## 3 Self-Triggered Control for Islanded Microgrids under DoS Attacks
### Model of DoS Attacks  
In cyber physical environments, the adversaries can conduct DoS attacks by jamming communication channels, disabling sending devices, and distorting communication protocols to make parts or all components of the control system inaccessible, which would cause data loss and control failure. In general, DoS attacks can be categorized into node-based attacks and link-based attacks [20]. Node-based attacks can prevent a node from sending its state to its neighbors, while link-based attacks block information exchange between two neighboring nodes. The MAS-based secondary control scheme of microgrid requires the communication topology to maintain connectivity. DoS attacks may destroy communication connections among neighboring DGs and even ruin the stability and convergence of the MAS-based secondary control strategy. Figure 3 shows the communication network of the microgrid system under DoS attacks.  
As illustrated in Figure 3, the adversaries launch DoS attacks on the communication link \((i,j)\in\Xi\) to block the data transmission between \(DG_{i}\) and \(DG_{j}\). Since the adversaries with limited resources cannot continuously jam the channels for a long time, the sequence of DoS attacks repeats the cycles of jamming and sleeping. The set \(\left\{a_{k}^{ij}\geq 0\right\}\) and the set \(\left\{\tau_{k}^{ij}\geq 0\right\}\) are defined to represent the starting time instants and durations of the DoS attacks on the communication link \((i,j)\in\Xi\), in which \(k\in N_{0}\) and \(a_{k+1}^{ij}>a_{k+1}^{ij}+\tau_{k}^{ij}\) is satisfied. The \(k\)th attack interval can be expressed as \(A_{k}^{ij}=\left[a_{k+1}^{ij},a_{k+1}^{ij}+\tau_{k}^{ij}\right]\), during which the information sharing is prevented between agents \(i\) and \(j\). In the time interval \([t_{0},t]\), the total duration of the attacks \(\left|A_{(t_{0},t)}^{ij}\right|\) on the communication link \((i,j)\in\Xi\) can be expressed as  
\[\left|A_{(t_{0},t)}^{ij}\right|\leq\lambda^{ij}+\mu^{ij}(t-t_{0}) \tag{8}\]  
where the scalar \(\lambda^{ij}\geq 0\) and \(\mu^{ij}\in(0,1)\). Note that Equation (8) implies that \(\lim_{t\rightarrow\infty}\left|A_{(0,t)}^{ij}\right|/t\leq\mu^{ij}\), \(\mu^{ij}\) can be considered as the upper bound on the ratio of the total duration of DoS attacks in a long time interval.  
Since the jamming period cannot span the entire time, the longest duration for jamming period cannot be larger than \(\frac{ij}{1-\mu^{ij}}\), and \(\lambda^{ij}\) can be used to model the attacker's capability for continuous jamming.  
**Remark 2:**_In a wireless communication network, the adversaries require energy for sending radio inference to block data transmissions. Due to resources and energy constraints, the adversaries cannot constantly jam the communication channels. Our DoS attack model can capture different scenarios, such as reactive, random, and periodic jamming attacks [29]. In a wired communication network, DoS attacks can be conducted with a flood of internet traffic to the targeted devices. The adversaries may keep the number of attacks small to make them less detectable [30]. Thus, DoS attacks cannot span the entire time and repeat the cycles of jamming and sleeping. Note that node-based DoS attacks can be considered as some or all of the links connected to the node subject to attacks. Our DoS attacks model can capture a more general scenario, in which communication links can be attacked independently._