[Paper Review] Near Optimal Energy Control and Approximate Capacity of Energy Harvesting Communication

Yishun Dong, Farzan Farnia+1 more|arXiv (Cornell University)|May 6, 2014

Energy Harvesting in Wireless Networks9 references4 citations

TL;DR

This paper proposes a near-optimal energy control policy and a constant-gap approximation to the information-theoretic capacity of an energy-harvesting communication system with a finite battery. For an i.i.d. Bernoulli energy arrival process, it shows that the capacity is approximately $ C \approx \frac{1}{2}\log(1 + pE) $ when $ B_{\text{max}} > E $, with a bounded approximation gap of 2.58 bits regardless of system parameters.

ABSTRACT

We consider an energy-harvesting communication system where a transmitter powered by an exogenous energy arrival process and equipped with a finite battery of size $B_{max}$ communicates over a discrete-time AWGN channel. We first concentrate on a simple Bernoulli energy arrival process where at each time step, either an energy packet of size $E$ is harvested with probability $p$, or no energy is harvested at all, independent of the other time steps. We provide a near optimal energy control policy and a simple approximation to the information-theoretic capacity of this channel. Our approximations for both problems are universal in all the system parameters involved ($p$, $E$ and $B_{max}$), i.e. we bound the approximation gaps by a constant independent of the parameter values. Our results suggest that a battery size $B_{max}\geq E$ is (approximately) sufficient to extract the infinite battery capacity of this channel. We then extend our results to general i.i.d. energy arrival processes. Our approximate capacity characterizations provide important insights for the optimal design of energy harvesting communication systems in the regime where both the battery size and the average energy arrival rate are large.

Motivation & Objective

• To provide a simple, universal approximation to the information-theoretic capacity of energy-harvesting communication systems with finite battery size.
• To identify the dependence of capacity on key system parameters such as average energy arrival rate, battery size, and energy packet size.
• To offer engineering insights into optimal battery sizing and energy harvesting profile selection in practical systems.
• To extend results from the simple Bernoulli energy arrival model to general i.i.d. energy harvesting processes.
• To establish a constant-gap approximation that is universal across all system parameters, enabling scalable system design insights.

Proposed method

• Proposes a near-optimal online energy control policy that dynamically allocates transmit power based on current battery level and energy arrival process.
• Derives a closed-form approximation for capacity: $ C \approx \frac{1}{2}\log(1 + pB_{\text{max}}) $ for $ B_{\text{max}} \leq E $, and $ \frac{1}{2}\log(1 + pE) $ for $ B_{\text{max}} > E $, under i.i.d. Bernoulli energy arrivals.
• Uses a constant-gap approximation framework, bounding the difference between upper and lower capacity bounds by a universal constant (2.58 bits).
• Applies the Verdu-Han framework to characterize capacity and derives bounds using the expected energy surplus and tail distribution of the energy process.
• Extends results to general i.i.d. energy arrival distributions by analyzing the ratio of the integral of the survival function to the maximum of $ x(1-F(x)) $, which controls the approximation gap.
• Constructs counterexamples with unbounded approximation gaps to show limitations of the approach under certain heavy-tailed energy profiles.

Experimental results

Research questions

• RQ1 What is the approximate capacity of an energy-harvesting AWGN channel with a finite battery and i.i.d. energy arrivals?
• RQ2 How does the capacity depend on the average energy arrival rate, energy packet size, and battery size?
• RQ3 Is there a universal constant-gap approximation to the capacity that is independent of system parameters?
• RQ4 What is the optimal battery size for extracting the maximum capacity in energy-harvesting systems?
• RQ5 Which energy harvesting profiles lead to bounded versus unbounded approximation gaps in capacity estimation?

Key findings

• The capacity of the energy-harvesting AWGN channel is approximately $ \frac{1}{2}\log(1 + pE) $ when $ B_{\text{max}} > E $, with a constant approximation gap of 2.58 bits regardless of $ p $, $ E $, or $ B_{\text{max}} $.
• For $ B_{\text{max}} \leq E $, the capacity is approximately $ \frac{1}{2}\log(1 + pB_{\text{max}}) $, indicating that battery size directly limits capacity in this regime.
• A battery size of $ B_{\text{max}} \geq E $ is (approximately) sufficient to achieve the infinite-battery capacity, suggesting that larger batteries provide diminishing returns beyond this threshold.
• The approximation gap remains bounded at 2.58 bits for all i.i.d. Bernoulli energy arrival processes, enabling universal design insights.
• For certain heavy-tailed energy distributions, such as those with $ 1-F_n(x) = 1/x $ for $ x \in [1,n) $, the approximation gap grows unboundedly with $ n $, showing the limits of the constant-gap approach.
• Discrete approximations of such heavy-tailed profiles (e.g., with $ A_i = i $, $ p_i = \frac{1}{i} - \frac{1}{i+1} $) also yield unbounded gaps, confirming the instability of the approximation under extreme energy profiles.