Geometric Understanding of the Stability of Power Flow Solutions

A grand challenge for power grid management lies in how to plan and operate with increasing penetration of distributed energy resources (DERs), such as solar photovoltaics and electric vehicles, which disturb the power grid stability. Existing approaches are unable to verify if a point is on a loadability boundary or characterize all loadability boundary points exactly. This inability leads to a poor understanding of locational hosting capacity for accommodating distributed resources. To solve these problems, we compare existing approaches and propose a rectangular coordinatebased analysis, which drew less attention in the past. We demonstrate that such a coordinate (1) provides an integrated geometric understanding of active and reactive power flow equations, (2) enables linear representation of elements in the Jacobian matrix, (3) verifies if an operating point is on the loadability boundary and what is the margin, and (4) characterizes the power flow feasibility boundary points. Finally, IEEE standard test cases demonstrate the capability of the new method. Index Terms Power flow, loadability, rectangular coordinate, geometric understanding, maximum power output