## 2 Preventive Control Model
For example, a failure \(X_{00}\) in a source node Bus\({}_{3}\) of an ADN occurs in a steady state \(g_{0}\) = <\(V_{0}\), E\({}_{0}\) > \(\in\)G, as shown in Figure 2a, its operation states are described as \(X_{00}\) = (\(x_{\rm{PC1}}\), \(x_{\rm{Bus3}}\), \(x_{\rm{DN1}}\), \(x_{\rm{EN1}}\), \(x_{\rm{Bus1}}\), \(x_{\rm{Bus2}}\), \(x_{\rm{YC2}}\), \(x_{\rm{SN1}}\), \(x_{\rm{KI1}}\), \(x_{\rm{CN1}}\), \(x_{\rm{IR2}}\), \(x_{\rm{AN1}}\), \(x_{\rm{CN2}}\), \(x_{\rm{NR3}}\), \(x_{\rm{SN2}}\), \(x_{\rm{AN2}}\))\({}^{T}\) =(0,1,0,0,0,0,0,0,0,0,0,0,0,0)\({}^{T}\). During the CCF propagation from the failure in a source node Bus\({}_{3}\) to a failure \(X_{in}\) in a sink node, there are three failures occurring, and \(s_{0}\) = \(X_{00}\)\(\rightarrow\)\(X_{01}\)\(\rightarrow\)\(X_{03}\) = (0,1,0,0,0,0,0,0,0,0,0,0,0,0)\({}^{T}\)\(\rightarrow\) (1,0,0,0,0,0,0,0,0,0,0,0,0,0)\({}^{T}\)  
Figure 2: State transitions of an ADN during the propagation of CCF. (0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)\({}^{T}\). When a PCP is enabled to prevent the CCF propagation sequence \(s_{0}\), the ADN is transferred from \(g_{0}\) to \(g_{1}\), as shown in Figure 2. In the next state \(g_{1}\), there could be CCF propagation sequences from other source nodes of an ADN._  
_(3) \(A\subseteq\)\(R^{m}\) represents a set of control actions, which is composed of some control actions following a PCP to prevent the CCF propagation. A PCP is a mapping (\(l_{\perp}\): S\(\rightarrow\)\(A\)) from the CCF set to the control action set. For example, in order to prevent the instability of an ADN caused by the failure of the source node \(Bus_{3}\), a backup power \(Bus_{1}\) is activated to enable alternative power edges to the nodes in the CN primarily powered by \(Bus_{3}\), and all connections from nodes (\(PG_{1}\), \(Bus_{3}\) and \(EN_{1}\)) to other devices are cut off, the topology of the ADN is changed, as shown in Figure 2b._  
_(4) \(F\): G\(\times\)S\(\rightarrow\)G represents a steady state transition function between two steady states of an ADN. For example, the CCF propagation sequence \(s_{0}\) of a state \(g_{0}\) = \(<\)\(V_{0}\), \(E_{0}\) > \(\in\)G in an ADN is blocked by a control action l(\(s_{0}\))\(\in\)\(A\), so that the ADN can quickly reach a new steady state \(g_{1}\) = \(<\)\(V_{1}\),\(E_{1}\) >\(\in\)G, it is expressed as \(<V_{0}\), \(E_{0}\) >\(\rightarrow\)\({}^{g_{0}\)\(\in\)\((g_{0})}\)\(<V_{1}\), \(E_{1}\) >._  
_(5) \(Pr\): F\(\rightarrow\)R represents a state transition probability function under a certain action during the propagation process of CCF, R represents the set of real numbers. The propagation process of CCF has the Markov property [22], and this process is described as a Markov stochastic process. So, the state transition probability function is only related with the current state and action._  
_(6) \(r\): F\(\rightarrow\)R is an immediate reward function after taking a certain action on a transition._  
The preventive control model is a finite automaton, which can be described as a transition system. The state transition of an ADN shown in Figure 3 can be described via a finite automaton, which corresponds to Figure 2.