Heat pumps coupled with thermal energy storage (TES) systems are seen as a promising technology for load management that can be used to shift peak loads to off-peak hours. Most of the existing model predictive control (MPC) studies on tariff-based load shifting deploying hot water tanks use simplified tank models. In this study, an MPC framework that accounts for transient thermal behavior (i.e., mixing and stratification) by applying energy (EMPC) and exergy (XMPC) analysis is proposed. A case study for an office building equipped with an air handling unit (AHU) revealed that the MPC strategy had a high load-shifting capacity: over 80% of the energy consumption took place during off-peak hours when there was an electricity surplus in the grid. An analysis of a typical day showed that the XMPC method was able to provide more appropriate stratification within the TES for all load characteristics. An annual exergy analysis demonstrated that, during cold months, energy degradation in the TES is mainly caused by exergy destruction due to irreversibility, while, during the transition to milder months, exergy loss dominates. Compared to the EMPC approach, the XMPC strategy achieves additional reductions of 18% in annual electricity consumption, 13% in operating costs, and almost 17% in emissions.