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Optimal Transport Theory for Power-Efficient Deployment of Unmanned Aerial Vehicles

Mohammad Mozaffari, Walid Saad, Mehdi Bennis, Merouane Debbah

arXiv:1602.01532v1cs.IT

TL;DR

The paper asks how multiple UAV base stations can minimize downlink transmit power while satisfying all users’ rate requirements. It jointly optimizes UAV locations and cell boundaries using facility location and optimal transport theory, while also studying altitude. The proposed deployment substantially reduces power, with combined optimization improving power efficiency by a factor of 20 over fixed-location Voronoi association.

  • Problem

    Multiple-UAV deployment must account jointly for user distributions, coverage boundaries, mobility, altitude, and guaranteed rate requirements under limited UAV energy.

  • Method

    The paper alternates between facility-location optimization of UAV positions and optimal-transport optimization of cell boundaries, while evaluating adjustable altitudes.

  • Results

    A factor of 20 improvement in power efficiency is reported over the Voronoi case with fixed UAV locations.

  • Takeaways & Limitations

    Power consumption is minimized by adapting coverage regions and UAV locations to users’ distribution and selecting appropriate altitudes.

Abstract

from arXiv · show

In this paper, the optimal deployment of multiple unmanned aerial vehicles (UAVs) acting as flying base stations is investigated. Considering the downlink scenario, the goal is to minimize the total required transmit power of UAVs while satisfying the users' rate requirements. To this end, the optimal locations of UAVs as well as the cell boundaries of their coverage areas are determined. To find those optimal parameters, the problem is divided into two sub-problems that are solved iteratively. In the first sub-problem, given the cell boundaries corresponding to each UAV, the optimal locations of the UAVs are derived using the facility location framework. In the second sub-problem, the locations of UAVs are assumed to be fixed, and the optimal cell boundaries are obtained using tools from optimal transport theory. The analytical results show that the total required transmit power is significantly reduced by determining the optimal coverage areas for UAVs. These results also show that, moving the UAVs based on users' distribution, and adjusting their altitudes can lead to a minimum power consumption. Finally, it is shown that the proposed deployment approach, can improve the system's power efficiency by a factor of 20 compared to the classical Voronoi cell association technique with fixed UAVs locations.

I. INTRODUCTION

The paper addresses power-efficient deployment of multiple UAV base stations by jointly optimizing their locations, coverage cells, and altitudes according to user distributions.

  • Motivation: UAVs can provide on-demand coverage and rate support because mobility and line-of-sight links improve service for ground users.They are especially suited to temporary events and hotspot areas.
  • Related work: Prior work largely studied air-to-ground channel modeling, including LOS probability, path loss, and altitude-dependent channel characteristics.These studies established that altitude affects aerial-base-station channel behavior through path loss and shadowing.
  • Research gap: Existing deployment studies did not jointly address multiple UAVs, UAV mobility, and user-distribution effects.Earlier work considered either a single static UAV or two UAVs without incorporating users’ distribution.
  • Research gap: Energy constraints are important because UAV power must support both transmission and mobility, while prior energy studies omitted multiple access points or guaranteed user rates.Other work also assumed known target locations or did not model network randomness.
  • Contribution: The proposed approach jointly optimizes UAV locations and cell boundaries, using facility location for locations and optimal transport theory for coverage areas.The framework also considers adjustable UAV altitude and mobility based on users’ distribution.

II. SYSTEM MODEL

The system divides a service area into subareas, assigns each to one UAV flying as a base station, and uses FDMA downlink transmission across noninterfering frequency bands.

  • System setup: A geographical area is divided into K subareas containing N users distributed according to an arbitrary density f(x, y).Each subarea is served by one UAV at location (x_i, y_i, h_i).
  • Cell association: Each UAV serves users inside its associated cell, whose initial boundary is not necessarily optimal.The deployment objective is to optimize cell boundaries so total transmit power is minimized.

A. Air-to-ground channel model

The channel model combines free-space propagation with probabilistic LOS and NLOS effects, making path loss depend on distance, attenuation, environment, and elevation angle.

  • Propagation model: NLOS links experience higher attenuation from shadowing and diffraction, so the model adds an NLOS path-loss term to free-space propagation loss.Both LOS and NLOS links are included in the air-to-ground model.
  • Propagation model: Path loss depends on user-UAV distance, the path-loss exponent α, and the additional NLOS attenuation factor η.These quantities characterize the average loss for a user served by UAV i.
  • LOS probability: The LOS probability depends on environment, building density and height, user and UAV locations, and their elevation angle.Constants C and D vary across rural, urban, and dense-urban environments.
  • LOS probability: The NLOS probability is 1 − P_LOS, while the LOS probability increases with the elevation angle between user and UAV.Thus, elevation angle directly affects the balance between LOS and NLOS propagation.
  • Average path loss: The model concludes by forming the average path loss from the probabilistic LOS and NLOS components.This average loss is then used in subsequent rate and power calculations.

B. Problem formulation

The formulation minimizes total UAV transmit power while maintaining every user’s rate requirement, jointly choosing cell boundaries and UAV locations despite their mutual dependence.

  • Rate and power model: The achievable user rate depends on UAV transmission bandwidth, transmit power, average path loss, and noise power.Each user is associated with a UAV through its cell boundary.
  • Rate and power model: Each UAV’s bandwidth per user is W_i = B_i/M_i, where M_i is the number of users in its cell.The served-user count depends on the user distribution and cell boundary.
  • Rate and power model: As the number of users increases, bandwidth per user decreases and higher transmit power is required to satisfy the rate requirement β.The minimum power is derived by imposing R(x, y) ≥ β.
  • Optimization objective: The optimization jointly finds UAV locations and associated cell boundaries to minimize total required transmit power.The objective includes the average power contributions over the users served by each UAV.
  • Optimization challenge: Solving the optimization is challenging because user counts, UAV locations, altitudes, and cell boundaries are mutually dependent over a continuous search space.The formulation involves infinitely many possible locations and boundaries.

III. POWER OPTIMIZATION: OPTIMAL TRANSPORT THEORY AND FACILITY LOCATION

The power-minimization problem is solved sequentially by optimizing UAV locations for fixed cells and cell associations for fixed UAV locations.

  • The first optimization fixes cell boundaries and finds UAV locations that minimize transmit power.
  • The second optimization fixes UAV locations and derives cell associations that minimize total required transmit power.

A. Optimal UAVs location given cell boundaries

Given cell boundaries, the paper positions each UAV to minimize transmit power over its assigned subarea, using facility-location methods and altitude-dependent approximations.

  • A. Optimal UAVs location given cell boundaries: Facility location determines each UAV’s optimal position within its assigned subarea based on the users’ distribution.The objective is to minimize transportation costs between facilities and clients, here corresponding to UAV placement over user regions.
  • A. Optimal UAVs location given cell boundaries: For a fixed cell boundary, minimizing total transmit power is equivalent to minimizing each UAV’s transmit-power contribution.The number of users inside each cell is fixed in this subproblem, so optimization does not depend on that user count.
  • A. Optimal UAVs location given cell boundaries: At high or low altitudes relative to the subarea size, closed-form expressions are derived for the power-minimizing UAV location.The analysis uses different altitude regimes to obtain an approximate location solution.
  • A. Optimal UAVs location given cell boundaries: The optimal location corresponds to the centroid of the area when the user distribution is represented by f(x, y).
  • A. Optimal UAVs location given cell boundaries: At arbitrary altitudes, the optimal location is characterized by a system of equations derived from an approximation of the LOS probability.The approximation is stated for 100 m ≤ h_i ≤ 2000 m and horizontal distances up to 1000 m.
  • A. Optimal UAVs location given cell boundaries: The resulting system can be solved with Newton-Raphson, and continuity and differentiability ensure that an optimal solution exists.

B. Optimal cell boundaries given UAVs location

With UAV locations fixed, the paper derives user cell boundaries that minimize total transmit power by accounting for both channel gain and the number of users assigned to each cell.

  • B. Optimal cell boundaries given UAVs location: The fixed-location subproblem derives optimal cell boundaries whose association depends on the users’ distribution.
  • B. Optimal cell boundaries given UAVs location: Because each UAV’s transmit power increases exponentially with its served-user count, cell association must consider user count alongside channel gain.
  • B. Optimal cell boundaries given UAVs location: The paper maps this association problem to optimal transport, treating UAV-to-user assignments as transporting data at a cost equal to required transmit power.The optimal transport map yields the best users associated with each UAV and therefore the cell boundaries.
  • B. Optimal cell boundaries given UAVs location: A lemma for continuous F and differentiable S gives the solution to the reformulated transport optimization problem.
  • B. Optimal cell boundaries given UAVs location: The resulting boundaries minimize total transmit power and apply to arbitrary user distributions without a specific distributional assumption.Uniform and truncated Gaussian distributions are then considered for numerical modeling, including hotspot areas.
  • B. Optimal cell boundaries given UAVs location: The full deployment alternates between optimizing UAV locations for current cells and updating cell boundaries for the new locations.

IV. NUMERICAL RESULTS

The numerical results show that adapting UAV coverage boundaries, locations, and altitudes to user distribution substantially reduces required transmit power. The combined optimization outperforms separate optimization strategies and fixed Voronoi association.

  • 0.12 W versus around 0.3 W: optimal cell boundaries require less average transmit power than Voronoi boundaries.The comparison assumes UAVs at subarea centers and altitude 200 m.
  • ρ ≈0.02 yields the maximum power improvement of the proposed optimal cell boundaries over Voronoi association.At low density, users are more spread out and the two approaches perform similarly; at very high density, they become closer again.
  • 400 m minimizes total average transmit power when both UAVs share the same altitude.The individually optimal altitudes are around 320 m for UAV1 and 500 m for UAV2, while the joint minimum occurs at h1 = h2 = 400 m.
  • 0.12 W is the minimum total average transmit power achieved at h1 = 310 m and h2 = 530 m when altitude combinations are varied.UAV2 is placed higher because UAV1 is closer to the hotspot center and has a higher average chance of LOS links.
  • 20×: combined optimal cell association and UAV location improves power efficiency over fixed-location Voronoi association.The combined method outperforms UAV-location optimization and cell-association optimization by factors of 3 and 10, respectively.

V. CONCLUSIONS

The paper proposes an optimal deployment framework for UAV flying base stations that minimizes transmit power while satisfying ground-user rate requirements. It combines optimal transport for cell association, facility location for UAV placement, and altitude adjustment to reduce power consumption.

  • The framework formulates UAV deployment as transmit-power minimization subject to satisfying every ground user's rate requirement.
  • Optimal transport determines cell association, while the facility location framework derives UAV locations based on users' distribution.
  • Determining optimal coverage regions significantly decreases total required transmit power.
  • Placing UAVs according to user distribution and adjusting their altitudes to optimal values yields minimum power consumption.
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