# Fault Location in Double Circuit Medium Power Distribution Networks Using an Impedance-Based Method
## 3 The Developed Fault Location Algorithm  
The developed method in this paper considers the mutual effects of the lines on each other. It also considers the existence of central loads and the main and subsidiary branches for a double circuit. The details of the algorithm are given below:  
1. Fault detection. 2. Determining the fault type. 3. Estimation of the fault current using the equation below: \[I_{F}=I_{S_{F}}-I_{S},\] (14) where \(I_{S_{F}}\) and \(I_{S}\) are the measured currents during and before the fault in the local relay. 4. Determining the fault's distance from the general fault location equation (Equation (13)). 5. Determining the exact physical location of the fault. 6. Checking the convergence of \(x\) using Equation (15): \[|x(n)-x(n-1)|<\delta.\] (15) For \(n>1\), the tolerance \(\delta\) is predefined and \(n\) is the number of repetitions. 7. If \(x\) is convergent with the analyzed part of the last section, then \(x\) is the fault's location and go to the next step; if it is convergent with a location beyond the current location, then we update \(V_{S}\) and \(I_{S}\) in the system's next bus (changing the reference bus) and return to section one. 8. Calculating the fault's point voltage by using Equation (5), \(x(n)\), voltages and currents of the bus in the correct analyzed upstream section (\(k\) in \(V_{S_{k}}\) and \(I_{S_{k}}\) refers to the reference bus). Updating the fault's downstream current in the faulted phases using the calculated voltage of the fault point and based on Equation (16), finally \(I_{D_{\text{{dc}}}}\) is obtained using [PERSON]'s conversion as seen in Equation (20). \[\left\{ \begin{array}{l}I_{D_{1}}=\left\{Z_{T}^{-1}+0.5(1-x)y_{1}\right\}V_{ \text{F}_{1}}\\ I_{D_{2}}=\left\{Z_{T}^{-1}+0.5(1-x)y_{2}\right\}V_{\text{F}_{2}}\\ I_{D_{0}}=\left\{\left(\frac{\zeta}{d}\right)+e+f\right\}V_{\text{F}_{0}}-\left\{ \left(\frac{b}{d}\right)+f\right\}V_{\text{F}_{2_{0}}}\\ \end{array} \right.\] (16) In the above equation, in series state one and two, is obtained from Equation (17), in which is the impedance connected to the -bus. Additionally, for obtaining the equivalent impedance in zero series state, Equations (18) and (19) are obtained from Y-\(\Delta\) and \(\Delta\)-Y conversions, and finally is derived: \[Z_{T}=(1-x)z_{1}+\left[Z_{k+1}^{-1}+0.5(1-x)y_{1}\right]^{-1},\] (17) \[\left\{ \begin{array}{l}a=(1-x)Z_{0}+Z_{5}+Z_{3}\\ b=(1-x)Z_{m_{0}}+Z_{3}\\ c=(1-x)Z_{0}+Z_{4}+Z_{3}\\ d=(ac)-b^{2}\\ e=0.5(1-x)y_{0}\\ f=0.5(1-x)y_{m_{0}}\end{array} \right.,\] (18) \[\left\{ \begin{array}{l}z_{1}=Z_{L}+\left[0.5(1-x)y_{0}\right]^{-1}\\ z_{2}=Z_{L}+\left[0.5(1-x)y_{0}\right]^{-1}\\ z_{3}=\frac{Z_{1}z_{2}}{Z_{1}+z_{2}+\left[0.5(1-x)y_{m_{0}}\right]^{-1}}\\ z_{4}=\frac{\left[0.5(1-x)y_{m_{0}}\right]^{-1}z_{2}}{Z_{1}+z_{2}+\left[0.5(1- x)y_{m_{0}}\right]^{-1}}\\ z_{5}=\frac{\left[0.5(1-x)y_{m_{0}}\right]^{-1}z_{1}}{Z_{1}+z_{2}+\left[0.5(1- x)y_{m_{0}}\right]^{-1}}\\ \end{array} \right.,\] (19) \[I_{D}=AI_{D_{012}}.\] (20)
1. Updating the faults current by using Equation (21): \[I_{\text{F}}=I_{U}-I_{D},\] (21) where \(I_{U}\) is the upstream current of the faulted point according to Figure 1. \[\left\{ \begin{array}{l}I_{U_{1}}=V_{S_{1}}\big{(}-xy_{1}-0.25x^{3}y_{1}z_{1}y _{1}\big{)}\\ +I_{S_{1}}\big{(}1+0.25x^{2}y_{1}z_{1}y_{1}\big{)}\\ I_{U_{2}}=V_{S_{2}}\big{(}-xy_{1}-0.25x^{3}y_{1}z_{1}y_{1}\big{)}\\ +I_{S_{2}}\big{(}1+0.25x^{2}y_{1}z_{1}\big{)}\\ I_{U_{0}}=V_{S_{0}}(-0.5 xy_{0}-0.5 xy_{m_{0}})+I_{S_{0}}\\ +V_{S_{0}}\big{(}0.5 xy_{m_{0}}\big{)}-V_{\text{F}}(0.5 xy_{0}+0.5 xy_{m_{0}})\\ +V_{\text{F}_{2}}(0.5 xy_{m_{0}})\end{array} \right.\] (22) \(I_{U_{\text{dc}}}\) is obtained using [PERSON]'s conversion. 1. Return to step four. It is known that the distribution feeders are distributed in single circuit or double circuit. This depends upon the load demands and its peak. These loads can be unbalanced and distributed in the feeder. Therefore, in the proposed method, the KVL and KCL matrices are used.