In this paper, we consider the manifold that describes all feasible power flows in a power system as an implicit algebraic relation between nodal voltages (in polar coordinates) and nodal power injections (in rectangular coordinates). We derive the best linear approximant of such a relation around a generic solution of the power flow equations. Our linear approximant is sparse, computationally attractive, and preserves the structure of the power flow. Thanks to the full generality of this approach, the proposed linear implicit model can be used to obtain a fast approximate solution of a possibly unbalanced three phase power system, with either radial or meshed topology, and with general bus models. We demonstrate how our approximant includes standard existing linearizations, we validate the quality of the approximation via simulations on a standard testbed, and we illustrate its applicability with case studies in scenario-based optimization and cascading failures.

Fast power system analysis via implicit linearization of the power flow manifold

Several exact fast static structural reanalysis techniques, introduced by researchers mostly for truss structures and some for frames and plate structures, are reviewed. Most utilize the property that the solution of a system of linear equations can be updated inexpensively when the matrix is changed by a low-rank increment. This paper shows that these methods are variants of the well-known Sherman–Morrison and Woodbury (SMW) formulas for the update of the inverse of a matrix. In addition, the paper extends the low-cost linear reanalysis in the spirit of the SMW formulas to some non-linear reanalysis problems. For a linear reanalysis, the extension reduces to the SMW formulas. Copyright © 2001 John Wiley & Sons, Ltd.

Fast exact linear and non‐linear structural reanalysis and the Sherman–Morrison–Woodbury formulas

In this paper a new method for computing the solution of a linear system having a symmetric circulant tridiagonal matrix is presented. This special kind of system appears in many applications. After an appropriate partition of the system and the elimination of the last unknown, we apply the Woodbury formula to arrive at a very efficient and stable method, which is quite competitive with Gaussian elimination and with the modified double sweep method.

A new method for solving symmetric circulant tridiagonal systems of linear equations

Effective algorithms with circulant-block matrices

In this paper, a fast algorithm for solving the special tridiagonal system is presented. This special tridiagonal system is a symmetric diagonally dominant and Toeplitz system of linear equations. The error analysis is also given. Our algorithm is quite competitive with the Gaussian elimination, cyclic reduction, specialLU factorization, reversed triangular factorization, and Toeplitz factorization methods. In addition, our result can be applied to solve the near-Toeplitz tridiagonal system. Some examples demonstrate the good efficiency and stability of our algorithm.

A fast algorithm for solving special tridiagonal systems

The solution of linear systems having circulant coefficient matrices iss considered in this paper. This kind of system occurs in many applications: prediction, time series analysis, spline approxim...

On the solution of circulant linear systems

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On the solution of circulant linear systems