## 5 Simulation Results
### FDI Attack Results  
Next, we evaluate the effects of FDI attacks on the three LFC system models. To begin with, stationary univariate and multivariate attacks are launched. The frequency deviation results under a univariate attack on the frequency measurement of Area 2 are shown in Figure 7. We can see that regarding the attack impact index of MFD (maximum frequency deviation) during the transients, there would be a larger MFD in the LFC of the system interconnected by AC/HVDC lines and equipped with ESS to emulate inertia. This observation is consistent with the result of [32], and we refer to [32] for a more detailed analysis of univariate attacks on the other wide-area measurements (e.g., frequency  
Figure 5: Case 2: results of both areas under a step-load change at \(5\,\mathrm{s}\) and also DoS attacks that start from \(6\,\mathrm{s}\). (**a**) Frequency deviation of Area 1. (**b**) Frequency deviation of Area 2. Figure 6: Case 3: results of both areas under a step-load change at \(5\,\mathrm{s}\) and also DoS attacks that start from \(12\,\mathrm{s}\). (**a**) Frequency deviation of Area 1. (**b**) Frequency deviation of Area 2.  
of Area 1, AC/DC power flow). Then, we move to stationary multivariate attacks where multiple measurements are attacked simultaneously to be disruptive and undetectable (with respect to data quality checking programs). The optimal strategies for these attacks can be obtained from (31) by solving the resulted MILP using the solver CPLEX. It turns out that a multivariate attack (\(\kappa_{i}^{*}=2\)) that can attack power flows on both AC and HVDC lines with a attack magnitude vector \(f=[0.44\,-0.39]^{\top}\) (in p.u.) is able to disrupt the LFC system and avoid data quality alarms. The frequency deviations under this multivariate attack are shown in Figure 8. The MFD of Area 1 in the AC/DC interconnected system with emulated inertia arrives at \(-0.8\,\mathrm{Hz}\) after the occurrence of multivariate attack, which may mislead wrong system operations of load shedding. To be noted, when solving solving (31) for the normal AC system, there is no such kind of multivariate FDI attack that can have enough impact regarding the index MFD but remain undetected from data quality checking programs. From the observations above, the inertia emulation functionality plays a key role in affecting the dynamic behavior of the test LFC system under FDI attacks. Due to the frequency variations caused by FDI attacks on supervisory controls, the inertia emulator is also being \"misled\" as it calculates rate of change of frequency (ROCOF) from local frequency information in its derivative control loop (recall Section 2.3), which in turn would contribute to a larger MFD. The stationary FDI attacks above can be detected by an advanced anomaly detector. However, as illustrated in Section 4.2, the so-called zero-dynamics attack can lead to zero system outputs and thus remain hidden with respect to arbitrary anomaly detectors. To the end, we move to the simulations of such an attack scenario. From the calculations of system zeros, we see that for the LFC of normal AC  
Figure 8: Results of both areas under multivariate attacks on power flow measurements of both AC and HVDC lines with magnitudes of \(0.44\,\mathrm{p.u. }\) and \(-0.39\,\mathrm{p.u. }\), respectively, at \(t=10\,\mathrm{s}\). (**a**) Frequency deviation of Area 1. (**b**) Frequency deviation of Area 2. Figure 7: Results of both areas under a univariate attack with a magnitude of \(0.1\,\mathrm{Hz}\) on the measurement of frequency in Area 2, at \(t=10\,\mathrm{s}\). (**a**) Frequency deviation of Area 1. (**b**) Frequency deviation of Area 2.  
system, there are four system zeros (0.5850, 0.5908, 0.6927, 1) of real values and correspondingly four unitary vectors that are solutions to (34). For the LFC of AC/DC system but without virtual inertia, there are five system zeros (\(-\)0.0250, 0.5856, 0.5908, 0.8087, 1) of real values and correspondingly five unitary vectors that are solutions in the null-space of \(\mathbf{P}(z_{0})\).