# Model Based Open-Loop Wind Farm Control Using Active Power for Power Increase and Load Reduction
## 2 Methodology
### Wind Farm Controller  
The wind farm controller constructed and imbedded to the wind farm simulation tool in this study means a supervisory controller that collects wind turbine states and sends commands to individual wind turbines to maximize the farm power output. It receives the blade pitch angle, generator speed and electrical power from individual wind turbines and sends power command to individual turbines. The wind farm controller consists of a wind speed estimator, a power command calculator, and a wind farm model. Figure 2 is a block diagram of the relation between the simulation tool for the verification of the control algorithm and the wind farm developed through this study. The wind speed estimator estimates the averaged wind speeds at the rotor center of individual wind turbines from the known properties and measured signals from wind turbines. Firstly, it calculates the aerodynamic torque from Equation (3) which is a two-mass model equation of motion for the drive train available in the literature [28]. Figure 2: Structure of wind farm controller. \[\left(J_{r}+N^{2}J_{g}\right)\frac{\mathrm{d}\dot{\theta_{r}}}{\mathrm{d}t}= \mathrm{T}_{a}-N\mathrm{T}_{g}-\left(\mathrm{B}_{r}+N^{2}\mathrm{B}_{g}\right) \dot{\theta_{r}} \tag{3}\]  
In Equation (3), \(J\) is the mass moment of inertia, \(N\) is the gear ratio, \(\dot{\theta_{r}}\) is the rotational speed, \(\mathrm{T}\) is the torque, \(\mathrm{B}\) is the damping coefficient, and \(\mathrm{d}/\mathrm{d}t\) is the time derivative. In addition, the subscripts \(r\), \(g\), and a represent rotor, generator and aerodynamic, respectively. After finding the aerodynamic torque, \(\mathrm{T}_{a}\), from Equation (3), the averaged wind speed \(V\) is obtained from Equation (4) [29]. \[\mathrm{T}_{a}=\frac{1}{2}\rho\pi\mathrm{R}^{3}\left(\frac{C_{P}(\lambda, \beta)}{\lambda}\right)V^{2} \tag{4}\]  
In Equation (4), \(\rho\) is the density of air, \(\mathrm{R}\) is the rotor radius, \(C_{P}\) is the power coefficient, \(\lambda\) is the tip speed ratio, \(\beta\) is the pitch angle, and \(V\) is the averaged wind speed. The power command calculator from the wind farm controller calculates available powers of the wind turbines based on the estimated wind speed obtained using Equations (3) and (4) by using  
\[\mathrm{P}_{av}=\mathrm{T}_{a}\cdot\dot{\theta_{r}}\cdot\eta_{m}\cdot\eta_{g} \tag{5}\]  
where \(\mathrm{P}\) is the electrical power, and \(\eta\) is the efficiency. The subscripts \(av\), and \(m\) represent available and mechanical, respectively [21]. The power command calculator of the wind farm controller then evaluates the total available power from the wind farm by adding all the available powers and finds individual power commands to wind turbines to maximize the wind farm power as shown in Equation (6). It is a function minimization problem to find power commands to wind turbines to maximize the total power output, and the wind farm model is used for the function minimization process. In Equation (6), \(J\) is the objective function to be used for the function minimization process. \[J=\left|\mathrm{TSO}_{\mathrm{CMD}}-\sum\mathrm{WFC}_{\mathrm{gen},i}\right| \tag{6}\]  
The function minimization algorithm is an optimization algorithm that finds optimal variable values that can minimize the objective function consisting of a single or multiple variables. Any function minimization algorithms can be used, but, in this study, the Nelder-Mead simplex algorithm was used for convenience in coding. The algorithm is a multidimensional unconstrained optimization algorithm for nonlinear optimization problems. In addition, it is one of the most widely used function minimization algorithms as a direct search method. The method uses a simplex known as a polytope which has \(n+1\) vertices (or \(n+1\) test points) in \(n\) variables of objective function.