## 3 Current Differential Protection Based on Adaptive Angle Compensation Coefficient
### Principle of Current Differential Protection Based on Adaptive Phase Angle Compensation Coefficient  
The TCDP based on the PSCFC can trip reliably and sensitively in most fault scenarios. However, as described in Section 3.1, in some internal fault scenarios, the sensitivity of TCDP decreases or even the TCDP cannot trip. To solve this problem, the adaptive phase angle compensation coefficient is introduced in the TCDP to correct the phase of the measured current so that the improved CDP can trip sensitively and reliably in any fault scenario. The improved criterion can be expressed as  
\[\left|\Delta\dot{I}_{\text{m}}+\Delta\dot{I}_{\text{n}}\cdot e^{jg(\varepsilon )\delta}\right|>K\left|\Delta\dot{I}_{\text{m}}-\Delta\dot{I}_{\text{n}}\cdot e ^{jg(\varepsilon)\delta}\right| \tag{6}\]  
where \(e^{j}\) is the twiddled factor; \(g(\varepsilon)\) is the step function; and \(\delta\) is the phase difference of the PSCFCs on both sides of the protected feeder. Their values can be expressed as Equation (7). \[\left\{ \begin{array}{l}\delta=\arg(\Delta\dot{I}_{\text{m}})-\arg(\Delta\dot {I}_{\text{n}})\\ g(\varepsilon)=\left\{\begin{array}{l}0\,,\varepsilon\leq a\\ 1\,,\varepsilon>a\end{array}\right.\end{array} \right. \tag{7}\]  
In Equation (6), \(e^{jg(\varepsilon)\delta}\) is the adaptive phase angle compensation coefficient, which is used to compensate for the phase of the acquired PSCFC \(\Delta\dot{I}_{\text{n}}\) on the DG side. As can be seen from Equation (7), the value of the adaptive phase angle compensation coefficient is adaptively adjusted according to the magnitude ratio \(\varepsilon\) and phase angle difference \(\delta\) of the PSCFC on both sides of the protected feeder. When \(\varepsilon>a\), \(g(\varepsilon)=1\), the acquired PSCFC \(\Delta\dot{I}_{\text{n}}\) on the DG side is compensated so that the phase of \(\Delta\dot{I}_{\text{n}}\) after compensation is equal to the phase of the PSCFC \(\Delta\dot{I}_{\text{m}}\) on the GRID side; when \(\varepsilon\leq a\), \(g(\varepsilon)=0\), the acquired PSCFC \(\Delta\dot{I}_{\text{n}}\) is not compensated, and Equation (6) is the same as Equation (4). For the external fault of the protected area, under ideal conditions, the amplitude ratio of the PSCFCs on both sides of the protected feeder is 1, and the phase difference is 180\({}^{\circ}\). However, in practical engineering, the measurement error of the CT and the lock angle of the CDP need to be considered. The maximum measurement error of CT is usually \(\pm\)10%, and the amplitude ratio of the PSCFCs is taken in the range of [0.82,1.22]. The lock angle of the CDP can be expressed as [18]  
\[\varphi_{\text{b}}=7^{\circ}+15^{\circ}+\frac{L}{100}\times 6^{\circ}+\varphi_{y} \tag{8}\]  
where \(7^{\circ}\) denotes the maximum error of the CT secondary current; \(15^{\circ}\) denotes the error caused by the sequence filter and transceiver operation loop in the protection device; \(L\) denotes the protected line length; this part of the error is less than \(1^{\circ}\) due to the short feeders in the distribution network; \(\varphi_{y}\) indicates the margin angle, generally taken as \(15^{\circ}\)but taking into account the influence of unmeasured load branches, the margin angle in this paper is taken as 22\({}^{\circ}\). In summary, the lock angle \(\varphi_{\text{b}}\) is 45\({}^{\circ}\). According to the boundary conditions of the fault within the protected feeder as described in Section 2 and considering a certain margin, the value of \(a\) in Equation (7) can be determined as 1.5. The specific criterion of the adaptive current differential protection (ACDP) based on the phase angle compensation coefficient can be obtained as  
\[\left\{ \begin{array}{l}\left|\Delta\dot{I}_{m}+\Delta\dot{I}_{n}\cdot e^{j( \varepsilon)\delta}\right|>K\left|\Delta\dot{I}_{m}-\Delta\dot{I}_{n}\cdot e^{ j(\varepsilon)\delta}\right|\\ \delta=\arg(\Delta\dot{I}_{m})-\arg(\Delta\dot{I}_{n})\\ g(\varepsilon)=\left\{\begin{array}{l}0\,,\varepsilon\leq 1.5\\ 1\,,\varepsilon>1.5\end{array}\right.\end{array} \right.