Matt Davison

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.

A Hybrid System-Econometric Model for Electricity Spot Prices: Considering Spike Sensitivity to Forced Outage Distributions

With the current trend in deregulation, all electricity markets have been subject to volatile electricity prices, typically in peak season. As markets mature, new financial and operational risk management instruments are becoming available. In order to price such instruments, a model for the underlying price process is required. In this paper a hybrid model is described that contains aspects of the power system, as well as the historical time-series of spot prices observed in the market. By including both aspects of the problem, a model with both economic and system-based aspects is created. Its modular design allows easy adaptation to different markets based on generating system reliability, load patterns, and price histories. The spot price histories are manifest in two probability distributions for these prices, whereas the system specifics are included via load and generation models which are calibrated to the climate and system of interest.

Choquet expectation and Peng’s g-expectation

In this paper we consider two ways to generalize the mathematical expectation of a random variable, the Choquet expectation and Peng’s g-expectation. An open question has been, after making suitable restrictions to the class of random variables acted on by the Choquet expectation, for what class of expectation do these two definitions coincide? In this paper we provide a necessary and sufficient condition which proves that the only expectation which lies in both classes is the traditional linear expectation. This settles another open question about whether Choquet expectation may be used to obtain Monte Carlo-like solution of nonlinear PDE: It cannot, except for some very special cases.

Fractional diffusion, irreversibility and entropy

Abstract Three types of equations linking the diffusion equation and the wave equation are studied: the time fractional diffusion equation, the space fractional diffusion equation and the telegraphers equation. For each type, the entropy production is calculated and compared. It is found that the two fractional diffusions, considered as linking bridges between reversible and irreversible processes, possess counter-intuitive properties: as the equation becomes more reversible, the entropy production increases. The telegraphers equation does not have the same counter-intuitive behavior. It is suggested that the different behaviors of these equations might be related to the velocities of the corresponding random walkers.

The similarity group and anomalous diffusion equations

A number of distinct differential equations, known as generalized diffusion equations, have been proposed to describe the phenomenon of anomalous diffusion on fractal objects. Although all are constructed to correctly reproduce the basic subdiffusive property of this phenomenon, using similarity methods it becomes very clear that this is far from sufficient to confirm their validity. The similarity group that they all have in common is the natural basis for making comparisons between these otherwise different equations, and a practical basis for comparisons between the very different modelling assumptions that their solutions each represent. Similarity induces a natural space in which to compare these solutions both with one another and with data from numerical experiments on fractals. It also reduces the differential equations to (extra-) ordinary ones, which are presented here for the first time. It becomes clear here from this approach that the proposed equations cannot agree even qualitatively with either each other or the data, suggesting that a new approach is needed.

COMPETITIVE MODES AND THEIR APPLICATION

We investigate nonlinear dynamical systems from the mode competition point of view, and propose the necessary conditions for a system to be chaotic. We conjecture that a chaotic system has at least two competitive modes (CMs). For a general nonlinear dynamical system, we give a simple, dynamically motivated definition of mode suitable for this concept. Since for most chaotic systems it is difficult to obtain the form of a CM, we focus on the competition between the corresponding modulated frequency components of the CMs. Some direct applications result from the explicit form of the frequency functions. One application is to estimate parameter regimes which may lead to chaos. It is shown that chaos may be found by analyzing the frequency function of the CMs without applying a numerical integration scheme. Another application is to create new chaotic systems using custom-designed CMs. Several new chaotic systems are reported.

Using entropy measures for comparison of software traces

The analysis of execution paths (also known as software traces) collected from a given software product can help in a number of areas including software testing, software maintenance and program comprehension. The lack of a scalable matching algorithm operating on detailed execution paths motivates the search for an alternative solution. This paper proposes the use of word entropies for the classification of software traces. Using a well-studied defective software as an example, we investigate the application of both Shannon and extended entropies (Landsberg-Vedral, Renyi and Tsallis) to the classification of traces related to various software defects. Our study shows that using entropy measures for comparisons gives an efficient and scalable method for comparing traces. The three extended entropies, with parameters chosen to emphasize rare events, all perform similarly and are superior to the Shannon entropy.

Modelling assumptions and requirements in the context of project risk

Many researchers have emphasized the importance of documenting assumptions (As) underlying software requirements (Rs). However, As and Rs can change with time for reasons such as: (i) an A or R was elicited incorrectly and subsequently needs to be changed; (ii) operational domain changes induce changes in the A and R sets; and (iii) the change in validity of an A, or desirability of an R, respectively, causes the validity of another A or desirability of an R to change. In Section 2, we describe our model and how it works. To put such a model into practice, we need to consider at least two scenarios. One is intra-release cycle-time, where invalidity risk is predicted at the start of the project for times between the inception and completion of the project. This would give us intra-release risk trends. The second scenario is prediction over multiple releases. This would give us a risk trend over a longer period of time. The full paper describes an algorithm to cover both of these scenarios and gives an example (from a banking application) of how the model could apply in practice. Here, we consider only the first scenario due to limitation of space.

Extended Entropies and Disorder

Landsbergs notion of disorder, entropy normalized to maximum entropy, was originally proposed for the Shannon information-theoretic entropy to overcome extensivity-based deficiencies of entropy as a measure of disorder. We generalize Landsbergs concept to three classes of extended entropies: Renyi, Tsallis and Landsberg–Vedral. We show an intimate connection between the Renyi disorders and the spectrum of dimensions known as multifractals. Three examples are treated, including one for power law distributions and one based on the logistic map. On the basis of the three examples, it is demonstrated that all three classes of extended disorder are required to fully characterize the corresponding properties of a system. We conjecture, and sketch a proof to support, that all three extended disorders are also sufficient to completely determine a dimension spectrum.

Correcting the Bias in Monte Carlo Estimators of American-style Option Values

Existing Monte Carlo estimators of American option values are consistent but biased. This article presents a general bias reduction technique which corrects the bias due to making suboptimal exercise decisions. The derived asymptotic expression for the bias is independent of dimensionality, holds for very general underlying processes and option payoffs, and is easily evaluated. The bias is subtracted from the estimators at each exercise opportunity in order to produce bias-corrected estimators. We illustrate how to apply this technique to three methods of generating estimators — stochastic tree, stochastic mesh and least-squares Monte Carlo. Numerical results demonstrate that for a fixed sample size this technique significantly reduces the relative error for both high- and low-biased estimators.