Introduction to linear and nonlinear programming

A dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition (POD) method for constructing reduced-order models for time dependent and/or parametrized nonlinear partial differential equations (PDEs). In the presence of a general nonlinearity, the standard POD-Galerkin technique reduces dimension in the sense that far fewer variables are present, but the complexity of evaluating the nonlinear term remains that of the original problem. The original empirical interpolation method (EIM) is a modification of POD that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by POD. We propose a discrete empirical interpolation method (DEIM), a variant that is suitable for reducing the dimension of systems of ordinary differential equations (ODEs) of a certain type. As presented here, it is applicable to ODEs arising from finite difference discretization of time dependent PDEs and/or parametrically dependent steady state problems. However, the approach extends to arbitrary systems of nonlinear ODEs with minor modification. Our contribution is a greatly simplified description of the EIM in a finite-dimensional setting that possesses an error bound on the quality of approximation. An application of DEIM to a finite difference discretization of the one-dimensional FitzHugh-Nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long-time integration that fully captures nonlinear limit cycle behavior. We also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results.

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Nonlinear Model Reduction via Discrete Empirical Interpolation

With construction of new large-scale water storage projects on the wane in the U.S. and other developed countries, attention must focus on improving the operational effectiveness and efficiency of existing reservoir systems for maximizing the beneficial uses of these projects. Optimal coordination of the many facets of reservoir systems requires the assistance of computer modeling tools to provide information for rational management and operational decisions. The purpose of this review is to assess the state-of-the-art in optimization of reservoir system management and operations and consider future directions for additional research and application. Optimization methods designed to prevail over the high-dimensional, dynamic, nonlinear, and stochastic characteristics of reservoir systems are scrutinized, as well as extensions into multiobjective optimization. Application of heuristic programming methods using evolutionary and genetic algorithms are described, along with application of neural networks and fuzzy rule-based systems for inferring reservoir system operating rules.

Optimal Operation of Multireservoir Systems: State-of-the-Art Review

The objective of this paper is to review the state-of-the-art of mathematical models developed for reservoir operations, including simulation. Algorithms and methods surveyed include linear programming (LP), dynamic programming (DP), nonliner programming (NLP), and simulation. A general overview is first presented. The historical development of each key model is critically reviewed. Conclusions and recommendations for future research are presented.

Reservoir Management and Operations Models: A State‐of‐the‐Art Review

The purpose of this survey is to review parameter identification procedures in groundwater hydrology and to examine computational techniques which have been developed to solve the inverse problem. Parameter identification methods are classified under the error criterion used in the formulation of the inverse problem. The problem of ill-posedness in connection with the inverse problem is addressed. Typical inverse solution techniques are highlighted. The review also includes the evaluation of methods used for computing the sensitivity matrix. Statistics which can be used to estimate the parameter uncertainty are outlined. Attempts have been made to compare and contrast representative inverse procedures, and direction for future research is suggested.

Review of Parameter Identification Procedures in Groundwater Hydrology: The Inverse Problem

A practical monthly optimization model, called SISOPT, is developed for the management and operations of the Brazilian hydropower system. The system, one of the largest in the world, consists of 75 hydropower plants with an installed capacity of 69,375 MW, producing 92% of the nation's electrical power. The system size and nonlinearity pose a real challenge to the modelers. The basic model is formulated in nonlinear programming ~NLP!. The NLP model is the most general formulation and provides a foundation for analysis by other methods. The formulated NLP model was first linearized by two different linearization techniques and solved by linear programming ~LP!. A comparative analysis was made of the results obtained from the linearized and the NLP models. The results show that the simplest linearized model ~referred to as the LP model! without iteration is suitable for planning purposes. For example, the LP model could be used in system capacity expansion studies or to explore various design parameters in connection with feasibility studies, where details in storage variation are not as important as the power production. With a good initial policy provided by the LP model, the successive linear programming ~SLP! model produced excellent results with fast convergence. The NLP model, the most complex and accurate model in the suite, is particularly suited for setting up guidelines for real-time operations using inflow forecast with frequent updating. The performance of the NLP model was checked against the historical operational records, and the comparison yields indica- tions of superior performance.

Optimization of Large-Scale Hydropower System Operations

Some basic concepts and methods for solving the coupled inverse problem in groundwater modeling are presented in a series of two papers. Because there are crossover effects between state variables and parmeters, coupled inverse problems are different from single inverse problems in several aspects, especially in the selection of objective functions, identifiability, and experimental design. This paper discusses the general definition of coupled inverse problems posed in the framework of multiobjective optimization. In order to obtain gradient vectors of objective functions as well as all sensitivity coefficients, a general procedure for deriving the adjoint state equations and their associated conditions is described. Using the rules of adjoint operation presented in this paper, the derivation of adjoint problems becomes very easy, even for nonlinear and transient coupled problems. As examples, the adjoint problem of flow-mass transport (or flow-heat transport), the adjoint problem of saltwater intrusion, and the adjoint problem of two-phase flow are derived. Numerical experiments on a two-dimensional flow-mass transport problem are given to explain the procedure of sensitivity analysis and parameter identification for coupled problems. The hydraulic conductivity can be identified by using observations of head and/or solute concentration.

Coupled inverse problems in groundwater modeling: 1. Sensitivity analysis and parameter identification

This research develops a mixed integer linear programming (MILP) model that considers simultaneously both the traditional reservoir rule curves and the hedging rules to manage and operate a multipurpose, multireservoir system. During normal periods of operation, when inflows are plentiful, this optimization model efficiently distributes the available stored water from different reservoirs to meet the planned demands imposed by competing users. However, during periods of drought, or when anticipating a drought, the planned demands cannot be fully met, and a water shortage occurs. By considering the hedging rules along with the rule curves, guidelines are provided for reservoir releases. To minimize the impact of drought, the hedging rules effectively reduce the ongoing water supply to balance with the target storage requirement. The MILP model is applied to a multireservoir system in the southern region of Taiwan, where the results obtained demonstrate the applicability and utility of the model.

William W.-G. Yeh

Papers

Reservoir Management and Operations Models: A State‐of‐the‐Art Review

Review of Parameter Identification Procedures in Groundwater Hydrology: The Inverse Problem

Optimization of Large-Scale Hydropower System Operations

Coupled inverse problems in groundwater modeling: 1. Sensitivity analysis and parameter identification

Optimization of Reservoir Management and Operation with Hedging Rules