## 2 Modeling of Load Frequency Control in New Energy Power Systems  
The LFC system model of the \(i\)-th area is presented in Figure 1. The parameters of the \(i\)-th control area are listed in Table 1. The LFC system model of the multi-area power system can be described as shown in Equation (1) [30]:  
\[\left\{ \begin{aligned} \dot{x}(t)&=Ax(t)+Bu(t)+F \omega(t)\\ y(t)&=Cx(t)\end{aligned} \right. \tag{1}\]  
where the following is true:  
\[x_{i}(t) =\left[\Delta f_{i}\quad\Delta P_{mi}\quad\Delta P_{vi}\quad \Delta P_{tie-i}\quad\int ACE_{i}\quad\Delta P_{bi}\quad\Delta P_{pvi}\quad \Delta P_{pi}\right]^{T},\] \[x(t) =\left[x_{1}(t)\quad x_{2}(t)\quad x_{3}(t)\quad\ldots\quad x_{n} (t)\right]^{T},\,\omega_{i}(t)=\Delta P_{di},\] \[\omega(t) =\left[\omega_{1}{}^{T}(t)\quad\omega_{2}{}^{T}(t)\quad\omega_{ 3}{}^{T}(t)\quad\ldots\quad\omega_{n}{}^{T}(t)\right]^{T},\,y_{i}(t)=\left[ \Delta f_{i}\quad\int ACE_{i}\right]^{T}\] \[y(t) =\left[y_{1}(t)\quad y_{2}(t)\quad y_{3}(t)\quad\ldots\quad y_{n} (t)\right]^{T},\] \[u(t) =\left[u_{1}{}^{T}(t)\quad u_{2}{}^{T}(t)\quad u_{3}{}^{T}(t) \quad\ldots\quad u_{n}{}^{T}(t)\right]^{T},\] \[A =\left[ \begin{matrix}A_{11}&\ldots&A_{1n}\\ \vdots&\ddots&\vdots\\ A_{n1}&\ldots&A_{nm}\end{matrix} \right]\!\!,\,B=[0\quad 0\quad\frac{k_{2}}{T_{gi}} \quad 0\quad 0\quad\frac{k_{1}\lambda_{bi}}{T_{bi}}\quad\lambda_{p}k_{3} \quad\lambda_{pi}k_{3}]^{T},\] \[A_{ii} =\left[ \begin{matrix}-\frac{D_{i}}{M_{i}}&\frac{1}{M_{i}}&0&- \frac{1}{M_{i}}&0&\frac{1}{M_{i}}&\frac{1}{M_{i}}&0\\ 0&-\frac{1}{T_{chi}}&\frac{1}{T_{chi}}&0&0&0&0&0\\ -\frac{1}{R_{i}T_{gi}}&0&-\frac{1}{T_{gi}}&0&0&0&0&0\\ 2\pi\sum\limits_{j=1,j\
eq i}^{n}T_{ij}&0&0&0&0&0&0&0\\ \beta_{i}&0&0&1&0&0&0&0\\ 0&0&0&0&0&-\frac{1}{T_{bi}}&0&0\\ 0&0&0&0&0&0&-c&b-a\\ 0&0&0&0&0&0&0&-a\end{matrix} \right],\] \[A_{ij} =\left[ \begin{matrix}0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 2\pi\sum\limits_{j=1,j\
eq i}^{n}T_{ij}&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\end{matrix} \right],\] \[C =\left[ \begin{matrix}1&0&0&0&0&0&0&0\\ 0&0&0&0&1&0&0&0\end{matrix} \right]\!\!,\,F=\left[-\frac{1}{M_{i}}0&0&0&0&0&0&0 \end{matrix}\right]^{T}\]The output power of the governor is described as follows [40]:  
\[\Delta\dot{P}_{vi}=\frac{k_{2}}{T_{gi}}u(t)-\frac{1}{T_{gi}R_{i}}\Delta f_{i}- \frac{1}{T_{gi}}\Delta P_{vi} \tag{2}\]  
The output power of the turbine is expressed as follows [40]:  
\[\Delta\dot{P}_{mi}=\frac{1}{T_{chi}}\Delta P_{vi}-\frac{1}{T_{chi}}\Delta P_{mi} \tag{3}\]  
The transfer function of the PV system is described as follows [40]:  
\[G_{pvi}(s)=\frac{\lambda_{pi}}{s+a}\cdot\frac{s+b}{s+c} \tag{4}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Description** \\ \hline \(\Delta P_{di}\) & Load deviation \\ \(\Delta P_{mi}\) & Generator mechanical output deviation \\ \(\Delta P_{vi}\) & Valve position deviation \\ \(\Delta P_{bi}\) & Battery output power deviation \\ \(\Delta T_{tie-i}\) & Tie-line active power deviation \\ \(\Delta P_{pvi}\) & PV output deviation \\ \(\Delta f_{i}\) & Frequency deviation \\ \(M_{i}\) & Moment of inertia of the generator \\ \(D_{i}\) & Generator damping coefficient \\ \(T_{gi}\) & Time constant of the governor \\ \(T_{chi}\) & Time constant of the turbine \\ \(R_{i}\) & Speed drop \\ \(\beta_{i}\) & Frequency bias factor \\ \(T_{ij}\) & Tie-line synchronizing coefficient \\ \(\lambda_{pi}\) & PV gain factor \\ \(\lambda_{bi}\) & ESS gain factor \\ \(k_{1}\) & ESS proportional factor \\ \(k_{2}\) & Turbine proportional factor \\ \(k_{3}\) & PV proportional factor \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Explanation of symbols for the multi-area LFC system. Figure 1: The model of the \(i\)-th area in a multi-area LFC, including PV and ESSs.