## 2 Layered Fault Tree Model for Smart Grids
### Importance Measures  
Importance measures are used to describe the contributions and effect of the components to the occurrence of the top event in a fault tree. They give a characterization of importance to the components, in order to find the weak parts of the system and to improve the system design. In this paper, networked importance measures are chosen and revised to adapt to the features of smart grids.  
The weighted failure probabilities of power delivery to all the loads are considered to get the Loss of load probability(LOLP) measure of the power system [17]:  
Figure 4: Comparison between layered fault tree model and traditional fault tree model: **(a)** a circuit example; **(b)** layered fault tree for the load point; **(c)** traditional fault tree for the load point.  
\[LOLP=\sum_{i=1}^{N_{L}}Q_{GDi}\frac{Ki}{K} \tag{7}\]  
\[K=\sum_{i=1}^{N_{L}}Ki \tag{8}\]  
Risk achievement worth (RAW) and risk reduction worth (RRW) have been proposed as importance measures for fault tree model [27]. Risk achievement worth identifies components that should be maintained well, in order the reliability of the system is not significantly reduced. Risk reduction worth identifies those components probably redundant, for their reliability significantly increasing system reliability:  
\[RAW_{GDi}^{k}=\frac{Q_{GDi}(Q_{k}=1)}{Q_{GDi}} \tag{9}\]  
\[RRW_{GDi}^{k}=\frac{Q_{GDi}}{Q_{GDi}(Q_{k}=0)} \tag{10}\]  
The network importance risk measures, namely network risk achievement worth (NRAW) and network risk reduction worth (NRRW), are defined using the definition of the importance measures from a single fault tree and can be applied to power systems [17]:  
\[NRAW^{k}=\frac{LOLP(Q_{k}=1)}{LOLP}=\frac{\sum_{i=1}^{N_{L}}Q_{GDi}(Q_{k}=1) Ki}{\sum_{i=1}^{N_{L}}Q_{GDi}Ki} \tag{11}\]  
\[NRRW^{k}=\frac{LOLP}{LOLP(Q_{k}=0)}=\frac{\sum_{i=1}^{N_{L}}Q_{GDi}Ki}{\sum_{i =1}^{N_{L}}Q_{GDi}(Q_{k}=0)Ki} \tag{12}\]  
With the pre-defined priorities of loads set by the user, when assessing reliability and component importance, it is essential to take the priorities into consideration. In order to get more practical results for smart grids, in this paper, the priority factor is proposed to revise the previous procedure. Because the priority factor is developed only with the priority ranking of the loads, without detailed preference information about each load point, isometric measures may be the most objective. An isometric priority factor is given as:\[Pi=\frac{NL-i+1}{NL} \tag{13}\]  
where \(Pi\) is the priority factor; and \(Pi\in(0,1]\). \(Pi\) describes the decreasing importance of loads with the order of priorities. The modified Loss of load probability measure of a local smart grid framework can be revised as:  
\[LOLP^{*}=\sum_{i=1}^{NL}Q_{GDi}\frac{Ki}{K}Pi \tag{14}\]  
\(Pi\)_*Ki\(/K\)_ is the compound weighting factor corresponding to load \(i\).  
With the priority factor considered for smart grid system, substitute modified Loss of load probability \(LOLP^{*}\) for \(LOLP\), and the revised network risk achievement worth (NRAW\({}_{+}\)) and network risk reduction worth (NRRW\({}_{+}\)) for smart grid system are as:  
\[\begin{split} NRAW_{+}^{k}=\frac{LOLP^{*}(Q_{k}=1)}{LOLP^{*}}= \frac{\sum_{i=1}^{NL}Q_{GDi}(Q_{k}=1)Ki*Pi}{\sum_{i=1}^{NL}Q_{GDi}Ki*Pi}\\ =\frac{\sum_{i=1}^{NL}Q_{GDi}Ki*RMW_{GDi}^{k}*Pi}{\sum_{i=1}^{NL}Q_ {GDi}Ki*Pi}\\ \end{split} \tag{15}\]  
\[\begin{split} NRRW_{+}^{k}=\frac{LOLP^{*}}{LOLP^{*}(Q_{k}=0)}= \frac{\sum_{i=1}^{NL}Q_{GDi}Ki*Pi}{\sum_{i=1}^{NL}Q_{GDi}(Q_{k}=0)Ki*Pi}\\ =\frac{\sum_{i=1}^{NL}Q_{GDi}Ki*Pi}{\sum_{i=1}^{NL}\underbrace{Q_ {GDi}Ki*Pi}_{GRW_{GDi}^{k}}}\\ \end{split} \tag{16}\]  
System importance measures for components groups corresponding to specific combination are defined similarly as the above importance measures.