Time-of-use electricity tariffs are gradually being introduced around the world to expose consumers to the time-dependency of demand, however their effects on peak flows in distribution networks, particularly in areas with domestic energy storage, are little understood. This paper presents investigations into the impact of time-of-use and time-of-export tariffs in residential areas with various penetrations of battery storage, rooftop solar PV, and heat pumps. By simulating battery operation in response to high resolution household-level electrical and thermal demand data, it is found that home batteries operating to maximise cost savings in houses signed up to time-dependent tariffs cause little reduction in import and export peaks at the low voltage level, largely because domestic import and export peaks are spread out over time. When operating to maximise savings from the first three-tier time-of-use tariff introduced in the UK, batteries could even cause increases in peak demand at low voltage substations, if many batteries in the area commence charging at the start of the overnight off-peak price band. Home batteries operating according to time-dependent electricity tariffs significantly miss out on the potential peak shaving that could otherwise be achieved through dedicated peak shaving incentives schemes and smarter storage control strategies.