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Energy-Efficient Resource Allocation for Wireless Powered Communication Networks

Qingqing Wu, Meixia Tao, Derrick Wing Kwan Ng, Wen Chen, Robert Schober

arXiv:1511.05539v1cs.ITmath.ITmath.OC

TL;DR

The paper studies energy-efficient time and power allocation in wireless powered communication networks, including initial user battery energy and a minimum throughput requirement. It derives scheduling structures for best-effort communication and an iterative optimization method for the throughput-constrained case.

  • Problem

    The paper studies how to jointly allocate time and control power to maximize WPCN energy efficiency while accounting for user energy and, in one case, minimum system throughput.

    Source: §III · pp. 8–9, B. Transformation of the Objective… · p. 16

  • Method

    For best-effort communication, the paper reduces the problem to two simplified networks; for throughput-constrained communication, fractional programming and KKT analysis yield a convex formulation and an iterative algorithm.

    Source: §I · p. 4

  • Results

    Best-effort energy efficiency is equivalent to that of either an initial-energy-limited or purely wireless-powered network; under a throughput constraint, long transmission time favors each user’s maximum EE, while short time requires sacrificing EE to meet throughput.

    Source: §I · p. 4

  • Takeaways & Limitations

    The optimal allocation structure depends on the communication regime: best-effort scheduling follows the simplified-network structures, while throughput-constrained allocation balances EE against the minimum throughput.

    Source: §I · p. 4

Abstract

from arXiv · show

This paper considers a wireless powered communication network (WPCN), where multiple users harvest energy from a dedicated power station and then communicate with an information receiving station. Our goal is to investigate the maximum achievable energy efficiency (EE) of the network via joint time allocation and power control while taking into account the initial battery energy of each user. We first study the EE maximization problem in the WPCN without any system throughput requirement. We show that the EE maximization problem for the WPCN can be cast into EE maximization problems for two simplified networks via exploiting its special structure. For each problem, we derive the optimal solution and provide the corresponding physical interpretation, despite the non-convexity of the problems. Subsequently, we study the EE maximization problem under a minimum system throughput constraint. Exploiting fractional programming theory, we transform the resulting non-convex problem into a standard convex optimization problem. This allows us to characterize the optimal solution structure of joint time allocation and power control and to derive an efficient iterative algorithm for obtaining the optimal solution. Simulation results verify our theoretical findings and demonstrate the effectiveness of the proposed joint time and power optimization.

I. INTRODUCTION

The paper targets energy efficiency in multiuser WPCNs, where prior work largely prioritizes throughput and existing EE methods do not directly address the network’s coupled time and power allocation, circuit consumption, and initial user energy. It models a harvest-then-transmit network and develops EE optimization for best-effort and throughput-constrained operation.

I. INTRODUCTION

  • Most prior WET work emphasizes throughput, while existing EE approaches do not directly handle the WPCN’s distinct architecture, coupled time and power allocation, and non-negligible circuit and WET energy costs.The paper therefore optimizes system EE while accounting for initial user energy and minimum-throughput requirements.Source: §I · pp. 2–4
  • For best-effort communication, the EE problem reduces to either an initial-energy-only network, where only the highest-EE user is scheduled, or a WET-only network with structured scheduling.In the WET-only case, the power station uses maximum power, scheduled users deplete their energy, and all available time is used; the system EE has a closed-form expression based on user EEs.Source: §I · p. 4
  • For throughput-constrained communication, fractional programming yields a convex optimization problem whose KKT analysis characterizes optimal time and power allocation and supports an efficient iterative algorithm.With sufficiently long transmission time, each user achieves its own maximum EE; with short time, users meet minimum throughput at the cost of system EE.Source: §I · p. 4

A. System Model

  • The system uses a harvest-then-transmit protocol: multiple users harvest power-station RF energy in the downlink, then transmit in time division to a separately located information receiver.Users have initial battery energy Q_k, which may come from earlier transmission blocks and can be used for current-block transmission.Source: A. System Model · p. 5

B. Power Consumption Model

  • The energy model includes transmitter circuit consumption as well as over-the-air power during downlink energy transfer and uplink information transmission.The power station’s circuit energy during downlink transfer is P_cτ_0; user energy consumption includes power-amplifier and circuit consumption.Source: B. Power Consumption Model · p. 7

C. User Energy Efficiency

The paper defines user energy efficiency for WPCN transmission and uses it to analyze best-effort system energy efficiency. The best-effort problem jointly optimizes downlink and uplink time allocation and power control while accounting for available energy.

C. User Energy Efficiency

  • User energy efficiency is each user's achievable throughput divided by its energy consumed during wireless information transmission.The consumed energy includes power-amplifier and electronic-circuit energy.Source: C. User Energy Efficiency · p. 8
  • A user's energy efficiency is strictly quasiconcave in transmit power and has a unique stationary point that is its maximum.The optimal transmit power is obtained by setting the derivative of user energy efficiency to zero.Source: C. User Energy Efficiency · p. 8

III. ENERGY-EFFICIENT RESOURCE ALLOCATION FOR BEST-EFFORT WPCN

  • The best-effort WPCN objective is system throughput divided by total energy, optimized jointly over downlink and uplink time allocation and power control.The formulation accounts for harvested energy and users' initial energy, along with the total transmission-time constraint.Source: §III · pp. 8–9

A. Equivalent Optimization Problems

For best-effort WPCN, the paper reduces energy-efficiency maximization to one of two simplified networks, then characterizes optimal resource use in the purely wireless-powered case. These results identify which users to schedule and how system parameters affect that choice.

A. Equivalent Optimization Problems

  • Best-effort WPCN energy-efficiency maximization is equivalent to solving either a purely wireless-powered network or an initial-energy-limited network.The optimal solution to the original problem is provided by whichever simplified problem yields the larger system energy efficiency.Source: A. Equivalent Optimization Problems · pp. 9–10

B. Properties of Energy-Efficient PWPCN

  • In the purely wireless-powered network, the power station transmits at maximum allowed power and the full available transmission time is used.Using the full time can maintain system energy efficiency while improving throughput.Source: B. Properties of Energy-Efficient PWPCN · p. 11
  • A user is scheduled in the purely wireless-powered network only when its user energy efficiency is at least the system energy efficiency; users with strictly higher user energy efficiency use all their energy.The paper contrasts this energy-efficiency-oriented selection with throughput maximization, where all users may be scheduled despite weak channels and added circuit consumption.Source: B. Properties of Energy-Efficient PWPCN · p. 12
  • The optimal system energy efficiency for the purely wireless-powered network is expressed using the optimal scheduled-user set and user energy efficiencies.Once the scheduled set is determined, the optimal allocation follows from the expression and associated solution.Source: B. Properties of Energy-Efficient PWPCN · pp. 12–13
  • The number of scheduled users increases with power-station circuit power and maximum transmit power, and decreases with energy conversion efficiency.These trends connect scheduling to power-station parameters and user-terminal energy conversion.Source: B. Properties of Energy-Efficient PWPCN · p. 13

C. Properties of Energy-Efficient IELCN

In the initial-energy-limited network, optimal energy efficiency is achieved by scheduling only the user with the highest user energy efficiency. The resulting value can be obtained from the user-energy-efficiency definition and combined with the purely wireless-powered solution to solve the original best-effort problem.

C. Properties of Energy-Efficient IELCN

  • The initial-energy-limited problem is equivalent to an optimization whose optimal solution is given by the paper's stated allocation.The theorem identifies the initial-energy-limited network as one of the two simplified alternatives for the original best-effort problem.Source: C. Properties of Energy-Efficient IELCN · p. 14
  • Only the user with the highest user energy efficiency is scheduled in the initial-energy-limited network.The maximum system energy efficiency can therefore be obtained using the user energy efficiencies defined earlier.Source: C. Properties of Energy-Efficient IELCN · p. 14
  • The original best-effort problem's optimal solution is the one of the initial-energy-limited and purely wireless-powered solutions with larger system energy efficiency.This follows from the two-network equivalence result.Source: C. Properties of Energy-Efficient IELCN · p. 14

IV. ENERGY-EFFICIENT RESOURCE ALLOCATION FOR WPCN WITH A QOS CONSTRAINT

With a minimum throughput requirement, the paper first tests feasibility and then transforms the energy-efficiency problem for solution by convex optimization and an iterative algorithm. The optimal policy depends on available transmission time and the users' energy constraints.

A. Feasibility of Problem (20)

  • The throughput-constrained energy-efficiency problem is feasible exactly when the required throughput does not exceed the maximum throughput R*.R* is the maximum objective value of a concave optimization problem; feasibility can be checked using standard optimization methods.Source: A. Feasibility of Problem (20) · p. 15

B. Transformation of the Objective Function

  • The minimum-throughput requirement can force use of both wireless energy transfer and users' initial energy, so the problem cannot generally be reduced to either simplified network.The original problem is non-convex because of its fractional objective and non-convex constraints.Source: B. Transformation of the Objective… · p. 16
  • Dinkelbach's iterative subtractive-form transformation updates the energy-efficiency value, and substituting each user's consumed energy for power-times-time yields a standard convex optimization problem.The paper analyzes the convex problem with KKT conditions to obtain an optimal solution.Source: B. Transformation of the Objective… · pp. 16–17

C. Iterative Algorithm for Energy Efficiency Maximization

  • The proposed algorithm has complexity O(K), linear in the number of users.The per-user steps scale linearly with K, while Dinkelbach and bisection updates are independent of K.Source: C. Iterative Algorithm for Energy… · p. 20
  • With sufficiently long transmission time, each scheduled user transmits at the power maximizing its own energy efficiency; short available time may require higher power that sacrifices system energy efficiency to meet throughput.Users not scheduled without the throughput requirement may also need to be scheduled to satisfy it.Source: C. Iterative Algorithm for Energy… · pp. 20–21
  • When a scheduled user has unused initial energy and wireless energy transfer is active, higher required throughput keeps scheduled users' powers constant until that user's energy is exhausted, while shifting time toward that user.Energy-transfer time and other scheduled users' transmission times decrease; using substantial initial energy can be preferable to prolonging downlink energy transfer because transfer incurs circuit energy and reduces uplink time.Source: C. Iterative Algorithm for Energy… · p. 21

V. NUMERICAL RESULTS

The numerical evaluation uses a five-user WPCN with specified channel, circuit-power, and transmission parameters. Figure 2 reports system EE against outer-layer iterations for different minimum-throughput settings.

V. NUMERICAL RESULTS

  • The simulations use five uniformly distributed users, a 20 kHz bandwidth, and a 300 m separation between the power station and information receiving station.Users are distributed within a 15 m maximum service distance, with a 2 m reference distance.Source: §V · p. 21
  • The default settings are η = 0.9, Tmax = 1 s, and Pmax = 43 dBm, with different throughput setups used in Figures 2 and 7.The model also sets power-station and user-terminal circuit consumption to 500 mW and 5 mW, respectively.Source: §V · p. 21
  • Figure 2 plots system EE against the proposed algorithm’s outer-layer iteration count for different minimum system throughput requirements.The supplied caption does not specify the throughput values.Source: Figure 2 · p. 22

A. Convergence of Proposed Algorithm

The numerical results support the algorithm’s fast convergence and illustrate how energy efficiency and scheduling respond to system parameters and throughput requirements. The conclusions distinguish best-effort operation from throughput-constrained operation and describe the corresponding optimal strategies.

A. Convergence of Proposed Algorithm

  • At most six outer-layer iterations are needed on average to reach the optimal solution, and the inner bisection optimization also converges quickly.The authors state that the proposed algorithm is guaranteed to converge quickly.Source: A. Convergence of Proposed Algorithm · p. 22

B. System EE of WPCN: PWPCN versus IELCN

D. Number of Scheduled Users versus Energy Harvesting Efficiency

  • The number of scheduled users is nondecreasing as energy-harvesting efficiency η rises, and more users are scheduled when Pc is larger.The authors attribute the η trend to reduced energy loss and higher system EE, which favors more conservative scheduling.Source: D. Number of Scheduled Users versus… · p. 24, Figure 6 · p. 25

E. System EE versus Minimum Throughput Requirement

  • As the minimum throughput Rmin rises, system EE first stays constant and then decreases; EE increases with user count K, but falls faster at higher K.The paper attributes the K effect to reduced relative DL energy loss and greater UL multiuser diversity; at high throughput requirements, energy use is more sensitive to throughput changes for larger K.Source: E. System EE versus Minimum Throughput… · pp. 24–25, Figure 7 · p. 25

VI. CONCLUSIONS

  • For sufficiently long transmission time, the system maximizes EE when each user achieves its own maximum user EE; with too little time, meeting the throughput requirement costs system EE.The paper reports an efficient algorithm for the throughput-constrained problem.Source: §VI · p. 25
  • For best-effort WPCN, the EE problem reduces to either PWPCN or IELCN: PWPCN scheduling grows with circuit power and falls with user-side energy-conversion efficiency, while IELCN schedules only the highest-user-EE user.Under a minimum throughput constraint, short transmission time requires sacrificing system EE to meet the requirement.Source: §VI · p. 25

APPENDIX A: PROOF OF THEOREM 1

The proof establishes that best-effort WPCN EE is attained by one of two simplified networks, PWPCN or IELCN. It then supports the corresponding conclusions about the optimizer and feasible operating time.

APPENDIX A: PROOF OF THEOREM 1

  • The proof separates the original problem into cases where only WET-powered users transmit, only initially powered users transmit, or both groups transmit.In the first two cases, the original problem reduces to PWPCN or IELCN, respectively; the mixed case is compared against their optima.Source: Appendix A · p. 26, PWPCN, EE∗ · p. 27

PWPCN, EE∗

  • The mixed case cannot exceed the better simplified-network optimum; equality is possible only when the PWPCN and IELCN maximum EEs are equal.The proof concludes that either PWPCN or IELCN is optimal.Source: PWPCN, EE∗ · p. 27

APPENDIX B: PROOF OF LEMMA 1

  • Lemma 1’s proof constructs a feasible alternative with P0 = Pmax and argues that it would have higher EE, contradicting optimality of the assumed solution.The construction uses the actual energy transmitted by the power station, E0 = P0τ0.Source: Appendix B · p. 28

APPENDIX C: PROOF OF LEMMA 2

The appendix proves that an optimal solution can use the full available time and supports a structural claim about scheduled users in PWPCN. The proof of Lemma 3 uses a constructed alternative to test whether the assumed solution can be improved.

APPENDIX C: PROOF OF LEMMA 2

  • The proof of Lemma 2 constructs a feasible solution using all available time, Tmax, with the same optimal system EE.Source: Appendix C · p. 28

APPENDIX D: PROOF OF LEMMA 3

  • For Lemma 3, the proof tests an assumed PWPCN optimum by changing energy-transfer and user-transmission times, then compares the resulting EE using an auxiliary lemma.The proof states that the remaining parts follow by a similar procedure.Source: Appendix D · p. 29

APPENDIX E: PROOF OF THEOREM 2

The proofs reduce energy-efficiency optimization to user-level power choices and establish that the maximum system EE is achieved by scheduling a single user.

APPENDIX E: PROOF OF THEOREM 2

  • For a given scheduled-user set, maximizing network EE reduces to maximizing each user's EE through its transmit power.The proof also states that scaling τ0 within the feasible region does not change PWPCN system EE.Source: Appendix E · p. 30

APPENDIX F: PROOF OF THEOREM 3

  • The maximum system EE is always achieved by scheduling a single user.The proof then applies the optimal power in the time and energy-harvesting constraints; IELCN system EE is unaffected by τk.Source: Appendix F · p. 30

APPENDIX G: PROOF OF THEOREM 6

The proof uses the monotonicity of f(γk, µk) to characterize time and energy allocation by users' channel gains relative to a common threshold.

APPENDIX G: PROOF OF THEOREM 6

  • Lemma 6 states that f(γk, µk) increases as γk increases and decreases as µk increases; the cited passage leaves its qualifying condition unspecified.Source: Appendix G · p. 32
  • The threshold x* is identical across users because the parameters defining it do not depend on user index k.It is characterized by f(x*, 0) = 0.Source: Appendix G · p. 32
  • Users with γk below x* receive zero transmission time.The proof states τk = 0 for this case.Source: Appendix G · p. 32
  • For γk above x*, scheduled users use all their energy, while users at x* may use any portion of their energy.The proof derives these cases from the complementary-slackness conditions.Source: Appendix G · p. 32

APPENDIX H: PROOF OF COROLLARY 2

The proofs characterize optimal transmit powers and show how a tighter throughput requirement changes time allocation and energy harvesting.

APPENDIX H: PROOF OF COROLLARY 2

  • When available transmission time is not fully used, the proof obtains the optimal transmit power p* from the unique solution of an increasing function D(pk).D(pk) is increasing, with a negative value at pk = 0 and a limit of +∞ as pk grows.Source: Appendix H · p. 33

APPENDIX I: PROOF OF COROLLARY 3

  • When a user's energy is not exhausted, scheduled-user powers depend only on system parameters and that user's channel gain.In this case, the transmit power remains constant.Source: Appendix I · p. 34
  • When a user's energy is exhausted, the transmit power varies with its energy multiplier µm.The proof introduces µm in the power condition for this case.Source: Appendix I · p. 34
  • When WET is active and a user's energy is not exhausted, the total available time is used and the required system throughput is satisfied.The stated conditions are τ0 > 0 and unused energy for the user.Source: Appendix I · p. 34
  • As Rmin increases, the stricter time constraint reduces τ0 and the energy harvested by each user.The proof also states that the information-transmission time increases.Source: Appendix I · p. 34

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