## 3 Distributed Residual Current Protection Utilizing Closed-Section Methods
### Protective Criterion Design
The residual current matrix measured by each LTU is defined as \(\dot{I}_{\text{rLTU}}\), and the element in it, which is the residual current phasor measured by each LTU, is recorded as \(\dot{I}_{m}\) (\(n=1,2,\ldots,N\)), which can be expressed as  
\[\dot{I}_{\text{rLTU}}=\left[\dot{I}_{\text{r1}}\quad\cdots\quad\dot{I}_{\text{ rN}}\right]^{\text{T}} \tag{12}\]  
The residual current matrix of the closed section is recorded as \(\Delta\dot{I}_{\text{CSm}}\) (\(m=1,2,\ldots,\)\(M\)), which is given by  
\[\Delta\dot{I}_{\text{CSm}}=\left[\Delta\dot{I}_{\text{CS1}}\quad\cdots\quad \Delta\dot{I}_{\text{CSM}}\right]^{\text{T}} \tag{13}\]  
According to the definition of the relationship matrix, the matrix \(R\), \(\dot{I}_{m1}\), \(\Delta\dot{I}_{m1}\) satisfies the following:  
\[\Delta\dot{I}_{\text{CSm}}=R\dot{I}_{\text{rLTU}} \tag{14}\]  
Taking the absolute value (modulus value) of the elements in the matrix \(\Delta\dot{I}_{\text{CSm}}\) as the current action matrix, denoted as \(\Delta\dot{I}_{\text{CSd}}\), it is expressed as  
\[\Delta I_{\text{CSd}}=\left[\left|\Delta\dot{I}_{\text{CS1}}\right|\quad \cdots\quad\left|\Delta\dot{I}_{\text{CSM}}\right|\right]^{\text{T}} \tag{15}\]  
The braking relationship matrix is defined as \(R_{\text{res}}\). \(R_{\text{res}}\) is the absolute value of the elements in the relationship matrix \(R\), which can be expressed as  
\[R_{\text{res}}=\left[ \begin{array}{cccc}\left|\text{r}_{11}\right|&\left| \text{r}_{12}\right|&\cdots\quad\left|\text{r}_{1N}\right|\\ \left|\text{r}_{21}\right|&\left|\text{r}_{12}\right|&\cdots\quad\left|\text{ r}_{2N}\right|\\ \vdots&\vdots&\cdots\quad\vdots\\ \left|\text{r}_{M1}\right|&\left|\text{r}_{M2}\right|&\cdots\quad\left|\text{ r}_{MN}\right|\\ \end{array} \right] \tag{16}\]  
According to the definition of the braking relation matrix, the braking current matrix is expressed as  
\[I_{\text{CSres}}=R_{\text{res}}\left|\dot{I}_{\text{CSm}}\right| \tag{17}\]  
Defining the protection minimum operating current matrix as \(I_{\text{CSset}}\), the elements in it are recorded as \(I_{\text{CSset}}\) (\(m=1,2,\ldots,M\)) and \(I_{\text{CSset}}\) can be expressed as \[I_{\text{CSset}}=\left[I_{\text{CSset1}}\quad\cdots\quad I_{\text{CSset}M}\right]^{T} \tag{18}\]  
The elements in Equation (18) are set according to Equations (9) and (10). Each row in the above matrix represents a closed section, and (15), (17), (18) are put into (6), and the matrix representing whether the operating current exceeds the limit can be obtained, which is defined as \(D\). The limit matrix \(D\) is expressed as  
\[D=\left[D_{1}\cdots D_{i}\cdots D_{M}\right] \tag{19}\]  
The element \(D_{i}\) (\(i=1,\,2,\ldots,M\)) in the matrix has two states according to whether the closed-section action current exceeds the limit:  
\[\left\{ \begin{array}{c}D_{i}=1\\ D_{i}=0\end{array} \right. \tag{20}\]  
\(D_{i}=1\) indicates that the operating current exceeds the limit; \(D_{i}=0\) means the operating current does not exceed the limit. Under normal operation, the elements in the limit-crossing matrix \(D\) are all 0. After the residual current of the closed section exceeds the limit, the corresponding element becomes 1.