  We derive new exact charged $d$-dimensional black hole solutions for
quadratic teleparallel equivalent gravity, $f({\cal T})=a_0+a_1{\cal
T}+a_2{\cal T}^2$, where $\cal T$ is the torsion scalar, in the case of
non-linear electrodynamics. We give a specific form of electromagnetic function
and find out the form of the unknown functions that characterize the vielbeins
in presence of the electromagnetic field. It is possible to show that the black
holes behave asymptotically as AdS solutions and contain, in addition to the
monopole and quadrupole terms, other higher order terms whose source is the
non-linear electrodynamics field. We calculate the electromagnetic Maxwell
field and show that our d-dimensional black hole solutions coincide with the
previous obtained one \cite{2017JHEP...07..136A}. The structure of the
solutions show that there is a central singularity that is much mild in
comparison with the respective one in General Relativity. Finally, the
thermodynamical properties of the solutions are investigated by calculating the
entropy, the Hawking temperature, the heat capacity, and other physical
quantities. The most important result of thermodynamics is that the entropy is
not proportional to the area of the black hole. This inanition points out that
we must have a constrain on the quadrupole term to get a positive entropy
otherwise we get a negative value.
