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In this paper, we investigate the problem of dynamic power allocation for a multiuser transmitter supplied by hybrid energy sources in details. Specifically, we focus on the hybrid energy sources which include both the traditional power grid and various renewable sources whereby there are a few issues in considerations: (1) The energy harvested jointly from various renewable sources is time-varying and possibly unpredictable and is stored in a limited capacity buffer with battery leakage. (2) At the meantime, the data arrives randomly to the transmitter and queues according to the individual receivers to wait to be transmitted. (3) In addition, the wireless channels fluctuate randomly due to fading. Taking into account the time variant and dynamic features of this system, we develop a dynamic power allocation algorithm for the transmitter with the aim of minimizing the average amount of energy consumption from the power grid over an infinite horizon, subject to all data in queues cannot exceed a given deadline of receivers. The research question is formulated as a stochastic optimization problem, then we utilize Lyapunov optimization to exploit an online algorithm with low complexity, and it does not require prior statistical knowledge of the stochastic processes. Performance analysis of the proposed algorithm is carried out in theory, which shows that the proposed algorithm performs arbitrarily close to the optimal objective value; meanwhile, the algorithm ensures that the maximum delay of all data queues cannot exceed a given value. Finally, performance comparison shows that our proposed algorithm provides not only better performance but also less time delay than other two algorithms. Keywords: Energy harvesting, Power allocation, Hybrid energy sources, Lyapunov optimization, Wireless communication 1 Introduction On the other hand, as an economical and environmental- As the vast energy consumption of the devices in friendly supply of energy for communication nodes com- wireless communication systems has recently raised pared to traditional sources of energy, energy harvesting considerable environmental concerns, eco-friendly green (EH) has recently attracted a large amount of attention of communication, aiming at maximizing energy efficiency researchers [7–9]. EH nodes can harvest energy from nat- (bit-per-Joule), have drawn considerable research ural resources, such as solar, wind, vibration, electromag- interests [1–3]. A large number of green technologies/ netic, and thermoelectric, thereby the harvested energy is methods for different wireless communication systems substantially free of cost and can be unlimitedly available. have been reported in the literatures [4–6]. Most of As such, wireless networks composed of EH nodes can be these works assume that the communication systems are energy self-sustained and reduce the use of conventional powered by a constant energy source (such as traditional energy and accompanying carbon footprint. In addition, power grid, and diesel generator) or a rechargeable bat- EH devices do not require conventional recharging; it en- tery, such that the energy can be continuously used for ables untethered mobility and therefore can be deployed system operations whenever needed. in hard-to-reach places such as remote rural areas, even within the human body [7, 10, 11]. However, the energy * Correspondence: ldd866@mailbox.gxnu.edu.cn that can be harvested from the environment is unstable School of Telecommunication Engineering, Xidian University, Xi’an 710071, and varies over time, e.g., energy fluctuation caused by China 2 time-dependent solar and wind patterns. Therefore, EH College of Electronic Engineering, Guangxi Normal University, Guilin, Guangxi 541004, China brings new problems of intermittency and randomness of Full list of author information is available at the end of the article © The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Liu et al. EURASIP Journal on Wireless Communications and Networking (2017) 2017:203 Page 2 of 12 available energy. As a result, all wireless nodes powered by model and proposed an iterative algorithm using the renewable energy are subject to the EH constraints over Proximal Jacobian ADMM. For the deterministic EH time, i.e., the total energy consumption up to any time model, [26] proposed the optimal offline transmission must be less than the energy harvested by that time. scheme for the point-to-point transmission to reduce Within the past few years, a large body of research works the grid energy consumption. Considering random en- on power management has been done [12–19] in EH ergy and data arrivals, Gong et al. [27] explored the wireless communication systems including single-user set- structure of the optimal power allocation policy based ting, broadcast channels, relay channels, interference on the statistical EH model. At the same time, paper channels, and multiple access channels. [28] studied the energy-efficient resource allocation However, the above works with EH capability [12–19] problem in an interference-free network and proposed assume that EH is the only source of energy for the the optimal offline and online algorithms based on the transmitter, the proposed schemes just apply to the com- two EH, respectively. Similarly, the authors in [29] munication system with low traffic demands. As a mat- exploited an optimal resource allocation scheme to meet ter of fact, the productions of renewable energy, strongly outage probability constraint by using dynamic program- influenced by weather conditions, are intermittent and ming (DP) approach. cannot be forecasted accurately [6]. Therefore, a sole EH All these works [24–29] based on both models pro- source may not be able to maintain stable operation or vided many important references for our research. In guarantee a certain quality of service (QoS) of the sys- practice, it is difficult to know the energy profile a priori tem. To achieve both reliable and green communication, at the transmitter [6]. Especially, it is more difficult to the concept of hybrid energy sources, i.e., using different obtain the statistical knowledge of the energy generated energy sources in a complementary manner, has also jointly by both solar and wind energy sources, even drawn interests from both industry and academia [20–23]. more renewable sources. Besides, both the time-varying For instance, Huawei Pty Ltd. has already developed base channel conditions and the dynamic mobile traffic have stations which draw energy from both solar panels and a the common features of randomness and unpredictability, wind energy harvester [20], and power grid as a supple- resulting in that their statistical properties are uncertain ment, as shown in Fig. 1. or hard to obtain in a longtime. However, all works in With the hybrid energy sources [23–29], most of the [24–26, 28] only consider full buffer networks without researches focus on two categories: (1) deterministic EH taking into account the dynamics of the data queues. model and (2) statistical EH model. The first category Although Han and Ansari [27] took into account this refers to the model that the energy arrival times and the factor, just focused on the single-link scenario and their pro- amount of harvested energy are to be known as a priori posed algorithms are not suitable for networks with mul- at the transmitter. And the second model is referred to tiple users. As such, the algorithms proposed in [24–29] that the prior knowledge of the statistical distribution of are hardly implemented in practice because they require the EH process is known. Paper [24], based on the prior knowledge of the EH process, data arrival process deterministic EH model, developed an energy efficient and the channel state process. resource allocation scheme for timesharing multiuser In this paper, we develop a dynamic power allocation systems by Lagrange dual decomposition method. And algorithm for the multiuser transmitter with hybrid [25] focused on the joint energy-bandwidth allocation energy sources, which are independent of the prior problems in multiuser channels based on the first EH knowledge of any stochastic events, with the goal of minimizing the time average energy drawn from the power grid over an infinite horizon under certain delay requirement. We consider hybrid energy sources in- cluding both the traditional power grid and various re- newable sources. The energy harvested from various renewable sources is stored in a buffer (battery) with limited capacity, and the harvested energy is time- varying and possibly unpredictable. Moreover, the battery is not perfect, such as storage loss and energy leakage, which degrade the efficiency of the renewable energy. The data arrives randomly to the transmitter and queues according to the individual receivers, and the wireless channel fluctuates randomly due to fading. Taking into account the time variant and dynamic Fig. 1 A base station (BS) with hybrid energy sources features of this system, we formulate the problem as a Liu et al. EURASIP Journal on Wireless Communications and Networking (2017) 2017:203 Page 3 of 12 stochastic network optimization problem and solved by 2 System model Lyapunov optimization approach initially developed in We consider a multiuser EH transmitter supplied by hy- [30, 31]. Researchers show in [30, 31] that the brid energy sources (composed of both power grid and optimization technique is well-suited for the queuing multiple renewable energy sources) in fading channels, model in the scheduling problem for renewable energy as shown in Fig. 2. The energy harvested from various supply and present a simple algorithm that does not re- renewable sources is first stored in a limited capacity quire prior statistical information and is provably close buffer with imperfection before it can be used by the to optimal. The authors in [32, 33] applied the technique transmitter. Without loss of generality, we assume that in smart grid to solve the problems of power manage- the energy harvested is used only for transmission and ment and energy trading respectively. Paper [34] studied the energy consumed by circuit or for signal processing the issue of electric vehicles charging with renewable en- is supplied by the power grid. The system operates in ergy based on Lyapunov optimization. slotted time t ∈ {0, 1, 2⋯} with fixed time slots, the The work in this paper is an extension of our earlier interval Δt is given at 1 s. work in [35] that only considered a single user in point- The transmitter has N receiver users; in every slot, the to-point communication system. Note that our problem new data randomly arrives at the transmitter and queues formulation is different from that in [30–35]; we apply according to individual receivers to await transmission the approach to the multiuser transmitter with hybrid through individual wireless channels. Let a(t)=[a (t), energy sources in fading channels, which have multiple a (t), ⋯, a (t)] be the vector of new data arrivals on slot 2 N data queues to individual receivers. What is more, t; here, a (t), n ∈ {1, 2, ⋯, N}, is the rate of data in- different receivers have individual tolerable delay times. coming to the n-th data queue on slot t. We assume that As such, multiple queues will compete the limited re- 0 ≤ a (t) ≤ a , ∀n, t, a is the maximum arrival rate n max max sources with each other in the case of limited energy for every data queue. Let μ(t)= [μ (t), μ (t), ⋯, μ (t)] 1 2 N harvested by the transmitter, while at the same time sat- denotes the vector of departure rate from data queues; isfying the maximal transmission power constraint and here, μ (t), n ∈ {1, 2, ⋯, N}, in practice, is the transmis- the rate-power relationship constraint. The work in this sion rate over corresponding wireless channel. Thus, the paper is not just only incremental with respect to our data queue is updated by earlier work in [35]. The research problem is now more complicated and practical; a simple algorithm for power QðÞ t þ 1 ¼ max½ Q ðÞ t −μ ðÞ t ; 0 n n allocation cannot resolve the problem anymore. þ a ðÞ t ; ∀n; t ð1Þ Our major contributions for this research are three- fold: (1) No need to know the statistical information of where Q (t) expresses the backlog of n-th data queue. the EH process, data arrivals, and channel states; we de- We assume Q (0) = 0 for all n, that is, each data queue velop a dynamic power allocation algorithm for a multi- is empty before transmission. user transmitter with the aim of minimizing the energy For every slot, the transmission rate μ (t) depends on consumption from the power grid, taking into account transmission power allocated by the transmitter and the battery imperfection. (2) The proposed dynamic current channels condition. We assume that the wireless algorithm can be easily implemented in practice, just channels fluctuate randomly due to fading and all chan- according to the current queue backlogs, channel states, nels are orthogonal. Let h(t)=[h (t), h (t), ⋯ , h (t)] 1 2 N and EH condition. In addition, we reveal the tradeoff be the vector of channels condition between the trans- between performance and delay by theoretical analysis. mitter and receivers, and h (t) represents the attenu- (3) The solution of the optimization problem considered ation value and/or noise level of the n-th channel state in this paper provides a universal power allocation policy on slot t. Suppose that the channel state information for multiuser transmitter with hybrid energy sources (CSI) at the beginning of every timeslot is known at the over an infinite horizon and facilitates the design of reli- transmitter via channel monitoring and feedback link, and able green communication. the overhead incurred by channel monitoring is neglected The remainder of this paper is organized as follows. In for simplicity [15, 29]. The channel conditions remain Section 2, the model of a multiuser communication constant for the duration of each slot but may change at system where the transmitter is powered by hybrid slot boundaries. For any n and t, the value of h (t)isde- energy sources is described. In Section 3, minimization terministically bounded by constants, h ≤ h (t) ≤ h . min n max problem of the time average energy consumption from The transmission rate μ over the wireless link (a, b) ab the power grid is formulated and the dynamic power depends on the channel state h and transmission ab allocation policy is elaborated. Simulation results are power P ; the rate-power curve is shown in Fig. 3. ab presented in Section 4. In Section 5, some concluding Further, the relationship between the channel state, remarks are given. transmission power, and rate on slot t can be described Liu et al. EURASIP Journal on Wireless Communications and Networking (2017) 2017:203 Page 4 of 12 Fig. 2 Multiuser EH transmitter supplied by hybrid energy sources in fading channels by the function g(p (t), h (t)) given by Shannon’s cap- n n transmission power on slot t is p ðÞ t , which is sup- n¼1 acity formula [28, 36]. plied by both the EH sources and the power grid. μ ðtÞ¼ gðp ðtÞ; h ðtÞÞ ¼ log ð1 þ p ðtÞ⋅h ðtÞÞ; ∀n; t n n p ðÞ t ¼ P ðÞ t þ P ðÞ t ; ∀t ð3Þ n n 2 n p h n¼1 ð2Þ where P (t)and P (t) are supplied by the power grid and p h where n ∈ {1, 2, ⋯ , N}, the rate-power function g(⋅) the energy queue buffered the energy harvested from EH is assumed to be monotonically non-decreasing, deter- sources, respectively. Furthermore, the total power mining the number of bits in data queue that can be consumed from the two types of energy sources is given transferred over the wireless link. However, the data in by ρ p ðtÞ.Here, ρ ≥ 1 is a constant, which accounts n¼1 queues may be packets; we allow arbitrary fragmentation for the inefficiency of the non-ideal transmitter [26]. of packets during transmission. We assume that b(t) Joules of energy is collected From the above discussion, in order to finite backlog jointly from various renewable sources at the end of the of all data queues (i.e., the data queues are all stable) t-th interval, the harvested energy is buffered in the bat- [30], the transmitter must make a decision of transmis- tery before it can be used in the next time slot, b(t) ≤ sion power on each slot according both the backlog of B , where B represents the maximum capacity of max max each data queue and current channels condition. the buffer, i.e., the rechargeable battery can store at most Assume that the transmission power vector on slot t is B Joules of energy. Due to the battery defects, such max denoted as p(t)= [p (t), p (t), ⋯ , p (t)]. The total as energy leakage, supposing that a factor of 1 − β of the 1 2 N stored harvested energy is leaked per time interval due to the inefficiency of the battery [37], where 0 < β <1 represents the efficiency of the battery per time slot. Let B(t) be the amount of the available energy in the re- chargeable battery (energy queue), thus we have the fol- lowing update equation of energy queue: Bðt þ 1Þ¼ min max ½βðBðtÞ−ρP ðtÞΔtÞ; 0þ bðtÞ; B h max ð4Þ We assume B(0) = 0, which denotes the available energy before transmission. The optimization goal is to minimize the time average energy consumption from the power grid over a long time, subject to the constraints of the stability of all data queues. Due to the finite storage capacity and the pos- Fig. 3 Set of rate-power curassumed to be monotonicallyions sible leakage of the battery, it is beneficial to draw the h <h <h 1 2 3 energy as quickly as possible from the battery so that Liu et al. EURASIP Journal on Wireless Communications and Networking (2017) 2017:203 Page 5 of 12 more harvested energy can be stored in the future, and maximum transmission power of the transmitter P , i.e., max max thus the amount of possibly wasted harvested energy is p ≤P . max n¼1 n minimized. Based the above discussion and the objective, we expect to find a dynamic power allocation scheme in 3.1 The delay-aware virtual queue the next section, which provides insight into how to effi- To ensure that the optimization objective satisfies the ciently utilize the energy supplied by the EH source, sav- delay constraints (7), we utilize virtual queues account- ing the traditional energy. ing for the constraints, which initially introduced in [30]. Let Z (t), n ∈ {1, 2, ⋯ , N} be the virtual queues. Fix 3 Problem formulation and solution any parameter σ > 0, define Z (0) = 0 for all n, and the n n Our objective is to minimize the energy consumption virtual queues update according to the following: from the power grid by making a decision of transmis- sion power p(t) in every slot; the problem can be formu- ZðÞ t þ 1 ¼ max Z ðÞ t þ σ ⋅1 −μ ðÞ t ; 0 ð10Þ n n n fg Q ðÞ t >0 n n lated as a stochastic optimization problem as follows: where 1 is an indicator function that is 1 if Q ðÞ t >0 fg ( " #) t−1 X X Q (t) > 0 and 0 else. We can see from Eq. (9), the virtual min : lim E max ρ p ðτÞ−BðτÞ; 0 queue Z (t) has an arrival process that add σ whenever n n t→∞ t τ¼0 n¼1 the backlog of the actual queue Q (t) is non-empty. This ð5Þ ensures that Z (t) grows when there is unserved data in the actual data queue Q (t). The constant σ can adjust n n the growth rate of the virtual queue Z (t). If we can con- trol the transmitter to guarantee that the queues Q (t) s:t : Q < ∞; ∀n ð6Þ and Z (t) have finite upper bounds, then we can ensure max that all bits in the n-th data queue are served within Tolerable delay for the n‐th user ≤D ; ∀n ð7Þ max maximum delay of D slots, which is given in the fol- lowing lemma. μ ðtÞ¼ log ð1 þ p ðtÞ⋅h ðtÞÞ; ∀n; t ð8Þ n 2 n Lemma 1 Suppose the system is controlled so that the queue Q (t)and Z (t) have finite upper bounds, e.g., n n max max max 0≤p ðÞ t ≤p ð9Þ n n Z ðÞ t ≤Z and Q ðÞ t ≤Q for all t,thenall bits in n n n data queue n are served with a maximum delay of where the optimization goal (5) shows that the time max D slots, which is defined as: average expected energy consumed by the transmitter max max from the power grid is minimized over an infinite Q þ Z max n n D ¼ ð11Þ N n horizon, therein max ½ρ p ðτÞ−BðτÞ; 0 represents n n¼1 the energy consumption from the power grid on slot The proof of Lemma 1 follows the approach of Lyapu- τ, E{⋅} denotes statistical expectation. Constraint Eq. (6) nov optimization in [30]. guarantees that all the queues are stable which de- t−1 3.2 Lyapunov optimization fined as: Q ≜ lim sup EQfg ðÞ τ < ∞.Con- n t→∞ n t τ¼0 Define Θ(t) ≜ (Q(t), Z(t)) as the concatenated vector of the straint Eq. (7) shows that the data queues have real and virtual queues, here Q(t)=[Q (t), Q (t), ⋯ , 1 2 individual delay requirements, namely, the data in each Q (t)], Z(t)=[Z (t), Z (t), ⋯ , Z (t)]. As a scalar meas- N 1 2 N ure of the congestion in all queues, we define the following queue waits for be transmitted to corresponding user, and Lyapunov function: the waiting time cannot exceed a deadline of cor- N 2 2 LðÞ ΘðÞ t ≜ Q ðÞ t þ Z ðÞ t . Define the condi- max n¼1 n responding user. In constraint Eq. (9), p is the max- tional 1-slot Lyapunov drift as follows: imum transmission power allocated by the transmitter to ΔLðÞ ΘðÞ t ≜ELfg ðÞ ΘðÞ t þ 1 −LðÞ ΘðÞ t jΘðÞ t ð12Þ the n-th user. To ensure that the problem Eqs. (5)–(9)are always feasible, we assume that the set of data arrivals Making a decision p(t) to minimize ΔL(Θ(t)) alone vector is in the feasible region of the problem. The authors would push all queues towards lower backlog (i.e., delay) [30] but which incur more energy consumption from in [38, 39] defined the set of data arrivals vector that can the power grid. Considering both the energy consump- be transmitted reliably under some power-allocated algo- tion from the power grid (5) and queues backlog growth rithm. In addition, the sum of the maximum allocation (1) and (10), our objective is then to minimize the fol- power for each user is assumed no more than the lowing function in each timeslot t: Liu et al. EURASIP Journal on Wireless Communications and Networking (2017) 2017:203 Page 6 of 12 " # ( " # ) N X min: ΔLðΘðtÞÞ þ V⋅E max ρ p ðtÞ−BðtÞ; 0 ΘðtÞ min: V max ρ p ðtÞ−BðtÞ; 0 n¼1 n¼1 N N X X ð13Þ þ Q ðtÞ½a ðtÞ−μ ðtÞ þ Z ðtÞ½σ −μ ðtÞ n n n n n n n¼1 n¼1 ð17Þ Note that the left part of Eq. (13) is the growth of the queues and the right part of Eq. (13) is the expected en- Simplifying Eq. (17) and removing the parts which ergy consumption from the power grid (be called pen- have nothing with our decision variable vector p(t), then alty), and Eq. (13) is called drift-plus-penalty expression we obtain: [30]. The parameter V > 0 is used to tune performance- N N X X delay tradeoff between performance and queue backlog min: ρV p ðtÞ− ½Z ðtÞþ Q ðtÞ μðp ðtÞ; h ðtÞÞ n n n n n (i.e., delay). So our approach minimizes a weighted sum n¼1 n¼1 of drift and penalty, which can be proven bounded. ð18Þ Lemma 2 The drift-plus-penalty expression for all slots t satisfied: 4.1 Real-time optimization algorithm ( " # ) Our online optimization algorithm is described as follows: Step 1. Every slot t,observe Z(t), Q(t), h(t), a(t) ΔLðΘðtÞÞ þ V⋅E max ρ p ðtÞ−BðtÞ; 0 ΘðtÞ and b(t), then choose p(t)= [p (t), p (t), ⋯ , p (t)] n¼1 1 2 N ( " # ) to minimize Eq. (18), subjecting to the constraints Eqs. (7)–(9); ≤C þ V⋅E max ρ p ðtÞ−BðtÞ; 0 ΘðtÞ n¼1 Step 2. Update the real queues, virtual queues, and X energy queue according to Eq. (1), Eq. (10), and Eq. (4), þ Q ðtÞEfa ðtÞ−μ ðtÞjΘðtÞg n n n respectively. n¼1 The optimization solution of Eq. (18) can be solved by examining each vertex formed by the solution space. We þ Z ðtÞEfσ −μ ðtÞjΘðtÞg n n denote the optimal power allocated to n-th data queue n¼1 in timeslot t as p ðÞ t , ð14Þ n p ðtÞ¼ arg min ½ρVp ðtÞ where the constant C is defined as: − ½Z ðtÞþ Q ðtÞ log ð1 þ p ðtÞh ðtÞÞ ∀n n n n 2 n 2 2 ½a þ μ max max n¼1 C ¼ To solve p ðÞ t , substituting the rate-power function X ð15Þ Eq. (2) into Eq. (18), then differentiating with respect to 2 2 max½σ ; μ n max the transmit power p (t) (decision variable), we will obtain: n¼1 Q ðÞ t þ Z ðÞ t 1 p ðÞ t ¼ − ∀n; t ð19Þ 2 ln2⋅pV h ðÞ t The proof of Lemma 2 follows the approach of drift- max However, subjected to the constraints 0≤p ðÞ t ≤p plus-penalty in [30] using the following inequality: for any n, t, the actual transmit power allocation p (t)to n-th user in slot t can be obtained according to: 2 2 2 2 ½ maxðÞ b−c; 0þ a ≤b þ c þ a þ 2baðÞ −c ð16Þ max p ðÞ t ¼ min p ; max 0; p ðÞ t ð20Þ n n which holds for a ≥ 0, b ≥ 0, and c ≥ 0, then we can yield P If p p ðÞ t ≤BtðÞ holds, the transmitter does not n¼1 n Eq. (14). need to draw additional energy from the power grid in slot t; otherwise, the transmitter needs to draw the 4 Real-time power allocation algorithm amount of p p ðÞ t −BtðÞ additional energy in slot t. By referring to Lyapunov optimization approach, we n n¼1 transform the problem Eqs. (5)–(9) to minimize the 4.2 Performance analysis drift-plus-penalty expression in each slot, thus it is max max Theorem 1 Suppose g p ; h ≥a , ∀n ∈ {1, 2, ⋯ , equivalent to minimizing the right-hand-side of the min n n drift-plus-penalty bound Eq. (14) in each slot t, N}. If Q (0) = Z (0) = 0, then for any fixed parameter n n Liu et al. EURASIP Journal on Wireless Communications and Networking (2017) 2017:203 Page 7 of 12 max σ , 0≤σ ≤a and V >0 for all t, the proposed algo- 4.3 Comparison between the proposed algorithm and DP rithm has the following properties for each queue n: The optimization problems considered in related works [26, 28] are based on dynamic programming (DP); the 1. In all slots, for all queues, Q (t) and Z (t) are upper obtained algorithms can achieve the optimal objective n n max max bounded by Q and Z respectively, where: value. While the power allocation algorithm based on n n Lyapunov optimization in this paper performs asymptot- ically close to the optimal objective value by tuning the max max value of V, as shown in Eq. (23). However, DP requires Q ¼ 2ln2⋅ρV þ p þ a ð21Þ max n n h min more stringent system modeling assumptions, i.e., re- quiring the prior knowledge of the probabilistic charac- teristics of the EH process, data arrivals, and channel 1 states. In contrast, the Lyapunov optimization technique max max Z ¼ 2ln2⋅ρV þ p þ σ ð22Þ n n does not need the prior knowledge of these stochastic min events. If the prior knowledge of energy harvesting, data arrivals and channel state values (a(t), b(t), h(t)) were known in advance, one could in principle make p(t) de- 2. The maximum delay of the data queue n can be cisions that minimize average energy consumption from calculated according to (11) given by: the power grid. One of the contributions of this paper is to provide an efficient algorithm without knowing the prior knowledge of any stochastic events. So our pro- posed algorithm is suitable for broader applications. 1 max 4ln2 ⋅ Vρ þ p þ a þ σ max n n Besides, our proposed algorithm just needs the obser- min max D ¼ ð23Þ vations of the current system states firstly, i.e., Z(t);Q(t); h(t); a(t), and b(t), then make p(t) decisions according to Eq. (18). So the proposed algorithm is simple to imple- ment, the complexity is linear with the number of 3. Given that σ ≤ E{a }, the time average expected n n queues. In contrast, the algorithms in [26–28] based on additional energy drawn from the power grid using DP showed that the complexity increase exponentially the proposed algorithm is upper bounded with C/V with the number of time intervals. DP approach involves of the optimal value Topt, i.e., computation of value function that can be difficult when the state space of the system is large and suffers from a curse of dimensionality when being applied to large- ( " #) dimensional systems (such as systems with many queues) t−1 X X 1 C [31]. Therefore, as aforementioned, our proposed algo- lim E max ρ p ðτÞ−BðτÞ; 0 ≤T þ opt t→∞ t V rithm has better scalability and easy to use. τ¼0 n¼1 ð24Þ 5 Simulation results where T is the optimal value of minimizing average To evaluate the performance of the proposed dynamic opt energy drawn from the constant source, and C is given power allocation algorithm, we assume that there are in Eq. (15). three users and the energy is harvested from both solar The proof further conducts the performance analysis and wind energy, the energy output characteristic of the Lyapunov optimization as described in [30], and follows an i.i.d. Poisson process. We evaluate the per- the proof of Theorem 1 is given in the Appendix of formance of the proposed algorithm on daily data set, this paper. i.e., in 3600 timeslots (the time interval is fixed as 1 s). The performance analysis shows that the congestions Note that we adopt the distribution just for exposition of the queues grow linearly with V, while our goal de- purpose; the analysis in the previous section does not creases with increased V value, which is a tuning param- depend on the distributions. The related simulation set- eter to balance performance and delay. The performance tings are summarized in Table 1. can be pushed arbitrarily close to the optimum by tun- To better evaluate the performance of our proposed al- ing V, but the queues backlog may be longer. Thus, we gorithm, three strategies are considered in the simulations. max should choose appropriate V value. To reduce D The first strategy uses Lyapunov optimization algorithm. value, we should use σ as large as possible while still The latter two strategies (second and third strategies) use meet σ ≤ E{a }. We can choose σ = E{a } if this expect- simple greedy algorithm. The second strategy deploys n n n n ation is given. “absorb-upon-arrival” policy, which describes such Liu et al. EURASIP Journal on Wireless Communications and Networking (2017) 2017:203 Page 8 of 12 Table 1 Simulation setting grid even if the EH source cannot meet the need of the transmitter until any data delay exceed the maximum, Parameters Value where the deadline is set to 25 slots. Number of users 3 The performance comparison of three strategies is Timeslot length 1 s shown in Fig. 4. Among the figures, Fig. 4a shows the EH sources Solar and wind energy energy drawn by the transmitter from power grid in Harvest process i.i.d. Poisson process every timeslot, and Fig. 4b shows the accumulated Data arrival process Uniform distribution amount of energy drawn from the power grid over a day Inefficiency factor ρ 1.2 (i.e., 3600 timeslots). From Fig. 4, we can see that Lyapunov optimization algorithm (our proposed algo- Energy leaked factor β 0.9 rithm) achieves the best performance (the minimum Bandwidth B 1 M/Hz amount of energy consumption from the power grid) Channel Fading Gaussian among the three strategies, that is, more traditional en- Average SNR 10 dB ergy is saved using our proposed algorithm. Under this Max transmit power P 6W max condition of parameter setting, about 2337 J of trad- max p For every user 2 W itional energy can be saved using the proposed algorithm 3 4 3 in comparison with using the strategy of absorb-upon- σ , n=1,2,3 Eafg; Eafg; Eafg n 1 2 3 4 5 4 arrival only over a day (3600 timeslots), and about max a n = 1,2,3 4.4, 2.9, 3.9 bit/slot 1225 J of traditional energy can be saved in comparison with using the strategy of absorb-at-deadline over a day. Here, V is set to 80 by trail and error. scenario: when the energy in rechargeable battery cannot Figure 5 shows the accumulated amount of the energy meet the need of the transmitter, the transmitter immedi- consumption from power grid in different cases of aver- ately draws energy from the power grid for sending data, age amount of energy harvested b (b ≤ b ≤ b ). av av1 av2 av3 which results in the least data delay time, but possibly From Fig. 5, we can see that no matter under which more energy consumption from the power grid. The third case, Lyapunov optimization algorithm can achieve the strategy deploys the policy “absorb-at-deadline,” which best performance among the three strategies, and means that the transmitter uses only renewable energy be- absorb-upon-arrival policy provides the worst per- fore deadline, and does not draw energy from the power formance. The reason is that Lyapunov optimization Fig. 4 Performance comparison of three strategies. a Energy consumption from the power grid in every timeslots. b Accumulated amount of energy consumption from power grid over a day Liu et al. EURASIP Journal on Wireless Communications and Networking (2017) 2017:203 Page 9 of 12 Lyapunov optimization algorithm results in much smaller delay than the deadline. Most of the arrival data wait about 5 slots used our proposed algorithm, while the strategy absorb-at-deadline wait mostly 24 slots. Based on Lyapunov optimization algorithm, the max- max imum delay D (time-slots) computed by formula (11) actu and the actual average delay D of 3 data queues by dead simulate, and the average delay D based on absorb- at-deadline strategy of 3 data queues are shown in Table 2. In order to study the impact of parameter V on the additional energy cost from the power grid and average delay of the data in the data queues, we have plotted Fig. 7 showing the relationship between the energy cost and the value of V and the relationship between the Fig. 5 Comparison of accumulated amount of energy consumption from power grid using three strategies (average amount of energy average delay time and the value V. We can see that as harvested b : b ≤ b ≤ b ) av av1 av2 av3 we expected, the average delay increases non-linearly with the value of V while the energy cost decreases with V. The energy cost and average delay reach saturation when V is larger than a certain value (V = 80, seen from algorithm enables the transmitter to send more data Fig. 7), which illustrates that when V is large enough, when channel state is better, while absorb-upon-arrival the average delay will reach its maximum and the energy policy can provide the least data delay time, which cost is close the optimal value (T ). opt resulting in the worst performance. However, no matter which strategy is adopted among three strategies, the smaller the amount of energy harvested by the trans- 6 Conclusions mitter from the renewable energy sources is, the In this paper, we develop a dynamic power allocation more energy from the power grid is supplied for the algorithm for a multiuser transmitter powered by hybrid transmitter. Here, V is set to 80. energy sources (including the traditional power grid and To have a better insight of the delay time reduction, EH sources). The proposed algorithm provides insight two strategies (Lyapunov optimization algorithm and into how to efficiently utilize the energy supplied by the absorb-at-deadline policy) has been compared. Simula- EH sources, namely how to minimize the time average tion results on the fraction of waiting data of three energy consumption from the power grid at the same queues are shown in Fig. 6. Seen from Fig. 6, using time ensure the QoS of communication. Firstly, we Fig. 6 Histogram of delay of data waiting in three data queues Liu et al. EURASIP Journal on Wireless Communications and Networking (2017) 2017:203 Page 10 of 12 Table 2 Simulation results of two algorithms max max Q ¼ 2ln2⋅ρV þ p þ a ; ∀n; t max n n Algorithms Queue 1 Queue 2 Queue 3 min max Lyapunov D 31.7638 20.7181 23.8009 actu Lyapunov D 4.6008 4.5276 4.8873 dead absorb-at-deadline D 14.5031 12.7796 15.6809 It holds clearly for t = 0 (because Q (0) = 0). Next we assume max Q ðtÞ≤2ln2⋅ρV þ p þ a ; ∀n; t max n n model this kind transmitter in fading channels. The data min arrivals and energy harvested from surrounding both what we can do is to prove it also true for slot t +1. If randomly arrived at the transmitter, in addition the wire- less channels fluctuate randomly, without knowing their max statistical probabilities. Secondly the issue is formulated Q ðtÞ≤2ln2⋅ρV þ p min as a stochastic optimization problem, and a real-time power allocation algorithm is exploited with low com- the maximum queue backlog growth is a , then max plexity. The theoretical performance analysis shows that the proposed algorithm outperforms the state-of-the-art max Q ðtÞ≤2ln2⋅ρV þ p þ a max n n algorithms in terms of achieving a near optimal value by min tuning the parameter V, while ensure the time delay of max 1 max data queues would not exceed the maximum delay D . n If Q ðtÞ≥2ln2⋅ρV þ p ,since Z (t) ≥ 0, we have: n n min A further comparison of the proposed algorithm with other two greedy algorithms demonstrates the proposed Q ðtÞþ Z ðtÞ≥2ln2⋅ρV n n algorithm can consume much less energy from the max h þ p min power grid. Moreover, the algorithm of the optimization problem in this paper does not require the knowledge of ≥2ln2⋅ρV max h ðtÞ þ p statistical probabilities of the random processes; thereby, it n provides a universal power allocation policy for multiuser In this case, according to the algorithm proposed transmitter with hybrid energy sources and facilitates the max above we will have p ðÞ t > p by formula (19). Then n n design of reliable green communication paradigm. max we will choose p ðÞ t ¼ p on slot t according to (20), n n thus the data queue is served by at least a , because max max 7 Appendix gp ; h ≥ max½ a ; σ min max n 7.1 Proof of Theorem 1 Hence the data queue backlog cannot grow on the 1. We use induction method to show that: next slot, i.e., Fig. 7 Energy from power grid and average delay for different V values using our proposed algorithm Liu et al. EURASIP Journal on Wireless Communications and Networking (2017) 2017:203 Page 11 of 12 LðΘðTÞÞ−LðΘð0ÞÞ max Q ðt þ 1Þ≤Q ðtÞ≤2ln2⋅ρV þ p þ a max n n ( " #) min þ VE max ρ p ðtÞ−BðtÞ; 0 n;pro n¼1 Therefore, we have ≤CT þ VT⋅T opt ð26Þ max Q ðtÞ≤2ln2⋅ρV þ p þ a n max min Using the fact that L(Θ(T)) ≥0and L(Θ(0)) = 0, dividing both sides of (26) by VT and letting T→ ∞ results in: for all slot t. ( " #) T−1 X X 1 max 1 C The proof that Z ðtÞ≤2ln2⋅ρV þ P þ σ is n n n lim E max ρ p ðtÞ−BðtÞ; 0 ≤T þ h opt min n;pro T→∞ T V n¼1 t¼0 similar above. Acknowledgements 2. It is very easy to prove according to Lemma 1 and The authors also gratefully acknowledge the helpful comments and the conclusion of Theorem 1. suggestions of the reviewers, which have improved the presentation. 3. Since the proposed algorithm will always try to Funding minimize the right-hand-side part of the inequality The research was supported by the National Natural Science Foundation of (14) among all feasible solutions, even the optimal China (Grant No. 61571143), Guangxi Youth Talent Program (F-KA16016), solution, assume the solution given by the proposed Guangxi Natural Science Foundation (2014GXNSFAA118387), and Foundation of Key Scientific Research Project of Guangxi Normal University (2016ZD008). algorithm and optimal solution are p , (t)and p n pro n, (t) respectively, and the optimal result for opt Authors’ contributions minimizing average energy drawn from the constant DL, JL, and JW conceived and designed the study. DL and JW performed the experiments. DL provided the performance analysis of the proposed algorithm source is T , then by plugging the solution into the opt in theory. DL and JL wrote the paper. FJ reviewed and edited the manuscript. inequality (14), we can have the following: All authors read and approved the manuscript. Authors’ information Didi Liu received her BS degree in electronic and information engineering from Guilin University of Technology, China, in 2003, and her MS degree in ( " # ) N communication and information system from Guilin University of Electronic Technology, China, in 2006. From 2006 to 2013, she was a researcher in ΔLðΘðtÞÞ þ VE max ρ p ðtÞ−BðtÞ; 0 ΘðtÞ n;pro Guangxi Normal University. She is currently pursuing the Ph.D. degree in the n¼1 School of Telecommunication Engineering, Xidian University, China. Her research ( " # ) interests include stochastic network optimization and signal processing. ≤C þ VE max ρ p ðtÞ−BðtÞ; 0 ΘðtÞ Jiming Lin received his BS degree in electronic engineering from Harbin n;opt n¼1 Engineering University, China, in 1992, and his MS degree in communication and X information system from the University of Electronic Science and Technology of þ Q ðtÞEfa ðtÞ−μ ðp ðtÞ; h ðtÞÞ jΘðtÞg China in 1995. In 2001, he received his PhD degree in acoustics from Nanjing n n n n;opt University, China, in 2002. Subsequently, he held a half-year postdoctoral fellowship n¼1 N at State Key Laboratory for Novel Software Technology at Nanjing University. Since 2004, he has been a professor with the school of information and communications, þ Z ðtÞEfσ −μ ðp ðtÞ; h ðtÞÞ jΘðtÞg n n n n n;opt Guilin University of Electronic Technology. His research interests are in n¼1 synchronization and localization in WSNs, ultra-wideband communication. Junyi Wang received his BS and MS degrees in Mathematics from Heibei ≤C þ VT opt University and Xiangtan University, China, in 1999 and 2003, respectively. In 2008, ð25Þ he received his PhD degree in signal and information processing from Beijing University of post, Beijing. He has been a professor with the school of information and communications, Guilin University of Electronic Technology. His The result of (25) is based on the facts that research interests are in stochastic network optimization and signal processing. Dr. Frank Jiang completed his PhD degree in communication engineering and software engineering at University of Technology, Sydney, and he was the T−1no X winner of a prestigious UNSW Vice-Chancellor’s Postdoctoral Research Fellowship lim Ea ðÞ t −μ p ðÞ t ; h ðÞ t ΘðÞ t ≤0 from University of New South Wales (successful rate 4.8). His current research n n n n;opt T→∞ T t¼0 interests include bio-inspired algorithms and meta-heuristics, big data-driven cyber security, cloud-based communication, AI, network protocols, and mesh networks. Up to date, he has published over 80 international journal and T−1no 1 conference papers in the fields; his work is mainly published in the journals— lim E σ −μ p ðÞ t ; h ðÞ t ΘðÞ t ≤0 n n n n;opt Systems and Control Letter, IEEE Transactions on Network and Systems T→∞ T t¼0 Management, Engineering Applications Of Artificial Intelligence, Physics Letters A, International Journal of Computational Intelligence and Applications, and Journal of Network and Systems Management. He has regular services as journal Summing inequality (25) over slots t ∈ {0, ⋯ , T}, we reviewers such as for IEEE Transactions on Parallel and Distributed Computing, can have: IEEE/ACM Transactions on Networking, and IEEE Transactions on Neural Networks Liu et al. EURASIP Journal on Wireless Communications and Networking (2017) 2017:203 Page 12 of 12 and Learning Systems. His contributions in bio-inspired computing earned him 19. A Imran, A Imtiaz, H Jahangir, Optimal stochastic power allocation and relay international reputations in NOMs 2007 in Vancouver with IEEE-IFIP awards; his selection for energy harvesting systems. IEEE Wireless Communications work was nominated as the best paper in CEC 2012. Letters 6(4), 546–549 (2017) 20. ”Green Energy Solution by Huawei.” Available: http://www.huawei.com/en/ solutions/go-greener/hw-076723-green-hybrid -powercube.htm.UdxZdxbRkRk. Competing interests Acccessed 6 Nov 2017. The authors declare that they have no competing interests. 21. N Ansari, T Han. Optimizing green energy utilization for mobile networks with hybrid energy supplies. Green mobile networks: a networking perspective. 2016. Publisher’sNote 22. T Han, N Ansari. 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EURASIP Journal on Wireless Communications and Networking – Springer Journals
Published: Dec 1, 2017
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