## 2 System Model of Power Grid
### Exponential Stability and Cost Function  
Before elaborating the control scheme for power grid, the exponential stability and cost function are introduced. Exponential stability [43,44] is a kind of asymptotic stability. According to the exponential stability, the state converges to zero with an exponential rate, and \(x(t)=x(0)e^{-t}\). If a Lyapunov function satisfies \(\kappa_{1}|x|\leq V(x)\leq\kappa_{2}|x|\), we have _Proposition 1_.  
**Proposition 1**[45]. _Let \(V:R^{n}\to R^{+}\) be a quadratic Lyapunov candidate function satisfying \(V(x)=x^{T}Px\), \(\forall x\in R^{n}\), with \(P=P^{T}>0\). If the condition:_  
\[\dot{V}(x)+2\beta V(x)\leq 0 \tag{7}\]  
is satisfied for all trajectories of (4), for a given scalar \(\beta>0\), the system origin is globally \(\beta\)-stable (i.e., there exists a scalar \(\beta\) and \(\alpha\), such that the trajectories satisfy \(\|x(t)\|\leq\alpha e^{-\beta t}\|x_{0}\|\) for any initial condition \(x_{0}\)). Proposition 1 addresses the relationship between the time derivation of Lyapunov function \(V(x)\) and the \(V(x)\) under the restriction of exponential stability.  
As both physical cost and cyber cost are considered, the cost of system consists of two parts. The first part, the physical state cost expressed by the following equation, is a general form that is widely used in optimal control theory [46].  
\[p(t)=x(t)^{T}Qx(t)=\|x(t)\|_{Q}^{2}\]  
where \(Q\) is the weight matrix for state cost. The communication cost as the second part is essentially related to the triggering rate. High triggering rate means high communication bandwidth cost in data transmission within a time unit. Therefore, the communication cost is proportional to the sampling rate.  
**Definition 1**.: _At the \(k\)th triggering period, the communication cost is defined as:_  
\[c(t)=\frac{\varrho}{\tau(k)}.\]  
where \(\varrho\) is the weight of communication cost, and \(\tau(k)\) is the sampling interval between number \(k\) sampling and the number \(k+1\) sampling.  
Integrating physical state cost with communication cost, the cost function is shown as:  
\[\begin{array}{l}J=\int_{0}^{T_{f}}[q(t)+c(t)]dt\\ =\int_{0}^{T_{f}}[\|x(t)\|_{Q}^{2}+\frac{\varrho}{\tau(k)}]dt\end{array} \tag{8}\]  
Figure 3: Self-triggered control scheme.  
where \(T_{f}\) is the terminal time. When \(t=T_{f}\), the system state converges to a small value that is close to zero. Moreover, when \(t=T_{f}\), the Lyapunov function value is VT, which is much smaller than the initial Lyapunov function value \(V_{0}\). \(c\) is the number of control actions, which have \(T_{f}=\sum_{i=0}^{n}\tau(i)\). This model considers two parts. The first is a function \(\sigma(x(t_{k}))=t_{k+1}-t_{k}\) for calculating self-triggered time, which is adaptive to state \(x(t_{k})\), to reduce costs in communication; the second is the determination of control gain \(K=(-L_{ij})_{n\times n}\). Hence, the cost of system \(J\) is guaranteed under a specified upper bound. The following assumptions are made to calculate the upper bound.  
**Assumption 1**.: _For a given Lyapunov function \(V(x)=x^{T}Px\), \(\varepsilon\) and initial state \(x_{0}\), the system is exponentially stable. The terminal time of control process is \(T_{f}\) defined by \(dV(x(T_{f}))/dt=\
u\), where \(\
u\) is a small number. The Lyapunov function is \(V(x(T_{f}))=V_{T}\), and its initial value is \(V(x(0))=V_{0}\), which has \(V_{T}<V_{0}\)._  
Assumption 1 is applied to make the value of Lyapunov function close to zero at terminal time \(T_{f}\), so that the terminal time \(T_{f}\) for cost function can be determined. It should be noted that the variation of the Lyapunov function converges to zero, when the system converges to a stable state. Therefore, \(\
u\) should be a small number.