Novel Homotopy Theory for Nonlinear Networks and Systems and Its Applications to Electrical Grids

Homotopy methods have several useful applications in biology, economics, nonlinear circuit design, and electric power networks, among others. A novel homotopy theory and a convergence theorem are presented in this paper for general-purpose homotopy methods. It is shown that the derived analytical results are applicable to a class of large-scale integrated-circuit designs and protein interaction networks. In addition, an application of this theory is illustrated to prove the convergence of the homotopy-enhanced implicit Z-bus power-flow method. The existence of power-flow solutions for general distribution networks with distributed generators modeled as photovoltaic nodes is also justified. Numerical studies on the IEEE 37-bus, 123-bus, and 8500-node distribution networks are conducted to illustrate the set of sufficient conditions for the main theorem.