Sum-Rate Maximization for UAV Aided Wireless Power Transfer in Space-Air-Ground Networks

The Internet of things (IoT) has become a prominent platform which bridges diverse technologies in order to meet the ever-increasing application requirements of various industries. However, the IoT devices, especially in remote areas that lack infrastructures, are featured by the restricted energy and pose great challenges on network access and sustainable communication. In this paper, we investigate the unmanned aerial vehicle (UAV) aided wireless power transfer under a space-air-ground (SAG) network, where the UAV is exploited as an aerial relay to assist in uploading information generated by ground nodes (GNs), and mounted with energy transmitter to deliver wireless energy for GNs. The goal is to maximize the system sum rate while satisfying the proportional rate for GNs and the sustainability of the ground network. To this end, by adopting decode and forward (DF) and amplify and forward (AF) protocols, two sum rate maximization problems are formulated via jointly optimizing power control, time allocation as well as UAV trajectory. The resource allocation problems are both nonconvex, which are difficult to solve directly. To tackle them, two near-optimal iterative algorithms are proposed by leveraging the successive convex approximation technology and the alternating optimization method. Extensive simulations are provided to demonstrate the effectiveness of the proposed algorithms and evaluate the impacts of various parameters on DF and AF relays.


I. INTRODUCTION
With the rapid development of sensing and wireless communication technologies, Internet of things (IoT), as a pivotal platform for information collection and exchange among physical devices, has been extended to various fields. IoT applications deployed in urban areas (e.g. smart cities, smart homes, etc.) can make full use of advantages provided by mature terrestrial networks, such as high speed, ultrareliability, and low latency [1]. However, as for Internet of remote things (IoRT) located in geographies such as oceans, forests, and polar regions, the costs of deployment and maintenance of the terrestrial networks are very high, which imposes great challenges on providing cost-effective network access [2]- [4]. In addition, low power consumption and long battery life on IoRT devices have become paramount requirements of the industry [5]. Battery recharging or manual battery replacement and envisioned as two feasible solution to The associate editor coordinating the review of this manuscript and approving it for publication was Xiaofan He .
prolong the battery life. However, it is inefficient and inconvenient for widely distributed IoRT devices.
To tackle the above mentioned issues, satellite communications (SatCom) and wireless energy transfer (WPT) have become the focus in both academia and industry. Recently, SatCom have attracted more and more organizations due to the inherent advantages of large coverage and high throughput. Aerospace manufacturer SpaceX, for example, plans to launch about 12000 LEO satellites with the aim of providing ubiquitous network coverage [6]. As a supplement to the terrestrial networks, SatCom have emerged as the key enabler of wireless communications to provide seamless connectivity in various scenarios such as military operations, disaster scenarios, as well as the remote areas such as oceans, mountains, and forests [7]. However, the severe propagation loss of satellite-to-ground links will further aggravate the energy consumption of IoT devices. To this end, relay communication is regarded to be an effective and indispensable method, deriving an emerging architecture termed as space air ground (SAG) networks, which have been extensively VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ investigated [8]- [11]. For instance, [12] employed multi UAVs as low altitude platform to forward the data generated by IoRT devices to satellite based on amplify and forward (AF) protocols. On the other hand, the radio frequency (RF) wireless power transmission (WPT) is envisioned as a promising scheme to provide low-power wireless devices with reliable and convenient energy [13]- [15]. Different from the traditional coupling charging, WPT employs RF signal as a carrier for energy transmission so as to extend the coverage and charge multiple devices simultaneously, which is suitable for widely distributed devices. Additionally, WPT is appealing in comparison with the solar or wind charging since it has the advantages in terms of controllability and stability [16]. To reap the benefits of relay techniques and WPT, by integrating WPT into the relay communication system can prolong the battery life of the source node and assist communication among far-apart devices [17]- [19]. However, the amount of harvested energy will be significantly affected by the severe propagation loss. Recently, UAV-aided WPT becomes a beneficial remedy to improve the efficiency of energy transmission, where the ground devices are powered by a UAV equipped with an energy transmitter. It was shown that the amount of energy harvested by ground devices has been significantly improved due to the existence of high probability line of sight (LoS) links and the controllable mobility of UAV [20], [21].
To our best knowledge, there are few researches on the energy transmission problem under SAG networks. In order to further explore the potential of UAV in the SAG networks, this paper studies the UAV-assisted WPT in the SAG network to meet the sustainable communication needs of IoRT devices. Specifically, in the space layer, low-orbit satellite provides seamless network access and collects data generated by IoRT ground nodes (GNs). As for air layer, the UAV is deployed as the low altitude platform to forward ground data to satellites by adopting either DF or AF protocol, and employs WPT to power GNs in the downlink with the aim of maintaining the battery level of GNs.
In addition, the max-sum criterion or the max-min criterion is generally considered as the optimization criterion when the objective is related to data rate. The former aims to maximize the sum rate of the system, nevertheless, may sacrifice the nodes with poor links [22]. By contrast, the latter is to maximize the minimum achieveable rate of nodes to improve the fairness among the nodes [2]. It is worth noting that the GNs in IoRT are in general deployed with different tasks, which leads to different rate requirements for each GN [23]. With such a motivation, we aim to maximize the system sum rate, subject to the different rate requirements by taking account into the proportional rate constrains (PRC) among the GNs, where PRC can be considered as a fairness-level configuration to guarantee that each GN gets a predetermined proportion of system sum rate [24].
The main contributions of this paper are summarized as follows: • This paper studies the UAV aided WPT in the SAG network, in which the UAV is employed as a mobile relay to forward the data generated by the GNs to the satellite, and charges GNs with the dedicated signal in the downlink. The system sum rate maximization problems are investigated in both DF and AF relaying modes via jointly optimizing the power of GNs and UAV, time allocation and UAV trajectory, subject to the energy neutral constraints (ENC) and PRC, which guarantee the proportional data rates for GNs and the sustainability of the ground network.
• It can be verified that the formulated problems are both nonconvex due to the highly coupled variables. To tackle them, for DF, an iterative algorithm is proposed to obtain a near-optimal solution by leveraging the successive convex approximation (SCA) technique. For AF relay, an alternating iterative algorithm is developed by resorting to the SCA technique and the alternating optimization method. Specifically, in order to break the couplings among the variables, the problem is decoupled into three sub-problems, which are solved in an alternating manner, and an near-optimal solution is obtained.
• Simulation results are provided to evaluate the effectiveness and superiority of the proposed scheme. The performance of DF and AF protocols are compared in different power settings and flying altitudes via simulations, which shows that DF is superior to AF at a certain height. Compared with the traditional static relay, the proposed scheme significantly improves the system sum rate. In addition, compared with the dynamic UAV with a circular trajectory, the proposed schemes bring significant 122.79% and 149.93% gains to the system sum rate in DF and AF modes, respectively.

II. SYSTEM MODEL
The architecture of UAV aided WPT in SAG networks is shown in Fig. 1. We focus on the uplink of the network containing a LEO satellite, K GNs and one UAV. The communication between the satellite and the GNs is assisted by the UAV relay due to the channel attenuation and the constrained energy of the GNs. Considering the limited energy of the GNs, suppose that the GNs stay stationary relative to the ground. The UAV and each GN are both equipped with a single antenna for receiving or transmitting signals. Besides, each GN is equipped with a supercapacitor storing the harvested energy transmitted by the UAV via the RF signal. In order to reduce the UAV energy consumption, it is assumed that the UAV flying altitude is fixed at H U during the time interval T , which is the minimum altitude that can avoid the terrain and blockage. Besides, the time interval T is divided into N equal slots, i.e., T = N δ, where δ represents the slot length which is chosen to be sufficiently small so that the distance between the UAV and the GNs is constant within δ. Herein, the UAV horizontal coordinate position at each slot n ∈ N = {1, . . . , N } can be approximated by maximum flying speed of the UAV, and the flying distance within δ is constrained, which can be expressed as In addition, this paper considers cycle flight mode, which can be expressed as The location of the GN k ∈ K = {1, . . . , K } is denoted by w k = [x k , y k ] T , which can be acquired by the UAV. Hence, the Euclidean distance between the UAV and the GN k at slot n can be given by Without loss of generality, the links between the GNs and the UAV are dominated by LoS, and the free space path loss model is considered [21], [25]. The channel power gain between the GN k and the UAV at slot n can be expressed as where β 0 denotes the channel gain at the reference distance d 0 = 1m. The satellite altitude H S is so high that the change of the UAV position is negligible during the mission duration. According to [26], the channel power gain between the UAV and the LEO satellite at slot n can be simplified as where G tr u and G re s denote the transmit antenna gain of the UAV and the receiving antenna gain of the satellite, respectively. c is the speed of light and f is the carrier frequency.

A. DECODE AND FORWARDE RELAY
In the downlink, the UAV broadcasts dedicated RF signals to charge the GNs, i.e., WPT mode. Subsequently, the harvested energy of the GNs is utilized to transmit information to the UAV. We assume that the relay has adequately large buffer to store the received information. In the UAV uplink, the ground information uploaded to the UAV is decoded and forwarded to the satellite.
In this paper, we adopt the time division multiple access (TDMA) providing orthogonal time domain resources for energy transmission and information transmission. The discretized N identical slots are further divided into at most 2K + 1 sub-slots. As shown in Fig. 2, t DF k [n] δ indicates the time interval at which the GN k communicates with the UAV at slot n. Besides, τ DF k [n] δ denotes the time allocated to the UAV at slot n to forward the information of the GN k to the satellite. τ DF 0 [n] δ means that the UAV works in the WPT mode at slot n. Herein, the allocation of time resources should be constrainted as following Denote the transmit power of the UAV to forward the information of the GN k at slot n by P DF k [n] and define P DF 0 [n] as the energy transmit power of the UAV at slot n. In order to prevent a surge in transmit power caused by sending a large amount of information in the limited duration, we consider a VOLUME 8, 2020 maximum power constraint on the UAV transmit power, i.e., Similarly, let p DF k [n] be the transmit power of the GN k at slot n, we have the following maximum power constraint Additionally, denote the average transmit power of the UAV as P ave , and the average transmit power constraints at the UAV can be expressed as This paper adopts the linear regime for energy conversion [21] and the amount of energy harvested by the node k at slot n can be expressed as where η k ∈ (0, 1] represents the energy conversion efficiency of the GN k, which depends on the condition of the received energy signal [27], [28]. For ease of analysis, we assume that the energy conversion efficiency of each GN is the same [21], that is, η = η k , ∀k ∈ K.
To ensure the terrestrial network has self-sustainable operation capabilities, we set the ENC of each GN [21], i.e., the energy consumed by each GN for transmitting information cannot exceed the energy harvested from the UAV. Hereby, we have the following ENC for each GN In addition, we assume that the noise at the UAV and the satellite receivers is additive white Gaussian noise (AWGN) with a mean of zero and variances of σ 2 u and σ 2 s , respectively. For convenience, the system bandwidth is normalized to be unit. Suppose that data bits are transmitted instead of packets, the achievable rate from the GN k to the UAV at slot n can be given by At slot n, the rate of UAV forwarding the information generated by GN k can be expressed as For the DF protocol, the UAV in any slot can only forward the information that has been received from the GNs. We consider that the received information will have a processing delay of one time slot [29], and will be forwarded to the satellite in the next time slot in a first in first out (FIFO) manner. As a result, we have the following information causality constraints (ICC) Denote the average rate of the GN k over the period T by R DF k , which is equivalent to that of satellite reception while subjecting to the ICC. Consequently, the system sum rate over the period T is expressed as , ∀k, n . Our objective is to maximize the system sum rate by jointly optimizing the time allocation (i.e., τ , t), power control (i.e., p, P) as well as the trajectory of UAV (i.e., q). The optimization problem can be formulated as follows where (17e) represents the PRC, and {φ 1 , φ 2 , · · · , φ K } denotes the rate proportion among GNs. For GN k, the proportion of system sum rate can be written as φ k = φ k / K k=1 φ k . Hence, the PRC (17e) can be rewritten as It can be observed that the objective function of P1 is nonconvex as well as its constraints (e.g., ENC, ICC, PRC). Therefore, problem P1 is a nonconvex problem, and the corresponding solution is proposed in section III.

B. AMPLIFY AND FORWARD RELAY
In the following, the framework of AF protocol is introduced. As depicted in Fig. 2 δ represents the time taken by the GN k to forward the information to the satellite at slot n, where the time allocated to the first hop and the second hop is set to be equivalent [30]. The phase t AF 0 [n] δ indicates the duration of UAV working in WPT mode at slot n. Herein, the time allocation should be constrainted as following Besides, we define P AF 0 [n] as the energy transmit power of UAV at slot n and denote P AF k [n] as the transmit power of UAV to forward the information of the GN k at slot n. Similar with DF protocol, the average transmit power constraint in AF protocol can be expressed as The linear regime for energy conversion is considered in this paper, thus the amount of energy harvested by the GN k at slot n can be written as To guarantee the self-sustainability of the terrestrial network, we have the following ENC for the AF protocol where p AF k [n] represents the information transmit power of the GN k at slot n.
According to [31], the end-to-end signal noise ratio (SNR) for the GN k to the satellite at slot n can be derived as follows with Consequently, the average rate of the GN k within the time period T can be expressed as where denotes the achievable rate of the GN k at slot n. As a result, the system sum rate for AF protocol can be expressed as Our goal is to maximize the system sum rate by varying the time allocation, transmit power control of UAV and GNs as well as UAV trajectory.
, ∀k, n , the optimization problem in AF protocol can be formulated as where (29b) and (29c) denote the maximize power constraints for GNs and UAV, respectively. (29f) represents the PRC in AF protocol. In a similar manner to (18), (29f) can be rewritten as Since the objective function is nonconvex and constraints (29d)-(29f) constitute a non-convex set, P2 is a nonconvex optimization problem. The corresponding solution is proposed in section IV.

III. PROPOSED SOLUTION FOR DF PROTOCOL
In this section, we investigate the optimization problem P1 and obtain the solutions of power control, time allocation and UAV trajectory for maximizing the system sum rate with the DF relay. As discussed earlier, problem P1 is nonconvex. To tackle this issue, the primal problem is equivalently rendered into P3 by introducing the slack variables. With the help of appropriate variable substitution, P3 is further transformed into P4. Then P4 is addressed by iteratively solving a series of the convex approximation problem P5.
To deal with the equality constraints in (18), we resort to the following lemma. Lemma 1: The equalities in (18) can be replaced by the inequalities as follows When the constraints in (31) are not violated, the constraint (31) must be satisfied with the equality. Proof: We prove Lemma 1 by contradiction. Suppose that there exists at least one GN k that the corresponding constraint in (31) is satisfied with strict inequality. As a result, we have the following inequalities which contradicts the fact that R DF sum = K k=1 R DF k . The proof is completed. VOLUME 8, 2020 Relying on Lemma 1, the equalities in (18) can be equivalently converted into the inequalities in (31). Then, we proceed to solve the nonconvex constraints ICC and PRC. We introduce the slack variables π k [n], where π k [n] ≤ R DF k,2 [n]. The PRC and ICC are separately arranged as follows Consequently, P1 is reformulated as the following problem Theorem 1: Solving problem P3 is equivalent to solving problem P1.
Proof: For the optimal solution of problem P3, if there exists a GN k at slot n such that the constraint π k [n] ≤ R DF k,2 [n] does not hold the equality, we can always enable the inequality to be equality by reducing the corresponding UAV information transmit power P DF k [n] or the allocated time τ DF k [n], yet without changing the objective value of P3. Hence, we always have an optimal solution so that all the inequalitiy constraints in (35c) are active, which completes the proof.
Note that P3 is still nonconvex due to the tightly coupled variables. To address the couplings in the achievable rate expressions, we introduce the auxiliary variables . Then the righthand side (RHS) of (35b) and (35d) can be rewritten as  [32]. Moreover, the summation of concave functions still has the concave property [32]. As a result, the right-hand side (RHS) of (35d) is equivalently rendered into a concave function.
It can be observed thatR DF k,1 [n] is still a nonconcave function. By introducing the auxiliary variables .
With the help of the foregoing manipulations, problem P3 can be reformulated as ∀k ∈ K, n ∈ N , (40j) (1), (2), (6), (7). (40k) It can be seen that the nonconvex PRC, ICC and the average power constraint have been transformed into convex constraints and two additional constraints are introduced. The ENC (40e) is still nonconvex after replacing theẼ DF k [n] into the ENC due to the fact thatẼ DF k [n] is a jointly convex function over ρ 0 [n] and ω k [n] instead of the concave function.
Besides, the greater side of additional constraint (40f) is also a convex function, which results in the nonconvex constraint.
To handle the foregoing issues, we resort to the Taylor series approximation, also namely SCA technique. Assume f (x) is a differentiable convex function of x, and there is a tangential function g (x,x) of f (x) at tangency pointx. Then, we have inequality f (x) ≥ g (x,x), which holds the strictly inequality for all the feasible points of f (x) exceptx and the equality holds when x =x. Moreover, g (x,x) can be obtained by the first-order Taylor expansion of f (x) at tangency pointx. As a result, we have the following inequality Accordingly, to deal with the the non-convexity of constraints in (40e), with the given points ρ l 0 [n] and ω l k [n] in the l-th iteration,Ẽ DF k [n] satisfies the following inequality The equality in (42)  Then, we proceed to tackle the additional constraint (40f). In a similar way, with given points θ l k [n] and ω l k [n] in the l-th iteration, the lower bound of the introduced auxiliary constraint (40f) can be derived as follows in which the equality in (43)  , the following problem need to be solved in the l-th SCA iteration which is a standard convex optimization problem and its optimal solution can be obtained by the mature convex optimization method, such as interior point method [32]. It is worth mentioning that the solution obtained by the approximated problem P5 is always feasible for problem P1, but Update l = l + 1. 5: until (Obj (l) − Obj (l − 1)) ≤ ε. Output: The sub-optimal solution for maximizing the sum rate of optimization problem P1.
the reverse dose not hold generally. As a result, the optimal solution obtained from problem P5 is generally the lower bound of problem P1. The proposed SCA-based solution for the problem P1 is summarized in Algorithm 1, where ε represents the convergence threshold of the algorithm.
Since the objective value of Algorithm 1 obtained by SCA technique is non-decreasing over each iteration, and the objective value obtained from Algorithm 1 is upperbounded by the optimal value of P1, which is obviously a finite value, Algorithm 1 can be guaranteed to converge to at least one locally optimal solution [33]. Furthermore, the complexity of Algorithm 1 is mainly derived from the convex optimization problem P5. The complexity of solving P5 by utilizing the interior point method is O (KN ) 3.5 , where K and N denote the number of GNs and time slots, respectively [32]. Further, let r max be the maximum iteration number that allows Algorithm 1 to converge. Thus, in the worst case, the computational complexity of Algorithm 1 can be estimated by O r max (KN ) 3.5 .

IV. PROPOSED SOLUTION FOR AF PROTOCOL
To address the nonconvex equality constraint PRC of P2, in the light of Lemma 1, we replace the equalities in (30) with the inequalities as follows Then, we introduce the relaxation variable ≤ R AF sum and obtain the following optimization problem ≤ R AF sum , (46c) (29b), (29c), (29d), (29e), (29g). (46d) Theorem 2: Problem P6 is equivalent to problem P2. Proof: We prove Theorem 2 by contraction. Suppose that there exists an optimal solution * that satisfies the strict inequality in (46c), we can always increase * such that the objective value of P6 can be further increased and the constraint in (46c) holds the equality, which contradicts the assumption that * is the optimal solution for P6. Therefore, we always have an optimal solution such that the constraint in (46c) hold the equality, which completes the proof.
Though the objective function has been transformed into a scalar variable , it is still nonconvex due to the coupled variables of the constraints. Next, with the intention of breaking the couplings, we decompose the problem into three subproblems by resorting to the alternating optimization method. The main idea is to obtain the optimal value of the current block variables when the variables of the other two blocks are fixed, and the optimal value of each block is then used to update the variables of other blocks until achieving preset error accuracy.

A. TIME ALLOCATION
In this section, the power control (i.e.,P,p) and UAV trajectory (i.e., q) are given to obtain the optimal time allocation variables. The optimization problem can be expressed as It can be seen that both the objective function and the constraints of P7.1 are affine. As a result, the aforementioned optimization problem is a typical linear programming problem, which can be optimally solved by the off-the-shelf optimization solvers, such as CVX [32].

B. POWER CONTROL
In this section, we fix the time allocation and UAV trajectory (i.e.,t, q), then the power control optimization problem can be expressed as where Hence, with Eq. (50) in hand, problem P7.2 is approximated by the following optimization problem . It is noticeable that problem P7.2A is an approximation of the primal problem P7.2 and the optimal solution obtained by the problem P7.2A is the lower bound of P7.2.

C. UAV TRAJECTORY OPTIMIZATION
In this section, given the time allocation (i.e.,t) and power control (i.e.,P,p), the trajectory optimization problem can be written as where f l k, 3 [n], as shown at the bottom of the page, and Z l k [n] = q l [n] − w k [n] 2 + H 2 U . To tackle the nonconvex constraint (55b), in a similar manner, with the attained trajectory q l [n] in the previous iteration, we can derive the tight lower bound for E AF k [n], which can be expressed as To this end, in the (l + 1)-th iteration, trajectory optimization problem P7.3 is approximated as follows which is satisfied with the requirements of convex optimization. It is worth mentioning that the objective value achieved by problem P7.3A is always the lower bound of that of P7.3. Algorithm 2 summarizes the solution to solve problem P2. Though the second and the third sub-problem (i.e., P7.2 and P7.3) are solved by the SCA techniques that cannot guarantee to achieve the global optimal values, it can be shown that the objective value obtained by each sub-problem is nondecreasing, which results the objective value in the outer loop (i.e., P2) is also non-decreasing after each iteration. Moreover, the objective value is upper-bounded by the optimal value of P2, which is obviously a finite value. As a result, the proposed Algorithm 2 is guaranteed to converge [33].
Note that the primal problem is decomposed into three subproblems by using alternating optimization method. According to the analysis in [34], the computational complexity of Algorithm 2 can be expressed as O l max (KN ) 3.5 , where l max indicates the maximum iteration number that allows Algorithm 2 to converge.

V. SIMULATION RESULTS
In this section, simulation results are presented to verify the performance of the proposed algorithms, which are both implemented by the CVX. In the simulations, K = 8 GNs are randomly distributed in a rectangular area of 400 m × 400 m. The flying altitude of the UAV is set to H U = 20 m. The maximum speed of UAV is set to be V max = 50 m/s. The flight cycle is T = 25 s and the slot length is δ = 0.5 s. The system bandwidth is set to 1 MHz, and the noise power spectral density at the receiver of the UAV is −174 dBm/Hz. The channel gain between the GNs and the UAV at the reference distance d 0 = 1 m is 10 −3 . The height of the LEO satellite is H S = 780 km. The noise power at the receiver of Algorithm 2 Solution for Solving Problem P2 Input: The tolerance error . 1: Initialize: l = 0, and q (0) ,p (0) ,P (0) . 2: repeat 3: Solve problem P7.1A according to the given q (l) ,p (l) ,P (l) , and denote the optimal time resource allocation solution ast (l+1) . 4: Solve problem P7.2A according to the given t (t+1) , q (l) , and denote the optimal power control solution as p (l+1) ,P (l+1) .
In order to verify the effectiveness of the proposed algorithm, the following baselines are used. Baseline 1 is designed as a circular flight trajectory, in which the center of the circle is the geometric center of the IoRT nodes, and the radius of the circle is the distance between the geometric center and the farthest point. Baseline 2 is set as UAV fixed in the geometric center of the IoRT nodes. It can be found that baseline 1 and baseline 2 are resource optimization problems under a given flight trajectory. The corresponding optimal solution of the DF relay protocol is detailed in the Appendix B. The solution of AF relay can be obtained by solving the first two subproblems of problem P7 via the alternating optimization method.
In the Fig. 3, the rate proportion of the GNs is adjusted to visually observe the effect of the rate proportion on the trajectory, where the proportion is set to φ 1 : φ 2 : φ 3 : φ 4 = 1 : 1 : 5 : 5 and φ 1 : φ 2 : φ 3 : φ 4 = 5 : 5 : 1 : 1, respectively. It can be intuitively observed that for DF and AF, the flight trajectory is always close to the node with high proportion. This is expected because the distance between the UAV and the node will directly affects the channel gain, which has a significant impact on energy harvesting and signal transmission. Therefore, the channel state between the nodes with high proportion and the UAV is improved by adjusting the trajectory of the UAV, so as to improve the achievable rate and the harvested energy of the nodes with high proportion.  Furthermore, Fig. 4 and Fig. 5 show the flight trajectory of the UAV. Fig. 4 depicts the optimized trajectory with the flight cycle T = 25 s. It can be seen that for both DF and AF, the initial circular trajectory is adjusted to be as close to each GN as possible. Fig. 5 depicts the optimal UAV trajectory with different flight cycles. It can be observed that when the flight cycle is shorter, the UAV cannot traverse each node due to the flight speed constraints. Additionally, the trajectory of the UAV is closer to the nodes with high proportion. For instance, when T = 10 s, it can be clearly observed that the UAV flight trajectory is closer to the GNs 1, 2, 3 and 4. This is consistent with the observation of the Fig. 3. Moreover, as the flight duration increases, the UAV expands the trajectory to each node as far as possible, and the flight trajectory gradually stabilizes. Fig. 6 evaluates the performance of the proposed scheme by varying the average power P ave and comparing with the baselines. It can be observed that due to the mobile characteristics of the UAV, the system sum rate of the baseline 1 is improved compared to the baseline 2. In addition, by optimizing the trajectory of the UAV, the sum rate of the system has been further increased. For instance, when P ave = 26 dBm, in comparison with the baseline 1, the proposed scheme brings significant gains of 122.79% and 149.93% with respect to the system sum rate in DF and AF modes, respectively. In addition, due to the decoding mechanism of DF mode and the more flexible time allocation scheme compared with AF mode, it can be intuitively observed that the sum rate of DF is always larger than that of AF in corresponding scheme at all given average power values. Fig. 7 depicts the energy harvested by each GN with different schemes. Firstly, it can be intuitively observed that the amount of energy harvested by GN has been significantly improved compared with the baselines. By adjusting  the trajectory of the UAV, the proposed scheme can improve the channel state and flexibly allocate the charging time and power at the same time, thus improving the efficiency of energy transmission. In addition, relying on the flight characteristics of the UAV, GNs 1-4 with a higher proportion harvested more energy than GNs 5-8 under the proposed scheme and baseline 1, which enables GNs with high proportion to transmit more information. By contrast, the harvested energy of the baseline 2 depends entirely on the distance between the GN and the energy transmitter. Fig. 8 draws the effect of UAV flying altitude H S on the system sum rate. It can be seen that the performance of the proposed scheme at low flying altitude delivers a superior performance over the baseline in both DF and AF modes. Besides, we can observe that increasing the UAV flying altitude induces the reduction of the system sum rate, and the substantial performance gap between the proposed and the baseline schemes gradually deteriorates. The reason is that increasing the altitude of the UAV incurs a weaker link between the UAV and the GN, which will significantly VOLUME 8, 2020  decrease the amount of harvested energy and attenuate the signal strength from the GN to the UAV. Due to the existence of ENC, the transmitted signal strength of the GNs will also be reduced. When the altitude of UAV becomes sufficiently high, the performance gain brought by the UAV maneuverability even can be basically neglected. In addition, since the performance of DF protocol is limited by the weaker hop, the system sum rate performance of all schemes tends to the same level in the case of excessively worse link between the UAV and the GN.
In Fig. 9, we describe the performance in terms of the system sum rate versus the flight cycle T of UAV. Observing the system sum rate of the baseline 1 and baseline 2, we can find that the sum rate remains stable after the trajectory is determined. In addition, we can clearly visualize that the system sum rate of the proposed scheme is gradually amplified with the increases of flight cycle. This is because with the increase of flight cycle, the UAV can adjust the flight trajectory to traverse each GN as far as possible, so as to improve the channel state between the UAV and the GN. Further, the system sum rate of the proposed scheme tends to be saturated as time becomes sufficiently long. This is because the UAV flight trajectory becomes stable over time, which is consistent with the results observed in Fig. 5.

VI. CONCLUSION
In this paper, we have studied the UAV aided WPT in the SAG network, where the UAV has been dispatched to charge the GNs and assist the communication between the GNs and the satellite. Both AF and DF protocols have been employed to support the studied network. Moreover, the system sum rate optimization problems with PRC and ENC have been established, which enable the network to provide proportional rate for the GNs and self-sustainable operation for the ground network. To address the non-convexity of the optimization problems, the problem based on the DF protocol has been effectively solved by leveraging the SCA technique. In addition, the optimization problem based on the AF protocol has been divided into three sub-problems (i.e., time division, power control, and UAV trajectory optimizing) by utilizing the alternating optimization method to break the couplings. It is revealed that the proposed schemes significantly improve the system sum rate as well as the energy harvested level of each GN by optimizing the UAV trajectory.

APPENDIX A
The convexity of R k [n] can be validated by definition of the convex function. We first obtain the Jacobian matrix of R k [n] which can be expressed as follows the Hessian matrix of R k [n] can be readily found semidefinite since all the principal minor are nonnegative. Therefore, the proof is completed.

APPENDIX B
To handle the optimization problem in static state with DF protocol, we introduce the auxiliary variables ρ k [n] = τ k [n] P k [n] , n ∈ N , k ∈ {0, K} and θ k [n] = t k [n] p k [n] , n ∈ N , k ∈ K. The first and second hop achievable rate of GN k at slot n can be equivalently written as She took part in performance evaluation work of the Chinese Evaluation Group, as a Representative of BUPT. She has authored over 100 papers in international journals and conferences proceedings. Her research interests include the cooperative and cognitive systems, radio resource management, and mobility management in 5G systems.