of the tunnel and thus entirely within the boundary layer. Further details on how the boundary layer generated in the L1-B tunnel replicates atmospheric flow conditions can be found in (Conan, Reference Conan2012).

A schematic of the experimental setup is shown in Figure 6. The turbine models were arranged with spanwise and streamwise spacings of 3D and 5D, respectively. For the nonlinear identification test, three turbines from the central row of the array were selected: one located in the freestream and two positioned in the wake region. This setup targets turbines operating near their optimal performance point under steady turbulent inflow, consistent with typical wind turbine control strategies. Notably, the freestream wind velocity was not measured directly at the first turbine. Instead, as shown in Figure 6, a Prandtl tube was positioned at a streamwise distance $ {d}_1=1.2\hskip0.22em \mathrm{m} $ upstream of the first turbine row, and at a spanwise offset $ {d}_2=1\hskip0.22em \mathrm{m} $ from the row centerline. Although the measurement is performed at hub height, it does not provide a rotor-equivalent wind speed representative of the inflow experienced by the turbine, a quantity that is particularly relevant for modeling.

Figure 6.

Wind farm configuration in the VKI Wind Engineering Facility L-1B, with the three identified wind turbines highlighted in red.

To maintain a consistent operating point, the resistance $ {R}_{\mathrm{var}} $ connected to the DC generator was fixed at $ {R}_{\mathrm{val}}=1\Omega $ , corresponding to a tip-speed ratio of $ {\lambda}_1=4.5 $ for a reference inflow velocity of $ {u}_1\approx 8.5\mathrm{m}/\mathrm{s} $ . This value approximately maximizes the power coefficient, as shown in (Coudou, Reference Coudou2021). The experimental setup and measurement procedure matched those described in previous sections. All signals were sampled at $ {f}_s=20\mathrm{Hz} $ over episodes lasting approximately $ {T}_0=20\mathrm{s} $ , with 80% of each dataset used for training and the remaining 20% reserved for testing.

4.4. Control under low turbulence conditions

To evaluate the accuracy of the identified models for control purposes, we return to the tandem configuration in Section 4.2. For both turbines, we consider the classic $ K{\omega}^2 $ law, commonly used in Region 2 to maximize power extraction (Pao and Johnson, Reference Pao and Johnson2009). This controller gives a quadratic torque law

(4.2) $$ {\tau}_g=K{\omega}^2, $$

where the constant $ K $ is defined as (Pao and Johnson, Reference Pao and Johnson2009):

(4.3) $$ K=\frac{1}{2}\rho {AR}^3\frac{C_{P_{\mathrm{max}}}}{\lambda_{\mathrm{max}}^3}, $$

where $ {\lambda}_{\mathrm{max}} $ denotes the tip-speed ratio corresponding to the maximum power coefficient $ {C}_{P_{\mathrm{max}}} $ .

In this work, the $ K{\omega}^2 $ controller is employed in terms of electrical generator resistance. Combining Equations (4.2) and (4.3) with (4.1) gives

(4.4) $$ {R}_{\mathrm{v}}=\frac{k_{\tau }{k}_{\omega }}{K\omega}-{R}_{tot}. $$

This model-based control strategy offers a valid test to assess the fidelity of the assimilated dynamic model, as the controller performance is inherently tied to the accuracy of the assumed $ {C}_P\left(\lambda, \mathit{\operatorname{Re}}\right) $ distribution. Moreover, in the investigated scenario, the controller is not used under typical operational assumptions that is to track the tip-speed ratio maximizing power production. Instead, the setpoint in terms of tip-speed ratio is intentionally varied to stress-test the model across a wider operating range.

To ensure reliability of the sensor measurements, all input signals are filtered using a first-order Butterworth low-pass filter with a cutoff frequency of $ {f}_{\mathrm{c}.\mathrm{o}.}=2 $ Hz. The performance of the controller is evaluated using three different modeling approaches: (1) the model identified from data, as described in Section 3.4; (2) the parametric model obtained from steady-state data regression through Equation (3.7), which provides the initial guess for the adjoint-based refinement; and (3) the model obtained by regression of Blade Element Momentum (BEM) simulation results, as described by Equation (3.9). This comparison enables an assessment of controller performance under different modeling assumptions.