## 5 Case Study
### Influence of the AC Infeed  
Taking into account the AC infeed during the fault, the energy is well balanced between the AC side and the DC side:  
\[E_{\mathrm{dc}}=E_{\mathrm{ac}}+EL_{\mathrm{CONV}} \tag{23}\]where, \(E_{\rm dc}\) represents the energy storage of DC side including the energy in lines and reactors. \(E_{\rm ac}\) represents the energy input from the AC system, and \(E_{\rm CONV}\) represents the released energy of an MMC converter. \(E_{\rm L}\) is the released energy of the phase reactors. \(E_{\rm dc}\) and \(E_{\rm ac}\) can be calculated as follows:  
\[ \begin{array}{l}E_{\rm dc}=\int_{t_{0}}^{t_{0}+\Delta t}p_{\rm dc}dt\\ E_{\rm ac}=\int_{t_{0}}^{t_{0}+\Delta t}p_{\rm ac}dt\end{array}  \tag{24}\]  
where, \(t_{0}\) is when the fault occurs. \(\triangle t\) is the calculation time. Since the equivalent resistance of the arm is usually small, the energy change of the resistance can be ignored. The energy storage components of the MMC mainly include the sub-module (SM) capacitors. The total stored energy of the SM capacitors and the inductors are respectively listed as follows. \[E\frac{1}{2}\sum_{j=a,b,c}\frac{2C_{0}}{N}\Big{(}\Big{(}u_{pj}^{2}(t_{0}+ \Delta t)-u_{pj}^{2}(t_{0})\Big{)}+\Big{(}u_{ij}^{2}(t_{0}+\Delta t)-u_{ij}^{2} (t_{0})\Big{)}\Big{)}_{\rm CON} \tag{25}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Items** & **Values** \\ \hline \hline \multicolumn{2}{c}{Converter Stations} \\ Rated AC voltage (kV) & 230 (s\({}_{1,}\)s\({}_{3}\))/525 (s\({}_{2,}\)s\({}_{4}\)) \\ Transformer impedance(p.u.) & 0.0176 (s\({}_{1,}\)s\({}_{3}\))/0.0088 (s\({}_{2,}\)s\({}_{4}\)) \\ Smoothing reactor(mH) & 300 \\ Rated capacity(MW) & 1500 (s\({}_{1,}\)s\({}_{3}\))/3000 (s\({}_{2,}\)s\({}_{4}\)) \\ Rated DC voltage & \(\pm\)500 kV \\ Arm inductor(mH) & 40 (s\({}_{1,}\)s\({}_{3}\))/75 (s\({}_{2,}\)s\({}_{4}\)) \\ SM capacitor(uF) & 10000 (s\({}_{1,}\)s\({}_{3}\))/15000 (s\({}_{2,}\)s\({}_{4}\)) \\ Number of SMs per arm & 218 \\ \multicolumn{2}{c}{DC transmission lines} \\ Resistance per unit length & 0.009735 \(\Omega\)/km \\ Inductance per unit length & 0.0176 mH/km \\ Capacitance per unit length & 0.001367 \(\mu\)F/ km \\ Control mode & \\ s\({}_{1}\) & \(Q=0\) MVar, \(P=-1500\) MW \\ s\({}_{2}\) & \(Q=0\) MVar, \(P=-3000\) MW \\ s\({}_{3}\) & \(Q=0\) MVar, \(P=1500\) MW \\ s\({}_{4}\) & \(Q=0\) MVar, \(U_{\rm dc}=\pm 500\) kV \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main Circuit Parameters of the Test System. Figure 7: Schematic Diagram of the Zhangbei \(\pm\)500 kV bipolar DC grid. \[E=\frac{\frac{\text{Larm}}{2}}{2}\left(\sum\limits_{j=a,b,c}\left(i_{pj}^{2}(t_{0}+ \Delta t)-i_{pj}^{2}(t_{0})\right)+\sum\limits_{j=a,b,c}\left(i_{j}^{2}(t_{0}+ \Delta t)-i_{jj}^{2}(t_{0})\right)\right)_{L} \tag{26}\] \[+\frac{\frac{L_{dc}}{2\left(i_{dc}^{2(t_{0}+\Delta t)}i_{dc}^{2(t _{0})}\times 2\right)}}{2\left(i_{dc}^{2(t_{0}+\Delta t)}i_{dc}^{2(t_{0})} \times 2\right)}\]  
According to the analysis of power transfer among the AC side, DC side and MMC's internal components, the influence of the AC active power on the fault currents and voltages could be studied. Based on Equations (23)-(26), the energy changes of the DC side, MMC internal components and AC side of each converter station within 10 ms after the fault occurs with and without AC infeed are shown in Figure 8.  
It could be seen from Figure 8 that whether the AC infeed is taken into account or not, the energy of the SM capacitors changes the most. For sending end converter stations (such as s\({}_{1}\) and s\({}_{2}\)), the energy released by the SM capacitors is reduced when there is AC infeed. For receiving end converter stations (such as s\({}_{3}\) and s\({}_{4}\)), the energy released by the SM capacitors increases when the AC side absorbs energy from the DC system. As a dynamic energy balance between the AC side and the SMs is maintained in the whole process, the energy released to the DC side changes slightly.