The increasing penetration of residential photovoltaics (PV) comes with numerous challenges for distribution system operators. Technical difficulties arise when an excess of PV energy is injected into the grid, causing voltage rise or overloading of the lines. Economic challenges appear because PV owners and consumers are not participating equally in the grid costs. Indeed, PV owners benefit by self-consuming their PV production and by gaining additional revenues when they sell their PV surplus to the grid. Hence, they lower their grid costs. In this paper, we propose a mixed-integer-linear programming approach to solve the design and operation of a PV and battery system efficiently. We use this tool to benchmark five different tariff scenarios, which include real-time pricing, a capacity-based tariff, and a block rate tariff, and evaluate their effect on the design and operation of the system. Carefully tailored metrics show the impact of these tariff structures on the trade-off between the economic viability of privately owned energy systems and their grid usage intensity. Considering both aspects, we show that a block rate tariff is the most promising approach and that capacity-based tariffs rely on PV curtailment alone to curtail the generation peaks.