Numerical polynomial homotopy continuation method to locate all the power flow solutions

Abstract

Abstract-The manuscript addresses the problem of finding all solutions of power flow equations or other similar nonlinear system of algebraic equations. This problem arises naturally in a number of power systems contexts, most importantly in the context of direct methods for transient stability analysis and voltage stability assessment. We introduce a novel form of homotopy continuation method called the numerical polynomial homotopy continuation (NPHC) method that is mathematically guaranteed to find all the solutions without ever encountering a bifurcation. The method is based on embedding the real form of power flow equation in complex space, and tracking the generally unphysical solutions with complex values of real and imaginary parts of the voltage. The solutions converge to physical real form in the end of the homotopy. The so-called γ-trick mathematically rigorously ensures that all the paths are wellbehaved along the paths, so unlike other continuation approaches, no special handling of bifurcations is necessary. The method is embarrassingly parallelizable and can be applied to reasonably large sized systems. We demonstrate the technique by analysis of several standard test cases up to the 14-bus system size. Finally, we discuss possible strategies for scaling the method to large size systems, and propose several applications for transient stability analysis and voltage stability assessment.

An efficient method to locate all the load flow solutions - revisited

“…This revised method succeeds in finding all realvalued solutions to those system for which we know the solutions including the counter example of [2], and the systems presented in [8][9][10]. We also analyze the IEEE 14 bus system for which we find 30 solutions, consistent with a report in [12].…”

“…The Bezout bound for which initial conditions can be readily created is 67,108,864 for this case. In [12] the authors use a different bounding technique and find the 30 solutions using 49,283,072 traces.…”

“…At the time [10] was written the authors were not able to contemplate solving 14-bus systems. The authors of [12] used high performance computing to complete this task. This brute force technique appears to still be impractical for most systems.…”

“…Because of this counterexample, the only algorithms that can provably find all solutions, such as the homotopy types used in [10] and [12], require the calculation of all real and complex solutions. Given the that the number of realvalued solution is very small compared to the number of complex solutions, these approaches are not efficient.…”

Algebraic geometrization of the Kuramoto model: Equilibria and stability analysis

“…The method has also recently used in solving the power-flow systems [30,31]. For example, consider solving a system of m polynomial equations in m variables, f (x) = (f 1 (x), .…”