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This study systematically proposes and defines a set of new quantitative indices named total supply capability (TSC) and its associated indices, such as available supply capability, substation supply capability, network transfer capability, maximum supply capability and expandable supply capability. The concepts in the TSC family provide new approaches to analyse and optimise distribution networks. TSC for distribution systems is comparable to the total transfer capability for transmission systems, but there are significant differences, such as the radial configuration and service restoration in TSC consideration. TSC is defined as the maximum load to serve under the expanded N-1 security guideline for the distribution system, taking into account the capacities of the substation transformers, network topology, link capacity and other constraints. Here, the expanded N-1 security means a successful, near-immediate service restoration to all loads except for the faulted section. A mathematical model for TSC is then set up and an accurate method is presented to calculate the TSC with lingo. Sample applications are provided to illustrate a new TSC-based approach to evaluate the link capacity and obtain the reconductoring proposal. The results from a real distribution system demonstrate the effectiveness of the proposed TSC modelling and TSC-based approach.