## 2 Problem Formulation  
### Power System Model  
Interconnected power systems comprise multiple regions interconnected via tie lines. Frequency variations in one region can affect neighboring regions through propagation. To keep frequency and tie-line power deviations within specified limits, we term the overall output signal of each control system the Area Control Error (ACE):  
\[\text{ACE}_{i}=\beta_{i}\Delta f_{i}+\Delta P_{tie-i} \tag{1}\]  
where \(\beta_{i}\) represents the frequency deviation factor, \(\beta_{i}=1/R_{i}+D_{i}\), and \(D_{i}\) denote the generator damping coefficients, and \(R_{i}\) corresponds to the bias coefficient. The linear model representing interconnected power systems was introduced in reference [14]. The dynamic behavior of this model is described by the following equations:  
\[\left\{ \begin{array}{l}\Delta f_{i}(t)=\frac{1}{M_{i}}(\Delta P_{mi}(t)- \Delta P_{di}(t)-\Delta P_{tie-i}(t)-D_{i}\Delta f_{i}(t))\\ \Delta\dot{P}_{mi}(t)=\frac{1}{T_{di}}(\Delta P_{vi}(t)-\Delta P_{mi}(t))\\ \Delta\dot{P}_{gi}(t)=\frac{1}{T_{gi}}\left(\Delta P_{ci}(t)-\frac{1}{R_{i}} \Delta f_{i}(t)-\Delta P_{vi}(t)\right)\\ \Delta\dot{P}_{tie}(t)=2\pi\sum_{j=1,j\
eq i}^{n}T_{ij}\left(\Delta f_{i}(t)- \Delta f_{j}(t)\right)\end{array} \right. \tag{2}\]  
The relevant parameters and associated signals in the equation are defined as shown in Table 1.  
Defining \(x_{i}(t)=\left[ \begin{array}{cccc}\Delta f_{i}&\Delta P_{mi}&\Delta P_{gi}& \Delta E_{i}&\Delta P_{tie-i}\end{array} \right]^{\top}\), \(y_{i}(t)=ACE_{i}(t)\), and \(u_{i}(t)=\Delta P_{ci}(t)\) as system inputs, and \(\theta_{i}^{T}(t)=\left[\Delta P_{di}(t)\sum_{j=1,j\
eq i}^{N}T_{ij}\Delta f_{ j}(t)\right]\) as the disturbance vector, the dynamic model (1) can be transformed into the following state-space equation:  
\[\left\{ \begin{array}{l}\dot{x}(t)=Ax(t)+Bu(t)+F\dot{\theta}(t)\\ y(t)=Cx(t)\end{array} \right. \tag{3}\]  
where  
\[x(t)=\left[ \begin{array}{cccc}x_{1}(t)&x_{2}(t)&\ldots&x_{N}(t)\end{array}  \right]^{T}\!\!\!,\]  
\[y(t)=\left[ \begin{array}{cccc}y_{1}(t)&y_{2}(t)&\ldots&y_{N}(t)\end{array}  \right]^{T}\!\!\!,\]  
\[u(t)=\left[ \begin{array}{cccc}u_{1}(t)&u_{2}(t)&\ldots&u_{N}(t)\end{array}  \right]^{T}\!\!\!,\]  
\[\Delta P_{d}(t)=\left[ \begin{array}{cccc}\Delta P_{d1}(t)&\Delta P_{d2}(t)& \ldots&\Delta P_{dN}(t)\end{array} \right]^{T}\!\!\!,\]\[A=\left[ \begin{array}{ccccc}A_{11}&A_{12}&\ldots&A_{1N}\\ A_{21}&A_{22}&\ldots&A_{2N}\\ \vdots&\vdots&\vdots&\vdots\\ A_{N1}&A_{N2}&\ldots&A_{NN}\end{array} \right],\]  
\[B=\text{diag}\big{[} \begin{array}{ccccc}B_{1}&B_{2}&\ldots&B_{N}\end{array}  \big{]},\]  
\[C=\text{diag}\big{[} \begin{array}{ccccc}C_{1}&C_{2}&\ldots&C_{N}\end{array}  \big{]}\]  
\[F=\text{diag}\big{[} \begin{array}{ccccc}F_{1}&F_{2}&\ldots&F_{N}\end{array}  \big{]}.\]  
The state and matrix representation of the region are as follows:  
\[A_{ii}=\left[ \begin{array}{ccccc}-\frac{D_{i}}{M_{i}}&\frac{1}{M_{i}}&0&0& \frac{-1}{M_{i}}\\ 0&\frac{-1}{T_{di}}&\frac{1}{T_{di}}&0&0\\ \frac{-1}{RT_{gi}}&0&\frac{-1}{T_{gi}^{i}}&0&0\\ \beta_{i}&0&0&0&1\\ 2\pi\sum_{j=1,j\
eq i}^{n}T_{ij}&0&0&0&0\end{array} \right],A_{ij}=\left[ \begin{array}{ccccc}0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ -2\pi T_{ij}&0&0&0\end{array} \right]\]  
\[B_{i}=\left[ \begin{array}{ccccc}0&0&\frac{1}{T_{gi}}&0&0\end{array} \right]^{ T},C_{i}=\left[ \begin{array}{ccccc}0&0&0&0&1\\ \beta_{i}&1&0&0&0\end{array} \right]F_{i}=\left[ \begin{array}{ccccc}\frac{-1 }{M_{i}}&0&0&0&0\end{array} \right]^{T},\]  
To accurately represent the operational behavior of power systems, this research employs a discretization process on the continuous state-space equation. Given a specific sampling period \(T\), the discrete representation of the power system model is articulated as follows:  
\[\left\{ \begin{array}{l}x(k+1)=Gx(k)+Hu(k)+W\vartheta(k)\\ y(k)=Cx(k)\end{array} \right.