Tether Drag ODE Solver

Description

This code improved on varius simplistic tether models for airborne wind energy [AWE] that maybe assume that the tether is straight and without mass. It does not go all the way though, implementing a full segmented time simulated model. Instead, we try to strike a better balance between simplicity and accuracy by solving the 3D equations governing the tether shape subjected to drag and centrifugal forces in a quasi-static manner. Note only apparent wind is used and not actual wind, to make the solution independent on shaft orientation.

After doing that, one may perhaps opt to go back to the straight line models after adjusting the effective drag coefficient if necessary. The code nicely verifies the 1/4 tether drag assumption for conventional (non-TRPT) AWE.

The code may be used for TRPT [Tensile Rotary Power Transmission] AWE plants that rely on the shaft to transfer power from the kites to the ground, or also for more traditional designs such as Yoyo or Flygen [bounding and hovering flight patterns of a single kite].

More information about the math is found in this PDF.

This runs entirely in Julia.

Getting Started

Installing

pkg> add https://github.com/tallakt/TetherDragODESolver#main

Executing program

To run a simple shaft configuration you could do

julia> using TetherDragODESolver
julia> ode = let v = 50.0, d = 0.003, r0 = 10.0, r1 = 30.0, omega = v / r1, tension = 5000.0, l = 200.0, theta = deg2rad(90)
  solve_tether(omega, tension, r0, r1, l, theta, d)
end;
julia> solved_efficiency_ratio(ode)
0.4948173963867654
julia> solved_lambda(ode)
0.047844989963104156
julia> solved_lambda(ode)
0.047844989963104156
julia> solved_drag_coefficient_multiplier(ode)
0.26583523704150247

For a non TRPT plant, you may have done

julia> using TetherDragODESolver
julia> ode = let v = 50.0, d = 0.003, r0 = 10.0, r1 = 30.0, omega = v / r1, tension = 5000.0, l = 200.0, theta = deg2rad(90)
  solve_tether_for_non_trpt(omega, tension, r1, l, d)
end;
julia> solved_drag_coefficient_multiplier(ode)
0.25054208983609066

To plot a sweep from zero to 180 degrees in shaft twist vs effective drag coefficient

julia> using Plots
julia> let twist = 0:5:180, v = 50.0, d = 0.004, r0 = 10.0, r1 = 30.0, omega = v / r1, tension = 5000.0, l = 150.0
         plot(twist, [solved_drag_coefficient_multiplier(solve_tether(omega, tension, r0, r1, l, deg2rad(tw), d)) for tw in twist], ylims = (0, Inf), lab = "solved")
         plot!(twist, [1//4 for _ in twist], lab = "simplistic")
         plot!(xlabel = "shaft twist [deg]", ylabel = "C_D multiplier")
       end

To plot a sweep from zero to 180 degrees in lambda value actual vs straight tether model

julia> using Plots
julia> let twist = 0:5:180, v = 50.0, d = 0.004, r0 = 10.0, r1 = 30.0, omega = v / r1, tension = 5000.0, l = 150.0
         plot(twist, [solved_lambda(solve_tether(omega, tension, r0, r1, l, deg2rad(tw), d)) for tw in twist], ylims = (0, Inf), lab = "solved")
         plot!(twist, [calculate_non_drag_lambda(tension, r0, r1, l, deg2rad(tw)) for tw in twist], ylims = (0, Inf), lab = "simplistic")
         plot!(xlabel = "shaft twist [deg]", ylabel = "Lambda")
       end

Or just plot the shape of the tether as seen down the centerline

julia> let v = 50.0, d = 0.003, r0 = 10.0, r1 = 30.0, omega = v / r1, tension = 2000.0, l = 400.0, theta = deg2rad(80)
  ode = solve_tether(omega, tension, r0, r1, l, theta, d)
  plot(aspect = :equal, xlims = (-40, 40), ylims = (-40, 40), size = (700, 700), legend = false)
  plot!(r0 .* cos.(deg2rad.(0:360)), r0 .* sin.(deg2rad.(0:360)), lc = :gray70)
  plot!(r1 .* cos.(deg2rad.(0:360)), r1 .* sin.(deg2rad.(0:360)), lc = :gray70)
  plot!([x[1] for x = ode.solution.u], [x[3] for x = ode.solution.u], lc = :blue)
end

Authors

@tallakt