PaperlayerView exact revision on arXiv

Source-linked AI summary

Energy Cooperation in Energy Harvesting Communications

Berk Gurakan, Omur Ozel, Jing Yang, Sennur Ulukus

arXiv:1303.2636v1cs.ITcs.NI

TL;DR

Energy-harvesting communication traditionally depends on the timing of harvested energy, while energy cooperation enables users to transfer part of that energy wirelessly. The paper optimizes this cooperation across relay, two-way, and multiple access channels using Lagrangian/KKT methods and a two-dimensional directional water-filling algorithm, with a generalized version achieving the two-way channel capacity-region boundary.

  • Problem

    Energy-harvesting communication requires energy management based on energy arrival profiles, motivating control of those arrivals through energy cooperation.

  • Method

    The paper uses Lagrangian optimization and KKT conditions to derive transmit-power and energy-transfer policies for relay, two-way, and multiple access channels.

  • Results

    A two-dimensional directional water-filling algorithm optimally controls energy flow over time and among users, and its generalized version achieves the two-way channel capacity-region boundary.

  • Takeaways & Limitations

    Wireless energy transfer provides a battery-level energy-allocation degree of freedom that can improve multi-user energy-harvesting communication.

Abstract

from arXiv · show

In energy harvesting communications, users transmit messages using energy harvested from nature during the course of communication. With an optimum transmit policy, the performance of the system depends only on the energy arrival profiles. In this paper, we introduce the concept of energy cooperation, where a user wirelessly transmits a portion of its energy to another energy harvesting user. This enables shaping and optimization of the energy arrivals at the energy-receiving node, and improves the overall system performance, despite the loss incurred in energy transfer. We consider several basic multi-user network structures with energy harvesting and wireless energy transfer capabilities: relay channel, two-way channel and multiple access channel. We determine energy management policies that maximize the system throughput within a given duration using a Lagrangian formulation and the resulting KKT optimality conditions. We develop a two-dimensional directional water-filling algorithm which optimally controls the flow of harvested energy in two dimensions: in time (from past to future) and among users (from energy-transferring to energy-receiving) and show that a generalized version of this algorithm achieves the boundary of the capacity region of the two-way channel.

I. INTRODUCTION

The paper introduces energy cooperation, in which harvested energy is transferred between users to shape energy arrivals and optimize communication in several multi-user networks. It formulates offline throughput maximization and develops policies for relay, two-way, and multiple access channels.

  • Contribution: Energy cooperation transfers battery energy between harvesting users, unlike conventional cooperation at the signal level.The transfer uses a separate wireless energy transfer unit.
  • Problem formulation: The paper studies offline optimal energy management for multi-user transmitters with energy harvesting and one-way wireless energy transfer.The objective is throughput maximization under energy causality and transfer constraints.
  • Problem formulation: Energy transfer creates directional energy flow both over time and among users, requiring joint management in two dimensions.Energy can move from past to future for each user and between users at a given time.
  • Network models: The analysis covers additive Gaussian relay, two-way, and multiple access channels with one-way energy transfer.For the two-way and multiple access models, the paper determines simultaneously achievable throughput regions.
  • Optimization method: A Lagrangian formulation and KKT conditions determine optimum transmit powers and energy-transfer policies.The resulting directional water-filling algorithm controls energy flow across time and users.
  • Implications: The resulting solutions provide insights into battery-level energy cooperation in wireless energy-transfer networks.The paper frames wireless energy transfer as a new degree of energy allocation in multi-user scenarios.

II. TWO-HOP RELAY CHANNEL WITH ONE-WAY ENERGY TRANSFER

The paper models a slotted two-hop relay channel in which both source and relay harvest energy, while the source can wirelessly transfer energy to the relay. Data and energy queues evolve with one-slot relay delay and are governed by energy and data causality.

  • The system comprises an energy-harvesting source, relay, and destination connected by two AWGN hops.The source-to-relay and relay-to-destination channel coefficients are normalized to one, with unit-variance Gaussian noise.
  • Time is divided into T unit-length slots, with energy arrivals E_i at the source and ¯E_i at the relay.The source and relay harvest their respective energy amounts in each slot.
  • The source selects transmission powers P_i and transfer amounts δ_i, while the relay selects powers ¯P_i.Transferring δ_i energy in slot i delivers αδ_i to the relay in the next slot.
  • Source transmissions increase the relay’s data queue by 1/2 log(1 + P_i) bits in the next slot, and relay transmissions serve that queue.The relay operates full-duplex, updating its data and energy queues simultaneously.
  • Feasible policies satisfy temporal energy causality and relay data causality, because harvested energy is unavailable before arrival and the relay forwards source data.The relay and source slot indices are aligned despite the relay’s one-slot queue-update delay.

III. END-TO-END THROUGHPUT MAXIMIZATION FOR THE RELAY CHANNEL

The relay throughput problem is formulated as an offline optimization over source powers, relay powers, and transferred energy under energy and data causality. Necessary conditions characterize energy exhaustion, rate matching, and when transfer is required, while specific scenarios yield constructive policies.

  • The offline objective maximizes end-to-end relay throughput subject to energy causality at both nodes and data causality at the relay.The formulation is equivalent to a convex optimization problem and can be solved using a Lagrangian and KKT conditions.
  • The source must send as many bits as the relay can forward, and the relay must finish with no data remaining in its buffer.These conditions express rate matching and end-of-session data clearance in the optimal policy.
  • If a separable policy already satisfies the necessary rate condition, it is optimal and no energy transfer is needed.When the relay energy profile is sufficient to forward all source bits, the optimal policy is trivial.
  • The source exhausts all harvested energy through data transmission or wireless transfer, while any nonzero-transfer optimum exhausts the relay’s energy.The total relay energy expenditure must exceed the total source energy expenditure in the optimal policy.
  • Optimal source and relay rate allocations are monotone non-decreasing, and equality of their total rates requires matching rates in every slot.Majorization and strict Schur convexity provide the proof mechanism.
  • When the relay’s own energy is insufficient to forward the source stream, strictly positive source-to-relay energy transfer is required.This follows from the condition comparing the relay’s total harvested energy with the source’s total energy.
  • The necessary conditions do not provide detailed structural properties sufficient for an algorithmic solution, motivating analysis of specific scenarios.The paper then examines practically relevant cases, including one-node energy harvesting.

C. Specific Scenario: Source Energy Available at the Beginning

When the source has all energy initially and the relay harvests over time, the source throughput is independent of the transfer slot, but earlier transfer improves later relay transmission opportunities. Thus, transfer occurs in the first slot.

  • The scenario assumes source energy is available only initially while the relay harvests energy throughout communication.The relay’s energy profile is insufficient to forward the source’s constant-rate data stream without transfer.
  • The source throughput is independent of the slot in which energy is transferred because its total available energy is reduced by the same amount.The resulting source transmission power depends on E_1 − δ_i, not the transfer time index.
  • Transferring energy earlier can increase relay transmit powers after the transfer slot, so optimal transfer occurs as early as possible, in the first slot.The paper therefore sets the transfer timing to the first slot in this scenario.

IV. GAUSSIAN TWO-WAY CHANNEL WITH ONE-WAY ENERGY TRANSFER

The Gaussian two-way channel lets one user transfer energy unidirectionally to the other, coupling their achievable throughputs through the transfer vector. The capacity region is obtained by optimizing over feasible transfers and tracing weighted-rate boundaries.

  • The two-way model has two data and energy-harvesting users communicating over a memoryless Gaussian channel.Each user’s input contributes to both received signals, with independent unit-variance Gaussian noises.
  • User 1 transfers energy to user 2 with efficiency α, and its policy includes transmission powers and transfer amounts.User 2 has its own power sequence ¯P_i.
  • For a fixed transfer vector and feasible power policies, each user’s achievable rate is bounded by its maximum throughput under the feasibility constraints.The resulting achievable regions are rectangles determined by individual throughputs C_i.
  • As energy transfer from user 1 to user 2 increases, C1 decreases while C2 increases.This describes the trade-off induced by reallocating energy between the users.
  • The Gaussian two-way capacity region is the union of these rate regions over all feasible energy-transfer vectors and power policies.Weighted-rate maximization problems trace the boundary of the region.

V. CAPACITY REGION OF THE GAUSSIAN TWO-WAY CHANNEL

The Gaussian two-way channel capacity region is characterized through convexity, weighted rate maximization, and KKT-based energy and power allocation. A generalized directional water-filling interpretation jointly allocates energy across time and users.

  • The capacity region is convex, so each boundary point can be obtained by solving a weighted rate maximization problem.
  • The optimization is convex because the objective is concave and the feasible set is convex.
  • At the extreme objectives, maximizing only user 1 makes energy transfer strictly suboptimal, whereas maximizing only user 2 requires transferring all of user 1’s energy.
  • For fixed energy transfer profiles, the two users’ power-control problems can be separated and solved independently.
  • The KKT conditions provide necessary optimality conditions for non-zero energy transfer and determine the optimal power allocation.
  • Optimal energy must be allocated jointly across time and users, motivating a two-dimensional directional water-filling algorithm.

A. Two-Dimensional Directional Water-filling Algorithm

The two-dimensional directional water-filling algorithm represents energy movement both forward in time and downward between users. Metered permeable taps coordinate these flows until the water levels satisfy the required balance conditions.

  • Right permeable taps model future energy use, while down permeable taps model energy transferred from user 1 to user 2.
  • Meters track energy already passed through taps and permit reverse flow when allocation updates require it.
  • The algorithm first performs directional water-filling over time for each user, then opens down taps backward to transfer energy among users.
  • The allocation is optimal when the water levels are balanced while satisfying the proportionality relation associated with non-zero transfer.

B. A Specific Run of the Algorithm

A numerical run illustrates why the prescribed tap-opening order matters. Opening horizontal taps before vertical taps produces the balanced optimal profile, whereas alternative orders violate optimality conditions.

  • For E = [0, 12, 0] mJ, Ē = [6, 6, 0] mJ, and α = 1, the optimal profiles are P = [0, 4.8, 4.8] and P̄ = [4.8, 4.8, 4.8].
  • The optimal profile spreads energy as equally as possible across users and time slots subject to energy causality.
  • Opening horizontal taps alone yields P = [0, 6, 6] and P̄ = [4, 4, 4], while subsequent transfer produces a non-optimal profile because user 2 changes power with a non-empty battery.
  • Opening vertical taps before horizontal taps yields P = [0, 4.5, 4.5] and P̄ = [5, 5, 5], violating Lemma 7.
  • Opening horizontal taps first and then vertical taps establishes the optimal balance across the slots and users.

VI. MULTIPLE ACCESS CHANNEL WITH ONE-WAY ENERGY TRANSFER

The Gaussian multiple access channel capacity region is obtained by optimizing over feasible power allocations and one-way energy-transfer profiles. Its boundary reflects different transfer regimes, including no transfer, full transfer, and intermediate weighted objectives.

  • For fixed energy transfer and feasible power policies, the achievable-rate region is a pentagon.
  • The capacity region is the union of these pentagons over all feasible power allocations and energy-transfer profiles.
  • The capacity region and optimal power allocation are characterized using convexity, weighted optimization, and KKT conditions.
  • When θ1 ≥ θ2, no energy transfer from user 1 to user 2 is needed for the relevant boundary points.
  • When α = 1, multiple optimal energy-transfer profiles may exist, including no transfer, and points 2, 3, and 4 lie on the 45° line.
  • No analytical closed-form solution is available for the stated KKT equations, so standard numerical convex-optimization methods may be used.
  • When α < 1 and θ2 is sufficiently large, user 1 transfers all its energy to user 2, and the boundary has slope at most 1.

VIII. NUMERICAL RESULTS

Numerical examples illustrate optimal energy-transfer and power-allocation policies for relay channels under specified energy-arrival profiles and transfer efficiency. The examples show matched optimal powers in the studied cases, while one example notes that the transfer profile may be nonunique.

  • The numerical studies use one-second slots, N0 = 10^-19 W/Hz, 1 MHz bandwidth, and 100 dB path loss on each link.
  • For the first relay example, the specified source and relay energy profiles produce matched optimal rate profiles.The relay profile is higher initially and lower at the end, crossing once in the third slot.
  • δ = [0; 0; 1.33; 3.33] mJ and P̄ = P = [2; 3; 4; 6.33] mW are the optimal transfer vector and resulting powers in that example.The optimal energy-transfer profile is not unique, but the resulting optimal powers are unique.
  • When the source is not energy harvesting, the optimal transfer vector is δ = [2.67; 0; 0; 0] mJ.The resulting source and relay powers both equal [2.33; 2.33; 2.33; 2.33] mW in this specific example.

B. Numerical Example for the Gaussian Two-Way Channel

The Gaussian two-way and multiple access examples compare capacity regions with and without wireless energy transfer. Energy transfer significantly improves the two-way region and improves the multiple access region when the second user has higher priority, while the two-way boundary is achieved by generalized directional water-filling policies.

  • B. Numerical Example for the Gaussian Two-Way Channel: The Gaussian two-way capacity region is computed by running the two-dimensional directional water-filling algorithm over all θ1, θ2 ≥ 0.The example uses energy profiles [5; 10; 5] mJ and [10; 5; 10] mJ with α = 0.7.
  • B. Numerical Example for the Gaussian Two-Way Channel: Wireless energy transfer significantly improves the Gaussian two-way channel capacity region relative to the no-transfer region.Without transfer, the capacity region is the rectangle formed by single-user optimal rates under individual energy arrivals.
  • B. Numerical Example for the Gaussian Multiple Access Channel: The multiple access capacity-region boundaries match with and without transfer when user 1 has higher priority.The example uses profiles [5; 2; 5] mJ and [1; 3; 1] mJ with α = 0.5.
  • B. Numerical Example for the Gaussian Multiple Access Channel: Wireless energy transfer significantly improves the multiple access capacity region when user 2 has higher priority than user 1.
  • IX. CONCLUDING REMARKS: For the two-way channel, a generalized two-dimensional directional water-filling algorithm achieves the capacity-region boundary.
  • IX. CONCLUDING REMARKS: The concluding analysis states that no transfer is needed when the first user has higher priority, whereas sufficiently high priority for the second user requires transferring all energy to that user.These results treat wireless energy transfer as an additional degree of freedom for multi-user energy allocation.

APPENDIX A

Appendix A establishes convexity of the achievable rate region by combining feasible policies and using the concavity of log(1 + p). The constructed policy attains rates at least as high as the line joining the original policies’ upper corners.

  • A convex combination of two feasible power and energy-transfer policies is constructed as λ(P1, P̄1, δ1) + (1 − λ)(P2, P̄2, δ2), for 0 < λ < 1.The appendix shows feasibility for the combined policy using the linearity of the relevant constraints.
  • Because log(1 + p) is concave in p, the combined policy achieves higher throughput for both users than the line joining the two original upper-corner points.
  • Therefore, the achievable rate region C(E, Ē) is convex.The same convexity conclusion is stated again after verifying the combined rate point satisfies the region’s boundary conditions.
  • The proof forms a combined rate point t3 = λt1 + (1 − λ)t2 from points selected in the regions generated by the two original policies.
Loading 1303.2636v1…