## 2 LFC Modeling in the Hybrid AC/DC System with Virtual Inertia
### The Conventional LFC Structure  
The LFC loop is a typical high-level control application. To regulate the power grid frequency, the LFC module in each controlled area receives wide-area measurements of frequency and tie-line power and sends control signals of set points for the output power of the participating generators in that area (e.g., GEN 1 and GEN 2 in Area 1 of Figure 1). To understand the LFC concept, we first introduce an area control error (ACE) signal as follows,  
\[ACE_{i}=\beta_{i}\Delta\omega_{i}+\Delta P_{tie_{ij}}\,, \tag{1}\]  
where \(\beta_{i}\), \(\Delta\omega_{i}\), and \(\Delta P_{tie_{ij}}\) are the frequency bias factor, the frequency deviation of Area \(i\), and the net tie-line power flow variation between Area \(i\) and Area \(j\), respectively; for a two-area power system, like the one in Figure 1, \(\Delta P_{tie_{ij}}=-\Delta P_{tie_{ji}}\). Thus, for a normal AC system where there are only pure AC transmission lines, let \(\omega_{0}\) be the nominal value of frequency, and we have  
\[\Delta P_{tie_{ij}}=\Delta P_{tie,AC}\,,\ \Delta\omega_{i}=\omega_{i}-\omega_{0}\,. \tag{2}\]  
To be mentioned, the ACE value in (1) defines the frequency to restore and the power to compensate in the event of load-generation imbalance. As noted earlier, the LFC system is a high-level control application, and we pay more attention to the collective performance of all generators [33]. Then, we can do certain levels of abstraction and suppose that each area consists of equivalent governors, turbines, and generators. The dynamics of  
Figure 1: The block diagram of the test system. The communication channels (the red line) for the transmission of wide-area measurements are very vulnerable, and therefore the cyber attacks of Denial of Service (DoS) and false data injection (FDI) in this article are mainly on the measurements side. To be noted, we assume that the channel for control signals is equipped with advanced encryption techniques and thus not attacked. each area is represented by a linearized model. In this regard, the frequency dynamics of Area \(i\) in the two-area system can be described in the Laplace domain,  
\[\Delta\omega_{i}(s) = \frac{K_{p_{i}}}{1+sT_{p_{i}}}\big{[}\sum_{g=1}^{G_{i}}\Delta P_{m_ {i,g}}-\Delta P_{d_{i}}-\Delta P_{tie,AC}\big{]}\, \tag{3}\]  
where \(K_{p_{i}}\) and \(T_{p_{i}}\) are the system gain and the time constant, respectively. The gain \(K_{p_{i}}\) is related to the damping coefficient. The time constant \(T_{p_{i}}\) is associated with both the equivalent inertia and the damping. \(\Delta P_{m_{i,g}}\) is the output power of each participated generator in Area \(i\), and \(G_{i}\) denotes the number of these generators. \(\Delta P_{d_{i}}\) represents the total load variation in Area \(i\). For the variables \(\Delta P_{m_{i,g}}\) and \(\Delta P_{tie,AC}\), we have  
\[\Delta P_{m_{i,g}}(s)=\frac{1}{1+sT_{cl_{ig}}}\big{[}\frac{\Delta\omega_{i}}{ R_{i,g}\times 2\pi}-\phi_{i,g}\Delta P_{qc_{i}}\big{]}\, \tag{4}\]  
\[\Delta P_{tie,AC}(s)=\frac{T_{AC_{ij}}}{s}\big{[}\Delta\omega_{i}-\Delta\omega _{j}\big{]}\, \tag{5}\]  
where \(R_{i,g}\) is considered as droop for each participated generator in Area \(i\). \(T_{cl_{ig}}\) is the time constant of the whole turbine-governor unit (we assume that each dynamic generator model consists of its turbine-governor model). \(\Delta P_{qgc_{i}}\) denotes the AGC signal generated by the LFC control loop in Area \(i\). \(\phi_{i,g}\) is an area participating factor satisfying \(\sum_{g=1}^{G_{i}}\phi_{i,g}=1\). \(T_{AC_{ij}}\) is the coefficient for the power flow on the AC transmission line between these two areas. The AGC signal \(\Delta P_{qgc_{i}}\) is used to regulate the set points of participated generators for active power output. The goal is to guarantee that the system frequency restores to nominal value in a load-generation imbalance event. Meanwhile, the tie-line power flow between controlled areas should act as the scheduled one.