## 2 Materials and Methods
### Optimization Model  
The seasonal autoregressive integrated moving average (SARIMA) model is employed specifically for data series with a seasonal component. This model is equivalent to the autoregressive integrated moving average (ARIMA) model, with the exception of the S  component for the seasonal factor. The model is obtained by combining autoregressive analysis and moving averages of the values and errors due to the randomness of the time series. In this way, a value of the time series considered is related or is due to the correlation of the previous values and with the moving averages of the statistical residuals of the time series considered to obtain the model. Once the model is adjusted, forecasts are made for the modeled series of values. Data modeling versus time is obtained following a Box-Jenkins approach consisting of four stages: model identification, parameter estimation, model diagnosis and forecast verification. The identification is performed with the autocorrelation function (_ACF_) and partial autocorrelation function (_PACF_) plots. However, the stationarity of the time series must also be checked, as it is a fundamental criterion for applying the proposed approach. The Dickey-Fuller technique (1979) [50] is one of the most widely used methods to demonstrate the stationarity of the series. It is based on a hypothesis test where the null hypothesis indicates that the series is not stationary and has unit roots, and the alternative hypothesis indicates that the series is stationary and has no unit roots. The autocorrelation function (_ACF_) represents the correlation of each value of the series considered with the lagged values at different lengths. The values to be estimated are those above the confidence limits. The autocorrelation is calculated using the following  expression:  
\[ACF(k)\;o\;AC(k)=\frac{Cov(y_{t},y_{t-k})}{Var(y_{t})} \tag{1}\]  
where _Cov_ is a first lag correlation in the relationship of each of the values of the time series, at an instant \(t\), with the values measured at the previous instant, \(t-k\) and without taking into account the intermediate values when the lags are greater. This function, in ARIMA models, indicates the order (q) of the moving averages. The partial autocorrelation function (PACF) is used to find the partial correlation between the separated residuals. As in the previous function, the lag of the correlation means the relationship, in this case, of the residuals calculated at an instant t with the immediately previous ones separated at a specific time instant \(t-k\), taking into account the intermediate residuals. \[PACF(k)\;o\;PAC(k)=\frac{Cov(y_{t},y_{t-k}|y_{t-1},y_{t-k+1})}{\sqrt{Var(y_{t} |y_{t-1},y_{t-k+1})\cdot Var(y_{t-k}|y_{t-1},y_{t-k+1})}} \tag{2}\] \begin{table}
\begin{tabular}{c c c} \hline  & **Working Days** & **Weekend** \\  & **(Monday to Friday)** & **(Saturday and Sunday)** \\ \hline
00:00 to 01:00 & 28 & 28 \\
01:00 to 02:00 & 28 & 22 \\
02:00 to 05:00 & 22 & 22 \\
05:00 to 06:00 & 24 & 23 \\
06:00 to 07:00 & 24 & 26 \\
07:00 to 08:00 & 32 & 26 \\
08:00 to 09:00 & 32 & 33 \\
09:00 to 12:00 & 33 & 33 \\
12:00 to 13:00 & 32 & 33 \\
13:00 to 14:00 & 32 & 32 \\
14:00 to 00:00 & 33 & 32 \\ \hline \end{tabular}
\end{table} 
Table 1: Critical point target pressure (m). The selection of the best model can be performed using the Akaike criterion (_AIC_). This test allows selecting the SARIMA model with the best estimation and with the smallest number of parameters, so that increasing the number of parameters for a better fit is detrimental to the result of the test. The best model will be the one with the lowest value. The value is obtained by the following function:  
\[AIC=-2\,ln(l)+2k \tag{3}\]  
where \(k\) is the number of model parameters and \(l\) is the maximum value of the likelihood function for the estimated model. Another criterion for selecting the SARIMA model is the Bayesian information criterion (_BIC_).