# Coverage Performance of UAV-powered Sensors for Energy-neutral Networks with
Recharging Stations
Oktay Cetinkaya1 Mustafa Ozger2 David De Roure1

1Oxford e-Research Centre (OeRC)
Department of Engineering Science
University of Oxford, UK
<EMAIL_ADDRESS>2School of Electrical Engineering
and Computer Science
KTH Royal Institute of Technology, Sweden
<EMAIL_ADDRESS>
###### Abstract

The projected number of Internet of Things (IoT) sensors makes battery
maintenance a challenging task. Although battery-less IoT is technologically
viable, the sensors should be somehow energized, either locally or remotely.
Unmanned aerial vehicles (UAVs) can respond to this quest via wireless power
transfer (WPT). However, to achieve energy neutrality across the IoT networks
and thus mitigate the maintenance issues, the UAVs providing energy and
connectivity to IoT sensors must be supplied by recharging stations having
multi-source energy harvesting (EH) capability. Yet, as these sensors rely
solely on UAV-transferred power, the absence of UAVs causes sensor outages and
hence loss of coverage when they visit recharging stations for battery
replenishment. Hence, besides the UAV parameters (e.g., battery size and
velocity), recharging duration and station density must be carefully
determined to avoid these outages. To address that, this paper uses stochastic
geometry to derive the coverage probability of UAV-powered sensors. Our
analysis sheds light on the fundamental trade-offs and design guidelines for
energy-neutral IoT networks with recharging stations in regard to the
regulatory organization limitations, practical rectenna and UAV models, and
the minimum power requirements of sensors.

###### Index Terms:

Unmanned Aerial Vehicles, Wireless Power Transfer, Stochastic Geometry,
Coverage Probability, IoT.

##  I Introduction

Unmanned aerial vehicles (UAVs) are becoming increasingly essential in
wireless networks, serving as either mobile users or base stations/access
points (APs) [1]. In addition to the numerous value-added services they
already provide, UAVs also play a critical role in addressing the battery
constraints of the Internet of Things (IoT) through wireless power transfer
(WPT) [2]. They are particularly beneficial for networks that rely on a
multitude of sensors, which require excessive maintenance, or that operate in
hard-to-reach areas, such as rainforests. By offering line-of-sight (LoS) air-
to-ground links, UAVs enable highly efficient WPT [3], which, in turn, enables
the battery-less operation of sensors deployed on the ground. After delivering
the energy required by sensors, UAVs can also collect sensory data by acting
as mobile APs, thereby offering an all-in-one solution. In these settings,
UAVs govern both energy and data flows with no human supervision for battery
maintenance or terrestrial APs for data collection, aiming for a certain level
of autonomy in network operation.

Since the primary goal of a WPT-enabled UAV setting is to minimize the battery
constraints of sensors, besides the prevailing energy scarcity across the IoT
domain, the energy required for UAV operation, including the power transferred
to sensors, has to be provisioned within the network. In this way, energy
neutrality can be enabled [4], mitigating the challenges mentioned above. One
approach for achieving this goal is to replenish UAV batteries via recharging
stations having energy harvesting (EH) capability. Here, multiple ambient
sources, such as solar and wind power, can be simultaneously exploited [5] to
minimize the variance and intermittency in the EH output for assured
reliability.

The literature has vast examples of UAV-based service provisioning for IoT
devices, including recharging stations. In most cases, the UAVs operate as
flying APs [6] to collect sensory data underpinned by terrestrial
counterparts. They sometimes deliver power [7] in addition to or aside from AP
functionality [8]. The recharging stations are usually deemed to have mains
connection, or their source of power is untold [9], both referring to a case
with an unlimited energy source, i.e., the total disregard of the energy
neutrality objective.

One domain that has been exhaustively studied in the literature is the UAV-
assisted IoT networks, in which the UAVs provide coverage to sensors. For
example, the authors in [10] optimized the trajectory of a UAV energized by a
solar-powered recharging station in consideration of data rate, energy
consumption, and fairness of coverage. However, they focused on a single UAV
operation without considering the effect of recharging station density,
limiting their application potential. Furthermore, the authors in [11]
proposed a distributed blockchain-based scheme to enable secure and reliable
energy exchange between UAVs and recharging stations. However, they did not
consider the energy-neutral operation of their system, leading to an
impractical solution. The authors in [12] proposed reinforcement learning
algorithms to jointly optimize the velocity and energy replenishment of UAVs
that collect data from sensors. Although they enabled efficient transfer
learning techniques to decrease the learning time and improve the overall
learning process, they did not take energy neutrality into consideration in
their setting, similar to other studies.

As discussed in [13] and the references therein, the literature on UAV-
assisted sensor coverage mainly focused on various other aspects, such as
clustering the sensor nodes for more energy-efficient data collection,
different flying modes of UAVs, and joint path planning and resource
allocation via graph-theory, optimization, machine learning, etc. Despite the
promising findings of these studies, the research must look towards a more
pressing and fundamental issue, i.e., how to achieve energy-neutral IoT at the
minimal sensor outage, occurring due to recharging station-driven UAV
operation. Hence, the following aspects need considerable attention for a more
realistic analysis of the coverage performance of sensors, facilitating energy
neutrality in the IoT domain: i) effective isotropic radiated power (EIRP)
limitations enforced by regulatory organizations; ii) practical rectenna
models with non-linear EH behavior; iii) minimum power requirements of
sensors; iv) individual duration of each UAV operation; and most importantly,
v) a limited source of power for WPT, i.e., the UAVs energized by multi-source
EH recharging stations.

The current gap in the research field being highlighted, this work aims to
mathematically analyze the coverage probability of sensors powered by multi-
source EH recharging stations through UAVs performing WPT with directive
antennae. Since the UAVs manage both energy and data flows in the envisioned
scenario, no service is available during their trip (towards sensors and
recharging stations) and battery replenishment, which refers to an outage or
no coverage of sensors. The goal is to maximize the coverage probability by
tweaking the crucial design parameters, such as the station density, UAV
velocity, battery capacity, and recharging time.

Following the agenda given above, we used stochastic geometry in order to
derive a tractable expression for the event of service guaranteeing a certain
level of coverage. Our analyses, based on practical rectenna and UAV
propulsion models, revealed the non-trivial relationships between the UAV
attributes (e.g., availability, velocity, descent altitude, antenna
directivity, energy budget, output power, transmission duration, and operating
frequency); sensor characteristics (e.g., sensitivity, antenna gain, and power
conversion efficiency); medium specifications (e.g., urban, suburban); and
application requirements (e.g., minimum reporting frequency). In the end, our
study provides an upper bound for the coverage probability of sensors, which
can be practically achieved by carefully selecting the design parameters in a
UAV-powered energy-neutral application scenario involving EH recharging
stations.

The remainder of this paper is organized as follows. We first introduce the
system model in Section II, where the event of service, incorporating
directivity, UAV propulsion, WPT and rectenna models, and FCC regulations, are
formulated. Then, in Section III, we derive the coverage probability of
sensors based on stochastic geometry. This is followed by the numerical
evaluation of the proposed model in Section IV to reveal the design guidelines
that must be followed for the best achievable coverage while meeting
application requirements. Finally, Section V discusses future research
directions and concludes the paper.

##  II System Model

As illustrated also in Fig. 1, we envision an energy-neutral network scenario,
which comprises: i) recharging stations, ii) UAVs, and iii) battery-less
sensors. The UAVs retrieve energy from multi-source EH recharging stations via
inductive power transfer, fly towards the sensors, energize them via radio
frequency (RF) power transfer, and collect their data. During power transfer
and data collection, which refers to service, the UAVs do not move in the 3D
space; they just hover at the centers of event areas, i.e., where sensors
reside. The sensors become active as soon as they intercept enough power from
a UAV. When active, they probe their vicinity for an application-defined
parameter, e.g., temperature, humidity, and/or noise level, and notify their
respective UAV with their readings. After collecting sensor data, the UAVs fly
back to the nearest recharging station to replenish their batteries.

As explained, the UAVs singlehandedly manage energy and data flows within the
network in an autonomous manner; there are no other authorized entities,
terrestrial APs, energy providers, etc. The service is provided at the centers
of event areas, i.e., randomly located circles with radius $r_{\Delta}$, which
are modeled as a Poisson point process (PPP). Within each circle, the sensors
are uniformly distributed, and finally, the locations of recharging stations
are modeled as a PPP $\Phi_{Ch}$ with density $\lambda_{Ch}$. Below, we
explain the service in detail.

###  II-A Service Provisioning

In our envisioned scenario, recharging stations refill the UAV battery with
energy that is just enough for i) a round trip to the event area, i.e., travel
to its center and descent/ascent to/from it, ii) providing service, and iii)
hovering when providing service. During the trip and getting its battery
replenished, the UAV cannot provide any service, i.e., it is unavailable.

Figure 1: Illustration of the envisioned network scenario.

The definition above confirms that the service is conditioned on the distance
between the point where it is provided and the nearest charging station,
$d_{\Delta}(=\\!R_{\Delta}+h_{l})$. However, only one of $d_{\Delta}$’s
components, namely $R_{\Delta}$, changes randomly due to the distribution of
recharging stations. Hence, the probability of UAV’s availability, i.e., the
event of service $\mathcal{E}$, is conditioned on $R_{\Delta}$, which can be
given as:

$\begin{split}P_{(\mathrm{e|R_{\Delta}})}&=\mathbb{P}(\mathcal{E}|R_{\Delta})\\\
&=\frac{t_{PT}+t_{AP}}{t_{PT}+t_{AP}+t_{Ch}+t_{J}},\end{split}$ (1)

where $t_{PT}$ is the time spent for power transfer, $t_{AP}$ is the time
spent for data collection, $t_{Ch}$ is the time spent for recharging the UAV
battery, and $t_{J}$ is the time spent for the round trip. Here, each of
$t_{J}$, $t_{PT}$, and $t_{AP}$ can be defined as:

$\displaystyle\begin{aligned}
t_{J}&=\frac{2(R_{\Delta}+h_{l})}{V}=\frac{2d_{\Delta}}{V},\\\
t_{PT}&=\frac{E_{PT}}{P_{T}},\\\
t_{AP}&=\frac{B_{\mathrm{UAV}}-t_{PT}(P_{h}+P_{T})-t_{J}P_{J}}{P_{h}},\end{aligned}$
(2)

where $P_{J}$ is the power consumption during the trip, $V$ is the UAV’s
velocity during the trip, $E_{PT}$ is the energy budget spared for power
transfer, $P_{h}$ is the power consumption during hovering at the center of an
event area, $P_{T}$ is the transmit power of the UAV, and $B_{\mathrm{UAV}}$
is the energy level of the UAV battery. We should note that the UAV battery
might not be fully charged always since it predominately depends on the power
transfer rate of the wireless pad, $\xi_{Ch}$, of the recharging station and
the time the UAV spends on it, $t_{Ch}$. From (1), we also know that $t_{Ch}$
is inversely proportional to UAV availability since it is on the denominator
of the equation, so it should be limited. That is also because the UAV battery
has a limited/maximum capacity, $B_{\mathrm{max}}$, so the UAV should not
reside on the charging pad beyond when the battery gets fully charged,
referring to $t_{Ch}^{sat}$. Note that the saturation time $t_{Ch}^{sat}$ can
be achieved sooner or later depending on the charging rate $\xi_{Ch}$ since
$B_{\mathrm{max}}$ is fixed. Considering all these, an accurate battery
charging model for the UAV can be given as:

$B_{\mathrm{UAV}}(t_{Ch})=\begin{cases}\xi_{Ch}\cdot
t_{Ch},&\leavevmode\nobreak\ t_{Ch}\in[0,t_{Ch}^{sat}],\\\
B_{\mathrm{max}},&t_{Ch}\geq t_{Ch}^{sat}.\end{cases}$ (3)

and finally, by taking the expectation of (1), the service probability of the
UAV can be calculated as:

$P_{\mathrm{e}}=\mathbb{E}_{\Phi_{Ch}}\bigg{[}\frac{t_{PT}+t_{AP}}{t_{PT}+t_{AP}+t_{Ch}+t_{J}}\bigg{]}.$
(4)

###  II-B Power Transfer

The UAV performs RF power transfer with a directional antenna having a pencil-
beam-like radiation pattern. For such an antenna, i.e. with one major lobe and
very negligible minor lobes of the beam, the gain $G_{T}$ can be approximated
by:

$G_{T}=\begin{cases}\approx\\!\dfrac{30000}{\theta_{B}^{2}},&\leavevmode\nobreak\
\frac{-\theta_{B}}{2}\leq\varphi\leq\frac{\theta_{B}}{2},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{(major lobe)}\\\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ g(\varphi),&\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\text{otherwise},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{(minor lobes)}\end{cases}$ (5)

where $\varphi$ is the sector angle, $\theta_{B}$ is the directional antenna
half-power beamwidth (HPBW) -both in degrees, $\approx\\!30000/\theta_{B}^{2}$
is the maximum gain, and $g(\varphi)$ is the gain outside of the major lobe
(including minor lobes), which can be neglected [14]. Note that (5) is for a
symmetrical radiation pattern, where the HPBWs in each plane are equal to each
other, i.e. $\theta_{1d}\\!=\\!\theta_{2d}$.

Contrary to expectations, the transmit power of the UAV, $P_{T}$ in (2),
cannot be altered casually; it is determined by regulatory organizations,
e.g., the Office of Communications (Ofcom) in the UK, the Federal
Communications Commission (FCC) in the US. For example, FCC Part 15.247 rules
[15] declare that the maximum $P_{T}$ fed into the (in our case, UAV’s)
antenna cannot exceed $30$dBm ($1$W) for the industrial, scientific, and
medical (ISM) bands, in which the maximum effective isotropic radiated power
($\mathrm{EIRP_{max}}$) is limited to $36$dBm ($4$W). This indicates that
increasing $G_{T}$ necessitates a proportional decrease in $P_{T}$, and vice
versa, such that the total RF power radiated by the antenna remains the same,
i.e., $4$W EIRP, where $\max(P_{T})\\!=\\!1$W for each case. For directional
dispersion, however, there are some exceptions to the $\mathrm{EIRP_{max}}$,
details of which can be found in [15].

Here, we should note the following: i) since the antenna size increases with
increasing $G_{T}$, the $G_{T}$ vs. $P_{T}$ balance must be maintained
regarding what a UAV can physically accommodate, ii) power transfer should be
administrated at a low frequency ($f$, preferably sub-GHz), as the power
received by sensors ($P_{R}$) is inversely proportional to the square of $f$,
i.e., $P_{R}\varpropto 1/f^{2}$, iii) since the UAV has a fixed budget for
power transfer ($E_{PT}$), decreasing $P_{T}$ means a longer $t_{PT}$, which
may affect the service probability of the UAV -from (1). The lengthened
coverage lifetime, $t_{PT}$, despite sounding attractive, will alter the duty
cycle of sensors, which cannot be tolerated always due to the certain
reporting frequency requirements of IoT applications [16]. Thus, these trade-
offs must be carefully considered during the system design to maximize the
performance metric defined by the application.

###  II-C Trip Power Consumption

The rotary-wing type UAVs that we have need fixed power during their trip,
which can be approximated as [17]:

$P_{J}(V)\approx
P_{0}\bigg{(}1+\frac{3V^{2}}{U^{2}_{tip}}\bigg{)}+\frac{P_{i}v_{0}}{V}+\frac{1}{2}d_{0}\rho
sAV^{3},$ (6)

where $U^{2}_{tip}$ is the tip speed of the rotor blade, $v_{0}$ is the mean
rotor-induced velocity when hovering, $d_{0}$ is the fuselage drag ratio,
$\rho$ is the air density, $s$ is the rotor solidity, $A$ is the rotor disc
area, and $P_{0}$ and $P_{i}$ are the UAV’s blade profile power and induced
power in hovering status, respectively. Here, $P_{h}$ can be defined as the
sum of $P_{0}$ and $P_{i}$, i.e.,:

$P_{h}=\underbrace{\frac{\delta}{8}\rho sA\Omega^{3}R^{3}}_{\text{$\triangleq
P_{0}$}}+\underbrace{(1+k)\frac{W^{3/2}}{\sqrt{2\rho A}}}_{\text{$\triangleq
P_{i}$}},$ (7)

where $\Omega$ is the blade angular velocity, $R$ is the rotor radius, $k$ is
the incremental correction factor to induced power, and $W$ is the UAV weight.
Using the respective values of each parameter given in [17], $P_{J}$ as a
function of $V$ is illustrated in Fig. 2(a), which is also used in our
calculations.

From (2) and (6), the energy that the UAV needs for a round trip, $E_{J}$, is
$t_{J}\,\times\,P_{J}\,=\,\tfrac{2(R_{\Delta}+h_{l})}{V}P_{J}$, where each leg
consumes the half, i.e., $E_{J}/2$ for travelling to or from the service
point. That is important, as the energy left in $B_{\mathrm{UAV}}$ after
providing service must be enough for UAV to make it to the nearest charging
station before depleting its battery, i.e.,
$\\!E_{\mathrm{left}}\\!\geq\\!E_{J}/2$. In our analyses, we evaluate the
effect of $V$ in minimizing $E_{J}$, ensuring that the UAV has more resources
for service.

(a)

(b)

Figure 2: (a) Trip power consumption, $P_{J}$, vs. UAV velocity, $V$, as
modelled in [17]; (b) rectenna efficiency, $\Psi$, as a function of
input/intercepted power, $P_{I_{\chi}}$, for the rectenna in [18]
($@f_{c}\\!=\\!868$MHz).

###  II-D Sensor Association

We assume that each event area, $\Delta$, is serviced by only one UAV at a
time, i.e., each sensor is associated with one UAV hovering at the center of
$\Delta$ that the sensor falls into. Otherwise, the sensor has no service,
which refers to the outage.

The power intercepted by the sensor, $P_{I}$, after the UAV initializes the
power transfer process is:

$P_{I_{\chi}}=\frac{\mathrm{EIRP^{+}}G_{R}G_{h}}{\mathrm{PL}_{\chi}},$ (8)

where $\mathrm{EIRP^{+}}\\!$ is equal to $P_{T}\,\times\,G_{T}$, $G_{R}$ is
the gain of sensor antenna, $G_{h}$ is exponentially distributed fading power
coefficient, $\mathrm{PL}$ is path loss as a function of distance between the
UAV and sensor, and finally, $\chi$ refers to $\mathrm{LoS}$ and
$\mathrm{NLoS}$, i.e., the line of sight (LoS) and Non-LoS, indicating the
condition of the air-to-ground link between the UAV and sensors.

To perform sensory operations, however, $P_{I_{\chi}}$ has to be converted
into utilizable DC power, $P_{R_{\chi}}$, using a rectifying antenna or
rectenna. Considering the sensitivity and saturation of rectennas, the
research field has agreed on the following piecewise linear function,
capturing $P_{R_{\chi}}$ as a high-order polynomial [19]:

$P_{R_{\chi}}(P_{I_{\chi}})\triangleq\begin{cases}0,&\leavevmode\nobreak\
P_{I_{\chi}}\in[0,P_{th}),\\\
(p_{w}+\sum\limits_{j=1}^{w-1}p_{j}P_{I_{\chi}}^{w-j})\\!\cdot\\!P_{I_{\chi}},&\leavevmode\nobreak\
P_{I_{\chi}}\in[P_{th},P_{sat}],\\\
P_{R_{\chi}}(P_{sat}),&\leavevmode\nobreak\ P_{I_{\chi}}\geq
P_{sat},\end{cases}$ (9)

where $P_{R_{\chi}}(P_{I_{\chi}})$ is non-decreasing and continuous for all
$P_{I_{\chi}}\\!\in\\!\mathbb{R},\,P_{I_{\chi}}\\!\geq\\!0$. In our analyses,
the rectennas are assumed to operate in the ideal region, i.e.,
$P_{I_{\chi}}\\!\in\\![P_{th},P_{sat}]$ for all sensors. Hence,
$(p_{w}\\!+\\!\sum_{j=1}^{w-1}p_{j}P_{I_{\chi}}^{w-j})$ is the rectenna (or
RF-to-DC conversion) efficiency as a function of $P_{I_{\chi}}$, i.e.,
$\Psi(P_{I_{\chi}})$ ($0\\!\leq\\!\Psi(P_{I_{\chi}})\\!<\\!1$), with $w$ being
the degree of polynomial and $\\{p_{j}\\}_{j=1}^{w}$ the respective
coefficients. Furthermore, $\Psi(P_{I_{\chi}})$ is calculated using real data
outsourced from [18], where Fig. 2(b) depicts the measured and simulated
behaviors of the rectenna design of authors.

$\mathrm{PL_{LoS}}$ and $\mathrm{PL_{NLoS}}$ in (8) can be given as [20]:

$\displaystyle\mathrm{PL_{LoS}}=$ $\displaystyle
20\mathrm{log_{10}}\bigg{(}\frac{4\pi
f_{c}d_{\mathrm{{3D}}}}{c}\bigg{)}+\eta_{\mathrm{LoS}},$
$\displaystyle\mathrm{PL_{NLoS}}=$ $\displaystyle
20\mathrm{log_{10}}\bigg{(}\frac{4\pi
f_{c}d_{\mathrm{3D}}}{c}\bigg{)}+\eta_{\mathrm{NLoS}},$ (10)

where $f_{c}$ is the carrier frequency, $c$ is the speed of light, and
$\eta_{\mathrm{LoS}}$, $\eta_{\mathrm{NLoS}}$ are average additional loss,
depending on the environment for LoS, and NLoS links, respectively.
Furthermore, the probability that the UAV has a LoS air-to-ground link with a
sensor can be formulated as [21]:

$P_{\mathrm{LoS}}=\frac{1}{1+\gamma\mathrm{exp}(-\delta(90-\theta_{B}/2-\gamma))},$
(11)

where $\gamma$ and $\delta$ are constant values that depend on the environment
(e.g., suburban, high-rise urban), and ($90-\theta_{B}/2$) is the elevation
angle of the UAV. Finally, the probability of NLoS is always
$P_{\mathrm{LoS}}=1-P_{\mathrm{NLoS}}$.

Based on the equations given above, we can define the coverage probability of
the UAV conditioned on $R_{\Delta}$ as:

$P_{\mathrm{cov|R_{\Delta}}}=P_{\mathrm{(e|R_{\Delta})}}P_{\mathrm{cov,s}},$
(12)

where $P_{\mathrm{(e|d_{\Delta})}}$ is given in (1). Hence, the unconditional
coverage probability can be expressed as:

$P_{\mathrm{cov}}=P_{\mathrm{e}}P_{\mathrm{cov,s}},$ (13)

in which:

$P_{\mathrm{cov,s}}=\mathbb{P}(P_{R_{\chi}}\geq\Gamma_{th}),$ (14)

where $\Gamma_{th}$ is the minimum power that has to be received by a sensor
to become active, referring to the sensitivity, probe its vicinity, and
deliver the data it collects to the UAV. Here, we should note that $d_{3D}$,
and so $R_{RF}$, can be assumed as equal to $h_{UT}$ due to directivity
($h_{UT}\\!=\\!\cos{(\theta/2)}\\!\times\\!R_{RF}$ actually, but $\theta/2$ is
quite small; hence, $R_{RF}\\!\approx\\!h_{UT}$). That leads to the assumption
that $P_{R_{\chi}}$ will be equal for each point in the event area, i.e., all
sensors in $\Delta$ will receive the same power irrespective of their
locations. Therefore, $P_{R_{\chi}}$ does not need to be averaged when
$P_{\mathrm{cov,s}}$ is calculated.

##  III Performance Metrics

In this section, we first derive the service probability conditioned on the
distance to the nearest charging station, i.e., $P_{(\mathrm{e|R_{\Delta}})}$,
to calculate the unconditioned service probability, $P_{\mathrm{e}}$. Then, we
find $P_{\mathrm{cov,s}}$, and hence, study the coverage probability,
$P_{\mathrm{cov}}$, for the envisioned energy-neutral network scenario.

###  III-A Service Probability

By substituting for (2) in (1), we can derive the service probability given
the value of $R_{\Delta}$ as:

$P_{(\mathrm{e|R_{\Delta}})}=\frac{\zeta-2(R_{\Delta}+h_{l})P_{J}}{\zeta-2(R_{\Delta}+h_{l})(P_{J}-P_{h})+Vt_{Ch}P_{h}},$
(15)

where $\zeta=VB_{\mathrm{UAV}}-Vt_{PT}P_{T}$. It should be noted that (15)
only holds if $R_{\Delta}\leq$$\leavevmode\nobreak\
\tfrac{VB_{\mathrm{UAV}}-2h_{l}P_{J}}{2P_{J}}$; otherwise,
$P(\mathrm{e|R_{\Delta}})\\!=\\!0$. If this condition is not satisfied, it
means that $B_{\mathrm{UAV}}$ is not large enough to support the energy
required for the round trip. Hence, there will not be enough power for the UAV
to provide service in $\Delta$. In addition, when $R_{\Delta}\\!=\\!0$,
$h_{l}$ is also $0$, because the UAV cannot descent when it is still on the
recharging station. In that case, the maximum service probability is achieved,
i.e., $P_{(\mathrm{e|R_{\Delta}\,=\,0})}$$\leavevmode\nobreak\
=\tfrac{\zeta}{\zeta+Vt_{CH}P_{h}}$.

Now, using (15), let’s calculate the CDF of the conditional service
probability, $F_{P_{(\mathrm{e}|\mathrm{R_{\Delta}})}}(x)$, as:

$\displaystyle F_{P_{(\mathrm{e}|\mathrm{R_{\Delta}})}}(x)$
$\displaystyle=\mathbb{P}(P_{(\mathrm{e}|\mathrm{R_{\Delta}})}\leq x)$
$\displaystyle=\mathbb{P}\bigg{(}\frac{\zeta-2(R_{\Delta}+h_{l})P_{J}}{\zeta-2(R_{\Delta}+h_{l})(P_{J}-P_{h})+Vt_{Ch}P_{h}}\leq
x\bigg{)},$

and given that $P_{(\mathrm{e}|\mathrm{R_{\Delta}})}$ is a decreasing function
of $R_{\Delta}$, the preimage can be obtained as:

$\begin{split}=\mathbb{P}\bigg{(}R_{\Delta}\geq\frac{\zeta(1-x)-h_{l}\kappa-
xVt_{Ch}P_{h}}{\kappa}\bigg{)},\end{split}$ (16)

where $\kappa=2[P_{J}(1-x)+xP_{h}]$.

Hence, the CDF becomes:

$F_{P_{(\mathrm{e|R_{\Delta}})}}(x)=e^{-\lambda_{Ch}\pi Q^{2}(x)},$ (17)

in which:

$Q(x)=\frac{\zeta(1-x)-h_{l}\kappa-xVt_{Ch}P_{h}}{\kappa}.$

Since the minimum value of $R_{\Delta}=0$, and its maximum value for a non-
zero availability probability is
$\tfrac{VB_{\mathrm{UAV}}-2h_{l}P_{J}}{2P_{J}}$, then:

$0\leq x\leq\frac{\zeta}{\zeta+Vt_{CH}P_{h}}$

Using these results, we can find the service probability as:

$\displaystyle P_{\mathrm{e}}$
$\displaystyle=\mathbb{E}_{\Phi_{Ch}}[P_{(\mathrm{e}|\mathrm{d_{\Delta}})}]$
$\displaystyle=\int_{0}^{\infty}(1-F_{P_{(\mathrm{e}|\mathrm{d_{\Delta}})}}(x))\,\mathrm{d}x$
$\displaystyle=\int_{0}^{\tfrac{\zeta}{\zeta+Vt_{Ch}P_{h}}}(1-e^{-\lambda_{Ch}\pi
Q^{2}(x)})\,\mathrm{d}x.$ (18)

###  III-B Coverage Probability

Based on the equations derived in the previous subsections, we can now work on
the coverage probability given in (14). First, we need to reexpress
$P_{\mathrm{cov,s}}$ as follows:

$\displaystyle P_{\mathrm{cov,s}}$
$\displaystyle\\!=\\!P_{{\mathrm{cov}}_{\mathrm{LoS}}}P_{\mathrm{LoS}}+P_{\mathrm{{cov}_{NLoS}}}P_{\mathrm{NLoS}}$
$\displaystyle\\!=\\!\mathbb{P}\big{(}P_{R_{\mathrm{LoS}}}\\!\geq\\!\Gamma_{th}\big{)}P_{\mathrm{LoS}}+\mathbb{P}\big{(}P_{R_{\mathrm{NLoS}}}\geq\Gamma_{th}\big{)}P_{\mathrm{NLoS}}$
$\displaystyle\\!=\\!\mathbb{P}\big{(}\eta_{\mathrm{LoS}}\tfrac{\mathrm{EIRP\\!^{+}}G_{R}G_{h}}{\mathrm{PL_{LoS}}}\geq\Gamma_{th}\big{)}P_{\mathrm{LoS}}+...$
$\displaystyle\hskip
82.51299pt...+\mathbb{P}\big{(}\eta_{\mathrm{NLoS}}\tfrac{\mathrm{EIRP\\!^{+}}G_{R}G_{h}}{\mathrm{PL_{NLoS}}}\geq\Gamma_{th}\big{)}P_{\mathrm{NLoS}}$
$\displaystyle\\!=\\!\mathbb{P}\bigg{(}\\!G_{h}\\!\geq\\!\tfrac{\Gamma_{th}\mathrm{{PL_{LoS}}}}{{}^{\\!\eta_{\mathrm{LoS}}\mathrm{EIRP\\!^{+}}\\!G_{R}}}\\!\bigg{)}\\!P_{\mathrm{\mathrm{LoS}}}\\!+\\!\mathbb{P}\bigg{(}\\!G_{h}\\!\geq\\!\tfrac{\Gamma_{th}\mathrm{{PL_{NLoS}}}}{{}^{\eta_{\mathrm{NLoS}}\mathrm{{EIRP\\!^{+}}}\\!G_{R}\\!}}\\!\bigg{)}\\!P_{\mathrm{\mathrm{NLoS}}}$
$\displaystyle\\!=\\!e^{-\Big{(}\frac{\Gamma_{th}\mathrm{{PL_{LoS}}}}{\eta_{\mathrm{LoS}}\mathrm{EIRP\\!^{+}}G_{R}}\Big{)}}\\!P_{\mathrm{\mathrm{LoS}}}\\!+\\!e^{-\Big{(}\frac{\Gamma_{th}\mathrm{{PL_{NLoS}}}}{\eta_{\mathrm{NLoS}}\mathrm{EIRP\\!^{+}}G_{R}}\Big{)}}\\!P_{\mathrm{\mathrm{NLoS}}}.$
(19)

Therefore, the coverage probability, $P_{\mathrm{cov}}$ in (13), can be
finally calculated by using (III-A) and (III-B).

##  IV Numerical Results

In this section, we calculate the coverage probability using the derived
equations to study the effect of recharging time and station density, maximum
battery size, and UAV speed. Unless otherwise stated, Table I provides the
parameter values, where $\eta_{\textrm{LoS}}$ and $\eta_{\textrm{NLoS}}$ are
for high-rise urban scenario.

Table I: Parameter values. Service-related |  |  |
---|---|---|---
$P_{T}$ |  $21$ [dBm] | $G_{T}$ |  $15$ [dBi]
$\theta_{B}$ |  $30.8$ [∘] | $f_{c}$ |  $868$ [MHz]
$c$ |  $3\times 10^{8}$ [m/s] | $\Gamma_{th}$ |  $1$ [$\mu$W]
Environment-related | [22] |  | [22]
$\eta_{\mathrm{LoS}}$ |  $1.6034$ [dB] | $\gamma$ | $27.1157$
$\eta_{\mathrm{NLoS}}$ |  $29.6462$ [dB] | $\delta$ | $0.1232$
UAV-related |  |  | [17]
$B_{\mathrm{max}}$ |  $770$ [Wh] | $P_{h}$ |  $168.48$ [W]
$V$ |  $10.36$ [m/s] | $P_{J}$ |  $126.395$ [W]
Others |  |  |
$\lambda_{Ch}$ |  $10^{-6}$ [m-2] | $G_{R}$ |  $9$ [dBi]
$h_{Ch}$ |  $100$ [m] | $h_{l}$ |  $80$ [m]

(a)

(b)

(c)

Figure 3: Coverage probability, $P_{cov}$, vs. recharging station density,
$\lambda_{Ch}$, for different $t_{Ch}$: (a) half-charged; (b) fully-charged;
(c) overflowed.

We first investigate the effect of varying recharging time on the coverage
probability for an increasing recharging station density. For this analysis,
using (3) for the considered $B_{\textrm{max}}$ and $\xi_{Ch}$ values, we know
that $t_{Ch}^{sat}$ should be $3600$ seconds, i.e., $1$ hour, which is typical
for commercial UAVs. Fig. 3(a) depicts what happens when the UAV stays at the
recharging station until its battery gets half-full gradually. Here, we
observe that, depending on the recharging level, certain discrepancies arise,
which are especially evident for lower densities of recharging stations. For
example, in the case of $0.001$ stations per km2, $P_{cov}$ gets $2.7$ times
better when $t_{Ch}$ is increased from $600$ to $1800$ seconds. This effect is
less apparent for higher $\lambda_{Ch}$ because finding a recharging station
becomes more likely for the UAV; hence, it may not need to depend heavily on
the charge in its battery due to the increased recharging possibility.
Numerically speaking, $P_{cov}$ increases only up to $1.3$ times for the same
increment in $t_{Ch}$ when $\lambda_{Ch}$ is ten times higher, i.e., $0.01$
per km2. When we look at the fully charged case shown in Fig.3(b), which can
be considered ideal in theory, we see that the performance gap increases for
lower-density values. Unsurprisingly, a fully-charged battery can help the UAV
achieve better coverage, especially at a low $\lambda_{Ch}$, compared to those
of partially-charged cases. Finally, in Fig.3(c), we analyze the case when the
battery overflowed, i.e., when the UAV continues to stay at the recharging
station after its battery gets fully charged. From (1), we know that
increasing $t_{Ch}$ beyond saturation is unsuggested since the service
probability, $P_{e}$, and hence $P_{cov}$, is inversely proportional to it.
However, Fig.3(c) reveals that this might not be the case always. As seen,
when the battery is charged for $4350$ seconds ($>\\!t_{Ch}^{sat}$), it is
still possible to obtain higher $P_{cov}$ compared to the partial charge
cases, depending on $\lambda_{Ch}$. For example, $t_{Ch}$ of $600$ seconds can
only outperform the overflowed case when $\lambda_{Ch}$ is $7.81\times
10^{-6}$m-2 or higher. Similar comments also hold for other $t_{Ch}$ values
(except for $t_{Ch}^{sat}$), as can be seen from the dashed red lines
perpendicular to the x-axis, where the $\lambda_{Ch}$ threshold significantly
reduces for increasing charge level towards the full battery, e.g., $2\times
10^{-9}$m-2 for $t_{Ch}$ of $2850$ seconds. The take-home message from this
analysis is that a relatively empty battery might be worse than overstaying at
the recharging station, e.g., for maintenance purposes, if the application
mandates a certain level of $\lambda_{Ch}$.
Next, we extend our discussion to the impact of using batteries of different
sizes on the coverage probability for varying charging times. As can be seen
from Fig. 4(a), in the case of a partial charge, all batteries show the same
performance until $t_{Ch}=1440$ seconds, i.e., when the battery of size
$308$Wh gets full. That is because all batteries are charged to the same level
at that $t_{Ch}$ value regardless of their total size, $B_{\textrm{max}}$. The
same phenomenon is also observed for the remaining batteries until the one
with the smallest size reaches its $t_{Ch}^{sat}$, where the respective
$P_{cov}$ starts its dramatic decay due to overflow. Fig. 4(a) and Fig. 4(b)
are for different densities of stations; $10^{-9}$ and $10^{-6}$ [m-2],
respectively. Although $t_{Ch}^{sat}$ of each battery remains the same, the
achievable $P_{cov}$ changes dramatically due to the clockwise rotation of the
behavior.

(a)

(b)

Figure 4: Coverage probability, $P_{cov}$, vs. recharging time, $t_{Ch}$, for
different $B_{\textrm{max}}$: (a) @$\lambda_{Ch}\\!=\\!10^{-9}\\!$ [m-2]; (b)
@$\lambda_{Ch}\\!=\\!10^{-6}\\!$ [m-2]. Figure 5: Coverage probability,
$P_{cov}$, as a function of UAV velocity, $V$, for varying $\lambda_{Ch}$
values.

We lastly focus on understanding how the UAV velocity affects the coverage
performance depending on the recharging station density. Before explaining the
results, we should note that Fig. 5 is produced for a $B_{\textrm{max}}$ of
$192.5$Wh, i.e., the quarter of what Table I mentions, which is only charged
until its half at the same rate. As seen, the impact of $V$ is negligible for
high $\lambda_{Ch}$ due to the same reason explained earlier in this section.
For low $\lambda_{Ch}$, on the other hand, the UAV should speed up to reach
the nearest station faster, although that means an increased $P_{J}$. Due to
the $P_{J}$ component, however, the increment in $V$ is only beneficial until
a certain point, as shown with the black dashed line marking the optimal
velocity values for each $\lambda_{Ch}$. Using a bigger battery and charging
it to the ideal level, i.e., $100\%$ -no overflow, will diminish this
behavior, forcing higher speeds only for low $\lambda_{Ch}$ values.

The results above suggest that for a given performance target, e.g., maximum
$P_{cov}$ with minimum $\lambda_{Ch}$, shorter $t_{Ch}$, or a fixed
$B_{\textrm{max}}$, the relevant design parameters can be tweaked as required.
Considering that most of these parameters affect each other, e.g., the EIRP
limit on $P_{T}$ alters not only $P_{\mathrm{LoS}}$ (due to the dictated
$\theta_{B}$) but also the size of the event area and hence the number of
sensors that can be powered (not within the scope of this study) and even
their reporting frequencies, the network requires a holistic design approach
as optimizing the trade-offs for the best performance achievable. We believe
this paper provides a practice-based showcase on this, providing guidance for
future efforts.

##  V Conclusions

This study investigates the factors affecting the coverage performance of
sensors energized by UAVs with directional antennae, constituting an energy-
neutral IoT network together with the multi-source EH recharging stations
replenishing UAVs batteries. With this goal, We first derived the service
probability as a function of UAV power consumption/velocity and battery size,
recharging time and station density, and WPT duration. That was then joined
with distance-conditioned coverage probability, incorporating the effects of
directivity, the LoS/NLoS connectivity, and the non-linear EH model. The
analyses revealed the design considerations for the best coverage with regard
to FCC regulations, realistic rectenna and UAV operation, and minimum power
requirements of sensors. Future works will focus on calculating the number of
sensors that can be powered by UAVs. We will also try maximizing the
communication throughput of sensors in the envisioned setting by optimizing
the parameters/trade-offs that have been discussed in this study.

##  VI Acknowledgement

This work has been supported by the PETRAS National Centre of Excellence for
IoT Systems Cybersecurity, funded by the UK EPSRC under grant number
EP/S035362/1.

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