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Energy-Efficient Scheduling and Power Allocation in Downlink OFDMA Networks with Base Station Coordination

Luca Venturino, Alessio Zappone, Chiara Risi, Stefano Buzzi

arXiv:1405.2962v1cs.ITcs.NI

TL;DR

The paper addresses energy-efficient resource allocation for coordinated downlink OFDMA networks, where both radiated and circuit power matter. It optimizes three energy-efficiency objectives through scheduling and power-allocation algorithms under transmit-power constraints, including a noise-limited special case. Results show that low-power energy-efficiency optimization approximately aligns with spectral-efficiency maximization, while higher-power operation can trade moderate rate reductions for substantial energy savings and increasing out-of-cluster interference narrows strategy differences.

  • Problem

    Environmental and economic concerns motivate accounting for energy efficiency in cellular access networks alongside capacity-oriented resource allocation.

  • Method

    The paper optimizes GEE, Sum-EE, and Prod-EE through coordinated user scheduling and power allocation under per-subcarrier or per-base-station power constraints.

  • Results

    At low maximum transmit powers, energy-efficiency maximization approximately matches spectral-efficiency maximization; at higher powers, moderate rate reductions can produce substantial energy savings.

  • Takeaways & Limitations

    Sum-EE and Prod-EE offer more design flexibility than GEE, with Prod-EE encouraging more balanced use of available subcarriers.

Abstract

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This paper addresses the problem of energy-efficient resource allocation in the downlink of a cellular OFDMA system. Three definitions of the energy efficiency are considered for system design, accounting for both the radiated and the circuit power. User scheduling and power allocation are optimized across a cluster of coordinated base stations with a constraint on the maximum transmit power (either per subcarrier or per base station). The asymptotic noise-limited regime is discussed as a special case. %The performance of both an isolated and a non-isolated cluster of coordinated base stations is examined in the numerical experiments. Results show that the maximization of the energy efficiency is approximately equivalent to the maximization of the spectral efficiency for small values of the maximum transmit power, while there is a wide range of values of the maximum transmit power for which a moderate reduction of the data rate provides a large saving in terms of dissipated energy. Also, the performance gap among the considered resource allocation strategies reduces as the out-of-cluster interference increases.

I. INTRODUCTION

The paper studies energy-efficient scheduling and power control in coordinated downlink OFDMA networks, considering multiple energy-efficiency objectives and transmit-power constraints. It develops optimization procedures and evaluates how energy efficiency, spectral efficiency, and interference affect resource-allocation performance.

  • System and motivation: The system uses coordinated base stations sharing channel-quality information to schedule users and control power over the same radio spectrum.Each user is associated with one base station, and each base station serves at most one user per subcarrier.
  • Energy-efficiency objectives: Three energy-efficiency objectives are considered: global energy efficiency (GEE), Sum-EE, and Prod-EE.They respectively emphasize network-wide sum-rate per power, weighted per-resource energy efficiency, and an exponentially weighted product across resource slots.
  • Optimization methods: The proposed procedures optimize these objectives under maximum transmit-power constraints applied either per subcarrier or per base station.GEE uses concave-convex fractional relaxations, Prod-EE uses concave relaxations, and Sum-EE uses an iterative method for the KKT conditions.
  • Numerical findings: At low maximum transmit powers, the considered objectives have similar performance, and energy-efficiency maximization approximately matches spectral-efficiency maximization.Numerical results identify this regime as one where radiated power is small relative to static consumption and interference is small relative to noise.
  • Numerical findings: At higher maximum transmit powers, a moderate spectral-efficiency reduction can yield substantial energy savings, while Sum-EE and Prod-EE provide more control over individual base-station efficiency than GEE.Prod-EE also promotes a more balanced use of available subcarriers, whereas optimized solutions can stop using excess power to increase rate.

B. Power model

The paper models energy efficiency using radiated amplifier power and constant circuit power, then optimizes scheduling and power allocation under coordinated-BS constraints. It considers three energy-efficiency objectives and develops iterative optimization procedures for per-BS or per-subcarrier power limits.

  • Power model: Consumed power combines amplifier dissipation, modeled as γp, with constant RF-circuit dissipation θ.The scaling coefficient γ accounts for amplifier and feeder losses.
  • Energy-efficiency objectives: The three objectives are global energy efficiency, weighted Sum-EE, and exponentially weighted Prod-EE.They capture different aspects of energy efficiency across the coordinated cluster.
  • Energy-efficiency objectives: Prod-EE forces every subcarrier to be used by every base station and produces more balanced subcarrier power allocation.This differs from GEE or Sum-EE, which may leave some subcarriers unused.
  • System design: The algorithms assume perfect instantaneous channel state information and use a centralized controller to assign users and transmit powers.Long-term channel-based allocation is outside the paper’s scope.
  • GEE optimization: For GEE, iterative lower-bound optimization and Dinkelbach’s procedure monotonically improve the objective and converge to a KKT point.The lower bound is tightened during alternating power-allocation and user-selection updates.

1) Noise-limited (NL) regime:

In the noise-limited regime, neglecting intercell interference simplifies GEE optimization. The proposed procedure alternates user selection and power optimization, with the latter solved through waterfilling-like problems and bisection.

  • Noise-limited formulation: Neglecting intercell interference simplifies GEE into the noise-limited objective GEE-NL.This regime is relevant when intercell interference is weak enough to be neglected.
  • Optimization result: Algorithm 3 monotonically improves GEE-NL and provides a globally optimal solution to the noise-limited problem.The result follows from the concave-linear fractional formulation and its associated procedures.
  • Algorithm: The algorithm iterates between solving for powers and recomputing scheduled users until convergence or the iteration limit Imax.Dinkelbach’s procedure solves the fractional power subproblem.
  • Power optimization: Power updates reduce to M waterfilling-like problems, with each nonnegative Lagrange multiplier obtained by bisection search.The decomposition is performed per base station.

B. Per-subcarrier power constraint

Under per-subcarrier power constraints, GEE no longer decouples across subcarriers, but the paper adapts its earlier derivations and solves the resulting power conditions through fixed-point iterations.

  • Constraint structure: With per-subcarrier constraints, GEE is not separable across subcarriers.The constraint limits the maximum power radiated by each base station on each subcarrier.
  • Adapted optimization: The per-BS GEE algorithms remain applicable after redefining the feasible set for the per-subcarrier power limits.Algorithms 1 and 2 are retained with a modified feasible set.
  • Fixed-point solution: The stationary condition is recast so that its right-hand side becomes a standard interference function.This structure enables an iterative fixed-point solution.
  • Fixed-point solution: The optimal power variables are obtained by iteratively solving the fixed-point equations from any feasible power allocation.The updates are applied for every base station and subcarrier.

IV. OPTIMIZATION OF SUM-EE

For Sum-EE, the paper derives alternating updates for user scheduling and power allocation under coordinated-base-station constraints. The method uses equivalent rate weights, marginal power costs, and interference-leakage penalties, but general convergence conditions remain unavailable.

  • Problem formulation: Sum-EE optimization is separable in the power variables, so the results extend straightforwardly to per-subcarrier power constraints.The section also treats the noise-limited scenario separately.
  • Alternating optimization: For fixed powers, user selection is separable across base stations and subcarriers.For fixed user selection, optimal powers satisfy the stated KKT conditions.
  • Power allocation: The power update allocates more power to subcarriers with larger equivalent weights and better channel conditions.Equivalent weights incorporate rate and power-consumption effects.
  • Power allocation: Marginal power costs and interference leakage reduce radiated power when transmission is expensive or harms other co-channel users.These taxation terms encode local power prices and leakage to undesired receivers.
  • Algorithm: The iterative procedure updates interference, powers, and scheduled users, with waterfilling-like subproblems solved by bisection.Lagrange multipliers are selected to satisfy power constraints and complementary slackness.
  • Convergence: General convergence conditions for Algorithm 5 appear intractable, although experiments always observed convergence without the monotonicity modification.If convergence occurs, the resulting variables satisfy the KKT conditions by construction.

A. Insights into Algorithm 5

The method updates each base station’s power allocation through a concave maximization incorporating rate, energy-efficiency, and interference costs. The resulting power levels can be obtained with convex optimization tools.

  • A. Insights into Algorithm 5: For fixed scheduled users, feasible power levels must satisfy the KKT conditions of one optimization problem per base station.The formulation keeps user scheduling fixed while determining power levels that satisfy the relevant constraints and stationarity conditions.
  • A. Insights into Algorithm 5: The objective combines weighted rates with costs for power consumption and interference caused to other co-channel users.The power-consumption cost represents energy efficiency, while the interference cost accounts for leakage to other users.
  • A. Insights into Algorithm 5: The power update lowers radiated power when the marginal power price or interference leakage becomes large.These two terms respectively penalize energy consumption and excessive interference to co-channel users.
  • A. Insights into Algorithm 5: Because the per-base-station problem is a concave maximization, its multiplier and power levels can be computed with any convex optimization tool.This provides an alternative way to obtain the quantities used in Algorithm 5.

B. Noise-limited regime

In the noise-limited regime, neglecting intercell interference simplifies Sum-EE and enables alternating optimization over user scheduling and power allocation. The relaxed power problem has a unique solution for fixed scheduling.

  • B. Noise-limited regime: Neglecting intercell interference simplifies Sum-EE and makes the resulting objective separable in power and user scheduling.The simplification applies specifically to the noise-limited regime.
  • B. Noise-limited regime: For fixed power, user scheduling can be optimized through the decoupled scheduling problem.The supplied passages state that the objective separates with respect to both power and scheduling variables.
  • B. Noise-limited regime: For fixed scheduling, the relaxed power-allocation problem has a unique solution obtained by separately maximizing each objective summand.The unique solution is denoted by ¯p.
  • B. Noise-limited regime: Lemma 2 recasts the noise-limited power-allocation problem as a concave problem.The concave reformulation supports tractable optimization in this special regime.
  • B. Noise-limited regime: The optimal resource allocation is found by alternating maximization over scheduling and power according to the stated update rules.Scheduling and power are optimized successively using the corresponding subproblem solutions.

V. OPTIMIZATION OF PROD-EE

Prod-EE optimization uses alternating updates for power, user selection, and a tightened bound under a per-base-station power constraint. The procedure monotonically improves Prod-EE and converges to a KKT point.

  • V. OPTIMIZATION OF PROD-EE: Prod-EE is optimized under a per-base-station power constraint, with per-subcarrier constraints handled straightforwardly because Prod-EE is separable in power.The section explicitly studies the per-base-station case.
  • V. OPTIMIZATION OF PROD-EE: The optimization first fixes the scheduling variables and derives the corresponding power-allocation problem and lower bound.The logarithm of the objective is used without loss of optimality, and a lower bound is introduced for power optimization.
  • V. OPTIMIZATION OF PROD-EE: The power variables are transformed using p = exp{q}, producing a relaxed power-allocation problem solved as a concave maximization.The algorithm computes q from the concave problem and then updates p.
  • V. OPTIMIZATION OF PROD-EE: Algorithm 6 alternates power optimization, best-user selection, and tightening of the bound until convergence.The power update uses problem (38), while user selection uses problem (36).
  • V. OPTIMIZATION OF PROD-EE: Algorithm 6 monotonically improves Prod-EE at each iteration and converges to a solution satisfying the KKT conditions for the original problem.The KKT conditions are first-order necessary under Slater’s constraint qualification.
  • V. OPTIMIZATION OF PROD-EE: In the noise-limited regime, the objective simplifies and the derivations from the earlier noise-limited analysis can be replicated.The same structural simplification is applied to the Prod-EE formulation.

VI. NUMERICAL RESULTS

The numerical study evaluates coordinated base stations on 16 OFDMA subcarriers under fading, shadowing, path loss, thermal noise, and out-of-cluster interference. Results are averaged over 1000 independent user drops, with GEE examined against power and interference conditions.

  • VI. NUMERICAL RESULTS: The simulated network has three coordinated base stations, 16 subcarriers, and three uniformly distributed users served by each base station.Each subcarrier has bandwidth B = 180 kHz.
  • VI. NUMERICAL RESULTS: The channel model includes Rayleigh fading, 8 dB Log-Normal shadowing, and a fourth-power path-loss model at a 1800 MHz carrier frequency.The reference distance is d0 = 100 m.
  • VI. NUMERICAL RESULTS: Noise variance accounts for both thermal noise and out-of-cluster interference from uncoordinated base stations.The interference model uses the average per-subcarrier radiated power Pout of those base stations.
  • VI. NUMERICAL RESULTS: Pout = 0 represents an isolated cluster of coordinated base stations.Nonzero Pout models interference from uncoordinated base stations outside the cluster.
  • VI. NUMERICAL RESULTS: The plots use a per-subcarrier power constraint and average results over 1000 independent user drops.A per-base-station constraint showed similar behavior in the experiments and is not illustrated.
  • VI. NUMERICAL RESULTS: Figure 2 plots GEE against Pmax when Pout = 0 and against Pout when Pmax = 35 dBm.The two relationships are shown in the left and right panels, respectively.

A. Implementation of the proposed algorithms

The evaluation compares energy-efficiency, sum-rate, and radiated-power outcomes across optimization strategies, cluster settings, and interference levels. The algorithms converge in few iterations, while Prod-EE produces more balanced subcarrier energy efficiencies.

  • Performance comparison: All considered resource-allocation strategies perform similarly for Pmax ≤ 10 dBm in an isolated cluster.At low maximum transmit power, radiated-power consumption is negligible relative to static consumption and cochannel interference is small.
  • Performance comparison: For increasing Pmax, energy-efficiency optimizers reach a performance floor because they do not use excess power to further increase the rate.The corresponding per-BS radiated-power behavior is shown in Figure 6.
  • Interference effects: Increasing Pout degrades GEE, Sum-EE, Prod-EE, and sum-rate, while reducing the performance gap among the considered solutions.As out-of-cluster interference becomes dominant, coordinated resource allocation becomes less beneficial.
  • Interference effects: GEE-opt, Sum-EE-opt, and Prod-EE-opt progressively use a larger fraction of available power as out-of-cluster interference increases.This behavior counteracts the increased interference level.
  • Subcarrier balance: Prod-EE-opt yields less dispersed energy efficiency across individual subcarriers than GEE-opt and Sum-EE-opt.This confirms that Prod-EE maximization provides a more balanced use of the available subcarriers.
  • Convergence: All algorithms reach a steady value in few iterations across isolated and non-isolated scenarios.Convergence slows as Pmax increases and as Pout decreases.

C. Influence of the weights

The paper examines how weighting energy-efficiency objectives affects coordinated base-station resource allocation, alongside scalability and convergence behavior. Higher priority weights can rebalance per-BS efficiency, while the proposed algorithms converge in few iterations and performance changes only marginally with proportionally scaled subcarriers.

  • Influence of the weights: Higher priority weights can shift Sum-EE resource allocation toward selected base stations and produce a more balanced allocation.Increasing BS3's priority and decreasing BS1's priority improves balance, while BS2 remains approximately unchanged when its weight is fixed.
  • Impact of the number of subcarriers and users: With linearly scaled maximum transmit power, increasing the number of subcarriers only marginally affects system performance.The sum-rate and power consumption increase proportionally, leaving GEE and per-subcarrier energy efficiencies approximately unchanged.
  • Impact of the number of subcarriers and users: Increasing the number of users tends to improve network-wide performance through multiuser diversity, while reducing average resources per user.
  • Algorithms' complexity: The proposed algorithms converge to steady values in only 5–15 iterations, depending on the operating scenario.A single iteration's complexity is mainly tied to transmit-power optimization, and complexity generally grows with coordinated base stations and active users.
  • Conclusions: Sum-EE and Prod-EE provide weight-based design flexibility, while Prod-EE promotes more balanced use of available spectrum than GEE.

APPENDIX

The appendix develops optimization and convergence arguments for the proposed energy-efficiency algorithms. It uses concavity, pseudo-concavity, tight relaxations, and Dinkelbach-based updates to establish monotonic improvement or first-order optimality properties under stated conditions.

  • Relaxation properties: The concavity and convexity used in the relaxations follow from the convexity of the log-sum-exp function.
  • GEE optimization: The GEE procedure converges because each update does not decrease the objective, while GEE is bounded above.
  • Optimality conditions: At convergence, Algorithm 1 satisfies the first-order optimality conditions through a tight relaxation and suitable Lagrange multipliers.
  • Noise-limited regime: GEE-NL is strictly pseudo-concave in transmit power, enabling alternating optimization of user selection and power allocation.
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