## 3 Model of the Variable-Speed Pumped Storage Unit
At this point, the transient circulating current of each phase measured by the split-phase transverse differential protection is as shown in Figure 4. Taking phase A as an example, FFT transformation is performed on the circulating current of phase A to obtain the spectrum diagram, as shown in Figure 5. In Figures 4 and 5, the normalized reference values are both the amplitude of the 50 Hz stator phase current. The characteristic frequencies are consistent with the theoretical analysis values in Table 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**The Harmonic Magnetic Field with** & **Frequency of the Circulating** & **Frequency of the Circulating** \\
**Order \(k\)** & **Current/Hz** & **Current/Hz** \\  & **50[k(1\(-\)s)+s]** & **50[k(1\(-\)s)\(-\)s]** \\ \hline
1/7 & 11.43 & 1.43 \\
2/7 & 17.86 & 7.86 \\
3/7 & 24.28 & 14.28 \\
4/7 & 30.71 & 20.71 \\
5/7 & 37.14 & 27.14 \\
6/7 & 43.57 & 33.57 \\
8/7 & 56.43 & 46.43 \\
9/7 & 62.86 & 52.86 \\
10/7 & 69.29 & 59.29 \\
11/7 & 75.71 & 65.71 \\
12/7 & 82.14 & 72.14 \\
13/7 & 88.57 & 78.57 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The theoretical characteristic frequency of the circulating current between stator branches. Figure 4: The transient circulating current of each stator phase (the inter-turn short circuit fault occurs at \(t=35.0\) and in rotor branch A2). Figure 5: Spectrum diagram of the circulating current of phase A obtained by the multi-loop method. Figure 3: The transient current of waveform stator branches (the inter-turn short circuit fault occurs at \(t=35.0\) and in rotor branch A2). The finite element method is another recognized effective simulation method that is different from the multi-loop method used in this paper. The finite element analysis method is used to model the actual variable-speed pumped storage unit. Considering the boundary conditions, the equation for solving the nonlinear time-varying electromagnetic field is:  
\[ \begin{array}{c}\frac{\partial}{\partial x}\left(\frac{1}{\mu}\frac{\partial A _{z}}{\partial x}\right)+\frac{\partial}{\partial y}\left(\frac{1}{\mu}\frac{ \partial A_{z}}{\partial y}\right)=-J_{z}+\sigma\frac{\mathrm{d}A_{z}}{ \mathrm{d}I}\\ S:A_{Z}=0\end{array}  \tag{20}\]  
where \(A_{z}\) is the \(z\)-axis component of the vector magnetic potential, \(J_{z}\) is the source current density, \(\mu\) is the permeability of the material, \(\sigma\) is the electrical conductivity of the material, and \(S\) is the outer diameter of the stator core. The stator winding has four branches in each phase that are connected by double-layer winding. The rotor winding is connected by double-layer wave winding with two branches per phase. The winding connection form is complex, not symmetrical. In addition, the short-circuit fault destroys the symmetry of the magnetic field when the rotor winding short-circuit fault occurs. Therefore, the entire electrical machine must be simulated; it cannot be decomposed into 1/2 units or 1/4 units for simulation. To ensure that the coil connection sequence of stator and rotor windings in the simulation model is consistent with the actual unit, the Maxwell circuit design is used to set the coil connection sequence of stator and rotor windings. In addition, the rotor winding short-circuit fault is also based on the switch element setting in Maxwell circuit design. The materials of stator and rotor cores are silicon steel sheets, and the model is \"DW315_50\". The materials of stator and rotor windings are copper, and the model is \"copper_75C\". The material of the air gap is the vacuum, and the model is \"vacuum\". The solution algorithm is Backward Euler, which is an implicit solution method with better convergence. The nonlinear residual is 0.0001, which ensures that the simulation results have higher accuracy. During the normal operation of the variable-speed pumped storage unit, the distribution of magnetic density and the magnetic force line is as shown in Figure 6a.