## 2 Methodologies
### Precipitation Data Assessment and Models for Water Availability in Hydropower Plants  
Since precipitation is the prime driver of a catchment's hydrologic cycle and water availability for hydropower production, it is useful to conduct an analysis of historical hourly precipitation profiles for the geographic locations of interest. The months constituting the seasons are aggregated as usually done in the literature [43]: December-January-February (DJF) for winter, March-April-May (MAM) for spring, June-July-August (JJA) for summer, September-October-November (SON) for autumn. This analysis aims to understand whether the trend of hourly rainfall in recent years may positively or negatively impact the availability of water for hydropower generation. In addition, it may be interesting to observe what the effects of climate change may be on trends in extreme precipitation events. There is no univocal definition of extreme precipitation in the literature, and this paper considers extreme precipitation to be precipitation intensity greater than 3.6 mm/h, as defined in [44]. Direct runoff is proportional to the amount of water that precipitates. Thus, analysis of precipitation trends over the past twenty years also allows conclusions to be drawn about the amount of direct runoff. There are two main methods for evaluating direct runoff from precipitation: the Rational method, and the Natural Resources Conservation Service (NRCS) method [31]. The Rational method considers the intensity of precipitation (mm/h), the recurrence time of precipitation with certain intensities, and the specific catchment area considered (km\({}^{2}\)), as shown by the following equation [31]:  
\[\mathrm{Rf}=\mathrm{\ c}\cdot\mathrm{Pi}\cdot\mathrm{A}\left(\frac{mm\cdot km ^{2}}{h}\right) \tag{8}\]  
where Rf is the direct runoff; c is called the _runoff coefficient_ and varies according to the type of land cover and slope of the land (dimensionless); Pi is the intensity of precipitation (mm/h); and A is the extent of the drainage area (km\({}^{2}\)). To obtain the direct runoff in Equation (8) with units equal to m\({}^{3}\)/s, one must multiply (8) by 0.278, which is thus a conversion factor. Regarding the runoff coefficient, the values of c referring to grassland (land cover assumed for ET\({}_{0}\) evaluation) and for different land slopes, are given in the following Table 1[31]: \begin{table}
\begin{tabular}{c c} \hline
**Slope Grass Cover (\%)** & **c Range** \\ \hline \(0\div 5\) & \(0.10\div 0.40\) \\ \(5\div 10\) & \(0.16\div 0.55\) \\ \(10\div 30\) & \(0.22\div 0.60\) \\ \hline \end{tabular}
\end{table} 
Table 1: Runoff coefficient values in the case of grassland. The main limitation of the Rational method is that the considered drainage area should not exceed 0.8 km\({}^{2}\), so this value is here set for A. The NRCS method estimates, not the flow rate, but the amount of water available as runoff (mm) [45]. This method, unlike the Rational one, uses the cumulative amount of precipitation (mm) as input. The equation that 'converts' this amount to runoff is as follows [45]:  
\[\mathrm{Rf}^{\prime}=\frac{\left(\mathrm{Pr}-\mathrm{I_{a}}\right)^{2}}{ \mathrm{Pr}-\mathrm{I_{a}}+\mathrm{S}}(\mathrm{mm}) \tag{9}\]  
where \(\mathrm{Pr}\left(\mathrm{mm}\right)\) is the cumulative precipitation of a day; \(\mathrm{I_{a}}\left(\mathrm{mm}\right)\), known as initial abstraction, is the loss of water to runoff due to infiltration, interception, and storage of water in surface depressions; \(\mathrm{S}\left(\mathrm{mm}\right)\) is the maximum potential retention after runoff begins.