## 3 Method and Results of Modeling
In addition, on each side of the conductor system, additional branches are formed with impedance values calculated by half the values of the corresponding mutual capacitances. The self-impedances of power transmission line conductors are calculated from [PERSON]'s equations for the near range, with the addition of the internal resistance of the conductors. The external impedance is calculated as per the following expressions:  
\[\underline{Z}_{\mathrm{ext}}=0.001f+jf[0.01148-0.001256\ln(r\sqrt{0.02\sigma f })],\ \mathrm{Ohm/km}; \tag{4}\]  
where \(f\) is the frequency, measured in Hz; \(r\) is the equivalent conductor radius (for steel-reinforced aluminum conductors, it is assumed to be 0.95 times the external radius of the conductor cross-section), measured in cm; and \(\sigma\) is the specific conductance of the homogeneous earth (or equivalent weighted average conductance), measured in S/m. The internal resistance is different for different types of conductors. For steel-reinforced aluminum conductors, approximating approximate relationships are used:  
\[R_{\mathrm{in}}=R_{0}(0.9~{}+~{}0.0063f^{0.755}),\ \mathrm{Ohm/km}; \tag{5}\]  
\[X_{\mathrm{in}}=0.001\Big{[}\Big{(}0.033~{}-~{}0.00107f^{0.83}\Big{)}S~{}+~{} 1.07f^{0.83}~{}-~{}13.5\Big{]},\ \mathrm{Ohm/km}; \tag{6}\]where \(R_{0}\) is the resistance of 1 km of the conductor to direct current, measured in Ohm /km; \(f\) is the frequency, measured in Hz; and \(S\) is the cross-sectional area of the conductor, measured in mm\({}^{2}\). For solid aluminum and copper conductors with a cylindrical cross-section, the internal resistance is calculated by approximate formulas, provided that \(|qr|\leq 4\):  
\[R_{\text{in}}=R_{0}(1\ +\ 0.0049x^{4}\ -\ 0.000035x^{7}),\ \text{Ohm/km}; \tag{7}\]  
\[X_{\text{in}}\ =\ R_{0}(0.125x^{2}\ -\ 0.000613x^{5}),\ \text{Ohm/km}; \tag{8}\]  
\[\underline{q}=\sqrt{-j\omega\gamma_{\text{TI}}\mu_{0}};\ x=0.01r\sqrt{\frac{78 96f}{R_{0}S}} \tag{9}\]  
where \(\gamma_{\text{TI}}\) is the specific conductance of conductor material; \(r\) is the conductor radius, measured in cm; and \(S\) is the cross-sectional area of the conductor, measured in mm\({}^{2}\). For large values of \(x\) we use the ratio  
\[R_{\text{in}}=X_{\text{in}}=\frac{R_{0}x}{2\sqrt{2}} \tag{10}\]  
For steel wires and rails, we use the approximate expression  
\[R_{\text{in}}=R_{50}\sqrt{0.02f},\ X_{\text{in}}=0.75R_{\text{in}} \tag{11}\]  
where \(R_{50}\) is the ohmic resistance for a frequency of 50 Hz. The technique for load flow analysis of complex EPSs properly calculates the mutual electromagnetic effects in the near, intermediate, and far ranges of Carson's integral [34]. Mutual inductance \(M_{ik}\) is the central concept from which inductive effects are determined. It involves the instantaneous transmission of load flow changes of individual conductors throughout a multi-conductor system. When determining \(M_{ik}\), we consider the conductor-to-ground circuits. For sine wave processes, the mutual inductance yields a proportional relationship between the current in the circuit that creates the magnetic field and the EMF induced in the other circuit. \[\hat{\mathcal{E}}=-j\omega M_{ik}Il=-Z_{ik}\hat{l}l \tag{12}\]  
where \(Z_{ik}\) represents the impedance of the mutual-inductance coupling between circuits \(i\) and \(k\), referred to 1 km of the length of the interacting conductors of length \(l\); and \(\omega\) is the angular frequency. [PERSON]'s formulas [34] are based on the correct solution of [PERSON]'s equations and allow the self- and mutual inductances of conductors to be determined. They can be used to find the external self-impedance of a straight-line conductor over a flat surface of the homogeneous earth, as well as the mutual impedance of two conductors, taking into account the ground return currents. There are approximate expressions for the near and far ranges that do not apply to the intermediate range.