# Energy Efficient Altitude Optimization
of an Aerial Access Point
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31st PIMRC’20 - Workshop on UAV Communications for 5G and Beyond. This work is
supported by the project PAINLESS which has received funding from the European
Union’s Horizon 2020 research and innovation programme under grant agreement
No 812991.

Nithin Babu  Research, Technology and Innovation Network (RTIN), The American
College of Greece, Greece  Department of Electronic Systems, Aalborg
University, Denmark  Konstantinos Ntougias  University of Cyprus  Constantinos
B Papadias  Research, Technology and Innovation Network (RTIN), The American
College of Greece, Greece  Department of Electronic Systems, Aalborg
University, Denmark  Petar Popovski  Department of Electronic Systems, Aalborg
University, Denmark

###### Abstract

In this paper, we propose an energy-efficient optimal altitude for an aerial
access point (AAP), which acts as a flying base station to serve a set of
ground user equipment (UE). Since the ratio of total energy consumed by the
aerial vehicle to the communication energy is very large, we include the
aerial vehicle’s energy consumption in the problem formulation. After
considering the energy consumption model of the aerial vehicle, our objective
is translated into a non-convex optimization problem of maximizing the global
energy efficiency (GEE) of the aerial communication system, subject to
altitude and minimum individual data rate constraints. At first, the non-
convex fractional objective function is solved by using sequential convex
programming (SCP) optimization technique. To compare the result of SCP with
the global optimum of the problem, we reformulate the initial problem as a
monotonic fractional optimization problem (MFP) and solve it using the
polyblock outer approximation (PA) algorithm. Numerical results show that the
candidate solution obtained from SCP is the same as the global optimum found
using the monotonic fractional programming technique. Furthermore, the impact
of the aerial vehicle’s energy consumption on the optimal altitude
determination is also studied.

###### Index Terms:

Global energy efficiency, UAV communication, altitude Optimization, sequential
convex programming, monotonic optimization.

##  I Introduction

The role of uninhabited AAP in the deployment of emergency networks such as
deploying aerial base stations to provide reliable connectivity in disaster
areas [1] or in social events such as concerts is vital. In Japan, earthquake
affected areas were provided with internet access with the help of unmanned
aerial vehicles (UAV) [2]. Cellular coverage extension using drone deployed
base stations by Nokia’s F-cell technology is another proven application of
portable access points [3]. The mobility and ability of aerial vehicles to
adjust their altitude to improve the probability of line-of-sight (LoS)
communication channel to the ground UEs makes them suitable for acting as
relays in the internet of things (IoT) applications [4]. Despite all these
applications, the efficiency of an aerial communication system (ACS) is highly
dependent on the limited energy available at the aerial vehicle [5]. Any
improvement in the energy efficiency of ACS implies longer aerial vehicle
hovering, hence more information bits transmitted to UEs.

Compared to the conventional cellular communication systems, the total energy
required by ACS is very high. This is because, in ACS, in addition to the
communication-related energy, the aerial vehicle consumes energy during
vertical climb and hovering. Most of the works in the literature only consider
communication-related energy, which is suboptimal in the case of an ACS. In
[6], the authors present an analytical approach to optimize the altitude of
low altitude aerial platforms to maximize the radio coverage area. The authors
of [7] jointly optimize the flying altitude and the antenna beamwidth for
throughput maximization. A new 3-dimensional deployment plan for the drone-
base station to serve the users based on their service requirements, while
minimizing the number of drones, is presented in [8]. The work in [9] proposes
a new polynomial-time complex spiral mobile base station placement algorithm
in UAV-UE communications. The works in [10],[11] find the optimal altitude for
UAV-base stations that maximizes the number of covered users using the minimum
transmit power.

None of the above works consider the energy consumption of the aerial vehicle
in the optimization problem. Since the ratio of communication energy to the
total energy consumed by the aerial vehicle is negligible, the results
proposed in the above works are suboptimal for the GEE maximization of ACS.
When the altitude of an AAP increases, the LoS coverage area increases, the
LoS channel gain decreases and the energy consumed by the aerial vehicle also
increases. With these facts, we can say that the GEE of an ACS, defined as the
ratio of the total number of data bits transmitted to the total energy
consumed, will not be maximum either at maximum or minimum permitted AAP
altitudes. We exploit this tradeoff between the total number of bits
transmitted and the energy consumed to determine an energy-efficient hovering
altitude for the AAP. Some of the works which consider the aerial vehicle’s
energy consumption include [12], [13], [14]. An energy-efficient 3D trajectory
of a UAV deployed to serve a set of IoT nodes is investigated in [15]. Optimal
trajectories, which minimize the fixed and rotary-wing UAV associated energy
are designed in [13] and [14] respectively. The authors in [12] maximizes the
minimum average rate and energy efficiency through joint optimization of
trajectory, velocity, and acceleration of UAV flying at a fixed altitude. An
altitude-dependent energy consumption model is used by the authors of [16] to
find drone locations that minimize the cost while ensuring the surveillance of
all the targets.

To the best of our knowledge, we are the first to determine an optimal
altitude which maximizes the GEE for an ACS considering both the energy
required for communication and energy consumed by the aerial vehicle. The rest
of the paper is organized as follows. In section II, we model the system. The
energy consumption of the aerial vehicle and GEE are explained in section III.
Section IV formulates the optimization problem and solves it using SCP and MFP
techniques. The numerical results are discussed in section V. Finally, our
findings are concluded in section VI.

In this paper, scalars are represented by lowercase letters. Boldface
lowercase letters are used to denote vectors and $\mathbb{R}^{M}$ denotes the
set of $M$ dimensional real-valued vectors.

##  II System Model

###  II-A System Model

We consider an orthogonal multiple access downlink broadcast transmission
scenario enabled by an AAP acting as a flying base station, where each user is
allocated a fixed bandwidth. We assume there is always a sufficient number of
orthogonal channels (e.g., narrowband frequency division multiple access
systems [17]). As shown in Fig.1, we assume a uniform distribution of $N$ UEs
in the AAP coverage area $A_{ue}=\pi\overline{r}^{2}$ such that
$N=\rho_{ue}\ast A_{ue}$, where $\rho_{ue}$ and
$\overline{r}=h_{A}\text{cot}(\phi)$ represent the density of UEs and the
radius of the AAP coverage area respectively, and $\phi$ represents the
minimum elevation angle required for the LoS channel between the edge UE and
the AAP [6]. The AAP is employed at an altitude of $h_{A}$ meters (m) with the
horizontal plane coordinates the same as the center of $A_{ue}$. In addition
to this, we consider the deployment of this system in rural areas where the
channel between the AAP and UE is dominated by the LoS link. In real life,
this represents the access segment of an ACS in which an AAP is deployed for
cellular coverage extension in a rural area. Given this, the LoS channel gain
between the UE located at a distance $r$ from the center of the coverage area
and the AAP is given by

$\displaystyle h(r)=\dfrac{h_{0}}{r^{2}+h_{A}^{2}}$ (1)

where $h_{0}$ represents the channel gain at a reference distance of 1m.

Figure 1: AAP enabled downlink broadcast transmission scenario.

The signal-to-noise ratio (SNR) $\gamma_{\overline{r}}$, at the edge UE is
given by

$\displaystyle\gamma_{\overline{r}}=\dfrac{P_{T}h_{0}}{N(\overline{r}^{2}+h_{A}^{2})\sigma^{2}}$
(2)

where the total data transmission power $P_{T}$ available at the AAP is
divided equally among the $N$ UEs and $\sigma^{2}$ represents the variance of
the zero-mean additive white Gaussian noise at the corresponding receiver.

Hence, the total number of bits transmitted per unit Hz of bandwidth from the
AAP to the considered UE through a channel of bandwidth $W$ in $T$ seconds is
expressed as

$\displaystyle
R(\overline{r})=T\text{log}_{2}(1+\gamma_{{\overline{r}}})\,\,\text{bits/Hz}$
(3)

Through (3) and (2), the data rate of a UE depends on distance $r$ from the
center of the coverage area. Because of the inverse relationship between
$\gamma_{r}$ and $r$, the data rate of any UE is lower bounded by the data
rate of the edge UE. That is

$\displaystyle R(r)\geq
R(\overline{r})=R(h_{A})\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall
r\,\,\leq\,\,\overline{r}$ (4)

Since maximizing $R(r)$ is equivalent to maximizing $R(\overline{r})$ and for
ease of explanation, we consider the sum of minimum rate, $R(\overline{r})$,
in the definition of the GEE of the considered ACS in section III. The
algorithm developed in section IV applies to the maximization of GEE defined
in terms of the sum of actual rate $R(r)$.

##  III Global Energy Efficiency of the ACS

The global energy efficiency of the considered ACS is given by

$\displaystyle\text{GEE}[\text{bits}/{\text{Joule.Hz}}]=\dfrac{\overline{R}(h_{A})[\text{bits/Hz}]}{E(h_{A},T)[\text{Joule}]}$
(5)

where $\overline{R}(h_{A})$ is the sum of the minimum number of data bits
transmitted per Hz from the AAP to the $N$ UEs in $T$ seconds;
$E(h_{A},T)=E_{A}(h_{A},T)+E_{C}(T)$ is the total energy consumed by the AAP,
in which $E_{C}(T)$ is the energy required for data communication and
$E_{A}(h_{A},T)$ given by (9), is the total energy consumed by the mechanical
parts of the aerial vehicle during vertical climb and hovering. We consider a
climb-hover communicate scheme in which the AAP climbs at a specific altitude
and then communicates with $N$ UEs while hovering.

###  III-A Sum of the minimum number of data bits transmitted,
$\overline{R}(h_{A})$

Considering the uniform distribution of UEs over $A_{ue}$, the sum of the
minimum number of data bits transmitted per Hz from the AAP to the $N$ UEs in
$T$ seconds through orthogonal channels of bandwidth $W$ Hz is expressed as

$\displaystyle\overline{R}(h_{A})$ $\displaystyle=$ $\displaystyle
T\int_{0}^{2\pi}\int_{0}^{\bar{r}}\rho_{ue}R(\overline{r})rdrd\theta$
$\displaystyle=$ $\displaystyle T\rho_{ue}\pi
h_{A}^{2}\text{cot}^{2}\phi{\text{log}_{2}\left(1+\dfrac{P_{T}h_{0}}{N({\overline{r}}^{2}+h_{A}^{2})\sigma^{2}}\right)}$
$\displaystyle=$ $\displaystyle T\rho_{ue}\pi
h_{A}^{2}\text{cot}^{2}\phi\text{log}_{2}\left(1+\frac{\beta}{h_{A}^{4}}\right)$

where
$\beta=\dfrac{P_{T}h_{0}\text{sin}^{2}\phi}{\pi\rho_{ue}\text{cot}^{2}\phi\sigma^{2}}$.

###  III-B Aerial vehicle energy consumption

The total energy consumed by an aerial vehicle ($E(h_{A},T)$) is composed of
three main parts:

  1. 1.

Energy required for data communication ($E_{C}(T)$).

  2. 2.

Energy consumed by the rotor of the aerial vehicle during climbing from ground
to an altitude of $h_{A}$ $(E_{cl}(h_{A}))$.

  3. 3.

Energy consumed by rotor during hovering at altitude $h_{A}$
$(E_{ho}(h_{A},T))$.

The energy required for data communication is given by

$\displaystyle E_{C}(T)=(P_{T}+P_{H})T$ (6)

where $P_{T}$ is the total power used for the symbol transmission and $P_{H}$,
is the total power consumption by all the hardware circuits in the transmitter
section of the AAP. The energy parts $E_{cl}(h_{A})$ and $E_{ho}(h_{A},T)$
follow the energy consumption model presented by the authors of [18]. In [18],
the authors presented different power/energy consumption factors based on the
field experiments performed on the Intel Aero Ready to Fly Drone. Unlike fixed
and rotary-wing unmanned aerial vehicles [13],[14], the energy consumed by the
rotor of a quadropter/drone during hovering is dependent on the hovering
altitude [18],[16]. According to [18], the energy consumed by the quadropter
during climbing from the ground to an altitude of $h_{A}$ with a constant
climb rate is given by

$\displaystyle E_{cl}(h_{A})=\alpha_{cl}h_{A}+\beta_{cl}$ (7)

and the energy consumed during hovering at an altitude $h_{A}$ for $T$ seconds
is given by

$\displaystyle E_{ho}(h_{A},T)=(\alpha_{ho}h_{A}+\beta_{ho})T$ (8)

where the constants $\alpha_{cl},\beta_{cl},\alpha_{ho},\beta_{ho}$ are
determined from the curve fitting performed on the measured power/energy
values from the field experiments.

Hence the total energy consumed by the rotor of the AAP to climb to an
altitude of $h_{A}$ m and hover for $T$ seconds is given by

$\displaystyle E_{A}(h_{A},T)=E_{cl}(h_{A})+E_{ho}(h_{A},T)$ (9)

Figure 2 shows the increasing nature of $E_{A}(h_{A},T)$ with altitude for a
fixed time of operation with constants
$\alpha_{cl}=315,\beta_{cl}=-211.261,\alpha_{ho}=4.917,\beta_{ho}=275.204$
[18] and $T=400s$. It is because as the altitude increases, the air
temperature and pressure decreases. The decrease in the air pressure reduces
the upward thrust provided by the air, to balance the downward force produced
by the weight of the aerial vehicle. Hence, at higher hovering altitudes, to
balance the weight, the propeller of the aerial vehicle needs to generate an
additional force, which results in increased energy consumption.

Hence the total energy consumed by the AAP is given by

$\displaystyle E(h_{A},T)=E_{cl}(h_{A})+E_{ho}(h_{A},T)+E_{C}(T)$ (10)

Figure 2: Total energy consumed by the rotor of the aerial vehicle.

##  IV Problem Formulation

Our objective is to find the optimum altitude for the AAP, which maximizes the
system’s global energy efficiency (GEE) subject to minimum data rate and
altitude constraints. By using (5),(III-A) and (10), our main objective is
formulated as an optimization problem and is expressed as follows:

(P1) $\displaystyle:$
$\displaystyle\underset{h_{A}}{\text{maximize}}\,\,\,\,\dfrac{\overline{R}(h_{A})}{E(h_{A},T)}$
s.t. $\displaystyle h_{min}\leq h_{A}\leq h_{max}$ (12) $\displaystyle
W\text{log}_{2}\left(1+\dfrac{\beta}{h_{A}^{2}(\overline{r}^{2}+h_{A}^{2})}\right)\geq
R_{0}$

where (12) represents the permitted AAP altitude range specified by the
aviation regulatory board and $R_{0}$ is the minimum data rate required by the
UE in bits-per-second (bps).

The objective function of (P1) belongs to the class of fractional programming
problems [19]. It can be globally solved using Dinkelbach’s algorithm [20],
provided $\overline{R}(h_{A})$ is concave, and $E(h_{A},T)$, (12), (12) are
convex functions of $h_{A}$. From (10), (12), we find that the denominator of
the objective function and AAP altitude constraint are a convex function of
$h_{A}$ while the numerator $\overline{R}(h_{A})$ in (III-A), is neither
convex nor concave in nature. Also, the minimum individual data rate
constraint (12) is non-convex in nature. Hence (P1) cannot be globally solved
with polynomial time complexity. As a means to obtain an efficient solution
that fulfills the Karush Kuhn Tucker (KKT) conditions of (P1), we use the
polynomial-time complex sequential convex programming (SCP) technique [21].
Besides, to obtain the global optimum of (P1), we exploit the monotonic
structure of the objective function in the general framework of monotonic
fractional programming (MFP) optimization [22][23].

###  IV-A GEE Maximization using SCP

In this part, we find the optimal altitude of the AAP, which maximizes the GEE
of the ACS using sequential convex programming. The fundamental idea of SCP is
to iteratively solve a sequence of convex approximated problems of the
original non-convex problem so that the feasible solution points converge to
the KKT point of the original non-convex problem [21]. Here we approximate the
non-concave numerator, $\overline{R}(h_{A})$ of (P1) as a concave function
using a first-order Taylor approximation technique.

For the $k^{th}$ iteration, let $h_{k}$ be the feasible solution from the
previous iteration. Then the first order Taylor approximation of
$\overline{R}(h_{A})$ about $h_{k}$ is

$\displaystyle\overline{R}(h_{A})$ $\displaystyle\approx$
$\displaystyle\overline{R}(h_{k})+{\overline{R}}^{\prime}(h_{k})(h_{A}-h_{k})$
(13)

where

$\displaystyle{\overline{R}}^{\prime}(h_{k})$ $\displaystyle=$ $\displaystyle
T\rho_{ue}\pi\text{cot}^{2}\phi
2h_{k}\text{log}_{2}\left(1+\frac{\beta}{h_{k}^{4}}\right)$ (14)
$\displaystyle-T\rho_{ue}\pi\text{cot}^{2}\phi\dfrac{4\beta
h_{k}}{\text{log}_{e}(2)\left({\beta}+{h_{k}}^{4}\right)}$

Similarly, the non-convex nature of (12) is tackled with the following Taylor
approximation:

$\displaystyle
W\text{log}_{2}\left(1+\dfrac{\beta}{h_{k}^{4}}\right)-\dfrac{4W\beta(h_{A}-h_{k})}{h_{k}(\beta+h_{k}^{4})\text{log}_{e}2}\geq
R_{0}$ (16)

Using (13) and (16), (P1) can be reformulated as

(P2) $\displaystyle:$
$\displaystyle\underset{h_{A}}{\text{maximize}}\,\,\,\,\dfrac{\overline{R}(h_{k})+{\overline{R}}^{\prime}(h_{k})(h_{A}-h_{k})}{E(h_{A},T)}$
s.t. $\displaystyle\eqref{c1},\eqref{c3}$ (17)

1 Initialize $h_{1}$, $l_{1}^{s}=\dfrac{S(h_{1},h_{1})}{E(h_{1},T)}$, $k=1$.

2 while  _$(1)$_ do

3 $h_{opt}^{s}=h_{k}$

4 Determine the optimal solution $h_{k}^{s*}$ by solving
$\displaystyle\underset{h_{A}}{\text{maximize}}\,\,\,\,S(h_{A},h_{k})-l_{k}^{s}E(h_{A},T)$
$\displaystyle s.t\,\,\,\eqref{c1},\eqref{c3}$

5 $l_{k+1}^{s}=\dfrac{S(h_{k}^{s*},{h_{k}})}{E(h_{k}^{s*},T)}$

6 if  _(( $l_{k+1}^{s}-l_{k}^{s})/{l_{k+1}^{s}})<\zeta$_ then

7 break;

8

9 $h_{k+1}=h_{k}^{s*}$

10 $k=k+1$

11

Output: Optimal AAP Altitude$=h_{opt}^{s}$

Algorithm 1 GEE Maximization using SCP

Note that (P2) is a single ratio fractional maximization problem with a
concave numerator
$S(h_{A},h_{k})=\overline{R}(h_{k})+{\overline{R}}^{\prime}(h_{k})(h_{A}-h_{k})$,
convex denominator $E(h_{A},T)=E_{A}(h_{A},T)+E_{C}(T)$ and convex
constraints. Therefore (P2) can be efficiently solved by using polynomial time
complex Algorithm 1. In every iteration of Algorithm 1, the optimal solution
in step 4 is determined by using standard convex optimization tools like CVX
[24]. In section V, we show that the efficient solution of (P1) obtained by
solving (P2) through Algorithm 1 matches the global optimum obtained using the
monotonic fractional program optimization technique.

###  IV-B GEE maximization using MFP

The candidate solution obtained from SCP cannot be considered as the global
optimum of (P1). Therefore, to obtain the global optimum of (P1), we exploit
the monotonic behavior of the objective function using the monotonic
fractional programming technique [22][23]. The key idea is that the global
optimum of an increasing objective function of a maximization problem lies in
the outer boundary of the feasible set formed by the constraints. Following
the fundamental definitions from [23], a maximization problem takes the
canonical form of a monotonic optimization problem, if it can be formulated as

(P3) $\displaystyle:$
$\displaystyle\underset{\mathbf{v}}{\text{maximize}}\,\,\,\,f(\mathbf{v})$
s.t. $\displaystyle\mathbf{v}\in\mathcal{G}\cap\mathcal{H}$

where $f:\mathbb{R}^{M}\rightarrow\mathbb{R}$ is an increasing function of
$\mathbf{v}$, $\mathcal{G}\subset[\mathbf{0},\mathbf{a}]$ is a compact normal
set with nonempty interior, and $\mathcal{H}$ is a closed conormal set on
$[\mathbf{0},\mathbf{a}]$. For exact definitions of monotonicity, normal and
co-normal sets please refer to [23].

The optimization problem (P1) fits in the class of fractional problems, which
can be globally solved by Algorithm 2. For a given positive $l_{k}^{m}$, in
every $k^{th}$ iteration of Algorithm 2, we need to solve the following
maximization problem in step $4$:

(P4) $\displaystyle:$
$\displaystyle\underset{h_{A}}{\text{maximize}}\,\,\,\,\overline{R}(h_{A})-l_{k}^{m}\left\\{E_{A}(h_{A},T)+E_{C}(T)\right\\}$
(18) s.t. $\displaystyle\eqref{c1}-\eqref{c2}$ (19)

It should be noted that, at first look, (P4) doesn’t take the canonical form
of monotonic optimization problem defined in (P3). However, (P4) can be
expressed as the maximization of differences of increasing functions of
$h_{A}$, which allows us to reformulate (P4) as a monotonic optimization
problem. For the ease of reformulation, we equivalently represent the minimum
individual data rate constraint as

$\displaystyle
h_{max}=\left[\dfrac{\beta}{2^{\dfrac{R_{0}}{W}}-1}\right]^{1/4}$ (20)

Note that (18) can be rewritten as

(P5) $\displaystyle:$
$\displaystyle\underset{h_{A}}{\text{maximize}}\,\,\,\,\overline{R}_{1}(h_{A})-\overline{R}_{2}(h_{A},l_{k}^{m})$
(21) s.t. $\displaystyle\eqref{c1}$ (22)

where

$\displaystyle\overline{R}_{1}(h_{A})$ $\displaystyle=$ $\displaystyle
T\rho_{ue}\pi\text{cot}^{2}\phi
h_{A}^{2}\text{log}_{2}\left(\beta+h_{A}^{4}\right)$
$\displaystyle\overline{R}_{2}(h_{A},l_{k}^{m})$ $\displaystyle=$
$\displaystyle T\rho_{ue}\pi\text{cot}^{2}\phi
h_{A}^{2}\text{log}_{2}\left(h_{A}^{4}\right)$ (23)
$\displaystyle+\>l_{k}^{m}(E(h_{A},T))$

are monotonically increasing functions of $h_{A}$, and $h_{max}$ of (12) is
given by (20). In order to write (P5) in canonical form, we introduce the
additional variable
$t=\overline{R}_{2}(h_{max},l_{k}^{m})-\overline{R}_{2}(h_{A},l_{k}^{m})$,
which allows (P5) to be reformulated as

(P6) $\displaystyle:$
$\displaystyle\underset{h_{A},t}{\text{maximize}}\,\,\,\,\overline{R}_{1}(h_{A})+t$
(24) s.t. $\displaystyle(h_{A},t)\in\mathcal{G}\cap\mathcal{H}$ (25)

where

$\displaystyle\mathcal{G}=\left\\{\begin{aligned} (h_{A},t):h_{A}\leq
h_{max},\\\
t\leq\overline{R}_{2}(h_{max},l_{k}^{m})-\overline{R}_{2}(h_{A},l_{k}^{m})\end{aligned}\right\\}$
(26) $\displaystyle\mathcal{H}=\left\\{(h_{A},t):h_{A}\geq h_{min},t\geq
0\right\\}$ (27)

By the monotonically increasing behavior of
$\overline{R}_{2}(h_{A},l_{k}^{m})$ we can relate

$\displaystyle\overline{R}_{2}(h_{min},l_{k}^{m})\leq\overline{R}_{2}(h_{A},l_{k}^{m})$
(28)

By [proposition 2, [22]], (26) defines a normal set and (27) defines a co-
normal set in the polyblock

$\displaystyle\left[h_{min},h_{max}\right]\times\left[0,\overline{R}_{2}(h_{max},l_{k}^{m})-\overline{R}_{2}(h_{min},l_{k}^{m})\right]$
(29)

with the vertex set $\mathcal{V}$. Hence by using (24)-(27) we represent (P4)
in the canonical form of monotonic optimization problem with
$f(\mathbf{v})=\overline{R}_{1}(\mathbf{v}(1))+\mathbf{v}(2)$,
$\forall\mathbf{v}\in\mathcal{V}$, which can be globally solved by using the
polyblock outer approximation algorithm as explained in Algorithm 3 [23]. Even
though the complexity of this global optimization algorithm is exponential in
the number of variables, it is much lower compared to other global
optimization techniques, which exhaustively search over the entire feasible
set. Hence the globally optimal AAP altitude is obtained by solving (P1) using
Algorithm 2 in which, at each iteration, step 4 is solved by using Algorithm
3.

1 Initialize $h_{1}$, $l_{1}^{m}=\dfrac{\overline{R}(h_{1})}{E(h_{1},T)}$,
$k=1$.

2 while  _$(1)$_ do

3 $h_{opt}^{m}=h_{k}$

4 Determine the optimal solution $h_{k}^{m*}$ by solving the monotonic
optimization problem (P6) using Algorithm 3

5 $l_{k+1}^{m}=\dfrac{\overline{R}(h_{k}^{m*})}{E(h_{k}^{m*},T)}$

6 if  _(( $l_{k+1}^{m}-l_{k}^{m})/{l_{k+1}^{m}})<\zeta$_ then

7 break;

8

9 $h_{k+1}=h_{k}^{m*}$

10 $k=k+1$

11

Output: Optimal AAP Altitude$=h_{opt}^{m}$

Algorithm 2 GEE Maximization using MFP

##  V Numerical Evaluation

In this section, we compare the optimal altitude values obtained through SCP
and MFP optimization techniques. Furthermore, the convergence behavior of the
PA algorithm; the impact of aerial vehicle’s energy consumption on GEE; the
variation of GEE with minimum data rate requirement are discussed. We consider
$h_{0}=1.42\times 10^{-4}$, $P_{T}=10$ dBm, $W=20$MHz,
$\sigma_{0}^{2}=-169\text{dBm}/\text{Hz}$ , $\phi=43^{\circ}$,
$\rho=0.005\text{UEs}/\text{m}^{2}$, $P_{H}=5$W, $T=400$s, $R_{o}=20$Mbps and
$h_{min}=10$m.

Figure 3 shows the accurate plot of GEE with the altitude of aerial vehicle,
$h_{A}$, along with the optimal points obtained through SCP and MFP
techniques. From the plot, it is observed that GEE decreases when $h_{A}$ is
very low or very high. The reason for this behavior is that, at low $h_{A}$,
the number of UEs covered ($N=\rho_{ue}\pi h_{A}^{2}\text{cot}^{2}\phi$) by
the AAP decreases with decreasing $h_{A}$, leading to a decrease in the total
number of bits transmitted, thereby to reduced GEE. At high altitude regions,
the LoS channel gain between the UE and AAP decreases, the number of UEs
covered by AAP increases, and $E(h_{A},T)$ increases. In addition to this,
with an increase in the number of users, power alloted for a single UE
decreases. So in the high altitude region, the increase in the number of UEs
is highly compensated by the combined effect of the decrease in channel gain,
decrease in power per UE and increase in $E(h_{A},T)$, which result in a low
GEE. Figure 3 also shows that the optimal AAP altitude obtained by the SCP is
very close to the globally optimal altitude obtained from the monotonic
fractional programming technique. Hence the global optimum of our objective
can be obtained by the polynomial-time complex sequential convex optimization
technique.

Figure 3: Optimal solutions obtained from SCP and MFP.  Figure 4: Convergence
behavior of the PA algorithm.

Figure 4 shows the convergence behavior of the PA algorithm in the last
iteration of MFP. The converging nature of upper($f_{max}$) and
lower($f_{min}$) bounds of the PA algorithm guarantees the evaluation of the
global optimum of GEE in a finite number of convex evaluations; with the
number of convex evaluations much greater than that required by SCP. The
optimal altitude plots of MFP and SCP says that the locally optimal altitude
value obtained using SCP is equal to the globally optimal altitude obtained
using MFP. In addition to this, Figure 4 shows the error in determining the
optimal altitude without considering the rotor energy consumption,
$E_{A}(h_{A},T)$. It is observed that the optimal altitude determined with
$E_{A}(h_{A},T)=0,$ is 18.23m higher than the actual optimal altitude value.
Hence, according to Figure 3, hovering at an altitude higher than the actual
optimal value yields low GEE. Therefore, to achieve the maximum GEE value, the
rotor energy consumption of the aerial vehicle should be considered while
formulating the optimization problem.

Figure 5: Global energy efficiency versus minimum individual data rate
requirement.

Figure 5 depicts the variation of GEE with the minimum data rate requirement.
The two GEE plots correspond to the variation of GEE with $R_{o}$ when the
aerial vehicle is hovering at the optimal altitude obtained: (a) with zero
energy consumed by the rotor of the aerial
vehicle($h_{hover}=h_{opt}(E_{A}(h_{A},T)=0)$ ); (b) with non-zero energy
consumed by the rotor of the aerial vehicle
($h_{hover}=h_{opt}(E_{A}(h_{A},T)\neq 0)$ ). As said before, the GEE of the
ACS with $h_{hover}=h_{opt}(E_{A}(h_{A},T)\neq 0)$ is more than that with
$h_{hover}=h_{opt}(E_{A}(h_{A},T)=0)$. This gain in GEE reflects the effect of
considering the aerial vehicle’s rotor energy consumption in altitude
optimization, showcasing the novel aspect of this paper. In the plot, the
value of GEE is constant for a range of $R_{o}$ and then it starts decreasing
with an increase in $R_{o}$. From (20), the value of the maximum allowed
altitude, $h_{max}$ decreases with increase in $R_{o}$. It is because a higher
minimum individual data rate is achieved by increasing the LoS channel gain
obtained by decreasing the hovering altitude of the aerial vehicle. When
$h_{max}(R_{o})$ is greater than the $h_{A}$ corresponding to the global
optimum of GEE ($\text{GEE}_{\text{global}}$), the optimal altitude is equal
to $h_{A}(\text{GEE}_{\text{global}})$ and GEE remains constant; for
$h_{max}(R_{o})\leq h_{A}(\text{GEE}_{\text{global}})$, optimal altitude is
equal to $h_{max}$, results in decrease in GEE with increase in $R_{o}$. The
decrease in GEE with increase in $R_{o}$(decrease in $h_{max}$) shows the
monotonically increasing property of GEE which is exploited in MFP.

##  VI Conclusion

In this work, we found the optimal energy-efficient altitude of an aerial
access point which acts as a flying base station for an orthogonal multiple
access downlink broadcast transmission scenario. The modeled energy
consumption is the sum of energy consumed by the aerial vehicle and the energy
required for the communication between the AAP and the UEs. An efficient
solution to the formulated GEE maximization problem with individual data rate
constraint and altitude constraint is obtained using sequential convex
programming and is compared to the global optimum achieved by the monotonic
fractional programming technique. One can see that the optimal altitude value
from the polynomial-time complex SCP matches the globally optimal altitude
value obtained from the monotonic fractional programming. Further, we observed
that there is a gain in the GEE when the aerial access point is hovering at an
optimal altitude determined by considering the non-zero rotor energy
consumption of the aerial vehicle. In addition to this, the optimal altitude,
and hence GEE, decrease with an increase in the minimum individual data rate
constraint. Joint altitude and power optimization in a non-orthogonal multiple
access transmission scheme with multiple AAPs is left as our future work.

1

2Initialize $i=1$, $\mathcal{V}_{i}$ as the vertexset of polyblock (29)

3 Set
$\mathbf{v}_{min}=argmin\\{f(\mathbf{v})\mid\mathbf{v}\in\mathcal{V}_{i}\\}$

4 Set
$\mathbf{v}_{max}=argmax\\{f(\mathbf{v})\mid\mathbf{v}\in\mathcal{V}_{i}\\}$

5 Set $f_{max}=max_{\mathbf{v}\in\mathcal{V}_{i}}f(\mathbf{v})$ and
$f_{min}=f(\mathbf{v}_{min})$

6 while  _$((f_{max}-f_{min})/f_{max} >e)$_ do

7 Obtain $\mathbf{v}_{o}$, the intersecting point of the line drawn from
$\mathbf{v}_{min}$ to $\mathbf{v}_{max}$ with the normal region $\mathcal{G}$
using bisection method [Algorithm 1 [25]]

8 Update the vertex set, $\mathcal{V}_{i+1}$ according to Lemma 2.16 of [25]

9 if  _$f(\mathbf{v}_{o}) >f_{min})$_ then

10 $f_{min}=f(\mathbf{v}_{o})$

11 $\mathbf{v}_{min}=\mathbf{v}_{o}$

12 set $i=i+1$

13 remove all $\mathbf{v}\in\mathcal{V}_{i}$ with $f(\mathbf{v})\leq
f_{min}+e$

14

15 Set $f_{max}=max_{\mathbf{v}\in\mathcal{V}_{i}}f(\mathbf{v})$

16

17Output: $h_{k}^{m*}=\mathbf{v}_{o}(1)$.

18

Algorithm 3 PA Algorithm [25]

Manuscript submitted April 2020. This work is supported by the project
PAINLESS which has received funding from the European Union’s Horizon 2020
research and innovation programme under grant agreement No 812991.

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