## 2 Pitch System Principles and Modeling  
The operational principle of a variable pitch system in wind turbines is depicted in Figure 1. The servo motor is aligned coaxially with the hydraulic pump, while the hydraulic cylinder is linked directly to the variable pitch mechanism. By adjusting the servo motor's speed, the hydraulic pump regulates the flow and generates sufficient pressure to drive the hydraulic cylinder. This pressure must exceed the external load applied to the hydraulic cylinder, accounting for any additional pressure losses along the pump-actuator line. This ensures that the hydraulic cylinder can reach the desired position and perform the necessary pitch adjustment. As depicted in the diagram, the components are labeled as follows: (1) motor, (2) fixed-displacement pump, (3) relief valve, (4) pilot-operated check valve, (5) solenoid directional valve, (6) throttle valve, (7) check valve and throttling valve, (8) accumulator, and (9) single-rod hydraulic cylinder. During pitch adjustment, solenoid valves 5.1, 5.2, and 5.3 are activated, while 5.4 remains closed. Accumulator 8.2 compensates for the asymmetrical flow between the two chambers of the single-rod hydraulic cylinder. In emergency situations, solenoid valve 5.4 is activated, and valves 5.1 and 5.2 are deactivated, allowing accumulator 8.1 to power the hydraulic cylinder for emergency feathering. The hydraulic cylinder is connected to the hub via a hinge, with its piston linked to the blade base. Pitch variation is achieved by controlling the piston rod's extension and retraction. A schematic of the variable pitch system mechanism is provided in Figure 2.  
The minimum distance of the hydraulic cylinder, denoted as L, is 1500 mm. H\({}_{\mathrm{L}}\) represents a fixed distance with a dimension of 1750 mm, and the rotational radius is 625 mm. The fixed angle \(\gamma\) is 56.633\({}^{\circ}\). The paddle pitch angle \(\Omega\) is adjusted according to the extension distance \(\mathrm{x_{p}}\) of the hydraulic cylinder. The pitch angle is calculated as follows:  
\[\Omega=\arccos\left[\frac{\mathrm{R_{e}^{2}+H_{L}}^{2}-\left(L+x_{p}\right)^{ 2}}{2\mathrm{R_{e}H_{L}}}\right]-\gamma \tag{1}\]  
Figure 1: Schematic diagram of a variable pitch system in wind turbines. Figure 2: Schematic of the variable pitch mechanism in a wind turbine system. The first-order derivative of \(\Omega\) is  
\[\dot{\Omega}=\frac{\left(\mathrm{L+x_{p}}\right)\dot{\mathrm{x}_{p}}}{\mathrm{R_{e }H_{L}}\sqrt{1-\left[\frac{\mathrm{R_{e}^{2}+H_{L}}^{2}-\left(\mathrm{L+x_{p}} \right)^{2}}{2R_{e}H_{L}}\right]^{2}}} \tag{2}\]  
The second-order derivative of \(\Omega\) is  
\[\ddot{\Omega}=\frac{\left(\mathrm{L+x_{p}}\right)\ddot{\mathrm{x}_{p}}+\dot{ \mathrm{x}_{p}^{2}}}{\mathrm{R_{e}H_{L}}\sqrt{1-\left(\frac{\mathrm{R_{e}^{2}+ H_{L}}^{2}-\left(\mathrm{L+x_{p}}\right)^{2}}{2R_{e}H_{L}}\right)^{2}}}-\frac{4 \left(\mathrm{L+x_{p}}\right)^{2}\dot{\mathrm{x}_{p}^{2}}\left(\mathrm{R_{e}^{ 2}+H_{L}}^{2}-\left(\mathrm{L+x_{p}}\right)^{2}\right)}{\mathrm{R_{e}^{3}H_{L }}^{3}\left(4-\frac{\left(\mathrm{R_{e}^{2}+H_{L}}^{2}-\left(\mathrm{L+x_{p}} \right)^{2}\right)^{2}}{\mathrm{R_{e}^{2}H_{L}}^{2}}\right)^{\frac{3}{2}}} \tag{3}\]  
In the variable pitch system, the servo motor speed serves as the input. Since the servo motor is coaxially connected to the dosing pump, its speed is equal to the dosing pump's speed. Therefore, the dosing pump speed is also considered the system input. Several assumptions were made when modeling the system, such as liquid incompressibility, ignoring hose deformation, and eliminating pump pulsation effects. The pump flow equation is as follows:  
\[\left\{ \begin{array}{l}\mathrm{q_{1}}=\mathrm{D_{p}}\omega_{\mathrm{p}}\\ \mathrm{q_{2}}=\mathrm{D_{p}}\omega_{\mathrm{p}}\end{array} \right. \tag{4}\]  
where \(\mathrm{q_{1}}\) is the flow rate of the left chamber of the system, \(\mathrm{q_{2}}\) is the flow rate of the right chamber of the system, \(\mathrm{D_{p}}\) is the displacement for the pump, and \(\omega_{\mathrm{p}}\) is the speed for the pump.