## 2 Model Building
### Design Constraints
#### 2.2.1 Constraints on Pitch Angle  
In the following analysis, a 1.5 MW variable speed and variable pitch (VSVP) wind turbine is taken as an example. The constraint conditions will be built by analyzing different load cases, which are simulated with a consideration of the aeroelastic property and the action of the control system by using PHATAS model in FOCUS software. Constraints between Pitch Angle and Wind Speed  
The minimum pitch angle for the wind speed is expressed by \(\beta_{v}\). For VSVP wind turbine, the blade pitch angle will get larger with the increase of the wind speed so as to reduce the rotor speed. Various pitch angles may occur while the wind turbine operates at the same wind speed due to the control system. Hence, obtaining the minimum pitch angle for a wind speed is very difficult, and also the key point of data that is requires so as to confirm the accuracy of the PSO-ELPM. A variety of load cases have been computed in compliance with GL standard [20], especially the load cases that make it easy to generate the extreme loads, such as the DLC 1.6 and DLC 4.2. Figure 2 gives the different pitch angles for different wind speeds during the simulation of DLC 1.\(X\) (\(X\) expressed by the number 3, 4, 5, 6, 7, 8, 9) and DLC 4.2, which are computed by FOCUS. From Figure 2, we can see that the \(\beta_{v}\) varies with wind speeds. When the wind speed is less than 15 m/s, \(\beta_{v}\) can reach the installation angle. But when the wind speed is above 15 m/s, \(\beta_{v}\) becomes larger and larger with the increase of wind speed. Practically, \(\beta_{v}\) will also change with the variation of the aerodynamic profile, blade layers and control strategy at the same wind speed. In this paper, a rough and versatile method is used to get \(\beta_{v}\). Because the extreme DLCs simulate the extreme condition suddenly occurring during the normal operation of wind turbine, the variation of the pitch angle will be not far away from the steady condition by using control system and safety system. Therefore, the \(\Delta\beta_{2}\) is set as the maximum offset value away from the pitch angle under the condition of steady operation. That is to say, if the pitch angle is 10 degrees when the wind turbine normally operates at 15 m/s, the minimum pitch angle is equal to \(10-\Delta\beta_{2}\) at 15 m/s for all load cases. Practically, \(\Delta\beta_{2}\) may be different for different wind speeds. In order to simplify it, \(\Delta\beta_{2}\) is set as a constant. The \(\Delta\beta_{2}\) can be obtained from the calculation results of all load cases or part of the load cases. For this 1.5 MW wind turbine, \(\Delta\beta_{2}\) is 10.86 degree from part of the load cases computed by FOCUS. Considering the limitation of the installation angle \(\beta_{l}\), the \(\beta_{v}\) for different wind speeds is shown with red triangles in Figure 2. Based on Figure 2, the \(\beta_{v}\) versus wind speed \(V_{1}\) is divided into three sections. For the first section, \(\beta_{v}\) is equal to \(\beta_{l}\) when the wind speed is less than \(V^{\prime}\), which is always the rated wind speed. So, it can be expressed as:  
\[\beta_{m}=\beta_{l}\quad(V_{1}\leq V^{\prime}) \tag{6}\]For the second section, \(\beta_{v}\) is deduced by the offset \(\Delta\beta_{2}\) from the steady operation value when \(V_{1}\) is between \(V^{\prime}\) and \(V_{out}\). For VSVP wind turbine, the steady curve can be expressed as:  
\[\beta_{v}=B_{1}(V_{1})^{2}+B_{2}V_{1}+B_{3} \tag{7}\]  
where \(B_{1}\), \(B_{2}\), and \(B_{3}\) are constant and can be obtained by quadratic polynomial fitting.