Nonlinear programming using an expanded Lagrangian function, a water resources management case study

Abstract

Optimal planning and operation of large hydro-power systems, when realistically considered, usually result in non-linear, non-convex optimization problems of high dimension which can be difficult to solve using most optimization techniques. Our goal is to use a special form of potential function called the Expanded Lagrangian Function cbombined with the trust region algorithm to solve large-scale optimization problems arising in the applications of water resources management problems.

Our trust region algorithm uses a linear combination of an inexact Newton's direction and a steepest descent direction, to obtain a feasible descent direction. A bi-dimensional trust region scheme is used to obtain fast convergence. The inexact Newton's direction is obtained by solving a linear system of equations using a preconditioned conjugate gradient method which uses drop-tolerance pre-conditioner with RCM ordering.

The proposed method is tested on real data of 90 years of information for the Great Lakes water resources problem. The same application is solved with LANCELOT, using two different features of this software.

The results of the studies have shown that both algorithms converge to optimum objective values within a 3.0% difference from each other with LANCELOT providing worser objective values in most cases. Computer time required by both algorithms are comparable, with LANCELOT being somewhat slower.

The optimal storage levels and releases obtained from the proposed method when compared with past operations provide a significantly better operation.