## 3 Distributed Residual Current Protection Utilizing Closed-Section Methods
### Protective Criterion Design  
Referring to the method of differential protection setting [3; 4; 5; 6; 7; 8], Equation (6) gives a distributed residual current protection criterion. \[\left\{ \begin{array}{l}I_{\rm d}>KI_{\rm res}\\ I_{\rm d}>I_{\rm set}\end{array} \right. \tag{6}\]  
In the formula, \(K\) is the ratio braking coefficient; \(I_{\rm d}\) and \(I_{\rm res}\) are the operating current and braking current, respectively; \(I_{\rm set}\) is the minimum operating current. The first criterion is the action criterion; the second is the protection initiation criterion. The expressions for \(I_{\rm d}\) and \(I_{\rm res}\) are as follows:  
\[\left\{ \begin{array}{l}I_{\rm d}=|\sum\limits_{i=1}^{N}\dot{I}_{i}|\\ I_{\rm res}=\sum\limits_{i=1}^{N}|\dot{I}_{i}|\end{array} \right. \tag{7}\]  
\(\dot{I}_{i}\) represents the residual current phasor measured by the device at the external contact point of the closed section, this paper sets the direction of the current flowing into the closed section as positive and the outflow as negative. \(I_{\rm set}\) is usually selected in two ways:  
(1) It must exceed the maximum unbalanced current when externally grounded. We obtain the following:  
\[I_{\rm set}=K_{\rm rel}K_{\rm rp}K_{\rm er}K_{\rm st}I_{\rm kmax} \tag{8}\]  
\(K_{\rm rel}\) is the reliability coefficient, which is taken as 1.2-1.3; \(K_{\rm rp}\) is the aperiodic component coefficient, which is taken as 1; \(K_{\rm er}\) is the 10% error coefficient of the current transformer; \(K_{\rm st}\) is the coefficient of the same type of transformer, and it is taken as 0.5 when the current transformer is of the same type; \(I_{\rm kmax}\) is the maximum short-circuit current flowing through the current transformer when it is externally grounded. (2) It must exceed the maximum inherent residual current in the closed section. We obtain the following:  
\[I_{\rm set}=K_{\rm rel}I_{\rm rmax} \tag{9}\]  
\(K_{\rm rel}\) is the reliability coefficient taken as 1.2-1.3; \(I_{\rm rmax}\) is the maximum residual current during normal operation in the closed-section protection. The design of the distributed residual current protection criterion is described in matrix form. First, the number of LTUs in the protection and the number of closed sections  
Figure 4: Schematic of distributed residual current protection of TN-C-S configuration. are determined. It is assumed that there are \(N\) LTUs and \(M\) closed sections in the protection. A relationship matrix \(R\) is defined to represent the relationship between the closed-section CS and LTU, and it is expressed as  
\[R=\left[ \begin{array}{cccc}\text{r}_{11}&\text{r}_{12}&\cdots&\text{r}_{1N}\\ \text{r}_{21}&\text{r}_{12}&\cdots&\text{r}_{2N}\\ \vdots&\vdots&\cdots&\vdots\\ \text{r}_{M1}&\text{r}_{M2}&\cdots&\tau_{MN}\\ \end{array} \right] \tag{10}\]  
In the matrix, the number of rows is equal to the number of closed sections, the number of columns is equal to the number of LTUs, and r\({}_{ij}\) (\(i=1,2,\ldots,N;j=1,2,\ldots,M\)) represents the relationship of each LTU in the closed section; r\({}_{ij}\) has three states:  
\[\left\{ \begin{array}{l}\text{r}_{ij}=1\\ \text{r}_{ij}=0\\ \text{r}_{ij}=-1\\ \end{array} \right. \tag{11}\]  
r\({}_{ij}=1\) means that the LTU is associated with the closed section, that is, the LTU is the external contact point of the closed section, and the current is for the inflow; r\({}_{ij}=0\) means that the LTU is not associated with the closed section; r\({}_{ij}=-1\) means that the LTU is associated with the closed section, and the current is for the outflow.