# Energy-Efficient Power Allocation in Non-Linear Energy Harvesting Multiple Relay Systems
## 1 Introduction  
According to statistics, the average annual energy consumption of the information and communications industry accounts for 3% of the global total energy consumption, and carbon emissions exceed 2% of the global total [1, 2]. Collaborative relay technology and energy harvesting technology have emerged to meet the quality of communication while reducing energy consumption. In the post-5G era, because wireless networks are becoming more and more complex, how to design appropriate energy allocation strategies to improve system performance is a problem that needs further research [3]. Collaborative relay technology based on the energy harvesting (EH) model has been studied. In [4], the throughput maximization problem was solved via a successive convex approximation approach, in which the relay harvests energy in the natural environment. In [5], the authors analyzed the difference between an EH relay and a normal collaborative relay. The authors in [6] presented a power allocation algorithm in which the relay could switch between amplify and forward (AF) and decoding and forwarding (DF) modes to obtain the optimal transmit power and operational mode of the relay in order to maximize the energy efficiency (EE). The power allocation problem for a two-hop two-way single-relay system to maximize the EE was studied in [7]. Both source nodes and relay nodes use EH technology, and it was the aim of the authors to maximize the system capacity in [8]. The authors of [9] mainly analyzed and compared the advantages and disadvantages of the two schemes of simultaneous wireless information and power transmission (SWIPT). However, the above EH model assumes a linear EH model, in which the power conversionefficiency from the radio frequency (RF) to the direct current (DC) is a fixed constant. In fact, because the diodes, inductors and capacitors in the actual energy harvester [10] are nonlinear, the power conversion efficiency from the RF to the DC will vary with the input power of the energy harvester circuit. There are also preliminary studies on the application of nonlinear EH models in relay systems. In [11], a power allocation algorithm for a nonlinear EH model that maximized the total harvesting power of the energy harvester was presented. The problem of maximizing the EE of a two-way DF single-relay network was studied in [12]. The simulation results in [13] show that the power allocation based on the nonlinear EH model has higher throughput than that based on the existing linear model. In [14], the authors investigated the interrupt performance of nonlinear EH relay networks with a PS scheme. However, the above works only studied power allocation for single-relay systems and not for multi-relay systems. This paper presents a power allocation algorithm based on the nonlinear EH model for maximizing the EE in a two-hop multi-relay cooperative forwarding system. For clarity, we have listed the contributions as follows:  
* In this paper, a piecewise linear EH model is used to approximate the nonlinear EH model. Based on that, the EE maximization problem with quality of service (QoS) guaranteed is a complex, nondeterministic polynomial (NP) problem and can be decoupled into a single-relay power allocation problem. It is optimized based on the Dinkelbach iteration algorithm, and the optimization problem after iteration simplification is proven to be a convex programming problem. * Because the optimization problem is a maximum-minimum function problem, it is decomposed into two maximization problems to obtain the optimal solution by adding the corresponding constraints. For each maximization problem, simplified expressions are obtained according to the different polylines of EH models, whose closed-form solutions of the optimal relay transmit power and optimal power-splitting (PS) ratio are obtained by using mathematical methods such as monotonicity, Lagrange multipliers, the KKT condition and the Cardan formula.