## 3 Active Distribution Networks Modeling
A sensor node \(n_{L1}\)\(\in V_{I1}\) in the CN can collect \(L_{1}\) nodes in the PN, and the average information collecting fan in of the node \(n_{L1}\) is \(<\)\(L_{1}\)\(>\) = (1/\(|V_{I1}|)\times\Sigma L_{1}\). Conversely, a node \(n_{L2}\) in the PN is perceived by \(L_{2}\) sensor nodes in the CN and the average information collecting fan out of the node \(n_{L2}\) is \(<\)\(L_{2}\)\(>\) = \(|V_{I1}|\)\(\times\)\(<\)\(L_{1}\)\(>\)/\(NON_{P}\). Correspondingly, the average power fan out of a node in the PN is \(<\)\(O_{2}\)\(>\) = (1/\(|V_{P2}|\)) \(\times\)\(\Sigma\)\(O_{2}\), and a node \(n_{O1}\) in the CN can be power supplied by \(O_{1}\) nodes of the PN, then the average power fan in of the node \(n_{O1}\) is \(<\)\(O_{1}\)\(>\) = \(|V_{P2}|\)\(\times\)\(<\)\(O_{2}\)\(>\)/\(NON_{I}\). For example, as shown in Figure 2a, \(|V_{P1}|\) = 2, \(|V_{P2}|\) = 3, \(|V_{P3}|\) = 1, \(|V_{P4}|\) = 3, \(|V_{P5}|\) = 1, \(|V_{I1}|\) = 3, \(|V_{I2}|\) = 5, \(|V_{I3}|\) = 4, and \(|V_{I4}|\) = 3. Therefore, \(<\)\(K_{1}\)\(>\) = 10/4 = 2.5, \(<\)\(K_{2}\)\(>\) = 1, \(<\)\(L_{1}\)\(>\) = 10/3, \(<\)\(L_{2}\)\(>\) = 1, \(<\)\(O_{1}\)\(>\) = 1, and \(<\)\(O_{2}\)\(>\) = 15/3 = 5. (In Appendix D, a special case about how to find \(<\)\(O_{1}\)\(>\) and \(<\)\(O_{2}\)\(>\) is shown)  
According to the aforementioned Definitions 2 and 3, the definition of the ADN is given below. **Definition 4**. : _ADN is a combination of the PN and the CN, so it is also a sextuple system \(G_{A}\) = (\(V_{A}\), \(E_{A}\), Load\({}_{A}\), \(W_{A}\), Trans\({}_{A}\), Thres\({}_{A}\)), where_  
1. \(V_{A}\) _represents the set of nodes in the ADN, and_ \(V_{A}\) = \(V_{P}\cup V_{I}\)_._
2. \(E_{A}\) _represents the edge set in the ADN, and_ \(E_{A}\) = \(E_{P}\cup E_{I}\cup E_{PI}\cup E_{IP}\)_. The edge set of the ADN includes the edge set of the PN and the edge set of the CN. In addition, the edge set formed by the interdependence between the nodes of the PN and CN is added. Where_ \(E_{PI}\) = \(E_{PI}\)\(\sim\)\(\cup\)\(E_{PI}\)_, it includes the virtual edge set_ \(E_{PI}\)_-formed by the sensor nodes in the CN perceiving the corresponding nodes in the PN, and it indicates the information gathering relationship between a sensor node_ \(n_{II}\) \(\in\)\(V_{II}\) _in the CN and a node in the PN. The set_ \(E_{PI}\) _also includes the solid edge set_ \(E_{PI}\) _of the nodes in the PN supplying power to the nodes in the CN._ \(E_{IP}\) _represents the virtual edge set formed by the actuator nodes in the CN acting on the nodes in the PN._
3. \(Load_{P}\)_:_ \(V_{A}\)\(\times\)_Int\(\rightarrow\)\(C\) _represents the load of a node_ \(n\) _in the ADN at step_ \(N\) _after the CCF occurs. Where Int represents a set of positive integer numbers,_ \(C\) _represents the set of complex numbers. It is assumed that the load of a failure node is only redistributed to its neighbor node of the same network following the nearest neighbor rule._
4. \(W_{A}\) _represents the edge weight mapping in the ADN, and_ \(W_{A}\)_:_ \(E_{A}\)\(\rightarrow\)\(C\)_._ \(C\) _is a set of complex numbers._
5. \(Trans_{A}\) _represents a subgraph generated by the load redistribution after a node or an edge fails in the ADN at step_ \(N\) _during the CCF, and_ \(Trans_{A}\) = (\(Trans_{P}\), \(Trans_{I}\), \(E_{PI}\)-\(N\), \(E_{IP}\)-\(N\))_._ \(E_{PI}\)_-\(N\) _represents the interdependence edges set from a node in the PN to a node in the CN._ \(E_{IP}\)_-\(N\) _represents the interdependence edges set from a node in the CN to a node in the PN._
6. \(Thres_{A}\) = \(Thres_{P}\cup Thres_{I}\cup Thres_{PI}\cup Thres_{IP}\) _represents the threshold set of nodes and edges in the ADN._ \(Thres_{PI}\) _represents the threshold set of edges in the set_ \(E_{PI}\)_,_ \(Thres_{IP}\) _represents the threshold set of edges in the set_ \(E_{IP}\)_._  
In order to analyze the robustness of the ADN against the CCF, we need to show the normal operation conditions of nodes and edges in the PN and CN respectively.