Shares of renewables continue to grow in the European power system. A fully renewable European power system will primarily depend on the renewable power sources of wind and photovoltaics (PV), which are not dispatchable but intermittent and therefore pose a challenge to the balancing of the power system. To overcome this issue, several solutions have been proposed and investigated in the past, including storage, backup power, reinforcement of the transmission grid, and demand side management (DSM). In this paper, we investigate the potential of DSM to balance a simplified, fully renewable European power system. For this purpose, we use ten years of weather and historical load data, a power-flow model and the implementation of demand side management as a storage equivalent, to investigate the impact of DSM on the need for backup energy. We show that DSM has the potential to reduce the need for backup energy in Europe by up to one third and can cover the need for backup up to a renewable share of 67%. Finally, it is demonstrated that the optimal mix of wind and PV is shifted by the utilisation of DSM towards a higher share of PV, from 19% to 36%.