## 4 Experiment Environment
### Case 1: Frequency Control of the IEEE 14-Bus Testing System under Different DoS Attack Intensities  
IEEE14 bus testing system is used to verify the effectiveness of the proposed control strategy under load change disturbances and different data packet loss rates. \(\eta_{ji}\in\{0.05,0.1,0.2,0.4,0.6\}\) is set. Both DoS attack period and total attack time under different \(\eta_{ji}\) are shown in Table 2.  
As shown in Table 3, the values of ADES controller parameters are initialized, including the learning rate of convex neural network weight update \(\beta\) and \(\gamma\), the number of subsystems \(N\), the number of network layers of convex neural networks \(Q_{N}\), the maximum absolute value \(|\lambda_{max}(\mathbf{L}_{i})|\) of the eigenvalue of matrix \(\mathbf{L}_{i}\), \(\left\|\mathbf{B}_{i,\max}\right\|^{2}\), Lipschitz constant \(L_{i}^{\prime}\), the weight \(a_{1}\) of the state \(\mathbf{x}_{i}(k)\), the weight \(a_{2}\) of the strategy \(u_{i}(k)\), the attenuation factor \(\alpha\), the number of long-term evaluation time windows \(S\), the Lipschitz constant of the function \begin{table}
\begin{tabular}{c c c} \hline \hline \(\eta_{ji}\) & **Attack Time** & **Total Attack Time** \\ \hline
0.05 & 1–1.3 s. & 0.3 s \\
0.1 & 1–1.3 s, 7.2 s–7.4 s. & 0.5 s \\
0.2 & 1–1.3 s, 2 s–2.2 s, & 1 s \\  & 3 s–3.5 s. & \\
0.4 & 1–1.3 s, 1.6 s–2 s, & 1.1 s \\  & 3 s–3.2 s, 7.2 s–7.4 s. & \\  & 1.3 s–1.5 s, 2 s–2.5 s, & \\
0.6 & 3 s–4 s, 6.3 s–6.5 s, & 1.9 s \\  & 7.2 s–7.4 s. & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: DoS attack period and total attack time under different data packet loss rates. Figure 5: Frequency deviation of the bus without ADES controller. Figure 6: Generator power deviation curve. \(\tanh x\), the maximum constraint of the control output \(u_{lim}\) and the maximum constraint of \(\Delta P_{B_{j}}\). These initialization parameters all satisfy the power grid stability and the sufficient conditions for the convergence of the actor-critic network weights Equation (32). \(\mathbf{w}_{a,i}\) and \(\mathbf{w}_{c,i}\) are initialized as vectors that satisfy the Assumption 2, and the simulation time is 8 s. The experimental results are shown in Figures 7-12. As shown in Figures 7-11, subfigure (a) shows the frequency deviation curve, subfigure (b) shows the change curve of the control output \(u_{i}\), subfigure (c) shows the deviation curve of the BESS power output. The power output of BESS is determined by the control output \(u_{i}\). There is a first-order inertia relationship between BESS and \(u_{i}\). Therefore, the output of Bess can be expressed through the first-order inertia link. Subfigure (d) shows the generator power output curve. Figure 12 shows the average loss curve of the weight training of convex neural networks under different data packet loss rates. The stabilization time is defined as the time required for the frequency deviation to remain within a fixed interval after being adjusted by the ADES controller. Assuming that the stable interval is \(\pm\)0.1 rad/s (\(\pm\)0.1 Hz), when the frequency deviation of all buses is maintained between \(\pm\)0.01 Hz, the system has been recovered to its nominal operating state [56]. As shown in Figure 7a,b, the frequency deviations of buses 1, 5, 9, and 14 exceed the frequency deviation threshold Hz at 0.4, 0.3, 0.7, and 0.5 s, respectively. At this time, the ADES controller starts and generates \(u_{i}\) (shown in Figure 7b) to control the active power output of BESS. The power changes of BESS are shown in Figure 7c. At the same time, the generator end also participates in frequency control, and the power change curve of the generator is shown in Figure 7d. In the simulation process, the maximum frequency deviation is 0.1 Hz, and each bus enters a stable state after about 4.3 s and finally shows small fluctuations in the stable interval. As shown in Figure 8a,b, the frequency deviations of buses 1, 5, 9, and 14 exceed the frequency deviation threshold Hz at 0.3, 0.3, 0.4, and 0.3 s, respectively.