## 2 Model Building
### Design Objects  
For the large-scale blade used in the low wind speed area, the extreme load, especially the extreme bending moment at the blade root plays a key role in the blade design. It not only influences the blade reliability, but also involves the security of other equipment, such as the pitch motor and connectingbolts. Hence, for the single objective optimization, the extreme moment in one direction at a given section is selected as the design object. For the multi-objective optimization, the design objects can be defined as the different direction bending moments at the same section, or the same direction bending moment at different sections, or the different direction bending moments at different sections.  
To compute the bending moments at different sections, the blade element moment (BEM) theory is chosen to obtain the aerodynamic force \(F_{a}\), which are \(F_{n}\) and \(F_{t}\) in Equation (1). From Equations (1)-(5), we can see that the induction factors of \(a\) and \(b\) are the only unknown numbers, while the other parameters are known or the design variables. In this article, the traditional iteration algorithm for BEM is used to obtain them with considering the tip loss correction and root loss correction. It is as follows.  
Step 1: Supposing initial values for \(a\) and \(b\), such as \(a=0\), \(b=0\);  
Step 2: Using Equation (4) to get the inflow angle;  
Step 3: Using Equation (3) to get the attack angle;  
Step 4: Computing the blade loss factor \(F=F_{tip}F_{hub}\);  
Where \(F_{tip}\) is tip loss factor, \(F_{hub}\) is root loss factor. They are shown as  
\[F_{tip}=\frac{2}{\pi}\cos^{-1}\mathrm{e}^{-\frac{(\frac{8(R-r)}{2}\sin\phi})} \tag{21}\]  
Figure 4: (**a**) Power and (**b**) pitch angle changing with wind speed in steady simulation.  
Figure 5: Calculation method of \(\Delta\beta_{2}\).  
\[F_{hub}=\frac{2}{\pi}\cos^{-1}\mathrm{e}^{-(\frac{B(r-R_{hub})}{R_{hub}\sin\pi})} \tag{22}\]  
where \(B\) is the number of the blade, \(R\) is the rotating radius of rotor, \(R_{hub}\) is the rotating radius of hub, \(r\) is the rotating radius of blade element.  
Step 5: Computing the thrust coefficient of the rotor \(C_{T}\);  
\[C_{T}=\frac{\sigma(1-a)^{2}C_{n}}{\sin^{2}\phi} \tag{23}\]  
where \(\sigma\) is rotor solidity and it is shown as  
\[\sigma=\frac{Bc}{2\pi r} \tag{24}\]  
Step 6: If \(C_{T}\geq 0.96F\), the new induction factor \(a^{\prime}\) will be computed by Equation (25);  
\[a^{\prime}=\frac{1-k_{1}+\sqrt{k_{1}{}^{2}-4k_{2}(k_{0}-C_{T})}}{2k_{2}} \tag{25}\]  
where \(k_{2}=1/0.18-4F\), \(k_{1}=-(0.8/0.18-4F)\), \(k_{0}=0.16/0.18\) or else, the new induction factor \(a^{\prime}\) will be computed by Equation (26).  
\[a^{\prime}=\left[1+\frac{4F\sin^{2}\phi}{\sigma C_{n}}\right]^{-1} \tag{26}\]  
Step 7: The new induction factor \(b^{\prime}\) will be computed by Equation (27);  
\[b^{\prime}=\left[\frac{4F\sin\phi\cos\phi}{\sigma C_{t}}-1\right]^{-1} \tag{27}\]  
Step 8: Computing the error \(\Delta a=|a-a^{\prime}|\) and \(\Delta b=|b-b^{\prime}|\);  
Step 9: If \(\Delta a\leq 0.001\) and \(\Delta b\leq 0.001\), end the iteration and use \(a^{\prime}\), \(b^{\prime}\) and Equations (1)-(5) to get the \(F_{n}\) and \(F_{t}\), that is \(F_{a}\) or else, return to step 2 use new induction factor \(a^{\prime}\) and \(b^{\prime}\).  
Moreover, the gravity \(F_{g}\) and centrifugal force \(F_{c}\) at different section is also taken into account. They are shown as follows  
\[F_{c}=\dot{m}_{k}\Omega^{2}r \tag{28}\]  
\[F_{g}=\dot{m}_{k}g \tag{29}\]  
where \(\dot{m}_{k}\) is section mass per length, \(g\) is gravity acceleration.  
Then, we can obtain the force \(F_{k}\) at a certain section by coordinate transformation.  
\[F_{k}=F_{c}+F_{g}+F_{a} \tag{30}\]  
Supposing that \(F_{k}\) is not changing along one blade element, the forces and moments at a section in a certain coordinate can be solved by coordinate transformation and integration.  
Finally, the flow chart of the blade load calculation model is given below (see Figure 6).