Negative Reactance Impacts on the Eigenvalues of the Jacobian Matrix in Power Flow and Type-1 Low-Voltage Power-Flow Solutions

It was usually considered in power systems that power flow equations had multiple solutions and all the eigenvalues of Jacobian matrix at the high-voltage operable solution should have negative real parts. Accordingly, type-1 low-voltage power flow solutions are defined in the case that the Jacobian matrix has only one positive real-part eigenvalue. However, an important issue that has not been well addressed yet is that the "negative reactance" may appear in the practical power system models. Thus, the negative real-part eigenvalues of the Jacobian matrix at the high-voltage operable solution may be positive and also the type-1 low-voltage solutions could have more than one positive real-part eigenvalues, being a major challenge. Therefore, in this paper, the recognition of the type-1 low-voltage power flow solutions is re-examined with the presence of negative reactance. Selected IEEE standard power system models and the real-world Polish power systems are then tested to verify the analysis. The results reveal that the negative reactance in the practical power systems has a significant impact on the negative real-part eigenvalues of the Jacobian matrix at the high-voltage operable solution as well as the number of positive real-part eigenvalues at the type-1 low-voltage power flow solutions.