## 6 Grid and SDG Current Estimations
### Fault in the Section behind the SDG (Point C)  
The grid current is estimated from the relationship between the substation voltage (\(V_{S}\)) and the voltage of IBDG1 (\(V_{IBDG1}\)). The estimated grid current is expressed as  
\[I_{S-est}=[Z_{1}l_{1}]^{-1}[V_{S}-V_{IBDG1}], \tag{14}\]  
where \(l_{1}\) is the line length between the substation and IBDG1. On the other hand, the SDG current in the stator, field, and damper windings can be computed from a state-space (SS) equation of a synchronous machine (SM). The SS equation using the \(dq\)-axis current (\(I_{SDG,dq}\)) as a state variable is expressed as the following equations:  
\[\dot{I}_{SDG,dq}=L_{SDG}^{-1}[R_{SDG}+\omega L_{SDG}]I_{SDG,dq}-L_{SDG}^{-1}V_{ SDG,dq}, \tag{15}\]  
\[I_{SDG,dq}=\left[\begin{array}{c}I_{sq}\\ I_{sd}\\ I_{fd}\\ I_{ld}\\ I_{kq}\end{array}\right] \tag{16}\]  
\[V_{SDG,dq}=\left[\begin{array}{c}V_{tq}\\ V_{td}\\ V_{fd}\\ V_{kd}\\ V_{kq}\end{array}\right], \tag{17}\]  
\[R_{SDG}=\left[\begin{array}{cccc}R_{s}&0&0&0&0\\ 0&R_{s}&0&0&0\\ 0&0&R_{fd}&0&0\\ 0&0&0&R_{kd}&0\\ 0&0&0&0&R_{kq}\end{array}\right], \tag{18}\]  
\[L_{SDG}=\left[\begin{array}{cccc}L_{q}&0&0&0&L_{mq}\\ 0&L_{d}&L_{md}&L_{md}&0\\ 0&L_{md}&L_{fd}&L_{md}&0\\ 0&L_{md}&L_{md}&L_{kd}&0\\ L_{mq}&0&0&0&L_{kq}\end{array}\right], \tag{19}\]  
\[\omega=\left[\begin{array}{cccc}0&\omega_{r}&0&0&0\\ -\omega_{r}&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\end{array}\right]. \tag{20}\]  
Figure 6: Radial distribution system.  
The dynamics of the stator, field, and damper windings are taken into account. The SS equation can represent the dynamic behavior of the SM. The state variables of the SS include a _dq_-axis stator current (\(I_{sd}\) and \(I_{sq}\)), a _d_-axis field current (\(I_{fd}\)), and a _dq_-axis damper winding current (\(I_{ld}\) and \(I_{sq}\)). The _dq_-axis voltage (\(V_{SDG,dq}\)) matrix consists of a _dq_-axis terminal voltage (\(V_{td}\) and \(V_{sq}\)), _d_-axis field voltage (\(V_{fd}\)), and _dq_-axis damper winding voltage (\(V_{kd}\) and \(V_{sq}\)). The resistance (\(R_{SDG}\)) is the diagonal matrix, including stator resistance (\(R_{s}\)), _d_-axis field resistance (\(R_{fd}\)), and _dq_-axis damper winding resistance (\(R_{kd}\) and \(R_{kq}\)). The inductance (\(L_{SDG}\)) matrix consists of _dq_-axis inductance (\(L_{d}\) and \(L_{q}\)), _dq_-axis field inductance (\(L_{fd}\)), _dq_-axis damper winding inductance (\(L_{kd}\) and \(L_{kq}\)), _dq_-axis mutual inductance (\(L_{md}\) and \(L_{mq}\)), and zero elements. The angular velocity (\(\omega\)) matrix is built from angular rotor speed (\(\omega_{r}\)) and zero elements. According to Equation (15), _dq_-axis terminal voltage, _dq_-axis damper winding voltage, _d_-axis field voltage, angular rotor speed, resistance, and inductance are needed. The three-phase terminal voltage of SDG (\(V_{SDG,t}\)) can be calculated from  
\[V_{SDG,t}=V_{IBDG1}-[Z_{2}l_{2}]\Big{[}I_{g-est}+I_{IBDG1}\Big{]}, \tag{21}\]  
where \(l_{2}\) defines the line length between IBDG1 and the SDG. \(I_{IBDG1}\) is the current of IBDG1. When the three-phase terminal voltage is obtained, it is transformed from its _a_-_b_-_c_ frame to its _d_-_q_ frame. The _dq_-axis damper winding voltage is defined as zero due to the closed conducting path on the rotor. Otherwise, the _d_-axis field voltage and angular rotor speed are set at a nominal value. The resistance and inductance are known parameters. When the _dq_-axis stator current is achieved from the SS equation, it is transformed back from a _d_-_q_ frame into an _a_-_b_-_c_ frame. This transformed current is used as the estimated SDG current. Finally, both the estimated grid and SDG currents are sent to the FIM strategy, as shown in Figure 3.