## 1 Introduction  
Recently, cooperative communication has become an effective method to achieve spatial diversity by establishing a virtual multi-antenna structure. In a cooperative communication system, relay is used to assist the communication from the source node to its far-end destination node. Due to the communication nodes operated at a half-duplex mode, the information exchanging between two nodes in the one-way relaying network requires four time slots. Thanks to the physical network coding technology, two-way relaying network (TWRN) can complete the bidirectional information transmission in two phases, which effectively mitigates the loss of spectral efficiency [1; 2; 3]. Although cooperative communication has many advantages, the node power constraint is still a tricky problem. Since relay nodes are always placed in harsh or dangerous places, recharging or replacing batteries of relay nodes may be inconvenient and can incur a high cost [4]. Energy harvesting technology has recently emerged as a promising approach to prolong the lifetime of energy-constrained wireless network from ambient energy, such as wind, solar, etc. However, the received energy at nodes is random and intermittent, which motivates an upsurge of research interest on radio frequency (RF)-based simultaneously wireless information and power transfer [5; 6; 7]. For RF energy harvesting relay nodes, there are two major energy harvesting relay protocols, which are named as \"time switching relay (TSR) protocol\" and \"power splitting relay (PSR) protocol\". For the TSR protocol, the receiver switches two statuses between information decoding and energy harvesting. For the PSR protocol, the receiver splits the signal into two streams of a different power for providing the transmission power and decoding information separately to complete the whole transmission. In one study [8], a dynamic PS scheme was proposed to split the received signal with adjustable power levels based on the instantaneous channel condition, which could obtain the optimal PS rule at the receiver to achieve various trade-offs between the maximum ergodic capacity and the maximum average harvested energy. For another performance standard, outage probability was investigated in PSR asymmetric two-way relay system [9]. In addition, the system outage performance was studied when the channel state information was imperfect [10]. Additionally, a multi-relay cooperation system could be used to improve the sum rate with PSR protocol [11]. Thus, some paper propose that PSR theoretically achieves better performance [12]. However, PS requires appropriate power split circuits to decide how much of the signal is harvested as energy and how much of it is used for extracting information. This in turn, increases the complexity/cost of the hardware. On the other hand, hardware non-idealities can significantly reduce the efficiency of PS technology [13]. Therefore, TS can be widely investigated due to its low implementation complexity [14; 15]. In another study [16], the throughput of the TSR one-way relay system was investigated. In this paper, we employ the TS protocol in TWRN. The TSR relay nodes need to use additional time slots for energy transmission, which will reduce the performance of the system. Relay selection and power allocation are two effective ways to improve system performance in the TWRN. Traditional relay selection scheme is based on the channel status information between the relay and the user [17]. In one study [18], [PERSON] theorem was applied in the relay selection scheme, which improved the throughput significantly. Combining power allocation, the algorithm in another study [19] selected the relay with minimum symbols error probabilities as the cooperative relay and obtained the closed-form expression of the power. In one finding [20], a joint relay selection and power allocation for TWRN was proposed to maximize the smaller of the received SNRs of the two transceivers under the total transmitted power budget.