# Power System State Estimation Approach Considering Transmission Line Temperature
## 2 ETC-SE Model  
### Measurement and State Vector of the ETC-SE Model  
The lumped parameter model of the transmission line \(l\) is shown in Figure 1, supposing that the two terminal ends are bus \(i\) and \(j\), respectively. \(P_{i}\) and \(Q_{i}\) are the active and reactive power injection of bus \(i\), respectively; \(P_{j}\) and \(Q_{ij}\) are the active and reactive power injection of bus \(j\), respectively; \(P_{ij}\) and \(Q_{ij}\) are the active and reactive power flow of the line \(l\), respectively; \(U_{i}\) and \(U_{j}\) are the voltage amplitudes of bus \(i\) and \(j\), respectively; \(I_{ij,real}\) and \(I_{ij,imag}\) are the real and image parts of the current from bus \(i\) to bus \(j\), respectively; \(I_{ij,real}\) and \(I_{ij,imag}\) are the real and image parts of the current from bus \(j\) to bus \(i\), respectively; \(g_{ij}\) and \(b_{ij}\) are the mutual conductance and susceptance of the line \(l\), respectively; and \(g_{ij,c}\) and \(b_{ij,c}\) are the self-conductance and susceptance of the line \(l\), respectively.  
\begin{table}
\begin{tabular}{c c c} \hline
**Reference** & **Method/Model** & **Characteristic/Application** \\ \hline
[15] & State space model & Thermal rating with demand response \\
[16] & HBE & Thermal rating to detect line overload \\
[17] & Transient thermal rating equation & Thermal rating with tripping scheme \\
[18] & Newton-Raphson method & Temperature-dependent power flow \\
[19] & ETC & Power flow analysis for system security \\
[20] & Primal-dual interior point method & Weather-based optimal power flow \\
[21] & ETC & Influence analysis of temperature on SE \\
[22] & Monte Carlo method & Probabilistic simulation of line temperature \\
[23] & Analytical solution method & Simulation of line temperature \\
[24] & Predictor-corrector interior point algorithm & Line temperature estimation \\
[25] & Weighted least square SE & SE considering bad data detection \\
[26] & Robust optimization & SE considering parameter errors \\
[27] & Bilinear optimization & SE with nonlinear measurements \\
[28] & Iterative reweight least squares method & SE with bad data \\
[29] & WLS & SE for the placement of PMUs \\
[30] & Modified Java algorithm & SE for the placement of PMUs \\
[31] & Unscented Kalman filter & SE for PMSM \\
[32] & Data-driven Kalman filter & Multi-area and distributed SE \\
[33] & Adaptive smooth variable structure filter & SE for EVBs \\ \hline \end{tabular}
\end{table}
Table 1: Comparative literature survey on state-of-the-art SE methods.  
Figure 1: Lumped model of the transmission line.  
Considering the influence of line temperature, the vectors of measurements and states are augmented in the ETC-SE model. The augmented measurement vector \(\mathbf{z}_{aug}\) can be represented as,  
\[\mathbf{z}_{aug}=\left[\mathbf{z}_{elec}\ \mathbf{z}_{meteo}\right]^{T} \tag{1}\]  
with  
\[\mathbf{z}_{elec}=\left[U_{i}\ \ P_{i}\ \ Q_{i}\ \ P_{ij}\ \ Q_{ij}\right]^{T} \tag{2}\]  
\[\mathbf{z}_{meteo}=\left[Q_{solar}\ \ \ \theta\ \ V_{\mathbf{w}}\ \ T_{\mathbf{a}}\ \ \phi\right]^{T} \tag{3}\]  
where \(\mathbf{z}_{elec}\) and \(\mathbf{z}_{meteo}\) are the measurement vectors of electrical and meteorological quantities, respectively; \(Q_{solar}\), \(\theta\), \(V_{w}\), \(T_{a}\), and \(\phi\) are the solar irradiation intensity, solar incidence angle, wind speed, ambient temperature, and wind directions, respectively.  
Then, the augmented measurement function vector \(\mathbf{h}_{aug}\) can be represented as,  
\[\mathbf{h}_{aug}=\left[\mathbf{h}_{elec}\ \ \mathbf{h}_{HBE}\right]^{T} \tag{4}\]  
where \(\mathbf{h}_{elec}\) and \(\mathbf{h}_{HBE}\) are the measurement function vectors of electrical and HBE, respectively.  
In addition, the augmented state vector \(\mathbf{x}_{aug}\) can be represented as,  
\[\mathbf{x}_{aug}=\left[\mathbf{U}\ \mathbf{\theta}\ \mathbf{T}\right]^{T} \tag{5}\]  
where \(\mathbf{U}\), \(\mathbf{\theta}\), and \(\mathbf{T}\) are the voltage magnitude, voltage angle, and line temperature vectors, respectively.