Given the historically static nature of low-voltage networks, distribution network companies do not possess tools for dealing with an increasingly variable demand due to the high penetration of distributed energy resources (DERs). Within this context, this paper proposes a probabilistic framework for tariff design that minimises the impact of DER on network performance, stabilise network company revenue, and improves the equity of network costs allocation. To address the issue of the lack of customers' response, we also show how DER-specif-ic tariffs can be complemented with an automated home energy management system (HEMS) that reduces peak demand while retaining the desired comfort level. The proposed framework comprises a nonparametric Bayesian model which statistically generates synthetic load and PV traces, a hot-water-use statistical model, a novel HEMS to schedule customers' controllable devices, and a probabilistic power-flow model. Test cases using both energy- and demand-based network tariffs show that flat tariffs with a peak demand component reduce the customers' cost, and alleviate network constraints. This demonstrates, first, the efficacy of the proposed tool for the development of tariffs that are beneficial for networks with a high DER penetration, and second, how customers' HEM systems can be part of the solution.