# A Service-Constrained Positioning Strategy for an Autonomous Fleet of Airborne Base Stations
## 5 Energy Consumption
### Forward Flight  
Regarding the forward flight, it has a single power state that is for positive \(V_{F_{i}}\), as neither the UAVs nor the helicopters can fly backward. Moreover, the state \(V_{F_{i}}=0\) (hovering) is not contemplated here as it is already done in the vertical flight. As defined in [35], this model has two components, namely the parasitic power and the induced power:  
\[P_{F}=P_{par}+P_{ind}. \tag{41}\]  
The parasitic power is the power required for overcoming the parasitic drag due to the skin friction of the UAV, form drag, etc. Its expression, extracted from [35], is the following:  
\[P_{par}=\frac{1}{2}\cdot\rho\cdot C_{D}\cdot\frac{a}{n}\cdot V_{F_{i}}^{3}, \tag{42}\]  
where \(C_{D}\) is the drag coefficient, and \(a\) is the top area of the UAV, which is divided by the number \(n\) of propellers, as it has been done with the weight. The induced power is the power required to overcome the lift-induced drag and its expression has been extracted from [32]. It depends on the induced velocity \(V_{I}\) at the rotor, whose value in the forward flight has to be found as the solution to the following non-linear equation that does not admit a closed-form solution:  
\[V_{I}=\frac{V_{Hov}^{2}}{\sqrt{(V_{F_{i}}\cdot\cos(\alpha_{P}))^{2}+(V_{F_{i} }\cdot\sin(\alpha_{P})+V_{I})^{2}}}, \tag{43}\]  
where \(\alpha_{P}\) is the tilt of the UAV, which, for simplicity, is considered constant and always the same in the forward flights. Consequently, a solution has to be calculated numerically. To do it, [PERSON] [32] introduced the auxiliary variable \(\lambda\), called induced velocity ratio, which is related with the velocity according to the following expression:  
\[\lambda=\frac{V_{F_{i}}\cdot\sin(\alpha_{P})+V_{I}}{\Omega\cdot r}, \tag{44}\]  
where \(\Omega\) is the angular velocity of the propeller, which, for simplicity, has been assumed constant and the same for all the propellers. The numerical value of \(\lambda\) has also to be calculated numerically as the solution to the following equation (which is simpler than the equation formulated in terms of the velocity):  
\[\lambda=\mu_{P}\cdot\tan(\alpha_{P})+\frac{\lambda_{H}^{2}}{\sqrt{\mu_{P}^{2}+ \lambda^{2}}}, \tag{45}\]  
where \(\mu_{P}\) is the advance ratio and \(\lambda_{H}\) is the induced velocity ratio in hover, which are defined as follows:\[\mu_{P} = V_{F_{i}}\cdot\frac{\cos(\alpha_{P})}{\Omega\cdot r}, \tag{46}\] \[\lambda_{H} = \frac{V_{H\mathrm{o}}}{\Omega\cdot r}, \tag{47}\]  
Therefore, to calculate the numerical value of \(\lambda\) from Equation (45), [PERSON] [32] proposed to apply the Newton-Raphson method. This iterative procedure is carried out at each iteration of Algorithm 1 to obtain the instantaneous inductive power:  
\[P_{ind}=W^{\prime}\cdot V_{H\mathrm{o}v}\cdot\frac{\lambda}{\lambda_{H}}. \tag{48}\]  
Therefore, once the models for the vertical and forward flights have been defined, the final instantaneous supplied power model proposed for the mechanical branch (\(P_{mech}\)) is the one detailed in Equation (37) with \(P_{hel}=P_{V}+P_{F}\), but taking into account two important points. On the one hand, \(P_{F}\) is only added for \(V_{F_{i}}>0\) to avoid the duplication of the hovering power and, on the other hand, when \(V_{F_{i}}>0\) and \(V_{V_{i}}=0\), \(P_{V}\) has to be neglected as the UAV is doing a forward flight instead of hovering. There is one last concept that has not been detailed yet, which is the dependence of the model with the altitude of the UAV. As mentioned above, this dependence is modeled by the air density as it changes depending on the altitude [36]:  
\[\rho=\frac{M_{m}\cdot p_{0}\cdot(1-\frac{L\cdot h}{T_{0}})\frac{\xi^{M_{m}}}{ kT}}{R\cdot(T_{0}-L\cdot h)}, \tag{49}\]  
where \(M_{m}\) is the molar mass of dry air (0.0289644 kg/mol), \(p_{0}\) is the sea level standard atmospheric pressure (101325 Pa), \(L\) is the temperature lapse rate (0.0065 K/m), \(T_{0}\) is the sea level standard temperature (288.15 K) and \(R\) is the ideal gas constant (8.31447 J/(mol \(\cdot\) K)).