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Optimum Transmission Policies for Battery Limited Energy Harvesting Nodes

Kaya Tutuncuoglu, Aylin Yener

arXiv:1010.6280v2cs.IT

TL;DR

The paper addresses optimal transmission for energy-harvesting nodes whose replenishment and finite battery storage constrain energy use. It derives optimal policies for deadline-constrained throughput and completion-time minimization, showing the two solutions are equivalent and validating them numerically.

  • Problem

    The paper asks how to optimize transmission when energy harvesting and finite battery storage impose energy-causality and storage constraints.

  • Method

    The paper models discrete noncausal energy arrivals and derives necessary optimality conditions and algorithms for both power-allocation problems.

  • Results

    The throughput-maximization and minimum-completion-time solutions are identical for the same parameters, and numerical curves match as predicted.

  • Takeaways & Limitations

    The proposed algorithms yield optimal solutions for both problems under limited energy storage capacity.

Abstract

from arXiv · show

Wireless networks with energy harvesting battery powered nodes are quickly emerging as a viable option for future wireless networks with extended lifetime. Equally important to their counterpart in the design of energy harvesting radios are the design principles that this new networking paradigm calls for. In particular, unlike wireless networks considered up to date, the energy replenishment process and the storage constraints of the rechargeable batteries need to be taken into account in designing efficient transmission strategies. In this work, we consider such transmission policies for rechargeable nodes, and identify the optimum solution for two related problems. Specifically, the transmission policy that maximizes the short term throughput, i.e., the amount of data transmitted in a finite time horizon is found. In addition, we show the relation of this optimization problem to another, namely, the minimization of the transmission completion time for a given amount of data, and solve that as well. The transmission policies are identified under the constraints on energy causality, i.e., energy replenishment process, as well as the energy storage, i.e., battery capacity. The power-rate relationship for this problem is assumed to be an increasing concave function, as dictated by information theory. For battery replenishment, a model with discrete packets of energy arrivals is considered. We derive the necessary conditions that the throughput-optimal allocation satisfies, and then provide the algorithm that finds the optimal transmission policy with respect to the short-term throughput and the minimum transmission completion time. Numerical results are presented to confirm the analytical findings.

I. INTRODUCTION

The paper develops optimal transmission policies for energy-harvesting wireless nodes with finite batteries, addressing throughput by a deadline and completion time for a given data amount.

  • I. INTRODUCTION: The work aims to identify fundamental performance limits for wireless nodes with replenishing energy sources and finite batteries.This motivation follows the need to use both currently available and future harvested energy efficiently.
  • I. INTRODUCTION: The primary objective is maximizing short-term throughput, defined as data transferred under a deadline constraint.The paper seeks the optimum transmission policy for this finite-horizon problem.
  • I. INTRODUCTION: Finite battery storage is incorporated alongside energy causality, so overflow energy is lost and harvested energy cannot be spent before arrival.This extends prior work that assumed an infinite-capacity battery.
  • I. INTRODUCTION: The throughput-maximization solution is identical to the minimum-completion-time solution for the same parameters, extending the latter problem to battery storage constraints.The paper therefore provides a solution to both related optimization problems.
  • I. INTRODUCTION: Algorithms are developed to find optimum power allocations for both throughput maximization and transmission completion-time minimization.The throughput algorithm is developed first, followed by a related completion-time algorithm.

II. SYSTEM MODEL AND PROBLEM DEFINITION

The model considers continuous-time transmission with a finite-capacity battery, discrete noncausal energy arrivals, and an increasing concave power-rate relationship. The optimization maximizes transmitted data by a deadline over energy-feasible power allocations.

  • II. SYSTEM MODEL AND PROBLEM DEFINITION: The power-rate function is increasing and strictly concave, mapping transmission power p(t) to rate r(p(t)).The form is stated to be valid for the AWGN channel, while processing power and leakage can be incorporated through the function.
  • II. SYSTEM MODEL AND PROBLEM DEFINITION: The battery stores at most Emax, remains within [0, Emax], and has constant capacity throughout the transmission horizon.Battery wear or fatigue is assumed slower than the problem's time scale.
  • II. SYSTEM MODEL AND PROBLEM DEFINITION: Energy arrives as discrete packets En at times sn, with arrival sizes and times known noncausally in the offline model.The initial arrival at s0 = 0 represents the battery's initial energy.
  • II. SYSTEM MODEL AND PROBLEM DEFINITION: Battery overflow discards excess harvested energy, while energy causality prevents consumption before harvesting.All harvested energy must first be stored before consumption.
  • II. SYSTEM MODEL AND PROBLEM DEFINITION: The objective maximizes total bits departed during [0, T] over bounded nonnegative power allocations that keep battery energy feasible.The deadline is T and feasibility requires the battery state to stay within [0, Emax].

III. SHORT-TERM THROUGHPUT MAXIMIZATION

The paper characterizes necessary properties of the throughput-optimal policy and then constructs and proves optimal an algorithm that generates a feasible policy for finite-horizon transmission.

  • III. SHORT-TERM THROUGHPUT MAXIMIZATION: The throughput problem assumes an infinite backlog and a finite transmission duration.The policy is optimized for an energy-harvesting node under the model's feasibility constraints.
  • III. SHORT-TERM THROUGHPUT MAXIMIZATION: Necessary properties of the optimal policy are established before constructing the transmission policy.These properties provide the basis for the algorithmic solution.
  • III. SHORT-TERM THROUGHPUT MAXIMIZATION: The algorithm generates a policy satisfying all necessary conditions, and the paper proves that this policy is optimal.Thus, the section supplies both a construction procedure and an optimality argument.

A. Optimality conditions

The optimal allocation is characterized by uniform power between energy arrivals, no battery overflow, and power changes only when battery constraints are active. Strict concavity makes equalized power strictly throughput-improving, while battery state determines the permitted direction of changes.

  • A. Optimality conditions: Constant power between energy arrivals is optimal for any fixed feasible energy expenditure.Strict concavity means equalizing power over an interval strictly increases transmitted bits.
  • A. Optimality conditions: Any policy causing battery overflow is strictly suboptimal because the excess energy could increase earlier transmission power.The additional energy can be recovered at the overflow arrival without changing the remaining schedule.
  • A. Optimality conditions: Optimal transmission power changes only when the battery is full or completely depleted.Otherwise, replacing a local power change with constant power preserves feasibility and strictly improves throughput.
  • A. Optimality conditions: At an energy arrival, power can increase when the battery is depleted and decrease when the battery is full.The direction follows from preserving feasibility at the active battery constraint.
  • A. Optimality conditions: The optimal policy expends all harvested energy by the end of transmission.Using leftover energy immediately before termination strictly improves throughput because the power-rate function is increasing.

B. Throughput Maximizing Policy

The throughput-maximizing policy constructs feasible constant-power epochs within an energy tunnel and recursively solves shifted residual problems. Algorithm A1 is proved optimal under initial energy, discrete arrivals, finite battery capacity, and deadline constraints.

  • B. Throughput Maximizing Policy: After each epoch, the remaining power levels are found recursively by applying the same algorithm to shifted arrival times and updated parameters.The shifted problem preserves the structure of the residual optimization.
  • B. Throughput Maximizing Policy: Feasible constant power levels at each arrival form intervals P[n] = [pmax[n], p0[n]], and a transmission must remain feasible across all intervals it spans.The interval bounds represent power levels compatible with feasibility at the corresponding arrival when earlier constraints are considered.
  • B. Throughput Maximizing Policy: When the next feasible interval lies below or above the current power, the next epoch selects a boundary value that respectively decreases or increases power.The choice is determined by whether the current constant transmission would over-deplete or overflow the battery at the next arrival.
  • B. Throughput Maximizing Policy: Algorithm A1 determines the throughput-maximizing power allocation for a node with initial energy, discrete arrivals, battery capacity Emax, and deadline T.The algorithm is explicitly stated for initial energy E0, arrivals En at times sn, capacity Emax, and deadline T.
  • B. Throughput Maximizing Policy: The first algorithmic step selects the longest constant-power transmission that remains inside the energy-feasibility tunnel.The tunnel is bounded by cumulative harvested energy and the same curve shifted downward by Emax.
  • B. Throughput Maximizing Policy: Theorem 1 states that Algorithm A1 yields the optimal power allocation policy.Any different energy-feasible allocation is shown to be suboptimal relative to the algorithm’s constant-power construction.

IV. TRANSMISSION COMPLETION TIME MINIMIZATION PROBLEM

The paper relates finite-deadline throughput maximization to minimum-time transmission for a specified bit volume. The two problems produce identical policies for matching time and bit constraints, enabling a modified algorithm for completion-time minimization.

  • IV. TRANSMISSION COMPLETION TIME MINIMIZATION PROBLEM: For a given energy-arrival scenario, throughput maximization and completion-time minimization yield identical power allocation policies for matching time and bit constraints.A policy transmitting B bits by deadline T solves both corresponding formulations.
  • IV. TRANSMISSION COMPLETION TIME MINIMIZATION PROBLEM: The equivalence follows because the completion-time problem’s inner maximization is identical to the throughput maximization problem.The optimal power allocation for the minimum completion time therefore matches the throughput-optimal allocation at T*.
  • IV. TRANSMISSION COMPLETION TIME MINIMIZATION PROBLEM: The modified completion-time algorithm changes the throughput algorithm mainly through its termination condition, while earlier power allocations remain identical.The time-minimization procedure terminates when a feasible constant-power step can transmit all remaining bits.
  • IV. TRANSMISSION COMPLETION TIME MINIMIZATION PROBLEM: The completion-time formulation uses a virtual arrival point as a candidate transmission endpoint when a constant-power transmission can send all bits.This point is selected when the resulting constant-power step is feasible.
  • IV. TRANSMISSION COMPLETION TIME MINIMIZATION PROBLEM: Algorithm A2 yields the optimal power allocation scheme for the transmission completion-time minimization problem.Its optimality is established using the correspondence with the throughput-maximizing allocation at the resulting completion time.

V. SIMULATION RESULTS

Simulations confirm the analytical relationship between the two optimization problems and show that the optimal offline policy improves over simpler alternatives. Benefits are especially notable with limited battery capacity and variable energy arrivals.

  • V. SIMULATION RESULTS: In the sample deadline-T = 12 run, the algorithm terminates after selecting a feasible constant-power transmission in the shifted problem.The resulting policy is shown to be no worse than any other allocation within the energy-feasible tunnel.
  • V. SIMULATION RESULTS: The throughput-maximization and completion-time-minimization curves match exactly across deadline and packet-size parameters.Each point (B1, T1) represents one identical policy solving both formulations.
  • V. SIMULATION RESULTS: More time is required to transmit a longer packet, while more bits can be transmitted under a more lenient deadline.Both simulated objectives are strictly increasing with their respective parameters.
  • V. SIMULATION RESULTS: Energy-causality and battery constraints reduce performance relative to perpetual energy, but arrival information and variable power recover a major portion of the loss.The comparison uses a deadline of T = 10000 sec and contrasts the optimal policy with a perpetual-energy upper bound.
  • V. SIMULATION RESULTS: The optimal offline algorithm significantly outperforms the greedy on-off algorithm.The on-off policy transmits at a constant level when energy is available and shuts off when depleted.
  • V. SIMULATION RESULTS: Optimal power allocation provides especially notable improvements with small Emax, large energy packets, and high arrival-process variability.These trends were observed across simulations with varied battery capacities and arrival statistics.

VI. CONCLUSION

The paper solves optimal transmission problems for energy-harvesting nodes with limited storage, showing equivalence between deadline-constrained throughput maximization and completion-time minimization. It provides optimal algorithms and identifies extensions toward causal, stochastic, and multiterminal systems.

  • The paper solves short-term throughput maximization for a link with an energy-harvesting transmitter and limited energy storage.
  • Maximum data transfer by a given deadline is equivalent to minimizing completion time for a given amount of data.
  • The proposed algorithms yield optimal solutions for both optimization problems, supported by proofs.
  • The findings provide insight into transmission policies for nodes with some knowledge of when and how much energy they can harvest.
  • Future work includes online power allocation for causal or stochastic future energy harvests and extensions to multiterminal systems.
  • The study's model is a single-link model, motivating extensions beyond that setting.

snub1

The figures cover the optimal first-step structure, an energy-arrival simulation, throughput and completion-time behavior, and policy comparisons under a specified random-arrival setting.

  • Figure 4 depicts the optimal first step and suboptimal regions.
  • Figure 5 presents an arrival scenario and simulation results for a sample run.
  • Figure 6 overlays throughput by deadline with completion time by packet size for the Figure 5 energy-harvest scenario.
  • Figure 7 compares transmission policies for T = 10000s and Emax = 100, with uniform energy arrivals and exponentially distributed interarrival times averaging 5 seconds.
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