The growing use of intermittent renewable energy sources requires an increased amount of storage capacity to match uncertain generation with uncertain demand. A possible solution is the use of thermal and electrical storages. This paper compares several model formulations: mixed integer linear programs (MILPs), nonlinear programs (NLPs), mixed integer nonlinear programs (MINLPs) for optimizing the operation of a multi-modal home energy system comprising heating and electricity subsystems. The respective optimization problems are then resolved within a model predictive control scheme and the final solutions are compared in terms of runtime and optimality. The results indicate that a thermocline-based thermal storage model leads to the overall lowest costs while not significantly impeding computing times. Additionally, the results show that a continuous heat pump model leads to reduced computing times without affecting the modelling accuracy.