## 4 Conclusions  
Three-dimensional transient numerical simulations were carried out on the runaway processes of a low specific-speed pump-turbine at four GVOS. The evolution of BFVSs at the runner inlet was investigated in depth. Further, entropy production theory was used to analyze the dissipation characteristics of different types of BFVSs and their impacts on the runaway stability. The internal mechanism of runaway oscillation instability was obtained. A design criteria is proposed to extend the stable runaway operating range.  
During the runaway oscillation process, BFVSs occur at the runner inlet for all GVOs, which results in severe separations and complicated vortices in the guide-vane and runner domains. For the large GVO condition, the energy dissipations have a sharp change around the no-load points with the location of BFVSs at the runner inlet changing between hub side and midspan, which makes the discharge change rapidly. As a result, the slopes of dynamic trajectories at the no-load point become positive, so the runaway oscillation is self-excited. Conversely, for the small GVO condition, the BFVSs only occur at the midspan of the runner inlet, and the change rate of energy dissipation is very small around the no-load points, which makes the discharge change slightly. Consequently, the slopes of dynamic trajectories at the no-load points become negative, and the runaway oscillation is therefore damped. If the occurrence of BFVSs at hub side of the runner inlet the runner is suspended, a more stable pump-turbine for runaway process can be obtained. Experimental verifications will be conducted in future work.  
Figure 16: The dynamic characteristic between the improved and the original cases during runaway process at 9\({}^{\circ}\) GVO. (**a**) Transient parameters; (**b**) transient radial velocities; (**c**) dynamic trajectories.  
**Author Contributions:** [PERSON] conceived and designed the study and wrote the paper; [PERSON] performed the study; [PERSON] analyzed the data.  
**Funding:** This research received no external funding.  
**Acknowledgments:** This work was supported by the National Natural Science Foundation of China (Grant No. 51579187, 51839008), the Technology Research Program of the National Science and Technology Research Center for Inland Waterway Regulation Engineering (grant number NERC2014A05), the Chongqing Research Program of Application Foundation and Advanced Technology (grant number cstc2016 icyjA1935, cstc2016 icyjA1937), the Science and Technology Project of the Chongqing Education Committee (grant number KJ1600514, KJ1600535, KJQN201800712), and the Open Fund Project of the Key Laboratory for Inland Waterway Regulation Engineering of Ministry of communication in Chongqing Jiaotong University (grant number NHHD-201505, NHHD-201513).  
**Conflicts of Interest:** The authors declare no conflict of interest.  
**Nomenclature**  
\begin{tabular}{l l} \(T\) & Torque (N\(\cdot\)m) \\ \(n\) & Rotational speed (rpm) \\ \(Q\) & Rotational inertia (kg\(\cdot\)m\({}^{2}\)) \\ \(g\) & Gravitational acceleration (m\(\cdot\)s\({}^{-2}\)) \\ \(D_{1}\) & Runner inlet diameter (m) \\ \(H\) & Water head (m) \\ \(\rho\) & Water density (kg\(\cdot\)m\({}^{-3}\)) \\ \(t\) & Time (s) \\ \(\mu_{\text{ED}}\) & Speed factor (\(nD_{1}/\left(60\sqrt{gH}\right)\)) \\ \(Q_{\text{ED}}\) & Discharge factor (\(Q/\left({D_{1}}^{2}\sqrt{gH}\right)\)) \\ \(T_{\text{ED}}\) & Torque factor (\(T/\left(\rho D_{1}^{3}gH\right)\)) \\ \(P_{\text{T}}\) & Total pressure (Pa) \\ \(P_{\text{N}}\) & Pressure coefficient (\(P/0.5\rho\) (\(\tau nD_{1}/60\))\({}^{2}\)) \\ \(GVO\) & Guide vane opening \\ \(n_{\text{s}}\) & Specific speed \(nQ^{0.5}/H^{0.75}\) \\ BFVS & Backflow vortex structure \\ \(V_{r}\) & Radial velocity (m\(\cdot\)s\({}^{-1}\)) \\ \(V_{rN}\) & Normalized radial velocity (\(V_{r}/\pi n_{0}D_{1}/30\)) \\ \end{tabular}