In this paper, we investigate the Resource-Constrained Project Scheduling Problem (RCPSP) with time-of-use energy tariffs (TOU) and machine states, a variant of RCPSP for production scheduling where energy price is part of the criteria and one machine is highly energy-demanding and can be in one of the following three states: proc, idle, or off. The problem involves scheduling all tasks, respecting precedence constraints and resource limitations, while minimizing the combination of the overall makespan and the total energy cost (TEC), which varies according to the TOU pricing, which can take negative values. We propose two novel approaches to solve it: a monolithic Constraint Programming (CP) approach and a Logic-Based Benders Decomposition (LBBD) approach. The latter combines a master problem dealing with energy cost solved using Integer Linear Programming (ILP) with a subproblem handling the RCPSP resolved using CP. Both approaches surpass the monolithic compact ILP approach, but the LBBD significantly outperforms the CP when the ratio of energy-intensive tasks over the overall tasks is moderate, allowing for solving instances with up to 1600 tasks in sparse instances. Finally, we put forth a way of generalizing our LBBD approach to other problems sharing similar characteristics, and we applied it to a problem based on an RCPSP problem with blocking times & total weighted tardiness criterion and a flexible job shop.