# A Comparative Performance Analysis of Four Wind Turbines with Counter-Rotating Electric Generators
## 4 Numerical Simulations and Discussions
### Transmission Input Powers and Efficiency  
The analysis of the diagrams depicted in Figures 3-6 allows the following particularities to emerge:  
* The ratios \(k_{\omega}\) and \(k_{t}\) directly influence the operation state of the secondary wind rotors \(R2\) and \(R3\) and the mechanical power flow through the planetary transmission implicitly; thus, for Case A, the variation ranges of these ratios for which the wind system operation makes sense are further analyzed, i.e., the secondary wind rotors \(R2\) and \(R3\) are motor sources (\(P_{R2}>0\) and \(P_{R3}>0\)).
* In the premise \(P_{R1}>0\), the angular speed and the torque of the rotor \(R2\) (controlled by \(k_{\omega}\) and \(k_{t}\)) also influence the power of the secondary wind rotor \(R3\) (Figures 3 and 4).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Planetary Gear Set** & \(\mathbf{i_{0}}\) & \(\eta_{0}\) & \(\mathbf{k_{\omega}}\)=\(\frac{\omega_{R2}}{\omega_{R1}}\) & \(\mathbf{k_{t}}\)=\(-\frac{T_{R1}}{T_{R1}}\) \\ \hline I & \(-1.5\) & \(0.8836\) (=0.942) & \(-0.5\dots\) 1 & - \\ II & \(-5\) & \(0.925\) (=0.952) & - & \(0\dots\) 1.5 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Input data for the numerical simulations of the analytical model.  
Figure 4: Reduced input power variations depending on the ratio \(k_{\omega}\) for various values of ratio \(k_{t}\): (**a**) reduced power of the wind rotor R2; (**b**) reduced power of the wind rotor R3 (Case A).  
Figure 5: Variations of the electric generator reduced power, depending on: (**a**) the ratio \(k_{t}\); (**b**) the ratio \(k_{\omega}\) (Case A).  
Figure 3: Reduced input power variations depending on the ratio \(k_{t}\) for various values of ratio \(k_{\omega}\): (**a**) reduced power of the wind rotor R2; (**b**) reduced power of the wind rotor R3 (Case A).  
3. Rotor R2 is the most reactive to the variations of \(k_{\omega}\) and \(k_{t}\), which can change its state from the motor mode (\(P_{R2}>0\)) to the brake mode (\(P_{R2}<0\)) (Figures 3a and 4a); additionally, the power of rotor \(R2\) is cancelled for \(k_{t}=1\), irrespective of the \(k_{\omega}\) value. 4. At normal operations (\(k_{\omega}>0\)), rotor \(R2\) works in the motor mode only for \(k_{t}<1\), becoming a brake for \(k_{t}>1\); for \(k_{\omega}<0\) (i.e., rotor \(R2\) rotates in the same direction as the primary rotor \(R1\)), \(R2\) becomes the motor only if \(k_{t}>1\) (see the line \(k_{\omega}=-0.5\) in Figure 3a). 5. Rotor \(R3\) always operates in the motor mode (\(P_{R3}>0\)), irrespective of the \(k_{t}\) and \(k_{\omega}\) values (Figures 3b and 4b); the power \(P_{R3}\) increases linearly with the increase of \(k_{t}\), while the slope of line \(P_{R3}\) increases with \(k_{\omega}\). 6. The maximum power generated by rotor \(R3\) (\(P_{R3}=6P_{R1}\) for \(k_{t}=1.5\) and \(k_{\omega}=1\), Figure 3b) is significantly higher than the maximum power of rotor \(R2\) (\(P_{R2}=P_{R1}\) for \(k_{t}=0\) and \(k_{\omega}=1\), Figure 3a). 7. The mechanical power received by the generator increases linearly with the increase of the ratios \(k_{t}\) and \(k_{\omega}\) (Figure 5), even in the brake mode of rotor \(R2\) (\(P_{R2}<0\) for \(k_{t}>1\) and \(k_{\omega}>0\), Figure 3a). 8. The transmission efficiency does not depend on the ratio \(k_{\omega}\) (see Table 3), but it is influenced by ratio \(k_{t}\), with a maximum value of 0.917 for \(k_{t}=1\) (Figure 6), i.e., when the energy contribution of rotor \(R2\) is zero (\(P_{R2}=0\), Figure 3a).  
In conclusion, as \(k_{t}\) and \(k_{\omega}\) increase, the mechanical power supplied to the electric generator increases as well, with a higher input brought about by wind rotor \(R3\) (Figures 3 and 4). The diagrams depicted in Figures 3-6 allow highlighting the particular cases in which the ratios \(k_{t}\) and \(k_{\omega}\) become null; by customization, the type A systems become types B (\(k_{\omega}=0\)), C (\(k_{t}=0\)) and D (\(k_{\omega}=0\), \(k_{t}=0\)).