# Height Control System for Wind Turbines Based on Critical Wind Speed Calculation
## 4 Mathematical Model of the Control System
The dynamics of the linear actuator are described by the following second-order differential equation [32]:  
\[V_{a}(t)=K_{a}V_{w}(t)\left(1-{}^{-t/\tau_{a}}\right) \tag{7}\]0  
where \(m\) = 10 kg is the mass of the actuator; \(\mathrm{c}\) = 50 Nm/c is the damping coefficient; \(k\) = 200 N/m is the spring stiffness; \(y(t)\) is the displacement of the actuator at a given moment in time \(t\); and \(F(t)\) is the force applied to the actuator at a given moment in time \(t\)The Laplace transform of (21) is as follows:  
\[V_{a}(t)=K_{a}V_{w}(t)\left(1-{}^{-t/\tau_{a}}\right) \tag{7}\]1  
Solving for \(Y(s)\), the equation obtained is as follows:  
\[V_{a}(t)=K_{a}V_{w}(t)\left(1-{}^{-t/\tau_{a}}\right) \tag{7}\]2  
The inverse Laplace transform provides the solution in the time domain [33]:  
\[V_{a}(t)=K_{a}V_{w}(t)\left(1-{}^{-t/\tau_{a}}\right) \tag{7}\]3  
A potentiometer is used to measure the current position of the linear actuator and provide feedback to the control system. The model of the potentiometer is described by the following linear equation [34]:  
\[V_{a}(t)=K_{a}V_{w}(t)\left(1-{}^{-t/\tau_{a}}\right) \tag{7}\]4  
where \(V_{p}(t)\)--the output voltage of the potentiometer at a given moment in time \(t\); \(K_{\wp}=0.5\) V/m is the conversion coefficient of displacement to voltage; and \(y(t)\) is the displacement of the linear actuator at a given moment in time \(t\). The connections between components through their output and input parameters should be considered to create a complete model of the control system. The wind speed \(V_{\varpi}(t)\) is measured by the anemometer, which converts it into the output voltage \(V_{\mathrm{a}}(t)\). This voltage is fed to the PI controller, which calculates the control signal \(U(t)\) based on the regulation error \(E(t)\). The control voltage \(U(t)\) is applied to the frequency converter, which changes the frequency \(F(t)\) according to the input signal. The frequency determines the force applied to the linear actuator. The linear actuator, in turn, moves the rotor by a distance \(y(t)\), which is measured by the potentiometer. The potentiometer converts the displacement into an output voltage \(V_{p}(t)\), which is fed back to the controller to adjust the control signal based on the current rotor position, thereby ensuring a closed control loop. By combining the mathematical model equations of each component, the complete system equation was obtained:  
\[V_{a}(t)=K_{a}V_{w}(t)\left(1-{}^{-t/\tau_{a}}\right) \tag{7}\]5  
The presented mathematical model of the wind turbine control system demonstrates the importance of accurate measurement and proper parameter control to ensure the system's efficient operation and safety. Modeling the interactions between the anemometer, PI controller, frequency converter, and linear actuator allows for the precise adjustment and control of the process. Integrating all the components into a single system with a closed-loop control ensures the stable and reliable operation of the wind turbine, minimizing the risk of damage and optimizing performance.