# A Comparative Performance Analysis of Four Wind Turbines with Counter-Rotating Electric Generators
## 2 Problem Formulation  
The wind systems with counter-rotating generators (_WSCGs_) have superior conversion efficiencies compared to conventional generators (with fixed stator) due to the branched transmission of mechanical power and higher relative speed between the rotor (GR) and the stator (GS) of the electric generator (G). A counter-rotating generator has both armatures (GR and GS) movable and rotates in opposite directions; usually, due to inertial reasons, the generator rotor has a higher speed than the stator. However, the energy performance of _WSCGs_ is significantly influenced by the number of WRs, as well as by the speed increase type. In order to highlight the influence of the number of WRs (inputs) and of the structural degree of freedom, a comparison between _WSCGs_ with 1-DOF or 2-DOF speed increasers with one input (i.e., with one _WR_ and the total number of external connections \(L=3\), implicitly), two inputs (two _WRs_ and \(L=4\)) and three inputs (three _WRs_ and \(L=5\)) is further approached. To this end, a planetary transmission with a variable structure, based on three clutches and derived from a solution for which the authors have a patent application [54], is proposed; by appropriate combinations of clutch engaging and disengaging, the transmission can operate in various structures, of which four cases were selected (Figure 1):  
* Case A: 2-DOF (differential) transmission with three inputs (2-DOF, \(L=5\), and three _WRs_);
* Case B: 1-DOF (monomobile) transmission with two inputs (1-DOF, \(L=4\), and two _WRs_);
* Case C: 2-DOF transmission with two inputs (2-DOF, \(L=4\), and two _WRs_);
* Case D: 1-DOF transmission with one input (1-DOF, \(L=3\), and one _WR_). As a result, the energy performances of _WSCGs_ with two or three wind rotors can be directly compared with those of conventional wind turbines (with one wind rotor), highlighting the specific differences of cases with 1-DOF vs. 2-DOF transmissions. The approach starts from a \"variable\" structure, based on setting the clutches C1... C3 (Figure 1); this structure contains: three wind rotors (a permanent primary rotor \(R1\) and two secondary rotors \(R2\) and \(R3\) activated by clutches C2 and C3, respectively), a planetary gear set with level gears \(1\equiv 2\)-\(1\)-\(1\)-\(1\)-\(3\)-\(H_{1}\) (equipped with clutch C1 for blocking the satellite carrier \(H_{1}\)) and a differential cylindrical planetary gear set \(\Pi\equiv 4\)-\(5\)-\(6\)-\(H_{2}\) with two counter-rotating outputs, \(6\) and \(7\), secured to the rotor \(GR\) and the stator \(GS\), respectively. The primary rotor \(R1\) is coupled to the level gear (2), which is assembled on the carrier \(H_{2}\). The power generated by \(R1\) is distributed on two branches to the planetary gear set \(\Pi\) through the level sun gear (3), assembled into the ring sun gear (4) and directly to the carrier \(H_{2}\); further, the power is transmitted by gear set \(\Pi\) to the rotor \(GR\equiv 6\) and stator \(GS\equiv 7\), respectively. Denoting a disengaged clutch by \(Ci=0\) and an engaged clutch by \(Ci=1\) (\(i=1\), \(2\), \(3\)), the structures related to Cases A, B, C and D can be described as follows:  
* Case A, Figure 1a: \(C1=0\), \(C2=C3=1\), \(2\)-DOF transmission with three inputs and two outputs (\(L=5\));
* Case B, Figure 1b: \(C1=C3=1\), \(C2=0\), \(1\)-DOF transmission with two inputs and two outputs (\(L=4\)); for \(C1=1\), the level transmission I is a \(1\)-DOF gear set with fixed axes, and, for \(C2=0\), the rotor \(R2\) idles and torque \(T_{R2}=0\), implicitly.