# Coverage Analysis and Efficient Placement of Drone-BSs in 5G Networks +
## 4 Optimal Drone-BSs Placement  
In this section we propose to solve the 3D deployment of drone-BSs using GWO [10], where the goal is to maximize coverage by increasing the number of users covered by drone-BSs. Generally, the drone-BSs placement constitutes a NP-hard problem. For this reason, evolutionary algorithms can be a suitable alternative. Mathematically, the coverage problem can be formulated using Mixed Integer Non-Linear Programming (MINLP), as follows:  
\[\max_{x_{i},\ y_{i},\ h_{i},\ \varphi_{j}^{i}}\sum_{j=1}^{N}\sum_{i=1}^{M} \varphi_{j}^{i} \tag{13}\]  
Subject to:  
\[\min\ \sum_{i=1}^{M}\omega_{xjh}^{i} \tag{14}\] \[h_{min}\leq h\leq h_{max}\] (15) \[\sum_{j=1}^{N}b_{j}\ \varphi_{j}^{i}\leq BW \tag{16}\]  
The first constraint assumes that the coverage must be met with a minimum number of drone-BSs deployed. Indeed, a considerable deployment of drones leads to a reduction in the distance between them and hence, the interference from the adjacent drone-BSs increases [4]. The variables \(h_{min},\ h_{max}\) in the second constraint represent the minimum and maximum drone-BS's altitude, respectively. Nevertheless, as the altitude of drone-BS increases, the coverage radius also increases, and energy performance of drone-BSs decreases. The last proposed constraint indicates the total limiting available bandwidth, where \(bj\) is the bandwidth required by user \(j\) and \(BW\) denotes the total bandwidth of the drone-BS [7]. Now we apply GWO in order to find the optimal 3D drone-BSs placement. Proposed by [10], GWO is a meta-heuristic algorithm inspired by the Grey Wolf hunting mechanism. The GWO algorithm takes into consideration the search, encircling and attacking prey by Grey Wolf. Notwithstanding, the algorithm classifies wolf into four groups to find leadership: \(\alpha\) is the most optimal solution and \(\beta\), are the second andthird best solutions, respectively. \(X\) represents the alternative solutions. The process is mathematically modelled as follows:  
\[X(t+1)=\frac{1}{3}(X_{1}(t)+X_{2}(t)+X_{3}(t)) \tag{17}\]  
where:  
\[X_{1}(t)=X_{a}-A_{1}\times D_{a} \tag{18}\]  
\[X_{2}(t)=X_{\beta}-A_{2}\times D_{\beta} \tag{19}\]  
\[X_{3}(t)=X_{\theta}-A_{3}\times D_{\theta} \tag{20}\]  
From the equations above, \(X_{1}(t),\ X_{2}(t),\ X_{3}(t)\) are the position vectors of the prey, and \(X(t+1)\) represents the grey wolf position vector. Grey wolves encircle prey during the hunt. The mathematically model-encircling behaviour is given by the following formula:  
\[D=\left|C\times X_{p}(t)-X(t)\right| \tag{21}\]  
where: \(C=2\times r_{2}\) and \(A=2a\times r_{2}-a\) are two coefficient vectors, and \(r_{1},r_{2}\) are vectors randomly generated between \(\ [0,\ 1]\). The Algorithm 1 of GWO is given by:  
```
Initializing the drone-BSs population \(X_{i}=(i=1,2,\ldots n)\)  Initialize \(\alpha,\ A\) and \(C\)  for \(i=1:n\) do \(\text{j}=1:\text{m}\) do  Calculate a distance, pathloss and evaluate coverage probability in each user  end for  Find the fitness of each search agent \(X_{a},X_{\beta},X_{\theta}\)  while \(t<Max_{iter}\)  for each search agent \[X(t+1)=\frac{1}{3}(X_{1}(t)+X_{2}(t)+X_{3}(t))\]  end for  Update \(\alpha,A\) and \(C\)  Calculate the fitness of all search agents  Update \(X_{a},X_{\beta},X_{\theta}\) \(t=t+1\)  end while  return \(X_{\alpha}\)
```  
**Algorithm 1**GWO for drone-BS deployment.