# UAV-BS Site Planning Based on Circular Coverage Strategy
## 4 Methodology
### Circle Coverage Power Optimization Strategy
subject to_  
C1: \(0<P_{g}<P_{\max}\);  
C2: \(S_{c}(n)-S\geq 0\);  
C3: \(n\leq N\);  
C4: \(n\in\mathbb{N}^{+}\);  
where \(R\) represents the radius of the target area, and \(r_{0}(n)\) represents the proportional relationship between the coverage radius of each UAV-BS and the radius of the target area when \(n\) UAV-BSs are deployed. The value of \(r_{0}(n)\) satisfies Table 2. Due to the integer constraint included in the problem, the objective function becomes an equation with integer variables, which makes the problem difficult to solve. To address this, this paper attempts to use curve fitting to relax the integer constraints of the problem, thus finding a continuous solution. This solution is then rounded and compared to obtain the optimal number of UAV-BSs for the target area. This paper utilizes nonlinear least squares to fit the data in Table 2 and employs a binomial power function for approximation. The fitting function for the relationship between the number of small circles and the radius of small circles under the CCP model is given as follows:  
\[r_{0}(n)=an^{-b}+c, \tag{17}\]  
where \(a\), \(b\), and \(c\) are constants. Under a 95% confidence level, the confidence intervals for \(a\), \(b\), and \(c\) are \(\{(1.725,1.851),(-0.8055,-0.7105),(0.06404,0.1085)\}\), and the values are taken as \((1.788,0.758,0.08626)\). For the deployment of \(n\) UAVs, the maximum distance each UAV can connect to is as follows:  
\[d_{n}=\sqrt{r(n)^{2}+H^{2}}. \tag{18}\]  
The maximum radius of the projection circle covered by each UAV-BS is as follows:  
\[r(n)=Rr_{0}(n). \tag{19}\]  
Equation (16) can be replaced with the following:  
\[P_{\rm s}(n)=An(r_{0}(n)^{2}+\frac{H^{2}}{R^{2}})^{a/2}+(n-1)P_{u}, \tag{20}\]  
where,  
\[A=R^{a}10\frac{\gamma+P_{\rm noise}+20\log_{10}(\frac{4\pi f}{R})+\varepsilon} {10}. \tag{21}\]  
Considering the optimization problem for the minimum communication energy consumption required for the UAV-BS to provide full coverage to the ground, it can be expressed as follows:  
**P3:**  
\[n^{*}=\arg\min(P_{\rm s}(n))\]  
_subject to:_  
C1: \(0<P_{g}<P_{\max}\);  
C2: \(S_{c}(n)-S\geq 0\);  
C3: \(n\leq N\). **Theorem 1. **_The total transmission power \(P_{\rm s}(n)\) is a convex function with respect to \(n\)._Proof of Theorem 1.: The second part of the objective function \(P_{s}(n)\) is convex for \(n>0\). If the first part is also convex, then the objective function, being a weighted sum of convex functions, maintains convexity. To prove that \(A\cdot n\Big{(}r_{0}(n)^{2}+\frac{H^{2}}{R^{2}}\Big{)}^{\alpha/2}\) is a convex function, we consider the general form \(f(x)=x(g(x)^{2}+k)^{\alpha/2}\). The first and second derivatives are given by the following:  
\[f^{\prime}(x)=\Big{(}g(x)^{2}+k\Big{)}^{\frac{\alpha}{2}}+\alpha xg(x)g^{\prime }(x)\Big{(}g(x)^{2}+k\Big{)}^{\frac{\alpha}{2}-1}. \tag{22}\]  
Figure 2:0 Figure 2:1 Figure 2:2 Figure 2:3  
where  
Figure 2:4  
Figure 2:5  
Figure 2:6  
To ensure that \(f^{\prime\prime}(x)\geq 0\), the value of \(x\) must satisfy  
Figure 2:7  
When \(\alpha\geq 2\), \(a=0.785\), \(b=0.08626\), and \(0<k<1\), using mathematical solving tools, it can be determined that \(x\) is within \((0,50]\) and the second derivative holds true. Therefore, it is known that \(An\big{(}r_{0}(n)^{2}+H^{2}\big{)}^{\alpha/2}\) is convex, meaning \(P_{s}(n)\) is a convex function. All the above constraints are convex sets. According to the definition of convex optimization problems, the UAV deployment optimization problem modeled in this paper is a convex problem, and there exists an optimal solution \(n^{*}\) that minimizes the total transmission power required by the system. P3 is reformulated into a standard convex problem:  
**P4:**  
Figure 2:8  
_subject to_  
Figure 2:9 Table 2.0 Table 2.1 Table 2.2 Table 2.3  
This paper uses the method of Lagrange multipliers to transform the above problem into an unconstrained optimization problem to solve for the solution that satisfiesthe inequality constraints.