# A Comparative Performance Analysis of Four Wind Turbines with Counter-Rotating Electric Generators
## 4 Numerical Simulations and Discussions
### Power Flow
Since ratio \(k_{t}\) intervenes in the relation of the torques, powers and transmission efficiency (see Tables 2 and 3), Case C (\(k_{t}=0\)) is characterized by the following particularities (Figure 9 and Table 5):  
* the torque of the rotor \(R2\) is equal, in absolute value, to that of the primary rotor \(R1\); as a result, the power generated by \(R2\) increases with the increase of the \(k_{\omega}\) ratio;
* the input powers (\(P_{R1}\) and \(P_{R2}\)) flow into three branches directly toward the rotor \(GR\) and stator \(GS\);
* the transmission efficiency is constant (\(\eta\) = 0.85), regardless of the value of the ratio \(k_{\omega}\);
* the power supply to the stator \(GS\) increases with the increase of \(k_{\omega}\): from 12.1% \(P_{G}\) for \(k_{\omega}=0.5\) to 13.3% \(P_{G}\) for \(k_{\omega}=1\). * Case D  
Case D (Figure 2d) is derived from Case A by removing rotors \(R2\) and \(R3\), i.e., \(k_{\omega}=0\), \(k_{t}=0\); rotor \(R1\) delivers a power whose flow (Figure 10) is similar to the homologous flow from Case C (see Figure 9). The system ensures a power supply of the stator \(GS\) of 10% of \(P_{G}\) and a constant efficiency (\(\eta\) = 0.85). \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline
**Case** & **DOF** & **No WRs** & \(k_{t}\) & \(k_{\omega}\) & \(i_{a}\) & \(P_{R2}/P_{R1}\) & \(P_{R3}/P_{R1}\) & \(P_{int}/P_{R1}\) & \(1P_{G}\) /\(1P_{R1}\) & \(\eta\) & \(P_{GS}/P_{G}\) \\ \hline A & 2 & 3 & 0.5 & 1 & 30 & 0.500 & 2.000 & 3.500 & 3.143 & 0.898 & 13.3\% \\  & & 1 & 1 & 30 & 0 & 4.000 & 5.000 & 4.587 & 0.917 & 13.3\% \\  & & 1 & 0.5 & 22.5 & 0 & 2.750 & 3.750 & 3.440 & 0.917 & 12.2\% \\ B & 1 & 2 & 0.5 & 0 & 15 & 0 & 0.750 & 1.750 & 1.572 & 0.898 & 10.0\% \\  & & 1 & 0 & 15 & 0 & 1.500 & 2.500 & 2.293 & 0.917 & 10.0\% \\ C & 2 & 2 & 0 & 0.5 & 22.5 & 0.500 & 0 & 1.500 & 1.275 & 0.850 & 12.2\% \\  & & 0 & 1 & 30 & 1.000 & 0 & 2.000 & 1.700 & 0.850 & 13.3\% \\ D & 1 & 1 & 0 & 0 & 15 & 0 & 0 & 1.000 & 0.850 & 0.850 & 10.0\% \\ \hline \end{tabular}
\end{table} 
Table 5: Performances of the four _WSCGs_ for certain representative numerical scenarios. Figure 9: Reduced power flows of the 2-DOF and \(L\) = 4 system (Case C) for \(k_{\omega}=0.5\) (blue) and \(1\) (red). According to Table 5, both the structure's degree of freedom and the number of wind rotors have a major influence on the output power (to the generator) and on the input powers, as well as on the power distribution between secondary rotors \(R2\) and \(R3\), with a consistent increase for \(R3\) in Case A. It should be noted that the 2-DOF transmissions double the speed amplification ratio (\(i_{a}=30\) for \(k_{\omega}=1\)) compared to the 1-DOF ones (\(i_{a}=15\) for \(k_{\omega}=0\)); a similar trend has the energy supply brought about by mobile stator \(GS\) (see the \(P_{GS}/P_{G}\) ratio in Table 5). Being independent of the ratio \(k_{\omega}\) and highly dependent on \(k_{t}\), the transmission efficiency is higher in Cases A and B (\(\eta=0.917\), \(k_{t}=1\)) vs. Cases C and D (\(\eta=0.850\), \(k_{t}=0\)). The performances of the turbines with two wind rotors and 1-DOF (Case B) or 2-DOF (Case C) speed increasers significantly depend on \(k_{t}\) and \(k_{\omega}\); for the considered data, the 1-DOF transmission (Case B) shows a superior performance when \(k_{t}=1\) and \(k_{\omega}=1\).