## 3 Proposed Scheme
### Power Control Scheme
#### 3.2.2 Open-Loop Power Control  
In open-loop power control, the power is adjusted by calculating the PL. In order to obtain the PL, we need to calculate the position of the satellite in the orbit according to the six elements of the orbit so as to estimate the path loss of the satellite to the ground station. Select the Earth inertial coordinate system (ECI) as the reference system to analyze the various characteristics of each parameter. Based on the [PERSON] constellation [32], the distribution of satellites in the constellation can be expressed as \(N/P/F\), where \(P\) denotes the total number of orbits contained in the constellation, \(F\) represents the phase factor of the constellation and \(S\) denotes the number of satellites in each orbit. \(N=S\times P\) is the total number of satellites in the constellation. The phase offset between the nearest two satellites between adjacent orbits is determined by \(2\pi F/N\), while the value range of \(F\) is \([0,P-1]\). According to the spatial-geometric relationship, the position of the \(k\)-th satellite in the \(i\)-th orbit of the constellation can be represented by  
\[ \begin{array}{l}x_{ik}(t)=-R\cos\theta\sin(\frac{2\pi i}{P})\sin(\omega t+2 \pi(\frac{k}{5}+\frac{iF}{PS}))+\\ R\cos(\frac{2\pi i}{P})\cos(\omega t+2\pi(\frac{k}{5}+\frac{iF}{PS}))\\ y_{ik}(t)=R\cos\theta\cos(\frac{2\pi i}{P})\sin(\omega t+2\pi(\frac{k}{5}+\frac {iF}{PS}))+\\ R\sin(\frac{2\pi i}{P})\cos(\omega t+2\pi(\frac{k}{5}+\frac{iF}{PS}))\\ z_{ik}(t)=R\sin\theta\sin(\omega t+2\pi(\frac{k}{5}+\frac{iF}{PS}))\end{array}  \tag{34}\]  
Figure 5: Diagram of closed-loop power control. where \(0\leq i\leq P-1,0\leq k\leq S-1\). \(R\) denotes the distance between the satellites and the earth's center, \(\omega\) is the orbital angular velocity of the satellites and \(\theta\) represents the orbital inclination of the satellites. The six orbital elements include semi major axis, eccentricity, inclination, the longitude of ascending node, the argument of periapsis and true anomaly. When the orbit is a circular orbit, the argument of periapsis and true perigee angle can be merged into the argument of latitude, and the eccentricity is zero. (34) can be expressed by the satellite orbital elements as  
\[ \begin{array}{l}x_{ik}(t)=-R\cos\theta\sin\Omega\sin(\alpha+\varphi)+R\cos \Omega\cos(\alpha+\varphi)\\ y_{ik}(t)=-R\cos\theta\cos\Omega\sin(\alpha+\varphi)+R\sin\Omega\cos(\alpha+\varphi) \\ z_{ik}(t)=R\sin\theta\sin(\alpha+\varphi)\end{array}  \tag{35}\]  
where \(R\) represents semi major axis, \(\theta\) denotes the inclination. \(\Omega\) is the longitude of ascending node. \(\alpha\) is the argument of periapsis, and \(\phi\) denotes the true anomaly. Set the coordinates of ground terminal \(\left[x_{0}(t),y_{0}(t),z_{0}(t)\right]\), the distance between satellites and ground terminal can be expressed  
\[d_{k}=\sqrt{\left(x_{k}(t)-x_{0}(t)\right)^{2}+\left(y_{k}(t)-y_{0}(t)\right) ^{2}+\left(z_{k}(t)-z_{0}(t)\right)^{2}} \tag{36}\]  
According to [33], PL can be modeled by  
\[PL_{b}=FSPL(d,f_{c})+SF+CL(\alpha,f_{c}) \tag{37}\]  
where \(FSPL(d,f_{c})\) denotes the free-space path loss (FSPL) in dB, which is given by  
\[FSPL(d,f_{c})=32.45+20\text{log}_{10}(f_{c})+20\text{log}_{10}(d) \tag{38}\]  
\(d\) is the distance between the satellite and ground terminal, \(f_{c}\) is the carrier frequency. \(SF\) denotes the shadow fading, \(CL(\alpha,f_{c})\) represents the cluster loss. The open-loop power control process is described as follows:  
Step 1: Determine the target SINR of satellite 1 and satellite 2, calculate the transmission power \(P_{sat1}^{1},P_{sat2}^{1}\) at the initial time according to the ephemeris information and Formulas (29)-(38). \(P_{sat1}^{1},P_{sat2}^{1}\) represent the transmission power of satellite 1 and satellite 2 at the initial time. Step 2: Obtain ephemeris information at time \(t_{0}+T\). Calculate the two satellites' transmission power at time \(t_{0}+T\) according to Formulas (29)-(38), where \(T\) is the power adjustment cycle.