An increase in molecular weight of the polymer generally impedes solubility in common solvents and may influence the polymer optoelectronic properties as well. Indeed, higher molecular weights are expected to increase charge carrier mobilities and therefore give rise to better photovoltaic performances of bulk heterojunction solar cells. In this work, we use copolymers based on 2,1,3-benzothiadiazole, thiophene and thieno[3,2-b]thiophene units of various fractions differing in molecular weights almost by a factor of 4 with a fullerene based acceptor material Indene-C60 Bisadduct (ICBA) to elaborate bulk heterojunction solar cells. We investigate the influence of post-deposition annealing temperatures and polymer:fullerene ratios on the final cell performances. We use ICBA as an acceptor to enhance the open circuit voltage due to its high lying LUMO level . Additionally, charge carrier mobilities were probed using bottom contact organic field-effect transistors. As expected, higher molecular weights (as long as homogeneity was maintained) resulted in an increase of the hole field-effect mobility (up to 7x10-3 cm2V-1s-1). Consequently, the power conversion efficiencies of bulk heterojunction solar cells could be improved by increasing the copolymer molecular weight. A power conversion efficiency of 2.4% with an open circuit voltage of 0.82V was reached in a standard device configuration with aluminum as a cathode after post-deposition thermal annealing.