# A Review on Wind Turbine Deterministic Power Curve Models +
## 3 Polynomial Models
### Cubic Models  
Among the polynomial models, cubic ones (CM) are those with values of \(n_{j}\) equal to or lower than 3, for all intervals. The most used one is that which is based on three intervals according to Equation (11) [14].  
\[P=\left\{\begin{array}{cc}a_{01}&v_{1}\leq v<v_{2}\\ a_{02}+a_{32}v^{3}&v_{2}\leq v<v_{3}\\ a_{03}&v_{3}\leq v<v_{4}\end{array}\right. \tag{11}\]  
Table 3 shows the right assignation of parameters.  
For \(j=4\), there is no interval, but \(v_{4}=v_{co}\). The errors in the accuracy of this model in the area of the rated wind speed are very important.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \(j\) & \(n_{j}\) & \(a_{0j}\) & \(a_{1j}\) & \(a_{2j}\) & \(v_{j}\) \\ \hline
1 & 0 & 0 & 0 & 0 & 0 \\
2 & 2 & \(-v_{ci}^{2}\): \(P_{r}/\left(v_{r}^{2}-v_{ci}^{2}\right)\) & 0 & \(P_{r}/\left(v_{r}^{2}-v_{ci}^{2}\right)\) & \(v_{ci}\) \\
3 & 0 & \(P_{r}\) & 0 & 0 & \(v_{r}\) \\
4 & - & - & - & - & \(v_{co}\) \\ \hline \end{tabular}
\end{table}
Table 2: Relationship among parameters of the quadratic models and the parameters of the WT.  
A second option consists of using the CM with \(a_{02}=0\) as shown in [12], but it operates without avoiding the errors of the previous one, and it involves some other errors in the area of the cut-in wind speed.  
One of the most used CMs for WTPCs that can be considered the most accurate is the spline model (SP) of order three. It consists of using as many intervals as the double of the value of its \(v_{co}\) in m/s. For instance, if \(v_{co}=25\) m/s, the model has 50 intervals, i.e., one interval every 5 m/s. On each interval, the model is as expressed in Equation (12) [49]. Therefore, this model establishes a difference with respect to all those commented above, which only considered the four intervals described earlier.  
\[P= a_{0j}+a_{1j}v+a_{2j}v^{2}+a_{3j}v^{3}\quad\frac{j}{2}-0.5\leq v<\frac{j}{2} \tag{12}\]  
Notice that the systematization in the implementation of the model is related with the interval limits \(j/2-0.5\) and \(j/2\). In addition, this model assures the continuity and derivability of the function in all its range of applicability. The continuity of the model provides accuracy in the limits of the intervals. Its derivability facilitates the combination of the model with other functions, i.e., the wind speed probability density function. In this case, there is no relationship between both types of parameters.  
The SP can be simplified if polynomials of order three are just used in the interval \([\,v_{ci},\,v_{r}]\), according to Equation (13), which is a proposal of this paper:  
\[P=\left\{\begin{array}{ccc}a_{01}&v_{1}\leq v<v_{2}\\ \cdots&\cdots\\ a_{0k}+a_{1k}\cdot v+a_{2k}\cdot v^{2}+a_{3k}\cdot v^{3}&v_{k}\leq v<v_{k}+0.5 \\ \cdots&\cdots\\ a_{0m}&v_{m}\leq v<v_{m+1}\end{array}\right. \tag{13}\]  
where the range of \(k\) is from 2 to \(2\cdot(v_{r}-v_{ci})+1\) and \(m\) equals \(2\cdot(v_{r}-v_{ci})+2\).  
For a better understanding of the model, several lines are given in Table 4 with related information. The only drawback of this model is that the derivability of the function is not guaranteed in two points, i.e., \(v_{ci}\) and \(v_{r}\), but it is continuous in the full range \([\,0,\,v_{co}]\). The main advantage of this model over the traditional SP is the number of parameters; in the latter case, this number is reduced to \(8\cdot(v_{r}-v_{ci})+2\), which is around the half for most WTs.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(j\) & \(n_{j}\) & \(a_{0j}\) & \(a_{1j}\) & \(a_{2j}\) & \(a_{3j}\) & \(v_{j}\) \\ \hline \(1\) & \(0\) & \(0\) & \(0\) & \(0\) & \(0\) & \(0\) \\ \(2\) & \(3\) & \(-v_{ci}^{3}\cdot P_{r}/\left(v_{r}^{3}-v_{ci}^{3}\right)\) & \(0\) & \(0\) & \(P_{r}/\left(v_{r}^{3}-v_{ci}^{3}\right)\) & \(v_{ci}\) \\ \(3\) & \(0\) & \(P_{r}\) & \(0\) & \(0\) & \(0\) & \(v_{r}\) \\ (4) & - & - & - & - & - & \(v_{co}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Relationship among parameters of the cubic ones and the parameters of the WT.