An application of Karmarkar's interior-point linear programming algorithm for multi-reservoir operations optimization

Optimization of multi-reservoir systems operations is typically a very large scale optimization problem. The following are the three types of optimization problems solved using linear programming (LP): (i) deterministic optimization for multiple periods involving fine stage intervals, for example, from an hour to a week (ii) implicit stochastic optimization using multiple years of inflow data, and (iii) explicit stochastic optimization using probability distributions of inflow data. Until recently, the revised simplex method has been the most efficient solution method available for solving large scale LP problems. In this paper, we show that an implementation of the Karmarkar's interior-point LP algorithm with a newly developed stopping criterion solves optimization problems of large multi-reservoir operations more efficiently than the simplex method. For example, using a Micro VAX II minicomputer, a 40 year, monthly stage, two-reservoir system optimization problem is solved 7.8 times faster than the advanced simplex code in MINOS 5.0. The advantage of this method is expected to be greater as the size of the problem grows from two reservoirs to multiples of reservoirs. This paper presents the details of the implementation and testing and in addition, some other features of the Karmarkar's algorithm which makes it a valuable optimization tool are illuminated.     

Storage-yield analysis of surface water reservoirs: the role of reliability constraints and operating policies

Stochastic optimization methods are used for optimal design and operation of surface water reservoir systems under uncertainty. Chance-constrained (CC) optimization with linear decision rules (LDRs) is an old approach for determining the minimum reservoir capacity required to meet a specific yield at a target level of reliability. However, this approach has been found to overestimate the reservoir capacity. In this paper, we propose the reason for this overestimation to be the fact that the reliability constraints considered in standard CC LDR models do not have the same meaning as in other models such as reservoir operation simulation models. The simulation models have fulfilled a target reliability level in an average sense (i.e., annually), whereas the standard CC LDR models have met the target reliability level every season of the year. Mixed integer nonlinear programs are presented to clarify the distinction between the two types of reliability constraints and demonstrate that the use of seasonal reliability constraints, rather than an average reliability constraint, leads to 80–150 % and 0–32 % excess capacity for SQ-type and S-type CC LDR models, respectively. Additionally, a modified CC LDR model with an average reliability constraint is proposed to overcome the reservoir capacity overestimation problem. In the second stage, we evaluate different operating policies and show that for the seasonal (average) reliability constraints, open-loop, S-type, standard operating policy, SQ-type, and general SQ-type policies compared to a model not using any operation rule lead to 190–460 % (200–550 %), 100–200 % (80–300 %), 0–90 % (0–60 %), 30–90 % (0–20 %), and 10–90 % (0–10 %) excess capacity, respectively.     

Flow modelling in gravel‐bed rivers: rethinking the bottom boundary condition

A key problem in computational fluid dynamics (CFD) modelling of gravel‐bed rivers is the representation of multi‐scale roughness, which spans the range from grain size, through bedforms, to channel topography. These different elements of roughness do not clearly map onto a model mesh and use of simple grain‐scale roughness parameters may create numerical problems. This paper presents CFD simulations for three cases: a plane bed of fine gravel, a plane bed of fine gravel including large, widely‐spaced pebble clusters, and a plane gravel bed with smaller, more frequent, protruding elements. The plane bed of fine gravel is modelled using the conventional wall function approach. The plane bed of fine gravel including large, widely‐spaced pebble clusters is modelled using the wall function coupled with an explicit high‐resolution topographic representation of the pebble clusters. In these cases, the three‐dimensional Reynolds‐averaged continuity and Navier–Stokes equations are solved using the standard k ? ε turbulence model, and model performance is assessed by comparing predicted results with experimental data. For gravel‐bed rivers in the field, it is generally impractical to map the bed topography in sufficient detail to enable the use of an explicit high‐resolution topography. Accordingly, an alternative model based on double‐averaging is developed. Here, the flow calculations are performed by solving the three‐dimensional double‐averaged continuity and Navier‐Stokes equations with the spatially‐averaged 〈k ? ε〉 turbulence model. For the plane bed of fine gravel including large, widely‐spaced pebble clusters, the model performance is assessed by comparing the spatially‐averaged velocity with the experimental data. The case of a plane gravel bed with smaller, more frequent, protruding elements is represented by a series of idealized hypothetical cases. Here, the spatially‐averaged velocity and eddy viscosity are used to investigate the applicability of the model, compared with using the explicit high‐resolution topography. The results show the ability of the model to capture the spatially‐averaged flow field and, thus, illustrate its potential for representing flow processes in natural gravel‐bed rivers. Finally, practical data requirements for implementing such a model for a field example are given. Copyright © 2011 John Wiley & Sons, Ltd.     

Predicting cumulative sediment depostition in large reservoirs

Sediment deposition and its accumulation in a large resorvoir depends on the inflow and reservoir storage content, respectively. Because of this fact it is possible to model the cumulative deposition of sediment as an additive process defined on a bivariate Markov chain. Using the bivariate Markov chain model the mean and variance of the cumulative deposition of John Martin Reservoir, Colorado, U.S.A. are estimated and compared with observed sedimentation data.     

Application of the distributed parameter filter to predict simulated tidal induced shallow water flow

Distributed parameter filtering theory is employed for estimating the state variables and associated error covariances of a dynamical distributed system under highly random tidal and meteorological influences. The stochastic-deterministic mathematical model of the physical system under study consists of the shallow water equations described by the momentum and continuity equations in which the external forces such as Coriolis force, wind friction, and atmospheric pressure are considered. White Gaussian noises in the system and measurement equations are used to account for the inherent stochasticity of the system. By using an optimal distributed parameter filter, the information provided by the stochastic dynamical model and the noisy measurements taken from the actual system are combined to obtain an optimal estimate of the state of the system, which in turn is used as the initial condition for the prediction procedure. The approach followed here has numerical approximation carried out at the end, which means that the numerical discretization is performed in the filtering equations, and not in the equations modelling the system. Therefore, the continuous distributed nature of the original system is maintained as long as possible and the propagation of modelling errors in the problem is minimized. The appropriateness of the distributed parameter filter is demonstrated in an application involving the prediction of storm surges in the North Sea. The results confirm excellent filter performance with considerable improvement with respect to the deterministic prediction.     

Application of the distributed parameter filter to predict simulated tidal induced shallow water flow

Distributed parameter filtering theory is employed for estimating the state variables and associated error covariances of a dynamical distributed system under highly random tidal and meteorological influences. The stochastic-deterministic mathematical model of the physical system under study consists of the shallow water equations described by the momentum and continuity equations in which the external forces such as Coriolis force, wind friction, and atmospheric pressure are considered. White Gaussian noises in the system and measurement equations are used to account for the inherent stochasticity of the system. By using an optimal distributed parameter filter, the information provided by the stochastic dynamical model and the noisy measurements taken from the actual system are combined to obtain an optimal estimate of the state of the system, which in turn is used as the initial condition for the prediction procedure. The approach followed here has numerical approximation carried out at the end, which means that the numerical discretization is performed in the filtering equations, and not in the equations modelling the system. Therefore, the continuous distributed nature of the original system is maintained as long as possible and the propagation of modelling errors in the problem is minimized. The appropriateness of the distributed parameter filter is demonstrated in an application involving the prediction of storm surges in the North Sea. The results confirm excellent filter performance with considerable improvement with respect to the deterministic prediction.     

Measuring Flume Surfaces for Hydraulics Research Using a Kodak DCS460

A critical problem in hydraulics research is accurate measurement of fluvially worked sediments, both in the field and in scaled representations of field situations in laboratory flumes. Such measurement must provide information on individual grain characteristics, and their organisation into structures referred to as bedforms. Existing measurement approaches are based upon mechanical or laser profiling devices, which are both expensive and take considerable time to acquire data, particularly where information is required at very high densities. This paper demonstrates how conventional automated terrain model extraction software, combined with image acquisition using a Kodak DCS460 digital camera, has been effective in generating digital elevation models of complex bed morphology. This has reduced time spent collecting data in the flume and has allowed data collection at much higher spatial and temporal densities. Application of the method is illustrated by research carried out at Hydraulics Research Wallingford. Issues discussed include configuration of photographs and control coordinates; appropriate camera calibration methods; stability of inner orientation of the Kodak DCS460; and accuracies obtained. Comparisons with independent check data reveal that accuracies of ±2.5mm have been achieved using a camera-to-object distance of 4.2 m.     

Tsunami Travel Time Atlas for the Atlantic Ocean

Compared to the Pacific Ocean, tsunamis are rare both in the Atlantic and Indian Oceans. However, the December 26, 2004, tsunami demonstrated that, no matter how rare they may be, when a major tsunami occurs, it could be very disastrous. The most basic information in tsunami warning center requires are charts showing tsunami travel times to various locations around the rim of the ocean. With this in mind, a tsunami travel time atlas for the Atlantic Ocean is in preparation. The Caribbean Sea is also included in this Atlas, as it is more or less a part of the Atlantic Basin.