An acoustic black hole (ABH) beam termination can be achieved by decreasing its thickness according to a power-law profile. Waves entering the ABH slow down and vibrational energy strongly concentrates at the tip of the beam. This can be exploited for energy harvesting, as suggested in some recent works. The finite element method (FEM) is commonly used to carry out the simulations, which hampers long parametric analyzes. In this paper, we develop a semi-analytical approach to characterize the performance of a piezolectric bimorph cantilever with an ABH termination. The method can be easily extended to further configurations and allows one to determine ABH harvesting capabilities when varying system parameters, in a fast and efficient way. The Lagrangian of the ABH beam plus piezoelectric layers is constructed and the coupled equations for the flexural vibrations and voltage are derived from it. The flexural displacement field is expanded in terms of Gaussian basis functions. Vibration shapes and harvested power are computed with the proposed method and validated against FEM simulations. The ABH piezolectric bimorph cantilever is shown to substantially enhance the harvesting capabilities of a cantilever with uniform cross-section. The semi-analytical approach is then used to examine the influence of several ABH and piezoelectric layer parameters on energy harvesting efficiency. As regards the former, the effects of the tip truncation thickness and ABH order are explored. In what concerns the piezoelectric layer, we investigate the effects of its location, thickness, splitting it into several patches and varying the load resistors to enhance its performance in a broad frequency range. The proposed method constitutes a valuable tool for the design of ABH energy harvesting devices.