Method for direct calculation of quadratic turning points

For a given one-parameter nonlinear system, the simplest bifurcation is the quadratic turning bifurcation where the Jacobian matrix becomes singular due to rank deficiency 1. To overcome the difficulty in solving the quadratic turning point caused by the singularity of the Jacobian matrix, the conventional Newton method can be applied to the so-called Moore-Spence determination system to solve for the quadratic turning point. However, the Moore-Spence system has much higher dimensions and causes much more complexity in factorisation of the extended Jacobian matrix. In the paper, by introducing an auxiliary variable and an auxiliary linear equation into Newton iterations in solving the Moore-Spence determination system, a matrix reduction technique can be worked out to solve the Moore-Spence extended equations much more efficiently. The high dimensions of the matrix can thus be reduced and the complexity involved in matrix factorisation can be reduced noticeably. The technique is proposed for general nonlinear systems. Formulation is derived for applying this technique to solving quadratic turning points, or say nose points, on load-flow solution curves of power systems. Computer tests on the IEEE 30-busbar system and a 2416-busbar East China power system are reported to show the effectiveness of the suggested technique.