## 3 Model of the Variable-Speed Pumped Storage Unit
It is obvious that the unit has seven pairs of poles, and the magnetic field is symmetrically distributed on the circumference. After the rotor winding has a short-circuit fault, the distribution of magnetic density and magnetic force line is as shown in Figure 6b. The short-circuit fault is located in the upper right corner of Figure 6b. It can be found that the magnetic field at the short circuit will cause serious distortion, and the magnetic line is no longer symmetrical. Take a short-circuit fault occurring in the 32 nd turn coil of branch A2 and the 34 th turn coil of branch A2 as an example. Using the finite element method for modeling and simulation, the spectrum diagram of the circulating current of phase A is obtained by FFT transform, as shown in Figure 7. Comparing Figures 5 and 7, it can be found that the characteristic frequencies of the two simulation results are the same, which is consistent with the theoretical analysis. The characteristic frequency amplit  
Figure 6: The finite element analysis model; (**a**) the distribution of magnetic density and magnetic force line during normal operation; (**b**) the distribution of magnetic density and magnetic force line after the rotor winding has a short-circuit fault. simulation methods are compared, as shown in Table 3. The maximum relative error of each harmonic component amplitude is \(-14.0553\%\). The accuracy of the multi-loop method is not as good as that of the finite element method, but we still chose the multi-loop method for the following considerations. According to the statistics of the number of rotor winding short-circuit faults, there are as many as thousands or even tens of thousands possible faults for an actual unit. It is inevitable to consider the time cost required for simulation when simulating these faults. The computer configuration for simulation included an Intel Core i7-11700 processor and 16.0 GB of RAM. For the multi-loop method, it takes about \(34\) s to simulate a fault. For the finite element method, it takes about \(2812\) s to simulate a fault. In addition, each fault simulated by the finite element method needs to change the wiring of the fault location in the Maxwell circuit design, which requires the participation of personnel. It can be estimated that it is difficult to simulate all faults by the finite element method. The influence of the simulation error of the multi-loop method on the proposed protection is further analyzed. Combined with the data shown in Table 3, the protection action values corresponding to the two simulation methods are calculated by Equation (8). The protection action value corresponding to the multi-loop method is \(0.163\) p.u, and that corresponding to the finite element method is \(0.169\) p.u. The relative error is \(-3.55\%\), and the error is small. In addition, the protection action value of the multi-loop method is  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**The Characteristic** & **Amplitude Obtained by** & **Amplitude Obtained by Finite** & **Relative Error** \\
**Frequency (Hz)** & **Multi-Loop Method (p.u)** & **Element Method (p.u)** & **(\%)** \\ \hline
1.43 & 0.0729 & 0.0738 & \(-1.2195\) \\
7.86 & 0.0839 & 0.0841 & \(-0.2378\) \\  
smaller than that of the finite element method, which makes the analysis result of the multi-loop method more conservative than that of the finite element method. In the multi-loop method, the calculation formulas of the inductance parameters of stator and rotor windings are Equations (13)-(19). The inductance parameter is inversely proportional to the equivalent air gap length. In the modeling, the actual air gap length is set as the equivalent air gap length. That is, the multi-loop method ignores the core reluctance and only considers the air gap reluctance. However, both the core reluctance and air gap reluctance are considered in the finite element method. This makes the equivalent air gap length in the multi-loop method smaller, which leads to the calculated inductance parameters being larger.