A new approach to load flow analysis using Krylov subspace methods for well conditioned systems

Power system load flow analysis mainly utilizes the Gauss-Seidel method, the Newton-Raphson method, and the Fast Decoupled Load Flow method. All these stationary iterative algorithms assure convergence for a limited class of well-conditioned matrices, and require a good enough estimate of nodal voltages at all system busbars under consideration, to provide assured convergence. The Krylov subspace methods are widely generalized in their approach, and work by forming an orthogonal basis of the sequence of successive matrix powers times the initial residual (the Krylov sequence). The prototypical method in this class is the conjugate gradient method (CG). In this work, we propose to apply the conjugate gradient algorithm to the sparse systems; we encounter these in the system admittance matrices, and we will search for a numerical solution to this system using the locally optimal steepest descent method. The system admittance matrices for an IEEE 30-bus or 57-bus system(s) are too large to be handled by direct methods like the Cholesky decomposition method. Hence, we will make use of the flexible preconditioned conjugate-gradient method, which makes use of sophisticated preconditioners, leading to variable preconditioning that change between successive iterations. The Polak-Ribière formula, a highly efficient preconditioner, is applied to the system, to yield drastic improvements in convergence. Our experimental results include a comparison of the Krylov subspace method with traditional methods, assuming the IEEE five-busbar, seven-line reference system as the common basis for all load-flow analysis. The system base quantities are VA_base = 100 MVA and V_base = 132 kV. The results show an overall better assurance of convergence for all general systems, a lesser dependence on starting voltage profiles assumption and a robustness and efficiency of computation for well-conditioned systems.