Propositions on the number of confluence vertices in distribution networks

In this paper, confluence vertices in distribution networks are discussed from the point of view of network topology, to solve the problems that require a number of confluence vertices before hydraulic calculation. First, open and closed network models are introduced to represent the different types of distribution networks and constructed based on the general assumptions. Six propositions are proven according to graph theory to reveal the relationship between the number of confluence vertices and the network topology, not only based on the general assumptions but also the strict assumptions with additional limitation on the degree of vertices. Besides, the equations also reveal the relationship between the number of confluence vertices and the number of independent loops in the network models with the strict assumptions. Through discussion of hydraulic variations, the number of confluence vertices in the network model that follows the strict assumptions is dependent only on the network topology and can be calculated by the number of pipes, vertices, suppliers, and consumers; whereas in the network model with the general assumptions determination of the number of confluence vertices requires the support of hydraulic analysis.