For the boltaztrap we used the formula for calculating the Seebeck coefficient using a different formula mentioned in doi:10.1016/j.cpc.2006.03.007 and the formula we are using in the experiment is gradV/gradT. How these formulae are connected?
If you read the article you cited you would see they are indeed the same. The only difference is that in the article the they have replace \grad V with the electric field E. The formula in Eq. 6 is only the expression of the electric field and thermal gradient in terms of the the velocity correlation functions that are calculated in the Boltzmann transport equation (indeed, you do not know the electric field generated by a thermal gradient unless you solve the equation of motion).
However, I would say that in the experiments, one use more \Delta V/\Delta T where the \Delta refers to the variation of a quantity in different contacts. Indeed, \grad would be a local quantity.
Through eq. 11 where they sum up over all indexes to get the energy and direction dependence.