# A Service-Constrained Positioning Strategy for an Autonomous Fleet of Airborne Base Stations
## 5 Energy Consumption  
Once the positioning strategy has been detailed, in this section, an energy consumption model is introduced. According to [10], in the context where the positioning strategy has been implemented where UAVs are used as ABBSs, two types of energies have to be defined: communication energy (\(E_{comm}\)) and mechanical energy (\(E_{mech}\)).  
The communication energy is the energy required for a UAV to serve all its associated users and communicate with the ground station through the corresponding backhaul link, and the mechanical energy is associated with the flying mechanisms that keep a UAV in the air performing the maneuvers imposed by the positioning strategy. Each one can be computed as  
\[E(t)=\sum_{\tau<=t}P(\tau)\cdot T_{it}, \tag{34}\]  
where \(E(t)\) is the total energy spent until the \(t\)th iteration, \(P(\tau)\) is the power consumption at the \(\tau\)th iteration and \(T_{it}\) is the time interval associated to each iteration. The battery will be completely spent whenever \(E(t)\) exceeds the energy available initially in the battery.  
Regarding the communication energy, in the ABBS framework, nano small cells are needed, being particularly characterized by the fact that the consumed power is almost independent of the radiated power, as assessed by the model in [28]. However, the purpose of this paper is the development of a positioning strategy focusing on the movement of the fleet of UAVs. Thus, we neglect in this section the energy consumption associated to communication, assuming that it is supported by dedicated airborne batteries.  
Regarding the mechanical energy, an instantaneous mechanical power model (\(P_{mech}\)) is defined corresponding to the flying maneuvers of a UAV. This power depends on the vertical and forward components of the UAV movement (\(V_{V_{i}}\) and \(V_{F_{i}}\), respectively):  
\[V_{F_{i}} = \frac{\sqrt{(v_{i_{x}}[n]-v_{i_{x}}[n-1])^{2}+(v_{i_{y}}[n]-v_{i_ {y}}[n-1])^{2}}}{T_{it}}, \tag{35}\] \[V_{V_{i}} = \frac{v_{i_{x}}[n]-v_{i_{x}}[n-1]}{T_{it}}, \tag{36}\]  
where **v** is the vector detailed in Equation (27), and \(i_{x}\), \(i_{y}\) and \(i_{z}\) denote the \(x\), \(y\) and \(z\) coordinates of the \(i\)th UAV.  
As stated in Section 2, the required type of UAV is rotary-blade. However, only a few works have studied an instantaneous mechanical power model of this type of UAVs. Some of them only provide practical data extracted from real UAV flights but without an analytical model (e.g., [29]), while others provide models that do not fit our task (e.g., [30]), where the model depends on the instantaneous angular velocity of each propeller instead of the instantaneous velocity of the UAV. Thus, in this paper, we have followed the same methodology as in [31], that is, we adapt the well-known instantaneous mechanical power model of an helicopter (\(P_{hel}\)) using the flight parameters of a rotary-blade UAV.  
This adaptation consists in dividing the weight \(W\) of the UAV by the number \(n\) of propellers, as a helicopter has a single propeller, and multiplying the obtained power, which belongs to a single propeller, by \(n\) to obtain the total instantaneous mechanical power. Moreover, a figure of merit \(F_{m}\) is added to model the efficiency of the engines, which typically takes values between 0.7 and 0.8:  
\[P_{mech}=\frac{n}{F_{m}}\cdot P_{hel}(W^{\prime}), \tag{37}\]  
with  
\[W^{\prime}=\frac{W}{n}=\frac{(m+M)\cdot g}{n}, \tag{38}\]  
where \(g\) is the gravity constant, \(m\) is the mass of the UAV and \(M\) is the load, which in the ABBS context is the mass of the transmission equipment.  
The mathematical models for the instantaneous power supplied during the flight have been mainly extracted from [32]. They can be characterized in a different way for the cases of vertical and forward flights.