## 2 Problem Formulation
Only the gearing friction losses are considered, neglecting the friction in bearings;
4. The pitch angle of WR blades does not change during operation; therefore, the adjustment parameters of the wind rotor remain constant during operation;  
Figure 1: Single-rotor wind turbine with counter-rotating electric generator: (**a**) conceptual scheme; (**b**) block scheme. As 1-DOF transmission is employed, the system has one independent motion attributed to the input element, i.e., the wind rotor;
6. A direct current (DC) generator is used, and implicitly, its mechanical characteristic is a linear function with constant coefficients; during generator operation, the balancing condition of the torques of the rotor \(GR\) (\(T_{GR}\)) and of the stator \(GS\) (\(T_{GS}\)) is described by: \(T_{GR}+T_{GS}=0\);
7. The mechanical characteristic of the wind rotor is modeled over four rotational speed intervals by linear functions with constant coefficients, obviously at a constant wind speed. Dynamic modeling of the WT mechanical system from Figure 1 is based on the block scheme depicted in Figure 2a, in which the planetary speed increase is modeled by two planetary units (PU) I (\(H\)-5-4-3-1) and II (\(H\)-5-4-3-2), connected in parallel. By decomposing the wind system from Figure 2a, the six structural components represented in Figure 2b-e are obtained, characterized by specific kinematic and dynamic equations, according to the methodology detailed in Section 3. In the proposed approach, the axial moments of inertia of the components are reduced on the outer shafts of the planetary transmission (i.e., the shafts of the \(R\), \(GR\), and \(GS\) bodies) [37]. Thus, the torque equations for the components in Figure 2c,d can be described by the classical relations established un  
Figure 2: (**a**) Block scheme of the wind turbine and its decomposition into components: (**b**) wind rotor; (**c**) intermediate shaft; (**d**) planetary units; and (**e**) generator rotor and stator. der steady-state conditions and for the other components, depicted in Figure 2b,e,--under dynamic conditions. The axial moment of inertia of the \(n_{s}\) satellites 3-4, mounted in parallel, is reduced to the satellite-carrier \(H\) axis based on the principle of equalizing their kinetic energy. The satellite body has a combined rotation (around its own axis) and revolution (around the fixed sun axis) motion; its kinetic energy is considered equal to that of a virtual body with motion around the sun axis having an equivalent moment of inertia \(J_{sH}\):  
\[K_{s}=\frac{1}{2}n_{s}\Big{(}m_{s}v_{Gs}^{2}+J_{s}\omega_{s}{}^{2}\Big{)}= \frac{1}{2}J_{sH}\omega_{H}^{2}, \tag{3}\]  
\[J_{sH}=n_{s}m_{s}\Big{(}\frac{v_{Gs}}{\omega_{H}}\Big{)}^{2}+n_{s}J_{s}\bigg{(} \frac{\omega_{s}}{\omega_{H}}\Big{)}^{2}, \tag{4}\]  
in which  
\[\frac{\omega_{s}}{\omega_{H}}=\frac{\omega_{H}+\omega_{sH}}{\omega_{H}}=1+ \frac{\omega_{sH}}{\omega_{H}}=1+\frac{\omega_{sH}}{\omega_{H5}}=1-\frac{ \omega_{sH}}{\omega_{5H}}=1-i_{45},\ v_{Gs}=\omega_{H}\cdot r_{H}, \tag{5}\]  
yielding to  
\[J_{sH}=n_{s}\Big{(}m_{s}r_{H}^{2}+J_{s}(1-i_{45})^{2}\Big{)}, \tag{6}\]  
where \(m_{s}\) and \(J_{s}\) are the mass and the axial mechanical moment of a satellite 3-4, respectively (\(J_{s}\) is established with respect to the own axis of rotation), \(r_{H}\) is the radius of satellite axis arrangement on the carrier \(H\), and \(i_{45}\) is the kinematic ratio of the gear pair with fixed axes 4-5 (i.e., \(i_{45}=z_{5}/z_{4}\), where \(z_{4}\) and \(z_{5}\) are the numbers of teeth of gears 4 and 5, respectively). Next, the main steps of the dynamic modeling algorithm are presented, based on dynamic equations of the transmission external shafts, modeled by the Newton-Euler method, the kinematic and static equations of both planetary units I and II, and the mechanical characteristics of the wind rotor and counter-rotating electric generator.