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arXiv:2202.03240v1 [cs.NI] 7 Feb 2022
1
Minimization of the Worst-Case Average Energy
Consumption in UAV-Assisted IoT Networks
Osmel Mart´ınez Rosabal, Onel Alcaraz L ´opez, Dian Echevarr´ıa P´erez, Mohammad Shehab, Henrique Hilleshein,
and Hirley Alves
Abstract—The Internet of Things (IoT) brings connectivity
to a massive number of devices that demand energy-efficient
solutions to deal with limited battery capacities, uplink-dominant
traffic, and channel impairments. In this work, we explore
the use of Unmanned Aerial Vehicles (UAVs) equipped with
configurable antennas as a flexible solution for serving low-
power IoT networks. We formulate an optimization problem to
set the position and antenna beamwidth of the UAV, and the
transmit power of the IoT devices subject to average-Signal-to-
average-Interference-plus-Noise Ratio (
¯
SINR) Quality of Service
(QoS) constraints. We minimize the worst-case average energy
consumption of the latter, thus, targeting the fairest allocation
of the energy resources. The problem is non-convex and highly
non-linear; therefore, we re-formulate it as a series of three
geometric programs that can be solved iteratively. Results reveal
the benefits of planning the network compared to a random
deployment in terms of reducing the worst-case average energy
consumption. Furthermore, we show that the target
¯
SINR is
limited by the number of IoT devices, and highlight the dominant
impact of the UAV hovering height when serving wider areas. Our
proposed algorithm outperforms other optimization benchmarks
in terms of minimizing the average energy consumption at the
most energy-demanding IoT device, and convergence time.
Index Terms—IoT, UAV, energy efficiency, worst-case average
energy consumption, reconfigurable antennas, geometric pro-
gramming.
I. INTRO DUC TI ON
The fifth generation of cellular networks (5G) is introducing
for the first time, in addition to the traditional human-centric
broadband communication services, new service classes re-
lated to the Internet of Things (IoT) [1]. IoT use cases are
usually characterized by the deployment of numerous low-cost
low-power devices, for which novel energy-efficient strategies
are increasingly needed as the network densifies [2]–[4].
Furthermore, the information and communication technology
industry currently contributes to 6% of global CO2emissions
[5]. As a consequence, energy-efficient technologies and solu-
tions are relentlessly pursued by industry and academy. We
need to consider myriad of different approaches to obtain
environment friendly IoT deployments including, but not lim-
ited to, reducing energy consumption, greener materials in the
production of IoT devices, proper waste disposal, and sharing
infrastructure [6], [7]. Focusing on the energy consumption
component, IoT devices should just transmit necessary data
while using efficient wake up protocols, sleep scheduling,
All authors are with the Centre for Wireless Communications (CWC),
University of Oulu, Finland. Email: firstname.lastname@oulu.fi
This work is partially supported by Academy of Finland 6Genesis Flagship
(Grant no. 318927).
collision/congestion avoidance schemes and other possible
energy saving improvements [6].
The IoT traffic is uplink-dominant [8], thus, significant en-
ergy saving is attained from reducing the IoT devices’ transmit
energy1[9]. Besides, with a network-wide reduced transmit
energy, interference is reduced, which may help to sustain the
desired Quality of Service (QoS) with fewer resources. There-
fore, it is important to reduce the uplink transmit energy to
reach more energy-efficient IoT solutions. However, depending
on the distance and position of the IoT device with respect
to its associated terrestrial Base Station (BS), this may be
extremely difficult to achieve due to shadowing and blockage
effects. Uplink channel may require to be compensated with
greater transmit power [10]. In that sense, it is desirable to
have a BS that can dynamically change its coverage based on
the position and traffic pattern of the IoT devices. This goal
can be achieved by using aerial BSs, such as Unmanned Aerial
vehicles (UAVs) [11].
To this end, some authors recognize UAVs as a promising
technology due to their potential to provide provisional com-
munication infrastructure in disaster scenarios, as opportunistic
relays to serve blocked links, or as flying BSs to boost cov-
erage in certain areas [12]. Considering the foreseen massive
number of IoT devices, UAVs are an attractive solution for
energy efficiency and QoS improvements due to the enhanced
coverage resulting from their high mobility and ability to
hover as discussed in [13]. In fact, it is challenging to obtain
Line of Sight (LOS) using terrestrial BSs in urban canyon
environments [14], [15], and it is hard to envision smart cities
without the assistance of UAVs [13], [16]. Furthermore, by
equipping the UAVs with reconfigurable antennas [17], more
degrees of freedom could be attained since it is possible to
adjust the beam footprint of the UAV by means of electrical,
optical, mechanical, and material change techniques to boost
even more the coverage with QoS guarantees 2. These features
are precisely exploited in our work to reduce the worst-case
average energy consumption of the IoT devices.
1Note that herein we refer to transmit energy to highlight that both, the
transmit power and transmission duration, influence the energy consumption,
and consequently the lifetime of the IoT device battery.
2We acknowledge that some design challenges impact the performance
of reconfigurable antennas such as the proper design of a biasing network,
difficult integration, high power consumption, and mechanical stress of
moving parts. However, recent works have shown how to ease the design
process of reconfigurable antennas. Please, see [18] and references therein for
more information
2
A. Related Literature
Many recent works have addressed problems such as the
optimal UAVs’ positioning and trajectory, and device asso-
ciation for reliable and energy-efficient communications. For
instance, the work in [19] maximized the energy efficiency
of the IoT network by associating UAVs and IoT devices
using regret-matching learning. The authors of [20] derived
the optimal hovering height of low-altitude UAVs to achieve
the maximum coverage based on the path loss, elevation angle
and statistical parameters of the urban environment. In [21], an
approach for maximizing the coverage area was derived based
on the optimal deployment of multiple UAVs at a fixed altitude
with directional antennas. An energy-efficient 3D placement
study was carried out in [22] to maximize the number of
covered users with a minimum transmit power. Meanwhile,
UAVs are shown in [23] to save valuable energy resources
to ground devices in hostile or inaccessible places. Authors
discussed the trade-off between system efficiency and energy
efficiency considering the UAV altitude and speed, and also the
frame length at the MAC layer. In [24], authors used Artificial
Neural Networks (ANNs) to predict the coverage and the
received signal strength of IoT devices based on the altitude
and distance from the UAV. They demonstrated that ANNs can
predict better the path loss of the link between the IoT devices
and the UAV than the empirical Hata model. Moreover, a better
UAV altitude prediction leads to satisfy the QoS requirements
more easily. Also, authors estimated the probability of LOS
for suburban, urban and dense urban environments in function
of the elevation angle, and showed that given an elevation
angle closer to 90o, the probability of LOS tends to be 1
for any of the studied urban environments. Authors in [25]
compared the performance of exhaustive search (ES) and
maximal weighted area (MWA) in finding the optimal altitude
for UAVs to provide coverage to devices with different QoS
requirements. Their numerical simulations showed that MWA
algorithm has a close performance to ES algorithm while it has
lower complexity. Authors in [26] proposed a reinforcement
learning (RL) based approach to jointly optimize the IoT
nodes’ transmission power and the UAV positioning. Authors
in [27] studied the joint age of information and IoT devices’
energy consumption minimization in a UAV-assisted data
collection problem. Therein, the authors trained an ANN to
find the optimal trajectory of the UAV as well as the resource
(bandwidth and transmit power) allocation for data collection.
An energy-efficient UAV-enabled solution for massive IoT
shared spectrum access was proposed in [28], where the IoT
devices’ transmit power was optimized while the interference
constraint for the closest primary user is respected.
Moreover, UAVs may be used as data aggregators. For
instance, the authors of [29] proposed an energy-efficient solu-
tion for IoT data collection at cell edges. Therein, the trajectory
of the UAV and the IoT devices transmission schedule is
optimized. In [30], UAVs were used as communication relays
to assist the links between smart devices and a low-orbiting
satellite in scenarios without coverage from terrestrial BSs.
The authors optimized the subchannel selection, UAV relays
deployment and uplink transmission power control of smart
devices to maximize the energy efficiency of the system.
The authors of [31] minimized the overall transmit power
of IoT devices considering the radio resource allocation, 3D
placement and user association to the UAV BS. They used
semi-definite relaxation and Geometric Programming (GP) to
solve the corresponding optimization problem. However, the
works in [19]–[26], [28]–[31] do not consider the activation
pattern of the IoT devices, which considerably influences
the optimum system setup. Conversely, the authors of [32]
jointly determined the optimal UAV’s location, device asso-
ciation and uplink power control considering the activation
patterns of IoT devices and a channel assignment strategy.
They transformed this non-convex optimization problem into
convex by decomposing it in two sub-problems. Firstly, they
considered that the UAV BSs are in a fixed position to find
the jointly optimal device associations and devices’ transmit
power. Secondly,the positioning of the UAVs was optimized
given the device associations. However, they minimized the
total transmission power, which could cause unfairness among
IoT nodes because UAV BSs would focus on big clusters of
IoT nodes to reduce the overall transmission power.
Recently, in [33] the authors considered the use of UAVs
as providers of computational resources for terrestrial devices
that can either upload their tasks to the UAVs or compute
them locally. Therein, they jointly minimize the total energy
consumption spent for uplink transmissions at the served
devices and the trajectory of the UAVs using a deep RL ap-
proach. Finally, in [34] the authors studied a multiuser network
served by a UAV which is equipped with a reconfigurable
antenna. Therein, the authors jointly optimize the UAV’s
hovering height and antenna beamwidth to maximize the
throughput under different multiuser communication models.
They divided the network into non-overlapping clusters and
then proposed a fly-hover-and-communicate protocol for the
UAV to sequentially serve each cluster.
B. Contributions
Different from the above works, herein, we propose a fair
energy-efficient UAV-assisted IoT network, where we reduce
the transmit energy consumption for the worst-case IoT nodes.
In this case, the UAV BS is re-positioned considering the
worst-case average energy consumption of the IoT nodes’
uplink transmission and their activation patterns. In order
to achieve this, we consider a UAV BS equipped with a
reconfigurable antenna and serving multiple IoT devices. This
is motivated by the increasing interest on incorporating recon-
figurable antennas at UAVs [35] and/or creating antenna arrays
via UAV swarms [36]. By using a reconfigurable antenna,
the UAV can properly vary the beamwidth to optimize the
system performance. In this case, the optimization is in terms
of the energy efficiency measured as the average energy
consumption at the most energy-demanding IoT device when
transmitting in the uplink. The optimal UAV 3D position and
devices’ transmit power are found based on their activation
pattern and subject to average-Signal-to-average-Interference-
plus-Noise Ratio (¯
SINR) QoS constraints.
Our main contributions are three-fold:
3
•Instead of the traditional total power minimization prob-
lem as in [32], we aim at reducing the average energy
consumption at the most energy-demanding device in the
network. This conduces to fair allocation of the power
resources, which allows synchronizing the devices’ life-
time so that maintenance (e.g., for battery replacement)
can be efficiently planned;
•The resulting optimization problem, which is not convex
and highly non-linear, is approximately re-cast as a series
of three GPs that can be efficiently solved.
•The proposed algorithm reaches near-global optimal so-
lutions for the worst-case average energy consumption,
and considerably outperforms other benchmark schemes
based on Interior-Point Methods (IPMs) and Genetic
Algorithms (GA) in terms of minimizing the worst-case
average energy consumption of the IoT devices and the
computation time;
•Results show that the number of IoT devices limits
the achievable QoS due to interference, and that the
worst-case average energy consumption does not depend
significantly on the density of the obstacles but on their
height. The optimal hovering height of the UAV increases
linearly with respect to the coverage area, and hence
the worst-case average energy consumption of the uplink
transmissions grows with the distance between the UAV
and the IoT devices.
C. Outline
The remainder of this paper is structured as follows. Section
II describes the system model and presents the problem
formulation. Section III discusses some insights on the prob-
lem feasibility conditions and reformulates the problem as a
series of GPs. Section IV presents the proposed optimization
algorithm, while simulation and numerical results are analyzed
in Section V. Finally, we draw conclusions and make final
remarks in Section VI. To make the paper more tractable, we
summarize the key abbreviations and symbols that will appear
throughout the paper in Table I.
II. SY S TE M LAYOUT A ND PROB LE M FO RM ULATI O N
A. System Layout
We consider a wireless system consisting of a set K=
{1,2,···, K }of Klow-power single-antenna IoT nodes,
whose deployment in the 2D plane is given by {(xk, yk)|k∈
K}. An IoT node is considered to be active when it has
data to transmit [37]. Note that not all devices are active
at the same time, thus, hereafter, ck∈(0,1) denotes the
probability of device k∈ K being active. This information
could be acquired beforehand by applying a traffic prediction
algorithm that depends on correlated devices activity or prior
knowledge of devices activation patterns as in [37]–[39]. The
activation pattern is herein exploited to efficiently allocate
resources to IoT nodes [37]. The same applies for downlink
communications in temporally crowded places due to major
events (e.g. sport matches and concerts) where UAVs could be
sent to offload the existing permanent wireless network [40].
In both uplink and downlink cases for UAV-assisted wireless
TABLE I
IMP ORTANT A BBRE VIATI ON S AN D SY MB OLS.
GA Genetic Algorithms
GP Geometric Programming
IoT Internet of Things
IPMs Interior-point methods
LAP Low-altitude platform
LOS Line of Sight
QoS Quality of Service
¯
SINR average-Signal-to-average-Interference-plus-Noise Ratio
UAV Unmanned Aerial Vehicle
inf Infimum
sup Supremum
ckActivation probability of device k
GkAntenna gain seen by device k
LkPath loss between the UAV and device k
O(·)Order of the function
pkTransmit power of device k
β, ψ Propagation parameters
θBUAV’s antenna half beamwidth
ηPath loss coefficient
ζConvergence parameter
KNumber of IoT devices
γk¯
SINR of device k
Fig. 1. The system model comprises a set Kof IoT nodes served by a
rotary-wing UAV.
network, we can use the data traffic prediction to efficiently
allocate resources and to re-position the UAVs.
We analyze an uplink scenario as illustrated in Fig. 1, where
4
active devices communicate over the same frequency band3
with a rotary-wing UAV at height4h. We assume the use of
a low-altitude platform (LAP) such as a quadrotor UAV. The
position of the UAV is then fully given by (xuav, yuav , h).
Additionally, we denote the UAV’s directional antenna half
beamwidth by θB, thus, the antenna gain seen by the k−th
IoT node transmissions can be approximated by
Gk=(G3dB,−θB
2≤ϕk≤θB
2,
0,otherwise,(1)
where ϕkis the corresponding sector angle. G3dB is the main
lobe gain, which we consider as a null gain outside the main
lobe, and is given approximately by G3dB ≈8.83
θ2
B
with θBin
radians [21], [43]. Additionally, the UAV is equipped with
a reconfigurable antenna such that it is capable of tuning
θB≥θ0as it sees fit, where θ0is the minimum antenna half
beamwidth. The reader can refer to [44] and the references
therein for more information about different techniques for
implementing reconfigurable antennas.
B. Channel model
The ground-to-air channel depends greatly on the type of
environment (e.g., rural, suburban, urban, highrise urban, etc).
Notice that in such practical scenarios one may not have any
additional information about the exact locations, heights, and
number of the obstacles. Therefore, it is advisable to con-
sider the randomness associated with the LOS and non-LOS
(NLOS) links when designing the UAV-based communication
system.
For ground-to-air communications, each device will typ-
ically have a LOS view towards the UAV with a given
probability. This LOS probability depends not only on the
environment but also on the elevation angle, and for the k−th
IoT device it is commonly modeled as [32]
Plos
k=1
1 + ψe−β(θk−ψ),(2)
where ψand βare constant values, which depend on the
carrier frequency and type of environment, while
θk= tan−1h
rk
= sin−1h
dk
(3)
is the elevation angle. Notice that
dk=qr2
k+h2,(4)
rk=p(xk−xuav)2+ (yk−yuav)2,(5)
denote the distance from k∈ K to the UAV and to its projec-
tion on ground, respectively. Then, the NLOS probability is
given by Pnlos
k= 1 −Plos
k. As expected, the LOS probability
in (2) models practical phenomena since by increasing the
3The extension to a multi-channel scenario with random or deterministic
channel allocation is straightforward.
4The term “height” refers to the vertical distance from the surface where
the IoT nodes are deployed to the UAV. However, the regional operational
rules for flying small UAVs define, in most cases, the hovering height limits
in terms of the absolute altitude, which is the height of the UAV above sea
level. Please, see [41], [42] for more information.
elevation angle and/or the UAV altitude, the chances of LOS
are greater.
The path loss model for LOS and NLOS links between
device kand the UAV is given by [45]
Lk=η4πfcdk
cα,(6)
where fcis the carrier frequency, αis the path loss exponent,
cis the speed of light, while η∈ {η1, η2}, where η1and η2
(η2> η1>1) are the excessive path loss coefficients under
LOS and NLOS conditions, respectively. Now, leveraging (2)
and (6), the average path loss between device kand the UAV
can be expressed as
¯
Lk=Plos
kη1+Pnlos
kη2(κdk)α,(7)
where κ= 4πfc/c. Then, the average channel power gain
is given by ¯gk= 1/¯
Lk. Finally, the per-link communication
performance is measured through its ¯
SINR5, which for the
k−th IoT device it is given by
γk=Gkpk¯gk
Pj∈K\kcjGjpj¯gj+σ2,(8)
where pkdenotes its transmit power and σ2is the additive
white gaussian noise power at the UAV receiver.
C. Problem Formulation
Herein, we are interested in minimizing the worst-case
average energy consumption per device by optimizing not only
their transmit power but also the UAV position xuav, yuav, h
and antenna beamwidth θB. To this end, we cast the min-max
optimization problem as follows:
P1: minimize
xuav,yuav ,h,{pk},θB
max
k∈K {ckpk}(9a)
subject to γi≥γ0,∀i∈ K (9b)
pmin ≤pi≤pmax,∀i∈K,(9c)
h≥hmin,(9d)
θB≥θ0.(9e)
Herein, we consider that the transmission time is normalized,
which allows us to use maxk∈K {ckpk}as the average energy
consumption of the most energy-demanding device of the net-
work. By minimizing this quantity, we limit the average energy
consumption at each device which ultimately prevent long-
lasting peaks in their energy consumption profile that quickly
reduce their batteries’ lifetime [46]. Besides, we assume that
the IoT devices contend for the same uplink resource block
using a grant-free random access protocol, whose recovery
performance at the UAV depends on γ0in (9b) [47], i.e., an
5The term ¯
SINR here is used to highlight the fact that instead of using
Lkseparately for LOS and NLOS links we utilize ¯
Lk, while accounting also
for the average activation probabilities. By doing this, the SINR expression
becomes more tractable than the average SINR which is more naturally linked
to decoding success. Hence, γ0is chosen such that when ¯
SINR ≥γ0, the
chances of outage are negligible. Such approach has been also adopted in
several works, e.g., [20], [22], [32], [45].
5
¯
SINR target that must be achieved by all devices6. Power
restrictions in (9c) are due to hardware limitations and/or
spectrum regulations, while the UAV altitude restriction in (9d)
is due to aviation regulations [10], and the antenna beamwidth
constraint in (9e) is inherent to the antenna hardware. Finally,
we assume that the system dynamics are quasi-static.
Since neither the objective function (9a) nor the inequality
constraints (9b) are convex, P1 is obviously not convex. This
non-convexity and the extreme non-linearity of (9b) on all
the optimization variables, render P1 extremely hard to solve
efficiently in its current form. Therefore, in the following
section, we aim at solving such issue by approximately casting
P1 as a series of GP problems, which in turn can be efficiently
solved.
III. PRO BL E M RE FOR MU L ATION
A. Insights on problem feasibility
We start by studying the feasibility of the problem. We
observe that γkin (8) is a step function of G3dB since we
assume null gain outside the main antenna lobe. Then, for
guaranteeing (9b) all the IoT devices must be in the ground
footprint of the UAV’s antenna main lobe. Such practical
constraint can be geometrically given as
θB≥2 tan−1maxkrk
h,(10)
and then, we are free to modify (8) as follows
γk=G3dBpk¯gk
G3dB Pj∈K\kcjpj¯gj+σ2.(11)
The next Proposition establishes the maximum target ¯
SINR
for any IoT node in our setup.
Proposition 1. For feasibility, the target ¯
SINR is required to
satisfy the following constraint in the considered scenario
γ0≤1
(K−1) maxkck+σ2θ2
0
8.83pmax
,(12)
<1
(K−1) maxkck
.(13)
Proof. Let us proceed as follows. From (11), we have
γ0≤inf
k∈K sup
pk,ck,gk,θB{γk}
(a)
= sup
pk,ck,gk,θBnG3dBpk∗ck∗¯gk∗/ck∗
G3dBck∗pk∗Pj∈K\k∗¯gj+σ2o
(b)
= sup
θB1
(K−1) maxkck+σ2
G3dBpmax
(c)
=1
(K−1) maxkck+σ2θ2
0
8.83pmax
,(14)
where the inf operation exists because all nodes are required
to satisfy the QoS constraint γ≥γ0. Then, (a)comes from
6Herein, we consider a common ¯
SINR target constraint for all devices—
possibly corresponding to the same application—for solving the minimization
of the worst-case average energy consumption of the massive IoT deploy-
ment. However, we acknowledge that IoT networks have heterogeneous QoS
requirements, and thus we left this problem for future works.
assuming that k∗∈ K is the IoT node with the weakest ¯gk
and using the fact that in the best scenario the solution of P1
leads to ckpk=µ, ∀k∈ K. Next, (b)comes from assuming
all nodes with equal path loss and maximum transmit power,
which maximizes the expression given in (a). Then, we attain
(c)after letting θB→θ0, which matches (12); while (13) is a
relaxed result that allows for establishing a preliminary bound
without the knowledge of σ2, θ0and pmax.
Note that (13) becomes tight as K,maxkck,pmax increase
and/or σ2,θ0decrease.
B. Geometric program formulation
At first, let us assume that P1 can be partitioned into two
GP sub-problems as follows:
•P1-1, which is P1 given xuav, yuav. This requires initial-
izing xuav, yuav , which can be done by simply choosing
it such that [xj, yj]T[xuav, yuav ]T[xk, yk]T, for
certain j, k ∈ K. Intuitively, the UAV’s projection on
ground is expected to be close to a centroid determined by
all IoT nodes position. Therefore, and given the devices
activation probability a good initialization for xuav, yuav
can be
x(0)
uav =PK
k=1 ckxk
PK
k=1 ck
, y(0)
uav =PK
k=1 ckyk
PK
k=1 ck
.
(15)
•P1-2, which is P1 with optimization variables xk, yk, pk,
∀k∈ K, where hand θBare outputs of P1-1.
Notice that if such problem partition exists as will be shown
later, then P1 can be solved by mutually projecting the
subproblems’ solutions in an iterative way. This is because
each GP subproblem can be readily transformed to a convex
problem (see Section IV), and then according to Newmann’s
alternating projection Lemma, the resulting iterative procedure
always converges to the global optimum [48].
1) P1-1:We depart from P1 (9) by noticing that the
unconstrained optimization part in (9a) can be alternatively
stated in the GP standard form as
minimize max
k∈K {ckpk} →minimize t, (16a)
subject to ckpkt−1≤1,∀k∈ K,
(16b)
while the constraints (9c)−(9e) transform to
pminp−1
i≤1,∀i∈ K (17a)
pip−1
max ≤1,∀i∈ K (17b)
hminh−1≤1,(17c)
θ0θ−1
B≤1.(17d)
Afterwards, we are required to deal just with (9b), which is
very intricate. As commented in the previous subsection, such
constraint can be divided into two constraints, which are given
in (10) and (11). Although, the first one already involves a
tangent function, which is not allowed in a GP environment,
we can take advantage of the limited range of θBvalues, e.g.,
θB∈θ0,2 tan−1maxi∈K ri
hmin , to use standard curve fitting
6
tools and write tan(θB/2) ≈q1θq2
B. Notice that θB< π since
hmin >0, which favors the adopted power approximation.
Also, q1and q2are positive since for a feasible θBthe function
is increasing and positive. The approximation can be tight for
maxi∈K ri≤2hmin, which should hold in practical setups.
Therefore, (10) is relaxed to
q−1
1h−1θ−q2
Bmax
i∈K ri≤1.(18)
Regarding the constraints related to γi≥γ0, where γiis given
in (11), we proceed as follows
G3dBpk¯gk
G3dB Pj∈K\kcjpj¯gj+σ2≥γ0
γ0X
j∈K\k
cjpj¯gjp−1
k¯g−1
k+γ0σ2G−1
3dBp−1
k¯g−1
k≤1
γ0Pj∈K\kcjpj¯gjp−1
kuk+γ0σ2G−1
3dBp−1
kuk≤1
¯g−1
ku−1
k≤1,(19)
where the last transformation comes from introducing the
auxiliary variables {uk}. Then, now it is just a matter of
expressing ¯gkin a posynomial form [49], which we address
as follows
¯gk=1
¯
Lk
=κ−αd−α
k
Plos
kη1+Pnlos
kη2
(a)
=κ−αd−α
k1 + ψeψβ e−β θk
η1+η2ψeψβ e−β θk
(b)
≤(rkh)−α/2δ, (20)
where δ=2−α/2κ−α(1+ψeψβ )
η1+η2ψeψβ . Notice that (a)comes from
using (7) and (2), while (b)follows after using the inequality
between the arithmetic and geometric means: d−α
k= (r2
k+
h2)−α/2≤(2rkh)−α/2, and taking advantage of
1 + ψeψβ e−β θk
η1+η2ψeψβ e−β θk≤1 + ψeψ β
η1+η2ψeψβ = 2α/2καδ.
Notice that by using the upper bound of ¯gkprovided in
(20), we still guarantee that all the constraints of the original
problem are satisfied. However, since we constrained further
the feasibility set, we may find a non-global optimum solution,
which is the cost paid for our simplifications. Now, we can
state P1-1 as a GP as given next
P1-1: minimize
h,{pk},θB,t,{uk}t(21a)
subject to cipit−1≤1,∀i∈ K,(21b)
pminp−1
i≤1,∀i∈ K,(21c)
pip−1
max ≤1,∀i∈ K,(21d)
hminh−1≤1,(21e)
θ0θ−1
B≤1,(21f)
q−1
1h−1θ−q2
Bmax
i∈K ri≤1,(21g)
ωh, {pi}, θB, ui≤1,∀i∈ K,(21h)
u−1
irα/2
ihα/2δ−1≤1,∀i∈ K,(21i)
where (21h) and (21i) come from substituting (20) into (19).
Then, we have
ωh, {pi}, θB, ui=γ0h−α/2δp−1
iuiX
j∈K\i
cjpjr−α/2
j
+γ0σ2G−1
3dBp−1
iui.(22)
Finally, notice that P1-1 is a GP problem with 2K+3 variables
and 5K+ 3 inequality constraints.
2) P1-2:Now, given the optimization results from P1-1
and departing from P1 (9), we formulate P1-2 in order to
find the optimum UAV position and power allocation profile.
Notice that similar to P1-1, herein (16) and (17) also hold.
Meanwhile, without loss of generality we assume positive
coordinates7, e.g., xk, yk≥0,∀k∈ K such that obviously
xuav, yuav ≥0holds as well. Then, we define ˜xk=|xk−xuav |
and ˜yk=|yk−yuav|such that for given θBand hand with
the help of (5), constraint (10) can be re-written as
tan θB
2≥rk
h
htan θB
2−1q˜x2
k+ ˜y2
k≤1
htan θB
2−2
˜x2
k+htan θB
2−2
˜y2
k≤1,(23)
while instead of ˜xk=|xk−xuav |, we use
˜xk=|xk−xuav | → n˜xk≥xk−xuav
˜xk≥xuav −xk
→n˜xk
xk+xuav
xk≥1
˜xk
xuav +xk
xuav ≥1
∼
→8n2x−1
k˜x1/2
kx1/2
uav ≥1
2x−1
uav ˜x1/2
kx1/2
k≥1
→n1
2xk˜x−1/2
kx−1/2
uav ≤1
1
2xuav ˜x−1/2
kx−1/2
k≤1,(24)
and the same applies for ˜yk=|yk−yuav|.
Regarding the constraints related with γi≥γ0, where γiis
given in (11), notice that (19) still holds, while (20) can be
further transformed to
¯gk≤(2˜xk˜yk)−α/4h−α/2δ, (25)
by using r−α/2
k= (˜x2
k+ ˜y2
k)−α/4≤(2˜xk˜yk)−α/4. Then, we
7It can be straightforwardly performed by properly setting the origin of
coordinates.
8Herein, we use the inequality between the arithmetic and geometric means.
Notice that this somewhat reduces the constraint set, but we deal with it later
in Section IV.
7
can state P1-2 as a GP as given next
P1-2: minimize
{xuav,yuav ,˜xk,˜yk,pk,uk},t t
subject to cipit−1≤1,(26a)
pminp−1
i≤1,(26b)
pip−1
max ≤1,(26c)
htan θB
2−2
˜x2
i+htan θB
2−2
˜y2
i≤1,(26d)
1
2xk˜x−1/2
kx−1/2
uav ≤1,(26e)
1
2xuav ˜x−1/2
kx−1/2
k≤1,(26f)
1
2yk˜y−1/2
ky−1/2
uav ≤1,(26g)
1
2yuav ˜y−1/2
ky−1/2
k≤1,(26h)
˜ω(˜xi,˜yi, pi, ui)≤1,(26i)
(2˜xi˜yi)α/4u−1
ihα/2δ−1≤1,
(26j)
where the constraints are ∀i∈ K and ˜ωis given by ωin (22),
but for fixed h,θBand by substituting r−α/2
iby (2˜xi˜yi)−α/4.
Finally, notice that P1-2 is a GP problem with 6K+1 variables
and 10Kinequality constraints.
IV. OPT IMI ZATIO N ALG ORI TH M
Although GP problems are not in general convex, they can
be transformed straightforwardly to convex problems [49].
For P1-1 and P1-2, it is just a matter of changing each
variable “var” by ln(“var”) and taking the logarithm of the
constraint functions (i.e., the posynomials are transformed
into log-sum-exp functions, which are convex [49]). After
such transformation, each sub-problem can be solved by any
convex optimization algorithm, by taking advantage of the
KKT conditions.
Algorithm 1 details the steps for solving the general opti-
mization problem P1 through the two proposed subproblems.
Specifically, lines 1-3 deal with initialization, while lines 4-9
deal with the process of solving P1-1 and P1-2 consecutively
until the objective function at each iteration decreases at most
by ξ, which is a convergence parameter given as input to the
algorithm. Notice that after solving P1-1, which uses (18) as an
approximation for (10), we project θBback to the edge of the
original constraint as captured in line 7. After the approximate
GP subproblems have been solved, we can still refine (at
least some) of the optimization variables to reduce further the
objective function. Notice that this may be possible since the
optimization problem determined by the two GP subproblems
operates over a feasible set that is a subset of the original
feasible set given by (9b)-(9c). We perform this in line 10 of
the optimization algorithm by solving P1-3, which is nothing
but P1 with a fixed input {h, θB, xuav, yuav}and optimization
Algorithm 1 Optimum UAV position and IoT nodes’ transmit
power
1: Input: {xk, yk, ck}∀k∈K, γ0, pmin , pmax, hmin, ξ
2: it = 0 (iteration index)
3: Set t(0) =∞and x(0)
uav,y(0)
uav according to (15)
4: repeat
5: it ←it + 1
6: Solve P1-1 given x(it−1)
uav ,y(it−1)
uav ,output:
h(it),{p(it)
k}, θ(it)
B
7: Update: θ(it)
B←max 2 tan−1maxkrk
h(it) , θ0
8: Solve P1-2 given h(it), θ(it)
B,output: x(it)
uav,
y(it)
uav,{p(it)
k},t(it)
9: until t(it−1) −t(it) ≤ξ
10: Solve P1-3 given h(it), θ(it)
B, x(it)
uav, y(it)
uav,output: {p∗
k}
11: Output: {p∗
k},x∗
uav =x(it)
uav, y∗
uav =y(it)
uav,θ∗
B=θ(it)
B,
t= max{ckp∗
k}
variable set {pk}as given next
P1-3: minimize
{pk},t t(27a)
subject to cipi≤t, (27b)
pi≤pmax,(27c)
−pi≤pmin,(27d)
γ0X
j∈K\i
cjGj¯gjpj−Gi¯gipi+σ2≤0,(27e)
where the constraints are ∀i∈ K and (27e) comes form writing
(9b) as a linear equation in piand pj. Note that P1-3 is written
in linear programming (LP) form, and since no approximation
was used, we can claim global optimality for this particular
convex sub-problem.
In practice, Algorithm 1 iteratively optimizes the UAV’s
(position and antenna beamwidth), and the IoT devices’ (trans-
mit power) parameters to minimize the energy consumption
of the latter. The optimization relies on the known average
channel statistics, which depend on the average propagation
parameters for different environments, and also on the esti-
mated activation probabilities and relative positions of the IoT
devices. Therefore, it can be carried out at the UAV, which
periodically reports the updated decisions to the IoT devices.
Hence, our proposed solution can be implemented in real-time
as long as the system dynamics hold within the convergence
time of Algorithm 1. Finally, notice that all simplifications that
lead to the Algorithm 1 aim at finding a good local optimum
of P1, for which a convex equivalent form does not exist to the
best of the authors knowledge, and hence, its global optimum
cannot be guaranteed by any solver.
Using barrier-based IPM, each GP sub-problem in Algo-
rithm 1 can be efficiently solved with accuracy error ǫin a
worst-case polynomial time complexity [50]. The number of
per-GP required iterations is in the order of
C1=O√n+mln (n+m)∆
ǫ,(28)
8
TABLE II
VALU ES O F m,n,AND sF OR EACH OP TI MIZAT IO N SUB-P RO BL E M.
Parameter P1-1 P1-2 P1-3
m5K+ 3 10K4K
n K2+ 4K+ 3 K2+ 9K−
s2K+ 3 6K+ 1 K+ 1
5 10 15 20 25 30 35 40 45 50
0
100
200
300
400
500
600
700
800
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2109
Fig. 2. Complexity analysis vs Kof both P1-1 and P1-2, for ǫ= 10−2.
while each iteration demands
C2=O(m+s)(s+n)√m+nln (m+n)∆
ǫ(29)
arithmetical operations, where m,n,sdenote the number of
constraints, monomial terms, and variables respectively [51].
The term ∆is related to a perturbation in the feasible set
when solving the problem. Without loss of generality, we
may assume ∆ = 1 since the complexity scales linearly with
ln(∆). Finally, P1-3 requires O√mlog (1/ǫ)iterations each
computed in Os2marithmetic operations when using the
same IPM [52]. Table II presents the values of m,n, and sfor
each individual GP problem. Fig. 2 shows that as Kincreases,
the number of iterations for solving the GPs grows linearly,
whereas the number of operations grows exponentially. The
readers can observe that P1-2 is the most costly sub-problem in
Algorithm 1, so it roughly determines the final computational
time.
V. N UME RI CAL ANALYS IS
In this section, we elucidate numerical results regarding the
solution of P1 throughout three methods. The main approach
applies Algorithm 1, where each optimization sub-problem
is solved with the help of the specialized MOSEK solver
[53]. As a benchmark, we solve directly P1 using IPMs
based on logarithmic barrier function and GA. The former
solves a sequence of approximate minimization problems by
either solving the KKT system or using the conjugate gradient
method [54], whereas the latter relies on stochastic derivative-
free techniques that mimic the evolution of living species [55].
As Fig. 3 depicts, we consider both: i) a cellular-like
deterministic deployment, and ii) random deployments. In
both cases the devices are uniformly distributed in the area,
although deterministically/randomly in case of deployment
TABLE III
PROPA GATI ON PARA ME TERS FOR DIFF ER ENT E NV IR ON MENT S [45].
Environment Parameters {ψ, β, η1[dB], η2[dB]}
Suburban {4.88,0.43,0.1,21}
Urban {9.61,0.16,1,20}
Dense Urban {12.08,0.11,1.6,23}
Highrise {27.23,0.08,2.3,34}
0 10 20 30 40
0
5
10
15
20
25
30
35
40
0 10 20 30 40
0
5
10
15
20
25
30
35
40
Fig. 3. Analyzed scenarios: (a) deterministic and (b) a random deployment.
The green squares represent the IoT devices, whereas the coverage area is
delimited with the blue circumference. We set R= 20 m and K= 40.
TABLE IV
DEFAU LT S IM UL AT IO N PARAM ETER S .
Parameter Value Parameter Value
R20 mpmin 1mW
γ0-16 dB pmax 500 mW
θ0
π
18 hmin 40 m
fc2.5GHz hmax 1000 m
i)/ii). Additionally, we assume that the activation probabil-
ities of devices are uniformly distributed such that ck∼
U(0,0.5),∀k∈ K, regardless of the deployment strategy. In
order to study our problem for different propagation condi-
tions, we consider different urban scenarios namely suburban,
urban, dense urban, and highrise urban, for which the set of
parameters {ψ, β , η1, η2}is given in Table III. Finally, unless
we state the contrary, the numerical simulations are based
on the parameters listed in Table IV. Note that the maxi-
mum transmit power meets the typical values of low-power
transceivers, e.g., [56], whereas the value of γ0corresponds
to low-rate transmissions as typical in IoT devices. Besides,
the carrier frequency matches with the numerical setup in [57]
and the hovering height limits lie within the allowed range for
a LAP [58].
A. On the Impact of the Target ¯
SINR
Fig. 4(a) illustrates the objective function as a function of
the target ¯
SINR γ0for a scenario where 25 and 50 IoT devices
are deployed. Notice that the maximum γ0is subject to Propo-
sition 1’s result (12), and it strongly depends on the number of
IoT devices. Although the problem was solved using the three
previously mentioned optimization tools; Fig. 4(a) shows only
the solution provided by Algorithm 1 since IPMs and GA did
not often return feasible solutions when varying γ0. We can
note the benefits of a deterministic homogeneous deployment
enabled via proper network planning in terms of reducing the
average energy consumptionat the most energy-demanding IoT
9
-17 -16 -15 -14 -13 -12 -11 -10
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
-31 -24 -17 -11
0
50
100
150
200
250
300
350
Fig. 4. (a) Average maxkckpkfor K∈ {25,50}and (b) UAV hovering
height for K= 25, vs γ0, in dense urban environment using Algorithm 1.
device. However, the gain with respect to random deployments
becomes small as the number of IoT devices decreases, thus
it is not worth restricting their positions; instead, random
deployments can provide a quite similar performance with a
slight increase of the worst-case average energy consumption.
Meanwhile, Fig. 4(b) shows the optimal hovering heights of
the UAV when 25 devices are randomly and deterministically
deployed. As we can notice, lower altitudes are available for
the UAV in deterministic deployments when using Algorithm 1
and IPMs. Height results under IPMs exhibit a decreasing
behaviour with γ0, while GA does not show a monotonic
behaviour with the target ¯
SINR and Algorithm 1 tends to fix
the height. The latter may benefit the UAV by saving valuable
energy from propelling power.
B. On the Impact of the Number of IoT Devices
The impact of the number of devices on the worst-case
average energy consumption is depicted in Fig. 5. Clearly,
Algorithm 1 solution outperforms the other methods in terms
of the objective function value, whereas IPMs exhibit some
irregularities mainly caused by the non-derivable objective and
the highly nonlinear constraint (9b). The worst-case average
energy consumption has an increasing behaviour as a function
5 10 15 20 25 30 35 40 45 50
-5
0
5
10
15
20
25
Fig. 5. Average maxkckpkfor both deterministic and random deployment
as a function of the number of IoT nodes Kin a dense urban environment.
Suburban Urban Dense Urban Highrise Urban
-4
-3
-2
-1
0
1
2
3
4
Fig. 6. Average maxkckpkevaluation through Algorithm 1 for different
urban environments and K∈ {25,50}. The optimum height is approximately
236 m for all scenarios.
of the number of IoT devices K. This is because more energy
is needed to overcome the interference which results from the
higher number of devices when the network becomes more
dense. Herein, the difference of the solution for the determin-
istic and the random deployments is negligible, showing that
there is no additional gain in carefully planning the network.
From the results in Fig. 5 we can estimate the battery life of
the worst IoT device for each algorithm, which is a common
metric when evaluating the network performance. Assuming a
low dense scenario with K= 10 IoT devices equipped with
a fully charged battery (for instance, CR MULTICOMP 2032,
3V, 210 mAh), we have that on average the battery life of
the fist device running out of energy is 8 hours for IPMs,
1 day and 11 hours for GA and 65 days and 22 hours for
Algorithm 1. The reader can notice that our strategy extends
the devices’ battery lifetime by minimizing the peaks in their
energy consumption profile.
C. On the Impact of the Environment and Coverage Area
In Fig. 6, we show the devices’ worst-case average energy
consumption for different urban environments. In the first three
10
5 10 15 20 25 30 35 40 45 50
-4
-2
0
2
4
6
8
10
12
14
16
5 10 15 20 25 30 35 40 45 50
0
100
200
300
400
500
600
15 20 25 30
40
45
50
55
Fig. 7. (a) Average maxkckpk, and (b) Optimal UAV hovering height vs
R. We consider a dense urban environment, and K∈ {25,50}. The attained
optimum beamwidth remains constant over R:θ∗
B=θ0=π
18 .
environments, the gap among the plots is just appreciated
for different number of devices, and not because of the
propagation conditions. However, the highrise environment
pushes the energy consumption to larger values to overcome
the blockage caused by surrounding objects. In all the cases,
the UAV’s altitude, which is not shown in the figure, remains
nearly constant, and the main distinction is on the transmit
power allocation.
On the other hand, the impact of the coverage region
dimensions is presented in Fig. 7(a) assuming a dense urban
environment. Note that for small coverage areas, the per-
formance remains steady but as the radius increases from
certain point (R > 20 m and R > 25 m for K= 50 and
K= 25, respectively), all curves start to steadily increase.
This behaviour is supported by the fact that the UAV goes
higher in order to serve larger areas, as shown in Fig. 7(b),
which demands greater energy resources from the IoT devices.
We have shown the results delivered by Algorithm 1, since
IPMs and GA do not converge smoothly to the final solution.
Note that for R≥30 m, the UAV requires to fly 15 m
higher on average, to serve the IoT devices in the random
deployment over the case when the network has been planned,
5 10 15 20 25 30 35 40 45 50
100
101
102
103
Fig. 8. Average computation time vs Kin a dense urban environment.
which costs a slight increase in the UAV’s propelling power.
As a benchmark, we also present the results of using antennas
with fixed beamwidth at the UAV. The reader can notice
the performance degradation in terms of worst-case average
energy consumption as θBincreases (Fig. 7(a)), despite the
fact that the hovering height has decreased (Fig. 7(b)) with
respect to the scenario with reconfigurable antennas. This is
because the antenna gain has significantly deteriorated and the
IoT devices must transmit with higher power. In particular, the
optimization problem is no longer feasible for R > 25 m,
when θB=π. We can also notice that the optimal UAV
hovering heights in both Fig. 4(b) and Fig. 7(b) fall within
the allowed range for the operation of LAPs [59].
Fig. 8 shows the average computation time (using parallel
quad-core processing) for all methods as a function of the
number of devices. As observed, the average computation
time grows exponentially as a function of the number of
devices Kwhen using IPMs and GA methods, while the
increase is closely linear when using Algorithm 1, which is
expected according to our discussions around Fig. 2. Notice
that Algorithm 1 is the least time-consuming method for
every K—it converges in around three iterations—, while it is
worth mentioning that IPMs have the highest computation time
because of the extreme non-linearity of (9b), which does not
impact significantly the other approaches. Finally, computing
the solution under random deployment costs slightly less time
when compared to deterministic deployment.
VI. CO NCL USI ON S
In this work, we proposed a GP-based algorithm for min-
imizing the worst-case average energy consumption of IoT
devices served by a hovering UAV. The proposed algorithm
not only controls the uplink transmit power of the devices
but also the UAV’s position and antenna configuration. We
quantified the maximum attainable QoS in terms of per-
link ¯
SINR as a function of the number of IoT devices and
their activation probabilities, and showed how it diminishes
in crowded deployments due to excessive interference. Our
proposed optimization algorithm stands out when compared
to two different benchmark schemes relying on IPMs and
11
GA solvers, with clear reduction in the computer process-
ing time. Additionally, we showed the marginal benefits of
planning the network compared to the random deployment in
terms of reducing the worst-case average energy consumption,
especially in dense deployments. Among the analyzed urban
environments, the highrise environment was shown to be the
most demanding in terms of energy consumption. Finally,
the worst-case average energy requirements were shown to
increase as the UAV flies higher to serve wider areas.
As an interesting future work, we could also consider energy
consumption and maximum flying speed at the UAV, which are
limiting factors of UAV-assisted networks [60]. Besides, we
can include specific regulations on the maximum hovering al-
titude in the worst-case average energy minimization problem.
Finally, the trajectory of a UAVs’ swarm can be optimized for
serving the IoT network as an extension for this work instead
of utilizing a single UAV.
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