1 (19) Hannele Holttinen, Jussi Ikäheimo

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1 Hannele Holttinen, Jussi Ikäheimo 1 (19) Wind predictions and bids in Denmark Wind power producers need a prediction method to bid their energy to day-ahead market. They pay balancing costs for the energy that has been forecasted wrong. If the prices for up- and down-regulation are asymmetric, it is worthwhile to bid a different amount than the forecast. Minimising the costs results in different bid than minimising the errors in energy. In this way the balancing costs are minimised. This is the case now in Denmark (both in West and in East), as the price difference of down-regulation and spot price is higher than for up-regulation. The question is, whether this kind of behaviour could result in wrong information for the TSO and cause more critical situations with larger errors in wind power production. This has been studied by a case of 12 months, for 2 wind farms in Denmark. They are both 2 MW wind farms, one is in East Denmark (Middelgrunden) and one in West Denmark (Klim). The results for the analysis are described in this document and more detailed results are in table 1. It is shown that for large errors downward (forecast production larger than realised production) there will be larger errors when using the bidding strategy. In those situations the information that the TSO receives from the bids would result in larger errors and more up-regulation would be needed. Data Data for actual production and predictions at : hours were obtained from DTU. This data was used together with the price data at Nordpool, West and East Denmark, for the spot prices and the balancing prices (up- and down-regulation). Also Swedish price data was used. Using the prediction data from : hours will overestimate the forecast errors compared with real life, as there is still time to get a net set of forecast in the morning before giving the bids to Nordpool market. However, for this analysis the data is adequate. 12 months were chosen from to West DK data had 62 hours missing and East DK data 887 hours missing. These were excluded from the calculations. The sum of the two could be calculated for 7822 hours (962 hours missing). Correlations The production time series of East and West Denmark are correlated (about.6), but the forecast error time series for the two wind farms are correlated only very weakly (about.2). The correlation of wind power prediction errors and regulation price was checked this is the case for overall aggregated wind power prediction errors and regulation price in West Denmark. However, for this one wind farm only, there was only slight correlation (.11.12). The correlation with Swedish and East DK prices was.6. For East Denmark wind farm, the correlation was.7.8 for all the prices (West/east DK and Sweden), as expected. Bidding model Bidding model minimising the balancing costs has been made, according to the asymmetric up/down regulation prices. The down-regulation is penalised more than up-regulation, this is why the optimal bids will overestimate the production more than forecasted production. As the regulating power market was still new in 23, the historical up/down regulation prices before the analysed period were too volatile to be taken as input values. That is why the up/down regulating prices during the same period were used in the model. The prediction error analysis and the bidding model is described in Appendix 1. VTT Energia /home/han/proj/pso411/report/final/collect/dkvind_bids_and_balance_vtt_26_update1.doc

2 2 (19) Balancing costs for the two wind farms Calculating from the forecasted production, the balancing costs for the individual wind farms, for East DK prices would have been 5.8 DKK/MWh for the analysed period. For the sum of the two, the balancing costs would have been 4.8 DKK/MWh. For West DK prices the balancing costs would have been 16.8 DKK/MWh. Using optimised bidding according to East DK prices, the costs would drop to 5.4 DKK/MWh (about 7 %). A similar 7 % benefit would be achieved for the sum of two wind farms. For West DK prices, the optimised bidding would not help much (2 %). This is probably due to wind power affecting the regulation prices in West DK. For West Denmark, wind power prediction errors are to the same direction as regulating market prices nearly 6 % of time, where as for other prices (East DK and Sweden) more than 5 % of time wind power does not pay extra for imbalances. For the sum of the wind farms, only East DK prices were used. This is because the West DK prices have been affected by the wind power production in West DK so the East DK wind farm prediction errors do not correspond to that correlation. As wind power in East DK is not affecting the regulating prices at least to a large extent like in West DK, there is not a correlation pattern in the errors and prices. When the cable between West and East DK is built, it is possible that prices in both areas will follow the East DK prices. Also Swedish price data was used to reflect the situation where wind prediction errors would not affect the prices when no bottlenecks in Nordic system would prevail, as may be the case in future when transmission is improved. For Swedish prices the balancing costs would be 7.7 DKK/MWh for Middelgrunden and 8.8 DKK/MWh for Klim. This is a bit more than with East Denmark prices. The bidding strategy works quite similarly for Swedish and East Denmark prices. NOTE: the fixed fee for balancing (the volume fee for each MWh of balancing,.2 /MWh in Denmark) has not been used here! This makes it harder for the producer to gain by optimising the bids.

3 3 (19) TABLE 1. Forecast error and balancing cost calculation for case Denmark. Middelgrunden SUM 2 wind Middelgrunden location 2 MW Klim 2 MW farms Klim 2 MW 2 MW Klim 2 MW price East DK West DK East DK East DK Sweden Sweden WIND DATA: total data begin total data end analysis data begin analysis data end available data points average power MW sum production MWh Correlation of production time series.58 WIND PREDICTION DATA: average power MW sum production MWh Correlation of prediction time series.67 WIND PREDICTION ERRORS (PROD FOREC) average error MW mean absolute error MW correlation between errors.22 PRODUCTION ERRORS (PROD - BID) average error MW mean absolute error MW correlation between errors PRICE DATA: average spotprice DKK/MWh average up-reg price - spot price DKK/MWh average spot price - down-reg price DKK/MWh wind income from spot - production data (perfect forecast) DKK DKK/MWh Penalty prices seen by producer when bidding according to forecast average up-reg penalty seen by producer DKK/MWh average down-reg penalty seen by producer DKK/MWh Penalty prices seen by producer when bidding optimally average up-reg penalty seen by producer DKK/MWh average down-reg penalty seen by producer DKK/MWh average up-reg penalty seen by producer, weighed by regulation need DKK/MWh average down-reg penalty seen by producer, weighed by regulation need DKK/MWh % time zero penalty % 54.7 % 41. % 54. % 55. % 54.3 % 54.3 % Balance settlement costs when using forecasts as bids DKK when using optimal bids DKK for predicted time series DKK/MWh for bid time series DKK/MWh IMPROVEMENT by bidding strategy % 98.1 % % % 93.7 % % Bidding strategy affecting the system balance The question is, to what extent the bidding strategy resulting from individual producers optimizing their bids according to prices will affect the system balance? Does this behaviour make the critical situations of large wind power prediction errors worse? The comparison of errors from bids and forecasts, resulting from the bidding strategy used in this case study analysis is shown in Fig 1. This shows that for large errors downward (forecast production larger than realised production) there will be larger errors when using the bidding strategy. This means more up-regulation would be needed in those situations. Whether this is critical depends on the available upregulation capacity.

4 4 (19) bid error MW forecast error MW Figure 1: Summed errors for both parks calculated from optimal bids (on ordinate) plotted against summed errors for both parks calculated from power forecasts (abscissa).

5 APPENDIX 1 Jussi Ikäheimo 1 (19) 1. POWER FORECASTS FOR THE WIND PARKS MIDDELGRUNDEN AND KLIM General observations Hourly power forecasts were available 1 to 48 hours ahead of a nominal analysis preparation time, which was either at : UTC or 12: UTC each day. In other words, : UTC analysis gave forecasts for all hours of the beginning day as well as the next day. The 12: UTC analysis gave forecasts for hours of the current day, hours 1 24 of the next day and hours 1 12 of the third day. The power forecasts need wind forecasts as input. In practise the wind forecast becomes available for use a certain delay period after the stated analysis time. This delay may be 3 6 hours and depends on the calculation method and hardware. In Figure 1 the relationship between the sample mean values of observed production P and the corresponding forecasts Pˆ are shown together with 95 % confidence intervals. The curve has been calculated by binning the time points according to the forecast power Pˆ. If the curve deviated from the diagonal, this would indicate bias in the forecasts. There is some dipping of the curve around forecast power 7 MW but this may well be a coincidence. Also there seems to be some bias at very low (< 5 kw) and high forecasts. The corresponding calculation has been done for the park Klim in Figure 2. 2 x observation forecast Figure 1: Averages of forecasts of production power at Middelgrunden plotted against realized values. The solid curve is the average of observations within 1 MW forecast interval, and the dashed curves are ±2 standard deviations confidence intervals. VTT Energia /home/han/proj/pso411/report/final/collect/dkvind_bids_and_balance_vtt_26_update1.doc

6 2 (19) Figure 2: Forecasts of production power at Klim plotted against realized values. The solid curve is the average of observations within 1 MW forecast interval, and the dashed curves are ±2 standard deviations confidence intervals. Distribution of forecast errors Let us define the forecast error. Here this is defined the same way as Madsen et al (25) as the observed power minus forecast power: e ( t k t) = P( t + k) Pˆ( t + k t) +, (1) where the notation (t + k t) refers to the forecast made for hour t + k at time t. Because the realized power is always between zero and the installed capacity, the distribution of forecast errors is normally skewed. When the forecast power Pˆ is small, large positive forecast errors are possible, where as negative ones are not. In other words, the support 1 of the distribution is a closed region, and varies according to the power forecast Pˆ. It seems that a scaled beta distribution would describe the forecast errors well as is discussed below. There is a slight dependency between the forecast errors of different parks as is shown in Figure 3. 1 Support is the region where probability distribution is positive.

7 3 (19) x x 1 4 Figure 3: Forecast errors of Middelgrunden (abscissa) and Klim plotted against each other. Pearson's correlation coefficient was.21. Estimating the conditional distribution of realized power Estimating distribution of forecast errors is equal to estimating the conditional distribution of realized (observed) power. In this estimation, when forecast has been given, there are two possibilities. One is to use a nonparametric method so that no distribution, which is defined by some formula, is fitted into the available data points (samples). Percentiles can be then extracted from the conditional distribution by sorting the sample points and taking the n'th largest sample point, where n is defined by the percentile. The drawback with this method is that there are normally no sample points which lie in the tail regions of the distribution and the results are thus flawed. Another possibility is to fit a distribution into the available data points, which also extrapolates the regions beyond the smallest and largest data point. The problem with this method is the practical task of finding the best distribution parameters, as well as the question whether the particular distribution is a good description of the true distribution. Beta distribution is a continuous probability distribution, whose support is finite, i.e. the region into which it is confined can be defined. In the basic case the distribution lies between and 1 and its density function is given by x f ( x; α, β) = 1 u α 1 α 1 (1 x) (1 u) β 1 β 1 du, x [,1 ] (2)

8 4 (19) Of course, the distribution can be scaled and shifted to lie between a lower limit L and upper limit H. When we talk about beta distribution, we normally mean the scaled and shifted distribution. In this case the density function is 1 x L f ( x; α, β, L, H ) = f ( ; α, β), x, H L H L [ L H ] The reason for using beta distribution was that it commonly in widespread use, allows limiting the distribution to a specific region (thus negative values or values greater than installed power are never produced) and most importantly the sample conditional distributions of realized power, conditional on power forecast, seemed to follow the beta distribution quite well as noted below. Beta distribution matches sample data relatively well. This is shown in Figure 4 for Middelgrunden park. The different forecast powers Pˆ have been stacked on the vertical axis, whereas the distribution of the corresponding observed powers has been placed on the horizontal axis. The error between true observations and that predicted by the beta distribution of shown with a color code. The corresponding illustration for Klim park is shown in Figure 5. In other words, histograms of observed powers for different forecast ranges were made and they were compared to values given by beta distributions fitted to each sample of observed powers. (3) 4 forecast bin observation error bin bin Figure 4: Discrepancy between the number of true observations and that predicted by a scaled beta distribution for different parts of the distribution of observed powers (horizontal axis) and different values of forecast power (vertical axis). The discrepancy is shown as proportion of the total points belonging to each forecast bin. The size of the bins of on the horizontal axis was 2 MW.5 MW on the vertical axis. This picture is for wind park Middelgrunden. For example value.8 indicates 8 % difference in sample data and theoretical value predicted by beta distribution.

9 5 (19) forecast bin observation error bin bin Figure 5: Discrepancy between the number of true observations and that predicted by a scaled beta distribution for different parts of the distribution of observed powers (horizontal axis) and different values of forecast power. The discrepancy is shown as proportion of the total points belonging to each forecast bin. The size of the bins of on the horizontal axis was 2 MW.5 MW on the vertical axis. This picture is for wind park Klim. Thus, given a power forecast, we assumed that the distribution of the realized power is the beta distribution with parameters α and β. Naturally these parameters depend on the power forecast. Therefore they were assumed to be functions of the power forecast: α = α( Pˆ ) and β = β( Pˆ ). These functions were not explicitly specified. Instead, the dependency of conditional mean and variance of realized power, conditional on power forecast, of power forecast was sought. The conditional mean was assumed to be equal to the power forecast, in other words the forecasts were unbiased. Conditional standard deviation was estimated for different power forecasts by first binning the data so that for each bin forecast power was in a specified range Pˆ Pˆ Pˆ. Bin width of 5 kw for the power forecast was used, which means that there were 4 bins. A larger bin would be desirable for high values though, because the power forecasts are concentrated into low values. A fourth-degree polynomial was then fitted into the values of conditional standard deviation. Also an estimate of the uncertainty of the estimate of standard deviation was calculated and it was used as a weight in polynomial fitting. L H

10 6 (19) 6 Mgrund 5 stdev of error forecast MW Figure 6: Fourth-degree polynomial fitting (dashed line) of forecast error standard deviations as function of forecast power for Middelgrunden park. Note that for high forecasts the reliability of sample standard deviations becomes lower. Given a power forecast conditional mean and variance of realized power can now be found. If E ( x Pˆ ) is the conditional mean of realized power and Var ( x P ˆ) is the conditional variance, the parameter estimates for α and β, given by method of moments, are E( x Pˆ)(1 E( x Pˆ)) ˆ = E( x Pˆ) 1 Var( x Pˆ) α and (4) E( x Pˆ)(1 E ( x Pˆ)) ˆ = (1 E( x Pˆ)) 1 Var( x Pˆ) β. (5) In some cases it was not possible to create a beta distribution which has the same variance as the sample. This was because at low forecast powers the forecasts are not unbiased as shown below. Also the interpolation polynomial did not go to zero at minimum and maximum forecasts.

11 7 (19) 12 1 realized kw Mgrund forecast MW Figure 7: Sample mean of observed power plotted against forecast power for Middelgrunden park (all lead times included). This figure is similar to Figure 1 above but a different region is shown and different binning based on forecast powers is used. The requirement for variance of beta distribution is that (when scaled between and 1), Var ( x Pˆ) ( x Pˆ) ( 1 E( x Pˆ) ) E. (6) Actually this holds for any distribution, i.e., beta distribution is able to have the highest possible variance of all distributions whose support is from a lower limit L to upper limit H. Because of this variance limitation the polynomial interpolation curve should pass points (L,) and (H,) where L= and H is the installed power. At present it does not. It is possible that some other form of trendline is better than polynomial if the curve is forced to pass through these points. The unbiasedness of low-value forecasts was not corrected but the forecast was still assumed unbiased and the beta distribution with highest possible variance was chosen.

12 8 (19) a = 2 b = 4 a = 4 b = 4 a = 1.2 b = Figure 8: Different beta distributions in the region [,1]. If α decreased below one, the function would be strictly decreasing, i.e., it would not leave from the origin (here "a" refers to α and "b" to β). Switching α and β makes the distribution its mirror image. After these steps, the conditional distribution of realized power is ready. A suitable percentile can be drawn from it for the purpose of making a bid to electricity exchange. Correlation between forecast errors and regulation price An interesting question is the relationship between the forecast errors and regulation price, or more exactly the penalty price, defined as follows: p p u d = p = p upreg spot p p spot downreg, (7) where p u is the upregulation penalty, p d is the downregulation penalty, p spot is the spot price, p upreg is the upregulation price and p downreg is the downregulation price. Because of the way the up- and downregulation prices are calculated, one of the penalties p u and p d is always zero.

13 9 (19) 8 6 penalty price crowns forecast error MW Figure 9: Penalty prices p d (on the positive half of y-axis) and -p u (on the negative half of y-axis) plotted as a function of the forecast error e for wind park Middelgrunden. Regulation price data from Denmark East, to penalty price crowns forecast error MW Figure 1: Penalty prices p d (on the positive half of y-axis) and -p u (on the negative half of y-axis) plotted as a function of the forecast error e for wind park Klim. Regulation price data from Denmark West, to

14 1 (19) 4 35 penalty price crowns forecast error MW Figure 11: Penalty prices p d and -p u (on the y-axis) plotted as a function of the forecast error e for wind park Middelgrunden. The dashes line is 95 % confidence interval. Regulation price data from Denmark East, to penalty price crowns forecast error MW Figure 12: Penalty prices p d and -p u (on the y-axis) plotted as a function of the forecast error e for wind park Klim. The dashes line is 95 % confidence interval. Regulation price data from Denmark West, to OPTIMAL BIDS Let us define the production error as follows: B( t) = P( t) B( t), (8) where B(t) is the bid placed for hour t. When the producer faces a linear penalty function for both negative (less than the bid) and positive (more than the bid) production errors, the optimal bid may be

15 11 (19) calculated based on the slopes of the linear penalties as well as the quantiles of the distribution of future observed power (=forecast error + forecast). Specifically Bremnes (24) presents a simple formula for the optimal bid B * : B * = F 1 pd pu + p d, (9) where F -1 () represents the inverse cumulative distribution function for the future observed power, in this case the inverse cumulative beta distribution. In other words, the fraction p d / (p u + p d ) the quantile which must be taken from the distribution of future observed power. If the penalty prices were equal, the optimal bid would be equal to the median of the distribution. Unfortunately there is not analytic formula for the inverse cumulative beta distribution but each value must be solved numerically. The penalty Q the producer has to pay is equal to B p Q( B) = B p d u,, B >. (1) B < Given the optimal bid, the expected penalty is equal to P E( Q) = max Q( P B * ) f ( P) dp = P max Q P F 1 p u pd + p d f ( P) dp, (11) where f(p) is the probability distribution function for the future observed power. Especially one should notice that the optimal bid is a quantile of the distribution. Therefore for a rightskewed distribution the optimal bid would tend to be lower than the expected value of the distribution (which for unbiased forecast is equal to the actual forecast power). This is important because most of the time wind parks produce a low power output and the forecast power is also low. Therefore most of the time the distribution of forecast errors is skewed to the right. Of course, the relationship between the upand downregulation penalties affects the optimal bid. For example, if the downregulation penalty is higher, this would counteract the effect of right-skewed distribution. Forecasts for Elspot market Bids to Elspot system must be fed in before 12: Norwegian time and they are given for the hours 1 24 of the following day. In UTC time this means during winter time the period from 23: UTC of the same day to 23: UTC of the following day. When daylight saving time is used, this means the period from 22: UTC of the same day to 22: UTC of the following day. On the day when Norway shifts from winter time to daylight saving time, it means the period from 23: UTC of the same day to 22: UTC of the following day. On the day when Norway shifts from daylight saving time to winter time, it means the period from 22: UTC of the same day to 23: UTC of the following day. Normally an analysis from 6: UTC is used for this purpose. For this work this analysis was not available. Instead the : UTC analysis was used. Somewhat larger forecast errors can be expected when this earlier analysis is exploited.

16 12 (19) Values for the regulation penalties p u and p d The natural question is, where to get the regulation penalties p u and p d? They are not known at the time of placing the bids. Therefore, they are also random variables. In Bremnes's (24) model, however, they are assumed deterministic. Allowing them to follow some random distribution would invalidate the analytic formula and require a numerical solution of the optimal bid. Linnet (25) shows that if the regulation penalties are independent of the wind power production, we can use their expected values (e.g. past averages) in calculating the optimal bid according to Formula 9. In this work, it was chosen to use the average prices from the same year This is because the prices in were not representative as the regulating power market had just started. In practise there is some correlation between the production errors and penalty prices. For the producer it is important to know a "good value" for both the up- and downregulation penalty, given his particular distribution of production errors. The problem is of course that production errors depend on bids, which (if calculated as optimal bids as above) again depend on the regulation penalties. The situation is as shown in Production Forecast Difference Bid Calc bids Prod error Penalties seen by producer Penalties Figure 13: If the penalties, which the producer himself faces are used in bid calculation, we end up into an iterative process where bids are calculated from these, then the penalties faced by the producer are again calculated from production errors, from which a second generation of bids are calculated. However, the penalties seen by the producer, when calculated from the first generation of bids, are not too much different from those calculated from power forecasts. Therefore the iterative process can be expected to converge quickly. Much bigger problem is if past values of penalty prices are good estimates for the next day's values.

17 13 (19) 3. RESULTS Below shown are the hourly production errors, assuming that the producers bid according to the power forecasts, or according to the optimal bid formula. One can see that the differences in production errors according to the optimal bid vary the most when production error according to the forecast power is around zero. Actually, the upper and lower bounds of the cloud of points at zero x-value show how much the optimal bid can differ from the power forecast Figure 14: errors for Middelgrunden calculated from optimal bids (on ordinate) plotted against errors calculated from power forecasts (abscissa).

18 14 (19) Figure 15: errors for Klim calculated from optimal bids (on ordinate) plotted against errors calculated from power forecasts (abscissa) bid error MW forecast error MW Figure 16: Summed errors for both parks calculated from optimal bids (on ordinate) plotted against summed errors for both parks calculated from power forecasts (abscissa).

19 15 (19) 4. REFERENCES John Bjørnar Bremnes (24). "Probabilistic Wind Power Forecasts Using Local Quantile Regression", Wind Energy 24; Vol 7 pp Ulfar Linnet (25). " Tools supporting wind energy trade in deregulated markets ". Master's thesis. Technical University of Denmark, dep. of Informatics and Mathematical Modelling. H. Madsen, P. Pinson, G. Kariniotakis, H. A. Nielsen, T. S. Nielsen (25). Standardizing the Performance of Evaluation of Short-Term Wind Power Prediction Models. Wind Engineering, Vol. 29, No. 6, pp C.S Nielsen, H.F.Ravn, (23). Criteria in short term wind power prognosis. In CD proceedings of European Wind Energy Conference EWEC 3, June 23, Madrid, Spain.