Source-linked AI summary
A General Framework for the Optimization of Energy Harvesting Communication Systems with Battery Imperfections
Bertrand Devillers, Deniz Gunduz
TL;DR
Energy-harvesting communication systems need transmission policies that maximize data under deadline, harvesting, and battery constraints. The paper develops an offline cumulative-curve framework, extends it to continuous arrivals and broadcast systems, and characterizes constant-leakage operation. Its conclusion is that optimal schemes can be identified across these models, including constant leakage at a fixed rate.
Problem
The paper studies how to maximize data transmitted by a deadline when energy harvesting and battery limitations constrain communication.
Method
It uses offline cumulative harvested-energy and minimum-energy curves to derive optimal transmission policies, including continuous arrivals, battery constraints, broadcast systems, and leakage.
Results
The framework characterizes optimal transmission schemes for continuous harvesting, time-varying battery constraints, broadcast systems, and constant-rate battery leakage.
Takeaways & Limitations
The curve-based framework provides a common way to optimize harvested-energy communication under several battery models and arrival processes.
Takeaways & Limitations
Battery leakage is not directly handled by the minimum-energy-curve framework because the resulting maximum-energy curve depends on the transmitted-energy curve.
Abstract
from arXiv · showhide
Energy harvesting has emerged as a powerful technology for complementing current battery-powered communication systems in order to extend their lifetime. In this paper a general framework is introduced for the optimization of communication systems in which the transmitter is able to harvest energy from its environment. Assuming that the energy arrival process is known non-causally at the transmitter, the structure of the optimal transmission scheme, which maximizes the amount of transmitted data by a given deadline, is identified. Our framework includes models with continuous energy arrival as well as battery constraints. A battery that suffers from energy leakage is studied further, and the optimal transmission scheme is characterized for a constant leakage rate.
I. INTRODUCTION
The paper addresses offline maximization of transmitted data by a deadline under diverse harvesting and battery constraints. It develops a general curve-based framework and separately characterizes transmission with constant battery leakage.
- The objective is to maximize transmitted data by a deadline under varied energy-harvesting models and battery limitations.
- The offline formulation assumes the energy arrival profile is known in advance at the transmitter.
- The framework includes continuous energy harvesting, extending earlier packetized-arrival models.
- Cumulative harvested-energy and minimum-energy curves model harvesting and battery constraints for deriving the optimal transmission policy.
- For a battery with constant leakage, the optimal transmission strategy is characterized for packetized energy arrivals.
II. SYSTEM MODEL
The system is modeled in continuous time using harvested-energy and transmitted-energy curves. Feasible policies lie between cumulative harvested energy and a minimum-energy curve, and the goal is to maximize data offline over a finite interval.
- Harvested energy is represented as a continuous-time process, while transmission power and rate can be adjusted instantaneously.
- The harvested energy curve H(t) records energy harvested during [0, t], while the transmitted energy curve E(t) records energy used for transmission.
- A feasible transmitted-energy curve satisfies M(t) ≤ E(t) ≤ H(t) at every time.
- The optimization selects, among feasible curves, the policy transmitting the most data over [0, T] with the curves known in advance.
- The transmission rate is a non-negative, strictly concave, increasing function of instantaneous power, with r(0) = 0.
III. OPTIMAL TRANSMISSION SCHEME UNDER BATTERY SIZE CONSTRAINTS
Under battery-size constraints, feasible energy policies are bounded by harvested and minimum-energy curves, and strict concavity determines a unique optimal curve. The framework also covers time-varying capacities, battery lifetimes, and continuous solar harvesting.
- Battery-size constraints: The minimum-energy curve models battery-size constraints by requiring surplus energy to be transmitted before it is discarded.
- Extensions: A time-varying battery capacity b(t) models degradation through a continuously changing minimum-energy curve.
- Optimality conditions: Jensen’s inequality is the main tool used to derive optimality conditions for transmitted-energy curves.
- Optimality conditions: A strictly concave rate function makes the feasible curve with no improvable straight-line replacement unique and data-maximizing.
- Optimality conditions: At any transmission-power change, the optimal curve intersects H(t) or M(t); slopes increase at H(t) and decrease at M(t).
- Extensions: The same curve framework handles batteries that die at different times by treating each battery’s death time as a deadline for using its stored energy.
- Continuous energy arrival: For solar harvesting, the optimal policy follows the harvested-energy curve until a tangent point, then uses constant-power transmission.
IV. OPTIMAL BROADCAST SCHEME WITH BATTERY CONSTRAINT
The general energy-harvesting framework extends to broadcast channels by jointly optimizing total transmit power over time and its allocation between receivers. Strict concavity enables the point-to-point optimization approach, including continuous arrivals and battery constraints.
- Scope and contribution: Compared with earlier broadcast-channel solutions, this approach generalizes optimal policies to continuous energy arrivals and transmitters with battery constraints.
- Problem formulation: The weighted-sum objective maximizes µ1B1(T) + µ2B2(T) for nonnegative receiver weights µ1 and µ2.
- Problem formulation: The broadcast-channel transmitter optimizes both the transmitted energy curve and the instantaneous power allocation between two receivers.The objective is a weighted sum of bits delivered to the receivers by deadline T.
- Optimization structure: The optimization decouples into selecting total power over time and allocating that power between receivers according to the broadcast-channel rate function.Once total power is characterized, per-user allocation follows from the rate function.
- Power allocation: For intermediate weight ratios, low total power goes entirely to receiver 1, while power above pth is split with receiver 1 fixed at pth.The point-to-point setting results when µ > N2/N1 or µ ≤ 1.
- Optimization structure: Strict concavity of the broadcast rate function permits direct application of the point-to-point energy-harvesting results.The rate function is continuous, differentiable, and has a derivative decreasing with power.
V. OPTIMAL TRANSMISSION SCHEME WITH BATTERY LEAKAGE
The paper models battery leakage through a leakage curve and a resulting maximum feasible transmitted-energy curve. Unlike a battery-size constraint, leakage makes that maximum curve depend on the transmission policy, so the earlier framework does not directly apply.
- Leakage model: Battery leakage is modeled under a constant leakage-rate assumption, although actual leakage depends on battery type, age, usage, and temperature.
- Leakage model: The energy leakage curve L(t) records energy leaked during [0,t] and is continuous and non-decreasing under the constant-rate model.
- Feasibility: With discrete energy arrivals and no minimum energy curve, feasibility is expressed by 0 ≤ E(t) ≤ U(t), where U(t) = H(t) − L(t).The section focuses on packetized arrivals and a finite transmission interval.
- Framework limitation: The leakage model differs from a battery-size constraint because it produces a maximum, rather than minimum, energy curve.The maximum curve removes total leaked energy from harvested energy.
- Framework limitation: Because leakage depends on the transmitted energy curve, the maximum energy curve also depends on the transmission policy.Therefore, the solution framework developed for battery constraints does not directly extend to leakage.
A. The Single-Packet Problem
For a single harvested energy packet, the optimal leakage-aware policy uses constant power. With an infinite deadline it transmits at a finite power p*, while a fixed deadline uses the larger of p* and the deadline-implied slope s.
- Setup: The single-packet problem uses one energy packet harvested at t = 0 as the building block for the general N-packet problem.
- Infinite deadline: For an infinite deadline, the optimal policy transmits at constant power until the battery is empty, then remains silent.
- Infinite deadline: Leakage creates a trade-off: lower power improves energy efficiency, but longer transmission increases energy wasted through leakage.
- Infinite deadline: The optimal constant power p* maximizes f(p) = r(p)/(p + ǫ) and is finite for a strictly concave increasing rate function.The value p* is independent of the packet energy E.
- Fixed deadline: For a fixed deadline, if s < p*, the transmitter uses p* until the battery empties; otherwise it transmits at constant power s throughout [0,T].Equivalently, the policy uses constant power p̃ = max(p*, s).
B. The N-Packet Problem
The N-packet problem maximizes transmitted data by a deadline when energy arrives in packets, and it can be related to an equivalent storage-throughput problem. Equivalence holds under a sufficient geometric condition, but not generally.
- Problem formulation: The N-packet problem seeks the maximum data transmitted by deadline T under packetized energy arrivals and battery leakage.The optimal solution is evaluated against an equivalent ST problem with total energy E = Σ_n E_n.
- Equivalence results: The equivalent ST problem can achieve at least as much data as the N-packet problem: DST ≥ DNT.An optimal N-packet schedule can be emulated in the ST setting, establishing achievability and the inequality.
- Problem formulation: The optimal N-packet policy is piecewise constant-power transmission, with silence when the battery is empty.The construction divides time into intervals between energy arrivals and allows silent intervals when stored energy runs out.
- Equivalence results: Equivalence is not guaranteed: with two packets, the battery may empty before the second arrival, preventing constant-power transmission throughout [0,T].In the counterexample, the battery runs out at T/2 and transmission must remain silent until t2.
- Equivalence results: If the line from the origin to AN intersects eU(t) only at A1,...,AN, then DNT = DST and constant power is used whenever the battery is non-empty.The sufficient condition is expressed through N − 1 slope inequalities; Theorem 5.3 gives the equivalence and policy characterization.
- Algorithm and special case: Algorithm 5.1 solves general N-packet problems by selecting the rightmost admissible endpoint and assigning constant powers per inter-arrival interval.For each interval, transmission uses power p̃_n while the battery remains non-empty; without a deadline, all intervals use p* whenever energy is available.
VI. CONCLUSION
The paper develops a general framework for optimizing energy-harvesting transmission under battery constraints and characterizes optimal schemes for continuous arrivals and constant leakage.
- The framework optimizes transmitted data by a deadline under energy-harvesting and battery-size constraints, extending prior models to continuous energy arrivals and time-varying battery sizes.
- The framework also applies to energy-harvesting broadcast channels and incorporates battery constraints into the problem formulation.
- For a battery with constant leakage, the optimal transmission scheme is characterized.
A. Properties of f(p) ≜r(p)
The function f(p) has behavior determined by the rate function's strict concavity and the parameter ǫ: its maximizer is finite for ǫ > 0, while for ǫ = 0 the maximum occurs at p = 0.
- At p = 0 and ǫ > 0, l’Hôpital’s rule gives lim p→0 f′(p) = r′′(0)/2 < 0.
- For ǫ = 0, f(p) is maximized at p∗ = 0 and strictly decreases for p > 0.
- The numerator n(p) is strictly decreasing for p ≥ 0 because its first term decreases and its concavity term is negative and strictly decreasing.
- For ǫ > 0, f(p) has a unique maximum at a finite p∗, and f(p) strictly decreases for p > p∗.
- The maximizer p∗ decreases as ǫ decreases.
B. Proof of Optimality of Algorithm 5.1
Algorithm 5.1 is optimal by partitioning the general packet-arrival problem when the relevant inequalities fail at the full packet horizon, using battery-emptying and recursive decomposition.
- When the inequalities hold for k = N, Algorithm 5.1 directly produces the optimal solution given by Theorem 5.3.
- If the inequalities fail for k = N, the optimal energy curve empties the battery no later than t_k+1, before the next energy packet arrives.
- Waiting until t′ ≥ t_k+1 to increase the transmission slope is suboptimal because it violates Theorem 3.3.
- After the battery empties, the first k-packet subproblem is solved independently, and the remaining N−k packets are handled recursively from an empty battery.