Optimal power allocation on discrete energy harvesting model

Wang et al. EURASIP Journal on Wireless Communications and Networking Optimal power allocation on discrete energy harvesting model Xiaolei Wang 0 2 Jie Gong 0 2 Congshi Hu 1 Sheng Zhou 0 2 Zhisheng Niu 0 2 0 Tsinghua National Laboratory for Information Science and Technology, Department of Electronic Engineering, Tsinghua University , 100084 Beijing , People's Republic of China 1 China Mobile Communication Corporation (CMCC) , No. 29, Financial Street, Xicheng District, Beijing 100033 , People's Republic of China 2 Tsinghua National Laboratory for Information Science and Technology, Department of Electronic Engineering, Tsinghua University , 100084 Beijing , People's Republic of China This paper studies the power allocation problem in energy harvesting systems with finite battery. We adopt the discretized energy arrival and power allocation model. Hence, the service process can be modeled as a finite state Markov chain. Based on the discretized model, we analyze the stationary distribution of the Markov chain and formulate the utility maximization problem, which is then reformed as a linear programming problem. By analyzing the linear programming problem, we provide some intuition on the structure of the optimal power allocation policy and find the condition in which the greedy power allocation is optimal. Numerical simulations show the influence of the energy arrival process on the optimal power allocation policy, and the results are consistent with our analysis. Energy harvesting; Markov chain; Power allocation - infinite battery capacity in non-fading channel is studied in [2] for two scenarios, i.e., all packets are ready before transmission and packets arrive during transmission. Tutuncuoglu [3] finds the optimal transmission policy to maximize the short-term throughput with limited energy storage capacity, and exploits the relation between the throughput maximization and the transmission completion time minimization. For the fading channel, authors in [4] propose the directional waterfilling (WF) algorithm which is proved throughput optimal for greedy source. Similar result is obtained in [5], which further considers the optimal solution with causal information. The algorithm is then extended to multiple antennas scenario in [6], where the spatial-temporal WF is proposed. Further, considering the dynamic data arrival with hybrid energy harvesting and power grid supplies, [7] proposes the optimal reverse multi-stage WF policy. Considering the circuit power consumption, a two-phase transmission policy is shown to be optimal [8]. In [9], the authors study the throughput maximization problem for the orthogonal relay channel with energy harvesting source and relay nodes under the deterministic model and show the structure of the optimal source and relay power allocation. Although the above algorithms give some insights about the optimal solution, they assume that all the energy arrival, the channel fading, and the data arrival must be explicitly known before transmission, which is called the offline condition. Since the solutions based on the offline condition require accurate predictions for the system states, they are not always applicable in real communication systems. Based on the online condition that only the past and current system states can be known, researchers have studied the optimal and sub-optimal power allocation policies in some special scenarios. Sharma [10] identifies throughput optimal and mean delay optimal energy management policies and shows a greedy policy to be optimal in low SNR regime with infinite battery capacity. And a throughput maximization algorithm in point-topoint communications with causal information based on Markov decision process (MDP) [11] approach is proposed in [12]. Recent work [13] studies the finite-horizon scheduling problem with discrete rates and proposes a low complexity threshold-based policy. However, the properties of the optimal solution can not be directly obtained via MDP approach. In addition, the MDP approach experiences very high computational complexity due to the curse of dimensionality, hence may not be applicable when the system state space grows large. From the information theory perspective, [14] studies the channel capacity of energy harvesting links with finite battery capacity and proves that the Markovian energy management policies are sufficient to achieve the capacity. Besides the throughput maximization problems, some other issues on the energy harvesting systems, such as the quality of service (QoS), the energy efficiency, and etc. are also studied. Huang [15] studies the utility optimization problem in energy harvesting networks under limited average network congestion constraint and develops a close-tooptimal algorithm using the Lyapunov optimization theory, which jointly manages the power allocation and the data access control. As the renewable energy is usually distributed asymmetrically in space domain, there are some papers considering the energy cooperation problem to balance the harvested energy in different places, including cellular network planning [16] and power grid energy saving [17], so that the overall system energy efficiency can be improved. But still, under the dynamic property of the energy harvesting process, how to allocate the energy to achieve the optimal system performance in general case is still an open question. It is desirable to explore the closed-form analytical solution for the online condition with some statistic characteristic of the energy harvesting process. In this paper, we consider the power allocation problem in energy harvesting capacity to achieve the optimal system utility. Specifically, we study a single link with renewable energy transmitter, which only has the casual state information, including the distribution of the energy harvesting process, the past, and the current battery energy state. We model the energy arrival, storage, and usage as a discrete model and derive the optimal solution with closed-form expressions. The main contributions of this paper are presented as follows. We propose the discrete model for the energy harvesting system analysis. On one hand, the digital equipment has been widely used in modern communication systems, and it is feasible to give a discrete model for the energy harvesting process. On the other hand, the discrete model enable us to give a Markovian analysis and get some interesting closed-form analytical solution. For the independent identically distributed (i.i.d.) energy arrival process, we show the optimal solution can be obtained by solving a linear programming problem. Based on the linear programming formulation, we get some properties of the optimal power allocation policy and find the condition under which the greedy policy is optimal. Through extensive numerical simulations, we discuss the influence of the statistics of the energy arrival process on the optimal power allocation policy, which is (...truncated)