Applications of Homotopy for solving AC Power Flow and AC Optimal Power Flow

Abstract

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.

Simple certificate of solvability of power flow equations for distribution systems

“…Another important application can be developed from simple the solvability criterion described in (9). The simple criterion can be incorporated as a security constraint in planning and operational problems such as OPF.…”

“…In order to overcome such handicaps, numerous attempts have been made to design more efficient and accurate ways to solve or certify existence of solutions to those equations [7]- [9]. Among them, in [10], Bolognani proposed a method of certifying the solvability of power flow equations using the Banach Fixed Point Theorem.…”

Robust Convergence of Power Flow Using TX Stepping Method with Equivalent Circuit Formulation

“…To tackle these challenges we propose the use of homotopy methods to ensure convergence for the system to the correct physical solution independent of its complexity or scale. Homotopy methods that have been proposed in the past [2], [8] have suffered from convergence to low voltage solutions [2] and divergence. Furthermore, none of the previously proposed homotopy methods are known to scale up to test systems [9] that are of the scale of European or the US grids, which is essential for secure operation and operation of these systems.…”

“…In this paper, we propose a homotopy continuation method that we refer to as "Tx Stepping" to achieve robust convergence for large scale power flow cases from a set of arbitrary initial guesses. Homotopy methods have been previously studied for the power flow problem [2], [8] but have mostly been unsuccessful due to convergence to low voltage solutions or their inability to scale to large cases [9]. Tx stepping is based on the physics of the grid and takes inspiration from the "gmin stepping" method in the circuit simulation domain.…”

Emulating AC OPF solvers for Obtaining Sub-second Feasible, Near-Optimal Solutions

“…In the 1,354-bus test case, load profiles were generated that challenged the MIPS solver. Typically for these difficult cases, continuation methods can be used to robustly solve the AC OPF by solving a series of simpler OPF problems [8]. While robust, these methods can be very time consuming and may not be suitable for real-time operation.…”