#  Energy Minimization for Active RIS-Aided UAV-Enabled SWIPT Systems

Zhangjie Peng, Ruijing Liu, Cunhua Pan, , Zhenkun Zhang, and Jiangzhou Wang Z.
Peng and R. Liu are with the College of Information, Mechanical and Electrical
Engineering, Shanghai Normal University, Shanghai 200234, China. (e-mail:
<EMAIL_ADDRESS>1000511821@smail.shnu.edu.cn).C. Pan and Z. Zhang
are with the National Mobile Communications Research Laboratory, Southeast
University, Nanjing 210096, China. (e-mail<EMAIL_ADDRESS>zhenkun
zhang@seu.edu.cn).J. Wang is with the School of Engineering, University of
Kent, CT2 7NT Canterbury, U.K. (e-mail: j.z.wang@kent.ac.uk).

###### Abstract

In this paper, we consider an active reconfigurable intelligent surface
(RIS)-aided unmanned aerial vehicle (UAV)-enabled simultaneous wireless
information and power transfer (SWIPT) system with multiple ground users.
Compared with the conventional passive RIS, the active RIS deploying the
internally integrated amplifiers can offset part of the multiplicative fading.
In this system, we deal with an optimization problem of minimizing the total
energy cost of the UAV. Specifically, we alternately optimize the
trajectories, the hovering time, and the reflection vectors at the active RIS
by using the successive convex approximation (SCA) method. Simulation results
show that the active RIS performs better in energy saving than the
conventional passive RIS.

###### Index Terms:

Reconfigurable intelligent surface (RIS), active RIS, unmanned aerial vehicle
(UAV), simultaneous wireless information and power transfer (SWIPT),
successive convex approximation (SCA).

##  I Introduction

As one of the potential technologies driving the development of 6G, the
reconfigurable intelligent surface (RIS), which consists of a large number of
reflecting elements, has become a promising technology to enhance the
communication quality of wireless networks. With the characteristics of low
cost and easy deployment, researchers have studied the deployment of the RIS
into wireless communication systems [1, 2, 3].

Due to the high flexibility, unmanned aerial vehicle (UAV) has been proposed
to assist the signal transmission in the communication system [4, 5].
Meanwhile, simultaneous wireless information and power transfer (SWIPT), which
can enhance the energy and information transmission, has emerged as a
promising communication transmission approach for internet-of-things (IoT)
networks [6]. By integrating the RIS, UAV and SWIPT, the performance of the
communication system can be further enhanced. In [7], an RIS was deployed in
SWIPT systems to enhance both information and energy transmission. The authors
in [8] considered a problem for maximizing the average achievable rate, and
showed that the performance of SWIPT systems with a single IoT device can be
improved with the deployment of both the UAV and RIS.

However, the performance of the RIS-aided systems is affected by the
significant multiplicative fading on the cascaded channels of the reflective
links. To solve this problem, the active RIS integrates the amplifiers into
the reflecting elements to offset the multiplicative fading [9]. The authors
in [10] theoretically compared the active RIS-aided system with the passive
RIS-aided system, and demonstrated that the better performance can be achieved
by the active RIS-aided system when the power budget is adequate. In [11], the
authors investigated an active RIS-aided system and proposed a joint computing
and communication design using the successive convex approximation (SCA)
method.

Against the above background, we consider an active RIS-aided UAV-enabled
SWIPT system with multiple ground users, where the UAV serves as a
transmitter. In this paper, we aim to minimize the total energy cost of the
UAV while fulfilling the energy reception and information transmission target
for all users. The problem is solved by reformulating the original problem
into two subproblems, and applying the SCA method to handle the subproblems,
and the original problem is solved by alternately optimizing two subproblems.
Simulation results show that the active RIS performs better in terms of energy
saving than the passive RIS under the same total energy supply for the UAV.

Figure 1: Active RIS-aided UAV-enabled SWIPT system.

##  II System Model And Problem Formulation

###  II-A System Model

As shown in Fig. 1, we consider an active RIS-aided UAV-enabled SWIPT system
with ${K}$ ground users, where the active RIS consists of ${M}$ reflecting
elements, and both the UAV and each user are equipped with one antenna. To
facilitate the representation of distances between devices, we use a complex
number ${q=x+jy}$ to denote the horizontal coordinate $(x,y)$ of the device.
In this case, the position of the active RIS and user ${k}$ are respectively
denoted by ${q_{\rm R}=q_{\rm R}^{x}+jq_{\rm R}^{y}}$ with the height ${H_{\rm
R}}$ and ${q_{{\rm S},k}=q_{{\rm S},k}^{x}+jq_{{\rm S},k}^{y}}$ for
${k\in\mathcal{K}\triangleq\left\\{1,...,K\right\\}}$ with zero height.
Meanwhile, we set the UAV flying at the same height, and each flight consists
of ${L}$ straight path segments starting from ${q_{{\rm V},0}}$ and ending at
${q_{{\rm V},L}}$. Besides, the hovering position of the UAV is denoted by
${q_{{\rm V},l}=q^{x}_{{\rm V},l}+jq^{y}_{{\rm V},l}}$ for
${l\in\mathcal{L}\triangleq\left\\{1,...,L-1\right\\}}$ with the height
${H_{\rm V}}$. Then, the trajectory of the UAV is expressed as
${\mathbf{q}_{\rm V}\\!\\!=\left[q_{{\rm V},0},...,q_{{\rm V},l},...,q_{{\rm
V},L}\right]^{\rm T}\\!\\!\\!\in\mathbb{C}^{\left(L+1\right)\times 1}}$, and
the hovering time is denoted by ${\mathbf{t}=[t_{1},t_{l},...,t_{L-1}]^{\rm
T}}$.

When the UAV is hovering at the ${l}$-th hovering position, the reflection
vector and the diagonal reflection matrix are respectively denoted by
${\boldsymbol{\phi}_{l}\\!\\!=\\!\\!\left[b_{1,l}e^{j{\theta}_{1,l}},...,b_{M,l}e^{j{\theta}_{M,l}}\right]^{\rm
T}}$ and ${\mathbf{\Theta}_{l}\\!={\rm
diag}\left(\boldsymbol{\phi}_{l}\right)=\\!{\rm
diag}\\!\left(b_{1,l}e^{j{\theta}_{1,l}}\\!,...,b_{M,l}e^{j{\theta}_{M,l}}\right)}$,
where ${{\theta}_{m,l}}$ and ${b_{m,l}}$ are the phase shift and the amplitude
of the ${m}$-th reflecting element at the active RIS, respectively. The
channels from the UAV to user ${k}$, from the UAV to the active RIS, and from
the active RIS to user ${k}$ are respectively denoted by ${g_{{\rm
d},k,l}\\!\in\\!\mathbb{C}^{1\\!\times\\!1}}$, ${\mathbf{g}_{{\rm
t},l}\\!\in\\!\mathbb{C}^{M\\!\times\\!1}}$ and ${\mathbf{g}_{{\rm
r},k}\\!\in\\!\mathbb{C}^{M\\!\times\\!1}}$. The Rician fading channels
${g_{{\rm d},k,l}}$, ${\mathbf{g}_{{\rm t},l}}$ and ${\mathbf{g}_{{\rm r},k}}$
are respectively modeled as

$\displaystyle g_{{\rm d},k,l}=$ $\displaystyle\sqrt{\beta_{{\rm
d},k,l}}\left(\sqrt{\frac{\mu_{\rm d}}{\mu_{\rm d}+1}}g_{{\rm d},k,l}^{\rm
LoS}+\sqrt{\frac{1}{\mu_{\rm d}+1}}g_{{\rm d},k,l}^{\rm NLoS}\right),$ (1)
$\displaystyle\mathbf{g}_{{\rm t},l}\\!\\!\\!\\!\\!\quad=$
$\displaystyle\sqrt{\beta_{{\rm t},l}}\\!\quad\left(\sqrt{\frac{\mu_{\rm
t}}{\mu_{\rm t}+1}}\mathbf{g}_{{\rm t},l}^{\rm
LoS}\\!\\!\\!\\!\quad+\sqrt{\frac{1}{\mu_{\rm t}+1}}\mathbf{g}_{{\rm
t},l}^{\rm NLoS}\right),$ (2) $\displaystyle\mathbf{g}_{{\rm r},k}=$
$\displaystyle\sqrt{\beta_{{\rm r},k}}\\!\\!\quad\left(\sqrt{\frac{\mu_{\rm
r}}{\mu_{\rm r}+1}}\mathbf{g}_{{\rm r},k}^{\rm
LoS}\\!\\!\\!\\!\quad+\sqrt{\frac{1}{\mu_{\rm r}+1}}\mathbf{g}_{{\rm
r},k}^{\rm NLoS}\right),$ (3)

where each element of the non-line-of-sight (NLoS) components ${g_{{\rm
d},k,l}^{\rm NLoS}}$, ${\mathbf{g}_{{\rm t},l}^{\rm NLoS}}$ and
${\mathbf{g}_{{\rm r},k}^{\rm NLoS}}$ are i.i.d. circularly symmetric complex
Gaussian distribution with zero mean and unit variance, ${\mu_{\rm d}}$,
${\mu_{\rm t}}$ and ${\mu_{\rm r}}$ are the Rician factors, ${g_{{\rm
d},k,l}^{\rm LoS}}$, ${\mathbf{g}_{{\rm t},l}^{\rm LoS}}$ and
${\mathbf{g}_{{\rm r},k}^{\rm LoS}}$ are the line-of-sight (LoS) components.
Under the fly-hover-broadcast (FHB) protocol [5], it is assumed that the UAV
only transmits signals during hovering mode, and thus the large-scale fading
coefficients ${\beta_{{\rm d},k,l}}$, ${\beta_{{\rm t},l}}$ and ${\beta_{{\rm
r},k}}$ that depend on the hovering position are respectively given by

$\displaystyle\beta_{{\rm d},k,l}$
$\displaystyle=\frac{\beta_{0}}{(\left|q_{{\rm V},l}\\!-\\!q_{{\rm
S},k}\right|^{2}\\!\\!+\\!\\!H_{\rm V}^{2})^{\tau_{\rm
d}/2}}\triangleq\frac{\beta_{0}}{d_{{\rm d},k,l}^{\tau_{\rm d}}},$ (4)
$\displaystyle\beta_{{\rm t},l}$
$\displaystyle=\frac{\beta_{0}}{(\left|q_{{\rm V},l}-q_{\rm
R}\right|^{2}+\left(H_{\rm V}-H_{\rm R}\right)^{2})^{\tau_{\rm
t}/2}}\triangleq\frac{\beta_{0}}{d_{{\rm t},l}^{\tau_{\rm t}}},$ (5)
$\displaystyle\beta_{{\rm r},k}$
$\displaystyle=\frac{\beta_{0}}{(\left|q_{{\rm S},k}-q_{\rm
R}\right|^{2}+H_{\rm R}^{2})^{\tau_{\rm
r}/2}}\triangleq\frac{\beta_{0}}{d_{{\rm r},k}^{\tau_{\rm r}}},$ (6)

where ${\beta_{0}}$ is the channel gain of ${1\,\rm m}$, ${\tau_{\rm d}}$,
${\tau_{\rm t}}$ and ${\tau_{\rm r}}$ are the path-loss coefficients,
${d_{{\rm d},k,l}}$, ${d_{{\rm t},l}}$ and ${d_{{\rm r},k}}$ are the distances
from user ${k}$ to the UAV, from the UAV to the active RIS and from the active
RIS to user ${k}$, respectively. ${g_{{\rm d},k,l}^{\rm LoS}}$,
${\mathbf{g}_{{\rm t},l}^{\rm LoS}}$ and ${\mathbf{g}_{{\rm r},k}^{\rm LoS}}$
represent the LoS components under the uniform linear array (ULA) model, which
are respectively expressed as

$\displaystyle\\!\\!\\!g_{{\rm d},k,l}^{\rm LoS}$
$\displaystyle=e^{-j\frac{2\pi}{\lambda}d_{{\rm d},k,l}},$ (7)
$\displaystyle\\!\\!\\!\mathbf{g}_{{\rm t},l}^{\rm LoS}$
$\displaystyle=\\!e^{-j\frac{2{\pi}d_{{\rm
t},l}}{\lambda}}\\!\left[\\!1,\\!e^{-j\frac{2{\pi}d}{\lambda}\cos{\omega_{{\rm
t},l}}},\\!...,\\!e^{-j\frac{2{\pi}\left(M-1\right)d}{\lambda}\cos{\omega_{{\rm
t},l}}}\\!\right]^{\rm T}\\!\\!\\!\\!,$ (8)
$\displaystyle\\!\\!\\!\\!\mathbf{g}_{{\rm r},k}^{\rm LoS}$
$\displaystyle=\\!e^{-j\frac{2{\pi}d_{{\rm
r},k}}{\lambda}}\\!\\!\left[\\!1,\\!e^{-j\frac{2{\pi}d}{\lambda}\cos{\omega_{{\rm
r},k}}}\\!,\\!...,\\!e^{-j\frac{2{\pi}\left(M-1\right)d}{\lambda}\cos{\omega_{{\rm
r},k}}}\\!\right]^{\rm T}\\!\\!\\!\\!,$ (9)

where ${\lambda}$ and ${d}$ respectively represent the wavelength and the
element spacing of the active RIS, ${\cos{\omega_{{\rm t},l}}=\frac{q_{\rm
R}^{x}-q_{{\rm V},l}^{x}}{d_{{\rm t},l}}}$ and ${\cos{\omega_{{\rm
r},k}}=\frac{q_{{\rm S},k}^{x}-q_{\rm R}^{x}}{d_{{\rm r},k}}}$ are the cosine
of angle-of-arrival (AoA) and angle-of-departure (AoD), respectively.

In our system, the information transfer and energy supplication can be
obtained simultaneously, and the power split ratio for decoding information is
${\eta\in[0,1]}$ and for harvested power is ${1-\eta}$. Then, the received
signal of user ${k}$ is given by

$\displaystyle y_{k}=$ $\displaystyle\sqrt{p_{k}}(\mathbf{g}_{{\rm r},k}^{\rm
H}\mathbf{\Theta}_{l}\mathbf{g}_{{\rm t},l}\\!+\\!g_{{\rm
d},k,l})x_{k}+\mathbf{g}_{{\rm r},k}^{\rm H}\mathbf{\Theta}_{l}\mathbf{n}_{\rm
RIS}+n_{\rm r}$ $\displaystyle+\\!\\!\\!\\!\\!\\!\sum\limits_{j\neq
k,j\in\mathcal{K}}\\!\\!\\!\sqrt{p_{j}}(\mathbf{g}_{{\rm r},k}^{\rm
H}\mathbf{\Theta}_{l}\mathbf{g}_{{\rm t},l}\\!+\\!g_{{\rm d},k,l})x_{j}$
$\displaystyle=$
$\displaystyle\sqrt{p_{k}}g_{k,l}x_{k}+\\!\\!\\!\\!\\!\\!\sum\limits_{j\neq
k,j\in\mathcal{K}}\\!\\!\\!\sqrt{p_{j}}g_{k,l}x_{j}+\mathbf{g}_{{\rm
r},k}^{\rm H}\mathbf{\Theta}_{l}\mathbf{n}_{\rm RIS}+n_{\rm r},$ (10)

where ${g_{k,l}\\!=\\!\mathbf{g}_{{\rm r},k}^{\rm
H}\mathbf{\Theta}_{l}\mathbf{g}_{{\rm t},l}\\!+\\!g_{{\rm d},k,l}}$, ${p_{k}}$
is the transmit power for user ${k}$ from the UAV,
${x_{k}\\!\sim\\!\mathcal{C}\mathcal{N}\\!\left(0,1\right)}$ is the transmit
signal for user ${k}$, ${n_{\rm
r}\sim\mathcal{C}\mathcal{N}\left(0,\delta_{\rm r}^{2}\right)}$ is the noise
at user $k$, and ${\mathbf{n}_{\rm
RIS}\sim\mathcal{C}\mathcal{N}\left(0,\delta_{\rm RIS}^{2}\mathbf{I}\right)}$
is the noise at the active RIS. When the UAV is hovering at the ${l}$-th
hovering position, the ergodic rate of user ${k}$ is given by

$\displaystyle\\!\\!\\!R_{k,l}\\!=\\!\mathbb{E}\\!\left\\{\\!\\!{\rm
log}_{2}\\!\\!\left(\\!\\!1\\!\\!+\\!\frac{\eta
p_{k}|g_{k,l}|^{2}}{\\!\\!\\!\\!\sum\limits_{j\neq
k,j\in\mathcal{K}}\\!\\!\\!\\!\\!\eta p_{j}|g_{k,l}|^{2}+\delta_{\rm
RIS}^{2}\|\mathbf{g}_{{\rm r},k}^{\rm H}\mathbf{\Theta}_{l}\|^{2}+\delta_{\rm
r}^{2}}\\!\\!\right)\\!\\!\\!\right\\}\\!.\\!$ (11)

By utilizing the derivations in [12, Lemma 1], ${{R}_{k,l}}$ of user ${k}$ can
be approximated as

$\displaystyle R_{k,l}\approx\tilde{R}_{k,l}={\rm log}_{2}\left(1+{\rm
SINR}_{k,l}\right),$ (12)

where ${{\rm SINR}_{k,l}}$ is given by

$\displaystyle\\!\\!\\!\\!\\!\\!{\rm SINR}_{k,l}\\!=\\!\frac{\eta
p_{k}\mathbb{E}\\{|g_{k,l}|^{2}\\}}{\\!\\!\\!\\!\\!\\!\\!\sum\limits_{j\neq
k,j\in\mathcal{K}}\\!\\!\\!\\!\\!\\!\\!\eta
p_{j}\mathbb{E}\\{|g_{k,l}|^{2}\\}\\!\\!+\\!\delta_{\rm
RIS}^{2}\mathbb{E}\\{\|\mathbf{g}_{{\rm r},k}^{\rm
H}\mathbf{\Theta}_{l}\|^{2}\\}\\!\\!+\\!\delta_{\rm r}^{2}}.\\!\\!\\!\\!\\!$
(13)

Furthermore, the harvested power ${\tilde{P}_{k,l}}$ at user ${k}$ is given by

$\displaystyle\tilde{P}_{k,l}=(1-\eta)p_{k}\mathbb{E}\\{\left|g_{k,l}\right|^{2}\\}.$
(14)

When the UAV flies at a speed of ${v}$, the propulsion power is modeled as

$\displaystyle\\!\\!\\!P_{\rm
p}(v)\\!=\\!P_{0}\\!\\!\left(\\!\\!1\\!+\\!\frac{3v^{2}}{U_{\rm
tip}^{2}}\right)\\!\\!+\\!\\!P_{\rm
i}\\!\left(\\!\\!\sqrt{1\\!\\!+\\!\\!\frac{v^{4}}{4v_{0}^{4}}}\\!-\\!\frac{v^{2}}{2v_{0}^{2}}\right)^{\frac{1}{2}}\\!\\!\\!\\!+\\!\frac{1}{2}d_{0}{\rho}sAv^{3}\\!\\!,\\!\\!\\!$
(15)

where the constants such as ${P_{0}}$, ${P_{\rm i}}$ and ${v_{0}}$ represent
the configuration of the UAV, and their definitions are detailed in [5].

Then, the total energy cost of the UAV can be given by

$\displaystyle\\!\\!\\!\\!\\!\\!\\!E_{\rm V}\\!\left(\mathbf{q}_{\rm
V}\\!,\mathbf{t}\right)\\!=\\!P_{\rm
p}(v)\\!\sum\limits_{l\in\mathcal{L}}\\!\frac{\left|q_{{\rm
V},l}\\!-\\!q_{{\rm
V},l-1}\right|}{v}\\!+\\!\\!\\!\sum\limits_{l\in\mathcal{L}}\\!\\!\left(P_{\rm
p}^{\rm hov}\\!\\!+\\!P_{\rm t}\right)\\!t_{l},\\!\\!$ (16)

where ${P_{\rm p}^{\rm hov}=P_{\rm p}(0)=P_{0}+P_{\rm i}}$ is the hovering
power, and ${P_{t}}$ is the radiated power of the UAV.

###  II-B Problem Formulation

We consider a problem of minimizing the total energy cost of the UAV, subject
to the SINR threshold ${\gamma_{k}}$ at user ${k}$, the energy requirement
${E_{k}^{\rm req}}$ at user ${k}$, and the energy limitation ${E_{\rm
RIS}^{\rm act}}$ at the active RIS. Then, the problem is formulated as

$\displaystyle\min_{{\mathbf{q}_{\rm
V}},{\\{\boldsymbol{\phi}_{l}\\}},\mathbf{t}}$ $\displaystyle\quad E_{\rm
V}\left(\mathbf{q}_{\rm V},\mathbf{t}\right)$ (17a)
$\displaystyle\mathrm{s}.\mathrm{t}.\\!\quad$ $\displaystyle\quad{\rm
SINR}_{k,l}\geqslant\gamma_{k},\,k\in\mathcal{K},\,l\in{\mathcal{L}},$ (17b)
$\displaystyle\quad\sum_{l\in{\mathcal{L}}}{t_{l}\tilde{P}_{k,l}}\geqslant
E_{k}^{\rm req},\,k\in\mathcal{K},$ (17c)
$\displaystyle\quad\mathbb{E}\\{\left\|\mathbf{\Theta}_{l}\mathbf{s}_{l}\right\|^{2}\\}t_{l}\leqslant
E_{\rm RIS}^{\rm act},\,l\in{\mathcal{L}},$ (17d)

In (17d),
${\mathbf{s}_{l}\\!=\\!\sum_{k\in\mathcal{K}}\\!\\!\sqrt{p_{k}}\mathbf{g}_{{\rm
t},l}x_{k}\\!+\\!\mathbf{n}_{\rm RIS}}$ and ${\mathbf{\Theta}_{l}\\!={\rm
diag}\left(\boldsymbol{\phi}_{l}\right)}$.

##  III Joint Reflection Coefficients And Trajectory Design

In this section, we propose an effective SCA-based algorithm for solving
Problem (17). Specifically, we reformulate the original problem into two
subproblems, and solve the subproblems by using SCA method. Then, the original
problem is solved by alternately optimizing two subproblems.

###  III-A The Design of Reflection Coefficient Vectors

In this subsection, we first optimize
${\left\\{\boldsymbol{\phi}_{l}\right\\}}$ with given ${\mathbf{q}_{\rm V}}$
and ${\mathbf{t}}$ when the UAV is at the ${l}$-th hovering position. Since
(17a) is not a function of ${\left\\{\boldsymbol{\phi}_{l}\right\\}}$, the
corresponding subproblem is a feasibility-check problem. To improve the
convergence performance, the optimization objective can be enhanced by
maximizing the total oversupplied energy of all users [1]. Since the
subproblem may not be feasible, it may cause the algorithm to converge early.
To solve this, we set the objective function to maximize the minimum charged
energy of all users instead [1], and the subproblem is formulated as

$\displaystyle\max_{{\\{\boldsymbol{\phi}_{l}\\}},\varepsilon}$
$\displaystyle\quad\varepsilon$ (18a) $\displaystyle\mathrm{s}.\mathrm{t}.$
$\displaystyle\quad(1/E_{k}^{\rm
req})\sum_{l\in{\mathcal{L}}}{t_{l}\tilde{P}_{k,l}}\geqslant\varepsilon,\,k\in\mathcal{K},$
(18b) $\displaystyle\quad\eqref{swipt_Problem1-b},\,\eqref{swipt_Problem1-d},$

where ${\varepsilon}$ is an auxiliary variable.

However, Problem (18) is still non-convex due to the non-convex constraints
(17b) and (18b). In the following, we formulate constraints (18b) and (17d)
into the more tractable forms, and then derive constraints (17b) and (18b)
into the convex constraints.

Firstly, we need to reformulate constraint (18b) into a more tractable form.
By substituting (1), (2) and (3) into the term
${\mathbb{E}\\{\left|g_{k,l}\right|^{2}\\}}$ of ${\tilde{P}_{k,l}}$ in (14),
we have

$\displaystyle\\!\\!\\!\\!\\!\mathbb{E}\left\\{\left|\left(\mathbf{g}_{{\rm
r},k}^{\rm H}\mathbf{\Theta}_{l}\mathbf{g}_{{\rm t},l}\\!+\\!{g}_{{\rm
d},k,l}\right)\right|^{2}\right\\}\\!\\!$
$\displaystyle=\\!\\!\mathbb{E}\\!\left\\{\left|a_{0,k,l}\right|^{2}\right\\}\\!\\!+\\!\mathbb{E}\\!\left\\{\left|a_{1,k,l}\right|^{2}\right\\}$
$\displaystyle+\mathbb{E}\left\\{\left|a_{2,k,l}\right|^{2}\right\\}\\!\\!$
$\displaystyle+\\!\mathbb{E}\\!\left\\{\left|a_{3,k,l}\right|^{2}\right\\}\\!\\!+\\!\mathbb{E}\\!\left\\{\left|a_{4,k,l}\right|^{2}\right\\}\\!\\!,$
(19)

where

$\displaystyle\\!\\!\\!a_{0,k,l}\\!=\\!\\!\sqrt{\\!\\!\frac{\mu_{\rm
r}\beta_{{\rm r},k}}{\mu_{\rm r}\\!+\\!1}}\\!\sqrt{\\!\\!\frac{\mu_{\rm
t}\beta_{{\rm t},l}}{\mu_{\rm t}\\!+\\!1}}\\!\\!\left(\\!\mathbf{g}_{{\rm
r},k}^{\rm LoS}\right)^{\rm H}\\!\\!\mathbf{\Theta}_{l}\mathbf{g}_{{\rm
t},l}^{\rm LoS}\\!\\!+\\!\\!\sqrt{\\!\\!\frac{\mu_{\rm d}\beta_{{\rm
d},k,l}}{\mu_{\rm d}+1}}{g}_{{\rm d},k,l}^{\rm LoS},\\!\\!$ (20)
$\displaystyle\\!\\!\\!a_{1,k,l}\\!=\\!\\!\sqrt{\frac{\beta_{{\rm
d},k,l}}{\mu_{\rm d}+1}}{g}_{{\rm d},k,l}^{\rm NLoS},$ (21)
$\displaystyle\\!\\!\\!a_{2,k,l}\\!=\\!\\!\sqrt{\frac{\beta_{{\rm
r},k}}{\mu_{\rm r}+1}}\sqrt{\frac{\mu_{\rm t}\beta_{{\rm t},q}}{\mu_{\rm
t}+1}}\left(\mathbf{g}_{{\rm r},k}^{\rm NLoS}\right)^{\rm
H}\\!\\!\mathbf{\Theta}_{l}\mathbf{g}_{{\rm t},l}^{\rm LoS},$ (22)
$\displaystyle\\!\\!\\!a_{3,k,l}\\!=\\!\\!\sqrt{\frac{\mu_{\rm r}\beta_{{\rm
r},k}}{\mu_{\rm r}+1}}\sqrt{\frac{\beta_{{\rm t},l}}{\mu_{\rm
t}+1}}\left(\mathbf{g}_{{\rm r},k}^{\rm
LoS}\right)^{H}\mathbf{\Theta}_{l}\mathbf{g}_{{\rm t},l}^{\rm NLoS},$ (23)
$\displaystyle\\!\\!\\!a_{4,k,l}\\!=\\!\\!\sqrt{\frac{\beta_{{\rm
r},k}}{\mu_{\rm r}+1}}\sqrt{\frac{\beta_{{\rm t},l}}{\mu_{\rm
t}+1}}\left(\mathbf{g}_{{\rm r},k}^{\rm NLoS}\right)^{\rm
H}\\!\\!\mathbf{\Theta}_{l}\mathbf{g}_{{\rm t},l}^{\rm NLoS}.$ (24)

$\displaystyle a_{0,k,l}=$ $\displaystyle\sqrt{\frac{\mu_{\rm r}\beta_{{\rm
r},k}}{\mu_{\rm r}+1}}\sqrt{\frac{\mu_{\rm t}\beta_{{\rm t},l}}{\mu_{\rm
t}+1}}\left(e^{-j\frac{2\pi d_{{\rm r},k}}{\lambda}}\left[1,e^{-j\frac{2\pi
d}{\lambda}\cos\omega_{{\rm
r},k}},\cdots,e^{-j\frac{2\pi\left(M-1\right)d}{\lambda}\cos\omega_{{\rm
r},k}}\right]^{\rm T}\right)^{\rm H}\cdot{\rm diag}(b_{1},...,b_{M})$
$\displaystyle\cdot{\rm diag}(e^{j\theta_{1,l}},...,e^{j\theta_{M,l}})\cdot
e^{-j\frac{2\pi d_{{\rm t},l}}{\lambda}}\\!\\!\left[1,e^{-j\frac{2\pi
d}{\lambda}\cos\omega_{{\rm
t},l}},\cdots,e^{-j\frac{2\pi\left(M-1\right)d}{\lambda}\cos\omega_{{\rm
t},l}}\right]^{\rm T}\\!\\!\\!+\\!\sqrt{\frac{\mu_{\rm d}\beta_{{\rm
d},k,l}}{\mu_{\rm d}+1}}e^{-j\frac{2\pi d_{{\rm d},k,l}}{\lambda}}$
$\displaystyle=$ $\displaystyle\sqrt{\frac{\mu_{\rm r}\mu_{\rm t}\beta_{{\rm
r},k}\beta_{{\rm t},l}}{\left(\mu_{\rm r}+1\right)\left(\mu_{\rm
t}+1\right)}}\sum_{m=1}^{M}{b_{m}e^{j(\frac{2\pi(d_{{\rm r},k}-d_{{\rm
t},l})+2\pi(m-1)d(\cos\omega_{{\rm r},k}-\cos\omega_{{\rm
t},l})}{\lambda}+\theta_{m,l})}}\\!\\!\\!\quad+\sqrt{\frac{\mu_{\rm
d}\beta_{{\rm d},k,l}}{\mu_{\rm d}+1}}e^{-j\frac{2\pi d_{{\rm
d},k,l}}{\lambda}}$ $\displaystyle=$ $\displaystyle e^{-j\frac{2\pi d_{{\rm
d},k,l}}{\lambda}}\left(\sqrt{\frac{\mu_{\rm d}\beta_{{\rm d},k,l}}{\mu_{\rm
d}+1}}+\sqrt{\frac{\mu_{\rm r}\mu_{\rm t}\beta_{{\rm r},k}\beta_{{\rm
t},l}}{\left(\mu_{\rm r}+1\right)\left(\mu_{\rm
t}+1\right)}}\sum_{m=1}^{M}{b_{m}e^{j\left(\psi_{k,m,l}+\theta_{m,l}\right)}}\right),$
(25)

$\displaystyle\\!\\!\mathbb{E}\\{\left|a_{0,k,l}\right|^{2}\\}$
$\displaystyle=\frac{\mu_{\rm d}\beta_{{\rm d},k,l}}{\mu_{\rm
d}+1}\\!+\\!\frac{\mu_{\rm r}\mu_{\rm t}\beta_{{\rm r},k}\beta_{{\rm
t},l}}{\left(\mu_{\rm r}\\!+\\!1\right)\left(\mu_{\rm
t}\\!+\\!1\right)}\left|\sum_{m=1}^{M}{b_{m}e^{j\left(\psi_{k,m,l}+\theta_{m,l}\right)}}\right|^{2}\\!\\!\\!+\\!2\mathrm{Re}\left\\{\sqrt{\frac{\mu_{\rm
r}\mu_{\rm t}\beta_{{\rm r},k}\beta_{{\rm t},l}}{\left(\mu_{\rm
r}+1\right)\left(\mu_{\rm
t}+1\right)}}\sum_{m=1}^{M}{b_{m}e^{j\left(\psi_{k,m,l}+\theta_{m,l}\right)}}\right\\}$
$\displaystyle=\frac{\mu_{\rm r}\mu_{\rm t}\beta_{{\rm r},k}\beta_{{\rm
t},l}}{\left(\mu_{\rm r}+1\right)\left(\mu_{\rm
t}+1\right)}\boldsymbol{\phi}_{l}^{\rm
H}\boldsymbol{\psi}_{k,l}\boldsymbol{\psi}_{k,l}^{\rm
H}\boldsymbol{\phi}_{l}+2\sqrt{\frac{\mu_{\rm r}\mu_{\rm t}\beta_{{\rm
r},k}\beta_{{\rm t},l}}{\left(\mu_{\rm r}+1\right)\left(\mu_{\rm
t}+1\right)}}\mathrm{Re}\left\\{\boldsymbol{\psi}_{k,l}^{\rm
H}\boldsymbol{\phi}_{l}\right\\}+\frac{\mu_{\rm d}\beta_{{\rm
d},k,l}}{\mu_{\rm d}+1}.$ (26)

As ${\mathbb{E}\\{\mathbf{g}_{{\rm r},k}^{\rm NLoS}(\mathbf{g}_{{\rm
r},k}^{\rm NLoS})^{\rm H}\\}=\mathbf{I}_{M}}$, we have
${\mathbb{E}\\{\left|a_{1,k,l}\right|^{2}\\}=\frac{\beta_{{\rm
d},k,l}}{\mu_{\rm d}+1}}$,
${\mathbb{E}\\{\left|a_{2,k,l}\right|^{2}\\}=\frac{\mu_{\rm t}\beta_{{\rm
r},k}\beta_{{\rm t},l}}{\left(\mu_{\rm r}+1\right)\left(\mu_{\rm
t}+1\right)}\boldsymbol{\phi}_{l}^{\rm H}\boldsymbol{\phi}_{l}}$,
${\mathbb{E}\\{\\!\left|a_{3,k,l}\\!\right|^{2}\\!\\}\\!\\!=\\!\\!\frac{\mu_{\rm
r}\beta_{{\rm r},k}\beta_{{\rm t},l}}{\left(\\!\mu_{\rm
r}\\!+1\\!\right)\left(\mu_{\rm t}\\!+1\\!\right)}\boldsymbol{\phi}_{l}^{\rm
H}\\!\boldsymbol{\phi}_{l}}$, and
${\mathbb{E}\\{\\!\left|a_{4,k,l}\\!\right|^{2}\\!\\}\\!\\!=\\!\\!\frac{\beta_{{\rm
r},k}\beta_{{\rm t},l}}{\left(\\!\mu_{\rm r}\\!+1\\!\right)\left(\\!\mu_{\rm
t}\\!+1\\!\right)}\boldsymbol{\phi}_{l}^{\rm H}\\!\boldsymbol{\phi}_{l}}$. The
expansion of term ${a_{0,k,l}}$ and the derivation of
${\mathbb{E}\\{\left|a_{0,k,l}\right|^{2}\\}}$ are respectively shown in
(III-A) and (III-A), and ${\boldsymbol{\psi}_{k,l}}$ in (III-A) is given by

$\displaystyle\boldsymbol{\psi}_{k,l}\triangleq$
$\displaystyle\left[\begin{matrix}e^{-j\psi_{k,1,l}},&\cdots&,\\\
\end{matrix}e^{-j\psi_{k,M,l}}\right]^{\rm T},$ (27)

where

$\displaystyle\\!\\!\\!\\!\\!\psi_{k,m,l}\triangleq$
$\displaystyle\frac{1}{\lambda}(2\pi\left(d_{{\rm d},k,l}+d_{{\rm
r},k}-d_{{\rm t},l}\right)$
$\displaystyle+2\pi\left(m-1\right)d\left(\cos\omega_{{\rm
r},k}-\cos\omega_{{\rm t},l}\right)).$ (28)

Then, ${\tilde{P}_{k,l}}$ in (14) can be reformulated as

$\displaystyle\tilde{P}_{k,l}=\left(1-\eta\right)p_{k}\mathcal{D}_{k,l},$ (29)

where

$\displaystyle\mathcal{D}_{k,l}\\!\triangleq$
$\displaystyle\mathbb{E}\\{\left|g_{k,l}\right|^{2}\\}=\boldsymbol{\phi}_{l}^{\rm
H}\mathbf{A}_{k,l}\boldsymbol{\phi}_{l}+2\mathrm{Re}\left\\{\mathbf{a}_{k,l}^{\rm
H}\boldsymbol{\phi}_{l}\right\\}+\beta_{{\rm d},k,l},$ (30)
$\displaystyle\\!\\!\\!\\!\mathbf{A}_{k,l}\\!\triangleq$
$\displaystyle\frac{\mu_{\rm r}\mu_{\rm t}\beta_{{\rm r},k}\beta_{{\rm
t},l}}{\left(\mu_{\rm r}\\!\\!+\\!1\right)\\!\left(\mu_{\rm
t}\\!\\!+\\!1\right)}\boldsymbol{\psi}_{k,l}\boldsymbol{\psi}_{k,l}^{\rm
H}\\!\\!+\\!{\frac{\left(\mu_{\rm r}\\!\\!+\\!\\!\mu_{\rm
t}\\!\\!+\\!\\!1\right)\\!\beta_{{\rm r},k}\beta_{{\rm t},l}}{\left(\mu_{\rm
r}\\!+\\!1\right)\\!\left(\mu_{\rm t}\\!+\\!1\right)}}\\!\mathbf{I}_{M},$ (31)
$\displaystyle\mathbf{a}_{k,l}\triangleq$ $\displaystyle\sqrt{\frac{\mu_{\rm
d}\mu_{\rm r}\mu_{t}\beta_{{\rm d},k,l}\beta_{{\rm r},k}\beta_{{\rm
t},l}}{\left(\mu_{\rm d}+1\right)\left(\mu_{\rm r}+1\right)\left(\mu_{\rm
t}+1\right)}}\boldsymbol{\psi}_{k,l}.$ (32)

Therefore, by defining ${h_{k}(\boldsymbol{\phi})\\!\triangleq\\!(1/E_{k}^{\rm
req})\\!\sum_{l\in\mathcal{L}}{\\!t_{l}\tilde{P}_{k,l}}}$, constraint (18b)
can be reformulated as the following tractable form.

$\displaystyle
h_{k}(\boldsymbol{\phi})\geqslant\varepsilon,\,k\in\mathcal{K}.$ (33)

where

$\displaystyle\\!\\!\\!h_{k}\\!\left(\boldsymbol{\phi}\right)\\!=$
$\displaystyle\boldsymbol{\phi}^{\rm
H}\hat{\mathbf{A}}_{k}\boldsymbol{\phi}+2\mathrm{Re}\left\\{\hat{\mathbf{a}}_{k}^{\rm
H}\boldsymbol{\phi}\right\\}+{\zeta}_{1,k},$ (34)
$\displaystyle\boldsymbol{\phi}\\!\\!\\!\\!\quad\triangleq$
$\displaystyle\left[\boldsymbol{\phi}_{1}^{\rm
T},\cdots,\boldsymbol{\phi}_{L-1}^{\rm T}\right]^{\rm T},$ (35)
$\displaystyle{\zeta}_{1,k}\triangleq$
$\displaystyle\sum_{l\in{\mathcal{L}}}{\frac{(1-\eta)p_{k}t_{l}}{E_{k}^{\rm
req}}\beta_{{\rm d},k,l}},$ (36)
$\displaystyle\\!\\!\\!\\!\hat{\mathbf{A}}_{k}\triangleq$ $\displaystyle{\rm
diag}\left(\\!\\!\frac{\left(1\\!-\\!\eta\right)\\!p_{k}t_{1}A_{k,1}}{E_{k}^{\rm
req}},\\!...,\\!\frac{\left(1\\!-\\!\eta\right)\\!p_{k}t_{L-1}A_{k,L-1}}{E_{k}^{\rm
req}}\\!\\!\right)\\!\\!,\\!\\!$ (37)
$\displaystyle\hat{\mathbf{a}}_{k}\triangleq$
$\displaystyle\left[\frac{\left(1-\\!\eta\right)p_{k}t_{1}}{E_{k}^{\rm
req}}a_{k,1}^{\rm
T},\cdots,\\!\frac{\left(1-\eta\right)p_{k}t_{L-1}}{E_{k}^{\rm
req}}a_{k,L-1}^{\rm T}\right]^{\rm T}\\!\\!\\!\\!.$ (38)

As the term
${\mathbb{E}\\{\left\|\mathbf{\Theta}_{l}\mathbf{s}_{l}\right\|^{2}\\}}$ in
(17d) is difficult to handle, we reformulate it into a more tractable form as

$\displaystyle\mathbb{E}\left\\{\left\|\mathbf{\Theta}_{l}\left(\sum_{k\in\mathcal{K}}\sqrt{p_{k}}\mathbf{g}_{{\rm
t},l}x_{k}+\mathbf{n}_{\rm RIS}\right)\right\|^{2}\right\\}$
$\displaystyle=\Big{(}\sum_{k\in\mathcal{K}}{p_{k}}\mathbb{E}\Big{\\{}\\!\\!\sqrt{\beta_{{\rm
t},l}}\left(\sqrt{\frac{\mu_{\rm t}}{\mu_{\rm t}+1}}\mathbf{g}_{{\rm
t},l}^{\rm LoS}+\sqrt{\frac{1}{\mu_{\rm t}+1}}\mathbf{g}_{{\rm t},l}^{\rm
NLoS}\right)^{\rm H}$ $\displaystyle\\!\\!\\!\\!\quad\mathbf{\Theta}_{l}^{\rm
H}\\!\mathbf{\Theta}_{l}\sqrt{\\!\beta_{{\rm
t},l}}\left(\\!\sqrt{\\!\frac{\mu_{\rm t}}{\mu_{\rm
t}\\!+\\!1}}\mathbf{g}_{{\rm t},l}^{\rm
LoS}\\!\\!+\\!\sqrt{\\!\frac{1}{\mu_{\rm t}\\!+\\!1}}\mathbf{g}_{{\rm
t},l}^{\rm N\\!LoS}\right)\\!\\!\Big{\\}}\\!+\\!\delta_{\rm
RIS}^{2}\boldsymbol{\phi}_{l}^{\rm H}\boldsymbol{\phi}_{l}\\!\\!\Big{)}\\!$
$\displaystyle=\left(\sum_{k\in\mathcal{K}}{p_{k}}\left(\frac{\beta_{{\rm
t},l}\mu_{\rm t}}{\mu_{\rm t}+1}\boldsymbol{\phi}_{l}^{\rm
H}\boldsymbol{\phi}_{l}+\frac{\beta_{{\rm t},l}}{\mu_{\rm
t}+1}\boldsymbol{\phi}_{l}^{\rm H}\boldsymbol{\phi}_{l}\right)+\delta_{\rm
RIS}^{2}\boldsymbol{\phi}_{l}^{\rm H}\boldsymbol{\phi}_{l}\right)$
$\displaystyle=\left(\sum_{k\in\mathcal{K}}{p_{k}}\beta_{{\rm
t},l}+\delta_{\rm RIS}^{2}\right)\boldsymbol{\phi}_{l}^{\rm
H}\boldsymbol{\phi}_{l}.$ (39)

Then, constraint (17d) can be reformulated as

$\displaystyle\left(\sum_{k\in\mathcal{K}}{p_{k}}\beta_{{\rm t},l}+\delta_{\rm
RIS}^{2}\right)\boldsymbol{\phi}_{l}^{\rm
H}\boldsymbol{\phi}_{l}t_{l}\leqslant E^{\rm act}_{\rm
RIS},\,l\in{\mathcal{L}}.$ (40)

Note that constraints (17b) and (33) are still non-convex. By using the
similar processing procedure in (39), we can obtain
${\mathbb{E}\\{\|\mathbf{g}_{{\rm r},k}^{\rm
H}\mathbf{\Theta}_{l}\|^{2}\\}=\beta_{{\rm r},k}\boldsymbol{\phi}_{l}^{\rm
H}\boldsymbol{\phi}_{l}\delta_{\rm RIS}^{2}}$, and reformulate (17b) as

$\displaystyle\frac{\eta p_{k}\mathcal{D}_{k,l}}{\\!\\!\\!\sum\limits_{j\neq
k,k\in\mathcal{K}}{\\!\\!\\!\eta p_{j}\mathcal{D}_{k,l}}\\!+\\!\beta_{{\rm
r},k}\boldsymbol{\phi}_{l}^{\rm H}\boldsymbol{\phi}_{l}\delta_{\rm
RIS}^{2}\\!+\\!\delta_{\rm
r}^{2}}\\!\geqslant\\!\gamma_{k},\,l\in{\mathcal{L}}.$ (41)

Then, we can rewrite (41) as

$\displaystyle\boldsymbol{\phi}_{l}^{\rm
H}\mathbf{F}_{k,l}\boldsymbol{\phi}_{l}+2\mathrm{Re}\left\\{\mathbf{f}_{k,l}^{\rm
H}\boldsymbol{\phi}_{l}\right\\}\geqslant\varGamma_{k,l},\,l\in{\mathcal{L}},$
(42)

where

$\displaystyle\mathbf{F}_{k,l}$
$\displaystyle=\eta\big{(}p_{k}-\gamma_{k}\sum_{j\neq
k,j\in\mathcal{K}}{p_{j}}\big{)}\mathbf{A}_{k,l}-\gamma_{k}\beta_{{\rm
r},k}\delta_{\rm RIS}^{2}\mathbf{I}_{M},$ (43) $\displaystyle\mathbf{f}_{k,l}$
$\displaystyle=\eta\big{(}p_{k}-\gamma_{k}\sum_{j\neq
k,j\in\mathcal{K}}{p_{j}}\big{)}\mathbf{a}_{k,l},$ (44)
$\displaystyle\varGamma_{k,l}$ $\displaystyle=\gamma_{k}\delta_{\rm
r}^{2}-\eta\big{(}p_{k}-\gamma_{k}\sum_{j\neq
k,j\in\mathcal{K}}{p_{j}}\big{)}\beta_{{\rm d},k,l}.$ (45)

Since we can obtain a lower bound of a convex function by using its first-
order Taylor expansion, the convex lower bound of ${\boldsymbol{\phi}_{l}^{\rm
H}\mathbf{F}_{k,l}\boldsymbol{\phi}_{l}}$ is expressed as

$\displaystyle\boldsymbol{\phi}_{l}^{\rm
H}\mathbf{F}_{k,l}\boldsymbol{\phi}_{l}$
$\displaystyle\geqslant({\boldsymbol{\phi}_{l}}^{n})^{\rm
H}\mathbf{F}_{k,l}{\boldsymbol{\phi}_{l}^{n}}+2\mathrm{Re}\\{({\boldsymbol{\phi}_{l}}^{n})^{\rm
H}\mathbf{F}_{k,l}\\}(\boldsymbol{\phi}_{l}-{\boldsymbol{\phi}_{l}^{n}})$
$\displaystyle=2\mathrm{Re}\\{({\boldsymbol{\phi}_{l}}^{n})^{\rm
H}\mathbf{F}_{k,l}\boldsymbol{\phi}_{l}\\}-({\boldsymbol{\phi}_{l}}^{n})^{\rm
H}\mathbf{F}_{k,l}{\boldsymbol{\phi}_{l}^{n}},$ (46)

where ${{\boldsymbol{\phi}_{l}^{n}}}$ is the value of
${\boldsymbol{\phi}_{l}}$ at the ${n}$-th iteration.

Finally, constraint (42) can be reformulated as

$\displaystyle\\!\\!\\!\\!\\!\\!2\mathrm{Re}\\{({\boldsymbol{\phi}}_{l}^{n})^{\rm
H}\\!\mathbf{F}_{k,l}\boldsymbol{\phi}_{l}\\}\\!\\!-\\!({\boldsymbol{\phi}}_{l}^{n})^{\rm
H}\\!\mathbf{F}_{k,l}{\boldsymbol{\phi}_{l}^{n}}\\!\\!\\!+\\!\\!2\mathrm{Re}\\!\left\\{\\!\mathbf{f}_{k,l}^{\rm
H}\boldsymbol{\phi}_{l}\\!\right\\}\\!\\!\\!\geqslant\\!\\!\varGamma_{k,l},\,\\!\\!l\\!\in\\!{\mathcal{L}}.\\!\\!\\!\\!\\!$
(47)

For non-convex constraint (33), we similarly derive a lower bound of
${\boldsymbol{\phi}^{\rm H}\hat{\mathbf{A}}_{k}\boldsymbol{\phi}}$, and
reformulate (33) as

$\displaystyle h_{k}\left(\boldsymbol{\phi}\right)$ $\displaystyle\geqslant
2\mathrm{Re}\\{({\boldsymbol{\phi}}^{n})^{\rm
H}\\!\hat{\mathbf{A}}_{k}\boldsymbol{\phi}\\}\\!-\\!({\boldsymbol{\phi}}^{n})^{\rm
H}\\!\hat{\mathbf{A}}_{k}{\boldsymbol{\phi}^{n}}\\!\\!\\!+\\!2\mathrm{Re}\\{\hat{\mathbf{a}}_{k}^{\rm
H}\boldsymbol{\phi}\\}\\!\\!+\\!\\!{\zeta}_{1,k}$
$\displaystyle=2\mathrm{Re}\\{\mathbf{b}_{k}^{\rm
H}\boldsymbol{\phi}\\}+\zeta_{2,k}\triangleq
f_{k}(\boldsymbol{\phi})\geqslant\varepsilon,$ (48)

where ${\mathbf{b}_{k}^{\rm H}\\!=({\boldsymbol{\phi}}^{n})^{\rm
H}\hat{\mathbf{A}}_{k}+\hat{\mathbf{a}}_{k}^{\rm H}}$ and
${\zeta_{2,k}={\zeta}_{1,k}-({\boldsymbol{\phi}}^{n})^{\rm
H}\hat{\mathbf{A}}_{k}{\boldsymbol{\phi}^{n}}}$.

Then, by replacing (17b), (17d) and (18b) with (47), (40) and (48),
respectively, Problem (18) is reformulated as

$\displaystyle\max_{{\boldsymbol{\phi}},\varepsilon}$
$\displaystyle\quad\varepsilon$ (49a) $\displaystyle\mathrm{s}.\mathrm{t}.$
$\displaystyle\quad
f_{k}(\boldsymbol{\phi})\geqslant\varepsilon,k\in\mathcal{K},$ (49b)
$\displaystyle\quad\eqref{ris-energy},\eqref{sinr2},$

which can be solved using CVX. By solving Problem (49) at each SCA iteration,
the generated solutions are guaranteed to converge to the Karush-Kuhn-Tucker
solution of Problem (18) [13].

###  III-B The Trajectory Design of UAV

From Problem (17), the subproblem for ${\mathbf{q}_{\rm V}}$ and
${\mathbf{t}}$ with given ${\\{\boldsymbol{\phi}_{l}\\}}$ can be formulated as

$\displaystyle\min_{{\mathbf{q}_{\rm V}},\mathbf{t}}$ $\displaystyle\quad
E_{\rm V}\left(\mathbf{q}_{\rm V},\mathbf{t}\right)$ (50a)
$\displaystyle\mathrm{s}.\mathrm{t}.$ $\displaystyle\quad t_{l}\geqslant
0,\,l\in\mathcal{L},$ (50b)
$\displaystyle\quad\eqref{swipt_Problem1-b},\,\eqref{swipt_Problem1-c},\,\eqref{ris-
energy}.$

As constraints (17c) and (40) are non-convex for this subproblem, Problem (50)
cannot be solved by CVX directly. Due to the strong coupling between
${\tilde{P}_{k,l}}$ and ${\mathbf{q}_{\rm V}}$ in (29), (17c) is difficult to
handle. Since the value of ${\mathbf{q}_{\rm V}}$ generally changes little in
each iteration, by denoting ${\boldsymbol{\psi}_{k,l}^{n}}$ as the value of
${\boldsymbol{\psi}_{k,l}}$ at the ${n}$-th iteration, $\tilde{P}_{k,l}$ can
be approximated as

$\displaystyle\tilde{P}_{k,l}\approx$
$\displaystyle\left(1-\eta\right)p_{k}((U_{1,k,l}^{n}+U_{3,k,l})\beta_{{\rm
t},l}$ $\displaystyle+U_{2,k,l}^{n}\sqrt{\beta_{{\rm d},k,l}\beta_{{\rm
t},l}}+\beta_{{\rm d},k,l}),$ (51)

where

$\displaystyle U_{1,k,l}^{n}=\frac{\mu_{\rm r}\mu_{\rm t}\beta_{{\rm
r},k}}{\left(\mu_{\rm r}+1\right)\left(\mu_{\rm
t}+1\right)}\boldsymbol{\phi}_{l}^{\rm
H}\boldsymbol{\psi}_{k,l}^{n}(\boldsymbol{\psi}_{k,l}^{n})^{\rm
H}\boldsymbol{\phi}_{l},$ (52) $\displaystyle
U_{2,k,l}^{n}=2\sqrt{\frac{\mu_{\rm d}\mu_{\rm r}\mu_{\rm t}\beta_{{\rm
r},k}}{\left(\mu_{\rm d}\\!+1\right)\left(\mu_{\rm
r}\\!+1\right)\left(\mu_{\rm
t}\\!+1\right)}}\\!\mathrm{Re}\\!\left\\{\\!(\boldsymbol{\psi}_{k,l}^{n})^{\rm
H}\boldsymbol{\phi}_{l}\\!\right\\}\\!\\!,$ (53) $\displaystyle
U_{3,k,l}=\frac{\left(\mu_{\rm r}+\mu_{\rm t}+1\right)\beta_{{\rm
r},k}}{\left(\mu_{\rm r}+1\right)\left(\mu_{\rm
t}+1\right)}\boldsymbol{\phi}_{l}^{\rm H}\boldsymbol{\phi}_{l}.$ (54)

Hence, we approximate (17c) as

$\displaystyle\frac{\left(1-\eta\right)}{E_{k}^{\rm
req}}\sum_{l\in{\mathcal{L}}}p_{k}((U_{1,k,l}^{n}+U_{3,k,l})\beta_{{\rm t},l}$
$\displaystyle+U_{2,k,l}^{n}\sqrt{\beta_{{\rm d},k,l}\beta_{{\rm
t},l}}+\beta_{{\rm d},k,l})t_{l}\geqslant 1.$ (55)

In order to transform (III-B) into a convex constraint, we derive the lower
bounds of ${\beta_{{\rm d},k,l}}$ and ${\beta_{{\rm t},l}}$ with its first-
order Taylor expansion as follows

$\displaystyle\\!\\!\\!\\!\\!\\!\beta_{{\rm d},k,l}\geqslant$
$\displaystyle\frac{\beta_{0}}{(|q_{{\rm V},l}^{n}-q_{{\rm S},k}|^{2}+H_{\rm
V}^{2})^{\frac{\tau_{\rm d}}{2}}}$ $\displaystyle-\frac{\tau_{\rm
d}\beta_{0}(|q_{{\rm V},l}-q_{{\rm S},k}|^{2}-|q_{{\rm V},l}^{n}-q_{{\rm
S},k}|^{2})}{2(|q_{{\rm V},l}^{n}-q_{{\rm S},k}|^{2}+H_{\rm
V}^{2})^{\frac{\tau_{\rm d}}{2}+1}}\triangleq\bar{\beta}_{{\rm d},k,l},\\!\\!$
(56) $\displaystyle\beta_{{\rm t},l}\geqslant$
$\displaystyle\frac{\beta_{0}}{(|q_{{\rm V},l}^{n}-q_{\rm R}|^{2}+(H_{\rm
V}-H_{\rm R})^{2})^{\frac{\tau_{\rm d}}{2}}}$ $\displaystyle-\frac{\tau_{\rm
t}\beta_{0}(|q_{{\rm V},l}-q_{\rm R}|^{2}-|q_{{\rm V},l}^{n}-q_{\rm
R}|^{2})}{2(|q_{{\rm V},l}^{n}-q_{\rm R}|^{2}+(H_{\rm V}-H_{\rm
R})^{2})^{\frac{\tau_{\rm t}}{2}+1}}\triangleq\bar{\beta}_{{\rm t},l},$ (57)

where ${q^{n}_{{\rm V},l}}$ is the value of ${q_{{\rm V},l}}$ at the ${n}$-th
iteration. By setting the slack variables ${\\{{\tilde{z}}_{1,l}\\}}$,
${\\{{\tilde{z}}_{2,k,l}\\}}$, ${\\{{\tilde{z}}_{3,k,l}\\}}$ and
${\\{{\tilde{y}}_{k,l}\\}}$, constraint (III-B) can be transformed as

$\displaystyle W_{k,l}\geqslant\frac{E_{k}^{\rm
req}{\tilde{y}}_{k,l}^{2}}{\left(1-\eta\right)p_{k}t_{l}},$ (58)

where

$\displaystyle
W_{k,l}=(U_{1,k,l}^{n}+U_{3,k,l}){\tilde{z}}_{1,l}+U_{2,k,l}^{n}{\tilde{z}}_{3,k,l}+{\tilde{z}}_{2,k,l},$
(59) $\displaystyle\bar{\beta}_{{\rm t},l}\geqslant{\tilde{z}}_{1,l}\geqslant
0,\,l\in\mathcal{L},$ (60) $\displaystyle\bar{\beta}_{{\rm
d},k,l}\geqslant{\tilde{z}}_{2,k,l}\geqslant
0,\,l\in\mathcal{L},\,k\in\mathcal{K},$ (61)
$\displaystyle{\tilde{z}}_{1,l}\geqslant(1/{\tilde{z}}_{2,k,l}){\tilde{z}}_{3,k,l}^{2},\,l\in\mathcal{L},\,k\in\mathcal{K},$
(62) $\displaystyle\sum_{l\in{\mathcal{L}}}{{\tilde{y}}_{k,l}^{2}}\geqslant
1,\,k\in\mathcal{K}.$ (63)

Similarly, constraint (63) is transformed as

$\displaystyle\sum_{l\in{\mathcal{L}}}{(2{\tilde{y}}_{k,l}^{n}{\tilde{y}}_{k,l}-({\tilde{y}}_{k,l}^{n})^{2})\geqslant
1},\,k\in\mathcal{K},$ (64)

where ${{\tilde{y}}_{k,l}^{n}}$ is the value of ${{\tilde{y}}_{k,l}}$ at the
${n}$-th iteration.

Algorithm 1 SCA-based algorithm for solving Problem (13)

Initialize: the feasible ${\boldsymbol{\phi}^{(1)}}$, ${\mathbf{q}_{\rm
V}^{(1)}}$ and ${\mathbf{t}^{(1)}}$, the maximum iterations ${x_{\rm max}}$,
${n_{\rm max}}$ and ${r_{\rm max}}$, error tolerance ${\sigma}$.

1:for $x=1$ to $x_{max}$ do

2: Set $n=1$, $r=1$;

3: while ${n\leqslant n_{\rm max}}$ do

4: Given ${\boldsymbol{\phi}^{(1)}}$, calculate ${\mathbf{q}_{\rm V}^{(n+1)}}$
and ${\mathbf{t}^{(n+1)}}$ by solving

5: Problem (70);

6: Set ${n+1\to n}$

7: end while

8: Output ${\mathbf{q}_{\rm V}^{[1]}=\mathbf{q}_{\rm V}^{(n+1)}}$ and
${\mathbf{t}^{[1]}=\mathbf{t}^{(n+1)}}$

9: while
$|f(\boldsymbol{\phi}^{(r\\!+1)})\\!-\\!\\!f(\boldsymbol{\phi}^{(r)})|>\\!\sigma
f(\boldsymbol{\phi}^{(r)})$ and ${r\\!\leqslant r_{\rm max}}$ do

10: Given ${\mathbf{t}^{[1]}}$ and ${\mathbf{q}_{\rm V}^{[1]}}$, calculate
${\boldsymbol{\phi}^{(r+1)}}$ by solving

11: Problem (49);

12: Set ${r+1\to r}$

13: end while

14: Output ${\boldsymbol{\phi}^{[1]}=\boldsymbol{\phi}^{(r+1)}}$

15: if ${\rm min}\\{f(\boldsymbol{\phi}^{[1]})\\}<{\rm
min}\\{f(\boldsymbol{\phi}^{(1)})\\}$ then

16: Set $\boldsymbol{\phi}^{(1)}\to\boldsymbol{\phi}^{[1]}$;

17: else Set ${\boldsymbol{\phi}^{[1]}\to\boldsymbol{\phi}^{(1)}}$;

18: end if

19:end for

To address the non-convex constraint (40), we first derive a lower bound of
${\beta_{{\rm t},l}}$. By defining $\mathcal{Q}_{l}\triangleq|q_{{\rm
V},l}-q_{\rm R}|^{2}=(q_{{\rm V},l}^{x}-q_{{\rm R}}^{x})^{2}+(q_{{\rm
V},l}^{y}-q_{{\rm R}}^{y})^{2}$, and utilizing the first-order Taylor
expansions to ${q_{{\rm V},l}^{x}}$ and ${q_{{\rm V},l}^{y}}$, we have

$\displaystyle\mathcal{Q}_{l}\geqslant$ $\displaystyle((q_{{\rm
V},l}^{x})^{n}-q_{{\rm R}}^{x})^{2}\\!\\!+((q_{{\rm V},l}^{y})^{n}-q_{{\rm
R}}^{y})^{2}$ $\displaystyle+2((q_{{\rm V},l}^{x})^{n}-q_{{\rm
R}}^{x})(q_{{\rm V},l}^{x}-(q_{{\rm V},l}^{x})^{n})$ $\displaystyle+2((q_{{\rm
V},l}^{y})^{n}-q_{{\rm R}}^{y})(q_{{\rm V},l}^{y}-(q_{{\rm
V},l}^{y})^{n})\triangleq\bar{\mathcal{Q}_{l}},$ (65)

where ${(q_{{\rm V},l}^{x})^{n}}$ and ${(q_{{\rm V},l}^{y})^{n}}$ are the
values of ${q_{{\rm V},l}^{x}}$ and ${q_{{\rm V},l}^{y}}$ at the ${n}$-th
iteration, respectively.

Then, we can reformulate constraint (40) as

$\displaystyle\\!\\!\\!\\!\big{(}\sum_{k\in\mathcal{K}}{p_{k}}\frac{\beta_{0}}{(\bar{\mathcal{Q}_{l}}\\!+\\!\left(H_{\rm
V}\\!-\\!H_{\rm R}\right)^{2})^{\frac{\tau_{\rm t}}{2}}}+\delta_{\rm
RIS}^{2}\big{)}\boldsymbol{\phi}_{l}^{\rm
H}\boldsymbol{\phi}_{l}t_{l}\leqslant E^{\rm act}_{\rm RIS}.$ (66)

By utilizing the first-order Taylor expansion with respect to ${\\{t_{l}\\}}$,
constraint (66) can be reformulated as

$\displaystyle\big{(}\sum_{k\in\mathcal{K}}{p_{k}}\frac{\beta_{0}}{(\bar{\mathcal{Q}_{l}}\\!+\\!\left(H_{\rm
V}\\!-\\!H_{\rm R}\right)^{2})^{\frac{\tau_{\rm t}}{2}}}+\delta_{\rm
RIS}^{2}\big{)}\boldsymbol{\phi}_{l}^{\rm H}\boldsymbol{\phi}_{l}$
$\displaystyle\leqslant E^{\rm act}_{\rm
RIS}(\frac{1}{t_{l}^{n}}-\frac{1}{(t^{n}_{l})^{2}}(t_{l}-t^{n}_{l})),\,l\in\mathcal{L},$
(67)

where ${t_{l}^{n}}$ is the value of ${t_{l}}$ at the ${n}$-th iteration.

For constraint (17b), we reformulate it into

$\displaystyle\frac{\eta p_{k}W_{k,l}}{\sum\limits_{j\neq
k,j\in\mathcal{K}}{\eta p_{j}W_{k,l}}+\beta_{{\rm r},k}\delta_{\rm
RIS}^{2}\boldsymbol{\phi}_{l}^{\rm H}\boldsymbol{\phi}_{l}+\delta_{\rm
r}^{2}}\geqslant\gamma_{k}.$ (68)

With some basic transformations, we can rewrite (68) as

$\displaystyle\eta\big{(}\\!p_{k}-\gamma_{k}\sum\limits_{j\neq
k,j\in\mathcal{K}}{p_{j}}\\!\big{)}\\!W_{k,l}\\!\geqslant\\!\gamma_{k}(\beta_{{\rm
r},k}\delta_{\rm RIS}^{2}\boldsymbol{\phi}_{l}^{\rm
H}\boldsymbol{\phi}_{l}+\delta_{\rm r}^{2}).$ (69)

Finally, by defining ${\Upsilon}$ as the set of slack variables
${\\{{\tilde{z}}_{1,l}\\}}$, ${\\{{\tilde{z}}_{2,k,l}\\}}$,
${\\{{\tilde{z}}_{3,k,l}\\}}$ and ${\\{{\tilde{y}}_{k,l}\\}}$, we obtain the
problem to be solved at the $n$-th SCA iteration for Problem (50) as follows

$\displaystyle\min_{{\mathbf{q}_{\rm V}},\mathbf{t},\Upsilon}$
$\displaystyle\quad E_{\rm V}\left(\mathbf{q}_{\rm V},\mathbf{t}\right)$ (70a)
$\displaystyle\mathrm{s}.\mathrm{t}.\\!\\!\\!\\!\quad$
$\displaystyle\quad\eqref{fort},\,\eqref{function2},\eqref{set2}-\eqref{set4},\eqref{set5},\eqref{ris-
energy3},\eqref{sinr3}.$

Problem (70) is convex and thus can be solved using CVX.

###  III-C Algorithm Development

Based on the above discussions, we propose an SCA-based algorithm for solving
Problem (17), of which details are summarized in Algorithm 1. Furthermore, we
update the value of ${\boldsymbol{\psi}_{k,l}}$ when solving Problem (70) to
limit the approximation gap in (III-B). Since the optimal solution to Problem
(70) may not satisfy constraint (58), the objective function value of (17)
generated by Algorithm 1 may not decrease monotonically. However, the
simulation results show the fluctuations are not significant.

Figure 2: The total energy cost versus the number of reflecting elements.

Figure 3: The optimized trajectory of UAV under FHB protocol.

##  IV Simulation Results

To provide numerical results of our system, 5 users are assumed in a
semicircular area with a radius of ${30\,\rm m}$, and the positions of user
S1-S5 are ${(-30,0)}$, ${(-15\sqrt{2},15\sqrt{2})}$, ${(0,30)}$, ${(30,0)}$
and ${(15\sqrt{2}/2,15\sqrt{2}/2)}$, respectively, i.e., user S1-S4 are placed
on the circle, and user S5 is placed at the midpoint of the radius. The
position of the active RIS is ${(q_{\rm R}^{x},q_{\rm R}^{y})=(0,0)}$ with
${H_{\rm R}=10\,{\rm m}}$, the initial position ${q_{{\rm V},0}=(-35,0)}$ and
the final position ${q_{{\rm V},L}=(35,0)}$. The UAV’s flight height is set as
${H_{\rm V}=20\,{\rm m}}$, and the transmit power ${P_{k}=0.2\,{\rm w}}$. The
SINR threshold of the users and the energy constraint of the active RIS are
set as ${\gamma\\!=\\!-10\,{\rm dB}}$ and ${E_{\rm RIS}^{\rm
act}\\!=\\!20\,{\rm J}}$, respectively. The parameters related to the
propulsion power of the UAV are set as the same in [5], where the maximum-
range speed is ${v=18.3\,{\rm m/s}}$. Unless stated otherwise, we set the
Rician factors ${\mu_{\rm t}=\mu_{\rm r}=\mu_{\rm d}=10}$, the path-loss
exponents ${\tau_{\rm t}=\tau_{\rm r}=2.3}$ and ${\tau_{\rm d}=2.4}$, the
energy requirement for each user ${E_{\rm req}=0.04\,{\rm mJ}}$, ${\eta=0.5}$,
${\lambda=1\,{\rm m}}$, and ${d=0.5\,{\rm m}}$.

To compare the performance of two types RIS-aided model, the energy resources
of UAV are assumed to be equal in both the active RIS-aided and the passive
RIS-aided systems. Specifically, the total energy cost models are expressed as

$\displaystyle E_{\rm pas}$ $\displaystyle=E^{\rm pas}_{\rm V},$ (71)
$\displaystyle E_{\rm act}$ $\displaystyle=E_{\rm V}^{\rm act}+(L-1)E_{\rm
RIS}^{\rm act}.$ (72)

Fig. 3 shows the total energy cost versus the number of reflecting elements.
It is seen from the figure that the total energy cost decreases with the
increase of the number of reflecting elements, which verifies the
effectiveness of the RIS-aided system in energy saving. From the passive RIS-
aided scheme with ${M=32}$, the active RIS performs better in terms of energy
saving under the same energy resources in the UAV. Moreover, the convergence
of proposed algorithm, indicates that it converges within a few iterations and
exhibits no noticeable fluctuations.

Fig. 3 shows the optimized trajectory of both the active and passive RIS-aided
systems when ${M=32}$. Note that the trajectory of the UAV in the active RIS-
aided system is closer to the RIS than in the passive RIS-aided system, which
shows a decreased flight range, leading to a corresponding decrease in the
duration available for signal transmission.

##  V Conclusions

In this paper, an active RIS-aided UAV-enabled SWIPT system was considered. We
aimed to minimize the total energy cost of the UAV by alternately optimizing
the trajectories, the hovering time and the reflection vectors at the active
RIS based on the SCA method. The simulation results showed that the active
RIS-aided system performs better in energy saving than the passive RIS-aided
system under the same energy resources in the UAV-enabled systems.

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