Study of multisolution quadratic load flow problems and applied Newton-Raphson like methods

A number of facts about quadratic load flow problems y=f(x)=0, x∈R_xⁿ, y∈R_y ⁿ is proved. The main results are the following: If any point x belongs to a straight line connecting a pair of distinct solutions in the state space R_xⁿ, the Newton-Raphson iterative process goes along this line. If a loading process y(β) reaches a singular point of the problem, the corresponding trajectory of state variables x(β) in R_xⁿ tends to the right eigenvector nullifying the Jacobian matrix at the singular point. In any singular point of the quadratic problem, there are two solutions which merge at this point. The maximum number of solutions on any straight line in state the space R_xⁿ is two. Along a straight line through two distinct solutions of a quadratic problem, this problem can be reduced to a single scaler quadratic equation which locates these solutions. In addition, a number of other properties is reported. New proofs of them are given. There is a point of singularity in the middle of a straight line connecting a pair of distinct solutions in the state space R_xⁿ. A vector co-linear to a straight line connecting a pair of distinct solutions in R_x ⁿ nullifies the Jacobian matrix at the point of singularity in the middle of the line