Calculation of solar irradiation prediction intervals combining volatility and kernel density estimates

Highlights

• This work explores uncertainty forecasting models to build prediction intervals.

• Kernel density estimators, exponential smoothing and GARCH models are compared.

• An optimal combination of methods provides the best results.

• A good compromise between coverage and average interval width is shown.

Abstract

In order to integrate solar energy into the grid it is important to predict the solar radiation accurately, where forecast errors can lead to significant costs. Recently, the increasing statistical approaches that cope with this problem is yielding a prolific literature. In general terms, the main research discussion is centred on selecting the “best” forecasting technique in accuracy terms. However, the need of the users of such forecasts require, apart from point forecasts, information about the variability of such forecast to compute prediction intervals. In this work, we will analyze kernel density estimation approaches, volatility forecasting models and combination of both of them in order to improve the prediction intervals performance. The results show that an optimal combination in terms of prediction interval statistical tests can achieve the desired confidence level with a lower average interval width. Data from a facility located in Spain are used to illustrate our methodology.

Introduction

Solar power generation has been steadily increasing worldwide as a response to environmental concerns. Unfortunately, the integration of solar energy into the energy mix of a country brings new challenges. The main problem is due to the variability of the solar energy, which is not available “on demand”. Therefore, reasonably accurate forecasts are required to make this kind of energy economically viable. Depending on the objective, different forecasting horizons are needed, for instance, long-term forecasts are useful for locating potential solar power plants, energy resource planning and scheduling employs mid-term (up to 48 h) solar forecasts, whereas intraday forecasts are required for load following and predispatch [20]. Kraas et al. [21] economically quantified the impact of forecasting errors in the Spanish electricity system for a concentrating solar power plant. In Spain, forecasts of production have to be provided to the transmission system operator (TSO). In case of deviations from the scheduled production, the TSO applies a cost penalty. If the forecast is higher than the real production, the TSO charges falling penalties, and conversely, charges rising penalties for a production dispatch above the forecasted value. In that reference, forecasting improvement may significantly reduce penalty charges by 47.6% compared to the simple persistence forecasts.

In general terms, most of the published literature on solar energy forecasting is based on the application of different techniques in order to provide point forecasts. Among the diverse techniques provided, the selection of the best technique is usually based on choosing the one with the lowest forecast error. Nevertheless, the need of the users of such forecast (or stakeholders) require more information apart from the point forecast. In particular, they require both uncertainty and variability forecasts (chapter 14 [20]), that they can be even more useful than typical point forecasts [10]. In this context, point forecasts and its associated uncertainty are usually given as hourly-average time series, whereas irradiance variability or ramp events are measured in an intra-hour time scale, for example, minutes. Note that this difference between variability and uncertainty might not be unanimous in other research disciplines out of the solar energy literature. For instance, in supply chain applications, variability is commonly employed as a measure of uncertainty. Nonetheless, in this work we follow the distinction made in (chapter 14, [20]).

The aim of this work is twofold. Firstly, to bridge the gap in the solar energy forecasting literature by focusing on uncertainty forecasting, which has remained overlooked in comparison with point forecasting. Secondly, to use such uncertainty forecasts to compute prediction intervals on the basis of a novel methodology that combines them in an optimal manner.

Essentially, uncertainty can be quantified by the standard deviation of the forecast error, also known as volatility in finance terms, and this is assumed to be homoskedastic, independent and normally distributed. However, since the solar irradiance time series involves very complex processes, it is expected that those assumptions may be violated. Therefore, this work explores different models depending on the assumption that might not be fulfilled. For instance, if the forecast error is not normal a potential solution is to use Kernel Density estimates [31]. On the other hand, if the forecast error is neither homoskedastic nor independent, then, volatility estimators as either Generalized Autoregressive Conditional Heteroskedastic (GARCH) [4] or exponential smoothing models [34] can be employed. Since it is also possible that any of the aforementioned assumptions may be fulfilled, the novel methodology proposed intend to compute prediction intervals by combining Kernel Density estimates and volatility models. Furthermore, the combination suggested is optimal in the sense that maximizes the conditional coverage Christoffersen test p-value [9].

The results show that such a combination provides a robust performance by achieving a compromise between prediction interval coverage and average interval width.

In order to illustrate the methodology employed, we are going to focus on one-step-ahead uncertainty forecasts obtained from Global Horizontal Irradiation (GHI) data that is crucial for photovoltaic generators. Note that this study can be extended in a straightforward manner to analyze the Direct Normal Irradiation (DNI) that is more relevant for Concentrated Solar Power applications.

The rest of the paper is organised as follows: Section 2 reviews the literature on prediction intervals and it describes the models employed in this article. Section 3 describes the case study dataset. Section 4 lays out the experimental setup and the discussion of initial results. Section 5 defines a new approach to compute prediction intervals based on combining previous models, and finally, Section 6 presents the concluding remarks.