Using sigmoid functions for representing limits of generators and static Var compensators and their impact on the voltage stability study

Abstract

This work presents new equipment’s representation methodologies in the steadystate power flow analysis and their impacts on the voltage stability of electrical power systems. With special focus on system’s voltage collapse scenario, characterized by a stability to instability transition, the traditional power flow method is reformulated with the introduction of smooth functions. As an option to introduce the desired smoothness, it is opted to incorporate the sigmoid function into the power flow formulation. The sigmoid function can be implemented within the traditional Newton-Raphson power flow formulation in the modeling of various control equipments. For the purpose of this work, it will be proposed new modelings for generators and Static VAr Compensators (SVCs). On account of being able to actively participate in voltage regulation, by means of controling reactive power injection, absorption and flow, these equipments received an exceptional attention. In order to analyze the condition of voltage collapse, power systems are stressed with gradual increases in generation and load. The continuous power flow tool is, therefore, applied, considering equipment’s limits and saturation characteristics under study. On a general point of view, the voltage collapse is considered as a power system’s point of maximum loadability, on the threshold of voltage stability. In a mathematical point of view, however, the voltage collapse is characterized as a point where a bifurcation occurs. Since power flow solves a system of nonlinear equations, and the continuous power flow determines a sequence of nonlinear equations solutions, bifurcations can be present throughout power flow analysis and simulations. Hence, several types of bifurcations can be observed, being characterized by dierent mathematical conditions. The use of sigmoid function in the modeling of control equipment proposes a new condition of voltage stability for electrical systems. The bifurcations found within the system, that once may have had dierent characteristics, now have the same common characteristic. This condition is translated as an advantage for voltage stability of electrical power systems in terms of easily identification of critical scenarios, which are responsible for admitting collapse. All propositions were tested by means of several simulations on dierent test-systems. A Python-based program was developed and the result simulations were validated by CEPEL’s (Electric Energy Research Center) production-grade academic version software.