# A Review on Wind Turbine Deterministic Power Curve Models +
## 2 Wind Turbine Deterministic Power Curve Models  
According to the related literature, two types of WTPC models have been generally used in wind energy studies: deterministic and probabilistic models. The focus of this paper is exclusively on deterministic models, with the aim of establishing connections among them and, additionally, analyzing a possible reduction of their number. In addition, there is a relationship between the parameters of a WT and those involved in its WTPC model. Hence, the proposal consists of deriving a general model for each case in order to study such a relationship. As stated in the previous paragraph, it is not the aim of this paper to analyze probabilistic models. However, for those readers who may be interested in deepening their study, some papers can be of interest, such as [43; 44; 45; 46; 47]. Coming back to the deterministic approach, it must be reminded that an important objective of these models is to represent WTPCs by means of functions. On a given terrain, i.e., out of laboratory conditions, the same value of wind speed can give rise to different values of power, depending on a survey of conditions, such as those mentioned earlier. However, the mathematical meaning of a function is involved in giving a single output value for a given input one, and this is in part what is pursued with the use of these models. It can be considered there are two main types of WTPC deterministic models, i.e., those described by means of polynomial functions and those other described by means of sigmoids. In both cases, the models can be used for studying the whole operating range of the WT or only a certain wind speed interval or intervals of it. Additionally, all of these models can be described by means of continuous nonpiecewise functions or by means of piecewise ones, which in general are not continuous. The reason for this is explained in the following paragraph. This last circumstance means that, in most cases, the derivability of the whole function is not guaranteed by the model. In general, in the case of three-bladed HAWTs, the whole range of possible wind speeds is divided into four intervals. The first one covers from a wind speed equal 0 up to the cut-in wind speed. The cut-in wind speed is the value for which a WT starts operating. Below this wind speed, the WT does not generate any power. The second interval covers the range between the cut-in and the rated wind speeds, and in this interval, the WT increases its generated power as wind speed increases. The third interval depends on the type of WT. In old stall-regulated WTs, due to aerodynamic losses, WT power output decreases as wind speed increases. However, most of the WTs currently used are of the type called pitch-regulated, and in this interval the power output keeps constant as wind speed increases. In both types of WTs, this operating mode happens up to the so-called cut-out wind speed, which is the limit between this third interval and the fourth one, in which the WT is forced to finish its operation due to safety reasons. A typical graphical representation of a WTPC reveals that in the real case, there are two points where the curve is not continuous. The first one corresponds to the cut-in wind speed, where there is a step from power 0, at a point when the wind is below the cut-in wind speed, to a small amount close to but greater than 0 for wind speeds greater than the cut-in one. The second one corresponds to the cut-out wind speed. There is a value of power for wind speeds close to but below this wind speed, and then there is a negative step down to 0 for wind speeds greater than the cut-out one, in this case due to safety reasons as mentioned. The power generated by a WT \(P\) not only depends on the wind speed \(v\) but also on the air density \(\rho\); the area swept by the rotor \(A\); and the power coefficient \(c_{p}(\lambda,\beta)\), where \(\lambda\) is the tip speed ratio and \(\beta\) the WT blade angle of attack, according to the well-known Equation (1).