# Power System State Estimation Approach Considering Transmission Line Temperature
## 5 Case Studies
### Estimation Results of Proposed ETC-SE Approach  
The simulation and estimation of the three systems are conducted 10,000 times. The probability density functions (PDFs) of the estimated voltage \(U_{1}\) with the proposed AJ- and ITS-ETC-SE approaches are depicted in Figure 5. For comparison, the PDF of the WLS-SE method is also depicted.  
Figure 4: Flowchart of the proposed ITS-ETC-ES approach.  
Table 2 shows the RMS and maximum estimated errors of the 10,000 samples with different methods. The errors of the proposed ETC-SE approaches are almost the same and smaller than the errors of the WLS-SE method, whose voltage amplitude and phase angle errors are merely 0.13% and 6.18%, respectively. In addition, the averaged voltage errors of the IEEE 39-bus system with the proposed ITS-ETC-SE approach and WLS-SE method can be depicted in Figure 6. The temperatures are considered in the ITS-ETC-SE method but not in the WLS-SE method. Due to the influence of variable temperatures, the fluctuations of the WLS-SE method are more serious than the ITS-ETC-SE method.  
Furthermore, the estimated temperature and power loss of transmission lines in the IEEE 39-bus system are demonstrated in Table 3. It is obvious that the temperature varies greatly on different lines, ranging from 34 \({}^{\circ}\)C to 68 \({}^{\circ}\)C, which is affected by branch power  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Method** & **ITS-ETC-SE** & **AJ-ETC-SE** & **WLS-SE** \\ \hline Voltage & \(e_{l,\text{RMS}}\) (\%) & 0.13 & 0.13 & 0.41 \\ amplitude & \(e_{l,\text{max}}\) (\%) & 0.63 & 0.58 & 1.54 \\ \hline \multirow{2}{*}{Phase angle} & \(e_{\theta,\text{RMS}}\) (\%) & 6.18 & 6.23 & 6.48 \\  & \(e_{\theta,\text{max}}\) (\%) & 27.66 & 28.03 & 44.20 \\ \hline Line & \(e_{T,\text{RMS}}\) (\%) & 0.86 & 0.85 & \(\backslash\) \\ temperature & \(e_{T,\text{max}}\) (\%) & 2.26 & 2.23 & \(\backslash\) \\ \hline Active power & \(e_{P,\text{RMS}}\) (\%) & 4.99 & 4.98 & 5.32 \\  & \(e_{P,\text{max}}\) (\%) & 39.70 & 32.88 & 42.02 \\ \hline Reactive power & \(e_{Q,\text{RMS}}\) (\%) & 5.43 & 5.40 & 6.51 \\  & \(e_{Q,\text{max}}\) (\%) & 31.36 & 37.73 & 36.57 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Estimated errors with different methods.  
Figure 5: PDF of estimated voltage \(U_{1}\) with different methods. **(a)** ITS-ETC-SE approach, **(b)** AJ-ETC-SE approach, **(c)** WLS-SE method.  
Figure 6: Averaged voltage errors of the IEEE 39-bus system.  
flow. Similar results have been obtained in power loss, with an increase over 10% in the proposed ITS-ETC-SE approach compared with the WLS-SE method.  
Estimated errors of line resistance and temperature in the IEEE 39-bus system are demonstrated in Figure 7. Line resistance errors refer to the deviation between the actual resistance and its rated value. Compared with its rated values, the actual values of most line resistance increase over 10% when taking ETC into account. The resistance errors of the WLS-SE method are over 10% and the maximum error even reaches 19.1%. Estimated temperature error of most lines with the proposed ETC-SE approaches are less than 1%. Furthermore, the voltage and temperature errors of the IEEE 118-bus system are depicted in Figure 8.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Branch**} & **Power Loss of** & **Power Loss of** & **Power Loss** & \multirow{2}{*}{**T (\({}^{\circ}\)C)**} \\  & **WLS-SE (pu)** & **ITS-ETC-SE (pu)** & **Change (\%)** & \\ \hline  
\hline \hline \end{tabular}
\end{table}
Table 3: Estimated results of the IEEE 39-bus system.  
Figure 8: Resistance and line temperature error of the IEEE 118-bus system. (**a**) Voltage amplitude error, (**b**) phase angle error, (**c**) line temperature error.  
Figure 7: Resistance and line temperature error of the IEEE 39-bus system. (**a**) Resistance error, (**b**) temperature error.