## 3 Framework
### Preliminaries  
Consider a transmission line connecting bus \(k\) and bus \(m\), and represented in a \(\pi\)-model. With the line admittance \(y_{km}\), the conjugate of the complex power flow through that line can be written as [27]:  
\[S_{km}^{*}=P_{k}-jQ_{k}=E_{k}^{*}I_{km}. \tag{7}\]  
where \(E_{k}\) is the complex voltage at bus \(k\), \(I_{km}\) is the complex current flowing from bus \(k\) to bus \(m\), and the \(*\) indicates the conjugate of the complex quantity. Using \(I_{km}=(E_{k}-E_{m})y_{km}\), we can write the complex power as:  
\[S_{km}^{*}=V_{k}^{2}y_{km}V_{k}e^{-j\theta_{k}}(V_{k}e^{i\theta_{k}}-V_{m}e^{i \theta_{m}})+jb_{km}^{sh}. \tag{8}\]  
where \(y_{km}\) is the admittance between bus \(k\) and bus \(m\), \(V_{k}\) and \(V_{m}\) are the magnitudes of the complex voltages at bus \(k\) and \(m\), respectively, \(\theta_{k}\) and \(\theta_{m}\) are the angles of the complex voltages at bus \(k\) and \(m\), respectively, and \(b_{km}^{sh}\) is the shunt admittance of the line connecting bus \(k\) and bus \(m\). Expanding the right hand side of (7) and decomposing the expression into real and imaginary parts, one can obtain the following:  
\[P_{km}=(V_{k}^{2}-V_{k}V_{m}\cos\theta_{km})g_{km}-(V_{k}V_{m}\sin\theta_{km}) b_{km}. \tag{9}\]  
\[Q_{km}=(-V_{k}V_{m}sin\theta_{km})g_{km}+(-V_{k}^{2})b_{km}^{sh}+(V_{k}V_{m}cos \theta_{km}-V_{k}^{2})b_{km}. \tag{10}\]where \(g_{km}\) is the real part of the line admittance connecting bus \(k\) and bus \(m\), i.e., \(\Re\{y_{km}\}\), and \(b_{km}\) is the imaginary part of the line admittance connecting bus \(k\) and bus \(m\), i.e., \(\Im\{y_{km}\}\). Equations (9) and (10) represent the real and reactive power flows in the line connecting bus \(k\) and bus \(m\), respectively. With the real power flow from bus \(m\) to bus \(k\), i.e., \(P_{mk}\) (by changing bus index), one can express the real power loss of the same line as:  
\[\begin{split} P_{km}^{loss}&=P_{km}+P_{mk}\\ &=g_{km}(V_{k}^{2}+V_{m}^{2}-2V_{k}V_{m}cos\theta_{km})\\ &=g_{km}|E_{k}-E_{m}|^{2}.\end{split} \tag{11}\]  
where \(E_{k}\) and \(E_{m}\) are complex voltages at bus \(k\) and bus \(m\), respectively. Similar procedure to (11), an expression of the reactive power loss in the line can expressed as:  
\[\begin{split} Q_{km}^{loss}&=Q_{km}+Q_{mk}\\ &=-(V_{k}^{2}+V_{m}^{2})b_{km}^{sh}-(V_{k}^{2}+V_{m}^{2}-2V_{k}V_ {m}cos\theta_{km})b_{km}\\ &=-(V_{k}^{2}+V_{m}^{2})b_{km}^{sh}-(|E_{k}-E_{m}|^{2})b_{km}. \end{split} \tag{12}\]  
Equations (9)-(12) are the basic equations that govern a line connecting bus \(k\) and bus \(m\) from the SE perspective. In AC SE, Equations (9) and (10) are used when \(P_{km}\) and/or \(Q_{km}\) are present in the measurements set. The relationship between \(g_{km}\) and \(b_{km}\) can be derived from the actual impedance of the line as:  
\[y_{km}=g_{km}+jb_{km}=\frac{1}{z_{km}}=\frac{1}{R_{km}+jX_{km}}. \tag{13}\]  
\[g_{km}=\Re\{\frac{1}{z_{km}}\}=\frac{R_{km}}{R_{km}^{2}+X_{km}^{2}}. \tag{14}\]  
\[b_{km}=\Im\{\frac{1}{z_{km}}\}=-\frac{X_{km}}{R_{km}^{2}+X_{km}^{2}}. \tag{15}\]  
If (14) is divided by (15), the following expression is obtained:  
\[\frac{b_{km}}{g_{km}}=-\frac{X_{km}}{R_{km}}. \tag{16}\]  
Therefore, (16) correlates line conductance to its susceptance. Further, the term \(\left(\frac{X_{km}}{R_{km}}\right)\) is known as \(X/R\) ratio and commonly used in short circuit studies. In transmission systems, this ratio is higher compared to distribution systems. In addition, this ratio is a characteristic of the line that indicates the tangent angle between line resistance and line inductance. Hence, having this factor will eliminate the non-linearity in retrieving the original measurement model parameters that will be presented in Section 3.3.