# Height Control System for Wind Turbines Based on Critical Wind Speed Calculation
## 6 Results and Discussion  
This chapter presents the results of the numerical calculation of the wind turbine control system based on the mathematical model described in Section 4. The model includes the interaction of the anemometer, PI controller, frequency converter, and linear actuator. Additionally, the experimentally obtained PI controller coefficients from Section 5 were used. For the complete control system model, the equation is as follows:  
\[Y(s)=\frac{k_{f}\Big{(}k_{p}(R(s)-Y(s))+\frac{k_{i}}{s}(R(s)-Y(s))\Big{)}}{ \Big{(}\tau_{f}s+1\Big{)}(ms^{2}+cs+k)} \tag{22}\]  
The following component parameters were used for substitution: \(k_{f}\)= Hz/V--conversion coefficient of voltage to frequency; \(k_{p}\) = 2.5--proportional coefficient of the PI controller; \(k_{i}\) = 5 s\({}^{-1}\) (calculated as \(k_{i}=\frac{1}{T_{i}}\) where \(T_{i}\) = 200 ms = 0.2, which corresponds to \(k_{i}\) = 5 s\({}^{-1}\))--integral coefficient of the PI controller; \(\gamma\)= 0.2 s--time constant of the frequency converter; \(m\) = 10 kg--mass of the linear actuator; \(c\) = 50 Ns/m--damping coefficient; and \(k\) = 200 N/m--spring stiffness. Substituting these values into Equation (23), the following equation was obtained:  
\[Y(s)=\frac{50\big{(}2.5(R(s)-Y(s))+\frac{5}{s}(R(s)-Y(s))\big{)}}{(0.2s+1)(10s ^{2}+50s+200)} \tag{23}\]  
or  
\[Y(s)=\frac{125R(s)+\frac{250R(s)}{s}}{(0.2s+1)(10s^{2}+50s+200)}-\frac{125Y(s) +\frac{250Y(s)}{s}}{(0.2s+1)(10s^{2}+50s+200)} \tag{24}\]  
The code designed to perform numerical calculations for the wind turbine control system is given in (Appendix A Figure A1) and the flowchart is shown in Figure 14. It is based on the mathematical model described in Section 4 and incorporates the experimentally obtained PI controller coefficients from Section 5. The code performs numerical simulations, modeling the interaction of the system components: the anemometer, PI controller, frequency converter, and linear actuator. The output of the code is a graph showing the displacement of the linear actuator. The code defines the time vector and the Laplace variable. To prevent division by zero, the value of the Laplace variable at zero frequency is set to 1e-9. Then, the individual components of the system are calculated: the numerator of the equation, which includes the conversion of the signal to frequency (num_R), and the denominator of the equation, which includes the damping and stiffness effects of the system (denom). The complete system response in the frequency domain (Y_s) is calculated by dividing the numerator by the denominator, after which the inverse Laplace transform is performed to obtain the time-domain displacement of the actuator (y_t). Finally, the code plots the graph, showing the displacement of the linear actuator over time. The graph demonstrates the system's adequate response to changes in the wind speed, proper PI controller tuning, and correct damping, which is important to prevent sharp oscillations and system damage. Figure 15 shows the displacement of the wind turbine's linear actuator. Figure 14: Flowchart of the mathematical model and the proposed program code. Figure 15: Wind turbine height control system. The analysis of Figure 15 shows the displacement of the linear actuator, which is responsible for lowering the wind turbine rotor. At the start of the graph, there is a small delay before the actuator begins to move. This delay occurs because the system is analyzing the incoming data to decide when to activate the actuator. Once the system triggers, the graph displays a smooth and controlled lowering of the rotor, where the actuator moves down by 40% of its full stroke over a period of approximately 9 s.  
This smooth and gradual motion demonstrates that the control system is operating correctly, avoiding sudden movements that could cause mechanical stress or instability. The key indication of the system's effectiveness is the absence of sharp oscillations, showing that the actuator's response is well damped and controlled.