## 3 Wind Turbine Model Validation and Disturbances
### Model Identification and Validation  
The state-space representation of the system can be derived from the equations of the theoretical model, Equations (4)-(6). Initially, the model identification is performed without including the TMD dynamics. Therefore, the state-space model consists of four first-order differential equations with constant coefficient matrices \(\mathbf{A}\), \(\mathbf{B}\), \(\mathbf{C}\), \(\mathbf{D}\), Equations (7) and (8). The output \(Y(t)\) is a vector that contains the barge and tower angular displacements, \(\theta_{t}\) and \(\theta_{b}\). During the model identification process, the inputs are considered to be zero, i.e., \(\mathbf{B}=\mathbf{D}=0\). A free decay test is performed; for this, the barge is initially displaced from its equilibrium (that is, the initial condition for the barge pitch angle is set to be different from zero). In this case, it is seen that both platform and tower exhibit unforced damped oscillations (decaying amplitude). \[\dot{\mathbf{X}}(t) =\mathbf{A}\mathbf{X}(t)+\mathbf{B}\mathbf{u}(t), \tag{4}\] \[Y(t) =\mathbf{C}\mathbf{X}(t)+\mathbf{D}\mathbf{u}(t),\] (5) \[X(t) = \begin{bmatrix}\dot{\theta}_{b}\\ \dot{\theta}_{t}\\ \theta_{b}\\ \theta_{t}\end{bmatrix} ,\;Y(t)= \begin{bmatrix}\theta_{b}\\ \theta_{b}\\ \theta_{t}\end{bmatrix} ,\] (6) \[\mathbf{A}= \begin{bmatrix}1/I_{b}(-d_{b}+d_{t})&1/I_{b}(d_{t})&1/I _{b}(-k_{b}-m_{b}gR_{b}+k_{t})&1/I_{b}(k_{t})\\ 1/I_{b}(d_{t})&1/I_{b}(-d_{t})&1/I_{b}(k_{t})&1/I_{b}(m_{t}gR_{t}-k_{t})\\ 1&0&0&0\\ 0&1&0&0\\ \end{bmatrix} ,\] (7) \[\mathbf{B}=0,\;\mathbf{C}= \begin{bmatrix}0&0&1&0\\ 0&0&0&1\end{bmatrix} ,\;\mathbf{D}=0, \tag{8}\]  
By simulating the turbine under similar operating conditions and the same inputs, OpenFAST helped to determine the spring stiffness, damping coefficient and inertia value of the model. This simulation program allows the generation of a wide variety of data sets needed for model identification and validation. In this study, the Levenberg-Marquardt algorithm was employed for parameter identification, using various initial platform angular displacement angles. These free decay experiments are also representative of wave-induced responses. The parameters obtained from the identification procedure are presented in Table 1, where subscripts \(b\) and \(t\) correspond to the barge and tower, respectively. The remaining parameters are predefined and can be found in [23]. The response of the simplified model is validated by comparing it with an OpenFAST FOWT simulation for a 100 s test run. In this comparison, the initial pitch angle of the platform is set to either \(3^{\circ}\) or \(5^{\circ}\), including the influence of the tuned mass damper (TMD). The TMD parameters are defined as follows: mass \(m_{TMD}\) = 20,000 kg, damping coefficient \(d_{TMD}\) = 9000 Ns/m, and stiffness coefficient \(k_{TMD}\) = 5000 N/m. This configuration is implemented to ensure the accuracy and validation of the theoretical model. It is worth noting at this point that the TMD setup utilized aligns with the configuration described in [11]. The results of both simulations are presented in Figure 2. The comparison between the OpenFAST reference model (blue line) and the reduced model simulation (red line) shows a satisfactory agreement, with a minor discrepancy that gradually increases over time. The pitch angle of the platform, referred to as PtfmPitch in OpenFAST, is depicted in the first plot. In the simplified model, this corresponds to the pitch angle \(\theta_{b}\), which is the angle between the platform and the vertical \(z\)-axis. The second plot illustrates the horizontal displacement of the tower top relative to the \(z\)-axis, represented by the Tower Top Displacement (TTD) in the Fore-Aft direction (TTDspFA). In the proposed model, this variable is expressed as \(L_{t}\cdot\theta_{t}\), where \(L_{t}\) denotes the length of the tower. Analyzing the behavior of these signals is important, as the oscillations induce loads on the tower, thereby reducing the wind turbine's operational lifespan.