# UAV-BS Site Planning Based on Circular Coverage Strategy
## 4 Methodology
### Circle Coverage Power Optimization Strategy
The Lagrange function for the above problem is constructed as follows:  
\[ \begin{split} L(n,\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})& =P_{\text{s}}+\sum_{i=1}^{i=4}\lambda_{i}\text{C}(i)\\ &=P_{\text{s}}-\lambda_{1}P_{g}+\lambda_{2}(P_{g}-P_{\text{max}} )+\lambda_{3}(S-S_{c})+\lambda_{4}(n-N)\end{split}  \tag{28}\]  
In order to facilitate the subsequent calculation, the above formula is simplified:  
\[ \begin{split} L(n,k_{i})&=\text{A}\cdot n(r_{0}(n) ^{2}+\frac{H^{2}}{R^{2}})^{\alpha/2}\\ &+(n-1)P_{u}-k_{1}n(r_{0}(n)^{2}+\frac{H^{2}}{R^{2}})^{\alpha/2} \\ &+k_{2}(n(r_{0}(n)^{2}+\frac{H^{2}}{R^{2}})^{\alpha/2}-\frac{P_{ \text{max}}}{A})\\ &+k_{3}(1-r_{0}(n)^{2})+k_{4}(n-N),\end{split}  \tag{29}\]  
where \(k_{1},k_{2},k_{3}\), and \(k_{4}\) are non-negative Lagrange multipliers. For Equation (28) to have an optimal solution, it must satisfy the KKT conditions in optimization theory. That is,  
\[ \begin{split}& C(i)\leq 0\\ & k_{i}\geq 0\\ & k_{i}\text{C}_{i}(x)=0\\ &\frac{\partial L(n,k_{1},k_{2},k_{3},k_{4})}{\partial n}=0.\\ &\text{for}\ i=1,2,3,4.\end{split}  \tag{30}\]  
To solve Equation (30) using [PERSON]'s method, let the \(i\)-th iteration value be denoted as \(n_{i}\), and the \((i+1)\)-th iteration value be the following:  
\[n_{i+1}=n_{i}-\frac{L^{\prime}(n_{i})}{L^{\prime\prime}(n_{i})}. \tag{31}\]  
\[ \begin{split} L(n_{i})&=\text{A}\cdot n_{i}(r_{0}(n _{i})^{2}+\frac{H^{2}}{R^{2}})^{\alpha/2}\\ &+(n_{i}-1)P_{\text{max}}-k_{1}n_{i}(r_{0}(n_{i})^{2}+\frac{H^{2} }{R^{2}})^{\alpha/2}\\ &+k_{2}(n_{i}(r_{0}(n_{i})^{2}+\frac{H^{2}}{R^{2}})^{\alpha/2}- \frac{P_{\text{max}}}{A})\\ &+k_{3}(1-r_{0}(n_{i})^{2})+k_{4}(n_{i}-N).\end{split}  \tag{32}\]  
\[L(n_{i})=(\text{A}-k_{1}+k_{2})D(n_{i})-2k_{3}r_{0}(n_{i})r_{0}^{\prime}(n_{i })+P_{\text{max}}+k_{4}, \tag{33}\]  
where,  
\[D(n_{i})=(1+\frac{2\alpha nr_{0}(n_{i})r_{0}(n_{i})}{r_{0}(n_{i})^{2}+\frac{H^ {2}}{R^{2}}})\times(r_{0}(n_{i})^{2}+\frac{H^{2}}{R^{2}})^{\alpha/2}. \tag{34}\]  
Then, n conforming to P2 can be obtained by solving Equation (31), and the minimum total transmit power of the system required to cover the target area with the proposed base station site planning strategy can be obtained by inserting Equation (19). Considering that the number of UAV-BSs is an integer value, the optimal number is rounded down and up at the same time, and the system transmit power required to cover the target area under the corresponding circular coverage strategy is calculated. Comparing the two calculation results, the UAV-BS site planning corresponding to the smaller value is selected. In summary, the specific steps for UAV-BS site planning using CCPO are in Algorithm 1:  
```
0: SNR threshold \(\gamma_{th}\), maximum transmission power of the base station \(P_{\max}\)
0: Optimal number of UAV-BSs \(n^{*}\) under CCPO, minimum system transmission power \(P^{*}\), and corresponding UAV-BS site planning
1: Initialize: Bandwidth \(B\), frequency \(f\), attenuation exponent \(\alpha\), target area radius \(R\), flight altitude \(H\), and the relationship between the number of small circles and their radius as shown in Table 2
2: Use Equation (17) to fit the relationship between the number of small circles and their radius in the CCP solution, and calculate the coverage radius of each base station in the target area according to Equation (18)
3: Based on the initialized parameters, calculate the transmission power of a single UAV-BS under the circular coverage strategy, and find the minimum number of UAV-BS deployments \(n_{\min}\) that meet the power constraint
4: Optimal number of UAV-BS deployments and minimum system transmission power 1. Set the initial value of the iteration \(n_{0}=n_{\min}\) and the error limit \(\eta\). 2. Calculate \(n_{i+1}=n_{i}-\frac{L^{\prime}(n_{i})}{L^{\prime}(n_{i})}\) using [PERSON]'s method. 3. Determine if the result from (b) satisfies \(|n_{i+1}-n_{i}|<\eta\). If it does, terminate the iteration and output \(n^{*}=n_{i+1}\). Otherwise, return to step (b) and continue the iteration.