## 3 Results and Analysis
### Influence of Discharge Change Rate on Runaway Stability  
In general, the water head of a pump-turbine during transient process were main influenced by the discharge change rate \(dQ/dt\) and the hydraulic loss [12]. In order to identify the cause of above distinctive differences on dynamic trajectories between large GVOs and \(6^{\circ}\) GVO, the transient variations of discharge \(Q\), the discharge change rate \(dQ/dt\), the total pressure, and static pressure for the GVOs at \(15^{\circ}\) and \(6^{\circ}\) are shown in Figure 8. As shown in the figure, the change laws of static pressure fluctuations are consistent with the change laws of the discharge change rate \(dQ/dt\) for the two cases. However, the variation amplitudes of discharge change rate \(dQ/dt\) and static pressure for the \(15^{\circ}\) GVO condition are both approximately 3.5 times higher than those for the \(6^{\circ}\) GVO condition, which means that the flow fields inside the pump-turbine change more rapidly at larger GVO. In addition, the change laws of total pressure and static pressure for \(15^{\circ}\) GVO are similar (Figure 8a), while they are significantly different for \(6^{\circ}\) GVO (Figure 8b). This difference demonstrates that the sharp change in discharge make the static pressure dominate the water head for \(15^{\circ}\) GVO. If the discharge changes slowly, the change in kinetic pressure will have more influence on the transient total pressure, i.e., the water head. When the pump-turbine operates at part load, the kinetic pressure in the turbine will be enhanced by the development of vortex structures, which leads to energy dissipation with vortex breakdown. The traditional method for evaluating energy dissipation, i.e., hydraulic loss, in a pump-turbine is through the evaluation of the pressure drop. However, the pressure drop cannot intuitively reflect the influence of energy dissipation caused by local unstable flow patterns. According to the second law of thermodynamics, the excess hydraulic energy is dissipated into internal energy with the development of vortices, which is irreversible and leads to increased entropy [30; 31; 32]. Therefore, the local entropy production rate can be used to access the influence of the development of local vortices in the pump-turbine on local flow energy dissipation. Figure 8: Transient variations of operating parameters for \(15^{\circ}\) and \(6^{\circ}\) GVOs. (**a**) \(15^{\circ}\) GVO; (**b**) \(6^{\circ}\) GVO. The temperature in the flow of the pump-turbine is constant, and the temperature change is neglected. [PERSON] and [PERSON] (2018) proposed a model to calculate the local entropy production rate induced by the velocity fluctuation. In [PERSON]'s \(k\)-\(\omega\) SST turbulence model, the local entropy production rate induced by velocity fluctuation can be calculated by Equation (8) as  
\[Ep=\beta\frac{\rho\omega k}{T_{\rm e}} \tag{8}\]  
where \(\beta\) is the SST closure constant, given as \(\beta\) = 0.09, \(\omega\) is the turbulent eddy frequency (s\({}^{-1}\)), \(k\) is the turbulent energy(m\({}^{2}\)/s\({}^{2}\)), \(\rho\) is the fluid density(kg/m\({}^{3}\)), and \(T_{\rm e}\) is the temperature (K). Figures 9 and 10 show the distributions of entropy production rate \(Ep\) and total pressure coefficient \(P_{\rm T}\) for 6\({}^{\circ}\) GVO condition at some typical instants. As the operating point moves forward to the no-load point, the entropy production rate mainly increases in the guide-vane, around the runner inlet and near the draft tube wall (Figure 9), in which the flow separations and vortex structures will gradually emerge and develop. The region with high total pressure in the draft tube (Figure 10) is in accordance with the region with the high entropy production rate; however, the total pressure distribution cannot reflect the energy dissipation in the runner and guide-vane domain. Figure 11 shows a new dynamic trajectory (green line) by neglecting the influence of kinetic pressure in the water head for the 6\({}^{\circ}\) GVO condition.