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

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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.

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  1. In 1991 important features of the current state support were already introduced by means of the Act on the Feed-in of Electricity from Renewable Energy Sources into the Grid; short title, Feed-In Act (see Jarass et al. 2009, p. 98).

  2. Germany is currently divided into four regional zones, each with a responsible TSO. The four TSOs exchange the EEG electricity among themselves in order to balance the uneven distribution of the EEG plants in the German Federal Republic (see § 34ff., EEG).

  3. In 2011 the levy amounted to 3.53 cents for each KWh consumed, in 2012 it was 3.592 cents for each KWh and in 2013 it is 5.277 cents for each KWh. Basically, every electricity end-user must pay the levy. Exceptions are especially energy-intensive companies.

  4. Some years ago a market premium was introduced in other European countries, for example, in Spain and Denmark. The idea for a moving market premium for the direct merchandising by the plants supported by the EEG in Germany goes back to the Fraunhofer Institut für System- und Innovationsforschung in Karlsruhe (‘Fraunhofer Institute for Systems and Innovations Research’) (see, for example, Sensfuss, Ragwitz 2009).

  5. Since the market premium is calculated on the basis of the average production of all plants of a specific source of renewable energy, a risk remains when the individual plant deviates in its delivery profile from the average value. As a result the sum from the moving market premium and power exchange revenues can turn out to be slightly higher or lower than the EEG compensation.

  6. The TSOs are instructed by legislators to carry out the trading as efficiently (see § 64 Paragraph 3, EEG), transparently and objectively (see § 2 ‘Equalisation Scheme Ordinance’) as possible and in accordance with the latest state of science and technology (see §§ 1, 3 ‘Equalisation Scheme Ordinance’). In addition, for the TSO an incentive mechanism for EEG marketing was developed. The mechanism for an efficient trading is regulated in § 7 der ‘Equalisation Scheme Ordinance’ and is based upon a consideration of costs. If the costs of the EEG trading decrease per marketed unit in comparison to a reference value, the TSO receives a bonus in the amount of 25 % of the costs saved.

  7. In accordance with the ‘Equalisation Scheme Ordinance’ it is also possible for a TSO to trade on a stock exchange in another market area in Europe, but the transport of the electricity abroad and the reservation of the necessary transport capacity by means, for example, of explicit capacity auctions, are more complex and carry additional risks. Moreover, the integration of the European domestic market for electricity through the current development of market coupling is increasing, so that a decline of price differences between the European market areas is to be expected (see A. Weber et al. 2010, pp. 303ff.). Thus in practice all German TSOs exclusively employ the EPEX for their marketing activities.

  8. Control reserves are required in the electricity network to continuously maintain a balance between supply and demand of electricity. By means of control reserves, power supply companies or electricity producers are offered balancing energy to offset short-term demand or production fluctuations. For this purpose power stations, which on short notice can increase or decrease production, receive contracts from TSOs according to a regulated process. The costs for the compensation of the power stations are paid by the TSOs.

  9. The balancing energy price is always the same for all market participants. In the event of an excess demand for electricity, the price is typically more expensive than on the spot market; in the case of an excess supply, electricity is typically less expensive. Thus for market players costs occur when their needs for balancing energy are adjusted to the needs in total.

  10. The values 0 and 1 here describe the merely theoretically possible lower and upper limits of the degree of utilisation of the installed capacity of a plant. Since as a rule there is never a complete absence of wind throughout the Federal Republic of Germany, the empirical minimum level lies at just under 0.05. Moreover, the upper limit, at just under 0.85 empirically, clearly lies under the theoretical maximum. The reason for this difference is that there is never a perfect wind situation for all plants at the same time, and on the basis of maintenance measures or defects a certain number of plants will always be at a standstill.

  11. It is \( \left( {{\varvec{E}^{\prime}_{h}} \varvec{E}_{h}} \right)_{r,s} \) the element of r-th row and s-th column of the matrix products \( \left( {{\varvec{E}^{\prime}_{h}} \varvec{E}_{h}} \right) \) with r,s = 1,…,S. Moreover \( \left( {{\varvec{B}^{\prime}_{h}} \varvec{B}_{h}} \right)_{r,s} \ne 0 \) is for all s and r assumed, i.e., at least at one time t, valid values must exist for s and r.

  12. When the total historical time frame is examined, the individual forecasts are almost completely unbiased. The very small bias that is to be noticed in the estimation time frame is not systematic and is not repeated in the validation time frame. Therefore, the condition \( \varvec{w}^{\prime}_{h} \varvec{1} = 1 \) is reasonable in making the estimate.

  13. For example, the Frank–Wolfe algorithm can be used (see, Domschke and Drexl 2011, pp. 195ff.). However, the use assumes a convex objective function, that is, that the matrix \( {\tilde{\varvec{K}}}_{h} \) must be positive semidefinite. From the derivation of \( {\tilde{\varvec{K}}}_{h} \) it is clear that this is not always guaranteed when data gaps occur. For the empirical data used in the process this, however, was always the case. Otherwise, various methods may serve to generate a ‘most similar’ positive semidefinite matrix (see, for example, Higham 2002).

  14. The introduction of a minimum trading volume has various operational reasons. For one thing, on the European Power Exchange EPEX a minimum volume of 0.1 MWh per transaction exists. Moreover, operational issues, such us the temporal need for a transaction or transaction costs, play a role that in addition demand a higher minimum volume. A minimum trading volume of 0.5 % of the installed capacity means for a company that markets 15 % of the nationwide wind energy an absolute value of approx. 20 MW and thus an average transaction value of approx. 1,200 €.

  15. A comparison of the systems is relatively straightforward, since all four TSOs merchandise a certain percentage of the total German wind energy generation, and a large part of the data regarding the merchandising is published in the Internet.



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Correspondence to Dietmar Graeber.



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:

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

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Graeber, D., Kleine, A. The combination of forecasts in the trading of electricity from renewable energy sources. J Bus Econ 83, 409–435 (2013).

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