## 2 Materials and Methods
\begin{bmatrix}Y_{0}&-Y_{TA}&-Y_{TB}&-Y_{TC}&0&0&0&0&0&0\\ -Y_{TA}&Y_{A1}&-Y_{AAB1}&-Y_{BAC1}&-Y_{L41}&0&0&0&0&0\\ -Y_{TB}&-Y_{KAB1}&Y_{B1}&-Y_{BAC1}&0&-Y_{B11}&0&0&0&0\\ -Y_{TC}&-Y_{RAC1}&-Y_{BBC1}&Y_{C1}&0&0&-Y_{LC1}&0&0&0\\ 0&-Y_{L41}&0&0&Y_{A2}&-Y_{KAB2}&-Y_{RAC2}&-Y_{L42}&0&0\\ 0&0&-Y_{L81}&0&-Y_{KAB2}&Y_{B2}&-Y_{RBC2}&0&-Y_{L82}&0\\ 0&0&0&-Y_{L81}&-Y_{RAC2}&-Y_{RBC2}&Y_{C2}&0&-Y_{L82}&0\\ 0&0&0&-Y_{LC1}&-Y_{RAC2}&-Y_{RBC2}&Y_{C2}&0&0&-Y_{LC2}\\ 0&0&0&0&-Y_{L42}&0&0&Y_{A3}&-Y_{KAB3}&-Y_{KAC3}\\ 0&0&0&0&-Y_{L82}&0&-Y_{RAB3}&Y_{B3}&-Y_{RBC3}\\ 0&0&0&0&0&-Y_{LC2}&-Y_{RAC3}&-Y_{RBC3}&Y_{C3}\end{bmatrix}\begin{bmatrix}-E_{ A}\cdot Y_{TA}-E_{B}\cdot Y_{TB}-E_{C}\cdot Y_{TC}\\ E_{A}\cdot Y_{TA}\\ E_{B}\cdot Y_{TB}\\ E_{C}\cdot Y_{TC}\\ 0\\ 0\\ V_{A3}\\ V_{B3}\\ V_{C3}\end{bmatrix}  \tag{2}\]  
where  
\[Y_{TA}=\frac{1}{Z_{TA}}\ Y_{TB}=\frac{1}{Z_{TB}}\ Y_{TB}=\frac{1}{Z_{TB}}\]  
\[Y_{LA1}=\frac{1}{Z_{LA1}}\ Y_{LB1}=\frac{1}{Z_{LB1}}\ Y_{LC1}=\frac{1}{Z_{LC1}}\]  
\[Y_{LA2}=\frac{1}{Z_{LA2}}\ Y_{LB2}=\frac{1}{Z_{LB2}}\ Y_{LC2}=\frac{1}{Z_{LC2}}\]  
\[Y_{RAB1}=\frac{1}{Z_{RAB1}}\ Y_{RBC1}=\frac{1}{Z_{RBC1}}\ Y_{RAC1}=\frac{1}{Z_{ RAC1}}\]  
\[Y_{RAB2}=\frac{1}{Z_{RAB2}}\ Y_{RBC2}=\frac{1}{Z_{RBC2}}\ Y_{RAC2}=\frac{1}{Z_{ RAC2}}\]  
\[Y_{RAB3}=\frac{1}{Z_{RAB3}}\ Y_{RBC3}=\frac{1}{Z_{RBC3}}\ Y_{RAC1}=\frac{1}{Z_{ RAC3}}\]  
\[Y_{0}=\frac{1}{Z_{0}}+\frac{1}{Z_{TA}}+\frac{1}{Z_{TB}}+\frac{1}{Z_{TC}}\]  
\[Y_{A1}=\frac{1}{Z_{TA}}+\frac{1}{Z_{RAB1}}+\frac{1}{Z_{RAC1}}+\frac{1}{Z_{L41}} +\mu\mathcal{C}_{A1}\ Y_{B1}=\frac{1}{Z_{TB}}+\frac{1}{Z_{RAB1}}+\frac{1}{Z_{ RBC1}}+\frac{1}{Z_{RBC1}}+\mu\mathcal{C}_{B1}\ Y_{C1}=\frac{1}{Z_{TC}}+\frac{1}{Z_{RAC1}}+ \frac{1}{Z_{RBC1}}+\frac{1}{Z_{LC1}}+\mu\mathcal{C}_{C1}\]  
\[Y_{A2}=\frac{1}{Z_{LA1}}+\frac{1}{Z_{RAB2}}+\frac{1}{Z_{RAC2}}+\frac{1}{Z_{L42} }+\mu\mathcal{C}_{A2}\ Y_{B2}=\frac{1}{Z_{LB1}}+\frac{1}{Z_{RAB2}}+\frac{1}{Z _{RBC2}}+\frac{1}{Z_{L2R}}+\mu\mathcal{C}_{B2}\ Y_{C2}=\frac{1}{Z_{LC1}}+\frac {1}{Z_{RAC2}}+\frac{1}{Z_{RBC2}}+\frac{1}{Z_{L2C}}+\mu\mathcal{C}_{C2}\]  
\[Y_{A3}=\frac{1}{Z_{L42}}+\frac{1}{Z_{RAB3}}+\frac{1}{Z_{RAC3}}+\mu\mathcal{C}_{ A3}\ Y_{B3}=\frac{1}{Z_{LB2}}+\frac{1}{Z_{RAB3}}+\frac{1}{Z_{RBC3}}+\mu \mathcal{C}_{B3}\ Y_{C3}=\frac{1}{Z_{LC2}}+\frac{1}{Z_{RAC3}}+\frac{1}{Z_{RBC3 }}+\mu\mathcal{C}_{C3}\]  
Using [PERSON]'s method, the system of equations can be solved by determining the values of the characteristic determinants:  
\[W= \begin{bmatrix}Y_{0}&-Y_{TA}&-Y_{TB}&-Y_{TC}&0&0&0&0&0&0\\ -Y_{TA}&Y_{A1}&-Y_{KAB1}&-Y_{RAC1}&-Y_{LA1}&0&0&0&0&0\\ -Y_{TB}&-Y_{RAB1}&Y_{B1}&-Y_{RBC1}&0&-Y_{LB1}&0&0&0&0\\ -Y_{TC}&-Y_{RAC1}&-Y_{RBC1}&Y_{C1}&0&0&-Y_{LC1}&0&0&0\\ 0&-Y_{LA1}&0&0&Y_{A2}&-Y_{RAB2}&-Y_{RAC2}&-Y_{LA2}&0&0\\ 0&0&-Y_{LB1}&0&-Y_{KAB2}&Y_{B2}&-Y_{RBC2}&0&-Y_{LB2}&0\\ 0&0&0&-Y_{LC1}&-Y_{RAC2}&-Y_{RBC2}&Y_{C2}&0&0&-Y_{LC2}\\ 0&0&0&0&-Y_{LA2}&0&0&Y_{A3}&-Y_{RAB3}&-Y_{RAC3}\\ 0&0&0&0&0&-Y_{LB2}&0&-Y_{RAB3}&Y_{B3}&-Y_{RBC3}\\ 0&0&0&0&0&-Y_{LC2}&-Y_{RAC3}&-Y_{RBC3}&Y_{C3}\end{bmatrix}  \tag{3}\]  
The analyses presented in this paper only include the case of symmetrical loads. Asymmetrical single-phase load arrangements have a significant effect on voltage asymmetry in low-voltage grids, while in medium-voltage grids the effect is negligible, as the authors showed in their paper [34]. It was assumed that the line is loaded symmetrically at three points, with a power of 0.667 MW (total 2 MW) and a power factor \(\cos\varphi=0.95\) of an inductive character. The load impedance taken in this case is \(Z_{R}=304.34-j100.43\ \Omega\). As a result of the analytical calculations, the values of the voltage asymmetry factors were obtained using the equivalent model (Figure 1) and the equations (\(1\div 20\)). Figures 2-4 show the dependence of the zero-sequence voltage asymmetry factor \(\alpha_{U0}\) as a function of the earth fault compensation detuning factor s for different line cross-sections and lengths.