## 3 Results and Discussion
### A Regulation of the Hydraulic System through the Defined Mathematical Model
After the system stabilised (after 1400 s), a repeated increase in \(\zeta_{\rm eq}\) was simulated at a rate of double the amount of the previous decrease. That change caused a short-term increase in the regulated pressure \(p_{\rm V2}\) and a  
Figure 6: The curves of the operational parameters in Model Example 1 with the value of coefficient \(\varepsilon\) = 0; (**a**) the curves of \(p_{1},p_{2}\) pressures and \(\zeta_{\rm eq}\); (**b**) the regulated pressure of water at the inlet into the facility in the V2 Branch; (**c**) flow rates \(Q_{1}\) and \(Q_{4}\); (**d**) the regulated flow rate \(Q_{3}\). decrease in pressures \(p_{1}\) and \(p_{3}\) and in flow rate \(Q_{1}\). After 1700 s, the flow rates were stabilised with an unchanged regulated pressure \(p_{\rm V2}\) and flow rate \(Q_{3}\). During the 20.81% decrease in flow rate \(Q_{1}\), there was a short-term change in the regulated pressure \(p_{\rm V2}\) by 6.5% and a \(-\)5.3% change in flow rate \(Q_{3}\). For the purpose of comparison, Figure 7 shows the curves of pressures and flow rates with the coefficient \(\varepsilon=5\), at which the regulation was carried out with the use of the difference between the measured and the required values of pressure \(p_{\rm V2}\). A significant difference that was observed with the coefficient \(\varepsilon\) being set to 5, compared to the previous example, was the change in the curves of pressures \(p_{1}\) and \(p_{3}\), which in the regulation apply the difference between the measured and the preset value of pressure \(p_{\rm V2}\). This resulted in a significant decrease in the short-term deviation of the regulated pressure \(p_{\rm V2}\), which amounted to \(-\)0.77% (after 827 s) and 1.18% (after 1430 s). At the flow rate \(Q_{3}\), only a negligible change in the short-term deviation was observed when compared to the previous case. In Model Example 2, the curve of the value of the equivalent local loss was simulated with a sine curve, representing the alternating changes in the flow rate in the V2 Branch and in time. The \(\zeta_{\rm eq}\) curve was simulated with an amplitude of 200 and an interval of 630 s. Figure 8 shows the curves of pressures and flow rates with the preset coefficient \(\varepsilon=0\). The deviation of pressure \(p_{\rm V2}\) in the Model Example 2 with coefficient \(\varepsilon=0\) was \(\pm\)2.3%, while the deviation of flow rate \(Q_{3}\) was \(\pm\)1.8%. When the coefficient \(\varepsilon\) was increased to 5, the deviation of pressure \(p_{\rm V2}\) decreased to \(\pm\)0.39% (Figure 9). Figure 7: The curves of the operational parameters in Model Example 1 with the value of coefficient \(\varepsilon=5\); (**a**) the curves of \(p_{1},p_{2}\) pressures and \(\zeta_{\rm eq}\); (**b**) the regulated water pressure at the inlet into the facility in the V2 Branch; (**c**) flow rates \(Q_{1}\) and \(Q_{4}\); (**d**) the regulated flow rate \(Q_{3}\). Figure 8: The curves of the operational parameters in Model Example 2 with the value of coefficient \(\varepsilon=0\); **(a)** the curves of \(p_{1}\), \(p_{2}\) pressures and \(\zeta_{\text{eq}}\); **(b)** the regulated water pressure at the inlet into the facility in the V2 Branch; **(c)** flow rates \(Q_{1}\) and \(Q_{4}\); **(d)** the regulated flow rate \(Q_{3}\). Figure 9: The curves of the operational parameters in Model Example 2 with the value of coefficient \(\varepsilon=5\); **(a)** the curves of \(p_{1}\), \(p_{2}\) pressures and \(\zeta_{\text{eq}}\); **(b)** the regulated water pressure at the inlet into the facility in the V2 Branch; **(c)** flow rates \(Q_{1}\) and \(Q_{4}\); **(d)** the regulated flow rate \(Q_{3}\). In order to point out the regulation possibility in the standard operation, local loss factor \(\zeta_{\text{eq}}\) was subjected to the stochastic signal with a maximum absolute value of 1343 and a minimal value of 1200 (Figure 10). The local loss factor ranged from +5.3% to \(-5.9\%\).