## 2 Linearization of the Power Deficit Calculation
### Comparison of Traditional Aggregation Methods and the Proposed Method  
The estimates of \(\mathbb{E}[P]\) derived from the three methods have been compared for Anholt OWF, based on the data series for the years 2013 and 2014 available in [21], from which the wind rose in Table 11 has been deduced. Using the obtained values, the capacity factor can be calculated as the ratio between the average power of the farm and the sum of the wind turbine-rated capacities \(P_{1}^{rated}\), i.e.,  
\[cf=\frac{\mathbb{E}[P]}{nt\,P_{1}^{rated}}\quad\text{with }nt=111\quad\text{and }P_{1}^{rated}=3.6\ MW \tag{20}\]  
The results are presented in Table 2 where the second column refers to \(\mathbb{E}[P]\); the third one corresponds to this same value multiplied by the availability factor (0.97 [22]); the fourth one is the resulting capacity factor and the fifth one is the observed capacity factor, as given by SCADA. According to [23], this capacity factor is 47.8% (for the year 2018), while [23] indicates that it fluctuates between 40% and 54% from 2013 to 2019. For the period from 2013 to 2014, a value of 47.66% was obtained from [23], which was practically identical to that from 2018. As expected, the LAPD results in higher deviations from the observed value, while the LAVD and QAVD result in closer estimations. This inaccuracy occurs because the hypothesis of aggregating power deficits instead of wind speed deficits for an accurate estimation of AEP is only valid in the speed range in which the turbine power curve grows linearly with speed. This range extends from about 2 m/s above the \(u_{ci}\) to about 3 m/s below the rated wind speed. In general, the wind speed frequently falls outside this range, so a precise estimate of the AEP is not expected. It is worth mentioning that for very high wind speeds, the thrust coefficient is low, and the expected deficit will also be low. Furthermore, if the wind speed value, once disturbed, remains above the rated speed, for most turbines (with variable pitch), the output power will not be affected. Consequently, all aggregation models will yield the same results. Although the value provided by the LAPD aggregation is not as accurate as the other two methods, it will be shown that it maintains a value distribution very similar to theirs, especially to that of LAVD. This is important because an optimization of the layout based on energy calculations using either method will yield similar solutions regarding the arrangement of the turbines. This affinity between the three types of aggregation has been tested by creating maps of power deficits in an area of 500 \(\times\) 428 locations, representing a surface of 3500 \(\times\) 3000 m under the wind conditions of Horns Rev 1 (HR1) given in Table 11. Figure 2 compares the maps for power deficits using the three types of aggregation, according to the estimation \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Aggregation** & \(\mathbb{E}[P]\) **(MW)** & **Availability (97\%)** & **cf 1** & **SCADA cf** \\ \hline \multirow{3}{*}{Anholt} & LAVD & 190.24 & 184.53 & 46.2\% & \\ \cline{2-6}  & QAVD & 203.12 & 197.03 & 49.3\% & 47.7\% \\ \cline{1-1} \cline{2-6}  & LAPD & 179.42 & 174.04 & 43.6\% & \\ \hline \hline \end{tabular} 1  
\end{table} 
Table 2: Comparison of capacity factor (cf) obtained from different ways of aggregation. using Equations (9), (11) and (13), respectively, for maps (a), (b), and (c). The evaluation took 29 min 43 s for LavD, 27 min 20 s for QAVD, and only 2.5 s for LAPD. Although a precise estimate of AEP is not reached using LAPD, a **scaling** effect in energy is detected by comparing maps (a) and (c), in the sense that the distribution of power deficits at all points of the concession zone follows a pattern similar to that obtained with LavD. This equivalence in the maps of power deficits is translated into an equivalence in the distribution of their gradients, which allows designing an optimization algorithm based on LavD that is **200 times faster** than on QAVD or LavD.