# Modeling of Voltage Fluctuations Generated by Arc Furnaces
## 3 Modeling of Voltage Fluctuations in Parallel Operation of Arc Furnaces
hence:  
\[\text{Dcos}\ \beta=-\text{U}_{\text{RT1}}\frac{\text{R}_{\text{L}}}{\text{R}_{ \text{T}}}\text{sin}\ \ \alpha_{1}+\text{U}_{\text{XTI}}\frac{\text{X}_{\text{L}}}{\text{X}_{\text{T}}} \text{cos}\ \alpha_{1}-\text{U}_{\text{RT2}}\frac{\text{R}_{\text{L}}}{\text{R}_{\text{T}}} \text{sin}\ \ \alpha_{2}+\text{U}_{\text{XTI2}}\frac{\text{X}_{\text{L}}}{\text{X}_{\text{T}}} \text{cos}\ \alpha_{2} \tag{25}\]  
\[\text{sin}\ \alpha_{1}=\frac{\text{U}_{\text{XTI}}}{\text{U}_{\text{S}}}\text{ cos}\ \alpha_{1}=\frac{\text{U}_{\text{A}\text{e}\text{T}}+\text{U}_{\text{RT1}}}{\text{U}_ {\text{S}}} \tag{26}\]  
\[\text{sin}\ \alpha_{2}=\frac{\text{U}_{\text{XTI2}}}{\text{U}_{\text{S}}} \text{cos}\ \alpha_{1}=\frac{\text{U}_{\text{A}\text{e}\text{T}}+\text{U}_{\text{RT2}}}{ \text{U}_{\text{S}}} \tag{27}\]  
\[\text{U}_{\text{XTI}}=\text{U}_{\text{S}}\text{sin}\ \alpha_{1}\text{U}_{\text{ XTI2}}=\text{U}_{\text{S}}\text{sin}\ \alpha_{2} \tag{28}\]  
we get (for \(\text{R}_{\text{T1}}=\text{R}_{\text{T2}}\) and \(\text{X}_{\text{T1}}=\text{X}_{\text{T2}}\)):  
\[\text{Dsin}\ \beta=\text{U}_{\text{S}}\frac{\text{X}_{\text{L}}}{\text{X}_{ \text{T}}}(\text{sin}^{2}\ \alpha_{1}+\text{sin}^{2}\ \alpha_{2})+\text{U}_{\text{S}}\frac{\text{R}_{\text{L}}}{\text{R}_{\text{T} }}(\text{sin}\ \alpha_{1}\text{cos}\ \alpha_{1}+\text{sin}\ \alpha_{2}\text{cos}\ \alpha_{2}) \tag{29}\]  
\[\text{Dcos}\ \beta=\text{U}_{\text{S}}\frac{\text{X}_{\text{L}}}{\text{X}_{ \text{T}}}(\text{sin}\ \alpha_{1}\text{cos}\ \alpha_{1}+\text{sin}\ \alpha_{2}\text{cos}\ \alpha_{2})-\text{U}_{\text{S}}\frac{\text{R}_{\text{L}}}{\text{R}_{\text{T} }}(\text{sin}^{2}\ \alpha_{1}+\text{sin}^{2}\ \alpha_{2}) \tag{30}\]  
The determined voltage \(\text{U}_{\text{S1}}\) (in the first approximation--Equation (31) is substituted for the equation):  
\[U_{L}^{2}=U_{S}^{2}+2U_{S}D\sin\beta+(D\sin\beta)^{2}+(D\cos\beta)^{2} \tag{31}\]  
Then, the supply voltage \(U_{L}\) for the first \(U_{\text{S}}\) approximation is determined. By correcting \(U_{\text{S}}\) in accordance with the relationship:  
\[\text{U}^{\prime}{}_{\text{S}}=\text{U}_{\text{S1}}\frac{\text{U}_{\text{L}} }{\text{U}^{\prime}{}_{\text{L}}} \tag{32}\]  
another approximation \(\text{U}_{\text{S}}\) is determined until the assumed accuracy of the supply voltage is obtained, e.g., \(\text{U}_{\text{L}}=100\%+0.000001\%\). Based on the calculated voltages \(\text{U}_{\text{SAA}}\), \(\text{U}_{\text{SBB}}\), \(\text{U}_{\text{SAB}}\), \(\text{U}_{\text{SBA}}\), the average value of the voltage on the rails of the steel plant is:  
\[\overline{\text{U}}_{\text{S2}}=\frac{\text{U}_{\text{SAA}}+\text{U}_{\text{ SBB}}+\text{U}_{\text{SAB}}+\text{U}_{\text{SBA}}}{4} \tag{33}\]  
and the mean of the squares of \(\text{U}_{\text{S}}\) voltages:  
\[\text{Dcos}\ \beta=-\text{U}_{\text{RT1}}\frac{\text{R}_{\text{L}}}{\text{R}_{ \text{T}}}\text{sin}\ \ \alpha_{1}+\text{U}_{\text{XTI}}\frac{\text{X}_{\text{L}}}{\text{X}_{\text{T}}} \text{cos}\ \alpha_{1}-\text{U}_{\text{RT2}}\frac{\text{R}_{\text{L}}}{\text{R}_{\text{T}}} \text{sin}\ \ \alpha_{2}+\text{U}_{\text{XTI2}}\frac{\text{X}_{\text{L}}}{\text{X}_{\text{T}}} \text{cos}\ \alpha_{2} \tag{25}\]0  
and standard deviation SD (\(\text{U}_{\text{S2}}\)):  
\[\text{Dcos}\ \beta=-\text{U}_{\text{RT1}}\frac{\text{R}_{\text{L}}}{\text{R}_{ \text{T}}}\text{sin}\ \ \alpha_{1}+\text{U}_{\text{XTI}}\frac{\text{X}_{\text{L}}}{\text{X}_{\text{T}}} \text{cos}\ \alpha_{1}-\text{U}_{\text{RT2}}\frac{\text{R}_{\text{L}}}{\text{R}_{\text{T}}} \text{sin}\ \ \alpha_{2}+\text{U}_{\text{XTI2}}\frac{\text{X}_{\text{L}}}{\text{X}_{\text{T}}} \text{cos}\ \alpha_{2} \tag{25}\]1  
Based on the knowledge of the standard deviation of the voltage in the operation of a single furnace \(\text{SD}(\text{U}_{\text{S1}})\) and the standard deviation of the voltage in the operation of two furnaces \(\text{SD}(\text{U}_{\text{S2}})\), it is possible to determine the coefficient of voltage fluctuation increase on the steel mill busbars caused by the connection of the second furnace--Equation (36).