## 2 Layered Fault Tree Model for Smart Grids
### The Layered Fault Tree
#### 2.2.2 Construction of the Secondary Fault Tree  
The secondary fault tree can be regarded as the local supply adequacy assessment tree. This part of the fault tree is constructed to assess the local power supply adequacy considering the flow paths. In fact, input parts of the flow paths connected to the loads have been decomposed into the primary fault tree, in order to ensure the utility supply for the load. In this section of the fault tree, only the output parts of flow paths for the DGs are taken into consideration, so the events included in the secondary fault tree are the failure events for devices of the remaining parts of the flow paths and the modular failure events of DGs in the islanded smart grid system. The secondary fault tree is actually the logical combination of failures of flow paths as well as DGs. Islanded operation mode of the local framework of a smart grid won't last long, generally no longer than one week [20]. This is another important reason to construct the fault tree in a layered form. It can make the evaluation process more realistic, for the difference of working time for the devices in the two layers and the islanded/grid-connected mode mutually change being describable. This advantage makes it efficient for simulation process for reliability evaluation. Figure 2: Logical operation connecting the two layers of fault tree. The thoughts of peer to peer and standardization of structure have been introduced to the design and use of smart grids for the consumers to access loads and power generating devices freely. This provides great convenience for the construction of the fault tree, in that all the loads are in the same appliance architecture according to a specific smart grid, and electricity from all the generators is accessible for all the loads in the local framework. Intelligent energy dispatching can be realized by the intelligent power management subsystem. However, the adequacy assessment is not a simple process especially in the islanded operation mode, which concerns the secondary fault tree chiefly. In the local framework of a smart grid, priorities of the loads are pre-defined by the users [21,22]. The power demand of an important load with a high priority should be guaranteed preferentially. Before introducing the construction strategy of the secondary fault tree, it is necessary to model the DGs and loads. Generally, the output power of the renewable DGs depends highly on the time and their locations. The power requested by a user is also related to the day's moment. Time-series models for renewable DGs [23,24] as well as the loads [25] have long been used to describe the power generated and requested, in order to assess the adequacy. With these methods, the power generated by renewable DGs and requested by the loads in a certain period of time can be calculated and used in adequacy assessment. Beyond models, real data can also be used directly into the process, without considering the difficulty of data collection. In this paper, the number of a load is set consistent with its priority. With the priorities of loads coincided with the labels, inadequacy judgment function for the load of a certain priority can be represented as:  
\[\phi_{\textit{hnode}}(t)= \begin{cases}1,\textit{POW}_{\textit{tot}}(t)<\sum \limits_{t=1}^{\textit{prio}}\textit{pow}D_{\textit{i}}(t)\\ 0,\textit{POW}_{\textit{tot}}(t)\geq\sum\limits_{t=1}^{\textit{prio}}\textit{ pow}D_{\textit{i}}(t)\end{cases}  \tag{3}\]  
\(prio\) is the priority of the load point concerned currently, with \(prio=1\) representing the most important load. When \(\phi_{\textit{hnode}}(t)=1\), the top event of the secondary fault tree is considered as occurred. In order to judge the inadequacy judgment function, standard strategy of fault tree modeling can be complex especially for the loads with low priorities. The fault tree will be built with the changing logical combinations of all the DGs' failure events repeatedly.