Mathematical Base and Deduction of Security Region for Distribution Systems With DER

This paper presents the mathematical definition and deduction for distribution system security region (DSSR), which lays a foundation for the DSSR theory. The DSSR is the set of all the N-1 secure operating points in the state space of a distribution system. First, this paper proposes the strict mathematical definition of DSSR, which defines that a set can be called a region only if three mathematical conditions are satisfied. Then, the N-1 security is deducted, based on which DSSR is proved existing and is mathematically deducted. As a basic feature of the future smart distribution system with distributed energy resources, the bidirectional power flow is fully considered in the deduction. In the conclusion, the observation and mathematical deduction both show the existence of security region for any distribution system.