Throughput Maximization for UAV-Enabled Wireless Powered Communication Networks

01/14/2018 ∙ by Lifeng Xie, et al. ∙ National University of Singapore 0

This paper studies an unmanned aerial vehicle (UAV)-enabled wireless powered communication network (WPCN), in which a UAV is dispatched as a mobile access point (AP) to serve a set of ground users. The UAV employs the radio frequency (RF) signals based wireless power transfer (WPT) to charge the users in the downlink, and the users use the harvested RF energy to send individual information back to the UAV in the uplink. Unlike the conventional WPCN with fixed APs, the UAV-enabled WPCN can exploit the mobility of the UAV via trajectory optimization, jointly with the wireless resource allocation, to improve the system performance. In particular, we aim to maximize the uplink common throughput among all ground users over a particular finite time period, subject to the UAV's maximum speed constraint and the users' energy harvesting constraints. The decision variables include the UAV's trajectory design, as well as the UAV's downlink power allocation for WPT and the users' uplink power allocation for wireless information transfer (WIT). This problem is non-convex and thus very difficult to be solved optimally. To overcome this issue, we first consider a special case when the maximum UAV speed constraint is ignored, and obtain the optimal solution to this relaxed problem. The optimal solution shows that the UAV should hover above a finite number of ground locations over time for downlink WPT, and then hover above each user at different time for uplink WIT. Next, based on the optimal multi-location-hovering solution to the relaxed problem, we propose the successive hover-and-fly trajectory, jointly with the downlink and uplink power allocation, to efficiently solve the original problem with the maximum UAV speed constraint. Numerical results show that the proposed UAV-enabled WPCN achieves much higher uplink common throughput than the conventional WPCN with fixed-location AP.

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I Introduction

Radio frequency (RF) wireless power transfer (WPT) has emerged as a promising solution to provide convenient and reliable energy supply to low-power Internet-of-things (IoT) devices such as sensors and RF identification (RFID) tags[2, 3, 4, 5]. Compared to the near-field WPT based on inductive coupling or magnetic resonant coupling, the far-field WPT via RF radiation is able to operate over a much longer range and charge multiple wireless devices (WDs) simultaneously even when they are moving and densely deployed, and yet with transceivers of significantly reduced form factor. There are in general two major applications of RF WPT in wireless communications, namely simultaneous wireless information and power transfer (SWIPT) and wireless powered communication network (WPCN), which unify the WPT and wireless information transfer (WIT) in a joint design framework over the same (downlink for both WPT and WIT) and opposite (downlink for WPT and uplink for WIT) transmission directions, respectively[6, 7]. In particular, WPCN enables dedicated wireless charging and information collection for massive IoT devices, thus significantly enhances the operation range and throughput of traditional backscattering-based wireless communications. However, in conventional WPCNs, access points (APs) are usually deployed at fixed locations, which cannot be changed once deployed [8, 9, 10].

The conventional WPCN with fixed APs faces several challenges. First, due to the severe propagation loss of RF signals over distance, the end-to-end WPT efficiency is generally low when the distance from the AP to a WD becomes large. Next, the conventional WPCN suffers from the so-called “doubly near-far” problem[8], i.e., far-apart WDs from the AP receive lower RF energy in the downlink WPT, but they need to use higher transmit power in the uplink WIT to achieve the same rate as nearby WDs. The doubly near-far problem result in a severe user fairness issue among WDs when they are geographically distributed over a large area. To overcome the above issues, various approaches have been proposed in the literature, such as adaptive time and power allocation [8], multi-antenna beamforming[11, 12, 13, 14, 15], and user cooperation [16, 17]. However, all these prior works focused on the wireless resource allocation designs to enhance the WPT/WIT performances of WPCNs with fixed APs. By contrast, in this paper we propose an alternative solution based on a new unmanned aerial vehicle (UAV)-enabled WPCN architecture with UAVs employed as mobile APs.

Fig. 1: Illustration of a UAV-enabled WPCN.

UAVs have found abundant applications such as for cargo delivery, aerial surveillance, filming, and industrial IoT. Recently, UAVs-enabled/aided wireless communications have attracted substantial research interests, due to their advantages in flexible deployment, strong line-of-sight (LoS) channels with ground users, and controllable mobility[18]. For example, UAVs can be utilized as mobile relays to help information exchange between far-apart ground users [19], or as mobile base stations (BSs) to help enhance the wireless coverage and/or the network capacity for ground mobile users [20, 21, 22, 23, 24, 25, 26, 27, 28]. Furthermore, UAV-enabled WPT has been proposed in [29, 30], in which UAVs are used as mobile energy transmitters to charge low-power WDs on the ground. By exploiting its fully controllable mobility, the UAV can properly adjust its location over time (a.k.a. trajectory) to reduce the distances with target ground users, thus improving the efficiency for both WIT and WPT.

Motivated by the UAV-enabled wireless communications as well as WPT, this paper pursues a unified study on both of them in a UAV-enabled WPCN as shown in Fig. 1. Specifically, a UAV following a pre-designed periodic trajectory is dispatched as a mobile AP to charge a set of ground users in the downlink via WPT, and the users use the harvested RF energy to send independent information to the UAV in the uplink. We investigate how to optimally exploit the UAV mobility via trajectory design, jointly with the wireless resource allocation, to maximize the uplink data throughput of the multiuser WPCN in a fair manner. To this end, we maximize the uplink common (minimum) throughput among all ground users over a given UAV’s flight period, by optimizing its trajectory, jointly with the downlink and uplink transmission resource allocations for WPT and WIT, respectively, subject to the UAV’s maximum speed and the users’ energy neutrality constraints. However, due to the complex data throughput and harvested energy functions in terms of coupled UAV trajectory and resource allocation design variables, our formulated problem is non-convex and thus difficult to be solved optimally.

To tackle this difficulty, we first consider an ideal case without considering the UAV’s maximum speed constraint. We show that the strong duality holds between this problem and its Lagrange dual problem, and thus it can be solved optimally via the Lagrange dual method. The optimal solution shows that the UAV should successively hover above a finite number of ground locations for downlink WPT, as well as above each of the ground users for uplink WIT, with the optimal hovering duration and wireless resource allocation for each location. Next, we address the general problem with the UAV’s maximum speed constraint considered. Based on the multi-location-hovering solution to the above relaxed problem, we first propose a heuristic

successive hover-and-fly trajectory design, jointly with the downlink and uplink resource allocations, to find an efficient suboptimal solution. The proposed solution is also shown to be asymptotically optimal as the UAV’s flight period becomes infinitely large. In addition, we further propose an alternating optimization based algorithm to obtain a locally optimal solution, which optimizes the wireless resource allocations and the UAV trajectory in an alternating manner, via convex optimization and successive convex programming (SCP) techniques, respectively. By employing the successive hover-and-fly trajectory as the UAV initial trajectory, the alternating-optimization-based algorithm iteratively refines the wireless resource allocations and the UAV trajectory to improve the uplink common throughput of all ground users until convergence. Finally, we present numerical results to validate the performance of our proposed UAV-enabled WPCN. It is shown that the joint trajectory and wireless resource allocation design significantly improves the uplink common throughput, as compared to the conventional WPCN with the AP at a fixed location.

It is worth noting that there is another line of related research that employs ground moving vehicles to wirelessly charge and/or collect information from ground sensors, e.g., [32, 34, 33, 35]. However, different from the UAV that can freely fly in the three-dimensional (3D) airspace, the ground vehicle can only move following a constrained path in the two-dimensional (2D) plane. Furthermore, unlike UAVs that have strong LoS links with ground users, the wireless channels from ground vehicles to users usually suffer from severe fading, thus limiting the performance for both WIT and WPT. As such, the joint trajectory design and wireless resource allocation in the UAV-enabled WPCN is a new study different from that with ground moving vehicles, which has not been investigated in the literature to our best knowledge.

The remainder of this paper is organized as follows. Section II presents the system model of the UAV-enabled WPCN, and formulates the uplink common throughput maximization problem of our interest. Section III considers an ideal case without the UAV’s maximum speed constraint and presents the optimal solution to the relaxed problem. Section IV and Section V present two efficient solutions to the general problem with the UAV’s maximum speed constraint considered. Section VI provides numerical results to validate the effectiveness of our proposed designs. Finally, Section VII concludes the paper.

Ii System Model and Problem Formulation

As shown in Fig. 1, we consider a UAV-enabled WPCN, in which a UAV is dispatched to periodically charge a set of ground users via WPT in the downlink, and each user uses its harvested energy to send independent information to the UAV in the uplink. Suppose that each user is at a fixed location on the ground in a 3D Cartesian coordinate system, where is defined as the horizontal coordinate of user . The users’ locations are assumed to be a-priori known by the UAV for its trajectory design and transmission resource allocation.

We focus on one particular flight period of the UAV, denoted by with finite duration in second (s), in which the UAV flies horizontally at a fixed altitude in meter (m). At any given time instant , let denote the location of the UAV projected on the horizontal plane. Accordingly, the distance between the UAV and each user is given by

(1)

where

denotes the Euclidean norm of a vector. By denoting the UAV’s maximum speed as

in m/s, we have , where and denote the first-derivatives of and with respect to , respectively. Note that we assume the UAV can freely choose its initial location and final location for performance optimization.

We consider that the wireless channels between the UAV and the ground users are dominated by LoS links. In this case, the free-space path loss model can be practically assumed, similarly as in[19, 29]. Accordingly, the channel power gain between the UAV and user at time instant is given by

(2)

where denotes the channel power gain at a reference distance of m.

We consider a time-division multiple access (TDMA) transmission protocol, in which the downlink WPT for all users and the uplink WIT of different users are implemented in the same frequency band but over orthogonal time instants. At any time instant , we use the indicators , to denote the transmission mode. We use and , to indicate the downlink WPT mode, in which the UAV transmits RF energy to charge the users simultaneously; while we use , and , to represent the uplink WIT mode for user , in which user sends its information to the UAV by using its harvested energy. As the TDMA protocol is employed, it follows that .

First, consider the downlink WPT mode at time instant , in which , and . Suppose that the UAV adopts a constant transmit power in the downlink WPT mode. Accordingly, the harvested power at each user is given by

(3)

where denotes the RF-to-direct current (DC) energy conversion efficiency at the energy harvester of each user.111Note that in practice, the RF-to-DC energy conversion efficiency is generally non-linear and depends on the received RF power level [36] and signal waveform [37]. For the purpose of exposition, in (3) we consider a simplified constant RF-to-DC energy conversion efficiency by assuming that each receiver operates at the linear regime for RF-to-DC conversion. Nevertheless, the design principles in this paper are also extendable to the scenario with non-linear RF-to-DC energy conversion efficiency, which, however, is left for future work. Therefore, the total harvested energy at user over each period of duration is given by

(4)

Next, consider the WIT mode for user at time instant with , and . Let denote the transmit power of user for the uplink WIT to the UAV. Accordingly, the achievable data rate from user to the UAV in bits/second/Hertz (bps/Hz) at time instant is given by

(5)

where denotes the noise power at the information receiver of the UAV, and is the reference signal-to-noise ratio (SNR).

Therefore, the average achievable rate or throughput of user over each period in bps/Hz is given by

(6)

Note that for the purpose of exposition, we consider that the energy consumption of each ground user is mainly due to the transmit power for its uplink WIT. In this case, the total energy consumption at user is

(7)

In order to achieve the self-sustainable operation for the WPCN, we consider the energy neutrality constraint at each user , such that the user’s energy consumption for uplink WIT (i.e., in (7)) cannot exceed the energy harvested from the downlink WPT (i.e., in (4)) in each period.222 Note that we assume that at the beginning of each period, each user has sufficient energy in its storage and the storage has sufficiently large capacity. In this case, as long as the energy neutrality constraints are satisfied over each period, the more stringent energy causality constraints in energy harvesting wireless communications [38] will be automatically ensured, and thus each of the users can sustain its operation without energy outage. As a result, we have the following energy neutrality constraints for the users,

(8)

In this work, our objective is to maximize the uplink common throughput among all users (i.e., ) subject to the UAV’s maximum speed constraint and the users’ energy neutrality constraints. The decision variables include the UAV trajectory , the transmission mode , and the transmit power for uplink WIT. As a result, the problem is formulated as

(P1)
(9)
(10)
(11)
(12)
(13)

where (13) denotes the UAV’s maximum speed constraint.

It is observed that for problem (P1), the objective function is non-concave and constraints (9) and (11) are non-convex, due to the complicated rate and energy functions with respect to coupled variables , , and , as well as the binary constraints on ’s. Therefore, (P1) is a non-convex optimization problem. Furthermore, (P1) contains an infinite number of optimization variables over continuous time. For these reasons, problem (P1) is difficult to be solved optimally. To tackle this problem, in Section III we first consider an ideal case by ignoring the UAV’s maximum speed constraint in (13), and solve the relaxed problem of (P1) as follows:

(9), (10), (11) and (12).

Note that problem (P2) also corresponds to the practical scenario when the UAV’s flight duration is sufficiently large for any given finite such that the flying time of the UAV becomes negligible as compared to its hovering time (see Section III for details). In Section IV and Section V, we propose efficient algorithms to solve the general problem (P1) with the UAV’s maximum speed constraint based on the optimal solution obtained for the relaxed problem (P2).

Iii Optimal Solution to Problem (P2)

In this section, we consider problem (P2). By introducing an auxiliary variable , problem (P2) can be equivalently expressed as

(14)
(9), (10), (11), and (12).

Although problem (P2.1) is still non-convex, one can easily show that it satisfies the so-called time-sharing condition in [39]. Therefore, the strong duality holds between (P2.1) and its Lagrange dual problem. As a result, we can optimally solve (P2.1) by using the Lagrange dual method.

Let and , , denote the dual variables associated with the -th constraints in (14) and (9), respectively. For notational convenience, we define and . The partial Lagrangian of (P2.1) is

(15)

The Lagrange dual function of (P2.1) is

(16)
Lemma iii.1

In order for the dual function to be upper bounded from above (i.e., ), it must hold that .

Proof:

Suppose that (or ). Then by setting (or ), we have . Therefore, must hold in order for to be bounded from above, and this lemma is proved.

Based on Lemma 3.1, the dual problem of problem (P2.1) is given by

For notational convenience, let denote the set of and specified by the constraints in (D2.1). As the strong duality holds between (P2.1) and (D2.1), we can solve (P2.1) by equivalently solving (D2.1). In the following, we first obtain by solving problem (16) under any given , and then solve (D2.1) by finding the optimal and to minimize .

Iii-1 Obtaining by Solving Problem (16) Under Given

First, consider problem (16) under any given . It is evident that problem (16) can be decomposed into the following subproblems.

(17)
(18)
(19)

, where

Here, problem (18) consists of an infinite number of subproblems, each corresponding to one time instant .

Note that the optimal value of problem (17) is always zero as (see Lemma 3.1). In this case, the optimal solution to problem (17) can be chosen as any arbitrary real number. Therefore, we only need to focus on problem (18). As the subproblems in (18) are identical for different time instants ’s, we can drop the index for notational convenience, and denote the optimal solution as , and .

As for problem (18), there are a total of feasible choices for due to the constraints in (19). In the following, we solve problem (18) by first obtaining the maximum objective value (and the corresponding optimal and ) under each of the feasible , and then comparing them to obtain the optimal .

First, consider that and . In this case, problem (18) can be re-expressed as

(20)

for which the optimal solution is given as , and , where

(21)

corresponds to the set of optimal hovering locations for downlink WPT, with denoting the number of optimal solutions to problem (21). Here, for the non-convex problem (21), we solve it by using a 2D exhaustive search over the region , where . Note that when the optimal solution to problem (21) is non-unique (or ), we can arbitrarily choose any one of ’s for obtaining the dual function . Accordingly, the optimal value of problem (18) is given as .

Next, consider that for any one and . In this case, problem (18) can be re-expressed as

(22)

Note that the objective function of problem (22) is concave with respect to , and therefore, problem (22) is convex. By checking the Karush-Kuhn-Tucker (KKT) conditions, we have the optimal solution as , , and , where . Therefore, the corresponding optimal value of problem (18) is .

By comparing the optimal values, i.e., and , , we have the following proposition, for which the proof is straightforward and thus omitted.

Proposition iii.1

The optimal solution to problem (18) is obtained by considering following two cases.

  • If , then the UAV operates in the downlink WPT mode, i.e.,

    (23)

    where is generally non-unique when .

  • Otherwise, we denote . Then the UAV operates in the uplink WIT mode for user , i.e.,

    (24)

Note that if any two of the optimal values (i.e., and ) are equal, then the corresponding solutions in (23) and (24) are both optimal for problem (18). Based on Proposition 3.1, problem (18) is solved, and thus the function is obtained.

Iii-2 Finding Optimal and to Solve (D2.1)

Next, we search over () to minimize for solving (D2.1). Since the dual problem (D2.1) is always convex but in general non-differentiable, we can use subgradient based methods, such as the ellipsoid method[41], to obtain the optimal and , denoted by and . Note that for the objective function in (D2.1), the subgradient with respect to () is

where is chosen for simplicity.

Iii-3 Constructing Optimal Primal Solution to (P2.1)

With and at hand, it remains to construct the optimal primal solution to (P2.1), denoted by and . Before proceeding, we have the following proposition.

Proposition iii.2

It must hold that at the optimal and .

Proof:

See Appendix -A.

By combining Propositions III.1 and III.2, it follows that under the optimal dual solution and to (D2.1), problem (18) has a total number of optimal solutions. Among them, the optimal solutions are given in (23) for downlink WPT, and the other solutions are given in (24) for uplink WIT (each for one user ). In this case, we need to time-share among these optimal solutions to construct the optimal primal solution to (P2.1).

More specifically, notice that the solutions in (23) correspond to hovering locations for downlink WPT, at which only the UAV transmits at constant power with ; on the other hand, the -th solution in (24), , corresponds to that the UAV hovers above user at location for uplink WIT, at which user transmits with and . With time-sharing, let and denote the hovering durations at the location , and , respectively. In this case, we solve the following uplink common throughput maximization problem to obtain the optimal hovering durations ’s and ’s for time sharing.

(25)
(26)
(27)

Note that problem (P2.2) is a linear program, which can be solved by standard convex optimization techniques in

[41]. The optimal solution to (P2.2) is denoted as , and . Accordingly, we can divide the whole period into sub-periods, where the first sub-periods, denoted by , are for downlink WPT, and the next sub-periods, denoted by , , are for uplink WIT of the users. As a result, we have the following proposition, for which the proof is omitted for brevity.

Proposition iii.3

The optimal solution to (P2.1) (and thus (P2)) is given as follows. During sub-period , the UAV hovers at the location for downlink WPT, i.e.,

(28)

. During sub-period , , the UAV hovers above user at , and user sends information to the UAV in the uplink, i.e.,

(29)

. The optimal uplink common throughput is given as (with denoting the optimal solution obtained for (P2.2)).

In summary, we present the overall algorithm for solving (P2) as Algorithm 1 in Table I, and we refer to such a solution as the multi-location-hovering solution. Notice that Algorithm 1 is guaranteed to converge to the globally optimal solution to problem (P2).

 

  • Initialization: Given an ellipsoid containing (), where () is the center point of and the positive definite matrix characterizes the size of .

  • Repeat:

    • Obtain to maximize in problem (21) via a 2D exhaustive search over the region ;

    • Obtain under given () by using Proposition III.1;

    • Compute the subgradients of , and then update and by using the ellipsoid method [41].

  • Until and converge with a prescribed accuracy.

  • Set ( ().

  • Obtain the optimal solution to problem (P2.1) or (P2) based on Proposition III.3.

 

TABLE I: Algorithm 1 for Solving Problem (P2)
Remark iii.1

It is worth noting that similar multi-location-hovering solutions have been proposed in the UAV-enabled multiuser WPT system [29] and the UAV-enabled multiuser communication system with TDMA transmission [31], when the UAV’s flight period becomes sufficiently long. In [29], the UAV successively hovers above a given set of locations to maximize all users’ minimum received energy; while in [31], the UAV successively hovers above each user to maximize the minimum throughput of all users. As a matter of fact, our derived multi-location-hovering solution to problem (P2) in Proposition III.3 unifies the results in [29] and [31], which consists of two sets of hovering locations: ones for WPT and the other ones for WIT. Nevertheless, note that the hovering locations to problem (P2) for WPT are generally different from those in [29], as they are designed based on different objective functions (max-min communication throughput versus max-min harvested energy).

Remark iii.2

To gain more insights, it is interesting to consider problem (P2) in the special case of users. Without loss of generality, we assume that the two users are located at and with , , and , where denotes the distance between the two users. Due to this symmetric setup, it can be shown that the optimal hovering locations for WPT to problem (P2) are actually identical to the optimal hovering locations to maximize the two users’ minimum harvested energy in the UAV-enabled WPT system [29, 30]. It follows from [29, 30] that the optimal hovering locations for downlink WPT are critically dependent on the UAV’s flying altitude and the users’ distance . In particular, when , there are = 2 hovering locations at and for WPT, where with ; while when there is only = 1 hovering location right above the middle point of two users. By combining the optimal hovering locations for WPT and those for WIT, it is evident that when , the UAV needs to hover above four locations for efficient WPCN, as shown by the example in Fig. 3; while when , the UAV needs to hover above three locations, as shown by the example in Fig. 3.

Fig. 2: The optimal hovering locations for WPT and WIT in a two-user WPCN, where  m and  m with .
Fig. 3: The optimal hovering locations for WPT and WIT in a two-user WPCN, where  m and  m with .

Iv Proposed Solution to Problem (P1) with Successive Hover-and-Fly Trajectory

This section considers problem (P1) with the UAV’s maximum speed constraint considered. First, we present a successive hover-and-fly trajectory motivated by the multi-location-hovering solution to the relaxed problem (P2), in which the UAV sequentially visits the hovering locations for efficient WPT and WIT, respectively. Next, under such a flying trajectory, we design the duration at each hovering location and the transmission resource allocation for (P1) by discretizing the time period. Finally, we discuss the case when the UAV’s flight duration is too small to visit all these hovering locations, for which the trajectory and transmission resource allocations are redesigned.

Iv-a Successive Hover-and-Fly UAV Trajectory

In the proposed successive hover-and-fly trajectory design, the UAV sequentially visits the optimal hovering locations that are obtained for (P2), i.e., for downlink WPT, and for uplink WIT. For notational convenience, we denote the hovering locations as , where , and . In order to maximize the time for efficient WPT and WIT, the UAV flies among these hovering locations by using the maximum speed , and the UAV aims to minimize the flying time by equivalently minimizing the traveling path among the locations.

Towards this end, we define a set of binary variables

, where or indicates that the UAV flies or does not fly from the -th hovering location to the -th hovering location . Hence, the traveling path minimization problem becomes determining to minimize , provided that each of the locations is visited once, where denotes the distance between and . Note that as shown in [29], the flying distance minimization is similar to the well-established traveling salesman problem (TSP), with the following differences. The standard TSP requires the salesman (or the UAV in this paper) to return to the origin city (the initial hovering location) after visiting all the other cities (or hovering locations here), while the flying distance minimization problem of our interest does not have such a requirement since the initial and final hovering locations can be optimized. As shown in [40], we can transform our traveling distance minimization problem to the standard TSP as follows. First, we add a dummy hovering location, namely the -th hovering location, whose distances to all the existing hovering locations are 0, i.e., , . Note that this dummy hovering location is a virtual node that does not exist physically. Then, we obtain the desirable traveling path by solving the standard TSP problem for the hovering locations, and then removing the two edges associated with the dummy location. As a result, we use the permutation over the set to denote the obtained traveling path, where the -th hovering location is first visited, followed by the -th, the -th, etc., until the -th hovering location at last. We denote the traveling distance and traveling duration from the -th hovering location to the -th hovering location