# Modeling of Water Losses in Hydraulic Tunnels under Pressure Based on Stepwise Regression Method
## 4 Case Study
### Model of Total Water Losses in the HT Based on Principal Components
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{1}{c}{**Type**} & \multicolumn{4}{c}{**Single Terms**} & \multicolumn{2}{c}{**Interaction Terms**} \\ \hline Principal components & \(\overline{H}_{st}\) & \(\overline{H}_{r}\) & \(hP\) & \(T_{wcc}\) & \(\overline{t}\) & \\ Regressors & \(\overline{H}_{st}\overline{H}_{st}\)2 & \(\overline{H}_{r}\overline{H}_{r}\)2 & \(hPhP\)2 & \(T_{wcc}\), \(T_{wcc}\)2 & \(e^{-\overline{t}}\), \(e^{-\overline{t}}\), \(e^{-\overline{t}/2}e^{-\overline{t}/4}\), \(e^{-\overline{t}/10}\) & \(T_{wcc}\overline{H}_{st}\), \(T_{wcc}\overline{H}_{st}\)2 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Potential regressors for the water losses model. The analytic form of the MLR model of HTs total water losses, based on PCs, is presented in Equation (10). \[ \begin{split} q=10.20945+4.22495\cdot\overline{H}_{st}-1.78695\cdot T _{twcc}-1.15069\cdot hP\\ -1.16897\cdot T_{twcc}\cdot\overline{H}_{st}-0.66310\cdot e^{- \overline{t}}+0.45082\cdot\overline{H}_{st}{}^{2}+0.40840\cdot T_{twcc}{}^{2}, \end{split}  \tag{10}\]  
The measured and predicted values of total water losses in the HT, based on the LDS, for the period 2005-2015 are shown in Figure 6.  
Various 3D surface plots can also represent the adopted mathematical model of HT total water losses. Such visualizations (Figure 7) can be useful for the interpretation of physical quantity effects, and the significance of individual predictors. In the diagrams shown, in addition to the predictors shown on the axes, all other predictors are constant and equal to their mean values. The diagram in Figure 7 presents a surface showing an increase in total water losses in HT of PSHPP \"Bajina Basta\", as a function of time and temperature in the extrados of the concrete lining. It can also be seen that the effect of reparation on water losses from the tunnel is the largest in the first few years after the tunnel reparation was carried out in 2005. After that, this effect slowly dissipates, following an exponential law. The diagram in Figure 7 presents the surface showing that water losses increase with the increasing water level in the surge tank and with the reduction of concrete temperature. The final adopted model of HT total water losses was verified using the dataset that was not used in the model development process. Figure 8 shows diagrams of predicted and measured values of total water losses during the test period. Figure 6: Measured and predicted values of HT total water losses for the LDS, for the period 2005–2015. Figure 7: Model of HT total water loss: examples (**a**) q = f(Tc, t) and (**b**) q = f(Tc, Hst). In Figure 8, \(q\) represents the measured values, \(\hat{q}\) represents the model values according to the model (10) in this paper, and \(\hat{q}\ast\) represents the model values of water losses according to the model given in paper [4]. Test sessions refer to the last four sessions, sessions 76-79, i.e., measured values 157-180. \(LMCI\), \(UMCI\), \(LICI\), and \(UICI\) represent the lower (L) and upper (U) limits of the confidence intervals for the expected mean (M) and individual values (I) of the model (10). The accuracy of the model for the test period is \(\text{RMSE}_{\text{Test}}=2.0161\) L/s. Model (10) is additive, and Figure 9a shows the aggregate effects of regressors grouped by their physical context. The additive component \(aHist\) represents the sum of linear and quadratic hydrostatic effects (\(H_{st}\) and \(H_{sl}^{2}\)), \(aTvec\) represents the sum of linear (\(T_{twcc}\)) and quadratic (\(T_{twcc}^{2}\)) thermal and size of crack openings effects, while component \(aH\) is the sum of hydrostatic effects of groundwater measured by piezometers P\({}_{3}\) and P\({}_{4}\). Component \(T_{twcc}\cdot H_{st}\) represents the interaction effects of appropriate PCs, and \(t\)--encompasses the effect of rheology on aging and deterioration of concrete (\(e^{-t}\)). Figure 8: Measured and modeled values of total losses during verification period 2015–2017. Figure 9: \(Cont.\)