## 3 Results and Discussion  
### Time-Based SARIMA Statistical Control Model  
The available data series covers almost the entire year 2021 with hourly intervals. The period from April to September was selected to obtain a model which can be used for any date of the year. This selection also allows enough data available for model validation. Data from October to December will be used for model validation purposes. The time series to be modeled is obtained from the quotient between the critical point pressure (_P3_) and the pressure at the outlet of the PRV (_P2_), measured every hour, between 0:00 h and 23:00 h on each of the days considered. It involves a total of 4392 datapoints. Before identifying the model, it is analysed whether the data series is stationary or not, a condition to be fulfilled to obtain the ARIMA model. The original data series is subjected to the Augmented Dickey-Fuller test using the EVIEWS program test to confirm the stationarity or non-stationarity of the data. The result obtained has a \(p\)-value greater than 0.05, specifically 0.0629, which means that the null hypothesis is fulfilled; thus, the series is non-stationary and it must be transformed to make it stationary. Figure 5 shows the original _P3/P2_ series (left) and the correlogram of the original series (right). According to the representation of the correlation coefficients (_AC_), the presence of seasonality can be deduced by the fact that the 1 st and 24 th autocorrelation coefficients take maximum values, which makes regular differentiation or taking logarithms unnecessary, as the original data do not present a trend. Knowing the non-stationarity condition, and the seasonality present in the data, a seasonal difference is made to the original data series. Figure 6 represents the series obtained after differentiation by removing the seasonal component (a) and a correlogram of the series with seasonal difference (b). The Dickey-Fuller test is then performed again, considering the cycle of 24, to check that the seasonal component has been eliminated. A significance level less than 0.05 is obtained, which leads to the conclusion that the new series is stationary, rejecting the null hypothesis. Once the stationarity has been confirmed, the model is built. The choice of model is made using the correlograms in Figure 6 and using the criteria of AKAIKE and BIC [52,53]. Figure 5: _P3/P2_ original data series (a) and correlogram of the series without transformation (b). Figure 6: _P3/P2_ series with seasonal differentiation (a) and correlogram of the series with seasonal difference (b) (EVIEWS). From the correlogram obtained with the seasonally differenced series (Figure 6b), it is estimated that the best fitting model would be an ARIMA (1,0,10)(1,1,1)\({}_{24}\) pending confirmation by the aforementioned statistical criteria. Table 2 shows a sample of the comparison of the estimated models, using the EVIEWS program, considering the information obtained from the correlogram in Figure 6b. The models shown have been compared using the differenced data series. The model and its coefficients are obtained using the IBM SPSS 28 software package based on the model estimated in the previous section [54]. The results obtained are shown in Table 3.  
An R\({}^{2}\) value of 0.973 indicates the goodness of fit and a value of 12.098 of the Ljung-Box Q with a significance level of 0.147 confirms the randomness of the residual errors. Figure 7 shows the plot of the model fit with the data considered in the modeling. To facilitate the visibility of the data, only two months have been represented: April, a month  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Model** & **LogL** & **AIC** & **BIC** & **HQ** \\ \hline (1,0,10)(1,1  
of low consumption (a), and August, a month of high consumption (b). It can be seen that the values obtained by the model for the _P3_/_P2_ ratio fit very well to the real data. Once the model has been obtained, it is validated by applying it to data which were not used to build the model, from October to December 2021.