## 3 DoS Attacks on the AC/DC Multi-Area LFC System with Virtual Inertia
### The Test LFC System under DoS Attacks  
First, we need to modify the LFC system model developed in Section 2 to include DoS attacks into the control loop. As illustrated in Figure 1, we consider the attack scenario where communication channels for the transmission of wide-area measurements are attacked by DoS. To help in model analysis, one needs to modify the state-space representation in (13) by defining the following new \"virtual\" state and output vectors,  
\[ \begin{split}\mathbf{x}:=&\left[\Delta\omega_{1}\ \Delta\omega_{2}\ \Delta P_{m_{1,1}}\ \Delta P_{m_{1,2}}\ \Delta P_{m_{2,1}}\ \Delta P_{m_{2,2}}\ \Delta P_{ii,AC}\ \Delta P_{DC}\ \Delta P_{ESS_{1}}\ \Delta P_{ESS_{2}}\ \int ACE_{1}\ \int ACE_{2} \right]^{\top},\\ \mathbf{y}:=&\left[\Delta\omega_{1}\ \Delta\omega_{2}\ \Delta P_{ ii,AC}\ \Delta P_{DC}\ \int ACE_{1}\ \int ACE_{2}\right]^{\top},\end{split}  \tag{18}\]  
where \(\int ACE_{i}\) is the integration of the ACE signal in Area \(i\). Note that \(\int ACE_{i}\) in \(\mathbf{y}\) is a virtual variable and the practical wide-area measurements in the output vector \(\mathbf{y}\) are frequencies (\(\Delta\omega_{i}\)) and AC/DC power flows (\(\Delta P_{tie,AC}\), \(\Delta P_{DC}\)). Then, the integral action in the supervisory AGC loop can be transformed into a static output feedback control problem [20]. We still use definitions of input vectors \(\mathbf{u}\) and \(\mathbf{d}\) in (14). Then, we can obtain the following \"modified\" dynamic model for the test LFC system  
Figure 2: Results of both areas when there is a step load change in Load 1 of Area 1 in the two-area power system. (**a**) Frequency deviation of Area 1. (**b**) Frequency deviation of Area 2.  
interconnected by AC/HVDC links and equipped with bulk ESS, by considering the \"virtual\" state vector \(\mathbf{x}\) and output vector \(\mathbf{y}\) in (18),  
\[ \begin{split}\dot{\mathbf{x}}(t)&=\ \mathbf{A}_{c}\mathbf{x}(t)\,+ \mathbf{B}_{c,u}\mathbf{u}(t)\,+\,\mathbf{B}_{c,d}\mathbf{d}(t)\,,\\ \mathbf{y}(t)&=\ \mathbf{C}\mathbf{x}(t)\,.\end{split}  \tag{19}\]  
Note that now \(\mathbf{u}\) represents the input signal from the resulted static output feedback control in the above open-loop LFC system model of (19). By using the \"virtual\" output vector \(\mathbf{y}\) in (18), the static output feedback control process can be expressed as \(\mathbf{u}=\mathbf{K}\mathbf{y}\) where \(\mathbf{K}\) is the gain of the static output feedback control and we can have  
\[\mathbf{K}= \begin{bmatrix}K_{1}&K_{2}&K_{AC}&0&0\\ 0&0&0&0&K_{I_{1}}&0\\ 0&0&0&0&0&K_{I_{2}}\end{bmatrix} \,. \tag{20}\]  
For the matrices \(\mathbf{A}\), \(\mathbf{B}_{c,u}\), \(\mathbf{B}_{c,d}\), and \(\mathbf{C}\) in the model (19), it is easy to derive their formulations based on \(\bar{\mathbf{A}}_{c}\), \(\bar{\mathbf{B}}_{c,u}\), \(\bar{\mathbf{B}}_{c,d}\) and \(\bar{\mathbf{C}}\) given in the original open-loop LFC model in Section 2.4. In a realistic framework, the wide-area measurements are applied to the supervisory AGC and SPMC controllers in discrete-time samples. Thus we would like to express the dynamic LFC system model in a discrete-time framework. To do that, (19) needs to be discretized. Taking a sampling period \(T_{s}\), we have the following discretization results for a zero-order hold (ZOH) [38],  
\[\mathbf{A}=e^{\mathbf{A}_{c}T_{s}}\,,\quad\mathbf{B}_{u}=\int_{t=0}^{T_{s}}e^{\mathbf{A}_{c}(T _{s}-t)}\mathbf{B}_{c,u}\mathrm{d}t\,,\quad\mathbf{B}_{d}=\int_{t=0}^{T_{s}}e^{\mathbf{A}_ {c}(T_{s}-t)}\mathbf{B}_{c,d}\mathrm{d}t\,. \tag{21}\]  
To be noted, (21) can be explained as the analytical solution of the ZOH discretization. Then, after discretization of (19), the discrete-time version of the open-loop LFC model for the two-are system equipped with AC/HVDC transmission lines and bulk ESS can be described by  
\[ \begin{split}\mathbf{x}[k+1]&=\ \mathbf{A}\mathbf{x}[k]\,+\mathbf{B}_{u}\mathbf{u}[k]\,+\, \mathbf{B}_{d}\mathbf{d}[k]\,,\\ \mathbf{y}[k]&=\ \mathbf{C}\mathbf{x}[k]\,.\end{split}