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Throughput Optimal Policies for Energy Harvesting Wireless Transmitters with Non-Ideal Circuit Power

Jie Xu, Rui Zhang

arXiv:1204.3818v2cs.IT

TL;DR

The paper asks how to maximize throughput while balancing energy and spectrum efficiency for an energy-harvesting AWGN transmitter with non-ideal circuit power. It develops offline and multichannel optimization methods, then proposes a causal-information online policy. The offline solution has a two-phase structure, and simulations show the online policy performs close to the offline policy while outperforming selected heuristics in stated settings.

  • Problem

    The paper addresses the EE–SE tradeoff and throughput-maximization problem for a finite-horizon, energy-harvesting AWGN transmitter with non-ideal circuit power.

  • Method

    It derives an offline policy with known energy arrivals, extends it to parallel AWGN channels through nested optimization, and designs a heuristic online policy using only causal energy state information.

  • Results

    The offline optimum generally has an EE-maximizing on-off phase followed by a non-decreasing SE-maximizing phase, while simulations show the online policy performs close to offline optimization and can outperform heuristics.

  • Takeaways & Limitations

    EE- and SE-oriented power allocation are unified in the offline energy-harvesting solution, providing a basis for practical online transmission with causal energy information.

Abstract

from arXiv · show

Characterizing the fundamental tradeoffs for maximizing energy efficiency (EE) versus spectrum efficiency (SE) is a key problem in wireless communication. In this paper, we address this problem for a point-to-point additive white Gaussian noise (AWGN) channel with the transmitter powered solely via energy harvesting from the environment. In addition, we assume a practical on-off transmitter model with non-ideal circuit power, i.e., when the transmitter is on, its consumed power is the sum of the transmit power and a constant circuit power. Under this setup, we study the optimal transmit power allocation to maximize the average throughput over a finite horizon, subject to the time-varying energy constraint and the non-ideal circuit power consumption. First, we consider the off-line optimization under the assumption that the energy arrival time and amount are a priori known at the transmitter. Although this problem is non-convex due to the non-ideal circuit power, we show an efficient optimal solution that in general corresponds to a two-phase transmission: the first phase with an EE-maximizing on-off power allocation, and the second phase with a SE-maximizing power allocation that is non-decreasing over time, thus revealing an interesting result that both the EE and SE optimizations are unified in an energy harvesting communication system. We then extend the optimal off-line algorithm to the case with multiple parallel AWGN channels, based on the principle of nested optimization. Finally, inspired by the off-line optimal solution, we propose a new online algorithm under the practical setup with only the past and present energy state information (ESI) known at the transmitter.

I. INTRODUCTION

The paper studies the EE–SE tradeoff for energy-harvesting transmitters with non-ideal circuit power, then develops offline, multichannel, and online throughput policies. Its offline solution uses a two-phase structure that unifies EE- and SE-oriented transmission.

  • Motivation: Energy harvesting communication is studied as a joint energy-efficiency and spectrum-efficiency design problem for point-to-point wireless links.The motivation combines bits-per-Joule efficiency with the need for high data rates.
  • EE–SE tradeoff: Non-ideal circuit power drastically changes the EE–SE tradeoff compared with the ideal case α = 0.When transmitting, total power includes transmit power plus circuit power α; the transmitter can switch to an off mode when P = 0.
  • Offline optimization: The offline throughput optimization is non-convex but has an efficient optimal solution with two transmission phases.The first phase uses EE-maximizing on-off power, while the second uses SE-maximizing power that is non-decreasing over time.
  • Multichannel extension: For multiple parallel AWGN channels, nested optimization converts vector power optimization into an equivalent scalar power optimization.The resulting scalar problem can be solved using the single-channel algorithm.
  • Online policy: A heuristic online algorithm uses only causal past and present energy state information and is evaluated against offline and other heuristic policies.The proposed online policy is reported to achieve a small gap from the offline throughput upper bound and to outperform other heuristics.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The system is a finite-horizon AWGN link powered by renewable energy arrivals, with an on-off transmitter and cumulative harvested-energy constraint. The paper formulates offline optimization with known arrivals while noting non-convexity when circuit power is nonzero.

  • System model: The model uses a constant-channel point-to-point AWGN link whose transmitter harvests renewable energy over a block of duration T.Energy arrives at discrete instants, and the interval between consecutive arrivals is an epoch.
  • System model: The achievable rate R(P(t)) is nonnegative, strictly concave, and monotonically increasing in transmit power.These properties include R(0) = 0 and cover the stated rate model assumptions.
  • Problem formulation: The finite-horizon objective maximizes throughput subject to cumulative consumed energy not exceeding cumulative harvested energy.The constraint is imposed over time rather than only at the end of the transmission block.
  • Offline and online information: Offline optimization assumes all energy-arrival times and amounts are known in advance, whereas causal online optimization can require high-complexity dynamic programming.The dynamic-programming approach suffers from the curse of dimensionality and provides limited structural insight.
  • Optimization structure: When α > 0, the objective remains concave but the energy constraint is generally non-convex because total consumed power is concave in transmit power.For α = 0, the offline problem can instead be shown to be convex, with a non-decreasing piecewise-constant allocation.

III. OFF-LINE OPTIMIZATION

The offline optimization section addresses throughput maximization with non-ideal circuit power, α > 0, under the energy-harvesting model.

  • Offline optimization: The paper solves the offline throughput optimization problem for the non-ideal circuit-power case α > 0.This section focuses on the optimal allocation when circuit consumption makes the optimization generally non-convex.

A. Reformulated Problem

The reformulation restricts each epoch to constant-power on-periods and zero-power off-periods, but the resulting optimization remains non-convex because power and on-duration are coupled.

  • During each epoch, the optimal allocation uses a positive constant power for part of the epoch and zero power for the remainder.
  • The original problem can therefore be reformulated using each epoch’s constant on-power and on-period length.
  • The reformulated problem remains non-convex because the on-power variables and on-duration variables are coupled.
  • The solution proceeds by first treating the single-epoch case and then generalizing to multiple epochs.

B. Single-Epoch Case with N = 1

For a single epoch, the optimization identifies the EE-maximizing power Pee as a central solution, obtainable efficiently because the relevant objective is quasi-concave.

  • The single-epoch formulation reduces the problem to choosing one transmit power and its associated on-period.
  • The optimal single-epoch power and on-duration are characterized by Proposition 3.1.
  • Pee is the power allocation that maximizes energy efficiency for the AWGN channel with non-ideal circuit power.
  • The solution uses on-off transmission when Pee exceeds the available single-epoch energy level, whereas the alternative case corresponds to continuous transmission.
  • Pee can be efficiently obtained by bisection because the relevant objective is quasi-concave and concave-over-linear in transmit power.

C. Multi-Epoch Case with N ≥1

The multi-epoch optimum has two phases: an EE-maximizing on-off phase followed by continuous transmission with a non-decreasing staircase power allocation, thereby combining EE and SE policies.

  • Multi-epoch optimal solution: The general multi-epoch solution is derived from the single-epoch structure and is expressed as an optimal vector of epoch powers and on-periods.
  • First phase: In the first phase, all on-periods use the constant EE-maximizing power Pee.
  • First phase: The first-phase on-periods may not be unique provided they satisfy the stated energy conditions.
  • Second phase: In the second phase, continuous transmission is optimal, with each epoch fully occupied by transmission and powers following a non-decreasing staircase allocation.
  • EE-SE unification: The two phases unify EE maximization through Pee with SE maximization through the staircase allocation under modified energy constraints.
  • Algorithm: The paper presents the resulting single-channel procedure as an optimal off-line policy in Table I.
  • Numerical illustration: The numerical example uses a two-phase allocation with Pee = 79.2mW before a continuous non-decreasing staircase allocation.

IV. MULTI-CHANNEL OPTIMIZATION

The paper extends the optimal off-line policy to multiple parallel AWGN channels with total energy harvesting and non-ideal circuit-power constraints. Nested optimization reduces vector power allocation to scalar optimization, enabling efficient solution through the single-channel algorithm.

  • The multi-channel problem considers parallel AWGN channels with a total energy harvesting power constraint and non-ideal transmitter circuit power.
  • With α > 0, the multi-channel throughput maximization is non-convex and cannot be solved by standard convex optimization techniques.
  • Nested optimization converts vector power optimization into an equivalent scalar power optimization problem.
  • The scalar problem is solved using the single-channel algorithm, while the resulting channel-power allocation can be obtained by convex optimization or water-filling for sum-throughput.
  • Table II summarizes the optimal off-line policy for the multi-channel case.

V. ONLINE ALGORITHM

This section addresses online transmission when the transmitter has only causal energy state information. It proposes an online policy based on the previously derived optimal off-line policy structure.

  • The online problem assumes that only past and present energy state information is available at the transmitter.
  • The proposed online policy is based on the structure of the optimal off-line policy derived earlier.
  • The section focuses on a practical online case rather than the non-causal-information setting used for the off-line policies.

A. Proposed Online Algorithm

The proposed online algorithm approximates the off-line policy using causal energy information and statistical knowledge of energy arrivals. Its allocation begins with an on-off phase and then follows a non-decreasing average power profile, with throughput remaining robust to assumed average harvested power.

  • The online policy uses stored energy, the remaining horizon, and known energy-arrival statistics to approximate the off-line allocation in real time.
  • The policy compares expected available transmit power with the EE-maximizing power Pee, transmitting at Pee below that level and using higher power to maximize SE above it.
  • The proposed online allocation resembles the off-line structure by starting with on-off transmission followed by a non-decreasing average power allocation.
  • 61.38Mbits and 61.60Mbits are obtained when the assumed average harvested power is 150mW and 200mW, respectively, indicating a very small loss between these settings.

VI. SIMULATION RESULTS

Simulations compare the proposed online policy with the optimal off-line upper bound and two heuristic policies under single-channel and multi-channel energy-harvesting settings. The proposed policy stays close to the off-line benchmark, while EEP and ENP are favored under different energy-arrival regimes.

  • Single-channel results: When λe is small, EEP and the proposed online policy resemble the optimal off-line policy, while ENP achieves almost zero throughput.The low-arrival regime makes transmission near the EE-maximizing power advantageous, whereas ENP deviates from that level and wastes circuit power.
  • Single-channel results: As λe increases, the off-line-to-online throughput gap enlarges, EEP degrades severely, and ENP approaches the proposed online policy.The proposed policy degenerates toward ENP when the stored-energy term is negligible relative to expected harvested energy.
  • Single-channel results: The proposed online policy performs close to the optimal off-line policy across block durations in both tested energy-arrival cases.Fig. 5 uses λe = 0.3/sec and 1/sec with ¯E = 0.5J.
  • Single-channel results: At λe = 0.3/sec, EEP performs much better than ENP, whereas at λe = 1/sec, ENP performs better than EEP.The paper attributes the reversal to the differing energy-efficiency behavior of the policies across arrival rates.
  • Multi-channel results: The multi-channel experiment evaluates average throughput versus λe in an OFDMA downlink with an energy-harvesting base station.The setup uses ¯E = 200J, T = 20secs, a 1000-meter cell, W = 5MHz, and BS circuit power α = 60Watt.

VII. CONCLUDING REMARKS

The paper concludes that non-ideal circuit power leads to a throughput-optimal two-phase policy that unifies EE and SE optimization. It extends the solution to multiple channels and derives an online policy that closely approaches the off-line upper bound.

  • Contributions: The single-channel optimal off-line solution has a two-phase structure unifying separate energy-efficiency and spectrum-efficiency optimizations.The result combines an EE-oriented initial phase with the later SE-oriented transmission structure established in the paper.
  • Contributions: Nested optimization extends the optimal off-line solution to multiple AWGN channels under a total energy-harvesting power constraint.The multidimensional vector optimization is transformed into an equivalent scalar optimization.
  • Contributions: The proposed online algorithm uses the closed-form off-line solution while assuming only causal energy-state information.Its design is motivated by the structure of the optimal off-line policy.
  • Contributions: The online algorithm has performance very close to the optimal off-line upper bound and outperforms other heuristic online algorithms in simulations.The conclusion reports this comparison without specifying a numerical gap.

APPENDIX C PROOF OF THEOREM 3.1

The appendix proves the two-phase off-line solution by separating the optimization before and after the EE phase boundary. It establishes optimality of the early and late subproblems and then combines them through decoupled energy constraints.

  • Proof structure: The proof partitions the optimization into P1 for the first iee epochs and P2 for the remaining N − iee epochs.The partition corresponds to the two phases of the proposed optimal transmission structure.
  • Early epochs: An auxiliary problem ¯P1 aggregates the early harvested energy at the first arrival and provides an upper bound for P1.Because later arrivals are removed in ¯P1, it becomes a single-epoch problem over the same horizon.
  • Early epochs: The solution specified by (12), (13), and (14) attains the auxiliary optimum, so it is optimal for P1.The appendix shows the feasible solution reaches objective value R(Pee), equal to the optimal value of ¯P1.
  • Late epochs: For P2, any off-period after tiee can be replaced by a higher-throughput allocation while preserving the consumed energy.The contradiction argument uses a power level above Pee and the decreasing behavior of R(x)/(x+α) for x > Pee.
  • Late epochs: The resulting late-phase solution is optimal because the late subproblem has the same structure as the established non-decreasing-power problem.The appendix invokes the corresponding theorem after showing that each late epoch uses its full on-period.
  • Combining phases: Since the energy constraint decouples before and after tiee, separately optimal P1 and P2 solutions are jointly optimal for the full problem.This completes the proof of Theorem 3.1.
  • Concavity result: The appendix also establishes that the effective rate function ¯R(P(t)) is strictly concave over P(t) ≥ 0.The proof relies on zero duality gap and uniqueness of the inner optimization solutions.
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