# Reliability of Supply and the Impact of Weather Exposure and Protection System Failures +
## 3 Method  
In this section, a general and compact method for calculating reliability indices including protection system misoperations is postulated, using both approximate methods and a time-series method, based on the definitions of fault types as originally proposed in [17]. The basis for approximate methods is [34] in Section 3.1, and that for a time-series approach is [10] in Section 3.2. In power systems, we normally look at buses as vertices and transmission lines as edges. However, the idea behind the proposed methodology in this paper is to see vertices as transmission lines and edges as dependencies. This structure can then be utilized to calculate the contribution to failure rates at a given transmission line, given failures at adjacent lines. When including protection system failures into the system, we consider two different types of lines in each case: the target line \(i\), for which we wish to calculate the failure rates, and the source line \(j\), which is adjacent to the target line and contributes to the failure rate of the target line through propagation of protection system failures. The target- and source lines are adjacent lines if they are connected to a common bus, as seen in Figure 2. The subscript \(l\) is used when it is not specific if the line is a target or a source line. A line is associated with a protection system on each end. These are referred to as the a-side and the b-side protection system, represented as the set \(s=\{a,b\}\) for simplicity. For a line \(l\), the failure rate of the line \(\lambda_{l}\), and the two protection systems, \(P_{l}^{s}\) are considered. The protection systems have three parameters: (1) a specific annual failure rate, \(\lambda_{l}^{s}\), (2) a conditional probability of missing operation, \(p_{l}^{s,m}\), if the line experiences a failure, and (3) upon a correctly cleared failure of an adjacent line, a conditional probability of unwanted non-selective tripping at the target line, \(p_{l}^{s,u}\). The associated outage duration, \(r\), is denoted with a subscript indicating which line is considered, and a superscript indicating which fault type it is applicable to. All line- and protection system specific information is represented as ordered column vectors. The primary concern is how misoperation of the protection system of one transmission line can cause an adjacent line to be isolated from the system. The system can initially be considered a graph \(G\), where each edge \(l=\{u,v\}\) represents a transmission line, and the buses are represented by vertices, which are unique observations of \(u\) and \(v\). Let \(u\) represent the a-side connecting bus of a transmission line, and \(v\) the b-side connecting bus of the line. An adjacency matrix indicates connections between vertices and is used to represent the system in a form where vertices are transmission lines and edges are directed paths of failure propagation between the lines. An adjacency matrix is constructed for  
Figure 1: Illustrative graph of a 4-bus power system. Figure 2: Two adjacent transmission lines, \(i\) and \(j\). each side of the source line, A\({}^{s}\) in (4), to take into account which side of the source line is connected to the target line. The adjacency matrix is an ordered \(l*l\)-matrix, where rows represent a target line \(i\) and columns represent a source line \(j\). If line \(j\) is connected to line \(i\) through its \(s\)-side, it is marked with a digit \(1\) in the appropriate element of the matrix, \(0\) otherwise. Matrices are typeset in uppercase regular font, vectors are typeset in lowercase bold italic, while scalar values such as specific elements of vectors or matrices are typeset with italic fonts. \[\text{A}^{s}=\left[a_{i,j}^{s}\right]=\left[a_{1,1}^{s}\quad\dots\quad a_{1,l} \right]\quad,where: \tag{4}\]  
\[a_{i,j}^{a}= \begin{cases}1&\text{if }u_{j}\in\{u_{i},v_{i}\}\\ 0&\text{Otherwise}\end{cases} \]  
\[a_{i,j}^{b}= \begin{cases}1&\text{if }v_{j}\in\{u_{i},v_{i}\}\\ 0&\text{Otherwise}\end{cases}