The combination of forecasts in the trading of electricity from renewable energy sources

Abstract

The legal support provided by the ‘Act on Granting Priority to Renewable Energy Sources’ (German Erneuerbare Energien Gesetz, EEG) and its precursor has in the last 20 years led to a marked growth in Germany in the use of renewable energies to generate electricity. As a result of the EEG amendment adopted in the summer of 2011 and in force since 1 January 2012, the market integration of electricity generated from renewable energy sources (RES-E) has become more important. Consequently, the economic importance of trading RES-E has also increased. A major role in determining costs in trading electricity from wind and solar energy on the wholesale markets plays the forecasting method used. If a forecast inaccurately predicts the amount of electricity actually generated, one result could be elevated costs in the trading process. In the beginning of this article we introduce the legal framework governing the trading of RES-E. Subsequently, we present a method for combining several individual forecasting methods. Finally, using empirical data, we show that in comparison to the best available individual forecast, the proposed combined forecast results in a clear improvement of forecasting quality as well as in a reduction in trading costs.

Appendix

Under the introduction of an index r that is equivalent to index s and the condition ${\tilde{\varvec{w}}^{\prime}_{h}} \varvec{1} = 1$, formula 8 can be derived as follows. In this case it is first assumed that at no point in time are forecasting data missing

$$ \begin{aligned} c^{id} & = \frac{1}{T}\mathop \sum \limits_{h = 1}^{H} \mathop \sum \limits_{n = 1}^{h - 1} P_{n}^{id} \mathop \sum \limits_{t = 1}^{T} \left( {p_{t,h} - p_{t,h - n} } \right)^{2} \\ & = \frac{1}{T}\mathop \sum \limits_{n = 1}^{h - 1} P_{n}^{id} \mathop \sum \limits_{t = 1}^{T} \left( {y_{t} + \mathop \sum \limits_{s = 1}^{{\tilde{S}}} w_{s,h} e_{t,s,h} - \left[ {y_{t} + \mathop \sum \limits_{s = 1}^{{\tilde{S}}} w_{s,h - n} e_{t,s,h - n} } \right]} \right)^{2} \\ & = \frac{1}{T}\mathop \sum \limits_{h = 1}^{H} \mathop \sum \limits_{n = 1}^{h - 1} P_{n}^{id} \mathop \sum \limits_{t = 1}^{T} \left( {\mathop \sum \limits_{s = 1}^{{\tilde{S}}} w_{s,h} e_{t,s,h} - \mathop \sum \limits_{s = 1}^{{\tilde{S}}} w_{s,h - n} e_{t,s,h - n} } \right)^{2} \\ & = \frac{1}{T}\mathop \sum \limits_{h = 1}^{H} \mathop \sum \limits_{n = 1}^{h - 1} P_{n}^{id} \mathop \sum \limits_{s = 1}^{{\tilde{S}}} \mathop \sum \limits_{r = 1}^{{\tilde{S}}} \left( {w_{s,h} w_{r,h} \mathop \sum \limits_{t = 1}^{T} e_{t,s,h} e_{t,r,h} + w_{s,h - n} w_{r,h - n} \mathop \sum \limits_{t = 1}^{T} e_{t,s,h - n} e_{t,r,h - n} - 2w_{s,h} w_{r,h - n} \mathop \sum \limits_{t = 1}^{T} e_{t,s,h} e_{t,r,h - n} } \right) \\ \end{aligned} $$

In the case of missing forecast data an estimate of the costs with the aid of the matrix $\tilde{\varvec{L}}_{h,h - n}$ is possible:

$\begin{array}{*{20}c} {c^{id} } & { \approx \mathop \sum \limits_{h = 1}^{H} \mathop \sum \limits_{n = 1}^{h - 1} P_{n}^{id} \cdot \left( {\tilde{\varvec{w}}^{\prime}_{h} \tilde{\varvec{K}}_{h} \tilde{\varvec{w}} + \tilde{\varvec{w}}^{\prime}_{h - n} \tilde{\varvec{K}}_{h - n} \tilde{\varvec{w}}_{h - n} - 2\tilde{\varvec{w}}^{\prime}_{h} \tilde{\varvec{L}}_{h,h - n} \tilde{\varvec{w}}_{h - n} } \right)} \\ \end{array}$

As a result it is valid for all h and all n < h:

$\tilde{\varvec{L}}_{h,h - n} = \left( {\begin{array}{*{20}c} {\frac{{\left( {\tilde{\varvec{E}}^{\prime}_{h} \tilde{\varvec{E}}_{h - n} } \right)_{1,1} }}{{\left( {\tilde{\varvec{B}^{\prime}_{h}} \tilde{\varvec{B}}_{h - n} } \right)_{1,1} }}} & \cdots & {\frac{{\left( {\tilde{\varvec{E}}^{\prime}_{h}\tilde{E}_{h - n} } \right)_{{1,\tilde{S}}} }}{{\left( {\tilde{\varvec{B}^{\prime}_{h}} \tilde{\varvec{B}}_{h - n} } \right)_{{1,\tilde{S}}} }}} \\ \vdots & \ddots & \vdots \\ {\frac{{\left( {\tilde{\varvec{E}}^{\prime}_{h} \tilde{\varvec{E}}_{h - n} } \right)_{{\tilde{S},1}} }}{{\left( {\tilde{\varvec{B}^{\prime}_{h}} \tilde{\varvec{B}}_{h - n} } \right)_{{\tilde{S},1}} }}} & \cdots & {\frac{{\left( {\tilde{\varvec{E}}^{\prime}_{h} \tilde{\varvec{E}_{h - n}} } \right)_{{\tilde{S},\tilde{S}}} }}{{\left( {\tilde{\varvec{B}^{\prime}_{h}} \tilde{\varvec{B}_{h - n}} } \right)_{{\tilde{S},\tilde{S}}} }}} \\ \end{array} } \right).$