This paper introduces a new paradigm for solving AC Power Flow (ACPF) and AC Optimal Power Flow (ACOPF) with improved convergence robustness. This approach exploits the globally convergent properties of continuation methods. Continuation methods achieve robustness by generating a sequence of nonlinear problems and repeatedly and consistently providing good initial guesses for locally convergent nonlinear solvers such as Newton-Raphson. The Homotopy implemented in this paper, (referred to as Power Flow Homotopy, PFH), is formulated in a way that gradually transforms the "easy" DC into the "difficult" AC Power Flow. Successive changes of the homotopy parameter modify the system of equations from fully linear and convex DC into non-linear and non-convex AC (optimal) power flow. As a result, the AC solution is obtained with increased robustness and multiple AC power flow solutions can also be detected. Similarly, Optimal Power Flow Homotopy (OPFH) is defined for solving AC Optimal Power Flow, by gradually transforming the convex DC OPF problem. Simulation results provide a comparison between the simple Newton-Raphson method and PFH in terms of performance and quality of detected solution. Comparisons are also performed between the Interior-Point method and OPFH.

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