Model For Electricity Spot Prices Research Articles

Econometric modeling of regional electricity spot prices in the Australian market

Wholesale electricity markets are increasingly integrated via high voltage interconnectors, and inter-regional trade in electricity is growing. To model this, we consider a spatial equilibrium model of price formation, where constraints on inter-regional flows result in three distinct equilibria in prices. We use this to motivate an econometric model for the distribution of observed electricity spot prices that captures many of their unique empirical characteristics. The econometric model features supply and inter-regional trade cost functions, which are estimated using Bayesian monotonic regression smoothing methodology. A copula multivariate time series model is employed to capture additional dependence — both cross-sectional and serial — in regional prices. The marginal distributions are nonparametric, with means given by the regression means. The model has the advantage of preserving the heavy right-hand tail in the predictive densities of price. We fit the model to half-hourly spot price data in the five interconnected regions of the Australian national electricity market. The fitted model is then used to measure how both supply and price shocks in one region are transmitted to the distribution of prices in all regions in subsequent periods. Finally, to validate our econometric model, we show that prices forecast using the proposed model compares favorably with those from some benchmark alternatives.

Forecasting performance of time series models on electricity spot markets: a quasi-meta-analysis

PurposeEmpirical publications on the time series modeling and forecasting of electricity prices vary widely regarding the conditions, and the findings make it difficult to generalize results. Against this background, it is surprising that there is a lack of statistics-based literature reviews on the forecasting performance when comparing different models. The purpose of the present study is to fill this gap.Design/methodology/approachThe authors conduct a comprehensive literature analysis from 2000 to 2015, covering 86 empirical studies on the time series modeling and forecasting of electricity spot prices. Various statistics are presented to characterize the empirical literature on electricity spot price modeling, and the forecasting performance of several model types and modifications is analyzed. The key issue of this study is to offer a comparison between different model types and modeling conditions regarding their forecasting performance, which is referred to as a quasi-meta-analysis, i.e. the analysis of analyses to achieve more general findings independent of the circumstances of single studies.FindingsThe authors find evidence that generalized autoregressive conditional heteroscedasticity models outperform their autoregressive–moving-average counterparts and that the consideration of explanatory variables improves forecasts.Originality/valueTo the best knowledge of the authors, this paper is the first to apply the methodology of meta-analyses in a literature review of the empirical forecasting literature on electricity spot markets.

Space-time Modeling of Electricity Spot Prices

Using data for the Nord Pool power grid, we derive a space-time Durbin model for electricity spot prices with both temporal and spatial lags. Joint modeling of temporal and spatial adjustment effects is necessarily important when prices and loads are determined in a network grid. By using different spatial weight matrices, statistical tests show significant spatial dependence in the spot price dynamics across areas. In fact, estimation of the model shows that the spatial dependence is as important as the temporal dependence in describing the spot price dynamics. We decompose price impacts into direct and indirect effects and demonstrate how price effects transmit to neighboring markets and decline with distance. A forecasting comparison with a non-spatial model shows that the space-time model improves forecasting performance for 7 and 30 days ahead forecasts. A model with time-varying parameters is estimated for an expanded sample period and it is found that the spatial correlation within the power grid has increased over time. We interpret this to indicate an increasing degree of market integration within the sample period.

Bayesian calibration and number of jump components in electricity spot price models

We find empirical evidence that mean-reverting jump processes are not statistically adequate to model electricity spot price spikes but independent, signed sums of such processes are statistically adequate. Further we demonstrate a change in the composition of these sums after a major economic event. This is achieved by developing a Markov Chain Monte Carlo (MCMC) procedure for Bayesian model calibration and a Bayesian assessment of model adequacy (posterior predictive checking). In particular we determine the number of signed mean-reverting jump components required in the APXUK and EEX markets, in time periods both before and after the recent global financial crises. Statistically, consistent structural changes occur across both markets, with a reduction of the intensity and size, or the disappearance, of positive price spikes in the later period. All code and data are provided to enable replication of results.

Modelling electricity prices: a time change approach

To capture mean reversion and sharp seasonal spikes observed in electricity prices, this paper develops a new stochastic model for electricity spot prices by time changing the Jump Cox-Ingersoll-Ross (JCIR) process with a random clock that is a composite of a Gamma subordinator and a deterministic clock with seasonal activity rate. The time-changed JCIR process is a time-inhomogeneous Markov semimartingale which can be either a jump-diffusion or a pure-jump process, and it has a mean-reverting jump component that leads to mean reversion in the prices in addition to the smooth mean-reversion force. Furthermore, the characteristics of the time-changed JCIR process are seasonal, allowing spikes to occur in a seasonal pattern. The Laplace transform of the time-changed JCIR process can be efficiently computed by Gauss–Laguerre quadrature. This allows us to recover its transition density through efficient Laplace inversion and to calibrate our model using maximum likelihood estimation. To price electricity derivatives, we introduce a class of measure changes that transforms one time-changed JCIR process into another time-changed JCIR process. We derive a closed-form formula for the futures price and obtain the Laplace transform of futures option price in terms of the Laplace transform of the time-changed JCIR process, which can then be efficiently inverted to yield the option price. By fitting our model to two major electricity markets in the US, we show that it is able to capture both the trajectorial and the statistical properties of electricity prices. Comparison with a popular jump-diffusion model is also provided.

Bayesian Calibration and Number of Jump Components in Electricity Spot Price Models

Bayesian Calibration and Number of Jump Components in Electricity Spot Price Models

Forecasting Electricity Spot Prices Using Lasso: On Capturing the Autoregressive Intraday Structure

In this paper we present a regression based model for day-ahead electricity spot prices. We estimate the considered linear regression model by the lasso estimation method. The lasso approach allows for many possible parameters in the model, but also shrinks and sparsifies the parameters automatically to avoid overfitting. Thus, it is able to capture the autoregressive intraday dependency structure of the electricity price well. We discuss in detail the estimation results which provide insights to the intraday behavior of electricity prices. We perform an out-of-sample forecasting study for several European electricity markets. The results illustrate well that the efficient lasso based estimation technique can exhibit advantages from two popular model approaches.

Modelling Electricity Prices: A Time Change Approach

To capture mean reversion and sharp seasonal spikes observed in electricity prices, this paper develops a new stochastic model for electricity spot prices by time changing the Jump Cox-Ingersoll-Ross (JCIR) process with a random clock that is a composite of a Gamma subordinator and a deterministic clock with seasonal activity rate. The time-changed JCIR process is a time-inhomogeneous Markov semimartingale which can be either a jump-diffusion or a pure-jump process, and it has a mean-reverting jump component that leads to mean reversion in the prices in addition to the smooth mean-reversion force. Furthermore, the characteristics of the time-changed JCIR process are seasonal, allowing spikes to occur in a seasonal pattern. The Laplace transform of the time-changed JCIR process can be efficiently computed by Gauss-Laguerre quadrature. This allows us to recover its transition density through efficient Laplace inversion and to calibrate our model using maximum likelihood estimation. To price electricity derivatives, we introduce a class of measure changes that transforms one time-changed JCIR process into another time-changed JCIR process. We derive a closed-form formula for the futures price and obtain the Laplace transform of futures option price in terms of the Laplace transform of the time-changed JCIR process, which can then be efficiently inverted to yield the option price. By fitting our model to two major electricity markets in the US, we show that it is able to capture both the trajectorial and the statistical properties of electricity prices. Comparison with a popular jump-diffusion model is also provided.

Efficient modeling and forecasting of electricity spot prices

The increasing importance of renewable energy, especially solar and wind power, has led to new forces in the formation of electricity prices. Hence, this paper introduces an econometric model for the hourly time series of electricity prices of the European Power Exchange (EPEX) which incorporates specific features like renewable energy. The model consists of several sophisticated and established approaches and can be regarded as a periodic VAR-TARCH with wind power, solar power, and load as influences on the time series. It is able to map the distinct and well-known features of electricity prices in Germany. An efficient iteratively reweighted lasso approach is used for the estimation. Moreover, it is shown that several existing models are outperformed by the procedure developed in this paper.

Risk Premium in Electricity Prices: Evidence from the PJM Market

In this article, we construct a stochastic model for electricity spot prices, derive a pricing formula for electricity forward contracts, and specify risk premia inherent in such contracts. Our spot price model accounts for seasonality, mean‐reversion, and time‐varying jump intensity. Empirically, the model with a sinusoidal seasonal function, different mean‐reversion rates for the diffusion and jumps, and constant volatility is most appropriate among six different models for the United States market. The modeling approach leads us to investigate the stochastic risk premium due to both the diffusion and the jump components explicitly. The empirical results reveal that both risk premia are time‐varying and state‐dependent. The diffusion risk premium is negatively correlated with the diffusion level and fluctuates about zero, and the jump risk premium is negatively correlated with the jump level and always negative. In magnitude, the jump risk premium is much larger than the diffusion risk premium. © 2014 Wiley Periodicals, Inc. Jrl Fut Mark 35:776–793, 2015

A Nonparametric Model for Spot Price Dynamics and Pricing of Futures Contracts in Electricity Markets

This paper deals with the estimation of continuous time diffusion processes describing the dynamics of electricity spot prices. Different parametric models have been proposed in the literature, each attempting to capture empirical characteristics and stylized facts of the electricity market like the spiky behavior of the spot prices. Although jump-diffusion and regime-switching models perform reasonably well, there is always a trade-off between model parsimony and adequacy. The results in the literature indicate that none of the models seem to consistently outperform its counterparts. This paper avoids making parametric assumption about the drift and the diffusion coefficient functions of the underlying electricity spot prices, and estimates these functions together with the market price of risk in a nonparametric way. The latter allows us to price futures contracts written on electricity spots. Using electricity spot prices and futures data from the regional electricity markets in Australia, we show that besides offering a convenient way of estimating the continuous-time models for electricity spot prices, our nonparametric estimation procedure performs well in- and out-of-sample when dealing with pricing of future contracts.

A semiparametric model for electricity spot prices

This article proposes a semiparametric single-index model for short-term forecasting day-ahead electricity prices. The approach captures the dependency of electricity prices on covariates, such as demand for electricity, amount of energy produced by intermittent sources, and weather-dependent variables. To obtain parsimonious models, principal component analysis is used for dimension reduction. The approach is tested on two data sets from different markets and its performance is analyzed in terms of fit, forecast quality, and computational efficiency. The results are encouraging, in that the proposed method leads to a good in-sample fit and performs well out-of-sample compared with four benchmark models, including a SARIMA model as well as a functional nonparametric regression approach recently proposed in the literature.

Filtering and Identification of Stochastic Risk Premium for Electricity Spot Price Models

Starting from the simple model for the spot price which is set as the jump augmented Vasicek model, we construct a factor model of the electricity futures as the stochastic hyperbolic systems with jumps. Representing the main spike phenomena of the electricity spot price from one observed futures data by proxy, the filtering of factor process and the related stochastic risk premium are formulated in a Gaussian frame work. After serving the likelihood functional, the systems parameter estimation problem is solved.

A nonparametric model for spot price dynamics and pricing of futures contracts in electricity markets

AbstractThis paper deals with the estimation of continuous time diffusion processes describing the dynamics of electricity spot prices. Different parametric models have been proposed in the literature, each attempting to capture empirical characteristics and stylized facts of the electricity market like the spiky behavior of the spot prices. Although jump-diffusion and regime-switching models perform reasonably well, there is always a trade-off between model parsimony and adequacy. The results in the literature indicate that none of the models seem to consistently outperform its counterparts. This paper avoids making parametric assumption about the drift and the diffusion coefficient functions of the underlying electricity spot prices, and estimates these functions together with the market price of risk in a nonparametric way. The latter allows us to price futures contracts written on electricity spots. Using electricity spot prices and futures data from the regional electricity markets in Australia, we show that besides offering a convenient way of estimating the continuous-time models for electricity spot prices, our nonparametric estimation procedure performs well in- and out-of-sample when dealing with pricing of future contracts.

Modeling and forecasting electricity spot prices: A functional data perspective

Classical time series models have serious difficulties in modeling and forecasting the enormous fluctuations of electricity spot prices. Markov regime switch models belong to the most often used models in the electricity literature. These models try to capture the fluctuations of electricity spot prices by using different regimes, each with its own mean and covariance structure. Usually one regime is dedicated to moderate prices and another is dedicated to high prices. However, these models show poor performance and there is no theoretical justification for this kind of classification. The merit order model, the most important micro-economic pricing model for electricity spot prices, however, suggests a continuum of mean levels with a functional dependence on electricity demand. We propose a new statistical perspective on modeling and forecasting electricity spot prices that accounts for the merit order model. In a first step, the functional relation between electricity spot prices and electricity demand is modeled by daily price-demand functions. In a second step, we parameterize the series of daily price-demand functions using a functional factor model. The power of this new perspective is demonstrated by a forecast study that compares our functional factor model with two established classical time series models as well as two alternative functional data models.

Goodness-of-fit testing for the marginal distribution of regime-switching models with an application to electricity spot prices

This paper complements a recently published study (Janczura and Weron in AStA-Adv Stat Anal 96(3):385–407, 2012) on efficient estimation of Markov regime-switching models. Here, we propose a new goodness-of-fit testing scheme for the marginal distribution of such models. We consider models with an observable (like threshold autoregressions) as well as a latent state process (like Markov regime-switching). The test is based on the Kolmogorov–Smirnov supremum-distance statistic and the concept of the weighted empirical distribution function. The motivation for this research comes from a recent stream of literature in energy economics concerning electricity spot price models. While the existence of distinct regimes in such data is generally unquestionable (due to the supply stack structure), the actual goodness-of-fit of the models requires statistical validation. We illustrate the proposed scheme by testing whether commonly used Markov regime-switching models fit deseasonalized electricity prices from the NEPOOL (US) day-ahead market.

A critical empirical study of three electricity spot price models

We conduct an empirical analysis of three recently proposed and widely used models for electricity spot price process. The first model, called the jump-diffusion model, was proposed by Cartea and Figueroa (2005), and is a one-factor mean-reversion jump-diffusion model, adjusted to incorporate the most important characteristics of electricity prices. The second model, called the threshold model, was proposed by Roncoroni (2002) and further developed by Geman and Roncoroni (2006), and is an exponential Ornstein–Uhlenbeck process driven by a Brownian motion and a state-dependent compound Poisson process. It is designed to capture both statistical and pathwise properties of electricity spot prices. The third model, called the factor model, was proposed by Benth et al. (2007). It is an additive linear model, where the price dynamics is a superposition of Ornstein–Uhlenbeck processes driven by subordinators to ensure positivity of the prices. It separates the modelling of spikes and base components. We calibrate all three models to German spot price data. Besides employing techniques similar to those used in the original papers we adopt the prediction-based estimating function technique (Sørensen, 2000) and the filtering technique (Meyer-Brandis and Tankov, 2008). We critically compare the properties and the estimation of the three models and discuss several shortcomings and possible improvements. Besides analysing the spot price behaviour, we compute forward prices and risk premia for all three models for various German forward data and identify the key forward price drivers.

An empirical comparison of alternate regime-switching models for electricity spot prices

One of the most profound features of electricity spot prices are the price spikes. Markov regime-switching (MRS) models seem to be a natural candidate for modeling this spiky behavior. However, in the studies published so far, the goodness-of-fit of the proposed models has not been a major focus. While most of the models were elegant, their fit to empirical data has either been not examined thoroughly or the signs of a bad fit ignored. With this paper we want to fill the gap. We calibrate and test a range of MRS models in an attempt to find parsimonious specifications that not only address the main characteristics of electricity prices but are statistically sound as well. We find that the best structure is that of an independent spike 3-regime model with time-varying transition probabilities, heteroscedastic diffusion-type base regime dynamics and shifted spike regime distributions. Not only does it allow for a seasonal spike intensity throughout the year and consecutive spikes or price drops, which is consistent with market observations, but also exhibits the ‘inverse leverage effect’ reported in the literature for spot electricity prices.

Nonlinear maximum likelihood estimation of electricity spot prices using recurrent neural networks

Electricity spot prices are complex processes characterized by nonlinearity and extreme volatility. Previous work on nonlinear modeling of electricity spot prices has shown encouraging results, and we build on this area by proposing an Expectation Maximization algorithm for maximum likelihood estimation of recurrent neural networks utilizing the Kalman filter and smoother. This involves inference of both parameters and hyper-parameters of the model which takes into account the model uncertainty and noise in the data. The Expectation Maximization algorithm uses a forward filtering and backward smoothing (Expectation) step, followed by a hyper-parameter estimation (Maximization) step. The model is validated across two data sets of different power exchanges. It is found that after learning a posteriori hyper-parameters, the proposed algorithm outperforms the real-time recurrent learning and the extended Kalman Filtering algorithm for recurrent networks, as well as other contemporary models that have been previously applied to the modeling of electricity spot prices.

On the market impact of wind energy forecasts

This paper presents an analysis of how day-ahead electricity spot prices are affected by day-ahead wind power forecasts. Demonstration of this relationship is given as a test case for the Western Danish price area of the Nord Pool's Elspot market. Impact on the average price behaviour is investigated as well as that on the distributional properties of the price. By using a non-parametric regression model to assess the effects of wind power forecasts on the average behaviour, the non-linearities and time variations in the relationship are captured well and the effects are shown to be quite substantial. Furthermore, by evaluating the distributional properties of the spot prices under different scenarios, the impact of the wind power forecasts on the price distribution is proved to be considerable. The conditional price distribution is moreover shown to be non-Gaussian. This implies that forecasting models for electricity spot prices for which parameters are estimated by a least squares techniques will not have Gaussian residuals. Hence the widespread assumption of Gaussian residuals from electricity spot price models is shown to be inadequate for these model types. The revealed effects are likely to be observable and qualitatively similar in other day-ahead electricity markets significantly penetrated by wind power.