Recently, unmanned aerial vehicles (UAVs) have been adopted in many applications such as weather monitoring and traffic control , and the usage of the UAV in wireless communication systems has drawn great attentions [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Compared to conventional networks where APs are fixed at given locations, wireless communication networks employing a UAV-mounted access point (AP) exhibit cost-efficiency and deployment flexibility. Moreover, the mobility of the UAV can provide an opportunity for the wireless networks to enhance the system capacity.
In [2, 3, 4], UAV-enabled relaying channels were studied where UAVs act as mobile relays which forward the information of sources to destinations located on the ground. For the UAV relay networks, deployment and direction control problems were investigated in , and the work in 
minimized the network outage probability when the UAV trajectory is given as a circular path. The authors in solved the throughput maximization problem by optimizing the source and the relay transmit power allocation along with the UAV relay trajectory. In addition, UAVs have been employed as mobile base stations in various wireless networks [5, 6, 7, 8, 9]. The mobile base station placement problems were investigated in  and  in order to maximize the overall wireless coverage. In , analytical expressions for the optimal UAV height were derived to minimize the outage probability of air-to-ground links. The authors in  focused on the theoretical energy consumption modeling for UAVs, and proposed trajectory optimization methods for maximizing the energy efficiency of a UAV. Also, the trajectories of multiple UAVs were examined in  to maximize the minimum throughput performance of multiple ground terminals (GTs). Moreover, UAV-aided caching and mobile cloud computing systems were researched in  and , respectively.
In the meantime, energy harvesting (EH) techniques based on radio frequency (RF) signals have been considered as promising solutions for extending the lifetime of battery-limited wireless devices [12, 13, 14, 15, 16, 17, 18, 19, 20]. By utilizing wireless energy transfer (WET) and wireless information transmission (WIT), the RF-based EH methods have been studied for traditional wireless communications, and wireless powered communication networks (WPCN) protocols have been widely investigated in recent literature [17, 18, 19, 20].
Particularly, in the WPCN, a hybrid access point (H-AP) sends wireless energy via the RF signals to energy-constrained devices in the downlink WET phase. In the subsequent uplink WIT phase, the devices transmit their information signals to the H-AP by using the harvested energy. In , throughput maximization problems were introduced for the WPCN by optimizing the time resource allocated to users under the harvest-then-transmit protocol. The authors in  proposed the multi-antenna energy beamforming and time allocation algorithms to maximize the minimum throughput performance. The sum-rate maximization problems with a full-duplex H-AP were investigated in  for orthogonal frequency division multiplexing, and the precoding methods for the multiple-input multiple-output WPCN was provided in . Note that these works were restricted to a static H-AP setup, and thus it would suffer from the ‘doubly near-far’ problem , which is induced by the doubly distance-dependent signal attenuation both in the downlink and the uplink.
Recently, there have been several works combining mobile vehicle techniques and the WPCN [21, 22, 23, 24, 25, 26, 27]. For the magnetic resonant based WET, [21, 22, 23, 24] considered wireless charging vehicles which travel the networks to supply power to wireless sensors. However, due to short charging coverage of the magnetic resonance technique, the vehicles should stay quite a while to transfer energy to nearby sensors. To overcome this limitation, the authors in  adopted the RF-based WET methods to UAV-aided WPCN where a UAV flies towards a GT to transmit the RF energy signal and receive uplink data. However, only a single GT case was considered in  under a fixed line trajectory setup without optimizing the traveling path of the UAV. The works in  and  also examined the UAV-enabled WET systems, but they did not take into account the communications of GTs.
In this paper, we investigate the UAV-aided WPCN where multiple energy-constrained GTs are served by UAVs with arbitrary trajectories. Depending on the roles of the UAVs, we classify the UAV WPCN into two categories: integrated UAV and separated UAV WPCNs. First, in the integrated UAV WPCN, a single UAV behaves as an H-AP which broadcasts the RF energy signal to the GTs in the downlink WET phase and decodes the information from the GTs in the uplink WIT phase. In contrast, in the separated UAV WPCN, the WET and WIT operations are assigned to two different UAVs separately. In both systems, we adopt a time division multiple access (TDMA) based harvest-then-transmit protocol in where the WET of the UAVs and the WIT at the GTs are performed over orthogonal time resources.
In our proposed systems, we jointly optimize the trajectories of the UAVs, the uplink power control at the GTs, and the time resource allocation with the aim of maximizing the minimum throughput performance among the GTs. Since the location of the UAVs changes continuously, the time resource allocation in the UAV WPCN is totally different from that of the conventional WPCN with static H-APs. Also, compared to  where the trajectory of the UAV is restricted to a straight line, our systems consider a general traveling path optimization problem without any constraints on the UAV trajectory.
As the problem is non-convex, we propose iterative algorithms to obtain the local optimal solution by applying the alternating optimization method. To be specific, we first jointly optimize the trajectory of the UAVs and the uplink power of the GTs with given time allocation, and then update the time resource allocation solution by fixing other variables. First, to find the trajectory and the uplink power control solution, the concave-convex procedure (CCCP) framework 
is employed which successively solves approximated convex problems of the original problem. Next, the time allocation solution can be determined by applying linear programming (LP). The convergence and the local optimality of the proposed algorithms are then mathematically proved. From numerical results, we demonstrate that the proposed algorithms substantially improve the performance of the UAV WPCN compared to conventional schemes.
The rest of this paper is organized as follows: Section II explains a system model for the UAV WPCN and formulates the minimum throughput maximization problems. In Sections III and IV, we propose efficient algorithms for the integrated UAV and the separated UAV systems, respectively. Section V presents simulation results for the proposed algorithms and compares the performance with conventional schemes. Finally, the paper is terminated in Section VI with conclusions.
Throughout this paper, normal and boldface letters represent scalar quantities and column vectors, respectively. We denote the Euclidean space of dimensionas , and indicates the transpose operation. Also, and stand for the absolute value and the 2-norm, respectively.
Ii System Model
As shown in Figure 1, we consider a UAV-aided WPCN where single antenna GTs are supported by single antenna UAVs which transmit and receive the RF signals. It is assumed that the GTs do not have any embedded power supplies, while the UAVs are equipped with stable and constant power sources. To communicate with the GTs, the UAVs travel through the area of interest while transferring energy to the GTs in the downlink. By utilizing the harvested energy from the UAV, the GTs send their information in the uplink. We assume that the UAVs fly at a constant altitude of with the maximum speed for the time period , whereas all the GTs are fixed at given locations.
Depending on the operations of the UAVs, we classify the UAV-aided WPCN into two categories. First, in the integrated UAV WPCN illustrated in Figure 1(a), a single UAV transmits energy and collects data of the GTs. Thus, the UAV in the integrated UAV WPCN acts as an H-AP in the conventional WPCN . Second, in the separated UAV WPCN in Figure 1(b), the WET and the WIT are independently performed at two different UAVs. Therefore, each UAV in the separated system is dedicated to the energy transferring (ET) or the information decoding (ID). In the following, we present the system model for both UAV WPCN systems.
Ii-a Integrated UAV WPCN
Let us denote as the position of the UAV at time instant and as the location of GT , which is assumed to be known to the UAV in advance. For ease of analysis, the total time period is equally divided into time slots as in , where the number of time slots is chosen as a sufficiently large number such that the distance between the UAV and the GTs within each time slot can be considered approximately static.
Therefore, the trajectory of the UAV can be represented by a sequence of locations at each time slot as
where indicates the length of the time slot. Since we consider the discrete time trajectory , the maximum speed constraint can be expressed as
For the air-to-ground channel between the UAV and the GTs, the deterministic propagation model is adopted in this paper which assumes the line-of-sight links without the Doppler effect ,,. Then, the average channel power gain between the UAV and GT at time slot is given by
where denotes the reference channel gain at distance of 1 meter.
Next, we explain the transmission protocol for the UAV-aided WPCN. As shown in Figure 2, we divide each time slot into subslots, where the 0-th subslot of duration is allocated to the dedicated downlink WET and the -th subslot of duration for is assigned to the uplink WIT of GT . Note that the variable accounts for the time durations at the -th subslot in time slot . Thus we have the following constraints on the time resource allocation variable as
Now, we describe the WET and the WIT procedures of the integrated UAV WPCN. At the 0-th subslot of each time slot, the UAV broadcasts the wireless energy signals with the transmit power . Then, the harvested energy of GT at time slot can be written as
where stands for the energy harvesting efficiency of GT . For simplicity, we assume that all the GTs have the same energy harvesting efficiency, i.e., for .
Due to the processing delay of EH circuits at the GTs, the harvested energy may not be available at time slot . Hence, GT only can utilize at the future time slots . Defining as the uplink transmit power of GT at time slot , the available energy at time slot of GT can be expressed as
where the first and the second terms represent the cumulative harvested energy and the consumed energy of GT during the past time slots for , respectively. As a result, the uplink power constraint for GT at time slot is given as
where we have due to the EH circuit delay.
Also, the instantaneous throughput of GT at time slot can be obtained as
where is a portion of the stored energy used for the uplink information transmission at GT . For simplicity, we assume for . Then, the average throughput of GT for the time period can be written by
In this paper, we aim to maximize the minimum average throughput of the GTs by jointly optimizing the UAV trajectory , the uplink power control at the GTs, and the time resource allocation variables . Denoting as the minimum throughput of the GTs, the optimization problem can be formulated as
where the uplink energy constraint in (9) is derived from (1), (9) indicates the periodical constraint that the UAV needs to get back to the starting position after one time period 111Depending on the application, one may want to determine the initial location and the final location of the UAV in advance. In this case, we can simply add constraints on and and discard the constraint in (9)., and (9) is the peak uplink power constraint. One can check that (P1) is non-convex due to the constraints in (9) and (9), and therefore it is not straightforward to obtain the globally optimal solution.
Ii-B Separated UAV WPCN
In the separated UAV WPCN, we design the trajectories of two different UAVs, i.e., ID UAV and ET UAV. Let us define and as the position of the ET UAV and the ID UAV at time slot , respectively. Similar to the integrated UAV WPCN, we adopt the TDMA protocol in Figure 2. Then, the uplink energy constraint of GT at time slot and the average throughput of GT can be respectively expressed as
where and stand for the flight altitude of the ID UAV and the ET UAV, respectively.
Thus, the minimum throughput maximization problem for the separated UAV WPCN is given as
where and represent the maximum speed of the ID UAV and the ET UAV, respectively. This problem is also non-convex due to the constraints (II-B) and (II-B). In the following sections, we present efficient approaches for solving (P1) and (P2).
Iii Proposed Solution for Integrated UAV WPCN
In this section, we propose an iterative algorithm for (P1) which yields a local optimal solution. To this end, we employ the alternating optimization framework which first finds a solution for the trajectory and the uplink power with given time resource allocation , and then computes by fixing and .
Iii-a Joint Trajectory and Uplink Power Optimization
For a given time resource allocation , (P1) can be simplified as
Problem (III-A) is still non-convex due to the constraints in (9) and (9). To tackle this difficulty, let us first introduce auxiliary variables such that for and . Then, the left hand side (LHS) of (9) and the right hand side (RHS) of (9) are respectively lower-bounded by
For these bounds, we can construct an equivalent problem for (III-A) based on the following lemma.
The optimal solution for the problem (III-A) can be obtained by solving the following optimization problem:
First, let and denote the optimal value of problem (III-A) and (P1.1), respectively. Then it can easily be checked that , where the equality holds when , and . Next, by contradiction, we will prove that the optimum of (P1.1) can be attained when . Suppose that there exists at least one satisfying at the optimum of (P1.1) and denote a set of such as . If the equality holds in (1) for , the minimum throughput can be increased by reducing so that constraints (1) and (1) hold with equality. This contradicts the assumption. Even if the equality does not hold in (1) for at the optimum, decreasing does not affect the minimum throughput . Therefore, for all these cases, we can always find the optimal for (P1.1) satisfying . As a result, the optimal solution of (III-A) can be equivalently obtained by solving (P1.1).
Still, (P1.1) is non-convex in general. Thus, we provide the CCCP  approach to address (P1.1). First, we consider the throughput constraint in (1). By using a first-order Taylor approximation at , we can derive a concave lower bound for the LHS of (1) as
Note that is a jointly concave function with respect to and , and gives a tight lower bound in which equality holds at . In a similar way, the RHS of constraint (1), which is convex with respect to , can be lower-bounded by
(P1.1A) can be solved by existing convex solvers, e.g., CVX . Since the feasible region of (P1.1A) is a subset of that of the original problem (P1.1), we can always obtain a lower bound solution for problem (P1.1) from its approximation (P1.1A).
As a result, a solution for (P1.1) can be calculated by iteratively solving (P1.1A) based on the CCCP. At the -th iteration of the CCCP algorithm, we compute the solution and of (P1.1A) by setting , where and are the solution determined at the -th iteration. In this algorithm, we set to for all and . It has been proved that this CCCP method converges to at least a local optimal point . Note that for solving (P1.1) with the CCCP, we need to carefully initialize . This will be clearly explained in Section III-C.
Iii-B Time Resource Allocation
Now, we identify a solution for the time resource allocation for given and . The problem is written as
where and . It can be shown that (P1.2) is a convex LP, which can be optimally solved by the standard LP optimization tools.
|: Proposed Algorithm for (P1)|
|Initialize and , and , and set .|
|Set , and , and .|
|Set , and .|
|Solve (P1.1A) for given by using the CVX.|
|Update and , and .|
|Compute and from (P1.2) for given and .|
As a result, a solution of (P1) can be obtained by employing the alternating optimization framework and the overall process is given in Algorithm 1. In this algorithm, (P1.1) and (P1.2) are iteratively solved by fixing and , , respectively. To be specific, at the -th iteration, Algorithm 1 first successively solves (P1.1A) for given based on the CCCP until the objective value converges. Note that we denote the solution obtained at the -th iteration of the CCCP method as . Then, a solution of (P1.2) is computed for given , and this procedure is repeated until convergence.
Now, we verify the convergence of Algorithm 1. Let us define and as the objective value from the CCCP for (P1.1) and the optimal value of (P1.2) at the -th iteration, respectively. Then, it is obvious that since the CCCP algorithm monotonically increases the objective value of (P1.1A) with respect to the iteration index . Also, due to the fact that is the global optimal value of (P1.2) for given and , it follows .
As a result, we have
which implies that is non-decreasing with respect to the iteration index . Because the minimum throughput is upper-bounded by a certain value, Algorithm 1 is guaranteed to converge. It is worth noting that the solutions at each iteration of Algorithm 1 are given by the local optimum and the global optimum for (P1.1) and (P1.2), respectively. For this reason, Algorithm 1 always yields at least a local optimal point for (P1).
Iii-C Trajectory Initialization
In this subsection, we present a simple initialization method for Algorithm 1. Although the time resource allocation satisfying (9) and (9) can be initialized without problems, it is not easy to determine the feasible initial trajectory due to the complicated constraints in (9) and (9). Thus, we apply the circular path scheme in  to our scenario whose center and radius on xy-plane are respectively set to
where represents the centroid of the GTs, and and indicate the mean distance between and the GTs and the maximum allowable radius of the path with given speed constraint , respectively. Thereby, the initial UAV trajectory becomes
Note that for Algorithm 1, the time resource allocation should also be initialized. The details will be discussed in Section V.
Iv Proposed Solution for Separated UAV WPCN
In this section, we present an efficient algorithm for (P2) based on the alternating optimization. Similar to the integrated UAV WPCN, we first finds a solution for the trajectories and and the uplink power for given , and then computes for fixed , , and . The details are described in the following subsections.
Iv-a Joint Trajectories and Uplink Power Optimization
In this subsection, we optimize , , and with given . In this case, (P2) can be simplified as
To solve the non-convex problem (IV-A), similar to (P1.1), we introduce new auxiliary variables and such that and for and .
Then, problem (IV-A) can be reformulated as
The equivalence between problem (IV-A) and (P2.1) can be easily verified by a similar approach in Lemma 1 since we have and at the optimal point of (P2.1).
As in (P1.1), we can check that (P2.1) with the non-convex constraints (IV-A) and (IV-A) is the difference of convex problem which can be handled by the CCCP method . Thus, at each iteration of the CCCP algorithm, we address the following approximated convex problem as