In many jurisdictions, the recent wave of rooftop PV investment compromised the equity of network cost allocations across retail electricity customer classes, in part due to poorly designed network, retail and feed-in tariffs. Currently, a new wave of investment in distributed energy resource (DER), such as residential batteries and home energy management systems, is unfurling and with it so does the risk of repeating the same tariff design mistakes. As such, distribution network service providers need improved tools for crafting DER-specific tariffs. These tools will guide the design of tariffs that minimise DER impacts on network performance, stabilise network company revenue, and improve the equity of network costs across the customer base. Within this context, this paper proposes a probabilistic framework to assess the impacts of different network tariffs on the consumption pattern of electricity consumers with flexible DER, such as thermostatically controlled loads and battery storage. The assessment tool comprises randomly-generated synthetic load and rooftop generation traces, which are fed into a mixed integer linear programming-based home energy management system to schedule residential customers' controllable devices connected to a low voltage network. Customer net loads are then used in low voltage power flow studies to assess the network effects of various tariff designs. In this work, assessments are made of energy-and demand-based tariffs. Simulation results show that flat tariffs with a peak demand component perform best in terms of electricity cost reduction for the customer, as well as in mitigating the level of binding network constraints. This demonstrates how the assessment tool can be used by distribution network service providers and regulators to develop tariffs that are beneficial for networks that play host to growing numbers of PV-battery systems and other DER.