Error Analysis of the Convex Hull Method for the Solution of the Distribution System Security Region

The convex hull method is a common approach for the solution of the distribution system security region (DSSR). For the first time, this paper identifies that this method is not applicable to solve many DSSRs. Firstly, the model of the DSSR and the convex hull based solving method for the DSSR are briefly introduced. Secondly, the concepts of the concave region and convex region in the DSSR are presented. Thirdly, theoretical analyses are separately conducted for concave and convex regions, which result in two theorems and one corollary, leading to the following conclusions: (1) The convex hull method is not suitable for solving concave regions, while concave regions are widely present in real-world distribution networks. (2) Error may also be produced by the convex hull method when solving convex regions. For the convex region, the condition for an error-free solution is proven, the error causes are analyzed, and error reduction measures are proposed. Finally, the theoretical analyses are validated through case studies. The validation shows that when solving concave regions, the convex hull method can produce significant error and thus cannot satisfy the requirements for a security analysis. When solving convex regions, measures should be taken to minimize or remove error. This paper has significant value in enhancing the fundamental theory of the DSSR and applying it correctly in practice.