## 2 Amplitude and Phase Characteristics of PSCFCs in Active Distribution Networks
### Amplitude and Phase Characteristics of PSCFCs in Active Distribution Networks with IBDGs
#### 2.2.2 Phase Characteristics of PSCFCs  
Based on the IEEE 1547 standard and the China national standard [31], the IBDGs in the actual distribution network adopt a constant power (PQ) control strategy that suppresses the negative-sequence component and has a low voltage ride-through (LVRT) function. After a fault occurs, the fault current provided by the IBDG during LVRT is [16]  
\[\left\{\begin{array}{l}I_{\text{DG:d}}=\min\{k(0.9-U_{\text{df}}^{+}),I_{ \text{max}}\}\\ I_{\text{DG:d}}=\min\{\frac{P_{\text{ref}}}{U_{\text{df}}^{+}},\sqrt{I_{\text{ max}}^{2}-I_{\text{DG:d}}^{2}}\}\\ I_{\text{DG:amp}}=\sqrt{I_{\text{DG:d}}^{2}+I_{\text{DG:d}}^{2}}\\ \alpha=\arctan\frac{I_{\text{DG:d}}}{I_{\text{DC:d}}}\end{array}\right. \tag{3}\]  
where \(I_{\text{DG:d}}\) and \(I_{\text{DG:q}}\) are the d- and q-axis components of the IBDG output currents during faults, representing the active and reactive currents, respectively; \(U_{\text{df}}^{+}\) is the positive-sequence component of the grid-connected point voltage; \(k\) is the reactive support coefficient; \(I_{\text{max}}\) is the maximum fault current allowed to be output by the IBDG, generally limited to 2 times the rated current; \(I_{\text{DG:amp}}\) is the amplitude of the IBDG output current; \(\alpha\) is the angle of the grid-connected point voltage ahead of the output current.  
From Equation (3), the phase relationship between the output current of the IBDG and the voltage of the grid-connected point can be obtained, as shown in Figure 4. \(\dot{U}_{\text{d0}}\) is the pre-fault grid-connected point voltage, and assuming that the post-fault grid-connected point voltage lags behind the pre-fault angle, the sector area in Figure 4 with \(I_{\text{max}}\) as the radius is the range of the current universal phasor output by the IBDG. When the voltage drop at the grid-connected point is small, corresponding to \(\dot{U}_{\text{d1}}\) in Figure 4, the reactive current \(I_{\text{DG:q}}\) and the active current \(I_{\text{DG:d}}\) increase slightly, the IBDG output current is \(I_{\text{DG1}}\), the PSCFC on the IBDG side is \(\Delta I_{\text{DG1}}\), and the angle between \(\Delta I_{\text{DG1}}\) and the pre-fault current \(I_{\text{DG:d}}\) is slighter and less than 90\({}^{\circ}\); similarly, when the voltage drop at the grid-connected point is significant, corresponding to \(\dot{U}_{\text{d2}}\) in Figure 4, the angle between \(\Delta I_{\text{DG2}}\) and the pre-fault current \(I_{\text{DG:d}}\) is greater than 90\({}^{\circ}\).  
In summary, the phase angle of the PSCFC on the IBDG side is affected by the amplitude drop and phase change degree of the grid-connected point voltage. Under different fault conditions, the amplitude drop and phase change are different, resulting in a significant difference in the value of the phase angle. In addition, when the fault point is far away from the IBDG and the fault resistance is larger, the crossing current is dominant in the short-circuit current on the IBDG side. At this time, the amplitude ratio of the PSCFCs on  
Figure 4: The phase relationship between the output current of the IBDG and the voltage of the grid-connected point.  
both sides of the feeder is significantly reduced, and the absolute value \(\mid\delta\mid\) of the phase difference is greatly increased, approaching nearly 180\({}^{\circ}\).