Unmanned Aircraft System Path Planning for Visually Inspecting Electric Transmission Towers

To detect faults on the power transmission and distribution systems, current electric utilities perform a visual inspection by dispatching line crews and helicopters. This practice has disadvantages such as high operation costs and safety concerns. To resolve these issues, power utilities are considering the use of an unmanned aircraft system (UAS). In this paper, we formulate an optimization model to find an efficient flight path for a UAS for visually inspecting a transmission tower. The objective of the model is to maximize a function involving three performance ratios, namely, flight time, image quality, and tower coverage. The optimization model is non-linear, non-differentiable, and multi-modal. We solve the problem by using a particle swarm optimization (PSO) based-algorithm and a simulated annealing (SA) based-algorithm and compare their results. We test the model under three inspection strategies. The experimental results show that the PSO-based algorithm outperforms the SA-based algorithm. They also show that the proposed model can provide a flight path that comprises a good balance over the three performance ratios.

i :

Particle (flight path) in the PSO

j :

Waypoint in the particle

t :

Time step

Φ:

Set of all coordinates within the flying area

Φ_S :

Set of surface coordinates of the tower

Φ_N :

Set of coordinates of the no-fly zone

Φ_C :

Set of candidates’ coordinates for a waypoint Φ_C = Φ −Φ_N

D :

Maximum x, y, and z of the flying area

Ω :

Weight vector of the objective function (ω₁,ω₂,ω₃)

e :

Unit vector of each coordinate in a search space

FS :

Flying speed of the UAS (m/s)

H :

Hovering time of the UAS at a waypoint (sec)

F T ^max :

Maximum flight time (sec)

F D ^max :

Maximum flight distance (m)

N ^s :

Total number of surface elements of the tower

(X ^tc,Y ^tc,0):

Tower center-base coordinates (m)

G ^max :

Maximum ground sampling distance (mm)

FL :

Focal length of the camera (mm)

L :

Height of the image sensor (mm)

N ^p :

Vertical number of pixels in the image

N ^max :

Maximum number of waypoints

N :

Number of waypoints

M :

Number of particles

C ₁,C ₂ :

Learning factors

W :

Inertial weight

T ₀ :

Initial temperature

N _S :

Number of cycles

N _T :

Number of step vector adjustments

N _ε :

Number of successive temperature reductions

N _{e v a l} :

Maximum number of objective function evaluations

r _T :

Reduction coefficient

ε:

Tolerance for stopping iterations

f p _i :

Position of flight path (particle) i

w p _{i, j} :

Coordinates of waypoint j in flight path (particle) i

f v _i :

Velocity of particle i

w v _{i, j} :

Velocity of waypoint j in particle i

R :

Rotation matrix

T :

Translation matrix

TR :

Transformation matrix

P _i :

Position of particle i

V _i :

Velocity of particle i

\({\mathbf {PB}}_{i}^{t}\) :

Best position of particle i at time t

G B ^t :

Position of the global best particle at time t

s :

Step vector

x :

Decision variable vector

\(\mathrm {\mathbf {x}}^{\prime }\) :

New decision variable vector

p r _T :

Total performance ratio of a flight path

p r ₁ :

Performance ratio for the total flight time

p r ₂ :

Performance ratio for the inspection coverage

p r ₃ :

Performance ratio for the image resolution

f t _i :

Flight time of particle i

f d _i :

Flight distance of particle i

w d _i :

Flight distance from the UAS home-base to the last waypoint of particle i

S ^rc :

Real coordinates (3D) of a tower surface element

S ^cc :

Camera coordinates (3D) of a tower surface element

S ^ic :

Image coordinates (2D) of a tower surface element

(x,y,z):

Real coordinates

(u,v,w):

Camera coordinates

\({(x}^{\prime },y^{\prime })\) :

Image coordinates

\({n_{i}^{s}}\) :

Total surface elements covered by the images taken during flight path (particle) i

g _{i, j} :

Ground sampling distance of the images taken at waypoint j in particle i

v ^max :

Maximum velocity of a particle

r,r ₁,r ₂ :

Random numbers in [0,1]

T :

Temperature

p :

Acceptance probability

m :

Number of accepting a new decision variable vector

Authors and Affiliations

1. Department of Industrial and Systems Engineering, Auburn University, Auburn, AL, 36849, USA

   Hyeoncheol Baik & Jorge Valenzuela

Corresponding author

Correspondence to Jorge Valenzuela.

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