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Optimal Save-Then-Transmit Protocol for Energy Harvesting Wireless Transmitters

Shixin Luo, Rui Zhang, Teng Joon Lim

arXiv:1204.1240v3cs.IT

TL;DR

Energy-harvesting transmitters must coordinate charging and transmission because storage devices cannot do both simultaneously. The paper proposes and optimizes a two-ESD save-then-transmit protocol, compares random and constant power, and extends the protocol to multi-transmitter networks. Random power considerably degrades outage performance, although its diversity order matches constant power over Rayleigh fading, while multi-transmitter outage depends on transmitter count relative to the optimal save-ratio threshold.

  • Problem

    The paper addresses practical outage optimization for wireless transmitters powered exclusively by unreliable harvested energy under the energy half-duplex constraint.

  • Method

    It uses two ESDs in a save-then-transmit protocol, optimizes save-ratio ρ with storage efficiency η and circuit power Pc, and extends the design through TDMA-ST.

  • Results

    Random power considerably degrades outage performance, but exponentially distributed random power has the same diversity order as constant power over Rayleigh fading.

  • Takeaways & Limitations

    Multi-transmitter system outage depends critically on whether transmitter count N is below or above 1/(1−ρ), with different behavior for independent and common data.

Abstract

from arXiv · show

In this paper, the design of a wireless communication device relying exclusively on energy harvesting is considered. Due to the inability of rechargeable energy sources to charge and discharge at the same time, a constraint we term the energy half-duplex constraint, two rechargeable energy storage devices (ESDs) are assumed so that at any given time, there is always one ESD being recharged. The energy harvesting rate is assumed to be a random variable that is constant over the time interval of interest. A save-then-transmit (ST) protocol is introduced, in which a fraction of time ρ (dubbed the save-ratio) is devoted exclusively to energy harvesting, with the remaining fraction 1 - ρ used for data transmission. The ratio of the energy obtainable from an ESD to the energy harvested is termed the energy storage efficiency, η. We address the practical case of the secondary ESD being a battery with η < 1, and the main ESD being a super-capacitor with η = 1. The optimal save-ratio that minimizes outage probability is derived, from which some useful design guidelines are drawn. In addition, we compare the outage performance of random power supply to that of constant power supply over the Rayleigh fading channel. The diversity order with random power is shown to be the same as that of constant power, but the performance gap can be large. Furthermore, we extend the proposed ST protocol to wireless networks with multiple transmitters. It is shown that the system-level outage performance is critically dependent on the relationship between the number of transmitters and the optimal save-ratio for single-channel outage minimization. Numerical results are provided to validate our proposed study.

I. INTRODUCTION

The paper develops a practical save-then-transmit design for energy-harvesting transmitters under the energy half-duplex constraint, then extends it to multi-transmitter networks. It analyzes outage effects of storage inefficiency, circuit power, random power availability, and transmitter count.

  • Energy harvesting offers potentially maintenance-free operation for inaccessible sensor nodes whose batteries cannot be replaced.
  • The ST protocol uses two ESDs because one device harvests energy while the other powers transmission under the energy half-duplex constraint.The MESD supplies transmission power, while the SESD charges and later transfers its stored energy to the MESD.
  • The save-ratio ρ reserves part of each frame for MESD harvesting, leaving (1−ρ)T for transmitting Q bits with circuit power Pc included.The SESD harvests during transmission, and its recovered energy is scaled by efficiency η.
  • The paper minimizes outage over ρ for arbitrary block-fading distributions and studies how η and Pc affect the resulting design.The main ESD is modeled as an ideal super-capacitor with η = 1, while the secondary ESD is a rechargeable battery with 0 ≤η ≤1.
  • Random power considerably degrades outage performance relative to constant power, although exponentially distributed random power has the same diversity order over Rayleigh fading.The corresponding outage curve may show the predicted diversity slope only at substantially higher SNRs.
  • The TDMA-ST extension supports independent or common data, with system outage depending on whether transmitter count N exceeds 1/(1−ρ).Below this threshold, all transmitters can achieve their individual minimum outage probability; above it, the two data types behave differently.

B. Outage Probability

The paper formulates outage under random harvested energy and fading, separating circuit and channel failures. It then optimizes the ST save-ratio ρ and characterizes how battery efficiency and circuit power affect minimum outage.

  • Harvested energy is modeled as a nonnegative random rate X with finite support 0 ≤X ≤PH, constant over each frame.
  • Outage is the union of mutually exclusive circuit outage and channel outage events under the ST protocol.Circuit outage occurs when stored energy cannot cover Pc for transmission; channel outage occurs when the channel cannot support Reff.
  • The effective transmission rate is Reff = Q/[(1−ρ)T], while outage depends jointly on ρ, η, and Pc because transmit power and channel gain are random.
  • The optimal save-ratio ρ*(η, Pc) minimizes average outage over ρ for fixed battery efficiency η and circuit power Pc.The optimization can always be solved numerically, although a general closed-form expression is difficult.
  • Pout is non-increasing in η and non-decreasing in Pc, while minimum outage is strictly decreasing in η and strictly increasing in Pc.Higher storage efficiency preserves more harvested energy, whereas higher circuit power leaves less energy for transmission.

A. Ideal System: η = 1 and Pc = 0

With η = 1 and Pc = 0, the optimal save-ratio is ρ = 0, so the transmitter continuously transmits under the idealized two-ESD model. When η < 1, the optimal save-ratio instead exhibits threshold behavior and varies with efficiency.

  • Ideal System: η = 1 and Pc = 0: ρ∗ = 0 when η = 1 and Pc = 0, so the transmitter should transmit continuously.This excludes negligible time needed to transfer energy from the SESD to the MESD.
  • Ideal System: η = 1 and Pc = 0: Perfect SESD efficiency makes harvesting during transmitter idle time unnecessary because both ESDs collect energy equally efficiently.The paper notes that idling wastes transmission time without increasing harvested energy.
  • Inefficient Battery: η < 1 and Pc = 0: When η < 1 and Pc = 0, the optimal save-ratio ρ∗(η, 0) exhibits a phase transition.Above a threshold, ρ∗ = 0; below it, some time is devoted to harvesting with the higher-efficiency MESD.
  • Inefficient Battery: η < 1 and Pc = 0: ρ∗(η, 0) is non-increasing in η for 0 ≤ η ≤ 1.Higher SESD efficiency reduces the need to reserve time for MESD harvesting.
  • Inefficient Battery: η < 1 and Pc = 0: At η = 0, the model effectively becomes a single-ESD system whose optimal save-ratio ρ∗(0, 0) is always greater than 0.A single ESD cannot harvest and transmit simultaneously, so ρ = 0 would leave no harvesting time.

C. Non-Zero Circuit Power: η ≤1, Pc > 0

With non-zero circuit power, outage depends on both the ability to power the transmitter and the power remaining for transmission. The paper also shows that random power changes high-SNR behavior even though its diversity order can match constant power.

  • Non-Zero Circuit Power: η ≤ 1, Pc > 0: Non-zero circuit power can prevent transmitter operation or increase outage because some harvested power runs the hardware.The first effect occurs when available harvested power is insufficient; the second occurs when circuit power reduces transmit power.
  • Non-Zero Circuit Power: η ≤ 1, Pc > 0: Pc < ηPH means the transmitter can theoretically recover enough energy to transmit with non-zero probability for every ρ ∈ [0, 1).The condition accounts for energy recovered through the SESD.
  • Non-Zero Circuit Power: η ≤ 1, Pc > 0: Small Pc and high η favor continuous transmission, with ρ∗(η, Pc) = 0.Larger circuit power can instead be compensated by greater ESD efficiency when the efficiency threshold is below 1.
  • Non-Zero Circuit Power: η ≤ 1, Pc > 0: When η = 1, a positive save-ratio is driven by sufficiently large circuit power rather than by harvesting with a higher-efficiency ESD.The transmitter then uses a shorter interval at higher power to minimize outage.
  • Diversity Analysis: Random power produces high-SNR outage decay proportional to ¯γ−1 ln(¯γ), rather than ¯γ−1 for constant power.Its diversity order is nevertheless, in principle, the same as under constant power, while convergence is slower.

V. MULTIPLE TRANSMITTERS

The ST protocol is extended to multiple-transmitter wireless networks through TDMA-ST, and system-level outage is quantified as a function of transmitter count.

  • V. MULTIPLE TRANSMITTERS: TDMA-ST extends the single-channel ST protocol to multiple transmitters and evaluates system-level outage as the network size changes.The protocol allocates orthogonal time slots to transmitters communicating with a common fusion center.

A. TDMA-ST

TDMA-ST assigns orthogonal slots to multiple energy-harvesting transmitters, constraining each save-ratio according to the number of transmitters. Outage remains individually optimal below a threshold but degrades when that constraint is exceeded.

  • A. TDMA-ST: TDMA-ST divides each frame into N orthogonal time slots for N transmitters reporting to a common fusion center.The extension supports both independent and common source data.
  • A. TDMA-ST: TDMA requires ρ ≥ 1 − 1/N, so each transmitter’s maximum transmit-ratio 1 − ρ cannot exceed 1/N.This constraint ensures orthogonal transmissions across transmitters.
  • A. TDMA-ST: If ρ∗ ≥ 1 − 1/N, every transmitter can use the single-channel optimal save-ratio and attain its individual minimum outage probability.The corresponding system minimum outage equals the single-transmitter optimum.
  • B. Independent Data: For independent data, symmetric system-level outage is equivalent to the outage of an individual transmitter because all transmitters have the same average performance.Packets are decoded separately at the fusion center.
  • B. Independent Data: If ρ∗ < 1 − 1/N, the TDMA constraint prevents all transmitters from operating at ρ∗, forcing each away from its individual minimum-outage point.For independent data, the best feasible choice is ρ = 1 − 1/N, and system outage degrades when N exceeds the threshold.

C. Common Data

The common-data multi-transmitter case uses diversity combining, with outage behavior governed by the number of transmitters relative to the single-channel optimal transmit threshold. Numerical results also show how efficiency, circuit power, save-ratio optimization, and random power affect outage performance.

  • Common Data: Selection combining decodes identical packets from multiple transmitters, and the system outage probability follows the corresponding common-data combining model.The paper focuses on selection combining while noting that similar results hold for other diversity techniques.
  • Common Data: The optimal transmit strategy for common data is the same as for independent data and minimizes system outage probability.
  • Common Data: When N is below the reciprocal of the single-channel optimal transmit-ratio, all transmitters can attain their individual minimum outage probability; beyond this threshold, common-data behavior differs.Increasing N improves selection-combining diversity but also moves transmitters farther from their individual minimum-outage save-ratios.
  • Numerical Examples: Larger circuit power and lower battery efficiency increase the optimal save-ratio, while circuit power has the stronger effect on both save-ratio and outage performance.The minimum outage decreases with η and increases with Pc; achieving outage below 0.05 requires small Pc and η close to 1.
  • Numerical Examples: With normalized circuit power Pc/PH = 0.5 and η = 0.9, the single-transmitter optimal save-ratio is ρ* = 0.7930 and the transmitter threshold is 1/(1−ρ*) = 4.83.
  • Numerical Examples: For common data, outage decreases as N increases through N = 7, then increases, implying an optimal number of transmitters.For independent data under the same setting, outage remains at the single-transmitter optimum through N ≤ 4 and rises dramatically for N > 4.

VII. CONCLUSION

The paper concludes that random power considerably degrades outage performance and extends save-then-transmit operation to multiple-transmitter networks. In TDMA-ST, the relationship between transmitter count and the single-transmitter optimal save-ratio determines whether transmitters retain or depart from individual optima.

  • Random power considerably degrades outage performance compared with constant power under Rayleigh fading.
  • TDMA-ST allocates orthogonal time slots to multiple transmitters periodically reporting to a fusion center.
  • The multiple-transmitter analysis examines independent-data and common-data source models.
  • Below the reciprocal of the single-transmitter optimal save-ratio, each transmitter should use its minimum-outage save-ratio.
  • Above that threshold, each transmitter must deviate from its individual optimal operating point.
  • Future work includes different battery/supercapacitor and MESD/SESD configurations and more sophisticated multiple-access techniques than TDMA.

APPENDIX A PROOF OF PROPOSITION 3.1

The proof establishes monotonicity properties for outage probability and its optimized value with respect to battery efficiency and circuit power. It uses derivative signs and case analysis under the stated circuit-power condition.

  • The derivative analysis shows outage probability decreases with battery efficiency and increases with circuit power.
  • The proof of efficiency monotonicity considers whether 1−ρ lies below, between, or above the two efficiencies.
  • For fixed nonzero circuit power and ρ ∈ [0, 1), outage probability is non-increasing in battery efficiency.
  • The optimized outage value inherits the efficiency monotonicity through comparison of optimal save-ratios.

APPENDIX B PROOF OF LEMMA 3.1

The proof reduces outage minimization to minimizing an auxiliary function of the save-ratio. It then characterizes boundary and interior optima using derivative monotonicity and a necessary-and-sufficient sign condition.

  • Because outage is non-decreasing in g(ρ), minimizing g(ρ) is equivalent to minimizing outage probability.
  • The derivative h(ρ) is increasing, so its minimum over 0 ≤ ρ ≤ 1 occurs at ρ = 0.
  • For Q > 0, the gradient of g(ρ) is positive throughout the feasible range, making g(ρ) increasing and minimizing it at ρ = 0.
  • For Problem (P3), the sign of g′(ρ) matches u(ρ), whose unique interior zero exists exactly when u(0) < 0.
  • Increasing secondary-ESD efficiency reduces the optimal save-ratio in the compared cases.

APPENDIX D PROOF OF LEMMA 3.3

The proof derives a sufficient condition for a positive optimal save-ratio by analyzing the derivative of an auxiliary function with respect to the relevant energy variable.

  • A sufficient condition for ρ*(η, Pc) > 0 is v(0) < 0 for all x in the specified interval, with Pc < ηPH.
  • The condition v(0) < 0 is converted into a condition on battery efficiency η and circuit power Pc.
  • The lemma follows after establishing that v(0) increases with x.

APPENDIX E PROOF OF LEMMA 4.1

The appendix proves Lemma 4.1 by deriving the distribution of Z = PΓ and using it to characterize outage probability through Bessel-function series expansions and evaluated integrals.

  • Distribution derivation: Z = PΓ is modeled as the product of exponential random variables, with its probability density derived in terms of a modified Bessel function.P and Γ have means λp and λγ, respectively.
  • Outage characterization: The derivative of F(z) is used to obtain the density expression needed for the outage-probability characterization.The derivation then applies equation (33) and the relationship in (15).
  • Series evaluation: Series expansions for the modified Bessel function reduce the resulting expressions to evaluable terms involving integrals and the digamma function.The two integrals in (38) are evaluated to complete the calculation.
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