# Small Magnus Wind Turbine: Modeling Approaches
## 2 Magnus Effect-Based Wind Turbine Simulation Methods  
The wind turbine model can be separated into different subsystems, including a model of the wind, aerodynamic model, mechanical model of the turbine and drive train, electromechanical model of the generator, the electrical model of the power converter, and the model of the control system. The main goal of this work was to perform an aerodynamic simulation of a wind turbine based on the Magnus effect. The aerodynamic simulation of the wind turbine is based on Blade Element Momentum theory and can be presented as a function of mechanical power extracted from the wind by the wind wheel, as shown in (1) [21]. \[P=\frac{1}{2}\cdot A\cdot p\cdot V^{3}\cdot C_{p} \tag{1}\]  
\(P\) is the power extracted from the flow, \(A\) is the rotor swept area, \(\rho\) is the air density, \(V\) is the wind speed, and \(C_{p}\) is the power coefficient of a wind energy conversion system, showing the ratio of extractable mechanical power \(P\) to the power contained in the air stream \(P_{0}\). The maximal possible \(C_{p}\) value is known as the Betz limit and has an estimated value of 0.593. Based on [22], the power coefficient \(C_{p}\) depends on the wind speed ratio and is calculated using the strip theory [23]. The book [22] held extensive discussions on the dependence between power coefficient \(C_{p}\), the rotor power curves, and the torque curves in wind turbines with various types of blades, including the one vertical rotation blade wind turbine. The main parameters of wind turbine dominating the \(C_{p}\), under [22], are the number of rotor blades, length of blades, blades aerodynamic profile, and gondola profile. According to (1) the power \(P\) captured from the wind is linear to the \(C_{p}\). This electric output power versus the tip speed ratio provides us with the power curve depending on wind turbine technical characteristics, wind profile, and turbine design. Due to the linear relationship between output power and power coefficient, further discussion will focus on \(C_{p}\). Building the \(C_{p}\) function for the selected wind turbine is required to create an accurate aerodynamic model of the wind wheel. The power coefficient can be presented as a function of a velocity ratio of the flow before and after interacting with the wind wheel. The wind wheel is presented in Figure 2. Extraction of mechanical energy from the free flow leads to a decrease in wind velocity \(V\) after passing through the rotor area; by the flow conservation law, this leads to an increase in area \(A\). The flow conservation law is shown in (2), where \(A\) is the rotor swept area, \(V^{\prime}\) is the wind speed at the windwheel, \(A_{1}\) and \(A_{2}\) are cross-sectional upstream and downstream, where \(V_{1}\) and \(V_{2}\) are upstream and downstream wind speed, respectively. \[A_{1}\cdot V_{1}=A\cdot V=A_{2}\cdot V_{2} \tag{2}\]  
The power coefficient can thus be specified as (3). \[C_{p}=\frac{P}{P_{0}}=\frac{1}{2}\cdot\left[1-\left(\frac{V_{2}}{V_{1}}\right) ^{2}\right]\cdot\left[1-\frac{V_{2}}{V_{1}}\right] \tag{3}\]  
Figure 2: Flow conditions due to the extraction of mechanical energy from a free-stream airflow, according to the elementary momentum theory. Determining the function of the power coefficient is an essential step in modeling wind energy conversion systems. According to [PERSON] et al. in [2], different methods of representing \(C_{p}\) can be used, including exponential models, sinusoidal models, and polynomial models. As it can be seen from the above-mentioned research work, one of the most commonly used models is an exponential model.