## 3 Dynamic Modelling
### Mechanical Characteristics  
The set of dynamic equations of the three bodies B1... B3 is completed with the linear mechanical characteristics with constant coefficients of the wind rotor and the counter-rotating electric generator, respectively:  
* the wind rotor \(R\) \[T_{R}=-a_{R}\omega_{R}+b_{R},\;0\leq\omega_{R}\leq\frac{b_{R}}{a_{R}},\;a_{R} \,>\,0,\;b_{R}\,>\,0\] (22)
* the electric generator \(G\) \[T_{G}=-a_{G}\omega_{G}+b_{G},\;\omega_{G}\leq\frac{b_{G}}{a_{G}},\;a_{G}\,>\, 0,\;b_{G}\,<\,0\] (23)where \(a_{R},\ b_{R},\ a_{G},\ b_{G}\) are the constant coefficients under the steady-state regime, and, by definition, \(T_{G}=T_{GR}\). Moreover, the functional specificity of an electric generator ensures the torque equilibrium condition between the rotor \(GR\) and the stator \(GS\):  
\[T_{GR}+T_{GS}=0 \tag{24}\]  
According to the chosen positive direction (see Figure 3), the mechanical characteristic of the wind rotor is active in quadrant I (\(T_{R}>0\) and \(\omega_{R}>0\)), while for the electric generator in quadrant II, (\(T_{G}>0\) and \(\omega_{G}<0\)) for absolute values of the angular speed \(\omega_{G}\) larger than the critical value \(|b_{G}/a_{G}|\), which delimits the operating modes as motor and generator of the \(DC\) machine, Figure 4. Therefore, in the dynamic analysis, the electric machine will be considered in idle functioning regime (i.e., \(T_{G}=0\)) until the required operating speed as a generator is reached:  
\[T_{G}=0\ \text{for}\ \omega_{G}>\frac{b_{G}}{a_{G}} \tag{25}\]  
In the present study, it was assumed that the electric generator was not adjusted during operation, and the pitch angle of the wind rotor blades was not changed. Instead, the mechanical characteristic of the wind rotor changes with the variation of the wind speed, as the mechanical power \(P_{R}\) generated at the wind rotor shaft depends on wind speed value [38]:  
\[P_{R}=\frac{1}{2}\tau pC_{p}r_{R}^{2}v_{w}^{3} \tag{26}\]  
where \(\rho\) is the air density, \(C_{p}\)--aerodynamic power coefficient, \(r_{R}\)--radius of the wind rotor, and \(v_{w}\)--the wind speed. In the general case, the coefficient \(Cp\) depends on the wind speed \(v_{w}\), the angular speed of the wind rotor \(\omega_{R}\), and the blade pitch angle \(\beta\). Since, in this study, the pitch angle was considered constant \(\beta=0^{\circ}\), the general relation of the power coefficient \(Cp\)[38; 39; 40] is obtained through particularisation in the following form:  
\[C_{p}=c_{1}\left(\frac{c_{2}}{\lambda_{i}}-c_{3}\right)\cdot e^{-\frac{c_{4}} {\lambda_{i}}}+c_{5}\lambda \tag{27}\]  
Figure 4: Linear mechanical characteristics of a wind rotor (\(R\)) and a \(DC\) generator (\(G\)). where \(\lambda\) is the tip speed ratio:  
\[\lambda=\frac{\omega_{R}r_{R}}{v_{w}} \tag{28}\]  
and  
\[\frac{1}{\lambda_{i}}=\frac{1}{\lambda}-c_{6} \tag{29}\]  
where \(c_{1}=0.5176\), \(c_{2}=116\), \(c_{3}=5\), \(c_{4}=21\), \(c_{5}=0.0068\), and \(c_{6}=0.035\). Thus, the mechanical characteristic of a wind rotor \(T_{R}=T_{R}(\omega_{R},\,v_{w})\) can be described analytically by Equation (30):  
\[T_{G}=-a_{G}\omega_{G}+b_{G},\;\omega_{G}\leq\frac{b_{G}}{a_{G}},\;a_{G}\,>\, 0,\;b_{G}\,<\,0\]0  
Considering the air density \(\rho=1.225\) kg/m\({}^{3}\), the mechanical characteristic of a wind rotor with the radius \(r_{R}=5\) m is illustrated in Figure 5 for different values of wind speed \(v_{w}\). There is a distinct evolution of the torque \(T_{R}\) on the three zones A, B and C (Figure 5a), with approximately linear shapes, rounded at the transition from one zone to another. Thus, the torque \(T_{R}\) remains constant with the increase in the angular velocity \(\omega_{R}\) up to a threshold value of the angular speed (which increases with the increase in the wind speed, zone A), followed by an ascending evolution to the maximum value (_max_\(T_{R}\), zone B) and a lower slope decrease on zone C--the normal operating area of the wind rotor.