# Power Flow Calculation Methods for Power Systems with Novel Structure UPFC
## 4 Case Studies
### WSCC 9-Bus Test System
When the UPFC power flow regulation effect is small, the iteration number of the improved method is  
Figure 5: The comparison of iteration numbers under different \(P_{c}\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(P_{c}\)/MW} & \multirow{2}{*}{**Original Method**} & \multirow{2}{*}{**Simplified Method**} & \multicolumn{4}{c}{**Improved Method**} \\ \cline{5-6}  & & & \(\lambda\) = \(\lambda_{\min}\) & \(\lambda_{\min}\) & \(\lambda\) = 0.1 \\ \hline
120 & 6 & 47 & 17 & 0.15 & 25 \\
90 & 6 & 19 & 12 & 0.1 & 12 \\
60 & 5 & 11 & 9 & 0.05 & 10 \\
30 & 5 & 8 & 9 & 0.05, 0.1 & 9 \\
0 & 5 & 8 & 9 & 0.05 & 10 \\ \(-30\) & 5 & 8 & 9 & 0.05 & 10 \\ \(-60\) & 5 & 8 & 8 & 0.05, 0.1 & 8 \\ \(-90\) & 5 & 11 & 9 & 0.05 & 10 \\ \(-120\) & 5 & 15 & 11 & 0.1 & 11 \\ \(-150\) & 5 & 22 & 13 & 0.1 & 13 \\ \(-180\) & 6 & 32 & 16 & 0.15 & 19 \\ \(-210\) & 6 & \(>\)50 (not converge) & 17 & 0.15 & 28 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Iteration numbers of the three methods under different \(P_{c}\). not much different from the simplified method; both methods can reach convergence in about 10 iterations. In addition, for each situation in Table 3, when the original method or the improved method is used, the computation time is about 0.01 s. In most cases, the computation time of the simplified method is also around 0.01 s, except when \(P_{c}\) is 120 MW or \(-\)210 MW, the computation time is 0.02 s. This is because the iterations of the three methods are all based on the Newton-Raphson algorithm, only the calculation method of the UPFC equivalent power injections and the Jacobian matrix are different. The calculation of the UPFC equivalent power injections and the modification of the Jacobian matrix do not need to be iterated, so their computation time is very short, and can almost be ignored. Therefore, the difference in the computation time of the three methods is mainly related to the number of iterations. When \(P_{c}\) is 120 MW or \(-\)210 MW, the number of iterations of the simplified method increases significantly, so the computation time also increases. When \(P_{c}\) is \(-\)150 MW, the variations of the largest voltage magnitude change and the largest voltage phase angle change of the three methods are shown in Figure 6. As shown in Figure 6, the original method maintains the square-convergence characteristic of the Newton-Raphson algorithm, while the convergence characteristic of the simplified method is approximately linear convergence. The convergence characteristic of the improved method is between the original method and the simplified method. For the system shown in Figure 4, if the general structure UPFC is changed to a conventional structure UPFC with the shunt converter connected to bus 4, the iteration numbers of the three methods under different UPFC control targets are shown in Figure 7. As shown in Figure 7, the iteration numbers are all about 6. According to the analysis in Section 3.2, the main reason for the above result lies in: for the conventional structure UPFC, if the series converter controls the active power and reactive power of the line, and the shunt converter controls the AC bus voltage, only one item is needed to be modified in the Jacobian matrix. The addition of the UPFC has little effect on the Jacobian matrix, so the iteration numbers of the three methods are very close. Figure 6: The iteration process of the three methods when \(P_{c}\) is \(-\)150 MW. (**a**) The largest voltage magnitude change; (**b**) the largest voltage phase angle change.