A Lyapunov Optimization Approach for Green Cellular Networks with Hybrid Energy Supplies

I. INTRODUCTION

• Dense wireless deployments increase energy use and carbon emissions, while variable harvested energy makes satisfactory QoS difficult for EH-only networks.HES combines harvesting with grid power to address renewable-supply instability.Source: §I · p. 1

A. Related Works and Motivations

• Prior HES studies simplify energy accumulation or assume non-causal energy information; practical online policies with causal information remain unknown, while MDP complexity and heuristic guarantees are obstacles.Lyapunov methods offer low complexity, little prior-information need, and quantifiable worst-case performance.Source: A. Related Works and Motivations · p. 2

B. Contributions

• The network service cost combines grid-energy use and QoS, and LBAPC provides low-complexity online base-station assignment and power control through an efficiently solvable per-slot problem.The per-slot problem is deterministic and uses an inner-outer optimization algorithm.Source: B. Contributions · p. 2
• LBAPC is shown asymptotically optimal without channel or energy-harvesting distribution information, and simulations report gains over greedy transmission in service cost, grid use, and QoS.The contribution passage also reports more efficient spectrum use.Source: B. Contributions · p. 2

II. SYSTEM MODEL

• The modeled network has one harvest-powered EH-BS and one grid-and-harvest-powered HES-BS serving users over separately allocated orthogonal channels.The numbers of channels assigned to the two base stations are fixed and predetermined.Source: §II · pp. 2–4

• The section introduces network service cost as the performance metric before formulating its minimization problem and identifying technical challenges.Source: §III · p. 4

A. The Network Service Cost Minimization Problem

• The network service cost is a weighted sum of grid-energy and packet-drop costs, balancing grid consumption against successful packet delivery and QoS.The relative weights determine whether grid-energy sensitivity or successful delivery receives greater emphasis.Source: A. The Network Service Cost… · p. 4

B. Problem Analysis

• The long-term average cost problem is an infinite-horizon MDP whose state space can exceed 2.5 × 10^11 states in the paper's K = 4 example.The example quantizes battery energy, harvestable energy, and channel gains, making standard MDP solution methods impractical in both computation and memory.Source: B. Problem Analysis · p. 5
• LBAPC uses low-complexity per-slot decisions without storing an optimal policy or requiring channel or renewable-energy distribution information.Its performance can be tuned arbitrarily close to P1's optimum, and the paper also derives an upper bound on required battery capacity.Source: B. Problem Analysis · p. 5

IV. ONLINE BS ASSIGNMENT AND POWER CONTROL: THE LBAPC ALGORITHM

• LBAPC is designed for the modified NSCM problem and provides a feasible solution to P1, with asymptotic optimality analyzed in the following section.The paper uses the modified problem to enable Lyapunov-based algorithm design.Source: §IV · p. 5

A. The Modified NSCM Problem

• The modified NSCM problem tightens battery-output constraints so per-slot decisions can ignore energy causality, while any feasible solution remains feasible for P1.The paper introduces non-zero lower bounds on battery output to eliminate coupling between time slots.Source: A. The Modified NSCM Problem · p. 5
• As the battery-output margins approach zero, the modified problem's optimum approaches P1's optimum; convergence is linear in ε_H2, while the ε_H1 rate depends on channel statistics.The modified problem's optimal cost is bounded above by P1's optimum plus a positive term that diminishes with the margins.Source: A. The Modified NSCM Problem · pp. 5–6

B. The LBAPC Algorithm

• Conventional Lyapunov decisions cannot be applied directly because battery-energy temporal correlation makes system decisions time-dependent.The paper uses a weighted perturbation method for the modified NSCM problem to address this issue.Source: B. The LBAPC Algorithm · p. 6
• The virtual energy queues are shifted versions of actual battery levels, and their weighted lengths are minimized alongside network service cost to stabilize battery levels around perturbations.The perturbation parameters are bounded constants; V is a control parameter with units of J² per cost unit.Source: B. The LBAPC Algorithm · p. 6
• Each slot's decisions depend on current channel information and virtual queue state, without prior knowledge of harvesting-process or channel statistics.The algorithm solves a deterministic per-time-slot problem using the current system state.Source: B. The LBAPC Algorithm · p. 6
• At each slot, LBAPC obtains queue and channel states, solves for harvesting, assignment, and power decisions, then updates the virtual queues.The algorithm repeats this process at the next time slot.Source: B. The LBAPC Algorithm · p. 6

C. Solving the Per-Time Slot Problem

• The per-slot problem has a closed-form optimal energy-harvesting decision, while an inner-outer optimization algorithm solves assignment and power control.These solutions support the LBAPC feasibility and performance analysis.Source: C. Solving the Per-Time Slot Problem · p. 6
• Exhaustive search for the assignment and power subproblem is prohibitively costly because it has 3^K assignments and non-convex power control for fixed assignments.The subproblem is a mixed-integer nonlinear program.Source: C. Solving the Per-Time Slot Problem · p. 7
• The inner-outer algorithm reduces the assignment search space and derives closed-form power allocations, avoiding repeated solution of the non-convex power-control problem.The inner optimization determines assignment and power choices for the HES-BS, and the outer optimization identifies the global optimum.Source: C. Solving the Per-Time Slot Problem · p. 7, 1) The Inner Problem · p. 7

1) The Inner Problem:

• An optimal inner solution can prioritize HES-BS service to users with lower channel-inversion powers, which corresponds to better channel conditions than those of unserved users.The result restricts which user subsets need to be considered.Source: 1) The Inner Problem · p. 7
• For one inner-problem case, HES-BS transmit power comes either from harvested energy or grid energy, not a mixture of both.When the relevant virtual-queue condition exceeds V times the grid-energy weight, the optimal HES-BS solution consumes no harvested energy.Source: 1) The Inner Problem · pp. 7–8

2) The Outer Problem:

• The outer problem obtains a globally optimal assignment by searching all subsets of F_H and determining associated power allocations.The algorithm updates its best assignment and power decision as it evaluates candidates.Source: 2) The Outer Problem · p. 8
• The IOO algorithm eliminates assignment combinations that cannot be optimal and derives closed-form power allocations for fixed assignments.This avoids solving the non-convex power-control problem during the outer search.Source: 2) The Outer Problem · p. 8
• Worst-case complexity remains exponential, but the paper says the search is faster than exhaustive search while maintaining optimality and is acceptable for a reasonable number of users.The paper characterizes exponential complexity as inevitable for base-station assignment problems, which are typically NP-hard.Source: 2) The Outer Problem · p. 8

V. PERFORMANCE ANALYSIS

• The LBAPC battery energy levels stay within bounds that ensure energy causality and require capacity of θ_j plus the maximum harvested energy per slot.Given available storage, the control parameter V can be determined from the capacity.Source: §V · pp. 8–9
• The Lyapunov drift is based on squared virtual-queue lengths, and its derived bound coincides with the objective of LBAPC’s per-slot problem.The virtual queues measure the distance between battery energy levels and perturbed energy levels.Source: §V · p. 9
• P3 replaces P2’s energy-causality constraint with average energy balance, is a relaxation of P2, and admits a stationary randomized policy arbitrarily close to its optimum.Any feasible P2 solution is also feasible for P3.Source: §V · pp. 9–10
• As V increases and ε_Hj decreases, LBAPC’s cost upper bound becomes arbitrarily tight, yielding asymptotic optimality for P1.Theorem 1 bounds the algorithm’s network service cost; the remark states this limiting result.Source: §V · p. 10
• Approaching optimal performance requires greater battery capacity and longer convergence time, because higher V or lower ε_Hj raises the stabilized battery energy level.The authors identify this as a tunable tradeoff between system performance and battery capacity/convergence time.Source: §V · p. 10
• The analysis studies two typical renewable-energy base-station types; extending the approach to larger networks increases per-slot optimization complexity.The paper calls for further low-complexity per-slot algorithms with performance guarantees.Source: §V · p. 10

VI. SIMULATION RESULTS

• 4 MUs are considered unless otherwise specified; harvested energy is uniform over the stated range, and channel gains are exponential with mean g0d^-4.The simulation parameters include g0 = −40 dB, d = 50 m, and τ = 1 ms.Source: §VI · p. 10
• The Cost-aware Greedy benchmark normalizes grid-energy and packet-drop costs to 1 per Joule and 1 per packet drop, respectively.It prioritizes harvested energy before considering grid-energy cost against packet-drop cost.Source: §VI · p. 10
• Cost-aware Greedy assigns users to the EH-BS and then the HES-BS by harvested-energy availability, prioritizing users with higher channel gains.It assigns remaining users to the HES-BS when the added grid-energy cost is less than the reduction in packet-drop cost.Source: §VI · p. 10

A. Theoretical Results Verification

• Battery energy accumulates initially and then stabilizes at the perturbed energy levels, supporting the algorithm’s feasibility verification.Fig. 2 shows battery energy levels under P_H1 = P_H2 = 30 mW and w_D = 0.01.Source: A. Theoretical Results Verification · p. 10, Figure 2 · p. 11
• Larger V or smaller ǫ_Hj improves long-term average performance but slows convergence; properly tuned parameters let LBAPC greatly outperform Cost-aware Greedy.The network service cost over time is shown in Fig. 3.Source: A. Theoretical Results Verification · p. 11, Figure 3 · p. 11
• As V increases, LBAPC’s network service cost decreases inversely toward the optimal value of P2, while required battery capacity grows linearly.The reported relationship verifies the algorithm’s asymptotic optimality; capacity growth follows the linear increase of θ_j with V.Source: A. Theoretical Results Verification · p. 11, Figure 4 · p. 11
• With 100 mJ batteries and V = 1.0×10^-4 J^2·cost^-1, LBAPC obtains a 47% performance gain over the benchmark.The paper presents this as one parameter choice balancing achievable performance, convergence time, and required battery capacity.Source: A. Theoretical Results Verification · p. 11

B. Performance Evaluation

• As PH1 rises, network service cost is non-increasing for both policies, and LBAPC significantly reduces cost versus the greedy algorithm.With NB1=1, increasing PH1 brings negligible benchmark benefit but continues to reduce LBAPC cost; increasing NB1 from 2 to 4 benefits the greedy policy more than LBAPC.Source: B. Performance Evaluation · p. 11, Figure 5 · p. 12
• Increasing PH2 lowers network service cost linearly, with no high-PH2 saturation when NB2=K; increasing PH1 yields a more obvious improvement than increasing PH2.The paper attributes the stronger PH1 improvement to diversity gain from independent channels at the two base stations.Source: Figure 6 · p. 12, B. Performance Evaluation · p. 12
• With increasing wD, grid energy consumption rises while packet drop ratio falls; at sufficiently large wD, LBAPC approaches zero drops while the benchmark remains at 1.0%.The paper says adjusting the grid-energy and packet-drop weights yields different tradeoffs.Source: Figure 7 · p. 12, B. Performance Evaluation · p. 12

VII. CONCLUSIONS

• The conclusion reports that performance analysis establishes LBAPC’s asymptotic optimality and describes the algorithm as low-complexity and requiring little prior knowledge.The paper identifies extensions to more general HES networks, realistic EH models, multi-antenna techniques, interference management, and user scheduling as future work.Source: §VII · pp. 12–13

B. Performance Evaluation

• As K increases, network service cost rises for both policies, while LBAPC’s performance gain over the benchmark expands.The paper links higher load with greater grid-energy use and packet drops when harvested energy is insufficient.Source: B. Performance Evaluation · p. 12, Figure 8 · p. 13

VII. CONCLUSIONS

• LBAPC significantly outperforms the greedy scheme in network service cost, grid energy consumption, and achievable QoS.The performance evaluation also reports that LBAPC suppresses grid consumption and packet drops more effectively.Source: B. Performance Evaluation · p. 12, §VII · p. 13

A. Proof of Proposition 1

• The proof constructs a feasible P2 solution by leaving the EH-BS unused, dropping its assigned users’ packets, and replacing HES-BS harvested energy with grid energy.The passages identify P2 as a tightened version of P1.Source: A. Proof of Proposition 1 · p. 13

B. Proof of Lemma 2

• When H2,k is positive, the proof states that zero battery output is optimal; it also states that channel-inversion power is optimal when the relevant grid-cost term is positive.The proof concludes that the solution in (25) is optimal.Source: B. Proof of Lemma 2 · p. 13

C. Proof of Proposition 3

• Below a battery-energy threshold, no harvested energy at that base station participates in transmission.The proof identifies this as a property of the optimal per-time-slot solution.Source: C. Proof of Proposition 3 · p. 13

D. Proof of Theorem 1

• The theorem proof takes expectations, sums the per-slot inequalities over t=0,…,T−1, divides by T, and lets δ approach zero.It then invokes Proposition 1 and Lemma 4 to complete the proof.Source: D. Proof of Theorem 1 · p. 14