## 3 Dynamic Modelling  
The dynamic equations of the wind system components in the mentioned working premises are linear differential equations of the second order with constant coefficients; they can be obtained by applying the Newton-Euler method considering the positive direction of angular velocity and torque vectors according to Figure 2.  
According to premise (2), the following kinematic and static equations can be written for the two planetary units PU-I and PU-II [16], see Figure 2d:  
\[\text{PU-I}:\ \left\{ \begin{array}{l}\omega_{1}=\omega_{H}(1-i_{01})\\ \omega_{H}T_{H1}\eta_{1}+\omega_{1}T_{1}=0\end{array} \right.\ \text{PU-II}:\ \left\{ \begin{array}{l} \omega_{2}=\omega_{H}(1-i_{02})\\ \omega_{H}T_{H2}\eta_{2}+\omega_{2}T_{2}=0\end{array} \right. \tag{7}\]  
where \(T_{x}\) is the resultant torque acting on the element \(x\); \(i_{01}\), \(i_{02}\), and \(\eta_{1}\), \(\eta_{2}\) are the internal kinematic ratio and the efficiency of PU-I and PU-II, respectively, and \(T_{H}=T_{H1}+T_{H2}\)--Figure 2b. The relations for the kinematic ratios and transmission efficiencies, specified in Table 1, can be easily derived from Equation (7). According to the block diagram in Figure 2 and Equation (2), Table 2 illustrates the dynamic schemes of the three WT components, resulting from the decomposition of the wind system into distinctive rigid bodies, as well as their related kinematic and dynamic equations. The set of equations in Table 2 is augmented by the linear mechanical characteristics with constant coefficients of the wind rotor (\(R\)) and generator (\(G\)):  
\[T_{R}=-a_{R}\omega_{R}+b_{R};\ T_{G}=-a_{G}(\omega_{GR}-\omega_{GS})+b_{G}, \tag{8}\]  
where \(a_{R},\ b_{R},\ a_{G},\ b_{G}\) are constant coefficients under steady-state conditions, and by definition \(T_{G}=T_{GR}\). The mechanical characteristic of a wind rotor is a nonlinear function [33; 35], which can, however, be acceptably approximated by straight line segments; in this study, four-zone nonlinear characteristic modeling was adopted (Figure 3): zones I and II are used at start-up, zone III includes the point of maximum power \(P_{max}\), and zones III and IV designate the working zones of the wind rotor. Obviously, the coefficients \(a_{R},\ b_{R}\) in Equation (8) are replaced specifically by \(a_{Ri},\ b_{Ri},i=1\ldots 4\) for each of the four zones I..IV. Obviously, linearizing the WT mechanical characteristic leads to inaccuracies in the calculated torque \(T_{R}\), as qualitatively shown in Figure 3. For the case study presented in Section 4, the most significant torque errors are recorded in zone III, with a relative error value of less than 3.4% (its maximum value near the midpoint of zone III, Figure 3), an acceptable level in numerical simulation of such a complex system. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**PU** & \(\dot{a}_{1,2}\) & \(\dot{a}_{1,2}\) & \(\dot{n}_{1,2}\) \\ \hline I & \(-\frac{25}{24}\cdot\frac{23}{2i}\) & \(1-i_{01}\) & \(\frac{1-i_{01}}{1-i_{01}}\) \\ II & \(\frac{25}{24}\cdot\frac{23}{2i}\) & \(1-i_{02}\) & \(\frac{1-i_{02}}{1-i_{02}}\) \\ \hline \(z_{j}\) is the no. of teeth of the gear \(j=1\ldots 5\) (see Figure 1); \(\eta_{01}\), \(\eta_{02}\)—internal efficiency of PU-I and PU-II, respectively; \(\eta_{01}=\eta_{02}=\eta_{G}^{2}\), where \(\eta_{G}\) is the efficiency of a gear pair with fixed axes. & \\ \end{tabular}
\end{table} 
Table 1: Transmission ratios and efficiencies. Substituting the gear ratio and efficiency relations from Table 1 into the kinematic, static, and dynamic equations from Table 2 and corroborating with Rels.