Residential PV systems combined with a battery, a heat pump and a heat storage can provide a great part of the energetic needs of the residents in a decentralized manner and without emitting carbon dioxide during operation. The management of this hybrid energy storage system's (HESS) degrees of freedom can be addressed by model predictive control (MPC), relying on an optimization model and forecasts, which are both never perfectly accurate. The aim of this paper is to systematically assess the impact of errors in the model's structure and parameters on the performance of an MPC for such a residential PV-battery-heat pump-heat storage system in order to lay the foundations for systematic improvements in the MPC's robustness. With this being a simulation-based study, the results will also depend on the plant model used for the simulation, which is why the sensitivity of the selected performance criteria towards the plant model was also studied. The results show that there are big differences in the sensitivities of operating costs, security of domestic hot water supply and PV curtailment towards changes in the plant and optimization models, with most being more sensitive towards the plant model. However, for big changes in the optimization model, the MPC's usage of the two storage devices changed fundamentally, resulting in big changes for grid relief and the security of domestic hot water supply. These results show that, at least in the case of this HESS, including the nonlinear system behavior in the formulation of the optimization model greatly improves the MPC's performance.