## 3 Protection Method Combining the Stator and Rotor Currents  
### Protection Criterion  
Taking the VSPSGM of Hebei Fengning Pumped Storage Power Station as an example, its capacity is 300 MW; the pole-pairs number is 7; the rated speed is 428.6 r/min; the rotor speed range is \(-10\%\)\(\rightarrow\)\(+10\%\); and the power factor is 0.9. Other parameters of the VSPSGM are shown in Table 1[16]. The mathematical analytical simulation model is established based on the multiloop method [20]:  
\[p\{\begin{bmatrix} \begin{bmatrix}\mathbf{L_{ss}}&\mathbf{L_{sr}}\\ \mathbf{L_{rs}}&\mathbf{L_{rr}}\\ \end{bmatrix}\begin{bmatrix}\mathbf{I_{s}}\\ \mathbf{I_{r}}\\ \end{bmatrix} \}+ \begin{bmatrix}\mathbf{R_{s}}\\ \mathbf{R_{r}}\\ \end{bmatrix}\begin{bmatrix}\mathbf{I_{s}}\\ \mathbf{I_{r}}\\ \end{bmatrix} = \begin{bmatrix}\mathbf{U_{s}}\\ \mathbf{U_{r}}\\ \end{bmatrix}  \tag{17}\]  
where \(p\) is a differential operator; \(\mathbf{U_{s}}\) and \(\mathbf{U_{r}}\) are the branch voltage matrices of the stator and rotor, respectively; \(\mathbf{I_{s}}\) and \(\mathbf{I_{r}}\) are the branch current matrices of the stator and rotor, respectively; \(\mathbf{R_{s}}\) and \(\mathbf{R_{r}}\) are the branch resistance matrices of the stator and rotor, respectively [21]. \(\mathbf{L_{ss}}\) is the stator inductance matrix, \(\mathbf{L_{rr}}\) is the rotor inductance matrix, \(\mathbf{L_{sr}}\) and \(\mathbf{L_{rs}}\) are the mutual inductance matrices of the stator and rotor [22]. The detailed calculation process of each parameter is shown in [23]. Equation (17) is a time-varying ordinary differential equation system, which can be solved by the fourth-order Runge-Kutta algorithm [24]. When the slip ratio \(s\) is 0.1, it is assumed that a short-circuit fault occurs between the 11 th coil of the first branch of Phase A and the 8 th coil of the second branch of Phase A. The coil turn difference is 3 and the fault time is 35 s. Through simulation calculation, the instantaneous waveform of stator and rotor three-phase currents are obtained, as shown in Figure 2a,b, respectively. After the fault, the stator three-phase current is dominated by fundamental frequency, containing a small amount of 1 \(\pm\) 2 \(s\) harmonic components, and the rotor three-phase current changes greatly. The \(f(i_{\mathrm{d,s}},i_{\mathrm{q}\mathrm{s}})\) and \(h^{2}f(i_{\mathrm{d,r}},i_{\mathrm{q}\mathrm{r}})\) obtained by the Clark transformation are shown in Figure 3. The current values in the figure are both per-unit values. The reference value is the peak current of the stator phase. \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Stator** & **Rotor** \\ \hline Slot number (\(Z\)) & 252 & 294 \\ Winding form & Double-layer lap winding & Double-layer wave winding \\ Number of parallel branches & 4 & 2 \\ Number of coils per branch & 21 & 49 \\ The first pitch (\(y_{1}\)) & 15 & 21 \\ The second pitch (\(y_{2}\)) & 14 & 21 \\ Rated current/ A & 12,317 & 6400 \\ \hline \end{tabular}
\end{table} 
Table 1: The basic parameters of the VSPSGM. Figure 2: Three-phase currents of stator side and rotor side: (a) stator side; (b) rotor side. The simulation results in Figure 3 are consistent with the analysis results. After the fault occurs, \(f(i_{\text{d}s},i_{\text{q}s})\) and \(h^{2}f(i_{\text{d}r},i_{\text{q}r})\) have large differences. However, the protection cannot simply use whether the difference between the two \(g(t)\) is greater than a certain value as the basis for judging whether the rotor winding short-circuit fault occurs. There are two reasons: (1) \(g(t)\) will change dramatically over time, and the protection setting value is difficult to set; (2) in normal operation, if there is a large measurement error in the current measurement value, \(g(t)\) at that time may be too large, resulting in protection misoperation. Aiming at the above problem, it is proposed to judge whether the fault occurs according to the size of the area around \(f(i_{\text{d}s},i_{\text{q}s})\) and \(h^{2}f(i_{\text{d}r},i_{\text{q}r})\).