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Optimal Energy Allocation for Wireless Communications with Energy Harvesting Constraints

Chin Keong Ho, Rui Zhang

arXiv:1103.5290v2cs.IT

TL;DR

The paper studies finite-horizon throughput maximization for wireless transmitters powered by time-varying harvested energy under causal or full side information. It uses dynamic programming and structural analysis to derive optimal allocations, including staircase water-filling with unlimited storage and full information. A causal-information heuristic performs relatively close to the full-information optimum in numerical studies.

  • Problem

    The paper addresses throughput-maximizing energy allocation when fading channels and harvested energy vary over a finite horizon and energy use is constrained by harvesting causality.

  • Method

    It uses dynamic programming and structural analysis to obtain optimal policies under causal and full side information, including offline lookup-table computation for causal information.

  • Results

    With full side information and unlimited storage, the optimal allocation has a water-filling interpretation whose multiple water levels form a staircase over time.

  • Takeaways & Limitations

    The derived structural properties enable efficient computation of optimal throughput, while a causal-information heuristic performs relatively close to the full-information optimum in numerical studies.

Abstract

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We consider the use of energy harvesters, in place of conventional batteries with fixed energy storage, for point-to-point wireless communications. In addition to the challenge of transmitting in a channel with time selective fading, energy harvesters provide a perpetual but unreliable energy source. In this paper, we consider the problem of energy allocation over a finite horizon, taking into account channel conditions and energy sources that are time varying, so as to maximize the throughput. Two types of side information (SI) on the channel conditions and harvested energy are assumed to be available: causal SI (of the past and present slots) or full SI (of the past, present and future slots). We obtain structural results for the optimal energy allocation, via the use of dynamic programming and convex optimization techniques. In particular, if unlimited energy can be stored in the battery with harvested energy and the full SI is available, we prove the optimality of a water-filling energy allocation solution where the so-called water levels follow a staircase function.

I. INTRODUCTION

The paper studies finite-horizon throughput maximization for wireless transmitters powered by unreliable harvested energy under causal or full side information. It derives structural properties and allocation schemes, including staircase water-filling with unlimited storage and a heuristic causal-information scheme.

  • Motivation: Energy-harvesting transmitters face causality constraints because each slot can use only currently stored energy, while future harvested energy is unavailable.This differs from conventional power or sum-energy constraints.
  • Problem and scope: The problem maximizes throughput over K finite slots while channel SNRs and harvested energy vary across slots.The study examines the structure of maximum throughput and optimal energy allocation.
  • Problem and scope: Two side-information settings are considered: causal information about past and present conditions, and full information including future conditions.Full side information is motivated for highly predictable environments.
  • Contributions: With causal side information and first-order Markov variations, dynamic programming yields an optimal allocation policy computable offline and stored in a lookup table.The paper also derives structural results characterizing the optimal solution.
  • Contributions: With full side information and unlimited storage, the optimal allocation has a water-filling interpretation with multiple non-decreasing water levels forming a staircase over time.A closed-form solution is also obtained for K = 2 slots.
  • Contributions: A proposed causal-information heuristic performs relatively close to the full-information optimal throughput in numerical studies.The comparison is against a naive scheme.

II. SYSTEM MODEL

The system is a point-to-point, flat-fading, single-antenna link whose transmitter uses harvested energy stored in a battery. Channel SNR and harvested energy evolve over slots under Markov assumptions, with throughput determined by mutual information.

  • Communication model: Each slot transmits one backlogged packet over n symbols, with rate determined by the mutual information I(γ_k, T_k).The channel is quasi-static within each slot and reliable decoding is assumed for sufficiently large n.
  • Energy model: The transmitter stores harvested energy in a battery and uses transmission energy T_k through the power amplifier, while other circuit consumption is negligible.Energy is measured per symbol, so energy and power are used interchangeably.
  • Communication model: The mutual information I(γ, T) is assumed increasing and concave in transmission energy T for each SNR γ.Gaussian signaling over a complex Gaussian channel is given as an example.
  • Energy model: Battery energy evolves from stored energy, harvested energy, and transmission energy, subject to storage efficiency, memory effects, and a maximum capacity B_max.The initial stored energy B_1 is known, and the storage process follows a first-order Markov model.
  • Channel and harvest dynamics: The channel SNR and harvested energy are modeled as first-order stationary Markov processes, with their joint distribution assumed known.The model includes independent and deterministic special cases.

4) Overall Dynamics:

The paper formulates energy allocation as a finite-horizon dynamic program in which current transmission trades off present mutual information against expected future throughput. It proves concavity and monotonicity properties and uses them to characterize efficient optimal allocation.

  • A. Problem Statement: Under causal side information, the transmitter chooses a feasible policy T_k(s_k) using the current state, with future states unknown.The policy can be optimized offline and implemented through a stored lookup table.
  • A. Problem Statement: Energy allocations across slots cannot generally be optimized independently because current transmission changes the battery energy available in later slots.For two slots, T_1 affects B_2 and therefore the feasible choice of T_2.
  • B. Optimal Solution: Dynamic programming solves the optimization by recursively replacing future decisions with value functions, yielding the optimal throughput J_1(s_1).The Bellman recursion balances present mutual information against expected future mutual information.
  • B. Optimal Solution: If mutual information is concave in transmission energy, the value functions and maximum throughput are concave in battery energy.This property simplifies computation of the optimal allocation.
  • B. Optimal Solution: The optimal transmission energy is non-decreasing in available battery energy under the same concavity assumption.This monotonicity restricts the numerical search space.

C. Numerical Computations

The numerical-computation discussion exploits concavity to solve per-slot maximizations efficiently and examines i.i.d., Rayleigh-fading, time-invariant, and AWGN cases. Even under i.i.d. conditions, battery history keeps the dynamic program coupled across slots.

  • Numerical solution: The unconstrained per-slot objective is concave, so its unique maximizer can be found numerically, for example by bisection search.Battery limits are then imposed by restricting the feasible maximization interval.
  • D. I.I.D. SNR and Harvested Energy: For i.i.d. SNR and harvested energy, the optimization remains coupled because current transmission affects future stored energy.The dependence on past harvested energy prevents full decoupling despite identical distributions across slots.
  • D. I.I.D. SNR and Harvested Energy: For Rayleigh fading with mean SNR γ̄, the SNR distribution is exponential over nonnegative SNR values.The expected mutual information is then evaluated under this distribution.
  • D. I.I.D. SNR and Harvested Energy: In a time-invariant channel, setting γ_k = γ̄ for every slot gives the corresponding expected mutual-information formulation.
  • D. I.I.D. SNR and Harvested Energy: In AWGN channels, the value function is independent of SNR but remains dependent on harvested energy, so recursive optimization is still required.The problem therefore does not reduce to decoupled per-slot optimizations.

IV. FULL SIDE INFORMATION: ARBITRARY Bmax

With full side information and arbitrary battery capacity, the paper characterizes optimal throughput and energy allocation using dynamic programming, including closed-form two-slot structure and larger-horizon properties.

  • Full side information yields the optimal throughput for arbitrary Bmax, computable recursively through Bellman equations.
  • Greedy allocation uses available stored energy, conservative allocation saves stored energy without wasting harvests, and balanced allocation trades stored energy across slots according to channel conditions.
  • For the final slot, or when K = 1, all available power is allocated for transmission; for K = 2, the first-slot allocation is given in closed form.
  • For K = 2 slots, the optimal first-slot energy allocation has greedy, balanced, or conservative modes determined by stored energy, harvested energy, and SNRs.The mode thresholds use a = Bmax − H1, b = H1 + 1/γ2 − 1/γ1, and c = 2Bmax − H1 + 1/γ2 − 1/γ1.
  • As Bmax approaches infinity, the two-slot solution simplifies, allocating roughly half the battery energy plus or minus a correction based on SNRs and harvested energy.
  • For larger K, closed-form expressions become unwieldy and less intuitive, motivating the infinite-storage analysis as an upper-bound and tractable special case.

V. FULL SIDE INFORMATION: INFINITE Bmax

Under full side information and unlimited battery capacity, the constrained throughput problem is compared with conventional water-filling and solved using a generalized staircase water-filling structure.

  • With infinite Bmax, feasibility requires nonnegative battery storage after every slot, and throughput maximization is formulated under these energy-harvesting constraints.
  • Relaxing all intermediate battery constraints produces the conventional sum-energy water-filling problem with total energy Pmax = B1 + P_K−1.
  • The conventional water-filling solution is an upper bound but is no longer optimal when intermediate energy-harvesting constraints are imposed.
  • A conventional water-filling algorithm is provided for numerical implementation, using a tolerance ǫ to approach the power constraint.
  • The constrained optimization is convex and is solved through its dual problem, yielding generalized water-filling with a staircase-like water level.

1) Structural Properties:

The optimal allocation has staircase water levels shaped by energy-harvesting constraints: levels rise across slots, transition slots empty the battery, and monotone SNRs imply monotone allocations.

  • Definitions: A transition slot is one after which the water level changes; the final slot is also defined as a transition slot.
  • Structural Properties: The optimal water level is non-decreasing across slots, forming staircase water-filling rather than the constant level of conventional water-filling.
  • Structural Properties: At every transition slot, the battery storage is empty, so water-level changes coincide with active energy-causality constraints.
  • Corollaries: If SNR is non-decreasing over slots, the optimal power allocation is also non-decreasing over slots.
  • Interpretation: For constant SNR, causal energy arrivals can force later-slot power increases because harvested energy becomes available only in later slots.
  • Limitations: The converse need not hold: arbitrary SNR patterns can place high-SNR slots away from later slots, so optimal power need not be non-increasing.

2) Efficient Implementation:

The efficient implementation decomposes staircase water-filling into conventional water-filling over slot intervals and identifies optimal transition slots through forward search. Algorithm 2 is proven optimal for finding the transition-slot set.

  • Staircase water-filling: The optimal allocation performs conventional water-filling separately within each slot interval, using the interval’s available sum power.Together, these interval allocations form the staircase water-filling solution.
  • Transition-slot search: The optimization reduces to searching for an optimal transition-slot set S⋆ with size between 1 and K.Each candidate set partitions the horizon into slot intervals whose power allocations must satisfy the original constraints.
  • Feasible search: A feasible-search procedure tests candidate first transition slots by computing water-filling allocations and retaining those satisfying the constraints.The procedure initializes S1, evaluates t1 from 1 through K, and admits feasible candidates.
  • Feasible search: The optimal first transition slot is the largest element of the feasible set S1.Lemma 2 establishes that all smaller feasible candidates are suboptimal.
  • Algorithm 2: Algorithm 2 uses forward outer searches and backward inner searches to find successive transition slots until the final transition reaches K.Theorem 4 states that this algorithm obtains the optimal transition-slot set.

3) Update Algorithm when New Slots Become Available:

When a new slot becomes available, the existing optimal transition-slot set can be updated rather than recomputed from scratch. The update checks the new slot during each outer iteration and requires at most K inner iterations overall.

  • Update procedure: The update algorithm incorporates a newly available slot into the existing optimal transition-slot set S⋆.It is intended for systems whose available slots can increase dynamically, including multi-user settings where another user relinquishes a slot.
  • Update procedure: Each outer iteration executes only the inner search for k = K + 1 and either accepts the new transition slot or continues with the known prior solution.This avoids rerunning Algorithm 2 from the beginning.
  • Complexity: At most K inner iterations are required in total because Algorithm 2 has at most K outer iterations.The update therefore provides an efficient way to revise S⋆ when the horizon expands.

VI. NUMERICAL RESULTS

Numerical experiments evaluate causal and full side information under i.i.d. channel and harvested-energy conditions for AWGN and Rayleigh fading channels. Throughput increases with the horizon, with larger gains from full side information when the horizon is short, although the difference becomes limited in the reported setting.

  • Experimental setup: The experiments assume i.i.d. SNRs and harvested energies, with initial and harvested energy values in {0, 0.5, 1} and Bmax →∞.Throughput per slot is evaluated as average SNR increases for AWGN and Rayleigh fading channels.
  • Policy computation: With full side information, Algorithm 2 produces the optimal policy and is significantly faster while remaining equivalent to standard optimization software.The reported throughput is averaged over 10^4 independent Monte Carlo runs.
  • Throughput trends: For K = 1, causal and full side information yield the same throughput because future slots cannot be exploited.The equality holds in both AWGN and Rayleigh fading cases.
  • Throughput trends: Throughput per slot increases as K increases, with a more substantial increment under full side information.The larger gain is attributed to better exploitation of information about future slots.
  • Throughput trends: The improvement from increasing K is significant for small K but becomes less significant when K is large.In the reported setting, the throughput difference between full and causal side information is not significant, possibly because exploitable full information is limited.

A. Heuristic Schemes with Causal SI

The paper proposes causal-SI heuristic schemes, especially power-halving, to allocate stored energy across slots and approach full-SI throughput. Numerical results show improved per-slot throughput as the horizon grows, with small gaps to full SI.

  • Performance: For K = 1, causal-SI and full-SI policies both use all stored energy, while the throughput is substantially worse than for K > 2.
  • Power-halving scheme: The power-halving scheme uses half the stored energy in intermediate slots and all remaining stored energy in the final slot.Its weights are w_k = 1/2 for other slots and w_k = 1 in the last slot.
  • Power-halving scheme: The scheme defers more stored energy to later slots, resembling the staircase water-level policy under full SI.
  • Performance: The power-halving scheme improves per-slot throughput as K increases and remains within about 0.2 bits of the full-SI case.
  • Performance: At lower SNR, the power-halving scheme shows an even smaller throughput degradation relative to full SI, with further gains possible from explicit channel-condition optimization.
  • Scope: The analysis considers finite-horizon energy allocation with varying harvested energy and develops structural results enabling efficient computation for causal and full SI.

APPENDIX A

The appendix establishes concavity and monotonicity properties of the dynamic-programming value functions and optimal transmission decisions. These properties support uniqueness and increasing optimal transmission with available battery energy.

  • APPENDIX A: Induction proves that the dynamic-programming value functions are concave in battery energy and post-harvesting stored energy.
  • APPENDIX A: Concavity follows through expectation preservation and the convolution structure of the Bellman recursion.
  • APPENDIX A: The optimal transmission amount T⋆(B) is unique because the objective function is concave.
  • APPENDIX A: The optimal transmission amount T⋆(B) is non-decreasing in the available battery energy B.
  • APPENDIX A: For K = 2 with full SI, the objective combines current-slot mutual information with future-slot mutual information subject to battery saturation.
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