# Reliability Assessment of Cyber-Physical Distribution Systems Considering Cyber Disturbances
## 2 Problem Description
### The Proposed Reliability Assessment Method
#### 2.2.3 Reliability Assessment Method--Solving the Minimum Cut Set Using Petri Nets  
The method of minimum cut set analysis essentially finds the minimum cut set for the load points in the network and calculates the reliability indices, which can reflect the least failure composition pattern of the system failure occurrence and is the necessary condition for system failure. The reliability analysis of the system can be easily performed based on the minimum cut set. Due to the reachability and boundedness of Petri nets, various common phenomena in complex systems are well described with abundant analysis methods [14]. Among them, the analysis method based on the correlation matrix is suitable for large-scale Petri net models with high complexity. In this paper, based on the Petri net model, the topological relations embodied in the Petri net model are calculated and transformed into the form of a correlation matrix, from which the minimum cut set is solved, and finally the top event occurrence probability is calculated according to the failure rate of each component in the minimum cut set; thus, the system reliability indices are solved.  
* Find the correlation matrix by Petri net:  
The structure of Petri nets can be represented by a matrix, which allows the introduction of a linear algebra approach to the analysis of Petri net properties.  
Let \(\sum=(\mathbf{P},\mathbf{T};\mathbf{F},\mathbf{M})\) be a Petri net, where \(\mathbf{P}=\{p_{1},p_{2},\cdots,p_{m}\}\), \(\mathbf{T}=\{t_{1},t_{2},\cdots,t_{n}\}\). \(\mathbf{P}\) is the set of places, representing the state of the system; \(\mathbf{T}\) is the set of transitions, representing the consumption, use of resources, and the changes that make the state of the system; \(\mathbf{F}\) is the set of input and output functions, called flow relations, establishing a unidirectional connection from places to transitions and from transitions to places; and \(\mathbf{M}\) is the identity vector on the set of places. Then, the structure of the Petri net \(\sum\) can be represented by an \(n\)-row and \(m\)-column matrix \(\mathbf{A}\):  
\[\mathbf{A}=\left[\text{a}_{ij}\right]_{n\times m}, \tag{8}\]  
\[\text{a}_{ij}=\text{a}_{ij}^{+}-\text{a}_{ij}^{-}, \tag{9}\]  
\[\text{a}_{ij}^{+}=\left\{\begin{matrix}1,\big{(}t_{i},p_{j}\big{)}\in\mathbf{F}_{ i}\\ 0,\text{else}\end{matrix}\right.i\in\{1,2,\ldots,n\},j\in\{1,2,\ldots,m\}, \tag{10}\]  
\[\text{a}_{ij}^{-}=\left\{\begin{matrix}1,\big{(}p_{j},t_{i}\big{)}\in\mathbf{F}_{ i}\\ 0,\text{else}\end{matrix}\right.i\in\{1,2,\ldots,n\},j\in\{1,2,\ldots,m\}, \tag{11}\]  
and \(\mathbf{A}\) is called the association matrix of \(\sum\).  
* Find the minimum cut set from the correlation matrix: Solving the system minimum cut set from the correlation matrix, the steps refer to [34].
* Find the top event probability from the minimum cut set: The probability of occurrence of the top event is determined using the probability of occurrence of the minimum cut set \(K_{h}(h=1,2,\cdots,n)\). The mathematical expression is as follows: \[\begin{array}{ll}P(T)&=P(K_{1}\cup K_{2}\cup\cdots\cup K_{n})\\ &=\sum\limits_{h=1}^{n}P(K_{h})-\sum\limits_{h<j=2}^{n}P\big{(}K_{h},K_{j} \big{)}+\sum\limits_{h<j<i=3}^{n}P\big{(}K_{h},K_{j},K_{i}\big{)}+\cdots+(-1)^{ n-1}P(K_{1},K_{2},\cdots,K_{n})\end{array},\] (12) where \(P(T)\) is the probability of the top event \(T\), \(P(K_{h})\) is the probability of the \(h\)-th cut set, and \(n\) is the maximum of the number of minimum cut sets.