## 5 Simulation Results
### DoS Attack Results  
We start with DoS attacks. In Section 3.2, we have introduced a constant \(\lambda\in[0,1]\) such that if an advanced attacker can choose a proper \(\lambda\), the \"equivalent\" closed-loop LFC system under DoS attacks on measurements (with system matrix \(\Phi(\lambda)=\Phi_{1}^{\lambda}\Phi_{2}^{(1-\lambda)}\)) can be made unstable. From the proof of Lemma 1, we would note that the smaller \(\lambda\) is, the earlier the DoS attack occurs. Then, we let \(\gamma_{m}\) denote the maximum real part of eigenvalues of \(\Phi(\lambda)=\Phi_{1}^{\lambda}\Phi_{2}^{(1-\lambda)}\). To study how the DoS attack would affect the stability of the underlying LFC systems, we compute \(\gamma_{m}\) with \(\lambda\) ranging between 0 and 1 for the three LFC system models. The results are shown in Figure 3. It can be seen form Figure 3 that \(\gamma_{m}>1\) when \(\lambda\) is close to 0 for all of these systems, which implies that there exists an eigenvalue with magnitude outside the unity circle and the systems are unstable. With the increase of \(\lambda\), \(\gamma_{m}\) may decrease and be smaller than 1 later. This result is straightforward since the LFC systems become unstable more easily when DoS attacks occur at an early period. For the LFC of normal AC system, \(\gamma_{m}<1\) when \(\lambda\geq 0.6\). For AC/DC system, \(\gamma_{m}<1\) when \(\lambda\geq 0.33\). For AC/DC system but with virtual inertia, \(\gamma_{m}\) is around 1 when \(\lambda\) is small, and is smaller than 1 when \(\lambda\geq 0.53\). We let the time interval in the proof of Lemma 1 be [0, 30) (in seconds). Then, three case studies are considered in the simulations of DoS attacks: (i) Case 1: \(\lambda=0.033\), the three LFC systems are under DoS attacks which start from \(t=1\) s; (ii) Case 2: \(\lambda=0.2\), the three LFC systems are under DoS attacks which start from \(t=6\) s; (iii) Case 3: \(\lambda=0.4\), the three LFC systems are under DoS attacks which start from \(t=12\) s. For all the cases, similar to Figure 2, we add a step load input to the test LFC systems that there is an increase of 0.03 p.u. in Load 1 of Area 1, at \(t=5\) s. The simulation results of the three case studies are presented in Figures 4-6. For Case 1 in Figure 4, the DoS attacks occur at \(t=1\) s, which is before the load change. It can be observed that there are large steady-state frequency deviations (SSFDs) because the controller is attacked by DoS completely. For Case 2 in Figure 5, the DoS attacks occur right after the event of step-load change and we still see large SSFDs. Comparing the results of Figures 4-6, it is reasonable to conclude that from the viewpoint of the attacker, it is optimal to launch DoS attacks as early as the dynamics of the LFC system does not converge. When the attacks occur in a late stage, the DoS attacks might not have big impact; see the results of Case 3 where \(\lambda=0.4\). It can be also expected that as long as the LFC system dynamics has converged, the DoS attacks would not have effects any more. Figure 4: Case 1: results of both areas under a step-load change at 5 s and also DoS attacks that start from 1s. (**a**) Frequency deviation of Area 1. (**b**) Frequency deviation of Area 2. Figure 3: The maximum real part of the eigenvalues of \(\Phi(\lambda)=\Phi_{1}^{\lambda}\Phi_{2}^{(1-\lambda)}\) for the three load frequency control (LFC) systems. When looking into the frequency deviations of the three system LFC models in Figures 4 and 5, we can also see that the impact of early DoS attacks on the LFC models of normal AC system and AC/DC system but without virtual inertia can be more significant from the perspective of SSFD, comparing with the one of the AC/DC interconnected system with virtual inertia. This is due to the fact that the control loop of inertia emulation is not attacked directly by DoS as it is using local frequency information only, while the DoS attacks are mainly on measurements for supervisory AGC and SPMC loops.