# Modeling of Voltage Fluctuations Generated by Arc Furnaces
## 3 Modeling of Voltage Fluctuations in Parallel Operation of Arc Furnaces
\begin{array}{l}\mathrm{U}_{\mathrm{L}}^{2}=\mathrm{U}_{\mathrm{S}}^{2}+2 \mathrm{U}_{\mathrm{S}}^{2}\frac{\mathrm{X}_{\mathrm{L}}}{\mathrm{X}_{ \mathrm{T}}}(2-\frac{\mathrm{U}_{\mathrm{Arc1}}^{2}+\mathrm{U}_{\mathrm{Arc2} }^{2}}{\mathrm{U}_{\mathrm{S}}^{2}})+\\ +\mathrm{U}_{\mathrm{S}}^{2}\frac{\mathrm{X}_{\mathrm{L}}^{2}}{\mathrm{X}_{ \mathrm{T}}^{2}}\left[\left(2-\frac{\mathrm{U}_{\mathrm{Arc1}}^{2}+\mathrm{U} _{\mathrm{Arc2}}^{2}}{\mathrm{U}_{\mathrm{S}}^{2}}\right)^{2}+\left(\frac{ \mathrm{U}_{\mathrm{Arc1}}}{\mathrm{U}_{\mathrm{S}}}\sqrt{1-\left(\frac{ \mathrm{U}_{\mathrm{Arc1}}}{\mathrm{U}_{\mathrm{S}}}\right)^{2}}+\frac{ \mathrm{U}_{\mathrm{Arc2}}}{\mathrm{U}_{\mathrm{S}}}\sqrt{1-\left(\frac{ \mathrm{U}_{\mathrm{Arc2}}}{\mathrm{U}_{\mathrm{S}}}\right)^{2}}\right)^{2} \right]\end{array}  \tag{20}\]  
After omitting the expressions \((\mathrm{X}_{\mathrm{L}}/\mathrm{X}_{\mathrm{T}})^{2}\), the above formula is simplified to the form:  
\[\mathrm{U}_{\mathrm{L}}^{2}=\mathrm{U}_{\mathrm{S}}^{2}+4\mathrm{U}_{\mathrm{ S}}^{2}\frac{\mathrm{X}_{\mathrm{L}}}{\mathrm{X}_{\mathrm{T}}}-2\frac{\mathrm{X}_{ \mathrm{L}}}{\mathrm{X}_{\mathrm{T}}}(\mathrm{U}_{\mathrm{Arc1}}^{2}+\mathrm{ U}_{\mathrm{Arc2}}^{2}) \tag{21}\]  
from which the first approximation of the tension on the steel rails can be determined:  
\[\mathrm{U}_{\mathrm{SI}}^{2}=\frac{\mathrm{U}_{\mathrm{L}}^{2}+2\frac{\mathrm{ X}_{\mathrm{L}}}{\mathrm{X}_{\mathrm{T}}}(\mathrm{U}_{\mathrm{Arc1}}^{2}+\mathrm{ U}_{\mathrm{Arc2}}^{2})}{1+4\frac{\mathrm{X}_{\mathrm{L}}}{\mathrm{X}_{\mathrm{T}}}} \tag{22}\]  
\[\mathrm{U}_{\mathrm{SI}}=\sqrt{\frac{\mathrm{U}_{\mathrm{L}}^{2}+2\mathrm{X}_{ \mathrm{T}}^{2}(\mathrm{U}_{\mathrm{Arc1}}^{2}+\mathrm{U}_{\mathrm{Arc2}}^{2} )}{1+4\frac{\mathrm{X}_{\mathrm{L}}}{\mathrm{X}_{\mathrm{T}}}}} \tag{23}\]  
Successive approximations to \(\mathrm{U}_{\mathrm{S}}\) are obtained by substituting to Equation (23) until a satisfactory accuracy is obtained, i.e., when the voltage is \(\mathrm{U}_{\mathrm{L}}=100\%\) (with an accuracy of 0.000001%). Due to the fact that \(\mathrm{U}_{\mathrm{S}}\) is determined by the method of successive approximations (iterative method), the accuracy presented in the paper was adopted 0.000001%. For example, the \(\mathrm{U}_{\mathrm{S}}\) determined in the first approximation is \(\mathrm{U}_{\mathrm{S}}(1)=97.80056145\%\). In the last step (with the accepted accuracy), the determined voltage is \(\mathrm{U}_{\mathrm{S}}=97.80876776\%\). Figure 13 shows a diagram of the power supply system of a steelworks with two arc furnaces, taking into account the resistance in the electric arc supply circuit. Based on the vector diagram presented in Figure 14, the following dependencies were determined:  
\[ \begin{array}{l}\mathrm{Dsin}\beta=\mathrm{U}_{\mathrm{RT1}}\mathrm{cos}\ \alpha_{1}+\mathrm{U}_{\mathrm{XT1}}\mathrm{sin}\ \alpha_{1}+\mathrm{U}_{\mathrm{RT2}}\mathrm{cos}\ \alpha_{2}+\mathrm{U}_{\mathrm{XT2}}\mathrm{sin}\ \alpha_{2}=\\ =\mathrm{U}_{\mathrm{RT1}}\frac{\mathrm{R}_{\mathrm{L}}}{\mathrm{RT}}\cos\ \alpha_{1}+\mathrm{U}_{\mathrm{XT1}}\frac{\mathrm{X}_{\mathrm{T}}}{\mathrm{ X}_{\mathrm{T}}}\mathrm{sin}\ \alpha_{1}+\mathrm{U}_{\mathrm{RT2}}\frac{\mathrm{R}_{\mathrm{L}}}{\mathrm{RT}} \mathrm{cos}\ \alpha_{2}+\mathrm{U}_{\mathrm{XT2}}\frac{\mathrm{X}_{\mathrm{L}}}{\mathrm{X}_{ \mathrm{T}}}\mathrm{sin}\ \alpha_{2}=\\ =\mathrm{U}_{\mathrm{XT1}}\frac{\mathrm{RT}}{\mathrm{RT}}\frac{\mathrm{R}_{ \mathrm{L}}}{\mathrm{RT}}\cos\ \alpha_{1}+\mathrm{U}_{\mathrm{XT1}}\frac{\mathrm{X}_{ \mathrm{L}}}{\mathrm{RT}}\frac{\mathrm{X}_{\mathrm{L}}}{\mathrm{RT}}\sin \alpha_{1}+\mathrm{U}_{\mathrm{XT2}}\frac{\mathrm{R}_{\mathrm{T}}}{\mathrm{ R}_{\mathrm{L}}}\frac{\mathrm{R}_{\mathrm{L}}}{\mathrm{RT}}\cos\ \alpha_{2}+\mathrm{U}_{\mathrm{XT2}}\frac{\mathrm{X}_{\mathrm{L}}}{\mathrm{X}_{ \mathrm{T}}}\mathrm{sin}\ \alpha_{2}\end{array}  \tag{24}\]  
Figure 13: Scheme of the power supply system of a steelworks with two arc furnaces including resistance.