Energy Efficiency Maximization for SWIPT Enabled Two-Way DF Relaying
Liqin Shi, Yinghui Ye, Rose Qingyang Hu, Hailin Zhang
TL;DR
This paper develops an optimal resource allocation scheme to maximize energy efficiency in a SWIPT-enabled two-way relay network, considering a non-linear energy harvesting model and multiple constraints.
Contribution
It introduces an iterative algorithm for joint optimization of transmit powers, power-splitting ratios, and transmission time in a non-convex setting.
Findings
The proposed scheme outperforms benchmark schemes in energy efficiency.
Simulation results confirm the effectiveness of the iterative optimization algorithm.
The resource allocation improves energy efficiency under rate and power constraints.
Abstract
This paper focuses on the design of an optimal resource allocation scheme to maximize the energy efficiency (EE) in a simultaneous wireless information and power transfer (SWIPT) enabled two-way decode-and-forward (DF) relay network under a non-linear energy harvesting model. In particular, we formulate an optimization problem by jointly optimizing the transmit powers of two source nodes, the power-splitting (PS) ratios of the relay, and the time for the source-relay transmission, under multiple constraints including the transmit power constraints at sources and the minimum rate requirement. Although the formulated problem is non-convex, an iterative algorithm is developed to obtain the optimal resource allocation. Simulation results verify the proposed algorithm and show that the designed resource allocation scheme is superior to other benchmark schemes in terms of EE.
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Energy Efficiency Maximization for SWIPT Enabled Two-Way DF Relaying
Liqin Shi, Yinghui Ye, Rose Qingyang Hu, and Hailin Zhang Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected] Shi, Yinghui Ye and Hailin Zhang are with the ISN of Xidian University, Xi’an, China. Rose Qingyang Hu is with the Department of ECE, Utah State University, USA. The corresponding author is Yinghui Ye ([email protected]).This work was supported by the scholarship from China Scholarship Council.
Abstract
This paper focuses on the design of an optimal resource allocation scheme to maximize the energy efficiency (EE) in a simultaneous wireless information and power transfer (SWIPT) enabled two-way decode-and-forward (DF) relay network under a non-linear energy harvesting model. In particular, we formulate an optimization problem by jointly optimizing the transmit powers of two source nodes, the power-splitting (PS) ratios of the relay, and the time for the source-relay transmission, under multiple constraints including the transmit power constraints at sources and the minimum rate requirement. Although the formulated problem is non-convex, an iterative algorithm is developed to obtain the optimal resource allocation. Simulation results verify the proposed algorithm and show that the designed resource allocation scheme is superior to other benchmark schemes in terms of EE.
I Introduction
Simultaneous wireless information and power transfer (SWIPT) has been recognized as a promising approach to solving the energy scarcity problem in energy-constrained wireless networks, e.g., the Internet of Things (IoT) networks[1, 2, 3, 4, 5]. Generally, there are two practical schemes to facilitate SWIPT, namely, time-switching (TS), and power-splitting (PS) schemes. For the TS scheme, the receiver switches the decoding information and the harvesting energy in time domain while for the PS scheme, the received signal is split into two different power streams in power domain, one for energy harvesting (EH) and the other for information decoding.
Moreover, wireless relaying has been promoted as an effective solution to improving system energy efficiency (EE), to extending coverage, and to enhancing capacity, particularly in the deep fading of wireless propagations. However, the limited energy supply in the relay node may discourage the relay from engaging information transmission. Inspired by this, SWIPT enabled wireless relaying was proposed and devoted to solving this problem [6, 7]. For the SWIPT enabled one-way relay network, the authors in [6] proposed an optimal transmission scheme of joint time allocation and PS to maximize the system capacity under the decode-and-forward (DF) protocol. In [7], the authors studied the optimal PS/TS ratio for the one-way amplify-and-forward (AF) relay network with a non-linear EH model. Since two-way relaying achieves a higher spectral efficiency, SWIPT enabled two-way relaying has attracted extensive interests [8, 9]. In [8], the authors studied the achievable throughput of SWIPT enabled AF two-way relaying under three wireless power transfer policies. Authors in [9] proposed an optimal asymmetric PS scheme for the SWIPT enabled product two-way AF relaying network to minimize the system outage probability. Recall that EE has become an emerging important metric in future wireless networks, the EE was maximized by optimizing both precoding matrices and the PS ratio for SWIPT enabled multiple-input-multiple-output (MIMO) two-way AF networks [10, 11]. By utilizing the statistical channel state information (CSI), the authors in [12] studied the optimal power allocation at sources to maximize the EE of SWIPT enabled two-way AF networks. Further, the EE fairness was investigated for multi-pair wireless-powered AF relaying systems in [13]. To the best of our knowledge, there has been no open work on the EE maximization for SWIPT enabled two-way DF relaying.
In this paper, we focus on the EE maximization for SWIPT enabled two-way DF relay networks under a practical non-linear EH model instead of the linear one adopted in [10, 11, 12, 13]. The optimization problem is formulated by jointly optimizing the transmit powers for the two source nodes, the PS ratios at the relay, and the time used for the source-relay transmission. The non-linear EH model makes our formulated problem much more challenging than that based on the linear EH model. Towards that end, we propose an iterative algorithm to obtain the optimal solution. Simulation results demonstrate the superiority of the proposed resource allocation scheme in terms of EE.
II System model
As shown in Fig. 1, we consider a SWIPT enabled two-way DF relaying network, where both the time division broadcast (TDBC) protocol111Since the low complexity of hardware is very vital to energy-constrained networks and the operational complexity of the TDBC protocol is lower than that of the Multiple Access Broadcast (MABC) protocol, this work considers the TDBC protocol to realize two-way relaying. and the “harvest-then-forward” scheme are employed. Two destination nodes and exchange information via an energy-constrained relay node . Assume that no direct link exists between nodes and due to severe path loss and/or shadowing and each node works in the half-duplex mode with a single antenna equipped. Let () denote the channel coefficient between nodes and . Each channel is assumed to undergo independent identically distributed (i.i.d) quasi-static block fading that is reciprocal in two directions. Here, we assume that the perfect CSI222The assumption of perfect CSI can obtain the upper bound of performance for our considered network. Thus, there are also many state-of-art works [9, 10, 11, 13], which are studied based on the assumption of perfect CSI. This motivates us to consider the assumption of perfect CSI. between the relay and two source nodes is available. Specifically, each source node needs to know the CSI between the source and the relay. The CSI is used to perform successive interference cancellation (SIC) at the source node. The relay needs to know the CSI between the relay and the two source nodes to determine the optimal resource allocation policy to maximize the EE of the whole system. Then the optimal resource allocation policy is known at each source node via the feedback before data transmission in each transmission block. Note that the CSI can be obtained following [14] and [15].
Let be the duration of the entire transmission block, which can be divided into three time slots based on time proportion factor 333In this work, it is assumed that the transmission time length of the - link equals that of the - link, the same as in [9, 15, 16]. It is worth pointing out that allowing different time lengths for the - and - links can further improve the performance of the SWIPT enabled two-way DF relaying network, which is beyond the scope of this paper and will be considered in our future work.. At the first or the second time slot of duration , or transmits the signal or to with the transmit power or and the received signal from () at is given by , where is the additive white Gaussian noise (AWGN); denotes the bandwidth of the system, and is the noise power spectral density. After receiving signal from (), splits it into two parts with ratio , where one part is used for energy harvesting and the other part is used for information processing.
For the energy harvesting, we employ a more practical piecewise linear EH model [17, 7] and the harvested power from can be computed as
[TABLE]
where is the received power from at ; with and are the thresholds on for linear segments; and are the scope and the intercept for the linear function in the -th segment, respectively, and denotes the maximum harvestable power when the circuit is saturated. According to (4), we have and .
For the information processing, the received achievable rate at from node is given by . During the first two time slots, the total harvested energy is calculated as
[TABLE]
where or () denotes the segment that or belongs to.
In the remaining block time of duration , combines the information from and as , where and denote the decoded signals for and , and broadcasts the combined signal with the harvested energy . Accordingly, the received signal at is written as
[TABLE]
where is the transmit power at ; is the AWGN at ; step (a) follows by using SIC [9, 8]; denotes the index of the other destination node. Based on (6), the achievable rate at from is given by , where , and .
Accordingly, the achievable rate of the link is . Then the total system achievable rate is given by . The total system energy consumption can be computed as , where denotes the power amplifier efficiency of the sources; is the constant circuit power consumption at nodes and as transmitters and is the constant circuit power consumed at nodes and as receivers.
III Energy Efficiency Maximization
III-A Problem Formulation
Considering that the non-linearity of the practical energy harvester always makes the optimization problem non-convex and difficult to solve, here we apply the piecewise linear EH model and the optimization problem to maximize the EE of the system can be formulated as .
[TABLE]
where constrains and ensure the minimum required rate for each end-to-end link; is the maximum transmission power at sources; and are the constraints that the energy harvester works in the -th and -th () linear regions, respectively.
In order to solve , there are three main steps as follows. In the first step, we compute the maximum number of segments that and may belong to, denoted by and (, ), respectively. Specifically, is the maximum number of segments which satisfies . In the second step, we solve the following optimization problem for given parameters and ():
[TABLE]
Note that for the case with , the total harvested energy is always [math]. Then the total system achievable rate is [math] and the constraints and can not be guaranteed. In this case, the optimization problem is infeasible. For the cases with , by solving , we can obtain resource allocation policies, denoted by , and compute the corresponding EE as . In the third step, we compare these values of EE and find the optimal solution to , denoted by . Specifically, the optimal solution to is determined by . Note that the main difficulty is to solve the non-convex fractional optimization problem due to the existence of coupling relationship among different optimization variables and the non-convex constraints, i.e., and .
III-B Solution to
To deal with the coupling relationship between and (), we let and can be rewritten as
[TABLE]
where
\min\!\big{(}{{{\log}_{2}}\!\left({1\!+\!\frac{{P_{\rm{A}}|{h_{\rm{A}}}{|^{2}}\left({1-{\rho_{\rm{A}}}}\right)}}{{{W\sigma^{2}}}}}\right),t{{\log}_{2}}\!\big{(}{1\!+\!\frac{{{K_{\rm{B}}}}}{t}\big{(}{\phi\left({{\rho_{\rm{A}}},{\rho_{\rm{B}}}}\right)\!+\!{K_{1}}}\!\big{)}}\!\big{)}}\!\big{)} and .
Let denote the optimal solution to . It is worth noting that the optimal solution to is given by , where The maximum EE for given and , denoted by , can be defined as
[TABLE]
where .
According to the Dinkelbach’s method [18], the maximum EE is achieved if and only if the following equation is satisfied.
[TABLE]
Thus, the problem in can be transformed by solving a parametric problem .
[TABLE]
where is a given parameter.
For the optimization problem , it is still non-convex due to the coupling between and () and non-convex constraints. We introduce four auxiliary variables into the problem. Specifically, we let , and . Then the optimization problem can be equivalently expressed as .
[TABLE]
where .
Proposition 1: The optimization problem in is convex.
Proof: See the Appendix.
III-C Algorithm
In order to solve , we develop a Dinkelbach-based iterative algorithm as shown in Algorithm 1. In Algorithm 1, we solve the optimization problem with a given in each iteration and obtain the optimal solution, denoted by . Given an error tolerance , when is satisfied, the solution to can be obtained. Using Algorithm 1, the optimal solution to can be obtained based on the three steps mentioned in Section III-A.
Assume that the interior point method is used to obtain the optimal solution to and that the number of iterations for Algorithm 1 is . Based on [19], the computational complexity for Algorithm 1 can be computed as , where denotes the number of the inequality constraints for . Then the computational complexity to solve can be computed as .
IV Simulations
In this section, we evaluate the effectiveness of our proposed EE maximization framework via computer simulations. In particular, the distance-dependent pathloss model () is adopted, where is the - channel coefficient, is the distance between node and the relay and denotes the pathloss exponent. According to [20, 11], the simulation parameters are set as follows: m, m, dBm, dBm, , kHz and Hz. The power amplifier efficiency is set to be . The minimum required rate is set as kbps. We employ the piecewise linear EH model with , where uW, and uW [17].
Fig. 2 demonstrates the convergence of the proposed Algorithm 1 under different sets of and . We set and . It can be observed that with any given and , the EE always converges to the optimal value within a limited number of iterations, which indicates that our proposed algorithm is computationally efficient.
Fig. 3 shows the average EE versus the maximum transmit power . To verify the effectiveness of the proposed scheme, we compare it with four other schemes under the same constraints. These four schemes are the equal transmit power allocation scheme (denoted as “Equal transmit power ”), the equal PS ratio scheme (denoted as “Equal PS ratio ”), the equal transmit power allocation and PS ratio scheme (denoted as “Equal transmit power and PS ratio ”), and the throughput maximization scheme. It can be observed that the throughput maximization scheme achieves a much lower EE than the proposed scheme due to the fact that the optimal resource allocation for SE maximization is not energy efficient, which illustrates the importance of considering the EE. Another observation is that our proposed scheme can achieve the highest EE among these schemes since the proposed scheme provides more flexibility to utilize the resource efficiently.
V Conclusions
In this paper, we have studied the EE optimization for SWIPT enabled two-way DF relay networks, where a non-linear energy harvester is equipped at the relay. We have formulated the EE maximization problem by jointly optimizing the transmit powers for two source nodes, the PS ratios at the relay, and the time for the source-relay transmission and proposed an iterative algorithm to achieve the optimal solution and the maximum EE. Simulation results illustrate the superiority of our proposed resource allocation scheme in terms of the EE.
Appendix
Since the objective function of is a linear function with respect to , and , and and are linear constraints, whether is convex or not depends on constraints and . That is, if both functions and , ( or ), are concave, then is convex. Taking the second-order derivative of , the Hessian matrix is given by \frac{{{\partial^{2}}{f_{1}}\left({P_{i},{x_{i}}}\right)}}{{\partial{{\left({P_{i},{x_{i}}}\right)}^{2}}}}=\left[{\begin{array}[]{*{20}{c}}{\frac{{-a_{i}^{2}}}{{{{\left({1+{a_{i}}\left({P_{i}-{x_{i}}}\right)}\right)}^{2}}\ln 2}}}&{\frac{{a_{i}^{2}}}{{{{\left({1+{a_{i}}\left({P_{i}-{x_{i}}}\right)}\right)}^{2}}\ln 2}}}\\ {\frac{{a_{i}^{2}}}{{{{\left({1+{a_{i}}\left({P_{i}-{x_{i}}}\right)}\right)}^{2}}\ln 2}}}&{\frac{{-a_{i}^{2}}}{{{{\left({1+{a_{i}}\left({P_{i}-{x_{i}}}\right)}\right)}^{2}}\ln 2}}}\end{array}}\right]\leq 0, where . Thus, is concave. As for , the Hessian matrix is given by (6) at the top of this page, where , and . Since the second and third order leading principle minors are 0, is negative semidefinite and is concave. Therefore, the optimization problem is convex.
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