This study explores the interactions between the national power system and prosumer households equipped with PV-battery systems. For this purpose, a dispatch model representing the electricity supply in five European countries is expanded by an additional household layer. This hard-coupled setup allows to assess the optimal operation of stationary household batteries with the objective of minimizing the national electricity supply cost. This is contrasted with the self-consumption maximization from the household's perspective under various tariffs. In all cases, the analysis focuses on changes in the cost of national power production due to storage operation, i.e. the household battery system value. Under the household perspective operation, this value can be negative, which means that prosumer storage increases the national power supply cost. This can be the case if wholesale price profiles are not sufficiently reflected in the electricity tariffs. The value increases if tariffs reflect the market conditions more closely. Enabling the option of battery grid-charging always reduces the cost of national power production. In most cases, storage value rises during future years. This is because of the naturally improved match of the household dispatch with the electricity market price profiles under higher national solar power shares. Major parts of the battery value stem from the replacement of less efficient pumped hydro storage. Consequently, battery storage value shows a strong sensitivity with respect to hydro power flexibility. Various other sensitivity analyses are presented. While the default case represents the household sector by 12 individual nodes, implications of smaller sample sizes as well as the aggregation of household nodes are discussed. While this does not affect battery value operated from the system's perspective, impacts on household perspective operation value are statistically significant. Further, we assess the impact of frozen fuel prices and power plant capacities, larger volumes of prosumer households, and lower time resolution of the household nodes.