Reijer Idema

Scalable Newton-Krylov Solver for Very Large Power Flow Problems

The power flow problem is generally solved by the Newton-Raphson method with a sparse direct solver for the linear system of equations in each iteration. In this paper, alternatives based on iterative linear solvers are presented that are faster and scale much better in the problem size, making them ready for the ever-growing power systems of the future. For the largest test problem, with around one million busses, the presented alternative is over 120 times faster than using a direct solver.

Towards Faster Solution of Large Power Flow Problems

Current and future developments in the power system industry demand fast power flow solvers for larger power flow problems. The established methods are no longer viable for such problems, as they are not scalable in the problem size. In this paper, the use of Newton-Krylov power flow methods is proposed, and a multitude of preconditioning techniques for such methods are discussed and compared. It is shown that incomplete factorizations can perform very well as preconditioner, resulting in a solver that scales in the problem size. It is further shown that using a preconditioned inner-outer Krylov method has no significant advantage over applying the preconditioner directly to the outer iterations. Finally, algebraic multigrid is demonstrated as a preconditioner for Newton-Krylov power flow and argued to be the method of choice in some scenarios.

Fast Newton load flow

The Newton-Raphson method is widely used to solve load flow problems. Traditionally a direct solver is used to solve the linear systems within this method. In this paper we explore the use of an iterative method to solve the linear systems, leading to an inexact Newton-Krylov method. The main parameters of this method are the preconditioner and the forcing terms. Several candidate choices for these parameters are discussed and tested. With the proper preconditioner, and forcing terms, the inexact Newton-Krylov method is shown to greatly improve on using a direct solver.

Computational Methods in Power System Analysis

This book treats state-of-the-art computational methods for power flow studies and contingency analysis. In the first part the authors present the relevant computational methods and mathematical concepts. In the second part, power flow and contingency analysis are treated. Furthermore, traditional methods to solve such problems are compared to modern solvers, developed using the knowledge of the first part of the book. Finally, these solvers are analyzed both theoretically and experimentally, clearly showing the benefits of the modern approach.

On the convergence of inexact Newton methods

The inexact Newton method is widely used to solve systems of non-linear equations. It is well-known that forcing terms should be chosen relatively large at the start of the process, and be made smaller during the iteration process. This paper explores the mechanics behind this behavior theoretically using a simplified problem, and presents theory that shows a proper choice of the forcing terms leads to a reduction in the non-linear error that is approximately equal to the forcing term in each Newton iteration. Further it is shown that under certain conditions the inexact Newton method converges linearly in the number of linear iterations performed throughout all Newton iterations. Numerical experiments are presented to illustrate the theory in practice.

Power Flow Test Cases

To conduct numerical experiments with power flow solvers, a test set of power flow problems is needed. Problems with up to a few hundred buses are readily available, but problems of realistic size are hard to come by. This chapter treats the test cases used in this book and explains how they were constructed by copying a smaller model and interconnecting the copies with new transmission lines.

Newton–Krylov Power Flow Solver

Newton power flow solvers traditionally use a direct method to solve the linear systems. For large linear systems of equations with a sparse coefficient matrix, iterative linear solvers are generally more efficient than direct solvers. Using a Krylov method to solve the Jacobian systems in the Newton-Raphson method, leads to a Newton-Krylov method. In this chapter, computational aspects of Newton-Krylov power flow solvers are discussed. It is shown that direct solvers, and other methods using the LU factorisation, scale very badly in the problem size. The alternatives proposed in this chapter are faster for all tested problems and have near linear scaling, thus being much faster for large power flow problems. The largest test problem, with a million buses, takes over an hour to solve using a direct solver, while a Newton-Krylov solver can solve it in less than 30 seconds. That is 120 times faster.

Solving Nonlinear Systems of Equations

It is not possible to solve a general nonlinear equation analytically, let alone a general nonlinear system of equations. However, there are iterative methods to find a solution for such systems. The Newton-Raphson algorithm is the standard method for solving nonlinear systems of equations. Most, if not all, other well-performing methods can be derived from the Newton-Raphson algorithm. In this chapter the Newton-Raphson method is treated, as well as some common variations.

Traditional Power Flow Solvers

As long as there have been power systems, there have been power flow studies. This chapter discusses the two traditional methods to solve power flow problems: Newton power flow and Fast Decoupled Load Flow (FDLF).

Solving Linear Systems of Equations

A solver for systems of linear equations can either be a direct method, or an iterative method. Direct methods calculate the solution to the problem in one pass. Iterative methods start with some initial vector, and update this vector in every iteration until it is close enough to the solution. Direct methods are very well-suited for smaller problems, and for problems with a dense coefficient matrix. For large sparse problems, iterative methods are generally much more efficient than direct solvers.