Henning Rasmussen

Valuation and Optimal Operation of Electric Power Plants in Competitive Markets

We present an algorithm for the valuation and optimal operation of hydroelectric and thermal power generators in deregulated electricity markets. Real options theory is used to derive nonlinear partial-integro-differential equations (PIDEs) for the valuation and optimal operating strategies of both types of facilities. The equations are designed to incorporate a wide class of spot price models that can exhibit the same time-dependent, mean-reverting dynamics and price spikes as those observed in most electricity markets. Particular attention is paid to the operational characteristics of real power generators. For thermal power plants, these characteristics include variable start-up times and costs, control response time lags, minimum generating levels, nonlinear output functions, and structural limitations on ramp rates. For hydroelectric units, head effects and environmental constraints are addressed. We illustrate the models with numerical examples of a pump storage facility and a thermal power plant. This PIDE framework can achieve high levels of computational speed and accuracy while incorporating a wide range of spot price dynamics and operational characteristics.

Left ventricular ejection: model solution by collocation, an approximate analytical method.

A differential equation solution method that does not utilize numerical integration is demonstrated on a physiologic model. A previously proposed model of left ventricular ejection incorporating time-varying elastance, internal resistance, aortic inductance, and a three-component windkessel is solved by collocation. Assuming internal resistance to be constant allows simplification to a third order linear differential equation in left ventricular volume. A trigonometric series is used to approximate the solution and the coefficients of the series as well as the duration of ejection and pre-ejection periods are chosen so that the governing differential equation is exactly satisfied at certain times during ejection (the collocation points), as well as the boundary conditions and steady state condition.

Local quaternion Fourier transform and color image texture analysis

Color images can be treated as two-dimensional quaternion functions. For analysis of quaternion images, a joint space-wavenumber localized quaternion S transform (QS) is presented in this study for a simultaneous determination of the local color image spectra. The QS transform uses a two-dimensional Gaussian localizing window that scales with wavenumbers. Rotation invariance, invertibility and computational aspects of the QS transform are discussed. The power map of the QS transform is presented here as a powerful tool in color image texture and pattern analysis. Examples are presented.

The trinion Fourier transform of color images

Any color may be represented in terms of three components (RGB or HSL) or four components (CMYK). For the four component color representation the use of quaternions, with one real and three imaginary components, is natural. By setting one component to zero, quaternions have been used in RGB or HSL representation of colors and color images. In this paper a new quantity, trinion, with one real and two imaginary components, is introduced and its use in color image representation is examined. The goal is to see if significant efficiencies in representation, analysis and computation involving three component color images accrue with the use of trinions. Two versions of the trinion Fourier transform (TFT) are introduced and it is shown that using TFT is preferable for combined analysis of three component color images rather than separate monochromatic analysis of each component and use of quaternions. Joint space-wavenumber localized trinion S (TS) transform with a two-dimensional Gaussian window function that scales with wavenumbers is also presented. Invertibility, rotation invariance, and computational aspects of the TS transform are discussed.

Competitive pricing with dynamic asymmetric price effects

We model the temporal pricing strategies for two firms with asymmetric costs and differing market power (i.e. big-box retailer versus smaller local merchant). A firms demand is a function of its price, a reference price and its competitors price. Price effects may be asymmetric, i.e. consumers respond differently if they perceive a good to be over-priced versus under-priced. We derive analytical results for optimal prices. We show through a series of numerical examples under what settings firms choose various pricing strategies as well as profit implications for firms with differing costs.

Analysis of a class of distributed delay logistic differential equations

In this paper, we consider the illustrative example of generalised logistic equations where the carrying-capacity effect is modelled by a distributed-delay effect (which may be over the infinite past). These distributed delay differential equations, though simple in structure, possess a rich array of solutions. If the delay is sufficiently large a supercritical Hopf bifurcation occurs, which finally disappears asymptotically when the delay becomes distributed infinitely. This mirrors the situation when there is just a point delay. Similar models with two or more state variables occur in pasture mixtures.

Assessing cellular automata based models using partial differential equations

This paper presents a novel method of assessing cellular automata based models. The cellular model considered in this paper was designed to simulate sediment transport and topographic changes in rivers. It is demonstrated how prior knowledge about partial differential equations and their solution schemes can be used to provide deeper insight into the performance of two numerical implementations for this cellular model. Assessment of the implementations as solution schemes of the derived partial differential equations showed that the cellular model is robust for two numerical implementations, one of which is superior in terms of accuracy, stability and convergence. In comparison with more sophisticated solution methods, both implementations return accurate results. This study encourages the further application of cellular automata based models to complex problems.

Nonreflecting boundary condition for the Helmholtz equation

Abstract To solve the Helmholtz equation in an infinite three-dimensional domain a spherical artificial boundary is introduced to restrict the computational domain Ω. To determine the nonreflecting boundary condition on ∂Ω, we start with a finite number of spherical harmonics for the Helmholtz equation. With a precise choice of (primary) nodes on the sphere, the theorem on Gauss-Jordan quadrature establishes the discrete orthogonality of the spherical harmonics when summed over these nodes. An approximate nonreflecting boundary condition for the Helmholtz equation follows readily upon solving the exterior Dirichlet problem. The accuracy of the boundary condition is determined using a point source, and the computational results are presented for the scattering of a wave from a sphere.

A compact method for second-order boundary value problems on nonuniform grids

Abstract A compact finite difference procedure is developed for the numerical solution of second-order two point boundary value problems with mixed boundary conditions. The procedure is applied to a series of linear and nonlinear problems on both uniform and nonuniform grids. In all cases the global estimates of the rate of convergence tend to four. In the last part of the paper, the procedure is extended to a nonlinear partial differential equation, Burgers equation.

A numerical study of unstable Hele-Shaw flow

Abstract A numerical procedure which is based on an integral equation for the normal velocity at the interface is developed for the unstable flow with surface tension in a Hele-Shaw cell. The procedure has been validated by comparing solutions obtained by it with published results. It has also been applied to a study of the effect of small changes in the initial data on the later profiles and it was found that even very small differences can lead to large differences in the profiles at later times.