Greater stress is exerted on the local electricity grid as more sectors are electrified, including transportation, in which passenger cars represent a significant share, and as electricity production becomes increasingly decentralized in terms of solar and wind power. A key challenge for the distribution grid is to connect new production and demand (e.g., large-scale home charging of electric vehicles, EV) in both space and time. While several studies have examined how hourly electricity prices affect EV charging behavior, less attention has been given to the role of network tariffs, fees paid for the use of transmission and distribution grids, in shaping EV charging patterns. Therefore, this study investigates how different network tariff designs influence both the cost-optimal charging of EVs and household peak power demand. Logged EV charging data and hourly electricity spot prices are used as input to a linear cost-minimization model that compares EV charging in three different scenarios, with and without power tariffs. The results show that the network tariff influences when it is cost-optimal to charge EVs, indicating that tariff design could play a critical role in aligning EV charging with grid capacity.