This paper presents a methodology for the solution of multistage stochastic optimization problems, based on the approximation of the expected-cost-to-go functions of stochastic dynamic programming by piecewise linear functions. No state discretization is necessary, and the combinatorial "explosion" with the number of states (the well known "curse of dimensionality" of dynamic programming) is avoided. The piecewise functions are obtained from the dual solutions of the optimization problem at each stage and correspond to Benders cuts in a stochastic, multistage decomposition framework. A case study of optimal stochastic scheduling for a 39-reservoir system is presented and discussed.

Multi-stage stochastic optimization applied to energy planning

This paper describes a mathematical framework for the solution of the economic dispatch problem with security constraints, which can take into account the system corrective capabilities after the outage has occurred. The solution algorithm is based on mathematical programming decomposition techniques that allow the iterative solution of a "base-case" economic dispatch and separate contingency analysis with generation rescheduling to eliminate constraint violations. The resulting dispatch has the same security level as the usual security-constrained dispatch, but with lower operating costs. Case studies with the IEEE 118-bus are presented. The extension of the methodology to the probabilistic case will be discussed in a forthcoming paper.

Security-Constrained Optimal Power Flow with Post-Contingency Corrective Rescheduling

An inexact two-stage stochastic programming (ITSP) model is proposed for water resources management under uncertainty. The model is a hybrid of inexact optimization and two-stage stochastic programming. It can reflect not only uncertainties expressed as probability distributions but also those being available as intervals. The solution meth od for ITSP is computationally effective, which makes it applicable to practical problems. The ITSP is applied to a hypothetical case study of water resources system operation. The results indicate that reasonable solutions have been obtained. They are further analyzed and interpreted for generating decision alternatives and identifying significant factors that affect the system's performance. The information obtained through these post-optimality analyses can provide useful decision support for water managers.

An inexact two-stage stochastic programming model for water resources management under uncertainty

This paper presents a new stochastic decomposition method well-suited to deal with large-scale unit commitment problems. In this approach, random disturbances are modeled as scenario trees. Optimization consists in minimizing the average generation cost over this "tree-shaped future". An augmented Lagrangian technique is applied to this problem. At each iteration, nonseparable terms introduced by the augmentation are linearized so as to obtain a decomposition algorithm. This algorithm may be considered as a generalization of price decomposition methods, which are now classical in this field, to the stochastic framework. At each iteration, for each unit, a stochastic dynamic subproblem has to be solved. Prices attached to nodes of the scenario trees are updated by the coordination level. This method has been applied to a daily generation scheduling problem. The use of an augmented Lagrangian technique, provides satisfactory convergence properties to the decomposition algorithm. Moreover, numerical simulations show that compared to a classical deterministic optimization with reserve constraints, this new approach achieves substantial savings.

Stochastic optimization of unit commitment: a new decomposition framework

An interval-parameter multi-stage stochastic programming model for water resources management under uncertainty

A computational scheme that is able to determined at each stage the most economical generation decision for each plant of a hydrothermal system is presented. The algorithm is based on the stochastic extension of Benders decomposition and can be implemented from already existing operation models. The results can be used in the weekly or monthly generation scheduling activities in real-time operation. A case study with 37 reservoirs of the Brazilian system is presented and discussed.

L. M. V. G. Pinto

Papers

Stochastic Optimization of a Multireservoir Hydroelectric System: A Decomposition Approach