# Calculation of Main Circuit Steady-State Parameters for Capacitor Commutated Converter System
## 4 Calculation Procedure of Main Circuit Parameters
### Detailed Calculation Procedure
Step 6: Calculate the firing angle \(\alpha\) at the rectifier side:  
\[\alpha=\arccos\left[\frac{U_{\text{dR}}}{p}+U_{\text{T}}+(d_{\text{acR}}+d_{ \text{FR}})\cdot\frac{I_{\text{d}}}{I_{\text{dN}}}\cdot U_{\text{dioNR}}}{U_{ \text{dioR}}}\right] \tag{53}\]  
Step 7: Assuming that the firing angle range is \(\alpha_{\text{Tmin}}\)-\(\alpha_{\text{Tmax}}\), update the tap position \(n_{\text{TR}}\) according to the calculated \(\alpha\). * If \(\alpha\) is in this range, go to the next Step. * If \(\alpha<\alpha_{\text{Tmin}}\), since \(\alpha\) is negatively correlated with \(n_{\text{TR}}\), reduce \(n_{\text{TR}}\) and recalculate \(\alpha\) until \(\alpha\) is within the specified range. If \(n_{\text{TR}}\) decreases to the minimum value, there is still \(\alpha<\alpha_{\text{Tmin}}\), then set \(\alpha=\alpha_{\text{Tmin}}\), solve the Equations (51) and (53) to obtain \(I_{\text{d}}\) and \(U_{\text{dR}}\). Step 8: According to \(n_{\text{TR}}\) and Equation (52), update \(U_{\text{dioR}}\). Calculate the overlap angle \(\mu_{\text{R}}\) and the reactive power consumption \(Q_{\text{R}}\) at the rectifier side. \[\mu_{\text{R}}=\arccos(\cos\alpha-2\cdot d_{\text{acR}}\cdot\frac{I_{\text{d }}}{I_{\text{dN}}}\cdot\frac{U_{\text{dioNR}}}{U_{\text{dioR}}})-\alpha \tag{54}\]  
\[Q_{\text{R}}=P_{\text{dR}}\frac{\sin(2\alpha)-\sin(2\alpha+2\mu_{\text{R}})+2 \mu_{\text{R}}}{\cos(2\alpha)-\cos(2\alpha+2\mu_{\text{R}})} \tag{55}\]  
Step 9: For the inverter, the default tap position of the converter transformer is set at the highest position. The corresponding AC voltage at the valve side of the CCC is:  
\[U_{\text{acL}}=\frac{U_{\text{acNI}}}{1+n_{\text{TI}}\cdot\Delta\eta} \tag{56}\]  
Step 10: Substitute \(U_{\text{acI}}\) into the steady-state mathematical model of the CCC to calculate \(\gamma_{\text{app}}\), \(\gamma_{\text{real}}\), \(\mu_{\text{I}}\), \(P_{\text{I}}\) and \(Q_{\text{I}}\). Step 11: Assume that the extinction angle range satisfies:  
\[ \begin{cases}\gamma_{\text{app}}\geq\gamma_{\text{appmin}}\\ \gamma_{\text{real}}\geq\gamma_{\text{realmin}}\end{cases}  \tag{57}\]  
To ensure the commutation margin, set \(\gamma_{\text{realmin}}\) = 17\({}^{\circ}\) and \(\gamma_{\text{appmin}}\) = \(\gamma_{\text{appN}}\). If \(\gamma_{\text{app}}<\gamma_{\text{appmin}}\) or \(\gamma_{\text{real}}<\gamma_{\text{realmin}}\), reduce \(n_{\text{TI}}\) and recalculate \(U_{\text{acI}}\). Then, return to Step 9 to recalculate \(\gamma_{\text{app}}\) and \(\gamma_{\text{real}}\) until the extinction angles are within the specified ranges. If \(n_{\text{TI}}\) is reduced to the minimum value, and \(\gamma_{\text{app}}\) (or \(\gamma_{\text{real}}\)) is still not within the constraints, then set \(\gamma_{\text{app}}=\gamma_{\text{appmin}}\) (or \(\gamma_{\text{real}}=\gamma_{\text{realmin}}\)). The operating parameters of the rectifier are now determined by those of the inverter. The calculating process is as follows:  
Substitute \(\gamma_{\text{app}}\) (or \(\gamma_{\text{real}}\)) into the steady-state mathematical model of the CCC to calculate the inverter side DC voltage \(U_{\text{dl}}\). Calculate the DC voltage \(U_{\text{dR}}\) according to (58) and update the DC current \(I_{\text{d}}\) according to (51). Then repeat the calculation to modify \(U_{\text{dl}}\), \(U_{\text{dR}}\) and \(I_{\text{d}}\). After several iterations, the accurate \(U_{\text{dl}}\), \(U_{\text{dR}}\) and \(I_{\text{d}}\) can be obtained. Finally, update the other operating parameters of the rectifier and the inverter by executing Step 4-Step 10. \[U_{\text{dR}}=U_{\text{dl}}+I_{\text{d}}R_{\text{d}} \tag{58}\]  
Step 12: Verify whether the reactive power consumption of the rectifier meets the requirements of the minimum reactive power consumption \(Q_{\text{Rmin}}\). The purpose is to avoid injecting too much reactive power into the AC system when the DC system is low loaded, especially under the minimum filter control mode.