## 2 Layered Fault Tree Model for Smart Grids
### Simulation Strategy  
With the procedure proposed to construct the fault tree, quantitative evaluation of the fault tree is difficult to carry out with conventional method. Three main reasons are as:  
1 The fault tree constructed reflects the operational mechanism of smart grids. The change of two different power supply modes should be embodied in the reliability assessment process with the proposed fault tree.
2 Start time of operation are different between devices in the two layers of fault tree, in that local supply is triggered by utility supply failure. Time requirement for islanded operation is not as high as for normal systems, so a short islanded operation time cycle should be integrated in the assessment.  
The proposed procedure for assessment of inadequacy judgment function for the secondary fault tree should be integrated in the reliability evaluation process.  
Monte Carlo simulation is a kind of numerical simulation method based on the theory of probability and statistics. It is feasible through computer programming. Based on the layered fault tree model proposed for smart grids, this paper presents a hierarchical simulation strategy to assess fault tree model built for the load points, in order to evaluate the reliability of the overall system.  
In a Monte Carlo run, the time to failure is generated for each component, then the components states are set to \"failed\", one at a time in order of increasing time, until the top event is produced [28]. Typically, the lives of electronic devices obey exponential distribution. Random sampling can be done to the occurrence time of each basic event which obeys exponential distribution during the simulation as the following process:  
\[F(t)=1-e^{-\lambda t}=\xi \tag{17}\]  
\[t=F^{-1}(\xi)=-\frac{\ln(1-\xi)}{\lambda} \tag{18}\]  
where \(F(t)\) is the life distribution function of the basic events; \(\lambda\), the fault rate of a basic event; \(\xi\), random sampling result of the uniform distribution in the interval of [0,1]. With this method, the occurrence time for each basic event can be calculated.  
Reliability system is a discrete event dynamic system (DEDS) and the basic simulation method for this kind system is simulation clock advancing method. Simulation procedure is advanced along with the event list in time order. Let \(\phi\big{[}X(t)\big{]}\) express the structure function of the fault tree, \(X(t)\) as the state variable of the system. Let \(N\) denote the total times of simulation, and \(j=1\), \(2\),..., \(N\). The state function of the system in the \(j\)th simulation at the moment \(t\) as:  
\[X_{j}(t)=[b_{1,j}(t),b_{2,j}(t),...,\,b_{k,j}(t),...,\,b_{n,j}(t)] \tag{19}\]  
\[b_{k,j}(t)=\begin{cases}1,&t\geq t_{k,j}\\ 0,&t<t_{k,j}\end{cases} \tag{20}\]  
So in the \(j\)th simulation, the state of the top event can be expressed as:  
\[\phi_{j}(t)=\phi_{j}[X(t)]=\begin{cases}1,&t\geq t_{GDi,j}\\ 0,&t<t_{GDi,j}\end{cases} \tag{21}\]  
The main steps and processes of the simulation are given by the flow chart in Figure 5. A fixed time cycle is set for islanded operation of the local smart gird framework in this simulation procedure. The assessment for inadequacy judgment function of the secondary fault tree is integrated in the simulation process. In each single time of simulation, the repetition of utility supply failure is ignored due to its bare possibility. The simulation strategy can be seen as a hierarchical process, taking the simulation for the secondary fault tree as a conditioned sub-process of simulation. With this simulation strategy proposed, quantitative assessment of the layered fault tree for smart grid can be realized.