1. Introduction
The installed capacity of wind power has increased in the Danish power market over the past 10–15 years, resulting in an increasing share of renewable power production in the Danish power market [
1]. In 2005, renewable power production accounted for roughly 18% of the total power production in Denmark, and increased to more than 47% of the total power production in late 2019. The development of power production in Denmark over the period from 2005 to 2018 is illustrated in
Figure 1.
The rapid growth in power production from renewable energy sources has increasingly urged producers to sell their electricity generation through the market, due to renewable intermittency. Consequently, interest has grown in developing optimal bidding strategies for renewable energy producers in the deregulated electricity markets [
3]. Forecasting electricity prices is an emerging topic in modern research. The number of proceedings papers, articles and citations on the topic of energy price forecasting has increased significantly over the past 20 years [
4,
5]. Producers and utility companies in the power markets have been forecasting electricity spot prices over longer time horizons, for the purposes of risk management and asset valuation [
6,
7].
As modern research is building on increasingly sophisticated theory, the complexity of forecasting models increases; therefore, the objective of this paper is to identify and demonstrate how electricity producers and utilities can use more simplistic models to predict long-term electricity prices.
Electricity can be classified as a flow commodity [
8], characterized by storage and transportation limitations [
4,
8]. A balance between production and consumption is essential to ensure the reliability of the power system [
7]. Large-scale electricity storage applications, storing energy and delivering electricity at different times on different dates, have yet to be implemented. Electricity prices therefore depend strongly on the demand for electricity [
8]. This knowledge can be used when developing forecasting models for electricity spot prices.
Weron and Misiorek [
9] have presented various autoregressive models, such as ARX, an autoregressive model with exogenous variables, a regime-switching model with autoregressive processes, a threshold variable with exogenous variables and a semiparametric autoregressive model for forecasting electricity spot prices 24 h ahead in four five-week periods.
Kristiansen [
10] elaborated on the Weron and Misiorek [
9] findings, but with a simpler form of the autoregressive model with exogenous variables. Based on big data analysis from Nord Pool, Kristiansen [
10] used Danish wind power production and Nordic power demand as exogenous variables in the ARX model with an R-squared of 0.87–0.88 and out-of-sample results featuring a weekly-weighted mean absolute error (WMAE), as well as an hourly mean absolute percentage error (MAPE) of approximately 5% for day-ahead prices.
The Weron and Misiorek [
10] paper also presented a mean-reverting jump diffusions model, which has been widely used to model electricity for an extensive period, as well as to forecast hourly electricity spot prices and volatility.
To forecast weekly spot prices, Kristiansen [
11] presented an autoregressive model with the natural logarithm of previous week spot prices, the natural logarithm of the previous inflow levels to hydro reservoirs and the natural logarithm of the change in hydro reservoir levels, as independent variables. Using a regression analysis, the model achieves an R-squared of 94.9% and, when forecasting the weekly spot prices, the model achieves a MAPE of 7.5% and under-forecasts the spot price by 1.4 NOK/MWh [
11].
Knapik [
12] used three models to model and forecast electricity price jumps in the Nordic power market: an autoregressive ordered probit with and without explanatory variables, homogenous and non-homogenous Markov models with and without explanatory variables [
13,
14], and autoregressive conditional multinomial models with and without explanatory variables. The explanatory variables consist of electricity spot prices, heating degree days, consumption, wind power production and water reservoir levels. Knapik [
12] determined the best-fit method to be an autoregressive conditional multinomial model with explanatory variables [
15,
16]; that is, an ACM (1.1), which captured 81% of the jumps in the period. Knapik [
12] also made use of the Markov switching-regime model and a stochastic volatility model. When combining the models (the MS-ARX-SV model), however, the results were worse than the estimations from the individual models.
In recent decades, computer intelligence-based models have emerged as alternatives to the more traditional statistics-based forecasting methods [
17], and several studies [
4,
18] have presented different types of models for the forecasting of electricity prices and electricity demand. Using machine-learning approaches, Kurbatsky et al. [
17] presented two hybrid models, based on artificial neural network and support vector machine approaches, both using the Hilbert-Huang transform [
19]. These hybrid models are used to forecast the electricity price in the Nord Pool market one hour ahead. The artificial neural network performs best with a MAPE of 3.12%, a mean average error (MAE) of 1.44 and a root mean squared error (RMSE) of 1.92 [
17]. This is generally 0.5–1% better than the support vector machine approach [
20,
21]. Saâdaoui introduced a seasonal autoregressive neural network to forecast the electricity price in the Nord Pool market in a 72-h horizon. The approach performs well, with an overall RMSE of 1.46 and a MAPE of 3.65% in the 72-h ahead horizon, which is significantly lower than the benchmark models [
22].
The stochastic nature of the wind introduces numerous challenges to the grid operator, since the power produced by a wind farm is critically dependent on the volatility of the wind. Moreover, unexpected variations in wind power production may increase the operating costs of the electrical grid, by the growing requirements of the primary reserves and potential risks to the supply reliability [
23].
Wind power has proven to be a viable renewable energy source in the electricity mix, and the successful deployment of wind power capacities increases the need for reliable wind power forecasts as input to decision-making problems (transmission system operators and participation in the electricity market) [
24,
25]. The increased deployment of renewable energy sources has made the dynamics of the electricity spot prices even more complex with more extreme prices that are very difficult to predict [
26].
The Danish price area DK1 may be seen as a representation of the future deregulated electricity market with a high share of renewable energy sources, although the share of wind power in this area is larger than anywhere else in the world. The effects of the high wind power penetration can plausibly be detected in other markets, such as Germany or Spain [
26]. Studying the impact of the high penetration of wind power on the power grid, evidence shows that the effect of intermittent wind power decreases the electricity prices in Germany, while increasing the price volatility [
27]. This effect is induced by the merit-order effect, as the demand for electricity is supplied by sources with a lower marginal cost [
27].
2. Research Materials and Methods
An initial literature review was conducted to identify best-practice methods for high-resolution electricity spot price forecasting for the Danish power market. The literature review concluded that academia and industry remain challenged by the stochastic nature of wind power when developing methods and models for power forecasting. It was also found that the literature focusing on the Nordic power market presents different metrics for evaluating the performance of the model. The literature presents metrics such as MAE, WMAE, RMSE and MAPE. As seen in the introduction, there are different methods for evaluating forecasting models, including the four methods stated above. This paper uses MAE, RMSE and MAPE to evaluate the forecasting accuracy. MAE is defined as the sum of errors divided by the number of fitted points:
RMSE is commonly used in statistics to evaluate the accuracy of statistical models. RMSE is defined as the square root of the sum of errors divided by the number of fitted points:
where
denotes the actual value of the electricity price,
denotes the forecasted value of the electricity price and n denotes the number of fitted points.
MAPE is defined as the sum of the absolute value errors divided by the actual value divided by the number of fitted points.
The literature presents various types of forecasting models. These models can be divided into six overall types of models, including multi-agent models, fundamental models, reduced-form models, statistical models, computational intelligence models and models combining two or more of the above into hybrid models (Weron, 2014).The simulations are based on big data collected from the Danish transmission system operator
Energinet and Nord Pool, the Nordic power exchange. The data used in the empirical study is collected from hourly data from the period 1 January 2014 to 31 December 2015. Data from 1 January 2016 to 31 December 2016 is reserved to evaluate the out-of-sample performance of the models. The first part of the empirical study aims at finding the main price drivers of the electricity price development in one month. The data consists of production data, consumption data, price data from the year before the month of interest, exchange between the price area as well as the surrounding price areas. The data available for the regression model is shown in
Table 1.
This study underscores the need for complex and sophisticated models, by testing simplistic statistical models with fundamental variables in electricity spot price forecasting. Few, if any, have developed forecasting algorithms that sufficiently meet the increasing demands from grid operators and the electricity consumers toward the sustainability of the system. Furthermore, the current paper tests the three forecasting error evaluation metrics shown in formulas (1–3) above, to find the most suitable way of evaluating high-resolution forecasts.
2.1. Multivariable Linear Regression Analysis
A multivariable linear regression analysis is applied to determine which factors to use in modeling and forecasting the electricity spot price in DK1. The multivariable linear regression analysis provides measures for goodness of fit and statistical significance. The coefficients of the regression analysis are estimated using ordinary least squares (OLS).
2.2. Goodness of Fit
The coefficient of determination (denoted as R-squared, which is an analysis of the residuals) is used as a check of the goodness of fit of the model. R-squared is defined as the sum of squares due to regression, divided by the total sum of squares corrected by the mean:
where
denotes the
tth predicted value of
y and
denotes the
tth value of
y,
denotes the average of the
[
28].
2.3. Statistical Significance
T-tests are used to check the statistical significance by testing the overall fit and individual parameters. To determine which variables can be used as explanatory variables for the electricity spot price in DK1, a multivariable linear regression analysis is conducted using Stata. The t-statistics is used to test whether the given variable is statistically significant from zero, values below −1.96 and greater than 1.96 being defined as statistically significant when using a 95% confidence interval.
4. Discussion
The complexity of the forecasting models available for forecasting electricity spot prices generally increases when using more sophisticated methods, such as neural networks and support vector machines. This complexity leads to a greater demand for computational power, as well as an increased demand for data points to be considered.
This paper presents a simple statistical method with explanatory variables for forecasting the electricity spot prices based on historical data used to estimate parameters and a wind power production forecast. The forecasting model can predict the electricity spot price based on a wind power production forecast for DK1, an electricity consumption forecast for DK1 and previous prices for the year before. The wind power production and electricity consumption forecasts are used to predict the DK1 thermal power production. The estimation of parameters was performed based on the coefficients of a linear regression analysis from the same month of interest the two previous years.
From the analysis of the electricity spot price with power production, power consumption and previous prices as explanatory variables, the linear regression model provides an R-squared on average of 0.73 and an RMSE of 4.26. When forecasting the electricity spot prices over a 12-day period, the model provides an in-sample RMSE of 4.52 for January 2014 and an out-of-sample RMSE of 6.22 for January 2016. When using an ARX model, the regression results reach 0.87–0.88 for a day-ahead forecasting [
10]. Using a seasonal neural network approach for 72-h-ahead forecasting [
22] presented a model with an overall RMSE of 1.46 for the period, which is more than four times greater than the simple linear model, but the forecasting period is also four times shorter than the forecasting period for the simple linear model.
The artificial neural network approach provides a one-hour-ahead forecast with a MAE of 1.44 and an RMSE of 1.92 [
17]. The simple linear model provides an in-sample MAE of −0.12 and out-of-sample MAE of −4.04. The results from the comparative analysis are interesting, as: (1) they provide an overview of which forecasting models to use for which scenarios; (2) improved forecasting models are likely to increase the accuracy in energy supply assessments, thereby improving the grid operator’s ability to plan for additional renewable energy input and, finally, (3) they support the continuation of new wind farm installations aimed at meeting the global climate target.
5. Conclusions
The increase of power production based on renewable energy sources—wind especially—has made it more complicated to forecast electricity prices. This study has examined the country with the highest share of wind power production in the combined energy system, namely Denmark. The Danish electricity market is highly dependent on the performance of the country’s wind farms, which naturally affects electricity prices. Various forecasting models have been considered for best-practice forecasting, and a method requiring little to no computational resources has been proposed.
Many of the models are linear models based on statistics that are more sophisticated, whereas this paper tests a simple linear regression model to perform electricity spot price forecasts for a 12-day period. From the linear regression analysis, the main price drivers were found to be exchange rates between DK1 and the neighbor areas, as well as power production in DK1, such as wind power, thermal power and the demand for electricity in DK1.
This paper proposes a simple linear model to forecast the electricity prices 12 days ahead with hourly resolution, with significantly larger errors than the models presented in the literature, but the simple model provides hourly price forecasts for 4–12 times longer forecasting horizons than the models in the literature.
These results mean that grid operators, energy traders etc. can estimate prices with an hourly resolution up to 12 days ahead, which supports the continuous development of wind power, as such knowledge can guide storage technologies and other ancillary services.