## 4 Conclusions  
PM includes a different control model for the optimization of WDN. CPC is identified as the most advantegeous method under variable and seasonal demand pattern since it allows to reduce leakage in the network while providing the right level of service to the customer. However, current CPC methods rely on existing flow measurements or on the connection of the PRV with the critical point of the system. Sometimes, these two issues are not possible from a technical point of view or imply massive cost.  
In this work, a time series of a ratio of pressures measured downstream of the PRV and the critical point on a real location in Noja village was modeled to create and validate a new open-loop control scheme for CPC based on time. Hourly pressure values were taken from February to December 2021. Due to the influence in the demand of the holiday periods in the area, it is considered to estimate, for the modeling, data from three months of a non-holiday season, with lower consumption (April, May, June), and another three months of a holiday period, with higher consumption (July, August, September), in order to obtain a useful model regardless of the time of the year. In this way, 4392 observations were considered to estimate the model. The data from 1 October to 31 December, precisely 2208 observations, were used to validate the model.  
A SARIMA model has been built and implemented as an alternative to the current PM model based on the flow reading that the Noja utility is using. The SARIMA model has been validated by comparing the values predicted by the model with the data taken between October and December 2021. The first test was conducted by calculating the RMSE, giving a value of 0.018. The coefficient of determination of the comparison was also calculated, giving a value of 0.9648. Both tests support the goodness of fit of the model to predict pressure values at the valve outlet and at the critical point of the zone. The SARIMA model was also tested in real field conditions for eleven days in December 2022. The hydraulic variables of the area were analyzed, showing that there is no difference when working under the previous model and that the PRV is under the SARIMA model control. The real field measurements at the critical point obtained when applying the SARIMA model were compared with the critical point target pressure by means of an ANOVA analysis with a \(p\) value of 0.412, greater than 0.05, thus confirming the null hypothesis of this analysis and that there is no statistical difference between the values compared.  
The result of the field test and the statistical analysis performed concludes that SARIMA is a useful tool to modulate pressure at a PRV outlet to achieve a target value at the critical point of the area supplied. The new model can be implemented as an alternative for CPC in situations where no flow signal is available or where it is not possible to establish real-time communication between the critical point and the PRV.  
Conceptualization, A.O.-B., D.M.-R., M.-J.-A.-U.; F.J.d.I.S.-Z. and A.-J.P.-M.; methodology, A.O.-B., D.M.-R., M.-J.A.-U.; F.J.d.I.S.-Z. and A.-J.P.-M.; investigation, A.O.-B., D.M.-R., M.-J.A.-U.; F.J.d.I.S.-Z. and A.-J.P.-M.; resources, A.O.-B., D.M.-R., M.-J.A.-U.; F.J.d.I.S.-Z. and A.-J.P.-M.; writing--original draft preparation, A.O.-B., D.M.-R., M.-J.A.-U.; F.J.d.I.S.-Z. and A.-J.P.-M. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.