## 1 Introduction  
Wind turbines can shut down during operation, typically due to lack of wind, high wind speeds, or need for maintenance, and followed afterward by their transition from rest to operation state. The start-up of medium-large wind turbines is carried out automatically and in a controlled manner, and prior knowledge of their dynamics is typically employed in controllers. Thus, dynamic behavior in transient regimes represents a challenge for researchers and an advantage for designers in the optimization of the control system and wind system design [1, 2, 3]. An important issue in approaching the wind system dynamics concerns the assumption of variable wind speed and identification of system dynamic behavior in transient regimes owing to the change in wind speed or starting from rest. Dynamic studies refer to both the wind system as a whole [2, 3, 4, 5, 6, 7] and its component subsystems, such as electric generators [8] or mechanical gear transmission [9, 10, 11] with the role of speed increasers with fixed axes [2; 12], mobile axes [7; 13; 14; 15; 16; 17] or combined [18; 19; 20; 21; 22; 23; 24; 25; 26]. The overwhelming majority of speed increasers are monomobile (single degree of freedom, 1-DOF) mechanical transmissions [2; 27; 28; 29; 16; 20; 23; 24; 2] and rarely differential (2-DOF) transmissions [5; 17]. The dynamic response of wind systems can be electrical or mechanical, depending on the type of power pursued: electrical or mechanical power, respectively. Aiming at improving the performance of electric generators, in conjunction with their overall size reduction, various counter-rotating type generators [2; 17; 20; 29; 30] are addressed in literature: with permanent magnets [31], with liquid metals [32], DFIG (Double Fed Induction Generator) that uses a back-to-back Pulse-Width converter Modulation (PWM) for bidirectional control [8], etc. Dynamic analysis of wind systems and their subsystems requires the use of specific software such as FAST v8.16 (Fatigue, Aerodynamics, Structures, and Turbulence) for aero-elastic dynamic modeling [1; 33; 34], MATLAB-Simulink [1; 2; 18; 35; 36], with results having errors below 2% compared to FAST, Ansys 15.0 [15], SIMPACK [33; 34] based on dynamic multibody modeling, PSCAD/EMDTC [37], etc. These software packages allow the identification of various representative dynamic parameters related to mechanical efficiency [2; 16; 38], shaft speeds [3; 36; 38] and torques [38; 39; 40], mechanical powers [3; 29; 38], and for electrical response: current intensity [36; 38] and voltage [38]. Modeling the dynamic response of a wind system requires also knowledge of the mechanical characteristics of both the wind rotor and the electric generator. In literature, these features are modeled as nonlinear [8; 27; 33] or linear [2; 20] functions. Thus, [PERSON] et al. [2] addressed the dynamic modeling and simulation of a 10 kW wind turbine with a single wind rotor and counter-rotating generator, equipped with a classical transmission with fixed axes, by considering only one linearized zone of the WR mechanical characteristic: the working zone. The dynamic numerical simulations cover scenarios of wind speed changes during operation by maintaining the operating point on this working zone. However, this research did not consider the starting phase from rest or optimizing the generator's load entry. Similarly, a single linear function for the working zone is proposed for power flow modeling in the steady state of a dual-rotor wind turbine with a counter-rotating generator [20]. In most cases, the dynamic response is an analytical or grapho-analytical result, the most common dynamic modeling methods applied for wind turbines being Newton-Euler [1; 2; 13; 41; 42; 43], Lagrange + Runge-Kutta [28; 39; 40; 6], lumped parameter theory [10; 26; 44] or polynomial chaos [12]. Generally, the approaches of WT dynamics consider the gearbox effect in a simplified model by neglecting the rotation of the satellite gears in planetary transmissions.