The ac power flow problem can be solved efficiently by Newton's method. Only five iterations, each equivalent to about seven of the widely used Gauss-Seidel method, are required for an exact solution. Problem dependent memory and time requirements vary approximately in direct proportion to problem size. Problems of 500 to 1000 nodes can be solved on computers with 32K core memory. The method, introduced in 1961, has been made practical by optimally ordered Gaussian elimination and special programming techniques. Equations, programming details, and examples of solutions of large problems are given.

Power Flow Solution by Newton's Method

A survey is presented on the currently available numerical techniques for power-system load-flow calculation using the digital computer. The review deals with methods that have received widespread practical application, recent attractive developments, and other methods that have interesting or useful characteristics. The analytical bases, computational requirements, and comparative numerical performances of the methods are discussed. Attention is given to the problems and techniques of adjustments in load-flow solutions, and the suitabilities of various methods for modern applications such as security monitoring and optimal load flow are examined.

Review of load-flow calculation methods

The conventional Newton-Raphson power flow study has been reformulated to permit the inclusion of nonlinear loads. These loads give rise to harmonic signals which propagate throughout the power system. The reformulation is based on the reduction to zero of the mismatch active power and reactive voltamperes, the imbalance current at harmonic frequencies, and the mismatch apparent Voltamperes. Conclusions on the existence of positive, negative, and zero sequence signals are made for harmonic frequencies. The harmonic power flow study formulation is illustrated for a three phase full wave bridge rectifier. A companion paper presents a discussion of the actual computer implementation and several practical examples.

Harmonic Power Flow Studies Part I - Formulation and Solution

Automatic Calculation of Load Flows

A comparison in the context of sparse matrices is made between the Product Form of the Inverse PFI (a form of Gauss-Jordan elimination) and the Elimination Form of the Inverse EFI (a form of Gaussian elimination). The precise relation of the elements of these two forms of the inverse is given in terms of the nontrivial elements of the three matrices L, U, U-1 associated with the triangular factorization of the coefficient matrix A; i.e., A = L. U, where L is lower triangular and U is unit upper triangular. It is shown that the zero- nonzero structure of the PFI always has more nonzeros than the EFI. It is proved that Gaussian elimination is a minimal algorithm with respect to preserving sparseness if the diagonal elements of the matrix A are nonzero. However, Gaussian elimination is not nec- essarily minimal if A has some zero diagonal elements. The same statements hold for the PFI as well. A probabilistic study of fill-in and computing times for the PFI and EF sparse ma- trix algorithms is presented. This study suggests quantitatively how rapidly sparse matrices fill up for increasing densities, and emphasizes the necessity for reordering to minimize fill-in. I. Introduction. A sparse matrix is a matrix with very few nonzero elements. In many applications, a rough rule seems to be that there are O(N) nonzero entries; typically, say 2 to 10 nonzero entries per row. If the dimension N of the matrix is not large, then there is no compelling reason to treat sparse matrices differently from full matrices. It is when N becomes large and one attempts computations with the sparse matrix that it becomes necessary to take advantage of the zeros. The reason for this is obvious: there is a storage requirement of the order of N2 and arithmetic operations count of the order of N3 for many matrix algorithms using the full matrix. On the other hand, by storing only nonzero quantities and using logical operations to decide when an arithmetic operation is necessary, the storage requirement and arithmetic operations count can be reduced by a factor of N in many instances. Of course, this not only becomes a sizable savings of computer time, but also dictates whether or not some problems can be attempted. Computations with sparse matrices are not new. Iterative techniques for these matrices, especially those related to the solution of partial differential equations have been extensively developed (1). Sparse matrix methods for solving linear equa- tions by direct methods have been used for a long time in linear programming and there is a large body of literature, computational experience, programs, and artfulness, which has been built up in this area (2)-(15). In most linear programming codes, the product form of the inverse (PFI) is the method used to solve linear equations (16)-(19), although there are exceptions (20)-(21). Methods for scaling, pivoting for accuracy and sparseness, structuring data, and handling input-output have been extensively developed (22)-(72). However, there do not seem to exist rigorous results

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Rodney J. Brown

Papers

Digital Solutions for Large Power Networks