## 3 Energy Efficiency Problem Model for SONs
### Energy-Efficient Problem Model  
This section presents how energy-efficient routing problems in edge-AI-driven SONs are solved through topology planning. The input consists of pairs \((G,T)\in Sp_{k}\), with the network's state over a period described by a series of snapshots \(Sp_{1}\sim Sp_{N}\). Each snapshot \(Sp_{k}\) captures a fixed topology within a specific time interval \([t_{1},t_{2}]\). The pair \(G\) includes comprehensive topology information and satellite power status, while \(T\) represents the network's traffic matrix, calculated at time \(t=(t_{1}+t_{2})/2\) according to Equation (11), depicting the network's continuous state over time. When satellites enter dormancy, they maintain operations at a fixed low power, as analyzed in Section 2.2, composed of a baseline power consumption and variable consumption based on node throughput. As satellites go dormant, they cease communications tasks, directly shutting down ground links, and all optical terminals for inter-satellite links are also shut, due to the requirement for simultaneous activation with neighbor satellites for  
Figure 2: The global traffic demand. laser link establishment. This results in a network-wide deactivation of optical terminals during the snapshot, represented by \(S_{k}^{close}\), transforming the original topology to \(G_{k}(V_{k},E_{k})\). If satellite \(i\) enters dormancy, and \(OT(i,j)\) denotes its optical terminal for the link to satellite \(j\), then \(\forall j\) belongs to the neighbors of satellite \(i,OT(i,j)\in S_{k}^{close}\), and \(OT(j,i)\in S_{k}^{close}\). However, due to the challenge of aligning lasers, switching an optical terminal from standby to active mode requires a time \(t^{\prime}\) to align the laser. Therefore, if an optical terminal is in standby during \(Sp_{k}\) but active in \(Sp_{k+1}\), it needs to be activated before \(t_{2}-t^{\prime}\) to ensure proper function in the next period, indicating that longer dormancy periods can enhance energy savings. Equations (16)-(18) describe the power consumption variations of optical terminals during \(Sp_{k}\). \(S_{k}^{open}\) represents the collection of optical terminals active network-wide during \(Sp_{k}\), \(S_{OT}^{i}\) denotes the set of optical terminals owned by satellite \(i\), and \(S_{i}^{open1}\) represents those active during \([t_{1},t_{2}-t^{\prime}]\). Given some optical terminals reverse status in \(Sp_{k+1}\), the set \(S_{i}^{open2}\) represents those active during \([t_{2}-t^{\prime},t_{2}]\), with \(S_{i}^{open1}\in S_{i}^{open2}\). The fixed power consumption \(P_{const}^{OT}(t)\) of satellite \(i\)'s optical terminals is proportional to the number of active terminals \(|S_{i}|\). Finally, Equation (19) expresses the satellite's power consumption. \[S_{i}^{open1}=S_{OT}^{i}\cup S_{k}^{open} \tag{16}\]  
\[S_{i}^{open2}=S_{OT}^{i}\cup S_{k+1}^{open} \tag{17}\]  
\[P_{const}^{OT}(t)= \begin{cases}T_{a}\cdot\left|S_{i}^{open1}\right|,t\in[t_{1},t_{2}-t^{\prime}]\\ T_{a}\cdot\left|S_{i}^{open2}\right|,t\in[t_{2}-t^{\prime},t_{2}]\end{cases}  \tag{18}\]  
\[P_{i}= \begin{cases}P_{OS}+P_{const}^{OT}(t)+P_{available}+P_{base}&(workmode) \\ \varphi P_{OS}+P_{base}&(sleepmode)\end{cases}  \tag{19}\]  
The problem is formulated by considering a series of consecutive operational snapshots, labeled \(Sp_{1}\sim Sp_{N}\). Each snapshot, \(Sp_{i}\), encapsulates the network's physical connectivity, \(G(V,E)\), and documents the unique residual battery levels for all satellites at that specific moment. Associated with each snapshot is a traffic matrix, \(T\), which remains consistent throughout its duration. The analysis further incorporates quantitative formulas to define satellite power consumption (Equations (2)-(5) and (16)-(19)) and battery life degradation (Equations (6)-(8)).