## 3 Energy Efficiency Problem Model for SONs
### GreenRP Algorithm
Normalization issues are addressed by using positive benchmarks, \(L_{SP}\) and \(D_{SP}\), as denominators, while packet loss \(B_{SP}\), which may be zero under light loads, is handled through an exponential form to highlight the optimization goal and prevent excessive hibernation from causing network congestion. \[score=\left\{ \begin{array}{ll}X\cdot\frac{L}{L_{SP}}+Y\cdot\frac{B}{B_{SP}}+ Z\cdot\frac{D}{D_{SP}},B_{SP}>0&\\ X\cdot\frac{L}{L_{SP}}+Y\cdot\mathbf{e}^{B}+Z\cdot\frac{D}{D_{SP}},B_{SP}=0& \end{array} \right.\quad L_{SP},D_{SP}>0 \tag{21}\]  
During the population evolution process, the algorithm treats lower fitness scores as indicative of superior genetic traits, ensuring that such individuals have a higher likelihood of reproducing. Crossover is performed using a single-point method, where a gene index is randomly selected and genes are swapped to simulate chromosome crossover. Mutation consists of randomly toggling link states. Fitness influences genetic inheritance, with crossover and mutation probabilities adaptively adjusted according to the iteration count, as detailed in Equations (22) and (23). The iteration count \(iter\) and the maximum limit \(iter_{\max}\) influence these probabilities; higher crossover probabilities early in the iterations facilitate global search, while increased mutation probabilities later enhance local search. To ensure stable convergence of the algorithm and protect superior genes from premature elimination, an elite preservation strategy is implemented. An elitism retention strategy is implemented to stabilize the algorithm and prevent premature loss of superior genes. By preserving the top-performing individuals before each generation's operations, this strategy not only maintains population diversity but also ensures that high-fitness genes are retained, promoting convergence to optimal solutions. \[P_{c}=0.5*\cos(\frac{iter}{iter_{max}}*\frac{\pi}{2})+0.1 \tag{22}\]  
\[P_{m}=0.03*\sin(\frac{iter}{iter_{max}}*\frac{\pi}{2})+0.02 \tag{23}\]  
Traditional genetic algorithms typically terminate once performance metrics meet a predetermined threshold, outputting the best gene found to that point. Due to the NP-hard nature of the problem, convergence thresholds are inestimable; setting too high a target value may prevent the algorithm from stopping, while too low a value may terminate iterations prematurely. Consequently, this study employs a dynamic termination criterion based on the gradient change of performance metrics. We focus on the best gene and its score produced in each iteration, comparing these with the historically best solution. Ifthe newly obtained best solution does not surpass the historical best in three consecutive iterations, it is considered unlikely that further iterations will converge to a better solution, prompting termination of the algorithm. This strategy, based on gradient changes and comparisons with historical bests, allows the algorithm to flexibly determine and find the appropriate time to terminate without a clear threshold. Finally, to ensure the network connectivity of the final energy-efficient topology, we implemented a supplementary strategy: upon establishing the energy-efficient topology, we applied the shortest path algorithm pairwise to the 155 terrestrial regions. All links that appeared in the path set were activated to ensure that any terrestrial area could communicate via the satellite network. This approach guarantees comprehensive connectivity across all regions, maintaining robust network performance within the energy-efficient framework.