A practical method is given for solving the power flow problem with control variables such as real and reactive power and transformer ratios automatically adjusted to minimize instantaneous costs or losses. The solution is feasible with respect to constraints on control variables and dependent variables such as load voltages, reactive sources, and tie line power angles. The method is based on power flow solution by Newton's method, a gradient adjustment algorithm for obtaining the minimum and penalty functions to account for dependent constraints. A test program solves problems of 500 nodes. Only a small extension of the power flow program is required to implement the method.

Optimal Power Flow Solutions

The finite element displacement method of analyzing structures involves the solution of large systems of linear algebraic equations with sparse, structured, symmetric coefficient matrices. There is a direct correspondence between the structure of the coefficient matrix, called the stiffness matrix in this case, and the structure of the spatial network delineating the element layout. For the efficient solution of these systems of equations, it is desirable to have an automatic nodal numbering (or renumbering) scheme to ensure that the corresponding coefficient matrix will have a narrow bandwidth. This is the problem considered by R. Rosen1. A direct method of obtaining such a numbering scheme is presented. In addition several methods are reviewed and compared.

Reducing the bandwidth of sparse symmetric matrices

The Dommel-Tinney approach to the calculation of optimal power-system load flows has proved to be very powerful and general. This paper extends the problem formulation and solution scheme by incorporating exact outage-contingency constraints into the method, to give an optimal steady-state-secure system operating point. The controllable system quantities in the base-case problem (e.g. generated MW, controlled voltage magnitudes, transformer taps) are optimised within their limits according to some defined objective, so that no limit-violations on other quantities (e. g. generator MVAR and current loadings, transmission-circuit loadings, load-bus voltage magnitudes, angular displacements) occur in either the base-case or contingency-case system operating conditions.

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Optimal Load Flow with Steady-State Security

This paper describes a simple, very reliable and extremely fast load-flow solution method with a wide range of practical application. It is attractive for accurate or approximate off-and on-line routine and contingency calculations for networks of any size, and can be implemented efficiently on computers with restrictive core-store capacities. The method is a development on other recent work employing the MW-?/ MVAR-V decoupling principle, and its precise algorithmic form has been determined by extensive numerical studies. The paper gives details of the method's performance on a series of practical problems of up to 1080 buses. A solution to within 0.01 MW/MVAR maximum bus mismatches is normally obtained in 4 to 7 iterations, each iteration being equal in speed to 1? Gauss-Seidel iterations or 1/5th of a Newton iteration. Correlations of general interest between the power-mismatch convergence criterion and actual solution accuracy are obtained.

Fast Decoupled Load Flow

Optimal power flow using particle swarm optimization

State estimation is a digital processing scheme which provides a real-time data base for many of the central control and dispatch functions in a power system. The estimator processes the imperfect information available and produces the best possible estimate of the true state of the system. The basic theory and computational requirements of static state estimation are presented, and their impact on the evolution of the data-acquisition, data- processing, and control subsystems are discussed. The feasibility of this technique is demonstrated on network examples.

State Estimation in Power Systems Part I: Theory and Feasibility

The compensation theorem is applied in conjunction with ordered triangular factorization of the nodal admittance matrix to simulate the effect of changes in the passive elements of the network on the solution of a problem without changing the factorization. The scheme includes network elements with mutual impedances. Compared with impedance matrix methods which are ordinarily used for power system applications, this method, which permits exploitation of matrix sparsity, always requires less computer storage and, with few exceptions, is much faster.

Compensation Methods for Network Solutions by Optimally Ordered Triangular Factorization

This paper describes some computer programing techniques for taking advantage of the sparsity of the admittance matrix. The techniques are based on two main ideas; (1) determination of a sequence of operations. which results in a near minimum of memory and computing, (2) preservation of these operations for repetition. Use of these techniques makes it possible to obtain significant reductions in memory and processing time for many network analysis programs. Claims are substantiated by actual results.

Techniques for Exploiting the Sparsity or the Network Admittance Matrix

The general problem of minimizing the operating cost of a power system by proper selection of the active and reactive productions is formulated as a nonlinear programming problem in accordance with previous work by Carpentier of Electricitede France. This general problem is particularized to the minimization of transmission line losses by suitable selection of the reactive productions and transformer tap settings. An efficient computational procedure based on the Newton-Raphson method for solving the power-flow equations and on the dual (Lagrangian) variables of the Kuhn and Tucker theorem is discussed. This minimization procedure has been applied successfully to a 500-node system studied by the Bonneville Power Administration for which an effective power-flow program had been developed previously. The dual variables associated with the primary (electrical) variables are obtained in the course of the computation, and their engineering significance for power system design and tariffication is emphasized. The procedure has been extended to the general case of combined active and reactive optimization.

William F. Tinney

Papers

Optimal Power Flow Solutions

State Estimation in Power Systems Part I: Theory and Feasibility

Compensation Methods for Network Solutions by Optimally Ordered Triangular Factorization

Techniques for Exploiting the Sparsity or the Network Admittance Matrix

Optimum Control of Reactive Power Flow