Joint Resource and Trajectory Optimization for Energy Efficiency Maximization in UAV-Based Networks

Abstract

The explosive growth of unmanned aerial vehicles (UAVs)-based networks has accelerated in recent years. One of the crucial tasks of a UAV-based network is managing and allocating resources, including time, power, fly trajectory, and energy resources. We investigate a UAV-based network that gathers information from smart devices, sensor devices, and IoT devices (IDs) with respect to energy efficiency (EE) maximization. The EE of users served by the UAV over the time slots of a cycle is maximized through three categories: UAV trajectory optimization, power allocation, and time slot assignment. However, these are non-convex problems that are very difficult to solve. To solve the problem efficiently, we divide it step by step and convert the non-convex optimization problem into an equivalent convex optimization problem, optimizing each equivalent problem over each variable while other variables are fixed. Firstly, we perform a UAV trajectory optimization with a different number of ground users. Secondly, the Dinkelbach method is used to construct a non-convex fractional power allocation problem. In addition, we develop an algorithm to distribute time slots to all users, which continually raises the EE value. Eventually, a scheme is provided to sequentially update the method to each equivalent problem. The numerical results provide evidence that by solving the proposed sum-rate maximization problem, the performance of ground users has significantly improved with the support of the UAV-based network.

1. Introduction

The development of wireless communication systems, such as 5G, 6G, IoT, and Big Data, has explosively grown. In recent years, the public has been drawn to the emergence of intelligent devices such as wearable technology and other sensor devices. Several extensively used smart devices and sensor nodes exist in various systems’ networks. Intelligent devices are typically widely used in metropolitan locations, such as the smart home city. Additionally, smart devices are dispersed in remote locations that cannot be monitored or sensed by terrestrial access networks, including deserts, woodlands, and seas. It can be challenging to provide reliable uplink connections because terrestrial access networks do not serve smart devices or sensor devices. Therefore, unmanned aerial vehicles (UAVs) or drones emerge as comprehensive solutions thanks to their mobility. There have been many applications of UAVs in daily life to meet users’ needs. Especially, the leading companies in the telecommunications sector have also entered this potential field. Nokia has successfully launched telecommunications solutions for remote areas in Spain through drones. In addition, they also have Nokia drone network solutions for public safety and industries [1]. Amazon has developed UAV delivery systems that increase the overall safety and efficiency of the transportation system [2]. Furthermore, in the aspect of UAVs fire alarm monitoring, we can provide forest fire alarm monitoring as an instance. Since the geographical area of forests is usually far away, it is challenging to deploy base stations on the ground. However, the management of forest fires is essential. The overall strategy is to primarily deploy some intelligent devices in the woodland to collect and observe statements such as humidity and temperature, enhancing situational understanding and sustaining resource planning, hazard evaluation, and fast decision making, which is a critical task in forest fire deterrence. Consequently, fire alarm monitoring is a typical scenario for UAV-based networks. In the current approach, the UAV is integrated into the mobile network as an aerial base station to achieve wireless coverage for service areas or quick troubleshooting of natural disasters [3,4,5]. Such UAV applications can significantly improve data transmission efficiency, security, and transient scenarios in order to meet the needs of ground users because of their flexibility and mobility [6]. As a result, it can significantly improve the service quality of users.

In order to optimize the system performance, the amount of transmitted data per power consumption unit is usually assessed, which refers to the energy efficiency of the system. There were many works that have investigated solving the EE in wireless systems [7,8,9,10]. It is also crucial for the UAV-based networks to preserve EE to keep the maximum operating status and efficiently provide services to ground users. The authors in [11,12,13,14,15] investigated a system that is related to the EE of massive multiple-input–multiple-output (MIMO) systems with considerable Industrial Internet of Things (IIoT) devices. Ruan et al. [16] considered a UAV coverage concern and proposed a multi-UAV coverage model based on energy-efficient transmission. Qi et al. [17] focus on maximizing the overall EE while fulfilling the quality-of-service essentials of the cellular nodes via optimizing the scheduling strategies of the user, transmitting power of the UAV and D2D-TX nodes, and UAV trajectory. Nie et al. [18] investigated the EE optimization issue as a Markov decision process (MDP) that was based on the deep reinforcement learning (DRL) algorithm to develop the UAV trajectory with the constraints of the backscatter appliances scheduling, the power deliberation coefficients, the transmission power, and the fairness among backscatter devices [19]. The impact of a binary determination variable designated to organize UAV-to-user communication in the circumstance of the base stations’ malfunctioning was also considered in [20]. The authors in [21,22] studied EE UAV communications, extending the batteries and reliable energy. Ref. [23] studied the accompanying wireless data and energy transfer for the unmanned aerial vehicle (UAV) enabled Internet-of-Things (IoT) networks. However, this method also causes some problems, such as the management of the energy consumption for the activity of the UAV or the control of the movement of UAVs. In order to optimize the use of the system resources, there are two main criteria, including the total energy consumption and the achievable rate, which are used to estimate the method’s performance. Moreover, leaning on precise essentials from different networks, the system also needs to guarantee several minimum thresholds, including the association, energy allocation, communication coverage, and latency from terrestrial users [24,25]. Zhan et al. [26] strived to minimize the EE of cellular-connected UAVs via jointly developing the mission fulfillment time and UAV trajectory via the development of a DRL algorithm based on multi-step understanding and dual Q-learning over a dueling Deep Q-Network (DQN) architecture. Zeng and Hu et al. [27] studied hoe to meet the quality of service of all users with narrow strategy resources and limited UAV power by optimizing user communication scheduling, UAV trajectory, transfer power, and resource allocation to maximize EE and meet user QoE essentials. In particular, Zhang and Yan et al. [28] aim to maximize the end-to-end throughput through joint UAV trajectory and transmit power optimization. Ref. [29] studied the objective of maximizing the minimum average rate among all users via user scheduling and association; the UAV trajectories and transmit power are jointly optimized. Zeng in [6] manipulated the controllable channel variation induced by relay mobility; the end-to-end throughput is maximized via optimizing both the relay trajectory as well as the source/relay power allocation, which can achieve significant throughput gains over the conventional static relaying. Some literature has shown that owing to a more satisfactory air-terrestrial channel, UAVs can be widely embraced as base stations (BSs) and support for BSs in wireless networks. Zhao in [30] studied that UAVs can support small-cell base stations (SBSs) that offload traffic via wireless backhaul to enhance coverage and improve the rate. Nouri and Abouei et al. [31] consider the problem of a UAV-enabled cloud network under a partial estimation offloading technique, where numerous UAV-mounted aerial base stations are used to serve a gathering of remote Internet-of-Things ground-based smart devices (ISDs). In addition, some of the important challenges in UAV wireless networks are energy conservation and security, which are also considered in [32,33,34,35]. The authors in [36] studied the application of UAV for data collection with wireless charging, which is crucial for delivering a seamless range and enhancing system performance in the next-generation wireless networks by considering UAV as a Markov decision problem, and they manipulate Q-learning to discover the optimal policy. At the same time, the reward function considers the energy efficiency of UAV flight and data collection. In recent years, federated learning (FL) has also been a key vision of artificial intelligence [37,38]. The authors in [39] consider UAVs and wireless-powered communications (WPC) for FL networks to minimize the total energy consumption of the aerial server and users.

Although EE and UAVs have been considerably investigated in the prior works, very few studies have combined these three favorable factors to overcome their respective deficiencies. In this work, we have developed a UAV-based network scenario to collect data and information to meet users’ requirements. In scenarios where the distance between the UAV relay and the ground base station is far, it causes high delay, signal loss, and poor user quality. This inspired us to propose a UAV-base network solution that directly receives and transmits information to users without going through a transit point to save costs, reduce power consumption, and improve efficiency and high-security capabilities. As a performance metric for UAV-based networks, we use EE, which is the ratio of the total achievable rate to the total energy consumption. The EE maximization problem is essential for every network to ensure that the whole data from ground users can be transmitted to the UAV, given the amount of energy consumption. Therefore, this paper studies the EE maximization in a UAV-based network by jointly optimizing the UAV trajectory, power allocation, and time slot assignment to maximize uplink throughput for the terrestrial user in hard-to-reach areas. The primary contributions of this work are summarized as follows.

• We propose a UAV-based network where orthogonal frequency-division multiplexing (OFDM) is adopted for UAV gathering information. We aim to maximize EE, and the optimization problems of time slot assignment, UAV trajectory, and power allocation are decomposed into three parts to solve.
• First, the UAV optimization problem is formulated to maximize the uplink throughput of UAV-served users, with the flight trajectory of the UAV also considered. The Taylor theory we have put forward for this problem turns the non-convex problem into an approximated problem that becomes a convex problem.
• Second, the power allocation is one of the decisive factors for increasing EE. The Dinkelbach method is considered as an efficient algorithm in the infraction form problem to convert to a non-infraction form.
• Third, the system schedule of the number of time slots to allocate N users. We construct the price matrix in case the number of users is smaller than the number of time slots. Therefore, the Hungarian algorithm can be applied to obtain the optimal solution for the assignment problem.
• Finally, we show that the EE applied to our alternating optimization algorithm has been significantly improved compared to other algorithms. The flight trajectory of the UAV also shows us that it tends to always focus on serving ground user best.

The paper is organized as follows. The system model and problem formulation are described in Section 2. The UAV trajectory optimization, power allocation, and time slot assignment sub-problem are described and solved in Section 3. Numerical results for the verification of the proposed alternating algorithm and those related to the discussion are provided in Section 4, and concluding remarks are given in Section 5.

All the assumed notations used in this paper are included in

Table 1. Abbreviations.

Symbols	Descriptions
N	Number of users
L	Number of time slots
$P_n$	Allocated power at n user
$x_n$	Transmitted signal from the n-th user
$P_n^{\mathrm{thr}}$	Maximum transmit power of the n-th users
$y\left[l\right]$	Received signal at the UAV at the l-th time slot
$n_{\left[l\right]}$	Additive white Gaussian noise at the l-th time slot with zero mean
${\tau }_n\left[l\right]$	Integer variable
$h_n\left[l\right]$	Channel gain between the UAV and the n-th user in the l-th time slot
${[x\left[l\right],y\left[l\right]]}^T$	Position of the UAV at the l-th time slot.
$\mathbf{\Theta }$	Time slot assignment
$\mathit{p}$	Power allocation
$\mathit{W}$	Trajectory of UAV
${\rho }_0$	Reference channel power gain
r	Cellular radius
$\mathcal{T}$	Cycle duration
H	Height of UAV
$\mathit{v}$	Maximum speed of UAV
$\sigma$	Power spectral density of the thermal noise
$R_n\left[l\right]$	Achievable rate of the n-th user using the l-th time slot
$\eta$	Maximal EE
B	Bandwidth
$ϵ$, ${ϵ}_2$	Terminated conditions

2. System Model and Problem Formulation

2.1. System Model

Consider a system where a UAV network is used to gather information from users in a certain area for the purposes of monitoring devices and smart home systems, as shown in Figure 1. The system allows users to communicate with the UAV via different time slots. There are a total of N users in the system, while the number of time slots is L. Moreover, in this scenario, we assume all ground users and the UAV are equipped with a single antenna. During the uplink transmission, the n-th users use the allocated power $P_n$ to send the signal $x_n$ to the UAV. The transmit power of each user should satisfy the following constraints.

$$P_n\le P_n^{\mathrm{thr}},\forall n,$$

(1)

where $P_n^{\mathrm{thr}}$ is the maximum transmit power of the n-th users. We assume that $L\ge N$; then, each time slot is occupied by only one user, and a user could use more than one time slot. Thus, there is no co-channel interference between users during the uplink transmission.

The received signal at the UAV at the l-th time slot can be expressed as:

$$y\left[l\right]=\sum _{l=1}^L{\tau }_n\left[l\right]h_n\left[l\right]\sqrt{P_n\left[l\right]}{x}_n+n\left[l\right],$$

(2)

where $x_n$ is the transmitted signal from the n-th user. Here, $n_{\left[l\right]}$ represents the additive white Gaussian noise (AWGN) at the l-th time slot with zero mean. The integer variable is defined as:

$${\tau }_n\left[l\right]=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\mathrm{the}n\text{-}\mathrm{th}\mathrm{user}\mathrm{occupies}\mathrm{the}l\text{-}\mathrm{th}\mathrm{time}\mathrm{slot},\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(3)

the channel gain between the UAV and the n-th user in the l-th time slot is given as:

$$h_n\left[l\right]=\sqrt{\frac{{\rho }_0}{H^2+{\parallel \mathit{w}\left[l\right]-{\mathit{L}}_n\parallel }^2}}$$

(4)

where ${\rho }_0$ denotes the reference channel power gain at the distance $d_0=1$ m, $\mathit{w}\left[l\right]={[x\left[l\right],y\left[l\right]]}^T$ is the position of the UAV at the l-th time slot. The height of the UAV is fixed at H (meters). The location of the n-th user is denoted by ${\mathit{L}}_n$. As a result, the trajectory of the UAV should satisfy the following constraints.

$$\begin{array}{cccc}\hfill & \hfill & & \mathit{w}\left[1\right]=\mathit{w}\left[L\right],\hfill \\ \hfill & \hfill & & {\parallel \mathit{w}[l+1]-\mathit{w}\left[l\right]\parallel }^2\le {\left(\frac{v\mathcal{T}}{L}\right)}^2,\forall l,\hfill \end{array}$$

(5)

where Equation (5) indicates that the UAV will go back to its beginning position at the end of each fly cycle duration $\mathcal{T}$, the maximum speed of the UAV is denoted by $\mathit{v}$, and the expression ($\mathit{v}\mathcal{T}/L$) in Equation (5) shows the maximum distance that the UAV could fly during each time slot. For convenient expression, we denote ${\mathit{\tau }}_n=[{\tau }_n\left[1\right],\dots ,{\tau }_n\left[L\right]]$, $\mathbf{\Theta }={[{\mathit{\tau }}_1,\dots ,{\mathit{\tau }}_N]}^T$, $\mathit{p}=[P_1,\dots ,P_N]$, and $\mathit{W}=\left[\mathit{w}\right[1],\dots ,\mathit{w}[L\left]\right]$. The achievable rate of the n-th user using the l-th time slot is given as:

$$R_n\left[l\right](\mathit{p},\Theta ,\mathit{W})=Blog\left(1+\frac{P_n\left[l\right]{\left|h_n\left[l\right]\right|}^2}{{\sigma }^2}\right),$$

(6)

The average rate of the n-th user served by UAV over the L time slots is given by:

$$R_n(\mathit{p},\Theta ,\mathit{W})=\frac{1}{L}\sum _{l=1}^L{R}_n\left[l\right].$$

(7)

Here, the index $\frac{1}{L}$ comes from the fact that each user occupies only one time slot over the total L time slots.

2.2. Problem Formulation

The objective of this work is to optimize the total data that could be transmitted from users to the UAV given in a unit power consumption, and thus, the EE is introduced as:

$$\eta (\mathit{p},\Theta ,\mathit{W})=\frac{{\sum }_{n=1}^N{R}_n}{{\sum }_{n=1}^N{P}_n}$$

(8)

We now formulate the EE maximization problem, which can be represented as follows:

$$\begin{array}{cccc}\hfill & \hfill & \hfill \underset{\{\mathit{p},\Theta ,\mathit{W}\}}{\mathrm{maximize}}& \eta (\mathit{p},\Theta ,\mathit{W})\hfill \\ \hfill & \hfill & \hfill \mathrm{Subject} \mathrm{to}& P_n\le P^{\mathrm{thr}},\forall n,\hfill \\ \hfill & \hfill & & \mathit{w}\left[1\right]=\mathit{w}\left[L\right],\hfill \\ \hfill & \hfill & & {\parallel \mathit{w}[l+1]-\mathit{w}\left[l\right]\parallel }^2\le {\left(\frac{v\mathcal{T}}{L}\right)}^2,\forall l,\hfill \\ \hfill & \hfill & & \sum _{n=1}^N{\tau }_n\left[l\right]\le 1,\forall l.\hfill \end{array}$$

(9)

where the last constraint in Equation (9) is to ensure that each time slot has only one user who can use it. We now observe that the problem in Equation (9) is difficult to solve due to the non-convex form of the objective function and the existence of the integer variables. Therefore, we suggest decomposing the problem into three sub-problems with respect to three variables. The details are given in the next section.

3. Solution Approach

In this section, we will provide a solution to the problem in Equation (9). In detail, instead of simultaneously solving three variables, which requires a lot of computations, our solution is to divide the problem into three sub-problems with respect to three variables. Specifically, we iteratively solve each sub-problem while considering the remaining ones as fixed. The solution of each sub-problem will be used as fixed terms to solve the next sub-problem. The detailed procedure is provided as below:

1

Solving the UAV trajectory while fixing the power allocation and the time slot assignment of all users.

2

Solving the power allocation while fixing the UAV trajectory and the time slot assignment of all users.

3

Solving the time slot assignment while fixing the power allocation of all users and the UAV trajectory.

We repeat the above three steps until the EE value cannot be increased. The final UAV trajectory, the final time slot assignment, and the final power allocation are given after convergence. Detailed solutions to three sub-problems are provided in the following parts.

3.1. UAV Trajectory Optimization

Given the power allocation and the time slot assignment of all users, the problem in Equation (9) now becomes the total achievable rate maximization problem as below:

$$\begin{array}{cccc}\hfill & \hfill & \hfill \underset{\left\{\mathit{W}\right\}}{\mathrm{maximize}}& \sum _{n=1}^N{R}_n\left(\mathit{W}\right)\hfill \\ \hfill & \hfill & \hfill \mathrm{Subject} \mathrm{to}& \mathit{w}\left[1\right]=\mathit{w}\left[L\right],\hfill \\ \hfill & \hfill & & {\parallel \mathit{w}[l+1]-\mathit{w}\left[l\right]\parallel }^2\le {\left(\frac{v\mathcal{T}}{L}\right)}^2,\forall l.\hfill \end{array}$$

(10)

We see that the the achievable rate is not convex on $\mathit{w}\left[l\right]$ but convex on ${\parallel \mathit{w}[l+1]-\mathit{w}\left[l\right]\parallel }^2$. We can obtain the lower bound of the achievable rate by applying the first-order Taylor expansion. We then have the following result.

Lemma 1.

Given the power allocation and the time slots assignment of all users, the lower bound of the achievable rate at the t-th iteration is given as:

$$\begin{array}{cc}\hfill R_n\left[l\right]\left(\mathit{W}\right)\ge {\tilde{R}}_n\left[l\right]\left(\mathit{W}\right)=& \frac{\frac{-p{\rho }_u{log}_2\left(e\right)}{{\sigma }^2(H^2+\parallel {\mathit{w}}^{(t-1)}\left[l\right]-{\mathit{L}}_n{{\parallel }^2)}^2}}{1+\frac{p{\rho }_u}{{\sigma }^2(H^2+\parallel {\mathit{w}}^{(t-1)}\left[l\right]-{\mathit{L}}_n{\parallel }^2)}}\left(\parallel \mathit{w}\left[l\right]-{\mathit{L}}_n{\parallel }^2-{\parallel {\mathit{w}}^{(t-1)}\left[l\right]-{\mathit{L}}_n\parallel }^2\right)\hfill \\ \hfill & +{log}_2\left(1+\frac{p{\rho }_u}{{\sigma }^2(H^2+\parallel {\mathit{w}}^{(t-1)}\left[l\right]-{\mathit{L}}_n{\parallel }^2)}\right).\hfill \end{array}$$

(11)

where $w^{(t-1)}\left[l\right]$ is the obtained trajectory from the previous iteration.

Proof. 

The first-order Taylor expansion is given as:

$$f\left(x\right)=f^{\prime }\left(a\right)(x-a)+f\left(a\right),$$

(12)

The derivation of the achievable rate with respect to ${\parallel \mathit{w}[l+1]-\mathit{w}\left[l\right]\parallel }^2$ is calculated as:

$$\frac{\partial \left(R_n\left[l\right]\left(\mathit{W}\right)\right)}{\partial (\parallel \mathit{w}[l+1]-\mathit{w}\left[l\right]{\parallel }^2}=\frac{\frac{-p{\rho }_u{log}_2\left(e\right)}{{\sigma }^2(H^2+\parallel \mathit{w}\left[l\right]-{\mathit{L}}_n{{\parallel }^2)}^2}}{1+\frac{p{\rho }_u}{{\sigma }^2(H^2+\parallel \mathit{w}\left[l\right]-{\mathit{L}}_n{\parallel }^2)}}.$$

(13)

Therefore, we can obtain the lower bound of the achievable rate at $\parallel {\mathit{w}}^{(t-1)}\left[l\right]-{\mathit{L}}_n{\parallel }^2$ as in Equation (11). The proof is completed. □

We can now provide the relaxation of the problem in Equation (10) as:

$$\begin{array}{cccc}\hfill & \hfill & \hfill \underset{\left\{\mathit{W}\right\}}{\mathrm{maximize}}& \sum _{n=1}^N{\tilde{R}}_n\left[l\right]\left(\mathit{W}\right))\hfill \\ \hfill & \hfill & \hfill \mathrm{Subject} \mathrm{to}& \mathit{w}\left[1\right]=\mathit{w}\left[L\right],\hfill \\ \hfill & \hfill & & {\parallel \mathit{w}[l+1]-\mathit{w}\left[l\right]\parallel }^2\le {\left(\frac{v\mathcal{T}}{L}\right)}^2,\forall l.\hfill \end{array}$$

(14)

The problem in Equation (14) now is convex, and thus, we can make use of the cvx toolbox [40] to solve and obtain the optimal solution.

3.2. Power Allocation

In this subsection, we now provide a solution to the power allocation given the information, the time slot assignment, and the trajectory of the UAV. The problem in Equation (9) can be rewritten as:

$$\begin{array}{cccc}\hfill & \hfill & \hfill \underset{\left\{\mathit{p}\right\}}{\mathrm{maximize}}& \eta \left(\mathit{p}\right)=\frac{{\sum }_{n=1}^N{R}_n\left(\mathit{p}\right)}{{\sum }_{n=1}^N{P}_n}\hfill \\ \hfill & \hfill & \hfill \mathrm{Subject} \mathrm{to}& P_n\le P^{\mathrm{thr}},\forall n.\hfill \end{array}$$

(15)

The above problem is a non-convex problem and in fractional form, and thus, it is hard to obtain the optimal solution. We suggest applying the Dinkelbach method [41] to transform the problem into the non-fractional form as:

$$\begin{array}{cccc}\hfill & \hfill & \hfill \underset{\left\{\mathit{p}\right\}}{\mathrm{maximize}}& \sum _{n=1}^N{R}_n-{\eta }^{(i-1)}\sum _{n=1}^N{P}_n\hfill \\ \hfill & \hfill & \hfill \mathrm{Subject} \mathrm{to}& P_n\le P^{\mathrm{thr}},\forall n.\hfill \end{array}$$

(16)

where ${\eta }^{(i-1)}$ is a non-negative parameter, which is calculated as:

$$\begin{array}{c}\hfill {\eta }^{(i-1)}\left({\mathit{p}}^{(i-1)}\right)=\frac{{\sum }_{n=1}^N{R}_n\left({\mathit{p}}^{(i-1)}\right)}{{\sum }_{n=1}^N{P}_n^{(i-1)}}.\end{array}$$

(17)

therein, $R_n\left({\mathit{p}}^{(i-1)}\right)$ is calculated at the achieved power allocation ${\mathit{p}}^{(i-1)}$ of the previous iteration. After obtaining the power allocation ${\mathit{p}}^{\left(i\right)}$, we can update the EE value

$${\eta }^{\left(i\right)}\left({\mathit{p}}^{\left(i\right)}\right)=\frac{{\sum }_{n=1}^N{R}_n\left({\mathit{p}}^{\left(i\right)}\right)}{{\sum }_{n=1}^N{P}_n^{\left(i\right)}}$$

(18)

for the next iteration. The optimal ${\eta }^{*}$ can be achieved if and only if

$$\begin{array}{c}\hfill \sum _{n\in \mathcal{N}}{R}_n^{*}-{\eta }^{*}\sum _{n\in \mathcal{N}}{P}_n^{*}=0\end{array}$$

(19)

with ${\eta }^{*}={\displaystyle \underset{P_n}{max}}\frac{{\sum }_{n\in \mathcal{N}}{R}_n}{{\sum }_{n\in \mathcal{N}}{P}_n}$ where $R_n^{*}$ and $P_n^{*}$ are optimal values of $R_n$ and $P_n$ when $\eta$ is optimal. The problem in Equation (16) now becomes a convex problem on $P_n$. The CVX toolbox is used to obtain the optimal solution. The detail of the Dinkelbach procedure for the problem in Equation (15) is described in Algorithm 1.

Algorithm 1: Power allocation based on the Dinkelbach method.

3.3. Time Slot Assignment

Given the power allocation and the trajectory of the UAV, we can rewrite the problem in Equation (9) as the total achievable rate maximization with the time slot scheduling variable as follows:

$$\begin{array}{cccc}\hfill & \hfill & \hfill \underset{\{\Theta \}}{\mathrm{maximize}}& \sum _{n=1}^N{R}_n\hfill \\ \hfill & \hfill & \hfill \mathrm{Subject} \mathrm{to}& \sum _{n=1}^N{\tau }_n\left[l\right]\le 1,\forall l,n.\hfill \end{array}$$

(20)

It is obvious that the problem in Equation (20) is less complicated compared to the problem in Equation (9). However, this problem becomes intractable when the parameters N and L increase, and thus, the exhaustive search is not suitable, even though we observe that the achievable rate of each user does not depend on the choice of the others. Therefore, the Hungarian method then can be used to solve the time slots assignment problem. We first calculate a matrix containing all achievable rates of all users’ overall time slots as:

$${\mathbb{R}}^{\mathrm{pr}}=\begin{array}{cc}\begin{array}{ccccc}1& \hspace{1em}\dots & \hspace{1em}n& \hspace{1em}\dots & \hspace{1em}N\end{array}& \\ \left[\begin{array}{ccccc}{R}_1\left[1\right]& \dots & R_n\left[l\right]& \dots & R_n\left[l\right]\\ ⋮& \ddots & ⋮& \ddots & ⋮\\ R_1\left[l\right]& \dots & R_n\left[l\right]& \dots & R_N\left[l\right]\\ ⋮& \ddots & ⋮& \ddots & ⋮\\ R_1\left[L\right]& \dots & R_n\left[L\right]& \dots & R_N\left[L\right]\end{array}\right]& \begin{array}{c}1\\ ⋮\\ l\\ ⋮\\ L\end{array}\end{array}$$

However, in this case, the number of users is smaller than the number of time slots, and thus, we suggest using dummies as pseudo users. The price matrix is then rewritten as:

$${\mathbb{R}}^{\mathrm{pr}}=\begin{array}{cc}\begin{array}{cccccc}\hspace{1em}1& \hspace{1em}\dots & \hspace{1em}N& \dots & \dots & L\end{array}& \\ \left[\begin{array}{cccccc}{R}_1\left[1\right]& \dots & R_N\left[l\right]& 0& \dots & 0\\ ⋮& \ddots & ⋮& ⋮& \ddots & ⋮\\ R_1\left[l\right]& \dots & R_N\left[l\right]& 0& \dots & 0\\ ⋮& \ddots & ⋮& ⋮& \ddots & ⋮\\ R_1\left[L\right]& \dots & R_N\left[L\right]& 0& \dots & 0\end{array}\right]& \begin{array}{c}1\\ \\ l\\ \\ L\end{array}\end{array}$$

In order to optimize EE, all time slots from the system should be used. Therefore, the constraint in Equation (20) will be rewritten as:

$$\sum _{n=1}^N{\tau }_n\left[l\right]=1.$$

(21)

Given the price matrix as in Section 3.3, we apply the Hungarian algorithm, which is adopted from [42], to obtain the optimal time slot assignment. The detailed solution is provided in Algorithm 2.

Algorithm 2: Hungarian time slot assignment algorithm.

3.4. Alternating Optimization Algorithm

In order to provide a solution to our principal problem in Equation (9), we suggest an alternating algorithm to provide a solution to the problem in Equation (9). At the initial $t=0$, we initialize the power allocation ${\mathrm{p}}^{\left(0\right)}={\mathrm{p}}^{\max }$ and the initial time slots assignment ${\Theta }^{\left(0\right)}$. Therein, to the time slots scheduling ${\Theta }^{\left(0\right)}$, the system will assign time slots to users in favor until the time slots run out. At the t-th round, we consecutively solve the UAV trajectory, the time slot assignment, and the power allocation while considering the remaining ones as constant. We stop the repetition until the EE value cannot increase. The detailed procedure of the alternating algorithm is provided in Algorithm 3.

Lemma 2.

Given the initial power allocation and the time slot scheduling satisfying the constraints in Equations (9), respectively, Algorithm 3 always improves the EE value, and Algorithm 3 always converges.

Proof. 

At the t-th iteration of Algorithm 3, given the power allocation ${\mathrm{p}}^{\left(t\right)}$, and the time slots assignment ${\Theta }^{\left(t\right)}$, from Section 3.1, we can optimize the trajectory of UAV to improve the EE value. Next, by applying the Dinkelbach method, we can obtain the optimal power allocation given the trajectory and the time slot assignment. Finally, the Hungarian method consistently provides the optimal solution to the time slot assignment problem. Therefore, the EE value always increases at every iteration of Algorithm 3. Furthermore, all variables are limited by the constraints, and thus, the EE value is bounded. Algorithm 3 always converges. Note that three variables are obtained alternatively, and thus, we will achieve a sub-optimal solution to the problem in Equation (9). □

Algorithm 3: Alternating algorithm.

4. Simulation Results and Discussion

In this section, we analyze the system performance by comparing it to different baseline schemes. Therein, all simulation results are carried out using MATLAB software, MathWorks, Inc. The problems in Equations (14) and (16) are obtained by using the CVX toolbox. For comparison, we compare the effectiveness of three proposed solutions on the alternating Algorithm 3. In detail, we replace the proposed solutions with other methods, including:

(1)

The circular trajectory is used to compare with the proposed trajectory optimization.

(2)

The maximum power will be set to all users to compare with the proposed power allocation.

(3)

Finally, the fairness time slot allocation will be used to replace the Hungarian with the allocation time slot for all users, in which the system will divide the time slots resource evenly to all users.

In this work, we considered a specific area such as a smart home city, a small town, or forest fire alarm monitoring management. Therefore, the communication location is a circle with a radius of 250 (m), where N users are served by a UAV-based network within this location. The UAV flies periodically at a fixed altitude H = 50 m above the location with the maximum speed of UAV denoted by v set as $\mathbf{v}\left(max\right)=50$ m/s. The general parameters of the system setting and propagation channels are listed in

Table 2. Simulated network parameters.

Parameters	Value
Cellular radius	250 (m)
Cycle duration	100 (s)
Reference channel power gain	${10}^{(-6)}$
Height of UAV	50 (m)
Maximum Speed of UAV	50 (m/s)
Operating frequency	1.9 (GHz)
Maximum transmit power of users	23 (dBm)
Power spectral density of the thermal noise	−174 (dBm)
Bandwidth	180 (kHz)
Terminated conditions	0.0001

, while several specific parameters are presented in each figure. In addition, in order to demonstrate the higher performance of the suggested methods, Algorithm 3 will involve different baseline methods for comparison. The proposed solutions for sub-problems will be compared to different benchmarks. Especially, each benchmark will be combined with other proposed solutions in Algorithm 3, so each suggested solution to each sub-issue in the proposed algorithm can be estimated.

As the result of the UAV trajectory optimization in Figure 2, we consider a scenario with $N=5$ for Figure 2a, N = 10 for Figure 2b, and N = 15 for Figure 2c. The optimal UAV trajectories are demonstrated with the various numbers of users. We set ${\mathit{P}}^{\mathrm{thr}}$ = 20 dBm and $\mathcal{T}=100 \mathbf{s}$. We can observe that the UAV tends to fly straight to the users’ location instead of flying on a circular trajectory to provide users with the most satisfactory achievable rate. In addition, the number of users gradually increased from 5 to 15. The flying trend of UAVs is also almost unchanged so that users can achieve the highest average rate. The UAV is usually encountered above the coordinate of the users. Similarly, by checking the time slot assignment, the trajectory of the UAV tends to follow the time slot assignment of users. Meanwhile, the circular trajectory will ignore users’ location and time slot assignment. Accordingly, the system could not optimize the amount of collected data efficiently.

In Figure 3, we compare EE versus trajectory optimization and UAV with circular trajectory. Figure 3 demonstrates that the EE of a circular trajectory when the cycle duration $\mathcal{T}=100 \mathbf{s}$ can achieve a maximum of EE = 2.3 (bit/Hz). Meanwhile, the method given in Algorithm 3 indicates that the performance achieved by EE significantly exceeds the circular trajectory. Even when we adjusted the transmission power of the UAV to half, the proposed scheme we came up with was still $7.6$% more useful than the circular trajectory. Furthermore, the circular trajectory only focuses on maximizing the sum rate of UAV-served users without considering the power allocation and time slot assignment for users. On the contrary, the power allocation and time slot assignment are bounded to a few constraints in our proposed scheme, which can guarantee the throughput of UAV-served users. Thus, our proposed method is most appropriate for this scenario.

Figure 4 illustrates the change of the EE value when the circuit power increases. It is evident that when the circuit power increases, the total power consumption increases, and it leads to a reduction in the EE value. However, the proposed algorithm always shows superior performance compared to the conventional methods around 7% to 12%. For instance, when the circuit power is 50 (mW), the proposed algorithm provides the EE value at $2.9$ (bits/joule). At the same time, the maximum power allocation and circular trajectory are around $2.65$ and $2.49$ (bits/joule), respectively. Eventually, when the circuit power is 130 (mW), the maximum EE value of the proposed algorithm, maximum power allocation, and circular trajectory is around 1.2 to 1.4 (bit/joule). This demonstrates the inverse ratio in the EE formula.

Figure 5 represents the CDF of the maximum achieved EE value when $N=5$, $L=5$. We observe that the proposed algorithm always provides the highest percentage for which EE could take a value. Specifically, in 100% of the cases, the proposed algorithm provides the maximum EE value higher than $2.7$ (bits/joule). Meanwhile, the fairness time slot allocation shows a similar trend. However, the EE value is only $2.5$ (bits/joule). In the case of a circular trajectory, the EE value is higher than $2.5$ (bits/joule) in only 20% of cases. Finally, the maximum power allocation and all benchmarks are trivial; in 100% of cases, the EE value is smaller than $0.2$ (bits/joule).

Figure 6 shows the increase of the EE value according to Algorithm 3. For comparison, each proposed solution is replaced by a baseline. The proposed solution is a blue line. It provides the highest value of EE and converges the quickest at $2.7$ (bits/s/hz). Meanwhile, the fairness time slot is a brown line, which describes the number of the time slots of users divided evenly. This method can provide the highest EE value up to $2.5$ (bits/s/hz). Meanwhile, the circular method only can provide the value of EE at around $1.7$ (bits/s/hz). The justification for the huge gap between the proposed method and the circular trajectory is that the circular trajectory does not take the time slot assignment of users into account to optimize the trajectory. Meanwhile, the proposed trajectory optimization will consider the time slot and locations of all users in the system, and thus, the achievable rate of users will be maximized effectively. Finally, the maximum power allocation and the benchmarks show inefficiency, since the amount of power consumption is too high. Therefore, it leads to an extremely small value of EE for both these methods.

Figure 7 illustrates the change in EE value when the number of time slots in the system increases. We first observe that the decreasing trend has come from the fraction $\frac{1}{L}$ in the achievable rate equation. However, the proposed solution always provides better performance compared to other methods. The number of the proposed declines slowly from 4 (bits/s/hz) to $1.4$ (bits/s/hz) when the number of time slots rises from 10 to 30. The closest line to the proposed solution is the fairness of time slot allocation, but the value of this method is smaller than the proposed method, around 5∼6%. The purple line representing the circular trajectory shows us that the value of EE decreases from 3 (bits/s/hz) to $1.3$ (bits/s/hz), respecting the increase in the number of time slots. Finally, the maximum power allocation and all benchmarks cases almost hit the zero value, since the amount of power consumption is too high while the achievable is trivial.

Figure 8 reflects the change in EE value when the number of users in the system increases. We can easily observe that the proposed solution always gives the maximum EE value of around 2 (bits/s/hz) in all cases. Meanwhile, the fairness time slots allocation provides the EE value up to $1.8$ (bits/s/hz), which is about 10% lower than the proposed method. The value of the circular trajectory is $1.7$ (bits/s/hz) and smaller than the figure of the proposed solution around 15% at all schemes. The maximum power allocation method decreases slightly from $0.2$ (bits/s/hz) to approximately $0.05$ (bits/s/hz) when the number of users increases from 3 to 10. Likewise, the value of EE approaches zero for the all benchmarks method.

5. Conclusions

In this work, we considered a system where multiple users are scattered in a wide area. A UAV is used to gather information from these users. In order to optimize the system performance, we investigated the energy efficiency of the system. Therein, the trajectory of the UAV, the power allocation of all users, and the time slot assignment were jointly optimized. The formulated problem was extremely challenging to solve, and we suggest decomposing it into three sub-problems respective to the three considered variables. First, the UAV trajectory is optimized to maximize the EE of the UAV-based network. Second, the power allocation optimization is solved by applying the Dinkelbach method. Next, the time slot assignment is optimally achieved by using the Hungarian method. All three sub-problems are ensured to continuously improve the EE value. An alternating algorithm was provided to connect and provide a sub-optimal solution to the initial problem. Eventually, the simulation results demonstrated the higher performance of our proposed algorithms compared to different baseline methods in various schemes.