# Power Control Method for Energy Efficient Buffer-Aided Relay Systems
## 5 Proposed Power Control Method for Energy Efficient RN  
The EE of the network with respect to RN is defined as a ratio of the average achievable rate and the power consumption at RN as follows [32; 33; 34; 36; 37; 38]. \[\mathsf{EE}(P_{r})=\frac{R_{B}(P_{r})}{\eta P_{r}+P_{c}}, \tag{9}\]  
where \(\eta\) represents system inefficiency (\(\eta>1\)) that is caused by overhead power consumption at radio frequency circuits, and \(P_{c}\) is the power consumption which is independent of transmit power. The first term in the denominator of Equation (9) is thus the power consumption at the power amplifier of RN. On the other hand, the second term in the denominator of Equation (9) includes a part of power consumption for communication at, for example, a power supply, an alternating current to direct current (AC/DC) converter, a DC/DC converter, and an active cooling system, and the propulsion power consumption for hovering [39]. To design \(P_{r}\), such that EE in Equation (9) is maximized, we evaluate the EE over \(P_{r}\). In Figure 5, the EEs of two relay systems with and without a 10-bit buffer are compared when \(P_{s}=23\) dBm, \(d_{sr}=0.6\) km, and \(d_{rd}\in\{0.3,0.6,0.9\}\). For the power consumption parameters, we set them as \(\eta=5.26\) and \(P_{c}\in\{10,20,30,40\}\) dBm. Here, the values of simulation parameter are typical for the wireless communication systems (refer to works by the authors of [32; 33; 34; 36; 37; 38] and references therein), and they can be adjusted according to the application systems. Note the the proposed power control framework, which is introduced shortly, is independent of the values of parameters. From the results in Figure 5, we can verify that the EE can be improved by using a buffer at RN, regardless of \(d_{rd}\) and \(P_{c}\)  
Figure 4: Average achievable rate when \(P_{s}=23\) dBm, \(b=10\), and \(d_{sr}=0.6\) km. It is also observed from Figure 5 that the EE function is a unimodal function having a unique maximum of EE. However, to analytically find the optimal \(P_{r}\) maximizing EE is challenging. Using Equations (1), (5), and (7)-(9), we see that the EE function is intractable, i.e., there is no closed form of the first derivative of EE with respect to \(P_{r}\). One can immediately consider a one-dimensional numerical search algorithms, such as golden section search, quadratic interpolation method, and inexact line searches, to find the local optimal solution [40], or machine learning based algorithm, e.g., reinforcement learning [41; 42; 43]. However, since the average achievable rate needs to be fed back from DN to RN for each iteration with adapted \(P_{r}\) to estimate the EE, the iterative approaches require significant overhead of the networks. In the work by the authors of [24], the EE is analytically maximized by optimally allocating the frequency and transmit power resources. To this end, however, the transmitter, namely SN and RN, should know the channel state information among the nodes. Though the analytical approach can provide the optimal EE performance, this analytical strategy is challenging to be practically implemented, due to the network overhead. Thus, in this study, we employ a two-dimensional surface fitting method to model the optimal transmit power of RN as a function of two essential variables of EE, namely \(d_{rd}\) and \(P_{c}\). In Figure 6, the optimal \(P_{r}^{o}\)'s are shown for \(d_{rd}\in\{0.3,0.6,0.9\}\) and \(P_{c}\in\{10,20,40,40\}\), which are obtained from Figure 5. The optimal transmit power of RN, i.e., \(P_{r}^{o}\) increases as \(P_{c}\) or \(d_{rd}\) increases. To fit the optimal points of \(P_{r}^{o}\) to a surface, the multidimensional regression methods are employed [44]. In this study, we employ a polynomial surface fitting method, which is simple to design based on the least squares method and requires a small amount of memory at the RN. The quadratic polynomial surface is modeled as follows.