## 2 Dynamic Area Optimization Using Search Methods  
### System Model  
To achieve wide coverage and high capacity, the antenna parameters are optimized for user distributions. We utilize phased array antennas [26], which can control beam directions and beamwidths vertically and horizontally. Figure 2 illustrates the beam parameters for a cell. The antenna pattern is based on the model described in [23], which refers to the ITU-R recommendations [27]. The antenna gains for angle \(\Psi\) are defined as follows:  
Figure 2: Beam parameters for a cell. \[G(\Psi)=\left\{ \begin{array}{ll}-3(\Psi/\Psi_{b})^{2};&0^{\circ}\leq\Psi\leq\Psi_ {1}\\ L_{N};&\Psi_{1}\leq\Psi\leq\Psi_{2}\\ X-A\log_{10}(\Psi);&\Psi_{2}\leq\Psi\leq\Psi_{3}\\ L_{F};&\Psi_{3}\leq\Psi\end{array} \right. \tag{1}\]  
where \(\Psi_{b}\) is the half-power beamwidth divided by 2, \(\Psi_{1}=\Psi_{b}\)\(\sqrt{-L_{N}/3}\), \(\Psi_{2}=3.745\Psi_{b}\), \(X=L_{N}+A\log_{10}(\Psi_{2})\), and \(\Psi_{3}=10^{((X-L_{F})/A)}\). In this study, \(\Psi_{b}\) corresponds to the vertical beamwidth \(\theta_{\rm 3 dB}\) and the horizontal beamwidth \(\phi_{\rm 3 dB}\) in Figure 2. Considering a practical antenna pattern achievable for a planar patch array antenna, we set the near-inside-lobe level (\(L_{N}\)), the far side-lobe level (\(L_{F}\)), and \(A\) to \(-20\) dB, \(-30\) dB, and \(20\), respectively. Consequently, the antenna gain is expressed as follows:  
\[G=\max(G_{v}+G_{h},L_{F})+G_{p}, \tag{2}\]  
where \(G_{v}\) and \(G_{h}\) represent the antenna gains for vertical and horizontal planes, respectively, which can be calculated by substituting \(\Psi_{b}\) in Equation (1) with \(\theta_{\rm 3 dB}\) and \(\phi_{\rm 3 dB}\). Importantly, \(\Psi\) is excluded from the expressions of \(G\), \(G_{v}\), and \(G_{h}\). The peak of the antenna gain, denoted by \(G_{p}\) in dB, can be expressed as follows:  
\[G_{p}=10\log_{10}\left(\frac{80^{2}}{\theta_{\rm 3 dB}\theta_{\rm 3 dB}}\right)+6. \tag{3}\]  
The expression is derived considering an antenna with a gain of 6 dBi and a beamwidth of \(80^{\circ}\) for both the horizontal and vertical planes. Subsequently, we describe the calculations for the signal-to-noise ratio (SNR) and signal-to-interference-plus-noise ratio (SINR). The downlink (DL) received signal level for the \(u\)-th UE from the \(c\)-th cell in dB is as follows:  
\[R_{u,c}=P_{t}+G_{u,c}-L_{u}+G_{UE}. \tag{4}\]  
Here, \(P_{t}\), \(G_{u,c}\), and \(G_{UE}\) denote the BS transmission power, the BS antenna gain of the \(c\)-th cell for the \(u\)-th UE, which follows Equation (2), and the antenna gain for UE, respectively. \(L_{u}\) denotes the free-space path loss in dB between the \(u\)-th UE and HAPS, which is expressed as  
\[L_{u}=20\log_{10}\left(\frac{4\pi r_{u}}{\lambda}\right). \tag{5}\]  
Here, \(r_{u}\) and \(\lambda\) denote the distance between the \(u\)-th UE and HAPS, and the wavelength, respectively. In this case, we calculate \(N\) values for the \(u\)-th UE, where \(N\) denotes the number of cells in a HAPS. We define the cell to which the \(u\)-th UE is attached as the cell with the highest receiver level among the \(N\) cells. Let \(c_{u}\) represent the cell index to which the \(u\)-th UE attaches. In such a case, the total interference power of the \(u\)-th UE, which is attached to the \(c_{u}\)-th cell, can be expressed in dB as follows:  
\[l_{u}=10\log_{10}\left(\beta\sum_{j=1,j\
eq c_{u}}^{N}10^{\frac{R_{u,j}}{10}} \right), \tag{6}\]  
where \(\beta\in[0,1]\) stands for an activation factor, denoting the number of resource blocks (RBs) used in neighboring cells. We assume \(\beta\) to be 0.5, suggesting that 50% of the RBs in the neighboring cells are used for interference calculations.