Preface. Acknowledgments. 1 Complex Electricity Markets. 1.1 Liberalization. 1.2 The Marketplace. 1.2.1 Power Pools and Power Exchanges. 1.2.2 Nodal and Zonal Pricing. 1.2.3 Market Structure. 1.2.4 Traded Products. 1.3 Europe. 1.3.1 The England and Wales Electricity Market. 1.3.2 The Nordic Market. 1.3.3 Price Setting at Nord Pool. 1.3.4 Continental Europe 13. 1.4 North America. 1.4.1 PJM Interconnection. 1.4.2 California and the Electricity Crisis. 1.4.3 Alberta and Ontario. 1.5 Australia and New Zealand. 1.6 Summary. 1.7 Further Reading. 2 Stylized Facts of Electricity Loads and Prices. 2.1 Introduction. 2.2 Price Spikes. 2.2.1 Case Study: The June 1998 Cinergy Price Spike. 2.2.2 When Supply Meets Demand. 2.2.3 What is Causing the Spikes?. 2.2.4 The Definition. 2.3 Seasonality. 2.3.1 Measuring Serial Correlation. 2.3.2 Spectral Analysis and the Periodogram. 2.3.3 Case Study: Seasonal Behavior of Electricity Prices and Loads. 2.4 Seasonal Decomposition. 2.4.1 Differencing. 2.4.2 Mean or Median Week. 2.4.3 Moving Average Technique. 2.4.4 Annual Seasonality and Spectral Decomposition. 2.4.5 Rolling Volatility Technique. 2.4.6 Case Study: Rolling Volatility in Practice. 2.4.7 Wavelet Decomposition. 2.4.8 Case Study: Wavelet Filtering of Nord Pool Hourly System Prices. 2.5 Mean Reversion. 2.5.1 R/S Analysis. 2.5.2 Detrended Fluctuation Analysis. 2.5.3 Periodogram Regression. 2.5.4 Average Wavelet Coefficient. 2.5.5 Case Study: Anti-persistence of Electricity Prices. 2.6 Distributions of Electricity Prices. 2.6.1 Stable Distributions. 2.6.2 Hyperbolic Distributions. 2.6.3 Case Study: Distribution of EEX Spot Prices. 2.6.4 Further Empirical Evidence and Possible Applications. 2.7 Summary. 2.8 Further Reading. 3 Modeling and Forecasting Electricity Loads. 3.1 Introduction. 3.2 Factors Affecting Load Patterns. 3.2.1 Case Study: Dealing with Missing Values and Outliers. 3.2.2 Time Factors. 3.2.3 Weather Conditions. 3.2.4 Case Study: California Weather vs Load. 3.2.5 Other Factors. 3.3 Overview of Artificial Intelligence-Based Methods. 3.4 Statistical Methods. 3.4.1 Similar-Day Method. 3.4.2 Exponential Smoothing. 3.4.3 Regression Methods. 3.4.4 Autoregressive Model. 3.4.5 Autoregressive Moving Average Model. 3.4.6 ARMA Model Identification. 3.4.7 Case Study: Modeling Daily Loads in California. 3.4.8 Autoregressive Integrated Moving Average Model. 3.4.9 Time Series Models with Exogenous Variables. 3.4.10 Case Study: Modeling Daily Loads in California with Exogenous Variables. 3.5 Summary. 3.6 Further Reading. 4 Modeling and Forecasting Electricity Prices. 4.1 Introduction. 4.2 Overview of Modeling Approaches. 4.3 Statistical Methods and Price Forecasting. 4.3.1 Exogenous Factors. 4.3.2 Spike Preprocessing. 4.3.3 How to Assess the Quality of Price Forecasts. 4.3.4 ARMA-type Models. 4.3.5 Time Series Models with Exogenous Variables. 4.3.6 Autoregressive GARCH Models. 4.3.7 Case Study: Forecasting Hourly CalPX Spot Prices with Linear Models. 4.3.8 Case Study: Is Spike Preprocessing Advantageous?. 4.3.9 Regime-Switching Models. 4.3.10 Calibration of Regime-Switching Models. 4.3.11 Case Study: Forecasting Hourly CalPX Spot Prices with Regime-Switching Models. 4.3.12 Interval Forecasts. 4.4 Quantitative Models and Derivatives Valuation. 4.4.1 Jump-Diffusion Models. 4.4.2 Calibration of Jump-Diffusion Models. 4.4.3 Case Study: A Mean-Reverting Jump-Diffusion Model for Nord Pool Spot Prices. 4.4.4 Hybrid Models. 4.4.5 Case Study: Regime-Switching Models for Nord Pool Spot Prices. 4.4.6 Hedging and the Use of Derivatives. 4.4.7 Derivatives Pricing and the Market Price of Risk. 4.4.8 Case Study: Asian-Style Electricity Options. 4.5 Summary. 4.6 Further Reading. Bibliography. Index.

Modeling and Forecasting Electricity Loads and Prices: A Statistical Approach

The role of fractional calculus in modeling biological phenomena: A review

The Smoluchowski-Chapman-Kolmogorov functional equation is applied to the electric signals that precede rupture. The results suggest a non-Markovian character of the analyzed data. The rescaled range Hurst and detrended fluctuation analyses, as well as that related with the "mean distance a walker spanned," lead to power-law exponents, which are consistent with the existence of long-range correlations. A "universality" in the power spectrum characteristics of these signals emerges, if an analysis is made (not in the conventional time frame, but) in the "natural" time domain. Within this frame, it seems that certain power spectrum characteristics of ion current fluctuations in membrane channels distinguish them from the electric signals preceding rupture. The latter exhibit a behavior compatible with that expected from a model based on the random field Ising Hamiltonian at the critical point.

/pdf/long-range-correlations-in-the-electric-signals-that-precede-1tu6g0cmln.pdf

Long-range correlations in the electric signals that precede rupture.

Fractional Brownian motion with Hurst index less then $1/2$ and continuous-time random walk with heavy tailed waiting times (and the corresponding fractional Fokker-Planck equation) are two different processes that lead to a subdiffusive behavior widespread in complex systems We propose a simple test, based on the analysis of the so-called $p$ variations, which allows distinguishing between the two models on the basis of one realization of the unknown process We apply the test to the data of Golding and Cox [Phys Rev Lett 96, 098102 (2006)], describing the motion of individual fluorescently labeled mRNA molecules inside live E coli cells It is found that the data does not follow heavy tailed continuous-time random walk The test shows that it is likely that fractional Brownian motion is the underlying process

/pdf/fractional-brownian-motion-versus-the-continuous-time-random-4z814n8vuf.pdf

Fractional brownian motion versus the continuous-time random walk: a simple test for subdiffusive dynamics.

Approaching complexity by stochastic methods: From biological systems to turbulence

A canonical decomposition of H-self-similar Levy symmetric alpha-stable processes is presented. The resulting components completely described by both deterministic kernels and the corresponding stochastic integral with respect to the Levy symmetric alpha-stable motion are shown to be related to the dissipative and conservative parts of the dynamics. This result provides stochastic analysis tools for study the anomalous diffusion phenomena in the Langevin equation framework. For example, a simple computer test for testing the origins of self-similarity is implemented for four real empirical time series recorded from different physical systems: an ionic current flow through a single channel in a biological membrane, an energy of solar flares, a seismic electric signal recorded during seismic Earth activity, and foreign exchange rate daily returns.

/pdf/complete-description-of-all-self-similar-models-driven-by-4dxgw90h11.pdf

Complete description of all self-similar models driven by Lévy stable noise.

The statistical analysis of an ionic current signal recorded from a single channel of a biological membrane is presented. We find the main characteristics of the ionic current probability density, the closed- and open-state distributions, and the autocorrelation function of the current recordings by using procedures based on the kernel and tail estimators, the bootstrap methodology; and the Zipf plots. The results provide evidence for the non-Markovian character of the channel kinetics of the investigated data.

Statistical analysis of ionic current fluctuations in membrane channels.

An explicit stochastic representation of a stationary ionic current signal recorded from a single channel of a biological membrane is presented. In the framework of the proposed approach we show how the dichotomous time structure of the signal leads to the non-Markovian character of the channel current. The rescaled range Hurst and detrended fluctuation analyses confirm the theoretical result. To investigate the ionic current fluctuations we introduce the Orey index as a statistical method providing additional information on the properties of stochastic processes. In order to reveal any differences between the experimental and reconstructed signals, we apply also the statistical tests to the model-based simulations of the channel action.

Stochastic origins of the long-range correlations of ionic current fluctuations in membrane channels.

Enigma of Self-Similarity of Fractional Levy Stable Motions

The time series of successive closed- and open-states durations determined for a voltage-dependent big conductance locust potassium channel (BK channel) have been analyzed. The problem of correlation between the states have been discussed by means of the autocorrelation function, Hurst and detrended fluctuation analysis (DFA) as well as surrogate data tools. We showed that the existence of correlation between closed and open times can be detected only by simultaneous application of several different tools. All techniques pointed at a stronger correlation, therefore longer memory, in the series of closed states. Persistent character of both examined dwell-time series has been shown by the Hurst and DFA analyses, whose exponents were found significantly larger than 0.5—a value typical for random walk. It may originate from the externally applied electric field, which directs ion movement and influences channel kinetics. DFA detected two ranges of different correlations in the series of closed states durations. This result points to the existence of two processes with different correlation characteristics which govern the channel functioning. The DFA exponent equal to 0.86±0.07 for long-range scaling of closed times suggests that the long memory of the BK channel action arising from the dichotomous nature of the signal and the long-tail properties of the closed-times distribution can be also governed by the correlation between closed states. The possible biological implications of the findings have been also discussed.

Application of dwell-time series in studies of long-range correlation in single channel ion transport: analysis of ion current through a big conductance locust potassium channel

Szymon Mercik

Papers

Complete description of all self-similar models driven by Lévy stable noise.

Statistical analysis of ionic current fluctuations in membrane channels.

Stochastic origins of the long-range correlations of ionic current fluctuations in membrane channels.

Enigma of Self-Similarity of Fractional Levy Stable Motions

Application of dwell-time series in studies of long-range correlation in single channel ion transport: analysis of ion current through a big conductance locust potassium channel