## 3 System Model and Problem Modeling
### Communication Model and Energy Consumption Model  
In this paper, Frequency Division Multiple Access (FDMA) technology is employed for communication to prevent interference between signals from different devices [15]. Binary variables \(a_{n,m}(t)\) are used to represent the association between GPF \(n\) and AP \(m\) within each time slot. Binary variables \(a_{n,s}(t)\) are used to represent the association between GPF \(n\) and LEO satellite \(s\) within each time slot. \(a_{n,m}(t)=1\) denotes that in the current time slot, GPF \(n\) chooses to upload data to AP \(m\); otherwise, \(a_{n,m}(t)=0\). \(a_{n,s}(t)=1\) indicates that in the current time slot, GPF \(n\) opts for a direct connection to LEO satellite \(s\) for data upload; otherwise, \(a_{n,s}(t)=0\). Each device can choose only one data transmission mode in the same time slot, either connecting to the LEO satellite or a specific AP, and each device is subject to constraints as follows:  
\[\sum_{m=1}^{M}a_{n,m}(t)+a_{n,s}(t)\leq 1,\ \forall t\in\mathcal{T} \tag{1}\]  
According to [12], the connection between GPFs and LEO satellites, as well as APs, is characterized by Line-of-Sight (LoS) transmission links. Therefore, the channel gain between GPF n and LEO satellite s is defined as  
\[h_{n,s}=g_{0}\cdot(H_{s})^{-\theta}, \tag{2}\]  
where \(g_{0}=G_{n}^{tr}.G_{s}^{re. }\left(\frac{\lambda}{4\pi d_{0}}\right)^{-\theta}\) represents the channel power gain between GPF \(n\) and LEO satellite \(s\) at a unit distance \(d_{0}=1m\). \(G_{n}^{tr}\) denotes the transmit antenna gain of GPF \(n\), \(G_{s}^{re}\) denotes the receive antenna gain of LEO satellite \(s\), \(\lambda=\frac{c}{f}\) represents the carrier wavelength, \(c\) represents the speed of light, \(f\) represents the carrier frequency, and \(\theta\) represents the distance attenuation factor. Furthermore, since the distance between the GPFs and the LEO satellite is far away, the distance between LEO satellite \(s\) and GPF \(n\) is simplified to the altitude of the LEO satellite, denoted as \(H_{s}\). The direct communication transmission rate between GPF \(n\) and LEO satellite s is given by  
\[R_{n,s}(t)=W\cdot\log_{2}\left(1+\frac{p_{n}(t)\cdot h_{n,s}}{\sigma^{2}} \right), \tag{3}\]  
where \(W\) represents the allocated fixed bandwidth for communication between GPFs and LEO satellites and \(\sigma^{2}\) denotes the variance of Additive White Gaussian Noise (AWGN). \(p_{n}(t)\) represents the transmission power of GPF \(n\) during time slot \(t\), which is subject to the constraint imposed by the maximum transmission power \(p_{max}\) of the power facility as follows:  
\[p_{n}(t)\leq p_{\max},\ \forall n\in\mathcal{N},t\in\mathcal{T}, \tag{4}\]when the immediate communication workload \(\mu_{n,\text{immediate}}\), required by GPF \(n\) is substantial, due to the constraints on the transmission rate \(p_{\text{max}}\), the direct transmission rate \(R_{n,s}\) may not be able to complete the communication task of GPF \(n\) within the coverage time. In such cases, the GPFs will consider completing the communication task using the relay forwarding assistance of APs. In time slot \(t\), the channel gain between GPF \(n\) and AP \(m\) is defined as  
\[h_{n,m}(t)=g_{1}\cdot(d_{n,m}(t))^{-\theta}=g_{1}\cdot\left(\frac{1}{\sqrt{ \left\|\mathbf{q}_{m}(t)-\mathbf{q}_{n}\right\|^{2}+H_{it}^{2}}}\right)^{\theta}, \tag{5}\]  
where \(g_{1}=G_{n}^{tr}\cdot G_{m}^{re}\cdot\left(\frac{\lambda}{4\pi d_{0}}\right)^{ -\theta}\) represents the channel gain at a unit distance. \(G_{m}^{re}\) denotes the receive antenna gain of AP \(m\). According to [PERSON]'s formula, the transmission rate for communication between GPF \(n\) and AP \(m\) is defined as  
\[R_{n,m}(t)=W\cdot\log_{2}\left(1+\frac{p_{n}(t)\cdot h_{n,m}(t)}{\sigma^{2}} \right), \tag{6}\]  
and relays through AP \(m\).