## 3 Active Distribution Networks Modeling
\(Trans_{P}=(V_{P}(N)\), \(E_{P}(N))\) represents the subgraph generated by the load (power) redistribution after a node or an edge fails in the PN at step N during the CCF. For example, if the subgraph is generated by a node \(n_{Pj}\in V_{P}\) failure, then \(V_{P}(N)=V_{P}(N-1)-Over_{P}(Neigh_{P}(n_{Pj}))\), \(E_{P}(N)=E_{P}(N-1)-Edg_{P}(n_{Pj})\). Where \(Over_{P}(Neigh_{P}(n_{Pj}))=\{n\in Neigh_{P}(n_{Pj})\|load_{P}(n,N)\geq T_{Pp}(n)\}\) represents the set of overloaded nodes in the set of neighbor nodes at step N. \(T_{Pp}(n)\) represents the load threshold of a node \(n\) in the PN, when the load (power) of a node \(n\) in the PN is greater than its threshold, then the node will fail. The process of an edge failure in the PN is similar. 6. \(Thres_{P}=Thres_{P}\cup Thres_{P1}\) represents the thresholds set of nodes and edges in the PN. Where \(Thres_{P1}\) and \(Thres_{P1}\) represent the thresholds sets of nodes and edges in the PN respectively. If the load flowing through an edge of the PN is greater than its threshold, then the edge will fail. The node situation is similar to the edge situation. Since the ADN includes the PN and the CN, they are interdependent with each other. Thereinafter, the definition of the CN in the ADN is introduced, and the model of interdependence between the PN and the CN is also given. **Definition 3**. : _The CN of the ADN is also a sextuple system \(G_{I}=(V_{I},E_{I},Load_{I},W_{I},Trans_{I},Thres_{I})\), where_  
1. \(V_{I}\) _represents the set of nodes in the CN, and_ \(V_{I}=V_{I1}\cup V_{I2}\cup V_{I3}\cup V_{I4}\)_._
2. \(E_{I}\) _represents the edge set in the CN, and_ \(E_{I}{\subseteq}V_{I}\times V_{I}\)_._
3. \(Load_{P}\)_:_ \(V_{I}\times Int{\rightarrow}Int\) _represents the load (data packets) of a node_ \(n\) _in the CN at step N after the CCF occurs. When the neighbor node_ \(n_{li}\) _of a node_ \(n_{lj}\) _is failed, the original load (data packets) of this node_ \(n_{li}\) _is redistributed to the node_ \(n_{lj}\)_, and the load (data packets) of the node_ \(n_{lj}\) _changes according to the following recursive Equation (3)._ \[Load_{I}(n_{lj},N+1)\ =\ \left\{ \begin{array}{l}Load_{I}(n_{lj},N)+\frac{Load_{I}(n_{ li},N)\times W_{I}(n_{lj},n_{li})}{\sum\limits_{n_{l}\in Neigh_{I}(n_{li})}W_{I}(n_{li} )},n_{li}\ \in Fail_{I}\\ Load_{I}(n_{lj},N),n_{li}\ \
otin Fail_{I}\end{array} \right.\] (3) _where Neigh\({}_{I}\):_ \(V_{I}{\rightarrow}2^{Vl}\) _is a mapping that represents the neighbor nodes set of a node in the CN._ \(Fail_{I}\) _is the set whose elements are the contiguous failed nodes of the node_ \(n_{lj}\)_, that is_ \(Fail_{I}=\{n_{li}\mid n_{li}\in Neigh_{I}(n_{lj})\wedge\
eg Run(n_{li})\}\)_._
4. \(W_{I}\)_:_ \(E_{I}{\rightarrow}Int\) _represents the edge weight mapping in the CN._
5. \(Trans_{I}=(V_{I}(N)\)_,_ \(E_{I}(N))\)_represents the subgraph generated by the load (data packets) redistribution after a node or an edge fails in the CN at step N during the CCF._
6. \(Thres_{I}=Thres_{IN}\cup Thres_{IL}\)_represents the thresholds set of nodes and edges in the CN._  
We modeled the interdependence between the CN and the PN in the following ways. The symbol \(<>\) represents the average value of the corresponding variable. In the CN, a sensor node \(n_{II}\in V_{I1}\) collects information of the nodes in the PN, and an actuator node \(n_{I4}\in V_{I4}\) executes the commands from a control node \(n_{I3}\in V_{I3}\) to control the physical process of the PN. Abstractly, it can be seen that a control node \(n_{K1}\in V_{I3}\) in the CN can control \(K_{1}\) nodes of the PN, and the average control fan out of the node \(n_{K1}\) is \(<\)K\({}_{1}\)\(>\) = (1/\(|V_{I3}|)\times\Sigma K_{1}\). A node \(n_{K2}\) in the PN is controlled by \(K_{2}\) control nodes in the CN, and the average control fan in of the node \(n_{K2}\) is \(<\)K\({}_{2}\)\(>\) = \(|V_{I3}|\times\)\(<\)K\({}_{1}\)\(>\)/\(NON_{P}\).