A geographical division clustering algorithm for multiple flying base stations

Unmanned Aerial Vehicles (UAVs) can play a significant role as flying base station (FBSs) in assisting terrestrial base stations (BSs) to increase overall network capacity by providing localized transmission to a set of users. In that respect, FBSs can be deployed from a terrestrial macro BS which can act as the depot. In this letter, we propose two flavors of a Geographical Division (GD) clustering algorithm to assign FBSs located at the same terrestrial BS to a set of end users. Numerical investigations demonstrate that the proposed algorithm outperforms the two most widely used and best known clustering algorithms for this specific problem, namely K‐means and Hierarchical clustering algorithms.


INTRODUCTION
Unmanned Aerial Vehicles (UAVs) utilized as flying base stations (FBSs)* are regarded as a promising solution to increase network capacity especially for elastic services. FBSs are envisioned to fly over an area of interest and by hovering at defined locations they can provide localized transmissions achieving very high data rates. [1][2][3] The FBSs are deployed from a terrestrial base station (BS), to serve ground users at different cluster head (CH) points and subsequently return to the BS for recharging. In this case there are two underlying problems; the first one is to decide which CH each FBS will visit (clustering) and the second is for the selected set of CHs to provide the FBS trajectory. The second problem resembles the well-known traveling salesman problem (TSP) and has received significant attention. However, the focus in this letter relates to the first problem which has received much less attention. Note that the clustering problem is equally important in order to enable for an efficient trajectory of the FBS visiting those allocated CH points to allow for increased network scalability and energy efficiency. 4 In this regard, previous work 4 proposed a dynamic and distributed clustering approach which addresses how a FBS covers ground nodes and selects cluster heads when serving as a mobile sink in a large-scale wireless sensor network in order to reduce the energy consumption for the FBS in comparison with two other different benchmarks. A relevant work 5 presented and studied two conventional transformed algorithms for clustering nodes to be visited by a FBS in a wireless sensor network and compared their network performance such as the energy consumption and the bit error rate for the FBS when varying the number of CHs.
In this paper, we propose an efficient clustering algorithm where FBSs deployed from a single terrestrial BS and serve end users in a geographical division way. Furthermore, within each cluster the FBS trajectories are implemented using the 2-Opt heuristic which is the most well-known approximation algorithm within each cluster to solve the TSP. 6 In addition, we also propose an enhanced approximation version of the clustering algorithm in order to improve the computational complexity. We compare the proposed scheme with the two best known (and most widely used) clustering algorithms; namely the K-means and the Hierarchical clustering in terms of total tour (path) length, the consumed time that the FBS travels and serves all CHs, and overall energy consumption. These are important performance metrics for FBS assisted networks that can be added to more traditional network performance criteria.

SYSTEM MODEL
We consider a set of FBSs hosted at a terrestrial macro BS which act as the depot and a set of randomly distributed CHs across a pre-defined area that need to be served. The UAV-assisted wireless network is modeled as an undirected graph = ( , ), where indicates the set of CHs, and denotes the set of links created by a FBS traveling between two CHs. = {1, 2, … , K} denotes the set of UAVs and = {Ψ 1 , Ψ 2 , … , Ψ K } is the set of K clusters to be formed. Each cluster set Ψ k can be written as Ψ k = { k1 , k2 , … , kN k } where N k is the number of CHs in the set Ψ k and ki = [x ki , y ki ] denotes the Euclidean location in the 2-D defined serving area of a CH i in set Ψ k . An example of this single macro BS (ie, single cell) scenario is shown in Figure 1. When a FBS arrives at a CH, it will hover and serve end users, creating in essence a small-cell. The FBS will be then flying to the next CH once completing the service at the current serving CH point. Finally, all FBSs return eventually to the macro BS, where FBSs can autonomously refuel/recharge. Hereafter, without loss of generality, we consider that the FBSs are flying with constant velocity and at the same altitude. In the assumed scenario, FBSs receive and/or transmit data from the users on the ground and offload them to the terrestrial macro BS.

CLUSTERING ALGORITHMS
The first flavor of the clustering scheme, which is denoted as Geographical Division (GD), divides equally the plane into K regions using linear line segments that intersect at the location of the macro-BS. The segments are then rotated by a pre-defined angle and for each rotation a new clustering is created and the overall route length is calculated as shown in Algorithm 1. A simple illustration of such clustering is depicted in Figure 1A for the case of 4 FBSs. In Figure 1A, observe that the linear line segments divide the coordinate plane in 4 sections and each FBS will be serving CHs falling within the 4 quadrants where the FBS will be operating. Then, the linear line segments are rotated anti-clockwise by 1 • ( ) and for each rotation the overall trajectory length for the specific clustering is calculated using the 2-Opt local heuristic algorithm. To reduce the computational time and generalize the previous scheme, we detail the Approximation-GD (A-GD) clustering algorithm that provides an estimate of the angle as shown in Figure 1B. To do so, we compute, as shown in Equations (1), the average distance D k between CHs in each cluster Ψ k where we have N k CHs.
We then calculate a parameterized average distance across all K clusters in the plane as follows, where the parameter Δ ∈ Q called a resolution coefficient is extracted from numerical experimentation. The distance is transformed to an angle ( , r) in degrees (it is also called the resolution) which depends on the radius r of the terrestrial macro cell as follows, The above procedure can significantly enhance the computational time compared to the use of a pre-defined static angle . More formally, the clustering algorithm is presented in a pseudocode format in Algorithm 1. We note that the complexity of the GD and A-GD algorithms is ( f logK). We also introduce two general clustering algorithms i,e,. K-means and Hierarchical clustering algorithms. The K-means clustering algorithm partitions in a iterative manner the set of CHs into'K' clusters to be served by the FBSs. 7 In a nutshell, the algorithm firstly randomly assigns CHs to K clusters and determines initial centroids in each cluster. Then, it calculates the sum of the squared distances between the CHs and the determined centroid in each cluster. Based on that information it proceed iteratively to find the minimum possible distance of the CHs to their assigned centroids. The Hierarchical clustering algorithm is also another very commonly used clustering heuristic. 7 In the nominal agglomerated hierarchical clustering method, each CH is initially assigned its own cluster. Then, a similarity index (eg, distance) between each of the clusters is calculated and the two most similar clusters are merged. Finally, the algorithm repeats the above steps until there is only a single cluster left. When we have K available FBSs we are using the hierarchical clustering at level K (linkages). 7
← Generate and locate CHs randomly distributed on the plane 5: d(Ψ k ) ← Calculate Euclidean distances between CHs in each set Ψ k 6: ← Calculate the parameterized average distance by Equations (1)

NUMERICAL INVESTIGATIONS
In this section, we provide an extensive set of Monte Carlo based numerical investigations to assess the performance of the proposed geographical based clustering algorithm with linear segments compared to the aforementioned commonly used clustering algorithms. Since computational times of the different clustering algorithms are also reported and discussed, it is worth mentioning that all simulations run on MATLAB with Quad Core CPU and 16GB RAM. Table 1 shows the simulation parameters. The widely used Penguin C UAS has been adopted as a UAV model. † The map size is 1000m × 1000m assumed as a nominal size of a cell. This can represent a sub-urban environment where multiple UAVs serving ground end users at different CHs might be required. We also assumed that there is a BS in the center of the cell. We varied the number of UAVs and the number of CHs and fixed the service time T serv as 10s without loss of generality. For the Hierarchical algorithm, the number of linkages is the same as the number of UAVs K and its clustering mode is 'average'. 7 It is also assumed that the altitude of UAVs is identical. The resolution coefficient Δ was extracted from

F I G U R E 2 An example of achieved CH clustering by the different algorithms and the associated UAV trajectories for the case of K=4
and N k =30 in comparison with Total tour length (L), Energy consumption (E), Computation time (C t ). Note that the blue lines represent FBS trajectory, different colors of the CHs represent different clusters, and the four black cross points in Fig (d) indicate the centroid numerical experimentation and fixed. In order to compute the consumed energy consumption for the UAVs, we assumed that the velocity of all UAVs is identical; furthermore, the aspect related to acceleration/deceleration is not considered. In that respect, the straight-and-level flight (SLF) version of the energy consumption model detailed in 8 is used, where the energy consumption in Joules (J) can be expressed as follows, where T represents the total flying time (in seconds) between different CH points and the hovering time when serving different CHs. The velocity of the UAVs is denoted by V (m/s) and c 1 and c 2 are constant parameterized values related to flying aerodynamics and keeping the UAV aloft; detailed treatment of those parameters can be found in Appendix A [8].
In addition, to provide a fair comparison between clustering schemes, we also assume that FBSs spend an equal time T serv to serve ground users and therefore the hovering time per CH is constant and the same for all FBSs. Based on the above, the travel time can be expressed as the total tour length L divided by the velocity V. Therefore, the total travel time denoted as T, can be written as follows, Note that since the goal is to compare the effect of the different clustering algorithms on the overall trajectory, as already eluded we fixed the service time to be consumed at each serving CH. Observe from Equations (4), that the energy

F I G U R E 3 Tour length and energy consumption vs number of UAVs
consumption strongly depends on the velocity V. To this end, an optimal velocity V • can be calculated using the actual values for the parameters c 1 and c 2 as shown in Table 1. The optimal velocity V • can be readily calculated by solving the energy consumption minimization problem from Equations (4), where T is omitted. This is simply solving a third degree polynomial and it can be shown that the energy consumption is minimized when V • = 14.9 m/s; in the numerical investigations discussed in the sequel, the assumed velocity for all UAVs serving different CH will be equal to V • . Figure 2 shows an example of the trajectory of UAVs for all clustering algorithms in the scenario when K=4 and N k =30. As can be seen the figure, different generated clusters entail different trajectories for the FBSs. The GD clustering algorithm has the lowest total tour length L and the lowest energy consumption E when the optimal Opt =46 • is used, while the A-GD clustering algorithm (with Opt =33.9 • ) has also lower values of L and E compared to the Hierarchical and K-means algorithms. Also note that in terms of C t the A-GD is faster than the GD clustering scheme. Figure 3A,B shows the average total tour length L for all algorithms. K-means and Hierarchical algorithms showed approximately similar performance in terms of achievable tour length throughout the simulations. On the other hand, observe that the GD algorithm has the lowest tour length L and the A-GD algorithm has a similar performance. It is worth pointing out that the GD algorithm gained more than 16.3% compared to both K-means and Hierarchical algorithms when the number of UAVs K is 10 and for the case where we have N k =15 as shown in Figure 3A. Figure 3C,D shows the average energy consumption E for all clustering algorithms. K-means and Hierarchical algorithms also showed similar performance in terms of E values throughout the simulations, while GD and A-GD algorithms manage to achieve significantly lower values of energy consumption compared to K-means and Hierarchical clustering approaches; also note that both the GD and A-GD algorithms have approximately the same performance. More specifically, the proposed GD algorithm decreased the overall energy consumption E at around 12.7% compared to both K-means and Hierarchical algorithms when N k =15 as shown in Figure 3C. Figure 3E,F shows the average computational time C t of all different clustering algorithms for the entire simulations. As can be seen from the figure, the proposed GD algorithm showed relatively higher C t because of its requirement to compute UAV trajectories for each rotation of the linear segments by = 1 • . Using the calculated parameter , the A-GD algorithm has 6 times lower computational time than GD algorithm when K=2 and N k =30, while its performance that has been discussed previously in terms of L and E, was approximately the same with the GD algorithm. From the figure, it can be seen that the Hierarchical algorithm has the lowest computational complexity followed by the K means algorithm. It is also important to observe that the C t cost for the A-GD is comparable to the K-means algorithm when K>8 in the case when N k =15 and N k =30. Therefore, the proposed scheme achieved better performance in terms of L and E from those two traditional clustering algorithm whilst having similar computational complexity.
Finally, Table 2 shows the maximum gains of the proposed GD algorithm compared to the two traditional algorithms when varying the number of FBSs and CHs. The above results described in Figure 3 reflected average gains. However, it is worth pointing out that in terms of maximum gains, the proposed scheme achieved up to 31.5% for the tour length in comparison with K-means scheme.

CONCLUSIONS
This letter proposes a clustering method for multiple UAVs operating as FBSs which are located at a terrestrial BS acting as their depot. Numerical investigations revealed that the proposed clustering algorithms have maximum gains of up to 31.5% and approximately 20% for the total tour length and the energy consumption respectively compared to the two classical clustering algorithms, namely the K-means and Hierarchical algorithm. In addition, the A-GD clustering scheme is amenable for real-time implementation since it has a low computational complexity, which is comparable with both classical clustering algorithms while maintaining its gain in terms of the total tour length and energy consumption.