# Calculation of Main Circuit Steady-State Parameters for Capacitor Commutated Converter System
## 3 Calculation Model and Method of the Main Circuit Parameters
### Determination of the Capacitance Value
#### 3.1.2 Steady-State Mathematical Model of the CCC
\tag{18}\]  
The average value of the DC voltage over a period is:  
\[U_{\rm d}=\frac{3}{\pi}\big{(}\int_{0}^{\mu}u_{\rm d}d\theta+\int_{\mu}^{\pi/ 3}u_{\rm d}d\theta\big{)} \tag{19}\]Substitute (18) into (19) to obtain:  
\[U_{\rm d}=-\frac{3\sqrt{2}}{2\pi}E[\cos\alpha+\cos(\alpha+\mu)]+(1-\frac{3\mu}{4 \pi})(\Delta U_{1}-\Delta U_{2}) \tag{20}\]  
In the CCC, the apparent extinction angle \(\gamma_{\rm app}\) is defined as the electrical angle corresponding to the time at which the valve turns off to the positive zero-crossing of the line-to-line voltage at the AC valve side [23]. It is easily obtained that the firing angle, the overlap angle and the apparent extinction angle in each period add up to \(\pi\). \[\alpha+\mu+\gamma_{\rm app}=\pi \tag{21}\]  
The real extinction angle \(\gamma_{\rm real}\) is defined as the electrical angle corresponding to the time at which the valve turns off to the positive zero-crossing of the line-to-line commutation voltage. The commutation voltage lags the valve side voltage due to the voltage drop across the series commutation capacitor. Thus, the real extinction angle \(\gamma_{\rm real}\) is larger than the apparent extinction angle \(\gamma_{\rm app}\). When \(\omega t=\mu+\gamma_{\rm real}\), the voltage drop across the valve 5 is 0, satisfying:  
\[v_{5}=(u_{\rm c}-u_{\rm cc})-(u_{\rm a}-u_{\rm ca})=0 \tag{22}\]  
The corresponding boundary conditions are:  
\[u_{\rm cc}=\frac{\pi I_{\rm d}}{3\omega C}\qquad\quad,\qquad u_{\rm ca}=\frac{ \gamma_{\rm real}I_{\rm d}}{\omega C}+\Delta U_{1}-\frac{\pi I_{\rm d}}{3 \omega C} \tag{23}\]  
Substitute (23) into (22) to obtain:  
\[\sqrt{2}E\sin(\alpha+\mu+\gamma_{\rm real})+(\frac{2\pi}{3}-\gamma_{\rm real}) \frac{I_{\rm d}}{\omega C}-\Delta U_{1}=0 \tag{24}\]  
In addition, the power characteristics of the CCC are the same as those of the conventional LCC. Thus, the active power and reactive power can be described as:  
\[P_{\rm d}=U_{\rm d}I_{\rm d} \tag{25}\]  
\[Q_{\rm d}=P_{\rm d}\frac{\sin(2\gamma_{\rm app})-\sin(2\gamma_{\rm app}+2\mu)+ 2\mu}{\cos(2\gamma_{\rm app})-\cos(2\gamma_{\rm app}+2\mu)} \tag{26}\]  
The commutation inductance is derived as:  
\[L=\frac{u_{\rm k}E}{\sqrt{2}\omega I_{\rm d}} \tag{27}\]  
where \(u_{\rm k}\) is the short-circuit impedance of the converter transformer, which is available from the equipment instruction. According to the equivalent circuit of the CCC, the peak voltage across the valves [24] can be derived as:  
\[u_{\rm VTmaxCCC}=\sqrt{2}E\sin(\frac{\pi}{3}+\gamma_{\rm app})+\frac{\pi I_{ \rm d}}{3\omega C}+\Delta U_{1} \tag{28}\]  
Equations (7), (12), (14)-(17), (20), (21), (24)-(28) constitute the steady-state mathematical model of the six-pulse CCC. There are 15 state variables in the above equations, including \(\gamma_{\rm real}\), \(\mu\), \(\alpha\), \(\Delta U_{1}\), \(\Delta U_{2}\), \(A\), \(B\), \(P_{\rm d}\), \(Q_{\rm d}\), \(\gamma_{\rm app}\), \(C\), \(\omega_{0}\), \(L\), \(E\), \(u_{\rm VTmaxCCC}\). However, only 13 equations are available. If any two of these state variables are determined, the rest of the state variables can also be calculated.