## 3 Results and Discussion
### Error Analysis On-Site
#### 3.3.2 Error Analysis  
The less the difference between the estimated and the real value, the better the estimation. Two kinds of errors were applied, where \"\(P_{m}\)\" was the measured power, \"\(P_{e}\)\" was the estimated power, \"\(s\)\" was the sample in consideration, and \"\(N\)\" was the total amount of samples. The first one was called Mean Absolute Error (MAE) defined by Equation (11) and was used as a standard statistical metric to measure the model performance in meteorology, air quality, and climate research studies [42; 51; 52]. The second one was the percentage of the MAE known as Mean Absolute Percentage Error (MAPE) defined by Equation (12).  
Given that the results shown in Figures 9 and 16 resemble a sinusoidal behavior, \(\left(P_{m}-P_{e}\right)/P_{m}\) does not present the real error between both signals. Considering Figure 19a, if the measured value is on a high level, then the rate between the \(P_{m}-P_{e}\) and \(P_{m}\) values will be low; however, if the measured value is on a low level, then the same rate will be high or close to 1, as seen in Figure 19b. According to this, the rate is changed regarding the full range of the measured signal (max-min) instead in order to obtain a reliable value. Equation (13) represents the modified MAPE.  
\[MAE=\frac{\sum\limits_{s=1}^{N}\left|P_{m}-P_{e}\right|}{N} \tag{11}\]  
\[MAPE_{\%}=\frac{\sum\limits_{s=1}^{N}\left|\frac{P_{m}-P_{e}}{P_{m}}\right|}{ N}\cdot 100 \tag{12}\]  
\[MAPE_{\left(range\right)_{\%}}=\frac{\sum\limits_{s=1}^{N}\left|\frac{P_{m}-P_ {e}}{\max\left(P_{m}\right)-\min\left(P_{m}\right)}\right|}{N}\cdot 100 \tag{13}\]  
With respect to the data from HS and MCS, Table 9 shows the values of the errors mentioned in Equations (11) and (13).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Location** & **Month** & **Fitted Curve** & R\({}^{2}\) \\ \hline \multirow{6}{*}{HS} & Aug & \(f\left(x\right)=1.005x+84.92\) & 0.9916 \\  & Sep & \(f\left(x\right)=1.032x+20.92\) & 0.9981 \\  & Oct & \(f\left(x\right)=1.031x+44.45\) & 0.9877 \\  & Nov & \(f\left(x\right)=1.022x-29.74\) & 0.9926 \\  & Dec & \(f\left(x\right)=1.05x-37.57\) & 0.9849 \\  & Jan & \(f\left(x\right)=1.009x+6.707\) & 0.9982 \\  & TOTAL & \(f\left(x\right)=1.025x-4.633\) & 0.9763 \\ \hline \multirow{6}{*}{MCS} & Jul & \(f\left(x\right)=1.003x+91.51\) & 0.9989 \\  & Aug & \(f\left(x\right)=1.01x+184.3\) & 0.9961 \\  & Sep & \(f\left(x\right)=0.9843x+525.7\) & 0.9946 \\ \cline{1-1}  & Oct & \(f\left(x\right)=1.003x+140.6\) & 0.9997 \\ \cline{1-1}  & Nov & \(f\left(x\right)=1.007x-40.05\) & 0.9982 \\ \cline{1-1}  & Dec & \(f\left(x\right)=1.004x-44.28\) & 0.9995 \\ \cline{1-1}  & TOTAL & \(f\left(x\right)=1.014x+19.18\) & 0.9966 \\ \hline \hline \end{tabular}
\end{table}
Table 8: The monthly and total fitted curve characteristic equation and determination coefficient values of estimation for both locations.  
Table 9 shows the errors for both locations, where the MAE result values for HS have a range from 138.1913 W to 299.7928 W and an overall of 200.6071 W. Considering MCS, its MAE goes from 590.6460 W to 1192.3 W, with an overall of 1130.4 W. From these results we can highlight that the values are low according to the total value range for each location, being up to 2500 W and 45 kW for HS and MCS, respectively. This statement is proven by column 4 from Table 9, achieving a MAPE of 6.9036% in general for HS; meanwhile, for MCS, its respective results is a MAPE of 1.9291%. The above demonstrates the estimation robustness of the statistical model.