# Modeling of Water Losses in Hydraulic Tunnels under Pressure Based on Stepwise Regression Method
## 6 Conclusions  
Most pumped-storage hydroelectric power plants in Europe and the world are 30-50 years old. Within these complex hydro-technical facilities, there are HTs where monitoring and forecast of static instability in terms of failure and functional instability in terms of water losses from the tunnel are of enormous significance. The assessment of SaFE indicators in HT is generally conducted in two ways: (a) by emptying the tunnel and conducting physical inspection (b) by mathematical modeling of significant variables in HT with different types of analytical, numerical, statistical, or ML models. The literature review shows that there are not many articles dealing with problems of HT safety and functionality, in particular with problems of water losses from the tunnel. For these reasons, this paper presents a novel methodology for the mathematical modeling of water losses in HT. In addition, the goal was to validate this methodology by building an MLR model of water losses with greater accuracy and reliability, compared to previously developed models of HT water losses.  
The methodology in this paper included the following modeling phases: correlation analysis, dimension reduction by PCA, generation of MLR models based on stepwise method, regression analysis, as well as iterative reduction of model complexity achieved by avoiding multicollinearity of the model, thus improving model stability.  
A model of HT water losses developed based on the case study was carried out for the hydraulic tunnel at PSHPP \"Bajina Basta\", located on the Drina river, in the Republic of Serbia for the period 2005-2017. The most accurate model, MLR\({}_{\text{Best}}\), which satisfies the condition of non-multicollinearity, was chosen as appropriate. A detailed regression analysis indicated that the regressors which have the greatest impact on HT water losses are water level in the surge tank, and the regressor that includes the impact of water temperature, concrete temperature and size of crack openings.  
By comparing the model adopted in this paper with other literature for the same tunnel, it was concluded that the new model has better accuracy. It should be emphasized that the model developed in this paper also takes into account more input variables (size of crack openings and tunnel aging) than the previous model.  
The analyses of water loss in hydrotechnical tunnels are of great significance because the loss of water implies economical losses, and on the other hand can point out that damages or changes in the tunnel have occurred, which can lead to loss of tunnel stability. Based on the proposed model in this paper, it is possible to determine, in a fast, efficient, and economically acceptable manner, any possible problems in the observed tunnel and provide indications for their resolution. Situations where there are significant deviations between the measured and model predicted values of water losses should demand a more detailed analysis, even including tunnel opening and its thorough inspection. Based on this analysis it can be determined if there is a need for new HT reparation.  
Due to the immense importance that the SaFE paradigm for HT has in the PSHPP, the authors intend to further improve the model of total water losses. The improvement of the model would be mostly based on the introduction of new modeling methods, such as artificial neural networks, from which higher model accuracy can be expected.  
Data curation, S.R. and S.O.; Formal analysis, S.R. and S.O.; Methodology, S.R., D.D. and N.M.; Supervision, M.M., B.S., D.D. and N.M.; Validation, M.M. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The authors declare no conflict of interest.