# Fault Location in Double Circuit Medium Power Distribution Networks Using an Impedance-Based Method
## 2 The Proposed Method
\begin{array}{c}R_{F_{k}}I_{F_{k_{r}}}+R_{F_{g}}I_{F_{r}}-X_{F_{g}}I_ {F_{i}}=\\ 0.5x^{2}E_{r}-x(G_{r}+H_{r})+L_{r}=M_{k_{r}}\\ \\ R_{F_{k}}I_{F_{k_{i}}}+R_{F_{g}}I_{F_{i}}-X_{F_{g}}I_{F_{i}}=\\ 0.5x^{2}E_{i}-x(G_{i}+H_{i})+L_{i}=M_{k_{i}}\end{array} \right.. \tag{8}\]  
In the above equation, index \(r\) indicates the real part and index \(i\) indicates the imaginary part of these variables. Equation (8) can be used for equation matching and to separate the fault resistance \(R_{F_{k}}\) from each faulted phase; thus, a group of \(n\) equations are obtained which are independent of \(R_{F_{k}}\) and dependent on \(x\), \(R_{F_{g}}\), and \(X_{F_{g}}\). \[R_{F_{g}}\bigg{[}-\frac{I_{F_{r}}}{I_{F_{k_{r}}}}+\frac{I_{F_{i}}}{I_{F_{k_{i} }}}\bigg{]}+X_{F_{g}}\bigg{[}-\frac{I_{F_{i}}}{I_{F_{k_{r}}}}+\frac{I_{F_{r}}}{ I_{F_{k_{i}}}}\bigg{]}+\bigg{[}-\frac{M_{K_{r}}}{I_{F_{k_{r}}}}+\frac{M_{K_{i}}}{I_{F_{ k_{i}}}}\bigg{]}=0. \tag{9}\]  
By multiplying \(I_{F_{k_{r}}}\cdot I_{F_{k_{r}}}\) to Equation (9) and completing the algebraic calculations, the below equation is obtained:  
\[R_{F_{g}}\Im\big{\{}I_{F_{k}}I_{F}^{*}\big{\}}-X_{F_{g}}\Re\big{\{}I_{F_{k}}I_ {F}^{*}\big{\}}+\Big{[}M_{k_{r}}I_{F_{k_{i}}}-M_{k_{i}}I_{F_{k_{r}}}\Big{]}=0. \tag{10}\]  
In this equation, \(\Re\) shows the real part, \(\Im\) shows the imaginary part, and \(*\) indicates the conjugate of the complex numbers. For each \(k\)-phase, Equation (10) can be re-written:  
\[R_{F_{g}}\Im\{I_{F}^{*}\big{\}}-X_{F_{g}}\Re\{I_{F}I_{F}^{*}\}+\sum_{k\in \Omega_{k}}\Big{[}M_{k_{r}}I_{F_{k_{i}}}-M_{k_{i}}I_{F_{k_{r}}}\Big{]}=0. \tag{11}\]  
Figure 2: General view of ground faults (single phase, two phase, and three-phase fault to the ground). Based on the equation below  
\[\Im\{I_{F}I_{F}^{*}\}=\Im\Big{\{}|I_{F}|^{2}\Big{\}}=0,\]  
and if we consider \(X_{F_{g}}=0\) as the fault's resistance, the next equation is derived:  
\[\sum\
olimits_{k\in\Omega_{k}}\Big{[}M_{k_{r}}I_{F_{k_{i}}}-M_{k_{ i}}I_{F_{k_{r}}}\Big{]}=0, \tag{12}\]  
where \(\Omega_{k}\) is a group of faulted phases that consist of the system's (a), (b), and (c) phases. In a three-phase system, there are seven different states of faults, including single-phase, two-phase, and three-phase. By replacing Equation (8) into Equation (12) and by making the algebraic changes in the mentioned equation, the final fault location equation for ground-faults, named the general ground-fault equation, is obtained:  
\[x^{2}\Bigg{[}0.5\sum\limits_{k\in\Omega_{k}}\Im\Big{\{}E_{k}I_{F_{k}}^{*}\Big{\}} \Bigg{]}-x\left[\sum\limits_{k\in\Omega_{k}}\Im\Big{\{}(G_{k}+H_{K})I_{F_{k}}^{* }\Big{\}}\right]+\left[\sum\limits_{k\in\Omega_{k}}\Im\Big{\{}L_{k}I_{F_{k}}^{* }\Big{\}}\right]=0. \tag{13}\]  
It is important to mention that this is a fault location equation, and it is used for determining the fault distance. Therefore, it needs the three-phase voltage and current in the post, line parameters (series impedance and parallel admittance), and the faults current (because the fault current cannot be determined in local station, Equation (4) is used to calculate the coefficients of Equation (13)). As a result, an equation is needed to estimate the fault current, which will be provided in Section 3.