## 8 Results and Discussion  
In this section, we evaluate the performance of the proposed joint user scheduling and hybrid beamforming design algorithm by numerical simulations. In the numerical simulations, we set the number of multicast groups to \(L=7\), which cover 150 active users, and set the SINR constraint threshold of each multicast group to \(\text{SINR}_{0}=1\). To facilitate analysis, we assume that the CSI errors of different multicast groups are the same, whichare expressed as \(\sigma^{2}_{f_{d,k,l}^{red}}=\sigma^{2}_{f_{d}^{red}}\) and \(\sigma^{2}_{\phi_{T_{k,l}}}=\sigma^{2}_{\phi_{t}}\). The value of \(P_{0}\) can be calculated by [31]. In addition, the system parameters used in the numerical simulations are shown in Table 1.  
Figure 7 shows the convergence trajectory of the EE of the massive MIMO LEO satellite multigroup multicast communication system, versus the number of iterations for different CSI errors, different numbers of preselected users and different scheduling algorithms. In this simulation, two groups of channel errors are set according to Refs. [23; 29], i.e., \(\sigma^{2}_{f_{d}^{red}}=25\), \(\sigma^{2}_{\phi_{t}}=10\) and \(\sigma^{2}_{f_{d}^{red}}=20\), \(\sigma^{2}_{\phi_{t}}=5\). In addition, we set \(U_{s}=2\) and \(P_{T}=50\) W. Meanwhile, we set two different numbers of the preselected users, i.e., \(U_{l,p}/U_{s}=2,U_{l,p}=4\) and \(U_{l,p}/U_{s}=3,U_{l,p}=6\). It can be seen that the proposed robust algorithm has higher performance gain than the traditional nonrobust algorithm, which shows the effectiveness of the robust algorithm. Meanwhile, it can be seen that when \(\sigma^{2}_{f_{d}^{red}}=25\), \(\sigma^{2}_{\phi_{t}}=10\) and \(\sigma^{2}_{f_{d}^{red}}=20\), and \(\sigma^{2}_{\phi_{t}}=5\), the EE performance gain of the proposed robust algorithm is improved by 9.8% and 6.7%, respectively, compared with the traditional nonrobust algorithm. The system EE of the proposed joint user scheduling and hybrid beamforming design algorithm is higher than that of the decoupling design algorithm, and the more preselected users, the higher performance improvement. Figure 8 compares the EE performance of the proposed algorithm and the traditional algorithm under different system parameters, versus different transmission power thresholds \(P_{T}\). It can be seen that with the increase in transmission power, the EE performance shows a trend of first rising and then falling. The reason is that the growth rate of the system rate is lower than that of the power consumption. Meanwhile, we can see that the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Values** & **Parameters** & **Values** \\ \hline \(N\) & \(8\times 8=64\) & \(\kappa\) & \(1.38\times 10^{-23}\) J\(\cdot\)K\({}^{-1}\) \\ \(L\) & 7 & \(P_{0}\) & 21.5 W \\ \(\kappa_{k,l}\) & 10 & \(T\) & 300 K \\ Bandwidth & 50 MHz & \(G_{Loo}\) & 3 dB \\ Orbit altitude & 1000 km & \(G_{ut}\) & 3 dB \\ Beam radius & 250 km & \(f\) & 20 GHz \\ \(LP_{at}\) & 0.017 dB & \(K\) & 150 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. Figure 7: Convergence trajectory of system EE relative to different CSI errors, different number of preselected users and different scheduling algorithms. EE of the joint user scheduling and hybrid beamforming design algorithm is higher than that of the decoupling design algorithm versus different transmission power thresholds \(P_{T}\). Under the conditions of \(\sigma_{f_{d}}^{2}=25\), \(\sigma_{\phi_{\tau}}^{2}=10\) and \(P_{T}=15\) W, when \(U_{l,p}/U_{s}=2,U_{l,p}=4\) and \(U_{l,p}/U_{s}=3,U_{l,p}=6\), the EE performance gain of the proposed joint design algorithm is 28.41% and 45.19% higher than that of the traditional decoupling design algorithm. Figure 9 indicates the change trend of the SE of the massive MIMO LEO satellite multigroup multicast communication system versus different transmission power thresholds \(P_{T}\). It can be seen that the SE increases with the increase in the transmission power.