Formal notion for energy complexity of computational problems
Is there a well studied adaptation of Turing machines that account for the energy consumed during the execution of algorithms?  No!

But maybe you could come up with one. It's possible you could divide the Turing machine steps into reversible and non-reversible (the non-reversible ones are where information is lost). Theoretically, it is only the non-reversible steps that cost energy. A cost of one unit of energy for each bit that is erased would theoretically be the right measure. 

There is a theorem of Charles Bennett that the time complexity increases by at most a constant when a computation is made reversible (C.H. Bennett, Logical Reversibility of Computation), but if there are also limits on space, then making the computational reversible might incur a substantial increase in time (Reference here). Landauer's principle says that erasing a bit costs $kT\, \ln 2$ of energy, where $T$ is temperature and $k$ is Boltzmann's constant. In real life, you cannot come anywhere close to achieving this minimum. However, you can build chips which perform reversible steps using substantially less energy than they use for irreversible steps. If you give reversible steps a cost of $\alpha$ and irreversible steps a cost of $\beta$, this seems like it may give a reasonable theoretical model.

I don't know how Turing machines with some reversible steps relate to chips with some reversible circuitry, but I think both models are worth investigating.