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UAV-Enabled Wireless Power Transfer: Trajectory Design and Energy Optimization

Jie Xu, Yong Zeng, Rui Zhang

arXiv:1709.07590v1cs.IT

TL;DR

The paper studies UAV-enabled WPT using UAVs as mobile energy transmitters and addresses trajectory optimization for multiuser energy transfer. It contrasts single-location hovering for sum-energy maximization with multiple fixed locations and optimized hovering times, reporting enhanced WPT performance over conventional systems.

  • Problem

    Near energy receivers can receive significantly more energy than distant receivers, creating a near-far fairness issue in multiuser WPT.

  • Method

    The paper uses UAVs as mobile energy transmitters and develops trajectory designs based on hovering over multiple fixed locations with optimized hovering-time allocations.

  • Results

    The optimal sum-energy solution has the UAV hover at only one fixed location during the whole charging period, while the proposed trajectory significantly enhances WPT performance over conventional WPT.

  • Takeaways & Limitations

    UAV mobility provides trajectory designs that improve multiuser WPT performance while supporting energy allocation across multiple fixed hovering locations.

Abstract

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This paper studies a new unmanned aerial vehicle (UAV)-enabled wireless power transfer (WPT) system, where a UAV-mounted mobile energy transmitter (ET) is dispatched to deliver wireless energy to a set of on-ground energy receivers (ERs). We investigate how the UAV should optimally exploit its mobility via trajectory design to maximize the energy transferred to all ERs during a finite period. First, we consider the maximization of the sum energy received by all ERs by optimizing the UAV's trajectory subject to its maximum speed constraint. We obtain its optimal solution, which shows that the UAV should hover at one single fixed location during the whole period. However, the sum-energy maximization incurs a "near-far" fairness issue. To overcome this issue, we consider a different problem to maximize the minimum received energy among all ERs. We first consider an ideal case by ignoring the UAV's maximum speed constraint, and show that the relaxed problem can be optimally solved via the Lagrange dual method. Then, for the general case with the UAV's maximum speed constraint considered, we propose a new successive hover-and-fly trajectory motivated by the optimal trajectory in the ideal case, and obtain efficient trajectory designs by applying the successive convex programing (SCP).

I. INTRODUCTION

The paper introduces UAVs as mobile energy transmitters for wireless power transfer and studies trajectory design to maximize energy delivery fairly among distributed ground receivers. It addresses both sum-energy and minimum-energy objectives under UAV mobility constraints.

  • Motivation and novelty: Unlike link-level methods such as beamforming, scheduling, and waveform optimization, this work approaches WPT improvement at the system level through UAV mobility.The proposed architecture is intended to improve WPT performance while reducing the number of required energy transmitters compared with fixed-location systems.
  • System architecture: UAVs are proposed as mobile energy transmitters that fly above a serving area to cooperatively charge distributed ground energy receivers.The receivers have fixed, a priori known locations for trajectory design.
  • Problem formulation: The central problem is jointly optimizing UAV trajectories to maximize energy transferred to all receivers fairly during a finite charging period.The paper focuses on one UAV/ET and K > 1 receivers, subject to a maximum speed constraint.
  • Sum-energy maximization: For sum-energy maximization, the optimal trajectory keeps the UAV hovering at one fixed location throughout the charging period.The location can be found by 2D exhaustive search, with a closed-form characterization for K = 2 receivers.
  • Sum-energy maximization: When K = 2, the optimal sum-energy hovering point is midway between receivers below a distance threshold, but shifts closer to one receiver beyond that threshold.This objective can create a severe near-far fairness issue, especially across large networks.

II. SYSTEM MODEL

The system models a UAV-mounted energy transmitter delivering wireless energy to fixed ground receivers during a finite charging period. The UAV flies at fixed altitude under a maximum-speed constraint, while received RF power and energy depend on trajectory-based channel distances.

  • The UAV delivers wireless energy to K ≥ 2 ground energy receivers during a finite charging period.
  • Each receiver has a known fixed ground location, and the UAV’s initial and final locations are freely optimized.
  • The UAV maintains fixed altitude H > 0 and must satisfy a maximum-speed constraint V.
  • Under a LOS-dominated channel and free-space path loss, the channel power gain decreases with the squared UAV–receiver distance.
  • Because generic nonlinear RF-to-DC conversion modeling is unavailable, the paper evaluates received RF power and pre-conversion received energy.

III. SUM-ENERGY MAXIMIZATION

The sum-energy problem optimizes the UAV trajectory under a speed constraint, but its optimal solution is single-location hovering. For two receivers, the location shifts from the midpoint toward one receiver as their separation grows, exposing a near-far fairness issue.

  • III. SUM-ENERGY MAXIMIZATION: The sum-energy objective is generally difficult because it has infinitely many trajectory variables and a non-concave objective.
  • A. Optimal Solution to Problem (P1): The optimal hovering location can generally be non-unique and is obtained through a 2D exhaustive search.
  • A. Optimal Solution to Problem (P1): The optimal trajectory is single-location hovering: x⋆(t) = x⋆ and y⋆(t) = y⋆ throughout the charging period.
  • III. SUM-ENERGY MAXIMIZATION: To address this issue, the paper reformulates the objective as maximizing the minimum received energy among all receivers.
  • B. Special Case with K = 2 ERs: For K = 2, the UAV hovers above the midpoint when receiver separation is below a threshold, otherwise hovering locations shift toward either receiver.
  • B. Special Case with K = 2 ERs: As D increases, the optimal location approaches D/2, so energy delivered to the other receiver becomes negligible and near-far fairness deteriorates.

IV. MIN-ENERGY MAXIMIZATION WITHOUT UAV SPEED CONSTRAINT

The paper reformulates fairness-aware WPT as maximizing the minimum received energy and solves the speed-unconstrained relaxation optimally using Lagrangian duality. The resulting trajectory generally time-shares among multiple hovering locations to balance receivers’ energy.

  • The min-energy objective maximizes the minimum received energy among all K ERs, addressing the fairness issue of sum-energy maximization.
  • Ignoring speed constraints yields relaxed problem (P3), which can be solved optimally through the Lagrange dual method.
  • When the dual subproblem has non-unique optima, time-sharing among those locations is required to construct an optimal primal solution.
  • The optimal relaxed trajectory generally makes the UAV hover over multiple fixed locations with selected time allocations to balance energy transferred to the K ERs.
  • For two ERs, the optimal hovering locations are also optimal for the corresponding sum-energy problem, and the achieved sum energy is identical without speed constraints.

V. MIN-ENERGY MAXIMIZATION WITH UAV SPEED CONSTRAINT

With the UAV’s maximum speed constraint included, the general min-energy problem is difficult to solve globally for more than two ERs. The paper therefore proposes two suboptimal solutions based on the ideal speed-unconstrained solution.

  • The general speed-constrained min-energy problem is difficult to solve globally optimally in general when K > 2.
  • The proposed solutions are inspired by the optimal trajectory obtained for the ideal problem without the UAV maximum speed constraint.

A. Successive Hover-and-Fly Trajectory Design for Problem (P2)

The successive hover-and-fly design visits the ideal problem’s hovering locations while minimizing travel distance and allocating remaining time for hovering. For sufficiently large charging durations, this design is asymptotically optimal.

  • The resulting trajectory sequentially hovers at the selected locations and flies between them at maximum speed.
  • The design first determines a minimum-distance path through the Γ ideal hovering locations, then allocates the remaining charging time among them.
  • The path-ordering problem resembles a traveling salesman problem but does not require returning to the initial hovering location.
  • A dummy zero-distance location transforms the open-path problem into a standard TSP, after which the dummy-associated edges are removed.
  • When T ≫ Tfly, the successive hover-and-fly trajectory is asymptotically optimal for problem (P2).

3) Trajectory Redesign When T < Tfly:

When the charging duration is shorter than the time needed to visit all ideal hovering locations, the trajectory is redesigned by linearly compressing travel toward a fixed location. For two ERs, the optimal policy changes across distance and duration regimes.

  • Trajectory Redesign When T < Tfly: For T < Tfly, the charging duration is insufficient for the UAV to visit all ideal hovering locations.
  • Trajectory Redesign When T < Tfly: As T → 0, the UAV hovers at one fixed location (xfix, yfix, H), whose horizontal coordinates are found by 2D exhaustive search.
  • Trajectory Redesign When T < Tfly: The redesigned trajectory uses the scaling factor κ = T/Tfly < 1 to linearly move the travel path toward the fixed location.
  • Two-ER Special Case: For two nearby ERs, the UAV hovers at the midpoint (0, 0, H) throughout the charging period.
  • Two-ER Special Case: For distant ERs with T ≤ 2ξ/V, the UAV flies at maximum speed from (−VT/2, 0, H) to (VT/2, 0, H).
  • Two-ER Special Case: For distant ERs with T > 2ξ/V, the UAV hovers at symmetric locations, flies between them at maximum speed, and uses equal hovering durations.

B. SCP-Based Trajectory Design for Problem (P2)

The SCP-based design discretizes the UAV trajectory, replaces the non-concave objective with tight concave lower bounds, and iteratively solves convex approximations under speed constraints. The objective increases monotonically, and the algorithm converges to a locally optimal solution, although performance depends on initialization.

  • Trajectory discretization: The charging duration is divided into N slots, with the UAV location approximated as constant within each slot and adjacent locations constrained by maximum speed.The discretized locations are denoted (x[n], y[n], H), with Δ = T/N and squared-distance constraints bounded by V^2Δ^2.
  • Optimization structure: The discretized problem is non-convex because its objective is not concave, despite convex maximum-speed constraints.This motivates the successive convex programming approach.
  • Successive convex programming: At each iteration, SCP replaces each received-energy term with a tight lower bound around the current trajectory and maximizes the resulting surrogate.The lower bound is tight at the current iterate, preserving the current point’s objective value in the approximation.
  • Successive convex programming: Each surrogate problem is convex because the lower-bounded energy function is jointly concave in the trajectory coordinates and can be solved using standard convex optimization techniques.The paper specifically mentions the interior point method as one solution technique.
  • Convergence: The original objective increases monotonically across iterations, and the SCP algorithm converges to a locally optimal solution.The convergence argument uses the lower-bound property and the finite optimal value of the discretized problem.
  • Initialization and comparison: SCP performance depends on the initial trajectory; initializing with the successive hover-and-fly design ensures performance no worse than that design.The paper uses the discretized successive hover-and-fly trajectory as its initialization.

VI. NUMERICAL RESULTS

The numerical evaluation measures average received power by normalizing each ER’s total received energy by the charging duration. Simulations use fixed channel, altitude, and transmit-power settings.

  • Evaluation metric: The simulations evaluate the average received power received by each energy receiver.Average received power is the principal metric used in the numerical results.
  • Simulation settings: The simulations set β0 = −30 dB, H = 5 m, and P = 40 dBm.These parameters are used for all simulations described in the numerical-results section.
  • Evaluation metric: Average received power is computed by normalizing total received energy by the charging duration T.This normalization converts the accumulated energy over the charging period into an average-power measure.

A. Sum-Energy Maximization

For sum-energy maximization, the optimal UAV trajectory is single-location hovering, but the resulting received power becomes increasingly unfair as ER distances grow. Numerical results show that the UAV favors nearer receivers in this regime.

  • Two-ER evaluation: The sum-energy experiments compare average received power against the distance D between two ERs, including a static-UAV baseline.The two ERs are positioned at x1 = −D/2 and x2 = D/2, with D their separation.
  • Two-ER evaluation: For small ER separation, the optimal hovering location is the midpoint, producing identical average received power at ER 1 and ER 2.The UAV hovers above the midpoint throughout the charging period.
  • Fairness behavior: As D increases beyond the threshold, the optimal location moves toward the nearer ER, increasing its received power while decreasing the farther ER’s power.The resulting sum power is dominated by the nearer ER, worsening near-far fairness.
  • Ten-ER evaluation: With K = 10 ERs, the optimal sum-energy hovering location is close to ERs 7–10 and far from the other ERs, especially ER 1.The corresponding individual-ER powers show much higher energy for ERs 7–10 than for the remaining receivers.

B. Min-Energy Maximization

Min-energy maximization addresses the fairness problem by allocating service across multiple hovering locations. With speed constraints, successive hover-and-fly and SCP trajectories improve max-min received power, with SCP approaching the unconstrained upper bound for long charging durations.

  • Speed sensitivity: Increasing the UAV’s maximum speed increases average max-min received power because travel time between hovering locations decreases.This effect is reported for the two-ER min-energy problem.
  • Trajectory designs: For the speed-constrained problem, SCP iteratively optimizes a convex lower-bound approximation, while successive hover-and-fly provides a simpler trajectory design.The two designs are evaluated against fixed-location and hover-over-all-ER benchmarks.
  • Unconstrained design: The unconstrained min-energy solution uses Γ = 4 optimal hovering locations near ER groups 1–2, 3, 4–6, and 7–10.Receivers close to one another can share a single hovering location.
  • Speed-constrained design: The proposed constrained designs visit the optimal hovering locations while accounting for UAV travel between them.Both successive hover-and-fly and SCP trajectories visit the Γ locations, while SCP may deviate during flights.
  • Numerical comparison: When T ≥ 15 s, both proposed designs outperform successive hover-and-fly over all ERs.The all-ER benchmark fixes hovering locations above the individual receivers rather than optimizing grouped hovering locations.
  • Numerical comparison: Both proposed designs outperform single-location hovering and achieve higher max-min average power as T becomes large.The result is reported for the 10-ER system.
  • Numerical comparison: SCP performs better than successive hover-and-fly and converges to the unconstrained upper bound as T becomes large.The upper bound is the optimal value obtained when UAV speed constraints are ignored.
  • Conclusion: The optimized UAV trajectory significantly enhances WPT performance over fixed ETs while achieving fair energy delivery to ERs.The conclusion states this as the main numerical outcome of the proposed system.

APPENDIX

The appendix establishes optimality results for relaxed and speed-constrained UAV trajectory problems, including fixed-location hovering and symmetric hover-and-fly structures.

  • Ignoring speed constraints yields a relaxed problem whose optimal value upper-bounds that of (P1).
  • The relaxed problem decomposes across time, and the fixed location x(t) = x⋆, y(t) = y⋆ is optimal at every time instant.
  • The trajectory {x⋆(t), y⋆(t)} is feasible for (P1) and achieves the relaxed problem's optimal value, proving optimality for (P1).
  • Two-ER symmetry: For two ERs, symmetry requires equal received energies and permits restricting attention to trajectories that use symmetric locations for equal durations.
  • Two-ER trajectory: When ξ ≤ VT/2, the speed-constrained optimum maximizes hovering at (−ξ, 0, H) before flying to the midpoint.
  • Two-ER trajectory: When ξ > VT/2, the UAV flies at maximum speed from (−VT/2, 0, H) toward the midpoint during the first half of the period.
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