Singularities Analysis of the Jacobian Matrix Modified in the Continuation Power Flow: Mathematical Modeling

In recent years, the concern for voltage stability issue has gained global highlighted when referring to the energy sector industry, this is because this issue is related to the operation and planning of electrical power systems. Factors such as the increasing energy demand, the transfer of large amounts of power to meet the consumption, combined with the economic and environmental requirements has led the systems operate in stressful conditions (close to their limits), i.e., with small margins of security that is a threat to its stable operating condition. The combination of these factors can be disastrous, enabling the vulnerability of electrical power systems, i.e., exposed to risk of a situation of instability. In the literature, a study to analyze stability and voltage instability is related to the P-V curve (power versus voltage magnitude) and the maximum loading point (MLP) (point on the curve that separates the stable operation of the unstable). The maximum loading point may be consequent to a saddle node bifurcation (SNB) related to transmission capacity limit in an electrical system where the Jacobian matrix is singular, or limit induced bifurcation (LIB), related the reactive power limit of the generator, where the matrix is not singular. In this sense, it is presented in this first part of the paper, an analysis of the modified Jacobian matrices (Jm) of the methods of continuation power flow (CPF) reported in the literature (parameterization methods), the study was developed in order to analyze the changes that the matrix conventional Jacobian (J) have to eliminate the singularity problems in the MLP and in the bifurcation points of each method