# Wind Turbines Optimal Operation at Time Variable Wind Speeds
## 1 Introduction
The power curve models based on polynomial approximation are simple to use, but only for small systems [13]. Most of the models are developed from the manufacturer's curves of wind turbines [14], which are turbine-specific, and they can be applied in the power prediction of single turbines and in sites with steady winds [15]. As the wind turbine's power is strongly dependent on the wind speed at the wind farm site, the selection of an appropriate wind speed model, along with the power curve model, is the main requirement for accurate prediction of the wind farm's output. Therefore, improved models are required, which should be able to minimize the prediction errors. In consequence, the method proposed in this paper belongs to the WTPC approach, which express the shape of the PMSG's power curve by a set of mathematical equations. Determination of the coefficients of these equations requires fitting the data to the mathematical model [13]. Furthermore, all these methods, PSF, OT, TSR, P&O, artificial intelligence techniques and WTPC, at some level fail in the proper detection of the maximum power point. Furthermore, the problems become more complex for time-varying wind speeds, due to the high value inertia moments [19; 20]. To cope with these problems, the moment of inertia has to be considered, making it possible to accurately detect the MPP, even under rapid wind variations. In the literature, there are only a few studies on methods to eliminate the inertial effect at variable wind speeds, when the optimal operation of the wind turbine is very hard to achieve [14]. However, as regards the WTPC method, there is no study taking into account the inertia. Therefore, the method proposed in this paper is based on the mathematical model of the wind turbine, which includes its moment of inertia, too. The paper is comprised of seven sections. Section 1 presents an overview of the arising problems regarding reaching a good wind turbine efficiency; Section 2 presents the wind turbines' mathematical model, on which the study is based; Section 3 describes the methodology of determining the PMSG's optimal power; Section 4 presents the implementation of a proportional integral (PI) type control applied to the PMSG's optimal power; in Section 5, a proportional integral derivative (PID)-type control is applied to the PMSG's optimal power; Section 6 presents a case study, and in Section 7, the conclusion is drawn. The case study is based on a real wind turbine's data, collected from a real wind farm and provided in [21]. The installed wind turbines are of 2.5 (MW) nominal output power, and of the three blades type, are manufactured by General Electric company and have the characteristics presented in Table 11, which can be found in the Appendix paragraph. Along with the case study presented in this paper, the operation of one wind turbine is analyzed for wind speeds measured with a sampling step of 30 (s). Based on the values of the wind speed, the maximum values of the chosen wind turbine's power are determined, \(P_{WTmax}\). Then, the operation of the wind turbine, at MPP, is analyzed, meaning that the optimal values of the PMSG's power can be determined based on the measured wind speed, so that the PMSG's angular speed tends towards the PMSG's optimal angular speed. Two control methods for constraining the PMSG's power to follow the optimal one are in this paper analyzed: one based on the PI-type regulator, and the second one based on the PID-type regulator. The values of the regulator coefficients were determined by the trial and error method. Then, finally, the optimal power values of the PMSG are compared to the ones obtained by implementing two types of control, based on either the PI or PID regulators. The simulations are actually represented by the numerical solutions obtained by using the MAPLE core of the Scientific Workplace simulation environment, and they are based only on the measured wind speed from the experimental data collected from the wind farm.