## 3 Model of the Variable-Speed Pumped Storage Unit
The calculation methods of inductance matrices \(\mathbf{L_{ss}}\), \(\mathbf{L_{sr}}\), \(\mathbf{L_{sr}}\), and \(\mathbf{L_{rr}}\) are as follows. Assume that the air gap is uniform. According to the analysis of air-gap permeance and taking the single-turn coil as the research object, the mutual inductance coefficient between the stator coil \(i\) and the coil \(j\) is  
\[M_{i,j}^{\mathrm{s}}=\frac{4w_{s}^{2}\tau l\mu_{0}}{P\pi^{2}\delta}\sum_{k} \frac{(\kappa_{sk})}{k}^{2}\cos[k(i-j)\alpha_{s}] \tag{13}\]  
where \(w_{s}\) is the number of turns of a single stator coil, \(\tau\) is the pole pitch, \(l\) is the core length, \(\mu_{0}\) is the permeability of vacuum, \(P\) is the number of pole-pairs, \(k\) is the harmonic order, \(k=1/P,2/P,3/P,\ldots,\delta\) is the equivalent air gap length, \(\alpha_{s}\) is the stator electric angle, and \(k_{sk}\) is the stator side short distance coefficient of \(k\)th harmonic. In particular, when \(i=j\), Equation (13) is converted to the self-inductance coefficient of the coil \(i\), which is expressed as:  
\[M_{i,i}^{\mathrm{s}}=\frac{4w_{s}^{2}\tau l\mu_{0}}{P\pi^{2}\delta}\sum_{k} \left(\frac{k_{sk}}{k}\right)^{2} \tag{14}\]  
If \(M_{H(x),J(t)}^{\mathrm{s}}\) is used to represent the mutual inductance between the coil \(x\) of the loop \(H\) and the coil \(t\) of the loop \(J\), then the mutual inductance \(M_{HJ}^{\mathrm{s}}\) between the stator loop \(H\) and the loop \(J\) is expressed as:  
\[M_{HJ}^{\mathrm{s}}=\sum_{r=1}^{m}\sum_{t=1}^{n}M_{H(x),J(t)}^{\mathrm{s}} \tag{15}\]  
where \(m\) is the number of coils in loop \(H\), and \(n\) is the number of coils in loop \(J\). When loop \(H\) and loop \(J\) are the same loop, the calculation result is the self-inductance coefficient of this loop. Since the rotor is also winding, the calculation method of the inductance coefficient of the rotor side is similar to that of the stator side. The mutual inductance coefficient between the rotor coil \(i\) and coil \(j\) is:  
\[M_{i,j}^{\tau}=\frac{4w_{r}^{2}\tau l\mu_{0}}{P\pi^{2}\delta}\sum_{k}\left( \frac{k_{rk}}{k}\right)^{2}\cos[k(i-j)\alpha_{r}] \tag{16}\]  
where \(w_{r}\) is the number of turns of a single rotor coil, \(\alpha_{r}\) is the rotor electric angle, and \(k_{rk}\) is the rotor side short distance coefficient of \(k\)th harmonic. Similarly, the mutual inductance coefficient \(M_{HJ}^{\mathrm{r}}\) between the rotor loop \(H\) and loop \(J\) is expressed as:  
\[M_{HJ}^{\mathrm{r}}=\sum_{r=1}^{m}\sum_{t=1}^{n}M_{H(x),J(t)}^{\mathrm{r}} \tag{17}\]  
where \(m\) is the number of coils in loop \(H\), and \(n\) is the number of coils in loop \(J\). Similarly, the mutual inductance between stator coil \(i\) and rotor coil \(j\) is:  
\[M_{i,j}^{sr}=\frac{4w_{s}w_{r}\tau l\mu_{0}}{P\pi^{2}\delta}\sum_{k}\frac{k_{sk}} {k}\frac{k_{rk}}{k}\cos[k(\gamma+i\alpha_{s}-j\alpha_{r})] \tag{18}\]  
where \(\gamma\) is the rotor position angle, which varies with the rotor rotation. The mutual inductance coefficient \(M_{HJ}^{sr}\) between the stator loop \(H\) and the rotor loop \(J\) is expressed as:  
\[M_{HJ}^{sr}=\sum_{r=1}^{m}\sum_{t=1}^{n}M_{H(x),J(t)}^{sr} \tag{19}\]  
The above inductance parameters \(M_{HJ}^{s}\), \(M_{HJ}^{r}\) and \(M_{HJ}^{sr}\) are the elements in the inductance matrices \(\mathbf{L_{ss}}\), \(\mathbf{L_{sr}}\), \(\mathbf{L_{sr}}\), and \(\mathbf{L_{rr}}\). Take a 300 MW variable-speed pumped storage unit with seven pairs of poles (\(P=7\)) as an example. The basic parameters of the unit are shown in Table 1, and the stator branch wiring diagram is shown in Figure 2. Each phase of the stator consists of four branches in parallel. In the figure, A, B, and C are phases, \(1,2,3,4\) are branch numbers. CT represents a current transformer. CT2 measures the sum current of branch A1 and branch A2, and CT3 measures the sum current of branch A3 and branch A4. The current difference between CT2 and CT3 is the circulating current of Phase A, and the split-phase transverse difference current of Phase A.