## 3 Setting and Theoretical Prediction
### Theoretical Prediction  
The rotor-stator interaction between the runner and the guide vanes is the main reason for the high frequency fluctuation in the runner. In this paper, according to the diameter modulus theory proposed by [PERSON] [24] and the method for hydraulic excitation in the pump-turbine, analyzed by [PERSON] [25], considering the special structure of the long and short blades, the relatively prominent rotor-stator interaction frequency in the pump-turbine with long and short blades is predicted from the theoretical level. Low-order and high-order hydraulic excitation modes can be induced by the rotor-stator interaction effect, dominated by potential flow. The modes can be expressed by the node diameter number \(k\). The expressions of the minimum and maximum node diameters are as follows:  
\[k_{1} = m\cdot Z_{r}-n\cdot Z_{g} \tag{3}\] \[k_{2} = m\cdot Z_{r}+n\cdot Z_{g} \tag{4}\]  
where \(m\) and \(n\) are arbitrary integers, \(Z_{r}\) is the number of runner blades (\(5+5\)), and \(Z_{g}\) is the number of guide vanes (16). The number of high-pressure and low-pressure regions in the circumferential direction of a specific frequency is represented by \(k\), whose pressure distribution is affected by the rotation of the runner. It also has a specific rotation speed. The angular velocity is calculated as follows. In a stationary system, the calculation is  
\[\omega_{1} = m\cdot Z_{r}\cdot\omega/k_{1} \tag{5}\] \[\omega_{2} = m\cdot Z_{r}\cdot\omega/k_{2} \tag{6}\]  
In a rotating system, the calculation is  
\[\omega_{1} = n\cdot Z_{g}\cdot\omega/k_{1} \tag{7}\] \[\omega_{2} = n\cdot Z_{g}\cdot\omega/k_{2} \tag{8}\]  
where \(\omega\) is the speed of the runner. It can be seen from Equations (5-8) that the positivity of \(k\) determines the rotation direction of the pressure field. When \(k\) is positive, the rotation direction is consistent with the rotation direction of the runner, and vice versa. The action frequency of hydraulic excitation, caused by rotor-stator interaction on the runner, is shown as the guide van overcurrent frequency \(f_{r}\) and its frequency multiplication,  
Figure 7: Monitoring points in the volute. while the action frequency in the guide vane stationary domain is shown as the overcurrent frequency \(f_{\rm s}\) and its frequency multiplication of the runner blade. The expression is as follows:  
\[f_{r} = n\cdot Z_{g}\cdot f_{n} \tag{9}\] \[f_{\rm s} = m\cdot Z_{r}\cdot f_{n} \tag{10}\]  
Table 3 shows the rotor-stator interaction frequencies at different node-diameter numbers. In general, the greater the absolute value of \(k\), the smaller the corresponding amplitude, and the smaller the influence of rotor-stator interaction. The frequency corresponding to the smaller absolute value k\({}_{1}\) is listed in the table, and the minimum diameter k\({}_{1}\) was 2, \(-\)2, 4, and \(-\)6, in order from the smallest absolute value to the largest. Although the absolute value was smallest when k\({}_{1}=\pm 2\), the corresponding frequency was larger, and the amplitude was lower than that when k\({}_{1}=4\) and \(-\)6. Therefore, the rotor-stator interaction frequency acting in the static system was mainly 10 f\({}_{\rm n}\) and 20 f\({}_{\rm n}\), and the rotation speeds of their respective pressure fields were \(-\)1.67\(\omega\) and 5\(\omega\), respectively. Its amplitude cannot be determined in theory, so we need to judge it according to the calculation results. The rotor-stator interaction frequency in the rotation system was mainly 16 f\({}_{\rm n}\), but the pressure field under a different node-diameter number had two different kinds of rotation speed, \(-\)2.67\(\omega\) and 4\(\omega\). In addition, due to the special design of long and short blades, flows at the outlet of long blades and short blades are significantly different, leading to differences in rotor-stator interaction between stay guide vanes. Thus, 5 f\({}_{\rm n}\) and its frequency doubling of a single type of blade can be generated.