Various power control strategies in variable-speed wind turbines assume that the whole system parameters are known and accurate wind speed measurements are available, which is not necessarily satisfied in practice. In this article, two nonlinear adaptive control strategies, which are independent on the knowledge of the system parameters and the wind speed measurements, are proposed for power regulation and tracking, in variable-speed wind turbines. One of these strategies does not even require the rotor acceleration, differently from other works. Both control strategies are formally proven by the Lyapunov method. The effectiveness of the proposed controllers is illustrated using the FAST wind turbine simulator under a turbulent wind profile.