# Power Flow Calculation Methods for Power Systems with Novel Structure UPFC
## 4 Case Studies
### IEEE 39-Bus Test System  
In order to further verify the conclusions obtained by the foregoing test case, and to investigate the convergence characteristics of the three power flow calculation methods in a larger system, the Institute of Electrical and Electronics Engineers (IEEE) 39-bus test system is modified. As shown in Figure 8, the shunt converter is connected with bus 5 and the series converters are connected in the double-circuit lines between buses 4 and 14.  
Similar to the previous case, when the reactive power of the double-circuit lines is both controlled at 25 MVAr and the voltage of bus 5 is controlled at 1 pu, the iteration numbers of the three methods under different active power control targets are shown in Table 4 and Figure 9.  
Figure 8: The Institute of Electrical and Electronics Engineers (IEEE) 39-bus test system with a novel structure UPFC.  
Figure 7: The iteration numbers of the test system with the conventional structure UPFC.  
As shown in Table 4 and Figure 9, the original method requires the least iterations. Under different \(P_{c}\), the iteration numbers of the original method are always kept at about 5; When the power flow control target is close to the natural power flow (which is about \(-125\) MW), that is, \(P_{c}\) changes in the range of \(-500\) MW to \(500\) MW, the iteration numbers of the simplified method and the improved method are similar, both are about 10, but when \(P_{c}\) exceeds the above range, the number of iterations of the simplified method increases rapidly, and the number of iterations of the improved method is less than that of the simplified method, which verifies the effectiveness of the improved method.  
In addition, for all situations in Table 4, when the number of iterations is less than 33, the computation time of the three methods is around 0.01 s; when the number of iterations is greater (equal to or more than 33), the computation time will reach 0.02 s. Overall, the computation time of the three methods is relatively short.  
Compared with the WSCC 9-bus test case, we can also find that the number of iterations of the three methods is independent of system scale. After the system scale increases, the number of iterations of the original method can still be kept at about 5. When the UPFC power flow regulation effect is small, the iteration numbers of the simplified method and the improved method can also be kept at about 10.  
Moreover, when the shunt converter of the UPFC is reconnected to bus 14 from bus 5, the iteration numbers of the three methods under different UPFC control targets are always about 6. The reason for this result is that the UPFC series converters are connected in double-circuit lines, and the shunt converter is connected to one terminal of the lines. Similar to the conventional structure UPFC,  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{\(P_{c}\)/MW} & \multirow{2}{*}{**Original Method**} & \multirow{2}{*}{**Simplified Method**} & \multicolumn{2}{c}{**Improved Method**} \\ \cline{3-5}  & & & \(\lambda\) = \(\lambda_{\min}\) & \(\lambda\) = 0.1 \\ \hline
1250 & 6 & 39 & 17 & 23 \\
1000 & 6 & 24 & 15 & 15 \\
750 & 5 & 17 & 13 & 13 \\
500 & 5 & 12 & 11 & 11 \\
250 & 5 & 8 & 9 & 9 \\
0 & 5 & 5 & 5 & 6 \\ \(-250\) & 5 & 8 & 8 & 9 \\ \(-500\) & 5 & 10 & 10 & 11 \\ \(-750\) & 5 & 16 & 13 & 13 \\ \(-1000\) & 5 & 23 & 15 & 15 \\ \(-1250\) & 6 & 36 & 17 & 21 \\ \(-1500\) & 6 & \(>\)50 (not converge) & 19 & 33 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Iteration numbers of the three methods under different \(P_{c}\).  
Figure 9: The comparison of iteration numbers under different \(P_{c}\).  
when the series converters control the active power and reactive power of the lines, and the shunt converter controls the AC bus voltage, only one item in the Jacobian matrix needs to be modified, thus the addition of the UPFC has little effect on the Jacobian matrix. Therefore, the three power flow calculation methods are not much different.