## 2 Energy Model
### TSTW Model  
To increase the energy balance performance of the NTSN system, a satellite illumination model is necessary. UsingSTK to simulate the above walker-delta constellation, it can be indicated that 40% of the orbit cycle is the \"night side\" and 60% of the orbit cycle is the \"day side\" for a satellite. In an orbit, a part of satellites are in the \"day side\", and the rest are in the \"night side\", as shown in Figure 3. Each point represents a satellite, and different orbital planes are represented by different colors. It can be calculated by the \"Light Time\" function in STK that satellites will enter \"day side\" in turn. On one hand, the charging interval between two adjacent satellites in an orbit is about 240 s. The average charging duration is about 3600 s, and the average umbra duration is about 1600 s. On other hand, the charging interval between two adjacent satellites in adjacent orbits is about 120 s. As shown in Figure 4, satellites enter the \"day side\" from \"night side\" in tune. For a satellite, the \"day side\" and \"night side\" alternate periodically. This figure shows the chargeable time and non-chargeable time, respectively. However, there is about 400 s of fuzzy duration between the alternations. In this duration, the charging power rating fluctuates with orbital status, solar storms, etc. Therefore, the role of this region is ignored in the theoretical model. \begin{table}
\begin{tabular}{c c c} \hline
**Item** & **Starlink** & **OneWeb** \\ \hline \(\theta\) & 70\({}^{\circ}\) & 90\({}^{\circ}\) \\ \(S\) & 20 m\({}^{2}\) & 2.5 m\({}^{2}\) \\ \(\eta\) & 25\% & 25\% \\ \(SC\) & 1250 w/m\({}^{2}\) & 1250 w/m\({}^{2}\) \\ \(CP\) & 5500 w & 650 w \\ \hline \end{tabular}
\end{table} 
Table 1: Power system parameters. Figure 3: Constellation lighting diagram. In order to introduce the energy balance into the optimization objective, an energy prediction model is necessary. According to \"Light Time\" analysis, 13 adjacent satellites are in the \"day side\" for any orbit. A satellite remains in the fuzzy duration and the charging state is ignored. A tubular sliding time window model (TSTW) can imply this property, as shown in Figure 5. In this model, a ring represents 24 satellites in an orbit, and a tubular sliding window represents the light duration. All satellites in the \"day side\" are surrounded by tubes. The tube will rotate clockwise. By rotating at a specific angle, the illuminating state of the satellite can be predicted. The illuminating state of an orbit can be represented by a single ring. A constellation applies 48 rings, and there is a one-bit delay between adjacent rings, as shown in Figure 6. With the tube rotating clockwise, satellites in different orbits are surrounded in order. Figure 4: Satellite lighting schematic. Figure 5: Charging prediction model for orbit. The main function of the model is an energy balance data structure suitable for NTSN. Utilizing the compass model with cyclic coverage, the state information can be obtained efficiently when the node energy changes. This information can include the past, present and predicted remaining energy. Especially for mega satellite constellations, efficient management of node information is always a key technology for on-orbit processing. In the routing algorithm, the node energy prediction hash table is periodically maintained. This table is updated synchronously with the charge prediction model. At the same time, we use the pointer address to select the charging node. Regular node charge and discharge information can be automatically completed by the model. The updated information between nodes only needs to contain the energy consumption and failure of each node. As shown in Figure 7, the data structure of the model is a hash table. Hash function and TSTW synchronous cycle transformation. Both node update and routing information are completed through the key-value pairs of the hash table.