# Power System State Estimation Approach Considering Transmission Line Temperature
## 5 Case Studies
### Performance Analysis  
A comparison of the computation time and iteration numbers on the proposed ETC-SE approaches and WLS-SE method was performed, as shown in Table 4. The ETC-SE approaches need more iteration numbers than the WLS-SE method. Line temperature converges more slowly than state variables. As for the computation time, the proposed ITS-ETC-SE approach is shorter than the AJ-ETC-SE approach.  
Table 5 shows the computational complexity of the AJ-ETC-SE approach and WLS-SE method. With the consideration of line temperature, the number of measurements and state variables, and the dimension of the Jacobian matrix increase sharply and even over 100% compared with the WLS-SE method.  
Figure 9 shows the maximum unbalance of state variables \(\|\Delta x\|_{\infty}\) during each iteration. It can be seen that the index \(\|\Delta x\|_{\infty}\) first declines linearly, and then nearly remains constant.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & \multicolumn{2}{c}{**ITS-ETC-SE**} & **AJ-ETC-SE** & **WLS-SE** \\ \hline \multirow{3}{*}{Iteration numbers} & 14-bus & 6 & 6 & 4 \\  & 39-bus & 6 & 6 & 4 \\  & 118-bus & 7 & 8 & 4 \\ \hline \multirow{3}{*}{Computation time (s)} & 14-bus & 0.057 & 0.141 & 0.018 \\  & 39-bus & 0.089 & 0.264 & 0.037 \\ \cline{1-1}  & 118-bus & 0.966 & 3.432 & 0.316 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of iteration numbers and computation time.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Index** & **14-Bus** & **39-Bus** & **118-Bus** \\ \hline \multicolumn{3}{c}{Number of ETC lines} & 15 & 34 & 170 \\ \hline \multirow{3}{*}{Number of States} & WLS-SE & 27 & 77 & 235 \\  & AJ-ETC-SE & 42 & 111 & 405 \\  & Change (\%) & 55.6 & 44.2 & 72.3 \\ \hline \multirow{3}{*}{Number of measurements} & WLS-SE & 42 & 165 & 403 \\  & AJ-ETC-SE & 57 & 199 & 573 \\  & Change (\%) & 35.7 & 20.6 & 42.2 \\ \hline \multirow{3}{*}{Dimension of Jacobian matrix} & WLS-SE & \(42\times 27\) & \(165\times 77\) & \(403\times 235\) \\  & AJ-ETC-SE & \(57\times 42\) & \(199\times 111\) & \(573\times 405\) \\ \cline{1-1}  & Change (\%) & \(111.1\) & 73.9 & 145.0 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of computational complexity of the AJ-ETC-SE approach and WLS-SE method.  
Figure 9: Maximum unbalance of state variables.  
Furthermore, different scenarios are conducted in order to explore the convergence performance of the proposed ETC-SE approaches, including varying environmental conditions and the existence of ill-conditioned branches. For each scenario, 1000 simulations are performed in the IEEE 118-bus system.  
The environmental conditions vary with the ambient temperature \(T_{a}\) and wind velocity \(V_{w}\). More specially, the ambient temperature \(T_{a}\) increases from 25 \({}^{\circ}\)C to 50 \({}^{\circ}\)C and the wind velocity \(V_{w}\) increases from 1 m/s to 20 m/s, which will lead to changes in convective heat loss \(q_{c}\). The iteration numbers of the proposed ETC-SE approaches can be summarized in Tables 6 and 7, respectively.  
Furthermore, the ill-conditioned branches are added to the IEEE 118-bus system. Moreover, the ill-conditioned branches refer to the high-resistance branches whose \(R=0.5\)X. The iteration numbers of the proposed ETC-SE approaches are shown in Table 8.