## 3 Adaptability Analysis of Traditional Current Differential Protection  
### Full Current Differential Protection  
The commonly used protection criteria for full current differential protection are as follows:  
\[\left\{ \begin{array}{c}I_{d\varphi}\ >\ kI_{r\varphi}\\ I_{d\varphi}\ >\ I_{o}\\ I_{d\varphi}\ =\left|\mathbf{I_{mp}}\ +\ \mathbf{I_{mp}}\right|\\ I_{r\varphi}\ =\left|\mathbf{I_{mp}}\ -\ \mathbf{I_{np}}\right|\\ \end{array} \right. \tag{6}\]  
where \(I_{\Phi\varphi}\), \(I_{r\varphi}\) respectively denote the differential and restraint currents for phase \(\varphi\); \(I_{o}\) denotes the action threshold; and \(\mathbf{I_{m\varphi}}\), \(\mathbf{I_{n\varphi}}\) respectively denote the phase currents measured at the protection installation points on the PV side and grid side of the transmission line. Considering an internal phase-A-to-phase-B fault as an example, due to the negative-sequence suppression by the inverters, the initial phase angle difference of the fault currents in the two affected phases remains fixed at \(120^{\circ}\) on the PV side. Crucially, this characteristic remains invariant across varying fault conditions. According to [PERSON]'s current law, the fault currents of phase A and phase B satisfy the following equation:  
\[\mathbf{I_{na}}+\mathbf{I_{nb}}\ =\ \mathbf{I_{mc}}\ +\ \mathbf{I_{e}} \tag{7}\]  
where \(\mathbf{I_{na}}\), \(\mathbf{I_{nb}}\), and \(\mathbf{I_{mc}}\) are the corresponding phase currents at the protection installations on the grid side and the PV side, respectively, and \(I_{e}\) denotes the line distributed capacitance current. Since the phase current on the system side always maintains a phase difference of \(120^{\circ}\) between the two phases, \(\mathbf{I_{mc}}=-(\mathbf{I_{na}}+\mathbf{I_{nb}})\). As shown in Equations (2)-(5), the transient fault current amplitude coefficient of grid-forming inverters depends on several factors, including the virtual impedance X/R ratio, the phase angle advance of the virtual internal voltage relative to the PCC voltage, and the magnitude of the PCC voltage pre- and post-fault. This behavior exhibits greater complexity and uncertainty compared to the fault current characteristics observed in traditional power systems. Furthermore, due to the presence of distributed capacitance, the phase angle difference between fault currents of phase A and phase B on the grid side deviates substantially from the conventional \(180^{\circ}\). Taking \(\mathbf{I_{nb}}\) as the reference, the phase diagrams are illustrated in Figure 3.  
The fault current phasors are analyzed under two scenarios. As shown in Case a, \(\mathbf{I_{na}}\) leads \(\mathbf{I_{mc}}+\mathbf{I_{e}}\), and they all reside in the second or third quadrant. Due to the small magnitude of \(\mathbf{I_{e}}\), the phase angle difference between \(\mathbf{I_{mc}}+\mathbf{I_{e}}\) and \(\mathbf{I_{mc}}\) is small. Thus, \(\mathbf{I_{na}}\) also  
Figure 3: The phasor diagrams of full current on both sides of the transmission line. leads \(\mathbf{I}_{\rm mc}\). The phase angle difference between \(\mathbf{I}_{\rm na}\) and \(\mathbf{I}_{\rm mc}\) ranges from 0\({}^{\circ}\) to 180\({}^{\circ}\). Given that the phase angle differences among \(\mathbf{I}_{\rm ma}\), \(\mathbf{I}_{\rm mb}\) and \(\mathbf{I}_{\rm mc}\) are mutually offset by 120\({}^{\circ}\), \(\theta_{\rm a}\) ranges from 120\({}^{\circ}\) to 180\({}^{\circ}\), and \(\theta_{\rm b}\) ranges from 60\({}^{\circ}\) to 180\({}^{\circ}\). When \(\theta_{\rm a}\) or \(\theta_{\rm b}\) exceeds 90\({}^{\circ}\) under specific fault scenarios, the restraint current exceeds the differential current in that phase, reducing protection sensitivity and significantly increasing the risk of failure to operate. As shown in Case b, \(\mathbf{I}_{\rm na}\) lags \(\mathbf{I}_{\rm mc}+\mathbf{I}_{\rm e}\), and they all reside in the first or fourth quadrant. Due to the small magnitude of \(\mathbf{I}_{\rm e}\), the phase angle difference between \(\mathbf{I}_{\rm mc}+\mathbf{I}_{\rm e}\) and \(\mathbf{I}_{\rm mc}\) is small.