## 3 Active Distribution Networks Modeling  
There were five types of nodes in the PN, including power generation nodes (distributed generators) \(V_{P1}\), substation nodes \(V_{P2}\), distribution nodes \(V_{P3}\), load nodes \(V_{P4}\), and external nodes (the power from transmission grid) \(V_{P5}\). There were four kinds of nodes in the CN, including sensor nodes \(V_{I1}\), information relay nodes \(V_{I2}\), control nodes \(V_{I3}\), and actuator nodes \(V_{I4}\). They were mutually disjoint sets. **Definition 1**. : _An effective control \(Econ_{I}\) in the CN refers to an effective control of the PN nodes by the control nodes of the CN, and it satisfies three conditions._  
1. _There is at least one complete simple directed path of length_ \(k-1\) _existing in the CN, that is path(_\(n_{II}\)_,_ \(n_{I4}\)_) =_ (_\(n_{1}\)_,_ \(\ldots\)_,_ \(n_{k}\)_). The path has_ \(k\) _nodes_ \(n_{1}\)_,_ \(\ldots\)_,_ \(n_{k}\)_. Where the source node_ \(n_{\Pi}=n_{1}\in V_{\Pi}\) _is the sensor node, the destination node_ \(n_{I4}=n_{k}\in V_{I4}\) _is the actuator node, and at least one of the remaining nodes belongs to the_ \(V_{I3}\) _nodes set. A sensor node_ \(n_{\Pi}\in V_{\Pi}\) _in the CN is applied to detect failure events in the PN, and then transmit the event information to a control node_ \(n_{I3}\in V_{I3}\) _through one or more information relay nodes_ \(n_{I2}\in V_{I2}\)_. After that, the control node_ \(n_{I3}\in V_{I3}\) _generates the response information based on specific algorithms and subsequently the response information is transmitted to an actuator node_ \(n_{I4}\in V_{I4}\) _through one or more information relay nodes_ \(n_{I2}\) _to control the physical process._
2. _All nodes and edges in the path path(_\(n_{\Pi}\)_,_ \(n_{I4}\)_) run normally._
3. \(t_{delay}+t_{ract}<t_{interval}\)_._ \(t_{delay}\) _denotes the time interval from the occurrence of a failure event to the time when the response information is generated by a node_ \(n_{I3}\)_._ \(t_{ract}\) _denotes the time interval from the time when the response information is generated to the time when the PN has been changed by actuators._ \(t_{interval}\) _denotes the minimum time interval between two adjacent failure events._  
**Definition 2**. : _The PN of the ADN is a sextuple system \(G_{p}\) = (\(V_{P}\), \(E_{P}\), \(H_{\text{NP}}\), Load\({}_{P}\), \(W_{P}\), Trans\({}_{P}\), Thresp), where (1) \(V_{P}\) represents the set of nodes in the PN, and \(V_{P}=V_{P1}\cup V_{P2}\cup V_{P3}\cup V_{P4}\cup V_{P5}\)._
2. \(E_{P}\) _represents the edge set in the PN, and_ \(E_{P}\subseteq V_{P}\times V_{P}\)_._
3. \(Load_{P}\)_:_ \(V_{P}\times Int\rightarrow\mathbb{C}\) _represents the load (power) of a node_ \(n\) _in the PN 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 (power) of a failure node is redistributed to its neighbor node following the nearest neighbor rule. When the neighbor node_ \(n_{Pi}\) _of a node_ \(n_{Pj}\) _in the PN is failed, the original load (power) of this node_ \(n_{Pi}\) _is redistributed to the node_ \(n_{Pj}\)_, and the load (power) of this node_ \(n_{Pj}\) _changes according to the following recursive Equation (_2_)._ \[Load_{P}(n_{Pj},N+1)\ =\ \left\{ \begin{array}{l}Load_{P}(n_{Pj},N)+\frac{Load_ {P}(n_{Pj},N)\times W_{P}(n_{Pj},n_{Pj})}{\sum\limits_{n_{P}\in NoS_{P}(n_{Pj},n_{Pj})}W_{P}(n_{Pj},n_{Pj})},n_{Pi}\in\textit{Fail}_{P}\\ Load_{P}(n_{Pj},N),n_{Pi}\
otin\textit{Fail}_{P}\end{array} \right.\] (2)where Neigh\({}_{P}\): \(V_{P}{\rightarrow}2^{Vp}\) is a mapping that represents the neighbor nodes set of a node in the PN. \(Fail_{P}\) is the set whose elements are the contiguous failed nodes of the node \(n_{Pj}\), that is \(Fail_{P}=\{n_{Pj}\mid n_{Pj}\in Neigh_{P}(n_{Pj})\wedge\
eg Kun(n_{Pj})\}\). 4. \(W_{P}\): \(E_{P}{\rightarrow}C\) represents the edge weight mapping in the PN. 5.