## 3 Distributed Residual Current Protection Utilizing Closed-Section Methods  
### Protection Principle  
The application of graph theory serves to streamline the topological connectivity within the low-voltage distribution network, facilitating the creation of a minimal closed-section area on the topology diagram. Within this designated area, all external contact points are equipped with Line Terminal Units (LTUs) and are incapable of further subdivision. The residual current at these external contact points is quantifiable, with all currents at these points adhering to [PERSON]'s law. This concept is visually represented in Figure 3.  
\[\Delta\dot{l}_{\text{CS}_{0}}=\dot{l}_{\text{r0}}+\dot{l}_{\text{r11}}+\dot{l}_ {\text{r12}}+\ldots\ldots+\dot{l}_{\text{r1m}} \tag{1}\]  
\[\Delta\dot{l}_{\text{CS}_{1}}=\dot{l}_{\text{r11}}+\dot{l}_{\text{r12}}+\ldots \ldots+\dot{l}_{\text{r1m}}+\dot{l}^{\prime}_{\text{r11}}+\dot{l}^{\prime}_{ \text{r12}}+\ldots\ldots+\dot{l}^{\prime}_{\text{r1m}} \tag{2}\]  
\[\Delta\dot{l}_{\text{CS}_{2}}=\dot{l}_{\text{r21}}+\dot{l}_{\text{r22}}+\dot{l }_{\text{r23}}+\dot{l}_{\text{r24}}+\dot{l}^{\prime}_{\text{r21}}+\dot{l}^{ \prime}_{\text{r22}}+\dot{l}^{\prime}_{\text{r23}}+\dot{l}^{\prime}_{\text{ r24}} \tag{3}\]  
\[\Delta\dot{l}_{\text{CS}_{3}}=\dot{l}_{\text{r31}}+\dot{l}_{\text{r32}}+\dot{ l}_{\text{r33}}+\dot{l}_{\text{r34}}+\dot{l}^{\prime}_{\text{r31}}+\dot{l}^{ \prime}_{\text{r32}}+\dot{l}^{\prime}_{\text{r33}}+\dot{l}^{\prime}_{\text{ r34}} \tag{4}\]  
It is evident that during normal operations, the residual current within the closed section is limited to the residual current within its designated protection range. However, following an earth fault (leakage), the residual current increases significantly, with almost all of it representing the fault branch current. Consequently, the protection settings can be set at a very low threshold, significantly enhancing sensitivity.  
For instance, considering general protection at the transformer outlet, its residual current comprises the cumulative inherent residual of the entire substation area. In contrast, for CS\({}_{0}\) utilizing data measured at the same location, its residual current is solely attributed to the inherent residual current on the bus within the substation area. In scenarios involving the TN configuration with repeated earthing of the neutral point on the load side, a judicious selection of the closed section can effectively address the challenge of installing residual current protection. An illustrative example is provided in Figure 4, where A, B and C represent three-phase power lines, N is a neutral line, PE represent protecting earthing line.  
The external connection point of the closed section is set before the repeated earthing point of the TN-C-S configuration. The residual current of the closed section composed of the residual current measured by the two points P1, P2, and P3 can be calculated as  
\[\Delta\dot{l}_{\text{P}}=\dot{l}_{\text{rP1}}+\dot{l}_{\text{rP2}}+\dot{l}_{ \text{rP3}} \tag{5}\]  
Figure 3: Schematic of distributed residual current protection.  
In Equation (5), during normal operation, \(\Delta I_{\rm P}\) does not include the current through the grounding point of the N line and the residual current generated by the load. Repeated earthing does not affect the sensitivity of the distributed residual current protection and can achieve reliable earth (leakage) fault protection.