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2 What Makes a Good Hourly Price Forward Curve? Marcus Hildmann Power Systems Laboratory, ETH Zürich, Switzerland hildmann@eeh.ee.ethz.ch Göran Andersson Power Systems Laboratory, ETH Zürich, Switzerland andersson@eeh.ee.ethz.ch Grégoire Caro swissquant Group AG, Zürich, Switzerland caro@swissquant.ch Donnacha Daly swissquant Group AG, Zürich, Switzerland daly@swissquant.ch Sebastiano Rossi swissquant Group AG, Zürich, Switzerland rossi@swissquant.ch Abstract While an Hourly Price Forward Curve (HPFC) plays a defining role in the profitability of all power trading, there is a lack of consensus on the a priori requirements of a suitable HPFC, as opposed to the a posteriori quality measure that executed trades be profitable. We attempt to address this issue by reviewing the methodology of HPFC construction, and recommending a suite of quality checks for the resulting prices, based on their intended application. I. INTRODUCTION Electricity spot prices display seasonality at the yearly, weekly, and intra-day -scales which would otherwise be arbitraged from the market in the case of a storable good. In order to trade profitability in energy markets, and particularly for structuring and pricing long-term over-thecounter contracts, it is therefore necessary to capture these price dynamics in the forward curve term-structure. The standard method for determining the hourly price profile is the arbitrage-free Hourly Price Forward Curve (HPFC) [1]. There is an abundance of recent studies on the appropriate construction of the HPFC for regional energy markets. However, there is a remarkable dearth of analysis on the correct evaluation of HPFC quality and suitability for different applications. Given the central role played by the HPFC in the profitability of power market participants, the current paper attempts to rectify this situation by quantitatively answering a simple question: What makes a good HPFC? The paper begins with a review of HPFC construction methodology. The core concern of this work is, however, judgment of the quality of a HPFC once it is constructed. Examples include validation of seasonal profile shape, holiday coverage and the impact of renewable energy effects. Since the HPFC is the average seasonal price profile and therefore will only be realized at the market in average, it is not trivial to quantify this HPFC quality. In the report we propose a detailed battery of specific measures which can be used as a general benchmark of HPFC quality. The suitability of the HPFC for different applications is also recommended and explained; for example it should not be used for short term spot forecasting. A number of examples from the German, French and Swiss markets are used to illustrate the presented arguments. It is hoped that this work will lead to a more consistent and standardized benchmarking framework for deployed HPFCs in regional energy markets. In this paper we use the following notation: the index for realized is t while the index for future is denoted by τ. Daily data is denoted by d, while hourly data is denoted by h. Future and forward products are denoted by F, and k futures build the Futures chain F k. The day-ahead spot is denoted by S a with the index a on hourly basis and the intra-day price by S i with the index i. A contract price in general is denoted by P. The daily profile is denoted by D with the day index d and the hourly profile by H with the hourly index h. The load is denoted by L t also with t as index. The paper outline will be as follows: Section II discusses the use and the general construction of the HPFC, Section III describes the proposed quality measures and finally Section IV concludes the main results of the paper. II. APPLICATION AND CONSTRUCTION OF THE HOURLY PRICE FORWARD CURVE The purpose of the Hourly Price Forward Curve (HPFC) is to calculate an arbitrage-free average price profile at an hourly resolution. The hourly basis is the standard resolution of most electricity markets for the day-ahead spot price in contrast with the load frequency, which is on 15 minute basis. The idea of the HPFC is an hourly prediction of the average price based on normal weather and load information. In contrast to a spot forecast, the HPFC does not usually consider extreme weather events or spikes but focuses on normal weather and structural market conditions. For several years an intra-day market has existed in some countries in Europe and, especially in Germany, intra-day volume is increasing [2]. Nevertheless, for the purposes of the HPFC, the intra-day market can be neglected for two reasons. Firstly, the intra-day market is used to settle the wind in-feed prediction error, which is quite significant on a day-ahead basis but rather small a few hours ahead. Secondly, statistical properties support the physical

3 explanation by the arbitrage-free property S a t = E t[s i t t Si ], (1) where S i denotes the cluster of the four 15 minutes intra-day blocks of the corresponding hour d. A. Usage of the Hourly Price Forward Curve The HPFC has many uses in electricity pricing, but the three main uses of the HPFC which are covered by this paper are: pricing of non standard products, average for price scenario simulations, power plant valuation. Since these three uses are rather different and impose different requirements on the HPFC, we discuss the necessary characteristics in the following. 1) Valuation of Non-Standard Products: While standard product prices are achieved in liquid and transparent markets (e.g. NordPool, European Energy Exchange (EEX)), all non-standard products have to be evaluated by each market participant individually. Product valuation has to be separated between linear and non-linear products. Linear products such as any hourly load profile can be valued by P = τ HPFC τ L τ. (2) Using this method, any hourly load profile without embedded optionality can be valued. The main property of a good HPFC in this case is the correct replication of the intra-day and intraweek profile, especially weekends and holidays. 2) Average for Scenario Valuation: Pricing any product with embedded optionality as well as the calculation of statistical risk measures such as Value-at-Risk (VaR) and conditional Value-at-Risk (CVaR) requires the use of scenarios for stochastic valuation. Scenario generation is not the focus of this paper, but even still, a set of scenarios fulfill some characteristics with relation to the HPFC: the scenarios be consistent with respect to the HPFC for every hour, the HPFC can be used to calculate the residuals from realized hourly prices. Since a good HPFC should include all seasonal information the residuals should be free of seasonal effects [1]. 3) Power Plant Valuation: The third important use is the valuation of flexible power plants on hourly basis. In this case, the valuation against the HPFC serves as the deterministic valuation of the plant. This is done by calculating an optimal operation scheme with the HPFC as price signal. However, the deterministic valuation is only the lower bound of the valuation, since it does not cover the optional value of the power plant [3]. The optional value can be calculated by the valuation against price scenarios as show in Section II-A2. B. Challenges In general, the use of several HPFCs is possible. For consistency reasons however, it is best-practice to use the same HPFC for trading, risk management and valuation of the plants. A trader s risk be measured on the same basis as his exposure and the power plant also be evaluated within the same framework the trader uses to operate the power plant. Therefore, a restriction to one HPFC is necessary and it has to fulfill all the needs for the uses given in Section II-A. Since the average hourly price profile of the HPFC will never be realized at any market, determining the HPFC quality is difficult, as a quantitative historical simulation is not possible due to data limitation. This results in the following challenges in construction and quality checks of the HPFC: 1) How to benchmark the HPFC, what are reliable and quantitative quality measures? 2) How to build an HPFC which replicates the correct seasonality? 3) How to build an HPFC, if no liquid Futures are available? 4) How to estimate profiles in changing markets (e.g Germany with large renewable energy sources (RES) deployment)? 5) How should the short end of the HPFC be handled in comparison to the spot market? C. Construction Methods The standard commodity forward pricing method for the forward maturity F t which is found by discounting the spot price S t=0 with interest rate r, storage costs U and to maturity T t as F t = S 0 e ( r+u)(t t) only works for storable goods [4]. As electrical energy is non-storable, this method cannot be applied. Another pricing method is contract pricing based on the ability to produce power [5]. This method, however cannot replicate seasonality and requires the full knowledge of the power plant infrastructure which is often unavailable. Because transparent high volume exchange markets provide fair prices for standardized long term products and spot, and also because the load is known, hourly pricing profiles can be calculated based on the known market and grid information. The pricing profiles replicate the yearly seasonality as well as the intra-week and intra-day patterns of electricity prices, taking into account fundamental factors such as temperature while also remaining arbitrage-free to long term product prices. On the market the most widely used method by all market participants for arbitrage free hourly pricing of electricity contracts is the HPFC [1], [6], [7]. The HPFC is an hourly price profile which is arbitrage free to the liquid exchange traded Future products. Since the Future products exist only for longer durations (the number of and delivery specification for available Future products differ from country to country but the general framework stays the same), all hours in the period of a Future product be the Future product price in average: 1 T T [HPFC t t F k ] = F k, (3) t=1

4 where F k denotes the cluster of the corresponding exchange traded Future products. The standard procedure of constructing the HPFC consists of two steps which are performed independently from one another (the calculation can also be done in one step an example is given in [8]): 1) hourly profile estimation, 2) application of Future products to hourly profile. D. Profile Construction The overall hourly profile y t weights every hour in the period of the HPFC. This profile carries the seasonality and also additional information such as wind and photovoltaic (PV) in-feed. This profile only represents the weighting of the different hours, the overall level is defined by the Future products as shown in Section II-E. There are generally three ways to estimate the shape of the hourly profile: 1) statistical model: calculation of the profile based on average calculations of historical spot price series. 2) fundamental model: calculation of the profile based on supply and demand curve (i.e merit order curve & load). 3) combined model: combination of the two models to calculate the hourly profile. Also known as hybrid model. While the combined model and the fundamental model use the underlying physical relationships for the estimation, the necessary information is considerable and also not always available. Statistical models are less data dependent but react more slowly to fundamental market changes. Several approaches for purely statistical modeling exist [7], [1]. The profile consists of a daily profile D t t d and an hourly profile H t t h. The combined profile y t can be calculated as ŷ τ = 1+ ˆD h τĥτ (4) where D h t is the same daily profile D t t d for all hours of the corresponding day d. To ensure the arbitrage free condition (3) after the application of the Future products the condition hold. This can be ensured by E τ [y τ ] = 1 (5) E τ [D τ ] = 0 and E t [H t ] = 0. (6) Since the HPFC has to represent an average price, the profiles are calculated by averaging over past data. The profile D τ is calculated by D t t d = D t t d +ε t t d, (7) with the expected value D t t d and where ε t defines a white noise (WN) process with zero expectation and a finite variance σ 2, i.e. ε t wn(0,σ 2 ). The average value D t t d is calculated as D t t d = E h [St t d h ]. (8) In the proposed model, all external information is carried in the daily profile. Therefore, the daily average can be modelled as a factor model. The estimation of the average daily profile ˆ D t t d is ˆ D t t d = f t t d +ϑ t t d (9) withf t t d = α+βx t t d. The hourly profileh t t h is defined as H t t h = H t t h +ǫ t t h, (10) with the expected value H t t h and where ǫ t defines a WN process with zero expectation and a finite variance σ 2. The average value H t t h is given by H t t h = E h [St t U h d,k ]. (11) where U d,k denotes the appropriate cluster of all hours h of day d and comparable days k. E. PFC and HPFC Relationship While the hourly profile represents the weighting of every hour compared to the other hours, the overall average level of the profile will be given by the Future products. Given (4), the HPFC can be calculated by: HPFC τ τ Fc,j j F c = ˆf τ τ Fc ˆ Hτ τ Fc,j j F c F τ τ Fc, (12) where F c is the set of days which are the days during the delivery of the Future product c, e.g. all days of the year 2016 belong to the Future contract with delivery in the year The sets F c together form the entire set F. Because of (5) the HPFC arbitrage free condition (3) holds by construction. In most countries, peak, off-peak and base Future products are available. The set F should therefore reflect both peak and off-peak PFC (The off-peak Future products are often illiquid, but can be artificially extracted from the peak and base Future products). F. New Markets with a Lack of Time Series In the case of new markets, or markets with a strong structural break, historical series may be unavailable or unreliable. In those cases statistical methods for the profile calculation cannot be used. The only way to model hourly profile and Future product level is the use of fundamental models. These model supply side prices using marginal costs given by the power plant [9]. The demand curve can be modeled using existing data. III. QUALITY MEASURES The Hourly Price Forward Curve (HPFC) is an artificial construction of an hourly price profile which will not be realized by market products. Therefore, specifying quality measurements for the HPFC is very difficult, since the classic backtesting and prediction error measurements such as mean square error (MSE) cannot be used because of the absence of a realized price. The quality measurements for the HPFC are separated in four aspects 1) Difference between HPFC and spot price prediction 2) Hourly profile characteristics

5 3) Effects of uncertain RES with FIT 4) Future Product application A. Difference between HPFC and Spot Price Forecast A key aspect of the HPFC is that, in contrast to spot price prediction, it reflects prices based on normal weather expectations and structural considerations on the market. This holds for the full period of the HPFC but is most important for the short end of the curve since this area is closest to the spot price realization. Because of the nonexistent or highly illiquid Forward market for a few days ahead, short-term weather forecasts are usually not used within the construction of the HPFC. Although it is possible and possibly more elegant to make the short end conditional on weather or load forecast (rather than norm weather), to incorporate the full dynamics of the Spot market still does not make sense. Forwards are most of all hedging instruments: the prices incorporate risk premia, and the trading is performed differently than with the Spot market. The HPFC and spot price predictions have thus very different use cases and therefore, the realized spot price cannot be used as benchmark for the HPFC, nor can the HPFC replace a dedicated spot price forecast model, which should be fully conditional on most up to date information. B. Hourly Profile Characteristics The hourly profile carries all the information about the shape of the HPFC without a price level, which is carried by the Future products as shown in Figure 1. The hourly profile fulfill several requirements. price [e/mwh] German HPFC Base German PFC Peak German PFC Jun2013 Dec2013 Jun2014 Dec2014 Fig. 1. German HPFC on 28 th of January. For the yearly Future determine 2014, the seasonal pattern is fully covered by historical profiles. 1) Seasonal Patterns: The most important property of the hourly profile is the seasonal pattern. The three aspects intraday, weekly and yearly seasonality be incorporated. The intra-day seasonality capture the peak and off-peak difference. The peak and off-peak definition differs between countries. While off-peak hours variation is rather small during the year, the peak hours show a notable seasonal behavior. Figure 2 shows the difference of the profile for Germany in winter and summer. The peak behavior changes quite significantly. While in the summer the noon peak is quite strong and the evening peak almost non-existant, in the winter the noon peak is lower than the evening peak. While the noon peak is a result mainly of cooking, the evening peak is a result of additional consumption for cooking, light and so on during the evening. Since the profile is built based on norm weather, a winter peak should reflect an average winter condition and not extreme events in certain winters. The weekly seasonality differ mainly between weekday weight Winter profile Summer profile Fig. 2. Winter and Summer weekly profiles for Germany, respectively in December and June. They contrast by a spread peak/off-peak more important in Winter and a drop in price in the middle of the day for the Summer profile, related to the little amount of cooling in Germany. and weekends/holidays. Figure 3 shows a weekly pattern in Germany during the Christmas period, where the holidays on 25 th and 26 th are shown. This also holds for other holidays such as Easter, since the additional consumption of electricity during weekdays is mainly related to working hours. Going further, bridge days can be integrated by lowering the prices for the specific days when a bridge day is expected (As a rule of thumb, there is a bridge whenever there is one working day contained between a holiday and the week-end. For Christmas, it can be extended to a two working day bridge). The weekly pattern is important for pump storage plant valuation since the operators have to fill the storage lakes during the weekend with cheaper electricity. The yearly seasonality, especially the higher demand in winter, because of heating and less daylight, has to be reflected in the profile. This seasonality is mainly linked to the temperature during the year. The yearly seasonality becomes more important when Future products cover longer periods such as quarters and years, since then all seasonal information has to be covered by the profile, while as long as Future products in monthly duration exist, they also carry the higher prices in the winter in their price. Since the yearly seasonality is mainly driven by the temperature, the profile shape during the year will depend on the temperature.

6 price [e/mwh] price [e/mwh] Fig. 3. Profiles are adapted for national holidays and bridge days: Christmas and St. Stephen Day 2013 (the 25 th and 26 th of December) are handled like Sundays. The Monday and Tuesday before Christmas (the 23 rd and 24 th ) and the Friday (27 th ) have the profile of a regular week day but at a lower level (i.e bridging days). 2) Country Properties: The general framework of seasonality given in section III-B1 holds. Additionally since electricity is significantly influenced by external factors, the profiles can change in some aspects at a regional level. Since yearly seasonality is linked mainly to temperature, there are no typical country effects in the same climate zone. The weekly seasonality is given mainly by working days. While in most of the countries Saturdays and Sundays are non-working days, the hourly profile has to reflect the national holidays. The main difference between several countries lies in the intra-day profile. While the classic noon and evening peak exist in most countries, France is an example to illustrate a deviation from the standard intra-day pattern as a result of national policies. While both standard peaks exist, in France we observe an additional peak at 10pm. This peak is a result of a large amount of electric heating installed to increase the consumption during night to support the large amount of inflexible nuclear power plants. The hourly profile has to reflect this structural peak. 3) Continuous Hourly Profile: The overall price level for peak and off-peak or baseare provided independently. Therefore there is a discontinuity in the series at the change from peak to off-peak and off-peak to peak. The hourly profile construction ensure that the peak hours are always higher than the off-peak hours. Figure 4 shows an example of a discontinuous profile during the change from off-peak to peak, where the last off-peak hour is higher than the first peak hour, because of a surplus weighting of the off-peak hour. This be avoided. 4) Negative Prices: The HPFC should not show negative prices, at least not in clusters. Some negative hours during some weekend days over the year are a result of low demand and the large supply of renewable energy in-feed. As long as the negative hours are single hours it is not worth to shut down large steam cycle plants such as lignite power plants Fig. 4. Example of a poor construction: shifted profiles between the off-peak and the peak hours. The 8-am discontinuity does not belong to a structural peak but to the catenation of two disconnected profiles. or nuclear power plants. The HPFC may show some negative price hours during Christmas holidays or as a statistical relic, but clusters of negative prices in average means, that the prices are systematically negative during that period and since it is clustered, power plant operators would shut the power plants. This situation is economically not feasible and should therefore not be observable in the HPFC. One exception may be Christmas in countries with a large amount of renewable energy sources (RES) as Germany as discussed in the next section. 5) Robust Against Outliers and Overfitting: The generation methodology of the HPFC should be robust against outliers and overfitting of unique events. This contains two aspects: 1) robust against heavy tailed price distributions on hourly basis, 2) robust against overfitting of exceptional effects on longer basis. The methodologies to calculate the profile averages D and H have to be robust against non-normal distribution effects such as skewness and outliers as a result of heavy tails [1]. Since the hourly profile should represent the average, the profile should not show any spikes. The spikes in the spot price are results of large unexpected external inputs such as very low temperature, power plant shutdowns and so on and are part of the residuals of the prices and not of the average value. The second aspect is the robustness against single longer period events. A good example is September 2007 in Germany, where the prices were higher than usual and this behavior cannot be explained by standard external effects such as weather. The HPFC reflect the average September and not an outlier September. C. Effects of Uncertain RES with FIT The deployment of RES subsidized by feed-in tariff (FIT) results in special price profile structures. Since the design of

7 the FIT is national, the portfolio RES depends from country to country. The two main electricity sources of RES are wind and photovoltaic (PV). While in most countries in Europe wind represents the majority of the installed RES capacity in Germany, the installed capacity of wind and PV is almost equal. Since one of the FIT mechanisms is privileged in-feed over other power sources, the generated electricity has to be bought by the transmission system operator (TSO). Since wind and PV are the most important in-feed, we will discuss the effects on market prices. PV: As already mentioned the demand during the day is higher than night. Since PV production occurs exclusively during daylight hours there is additional production during the peak periods. This results in a decreasing peak base spread [10], particulary during the summer. This effect be reflected in the HPFC. Wind: Compared to PV, wind is not coupled to daylight hours. Because of FIT and the privileged in-feed, a large wind in-feed lowers the prices in general independently from the season. As a result, several hours with a price of zero or even negative occur during days with low consumption such as Sundays. While during the year only a couple of negative hours happen during the day, during holidays, especially during Christmas, full days of negative hours can be seen in the spot price. While negative prices in general should not be part of the HPFC, negative prices during the Christmas season can be reasonable, because the demand is much lower than during other weekends and holidays, and many of the industry facilities are also shut down for maintenance. D. Future Product Application The second step of the HPFC calculation is the application of the Future products to the hourly profile. 1) Arbitrage Free: The most important property of the HPFC is, that the HPFC is arbitrage free to the Future products (12). This is a hard condition on the HPFC and the HPFC is incorrect if the condition does not hold. For most of the periods of the HPFC more than one overlapping product exist. E.g. for the next quarterly product, the individual months also exist as products. Since the monthly products be arbitrage free to the quarterly product, both three monthly or one quarterly product can be generally used to build the HPFC. Since the three products carry more information than one quarterly products (which is just the weighted average of the monthly products) it is recommended to use the higher resolution products, since they carry more information. 2) Overlapping Products and Product Liquidity: An issue for several products is the liquidity of the products. As stated in Section I the products result from market mechanics. The market becomes more efficient with higher liquidity of the traded products. For the products at the long end of the curve, this does not matter, since they do not have overlapping products. For the short end of the curve this is different, since several overlapping products are available as shown in Figure 5. Because the market prices are more reliable for more liquid products, we recommend to favor more liquid products over higher resolution products. Both methods are correct, there is no hard criteria which one is the better one. 3) Artificial Future Products and over-the-counter (OTC) Products: In some countries, e.g. Switzerland, no exchange traded Future products exist. Since they still need an HPFC, the overall price level the taken from other sources. Three methods are common: use foreign products plus transfer capacity cost, use OTC products, build artificial products. The use of foreign Future products is the most common approach. As an example, for the calculation of the Swiss HPFC, German Future products are often used with the cross border transfer capacity price as an additional price offset. This method is based on physical and economic setup of the electricity markets. The downside is, that the Future products and the transfer capacity auctions 1) do not match in the duration and 2) this method only works if transfer capacity is auctioned independently and if physical Forwards exists in sufficient quantity. This is the case in continental Europe at the moment. The second method is the use of OTC products. Since OTC markets are not transparent and non-standardized products are traded, it is somes difficult to find reliable OTC prices. Even for the same product, several prices may be available from different brokers. Therefore the reliability of OTC data has to be checked carefully and a possible liquidity weighted average of quoted OTC prices may be used. The third method is the construction of artificial products based on the traded Futures in neighboring markets [11], for example by regression on the Spot. The constructed Future products satisfy boundary constraints in relation to prices quoted in the surrounding countries. The artificial Future be in between the cheapest and the most expensive country connected including cross border capacity prices. If no congestion is present, the prices are equal. 4) Testing the Future Products Application: A simple quality test is to actually price a forward contract using the HPFC, and to compare it with the market data. The resulting difference should have the same order of magnitude as the difference between Forward and Futures mentioned above. Going further, the quality of the HPFC can be assessed when there is a cascading of the Future products. A cascading occurs typically at the end of quarters and years, when shorter maturity contracts are created and overlap with a pre-existing contract (For instance three monthly contracts created when a quarterly product comes closer to expiration). If the seasonal profile is correctly captured, the transition between the two forward structure should be smooth. E. Summary We have summarized the conclusions of this section in Table I, which lists the recommended checks on the HPFC quality which should be concluded. This constitutes the main contribution of the current work.

8 price [e/mwh] Weekly Base Futures 36 Monthly Base Futures Quaterly Base Futures Yearly Base Futures 34 Mar13 Jun13 Sep13 Jan14 Apr14 Jul14 Oct14 Fig. 5. Baseload Futures curve for Germany on the 28 th of January The short-term weekly Futures have a low liquidity and may therefore not be used as an arbitrage-free condition (see Figure 1). The average quarterly price covering 2015 is set to 0.01e higher than the yearly Futures, illustrating the no-arbitrage-free condition between the different Futures granularity, and better suitability for matching seasonal demand. (The curve is truncated at 2015, where the quarterly Futures availability ends.) Test Arbitrage free condition TABLE I RECOMMENDED TESTS FOR A GOOD HPFC Holiday and weekend pattern Correct seasonal profile Continuity of profiles and futures Full independence from past extreme events Reflects prices in neighboring markets when no local exchange-traded market prices exist Prefer Future products with higher frequency or liquidity Accurate pricing of existing OTC products Negligible impact on the profile of cascading IV. CONCLUSION Recommendation should should should should There is no established consensus among power market participants (traders, plant-managers, risk officers) as to what constitutes a good Hourly Price Forward Curve (HPFC). In this paper we have therefore introduced a-priori quality criteria for the HPFC. We present qualitative and quantitative quality measures for both the hourly profile and for the application of Future products. Some of the measures are hard measures and hold, such as the arbitrage free condition. Some are softer measures which cannot be quantified by exact numbers but reflect fundamental energy economic factors. The quality measures hold for any HPFC independent of the construction method. It is hoped that these criteria will will serve as a useful industrial benchmark on the suitability of deployed HPFC. ACKNOWLEDGMENT Financial support from Swiss Innovation Promotion Agency (KTI), project , is gratefully acknowledged. The authors thank Florian Herzog for his help finishing the paper. REFERENCES [1] M. Hildmann, F. Herzog, D. Stokic, J. Cornel, and G. Andersson, Robust Calculation and Parameter Estimation of the Hourly Price Forward Curve, in 17th Power Systems Computation Conference (PSCC), Stockholm, [2] EEX, EEX European Energy Exchange Transparency, [Online]. Available: [3] D. P. Bertsekas, Dynamic Programming and Optimal Control. Athena Scientific, 1995, vol. I. [4] J. Hull, Options, futures, and other derivatives, 7th ed. Upper Saddle River, N.J.: Pearson Prentice Hall, [5] J. Hinz and M. Wilhelm, PRICING FLOW COMMODITY DERIVATIVES USING FIXED INCOME MARKET TECHNIQUES, International Journal of Theoretical and Applied Finance (IJTAF), vol. 09, no. 08, pp , [Online]. Available: httpeconpapers.repec.orgrepecwsiijtafxv09y2006i08p er [6] A. Abdolkhalig, Optimized Calculation of Hourly Price Forward Curve (HPFC), World Academy of Science, Engineering and Technology, vol. 45, [7] L. Bloechlinger, K. Frauendorfer, and F. Trojani, Power Prices - A Regime-Switching Spot/Forward Price Model with Kin Filter Estimation, Ph.D. dissertation, St. Gallen, [8] M. Hildmann, E. Kaffe, Y. He, and G. Andersson, Combined Estimation and Prediction of the Hourly Price Forward Curve, in 2012 IEEE POWER & ENERGY SOCIETY GENERAL MEETING, [9] P. L. Skantze and M. D. Ilic, Valuation, Hedging and Speculation in Competitive Electricity Markets. Kluwer Academic Publishers, [10] M. Hildmann, A. Ulbig, and G. Andersson, Electricity Grid In-feed from Renewable Sources: A Risk for Pumped-Storage Hydro Plants, in 8th International Conference on the European Energy Market (EEM11), Zagreb, Croatia, [11] G. Caro, J. Cornel, M. Hildmann, D. Daly, and F. Herzog, A Quantitative Analysis of the Electricity Market in Switzerland, in 8th International Conference on the European Energy Market (EEM11), Zagreb, Croatia, 2011.