## 2 Problem Formulation  
Wind turbines with counter-rotating electric generator are characterised by a mechanical system with higher inertia versus the ones with conventional electric generator, due to the rotation of the generator stator, which usually is characterised by a higher mechanical moment of inertia than the generator rotor. Therefore, identifying the transient dynamic behaviour of these wind turbines when changing the wind speed is a challenge in expanding the implementation of wind turbine solutions with counter-rotating electric generators and also for their better operation. The modelling of the dynamic response of wind turbines with counter-rotating electric generator was further developed for a generic wind turbine including a simple gearbox. The analysed wind turbine consisted of a wind rotor \(R\), a fixed axes speed increase \(SI\) with one input and two outputs, and a counter-rotating electric generator \(G\), characterised by the rotation of the rotor \(GR\) and stator \(GS\) in opposite directions, Figure 1.  
The speed increase \(SI\) was derived from a cylindrical planetary gear set with one satellite gear, which functioned as a fixed axes mechanism obtained by connecting the satellite carrier \(H\) to frame 0. Thus, the mechanical power input in the speed increase was achieved through a ring gear 2, which was connected to the wind rotor \(R\) and to the generator stator \(GS\), which engaged with two or more evenly distributed gears 3. The gears 3 engaged simultaneously with gear 1, which ensured the transmission of mechanical power to the generator rotor \(GR\). The ring gear 2 and gear 1 had the same revolute axis, namely the _central axis_. As a result, the input mechanical power was distributed on two branches to (Figure 1c):  
* the generator rotor \(GR\), through the speed increase \(SI\), with speed amplification;
* the generator stator \(GS\), without changing the input speed due to the direct connection between the wind rotor \(R\) and the stator \(GS\). The angular speeds of the gears and the kinematic ratios, under the conditions of the branched transmission of mechanical power from the input \(2\equiv R\) to the outputs \(1\equiv GR\) and \(2\equiv GS\), can be obtained based on the scheme from Figure 1b:  
\[\omega_{1}=\frac{-v_{1}}{r_{1}}=-\tan\delta_{1};\ \omega_{1}=\omega_{GR} \tag{1}\]  
\[\omega_{2}=\frac{v_{2}}{r_{2}}=\tan\delta_{2};\ \omega_{2}=\omega_{R}=\omega_{ GS} \tag{2}\]  
\[i_{GR\_R}=\frac{\omega_{GR}}{\omega_{R}}=\frac{\omega_{1}}{\omega_{2}}=-\frac {\tan\delta_{1}}{\tan\delta_{2}}=-\frac{r_{2}}{r_{1}}=-\frac{z_{2}}{z_{1}} \tag{3}\]  
\[i_{GS\_R}=\frac{\omega_{GS}}{\omega_{R}}=1 \tag{4}\]  
where \(\omega_{x}\) is the angular speed of gear \(x=1\dots\) 3; \(v_{x}\)--the peripheral linear speed on the pitch circle of radius \(r_{x}\) of gear \(x=1\dots\) 3; \(\delta_{1,2}\)--the angle between the straight lines given by the center of rotation of gears 1, 2 and the origin and the tip, respectively, of the speed vector \(v_{1,2};z_{1,2}\)--the teeth number of the gears 1, 2; \(i_{x\_y}\)--the kinematic ratio (i.e., the ratio of angular speeds) when the rotation motion is transmitted from body \(x\) to body \(y\).