The continuous introduction of renewable electricity and increased consumption through electrification of the transport and heating sector challenges grid stability. This study investigates load shifting through demand side management as a solution. We present a four-month experimental study of a low-complexity, hierarchical Model Predictive Control approach for demand side management in a near-zero emission occupied single-family house in Denmark. The control algorithm uses a price signal, weather forecast, a single-zone building model, and a non-linear heat pump efficiency model to generate a space-heating schedule. The weather-compensated, commercial heat pump is made to act smart grid-ready through outdoor temperature input override to enable heat boosting and forced stops to accommodate the heating schedule. The cost reduction from the controller ranged from 2-17% depending on the chosen comfort level. The experiment demonstrates that load shifting is feasible and cost-effective, even without energy storage, and that the current price scheme provides an incentive for Danish end-consumers to shift heating loads. However, issues related to controlling the heat pump through input-manipulation were identified, and the authors propose a more promising path forward involving coordination with manufacturers and regulators to make commercial heat pumps truly smart grid-ready.