Simplified equations for the rotational speed response to inflow velocity variation in fixed-pitch small wind turbines

We propose simplified equations for the rotational speed response to inflow velocity variation in fixed-pitch small wind turbines. The present formulation is derived by introducing a series expansion for the torque coefficient at the constant tip-speed ratio. By focusing on the first- and second-order differential coefficients of the torque coefficient, we simplify the original differential equation. The governing equation based only on the first-order differential coefficient is found to be linear, whereas the second-order differential coefficient introduces nonlinearity. We compare the numerical solutions of the three governing equations for rotational speed in response to sinusoidal and normal-random variations of inflow velocity. The linear equation gives accurate solutions of amplitude and phase lag. Nonlinearity occurs in the mean value of rotational speed variation. We also simulate the rotational speed in response to a step input of inflow velocity using the conditions of two previous studies, and note that the form of this rotational speed response is a system of first-order time lag. We formulate the gain and time constant for this rotational speed response. The magnitude of the gain is approximately three when the wind turbine is operated at optimal tip-speed ratio. We discuss the physical meaning of the derived time constant.