Newton-Krylov Methods in Power Flow and Contingency Analysis

Abstract

A power system is a system that provides for the generation, transmission, and distribution of electrical energy. Power systems are considered to be the largest and most complex man-made systems. As electrical energy is vital to our society, power systems have to satisfy the highest security and reliability standards. At the same time, minimising cost and environmental impact are important issues. Steady state power system analysis plays a very important role in both operational control and planning of power systems. Essential tools are power flow (or load flow) studies and contingency analysis. In power flow studies, the bus voltages in the power system are calculated given the generation and consumption. In contingency analysis, equipment outages are simulated to determine whether the system can still function properly if some piece of equipment were to break down unexpectedly. The power flow problem can be mathematically expressed as a nonlinear system of equations. It is traditionally solved using the Newton-Raphson method with a direct linear solver, or using Fast Decoupled Load Flow (FDLF), an approximate Newton method designed specifically for the power flow problem. The Newton-Raphson method has good convergence properties, but the direct solver solves the linear system to a much higher accuracry than needed, especially in early iterations. In that respect the FDLF method is more efficient, but convergence is not as good. Both methods are slow for very large problems, due to the use of the LU decomposition. We propose to solve power flow problems with Newton-Krylov methods. Newton-Krylov methods are inexact Newton methods that use a Krylov subspace method as linear solver. We discuss which Krylov method to use, investigate a range of preconditioners, and examine different methods for choosing the forcing terms. We also investigate the theoretical convergence of inexact Newton methods. The resulting power flow solver offers the same convergence properties as the Newton-Raphson method with a direct linear solver, but eliminates both the need for oversolving, and the need for an LU factorisation. As a result, the method is slightly faster for small problems while scaling much better in the problem size, making it much faster for very large problems. Contingency analysis gives rise to a large number of very similar power flow problems, which can be solved with any power flow solver. Using the solution of the base case as initial iterate for the contingency cases can help speed up the process. FDLF further allows the reuse of the LU factorisation of the base case for all contingency cases, through factor updating or compensation techniques. There is no equivalent technique for Newton power flow with a direct linear solver. We show that Newton-Krylov power flow does allow such techniques, through the use of a single preconditioner for all contingency cases. Newton-Krylov power flow thus allows very fast contingency analysis with Newton-Raphson convergence.