# Small Magnus Wind Turbine: Modeling Approaches
## 2 Magnus Effect-Based Wind Turbine Simulation Methods
\[C_{p}(\lambda,\beta)=C_{0}+\left(C_{1}\gamma-C_{2}\beta-C_{3}\beta^{C_{4}}-C_{ 5}\right)e^{-C_{6}\gamma}+C_{7}\lambda \tag{4}\]  
Here, \(\lambda\) is the tip-speed ratio, determined by (5), \(\beta\) is the pitch angle, \(\gamma\) is determined by (6), \(R\) is the rotor radius, \(\omega\) is the rotational speed of the rotor, \(C_{0-7}\) and \(d_{0-2}\) are coefficients of the exponential model, which depend on mechanical and aerodynamic characteristics of the rotor and can be obtained via experiment or simulation. \[\lambda=\frac{\omega R}{V} \tag{5}\]  
\[\gamma=\frac{1}{\lambda+d_{0}\beta+d_{1}}-\frac{d_{2}}{1+\beta^{3}} \tag{6}\]  
Previous equations show that wind turbine \(C_{p}\) can be varied by tuning the rotor speed and blade pitch. However, determining constructional coefficients requires prior knowledge of the blade shape parameters of the rotor. Thus, obtaining these coefficients in the initial stage of modeling presents a challenging task. Moreover, in horizontal-axis MWT, controlled via the cylinder rotational speed, should be used instead of turning the blade pitch. Therefore, modeling the wind turbine based on the Magnus effect requires using a \(C_{p}\) function different from the one used for conventional bladed turbines. During this work, three methods of building a \(C_{p}\) function for horizontal-axis MWT were studied. The first approach, presented in [24], is based on the blade element momentum theory. This model is further referred to as the analytical model. This work aimed to estimate the wind power coefficient function for the MWT via an analytical solution. The authors present the power coefficient as the function of three parameters: tip rotor solidity \(\sigma_{T}\), tip speed ratio (TSR) \(\lambda_{1}\), and relative speed of the cylinder's rotation (or cylinder speed ratio, CSR) \(\lambda_{2}\). These parameters are obtained in (7)-(9) and depend on constructional features of each wind turbine, such as the radius of the rotor swept area and the cylinder. The analytical solution itself is based on calculating the lift force of each cylinder blade element. The application of blade element momentum theory allows deriving the \(C_{p}\) function, which is shown in (10). The presented function was modified following data obtained by numerical Boundary Element Method (BEM) solutions. Here, \(\omega_{1}\) is the rotational speed of the rotor, \(\omega_{2}\) is the rotational speed of the cylinder, \(R\) is the rotor radius, \(r\) is the cylinder radius, and \(N\) is the number of cylinders. The derived function shows that in the case of MWT, the MPPT can be achieved by varying the speed of the rotor and rotational speed of the cylinder. The maximal achievable value of \(C_{p}\), which can be obtained using this function, is 0.593, which is the same value as the Betz limit. The presented method of estimating \(C_{p}\) does not take the drag of the cylinder into account, however, it can be used at a preliminary stage of the development of horizontal-axis MWT. \[\lambda_{1}=\frac{\omega_{m}R}{V} \tag{7}\]  
\[\lambda_{2}=\frac{\omega_{c}r}{V} \tag{8}\]  
Figure 2.0  
Figure 2.1Another method is proposed in [20]. At the first stage of work, a mathematical model of the wind turbine was developed using data obtained by [PERSON]. For the sake of clarity, this model is referred to as the regression model. The authors used BEM and symbolic regression to create simplified models for different horizontal-axis MWTs with different cylinder numbers. Then, a direct method was applied to the model. Combinations of dimensionless parameters, such as the earlier-mentioned tip speed ratio, the relative speed of the cylinder's rotation, the aspect ratio, and Reynolds number, were generated using symbolical regression. As a result, the optimum function with the most significant correlation coefficient was selected for each configuration (number) of the cylinder. Experimental horizontal-axis MWT uses only two cylinders, thus, a function for a two-cylinder model is presented in (11).