# Application of MMC-RPC in High-Speed Railway Traction Power Supply System Based on Energy Storage
## 3 VSG Control
### Small Signal Analysis of VSG  
VSG control has a good effect on improving the low inertia for power electronic converters. In order to observe the im-pact of control parameters on stability, small-signal modeling and analysis of the VSG control under small disturbance are carried out. Figure 10 shows the VSG small signal analysis model, in which \(\Delta P_{\text{m}}\) is the input, and \(\Delta P\) is the output. The small signal disturbance of VSG variables near the steady-state value is substituted into the VSG mathematical model, the steady-state quantity and the disturbance above  
Figure 8: Improved stator electrical equation. Figure 10: VSG small signal analysis model. the second order are omitted, and then the Laplace transform is carried out to obtain the following equations:  
\[\left\{ \begin{array}{l}\int\omega s\Delta\omega=\hat{p}_{\text{m}}-\hat{p}-D \hat{\omega}\\ \Delta E=-k_{\text{PI}}\hat{q}\\ \hat{p}\hat{\bar{m}}=\hat{\omega}/\text{s}\\ \bar{p}=\hat{T}_{\text{e}}\omega_{0}\end{array} \right., \tag{18}\]  
When the active power of VSG acts alone, the transfer function of VSG small signal analysis model can be obtained by combining with Equation (18):  
\[G(\text{s})=\frac{\Delta P}{\Delta P_{\text{s}}}=\frac{\text{LU}_{\text{s}}} {\frac{\text{LU}_{\text{s}}+\text{l}}{\text{LU}_{\text{s}}+\text{l}(\omega \text{L})\omega_{\text{n}}\text{s}^{\text{s}}+D(\omega\text{L})\omega_{\text{n }}\text{s}}}\, \tag{19}\]  
According to Equation (19) and its constraint equation, the Bode diagram of the active power transfer function is drawn. As shown in Figure 11:  
It can be seen that with the increase in the \(J\) parameter, margins of both magnitude-frequency and phase-frequency diagrams are becoming larger, showing better stability. However, if the inertia \(J\) is too large, the dynamic performance of the system will be degraded and the adjustment speed will be slowed down, which is not conducive to system stability. According to Section 3.2, an improved stator electrical equation is proposed in this study. The voltage outer loop adopts PI-QPR compound control, in which the PI controller is used to realize the error-free tracking of the DC component in the d-axis and q-axis voltage components, and the output current command is the DC component. QPR controller is used to realize the zero-error tracking of the double frequency AC component, and the output current command is the AC component. Since the dq axis component of the inductance current and the dq axis output current command by the voltage outer loop are the same in the steady state, the voltage loop output current command can be directly used as the current input of the stator electrical equation. Taking d axis as an example, \(u_{\text{dref}}\) is the reference voltage of the d axis, \(i_{\text{dref}}\) is the reference value of inductance current at the inverter side, \(u_{\text{d}}\) is the output voltage of the d axis, \(i_{\text{d}}\) is the output current of d axis, and \(i_{\text{L},\text{D}}\) is the feedback value of inductance current, as shown in Figure 12[26]. Figure 11: Bode diagram of the active power transfer function. In Figure 12, the transfer function of PI control is [26]:  
\[G_{\rm I}(s)=K_{\rm P}+\frac{K_{\rm I}}{s}, \tag{20}\]  
where:  
\(K_{\rm P}\), \(K_{\rm I}\) are the proportional integral regulation parameters. The transfer function of QPR controller is:  
where [26]:  
\[G_{\rm QPR}(s)=K_{\rm P}+\frac{2K_{\rm R}\omega_{\rm c}s}{s^{2}+2\omega_{\rm c }s+\omega_{\rm o}^{2}}, \tag{21}\]  
where:  
1. \(\omega_{\rm c}\) is the cut-off frequency of QPR;
2. \(\omega_{\rm o}\) is the resonant frequency.