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Energy Efficient UAV Communication with Energy Harvesting

Zhaohui Yang, Wei Xu, Mohammad Shikh-Bahaei

arXiv:1911.00351v2eess.SP

TL;DR

The paper addresses minimizing UAV energy consumption while meeting users’ minimum data demands in an energy-harvesting communication system. It decomposes the problem into path planning and fixed-path energy minimization, optimizing user communication and UAV height. Simulations show that low altitude and half-duplex operation are preferred under the studied model, while broader 3D trajectory optimization remains future work.

  • Problem

    The paper asks how to minimize total UAV energy consumption while fulfilling users’ minimum data-transmission requirements in an energy-harvesting communication system.

  • Method

    The optimization is decomposed into path planning and fixed-path energy minimization, with trajectory, communication allocation, and UAV height optimized using dual, successive-convex-approximation, and one-dimensional search methods.

  • Results

    The UAV should hover directly above users at low height, and half-duplex operation is recommended over full-duplex operation.

  • Takeaways & Limitations

    Under the studied model, propulsion-aware height and trajectory design are central to reducing UAV energy consumption while meeting user data demands.

Abstract

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This paper investigates an unmanned aerial vehicle (UAV)-enabled wireless communication system with energy harvesting, where the UAV transfers energy to the users in half duplex or full duplex, and the users harvest energy for data transmission to the UAV. We minimize the total energy consumption of the UAV while accomplishing the minimal data transmission requests of the users. The original optimization problem is decomposed into two subproblems: path planning subproblem and energy minimization subproblem with fixed path planning. For path planning subproblem, the optimal visiting order is obtained by using the dual method and the trajectory is optimized via the successive convex approximation technique. For energy minimization subproblem with fixed path planning, we firstly obtain the optimal portion of data transmission time within the entire procedure and the optimal transmission power of each user. Then, the the energy minimization subproblem is greatly simplified and it is efficiently solved via a one-dimensional search method. Simulation results are illustrated to verify the theoretical findings.

I. INTRODUCTION

The paper studies energy-efficient UAV communication with energy harvesting, focusing on propulsion-aware trajectory, height, and communication design. It models a fly-hover-communicate mission in which the UAV collects data from multiple users while transferring wireless energy.

  • System model: The system uses a rotary-wing UAV as a data collector that wirelessly charges users before their uplink transmissions.The UAV serves multiple users and gathers their information data.
  • Problem formulation: The model jointly optimizes UAV trajectory, user transmission power, and mission completion time under half-duplex and full-duplex communication.The paper formulates energy minimization while meeting users’ data requirements.
  • Mission procedure: The mission visits users sequentially, returns to the initial point, and repeats the collection procedure as needed.The visiting order is denoted by π1 through πK, followed by a return stage.
  • Fly-Hover-Communicate Protocol: The fly-hover-communicate protocol flies near each user, broadcasts energy while collecting data, then changes altitude for the next user.Each stage includes straight flight, descent, hovering communication, and ascent.
  • Energy model: Straight flight is used between points because the straight line is the shortest path and is therefore energy-saving for propulsion.The paper applies this assumption to the UAV’s inter-user movement.
  • Energy model: Propulsion energy dominates communication-related energy in the adopted model, so communication-related energy is ignored.The propulsion model includes straight flight, vertical descent, hovering, and vertical climb components.

1) HD Mode:

The HD mode separates wireless energy transfer from uplink data transmission within each user’s communication interval. Its model accounts for channel randomness, harvested energy, user power consumption, and an approximate data-collection constraint.

  • HD operation: In HD mode, the UAV transfers energy for ρktk3, followed by user transmission for (1 −ρk)tk3.ρk is the time-splitting factor for user πk.
  • Channel model: The UAV–user channel models LoS and non-LoS attenuation together with exponentially distributed small-scale fading.The expected channel power gain averages over fading and LoS occurrence.
  • Energy harvesting: Harvested energy depends on the wireless charging interval and energy-harvesting efficiency ζ.The efficiency satisfies 0 < ζ < 1.
  • Uplink transmission: The user’s uplink data depends on its average transmit power during the transmission interval and the modeled random channel gain.The data expression uses the HD transmission duration and channel statistics.
  • Optimization constraint: The target data constraint is approximated by requiring rHDπk ≥ Dπk, where Dπk is user πk’s minimum required data.The paper reports satisfactory accuracy for this approximation based on numerical results in prior work.
  • Comparison with FD: Unlike HD, FD transfers energy and uploads data simultaneously, while its model includes self-interference and processing delay.The effective self-interference coefficient is γ and the user processing delay is δπk.

B. Problem Formulation

The paper formulates UAV energy minimization under minimal user upload demands, jointly considering trajectory, transmission, timing, height, and HD/FD operation. The formulation separates path planning from fixed-path energy minimization while enforcing flight and energy-harvesting constraints.

  • B. Problem Formulation: The objective minimizes total UAV energy while satisfying each user's minimal data-collection demand.
  • B. Problem Formulation: The formulation models UAV velocity and acceleration through trajectory derivatives, bounded by Vmax and Amax.
  • B. Problem Formulation: Constraints require successful data collection and ensure each user's consumed energy does not exceed harvested energy.
  • B. Problem Formulation: The discrete visiting order affects only first-step trajectories, enabling separate path-planning and fixed-path energy-minimization subproblems.
  • III. PATH PLANNING: The path-planning subproblem is reformulated using inter-point distances because each trajectory depends on dπkπk−1.
  • III. PATH PLANNING: Figure 3 presents velocity versus time for the path-planning case.
  • III. PATH PLANNING: The formulation includes boundary constraints for the UAV trajectory.

A. Solution for Problem (20)

The path-planning problem is converted from a nonlinear integer formulation into a convex-relaxed problem whose optimal visiting order remains integral. A dual iterative algorithm updates path variables and multipliers, with O(LK^3) total complexity.

  • The nonlinear objective is transformed by introducing vkli = w(k−1)lwki to represent consecutive visitation decisions.
  • The transformed problem becomes convex after temporarily relaxing binary visitation variables to wkl ∈ [0, 1].
  • The path-planning theorem expresses the optimal visitation order and auxiliary vector using the problem's dual solution.
  • The dual method obtains optimal path planning while the relaxed visitation variables remain exactly 0 or 1.
  • The algorithm alternates updates of primal path variables and dual variables until the objective converges.
  • O(K^3) complexity arises from updating the auxiliary vector, giving O(LK^3) total complexity over L iterations.

B. Solution for Problem (21)

The trajectory subproblem is made tractable by bounding the optimal flight time, discretizing the trajectory, and applying successive convex approximation. The resulting convex subproblems are solved iteratively until convergence.

  • The path-planning problem is difficult because flight time is unknown and the continuous trajectory contains infinitely many optimization variables.
  • A lemma provides an upper bound on the optimal flight time, allowing the problem to be solved with a fixed feasible duration without loss of optimality.
  • Discretizing flight time into N + 1 slots represents the continuous trajectory by discrete UAV locations, velocities, and accelerations.
  • The discretized problem remains non-convex because its objective contains a non-convex propulsion-energy term.
  • Slack variables and inequality reformulations convert the propulsion-energy expression into a form suitable for successive convex approximation.
  • The SCA algorithm initializes a feasible trajectory, repeatedly solves the convex approximation, and stops when the objective converges.
  • Each SCA iteration replaces non-convex terms with bounds based on first-order expansions, producing a convex problem solved by an interior-point method.

IV. ENERGY MINIMIZATION WITH FIXED PATH PLANNING

With path planning fixed, the energy-minimization problem is decomposed into per-user components and reduced to one-dimensional searches after deriving optimal time and power decisions.

  • The fixed-path energy-minimization problems are solved separately for HD and FD modes.
  • For a specific user, objective and constraint terms decouple, reducing the problem to a user-level energy minimization.
  • The fixed-path problems are represented through optimal objective values V1(hk), V2(hk), and V3(hk).
  • Problems (48) and (51) involve one variable and are solved using a one-dimensional exhaustive search.
  • The feasible solution for problem (49) follows from earlier feasibility relations, while problems (50) and (52) have closed-form optimal solutions.

A. Vertical Trajectory Optimization for Problem (49)

Problem (49) is discretized into time slots and reformulated with auxiliary variables to address its non-convex objective. Successive convex approximation is then used to solve the resulting trajectory optimization problem.

  • A. Vertical Trajectory Optimization for Problem (49): The time durations t_k2 and t_k4 are discretized into N + 1 slots, representing the UAV trajectory through discrete-time locations.The slot step sizes are δ2 = t_k2/N4 and δ4 = t_k4/N4 as indicated in the passage.
  • A. Vertical Trajectory Optimization for Problem (49): The objective formulation is derived from expressions (A.13) and (A.15), preserving the corresponding energy-related terms in the discrete problem.The supplied formulation includes terms involving P2(t_k2 + t_k4), W h_k, and acceleration-related quantities.
  • A. Vertical Trajectory Optimization for Problem (49): Problem (53) is non-convex because of its objective function, so auxiliary variables X_n and Y_n are introduced.These variables support an equivalent reformulation of the optimization problem.
  • A. Vertical Trajectory Optimization for Problem (49): Non-convex products and constraint left-hand sides are approximated using convexification and first-order Taylor expansions, yielding an SCA-based solution algorithm.Constraints (56b) and (56c) are formed by replacing equalities with inequalities before these approximations are applied.

B. HD Mode for Problem (50)

For half-duplex mode, the optimal solution of problem (50) is obtained in closed form. The resulting condition requires the harvested power at each user to exceed its receive power, producing a constraint on UAV height.

  • B. HD Mode for Problem (50): Theorem 2 states that the optimal solution of problem (50) for HD is available in closed form.The theorem identifies the optimal solution associated with the half-duplex formulation.
  • B. HD Mode for Problem (50): The closed-form expression involves the Lambert-W function W(·).The function appears in the expression labeled (61).
  • B. HD Mode for Problem (50): The harvested power at user π_k must be greater than its receive power, which yields a constraint on the UAV height.This condition follows from π_k > 0 and the relation involving ζP_b h^−α.

C. FD Mode for Problem (52)

The numerical results characterize how altitude, data demand, bandwidth, and transmission power affect UAV energy consumption, while comparing HD and FD operation. They also report that HD generally consumes less energy than FD and that low-height operation is favored.

  • Numerical results: The proposed path-planning algorithm achieves the same total energy consumption as exhaustive search, indicating optimal path planning in the comparison.Total energy consumption increases linearly with UAV altitude because higher altitude requires longer energy-broadcast and information-reception changes, increasing propulsion energy.
  • Numerical results: Optimal height decreases as minimal required data D increases, with HD decreasing faster than FD and the HD–FD ordering changing at D = 3 Mbits.For D ≤ 3 Mbits, HD has higher optimal height; for D > 3 Mbits, HD has lower optimal height. The reported optimal height is below 2 m.
  • Numerical results: Energy consumption decreases rapidly and then slowly as system bandwidth increases.At high bandwidth, hovering time is short, so propulsion energy for changing altitude dominates consumption.
  • Numerical results: Total energy consumption increases monotonically with minimal data demand D because higher demand lengthens UAV hovering time.HD is reported as superior to FD, while FD with γ = −120 dB slightly outperforms FD with γ = −100 dB.
  • Numerical results: Energy consumption first decreases dramatically and then decreases smoothly as maximal wireless transmission power P increases, with HD always outperforming FD.The paper attributes FD’s higher consumption to high user transmit-circuit power, which lowers user transmit power, lengthens hovering time, and raises UAV energy use.
  • Conclusion: The conclusion recommends that the UAV stay directly above each user at low height and favors HD over FD for energy minimization.The paper leaves general 3D trajectory optimization with beamforming, instability error, and delay requirements for future work.

APPENDIX A ROTARY-WING UAV PROPULSION ENERGY CONSUMPTION

The appendix derives propulsion-energy models for rotary-wing UAVs in fixed-height straight flight and vertical flight. These models account for aerodynamic forces, acceleration, velocity, and flight-specific power requirements.

  • Overview: The appendix defines the main propulsion-energy notation and derives models for both straight and vertical UAV flight.The appendix begins by listing notation and typical values, then treats fixed-height straight flight, vertical descent, and vertical climb.
  • Fixed-height straight flight: In fixed-height straight flight, rotor thrust, fuselage drag, and aircraft weight determine the longitudinal force balance.The force relations include T sin θ − D = ma and W − T cos θ = 0.
  • Fixed-height straight flight: The straight-flight propulsion model uses speed, acceleration, rotor thrust, and fuselage drag under a constant blade-section drag-coefficient assumption.The resulting total propulsion energy is obtained by integrating the propulsion power over the trajectory duration.
  • Vertical flight: Vertical flight is modeled separately by assuming velocity is nonzero only in the vertical direction.The appendix derives required power and total propulsion energy for both vertical descent and vertical climb.
  • Vertical flight: Vertical-descent and vertical-climb models incorporate weight, acceleration, velocity, and rotor-related power terms.For vertical descent, the acceleration is assumed smaller than gravitational acceleration g = 9.8 m/s2.

APPENDIX B PROOF OF THEOREM 1

The appendix derives optimal solutions by relaxing the dual problem, selecting minimum-cost associations, and solving continuous variables through stationarity and convex reductions. Equality of active constraints then yields closed-form optimal solutions for the subsequent problems.

  • Dual optimization: The relaxed dual problem is formulated with association, nonnegative-variable, and multiplier constraints.The formulation includes w_kl, v_kli, and the multiplier sets β, γ, λ, and µ.
  • Dual optimization: For each l, the optimal association sets w_kl to one for the k with the smallest association coefficient.This follows because the dual objective is a linear combination of the association variables.
  • Dual optimization: The optimal continuous variables v_kli are obtained by setting the objective derivative to zero while enforcing nonnegativity.The resulting solution is identified as the optimal solution to problem (27).
  • Convex reduction: The reduced problem is shown to be convex, and the optimal t_k32 is obtained by solving f′(t_k32) = 0.This reduction leads to the optimal solution of problem (50) stated in Theorem 2.
  • Optimality conditions: For problem (52), constraints (52b) and (52c) hold with equality at the optimum, producing the stated optimal solution.Otherwise, the objective could be decreased further, contradicting optimality.
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