## 2 Characteristics Analysis of Stator and Rotor Current
### Normal Operation  
Without considering the internal structural asymmetry of the generator-motor, the stator three-phase voltages are symmetrical. The stator three-phase currents also are symmetrical and only contain the fundamental-frequency positive-sequence component. It can be recorded as:  
\[ \begin{array}{l}i_{\rm a,s}(t)=I_{\rm s}\cos(\omega_{\rm s}t+\varphi_{\rm as}) \\ i_{\rm b,s}(t)=I_{\rm s}\cos(\omega_{\rm s}t-\frac{2\pi}{3}+\varphi_{\rm as})\\ i_{\rm c,s}(t)=I_{\rm s}\cos(\omega_{\rm s}t+\frac{2\pi}{3}+\varphi_{\rm as}) \end{array}  \tag{1}\]  
where \(I_{\rm s}\) is the amplitude of the stator phase current, \(\omega_{\rm s}\) is the stator angular frequency, and \(\varphi_{\rm as}\) is the stator A-phase current angle. Clark transformation is a commonly used coordinate-transformation method in the electrical analysis, which can convert the three-phase abc coordinate system to the two-phase stationary dq coordinate system. The transformation matrix is [17]:  
\[\left[ \begin{array}{c}i_{\rm a,s}(t)\\ i_{\rm q,s}(t)\end{array} \right]=\frac{2}{3}\!\left[ \begin{array}{ccc}1&- \frac{1}{2}&-\frac{1}{2}\\ 0&\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}\end{array} \right]\!\left[ \begin{array} []{c}i_{\rm a,s}(t)\\ i_{\rm b,s}(t)\\ i_{\rm c,s}(t)\end{array} \right] \tag{2}\]  
Substituting Equation (1) into Equation (2), the stator-side currents \(i_{\rm d,s}\) and \(i_{\rm q,s}\) of the variable-speed generator-motor in the dq coordinate system can be obtained as follows:  
\[ \begin{array}{l}i_{\rm d,s}(t)=I_{\rm s}\cos(\omega_{\rm s}t+\varphi_{\rm as })\\ i_{\rm q,s}(t)=I_{\rm s}\sin(\omega_{\rm s}t+\varphi_{\rm as})\end{array}  \tag{3}\]  
From Equation (3), the amplitudes of \(i_{\rm d,s}(t)\) and \(i_{\rm q,s}(t)\) are equal, and the phase difference is 90 degrees, so the following can be obtained:  
\[f(i_{\rm d,s},i_{\rm q,s})=i_{\rm d,s}^{2}(t)+i_{\rm q,s}^{2}(t)=I_{\rm s}^{2} \tag{4}\]  
It can be inferred that when the generator-motor is in normal operation, the corresponding dq current instantaneous value can be obtained by performing the Clark transformation on the instantaneous value of the stator three-phase current at any time. The \(f(i_{\rm d,s},i_{\rm q,s})\) obtained by Equation (4) is always equal to the square of the amplitude of the stator-side phase current. Similarly, the three-phase currents on the rotor side of the VSPSGM under normal operation only contains a positive sequence component, which can be denoted as:  
\[ \begin{array}{l}i_{\rm a,r}(t)=I_{\rm r}\cos(\omega_{\rm r}t+\varphi_{\rm ar })\\ i_{\rm b,r}(t)=I_{\rm r}\cos(\omega_{\rm r}t-\frac{2\pi}{3}+\varphi_{\rm ar}) \\ i_{\rm c,r}(t)=I_{\rm r}\cos(\omega_{\rm r}t+\frac{2\pi}{3}+\varphi_{\rm ar}) \end{array}  \tag{5}\]  
Figure 1: Schematic diagram of VSPSGM connected to the power grid. where \(I_{\rm r}\) is the amplitude of the rotor phase current, \(\omega_{\rm r}\) is the rotor angular frequency, \(\omega_{\rm r}=s\omega_{\rm s},s\) is the slip ratio, the slip ratio range of the VPSGM is usually between \(\pm 0.1\), and \(\varphi_{\rm ar}\) is the rotor A-phase current angle. After the Clark transformation, the rotor-side currents \(i_{\rm d.r}\) and \(i_{\rm q.r}\) can be obtained. Similar to Equation (4), \(f(i_{\rm d.r},i_{\rm q.r})\) can be constructed. When the generator-motor is in normal operation, \(f(i_{\rm d.r},i_{\rm q.r})\) is equal to the square of the amplitude of the rotor-side phase current. \[f(i_{\rm d.r},i_{\rm q.r})=i_{\rm d.r}^{2}(t)+i_{\rm q.r}^{2}(t)=I_{\rm r}^{2} \tag{6}\]  
The VPSGM is essentially an induction motor. Its operating principle is similar to the transformer. Therefore, the voltage and current at the stator and rotor sides satisfy the variable ratio relationship [18]:  
\[ \begin{array}{l}\frac{E_{\rm s}}{E_{\rm r}}=\frac{1}{s}\frac{N_{\rm s}k_{\rm w }}{N_{\rm e}k_{\rm w}}\\ \frac{I_{\rm s}}{I_{\rm r}}=\frac{N_{\rm e}k_{\rm w}}{N_{\rm e}k_{\rm w}}\end{array}