# Height Control System for Wind Turbines Based on Critical Wind Speed Calculation
## 4 Mathematical Model of the Control System  
This section presents the mathematical model of the control system based on the interaction of the various components such as the anemometer, PI controller, frequency converter, and linear actuator. Each of these components is described by differential equations, allowing the creation of a comprehensive system model to ensure effective process control. The anemometer is used to measure wind speed and transmit these data to the control system. It plays a crucial role because without accurate wind speed measurement, proper system control is impossible. The mathematical model of the anemometer is represented by a first-order linear differential equation [29]:  
\[\tau_{a}\frac{dV_{a}(t)}{dt}+V_{a}(t)=K_{a}V_{w}(t), \tag{4}\]  
where \(\tau_{a}=0.1\) s--the time constant of the anemometer; \(V_{a}(t)\) is the output voltage of the anemometer at a given moment in time \(t\); \(K_{a}=2\) V/(m/s) is the conversion coefficient of wind speed to voltage; and \(V_{w}(t)\)--the wind speed at a given moment in time \(t\). Next, the Laplace transform of the equation was performed:  
\[\tau_{a}V_{a}(s)+V_{a}(s)=K_{a}V_{w}(s) \tag{5}\]  
Solving Equation (10) for \(V_{a}\)(s), the following was obtained:  
\[V_{a}(S)=\frac{K_{a}}{\tau_{a}S+1}V_{w}(s) \tag{6}\] \begin{table}
\begin{tabular}{c c c c c} \hline \(P_{wind}\), **kW** & **\(A\), m\({}^{2}\)** & \(P_{gen}\), kW** & **\(v\), m/s** & **\(F\), N** \\ \hline
25 & 3.14 & 10 & 23.5 & 1058.65 \\ \hline \end{tabular}
\end{table} 
Table 1: Calculation of the wind force and speed optimal for the studied wind turbine. The inverse Laplace transform provides the solution in the time domain:  
\[V_{a}(t)=K_{a}V_{w}(t)\left(1-{}^{-t/\tau_{a}}\right) \tag{7}\]  
The controller implementing the PI regulator adjusts the control signal based on the regulation error. The PI regulator combines proportional and integral actions to achieve optimal control. In the Laplace domain, the equation of the PI regulator is as follows [30]:  
\[\bigcup(s)=k_{p}E(s)+\frac{k_{i}}{s}E(s), \tag{8}\]  
where \(k_{p}\) is the proportional gain; \(k_{i}\) is the integral gain; and \(E(s)\) is the regulation error in the Laplace domain. \[E(s)=R(s)-Y(s), \tag{9}\]  
where \(R(s)\) is the setpoint and \(Y(s)\) is the system output. In the time domain, the equation of the PI regulator is as follows [31]:  
\[u(t)=k_{p}e(t)+k_{i}\int_{0}^{t}\left(\tau\right)d\tau, \tag{10}\]  
where \(u(t)\) is the control signal at a given moment in time \(t\) and \(e(t)\) is the regulation error. \[e(t)=r(t)-y(t), \tag{11}\]  
where \(r(t)\) is the setpoint and \(y(t)\) is the system output. The frequency converter controls the frequency and, consequently, the speed of the linear actuator. It allows the precise control of the actuator's movement, ensuring smooth and accurate operation. The model of the frequency converter is described by the following differential equation:  
\[\tau_{f}\frac{dF(t)}{dt}+F(t)=K_{f}U(t), \tag{12}\]  
where \(\tau_{f}\) = 0.2 s is the time constant of the frequency converter; \(F(t)\) is the output frequency of the converter at a given moment in time \(t\); \(K_{f}\) = 50 Hz/V is the conversion coefficient of voltage to frequency; and \(U(t)\)--the control voltage from the controller at a given moment in time \(t\). After applying the Laplace transform, the equation obtained is as follows:  
\[\tau_{f}sF(s)+F(s)=k_{f}U(s) \tag{13}\]  
Solving for \(F(s)\):  
\[F(s)=\frac{K_{f}}{\tau_{f}s+1}U(s) \tag{14}\]  
The inverse Laplace transform provides the solution in the time domain:  
\[F(t)=K_{f}U(t)\left(1-e^{-t/\tau_{f}}\right) \tag{15}\]  
The linear actuator is responsible for the physical movement of the wind turbine rotor, adjusting its position based on wind speed to prevent damage and optimize operation.