# A Review on Wind Turbine Deterministic Power Curve Models +
## 4 Sigmoid Models
### Exponential Models  
EMs are sigmoids where \(f(v,b_{0},\ b_{1})=\mathrm{c}^{-b_{1}(v-b_{0})}\), being \(c\), \(b_{0}\), and \(b_{1}\) as three constants. A classical value for \(c\) is the base of the natural logarithms \(e\), although different values could be used instead. However, the fact of assuming \(c=e\) is generally accepted, and other values are seldom considered. Traditionally, EMs have used between three and six parameters. Modeling them with fewer than three parameters means very low accuracy, and using the six parameters makes the model clearly valid.  
The traditional EM with six parameters, also known as 6 PLE, can be expressed as in Equation (18). It is the result of the combination of Equation (17), where \(b_{6}=0\) and the function \(f(v,b_{0},\ b_{1})=\mathrm{c}^{-b_{1}(v-b_{0})}\), where \(c=e\).  
\[P=b_{5}+(b_{2}-b_{5})\frac{1}{\left(b_{3}+\mathrm{e}^{-b_{1}(v-b_{0})}\right)^ {1/b_{4}}} \tag{18}\]  
The 5 PLE model is given by Equation (19). It corresponds to the EM with five parameters, and it can be obtained from the 6 PLE, by doing \(b_{5}=0\):  
\[P=\frac{b_{2}}{\left(b_{3}+\mathrm{e}^{-b_{1}(v-b_{0})}\right)^{1/b_{4}}} \tag{19}\]  
A further simplification of the model converts it into the 4 PLE and is one of the two options for the EM with four parameters. It is obtained from the 5 PLE with \(b_{4}=1\). Its expression is given in Equation (20).  
\[P=\frac{b_{2}}{b_{3}+\mathrm{e}^{-b_{1}(v-b_{0})}} \tag{20}\]  
The alternative EM with four parameters, denoted as 4 PLE, is the one that corresponds to Equation (21) and is the result of the combination of Equation (17) with \(b_{5}=0\), \(b_{4}=1\), and \(b_{3}=1\); the function \(f(v,b_{0},\ b_{1})=\mathrm{c}^{-b_{1}(v-b_{0})}\); and \(c=e\).  
\[P=b_{2}\frac{1+b_{6}\mathrm{e}^{-b_{1}(v-b_{0})}}{1+\mathrm{e}^{-b_{1}(v-b_{ 0})}} \tag{21}\]  
The simplest EM is the one based on the 4 PLE with \(b_{3}=1\). It is known as 3 PLE, and its mathematical expression is given by Equation (22).  
\[P=\frac{b_{2}}{1+\mathrm{e}^{-b_{1}(v-b_{0})}} \tag{22}\]  
In Table 6, the relationships among the parameters of the models and the parameters of the WT are shown. Some of the models do not have a strong relationship with the parameters of the WT, i.e., those denoted as 6 PLE, 5 PLE, and 4 PLE. However, these relationships can be found in the other two, and these models are recommended due to their good balance between performance and the number of parameters [52], specially the 3 PLE.  
The parameters of the WT used in Table 6 are as follows: \(s\) as the slope of the WTPC, and \(v_{ip}\) and \(P_{ip}\) as the wind speed and power at the inflection point of the WTPC, respectively. The 3 PLE model may depend on three or four parameters of the WT. Both are included in Table 6, and the simplest option is under the consideration of \(P_{ip}=P_{r}/2\). In addition to this, the conversion from the 4 PLEE model into the 3 PLE one is simply achieved by making \(b_{6}=0\). Notice that some of these models, i.e., the 4 PLE and the 3 PLE, are known as logistic models because they are generally used in the field of logistics.