A comprehensive study is made on the properties of total supply capability（TSC）model solutions for distribution networks from a mathematical point of view.First，the TSC model is simplified to a general form of linear program，with its mathematical and physical meaning explained.Then，mathematical expressions for parameter condition are given when there is no or a unique solution.It is concluded that the TSC model has infinitely many solutions in actual distribution networks.Then，in order to find out the optimum operation from the solutions，the equilibrium degree of load rate（EDLR）is calculated and analyzed.Parameter matching conditions for network and main transformer capacity to make load more balanced are summarized for distribution systems.Finally，a case study is given to verify the effectiveness of the above conclusions.