## 2 Wind Turbine and Wind Speed Modelling
### Linear Model of Wind Turbine Using DNV-Bladed  
The linear model is obtained in state-space form by using the Bladed model of the Supergen 5 MW exemplar turbine in each operating mode as follows:  
\[\Delta\dot{\mathbf{x}}(t) =A\Delta\mathbf{x}(t)+B\Delta\mathbf{u_{t}}(t)\] \[\Delta\mathbf{y}(t) =C\Delta\mathbf{x}(t)+D\Delta\mathbf{u_{t}}(t) \tag{1}\]  
where \(A\), \(B\), \(C\), and \(D\) represent the state space matrices. \(\Delta\mathbf{x}(t)\in R^{p}\), \(\Delta\mathbf{u_{t}}(t)\in R^{n}\), and \(\Delta\mathbf{y}(t)\in R^{m}\) (where n and m are three and two, respectively, at each mean wind speed, and p is 240) are defined as the following:  
\[\Delta\mathbf{x}(t) =\mathbf{x}(t)-\mathbf{x_{op}}(t)\] \[\Delta\mathbf{u_{t}}(t) =\mathbf{u_{t}}(t)-\mathbf{u_{t,op}}(t)\] \[\Delta\mathbf{y}(t) =\mathbf{y}(t)-\mathbf{y_{op}}(t)\]  
where \(\Delta\mathbf{y}(t)\), \(\Delta\mathbf{u_{t}}(t)\), and \(\Delta\mathbf{x}(t)\) denote the output, input, and state, respectively. \(\mathbf{u_{t,op}}(t)\), \(\mathbf{y_{op}}(t)\), and \(\mathbf{x_{op}}(t)\) are the operating points around which the models are linearized. The inputs for the linear models are as follows:  
1. Generator torque;
2. Pitch angle;
3. Wind speed. The outputs are as follows:  
Figure 3: Point wind speed (green) vs. effective wind speed (black) at a mean wind speed of 10 m/s. Generator speed;  
2. Tower fore-aft acceleration. As aforementioned, for the linearized models containing 240 states, making the realization and tuning of MPC, \(H_{\infty}\), and LQG controllers could be challenging. Consequently, in order to design optimal controllers, the models must be initially reduced. Therefore, Match-DC-Gain model reduction algorithm, which eliminates the states with the smallest Hankel singular values, is employed. The Match-DC-Gain method subsequently modifies the remaining states to retain the DC gain. For the following continuous model the procedure for the Match-DC-Gain method is as follows:  
\[\dot{x} =Ax+Bu\] \[y =Cx+Du \tag{3}\]  
where the state vector is partitioned as follows:  
\[ \begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix}  = \begin{bmatrix}A_{11}&A_{12}\\ A_{21}&A_{22}\end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix} + \begin{bmatrix}B_{1}\\ B_{2}\end{bmatrix} u\] \[y = \begin{bmatrix}C_{1}&C_{2}\end{bmatrix} x+Du \tag{4}\]  
where \(x_{1}\) must be retained, and \(x_{2}\) is discarded. Thus, the solution \(x_{1}\) is obtained by setting the derivative of \(x_{2}\) to zero. The resulting reduced-order dynamic model is as follows. \[\dot{x} = \begin{bmatrix}A_{11}-A_{12}A_{22}{}^{-1}A_{21}\end{bmatrix} x_{1 }+ \begin{bmatrix}B_{1}-A_{12}A_{22}{}^{-1}B_{2}\end{bmatrix} u\] \[y = \begin{bmatrix}C_{1}-C_{2}A_{22}{}^{-1}A_{21}\end{bmatrix} x+ \begin{bmatrix}D-C_{2}A_{22}{}^{-1}B_{2}\end{bmatrix} u \tag{5}\]  
Figures 4-6 show the frequency responses of the original linear models, containing 240 states in modes 1, 2, and 3, respectively, in comparison with the frequency response of the resulting reduced-order models, which now contain only 20 states due to model reduction. The results demonstrate that the original and reduced-order models exhibit identical characteristics within the meaningful frequency range, i.e., \(10^{-2}\) to \(10^{2}\) rad/s. However, the responses diverge significantly outside of this range. Figure 4: Frequency response of Bladed and reduced–order models in mode 1. When a controller is designed in modes 1 and 2, the generator torque becomes the only input while wind speed is considered process disturbance, \(d(t)\in R^{1}\), and the pitch angle demand is set to zero altering Equation (1) because pitch control is not used in below-rated wind speed as follows:  
\[\Delta\dot{\mathbf{x}}(t) =A\Delta\mathbf{x}(t)+B_{u}\Delta u_{t}(t)+B_{dd}\Delta d(t)\] \[\Delta\mathbf{y}(t) =C\Delta\mathbf{x}(t)+D\Delta u_{t}(t)+D_{dd}\Delta d(t) \tag{6}\]  
where the generator torque demand is now the only input, \(u(t)\in R^{1}\).