Transmit power allocation of energy transmitters for throughput maximisation in wireless powered communication networks
2019; Institution of Engineering and Technology; Volume: 13; Issue: 9 Linguagem: Inglês
10.1049/iet-com.2018.6045
ISSN1751-8636
AutoresZhanwei Yu, Kaikai Chi, Kechen Zheng, Yanjun Li, Zhen Cheng,
Tópico(s)Wireless Power Transfer Systems
ResumoIET CommunicationsVolume 13, Issue 9 p. 1200-1206 Research ArticleFree Access Transmit power allocation of energy transmitters for throughput maximisation in wireless powered communication networks Zhanwei Yu, Zhanwei Yu School of Computer Science and Technology, Zhejiang University of Technology, Hangzhou, 310023 People's Republic of ChinaSearch for more papers by this authorKaikai Chi, Corresponding Author Kaikai Chi kkchi@zjut.edu.cn orcid.org/0000-0003-4751-2049 School of Computer Science and Technology, Zhejiang University of Technology, Hangzhou, 310023 People's Republic of ChinaSearch for more papers by this authorKechen Zheng, Kechen Zheng orcid.org/0000-0003-3886-4288 School of Computer Science and Technology, Zhejiang University of Technology, Hangzhou, 310023 People's Republic of ChinaSearch for more papers by this authorYanjun Li, Yanjun Li orcid.org/0000-0002-3976-3828 School of Computer Science and Technology, Zhejiang University of Technology, Hangzhou, 310023 People's Republic of ChinaSearch for more papers by this authorZhen Cheng, Zhen Cheng School of Computer Science and Technology, Zhejiang University of Technology, Hangzhou, 310023 People's Republic of ChinaSearch for more papers by this author Zhanwei Yu, Zhanwei Yu School of Computer Science and Technology, Zhejiang University of Technology, Hangzhou, 310023 People's Republic of ChinaSearch for more papers by this authorKaikai Chi, Corresponding Author Kaikai Chi kkchi@zjut.edu.cn orcid.org/0000-0003-4751-2049 School of Computer Science and Technology, Zhejiang University of Technology, Hangzhou, 310023 People's Republic of ChinaSearch for more papers by this authorKechen Zheng, Kechen Zheng orcid.org/0000-0003-3886-4288 School of Computer Science and Technology, Zhejiang University of Technology, Hangzhou, 310023 People's Republic of ChinaSearch for more papers by this authorYanjun Li, Yanjun Li orcid.org/0000-0002-3976-3828 School of Computer Science and Technology, Zhejiang University of Technology, Hangzhou, 310023 People's Republic of ChinaSearch for more papers by this authorZhen Cheng, Zhen Cheng School of Computer Science and Technology, Zhejiang University of Technology, Hangzhou, 310023 People's Republic of ChinaSearch for more papers by this author First published: 01 June 2019 https://doi.org/10.1049/iet-com.2018.6045Citations: 4AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Radio-frequency (RF) energy harvesting is one promising technology to power the nodes in wireless networks. This study focuses on large-scale wireless powered communication networks having multiple RF energy transmitters (ETs) and sinks, which almost have not been investigated previously. The authors aim to optimise the throughput via optimizing the transmit power allocation of ETs subject to a total power budget. Specifically, for the sum-throughput maximisation (STM) problem, they firstly formulate it to be a non-linear optimisation problem, then prove its convexity and finally propose an efficient dual sub-gradient algorithm to solve it. Owing to the throughput unfairness among nodes of the STM approach, they further consider the common-throughput maximisation (CTM; i.e. the worst node's throughput) and propose a very efficient algorithm for it. This algorithm divides the CTM problem into a master problem and a subproblem. The subproblem of determining the feasibility of a given common-throughput is solved by transforming it to a linear problem whose optimal solution indicates the feasibility. The master problem of determining the maximal common-throughput is solved by using the bisection search method. Simulation results demonstrate the effectiveness of the CTM approach to mitigate the throughput unfairness problem at the cost of decreased sum-throughput. 1 Introduction Conventionally, the nodes in wireless networks (like wireless sensor networks) usually operate on batteries [1, 2]. However, replacing or recharging the batteries is usually inconvenient, costly, and even dangerous, leading to quite limited node lifetime. Recently, the energy harvesting techniques have attracted a great deal of attention as a very promising technology for solving the limited lifetime problem of wireless nodes [3, 4]. The nodes can collect the energy from the ambient environment with energy harvesting techniques. However, since most renewable energy sources (such as wind) are unpredictable and intermittent, it is challenging to offer guaranteed network performance and these unstable energy sources are not applicable in some applications [5]. Different from these types of energy harvesting technologies, harvesting radio-frequency (RF) energy from the wireless RF signal transmitted by the dedicated RF energy transmitter (ET) is controllable and stable and thus has attracted considerable attention recently in the research communities and industries [6]. The research on wireless networks using RF-energy-harvesting technology can be classified into two directions, depending on whether or not the information and energy are retrieved at the receivers at the same time from wireless RF signals. Specifically, one direction aims the simultaneous wireless information and power transfer (SWIPT), which conducts wireless information transmission (WIT) and wireless energy transfer (WET) concurrently by using the same RF signal [7-17]. The other direction aims the wireless powered communication network (WPCN), which conducts the WET and WIT in separate RF signals [18-28]. SWIPT networks have drawn significant attention and have been considerably investigated so far. The idea of transmitting information and energy simultaneously was first proposed in [7]. So far, the SWIPT networks have been widely investigated for various channel setups, such as the point-to-point additive white Gaussian noise (AWGN) channel [10, 11], the fading AWGN channel [12, 13], the multi-antenna channel [14, 15], and the relay channel [16, 17]. For a more detailed introduction about SWIPT networks, please refer to [29]. Much groundwork research has already been done on WPCNs also. WPCNs have lower implementation cost than SWIPT networks. So far, most works focused on the star-topology WPCNs with single ET and sink [18-28]. In [18], Ju et al. presented the harvest-then-transmit (HTT) data collection strategy and achieved the sum-throughput maximisation (STM) and the common-throughput maximisation (CTM; i.e. the worst node's throughput) by optimising the durations of WET and WITs. In [20], we considered the scenario where each node needs to transmit a number of bits to hybrid sink (H-sink), which conducts both the WET and data receiving, and minimised the transmission completion time (TCT). This TCT minimisation problem covers the CTM problem as a special case where all nodes have the same amount of bits to transmit. Liu et al. [19] further considered the STM problem with the quality of service consideration in terms of communication reliability and diverse data traffic demands. Abd-Elmagid et al. [21] studied the STM and CTM problems of WPCNs where nodes have constant energy supply and can harvest RF energy also. Pejoski et al. [22] studied the WPCNs where the channel power gains during the subsequent time division multiple access (TDMA) frames are known by H-sink in advance and optimised H-sink's transmission power and the time allocation of WET/WITs inside each frame to maximise the sum of the logarithmic rates achieved by the network users and the STM, respectively. In [23], the non-orthogonal multiple-access (NOMA)-based WPCN was considered and the energy efficiency was maximised. The unmanned aerial vehicle (UAV)-enabled WPCN was considered in [24] where the UAV is dispatched as an H-sink to serve ground nodes periodically. The periodic trajectory and transmission resource allocation were jointly optimised to achieve the CTM. In addition, NOMA has attracted a lot of attention as a promising technology for the wireless network [30]. NOMA has been proposed to improve the spectrum efficiency as well as nodes fairness by receiving multiple nodes simultaneously to access the same spectrum. There has been some available works about NOMA [30-32]. In [31], the WPCN throughput maximisation was considered with NOMA mode and an efficient algorithm was proposed for time allocation. In [32], it is shown that NOMA outperforms TDMA in terms of sum-throughput and common-throughput. It is worth noting that most of the related works focus on the small-scale WPCNs having single ET and single sink. However, in many practical applications, a number of nodes are deployed in a large area and multiple ETs and sinks are needed in the network area. Therefore, it is important to investigate the design of large-scale WPCNs. So far, little research work has been carried out on the large-scale WPCNs. In [33], for the large-scale WPCNs, where WET and WIT use the same spectrum band and nodes transmit data to sinks based on the TDMA protocol, the nodes' spatial throughput was maximised by optimising the proportion of WET time. Bi and Zhang [34] studied the placement of ETs and sinks and aimed to minimise the number of ETs and sinks so as to minimise the deployment cost while satisfying the energy harvesting and communication performance requirements. In this study, we consider the large-scale WPCNs, where WET and WIT use different spectrum bands and the nodes transmit data to sinks based on the orthogonal frequency division multiplexing (OFDM) protocol, and aim to maximise the throughput via optimising the energy transmit power allocation of ETs subject to a total power budget. Notice that it is very important to optimise the transmit power allocation of ETs because we prefer to let the nodes with good WIT channel qualities have relatively large energy harvesting power in STM whereas letting the nodes with bad WIT channel qualities have relatively large energy harvesting power in CTM. To the best of our knowledge, this is the first attempt to investigate the transmit power allocation of ETs in large-scale WPCNs. The main contributions of this study are summarised as follows: • For the STM problem of the large-scale WPCNs, we firstly formulate it to be a non-linear optimisation problem, then prove that it is convex and finally propose an efficient dual sub-gradient algorithm to obtain the optimal transmit power allocation of ETs. • Due to the throughput unfairness among nodes of the STM approach, we further consider the CTM problem. After proving that it is convex, to efficiently find the optimal transmit power allocation of ETs, we firstly solve the feasibility problem (FP) of a given common-throughput by transforming it to be a linear problem whose optimal solution indicates whether the given common-throughput is feasible. Then the maximal common-throughput is found by using the bisection search method. • We demonstrate through simulations that, compared to the STM approach, the CTM approach is able to achieve a greatly larger common-throughput at the cost of decreased sum-throughput. The rest of this paper is organised as follows. Section 2 introduces the system model of the large-scale WPCNs having multiple ETs and multiple sinks. Sections 3 and 4 study the STM and CTM problems, respectively. Section 5 presents simulation results and discussions. Finally, Section 6 concludes the paper. 2 System model The topology of the considered large-scale WPCNs is shown in Fig. 1, which consists of M ETs, N sinks, and K nodes. In this topology, each node is associated with a sink (usually its nearest sink) for WIT, and thus each sink and its associated nodes constitute a star network. The mth ET transmits RF energy to the kth node with downlink channel coefficient and the kth node transmits its information to its associated sink with uplink (UL) channel coefficient . Thus, their corresponding channel power gains are and , respectively. In addition, denote the transmission power of the mth RF ET by , and let represent the maximum allowable transmit power. Clearly (1) Fig. 1Open in figure viewerPowerPoint Illustration of large-scale WPCNs consisting of multiple ETs and sinks In addition, the total energy transmit power of ETs cannot exceed a given budget, denoted by . Then it can be expressed as (2) Below we present the energy harvesting model and the communication model, respectively. 2.1 Energy harvesting model All ETs broadcast energy and nodes harvest the energy from all ETs. The transmitted energy signal of the mth ET is denoted by , which is a complex random signal satisfying . The received signal at the kth node is then expressed as (3)where and denote the received signal and AWGN at the kth node, respectively. It is assumed that is sufficiently large such that the energy harvested due to the receiver noise is negligible. Since the noise power is usually at the level of W, and it is negligible compared with the RF energy power. Thus, the energy harvesting power of the kth node can be expressed as (4)where is the energy harvesting efficiency. 2.2 Communication model Each node transmits the date to its associated sink using the harvested energy. Since the date transmission is performed on orthogonal frequency bands, there is no interfering with each other. Similarly, let be the complex signal transmitted by the kth node in its WIT duration such that . Then, the received signal at its associated sink is (5)where is the complex AWGN at its associated sink with the zero-mean and variance . Thus, the UL throughput of the kth node in bps/Hz can be expressed as (6)where and . 3 Sum-throughput maximisation In this section, we focus on the STM problem and present an efficient algorithm for it to obtain the optimal power allocation of ETs subject to the given total power budget. 3.1 Problem formulation of STM Based on the analysis in Section 2, we can formulate the STM problem as follows: (7a) (7b) (7c) Lemma 1. is a concave function of . Proof.Clearly, for any , is a concave function and is an affine function. According to the operations that preserve convexity, the composition is also concave of for any [35]. Thus, is also a concave function of . Thus, problem (P1) is a convex optimisation problem. Since there exists a feasible solution making inequality constraints strictly hold, the Slater's condition holds. So the strong duality holds for this convex optimisation problem. Therefore, we obtain the optimal solution of (P1) by solving its Lagrangian dual problem. 3.2 Design of efficient algorithm for STM In this subsection, we solve the Lagrangian dual problem by designing a dual sub-gradient algorithm. The Lagrange associated with problem (P1) is given by (8)where is the Lagrange multiplier associated with the constraint (7b). The Lagrange dual function is given by (9)where D is the feasible set specified by (7c). Meanwhile, the Lagrange dual problem is given by (10) It is well known that the Karush–Kuhn–Tucker (KKT) conditions are both necessary and sufficient to be primal and dual optimal, which are given by (11) (12) (13) (14) (15)where and represents the optimal primal and dual solutions. We can find that it is intractable to utilise (11-15) for obtaining the closed-form expression of . So we turn to design an iterative algorithm to obtain . Here, we develop the dual sub-gradient algorithm to solve the Lagrange dual problem based on the above KKT conditions. The algorithm iterates with two steps S1 and S2 until it converges: (S1) the Lagrangian maximisation with respective to given the current dual variable in the same fashion as (9) and (S2) the dual variable update through the sub-gradient-based algorithm that uses the results of the Lagrangian maximisation. (S1) Lagrangian maximisation aims to determine the optimal primal solutions for a given according to KKT conditions as shown below. We first present the following lemma. Lemma 2.For any (16)is a monotonically decreasing function with . Proof.Since is a monotonically decreasing function with . According to Lemma 2, we can search for satisfying (the ith equation in (15)) by using the bisection search method if each is given. Then the iterative method to obtain 's under a given is designed as follows. First, we initialise to be any positive value for . Then we conduct the following operation iteratively: in the order from i=1 to M, we use the bisection search method to find the solution of under the current 's (). The above operation is conducted iteratively until 's converge. (S2) Dual variable update aims to obtain the optimal dual variable , which is used to determine the optimal primal solutions. can be efficiently found by a sub-gradient-based algorithm, e.g. the ellipsoid method [36], as shown below: (17)where , is the small step size and denotes the index of iteration with the sub-gradient of given by (18) To summarise, the algorithm to solve (P1) is given in Algorithm 2 (see Fig. 2). Fig. 2Open in figure viewerPowerPoint Algorithm 1. Determining the optimal transmit power allocation of ETs for STM The computation complexity of this algorithm is , as explained below. Computing by using (16) has the complexity and the iteration times of steps 6–13 for obtaining using the bisection search method is which is constant. Thus, steps 4–13 have the complexity . It is observed in simulations that the iteration times of steps 3–14 are quite limited (usually several times or dozens of times). About the outer loop (from step 2 to step 16), the ellipsoid method has the complexity of to converge [36]. 4 Common-throughput maximisation It should be emphasised that maximising the sum-throughput results in the throughput unfairness among nodes although the achieved sum-throughput can be quite large, i.e. the nodes that are far from each ET or the nodes with bad WIT channel qualities may achieve very low throughput. Therefore, we further consider the CTM problem to maximise the smallest throughput of all nodes by optimising the power allocation of ETs subject to the given total power budget. 4.1 Problem formulation for CTM From (1-2) and (6), the CTM problem is formulated as (19a) (19b) (19c) Note that (P2) can be transformed to the following problem (P2B) by introducing an intermediate variable . (20a) (20b) (20c) (20d) It is not hard to know that this is a convex optimisation problem. Below we design a very efficient algorithm for it. 4.2 Design of efficient algorithm for CTM The maximum common-throughput is the largest one of all the feasible common-throughput that satisfies the rate inequalities (20d). To solve (P2), we first solve the following FP for any given positive : It is easy to know that determining the feasibility of (P2) is equivalent to determining whether the optimal solution of the following problem (P3B) satisfies Obviously, (P3) is feasible if and only if the minimised of (P3B) is smaller than or equal to . We transform the node throughput constraint (20d) in (P3B) to be (21) Notice that (21) is a linear constraint. Thus, problem (P3B) is a linear problem, and there are many available polynomial time algorithms (e.g. simplex algorithm) to solve it. It is obvious that as the common-throughput increases, the corresponding minimised total power of (P3B) also increases. Thus, we can search for by using the bisection search method. More specifically, for any given , we solve (P3B) to obtain the minimised . Then, if the minimised is larger than , the current common-throughput is too large (i.e. it is not achievable); otherwise, the current is smaller than the optimal common-throughput . To summarise, the algorithm to solve (P2) is given in Algorithm 2. Fig. 3Open in figure viewerPowerPoint Algorithm 2. Determining the optimal transmit power allocation of ETs for CTM The computation complexity of this algorithm is , which is explained as follows. The time complexity of steps 2–9 (the bisection method) conducts iterations, which is constant. Using the simplex method (step 4) to solve the linear programming problem (P3B) has the complexity of . Notice that the dual sub-gradient algorithm similar to that of the STM problem can be designed also to solve (P2B), i.e. (P2). However, it can be known that its computation complexity is , which is higher than Algorithm 2 (see Fig. 3). 5 Performance evaluation and discussions In this section, we compare the maximum sum-throughput with the maximum common-throughput of the large-scale WPCN. In the simulations, all ETs, sinks, and nodes are randomly distributed in a disk whose radius is 100 m. Each node is associated with its closest sink. The WET and WIT channel power gains are modelled as and , respectively, where is the distance between the kth node and the mth ET and is the distance between the kth node and its associated sink. The bandwidth is set as 1 MHz. The AWGN at all sinks is assumed to have a white power spectral density of . For each node, the energy harvesting efficiency is set to be . Fig. 4 shows the average sum-throughput and common-throughput of two schemes under different total power budget , with three ETs, three sinks, and 25 nodes. From this figure, we can draw the following conclusions. Fig. 4Open in figure viewerPowerPoint Average throughput and common-throughput versus the total transmit power budget (a) Average sum-throughput, (b) Average common-throughput First, as the total power budget increases, both the average sum-throughput and the common-throughput increase, no matter for the STM scheme or the CTM scheme. This is because as increases, each node can harvest more energy to achieve a larger throughput. Second, the STM scheme achieves a greatly larger sum-throughput than the CTM scheme. However, the CTM scheme greatly outperforms the STM scheme in terms of the common-throughput. This can be explained as follows. In the STM scheme, the nodes' throughputs are quite imbalanced. For example, we observed that, in a certain simulation topology, the best node with quite good WET/WIT channel qualities achieves 12.4 Mbps while the worst node with quite bad WET/WIT channel qualities achieves 0.2 Mbps only. The nodes with good WET/WIT channel qualities not only have high energy-provision efficiency but also can achieve more throughput per unit of energy consumption, i.e. they can contribute more throughput per unit of ET's energy provision. Therefore, the ETs near to these nodes are allocated with high transmit power such that these nodes achieve high throughputs. In the CTM scheme, the nodes' throughputs are quite balanced. The ETs nearest to the nodes with quite bad WET/WIT channel qualities are allocated with high transmit power such that there nodes have relatively large energy harvesting power to compensate their bad WIT channel qualities (i.e. these nodes' throughputs can be near to those nodes with good WET/WIT channel qualities). Fig. 5 shows the average throughput and common-throughput under different numbers of nodes, with three ETs, three sinks and . We can see that the average sum-throughput increases approximately linearly with the number of nodes no matter for the STM scheme or the CTM scheme. This is because the nodes operate in the OFDM mode and more nodes lead to a larger sum-throughput. Additionally, Fig. 5b shows that the average common-throughput is reducing with the increasing number of nodes, i.e. because as the number of nodes increases, the proportion of nodes located edge of topology would be greater. Thus, the worst WET/WIT channel power gain in this topology would be worst and this leads to the common-throughput reduction. Fig. 5Open in figure viewerPowerPoint Average throughput and common-throughput versus the number of nodes (a) Average sum-throughput, (b) Average common-throughput Fig. 6 shows that the average sum-throughput and common-throughput under different numbers of ETs, with three sinks, 25 nodes and . We can see that both the average throughput and the common-throughput increase with the number of ETs. When the number of ETs increases, the number of feasible power-allocation solutions increases also and thus a better solution of can be obtained. In addition, the average common-throughput is slowly increasing when the STM scheme is applied, i.e. because the STM scheme aims to maximise the sum-throughput rather than the common-throughput. Fig. 6Open in figure viewerPowerPoint Average throughput and common-throughput versus the number of ETs (a) Average sum-throughput, (b) Average common-throughput 6 Conclusions This study focused on large-scale WPCNs, which have multiple RF ETs and sinks, and aimed to optimise the throughput via optimising the power allocation of ET subject to a total power budget. We present efficient algorithms for the STM problem and the CTM problem, respectively. 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