# Sum-Rate Maximization for UAV-assisted Visible Light Communications using
NOMA: Swarm Intelligence meets Machine Learning

Quoc-Viet Pham, Thien Huynh-The, Mamoun Alazab, Jun Zhao, and Won-Joo Hwang
Quoc-Viet Pham is with the Research Institute of Computer, Information and
Communication, Pusan National University, Busan 46241, South Korea. Thien
Huynh-The is with the ICT Convergence Research Center, Kumoh National
Institute of Technology, Gyeongsangbuk-do, 39177 Korea. Mamoun Alazab is with
the College of Engineering, IT and Environment, Charles Darwin University,
Casuarina, NT 0810, Australia. Jun Zhao is with the School of Computer Science
and Engineering, Nanyang Technological University, 50 Nanyang Avenue, 639798
Singapore. Won-Joo Hwang is with the Department of Biomedical Convergence
Engineering, Pusan National University, Busan 46241, South Korea. Email:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

As the integration of unmanned aerial vehicles (UAVs) into visible light
communications (VLC) can offer many benefits for massive-connectivity
applications and services in 5G and beyond, this work considers a UAV-assisted
VLC using non-orthogonal multiple-access. More specifically, we formulate a
joint problem of power allocation and UAV’s placement to maximize the sum rate
of all users, subject to constraints on power allocation, quality of service
of users, and UAV’s position. Since the problem is non-convex and NP-hard in
general, it is difficult to be solved optimally. Moreover, the problem is not
easy to be solved by conventional approaches, e.g., coordinate descent
algorithms, due to channel modeling in VLC. Therefore, we propose using harris
hawks optimization (HHO) algorithm to solve the formulated problem and obtain
an efficient solution. We then use the HHO algorithm together with artificial
neural networks to propose a design which can be used in real-time
applications and avoid falling into the “local minima” trap in conventional
trainers. Numerical results are provided to verify the effectiveness of the
proposed algorithm and further demonstrate that the proposed algorithm/HHO
trainer is superior to several alternative schemes and existing metaheuristic
algorithms.

###### Index Terms:

Artificial Neural Network, Harris Hawk Optimization, Non-Orthogonal Multiple
Access, Unmanned Aerial Vehicles, Sum-Rate Maximization, Visible Light
Communications, Swarm Intelligence.

##  I Introduction

Visible light communications (VLC), unmanned aerial vehicles (UAV), and non-
orthogonal multiple access (NOMA) are envisioned as three key technologies in
5G and beyond [1, 2, 3]. On the one hand, VLC using light-emitting diodes
(LEDs) has emerged as a promising technology in optical wireless communication
and recently received much attention from both industry and academia due to
lots of offered advantages, such as easy installation, high data rate
communication, low cost and power consumption, and high security [1, 4]. On
the other hand, NOMA has been regarded as one of the key-enabler technologies
in 5G networks and beyond. Unlike orthogonal multiple access (OMA)
technologies, NOMA can serve many users using the same time-frequency
resource. Thus NOMA is suitable for applications with massive connectivity,
e.g., the Internet of Things (IoT) [5, 2]. Besides VLC and NOMA, UAV has been
recently exploited as a means to boost the capacity and coverage of wireless
networks [3]. Two key features of UAV are high mobility and the provision of
line-of-sight (LoS) connections.

###  I-A Related Works

Researches on VLC and NOMA have recently attracted significant attention. The
first integration between NOMA and downlink VLC was studied in [1] to enhance
the achievable data rate. The authors discussed some reasons for a good
interplay between NOMA and VLC. For instance, NOMA performs well at high
signal-to-noise ratio (SNR), which can be guaranteed in VLC systems thanks to
the short distance from the LED transmitter to users and LoS propagation.
Moreover, the channels stay almost constant in VLC systems, which is vital to
NOMA functionalities in performing successive interference cancellation (SIC)
operation. Besides, the LED as a small access point can provide services to a
few users, while NOMA typically multiplexes a few number of users in a
cluster. Moreover, motivated by the fact that users with poor channel
conditions should utilize higher transmit power due to severe inter-cluster
interference, a gain ratio power allocation (GRPA) scheme was proposed in [1].
The sum logarithmic data rate for NOMA-enabled downlink VLC networks was
considered in [6]. Power allocation schemes were investigated in [7, 8] to
maximize the sum rate of all the users. These studies showed that VLC systems
using NOMA can offer higher spectral efficiency compared with OMA
counterparts. An experimental demonstration of VLC systems using optical
power-domain NOMA was recently reported in [9].

The integration of NOMA and UAV has been recently studied as a key-enabler
technology for massive connectivity and coverage enhancement in 5G and beyond.
In [10], a joint optimization problem of bandwidth allocation, power
allocation, UAV altitude, and antenna beamwidth was formulated for NOMA-
enabled UAV downlink networks, which was solved using a path-following
algorithm. It was shown in [11] that NOMA can be applied for uplink cellular
networks to lower the UAV completion time while satisfying the quality of
service (QoS) requirement of ground users (GUs). A NOMA random access scheme
for uplink UAV-assisted wireless communications was studied in [12], where the
proposed access algorithm is compared with the existing slotted ALOHA. Joint
optimization of UAV’s placement and power allocation was considered in [13]
and [14] for the network sum rate and energy efficiency, respectively. NOMA
precoding vectors at the BS and UAV were optimized for UAV-assisted cellular
communications in [15] and for cellular-connected UAV networks in [16].
Recently, the authors in [17] developed an iterative algorithm to minimize the
total power consumption by optimizing user association and UAV location.

To optimize UAV/NOMA/VLC systems, there have been some tools like network
optimization [10, 15, 6], game theory [5], and machine learning (ML) [18].
Especially, due to the emergence of new applications and technologies, ML has
found many applications in 5G wireless and communication networks. For
example, deep learning was considered in [18] for an extended version of the
previous work [17], federated learning was used in [19, 20] for several
problems at the wireless edge, and deep reinforcement learning was
comprehensively reviewed in [21], where it can be used to solve many problems,
e.g., wireless caching, edge computing, and network security. Motivated by a
competitive performance with high reliability and fast convergence, swarm
intelligence is considered as a key approach for optimizing the 5G and beyond
networks. Harris Hawks Optimizer (HHO), as a swarm intelligence technique, is
one of the most recent algorithms that has been popular since the proposal
[22]. It has been considered in many engineering problems, e.g., the control
chart patterns recognition for the manufacturing industry in [23], the design
of microchannel heat sinks for the minimization of entropy generation in [24],
and resource allocation in wireless networks [25, 26], which all show
supervisor performance of the HHO algorithm. In the following parts of this
paper, we will show that HHO is considered suitable for our problem and can
achieve competitive performance compared with several state-of-the-art
alternatives.

###  I-B Motivation and Contributions

To the best of our knowledge, this work is the first attempt to study the
integration of UAV, NOMA, and VLC, which can offer many benefits compared with
existing studies. First, a UAV-assisted VLC network with NOMA can provide
illumination and communication services simultaneously for more users than VLC
with conventional OMA schemes, thus realizing massive connectivity
requirements for 5G and beyond. Second, UAVs using radio frequency (RF)
resources for communications are energy-consuming, which is opposed to the
fact that UAVs typically have a finite battery. This challenge can be
addressed by powering UAVs through energy harvesting and wireless power
transfer [27], and now can be further improved by equipping UAV with VLC
capabilities. Next, UAVs possess the features of high mobility and flexibility
to ensure LoS wireless links for VLCs. Therefore, the SIC operation in NOMA
can be enhanced and the network performance increases accordingly. Finally,
some concepts and testbed experiments have been conducted to verify the
applications and practicability of UAVs in VLCs [28, 29].

In this work, we consider a UAV-assisted VLC network using NOMA, where a UAV
is equipped with a LED to provide both illumination and communications, and
downlink transmissions between the UAV and GUs follow the NOMA principle. In
summary, features and contributions offered by our work can be summarized as
follows:

  * •

We formulate an optimization problem to maximize the sum rate of all GUs
subject to constraints on power allocation (e.g., budget at the UAV, peak
optical intensity, and non-negative signals), QoS requirement of GUs, and
UAV’s placement.

  * •

Since the formulated optimization problem is non-convex and NP-hard in
general, we propose using a swarm algorithm, namely HHO [22], to solve the
problem. Compared with conventional methods such as gradient-based approaches,
the HHO does not depend on gradient information of the objective functions and
constraints, and is easy to implement because of its simplicity. Moreover, the
HHO is capable of keeping a balance between exploration and exploitation
phases, so it can avoid the possibility of being trapped in local optima and
can be considered as a global optimizer.

  * •

Moreover, common approaches, e.g., path-following procedure and coordinate
descent method, are not applicable to solve the problem considered in this
paper. Our proposed HHO-based algorithm finds solutions for UAV’s placement
and power allocation simultaneously, whereas the coordinate descent method
needs to decompose the original problem into subproblems and solve them
alternatively in an iterative manner. For example, [15] proposed optimizing
the joint trajectory and precoding problem by alternatively solving the user
scheduling and trajectory optimization. Besides, successive convex
approximation (SCA) technique is usually used to solve such subproblems (e.g.,
power allocation and UAV trajectory) [16, 10, 11]. However, the SCA technique
is workable only if the channel gains follow an exponential-distance model,
i.e., the channel gain is given as $h=\psi d^{\alpha/2}$, where $\psi$ is a
constant, $d$ is the distance, and $\alpha$ is the path loss exponent.
Therefore, this technique is not amenable to channel models in VLC, which will
be presented in Section II-A. Our proposed HHO-based algorithm relaxes this
assumption and is thus suitable for channel modeling in VLC.

  * •

For the first time, we adopt the HHO algorithm to optimize the connection
weights and biases of artificial neural networks (ANNs). The inputs to the ANN
are the locations of GUs, while the outputs are the UAV’s placement and power
allocation. Our proposed trainer is named HHOFNN, can avoid local solutions as
in conventional algorithms (e.g., back propagation and conjugate gradient),
and is able to achieve higher sum-rate than well-known metaheuristic
optimizers: particle swarm optimization (PSO), evolution strategy (ES), and
genetic algorithm (GA).

  * •

Through numerical simulation results, it is shown that the proposed HHO for
power allocation and placement (HHOPAP) algorithm outperforms several existing
algorithms under various simulation settings, including two NOMA schemes -
gain ratio power allocation (GRPA) proposed in [1] and random placement
(RandP), and a conventional orthogonal frequency-division multiplexing access
(OFDMA) scheme.

The remaining of this paper is organized as follows. The system model and the
problem formulation are presented in Section II. The HHO-based algorithm is
proposed in Section III. Our proposed trainer for ANNs is explained in Section
IV. In Section V, numerical results are provided and discussed. Finally,
conclusions are drawn in Section VI.

Figure 1: Illustration of a VLC network with UAV and NOMA.

##  II Network Setting and Problem Formulation

###  II-A Network Setting

We consider a network setting as in Fig. 1, where the rotary-wing
UAV111Generally, UAVs can be classified as rotary-wing UAVs and fixed-wing
UAVs. Rotary-wing UAVs have the ability to hover, whereas fixed-wing UAVs need
to move forward to remain aloft. Extension of the current work to consider
rotary-wing UAVs is interesting and will be considered in our work. is
equipped with one LED transmitter to provide illumination and communication
for $N$ randomly positioned GUs under its coverage area. Denote by
$\mathcal{N}=\\{1,\dots,N\\}$ the set of GUs and by $h_{i}$ the channel gain
between the UAV and GU $i$. We assume that the UAV-mounted LED transmitter
simultaneously transmits the data to $N$ GUs using the same time-frequency
resource by appropriately adjusting its transmit power (i.e., power-domain
NOMA). GUs apply the successive interference cancellation (SIC) technique to
remove the messages transmitted for other GUs. In particular, in downlink
NOMA, the decoding order follows the increasing order of the channel gains.
Without loss of generality, we assume that $h_{1}<\dots<h_{N}$, i.e., the
first and last users are considered as the weakest and strongest GUs,
respectively. With this assumption, GU $i$ first decodes the message
transmitted for GU $1$ and subtracts it from the received signal, until GU
$(i-1)$.

For the channel model, we consider the LoS propagation path and use the
generalized Lambertian emission model, where the channel direct current (DC)
gain is proportional to the inverse of the squared distance between the LED
transmitter and user [30, Eq. (10)]. Specifically, the DC channel gain of GU
$i$ is expressed by

$h_{i}=\frac{A_{i}}{d_{i}^{2}}R_{0}(\varphi_{i})T_{s}(\phi_{i})g(\phi_{i})\cos(\phi_{i})$
(1)

for $0\leq\phi_{i}\leq\Phi_{i}$ and $h_{i}=0$ otherwise, where $\Phi_{i}$ is
the field of view (FoV) at GU $i$, $\varphi_{i}$ and $\phi_{i}$ are the angles
of irradiance and incidence, respectively. In (1), $A_{i}$ is the detection
area, $d_{i}$ the distance between the UAV/LED transmitter and GU $i$,
$T_{s}(\phi_{i})$ is the gain of the optical filter, $g(\phi_{i})$ denotes the
concentrator gain and is given by

$g(\phi_{i})=\frac{n^{2}}{\sin^{2}\Phi_{i}},\;0\leq\phi_{i}\leq\Phi_{i},$ (2)

with $n$ being the internal refractive index. Also, $R_{0}(\varphi_{i})$ is
the Lambertian radiant intensity and is given as

$R_{0}(\varphi_{i})=\frac{m+1}{2\pi}\cos^{m}\varphi_{i}$ (3)

where $m$ is related to the transmitter semiangle at half power and $m=-\ln
2/\ln\left(\cos\varphi_{1/2}\right)$. In this paper, we fix the value of
$\varphi_{1/2}$ to $60^{\circ}$, i.e., $m=1$.

We assume that the UAV, flying within a disc with the radius of $R$, is fixed
at the altitude $h$. As we know from the literature on VLC, VLC can be used
for location prediction, so we can assume that the UAV knows perfectly the
locations of GUs. All GUs are positioned at the ground (their altitude is
zero). The distance between the UAV and GU $i$ is given as

$d_{i}=\sqrt{\left(x_{u}-x_{i}\right)^{2}+\left(x_{u}-x_{i}\right)^{2}+h^{2}},$
(4)

where $(x_{u},y_{u},h)$ and $(x_{i},y_{i},0)$ are the coordinate vectors of
the UAV and GU $i$, respectively. We note that the values of the angles of
irradiance and incidence can be achieved from
$\cos\varphi_{i}=\cos\phi_{i}=h/d_{i}$.

Denote by $p_{i}$ and $s_{i}$ the transmit allocated power and the message for
GU $i$, respectively. The composite signal at the UAV can be expressed as
follows:

$s=\sum\nolimits_{i\in\mathcal{N}}\sqrt{p_{i}}s_{i}+A,$ (5)

where $A$ is a DC offset/bias added to ensure the positive instantaneous
intensity of the transmitted signal. To maintain non-negativity of the
transmitted signal $s$, the following constraint has to be satisfied:

$\sum\limits_{i\in\mathcal{N}}\sqrt{p_{i}}\leq\frac{A}{\delta},$ (6)

where $\delta$ is a coefficient, which is determined by modulation order of
the pulse amplitude modulation (PAM) employed in the network. Power allocation
of the UAV should be limited due to its power budget and eye safety [30]. To
consider eye safety, the transmitted optical intensity should be limited by a
peak optical intensity $B$ [6], as the following constraint

$\sum\limits_{i\in\mathcal{N}}\sqrt{p_{i}}\leq\frac{B-A}{\delta},$ (7)

According to the downlink NOMA principle, the achievable data rate of GU $i$
can be given as follows [2, 31]:

$R_{i}=\log_{2}\left(1+\frac{h_{i}p_{i}}{n_{0}+\sum\limits_{j\in\mathcal{N}:h_{j}>h_{i}}h_{i}p_{j}}\right),$
(8)

where $p_{i}$ is the transmit power allocated to GU $i$, $h_{i}$ is the
channel gain between the UAV and GU $i$, and $n_{0}$ is the noise power at
GUs. The NOMA rate in (7) is attained through implementations of the
superposition of signals at the transmitter side (i.e., BS) and successive
interference cancellation (SIC) at the receiver side (i.e., GUs). To perform
SIC operation at GUs successfully, the transmit power for each GU should be
assigned properly, which needs to satisfy the following constraints [32]:

$p_{i}\bar{h}_{i+1}-\sum\nolimits_{j=i+1}^{N}p_{j}\bar{h}_{i+1}\geq\theta,\;i=1,\cdots,N-1,$
(9)

where $\bar{h}_{i}=h_{i}/n_{0},\;\forall i\in\mathcal{N}$ and $\theta$ is the
minimum power difference required to distinguish between the signal to be
decoded and the remaining non-decoded signals.

###  II-B Problem Formulation

With the objective of maximizing the sum rate of all GUs222Note that we do not
limit this study to a specific objective, which means other objective
functions can be used instead of the sum rate in this paper. Moreover,
fairness among GUs can be studied by considering a well-known family of
$\alpha$-utility functions that has been discussed in [33]. , the optimization
problem can be formulated as follows:

$\displaystyle\underset{\\{\boldsymbol{w},\boldsymbol{p}\\}}{\max}$
$\displaystyle\sum\limits_{i\in\mathcal{N}}\log_{2}\left(1+\frac{h_{i}p_{i}}{n_{0}+\sum\limits_{j\in\mathcal{N}:h_{j}>h_{i}}h_{i}p_{j}}\right)$
(10a) s.t. $\displaystyle p_{i}\geq 0,\forall i\in\mathcal{N},$ (10b)
$\displaystyle\sum\nolimits_{i\in\mathcal{N}}p_{i}\leq P_{\max},$ (10c)
$\displaystyle\sum\nolimits_{i\in\mathcal{N}}\sqrt{p_{i}}\leq C,$ (10d)
$\displaystyle
p_{i}\bar{h}_{i+1}-\sum\limits_{j=i+1}^{N}p_{j}\bar{h}_{i+1}\geq\theta,\;i=1,\cdots,N-1,$
(10e) $\displaystyle R_{i}\geq R_{i}^{\text{th}},\;\forall i\in\mathcal{N},$
(10f) $\displaystyle x_{u}^{2}+y_{u}^{2}\leq R^{2},$ (10g)

where $\boldsymbol{w}=\\{x_{u},y_{u}\\}$ (placement vector),
$\boldsymbol{p}=\\{p_{1},\dots,p_{N}\\}$ (power allocation), and
$C=\delta^{-1}\min\\{A,B-A\\}$. The optimization variables
$(\boldsymbol{w},\boldsymbol{p})$ are subject to several constraints. First,
the transmit power is allocated to GUs on condition that the UAV power
limitation and eye safety are guaranteed, as illustrated in (10b)-(10e). Next,
the QoS requirements of GUs are imposed in (10e). Finally, the constraint
(10f) indicates that the UAV should be deployed within a disc with the radius
of $R$.

As the sum-rate maximization problem is non-convex NP-hard in general [33],
the problem (10) is difficult to be solved optimally. Moreover, the channel
model in VLC systems, illustrated in (1), makes conventional convex
approximation techniques not a proper tool to solve the problem (10). In the
next section, we will present the fundamentals of the HHO and utilize it as an
optimizer of the problem (10).

##  III Harris Hawks Optimization

The HHO is a population-based metaheuristic algorithm, which was proposed by
Heidari [22] in 2019 based on the cooperative behaviors Harris’ Hawks in
hunting escaping preys, e.g., rabbits, mice, voles, and squirrels. Originally,
the HHO can be applied to any continuous and unconstrained optimization
problem. Mathematically, the hunting behavior of Harris’ hawks can be modeled
in three phases: 1) exploration, 2) exploitation, and 3) transition between
exploitation and exploration. However, since there are several constraints,
the problem (10) should be adjusted accordingly. Thus, we will also present a
constraint-handling technique, which is to transform the problem (10) into a
proper form.

###  III-A Exploration phase

In this phase, Harris’ hawks wait, observe, and minor the prey by their
powerful eyes. If the hawks detect the prey, they can use two strategies to
attack. In the first strategy, each hawk selects another random one from the
family and decides the location to peach accordingly, whereas in the second
strategy, the hawks perch on some high positions based on the cooperation with
all the other family members. These strategies are considered with equal
probability. In other words, if we consider a random $q$ uniformly distributed
in $[0,1]$, the first and second strategies are selected under the conditions
$q<0.5$ and $q\geq 0.5$, respectively. Mathematically, these strategies are
expressed by the following system.

$\displaystyle
X(t+1)=X_{r}(t)-r_{1}\left|X_{r}(t)-2r_{2}X(t)\right|,\;\text{if }q\geq 0.5,$
(11) $\displaystyle X(t+1)=(X_{p}(t)-X_{a}(t))-r_{3}(LB+r_{4}(UB-LB)),$
$\displaystyle\phantom{cccccccccccccccccccccccccccccccccccccccccc}\text{if
}q<0.5,$ (12)

where $X(t)$ is the position vector of the hawks in the iteration $t$ (i.e.,
$X(t+1)$ is the position vector in the next iteration) and $X_{p}(t)$ is the
position of the prey, which is considered as the solution in the iteration
$t$. Here, $r_{1}$, $r_{2}$, $r_{3}$, $r_{4}$, and $q$ are numbers in the
range $[0,1]$ that are created randomly in each iteration, $LB$ and $UB$
denote the upper and lower bounds of variables, respectively. In our problem,
$UB$ and $LB$ of power allocation are set to be $P_{\max}$ and $0$,
respectively. Similarly, those are $R$ and $0$ for the UAV’s coordinate.

In (12), $X_{r}(t)$ is the position vector of a hawk selected randomly from
the current population and $X_{a}$ is the average position of the current
population of hawks, which is given as follows:

$X_{a}(t)=\frac{1}{N}\sum_{i=1}^{N}X_{i}(t),$ (13)

where $X_{i}(t)$ is the location of the $i$-th hawk in the iteration $t$ and
$N$ is the population dimensionality. Thanks to the use of different random
coefficients (in each iteration), the solution space can be diversified and
thus the HHO algorithm can avoid the trap into local optima.

###  III-B Transition from exploration to exploitation

To model the transition from exploration to exploitation, the energy of the
prey is given as

$E=2E_{0}(1-\frac{t}{T}),$ (14)

where $t$ denotes the iteration index, $T$ is the maximum number of
iterations, and $E_{0}$ is the initial energy of the prey, which is created
randomly inside the range $[-1\;1]$ in each iteration. When $E_{0}$ decreases
from $0$ to $-1$, the prey is physically weakening; otherwise, the prey is
strengthening.

When the value of $E_{0}$ decreases from $0$ to $-1$, the rabbit is physically
flagging, whilst when the value of $E_{0}$ increases from 0 to 1, it means
that the rabbit is strengthening. Clearly, the value of $E$ tends to decrease
over the course of iterations. When $|E|\geq 1$, the exploration phase gets
started, i.e., the algorithm tries to explore different regions in the
solution space, whilst the exploitation phase is carried out when $|E|<1$,
i.e., the HHO algorithm exploits the neighborhood of solutions to update the
position vectors for the next iteration. An example of the escaping energy $E$
is showed in Fig. 2, where the maximum number of iterations $T$ is set to be
$500$.

Figure 2: Evolution of the escaping energy over 500 iterations.

###  III-C Exploitation phase

In this phase, four attacking strategies are modeled according to the escaping
probability of the prey and the chasing patterns of the hawks. Here, we define
$r$ as the escaping probability of the prey, where $r\geq 0.5$ and $r<0.5$
indicate that the prey escapes from the attack unsuccessfully and
successfully, respectively. Depending on the retained energy $E$, the
algorithm decides to besiege either hard or softly. Particularly, when
$|E|\geq 0.5$, the soft siege occurs, whereas when $|E|<0.5$, the hard besiege
happens.

####  III-C1 Soft besiege

This tactic takes place when $r\geq 0.5$ and $|E|\geq 0.5$, i.e., the prey has
enough energy to escape but finally it fails to do, and the hawks encircle the
prey softly to make it exhausted. This process is stimulated as follows:

$\displaystyle X(t+1)=\Delta X(t)-E\left|JX_{p}(t)-X(t)\right|,$ (15)
$\displaystyle\Delta X(t)=X_{p}(t)-X(t).$ (16)

Here, $\Delta X(t)$ represents the difference between the position vectors of
the prey and hawks in the iteration $t$, $J=2(1-r_{5})$ indicates the jump
strength the prey, and $r_{5}$ is a random number inside the range $[0,1]$.
Similar to random coefficients mentioned earlier, the value of $r_{5}$ (and
$J$) is updated in each iteration to imitate the sudden movement of the prey.

####  III-C2 Hard besiege

This hunting tactic happens when $r\geq 0.5$ and $|E|<0.5$, , i.e., the prey
does not have sufficient energy to escape and the hawks performs a hard siege
before a surprise pounce (also known as seven kills, i.e., several hawks come
from different directions and attack the prey [34]). In this tactic, the
position vector of the hawks is updated as

$X(t+1)=X_{p}(t)-E\left|\Delta X(t)\right|,$ (17)

where $\Delta X(t)$ is given in Eq. (16).

####  III-C3 Soft besiege with progressive rapid dives

This tactic happens when $|E|\geq 0.5$ and $r<0.5$, i.e., the prey has enough
energy and escapes successfully, and the hawks softly besiege the prey to
finally perform the surprise pounce. Inspired by the behavior of Harris’s
hawks in nature, it is supposed that the hawks can progressively select the
best possible dive when they want to grab the prey. In this case, the next
move of the hawks can be expressed as follows:

$Y=X_{p}(t)-E\left|JX_{p}(t)-X(t)\right|.$ (18)

Additionally, The hawks perform a corresponding dive when they observe a
deceptive movement from the prey. To model such leapfrog movements, the Lévy
flight (LF) concept is utilized in the HHO algorithm. An LF is a special type
of arbitrary walk, where the step lengths follow a heavy-tailed probability
distribution [35]. Specifically, the LF-based patterns can be attained by the
following equation

$Z=Y+S\times LF(D),$ (19)

where $D$ is the population dimensionality and $S$ is a random vector of size
$D$, and $LF(\cdot)$ denotes the LF function. According to [22, 36], the LF
walk is modeled as follows:

$LF(x)=0.01u\sigma/\left|v\right|^{1/\beta},$ (20)

where $u$ and $v$ are random values inside the range $[0,1]$, $\beta$ is
generally set to be $1.5$, and $\sigma$ is given as

$\sigma=\left(\frac{\Gamma(1+\beta)\sin(\pi\beta/2)}{\Gamma((1+\beta)/2)\beta\times
2^{(\beta-1)/2})}\right)^{\frac{1}{\beta}}.$ (21)

Here, $\Gamma(\cdot)$ denotes the standard _Gamma_ function and is expressed
as $\Gamma(z)=\int_{0}^{\infty}t^{z-1}e^{t}dt$.

The final step of this strategy is updating the position vectors of the hawks,
which follows the rule below

$X(t+1)=\begin{cases}Y,&\text{if }F(Y)>F(X(t)),\\\ Z,&\text{if
}F(Z)>F(X(t)),\\\ \end{cases}$ (22)

where $F(\cdot)$ represents the fitness function. In the context of our
problem, the fitness value is the total sum rate of all GUs plus the penalty
imposed by the constraints, which will be shown in the following subsection.

####  III-C4 Hard besiege with progressive rapid dives

This hunting tactic happens when $|E|<0.5$ and $r<0.5$. In this case, the prey
does not have enough energy to escape and the hawks carry out a hard siege to
finally kill the prey. This tactic is similar to the soft besiege with
progressive rapid dives except that the next move of the hawks is evaluated
using the following rule

$Y=X_{p}(t)-E\left|JX_{p}(t)-X_{a}(t)\right|,$ (23)

where the average position vector $X_{a}$ is calculated according to Eq. (13).
Different from (18), where the hawks try to reduce the distance between the
position of the prey and their current locations, in (23) the hawks attempt to
decrease the distance between the position of the prey and the average
location of Harris’ hawks. Hence, the LF-based patterns and the next positions
are updated using (19) and (22), respectively.

In summary, details of the HHO algorithm are summarized in Alg. 1. As a
population-based approach, a number of solutions evolve during the
optimization process. In the experiment, we set $S=30$, i.e., the population
size is $30$. Among solutions in the current population, the one that achieves
the best fitness value is considered as the best solution, i.e., the position
of the prey $X_{p}$ (line 6). Either of two phases, i.e., exploration and
exploitation, is executed in an iteration, and the algorithm runs at most $T$
iterations (line 4). Using numerical results, we will show that the algorithm
can converge to a final solution after a few tens of iterations. Moreover,
thanks to a good balance between exploration and exploitation capabilities,
the HHO algorithm can be regarded as an efficient global optimizer.

###  III-D Mechanism to handle constraints

The HHO in its original form was proposed for meta-heuristically solving
unconstrained optimization problems. Since there are several constraints in
our problem formulated in (10), a natural question is how to incorporate these
constraints. In fact, there are mechanisms to deal with constraints [37],
e.g., penalty method, decoders, feasibility rules, stochastic ranking,
$\epsilon$-constrained method, multi-objective approach, and ensemble of
constraint-handling techniques. As the most common approach in the
evolutionary algorithm community, the penalty method is presented and used in
this work. For more details of other methods, we invite interested readers to
read [38, 39], and references therein.

The main idea of the penalty method is to define a fitness function so that
the constrained problem (10) is transformed into an unconstrained problem. In
particular, the penalty function can be expressed as follows:

$F(\boldsymbol{X})=\sum\limits_{i\in\mathcal{N}}\log_{2}\left(1+\frac{h_{i}p_{i}}{n_{0}+\sum\limits_{j\in\mathcal{N},j\neq
i}h_{i}p_{j}}\right)+P(\boldsymbol{X}),$ (24)

where $P(\cdot)$ is the penalty and
$\boldsymbol{X}=\\{x_{u},y_{u},p_{1},p_{2},\dots,p_{N}\\}$ is the optimization
vector, which has the size of $(2+N)$. The penalty term can be calculated as
follows:

$\displaystyle P(\boldsymbol{X})=$
$\displaystyle-\mu_{1}\left(\sum\limits_{i\in\mathcal{N}}p_{i}-P_{\max}\right)^{2}H\left(\sum\limits_{i\in\mathcal{N}}p_{i}-P_{\max}\right)$
$\displaystyle-\mu_{2}\left(\sum\limits_{i\in\mathcal{N}}\sqrt{p_{i}}-C\right)^{2}H\left(\sum\limits_{i\in\mathcal{N}}\sqrt{p_{i}}-C\right)$
$\displaystyle-\sum\limits_{i=1}^{N-1}\mu_{i+2}\left(\theta-
p_{i}\bar{h}_{i+1}+\sum\limits_{j=i+1}^{N}p_{j}\bar{h}_{i+1}\right)^{2}$
$\displaystyle\;\;\;\times H\left(\theta-
p_{i}\bar{h}_{i+1}+\sum\limits_{j=i+1}^{N}p_{j}\bar{h}_{i+1}\right)$
$\displaystyle-\sum\limits_{i=1}^{N}\mu_{i+N+1}\left(R_{i}^{\text{th}}-R_{i}\right)^{2}H\left(R_{i}^{\text{th}}-R_{i}\right)$
$\displaystyle-\mu_{2N+2}\left(x_{u}^{2}+y_{u}^{2}-R^{2}\right)^{2}H\left(x_{u}^{2}+y_{u}^{2}-R^{2}\right),$
(25)

where $\mu_{j},\;j=\\{1,\dots,2N+2\\}$ are position constraints called penalty
factors. Here, $H(f_{j}(x))$ denotes the indicator function of
$f_{j}(x),\;j=\\{1,\dots,2N+2\\}$. More specifically, $H(f_{j}(x))=0$ if
$f_{j}(x)\leq 0$, whereas $H(f_{j}(x))=1$ if $f_{j}(x)>0$. The minus sign in
the penalty term (III-D) is used since we consider maximization of the total
sum rate. If the problem (10) is modeled as a minimization problem, the
positive sign should be used instead.

It is worth mentioning that the penalty method is well applicable to many
problems; however, selecting appropriate penalty factors turns out to be
problem-specific. If the penalty factors are too small, an infeasible may not
get enough penalty. Thus, a feasible solution may be evolved in the
optimization process. If too large values are used, a feasible solution can be
of low quality and exploration over infeasible spaces is desirable. According
to [39], the penalty factors $\mu_{j},\;j=\\{1,\dots,2N+2\\}$ are typically in
the range of $10^{13}$ to $10^{15}$ for most problems. For experiments in this
paper, we set all the penalty factors to be $10^{14}$ for the sake of
simplicity.

In Alg. 1, the computation complexity of calculating the fitness values is
$\mathcal{O}(SD)$, where $S$ is the population size and $D$ is the dimension
of a hawk. In the context of our problem in (10), there are $2$ coordinates
$\boldsymbol{w}=\\{x_{u},y_{u}\\}$ and $N$ transmit power values corresponding
to $N$ GUs, thus $D=2+N$. Updating the position vectors of all the hawks
requires a computational complexity of $\mathcal{O}(ND)$. Moreover, the
computation of index functions for $(2N+2)$ inequality constraints requires
$\mathcal{O}(S(2N+2))$ times and the algorithm runs at most $T$ iterations. As
a results, in solving our problem formulated in (10) the HHO in Alg. 1 has the
computational complexity level of $\mathcal{O}(ST(D+2N+2))$.

Alg. 1 Pseudo-code of the HHO algorithm

1:Inputs: The population dimensionality $S$ and the maximum number of
iterations $T$.

2:Outputs: The position of the prey $X_{p}$ and corresponding fitness value.

3:Initialize the random population $X_{i}\;(i=1,2,\ldots,S)$, and set the
iteration index $t=0$.

4:while $(t\leq T)$ do

5: Increase the iteration index $t=t+1$ and calculate the cci fitness values
of the hawks.

6: Select $X_{p}$ with the highest fitness value as the position of the prey.
---

7: for $(i=1:S)$ do

8: Update the initial energy as $E_{0}=2\text{rand()}-1$.

9: Update the jump strength as $J=2(1-\text{rand()})$ and cccccc escaping
energy $E$ via Eq. (14).

10: if ($|E|\geq 1$) then $\triangleright$ Exploration phase

11: Create a random number $q$ in $[0,1]$.

12: if ($q\geq 0.5$) then

13: Update the position $X_{i}(t+1)$ via Eq.(11).

14: else if ($q>0.5$) then

15: Update the position $X_{i}(t+1)$ via Eq.(12).

16: end if

17: end if

18: if ($|E|<1$) then $\triangleright$ Exploitation phase

19: Create a random probability $r$.

20: if ($r\geq 0.5$ and $|E|\geq 0.5$ ) then

21: Update the position $X_{i}(t+1)$ via Eq. (15).

22: else if ($r\geq 0.5$ and $|E|<0.5$ ) then

23: Update the position $X_{i}(t+1)$ via Eq. (17).

24: else if ($r<0.5$ and $|E|\geq 0.5$ ) then

25: Calculate the next move via (18) and ccccccccccccc update the position
$X_{i}(t+1)$ via Eq. (22).

26: else if ($r<0.5$ and $|E|<0.5$ ) then

27: Compute the next move via (23) and update ccccccccccccc the position
$X_{i}(t+1)$ via Eq. (22).

28: end if

29: end if

30: end for

31:end while

32:Return The position of the prey $X_{p}$

##  IV An HHO-based Optimizer for Artificial Neural Networks

In this section, we propose an optimizer for ANNs based on the HHO algorithm
and then apply it for joint power allocation and UAV’s placement in UAV-
assisted VLC systems using NOMA. The input to our proposed feedforward neural
network (FNN) is only the normalized locations of all GUs and the outputs are
normalized power allocation and UAV’s placement vectors. After the training,
the UAV’s placement and transmit power can be computed in real-time via one-
to-one mapping from the normalized locations of GUs using the trained FNN.

###  IV-A Feedforward Neural Networks for Joint Power Allocation and UAV’s
Placement

FNNs are computational models consisting of many neurons (nodes), which are
organized in layer-by-layer basis. The first and last layers are called the
input layer and output layer, respectively, whereas ones between the input and
output layers are called hidden layers. The term “feedforward” in FNN means
that the input being evaluated $x$ goes through intermediate computations to
finally reach the output $y$, and there is no feedback connections from the
output [40]. Thanks to the structural representation, a single hidden layer
FNN with a finite number of neurons has the capabilities to approximate any
continuous function [41].

Figure 3: An FNN for joint power allocation and UAV’s placement from a set of
GUs’ locations, where the input layer has $I$ input nodes, the hidden layer
has $H$ activation functions, and the output layer has $O$ nodes.

The illustration of a fully-connected FNN with only one single hidden layer
for a VLC system is shown in Fig. 3. Mathematically, the output $y_{j}$ of a
node (indicated as $j$) in the hidden layer can be computed as follows:

$z_{j}=\varphi_{i}\left(S_{j}\triangleq\sum\limits_{i=1}^{I}w_{ij}x_{i}+b_{j}\right),$
(26)

where $I$ is the number of input nodes, $x_{i}$ is the $i$-th input, $w_{ij}$
is the connection weight from the $i$-th node in the input layer to the $j$-th
node in the hidden layer, $b_{j}$ is the bias of the $j$-th hidden node, and
$\varphi_{j}(\cdot)$ is the activation function of the $j$-th hidden node. As
mentioned earlier, the optimization vector has the size of $O=2+N$ (i.e., the
number of neurons in the output layer is $N+2$), and the input is the
locations of GUs, thus the input size is $I=2\times N$. Since there is no
standard method for selecting the number of neurons in the hidden layer, we
adopt a rule-of-thumb method [42], which advises that the number of neurons in
the hidden layer should be between the numbers of neurons in the input and
output layers. Additionally, the number of GUs supported simultaneously by
NOMA is typically not less than two (i.e., $N\geq 2$), so we fix the number of
neurons in the hidden layer as the size of the input layer, i.e., $H=2+N$.

Many activation functions can be used in the ANN, which is to limit the output
amplitude of a neuron. Some commonly used activation functions are uni-polar
sigmoid function, bipolar sigmoid function, hyperbolic tangent function, and
rectified linear unit (ReLU) function. For example, the uni-polar sigmoid
function can be expressed as [40]

$z_{j}=\varphi_{j}(S_{j})=\frac{1}{1+\exp(-S_{j})}.$ (27)

After calculating the outputs of all neurons in the hidden layer, the output
of the $k$-th node can be calculated as follows:

$y_{k}=\sum\nolimits_{j=1}^{H}w_{jk}z_{j}+b_{k},\;k=\\{1,\dots,O\\}.$ (28)

It is worth mentioning that the input and output of the FNN are normalized
values. More specifically, the normalized locations of GUs are computed as

$\hat{x}_{i}=\frac{x_{i}-\operatorname{\mathbb{E}}[x_{i}]}{\sqrt{\left(x_{i}-\operatorname{\mathbb{E}}[x_{i}]\right)^{2}}}\text{
and
}\hat{y}_{i}=\frac{y_{i}-\operatorname{\mathbb{E}}[y_{i}]}{\sqrt{\left(y_{i}-\operatorname{\mathbb{E}}[y_{i}]\right)^{2}}},$
(29)

where $x_{i}$ and $y_{i}$ are $x$-coordinate and $y$-coordinate of GU $i$,
respectively. The outputs of the FNN are the normalized transmit power,
denoted as $\hat{\boldsymbol{p}}$, and the UAV’s placement, denoted as
$\hat{\boldsymbol{w}}$. Then, the transmit power and UAV’s placement can be
calculated as $\boldsymbol{p}=P_{\max}\hat{\boldsymbol{p}}$ and
$\boldsymbol{w}=R\hat{\boldsymbol{w}}$, respectively.

###  IV-B HHO for training FNNs

Our purpose in this section is to utilize the HHO algorithm for training FNN
so as to obtain the solution for joint power allocation and UAV’s placement.
For this purpose, two important points are how to encode the FNN optimization
vector as the solution in the HHO algorithm and how to present the fitness
function (also known as loss function in the FNN). Actually, several
components of the FNN can be optimized, such as the connection weights,
architecture (e.g., the number of hidden layers and the number of neurons in a
hidden layer), training parameters, and node optimization (e.g., choosing the
activation nodes) [43]. Typically, scholars try to optimize the connection
weights while fixing the other values. We follow this way to train the FNN for
optimizing the problem formulated in this paper.

In the HHO algorithm, the position of each hawk needs to be encoded into a
one-dimensional vector, which represents a candidate solution for the FNN. The
vector includes three components: 1) the set of connection weights from the
input layer to the hidden layer, 2) the set of weights connecting the hidden
layer with the output layer, and 3) the set of biases. Thus, the size of each
solution in the HHO is equivalent to the total number of weights and biases,
i.e., $D=IH+HO+H+O$. More specifically, the position of a hawk can be encoded
as the following vector

$\displaystyle X=[$
$\displaystyle\underbrace{w_{11},\dots,w_{IH}}_{\text{input-hidden
weights}},\underbrace{w_{11}^{h},\dots,w_{HO}^{h}}_{\text{hidden-output
weights}},$ $\displaystyle\underbrace{b_{1}^{h},\dots,b_{H}^{h}}_{\text{hidden
layer biases}},\underbrace{b_{1},\dots,b_{O}}_{\text{output layer biases}}].$
(30)

Here, $w_{ij}$ denotes the weight connecting the $i$-th node in the input
layer with the $j$-th node in the hidden layer, $w_{jk}^{h}$ denotes the
connection weight from the $j$-th node in the hidden layer to the $k$-th node
in the output layer, and $b_{j}^{h}$ ($b_{k}$) denotes the bias associated
with the $j$-th ($k$-th) node in the hidden layer (output layer).

Another important point is the selection of the fitness function, which is
used to evaluate the positions of all the hawks. To this end, two following
fitness functions, labeled as $L_{T}$ and $L_{L}$, are taken into account for
optimizing the FNN:

$\displaystyle
L_{T}=-\sum\limits_{i\in\mathcal{N}}\log_{2}\left(1+\frac{h_{i}p_{i}}{n_{0}+\sum\limits_{j\in\mathcal{N}:h_{j}>h_{i}}h_{i}p_{j}}\right),$
(31) $\displaystyle
L_{L}=\left(SR_{\text{HHOPAP}}-\sum\limits_{i\in\mathcal{N}}\log_{2}\left(1+\frac{h_{i}p_{i}}{n_{0}+\sum\limits_{j:h_{j}>h_{i}}h_{i}p_{j}}\right)\right)^{2},$
(32)

where $SR_{\text{HHOPAP}}$ is the sum rate achieved by the HHOPAP algorithm as
in Alg. 1. The first fitness function represents the minus of the total sum
rate of all the GUs, i.e., the purpose is to train the FFN so that sum rate of
all GUs, whereas the second fitness function denotes the squared difference
between the sum rates obtained by the HHO trainer and HHOPAP algorithm, i.e.,
square deviation.

We recall that for conventional algorithms (e.g., back propagation, Quickpro,
Rprop, and conjugate gradient), information on the gradient of the loss
function with respect to the optimization variables is required for training
the FNN. For instance, a power allocation scheme was reported in [44] to
maximize the overall spectral efficiency (SEE) of underlay device-to-device
(D2D) users, where the loss function is defined as the negative overall SE
plus the violation in the interference threshold imposed to the cellular user.
However, the gradient-based method may fall to local minima, has a slow
convergence speed, and highly depends on initial parameters [45]. Different
from these algorithms, as a metaheuristic algorithm, the HHO relaxes the need
for gradient information and offers high efficiency in avoiding trapping into
local minima. To illustrate the effectiveness of the proposed HHO-based
algorithm, we will provide comparisons with three well-known metaheuristic
algorithms: PSO, ES, and GA.

Alg. 2 Pseudo-code of the HHOFNN trainer

1:Initialization: Create a random population $X_{i}\;(i=1,2,\ldots,S)$,
initialize the stopping tolerance $\epsilon$, and set the iteration index
$t=1$.

2:while $|L_{T}(t)-L_{T}(t-1)|\geq\epsilon$ do

3: The positions of the hawks are assigned to the weights and biases as
potential FNNs.
---

4: Evaluate the fitness values corresponding to the FNNs. The FNN with the
lowest fitness value is then selected as the best solution.
---

5: Update the positions of the hawks based on the HHO 11i algorithm as shown
in Alg. 1.

6: Set $t=t+1$.

7:end while

8:Return the connection weights and biases of the FNN.

In summary, the HHO algorithm for training the FNN, named as HHOFNN, is
described in Alg. 2. Step 3 is to assign the outputs of the HHOFNN algorithm
to the vectors of power allocation and UAV’s placement. Then, these vectors
are used to calculate the fitness values via either (31) or (32). In Step 4,
the FNN with the lowest fitness value is considered as the best FNN in the
current iteration, which corresponds to Step (6) in Alg. 1. Step 5 refers to
the main loop, lines 7-30, in Alg. 1, i.e., this step is to update the FNNs,
which are then used in the next iteration. The algorithm is executed until the
difference between loss values in two consecutive iterations is less than a
stopping tolerance $\epsilon$, i.e., the sum rate stays almost unchanged for
two consecutive training times. As analyzed earlier, the computational
complexity of Alg. 2 is $\mathcal{O}(SDT)$, where $S$ is the size of hawks and
$T$ is the number of iterations needed to train the FNN.

##  V Simulation Results

In this section, we present numerical results to evaluate the performance of
the proposed algorithm. For simulation settings, we consider a VLC system with
a LED transmitter, which is mounted on a UAV, and 20 GUs (i.e., $N=20$), which
are randomly distributed in a coverage of 10 m $\times$ 10 m $\times$ 3 m. The
maximal transmit power $P_{\max}=20$ mW, the DC-offset $A=20$, the peak
optical intensity $B=30$, and the noise power is $n_{0}=-104$ dBm. Moreover,
the bandwidth $B=20$ MHz, the disc radius $R=10$ m, the physical area of the
detector is $1$ cm2, the gain of the optical filter is $1.0$, and
$\delta=3\sqrt{5}/5$ (for VLC systems with 4-ary PAM).

Figure 4: Convergence of the HHOPAP algorithm.

First, to investigate the convergence of the proposed algorithm, we set the
maximum number of iterations to $350$ (used to update the escaping energy of
the prey, as in Eq. (14)), the minimum required rate of each GU to $200$ kbps,
and the number of hawks to $S=30$. As can be seen from Fig. 4, the sum rate
increases progressively for about $250$ iterations and becomes constant for
the remaining iterations. This shows that the proposed HHOPAP algorithm
converges to the final solution within a reasonable time. Moreover, as the
number of GUs increases, the sum rate decreases and experiences an increasing
drop at the convergent solution. It is reasonable since GUs may be affected by
more strong GUs (users with better channel conditions), thus even though SIC
operation in NOMA works properly, GUs can receive much more interference from
other strong GUs and the sum rate reduces accordingly.

To show advantages of the proposed algorithm (labeled as HHOPAP), the
following schemes are used for comparison:

  * •

GRPA: this NOMA scheme was proposed in [1]. The power allocation factor is set
to be $0.4$ and the UAV’s placement is optimized by our algorithm.

  * •

RandP: the UAV’s position is randomly created within a disc as constrained in
(10g) and transmit power allocated to GUs is optimized by the HHO algorithm.

  * •

Conventional OFDMA [46, Section 6.1.3]: the bandwidth is equally divided among
GUs.

Figure 5: Sum rate against the maximum transmit power.

Second, we evaluate the sum of data rates performance of the proposed HHOPAP
and compare algorithms versus the maximum transmit power of the LED. Note that
each plot below is obtained by averaging over 100 random network realizations,
in which GUs are positioned randomly at each realization. It is observed from
Fig. 5 that the sum rate experiences a steady increase as the maximum transmit
power of the LED $P_{\max}$ increases. It is because GUs can be allocated more
power by the UAV and their achievable rate increases accordingly. However, the
sum rate improves at a slower speed when the maximum transmit power increases.
For instance, the proposed HHOPAP algorithm achieves an increase of $27.25$
Mbits/s when the maximum transmit power grows from $20$ to $40$ mW, whereas
the increase is only $11.38$ ($6.67$) Mbits/s for the values $60$ and $80$ mW
($80$ and $100$ mW). Another observation from the figure is that all the NOMA
schemes (HHOPAP, GRPA, and RandP) outperform the conventional OFDMA scheme as
they achieve higher sum rates. The figure also depicts that the proposed
algorithm is better than alternative NOMA schemes (GRPA and RandP). This is
because the GRPA considered a simple power allocation scheme, where the power
is allocated to GUs based on the orders of their channel gains, and the RandP
only optimizes the power allocated to GUs, whereas the UAV’s placement is
randomly selected. Above observations _justify the importance of optimizing
UAV’s placement and NOMA power allocation_ as NOMA schemes are superior to the
OFDMA scheme, and the HHOPAP outperforms the GRPA (power is not optimized) and
the RandP (UAV’s placement is not optimized).

Figure 6: The sum-rate performance versus the GUs’ FoV.

Next, Fig. 6 shows the effect of the GU’s FoV on the sum rate function. As can
be seen from the figure, decreasing the GUs’ FoV increases the sum rate, i.e.,
the FoVs of $40^{\circ}$ and $45^{\circ}$ achieve yield higher sum rate than
the FoVs of $45^{\circ}$ and $50^{\circ}$, respectively. The reason for this
is that the concentrator gain increases when the FoV $\Phi$ varies from $0$ to
$90^{\circ}$, as justified in Eq. (2), thus increasing the channel gain of GUs
and the sum rate accordingly. This result is highly interesting since if
users, especially those who are using a smartphone, tend to keep the devices
with their preferred direction. In that case, the FoV may be adjusted and the
channel gain to the UAV-mounted LED may change significantly. Moreover,
decreasing the FoV leads to increasing the coverage probability [47]. As a
result, if the FoV is adjusted reasonably (i.e., decreasing the FoV’s value
while ensuring that the UAV is within the GUs’ FoV), the sum rate can be
enhanced. Thanks to its high flexibility, the UAV can be flexibly placed at a
proper position, where it is under the FoV of all the GUs. Again, the NOMA
schemes are superior to the conventional OFDMA scheme and the proposed
algorithm achieves better sum-rate performance than all the alternative
schemes.

Figure 7: Comparison of the sum rate for different cell radius values.

Fig. 7 studies the effect of the cell radius $R$ on the sum-rate performance.
The figure shows that increasing the cell radius leads to reducing the sum-
rate performance. It is reasonable since for a given network setting, the
larger the cell radius is, the longer the distance $d_{i},i\in\mathcal{N}$
from GUs to the UAV-mounted LED can be. Therefore, the channel gains between
the UAV and GUs may become weaker (as shown in Eq. (1)), and the achievable
rate reduces accordingly. What is more, the OFDMA scheme can surpass the RandP
approach when the cell radius is sufficiently large (it is $R=12$ meters for
the simulation setting in this paper). This is because as the cell radius is
larger, the UAV may be placed at a location very far from most GUs and thus
the performance cannot be well compensated by NOMA power allocation in the
RandP scheme. Again, this observation justifies the desirability of optimizing
UAV’s placement in UAV-assisted VLC systems. Furthermore, the sum rate
obtained by the proposed HHOPAP algorithm is the highest among all the
alternative schemes.

Figure 8: The sum-rate performance for different UAV’s altitudes.

In Fig. 8, the sum rates achieved by different algorithms are compared versus
the UAV’s altitude (also known as hovering height in the literature). From the
figure, it can be seen that the sum rate first increases when the UAV’s
altitude gets larger, but the sum rate starts to decrease when the UAV’s
altitude is large enough. This is due to the fact that increasing the UAV’s
altitude results in an increase in the channel gains. To illustrate this
point, we consider an example, where the UAV and a GU $i$ have the coordinates
of $(3,1,h)$ and $(1,2,0)$, i.e., the distance is given as
$d_{i}=\sqrt{5+h^{2}}$. When $h=1.5$, $d_{i}=\sqrt{6.25}$,
$\cos\phi_{i}=\cos\varphi_{i}=1.5/\sqrt{6.25}$, and the channel gain is
$h_{i}=1.5\zeta/(6.25)^{3/2}$ (here $\zeta$ is a constant and the value is
computed via Eq. (1)), and similarly when $h=2.5$, the channel gain is
$h_{i}=2.5\zeta/(11.25)^{3/2}$. Obviously, the altitude $h=2.5$ offers a
higher channel gain value than $h=1.5$, and the GU’s achievable rate increases
as a consequence. However, the channel gain and sum-rate performance become
worse if the altitude keeps increasing. Using the same example as above, the
channel gains for $h=4$ and $h=5$ are $h_{i}=4\zeta/(29)^{3/2}$ and
$h_{i}=5\zeta/(38)^{3/2}$, respectively. Moreover, the OFDMA system is better
than two NOMA schemes (GRPA and RandP) at lower values of the UAV altitude.
The main reason is that the NOMA schemes typically achieve a better sum rate
than OFDMA when the channel conditions of GUs are sufficiently distinctive,
but this is not the case when the UAV altitude is small [2]. Besides, the
figure shows that the proposed HHOPAP algorithm provides the best sum-rate
performance in all cases of the altitude.

(a) MSE convergence curve for the Iris dataset.  (b) MSE convergence curve for
the Cancer dataset.

(c) Sum rate performance of different trainers.

Figure 9: Performance comparison of the HHO trainer.

Finally, we show the performance of the HHO trainer and compare it with that
of PSO, ES, and GA trainers. To verify the performance of the proposed HHO
trainer on existing datasets, two popular ones (Iris and Cancer) are
selected333Due to the space limitation, we only present two datasets, which
are available at https://archive.ics.uci.edu/ml/datasets.php.. From Figs. 9a
and 9b, the HHO achieves the lowest mean square error (MSE), which are
$0.027562$ and $0.0016149$ for the Iris and Cancer datasets, respectively.
This result _demonstrates the capability of the HHO in avoiding falling into
the local minima trap_ , which usually happens in conventional trainers like
PSO and ES. As can be observed from Fig. 9c, the HHO trainer achieves almost
the same sum-rate performance with two types of fitness functions. In
particular, the first kind (labeled as HHOT) can achieve the sum rate of
$34.4156$ Mbits/s, whereas the second one (labeled as HHOL) achieves the sum
rate of $324.9810$ Mbits/s. Compared with the other trainers (ES, PSO, and
GA), the HHO offers a better performance in terms of the sum rate. Another
observation is that the sum rate obtained by the HHO trainer is closest to the
proposed HHOPAP with only about 10% loss, which shows that the HHO trainer is
a promising solution for real-time applications.

##  VI Conclusion and Future Works

In this paper, we have studied the integration of UAV and NOMA into VLC
systems. We have investigated the sum-rate maximization problem, which was
solved by facilitating the HHO algorithm. In addition, we have utilized the
HHO for training an FNN so that the UAV’s placement and power allocation can
be computed in real-time. Various simulation settings and performance metrics
have been tested to evaluate the effectiveness and superiority of the proposed
HHO algorithm and HHOFNN trainer. Furthermore, we have observed an obvious
gain of jointly optimizing UAV’s placement and NOMA power allocation, when
compared with the schemes of optimizing either UAV’s placement (i.e., GRPA) or
power allocation (i.e., RandP).

Many issues and open problems stemmed from this paper should be investigated
in the future. Firstly, since the performance loss is currently about 10%, it
is a valuable contribution to further improve the performance of the HHO
trainer. Secondly, when fixed-wing UAVs are considered, the joint three-
dimensional trajectory optimization and power allocation problem is an
important issue. Thirdly, because of the limited on-board energy of UAVs, the
energy-efficient design of UAV-assisted VLC is desirable to study. Next, the
LoS connection between the UAV and GUs may pose physical layer security
issues, which demand more efforts in the future. Besides, it is interesting to
investigate an algorithm, which can find the globally optimal solution for the
original problem (10). Finally, it is desirable to examine the HHO algorithm
with other ANN architectures and apply the HHO algorithm to other optimization
problems of resource allocation in wireless and communications.

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