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Energy Cooperation in Cellular Networks with Renewable Powered Base Stations
Yeow-Khiang Chia, Sumei Sun, Rui Zhang
TL;DR
The paper studies how cellular BSs with renewable generation, limited storage, conventional backup, and resistive interconnections can reduce conventional-energy use despite variable profiles. It formulates offline cooperation as a linear program, develops online and hybrid policies, and compares their performance. Under the sinusoidal profiles considered, the greedy online algorithm loses at most 2.3% relative to the offline algorithm.
Problem
Renewable-energy variability, limited storage, and transfer losses complicate reducing conventional energy consumption across cooperating cellular BSs.
Method
The paper models energy cooperation between hybrid-powered BSs, solves the fully known-profile case by linear programming, and proposes greedy online and partially informed hybrid algorithms.
Results
2.3% maximum loss: the greedy online algorithm compared with the offline algorithm under the sinusoidal energy profiles assumed in (9).
Takeaways & Limitations
Energy cooperation can be evaluated across offline, online, and hybrid information regimes, with the hybrid approach leveraging available offline profile information while operating online.
Takeaways & Limitations
The offline policy assumes the energy profile is known entirely for the duration of interest, while the partially informed setting assumes only some information is known.
Abstract
from arXiv · showhide
In this paper, we propose a model for energy cooperation between cellular base stations (BSs) with individual hybrid power supplies (including both the conventional grid and renewable energy sources), limited energy storages, and connected by resistive power lines for energy sharing. When the renewable energy profile and energy demand profile at all BSs are deterministic or known ahead of time, we show that the optimal energy cooperation policy for the BSs can be found by solving a linear program. We show the benefits of energy cooperation in this regime. When the renewable energy and demand profiles are stochastic and only causally known at the BSs, we propose an online energy cooperation algorithm and show the optimality properties of this algorithm under certain conditions. Furthermore, the energy-saving performances of the developed offline and online algorithms are compared by simulations, and the effect of the availability of energy state information (ESI) on the performance gains of the BSs' energy cooperation is investigated. Finally, we propose a hybrid algorithm that can incorporate offline information about the energy profiles, but operates in an online manner.
I. INTRODUCTION
The paper addresses renewable-energy variability and limited storage at cellular base stations by enabling energy sharing between connected BSs. It develops offline, online, and hybrid cooperation policies using available energy-profile information to reduce conventional-energy consumption.
- Motivation: Renewable sources reduce energy costs but remain variable, so BSs still require conventional power and storage support.Storage deployment is costly, making available storage limited.
- Energy cooperation: The proposed approach mitigates renewable variability through geographical diversity by transferring energy between BSs with excess and deficit energy.The model includes resistive power losses and storage inefficiency.
- Evaluation: The study compares energy-saving performance across the developed policies and investigates how energy-state information affects cooperation gains.The power line may also support coordinated multipoint transmission as a communication backhaul.
- Algorithms: The paper studies deterministic profiles with an offline policy, arbitrary profiles with an online policy, and partial offline information with a hybrid algorithm.The offline policy is evaluated as a lower bound for the online policy, while the hybrid algorithm incorporates some offline profile information.
II. SYSTEM MODEL
The system model focuses on two cellular BSs, each equipped with renewable generation, conventional energy, and storage, and connected by a power line for energy cooperation.
- System scope: The paper models two BSs with individual renewable generators, conventional energy sources, storage devices, and an interconnecting power line.The formulation can be generalized to more than two BSs, but two BSs are considered for simplicity.
- Time model: The system operates over a finite-horizon, time-slotted interval indexed by t, with 1 ≤ t ≤ N.The model defines elements separately for BS 1 and BS 2.
A. Model Elements
The model tracks renewable generation, demand, conventional energy, storage, and directed inter-BS transfers at each time slot.
- Energy quantities: Renewable generation RE_i(t), demand DE_i(t), and conventional energy w_i(t) are defined for each BS and time slot.These quantities are constrained to be nonnegative.
- Energy quantities: Net energy E_i(t)=RE_i(t)−DE_i(t) is positive for a surplus and negative for a deficit.The sign of net energy distinguishes excess generation from unmet demand before cooperation decisions.
- Storage: Stored energy s_i(t) is bounded by Smax, while charging c_i(t) and discharging d_i(t) obey d_i(t)≤s_i(t).Charging and discharging cannot both be strictly positive in the same slot.
- Energy transfer: Directed transfers x_12(t) and x_21(t) represent energy sent between the BSs, with at most one direction active in a time slot.The model prevents simultaneous positive transfers in opposite directions.
B. System Dynamics
Storage evolves through charging and discharging, while energy-balance constraints ensure each BS can satisfy demand using local, stored, transferred, renewable, or conventional energy.
- Storage dynamics: Storage follows s_i(t+1)=s_i(t)+αc_i(t)−d_i(t), where α captures charging inefficiency.The storage state remains bounded between zero and Smax.
- Storage dynamics: The model initializes both storage states at zero and imposes constraints that preserve storage feasibility over time.The combined dynamics produce bounds on the net storage change.
- Energy balance: Each BS must satisfy its demand through available renewable energy, storage discharge, transferred energy, or conventional energy.The balance constraints maintain energy neutralization at every time slot.
- Losses: The model accounts for inefficiency when drawing from storage and resistive loss when transferring energy between BSs.The transfer-loss factor β satisfies 0≤β≤1.
C. Control Policy and Objective Function
The section defines history-dependent control policies for the two-base-station system and formulates minimizing expected average conventional energy consumption over a finite horizon. Although a long-run average criterion is noted, the paper restricts attention to finite N for simplicity.
- System representation: The system state is the vector of stored energies, while controls specify grid usage, charging, discharging, and energy transfers between base stations.Controls depend on past energy profiles and their joint distribution.
- Objective: The objective is to minimize expected average conventional energy consumed under the optimal control policy.The expectation is taken with respect to the joint distribution of energy profiles.
- Objective: A long-run expected average conventional-energy criterion is possible, but the paper uses a finite horizon because it provides simpler intuition.The long-run criterion is described as insensitive to the starting state.
- Problem status: The optimal control policy is open in the general formulation, motivating analysis of special cases with useful structure.The paper proceeds by studying restricted settings where insight can be obtained.
III. OFFLINE ALGORITHM WITH DETERMINISTIC ENERGY PROFILE
With deterministic energy profiles known over the horizon, offline cooperation becomes a linear-programming problem. The resulting simulations show that cooperation, storage, and anti-correlation can reduce conventional energy consumption.
- Interpretation: The deterministic-profile assumption represents accurately predictable demand and renewable generation and helps identify when cooperation is beneficial.The paper highlights sufficiently uncorrelated or anti-correlated net energy profiles as favorable.
- Offline formulation: When deterministic net energy profiles are known for all times, the optimal control policy is obtained by solving a linear program.The objective reduces to the sum of conventional energy required at each time.
- Offline formulation: The linear-program optimum automatically satisfies the constraints that a base station cannot charge and discharge simultaneously or transfer energy in both directions.Removing these explicit constraints therefore causes no loss of optimality.
- Offline formulation: A secondary optimization can maximize energy stored at time N + 1 while preserving minimum conventional-energy consumption.This supports future reduction of grid energy use.
- Simulation results: Cooperating base stations consume less average conventional energy per base station than a single base station, and the cost decreases as θ increases from π/4 to π.Greater θ corresponds to increasingly anti-correlated profiles in the reported comparison.
- Simulation results: Increasing storage lowers cost by preserving excess energy for later deficits, but its benefit decreases as θ approaches π.At θ = π, increasing storage provides little additional benefit because one base station can compensate the other directly.
- Simulation results: With Smax = 1, savings increase from θ = 0 to π and then decline toward 2π as the profiles become correlated again.Anti-correlation at θ = π creates more opportunities for energy transfer between base stations.
IV. ONLINE ALGORITHM WITH STOCHASTIC ENERGY PROFILE
For stochastic net energy profiles that are not known ahead of time, the paper proposes an online cooperation algorithm based on a greedy heuristic. It analyzes optimality under specific profiles and compares online and offline performance by simulation.
- Online setting: The stochastic setting treats net energy at both base stations as unknown ahead of time and causally observed.This is presented as the more practical case relative to deterministic profiles.
- Online approach: The proposed online algorithm greedily minimizes conventional energy usage using current information.Its construction is based on a single-snapshot version of the offline linear programs.
- Analysis and evaluation: The paper analyzes properties of the online algorithm and states optimality results under specific energy profiles.It also compares online performance with the optimal offline algorithm through simulations.
A. Greedy Online Algorithm
The greedy online policy solves a small linear program at each time and has an equivalent case-based implementation for two base stations. It prioritizes energy balancing, storage use, transfers, and finally conventional generation according to the current net-energy states.
- Algorithm construction: The greedy policy is derived by combining the single-time linear programs and solving a small linear program at each time.For two base stations, an equivalent direct algorithm is provided.
- Two-step optimization: A second optimization step maximizes storage at the next time after fixing the minimum conventional-energy value from the first step.This preserves the primary objective while selecting among equivalent minimum-energy policies.
- Case-based policy: With nonnegative energy at both base stations, excess energy is transferred to the other station when its storage has room.Transfers stop when both storages are full or both remain below capacity.
- Case-based policy: When both base stations have deficits, each first uses its individual storage before the remaining deficit is supplied conventionally.The same logic is applied after intermediate transfers and storage actions update the net profiles.
- Special cases: With no storage, the modified greedy policy is optimal for arbitrary energy profiles.With no cooperation, transfers are removed and the corresponding optimality discussion is handled separately.
B. Optimality Properties
The greedy algorithm has broad structural optimality properties, including exact optimality under boundary transmission efficiencies and several restricted energy-profile conditions. These results also clarify when local storage or inter-base-station transfer is preferable.
- The greedy algorithm is optimal for arbitrary energy profiles when β = 0 or β = 1.
- Higher initial stored energy cannot increase optimal cost, and an optimal policy never draws grid energy solely to increase storage.
- When storage is available, excess energy is optimally stored locally first, while deficits are optimally compensated using local storage before the other base station's storage.
- If β > α, transferring energy to help the other base station is more cost-efficient than storing it for later use because storage incurs the larger proportional loss.
- When β > α and one base station has nonnegative net energy throughout the horizon, the greedy policy is optimal.
- If one base station is always in surplus while the other is always in deficit, the modified greedy policy that transfers energy is optimal, even when β ≤ α.
C. Numerical Results
The simulations compare greedy online and optimal offline cooperation under sinusoidal profiles and varying phase correlation. The greedy method can approach offline performance, but its gap depends on correlation and storage availability.
- Without future energy-profile information, the greedy algorithm cannot generally match the offline algorithm except under the identified special conditions.
- 2.3% maximum energy loss separates the greedy online algorithm from the offline algorithm under the assumed sinusoidal energy profiles.The plotted loss is the percentage increase in energy consumption relative to the optimal offline algorithm over horizon N.
- When energy profiles are anti-correlated or highly positively correlated, the greedy algorithm is almost as efficient as the offline algorithm.
- The offline algorithm generally achieves higher energy savings as maximum storage increases because it uses storage more strategically.
V. HYBRID MODEL AND ALGORITHM
The hybrid algorithm combines offline planning for a known deterministic energy component with online compensation for unknown deviations. It can outperform greedy online cooperation at moderate correlation phases, while greedy can prevail where future information adds little.
- The offline component is obtained by solving the linear program that minimizes energy consumption and maximizes storage at time N.
- The hybrid algorithm uses offline information for the deterministic profile and the greedy algorithm to compensate for random deviations.
- At each time t, the hybrid model assumes the deterministic profiles are known for all t, while realized profiles are known only through time k.
- The hybrid policy partitions storage between offline and online components, using residual maximum storage states for online optimization.
- For moderate θ values, the hybrid algorithm outperforms the greedy online algorithm because it uses knowledge of the deterministic energy profiles.
- When θ is close to 0 or π, greedy online cooperation can outperform the hybrid algorithm because future-profile information provides little additional saving.
VI. CONCLUSION AND FUTURE DIRECTIONS
The paper models energy cooperation between cellular base stations and develops offline, online, and hybrid policies for different levels of energy-profile information. It establishes a linear-program solution for fully known profiles, analyzes a greedy online algorithm under conditions, and reports simulation-based comparisons, while identifying extensions beyond the two-BS setting.
- Model and scenarios: The model covers two BSs with conventional and renewable supplies, limited storage, and a connecting power line for energy transfer.The paper frames the cooperation problem around hybrid power supplies, storage constraints, and inter-BS energy sharing.
- Offline and online policies: For fully known energy profiles, the optimal policy can be found by solving a linear program.This is the offline-information scenario considered in the paper.
- Offline and online policies: For profiles without future statistical information, the paper proposes a greedy online algorithm and analyzes its optimality properties under some conditions.The online policy addresses the opposite information extreme from the offline linear-program formulation.
- Evaluation: Simulations compare offline and online algorithms, and the hybrid algorithm can outperform the online algorithm in some regimes by using offline information.The reported comparison concerns the hybrid and greedy online policies under simulated prediction-error conditions.
- Hybrid information: The paper also considers partial future information and develops a hybrid algorithm that operates online while leveraging available energy-profile information.The hybrid setting lies between complete profile knowledge and no future statistical information.
- Future directions: The presented two-BS model and algorithms can be extended to multiple BSs and could incorporate time-varying conventional-energy prices, requiring algorithm changes.These are identified future directions rather than results established by the current model.
APPENDIX B
Appendix B derives the optimal energy-cooperation policy by partitioning net-energy cases and comparing transfer, storage, discharge, and conventional-energy decisions. The resulting policy prioritizes deficit compensation and reduces to simpler forms under conditions on line efficiency and storage states.
- Case analysis: The E1 ≥ 0, E2 < 0 case is divided into four sub-cases according to the size of BS 2's deficit and available storage.The first sub-case transfers all available energy to overcome the deficit, while the remaining cases analyze mixtures of transfer, charging, and discharge.
- Efficiency threshold: If β ≥ α^2, the optimal policy sets Δ = 0, whereas if β < α^2, it sets Δ to its maximum possible value.This threshold follows from the sign of the change in total storage levels as energy is allocated between transfer and storage.
- Transfer and storage priority: Transferred energy is used to compensate for deficit before charging storage, because charging can require additional discharge and reduce total storage.When βx12 ≤ |E2| and w2 = 0, charging BS 2 is suboptimal; the net storage change from diverting δ units is αδ − δ/α ≤ 0.
- Combined policy: For β ≥ α^2, the combined policy first transfers energy to cover all deficit, then charges storage at BS 1 followed by storage at BS 2.The policy is reduced to the stated form after combining the four sub-cases; for β < α^2, it reduces to the corresponding Case 2B form.
- Both-deficit case: When both BSs have deficits, each BS uses its own storage before helping the other because cross-BS storage use is less efficient while local storage remains.If residual net energy remains negative at both BSs, conventional energy generation compensates the residual deficits.
- Optimality proof: The proof also establishes that the greedy policy minimizes the cost-to-go by induction, with the terminal-time policy serving as the base case.The appendix states that the greedy policy is optimal at time t = N and that induction over t yields equality with the optimal policy.