## 2 LFC Modeling in the Hybrid AC/DC System with Virtual Inertia
### LFC System Model in the State-Space Form  
As shown in Figure 1, the wide-area measurements are mainly frequencies in the two areas and power flows on both AC and HVDC lines. These measurements would act as inputs for supervisory controllers in LFC, i.e., the AGC and SPMC loops; recall Sections 2.1 and 2.2. For the virtual inertia emulation part, it uses local frequency information only for a relatively faster response, as indicated in Section 2.3. Given the above explanations and the system descriptions in Sections 2.1-2.3, the open-loopLFC model for the test two-area system interconnected by AC and HVDC transmission lines and equipped with added ESS can be compactly described by a continuous-time state-space form:  
\[ \begin{split}\dot{\bar{\mathbf{x}}}(t)&=\ \bar{\mathbf{A}}_{c}\bar{\mathbf{x}}(t)\,+\bar{\mathbf{B}}_{c,\mathbf{u}}\mathbf{u}(t)\,+\ \bar{\mathbf{B}}_{c,\mathbf{d}}\mathbf{d}(t)\,,\\ \bar{\mathbf{y}}(t)&=\ \bar{\mathbf{C}}\bar{\mathbf{x}}(t)\,, \end{split}  \tag{13}\]  
where the state vector \(\bar{\mathbf{x}}\), the control input vector \(\mathbf{u}\), the disturbance input vector \(\mathbf{d}\), and the output vector \(\bar{\mathbf{y}}\) of wide-area measurements can be expressed as  
\[ \begin{split}\bar{\mathbf{x}}:=&\left[\Delta\omega_{1} \ \Delta\omega_{2}\ \Delta P_{m_{1,1}}\ \Delta P_{m_{1,2}}\ \Delta P_{m_{2,1}}\ \Delta P_{m_{2,2}}\ \Delta P_{tie,AC}\ \Delta P_{DC}\ \Delta P_{ESS_{1}}\ \Delta P_{ESS_{2}}\right]^{\top}, \\ \mathbf{u}:=&\left[\Delta P_{DC_{ref}}\ \Delta P_{agc_{1}}\ \Delta P_{agc_{2}} \right]^{\top},\\ \mathbf{d}:=&\left[\Delta P_{d_{1}}\ \Delta P_{d_{2}} \right]^{\top},\\ \bar{\mathbf{y}}:=&\left[\Delta\omega_{1}\ \Delta\omega_{2}\ \Delta P_{tie,AC}\ \Delta P_{DC}\right]^{\top}. \end{split}  \tag{14}\]  
We note that the control input vector \(\mathbf{u}\) consists of control signals from supervisory AGC and SPMC controllers. Besides, the disturbance input vector \(\mathbf{d}\) corresponds to load changes in each area. \[\bar{\mathbf{A}}_{11}= \begin{bmatrix}\frac{-1}{T_{P_{1}}}&0&\frac{K_{P_{1}}}{T_{ P_{1}}}&\frac{K_{P_{1}}}{T_{P_{1}}}&0&0&\frac{-K_{P_{1}}}{T_{P_{1}}}\\ 0&\frac{-1}{T_{P_{2}}}&0&0&\frac{K_{P_{2}}}{T_{P_{2}}}&\frac{K_{P_{2}}}{T_{ P_{1}}}\\ \frac{-1}{2\pi K_{1,1}T_{ch_{1,1}}}&0&\frac{-1}{T_{ch_{1,1}}}&0&0&0\\ \frac{-1}{2\pi K_{1,2}T_{ch_{1,2}}}&0&0&\frac{-1}{T_{ch_{2,1}}}&0&0\\ 0&\frac{-1}{2\pi K_{2,1}T_{ch_{2,1}}}&0&0&\frac{-1}{T_{ch_{2,1}}}&0&0\\ 0&\frac{-1}{2\pi K_{2,2}T_{ch_{2,1}}}&0&0&0&\frac{-1}{T_{ch_{2,2}}}&0\\ 0&0&0&0\end{bmatrix} _{(6\times 7)},\]  
\[\bar{\mathbf{A}}_{12}= \begin{bmatrix}\frac{-K_{P_{1}}}{T_{P_{1}}}&\frac{-K_{P_{1}}} {T_{P_{1}}}&0\\ \frac{K_{P_{2}}}{T_{P_{2}}}&0&\frac{-K_{P_{2}}}{T_{P_{2}}}\\ 0&0&0\\ \vdots&\vdots&\vdots\\ 0&0&0\end{bmatrix} _{(6\times 3)},\quad\bar{\mathbf{A}}_{21}= \begin{bmatrix}\frac{T_{AC_{1,2}}}{2 \pi}&\frac{-T_{AC_{1,2}}}{2\pi}&0&0&0&0\\ 0&0&0&0&0&0\end{bmatrix} _{(2\times 7)}, \tag{15}\]  
\[\bar{\mathbf{A}}_{22}= \begin{bmatrix}0&0&0\\ \frac{-1}{T_{DC}}&0&0\end{bmatrix} _{(2\times 3)},\quad\bar{\mathbf{A}}_{32}= \begin{bmatrix}\frac{I_{em_{1}}K_{P_{1}}}{T_{ES_{1}}T_{P_{1}}}&-\frac{1}{T_{ES _{1}}}-\frac{J_{em_{1}}K_{P_{1}}}{T_{ES_{1}}T_{P_{1}}}&0\\ -\frac{I_{em  
The matrices \(\bar{A}_{c}\), \(\bar{B}_{c,u}\), \(\bar{B}_{c,d}\), and \(\bar{C}\) in (13) are constant with appropriate dimensions. For a better illustration of the system state matrix \(\bar{A}_{c}\), we use the following expression,  
\[\bar{A}_{c}= \begin{bmatrix}\bar{A}_{11}&\bar{A}_{12}\\ \bar{A}_{21}&\bar{A}_{22}\\ \bar{A}_{31}&\bar{A}_{32}\end{bmatrix} _{(10\times 10)}\.\]  
Each sub-matrix of \(\bar{A}_{c}\) is presented in Equation (15). In addition, the system input matrices \(\bar{B}_{c,u}\) and \(\bar{B}_{c,d}\) that relate control signals and load changes to the system states are given in (16) and (17), respectively.